Role of electronic correlations in Ga
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Role of electronic correlations in Ga
Zhiyong Zhu, Xuhui Wang, and Udo Schwingenschl¨ogl ∗ Physical Sciences & Engineering Division, KAUST,Thuwal 23955-6900, Kingdom of Saudi Arabia
Abstract
An extended around mean field (AMF) functional for less localized p electrons is developed toquantify the influence of electronic correlations in α -Ga. Both the local density approximation(LDA) and generalized gradient approximation (GGA) are known to mispredict the Ga positionalparameters. The extended AMF functional together with an onsite Coulomb interaction of U eff =1 . PACS numbers: 61.66.Bi, 71.27.+a, 71.15.Mb, 71.20.Gj ∗ [email protected] d and f states [1–4]. The localized nature of these orbitals results in a non-negligible onsite Coulomb interaction U [5–9]. The importance of correlations in partiallyfilled p orbitals recently has been pointed out in the context of “ d magnetism”, where themagnetic order arises from the p states instead of the conventional d and f states [10, 11].First-principles calculations have shown that correlations in open p shells exist not only forfirst-row elements, like N and O (2 p ), but also for heavier atoms, like Te (5 p ) [12, 13]. Upto now, the interest in p orbital electronic correlations was limited to pure and doped ionicsemiconductors.Great interest in α -Ga was triggered by the common belief that it is the only elementalsolid in which metallicity and covalency coexist [14]. Showing orthorhombic crystal symme-try with space group Cmca , α -Ga is the most stable Ga phase at ambient conditions [15, 16].Besides the lattice constants a = 4 . b = 4 . c = 7 . u = 0 . v = 0 . ref in Fig. 1) hasone nearest neighbor (Ga ) and six neighbors in the next three coordination shells (Ga toGa ). Partial covalency of α -Ga, indicated by the short Ga ref -Ga bond, highly anisotropicelectronic and thermal conductivities, and a steep pseudogap around the Fermi level have bc Ga ref Ga Ga Ga Ga a FIG. 1. (Color online) Local environment of a reference atom Ga ref . Ga i ( i = 1 , , ,
4) denotesthe atoms in the i -th coordination shell of Ga ref . Notice that there are two Ga atoms due to theperiodicity along the b -axis. ref and Ga i ( i = 1 , , ,
4) constitute a special local environment,which is crucial to the coexistence of metallicity and covalency in α -Ga. As it is reportedin Ref. [17] and confirmed by our calculations, however, the LDA/GGA fails miserably toreproduce the experimental Ga positional parameters. It is our aim to evaluate to whichextent electronic correlations account for this deficiency.The standard way to treat electronic correlations in a DFT calculation is to consider anonsite parameter U and to embed a Hubbard-like Hamiltonian into the LDA/GGA Kohn-Sham equations (LDA/GGA+ U method) [2]. However, usually the onsite U is applied onlyto the electrons inside the non-overlapping muffin-tin spheres. By the neglection of effectsfrom the interstitial electrons, any result will depend on the muffin-tin radius R mt . Majorinaccuracies are expected for more delocalized states, like the 4 p electrons of α -Ga. As aconsequence, an extended LDA/GGA+ U functional is required which includes the effectsof the interstitial charge. Due to the reduced localization of the p electrons, a less stronglycorrelated state is expected. Hence, the around mean field (AMF) approximation is chosenfor modeling α -Ga [2, 24].Based on the fact that the LDA corresponds to the homogeneous solution of the mean-field Hartree-Fock equations with equal occupancy of all sub-orbitals with the same spin,the AMF energy functional is obtained by supplementing the LDA/GGA functional by theadditional term [2, 4] E AMF = − U eff X σ T r ( n σ − ¯ n σ I ) . (1)In this relation U eff = U − J is the effective interaction, where U and J denote the onsiteCoulomb and exchange interactions, respectively, and n σ the density matrix for spin σ . Inaddition, ¯ n σ = T r ( n σ ) / (2 ℓ + 1) is the average occupation number of the sub-orbitals withspin σ (and orbital quantum number ℓ ) and I the identity matrix. The double counting isalready corrected in Eq. (1).We assume a linear dependence n σ = λ σ n mt σ with an orbital dependent parameter λ σ .Because n σ and n mt σ can be obtained from the Mulliken method and the DFT calculations,respectively, λ σ = n σ ( n mt σ ) − can be evaluated. For the Mulliken analysis we will apply theGaussian code [25]. In general, λ σ will depend on the choice of the basis set. However, wehave checked a series of basis sets to ensure that the dependence is sufficiently weak. To fix3he dependence on R mt , we choose T r λ σ / (2 l +1) = 2 .
