Finite temperature density functional theory investigation to the nonequilibrium transient warm dense state created by laser excitation
Hengyu Zhang, Shen Zhang, Dongdong Kang, Jiayu Dai, M. Bonitz
aa r X i v : . [ phy s i c s . a t o m - ph ] S e p Finite temperature density functional theory investigation to the nonequilibriumtransient warm dense state created by laser excitation
Hengyu Zhang, Shen Zhang,
1, 2, ∗ Dongdong Kang, Jiayu Dai, † and M. Bonitz Department of Physics, National University of Defense Technology, Changsha, Hunan 410073, China Institut f¨ur Theoretische Physik und Astrophysik,Christian-Albrechts-Universit¨at zu Kiel, Leibnizstraße 15, 24098 Kiel, Germany
We present a finite-temperature density functional theory investigation of the nonequilibriumtransient electronic structure of warm dense Li, Al, Cu, and Au created by laser excitation. Photonsexcite electrons either from the inner shell orbitals or from the valence bands according to the photonenergy, and give rise to isochoric heating of the sample. Localized states related to the 3d orbitalare observed for Cu when the hole lies in the inner shell 3s orbital. The electrical conductivityfor these materials at nonequilibrium states is calculated using the Kubo-Greenwood formula. Thechange of the electrical conductivity, compared to the equilibrium state, is different for the caseof holes in inner shell orbitals or the valence band. This is attributed to the competition of twofactors: the shift of the orbital energies due to reduced screening of core electrons, and the increaseof chemical potential due to the excitation of electrons. The finite temperature effect of both theelectrons and the ions on the electrical conductivity is discussed in detail. This work is helpful tobetter understand the physics of laser excitation experiments of warm dense matter.
PACS numbers: 52.27.Gr, 52.50.Jm, 71.15.Mb, 72.80.-r
I. INTRODUCTION
Warm dense matter (WDM), referring to the inter-mediate state between condensed matter and ideal plas-mas, has attracted numerous studies in recent years.It is not only because WDM bridges the gap betweenatomic physics, condensed matter physics and plasmasphysics, but also because these extreme WDM condi-tions are found in a wide range of applications [1, 2].WDM can be characterized by the ion coupling param-eter, Γ = e / (¯ rk B T ) ∼
1, the electron degeneracy pa-rameter Θ = k B T /E F ∼
1, and the quantum couplingparameter r s = ¯ r/a B ∼
1, where Γ is the ratio of theCoulomb energy to the thermal energy, Θ is the ratio ofthe thermal energy to the Fermi energy, and r s is theratio of the mean interparticle distance to the Bohr ra-dius, respectively. The WDM regime typically covers awide range of temperature from several thousand Kelvinto tens of millions of Kelvin, and a density range fromnear solid density to a thousand times of solid density.The properties of WDM, such as equation of state (EOS),thermal and electrical conductivity, etc., are of great im-portance to model the structure and understand the for-mation of celestial bodies [3–6], such as the earth, browndwarfs and giant planets. In addition, the deuterium-tritium fuel for inertial confinement fusion (ICF) passesthrough the WDM regime before ignition [7, 8].One of the widely used techniques to produce WDMin the laboratory is isochoric heating of a solid sample,e.g. [9], with the choice of available sources such as opticallasers [10–12], ion beams [13–15], or X-ray free electron ∗ [email protected] † [email protected] lasers (XFEL) [16–18]. With the development of intenseultrafast optical lasers, the intensity can be as high as10 W/cm and the pulse duration can be less than 50fs [19–22]. During the pulse, the laser energy is trans-ferred to the electrons via photoexcitation of valence elec-trons to the conduction band with negligible movement ofthe much heavier ions. In order to ensure spatial homo-geneity, the thickness of the target sample is usually verysmall, typically on the order of 10 nm. The intensity ofthe laser beam is usually on the order of 10 W/cm , forsuch isochoric heating experiments. Compared to highintensity ultrafast optical lasers, the XFEL has a muchhigher photon energy, which delivers the XFEL energy tothe sample via photoionization of core electrons. XFELradiation can also penetrate deeper into the sample (onthe order of 10 µ m), and the intensity of the beam canbe as high as 10 W/cm [23], allowing even severe de-pletion of certain inner shells [16, 24]. The holes thatare created by photoexcitation or photoionization willbe refilled via Auger decay or other collisional processes.However, the lifetime of such a vacancy is typically com-parable to the pulse duration (for example, the lifetimeof