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PHYSICAL REVIEW LETTERS

Density-Functional Theory of Thermoelectric Phenomena 1

F. G. Eich,1,* M. Di Ventra,2 and G. Vignale1 Department of Physics, University of Missouri-Columbia, Columbia, Missouri 65211, USA 2 University of California, San Diego, La Jolla, California 92093, USA (Received 12 August 2013; published 14 May 2014)

We introduce a nonequilibrium density-functional theory of local temperature and associated local energy density that is suited for the study of thermoelectric phenomena. The theory rests on a local temperature field coupled to the energy-density operator. We identify the excess-energy density, in addition to the particle density, as the basic variable, which is reproduced by an effective noninteracting Kohn-Sham system. A novel Kohn-Sham equation emerges featuring a time-dependent and spatially varying mass which represents local temperature variations. The adiabatic contribution to the Kohn-Sham potentials is related to the entropy viewed as a functional of the particle and energy density. Dissipation can be taken into account by employing linear response theory and the thermoelectric transport coefficients of the electron gas. DOI: 10.1103/PhysRevLett.112.196401

PACS numbers: 71.15.Mb, 05.70.Ln, 72.20.Pa

Introduction.—Thermoelectric phenomena have long been the subject of intense research activity. More recently, renewed interest in these phenomena has surfaced due to their implications in the development of sustainable energy sources [1,2]. Besides its practical importance, thermoelectricity raises a host of fundamental questions and challenges. For instance, the thermopower of a given system is defined as the electric potential difference (at zero electrical current) that is induced by a thermal gradient across it. In this case, the electronic system is in mechanical equilibrium, and yet there is a steady flow of heat. At the microscopic level, we could argue that local temperature variations appear, which must be related to the heat-current density, ȷq . Unfortunately, neither concept has an unambiguous microscopic definition [3]. For the heatcurrent density the problem is that a unique local energy-density operator does not exist [4–6]. Similarly, the standard thermodynamic definition of temperature fails as soon as we leave the regime of local quasiequilibrium (for an operational definition of local temperature based on scanning thermal microscopy see, e.g., Refs. [7–9]). Achieving a clearer understanding of these quantities is important not only from the conceptual point of view but also for the practical calculation of familiar quantities, such as the electrical resistance. In this Letter, we propose a definition of the microscopic energy density and the associated temperature field, and we show that these quantities can be computed through a theoretical scheme that directly generalizes the well-known time-dependent density-functional theory (TDDFT) [10,11]. Our work is inspired by Luttinger’s seminal paper on the thermoelectric transport coefficients of the homogeneous interacting electron gas [12]. In the process of adapting the Kubo linear response formalism to thermal transport, Luttinger identified the “gravitational field”—a 0031-9007=14=112(19)=196401(5)

field that couples linearly to the energy density—as the mechanical proxy [13] of local temperature variations. In particular, for small fields applied to an initially homogeneous electron liquid, one can write the linear response relations [14],

−eȷn ȷq

¼

L11 L21

L12 L22

1

e ∇μ − ∇ϕ − ∇T T − ∇ψ

;

(1)

where −e is the charge of an electron, and ȷn and ȷq are the particle and the heat current, respectively. The Lij are transport coefficients that describe the thermoelectric properties of the system under investigation and they obey Onsager reciprocity relations, Lij ¼ Lji . From Eq. (1) we can see that a charge current is driven by a difference in chemical potential μ or an electric field E ¼ −∇ϕ. Similarly, a heat current is induced by a gradient in temperature or a gradient of Luttinger’s gravitational field, ψ. The fields ϕ and ψ are the mechanical counterparts of the chemical potential and the temperature, respectively [15]. The virtue of including these fields in the Hamiltonian is that they enable us to obtain the transport coefficients from a microscopic calculation. Luttinger’s study was limited to linear response about a homogeneous liquid state. Here we develop Luttinger’s idea into a full-fledged density-functional theory (DFT) of inhomogeneous systems (such as, e.g., nanojunctions) that are driven out of equilibrium by time-dependent temperature fields and potentials. Similar to ordinary DFT, we introduce the “Kohn-Sham system”—a fictitious noninteracting system that reproduces the exact density and the exact excess-energy density (defined below) caused by the varying temperature field. The resulting Schrödinger-like equation for this system (Kohn-Sham equation) includes a spatially and temporally varying mass term of a form often

