ISSN 10637834, Physics of the Solid State, 2014, Vol. 56, No. 6, pp. 1081–1086. © Pleiades Publishing, Ltd., 2014. Original Russian Text © Yu.Yu. Tsiovkin, 2014, published in Fizika Tverdogo Tela, 2014, Vol. 56, No. 6, pp. 1041–1045.

METALS

Residual Electrical Resistivity of Dilute Nickel Alloys Yu. Yu. Tsiovkin Institute for Educational Development, ul. Akademicheskaya 16, Yekaterinburg, 620066 Russia email: [email protected] Received September 23, 2013; in final form, December 6, 2013

Abstract—A quantitative calculation of the residual electrical resistivity of dilute ferromagnetic nickelbased alloys has been performed in the framework of the fourcurrent conduction model using the kinetic equation and ab initio approaches for and ab initio approaches for the determination of the scattering potential. The contributions to the residual electrical resistivity from scattering by inhomogeneities of the impurity Coulomb potential and the exchange interaction have been separated by comparing the calculated and experimental data. DOI: 10.1134/S1063783414060365

1. INTRODUCTION For many years, the calculation of residual electri cal resistivity (RR) of magnetic alloys has attracted the particular attention of various research groups. One of the first approaches was formulated within the two current conduction model by Fert and Campbell [1, 2]. The result of the performed calculations was the introduction of two conduction channels for each of the spin orientations (ρ↑ and ρ↓) and the mixed con duction channel ρ↑↓ corresponding to possible pro cesses of spinflip scattering of conduction electrons. However, the formal solution of the kinetic equation obtained in [1, 2] without taking into account the real scattering mechanism does not allow one to specify the resistivity ρ↑↓. For example, at zero temperature, the electron spin flip is possible to occur only as a result of the electron scattering by inhomogeneities of the spin–orbit interaction potential. However, this potential for transition metals is vanishingly small [3] and cannot lead to a significant contribution to RR. Therefore, in the calculations of RR of ferromagnetic alloys, the value of ρ↑↓ usually is taken to be zero ρ↑↓ 0. Attempts to more consistently take into account the mechanisms of electron scattering in magnetic alloys have been made within both the model and ab initio approaches. In particular, using the Friedel sum rule in the framework of the s–d exchange model and the ab initio data on the band structure of metals, V.Yu. Irkhin and Yu.P. Irkhin [4] calculated RR for a number of dilute alloys based on iron and nickel. The ab initio approach to the calculation of RR (when the singleelectron relaxation time is equal to the trans port time) of magnetic alloys was used in a series of studies summarized in the review by Mertig [5]. In the aforementioned approaches, attempts to specify and calculate ρ↑↓ have been unsuccessful.

Hence, when comparing the results of the performed calculations of RR with experimental data, it was nec essary to introduce an additional parameter of the the ory [1]: α = ρ↓/ρ↑. However, in the determination of the value of this parameter for the same alloys but in the framework of different modeling approaches, including ab initio approaches [2, 4–6], there arise significant differences (by a factor of 8–10), which indicates clear contradictions inherent in the pro posed calculation schemes. A reliable determination of the parameter α from the experimental data has also been impossible, because no direct or indirect meth ods for the relevant measurements of RR in different conduction channels in the alloys have been proposed. A somewhat different approach to the solution of the problem of calculating RR of magnetic alloys is based on the Mott twoband conductivity model [7]. According to Mott, scattered conduction electrons in transition metals can be transferred in a partially filled dband and cease to participate in transfer processes. This assumption provides an explanation for the sig nificant difference in RR of transition and nontransi tion metal alloys. Indeed, since the transition proba bility of a scattered electron is directly proportional to the density of states in the accepting band, the transi tions of mobile selectrons to the d band become most probable, because the ratio of the densities of states in the d and s bands at the Fermi level is of the order of 10–20. Consequently, the observed values of the resis tivity in transition metals are 10–20 times higher than those in ordinary metals. Earlier, the Mott conductivity model was success fully used for direct calculations of RR of all dilute nonmagnetic alloys with 3d, 4d, and 5d transition metals [8, 9]. This made it possible, for the first time, to prove the linear dependence of the reduced RR of the alloys on the square of the modulus of the s–d ele ment of the scattering Tmatrix, which was predicted

