Quantum Wires and Quantum Point Contacts. Quantum conductance

Transcription

1 Quantum Wires and Quantum Point Contacts Quantum conductance

2 Classification of quasi-1d systems 1. What is quantum of resistance in magnetic and transport measurements of nanostructures? Are these quanta different and if yes why? Give examples of quasi onedimensional nanostructures. What characteristic length scales are most important in nanophysics of Quantum Wires and Quantum Point Contacts, and how to classify them? Is effective mass approximation suitable to describe quantum wires? What is the exception from this rule? Quasi-1D systems with width w comparable to the de Broglie wavelength Quantisation in units R K =h/2e 2 / Quantum wires and point contacts 2

3 Quantization of conductance vs. gate voltage! Conductance quantum: S New universal unit of resistance h/2e 2 Absent in classical theory KOhm Introduction 3

4 Quantum Hall effect Klaus von Klitzing The accuracy δρ xy /ρ xy is of the order of Quantisation in units R K =h/e 2 Magnetotransport in 2DEG

5 Klaus von Klitzing: the new SI Since h/e2 and 2e/h are the only fundamental constants which can be measured with such a high precision, that conventional values were introduced for metrological applications, one may think to fix these numbers instead of fixing e and h which are not directly accessible by experiment with high accuracy. From presentation in The Royal Society, London, 2011 Magnetotransport in 2DEG

6 Cleaved edge overgrowth (CEO) (a) The layer growth by MBE is interrupted and the wafer is cleaved inside the chamber. Cleaving includes scratching the wafer at its edge and subsequently breaking it by mechanical pressure. The GaAs wafer breaks at the scratched position along a single crystal plane. (b) MBE growth is continued on top of this freshly cleaved surface. (c) The electrical connections to the different elements. Voltages can be applied to the two gates T and S, and a current is applied between the two areas separated by gate T. (d) (f) Sketches of how different gate voltage combinations shape the electron gas. In particular, a one-dimensional wire is formed in (f), which extends along the quantum well at the cleaved and regrown interface. Experimental Techniques 6

7 Angle evaporation Preparation of a small Al island coupled to two leads via smallarea tunnel barriers. A layer of electron beam soft resist (with low illumination dosage) is covered by a hard resist. This leads to cage formation as an extreme version of an undercut profile. The upper resist layer is free-standing over a certain area. After angle evaporation a sandwich structure with small overlap areas results, which can be below the pattern sizes in the resist mask. Resist here is used as a shadow mask. The overlap areas as small as 30 nm 30 nm can be prepared routinely by angle evaporation. Experimental Techniques 7

8 Diffusive quantum wires Top view of a diffusive quantum wire (L = 40 μm, geometric width w g = 150 nm), patterned into a Ga[AI]As- HEMT by local oxidation. The bright lines define the walls (a close-up is shown in the inset). The electrostatic wire width w can be tuned by applying voltages to the two in-plane gates IPG 1 and IPG2. TH Quantum wires and point contacts 8

9 Carbon nanotubes applications Fiber Spinning Nanotubes Electronics Pick and place Nanotechnology Types Properties, applications

10 A quasi-1d system 6.45 k Superconductivity is suppressed when the room temperature resistance is greater than the superconducting resistance quantum, R q =h/4e 2 Quantum wires and point contacts 10

11 Focused Ion Beam quantum wires fabrication YBa 2 Cu 3 O 7 x Ga - ions Quantum wires and point contacts 11

12 Wave functions and eigenvalues A strip of conducting material about 10 nanometers or less in width and thickness that displays quantum-mechanical effects. Split gate Interface 3. What is parabolic quantum well wire and how to define its density of states? How reflects the density of states one-dimensional character of the wire? Describe parabolic confinement model. Simple, but realistic model: Effective potential Electron density Parabolic potential well Quantum wires and point contacts 12

13 Parabolic confinement model Schrödinger equation: Potential: Solution: Harmonic oscillator! Quantum wires and point contacts 13

