Drude theory & linear response

Transcription

1 DRAFT: run through L A TEX on 9 May 16 at 13:51 Drude theory & linear response 1 Static conductivity According to classical mechanics, the motion of a free electron in a constant E field obeys the Newton equation m dv = ee, (1) dt whose solution is v(t) = v(0) eet/m; the electronic current density is j(t) = env(t), where n = N/V is the electron density. The macroscopic current obeys therefore the equation: d j dt = (n/m) e2 E. (2) Analogous results are retrieved in quantum mechanics (QM) and the macroscopic current obeys the equation: d j dt = (n/m) eff e 2 E. (3) The quantity (n/m) eff measures the density of free carriers and their inertia. QM linear response theory (Kubo formula) provides indeed the value of (n/m) eff ; this is discussed below, Sec. 5. In order to retrieve Ohm s law, Drude introduces by hand a phenomenological dissipation term; we can do the same within QM. The equation of motion is thus ( d j dt + j ) = (n/m) eff e 2 E. (4) τ For any given initial conditions, j(t) has a transient which decays exponentially with lifetime τ. After this, the steady state solution is j = σ Drude E = (n/m) eff e 2 τ E, (5) where σ Drude is the dc (i.e. static) conductivity in the Drude model. Notice that a dissipative system forgets its past, and the same steady state is reached independently of the initial conditions. 2 Linear response To start with, we fix our conventions about Fourier transforms: f(ω) = dt e iωt f(t) f(t) = 1 2π 1 dω e iωt f(ω). (6)

2 Suppose we have a general input signal f input (t) and the corresponding output f output (t), which is due to the response of a time-independent physical system. The most general linear response is given by a convolution f output (t) = dt χ(t t )f input (t ). (7) The response χ, also known as generalised susceptibility, can be equivalently written as the functional derivative χ(t t ) = δf output(t) δf input (t ), (8) evaluated at equilibrium. The response is therefore an equilibrium property of the system in absence of perturbation. The convolution theorem yields 3 Kramers-Kronig relationships f output (ω) = χ(ω) f input (ω). (9) If the input signal is chosen as f input (t) = δ(t), then we have f output (t) = χ(t): in a causal system therefore χ(t) = 0 for t < 0. We pause to notice that not all linear systems must be causal: an obvious example of a noncausal system is an amplifier. An amplifier can start self oscillations (Larsen effect), which quickly become anharmonic; however it needs to get energy from some power source. We also stress that causality and dissipation are not synonymous. While a dissipative system is causal, one can address causal systems where no mechanism allows for dissipation: the paradigmatic example is the (undamped) harmonic oscillator in classical mechanics. The Kramers-Kronig relationships are most frequently proved by addressing the Fourier transform of the response function in the complex ω plane (e.g. Kittel Ch. 15). Here we provide a less common and to my taste more elegant proof, found in the literature. The Fourier transform of a causal χ(t) is χ(ω) = χ (ω) + iχ (ω) = 0 dt e iωt χ(t). (10) We only address a response χ(t) which is real in the time domain: therefore χ (ω) is even and χ (ω) is odd. It is expedient to write χ(t) = χ(t) χ( t) 2 + sgn(t) χ(t) χ( t), (11) 2 2

3 where sgn is the signum function, and to extend the integral to (, ): iχ (ω) = 1 4 χ (ω) = 1 4 dt e iωt [χ(t) χ( t)] (12) dt e iωt sgn(t) [χ(t) χ( t)]. (13) We notice then that the Fourier transform of sgn(t) is the distribution 2i/ω, understood as a principal part. Using then the convolution theorem in Eq. (13), and exploiting Eq. (12) we have χ (ω) = 2i dω iχ (ω ) 2π ω ω = 1 π dω χ (ω ) ω ω. (14) The real and imaginary parts of χ(ω) are related via an Hilbert transform. The inverse relationship reads χ (ω) = 1 dω χ (ω ) π ω ω. (15) Finally, we may take advantage once more of the fact that χ (ω) is even and χ (ω) is odd, arriving at the Kramers-Kronig relationships in the equivalent form χ (ω) = 2 π χ (ω) = 2ω π 0 dω ω χ (ω ) ω 2 ω 2 0 dω χ (ω ) ω 2 ω 2. (16) The power of the Kramers-Kronig relationships is that one can measure e.g. χ (ω) only, and then obtain χ (ω) by Hilbert transform, under the very general hypothesis of linearity and causality. 4 Drude theory (ω-dependent) We now identify f input (t) with a time-dependent electric field E(t) and f output (t) with the linearly induced current density j(t): the generalized susceptibility χ coincides in this case with the conductivity (scalar in isotropic systems). Switching then to the frequency domain, the conductivity σ(ω) measures the current linearly induced by an electric field at frequency ω j(ω) = σ(ω)e(ω). (17) We use the above conventions about Fourier transforms; other conventions may change the sign of the imaginary parts in the response functions. 3

