Ultra-nonlocality in density functional theory for photo-emission spectroscopy
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Ultra-nonlocality in density functional theory for photo-emission spectroscopy
A.-M. Uimonen, G. Stefanucci,
2, 3, 4 and R. van Leeuwen
1, 4 Department of Physics, Nanoscience Center, University of Jyv¨askyl¨a, Survontie 9, 40014 Jyv¨askyl¨a,Finland Dipartimento di Fisica, Universit´a di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Rome,Italy INFN, Laboratori Nazionali di Frascati, Via E. Fermi 40, 00044, Frascati,Italy European Theoretical Spectroscopy Facility (ETSF) (Dated: 26 April 2018)
We derive an exact expression for the photocurrent of photo-emission spectroscopy using time-dependentcurrent density functional theory (TDCDFT). This expression is given as an integral over the Kohn-Shamspectral function renormalized by effective potentials that depend on the exchange-correlation kernel of currentdensity functional theory. We analyze in detail the physical content of this expression by making a connectionbetween the density-functional expression and the diagrammatic expansion of the photocurrent within many-body perturbation theory. We further demonstrate that the density functional expression does not provideus with information on the kinetic energy distribution of the photo-electrons. Such information can, inprinciple, be obtained from TDCDFT by exactly modeling the experiment in which the photocurrent is splitinto energy contributions by means of an external electromagnetic field outside the sample, as is done instandard detectors. We find, however, that this procedure produces very nonlocal correlations between theexchange-correlation fields in the sample and the detector.PACS numbers: 31.15.E-, 31.15.ee, 31.15.eg
I. INTRODUCTION
The photo-electric effect, in which electrons are emit-ted from a material by applying light, has played an im-portant conceptual role in quantum mechanics. Alreadyin 1905 Einstein established the famous relation E K = ~ ω − Φwhere E K is the maximum kinetic energy of the emittedphoto-electrons, ~ ω the energy of the incoming photonsand Φ the work function of the material (which is equalto minus the chemical potential provided we use a gaugein which the potential is zero at infinity ). Presentlyphoto-emission spectroscopy is a well-developed tool forthe study of bandstructures and surface properties ofmaterials (for a review see e.g. Ref. [3]) in which,apart from the kinetic energy, also the angular distri-bution of the photo-electrons is measured. The photo-emission spectrum is closely related to the spectral func-tion of the material which can exhibit a wide rangeof many-body features such as quasi-particle broaden-ing and plasmon satellites. Furthermore there are so-called extrinsic effects describing the energy losses ofthe photo-electron within the material on its way tothe surface. The proper treatment of all these phe-nomena requires a many-body description. The under-lying theory has been described in a number of classicreferences . Although these many-body approachescan deal with complex many-body processes they arecomputationally expensive. One may therefore wonderwhether one could develop a computationally more ef-ficient approach based on density-functional theory .As the photo-emission process is a time-dependent phe- nomenon we need a time-dependent version of densityfunctional theory . Since the outgoing photocur-rent density j ( r , t ) is a key variable in time-dependentcurrent-density functional theory (TDCDFT) anapproach based on this formalism appears the mostpromising. In terms of the time-dependent many-bodystate | Ψ( t ) i the current density is given by j ( r , t ) = h Ψ( t ) | ˆ j p ( r ) | Ψ( t ) i + h Ψ( t ) | ˆ n ( r ) | Ψ( t ) i A ( r t ) (1)where A is the applied vector potential, ˆ n ( r ) is the den-sity operator andˆ j p ( r , t ) = 12 i X σ [ ˆ ψ † ( x ) ∇ ˆ ψ ( x ) − ∇ ˆ ψ † ( x ) ˆ ψ ( x )]is the paramagnetic current operator expressed in termsof the field operators ˆ ψ and ˆ ψ † , where x = r , σ is aspace-spin index. In TDCDFT this current density iscalculated instead from a Kohn-Sham state | Ψ s ( t ) i witha time-evolution determined by a non-interacting Kohn-Sham Hamiltonian ˆ H s ( t ). This Hamiltonian contains anexternal Kohn-Sham vector field A s (in a gauge where weabsorb the scalar potentials in a vector potential)whichis a functional of the current density. In this way thephoto-emission experiment can be modelled theoreticallyby time-propagation of Kohn-Sham orbitals after a suit-able approximation for the Kohn-Sham vector potential A s has been chosen. Indeed some first calculations ofthis kind have been carried out recently .There are, however, two issues that remain unresolved.The first issue is the question whether TDCDFT allowsfor the determination of the kinetic energy distribution ofthe photo-electrons. The second issue is what the qualityof the corresponding exchange-correlation kernels mustbe in order to account for many-body features such asplasmon losses. These are the two issues that we willaddress in this paper. The paper is divided as follows.In section II we briefly review the many-body approachto photo-emission where we stress the equations that arerelevant for the connections to density-functional theory.In Section III we give a description of TDCDFT in thelanguage of Keldysh theory in order to make connectionwith the standard many-body approaches. We furthergive a discussion of the calculation of the kinetic energydistribution and the related very long range nonlocalitiesthat are required in TDCDFT to calculate this propertyexactly. Finally in Section IV we give our outlook andconclusions. II. MANY-BODY THEORY OF PHOTO-EMISSIONA. The photocurrent
