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Hall effect, magnetoresistance, resistivity and size effect in Ni
J. Phys. Chem. Solids
HALL
Pergamon Press 1968. Vol. 29, pp. 1293-1304.
EFFECT,
Printed in Great Britain.
MAGNETORESISTANCE, RESISTIVITY AND SIZE EFFECT IN Ni
A. C. EHRLICH* and D. RIVIER lnstitut de Physique Experimentale de l’llniversite de Lausanne, Lausanne, Switzerland (Received
19 June 1967; in revisedform
5 February
1968)
Abstract-The low temperature Hall effect, transverse magnetoresistance (with B both parallel and perpendicular to the surface of the plate sample) and electrical resistivity has been measured as a function of temperature and sample thickness on a high purity (residual resistivity ratio = 2200/l) polycrystalline Ni plate. The thickness was varied from 1%I5 to 0.041 mm by electropolishing and the measurements were made at 185, 4.15, 14.1 and 20,l”K. The electrical resistivity was studied in greater detail between 1~75-4~15”K and 14.1-20.1”K. Above magnetic saturation the Hall resistivity is linear in B at 185 and 4.15”K but is not linear at 14.1 and 20.1”K. At the lowest temperatures the electrical resistivity varies as T2. The only appreciable size effect appeared in the magnetoresistance at 1.85 and 4.15”K with B perpendicular to the plate. Four other samples of varying thickness were prepared by cold rolling. This mechanical work is shown to influence the galvanomagnetic properties. The results-are discussed. 1. INTRODUCTION
effect of the magnetoresistance and Hall effect should be one of the most direct ways of studying the mean free path of conduction electrons in metals. By size effect we mean the deviations from bulk behavior that occur when sample dimensions become comparable with the mean free path of the conduction electrons. Sondheimer [ 1] has reviewed the experimental and theoretical situation for this phenomena, and more recently, a great deal of experimental data has been reported [2]. Because of the apparent similarities between Ni and Cu (f.c.c. structure, open Fermi surface, comparable electron mobility) Berlincourt’s [3] study of the latter metal is of particular relevance to this paper. He has reported considerable size effect in samples which had a thickness as large as about ten times the mean free path. We have chosen to study the metal Ni for three reasons: (i) it is a ferromagnetic metal and these are not as well understood as the ordinary metals; (ii) it has been the most
THE SIZE
*Present address: U.S. Naval Research Washington, D.C. 20390, U.S.A.
Laboratory,
intensely investigated of the ferromagnetic metals; (iii) it has certain similarities to Cu whose size effect has already been studied. There are several methods of preparing specimens sufficiently thin to detect size effect. Deposited films provide the thinnest specimens but are not ideal because the results of galvanomagnetic measurements seem to depend on their method of production. The cold rolling of several samples to different thicknesses necessitates comparing results between samples that have undergone different amounts of mechanical treatment and thus requires distinguishing the effect of this treatment from true size effect. This distinction is often difficult to make. We have, therefore, made a series of measurements on a high purity ( p2y3/p4.15= 2200) polycrystalline Ni plate which was electropolished in steps from a thickness, t, of lXK&O~O41 mm. Galvanomagnetic measurements were made at these and several intermediate thicknesses at l-85, 4.15, 14.1 and 20el”K. For the sake of comparison we have also measured, at 4*15”K, a second samples which had p293/p4.15= 1200, t = 1.002 mm, and three other samples (samples 3-5) which were prepared by cold 1293
1294
A. C. EHRLICH
rolling to thicknesses 0.10 mm. 2. THEORETICAL (A)
of about O-45, 0.25 and CONSIDERATIONS
Size efect
Several theoretical treatments, all of which use the same basic idea, have been given for the size effect for different sample geometries. Sondheimer’s [4] treatment for the flat plate in a transverse magnetic field is most relevant to this paper since we have used samples of this shape. The fundamental idea is to take account of the electron scattering at the surface of the sample by taking the Fermi distribution function, f, as a function of the spatial coordinate z (the surfaces of the plate are parallel to the x-y plane) in addition to k, the wave vector of the electron states. Boundary conditions are then imposed on f at the surface of the plate corresponding to complete loss of drift velocity by the electrons when they are scattered by the plate surface. Results are also given for the more general case where a certain fraction, p, of the are specularly reflected, thus electrons retaining their drift velocity. These ideas have been extended to cases where p depends on the surface smoothness, and the angle of incidence of the reflected electrons [5]. Using a single spherical band-isotropic scattering time model, Sondheimer is able to derive an expression in closed form for the deviation for the Fermi surface from its equilibrium position. The essential parameters in this expression are k = a/l and p = a/r where a is the thickness of the sample, 1 the mean free path of the electrons and r their cyclotron radius. Except for certain limiting cases, the calculation of the Hall coefficient and magnetoresistivity must be done numerically. (Sondheimer actually calculates the magnetoresistivity and not the reciprocal of the magnetoconductivity as his notation might imply). The results of the numerical calculation are the following: (i) the magnetoresistance shows oscillations as a function of p for 1 < p < 10 but outside
and D. RIVIER
this range is very small. (In the absence of size effect his free electron model has no magnetoresistance); (ii) the Hall coefficient shows a smaller size effect than the resistivity and also has oscillations in the same p interval (see Zebouni et al. [6], who have carried out a more precise numerical calculation than appears in [4]. (B) Highjield
effects
For real metals whose Fermi surfaces are not, in general, spherical, a theory has been developed by Lifshitz, Azbel and Kaganov [7] for the galvanomagnetic effects in the high field limit, i.e. when o+- 9 1, where wC is the cyclotron frequency (proportional to B, the magnetic induction) and r a characteristic relaxation time. This theory was developed, strictly speaking, for nonmagnetic materials and predicts certain characteristic behavior for the various components of the resistivity tensor, in particular, for j+, the Hall resitivity and pI the transverse magnetoresistivity. These characteristics depend on whether or not there are open electron orbits in the particular metal for a given direction of the magnetic induction, B, and the number of hole states, &, and electron states, 12,. As will be seen in Section 5 this theory, for the case of closed orbits and n, # nh, plays an essential role in the discussion of our results. Starting from a Boltzman equation Lifshitz et al. calculate the deviation of the Fermi distribution from equilibrium in powers of l/B (rather than B as in the case of the low field limit). They use the series to calculate the lowest order non-vanishing term for each of the components in the conductivity tensor. For the case we are considering they find the following dependence of the conductivity tensor components on B: yI u B-2 WICCB” yij x B-‘, i # j
where yI is the transverse
magnetoconduc
(1
EFFECTS
IN Ni
1295
tivity, YIIthe longitudinal magnetoconductivity and yu the other components of the tensor. Furthermore, yH = yZ1, the Hall conductivity (for closed orbits y,,(B) = -y,,(B)) is the only component which does not depend on the scattering anisotropy but, for a given B, only on the Fermi surface geometry. The functional form, yI CQB+ does not depend on T either, but the absolute value of yI does. Inverting the tensor for the case we are discussing gives, for the Hall resistivity and transverse magnetoresistivity
to an equation of the form
where only lowest order terms in l/B have been kept.
