ISSN 10637834, Physics of the Solid State, 2010, Vol. 52, No. 1, pp. 1–5. © Pleiades Publishing, Ltd., 2010. Original Russian Text © Yu.Yu. Tsiovkin, V.V. Dremov, E.S. Koneva, A.A. Povzner, A.N. Filanovich, A.N. Petrova, 2010, published in Fizika Tverdogo Tela, 2010, Vol. 52, No. 1, pp. 3–7.

METALS AND SUPERCONDUCTORS

Theory of the Residual Electrical Resistivity of Binary Actinide Alloys Yu. Yu. Tsiovkina, V. V. Dremovb, E. S. Konevac, A. A. Povznerc, *, A. N. Filanovichc, and A. N. Petrovac a

b

Institute of Physics of Metals, Ural Branch, Russian Academy of Sciences, ul. S. Kovalevskoі 18, Yekaterinburg, 620219 Russia

Russian Federal Nuclear Center—Zababakhin AllRussia Research Institute of Technical Physics, ul. Vasil’eva 13, Snezhinsk, Chelyabinsk region, 456770 Russia c Ural State Technical University (UPI), ul. Mira 19, Yekaterinburg, 620002 Russia * email: [email protected] Received April 27, 2009

Abstract—A system of selfconsistent equations has been proposed for the coherent potential approximation of the multiband conductivity model for the case of conduction electron scattering from chaotic electric fields of ions of disordered binary alloy components at zero temperature. It has been qualitatively demonstrated that the deviation of the concentration dependence of the residual electrical resistivity of actinide alloys with multiband conductivity from the Nordheim rule is caused by the explicit dependence of the electrical resistivity of the alloy on the magnitude and sign of the real part of the Green’s function at the Fermi level. The derived system of equa tions for the multiband coherent potential approximation has been used to calculate the concentration depen dence of the density of states and the residual electrical resistivity of the alloys of neptunium and plutonium. The results of the calculations have been compared with the available experimental data. DOI: 10.1134/S1063783410010014

1. INTRODUCTION

trons of actinides and qualitatively describe the transi tion from the quasimetallic type of behavior of the band electrons in Th, U, and Np to the localized type in the case of Am, Cm, and Bk. Substantial advances have been made in understanding properties of the ground state of Pu, and the nonmagnetic state of this metal has been explained [6–9]. A quite reliable reproduction of the main experimental results (ground state energy, volume, density of states, etc.) that has been obtained in different ab initio calculations and approaches for actinides opens up possibilities for tak ing into account actual densities of electronic states of alloy components when calculating their resistive properties.

Investigations of resistive properties of actinides and their alloys revealed a number of important fea tures and anomalies both in the temperature and con centration dependences of the electrical resistivity [1– 3]. Only recently, a consistent explanation for the anomalous temperature dependence of the electrical resistivity of delta plutonium has been obtained over a wide temperature range owing to the interference con ductivity model [4, 5]. At the same time, there is no modern theoretical study of the concentration depen dence of the residual resistivity for actinide alloys and experimental data are interpreted mainly by qualita tive estimates obtained using the perturbation theory, which provide only a coarse description for the observed dependences. Therefore, it seems important and interesting to develop an approach to the calcula tion of the residual electrical resistivity of actinide based alloys within the multiband conductivity model without employing the main simplifying assumptions of the perturbation theory, which would take into account modification of the electronic band structure of the alloys with a variation in the concentrations of their components. This method of calculation of the electrical resistivity of the actinide alloys can be imple mented within the coherent potential approximation for the multiband conductivity model. Modern ab initio calculations of the ground state of actinides can reproduce the main features of 5f elec

In order to explain the observed concentration dependence of the residual electrical resistivity of binary actinide alloys, it is reasonable to assume, fol lowing Mott [10], that scattered sband conduction electrons are transferred to vacant d and f bands, as in conventional transition metals, with probabilities pro portional to the corresponding densities of states at the Fermi level of the target band. These transitions exclude some of conduction electrons from the pro cess of charge transfer and, consequently, significantly increase the electrical resistivity of metals and alloys. This assumption can explain qualitatively typical mag nitudes of the residual resistivity of the alloys in ques tion. However, ab initio calculations for the ground state of actinides demonstrate that the electron densi 1

