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Oscillatory bilinear exchange coupling in Fe/noble metal/Fe sandwiches

D. ERRANDONEA(*)

Departamento de Microelectrónica, CITEFA

Zufriategui 4380, (1603) Villa Martelli, República Argentina

(ricevuto il 26 Novembre 1996; revisionato il 10 Giugno 1997; approvato il 7 Luglio 1997)

Summary. — This work presents the results of a systematic calculation of the indirect exchange coupling (J(x)) between ferromagnetic monolayers in Fe(100)OX(100)OFe(100) and Fe(110)OX(111)OFe(110) sandwiches being X=Cu,Ag and Au. This calculation was done using the self-consistent real-space tight-binding method in the unrestricted Hartree-Fock approximation of the Hubbard Hamiltonian. The results obtained, which account for the well-known oscillatory behaviour of J(x), were fitted including only a bilinear exchange term. The calculated periods and asymptotic decay of the envelopes are in good agreement with both experimental and theoretical previous results.

PACS 75.70 – Magnetic films and multilayers. PACS 75.50.Bb – Fe and its alloys.

PACS 73.20.At – Surface states, band structure, electron density of states.

1. – Introduction

In recent years the use of molecular-beam epitaxy (MBE) systems for films growth has made it possible to produce high quality, ultrathin metallic films [1]. This has enabled researchers to search for the presence of an indirect magnetic coupling, between ferromagnetic components across a non-magnetic interlayer, in various multilayer systems. All the systems studied thus far have long-period oscillations in the exchange coupling

(

J(x)

)

between ferromagnetic layers that are function of the non-magnetic spacer layer thickness (x) [2-10]. This oscillatory dependence of the magnetic-coupling strength has created a particular interest, due to the potential practical applications, e.g., magnetic storage, as well as from the physical point of view.

Previously this physical phenomenon, has been related to Ruderman-Kittel-Kasuya-Yosida(RKKY) interactions [11 ,12] and to the presence of quantum well states in the noble-metal spacer layer [3, 9, 13]. Within the framework of the RKKY picture,

(*) Present address: Departamento de Física Aplicada, Universidad de Valencia, Dr. Moliner 50, (46100) Burjassot, España.

Bruno and Chappert [12] have shown that the oscillatory exchange coupling’s period is directly determined by wave vectors that are perpendicular to the surface and span nearly parallel faces of the spacer layer’s bulk Fermi surface. In addition, a variety of theoretical models [13-23] have been developed to explain the oscillatory behaviour of

J(x). Some of these theories of the interlayer magnetic coupling are based on the

tight-binding method [14-16], on ab initio calculations [17], on the Green’s function formulation [18, 19] or on the jellium model [14]. Unified theories [20-22], which reconcile the quantum well [9, 13] and RKKY [12] pictures as limiting cases of a common approach, have been developed too.

In a previous communication [24] we used the real-space recurrence method [25] to study the electronic and magnetic properties of two parallel Fe monolayers embedded in Ag. We chose this relative simple but powerful method, which has proved to give successful results in different magnetic systems [26-28], because it has the advantage that it is not so time and computer resources consuming as the state-of-the-art methods. Here, we extend the previous work considering systematically Cu, Ag and Au spacers in both [111] and [100] orientations, for different numbers of spacer monolayers, x 41, R , 25 monolayers (ML). The sandwiches studied are of the form Fe(100)OX(100)OFe(100) (case 1) and Fe(110)OX(111)OFe(110) (case 2), where X stands for Cu, Ag or Au. In particular, we have investigated the influence of the interspacing layer thickness on the indirect exchange coupling between the Fe monolayers. In addition, when chosing Ag and Au as noble metals, by varying the number of Fe monolayers, y 41, R , 5, we have verified that changes on the Fe thickness (y) qualitatively do not affect the behaviuor of J(x).

In a first step, we employ the self-consistently calculated electronic occupations and local magnetic moments to obtain numerically the interlayer exchange coupling. This is calculated, for each x, from the energy difference between ferromagnetic and antiferromagnetic solutions. In all the systems studied we have obtained that J(x) exhibits an oscillatory behaviour and decreases as x increases. In a second step, we fit the numerically computed exchange coupling with a phenomenological expression. We find out that in this expression it is not necessary to include a biquadratic term in order to reproduce the noble-metal thickness dependence of our calculated J(x). The reason for this is that the presence of a biquadratic term in the indirect exchage coupling mainly arises from the existence of an interfacial roughness [2, 10], and our calculations are carried out under the assumption of perfect interfaces. The fitted periods of oscillation and the fitted asymptotic decay envelopes are consistent with the experimentally observed [8-10] and theoretically predicted [12, 17, 23] results.

