Low-Dimensional Disordered Electronic Systems (Experimental Aspects)

Transcription

1 Low-Dimensional Disordered Electronic Systems (Experimental Aspects) or ГалопомпоРазупорядоченнымСистемам Michael Gershenson Dept. of Physics and Astronomy Rutgers, the State University of New Jersey Low-Temperature Nano-Physics Chernogolovka, August 007

2 Roadmap Quantum Interference (single-particle effects) Electron-Electron Interactions Macroscopically Inhomogeneous Systems (Percolation) Mesoscopic (0D) systems etc., etc., etc. Lecture 1. Weak and Strong Localization in 1D and D Conductors (single-particle effects). Lecture. Electron-electron Interactions and Dephasing Processes in Disordered Conductors Lecture 3. Quantum Corrections in the Conductivity of Si MOSFETs

3 Lecture 1: WL and SL in 1D and D conductors (non-interacting electrons) Intro: Anderson Localization, Scaling Theory Weak Localization Corrections - Non-locality of WL effects - Interplay between WL and percolation WL-SL crossover Hopping in Systems with Large Localization Length

4 Disordered Conductors: Characteristic Scales Quenched disorder, degenerate Fermi gas Structure size: L Fermi wavelength: λ F < 1 nm (metals), ~10 nm (semiconductors) Elastic mean free path: l =v F τ nm (metals), 0.1-1μm (semiconductors) Localization length: ξ Phase coherence length: L ϕ = Dτ ϕ l << ξ l <L ϕ < ξ, typically μm Thermal dephasing length: L T = / hd k B T l <L T < ξ, typically μm The magnetic length: Percolation correlation length ξ P Φ 0 L H = πh 1 0G (macroscopically non-homogeneous systems with percolation) Dimensionality: with respect to the quantum lengths ξ, L ϕ, L T, L H

5 Anderson Localization (3D) Non-interacting electrons in a 3D ordered system delocalized electron states What happens if we crank up the disorder? P.W. Anderson ( 58): Quantum interference can completely suppress the diffusion of a particle in random potential (disorder can localize a particle despite of tunneling). At T=0, there is a quantum phase transition in the system at a critical strength of disorder, x C. At x > x C, the electron states at the Fermi level are localized and look like that: More realistically, like that:

6 Anderson MIT Anderson insulator: finite density of states at the Fermi level. Realization of the Anderson MIT by tuning either disorder or electron concentration mobility edge E C ε ε delocalized DoS σ = e h nl k F, n = k 3π 3 F l 1 ~ k F e σ3d 10 h e σ D ~ ~ 6kΩ h 1/ /3 ~ n ~ Ω n ( ) 1 -E C localized Fermi level Anderson MIT continuous, no min. metallic conductivity! Wegner ( 76, 79): a close connection between the Anderson transition and the scaling theory of critical phenomena; field description of the localization problem in terms of a nonlinear σ-model. Efetov ( 80): a microscopic derivation of the σ-model.

7 Localization Length in 3D The length scale: the localization length ξ r r r ψ exp 0 /ξ ( r ) ( r r ) ξ Scaling at the Anderson MIT: ξ ν s ( E E) σ ( E E ) s = ν ( d ) C C 1/ξ 3D x C disorder strength Wigner ( 51) and Dyson ( 6): classification of ensembles of random Hamiltonians on the basis of invariance of the system under time reversal and spin rotations: unitary, orthogonal, and symplectic symmetries. Important: ξ depends on the disorder (proximity to the critical point) and the underlying symmetry of the system.

8 Scaling Ideas The conductivity σ is convenient when it is a local quantity that does not depend on the sample dimensions. The development of scaling ideas (Thouless, 74,77, Abrahams, Anderson, Licciardello, Ramakrishnan, 79, Wegner, 79) showed that more useful quantity is the dimensionless conductance g(l) of a system of the size L (in units of e /h). Scaling of the conductance g with the size of a disordered system (T=0, non-interacting electrons): Δ ξ ħ/τ T L, ξ

9 Scaling Theory Abrahams et al. ( 79): a scaling theory of localization which describes the flow of the dimensionless conductance g with the system size L. Evolution of the conductance with the system s size is determined by the conductance itself: MIT: β(g=g crit )=0 3D (orthogonal and unitary symmetry classes): Anderson MIT D and 1D: all states are localized, even if the disorder is weak. D: the geometrical factor in the conductance, g ~ (L/l) d-, disappears, and the equation β(g) does not depend explicitly on L.