67 with R mt = 2 . E AMF with respect to n mt σ , V AMF, mt σ = − U eff λ σ ( n σ − ¯ n σ I ) . (2)We have implemented the extended AMF approach, given by Eqs. (1) and (2), for theWIEN2k code [24]. Like the original, the modified one-electron potential is applied to thispart of a sub-orbital that is inside the muffin-tin sphere. Yet, there are two improvements:First, the potential exerted by a particular sub-orbital arises from the Coulomb repulsion ofelectrons both inside and outside the muffin-tin sphere, n m,σ , instead of only n mt m,σ . Second,the parameter λ m,σ mimicks the energetical effects which would be obtained if both partsof the sub-orbital (inside and outside the muffin-tin sphere) were subject to the potential.I.e., the extended AMF functional includes effects of charge not only inside the muffin-tin TABLE I. Fully optimized structural parameters ( a , b/a , c/a , u and v ) of α -Ga obtained fromstandard LDA/GGA calculations. Different XC functionals, including (1) LDA-PW92, (2) GGA-SOGGA, (3) GGA-AM05, (4) GGA-PBEsol, (5) GGA-WC, (6) GGA-PBEalpha, (7) GGA-PBE,(8) GGA-PW91, (9) GGA-BPW91, and (10) GGA-RPBE, are used [24]. The experimental valuesare taken from Ref. [17]. a (˚A) b/a c/a u v δ Vol (%)Exp 4.5102 1.695 1.0013 0.0785 0.1525 0(1) 4.4262 1.695 1.0014 0.0843 0.1559 − . − . − . − . − . R mt , making it suitable to study electronic correlations in less localized p orbitals.Using standard LDA/GGA, we have fully optimized the structure of α -Ga against all 5structural parameters, applying Pulay corrections to the forces. We use in all calculations R mt = 2 . R mt K max = 8 .
5, and l max = 10 as well as the same 23 × × k -mesh.The validity of the LDA/GGA is tested for ten exchange-correlation (XC) functionals, seeTable I. In order to provide a quantitative comparison between the experimental findingsand our results for the unit cell volume (Vol) we study δ Vol = (Vol − Vol
Exp ) / Vol
Exp . The a lattice parameter depends strongly on the choice of the XC functional, while both ratios b/a and c/a are almost constant. As a consequence, δ Vol varies within a wide range from − .
48% to 8 . a and thus thesmallest volume, which reflects the common insight that the LDA, in contrast to the GGA,underestimates the volume [17, 26]. The overestimation of u and v in all our data indicatesa systematic failure of the LDA/GGA to describe α -Ga. This may be due to the influenceof electronic correlations, which we probe in the following.We apply our extended AMF functional to the outer 4 p orbitals to optimize the crystalstructure, choosing the GGA-WC XC functional for which δ Vol is minimal. The computa-tional details are the same as in the LDA/GGA calculations. To establish the strength ofthe onsite Coulomb interaction, we employ the constraint LDA method [27], which leads to U eff = 1 . u and v are optimizedin the extended AMF scheme. The anisotropies of the electronic conductivity ( σ c : σ b : σ a ) andthermal conductivity ( κ c : κ b : κ a ) are obtained [28], see Table II. The extended AMF calcu-lations lead to a qualitative improvement of both the structural and transport properties.Most remarkably, u is improved by about 20% with respect to the experimental value. The TABLE II. Experimental [17, 18] and calculated positional parameters and transport anisotropies.Exp U eff = 0 eV U eff = 1 . u v σ c : σ b : σ a κ c : κ b : κ a .00.10.20.30.000.050.100.000.040.08-12-10 -8 -6 -4 -2 0 2 4 60.000.040.080.000.040.08 (a) Total (b) s (d) p y (e) p z E - E F (eV) (c) p x DO S ( e V - a t o m - ) (f) n ( x -3 e/a.u. ) -0.8 -0.4 0 0.4 0.8 FIG. 2. (Color online) (a)-(e) Total and partial DOS obtained for the GGA-WC (black solid line)and extended AMF (red dotted line) methods. (f) Charge density difference ∆ n in the (100) planeat x = 0 .
5. The positions of the Ga atoms are indicated by black dots. extended AMF calculations hence predicts a shorter Ga ref -Ga bond and a slightly higheranisotropy of the electronic and thermal conductivities. I.e., the partial covalency of α -Ga,which is underestimated by the GGA-WC, is improved. Still, the total and partial s , p x , p y ,and p z densities of states (DOS) in Figs. 2(a)-(e) reveal only little changes.On the other hand, an enhanced covalency along the c -axis due to the onsite interactionis reflected by the charge density. In Fig. 2(f) we show the charge density difference withinthe (100) plane, at x = 0 .
