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Electric field gradient in dilute transitional alloys of copper
Volume 107A, number 3
PHYSICS LETTERS
21 January 1985
ELECTRIC FIELD GRADIENT 1N DILUTE TRANSITIONAL ALLOYS OF COPPER J. SINGH 1 Department of Physics, Pun/ab Agricultural University, Ludhiana 141004, India
and S.K. RATTAN and S. PRAKASH Department of Physics, Pan/ab University, Chandigarh 160014, India Received 23 July 1984 Revised manuscript received 26 November 1984
The electric field gradient (EFG) and asymmetry parameter have been calculated for dilute CuNi, CuPd and CuPt alloys using a dielectric screening approach in the pseudopotential theory. The major contribution to EFG is found to arise from the valence effect as compared to the size effect, a novel feature.
In a cubic metal the EFG arises due to the presence of an impurity which disturbs the cubic symmetry. The EFG comprises two contributions: First due to the valence difference between the host and the impurity (called valence effect) and second due to the size difference of the impurity and host atoms (called size effect). A number of authors [1-3] have used a partial wave (PW) method to calculate the valence effect EFG in simple metal alloys with reasonable success but this approach gives poor results in dilute alloys with transition metal impurities [4--6]. Even the validity of the PW analysis for transitional alloys is questionable. Using the dielectric screening theory in the pseudopotential approach due to Fukai and Watanabe [7] one can include the d-band effects of the host and impurity consistently. We use this approach in the present calculation of the EFG. The scattering potential.8 V(r), which is the excess impurity potential screened by the conduction electrons of the host metal, is given as
I f ~ Vb(q) V(r) = (2=~ J ~ exp(iq ° r) dq,
(I)
where 1 Present address: Department of Physics, Guru Nanak Dev University, Amritsar, India.
0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North.Holland Physics Publishing Division)
Vb(q) = e(q) = ] -
Vb(q)- Vb(q), v(q)x(q)
v(q) = (4~'e* 2/q2) x(q) = ~ k
(2)
(3)
,
[1 -- fxc(q)] ,
(4)
f(Ek)-f(Ek+q ) Ek -
k+q
X ]( ~k(r)l exp(-iq • r) J~k+q(r)) j2.
(5)
Vb(q) and Vb(q) are the bare ion potentials of the impurity and host atoms, respectively, and e(q) is the dielectric function of the host metal, dJk(r)is the Bloch function with energy E k. f(Ek) is the Fermi distribution function, fxc(q) is the exchange correction function, e* is the effective electronic charge and k F is the Fermi momentum. Knowing 8 V(r) the component of valence effect EFG along the line joining the host and impurity atoms (taken as Z-axis) is [8] v 2 qll(r) = ~(1 - 7.)(d28 V/dr2 - r-1 d8 V/dO ,
(6)
where 7~ is the Sternheimer antishielding factor. To fred the EFG, both 5 Vb(q) and e(q) are needed which axe obtained in the pseudopotential approach. Here we use the Heine-Abarenkov model potential
rM(r), 129
Volume 107A, number 3
PHYSICS LETTERS qxSx = q± _s -_ -~qll 1 _ s = C(1/3e)
VM(r) = - ~ A 1 P l
for
t,
for
r>R M .
(7)
Here A l and P/are the well depth and projection operators for orbital quantum number l ; Z and R M are the valency and model potential radius. Ammalu [9] obtained the transition metal model potential (TMMP) by solving eq. (7) in the semi-local approximation. For both the host and impurity atoms we use Animalu's TMMP with normal valency for the host atom and Blatt's corrected effective valency for the impurity atom. Using the above model potential we have recently obtained the dielectric function for transition metals
tl0l me* 2
×
[1 - fxc(q)]
27rkF•2X 2
l + ~ l n
~
[l+3'(q)],
(8)
where (9)
X = q/Zk F ,
7(q) = ~3(q) + ¼ I~(q)12 ,
(10)
kF
0 dA l
× ~ (2/* 1)-~f 1
RM
[/l([k+q[r)]2r2 dr,
0
(I1)
Z h is valency of the host metal,/l (kr) are spherical Bessel functions and d A t / d E (per au of energy) are energy derivatives of the well depths. Here ?(q) is the depletion-hole contribution and contains s-, p- and d-characters of the host metal. Sagalyn and Alexander [11] gave the size effect EFG qS for cubic alloys. Recently Pal et al. [ 12] have modified qS assuming the crystal to be made up of dressed ions. The principal components o f q s at the first nearest neighbour (INN) distance are [12] q~ = ~qxxl s = _~qSx = C ( I / 6 e ) G I ( F 1 1 _ F12) ,
(12)
and at the second nearest neighbour (2NN) distance are
130
G2(Fll - F12 ) ,
(13)
r < R M, where
= -Ze*/r,
e(q)= 1 ÷
21 January 1985
G n = (q/16ZrVE)Rn 3 (a -1 d a / d c ) .