10 Localization Length in 1D and D Quasi-1D conductors: Berezinsky, 74; Efetov and Larkin, 83 Dorokhov, 83 Orthogonal quenched disorder, no mag. field (time reversal symmetry) Unitary strong mag. field breaks the time reversal symmetry D conductors: O ξ 1D N D l = 1 U ξ 1D N l = 1D W λ F <<W <<ξ N 1D = W/λ F >>1 the # of conducting channels Φ B >> 0 ξ W d d - flux quantum in the area of a localized state <<ξ Orthogonal Unitary ξ D O ξ D U = l exp π kfl π = l exp k F l Φ B >> ξ 0

11 Finite Temperatures Dephasing Processes All electronic states in 1D and D are localized. However, some of these systems are very good conductors at high temperatures. Why? If the disorder is not too strong and the temperature is not too low, the electron wavefunction will be mutilated before an electron has a chance to diffuse over the localization length. Because of these dephasing processes, an electron will be frequently scattered between localized states, and it will diffuse ALMOST as if its wavefunction is not localized. Sources of dephasing: Internal ( the enemies within ): phonons, other electrons, all dynamic degrees of freedom ( ons ) External ( the enemies without ): spin-spin scattering, external high-frequency E&M fields (to be discussed at Lecture ) Length scale: the phase coherence length L ϕ = Dτ ϕ Typical value of τ ϕ (1K) in metals ~ s, D ~ cm /s, L ϕ (1K) ~ μm L ϕ ( T ) ξ Weak Localization: ( ) ξ λ << l < Lϕ T << Strong Localization: λ < l < ξ < L ( T ) F F ϕ

12 Weak Localization Corrections Gor kov, Larkin, Khmelnitskii ( 79): resummation of singularities in perturbation theory in D and formulation of the scaling in the systematic form of a renormalization group theory. The total probability to get from A to B: w AB = The phase gain: Ai = Ai + i probabilities 1 Δϕ = h i ij A A The interference term vanishes A = 0 B A r pdl r i >> 1 ij j i A j A interference of amplitudes O p the Fermi momentum B due to the disorder averaging, but NOT for the time-reversed paths (loops) r r r p l l Δϕ = r The self-intersecting (loop-like) trajectories: p 0 0 = A1 + A + Re A1 A 4 w = A 1 As a result, the total scattering probability at site O increases, and the conductance decreases. The corresponding corrections to the transport properties are known as the weak localization corrections.

13 Different dimensionalities: The dimensionality of a conductor: L L ϕ (T) (<< ξ) λ F v F dt 3D: (Dt) 3/ Lϕ < all dimensions,ξ Δσ WL λ F the upper cutoff - τ ϕ, the loops with L > L ϕ (T) do not contribute to Δσ WL τ ϕ τ λ F v F e 1 dt const ( ) ( ) 3/ Dt h Lϕ T D: d < L ϕ <<ξ ddt d τ ϕ τ λ F v F dt ddt e h L ln ϕ ( T ) l Quasi-1D: d,w < L ϕ <<ξ dw(dt) 1/ W τ ϕ τ λ dw F v F ( Dt) 1/ dt e h L ϕ ( T ) Though these QC are small at L ϕ << ξ, they diverge in low dimensions as T decreases (and L ϕ increases), and eventually they drive the system into the SL regime.

