Exploring positron characteristics utilizing two new positron-electron correlation schemes based on three electronic-structure calculation methods
Wen-Shuai Zhang, Bing-Chuan Gu, Xiao-Xi Han, Jian-Dang Liu, Bang-Jiao Ye
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J un Exploring positron characteristics utilizing two new positron-electron correlation schemes based onmultiple electronic-structure calculation methods
Wen-Shuai Zhang, ∗ Bing-Chuan Gu,
1, 2, † Xiao-Xi Han,
1, 2, ‡ Jian-Dang Liu,
1, 2, § and Bang-Jiao Ye
1, 2, ¶ Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China State Key Laboratory of Particle Detection and Electronics, USTC, Hefei 230026, China
We make a gradient correction to a new local density approximation form of positron-electron correlation.Then the positron lifetimes and a ffi nities are probed by using these two approximation forms based on threeelectronic-structure calculation methods including the full-potential linearized augmented plane wave (FLAPW)plus local orbitals approach, the atomic superposition (ATSUP) approach and the projector augmented wave(PAW) approach. The di ff erences between calculated lifetimes using the FLAPW and ATSUP methods areclearly interpreted in the view of positron and electron transfers. We further find that a well implemented PAWmethod can give near-perfect agreement on both the positron lifetimes and a ffi nities with the FLAPW method,and the competitiveness of the ATSUP method against the FLAPW / PAW method is reduced within the bestcalculations. By comparing with experimental data, the new introduced gradient corrected correlation form isproved competitive for positron lifetime and a ffi nity calculations. PACS numbers: 78.70.Bj, 71.60. + z, 71.15.Mb I. INTRODUCTION
In recent decades, the Positron Annihilation Spectroscopy(PAS) has become a valuable method to study the microscopicstructure of solids [1–3] and gives detailed information on theelectron density and / or momentum distribution [4] in the re-gions scanned by positrons. An accompanying theory is re-quired for a thorough understanding of experimental results.A full two-component self-consistent scheme [5, 6] has beendeveloped for calculating positron states in solids based onthe density functional theory (DFT) [7]. Especially in bulkmaterial where the positron is delocalized and does not af-fect the electron states, the full two-component scheme can bereduced without losing accuracy to the conventional scheme[5, 6] in which the electronic-structure is determined by com-mon one-component formalism. However, there are variouskinds of approximations can be adjusted within this calcu-lations. To improve the analyses of experimental data, oneshould find out which approximations are more credible toproduce the positron state [8–10] . In this short paper, we fo-cus on probing the positron lifetimes and a ffi nities by usingtwo new positron-electron correlation schemes based on threeelectronic-structure calculation methods.Recently, N. D. Drummond et al. [11, 12] made two cal-culations for a positron immersed in a homogeneous elec-tron gas, by using the Quantum Monte Carlo (QMC) methodand a modified one-component DFT method, and then twoforms of local density approximations (LDA) on the positron-electron correlation are derived. Kuriplach and Barbiellini ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] supported by National Natural Science Foundation of China (GrantNos. 11175171 and 11105139). [8, 9] proposed a fitted LDA form and a generalized gradi-ent approximation (GGA) form based on previous QMC cal-culation, and then applied these two forms to multiple cal-culations for positron characteristics in solid. However, theLDA form based on the modified one-component DFT calcu-lation has not been studied. In this work, we make a gradi-ent correction to the IDFTLDA form and validate these twonew positron-electron correlation schemes by applying themto multiple positron lifetimes and a ffi nities calculations.Besides, we probe in detail the e ff ect of di ff erent electronic-structure calculation methods on positron characteristics insolid. These methods include the full-potential linearized aug-mented plane-wave (FLAPW) plus local orbitals method [13],the projector augmented wave (PAW) method[14], and theatomic superposition (ATSUP) method [15]. Among thesemethods, the FLAPW method is regarded as the most ac-curate method to calculate electronic-structure, the ATSUPmethod performs with the best computational e ffi