Variation of the LowTemperature Resistivity in Dilute Magnetic AlloysMichael W. Klein and Ralph J. Harrison Citation: Journal of Applied Physics 38, 1135 (1967); doi: 10.1063/1.1709513 View online: http://dx.doi.org/10.1063/1.1709513 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/38/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Lowtemperature spin correlations and spin dynamics in diluted magnetic semiconductors J. Appl. Phys. 57, 3415 (1985); 10.1063/1.335061 LowTemperature Electrical Resistivity and Magnetoresistance of Dilute Solutions of Fe in Cu–NiAlloys J. Appl. Phys. 42, 1549 (1971); 10.1063/1.1660336 LowTemperature Susceptibility in Dilute Systems of Magnetic Impurities in Noble Metals J. Appl. Phys. 39, 846 (1968); 10.1063/1.2163640 Does Logarithmic Accuracy Uniquely Define the LowTemperature Properties of Dilute MagneticAlloy Systems? J. Appl. Phys. 39, 708 (1968); 10.1063/1.2163592 LowTemperature Magnetization in Very Dilute PalladiumIron Alloys J. Appl. Phys. 39, 961 (1968); 10.1063/1.1656343

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded

to ] IP: 129.22.67.7 On: Wed, 03 Dec 2014 07:16:52

JOURNAL OF APPLIED PHYSICS VOLUME 38, NUMBER 3 1 MARCH 1967

Kondo Effect; Hyperfine Fields in Metals A. M. CLOGSTON, Chairman

Variation of the Low-Temperature Resistivity in Dilute Magnetic Alloys

MICHAEL W. KLEIN*

Sperry Rand Research Center, Sudbury, Mass.

AND

RALPH J. HARRISON

U. S. Army Materials Research Agency, Watertown, Mass.

We examine the effect of internal fields upon resistivity of dilute magnetic impurities in nonmag· netic metals, using the second Born approximation. Several distributions of internal fields are con­sidered. We find that the internal field effects suppress the Kondo log T term at moderately low tem­peratures (several orders of magnitude above the Suhl-Abrikosov resonance temperature Tr). This low· temperature behavior is found to be sensitive to the variation of the probability distribution of the fields P (H) near zero field. When P (H) is proportional to Hn-l for small H, the change in the low­temperature resistivity tJ.p (T) is found to be proportional to Tn (l-a log T). a is about 0.06 for copper and gold alloys. More generally, we find that, except for the small log T term, tJ.p (T) is proportional to that part of the low-temperature specific heat which arises from the magnetic disordering of the im­purities. For an Ising model, n= 1, and the resistivity is thus approximately linear in T for 0.01 < T <1 0K. This linearity has been observed for a 0.1 % Au-Fe alloy where the resistivity has been measured to suffi­ciently low temperatures, thus giving additional support to the Ising-like internal field model of the specific heat and tJ.p (T). The theory predicts the resistivity maximum as well as the disappearance of the minimum as a function of the impurity concentration in reasonable agreement with experiment. It is suggested that the low·temperature resistivity measurements be used to probe the distribution of the internal fields in the dilute alloy system.

DILUTE concentrations of magnetic impurities dis­solved in a nonmagnetic host exhibit various re­

sistive anomalies.! There is a minimum in the resistivity at about 10-20°K. As the temperature is further lowered some of the alloys show a maximum in the resistivity, where T max is approximately proportional to the im­purity concentration. For higher concentrations (for eu-Mn of the order of 1 %) the maximum as well as the minimum disappears, and the resistivity decreases monotonically as the temperature is lowered from high temperatures.

Kondo2 has explained the low-temperature minimum in terms of an s-d interaction between conduction elec­trons and the impurity spins. Using the second Born approximation, he found a logT term in the resistivity, which, combined with the rs term arising from the phonon contribution, gives the minimum in good agree­ment with experiment. The perturbation method used breaks down at lower temperatures, as indicated by

* This work supported in part by the U.S. Army Materials Research Agency.

