Carbon nanotubes with ferromagnetic and semiconducting contacts

Transcription

1 Carbon nanotubes with ferromagnetic and semiconducting contacts Ph.D. Thesis Jonas Rahlf Hauptmann May 2008 Niels Bohr Institute Nano-Science Center University of Copenhagen Denmark

2 Carbon nanotubes with ferromagnetic and semiconducting contacts Ph.D. Thesis c Jonas Rahlf Hauptmann rahlf@fys.ku.dk Niels Bohr Institute Nano-Science Center Faculty of Science University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen O Denmark ii

3 Preface This thesis is submitted to the faculty of science at the University of Copenhagen as a partial fulfillment of the requirements for the Ph.D. degree in physics. The experimental work presented in this thesis have primarily been performed at the Ørsted laboratory, Niels Bohr Institute and Nano-Science Centre. The aim was to study carbon nanotubes with different type of contacts, such as the ferromagnetic material nickel and the semiconducting materials galliumarsenide and silicon. These two rather different type of materials was experimental studied in parallel. The work was done in the nano-physics group that during my Ph.D. studies changed name to the nano-electronic group even though the focus of the group remained the same namely to study mesoscopic systems such as carbon nanotubes or nanowires. I am gratefully to my supervisor professor Poul Erik Lindelof for giving me the opportunity to study the fascinating topic carbon nanotubes. I would like to thank Skou stipendiat Jens Paaske for numerous discussion on the topic of Kondo effect. The semiconducting material have been grown in molecular beam epitaxial systems. The in house growth of galliumarsenide have been carried out thanks to Claus Sørensen and Matin Aagensen, while the silicon growth have taken place at Aarhus University by Arne Nylandsted and John L. Hansen, without the expertise of these four persons the second half of my thesis would have been very hard, if not impossible, to make. For frequently discussion of and on topic, at the journal club and lunches I would like to thank Thomas Sand Jespersen, Kasper Grove-Rasmussen, Mathias Lunde, Henrik Ingerslev Jørgensen, Jeppe Holm, Søren Erfurt, Anders Eliassen, Jesper Nygård, Finn Berg Rasmussen and Peter Nissen. The long hours in the office were made more enjoyable thanks to Martin Aagesen, Henrik Ingerslev Jørgensen and Magdalena Utko. For a lot of help and for a nice working atmosphere I would like to thank Nader Payami, Claus Sørensen, Thomas Sand Jespersen, Kasper Grove-Rasmussen, Mathias Lunde, Henrik Ingerslev Jørgensen, Jeppe Holm, Søren Erfurt, Anders Eliassen, Jesper Nygård, Finn Berg Rasmussen, Peter Nissen, Martin Aagesen, Søren Stobbe, Ane Jensen, Pawel Utko, Magdalena Utko, Brian Sørensen, Peter Bidstrup, Gitte Michelsen and Poul Erik Lindelof. All people that have been part of the group during my Ph.D. For help with the written word I would like to thank, Poul Erik Lindelof, Jens Paaske, Thomas Sand Jespersen, Claus Sørensen, Stefan Mabit, Søren Bredmose Rasmussen, Katrine Søndergård and Andreas Rahlf Hauptmann. iii

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5 Contents 1 Introductory comments 5 I Carbon nanotubes with magnetic contacts 7 2 Introduction Carbon nanotubes CNT geometry and band structure Transport properties Quantum dots Introduction Theory Quantum dots in carbon nanotubes Kondo Introduction History and Kondo in metals Kondo effect in a quantum dot Elastic co-tunneling and Kondo effect Finite bias and temperature dependence of the Kondo peak Kondo with ferromagnetic contacts Introduction Andersons model Renormalization with spin polarized electrodes Two contacts Parallel and anti-parallel domain magnetization Experimental methods Sample fabrication Chemical Vapour Deposition (CVD) Lithography Steps to make a sample Measurement setup Electrical setup

6 4.2.2 Cryogenics Electric-field controlled spin reversal in a quantum dot with ferromagnetic contacts Introduction Magnetic domain size in the electrodes Devices and quantum dots in the Kondo regime Spin-splitting of the Kondo state at fixed gate-voltages Spin-splitting and compensating field Parallel and anti-parallel magnetization of the contact domains Spin-splitting as a function of gate Movement of the degenerated state as a function of B Temperature dependence of the splitting Discussion Other Kondo effects Conclusion Hysteresis Introduction Domain flip before zero field Hysteresis at high fields Hysteresis at small fields with the Kondo effect Conlusion II Carbon nanotubes with semiconducting contacts 83 7 Carbon nanotubes incorporated in MBE grown structures Introduction Carbon nanotubes incorporated in GaAs heterostructures Gallium Arsenide (GaAs) Molecular Beam Epitaxy (MBE) system Sample preparation Dispersing carbon nanotubes Samples and results Analysis and discussion A short description of the growth Electrical abilities Conclusion and outlook Carbon nanotubes incorporated in Si Introduction Experimental techniques Different Si-CNT configurations

7 9.2.2 Experimental setup Electrical setup AFM characterization and manipulation AFM manipulation of CNTs Suspended CNT Low temperature measurement Measures taken to lower the contact resistance and results Annealing Samples with CVD-grown CNT Samples and sample behavior Samples produced Samples where the height of the CNTs has been measured Sample with the lowest resistance Semiconducting samples Conclusion and outlook A Plots for D1K2 and D1K3 125 B Magneto resistance plots 129 C Recipe for Carbon nanotubes with Ni contacts 137 3

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9 Chapter 1 Introductory comments The work presented in this thesis was made during my PhD study and four month working as a process engineer at the private company Hytronics. The work is presented in two parts that on a first glance can seem rather separated. They have the subtitles Ferromagnetic contacts to carbon nanotubes and Semiconducting contacts to carbon nanotubes. The offspring of these two separated parts was carbon nanotubes (CNT) incorporated in the ferromagnetic semiconductor GaMnAs. Since GaMnAs is a ferromagnetic semiconductor, we tried to simplify the system by separating the ferromagnetic and the semiconducting part into two studies: 1. The study of carbon nanotubes with ferromagnetic contacts. Here Ni was chosen as the contacts material, since Ni have been shown to give a rather low contact resistance to tubes, especially if combined with Pd [32]. Beside of this, chemical vapour deposition (CVD) grown tubes can be contacted with Ni; this is an advantage since they can be grown in-house and have been shown to produce high quality tubes. 2. Carbon nanotubes with semiconducting contacts, here different materials were tried. First GaAs, that can be grown in-house and have for a long time been the play ground for people with interest in mesoscopic physics. One idea was to create two dimensional electron gasses that could contact the CNT, enabling a study of a two dimensional - one dimensional contact; unfortunately it turned out to be difficult to incorporate CNT in GaAs. Since the combination of semiconducting material and CNT also potentially could be interesting for the semiconducting industry, another obvious material choice would be Si. Therefore, Si was tried as the contact material; the Si was grown at Aarhus University. It was during the work with CNT incorporated in Si that I had a four months employment at Hytronics. It should be possible to read the two parts of the thesis separately and I have tried too avoid to many cross references, although a brief introduction to CNTs is given in section 2.1, which might be useful for a reader of part II of this thesis. 5

10 Part I is about ferromagnetic contacts to CNT and starts with chapter 2. Chapter 2 gives a brief description of CNTs and quantum dot behaviour, introducing effects such as Coulomb oscillations. In chapter 3, we look at the more open regime and begin the chapter by introducing the Kondo effect. Later in the chapter, a more thorough discussion of the theory determining the renormalization of the spin states in a CNT quantum dot with ferromagnetic contacts is presented, using second order perturbation theory and the Anderson s model. We now turn to the experimental part, starting chapter 4 with the sample fabrication before turning to the electrical setup and the cryogenics used for measurements in the milli-kelvin regime. In chapter 5, results from measurements upon CNT with ferromagnetic contacts are shown. First the exchange field induced in a CNT quantum dot with ferromagnetic contacts is studied and it is shown that it is possible to compensated this exchange field by applying an external magnetic field. The gate dependence of this exchange field are then studied, and the chapter is ended with a discussion and a conclusion. This part of the thesis ends in chapter 6 where the hysteretic behaviour of a CNT quantum dot with ferromagnetic contacts in the Kondo regime is studied and phenomena like hysteresis at high external field are discussed. In part II, CNT with semiconducting contacts is studied. After a short introduction in chapter 7, chapter 8 deals with CNT incorporated in molecular beam epitaxial (MBE) grown GaAs structures. The chapter starts with introducing the MBE system followed by a short description of how the samples are made. At the end of chapter 8, the overgrown structures are studied ending with the conclusion. In chapter 9, we turn to the incorporation of CNT in Si. This is started, after a short introduction, with a description of how and which samples and sample structures made in this thesis. This is before the etching technique is described. The different samples are then studied and characterized by atomic force microscopy and electrical measurements, showing that it is possible to get a Si-CNT contact. In the end, the different samples produced, measured and their resistances are listed. Some of the semiconducting conduction behaviours are shown, before a conclusion and outlook are given in the end. With this short thesis outline in mind, I hope you are looking forward to the interesting world of CNT. Please enjoy!! Jonas Rahlf Hauptmann May

11 Part I Carbon nanotubes with magnetic contacts 7

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13 Chapter 2 Introduction 2.1 Carbon nanotubes Carbon nanotubes (CNT) were discovered in 1991 by Iijima [12] as a by product of fullerene research. This discovery started an intense research in these interesting structures; an interest that has lasted until now. The strong interest is sparked by CNTs unique physical and electrical abilities. Physical: They are approximately six times lighter than steel yet five times stronger, making it the only known material with a high enough strength to weight ration to, in theory, make the creation of a space elevator possible [15]. Electrical: Their electrical abilities depend on the chirality, i.e. on the precise geometry of the tube; this will be discussed further in the next section. They can be both semiconducting and metallic and are in theory able to carry current densities more than a 1000 times greater than in normal metals such as e.g. copper. Due to their size and geometry, they are essentially one dimensional conductors making them interesting from a pure physics point of view CNT geometry and band structure CNT can be thought of as rolled up graphene 1, if the vectors a, b are primitive lattice vectors defined in figure 2.1, the chirality vector C h is defined by C h = na 1 + ma 2 (n, m) n, m are integers and 0 m < n (2.1) and is the vector you get, going once around the circumference of the nanotube, ending up in the same point where you started. The chirality vector and therefore the indices (n, m) uniquely define a CNT, disregarding handedness, i.e. if the tube is spiralling left or right. The grey area in figure 2.1 defines a CNT unit cell, in this case for a (10,4) tube. The unit cell is spanned by C h and T the translation vector. T determines how far along the tube one should move 1 Graphene is a single layer of graphite. 9

14 a ) n (n, b Ch T θ a2 a1 (n,0) c Figure 2.1: Geometry of a CNT. a, Shows the chiral vector Ch and the translation vector T for a (10,4) nanotube. These two vectors span the nanotube unit cell which is the gray area. The tube appears by cutting out the piece of the graphite sheet that lies between the two dotted lines parallel to the translation vector and then assemble these two lines by rolling the graphite strip into a cylinder resulting in the tube seen in b. c, A larger part of the same tube is seen from the side. The (n, n) and (n, 0) defines so called armchair and zig-zag tubes respectively. Tubes with other indices are called chiral. 10

15 a Γ b π*-bonding E[eV] 5 0 ε F -5 π-bonding K K M K Γ M K Figure 2.2: Electronic structure of graphene. a, Shows the dispersion relation for graphene, called the tent dispersion. The points Γ, K, K and M are high symmetry points, and the area spanned by the K and K equivalent points is the first Brillouin zone, corresponding to the grey area seen in figure 2.3a-c. b, Appears by cutting a in straight lines between the points K Γ M K here π-bonding is the valence band and π is the conduction band. The graphs are taken from [17]. before the lattice starts repeating itself. The tubes shown in figure 2.1b,c are (10,4) tubes seen from different angles. Figure 2.1 demonstrates how a CNT geometrically is defined from rolling up a graphene sheet. The electronic properties of a CNT can to a first approximation be understood by imposing periodic boundary conditions on the electronic properties of graphene. The dispersion relation for graphene shown in figure 2.2 is found by using the Local Combination of Atomic Orbital approximation scheme (for further details on this calculation see [17] or [14]). Due to the two-atomic lattice cell of graphene, the dispersion relation is filled with electrons up to the K and K points. Actually, there are six points where the conduction and valence band are touching each other as can be seen on figure 2.2a. These six points spans the first Brillouin zone, but only two of the six points are linearly independent, i.e. can not be connected by reciprocal lattice vectors, namely the K and K points. This touching of the valence and conduction band at the Fermi energy ε F makes graphene a so called semi-metal or zero-band-gap semiconductor. This peculiar electronic structure is also the cause of the many electronic species of CNT. The M and Γ in figure 2.2 defines high symmetry points. The electronic properties of CNTs are found by imposing periodic boundary conditions on the graphene dispersion relation seen in figure 2.2a. The requirement is that e ik r = e i(k+c h) r where k is the wave-vector and r is a real space vector in the graphene plane. This implies that C h r = 2πj where j Z. This requirement results in a number of line-cuts in the dispersion relation, but how these cuts are placed depends on C h, i.e. how the graphene sheets are rolled together. In figure 2.3a-c, line-cuts for three different kinds of tubes are shown, and underneath are the resulting CNT band structure. To a first approxima- 11

16 a b c k y k x K K' K reciprocal lattice points K b 2 b 1 K' K K' K K' K E[eV] k k K K' K' k k k k d e f E[eV] E[eV] K' E[eV] E[eV] 5 E[eV] 5 E[eV] π/ T π/ T -π/ T π/ T -π/ T k k k π/ T Figure 2.3: a-c, Illustrate three Brillouin zones of graphene (gray areas). The black lines show allowed wave vectors for three different kinds of tubes (in all the cases some of the lines have been translated with reciprocal lattice vectors making them as close to symmetrical around Γ as possible). d-f, Dispersion relation for the three different tubes, to a first approximation. a, Shows the states for a (10,10) tube. It can be seen that it is metallic since one of the lines touches the K and K points. d, The corresponding band diagram. b, States for a (12,0) tube which is also metallic since it is of the type (n, 0) it is called a zig-zag tube, e corresponding band diagram. c, States for a (11,0) tube, it can be seen that it is semiconducting because none of the lines touches the K and K points this can also be seen from the band gap in f. The insets in d-f are a zoom at the Fermi energy. Inset in a one of the crossings, in the inset in e are the lines seen degenerate, and in f there are two insets where the right is a zoom of the left. The lines seen in the right inset are degenerate. Illustrations and graphs are taken and modified from [17] 12

17 tion, the big question is if the line-cuts are touching a K or K point. If this is the case, the CNT will be metallic as in figure 2.3a,b. If this is not the case, the tube will be semiconducting with a band gap, depending on the distance between the line-cuts. The distance is proportional to one over the diameter of the CNT. With a little calculation it is seen that if (n m)/3 is an integer, the lines touch the K and K points and the CNT will be metallic. If (n m)/3 is not an integer the CNT will be semiconducting. Going beyond the first approximation, this turns out not to be completely true, since the dispersion in figure 2.2 is for a graphene sheet and the rolling of this sheet into a CNT imposes strain on the bindings. This strain introduces a slight distortion in the dispersion relation, moving the K and K points. This will open a small band gap in the otherwise metallic tubes, except those with indices (n, n) called armchair tubes. These tubes have a symmetry that makes them truly metallic, i.e. so beyond the first approximation a small band gap will open in the tube seen in figure 2.3b,e. So electrically, there are three categories of tubes: Truly metallic Semiconducting Semiconducting small band gap large band gap Armchair Zig-zag and chiral Zig-zag and chiral (n,n) (n-m)/3 Z\0 (n-m)/3/ Z All three kinds of species have been measured and can be distinguished by their transport properties [69, 72] Transport properties Due to the K and K points, there are two bands crossing the Fermi energy; these two bands are normally referred to as orbitals. Because of the orbital and the spin degeneracy there are four conduction modes in a CNT, giving a maximum conduction of G = 4e 2 /h corresponding to a minimum resistance of 6.5 kω in a CNT device with reflection-less contacts. The intrinsic resistance in a CNT is often negligible and ballistic transport have been observed in 60 nm long CNT at room-temperature [75], and at low temperatures in tube lengths close to 4 µm [76]. This very low intrinsic resistance implies for device lengths below 1 µm that the transport properties of CNT often are determined by the contact resistance. The contact resistance is to a large extend determined by the choice of contact material. Depending on the contact resistance between the electrodes and the CNT, one can roughly put the CNT devices into three categories: For high-resistance contacts, we are in the Coulomb blockade regime where the number of charges on the dot are well defined and the transport mechanism at low temperatures is dominated by single electron tunneling - this regime will be further discussed in section 2.2. When the CNT couples better to the contact electrodes we enter the Kondo regime. Here the charges on the dot are still well defined, but higher order tunneling effects starts to play a role and phenomena like the Kondo effect and level renormalization can be observed. This regime is studied in chapter 3. The last regime is the so called Fabry-Perot 13

18 where the charge can be continuously added, and the CNT so to speak behaves like a wave guide. This regime will not be further studied in this thesis. Spin-transport The low intrinsic resistance, long mean free paths combined with a high Fermi velocity and very low content of nuclear spin 2 have made CNT promising candidates in the field of spintronics. These abilities have until recently also been thought to suggest a very small spin orbit coupling. But theory by Bulaev et. al. [78] and experiments by Kuemmeth et. al. [77], shows that a coupling between the orbital states in the CNT and the electron spin can give a rather substantial spin-orbit term. This spin-orbit coupling are present in CNT where there is little or no scattering between the two orbitals. The two orbitals encircle the CNT circumference in a clock-wise and anti-clock-wise fashion. The spins couple to this clock-wise and anti-clock-wise motion, thereby splitting the otherwise degenerate orbital states. If the scattering between the orbitals are significant, the orbital state is not well defined and the electrons can be in a superposition of the two states, thereby obscuring the spin-orbit coupling. Despite this, spin-orbit coupling spin-transport measurements in CNT have been reported by different groups [31,33 40,40]. The measurements study CNT with ferromagnetic contacts. In most of the measurements, this is done by recording the current or conduction as a function of magnetic field at different bias voltages. A change in resistance is then observed, as the ferromagnetic electrodes turns from a parallel to an anti-parallel configuration, giving a spinvalve effect. Both positive and negative resistance changes have been observed, and a gate control of this sign change have been demonstrated [33] for devices in the Coulomb regime. 2.2 Quantum dots Introduction When metal grains or other conducting structures become smaller, the energy for adding an extra charge will increase. If the structure is so small that the energy needed to add the charge is larger than the available energy (bias and temperature) and the resistance between the structure and the contacts is large enough 3, the system is called a dot 4. Systems that are so small that the electrons wave function is confined in all three spatial dimensions are called quantum dots. In the rest of this thesis, dot and quantum dot will be used synonymously and will always refer to a quantum dot. The confinement causes the electron wave functions to quantize into a discrete energy spectrum. This is very similar to 2 There are different isotops but the most common carbon isotope 12 C have no nuclear spin. 3 Large enough that the charge on the structure will be a well defined quantity. A more precise definition for this will be given in the next subsection. 4 Classical since the transport behaviour can be described without the use of quantum mechanics. 14

19 source C s quantum dot Cd drain R s R d C g I V g V b Figure 2.4: Quantum dot electrical diagram. The dot couples weakly to the source and drain with the resistances R s and R d and the capacitances C s and C d respectively. The gate couples only capacitively to the dot with the capacitance C g. Adapted from [17]. what are seen in atoms, and a quantum dot is therefore sometimes referred to as an artificial atom. Quantum dots have been seen in different structures such as two-dimensional electrongasses where the electrons in the plane are confined by etching [55] or top gates [54], InAs wires with topgates [52] or intrinsic grown barriers [56], InAs dots grown by Stranski Krastanov [68], molecules such as C60 [25], graphene [53] carbon nanotubes singlewalled [69] and multiwalled [57] and the list goes on. The quantum dot structures mentioned above have one thing in common: they can all be connected to a source and drain electrode, by a tunnel or high resistance contact and therefore be studied by transport measurements Theory In more formal terms, the system looks like the diagram in figure 2.4. We think of the system as a quantum dot connected to a source and drain electrode with the resistances and capacitances R s, C s and R d, C d, respectively. Beside of this, there is also a capacitive coupling to a gate electrode C g. The typical energy scale for these kind of systems is the charging energy E C = e2 C, which will be derived later. This energy becomes important when it exceeds the thermal energy k B T. To show dot behaviour, there is a second requirement the system has to fulfill; the number of charges on the dot has to be well defined. This turns into a requirement for the resistances R s, R d. The smallest contact resistance R s,d gives a typical time to charge or discharge the dot δt R s,d C where C = C s +C d +C g is the total capacitance. From Heisenberg uncertainty relation we have h < δeδt. Combining this with the charging energy and the thermal requirement, we end up with the following two conditions for observing Coulomb 15

20 0.4 I [na] Vg [V] Figure 2.5: Coulomb oscillations. The plot shows the current as a function of gate voltage V g measured with a bias of 10 µv. The plot is measured on a CNT with two metal electrodes showing clear Coulomb oscillations. Note that in some of the valleys, signs of the Kondo effects can be observed; this is especially clear in the valley marked with the a red arrow. Some four-period oscillations are marked by the grey lines. For sample fabrication and experimental details, see chapter 4. oscillations R s,d h e 2 (2.2) e 2 C k BT (2.3) If the system fulfill these requirements the system will behave as a dot and Coulomb oscillations such as the ones seen in figure 2.5 can be observed. To understand these measurements we need to take a look at the potential landscape of the electrode-dot system. This is schematically shown in figure 2.6, where the transport mechanism with and without bias is described. Having looked at figure 2.6, we can now understand the measurement in figure 2.5 made on a CNT with Ni contacts. The plot shows the current I as a function of gate voltage V g at a bias V b 10 µv; a series of distinct Coulomb peaks can be seen as V g is tuned; these appear each time a level is aligned with the chemical potential in the source and drain. When the level is moved outside the bias window, transport is blocked and we are in a Coulomb blockade region until the next level appears in the bias window. A quantum dot is often characterized by a stability diagram, also called a bias spectroscopy plot. In such a plot, di/dv is measured as a function of gate and bias voltage; for a single quantum dot this results in features resembling those schematically shown in figure 2.6f. In the figure the grey areas are Coulomb blockade regions, the black lines mark when an energy level will enter the bias window and the current starts to flow. In di/dv, the onset of current is seen as a line since the current is constant until the bias becomes large enough for an excited state to enter the bias window, resulting in an increased current. The occurrence of this is marked with a dashed line. 16

