- Email: [email protected]
Temperature dependence of the contribution to the transport coefficients of nearly ferromagnetic metals from electron-paramagnon scattering
J. Phys. Chem.Solids, 1973,Vol.34, pp. 679-686. PergamonPress. Printedin Great Britain
T E M P E R A T U R E D E P E N D E N C E OF T H E C O N T R I B U T I O N TO T H E T R A N S P O R T C O E F F I C I E N T S OF N E A R L Y F E R R O M A G N E T I C M E T A L S F R O M ELECTRON-PARAMAGNON SCATTERING* D. L. MILLS Department of Physics, University of California, Irvine, California and Metal Physics Branch, U.S. Naval Research Laboratory, Washington, D.C., U.S.A. (Received 5 June 1972)
Abstract--We present calculations of the temperature dependence of the contribution from electronparamagnon scattering to the electrical and thermal resistivity of a simple model of a nearly ferromagnetic metal. The purpose of the work is to explore the behavior of these quantities when the temperature is the order of or greater than the spin fluctuation temperature T~e. As the temperature T is raised from zero through T~s, the electrical resistivity varies more slowly with temperature than the T 2 law characteristic of the regime T ,~ Tsr. When T ~> Tee,the electrical resistivity becomes proportional to T, although this asymptotic behavior is approached very slowly. The Lorenz number rises monotonically with temperature, and appears to approach the ideal Sommerfeld value when T >> Tss, although this limit is also approached slowly. 1. INTRODUCTION
IN NEARLY ferromagnetic metals and alloys, large amplitude spin fluctuations are present, since the system is close to a magnetic instability. Current carrying electrons may scatter from these enhanced spin fluctuations, and this scattering leads to contributions to the temperature dependent part of the electrical resistivity p[1-3] and thermal resistivity w [4, 5] of these systems. The fluctuations also affect a number of other thermodynamic and magnetic properties of pure metals and their alloys [6]. Most theoretical descriptions of these fluctuations have focused on one of two limiting cases, the dilute alloy where the fluctuations are spatially inhomogeneous in amplitude, with the spin-fluctuation amplitude near the dilute constituent very different from that in the host matrix, or the pure metal where the fluctuations have a wave-like character, with *Technical Report #72-28. 679
strong enhancement of the long wavelength fluctuations if the host is near an instability with respect to the ferromagnetic state. The early theoretical studies of the contribution to the transport coefficients from electronspin fluctuation scattering were confined to the study of the low temperature limit, where p contains a contribution to T z from this source[l,2], and w contains a contribution that varies linearly with T [4, 5]. Doniach and Kaiser[7] have recently extended the theory of the contribution to p and w from electronlocal spin fluctuation (LSF) scattering in dilute alloys to the case where the temperature T is comparable to or greater than the LSF characteristic temperature TLS v. In the paper by Schindler and Rice[3], a theory of the temperature dependence of p in pure metals was presented and applied to the case where T is comparable to or greater than the spin fluctuation temperature Tss characteristic of the metal. We feel this work employs an expression for the spectral density of the spin
680
D.L.
fluctuations that is rather oversimplified for the reasons described below. As a consequence, we have investigated the behavior of p and w for the pure metal when T is comparable to or larger than Ts~. While the model we use is still a very simple one, we feel it contains the principle qualitative features needed before one can obtain a realistic description of the contribution to p and w from electron-spin fluctuation (electronparamagnon scattering) in pure metals. Our model is sufficiently complex that most of the results reported below must be obtained by numerical computations, unfortunately. This study has been motivated in part by a recent series of experimental investigations of the properties of Ni3A1 and NiaGa[8], and binary alloys of these constituents with compositions close to stoichiometry. Stoichiometric Ni3A1 is ferromagnetic with a Curie temperature of roughly 40~ and the AI rich alloys are paramagnetic with very large susceptibilities near the critical composition. These materials have very large temperature dependent contributions to p at low temperatures, and while p varies roughly like T ~ for compositions far from the critical one, the temperature dependence weakens for c near the critical composition cr. A considerable body of data on this system and on the Ni3Ga system are presented and compared with the spin fluctuation theory in Ref. [8]. There is qualitative, but certainly not quantitative agreement. It therefore seems of value to carry out a more complete theoretical investigation o f the contribution to p and w from electron paramagnon scattering in the region T >t Tss, since no adequate theory has yet appeared. This may allow a more critical comparison between the data and the theory. We do not attempt a quantitative fit to the data here since, as noted in Ref. [8], it is not clear that paramagnon theory provides a complete description of the alloys. For example, the dependence of the coefficient 3/of the term linear in the specific heat increases with the susceptibility as expected from the theory,
MILLS
but the dependence is far weaker than predicted[8]. Also a recent study of p and w in stoichiometric Ni3AI[9] shows that the exponent n in the expression p = A T n decreases from a value of roughly 1.3 at 10~ to a lower value as the temperature decreases, The Lorenz number varies little with temperature. While the theory presented here is applicable only to the paramagnetic state, this behavior appears hard to understand within the simple form of the itinerant electron model of a uniform metal. Finally, Robbins and Claus[10] have interpreted specific heat and magnetization measurements on Ni3A1 on the basis of a model which ascribes the magnetism to the presence of Ni clusters, and the interaction between clusters. 2. T H E E L E C T R I C A L RESISTIVITY
This work is based on a model introduced and discussed in detail in Ref. [ 1]. T he transition metal is presumed to have two sets of conduction electrons. One set, the s-electrons, has a mass ms comparable to the free electron mass, and the second set, the d-electrons, has an effective mass ma ~> ms. The current is carried primarily by the s-electrons, while the large density of states associated with the d-electrons means that the magnetic properties of the metal and the specific heat receive their principal contribution from the d-electrons. Of course, the wave functions associated with the two sets of electrons need not have pure s and d character for the model to be valid. It is only necessary that the Fermi surface consist of distinct light and heavy mass portions. We consider in this section the contribution to the temperature dependent portion of p from scattering of the s-electrons from exchange enhanced spin fluctuations in the (paramagnetic) d band. For simplicity, the Fermi surface of the s and d bands will be assumed spherical, while ks, ka and ~s, ca are the Fermi wave vectors and Fermi energies of the two sets of electrons, respectively. If the s and d electrons interact via an s - d exchange interaction of strength J, from the results in
TEMPERATURE DEPENDENCE OF NEARLY FERROMAGNETIC METALS
reference (1) the contribution to p from electron-paramagnon scattering may be written in the form[11] (with h = l)
9Vcms f
P= 32e2k rknT
dO
X dEf(e) [ 1 --f(e-- O) ][ 1 + n(O) ]
• ffm dr/~ aA (ks'o, ~)IF(k,~)P.