ciency, thePAW method has greater computational e ffi ciency and closeaccuracy as the FLAPW method but has not been com-pletely tested on positron lifetimes and a ffi nities calcula-tions except some individual calculations [16–19]. More-over, our previous work [20] showed that the calculatedlifetimes utilizing the PAW method disagree with that uti-lizing the FLAPW method. However, within those PAWcalculations, the ionic potential was not well constructed.In this paper, we investigated the influences of the ionicpseudo-potential / full-potential and di ff erent electron-electronexchange-correlations approaches within the PAW calcula-tions. Especially, the di ff erence between calculated life-times by using the self-consistent (FLAPW) and non-self-consistent (ATSUP) methods is clearly investigated in theview of positron and electron transfers.This paper is organized as follows: In Sec. 2, we give abrief and overall description of the models considered hereas well as the computational details and the analysis meth-ods we used. In Sec. 3, we introduce the experimental dataon positron lifetime used in this work. In Sec. 4, we firstlyapply all approximation methods for electronic-structure andpositron-state calculations to the cases of Si and Al, andgive detailed analyses on the e ff ects of these di ff erent ap-proaches, and then assess the two new correlation schemesby using the positron lifetime / a ffi nity data in comparison withother schemes base on di ff erent electronic-structure calcula-tion methods. II. THEORY AND METHODOLOGYA. Theory
In this section, we briefly introduce the calculation schemefor the positron state and various appproximations investi-gated in this work. Firstly, we do the electronic-structure cal-culation without considering the perturbation by positron toobtain the ground-state electronic density n e − ( ~ r ) and Coulombpotential V Coul ( ~ r ) sensed by positron. Then, the positron den-sity is determined by solving the Kohn-Sham Eq.:[ − ∇ ~ r + V Coul ( ~ r ) + V corr ( ~ r )] ψ + = ε + ψ + , n e + ( ~ r ) = | ψ + ( ~ r ) | , (1)where V corr ( ~ r ) is the correlation potential between electronand positron. Finally, the positron lifetime can be obtainedby the inverse of the annihilation rate, which is proportionalto the product of positron density and electron density accom-panied by the so-called enhancement factor arising from thecorrelation energy between a positron and electrons [21]. Theequations are written as follows: τ e + = λ , λ = π r c Z d ~ rn e − ( ~ r ) n e + ( ~ r ) γ ( n e − ) , (2)where r is the classical electron radius, c is the speed of light,and γ ( n e − ) is the enhancement factor of the electron densityat the position ~ r . The positron a ffi nity can be calculated byadding electron and positron chemical potentials together: A + = µ − + µ + . (3)The positron chemical potential µ + is determined by thepositron ground-state energy. The electron chemical poten-tial µ − is derived from the Fermi energy (top energy of thevalence band) in the case of a metal (a semiconductor). Thisscheme is still accurate for a perfect lattice, as in this case thepositron density is delocalized and vanishingly small at everypoint thus does not a ff ect the bulk electronic-structure [6, 21]. TABLE I: Parameterized LDA / GGA correlation schemes. γ a a a / a / a / α IDFTLDA 4.1698 0.1737 -1.567 -3.579 0.8364 0IDFTGGA 4.1698 0.1737 -1.567 -3.579 0.8364 0.143fQMCLDA -0.22 1 / / / / In practice of this work, each enhancement factor is appliedidentically to all electrons as suggested by K. O. Jensen [22].These enhancement factors can be divided into two categories:the local density approximation (LDA) and the generalizedgradient approximation (GGA), and parameterized by the fol-lowing equation, γ = + (1 . r s + a r s + a r s + a / r / s + a / r / s + a / r / s ) e − αǫ , (4)here, r s is defined by r s = (3 / π n e − ) / , ǫ is defined by ǫ = |∇ ln( n e − ) | / q ( q − is the local Thomas-Fermi screeninglength), a , a , a / , a / , a / , a / and α are fitted parame-ters. We investigated five forms of the enhancement factor andcorrelation potential marked by IDFTLDA [12], fQMCLDA[8, 9], fQMCGGA [8, 9], PHCLDA [23] and PHCGGA [24],plus a new GGA form IDFTGGA introduced in this workbased on the IDFTLDA scheme. The fitted parameters ofthese enhancement factors are listed in Table I . The LDAforms of V corr corresponding to IDFTLDA, fQMCLDA, PH-CLDA are given in Refs. [12], [8] and [25], respectively.Within the GGA, the corresponding correlation potential takesthe form V GGA corr = V LDA corr e − αǫ/ [26, 27]. The electronic den-sity and Coulomb potential were calculated by using variousmethods including: a) the all-electron full potential linearizedaugmented plane wave plus local orbitals (FLAPW) method[13] as implemented in Ref.