21 n a b c µ(n+1) Γs Γd µ s µ d E add l E i n n+1 µ(n+1) d Γs n e Γs n f V b n-1 n n+1 V g ev b ev b Figure 2.6: Potential landscape in a quantum dot. µ s and µ d are the chemical potentials for the source and drain, respectively. µ(n + 1) is the chemical potential for the (n + 1) th electron on the dot. Dotted lines are empty chemical potentials, and full-drawn lines are occupied chemical potentials. The lines are separated with the energies E l i except at the Fermi surface where the separation is the addition energy E add. Γ s, Γ d are the tunnel rates to source and drain, respectively, giving a broadening of the levels corresponding to Γ = Γ s + Γ d. n shows the number of electrons on the dot. In a, transport is blocked and we are in the Coulomb blockade regime. In b,c, a chemical potential on the dot is aligned with µ s, µ d and transport is allowed, giving os a Coulomb peak. This is achieved by tuning the gate voltage and thereby adjusting the dots chemical potentials. In b, an electron can jump on the dot giving os c where the electron has to jump off the dot, returning the dot to its starting point b. In d, a bias V b is applied, changing µ s with ev b. If the chemical potential µ(n+1) is in the bias window between µ s and µ d, transport through the dot is allowed. Increasing the bias further such that the first excited state enters the bias window results in an increased current. f, schematically show a stability diagram also called bias-spectroscopy plot; where di/dv is plotted as a function of gate and bias voltage. The grey areas are Coulomb blockade area where no current can run, the black lines show the onset of current seen as a line and the dotted lines are excited states. 17

22 Energy levels and chemical potential in the quantum dot The total ground state energy of the electrons E n is the sum of the energy due to Coulomb interaction for n electrons on the dot U(n) and the energy of the different energy levels of the electrons Ei l. It is therefore given by E n = U(n) + n Ei l (2.4) U(n) can be found by summing over the different charges on dot and is given by U(n) = ( e(n n 0) C g V g ) 2 (2.5) 2C where (n n 0 ) is the number of electrons added from the equilibrium, n 0. From this we can deduce the chemical potential µ which is the energy for adding one more electron to the dot and is given by i=1 µ(n) = E n E n 1 = e2 C (n n ) C g C ev g + E l n. (2.6) The addition energy E add is the energy needed to go from one degeneration point to the next, i.e. E add = µ(n + 1) µ(n) = E + e2 C (2.7) where E is the energy difference between the level of the n th electron En l and the (n + 1) th electron En+1, l called the level spacing. Note that the energy difference can be zero if the n th and (n + 1) th electron are in the same energy level. E C = e2 C is called the charging energy. From the pattern in the addition energy, knowledge of the electronic shell structure of the quantum dot can be extracted Quantum dots in carbon nanotubes In a CNT the electrons are due to geometry confined in two of the dimensions, i.e. it is only along the tube that the electrons are unconfined and free to move with all energies, at least for infinite long tubes that are metallic. To create a quantum dot, we therefore only need a restriction in the third dimension. This is normally achieved by contacting the tube with a source and drain electrode. Most commonly, the placement of metal electrodes on top of the CNT will result in a quantum dot being defined between the electrodes with a length corresponding to their separation. Depending on the metal chosen for the electrodes, different contact resistances can be reached 5 and the dot will be more or 5 This might be to stretch the truth a little, it would be more correct to say that the choice of metal gives a contact resistance range, but with a large uncertainty in the actual contact resistance. 18

23 less open 6. Close to perfect contacts have been achieved by Biercuk et al. [70], where a locally gated defect in the tube behaves as a quantum point contact. Due to the two degenerated orbitals in a CNT, the different levels can, if spin is included, be fourfold degenerated. If the degeneration of the orbital states for some reason is lifted, the degeneration is only twofold. This is in Coulomb oscillations seen as a so called even odd effect, i.e. the distance between every second Coulomb peak is E add = E C + E while for every other it is only E add = E C. A fourfold degeneration of the region can be observed in some gate regions in figure 2.5. The periods are marked with grey lines. An estimate of the level spacing E in metallic CNT can be found by combining the linear dispersion relation close to the K and K points with a simple particle in the box picture 7, where the length of the box L should correspond to the distance between the metal electrodes. The level spacing in a metallic CNT quantum dot is then given by E = k de dk = v F π L where de/dk = ± v F k, v F = m/s is the Fermi velocity, and k is determined by the hard wall boundary conditions and can be found from k n = nπ L where n is a positive integer. This calculation is in case of zero orbital splitting, i.e. orbital and spin degeneration gives a four times degeneration of each level. If the orbital degeneracy is lifted, we will have twice as many states that are only two times degenerated (spin-up and spin-down) and have an average separation of E = π v F 2L. 6 For palladium, it is even possible to get into the Fabry Perot regime where the contacts are semi transparent, and positive or negative interference effects are the cause of conductance fluctuations. 7 Assuming hard wall boundary conditions. 19

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25 Chapter 3 Kondo 3.1 Introduction When a carbon nanotube is contacted with metal electrodes the resistance between tube and electrodes determines the transport behaviour of the system. For high resistance contacts, we are in the quantum dot regime. If the coupling between the CNT and the electrodes improves a little, the CNT will still behave as a quantum dot, i.e. the electrons are still localized on the dot. But the lower contact resistance allow for second order transport phenomena to be observed, and effects such as co-tunneling and Kondo appears. This is the so called Kondo regime, and it is placed in the crossing between the quantum dot regime, where the electrons are completely localized on the dot and the Fabry-Perot regime, where the CNT more behaves as a wave guide. In the Kondo regime the electrons are localized on the dot but the tails of the electrons wave function spreads into the electrodes allowing the electrons to feel the electrodes. One can imagine the extension of the wave function into the electrodes as the electron jumping out of the dot onto the electrode and back again within a time span allowed by Heisenbergs uncertainty relation; other names for this is charge fluctuation or virtual excursions. This extension of the dot electrons wave function into the electrodes will renormalize the dots energy levels. This is sometimes referred to as a tunnel induced exchange interaction. If there in the electrodes is a difference in the density of states for electrons with spin-up and spin-down as in the case of ferromagnetic electrodes, the tunneling induced exchange interaction will depend on spin, and the renormalization of spin-up and spin-down states in the dot will be different. In this chapter, we will take a short look at these effects. This is done by introducing the Kondo effect first in metals in section 3.2 and then in quantum dots in section 3.3. In this section is the splitting in a magnetic field and the temperature dependence of the Kondo peak also discussed. This is before turning to more comprehensive treatment of the renomalization of the energy level with normal and ferromagnetic electrodes in section

26 a b resistance temperature Figure 3.1: Illustration of the Kondo effect in metal. a Illustration of the resistance as a function of temperature. Red line is the curve predicted by J. Kondo, which is turned into the dotted line if renormalization group techniques are applied, the blue line was the anticipated behavior, and the green line is for superconducting materials. b Illustration of the Kondo cloud in metals. 3.2 History and Kondo in metals When a metal is cooled, the resistance of the metal drops due to the decrease of lattice vibrations. At low temperatures the resistance saturates due to static defects in the lattice. In some metals such as Nb, Al, Pb and others, the resistance can suddenly decrease to zero. This happens at a critical temperature when the electrons undergo a transition from a normal -conducting state to a superconducting state. In the 1930 s, scientist noticed that when quote ref [1] not very pure metals were cooled, there was an increased resistance at low temperatures. This remained a puzzling behavior until the 1960 s where the resistance increase was found to be connected to magnetic impurities in the metal [2]. This effect was explained in 1964 by Jun Kondo [3] who also gave name to the effect, that until then had been known as the resistance minimum. Kondo assumed an antiferromagnetic exchange coupling between the magnetic impurity and the itinerant electrons. Using perturbation theory, he discovered that the second term could be much larger than the first, giving a logarithmical increase in the resistance at low temperatures. This resulted in the following phenomenological description of the resistance ρ of a metal with magnetic impurities ρ(t ) = at 5 + cρ 0 c log T (3.1) where a, c and ρ 0 are constants. Although Jun Kondo managed to describe the resistance increase, the model gives an unphysical raise to infinity as the temperatures approaches zero. In real metals, the resistance saturates at very low temperatures, giving a curve as the red dotted line in figure 3.1a. This problem was solved in 1975 by the development of the renormalization group technique [5]. The calculation showed that the impurity spin forms a singlet with the itinerant electrons in a certain area around the impurity. The electrons thereby bound to the impurity are called a Kondo cloud and screens the impurity spin as illustrated in figure 3.1b. In 1966, Appelbaum [6] described how the Kondo effect leads to an enhanced 22

27 conduction in transport through tunnel junctions with magnetic impurities. In 1988 Glazman and Lee [43,71] showed separately that this enhanced conduction should also be observable in transport through a quantum dot with a spin. The quantum dot would then so to say behave as a magnetic impurity. 3.3 Kondo effect in a quantum dot As mentioned in the previous section, will the Kondo effect in metals with magnetic impurities give rise to an increased resistance. This is due to an increased scattering on the magnetic impurities which is enhanced as the temperature is lowered. A quantum dot with a spin connected to two electrodes is very similar to such a system. Here the dot plays the role of a magnetic impurity. In this case, the result of the Kondo effect is an enhance conductivity through the system. One could say that the Kondo effect enhance conduction in the quantum dot systems because the electrons needs to scatter on the dot ( the magnetic impurity ) to move through the system Elastic co-tunneling and Kondo effect In a Coulomb blockade region, electron transport is classically prohibited due to the finite energy an electron needs to pass through the dot. To put an extra electron on the dot, an energy E add + ε d is needed, and to take an electron from the dot and put it in the lead, an energy ε d is needed where E add is the addition energy and ε d is the (negative) energy difference between the Fermi energy and the highest occupied electron state on the dot. Quantum mechanically, it is allowed for an electron to tunnel onto the dot if it only stays there for a time t /E add before it leaves the dot again, or an electron can tunnel of the dot if an electron tunnel onto the dot within the same time t returning the dot to its initially state. This last process is illustrated in figure 3.2a-c. Such processes leaves the dot in the ground state and are called elastic co-tunneling processes. For sufficiently good tunnel coupling between dot and electrodes, the co-tunneling effect leads, to a non-zero conductance in the Coulomb diamonds where transport normally is prohibited. This process can take place for any bias voltages, contrary to inelastic co-tunneling which can only take place at finite bias and contrary to the Kondo effect which in its simplest version can only occur at zero bias. Both effects are described further down in the text. The Kondo effect is also a co-tunneling effect that is seen, in the most simple case, when there is an odd number of electrons on the dot. In a quantum dot, the electrons will normally pair up two and two with opposite spins (if the spin states are degenerate), and the last electron will then be placed in the highest unoccupied orbit on the dot, resulting in a quantum dot with a single unpaired spin as illustrated in figure 3.2d. In this case, one can see the dot as a magnetic impurity, and the system is in many ways analogue to the system of a metal with a magnetic impurity. The electron on the dot can tunnel off the dot, and an electron with opposite spin can tunnel onto the dot, or an electron with opposite 23

28 a b c d e f g DOS T k Γ Figure 3.2: Elastic co-tunneling and the Kondo effect. a-c, Illustrates an elastic co-tunneling process where an electron tunnels out of the dot, leaving the dot in an intermediate virtual excited state b that is only allowed to exist for a time t before an electron tunnels onto the dot, restoring the dot in its initial state c. d-f, Illustrates a Kondo process very similar to the co-tunneling effect seen in a-c, except that a spinflip of the electron on the dot has occurred. g, Illustrates the many-body ground state that the Kondo processes leads to, where the peak in density of states has a width proportional to the Kondo temperature T K and is fixed to the Fermi energy of the leads. 24

29 spin can tunnel onto the dot if the residing electron leave the dot, all within the time t, as with the co-tunneling effect. The result of these processes is that the electron spin on the dot have changed from up to down or down to up as illustrated in figure 3.2d-f. This spin exchange changes the energy spectrum of the system. When many of such processes are added together, we end up with a new correlated many-body ground state that extends into the electrodes and is called a Kondo state. This Kondo state has a peak in the density of states (DOS) right at the source and drain bias, i.e. right at the Fermi energy ε F with a peak width T K where T K is the characteristic binding energy of the Kondo state and is given by k B T K = 1 2 (ΓE C) 1/2 e πe C/2Γ (3.2) here Γ = Γ s + Γ d is the broadening of the energy level. An illustration of the DOS of state is shown in figure 3.2g. The result of this new state is a highly increased conduction at zero bias, seen as a conduction peak. Since we need the spin-flip to build the Kondo state this zero-bias conductance peak can only be seen in diamonds with an odd number of electrons. In diamonds with an even number of electrons, transport is prohibited due to the Coulomb blockade. The zero bias Kondo conduction peak is a resonance co-tunneling peak and the conduction can therefore be described with G = 2e2 h 4Γ s Γ d (Γ s + Γ d ) 2. (3.3) For symmetric coupling to the electrodes, we are in the unitary limit [16], and the saturation conductance of the Kondo effect is G = 2e 2 /h in the case of a single degenerated energy level on the dot. Note that the conductance only depends on the symmetry of the coupling to the electrodes, and not on the strength of the coupling, i.e. as long as the coupling is strong enough for the Kondo effect to appear Finite bias and temperature dependence of the Kondo peak For a finite bias equal to or greater than the level spacing in the dot, inelastic cotunneling can take place. The process is very similar to the elastic co-tunneling process except that the electron, tunneling onto the dot, tunnels into an excited state. This means that when the residing electron leaves the dot, the dot is left in an excited state, i.e. the electron tunneling out of the dot has a different energy than the one tunneling in. The process is illustrated in figure 3.3a-c. When a bias is applied over the quantum dot at a Kondo resonance, the peak in the density of states that this correlated many-body ground state has given rise to, as illustrated in figure 3.2g, will split into two, as illustrated in figure 3.3d. Each peak being locked to the chemical potential of the source and drain respectively. This splitting of the peak as a function of bias leads to a 25

30 a b c ev b ev b ev b d e di/dv ev b Γ ~4k B T K ev b Figure 3.3: Inelastic co-tunneling and the Kondo effect. a-c, Illustrates an inelastic cotunneling process where an electron tunnels onto dot, leaving the dot in intermediate virtual excited state b that is only allowed to exist for a time t before another electron tunnels of the dot, leaving the dot in an excited state c. This inelastic co-tunneling process becomes possible when ev b E. Note that the charging energy in the drawings in aand b have been neglected. d, illustrates how the Kondo peak splits into two, each peak locked to the source and drain, respectively, when a bias is applied. e, peak in di/dv due to the Kondo effect close to V b = 0, the full width at half maximum (FWHM) is roughly given by 4k BT K. 26

31 decrease in the conduction, resulting in a conduction peak centered around zero bias and with a full width at half maximum, roughly given by 4k B T K [81] as illustrated in figure 3.3e. In a magnetic field, the degenerated spin level will split with gµ B B due to the Zeeman energy. When a bias is applied over the dot, one will see a conduction peak when the bias energy ev b hits the Zeeman, i.e. when ev b = ±gµ B B ( 115 µev/t) (3.4) where g = 2 is the expected g-factor for a CNT, µ B = J/T, and B is the applied magnetic field. Later on we will use this peak in the conductance as a measurement tool to probe the spin-splitting in CNT quantum dots, note that the peak to peak distance in bias voltage is given by 2gµ B B/e, giving a peak to peak splitting of 230 µev/t. Even though the Kondo effect in quantum dots has been studied intensely for two decades, the conductance of a finite bias Kondo peak can still not be described with an analytic expression. The correlated many-body ground state that is the Kondo effect, is very sensitive to thermal fluctuation, and the temperature dependence is given by G(T ) = G 0 ( T 2 K T 2 K + T 2 ) s (3.5) where T K = T K /(2 1/s 1) 1/2, s = 0.22 is for a spin-1/2 system [45], G 0 is the maximum conductance of the Kondo peak, and G is the conductance as a function of the temperature T. Typical Kondo temperatures for CNT are in the range 1-3 K. 27

32 3.4 Kondo with ferromagnetic contacts Introduction The combination of ferromagnetism and the Kondo effect has long been an intriguing system and has recently received a lot of attention. From the Kondo model, it is clear that if the contact electrodes are made of a half-metal 1 and the spin direction of the electrons in the two electrodes are in a parallel configuration, the Kondo effect can not exist. In the case of a anti-parallel configuration, it is unclear if the Kondo effect exist. Since no metal is known to be a halfmetal an obvious question is, what if the spin polarization in the contacts are not 100 %? This has been studied intensively theoretically [26 30], but only a few experimental studies exist [25, 63]. In this section, we will not go into how the polarized electrodes change the Kondo effect 2, but instead study how the spin states in the dot are effected by the polarized contacts. This is done by studying the Anderson Hamiltonian and looking at how the energy levels are renormalized, going to second order in perturbation theory: First, in the case of unpolarized contacts in section 3.4.2, then turning to polarized contacts in section 3.4.3, adding the complication of a stoner gap. In the end, we will shortly touch the case of parallel and antiparallel magnetization direction of the contact domains Andersons model When an electron in a spin degenerate level in a quantum dot is brought into tunnel contact with a continuum of states in a metallic electrode, the wavefunction of the electron will be able to spread into the electrode. This extension of the wave function into the electrodes lowers the potential and kinetic energy of the electron, thereby renormalizing the energy of the original level. The Anderson model [50] can be used to describe such a system and calculate the energy renormalization of the dot level. The Anderson Hamiltonian is given by H = ε k,σ c k,σ c k,σ + ε d,σ n σ + Un n + k,σ σ k,σ }{{}}{{} H F S H dot (V k d σc k,σ + V k c k,σ d σ) } {{ } H (3.6) where H F S is the energy of the continuum, H dot is the energy of the dot, and H is the interacting part of the Hamiltonian. Here c k,σ, c k,σ are the creation and annihilation operators, respectively creating and annihilating an electron in the continuum with spin σ, wave-vector k and energy ε k,σ. ε d,σ is the energy of the single dot level relative to the Fermi energy where a negative value means that the level is placed below the Fermi energy of the contacts, n σ = d σd σ 1 A half-metal is a metal where the electrons at the Fermi surface only have one kind of spin. Even though they are easy to imagine and a lot of experimental work have been put into producing/finding such a material the work have so far been fruitless, at least in the sense that no half metal have yet been observed. 2 Later in this thesis the Kondo effect is used as a probe to study the spin-splitting. 28

33 is the number operator counting the electrons on the dot; d σ, d σ being the creation/annihilation operators for electrons on the dot. U is the Coulomb interaction energy for two electrons on the dot and V k is the tunneling amplitude between the energy level and the continuum, it cannot a priori be taken to be spin independent but we will make this approximation 3. Normally there are two electrodes coupling to the dot, the coupling to these two can be assumed to be symmetric and by a unitary transformation the coupling can be simplified to one, in such a way that we without lose of generality can work with the Hamiltonian in equation (3.6) [28]. Another less formal argument is that we could add an extra term H to the Hamiltonian looking as H but describing the coupling to the other electrode. This would lead to an extra term to the second order perturbation expansion 4, describing the coupling to the other electrode, but otherwise completely similar to the first second order correction (in the case of parallel magnetization this term would just add to the effect). Due to this, and for the sake of simplicity, we make the calculation with one electrode. In the end we will argue further why the result can be used straight away 5. As seen from equation (3.6) the non-renormalized energies of the system in the case of 0, 1 or 2 particles on the dot are given by: Particles on dot ( Total energy E (s) n ) Dot energy (E n ) 0 state 0, E (s) 0 = 0 + ε k,σ, E 0 = 0, 1 state 1, σ, E (s) 1,σ = ε d,σ, E 1,σ = ε d,σ, 2 state 2, E (s) 2 = ε d, + ε d, + U ε k,σ, E 2 = ε d, + ε d, + U, Where states 0, 1, σ and 2 represent both the state of the dot and the continuum of states in the Fermi sea. Note, that since it is energy differences that are interesting we can define that in the case of one electron on the dot the energy of the Fermi sea is zero, i.e. in the case of zero electrons on the dot there will be an extra in the Fermi sea, therefore the the energy ε k,σ for the state E (s) 0. Another important note is that since we are concerned about electron transport it is the chemical potentials µ(1, σ) = E 1,σ E 0 = ε d,σ which are interesting. As discussed in section 2.2 ε d,σ can be tuned by a gate. We will now look at how E 0 and E 1,σ renormalize to Ẽ0 and Ẽ1,σ respectively, when allowing for the coupling to the continuum. To second order in H we have Ẽ n = E n + I 1, σ H I I H 1, σ. (3.7) E n (s) E (s) I With the sum running over all intermediate states I. The only possible intermediate states I for E (s) 0 are one electron on the dot with either spin-up or 3 As can be seen further down in the text when we are discussing spin polarized leads, we put the spin dependence of the coupling Γ into the DOS ρ σ in the leads 4 Explicitly we would get an extra term in equation (3.7) with H instead of H. 5 At least at equilibrium, i.e. at zero bias voltage. 29

34 down and the only I-states for E (s) 1,σ are 0 or 2 electrons on the dot. E n is the energy of the bare level, the first order correction is zero since the initial and final state have to be identical. By writing out all the terms in the second order corrections we get Ẽ 0 = E 0 + k,σ Ẽ 1,σ = E 1,σ + k + k 0 V k c k,σ d σ 1, σ 1, σ V k d σc k,σ 0 E (s) 0 E (s) 1 1, σ V k d σc k,σ 0 0 V k c k,σ d σ 1, σ E (s) 1 E (s) 0 (3.8) 1, σ V k c k, σ d σ 2 2 V k d σc k, σ 1, σ. (3.9) E (s) 1 E (s) 2 Where σ is the spin opposite to σ. Changing from summation to integration k ρ σ (ε k )dε k where ρ σ (ε k ) is the density of states (DOS) and moving the spin indices σ we get Ẽ 0 = E π σ Ẽ 1,σ = E 1,σ + 1 π dε k f(ε k )Γ σ (ε k ) ε k ε d (3.10) dε k ( (1 f(εk ))Γ σ (ε k ) ε d ε k f(ε k)γ σ (ε k ) ε d + U ε k ). (3.11) Here Γ σ (ε k ) = πρ σ (ε k ) V k 2 and f(ε k ) is the Fermi function added due to c, c, acting on the Fermi sea, to take into account the probability of the states in the continuum being empty or occupied. ε d,σ/ σ is changed to ε d since, the bare dot level is spin-degenerate. The integral is over all available states; in real metals this will be restricted to the band from which the electrons tunnel onto the dot. For a flat/constant, electron-hole symmetric DOS, with an upper and lower bound of the DOS given by ±D called the band-width, under the assumption that D ε d, U and at T = 0, we get Ẽ 0 E 0 1 π Ẽ 1,σ E 1,σ 1 π σ Γ σ ln D ε d ( Γ σ ln D ε d + Γ σ ln ) D ε d + U (3.12) (3.13) Since we have D > ε d, ε d + U the states will be renormalized down in energy as illustrated in figure 3.4. Since transport is taking place through levels defined by chemical potentials it is the quantity µ(0 1, σ) = Ẽ1,σ Ẽ0 = ε d,σ which is interesting. An important point is that the energy Ẽ0 seen in equation (3.12) has no spin dependence, (the summation is over both spin states) Renormalization with spin polarized electrodes If the electrodes are spin polarized the energy levels Ẽ1, and Ẽ1, will renormalize differently due to the different DOS for spin-up and spin-down, in the 30