Y&B,
(C)
Tensor inversion in a polycrystal
For a polycrystal without appreciable texture, y13 = yal = yZ3= y3* = 0 and the expressions for the inversion of the tensor are simply
PI
=
(3)
_&YL YL2+YH2
YH”
since for sufficiently high fields YH % yI. The right hand sides of equations (3), then, are valid when two conditions hold
pf, = ROB + pH*.
(9
This is consistent with the assumption that PH@>
M)
= PHO(B)+ p,$(M)
(6)
where pH” is the ‘ordinary’ part of the Hall resistivity equivalent to that found in nonmagnetic metals, pHs the magnetization contribution, and M the average sample magnetization. As Ehrlich, Huguenin and Rivier [8] have already pointed out, however, when it is not true that pH Q p. (p. is the resistivity in the absence of an applied field) equation (6) is not equivalent to the assumption
MI = YHO(B)i- YH’(W
(7)
i.e. that it is the Hall conductivities which are additive. A consistent theory which treats the ordinary and spontaneous effects simultaneously does not exist, and whether equation (6), equation (7) or neither is correct has not been decided at this time. Since the conductivities are calculated in the theories, we feel equation (7) is a more plausible assumption that equation (6) and in the light of the results presented below (Section 4, particularly Figs. 2 and 3) it is interesting to examine the consequences of this assumption. If it is assumed, in addition to equation (7) that the magnetoconductivities can also be separated into an ordinary and spontaneous component, i.e.
(4a) (4b)
Y~(B,
M) = rlO(B) + YLYMI
(8)
and equations (3), (7) and (8) are combined, while the validity of equation (1) requires only we have that (4a) holds. Since in the high field limit YH 3/H'+ YH’ does not depend on T, but the absolute value of PH = (yH”+yHs)2+(y ‘+y ’ yI does, there is no reason, a priori, that both equations (4a) and (4b) become valid approxiYLO + YIS mations at the same value of B. (9) PI = Ferromagnetic Hall data are usually fitted ~~H"+~~s~2+~~~o+~~s~2'
1296
4. C. EHRLICH
When YH” ----_l;YIf”+YLo
YIS Gl YH”+ Y1°
(10)
the denominators of equations (9) may be expanded and, dropping 2nd order terms in the spontaneous conductivities yields PH = &O(B) + PH”(B, M);
PI = Plow>
+ pLoS(B, Ml
(11)
where 0
PHO = yHozy;I yIoz; PLO = yHo2 YHS
pHOs=
3/ffoz+ Yloz
[
1-
Y”O +
yloz’
2YHo’
yff”* + YIOZI
-yo~Y$;~oz
(124 PI
YIS
OS =
Yf?
+ ‘yffoz [
1_
2YL0” 3/Ifo2+ YIOZI
_
YHS 3/fJo4 + YIW
(12b) Regardless of the validity of the specific assumptions made here, it is clear that the existence of the spontaneous effects may very well complicate the interpretation of high field galvanomagnetic measurements in ferromagnetic metals. Even if assumptions (7) and (8) are correct, it may not be easy to take account of yHs and YL5if they are appreciable because the approximation given in equation (10) will fail for sufficiently high fields, i.e. yHo and yLo fall with increasing B while, by hypothesis, yHs and yL8 remain constant. 3. EXPERIMENTAL
METHODS
All the samples used in this investigation were cut from the same plate,, nominally 1 mm thick, of Johnson Matthey spectroscopically pure Ni. Samples 1 and 2 were cut at right angles to each other. Samples 3-5 were prepared by cutting a specimen from the plate
and D. RIVIER
parallel to Sample 1, cold rolling to a thickness of 0.435 mm and removing a section which was machined into Sample 3. The rest of the specimen was then cold rolled to a thickness of 0.243 mm, a section removed for preparing Sample 4, and so on for Sample 5. Samples 2-5 were annealed at the same time at a temperature of 1000°C for 1 hr. Sample 1 was annealed in the same manner except that the temperature control was better. For both anneals the cooling rate did not exceed 3” per min. Data were taken at five temperatures. The samples were in direct contact with the baths used which were liquid He (1.85 and 4*5”K), liquid H, (14.1 and 20el”K) and a high thermal conductivity silicon oil (room temperature). Wires about 0.1 mm in dia. were drawn from material cut from the same plate as the samples. Four such wires were spot welded onto the samples for measuring the Hall effect and magnetoresistance; two on opposite edges for measuring the former and two on the same edge for the latter. The separation of the magnetoresistance contacts was roughly 1 cm except for Sample 1 t = O-072, and Sample 1 t = O+tl where the separation was approximately 7 and 4 mm, respectively. The Hall experimental setup was the standard one; i.e. the electric current direction, magnetic field direction, and a line connecting the two Hall contacts formed 3 mutually perpendicular axes. The sign of the measured potential was kept constant, in spite of the sign reversal for the Hall potential with magnetic field, by introducing a bias voltage into one Hall potential lead. This total potential was measured with the usual potentiometer-photocell amplifier-wall galvanometer system. The incremental method[9] was used to measure the Hall effect and transverse magnetoresistance for fields up to 3.25 w/m”. No contributions to the Hall electric field were detected which were not odd in the magnetic induction. Current densities used
