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.