1 For a review of the experimental and some of the theoretical aspects of these resistivity anomalies, see G. J. Van den Berg, Progress in Low-Temperature Physics, edited by C. J. Gorter (North Holland, Amsterdam, 1964), V-4, page 104.

2 J. Kondo, Prog. Theor. Phys. 32,37 (1964).

the divergence of the logT term. Subsequent works have examined this divergence.3

Later works4,6 have discussed the low-temperature resistivity maximum as arising from the presence of internal fields.6 The treatment of Silverstein4 was limited to a range of temperatures close to T max, whereas our treatment5 produced a solution which is formally valid, subject to the limitations of the second Born approxi­mation theory, to temperatures low compared to the resistivity maximum. In our work5 we assumed a prob­ability distribution of the random internal fields of a form7,8 derived from an Ising model, and it was shown that this gave rise to an approximately linear depend­ence of the low temperature resistivity with T.

3 Y. Nagaoka, Phys. Rev. 138, Al112 (1965); A. A. Abrikosov, Physics 2, 5 (1965); H. Suhl, Physics 2, 39 (1965); also, K. Yosida and A. Okiji, Progr. Theoret. Phys. (Kyoto) 34, 505 (1965),

4 S. D. Silverstein, Phys. Rev. Letters 16, 466 (1966); also see Bull. Am. Phys. Soc. 11,237 (1966).

6 R. J. Harrison and M. W. Klein, Phys. Rev. (to be published); also R. J. Harrison and M. W. Klein, Bull. Am. Phys. Soc. 11, 237 (1966).

6 Also see A. A. Abrikosov, Physics 2, 61 (1965) and N. Miko­shiba and K. Yoshihiro, J. Phys. Soc. Japan 19, 2346 (1964).

7 W. Marshall, Phys. Rev. 118, 1520 (1960); M. W. Klein and R. Brout, Phys. Rev. 132, 2412 (1963).

8 M. W. Klein, Phys. Rev. 136, 1156 (1964); M. W. Klein, Phys. Rev. 141, 489 (1966).

1135

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded

to ] IP: 129.22.67.7 On: Wed, 03 Dec 2014 07:16:52

1136 M. W. KLEIN AND R. J. HARRISON

It is the purpose of the present paper to extend the results of the previous works to a more general prob­ability distribution, showing how the low-temperature resistivity is rather critically dependent on the details of the probability distribution near zero fields.

CALCULATION OF THE RESISTIVITY

We assume a model Hamiltonian of the form given in Eqs. (4) and (5) of Kondo.9 Using this Hamiltonian we obtain the resistivity in the second Born approxi­mation. Let p( T) be the expression for the resistivity and p(O) be its formal limit as T ---,,0. Also let !:J.p( T) = p( T) - p(O). !:J.p( T) is found to be6

!:J.p(T) = ACJ2i: P('Y!J.H)

X[B(kBTx)FI(x)+D{F2(x)+Fa(x) 1 Jdn ('Y!J.H) , (1)

where A = (37rmn) / (2e2 N EF'Ii) , c is the fractional im­purity concentration, J the exchange constant, N /n the number of sites per unit volume, EF the Fermi energy, 'Ii the rationalized Planck's constant, !J. the Bohr magneton, H the internal field, m the mass of the electron, 'Y the gyromagnetic ratio, kB the Boltzman constant, and

x='Y!J.H/(kBT) .

B(kBTx) = 1+D[2+1n 1 (xk BT/4EF) IJ,

D=6Jz/EF,

FI(X) = S-M(x) cothtx+M(x/2) csch2tx,

F2(x) =txM(x) csch2tx[I2(x) -In 1 tx IJ, F3(X) =M2(X) [II (x) -In 1 tx IJ,

Z is the number of conduction electrons per atom, and M and M2 are the thermodynamic averages of the Z

component of the impurity moment and its square, respectively. [1(x)~-0.43+0.09x2 for small x and Intx-1.63x-2 for large x. [2(x)~-0.43+0.03x2 for small x and -1.0+1n 1 tx 1+3.2x-2 for large x. P('Y!J.H)dn('Y!J.H) is the probability for a particular in­ternal field H in the interval dn ('Y!J.H). The subscript n in Eq. (1) indicates an n-dimensional differential vol­ume.

