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Large Capacitance Enhancement in 2D Electron Systems Driven by Electron Correlations

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Large Capacitance Enhancement in 2D Electron Systems Driven by Electron Correlations Large capacitance enhancement in 2D electron systems driven by electron correlations Brian Skinner, University of Minnesota with B. I. Shklovskii May 4, 2013 Capacitance as a thermodynamic probe The differential capacitance material of interest V insulator contains information about thermodynamic properties metal electrode Correlated and quantum behavior can have dramatic manifestations in the capacitance. Thermodynamic definition of C Capacitance is determined by the total energy U(Q) -Q V +Q electrons Define the “effective capacitor thickness”: 2 2 d* = ε0ε/C = ε0ε (d U/dQ ) Thermodynamic definition of C 2D material with electron density n chemical potential μ -Q d d * d 0 V e2 dn +Q 1 1 1 C Cg Cq Quantum capacitance in 2D (Noninteracting) 2DEG with parabolic spectrum: dn const. m /2 d * d a / 4 d B graphene: 1/ 2 1/ 2 n Cq n n1/ 2 d * d 8 [Ponomarenko et. al., PRL 105, 136801 (2010)] -1 Electron correlations can make Cq large and negative, so that C >> Cg and d* << d In the remainder of this talk: 1. A 2D electron gas next to d a metal electrode V 2. Monolayer graphene in a strong magnetic field 2’. Double-layer graphene Large capacitance in a capacitor 1. with a conventional 2DEG A clean, gated 2DEG at zero temperature: electron density n(V) d 2DEG insulator V metal gate electrode What is d* = ε0ε/C as a function of n? Semiclassical scaling behavior -1/2 The problem has three length scales: d, aB << d, and n -1/2 case 1: n << aB 2DEG E d 2DEG is a degenerate Fermi gas ++++++++++++++++++++ metal d* = d + aB/4 Semiclassical scaling behavior -1/2 The problem has three length scales: d, aB << d, and n -1/2 -1/2 ~ n case 2: aB << n << d 2DEG Electrons undergo crystallization: E 2 1/2 2 ++++++++++++++++++++ e n /ε0ε >> h n/m metal electron repulsion >> kinetic energy Wigner crystal has negative density of states: d µ ~ -e2n1/2/ε ε d * d 0 0 e2 dn d* = d – 0.12 n-1/2 < d [Bello, Levin, Shklovskii, and Efros, Sov. Phys.-JETP 53, 822 (1981)] [Eisenstein, Pfeiffer, and West, PRL 68, 674 (1992)] Semiclassical scaling behavior -1/2 The problem has three length scales: d, aB << d, and n -1/2 case 3: n >> d 2DEG Electrode charge is not uniform: capacitor consists of electron- 2d image charge dipoles with repulsion metal 2 2 3 u(r) = 2e d /ε0εr -1/2 Calculating classical energy of a n triangular lattice of dipoles gives d * 2.7n1/ 2d 2 d Absence of crystallization at n → 0 Screening of the electron-electron interaction means that strong correlations disappear at extremely small n. -1/2 2 case 4: n >> d /aB -1/2 2 2 3/2 2 u(n ) ~ e d n /ε0ε << h n/m electron repulsion << kinetic energy Wigner crystal melts, becomes a Fermi-liquid of electron-image dipoles. * d aB / 4 d Capacitance at different density regimes aB << d 1 1 -interacting dipoles -interacting electrons d* r 3 r d + aB/4 d (quantum) Fermi liquid (classical) Wigner crystal (quantum) Fermi liquid aB/4 2 4 2 2 n aB /d 1/d 1/aB Results at d/aB = 4 Theory: 1.0 [BS and M. Fogler, PRB 82, 201306(R) d d d (2010)] / / / * * * 0.5 d d d 0.0 0 0.2 0.4 0.6 0.8 1 1/2 Experiment: n d [Li and Ashoori, Science 332, 825 ~4 nm (2011)] 2. Can d* < d appear in graphene? E E = h v k V k aB → ∞ In graphene, Wigner crystallization does not occur: kinetic energy of localized states: ~ hv n1/2 2 1/2 Coulomb repulsion between localized states: ~ (e /ε0ε) n ...except in a strong magnetic field. Focus: single nondegenerate Landau level Crystallization at small filling factor ν << 1 B 2lB In a strong magnetic field, KE is quenched n-1/2 ~ l ν-1/2 l / eB B B d* vanishes in the limit ν → 0 1/2 2 d* = 1.1 (d/lB) ν at ν << 1, nd << 1 Electron-hole symmetry The lowest Landau Level has electron-hole symmetry: At small ν At small 1-ν rare electrons rare holes vE(v) E(1) (1 v)E(1 v) E(1) d* (the electron compressibility) is symmetric about ν = 1/2 General calculation of E(ν) using interpolation General parameterization of energy in the lowest Landau level: 2 k / 2 [Fano and