(1)
681
d band at the Fermi surface. Equation (3b) is the form appropriate to a free electron gas in the low frequency limit, and we neglect the frequency dependence of Re[xo(q,O) ]. These approximations should be reasonable for a material with strong exchange enhancement, since the spin fluctuation temperature T~, defined by
k~Tsf= (1--i)Ea
(4)
will be small compared to the Fermi energy. In this expression, Vc is the volume of the unit We then take the simple parabolic form for cell, kB is Boltzmann's constant, and the func- Re [x0(q, 12)] exhibited in equation (3a). If tions f(E) and n(O) are the Fermi-Dirac and Na(O) is the density of states in the d band/ Bose-Einstein functions, respectively. The spin direction at the Fermi energy, and i = quantity IF(k,;o)12 is a form factor which we 1Na(ea), then forA (q, l~) we have set equal to unity in the remainder of this note. The parameter ~,, is max (2,2ka/k,). The A (q, 1~) = 7rNd(O) critical part of equation (1) is the function ~q A(q,O), the spectral density of the spin • (5) qS[ (1 --[) + icr2q2]2 + [ (zr/2)l(O/Va)]2" fluctuations in the d-band. This quantity is related to the dynamic susceptibility x(q, O) of the d electrons: We feel this form represents the principle qualitative features of the spin fluctuation 1 A (q, f~) = 7[X (q, O - ic) --X (q, l~ + ie) ] (2a) spectrum of stronlgy exchange enhanced paramagnets. For small values of q, when [ is close to unity the spectral density exhibits a where in the paramagnon model one has peak at a frequency ~q = Vsq, where Vs X (q, l~) = X0(q, lq)/[ 1 -- IXo (q, 1~) ]. (2b) (1 -- D va. As q increases, the o-2q2 term shifts the peak to frequencies higher than Vsq. InIn equation (2b), • O) is the dynamic sus- deed, for o-q >~ 1, the position of the peak in ceptibility of the d electrons in the one A (q, O) is insensitive to [, when i is close to electron approximation, and I is a measure of unity. the strength of the electron-electron interWe can compare the expression in equation actions. Since xo(q, ll) is in general a complex (5) to the form for A (q, O) used earlier by function which depends on the details of the Rice[3]. In his work, Rice ignored the term electronic band structure of the material, we proportional to f~2 in the denominator of base our calculations on the simple model equation (5), as well as the or2q 2 tenn. He forms then cut the spectral density offat a frequency 12c the order of the spin fluctuation frequency Re [Xo(q, 1~) ] -- Re [x0(q, O) ] = X0[ 1 --o-2q2] ~ . The approximation used by Rice thus (3a) ignores the contribution to p from the high and frequency tail in equation (5), which falls off 3rr ~ only as ~-1. More importantly, the peak in Im [xo(q, O) ] -- 4Ca Vaq' Va < q < 2ka (3b) the spectral density occurs at a frequency ~,, proportional to Vsq. We feel this last feature where Va is the velocity of the electrons in the is particularly unrealistic, since for large q, the
682
D . L . MILLS
falloff of Xo(q, 0) with q causes the peak to occur at a much higher frequency. When the temperature is low, the term proportional to 1~2 in the denominator may be ignored. One then recovers the low temperature limiting form of p, where p is proportional to T 2. In this region, we write p = A (i) T 2.
(7)
is of interest, since the dependence o f the temperature dependent portion of p on the susceptibility X = aX0 of the host matrix has been discussed [8]. A simple integration gives
A(I) 3 tan-' (~:M)- [seMI(1 +~:M2) ] A(u) = 2 ~M3(I _ 1 ) 2 , (8) where
i 1/2
~:~t = o'ks~M ( 1 -- i) ,/2"
(9)
T h e variation of A with the Stoner enhancement factor o~ provided by the present model is exhibited in Fig. 1. T o compute the values of A (]), we have used the parameters employed in an earlier study of the Lorenz number of the Pd Ni system [4]: 1
= V~ka ks = 2ka.
I00.0
~
"
(6)
T h e complete expression or A ( i ) is of little interest, since the simple model is not expected to yield accurate estimates of the absolute value ofA. H o w e v e r , the dependence of A on i, or more precisely on the parameter 1 C~=l__ /
10oQo
I0.0
I'01.0
(10b)
All of the numerical calculations reported in the present paper employ these values for oand ks. We do not expect the qualitative features of the prediction of the model to depend sensitively on the precise values of oand ks.
IOOD
Ot
I000.0
Fig. 1. The variation of the coefficient A of the T 2 term in the electrical resistivity with the Stoner enhancement factor a, for the model described in the text.