[8] being regarded as the mostaccurate method to calculate electronic-structure, b) the pro-jector augmented wave (PAW) method [14] with reconstruc-tion of all-electron and full-potential performing with greatercomputational e ffi ciency and close accuracy as the FLAPWmethod, c) the non-self-consistent atomic superposition (AT-SUP) method [15] performing with the best computational ef-ficiency. B. Computational details
During the calculations for electronic-structure, three meth-ods mentioned above are implemented in this work. ForFLAPW calculations, the WIEN2k code [28] was used, thePBE-GGA approach [29] was adopted for electron-electronexchange-correlations, the total number of k-points in thewhole Brillouin zone (BZ) was set to 3375, and the self-consistency was achieved up to both levels of 0.0001 Ry fortotal energy and 0.001 e for charge distance. For PAW cal-culations, the PWSCF code within the Quantum ESPRESSOpackage [30] was used, the PBEsol-GGA [31] and PZ-LDA[32] approaches were also implemented for electron-electronexchange-correlations besides the PBE-GGA approach, thePAW pseudo-potential files named
PSLibrary 0.3.1 and gen-erated by A. D. Corso (SISSA, Italy) were employed [33],the k-points grid was automatically generated with the param-eter being set at least (333) in Monkhorst-Pack scheme, thekinetic energy cut-o ff of more than 100 Ry (400 Ry) for thewave-functions (charge density) and the default convergencethreshold of 10 − were adopted for self-consistency. For AT-SUP calculations, the electron density and Coulomb potentialfor each material were simply approximated by the superposi-tion of the electron density and Coulomb potential of neutralfree atoms [15], while the total number of the node pointswas set to the same as in PAW calculations. Besides, the2 × × ff erence method while the unit cell ofeach material was divided into about 10 mesh spaces per bohr in each dimension. All important variable parameters werechecked carefully to achieve that the computational precisionof lifetime and a ffi nities are the order of 0.1 ps and 0.01eV,respectively. C. Model comparison
An appropriate criterion must be chosen to make a com-parison between di ff erent models. The root mean squared de-viation (RMSD) is the most popular one and defined as thesquare root of the mean of the squared deviation between ex-perimental and theoretical results: RMSD = [ P Ni = ( X exp i − X theo i ) / N ] / , here N denotes the number of experimental val-ues. In addition, since the theoretical values can be treated tobe noise-free, the simple mean-absolute-deviation (MAD) de-fined by MAD = P Ni = [ | X modelA i − X modelB i | / N ] is much moremeaningful to quantify the overall di ff erences between calcu-lated results by using various models. It is obvious that theexperimental data favor models producing lower values of theRMSD. III. EXPERIMENTAL DATA
Up to five recent observed values from di ff erent literaturesand groups for 21 materials were gathered to compose a re-liable experimental data set. All the experimental values foreach material investigated in this work are collected basicallyby using the standard suggested in Ref. [57] and listed inTable II with their standard deviation. Furthermore, the mate-rials having less than five experimental measurements and / orthe older experimental data were avoided being adopted. Itis reasonable to suppose that these materials having insu ffi -cient and / or unreliable experimental data would disrupt thecomparison between models. Especially, the measurementsfor alkali-metals reported before 1975 are not suggested tobe treated seriously [8]. The deviations of experimental re-sults between di ff erent groups are usually much larger thanthe statistical errors, even when just the recent and reliablemeasurements are considered. That is, the systematic error isthe dominant factor, so that the sole statistical error is far fromenough and not used in this work. However, the systematic er-ror is di ffi cult to derive from single experimental result. So inthis paper, the average experimental values of each materialwere used to assess the positron-electron correlation models,and the systematic errors are expected