(14)
R n is the displaced distance of the nth nearest neighbour, a -1 da/dc is the fractional change in lattice
parameter a, vE = 3 (1 - o)/(l + o) is a function of the Poisson ratio o. Here qSxx and q~_ are the principal components of qS perpendicular to the parallel direction. The components of the fourth-order tensor F l l , F12 etc. are calculated assuming the interaction potential between the dressed ions to be of ThomasFermi type which is calculated by satisfying the bulk modulus of the alloy [12]. The constant C takes into account the deficiencies of the displacement in the elastic continuum model and of the point charge model. The EFG q and asymmetry parameter .7 are calculated for Cu__Ni,C_uuPdand C___~tdilute alloys. Here the first underlined symbol denotes the host metal and the second symbol the lmpunty. To evaluate 6 Vb(q) the TMMP parameters are taken from Animalu [9] and to evaluate e(q) the parameters are taken from Singh et al. [13] ,1. e ( q ) is modified by the Hubbard exchange corrections. The parameters a -1 da/dc and R n are taken from ref. [6]. The space denvatwes of 6 V(r), needed to calculate q~, are evaluated numerically. The value of the parameter C is adjusted to obtain reasonable agreement between the calculated and experimental values of .7 for C__uNi.The same value of C is used to calculate q and *7 for other alloys at both the INN and 2NN distances. The calculated q and *7at the 1NN and 2NN distances are compared with the corresponding experimental values [14] qexp and *Texp m tables 1 and 2 respectwely. The interesting features of the present results are described below. (i) Both the valence and size effect contributions to EFG at the INNs and 2NNs (see tables I and 2) mcrease as one goes from the 3d to 5d impurity x.e. from Ni to Pt through Pd. But the valence effect EFG for C_.__~dand C__~t at the 2NN is nearly the same (table 2). It has been found that the depletion hole 7(q) contributes significantly towards the valence effect EFG. It is important to note that the major con,1 In this reference the value of the d-band width is misprinted but here we use its correct value 0.22 Ry.
Volume 107A, number 3
PHYSICS LETTERS
21 January 1985
Table 1 Calculated and experimental values ofq (in 1024 em-3) and ~ at the 1NN distance for C.._.uuNi,C_.__~dand CuPt alloys. The underlined values are the maximum component ofq which are taken as the EFGs in the alloys. Impurity
EFG contribution
qxx
q.L
qll
Iq Ical
r/cal
qexp
r~exp
Ni
total size valence total size valence total size valence
-0 170 -0.020 -0.150 -0.585 0.084 -0.669 -0.696 0.096 -0.792
-0.120 0.030 -0.150 -0.795 -0.126 -0.669 -0.936 -0.144 -0.792
0.290 -0.010 0.300 1.380 0.042 1.338 1.631 0.048 1.583
0.29
0.18
0.43
0.19
1.38
0.19
1.21
0.19
1.631
0.15
1.80
Pd Pt
tribution to q, at both the 1NN and 2NN distances, comes from the valence effect which is in agreement with the prediction of Minier and Minier [14] and Beal-Monod [4].Furthermore the magnitudes ofq~I and q~ are in close agreement with those given by Minier and Minier [ 14] but these authors have not calculated explicitly the values of the q~l component. Our results contradict the theoretical results of Pal et al. [6] based on the PW method whose validity for transitional alloys is doubtful. (ii) It is important to note that the value of the parameter C in qS [eqs. (12) and (13)] is very close to the value predicted by Faulkner [ 15]. But other authors [6,11,14] varied arbitrarily the size strength parameter in different copper alloys to get reasonable agreement with qexp and r/ex p. One should note that C is quite different from the size strength parameter used by other authors [6,11 ]. (iii) The principal component of the EFG tensor is along the parallel direction which is in accordance with the theoretical and experimental results of Minier and Minier [14].
