Experiments on the magnetoresistivity and hall effect in Ni and Ni alloys. The validity of Kohler's rule

J. Phys. Chm. Sali&

Pergamon

EXPERIMENTS

Press 1967. Vol. 28, pp. 253-260.

Printed in Great Britain.

ON THE MAGNETORESISTIVITY

AND

HALL EFFECT IN Ni AND Ni ALLOYS. THE VALIDITY OF KOHLER’S A. C.

RULE

EHRLICI-I, R. HUGUENIN and D. RIVER

Institut de Physique Experimentale

(Rem&i

de 1’Universiti

de Lausanne

25 May 1966; in revised form 15 August 1966)

Abstract-The low temperature transverse magnetoresistivity and Hall resistivity in three high purity polycrystalline Ni samples and three very dilute Ni based alloys (Ni-Fe, Ni-Co and Ni-Cu) were measured. The results are reported on Kohler diagrams. The transverse magnetoc~nductivities and the Hall conductivities were calculated and are also reported on Kohler diagrams. It is found that: (1) for the purest samples the magnetoresistivity and the Hall conductivity show krge and systematic deviations from Kohler’s rule, while the Hall resistivity and the magnetoconductivity obey it very well; (2) for the alloys, the Hall resistivities and magnetoconductivities fall respectively on different Kohler curves than the pure samples, but among themselves are in agreement with Kohler’s rule. These results are discussed.

1. INTRODUCTION different temperature can be written as AT@; k) THE complexity of the exact expressions given where X may be a function of temperature or by the available microscopic theories for the impurity content, but not of n or of k. Kohler’s magnetoresistance and Hall effect in metals has led rule should then hold for all components of the to the interpretation of experimental results on the relative magnetoresistivity tensor. In the case of a basis of highly simplified models. perfect metallic polycrystalline sample and if the Among the predictions of these elementary magnetic induction B is taken in the “3 direction”, models is KOHLER’Srule,(l) which is concerned one has with the dependance of a given component of the resistivity tensor, pi,, of a metallic sample upon the ~0) = ~za(B) = P,(B) magnetic induction, B the temperature, T, and the (1) 1 P&4 = PzL--B) = PEP) purity of the sample. Kohler’s rule states that when these quantities are varied, any component of the relative magneturesistivity tensor-that is the Here p I and pi are the transverse resietivity and ratio of the change of component (pit(B) - 6,~~) to the Hall magnetoresistivity respectively. p,,, the resistivity in the absence of a magnetic In ferromagnetic metals both the magnetofield should be a function of B/p,, only. The resistivity and Hall resistivity have their origins in conditions under which Kohler’s rule is met can two distinct processes: the first, essentially the be summarized as follows: (a) the rigid band direct action of the magnetic induction on the approximation is valid and the volume of occupied kinetic behaviour of the conduction electrons, is electron states does not change appreciably from present in any metal abd is responsible for the 80 sample to sample. (b) If T (n; k) is the relaxation called “ordinary e&c&‘; the second, unique to the time for a sample of a given purity at some given ferromagnetic state, is a result of a more subtle temperature (k is the wave vector of the Block interaction with the magnetization. Concerning state and n the band index), then the relaxation these “magnetization” of “spontrtneous” contributime for a sample of another purity or at some tions to the magnetoresistivity tensor, SM@) has 253

254

A. C. EHRLICH,

R. HUGUENIN

proposed that the effect on the transverse magnetoresistivity should be due essentially to a dependence of the relaxation time on the state of magnetixation, while LUTTINGJ@) and IRKHINand SHAVREV have, among others, explained the spontaneous Hall efkct in terms of the spin-orbit interaction. These “magnetization” contributions to the Hall and magnetoresistivities must, of course, be subtracted out of the measured values before the latter can be compared with Kohler’s rule which is concerned only with the ordinary part of the relative magnetoresistivity tensor. This subtraction cannot always be done in an unambiguous manner. In sufficiently pure ferromagnetic metals and at low enough temperatures, however, the magnetization contributions become negligible compared to those of the ordinary effects. The most thoroughly investigated of the ferromagnetic metals, Ni, has a complicated electronic structure with at least three bands, twos sub-bands corresponding to the two spin directions and one or more d sub-bands. In at least one of these bands the Fermi surface touches the Brillouin zone boundALPHEN ary. (‘) According to recent DE HAAS--VAN measurements(6) and other arguments(‘) this is probably the case for one or perhaps both of the s bands. The existence in Ni of anisotropic bands and therefore surely anisotropic scattering times, makes very unlikely that the conditions under which Kohler’s rule is valid would be met. The presence of a non-negligible amount of electronmagnon scattering would decrease still further the likelihood of fulfilling condition (b). 2. SAMPLES In the course of an experimental study of the influence of impurities and phonons on the galvanomagnetic properties of Ni, we have also measured the transverse magnetoresistance and Hall resistivity in several pure Ni samples and dilute alloys of Fe, Co and Cu in Ni. All the samples measured were polycrystals. The preparation of the samples occurred in the following way: two of our three samples of pure Nickel, Ni I and Ni II, were cut from the same Johnson Matthey Ni sheet but annealed differently: Ni I at 1200°C and Ni II at 1000°C. Both were cooled to room temperature at the rate of S”K/min. The last sample of pure Nickel, Ni III, was cut from another sheet, annealed at 1oooOC and cooled to room