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encountered in theoretical studies of compositionally graded semiconductors [16]. We furthermore suggest two basic approximations for the Kohn-Sham potentials: the adiabatic local density approximation and the linear response approximation— the latter being the simplest approximation that allows us to introduce dissipative effects. Last, we show how the linear response approximation predicts thermal corrections to the electrical resistivity, in addition to the well-known viscosity corrections, which have attracted considerable attention in the recent literature [17,18]. Formulation.—One of the simplest ways to introduce thermal TDDFT is to start from the Keldysh action [19], which we slightly modify here to describe systems that evolve from an initial equilibrium state at temperature T. The Keldysh action can be viewed as the generalization of the thermodynamic potential, governing equilibrium phenomena, to the time-dependent domain of nonequilibrium processes. Its form is R R 3 ˆ ˆ A½v ≡ i ln TrfT τ e−i C ½Hþ d rvðr;τÞnðrÞ g; (2) where the exponential is contour ordered, as indicated by T τ in Eq. (2), along the path C ¼ tðτÞ (parametrized by the real τ, cf. Fig. 1) in the complex time plane and R variable R 0 ˆ ˆ ˆ C ¼ C dτt ðτÞ. In this formula H ¼ T þ W is the sum of ˆ Further, β ¼ kinetic energy Tˆ and interaction energy W. ˆ ðkB TÞ−1 (cf. Fig. 1) is the inverse temperature, nðrÞ the particle density operator, and vðr; τÞ ¼ Vðr; τÞ − μ the local time-dependent potential Vðr; τÞ, minus the chemical potential μ [20]. The time-dependent density is given by nðr; τÞ ¼ δA½v=δvðr; τÞ. If the potential is time independent, i.e., vðr; τÞ ¼ v0 ðrÞ at all times, then the action functional, Eq. (2), does indeed reduce to −iβΩ½v0 , where Ω½v0 is the grand-canonical thermodynamic potential. We now want to extend the conventional TDDFT to allow for space- and time-dependent temperature fields. The idea is to replace the global inverse temperature β, which couples to the entire Hamiltonian, by a local temperature field β½1 þ ψðr; tÞ, which couples to the local ˆ energy density hðrÞ. ψðr; tÞ (Luttinger’s notation, cf. Ref. [12]) is our “temperature field.” In the limit of slow spatial variation it naturally describes the coupling of the system to a local thermal reservoir at the corresponding temperature [7–9]. It can thus be used for a first-principles treatment of electronic systems connected to local reservoirs at different temperatures and chemical potentials. However, its microscopic significance is more general: indeed, in a generic nonequilibrium situation one can

FIG. 1. Integration contour for the Keldysh action of Eq. (2).

define the “instantaneous local temperature” in terms of the field ψ that, when coupled to the microscopic energydensity operator, yields, under equilibrium conditions, the instantaneous local energy density of the nonequilibrium system. ˆ The form of hðrÞ is not unique, since there are different operators that, integrated over r, produce the same ˆ However, different forms are equivalent Hamiltonian H. as far as their long-wavelength content is concerned. Here we choose ˆ ˆ hðrÞ ¼ ˆtðrÞ þ wðrÞ;

(3a)

1 ˆ ð∇ ϕˆ † ðrÞÞ · ð∇r ϕðrÞÞ; 2m r Z ˆ 0Þ 1 ϕˆ † ðr0 Þϕðr ˆ ˆ d3 r0 ϕˆ † ðrÞ wðrÞ ¼ ϕðrÞ; 2 jr − r0 j ˆtðrÞ ¼

(3b) (3c)

ˆ where ˆtðrÞ and hðrÞ are the kinetic energy-density and interaction energy-density operator, respectively. We are now ready to present our generalized action functional, which, extending Eq. (2), reads R R 3 ˆ ˆ ˆ ~ ψ ≡ i ln TrfT τ e−i C fHþ d r½ψðr;τÞhðrÞþ~vðr;τÞnðrÞg A½v; g: (4) ~ τÞ ≡ vðr; τÞ½1 þ ψðr; τÞ: physically, We have defined vðr; this describes the coupling of the temperature field to the potential-energy density. Equations (2) and (4) highlight that “density functionalization” may be viewed as giving the intensive variables μ → vðr; tÞ and β → ψðr; tÞ a space and time dependence. However, it is important to keep in mind that the corresponding densities, nðr; tÞ and hðr; tÞ, are, in general, not locally related to vðr; tÞ and ψðr; tÞ. The equations for the densities ðn; hÞ in terms of the potentials ð~v; ψÞ are nðr; τÞ ¼