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by Luttinger and Kohn [10]. The purpose of this work is to perform a similar calculation for alloys with mag netic order, the more that the twoband character of conductivity for ferromagnetic metals and their alloys was proved experimentally [11, 12], and the ratios σs/σd for pure Fe, Co, and Ni were found to be equal to 11.67, 19.10, and 25.27, respectively [12].

In order to calculate RR of ferromagnetic alloys, we use the method of the generalized kinetic equation [10] and the Swenson identity [14]. The equation for σ the nonequilibrium addition f l to the Fermi distribu 0σ

tion function f l in the approximation linear with respect to the external electric field (F) in the limit of low impurity concentration has the form

2. MODEL

2πc

We assume that the spin–orbit interaction is negli gible and the processes of electron spinflip scattering are actually improbable. It is also assumed that con duction selectrons are scattered by electric fields of the metal ions, which are randomly distributed over sites of the crystal lattice, and can undergo intraband and interband transitions. The Hamiltonian describ ing this scattering in ferromagnetic alloys at zero tem perature (without spin flip) can be written in the form [3, 4] ˆ = H

∑E a

σ + l lσ a lσ

l, σ

+ 1 N

∑e

( k – k', r n )

+

ν ( n )D ( n )a lσ a l'σ . (1)

σ

σ

σ

2

σ

– E k'j' ) T kjk'j' ( f kj – f k'j' ) (3) 0σ

∂f kj σ = e ( F, e kj )   v kj , ∂E kj σ

σ

where v l is the electron velocity and T ll' is the scatter ing matrix. The complete system of four equations for each of the bands is written similarly to equation (3) with the explicit indication of the band and spin indi ces and differs from the equations obtained earlier in [8, 9] only by the presence of the spin index. Using the substitution 0σ

σ σ − 1 ΔJ ll' ( n ) 〈 S z〉 . D ll' ( n ) = ΔV ll' ( n ) + 2

(2)

σ

The first term containing Δ V ll' (n) describes the con ventional impurity scattering due to the random alter nation of ions in the lattice sites, and the second term describes the exchange part of the impurity scattering. When writing the Hamiltonian, we took into account that the periodic (nonscattering) part of the σ σ Aσ interaction is included in E l ; Δ V ll' (n) = V ll' (n) – A

σ kj

k', j'

n, l, l'

Here, the quantum number l includes the band index j σ (j = s, d) and the wave vector k, and E l is the unper turbed energy of an electron with the quantum index l in the band with the spin orientation σ = ↑ (↓). The σ quantity D ll' (n) has the following form:

Bσ

∑ δ(E

B

V ll' (n) and ΔJll'(n) = J ll' (n) – J ll' (n); rn is the radius vector of the nth site. The factor ν(n) = cAαB(n) – cBαA(n) describes the random distribution of ions of the alloy over the lattice sites; αA(B)(n) = 1 if the nth site is occupied by the A(B) ion and αA(B)(n) = 0 in the opposite case; cA(B) is the concentration of the compo nent A(B) of the alloy. Hamiltonian (1) can explicitly include the terms describing the s–d hybridization. However, the hybridization effects do not change the scattering part of the Hamiltonian and can be included in the peri odic part of the Hamiltonian [13].