F r o m Fig. 1, one sees that the dependence of A on a is not described by a simple power law. F o r a near unity, A increases more rapidly than a, while when a >> 1, A increases much more slowly. We next consider the temperature dependence of p, for the case where the temperature T is the order of magnitude of Tss or greater. T h e integration over E in equation (1) is readily performed. After some manipulation, w e obtain p =
6//(i) Tz
rrZ{tan-' (~:M) -- [ ~M/(1 + sou2) ] } •
(10a)
I0.0
fo~ (e-7_-dxx2 eX f~u ~ [ 1 1) ~
sC'd~: +~]~+r~x~
(lO)
where the quantity z = ksT/Or
and
4 / 1 - - I'~ 3/2 lqr : ~--~.a ~--~-- ) 'a-
(11 a) (lib)
As mentioned in Section 1, Kaiser and D o n a c h have extended the theory o f electronL S F scattering contributions to p and w [7] to
TEMPERATURE
DEPENDENCE
OF
the case where T >~ TLsr, where TLS F is the local spin fluctuation temperature. In their theory, if the contribution of electron-LSF scattering to p at low temperatures is denoted by a ,4 T 2, they find the ratio p/,4 T 2 to be a function only o f the reduced temperature (T/TLsr). T h e expression in equation (10) cannot be written in a similar form. In general, in the pure metal, the ratio p/A (I) T z is not a function only of T/T.~s. T h e shape of a plot of p/A (T) T 2 vs T/T~f thus depends on the value of the enhancement parameter ~. H o w e v e r , for the parameters employed in the numerical calculations below, the ratio p / A ( I ) T 2 is remarkably insensitive to a. In the low temperature limit, the right hand side may be expanded in powers of z 2. One readily sees that the first correction to the low temperature limiting form A ( i ) T 2 of p is proportional to T 4, and not T 5 as Rice's approximate scheme predicts. T h e fact that this correction term is proportional to T 4 may be seen to be the case quite generally, and this functional form is not a feature of our particular model. T h e coefficient of the term has a complex dependence on i, and we do not exhibit its general form here. When i is very close to unity, we find
NEARLY
FERROMAGNETIC
METALS
683
the temperature T, with a coefficient insensitive to the value o f / , as I --~ 1. The preceding discussion indicates that as T increases, the quantity p / A ( i ) T 2 falls monotonically. T h e resistivity becomes proportional to the temperature at large values of T. We have evaluated the ratio p/A (I) T 2 from the expression in equation (12), for the three values o~ -- 5, 40 and 200 of the enhancement parameter (l --i)-1. T h e values for k~ and tr employed in this work are those exhibited in equation (10). T h e results are displayed in Figs. 2 and 3. Before we discuss the results, we remark on the method used to evaluate the integral. T h e double integral was first evaluated by a Gaussian quadrature scheme[12]. An independent program which directly evaluated the two-fold Riemann sum was designed, and the results compared. F o r large values of a the second program proved most accurate. All calculations were checked for convergence by varying the size of the grid used to evaluate the Riemann sums, and at least two digit accuracy was confirmed for all the results reported here. 1.0
p ( T ) -- 1 A(I)T 2
3/7.4
(Orkd)2 ( _
16 0 _---~ \ T,,,)
2
-F-'- (12) 0.8 / ~
We exhib{t this form to illustrate the point in the preceding paragraph that p/A ( i ) T z is not simply a function of (T/T,s). Our numerical calculations show that the p o w e r series expansion is useful only o v e r a limited range of temperatures when a ~ 1. T h a t this is so is evident from the form in quation (12), since the coefficient of (T/T~I)2 is proportional to a. As T ~ oo, the parameter z in equation (10) becomes very large. In the limit as ~- ~ 0o only small values of x are important. T h e n the factor of e~/(e x - 1) 2 may be replaced by unity in equation (10), and the integral o v e r x may be performed. T h e n in this limit, we find that the resistivity becomes proportional to
UPPER CURVE:~=5 LOWER CURVE:a=200
P
AT2
=
0.6
0.4
0.2
I
0 00
1.0
l
2.0 T/T=f
I
I
5.0
4.0
F i g . 2. T h e r a t i o p / A T z f o r a = 5 a n d a = 2 0 0 .
684
D.L. MILLS
2.0
1.8
aft) zoo
a= ,_Jy
1.6
1.4.
~=AO
1.2 ( Z : 5 ~
*%
i ,o
2'.0
T T=
3'o
(]:200 t
T h e exponent n(T) at first decreases very rapidly with temperature, and a p p r o a c h e s the high t e m p e r a t u r e value of unity very slowly. T h e numerical calculations were not extended beyond T = 4Tse because the a s y m p t o t i c limit is a p p r o a c h e d very slowly. T h e r e is a very wide t e m p e r a t u r e range characterized by values of n ( T ) b e t w e e n 1.2 and 1-6. In the w o r k of the A m s t e r d a m group on Ni3AI and Ni3GA, resistivities characterized with n ~ 1.5 were reported. H o w e v e r , these low values of n were o b s e r v e d e v e n in the liquid H e temperature range (this is true also of the recent w o r k on stoichiometric Ni3AI[9]). T h e spin fluctuation t e m p e r a t u r e of these materials would have to be v e r y low indeed to explain these observations.
,o
Fig. 3. The quantity n (T) in the expression p = A T~ , for the three values of a employed in this work. The method used to determine n (T) is described in the text. In Fig. 2, we plot the ratio p/A ( i ) T 2 for a = 5 and a = 200, for 0 < T < 4Ts t. Throughout this range, the two curves are quite similar, and the differences b e t w e e n t h e m are poorly represented on the figure, particularly for the larger values of T/T~s. F o r T ,~ T~t, it is quite evident that for a = 200, p / A T 2 falls off more rapidly than for ot = 40, as e x p e c t e d f r o m the p o w e r series in equation (12). Experimentalists often fit the t e m p e r a t u r e d e p e n d e n c e of p to a p o w e r law of the form p = A T n. We write p ( T ) = A (i) T ntT), and we h a v e extracted values of n ( T ) f r o m the numerical results by fitting the resistivity o v e r a small t e m p e r a t u r e range by a p o w e r law. I f pl and P2 are the values of p at t e m p e r a t u r e s T1 and T2, and T = (T1 + T2)/2, then we define n ( T ) = In (p,/o2)/ln (T,/T2). T h e values of n ( T ) determined b y this means are plotted in Fig. 3. One can see that the T z behavior which characterizes the regime T < T, i is replaced by a m o r e gradual d e p e n d e n c e on t e m p e r a t u r e as T increases.
