Finite temperature electronic structure of Diamond and Silicon
Vaishali Shah, Bhavik Sanghavi, Rahul Ramchandani, M. P. Gururajan, T. R. S. Prasanna
FFinite temperature electronic structure of Diamond and Silicon
Vaishali Shah ∗ Interdisciplinary School of Scientific Computing, Savitribai Phule Pune University, Pune, 411007, India
Bhavik Sanghavi, Rahul Ramchandani, M. P. Gururajan and T. R. S. Prasanna † Department of Metallurgical Engineering and Materials Science,Indian Institute of Technology, Bombay, Powai, Mumbai - 400076, India (Dated: February 21, 2018)The electron-phonon interaction contribution to the electronic energies is included in density func-tional total energy calculations with ab initio pseudopotentials via the formalism of Allen [Phys.Rev. B, 18 5217 (1978)] to obtain the temperature dependent electronic structure of diamond andsilicon. This method allows us to obtain the thermally-averaged ab initio electronic structure in astraightforward and computationally inexpensive way. Our investigations on the finite temperatureelectronic structure of diamond and silicon bring out that a new criterion, that of temperature trans-ferability, is required in the input ab initio pseudopotentials for temperature dependent studies. Thetemperature transferability of the Troullier-Martins pseudopotentials used in this work is stronglydependent on the cut-off radius and the inclusion of the unbound 3d state. The finite temperatureindirect band gaps are highly sensitive to the choice of cut-off radius used in the pseudopotentials.The finite temperature band structures and density of states show that thermal vibrations affectthe electron energies throughout the valence and conduction band. We compare our results on theband gap shifts with that due to the Debye-Waller term in the Allen-Heine theory and discuss theobserved differences in the zero point and high temperature band gap shifts. Although, the electronenergy shifts in the highest occupied valence band and lowest unoccupied conduction band enableto obtain the changes in the indirect and direct band gaps at finite temperatures, the shifts in otherelectronic levels with temperature enable investigations into the finite temperature valence chargedistribution in the bonding region. Thus, we demonstrate that the Allen theory provides a simpleand theoretically justified formalism to obtain finite temperature valence electron charge densitiesthat go beyond the rigid pseudo-atom approximation. I. INTRODUCTION
Most electronic structure studies are performed forstatic lattices that are implicitly assumed to be at 0 K.Experimental investigations and determination of mate-rial properties are however performed at finite tempera-tures. The two important effects of temperature on thematerial are the lattice expansion and lattice dynamics.The effect of the lattice expansion on electronic ener-gies is straightforward to obtain. However, the electron-phonon interaction which has a major contribution fromthe lattice dynamical behaviour is harder to calculate.Temperature affects the nuclear motion in materials lead-ing to lattice dynamics which alters the electronic ener-gies by 2-4 k B T in solids. The resistivity of metals,directional Compton profiles, infrared, Raman, opticalspectra, specific heats, heat conduction, band gaps etc.are affected by the electron-phonon interaction. In orderto compare the results of experiments with the theoret-ical simulations the effect of electron-phonon interactionneeds to be included in the calculations of these phenom-ena. .Although of significant importance, the electron-phonon interaction happens to be the most difficult ∗ [email protected] † Corresponding author: [email protected] to compute from first principles. Currently, there arethree major approaches , namely, the molecular dynam-ics method, the frozen phonon method and the per-turbation theory method to understand the effects ofthe electron-phonon interaction on material properties.Each of these approaches has its advantages and disad-vantages . The to-date developments to include theelectron-phonon interactions in ab initio calculations andtheir computational implementations is discussed in de-tail in a rigorous and elegant review by Giustino .In the context of temperature dependent semicon-ductor band gaps, the earliest attempt to include theelectron-phonon interaction was by Fan who calculatedthe self-energy (SE) contribution to the semiconductorband gap shifts. This term is also referred to as theMidgal term in superconductor literature . Soon after-wards, Antoncik calculated the Debye-Waller (DW) cor-rection to the semiconductor band gaps. Subsequently,several authors calculated the semiconductor band gapshifts due to thermal vibrations, mostly by incorporat-ing the Debye-Waller term . These investigations ledBaumann to suggest that both self-energy and DW cor-rections are necessary for a complete account of the roleof thermal vibrations on electron energies.Allen and Heine developed the formal basis for incor-porating electron-phonon interactions in the harmonicapproximation at constant volume using second-orderperturbation theory. In this formalism , the DW termand the SE term appear separately and the total electron a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b energy shift is a sum of the two terms. The implementa-tion of the formalism requires a concurrent calculation ofthe DW and SE term. The calculation of the DW termis approximated by a second order expansion with theneglect of the higher order terms. Further, by exploit-ing translational invariance the DW term (in the secondorder form) and the SE term is recast to have similarforms. All recent studies based on the Allen-Heine theoryuse the recasted DW term since calculating it directly inthe second order form is computationally complex.The Allen-Heine theory was earlier used with empiricalpseudopotentials to obtain the band gap shifts in semi-conductors . However, in the last decade, several abinitio studies based on it have been reported. Thesestudies report the band gap shifts at specific k pointsthat usually correspond to the indirect and direct bandgaps . In these studies, the band gaps at all highsymmetry points or the complete valence electron bandstructure along the symmetry lines in the Brillouin zonehave not been reported.Subsequent to the Allen-Heine theory , a more accu-rate theory was developed by Allen for the temperaturedependent band structure. In this theory, the DW termis considered to all orders instead of its truncated sec-ond order expansion as in the Allen-Heine theory. In theimplementation of the Allen theory with pseudopoten-tials, the first step is to correct the pseudopotential formfactor with the DW term and use this corrected pseu-dopotential to obtain the finite temperature electronicstructure. The second step is to calculate the SE termusing the wavefunctions and energies obtained from theDW corrected electronic structure.The Allen theory provided the theoretical justifica-tion for the earlier empirical studies that calculatedthe finite temperature band gaps based on the DW cor-rection step. It must be noted that the first step, the DWcorrection step, directly leads to the temperature depen-dent band structure. The band gap shifts obtained fromthe first DW correction step gave satisfactory band gapshifts in some cases and unsatisfactory results in othercases . The DW corrected empirical studies on thetemperature dependence of band gaps in PbSe and PbTe,while giving reasonable results for the direct gap, failed tocorrectly estimate the indirect gap . Allen and Heine and Allen discuss this aspect and show that for accu-rate values of the finite temperature electron energies andband gap shifts the contribution