We next evaluate Eq. (1) at low temperatures (Tr«T < T max) for several distributions of fields.

Case 1. Ferromagnetic or anti ferromagnetic long-range order. Since only even powers of H multiply P( 'Y!J.H) in the integrand in Eq. (1), there is no loss of generality in assuming that all sites experience the same constant internal field, i.e., PC'Y!J.H) is assumed to be a delta function; P('Y!J.H) = O('Y!J.H-rkBTO) , where r is ap-

9 J. Kondo, Prog. Theor. Phys. 34, 373 (1965).

proximately unity and To is the ordering temperature. For this case we obtain

!:J.p( T)~AcJ2{2SB(rkBTo) (rTo/T)

X exp( -rTo/T» -1.6S2D(T/rTO)21, (2)

where the T2-term arises from the asymptotic form of [1(X) .

Case 2. Short-range order. For a set of randomly distributed impurities interacting via a Ruderman­Kittel potential, this probability distribution has been discussed in an Ising modeJ.7·8 In this model only the Z

component of the internal field enters the calculation (n= 1), and the resulting P( 'Y!J.H) gives an excess low­temperature speciflc heat in good agreement with ex­periment.7.8 However, if one considers the spins to interact like a set of classical vectors (one could possibly argue for such, since the Ising model has not been rigorously justified) with arbitrary orientation, n=3, with dn('Y!J.H) = 47r('Y!J.) 3H2dH.1O We rewrite Eq. (1) in the form

!:J.p( T) = AcJ2un(k BT)n L: P(kBTx)

X[B(kBTx) • FI(X) +D{ F2(X) +F3(x) 1 Jxn-Idx, (3)

where un = 1 for n=O, un=47r for n=3. Let P('Y!J.H) be slowly varying near H = 0 (this is found to be the case for the Ising model).8 Then for very low temperatures (Tstill»Tr) we can replace P(kBTx) inEq. (3) bya constant P(O). When this is done !:J.p( T) is given by

!:J.p(T) IX SJ2P(O) T nc{1+ (6 JZ/EF)

X (2+ln(kB/4EF» Ill-a InTI, (4)

where S is the impurity spin and

a= (6 JZ/EF ) /Il + (6 JZ/EF ) (2+1n(kB/4EF» I. Thus a is approximately 0.06 for copper alloys if J = -0.15 eV. The factor P(O) Tnc also enters as a factor of the low-temperature specific heat arising from the magnetic disordering of the impurities.8 For the Ising model (n= 1), the predicted low T resistivity is approximately proportional to T, except for the small a InT-term. Also, since P(O) is inversely7.8 proportional to c, the increase of !:J.p( T) is approximately concen­tration independent as is the low temperature specific heat. The low-temperature resistivity of a 0.1 % Au-Fe alloyll exhibits this linear behavior for a temperature range of 0.1 to 10 K. We have evaluated !:J.p( T) for the Ising distribution of spins6 from Eq. (1) and find that !:J.p~0.8XlO-8T n-cm, assuming S=1, EF=5 eV and

10 The possibility for such a field dependence of the probability distribution has been suggested by W. Marshall, Proceedings of the 8th International Conference on Low Temperature Physics (Butterworth Publications, London, 1963), p. 215.

11 D. K. C. MacDonald et al., Proc. ;Roy. Soc. A266, 161 (1962).

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded

to ] IP: 129.22.67.7 On: Wed, 03 Dec 2014 07:16:52

LOW - T E M PER A T U R ERE SIS T I V I T YIN D I L UTE MAG NET I CAL LOY S 1137

J -0.15 eV. The very good quantitative agreement with the experimental valuell of 1 X 10-8T Q-cm we consider to be fortuitous in view of experimental and theoretical uncertainties. However we consider the linearity of Ap( T) to be additional evidence that the Ising-like distribution of internal fields is the mechanism for the large excess low-temperature specific heat in these alloys.

