Energy bands of Rb and Cs have been calculated for the first time using average or effective pseudopotentials obtained from non-local ones. It is shown that the results so obtained compare well to those obtained with full non-local calculations.Načinili smo prve račune energijskih vrpci Rb i Cs primjenom prosječnog ili efektivnog pseudopotencijala koji smo izveli iz nelokalnih pseudopotencijala. Pokazuje se dobra suglasnost ishoda tih računa i računa primjenom potpunih nelokalnih pseudopotencijala

Gohil, V. B.

Pandya, T. C.

Vyas, P. R.

English

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ISSN1330–0008CODENFIZAE4ON THE BAND STRUCTURE CALCULATIONS BY NON-LOCALPSEUDOPOTENTIALST. C. PANDYA, P. R. VYAS and V. B. GOHILDepartment of Physics, School of Sciences, Gujarat University, Ahmedabad 380009,IndiaReceived 15 January 1999; revised manuscript received 11 October 1999Accepted 4 May 2000Energy bands of Rb and Cs have been calculated for the first time using average oreffective pseudopotentials obtained from non-local ones. It is shown that the resultsso obtained compare well to those obtained with full non-local calculations.PACS numbers: 71.25.Cx UDC 538.915Keywords: band structure, Rb, Cs, average and effective pseudopotentials1. IntroductionA large number of calculations of the electronic band structure of simple metalsbased on local and non-local pseudopotentials with varying degree of success havebeen published in the literature in last 40 years.Kay [1] has carried out extensive band-structure calculations for several simplemetals. He used the Shaw’s optimized model potential in full non-local form. Re-cently Radwan [2], in the framework of full non-local calculations, has studied theeffect of model potential parameters on the band structure of Rb and Cs and foundthat such effect is insignificant. The full non-local calculations are more rigorousas compared to calculations based on local model potential. However, the calcu-lations based on non-local approach are comparatively difficult in the sense thateach matrix element of the Hamiltonian matrix depends on the electron state wavevector ~k and momentum transfer vector ~q. In the local approximation, it dependson ~q only. Hence, many local model potentials have been proposed to simplify thecalculations which may generate the meaningful results.The purpose of the present paper is to show, for the first time, that if oneuses the effective or average pseudopotential generated in the manner suggestedby Bertoni et al. [3] instead of a full non-local form of a pseudopotential, then theFIZIKA A (Zagreb) 9 (2000) 2, 55–60 55pandya et al.: on the band structure calculations by non-local . . .results of the band structure agree extremely well with those generated by the fullnon-local calculations.The application of effective or average potential was made to calculate the effectof three-body forces on the phonon dispersion curves of simple metals in Ref. [3].However, such effective potentials were not tested for their ability to yield theresults for other properties of metals.It is well known that the band-structure calculations are at the base of al-most all the electronic properties of metals. It is true that with the advent of highspeed computers, with sophisticated and lengthy programmes, band-structure cal-culations can be made self-consistently. But when we examine the results of suchcalculations by different methods (OPW, APW, LAPW, TB, etc.), we find thatthe results do differ appreciably from each other. It is also a fact that the self-consistency can be tested at a very few points in the Brillouin zone. In many suchcalculations, at one stage or other, some approximations and parametrization arealso made. In such circumstances, even today, the pseudopotential method hasa special importance for most of the comprehensive studies of a large number ofstatic, dynamic and electronic properties [4]. Also, the pseudopotential methodsare highly important for the non-crystalline materials [5].Recently, Pollack et al. [6] have shown that for b.c.c. sodium the band structuresobtained from local and non-local density-based pseudopotentials are found to bealmost identical.In the light of the above facts, we thought it worthwhile to examine the appli-cability of the effective pseudopotential to the band structure of simple metals.2. CalculationsAccording to Bertoni et al. [3], an average local form of pseudopotential can beobtained from a non-local form by using the following expressionv̄(q) =Gq{〈~k + ~q |v|~k〉}12 [1− ε(q)]/vc(q). (1)HereGq{φ(~k)}=2(2π)3∫k≤kFφ(~k)d3kk2 −(~k + ~q)2 . (2)ε(q) is dielectric function and vc(q) = 4πe2/q2 in the Hartree approximation. v̄(q),generated from Eq. (1), is better than in the Fermi-sphere approximation in thesense that it takes into account the average of all matrix elements weighted by thedenominator of Eq. (2). Once v̄(q) is obtained from Eq. (1), it can be used tocalculate all physical properties depending on v̄(q) in a simple manner as it is done56 FIZIKA A (Zagreb) 9 (2000) 2, 55–60pandya et al.: on the band