Ortolani, Phys. Rev. B E( ) E(1) k [ (1 )] 37, 8179 (1988)] k 3 The energy E(1) is known: n 2 2 2 E(1) d rV (r)g (r) g (r) 1 er / 2lB 2 1 1 E(ν) at ν << 1 can be calculated quasiclassically. Full range of E(ν) at 0 < ν < 1 can be estimated by fitting the coefficients αk. d * 4 l 2 d 2 0 B E( ) d 2e2d d 2 Checks of our interpolation approach 1. For the unscreened 2DEG, the quasiclassical calculation reproduces the Hartree-Fock result at small ν: E( ) 1/ 2 3/ 2 5/ 2 [Lam and Girvin, Phys. Rev. B 2 0.782 0.282 ( ) 30, 473 (1984)] e / 40lB 2. The Fano-Ortolani expression is reproduced everywhere to within 3.5%. “cohesive energy” E E( ) E(1) 3. Reproduces the energy of FQHE states coh n E( ) d 2rV (r)g (r) 2 gν(r) is known for FQHE states ν = 1/3 and ν = 1/5 (offset) [Girvin, MacDonald, and Platzman, PRB 33, 2481 (1986)] Results At small d/lB, d*/d ~ d/lB in the lowest Landau level Limitations 1. Doesn’t account for disorder 2. Doesn’t account for FQHE gaps d*/d 1/3 2/3 [Eisenstein, Pfeiffer, and West, PRB 50, 1760 (1994)] Disorder and evolution of d* with magnetic field Vary lB/d for a given device by changing B, with fixed disorder and thickness: large d/lB → large B 1 decreasing 1 d d / / magnetic Disorder d* d* field has a significant smaller effect only d/lB at small ν, 1-ν inverse capacitance, capacitance, inverse inverse capacitance, capacitance, inverse 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 filling factor, filling factor, small d/lB → small B 1 1 d d / / d* d* Disorder dominates across the LLL inverse capacitance, capacitance, inverse inverse capacitance, capacitance, inverse 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 filling factor, filling factor, Recent experimental results d*/d A double-layer graphene capacitor Similar capacitance enhancement arises in capacitor made with two graphenes ν = 0 system At d << lB electrons and holes in opposite layers bind to make weakly- V repelling excitons Ground state is Ψ1,1,1, ν = 0δν rotated in pseudospin ν = 1 system (which-layer) space [Yoshioka and MacDonald, J. Phys. Soc. Japan 59, 4211 (1990)] d * d V ν = 1 - δν d 8 lB [Sanchez-Yamagishi, Taychatanapat, Watanabe, Taniguchi, Yacoby, and Jarillo-Herrero, PRL 108, 076601 (2012)] The exciton condensate as a weakly- interacting Bose gas Ground state is a Bose condensate of excitons with a short-range interaction: 2 n chemical potential 2 effective mass of ~ m* ~ of a 2D gas of 2 excitons * 1 l e m ln hard-disk Bosons B 2 na2D [Berman, Lozovik, and Gumbs, PRB 77, 155433 (2008)] [Lieb and Yngavson, J. Stat. Phys. PRB 103, 509 (2001)] In 2D, scattering length is exponentially small: u(r) 2 * 2 2 2 2 3 / m l l e d /lB B B a2D ~ lB exp 2 2 3 ~ lB exp 2 e d / lB d lB r e2nl e2nd 2 ~ B ~ 1 d d 2 d * ~ ~ 2 2 1 lB 2 lB / d ln e dn lB Unbinding of excitons d/lB decoupled 2DEGs d* ~ d exciton WCs? exciton stripes? 1 excitonic condensate 2 d* ~ d /lB 0 0.5 1 ν [C. H. Zhang and Y. N. Joglekar, arXiv:0711.4847 (2007)] Conclusions Large capacitance C >> Cg can appear in 2D electron systems as a result of correlations and screening between discrete electrons. 1. [Phys. Rev. B 82, 155111 (2010)] d / 1 d* d/l = 2 0.8 B 0.6 d/l = 1/2 2. 0.4 B 0.2 0 0 0.2 0.4 0.6 0.8 1 [Phys. Rev. B 87, 035409 (2013)] inverse capacitance, filling factor, Thank you. Reserve Slides d* in the LLL, with approximations Ground state phase diagram of parallel 2DEGs in the Lowest Landau Level uncoupled WCs WC of dipoles excitonic dipole “stripes” condensate 0 0.1 0.2 0.3 0.4 0.5 ν [C. H. Zhang and Y. N. Joglekar, arXiv:0711.4847 (2007)] Bose condensate of indirect excitons Electron-hole pairs are dipole- interacting Bosons 2n 2 ~ * m ~ 2 * 1 l e chemical potential of a m ln effective mass of B 2 2D gas of hard-disk na2D excitons Bosons In 2D, scattering length is exponentially small: u(r) 2 2 3 2 * 2 2 e d /lB / m lB lB a2D ~ lB exp 2 2 3 ~ lB exp 2 e d / lB d lB r e2nl e2nd 2 ~ B ~ 1 d d 2 d * ~ ~ 2 2 1 lB 2 lB / d ln e dn lB A variational approach Goal is to estimate the ground state energy E0(n) for a 2D system with interaction law Estimate E0(n) from trial wavefunctions: Evar H E0 Our approach: use the eigenstates of the 1/r Hamiltonian [BS and M.
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