from the SE term needsto be included, in general.The second order Allen-Heine formalism continues tobe of interest and has been used in several recent ab initio calculation based studies to obtain the semiconduc-tor band gap shifts with temperature. In contrast, theAllen theory has not been used in ab initio studies, eventhough it is a more accurate approach compared to theAllen-Heine theory.In this article we demonstrate that the Allen theorycan be combined with ab initio pseudopotentials to ob-tain ab initio finite temperature electronic structures without any additional increase in the computationalcomplexity of these calculations. We investigate the finitetemperature electronic structure of diamond and siliconusing the ab initio pseudopotential implementation of theAllen theory. We then compare the results of our theoret-ical calculations with earlier ab initio studies on diamondand silicon based on the second-order Allen-Heine theory.The focus of this paper is to address the main issuesthat affect the implementation of the Allen theory using ab initio pseudopotentials. Thus, only the DW contri-bution, which is the first step in Allen theory, has beencalculated. The (Fan/Fan-Migdal) self-energy term, thatis to be calculated from the results of the DW step, hasnot been evaluated.However, calculation of the DW correction to the elec-tronic energies is important for three reasons, as will beseen in this study. Firstly, the DW step is sufficient toaddress the fundamental question, viz. the viability ofimplementing Allen theory with ab initio pseudopoten-tials. Secondly, only the DW term can be the basis for thecomparison of the Allen and Allen-Heine theories. And,thirdly, the DW step is sufficient to obtain finite temper-ature valence electron charge densities. Thus, the resultsof the present study have important implications for abinitio finite temperature electronic structure studies.In the next section, we briefly describe the theory ofAllen for the inclusion of the electron-phonon interac-tion and its implementation within the pseudopotentialmethod for band structure calculations. In Sec. III, wedescribe the computational implementation and give thedetails of our calculations of temperature dependent elec-tronic structure. Sec. IV discusses the results of thetemperature dependent band gap trends that were ob-tained using existing pseudopotentials and the need touse temperature transferable pseudopotentials in elec-tronic structure calculations that are to be used to studythe finite temperature properties. Our results on thetemperature dependent band gap trends, band gap shiftsat zero point vibration and higher temperatures for dia-mond (Sec. IV B) and silicon (Sec. IV C) are discussedin detail with implications. In the Sec. IV E, we showthat theoretically it is possible to obtain charge densitiesthat go beyond the rigid pseudo-atom approximation fora direct comparison with experimental data and this isthen followed by conclusions in Sec. V II. ALLEN’S THEORY AND ITSIMPLEMENTATION
In the theory of temperature dependence of the en-ergy bands given by Allen-Heine and Allen , a lattice ofidentical atoms is considered with the atoms undergoinga small thermal displacement u α about their equilibriumpositions α . In this system, the potential experienced byan electron due to phonon disorder is assumed to moverigidly with the atoms so that the perturbation to thesystem can be expressed as H e − p = (cid:88) l [ V ( r − R α ) − V ( r − α )] (1)where, V(r) is the atomic potential, R α is the displacedposition of the atom, α is the equilibrium position and u α = R α − α is the displacement. The thermal displace-ments of atoms are time dependent and are related tothe phonon frequency. The perturbed electron energycan be calculated in the adiabatic approximation by asecond order Taylor expansion of Eq. 1 as proposed byAllen-Heine , where the first two terms are H (1) e − p = (cid:88) l (cid:104) ∂V ( r − α ) ∂r n (cid:105) u αn (2) H (2) e − p = 12 (cid:88) l (cid:104) ∂ V ( r − α ) ∂r n ∂r m (cid:105) u αn u αm (3)Considering the thermal average (cid:104) H e − p (cid:105) , the only non-vanishing terms are the even powered terms like Eq. 3 inthe Taylor expansion. The thermal average of Eq. 3 is theself energy correction in a Bloch-wave basis set. However,in a plane wave basis set all the non-vanishing terms canbe considered and result in the Debye-Waller correctionsto the crystal potential and Eq. 1 can be rewritten as H e − p = (cid:88) kk (cid:48) V ( k (cid:48) − k ) s ( k (cid:48) − k ) c † k (cid:48) c k (4)In this equation the k , k (cid:48) span all the Brillouin zonesand s is the structure factor. The thermal average of theperturbed Hamiltonian in reciprocal space is (cid:104) H e − p (cid:105) = BZ (cid:88) k (cid:88) GG (cid:48) V ( G (cid:48) − G )( e − W ( G (cid:48) − G ) − c + k + G (cid:48) c k + G (5)This Hamiltonian is periodic and the calculation inFourier space requires the sum to be performed only onthe first Brillouin zone. When added to the unperturbedHamiltonian it leads to a reduction of the pseudopoten-tial form factors V(G) of the static lattice by DW factors e − W ( G ) to incorporate the effect of the DW term of theelectron-phonon interaction. This result of Allen pro-vided the theoretical basis for the earlier empirical stud-ies and is also the theoretical basis for the present abinitio study.The underlying assumptions in both the Allen-Heine and Allen theories are the adiabatic approximation andthe rigid atom approximation. The ab initio pseudopo-tentials are generated under the assumption of frozen-core approximation for the core electrons, which satisfiesthe rigid-atom approximation. Thus, Allen-Heine andAllen theories are valid for semiconductors and insula-tors at all temperatures. The adiabatic approximation condition, however, breaks down for metals at low tem-peratures since the self-energy term cannot be correctlyrepresented .From Eq. 5, the finite temperature pseudopotentialform factor for any ion can be written as V psi ( G , T ) = V psi ( G , e − W i ( G ,T ) (6)where the DW factor is given by, W i = (cid:104) u i (cid:105)| G | (cid:104) u i (cid:105) is the mean square displacement of atom i.In order to use Eq. 6 in ab initio studies, both thepseudopotential form factor, V(G) and the DW factor(W) need to be obtained from ab initio studies. Cur-rently, there are several ab initio methods to calcu-late DW factor from first principles. In these studies, abinitio DW factor has already been calculated for severalmaterials including many semiconductors.However, if necessary, even experimental DWF can beused in Eq. 6. We note that in several ab initio stud-ies experimental lattice parameters are used. Inparticular, Erba et. al provide the justification for theuse of experimental lattice parameters viz. that they canbe validated by separate ab initio studies. Therefore, ifnecessary, the same justification can be the basis for theuse of experimental DWF in ab initio studies based onAllen theory.The other requirement is that of ab initio pseudopo-tentials which can be used in Eq. 6 to obtain finite tem-perature band structure. As discussed in Sec.IV, this isa much more stringent condition than expected.It is of interest to compare both the zero-point andthe higher temperature band gap shifts obtained usingthe Allen and Allen-Heine theories since the latter usesthe second-order approximation to the DW term. Inboth, Allen and Allen-Heine theories, the band gap shiftsshould increase with temperature due to the increase inthe mean-square displacements or W. The main differ-ence in their formalism is that, in the Allen-Heine the-ory, the pseudopotential form factor V G e − W is approxi-mated by V G (1-W) neglecting higher order