Quantum transport in nanostructures

Transcription

1 Quantum transport in nanostructures About the manifestations of quantum mechanics on the electrical transport properties of conductors V At macro scale I = V/R (Ohm s law) = σv At nano scale I? V

2 Moore s Law The number of transistors per microchip doubles roughly every two years.

3 Nanotubes/wires Nanoscale electronics Atomic point contacts Molecular junctions Tans et al. (1997) Organic electronics Scheer et al. (1998) from Nitzan et al. (003) Fast DNA sequencing Poly(3-hexylthiophene) source polymer insulator drain Doped Si Z.Q. Li et al. (006) Lagerqvist et al. (006)

4 Expected effects for electrons in nanostructures Quantum confinement effect Tunneling effects Charge discreteness and strong electron-electron Coulomb interaction effects Strong electric field effects Ballistic transport effects

5 Important length scales Elastic mean free path (l e ): average distance the electrons travel without being elastically scattered l e = v F τ e. v F denotes the Fermi velocity of the electrons Phase coherent length (l Φ ): average distance the electrons travel before their phase is randomized l Φ = v F τ Φ. τ Φ denotes the dephasing time of the electrons Fermi wavelength (λ F ): de Broglie wavelength of Fermi electrons in d = 3: λ F = 3/ (π/3n) 1/3 in d = : λ F = (π/n) 1/ in d = 1: λ F = 4/n

6 Important mesoscopic regimes

7 Quantum wires and point contact

8 Typical length scale for mesoscopic regime Temperature (K) L* (nm) 4. (liquid helium) < (liquid nitrogen) < (room temperature) < 10 *The numbers just give an order of magnitude

9 Conduction at the macroscale Large number of states contribute to overall current Large number of electrons Resistivity, mobility, electric field, bias voltage, macrocopic currents are well-defined Quantum effects are averaged out by thermal effects

10 Conduction at the nanoscale Small number of states can affect the overall current Wavefunction coherence lengths are comparable to characteristic device dimensions Single electrons charging effects can be significant These can amount to overall macroscopic electronic properties that show deviations from bulk electronic properties

11 Bolztmann Transport Equation Based on the semiclassical transport theory, considering the distributions of carriers to energies and momenta, taking into account scatterings. The electrons obey the semiclassical equations of motion v(k) = (1/ħ) k ε(k) dk/dt = e/ħ (E +v(k) B) The general Boltzmann equation to first order approximation: v(k) φ(k,r,t) ee/ħ k φ(k,r,t) + φ(k,r,t)/ t = [ φ(k,r,t)/ t] scatter Current density equals to the conductance times electric field j = σe With simplified Bolztmann equation σ = ne τ D /m = ne µ the electron mobility µ e τ D /m.

12 Flow of electrons between two reservoirs Electrons obey the Fermi-Dirac distribution A metal/semiconductor electrode Two electrodes with some other material (states) in between As T ~ 0 K, this is a step function Availability of carriers on the left, and empty slots on the right, how fast the carriers tunnel from the left to the center and how fast the carriers tunnel from the center to the right basically determine the current.

13 Four point technique Make quick measurements of conductivity on novel materials where contacts are not ideal Bulk Sample t >> s Thin Sheet thickness t << s Typical probe spacing s ~ 1 mm

14 Capacitance Measurements Parallel plate capacitor Linear capacitor Parallel plate Dielectric constant can be measured if the dielectric thickness is known

15 Capacitance Spectroscopy When semiconductors are used (surface potential and electric field are not linearly dependent) the dielectric layer has electric field dependent conductivity (loss) There are traps (or states) that can be charged and discharged only at certain voltages The small signal capacitance can be measured as a function of DC bias, and interpret C-V curves to gain information about the system

16 C-V characterization of MOS structures Measurement of C-V characteristics Apply any dc bias, and superimpose a small (15 mv) ac signal Generally measured at 1 MHz (high frequency) or at variable frequencies between 1KHz to 1 MHz The dc bias V G is slowly varied to get quasicontinuous C-V characteristics

17 Measured C-V characteristics on an n-type Si N D = cm -3 x ox = µm

18 Doping dependence of a MOS capacitor Can tell you carrier concentration, dielectric thickness or constant, dielectric interface trap densities, carrier diffusion properties etc.