14 Observation of the WL corrections σ << σ 0 Δ WL Still, WL effects are observable because Δσ WL depends (in a non-trivial way) on T and B, in contrast to the «residual» Drude conductivity σ 0. Δσ WL (T) due to the T-dependence of phase relaxation (the lower T, the slower the relaxation). However, Δσ WL is not the only T-dependent quantum correction, the interaction effects result in a similar dependence (Lectures,3) Curiously, the first observations of the increasing resistivity in 1D and D metal films with cooling, which have been considered as a support for the WL theory, were in fact the observations of INT corrections (the WL correction was T- independent because of a strong spin-spin scattering). AuPd wires AuPd films It is easier to analyze Δσ WL (B) because INT corrections aren t sensitive to weak magnetic fields. Giordano, PRB ( 80) Dolan and Osheroff, PRL43 ( 79)

15 Δϕ H Φ r r t -t e = ch r r Adl e = ch hc Φ0 = = e 7 r r Hds G cm WL in external magnetic field Classically weak magnetic fields: no trajectory bending. Aharonov-Bohm phase acquired by the loops: w 0 r p Φ = π Φ r p 0 e c r A e r r Ψ Ψ exp i Adl ch A - the vector potential - the phase difference between CW and counter-cw loops increases with field. - the flux quantum (~0G in 1μm ) ( Δϕ ) = A [ 1+ ( Δϕ )] = A + A + A A cos cos 1 1 H H Φ 0 = - the magnetic length (the characteristic length at which two interfering waves propagating along a loop in opposite directions acquire the phase difference ~ π) L H πh

16 Aharonov-Bohm effect in disordered conductors w Φ = A 1 + cos π Φ 0 0 Li «tube» Ø1.3μm T=1.1K Sharvin and Sharvin, 81 Altshuler et al., 8 multi-wall carbon nanotubes r e = 8.6 nm Bachtold et al. ( 98)

17 L H L ϕ WL magnetoresistance l When the magnetic length L H ~(Φ 0 /B) 1/ becomes smaller than L ϕ (T) WL magnetoresistance L H = (dl ϕ ) 1/ L H = l L H = L ϕ Cu film d=7 nm T=10K WL magnetoresistance: negative (no SOI so far) anisotropic in 1D and D (purely orbital effect) observed in classically weak magnetic fields B, G WL MR measurements are extensively used to measure L ϕ. MG et al., 81

18 Weak Anti-Localization So far, we ve considered spinless electrons. However, the electrons have spin 1/, and the spin scattering modifies the WL corrections. Two scattering processes: r B SO 1 c = r r ( v E) Because the interfering electron waves propagate in the opposite directions (v -v), the B SO vectors are also oppositely directed, and the spins diffuse away from each other on the Bloch sphere. The spin-orbit length: (a) spin-spin scattering (e.g., by paramagnetic impurities) destroys constructive interference at t > τ S ( L ϕ (T) L S ). (b) spin-orbit interaction (as a relativistic effect, strongly depends on v) When electrons are scattered by a strong electric field of nuclei, they feel (in their reference frame) an effective magnetic field L SO = Dτ SO - T-independent, a strong function of the nucleus charge (Abrikosov & Gorkov, 6) τ SO ( α ) 4 = τ Z After averaging over all trajectories with dimensions L SO <L<L ϕ, the WL correction becomes positive weak anti-localization

19 WL and WAL magnetoresistance L ϕ L SO L ϕ L ϕ l l L SO l L SO Only WL WL+WAL Only WAL WL MR L H =W W=70 nm T=0.K L H = L ϕ (T) Si MOSFET D Ag film 1D Ag wire

20 Non-locality of Quantum Corrections D and 3D: due to the non-locality of QC, the disorder is averaged over a = large length (in comparison with the defect size) universality of QC. L ϕ 0D and 1D: Because of non-locality, QC can be strongly influenced by the phase coherent regions extending beyond the classical current paths. Umbach et al., 87 T=0.1K L ϕ =.3μm MG et al., μm 1.6μm 0.4μm 0.μm distance between side branches

21 WL in macroscopically non-homogeneous conductors (D and 3D) To be specific, we ll consider WL (though many conclusions apply to the INT corrections as well). New scale: the percolation correlation length ξ P L ϕ (L T ) metallic insulating At L >> ξ P the conductivity scales with the conductor dimensions as in a homogeneous system (R macro ). Far from the percolation threshold (L ϕ >> ξ P ) - universal corrections, ξ P universal QC nonuniversal QC At L < ξ P - the anomalous diffusion, the diffusion constant D(L) depends on the scale L. Close to the threshold (L ϕ < ξ P ): an interplay between QC and percolation. x C Δσ WL Δρ = ρ WL macro x