EFFECTS
for the low temperature measurements ranged from approximately 0.8 A/mm2 for the thickest samples to 20 A/mm2 for the thinnest. Limited data were taken with currents smaller by a factor of five for the thickest and thinnest specimens and several intermediate thicknesses. Within experimental error the results were always identical. Sample geometry factors (thickness, width, distance between the resistivity contacts) needed to reduce the data to values of the resistivity and Hall resistivity were measured with a precision micrometer and moving table microscope on Samples 2-5 and, prior to making galvanomagnetic measurements or electropolishing, on Sample 1. For the latter, the ordinary Hall coefficient, Ro, and electrical resistivity, pO, were determined at room temperature using the measured dimensions as well as the galvanomagnetic measurements. Thereafter, as the thickness of Sample 1 was changed by electropolishing, the galvanomagnetic measurements at room temperature were used to determine the dimensions under the assumption that R, and p,, do not vary with thickness at room temperature. Some inhomogeneities occurred in Sample 1 as a result of electropolishing. The regions near the ends of the sample were attacked somewhat more rapidly than the rest of the surface. However, the edges maintained a good geometry throughout the electropolishing processes. Thus this did not affect the region of the Hall and magnetoresistance contacts which were placed centrally on the edges of the sample. The length of Sample 1 varied from 50 to 30 mm during the course of the experiment, while the width varied from 10 to 7 mm. Most of the reduction in length took place in reducing the thickness below 0.100 mm. The only other measureable inhomogeneity was a shallow depression running lengthwise along the middle of each surface of the sample. The depression was about 2 mm wide and deepest for the t = 041 sample, where it was approximately
IN Ni
1297
0.01 mm deep. With the method used here for calculating the effective thicknesses this inhomogeneity does not affect our conclusions regarding the size effect in Ni. 4. RESULTS
Figure 1 shows pH as a function of B for Sample 1, f = 0~190. The results for PH on the other thicknesses, to the scale used in this figure, appear essentially the same. At the I-Ie 2.5
: 0
-
.
-
0
2.0
+
-
20.1 *K 14.1 'K
:
4.15 *K
P
1.85 'K
9
X P l
9 2
1.5
4
d:
P
8 'I e
; 1.0
4 D
0.5
? P
t
0 0
2
1
3
B(weber/m*)
Fig. 1. Hall resistivity vs. magnetic induction at the indicated temperatures for Sample 1 at a thickness of 0.190 mm.
temperatures
PH has a linear dependence on about 1.0 and 3.25 w/m2. The hydrogen data is slightly ‘S’ shaped up to 2.5 w/m2. It is concave upward at low fields, and then turns concave downward at about 1.2 w/m”. Both the 14.1 and 20el”K data are linear between 2.5 and 3.25 wlm2. These details are clearly seen in Fig. 2 where a much larger scale has been achieved by presenting (PH-0’7 lo-‘OB) as a function of B.
B between
1298
A. C. EHRLICH
and D. RIVIER
0.6
%':
0.4
B F: P s I h
0.2
: :
I
0 0
1
2
3
Fig. 2. Hall resistivity minus 0.7. 10-‘“B vs. magnetic induction at the indicated temperatures for Sample 1 at a thickness of 0.190 mm.
On this scale a vertical displacement equal to the dia. of one of the data point circles corresponds to a 0.2 per cent change in oH at 3.25 w/m2. The results for I = 0.041 mm are presented in Fig. 3. Some difference in the shape of the dependence of pH on B but not in its magnitude can be distinguished between Figs. 2 and 3. Figure 4 shows the Hall resis-
1
2
1
2
3
B (weber/m*)
6 (weber/m*)
0
0
’
3
B (weberlm')
Fig. 3. Hall resistivity minus 0.7. 10-u’ vs. magnetic induction at the indicated temperatures for Sample 1 at a thickness of 0.041 mm.
Fig. 4. Hall resistivity minus 0.7. 10-*“B vs. magnetic induction for Samples 2-5 at 4.15”K. The thicknesses of the three thinnest Samples (3-5) were obtained by cold rolling.