35 ε F Continuum Γ a b c E 0 E 0 εd E 1,σ ε d E 1,σ E 0 E 1 ε d E 1 Figure 3.4: Illustration of Andersons model. The three different areas a, b and c describe different regimes. a, The bare energy level of the quantum dot E 0 is the energy of the dot with zero electrons, E 1σ is the energy with one electron and ε d is the energy difference between these two levels. b, The renormalization of the energy states when the quantum dot is coupled to a continuum. c, the renormalization in the case of a spin-polarized continuum, note that the level E 0 does not split, while E 1 splits in a spin-up and spin-down component. electrodes. One could say that the ferromagnetic electrodes induce a local exchange field B ex in the dot by the tunnel coupling. Lets start by assuming the simplest case, where we have a flat electron-hole symmetric density of states but ρ > ρ as illustrated in figure 3.5a, with spin-up as the majority spin direction. What we want to calculate is the spin-splitting ε d δε d δε d where δε d = ε d ε d. Since Ẽ0 has no spin dependence we have ε d = 1 π = P Γ π ln ( Γ ln D ε d Γ ln ) ( εd ε d + U D ε d + U + Γ ln D ε d + Γ ln ) D ε d + U (3.14) Where Γ = Γ +Γ is the total coupling and P = Γ Γ Γ is the spin polarization. From this equation, it can be seen that there is a logarithmic divergence for ε d 0 and ε d U and that this divergence gives a spin-splitting with spinup as the lowest level for ε d close to zero and with spin-down as the lowest level for ε d close to U, due to electron-hole symmetry in this approximation the splitting for ε d = U/2 is zero. It can be seen that by tuning the gate such that ε d goes from 0 to U/2 to U the splitting is tuned and the ground state is changed from spin-up to degenerate to spin-down. The physical picture one should have in mind is different depending on ε d. For ε d close to 0 the renormalization is primary due to one particle processes, i.e. we are close to the 0 1 degeneracy point and the two particle state is far away in energy. Therefore the most likely process is that the electron on the dot jumps out and one jumps in, since the spin-up 31

36 a b c ε k +D -D ρ σ D -D ε k D+ /2 D- /2 ρ σ -D+ /2 -D- /2 Figure 3.5: Spin polarized electrodes. a, DOS for a flat band with spin-polarization. b, spin-splitting as a function of gate voltage or ε d. The black lines outline a Coulomb diamond as a function of bias and gate, the number indicates number of electrons on the dot, red and blue line show the Kondo ridges. The separation of the ridges corresponds to two times the spin-splitting. c, DOS for a flat band with spin-polarization and a stoner splitting. electron has a higher probability to do this (due to the higher density of states for spin-up in the electrodes), the spin-up electron state will be lowered the most. For ε d close to U we are close to the 1 2 degeneracy point, and far away from the zero particle state, in this case the most likely process is for an extra electron to tunnel onto the dot, into a two particle state, and then of. To maximize the lowering of energy of this two particle state, the tunneling process should happen as often as possible hence it should be the electrons with spin-up that tunnels on and then of the dot. This is only allowed when the electron residing on the dot is a spin-down, and consequently it is the spin-down state which is lowered the most. The situation is illustrated in figure 3.5b. Stoner splitting In the previous section we were studying how a flat spin polarized density of states was affecting the spin-up and spin-down states on the dot. We will now add the complication of a stoner gap. The stoner gap shifts the density of states for spin-up and spin-down as seen in figure 3.5c. If we take this into account and make the same calculations as in the previous section we get ε d = P Γ π ln ( ) εd ε d + U + Γ ( ) D + /2 + π ln εd + U Γ ( ) D /2 + D /2 ε d π ln εd + U (3.15) D + /2 ε d }{{} st Here it was assumed that the continuums DOS with spin-up is shifted down in energy by /2 and the DOS with spin-down is shifted up. The first term in equation (3.15) is identical to the one seen in the case with no stoner splitting. the last two terms are the result of the stoner splitting lets call them st, i.e. 32

37 st is the resulting splitting of the spin-states due to the stoner splitting. st can be rewritten as: st = Γ ( [ ] (D + /2 εd )(D + /2 + ε d + U) ln 2π (D /2 ε d )(D /2 + ε d + U) [ ] ) (D /2 + εd + U)(D + /2 + ε d + U) + P ln (3.16) (D /2 ε d )(D + /2 ε d ) where the last term is zero for ε d = U/2. Another observation is that we will normally have D, D /2 ε d, U. In this case we can simplify equation (3.16) to st Γ ( ) D + /2 π ln Γ (3.17) D /2 π D showing that the gate dependence due to the stoner splitting will be very small but a constant offset in the spin-splitting of the dot level can be expected due to the stoner splitting. This offset in the splitting would result in a displacement of the crossing of the spin states to another value than ε d = U/2. External magnetic field will perturb the bands similar to the stoner splitting, i.e. by shifting the spin-up and spin-down states relative to each other, although normally on a much smaller scale, since the internal magnetic field in a ferromagnet normally is much bigger than the external fields we impose on the system Two contacts We have in the discussion so far, only been looking at the system as consisting of only one contact. In the devices we measure on this is not the case, we have two contacts to the dot. In this case we have to take an extra look at equation (3.14) that describes a coupling to a single electrode, but a second coupling would look similar, so we can use the equation as stated just by noting that Γ = Γ s + Γ d where Γ s = Γ s + Γ s and Γ d = Γ d + Γ d describes the coupling to the source and drain, respectively. This is strictly speaking only valid in equilibrium at zero bias since the chemical potentials in source and drain have to be at the same value for ε d to be well defined. Although this should be taken into account as done in reference [65] we will, since the bias applied when observing the spin-splitting is small, make the approximation that equation (3.14) is valid as stated Parallel and anti-parallel domain magnetization In the case of parallel magnetization direction of the two contact domains, the splitting of the spin-levels due to the exchange field will point in the same direction. In the case of anti-parallel configuration will the exchange field from the contacts be directed in opposite directions. In the case of symmetric coupling to the two electrodes Γ s = Γ d (with the same polarization), are the exchange 33

38 fields from each electrode of equal size and will cancel each other, resulting in zero splitting of the spin states. In the case of asymmetric couplings is the resulting splitting a measure of the asymmetry of the couplings. Since the Kondo effect gives a peak in the conductance when the bias applied over the dot fits the splitting of the spin states we can use the Kondo effect as a probe to find the splitting. Since the splitting is due to a combination of the renormalization of the spin states and the external magnetic field, the peak to peak splitting in bias V is given by e V = 2 gµ B B + a P r Γ r (3.18) r=s,d Here a is a constant that can be determined theoretically with the help of equation (3.14). For two electrodes of the same material we get splittings for the parallel and anti-parallel configuration corresponding to: e V P = ±2gµ B B + 2aP (Γ s + Γ d ) (3.19) e V AP = ±2gµ B B + 2aP (Γ s Γ d ) (3.20) Here the Zeeman splitting is added or subtracted depending on whether the external magnetic field is compensating or strengthening the exchange field. Using these equations, the asymmetry of the coupling Γ s /Γ d can be determined. 34

39 Chapter 4 Experimental methods In this chapter is the experimental methods described. First the carbon nanotubes (CNT) fabrication is described followed by the lithographic details, before turning to how the samples are made. In the end the electrical setup and the cryogenic system used for the measurements will be described. 4.1 Sample fabrication There are three standard ways to produce CNT arc discharge [73], laser ablation [74] and Chemical Vapor Deposition (CVD). We will concentrate on the describing the CVD method since all the tubes measured in this part of the thesis is made by CVD. The reasons for choosing CVD grown tubes is: 1. They can be made in house, in figure 4.1 a picture of the CVD system can be seen. 2. They are grown right on the wafer i.e. no post-grown processing such as etching and sonication are necessary 1, these processes have a tendency to introduce defects in the CNT. 3. The CNT are single tubes, i.e. not bundled up into ropes as is often the case by tubes that are dispersed. 4. By placing the catalyst in desired position, the CVD growth also give some control over the position of the CNT. This makes it possible to avoid the time consuming step of finding the tubes by atomic force microscopy (AFM). 1 Tubes grown with arc discharge and laser ablation normally have to be purified, this is normally done by etching, and disentanglement of the ropes of tubes are done by sonication. 35

40 Figure 4.1: Chemical vapour deposition (CVD) oven Chemical Vapour Deposition (CVD) CVD grown tubes are made by placing a substrate with catalyst in a furnace, heat it to C and then pass a feedstock gas over the substrate. The catalyst decomposes the feedstock gas and catalyst the growth of the tubes. The substrate used for the growth is a highly doped Si chip capped with a 500 nm thick layer of SiO 2. The catalyst are made of iron nitrate, molybdenum acetate and alumina (support) particles and is by lithographic methods placed as islands on the sample, with a typical island size of µm. The sample is then placed in the furnace, which is flushed with Ar before heating it to C in a constant Ar and H flow, when the temperature is reached the feedstock gas methane CH 4 is added to the flow giving a flow that is a mixture of all three gasses (with flow rates 1, 0.1, 0.5 L/min for Ar, H 2, CH 4 respectively). After min the growth is interrupted by turning off the methane flow and the sample is cooled. Methane is chosen as the feedstock gas since it is the most stable of the hydrocarbons. The stability have to be high to increase the percentage of gas being decomposed by the catalyst contra the natural decomposition of the gas. After growth is the sample loaded in a scanning electron microscope (SEM) and a test area is searched for tubes. If it has the desired density further processing on the sample are initiated. If the density of tubes are to low another growth is made 2 with a slightly changed temperature. If the density is to high one have to start over with a new sample 3. It is very important to note that the quality of the CVD grown tubes are crucial for the measurements. Another procedure for producing CNT in the CVD is described in section 9.2. This method is not used for three reasons; 2 The density of tubes seems to depend on the exact growth condition such a precise island size and catalyst thickness, amount of gas in the pressure bottle. Exact control of these parameters are hard to achieve and so it is a trial and error process. 3 Sometimes it is enough to place the electrodes further away from the catalyst islands, since fewer tubes get far away from the islands. 36

41 a b c Sample Resist Exposure Double layer resist UV-light or E-Beam Si ++ backgate SiO 2 d e f Developed Metallization Lift-off Figure 4.2: Illustration of the lithography process. 1. It produces very long tubes, that sometimes short-circuits between the UV-electrodes making it hard to interpret what is measured. 2. Since the catalyse in this process is scattered all over the sample the tubes have to be found by atomic force microscope (AFM) to align the electrodes to them. This is a very time consuming. 3. It seems like the quality of these tubes are not as high as the method previous described, at least are the measurements made on these kind of tubes rarely produces nice results, this could be due to less optimization of this process or just bad luck, but nevertheless was the first CVD-method chosen for the samples produced and measured in this part of the thesis Lithography Basically two types of lithography is used; UV-lithography (UVL) where ultraviolet light shinned through a mask is used to expose the resist and electron beam lithography (EBL) where an electron beam microscope equipped with a Raith controller is used to draw patterns in the resist. An illustration of the lithography process can be seen in figure 4.2. We start with a sample that are covered by two layers of resist which are essential long polymers, that are either sensitive to UV-light or a electron beam (E-beam). The resist is then exposed to UV-light or E-beam which chop the long polymer chains into shorter pieces. The UV-light is shinned on the sample through a mask with the wanted features, while in the case of E-beam the wanted pattern is drawn in the resist by the beam. Normally the bottom layer of resist is most sensitive to the exposure, i.e. a larger area will be exposed. The sample is then dipped in a developer that removes the resist, but due to the shorter chains of polymer where the sample has been exposed the resist in these areas will be removed first, and due to the higher sensitivity of the bottom layer the result look as seen in figure 4.2d, where the top layer have created a overhang. This is an advantage at the lift-off 37

42 since there often during metallization is a misalignment of sample and source 4. After the metallization the unwanted metal is removed normally by immersing the sample in acetone or another liquid removing the resist and the metal on top of the this. It should be noted that before the metallization the sample is normally plasma etched to remove any resist residuals in the exposed areas. This is especially important with UV-lithography which for some reason have more resist residuals than electron beam lithography. A more detailed recipe for UV and E-beam lithography is given beneath. UV-lithography In UV-lithography we can use double layer and single layer of resist, both types was used in the processing. The UV-lithography machine used for this is a Karl Suss MJB3. Double layer resist UV-lithography is made by first a layer of LOR3B spun on at 4000 rpm for 45 s then baked (on a hot plate) at 185 C for 4 min (dehydration), one can to some extent control the undercut by the baking time, longer time less undercut. This is followed by a top layer of AZ1505 spun on at 4000 rpm for 45 s, then backed at 115 C for 45 s. Before exposure it is a good ide to remove beads at edge of the sample, by exposing it for 1 min and develop in AZ400K (thinned 1:4 in H 2 O) for 90 s and flush in water. This is done to be able to get the sample up close to the mask without any misalignment due to edge effects. The sample is then exposed through the desired mask for 5 s (Intensity: 7.5 mw/cm2, 365nm wavelength) and developed it for 25 s in AZ400K (thinned 1:4 in H 2 O) and flush in water. The sample is then plasma etching before metallization 25 s. The lift-off is done with remover PG. How long time this take can varies a lot but with the double layer it is normally pretty fast 5 20 min the lift-off works best if the remover PG is heated. Ultra sonication can be used to speed up the lift-off process but can only be used before there are tubes on the sample since it can introduce defect in these. Single layer UV-resist, spin on AZ4511 resist at a speed of 4000 rpm for 40 s (speed and time determines the thickness) bake it at 115 C for 45 s. Exposed the sample for 10 s develop in AZ400K for 60 s, flush with water and then plasma etch it for 30 s. After the metallization the lift-off process can take anywhere between 20 min to 24 hours where the sample is submerged in acetone. This process are used as the UV-lithography method when there are CNTs on the sample, experience have shown that the following measurements behave nicely. This is still unknown for the double layer resist process therefore this process have been avoided when there is CNTs on the sample. E-Beam lithography In E-beam lithography we always use a double layer of resist. The recipe given beneath is used for making contact electrodes to the 4 The overhang minimize the risk of a connection between the metal on the resist and on the sample. Such a connection will increase the risk of a failed lift-off. 38

43 CNTs. First a layer of 6% Copolymer is spun on at 4000 rpm for 45 s give a thickness of around 100 nm, backed afterwards on a hotplate at 185 C for 45 s (9% Copolymer can be used if the thickness should be great but with a lower resolution at the exposure). Second layer is 2% PMMA (poly-methyl-methaacrylate, also called plexiglass) spun on at 4000 rpm for 45 s giving a thickness of around 50 nm, backed afterwards on a hotplate at 185 C for 45 s (4% PMMA can be used for larger structures). Make a scratch in a corner of the sample for focussing in the SEM. The sample is then exposed using a JEOL field emission electron microscope (model JSM-6320F) with an acceleration voltage of 30 kv and a current around 30 pa exposing the sample with a sensitivity of 200 µc/cm 2, in a pre drawn pattern. The sample is then developed in mixture of MIBK and isopropanol (IPA) (MIBK:IPA 1:3) for 45 s. No plasma etch is used here since it would destroy the tubes we are trying to contact. The lift-off is done in acetone Steps to make a sample The list below shows the steps used to make the CNT samples with ferromagnetic Ni contacts. For technical and detailed recipe see appendix C. The substrates used for the samples are made of highly doped silicon used as a backgate covered with a 500 nm thick SiO 2 layer. 1. Step one is to make alignment marks, this is made by using standard 2- layers resist and UV lithography 5. The metal used are Cr, Au would be easier to see in the alignment procedures to come, but cannot withstand the elevated temperatures under the CVD-growth. The thickness of the evaporated Cr should be at least 70 nm thick, to make it visible in the scanning electron microscope through the double layer of resist used with electron beam lithography. 2. Catalyst islands can be made both with 2-layers UV lithography (UVL) and Electron-beam lithography (EBL) made in the SEM. The islands are more precisely defined with EBL. But it is faster and can be defined for a larger area with UV, so unless the following lithography steps have to be placed very precisely, i.e. with a position below nm UV-lithography is used. For normal CNT devices the alignment are not that crucial since the metal electrodes are placed 1 2 µm from the catalyst islands. The catalyst is dissolved in methanol which also have a tendency to dissolve the top layer of the UV-resist, fortunately the bottom layer withstands this treatment. 3. CVD-growth of the CNTs as described previously. 4. SEM check of the test areas to see the density of CNT. Have to be on the test site since high voltage>10 kv can damage the tubes. It have later 5 It is hard with UV-lithography to get really sharp edges, but as long as the further alignment do not have to be more precise than around nm UV will be sufficient, else one have to use electron beam lithography. 39

44 come to my attention that it should be possible to do with 1 kv acceleration voltage without damaging the tubes. Another possibility would be to scan the area with an AFM but this is very time consuming and a higher throughput (of samples) is obtained by just looking at the test areas in the SEM The contacts to the CNT are made by SEM lithography, the pattern used for ferromagnetic Ni contacts can be seen in figure 4.3. The 50 nm thick Ni contacts is evaporated, with a pressure in the metallization chamber of around torr and with a rate of around 1 3 Å/s. Note that Ni will have a tendency to coil up if the evaporated layer get to thick nm and will then largely be removed in the lift-off process. 6. Bonding pads are made by using a 1-layer UV-lithography and evaporating Cr/Au 15/120 nm where the Cr is used as a sticking layer, remember to plasma etch the sample, before metallization, to make bonding of the sample possible. 7. Probing the sample. 8. Gluing the sample to a chip carrier and bonding. The selection of which samples to bond are made from the electrical measurements made in the probe station. The resistance of the device makes it possible to make a educated guess of which conduction regime the device will be in at low temperatures i.e. dead, Coulomb blockade, Kondo or Fabry-Perot. Although a to high conductivity can also be a sign that there is more than one CNT lying between the leads. After the bonding the chip is cooled in a Kelvinox cryostat which is 3 He/ 4 He dilution cryostat, for a more detailed description see section From the electrical measurements it is then possible to determine if there is one or more CNT connecting the leads, in case of doubt this can later be checked in the AFM. The mask for the SEM-lithography can be seen in figure 4.3 the green area is the alignment mark the turquoise the catalyst and the red the electrodes. The CNTs grows from the catalyst islands and the electrodes are then placed nearby with the hope that they are placed on a single CNT. There are four electrodes placed in close proximity each with either different with or length. The plan was to make spin-transport measurements and with four electrodes there was also a chance to make four probe measurements if the devices behaved nicely enough as done in reference [37] where spin and charge separation are observed. The reason for using Ni as the ferromagnetic material was to get a good contact to the CNTs. In the device fabrication 4-6 areas, like the one just described, are processed in parallel. The center of each looks schematically as seen in figure 4.3. For pictures of a real device see figure Even with very optimized sample processes, is the yield low and therefore a high throughput is essential to get samples that works. 40

45 Alignment marks Catalyst Electrodes Contacts to bonding-pads electrodes Figure 4.3: SEM-pattern. The figure show the pattern the SEM-electrodes are exposed from. On the figure the placement of the alignment marks and catalyst islands can also be seen. Contact to bonding-pads electrodes are where the UV-electrodes are connected to the SEM-pattern. a b Bonding pads d c Electrodes Catalyst islands Figure 4.4: Optical microscope pictures of a real device. The connection between the pictures should be self-explaining. 41

46 DAC0 (DC bias Vb) ~ (AC signal) Opto coupler 1000: :1 Sample Ni B Ni SiO 2 Si ++ backgate Field direction Current amp. I DAC1 (Gate Vg) Lock-in amplifier 10 MΩ ADC0 (DC current I) ADC1 (di/dv) Figure 4.5: Schematically diagram of the electrically AC-setup used in the measurements. A AC signal is superimposed on a DC signal in the grey-box that divides the DC signal with 1000 and the AC signal with The signal goes through the sample and the current amplifier to the lock-in, where the differential conductance di/dv are detected. A corresponding voltage is then feet to the computer through the ADC1 port. Under many of the measurements a magnetic field was imposed on the sample, with a field direction along the electrodes. 4.2 Measurement setup The measurement setup are consisting of two parts; a part describing the electrical setup and one concerning the cryogenics Electrical setup Most of the electrical measurements presented in this thesis on CNTs with ferromagnetic contacts is voltage controlled using a standard lock-in AC-setup technique. A schematic drawing of the setup can be seen in figure 4.5. The measurements are controlled through a computer using the software lab-view, via a data acquisition (DAQ) card (National instruments BNC-2090). The input/output can handle voltages in the range ±10 V through the DAC output and ADC input channels. To reduce noise is the sample circuit isolated from the surroundings by the battery driven opto-couplers (isolation amplifiers with unity gain). The DC signal is applied via DAC0, the DAQ gives a voltage output in the [V] range. The source drain voltages used for the sample is in the [mv] range, so to improve the resolution and avoid mistakes is the signal divided 1:1000. This happens in the AC-DC adder box where the AC and DC signal is added together. Here the AC signal is also divided by since the excitation voltages used is in the [µv] range. The AC signal is supplied from the lock-in amplifier. The current is measured in a battery driven current amplifier, from which a corresponding voltage is feet to the ADC0 port in the computer and the lock-in amplifier. From the ADC0 port is the DC current logged and 42

47 DAC0 (DC bias Vb) Opto coupler 100 kω 100 Ω Sample Ni B Ni SiO 2 Si ++ backgate Field direction Current amp. I DAC1 (Gate Vg) 10 MΩ ADC0 (DC current I) Figure 4.6: Schematically diagram of the electrically DC-setup. the lock-in amplifier feet the differential conductance to the ADC1 port. The DAC1 directly controls the gate potential through a 10 MΩ resistance put there to protect the sample in the case of gate-leak. To minimize noise care is taken to keep the wires from the AC/DC-adder box to the sample, and from the sample to the current amplifier as short as possible. This is very important in spite of the fact that the wires are coax-cables and the measurements are done in a screened room. Some measurements were done with the DC setup shown in figure 4.6. The gray box is a 1:1000 voltage divider. Most os the measurements shown in this part of the thesis are made with AC-setup but the samples measured were first screened with a DC-setup 7. This primary is due to two things; less calibration and it seems that you can measure faster with the DC-setup, although the result will not be as nice as with the AC-setup Cryogenics The Kondo effect and other physical phenomena studied in this thesis are either only observable at low temperatures or are getting more pronounced/clearer as the temperature is lowered. The normal Kondo temperature in CNT are around 1-3 K while the charging energy for CNT quantum dots are typically around 4-10 mev with excitation energies around and below 1 mev, to observe features in this energy range and to get them as pronounced as possible most of the measurements presented in this part of the thesis are made in the millie Kelvin temperature range. 7 In our CNT quantum dot devices are measured in a gate voltage range between ±10 V there are several hundred Coulomb diamonds, and one have to do a screening to find the areas with interesting or nicely looking diamonds. 43