3. T H E T H E R M A L RESISTIVITY
W e have also c o m p u t e d the contribution to the thermal resistivity w f r o m e l e c t r o n - p a r a magnon scattering, or the model described in Section 2. It is quite straightforward to derive an expression for w valid for arbitrary temperatures by generalizing the t r e a t m e n t of Ref. [4] slightly. A convenient w a y to describe the results is by means o f the L o r e n z n u m b e r L formed f r o m the contribution from e l e c t r o n p a r a m a g n o n scattering to p and to w: L =-p--
wT "
(13)
F o r the case where the resistivity of a simple metal is controlled b y elastic scattering processes, the L o r e n z n u m b e r is t e m p e r a t u r e independent, and a s s u m e s the Summerfeld value L,=
3 \e]"
At low temperatures, in pure metals with strong exchange e n h a n c e m e n t , one also expects L to be t e m p e r a t u r e independent, with L ~ Ls[4, 5]. F o r our model, upon following the discussion of Ref. [4], we find L =
Ls
1 1+A'
(14a)
TEMPERATURE DEPENDENCE OF NEARLY FERROMAGNETIC METALS where
685
I
I-y
3 A = - - q.g2
1.0 a=5
X f ~ dxxAn (x) [ 1 + n (x) ] [8, (x'r) -- ~ 3 (X1") ] 0.8
f o dxx2n(x) [ 1 + n(x) ]f13(xz)
L/L,
0~=40
(14b) 0.6
with =
[ 1 -
i]'=-"2 ("
(crkfl 1/2)m J0
X
=
d~:~:m+l
0.4
(14c) 0.2
where ~:,,, and ~- are defined in Section 2. As T ---* 0, the factor of xr in the denominator of equation (14c) may be ignored, and equations (14) then are equivalent to equation (2.1 1) of Ref. [4]. F o r the parameters given in Section 3, we have computed the ratio (L[Ls) for the three values of the enhancement factor o~= 5, 40 and 200 employed in the electrical resistivity calculation. The results are displayed in Fig. 4. F o r T ~ Tst, the L o r e n z number L < Ls and is temperature independent, as pointed out in previous work[4, 5]. (Due to the neglect of the form factor IF(k,~)]z, the values computed here are a few per cent larger than those displayed in Ref. [4].) In all cases, L rises abruptly to a value the order of 0.5L,, then continues to rise gradually toward unity, although this limit is approached very slowly. N o t e that it is clear from these curves that the function (W/BT) is clearly not approximated by a universal function o f T/Tar. (In this ratio, the parameter B is the temperature coefficient of the e l e c t r o n - p a r a m a g n o n contribution to W when T ,~ Tar.) Thus, while our numerical calculations suggest that for the parameters explored here, p/A T 2 can be thought of as a function of only T~ T~f to a good approximation, the same is not true of the ratio W/BT. Notice that when T > Tar, the ratio p i T
JPCS VoL 34, No. 4-- H
1.0 '
2. o
5 0'
4 .'0
T/Ta
Fig. 4. The ratio L/L~ of the Lorenz number constructed from the electron-paramagnon scattering contribution to p and w, to the ideal Summerfeld value, for three values of the Stoner enhancement factor cr
varies extremely slowly with temperature, as does the Lorenz number itself. The thermal resistivity w is thus nearly temperature independent. In stoichiometric Ni3Al, Schriempf et al.[9] found that w varies quite slowly with temperature.
Acknowledgements--I would like to thank Dr. A. I. Schindler and Dr. T. J. Schriempf for a number of stimulating discussions which preceded this investigation. REFERENCES
1. MILLS D. L. and LEDERER P., J. Phys. Chem. Solids 27, 1805 (1966). 2. LEDERER P. and MILLS D. L., Phys. Rev. 165, 837 (1968). 3. SCHINDLER A. I. and RICE M. J., Phys. Rev, 164, 759 (1967). 4. SCHRIEMPF J. T., SCHINDLERA. 1. and MILLS D. L., Phys. Rev. 187,959 (1969). 5. RICE M. J., Phys. Lett. 26A,86 (1967). 6. As an example of an exception to this statement, see the work by HARRIS R. and ZUKERMANN M., Phys. Rev. 4B, 101 (1972). 7. KAISER A. B. and DONIACH S.,lnt.J. Magnetism
686
D. L. MILLS
1, 11 ~(1970); KAISER A. B., Phys. Rev. 3B, 3040 (1971). 8. DE C H A T E L P. F., DE BOER F. R., D E D O O D W., F L U I T M A N J. H. J. and S C H I N K E L C. J., J. Phys. 32, CI 999 (1971), and also F L U I T M A N J. H. J. et al., preprint entitled The low temperature resistivities of Ni3A1 and NiaGa alloys confronted with spin density fluctuation theories, J. Phys. C, Solid St. Phys., to be published. 9. S C H R I E M P F T. J. (private communication). A brief account of this work has been given by J. T.
S C H R I E M P F , W. M. M c l N N E S and A. I. S C H I N D L E R . See paper BD8, Bull. Am. phys. Soc. 17, 256 (1972). 10. ROBBINS C. G. and C L A U S H., A. I. P. Conf. Proc. No. 5, Magnetism and Magnetic Materials, Part I, p. 527(1971). 11. This result is displayed in equation (2.10) of Ref. [4]. A factor of ks2 has been omitted from the denominator of the expressions in Ref. [4]. 12. I am indebted to Dr. S. Cunningham for writing the program for this integration routine.