terms in theexpansion. The band gaps obtained by the DW correctedpseudopotentials, V G e − W in the Allen theory and V G (1-W) in the Allen-Heine theory, are to be compared withthe band gaps obtained with the static lattice pseudopo-tential form factor, V G , to obtain the finite temperatureband gap shifts.We can examine the effect of the second-order approx-imation used in the Allen-Heine theory. For higher val-ues of the mean square displacement, the pseudopoten-tial form factor in the Allen-Heine theory, V G (1-W), issmaller than that of the actual value, V G e − W , that isused in the Allen theory. This implies that the pseu-dopotential form factor used in the Allen-Heine theoryundergoes a larger amount of change from the static lat-tice value (V G ) than the pseudopotential form factorin the Allen theory. It follows that, at higher temper-atures, the Allen-Heine theory should give larger bandgaps shifts than Allen theory due to the neglect of higherorder terms.In the Allen-Heine theory, the self-energy (SE) termcontribution is calculated from the static lattice electronwavefunctions and energies. In contrast, in the Allentheory, the SE term contribution is to be calculated fromthe finite temperature electron wavefunctions and ener-gies obtained from the DW correction step. Thus, theSE term contributions in the Allen and the Allen-Heinetheories will be different.Therefore, the primary basis for the comparison of theAllen-Heine and Allen theories is the Debye-Waller (DW)term contribution, which, in principle, must give identicalband gap shifts for the same pseudopotential and samemean-square displacement values as long as the second-order approximation is valid. III. COMPUTATIONAL METHODOLOGY
We have performed ab initio total energy calculationsbased on density functional theory to obtain the bandstructure of diamond and silicon at different tempera-tures using the Quantum Espresso (QE) software pack-age . The equilibrium lattice constant for the staticlattice (0 K) was obtained by choosing the energy con-verged k-point mesh and kinetic energy cutoff. A 6x6x6k-point mesh and kinetic energy cutoff of 60 Ry for car-bon (diamond) and 40 Ry for silicon was used in all ourcalculations.To incorporate the effect of finite temperature on theelectronic structure, we modified the QE code so that thelocal part of the ab initio atomic pseudopotential of thesystem under investigation is altered in G-space with theDebye-Waller factor, i.e. e − W as in Eq. 6. The non-localpart of the atomic pseudopotential is unmodified to pre-serve the angular dependence of the scattering potentialas in the earlier work based on empirical pseudopoten-tials . With this alteration, the finite temperature pseu-dopotential form factor is obtained within the QE codefor a given ab initio static lattice pseudopotential of anyatom. Using a value of the mean square displacement (cid:104) u i (cid:105) appropriate for a chosen temperature, an electronicstructure calculation was performed to obtain the bandstructure at that temperature.All electronic structure calculations were performed forvarious temperatures up to 1000 K at constant volumeusing the equilibrium lattice constant obtained for thestatic lattice. The ab initio mean-square displacement (cid:104) u i (cid:105) values of C and Si listed in Table I at various temper-atures were taken from the ab initio studies of Schowalter et al. . Using these values in the modified QE code, theself-consistent total energy and the band structure at dif-ferent temperatures were obtained using different kindsof ab initio pseudopotentials. T (K) (cid:104) u i (cid:105) (˚A )C Si0.001 0.001611 0.002471100 0.001626 0.003196200 0.00169 0.004865300 0.001807 0.006788400 0.001968 0.008772600 0.002436 0.01287800 0.002962 0.0170221000 0.003529 0.021198TABLE I: Ab initio mean-square displacement values for di-amond and silicon at various temperatures . IV. RESULTS AND DISCUSSION
We began our investigations on the effect of temper-ature on the electronic structure of diamond and sili-con with several norm conserving pseudopotentials thatare available in the pseudopotential library on the QEwebsite . In addition, some optimized and ultrasoftpseudopotentials available on the QE website library, re-cently developed Optimized Norm Conserving Vander-bilt (ONCV) pseudopotentials , the GBRV pseudopo-tentials and PAW pseudopotentials were also investi-gated.Separate electronic structure calculations were per-formed to obtain the band structure of diamond and sil-icon for each temperature up to 1000 K as given in Ta-ble I using each pseudopotential listed in Table II and III.The band gaps obtained for each pseudopotential werethen compared across the range of temperatures used inthis study. Table II and III list the trends in the bandgaps obtained for diamond and silicon from our calcu-lations. The equilibrium lattice parameter obtained foreach pseudopotential is listed along with the experimen-tal value for comparison.For the static lattice case, almost all of the above pseu-dopotentials show excellent agreement in values of the in-direct and direct band gaps reported in theoretical stud-ies as all these pseudopotentials have hitherto been de-veloped for static lattice ab initio calculations.Several previous studies on diamond and silicon,based on the Allen-Heine theory, using both ab initio andempirical pseudopotentials, have shown that the indirectand direct band gaps decrease with temperature in agree-ment with experiments. In particular, for diamond and silicon , the DW term alone leads to a decreasein the band gaps with temperature as calculated in theAllen-Heine theory.Thus, the primary criterion that needs to be fulfilledwhen choosing ab initio pseudopotentials for tempera-ture dependent studies on carbon (diamond) and siliconis that after modification with the DW factor, the di-rect and indirect band gaps must decrease with increas-ing temperatures. Only after this primary criteria is met Pseudopotential a Band gap (eV)(PP) Filename a.u. Static Finite T trendIndirect Direct Indirect DirectC.pbe-mt − gipaw 6.73 4.246 5.645 Decrease IncreaseC.pbe-n-kjpaw 6.74 4.135 5.599 Decrease IncreaseC.pbe-mt − fhi 6.63 4.308 5.756 Decrease IncreaseC.pw-mt − fhi 6.63 4.32 5.757 Decrease IncreaseC.blyp-mt 6.79 4.371 5.679 Decrease IncreaseC.blyp-hgh 6.79 4.317 5.664 Decrease IncreaseC.pbe-hgh 6.77 4.068 5.589 Decrease IncreaseC.pbe-van − bm 6.64 4.234 5.722 Decrease IncreaseC − pbe − v1.2.uspp.F 6.71 4.176 5.629 Decrease IncreaseC − ONCV − PBE-1.0 6.54 4.378 5.846 Decrease IncreaseExpt ab initio pseudopotentials.The names of the pseudopotential files are the same as in thesource library with a .UPF extensionPseudopotential a Band gap (eV)(PP) Filename a.u. Static Finite T trendIndirect Direct Indirect DirectSi.pbe-mt − gipaw 10.31 0.633 2.572 Decrease DecreaseSi.pbe-n-kjpaw 10.33 0.605 3.122 Increase DecreaseSi.pbe-mt − fhi 10.33 0.614 2.561 Increase IncreaseSi.pw-mt − fhi 10.17 0.457 2.568 Increase IncreaseSi.blyp-hgh 10.41 0.905 2.849 Increase DecreaseSi.pbe-n-van 10.34 0.614 3.108 Increase IncreaseSi.pbe-rrkj 10.35 0.639 2.558 Increase IncreaseSi − pbe − v1.uspp.F 10.33 0.691 2.533 Increase IncreaseSi − oncv − pbe-1.0 10.37 0.62 2.548 Increase Dec-IncSi − oncv − pbe-1.1 10.35 0.60 2.546 Increase Dec-IncExpt ab initio pseudopotentials. The names ofthe pseudopotential files are the same as in the source librarywith a .UPF extension can further finite temperature studies be performed. Forexample, the self-energy contributions should only be cal-culated for pseudopotentials which show the correct be-haviour in the DW step.We notice that, for diamond (Table II) all the pseu-dopotentials show decreasing indirect band gaps as is ex-pected with increasing temperature. However, the