Notes on Electron Transport Chi-Feng Pai September 16, 2016 This is a \cheat sheet" on the electron transport in (mostly) metals, which covers the Drude model (classical free electron), Sommerfeld model (semiclassical quantum free electron), Bloch state dynamics (semiclassical quantum electron band dynamics), a bit of non-equilibrium approach, and lastly a brief introduction on Berry phase. The materials are mostly summarized from Solid State Physics by Ashcroft / Mermin. Timeline 1856 William Thomson (Lord Kelvin) AMR 1879 Edwin Hall Hall effect 1897 J. J. Thomson Discovery of electron 1900 Paul Drude Drude model 1913 Millikan and Fletcher Oil drop experiment 1927 Arnold Sommerfeld Free electron Fermi gas 1928 Felix Bloch Bloch theorem 1931 A. H. Wilson Band conductor/insulator 1 Drude model Drude model describes the motion of electrons with collisions in metals. The col- lision/scattering is characterized by the relaxation time τ. A quick (yet plausible) derivation of electron conduction goes like this p (−e)Eτ ne2τ J = (−e)nv = (−e)n = (−e)n = E = σE: (1) m m m Note that the v here is the drift velocity vd. The relaxation time τ depends on possibly different scattering mechanisms, and generally follows Matthiessen's rule: −1 −1 −1 −1 τ = τimpurities + τdefects + τ (T )lattice/phonon + ::: (2) 1.1 Equations of motion For a more rigorous approach, we need to consider the equations of motion. That is, dp(t) p(t) = f(t) − ; (3) dt τ where the force term f = (−e)[E + v × B] : (4) In a steady state, i.e., dp=dt = 0, if we consider a 2-dimensional transport case under an out-of-plane magnetic field B = Bz^, then above equations yield −eE − eB(p =m) − p /τ = 0 x y x : (5) −eEy + eB(px=m) − py/τ = 0 Alternatively, σ E = (eBτ=m) J + J 0 x y x (6) σ0Ey = − (eBτ=m) Jx + Jy 2 with Drude DC conductivity in the absence of magnetic field σ0 = ne τ=m. Note that in the Hall effect measurement scheme, Jy = 0. Therefore, E = σ−1J x 0 x (7) Ey = − (eBτ/mσ0) Jx = − (B=ne) Jx ≡ RH BJx with the Hall coefficient Ey −1 RH ≡ = (8) JxB(z) ne 1.2 Resistivity So, for a 2-dimensional case under an out-of-plane magnetic field B = Bz^, the relation between electric field E and current density J can be written as E = ρJ (9) E ρ ρ J x = xx xy x ; (10) Ey ρyx ρyy Jy 2 where the resistivity tensor is 1 1 ! τ ρ = c (11) σ0 −!cτ 1 where the so-called cyclotron frequency !c = eB=m. 1.3 Conductivity Alternatively, one can use J = σE (12) J σ σ E x = xx xy x ; (13) Jy σyx σyy Ey to describe conduction in metals. The conductivity tensor is then σ0 1 −!cτ σ = 2 : (14) 1 + (!cτ) !cτ 1 Note that above expression satisfies σ = ρ−1. 1.4 Hall angle If we consider a typical situation that only an electric field along x-direction is applied, namely Ey = 0, then the ratio between transverse and longitudinal current density components, Jy=Jx = !cτ: (15) This ratio is often called (tangent of) the Hall angle θH , which describes the deviation of electron motion from longitudinal direction under the influence of an out-of-plane external magnetic field. We can further write θH ≈ tan θH = Jy=Jx = !cτ; (16) provided that the Hall angle is small. 3 2 Sommerfeld model (Free electron Fermi gas) Again we consider free electrons, i.e., they are not affected by their mutual interac- tions (electron-electron) and their interactions with ions (electron-phonon), impuri- ties, or defects. So it is still na¨ıve, but with quantum mechanical concepts. 2.1 Quantum mechanical free electron model The most classical example is the particle (free electron) in a 3-dimensional box. The Schr¨odingerequation yields 2 H^ (r) = − ~ r2 (r) = " (r): (17) 2m Note that only the kinetic term is present. And the wave function corresponds to a periodic boundary condition is 1 (r) = p eik·r; (18) V 3 where V = L and ki = ni · (2π=L) with ni 2 Z. The eigenvalue, which is the energy of electron, then can be expressed as "(k) = ~2k2=2m. And from this we have the radius of the Fermi sphere in k-space, the Fermi wave vector kF , which satisfies (4π=3)k3 k3 N = 2 × F = F V; (19) (2π=L)3 3π2 where N is the number of electrons and the factor 2 represents two spins in one electronic state. Therefore, Fermi wave vector 2 1=3 kF = 3π (N=V ) (20) Fermi energy 2 " = ~ 3π2 (N=V )2=3 : (21) F 2m Sometimes N=V is denoted as n. Also note that Fermi velocity vF and Fermi 2 temperature TF relate to the Fermi energy through "F = mvF =2 = kBTF . 