T E M P E R A T U R E D E P E N D E N C E OF T H E C O N T R I B U T I O N TO T H E T R A N S P O R T C O E F F I C I E N T S OF N E A R L Y F E R R O M A G N E T I C M E T A L S F R O M ELECTRON-PARAMAGNON SCATTERING* D. L. MILLS Department of Physics, University of California, Irvine, California and Metal Physics Branch, U.S. Naval Research Laboratory, Washington, D.C., U.S.A. (Received 5 June 1972)
Abstract--We present calculations of the temperature dependence of the contribution from electronparamagnon scattering to the electrical and thermal resistivity of a simple model of a nearly ferromagnetic metal. The purpose of the work is to explore the behavior of these quantities when the temperature is the order of or greater than the spin fluctuation temperature T~e. As the temperature T is raised from zero through T~s, the electrical resistivity varies more slowly with temperature than the T 2 law characteristic of the regime T ,~ Tsr. When T ~> Tee,the electrical resistivity becomes proportional to T, although this asymptotic behavior is approached very slowly. The Lorenz number rises monotonically with temperature, and appears to approach the ideal Sommerfeld value when T >> Tss, although this limit is also approached slowly. 1. INTRODUCTION
IN NEARLY ferromagnetic metals and alloys, large amplitude spin fluctuations are present, since the system is close to a magnetic instability. Current carrying electrons may scatter from these enhanced spin fluctuations, and this scattering leads to contributions to the temperature dependent part of the electrical resistivity p[1-3] and thermal resistivity w [4, 5] of these systems. The fluctuations also affect a number of other thermodynamic and magnetic properties of pure metals and their alloys [6]. Most theoretical descriptions of these fluctuations have focused on one of two limiting cases, the dilute alloy where the fluctuations are spatially inhomogeneous in amplitude, with the spin-fluctuation amplitude near the dilute constituent very different from that in the host matrix, or the pure metal where the fluctuations have a wave-like character, with *Technical Report #72-28. 679
strong enhancement of the long wavelength fluctuations if the host is near an instability with respect to the ferromagnetic state. The early theoretical studies of the contribution to the transport coefficients from electronspin fluctuation scattering were confined to the study of the low temperature limit, where p contains a contribution to T z from this source[l,2], and w contains a contribution that varies linearly with T [4, 5]. Doniach and Kaiser[7] have recently extended the theory of the contribution to p and w from electronlocal spin fluctuation (LSF) scattering in dilute alloys to the case where the temperature T is comparable to or greater than the LSF characteristic temperature TLS v. In the paper by Schindler and Rice[3], a theory of the temperature dependence of p in pure metals was presented and applied to the case where T is comparable to or greater than the spin fluctuation temperature Tss characteristic of the metal. We feel this work employs an expression for the spectral density of the spin
680
D.L.
fluctuations that is rather oversimplified for the reasons described below. As a consequence, we have investigated the behavior of p and w for the pure metal when T is comparable to or larger than Ts~. While the model we use is still a very simple one, we feel it contains the principle qualitative features needed before one can obtain a realistic description of the contribution to p and w from electron-spin fluctuation (electronparamagnon scattering) in pure metals. Our model is sufficiently complex that most of the results reported below must be obtained by numerical computations, unfortunately. This study has been motivated in part by a recent series of experimental investigations of the properties of Ni3A1 and NiaGa[8], and binary alloys of these constituents with compositions close to stoichiometry. Stoichiometric Ni3A1 is ferromagnetic with a Curie temperature of roughly 40~ and the AI rich alloys are paramagnetic with very large susceptibilities near the critical composition. These materials have very large temperature dependent contributions to p at low temperatures, and while p varies roughly like T ~ for compositions far from the critical one, the temperature dependence weakens for c near the critical composition cr. A considerable body of data on this system and on the Ni3Ga system are presented and compared with the spin fluctuation theory in Ref. [8]. There is qualitative, but certainly not quantitative agreement. It therefore seems of value to carry out a more complete theoretical investigation o f the contribution to p and w from electron paramagnon scattering in the region T >t Tss, since no adequate theory has yet appeared. This may allow a more critical comparison between the data and the theory. We do not attempt a quantitative fit to the data here since, as noted in Ref. [8], it is not clear that paramagnon theory provides a complete description of the alloys. For example, the dependence of the coefficient 3/of the term linear in the specific heat increases with the susceptibility as expected from the theory,
MILLS
but the dependence is far weaker than predicted[8]. Also a recent study of p and w in stoichiometric Ni3AI[9] shows that the exponent n in the expression p = A T n decreases from a value of roughly 1.3 at 10~ to a lower value as the temperature decreases, The Lorenz number varies little with temperature. While the theory presented here is applicable only to the paramagnetic state, this behavior appears hard to understand within the simple form of the itinerant electron model of a uniform metal. Finally, Robbins and Claus[10] have interpreted specific heat and magnetization measurements on Ni3A1 on the basis of a model which ascribes the magnetism to the presence of Ni clusters, and the interaction between clusters. 2. T H E E L E C T R I C A L RESISTIVITY
This work is based on a model introduced and discussed in detail in Ref. [ 1]. T he transition metal is presumed to have two sets of conduction electrons. One set, the s-electrons, has a mass ms comparable to the free electron mass, and the second set, the d-electrons, has an effective mass ma ~> ms. The current is carried primarily by the s-electrons, while the large density of states associated with the d-electrons means that the magnetic properties of the metal and the specific heat receive their principal contribution from the d-electrons. Of course, the wave functions associated with the two sets of electrons need not have pure s and d character for the model to be valid. It is only necessary that the Fermi surface consist of distinct light and heavy mass portions. We consider in this section the contribution to the temperature dependent portion of p from scattering of the s-electrons from exchange enhanced spin fluctuations in the (paramagnetic) d band. For simplicity, the Fermi surface of the s and d bands will be assumed spherical, while ks, ka and ~s, ca are the Fermi wave vectors and Fermi energies of the two sets of electrons, respectively. If the s and d electrons interact via an s - d exchange interaction of strength J, from the results in
TEMPERATURE DEPENDENCE OF NEARLY FERROMAGNETIC METALS
reference (1) the contribution to p from electron-paramagnon scattering may be written in the form[11] (with h = l)
9Vcms f
P= 32e2k rknT
dO
X dEf(e) [ 1 --f(e-- O) ][ 1 + n(O) ]
• ffm dr/~ aA (ks'o, ~)IF(k,~)P.
