Calculation of electric field gradient in transitional and nontransitional dilute cubic alloys

Physica 144B (1987) 368-375 NorthrHolland, Amsterdam

CALCULATION OF ELECTRIC FIELD GRADIENT IN TRANSITIONAL AND NONTRANSITIONAL DILUTE CUBIC ALLOYS

J. SINGH* Department of Physics, Guru Nanak Dev University, Amritsar-143005 (Punjab), India

S.K. RATTAN, B. PAL and S. PRAKASH Department of Physics, Panjab University, Chandigarh-160014, India

Received 11 March 1986

The electric field gradient (EFG) due to transitional and nontransitional impurities in fcc metals has been studied. Valence effect EFG has been calculated using parameter free dielectric screening theory which takes into account the s-, p- and d-character of host and impurity atoms. Size effect EFG has been calculated in elastic continuum model treating the ions in the screened point-ion model. Calculations of the EFG are performed for a number of Cu and AI alloys with nontransition metal impurities and for A1 alloys with transition metal impurities. The calculated results agree reasonably with the experimental values and thus point out the importance of exact inclusion of pre-asymptotic contribution to the valence effect, The inadequacy of the elastic continuum model at least at the first nearest neighbor is emphasized. The dielectric screening approach is found to work reasonably well in both the transitional and nontransitional dilute metallic alloys.

1. Introduction

The presence of a substitutional impurity in a cubic metal disturbs both the electrostatic and structural symmetries, thereby producing finite electric field gradient (EFG) upto several nearest neighbors [1]. The study of EFG is of immense importance in understanding the electronic structure of dilute metallic alloys as it gives informa- tion about the distribution of electronic charge [1]. The cubic alloys can be classified into four categories: (i) simple metal (SM) impurity in a SM host, (ii) SM impurity in a transition metal (TM) host, (iii) TM impurity in a SM host and (iv) TM impurity in a TM host. The first cate- gory is usually named as nontransitional alloys while the other three are called transitional alloys. The situation in case of TM impurities in metals is quite interesting as the experimental

* Present address: Physics Department, Panjab Agricultural University, Ludhiana (Pb), India.

studies [2-6] show strongly enhanced quadrupole effects compared to the SM impurities. A number of authors [7-13] have attempted different calculations for EFG and asymmetry parameter ~/ in nontransitional alloys but in transitional alloys only a few theoretical attempts [2, 14-16] have been made to calculate EFG and ~7. But these calculations are unable to include the d- band effects properly because the s -d mixing of localized d-states of the impurity causes resonant scattering which gives rise to virtual bound state [17]. Furthermore, there does not exist any single theory which can explain satisfactorily the EFG and ~ in both the transitional and nontransitional dilute cubic alloys. Recently we [18] gave a scattering mechanism which takes into account s-, p- and d-character of both the host and impurity atoms and applied it successfully to calculate EFG and ~ in dilute transitional alloys of Cu [18] and of V [19].

In the present paper we think it worthwhile to test this scattering mechanism on dilute transitional alloys of A1 and dilute nontransitional

0378-4363/87/$03.50 t~) Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

J. Singh et al. / Electric field gradient in dilute cubic alloys 369

alloys of AI and Cu metals and examine its validity. In section 2, we outline the theoretical formalism of electrostatic screening and different contributions to the EFG needed to understand the calculations and results reported in section 3. The conclusions are given in the concluding section 4.

2. Theory

The EFG at the host ion in a substitutional dilute cubic alloy originates due to two physical effects. First, the impurity ion scatters the conduction electrons of the host metal due to its different electronic structure. The impurity induced redistribution of the conduction electrons contributes towards the EFG, usually called the valence effect EFG. Second, the different size of the impurity ion produces localized strain field in the host matrix. The impurity induced strain field displaces the host ions around the impurity thereby producing further redistribution of the conduction electrons and hence the EFG, usually called the size effect EFG. Both the valence and size effects are equally important [9, 11-13, 15, 16, 18, 19] in metallic alloys.

2.1. Valence effect EFG

The screened impurity potential in the dielectric screening approach is given as

1 ~ AVb(q) aV(r) = ~ - ~ _ eh(q ) e 'q'r dq . (1)

Here AVb(q) is unscreened excess impurity potential defined as

AVb(q) b b = V i (q)- - Vh(q) , (2)

where Vbi(q) and Vb(q) are bare-ion potentials for the impurity and host ions. eh(q) is the dielectric function of the host metal and is defined as

eh(q) = 1 + v(q)xh(q) , (3)

where v(q) and Xn(q) are the electron-electron

interaction and response function, respectively, of the host matrix and are defined as

4rre* 2 v(q) = ------7--[1- f~xq)], (4)

q

and

xh(q) = ~] f(Ek)-f(Ek+q) *~<kFh E~+q - E k

× I ( ~0k(r) le- i '"l~0~+,(r) l = • ( 5 )

~bk(r ) is Bloch wave for a state with wavevector k and energy Ek, f(Ek) is the Fermi distribution function and fxc(q) is a function giving exchange corrections, e* and kvh are the effective electronic charge and Fermi momentum of the host metal. From eq. (5) it is evident that Xh(q) consists of two parts, band structure part and overlap matrix elements.