48 3 He gas Condenser line 1K-pot 4 He Flow impedance 3 He liquid Mixing chamber Pump Still line Still Heat exchange 3 He/ 4 He liquid Figure 4.7: Schematic drawing of the closed 3 He/ 4 He cooling part of the Kelvinox system. The rotary pump circulate the 3 He/ 4 He mixture around in the closed system in the direction of the arrows. When the system is up and running it is mainly 3 He that are circulated. The 3 He gas is condensed in the 1 K-pot where the temperature is around 1.2 K, archived by pumping on 4 He, the pressure in the 1 K-pot is high, due to the pump and the flow impedance. The condensed liquid 3 He is pushed through the flow impedance down to the mixing chamber on the way passing the heat exchangers. In the mixing chamber is the liquid 3 He pumped into the mixed liquid 3 He/ 4 He phase, this crossing of the phase boundary between 3 He and 3 He/ 4 He is cooling the mixing chamber and thereby the sample down to a base temperature of 30 mk. The figure is adapted and modified from [10]. Kelvinox To reach such low temperatures a cryogenics system is needed, the system used in this thesis is a Kelvinox dilution refrigerator from Oxford Instruments, which is a 3 He/ 4 He dilution refrigerator, with a superconducting magnet with a maximum field of ±7 T. The kelvinox has a base temperature of 30 mk and a corresponding minimum electron temperature of 80 mk. The basic cooling mechanism is illustrated in figure 4.7. In a closed circuit is a mixture of the two Helium isotopes 3 He and 4 He pumped around. At low temperatures will the mixed liquid He phase separate into a 3 He phase and a 3 He/ 4 He mixture phase (with 6% 3 He). At low temperatures it is mainly 3 He that are pumped around in the system. The 3 He is circulated by pumping on the still (see figure 4.7) containing liquid 3 He/ 4 He mixture, due to the lower boiling point of 3 He it is mainly this species that are evaporated. To uphold the 6% concentration of 3 He in the mixture, 3 He diffuses from the pure 3 He phase to the mixture, it is this crossing of the phase boundary between 3 He and 3 He/ 4 He which is cooling the system. After the 3 He have been evaporated in the Still it flows to the 44

49 1 K-pot where it is condensed, and in the liquid phase running through a flow impedance and heat exchanger before ending in the mixing chamber where it again can cross the pure 3 He mixed 3 He/ 4 He phase boundary. The sample is placed in a inner vacuum chamber with a good thermal connection to the mixing chamber. The mixing chamber is made by plastic to avoid eddy currents when a magnetic field are applied. A disadvantage of this is that the plastic is transparent for He at room temperature so all the gas have to be pumped out when the system is heated. For detailed instructions for how to run the Kelvinox see [11], appendix A in [10] and the Kelvinox manual. 45

50 46

51 Chapter 5 Electric-field controlled spin reversal in a quantum dot with ferromagnetic contacts 5.1 Introduction Manipulation of the spin-states of a quantum dot by purely electrical means is a highly desirable property of fundamental importance for the development of spintronic devices such as spin-filters, spin-transistors and single-spin memory as well as for solid-state qubits [18 24]. An electrically gated quantum dot in the Coulomb blockade regime can be tuned to hold a single unpaired spin- 1/2, which is routinely spin-polarized by an applied magnetic field [36]. Using ferromagnetic electrodes, however, the properties of the quantum dot become directly spin-dependent and it has been demonstrated that the ferromagnetic electrodes induce a local exchange field which polarizes the localized spin in the absence of any external fields [25, 26]. In this chapter we report on the experimental realization of this tunneling-induced spin-splitting in a carbon nanotube quantum dot coupled to ferromagnetic nickel-electrodes. We study the intermediate coupling regime in which single-electron states remain well defined, but with sufficiently good tunnel-contacts to give rise to a sizable exchange field. Since charge transport in this regime is dominated by the Kondo-effect, we can utilize this sharp many-body resonance to read off the local spin-polarization from the measured bias-spectroscopy. We show that the exchange field can be compensated by an external magnetic field, thus restoring a zero-bias Kondoresonance [26], and we demonstrate that the exchange field itself, and hence the local spin-polarization, can be tuned and reversed merely by tuning the gate-voltage [27, 28]. This demonstrates a very direct electrical control over the spin-state of a quantum dot which, in contrast to an applied magnetic field, allows for rapid spin-reversal with a very localized addressing. 47

52 It is fortunate that the coupling regime where the Kondo effect exist is also the regime where there is a sizeable tunneling induced exchange field making it easy to use the Kondo effect to observe the splitting of the spin states. Since the discovery of carbon nanotubes they have been intensively studied for their unique electrical properties. Their high Fermi-velocity and low content of nuclear spins make them particularly well-suited for spintronics applications utilizing the transformation of spin-information into electrical signals. Spinvalve effects in nano-junctions with a carbon nanotube (CNT) spanning two ferromagnetic electrodes have already been observed [33 35], and a strong gatedependence of the tunnel magnetoresistance has been demonstrated for carbon nanotube quantum dots in both the Coulomb blockade, and the Fabry-Perot regime [33]. In the intermediate coupling regime, odd numbered quantum dots exhibit the Kondo effect seen as a pronounced zero-bias conductance peak at temperatures below a characteristic Kondo temperature, T K [43 45]. This effect relies on the conduction electrons being able to flip the spin of the dot during successive cotunneling-events and is therefore expected to be sensitive to spin-polarization of the electrodes. As pointed out by Martinek et al. [26] and discussed in section 3.4, will quantum charge-fluctuations of the dot renormalize the singleparticle energy-levels in a spin-dependent manner and thereby break the spindegeneracy on the dot, causing the zero-bias Kondo peak to split in two. This tunneling-induced exchange field splitting has since been seen by Patsupathy et al. [25] in an electromigrated Ni-gap holding a C 60 -molecule. But in this system there were no gate, making it impossible to enter the interesting area of gate controlled spin-splitting. The splitting of the spin states have also quite recently been observed in a InAs dot with Ni contacts but even though they had a back-gate, no clear gate dependence of the splitting [63] was observed. We start this chapter by looking briefly at the magnetic Ni electrodes, before showing a stability diagram of one of the measured devices. In section 5.3 we look at the exchange field in the CNT quantum dot induced by the tunnel contacts to the magnetic electrodes. We show that it is possible to compensate the exchange field in the case of collinear magnetization of the domains and the external field. In section 5.4 and section 5.5 we study the gate dependence of this exchange field. In section 5.6 the temperature dependence of the Kondo effect is briefly studied followed by a discussion of the results in section 5.7. Then a short offtrack is taken studying a singlet-triplet Kondo before the chapter is concluded. The results presented in this chapter is to a large extend also presented in [80] Magnetic domain size in the electrodes A ferromagnet electrode in nickel consist of a number of domains that at high field turn into one big domain aligned along the external magnetic field direction. As the field is lowered will the different domains turn in a direction dictated by the easy axis (which is weak in Ni so it is probably determined by geometry 48

53 a AFM - height b MFM - phase Ni electrodes Same four electrodes Figure 5.1: Scanning probe images. a, AFM-height images, the size of the image is µm and the height scale is 150 nm. The measured height of the electrodes are 65 nm. The CNTs measured upon are lying between electrodes similar to 1 and 2 (device 1) separation 415 nm and 2 and 3 (device 2) separation 200 nm. b, MFM-phase image of the same area as in a, the domains seems to be in the plane and have a size comparable to the electrode width giving a minimum size of roughly 375 nm. and surface states) and the interaction with the surrounding domains. At a given critical field it will be energetically favourable for some of the domains to change the primary magnetization direction to point in the opposite direction, this can be before or after zero field as discussed in the next chapter. As the external field is increased more domains will turn and in the end at high field the electrode will be one big domain pointing in the external field direction. In this section we will investigate if the typically domain size indicates that the CNT have contact to only one domain in each side. The ferromagnetic nickel electrodes are around 65 nm high and varies in thickness between 300 and 1000 nm AFM-images of electrodes similar 1 to those measured, can be seen in figure 5.1a. b is the corresponding magnetic force microscopy (MFM) image (note that the images are recorded on an electrode that have not been exposed to an external field). The MFM image is acquired using a magnetic AFM-tip, the scan is made by scanning one line at the time, first by standard tapping mode AFM recording the topography then retracing the same line again while keeping the tip at a fixed distance above the surface. In the second scan the phase difference between the driving frequency, which is at the resonant frequency of the cantilever, and the actually oscillation frequency of the tip is recorded. The phase depends on the force-gradient experienced by the cantilever thus on the long range interaction between the tip magnet and magnetic field lines from the sample. From the images it can be seen that 1 As can be seen in chapter 4 is there from the beginning 6 areas on each Si-chip, these are all treated in parallel and in the end, each sample area is cut out from the Si-chip and bonded up, so the samples have been through identical treatments. 49

54 the domains are in the plane of the Ni electrodes, the domain size is hard to determine but it seem like it is determined by the size of the electrode, i.e. for the thinnest electrode (number 3) the domain size is nm. For the other electrodes the domain size seems larger. Since CNTs have a diameter around 1-3 nm it is reasonable that the part of the Ni electrode contacting a CNT can be assumed to be a single domain. 5.2 Devices and quantum dots in the Kondo regime The CNTs measured on in this chapter were grown by chemical vapor deposition on a SiO 2 wafer with a highly doped Si back-gate. The ferromagnetic leads were made of pure Ni strips of thickness 60 nm, widths nm and separated by nm. All measurements were made in a 3 He/ 4 He dilution refrigerator, with a base-temperature of 30 mk, corresponding to a minimum electron temperature of 80 mk, and using a standard AC-setup with asymmetric bias. A magnetic field was applied in the direction of the Ni-leads in the plane of the electrodes. For a description of the experimental details see chapter 4. The measurement shown in this chapter is made on two different samples sharing a electrode and the back-gate. Different Kondos are measured on the two devices and they are named D1K1 for device 1 Kondo 1 and so on. In the devices we observed a clear even-odd effect with a zero-bias anomaly in every second Coulomb diamond indicating a spin-1/2 Kondo effect with a typical Kondo temperature, T K 1 K. Many of the observed Kondo anomalies showed a gate-dependent splitting and here we discuss different Kondos on the two devices for which this dependence is particularly clear. Figure 5.2a shows a typical measurement (stability diagram) of di/dv vs. V g and V b for such a device (device 2). 7 Coulomb diamonds can be observed and in every second a spin-1/2 Kondo can be seen, a splitting of the spin states can be observed in some of the Kondo resonances, note also the clear even odd effect. The four spin-1/2 Kondos is numbered and will be referred to as D2K1 to D2K4. The temperature dependence of the four Kondos can be observed in b showing di/dv as a function of V g for different temperatures, the arrows indicates that the conductance is increasing as the temperature is lowered. Actually this is not the completely true at very low temperatures around T 150 mk (depending on the size of the splitting) there is a decrease in the zero bias conductance due to the splitting of the spin states, this will be discussed further in section 5.6, it can be seen most clearly in the Kondo marked with the red arrow, probably due to the large zero field splitting of this Kondo. A bias cut of the Kondo D2K4 can be seen in c, showing a pronounced zero bias peak, black line, the full width at half maximum (FWHM) give a Kondo temperature of T K 1 K 2. The red line is made through a even diamond where the even number of electrons prevent 2 This is actually a very rough estimate since there might be a splitting of the Kondo peak, which is just not visible at this resolution. 50

55 a 4 di/dv [e2/h] Vb [mv] 0-2 di/dv [e2/h] - 4 b T=75mK T=100mK 7. 0 T=200mK T=400mK T=600mK T=800mK T=1K V g [V] V g [V] c di/dv [e2/h] T K V b [mv] Figure 5.2: Bias-spectroscopy plot(stability diagram) on device 2. a, Biasspectroscopy plot of device 2 showing differential conductance di/dv as a function of gate V g and bias V b voltage. 7 Coulomb diamonds can be observed with a spin-1/2 Kondo in every second diamond. c, Bias cuts at the gate voltages marked with arrows in a, black is a spin-1/2 Kondo hereford referred to as D2K4, the FWHM corresponds to a Kondo temperature on T K 1 K and red is at a even Kondo and no zero bias peak is observed. b, di/dv as a function of V g measured at different temperatures at zero bias voltage, the conductance can be seen to increase as the temperature is lowered in the Kondo areas, these are marked with arrows, the one marked with a red arrow is also a Kondo even though the zero bias conductance does not increase as the temperature is lowered, this is discussed in section 5.6. Note that the four Kondos observed in b are the same as the ones seen in a but are shifted in gate voltage due to gate switches. 51

56 the possibility of a spin-1/2 Kondo and no peak is observed at zero bias 3. The gate voltages where the bias cuts are made are marked with a black and red arrow in a. 5.3 Spin-splitting of the Kondo state at fixed gate-voltages As discussed in section 3.4 is a singly occupied level residing just below the Fermi-energy of the electrodes strongly shifted by virtual tunneling of electrons out of the dot, whereas a level deep below the Fermi-energy (by almost the charging-energy) is shifted by tunneling of electrons into the dot. For spinpolarized electrodes having a difference in the density of spin-up, and spindown states, this implies a spin-splitting of the dot level where the sign and magnitude depends on the applied gate-voltage, i.e. the position of the level below the Fermi-energy. At the particle-hole symmetric point right between the empty and doubly occupied states the spin-degeneracy is expected to be intact [30]. In a material like Ni, however, one expects the band structure to be energy dependent and to have a Stoner splitting. This will break the particlehole symmetry and therefore shift the spin-degeneracy point away from the middle of the diamond [27, 28] Spin-splitting and compensating field As discussed in section MFM images of devices similar to the ones measured, indicate that the CNT quantum dot is most likely coupled to one single domain in both source, and drain electrode. Applying a strong magnetic field serves partly to align the two contact-domains and partly to provide a Zeemansplitting of the local spin. Figure 5.3a,c shows the conductance vs. bias-voltage and external field, measured for gate-voltages tuned to the middle of an odd occupied Coulomb blockade valley in the Kondo regime (device D1K1 and D2K4). The splitting of the Kondo peaks in device D1K1 (figure 5.3a,b) exhibit a simple linear behaviour in which the single-domain magnetization and hence the exchange field, B ex, is aligned with the external field, B. Since the Kondo ridges is crossing at finite external field it can be seen, that at this gate voltage, the exchange field and the external field points in opposite directions. The exchange field can therefore be completely compensated and the zero-bias Kondo peak is seen to be restored at B ±1.12 T, giving an indirect measure of the exchange field at this gate-voltage. Thus in the region B > 1.12 T the external field is larger than the exchange field and the Zeeman splitting of the spin states will therefore be with a ground state spin along the external field direction (spinup). For B 1.12 T the spin states are degenerate and the zero-bias Kondo peak can be observed. For 0 T< B < 1.12 T the exchange field are stronger 3 The peak seen at V b 1.9 mv are due to singlet triplet Kondo and will be discussed further in section

57 a V b [mv] b V b [mv] di/dv [e 2 /h] c V b [mv] d V b [mv] B [mt] di/dv [e 2 /h] B [mt] Figure 5.3: Kondo peak splitting as a function of applied magnetic field at fixed gate-voltage. a, di/dv vs. V b and B measured on device 1 Kondo 1, i.e. called D1K1, sweeping B from -4 T to 4 T. The plot is recorded in the middle of an odd numbered spin-1/2 Kondo diamond. The observed splitting of the Kondo peak is due to the the exchange field B ex and external field B. At B ±1.12 T the external field is compensating the exchange field and restoring the zero-bias Kondo peak. A change in di/dv is observed at B 0 mt due to a switching of one of the two contactdomains. At B 80 mt the other domain has flipped and the device are back in a parallel configuration. b, Plot of the peak positions from a, offset with V b = 5 µv to symmetrize the plot. The black, and red lines are fits to the peak positions and have a slope of ± V/T corresponding to g 1.8. c, Plot as in a, measured on D2K4, sweeping from -2 T to 2 T. In this device the exchange field is never completely compensated, indicating a misalignment of the domain magnetization direction and the applied magnetic field, B. A domain-switch causes a jump in the splitting at B 40 mt. d, Peak positions from c, offset by V b = 12 µv. The black, and red lines are fits to the plots where the domain magnetization direction is at an angle of 25 to the external field (see text). 53

58 than the external field and the split states will therefore have a spin-down as the lowest energy state. In device 2 (figure 5.3c,d), on the other hand, the zero-bias Kondo peak is never fully restored and the splitting merely reaches a minimum at B ±0.6 T. The fact that the exchange field cannot be completely compensated by the external field in D2K4 is interpreted as a misalignment of B ex and B, and due to this it is impossible for the external field to compensate the exchange field. Since bulk Ni has only weak magnetic anisotropy, this is most likely due to the reduced geometry and surface roughness of the electrodes. The solid line in figure 5.3d is a fit to the B-dependence of the peak-splitting V by e V = gµ B B + B ex (5.1) where B ex has a fixed angle to the external field; we find this angle to be (B, B ex ) 25. We have also tried to make another fit with a finite anisotropy barrier, using a simple Stoner-Wohlfarth model where the magnetic energy of the system is given by E = K sin 2 (θ φ) BM cos(φ), (5.2) in terms of applied field B and domain magnetization M at a relative angle φ, together with an angle θ between an easy-axis of the electrodes and the applied field. The constant K parameterizes the anisotropy barrier, thus allowing the magnetization to align with B for sufficiently large values of BM/K. Minimizing this energy as a function of angle for a given applied field gives a hysteretic magnetization curve from which we can then infer the sum of applied, and exchange field to determine the spin-splitting as a function of B. Nevertheless, this fit performed no better than the far simpler fit to a fixed angle. The fit indicates that the exchange field has an angle of 25 degrees to the external field, which implies that the magnetization of at least one of the contact domain forms an angle 25 to B. The exchange field is given as the sum of the exchange fields from each electrode, i.e. B ex = ap s Γ s M s M s + ap dγ d M d M d (5.3) Where M s,d represent the magnetization vector of the source and drain contact domains, respectively. Since the two measured device share an electrode and the splitting of the Kondo peaks in D1K1 has a linear behaviour it seems plausible so assume that on device 2 one of the contacting domains are aligned along the field. The other contacting domain must then be at an angle 25 degrees to the field for the two contacting domains to give a resulting exchange field with an angle of 25 degrees to the external field Parallel and anti-parallel magnetization of the contact domains Figure 5.3 was recorded by sweeping the field from large negative to large positive values, a decrease(increase) of the splitting is clearly visible at small neg- 54

59 0.6 a B=-80mT b B=0mT c B=80mT di/dv [e 2 /h] V b [mv] Figure 5.4: Plots of di/dv as a function of bias voltage for B = 80 mt, B = 0 mt and B = 80 mt the plots are bias-cuts through the plot shown in figure 5.3a, measured on D1K1. The black curves are the measurements with a baseline subtracted. The green lines are Lorentzian fits to the peaks and the red lines are the resulting doublepeak fits. The center-to-center distance of the Lorentzians in each of the three plots gives the splitting of the spin-up and spin-down states on the dot. The splittings are 0.239, and mv for a, b and c, respectively, corresponding to domain configurations which are a parallel, b anti-parallel and c parallel. ative(positive) fields. A similar switching was observed in Ref. [25] and can be ascribed to a switching from parallel (P) to anti-parallel (AP) configuration of the contact-domains driven by domain interactions 4. In the AP-configuration the tunnel-induced exchange field is nearly canceled, unless the couplings to source, and drain-electrodes are very different. Following the discussion in section using equation (3.19) and (3.20) we get Γ s Γ d = V P 2gµ B B/e + V AP V P 2gµ B B/e V AP, (5.4) Where the magnetic field is subtracted in the parallel case since it is compensating the exchange field. Comparing now the exchange fields in P, and APconfiguration, from figure 5.4 using the average splitting at 80 mt and 80 mt for the parallel splitting, we deduce that Γ s /Γ d 3, this corresponds to a zerobias Kondo peak height of (2e 2 /h)4γ s Γ d /(Γ s + Γ d ) 2 = 1.5 e 2 /h roughly consistent to the value of 1.4 e 2 /h, measured on D1K1 in the middle of the Coulomb blockade diamond for a fully compensating magnetic field near B = ±1.12 T. 5.4 Spin-splitting as a function of gate In a Coulomb blockade diamond with an odd number of electrons, the left, and right charge-degeneracy points correspond to respectively emptying the dot or filling it by one extra electron. With a finite spin-polarization in the leads, tunneling of majority-spins, spin-up say, will be favoured by the higher-density of states and a dot-state of spin-up will therefore be shifted further down in 4 These domain interactions will be further discussed in section