tivities for Samples 2-5, all of which were measured at 4.15”K. For Samples 3-5 pH is not a linear function of B. The Hall effect and resistivity results for Sample 1 at all thicknesses and Sample 2 are presented in Table 1. The slope and the intercept have been deduced from a least square fit of the data to equation (5) for the intervals of B where pH is a linear function of B, i.e. 1.0-3.25 w/m2 for the He temperature results, and 2.5-3.25 w/m2 for the H, results. The Sample 1, r = O-041 (deformed) results were obtained on the t = O-041 mm specimen after it had been bent around a cylindrical bar 3 mm in dia. and straightened. The Hall coefficients, Ro, have an absolute error no greater than 2 per cent at He temperatures and 4 per cent at H, temperatures. We estimate the uncertainty in the intercepts as 0.05 X lo-” and 0.25 X lo-” mw/A-s for the He and Hz data respectively. The electrical resistivity measurements have an absolute error of approximately 2-5 per cent with the larger figure applying to the He temperature results on the thickest samples. The transverse magnetoresistivity is the error, for all same, within experimental
EFFECTS
1299
IN Ni
Table 1. The slope (R,,) and intercept (yS) of the linear portion of the Hall resistivity as a function of magnetic induction, and the electrical resistivity (p) at the indicated thicknesses and temperatures Intercept (IO-“mw/A-s)
Resistivity (p$) ( lO-1’%m)
Thickness (mm)
Temperature W)
1.005
1.85 4.15 14.10 20.10
0.700 0.699 0.708 0.781
1.05 1.15 1.89 0.33
3.32 3.79 8.14 13.12
I
0.571
1.85 4.15
0.706 0.704
0.79 0.87
3.27 3.68
1
0.401
1.85 4.15 14.10 20.10
0.689 0.689 0.710 0.775
0.60 0.69 1.09 -0.27
3.11 3.65 7.93 12.99
1
0.190
1.85 4.15 14.10 20.10
0.699 0.700 0.722 0.788
044 0.48 090 -0.52
3.36 3.75 8.03 13.11
1
0.100
1.85 4.15 14.10 20.10
0.692 0.694 0.724 0.795
0.43 0.52 0.88 -0.72
3.90 4.27 9.15 14.38
1
0.072
1.85 4.15 14.10 20.10
1
o*O41
1.85 4.15 14.10 20.10
0.692 0.702 0.736 0.801
0.53 0.58 0.71 -0.89
4.06 4.52 9.06 14.23
1 (deformed)
0.041
1.85 4.15
0.710
0.54
5.00 5.48
2
1.002
1.85 4.15
0.712
1.59
5.70 6.04
Sample 1
SlopeCR,)
(1O-?TP/A-s)
thicknesses of Sample 1 equal to or greater than t = 0.190 mm. The results are given for B perpendicular and B parallel to the plate in Figs. 5 and 6 respectively. In the latter, the abscissa is not the magnetic induction in the sample, but rather the applied induction. Fig. 7 shows the results for the smaller thicknesses of Sample 1 (B perpendicular to the plate) at 1 35°K. The results at 4.15”K are
3.99 4.38 9.19 14.63
essentially the same. H, temperature measurements and the He measurements with B parallel to the plate do not differ from those shown in Figs. 5 and 6. The magnetoresistance for Samples 2-5 at 4+15”K is shown in Fig. 8. The temperature dependence of the resistivity of Sample 1, t = 0.190 is shown in Fig. 9 where (p-p(O))/T vs. T is given. p(O) is the
A. C. EHRLICH
1300
and D. RIVIER
-I
0.8
+
-
0.041 mm(dcformrd)
0
-
0.041 mm
14.1 ‘K
.
-
0.072 mnl
‘.15*K
A
-
0.100 mm
a
_
20,l’K
0
-
.
-
o
-
1.85.K
.
bulk
0 t. t
mm
6
0
0.6 0
Q75
A
&
. A 0
61
.
^o Q_
v
0
0.4
0
* ,e
^g
0
.
k -
s
0.50
: 0
0.25 0.2
2
1
0
3
B (weberlm2) 0
2
1
3
B (weber/m2)
Fig. 5. Transverse magnetoresistivity (with B perpendicular to the plate) vs. magnetic induction at the indicated temperatures for Sample 1 at all thicknesses greater than or equal to 0.190 mm.
08
o
-
20.1 ‘K
A
-
14.1
‘K
.
-
4.15
‘K
v
-
1.85 ‘K
0.6
Fig. 7. Transverse magnetoresistivity (with B perpendicular to the plate) vs. magnetic induction at l+W’K for the indicated thicknesses of Sample 1,
residual resistivity and has been obtained by extrapolating to 0°K the data between T = 1.75”K and T = 4*15”K. The extrapolation was straightforward since p cc T* in this temperature range. Also shown in Fig. 9 is the He data extrapolated up to H, temperatures as well as the H, data extrapolated down to He temperatures. The results on the thinner samples are essentially the same as the data shown. 5. DISCUSSION OF RESULTS (A) Hall effect and magnetoresistance
0
0
1
2
3
B (wcber/m21
Fig. 6. Transverse magnetoresistivity (with /3 parallel to the plate) vs. applied magnetic induction at the indicated temperatures for Sample I at all thicknesses.
The results for Sample 1 for R, at He temperatures are in excellent agreement with the high field measurements reported by Fawcett and Reed[lO] on a single crystal for a magnetic field direction chosen to give only closed orbits. This is not surprising in view of the fact that in Ni, the contact area of the Fermi surface within the Brillouin zone boundary is small and thus open orbits are found in only a small fraction of magnetic
EFFECTS
IN Ni
1301
‘6
12: 0
0
,-
LOC
v
-
0.100
mm
o
-
0.243
mm
*
-
0.435
mm
+
-
1002
mm
0
.
0
. .
0 A
g
0 0
0
0.75
2 d. Q5C
,-
Q2!
>_ -
2
,’
,/’ _/ ,,,_ 0
,/’
9 0
.
