Electric field gradient in dilute vanadium alloys

Volume 114A, number 5 PHYSICS LETTERS 24 February 1986

ELECTRIC FIELD GRADIENT IN DILUTE VANADIUM ALLOYS

S.K. RATFAN, S. PRAKASH

Department of Physics, Panjab University, Chandigarh 160014, India

and

J. S I N G H

Department of Physics, Guru Nanak Dev University, Amritsar 143005, India

Received 5 July 1985; revised manuscript received 30 November 1985; accepted for publication 9 December 1985

The electric field gradient (EFG) and the asymmetry parameter due to transition-metal impurities Ti, Cr, Fe, Nb, Ta and W in a host metal V, have been calculated. The size effect contribution to EFG is estimated using the continuum theory of elasticity, and the valence-effect contribution is evaluated using the dielectric screening theory. It is found that the asymmetry parameter is zero in alloys with bcc structure and the valence-effect contribution dominates the size-effect contribution, which are novel features.

Most of the nuclear magnetic resonance experi- ments, for the study of the EFG, have been performed on dilute transitional alloys (alloys with transition metal impurities) with fcc crystal structure [1,2] and they exhibit strongly enhanced quadrupole effects. But the substitutional alloys with bcc structure are least studied both experimentally and theoretically. Von Meerwall and Schreiber [3] and yon Meerwall and Rowland [4] carried out systematic investigations of quadrupole interactions in dilute V based alloys with 3d, 4d and 5d transition metal (TM) impurities. The theoretical study of the EFG is of immense im- portance as it gives information about basic quantities like the impurity scattering potential and the charge perturbation. Only a few attempts have been made to calculate the EFG in dilute transitional alloys with fcc structure [1,2,5,6] and bcc structure [7]. Recently Singh et al. [6] gave an impurity scattering formalism for the transitional alloys which includes the d-band effects, explicitly, of both the impurity and the host atoms and yields reasonably good results for the fcc alloys. We think it worthwhile to apply the dielectric screening approach for calculating the EFG in the dilute V alloys.

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The presence of an impurity in a cubic crystal breaks both the structural symmetry and the electrostatic symmetry around the impurity. The first is usually called the size effect and the second the valence effect and both contribute to the EFG. The size effect contributes in three ways to the EFG. First the EFG should be calculated at the displaced positions of the neighbouring host ions, due to the lattice distortion caused by the impurity induced strain field. The displaced position of the nth nearest neighbour (nNN) d n is calculated using a spherically symmetric displacement field in the continuum elastic model [8]. Second, due to lattice distortion the effective charge on the impurity changes and this effect is taken into account by applying the Blatt correction to the impurity valency. The two effects are called indirect size effects as they form corrections to the valence effect EFG. Third, the strain field causes a redistribution of conduction electrons which gives rise to a f'mite EFG, usually called the direct size effect. Recently we [7] have calculated the size effect EFG, for dilute alloys with bcc structure, in the continuum elasticity theory. The principal components of the size effect EFG tensor qS for an alloy with bcc structure at the

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1NN are given as [7],

q ~ l l l ] s = s = -2q[ l iO ] - 2 q D l ~ ]

= _ 3 a 3 1 da 3 a d-e CsF4412rrVEdl ' (1)

where [111], [1 i0] and [I17.] are eigenvectors o f q s. and The [111 ] direction is usually called the parallel direction (along the line joining the host and impurity atoms) and the component of EFG along this direction is writ- ten as q~, v E = 3(1 - o)/(1 + o) is a function of the Poisson ratio o, a - I da/dc is the fractional change in the lattice parameter a per unit concentration of impurity. C s is the size strength parameter and is a mea- sure of the deficiency of the elastic continuum model and the point-charge model [12]. Thecomponen t s F l l , F12 and F44 of a fourth order tensor F are calculated in the screened point-ion charge model [7]. The size effect EFG at the 2NN site is given as

s 1 da Cs(Fl l - F12) q~ = -2qSx = -2q~,y = - 3 a 3 a dc 4rWEd3 . (2)

From eqs. (1) and (2) it is evident that qS is diagonal at both the 1NN and 2NN in an alloy with bcc structure contrary to the findings in an alloy with fcc structure [2,5,6].