Here we will present a short overview of the many-body approach to photo-emission in which we highlightthe aspects relevant to the density functional treatment.We will follow the approach outlined by Almbladh . Weassume that the many-body system is described by aHamiltonian of the formˆ H ( t ) = ˆ H + ˆ∆( t )where ˆ∆ describes the electromagnetic field applied fortimes t > t and ˆ H is the many-body Hamiltonian of thesample before the field is applied. The time-evolution ofthe many-body state is described by the time-dependentSchr¨odinger equation i∂ t | Ψ( t ) i = ˆ H ( t ) | Ψ( t ) i with initial condition | Ψ( t ) i = | Φ i . We take | Φ i to bethe ground state of ˆ H , i.e. , ˆ H | Φ i = E | Φ i where E isthe ground state energy. If we know the state | Ψ( t ) i thenwe can calculate all observables of interest. In the case ofphoto-emission the observable of interest is the currentdensity outside the sample which describes the emissionof photo-electrons. This amounts to the calculation ofa one-body observable. In non-equilibrium many-bodytheory such observables can be calculated directly fromthe lesser Green’s function defined as G < ( x t, x ′ t ′ ) = i h Φ | ˆ ψ † H ( x ′ t ′ ) ˆ ψ H ( x t ) | Φ i where ˆ A H ( t ) = ˆ U ( t , t ) ˆ A ˆ U ( t, t ) is the Heisenberg formof the operator ˆ A with respect to the full Hamiltonianand ˆ U is the evolution operator of the system which ingeneral is a time-ordered exponential. The current canthen be calculated from j p ( r , t ) = − X σ ( ∇ − ∇ ′ ) G < ( x t, x ′ t ′ ) | x = x ′ . (2) Here we concentrate on the paramagnetic part of the cur-rent as we will see that the diamagnetic part only con-tributes to higher order in the applied field. Let us seewhat we get if we expand in powers of the electromag-netic coupling ˆ∆. To do this we first expand the time-dependent many-body state in powers of ˆ∆ | Ψ( t ) i = ∞ X n =0 | Ψ ( n ) ( t ) i where | Ψ ( n ) i is the n -th order term. In particular | Ψ (0) ( t ) i = e − iE ( t − t ) | Φ i . The k -th order term in theexpectation value for the current is then given by j ( k ) ( r , t ) = X n + m = k h Ψ ( n ) ( t ) | ˆ j p ( r ) | Ψ ( m ) ( t ) i . (3)If we are interested in the photo-emission current outsidethe sample then any term with m = 0 or n = 0 does notcontribute since | Ψ (0) ( t ) i is localized to the sample in po-sition space and vanishes exponentially outside the sam-ple. The diamagnetic current n ( r t ) A ( r t ) is even smallersince the lowest order contribution not involving | Ψ (0) ( t ) i is third order in the applied field. The lowest order non-zero contribution to the photocurrent is therefore givenby j (2) ( r , t ) = h Ψ (1) ( t ) | ˆ j p ( r ) | Ψ (1) ( t ) i . (4)The other two lowest order terms h Ψ (2) ( t ) | ˆ j p ( r ) | Ψ (0) ( t ) i and h Ψ (0) ( t ) | ˆ j p ( r ) | Ψ (2) ( t ) i contributing to j (2) are zerosince we are assuming the photocurrent to be measuredfar outside the sample.The calculation of j (2) requires the knowledge of thefirst order change in the many-body state upon applica-tion of the field. This is readily calculated to be | Ψ (1) ( t ) i = − i Z tt dt ′ ˆ U ( t, t ′ ) ˆ∆( t ′ ) ˆ U ( t ′ , t ) | Φ i where ˆ U ( t, t ′ ) = e − i ˆ H ( t − t ′ ) is the time-evolution opera-tor of the unperturbed system. Using this expression wecan write the photocurrent as j (2) ( r , t ) = Z tt dt dt h Φ | ˆ∆ H ( t )ˆ j p ,H ( r t ) ˆ∆ H ( t ) | Φ i (5)where the operators are now in the Heisenberg represen-tation with respect to ˆ H . This is the starting expres-sion for all our considerations. In many-body perturba-tion theory this expression is expanded in powers of themany-body interactions and can be represented as a di-agrammatic series. To do this it will be convenient todefine the equal-time lesser Green’s function (or equiva-lently the one-particle density matrix) to second order inthe external perturbation as G (2) < ( x t, x ′ t ) = i Z tt dt dt h Φ | ˆ∆ H ( t ) ˆ ψ † H ( x ′ t ) ˆ ψ H ( x t ) ˆ∆ H ( t ) | Φ i (6)as the Green’s function has a well-known expansion inFeynman diagrams. B. Diagrammatic expansion
To expand Eq. (6) into diagrams it will be convenientto write it as follows G (2) < ( x t, x ′ t ) = i Z tt dt dt h Φ | ˆ∆ + ( t ) ˆ ψ † ( x ′ ) ˆ ψ ( x ) ˆ∆ − ( t ) | Φ i where we definedˆ∆ − ( t ) = ˆ U ( t, t ) ˆ∆( t ) ˆ U ( t , t ) , ˆ∆ + ( t ) = ˆ U ( t , t ) ˆ∆( t ) ˆ U ( t , t ) . The operator ˆ∆ − ( t ) can now be expanded in time-ordered powers of the many-body interaction, whereasˆ∆ + ( t ) can be expanded in anti-time-ordered powers ofthe interaction. Since it will not be our goal to a givean overview of many-body theory we restrict ourselvesto the minimum which is required for understanding theconnections to density functional theory. Within the lan-guage of Keldysh many-body theory we can say thatthe operator ˆ∆ − is situated on the forward branch of theKeldysh contour whereas ˆ∆ + is situated on the backwardbranch. This leads to an expansion of G (2) < in terms ofthe non-interacting Greens’ functions G −− ( x t, x ′ t ′ ) = − i h χ | T [ ˆ ψ H ( x t ) ˆ ψ † H ( x ′ t ′ )] | χ i G ++ ( x t, x ′ t ′ ) = − i h χ | ˜ T [ ˆ ψ H ( x t ) ˆ ψ † H ( x ′ t ′ )] | χ i G − + ( x t, x ′ t ′ ) = i h χ | ˆ ψ † H ( x ′ t ′ ) ˆ ψ H ( x t ) | χ i G + − ( x t, x ′ t ′ ) = − i h χ | ˆ ψ H ( x t ) ˆ ψ † H ( x ′ t ′ ) | χ i where T is the time-ordering operator, ˜ T is a the anti-time-ordering operator and | χ i is the ground state ofthe non-interacting system and the operators are in theHeisenberg picture with respect to the noninteractingsystem. The Green’s functions G − + and G + − are equiv-alently denoted as G < and G > . The vertices in the dia-grams are labeled by − or + depending on whether theylie on the forward ( − ) or backward (+) branch of theKeldysh contour. The bare Coulomb interactions willbe denoted by wiggly lines and since these interactionsare instantaneous they will always connect times on thesame branch of the contour. Often the