Nonlinear screening and percolation transition in 2D electron liquid. Michael Fogler

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1 Dresden 005 Nonlinear screening and percolation transition in D electron liquid Michael Fogler UC San Diego, USA Support: A.P. Sloan Foundation; C. & W. Hellman Fund

2 Tunable D electron systems MOSFET Heterostructure gate metallic gate AlGaAs donors electrons D electron layer GaAs semiconductor (Si) Inter-electron separation a = A& No important lattice commensurability effects

3 Experimental evidence for the D Metal-Insulator Transition Kravchenko et al.(1995) Electrical Resistance (h / e ) Lower electron densities Critical density n c Higher (but still quite low ) electron densities Electrical Resistance (h / e ) insulating branch metallic branch Temperature (K) T / T 0( n ) h e = Ω Si MOSFETs p-gaas and SiGe heterostructures quantum unit of resistance low-density strongly interacting D electron systems T 0 n n c b b 1.6 True quantum phase transition? Reviews: Abrahams et al., Rev. Mod. Phys. 73, 51 (001) Kravchenko and Sarachik, Rep. Prog. Phys. 67, 1 (004) Shashkin, cond-mat/

4 3D Disorder-Driven Metal-Insulator Transition T = 0 conductivity (1 / Ω cm) Critical doping, 11.5% Resistivity (Ω cm) x 100 Amorphous Nb x Si 1-x Nb % Temperature (K) Bishop, Spencer, and Dynes (1983) All curves are on the metallic side!

5 Common pre-1994 view on the phase diagram Disorder / max e, a E F 1 Amorphous insulator? Anderson insulator?? Weakly pinned Wigner crystal Fermi liquid Wigner crystal Inter-electron separation r s

6 Post-1994 views on the metallic phase So-called metallic behavior is a new trick of the old states of matter 1. It s a Fermi liquid effect in a restricted temperature range, where: (a) localization is yet weak (b) Drude conductivity is T-dependent, due to, e.g., T-dependent screening dσ / dt < 0 Altshuler et al. Hwang and Das Sarma Zala, Narozhny, Aleiner. Fermi liquid Wigner crystal co-existence is involved: T Spivak et al. FL FL / WC co-exist Pinned WC T loc? r s

7 Alternative view on the metallic phase D metallic phase in a new state of matter. The D MIT is a true QCP Disorder Metallic non- Fermi liquid Amorphous insulator Wigner glass Spin liquid Other work: Castellani et al.(1998) Dobrosavljevic et al. (1997) He, Shi, and Xie (1998) Inter-electron separation r? Wigner crystal s Chakravarty et al. (1999)

8 Scanned probe microscopy of the D electron liquid 1. Near-field optical spectroscopy. Scanning Single-Electron Transistors Yoo et al. (1997); Ilani et al. (004) Eytan et al. (1997) I SET dµ dv BG 3. Subsurface charge accumulation Finkelstein et al. (000)

9 What do imaging microscopy experiments tell us? Conventional starting point: In reality: weakly interacting electrons weak disorder uniform density distribution good metal strongly interacting electrons strong effects of disorder inhomogeneous density BAD metal Disorder Amorphous insulator =? Your favorite phase here!?? Wigner crystal s r

10 Questions addressed in this talk Realistic model of disorder Linear vs. nonlinear screening Structure of the inhomogeneous low-density DEG ( bad metal ) Percolation transition Compressibility anomaly Implication for the apparent D metal-insulator transition

11 Disorder in D electron systems Heterostructure MOSFET Ionized donors gate s = 0 00 nm s + SiO + + dilute impurities D electron channel Si rough interface Random potential s x, y The random potential is smooth on the scale of the inter-electron distance: s > > a

12 Depletion regions and the percolation transition Local electron density n(r) n 0 0 x, y depletion regions, n( r) 0 1. Rare, isolated depletion regions. Depletions percolate y y x x May be metallic Insulator but hardly amorphous Percolation point n p =?

13 Metal-Insulator Transition in a model of a heavily doped 3D semiconductor Efros and Shklovskii (late 70 s) n = ( 1 K) K is the compensation n i At high compensation, the doping needed to reach the metallic state in 3D is determined by the percolation threshold and is given by: n i a 3 B 1 (1 K) ~ 3 >> 1 K 1 3 (much larger than Mott s critical density n i a B 0.06) Critical doping GaMnAs n i a B

14 Origin of the depletion regions n i n i n i distant ionized donors Consider a periodic impurity distribution: U d It creates the periodic potential x s D electron channel which causes the modulation of electron density: U = e nid d >> s (for ) -either- n 0 n = n i x n 0 -or- depletion region x Depletion Regions appear when the low-density electron liquid is unable to adequately screen random fluctuations of impurity density