σ σ σ ∂f kj f kj = e ( F, v kj )C kj   ∂E kj

(4)

and reducing the obtained system of vector kinetic equations (3) to the algebraic system of equations, we solve the latter system of equations with respect to the σ desired coefficients C kj . As a result, we find σ

σ

σ

I kj'j' + K kj'j + L kjj' σ C kj =   , σ σ σ σ σ σ ( I kjj + K kjj' ) ( I kj'j' + K kj'j ) – L kjj' L kj'j

(5)

where 2 σ σ σ σ 3 Ω I kjj = 3 d k'δ ( E kj – E k'j ) T kk', jj ( 2π )

∫

× (1 –

σ cos v kj,

(6)

σ v k'j ),

2 σ σ σ σ 3 Ω K kj' = 3 d k'δ ( E kj – E k'j' ) T kj, k'j' , ( 2π )

∫

(7)

2 σ σ σ σ 3 Ω L kjj' = – 3 d k'δ ( E kj – E k'j' ) T kj, k'j' ( 2π )

∫

(8)

σ

k'j' cos ( v σ , v σ ). × v   kj k'j' σ v kj

Here, the first integral (6) describes the intraband scattering; the second integral (7) and the third inte gral (8) describe the interband scattering accompanied

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RESIDUAL ELECTRICAL RESISTIVITY OF DILUTE NICKEL ALLOYS

by the direct j j' and reverse j' j charge carrier transitions, respectively; and Ω is the unit cell volume.

velocity), after the integration with respect to k, for the conductivity in the s band we find

For the direct calculation of the coefficients Ckj, it only remains to determine the elements of the scatter ing Tmatrix. In the singlesite and singleelectron approximations, the summation of an infinite series for the scattering Tmatrix can be performed accu rately. As a result, we have

σ 2 1 σ 2 1 σ 2 = σ D sd , T ds = σ D ds , A A σ σ σ σ σ 2 1 σ = σ D dd ( 1 – F s D ss ) + D ds F s D sd , A

σ 2 T dd

↑

σ

σ

σ

σ

σ

σ

σ

σ

↑

( I ss + K sd ) ( I dd + K ds ) – L ds L sd

σ

(9)

σ

σ

σ

σ

S s = I dd + K ds + L sd ,

1

(10)

σ k, j

σ

σ

σ

σ

σ

σ 2

σ 2

σ 2 T ds

σ

σ 2

D ds =  , σ A σ

σ 2 T dd

σ 2

D dd =   , (11) σ A

σ 2

j, σ

σ

For the calculation of RR, we further assume that the probability of the reverse (d s) transitions of scattered electrons are small compared to the direct σ transitions and, therefore, L sd 0. On the other hand, since integrals (6)–(8) are proportional to the density of states in the accepting band, we can assume

(12)

σ 2 σ ↑

↓

↑

↓

the approximation g s (EF) ≈ g s (EF) and v xS ≈ v xS has the form

σ

σ

band, and v xkj is the x component of the electron Vol. 56

σ

that I ss  K sd ≈ T sd g d (EF). Then, RR of the alloy in

(where n j is the electron concentration, g j (EF) is the density of states at the Fermi level in the conduction j

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3. RESULTS AND DISCUSSION

σ

0σ

σ σ σ 2 ∂f kj σ n j dEC kj ( v xkj )  g j ( E F ) ∂E kj

∑ ∫

det [ Q sd ] 1 3 ρ s =   ,   2 2 2 ns e ( v S ) gs ( EF ) Ss

which coincides with the main result obtained in [8, 9].

Using the generalized definition of the current 2

↓

x

A = 1 – F d D dd .

e F J =  3

↓

It is easy to see that, by omitting the spin indices in expression (14), we obtain

F s D jj'  1

σ σ σ σ σ 2 1 σ = σ D ss ( 1 – F d D dd ) + D sd F d D ds , A

D sd =  , σ A

↑

↑

can significantly simplify expressions (9) for the ele ments of the scattering Tmatrix, so that T ss

↓

⎞ ⎟, ⎟ ⎠

det [ Q sd ]det [ Q ds ] 3 ρ s =     . (14) ↑ ↓ ↓ ↑ 2 2 n s e ( v xS ) g s ( E F ) S s det [ Q ds ] + S s det [ Q sd ]

The inclusion of the Mott model assumptions [7] σ

σ

and assume that g s (EF) ≈ g s (EF) and v xS ≈ v xS . Then, the expression for the electrical resistivity can be writ ten in the most compact form

is the electron Green’s function in the j band with spin σ.