to be cancelled as inRef. [57]. Because the observed values for defect state areinsu ffi cient and / or largely scattered, it is hard to make a clear TABLE II: The experimental values of lifetime τ exp , the related meanvalue τ ∗ exp and the corresponding standard deviation σ exp for eachmaterial involved in this work. Material τ exp τ ∗ exp σ exp Si 216.7[34] 218[34] 218[34] 222[34] 216[34] 218.1 2.323Ge 220.5[34] 230[34] 230[34] 228[34] 228[34] 227.3 3.931Mg 225[35] 225[34] 220[34] 238[34] 235[34] 228.6 7.569Al 160.7[34] 166[34] 163[34] 165[34] 165[34] 163.9 2.114Ti 147[35] 154[34] 145[34] 152[34] 143[34] 148.2 4.658Fe 108[34] 106[34] 114[34] 110[34] 111[34] 109.8 3.033Ni 109.8[34] 107[34] 105[34] 109[34] 110[34] 108.2 2.127Zn 148[35] 153[34] 145[34] 154[34] 152[34] 150.4 3.781Cu 110.7[34] 122[34] 112[34] 110[34] 120[34] 114.9 2.514Nb 119[34] 120[34] 122[34] 122[34] 125[34] 121.6 2.302Mo 109.5[34] 103[34] 118[34] 114[34] 104[34] 109.7 6.418Ta 116[35] 122[34] 120[34] 125[34] 117[34] 120.0 3.674Ag 120[34] 130[34] 131[34] 133[36] 131[35] 129.0 5.147Au 117[34] 113[34] 113[34] 117[34] 123[34] 116.6 4.098Cd 175[35] 184[34] 167[34] 172[34] 186[34] 176.8 8.043In 194.7[34] 200[34] 192[34] 193[34] 189[34] 193.7 4.066Pb 194[35] 200[34] 204[34] 200[34] 209[34] 201.4 5.550GaAs 231.6[37] 231[38] 230[39] 232[40] 220[41] 228.9 5.043InP 241[42] 240[43] 247[44] 242[45] 244[46] 242.8 2.775ZnO 153[47] 159[48] 158[49] 161[50] 171[51] 160.4 6.618CdTe 284[52] 285[53] 285[54] 289[55] 291[56] 286.8 3.033 discussion on the defect state by using these positron-electroncorrelation models in this short paper. Thus, except the de-tailed analyses in the cases of Si and Al based on three usu-ally applied approaches for electronic-structure calculations,we mainly focus on testing the correlation models by usingbulk materials’ lifetime data and positron-a ffi nity data. Theexperimental data of positron a ffi nity are listed in Table V. IV. RESULTS AND DISCUSSIONA. Detailed analyses in cases of Si and Al
Representatively, the panels (a) and (c) in Fig. 1 (Fig. 2)show respectively the self-consistent all-electron and positrondensities on plane (110) for Al (Si) based on the FLAPWmethod together with the fQMCGGA form of the enhance-ment factor and correlation potential. It is reasonable to ob-tain that the panel (a) in Fig. 2 shows clear bonding statesof Si while the panel (a) in Fig. 1 shows the presence ofthe nearly free conduction electrons in interstitial regions. Tomake a comparison between the FLAPW and ATSUP methodfor electronic-structure calculations, we also plot the ratio oftheir respective all-electron and positron densities in panel (b)and (d) in Fig. 1 (Fig. 2) for Al (Si). These four ratio panelsactually reflect the electron and positron transfers from densi-ties based on the non-self-consistent free atomic calculationsto that based on the exact self-consistent calculations. It con-firms the fact that the positron density follows the changes ofthe electron density which yield not a big di ff erence in anni- (a) ρ e- FLAPW (c) ρ e+ FLAPW (b) ρ e- FLAPW / ρ e- ATSUP (d) ρ e+ FLAPW / ρ e+ ATSUP T e - Al T e+ Al FIG. 1: Left panels: the self-consistent all-electron (a) and positrondensities (c) (in unit of a.u.) on plane (110) for Al based on theFLAPW method and the fQMCGGA approximation. Right panels:the ratios of all-electron (b) or positron densities (d) calculated byusing the FLAPW method to that by using the ATSUP method. (c) ρ e+ FLAPW (a) ρ e- FLAPW /(b) ρ e- FLAPW ρ e- ATSUP (d) ρ e+ FLAPW / ρ e+ ATSUP T e- Si T e+ Si FIG. 2: As Fig. 1, but for Si. hilation rate between these two calculations [15].Now, taking more subtle analyses, the change of lifetimewithin the FLAPW calculation from that within the ATSUPcalculation for Al is attributed to the competition between thefollowing two factors: a) the lifetime is decreased by the trans-lations of electrons (illustrated in Fig. 1 (b) as T e − Al ) fromnear-nucleus regions with tiny positron densities to intersti-tial regions with large positron densities, b) the lifetime is in-creased by the translation of positron (illustrated in Fig. 1 (d)as T e + Al ) from core regions with large electron densities to in-terstitial regions with small electron densities. However, inthe case of Si with bonding states, the change of lifetime de-pends conversely on the translations of electrons and positron:a) the lifetime is increased by the translations of electrons (il-lustrated in Fig. 2 (b) as T e − Si ) from interstitial