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TSIOVKIN et al.

ties of states at the Fermi level for empty d and f bands are comparable and much higher than the density of states of s conduction electrons. Moreover, the Mott s–d twoband conductivity model, which is usually employed to calculate the electrical resistivity of tran sition metals, assumes that only one unoccupied shell exists. In actinides and their alloys, because of the presence of two unoccupied bands with large and comparable magnitudes of the densities of states, it is possible and probable that both s d and s f transitions of conduction electrons take place. There fore, to perform the consistent calculation of the elec trical resistivity of the alloys, it is necessary, first, to generalize the twoband conductivity model renounc ing simplifying assumptions of the perturbation theory concerning weak interaction of electrons with scatter ing centers. Second, the calculations should take into account the renormalization of densities of states in s, d, and f bands of the alloy that takes place both due to the change in the concentrations of components and because of scattering. It is evident that the densities of states in the s, d, and f bands of the alloys turn out to be complex interdependent functions of the scattering amplitude. As it will be demonstrated below in the framework of the qualitative analysis and detailed cal culations performed within the suggested coherent potential approximation for the multiband conductiv ity model, this feature of the scattering leads to nonad ditive contributions to the electrical resistivity from s d and s f transitions, shift in the maximum of the residual electrical resistivity away from the equi atomic composition, and significant deviation of the electrical resistivity from the concentration depen dence predicted by the Nordheim rule.

+ l l al

l

1 +  N

∑e

– i ( k – k', R n )

+

B l, l' a l a l' ,

∑ exp ( –i ( k – k', R ) )Δ δ n

+ j jj' a l a l .

(3)

n, l, l'

0

Let us consider a system of s (p), d, and f electrons that undergo intraband and interband transitions (without spin flop) due to the scattering from electric fields of ions of alloy components randomly distrib uted over sites of the crystal lattice. The Hamiltonian of such a system can be represented as

∑E a

ˆ = 1 Δ N

Then, we introduce the complete resolvent of the ˆ in the form R ˆ = (z – H ˆ )–1 and separate its operator H ˆ = ˆ representation: G exactly diagonal part in the H

2. MODEL AND DERIVATION OF EQUATIONS OF MULTIBAND COHERENT POTENTIAL APPROXIMATION

ˆ = H

eters λll and λll' describe the amplitudes of intraband and interband transitions for scattered electrons, respectively. It is known that the hybridization leads to substan tial renormalization of the initial densities of states of the s (p), d, and f conduction bands. However, these corrections are important only for determining the ground state of the metals and do not affect the scat tering potentials [11]. Since below we use the initial densities of states obtained with allowance made for the effects of hybridization of the conduction bands, we assume that the corresponding corrections are ˆ , but, for convenience, we retain initial included in H 0 names for s (p), d, and flike conduction bands. Notice that the employment of the densities of states obtained in ab initio calculations makes it possible to ˆ that are due to partially include the alloy effects in H 0 the strong electron–electron interaction in 5f metals [12]. The quantummechanical requirement that the operator of shift and broadening of singleelectron levels should be exactly diagonal in the representation ˆ makes it possi of the Hamiltonian of ground state H 0 ble to obtain the system of selfconsistent equations of multiband coherent potential approximation accord ing to a relatively simple scheme. Let us define the operator of shift and broadening of singleelectron ˆ representation in level that is exactly diagonal in H 0 the form

(1)

n, l, l'

where El is the periodic part of the total energy of elec trons with the quantum number l, including the band index j (j = s(p), d, f ) and the wave vector k, Rn is the radius vector of the nth site of the crystal lattice, and B l, l' ( n ) = ν ( n ) [ λ kj, k'j δ j, j' + λ kj, k'j ( 1 – δ j, j' ) ]. (2) The factor ν(n) = αA(n)cB – αB(n)cA randomly distrib utes ions of alloy components over sites of the crystal lattice, αA(B)(n) = 1 if nth site is occupied by the A(B) ion, and αA(B)(n) = 0 in the opposite case. The param