a hole in the L shell is estimated to be in the range of10 fs, for rare gas atoms [25], 40 fs for Al [26], and 34.5 fsfor Cu [27].), which is short compared with any electron-phonon coupling time (on a picosecond scale) [28]. As aresult, the isochoric heating technique can create a va-riety of nonequilibrium transient states of matter in thelaboratory including nonequilibrium electronic density ofstates and non-thermal melting, e.g. [29].On the other hand, theoretical investigation of sucha nonequilibrium transient state faces great challenges.Atomic kinetic simulations are widely used to interpretthe experimental results [30, 31], but the quality of theirparameters under such exotic conditions is not well exam-ined. Nonequilibrium Green functions are a powerful toolto study the ultrafast dynamics of scattering processes inatoms and condensed matter, e.g. [32–34], but the highcomputational cost allows to study only simple modelsystems. Time-dependent DFT (TD-DFT) is a feasibletheoretical tool for the investigation to WDM [35, 36] in-cluding the nonequilibrium dynamics, because it explic-itly solves the time-dependent Kohn-Sham equations toobtain the temporal evolution of the wave function [37].The nonequilibrium dynamics of the electronic structureobtained by TD-DFT breaks down into two relaxationchannels, i.e. the density of states reorganization and theredistribution of the electron number [38]. However, TD-DFT calculation is also very expensive and the knowledgeof its exchange-correlation functional is limited. In ad-dition, the dynamics such as Auger decay and electrontemperature cannot be treated directly in TD-DFT.Here we concentrate on a simpler and much fasterapproach – finite-temperature density functional theory(FTDFT). It does not rely on any fitting parameters fromthe experiment and has been widely used to study theproperties of WDM [39–41]. However, FTDFT is con-structed for the calculation of equilibrium states, and thepresence of a hole is usually accounted for in FTDFT bya specially designed pseudopotential with the configura-tion of an ionized atom [42]. The hole is, therefore, fixedin the frozen core of a particular atom serving as an im-purity in the system [43, 44].In this work, we have modified the original Kohn-Sham-Mermin scheme to allow for a nonequilibrium elec-tron distribution function within FTDFT to study tran-sient exotic states that were described above. With themanually introduced redistribution of the electron oc-cupations, this modified FTDFT treatment of nonequi-librium states has a much better efficiency comparedto the more advanced TD-DFT counterpart. The elec-tronic structures of the metallic systems of lithium, alu-minum and copper with a variable number of holes intheir inner shell orbitals are investigated theoretically bythe modified code, corresponding to the nonequilibriumtransient state immediately after photoionization by anXFEL pulse. For comparison, we also present the elec-tronic structure of copper and gold with variable numbersof holes in the valence band, which can be realized viaphotoexcitation by means of a high intensity ultrafastlaser. The electrical conductivity – a widely used diag-nostic technique for WDM [45] – is also calculated todemonstrate the effect of such holes on transport prop-erties. Clearly such a treatment neglects relaxation pro-cesses and gives, at best, only intermediate states of thesystem after excitation. Nevertheless, we expect thatthis approach allows one to study important trends ofnonequilibrium behavior which has been confirmed inexperiments [24]. Support for such a separation of re-laxation phases is also based on similar approaches inother fields including the three-step model of photoion-ization, e.g. [46] and the modeling of so-called hollowatoms where core electrons are removed by the impact ofa highly charged ion, e.g. [47]. The paper is organized as follows: in Sec. II, the the-oretical method together with the numerical details usedin this work are described. In Sec. III, results and dis-cussions are presented including the electronic structureand electrical conductivity of metallic systems with vari-able numbers of holes in both, inner shell and valenceband, and for electron temperatures of 300K and 1eV,respectively. Our conclusions are summarized in Sec. IV. II. METHODOLOGY AND NUMERICALDETAILS
In this section we describe the methodology and nu-merical details of this work. FTDFT calculations areperformed with the population of electrons manually con-trolled in the self-consistent field (SCF) cycle to createholes in different bands. Then the Kubo-Greenwood for-mula is used to calculate the electrical conductivity withthe wave functions obtained by the FTDFT calculation.