The evaluation of AV(r) requires the know- ledge of eh(q) and AVb(q). It is difficult to calculate Xh(q), or, in other words, band structure part and overlap matrix elements using the explicit Bloch waves. But the problem is immen- sely simplified if one is able to transform the Bloch wave into some simple wavefunction. Shaw and Harrison [20] gave model wavefunction transformation defined as

( OVm O,(r) = 1 - OE / #~,(r), (6)

where ~b,(r) is a model wavefunction and Vm(r, E) is a nonlocal model potential. The transformation given by eq. (6) is defined for the Heine-Abarenkov model potential defined as

Vm(r , E ) = - E At(E)P, for r ~< Rm,

' (7) = - Z e * / r for r > R m .

Here Al(E ) and Pl are the potential well-depth and projection operator for orbital quantum number l . R m is the model radius and Z is the valency of metal ion. Using eqs. (6) and (7) Singh and Prakash [21] calculated the response function of a TM, the final expression for which is given as

370 J. Singh et al. / Electric field gradient in dilute cubic alloys

m * k F h xh(q) = ~ [1+ %(q)]

X 1 1 - (q/2kvh) 2 "~ + 4( q/2kvh )

1 + q/2kFh ] In 1 q/2kFh

where

kF h

1 f [ Th(q) = ~ flh( k + q) "n" Z H

1 2] + & ( k + q) k 2

and

(8)

dk, (9)

in the present calculations depending upon the nature of the host and impurity ions in an alloy. Therefore, the potential AVb(q) can be written as

aVb(q) = Vim(q) - Vh ( q) , (12)

where Vim(q) and Vhm(q) are the Fourier trans- forms of the potential given by eq. (7) for the impurity and host ions, respectively. The only change which we make is to replace the normal valency of the impurity ion Z I by the Blatt corrected impurity valency Z~ given as

3 (I d a ] z Z~= Z ' - -~E a dc/ H, (13)

2 i / d A t h \ flh(k + q) = 8~" ~ (2l + 1 ) [ - - d ~ )

1=0

Rmh

× f [jt(lk + qlr)]2r 2 dr. 0

(10)

Here dAth/dE are energy derivatives of the potential well-depth of the host metal with energy in atomic units, ]~(Ik + qlr) is the spherical Bessel function, m* the effective electronic mass and Z H the valency of the host metal. The derivatives d A o J d E and dA,h/dE are calculated from the technical report [22] assuming dAlh/ dE = AAth/AE at the Fermi surface while dA2h/ dE is calculated in the T-matrix approach [21] as

dAzh A2h(EFh)(EFh -- Edh ) dE (Edh -- E) 2 + (1Wdh) 2 '

(11)

where Evh is the Fermi energy and Edh and Wdn are the d-band energy and the d-band width, respectively, of the host metal. The resonance at E = Edh in eq. (11) shows that the major part of screening of the impurity ion occurs in the d- band of the host metal.

To achieve internal consistency both eu(q) and AVb(q) must be derived from the same source. Eq. (7) has been solved, in the semilocal approximation, by Animalu and Heine [23] for SMs and by Animalu [24] for the TMs and these are used

where TE = 3(1 - or)/(1 + ~r) with or as the Pois- son ratio and a-'(da/dc) the fractional change in lattice parameter due to one atomic percent (at. %) of impurity.

Knowing the excess impurity potential the principal component of valence effect EFG along the line joining the host and the impurity ions (taken as the z-direction and which is also called the parallel direction) is given as [18]

v 2 )[ d 2 eql I = eq~ z = ~ ( 1 - y~

d;r 2 rl d ] AV(r) '

(14/

where y~ is the Sternheimer antishielding factor and takes into account the polarization of core states of the host ion at which EFG is to be calculated. As the valence effect EFG is cylindri- cally symmetric, the components perpendicular to q~ are half of it in magnitude but have oppo- site sign.