ˆ )–1. By using the Dyson equation and def ˆ –Δ (z – H 0 ˆ –1 ( R ˆ )G ˆ –1 , after ˆ =G ˆ –G inition of T scattering matrix T several transformations and excluding compensating blocks [13] for the shift operator, we obtain ˆV ˆG ˆ V ˆ = [V ˆV ˆV ˆV ˆG ˆ +V ˆG ˆG ˆ + … ] =  . Δ (4) D ˆV ˆ) (1 – G The index D means that only irreducible, exactly ˆ representation terms that do not diagonal in the H 0 contain compensating blocks are retained in the sum […]D. Within the singleelectron and singlesite approxi mations, the summation of series (4) can be performed exactly if it is assumed additionally that the matrix ele ments weakly depend on the wave vectors. It is conve nient to perform the summation in the matrix form by defining the diagonal (according to the band index) matrices F and Δ for the Green’s function Fj =

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THEORY OF THE RESIDUAL ELECTRICAL RESISTIVITY OF BINARY ACTINIDE ALLOYS

∑

–1

1/N k ( z – E k, j ) and the operators Δj, respectively, and the interaction matrix. As a result of summation of the infinite matrix series, we obtain the system of three equations for the selfconsistent determination of the coherent potentials for s, d, and f electrons. However, the system of equations obtained by the direct summa tion of the series turns out to be very cumbersome, which makes it difficult to analyze. Therefore, it seems advisable to analyze a simpler system of equations, which is obtained using the simplifying assumption that does not detriment the physical essence of the model. Indeed, since the densities of states of d and f electrons are much higher than that of s band, it can be assumed that the conditions |λjFs| Ⰶ 1 and |λFs | Ⰶ 1 are valid. In this case, the complexity of the complete sys tem of equations is greatly reduced, so that after the averaging over the configuration, we have 2

∑

2

3

( cB λj ) Fj ( cA λj ) Fj (7) Δ j = c A   + c B  , 1 – cB λj Fj 1 + cA λj Fj which coincide with those obtained previously for the singleband conductivity model [11]. The obtained equations of the coherent potential approximation contain several parameters of the scat tering amplitude that have to be found. In the coherent potential approximation, these parameters are usually assumed to be equal to the difference between the average energies of the corresponding alloy compo nents. Taking into account that the calculations use the densities of states for metals obtained from the first principles, it is sufficient to define only one parameter, for example, for the f band. Then, the parameters λs and λd would be determined automatically. The addi tional assumption λjj' ≈ 1/2(λj + λj') reduces the prob lem to the oneparameter form. Below, we find the PHYSICS OF THE SOLID STATE

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=

∑λ F . 2

j

j

Δs

=

∑

( λ sj F j λ jj + λ sj' F j' λ j'j )F j λ js .

(8)

j ( j ≠ j' )

2

2

(2)

Δs

(3)

( c B λ d ) F d [ 1 – c B λ f F f ] + ( c B λ df ) F f [ 1 + c B λ d F d ] Δ d = c A    2 [ 1 – c B λ d F d ] [ 1 – c B λ f F f ] – ( c B λ df ) F d F f (6) 2 2 ( c A λ d ) F d [ 1 + c A λ f F f ] – ( c A λ df ) F f [ 1 – c A λ d F d ] + c B  . 2 [ 1 + c A λ a F d ] [ 1 + c A λ f F f ] – ( c A λ df ) F d F f Here, we introduce the notation λjj = λj and λ = λsdλsf λdf. The equations for the coherent potential of f electrons are obtained from Eq. (6) by substitution of indices d f. Evidently, if there are no interband transitions λjj' = 0, the system of equations breaks apart into three inde pendent equations for the coherent potentials of inde pendent bands 2

3. RESULTS AND DISCUSSION 3.1. Qualitative Consideration Before discussing the results of the numerical solu tion to the equations of the coherent potential approx imation and calculating the residual electrical resistiv ity, it is appropriate to analyze certain results obtained within the perturbation theory. Since the electrical resistivity ρ ~ ImΔs, in the weak limit, Eq. (5) can be expanded into a series in a small parameter and, to the second order of the perturbation theory, we have

(5)

( c A λ sj ) F j [ 1 + c A λ j' F j' ] – ( c A λ ) F j F j' + c B   , 2 [ 1 + cA λd Fd ] [ 1 + cA λf Ff ] – ( cA λ ) Fd Ff 2

magnitude of the variable parameter λf by the best fit to experimental data.