Table 2 Calculated and experimental values ofq (in 1024 cm-s) at the 2NN distance for CuNi, C..__uuPdand CuPt alloys. Impurity
q~
q~
Ic/Ical
Ni Pd Pt
-0.510 -0.825 -0.812
0.014 0 . 4 9 6 -0.060 0.885 -0.080 0 . 8 9 2
qexp 0.056 0.197 0.295
(iv) The calculated values o f q and r/at the INN shell are in close agreement with the experimental values [14] while at the 2NN distance the calculated values are larger but of the same order of magnitude as the experimental values. The trend predicted by the calculated results is in agreement with the experimental values. The dielectric screening approach in the pseudopotential theory, for calculating q and 7/, is better than the PW method due to many reasons. The Bloch character arising due to the s-, p- and d-electrons of the host metal is included explicitly through the host metal dielectric function and the host TMMP while that of the impurity metal is included through the impurity TMMP. This method makes the valence effect EFG parameter free. Furthermore qV calculated using eq. (6), is free from the asymptotic and preasymptotic approximations faced in the PW method [3]. It is interesting to compare the present method with the Green function formalism for the transition metal based alloys of Riedinger [16]. In the calculation of the change in density of electronic states as a function of energy for CuNi, Riedinger [16] included the d-band effects of Cu using the hybridized s- and dband scheme of Mueller [17] and those of Ni by considering its d-orbital. The resonant scattering from the d-orbital of Ni gives rise to a virtual bound state. In the present calculations the d-band effects of the host and impurity are incomorated through the d potential well-depth A2(E ) which exhibits resonant behavlour [9] as A2(E ) oc (E - Ed) -1. Although we have not investigated the presence of the virtual bound states 131
Volume 107A, number 3
PHYSICS LETTERS
in Cu___Ni,Cu___Pdand C____~tbut the large contribution to q~l comes from the d-band effect resulting in large values of total EFG. This may reflect the existence of virtual b o u n d state in transitional alloys of Cu. In conclusion we want to say that to the best of our knowledge, this is the first attempt towards the calculation of the EFG in which s-, p- and d-characters of both the host and the impurity atoms are included explicitly. It is interesting to investigate the relative magnitude of valence effect EFG in other dilute transitional alloys and such an investigation is in progress.
References [1] A. Blandian and J Fnedel, J. Phys. Radium 21 (1960) 689. [2] W. Kohn and S.H. Vosko, Phys. Rev. 119 (1960) 912
132
21 January 1985
[3] G. Gruner and M. Mmler, Adv. Phy~ 26 (1977) 231, and references therein [4] M.T. Beal-Monod, Phys. Rev. 164 (1967) 360 [5 ] J.M. BretteU and A.J. Heeger, Phys. Rev. 153 (1967) 319 [6] B. Pal, S.D. Raj, S Prakash and J Singh,Can. J Phys 61 (1983) 1064 [7] Y. Fukai andW. Watanabe, Phys Rev. B2 (1970) 2353 [8] C A. Sholl, Proc Phys. Soc 91 (1967) 130 [9] A.O E. Animalu, Phys. Rev. B8 (1973) 3542 [10] J. Smgh and S. Prakash, J. Phys F l l (1981) 2409. [11] P.L. Sagalyn and M.N. Alexander, Phys. Rev B15 (1977) 5581. [12] B. Pal, S Mahajan, S.D. Raj, J Stag and S. Prakash, Phys. Rev. B30, to be pubhshed. [13] R. Singh, S. Prakash and J. Singh, J. Phys. F10 (1980) 1231 [14] M. Mmier and C. Mmler, Phys. Rev. B22 (1980) 21 [15] E.A. Faulkner, Philos. Mag. 5 (1980) 843 [16] R. Rledinger, J Phys F1 (1971) 392. [17] F.M. Mueller, Phys. Rev 153 (1967) 659.