2.2. Size effect EFG

The second contribution to the EFG is due to the lattice distortion and is denoted by size effect EFG tensor qS. There are three types of size effects. First, the valence effect EFG should be calculated at the displaced positions of the nth nearest neighbour (nNN) d n defined as,


d. =R. + us(R.), (15)

where R n is the position of the nNN in a pure host matrix and Us(r ) is the displacement field. The values of d n can be calculated using spheri- cally symmetric displacement field in the continuum elastic model [25]. Second, the effective charge on the impurity ion changes due to the lattice distortion. This effect has already been taken into account through eq. (13). These two effects are called indirect size effects as they form corrections to the valence effect EFG.

The third and the last is the direct size effect which arises due to the redistribution of conduction electrons due to the impurity induced strain field. The contribution to the size effect EFG has been formulated in the elastic continuum theory using the unscreened point charge model [9, 11] for alloys with fcc structure. Recently Pal et al. [13] calculated qS for dilute alloys with fcc structure in the elastic continuum theory using the screened point charge model which is also being used here. For an alloy with fcc structure, the principal components of qS at the 1NN are given as [131

q~ = lq~ x = -½q~

1 = Cs ~e GI(F'I -- F12)' (16)

and at the 2NN,

s s 1 s

qx~ = q± = -~qll

1 = Us ~e G2(F11 -- F12)' (17)

where

9 G , = 16~/E a ~c " (18)

Here q~x and q~ are the principal components perpendicular to the parallel direction. The components F u, F12 and F44 of a fourth order tensor F can be calculated assuming a Thomas-Fermi- type interaction between the ions which is ob-

tained by satisfying the bulk modulus of the host metal [13]. C s is the size strength parameter and is a measure of the deficiencies of the elastic continuum model and the point charge model [13].

3. Calculations and results

Calculations of the EFG and 7/are performed for some selected nontransitional alloys of AI and Cu, namely, C__uuZn, C___0_uGa, C__~uCd, C___~uln, AIMg, AISi, AlGa, AICd, and transitional alloys of A1, namely, A1Cr, AIFe and AISc. Here the first underlined symbol denotes the host metal and the second symbol denotes the impurity. The parameters needed to evaluate the SM and TM model potentials and eh(q) are taken from Animalu and Heine [23], Animalu [24] and Singh et al. [26]. The exchange corrections due to Hubbard are applied in the calculations of eh(q). The values of a - l ( d a / d c ) for A1 and Cu due to different impurities are taken from Pear- son [27] and Ocko et al. [28]. For Al and Cu the values of Ye are 1.48 and 1.44 and those of y~ are -2.59 and -17.36, respectively. Using these parameters, AV(r) , the derivatives of AV(r) and hence q~ are calculated numerically in double precesion. The parameter C s is fitted to get an agreement between the calculated and experimental values of EFG and r/at the 1NN sites of the impurity. The total EFG components are obtained by adding the corresponding components of valence and size effect EFG contributions. The maximum component of the total EFG tensor is taken to be the calculated EFG. The calculated and experimental values of EFG and ~/, at the 1NN site, for nontransitional alloys of Cu and A1 are given in tables I and II, respectively, and for the transitional alloys of Al,

v s in table III. Table IV lists the values of qll, qll and calculated EFG I qcall, taking C s = 1, at the 2NN sites for all the transitional and nontransitional alloys of Cu and A1 under consideration and these values are compared with the experimental values of EFG qexp"

Some interesting features of transitional and nontransitional alloys of Cu and Al which


emerge out of the present calculations are de- scribed below:

(i) In Cu alloys with nontransitional impurities qV dominates over qS except in C.._~uCd. On the other hand, qS > qV in nontransitional AI alloys like AISi and A1Mg, while in others (AICd and AlGa) qV> q~. But Sagalyn and Alexander [9] and Raj et al. [12] showed, using partial wave method, that qS > qV in nontransitional alloys of AI and Cu. The dominance of qS in their results [9, 12] may be partially due to approximate inclusion of preasymptotic contribution to qV and partially due to the use of large size strength parameter to obtain theoretical fitting with the experimental results. The values of q~ and q~ in the present calculations are comparable to those found in the ab init io calculations [10, 13]. It should be noted that the values of EFG and ~/ are larger in Cu alloys as compared to those found in AI alloys with the same impurity, e.g. see A1Cd and CuCd. In transitional alloys of A1, at the 1NN site, qS > q~ except in AIFe, which is in agreement with Pal et al. [15] who obtained qV using charge perturbation obtained in muffin tin approximation. At the 2NN site, in some alloys of A1 and Cu, q~> q~, while in others it is the reverse, but it should be noted that both are small and of nearly the same order. Therefore, in general, one cannot specify a definite trend as to whether qV or qS should dominate in a particular class of alloys.