F s D jj' / F d D jj'  1,

σ

⎛ σ σ σ σ I + K sd L sd [ Q jj' ] = ⎜ ss ⎜ σ σ σ ⎝ L ds I dd + K ds

σ 2

 ∑  (z – E )

σ

S d = I ss + K sd + L ds ,

↑

k

.

Further, we define

where σ 1 F j =  N

(13)

↓ ↓ ↓ I dd + K ds + L sd ↓ ↓ 2 ↓ n s ( v xs ) g s ( E F )    ↓ ↓ ↓ ↓ ↓ ↓

A = ( 1 – F d D dd ) ( 1 – F s D s ) + F s D sd F d D sd ,

σ 2 T sd

↑

I dd + K ds + L sd   ×  ↑ ↑ ↑ ↑ ↑ ↑ ( I ss + K sd ) ( I dd + K ds ) – L ds L sd +

σ 2

T sd

2

e ↑ ↑ 2 ↑ σ s =  n s ( v xs ) g s ( E F ) 3

1 D σ ( 1 – F σ D σ ) + D σ F σ D σ 2 , =  ss d dd sd d ds σ A

σ 2

T ss

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↑

↓ 2 ↓

↑ 2

g d ( E F ) T sd g d ( E F ) T sd 1 ρ ≈     . ↑ gs ( EF ) g ( E ) T↓ 2 + g↓ ( E ) T↑ 2 d

F

sd

d

F

sd

(16)

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Absolute values of RR and REEC calculated by the direct method and the values of the scattering potential recovered from the experimental data Impurity Parameter V

Nb

Mo

Rh

Pd

Ta

W

Ir

Pt

ρexp, μΩ cm

4.4

5

6.4

1.8

0.2

5.2

6

3.8

0.8

ρtheor, μΩ cm

4.2

6.2

5.6

0.8

0.1

6.6

6.2

2.2

0.4

ρ, μΩ cm [4]

1.8

2.7

4.5

1.2

0.6

ΔDexp

1.61

1.81

2.15

0.99

0.33

1.85

2.05

1.51

0.65

ΔZtheor

1.08

2.07

1.67

0.29

0.22

2.17

2.13

0.69

0.45

For the direct calculation of RR and the elements of ↑(↓) 2 T sd ),

the scattering Tmatrix ( we use the simplest Thomas–Fermi approximation for the screening potential [15–17]. After simple algebra using expres sion (16), for RR of the alloys we obtain σ 3πc ρ s =  B 2 2 e n s v s g s

(17) ↑ ↑ 2 ↓ ↓ 2 g d ( E F ) ( ΔD ) g d ( E F ) ( ΔD ) ×    , ↑ ↓ ↑ 2 ↓ 2 ↓ 2 ↑ 2 g d ( E F ) ( ΔD ) 1 – M + g d ( E F ) ( ΔD ) 1 – M where 2

σ 1 e ⎞ , B =   ⎛  ⎝ 2 σ 2 ε Ω ⎠ q 4k F + ( q ) 0 0

σ

σ

σ

σ

M = B G d ΔD , 2

and q is the radius of the screening potential and k F is the square of the Fermi wave vector of the electron. The obtained expression contains a number of parameters required to be defined. Most of these parameters characterize the solvent matrix ↑(↓) 2 ( g s ( d ) )(EF), ns, v s ), and only one of them (ΔDσ) ρs(ΔD↓), arb. units 0.3 0.2 0.1