PHYSICS LETTERS
21 January 1985
ELECTRIC FIELD GRADIENT 1N DILUTE TRANSITIONAL ALLOYS OF COPPER J. SINGH 1 Department of Physics, Pun/ab Agricultural University, Ludhiana 141004, India
and S.K. RATTAN and S. PRAKASH Department of Physics, Pan/ab University, Chandigarh 160014, India Received 23 July 1984 Revised manuscript received 26 November 1984
The electric field gradient (EFG) and asymmetry parameter have been calculated for dilute CuNi, CuPd and CuPt alloys using a dielectric screening approach in the pseudopotential theory. The major contribution to EFG is found to arise from the valence effect as compared to the size effect, a novel feature.
In a cubic metal the EFG arises due to the presence of an impurity which disturbs the cubic symmetry. The EFG comprises two contributions: First due to the valence difference between the host and the impurity (called valence effect) and second due to the size difference of the impurity and host atoms (called size effect). A number of authors [1-3] have used a partial wave (PW) method to calculate the valence effect EFG in simple metal alloys with reasonable success but this approach gives poor results in dilute alloys with transition metal impurities [4--6]. Even the validity of the PW analysis for transitional alloys is questionable. Using the dielectric screening theory in the pseudopotential approach due to Fukai and Watanabe [7] one can include the d-band effects of the host and impurity consistently. We use this approach in the present calculation of the EFG. The scattering potential.8 V(r), which is the excess impurity potential screened by the conduction electrons of the host metal, is given as
I f ~ Vb(q) V(r) = (2=~ J ~ exp(iq ° r) dq,
(I)
where 1 Present address: Department of Physics, Guru Nanak Dev University, Amritsar, India.
0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North.Holland Physics Publishing Division)
Vb(q) = e(q) = ] -
Vb(q)- Vb(q), v(q)x(q)
v(q) = (4~'e* 2/q2) x(q) = ~ k
(2)
(3)
,
[1 -- fxc(q)] ,
(4)
f(Ek)-f(Ek+q ) Ek -
k+q
X ]( ~k(r)l exp(-iq • r) J~k+q(r)) j2.
(5)
Vb(q) and Vb(q) are the bare ion potentials of the impurity and host atoms, respectively, and e(q) is the dielectric function of the host metal, dJk(r)is the Bloch function with energy E k. f(Ek) is the Fermi distribution function, fxc(q) is the exchange correction function, e* is the effective electronic charge and k F is the Fermi momentum. Knowing 8 V(r) the component of valence effect EFG along the line joining the host and impurity atoms (taken as Z-axis) is [8] v 2 qll(r) = ~(1 - 7.)(d28 V/dr2 - r-1 d8 V/dO ,
(6)
where 7~ is the Sternheimer antishielding factor. To fred the EFG, both 5 Vb(q) and e(q) are needed which axe obtained in the pseudopotential approach. Here we use the Heine-Abarenkov model potential
rM(r), 129
Volume 107A, number 3
PHYSICS LETTERS qxSx = q± _s -_ -~qll 1 _ s = C(1/3e)
VM(r) = - ~ A 1 P l
for
t,
for
r>R M .
(7)
Here A l and P/are the well depth and projection operators for orbital quantum number l ; Z and R M are the valency and model potential radius. Ammalu [9] obtained the transition metal model potential (TMMP) by solving eq. (7) in the semi-local approximation. For both the host and impurity atoms we use Animalu's TMMP with normal valency for the host atom and Blatt's corrected effective valency for the impurity atom. Using the above model potential we have recently obtained the dielectric function for transition metals
tl0l me* 2
×
[1 - fxc(q)]
27rkF•2X 2
l + ~ l n
~
[l+3'(q)],
(8)
where (9)
X = q/Zk F ,
7(q) = ~3(q) + ¼ I~(q)12 ,
(10)
kF
0 dA l
× ~ (2/* 1)-~f 1
RM
[/l([k+q[r)]2r2 dr,
0
(I1)
Z h is valency of the host metal,/l (kr) are spherical Bessel functions and d A t / d E (per au of energy) are energy derivatives of the well depths. Here ?(q) is the depletion-hole contribution and contains s-, p- and d-characters of the host metal. Sagalyn and Alexander [11] gave the size effect EFG qS for cubic alloys. Recently Pal et al. [ 12] have modified qS assuming the crystal to be made up of dressed ions. The principal components o f q s at the first nearest neighbour (INN) distance are [12] q~ = ~qxxl s = _~qSx = C ( I / 6 e ) G I ( F 1 1 _ F12) ,
(12)
and at the second nearest neighbour (2NN) distance are
130
G2(Fll - F12 ) ,
(13)
r < R M, where
= -Ze*/r,
e(q)= 1 ÷
21 January 1985
G n = (q/16ZrVE)Rn 3 (a -1 d a / d c ) .