The remaining part of this paper is organized as follows: In sect. 2 a brief account of the theoretical background is given. The calculated interlayer exchange interactions are presented and discussed in sect. 3. Finally, sect. 4 is devoted to summarize our conclusions.

2. – Theory

In this section we describe our calculations. All of them are performed by considering that all the atoms on a given j-th layer have the same averages of local magnetic moments and number of electrons. Moreover, bcc Fe has been assumed in Fe/Ag(Au)/Fe systems while fcc Fe in Fe/Cu/Fe systems. The bcc Fe (100)((110)) surface is lattice matched within 1% (7%) of the fcc Ag and Au (100)((111)) surfaces

since the cubic lattice constant of Ag (4.08 Å) and Au (4.06 Å) are aboutk2 times that of Fe (2.86 Å). The metastable fcc phase of Fe has its energy minimum at a lattice constant close to that of fcc Cu (3.61 Å), according to first-principle calculations [29, 30]. Thus, because there is virtually no lattice mistmatch between Fe monolayers and X spacer layers, we decided to neglect crystalline anisotropy and assume perfect interfaces in order to simplify our calculations. It is worth commenting briefly on this hyphotesis since in real samples always some interfacial roughness is present [31]. It is well known that deviations from ideal layer-by-layer growth can obscure important features of the magnetic coupling. For example, the coupling strength can be reduced and the short period suppressed by the roughness [12, 32-34]. However, as the long-period and multulong-period oscillations are intrinsic to the fact that the non-magnetic spacer thickness is an integer number of interlayer spacings [9, 12, 13], perfect interfaces can be assumed as a first approximation to investigate how the interlayer thickness affects the magnetic coupling [15, 23, 24].

2.1. Calculation model. – Following ref. [26] we use the self-consistent real-space tight-binding method in the unrestricted Hartree-Fock approximation of the Hubbard Hamiltonian to determine theoretically the electronic properties of the systems studied. The spin-polarized local density of states (SLDOS) rKimsis calculated by means of the recurrence method [25] including s, p, and d electrons with hopping integrals up to next-nearest neighbors. We assume for these integrals, for Fe-Fe and X-X interactions, the canonical values calculated by Andersen [35] which are not spin dependent. While for the hopping integrals between Fe-X pairs, we employ the geometrical averages of the values of Fe-Fe and X-X pairs [36].

For the electron-electron interaction we use a single-site approximation which has been extensively discussed in the literature [26, 37]. Within the model we are dealing with, this interaction can be reduced with some approximations, to a single form on-site potential shift D EKims4

!

m 8 UKimm 8DnKim 82 s 2 Ji K mmKim, (1)

where iKis the index of the i-th atomic site and m and s indicate the atomic orbital and spin, respectively. In this spin dependent term we include only intra-atomic Coulomb

UKimm 8 and exchange JKim interactions. The values of the effective Coulomb repulsions are estimated as in the work by Sarma [38]. These parameters are summarized in table I. The exchange integrals are considered to be equal to zero except for Fe d electrons, whose value is: J_{Fe4 0.97 eV for bcc Fe and JFe4 1.19 eV for the fcc Fe,} respectively. As usually [24, 26], these values were chosen to recover the proper bulk magnetic moment of Fe (2.2mBfor bcc Fe and 2.5mBfor fcc Fe).

TABLEI. – The effective intra-atomic Coulomb repulsions UKimm 8, where m and m 8 stand for s or p and Umd4 Udm.

Fe Cu Ag Au

U_{mm 8} 0.754 eV 0.866 eV 0.745 eV 0.641 eV

Umd 0.971 eV 1.110 eV 0.958 eV 0.824 eV

In eq. (1), DnKim4 nKim2 n 0 i K

m is the difference between the electronic occupation

nKim4 anKim Hb 1 anKim Ib and the corresponding occupation in the paramagnetic solution of the bulk nK0im, and mKim4 anKim Hb 2 anKim Ib is the local magnetic moment. anKimsb is determined self-consistently by requiring

anKimsb 4



2Q EF

rKims(E) dE , (2)

where the Fermi level (EF) is determined by demanding global charge neutrality through the condition:

!

i K m nKim4

!

i K m nK0_i m. (3)

It is important to notice that, in this model, charge transfer between atomic planes is allowed.

