Single Electron Transistor

Transcription

1 Single-Electron Transistors u Reference book: Single Charge Tunneling Coulomb Blockade Phenomena in Nanostructures By Hermann Grabertand Michel H. Devoret, 1992 u Referred Journal Review Papers: Correlated discrete transfer of single electrons in ultrasmalltunnel juntions By K. K. Likharev, IBM J. Res. Develop. Vol.32, p.144, 1989 Single-Electron Devices and Their Applications By K. K. Likharev, Proceedings of the IEEE, vol.87, p.606, 1999 Single-Electron Memory for Gita-to Tera Bit Storage By K. Yano et al., Proceedings of the IEEE, vol.87, p.633,

2 Contents u Single Electron Phenomena: A General Introduction u u Fabrication and Analysis u Applications 2

3 Single Electron Phenomena: A General Introduction u.scaling prospects for various bit-addressable memories. DRAM is expected to be bottlenecked at the generation of 64 Gbitintegration (70 nm technology) due to the problems with the storage capacitor scaling. Nonvolatile memory is going to be the mainstream for 64Gbit-16Tbit memory. SET/FET would be feasible starting from ~ 3 nm minimum feature sizes. 3

4 SET Evolution u The manipulation of single electrons was demonstrated in the seminal experiments by Millikanat the very beginning of 20 century. u Single Electron Device: in which the addition or substractionof a small number of electrons to/from an electrode can be controlled with one-electron precision using the charging effect. They are not interesting not only as new physical phenomena in nanostructures but also because they offer new operating principles for future ICs. Application: Memory, switch, Thermal meter u Advantages: Good stability: is the strong incentive to explore the possibility of the devices. atomic scale physical dimension ULSI possible Ultralowpower operation: simply because they use very small number of electrons to accomplish basic operation. Fast operation: only a few electrons (<100) are transferred, therefore, the charge/discharge process might be faster than those of conventional devices. (100,000 electrons) 4

5 Schematic of Single Electron Devices u A quantum dot is weakly coupled by tunnel barriers to two electron reserviors. Oxide 5

6 A General Introduction u Recall that the motion of electrons in an infinite potential well is in a standing waveform. That means that the energy of the particle in the infinite potential well is quantized. That is, the energy of the particle can only have particular discrete values h n π E = E n = 2 2ma where n is a positive integer 6

7 Charging Energy u Let a small conductor (island) be initially electroneutral, i.e., have exactly as many (m) electrons as it has protons in its crystal lattice. In this state the island does not generate any appreciable electric field beyond its borders, and a weak external force may bring in an additional electron from outside. Now the net charge of the island is (-e), and the resulting E field repulses the following electrons which might be added. 7

8 Charging Energy u The charging energy of the island is E C, where C is the capacitance of the island: 2 E e C = C u When the size of the island becomes comparable with the de Broglie wavelength of the electron inside the island energy quantization E ( n π h ) h k = + N 2 2 mw 2 m u The energy scale of the charging effects is given by a more general notion, the electron addition energy (E a ). In most cases of interest, E a could be approximated by E = E + E a C Here E k is the quantum kinetic energy of the addition electron; for a degenerate electron gas E k = 1/g(ε F )V, where V is the island volume and g(ε F ) is the density of states on the Fermi surface. k 8

9 Electron Transfer in an quantum dot u The transport of electrons through the quantum dots is an interplay of resonant tunneling and Coulomb blockade effects. u In the absence of charging effect, a conductance peak due to resonant tunneling occurs when the Fermi energy E F in the source lines up with one of the energy levels in the dot. E F = E n 2 2 h n π = 2 2 ma 2, where E is the bare energy level u However, this condition is modified by the charging effect. The energy level is renormalized by the charging effect 2 * 1 e E N E n + N, where E n is the bare level 2 C 2 e u That is, the renormalized level spacing E = E + C is enhanced above the bare level spacing by the charging energy. n 9

10 Single-electron tunneling through a quantum dot Bare Energy level Normalized Energy level E * = E + e 2 C The energy levels, E N, are modified by the charging effect. That is the charging energy regulates the level spacing. The spin degeneracy is lifted by the charging energy. 10

11 Single-electron tunneling through a quantum dot (a) (b) E N 2 + e = E + e ( N 2 C F φ the dot. An electron has tunneled into the dot, 1 ), with N referring to the lowest unoccupied level in E N e 2 = E 2 C F + e φ ( N ) 11