regions with thelargest positron densities to bonding regions with tiny positron V e + [100] (Angstrom) V FP V PP V FLAPW ρ e + − − ρ e+ (V FP ) ρ e+ (V PP ) (a) V e + [100] (Angstrom) V FP V PP V FLAPW ρ e + − − ρ e+ (V FP ) ρ e+ (V PP ) (b) FIG. 3: The total Coulomb potential V e + (in unit of Ry) sensed bythe positron based on the ionic pseudo-potentials (V PP ) and recon-structed ionic full-potential (V FP ) and the corresponding calculatedpositron densities ρ e + (in unit of a.u.) along the [100] direction be-tween two adjacent atoms for Al (a) and Si (b), respectively. Tomake a further comparison, the full-potentials calculated by usingthe FLAPW method (V FLAPW ) are also plotted. densities, b) the lifetime is decreased by the translation ofpositron (illustrated in Fig. 2 (d) as T e + Si ) from interstitialregions with tiny electron densities to bonding regions withlarge electron densities. Taking note of the magnitude of scalerulers, these two figures state clearly that the translations ofelectrons ( T e − ) are dominant factors for both Al and Si. Con-sequently, the lifetimes within the FLAPW calculations be-come smaller (larger) for Al (Si). These variances are provedby calculated values of lifetimes listed in Table III. In addi-tion, the lifetimes of Si calculated by using three GGA formsof the enhancement factor show greater di ff erences since thelarge electron-density gradient terms in bonding regions giv-ing decreases of the enhancement factor can further weakenthe e ff ect of the translation T e + Si .We calculated the bulk lifetimes for Al and Si based on thePAW method. Within the Table III, the label ”PAW” withouta su ffi x indicates that the electron-structure is calculated byusing the PBE-GGA electron-electron exchange-correlationsapproach [29] and positron-state is calculated by using re-constructed ionic full-potential (FP), the su ffi x ”-PZ” indi-cates that the PBE-GGA approach is replaced by the PZ-LDAapproach [32] during electron-structure calculations, and thesu ffi x ”-PP” indicates that ionic full-potential (FP) is replacedby the ionic pseudo-potential (PP) during positron-state cal-culations. The ionic potential together with the Hartree po-tential from the valence electrons compose the total Coulombpotential in Eq. (1). It can be easily found that the better im-plemented PAW method by using reconstructed full-potentialcan give a startling agreement with the FLAPW method on thepositron-lifetime calculations for Al and Si. By comparing theresults of PAW and PAW-PP approach, the PAW-PP approachleads to smaller lifetimes with the di ff erences up to 3.8 ps and4.3 ps for Al and Si respectively. These decreases are causedby the fact that the softer potential within the PAW-PP ap-proach more powerfully attracts positron into the near-nucleusregions with much larger electron densities. This statement isillustrated by the Fig. 3 showing the total Coulomb potentialV e + sensed by the positron based on the ionic pseudo-potential(V PP ) and reconstructed ionic full-potential (V FP ) and the cor-responding calculated positron densities ρ e + along the [100]direction between two adjacent atoms for Al (a) and Si (b), re-spectively. To make a further comparison, the full-potentialscalculated by using the FLAPW method (V FLAPW ) are alsoplotted and found nearly the same as the reconstructed PAWfull-potentials. This figure indicates that a change in the ionicpotential approachs (FP or PP) can lead to a change of morethan one order of magnitude in the positron densities nearthe nuclei. It should be noted that, in cases of PAW calcu-lations with underestimated core / semicore electron densitesin the near-nucleus regions [58], the e ff ect of overestimatedpositron densities based on the pseudo-potentials can be can-celled, and then excellent quality on the calculated positronlifetimes is able to be achieved. It is clear that the di ff erencesbetween the results of PAW-PZ and PAW are of the order of0.1 ps, and therefore the e ff ect of di ff erent electron-electronexchange-correlations schemes is small. More than this, wealso calculated the lifetimes by using the PBEsol-GGA ap-proach [31] which is revised for solids and their surfaces, andthe similar di ff erences of the order of 0.1 ps are also obtainedcompared with the PBE-GGA approach. TABLE III: Calculated results of positron lifetimes (in unit of ps) forAl, Si, and ideal monovacancy in Al and Si based on various methodsfor electronic-structure and positron-state calculations.