and D. RIVIER

temperature at 3”/min. The alloy samples were prepared from Johnson Matthey spectrographically pure metal by melting under vacuum in an induction furnace; the ingots were then cold rolled and ma&&d to shape; finally they were annealed in a manner similar to that of Ni II. All the samples were in the form of flat plates, to allow Hall effect measurements. The temperatures at which the measurements were made and the “electrical purity”-that is the ratio of the resistivity at 293°K to the resistivity at the measuring temperature-are given in Table 1. (The word “purity” will hereafter mean “electrical purity’ in the sense given here.) Table 1. Smnple measured, tenrpetature at which the ~ments were camed out and “electrical purity”, i.e. the ratio of the resistiwity at 293°K to the mistivity at the meamrtkg temperature

Sample

Measuring temperature T(“W

Electrical PUfity (293°K) (T)

Ni III Ni III Ni III Ni III Ni II Ni I Ni-Co(O*lO%) Ni-Fe (O-07%) Ni-Cu(O*14%)

1.85 4.15 14.1 20.1 4.15 4.15 4.15 4.15 4.15

2250 2000 920 560 490 170 102 61 42

Among all the measurements, we shall present here only those where the “magnetization” or “spontaneous” contributions to the magnetoresistivity tensor are negligible.(s) For the transverse magnetoresistivity, samples with impurity concentration higher than O*14o/ohave been excluded, since considerable spontaneous contributions appear even at 4*K, as is also the case for the purest Ni-Ni III at the next readily available temperature (that is liquid nitrogen temperature). Concerning the Hall resistivity, examination of the results indicates that the magnetization contributions are small except for the three alloys. For those samples, although without significant role for what follows, the spontaneous part of the Hall

EXPERIMENTS

ON THE MAGNETORESISTIVITY

AND HALL EFFECT

resistivity has been subtracted out of the measurement under the assumption that the ordinary and the magnetization Hall resistivities are additive. Summarizing, it appears that-for the first time as far as we know-the experimental conditions have been chosen optimally for a complete study of Kohler’s rule in a ferromagnetic metal, since the temperature and the alloys content have been varied in such a way that the ordinary part of the magnetoresistivity can be measured directly. 3. ExpERIMENTAt

RESULTS

The measurements were made in magnetic induction intensities up to about 3 Vsec/ma. The experimental error in the magnetoresistivity is about 2% except for the Ni III measurements at helium temperature where, because of the low resistivity of thii sample and the fact that its crosssectional area is about 10 times that of the other samples, an error of about 5% is possible. The Hall

30

-10’

255

resistivity measurements have an absolute error no greater than 2%. In Figs. 1 and 2 the relative magnetoresistivity and the relative Hall resistivity respectively are presented on Kohler diagrams. The insets show the data for small values of Blps. From Fig. 1 it is clear that the deviations from Kohler’s rule for the magnetoresistivity are remarkably large and systematic. The curves all have the same general shape and are displaced from each other in a regular way as the samples become less pure. On the other hand for the Hall resistivity, it is seen from Fig. 2 that, the Ni II and four Ni III measurements results fall on the same curve: for these samples the Hall resistivity follows Kohler’s rule quite well. The three alloys and Ni however, fall on a second curve somewhat below the Ni II and Ni III data. The systematic deviation from Kohler’s rule shown by the relative magnetoresistivity measurements as well as the general theoretical approach

40 B/e,

IN Ni AND Ni ALLOYS

o

Nil

ld5’K

o

Nil

4~1 5-K

v

NilI

20~1 ‘K

+

Nil

4tlS.K

50 (weberl

60 ohm-

A Ni-Co(O*lO%) 4.15 ’ K T

Ni-Fe(O07%)

4.15

.