Green’s func-tion lines in the diagrams are dressed by self-energy in-sertions such that we can expand in skeleton diagrams( i.e. , diagrams with self-energy insertions removed) butwith dressed Green’s function lines. Similarly the inter-actions are often dressed to become screened interactions W which now can connect vertices on different branchesof the contour. Since the Green’s function G (2) < hasthe same time on the ingoing and outgoing vertex the G ( xt , x't ) = (2)< −+−+ t t ΔΔ x x' −+−+ + −− −+−+ + ++−+−+ + −+ −+−+ + + − −+−+ + +−−+−+ + −− −+−+ + ++ + .... (a) (b) (c)(d) (e) (f)(g) (h) FIG. 1. Skeleton expansion of G (2) < in G and W to the firstorder in W . Green’s function lines are commonly drawn closed backupon themselves to form triangles. In Fig. 1 we show theskeleton diagram expansion of G (2) < to lowest order inthe screened interaction W and the dressed Green’s func-tion G . Diagrams (a) - (c) are so-called no-loss diagramswhereas diagrams (d) - (f) describe energy losses of thephoto-electron while leaving the sample. Diagrams (g)and (h) describe the renormalization of the photon-fieldinside the sample. For a more in-depth discussion werefer to References [4] and [6]. C. Spectral representation of the photocurrent
Let us study the lowest order diagram in W of Fig. 1.The structure of this diagram will also be relevant for thedensity functional case. It has the explicit form G (2) < ( x t, x ′ t ) = − Z t −∞ dt dt × h x | ˆ G −− ( t, t ) ˆ∆( t ) ˆ G − + ( t , t ) ˆ∆( t ) ˆ G ++ ( t , t ) | x ′ i (7)where the minus sign originates from integration on the+ / − branch and we used the convenient notation G αα ′ ( x t, x ′ t ′ ) = h x | ˆ G αα ′ ( t, t ′ ) | x ′ i . Now the lesser Green’s function h x | ˆ G < | y i vanishes forspatial coordinates outside the sample as it depends onlyon the occupied states of the unperturbed system. Wecan therefore write in our case thatˆ G −− ( t, t ′ ) = θ ( t − t ′ ) ˆ G > ( t, t ′ ) = ˆ G R ( t, t ′ ) , ˆ G ++ ( t, t ′ ) = θ ( t ′ − t ) ˆ G > ( t, t ′ ) = − ˆ G A ( t, t ′ ) , where the retarded and advanced propagators are definedas ˆ G R ( t, t ′ ) = θ ( t − t ′ )[ ˆ G > − ˆ G < ]( t, t ′ ) , ˆ G A ( t, t ′ ) = − θ ( t ′ − t )[ ˆ G > − ˆ G < ]( t, t ′ ) . In terms of these propagators the expression (7) attainsthe form G (2) < ( x t, x ′ t ) = Z t −∞ dt dt × h x | ˆ G R ( t, t ) ˆ∆( t ) ˆ G < ( t , t ) ˆ∆( t ) ˆ G A ( t , t ) | x ′ i . (8)This expression is valid for general time-dependent per-turbations. Let us, however, restrict ourselves to a mono-chromatic perturbation for t > t of the formˆ∆( t ) = ˆ w e − i Ω t + ˆ w † e i Ω t = X ρ = ± ˆ w ρ e iρ Ω t where Ω > w − = ˆ w and ˆ w + = ˆ w † .Inserting this expression into Eq. (8) and assuming that t is very far into the past we then obtain that G (2) < ( x t, x ′ t ) = X ρ,η = ± e − i ( η + ρ )Ω t × Z dω π h x | ˆ G R ( ω + η Ω) ˆ w η ˆ G < ( ω ) ˆ w ρ ˆ G A ( ω − ρ Ω) | x ′ i (9)where we wrote the Green’s functions as Fourier trans-forms ˆ G ( t, t ′ ) = Z dω π e − iω ( t − t ′ ) ˆ G ( ω ) . We can now manipulate this expression in an expansionin terms of the free particle Green’s functions ˆ G R,A of thephoto-electron leaving the sample. After some manipu-lations which are presented in the Appendix we find thatoutside the sample the lesser Green’s function attains theform G (2) < ( x t, x ′ t ) = δ σσ ′ π | r || r ′ |× Z dω π e iq ( | r |−| r ′ | ) h ϕ q ˆ r | ˆ w † ˆ G < ( ω ) ˆ w | ϕ q ˆ r ′ i (10)where q / ω + Ω is the kinetic energy of the photo-electron and ˆ r = r / | r | is the unit vector pointing fromthe sample to the detector. If we further define q = q ˆ r then the state | ϕ q i is a quasi-particle state satisfying theequation [ˆ h + ˆΣ A ( ω + Ω)] | ϕ q i = ( ω + Ω) | ϕ q i (11)where ˆ h is the one-body part of ˆ H and ˆΣ A is the ad-vanced self-energy. We can now use Eq. (2) to calculatethe current density which gives j (2) ( r t ) = ˆ r π | r | F (ˆ r ) (12)where F (ˆ r ) = Z µ −∞ dω π q h ϕ q | ˆ w † ˆ A ( ω ) ˆ w | ϕ q i (13) where we neglected terms of order 1 / | r | , and used thefluctuation-dissipation relation ˆ G < ( ω ) = if ( ω − µ ) ˆ A ( ω )with f the Fermi function at chemical potential µ andˆ A ( ω ) = i [ ˆ G R ( ω ) − ˆ G A ( ω )] the spectral function.In the experiment one measures the flux of the currentthrough a space angle d ¯Ω J d ¯Ω = Z d ¯Ω j · d S through a spherical surface S of radius | r | . If we furtherdefine ǫ = q / ω + Ω to be the kinetic energy of thephoto-electron as a new variable, then we can write forthe current per space angle ∂J∂ ¯Ω (ˆ r ) = 14 π Z µ +Ω −∞ dǫ π √ ǫ h ϕ q | ˆ w † ˆ A ( ǫ − Ω) ˆ w | ϕ q i . (14)Now, in an experiment also the kinetic energy of thephoto-electron can be measured. In this way the pho-tocurrent can be split into energy contributions and wecan then write ∂ J∂ ¯Ω ∂ǫ (ˆ r ) = √ ǫ (2 π ) h ϕ q | ˆ w † ˆ A ( ǫ − Ω) ˆ w | ϕ q i . (15)By measuring both the direction and energy of the photo-electron the right hand side of this expression can bemeasured. III. DENSITY FUNCTIONAL THEORY FORPHOTO-EMISSIONA. Current density functional theory