60 energy than a spin-down state, as long as one is closer to the left hand side (l.h.s.) of the diamond. Due to the Pauli-principle, the majority-spins can only tunnel into the dot if the residing electron is in a spin-down state. It is therefore the spin-down state which is lowered the most near the right hand side (r.h.s.) of the diamond. This simple mechanism of level-renormalization, illustrated in figure 5.5a-c, is encoded in the exchange field, see section equation (3.14) and (3.15), given to good accuracy by ε ex (ε d ) = e 0 + (P Γ/π) ln( ε d / U + ε d ) (5.5) to second order in the tunneling-amplitude [28] and assuming a constant density of states. U is the charging-energy and ε d is the dot-level-position which is proportional to the gate-voltage. Notice the strong negative and positive logarithmic corrections for ε d close to 0 or U, respectively (corresponding to the left, and right borders of the diamond). In the middle of the diamond, ε d = U/2, the exchange field is zero except for a constant term, 0, which is ascribed partially to a Stoner-splitting between the spin-up and spin-down bands, corresponding to a movement of degeneracy point with the value st as found in section 3.4.3, and partially to the spin-splitting arising from spindependent interfacial phase shifts (SDIPS) picked up by dot electrons upon reflection on the ferromagnetic electrodes [40 42]. Actually there can be a gate dependence from SDIPS but to know the functional form of this a detailed knowledge of the interfaces are required, this is close to impossible to attain and since, as will shown later in this chapter, equation (5.5) fits the results very well we will neglect any gate dependence of the SDIPS. Notice that possible stray-fields from the magnetic contacts would also contribute to 0 with a gate-independent spin-splitting which can be of the order 100 mt [67], this will be discussed further in section 6.2. Depending on the magnitude of 0, equation (5.5) predicts a change of the ground state spin direction as the localized level is moved from ε d = 0 to ε d = U. Figure 5.5d-g show di/dv as a function of gate and bias voltage measured for four different external magnetic fields, the measurements are made on D1K1 5. Note that the domains magnetization direction at these high external magnetic fields B 1 T should be aligned with the external field as also observed in figure 5.3a. The white ridges seen on the plots figure 5.5d-g are due to the Kondo effect, mapping out the splitting of the spin states, a clear gate dependence can be observed. As predicted by equation (5.5) a spin reversal can be observed in figure 5.5d-g and occurs where the Kondo ridges cross, seen as a conductance peak (red dot). The movement of the conductance peak as function of magnetic field, observed in figure 5.5d-g, confirms that the ground-state spin can indeed be reversed by changing the gatevoltage. Furthermore, the direction of the motion, i.e. the conductance peak moves to the right as a function of increasing external magnetic field, shows that it is predominantly spin-up (i.e. the spin along the external magnetic field) 5 Note that these plots have been recorded prior to a switch in device 1, whereas all other measurements taken on device 1 were recorded after. We present these plots since they bring out most clearly the movement of the degeneracy point. The corresponding plots recorded after the gate-switch are shown in figure 5.6c-g and are, as can be seen, qualitatively similar. 56

61 a V b b 0 2 U 1 V g c V b [mv] d e f g V g [V] -1T -2T -3T -4T di/dv [e 2 /h] Figure 5.5: Gate dependence of the spin-states probed by the Kondo effect. a, Plot from d, with yellow dotted lines added to indicate the diamond edges. Numbers indicate the number of electrons in one orbital state. Arrows indicates the local spin ground-state which changes with the exchange field as the gate-voltage is varied. These spin ground-states on the dot are consistent with the observation in d-g, that the conductance peak (red spot) moves to the right as the field is increased. b, c, Illustration of the virtual tunneling-processes leading to a spin-up(down) ground-state in the left(right) side of the dot, the full drawn lines are filled states and the dotted lines are empty states. Upper and lower diagrams correspond to ground, and excited states. Red spin-up represents the majority s-electron-band in the leads, hybridizing the most with the dot-electrons. d-g, Plots of di/dv as a function of gate, and biasvoltage, measured on D1K1 in different external magnetic fields. The Kondo peak is clearly seen to have a gate-dependent splitting. Full compensation in the middle of the diamond is obtained close to B = 1T as seen in d. 57

62 electrons which tunnel on and off the dot, giving a ground-state with spin-up at the l.h.s of the diamond and spin-down at the r.h.s. This is not obvious from the band structure of Ni but confirms earlier observations in conventional superconducting tunnel-junctions [46, 48] indicating that the tunneling electrons in Ni have majority-spin, i.e. positive polarization. Most likely, the more mobile s-electrons taking part in the tunneling are spin-polarized by their hybridization with the localized d-electrons [47, 58]. 5.5 Movement of the degenerated state as a function of B In order to substantiate the detailed gate-voltage dependence of the exchange field, we have extracted the peak-positions from the plots in figures 5.6d-g, using only the most pronounced peak, and fitted them by ( ε ex (ε d ) gµ B B)/e with ε ex given by equation (5.5). The result is shown in figure 5.6a, the symbols are the peak positions and the solid lines are fit to these. The x-axis have been displaced and normalized with the charging energy. There are only three fits because B = ±1 T were placed on top of each other, the plot for B = 0 T is not included since the domain configuration at this magnetic field, as already discussed and further substantiated in chapter 6, probably is anti-parallel. Note that U have not been used as a fitting parameter but read of. The inset shows 0 gµ B B/e as a function of B, note that the point for B = 1 T is actually an average of B = ±1 T. The slope of the line corresponds to g = 2 as expected for a carbon nanotube. The measurement in figure 5.6b show di/dv as a function of gate and external magnetic field at zero bias, the plots seen in h are gate cuts, made at the two arrows seen on top of b. h black line are made at B = 4 T at this field, no sign of the Kondo effect is seen at zero bias due to the splitting of the spin states and the plot only show two Coulomb peaks, the red line is made at B = 1 T and there a pronounced Kondo peak can be observed between the two Coulomb peaks. It is this Kondo peak movement in gate voltage as a function of external magnetic field the white ridge in figure 5.6b tracks. Since the pronounced Kondo peak is a sign of degeneracy of the spin states, does the white ridge map out the movement of the degeneracy point between the two nearly horizontal lines, corresponding to two Coulomb peaks. The plot have a gratifying resemblance to the theory-plots presented in Refs. [27, 28]. Inverting equation (5.5) we get ε d = U/(e (e 0 gµ B B)π/P Γ + 1) c (5.6) c is a constant determined by the first Coulomb peak, U is the charging energy and using the fitting-parameters deduced from figure 5.6a, we arrive at the parameter free fit seen as the black line in figure 5.6b, tracing the degeneracy point of the spin state. 58

63 a V b [mv] b V g [mv] B [mt] gµ B B/e [mv] B=0.5T B=1T B=2T ε d /U di/dv [e 2 /h] V b [mv] di/dv [e 2 /h] h c d e f g -1T 0T 0.5T 1T 2T V g [V] di/dv [e 2 /h] B [mt] V g [V] Figure 5.6: Gate dependence of the exchange field for different applied magnetic fields. a, Scatter-plot of the dominant Kondo peak position, in bias-voltage, as a function of level-position ε d ( V g ) gate-voltage for D1K1. Peak-positions are read off from the plots c-g. The different colours in a correspond to different magnetic fields. The gate-axis has been displaced and normalized with the charging energy such that the dot is emptied at ε d = 0 and becomes doubly occupied at ε d = U. Full lines are fits to the data, using ( ε ex gµ BB)/e defined by equation (5.5) and with g = 2. All three curves are fitted by P Γ = π. The inset shows a fit to the gate-independent term 0 gµ BB/e with 0 = 0.18 mv. b, di/dv as a function of gate-voltage and magnetic field measured at zero bias voltage. The two nearly horizontal lines are Coulomb peaks and the white line moving from the lower to the upper peak as B is lowered, is a Kondo peak, thus mapping out where the spin states are degenerate. The black line is described by the function ε d = U/(e (e 0 gµ B B)π/P Γ + 1) c where c is a constant determined by the first Coulomb peak and U is the charging energy. c and U are read off from the plot while the other constants are determined from the fits in a. Notice that, like in figure 5.3, a domain switch is seen at B 80 mt. c-g, Are plots similar to the ones seen in figure 5.5d-g. h, Gate-cuts through b at B = 4 T and B = 1 T for the black and red line respectively. 59

64 di/dv[e 2 /h] 0.6 a T=75mK b 1.05 c T=90mK T=75mK T=100mK T=150mK T=90mK T=100mK 0.90 T=300mK T=200mK 0.6 T=150mK T=200mK T=400mK T=300mK 0.75 T=500mK T=600mK T=800mK 0.60 T=1K di/dv[e 2 /h] di/dv[e 2 /h] V b [mv] V b [mv] D1K1 D2K T [mk] Figure 5.7: Temperature dependence of the Kondo Peak. a, di/dv as a function of bias voltage for different temperatures at fixed gate voltage, it can be seen that the peak splits in two around T 150 mk. b, zoom on the splitting at low temperatures, note that the zero bias conductance decrease as a function of temperature, at low temperatures. c, temperature dependence for D1K1 (squares) and D2K4 (circles), the red lines are fits to the points using equation (3.5) with the parameters; D1K1, G 0 = 1 e 2 /h, T K = 2.2 K, D2K4, G 0 = 0.62 e 2 /h, T K = 1.5 K. The decrease in conductance at low temperatures are beyond this simple fit (see text). 5.6 Temperature dependence of the splitting Having established that there is a gate dependent splitting of the spin states we will now shortly return to temperature measurements of the Kondo peaks. Normally, for degenerated spin states, the temperature dependence of the Kondo peak is described with equation (3.5), giving an increasing conductance that in the end saturates as the temperature is lowered. For split spin states the behaviour follows the, normal temperature dependence for T > T t where T t is a threshold describing when the temperature is low enough that the Kondo peak will start to split in two. When T t is reached the Kondo peak starts to split in two this implies that the zero bias conductance decreases. According to [60] the Kondo peak will split at T = 0 when the magnetic field B s corresponding to the splitting of the spin states, fulfill B s > 0.5k B T K /gµ B. The splitting as a function of temperature can be observed in figure 5.7. In a the conductance can be seen to increase as the temperature is lowered for T > 150 mk as expected for a Kondo peak. For T < 150 mk the zero bias conductance decrease. The explicit temperatures dependence of the zero bias conduction is plotted in figure 5.7c for D1K1 and D2K4, an estimation of the threshold is T t = 250 mk and T t = 150 mk respectively. The red lines are fits to the temperature dependence using equation (3.5) giving a Kondo temperature of T K = 2.2 K for D1K1 and T K = 1.5 K for D2K4. 60

65 V b [mv] a b c -0.75T V g [V] -2T e -1T d -0.5T V g [V] 0 di/dv [e 2 1 /h] -0.65T Figure 5.8: Non-collinear domains. di/dv as a function of gate and bias voltage measured at different external magnetic fields. All the plots are measured on D2K4 a permanent splitting of the spin states can be observed due to the misalignment of the external and the exchange field. 5.7 Discussion Figure 5.8 show di/dv as a function of gate and bias voltage at different magnetic fields measured on D2K4 and even though the slope of the Kondo ridges can be seen to change sign going from B = 2 T to B = 0.5 T indicating that degeneracy point have moved from the l.h.s. of the diamond to the r.h.s. one can at all gate voltages observe a splitting of the spin states, even at B = 0.75 T and B = 0.65 T where the Kondo ridges looks like two parallel lines. Because of this permanent splitting of the spin states, ascribed to the non-collinearity of the exchange and external field, there is no zero bias differential conductance ridge and therefore plots like the one in figure 5.6b just gives two horizontal lines showing the Coulomb peaks. The validity of equation (5.5) have been tested on three different Kondos from device 1. The Kondos are called D1K1, D1K2 and D1K3 were D1K2 and D1K3 are offset with around 600 electrons compered to D1K1. In figure 5.9 measurements similar to the ones seen in figure 5.6a are shown for the three Kondos where the bias-voltage measuring the spin-splitting have been normalized with P Γ/eπ and translated with 0 (for a more complete data set on D1K2 and D1K3 see appendix A). The different fitting parameters can be found in table 5.1. D1K2 and D1K3 are two on each other following Kondo diamonds, the difference in P Γ can therefore be explained with different couplings to the two orbitals in the CNT and it can be seen, under the reasonable assumption that P are equal for the two Kondos, that Γ D1K2 < Γ D1K3. It is interesting 61

66 6 3 V b /(PΓ/eπ) 0-3 D1K2 D1K3 D1K1 ln( x / 1-x ) ε d /U Figure 5.9: Fits showing the performance of equation (5.5). The fits shown here are similar to the ones seen in figure 5.6b scaled such that P Γ/eπ is equal to 1 and translated with 0. The measurements are made on D1K1, D1K2 and D1K3, and each of these Kondos are measured at 4, 6 and 6 different external magnetic fields, i.e. there are graphs from 16 measurements in this plot. The full drawn line are the function ln( x / 1 x ). The normalization constants can be found in table 5.1 D1K1 D2K4 D1K2 D1K3 P Γ/π[emV] [emv] Slope[V/T] E C [mv] T K [K] Table 5.1: Results from the fits. D1K1, D1K2 and D1K3 are the three samples, that have been fitted, and the different values are the fitting parameters. P Γ/π and 0 are from equation (5.5) the slope are from the linear fit seen in the insert in figure 5.6a, and should be ev/t for g = 2. These constant could not be found for D2K4 due to non-collinearity -see text. E C is the charging energy and T K the Kondo temperature found from temperature measurements and fits to equation (3.5). 62

67 though that 0,D1K2 > 0,D1K3 since the part of the splitting due to the stoner gab is given by st Γ( /Dπ) as seen in equation (3.17) and the bandwidth D and stoner gab should be the same for the two Kondos, therefore should st be proportional to Γ, i.e. st,d1k2 < st,d1k3. Other contributions to 0 is the stray field which should also be the same for the two Kondos since a small change in gate voltage should not change the magnetization and thereby the stray field for the Ni electrodes. The last contribution to 0, as mentioned in section 5.4, is SDIPS, this effect depend drastically on the interfaces and can therefore change from one Kondo to the next. The above arguments therefore strongly suggest that there is a significant contribution to the constant splitting from the SDIPS. Even though SDIPS are needed to argument for 0 is it surprisingly how well equation (5.5) describes the measurements. In the derivation of this equation a flat band with electron hole symmetry approximation was used, this is obviously not true for a real ferromagnet. The usefulness of the equation indicates that it is only the band shape close to the Fermi energy that is important, and as long as the band shape are approximately constant in this area, the equation describes the splitting fairly well. By close to the Fermi energy is meant on a energy scale of the charging energy. 5.8 Other Kondo effects In this chapter we have so far only discussed spin-1/2 Kondos, but another Kondo effect was also present in the measurements. This is a so called singlettriplet(s-t) Kondo where the states creating the Kondo instead of being spinup and spin-down are singlet and triplet states. Since there is a finite energy difference between the singlet and triplet states this effect is seen at a finite bias. In the stability diagram in figure 5.2 panel a high conduction lines can be seen at finite bias in the diamonds with an even number of electrons. This is observed both between Kondo 1 and 2 and also in the diamond between Kondo 2 and 3. We will study the line between Kondo 1 and 2 marked by a red arrow in figure 5.2a. Figure 5.10 show a bias plots through that diamond made at different temperatures at a fixed gate-voltage. A temperature dependent peak can be observed at high bias V b 1.9 mv and from the temperature dependence (figure 5.10b) it can be seen that this is a singlet-triplet (S-T) Kondo peak where the excitations instead of being between spin-up and spin-down is between a singlet and a triplet state as illustrated in figure 5.2d1-3 and discussed in reference [59]. According to theory one should be able to split the S-T Kondo into three peaks corresponding to the three different triplet states. The splitting of the peaks are not simply given by the Zeeman splitting starting at zero field as in [59] because there due to the electrons exchange interaction with the electrodes is a non-collinearity magnetic exchange field B ex. Beside of this the exchange interaction would probably also renormalize the states differently. In 63

68 a Vb [mv] di/dv [e2/h] c di/dv [e2/h] V b [mv] V b [mv] T=75mK T=90mK T=100mK T=150mK T=200mK T=300mK T=400mK T=500mK T=600mK T=800mK T=1K di/dv [e2/h] b di/dv [e2/h] d T [mk] B [mt] di/dv [e 2 /h] 0 ev b Figure 5.10: Singlet-triplet (S-T) Kondo. a, di/dv as a function of bias for different temperatures, the inset is a zoom of the peak at V b 1.9 mv. b, Temperature dependence of the peak conductivity, the red line is a fit using equation (3.5) with G 0 = 1.6 e 2 /h and T K = 4.2 K. c, di/dv as a function of bias and magnetic field the two green lines show how the S-T Kondo splits. d1-3 illustrates the process leading to the S-T Kondo. d2 is a virtual/intermediate state, d1,3 are the start and end state respectively. Note that d1-3 illustrates just one of the three singlet triplet transitions that exist. 64

69 the measurement only two states can be observed, it is often very hard to see all three states and from these measurement it is impossible to determine if the state is just hard to see or if it does not exist. It could be interesting to study this S-T Kondo with magnetic contacts further both theoretically and experimentally, but this have due to time issues not been done in this thesis. 5.9 Conclusion It have been observed that CNT contacted with ferromagnetic nickel electrodes, in the Kondo regime, exhibits a splitting due to the tunneling induced exchange interactions between the dot and the electrodes. This splitting can be completely compensated by an external field, in the case of collinear magnetization of the contact domains and the external field, else a permanent splitting is observed. The splitting have been seen to be gate dependent and it is even possible to change the spin ground state from up to down by tuning the gate voltage and thereby changing the spin of a single electron from up to down. The gate dependence can be described surprisingly well with a simple logarithmic equation with a constant term added due to constant fields resulting from different factors such as; a Stoner gab in the ferromagnetic electrodes, stray-fields from the electrodes, external magnetic fields and spin-dependent interfacial phase shifts. 65

70 66

71 Chapter 6 Hysteresis 6.1 Introduction In CNTs with ferromagnetic contacts a hysteretic behaviour is often observed seen as a conduction change close to zero field where there is a resistance difference between the contacting domains being in a parallel and a anti-parallel configuration. This resistance difference between the domains being in a parallel or anti-parallel configuration are often called magnetoresistance (MR) and the phenomena is referred to as a spin-valve effect. The MR is explained by an increased interfacial spin scattering, between CNT and drain electrode, in the anti-parallel configuration leading to an increased resistance. A decrease in the resistance in the anti-parallel domain configuration have also been observed and Sahoo et. al. [33] have shown that it is possible to tune the MR by the gatevoltage in a CNT in the Coulomb blockade regime. CNT with ferromagnetic contacts was first studied in 1999 by Tsukagoshi et al. [31], and have later on received a lot of attention [33 42]. The CNTs with ferromagnetic contacts studied here are in the Kondo regime, contrary to earlier reported measurements which have all been in the Coulomb or Febry-Perot regime. The Kondo effect give rise to a large peak in conduction that can be split by a magnetic field. The Kondo effect is therefore very sensitive to the tunneling induced exchange field and thereby the contact domains precise magnetization direction. In the interpretation of the measurements presented in this chapter the focus is therefore on the Kondo effect and its dependents on the contact domains configuration. Spin transport considerations are not taken into account, i.e. in the anti-parallel magnetization is the change in conduction explained from a increase/decrease in the Kondo splitting and not by an resistance change due to enhanced scattering at the spin interfaces. In section 6.2 we look at domains changing from a parallel to an anti-parallel configuration before zero external field. Then high field hysteresis a new phenomena in CNTs with ferromagnetic contacts is studied in section 6.3 before in section 6.4 turning to the changes in conduction when the domain configuration 67

72 change from parallel to anti-parallel. 6.2 Domain flip before zero field To take a closer look at the switching effects observed in figure 5.3a,d and figure 5.6b magnetoresistance (MR) measurements were made in the two Kondos. Figure 6.1 show the differential conductance as a function of external magnetic field B at zero bias measured on a D1K1 and b D2K4. The measurements are made by sweeping the magnetic field B from -500 mt to 500 mt black line and back again red line. At B = 500 mt should the magnetization direction of the domains be more or less aligned with the external field, this demands that the domains are changing direction in between the two boundary points. Depending on where and how the domains change their direction this can give rise to different domains configurations at the same magnetic field, depending upon the history of the magnetic field. The change of magnetization direction are primary happen in the region B 125 mt and we will refer to B 125 mt as the high field region. According to theory, the Kondo effect should give rise to an increased conduction when the domains magnetization direction change from a parallel (P) to an anti-parallel (AP) configuration, due to a smaller exchange field. In figure 6.1 panel a and b an increase in differential conductance can be observed starting around B 125 mt, for B 0. This indicates that around this magnetic field does at least one of the domains start to turn ending up in an AP configuration 1 for figure 6.1a at B = 0 and for figure 6.1b after zero at B 10 mt staying in that position until B 35 mt where the other domain starts to turn. It can also be observed that the domain shifts in a is non-abrupt, i.e. the gradual change of di/dv indicates that instead of a sudden shift from P to AP-configuration the domains instead rotates gradual changing their positions from P at B 125 mt to AP at B 0 mt and back to P again at B 125 mt. This is contrary to b where the change from AP to P seems more abrupt. Why the domain configuration start to change to an AP configuration before zero magnetic field is not obvious. A little calculation is therefore added. Detailed knowledge of the single domains and there interaction are needed to predict exactly when and how the domains will start to flip. Since we do not have this kind of knowledge about the system we will from a very rough model substantiate the reasonable in a change before zero field. Lets imagine two domains with a size of roughly V dom = nm 3 placed 200 nm apart, since we are looking at single domains is the magnetization saturated and for Ni is the magnetization saturation M sat = 0.61 T/µ 0 [66]. This implies that the magnetic dipole from a domain can be estimated with m dom = V dom M sat. Another dipole parallel to the first a distance r away 2 will feel a field corresponding 1 Or a configuration close to AP. 2 Orthogonal to the dipole direction. 68

73 di/dv [e 2 /h] a di/dv [e 2 /h] 0.35 b B [mt] Figure 6.1: Hysteresis. Measurements made in the middle of two different spin-1/2 Kondos, showing the conductance di/dv as a function of magnetic field for zero bias. The field are swept from - to + black line and vice versa red line. The two devices measured are known as D1K1 a and D2K4 b. to B dip (r) = µ 0 m dom 4π r 3. (6.1) With a separation of 200 nm this gives 18 mt, i.e. if the two domain were completely isolated and only interacting with each other one of the domains would flip when the external field is below 18 mt. This model is of course much to simple, since anisotropy and intra domain interactions and screening is completely neglected, but it still indicates that domain domain interaction can cause the domains to flip/rotate before zero field. Stray fields The above estimate of a dipole dipole interaction could be used to crudely estimate the stray field B stray from the magnetic domain, assuming that the dot is effectively located half-way between the two electrodes the field using equation (6.1) must be a factor 2 3 larger than calculated, i.e. a stray-field of 150 mt. According to Ensslin et. al. [67] doing numerical simulations with Fe electrodes, he reached a maximum stray-field of B stray 100 mt, since the magnetization saturation M s for Fe is roughly 3.5 times larger than Ni, the stray field from Ni should be smaller than from Fe. From the above considerations a rough estimate would be that the stray-field can be of the order B stray 100 mt. Note that the stray-field should only be present at P domain configuration at AP the two domains cancel each other and the stray field should therefore not influence the Kondo effect in the AP domain configuration. 6.3 Hysteresis at high fields Normally in MR measurements no hysteresis is observed at high fields, all the hysteresis happens below a critical field where the domains configuration change 69