(1)
681
d band at the Fermi surface. Equation (3b) is the form appropriate to a free electron gas in the low frequency limit, and we neglect the frequency dependence of Re[xo(q,O) ]. These approximations should be reasonable for a material with strong exchange enhancement, since the spin fluctuation temperature T~, defined by
k~Tsf= (1--i)Ea
(4)
will be small compared to the Fermi energy. In this expression, Vc is the volume of the unit We then take the simple parabolic form for cell, kB is Boltzmann's constant, and the func- Re [x0(q, 12)] exhibited in equation (3a). If tions f(E) and n(O) are the Fermi-Dirac and Na(O) is the density of states in the d band/ Bose-Einstein functions, respectively. The spin direction at the Fermi energy, and i = quantity IF(k,;o)12 is a form factor which we 1Na(ea), then forA (q, l~) we have set equal to unity in the remainder of this note. The parameter ~,, is max (2,2ka/k,). The A (q, 1~) = 7rNd(O) critical part of equation (1) is the function ~q A(q,O), the spectral density of the spin • (5) qS[ (1 --[) + icr2q2]2 + [ (zr/2)l(O/Va)]2" fluctuations in the d-band. This quantity is related to the dynamic susceptibility x(q, O) of the d electrons: We feel this form represents the principle qualitative features of the spin fluctuation 1 A (q, f~) = 7[X (q, O - ic) --X (q, l~ + ie) ] (2a) spectrum of stronlgy exchange enhanced paramagnets. For small values of q, when [ is close to unity the spectral density exhibits a where in the paramagnon model one has peak at a frequency ~q = Vsq, where Vs X (q, l~) = X0(q, lq)/[ 1 -- IXo (q, 1~) ]. (2b) (1 -- D va. As q increases, the o-2q2 term shifts the peak to frequencies higher than Vsq. InIn equation (2b), • O) is the dynamic sus- deed, for o-q >~ 1, the position of the peak in ceptibility of the d electrons in the one A (q, O) is insensitive to [, when i is close to electron approximation, and I is a measure of unity. the strength of the electron-electron interWe can compare the expression in equation actions. Since xo(q, ll) is in general a complex (5) to the form for A (q, O) used earlier by function which depends on the details of the Rice[3]. In his work, Rice ignored the term electronic band structure of the material, we proportional to f~2 in the denominator of base our calculations on the simple model equation (5), as well as the or2q 2 tenn. He forms then cut the spectral density offat a frequency 12c the order of the spin fluctuation frequency Re [Xo(q, 1~) ] -- Re [x0(q, O) ] = X0[ 1 --o-2q2] ~ . The approximation used by Rice thus (3a) ignores the contribution to p from the high and frequency tail in equation (5), which falls off 3rr ~ only as ~-1. More importantly, the peak in Im [xo(q, O) ] -- 4Ca Vaq' Va < q < 2ka (3b) the spectral density occurs at a frequency ~,, proportional to Vsq. We feel this last feature where Va is the velocity of the electrons in the is particularly unrealistic, since for large q, the
682
D . L . MILLS
falloff of Xo(q, 0) with q causes the peak to occur at a much higher frequency. When the temperature is low, the term proportional to 1~2 in the denominator may be ignored. One then recovers the low temperature limiting form of p, where p is proportional to T 2. In this region, we write p = A (i) T 2.
(7)
is of interest, since the dependence o f the temperature dependent portion of p on the susceptibility X = aX0 of the host matrix has been discussed [8]. A simple integration gives
A(I) 3 tan-' (~:M)- [seMI(1 +~:M2) ] A(u) = 2 ~M3(I _ 1 ) 2 , (8) where
i 1/2
~:~t = o'ks~M ( 1 -- i) ,/2"
(9)
T h e variation of A with the Stoner enhancement factor o~ provided by the present model is exhibited in Fig. 1. T o compute the values of A (]), we have used the parameters employed in an earlier study of the Lorenz number of the Pd Ni system [4]: 1
= V~ka ks = 2ka.
I00.0
~
"
(6)
T h e complete expression or A ( i ) is of little interest, since the simple model is not expected to yield accurate estimates of the absolute value ofA. H o w e v e r , the dependence of A on i, or more precisely on the parameter 1 C~=l__ /
10oQo
I0.0
I'01.0
(10b)
All of the numerical calculations reported in the present paper employ these values for oand ks. We do not expect the qualitative features of the prediction of the model to depend sensitively on the precise values of oand ks.
IOOD
Ot
I000.0
Fig. 1. The variation of the coefficient A of the T 2 term in the electrical resistivity with the Stoner enhancement factor a, for the model described in the text.