directband gaps are increasing with temperature which doesnot agree with the trends reported in literature. Thus,none of the ab initio pseudopotentials in Table II can beused for finite temperature electronic structure studies.In the case of silicon only one pseudopotential, Si.pbe-mt − gipaw.UPF, gave the expected trend of decreasing indirect and direct band gaps with increasing tempera-ture. The lattice constant obtained with this pseudopo-tential also gives a better agreement with the experi-mental value. The calculated band gaps and the shiftsin the band gaps for this Si pseudopotential, labelledSi pbe − gipaw in our study, are listed in Table IV. Temp Band gap Band gap shift(K) (eV) (meV)Indirect Direct Indirect Direct0 0.633 2.5720.001 0.539 2.544 94 28100 0.512 2.537 121 35200 0.449 2.519 184 53300 0.376 2.494 257 78400 0.296 2.466 337 106600 0.136 2.396 497 176800 - 2.319 - 2531000 - 2.246 - 326TABLE IV: The indirect and direct band gaps with tempera-ture and the energy shifts in the direct and indirect band gapsfor silicon obtained with Si.pbe-mt − gipaw.UPF (Si pbe − gipaw )pseudopotential It is well known that the Si band gaps are underesti-mated in the density functional calculations. A conse-quence of this underestimation is that at higher temper-atures of 800 K and 1000 K the indirect band gap of Si isseen to vanish. As will be discussed in detail later (Sec-IV C), while the band gap trends are correct, the actualvalues of the band gap shifts for the above pseudopoten-tial, Si pbe − gipaw , appear to be excessive when comparedto literature values.The finite temperature band gap results show that, ingeneral, the ab initio pseudopotential developed for staticlattice applications cannot be directly used for temper-ature dependent electronic structure calculations. Theabove results thus imply that for finite temperature stud-ies, ab initio pseudopotentials need to be generated soas to satisfy an additional criterion, that of temperaturetransferability.Since the ab initio pseudopotential is initially gener-ated for static lattice conditions, it has to be generatedwith the usual criteria viz. the norm conservation criteria(present or absent) and the choice of the cut-off radius,r c , to ensure the transferability of the pseudopotential toother chemical environments. The additional criterionof temperature transferability will ensure that the pseu-dopotentials can be used for finite temperature studies,in addition to static lattice calculations.Semiconductors are the class of materials that are mostsuited to test for temperature transferability. This isbecause there is a stringent test of temperature trans-ferability - the indirect and direct band gaps must ex-hibit the “Varshni effect” of redshift with increasingtemperature . In addition, the band gap shifts mustbe of the same order of magnitude when compared withexisting experimental or theoretical results.However, in some semiconductors the band gaps ex-hibit a blueshift with temperature . For such semicon-ductors, the temperature transferability criterion standsappropriately modified. A. Pseudopotentials with temperaturetransferability
Considering that several of the currently availableopen source C and Si ab initio pseudopotentials (Ta-ble II and III) when used for finite temperature electronicstructure calculations did not reproduce the correct bandgap trends and band gap shifts, we generated pseudopo-tentials based on the Troullier-Martins method , withGGA/PBE and LDA/PZ exchange-correlationfunctionals, using the OPIUM pseudopotential genera-tion code.Table V and VI list the details of the carbon andsilicon pseudopotentials (PP) that were generated andtested for temperature transferability. The cut-offradius, the number of valence states and their electronoccupancies were varied to study their effect on tem-perature dependent band gap behavior and to identifypseudopotentials that exhibit temperature transferabil-ity. PP Parameters Band gapName Valence r c E xc Finite T trendelectrons (a.u.) Indirect DirectC . . . ab initio pseudopoten-tials generated with PBE and PZ exchange correlation func-tional to study finite temperature behavior. From Table V, the pseudopotentials C – C and C –C generated with a r c of 1.5. a.u. showed the expected cut B a nd ga p s h i f t( m e V ) IndirectDirect
FIG. 1: Variation of zero-point band-gap shifts with cut-offradius for diamond trend of decreasing direct and indirect band gaps withincreasing temperature. However, the band gap shifts inthe two sets, C – C and C – C , differ by about 4-7meV. We note that the pseudopotentials C – C , thatdiffer from C – C only in the cut-off radius, are nottemperature transferable highlighting the sensitivity tothis parameter.To further examine the sensitivity to the cut-off radius,pseudopotentials, C - C were generated with cut-offradius varying from 1.3 to 1.7 a.u. From Table V it isseen that the correct trends are seen only for cut-off ra-dius of 1.5 a.u. and above. Figure 1 plots the variationof the zero-point indirect and direct band gap shifts asa function of the cut-off radius for the carbon pseudopo-tentials, generated in the 2s configuration withthe PBE exchange-correlation functional.The zero-point direct band gap shift decreases weaklywith the cut-off radius. This is an important result forcomparison of the Allen and the Allen-Heine theories asdiscussed in Sec. IV B 1. In contrast, the indirect zero-point band gap shift varies strongly with the cut-off ra-dius. The zero-point band gap shift is positive for r c < c < c > – Si ,Si pbe − gipaw and Si - Si give the correct band gap trendwith temperature. The pseudopotentials Si -Si have thesame cut-off radius (1.7 a.u.) but different valence statesand their occupancies. These pseudopotentials give onlythe zero-point indirect band gap to be larger than that ofthe static lattice. For all other temperatures, the indirectband gaps decrease with increasing temperature and aresmaller than that of the the static lattice. In the case ofthe direct band gap, the zero-point band gaps are largerthan that of the static lattice. For all other temperatures,the direct band gaps decrease in comparison with thezero-point vibration gap but are higher than that of thethe static lattice up to temperature of 800 K. PP Parameters Band gapName Valence r c E xc Finite T trendelectrons (a.u.) Indirect DirectSi . . . pbe − gipaw ab initio pseudopotentials generatedwith PBE exchange correlation functional to study finite tem-perature behavior. For the pseudopotentials Si -Si , that were generatedwith a cutoff radius of 2.0 a.u, the zero-point indirectband gap shows the correct behaviour i.e. smaller thanthe indirect band gap of the static lattice. The trend inthe direct band gaps however is the same as that of Si -Si pseudopotentials up to a temperature of 600 K. Thus,the pseudopotentials Si -Si do not exhibit temperaturetransferability.Table VI shows that the pseudopotentials Si -Si dif-fer from pseudopotentials Si -Si in that the unbound3d state was incorporated in the generation condition.These pseudopotentials show the correct indirect and di-rect band gap behavior and are hence, temperature trans-ferable. This leads to the conclusion that incorporationof the unbound 3d state in the pseudopotential gener-ation configuration is essential for temperature transfer-ability of Si pseudopotentials. In addition, the pseudopo-tential Si was generated with the same conditions asSi pbe − gipaw except that it did not have the gipaw con-struction. Both Si and Si pbe − gipaw (Table IV) givevirtually identical indirect and direct band gap shifts atall temperatures. This suggests that the gipaw construc-tion does not affect temperature transferability.To examine the dependence of the band gap shifts onthe cut-off radius, pseudopotentials Si - Si were gen-erated in the 3s configuration with the cut-off cut B a nd ga p s h i f t( m e V ) IndirectDirect