4 Inserting E(t) = E(ω)e iωt in Eq. (4), we get ( iω + 1 ) j(ω) = (n/m) eff e 2 E(ω). (18) τ The Drude phenomenological formula is then σ Drude (ω) = ie2 (n/m) eff, (19) ω + i/τ where (n/m) eff is the quantum analogue of the original n/m in the classical theory. The dc limit is purely dissipative: We rewrite Eq. (19) as σ Drude (0) = e 2 (n/m) eff τ. (20) σ Drude (ω) = i π D ω + iη, (21) where D = πe 2 (n/m) eff is the Drude weight and η = 1/τ. Since η > 0 the conductivity has a pole in the complex ω plane at ω = iη and is analytic in the upper half plane. This fact ensures a causal response and guarantees the Kramers-Kronig relationships. The real and imaginary parts of σ denote in-phase (dissipative) and out-of-phase (reactive) response to the E field. Within the Drude model Re σ Drude (ω) = 1 π Dη ω 2 + η 2 ; Im σ Drude(ω) = 1 π Dω ω 2 + η 2. (22) In the nondissipative (η 0+), yet causal, limit we get Re σ(ω) = D δ(ω) ; Im σ(ω) = D π P 1 ω, (23) where P denotes the principal part. The dc (ω = 0) in-phase conductivity has a δ- like divergence: this accounts for the obvious fact that free electrons in a constant field undergo free acceleration, and the current does not reach a steady state limit. Equivalently, we may say that the system has an undamped normal mode at ω = 0. 5 Quantum mechanics The conductivity tensor in QM is defined via linear-response theory; its expression belongs to the family of Kubo formulas. In general conductivity is a Cartesian tensor σ αβ, and is the sum of a regular term and a Drude (δ-like) term: σ αβ (ω) = D αβ δ(ω) + σ (regular) αβ (ω). (24) 4

5 The Drude term accounts for free acceleration, in analogy to the classical case. Recalling our previous definition D = πe 2 (n/m) eff, the Kubo formula provides the QM expression for (n/m) eff. The Kubo formula for conductivity can be formally written even for correlated systems, and even for finite temperature. In the special case of noninteracting electrons in a periodic (mean-field) potential we define the α component of the velocity in the n-th band as v nα (k) = 1 ε n (k). (25) m k α The Kubo formula for the Drude weight is then D αβ = πe2 (2π) 3 n dk f ϵ ε n (k) v nα(k)v nβ (k), (26) where f ϵ is the Fermi distribution function. At zero temperature D is a pure Fermisurface property, i.e. D depends only on the shape of the Fermi surface and on the k-derivatives of the band structure ε n (k) at the Fermi surface. Clearly, these are the only ingredients which can account for free acceleration in a crystalline system. We obviously do not provide a proof of the above formula. Here we only content ourselves to apply Eq. (26) to the simple case of noninteracting electrons in zero potential (free electron gas), and at zero temperature. In this case we 5

6 have: f ϵ = 2 θ(ϵ F ϵ), f ϵ(k) = 2 δ(ε F 2 2m k2 ) = 2 m D = πe2 (2π) 3 ( 2 m 2 k F ) 4πk 2 F ε(k) = 2 2m k2 ; (27) 2 k F δ(k F k); 1 3 v2 = 2 k 2 F 3m 2. (28) 2 k 2 F 3m 2 = πe2 k 3 F 3π 2 m = πe2 n m. (29) It is remarkable that for noninteracting electrons in zero potential we get precisely (n/m) eff = n/m, i.e. the QM result coincides with the classical one, obtained by Drude in In other words Schrödinger equation, Pauli principle, and Fermi-Dirac statistics do not provide any correction to the original Drude result in this simple case. The reasons why this happens are pretty clear from a figure in Kittel, reproduced here. 6