Similarly, to the third order, we obtain

3

( c B λ sj ) F j [ 1 – c B λ j' F j' ] + ( c B λ ) F j F j' Δs = c A    2 [ 1 – cB λd Fd ] [ 1 – cB λf Ff ] – ( cB λ ) Fd Ff j ≠ j'

3

Therefore, taking into account the terms of the second and third orders in the interaction, we obtain the fol lowing equation for the resistivity: ρ

(2 + 3)

∼ cA cB

∑ [ λ ImF + ( c 2 j

j

2 B

2

– c A )β j ],

(9)

j

where 2

β j ( j ≠ j' ) = [ 2λ sj λ jj ImF j ReF j + λ sj λ jj' λ j's ( ReF j ImF j' + ReF j' ImF j ) ] E = EF . The first term in Eq. (8) leads to a classical parabolic dependence of the electrical resistivity on the concen tration. The second term can lead to significant changes in the concentration dependence and devia tions from the Nordheim rule. This is related not only to the alternating factor, which depends on the con centration, but also to the fact that it depends on the real part of the Green’s function. Figure 1, which pre sents the results of numerical simulations for fixed val ues of ImFj' (EF), ReFj' (EF), and |λjj | = |λjj' |, suggests possible deviations of the dependence of the resistivity from the parabolic dependence for the systems in question. The calculated curves have a maximum sig nificantly displaced from the equiatomic composition point and substantial portions of almost linear depen dence of the resistivity on the concentration. It is almost impossible to predict which way the maximum would be shifted in one alloy or another, because the real part of the Green’s function is a rapidly changing alternating function of the energy and its dependence on the component concentrations is complicated. Since the inclusion of the third term in the perturba tion theory leads to a substantial deviation from the Nordheim rule, it becomes clear why it is important to analyze the resistivity of specific alloys using results of

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TSIOVKIN et al.

assumed that, for occupied (nj) and vacant ( n j ) elec tronic states in the jth band of the alloy, we have

Residual resistivity, arb. units

1.0 1 0.8

2

c α n α, j

nj =

and

0.4

E F, αj

EF

∫ g ( E ) dE = ∑ j

0.2

cα

α = A, B

E 0j

0.4 0.6 Concentration

0.8

i

(α =

2

∑j λj ImFj ) and (5) the Nordheim curve.

the selfconsistent solution to the equations of the coherent potential approximation that are obtained without limitations on the interaction strength.

The individual properties of components of the alloys can be taken into account in the calculations by using real, determined by ab initio methods, densities of states of the metals. In order to obtain the initial density of states of the coherent potential approxima tion, one usually uses the weighted average density of states [14] g(E) =

∑

∑c α

The residual electrical resistivity for Np–Pu alloys is calculated using ab initio densities of states for Pu and Np and the Kubo formulas for the diagonal part of the conductivity tensor modified for the calculations within the coherent potential approximation [5]. The matrix element of the electron velocity squared was for simplicity approximated via the mean kinetic energy 2 of the electrons ν x ≈ 2 E /m. The estimate is quite coarse, but, as demonstrated by calculations using very different approximations for the matrix element of velocity squared, the approximation in question has no significant influence on the final result.

(11)

∫

g α, j ( E ) dE,

E F, α

where EF and EF, α are the Fermi energies of the alloy and its components, respectively, and E0j(Ecj) and Eα, 0j(Eα, cj) are the initial (final) points on the energy scale for the alloy and its component below (above) which the densities of states are zero. The solution to Eq. (11) leads to the determination of the density of states of the alloy via the known densities of states of its components; that is, gj ( E ) =

3.2. Numerical Results

cα

α = A, B

EF

Fig. 1. Comparison of model calculations of the residual resistivity for α/β = (1) 1.25, (2) 2, (3) –2, and (4) –1.25

g α, j ( E ) dE,

E α, cj

∫ g ( E ) dE = ∑

1.0

∫

E α, 0j

E c, j

0.2

(10)

c α n αj .