1
2
3
3 0
B (weberlm2)
Fig. 8. Transverse magnetoresistivity (B perpendicular to the plate) vs. magnetic induction at 4.1YK for Samples 2-5. The thicknesses of the three thinnest Samples (3-5) were obtained by cold rolling.
field directions. Therefore only a negligible number of the crystallites in our polycrystal are so oriented as to give rise to open orbits. It seems certain that had we been able to extend our measurements to 8 w/m2, as did Fawcett and Reed, we would have also found the same results as they for this range of magnetic field. What is remarkable in our data is that pH is a linear function of B at fields at low as 1.0 w/m”. Ifere, though condition (4a) may or may not be well satisfied, condition (4b) certainly is not. In fact, between I.0 and 3.25 w/m* (rH/yl)” goes from about 3-16 and 4-20 for 4.15 and 185”K, respectively. Put otherwise, l/y, is not proportional to B, so it is difficult to understand why oH is. Sample 2 has a resistivity 1.6 times that of Sample 1 and thus its linearity is even more difficult to understand. The same general remarks apply to Fawcett and Reed’s measure-
1
2
3
I
I
T(‘K)
Fig. 9. Temperature dependence of electrical resistivity for Sample 1 at a thickness of 0.190 mm. The lower dashed line is the 14-20°K data extrapolated to the O-4°K temperature range and the upper dashed line the 1.84.2”K data extrapolated to the 14-20°K temperature range. The circles and triangles indicate two separate experiments for each temperature range. The arrows show the appropriate scale.
ments, whose pH for their highest and lowest field points seem to fall on the same straight line. Table 1 shows a trend of pHs towards decreasing values for Sample 1 as the thickness is reduced from l-005 to O-190 mm. This is not a true size effect, but can be quantitatively accounted for by our having used the ‘infinite flat plate approximation’ in calculating R, and pH8; i.e. we have assumed that the magnetic induction inside the sample is equal to the applied magnetic induction. For the initial dimensions of Sample 1, 50 X 10 X 1 mm, this is not exact. A Ni ellipsoid of these dimensions would have a B(inside) = B(applied)+ 0.06 w/m” for an applied field sufficiently large to saturate the sample [ 1I].
A. C. EHRLICH
1302
When t = O-2 mm (the other dimensions remaining approximately constant), the approximation is better than the experimental error and no further change in pH8 should occur. This is consistent with the data presented in Table 1. There is, of course, no effect on the calculated value of &. A striking feature of the dependence of pH on B is the appearance of ‘knees’ in the curves at about O-6 w/m2, the saturation magnetization of Ni, for all temperatures measured (see Fig. 2). This is typical of a spontaneous magnetization contribution to the Hall effect. In the case of the H, data, the ‘S’ shaped results in Fig. 2 can be explained semi-quantitatively by combining just such a spontaneous contribution with equation (12a). In [8] it was shown that -yI obeys Kohler’s rule while yH does not (‘Ni III’ of [8] is the same as Sample 1 of this paper). It is consistent with this result to suppose that yIs = 0 or, at least, that y18/yH8 + 1. Then, dropping the terms in equations (12) containing yL8 gives PH
=
YHO + YH@+ Ylon
E
Y”O +
YHo2+ YIOZ p1
=
YLO YZ+
YIOP
1_
2YH0’ ‘yffw+ Y1091
(134
YHsm(B)
?I0 YHon+ YIW
C
YHS yHo2 + yloz [
YHS yz+
.
2Yl”rJf”
yLoz y2- + y102
+YHSS(B).
(13b)
m(B) and s(B) could be evaluated by substituting yH and yL for yHoand yIo since the error introduced this way is 2nd order in yH8. In view of the approximations already made, however, a semi-quantitative discussion of equations (13) is sufficient. For the value of B, Bo, such that yHo = yIo, m(B) = 0. (B,= 1.43 w/m* at 14ml”K and 2.72 w/m* at 20al”K in Sample 1.) When B < B. (B > B,), m(B) > 1 (m(B) < 1). Because it is sharply concave upward at about 0.6 w/m* the H2 temperature data
and D. RIVIER
shown in Fig. 2 appear to have a y$ which is negative, i.e. the opposite sign of yHo. Thus the 2nd term in equation (13a) makes a negative contribution to pH in the range B > B, and a positive contribution when B > Bo. This is just what is needed to explain the S shape of the curves of Fig. 2. Furthermore, a negative yHs subtracted out of the curves in Fig. 4 of [8] would bring them closer to the He curves on the same graph, and thus into better agreement with Kohler’s rule. It is interesting to note also that yHss(B) should be linear in B in the high field range. The He temperature magnetoresistivities given in Fig. 5 have the same form as that of Fawcett and Reed[lO] in the same interval of magnetic induction except that they are slightly larger. This is not surprising in view of the fact that the absolute value of the magnetoresistance depends on the details of the scattering process in the high field range, while its functional dependence does not. (B) Size effect
A crude estimate of the mean free path, 1, of the conduction electrons can be obtained from the resistivity by using a free electron model. Assuming 1 conduction electron per atom, at l%“K Sample 1 has an 1 = 0.02 mm. This is large enough to expect size effect in some of the thinner specimens in Table 1. Since size effect in the resistivity is practically absent in the three thinnest samples, however, the considerable change in resistivity between t = 0.190 and t = 0.100 must be attributed to damage done to the sample in handling. Examination of the data shows no detectable corresponding effect in the Hall resistivity or the magnetoresistance. In the high field limit Sondheimer’s spherical band theory predicts no size effect in the Hall effect. (Note that when 1 = a; i.e. when there is size effect, a/r S- 1 is a necessary condition for having the high field limit.) This will hold also for an arbitrarily shaped Fermi surface provided there are a negligible number of open orbits because size effect
EFFECTS
arises from scattering of the electrons at the surface of the sample and the scattering mechanism has no influence on the Hall effect in the high field limit. This is in agreement with our Sample 1 Hall effect results which show practically no size effect (compare Figs. 2 and 3). In the high field limit the transverse magnetoresistance does depend on the relaxation time and can be expected to show size effect when B is perpendicular to the sample plate. For B parallel to the plate, size effect is suppressed since the electrons will be prevented from making contact with the sample surfaces. Both of these points are supported by the data (see Figs. 5-7). Sondheimer’s model predicts a size effect in the resistivity considerably larger than we have found. An explanation for this may be found in the ferromagnetic character of Ni. Generally, in ferromagnetic polycrystalline materials the resistivity in the absence of an applied field is approximately [ 121
1303
IN Ni
our electropolished Sample 1, these effects are seen to be a result of the cold rolling rather than a true size effect. Cold rolling produces preferentially oriented crystallites in the polycrystal. Metals with open Fermi surfaces have highly anisotropic galvanomagnetic effects, and different amounts of preferential orientation in two otherwise identical specimens can be expected to produce appreciable differences in their galvanomagnetic properties. (C) Resistivity The results of the resistivity vs. temperature measurements can be expressed by the equation p(T) --p(O) = aT+ bT2
(15)
where a = 0;
b = 2.7, x 10-59; 1.75”K < T c 4.15”K
(15a)
a = -4.6 x 10-Sa+; where pl and pll are the resistivities in the absence of an applied field when the magnetization is perpendicular and parallel, respectively, to the current direction. That is, even in demagnetized Ni the electrons are moving through magnetic domains with a magnetic field intensity, M, of about O-6 w/m”. Thus the equivalent of 2/3 of the electrons experience a Lorentz force tending to prevent them from coming into contact with the surfaces of the sample. The results shown in Figs. 4 and 8 of the Hall effect and magnetoresistance, respectively, for the Ni samples prepared by cold rolling are very similar to Berlincourt’s[3] results on Cu samples prepared in the same manner; i.e. the Hall voltage is not linear in B and shows a systematic deviation with sample thickness while the magnetoresistance of the samples differs from each other but not in a systematic way. In the light of the results on
b = 2.6, x lo-“e; 14eO”K c T s 20.1”K.