A. Finite temperature density functional theoryand molecular dynamics
Since the energy transfer between the electrons and theions is much slower than the evolution of the electron sub-system, a two-temperature system [48–50] can be used,where the electrons are described by FTDFT with a fixedion configuration. FTDFT calculations are performed us-ing the plane-wave-pseudopotential open-source package
Quantum ESPRESSO (QE) [51, 52] with minor mod-ifications to realize the manual control of the occupationnumbers. In the Kohn-Sham-Mermin scheme [53, 54], theelectrons are considered to be in thermal equilibrium andtheir occupation numbers follow the Fermi-Dirac distri-bution. In order to investigate the nonequilibrium statesuch as the one of the WDM system created by isochoricheating, we force the occupation numbers of the relevantorbitals to be zero in every SCF step, which manuallycreates holes in the inner shell orbitals or the valenceband. The total number of electrons in the calculationsystem remains unchanged by adding the same numberof electrons as that of holes to the chemical potential.Similar to the original SCF process, the new chemicalpotential is computed self-consistently, which, in return,determines the occupation numbers of all Kohn-Shamorbitals, except for those that were forced to zero. Bythis way, we can go beyond the frozen core assumptionand create holes in the valence band directly in our DFTcalculations, contrary to the common practice of creat-ing holes during the generation of pseudopotentials andtreating the corresponding atom as an impurity in theDFT calculation.We consider the metallic systems of lithium, alu-minum, copper and gold as illustrating examples for ournonequilibrium FTDFT calculations. Cubic boxes withperiodic boundary conditions are used for the 4 elementsmentioned above, and the system sizes are set to 128atoms for body-centered cubic (BCC) Li, and 108 atomsfor the remaining face-centered cubic (FCC) materials toguarantee the convergence with respect to finite size ef-fect. Except for the molecular dynamics calculations,the atoms are fixed in their perfect lattice configura-tion with varying electronic temperature. We use theprojector augmented wave (PAW) formalism [55] for theFTDFT calculation with the local density approxima-tion (LDA) [56] to the exchange-correlation interactionthroughout. Finite temperature effects in the exchange-correlation functional are examined by the comparisonwith the newly constructed GDSMFB functional [57].The minimum threshold of the occupation numbers is10 − , in all our calculations, to ensure the convergencefor the number of included bands.For Li, both K-shell and L-shell electrons (i.e., 1s )are treated as valence electrons, and the plane wave cut-off energy is set to be 100 Ry. For Al, we use a pseu-dopotential with both L-shell and M-shell electrons (i.e.,2s ) included as valence electrons with a planewave cutoff energy of 100 Ry. For Cu, two different pseu-dopotentials are applied for comparison. One includes 11electrons as the valence electrons (i.e., 3d ), and theother includes 19 electrons (i.e., 3s ). We use80 Ry for the plane wave cutoff energy, for the formerpseudopotential, and 120 Ry, for the latter one. 11 va-lence electrons (i.e., 5d ) are considered for the calcu-lation of Au, and the cutoff energy for plane wave basis is100 Ry. Unshifted Monkhorst-Pack K-point meshes areapplied for the sampling of the Brillouin zone [58], with4 × × × ×
8, for 11-valence-electrons Cu and Aufor the better description of the electronic structure nearthe chemical potential. The convergence with respect toboth plane wave cutoff energy and K point grid size iscarefully checked for all our calculations.To demonstrate the effect of the ionic structure on theelectrical conductivity, we also perform FTDFT calcu-lations combining with molecular dynamics (MD) simu-lations based on Born-Oppenheimer approximation [59].The FTDFT-MD calculation of Cu is carried out usinga pseudopotential with 11 valence electrons, as an illus-trating example. We apply a canonical (NVT) ensemble,where the ions, controlled by an Andersen thermostat,share the same temperature as the electrons. A timestep of 1 fs is used, and the system is thermalized formore than 1 ps to reach equilibrium before the ionic tra-jectories of the last 2000 MD steps are kept for electronicstructure calculation.