19 -D nanostructures: graphene, metallic thin films, superlattices,. 1-D nanostructures: carbon nanotubes, quantum wires, conducting polymers,. 0-D nanostructures: semiconductor nanocrystals, metal nanoparticles, lithographically patterned quantum dots,. Gate electrode pattern of a quantum dot on DEG SEM image

20 D electron gas (DEG)

21 UHV cryopump bake million cm /Vsec Sample structure Historical landmarks of DEG mobility in GaAs. Electron Mobility (cm /V sec) Al, Ga, As SourcePurity 198 LN Shielded Sources Sample loadlock Single interface Undoped setback Si-modulation-doping Stormer-Gossard-Dingle Bulk Temperature (K)

22 Electronic Structure of -D Systems λ = de Broglie wavelength of electron a = thickness of metal film a >> λ M Substrate a a λ M Substrate a k z Fermi surface k y k F k z n=5 n=4 n=3 n= n=1 k y k x D(ε) D(ε) = m * /π! k x ε F ε

23 Classical Hall effect (1880 E.H. Hall) Lorentz-force on electron: stationary current: Hall resistance: R xy = E y /(j x B) = 1/ρe

24 D electrons in magnetic fields: Landau levels Hamiltonian: coordinate transformation: electron X R center of cyclotron motion radial vector of cyclotron motion commutation relations:

25 D electrons in magnetic fields: Landau levels mapping to oscillator: H =ħω c R² / l m = ħω c ( a + a + ½ ) Landau levels

26 D electrons in magnetic fields: Landau levels typical scales: length B B energy magnetic length cyclotron frequency

27 D electrons in magnetic fields: Landau levels degeneracy of Landau levels: center of cyclotron motion (X,Y) arbitrary à degeneracy D density of states (DOS) filling factor one state per area of cyclotron orbit # atoms / # flux quanta

28 Landau levels in Gaphene Conduction (blue) and valence (pink) bands meet at a conical point with a linear energy-momentum dispersion for graphene. (b) The graphene carriers condense into narrow energy levels (Landau levels) when placed in a perpendicular magnetic field, B. (c) Direct measurement of graphene Landau levels with high resolution scanning tunneling spectroscopy. D. L. Miller, K. D. Kubista, G. M. Rutter, M. Ruan, W. A. deheer, P. N. First, and J. A. Stroscio, Science 34, (009).

29 Quantum Hall effect Quantization of conductivity for a two-dimensional electron gas at very low temperatures in a high magnetic field. electron gas B I V electron gas AlGaAs GaAs AlGaAs Semiconductor heterostructure confines electron gas to two spatial dimensions.

30 R H = (1/ν)(h/e )

31 Quantum Hall states Landau levels B edge states Landau level degeneracy integer quantum Hall fractional quantum Hall orbital states filled level incompressible liquid partially filled level Coulomb repulsion incompressible liquid

32 Electronic Structure of 1-D Systems k ε = εi, j+ h ψ( xyz,, ) ψi, j( xye, ) m ikz = i, j = quantum numbers in the cross section 1-D subbands D ( ε ) = Di, j( ε ) D ( ε ) i, j i, j = dn dk i, j dk dε = Let there be n 1D carriers per unit length, then L m π h ε ε i j (, ) n = k = k π π 1 D F F 4L hv ε > ε = i, j 0 ε < ε i, j i, j Fermi surface consists of points at k = ±k F

33 Electrical Transport in 1-D Conductance Quantization & the Landauer Formula 1-D channel with 1 occupied subband connecting large reservoir. Barrier model for imperfect 1-D channel Let Δn be the excess right-moving carrier density, D R (ε) be the corresponding DOS. I =Δnqv = DR ( ε ) L qv qv = qvv hv e = V q = ±e h The conductance quantum 1 h Likewise the resistance quantum RQ = = G e G Q e = depends only on fundamental constants. h Q

34 If channel is not perfectly conducting, G e = T Landauer formula F h ( ε ) ( ε ) F T = transmission coefficient. For multi-channel quasi-1-d systems T ( ε ) T ( ε ) = F i, j F i, j i, j label transverse eigenstates. For finite T, e I V T d f ev f h ( ε,, ) = ε ( ε ) ( ε) T ( ε) F L R R = h e T = h e T ( 1 T) + T h h = + e e R T R = reflection coefficient. Channel fully depleted of carriers at V g =.1 V.