In this section we give a short overview of the basicequations of TDCDFT and its connection to many-bodyperturbation theory. A much more detailed expositioncan be found in references [21] and [22]. So far we didnot specify the precise form of the perturbation. In gen-eral its form will be given by that of an electromagneticfield described by a time-dependent scalar potential anda vector potential A . However, we can always choose agauge where the time-dependent fields are absorbed in avector potential. Static potentials, such as the potentialsdue to atomic nuclei, may still be described in terms ofa scalar potential absorbed in ˆ H . If we do this we canwrite the perturbation asˆ∆( t ) = Z d r ˆ j p ( r ) · A ( r , t ) + 12 Z d r ˆ n ( r ) A ( r , t ) . We can then define a functional ˜ F [ A ] of the vector po-tential by ˜ F [ A ] = i ln h Φ | T γ e − i R γ dz ˆ H ( z ) | Φ i where T γ denotes contour ordering on the Keldysh con-tour γ with contour time z . This functional has thederivative δ ˜ Fδ A ( r , z ) = j p ( r , z ) + n ( r , z ) A ( r , z ) = j ( r , z )where j p is the paramagnetic current and j is the physicalgauge-invariant current. This physical current is the cen-tral object of time-dependent current-density-functionaltheory . We can make a current functional F [ j ] by aLegendre transform F [ j ] = − ˜ F [ A ] + Z γ d r dz j ( r , z ) · A ( r , z )which has the property δFδ j ( r , z ) = A ( r , z ) . The whole derivation did not depend on the specific formof the many-body interactions in ˆ H . The only thing thatwe assumed was that there is a one-to-one relation be-tween the physical current and the vector potential inour specific gauge given the initial state | Φ i . Wecould therefore have done the same derivation for a non-interacting system with initial state | Φ ,s i and obtaina current functional which we call F s [ j ]. We now as-sume that the functionals F [ j ] and F s [ j ] have the samedomain. We then define the exchange-correlation (xc)current functional F xc as F xc [ j ] = F s [ j ] − F [ j ] − F H [ j ] (16)where F H [ j ] = 12 Z d r d r ′ Z γ dz n ( r , z ) n ( r , z ) v ( r − r ′ )where v is the two-body interaction and where the density n ( r , z ) is regarded a functional of the current throughthe continuity equation. Differentiation of Eq. (16) withrespect to j gives A xc = A s − A − A H (17)where we defined A xc = δF xc /δ j and A H = δF H /δ j .The potential A s is the vector potential that for a non-interacting system gives the current density j . This sys-tem is called the Kohn-Sham system and A s will becalled the Kohn-Sham vector potential. If we take theinitial state | Φ s, i to be a Kohn-Sham ground state thenthe current can be calculated by solving the Kohn-Shamsingle-particle equations (cid:20) (cid:0) − i ∇ + A s ( r , t ) (cid:1) + v ext ( r ) (cid:21) φ j ( r , t ) = i∂ t φ j ( r , t ) . where v ext is the static external field of the unperturbedsystem. If we differentiate Eq. (17) with respect to j weobtain δA Hxc ,µ ( r , z ) δj ν ( r ′ , z ′ ) = δA s,µ ( r , z ) δj ν ( r ′ , z ′ ) − δA µ ( r , z ) δj ν ( r ′ , z ′ ) where A Hxc is the sum of the Hartree and xc vector po-tentials. The indices µ and ν label the three componentsof the vectors. The quantity on the left hand side isusually called the Hartree and xc kernel f Hxc which canbe split naturally into a Hartree part f H and an xc part f xc . The derivatives δA µ /δj ν represent the inverse of thecurrent-current response function given by χ µν ( r z, r ′ z ′ ) = δj µ ( r , z ) δA ν ( r ′ , z ′ ) = δ µν n ( r ) δ ( r − r ′ ) δ ( z, z ′ ) − i h Φ | T γ n ∆ˆ j p ,µ,H ( r , z )∆ˆ j p ,ν,H ( r ′ z ′ ) o | Φ i where the first part arises from the diamagnetic currentand the second from the paramagnetic one and we furtherdefined the current fluctuation operator by ∆ˆ j p ,µ,H =ˆ j p ,µ,H − h ˆ j p ,µ,H i . We have a similar response function χ s = δj/δA s for the Kohn-Sham system. From Eq. (18)we see then that we can write χ = χ s + χ s · f Hxc · χ where the dot product indicates a matrix product withrespect to the indices and integration over space-timevariables on the contour. This is the central equationof density functional theory for linear response . Ap-proximations for f Hxc can be found by expanding F xc indiagrams. Some explicit examples of this will be givenbelow. B. Photo-emission in current-density functional theory
The photocurrent within TDCDFT can be calculatedas j ( k ) s ( r , t ) = X n + m = k h Ψ ( n ) s ( t ) | ˆ j p ( r ) | Ψ ( m ) s ( t ) i (18)where were have expanded the Kohn-Sham state in pow-ers of the variation of the Kohn-Sham field [cfr. Eq. (3)].The current density of TDCDFT is exactly the sameas the current density of the real system and therefore j ( r , t ) = j s ( r , t ). Since we are measuring the photocur-rent far outside the sample and the initial Kohn-Shamstate is a Slater determinant of Kohn-Sham orbitals thatvanish exponentially outside the sample the terms with m = 0 and/or n = 0 do not contribute. Therefore, as insection II A where we expanded in powers of the physi-cal vector potential, the lowest order contribution in theKohn-Sham field to the photocurrent is j (2) s ( r , t ) = h Ψ (1) s ( t ) | ˆ j p ( r ) | Ψ (1) s ( t ) i . (19)Since the Kohn-Sham field A s [ A ] is to lowest order linearin A we have that j (2) s ( r , t ) = j (2) ( r , t ) + O ( A ). Thedifference with Eq. (4) is that the initial state | Φ i is theKohn-Sham initial state | Φ ,s i and that the perturbationˆ∆ is replaced by a Kohn-Sham perturbation ˆ∆ s . Since AA = + ss μ AA μ χ μ + χ μ + χ μ + χ μ f xc f xc FIG. 2. Diagrammatic expansion for the photocurrent in TD-CDFT to linear order in f Hxc and χ (see also Fig. 1). Theexternal vertex has a label µ corresponding to the action of acurrent operator and the Hartree kernel f H is indicated by awiggly line. there are no many-body interactions in the Kohn-Shamsystem the diagrammatic form of the current is simplygiven by the left hand side diagram in Fig. 2.To write this diagram in terms of the applied field A we need to expand the Kohn-Sham field A s in terms of A . To lowest order this gives A s,µ (1) = X ν Z γ d K µν (1 , A ν (2) (20)where K µν (1 ,
2) = δA µ,s (1) δA ν (2)= δ µν δ (1 ,
2) + X ρ Z γ d f Hxc ,µρ (1 , χ ρν (3 ,