14 Energy spectrum in parabolic confinement set of 1D bands Density of states: Quantum wires and point contacts 14

15 Energy spectrum and density of states for parabolic confinement F F Quantum wires and point contacts 15

16 Diffusive quantum wires Top view of a diffusive quantum wire (L = 40 μm, geometric width w g = 150 nm), patterned into a Ga[AI]As- HEMT by local oxidation. The bright lines define the walls (a close-up is shown in the inset). The electrostatic wire width w can be tuned by applying voltages to the two in-plane gates IPG 1 and IPG2. TH Quantum wires and point contacts 16

17 Magnetic field effect in parabolic confinement Parabolic confinement: Then we have the equation very similar to that leading to Landau quantization, however with replacements: The resulting equation is: 4. Describe magnetic field effect in parabolic-confinement model. What is characteristic cyclotron frequency in this case and how is it related to characteristic frequency of 2DEG? How is effective mass renormalized in this case? Are oscillations of the density of states periodic in 1/B? What happens in very strong and very weak magnetic fields? Quantum wires and point contacts 17

18 Magneto-electric modes Magnetic field increases the confinement strength and can lead to depopulation of magneto-electric modes. (at Fermi surface) As a result, density of states oscillates as a function of magnetic field. However, the oscillations are not periodic in 1/B. In very strong magnetic fields, come back to the case of 2DEG., we Quantum wires and point contacts 18

19 Quantum wire: comparison with experiment The magnetoresistivity of the wire shown earlier. The most prominent feature is the oscillation as a function of B, which detects the magnetic depopulation of the wire modes. Due to quantum interference The positions of the oscillation minima are plotted vs. 1/ B as full circles in the inset. Here, the line is a least squares fit to theory, which gives an electronic wire width of 158 nm. 5. What is dependence of the resistance of quantum wire on magnetic field? Does magnetic field influence density of state in the wire? Are minima of resistance periodic in 1/B? How can the position of resistance minima be used to find electronic wire width? Quantum wires and point contacts 19

20 Boundary scattering Boundary scattering in wires is important. If the roughness scale is less or of the order of the Fermi wave length, then the scattering is diffusive. For smooth edges the scattering is mirror, it does not influence conductance very much. Magneto-transport is specifically sensitive to the boundary scattering. Depending on the ratio between the wire width and cyclotron radius electron spend different time near the edges. It influences the conductance, see next slide. Quantum wires and point contacts 20

21 Boundary scattering and wire peak r c = (ħ/eb) 1/2 Diffusive scattering at the wire boundaries causes a magnetoresistivity peak at w = O.55r c The main figure shows these peaks for wires of different widths, studied at QW made by ion beam implantation, which generates particularly rough, diffusive walls. 6. What is wire peak and how boundary scattering influences resistance of quantum wire? At what cyclotron radius is the fraction of electron trajectories close to the wire edge maximised? Describe behaviour of the system with rough edges. Wire peaks in QWRs defined by top gates of similar widths (shown in the inset) are much weaker (the reflection is more specular). The length of the wires was L = 12 μm. For QWRs defined by top gates, by cleaved edge overgrowth, or by local oxidation, one can safely neglect boundary scattering. Quantum wires and point contacts 21

22 Bohr-Sommerfeld quantization rule The number of wavelength along the trajectory must be integer. Only discrete values of the trajectory radius are allowed Energy spectrum: ω c τ 1 Landau levels Wave functions are smeared around classical orbits with r n = l B (n+1) 1/2 ; l B = (ħ/ c m) 1/2 l B is called the magnetic length. It is radius of classical electron orbit for n = 0. v/r; r v/ ; mv 2 /2= ħ c (n+1/2); v n 0 = (ħ c /m) 1/2 ; l B = r n 0 = (ħ/ c m) 1/2 =(ħ/eb) 1/2 Magnetotransport in 2DEG

23 Ballistic quantum wires First system that have been studied ballistic point contact GaAlAs-HEMT split gate Two traces correspond to different concentrations tuned by illumination. T = 60 mk Gate voltage Conductance is quantized in units of in the absence of magnetic field 7. Can conductance of quantum wires and point contacts be quantised in the absence of magnetic field and if yes, in what units? How to tune the electronic width of quantum point contacts? Does conductance quantisation resemble the quantum Hall effect in its electron density-change form? What are specific features and accuracy of conductance quantisation in QWR? Quantum wires and point contacts 23