5, between the GGA-WC and the extended AMF results: ∆ n = n AMF − n GGA-WC . An accumulation of charge is found in the region between Ga ref and Ga ,indicating a stronger Ga ref -Ga bond and, thus, an enhanced covalency. The accumulationcomes along with charge transfer from the p x and p y sub-orbitals to the p z sub-orbital. The p x and p y occupations decrease from 0.176 and 0.187 to 0.174 and 0.185, respectively, whilethe p z occupation grows from 0.204 to 0.208. The principal component of the electric fieldgradient, the direction of which is indicated by the arrow in Fig. 2(f), grows from 4 . · V / m to 5 . · V / m , whereas the asymmetry parameter decreases from 0.207 to 0.153.The experimental value is 5 . · V / m with an asymmetry parameter of 0.179 [29]. Thefinite U eff of the 4 p electrons is related to the anomalous spatial contraction of the valenceorbitals in α -Ga, which is a consequence of incomplete screening of the nuclei by a relativelyshallow Ga 3 d state [30]. For the same reason, also a localization of the 4 s electrons would6
20 40 60 80 100 1200.000.040.080.120.16 p a s y mm pressure (GPa) -Ga Ga-II Ga-V Ga-III Ga-IV FIG. 3. (Color online) Relation between asymmetry, characterized by p asymm , and stability, char-acterized by the pressure at which a phase occurs, for various Ga crystalline phases. The data aretaken from Refs. [15, 16, 31, 32]. The red dotted line is a guide to the eye. be expected. Unlike the 4 p electrons, however, the U eff has no effect on the 4 s electrons inthe AMF approximation [2].By Eq. (1), a symmetry lowering due to electronic correlations is expected. Because anasymmetric crystal structure gives asymmetric orbital populations, p asymm = { P m,σ [( n m,σ − ¯ n σ ) / ¯ n σ ] } , the electronic correlation energy is related to the asymmetry of the structure bythe relation E AMF = − U eff ¯ n p /
2, where ¯ n = ¯ n ↑ = ¯ n ↓ in non-magnetic α -Ga. Electroniccorrelations thus stabilize a crystal structure with lower symmetry, due to a smaller E AMF .This picture is confirmed by the successful prediction of the quadrupolar lattice distortionin the perovskite compound KCuF [3], where the introduction of the U parameter createsa “mexican-hat” shaped energy surface. This shifts the energy minimum towards a latticewith lower symmetry. The symmetry lowering due to electronic correlations is also reflectedby the orbital polarization of the extended AMF potential functional in Eq. (2). If n m,σ islarger (smaller) than ¯ n σ , it further increases (decreases) due to a smaller (larger) V AMF,mt m,σ .The orbital polarization manifests in the aforementioned occupation numbers of α -Ga.If electronic correlations are present in α -Ga, the symmetry effect should be seen in theGa phase diagram. The Ga-IV phase, which has a face-centered cubic structure, occurs onlyat a pressure above 120 GPa [31]. In contrast, the α -Ga phase, which has an orthorhombicstructure with the lowest symmetry, is the most stable phase at ambient conditions [15, 16].For quantitative description, we use the pressure at which a phase occurs as measure of itsstability, i.e., higher/lower pressure corresponds to lower/higher stability. In Fig. 3 we show p asymm as a function of pressure. The results confirm that structures with lower symmetry7ive rise to more stable phases.In conclusion, we have extended the AMF functional to describe correlation effects of lesslocalized p electrons. The partial covalency of α -Ga, which is underestimated in LDA/GGAcalculations, is improved by our approach. This shows that finite electronic correlations arepresent in the partially filled 4 p orbitals. They are reflected by the crystal symmetry of thedifferent phases in the Ga phase diagram. [1] V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B , 943 (1991).[2] M. T. Czy˙zyk and G. A. Sawatzky, Phys. Rev. B , 14211 (1994).[3] A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. Rev. B , R5467 (1995).[4] V. I. Anisimov, I. V. Solovyev, M. A. Korotin, M. T. Czy˙zyk, and G. A. Sawatzky, Phys. Rev.B , 16929 (1993).[5] B. S. Chun, H. C. Wu, M. Abid, I. C. Chu, S. Serrano-Guisan, I. V. Shvets, and D. S. Choi,Appl. Phys. Lett. , 082109 (2010).[6] S. J. Clark and J. Robertson, Appl. Phys. Lett. , 022902 (2009).[7] U. Schwingenschl¨ogl, C. Schuster, and R. Fr´esard, EPL , 27002 (2008).[8] U. Schwingenschl¨ogl and C. Schuster, Phys. Rev. Lett. , 237206 (2007).[9] I. Leonov, A. N. Yaresko, V. N. Antonov, U. Schwingenschl¨ogl, V. Eyert, and V. I. Anisimov,J. Phys.: Condens. Matter , 10955 (2006).[10] A. Droghetti, C. D. Pemmaraju, and S. Sanvito, Phys. Rev. B , 140404(R) (2008).[11] R. Kov´aˇcik and C. Ederer, Phys. Rev. B , 140411(R) (2009).[12] J. A. Chan, S. Lany, and A. Zunger, Phys. Rev. Lett. , 016404 (2009).[13] V. L. Campo Jr and M. Cococcioni, J. Phys.: Condens. Matter , 14277(R) (1991).[15] O. Degtyareva, M. I. McMahon, D. R. Allan, and R. J. Nelmes, Phys. Rev. Lett. , 205502(2004).[16] O. Schulte and W. B. Holzapfel, Phys. Rev. B , 8122 (1997).[17] M. Bernasconi, G. L. Chiarotti, and E. Tosatti, Phys. Rev. B , 9988 (1995).[18] R. W. Powell, M. J. Woodman, and R. P. Tye, Br. J. Appl. Phys. , 432 (1963).[19] O. Hunderi and R. Ryberg, J. Phys. F: Metal Phys.
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