366 Minal H. Patel, A.M. Vora, P.N. Gajjar, and A.R. Jani Vol. 38 tained from Ref. [2]. In the case of the second objective, the concentrations x = 0:

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1 Commun. Theor. Phys. (Beijing, China) 38 (2002) pp. 365{369 c International Academic Publishers Vol. 38, No. 3, September 15, 2002 Asphericity in the Fermi Surface and Fermi Energy of Na{K, Na{Rb and Na{Cs Binary Alloys Minal H. Patel, A.M. Vora, P.N. Gajjar, and A.R. Jani Department of Physics, Sardar Patel University, Vallabh Vidyanagar , Gujarat, India ( Received February 9, 2002 Revised April 10, 2002 ) Abstract Detailed theoretical investigations into asphericity in the Fermi surface (FS) and Fermi energy (FE) of Na 1;xK x, Na 1;xRb x, and Na 1;xCs x binary solid solutions are carried out for the rst time. The alloying behavior of the K, Rb, and Cs with the Na generates the Fermi surface distortion (FSD) of bcc simple metals. The FS of Na{K, Na{Rb, and Na{Cs solid solution is a distorted sphere with the largest deviation along [110]. We have found that the impact of local-eld correction function on FSD is maximum at [100] point and minimum at [111] point. The exchange and correlation eect is found to suppress the value of FE. PACS numbers: H, H, g Key words: asphericity in the Fermi surface, Fermi energy, pseudopotential, exchange and correlation eects, binary alloys 1 Introduction The experimental and theoretical study on the Fermi surface (FS) of pure alkali metals yielded qualitative and quantitative information about the anisotropy of the FS and is established that the FS of pure alkalis shows small but denite asphericity from a free electron sphere. [1] It is also known that among the alkali metals the Fermi surface distortion (FSD) increases as one moves from Na to Cs. [1] On alloying alkali metals into other alkali metals, the atomic matrix elements in the pure state are aected by lattice distortion and charging eects. Hence it becomes an interesting eld of studying the shape of the FS in the alkali-alkali alloys. But, not a single attempt has been made so far for systematic investigation on the shape of the FS in the solid solution of alkali alloys. Very recently we have reported the Fermi energy (FE) and FSD of Cs 0:7Rb 0:3, Cs 0:3Rb 0:7, Cs 0:7K 0:3, Cs 0:3K 0:7, Rb 0:71K 0:29, and Rb 0:3K 0:7 disordered alloys. [2] The study concluded with: 1) For the Cs{Rb and Cs{K systems, the FSD along the zone boundaries increases with the atomic volume, whereas for the Rb{K the trend is reverse. 2) The maximum anisotropy in the FS is observed at the [110] point. 3) Various local eld correction functions have in- uence of about 64 93% on the aforesaid properties. In the present paper we focus our attention to the detailed theoretical investigation for the asphericity in the FS and FE of Na 1;xK x, Na 1;xRb x, and Na 1;xCs x binary solid solutions. The objective of the present report is of threefold, namely a) To establish the applicability of pseudo-alloy-atom (PAA) model in the study of sodium bases alkali binaries. b) To judge the eect of concentration x which may provide a help in understanding the alloying behavior of alkali elements. c) To verify the impact of various local eld correction functions on the FSD and FE of Na based binary systems. In the pseudo-alloy-atom model approach a hypothetical monoatomic crystal is supposed to be composed of pseudo-alloy-atoms, which occupy the lattice sites and form a perfect lattice in the same way as pure metals. In this model the hypothetical crystal made up of PAA is supposed to have the same properties as the actual disordered alloy material and the pseudopotential theory is then applied to studying various properties of an alloy. [2 3] The complete miscibility in the alkali-alkalisystems is considered as a rare case. Therefore, in such alloys the atomic matrix elements in the pure state are aected by the characteristics of alloys such as lattice distortion eects and charging eects. In the PAA model, such eects are involved implicitly. In addition to this it also takes into account the self-consistent treatment implicitly. [2 3] Hence we thought it is worth while to apply PAA model to investigate the FE and FSD of the Na based solid solution. In the present investigation, the model potential used to explain the electron-ion interaction under the PAA consideration is given by [2] V (r) = ; Ze2 r 3 h 2 ; exp 1 ; rrc i r 2 r r c = ; Ze2 r r c : (1) r Here Z is the valency, e is the electronic charge and r c is the parameter of the potential. The detailed information regarding the potential can be ob-