15 Field-penetration experiments: a sensitive probe of the depletion regions Top gate Penetrated field fraction de p V V ~ de 0 BAD metal D electron layer E 0 A E p A n m D electron density Bottom gate Eisenstein (1994) good metal Electric field can leak through the Depletion Regions In certain samples, n m coincides with the metal-insulator transition point Compressibility Anomaly (?) Dultz and Jiang (000) Ilani et al. (000)

16 Screening radius Away from the depletion regions, the random potential is screened linearly r TF 1 π e dµ dn Thomas-Fermi screening radius At low electron density r ~ a TF y If sizes of both the depletion and metallic regions exceed, the screening of the random potential inside of them is almost perfect r TF r TF In the cases of interest this condition is satisfied if s > 0.1 n i Example: ni = cm s = 40nm x s 5 n i

17 Energy, compressibility, and screening radius of a clean D system Screening radius Inverse compressibility r TF 1 π e dµ dn χ 1 0 dµ dn a B Electron density n = 1/ a π e a B n rtf 0. 3a χ e a At low density the screening radius is negative! Chemical potential e µ ~ a >> ab a a B h me κ = Bohr radius a ~ 15a B = ~ nm 30 nm a

18 Over-screening a) Usual screening b) Over-screening Charge density Charge density next next n tot n tot Coordinate Coordinate In both cases, n tot < n ext

19 Formulation of the nonlinear screening problem Local electron density n(r) n 0 0 Electrostatic potential Φ(r) µ y x, y x, y Efros and Shklovskii (70 s) Efros, Pikus, and Burnett (199) For a given impurity density distribution n i (r) find the electron density distribution (r) such that: n ( r) > 0 eφ (r) = µ n( r) = 0, eφ( r) > If then otherwise; µ i Φ = Φi + Φ e Φ ( r) = e Φ e d r n n ( r ) i ( r r ) n( r ) n0 ( r) = e d r r r + s x

20 Present approach Exact solution does not seem possible in general. Attempt an approximate analytical solution: 1. Perfect screening condition: Physical meaning: w(r) ( r ) d r n ( r ) K( r r ) n ( r) = n0 + w( r) is smoothed impurity density that needs to be screened: K( r) = π [ r w i 3/ s + 4s Perfect screening is not possible if w(r) is negative and large In general, 4. Strategy: find the local functional that minimizes the total energy by the variational method The problem reduces to the integral equation with the constraint n (r) is an unknown nonlocal functional of w(r). Can we get an insight from simpler geometries (a round hole, a droplet, a chess-board)? 3. Yes, it seems that a purely local functional can work fine f w dwn( w) e = n 5. Final step: check against the numerical results and exact analytical asymptotics 0 ] n ( r) = n[ w( r)] dw G( w, w )[ n( w ) w ] = µ e f n n = n 0 + w? w 0 w

21 Prior results Efros, Pikus, and Burnett (199) Percolation threshold: Scaling with ni and s n p n s i ni s impurity concentration spacer width Fast growth of Depletion Regions at Shi and Xie (001) n < 0.4 n s i Based on numerical simulations 100 Penetrated field Electron density (10-3 / a B )

22 Comparison with prior results 1. Percolation threshold. Energetics n p = 0. 1 n (prior numerical result was 0.11 ) s i n = n p -1.5 Efros et al. (199) / Electron density h=n 0 /(n i /s) Shi & Xie Theory Uniform 0 n = n m Electron density (10-3 / a B )

23 Compressibility anomaly χ = χ DR + χ 0 D electron layer E 0 χ 1 DR ~ e s exp( 0.n 0 / n p ) From Depletion Regions E p D electron density n m n 0 Penetrated field fraction: χ 1 dµ dn 0 de p de 0 χ 1 inverse compressibility From Metallic Regions 1 χ e 0 ~ a s >> a spacer thickness

24 New theoretical results summary 1. Relation between the compressibility anomaly AND the percolation threshold. Depleted area fraction vs. electron density n m n p ln(400n s i ) 3.0n p 100% 50% 1 χ 6% n p nm 0.5 Electron density, in units of n i / s n m D electron density At n = n m typically only 6% of the total area is depleted: n p

25 Compressibility anomaly and the MIT 1. n-gaas bilayer Galaktionov et al., in preparation No MIT Nearly perfect fit to the theory!. p-gaas. Single-layer structure. n p np p-gaas p-gaas

26 Conclusions We offer a simple explanation of the compressibility anomaly (CA) The Metal-Insulator Transition and the CA are nearby but, in general, do not coincide, in contrast to findings of the UCLA group Percolation transition occurs at altogether much lower densities which are never probed in the experiments We seem to understand well the thermodynamics (compressibility) of the BAD METAL and its basic real-space structure, at least in p-gaas Our approach may be extended to other systems (DMS? Cuprates?) and other geometries (3D, bilayers, multilayers) References: 1. Percolation: M. F., Phys. Rev. B. 69, (R) (004). Exact solutions for nontrivial basic geometries: Phys. Rev. B 69, 4531 (004) 3. Detailed comparison with experiment: Galaktionov, Savchenko, M.F., in preparation