The self-consistent procedure is stopped when the absolute difference between mKim (output) and mKim (input) is less than 1024. The number of k levels of the continuous fraction expansion is chosen so that the results become independent of k. In our previous work [24] we have found that k 412 is sufficient to fulfil this requirement, but here we have decided to take k 430 with the purpose of getting more accurate results. Iterative calculations were repeated by adopting a variety of initial trial values, for which the same mKim and nKim were obtained.

2.2. Indirect exchange coupling. – Once computed mKim and nKim, considering both ferromagnetic and antiferromagnetic couplings, we have calculated separately the total energy in the Hartree-Fock approximation for each system using the following expression: ET4

!

i K , m

{



2Q EF

k

E

!

s ri K ms(E)

l

dE 2

!

m 8 UKimm 8 2 [ (ni K m 8)22 (n 0 i K m 8)2] 1 JKim 4 mi K m 2

}

. (4)

This total energy is the sum over the one-electron states corrected by the Coulomb and exchange energies counted twice in this sum. The non-intuitive second and third terms of eq. (4) are precisely due to the double counting correction. This form for the energy is sufficient to differentiate, at a fixed lattice point, between both magnetic phases.

Finally, subtracting the ferromagnetic

(

EFM(x)

)

and antiferromagnetic

(

EAFM(x)

)

total energies, we obtain the exchange coupling, J(x) 4EFM(x) 2EAFM(x), between the Fe layers as a function of the non-magnetic interspacing layer thickness. It is important to stress that J(x) is several orders of magnitude smaller than the huge total-energy corresponding to each magnetic configuration. Because of this, eq. (4) must be evaluated very precisely.

3. – Results and discussion

In the six systems considered we have found that an oscillatory exchange coupling is present. This coupling depends on the non-magnetic spacer layer thickness and decreases when it increases, up to disappearing when x B25 ML. As we have analyzed in our previous work [24], this indirect exchange coupling between Fe monolayers is

present in our model due to the sp-d hybridization of Fe and non-magnetic metal states at the interface between them. This hybridization occurs via the hopping integrals which connect states in Fe and X planes and polarize the non-magnetic layers, leading to an indirect coupling. This is precisely the situation addressed by the RKKY theory. The results for Fe/Cu/Fe, Fe/Ag/Fe and Fe/Au/Fe sandwiches in case 1 are plotted in fig. 1. It is evident from this figure that the calculated exchange coupling presents an oscillatory behaviour, changing alternatively from ferromagnetic coupling

(

J(x) E0

)

to antiferromagnetic coupling

(

J(x) D0

)

. In fig. 1 it can also be seen that the magnitude of the interlayer coupling slowly decays as the spacer layer thickness is increased, in qualitative agreement with the RKKY picture, and monotonically decreases along the series CuKAgKAu as was previously established [12]. In the same way, when considering case 2 a similar behaviour has been obtained in each system (see fig. 2). Futhermore, in all of the six systems studied we have also obtained that the initial sign of J(x) is negative indicating a ferromagnetic coupling between Fe layers.

It is important to remark that, when assuming Ag and Au as noble metals, by ranging y from 1 to 5 ML, we have observed that the oscillatory behaviour of J(x) is not very sensitive to changes in the Fe thickness. This provides supporting evidence to the fact that, as we have pointed out above, the interaction between the magnetic and non-magnetic layers occurs at the interface between the two, making our coupling independent of the magnetic-layer thickness, as was observed experimentally [7]. In our model, this interaction is related to the existence of sp-d hybridization at the interfaces as was stated before. This hybridization induces a charge transfer from Fe to X. The charge transfer depends on the interlayer thickness and mainly affects the interfacial monolayer of Fe. For example, when Ag(111) is assumed as spacer layer the charge transfer reaches a limiting value of 0.35 electrons with reference to the bulk electronic occupation for x 425 ML. The electronic occupations of the rest of Fe monolayers remains nearly insensitive to the change of the interlayer thickness and are very similar to the bulk value with the exception of the surfacial monolayer. Thus, the variation of the total energy of the system due to the increase of the number of Fe monolayers is independent of the spacer layer thickness. As a consequence of this, our results are qualitatively not affected by a change in the number of Fe monolayers considered. Based on this phenomenon, we have limited ourselves to analyze the results obtained for Fe monolayers.