35 Voltage Probes & the Buttiker-Landauer Formulism T (n,m) = total transmission probability for an e to go from m to n contact. 1, are current probes; 3 is voltage probe. For a current probe n with N channels, µ of contact is fixed by V. Net current thru contact is Setting I = 0, V = V n n n e ( nm, ) n = n n m h T m I N V V N = T n m ( nm, ) For the voltage probe n, V n adjusts itself so that I n = 0. V n 1 nm = T N n m (, ) V m = m m T ( nm, ) T V ( nm, ) m µ = n m T m ( nm, ) T µ ( nm, ) m

36 I n, V n depend on T (n,m) their values are path dependent. Voltage probe can disturb existent paths. Let every e leaving 1 always arrive either at or 3 with no back scattering. V = ( 3,1) T V V ( ) ( ) = T + T 3 3,1 3, if T ( 3,1) ( 3,) = T Current out of 1: ( 1,3) ( 3 ) e I = V T V h e 1 = V 1 T h ( 1,3) < e V h no probe

37 Conductance of a quantum point contact

38 Quantum point contact GaAs/AlGaAs interface : two-dimensional electron gas Quantum conductance G = G n 0 e G0 = = h n = 1,, Ω

39 Electron flow close to a quantum point contact Electrons are wave with wave vector k F Interference stripe with λ = 1 k F

40 Quantum point contact formed in STM

41 Molecular Break Junctions

42 Electronic Structure of 0-D Systems Quantum dots: Quantized energy levels. e in spherical potential well: εnlm,, ε, For an infinite well with V = 0 for r < R : β nl, nl, ε = h ( ) m* R β n, l = n th root of j l (x). = ψ ( r, θ, φ) = R ( r) Y ( θ, φ) nl β r Rnl, r = j l nlm,, nl, lm, nl, l R for r < R ( n l) j β =, 0 β 0,0 = π (1S), β 0,1 = 4.5 (1P), β 0, = 5.8 (1D) β 1,0 = π (S), β 1,1 = 7.7 (P)

43 CdSe nanocrystals Semiconductor Nanocrystals For CdSe: m * 0.13 c = m ε nl, nl, β0,0 R β.9ev = For R = nm, ε0,1 ε0,0 = 0.76 ev For e, ε 0,0 increases as R decreases. For h, ε 0,0 decreases as R decreases. E g increases as R decreases. Optical spectra of nanocrystals can be tuned continuously in visible region. Applications: fluorescent labeling, LED.

44 Metallic Dots Mass spectroscopy (abundance spectra): Large abundance at cluster size of magic numbers ( 8, 0, 40, 58, ) enhanced stability for filled e-shells. Small spherical alkali metallic cluster Na mass spectroscopy Average level spacing at ε F : 1 Δε = D ( ε ) F ε F 3N For Au nanoparticles with R = nm, Δε mev. whereas CdSe gives Δε 0.76 ev. ε quantization more influential in semiconductor.

45 Optical properties of metallic dots dominated by surface plasmon resonance. χ If retardation effects are negligible, P= E 4π ext 1+ χ 3 ne χ( ω) = m ω 1 3 P = E ext = m ω 4π 3 ω 4π 1 ne 3 ω p E ext Surface plasma mode at singularity: For Au or Ag, ω p ~ UV, ω sp ~ Visible. ωp ω sp = 3 indep of R. liquid / glass containing metallic nanoparticles are brilliantly colored. Large E just outside nanoparticles near resonance enhances weak optical processes. This is made use of in Surface Enhanced Raman Scattering (SERS), & Second Harmonic Generation (SHG).