24 Quantum Hall effect Klaus von Klitzing The accuracy δρ xy /ρ xy is of the order of Quantisation in units R K =h/e 2 Magnetotransport in 2DEG

25 Quantization in quantum point contacts The quantization seems to be similar to the quantum Hall effect. However: No Landau quantization, no impurities in the contact region The typical accuracy is Conductance quantization usually disappears at L > 2 μm while quantum Hall effect can be observed in samples of millimetre sizes. Principal questions: How finite conductance can exist without scattering? Where the heat is released? What is the reason for quantization? Why is the conductance quantized in units 2e 2 /h? Quantum wires and point contacts 25

26 Conductance quantization in QPC: a simple model A barrier with tunneling probability T, connecting 1D wires Let us calculate the current in a symmetric system: 8. Describe quantum point contact in an energy-dependent model of one-dimensional leads connected by barrier with transmission probability T. How to calculate the conductance in this model? What is the link between the one-dimensional density of states and the electron velocity? What is total conductance of the ballistic quantum point contact for a mode with T = 1 at zero temperature? Quantum wires and point contacts 26

27 Conductance quantization in QPC For a 1D system the density of states is inversely proportional to the velocity! f 9. What is the conductance of the ballistic quantum point contact for a mode with variable T at zero temperature? What information gives the full conductance of the system about the origin of the quantum of conductance? Quantum wires and point contacts 27 E

28 Conductance quantization in QPC At T=1, Here we have assumed that the distributions in the leads are described by the Fermi functions, i. e., the leads are in the equilibrium. Such an equilibrium is achieved only at the distances larger than the electron mean free path. Thus, the dissipation takes place in the leads close to the contacts. What we have calculated is the total resistance of the device, which can be understood as an intrinsic contact resistance. Quantum wires and point contacts 28

29 Multimode conductance in QPC What happens if the wire is quasi-1d, i. e., has several modes? Landauer formula, - transmission amplitudes If there are no inter-mode scatterings, then the transport is called adiabatic. This is the case for the wires with smooth modulation of the width. In case of pronounced backscattering the shape of the conductance steps is modified. 10. What is the Landauer formula for the conductance of a multimode quantum point contact? What is adiabatic quantum transport? How is it linked with the scale of the Fermi wavelength and the width of an electron channel? Quantum wires and point contacts 29

30 Conductance in QPC influence of shape Smooth Sharp A model for smooth point contacts leads to the following criterion for the transition regions to be narrow: 11. How does the shape of quantum point contact influence the shape of conductance steps? What is the effect of temperature on steps? At what characteristic temperature do steps disappear? Are steps infinitely sharp at zero temperature? What is the form of steps for the saddle-type potential V(x, y) = 1/2 m*(ω y2 y 2 ω x2 x 2 )? How behave the steps at different ω y2 y 2 and ω x2 x 2? Quantum wires and point contacts 30

31 Conductance in QPC saddle-like shape Conductance of a QPC with a saddle point potential, characterized by ω x and ω y, calculated for zero temperature. The steps vanish as soon as ω y becomes smaller than ω x. Quantum wires and point contacts 31

32 Conductance in QPC influence of temperature The conductance steps vanish at a characteristic temperature given by the energy spacing of the modes. Conductance steps tends to saturate at low Θ. The steps are thermally smeared as soon as the full width at halfmaximum of f / E equals Δ. This is the case for Δ = 2k B Θln(3+ 2 2) 3.52k B T. Quantum wires and point contacts 32

33 Conductance in QPC influence of magnetic field The modes of a QPC get depleted by perpendicular magnetic fields since the magnetic field increases the confinement and the energy separation between the modes becomes larger. Simultaneously, quantum Hall states are formed in the 2DEG, which influences the resistance measurements. Depopulation of QPC modes by a magnetic field. As B is increased, the width of the conductance plateaus increases. At B = 1.8 T and above, the spin degeneracy gets lifted and additional plateaus evolve at odd integers of e 2 /h. 12. What is the influence of perpendicular and parallel magnetic field on the conductance of quantum point contacts? What is the behaviour of QPC in weak perpendicular magnetic fields? Explain the nature of diamagnetic shift. Quantum wires and point contacts 33