(ii) At the INN site, the magnitude of the size strength parameter C~ lies between 0 and 7 for all the transitional and nontransitional alloys of Cu and A1 which is the range specified by Faulkner [29] for Cu. While at the 2NN site, C s = 1 (which means parameter free) gives reasonably good agreement between calculated and experimental values in almost all the alloys. The different values of C~ at the 1NN site may be due to two reasons. Firstly, the different electronic structures of impurities in a particular host metal may be important at small distances from the impurity. Secondly, it shows that the elastic continuum model does not work satisfactorily near the impurity. But at the 2NN and beyond, a single value of C~ works well which indicates that the elastic continuum model is reasonably valid and

different electronic structures of impurities do not matter at large distances from the impurity. In the previous calculations various authors [9, 11, 12, 15] used different values of the size strength parameter varying over a wide range; moreover, their size strength parameter is not well defined.

(iii) In all the Cu and AI alloys the principal component of EFG at the 1NN (see tables I, II, and III) is found to be either in the parallel or perpendicular direction which is consistent with the experimental results [1, 9]. The self-consistent calculations of Pal et al. [13] give the principal component of EFG for AlGa along the x-axis which is contrary to the experimental observa- tions.

(iv) The agreement between the calculated and experimental values of EFG and 77, at the INN and 2NN sites, on the whole is reasonably good in almost all the AI and Cu alloys except the value of ~/in C__u_uCd, A1Cd and AISc. At the 2NN site our results are parameter free and are of the same order of magnitude as obtained by Pal et al. [13] and Raj et al. [12] using finite size strength parameter.

The comparison of the present results with existing calculations is quite interesting. The good agreement between the calculated and experimental values for dilute nontransitional alloys of AI may be due to the fact that the nonlinear [10] and nonspherical effects [30] in the screening of impurities are negligible. But in the transitional alloys of AI the nonlinear effects may be important. In case of nontransitional alloys of Cu, the free electron calculations, e.g. the partial wave analysis [9, 12] and the self- consistent calculations in the Jellium model [13], also exhibit small nonlinear effects. But these calculations neglect the d-band effects which may give rise to appreciable nonlinear and nonspherical contributions [18, 21], The recent ab initio calculation of core-enhancement factor for Cu due to Das and co-workers [31] also emphasizes the need of including the band structure (in particular, the d-band) effects in the TMs but their rigorous inclusion in the screened impurity potential seems to be a formidable task. There- fore, the good agreement in case of Cu-based


Table I The components of valence and size effect EFGs along with the total calculated and experimental values of EFG (in 1024 cm -3) and "0 at the 1NN site for nontransitional dilute alloys of Cu. V, S and T denote valence, size and total EFG components. The underlined value is the maximum component of EFG and is taken as the calculated EFG, qca~

EFG Alloy C~ contribution q,x q± qll [ qcal[ T]cal qexp 'r/exp

C___~uZn 2.0 V -0.230 -0.230 0.450 0.482 0.33 0.534 0.27 S 0.064 -0.096 0.032 T -0.166 -0.326 0.482

C.__u_uGa -7 .0 V -0.369 -0.369 0.738 0.661 0.58 0.599 0.59 S -0.154 0.231 -0.077 T -0.523 -0.138 0.661

C._.u_uCd -2 .0 V 0.010 0.010 -0.019 0.580 0.27 0.556 0.036 S -0.380 0.570 -0.190 T -0.370 0.580 -0.209

C__uln 1.0 V -0.220 -0.220 0.430 0.471 0.43 0.607 0.32 S 0.082 -0.123 0.041 T -0.138 -0.343 0.471

Table II The components of valence and size effect EFGs along with the total calculated and experimental values of EFG (in 1024 cm -3) and r /at the INN site for nontransitional dilute alloys of AI metal. The description is the same as in table I

EFG Alloy Cs contribution q.x q± qll [ qcall r/ca, qe*p ~'/exp

AISi -5 .0 V -0.009 -0.009 0.018 0.102 0.039 0.282 0.03 S 0.062 -0.093 0.031 T 0.053 -0.102 0.049

-5 .0 V -0.024 -0.024 0.048 0.189 0.089 0.195 0.07 S 0.110 -0.165 0.055 T 0.086 -0.189 0.103

-1 .0 V -0.120 -0.120 0.239 0.268 0.54 0.228 0.03 S 0.058 -0.087 0.029 T -0.062 -0.207 0.268

-0.04 V +0.110 +0.110 -0.210 0.211 0.035 0.182 0.03 S -0.003 -0.0045 -0.0015 T 0.107 0.114 -0.211

AIMg

AlGa

AICd

Table III The components of valence and size effect EFGs along with the total calculated and experimental values of EFG (in 1024 cm -3) and r /at the INN site for transitional alloys of AI metal. The description is the same as in table I

