Theory of the resistance minimum in dilute paramagnetic alloys

J. Phys. Chem. Solids

THEORY

Pergamon Press 1960. Vol. 15. pp. 140-145.

OF THE

Printed in Great Britain.

RESISTANCE

MINIMUM

PARAMAGNETIC A. D.

BRAILSFORD

Scientific Laboratory,

and

A.

DILUTE

ALLOYS W. OVERHAUSER

Ford Motor Company, (Received

IN

11 December

Dearborn,

Michigan

1959)

Abstract-The observed anomalous temperature dependence of the resistivity of dilute paramagnetic alloys is attributed to s-d exchange scattering of conduction electrons by nearest neighbor pairs of ferromagnetically coupled solut ’ ions. Occurrence of a resistance minimum can be explained quantitatively only if the effective range of the s-d exchange potential is not too short; and only if the periodic structure of energy bands is taken into account. The size of the minimum is proportional to the square of the solute concentration in the limit of extreme dilution (less than 1 per cent). Deviations from this law at higher concentrations are explained satisfactorily.

1. INTRODUCTION

A MINIMUM in the resistance as a function of temperature has been observed in several dilute paramagnetic alloys of a transition metal in a noble metal.(I) At temperatures below that of the minimum the resistance increases monotonically with decreasing temperature. In addition, the anomalous change in resistivity with atomic per cent of solute has been found to vary with solute concentration near extreme dilution. Although not conclusively established, it appears very probable that these effects are associated with paramagnetic ions in random solid solution.(s)* The first attempt to explain the anomalous temperature dependence of the resistivity was due to KORRINGA and CERRITSENc31.They introduced the concept of localized impurity states, centered at each paramagnetic ion, with an energy very close to the Fermi energy of the conduction electrons. This model yielded some results in qualitative * Note addedin proof: It has now been established that the minimum which has been observed with various group B solutes (e.g. Ga, Ge, Sn) in Cu is not a direct consequence of the solute itself but arises only insofar as the solute is able to reduce any iron oxide present as an impurity and thus bring iron into solid solution (GOLD A. V.. MACDONALD D. K. C., PEARSONW. B. and ;I(EMPLETON I. M., Phil. Mug., to be published). We are indebted to these authors for details of this work in advance of publication.

agreement with experiment. However, no independent evidence for the existence of such impurity states has been found. Later, SCHMITT@) pointed out that a temperature dependent resistivity would occur if the spin degeneracy of the states of the paramagnetic ion were removed. This effect was explored in detail by YOSIDA(5) in a calculation of the resistivity of the antiferromagnetic Cu-Mn alloys, where the spin degeneracies of the manganese ions are split by a molecular field. He found that below the NCel temperature the resistivity decreased with decreasing temperature, whereas above it the resistance was independent of temperature. Consequently the resistance minimum can not be ascribed to residual effects of some cooperative magnetic transition at lower temperature. In this paper a new model for the resistance minimum is presented in detail. (The method has been outlined briefly in a previous note.@)) The mechanism invoked is similar to that of SCHMITT, who postulated the presence of scattering centers with closely spaced energy states. Such states do indeed arise when pairs of nearest neighbor paramagnetic ions interact. The magnetic states of the pair are split into a sequence of degenerate levels, the energy separation of which depends on an effective exchange integral W. We show that the average elastic s-d exchange scattering cross section of each state of a level is 140

THEORY

OF THE RESISTANCE

MINIMUM

proportional to the square of the total spin of that level. Thus, for ferromagnetic coupling, (IV > 0), the lower the energy of a level, the greater is its elastic cross section. This implies an increasing contribution to the resistivity as the temperature decreases, and so explains the resistance minimum, Of course, account must be taken of the opposing tendency of the resistivity contribution arising from inelastic processes, which become frozen out at low temperatures. It is found, however, that elastic processes always dominate at high temperatures (kT > W) and therefore, when the phonon resistivity is included, a resistance minimum usually results. One immediate consequence of the model outlined above is that the anomalous resistance change for a given temperature increment below the minimum should be proportional to the square of the concentration of the paramagnetic solute. This relation is expected to hold only for extreme dilution. At higher concentrations the resistivity will be affected both by larger clusters of neighboring ions, and by randomly distributed single solute ions interacting by means of the pseudo-exchange mechanism.(7) The latter effect will ordinarily be the more important. As a rest& of these interactions the spin degeneracy of a paramagnetic level is removed. Consequently scattering processes for which an electron spin is reversed become inelastic and are frozen out at low temperatures. The resulting resistivity decrease with decreasing temperature competes with the net increase arising from coupled nearest neighbor pairs, so that the minimum in the total resistivity varies less rapidIy with concentration than the quadratic dependence, characteristic of extreme dilution. We believe this effect to be the qualitative explanation of the concentration dependence of the minimum in dilute Cu-Co alloys, for example. In Section 2, the resistivity arising from scattering by coupled nearest neighbor solute pairs is calculated. A resistance minimum is obtained only if the range of the effective s-d exchange potential is not too short. Furthermore, in Section 3 it is found necessary to take into account the periodicity of energy bands in k space in order to obtain quantitative agreement between theory and experiment. In Section 4, the contribution of isolated paramagnetic ions to the anomalous low temperature resistivity is estimated and found adequate to