In order to compare quantitatively our results with previous ones, we have fitted the calculated J(x) modeling it by using the following phenomenological expression :

J(x) 4 J1sin [ ( 2 px/l1) 1f1] 1J2sin [ ( 2 px/l2) 1f2]

( 2 kFax)n

, (5)

where x is the noble-metal spacer thickness in ML, kF refers specifically to the Fermi wave vector in the direction perpendicular to the layers, a stands for the distance between two neighboring atomic planes and n, J1( 2 ), f1( 2 )and l1( 2 )are adjustable power law of the decay envelopes, amplitudes, phases and periods, respectively. In eq. (5) we have only included a bilinear coupling, which is modeled by the sum of two sine waves with different phases. The reason of this is that in our model we have perfect interfaces, and most of the authors who have observed a biquadratic term in the exchange coupling interaction agree in attribuiting its presence to the existence of an interfacial roughness [2, 10].

J( x )(1 0 -3 J/m 2 ) -0.4 -0.2 0.0 0.2 J( x ) ( 1 0 -3 J/m 2 ) -0.4 -0.2 0.0 0.2

spacer layer thickness (ML)

0 5 10 15 20 25 J( x ) ( 1 0 -3 J/m 2 ) -0.4 -0.2 0.0 0.2 (a) Cu (b) Ag (c) Au

Fig 1. – Indirect exchange coupling(J(x))between Fe monolayers separated by spacer layers in [100] orientation. (a) Fe/Cu/Fe(p), (b) Fe/Ag/Fe(j) and (c) Fe/Au/Fe(!). The numerical uncertainty in the evaluation of J(x) corresponds to the size of calculated points. Fits using eq. (5) are shown in solid lines.

Expression (5) fits well our results. Figures 1 and 2 show in solid lines the obtained fittings of J(x) for case 1 and 2, respectively. In the latter case, the fits were made taking J24 0 although it is well known that in this kind of systems J(x) has only one

J( x )(1 0 -3 J/m 2 ) -0.4 -0.2 0.0 0.2 J( x ) ( 1 0 -3 J/m 2 ) -0.4 -0.2 0.0 0.2

spacer layer thickness (ML)

0 5 10 15 20 25 J( x ) ( 1 0 -3 J/m 2 ) -0.4 -0.2 0.0 0.2 (a) Cu (b) Ag (c) Au

Fig. 2. – Same as fig. 1 but with spacer layers in [111] orientation. (a) Fe/Cu/Fe (p), (b) Fe/Ag/Fe (j) and (c) Fe/Au/Fe (!). The numerical uncertainty in the evaluation of J(x) corresponds to the size of calculated points. Fits using eq. (5) are shown in solid lines.

periodicity. Making the fit for the six systems considered, we have determined the periods l and the decay law n for each one. In table II the fitted values for l and n are summarized that give the best fit of eq. (5) to our numerical results. The error intervals

TABLEII. – Previously reported periods l0and fitted values for l and n using eq. (4), for all the

considered systems. The periods are given in number of X monolayers.

l n l(a)0 l(b)0 Cu (111) 4.9 6 0.10 1.40 6 0.20 4.50 (100) 5.6 6 0.20 1.20 6 0.20 5.88 5.80 2.4 6 0.04 1.30 6 0.10 2.56 2.60 Ag (111) 5.8 6 0.15 1.70 6 0.20 5.94 (100) 5.5 6 0.10 1.70 6 0.25 5.58 5.60 2.3 6 0.12 1.80 6 0.20 2.38 2.40 Au (111) 4.8 6 0.15 1.80 6 0.20 4.83 (100) 8.7 6 0.10 1.80 6 0.25 8.60 8.60 2.4 6 0.08 1.85 6 0.20 2.51 2.48 (a) Ref. [12].