IDFT IDFT fQMC fQMC PHC PHCGGA LDA GGA LDA GGA LDAAl ATSUP 160.778 152.470 173.347 169.357 163.036 156.438FLAPW 156.615 149.852 169.972 166.530 159.397 153.878PAW 156.649 149.898 170.016 166.584 159.432 153.925PAW-PP 154.113 146.814 166.507 162.798 156.574 150.587PAW-PZ 157.208 150.204 170.421 166.906 159.898 154.220Si ATSUP 201.770 186.634 213.260 207.345 201.363 190.484FLAPW 211.843 188.285 217.520 208.477 208.639 191.790PAW 211.779 188.245 217.466 208.431 208.586 191.752PAW-PP 208.407 184.675 213.320 204.125 205.060 187.976PAW-PZ 211.248 188.388 217.399 208.625 208.247 191.905V Al ATSUP 229.441 216.639 246.294 240.941 229.686 220.274PAW 212.176 201.245 229.481 224.429 214.050 205.570V Si ATSUP 227.458 208.972 239.524 232.309 225.922 212.690PAW 236.052 208.712 241.816 231.443 231.504 212.145
In addition, as shown in Table III, the positron lifetimesfor monovacancy in Al and Si are also calculated based onthe ATSUP and PAW methods for electronic-structure calcu-lations and six correlation schemes for positron-state calcula- tions. The ideal monovacancy structure is used in these cal-culations, which means that the positron is trapped into a sin-gle vacancy without considering the ionic relaxation from theideal lattice positions. Larger di ff erences between the resultsof ATSUP and PAW are found in monovacancy-state calcula-tions compared with that in bulk-state calculations. Besides,the IDFTGGA / IDFTLDA correlation schemes produce sim-ilar lifetime values compared with the PHCGGA / PHCLDAcorrelation schemes and produce much smaller lifetime val-ues compared with the fQMCGGA / fQMCLDA correlationschemes in both monovacancy-state and bulk-state calcula-tions. B. Positron lifetime calculations
In this subsection we firstly give visualized comparisons be-tween experimental values and calculated results based on dif-ferent methods for electronic-structure and positron-state cal-culations. Within the PAW, the positron lifetimes are all cal-culated by using the reconstructed full-potential and certainlyall-electron densities from now on. τ t h e o - τ * e x p ( p s ) − − − − S i G e M g A l T i F e N i Z n C u N b M o T a A g A u C d I n P b G a A s I n P Z n O C d T e FLAPW PAW ATSUPIDFTGGA IDFTLDA fQMCGGA fQMCLDA PHCGGA PHCLDA
FIG. 4: The deviations of the theoretical results based on variousmethods from the experimental values alongwith the standard devia-tion of experimental values for each material.TABLE IV: The MADs between the calculated results by using theATSUP / PAW method and that by using the FLAPW method. Andthe RMSDs between the theoretical results and the experimental data τ ∗ exp . MAD [ps] RMSD [ps]ATSUP PAW FLAPW PAW ATSUPfQMCGGA 2.503 0.303 4.503 4.591 6.309IDFTGGA 5.068 0.316 4.809 4.821 5.611PHCGGA 3.667 0.287 6.148 6.013 7.672fQMCLDA 2.184 0.290 11.36 11.19 10.35IDFTLDA 1.966 0.253 25.19 24.99 23.88PHCLDA 1.936 0.260 22.83 22.63 21.54
The deviations of the theoretical results from the experi-mental data alongwith the standard deviations of observed val-ues for all materials are plotted in Fig. 4. The scattering re-gions of calculated results by di ff erent forms of the enhance-ment factor are found much larger in the atom systems withbonding states compared with that in pure metal systems. Be-sides, the deviations of the results by using the ATSUP methodfrom those by using the FLAPW method are mostly larger inGGA approximations compared with those in LDA approxi-mations. Numerically, the MADs for di ff erent forms of theenhancement factor between the calculated lifetimes by usingthe ATSUP method and those by using the FLAPW methodare shown in Table IV. These MADs range from 1.936 ps(PHCLDA) to 5.068 ps (IDFTGGA). Moreover, the well im-plemented PAW method is found able to give nearly the sameresults as the FLAPW method. Numerically, the MADs be-tween the calculated lifetimes by the PAW method and thoseby the FLAPW method for di ff erent forms of the enhancementfactor are also shown in Table IV. These MADs range from0.253 ps (IDFTLDA) to 0.316 ps (IDFTGGA). This near-perfect agreement between the PAW method and the FLAPWmethod proves our calculations are quite credible.Table IV also presents the RMSDs between the theoreticalresults and the experimental data τ ∗ exp by using six positron-electron correlation schemes. Two interesting phenomenacan be found in this table. Firstly, the RMSDs produced bythe IDFTLDA scheme are always worse among the RMSDsbased on three electron structure approaches, but are similarto those produced by the PHCLDA scheme. Thus, the gra-dient correction (IDFTGGA) to this LDA form (IDFTLDA)is needed. It is clear that the corrected IDFTGGA schemelargely improves the calculations, and performs better thanthe PHCGGA scheme, but is still worse than the fQMCGGAscheme. The fQMCGGA scheme together with the FLAPWmethod produced the best RMSD. This fact indicates that thequantum Monte Carlo calculation implemented in Ref. [11]is more credible than the modified one-component DFT cal-culation [12] on the positron-electron correlation. Secondly,compared to the RMSD produced by using the FLAPW / PAWmethod, the RMSD produced by using the simple ATSUPmethod is a little smaller based on the LDA correlationschemes, but is distinctly larger based on the GGA (espe-cially fQMCGGA) correlation schemes. This phenomenonimplies that the benefit of the exact eletronic-structure cal-culation approach (PAW / FLAPW) is swamped by the in-accurate approximation of the enhancement factor. Mean-while, the competitiveness of the ATSUP approach against theFLAPW / PAW method is reduced based on the most accuratepositron-electron correlation schemes.