The principal component of the traceless valence effect EFG tensor along the parallel direction (taken as Z-axis) is defined as

1 d V)) (3) q~l (r) = ~-( 1 - % . ) ( ~ 2 ( A V ) - r ~ ( A ,

where A V(r) is the screened excess impurity potential and 7** is the Sternheimer antishielding parameter. The screened electrostatic impurity potential in the dielectric screening theory is defined as

1 AVb(q) e x p ( - i q . r ) d q ,~V(r)=(~)3fe--~-~-- . (4)

Here AVb(q) is the unscreened excess impurity poten- where tial defined as kF h

where Vb(q) and Vbh(q) are the bare-ion potentials ff2Zh 0 d

for the impurity and host ions. eh(q) is the dielectric and function of the host metal and is defined as

eh(q ) = 1 + v(q)×h(q), (6)

where o(q) and Xh(q) are the electron-electron interaction and the response function, respectively, of the host matrix and are defined as

v(q) = (4,re*2/q 2) [I - / x c ( q ) ] , (7)

Xh(q) = ~ f(Ek)--f(Ek+q ) k<kFh Ek+q--Ek

× 1( ~bk(r)l exp ( - i q "r) l •k+q(r)>[ 2 . (8)

~bk(r ) is the Bloch wave for a state with wave vec- tor k and energy Etc. f (Ek) is the Fermi distribution function and fxc(q) is the Hubbard exchange correlation function, e* and kFh are the effective electronic charge and the Fermi momentum of the host metal, respectively. To achieve internal consistency both A Vb(q) and eh(q) are evaluated using the Animalu transition metal model potential (TMMP) def'med as [9[

Vm(r ) = - C - (A 0 - C)P 0 - (A 1 - C)P 1 - (A 2 - C)p 2 ,

f o r r < R m ,

=-Ze* / r , f o r r > R m . (9)

Here A l and PI are the potential well depth and the projection operator for orbital quantum number l. C = Z/R m for A l with l ~> 3. R m is the model radius of the TMMP. We use the Animalu TMMP [9] for both the host and impurity ions using effective valency for the impurity. Using the Animalu TMMP [eq. (9)] Singh and Prakash [10] gave a formalism of the dielectric screening in the TMs and obtained the final expression for the response function as

Xh(q) = (rn*kFh/rr2h2) [1 + 7dep(q)]

X + - 4 q ~ I n 1--q[2kFh '

[3h(k + q) + ~ 13h(k + q)l 2] k 2 ~- , (11)

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2 2 dA lb

= l=0 dE

R m h

× f [jl(Ik+qlr)]2r 2 dr. (12) 0

h ( Ik + q l r) is the spherical Bessel function, m* is the effective electron mass. The derivatives of the potential well depth dAm/dE for l = 0 and 1 are taken from ref. [1 I] while dA2h/dE is calculated in the T-matrix approach,

dA2h A2h(EFh) (EFh -- Edh ) - ( 1 3 )

dE (Edh _ E)2 + ({ Wdh)2 '

where EFh is the Fermi energy and Edh and Wdh are the d-band energy and d-band width of the host metal. The resonance at E = Edh shows that the major part of the screening due to d-electrons comes from energies near to Edh. 7dep(q) is called the depletion hole in the model potential theory. The valence effect EFG is cy- lindrically symmetric, therefore

q~ _2q~l~0] = v _ . = --2qtl12] (14)

It is easy to prove with the help of eqs. (3), (4) and (10) that the asymptotic expression for q~ shows the standard oscillatory behaviour. The total EFG is obtained by the addition of the corresponding components of the qV and qS tensors.