2 366 Minal H. Patel, A.M. Vora, P.N. Gajjar, and A.R. Jani Vol. 38 tained from Ref. [2]. In the case of the second objective, the concentrations x = 0:0 0:2 0:4 0:5 0:6 0:8 and 1:0 of A 1;xB x are used. The local-eld corrections due to the Hartree (H), [4] Vashishta{Singwi (VS), [5] Taylor, [6] Icimaru{Utsumi (IU), [7] Farid et al. (F) [8] and Sarlar et al. (S) [9] are used to fulll the third objective. 2 Method of Computations From the second-order perturbation theory and in the PAA model, we propose to study the Fermi energy, E F by Refs. [1], [2], [10], and [11] E F (PAA) = h2 (k 0 F )2 2m m X jv PAA (q)j 2 ; 2h 2 k 0 q F q6=0 ln 2k0 + q F : (2) 2k 0 ; q F Here m is the electronic mass, h = h=2 is the Planck's constant, k 0 the free electron Fermi radius, V F PAA(q) is the screened-ion pseudopotential from factor of equation (1). [2] The FE for the binary system A 1;xB x is also evaluated by using the phase mixture of A 1;x+B x. E F (mix) = (1 ; x)e FA + xe FB : (3) The distortion of the FS from a free-electron sphere is expressed by Refs. [1], [2], [10], and [11] k F = k 0 F [1 + ( kq )] : (4) Here, kq is the angle between k and the reciprocal lattice vector q, and ( kq ) is the dilatation to the free-electron Fermi radius k 0. F Under the usual second-order perturbation theory, the distortion along the k direction is investigated by making use of [ ] X jv PAA (q)j 2 = k PAA = k F k 0 F q;pairs E 0 F h 1 m h 2 q 2 =2m ; 4E 0 cos ; F kq 0 ln 2kF 2h 2 k 0 q F 2k 0 F + q i ; q : (5) Here E 0 is the free-electron Fermi energy. The bcc F phase of A 1;xB x alkali-alkali systems are well known and form solid solution over the whole range of concentration. Hence in computing the FE and FSD, the summation in equations (2) and (5) is taken over 30 nearest neighbors in the reciprocal space of bcc structure. 3 Results and Discussion Table 1 shows the constants and parameters used in the present study. Tables 2{4 display the computed values of the FE for the disordered phase and the phase mixture of Na{K, Na{Rb, and Na{Cs binary alloys, respectively. For all the three systems, E F under the pseudo-alloyatom model consideration is higher than the E F (MIX) computed under phase mixture consideration. The nonlinear relationship of E F (PAA) with concentration x is predicted for the systems under investigation and obeys a polynomial relation of degree 2 of the type E F (x) = a 0 +a 1 x+a 2 x 2. Such relation tells us how the FE changes on alloying one metallic component into other metallic components. Among the employed exchange and correlation functions, [4;9] the H [4] screening gives the maximum value of E F while T [6] screening generates the minimum value of E F. Thus inclusion of exchange and correlation eects suppresses the Fermi energy. This also conrms our recently reported observation for the Cs{K, Cs{Rb, and Rb{K binary systems. [2] Table 1 Constants and parameters used in the present work. In the table, Z is the valency, a is the lattice constants. Metal Z r s (A) a (A) k F (A ;1 ) 0 (A 3 ) r c (A) Na K Rb Cs Table 2 Fermi energy (;E F in 10 ;12 ergs) for Na 1;xK x. Local eld correction function

3 No. 3 Asphericity in the Fermi Surface and Fermi Energy of Na{K, Na{Rb and Na{Cs Binary Alloys 367 Table 3 Fermi energy (;E F in 10 ;12 ergs) for Na 1;xRb x. Local eld correction funciton Table 4 Fermi energy (;E F in 10 ;12 ergs) for Na 1;xCs x. Local eld correction function Fig. 1 Asphericity in the Fermi surface of (a) Na 1;xK x (b) Na 1;xRb x (c) Na 1;xCs x. The asphericity in the FS of Na 1;xK x, Na 1;xRb x, and Na 1;xCs x along the three major symmetry directions is shown in Figs. 1(a){1(c). To infer the inuence of concentration x, the results due to the T [6] dielectric function only