12 Addition Energy, Kinetic Energy u For 100-nm-scale devices, E a is dominated by the charging energy E c and is of the order of 1 mev (~10 K). Since the thermal fluctuations suppress most single-electron effects unless E a 10kBT these device have to be operated in the sub-1-k range. Unpractical! u If the island size is below ~ 10 nm, E a approaches 100 mev, and some single electron effects become visible at RT. u However, digital SE devices require even higher values of E a to avoid thermally induced random tunneling events, so that minimum feature size of SET has to be smaller than ~ 1nm. RT operation E a = E C + E 12

13 u The resulting SET device is reminiscent of a usual MOSFET, but with a small conducting island embedded between two tunnel barriers, instead of the usual inversion channel. V g u The expression of the electrostatic energy W of this SET is W [ n C + n C ] C const 2 = ( ne Q e ) / 2 C total ev / total + n 1 and n 2 are the number of electrons passed through the tunnel barriers one and two, respectively, so that n = n 1 - n 2, while the total island capacitance C total is now a sum of C 0, C 1, C 2, and whatever stray capacitance the island may have. The external charge Q e = C 0 V g is just a convenient way to present the effect of the gate voltage V g. 13

14 (SET) u Operation principle: 14

15 I d -V d Characteristics u I d -V d is a function of V g. u Coulomb Blockade Threshold Voltage V th. Coulomb gap 15

16 Coulomb Gap u Large bias (V ds ) I d -V ds measurements generally probe the excitation spectrum of the dot. The conductance peaks are associated with the excited electron states in the QD, appearing whenever such an excitation is aligned with the Fermi level of one of the S/D leads. u The Coulomb-blockade gap is manifested by the flat region of the I d - V ds curve spanning V ds ~ 0V. At the edge of the gap, the large peak in differential conductance on either side marks the threshold voltage above which electrons can tunnel into the dot. Coulomb gap = V ds = ( C + C ) d e s 16

17 Coulomb Staircase u Unlike the Coulomb suppression of current in the neighborhood of V ds = 0 V (Coulomb gap), the staircase is not a universal feature of the Coulomb blockade. Rather, it is a special result of having very different tunneling rates through the two tunneling barriers. Coulomb staircase 17

18 Coulomb Staircase u Increasing V ds, the quasi-fermi level on Source lead is raised by the bias potential; initially no current flows because electrons at the quasi-fermilevel do not yet have enough energy to overcome the charging energy of the QD. u Eventually, V ds reaches the point at which an electron can tunnel from the Source lead onto the QD. current flow and a peak in G is observed. 18

19 Physics of Coulomb Staircase u Increasing gate voltage V g attracts more and more electrons to the island. The discreteness of electron transfer through low-transparency barriers necessarily makes this increase step like. u When one tunnel barrier is significantly more transmitting than the other tunnel barrier, the I-V behavior of the dot can exhibit the namely Coulomb staircase behavior, that is a stepwise curve. u What is surprising is that even such a simple device allows a reliable addition/subtraction of single electrons to/from an island with an enormous (and unknown) number of background electrons, of the order of one million in typical low-temperature experiments with 100-nm-scale aluminum islands. 19

20 I d -V g Characteristics u Coulomb-Blockade Oscillation in I d -V g and conductance-v g, where conductance I G = V d d 20

21 Conductance Positions u The gate-voltage position of the conductance peaks, corresponding to chargedegeneracy points, are determined at very low temperatures by the conditions E(N) = E(N+1), which leads to eϕ N+1 = (N+1/2)e 2 /C + ε N+1. (Recall that, where ε i represents the energy of the i th eigenstaterelative to the Fermi level in the QD and the summation is over the set of occupied states.) a u The spacing between conductance peaks V g = = +, where E a is the single-electron addition energy, e α C dot E is the Coulomb charging energy and is the quantized level separation. The gate-voltage position of the conductance peaks contains information about the single-particle energies. (addition energy, charging energy, and quantized level separation) E ( N ) = ( Ne) 2 C 2 ϕ Ne + ε 21 N i E α e e α C dot E α e

22 Conductance Positions u In principle, unless the single-particle levels ε I are equally spaced, the conductance peaks are not exactly periodic in V g. This is true as long as kt << E, for which peaks are narrow enough to resolve the contribution of. u However, as long as the charging energy is larger than the levelspacing, as is often the case in the planar QDs that are a few hundred nm in size, the deviations from periodicity are relatively small. 22 g