structure calculations by non-local . . .in the local approach. Now, the band structure is obtained by solving the secularequationDet∣∣∣∣{(~k − ~g)2−E(k)}δ~g~g ′ + S (~g − ~g ′)V (|~g − ~g ′|)∣∣∣∣ = 0 . (3)In this equation, E(k) is energy, and ~g and ~g′ are reciprocal lattice vectors. Thestructure factor S (~g − ~g′) = 1 for a perfect b.c.c. or f.c.c. crystal.In the present study, the solution of secular equation (3) of order 42 × 42 hasbeen obtained with an accuracy of about 10−4 Ry. The Fermi energy is calculatedusing the formula given by Inoue et al. [7]EF =(3E100+ 4E111+ 6E110)13. (4)Here E100, E111 and E110 are energies at∣∣∣~k∣∣∣ = kF in the symmetry directions (100),(111) and (110), respectively.3. Results and discussionThe band structure of Cs and Rb obtained presently using the same Shaw’smodel potential parameters as used by Radwan [2] are shown in Figs. 1 and 2.The energy values are given in Table 1. In this table, we also list the energy valuesobtained by Radwan [2]. To test the reliability of the pseudopotential method ascompared with those sophisticated methods, we also give the energy values obtainedby Ham [8] who used the quantum defect method and those obtained by Moruzziet al. [9] for Rb who used self-consistent APW method.Table 1. Electron energy values (in Ry) at symmetry points.Symm. Cs Rbpoints Ref. [2] Ref. [5] Present Ref. [2] Ref. [5] Ref. [9] PresentH12 0.267 0.169 0.286 0.318 0.246 0.233 0.338H15 0.303 0.374 0.308 0.354 0.410 - 0.360P1 0.218 0.302 0.243 0.263 0.334 0.253 0.282P4 0.218 0.203 0.221 0.256 0.253 - 0.260N1 0.140 0.123 0.156 0.168 0.163 0.170 0.182N ′1 0.154 0.211 0.146 0.180 0.226 0.224 0.172Fermi 0.116 0.172 0.136en. (EF)FIZIKA A (Zagreb) 9 (2000) 2, 55–60 57pandya et al.: on the band structure calculations by non-local . . .The results of the present paper show that if one calculates the band structure bythe effective pseudopotential, then they are as good as those obtained from full non-local pseudopotentials. Thus, our results suggest that effective pseudopotentialsretain all the advantages of full non-local pseudopotentials with the additionaladvantage of making calculations simple.0,000,060,120,180,240,300,360,420,48HH15N1'P4H15H1H NΓPΓH12H1N1P1H12EFGΣΛF∆Energy (Ry)Fig. 1. Energy bands of Cs.As can be seen from Table 1, our results are also comparable with those obtainedby more sophisticated techniques.4. ConclusionIn the present study, we have shown that the band-structure calculations basedon average pseudopotentials are as good as those obtained from considerations offull non-locality. The results so obtained are also comparable to those obtainedfrom more sophisticated methods which are intricate and which require lengthycalculations and large computer time. The approach we suggest is simpler and lesstime consuming as it does not require the computation of ~k-dependent value ofeach matrix element of the potential. Thus, the method can readily be used for the58 FIZIKA A (Zagreb) 9 (2000) 2, 55–60pandya et al.: on the band structure calculations by non-local . . .calculations of energy values at large number of ~k points in the Brillouin zone, with0,000,060,120,180,240,300,360,420,480,52HH15N1'P4H15H1H NΓPΓH12H1N1P1H12EFGΣΛF∆Energy (Ry)Fig. 2. Energy bands of Rb.relative ease and accuracy. The successful application of the effective or averagepotential also suggests that it might be extended to non-crystalline solids for whichmost of the calculations are made with local pseudopotentials [5,10,11].References[1] P. Kay, Ph.D. Thesis The Electronic Band Structure and the Vibrational Properties ofMetallic Elements, University of Essex, U.K. (1975);[2] A.M. Radwan, Fizika (Zagreb) 20 (1988) 179;[3] C.M. Bertoni, V. Bortolani, C. Clandra and F. Nizzoli, J. Phys. F : Metal Phys. 4(1974) 19;[4] V.N. Antonov, V. Yu. Milman, V. V. Nemoshkalenko and A. V. Zhalko-Titarenko, Z.Phys. B. Condensed Matter 79 (1990) 223-233;[5] K.A. Azez, Proc. SPIE - Int. Soc. for Optical Engineering 2780 (1996) 110;[6] L. Pollack, J. P. Perdew, J. He, M. Marques, F. Nogueira and C. Fiolhais, Phys. Rev.B 55 (1997) 15544;[7] S. T. Inoue, S. Asano and J. Yamasita, J. Phys. Soc. Japan 30 (1971) 1546;FIZIKA A (Zagreb) 9 (2000) 2, 55–60 59pandya et al.: on the band structure calculations by non-local . . .[8] F. S. Ham, Phys. Rev. 128 (1962) 82;[9] V. L. Moruzzi, J. F. Janak and A. R. Williams, Calculated Electronic Properties ofMetals, Pergamon Press Inc., New York (1978), p. 108.[10] N. Gupta, K. C. Jain and N. S. Saxena, phys. stat. sol. (b) 176 (1993) 81;[11] K. A. Azez and I. A. Al-Omari, Acta Phys. Pol. A 92 (1997) 563.RAČUNANJE VRPČASTE STRUKTURE NELOKALNIMPSEUDOPOTENCIJALIMANačinili smo prve račune energijskih vrpci Rb i Cs primjenom prosječnog ili efek-tivnog pseudopotencijala koji smo izveli iz nelokalnih pseudopotencijala. Pokazujese dobra suglasnost ishoda tih računa i računa primjenom potpunih nelokalnihpseudopotencijala.60 FIZIKA A (Zagreb) 9 (2000) 2, 55–60

Računanje vrpčaste strukture nelokalnim pseudopotencijalima

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