0

1

2

3 4 ΔD↓, arb. units

describes the type of impurity. In accordance with expression (2), the parameter ΔDσ can be represented − 〈 ΔJ σ〉 , where ΔZσ is the rel in the form ΔDσ = ΔZσ + ative excess electric charge (REEC) introduced by the impurity ion into the electrically neutral solvent σ matrix and 〈 ΔJ 〉 is the effective scattering potential induced by the inhomogeneity of the distribution of the exchange interaction potential over sites of the crystal lattice of the alloy. As was shown in [8, 9], the value of ΔZ can be easily determined using modern ab initio calculation meth ods. Indeed, assuming that the initial unit cell is elec trically neutral and using the LMTO–ASA methods [18–20] (for example, TB–LMTO–ASA47 program package), we can easily calculate REEC introduced by the impurity ion into the solvent matrix. The algorithm of this calculation is very simple. In the first step, the band structure and wave functions of the solvent metal are determined by a selfconsistent method. Next, one of the ions of the solvent matrix is replaced by the impurity ion and, using the obtained basis wave functions of the solvent metal, the charge flow through the unit cell containing the impurity ion is calculated, which makes it possible to determine REEC at a 1% impurity. This scheme of the calcula tion ignores both the effects associated with the distor tion of the initial lattice of the solvent metal due to the incorporation of the impurity ion and the effects asso ciated with the exchange interaction. However, the previously obtained results for nonmagnetic alloys [8, 9] are quite encouraging regarding the use of this sim ple scheme in the case of ferromagnetic alloys of nickel. Owing to the specific band structure of nickel, the problem can be further simplified. Since the spinup nickel dband is completely filled, the probability of the transition of scattered electrons into this band is vanishingly small. Therefore, the expression for RR can be written as ↓

Reduced RR as a function of the parameter channel σ = ↓ in the nickel matrix.

ΔD↓

for the

↓ 2

↓2 g d ( E F ) ( ΔD ) 3πc ρ s ≈   B  , 2 2 ↓ 2 e n s v s g s 1–M

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RESIDUAL ELECTRICAL RESISTIVITY OF DILUTE NICKEL ALLOYS

which makes it possible to perform a comparison of the effective scattering potentials obtained from the direct calculation and recovered from the experimen tal data. For nickel alloys with metals of the vanadium–tan talum group, the calculated and experimental values of RR are very close to each other (see table). For these alloys, the estimate of the fluctuating part of the exchange interaction is actually constant and amounts to ~30% of the total scattering potential. In these alloys, the introduction of an impurity atom does not lead to significant changes in the fluctuating part of the exchange interaction, because the lattice parameters of the solvent and impurities are not essentially differ ent form each other. Furthermore, good agreement between the obtained results is probably associated with the specific dependence of RR on REEC (see fig ure), because, for large values of ΔD, RR remains almost unchanged. For impurities of the platinum–rhodium group, the nickel matrix is characterized by the ratios ρexp/ρtheor ≈ 2–3. This indicates a significant underes timation in the elementary approach used for deter mining the scattering potential of the effects of lattice distortions in the solvent matrix due to the incorpora tion of impurities and, consequently, the changes in the fluctuating part of the exchange interaction. Of course, the calculated results can be signifi cantly improved by means of the introduction of an additional parameter of the theory, which character izes the intensity of scattering by inhomogeneities of the exchange potential. However, this would violate the ab initio approach applied in this work to the determination of the scattering potential. Meanwhile, the available ab initio methods for calculating the exchange interaction parameters lead to very contra dictory results and depend essentially on the adopted scheme of these calculations. The results obtained by direct calculation methods for RR of dilute nickel alloys can also be improved by consistently taking into account distortions of the crystal lattice of the solvent metal due to the introduc tion of the impurities. However, the implementation of the adequate computational scheme requires the use of a large (64 or 128 atoms) initial unit cell of the metal and complex matching techniques. The cost of such improvements is a manifold increase in the vol ume of necessary computations without changing the physical essence.