F r o m Fig. 1, one sees that the dependence of A on a is not described by a simple power law. F o r a near unity, A increases more rapidly than a, while when a >> 1, A increases much more slowly. We next consider the temperature dependence of p, for the case where the temperature T is the order of magnitude of Tss or greater. T h e integration over E in equation (1) is readily performed. After some manipulation, w e obtain p =
6//(i) Tz
rrZ{tan-' (~:M) -- [ ~M/(1 + sou2) ] } •
(10a)
I0.0
fo~ (e-7_-dxx2 eX f~u ~ [ 1 1) ~
sC'd~: +~]~+r~x~
(lO)
where the quantity z = ksT/Or
and
4 / 1 - - I'~ 3/2 lqr : ~--~.a ~--~-- ) 'a-
(11 a) (lib)
As mentioned in Section 1, Kaiser and D o n a c h have extended the theory o f electronL S F scattering contributions to p and w [7] to
TEMPERATURE
DEPENDENCE
OF
the case where T >~ TLsr, where TLS F is the local spin fluctuation temperature. In their theory, if the contribution of electron-LSF scattering to p at low temperatures is denoted by a ,4 T 2, they find the ratio p/,4 T 2 to be a function only o f the reduced temperature (T/TLsr). T h e expression in equation (10) cannot be written in a similar form. In general, in the pure metal, the ratio p/A (I) T z is not a function only of T/T.~s. T h e shape of a plot of p/A (T) T 2 vs T/T~f thus depends on the value of the enhancement parameter ~. H o w e v e r , for the parameters employed in the numerical calculations below, the ratio p / A ( I ) T 2 is remarkably insensitive to a. In the low temperature limit, the right hand side may be expanded in powers of z 2. One readily sees that the first correction to the low temperature limiting form A ( i ) T 2 of p is proportional to T 4, and not T 5 as Rice's approximate scheme predicts. T h e fact that this correction term is proportional to T 4 may be seen to be the case quite generally, and this functional form is not a feature of our particular model. T h e coefficient of the term has a complex dependence on i, and we do not exhibit its general form here. When i is very close to unity, we find
NEARLY
FERROMAGNETIC
METALS
683
the temperature T, with a coefficient insensitive to the value o f / , as I --~ 1. The preceding discussion indicates that as T increases, the quantity p / A ( i ) T 2 falls monotonically. T h e resistivity becomes proportional to the temperature at large values of T. We have evaluated the ratio p/A (I) T 2 from the expression in equation (12), for the three values o~ -- 5, 40 and 200 of the enhancement parameter (l --i)-1. T h e values for k~ and tr employed in this work are those exhibited in equation (10). T h e results are displayed in Figs. 2 and 3. Before we discuss the results, we remark on the method used to evaluate the integral. T h e double integral was first evaluated by a Gaussian quadrature scheme[12]. An independent program which directly evaluated the two-fold Riemann sum was designed, and the results compared. F o r large values of a the second program proved most accurate. All calculations were checked for convergence by varying the size of the grid used to evaluate the Riemann sums, and at least two digit accuracy was confirmed for all the results reported here. 1.0
p ( T ) -- 1 A(I)T 2
3/7.4
(Orkd)2 ( _
16 0 _---~ \ T,,,)
2
-F-'- (12) 0.8 / ~
We exhib{t this form to illustrate the point in the preceding paragraph that p/A ( i ) T z is not simply a function of (T/T,s). Our numerical calculations show that the p o w e r series expansion is useful only o v e r a limited range of temperatures when a ~ 1. T h a t this is so is evident from the form in quation (12), since the coefficient of (T/T~I)2 is proportional to a. As T ~ oo, the parameter z in equation (10) becomes very large. In the limit as ~- ~ 0o only small values of x are important. T h e n the factor of e~/(e x - 1) 2 may be replaced by unity in equation (10), and the integral o v e r x may be performed. T h e n in this limit, we find that the resistivity becomes proportional to
UPPER CURVE:~=5 LOWER CURVE:a=200
P
AT2
=
0.6
0.4
0.2
I
0 00
1.0
l
2.0 T/T=f
I
I
5.0
4.0
F i g . 2. T h e r a t i o p / A T z f o r a = 5 a n d a = 2 0 0 .
684
D.L. MILLS
2.0
1.8
aft) zoo
a= ,_Jy
1.6
1.4.
~=AO
1.2 ( Z : 5 ~
*%
i ,o
2'.0
T T=
3'o
(]:200 t
T h e exponent n(T) at first decreases very rapidly with temperature, and a p p r o a c h e s the high t e m p e r a t u r e value of unity very slowly. T h e numerical calculations were not extended beyond T = 4Tse because the a s y m p t o t i c limit is a p p r o a c h e d very slowly. T h e r e is a very wide t e m p e r a t u r e range characterized by values of n ( T ) b e t w e e n 1.2 and 1-6. In the w o r k of the A m s t e r d a m group on Ni3AI and Ni3GA, resistivities characterized with n ~ 1.5 were reported. H o w e v e r , these low values of n were o b s e r v e d e v e n in the liquid H e temperature range (this is true also of the recent w o r k on stoichiometric Ni3AI[9]). T h e spin fluctuation t e m p e r a t u r e of these materials would have to be v e r y low indeed to explain these observations.
,o
Fig. 3. The quantity n (T) in the expression p = A T~ , for the three values of a employed in this work. The method used to determine n (T) is described in the text. In Fig. 2, we plot the ratio p/A ( i ) T 2 for a = 5 and a = 200, for 0 < T < 4Ts t. Throughout this range, the two curves are quite similar, and the differences b e t w e e n t h e m are poorly represented on the figure, particularly for the larger values of T/T~s. F o r T ,~ T~t, it is quite evident that for a = 200, p / A T 2 falls off more rapidly than for ot = 40, as e x p e c t e d f r o m the p o w e r series in equation (12). Experimentalists often fit the t e m p e r a t u r e d e p e n d e n c e of p to a p o w e r law of the form p = A T n. We write p ( T ) = A (i) T ntT), and we h a v e extracted values of n ( T ) f r o m the numerical results by fitting the resistivity o v e r a small t e m p e r a t u r e range by a p o w e r law. I f pl and P2 are the values of p at t e m p e r a t u r e s T1 and T2, and T = (T1 + T2)/2, then we define n ( T ) = In (p,/o2)/ln (T,/T2). T h e values of n ( T ) determined b y this means are plotted in Fig. 3. One can see that the T z behavior which characterizes the regime T < T, i is replaced by a m o r e gradual d e p e n d e n c e on t e m p e r a t u r e as T increases.