B. Kubo-Greenwood formula for electricalconductivity
The electrical conductivity calculations are performedby using the KGEC code [60]. The wave functions withholes in the system obtained in the previous modified FTDFT calculation are plugged into the KGEC codewith similar modification of manually controlled occu-pation numbers for the corresponding orbitals. Usingthe Kubo-Greenwood linear response formula, the realpart of the electrical conductivity above the direct cur-rent (DC) limit can be calculated as σ ( ω ) = 2 πe ~ m e Ω Σ k w k Σ nn ′ ∆ f n ′ k ,n k ∆ ǫ n k ,n ′ k h Ψ n k |∇| Ψ n ′ k i×h Ψ n ′ k |∇| Ψ n k i δ (∆ ǫ n k ,n ′ k − ~ ω ) , (1)which, near the DC limit, takes the following form σ ( ω ) = 2 πe ~ m e Ω ω Σ k w k Σ nn ′ ∆ f n ′ k ,n k h Ψ n k |∇| Ψ n ′ k ih Ψ n ′ k |∇| Ψ n k i δ (∆ ǫ n k ,n ′ k − ~ ω ) . (2)Here σ is the real part of the electrical conductivity asa function of frequency ω . The constants e , ~ , m e andΩ represent the electron charge, Planck’s constant, elec-tron mass, and cell volume, respectively. The numbers n and n ′ denote the band index. Together with the Bril-louin zone wave vector k they become a pair index forBloch states. The coefficient w k is the integration weightof the respective k-point, and Ψ n k is the wave function.∆ ǫ n k ,n ′ k = ǫ n k − ǫ n ′ k and ∆ f n ′ k ,n k = f ( ǫ n ′ k ) − f ( ǫ n k ) arethe differences of Kohn-Sham eigenvalues and nonequi-librium occupation numbers obtained by the previousFTDFT calculation, respectively. δ is the Dirac deltafunction, which is broadened (we use a Lorentzian form)in all our calculations. III. RESULTS AND DISCUSSIONSA. Electronic structure calculations
The electronic structure of Li, Al, Cu and Au witha nonequilibrium distribution of electrons is investigatedusing the modified FTDFT calculation scheme describedin Sec. II A corresponding to a transient exotic state cre-ated by the isochoric heating with XFEL or ultrashortoptical laser as the source. The samples are supposed tobe at ambient conditions, before the impact of the XFELor the optical laser pulse. This means an electron tem-perature of 300K for BCC Li at a solid density of 0.535g/cm , FCC Al at a solid density of 2.70 g/cm , FCC Cuat a solid density of 8.92 g/cm , and FCC Au at a soliddensity of 19.3 g/cm respectively, are used.First we consider the nonequilibrium electronic struc-ture created by the XFEL, using Li, Al and Cu as il-lustrating examples. XFEL sources have a tunable pho-ton energy with a pulse duration on the order of 10 fs,whereas the intensity can be as high as 10 W/cm forsuch isochoric heating experiments [24]. Li has a K edgeof 54.7 eV [61], which lies in the ultraviolet range acces-sible to free electron lasers such as FLASH, LCLS, etc. Figure 1. Density of states of (a) BCC Li, (b) FCC Al and(c) FCC Cu with a perfect lattice structure at ambient con-dition. The black solid lines show the calculated DOS withthe occupied DOS shown as shades. The positions of theirchemical potential are marked with the vertical dashed blacklines and labelled in the legends. The red lines represent thesame quantity as the black lines but are calculated with 10electrons removed from the inner shell orbitals (1s for Li, 2sfor Al, and 3s for Cu). For the blue lines, the number of suchholes is increased to 32. For a better display, the DOS forthe inner shell orbitals and for the valence band use differentscales corresponding to the left and right axes.
The L I edge of Al is 117.8 eV, and the M I edge of Cuis 122.5 eV, both of which are in the soft X-ray rangeof an XFEL. With a careful choice of the photon energy,one can create holes by predominant photoionization of1s electrons of Li, 2s electrons of Al, or 3s electrons of Cuto their Fermi level (or chemical potential for the finite temperature case) with other transitions being of minorimportance. Also, electrons excited to higher energiesare found to relax to the chemical potential within a fewfemtoseconds [24].We calculate the density of states (DOS) of suchnonequilibrium transient states, as plotted in Fig. 1. Theblack solid lines show the calculated DOS without holes,whereas the red ones correspond to the DOS in which10 electrons were removed from the inner shell orbitals(1s for Li, 2s for Al, and 3s for Cu), leaving a hole be-hind on each of 10 atoms. Even though the electrons donot leave the sample which remains neutral, those atomswith a hole can be considered “ionized” carrying a lo-calized positive charge. The blue lines show the DOSwith 32 holes in the system. The fraction of (singly) ion-ized atoms, for the case of Li is, therefore, 7.8%, for thered lines, and 25.0%, for the blue lines. For both alu-minum and copper this fraction becomes 9.2%, for thered lines, and 29.6%, for the blue lines. The DOS of theoccupied states are shown by the shaded areas to demon-strate the positions of the holes. The positions of theircorresponding chemical potentials are marked with ver-tical dashed color lines and their values are also labelledin the legends. The peak features of the DOS far belowthe chemical potential correspond to the contribution ofinner shell orbitals.There is a noticeable shift of these atom-like featurestowards lower energy, due to the presence of the holes.We attribute this blue shift to the reduced screening ofthe core electrons, which makes the Coulomb potentialthat the remaining electrons feel stronger [24]. Note thatthe shift of the orbital energies cannot be observed bythe conventional FT-DFT calculations within an impu-rity model, because the energy reference is ill-defined fordifferent pseudopotentials. Moreover, The shift of the 1sorbital for Li is 22.61 eV, which is substantially smallerthan the shift of 34.1 eV estimated by a separate DFTcalculation for an isolated atom using the atomic codeld1.x distributed with QE. The same is observed for alu-minum (a shift of 16.85 eV for the 2s level in a solid,as compared to a shift of 25.7 eV