Non-adiabatic time-dependent density functional theory of the impurity resistivity of metals
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Non-adiabatic time-dependent density functional theory of the impurity resistivity ofmetals
V. U. Nazarov, G. Vignale, and Y.-C. Chang Research Center for Applied Sciences, Academia Sinica, Taipei 11529, Taiwan Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
We make use of the time-dependent density functional theory to derive a new formally exactexpression for the dc resistivity of metals with impurities. This expression takes fully into accountthe dynamics of electron-electron interactions. Correction to the conventional T -matrix (phase-shifts) theory is treated within hydrodynamics of inhomogeneous viscous electron liquid. As afirst application, we present calculations of the residual resistivity of aluminum as a function of theatomic number of the impurities. We show that the inclusion of many-body corrections considerablyimproves the agreement between theory and experiment. PACS numbers: 31.15.ee, 71.45.Gm, 72.15.-v
Scattering of carriers by impurities is one of the fun-damental mechanisms of resistivity in solids [1]. Withinthe single-particle theory, this problem can be efficientlyaddressed with the use of well established techniques ofpotential scattering theory. Electron-electron interac-tions can be included, at this level, within the ground-state density-functional theory [2, 3] as electrostatic andexchange-correlation effective static potential which scat-ters the electrons as single particles [4, 5]. However, thesingle-particle approach fails to account for the dynamical exchange and correlation effects which cannot be forcedinto the mold of a static mean field theory.A powerful theoretical tool has been devised to ac-count, in principle exactly, for dynamical electron-electron interaction effects in inhomogeneous systems.This is known as the time-dependent density functionaltheory (TDDFT) [6]. Historically, TDDFT was devel-oped to improve the calculation of atomic scatteringcross-sections and excitation energies in both bounded [7]and extended systems [8, 9]. However, in recent yearsthis theory has often been applied to the treatment ofstatic phenomena (e.g., the polarizability of polymerchains [10, 11]), and steady-state transport phenomena(e.g., the stopping power of metals for slow ions [12, 13],and the conductance of quantum point contacts [14–18]).In these applications one needs to take the zero-frequencylimit of a time-dependent process, which can be properlydescribed by TDDFT or its current generalization – thetime-dependent current density functional theory (TD-CDFT) [19, 20].Moving in the same direction, in this Letter we presentthe first complete TDDFT formulation for the impurityresistivity of metals. We derive an exact formula forthe frequency-dependent resistivity in terms of quantitiesthat can be calculated entirely within density functionaltheory. While the standard Kubo formula gives a for-mally exact expression for the conductivity , our formulagives an expression for the resistivity, and furthermoredoes not require that we calculate explicitly the currentdistribution. A major advantage of working with the re- sistivity rather than with the conductivity is that phys-ically distinct dissipative processes enter the resistivityas additive contributions (Matthiessen’s rule [21]). Inparticular, we find that our expression for the resistiv-ity naturally separates into a single-particle contributionand a dynamical many-body contribution. The former isshown to reduce to the classical potential-scattering for-mula for the resistivity; the latter takes into account theviscosity of the electron liquid [22].Our formula appears to be a promising tool for a sys-tematic improvement on the existing calculations of theresistivity of metals. We demonstrate this in a concreteapplication, namely the model calculation of the resistiv-ity of aluminum in the presence of random impurities.We show that dynamical corrections, calculated with thehelp of the available formulas for the visco-elastic con-stants of the uniform electron liquid [23], can consider-ably improve the agreement between the calculated andthe measured resistivity.We start by writing down the classical (single-particle)formula [1] for the resistivity of an electron gas of density¯ n with impurities randomly distributed with density n i : ρ = k F n i σ tr ( k F ) / ( e ¯ n ) . (1)Here k F is the Fermi wave-vector of the electron gas, σ tr ( k F ) is the transport cross-section of an electron atthe Fermi level scattered by the potential of an individ-ual impurity, and e is the