23 Temperature Behavior of Conductance Peaks u At low temperatures, the heights of successive peaks in V g vary non-monotonically and adjacent peaks are separated by broad minima. when kt << E, the conductance of any particular peak is entirely determined by the tunneling rates Γ i l and Γ i r of a specific single particle state i. Large-amplitude peaks are associated with states that are more strongly coupled to the source or drain leads; low-amplitude peaks with weakly coupled states. u As the temperature is increased, all thepeaks are broaden; their amplitudes decrease in some cases and increase in other cases; in certain cases (the peak shown furthest to the right), the amplitude first decreases then increases as the temperature is raised. As temperature is increased and kt ~ E, each conductance peak is influenced by contributions of tunneling through several discrete energy states, although still within the constraint of one electron at a time. The monotonic increase in peak amplitudes at high temperatures simply reflects the trend towards more strongly coupled levels at higher V g. The coefficient α can be determined from the temperature dependence of the peak s full width at halfmaximum: FWHM = 3.5kT/(αe) 2. u Eventually, at the highest temperature, peak overlap significantly and the amplitude of successive peaks in V g increases monotonically. 23

24 Rhombus Shapes in the Contour Plot of differential conductance S 1 V g / V d =C d /C g S 2 V g / V d =(C s +C g )/C g G < 0 Fine structure 24

25 Device Parameters Extracted from the Rhombus Shapes u The electronic structure in the QD could be extracted from the contour plot of the differential conductance as a function of V g and V d. u The ratio of the gate-dot (C g ), drain-dot (C d ), and source-dot (C s ) capacitances can be calculated from the slope S 1 and S 2. C g : C d : C s = 1: S 1 : S 2 u Then the gain modulation α C g /C total, C total = C g + C d + C s u The addition energy E a = α V g. 25

26 Electronic Structure extracted from I-V u Addition Energy u Charging Energy u Energy level spacing u Dot diameter 26

27 Minimum tunneling resistance for single-electron charging u Implicit in the formulation of the Coulomb-blockade model is the condition that the number of electrons localized in the dot island, N, is a well-defined integer. This is to say, well defined in the classical sense, as opposed to a quantum definition which describes N in terms of an average value <N>, which is not necessarily an integer, and time-averaged fluctuations <δn 2 >. u The Coulomb-blockade model requires that <δn 2 > << 1. Clearly, if the tunnel barriers are not present, or are insufficiently opaque, nothing will constrain a quantized electronic charge to be confined within a certain volume. A general view is that there is a minimum resistance which the barriers must exceed in order to have <δn 2 > << 1, and this resistance is of the order of the quantum resistance R K = h/e 2 = 25,813 Ω. u The condition <δn 2 > << 1 requires that the time that an electron resides on the dot, τ, be much greater than δτ, the quantum uncertainty in this time. The current I cannot exceed e/τ since no more than one extra electron resides on the dot at any instant. The energy uncertainty of the electron, δe, is no larger than the applied voltage, hence the condition that δτ << τ translates into macroscopic variables using I e/τ, δτδe h and δe < ev ds. Doing so gives the minimum tunneling resistance condition, R K = V ds /I h/e 2. 27

28 Co-tunneling u Even if the minimum resistance criterion is met and single-electron charging effects are manifested, small quantum fluctuations, or uncertainties, in N are not entirely ruled out. In the classical Coulomb-blockade model there is then a fixed number of electrons N on the QD and at T= 0 the charge on the QD does not fluctuate. u However, the fact that very small quantum fluctuation in N may be present corresponds to electrons momentarily tunneling onto the QD, with an energy deficit on the scale of the classical Coulomb charging energy. u Essentially, the tunneling electron resides on the QD in a virtual charge state for a sufficiently brief interval such that the energy uncertainty of this state is larger than its classical energy deficit, subsequently tunneling out. This process has been referred to as co-tunneling or macroscopic quantum tunneling of charge. u The rational behind is that the total charge of the system (a macroscopic variable) undergoes a transition through a classically forbiddenintermediate state, in apparent violation of the Coulomb blockade. 28

29 Elastic co-tunneling u The tunneling of an electron into a certain energy state and the tunneling of an electron from the same state out of the dot. The end result of the two tunneling events is that the state of the QD is unchanged, and as such, this is referred to as elastic co-tunneling, which contributes a linear term to the I-V curve. I el = h σ σ 8 π e 1 E E 2 where is the average energy separation between eigenstates in the QD and E 1 (E 2 ) is the charging energy associated with adding (removing) a single electron to (from) the dot. Note, in particular, that the resulting conductance scales roughly as the ratio between the level spacing and the Coulomb gap U e 2 /C. u The case of elastic co-tunneling depends, in principle, on the geometry of the QD. This is because the electron involved has to couple to both leads; thus in a sense it must traverse the dot in a virtual state. V 29