3. T H E T H E R M A L RESISTIVITY
W e have also c o m p u t e d the contribution to the thermal resistivity w f r o m e l e c t r o n - p a r a magnon scattering, or the model described in Section 2. It is quite straightforward to derive an expression for w valid for arbitrary temperatures by generalizing the t r e a t m e n t of Ref. [4] slightly. A convenient w a y to describe the results is by means o f the L o r e n z n u m b e r L formed f r o m the contribution from e l e c t r o n p a r a m a g n o n scattering to p and to w: L =-p--
wT "
(13)
F o r the case where the resistivity of a simple metal is controlled b y elastic scattering processes, the L o r e n z n u m b e r is t e m p e r a t u r e independent, and a s s u m e s the Summerfeld value L,=
3 \e]"
At low temperatures, in pure metals with strong exchange e n h a n c e m e n t , one also expects L to be t e m p e r a t u r e independent, with L ~ Ls[4, 5]. F o r our model, upon following the discussion of Ref. [4], we find L =
Ls
1 1+A'
(14a)
TEMPERATURE DEPENDENCE OF NEARLY FERROMAGNETIC METALS where
685
I
I-y
3 A = - - q.g2
1.0 a=5
X f ~ dxxAn (x) [ 1 + n (x) ] [8, (x'r) -- ~ 3 (X1") ] 0.8
f o dxx2n(x) [ 1 + n(x) ]f13(xz)
L/L,
0~=40
(14b) 0.6
with =
[ 1 -
i]'=-"2 ("
(crkfl 1/2)m J0
X
=
d~:~:m+l
0.4
(14c) 0.2
where ~:,,, and ~- are defined in Section 2. As T ---* 0, the factor of xr in the denominator of equation (14c) may be ignored, and equations (14) then are equivalent to equation (2.1 1) of Ref. [4]. F o r the parameters given in Section 3, we have computed the ratio (L[Ls) for the three values of the enhancement factor o~= 5, 40 and 200 employed in the electrical resistivity calculation. The results are displayed in Fig. 4. F o r T ~ Tst, the L o r e n z number L < Ls and is temperature independent, as pointed out in previous work[4, 5]. (Due to the neglect of the form factor IF(k,~)]z, the values computed here are a few per cent larger than those displayed in Ref. [4].) In all cases, L rises abruptly to a value the order of 0.5L,, then continues to rise gradually toward unity, although this limit is approached very slowly. N o t e that it is clear from these curves that the function (W/BT) is clearly not approximated by a universal function o f T/Tar. (In this ratio, the parameter B is the temperature coefficient of the e l e c t r o n - p a r a m a g n o n contribution to W when T ,~ Tar.) Thus, while our numerical calculations suggest that for the parameters explored here, p/A T 2 can be thought of as a function of only T~ T~f to a good approximation, the same is not true of the ratio W/BT. Notice that when T > Tar, the ratio p i T
JPCS VoL 34, No. 4-- H
1.0 '
2. o
5 0'
4 .'0
T/Ta
Fig. 4. The ratio L/L~ of the Lorenz number constructed from the electron-paramagnon scattering contribution to p and w, to the ideal Summerfeld value, for three values of the Stoner enhancement factor cr
varies extremely slowly with temperature, as does the Lorenz number itself. The thermal resistivity w is thus nearly temperature independent. In stoichiometric Ni3Al, Schriempf et al.[9] found that w varies quite slowly with temperature.
Acknowledgements--I would like to thank Dr. A. I. Schindler and Dr. T. J. Schriempf for a number of stimulating discussions which preceded this investigation. REFERENCES
1. MILLS D. L. and LEDERER P., J. Phys. Chem. Solids 27, 1805 (1966). 2. LEDERER P. and MILLS D. L., Phys. Rev. 165, 837 (1968). 3. SCHINDLER A. I. and RICE M. J., Phys. Rev, 164, 759 (1967). 4. SCHRIEMPF J. T., SCHINDLERA. 1. and MILLS D. L., Phys. Rev. 187,959 (1969). 5. RICE M. J., Phys. Lett. 26A,86 (1967). 6. As an example of an exception to this statement, see the work by HARRIS R. and ZUKERMANN M., Phys. Rev. 4B, 101 (1972). 7. KAISER A. B. and DONIACH S.,lnt.J. Magnetism
686
D. L. MILLS
1, 11 ~(1970); KAISER A. B., Phys. Rev. 3B, 3040 (1971). 8. DE C H A T E L P. F., DE BOER F. R., D E D O O D W., F L U I T M A N J. H. J. and S C H I N K E L C. J., J. Phys. 32, CI 999 (1971), and also F L U I T M A N J. H. J. et al., preprint entitled The low temperature resistivities of Ni3A1 and NiaGa alloys confronted with spin density fluctuation theories, J. Phys. C, Solid St. Phys., to be published. 9. S C H R I E M P F T. J. (private communication). A brief account of this work has been given by J. T.
S C H R I E M P F , W. M. M c l N N E S and A. I. S C H I N D L E R . See paper BD8, Bull. Am. phys. Soc. 17, 256 (1972). 10. ROBBINS C. G. and C L A U S H., A. I. P. Conf. Proc. No. 5, Magnetism and Magnetic Materials, Part I, p. 527(1971). 11. This result is displayed in equation (2.10) of Ref. [4]. A factor of ks2 has been omitted from the denominator of the expressions in Ref. [4]. 12. I am indebted to Dr. S. Cunningham for writing the program for this integration routine.