IN DILUTE

PARAMAGNETIC

explain the concentration sistance minimum.

ALLOYS

dependence

2. RESISTMTY CONTRIBUTION SCATTERING BY PAIRS

141

of the m-

FROM

In an electric field Ez the conduction electron gas reaches a steady state when the rate of change of the electron momentum P, produced by the field is balanced by the rate of change produced by collisions :

dP~ldt-neE~ = 0,

(1)

where n is the electron concentration, and dPJdt is the rate of change of momentum (per unit volume) produced by collisions, which can be calculated by transition probability theory if the distribution functionf(k) is known. For resistivity calculations it is well known that the distribution function can be taken to be f(k)

= fo(&) + n.G cos 0 dSold&

(2)

where ~~(~~} is the equilibria distribution, depending only on the electron energy Ek. The angle between k and the x axis is 8; and a can be taken to be a constant since only its value at the Fermi surface, ko, enters when one calculates the current density, j = neaE,/Ako.

(3)

The electrical resistivity is evaluated immediately from equations (1) and (3) after equation (2) is employed for calculating dP,,/dt. The problem of this section is to perform the latter calculation for a pair of nearest neighbor solute spins, & and Ss. The Hamiltonian for an electron with spin S interacting with a pair of nearest neighbor solute ions at 0 and R will be taken to be the sum of the following two expressions :

Ho = (p2/2m) “I-U(r) - wsr - s2,

(4)

Hl = V(r)+V(r-R)-2J(r)s.S1-

- 2.J(r- R)s . S,.

(5)

V(r) and J(r) are the spin-independent and spindependent potentials of the solute-electron interaction, U(r) is the periodic potential of the lattice, and Wis the effective exchange interaction between solute spins. The energy levels of HO are: R(k)

= .&+

Wr,

(6)

142

A.

D.

BRAILSFORD

and

where, wz = *w[2S(S+l)-I(I+l)].

(7) Each solute pair energy level, WI, is (21+1)-fold degenerate; and I assumes integral values between 0 and 2s. In evaluating the various transitions caused by the perturbation (5), it is necessary in principle to specify the spin of the initial and final conduction electron states and to distinguish the distribution functions for spin-up and spin-down. However, since the spins of the paramagnetic pairs are randomly oriented, f+(k) = f-(k), and the distinction may be neglected. The rate of change of momentum of the electron gas caused by elastic scattering (I+ I) is:

z

x [l -W>l

Pz i

I=0

iv=-z

{I Vkk,-t

+SMJ*k42+~(I--)(I+M+ l>lJfzk~12>, (8) where q = F-k, N is the number of atoms per unit volume, Vkk, and J,+ are the matrix elements of V and J between Bloch states normalized in an atomic volume, S(E) is a delta function, andpz is the probability that a given pair state of level WI is occupied. PZ=

A.

W.

OVERHAUSER

One should notice the factors ]l+ exp( - iq - R) 12 in equations (8) and (10) because they account for the interference between the waves scattered by the two solute ions of a pair. If q - R is not too large, the interference is constructive for elastic scattering and destructive for inelastic scattering. In the last analysis it is just this behavior that allows the elastic scattering to predominate over the inelastic, and so produce a resistance minimum. In order to evaluate expressions of the foregoing type it is customary to make the following approximations : V kk"- v(q); JkP_&). (11) This step is almost always made for the sake of expediency, and cannot be considered more than a crude approximation. Equations (8) and (10) must be multiplied by Nz,, the number of solute pairs per unit volume; and the interference factors must be averaged over all orientations of R. After inserting expression (2) for the distribution function, the sums over k, k’ and M can be carried out, and the elastic and inelastic contributions to the electrical resistivity derived from equations (1) and (3). They are: Pelastic