The present formalism is applied to calculate the EFG in dilute V-based alloys with Ti, Cr, Fe, Nb, Ta and W impurities. The model potential parameters needed in the calculation are taken from refs. [9,11]. The values of 1 - 7o0 = 12 and t) E = 1.5 for V metal are taken from ref. [4]. Fig. 1 shows the contribu- tions/~h(k + q) for l = 1 and l = 2 alongwith the contribution/3h(k + q). The contribution/30h(k + q) is zero as dA0b/dE = 0 for V metal. It is evident from the figure that/3 h (k + q) increases smoothly with Jkl while flh(k + q) exhibits a resonance characteristic. Further the major contribution to/~h(k + q) comes from the d-band. "Ydep(q) decreases smoothly with the increase o fq [10]. The value of 7dep(q) for V, at Iq I = 0, is 0.458, the d-band contribution is 0.283 which forms the major part of it. The positive depletion hole enhances eh(q) by 45.8% and the majorr part of it comes from the d-band of the host.

o~

0.3

..u

0.2

0.I

0 0 0.4 0.8 0 0.4 0.8 K r

Fig. 1. The first order contr ibut ion #h(k + q) and second order contr ibut ion I#h(k + q)12/4 to the depletion hole axe plot- ted as a funct ion of Ikl for Iql = 0.0 and 0.2 au. The contri- but ion #h(k + q) for l = 1 and 2 are also shown separately.

The values of q~ and ql~ are calculated for the dilute transitional alloys of V metal. The parameter C s is adjusted to obtain good agreement between the calculated and experimental values EFG for VCr alloy and the same value of C s is used for other V alloys. The calculated and experimental values of EFG at 1NN for the V alloys are given in table 1. The calculation of EFG at the 2NN is not presented as no experimental data exists for comparison. The following interesting features emerge out of the present calculations:

(1) In the present calculations ofq~ the band structure effects due to s-, p- and d-characters of the conduction electrons of the host and impurity atoms have been taken care of through the host and impurity atom TMMPs and the dielectric screening function of the host. Therefore, the Bloch enhancement is built in the theory and no separate factor is needed, as in the partial wave method, to include these effects [I ]. The large difference between the present values ofq~ and those in the asymptotic limit shows that the exact inclusion of the preasymptotic contribution is very important to get reliable values ofq~ in transitional alloys of V. The values ofq~ do not show any correlation with the variation of the excess impurity charge which is also indicated by the experimental satellite

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Table 1 The principal components of the valence effect EFG qV and the size effect EFG qS at the 1NN site (in units of 10 ~4 cm -3) for transitional alloy of V metal. The calculated EFG Iqcall and the corresponding experimental values qexp, derived from ref. [4], are also tabulated. The value of the parameter C s = -5 is used for all the V alloys. In brackets are the values of the valence effect EFG and Iqcall in the asymptotic limit using the same value of C s for the size effect EFG.

Impurity Components of EFG Iqcall qexp

qt

Ti(3d 2) 0.69(-0.12) 0 . 2 6 0.95(0.14) 0.94 Cr(3d 3) -0.28(0.06) -0.20 0.48(0.14) 0.48 Fe(3d 6) 1.35(0.04) -0.48 0.87(0.44) 0.75 Nb(4d 4) 0.47(-0.17) 0 . 5 7 1.04(0.40) 0.84 Ta(5d 3) 0.37(-0.18) 0 . 5 7 0.94(0.39) 1.00 W(5d 4) 1.22(-0.16) -0.31 0.91(0.47) 0.71

intensity loss in dilute V.alloys [4]. The absence of such a correlation shows that the partial wave method [1] is not suitable for alloys with TlVl impurities.

(2) ql~ is obtained using a single value of parameter C s in all the V-alloys and it lies within the range speci- fied by Faulkner [13]. We would like to point out that if C s is adjusted separately for different alloys to get agreement between the calculated and experimental values of the EFG the magnitude of C s exhibits a weak linear dependence on the Blatt corrected impurity valency except for the _VNb alloy. It may be due to the fact that the strain field at the 1NN depends upon the impurity valency. But at distances greater than the 1NN distance C s may be impurity indepen- dent due to heavy screening of impurities in a metal host. In the present calculations the same value of C s gives reasonably good agreement because the Blatt corrected impurity valency is nearly the same in all the V alloys except the VW alloy in which case the agreement is not so good. In previous calculations [1, 5] the elements F 11, F12 and F44 were assumed to be a property of the host metal but in the present calculations they are calculated taking into account the interaction of the impurity atom with the 1NN host atoms [12]. It is evident from table 1 that either q~l is comparable with q~ or it is greater than q~. The pos- sible reason for such a behaviour is that in the transitional alloys the d-band contribution is large due to the resonance behaviour ofA2(E ) asA2(E) ~ (E - Ed ) - 1. The large d-band contribution may lead to the formation of virtual bound states, under favourable circumstances, the peaks of which will exhibit large

splitting due to a large crystal field in the TMs [14]. But in the present study we have not investigated the occurrence of virtual bound states.