δA½~v; ψ ; δ~vðr; τÞ

hðr; τÞ ¼

~ ψ δA½v; : δψðr; τÞ

(5)

Inverting these equations yields (at least in the linear response regime [21]) a unique solution for the fields v~ ðr; τÞ and ψðr; τÞ as functionals of nðr; τÞ and hðr; τÞ. Legendre transformation of A½~v; ψ with respect to v~ and ψ leads to the universal action functional A½n; h [22]. The external potentials v~ and ψ associated with the densities n and h are given by the equations v~ ðr; τÞ ¼ −

δA½n; h ; δnðr; τÞ

ψðr; τÞ ¼ −

δA½n; h : δhðr; τÞ

(6)

In order to make an explicit connection to Mermin’s finite-temperature DFT (FT-DFT) [23] we now split A½n; h into two contributions: an equilibrium part, Aeq ½n, which is easily related to the universal free-energy functional Feq ½n

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¯ h, of Mermin’s equilibrium theory, and a remainder, A½n; which we refer to as excess action. Thus, we write ¯ h; A½n; h ¼ Aeq ½n þ A½n;

(7)

R where AeqR½n ¼ C Feq ½nðτÞ. Now, in view of the fact that Feq ½n ¼ d3 rheq ½nðrÞ − β1 Seq ½n, where heq ½nðrÞ is the equilibrium energy density and Seq ½n the equilibrium entropy for a given n, we find it natural to introduce a similar decomposition for the excess action, namely, ¯ h ¼ A½n;

Z Z C

¯ ¯ h; d rh½nðτÞ; hðτÞðrÞ − S½n; 3

(8)

where we introduced the excess-energy density ¯ h½nðτÞ; hðτÞðrÞ ≡ hðr; τÞ − heq ½nðτÞðrÞ (the energy density relative to the instantaneous equilibrium energy density) and the excess entropy functional ¯ h ¼ S½n; h − 1 S½n; β

Z C

Seq ½nðτÞ;

(9)

R R where S½n; h ≡ C d3 rhðr; τÞ − A½n; h. An important difference between the equilibrium entropy and the excess entropy is that the latter is nonlocal in time. This means that it encodes retardation effects since it depends on the history of the density and the energy density. Kohn-Sham scheme.—A key concept in DFTs is the mapping of the interacting system onto a noninteracting system, the so-called Kohn-Sham (KS) system. An important condition for the construction of the KS scheme in thermal DFT is that the usual KS scheme of Mermin’s FT-DFT is reproduced for equilibrium situations [vðr; tÞ ¼ vðrÞ, ψðr; tÞ ¼ 0]. In this way our theory is a true generalization of FT-DFT to nonequilibrium situations described in terms of the charge- and energy-density variations. The action functional As ½~vs ; ψ s for the KS system is defined in complete analogy to Eq. (4) by simply omitting the contribution due to the electron-electron ˆ ˆ ˆ and hðrÞ. interaction in H Note that the operator hðrÞ, yielding the energy density of the interacting system, differs from the operator 1 ˆ hˆ s ðrÞ ≡ ˆtðrÞ ¼ ð∇ ϕˆ † ðrÞÞ · ð∇r ϕðrÞÞ 2m r

δAs ½n; hs ; δnðr; τÞ

ψ s ðr; τÞ ¼ −

Moreover, As ½n; hs can be decomposed in the same way as A½n; h [cf. Eqs. (7)–(9)]. As in usual TDDFT the KS system reproduces the time-dependent density nðr; tÞ of the interacting system. The condition that the equilibrium limit of our theory coincides with FT-DFT is satisfied if the interacting and the KS system have the same excess-energy density [cf. Eq. (8) and below]. Defining the Hartreeexchange-correlation (Hxc) energy density E Hxc ½nðrÞ ≡ heq ½nðrÞ − heq s ½nðrÞ

δAs ½n; hs : δhs ðr; τÞ (11)

(12)

we formulate our theory in terms of the KS energy density, by viewing the interacting energy density as a functional of n and hs , i.e., h½n; hs ðrÞ ¼ hs ðrÞ þ E Hxc ½nðrÞ:

(13)