22 Consequences of macroscopic non-homogeneity Near the percolation threshold (L ϕ,l H < ξ P ), the quantum corrections are much smaller than that expected for a homogeneous system with the same total resistance. Δρ ρ ( T, B) WL macro = " Δσ WL " << Δσ WL ( Δσ WL ) measured ( Δσ WL ) universal Aronov, MG, and Zhuravlev ( 84) - experiment + theory Palevski and Deutscher ( 86) Dumpich and Carl ( 91) using the theory of WL corrections in conductors with percolation, one can extract ξ P from the WL magnetoresistance 3D granular films one should be careful about interpretation of L ϕ in systems with percolation [L ϕ (L ϕ ξ P ) 1/ ]. In particular, τ ϕ may be insensitive to R macro and the temperature dependence of τ ϕ is weakened. [see also Germanenko et al. PRB 64, ( 01)]

23 Man-made D percolation networks D network made of 0.05μm-wide Au wires. The unit cell size 0.4 μm, # of cells ~ In the process of e-beam writing, the probability of skipping a bond varied from 0 to μm The percolation network made of 1D wires behaves like a homogeneous D film with respect to the quantum interference effects if L ϕ,l H > ξ P. MG et al. PRL 74, 446 ( 95) тупик мертвый конец Close to the percolation threshold, when ξ P becomes much larger than L ϕ,l H, the QC become quasi-1d. These 1D QC are affected by the presence of dead ends of the infinite percolation cluster when L ϕ > a (a the unit cell size).

24 What happens when σ D approaches e /h? With diminishing σ, the second-loop corrections become important (approx. at R >5kΩ/). Because of these corrections, Δσ WL (B) is strongly diminished whereas Δσ WL (T) remains universal with pre-factor ~ e /h. Minkov et al. ( 04) GaAs e Δσ WL( B) = α π h f ( B) Gornyi ( 04) Caution: vanishing WL MR at σ e /h is expected for both homogeneous and macroscopically inhomogeneous systems. In the latter case, however, Δσ WL (T) should also be diminished (similar to the behavior of high-mobility Si MOSFETs). Rahimi et al. ( 03)

25 WL SL crossover in 1D and D What happens when L ϕ (T) becomes of the order of ξ with cooling? WL: L ϕ < ξ SL: L ϕ > ξ L ϕ, ξ ξ SL L ϕ WL The QC become of the order of the Drude conductivity: T e h D: σ 0 = k l F σ e h k F l Lϕ ln 0 at L D l ϕ ξ [ ξ l exp( k l) ] D F e h 1D: σ 0 = k lw F σ e h ( k lw L ) 0 at L ξ [ ξ lk W] F ϕ ϕ 1D 1D F

26 ξ L 1D ϕ ξ 1D WL SL crossover ( T ), Nyquist dephasing T ξ ~ DW Quasi-1D wires in Si δ-doped GaAs # of wires 470, L = 500 μm, W = 0.05 μm, N 1d = 7, ξ = 0.4 μm, Δ ξ =.1 K 1 m * ( ) Δ ξ R ξ = h / e MG et al., PRL 79, 75 ( 97) Khavin, MG et al, PRL 81, 1066 ( 98). Khavin, MG, Bogdanov, PRB 58, 8009 ( 98) Electrons are localized over many impurity states (ξ >> n -1/,l), and there is a diffusive motion within the localization envelope. The localized states strongly overlap in space [e.g., in our Si:GaAs samples, there are 10 3 localized electrons confined within an area of a size of ξ W]. E F ~ 10 3 K ξ Δ ξ ~ 1K ħ/τ φ W the inter-level spacing within the localization length ~ E F /(nwξ) in quasi-1d ξ

27 1D WL SL crossover (cont.) Bundles of carbon nanotubes L ϕ ~ ξ ~ 0.5 μm AuPd thickness 7.5 nm, width down to 5 nm Shea, Martel, Avouris, PRL84, 4441 ( 00) At the crossover : Natelson et al., SSC 115, 69 ( 00) Natelson et al., PRL 86, 181 ( 01) L ϕ (T ξ ) ~ ξ The conductance of a wire segment of the length ξ e /h At the crossover temperature T ξ, all energy scales are of the same order of magnitude: ħ/τ ϕ (T ξ )~Δ ξ ~ k B T ξ ~ ħd/ ξ level broadening Inter-level distance crossover temperature Thouless time ξ /D

