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Electron-Electron Interaction and the Fermi-Liquid Theory

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Electron-Electron Interaction and the Fermi-Liquid Theory PHZ 7427 SOLID STATE II: Electron-electron interaction and the Fermi-liquid theory D. L. Maslov Department of Physics, University of Florida (Dated: February 21, 2014) 1 CONTENTS I.Notations 2 II.Electrostaticscreening 2 A.Thomas-Fermi model 2 B.Effective strength of the electron-electron interaction. Parameter rs: 2 C.Full solution (Lindhard function) 3 D.Lindhardfunction 5 1.Adiscourse: properties of Fourier transforms 6 2.Endof discourse 7 E.Friedel oscillations 7 1.Enhancement of the backscattering probability due to Friedel oscillations 8 F.Hamiltonianof the jellium model 9 G.Effective mass near the Fermi level 13 1.Effective mass in the Hartree-Fock approximation 15 2.Beyond the Hartree-Fock approximation 15 III.Stonermodel of ferromagnetism in itinerant systems 16 IV.Wignercrystal 20 V.Fermi-liquid theory 22 A.Generalconcepts 22 1.Motivation 23 B.Scatteringrate in an interacting Fermi system 24 C.Quasi-particles 27 1.Interaction of quasi-particles 29 D.Generalstrategy of the Fermi-liquid theory 31 E.Effective mass 31 F.Spinsusceptibility 33 1.Free electrons 33 2.Fermi liquid 34 G.Zerosound 35 2 VI.Non-Fermi-liquid behaviors 37 A.Dirty Fermi liquids 38 1.Scatteringrate 38 2.T-dependence of the resistivity 40 A.Bornapproximation for the Landau function 43 References 44 I. NOTATIONS • kB = 1 (replace T by kBT in the final results) • ~ = 1 (momenta and wave numbers have the same units, so do frequency and energy) • ν(") ≡ density of states II. ELECTROSTATIC SCREENING A. Thomas-Fermi model For Thomas-Fermi model, see AM, Ch. 17. B. Effective strength of the electron-electron interaction. Parameter rs: The ratio of the Coulomb energy at a typical inter-electron distance to the Fermi energy is U e2=hri C = : EF EF hri is found from 4 3 1=3 πhri3n = 1 ! hri = n−1=3 3 4π U 4π 1=3 1 e2n1=3m 2 (2)2=3 e2m e2m C = = = : 34 : 2=3 1=3 1=3 EF 3 (3π2)2=3 =2 n 3 π n n Lower densities correspond to stronger effective interactions and vice versa. 3 Parameter rs is introduced as the average distance between electrons measured in units of the Bohr radius 2 hri = rsaB = rs=me : Expressing rs in terms of n and relating density to kF ; we find 3 1=3 1 9π 1=3 me2 rs = 1=3 = : 4π n aB 4 kF In terms of rs; e2 UC = rsaB and 1 9π 2=3 1 1 EF = 2 2 : 2 4 maB rs 2=3 UC 4 = 2 rs ≈ : 54rs: EF 9π C. Full solution (Lindhard function) In the Thomas-Fermi model, one makes two assumptions: a) the effective potential acting on electrons is weak and b) the effective potential (and corresponding density) varies slowly on the scale of the electron's wavelength. Assumption a) allows one to use the perturbation theory whereas assumption b) casts this theory into a quasi-classical form. In a full theory, one discards assumption b) but still keeps assumption a). So now we want to do a complete quantum-mechanical (no quasi-classical assumptions) form. Let the total electrostatic potential acting on an electron be φ = φext + φind; where φext is the potential of external charges and φind is that of induced charges. Corre- spondingly, the potential energy v = −eφ = −eφext − eφind: Because we are doing the linear-response theory, the form of the external perturbation does not matter. Let's choose it as a plane-wave 1 v (~r; t) = v ei(~q·~r−!t) + c:c: (1) 2 q 4 Before the perturbation was applied, the wavefunction was 1 Ψ = e(ik·~r−"kt): 0 L3=2 The wavefunction in the presence of the perturbation is given by standard expression from the first-order perturbation theory i(~q·~r−!t) i(~q·~r−!t) vq e vq e− Ψ = Ψ0 1 + + ; 2 "k − "k+~q + ! 2 "k − "k−~q − ! where the last term is a response to a c.c. term in Eq.(1). The Fourier component of the wavefunction 1 vq 1 vq Ψk = Ψ0k 1 + + : 2 "k − "k+~q + ! 2 "k − "k−~q − ! The induced charge density is related to the wavefunction is X 2 2 ρind = −2e fk jΨkj − jΨ0kj ; (2) k where fk is the Fermi function, factor of 2 is from the spin summation and the homogeneous (unperturbed) charge density was subtracted off. Keeping only the first-order terms in vq; Eq.