46 Discrete Charge States Thomas-Fermi approximation: µ N = εn e ϕ εn+ 1 U = interaction between e s on the dot = charging energy. α = rate at which a nearby gate voltage V g shifts φ of the dot. = + NU α e V g Neglecting its dependence on state, U = e C α = C g C C = capacitance of dot. C g = capacitance between gate & dot If dot is in weak contact with reservoir, e s will tunnel into it until the µ s are equalized. Change in V g required to add an e is 1 Δ Vg = εn+ 1 εn + α e e C

47 U depends on size &shape of dot & its local environment. For a spherical dot of radius R surrounded by a spherical metal shell of radius R + d, U e d = ε RR+ d For R = nm, d = 1 nm & ε = 1, we have U = 0.4 ev >> k B T = 0.06eV Thermal fluctuation strongly supressed. at T = 300K For metallic dots of nm radius, Δε mev ΔV g due mostly to U. For semic dots, e.g., CdSe, Δε 0.76 ev ΔV g due both to Δε & U. Charging effect is destroyed if tunneling rate is too great. Charge resides in dot for time δt RC. ( R = resistance ) h δε δ t h RC = e h 1 Ce R Quantum fluctuation smears out charging effect when δε U, i.e., when R ~ h / e.

48 Conditions for a Coulomb Blockade 1) The Coulomb energy e /C needs to exceed the thermal energy k B T. Otherwise an extra electron can get onto the dot with thermal energy instead of being blocked by the Coulomb energy. A dot needs to be either small (<10 nm at 300K) or cold (< 1K for a µm sized dot). ) The residence time Δt=RC of an electron on the dot needs to be so long that the corresponding energy uncertainty ΔE=h/Δt = h/rc is less than the Coulomb energy e /C. That leads to a condition for the tunnel resistance between the dot and source /drain: R > h/e 6 kω

49 Electrical Transport in 0-D For T < ( U + Δε ) / k B, U & Δε control e flow thru dot. Transport thru dot is suppressed when µ L & µ R of leads lie between µ N & µ N+1 (Coulomb blockade) Transport is possible only when µ N+1 lies between µ L & µ R. Coulomb oscillations of G( V g ).

50 Gate Voltage versus Source-Drain Voltage The situation gets a bit confusing, because there are two voltages that can be varied, the gate voltage V g and the source-drain voltage V s-d. Both affect the conductance. Therefore, one often plots the conductance G against both voltages (see the next slide for data). Schematically, one obtains Coulomb diamonds, which are regions with a stable electron number N on the dot (and consequently zero conductance). G V s-d 1 / 3 / 5 / 7 / V g V g

51

52 Single Electron Transistor (SET) e - e - dot C g V g A single electron transistor is similar to a normal transistor (below), except 1) the channel is replaced by a small dot. ) the dot is separated from source and drain by thin insulators. An electron tunnels in two steps: source dot drain source gate channel drain The gate voltage V g is used to control the charge on the gate-dot capacitor C g. How can the charge be controlled with the precision of a single electron? Kouwenhoven et al., Few Electron Quantum Dots, Rep. Prog. Phys. 64, 701 (001).

53 Designs for Single Electron Transistors Nanoparticle attracted electrostatically to the gap between source and drain electrodes. The gate is underneath.

54 Two Step Tunneling source dot drain dot empty empty N+1 filled source N (filled) drain (For a detailed explanation see the annotation in the.ppt version.)

55 Charging a Dot, One Electron at a Time e - e - dot C g V g Sweeping the gate voltage V g changes the charge Q g on the gate-dot capacitor C g. To add one electron requires the voltage ΔV g e/c g since C g =Q g /V g. The source-drain conductance G is zero for most gate voltages, because putting even one extra electron onto the dot would cost too much Coulomb energy. This is called Coulomb blockade. N-1 Electrons on the dot N-½ N N+½ ΔV g e/c g Electrons can hop onto the dot only at a gate voltage where the number of electrons on the dot flip-flops between N and N+1. Their time-averaged number is N+½ in that case. The spacing between these half -integer conductance peaks is an integer.