absolute value of the electroncharge. The basic assumptions underlying Eq. (1) are:(i) Electrons do not interact with each other while beingscattered by the impurities, and (ii) Electrons feel onlyone impurity at a time, i.e., the coherent scattering ofan electron from more than one impurity is neglected.Both assumptions will be relaxed in the treatment thatfollows.Let us consider a monochromatic and uniform externalelectric field E ext ( t ) = E ext e − iωt applied to electron gaswith impurities positioned at R k , k = 1 , , ... . We canwrite the current density averaged over the normalizationvolume V as [ ? ] j i ( ω ) = icωV Z V d r d r ′ ˆ χ ij ( r , r ′ , ω ) E ext,j , (2)where ˆ χ ij ( r , r ′ , ω ) is the current-density response func-tion of the inhomogeneous electron gas with impurities.A summation over the repeated Cartesian index j is im-plied. We transform Eq. (2) with the help of the sum-rule[ ? ] c (cid:0) ω − ω p (cid:1) Z ˆ χ ij ( r , r ′ , ω ) d r ′ = cm × Z ˆ χ ik ( r , r ′ , ω ) ∇ ′ k ∇ ′ j V ( r ′ ) d r ′ + e ω m n ( r ) δ ij , (3)where V ( r ) = X k v ( r − R k ) , (4) v ( r ) is the bare potential of one impurity centered atorigin, n ( r ) is the ground-state electron density, ω p = p πe ¯ n /m is plasma frequency of the homogeneouselectron gas without impurities, and c and m are thespeed of light in vacuum and the mass of electron, re-spectively. Applying Eq. (3) twice with respect to theintegration over r and r ′ in Eq. (2), and using the ex-pression for the density-response function χ ( r , r ′ , ω ) = − ce ω ∇ i · ˆ χ ij ( r , r ′ , ω ) · ∇ ′ j , together with the static sum-rule [12] Z χ ( r , r ′ , ∇ ′ i V ( r ′ ) d r ′ = ∇ i n ( r ) , (5)we eventually write the current-density as j i ( ω ) = ieωm ( ω − ω p ) ¯ n E ext,i + 1 m ( ω − ω p ) V Z V d r Z V d r ′ × [ ∇ i V ( r )][ χ ( r , r ′ , ω ) − χ ( r , r ′ , ∇ ′ j V ( r ′ )] (cid:3) E ext,j (cid:9) . (6)Independently of the foregoing considerations, justfrom Maxwell’s equations for the electromagnetic field,we find that the resistivity can be written as [ ? ] ρ ( ω ) = E i ( ω ) j i ( ω ) = 1 ω (cid:18) πie + ωE ext j i E ext,i (cid:19) . (7)From Eq. (6) we conclude that the expression in theparentheses of Eq. (7) is zero at ω = 0. Taking the limit ω → ρ = − e ¯ n V Z V [ ∇ V ( r ) · ˆ E ext ][ ∇ ′ V ( r ′ ) · ˆ E ext ] × ∂ Im χ ( r , r ′ , ω ) ∂ω (cid:12)(cid:12)(cid:12) ω =0 d r d r ′ , (8) where ˆ E ext is the unit vector parallel to E ext .Equation (8) is the formal solution to the problem ofexpressing the resistivity in terms of the density-densityresponse function χ of the interacting inhomogeneouselectron gas with impurities. Using the relation [6] χ − ( r , r ′ , ω ) = χ − KS ( r , r ′ , ω ) − f xc ( r , r ′ , ω ) − | r − r ′ | , (9)we can conveniently rewrite Eq. (8) in terms of theKohn-Sham (KS) density-density response function χ KS of non-interacting electrons and the dynamical exchangeand correlation kernel f xc [ ? ] ρ = ρ + ρ , (10) ρ = − e ¯ n V Z V [ ∇ V KS ( r ) · ˆ E ext ][ ∇ ′ V KS ( r ′ ) · ˆ E ext ] × ∂ Im χ KS ( r , r ′ , ω ) ∂ω (cid:12)(cid:12)(cid:12) ω =0 d r d r ′ , (11)where V KS ( r ) is the static KS potential, and ρ = − e ¯ n V Z V [ ∇ r n ( r ) · ˆ E ext ][ ∇ r ′ n ( r ′ ) · ˆ E ext ] × ∂ Im f xc ( r , r ′ , ω ) ∂ω (cid:12)(cid:12)(cid:12) ω =0 d r d r ′ . (12)The first equation (11) is the single-particle (KS) con-tribution to the resistivity. The second (12) is the dynam-ical exchange-correlations contribution. If the frequencydependence of f xc is neglected – as one does, for exam-ple, in the adiabatic approximation to TDDFT – thenEq. (12) yields ρ = 0.To establish the connection between ρ and the clas-sical potential-scattering result of Eq. (1), we must ne-glect in Eq. (11) the coherent scattering from multipleimpurities. To do this we replace the full KS potential V KS ( r ) by the KS potential associated with a single im-purity in the electron gas, and we interpret the KS re-sponse function χ KS ( r , r ′ , ω ) accordingly. The normal-ization volume is taken to be equal to the volume perimpurity, i.e., V = 1 /n i . It can be rigourously proved [ ? ] that Eq. (11), thus modified, is equivalent to Eq. (1).This result, combined with the discussion of the previousparagraph, leads to the important conclusion that theadiabatic approximation to TDDFT is equivalent to theclassical potential-scattering ( T -matrix) approach as faras the calculation of the resistivity is concerned.The single-particle contribution to the resistivity isconventionally obtained from Eq. (1) [ ? ], using the T -matrix (phase-shift) technique to calculate the scatter-ing cross-section from the static KS potential [4, 