(2) gives 1 X 1 1 ρind = −2e 3 fkvq + (3) L "k − "k+~q + ! "k − "k−~q − ! k Z 3 d k fk − fk+~q = −2evq 3 ; (4) (2π) "k − "k+~q + ! where we shifted the variables as k − ~q ! k and k ! k +q ¯ in the last term. The charge susceptibility, χ, is defined as q2 q2 ρ = e χv = −e2 χφ ; (5) ind 4π q 4π q where φq is the Fourier component of the net electrostatic potential. Comparing Eqs. (4) and (5), we see that Z 3 4π d k fk − fk+~q χ = 2 (−2) 3 : q (2π) "k − "k+~q + ! The meaning of χ becomes more clear, if we write down the Poisson equation (in a Fourier- transformed form) 2 q φq = 4π (ρext + ρind) : 5 External charges and potentials satisfy a Poisson equation on their own 2 q φext = 4πρext: Now, q2 q2φ = q2φ + 4πρ = q2φ − 4πe2 χφ ! q ext ind ext 4π q φ φ = ext : q 1 + χe2 Using a definition of the dielectric function φ φ = ext ; (6) q (q; !) we see that 2 Z 3 2 4πe d k fk − fk+~q (q; !) = 1 + χe = 1 + 2 (−2) 3 : (7) q (2π) "k − "k+~q + ! This is the Lindhard's expression for the dielectric function. a. Check Let's make sure that the general form of (q; !) [Eq.(7)] does reduce to the Thomas-Fermi one in the limit ! = 0 and q kF : 2 Z 3 4πe d k fk − fk+~q (q; 0) = 1 + 2 (−2) 3 q (2π) "k − "k+~q For small q; fk+~q = f ("k+~q) = f ("k+~q − "k + "k) @f = f ("k) + ("k+~q − "k) + ::: @"k and 4πe2 Z d3k @f (q; 0) = 1 + 2 2 3 − : q (2π) @"k @f At T = 0; − = δ ("k − EF ) : The density of states at the Fermi energy @"k Z d3k νF = 2 δ ("k − EF ) : (2π)3 Now, 4πe2 κ2 (q; 0) = 1 + ν = 1 + ; q2 F q2 2 2 where κ ≡ 4πe νF ; which is just the Thomas-Fermi form. 6 D. Lindhard function As shown in AM, the static form of the Lindhard's dielectric function is given by 4πe2 1 1 − x2 1 + x (q; 0) = 1 + + ln ; q2 2 4x j1 − xj where x ≡ q=2kF : Notice that the derivative of (q; 0) is singular for q = 2kF ; i.e., x = 1: This singularity gives rise to a very interesting phenomenon{Friedel oscillations in the induced charge density (and corresponding potentials). Mathematically, it arises because of the property of the Fourier transform. To find the net electrostatic potential in the real space we need to Fourier transform back to real space Eq.(6). Let's say that the external perturbation is a 2 single point charge Q: Then (in the q− space) , φext = 4πQ=q and Z d3q 4πQ φ (r) = e−i~q·~r (8) (2π)3 q2 (q; 0) : 1. A discourse: properties of Fourier transforms Fourier transforms have the following property. Suppose we want to find the large t limit of Z d! F (t) = e−i!tF (!) : (9) 2π If function F (!) is analytic, the integral in Eq.(6) can be done by closing the contour in the complex plane. F (t) for t ! 1 will be then given by an exponentially decaying 00 00 function exp(−!mint) ; where !min is the imaginary part of that pole of F (!) which is closest to the real axis. For example, if F (!) = (!2 + a2)−1 ;F (t) / exp (−at) : Thus, the large-t asymptotes of analytic functions decay exponentially in time. On the other hand, if F (!) is non-analytic, F (t) decays much slower{only as a power-law. For example, for F (!) = exp (−a j!j) ; we have Z d! Z 1 d! F (t) = e−i!t exp (−a j!j) = e−i!t + ei!t e−a! 2π 0 2π Z 1 d! −i!t −a! 1 1 1 a 1 = 2Re e e = 2Re = 2 2 / 2 for t ! 1: 0 2π 2π a + it π a + t t In addition, if F (!) has a divergent derivative of order n at finite !, e.g., for ! = !0; that F (t) oscillates in t: This can be seen by doing the partial integration in Eq.(6) n + 1 times Z Z !0 Z 1 2πF (t) = d!e−i!tF (!) = d!e−i!tF (!) + d!e−i!tF (!) = −∞ !0 7 1 1 1 Z d = e−i!tF (!) j!0 + e−i!tF (!) j1 − d!e−i!t F (!) + :::; −i! −∞ −i! !0 −i! d! n d −i!0t i.e., until the boundary terms gives the divergent expression d!n F (!0) e which oscillates as e−i!0t: 2. End of discourse Coming back to Eq.(9), we can now understand why the induced density around the point charge oscillates as cos 2kF r and falls off only as a power law of the distance cos 2k r ρ / F : ind r3 Both of these effects are the consequences of the singularity of (q; 0) at q = 2kF : E. Friedel oscillations The physics of Friedel oscillations is very simple: they arise due to standing waves formed as a result of interference between incoming and backscattered electron waves.
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