C. Positron a ffi nity calculations The positron a ffi nity A + is a important bulk property whichdescribes the positron energy level in the solid, and allows usto probe the positron behavior in an inhomogeneous material.For example, the di ff erence of the lowest positron energies be-tween two elemental metals in contact is given by the positron a ffi nity di ff erence, and determines how the positron samplesnear the interface region. Besides, if the electron work func-tion φ − is known, the positron work function φ + can be de-rived by the equation: φ + = − φ − − A + . The crystal (e.g., Wmetal) having a stronger negative positron work function canemit slow-positron to the vacuum from the surface and there-fore be utilized as a more e ffi cient positron moderator for theslow-positron beam.The theoretical and experimental positron a ffi nities foreight common materials by using the new IDFTLDA andIDFTGGA correlation schemes are listed in Table V. To makea comparison, the results corresponding to the PHCGGA andfQCMGGA schemes are also listed. During the electron struc-ture calculation, the ATSUP method was not implementedbecause the ATSUP method is inappropriate for positron en-ergetics calculations and gives much negative positron workfunctions [15]. Within the PAW calculations, both the PBE-GGA and PZ-LDA approaches are used for electron-electronexchange-correlations. The RMSDs between the theoreticaland experimental positron a ffi nities are also presented in Ta-ble V.As in previous lifetime calculations, the calculated positrona ffi nities by using the FLAPW method are also near the sameas that by using the PAW method. Besides, our calculatedpositron a ffi nities by using the fQMCGGA & PZ-LDA ap-proaches are in excellent agreement with that reported in Ref.[8] with a MAD being 0.06 eV. Moreover, the di ff erencesbetween the RMSDs produced by using the PBE-GGA andPZ-LDA approaches, are not negligible, and the PBE-GGAapproach performs mostly better than the PZ-LDA approachexcept the case related to fQMCGGA. In addition, the gradi-ent correction (IDFTGGA) to the IDFTLDA form is neededto improve the performance for positron a ffi nity calculations.Meanwhile, the IDFTGGA correlation scheme makes dis-tinct improvement upon positron a ffi nity calculations com-pared with the PHCGGA scheme which is similar to the casesof positron lifetime calculations of bulk materials. Neverthe-less, the best agreement between the calculated and experi-mental positron a ffi nities is still given by the fQMCGGA &PZ-LDA approaches. V. CONCLUSION
In this work, we probe the positron lifetimes and a ffi ni-ties utilizing two new positron-electron correlation schemes(IDFTLDA & IDFTGGA) based on three common electronic-structure calculation methods (ATSUP & FLAPW &PAW).Firstly, we apply all approximation methods for electronic-structure and positron-state calculations to the cases of Si andAl, and give detailed analyses on the e ff ects of these dif-ferent approaches. Especially, the di ff erence between cal-culated lifetimes by using the self-consistent (FLAPW) andnon-self-consistent (ATSUP) methods is clearly investigatedin the view of positron and electron transfers. The well imple-mented PAW method with reconstruction of all-electron andfull-potential, is found being able to give near-perfect agree-ment with the FLAPW method, which proves our calculations TABLE V: Theoretical and experimental positron a ffi nities A + (in unit of eV) based on four positron-electron correlation schemes and severalelectron structure calculation methods. The RMSDs between the theoretical and experimental positron a ffi nities are also presented. Here, thePZ-LDA approach is labeled by PZ, and the PBE-LDA approach is labeled by PBE for short. A + IDFTGGA IDFTLDA PHCGGA fQMCGGA Exp.FLAPW PAW FLAPW PAW FLAPW PAW FLAPW PAWPBE PBE PZ PBE PBE PZ PBE PBE PZ PBE PBE PZSi -6.481 -6.478 -6.683 -6.884 -6.881 -7.070 -6.728 -6.726 -6.926 -6.182 -6.179 -6.373 -6.2Al -4.497 -4.504 -4.683 -4.624 -4.631 -4.813 -4.641 -4.648 -4.828 -3.981 -3.988 -4.169 -4.1Fe -3.914 -3.877 -4.290 -4.323 -4.289 -4.707 -4.120 -4.084 -4.498 -3.544 -3.508 -3.925 -3.3Cu -4.381 -4.437 -4.932 -4.875 -4.933 -5.435 -4.614 -4.671 -5.168 -4.073 -4.130 -4.630 -4.3Nb -3.847 -3.841 -4.085 -4.112 -4.107 -4.355 -4.020 -4.014 -4.260 -3.399 -3.394 -3.641 -3.8Ag -5.147 -5.083 -5.577 -5.670 -5.615 -6.109 -5.398 -5.337 -5.831 -4.875 -4.817 -5.310 -5.2W -1.956 -1.982 -2.304 -2.225 -2.254 -2.580 -2.121 -2.149 -2.472 -1.491 -1.520 -1.844 -1.9Pb -5.954 -5.936 -6.305 -6.328 -6.305 -6.683 -6.186 -6.166 -6.538 -5.622 -5.601 -5.977 -6.1RMSD 0.285 0.283 0.546 0.570 0.566 0.899 0.431 0.427 0.740 0.314 0.314 0.272 - are quite credible. While for ATSUP method, its competitive-ness against the FLAPW method is reduced within calcula-tions utilizing the best positron-electron correlation schemes(fQMCGGA). Then, we assess the two new positron-electroncorrelation schemes: the IDFTLDA form and the IDFTGGAform by using a reliable experimental data on the positron life-times and a ffi nities of bulk materials. The gradient correction(IDFTGGA) to the IDFTLDA form introduced in this work isfound necessary to promote the positron a ffi nity and / or life-time calculations. Moreover, the IDFTGGA performs bet-ter than the PHCGGA scheme in both positron a ffi nity andlifetime calculations. However, the best agreement betweenthe calculated and experimental positron lifetimes / a ffi nities isobtained by using the fQMCGGA positron-electron correla- tion scheme. Nevertheless, the new introduced gradient cor-rected correlation form (IDFTGGA) is currently competitivefor positron lifetime and a ffi nity calculations. Acknowledgment
We would like to thank Han Rong-Dian, Li Jun and HuangShi-Juan for helpful discussions. Part of the numerical cal-culations in this paper were completed on the supercomputingsystem in the Supercomputing Center of University of Scienceand Technology of China. [1] Tuomisto F and Makkonen I 2013
Rev. Mod. Phys. Chin. Phys. Lett. (04) 46101[3] Li Y F, Shen T L, Gao X and et al. 2014 Chin. Phys. Lett. (03)36101[4] Makkonen I, Ervasti M M, Siro T and Harju A 2014 Phys. Rev. B (4) 041105[5] Nieminen R M, Boro´nski E and Lantto L J 1985 Phys. Rev. B Phys. Rev. B Phys. Rev.
A1133[8] Kuriplach J and Barbiellini B 2014
Phys. Rev. B J. Phys.: Conf. Ser.
Comput. Mater. Sci.
Phys. Rev. Lett.
Phys. Rev. B Solid State Com-mun.
Phys. Rev. B J. Phys. F: Met. Phys. Phys. Rev. B Phys. Rev. B Phys. Rev. B Phys. Rev. B Acta Phys. Sin (21) 217804 (in Chinese)[21] Boro´nski E and Nieminen R M 1986 Phys. Rev. B J. Phys.: Condens. Matter Phys. Rev. B Nukleonika Phys. Rev. B (11) 6215[26] Barbiellini B, Puska M J, Torsti T and Nieminen R M 1995 Phys. Rev. B Phys. Rev. B Phys. Rev. Lett. J. Phys.: Con-dens. Matter (39) 395502[31] Perdew J P, Ruzsinszky A, Csonka G I, Vydrov O A, ScuseriaG E, Constantin L A, Zhou X and Burke K 2008 Phys. Rev.Lett.
Phys. Rev. B Computational Materials Science Defect Di ff us. Forum Phys. Rev. B J. Phys.:Condens. Matter Phys. Status Solidi B Phys. Stat.Sol. (a)
Phys.Rev. B Phys. Rev. B Appl. Phys.A: Solids Surf. Phys. Rev. B Phys. Rev. B Appl. Phys. A: SolidsSurf. Phys. Rev. B Ann. Phys. (Leipzig) Appl. Phys. Lett. Mater Trans Phys. Rev. B J. Appl. Phys. (5) 2481–2485[50] Brunner S, Pu ff W, Balogh A G and Mascher P 2001
Mater.Sci. Forum
Phys.Rev. Lett. J. Phys.: Con-dens. Matter Solid State Com-mun. Electrochem.Solid-State Lett. (3) 150[55] Ge ff roy B, Corbel C, Stucky M, Triboulet R, Hautoj¨arvi P,Plazaola F L, Saarinen K, Rajainm¨aki H, Aaltonen J, MoserP, Sengupta A and Pautrat J L 1986 Defects in Semiconductors ,ed. H. J. von Bardeleben, Materials Science Forum (Trans TechPublications, Aedermannsdor ff , 1986) Vols10-12, p1241[56] Dannefaer S 1982 J. Phys. C J. Phys.:Condes. Matter Phys. Rev. B65