Here we rely upon recent progress in constructing approximations to E Hxc ½n derived from the uniform electron gas at arbitrary temperatures [24–28]. The KS energy density is readily computed from the KS orbitals, which are obtained by solving the time-dependent Schrödinger-like equation 1 þ ψðr; tÞ þ ψ¯ xc ðr; tÞ i∂ t ϕα ðr; tÞ ¼ −∇r ∇r þ v~ ðr; tÞ 2m ¯ þ v~ eq ðr; tÞ þ v ðr; tÞ ϕα ðr; tÞ; (14) xc Hxc starting from the noninteracting equilibrium state with inverse temperature β. This is the KS equation for our theory. Notice that the effective field ψ þ ψ¯ xc enters in this equation as a position- and time-dependent correction to the effective mass. In Eq. (14) v~ eq Hxc is the equilibrium potential of Mermin’s theory, veq , Hxc modified by the local temperature, Z eq v~ eq Hxc ðr;tÞ ¼ vHxc ðr;tÞ þ

d3 r0 ψðr0 ;tÞ

δE Hxc ½nðtÞðr0 Þ ; (15) δnðr;tÞ

and v¯ xc and ψ¯ xc are given as functional derivatives of the exchange-correlation (xc) excess entropy, i.e., δS¯ xc ½n; hs ; δhs ðr;tÞ

(16a)

¯ hs þ E Hxc ½n − S¯ s ½n; hs : S¯ xc ½n; hs ¼ S½n;

(16b)

v¯ xc ðr;tÞ ¼ −

(10)

representing the energy density of the KS system. In order to emphasize this difference we will denote the energy density of the KS system by hs. Switching to the action functional As ½n; hs we obtain the KS potentials v~ s ðr; τÞ ¼ −

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δS¯ xc ½n;hs ; δnðr;tÞ

ψ¯ xc ðr; tÞ ¼ −

A detailed derivation of the KS potentials is provided in the Supplemental Material [29]. From the time propagation of the KSPorbitals we can readily compute the densitiesPn ¼ α f α jϕα j2 and the energy density hs ¼ ð1=2mÞ α f α j∇r ϕα j2 , where f α are the equilibrium occupations of the KS orbitals in the initial ensemble. We have already mentioned that the relation between the potentials ðvðrÞ; ψðrÞÞ and the corresponding densities

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ððnðrÞ; hðrÞÞ is nonlocal. Nevertheless, a large part of this nonlocality is accounted for by the solution of the noninteracting KS equation. Indeed, the success of DFT is largely due to the fact that the xc potentials can be approximated (semi-)locally in space and time. Therefore, we propose the following two approximations for our xc potentials. Adiabatic local-density approximation.—The simplest approximation—the adiabatic local-density approximations (ALDA)—ignores retardation and the remaining adiabatic contribution is treated locally. Accordingly, the approximated potentials are functions of the instantaneous densities, i.e., v¯ ALDA ðr; tÞ ¼ − xc

1 ∂sxc ðnðr; tÞ; hs ðr; tÞÞ ; β ∂nðr; tÞ

ψ¯ ALDA ðr; tÞ ¼ − xc

1 ∂sxc ðn; hs Þ : β ∂hs ðr; tÞ

(17a)

∇¯vdyn xc ðr; tÞ ∇ψ¯ dyn xc ðr; tÞ

−1

2 ρ þ βΠκ ¼− βΠ

βΠ κ β κ

κ

;

(19)

where ρ is the electrical resistivity, κ −1 the thermal resistivity, and Π the Peltier coefficient. In a clean noninteracting electron gas both ρ and κ −1 vanish, reflecting the absence of scattering mechanisms. In the interacting electron gas, however, the situation is profoundly different: on the one hand, κ −1 acquires a nonzero value, reflecting the intrinsic decay of thermal currents caused by electronelectron interactions; on the other hand, the homogeneous resistivity ρ is replaced by viscous friction via the substitution ρ → −n−1 ∇η∇n−1 , where η is the electronic viscosity due to electron-electron interactions [17]. The final formula for the nonadiabatic potentials is

(17b)

∇¯vdyn xc ∇ψ¯ dyn xc

¼

2

− n1 ∇η∇ n1 þ βΠκ βΠ κ

βΠ κ β κ

ȷn ; ȷq

(20)

where we have omitted the arguments ðr; tÞ for brevity. These potentials must be added to the ALDA potential to constitute the full xc potentials [35]. Thermal corrections to the dc resistivity.—Post-ALDA corrections allow us to study dissipative effects, such as the thermal contributions to the dc resistance of a conductor. If I is the current, then the energy dissipated per unit time is W ¼ RI 2 , where R is the resistance of the conductor. To calculate R we perform a microscopic calculation of the dissipated power. This is the work done by the external fields on the currents. The dissipated power can be decomposed into a KS part and an xc correction [17], where the former part is well described by the LandauerBüttiker formalism [36–38]. The xc correction to the dissipation reads, Z W xc ¼

d3 rh ȷn ðrÞ · ∇¯vxc ðrÞ þ ȷq ðrÞ · ∇ψ¯ xc ðrÞi;