2) (21)and we used the short notation j = r j , z j . This expres-sion can be written diagrammatically as in Fig. 3. A = s μ A + χ f Hxc μ A μ ν ρ FIG. 3. Diagrammatic representation of the Kohn-Sham field A s in terms of the applied filed A (see Eq. (20)). When we insert this diagrammatic representation inthe diagrams for the current we obtain the graphical ex-pansion on the right hand side of Fig. 2, in which we onlydisplayed terms up to linear order in f Hxc and χ . Wenote that this gives a rather different expansion than theone that we found for the expressions in many-body the-ory. In particular we see that all the exchange-correlationcontributions to the current amount to an effective renor-malization of the photon field as there are no terms con-necting the different legs of the triangle. Only the dia-grams (g) and (h) in Fig. 1 have a direct correspondencewith the second and third diagram after the equal signin Fig. 2 as both represent a renormalization due to theHartree field.Let us express the Kohn-Sham current in a frequencydependent form. If we take the external vector potential to be of the monochromatic form A ( r , t ) = a ( r ) e − i Ω t + a ∗ ( r ) e i Ω t then we can writeˆ∆( t ) = ˆ w e − i Ω t + ˆ w † e i Ω t where we neglected terms of order A and definedˆ w = Z d r ˆ j p ( r ) · a ( r ) . Within linear response also the Kohn-Sham field has thisform A s ( r , t ) = a s ( r , Ω) e − i Ω t + a ∗ s ( r , Ω) e i Ω t where a µ,s ( r , Ω) = X ν Z d r ′ K Rµν ( r , r ′ ; Ω) a ν ( r ′ ) . and where K Rµν (Ω) is the Fourier transform of the re-tarded component of K µν evaluated at the photon fre-quency Ω. Then if we defineˆ w s = Z d r ˆ j ( r ) · a s ( r , Ω)we have that the function F (ˆ r ) of (13) has the followingexpression in DFT F (ˆ r ) = Z µ −∞ dω π q h ϕ s, q | ˆ w † s ˆ A s ( ω ) ˆ w s | ϕ s, q i . (22)Here ˆ A s ( ω ) is the Kohn-Sham spectral functionˆ A s ( ω ) = 2 π X j | φ j ih φ j | δ ( ω − ǫ j ) (23)where ǫ j and | φ j i are the Kohn-Sham one-particle ener-gies and eigenstates and the photo-electron orbital | ϕ s, q i satisfies the equationˆ h s | ϕ s, q i = q | ϕ s, q i with incoming plane wave boundary condition. Here ˆ h s is the one-body Kohn-Sham hamiltonian of the unper-turbed system. Since the highest occupied Kohn-Shamorbital energy is equal to minus the ionization energy, ǫ N = − I , (provided we choose a gauge where the po-tential is zero at infinity, see Ref. [23]) we have µ = − I and therefore µ is the same for the true and the Kohn-Sham system. If we insert the expression for the spectralfunction operator into Eq. (22) we find that F (ˆ r ) = X ǫ j ≤ µ q j |h φ j | ˆ w s | ϕ s,q j ˆ r i| where we defined ǫ j + Ω = q j /
2. This is an exact al-ternative expression for F (ˆ r ) of Eq. (13). To expose itsphysical content we have to study explicit approxima-tions to the xc-kernel f xc . This will be done in the nextsection using diagrammatic expansions. A xc μ ν μ = Σ xc FIG. 4. Diagrammatic representation of the integral equationfor A xc (see Eq. (25)). C. Diagrammatic approximations for f xc We will give here a brief discussion of the diagram-matic expansion of f xc24–32 . The starting point of thediscussion is the equation δF xc δA s,µ (1) = X ν Z γ d δF xc δj ν (2) δj ν (2) δA s,µ (1)= X ν Z γ d χ s,µν (1 , A xc,ν (2) (24)where we used the symmetry in µ and ν of χ s . We nowassume that F xc is given by an expansion in Kohn-ShamGreen’s functions G s . Then the left hand side can bewritten as δF xc δA s,µ (1) = Z γ d d δF xc δG s (2 , G s (2 , j µ ( r ) G s (1 , δG s (2 , δA s,µ (1) = G s (2 , j µ ( r ) G s (1 , j µ ( r ) = 12 i ( −→ ∂ µ − ←− ∂ µ ) . If we call δF xc δG s (2 ,
3) = Σ xc (3 , X ν Z γ d χ s,µν (1 , A xc ,ν (2) == Z γ d d xc (3 , G s (2 , j µ ( r ) G s (1 ,
3) (25)which has the structure of a linearized Sham-Schl¨uterequation as displayed in Fig. 4. If we differentiatethis equation once again with respect to A s we obtain anintegral equation for f xc given by X νλ Z γ d d χ s,µν (1 , f xc ,νλ (2 , χ s,λκ (4 , Q µκ (1 , − Z γ d χ (2) s,µνκ (1 , , A xc ,ν (2) (26) μ f xc λν κ − μ κ Q μ = A xc κ − μ A xc κ FIG. 5. Diagrammatic representation for the integral equa-tion for f xc (see Eq. (26)). where we defined Q µκ (1 ,
3) = δ F xc δA s,µ (1) δA s,κ (3) (27)as well as the second order Kohn-Sham response function χ (2) s,µνκ (1 , ,
3) = δχ µν (1 , δA s,κ (3) . The corresponding integral equation for f xc is displayeddiagrammatically in Fig. 5. The diagrammatic structureof Eq. (26) has been studied in detail in Ref. 29 in whichexplicit diagrammatic expansions were derived from aLuttinger-Ward functional. For the case of the simpleexchange approximation to Σ xc , for instance, we haveΣ xc (1 ,
2) = − iv (1 , G s (1 ,
2) (28)where v (1 ,
2) = δ ( z, z ′ ) v ( r − r ) is the bare many-bodyinteraction. The corresponding diagrammatic expressionfor Q µκ is displayed in Fig. 6. A more advanced ap-proximation will be discussed below. But before we dothat we first discuss the diagrammatic expansion of thephotocurrent within TDCDFT. FIG. 6. Exchange-only approximation to Q µκ . D. Diagrams for the photocurrent
We have seen that we can derive approximate expres-sion for the xc-kernel of TDCDFT on the basis of many-body perturbation theory. A natural question to ask atthis point would be how to relate these expressions tothe Feynman diagrams for the photocurrent derived di-rectly from many-body theory, such as the diagrams dis-played in Fig. 1. The situation is complicated by thefact that we do not have a direct diagrammatic expres-sion of f µν, xc but rather one that is convoluted with twoKohn-Sham response functions as in Eq. (26). This is a + = − δ Q δ A − s − − g xc − A xc − A xc − A xc − FIG. 7. Diagrammatic expansion of the photocurrent ob-tained from differentiating the integral equation of the xc-kernel and integrating with the external fields. consequence of the fact that we are working in the zero-temperature formalism where the memory of initial cor-relations and initial-state dependence is lost. This allowsus to work with time-ordered quantities that depend onthe time-difference only and, therefore, can be Fouriertransformed. However, as it was first realized by Mearnsand Kohn and recently discussed by Hellgren and vonBarth , there are frequencies at which χ s is not invert-ible, thus preventing a direct diagrammatic expansion of f xc to be insert into the diagrams of Fig. 2. In order to generate three-point diagrams, we can dif-ferentiate Fig. 5 another time with respect to A s . Ifwe do this and collect our results we find an expressionwhich we display graphically in Fig. 7, where after differ-entiation we integrated two of the external vertices withthe external field ˆ∆( t ). In this expression we defined thehigher-order xc-kernel g xc by g xc ,µντ (1 , ,