Then, using definitions of the numbers of electrons (holes) via the corresponding density of states func tions, we find

3

0

∑

α = A, B

α = A, B

4

0.6

∑

nj =

5

E F, αj – E α, 0j  α  E F – E 0j

E F, α – E α, 0j × g αj ⎛   ( E – E 0j ) + E α, 0j⎞ ⎝ E F – E 0j ⎠ gj ( E ) =

∑ α

for

E ≤ EF , (12)

E α, cj – E F c α   E cj – E F, α

E α, cj – E F, α × g αj ⎛   ( E – E F ) + E F, α⎞ ⎝ E cj – E F ⎠

for

E ≥ EF .

Using the condition of continuity of the density of states at EF,

∑c α

=

E F, α – E α, 0  g α ( E F, α ) α  E F – E0

∑ α

(13)

E α, c – E F c α   g α ( E F, α ), Ec – EF

it is easy to determine the Fermi level of the alloy

∑

c α ( E F, α – E α, 0 )g α ( E F, α ) EF – E0 α  =  . Ec – EF c α ( E α, c – E F, α )g α ( E F, α )

c α g α ( E ).

∑

α

However, in this work, the initial density of states for the alloys is determined using condition of conserva tion of the number of d and f electrons [5]. Indeed, since conduction bands are normalized, it can be

(14)

α

Finally, the initial Green’s function, which is used in the first cycle of the iteration process of the numer ical solution to Eqs. (5) and (6) of the multiband

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THEORY OF THE RESIDUAL ELECTRICAL RESISTIVITY OF BINARY ACTINIDE ALLOYS

actinide alloys substantially deviates from the predic tions of the Nordheim rule and exhibits significant features: the maximum is displaced from the equi atomic composition point, and quasilinear portions exist even for equal values of the densities of states at the Fermi level for the components of the alloy. The deviations from the Nordheim rule are determined not only by the dynamics of changes in the densities of states at the Fermi level of the alloy, but they also depend in a complex way on the change in the real part of the Green’s function with variations in the concen trations of the alloy components. The calculations lead to the qualitatively correct description of the local minimum observed on the residual resistivity curve for the Np–Pu alloys.

Resistivity, μΩ cm

200 150 100 50

0

20

40 60 80 Np concentration, at %

100

Fig. 2. Residual electrical resistivity of the Np–Pu alloy. Points indicate the experimental data taken from [1].

coherent potential approximation, can be easily deter mined using the Lehmann representation +∞

F(z) =

∫

–∞

g ( E )fE  . z–E

(15)

The only variable parameter of the theory for the Np– Pu alloy λff = 2.65 eV is determined by the best fit to experimental data. The local minimum of the residual electrical resis tivity of α modification of the Np–Pu alloy is found in the range of 10–20 at % Np [1]. The explanation for the observed minimum was based on the assumption of the influence of the structural modification of the alloy on its electrical resistivity and possible incorrect inclusion of radiation effects. The results of our calcu lations within the suggested multiband coherent potential approximation show that the minimum can be the consequence of a strong change in the density of states at the Fermi level in the alloys in this range of concentrations. Indeed, the calculations show a sub stantial decrease in the density of states at the Fermi level and the corresponding decrease in the residual resistivity (Fig. 2). These results are in qualitative agreement with experimental data and generally reproduce the observed dip in the residual resistivity curve. With an increase in the Pu concentration, the magnitude of the density of states at the Fermi level demonstrates an insignificant change and the configu ration factor (cAcB) determines a quasiparabolic form of the concentration dependence of the residual resis tivity in the concentration range 30–90% Np (Fig. 2). 4. CONCLUSIONS As a result of calculations performed within the suggested multiband conductivity model, it was dem onstrated that the behavior of the concentration dependence of the residual electrical resistivity of the

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Translated by G. Tsydynzhapov