(21)

where the angular brackets denote the time average over a period of oscillation of the fields, which tends to infinity (ω ¼ 0) at the end of the calculation. Only the part of the effective field that oscillates in phase with the current contributes to dissipation. The effect comes entirely from the dynamical contribution to the xc fields, ∇¯vdyn xc and ∇ψ¯ dyn xc , defined in Eq. (20), i.e., Z W xc ¼

ȷn ðr; tÞ −1 −1 ¼ ðLs − L Þ : ȷq ðr; tÞ

L

In Eqs. (17) sxc ðn; hs Þ is the difference in entropy of an interacting uniform electron gas with density n and energy density hs þ E xc ðnÞ and a noninteracting uniform electron gas with density n and energy density hs. Note that we also employ a local approximation for E xc ½n at finite temperature [26,28]. Furthermore, we show in the Supplemental Material [29] that in a local approximation v~ eq Hxc ≈ ð1 þ ψÞvHxc . There we also present a more detailed discussion of the adiabatic approximation. Beyond the adiabatic approximation.—The first step in going beyond the adiabatic approximation is to include the dependence of the potentials on the time derivatives of the densities. Since these time derivatives are related by continuity equations to the divergence of particle and energy currents, it is natural to try and express the postALDA corrections in terms of current densities [30–33]. The relevant currents are the particle current density P ȷn ¼ ð1=mÞℑm α f α ϕP α ∇ϕα , and the energy current density ȷhs ¼ ð1=2m2 Þℑm α;i f α ð∂ i ϕα Þ∇ð∂ i ϕα Þ, determining eq the full heat current density ȷq ≡ ȷhs þ ðveq 0 þ vHxc Þȷn [34]. In the spirit of the local density approximation we use the thermoelectric conductivity matrix L of the homogeneous electron gas (and its noninteracting version Ls ) to relate the gradients of the dynamical xc potentials (i.e., the xc electric and thermal gradient fields) to the particle and heat currents. Employing Eq. (1), we thus have (e2 ¼ 1):

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ȷn 2 β d r η∇ ðrÞ þ j ȷq ðrÞ þ Π ȷn ðrÞj2 : (22) κ n 3

(18)

The structure of the thermoelectric resistivity matrix L−1 for the homogeneous electron gas is well known [14]:

More precisely, the first term corresponds to the so-called “viscosity correction” (cf. Ref. [17]) and the second term creates a “thermal correction” to the Landauer-Büttiker result.

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Conclusion.—The main accomplishment of this Letter is formal: we have proposed a time-dependent density functional formalism for the study of thermoelectric phenomena and we have suggested two basic approximation strategies—the adiabatic LDA and the linear response formalism. We emphasize that the proposed thermal DFT focuses on the electrons. Phonons have not been included so far. We expect this to be adequate for nanoscale systems where the electron-electron interactions dominate over electron-phonon or electron-impurity scattering [39]. For weak electron-phonon couplings the phononic contribution to the heat current can be added to the electronic contribution perturbatively [3]. The effect of phonons on the electronic contribution may be taken into account by referring the local approximations proposed in this Letter to a uniform electron gas coupled to phonons. Corrections due to impurities have already been included in a similar manner [40]. We believe that our theory will enable the inclusion of electron-electron interaction effects in thermal transport with relative ease when the required inputs, i.e., the entropy and transport coefficients for the homogeneous electron gas, become available. We gratefully acknowledge support from DOE under Grants No. DE-FG02-05ER46203 (F. G. E., G. V.) and No. DE-FG02-05ER46204 (M. D.).