28 R Ω D WL SL crossover Qualitatively similar to the 1D WL-SL crossover h/e Minkov et al. The WL-SL crossover is sharper for 1D samples. L ϕ T (K) R e /h 1.6. gated Si:GaAs.9 MG et al., unpublished R, Ω 17.9 ( T ) ξ l exp ( k l ) F T (K) k F l R ξ the D crossover can be observed only if k F l ~ 1 R (more room for that in 1D)

29 Limits of D WL Description k F l = 4.3 k F l = 1. 6 k F l = 1. 3 σ e /h - where should we stop using the WL theory? Minkov et al. 07: δ-doped GaAs structures, n~1.5-.5x10 1 cm ξ = + * τ τ D ϕ ϕ

30 The Limits of D WL Description (cont.) Minkov et al. 07: down to the very low conductivity value, σ~10-1e /h, the experimental results are described by the same expressions which are valid in the WL regime. One should only replace the dephasing rate τ ϕ -1 by the quantity τ ϕ -1 + τ ξ -1. hopping

31 Hopping in Systems with Large Localization Length Low-T WL-SL crossover - a unique opportunity to explore electron hopping with a large ξ, up to a few μm in 1D and ~0.5μm in D. Hopping with large ξ in 1D and D systems a very interesting transport regime, which is qualitatively different in many respects from the conventional hopping with small ξ in lightly doped semiconductors ξ L ϕ, ξ ξ SL L ϕ WL T λ < l < ξ < F L ϕ ( T ) - the electron motion is still diffusive within the localization length Signatures of hopping with large ξ: orbital magnetoresistance, hotelectron effects in strong electric fields. Several experimental facts ( universality of R(T), hot-electron effects in SL regime) suggest that this hopping might be driven by electronelectron interactions rather than electron-phonon interactions

32 1D R(T) in Hopping Regime D D Si:GaAs MG et al, PRL 85, 1718 ( 00) Similar dependences in ultra-thin metal films (e.g., Goldman et al.) h/e Gated quasi-1d wires in Si δ-doped GaAs Khavin, MG, Bogdanov, PRB 58, 8009 ( 98) Holes in GaAs/AlGaAs Leturcq et al. ( 0) The Arrhenius dependences R(T) have been observed for both 1D and D conductors in the SL regime. The activation energy is close to the mean level spacing within the localization envelope, ~(0.-1) Δ ξ. Puzzle: for different D systems, the pre-factor is universal not expected if hopping is phonon-assisted.

33 MR in strongly-localized 1D systems R( T, B) Δξ ( B) = R0 exp Δξ = 1Dξ T ( ν ) 1 the pre-factor R 0 does not depend on B negative exponentially strong orbital: vanishes in B II Khavin, MG, Bogdanov, PRB 58, 8009 (1998) O ξ 1D N D l = 1 U ξ 1D N l = 1D Φ B >> 0 ξ W Δ Δ ξ ξ T 0 (B) / T 0 (0) ( B) ( 0) Doubling of ξ Cutting Δ ξ =(ν 1D ξ) -1 in half K 0.3K 0.4K 0.5K 0.6K B ξ the strong field limit for N = B,T B (T)

34 What does the theory say? Kolesnikov and Efetov, PRL 83, 3689 ( 99) Schomerus and Beenakker, PRL 84, 397 ( 00) Kettemann, PR B 6, R138 ( 00) Kettemann and Mazzarello, PR B 65, ( 0) Δ Δ ξ ξ ( B) ( 0) S. Kettemann, PRB 6, R138 (000) B ( G) - direct measurements of ξ in 1D (and D) conductors with a large ξ

35 R (Ω) B = 0 - B = 1T MR in the D SL regime The activation energy is decreased in the magnetic field to the plane of a D sample: Δξ ( B) R( T, B) = R0 exp Δξ = Dξ T ( ν ) R (Ω) T -1 (K -1 ) ξ D O R0 h / e - does not depend on B In D, one might expect more radical drop of the activation energy in B: = l exp π kfl ξ D U π = l exp k However, k F l ~1 for the D conductors which demonstrate the SL-WL crossover at low T (= large ξ). F l B (T) ξ ~ 0.1μm ( ~ 10l)