34 Effect of parallel magnetic field in 2DEG Parallel field tunes the spin splitting. It also adds to the electrostatic confinement, shifting position of the potential well and energies of subbands to higher values: diamagnetic shift. In addition it increases the effective mass of charge carriers in the direction perpendicular to B, but in the plane of the electron gas and enhances separation of energy levels. Left: A parabolic quantum well in perpendicular and parallel magnetic fields. Center: Fermi sphere in a magnetic field applied in the x direction (full line), in comparison to the Fermi sphere for B = 0 (dashed line). Right: The confining potential in the z-direction for B = 0 (dashed line) shifts and narrows in a parallel magnetic field (full line). This leads to a diamagnetic shift of the energy levels, and an increase in the subband separation. Magnetotransport in 2DEG

35 Two-mode QPC influence of magnetic field The conductance quantization in units of 2e 2 /h is expected. If two degenerate modes (belonging to different ladders) cross the chemical potential, the conductance will change by 4e 2 /h. 13. Describe the influence of perpendicular magnetic field on twomode quantum point contact. Is the change of conductance by 4e 2 /h possible here? The evolution of steps in two-mode QPC in perpendicular magnetic field. The channels in the y-direction are labelled by l and in the z-direction by m. Quantum wires and point contacts 35

36 Transconductance measurements of two-mode QPC Transconductance dg/dusg as a function of magnetic field. Dark regions correspond to low transconductance: here, the chemical potential lies in between two adjacent modes. The bright lines (high transconductance) reflect the mode change. As a function of B y, both ladders show a positive dispersion. For magnetic fields in the x-direction, the dispersion of the m = 1 states is reversed 14. Explain transconductance measurements of two-mode quantum point contact in perpendicular and parallel magnetic field. What is the lever arm term related to the effect of gate voltage on the performance of device? What is its typical value? Transconductance dg/du sg as a function of (a) B y and (b) B x. The bright lines represent the QPC modes. (c, d) The measurements are compared to the energy spectrum of an elliptical parabolic confinement. Quantum wires and point contacts 36

37 Two-mode QPC theoretical description Perpendicular field Parallel field (B y ) The energy in the QPC is proportional to U sg. The lever arm α = de/du sg can be estimated by temperaturedependent measurements, which give the energy spacing between adjacent modes. A value of α 0.02 ev/v is found. The calculated spectra agree very well with the experimental result for ω y = 2 mev and ω z = 5 mev. 12. What is the influence of perpendicular and parallel magnetic field on the conductance of quantum point contacts? What is the behaviour of QPC in weak perpendicular magnetic fields? Explain the nature of diamagnetic shift. Quantum wires and point contacts 37

38 The Landauer formula explaining steps is applicable only if nothing happens inside the wire no interactions, inelastic processes, etc. It fails to address some problems. Example: puzzle of 0.7 plateaus Conductance of a QPC as a function of the spit-gate voltage. The 0.7 structure (a)the "0.7 feature" becomes sharper with temperature increase. (b) In strong parallel magnetic fields, the spin degeneracy is removed and additional plateaus are visible at odd integers of e 2 /h. As B is reduced, the spin-split plateau at G = e 2 /h evolves into the 0.7 feature Quantum wires and point contacts 38

39 Four-terminal resistance measurements How one can avoid measuring contact resistance? Use four-probe measurements! Four-terminal resistance measurements on a ballistic quantum wire. A quantum well is grown in [100] direction, and three tungsten gate stripes are evaporated on top. The wafer is then cleaved, and a modulation-doped layer of Al 0.3 Ga 0.7 As is grown along the [110] direction. The wire extends along the cleave. Cleaved Edge Overgrowth 4-terminal experiments beautifully confirm the theory of 1D ballistic transport Quantum wires and point contacts 39

40 Four-terminal resistance measurements: result The two-terminal resistance of the wire (A) along gate 2 and its four-terminal resistance (B) are compared. A 15. What is the origin of quantised resistance in quantum wires? How to avoid experiencing contact resistance in an experiment? How do potential leads influence contact resistance? What is the result of resistance measurements in a typical arrangement of potential leads? B In four-terminal measurements resistance vanishes! The inset shows the ratio between the four-terminal resistance and the two-terminal resistance as a function of the probability T for electrons to be transmitted between the wire and the probe. The lower trace in the main figure has been performed at R wp = 250 kω, i.e. T 0.05 in the inset. Quantum wires and point contacts 40