3) = δf xc ,µν (1 , δj τ (3) . The appearance of g xc is not surprising given the factthat the photoemission problem is nonlinear in the ex-ternal field. The first filled triangle on the right handside of Fig. 7 represents half of the derivative δQ/δA s .For the exchange-only approximation these diagrams (in-tegrations with the external field) are shown in Fig. 8.The last two diagrams in Fig. 7 are exponentially smalloutside the sample as they contain the response functionwith a coordinate in the position of the detector.Before continuing our analysis we observe that in theproximity of the sample the last two diagrams in Fig. 7are not the only contributions to add to the photocurrent.In fact, the photocurrent has a first-order contribution aswell j (1) s (1) = h Ψ (0) s ( t ) | ˆ j p ( r ) | Ψ (1) s ( t ) i + c.c. = Z d χ s (1 , δ A s (2) (29)which can be discarded only for | r | → ∞ . Let us ex-pand this equation up to second order in the true external FIG. 8. The exchange-only graphs contribution to the pho-tocurrent. field A . We have δA s,µ (1) = X ν Z δA s,µ (1) δA τ (2) δA τ (2) d
2+ 12 X τρ Z δ A s,µ (1) δA τ (2) δA ρ (3) δA τ (2) δA ρ (3) d d δ A s,µ (1) δA τ (2) δA ρ (3)= X ζη Z d d δ A s,µ (1) δj ζ (4) δj η (5) δj ζ (4) δA τ (2) δj η (5) δA ρ (3)+ X ζ Z d δA s,µ (1) δj ζ (4) δ j ζ (4) δA τ (2) δA ρ (3)= X ζη Z d d g xc ,µζη (1 , , χ ζτ (4 , χ ηρ (5 , X ζ Z d f Hxc ,µζ (1 , χ (2) ζµτ (4 , , . (31)By inserting this back into the equation (30) and theninto Eq. (29) we obtain diagrams with the same structureas the last two diagrams of Fig. 7 but now the exactresponse function appears. Replacing the exact χ with χ s we see that the diagram with g xc cancels out whereasthe diagram with f xc is halved. As Eq. (31) contains f Hxc = v + f xc we also get a diagram like the last diagramof Fig. 7 in which f xc is replaced by the interaction v .In the many-body treatment this term arises from theexpansion of j (1) (1) too.Let us continue our analysis of the nonvanishing dia-grams for the photocurrent outside the sample. From theexample of Fig. 8 we see that the functional derivative of Q yields diagrams with interaction lines connecting dif-ferent legs of the triangle. If we insert these diagramsback into Fig. 7 and then into Fig. 2 we recover the ex-pansion at the exchange-only level of the photocurrent(see Fig. 1 with W → v ) provided we use χ s instead of χ (this is justified since the expansion is first order in theinteraction). The second and third diagram of Fig. 2 areproduced by a change in the Hartree field and are alsonaturally included in a lowest order many-body expan-sion in the bare interaction. If we want to compare to FIG. 9. Diagrams for Q µκ in a GW -type approximation for f xc . The wiggly lines denote screened interactions. the many-body diagrams of Fig. 1 where the interactionis screened then we also need an approximation to f xc interms of W . To lowest order in W this approximationcan be derived from the GW self-energyΣ xc (1 ,
2) = − iG s (1 , W (1 , , (32)where the screened interaction W is the solution of W (1 ,
2) = v (1 ,
2) + Z d d v (1 , P (3 , W (4 , P given by P (1 ,
2) = − iG s (1 , G s (2 , . The corresponding equation for Q is illustrated diagram-matically in Fig. 9. Such an expression was studied inreference from a Luttinger-Ward functional . Thediagrams also include two terms which are second or-der in W and are important to include if one insistson having a conserving approximation . It assures, forinstance, that the f xc satisfies the linearized zero-forcetheorem which states that the exchange-correlationfields do not exert a net force on the system. By a dif-ferentiation of the corresponding function Q and integra-tion with the external fields we obtain the diagrams thatcontribute to the photocurrent. These are displayed inFig. 10 and have the same structure as in Fig. 1. We rec-ognize all diagrams (a) - (f) of this figure. The only differ-ence is that we here still integrate over the two branchesof the Keldysh contour. We also note that we have somediagrams with self-energy insertions. This is because westill expand in terms of Kohn-Sham Green’s functionsrather than the fully dressed ones. The diagrams (g) and(h) of Fig. 1 which describe the change in the effectiveHartree field are not included in Fig. 10 since they arealready absorbed in the Hartree part f H of f Hxc and arerepresented by the second and third diagram after theequal sign in Fig. 2. The remaining diagrams in Fig. 10describe processes that are higher order in W . Such di-agrams would also appear in a many-body treatment ifwe had expanded to higher order in the screened inter-actions. In photo-emission from metallic systems theywould, for instance, describe processes in which there aremultiple excitations of plasmons present. . We haveseen that within TDCDFT we can make a clear connec-tion between the many-body expansion for the photocur-rent and the Kohn-Sham expression for it. The question FIG. 10. Diagrams for the photocurrent derived from a GW -type approximation for f xc . The wiggly lines describescreened interactions. that remains to be answered is whether knowledge ofthe photocurrent provides us with enough informationto calculate the kinetic energy distribution of the photo-electrons. This question will be addressed in the nextSection. E. Ultra-nonlocality