in an atomic calcula-tion). Finally, for copper a shift of 5.26 eV is found, forthe 3s level in the solid, as compared to a shift of 14.7eV, in an atomic calculation. Therefore, the frozen coreapproximation should be applied with caution for suchcalculations. Except for Li, the position of the chem-ical potential rises as the number of atoms with holesincreases, mainly as a consequence of the increasing num-ber of electrons (which equals the number of holes, sincethe system remains neutral) that are excited to the statesnear chemical potential. The increase of the number ofexcited electrons competes with the blue shift of orbitalenergies, resulting in the modification of band structure.As a result, the chemical potential first drops slightlywith the creation of ten 1s holes, and then rises with 32ionized atoms involved for Li (cf. the numbers inside thefigure).Note that the position of the atomic-like line shifts to- Figure 2. LDOS of Li, Al and Cu with perfect lattice structureat room temperature calculated with 10 atoms having a holein their inner shell. (a) LDOS of BCC Li of atoms with (redcolor) and without (black color) a hole in the 1s orbital. (b)LDOS of FCC Al of atoms with (red color) and without (blackcolor) a hole in the 2s orbital. (c) LDOS of FCC Cu of atomswith (red color) and without (black color) a hole in the 3sorbital. The LDOS for the inner shell orbitals and for thevalence band is shown with different scale (left and right axis,respectively) for better display. wards lower energy is almost insensitive to the fractionof atoms with core holes. This also suggests that theshift has to be understood as an atomic process. Wecalculate the local density of states (LDOS) of Li, Aland Cu with 10 holes in the system, as shown in Fig. 2,to better understand the effect of the holes on the elec-tronic structure. The LDOS is obtained by projecting the one-electron eigenstates onto the local atomic orbitals,which indicates the contribution to the electronic struc-ture of a specific atom. In Fig. 2 (a), 128 Li atoms areplaced in a BCC structure with an electron temperatureof 300K, among which 10 atoms have a core hole. Theatom which has a hole in its K shell is represented byred color, whereas the atom without any hole is repre-sented by black color. Solid lines show the LDOS for the1s orbitals, while the dashed ones show those of the 2sorbitals, and the dotted ones show those of the 2p or-bitals. In Fig. 2 (b) [(c)], 108 Al [Cu] atoms are packedwith FCC structure with electrons also at 300K, and 10atoms having a core hole. Similar to Li, the LDOS of Al[Cu] on the atom with a hole in the 2s [3s] orbital is plot-ted with red color, while the LDOS on the atom withoutholes is plotted with black color. Solid, dashed, dotted,and dash-dotted lines represent the LDOS for 2s [3s], 2p[3p], 3s [3d] and 3p [4s] orbitals, respectively. The shift ofthe 1s orbital of Li with (red solid line) or without a hole(black solid line) is clearly shown in Fig. 2(a). So are theshifts of both 2s and 2p orbitals of Al, in Fig. 2(b), andthe shifts of both 3s and 3p orbitals of Cu, in Fig. 2(c).As discussed above, this blue shift is because of the re-duced screening of the core electrons.More interestingly, we also observe the localizationof the valence band due to the reduced screening [62].For Li, the LDOS of the 2s band on an atom with a 1shole (red dashed line) has more peak structures thanthat on an atom without any hole (black dashed line),as shown in Fig. 2 (a). The peak feature is also moresignificant for the M band of Al (red dotted line) withthe presence of 2s hole compared to its counterpartwithout hole (black dotted line) in Fig. 2 (b). For Cu,the appearance of a sharp peak indicates the localizationof the 3d orbital (black and red dotted lines), as shownin Fig. 2 (c). In fact, experiments have taken advantageof such a shift and of the localization of the valence bandto circumvent the Inglis-Teller effect [63] and to studyionization potential depression (IPD) [64, 65].We now investigate the electronic structure of Cu andAu with a nonequilibrium distribution of electrons, whichcorresponds to the transient exotic state created by theisochoric heating by an ultrashort optical laser pulse. Weconsider a laser with a pulse duration shorter than 50 fsand an intensity of the order of 10 W/cm . An opticallaser with 400 nm or 800 nm wavelength has a photonenergy of 3.1eV or 1.55 eV, respectively. Both of themcan create holes in the valence band of Cu or Au byphotoexcitation. We calculate the DOS of such a non-equilibrium transient state immediately after the pulse,as plotted in Fig. 3.The black solid lines show the calculated DOS with-out holes, the red ones show the DOS with 10 electronsexcited from the valence band by the laser, and the blueones show the DOS with 32 excited electrons. In thiscase, the electron in the valence band is no longer local-ized within one particular atom, therefore we can only es- Figure 3. Density of states of (a) Cu and (b) Au with perfectlattice structure at room temperature. As in Fig. 1, the blacksolid lines show the calculated DOS, while the shaded areashows the corresponding occupied DOS. The positions of thechemical potential are marked with the vertical dashed linesand are labelled in the legend. The red [blue] lines representthe same quantity as the black lines but are calculated with10 [32] electrons removed from their valence band (3d for Cu,and 5d for Au). timate the average ionization degree of the system whichis 0.09, for the red lines, and 0.30 for the blue ones. Theoccupied DOS are shown as the shaded areas with thesame color as the DOS. The positions of their correspond-ing chemical potentials are marked with vertical dashedcolored lines, and their values are given in the legends.This time, without the competing effect of the shift ofinner shell orbitals towards lower energy, the rise of theposition of chemical potential is more evident.