FIG. 2: Variation of zero-point band-gap shifts with cut-offradius for silicon radius varying from 1.8 a.u to 2.2 a.u. Figure 2 plotsthe zero-point indirect and direct band gap shifts as afunction of the cut-off radius. It is again seen that thezero-point indirect band gap shift varies strongly withthe cut-off radius. In comparison, the zero-point directband gap shift varies much more slowly with the cut-offradius.To summarize, the temperature transferability ofTroullier-Martins pseudopotentials is dependent on thechoice of the cut-off radius and the inclusion/exclusionof the unbound 3d state in the generation configuration.The cut-off radius strongly affects the indirect band gap.These observations in combination with the usual cri-teria for pseudopotential generation indicate that onlyin a small subset of the generation parameter space ofTroullier-Martins pseudopotentials the new criterion oftemperature transferability is also satisfied. These ob-servations will also be helpful in generating temperaturetransferable pseudopotentials for other elements that areconstituents of compound semiconductors.The question of whether other category of pseudopo-tentials, Optimized, Ultrasoft and ONCV pseudopoten-tials can also be generated to exhibit temperature trans-ferability needs to be explored. B. Finite temperature band structure - Diamond
We calculated the finite temperature electronic struc-ture of diamond (a wide band gap material) at varioustemperatures listed in Table I from 0 – 1000 K by usingthe DW corrected pseudopotential form factors. Whilethe calculations were performed for all the pseudopoten-tials listed in Table V, we discuss here the results ob-tained for pseudopotentials that exhibit the correct bandgap trends.
PP a Band gapName Static lattice Zero point shift(a.u.) (eV) (meV)Indirect Direct Indirect DirectC Table VII gives the zero-point shifts for the pseudopo-tentials that exhibited temperature transferability. Itshows that the zero-point band gap shifts fall into twoclear sets for C -C and C -C where the former givesslightly higher shifts of 4-7 meV. All these pseudopoten-tials have the same cut-off radius (1.5 a.u).The main difference between the two sets is the incor-poration of the unbound 3d state in the pseudopotentialgeneration configuration in C -C . Thus, for the case ofdiamond, the incorporation of the unbound 3d state inthe generation configuration has a consistent, but minor,effect.The pseudopotentials C and C vary from C onlyin the cut-off radius. The choice of the cut-off ra-dius strongly influences the indirect zero-point band gapshifts. The direct zero-point band gap shifts vary weaklywith the cut-off radius. Comparing with the previous re-sults , based on the Allen-Heine theory, for the directand indirect band gap shifts due to the DW term, thepseudopotentials we generated with the cut-off radius of1.5 a.u. give the closest agreement. Hence, we restrictour discussion to the results obtained with C and C pseudopotentials that differ in the generation parametersonly in the exchange-correlation functional.For diamond, the lattice constants and the static lat-tice band gaps are very similar to that in previous abinitio studies . The band gaps are lower than the ex-perimental values due to the well known underestimationof band gaps in density functional theory .From Table VII it is seen that the lattice parametersfor pseudopotentials with the LDA(PZ) parameterization(C and C ) are lower than others with the GGA(PBE)parameterization. In addition, the zero-point band gapshifts for pseudopotentials with the LDA(PZ) parame-terization C and C (Table VII) are higher than oth-ers with the GGA(PBE) parameterization. Both these -8-6-4-2 0 2 4 Γ X W L Γ K E - E F ( e V ) FIG. 3: The calculated band structure (static lattice andthermally-averaged) of diamond at 0 K, 300 K and 1000 Kin the vicinity of the Fermi level. The violet line representsthe band structure at 0 K, the blue line at 300 K and the redline at 1000 K. trends are similar to results from previous studies . Inparticular, the lattice parameter for the pseudopotentialC is 6.64 a.u. and is very similar to 3.52 ˚A and 6.652Bohrs for the pseudopotential with the same generationconditions. In addition, the direct band gap for the pseu-dopotential C is 5.68 eV that is very close to the valueof 5.67 eV for the similar ‘reference’ pseudopotential inPonce et al. . The similarities on the lattice parameterand band gap for the pseudopotential C with earlierresults justifies further comparisons.Figure 3 plots the static lattice (0 K) band structureof diamond and also the thermally-averaged band struc-tures (at 300 K and 1000 K) for the pseudopotential C for the in the vicinity of the Fermi level. The band struc-ture is plotted so that the Fermi level at each temperatureis at 0 eV. Fig. 3 demonstrates that the finite tempera-ture thermally-averaged band structure can be obtainedvia an ab initio electronic structure calculation using abinitio pseudopotentials using the Allen theory formalism.The finite temperature band structures show that the va-lence and conduction bands shift by different amounts fordifferent k points. From Fig. 3 it is clear that the tem-perature dependent band gap shifts can be obtained forany k point.Earlier ab initio studies , based on the Allen-Heinetheory, report the band gap shifts only at special k points,corresponding to the direct and indirect band gap shifts.The temperature dependent band structure of diamondhas been obtained with path-integral molecular dynam-ics simulations . These simulations are computationallyintensive as the ensemble-averages are calculated over ∼ steps . Our results highlight the advantages of im-plementing the Allen theory to obtain finite temperatureband structures without any increase in the computa-tional expense. DO S ( a r b . un it s ) E-E F (eV) FIG. 4: The density of states of diamond at 0 K (violet) and300 K (blue)
To understand the effect of temperature on the com-plete band structure we plot the density of states (DOS)for the carbon pseudopotential C at 0 K and 300 K inFigure 4. The effect of temperature on the valence elec-tron energy diagram is noticeable from the change in theDOS at 300 K. The DW term causes a positive shift ofthe valence electron energy levels towards the Fermi en-ergy. The unfilled conduction band energy levels on theother hand experience a negative shift towards the Fermienergy. This feature is observed in the DOS of all finitetemperature band structures that we have calculated. Inthe filled as well as unfilled electron energy levels, theamount of shift experienced from the static lattice is dif-ferent for different levels indicating that the levels are notjust scaled uniformly by a temperature dependent scalingconstant.Figure 5 plots the indirect and direct band gap shiftsof diamond due to the DW term for the four carbonpseudopotentials, C -C , at various temperatures. Theindirect and direct band gap shifts show a non-linearbehavior at low temperatures and a linear behavior athigher temperatures consistent with experimental obser-vations . While we have calculated the shifts for allpseudopotentials at all temperatures under investigation,we plot the results only for the pseudopotentials thatexhibit good temperature transferability behavior andhence are of interest for further studies. We note that theindirect band gap shifts decrease more rapidly with tem-perature than the direct band gap shifts and the changesin the generation configuration, in terms of the occu-pancies of the 2p state, has only a minor effect at alltemperatures.