In the transition from Eq. (14) to (15) we made a con-ceptual step. The total angular distribution of the pho-tocurrent was written as an integral over separate con-tributions from the electron kinetic energies ǫ . This steprequires a physical interpretation since it is not justifiedfrom a mathematical point of view. The expression can,however, be derived alternatively from Fermi’s GoldenRule applied to the many-body system which amountsto a calculation of transition rates between many-bodystates. We could apply the same procedure to the Kohn-Sham system but then we would calculate transitionsbetween Kohn-Sham Slater determinant states ratherthan between physical states. The corresponding for-mula would be given by removal of the integral sign inEq. (22) after the substitution ω = ǫ − Ω. This gives ∂ J∂ ¯Ω ∂ǫ (ˆ r ) = √ ǫ (2 π ) h ϕ s, q | ˆ w † s ˆ A s ( ǫ − Ω) ˆ w s | ϕ s, q i . (33)It is not difficult to see that we would run into a contra-diction if we assumed that the right hand side of thisequation would be identical to the right hand side ofEq. (15). This becomes clearer when we insert in Eq. (33)the explicit form of the Kohn-Sham spectral function ofEq. (23) ∂ J∂ ¯Ω ∂ǫ (ˆ r ) = √ ǫ (2 π ) X ǫ j ≤ µ |h φ j | ˆ w s | ϕ s, q i| δ ( ǫ − ǫ j − Ω) . (34)0 Ω detector FIG. 11. Deflection of two different kinetic energy compo-nents of the current by a field in the detector.
If we took the example of a finite system then the spec-trum on the righthand side of the equation would onlyhave peaks at the Kohn-Sham energies, whereas the ex-pression (15) has peaks at the true removal energies ofthe system. We conclude that Eq. (34) is not the same asEq. (15) but that only the integrals over these functionsup to ǫ = µ + Ω are the same. While this is apparent fora finite system for an infinite system the spectral peaksmerge into a continuum and then it is not immediatelyobvious that the two expressions are different. However,there is no reason to assume that they are equal as the in-terpretation based on Fermi’s Golden Rule demonstrates.We therefore conclude that the kinetic energy distribu-tion cannot be directly calculated from knowledge of thecurrent-density. This is mathematically clear since themomentum distribution requires knowledge of the one-particle density matrix which is no simple functional ofthe current density. However, in the experiment the ki-netic energy is, in fact, measured by measuring the cur-rent at various positions in the detector. This is doneby deflecting the photo-electron current with an appliedelectric or magnetic field , as depicted graphically inFig. 11. Here we display the detection of different kineticenergy components in the current. To every position inthe detector plate there is assigned a corresponding ki-netic energy. This detection process could be modeledin TDCDFT as well. There exists an effective Kohn-Sham field A s in the region of the detector which wouldbend the path of the currents in exactly the same way asthe true electromagnetic field in the detector. Thereforethese kinetic energies could, in principle, also be mea-sured in a Kohn-Sham approach. However, we realizethat such a field must have knowledge of the true many-body spectral function in the sample in order to splitthe current in exactly the right way to produce peaks inthe kinetic energy spectrum where the Kohn-Sham sys-tem has none. This means that the exchange-correlationfield in the detector far away from the sample (in factat a macroscopic distance in a real experiment) must de-pend in a nontrivial way on the many-body correlations in the sample. This is another illustration of extremenonlocality of the exchange-correlation field for which wecan find several other instances in density-functional the-ory. Other examples are the step structures in chargetransfer processes in molecules , the macroscopicexchange-correlation field of molecular chains and thelead-dependence of the exchange-correlation potential inquantum transport . IV. CONCLUSIONS
We derived an exact expression within TDCDFT forthe photocurrent of photo-emission spectroscopy. Thisexpression involves an integral over the Kohn-Shamspectral function weighted with effective Kohn-Shamone-body interactions. Although this expression directlygives the angular dependence of the photocurrent itdoes not provide us directly with the kinetic energydistribution of the photo-electrons. This information canbe obtained from TDCDFT as well, but there is a priceto be paid for this. In order to do it we need to split thephotocurrent into various kinetic energy distributionsusing an external exchange-correlation field outside thesample which depends in a very nonlocal manner on themany-body states inside the sample.From a practical point of view we may wonder whetherthe derived expression of Eq. (34) could represent a suffi-ciently accurate, albeit non-exact, approximation to thekinetic energy distribution of the photo-electron spec-trum. This probably depends highly on the studiedsystem in question. For instance, for photo-emission ofmetallic systems the plasmon excitations are an impor-tant physical ingredient. Diagrammatically these plas-monic effects are incorporated well in terms of Green’sfunctions based on the GW approximation. It may wellbe that an xc-kernel based on a Sham-Schl¨uter scheme atthis level would give the required features in the photo-electron spectrum. These features then would come out,not by creating extra levels in the spectral functions, butby a redistribution of the intensities of the bare Kohn-Sham spectral function by the matrix elements involvingthe xc-kernel. The most difficult case for TDCDFT ismaybe provided by finite systems, such as molecules, inwhich non-trivial doubly or multiple excited states may contribute important features to the spectral func-tion.