4 2.2 Density of states and Fermi-Dirac distribution The density of states g(") describes the number of states that can be occupied within a interval of energy. By definition, dn 1 dN 1 dN dk g(") = = = ; (22) d" V d" V dk d" Therefore for the 3-dimensional case this yields Density of states 1 2m3=2 p g(") = " : (23) 2π2 ~2 Also, recall that electrons are fermions so they obey Fermi-Dirac statistics. The distribution function is Fermi-Dirac distribution 1 f(") = : (24) e("−µ)=kB T + 1 The (number) density of electrons is then evaluated through 1 Z dk Z dk Z n = 2 × f("(k)) = f("(k)) = d"g(")f("): (25) V (2π=L)3 4π3 2.3 Conduction of Fermi gas The Sommerfeld theory of electric conduction, i.e., the conduction of free electron Fermi gas, can be viewed as the shift of Fermi sphere under an applied electric field E (See Fig. 1). Using again the equation of motion: dp(t) p(t) p(t) = f(t) − = (−e)E − ; (26) dt τ τ and in k-space the steady state condition (dp=dt = 0) gives k(τ) − k(0) = ∆k = −eEτ=~: (27) So the current (along x-direction) can be expressed as 5 Figure 1: Displacement of the Fermi surface due to an applied electric field. Z dk J = (−e) v : (28) x 4π3 x shifted Using a spherical coordinate, we can rewrite above expression into Z (k2 sin θ)(2∆k sin θ cos φ) J = (−e) F (v sin θ cos φ)dθdφ. (29) x 4π3 F After some math, ne2τ J = E = σ E ; (30) x m x 0 x which yields a same form of conductivity as Drude model. Although the expressions are the same, the physics behind them are very different: Drude model J = (−e)nvd (31) Sommerfeld model vd J = (−e) n hvF i : (32) vF Also note that vF vd. 6 2.4 Justification Is it really ok that we ignore all the electronic or ionic interactions? Qualitatively speaking, (i) the Pauli exclusion principle prevents electrons from getting too close; (ii) the screening of inner electrons reduces ionic interaction. Nevertheless, the free electron model seems to be okey only for alkali metals. 3 Bloch model In 1928, Felix Bloch, a Ph.D. student in Heisenberg's group, came up with a quantum mechanical approach of the electron theory in crystal lattices (periodic potential). 3.1 Bloch theorem By considering a periodic potential such as V (r) = V (r + R). The Schr¨odinger equation yields 2 H^ (r) = − ~ r2 + V (r) (r) = " (r): (33) 2m Bloch theorem states that if V (r) is periodic, then 1 ik·r k(r) = p e uk(r) ; (34) Ω 3 where Ω is the total volume (Ω = L ) and the periodic part uk(r) satisfies uk(r) = uk(r + R) : (35) After conceiving the band index concept, the Bloch states are typically expressed as ik·r nk(r) = e unk(r) ; (36) where n is the band labeling index. 3.2 Semiclassical dynamics of electrons in Bloch bands The three rules of band dynamics: (i) Band index n is a constant, i.e., no interband transitions. This will be satisfied if 7 No dielectric breakdown 2 eEa ("gap(k)) ="F (37) and No magnetic breakdown 2 eB~=m = ~!c ("gap(k)) ="F (38) (ii) The dynamics itself 8 1 @" (k) > r_ = v (k) = n <> n @k ~ (39) > :> _ ~k = (−e)[E(r; t) + vn(k) × B(r; t)] (iii) Restating the features of quantum mechanical Bloch theory, such as the repre- sentation of a single primitive cell; And the Fermi distribution dk 1 dk f("n(k)) = (40) 4π3 e("n(k)−µ)=kB T + 1 4π3 where µ is the chemical potential and it relates to the density of state g(") through 2 0 π 2 g ("F ) µ = "F − (kBT ) : (41) 6 g("F ) 3.3 Filled band In 1929, Rudolf Peierls suggested that a filled band does not contribute to conduction. Later in 1931, A. H. Wilson suggested that band structure and the number of valence electron can be used to determine the system-of-interest is a insulator (even number; fully-filled) or a conductor (odd number; partially-filled). Anyway, for a filled band, Z Z dk dk 1 @"n(k) J = (−e) vn(k) = (−e) = 0; (42) 4π3 4π3 ~ @k BZ BZ since Green's theorem for a periodic function yields that R rf(r) = 0. On the other c hand, for a partially-filled band, 8 0 1 Z dk Z dk Z dk J = (−e) v (k) = (−e) v (k) + v (k) = 0: (43) 4π3 n @ 4π3 n 4π3 n A BZ filled unfilled Therefore, for a nearly-filled band, Z dk Z dk J = (−e) v (k) = (+e) v (k) : (44) 4π3 n 4π3 n filled unfilled This is the concept of holes.