B. Electrical conductivity
The electrical conductivity, often being linked to thethermal conductivity via the Wiedemann-Franz Law, isan important transport property of WDM [70], as wellas a useful diagnostic technique for the change of theelectronic structure of the system [45]. Based on linearrespond theory, the Kubo-Greenwood formula is widely
Figure 4. Calculated real part of electrical conductivity ofLi, Al, and Cu placed in their perfect lattice structure atroom temperature with variable number of holes in their innershell. Experimental measurements are plotted as purple dotsfor comparison. Inset: Zoom-in of the DC limit. (a) Li: theblack line corresponds to the case without holes, while thered and blue lines represent the one with 10 and 32 atomshaving an electron removed from 1s orbital, respectively. (b)The same for Al. The red and blue lines correspond to thecases of 10 and 32 atoms having an electron removed fromthe 2s orbital, respectively. (c) The same for Cu: The redand blue lines correspond to the cases with 10 and 32 atomshaving an electron removed from the 3s orbital, respectively.Experimental data are marked with purple dots [66–68].
Figure 5. Calculated real part of the electrical conductivityof Cu and Au placed in their perfect lattice structure at roomtemperature with variable number of holes in their valenceband . Experimental measurements are plotted as purple dotsfor comparison [68, 69]. Inset: Zoom into the DC limit. (a)Cu: the black line represents the AC conductivity withoutholes, while the red (blue) line represents the one with 10holes (32 holes) in the 3d orbital. (b) Au: red (blue) linerepresents the case with 10 (32) holes in the 5d orbital. used to study the electrical conductivity of WDM [71],because the probe laser is usually weak and will notchange the electronic structure significantly. The probepulse can have a duration that is similar to that of thepump laser (or the XFEL), on the order of 10 fs, whichis comparable to the estimated lifetime of the hole. Weshow in Fig. 4 the calculated electrical conductivity ofthe transient exotic state of Li, Al and Cu created bythe isochoric heating by an XFEL pulse correspondingto the electronic structure calculation in Sec. III A. Theblack lines represent the electrical conductivity of Li, Al,and Cu without holes, while the red and blue lines rep-resent the one with 10 atoms and 32 atoms having a holein their inner shell orbitals, respectively. For reference,experimental measurements for metal films at room tem-perature [66–68] are also shown as dots in the figure. Further, the width of the Dirac delta function in equa-tion (1) and (2) is chosen as to achieve the best agree-ment with the experimental data. For Li, a double-peakfeature appears between 2.8 eV and 3.4 eV due to themore localized L band with 10 holes in the system, andthe peak near 2.8 eV becomes even higher for the casewith 32 holes. The presence of holes in the inner shellof both Al and Cu, gives rise to a smoothening effect forthe conductivity, that is attributed to the broadening ofthe energy levels due to different degrees of ionization ofthe atoms.The electrical conductivity near the DC limit iszoomed into in the inset. An initial drop of the DC con-ductivity in the case of 10 ionized atoms is observed forall these three materials. For Li, the DC conductivity fur-ther decreases when the number of ionized atoms in thesystem is increased to 32, while it rises for both Al andCu. The contribution of the DC conductivity comes fromthe transition between valence states with energies closeto each other, and the different behavior is attributedto the competition between lowering of orbital energies,due to the presence of holes, which increases the energydifference of orbitals, and the rise of the chemical poten-tial, due to excited electrons, which means more valencestates with close energies near the chemical potential. Asa result, the change of the DC conductivity due to innershell holes depends sensitively on the electronic structureof the material near its chemical potential.In order to show the effect of holes created in the va-lence band , we also calculate the electrical conductivityof the transient exotic state of Cu and Au that is cre-ated via isochoric heating by an optical laser. The atomsremain in a FCC lattice structure with an electron tem-perature of 300K. The black lines in Fig. 5 represent theelectrical conductivity of Cu and Au without holes, whilethe red (blue) line corresponds to 10 (32) electrons in thevalence band excited by the pump laser. We also showexperimental data (dots) at room temperature for refer-ence [68, 69].We first notice that the electrical conductivity startsto rise at around 1.5 eV, for Cu without holes. When 10holes appear in the valence band, this rise shifts towardshigher energies, and this shift is larger when the num-ber of holes is increased to 32. This effect is attributedto the shift of the chemical potential because the majorcontribution to the rise of electrical conductivity at thatenergy range is the transition of 3d electrons to statesaround the chemical potential. A similar shift of theconductivity is observed for Au. The rise of the chemicalpotential is more significant for the photoexcitation byan ultrafast laser than for photoionization by an XFEL.Thus, a stronger shift of the conductivity towards higherfrequency is observed in the former case. Even thoughthe calculated electrical conductivity with holes in thevalence band shows better agreement with the experi-mental measurement, we cannot claim that the result is