NiCu(O.14%)

4.15.K

70

90

l

K

90

meter’)

FIG. 1. Kohler diagram of the relative magnetoreaistivityfor the indicated aamplea and temperatures. The inset shows the data for small valuea of B/p0 on a larger scale.

256

A. C. EHRLICH,

I

I

R. HUGUENIN

1

I

I

and R. RIVIER

I

I

I

I

I

-

0 00

6-

0 8

0 0

a 0

0 10

0

o NIP

M5’K

0 NilI

4.15

l

l

K

Nit

4*15*K

A Ni-Co(@lO%)

4-l S’K

o NiH

14.1 *K

v

Ni-Fe(OXP%)

4.1 5-K

v

Nilll

20.1 ‘K

I

Ni-Cu(014*/.)

4~1 5’K

+

Nilf

4~15°K

1

I

I

1

I

1

I

I

I

20

30

40

50

60

70

80

90

B/c,

1 J

10’ (weber/ohm-meter’)

Frc. 2. Kohler disgram of the relative Hall resistivity for the indicated samples and temperatures. The inset shows the data for smsll values of B/p0 on a larger scale.

to the question led us to study the transverse msgnet~onhtivity, y,(B) and tire Hall cmdwtivity, y&B) rather than the corresponding reaistivities. These are obtained from an inversion of the resistivity tensor which leads to the expressions yl=

pL

YE =

PA-PI?

PH

P.L2+PI.12

(2)

4 however, it is obvious that-in contrast to the HaU resistivity-the Hall conductivity does not follow Kohler’s rule. Moreover this Kohler diagram shows the following three striking features (1) all the curves have their maximum for the same value of yoB, (2) their divergence is greatest for this value and (3) the curves tend to converge as y,B increases.

given the particular choice for the coordinate axes.

The

relative

magnetoconductivity

relative Hall conductivity

and

the

are shown on Kohler

diagrams in Figs. 3 and 4 respectively. Again the insets show the results at small values of B/PO

=

BY,.

In Fig. 3, it can be seen that-in contrast to the transverse transverse magnetor~~ti~ty-me magnetoconductivity follows very well the predictions of Kohler’s rule except for the alloys. In Fig.

4. DISCUSSION

In the following discussion, we wish to consider separately the following questions : 1. The range of the effective magnetic inductions applied to the sample. 2. The “unmagnetized” state of the samples to which p. or ys should refer. 3. The complementary aspects of the magnetoresistivity and of the m~net~onducti~ty tensors.

EXPERIMENTS ON THE MAGNETORESISTIVITY AND HALL EFFECT IN Ni AND Ni ALLOYS

4. The possible contribution of the microscopic theories for explaining the experimental results.

257

may be roughly valid in the cases of our Ni samples, it is seen from Fig. 4 that in our measurements 25 covers approximatively the range. 10-l 5 b 5 10 (3) that is from the “low field approximation” to the “high field approximation”.

samples

The knowledge of this range is important for the interpretation of the measurements in the light of the available microscopic theories. A convenient way of evaluating the range of the magnetic induc- Unmagrsetiaedstate to sohichp. or yO should refer The study of Kohler’s rule in a ferromagnetic tion applied to the sample is to refer it to the chmacteristic i~~t~B’, defined as the magnetic induc- metal involves first the problem of defining in a tion for which w,., the characteristic cyclotron correct way the state to which p. refers, i.e. the frequency, times the characteristic relaxation time state for which the magnetic induction inside of T is unity; B’ = mc/er. So we introduce the reduced the sample, Btit, is zero. It should be recalled that i&&on b = B/B’ = ~~7. The range of variation even in a demagnetized sample the conduction of b can be estimated from the Hall conductivity electrons are passing through magnetic domains curves on the Kohler diagram of Fig. 4. For a and that, though the net magnetization of the spherical band with isotropic scattering time T, sample is zero, the average effect of the domain the standard model shows that yn has its maximal magnetization of the conduction electron is in value when b = 1. Assuming that this property general not. From that it is clear that the resistivity I

I

I

I

I

0 0

OS

a3: 02 O+

0

I

0

00

OD

I

0

0

0

06

04

I

I

. A A

0

h

Ok 0

.