Equation (1) may be regarded as an integral equation for P( 'Y/-IH) in terms of the experimentally determined Ap( T) over a wide temperature range. Although a formal solution can easily be given it is not clear that this is warranted by the data presently available. It should be noted however that we have found5 the Ising distribution of fields to give a low-temperature maxi­mum proportional to the impurity concentration, and further to predict the disappearance of the minimum

and maximum at concentrations of about 0.8% for Cu and Au alloys, in good agreement with experiment.

CONCLUSIONS

It is suggested that the extension of the resistivity measurements to lower temperatures may be useful to probe the distribution of the internal fields in a dilute alloy system. In particular, information regarding the variation of the probability distribution of the internal fields near zero field is obtainable from the low-temper­ature resistivity. This information can be correlated with low temperature specific heats. The predicted slope of Ap(T) contains a remnant of the Kondo 10gT term. Thus experiments seeking to determine the breakdown3

of the 10gT dependence at low temperatures are still, in principle, feasible in spite of the presence of large in­ternal fields.

JOURNAL OF APPLIED PHYSICS VOLUME H, NUMBER J 1 MARCH 1967

Hyperfine Fields in Fe-Si Alloys

J. I. BUDNICK, S. SKALSKI, AND T. J. BURCH

Fordham University,* Bronx, New York

AND

J. H. WERNICK

Bell Telephone Laboratories, Murray Hill, New Jersey

We have measured the hyperfine field at Fe and Si nuclei in Fe-Si alloys containing 18.4 to 26.5 at. % Si by the NMR spin-echo technique at 4.2°K. We observe Fe resonances from nuclei at sites with 3, 4, 5, 6, and 8 Fe nearest neighbors at frequencies which are in good agreement with the reported Mossbauer results. In FeaSi, we observe an additional line at about 31.5 Me/sec which we identify with the Si nuclei. In the 23.0 at. % sample, this line is diminished in intensity and another line appears at about 47 Me/sec which grows in intensity as the Si concentration is decreased below 23.0 at. %. At 22.3 at. % Si and below, the 31.S-Mc/sec Si line is not observed. In addition, our data shows that a significant change in the Fe hyper fine field at sites with 4 and 8 nearest-neighbor Fe atoms occurs between 22 and 24 at. % Si. These observations on the hyperfine fields correlate well with the anomalous behavior of the resistivity in this concentration range and indicate a change in the electronic structure of Fe-Si alloys that occurs in the vicinity of 23 at. % Si.

WE have studied FeaSi-type order in a series of Fe-Si solid solutions of concentrations between

18.4 and 26.5 at. % silicon using spin-echo and cw nuclear magnetic spectroscopy. The iron hyperfine field in this concentration range had been studied previously by the Mossbauer technique,! and several distinct iron hyperfine fields were observed and assigned to iron sites with a definite number of iron nearest neigh­bors. The experimental data was adequately fitted without invoking contributions from more distant neighbors. In ordered FeaSi, we observe a three-line spectrum, one line more than observed in the Mossbauer experiments. We identify this additional line as that arising from the 29Si nuclei. This assignment is based

* Work supported in part by the National Science Foundation. 1 M. B. Stearns, Phys. Rev. 129, 1136 (1963).

on the identification of the other two lines of our spectrum with the two-line Mossbauer spectrum. The frequency of this additional resonance is 31.5 Me/sec, and is essentially unshifted in the concentration range 26.5 to 23.0 at. % Si. In the 23.0-at. % Si sample another line, not previously observed in the Moss­bauer experiment, appears at about 47 Me/sec. In samples of 22.3 and 18.4 at.% Si, the 31.5-Mc/sec line is no longer observed but the 47-Mc/sec line per­sists and grows in intensity.

Our samples were powders of about 400 mesh which were annealed in a vacuum (10-6 mm Hg) for 2 h at 600°C and slowly cooled. In Fe3Si, which orders from the melt, such a heat treatment is adequate to insure long-range order and remove the strain in­duced in preparing the powders. AU of the results

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded

to ] IP: 129.22.67.7 On: Wed, 03 Dec 2014 07:16:52