56 SET as Extremely Sensitive Charge Detector At low temperature, the conductance peaks in a SET become very sharp. Consequently, a very small change in the gate voltage half-way up a peak produces a large current change, i.e. a large amplification. That makes the SET extremely sensitive to tiny charges. The flip side of this sensitivity is that a SET detects every nearby electron. When it hops from one trap to another, the SET produces a noise peak. Sit here:

57 Including the Energy Levels of a Quantum Dot Contrary to the Coulomb blockade model, the data show Coulomb diamonds with uneven size. Some electron numbers have particularly large diamonds, indicating that the corresponding electron number is particularly stable. This is reminiscent of the closed electron shells in atoms. Small dots behave like artificial atoms when their size shrinks down to the electron wavelength. Continuous energy bands become quantized (see Lecture 8). Adding one electron requires the Coulomb energy U plus the difference ΔE between two quantum levels (next slide). If a second electron is added to the same quantum level (the same shell in an atom), ΔE vanishes and only the Coulomb energy U is needed. The quantum energy levels can be extracted from the spacing between the conductance peaks by subtracting the Coulomb energy U = e /C.

58 Precision Standards from Single Electronics Count individual electrons, pairs, flux quanta Current I Coulomb Blockade I = e f Voltage V Josephson Effect V = h/e f V/I = R = h/e Resistance R Quantum Hall Effect (f = frequency)

59 Quantum interference In general: δg small, random sign β α t nm,α, t nm,β : amplitude for transmission along paths α, β

60 Three prototypical examples: Disordered wire Disordered quantum dot Ballistic quantum dot Quantum interference

61 Scattering matrix and Green function Recall: retarded Green function is solution of In one dimension: Green function in channel basis: ε k = ε and v = h -1 dε k /dk Substitute 1d form of Green function r in lead j; r in lead k If j = k:

62 Aharonov-Bohm (A-B) Effect Illustration of interference experiment for Aharonov-Bohm effect

63 A-B Effect Formulations φ Δ = φ q! = A dx P Δφ = qφ! qvt! (Magnetic A-B Effect) (Electric A-B Effect)

64 Ring Oscillations Ring Oscillation without E/B Field

65 Ring Oscillations Ring Oscillations with E/B Field

66 Ring Oscillation ΔR = πr B NWte

67 A-B Ring Applications A-B Ring in Semiconductor

68 A-B Ring Applications A-B Ring in Metal

69 A-B Ring Applications A-B oscillation in a Ring with a QD connected in series

70 Localization T t1 t = 1 r1 r cos ϕ * + r1 r R = h e T = h 1 r r cos ϕ * + r r 1 1 e t1 t = average over φ* = average over k or ε. R = h 1+ r r r r cos ϕ * 1 1 e t1 t h 1+ r r = larger than incoherent limit 1 e t1 t h 1 r r 1 e t1 t Consider a long conductor consisting of a series of elastic scatterers of scattering length l e. Let R >>1, i.e., R 1 & T << 1, ( R + T = 1 ). For an additional length dl, Setting r 1 = R t 1 dl dr = dr + dt = 1 l = T e r = dr t = dt h 1+ R dr R+ dr = e T dt = h 1+ e T 1 R dr ( dr ) ( R ) dr L R ( 1+ d +L ) = R = h 1 1 e T + + +L ( R dr )( dr ) dl R = R 1+ + L le

71 dl R+ dr = R 1+ l e ln R L = l dl dr = R l h where R = R = = RQ = e 0 R L 0 0 e e R h L = exp e le C.f. Ohm s law R L For a 1-D system with disorder, all states become localized to some length ξ. Absence of extended states R exp( a L / ξ ), a = some constant. For quasi-1-d systems, one finds ξ ~ N l e, where N = number of occupied subbands. For T > 0, interactions with phonons or other e s reduce phase coherence to length l φ = A T α. R h lϕ exp e le for each coherent segment. Overall R incoherent addition of L / l φ such segments. For sufficiently high T, l φ l e, coherence is effectively destroyed & ohmic law is recovered. All states in disordered -D systems are also localized. Only some states (near band edges) in disordered 3-D systems are localized.

72 Weak localization A = (A 1 + A ) = A 1 + A + A 1 A = 4 A 1 Interference effects double the classical contribution and (slightly) suppress the conductance.

73 Weak (anti-) localization In a system with the carrier s spin coupled to its momentum, the spin of the carrier rotates as it goes around a self-intersecting path, and the direction of this rotation is opposite for the two directions about the loop. Because of this, the two paths of any loop interfere destructively, which leads to a lower net resistivity. This is called weak antilocalization.