74 a b di/dv [e 2 /h] Vg=7.133V di/dv [e 2 /h] Vg=6.845V Vg=7.034V 0.4 Vg=6.872V B [mt] Figure 6.2: High field hysteresis. Magnetoresistance measurement on device 2, The plots shows di/dv as a function of external magnetic field B the black line are for B going from - to + and the red line is in the opposite direction. The two plots in each panel are made at two different gate voltages in the same Coulomb diamond. a is measured in the diamond between Kondo 2 and 3 in figure 5.2 and b is measured on Kondo 2 refereed to as D2K2. from P to AP, i.e. the hysteresis only occurs at low fields. As can observed in the measurements in figure 6.1 this is not true in these devices since a clear hysteresis can be observed at high field, indicating different domain orientations depending on the B-field history. The measurements in figure 6.2 are performed on device 2 described in the previous chapter and are made on a CNT quantum dot in the Kondo regime on two concurring diamonds, a in a diamond with an even number of electrons, (the diamond between Kondo 2 and 3 in panel a figure 5.2) and b in a diamond with a odd number of electrons showing the Kondo effect D2K2 (Kondo 2 in panel a figure 5.2). Both panel a and b show hysteresis at low external field B 125 mt this will for b be discussed further in section 6.4. Note that the two measurements in each panel are at different gate voltages but in the same diamond. The upper plots in both panels are measured close to a Coulomb peak (see appendix B). In a an increased in di/dv can be observed in the AP domain configuration at V g = which is close to a Coulomb peak, this have been observed and explained by Sahoo et. al. [33]. He assumed a transmission that can be described with a Breit-Wigner formula and showed that on resonance for asymmetric coupling to the two electrodes an increased current can be observed in the AP configuration. In the remaining of this section we will concentrate on the high-field behaviour, i.e. at B 125 mt. In figure 6.2a no hysteresis is observed at high field this is typically interpreted as a domain configuration where the two domains and the external magnetic field are aligned. Therefore no change in the conduction as a function of field are observed, but as can be seen by looking at b this is not true. In panel b one can see a clear hysteresis for B 125 mt. Hysteresis like this, can be more or less pronounced but have been observed in 70

75 a b c di/dv [e 2 /h] B [mt] Figure 6.3: Illustration of domain configuration. a, External magnetic field. b, Domain configuration red arrows are domain configurations for a external magnetic field swept from + to - and black is in the opposite direction. c, Measurement made on D2K2 with coloured arrows showing sweep direction. all MR measurement made in Kondo diamonds. While in the even diamonds with no Kondo the high field hysteresis is not observed. This hysteretic behaviour implies that the precise magnetization direction of the domains are not the same for B going towards high fields as for B coming from high fields. The visibility of high field hysteresis in a Kondo diamond must be, because a very small change in magnetization direction of the domains give a change in the exchange field. Thereby splitting the Kondo peaks more or less giving a relatively large change in the zero bias conduction. In the areas where there is no Kondo a small change in the domains magnetization direction might lead to a small decrase/increase in the interfacial spin scattering between tube and electrode but this change is not large enough to change the conduction in a visible way. Possible domain movements giving rise to the high field hysteresis observed in the lower plots in figure 6.2b could be, following the black differential conductance curve: Starting at B = 500 mt is the domains largely aligned with the external field. When the field is lowered the exchange field compensates the external field more and more 3 diminishing the splitting of the Kondo peak until B 250 mt, thereby increasing the conduction. At B 250 mt the exchange field is overcompensating the external field and the conduction start to decrease again as the external field is lowered. Between B ±125 mt the domains is turning their magnetization direction. At B 125 mt is the domain again mainly pointing in the direction of the external magnetic field, but they are not as aligned as at B 125 mt since at that point the magnetic field was coming from high fields. Therefore is the non-collinear part of the exchange field larger at B 125 mt and the external field can not compensate, resulting in a larger splitting and lower conduction. Note that the strength of the exchange 3 Remember, as discussed in chapter 5, that the exchange field and domain magnetization direction can point in opposite directions. 71

76 field itself have probably also changed since it depends upon how parallel the two contact domains are, weighted by the coupling strength. As the external field is increased the domains aligns more and more with the external field until we reach B = 500 mt where the field is swept back to -500 mt, and the process described repeats. Figure 6.3b illustrates this turning of the domains, black arrows is the domain configuration sweeping the external field from - to +, giving rise to the di/dv curve seen in c and red is vice versa. In figure 6.2b and figure 6.3c the hysteresis seem to be zero at the end points B = ±500 mt this is because the sweep is made between ±500 mt, i.e. if the field was swept for larger fields the hysteresis would be visible until either the domains are completely aligned with the field or the separation between the Kondo peaks get so large that the tail of the Kondo peaks do not overlap. Note that it is difficult to justify the upper curves in figure 6.3b with the just described domain movements. The measurement is made on the side of a Coulomb peak which allow for single electron tunneling, which might be the reason for the different behaviour. Note also that the measurements here is made on D2K2 which are two Kondos away from D2K4 described in the previous chapter where the domains, from the plot in figure 5.3c,d, are found to give rise to an exchange field having an angle of 25 degrees, i.e. the domains illustrated in figure 6.3 should have a larger angle to the external field, the figure is simplified to illustrates the basic principle that the domains are more aligned with the external field, when the external field is coming from high fields than from low fields. Before ending this section lets take a brief look at figure 6.1b that show an interesting high-field hysteresis, i.e. at B 350 mt there is a crossing of the differential conductance trace coming from high field and the one from low field. A plausible explanation for this is; that the exchange field, when coming from high external fields, are larger than coming from zero field, but the domains are also more aligned with the external field and therefore is the compensation bigger. As the external field is lowered is the compensation removed and the Kondo peaks splits more are more due to the exchange field. Since the exchange field is larger for B coming from high fields than from low fields the splitting of the Kondo peak in the high field branch will be larger than in the zero field branch and a lower conduction is therefore observed. Note that we also in this case have a large difference between the two B field branches at B 125 mt but here the low field branch have the highest di/dv indicating that the exchange field is smaller here resulting in a smaller splitting of the Kondo peaks. It can be seen that at B = 125 mt is the splitting of the Kondo states in some case larger for B coming from zero field than for B coming from high fields e.g. the plot in figure 6.1b and sometimes smaller e.g. the plot in figure 6.2b. Which must be due to different symmetry in the coupling since the two Kondos measured are separated by a gate difference of 0.6 V and the Ni domain movements should therefore be the same in the two measurements. From the measurements it can be seen that small changes in domain magnetization direction give rise to a relative large change in conduction, making 72

77 0.9 a 0.6 b di/dv [e 2 /h] di/dv [e 2 /h] B [mt] Figure 6.4: MR measurements showing a decrease in di/dv as the domain configuration turns from P to AP. a, and b, show di/dv as a function of magnetic field at zero bias voltage measured on D1K2 and D2K1 respectively. Black line are swept form - to + and red line vice versa. it possible, at least in theory to probe the detailed domain movement. The conductance change is due to changes in the external field. The tunnel induced exchange field can due to domain movements change in two ways: The size can change due to a different alignment of the two contact domains. And the exchange field component along the external field direction can change, due to the two contact domains orientation relative to the external field 6.4 Hysteresis at small fields with the Kondo effect The hysteresis at low fields are hard to explain explicitly since the Kondo effect and thereby the differential conductivity depends very sensitively on the exchange field given by the exact domain configuration as observed in the previous section. In the hysteresis measurements shown so far an increase in the conduction have been observed, when the domains magnetization direction turns to an AP-configuration around zero field. As observed in the MR measurements in figure 6.4 this is not always the truth, in some cases there is a decrease in conduction when the domains are in the AP configuration. Figure 6.4 show MR measurements in the Kondo regime a is made on device 1 and are called D1K2 (for more plots on this device se appendix A) and b are made on device 2 called D2K1 (first Kondo in figure 5.2a). a has a very detailed structure for B 125 mt 4 so we will focus on b trying to understand the decrease in differential conductance as the domain turns towards AP. For a given ε d using equation (3.15) should the splitting of the spin states ε d, due 4 In the same Kondo at a different gate voltage an increase in the conduction is observed see appendix B figure B.7, this is the only Kondo where this have been observed. 73

78 a 0.8 di/dv [e 2 /h] b mt -40 mt V b [mv] di/dv [e 2 /h] c V b [mv] B [mt] V b [mv] di/dv [e 2 /h] d mt 10 mt di/dv [e 2 /h] B [mt] V b [mv] Figure 6.5: Kondo splitting at zero field. a,c, di/dv as a function of bias voltage V b and external magnetic field B measured on D2K3 and D1K2 respectively. b,d, Are bias cuts taken a, c respectively at the smallest magnetic field before zero where the peak is unsplit, and at the maximum splitting closest to zero field. The corresponding splitting is V = mv and V = mv for b and d respectively. Note that the sweep direction of the magnetic field in c was from + to -. to the difference in DOS for spin-up and spin-down and the stoner gap st, in the P and AP case be given by ε P d = 1 ( P b + ) (Γ s + Γ d ) π D ε AP d = 1 ( P b + ) (Γ s Γ d ) (6.2) π D Where b = ln( ε d / ε d + U ) this clearly show that an increased splitting of the Kondo peak and a resulting decrease in di/dv when the domains turn from the P to AP configuration can not be explained with the tunnel induced exchange interaction. (Note that by setting b = a P D this equation becomes similar to equation (3.18)). Other effect giving a splitting of the Kondo peak is the stray field and SDIPS as discussed in section 5.4. The stray field would probably change when the magnetization of the domains changes from P to AP, but according to reference [67] there should be no field in the AP configuration, even if there is it would be of the order B stray 100 mt as already discussed. The splitting in the AP configuration or at B close to zero field have been investigated in figure 6.5, c,d D1K2 and a,b D2K3 also showing a decrease in 74

79 di/dv, as the domain configuration goes from P to AP (see appendix B) and are two Kondos away from D2K1. The splitting of the Kondo peaks close to zero external field for these two samples are V = mv corresponding to an exchange field of roughly 0.8 T for D1K2 and V = mv corresponding to 1 T for D2K3. The splitting observed in these two plots are much larger than the stray field estimate and therefore other effects are needed to explain the behaviour. SDIPS is a another effect that can cause splitting of the spin states and it is probably also sensitive to the exact domain configuration and could therefore be a possible explanation. An interesting observation in this connection is that Kondo D2K1 and D2K3 show a decrease in di/dv as the domain configuration turns from P to AP while in D2K2 and D2K4 an increase in di/dv is observed. Indicating that the SDIPS are not completely local but also depends upon the orbital state. It should be noted that D2K1 - D2K4 is the only four period measured in this way. 6.5 Conlusion Hysteretic behaviour have been observed in two different devices and are studied in the Kondo regime. High field hysteresis have been observed in the odd diamonds where the Kondo effect is present while in even diamonds with no Kondo effect no high field hysteresis was observed. This is explained by a high sensitivity of the Kondo splitting on the size of the exchange field and on the alignment between the exchange and the external field. In some Kondos a decrease in the conduction was observed when the domains turned from a parallel to an anti-parallel configuration this can not be explained by changes in the tunneling induced exchange field and must be due to other effects which could be SDIPS. More measurements and theory is needed to get a better understanding of these phenomena. 75

80 76

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87 Part II Carbon nanotubes with semiconducting contacts 83

88

89 Chapter 7 Carbon nanotubes incorporated in MBE grown structures 7.1 Introduction In the semiconductor industry the improvement of device performance in electronic systems has, for the last four decades, been driven by down-scaling the size of the components. This approach to device improvement will soon encounter huge if not impossible scientific and technological problems. Due to this, industry is exploring other possible paths to improvements; among these spintronics and molecular electronics. An obvious candidate for use in molecular electronics is carbon nanotubes (CNT) [10]. One may imagine complete circuits made of CNTs. This is feasible since CNTs can be both semiconducting and metallic depending on chirality. Although pure CNTs circuits might be possible there is still a long way before it may become technically realizable. Problems such as how to change between semiconducting and metallic tubes is still to be solved, to make just one single circuit. Even more demanding: to produce array after array of identical circuits, this request that the problem above have been solved and a control of the CNTs that have not yet been achieved. An alternative approach to molecular based electronics is to combine semiconductors and molecules, to make an intermediate step between full molecular electronics and standard semiconductor technology. One way to realize this combination could be to incorporate CNTs in semiconductors. This approach will be investigated in this part of the thesis. To achieve a semiconductor-cnt contact the CNTs are incorporate inside a semiconductor. This is done by dispersing the CNTs on a perfect crystalline substrate and then cover them with a semiconductor using a Molecular Beam Epitaxy (MBE) system. MBE is used to take advantage of the possibility of also 85

90 Figure 7.1: Schematically drawing of a CNT incorporated in a crystal lattice. having layered semiconducting heterostructures. Because the semiconductor is grown by MBE this approach presumably result in CNTs embedded in a perfect crystalline semiconductor 1. An advantage of this approach is that both the substrate on which the tubes are placed and the overgrown crystal layers can be designed with a precision down to a single mono layer. After the top layers are grown the crystal can be processed by standard lithographic processes. This has previously been done by Jensen et. al [2, 3] and in my Master thesis [1] where the CNTs are embedded in the ferromagnetic material GaMnAs [47]. Figure 7.1 show a schematically drawing, where a CNT is seen protruding from a perfect crystalline lattice. The crystal modelled are GaAs, but with a slight change in lattice constant this could also be Si since it has the same structure. In this part of the thesis (chapter 7, 8 and 9) the properties of CNTs incorporated in different semiconducting materials will be described. Chapter 8 focuses on the incorporation of CNTs in gallium arsenide (GaAs). First however the MBE-system is very shortly described, followed by a description of the sample processing. Then the results are presented ending with a conclusion and outlook. In chapter 9 CNTs overgrown with Si are described, starting with the different sample configurations followed by a description of the sample processing. Subsequently we turn to AFM measurements, where CNTs are manipulated and characterized electrically. Finally, measures taken to lower the Si-CNT contact resistance are discussed, an overview of the samples processed and the resistances obtained are listed and in the end the gate dependence of different devices is shown as well as the conclusion. 1 As can be seen in section this is not always the result. 86

91 Chapter 8 Carbon nanotubes incorporated in GaAs heterostructures 8.1 Gallium Arsenide (GaAs) GaAs have long been the play-ground for physicist studying mesoscopic systems, such as two dimensional electron gasses, quantum dots, quantumpoint contacts, etc. and it could be interesting to combine such systems and the high degree of system control with CNTs. This could possible lead to systems where one could look at the interplay between a one-dimensional and a two dimensional systems or a contact interface where the number of carriers in the GaAs could be controlled. The carrier control might make it possible to tune the contact resistance between the CNT and a GaAs contact. One way to try and achieve a CNT-GaAs contact is to incorporate CNT into GaAs as will be described in this chapter. GaAs is a III-V semiconductor with a zinc blende structure as shown in figure 8.1. The atoms are placed in a diamond structure with Ga and As atoms on alternating sites, so that each Ga atom has four As atoms as nearest neighbors and vice versa. A GaAs crystal can be grown in a Molecular Beam Epitaxy (MBE) system. A thorough description of the MBE-system and MBE growth is beyond the scope off this thesis, but a ultra short description of the most important parts is given in the next section Molecular Beam Epitaxy (MBE) system Molecular Beam epitaxy is a technique for growing high quality thin layer films with atomic layer precision. The growth is performed on top of a crystalline substrate in a MBE growth chamber (schematically shown in figure 8.2), kept at a base pressure around torr. To preserve the Ultra High Vacuum (UHV) 87

92 As Ga Figure 8.1: GaAs lattice. The figure shows the zinc blend lattice structure of GaAs. The Ga and As atoms are symbolized by grey and white spheres respectively. the substrate is cleaned in separate loading and buffer chambers, before being introduced into the growth chamber. The substrate is placed in a temperature controlled substrate-holder, which can rotate to compensate for the nonuniform position of the effusion cells. The growth materials are evaporated from effusions cells, called Knudsen cells. The shutters in front of the cells are mechanically controlled and are used to start and stop the growth by blocking the molecular beams. The beam pressure is controlled by the temperature of the Knudsen cells, and can be monitored by placing a ionization gauge at the sample space. The layer deposition can be monitored by a Reflection High Energy Electron Diffraction gun, which gives a diffraction pattern making it possible to determine the surface crystal structure and to monitor the number of atomic layers grown. Due to the UHV, the mean free path of atoms leaving the Knudsen cells is sufficiently long (many km) so the atoms will reach the substrate without encountering collisions with other particles. Depending on parameters such as atom velocity, substrate temperature the species of the atoms and the substrate, the atoms will stick or desorb from the surface. An atom that stick to the surface will diffuse around until it is either incorporated in the lattice or desorbed from the surface. The incorporation will usually happen at step edges in the crystal or by nucleation with other atoms diffusing on the surface. The ratio of atoms being incorporated in the crystal to those arriving at the surface is called the sticking coefficient and can be everything between 0 and 1 depending on growth parameters. Typically Ga has a sticking coefficient close to 1 whereas As has a very low sticking coefficient Sample preparation The GaAs samples were all made at NanoScience Center/Niels Bohr Institute in a Varian Gen-II MBE system placed in a class 100 clean room. The growth was performed by C.B. Sørensen. All of the GaAs samples described here have been grown on (100) GaAs surfaces. The CNTs was dispersed on two kinds of substrates before the overgrowth. 88

93 Figure 8.2: Schematic view of a MBE growth chamber, courtesy of C.B. Sørensen [6] The two substrates used were: 1. A bare (100) epi-ready GaAs substrate without any additional chemical treatment. The semi insulating substrate was used to study the overgrowth, but the lack of back-gateing possibilities makes them less interesting to study electrically afterwards. However, since there is a MBE growth sequence less, in producing, these kind of samples, they are cheaper and less time consuming to make. After the deposition of the CNTs the oxide was removed by baking (at 550 C in an As pressure) in the MBE-system 1, and the growth/overgrowth was subsequently initiated. 2. Substrate with an electrical barrier. These substrates were grown on a n-type (100) GaAs wafer and consisted of a buffer layer of n-doped GaAs, a GaAs/AlAs ((2 nm/2 nm) 100) superlattice layer, a 20 nm intrinsic GaAs layer (to separate the superlattice from the active region, i.e. the region were the CNTs are deposited) and then a cap layer of As to protect the top from oxidation. The CNTs are then dispersed on the As layer which is subsequently evaporated in the MBE growth chamber before the overgrowth [1, 2, 4] is initiated. The superlattice electrically separates the back-gate from the active CNT region, making it possible to back-gate the device 2. 1 During thermal oxide desorption the crystal have to be in a constant As flux not to decompose. 2 These kind of substrates was used until the overgrowth problems were discovered, as discussed in section

94 8.1.3 Dispersing carbon nanotubes The CNTs used for dispersion are Rice tubes made by laser ablation [5]. To disperse them, a suspension is made by immersing a small amount of tubes (size of a small dust corn) in 1,2-dichloro-ethane and sonicate for min. From the beginning the tubes are tied together in knots, bundles and ropes and basically looks like black powder. The sonication separates the large bundles, into single tubes or at least smaller bundles. The sonication time is a trade off; extended sonication gives a good separation of the tubes but introduces defects and chops the CNTs into short pieces. Conversely, short sonication time results in thicker ropes/bundles but longer and more defect free tubes. Longer tubes are important since the electrodes are placed by chance, hoping that a CNT is bridging the gap, i.e. longer tubes implies more possible connections. After sonication the solution is dispersed on the wafer by spinning the wafer at a speed of 1000 rpm while dripping droplets of the suspension onto the wafer. Afterwards the wafer is washed in isopropanol and blown dry with nitrogen. Since the wafer, after the dispersion of the CNTs, has to be loaded in the MBE chamber, care is taken to keep it as clean as possible. Alternatively, CNTs can be grown by chemical vapor deposition, however this requires temperatures around C and these high temperatures will destroy the GaAs crystal Samples and results After the CNTs are dispersed the substrate is loaded in the MBE system, where it is first degassed in both the entry/exit chamber and afterwards in the buffer chamber, before being introduced into the main growth chamber where the amorphous As/oxide layer is removed and the overgrowth is initiated. The first growth was made with a substrate temperature of 500 C and a Beam Equivalent Pressure (BEP) of torr and torr for As (As 2 ) and Ga, respectively. Due to the 10 times higher As pressure, it is the amount of Ga which is the restrain in the growth process. The substrate was overgrown with 60 nm GaAs n-doped with Si to a density of cm 3. The result was a wafer filled with cracks or canyons as seen in figure 8.3a, caused by the CNTs. This sample will from now on be referred to as W500. Figure 8.3b shows the topography along the white line seen in a. From the plot it can be seen that there is an enhanced growth of the GaAs crystal on the (100) surface right at the edge of the canyon. This shows that there is a net diffusion of Ga atoms from the side facet of the canyon to the top surface, indicating that the diffusion time for the side facet is longer than for the (100) surface 4. In an attempt to get rid of 3 Interestingly recent research by Engel-Herbert et al. [15] indicates that it is possible under the right conditions. 4 Since there is a diffusion of Ga atoms from the side facet to the top (100) facet, there must be more Ga atoms diffusing around on the side facet than on the top. The side has a larger angle to the Ga source than the top (100) surface therefore is the Ga flux on the side facet lower than on the top facet this implies that to get a larger number of Ga atoms to diffuse around on the side facet the diffusion time on this facet must be longer than on the 90

95 a b z [nm] x [µm] Figure 8.3: CNTs overgrown with 60 nm of GaAs at a substrate temperature of 500 C, the sample are referred to as W500. a, AFM-image of the GaAs surface; the observed canyons are due to GaAs inability to grow on CNTs. b, Line-trace along the white line in a. Note that the line-shape is a convolution of the features on the surface and the AFM tip shape. The tip shape prevent the AFM tip to probe the bottom of the canyon, this is the reason that the apparent dept is only 40 nm even though the grown GaAs layer is 60 nm thick. 91

96 the canyons and actually overgrow the CNTs a series of new growths were made, at different temperatures. AFM-images of the results are shown in figure 8.4a,e grown at 370 C (W370) and 410 C (W410) respectively at a As(As 2 )/Ga BEP of / torr. At these intermediate temperatures, the tubes are partially overgrown with a clear directional dependence: Figure 8.4d gives the direction of the two AFM-images seen in the figure. The line traces in b,c and f,g,h is colour codded to the lines in seen a and e respectively, emphasizing the directional dependence. Figure 8.5 shows an AFM-image of a sample grown at 250 C (W250) at an As(As 4 )/Ga BEP of / torr. Note that in the last growth As 4 instead of As 2 was used as the As source due to problems with the cracker 5. This could be important since the As molecular type have an influence on the atoms abilities to nucleate on the surfaces [24]. On W250 all the tubes are covered although hills from at least the bigger ropes can be seen on the surface. 8.2 Analysis and discussion Due to an inborn structural difference between GaAs and CNTs, it is apparently not possible to grow a GaAs lattice on top of CNT. Although the incompatibility between the lattices were well known it was a surprise that the small gap caused by the CNT was not bridged when 60 nm GaAs (as in sample W500) was grown on top. This is in contrast to silicon as described in the next chapter. Actually there could be some small CNTs on W500 that have been covered by the overgrowth and therefore are invisible on the surface. But, from the number of canyons seen on the image, and by comparing with overgrowth at lower temperatures where hills from some tubes can be seen on the surface it seems plausible, that all the tubes on W500 have given rise to a canyon. If this was not the case, the tubes with a size that could just barely be covered would leave some trace on the surface, and on W500 only normal canyons are seen. From measurements done by SEM the depths of the canyons on W500 are found to be around 60 nm, fitting the thickness of the overgrown GaAs layer. Since GaAs seems so unwilling to cover CNTs we assume, that when a MBE growth is initiated on a wafer with CNTs the lattice will after the first few monolayers look schematically as in the illustration in figure 8.6b. These facets can also be seen on the SEM image in figure 8.6a. To determine which facets forms the sides of the canyon, the two top points in line plots like the ones in figure 8.3b, 8.4b and 8.4f, have been measured i.e. their separation and their height. From this and the knowledge of the layer thickness and under the assumption that there is only one facet on the side, the angle θ of the facet to the (100) surface can be estimated. The results are shown in table 8.1. In the calculations any width of the CNT has been neglected. The facets are assumed top (100) facet. 5 The As source have a temperature of C at his temperature is the molecular form As 4 normally the As 4 molecules are passed through the cracker which heats the As molecules to C cracking the As 4 molecules to As 2. 92