Figure 6. Real part of the electrical conductivity of FCC Cuat an electron temperature of 1 eV with variable number ofholes in the inner shell. Back line: no holes; the red (blue)line: 10 (32) electron removed from the 3s orbital, resultingin 10 (32) atoms having one hole. Inset: zoom into the staticlimit. actually more accurate. The reason is that DFT is knownfor notoriously underestimating the band gap. Therefore,we can only predict the shift due to the presence of holeswithout the exact value of such a shift.Finally, we also zoom into the electrical conductivitynear the DC limit, see the inset. Contrary to the calcula-tion for the case of inner shell holes, the DC conductivityincreases when the number of holes in the valence bandgrows. Similar to what we have discussed for the DOS,this is caused by the rise of the chemical potential due tothe increasing number of excited electrons.
C. Finite temperature effect
In this subsection, the finite temperature effect onthe nonequilibrium transient WDM states is discussed.This includes the effect of the finite temperature of theelectronic subsystem, the finite temperature exchange-correlation (FTXC) interaction and the structure of theions. Using copper as an illustrating example, we firstperform the calculations of the electrical conductivitywith holes in its inner shell, at an electron temperatureof 1 eV, as shown in Fig. 6. Note that the atoms arestill placed in a perfect FCC lattice structure for thiscase. Compared to the case of an electronic temperatureof 300K, shown in Fig. 4 (c), the electrical conductivityat 1eV, with or without holes, has less structure. Par-ticularly, in all cases it rises up between 0.5 eV and 1.5eV, after which the curves become overall flatter. Atthe same time, the effect of holes in the inner shell or-bital on the electrical conductivity is similar at an elec-tronic temperature of both 300K and 1 eV. For the holes
Figure 7. Same as Fig 6, but for the case that the holes arecreated in the valence band. The holes are not bounded tospecific atoms in this case.Figure 8. Finite temperature effect of the exchange-correlation potential on the electrical conductivity of FCCCu at an electronic temperature of 1 eV. Blue curve: fi-nite temperature exchange correlation functional based on theGDSMFB parametrization [57]. Red curve: zero-temperaturefunctional with the PZ parametrization [56]. in the valence band, as shown in Fig. 7, however, thehigher electronic temperature of 1 eV wipes out nearlyall the effects caused by the holes, and only a small shiftof the electrical conductivity is observed. This differenceis due to the fact that the electronic temperature of 1 eVhas a negligible effect on the 3s orbital which lies ∼ E xc .In particular, if the exact functional would be used, onecould reproduce the exact solution of the original many-body problem of interest. For practical applications,however, this term has to be approximated. In practice,it turned out to be successful to solve the special case ofthe spatially uniform (but Coulomb interacting) electrongas (UEG) using exact quantum Monte Carlo (QMC)simulations, first provided by Ceperley and Alder [72].These data were then used to construct approximateexchange-correlation functionals E xc ( r s ) for ground stateDFT simulations of real, more complicated materials. Asa result, how the exchange-correlation is constructed iscrucial to the accuracy of the DFT method.However, previous functionals, until recently, werelimited to the case of zero temperature, which is ade-quate for many condensed matter applications, but isdoomed to fail when it comes to WDM. In this casefinite temperature and entropic effects in the exchangecorrelation functional are becoming important, and, in-stead of E xc , an accurate exchange correlation free en-ergy, F xc , has to be provided as an input for FTDFT.Based on the recently reported ab initio QMC data forthe UEG at finite temperature [73–75], an FTXC freeenergy functional has been parameterized by Groth etal. (GDSMFB) [57]. Here we compare the electri-cal conductivity calculated with this new functional tothe well-known zero-temperature functional parameter-ized by Perdew and Zunger [56]. The results are shownin Fig. 8 to explore the finite temperature effect on theexchange-correlation functional for the nonequilibriumtransient state. Only little differences are observed be-tween the two calculations. This is, of course, due to therelatively low temperature. We note that finite tempera-ture effects on the DC conductivity, as high as 15% werereported in Ref. [76] for cases where Θ ∼ IV. CONCLUSION