= a(A +P+B+),

pinelastic

= +--B-

where, u = m2Npk&me%3N2,

(12)

(13) (14)

exp(-Wz,k~~,z~o(21'+l)exp(-Wz~/kT)(9)

The first term in the brackets of equation (8) arises from scattering processes for which the conduction electron spin is not flipped, whereas the second term is associated with processes for which the spin is flipped. The corresponding expression for inelastic scattering (1-t I+ 1) is, dP - -rc ~Z11-exp(-iq*R)/2f(k)[l-f(k’)]x z - w k,R’ x IJwP

2

(S[El(k) - Ez+@‘)]

Pz $

Z=O

x

; V(q)l2[1+ (sin qR)/qR](l - cos 0)sin Bd 8, s (15)

A =

0

t3+ =zzo Pz(21+ l)I(I+

b- = 3

0

1)[(2S+ 1)2 - (I+ 1)2] +

X

4(1+ 1>s- 1 + a[Ez(k) - Ez-@‘)I

x

(21- l)(I-M)[(2S+

1)2-F]

X

4Is-

1

1)2-yi/(pz+pz-1),

I-1

(16) (17)

Bk = ~~J(q)~2[1f(sinqR)/qR](l--cos6)sin~d~. s

M=-Z

(2I+ 3)(I+M+

pzpz-1ws+

1),4,

’ (10)

(18)

With regard to the integrations in definitions (15) and (18), it should be noted that q is a function of the scattering angle. The character of this functional dependence will be discussed in considerable detail below, and it will be concluded that the conventional choice is in fact not the correct one.

THEORY

OF THE RESISTANCE

MINIMUM

The temperature dependence of the resistivity is contained in the coefficients /I*. From equations (9), (16) and (17) their asymptotic forms at high and low temperatures can be easily derived. For high T: p+ cz &S(S+ l)[l f WS(S+

1)/3KT];

(19)

and for low T with W > 0 :

p+ z &s(2s+

l)-

S[(4S-

1)/(4S+ 1)] x

x exp(-2WS/kT), jI_ z

2s exp (- 2 WS/kT) ;

(20) (21)

whereas for low T with W < 0: P+ z 312 exp (W/kT),

(22)

p E 4S(S+

(23)

1) exp (W/kT).

The net resistivity contribution arising from exchange scattering by pairs is, for high T, p~~Ss(S+l)x x [B++B_+

WS(S+

l)(B+-B-)/SkT].

(24)

Consequently, if W(B+-B_) > 0, the resistance will increase with decreasing temperature, and should yield a resistance minimum when added to the phonon resistance contribution. Except for paramagnetic systems which become antiferromagnetic at low temperatures (e.g. CuMn alloys), the anomalous resistance increase below the minimum appears to be monotonic all the way to T = O.(l) The foregoing analysis implies then that W > 0 ; that is, the spins of a pair are ferromagnetically coupled. For if W < 0, the temperature dependence of definitions (22) and (23) would yield a decreasing resistance with decreasing temperature near T = 0. For W> 0, the temperature dependence of expressions (20) and (21) yields an increasing resistance with decreasing temperature near T = 0 only if B+/B- > 2(4S+ 1)/(4S- 1).

(25)

Furthermore, we have verified by numerical evaluation of equations (16) and (17) for intermediate temperatures that the resistance increase is monotonic everywhere below the minimum as long as the criterion (25) is satisfied. It is clear from definition (24) that a minimum is possible as long

IN DILUTE

PARAMAGNETIC

ALLOYS

143

as B+/B_ > 1, but a resistance maximum (below the minimum) would occur if expression (25) were not also satisfied. 3. QUANTITATIVE

DISCUSSION

OF THE MODEL

There remains the task of estimating the values of the parameters, Wand B,, which occur in the model described above, and deciding whether the quantitative magnitudes necessary for agreement with experiment can be justified on an a priori basis. Very little can be said theoretically regarding the value of W. Experimentally W N 20°K. Mechanisms which contribute to W are direct overlap of the localized paramagnetic orbitals and pseudo-exchange interactions (via s-d exchange interactions), as treated by YOSIDA(7). The latter calculation considers an idealized problem, and one cannot expect it to be quantitatively reliable for a real situation, for which one cannot neglect s-s exchange interactions and the finite size of the paramagnetic orbitals, for example. The magnitudes of B+ and B_ determine the size of the anomalous resistance increase below the minimum, which is: Ap = 8 crS[SB+-(S+

l)B-1.