(b) Refs. [9, 10] and [8] for Cu, Ag and Au, respectively.

correspond to the 95 % statistical confidence level. The previously reported periods

l0[8-10, 12] are also tabulated for comparison. Unlike ref. [21], here using more steps in the continuous fraction and finer meshers in the energy integral (1_{), we have found} that J(x) presents two periods when we considered [100] orientation (see table II). In this way, we have confirmed the fact that the absence of the short-wavelength period in our previous results could be attributed to a possible precision problem as supposed.

In table II, it is possible to see that the present calculated periods, for Ag and Au spacer layers, are in good agreement with the previous ones measured in SEMPA [8], inverse photoemision [9], FMR [10] and SMOKE [10] experiments; as well as with the theorethically predicted by Bruno and Chappert [12] using realistic Fermi surfaces determined by means of Haas-Van Alphen and cyclotron resonance measurements. This agreement with ref. [12] suggests that the polarization induced via sp-d hybridization at the interface in our model should be similar to that induced by a contact interaction between the conduction electrons and the spin located on atomic positions of the magnetic layer in the RKKY picture. Nevertheless, when Cu is considered as spacer layer, our resulting periods present discrepances with theirs. We have also obtained that J(x) decays in a similar way to x22_{, as was predicted by} Johnson [23], Lang [17] and in RKKY theory [11], when we consider either Ag or Au spacer layers. Instead of that, we have obtained a different decay power law Bx21.3 when Cu is considered as interspacing layer.

We belive that our discrepancies with previously reported results in Fe/Cu/Fe systems may be attribuited to the expansion of the Fe lattice constant when Fe crystallizes in the fcc structure. It is well known that in fcc Fe, the lattice parameter is very close to that of Cu [29, 30]. Thus, fcc Fe lattice constant results to be expanded 25% compared to that of bcc Fe. The change of this parameter imply a change of the interatomic distances, which involves a modification of the hopping integrals. These

(1) In spite of the increase of numerical work, which grows when the number of levels increase, the CPU time requiered still favours this method.

integrals depend on the distance between i-th neighboring atoms [39]. Their variation modify the hybridization between Fe and X states at the interfaces and the magnetic properties of the Fe monolayers, leading to the inaccurate values obtained for l and n. In spite of the discrepancies obtained in Fe/Cu/Fe systems, we think that our model appears to be satisfactory in predicting the real-space periods and the asymptotic decay envelope of the indirect exchange coupling. This is encouraging and suggests that our method may be applied to a wide variety of magnetic systems as Co/Cu/Co and Fe/Cr/Fe sandwiches and superlattices.

4. – Conclusions

Summing up, in this paper we have presented a theoretical contribution to the problem of indirect exchage coupling

(

J(x)

)

between Fe monolayers across a non-magnetic interlayer. We have only focused our attention on the problem of magnetic sandwiches, but with slight modifications our model can be extended to calculate J(x) in magnetic superlattices. Considering Cu, Ag and Au as spacer layers, we have calculated J(x) using a simple mean-field model of itinerant electrons within a reasonable computing time. Subsequently, we have fitted quite well the J(x) behaviour considering only a bilinear coupling as was expected for samples without roughness at the interfaces.

Our calculations account for the fact that the indirect exchange coupling between Fe monolayers is an oscillatory function of the non-magnetic spacer layers thickness, which tends to disappear when x is approximately equal to 25 ML. We have also verified that changing the number of Fe monolayers does not significantly affect our results. Considering the numerous approximations involved in the present model, the numerical values obtained for l and n are in qualitatively good agreement with the previously reported results [8-10, 12, 17, 23].

We check that the differences between the results obtained in our previous work [24] and those predicted in RKKY calculations [12] were due to a precision problem. We have verified that, including more steps in the continuous fraction than in our previous work, although it takes much more computing time, it does modify these differences. In addition, we think that the faliure of our calculations in predicting the periods and the decay envelopes of J(x) in Fe/Cu/Fe systems may be related to the expansion of the Fe lattice constant. We are considering now to solve it in further calculations in order to improve the present model. However, for a deep understanding of the oscillatory magnetic behaviour of these systems more refined models are required. It would be also interesting to make a theoretical approach considering the possibility of aliasing between Fe and noble metals.

R E F E R E N C E S

[1] FARROW R. F. C., PARKIN S., DOBSON P., NEAVE J. and ARROT A., Thin Film Growth

Techniques for Low-Dimensional Structures (Plenum, New York) 1987.

[2] RU¨CKERU. et al., J. Appl. Phys, 78 (1995) 387.