(15b)
p (0) has been determined by assuming equations (15) and (15a) valid down to T = 0°K. Magnon scattering has been shown to give a quadratic dependence of the resistivity on temperature[ 131 and this is the most straightforward explanation of the observed T2 behavior. It should be noted, however, that s-d electron-electron scattering can also give rise to a T2 term in the resistivity as has been shown by Baber[l4] and more recently discussed by Appel[ 151. There are two problems raised by the resistivity data. The first is that previous low temperature measurements in Ni of p vs. T by Kondorski, Galkina and Tchernikova[ 161 and White and Woods[ 171 have also given a quadratic dependence but with b = 4-O X low5
1304
A. C. EHRLICH
for the former and 1*6X 10m5@-cm/?? for the latter. These results show no correlation with the purity of the specimen, since the residual resistivity of the specimen in [16] is about ten times that in [ 173, which in turn is about ten times that reported here. The differences are much too large to be attributed to combined experimental error or to ordinary divergences from Matthiesen’s ruleflS]. Second, the coefficients in (15b) represent an empirical fit to the data between 14.1 and 2O.l”K and are probably not valid very far beyond this range. It is gratifying that ‘b’ in (15a) and (1 Sb) are almost identical, but somewhat anomalous that the temperature dependence of the resistivity is weaker in the hydrogen temperature range than at He temperatures. (For continuity of the resistivity it must be weaker still somewhere between 4.15 and 14.1”K). While these two probtems are not easily explained, it should be noted that the temperature dependence of a ferromagnetic metal is related to its magnetoresistance, and if the latter is anomalous, the former may be expected to be also. (The magnetoresistance of Ni as a function of temperature is anomalous in its deviation from Kohler’s rule [S, 191). The temperature dependence of the resistivity of a non-magnetic metal in a constant magnetic field will be, in general, different than in the absence of a magnetic field. In a ferromagnetic metal the existence of the domain magnetization means that even without an applied field, the resistivity measured is a sort of magnetoresistance (see equation (14)). As Blatt[20] has pointed out, the magnetoresistance is more sensitive to the electronic relaxation time, T, than is the resistivity since the latter depends on an average of 7 over the Fermi surface, while the former involves power and derivatives of 7.
and D. RIVIER
It is to be expected, then, that the resistivity will be more complicated in ferromagnetic than ordinary metals. Acknowled~e~enis-We wish to thank the members of the Institut de Physique Experimentale de I’Universite de Lausanne for their most kind cooperation during the course of this work. We are particularly grateful to Dr. R. Huguenin for many stimulating discussions and useful suggestions, We are also indebted to Miss G. Wendt for her help with the numerical calculations. Finally, we wish to thank the Swiss Fonds National de la Recherche Scientifique for supporting this work and for the grant that one of us (A. C. Ehrlich) has received for his stay at this laboratory. REFERENCES E. H.,Adv. Phys. 1, I(1952). 1. SONDHEIMER B. N., JEEP 2. See, for example, ALEKSANDROV 16, 871 (1963); ibid., p. 286; COOPER J. N., COTTI P. and RASMUSSEN F. B., Phys. Leti. 19, 560 (1965): FORSVOLL K, and HOLWECH I., Phil. Mug. 9, 435 (1964); ibid., 10, 1961 (1964); HOLWECH I., Phil. Mug. 12,117 (1965). T. G., Phys. Rev. 112,381 (1958). 3. BERLINCOURT 4. SONDHEIMER E. H..Phvs. Rev. 80.401(1950). 5. For references, see SOFFER S. B.,J. appi.‘Phys., In press. N. H., HAMBURG R. E. and 6. ZEBOUNl MACKEY H. J.,Phys. Rev. Lett. 11,260 (1963). 7. LIFSHITZ 1. M., AZBEL M. 1. and KAGANOV M. l.,J. exp. them. Phys.4,41 (1957). R. and RIVIER 8. EHRLICH A. C., HUGUENIN D., J. Phys. Chem. Solids 28,253 (I 967). 9. FONER S.. Phvs. Rev. 101, 1648 ( 1956). IO. FAWCET’? E: and REED W. A., Phys. Rev. 131, 2463 (I 963). II. OSBORN J. A., Phys. Rev. 67,35 1 (1945). 12. VAN ELST H. C., Physic-a 25.708 (1959). 13. For references, see GOODfNGS D. A., Phys. Rev. 132,542 (1963). AI58 383 (1937). 14. BABER W. G.,Proc.R.Soc. 15. APPEL J., Phil. Mag.8, 1071 (1963). 0. S. and E. I., GALKINA 16. KONDORSKI TCHERNIKOVA L. A., J. appL Phys. 29, 243 (1958). 17. WHITE G. K. and WOODS S. B., Phil. Trans. R. Sot. A251.273 (1959). 18. WILSON A. E., The Theory of Metals. p. 310. Cambridge University Press, Cambridge (1954). 19. See also MARCUS S. N. and LANGENBERG D. N.,J.uppf.Phys.34, I367(1963). 20. BLATT F. J., Solid State Physics (Edited by F. Seitz and D. Turnbull) Vol. 4, p. 199. Academic Press, New York (I 957).