(26)

For a 8% Cu-Co alloy, Ap N O*Olp s2 cm. Equation (26) yields this magnitude provided the average s-d exchange matrix element, J, defined by

P

= ;

jl/W(l-

cos @sin 0 d 0 = k(B+ + B-)

0

(27)

has a value, N 0.3 eV. This figure equals the magnitude of the s-d exchange interaction constant of a transition element ion. Consequently it appears at first sight that the model is satisfactory. However, a real difficulty arises when one tries to satisfy equation (25) and at the same time maintain the value of definition (27). This problem can be appreciated by considering the integrals (18) which define B,. In order that B+/B_ be large enough to satisfy expression (25), it is necessary that the factor (sin pR)/qR be positive and not too small (> N 5) for most of the integration. In the free electron approximation it is appropriate to take, Q = 2ko sin(8/2).

(28)

144

A.

D.

BRAILSFORD

and A.

Then, sinqR becomes negative for scattering angles greater than 54”. Consequently J(q) must fall off quickly with increasing q so as to allow the predominance of small angle scattering. (It is true that sinqR becomes positive again for angles greater than 130”, but sin qR/qR remains very small for large angle scattering.) Although one can easily satisfy equation (25) by choosing](q) so as to allow only small angle scattering, the total effective cross section in expression (27) is seriously reduced (by two orders of magnitude) if J(O) is left unchanged. The foregoing difficulty can be alleviated by the following considerations which are pertinent to all calculations of scattering by lattice imperfections. In real life one is always interested in the scattering between two Bloch states $k and & caused by a perturbation, say J(Y). The wave functions have the usual form, & = exp(ik * r)uk(r).

(29)

In calculating matrix elements of J(r) it is customary to neglect the variation of Q(T) with k, if not to neglect U*(Y) altogether. In principle one can expand uk’(r) in a power series in q = k’- k and neglect, perhaps, the linear term in q. Such an approximation may be adequate as long as q is small. Now, it is well known that Bloch states are periodic in k space, and that all states 4~ are identical to #b, for the set of K which satisfy K = k+2n-G,

(30)

G being the reciprocal lattice vectors. Therefore, in performing a scattering calculation in which the above mentioned approximations are made, one should examine all equivalent final state wave vectors K’ and pick that K’ for a given scattering event which makes K’-k

= q+2rrG

(31)

have the smallest magnitude. We shall define this smallest vector, K’- k = qo. It is interesting to know that for a monovalent metal such as Cu, qs is smaller than q for about g of the scattering events. We should like to emphasize that if one were calculating matrix elements with exact Bloch functions, it would not matter which K’ is chosen to represent the final state. However, if one is going to use the Fourier transform J(q) of J(r) as

W.

OVERHAUSER

the matrix element of the scattering event q, one should always useJ(qs) instead ofJ(q). (It may be noted that go is a continuous function of q, so that the scattering probability remains a continuous function of angle.) The importance of this procedure with regard to quantitative answers may be easily understood. In a monovalent metal many areas of the Fermi surface are close to the Brillouin zone boundary. States corresponding to velocity reversal (180’ scattering) are only a small wave vector distance away in the direction of the zone face, whereas they are 2Kc away by the usual description. A scattering potential whose range is not too short will have a Fourier transform that falls off rapidly with increasing q. Consequently it is essential that go be used in order to obtain proper account of large angle scattering events, especially since they are the important ones in a resistivity calculation. The authors are aware that the foregoing remarks call into question the quantitative significance of most previous calculations of scattering arising from lattice imperfections in metals. They make no pretense that the suggested improvements will adequately approach the results of a “correct” calculation. But they submit that the proposed treatment incorporates contributions from large angle scattering that have frequently been overlooked. Returning now to the problem of the resistance minimum, one should replace q by go in the integrals (15) and (18). The indicated summations were carried out with a digital computer by considering all scattering events between (loo), (1 lo), (111) and (123) type wave vectors. The function, J(40)

= J(O)/(l +%02),

(32)

was used for matrix elements of the s-d exchange It was found possible to satisfy interaction. expression (25) with X N lattice constant; and the magnitude of definition (27) could be retained if J(0) - 1 eV. Such values of h and J(0) are not unreasonable. It must be remembered that the scattering center is a paramagnetic orbital together with a conduction electron spin density readjustment. The s-d exchange interaction attracts conduction electrons of spin parallel to that of the d orbitals into their locality, and the (attractive) s-s exchange interaction amplifies this effect by an appreciable factor. The effective (“renormalized”)