(14)
R n is the displaced distance of the nth nearest neighbour, a -1 da/dc is the fractional change in lattice
parameter a, vE = 3 (1 - o)/(l + o) is a function of the Poisson ratio o. Here qSxx and q~_ are the principal components of qS perpendicular to the parallel direction. The components of the fourth-order tensor F l l , F12 etc. are calculated assuming the interaction potential between the dressed ions to be of ThomasFermi type which is calculated by satisfying the bulk modulus of the alloy [12]. The constant C takes into account the deficiencies of the displacement in the elastic continuum model and of the point charge model. The EFG q and asymmetry parameter .7 are calculated for Cu__Ni,C_uuPdand C___~tdilute alloys. Here the first underlined symbol denotes the host metal and the second symbol the lmpunty. To evaluate 6 Vb(q) the TMMP parameters are taken from Animalu [9] and to evaluate e(q) the parameters are taken from Singh et al. [13] ,1. e ( q ) is modified by the Hubbard exchange corrections. The parameters a -1 da/dc and R n are taken from ref. [6]. The space denvatwes of 6 V(r), needed to calculate q~, are evaluated numerically. The value of the parameter C is adjusted to obtain reasonable agreement between the calculated and experimental values of .7 for C__uNi.The same value of C is used to calculate q and *7 for other alloys at both the INN and 2NN distances. The calculated q and *7at the 1NN and 2NN distances are compared with the corresponding experimental values [14] qexp and *Texp m tables 1 and 2 respectwely. The interesting features of the present results are described below. (i) Both the valence and size effect contributions to EFG at the INNs and 2NNs (see tables I and 2) mcrease as one goes from the 3d to 5d impurity x.e. from Ni to Pt through Pd. But the valence effect EFG for C_.__~dand C__~t at the 2NN is nearly the same (table 2). It has been found that the depletion hole 7(q) contributes significantly towards the valence effect EFG. It is important to note that the major con,1 In this reference the value of the d-band width is misprinted but here we use its correct value 0.22 Ry.
Volume 107A, number 3
PHYSICS LETTERS
21 January 1985
Table 1 Calculated and experimental values ofq (in 1024 em-3) and ~ at the 1NN distance for C.._.uuNi,C_.__~dand CuPt alloys. The underlined values are the maximum component ofq which are taken as the EFGs in the alloys. Impurity
EFG contribution
qxx
q.L
qll
Iq Ical
r/cal
qexp
r~exp
Ni
total size valence total size valence total size valence
-0 170 -0.020 -0.150 -0.585 0.084 -0.669 -0.696 0.096 -0.792
-0.120 0.030 -0.150 -0.795 -0.126 -0.669 -0.936 -0.144 -0.792
0.290 -0.010 0.300 1.380 0.042 1.338 1.631 0.048 1.583
0.29
0.18
0.43
0.19
1.38
0.19
1.21
0.19
1.631
0.15
1.80
Pd Pt
tribution to q, at both the 1NN and 2NN distances, comes from the valence effect which is in agreement with the prediction of Minier and Minier [14] and Beal-Monod [4].Furthermore the magnitudes ofq~I and q~ are in close agreement with those given by Minier and Minier [ 14] but these authors have not calculated explicitly the values of the q~l component. Our results contradict the theoretical results of Pal et al. [6] based on the PW method whose validity for transitional alloys is doubtful. (ii) It is important to note that the value of the parameter C in qS [eqs. (12) and (13)] is very close to the value predicted by Faulkner [ 15]. But other authors [6,11,14] varied arbitrarily the size strength parameter in different copper alloys to get reasonable agreement with qexp and r/ex p. One should note that C is quite different from the size strength parameter used by other authors [6,11 ]. (iii) The principal component of the EFG tensor is along the parallel direction which is in accordance with the theoretical and experimental results of Minier and Minier [14].