Using lithium, aluminum, copper and gold as illustrat-ing examples, we have presented a theoretical investiga-tion to the nonequilibrium transient states generated byisochoric heating technique. We developed a modifiedFTDFT method by introducing a nonequilibrium distri-
Figure 9. Effect of the ionic structure on the electrical con-ductivity of Cu at T = 1 eV. Red: FCC Cu at an electronictemperature of 1eV. Blue: Cu with a temperature of 1 eV forboth ions and electrons. The error bars are smaller than theline width. The electrical conductivity of Cu at 300K (thesame as the black line in Fig. 5 (a) ) is also plotted as blackline for reference. bution of electrons into the simulation. The effect of holescreated by the photoexcitation of electrons in either, theinner shells or the valence band, on the electronic struc-ture was discussed in detail. These cases can be realizedby choosing either an XFEL pulse or a short laser pulseas pump source.The presence of holes in the inner shell results in the lo-calization of the valence band due to the reduced screen-ing of core electrons. In contrast, the presence of holes inthe valence band mainly increases the chemical potentialbecause of the increased number of excited electrons. Theelectrical conductivity of such a nonequilibrium transientstate was calculated by applying the Kubo-Greenwoodformula, with a special treatment for the DC limit. Thetwo competing effects that were mentioned above, i.e.,the lowering of orbital energies due to the presence ofholes, and the rise of the chemical potential due to excitedelectrons, are the key factors for the change of the elec-trical conductivity. The hole in the valence band resultsin a shift of the conductivity towards higher frequencybecause of the dominating latter factor, while the effectof holes in the inner shell on the electrical conductivity ismaterial-dependent due to the competition of both fac-tors.We also discussed finite-temperature effects on theelectrical conductivity. A finite electronic temperatureresults in an increase of the conductivity, for low fre-quency. On the other hand, the disorder in the ionicstructure further appearing when the lattice is heated,also increases the conductivity. The hole in the valenceband is more easily affected by the finite temperaturecompared to that in the inner shell, because of its rela-0tively small energy with respect to the chemical potential.Finally the finite-temperature effect on the exchange-correlation energy is found to be small, for temperaturesup to 1 eV discussed in this work.We also note that our work is qualitatively differentfrom earlier calculations of optical excitation of holes insemiconductors, e.g. [31, 48, 77]. In our case a signif-icantly greater portion of excited (ionized) atoms maybe involved giving rise to strong (transient) deviations ofthe band structure from the ground state behavior. Ourresults can give insight into the physical processes thatcan be expected in WDM experiments such as isochoricheating by means of ultrashort laser pules or free elec-tron lasers. Of particular interest will be to extend theanalysis to preheated samples that are highly ionized orin the plasma phase [65]. Then our analysis will giveaccess to the modification of the band structure due toexcitation of core electrons in dense plasmas. While wehave discussed the manifestation of the nonequilibriumband structure in conductivity measurements, a more di-rect experimental probe should be possible via photoe-mission spectroscopy. Using an X-ray probe pulse witha controlled time delay a time-resolved investigation of the described nonequilibrium effects should be possibleincluding the relaxation processes that are not includedin the present study. ACKNOWLEDGMENTS
The authors thank Prof. Andrew Ng for his insight-ful discussion about experiments. We also acknowledgediscussions with Prof. Wei Kang about technical is-sues of FTDFT simulations. This work has been sup-ported by the NSAF via grant No. U1830206, the Na-tional Natural Science Foundation of China via grantsNo. 11774429, 11874424 and 11904401 and the ScienceChallenge Project of China via grant No. TZ2016001,the National Key R&D Program of China via grantNo. 2017YFA0403200, the Deutsche Forschungsgemein-schaft via grant BO1366/15, and by the HLRN via grantshp00023 for computing time. Shen Zhang acknowledgesfinancial support from the China Scholarship Council(CSC) and from the German Academic Exchange Ser-vice (DAAD). [1] F. Graziani, M. P. Desjarlais, R. Redmer, and S. B.Trickey,
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