97 a b z [nm] c z [nm] x [nm] x [nm] d [0,-1,-1] e [1,0,0] [0,-1,1] z [nm] f g h x [nm] Figure 8.4: CNTs overgrown with 50 nm GaAs at substrate temperatures 410 C and 370 C for a-c and e-h respectively. a,e, Show AFM-images of the (100) GaAs surface overgrown at 410 C and 370 C. b,c, Are line traces along the lines seen in a that point in the [011] and [011] direction, respectively. d, Shows the orientation of the two AFM-images a,e. f-h, Line traces along the lines in e the directions are [011] for f and [011] for g,h. 93

98 Figure 8.5: CNTs overgrown with 50 nm GaAs at a substrate temperature of 250 C. The figure shows an AFM-image of the overgrowth. Structures resembling CNTs can be seen on the surface. a b J (100) J (111) facet (1,1,1) [1,0,0] GaAs crystal Carbon nanotube θ Figure 8.6: GaAs-facet. a, shows an SEM-image sample W500 cleaved and tilted to show the side and surface. b, illustration of an attempted overgrowth, here J (100) and J (111) are the Ga flux on the (100) and the (111) surfaces respectively. 94

99 Sample growth temp. θ avg. θ min θ max W C W C W C Table 8.1: Measured angles for the sides in the canyons of the wafer W500, W410 and W370. Note that the measurements determine the angle of W370 and W410 have been made in the [011] direction while the ones for W500 are as orthogonal to the canyon as possible, since the direction of the AFM picture in figure 8.3 are unknown. to meet in a point at the interface where the CNTs have been dispersed. Two obvious sources of errors to this angles are; a finite width of the CNT would result in a larger angle than the one estimated, and any growth on the side surface would lead to an overestimation of the angle. It is important to note that the lineplots measured for W410 and W370 are in the [011] direction while the ones for W500 are measured orthogonal to the canyon. The angle between the (100) surface and the (111)A or B surfaces 6 is 54 which is close to the angle measured on W410 and W370. The next lines will be used to substantiate the assumption that the side walls in the canyons are (n11) facets where n are some whole number less than or equal to 8. From the AFM-images in figure 8.4 it can be observed that the canyons are seen most clearly in the [011], [011] direction while they at these temperatures are more or less covered in the [011], [011] direction. This indicates that facets with a surface-normal orthogonal to these directions are the side facets in the canyons and only the (n11)a, (n11)b, (011) and (011) 7 surfaces comply with this description. The (011) and (011) are orthogonal to the (100) surface and the width of the canyons, with these sidewalls, should therefore be the width of the CNT bundle in the bottom of the canyon, implying that the thickness of the CNT bundles should be around nm. In this case the CNTs would be thicker than the grown layer and should be observable on the top. Koshiba et al. [26] have studied MBE overgrowth on a (100) surfaces with mesas with vertical sidewalls etched out in the [011] and [011] direction. During growth (111)B and (311)A facets, depending on direction, developed on top of the mesas. On the vertically sides of the mesas there were negligible growth. Also MBE on GaAs wafers partially covered with SiO 2 for laterally crystal growth has been studied by Umeno et al. [45] and (111)A, (311)A and (111)B facets have been seen to develop. In our case the CNTs will form lines separating different GaAs growth regions. At the edge of these regions the side facets are free to develop and from the above considerations it seems plausible that it is (n11)a and B facets which develop. The canyons pointing in the [011] direction will then have As terminated 6 In the (n11) direction the GaAs crystal is either Ga or As terminated, meaning that the last layer of atoms are either only Ga atoms or only As atoms, the convention is that the Ga terminated surface is called A and the As terminated surface is called B. 7 The facets (011) and (011) also obeys these conditions but are due to symmetry identical to the (011) and (011) facets respectively and are therefore not mentioned. 95

100 a (100) Ga (111)B [011] b [011] (111)A Ga (100) GaAs CNT GaAs CNT Figure 8.7: Illustration of canyon geometry. The illustration show the canyons pointing in a, the [011] direction with (111)B sidewalls and in b, the [011] direction with (111)A sidewalls. Note that the [011] direction and the [011] direction are orthogonal on each other. sides and be (n11)b facets while in the canyons running in the [011] direction the sides would be Ga terminated (n11)a facets, as illustrated in figure 8.7. Selective MBE growth of GaAs crystals containing both (100) and (111) A and B surface facets has been studied in references [16 43] 8. It seems fairly consistent in the litterateur that the B-side is a (111)B [17 28] a few have seen other facets (n11)b, where n (2, 3, 4), [33, 34, 41, 42]. At the A-side other facets more commonly tend to develop as the growth progress, such as (n11)a, where n (2, 3, 4, 5, 8), most often the side becomes a combination with the (111)A at the lower part of the side and a (n11)a with n (2, 3, 4, 5, 8) on the top close to the (100) surface [29 40, 42]. From the line-traces in figure 8.3b and 8.4b,f it can be seen that there is a bump at the edges on the surface where the sidewalls ends, showing an enhanced growth in this region. This is due to inter-surface diffusion from the side wall to the (100) top surface i.e. a net diffusion of Ga atoms from the (n11)a and (n11)b to the (100) facet. At intermediate temperatures i.e. 370 C and 410 C the enhanced growth at the edge is only observed in the [011] direction, showing that there is a net diffusion (n11)b (100), but that it has been reversed in the [011] direction i.e. the inter-surface diffusion is (100) (n11)a at these temperatures. The determining parameter for the inter-surface diffusion is the diffusion time on the different surfaces. This parameter is again dependent on different parameters such as As-pressure, Ga-pressure, As type (i.e. As 2 or As 4 ) and temperature. The change in inter-surface diffusion direction, for different substrate temperatures, is presumably due to different temperature dependencies for the diffusion time on different surfaces. From the above discussion an understanding of the different θ s observed for the different growth temperatures can be achieved. For sample W500 the average θ is smaller than for W410 and W370 since it is an average of measurements made on both A and B sides, and since A sides at this high temperature is 8 References [17 28] are studying the inter-surface diffusion between the (100) surface and the (111)B facet, i.e. the diffusion of Ga atoms between the (100) surface and (111)B surface. References [29 32] are studying the inter-surface diffusion between the (100) surface and the (111)A facet. References [33 40, 42] are studying the inter-surface diffusion between the (100) surface and both of the facets (111)A and (111)B. 96

101 probably a multi facet side, i.e. could be comprised of both (111)A and (311)A facets, then the measured angle would be smaller. As the temperature is lowered the diffusion time and diffusion length is decreased. This leads to an enhanced probability for growth on the sides, which is increasing as the temperature is lowered. This can easily be observed on the A side where the inter-surface diffusion has changed, but this is probably also true on the B side. Here the changes are not big enough to dramatically change the inter-surface diffusion, but there is probably still an increased growth which could explain the steeper angle observed at the lower temperature A short description of the growth Even though it is not possible from the above data to get a complete understanding of how the growth is evolving, the next lines will be used to describe how we imagine the growth is takeing place, when CNTs are overgrown with GaAs in a MBE chamber. When the growth is initiated the crystal starts to grow. Due to the tubes there are lines where the growth cannot take place. Since the (111)B and (n11)a are stable surfaces, these will be the dominant side facets of the canyon. At high temperatures T > 500 C the diffusion time and length on the A and B sides is so large that the growth on these sides is negligible and instead an enhanced growth is observed on the (100) surface edge as illustrated in figure 8.7. As the temperature is lowered the diffusion time on the A sides become comparable to the (100) surface and growth are initiated on this side. From line-traces such as the ones in figure 8.4c,g,h it can be seen that there is no longer a bump or increased growth at the edge i.e. indicating that there is no longer a net inter-surface diffusion of Ga atoms from the A to the (100) surface. At the B side on the other hand, even though there probably is an increased growth on the surface, the inter-surface diffusion situation is the same as at higher temperatures, i.e. a net flow of Ga atoms from the B surface to the (100) surface. At low temperatures T < 250 C the diffusion time for both the A and B side is so small that growth takes place at both surfaces at a rate comparable to the (100) surface Electrical abilities The reason for trying overgrowth of CNTs was to use the GaAs as an electrical contact material for the CNTs, but as seen for the growth done at higher temperatures this were not possible since the CNTs were not covered. At T = 250 the tubes were covered but the resistivity of the crystal grown at this temperature was so high that electrical measurements on a GaAs-CNT-GaAs device would make no sense. The high resistivity of low temperature grown GaAs is well known and are due to As antisites giving midgap states, the hope was that the high density of Si dopants would make it more conductive, and although this might have been the case, the conductivity is still so small that electrical measurements on GaAs-CNT-GaAs devices are impossible. A problem in using Si as a dopant in these kind of structures is that Si is an amphoteric dopant in 97

102 III-V compounds. This implies that Si in a GaAs crystal when grown in MBE acts as a n-type dopant when grown on (100), (111)B and (n11)a for n 5 surfaces while on other (n11)a type surfaces act as a p-type dopant [41, 43], in our devices this would make some repeatedly n p n transitions in the crystal. 8.3 Conclusion and outlook The temperature at which CNTs were successfully overgrown with GaAs was so low that the GaAs film was highly resistive. At intermediate temperatures there is a direction dependence of the overgrowth due to different atom termination of the (111) planes. The growth around a CNT depends upon the orientation of the GaAs wafer relative to the CNT. At high temperatures the CNTs are not overgrown indicating that at these temperatures the growth rates on (111)A and (111)B sides are negligible and the diffusion times on these surfaces are larger than on the (100) surfaces giving a net inter-surface diffusion of Ga atoms from (111) to (100). The problems with the overgrowth of the CNTs might be possible to solve by playing with other growth parameters such as As pressure [21 23] at the intermediate temperatures. 98

103 Chapter 9 Carbon nanotubes incorporated in Si 9.1 Introduction In the last chapter CNT incorporated into GaAs was studied and while this an interesting system especially from a physics point of view another tempting material candidate to incorporate CNT into is silicon. Si is a tempting candidate especially if one are looking towards applications since the most common material used in the semiconductor industry is Si. In this chapter we will look at CNTs incorporated into Si. All the MBE-growing described in this section, has been carried out at Aarhus University in a cooperation with A. Nylandsted and J.L. Hansen. All other sample processing and measuring were performed at University of Copenhagen at the HC-Ørsted Institute, partly as an employe of Hytronics A/S (CEO P.E. Lindelof) 1. The chapter start with a description of the experimental techniques, where the sample preparation and processing is described together with a description of the different sample structures made. In section 9.3 are the previous incorporated but by etch exposed CNTs characterized by electrical measurements and atomic force microscopy. Then measures taken to lower the Si-CNT contact resistance is described before an overview of the different samples measured and there behaviour is shown. The chapter ends with a conclusion and outlook. 9.2 Experimental techniques Dispersing carbon nanotubes The dispersion of CNTs are described in section Hytronics Milestone III 1 & 2 report pp.1-41 (2007) 99

104 CNTs grown by Chemical Vapor Deposition (CVD) Another method of getting tubes on the substrate are to grow them in a CVDoven. The growth is performed by dispersing catalyst material on the substrate, in this case Fe 3 No 4 9(H 2 O). The catalyst is dissolved in Isopropanol (IPA) by sonication and then dispersed on the wafer. The wafer is then inserted in the CVD-oven which is heated to between 850 and 950 C 2 under a constant Ar flow. The growth is initiated by turning off the Ar flow and turning on of alternately hydrogen and methane. Afterwards the sample is cooled in a flow of Ag. An advantage of CVD-grown tubes is the avoidance of sonication. This is advisably since sonication as mentioned in section introduces defects in the tubes. A disadvantage is that the inner tube diameter of the furnace is only around 2.5 cm wide, see figure 4.1, why only a small part of the wafer can be grown at a time. Another disadvantage of this procedure turned out to be that structures made in this way, for some reason, had a huge gate leak Different Si-CNT configurations np-structures In the first batch there were basically two different categories of structures the np-structure and the SOI-structure. The np-structures are made by dispersing the tubes on a p-doped silicon (Si) wafer and overgrown with a layer of n-doped silicon, resulting in a np-structure where the CNTs are placed in the interface between the n- and the p-doped layer. A sketch of the structure is seen in figure 9.2a. Measurement of the diode characteristic of this structure is shown in figure 9.1. The plot show a small current at reverse-biasing (V b > 0) where the slope of the curve, in this region, corresponds to a resistance around 2.5 MΩ. In a CNT with two Si contacts this leakage-current at reverse biasing makes it hard to interpreted if a current is running through the substrate or the CNT 3, especially since the Si-CNT contact resistance is unknown. Due to this the work was focused on the SOI structures where the interpretation of the results is clearer. SOI-structures The SOI means Silicon on insulator and consist of a conducting substrate followed by a 40 nm thick insulating SiO 2 layer topped by a 100 nm thick undoped silicon layer also called the intrinsic layer. The structure is shown in figure 9.2b. On the basic structure is the CNTs dispersed and then overgrown by the wanted toplayer. The result is CNTs caught between the intrinsic Si layer and the doped top layer. An example of such a structure is shown in figure 9.2c. The advantage 2 The temperature at which the tubes are produced fluctuates, so a couple of growths are sometimes necessary before the right temperature, producing CNTs, are found. 3 It should be possible to make a np-diode with a reverse-bias resistance much higher than this but it would take time before it could be produced reliable, so it was easier to change to the SOI structure. 100

105 0,1 NP-diode characteristic 0,0-0,1 I [ma] -0,2-0,3-0,4-0, Vb [V] Figure 9.1: Four probe diode characteristic of the np-structure. The red dashed line is a fit to the I-V characteristic at reverse biasing corresponding to a resistance of 2.5 MΩ. of this structure is that if you etch all the way through the top layer and the intrinsic layer, you end up at the isolating SiO 2 layer. Measurements on samples with this kind of structure should be easy to interpret since in the etched region the current can only run through the CNT 4. Samples made Figure 9.3 show a schematic of all the Si samples configurations grown in a Molecular Beam Epitaxy (MBE) chamber in this thesis. The first step in the MBE growth is to remove the natural oxide layer, which always exist on top of a silicon wafer by thermal heating. Afterwards the wanted structure can be grown on top. In the 1st batch the following samples were made: np-structures 4 different samples 1. Substrate p-type doped to a resistance of R=1 Ωcm, 40 nm thick top layer doped with antimony, to a density of cm 3 with CNTs dispersed. The structure is shown in figure 9.3a. 2. Substrate p-type doped to a resistance of R=1 Ωcm, 40 nm thick top layer doped with antimony, to a density of cm 3 with CVDgrown CNTs. The structure is shown in figure 9.3b. 3. Substrate p-type doped to a resistance of R=50-80 Ωcm, 40 nm thick top layer doped with antimony, to a density of cm 3 with CNTs dispersed. The structure is shown in figure 9.3c. 4 This is under the assumption that there is no leakage current through the insolating SiO 2 layer, as can be seen in some of the measurements. Especially on the samples with CVD-grown CNTs, this is not always the case. 101

106 a NP-structure CNT interface Si N-doped 40nm Si P-doped substrate b SOI basic structure CNT interface Si-intrinsic 100nm SiO 2 40nm Si P-doped substrate c Complete SOI structure CNT interface 40 nm doped Si Si-intrinsic 100nm SiO 2 40nm Si P-doped substrate Figure 9.2: a, np-structure with CNTs caught in the interface between the n and p doped layer. b, The basic SOI structure where the CNTs are dispersed on the top. c, The SOI structure where the CNTs are placed in the interface between the intrinsic Si and the doped top layer. 4. Substrate p-type doped to a resistance of R=50-80 Ωcm, 40 nm thick top layer doped with antimony, to a density of cm 3 with CVD-grown CNTs. The structure is shown in figure 9.3d. SOI-structures 2 samples nm top layer n-type doped with antimony, to a density of cm 3 CNTs dispersed. The structure is shown in figure 9.3e nm top layer n-type doped with antimony, to a density of cm 3 CVD-grown tubes. The structure is shown in figure 9.3f. The structures described have all been grown at 300 C which is a low growth temperature but necessary to get the high level of dopants incorporated. Because of reasons described previously the work in this thesis have been focused on the SOI structures and the np-structures have been used as test samples, primarily in the etching process. The 2nd batch were made to try to minimize the contact resistance between the Si electrodes and the CNTs. In that batch all the structures were SOI structures. The following list describes the layers placed on top of the basic structure shown in figure 9.2b. 1. The first 30 nm is doped with bor to a density of cm 3, which in silicon acts as a p-type doping. On top of this is a 10 nm thick top layer doped with bor to a density of cm 3. The p-type dopants could possibly give a better contact to the CNTs since the semiconducting CNTs normally acts as p-type conductors. The highly doped top layer is there to make it easier to externally contact the silicon. The structure was grown at 450 C which could also give a different contact resistance compared to 102

107 a NP-structure CNT deposit 40nm Si N-doped 2-5 *10 19 cm -3 Si P-doped substrate R=1Ωcm c NP-structure CNT deposit 40nm Si N-doped 2-5 *10 19 cm -3 Si P-doped substrate R=50-80Ωcm e Complete SOI structure CNT deposit 40nm Si N-doped 2-5 *10 19 cm -3 Si-intrinsic 100nm SiO 2 40nm Si P-doped substrate Si-intrinsic 100nm SiO 2 40nm Si P-doped substrate i Complete SOI structure CNT deposit 40nm Si N-doped 1 *10 20 cm -3 Si-intrinsic 100nm SiO 2 40nm Si P-doped substrate Sample structures 1st batch NP-structures b NP-structure CNT CVD-grown 40nm Si N-doped 2-5 *10 19 cm -3 Si P-doped substrate R=1Ωcm d NP-structure CNT CVD-grown 40nm Si N-doped 2-5 *10 19 cm -3 Si P-doped substrate R=50-80Ωcm SOI-structures f Complete SOI structure b1 CNT CVD-grown 40nm Si N-doped 2-5 *10 19 cm -3 Si-intrinsic 100nm SiO 2 40nm Si P-doped substrate 2nd batch g Complete SOI structure h Complete SOI structure CNT deposit b2-1 CNT deposit b2-2 10nm Si P-doped 4 *10 19 cm -3 10nm Si P-doped 4 *10 19 cm -3 30nm Si P-doped 2 *10 19 cm -3 30nm Si P-doped 2 *10 19 cm -3 3nm undoped 30% Ge 70%Si Si-intrinsic 100nm SiO 2 40nm Si P-doped substrate j Complete SOI structure b2-3 CNT deposit b2-4 40nm Si N-doped 1 *10 20 cm -3 3nm undoped 30% Ge 70%Si Si-intrinsic 100nm SiO 2 40nm Si P-doped substrate Figure 9.3: The structures shown in this figure are the ten structures that have been made and worked on in this thesis for a description of each structure see section The b1, b2-1 to b2-4 is numbering 5 of the samples. 103

108 the 300 C where the silicon doped with antimony is grown. The structure is shown in figure 9.3g. 2. First 3 nm of Si 70 Ge 30 is grown followed by a structure similar to the one just described. The Si 70% Ge 30% layer has a smaller bandgap than pure Si this could possibly change the contact resistance. The doped bor layer is again grown at 450 C. The structure is shown in figure 9.3h nm silicon doped with antimony with a flux rate that should give a density of cm 1 but it is unknown how much there is incorporated and how much that float on the surface. The enhanced doping concentration should lead to more carriers in the sample which could give a lower contact resistance. The sample was grown at 300 C. The structure is shown in figure 9.3i. 4. First there is grown 3 nm of Si 70 Ge 30 followed by a structure similar to the one just described. The structure is shown in figure 9.3j Experimental setup This section describes how the samples are processed and some advantages/disadvantages with different processing procedures. We start with a structure as described in the previous section, i.e. where the CNT have been incorporated in the Si structure. Lithography To make the devices we have used three lithographic steps. First some alignment marks are made by UV-lithography. Then electrodes are made with Electron Beam Lithography (EBL), using the alignment marks. The electrodes are separated by gaps that are respectively 0.7 and 1.3 µm wide. At last the big bonding pads are made by UV-lithography. In figure 9.4 a microscope picture of a sample can be seen. The connections between the pictures should be obvious. Due to the nature of the potassium hydroxide (KOH) etch (described in the next section) it is very important to plasma etch the sample before metallization to remove any resist residues. Otherwise the KOH etch might remove the resistresiduals and thereby the metal lines. The results of such a process are observed in figure 9.5 a. Etch The two different kinds of etchant used in this work is described below. bhf:hno 3 :H 2 O in the ratios 1:1:20, where bhf is buffered hydrofluoric acid, in this case ammoniumfluorid is used. This etch is an isotropic etch that works in a two step process. First the HNO 3 oxidizes the silicon surface, then bhf removes the SiO/SiO 2 [8]. An advantage of this etch, is that normal resist 104