HALL
Pergamon Press 1968. Vol. 29, pp. 1293-1304.
EFFECT,
Printed in Great Britain.
MAGNETORESISTANCE, RESISTIVITY AND SIZE EFFECT IN Ni
A. C. EHRLICH* and D. RIVIER lnstitut de Physique Experimentale de l’llniversite de Lausanne, Lausanne, Switzerland (Received
19 June 1967; in revisedform
5 February
1968)
Abstract-The low temperature Hall effect, transverse magnetoresistance (with B both parallel and perpendicular to the surface of the plate sample) and electrical resistivity has been measured as a function of temperature and sample thickness on a high purity (residual resistivity ratio = 2200/l) polycrystalline Ni plate. The thickness was varied from 1%I5 to 0.041 mm by electropolishing and the measurements were made at 185, 4.15, 14.1 and 20,l”K. The electrical resistivity was studied in greater detail between 1~75-4~15”K and 14.1-20.1”K. Above magnetic saturation the Hall resistivity is linear in B at 185 and 4.15”K but is not linear at 14.1 and 20.1”K. At the lowest temperatures the electrical resistivity varies as T2. The only appreciable size effect appeared in the magnetoresistance at 1.85 and 4.15”K with B perpendicular to the plate. Four other samples of varying thickness were prepared by cold rolling. This mechanical work is shown to influence the galvanomagnetic properties. The results-are discussed. 1. INTRODUCTION
effect of the magnetoresistance and Hall effect should be one of the most direct ways of studying the mean free path of conduction electrons in metals. By size effect we mean the deviations from bulk behavior that occur when sample dimensions become comparable with the mean free path of the conduction electrons. Sondheimer [ 1] has reviewed the experimental and theoretical situation for this phenomena, and more recently, a great deal of experimental data has been reported [2]. Because of the apparent similarities between Ni and Cu (f.c.c. structure, open Fermi surface, comparable electron mobility) Berlincourt’s [3] study of the latter metal is of particular relevance to this paper. He has reported considerable size effect in samples which had a thickness as large as about ten times the mean free path. We have chosen to study the metal Ni for three reasons: (i) it is a ferromagnetic metal and these are not as well understood as the ordinary metals; (ii) it has been the most
THE SIZE
*Present address: U.S. Naval Research Washington, D.C. 20390, U.S.A.
Laboratory,
intensely investigated of the ferromagnetic metals; (iii) it has certain similarities to Cu whose size effect has already been studied. There are several methods of preparing specimens sufficiently thin to detect size effect. Deposited films provide the thinnest specimens but are not ideal because the results of galvanomagnetic measurements seem to depend on their method of production. The cold rolling of several samples to different thicknesses necessitates comparing results between samples that have undergone different amounts of mechanical treatment and thus requires distinguishing the effect of this treatment from true size effect. This distinction is often difficult to make. We have, therefore, made a series of measurements on a high purity ( p2y3/p4.15= 2200) polycrystalline Ni plate which was electropolished in steps from a thickness, t, of lXK&O~O41 mm. Galvanomagnetic measurements were made at these and several intermediate thicknesses at l-85, 4.15, 14.1 and 20el”K. For the sake of comparison we have also measured, at 4*15”K, a second samples which had p293/p4.15= 1200, t = 1.002 mm, and three other samples (samples 3-5) which were prepared by cold 1293
1294
A. C. EHRLICH
rolling to thicknesses 0.10 mm. 2. THEORETICAL (A)
of about O-45, 0.25 and CONSIDERATIONS
Size efect
Several theoretical treatments, all of which use the same basic idea, have been given for the size effect for different sample geometries. Sondheimer’s [4] treatment for the flat plate in a transverse magnetic field is most relevant to this paper since we have used samples of this shape. The fundamental idea is to take account of the electron scattering at the surface of the sample by taking the Fermi distribution function, f, as a function of the spatial coordinate z (the surfaces of the plate are parallel to the x-y plane) in addition to k, the wave vector of the electron states. Boundary conditions are then imposed on f at the surface of the plate corresponding to complete loss of drift velocity by the electrons when they are scattered by the plate surface. Results are also given for the more general case where a certain fraction, p, of the are specularly reflected, thus electrons retaining their drift velocity. These ideas have been extended to cases where p depends on the surface smoothness, and the angle of incidence of the reflected electrons [5]. Using a single spherical band-isotropic scattering time model, Sondheimer is able to derive an expression in closed form for the deviation for the Fermi surface from its equilibrium position. The essential parameters in this expression are k = a/l and p = a/r where a is the thickness of the sample, 1 the mean free path of the electrons and r their cyclotron radius. Except for certain limiting cases, the calculation of the Hall coefficient and magnetoresistivity must be done numerically. (Sondheimer actually calculates the magnetoresistivity and not the reciprocal of the magnetoconductivity as his notation might imply). The results of the numerical calculation are the following: (i) the magnetoresistance shows oscillations as a function of p for 1 < p < 10 but outside
and D. RIVIER