41 Current distribution in QPC An instrument to measure current distribution in a QPC is Scanning Gate Microscopy (SGM) The tip introduces a movable depletion region which scatters electron waves flowing from the quantum point contact (QPC). An image of electron flow is obtained by measuring the effect the tip has on QPC conductance as a function of tip position. G in total current Spatial distribution Explanation: the branching of current flux is due to focusing of the electron paths by ripples in the background potential. Topinka et al., Nature 2001 More sophisticated SGM technique allows analyzing e-e interaction. The experimental current distribution can be explained by account of nonequilibrium effects in QPC. Jura et al., preprint 2010 Quantum wires and point contacts 41

42 The Landauer Büttiker formalism Landauer formula: Adiabatic transport: Transition amplitudes Transition probabilities S Wire D To get accurate quantization of conductance T needs to be very close to 1 for occupied modes and 0 for empty modes t 2. What are diffusive and ballistic quantum wires? What kind of wires is described by the formalism of R. Landauer and M. Büttiker? Can you describe main features of this formalism? Quantum wires and point contacts 42

43 Edges states in Quantum Hall effect Calculated energy versus center coordinate for a 200- nm-wide wire and a magnetic field intensity of 5 T. The shaded regions correspond to skipping orbits associated with edge-state behavior. Only possible scattering is in forward direction chiral motion Schematic illustration showing the suppression of backscattering for a skipping orbit in a conductor at high magnetic fields. While the impurity may momentarily disrupt the forward propagation of the electron, it is ultimately restored as a consequence of the strong Lorentz force. Magnetotransport in 2DEG

44 Edge states 1D states in 2DEG What happens near edges of 2DEG in magnetic field? Skipping orbits, or edge states Quantum mechanics: squeezing of wave function near edges leads to increase in the Landau level energy Quantum wires and point contacts 44

45 Flow is strictly one-dimensional All electrons near one edge move in the same direction while at the opposite edge move in the opposite direction Current flow diagram of a Hall bar for two occupied Landau levels: To be scattered back an electron has to traverse the whole sample, T=1. Current flow in 2DEG Hall bar edge behaves as an ideal ballistic quantum wire! Quantum wires and point contacts 45

46 Landauer Büttiker formalism The transmission probability of contact p into contact q is T q p = T qp. It is possible to have T qp > 1, since more than one mode may connect the two contacts. T qp does not have to be an integer, T pp is a backscattering probability. The total current emitted by contact p is I p, while μ p is the electrochemical potential of contact p. An ideal contact absorbs all incoming electrons and distributes the emitted electrons equally among all outgoing modes, such that they are filled up to μ p, assuming zero temperature. The Landauer formula that generalizes to the Büttiker formula is: t 16. Introduce Landauer Büttiker formalism. What is it used for? Can it describe current flow in 2D electron gas? Please give an example. Quantum wires and point contacts 46

47 Landauer Büttiker matrix For the problem above the Landauer - Büttiker formula gives: Here G = j(e 2 /h). By choosing μ d = 0 as a reference potential, and after eliminating the drain current: The solution gives: V s = V 3 = V 4 V 1 = V 2 = 0 I s = GV s The accuracy of the quantization is so much more accurate than in a QPC because backscattering is greatly suppressed. Quantum wires and point contacts 47

48 Current flow in 2DEG If the edges are in an equilibrium, or there is no backscattering from contacts, then the potential difference between the edges is just the same as between the longitudinal contacts. Then each channel contributes to the Hall current as an ideal ballistic quantum wire. This picture can be checked by inserting a gate causing backscattering in some channels. Quantum wires and point contacts 48

49 Filling factor in Landau quantization Usually the so-called filling factor is introduced as For electrons, the spin degeneracy Magnetic field splits energy levels for different spins, the splitting being described by the effective g-factor - Bohr magneton For bulk GaAs, Magnetotransport in 2DEG