1. Zero-point band gap shifts
Zero-point vibrations represent the minimum pertur-bation to the static lattice system. Hence, it is of interest B a nd ga p s h i f t( m e V ) C -IC -DC -IC -DC -IC -DC -IC -D FIG. 5: The direct (D) and indirect (I) band gap shifts indiamond due to the DW term with increasing temperature fordifferent temperature transferable carbon pseudopotentials to examine the zero point band gap shifts in detail. Forzero-point shifts, we assume that the DW term can beapproximated by the second order expansion and the ne-glect of higher order terms in Allen-Heine theory is unim-portant. That is, V G e − W ≈ V G (1-W) and the Allenand Allen-Heine theories should give similar zero-pointband gap shifts.Of the several ab initio studies on the band gapshifts in diamond based on Allen-Heine theory, onlytwo give separately the contributions of the DW andthe SE terms. Hence, the DW contribution to the zero-point shifts in our study (Table VII) can be comparedwith only these studies . In particular, our C pseu-dopotential has the same generation conditions as thepseudopotential in Giustino et al. and the ‘reference’pseudopotential in Ponce et al. . These studies, how-ever, report the DW contribution only to the direct bandgap and hence, the comparison is restricted to it.The DW contribution to the direct band gap shiftsfor the pseudopotentials in the present study range from135-149 meV for a variety of pseudopotential generationconditions (Table VII). In contrast, the indirect band gapshifts of these pseudopotentials (Table VII) have a muchwider spread (95-205 meV) on account of the sensitivityto cut-off radius. For the direct as well as indirect bandgap shifts, the pseudopotentials generated with PBE ex-change correlation functional and that include the 3d state give comparatively lower shifts for a constant cut-off radius. When the direct band gap shifts are com-pared to the literature values of 105-121 meV for thezero-point shifts in the Allen-Heine theory, for Troullier-Martins pseudopotentials with a variety of generationconditions, our results are higher by about 20-30 meV.In particular, Giustino et. al report a zero-point shiftof 117 meV (19% of 615 meV) due to the DW term for0the direct band gap of diamond. Ponc´e et. al reportthat the DW contribution ranges from 105-121 meV forfour (Troullier-Martins) pseudopotentials. Of these, theyreport a direct band gap shift of 115 meV for the ‘ref-erence’ pseudopotential, with the same generation con-ditions as in Giustino et. al and in C in the presentstudy. This is virtually identical to the DW zero-pointshift of 117 meV reported in Giustino et. al . Thus,the differences in the total (DW+SE) zero-point shiftsbetween Giustino et. al and Ponc´e et. al is due tothe SE contribution. The DW contribution to the zero-point direct band gap shift for the C pseudopotential inthe present study is 143 meV. This value can be directlycompared and is ∼
26 meV higher than that reported inthe above studies .
2. Finite temperature band gap shifts
The DW contribution to the band gap shifts calculatedin the present study can also be compared with resultsfrom Allen-Heine theory at high temperatures, where thehigher order terms may become important. We comparethe the DW contribution to the direct band gap shiftsobtained with the C pseudopotential since it has thesame generation configuration as in literature .Our calculated DW band gap shift of 152 meV at 200K, 164 meV at 300 K and 231 meV at 600 K compareswell with the reported DW contribution to the directband gap shift of ∼
160 meV at 200 K, 190 meV at 300K and 280 meV at 600K in literature . However, aninversion must be noted. Our finite temperature DWband gap shifts are smaller while our zero-point shiftsare larger compared to literature values .As seen in the previous section, there is a difference of ∼
26 meV between the results obtained using the Allenand Allen-Heine theory in the DW zero-point band gapshifts. Further insight can be gained if the differencesare compared with respect to the zero-point shifts whicheliminates the effect of the different zero-point band gapshifts. This helps to examine the increase in the bandgap shifts due to the increase in the mean-square dis-placements with temperature.Eliminating the zero-point shifts, the additional DWband gap shift in our study is 9 meV at 200 K, 21 meVat 300 K and 87 meV at 600 K. These small increases inthe band gap shifts with temperature can be attributedto the fact that the (cid:104) u i (cid:105) changes very slowly with tem-perature for diamond (Table I). For example, (cid:104) u i (cid:105) variesfrom 0.00161 ˚A (0.001 K) to 0.00169 ˚A (200 K) to0.0018 ˚A (300 K) at low temperatures.In comparison, in Giustino et. al , after eliminat-ing the zero-point shifts, the additional DW band gapshifts are ∼
40 meV at 200 K, 70 meV at 300 K and160 meV at 600 K. Compared to our results above, therelatively large band gap shifts for very small increasesin the mean-square displacement values (especially upto300 K) appear excessive.