EFG Alloy C s contribution q,x q± qll I q¢,ll nca, qexp "qcxp

A1Cr -6 .0 V -0.130 -0.130 0.260 0.706 0.28 0.639 0.23 S 0.384 -0.576 0.192 T 0.254 -0.706 0.452

1.0 V -0.163 -0.163 0.327 0.296 0.52 0.337 0.57 S -0.062 0.093 -0.031 T -0.225 -0.070 0.296

-7 .0 V -0.017 -0.017 0.034 0.605 0.24 0.804 0.05 S 0.392 -0.588 0.196 T 0.375 -0.605 0.230

AI____Fe

A_JSc


Table IV The valence and size effect contributions to the EFG (in 1024 cm -3) along with calculated and experimental values of EFG at the 2NN site for nontransitional dilute alloys of Cu and A1 and transitional alloys of A1. C s = 1 is used

Contribution to EFG

Alloy q~ q~l Iqc.,[ = [q~ + q~l[ qexp

C_.._~uZn -0 .320 -0 .023 0.343 C_.uuGa -0 .780 -0.015 0.795 C__._uuCd -0 .002 -0 .140 0.142 C___uuln -0.210 -0.061 0.271 A1Si -0.001 0.008 0.007 AIMg 0.022 0.015 0.037 AICd 0.019 -0.052 0.033 AlGa -0 .067 0.042 0.025 A1Cr -0 .012 0.043 0.031 A1Fe -0 .022 0.042 0.020 AISc -0 .009 0.042 0.033

<0.3 <0.22(<0.3)

0.15 <0.16 <0.02 <0.02

<0.02 0.049 0.07 0.328

nontransitional alloys and Al-based transitional alloys may be viewed as fortuitous.

4. Conclusions

The present dielectric screening approach in conjunction with the model potential method includes the s-, p- and d-band effects of both the host and impurity atoms. The nonlocal effects are included partially by using a semilocal model potential [23, 24] and through energy-dependent potential well-depths in the screening function. The d-potential well-depth, A2(E ) o: (E - Ed) -1, is strongly dependent on energy and represents the s-d hybridization analogous to that found in the orthogonalized plane wave pseudopotential theory [32]. Such a resonance behaviour is re- sponsible for the existence of virtual bound state under favourable conditions. The exchange corrections take care, to some extent, of the nonlinear screening effects. There are two notewor- thy aspects of the present approach. First, the calculation of AV(r) is internally consistent in the sense that both AVb(q) and eh(q) are derived from the same model potential. Second, the band structure effects are included internally in the formalism itself. On the other hand, the previous calculations [7-15] include the band structure effects through a parameter usually

called the core-enhancement factor. But the validity of the way in which the core-enhancement factor is introduced is not known. One of the possible improvements in the present model is to include local field effects, arising from the non- uniform density of d-electrons, which contributes significantly towards screening process in the TMs [33, 34]. The dielectric tensor, the non- diagonal elements of which are mainfestations of the local field corrections, should be used in the screening of impurities in the TM-based alloys. One of the best ab initio calculations for the TM-based alloys will be to use the KKR band structure technique [35] in estimating AV(r) but such a calculation is still awaited in dilute alloys of Cu.

The present dielectric screening approach in- volves only a single parameter, Cs, contrary to the earlier calculations [9, 11-13, 15] having two parameters. Further, it is free from the deficiencies involved in calculating the phase shift, par- ticularly for the TM impurities, in the partial wave method [15, 16]. The present results em- phasize the need of calculating the displacement field from ab initio methods or from computer simulation techniques. The present formalism has also been applied to the dilute transitional alloys of Cu [18] and V [19] and yields good agreement between the calculated and experimental values of EFG and ~. Therefore it


can be conc luded that the p resen t impur i ty screening approach works fairly well in all the di lute t rans i t iona l and non t r ans i t i ona l cubic alloys with bo th the fcc and bcc crystal s t ructures.

Acknowledgements

We are thankfu l to D e p a r t m e n t of Physics, G u r u N a n a k Dev Univers i ty and D e p a r t m e n t of Physics, P a n j a b Univers i ty for provid ing neces- sary facilities to carry out this work.

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Calculation of electric field gradient in transitional and nontransitional dilute cubic alloys

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Transcript of Calculation of electric field gradient in transitional and nontransitional dilute cubic alloys