(3) Meerwall and Rowland [4] assume the asymmetry parameter ~ = 0, at the 1NN site, in analyzing the quadrupole spectra of V alloys but the present formalism shows analytically that rl = 0 in V alloys and in general in dilute alloys with bcc structure. The cylindrical symmetry o fq s is due to the fact that the 1NN and 2NN sites of impurity in a bcc crystal possess four-fold rotational symmetry.

(4) In all the V alloys the principal component of the EFG tensor is found to be in the parallel direction which is consistent with the experimental observations [1,4].

We would like to compare the present results with other calculations. The displaced position of the 1NN site d 1 is calculated using the isotropic elasticity theory [8], which is a crude approximation. Dedericks and Deutz [ 15 ] have compared the lattice statics method with the continuum theory and showed signif- icant differences between the two methods at the first few nearest neighbours in the Fe based alloys with bcc structure. But to our knowledge the lattice statics method has not been used to calculate the displacement of the 1NN in dilute V alloys. The use of a more realistic displacement of the 1NN atoms, calculated with the help of the lattice statics method [15], will change the values of q~ to some extent but the overall trends exhibited by q~ are expected to remain the same. The values of EFG at the 2NN (results not presented here) are quite large in dilute nonmagnetic V

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alloys. Similar results for EFG were also obtained recently by us [7] for the V alloys. The large EFG values show that the impurities are not screened in the impurity Wigner-Seitz cell which is consistent with the results of Podloucky et al. [14] for the dilute alloys o f TMs. Podloucky et al. [14] studied the electronic structure o f 3d and 4d TM impurities in Mo and Nb from ab initio describing the impurity by a single perturbed muffin-tin potential. They found strong violations of the Friedel sum rule indicating the important perturbation o f the neighbouring host atom potentials. Further the situation in the dilute alloys of bcc TMs is quite different from those o f fcc where the impurity is nearly screened within the Wigner-Seitz cell o f the impurity atom [16] and the EFG at the 2NN site is quite small [5,6].

We conclude that the present approach involves only a single parameter. As far as alloys with bcc structure are concerned this is one of the first attempts to calculate EFG which takes care, explicitly, of the Bloch character o f both the host and impurity atoms although in a simplified manner. The present approach is free from the asymptotic and preasymptotic approx- imations and is also free from the deficiencies involved in calculating the phase shifts in the partial wave method. Finally we want to point out that the present method is applicable to all sorts o f transitional and nontransitional alloys.

the GND University and Panjab University for pro- viding the necessary facilities.

References

[1] G. Griiner and M. Minier, Adv. Phys. 26 (1977) 231, and references therein.

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and D. Turnbull (Acadamic Press, New York, 1956) p. 79. [9] A.O.E. Animalu, Phys. Rev. B8 (1973) 3542.

[10] J. Singh and S. Prakash, J. Phys. F l l (1981) 2409. [11] R. Singh, J. Singh and S. Prakash, J. Phys. F10 (1980)

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Phys. Rev. B30 (1984) 3191. [13] E.A. Faulkner, Philos. Mag. 5 (1960) 843. [14] R. Podloucky, J. Deutz, R. ZeUer and P.H. Dedericks,

Phys. Stat. Sol. (b) 112 (1982) 515. [15] P.H. Dedericks and J. Deutz, in: Continuum models of

discrete systems (Univ. of Waterloo Press, 1980) p.329. [16] P.H. Dedericks, R. Zeller, H. Akai, S. Bliigel and

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We are grateful to the Departments of Physics o f

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Electric field gradient in dilute vanadium alloys