C. Silicon
Silicon, a small band gap semiconductor used ex-tensively in a wide range of technology applications iswell studied experimentally as well as theoretically andis of interest for its temperature dependent properties.We calculated the finite temperature thermally-averagedelectronic structure of silicon at various temperatureslisted in Table I by using the DW factor modified pseu-dopotential. We performed the finite temperature calcu-lations for all pseudopotentials listed in Table VI.Table VIII lists the zero-point shifts in the indirect anddirect band gap for all the temperature transferable pseu-dopotentials. It shows that pseudopotentials Si -Si ,which differ only in the valence state occupancies of pelectrons, have similar zero-point shifts. The band gapshifts are virtually identical for Si pbe − gipaw (Table IV)and Si not only for zero-point vibrations but also forall other temperatures in our study. The band gap shiftsincrease monotonically for Si - Si indicating a strongdependence on the cut-off radius as seen in Figure 2. PP a Band gapName Static lattice Zero point shift(a.u.) (eV) (meV)Indirect Direct Indirect DirectSi pbe − gipaw The static lattice direct (2.56 eV) and indirect (0.63eV) band gaps for these pseudopotentials are very closeto literature values of 2.55 eV and 0.62 eV respectively.Si , Si pbe − gipaw , Si with higher cut-off radius havehigher indirect band gap shifts and Si , Si with a lowercut-off radius have lower band gap shifts. As observed indiamond, the indirect band gap shifts are highly sensitiveto the cut-off radius used in the pseudopotentials.Previous literature values for the total (DW + SE) shiftof the indirect band gap due to zero-point vibrations are57 meV , 60 meV and 64.3 meV and the experi-mental values are 62-64 meV . For the direct band gap,the reported values are 22 meV and 47 meV and theexperimental values are 25 ± .Previous studies , though based on empirical pseu-dopotentials, suggest that in silicon the DW term overes-1 B a nd ga p s h i f t( m e V ) Si pbe − gipaw -ISi pbe − gipaw -DSi -ISi -DSi -ISi -DSi -ISi -D FIG. 6: The direct (D) and indirect (I) band gap shifts insilicon due to the DW term with increasing temperature fordifferent temperature transferable pseudopotentials timates the band gap shifts. The SE term is of oppositesign so that the total band gap shifts are less than thoseobtained from just the DW term.The temperature transferable pseudopotentials in Ta-ble VIII give different magnitudes of zero-point bandgap shifts than in literature . This is due tothe strong sensitivity of the band gap shifts to the cut-offradius. This comparison shows that only the pseudopo-tentials Si -Si and Si give, band gap shifts that arethe closest to previous results.None of the ab initio studies on silicon separately re-port the DW contribution. However, Allen and Car-dona , in their study based on the Allen-Heine theorywith empirical pseudopotentials, report the DW contri-bution to the direct band gap shift at different tempera-tures upto 600 K. Our DW direct band gaps shifts (Fig-ure 6) in the temperature range upto 600 K are very simi-lar to these reported values . Allen and Cardona , how-ever, report negligible zero-point direct band gap shift,while we report a finite value.Figure 6 plots the indirect and direct band gap shiftsfor the four silicon pseudopotentials, Si , Si , Si pbe − gipaw and Si , at various temperatures. The indirect and di-rect band gap shifts exhibit a nonlinear behavior at lowtemperatures and a linear behavior at higher tempera-tures as seen in experimental studies .The effect of temperature on Si electronic structure isinvestigated and in Figure 7 we plot the (static latticeand thermally-averaged) band structure of silicon for thepseudopotential Si for 0 K, 300 K and 600 K. The ef-fect of temperature on the Si electronic structure is quiteinteresting with differing positive and negative shifts inthe energy with respect to the static lattice. The energyshifts at the different high symmetry points in the Bril- -4-3-2-1 0 1 2 3 Γ X W L Γ K E - E F ( e V ) FIG. 7: The calculated band structure (static lattice andthermally-averaged) of silicon at 0K, 300K and 600K in thevicinity of the Fermi level. The violet line represents the bandstructure at 0 K, the blue line at 300K and the red line at 600K. louin zone are significantly different for different temper-atures. The shifts in the energy values at different pointswithin any high symmetry line of the Brillouin zone withrespect to temperature are observed to vary.Figure 8 plots the density of states for the silicon pseu-dopotential Si at 0 K and 300 K. As in the case of di-amond, the temperature dependent pseudopotential af-fects not only the valence band and conduction band en-ergies near the Fermi level but throughout the valenceand conduction bands. For finite temperatures, the filledbands and the unfilled bands shift towards the Fermi en-ergy nonuniformly as seen in our studies on diamond. Ingeneral, the density of states broaden and shift towardsthe Fermi level. The amount of broadening and shiftincreases with increased temperature. D. Discussion of Allen and Allen-Heine theoryresults
The DW term of the electron-phonon interaction ismore accurately evaluated in the implementation of theAllen theory, whereas, it is approximated to the secondorder in the Allen-Heine theory. Considering this somediscrepancies in the band gap shift values obtained fromthese theories is expected. However, if the second-orderapproximation is considered to be valid, then both theo-ries should give similar band gap shifts.In order to separate the effect of the approximation tothe DW term from other input parameters used in thecalculations, it is essential that the same ab initio pseu-dopotential and mean square displacements of the atomsbe used. In the present study, we have used the samepseudopotential for diamond as in previous studies DO S ( a r b . un it s ) E-E F (eV) FIG. 8: The density of states of silicon at 0K (violet) and300K (green) based on the Allen-Heine theory. The mean square dis-placement values as in previous studies however, couldnot be used since they have not been reported.The implementation of Allen theory is based on us-ing explicit values of the mean-square displacements,whereas in the Allen-Heine theory the mean-square dis-placement values are not directly used and are implicit.In our study of finite temperature properties of diamondand silicon explicit values of the mean-square displace-ments obtained from ab initio studies have been used. Ifthe values of the mean-square displacements are differentfrom that in the earlier studies, then discrepancies can beexpected in the results from the two theories.In the case of diamond, we have used (Table I) thezero-point mean-square displacement values reported inSchowalter et al. . A similar value, for the zero-point vi-brations, has also been reported in Yang et al. . In boththese studies , the mean-square displacement valueshave been obtained from density-functional perturbationtheory. The previous Allen-Heine based studies thatreported the DW contribution to the direct band gapshift also use the density-functional perturbation theoryto account for thermal vibrations, though explicit valuesof the zero-point mean-square displacements are not re-ported. Since there are some discrepancies in the DWband gap shifts for diamond (as seen earlier), it wouldbe of interest to compare the zero-point mean-square dis-placements implicit in the Allen-Heine theory based stud-ies with the explicit values reported in literature that are used in our study based on the Allen theory.If the mean-square displacement values used in the twotheories are the same, then the discrepancies, if any, arelikely due to the different numerical implementations ofthe electron-phonon contribution to the electronic ener-gies in the two theories.Clearly, further studies are needed to fully understand the results obtained using the Allen and Allen-Heinetheories. Such studies for any semiconductor, howevermust necessarily use temperature transferable pseudopo-tentials. E. Finite temperature valence charge density