36 Examples of strong orbital MR in D 8.4T B = 0 Interesting parallel: strong-field MR of D systems close to the disorder-driven Superconductor- Insulator Transition Baturina et al., TiN films Shahar et al., InO x films T = 40 mk 1.7 nm thick Be film MR: large, anisotropic, negative in strong fields Butko and Adams, Nature 409, 161 ( 01) Quantum metallicity strong reduction of the activation energy due to the growth of ξ in B>Φ 0 /ξ? Φ 0 B ln R h/e 0.5 ~ ξ ~ ξ B=0 SC? B >> Φ 0 /ξ T -1 Baturina et al., 06 h R( T, B) = e Δ exp k ξ ( B) B T Δ ξ (B) mean level spacing within ξ (B)

37 Aharonov-Bohm Effect in Hopping Poyarkov et al. 86: disordered PbTeO x with persistent low-t photoconductivity Never reproduced!

38 Non-Linear Effects in Hopping Regime WL SL MG et al., PRL 85, 1718 (000) Non-linearity of the hopping conductivity of D systems with a large ξ is due to hot-electron effects rather than the field effects; the electric field, which is sufficient to overheat electrons, does not depend on ξ. The thermal conductivity between electrons and phonons, G e-ph = C e /τ e-ph, is the same on both sides of the D WL-SL crossover (this might be expected, since the el. motion is still diffusive at L<ξ, and q T ξ ~ 1 even at T=50 mk).

39 Non-Linear Effects in Conductors with a large ξ P NL, the Joule heat power at which the E-induced nonlinearity develops: P In the systems with a large ξ, electron heating is expected to dominate over the field effects. Let s compare two values of the power P dissipated in a conductor: NL = ( EL) R = h / e kbtl eξ 1 exp kbtν - decreases exponentially with decreasing ξ for exp( )>> 1 P D eeξ kbt kbt E eξ ξ 1 1 ξ exp ξ el. field nonlinearity P NL P E-PH hot-electron nonlinearity P E-PH, the power that can be removed from an electron system due to electron-phonon coupling without significant overheating of electrons - does not depend on ξ ξ

40 What have we learned from the hot-electron experiments: D systems: the ρ(e) dependence can be attributed to carrier heating all along the crossover from the diffusive to the VRH regimes [n-gaas MG et al. ( 00), SiGe Leturcq et al., ( 03)]. This tendency has been observed for σ as small as 10-4 e /h. Recall: Minkov et al. ( 04) considered the hot-electron regime as a signature of diffusiveness rather than hopping down to σ~x10 - e /h. 3D Strongly Localized systems: strong evidence for the existence of separate temperatures for the electron and phonon systems analogous to the hot-electron effects in metals [doped Si and Ge - Zhang et al., ( 98); doped Ge Wang et al., ( 90, 99); Marnieros et al., ( 00); doped Si - Galeazzi et al, 07]. Hopping conductivity depends only on the electron temperature when electrons are removed from equilibrium with phonons. The effective electron temperature is established via interactions among the localized electrons. This suggests that electronelectron interactions play crucial role in hopping in conductors with a large localization length (electron-assisted hopping rather than phonon-assisted one). In the WL regime: the temperature dependences of Δσ WL and Δσ EEI are governed by electron-electron interactions In the SL regime: universal hopping R(T); hot-electron effects when R electrons are removed from equilibrium with phonons Temptation: to develop a unifying (EEI-based) approach that would describe R(T) over a wide range of resistances (WL+SL). R, Ω R ξ

41 Summary Single-particle effects in disordered conductors: quantum interference leads to WL corrections to the Drude conductivity in the regime of weak disorder/high T (L ϕ <ξ) or SL in the regime of strong disorder/low T (L ϕ >ξ) Due to non-locality, the WL corrections are very universal; WL magnetoresistance provides unique opportunity to study inelastic and spin scattering in disordered conductors. Realization of the low-t WL-SL crossover provides an access to the regime of (unconventional) hopping with a large localization length.