109 a Electrodes Lithography b c Alignment marks Bonding pads electrodes Bonding pads Figure 9.4: Microscope pictures of the devices. a a zoom on the electrodes some of the alignment marks can be seen. b The complete SEM-lithography structure can be seen. c The bonding pads that connect the SEM-electrodes to the outside world. can be used as etching mask. The disadvantage is that the etching rate depends quite strongly on doping type, doping density, and other parameters such as temperature and light. These dependencies on the etch rate made it impossible to etch through the 100 nm intrinsic Si in the SOI structure, without side etching several µm of the antimony doped Si. The result resembled the picture in figure 9.5b. Due to this etch, the mesa can in some spots be seen to have turned reddish, where this has happened the mesa is nonconducting. KOH (potassium hydroxide) is another possible wet etchant. This is an anisotropic etch, etching doped and undoped Si with approximately the same speed, but is highly selective in etching Si/SiO 2. The selectivity depends on concentration and etching temperature. The Si/SiO 2 selectivity gives a natural etch-stop in the SOI structures. In this project we have used the following mix KOH:H 2 O:IPA in the weight ratio 5:16:4. The IPA is added to the liquid to give a smoother surface. One have to pay attention when using this kind of etch mix since it phase separates 5. A disadvantage of this etchant is its ability to remove resist making standard lithographic procedures impossible to use. The normal way to circumvent this problem is to mask the surface by some materials most commonly SiN or SiO 2. Since we can not evaporate these materials in house Cr/Au was used as an etching mask. 5 A smooth surface is a huge advantage when the Atomic Force Microscope (AFM) is used to identify the CNTs. The IPA could probably have been avoided since the SiO 2 work as a very effective etch-stop and therefore the surface would probably have been smooth in all cases. 105

110 a b c Non-conducting area Figure 9.5: Etch. a, shows the electrodes after the sample has been etched in KOH. The arrow points to where the KOH etch has removed the metal part of the electrode from the sample surface. b, how the SOI-mesa looks after it has been etched in bhf:hno 3 :H 2 O. c, failed attempt to use the natural oxide as an etch mask for a KOH etch. To work efficiently the KOH etch needs to be heated. Temperatures used in industry is between 60 and 80 C. To use the elevated temperature one needs a reflux system to sustain the starting concentrations of the chemicals. In our case 40 C was used giving an etching rate around 40 nm/min. Keeping the etchant under a lid makes it possible to approximately sustain the starting concentration at this temperature during etching. Etching the SOI structure for 4.5 min are in most cases enough to reach the SiO 2 layer. Due to the selectivity of the KOH etch, the surface becomes completely smooth making it easy to observe CNTs on the surface. Figure 9.6a show an AFM height image of two electrodes and the gap in between. In b a line-trace showing the z-height can be seen. From the line scan the height of the electrodes is estimated to 250 nm. This height fits approximately with the combined height of the 100 nm intrinsic Si layer, 40 nm doped Si and 100 nm evaporated Cr/Au metal layer Electrical setup Most of the electrical measurements, with regard to CNTs incorporated in a semiconductor, were made in the probe station where you connect to the bonding pads by pushing a metal rod into the source and drain pad respectively. All measurements are voltage controlled DC-measurements (the low temperature measurements were AC-measurements). 106

111 a b Figure 9.6: AFM pictures of the end of an electrode gap. a, Shows a topographic picture. b, is a line scan along the white line in a that shows the height profile it can be seen that the trench/gap is 250 nm deep. 9.3 AFM characterization and manipulation Incorporation of single wall carbon nanotubes in Si depends on the the ability of the CNTs to keep their unique properties after a series of processing steps. Epitaxial Si growth by MBE, require removal of oxide and growth at elevated temperatures, it is therefore important to check that the CNTs can retain their original physical and electrical properties. The physical inspection are done by AFM and the results are presented in this section. AFM images of CNTs on the SiO 2 surface after KOH etch Figure 9.7 show a typical amplitude image, taken with an AFM, of a SOIstructure that has been etched down to the SiO 2 layer as already described and where CNT threads can be seen. The structure and height of these threads makes it likely that they are CNTS or ropes of a few CNTs AFM manipulation of CNTs Electrical measurements between electrodes connected by CNT gave resistances between 180 kω and infinity. The current pathway between the two electrodes was investigated by first measuring the resistance between two electrodes. Then by AFM manipulate the CNTs to see if this influenced the conductance. Figure 9.8 show an AFM amplitude image of the full gap between two electrodes. Three CNT or ropes of CNT can be seen crossing the 50 nm gap. The height 107

112 Figure 9.7: These two AFM amplitude images shows CNT bundles, they can be seen as threads on the surface. 8,0 nm 1,4 nm 1,6 nm Figure 9.8: AFM amplitude image of a complete electrode gap. It can be seen that there are three CNTs that crosses the trench. The numbers written at each CNT is the height of the CNT bundle, measured in a AFM topographic image taken simultaneous with the amplitude image. The amplitude image is shown since it is easy to see the CNTs on these kind of images. 108

113 Measurement before and after AFM manipulation 2 1 I[nA] before AFM R= 4,4 M after AFM V b [mv] Figure 9.9: The graph shows the current as a function of the bias applied between the electrodes seen in figure 9.8. The black line is measured before AFM manipulation of the tubes in the gap and give a resistance of 4.4 MΩ and the dashed red line is measured after the manipulation. of the CNT 6 are 1.4, 1.6 and 8.0 nm. The 8 nm implies that it is a rope of CNTs. It can be seen that all three CNTs or bundles of CNTs are lying on the floor of the gap i.e. directly on the SiO 2 layer. The tubes are not suspended all the way between the two electrodes, although it looks like the 8 nm thick bundle might be partially suspended. In figure 9.9 a graph of the current as a function of bias-voltage is seen. The plot is measured between the two electrodes shown in figure 9.8. The black line is measured before AFM manipulation of the CNTs and corresponds to a resistance of 4.4 MΩ. Having measured the resistance of the device we started manipulating the CNTs with the AFM. Figure 9.10a-e shows a sequence of AFM topographic images, where we have manipulated or rather tried to break the bundle of CNTs. These manipulations is performed by scanning an image in tapping mode, selecting a point on the image (light blue dot) where the AFM tip is pressed into the surface by a given force, then another point is selected (dark blue dot) and the tip moves to this point in a straight line all the way pressed into the surface. The resulting topography are shown in the image where the manipulation dots have been drawn. The following list describes the different manipulation steps performed on figure 9.10a-e. 6 Measured on an AFM topographic image the amplitude image is shown since the large height difference between electrodes 100 nm and CNTs 1 nm makes the CNTs easier observable in the amplitude images. 109

114 Figure 9.10a Is just a AFM topographic image (no manipulation). Figure 9.10b The bundle of CNTs have moved a little, bending the rope a bit. Figure 9.10c The tip has dragged the tubes even further - as can be seen on the image, the length of the CNT bundle has increased. Two possible schemes could result in this: 1. The CNT bundle is dragged out from the crystal. 2. The CNTs in the bundle are sliding against each other thereby lengthen the rope. Or it could be a combination of the above mentioned schemes. In this image one should also notice that there are three bumps on the rope and that it looks like there is still a bundle/tube lying where the big bundle was before. Figure 9.10d The bundle of CNTs have been dragged and lengthened even more. It is worth to notice that the three dots seen on the previous image are still seen, but the mutual distance between two of the dots has increased, while the other two approximately have kept their mutual distance. This indicates that the CNTs, in the rope, are sliding against each other. Another observation supporting this is that the rope in this image, in some spots, seems to have a smaller height/diameter than on the previous image. Figure 9.10e In this step the AFM tip has actually torn the bundle of CNTs apart. Due to the elasticity of the CNTs, the rope has curled up in a kind of whip leash effect 7. Notice also that in this step the tip has been started far enough back to try to break the bundle of CNTs that were lying where the big bundle of CNTs ended after the first manipulation (figure 9.10b). The result was a movement/breaking of the little bundle. Afterwards we tried a couple of times to remove the line/cnt that still seems to connect the two electrodes, with no success. Figure 9.10f A topographic image of the area where the rope of CNTs has been tugged apart. Two white lines can be seen indicating where line traces of the topography have been made. Figure 9.10g Two line scans, showing the height curves, made where the white lines in f are seen. The two bumps on the blue line have a height around 1.4 nm indicating that it is a single single wall CNT. The red line is a line scan through the loop made in the end of the broken CNT rope. At the end the rope has a height of 1.5 nm also indicating a single single wall CNT. 7 Note that this whip leash effect further substantiate the assumption that it is indeed CNTs that are measured upon. 110

115 a b c d e f g Figure 9.10: AFM topographic images, that shows a sequence where a bundle of CNTs are manipulated. The blue dots on the image describes from where to where the tip have moved while being pushed into the surface. The tip was pressed down at the light blue dot and then moved in a straight line to the dark blue dot.a, The bundle of CNTs before manipulation. b First attempt to break the CNTs resulting in a slight bend. c, The CNTs are lengthened as consequence of the AFM manipulation. d, The CNTs are lengthened even more. e The rope of CNTs have been broken/snaped and has curled up due to the elasticity of the CNT. f, AFM image showing two line scans. g, The height profile along the two line scans. 111

116 After the bundle of CNTs was broken, another electrical measurement on the same device were made. The result is seen on the graph in figure 9.9, the red dashed line. This shows that there is no longer a electrical connection between the two leads, demonstrating that the current was running through the rope of CNTs before it was broken. The same test were made on two other samples with the same result. Notice that in figure 9.8, two other tubes can be seen to cross the gap. These have not been touched by the manipulation, i.e. revealing that it is not all the tubes that traverse the different electrode gaps that are carrying a current Suspended CNT Most of the AFM images of the CNTs shows that they are lying on the SiO 2 surface, i.e. that the CNTs by surface tension after the etch or due to slack have been sucked down to the surface. However, in some cases the tubes are suspended as in figure 9.11a,b. The two images shows the trace a and retrace b where the scan direction describes which way the tip has been moving while the image was taken. In other samples part of the CNT rope is suspended as in figure 9.11d. Figure 9.11c shows an AFM amplitude image of the partly suspended tube (or bundle) that has a minimum height of 4.2 nm and is around 72 nm high close to the upper electrode. 9.4 Low temperature measurement To increase the knowledge of this system nine samples were bonded and measured at liquid helium temperatures 4.2 K. At this temperature unfortunately only one of them was conducting. Figure 9.12 show a bias spectroscopy plot of this device. The plot shows the conduction in color scale as a function of gate and bias voltage. In the bias spectroscopy plot seen in figure 9.12 a diamond shaped structure is sketched. The diamond shape is a mark of quantum dot behaviour and although they can be observed, the irregularity and negative differential conduction indicates that the CNT is not a single quantum dot. There is probably more than one dot both in parallel and in series. This indicates that the CNTs are not perfect i.e. there are defects or potential barriers in the CNT dot, probably related to the processing treatment. 9.5 Measures taken to lower the contact resistance and results In this section measures taken to lower the contact resistances will be described. The first batch of samples gave resistances between 180k Ω and infinity. First the samples were annealed. Potentially this could activate more of the donor atoms giving a higher carrier density. Then new samples with different doping 112

117 a Trace scan direction b Retrace scan direction c d Figure 9.11: Suspended tubes. a,b,d, AFM topographic images c AFM amplitude image. a,b, Trace and retrace respectively, of a suspended CNT that is moved a little by the tip. c, Partly suspended CNT and the area around. d, Zoom on the partly suspended tube, it is suspended in the part close to the upper electrode 113

118 -25 Bias spectroscopy plot measured at 4K -13 Vb[mV] di/dv [e2/h] Vg [V] Figure 9.12: Bias spectroscopy plot, showing the differential conductance in color code as a function of gate and bias voltage. The measurement is made at 4 K. Diamond like structures can be seen and one is sketched by the black lines indicating that the device behaves as a quantum dot. Before 200 MΩ 830 kω 650 kω 20 MΩ 830 kω 900 kω After 850 kω 60 MΩ 2.5 MΩ 2 MΩ Table 9.1: Resistances before and after annealing. The means no connection. and compositions were grown based on the experience of the first batches. These samples are descried in batch-2 in section Annealing The samples in batch-1 were grown in the MBE-system at 300 C. The low temperature is necessary to incorporate the high donor atom density, but it is too low to make certain that all the donor atoms are activated. An annealing should therefore help on this situation by activating more carriers. Annealing a processed sample To make a direct comparison we annealed a sample already processed. The sample was annealed at 500 C in a Ar atmosphere for a hour. The result can be seen in table 9.1. This has obvious not improved the conduction. There could be a couple of explanations: 1. After the annealing the metal electrodes looks as seen on figure 9.13 i.e. the annealing has been tough to the metal. 2. The resistance measurements before the annealing was made with a floating backgate which make it hard to make a direct comparison of the two measurements. Still we find it reasonable to say that the annealing did not improve the contact resistance. 114

119 Optical microscope picture of an annealed sample Figure 9.13: The images shows how the metal electrodes looks like after having been annealed in a hour at 500 C in Ar atmosphere. Annealing of unprocessed samples Another three samples were annealed, before they were processed. They were all annealed for 30 min in Ar atmosphere at 500, 650 and 800 C respectively. After the annealing they were processed like the other samples. The lowest resistance obtained in these samples is 130 kω obtained in a sample that have been annealed at 650 C. Due to time issues these samples have not been measured in the AFM, but from the measured resistances we find it fair to say that the annealing did not have a large effect on the resistance Samples with CVD-grown CNT The samples made with CVD-grown CNTs incorporated in SOI structures were also processed. The advantage with CVD-grown tubes are normally that they under the right growth procedure, produces single single-wall CNTs which is preferable if one want to se semiconducting behaviour in the CNT. With CVDgrown tubes one also avoids using sonication on the tubes. Defect free CNTs are properly more inert to KOH etch, which might be an issue if some of the resistance, measured on these devices, are due to defects in the tubes. Unfortunately the two batches of samples made on these kind of substrates both showed huge gate-leak, making interpretation of the measurements on these structures impossible. This leak current must be due too the CVD-treatment, where the sample were heated to 940 C under a hydrogen and methane flow. From the annealed samples we know that the substrate can be heated to 800 C without these kind of problems, so the question is if the leak is due to the extra 140, the gas flow or a combination of these two factors. 115

120 Sample batch b1 b1- b1- b1- b1- b2- b2- b2- b2- CVD a500 a650 a b4 Structure see fig 9.3 e f e e e g h i j Samples Produced Probed Conducting Metallic Semi p-type Semi n-type Lowest R [MΩ] Percentage 28% 24% 25% 9% 16% 14% 73% 38% Table 9.2: The samples are named with the following systematics b1 means batch 1, b2 batch 2, b1-a500 is a sample from batch 1 annealed at 500 C and the b2-1 are from batch-2 described as number one in the enumeration in section Samples produced are the number of samples made. Probed are the number of samples probed, all the produced samples have not been probed some due to lithography errors and some due to lack of time. Lowest R gives the lowest sample resistance value measured in each sample batch. Percentage is the percentage of the probed samples that are conducting. 9.6 Samples and sample behavior In this section we will look upon the number of samples produced and do some statistic on these numbers Samples produced 1364 samples were made disregarding some initial sample making. 772 samples were probed and of these 191 had a finite resistance. In this number samples that had a gate leak have been disregarded. Of the 191 conducting samples 152 shoved metallic behaviour and 39 were semiconducting, 32 were p-type and 7 were n-type. 9 Samples were bonded and cooled to 4K where unfortunately only one of them worked. 33 samples were measured in the AFM. In 5 of these the CNTs were manipulated/broken, 3 of these successfully. Of the 33 samples 2 had short circuit and in 13 of them the height of the bundles of CNTs traversing the gap were measured. Table 9.2 shows samples made and measured. From table 9.2 it can be seen that the samples with bor as the dopant have the lowest percentage of conducting devices, disregarding sample b1-a800 which have also been annealed. This could be due to the distribution of the CNTs, but it suggest that if you want low contact resistance to the CNTs you should not use bor as the dopant. In figure 9.14 a distribution of the resistances in all the samples are seen. 73 samples showed a resistance below 1 MΩ. In comparison there were 154 samples with a resistance below 1 GΩ. 116

121 Figure 9.14: Histogram of the resistances in two different scales for all the conducting samples Samples where the height of the CNTs has been measured From the height of the bundles of CNTs one can give an rough estimate of the number of single wall CNTs in the bundle. Lets say that the average diameter of the dispersed single wall CNT is around 1.4 nm 8. If we take the height of the rope, measured in the AFM, and assume that this is the diameter of the rope, then we can estimate how many single wall CNTs with a diameter of 1.4 nm we can fit into the rope. Under these assumptions we can estimate the average resistance of each single CNT. On the device measured this gives a spread of resistances per CNT between 1.7 MΩ and 3.7 GΩ i.e. the average resistance per CNT change from device to device with a factor This big difference clearly suggest that the contact resistance to each individual CNT varies a lot, probably most of the CNTs does not have contact to the silicon while a few tubes have a much better contact than the 1.7 MΩ. This is a very normal observation when working with CNTs. It is therefore impossible in this case, from AFM measurement of the height to determine the lowest contact resistance to a single single wall CNT Sample with the lowest resistance The lowest resistance was measured on a sample from b2-3. The sample resistance was 110 kω. The I-V curve can be seen in figure The estimated number of CNTs traversing the gap are 31 calculated in the same way as in the previous section. 8 Fitting the diameter of a (10,10) tube. 117

122 I [na] Vg=10V R=110kohm Vg= 0V R=110kohm Vg=-10V R=120kohm V b [mv] Figure 9.15: Plot of current as a function of bias voltage the three lines are measured at three different gate-voltages Semiconducting samples As seen on the table above many of the devices produced showed semiconducting behavior. Figure 9.16 display four plots of the current as a function of gatevoltage at constant bias voltage for four semiconducting devices. The four devices are from the following sample batches. a Are from b2-1, b are from b1, c are from b2-2 and d are from b1-a650. In a,b an increased current can be observed as the gate becomes more negative. This demonstrates that the tubes are behaving as p-type semiconductors. c show ambipolar behaviour i.e. it is both p and n-type. This is observed in small bandgap CNTs [48] it has a minimum resistance of 860 kω. d behave as a n-type conductor, which is unusual for a CNT device. This behavior is probably due to potassium [12 14], which in a CNT behave as a n-type dopant. The potassium is an ingredient in the KOH etch. 9.7 Conclusion and outlook A recipe for producing samples and using Cr/Au as etching mask was developed. 10 different types of sample-structures have been made, 6 of these are SOIstructures that have been processed and measured upon. There have been made 1364 samples. Off these around 772 were probed giving 191 conducting samples with 39 semiconducting. After the KOH etch CNTs have been observed between the n- or p-typ silicon electrodes. These structures, have been imaged by AFM and are shown to be conducting. Low temperature measurements has been performed and have shown quantum dot behaviour. It would be interesting to study if the KOH etch affects the CNTs. Unusual n-doping was observed in some of the devices. If the etch deteriorates the CNTs, different approaches might be tried; one possibility could be to use CVD-grown 118

123 15 a Vb=50mV Vb=100mV 60 b Vb=100mV Vb=5mV I [na] c d Vb=50mV Vb=10mV Vg [V] Figure 9.16: Four graphs showing the current as a function of gate-voltage at a constant bias-voltage. a, shows a p-type semiconducting behavior and is measured on a sample from b1. The sample has a minimum resistance of 2.6 MΩ. b, P-type semiconductor with a minimum resistance of 6.8 MΩ. c, Shows a ambipolar behavior, the device has minimum resistance of 860 kω. d, n-type semiconductor with a minimum resistance of 350 kω. 119

124 tubes; such samples were made but more experiments in this direction would be interesting. Other kind of etches would also be interesting; an obvious candidate is Reactive Ion Etching (RIE) where it has been shown that multiwalled carbon nanotubes can be used as etch mask. From the second batch of samples it looks like it is possible to get contact between the SiGe layer and the CNT. These kind of Si/SiGe heterostructures can be used to create two dimensional electron and hole gases. This could make it possible to study a two dimensional system connected to a one dimensional CNT system. One could also imagine that the optical properties of CNTs would be interesting in combination with silicon heterostructures. Light emitting CNT pn diodes between Bragg mirror resonators might eventually lead to leasing, which would be a significant achievement. 120

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129 Appendix A Plots for D1K2 and D1K3 Figure A.1 shows the conduction as a function of gate, and bias voltage. The measurement is made on device 1 and two Kondo ridges near zero bias can be observed and henceforth we shall refer to the corresponding odd-numbered Coulomb-blockade diamonds showing the Kondo effect as D1K2 and D1K3, respectively. These Kondos are measured on the same sample as D1K1 but off-set in gate voltage by roughly 20 V. Figure A.2c shows a zoom-in on the Kondo ridges, measured at B = 1 T. A gate dependent splitting of the Kondo resonance is clearly observed in both device. The different slopes indicate a change in Γ = Γ s + Γ d from one Kondo resonance to the next, most likely due to a difference in couplings to the two orbitals in the CNT. The splittings of the Kondo peaks have been read off and plotted in figure A.2a,b. The lines are fits to the scatter-plots and the different colors correspond to different external magnetic fields. The fitted values used for P Γ/π are and mev for the groups of fits in a and b, respectively. The insets in both figures show 0 + gµ B B/e as function of B. The two linear fits yield Vb [mv] Vg [V] di/dv [e2/h] Figure A.1: Five Coulomb diamonds with a Kondo resonance in every second. The plot show the di/dv as a function of gate and bias voltage. 125

130 a V b [mv] B [mt] c B=-0.1T -0.2 B=-0.2T B=-0.375T B=-0.5T B=-1T B=-2T gµ B B/e [mv] b V b [mv] ε d /U B [mt] B=-0.1T -0.3 B=-0.2T B=-0.375T B=-0.5T B=-1T B=-2T gµ B B/e [mv] di/dv [e 2 /h] 1.0 V b [mv] B= -1T V g [V] Figure A.2: Gate dependence of the exchange field for different applied magnetic fields measured on a D1K2 and b D1K3. c, Show the di/dv as a function of bias and gate voltage the measurement are made with an external field of -1 T. Four Coulomb peaks and two Kondo resonances are observed the 1st resonance corresponds to D1K2 and the 2nd to D1K3. a, b, Shows the scatter plot of the peak position read off from measurement like the one shown in c, where the gate voltage have been normalized with the charging energy. The different colours and shapes corresponds to different external magnetic fields. The lines are fits to the scatters. The insert shows the fitting constant 0 + gµ B B/e for the different magnetic fields and a linear fit to these points. 126

131 a V g [V] b V g [V] B [mt] di/dv [e 2 /h] B [mt] di/dv [e 2 /h] Figure A.3: Gate dependence of the exchange field at zero bias. The plots shows the di/dv as a function of gate voltage and external magnetic field for D1K2 a and D1K3 b. The two horizontal lines in each plot are due to Coulomb peaks. The white line crossing from one peak to the other a or halfway between them b maps out the Kondo peak movement (the point where the spin states are degenerate) as a function of magnetic field. The black line is a parameter free fit where the constant have been found from the fits in figure A.2. Similar to the parameter free fit shown in figure 5.6b. a slope of (a) and (b ), corresponding to a g-factor of 1.7 and 1.8, respectively. Figure A.3 shows di/dv as function of gate voltage and magnetic field. Panels a and b correspond to D1K2 and D1K3, respectively. The black line is plotted using the procedure described in the caption of figure 5.6b, now with parameters obtained from figure A