Finite temperature charge densities are of interest,both theoretically and experimentally, to study bond-ing in materials. At present, there is no simple ab ini-tio method to obtain finite temperature valence electronwavefunctions or charge densities. Currently, finite tem-perature charge densities are obtained from static latticecalculations with empirical approaches to incorporate theeffect of thermal vibrations as discussed below.The valence electron densities have been experimen-tally obtained for several materials from x-ray diffrac-tion and electron diffraction studies . In particular,silicon and diamond are considered to be the prototypematerials in experimental (valence) electron density stud-ies . In these studies, the ‘fundamental step’ is tocorrect for the observed intensities at finite temperatureswith the DW factor . Following this correction, aneffective static lattice charge density is obtained. Sub-sequently, these charge densities are compared with the-oretical ab initio calculations performed for static lat-tice .Alternately, the theoretical charge density from staticlattice calculations are altered by incorporating thermalvibrations through the DW factors and the finite tem-perature charge densities or structure factors so obtainedare compared with finite temperature experimental val-ues .Fundamental to these approaches is the assumption ofthe rigid pseudo-atom approximation , viz. the elec-tron density, including the valence electron density, canbe partitioned to individual atoms and each such segmentis frozen and moves rigidly with the thermal vibrations.This leads to the DW factor correction, the ‘fundamentalstep’ in the charge density studies. We note that Jonesand March had raised concerns about the validity of therigid pseudo-atom approximation for valence electrons.In the rigid pseudo-atom approximation, since the va-lence charge densities move rigidly with thermal motion,it implies that there is no change in the valence electronwavefunctions and energies. It follows that the band gapsmust be unchanged with temperature. This is contrary toexperimental and theoretical observations . Thisis a simple demonstration of the incorrectness of the rigidpseudo-atom approximation for valence electrons in sili-con and diamond. Hence, at present, there is no simpleand theoretically justified method to obtain finite tem-perature valence charge densities.In the Allen theory , the first step is to include theDW term in the pseudopotential form factor (Eq. 6) andthen calculate the electronic structure. That is, the wave-functions and charge densities of valence electrons so ob-3 (a)(b)0 K 300 K 600 K FIG. 9: The charge density along the (110) plane for (a) diamond and (b) silicon. The charge density of the static lattice (0K) is shown in the left panel, at 300 K in the middle panel and at 600 K in the right panel for both the materials tained are temperature dependent and not frozen. Thus,the Allen theory provides a simple way to go beyond therigid pseudo-atom approximation to theoretically obtainfinite temperature valence charge densities.Further, in the Allen theory , the SE corrections to theelectron energies are calculated from the results of theabove step. That is, there is no further correction of thewavefunctions. Hence, the first (DW) step of the Allentheory is sufficient to obtain finite temperature valencecharge densities.Figure 9 plots the (pseudo) valence charge densitiesof (a) diamond (C ) and (b) silicon (Si ) at 0 K, 300K and 600 K. It shows that the charge distribution inthe bonding region is enhanced with an increase in tem-perature. In general, the charge delocalization in siliconincreases with increasing temperature. It is interestingto note that the differences in the charge density distri-bution at 300 K and 600 K compared to that at 0 Kare more for silicon than diamond. This is due to themuch higher mean square displacements of silicon com-pared to diamond (Table I). In silicon, the charge densitydistribution at finite temperatures shows more accumu-lation of charges in the bonding region with a reductionin the charge density in the vicinity of the core region.With better exchange correlation functionals, which re-duce the underestimation of the band gaps, a more ac-curate charge density distribution would be obtained. Indiamond, a similar effect is observed with a minor gainin the charge densities in the covalent bonding region.As is well known , the pseudopotential method only gives pseudo-valence charge densities. They are incorrectnear the nuclei. However, away from the nuclei and inthe valence region, the pseudo-valence charge densitiesare similar to the true valence charge densities . Thefinite temperature charge density information in the va-lence region is valuable as this is the region where bond-ing effects are strongly manifested. The pseudo-valencecharge densities plotted in Fig. 9 thus go beyond therigid pseudo-atom approximation. Therefore, if the ex-perimental data can also be analyzed appropriately toobtain finite temperature experimental charge densities,they can be compared with the theoretical results ob-tained from the Allen theory, a comparison that goesbeyond the rigid pseudo-atom approximation. V. CONCLUSIONS
In this paper, we report the implementation of theAllen theory with ab initio pseudopotentials in den-sity functional total energy calculations to obtain thefinite temperature thermally-averaged electronic struc-ture without any additional increase in the computa-tional complexity and cost. Our results on diamond andsilicon show that both the direct and indirect band gapsexhibit the “Varshni effect” with temperature only whenthe ab initio pseudopotentials satisfy an additional cri-terion, that of temperature transferability. The tem-perature transferability criterion is satisfied only in asmall subset of the parameter space of static lattice in4Troullier-Martins pseudopotentials for diamond and sil-icon. In these materials, the temperature transferabil-ity is strongly affected by the choice of the cut-off radiusand inclusion/exclusion of unbound 3d state in the pseu-dopotential generation configuration. The finite temper-ature indirect band gaps in diamond and silicon are seento be highly sensitive to the choice of cut-off radius.Our results on finite temperature electronic structureshow that the thermal vibrations affect the electron en-ergies throughout the valence and conduction bands. Wehave calculated the zero-point and higher temperatureband gap shifts in diamond and silicon for temperaturesup to 1000 K. We compared our results on direct bandgap shifts with those obtained using the Allen-Heine the-ory for the contribution from the Debye-Waller term. Fordiamond, our zero-point shifts in the direct band gaps arehigher by 26 meV for the same ab initio pseudopoten-tials. The finite temperature direct band gap shifts alsoshow noticeable discrepancies. For silicon, we see a goodagreement in the band gap shifts with the only results re-ported in literature (for the DW contribution) using em-pirical pseudopotentials. Further studies using the sametemperature transferable ab initio pseudopotentials and mean square displacements are essential to understandthe discrepancies in the Allen and Allen-Heine theoryband gap shifts, especially since the former, besides beingmore accurate, has a simpler numerical implementation.The inclusion of Debye-Waller correction using theAllen theory provides a simple and theoretically justifiedformalism to obtain finite temperature valence electroncharge densities. The finite temperature charge densitiesobtained using Allen theory go beyond the rigid pseudo-atom approximation, a limitation of the present methodsused in charge density studies. VI. ACKNOWLEDGEMENT
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