ACKNOWLEDGMENTS
AMU would like to thank the Alfred Kordelin Founda-tion for support. RvL would like to thank the Academy ofFinland for support. GS acknowledges funding by MIURFIRB Grant No. RBFR12SW0J and financial supportthrough travel grants Psi-K2 5813 of the European Sci-ence Foundation (ESF).1
Appendix A: Derivation of Eq. (10)
We will in this Appendix give a derivation of Eq. (10).We define the retarded Green’s function ˆ G R for a freeparticle outside the sample as( i∂ t − ˆ t ) ˆ G R ( t, t ′ ) = δ ( t − t ′ )where ˆ t is the kinetic energy of a free particle. The re-taded Green’s function of the sample satifies( i∂ t − ˆ h ) ˆ G R ( t, t ′ ) = δ ( t − t ′ ) + Z d ¯ t ˆΣ R ( t, ¯ t ) ˆ G R (¯ t, t ′ )where ˆ h = ˆ t + ˆ v where ˆ v is the confining potential for theelectrons in the sample (the potential due to the atomicnuclei) and ˆΣ R is the retarded many-body self-energy.Then we can write the Green’s function of the sample inDyson form as ˆ G R = ˆ G R + ˆ G R (ˆ v + ˆΣ R ) ˆ G R where integrations over internal time variables are im-plied. If we now define the retarded T -matrix ˆ T R byˆ T R = ˆ v + ˆΣ R + (ˆ v + ˆΣ R ) ˆ G R ˆ T R then we can writeˆ G R = ˆ G R + ˆ G R ˆ T R ˆ G R . If we introduce the short notationsˆ X Rη = (1 + ˆ T R ˆ G R )( ω + η Ω)ˆ X Aη = (1 + ˆ G A ˆ T A )( ω + η Ω)then we can rewrite Eq. (9) as G (2) < ( x t, x ′ t ) = X ρ,η = ± e − i ( η + ρ )Ω t Z dω π Z d y d y ′ × h x | ˆ G R ( ω + η Ω) | y ih y | ˆ X Rη ˆ w η ˆ G < ( ω ) ˆ w ρ ˆ X A − ρ | y ′ i× h y ′ | ˆ G A ( ω − ρ Ω) | x ′ i . (A1)Now the matrix element of ˆ G R has the explicit form h x | ˆ G R ( ν ) | y i = − δ σσ ′ π e i √ ν r r ν > e −√− ν r r ν < r = | r − r | with x = r , σ and y = r , σ ′ .Since ˆ G < ( ω ) has only contributions for ω ≤ µ and Ω > r → ∞ when the argument of ˆ G R in Eq. (A1) is ω +Ω.This implies that the integral becomes G (2) < ( x t, x ′ t ) = 14 π Z dω π Z d r d r e iq | r − r | | r − r |× h r , σ | ˆ X R ˆ w ˆ G < ( ω ) ˆ w − ˆ X A | r , σ ′ i e − iq | r − r | | r − r | (A2) where we defined q > q / ω + Ω.If we are looking at point r far from the sample then wecan use the approximation e iq | r − r | | r − r | ≈ e iq ( | r |− ˆ r · r ) | r | . If we define q = q ˆ r and the plane wave state | q , σ i with h r , σ | q , σ ′ i = δ σσ ′ e i q · r then we can write G (2) < ( x t, x ′ t ) = 14 π δ σσ ′ | r || r ′ | Z dω π e iq ( | r |−| r ′ | ) × h q ˆ r , σ | ˆ X R ˆ w ˆ G < ( ω ) ˆ w † ˆ X A | q ˆ r ′ , σ i (A3)where we used that the Green’s function must be diagonalin the spin indices. If we then further define the state | ϕ q ˆ r i = ˆ X A | q ˆ r , σ i = (1 + ˆ G A ˆ T A )( ω + Ω) | q ˆ r , σ i then the desired equation (10) follows immediately fromEq. (A3). It remains to give a more explicit characteri-zation of the state | ϕ q ˆ r i . It satisfies the equation | ϕ q ˆ r i = | q , σ i + ˆ v + ˆΣ( ω + Ω) ω + Ω − ˆ t − iη | ϕ q ˆ r i (A4)which represent an advanced solution of the Lippmann-Schwinger equation with incoming plane wave boundaryconditions. Equivalently we can write Eq. (A4) as[ˆ h + ˆΣ A ( q | ϕ q ˆ r i = q | ϕ q ˆ r i and we see that it equivalently satisfies a quasi-particletype equation for a continuum state. We have recoveredexactly Eq. (11). A. Einstein, Annalen der Physik , 132 (1905). N. D. Lang and W. Kohn, Phys. Rev. B , 1215 (1971). A. Damascelli, Z. Hussain and Z-X Shen, Rev. Mod. Phys. ,473 (2003). C.-O. Almbladh, Physica Scripta , 341 (1985). C.-O. Almbladh and L. Hedin, in
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