5]. Tofind the many-body contribution to the resistivity fromEq. (12), we need a good approximation to the dynami-cal exchange and correlation kernel f xc . It is known [24]that f xc ( r , r ′ ) is strongly non-local (i.e. a long-rangedfunction of | r − r ′ | ) and this non-locality is crucial toa proper description of many-body effects in transportphenomena, even on a qualitative level [13]. This imme-diately poses the problem of constructing a reasonablyaccurate non-local approximation for f xc . In a recentpaper [13] we have shown how this can be done start-ing from an exact representation of the scalar f xc kernelin terms of the tensorial exchange and correlation kernelˆ f xc of time-dependent current density functional theory.This representation reads f xc = − e ω c ∇ − ∇· (cid:26) ˆ f xc + (cid:16) ˆ χ − KS − ˆ f xc (cid:17)h ˆ T (cid:16) ˆ χ − KS − ˆ f xc (cid:17) ˆ T i − × (cid:16) ˆ χ − KS − ˆ f xc (cid:17) − ˆ χ − KS (cid:16) ˆ T ˆ χ − KS ˆ T (cid:17) − ˆ χ − KS (cid:27) · ∇∇ − , (13)where ˆ χ KS is the KS current-density response functionand ˆ T is the projector operator onto the subspace oftransverse vector fields (i.e. divergence-free fields) [ ? ]. By making use of the local density approximation(LDA) for the tensorial ˆ f xc in the right-hand side ofEq. (13), we obtain a non-local approximation for thescalar f xc , which satisfies the zero-force sum-rule require-ments [24,13], and can, therefore, be considered a promis-ingly accurate approximation for transport problems.
12 14 16 18 20 22 24 26 28 30 32012345678 / n i ( c m / a t . % ) Z Aluminumr s = 2.07 FIG. 1. (Color online) Residual resistivity of aluminum dueto substitutional impurities of atomic number Z , as a func-tion of Z . The chained curve with circles (red online) is ourresult with inclusion of the dynamical exchange and correla-tions (the sum of ρ and ρ obtained with use of Eqs. (1) and(12), respectively). The chained curve with squares (blackonline) is the result of the single-particle theory ( ρ only).Solid squares are experimental data compiled from Refs. [25]. In Refs. [20, 26], the LDA to the exchange and cor-relation kernel ˆ f xc of the TDCDFT has been workedout within the framework of the hydrodynamics of in-homogeneous viscous electron liquid. We, therefore, useEq. (12) with f xc given by Eq. (13) and ˆ f xc as expressedin Ref. [26] through the viscoelastic constants of elec-tron liquid. In Fig. 1, we present results for resistivity for substitutional impurities of atomic number Z from11 through 32 in an aluminum host. The latter is mod-eled as a jellium with Wigner-Seitz radius r s = 2 . ρ calculated fromEq. (1) and represented by the chained curve with squares(black online) is found to be in agreement with earlier cal-culations [4, 5]. The total resistivity, including dynamicalexchange and correlation contributions, is represented bythe chained curve with circles (red online). An improvedagreement between theory and experiment can be clearlyseen from the figure. The effects left out by our calcu-lation that could possibly contribute to the remainingdisagreement between the theory and experiment are (i)the band structure and lattice distortion effects, (ii) thepossible spin-polarization, and (iii) the coherent scatter-ing by the impurities at different sites. Another poten-tially important source of error can be in the values ofthe visco-elastic constants of the electron liquid.Finally we note that Eq. (8) allows us to establish ageneral relation between the impurity resistivity and thefriction coefficient [ ? ] of the same host for the sametype of impurity atom. The latter can be written as [12] Q = − Z [ ∇ V ( r ) · ˆ v ][ ∇ ′ V ( r ′ ) · ˆ v ] × ∂ Im χ ( r , r ′ , ω ) ∂ω (cid:12)(cid:12)(cid:12) ω =0 d r d r ′ , (14)where v is the velocity of the atom, and, comparing withEq. (8), we can write ρ = n i Q/ ( e ¯ n ) . (15)We point out that the relation (15) quite generally holds within the many-body theory and is a stronger statementthan ρ = n i Q / ( e ¯ n ) , which is a simple consequence ofEq. (1) and the corresponding single-particle result forthe friction coefficient Q = ¯ n k F σ tr ( k F ) [27].In conclusion, we have developed the non-adiabatictime-dependent density functional formalism for a sys-tematic calculation of the dc residual resistivity of met-als with impurities. The contribution to the resistivityarising from the many-body interactions has been ex-pressed through the dynamical exchange and correlationkernel f xc . We have shown that all the dynamical effectsof the electron-electron interaction are contained in thefrequency dependence of f xc . 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