50 Meaning of filling factor An even filling factor, levels are fully occupied., means that j Landau An odd integer number of the filling factor means that one spin direction of Landau level is full, while the other is empty. How one can control chemical potential of 2DEG in magnetic field? By changing either electron density (by gates), or magnetic field. Magnetotransport in 2DEG

51 Gate controlled current flow in 2DEG The resistances depends on filling factor N in the ungated region and M in the gated region: 17. What is the origin of resistance quantisation in 2D electron gas? Describe gate-controlled current flow in 2DEG. How do resistances between different contacts depend on filling factors: N in the un-gated region and M in the gated region? Left: at suitable gate voltages, the inner one of the two edge states gets reflected. Right: For a 2DEG in the regime of filling vector 2, with spin-split edge states, a plateau at R xx = h/2e 2 is observed as a function of the gate voltage. The dip around a gate voltage of 0.2V can be explained within a trajectory network formed below the gate. Quantum wires and point contacts 51

52 Conductance quantization in 2DEG Quantum wires and point contacts 52

53 Edge states in quantum Hall effect regime 18. Describe the concept of edge channels. Why are the edge channels formed? Does electron screening in metallic regions play role in their formation? Structure of spinless edge states in the QHE regime. (a)-(c) One-electron picture of edge states. (a) Top view on the 2DEG plane near the edge. (b) Adiabatic bending of Landau levels along the increasing potential energy near the edge. (c) Electron density as a function of the distance to the boundary. (d)-(t) Self-consistent electrostatic picture. (d) Top view of the 2DEG near the edge. (e) Bending of the electrostatic potential energy and the Landau levels. (f) Electron density as a function of distance to the middle of the depletion region. Quantum wires and point contacts 53

54 Other one-dimensional systems A mechanically controlled break junction for observing conductance quantization in an Al QPC. The elastic substrate is bent by pushing rod with a piezoelectric element. The thin Al bridge, fabricated by electron beam lithography, can be broken and reconnected for many cycles. 19. What do other quantum-wire or point-contact systems, except semiconducting, you know? Does conductance quantisation take place there? Please give examples. Quantum wires and point contacts 54

55 Mechanically controlled break junction Top: Conductance as a function of the deformation time of a gold junction (an STM setup was used). The observed steps are usually not quantized in units of 2e 2 /h. Bottom: A histogram of many consecutive sweeps of the upper type, however, reveals that steps of the expected height dominate. Quantum wires and point contacts 55

56 Carbon nanotubes Discovered by S. Iijima, 1991 FET News Electrostatic potential Modelling Melting Space Tvisting 20. Can carbon nanotubes be used as quantum wires? How would you classify carbon nanotubes? What applications of carbon nanotubes do you know? Quantum wires and point contacts 56

57 Carbon nanotubes applications Fiber Spinning Nanotubes Electronics Pick and place Nanotechnology Types Properties, applications

58 Making carbon nanotubes Graphite sheet can be rolled in a tube, either single-wall, or multi-wall. The graphene lattice and lattice vectors a 1 and a 2. A wrapping vector n 1 a 1 +n 2 a 2 = 4a 1 +2a 2 is shown. The shaded area of graphene will be rolled into a tube so that the wrapping vector encircles the waist of the NT. The chiral angle φ is measured between a 1 and the wrapping vector. Carbon nanotubes 58

59 Carbon nanotubes classification The geometry of an unstrained NT is described by a wrapping vector. The wrapping vector encircles the waist of a NT so that the tip of the vector meets its own tail. A NT with wrapping vector 5a 1 +5a 2, φ = 30. Chrial angle φ can vary between 0 and 30 (any wrapping vector outside this range can be mapped onto 0 < φ < 30 by a symmetry transformation). Carbon nanotubes 59

60 Carbon nanotubes properties The most eye-catching features of these structures are their electronic, mechanical, optical and chemical characteristics, which open a way to future applications. For commercial application, large quantities of purified nanotubes are needed. Electrical conductivity. Depending on their chiral vector, carbon nanotubes with a small diameter are either semi-conducting or metallic. The differences in conducting properties are caused by the molecular structure that results in a different band structure and thus a different band gap Mechanical strength. Carbon nanotubes have a very large Young modulus in their axial direction. The nanotube as a whole is very flexible because of the large length. Therefore, these compounds are potentially suitable for applications in composite materials that need anisotropic properties. Quantum wires and point contacts 60