30 Inelastic co-tunneling u It corresponds to an electron tunnels into a certain state in the dot and a second electron, from a different state, tunnels out of the dot. The state of the dot is modified, leaving an electron-hole excitation. The resulting current is nonlinear in V ds and temperature-dependent. The case of inelastic co-tunneling gives the following well-known form I inel 2 h σ ev = 1 σ e E E π 2 2 ( kt ) + V 30

31 Cotunneling u The distinction is made between these two processes because their relative contributions to the total net cotunneling current depend on the density of states in the QD. u In metal QDs, in which the density of states is large, the elastic component of co-tunneling is usually overwhelmed by the inelastic component. u In semiconductor QDs, in which the density of states is much smaller than in metals, both elastic and inelastic terms can contribute to the co-tunneling current. u In practice, co-tunneling is expected to modify the classical picture of singleelectron charging in the form of excess current in the region of the Coulombblockade gap, in the case of I-V ds measurements, or excess tunneling current between conductance peaks in low-bias measurements. 31

32 Fabrication u To apply SETsfor low power ICs, (i) room-temp operation; (ii) uniformity and (iii) compatible with the LSI processes are required. u Task: For room temperature operation, the quantum dot diameter should be less than 10 nm, which corresponds to the total capacitance about 1 af. E-beam lithography: High cost and the following etching process is not easy to control Scanning probe microscopy (SPM) to place Au atoms in nanostructure. Only applicable in specific substrate u Metal/Superconductor SET u Semiconductor SET Epitaxial growth quantum dots (self-assembled) or 2DEG with side gate (depletion) E-beam + dry etching 32

33 Fabrication u In addition to advanced e-beam lithography technology, matured and controllable fabrication processes are needed to form small quantum dots (<10 nm). u Various approaches to achieve room-temp operation: Oxidation Isotropic/anisotropic wet etch Scanning tunneling microscope nano-oxidation process Side-gate 33

34 Self-aligned Floating Dot gate (1997) 34

35 Siquantum dot formed by 2D oxidation 35

36 Double-dot charge transport in SET/SHT 36

37 Geimplantation/Ge Segregation 37

38 Application u Major application fields: Memory Digital-data-storage Precision Measurement u Memory >> Logic We can use SE devices only in a memory cell, whereas keep using conventional CMOS technology in the peripheral circuitry. Memory cell technology has continuously changed, including the emergence of flash memory technology and ferroelectric-film memory technology. The way of storing information is rapidly changing from the old regime, relying on papers and other analog electronic means, to the digital regime in the multimedia era. New needs of storing information are different from the older specifications in bandwidth, storage capacity, power consumption. Fundamental difficulty in a logic functional unit since SE devices generally have poor current-drive capability. 38

39 Application-Data Storage System 39

40 Quantum Information- Qubit u Developing a quantum computer is a basic endeavor in science and technology. The advantage of Si-based quantum computer is Low cost Large scale integration u Contrast to classical bits, 0> or 1>, a quantum computer consists of Qubits, which could be represented by a superposition of 0> or 1>, i.e., α 0> +β 1> (where α 2 + β 2 =1) This huge parallelism makes it possible to solve some of the most difficult problems, such as integer factorization. 40

41 Quantum Computer Roadmap: development status After NTT Technical Review, June

42 Roadmap of Quantum Computer 42

43 Quantum Computer Roadmap: status of solid-state QC 43

44 Si-based Key Devices for QC Source Island Drain Si Figure 1. Schematic diagram for a single electron transistor and a coupled quantum dots. 利用單電子電晶體來檢測基本量子訊息 此地量子訊息是由耦合量子點建構而成 每一個量子點僅存在單一電子 因此耦合量子點的基態是自旋為零 (singlet), 利用電極與閘極電位使得單一量子點處在 0> 與 1> 的線性疊加狀態 0> 表示量子點只有單一電子 1> 表示量子點有兩個電子 此線性疊加態構成單位量子訊息 (quantum bit). 單電子電晶體的量子島耦合到耦合量子點 當電子從汲極穿隨進到量子島 將可以量測到量子點的狀態 整個元件佈局是一個非常難的技術 44

45 Accomplishments in NCU u Atomic-scale Ge QDsformed by thermal oxidation of SiGe-on-insulator. 20 min. Ge precipitation Buried Oxide 25 min. 45

46 Room-Temperature Characterization of Ge SETs Peak-to-valley current ratio (PVCR) of 1.92 is observed at room temperature 46 Clear offsets and plateaus are seen for gate voltages corresponding to the drain current valleys, while linear relations are obtained for gate voltages corresponding to the drain current peaks.