+aA 0

w .

m *

1 10

o

Ni‘ll

1:85’K

o

NiH

4:15*K

A

Nil

14:l lK

v

Nia

20.1 ‘K

+

Nil

4:15’K

.

Nil

4:lS’K

l

Ni-Co(OlOYd

v ‘Ni-Fe{0

m Ni-Cu(0.14%) I 20

4clS’K

07%)

I 30

4*1S’K 4.15-K I 40

I SO

B & -10’ (weber I ohm

I 60

I 70

I 80

I 90

meter*)

FIG. 3. Kohler diagram for the relative magnetoconductivity for the indicated samples and temperatures. The inset shows the data for smaii values of By0 on a larger scale.

A. C. EHRLICH,

258

,. 1

1-O

o NilU 1~65%

l

Nit

P’

u Nil

4.15.K

I

Ni-&o~O*lO*/*) 4.t5.K

o Nilll

14.1 lK

v N i-Fe (007%) 4:15 l K

. L

v Nil

20~1 ‘K

I

I

U

R. HUGUENIN

and D. RIYIER

4:15*K

Ni-Cu(O.14%) 4:lS’K

+ NiH 4.15.K I .^

10

I __

zu

I __ XJ

I 40

I

I

I

I

I

50

60

70

90

90

~_

B & * 10"fweber I ohm-meter’) FIG. 4. Kohler diagram of the relative Hall conductivity for the indicated samples and temperatures. The inset shows the data for small values of By0 on a larger scale. The peaks in y&o have been used to estimate the values of By0 at which UJ = 1.

ps, measured when the extend induction B is zero is not necessarily the proper one for checking the validity of Kohler’s rule in a ferromagnetic metal. But this remark in our case appears rather academic in view of the fact that the replacement of ps by some more “correct” value would lead, in the relative magnetoresistivity curves of Fig. 1 only to a change of scale for p/p0 and forB/po, which would be the same for a given sample but perhaps different from one of our samples to the other. This could not, however bring the curves of Fig. 1 together. Furthermore in the very unlikely case where this change of scale would be big enough to bring the maxima of the curves in Fig. 4 to the same “height”, i.e. to the same value of y&o, the coincidence of the curves would not be improved since s~ult~~usly the abscissae of the maxima, i.e. the value of By0 would be correspondingly changed, differently for each sample, which would

destroy the striking property of having maxima occur for the same value of b.

all these

te??lsoYs In galvanomagnetic transport phenomena, experiments give data on well defined resistivity tensor components, whereas theoretical modela lead to evaluation of conductivity tensor components. This well known fact entails some difliculties in analysing experimental results since each component of the former involves several components of the latter. In our case, one should note first that the transition from the resistivity tensor pu to the conductivity tensor yu reduces to the particular relations given in equation (2) only if the pls and pas components of the resistivity tensor are negligible, an assumption which is very plausible in a cubic polycrystal.

EXPERIMENTS

ON THE MAGNRTORESISTIVITY

Secondly, it is easy to see that if all the components of the resistivity (or conductivity) tensor obey Kohler’s rule then all the components of the inverse conductivity (or resistivity) tensor must also obey it. Concerning our Ni II and Ni III measurements, therefore, from the fact that, as can be seen in Fig. 1, the relative transverse magnetoresistivity is not a single function ofB/ps (i.e. does not follow Kohler’s rule) whereas the relative transverse magnetoconductivity does as shown in Fig. 3, it follows immediately that the Hall conductivity cannot obey Kohler’s rule as Fig. 4 shows.* As already mentioned in the introduction, the problem in checking Kohler’s rule for the magnetoresistivity in the ferromagnetic metals is complicated by the existence of the magnetization or spontaneous effects. They can also complicate the process of inverting the ordinary resistivity tensor to obtain the ordinary conductivity tensor, in so far as these quantities exist. It is now generally admitted that the Hall resistivity in ferromagnetic metals could be separated into the sum of two components one of which depends only on the magnetic induction B while the other depends only on the magnetization, M, i.e. PH = PEIOW +f%Isw

(4)

where pa” and px* are the ordinary and the spontaneous contributions to the Hall resistivity. This relationship has been verified many times in experiments where PO 9 PH

(5)

and cannot be seriously questioned. From this it follows that

AND HALL EFFECT

IN Ni AND Ni ALLOYS

259

where

In this case any uncertainty introduced in subtracting out the spontaneous part of the Hall resistivity in equation (4) is not transferred to the magnetoconductivity in inverting the resistivity tensor. Condition (5) holds for our three alloy samples, which explains why both the magnetoconductivity measurements in Fig. 3 and the magnetoresistivity measurements in Fig. 1 fall on a single curve. In the case of the purest samples, condition (5) is not satisfied and the separation of the Hall resistivity into two additive parts is no longer equivalent to an analogous separation of the Hall conductivity. However in this case we still have pns + pHo with

P’” Q PH(I. PJ.