61 Quantization around a graphene cylinder In a cylinder such as a NT, the electron wave number perpendicular to the cylinder s axial direction,, is quantized. For zero chiral angle Electron states near E F are defined by the intersection of allowed k with the dispersion cones at the K points. Carbon nanotubes 61

62 Momentum quantization in carbon nanotubes The quantized are determined by the boundary condition where j is an integer and D is the NT diameter The exact alignment between allowed k values and the K points of graphene is critical in determining the electrical properties of a NT Consider NTs with wrapping indices of the form (n 1,0). When n 1 is a multiple of 3 (n 1 = 3q where q is an integer) there is an allowed that coincides with K. Setting j = 2q one gets = 2 j/n1 = 4 q/3q = 4 /3 K points are always at 1/3 (or 2/3) position Carbon nanotubes 62

63 Metallic and semiconducting carbon nanotubes n 1 = 3q+p Depending on the direction of the CN axis and the circumference of the CN, the k x of one mode may or may not hit a K-point of the graphite sheet s Brillouin zone, and hence a metallic or a semiconducting CN results. It can be shown that CNs are metallic if (2n 1 + n 2 ) is an integer multiple of 3. Carbon nanotubes 63

64 Carbon nanotubes: density of states and energy bands Energy bands E(k) for the metallic nanotubes (9,6) and (7,4). The Fermi level is ate=0. In both cases two electron bands cross the Fermi level at k 0, each corresponding to a conducting channel in the nanotube. [Saito'98] The calculated density of states of (a) semiconducting and (b) metallic carbon nanotubes. The Fermi energy is at E = 0. The bandgap equals 0.7 ev in the semiconducting CN.

65 Carbon nanotubes: conductance quantization In CNT, conductance is quantized in units G 0 Transport in nanotubes is ballistic. The nanotubes were not damaged even in the relatively high applied voltage such as 6 V that should correspond to a current density of more than 10 7 A/cm 2. Typically, such current would heat a 1 μm long nanotube with 20 nm in diameter to a temperature T max = K. [Frank'98] Quantum wires and point contacts 65

66 The way in which nanotubes are formed is not exactly known. The growth mechanism is still a subject of controversy, and more than one mechanism might be operative during the formation of CNTs. Hydrogen storage Composite materials Potential applications of CNTs Li intercalation (for Li-ion batteries) Electrochemical capacitors Molecular electronics -field emitting devices: flat panel displays, gas discharge tubes in telecom networks, electron guns for electron microscopes, AFM tips and microwave amplifiers; - transistors - nano-probes and sensors (AFM tips, etc) Quantum wires and point contacts 66

67 Carbon nanotube transistor Gate capacitance per unit width of CNTFET is about double that of p- MOSFET. The compatibility with high- k gate dielectrics is definite advantage for CNTFETs. About twice higher carrier velocity of CNTFETs than MOSFETs comes from the increased mobility and the band structure. CNTFETs, in addition, have about four times higher transconductance. Carbon nanotubes 67

68 Circuitry of ballistic wires and QPCs 21. Can you give examples of the use of quantum wires and quantum point contacts in an electronic circuitry? What kind of effects do appear in these circuits? How to describe circuit s behaviour and what formalism is useful in this description? Quantum wires and point contacts 68

69 Series connection Two QPCs in series is a non- Ohmic system: the series resistance is much smaller than the sum of the individual QPC resistances One needs quantum mechanics to understand such behavior after the first QPC the electron beam gets collimated, and then it is prepared to pass through the second QPC without backscattering. Quantum wires and point contacts 69

70 Parallel connection If two QPCs are connected in parallel, special effect in magnetoresitance occur. Shown are the effects of commensurability between the cyclotron orbits and the QPC spacing Quantum wires and point contacts 70

71 Open questions We avoided most interesting questions posed by electron-electron interaction in combination with quantum mechanical correlations. In particular, we did not discuss: Fractional quantum Hall effect Kondo effect in QPC and quantum dots Luttinger liquid effects, important for 1D systems These effects belong to cutting edge of modern nanoscience and nanotechnology. Quantum wires and point contacts 71