Table 2 Calculated and experimental values ofq (in 1024 cm-s) at the 2NN distance for CuNi, C..__uuPdand CuPt alloys. Impurity
q~
q~
Ic/Ical
Ni Pd Pt
-0.510 -0.825 -0.812
0.014 0 . 4 9 6 -0.060 0.885 -0.080 0 . 8 9 2
qexp 0.056 0.197 0.295
(iv) The calculated values o f q and r/at the INN shell are in close agreement with the experimental values [14] while at the 2NN distance the calculated values are larger but of the same order of magnitude as the experimental values. The trend predicted by the calculated results is in agreement with the experimental values. The dielectric screening approach in the pseudopotential theory, for calculating q and 7/, is better than the PW method due to many reasons. The Bloch character arising due to the s-, p- and d-electrons of the host metal is included explicitly through the host metal dielectric function and the host TMMP while that of the impurity metal is included through the impurity TMMP. This method makes the valence effect EFG parameter free. Furthermore qV calculated using eq. (6), is free from the asymptotic and preasymptotic approximations faced in the PW method [3]. It is interesting to compare the present method with the Green function formalism for the transition metal based alloys of Riedinger [16]. In the calculation of the change in density of electronic states as a function of energy for CuNi, Riedinger [16] included the d-band effects of Cu using the hybridized s- and dband scheme of Mueller [17] and those of Ni by considering its d-orbital. The resonant scattering from the d-orbital of Ni gives rise to a virtual bound state. In the present calculations the d-band effects of the host and impurity are incomorated through the d potential well-depth A2(E ) which exhibits resonant behavlour [9] as A2(E ) oc (E - Ed) -1. Although we have not investigated the presence of the virtual bound states 131
Volume 107A, number 3
PHYSICS LETTERS
in Cu___Ni,Cu___Pdand C____~tbut the large contribution to q~l comes from the d-band effect resulting in large values of total EFG. This may reflect the existence of virtual b o u n d state in transitional alloys of Cu. In conclusion we want to say that to the best of our knowledge, this is the first attempt towards the calculation of the EFG in which s-, p- and d-characters of both the host and the impurity atoms are included explicitly. It is interesting to investigate the relative magnitude of valence effect EFG in other dilute transitional alloys and such an investigation is in progress.
References [1] A. Blandian and J Fnedel, J. Phys. Radium 21 (1960) 689. [2] W. Kohn and S.H. Vosko, Phys. Rev. 119 (1960) 912
132
21 January 1985
[3] G. Gruner and M. Mmler, Adv. Phy~ 26 (1977) 231, and references therein [4] M.T. Beal-Monod, Phys. Rev. 164 (1967) 360 [5 ] J.M. BretteU and A.J. Heeger, Phys. Rev. 153 (1967) 319 [6] B. Pal, S.D. Raj, S Prakash and J Singh,Can. J Phys 61 (1983) 1064 [7] Y. Fukai andW. Watanabe, Phys Rev. B2 (1970) 2353 [8] C A. Sholl, Proc Phys. Soc 91 (1967) 130 [9] A.O E. Animalu, Phys. Rev. B8 (1973) 3542 [10] J. Smgh and S. Prakash, J. Phys F l l (1981) 2409. [11] P.L. Sagalyn and M.N. Alexander, Phys. Rev B15 (1977) 5581. [12] B. Pal, S Mahajan, S.D. Raj, J Stag and S. Prakash, Phys. Rev. B30, to be pubhshed. [13] R. Singh, S. Prakash and J. Singh, J. Phys. F10 (1980) 1231 [14] M. Mmier and C. Mmler, Phys. Rev. B22 (1980) 21 [15] E.A. Faulkner, Philos. Mag. 5 (1980) 843 [16] R. Rledinger, J Phys F1 (1971) 392. [17] F.M. Mueller, Phys. Rev 153 (1967) 659.