4 368 Minal H. Patel, A.M. Vora, P.N. Gajjar, and A.R. Jani Vol. 38 are shown in the gures. It is noticed that as the concentration x increases, the asphericity in the FS increases through the region considered. This is because FSD increases in going from Na to Cs. [1] The increase in the asphericity of binary systems with concentration x is also associated with the increase in atomic volume of the binary alloys. The alloying behavior of the alkali elements into the alkali elements generates the FSD like bcc metals. Thus the study conrms the metallic behavior of the sodium-based alkali binary alloys. It can be seen that the FS of Na{K, Na{Rb, and Na{Cs solid solution is a distorted sphere with the largest deviation from free-electron sphere when observed along [110]. The FSD along symmetry direction increases with the atomic volume of the alloy. Figures 2(a){2(c) are plotted to infer the impact of various local-eld correction functions [4;9] on the asphericity in the FS of equiatomic NaK, NaRb, and NaCs solid solutions. The results due to VS, [5] IU, [7] F, [8] and S [9] dielectric functions are lying between the results obtained by employing H [4] and T [6] dielectric functions. In the case of Na 0:5K 0:5, relative to the H dielectric function [4] (static only), the inclusion of the T [6] dielectric function inuences the FSD by 73:1% at [100], 70% at [110], and 54:1% at [111] directional points. For Na 0:5Rb 0:5 this inuence is found to be 79:38% at [100], is 75:53% at [110], and is 56:51% at [111]. In the case of Na 0:5Cs 0:5 the predicted inuence of the T [5] function is 80:01% at [100], 77:79% at [110], and 66:82% at [111] symmetric points. These values also suggest that the eect of exchange and correlation is more prominent in the case of larger values of r s. Fig. 2 Impact of various local-eld correlation functions on the asphericity in the Fermi surface of (a) Na 0:5K 0:5 (b) Na 0:5Rb 0:5 (c) Na 0:5Cs 0:5. 4 Conclusions It is found from the present computation that the FSD in the Na{K and Na{Rb systems are very similar. This may be due to the fractional dierence between the values of r c (Na)=r c (K) and r c (Na)=r c (Rb). Hence on replacing K by Rb, the replacement aects very little to the atomic matrix element. Besides these two systems, the Na{Cs shows bigger changes because of the large deviation in the value of r c (Na)=r c (Cs) in comparison to Na{K and Na{Rb. The large FSD of Cs among the K, Rb, and Cs also plays important contribution for such the changes in the nding of Na{Cs systems. As direct measurements of the asphericity in the FS for these alloys are not available, the estimated experimental data from the pure metallic components may serve to asses the present investigation only quantitatively. The nature of the asphericity in the FS resembles to that of the FSD in the bcc metallic component and hence conrms that on alloying the alkali-alkali elements, the resultant solid solution can still retain the metallic behavior over the full range of concentration. The maximum FSD is observed at [110] point for all the binaries. The non-linearity in E F (PAA) as a function of concentration favors the PAA model. The impact of local-eld correction on FSD is maximum at [100] point whereas minimum at [111] point. To obtain the full description of the FSD and FE of alkali binaries, the investigations on Li 1;xA x (A=Na, K, Rb, and Cs) are in progress and results will be communicated in near future.

5 No. 3 Asphericity in the Fermi Surface and Fermi Energy of Na{K, Na{Rb and Na{Cs Binary Alloys 369 Acknowledgement One of the authors (MHP) is thankful to the University Grant Commission (UGC), India for providing a Teacher Fellowship. We are also grateful to acknowledge UGC for providing nancial support under the UGC project and DRS/SAP. References [1] P.N. Gajjar, B.Y. Thakore, J.S. Luhar, and A.R. Jani, Ind. J. Pure & Appl. Phys. 38 (1995) 309, and references therein. [2] Minal H. Patel. A.M. Vora, P.N. Gajjar, and A.R. Jani, Physica B304 (2001) 152. [3] P.N. Gajjar, B.Y. Thakore, and A.R. Jani, Acta Phys. Polo. A99 (2001) 565. [4] W. Harrison, Elementary Electronic Structure, World Scientic, Singapore (1999) p [5] P. Vashishta and K.S. Singwi, Phys. Rev. B6 (1972) 875. [6] R. Taylor, J. Phys. F8 (1978) [7] S. Ichimaru and K. Utsumi, Phys. Rev. B24 (1981) [8] B. Farid, V. Heine, G. Engel, and I.J. Robertson, Phys. Rev. B48 (1993) [9] A. Sarkar, D. Sen, S. Haldar, and D. Roy, Mod. Phys. Lett. B12 (1998) 639. [10] P.N. Gajjar, B.Y. Thakore, and A.R. Jani, Solid State Commun. 100 (1996) 785. [11] D.C. Wallace, Thermodynamics of Crystals, Wiley, New York (1972).