essentially on how much the impurity atom distorts the solvent matrix and on whether the impurity is undersize or oversize. The significant discrepancy between the calculated and experimental data for some alloys, most likely, is explained by the fact that the lattice distortions of the solvent matrix due to the introduction of impurity atoms are ignored. On the other hand, it is of interest to consider the possibility of evaluating the contribution to RR from scattering of conduction electrons due to the inhomogeneity of the distribution of the exchange potential, because reli able calculations of the corresponding potential in the framework of the existing ab initio methods are actu ally impossible. The direct calculations of RR of dilute ferromag netic alloys demonstrated that the impedance of con duction channels “connected in parallel” with differ ent spin orientations cannot be considered in terms of a circuit of parallelconnected conductors or a primi tive “equivalent” circuit. This conclusion follows directly from the obtained system of kinetic equations and from the impossibility of introducing the classical transport time for multiband scattering processes in transition metals and alloys. In conclusion, it should be noted that the pro posed fourcurrent conduction model is largely uni versal and can be used for relevant calculations of RR not only of magnetic alloys with transition metals but also alloys with rareearth metals. REFERENCES 1. A. Fert and I. A. Campbell, J. Phys. F: Met. Phys. 6 (5), 849 (1976). 2. I. A. Campbell and A. Fert, in Handbook of Ferromag netic Materials, Ed. by E. P. Wohlfarth (NorthHolland, Amsterdam, 1982), Vol. 3, p. 747. 3. S. V. Vonsovskii, Magnetism (Nauka, Moscow, 1971; Wiley, New York, 1974). 4. V. Yu. Irkhin and Yu. P. Irkhin, J. Magn. Magn. Mater. 164, 119, (1996). 5. I. Mertig, Rep. Prog. Phys. 62, 237 (1999). 6. J. W. F. Dorleijn and A. R. J. Miedema, J. Phys. F: Met. Phys. 7 (1), L23 (1977). 7. N. F. Mott, Adv. Phys. 13 (51), 325 (1964). 8. Yu. Yu. Tsiovkin, A. N. Voloshinskii, V. V. Gapontsev, and V. V. Ustinov, Phys. Rev. B: Condens. Matter 71, 184206 (2005).

4. CONCLUSIONS The proposed model for calculating RR of dilute ferromagnetic alloys made it possible, with good accu racy, to determine the absolute values of RR for a series of Nibased alloys without introducing additional arti ficial parameters of the theory. It was shown that the fluctuating part of the exchange interaction depends PHYSICS OF THE SOLID STATE

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9. Yu. Yu. Tsiovkin, A. N. Voloshinskii, V. V. Gapontsev, and V. V. Ustinov, Low Temp. Phys. 32 (8–9), 863 (2006). 10. J. M. Luttinger and W. Kohn, Phys. Rev. 109, 1892 (1958).

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11. P. O. Nilsson, in Solid State Physics, Ed. by F. Seitz, D. Turnbull, and H. Ehrenreich (Academic Press, New York, 1974), Vol. 29, p. 139. 12. M. M. Noskov, Preprint IFM AN SSSR (Institute of Metal Physics, Ural Branch of the Academy of Sci ences of the Soviet Union, Sverdlovsk, 1969). 13. F. Brouers and A. V. Vedyayev, Phys. Rev. B: Solid State 5, 348 (1972). 14. R. J. Swenson, J. Math. Phys. 3 (5), 1017 (1962). 15. J. Ziman, The Theory of Transport Phenomena in Solids (Oxford University Press, New York, 1960). 16. K. N. R. Taylor and M. I. Darby, Physics of RareEarth Solids (Chapman and Hall, London, 1972).

17. W. A. Harrison, Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond (Dover, New York, 1989). 18. W. Kohm and L. J. Sham, Phys. Rev. A 140 (4), 11133 (1965). 19. O. K. Andersen, Phys. Rev. B: Condens. Matter: Solid State 12, 3060 (1975). 20. R. O. Jones and O. Gunnarsson, Rev. Mod. Phys. 61, 681 (1989).

Translated by O. BorovikRomanova

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