PH0

(8)

Then, if we assume (see next section remark l), that equation (4) is still valid, we find that yn can be expressed as (dropping terms of second order in &Ypn’) YH = y=O+ ya""

where YEI0 = yi”

=

PHO P .Loa+ pIp= PliB P J_“=f pIso (

1

2pEloa P I”= + PH* )

4 YHO’

(9)

Remarks in tke light of existing microscopic theories 1. When condition (5) does not hold, it is at 1 present not possible to decide whether it is the Hall yI = p, Yn = Yn”+Yn8 (6) conductivities, the Hall resistivities or neither which is additive. There are very few experimental results for this case and there is not yet a consistent * It is perhaps interesting to note that the two tensor components which follow Kohler’s rule, pH/fo and y,_/ro, theory which treats the ordinary and spontaneous are connected to the two corresponding components, Hall effects simultaneously. Y&O and pL/po, that do not by the same quantity CL 2. The fact that the magnetoconductivities tend where to converge as the effective magnetic induction i.e. increases is not surprising in view of the fact that, YO’ Q - pla+pHa _ according to the theory of LIFSHITZ,AZBELand PO= ‘Yls+Y2 KAGANOV,(~~)the Hall conductivity becomes PH ___=,E. fiindependent of the relaxation time in the “high - a-.fJl PO YO ’ YO PO field limit” (WIT = b 9 1).

260

A. C. EHRLICH,

R. HUGUENIN

and D. RIVIER

This remark is based on the hypothesis that in work and for the grant that one of us (A.C.E.) has our samples there are a negligible number of open received for his stay at the Institut de Physique Experimentale de l’Universit6 de Lausanne. orbits. This hypothesis is very plausible for the following two reasons. First, the contact area of the Fermi surface with the Brillouin zone boundary is REFERENCES small in Ni and it occurs in only one or two of the 1. KOHLBRM., Ann. Phys, 35 211 (1938). 2. SMIT J., Physica 17, 612 (1951). sub-bands. Secondly, our measurements of the 3. LUTTINGIUt J. M., Pkys. Rev. 112,739 (1958). Hall resistivity in Ni III at liquid He temperatures 4. hIiIN Yv. P. and SHAVROV V. G., Soviet Phys. J. are in excellent agreement with those of FAWCETT Exp. theor. Phys. 15, 854 (1962). and REED(~) who have reported measurements on a 5. FAWCI~~T E. and RXEDW. A., Phys. Rev. L&t. 9,336 single crystal Ni sample where the direction of the (1962). See also RBEDW. A. and FAWIZIT E., J. appl. Phys. 35,754 (1964). magnetic field was taken so as to allow only closed 6. JOSJXPH A. S. and THORSJZN A. C. Phys. Rev. orbits. Lett. 11, 554 (1963). 3. Since the three alloys among themselves obey 7. PHILLIPSJ. C., Phys. Rev. 133, Al020 (1964). Kohler’s rule, it is reasonable to assume that the 8. For the Hall constants of these samples, see Huou~~IN R. and RIVIBRD., Helv. Phys. Acta 38, 900 relaxation times associated with the electron (1965). scattering induced by Co, Fe and Cu in Ni have 9. This same regularity in samples of Ni of decreasing the same anisotropy. purity has been noticed by S. N. MARCUSand Acknowledgements-We wish to thank Miss M. BLUNCK and Miss G. WENDTfor their help with the numerical calculations. We are grateful also to the Swiss Fonds National de la Recherche Scientifique for supporting this

D. N. LANGRNBE~G. [J. appl. Phys. 34,1367 (1963)] by comparing some of their own results with those of other workers. 10. LIFSHITZI. M., AZBBLM. I. and KAGANOV M. I., J. Exp. theor. Phys. 4, 41 (1957).