[3] ORTEGAJ. and HIMPSELF., Phys. Rev. Lett, 69 (1992) 844.

[4] CELINSKIZ. and HEINRICHB., J. Magn. Magn. Mater., 99 (1991) 125. [5] PARKINS., Phys. Rev. Lett., 67 (1991) 3598.

[6] UNGURISJ., CELOTTAR. and PIERCED., Phys. Rev. Lett., 67 (1991) 140. [7] PARKINS., MOREN. and ROCHEK., Phys. Rev. Lett., 64 (1990) 2304. [8] UNGURISJ., CELLOTAR. and PIERCED., J. Appl. Phys., 75 (1994) 6437.

[9] ORTEGAJ., HIMPSELF., MANKEYG. and WILLISR., J. Appl. Phys., 73 (1993) 5771. [10] CELINSKIZ., HEINRICHB. and COCHRANJ., J. Appl. Phys., 73 (1993) 5966. [11] YAFETY., Phys. Rev. B, 36 (1987) 3948.

[12] BRUNOP. and CHAPPERTC., Phys. Rev. Lett., 67 (1991) 1602. [13] EDWARDSD. et al., Phys. Rev. Lett., 67 (1991) 493; 67 (1991) 1476(E).

[14] EDWARDS D., MATHON J., MUNIZ R. and PHAM M., J. Phys.: Condensed Matter, 3 (1991) 3941.

[15] HASEGAWAH., Phys. Rev. B, 42 (1990) 2368. [16] STILESM., Phys. Rev. B, 48 (1993) 7238. [17] LANGP. et al., Phys. Rev. Lett., 71 (1993) 1927.

[18] SHIZ.-P, LEVYP. M. and FRYJ. L., J. Appl. Phys., 73 (1993) 5975. [19] SHIZ.-P., LEVYP. M. and FRYJ. L., Phys. Rev. Lett., 69 (1992) 3678. [20] FERREIRAM. et al., J. Phys.: Condensed Matter, 6 (1994) L619. [21] D’ALBUQUERQUEand CASTROJ. et al., Phys. Rev. B, 49 (1994) 16062. [22] BRUNOP., Phys. Rev. B, 52 (1995) 6647.

[23] JOHNSONM. et al., Phys.Rev. B, 44 (1991) 5977.

[24] ERRANDONEAD., J. Phys.: Condensed Matter, 7 (1995) 9439.

[25] HAYDOCK R., Solid States Physics, Vol. 35, edited by H. EHRENREICH, F.: SEITZ and D. TURNBULL(Academic, New York) 1981, p. 215.

[26] PASTORG., DORANTES-DA´VILAJ. and BENNEMANNK., Phys. Rev. B, 40 (1989) 7642. [27] FABRICIUSG., LLOISA. M. and WEISSMANNM., Phys. Rev. B, 44 (1991) 6870. [28] BOUARABS., VEGAA., ALONSOJ. A. and IN˜IGUEZM. P., Phys. Rev. B, 54 (1996) 3003. [29] KU¨BLERJ., Phys. Lett. A, 81 (1981) 81.

[30] WANGC., KLEINB. and KRAKAUERH., Phys. Rev. Lett., 54 (1985) 1854.

[31] SCHULLER I. K., Physics, Fabrication and Applications of Multilayered Strucutres, edited by D. DHEZand C. WEISBUSH(Plenum, New York) 1988, p. 139.

[32] WANGY., LEVYP. M. and FRYJ. L., Phys. Rev. B, 65 (1990) 2732. [33] LACROIXC. and GAVIGANJ. P., J. Magn. Magn. Mater., 93 (1991) 413. [34] UNGURISJ., CELLOTAR. and PIERCED., Phys. Rev. Lett., 67 (1991) 140.

[35] ANDERSENO. K., Highlihgts of Condensed Matter Theory, edited by F. BASSANI, F. FUMIand M. TOSSI(North Holland, Amsterdam) 1985.

[36] PETTIFORD., J. Phys. F., 7 (1977) 613.

[37] VICTORAR. and FALICOVL., Phys. Rev. B, 31 (1985) 7335. [38] SARMAD. D. et al., Phys. Rev. B, 39 (1989) 3517.

[39] HARRISON W., Electronic Structure and The Properties of Solids. The Physics of The