*

[email protected] [1] G. S. Nolas, J. Sharp, and J. Goldsmid, Thermoelectrics: Basic Principles and New Materials Developments (Springer, New York, 2001). [2] Y. Dubi and M. Di Ventra, Rev. Mod. Phys. 83, 131 (2011). [3] M. Di Ventra, Electrical Transport in Nanoscale Systems (Cambridge University Press, Cambridge, England, 2008). [4] N. Chetty and R. M. Martin, Phys. Rev. B 45, 6074 (1992). [5] S. Lepri, R. Livi, and A. Politi, Rep. Phys. 377, 1 (2003). [6] L.-A. Wu and D. Segal, J. Phys. A 42, 025302 (2009). [7] Y. Dubi and M. Di Ventra, Nano Lett. 9, 97 (2009). [8] A. Caso, L. Arrachea, and G. S. Lozano, Phys. Rev. B 83, 165419 (2011). [9] J. P. Bergfield, M. A. Ratner, C. A. Stafford, and M. Di Ventra, arXiv:1305.6602. [10] E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984). [11] C. A. Ullrich, Time-Dependent Density-Functional Theory: Concepts and Applications (Oxford University Press, Oxford, 2012). [12] J. M. Luttinger, Phys. Rev. 135, A1505 (1964). [13] B. S. Shastry, Rep. Prog. Phys. 72, 016501 (2009). [14] N. W. Ashcroft and D. N. Mermin, Solid State Phys. (Thomson Learning, Toronto, 1976) Chap. 13, p. 253, 1st ed. [15] Strictly speaking ψ is the mechanical proxy for δT=ðT þ δTÞ. For a small temperature variation this implies ∇ψ ∼ T −1 ∇T.

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[16] M. R. Geller and W. Kohn, Phys. Rev. Lett. 70, 3103 (1993). [17] G. Vignale and M. Di Ventra, Phys. Rev. B 79, 014201 (2009). [18] D. Roy, G. Vignale, and M. Di Ventra, Phys. Rev. B 83, 075428 (2011). [19] R. van Leeuwen, Phys. Rev. Lett. 80, 1280 (1998). [20] It is assumed that vðr; tÞ tends to a constant v0 ðrÞ when t → −∞. Then, definition (2) assumes that the system is initially prepared in a thermal equilibrium state in the presence of the static potential v0 ðrÞ at temperature T. [21] G. W. Fernando, in Metallic Multilayers and their Applications Theory, Experiments, and Applications Related to Thin Metallic Multilayers, Handbook of Metal Physics Vol. 4, edited by G. W. Fernando (Elsevier, New York, 2008), p. 131. [22] A½n; h is defined as the negative of the Legendre transform of A½~v; ψ. [23] N. D. Mermin, Phys. Rev. 137, A1441 (1965). [24] F. Perrot and M. W. C. Dharma-wardana, Phys. Rev. B 62, 16536 (2000). [25] E. W. Brown, B. K. Clark, J. L. DuBois, and D. M. Ceperley, Phys. Rev. Lett. 110, 146405 (2013). [26] E. W. Brown, J. L. DuBois, M. Holzmann, and D. M. Ceperley, Phys. Rev. B 88, 081102 (2013). [27] T. Sjostrom and J. Dufty, Phys. Rev. B 88, 115123 (2013). [28] V. V. Karasiev, T. Sjostrom, J. Dufty, and S. B. Trickey, Phys. Rev. Lett. 112, 076403 (2014). [29] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.112.196401 for the derivation of the KS potentials and a detailed discussion of the adiabatic approximation. [30] G. Vignale, in Fundamentals of Time-Dependent Density Functional Theory, Lecture Notes in Physics Vol. 837, edited by M. A. L. Marques, N. T. Maitra, F. M. S. Nogueira, E. K. U. Gross, and A. Rubio (Springer, Berlin, Heidelberg, 2012), p. 457. [31] G. Vignale and W. Kohn, in Electronic Density Functional Theory: Recent Progress and New Directions, edited by J. F. Dobson, G. Vignale, and M. P. Das (Plenum, New York, 1996), p. 199. [32] G. Vignale and W. Kohn, Phys. Rev. Lett. 77, 2037 (1996). [33] G. Vignale, C. A. Ullrich, and S. Conti, Phys. Rev. Lett. 79, 4878 (1997). [34] These expressions neglect certain nonlinear terms involving the ψ field and are therefore only appropriate in the linear response regime. [35] Notice that Eq. (20) agrees with the Vignale-Kohn formula (cf. Ref. [32] and [33]) for the dynamical xc potential for the ordinary (longitudinal) current channel. [36] R. Landauer, IBM J. Res. Dev. 1, 223 (1957). [37] M. Büttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B 31, 6207 (1985). [38] R. Landauer, J. Phys. Condens. Matter 1, 8099 (1989). [39] A. V. Andreev, S. A. Kivelson, and B. Spivak, Phys. Rev. Lett. 106, 256804 (2011). [40] C. A. Ullrich and G. Vignale, Phys. Rev. B 65, 245102 (2002).

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