THEORY

OF THE RESISTANCE

MINIMUM

s-d exchange interaction includes not only the “bare” sil exchange interaction, but also an s-s contribution from scattering by the conduction electron spin density readjustment. Since the s-s exchange interaction of a free electron gas is - 5 eV /electron, a value for J(0) of N 1 eV does not seem inordinately large. We conclude that the present theory of the resistance minimum in dilute paramagnetic alloys can yield quantitative agreement with experiment without too much strain on the imagination. 4. EFFECTS OF SCATI’ERXNG BY SINGLE IONS multiple clusters of paramagnetic solute ions should modify the quadratic dependence of the resistance minimum at high solute concentrations. However, one would not expect such effects to be important for c N 1 per cent, where indeed the cs dependence appears to break down in the case of Cu-Co alloys.@) This failure can be attributed to long range spin dependent interactions between isolated solute ions, which split the magnetic egeneracy of solute levels and reduce thereby the inelastic contribution of spin disorder scattering when the temperature is smaller than the characteristic splitting. We shall estimate this splitting by assuming the pseudo-exchange interaction between any two solute spins 5’1 and Ss to be H = We exp ( - rla/A)Sl S2. (33) l

(The oscillatory distance dependence ascribed to this interaction by YOSIDA(~) arises by imposing an abrupt cut off in certain integrations instead of allowingJ((p) to be a smoothly diminishing function with increasing 9. If a smoothJ(q) is employed, we find the interaction to have approximately the form given in equation (33).) For a random distribution of solute spins the root mean square splitting of adjacent magnetic states is, lP2 = l’z,S(S+

l)iexp n

12(R-r,)IA],

(34)

where the prime on the summation sign indicates that the nearest neighbors are excluded from the sum over all neighbors (at distance rla) of a given ion. W is the nearest neighbor interaction constant, as defined by equation (4), and R the nearest neighbor distance. The parameter X is essentially the same as that employed in equation (32). The K

IN DILUTE

PARAMAGNETIC

ALLOYS

su~ation is easily evaluated numerically a value N 4. Consequently, lP -

2 W[cS(S+

1) ]%

145

and has

(35)

For a 1% alloy with S = 512, @~0*6W, which should be compared with the splitting 5 W between the I = 2S and I = 2S- 1 levels of a nearest neighbor pair. The spin disorder scattering arising from “isolated” ions is given by, p = y~%S’[S’+ 1 -Bs(~~~T)tanh(WlZ~T)],

(36)

where y = ~c~~~~ and Bs(x) is the standard Brillouin function. (The second term in expression (36) differs from that which would obtain were @ caused by an applied magnetic field, because it is assumed here that the effective fields associated with W are randomly oriented.) Since &(x) and tanh(x) are both increasing functions of X, this resistivity contribution decreases with decreasing temperature. Fortunately this decrease does not cancel the increase derived previously for nearest neighbor pairs. The high temperature expansion of equation (36) is p 2 &Y5$S+

l)[l-

1~6(w~~~)2],

(37)

which varies quadratically with @[k T. The corresponding result, equation (24), for the contribution from nearest neighbor pairs varies linearly with W/kT, so that a resistance minimum phenomenon will remain. However, the increase in W with c causes the low temperature effects of expression (36) to move to higher temperature so as to diminish the total size of the minimum. Quantitative estimates of definition (36) indicate that departure of the minimum from its cs dependence in the 1% range can be explained.

1. GERRITSEN A. N., Physica, s’Grav. 25,489 (1959). 2. PEARSON W., RIMEK D. and TEMPLETON I., Phil. &kg. 4, 612 (1959). 3. KORRINGA J. and GERRITSEN A. N., Physica, s’Grao. 19. 457 (1953). 4. SCHMITT R‘.W.; Phys. Rev. 103,83 (1956). 5. YOSIDAK., Phys. Rew. 107, 396 (1957). 6. BRAILSFORD A. D. and OVERHAWSER A. W., Phys. Rev. Letters 3,331 (1959). 7. YOSIDAK., Phys. Rm. 106,893 (1957). 8. JACOBSI. S. and SCHMITT R. W., Phys. Rm. 113, 459 (1959).