this range is very small. (In the absence of size effect his free electron model has no magnetoresistance); (ii) the Hall coefficient shows a smaller size effect than the resistivity and also has oscillations in the same p interval (see Zebouni et al. [6], who have carried out a more precise numerical calculation than appears in [4]. (B) Highjield
effects
For real metals whose Fermi surfaces are not, in general, spherical, a theory has been developed by Lifshitz, Azbel and Kaganov [7] for the galvanomagnetic effects in the high field limit, i.e. when o+- 9 1, where wC is the cyclotron frequency (proportional to B, the magnetic induction) and r a characteristic relaxation time. This theory was developed, strictly speaking, for nonmagnetic materials and predicts certain characteristic behavior for the various components of the resistivity tensor, in particular, for j+, the Hall resitivity and pI the transverse magnetoresistivity. These characteristics depend on whether or not there are open electron orbits in the particular metal for a given direction of the magnetic induction, B, and the number of hole states, &, and electron states, 12,. As will be seen in Section 5 this theory, for the case of closed orbits and n, # nh, plays an essential role in the discussion of our results. Starting from a Boltzman equation Lifshitz et al. calculate the deviation of the Fermi distribution from equilibrium in powers of l/B (rather than B as in the case of the low field limit). They use the series to calculate the lowest order non-vanishing term for each of the components in the conductivity tensor. For the case we are considering they find the following dependence of the conductivity tensor components on B: yI u B-2 WICCB” yij x B-‘, i # j
where yI is the transverse
magnetoconduc
(1
EFFECTS
IN Ni
1295
tivity, YIIthe longitudinal magnetoconductivity and yu the other components of the tensor. Furthermore, yH = yZ1, the Hall conductivity (for closed orbits y,,(B) = -y,,(B)) is the only component which does not depend on the scattering anisotropy but, for a given B, only on the Fermi surface geometry. The functional form, yI CQB+ does not depend on T either, but the absolute value of yI does. Inverting the tensor for the case we are discussing gives, for the Hall resistivity and transverse magnetoresistivity
to an equation of the form
where only lowest order terms in l/B have been kept.
Y&B,
(C)
Tensor inversion in a polycrystal
For a polycrystal without appreciable texture, y13 = yal = yZ3= y3* = 0 and the expressions for the inversion of the tensor are simply
PI
=
(3)
_&YL YL2+YH2
YH”
since for sufficiently high fields YH % yI. The right hand sides of equations (3), then, are valid when two conditions hold
pf, = ROB + pH*.
(9
This is consistent with the assumption that PH@>
M)
= PHO(B)+ p,$(M)
(6)
where pH” is the ‘ordinary’ part of the Hall resistivity equivalent to that found in nonmagnetic metals, pHs the magnetization contribution, and M the average sample magnetization. As Ehrlich, Huguenin and Rivier [8] have already pointed out, however, when it is not true that pH Q p. (p. is the resistivity in the absence of an applied field) equation (6) is not equivalent to the assumption
MI = YHO(B)i- YH’(W
(7)
i.e. that it is the Hall conductivities which are additive. A consistent theory which treats the ordinary and spontaneous effects simultaneously does not exist, and whether equation (6), equation (7) or neither is correct has not been decided at this time. Since the conductivities are calculated in the theories, we feel equation (7) is a more plausible assumption that equation (6) and in the light of the results presented below (Section 4, particularly Figs. 2 and 3) it is interesting to examine the consequences of this assumption. If it is assumed, in addition to equation (7) that the magnetoconductivities can also be separated into an ordinary and spontaneous component, i.e.
(4a) (4b)
Y~(B,
M) = rlO(B) + YLYMI
(8)
and equations (3), (7) and (8) are combined, while the validity of equation (1) requires only we have that (4a) holds. Since in the high field limit YH 3/H'+ YH’ does not depend on T, but the absolute value of PH = (yH”+yHs)2+(y ‘+y ’ yI does, there is no reason, a priori, that both equations (4a) and (4b) become valid approxiYLO + YIS mations at the same value of B. (9) PI = Ferromagnetic Hall data are usually fitted ~~H"+~~s~2+~~~o+~~s~2'
1296
4. C. EHRLICH
When YH” ----_l;YIf”+YLo
YIS Gl YH”+ Y1°
(10)
the denominators of equations (9) may be expanded and, dropping 2nd order terms in the spontaneous conductivities yields PH = &O(B) + PH”(B, M);
PI = Plow>
+ pLoS(B, Ml
(11)
where 0
PHO = yHozy;I yIoz; PLO = yHo2 YHS
pHOs=
3/ffoz+ Yloz
[
1-
Y”O +
yloz’
2YHo’
yff”* + YIOZI
-yo~Y$;~oz
(124 PI
YIS
OS =
Yf?
+ ‘yffoz [
1_
2YL0” 3/Ifo2+ YIOZI
_
YHS 3/fJo4 + YIW
(12b) Regardless of the validity of the specific assumptions made here, it is clear that the existence of the spontaneous effects may very well complicate the interpretation of high field galvanomagnetic measurements in ferromagnetic metals. Even if assumptions (7) and (8) are correct, it may not be easy to take account of yHs and YL5if they are appreciable because the approximation given in equation (10) will fail for sufficiently high fields, i.e. yHo and yLo fall with increasing B while, by hypothesis, yHs and yL8 remain constant. 3. EXPERIMENTAL
METHODS
All the samples used in this investigation were cut from the same plate,, nominally 1 mm thick, of Johnson Matthey spectroscopically pure Ni. Samples 1 and 2 were cut at right angles to each other. Samples 3-5 were prepared by cutting a specimen from the plate
and D. RIVIER
parallel to Sample 1, cold rolling to a thickness of 0.435 mm and removing a section which was machined into Sample 3. The rest of the specimen was then cold rolled to a thickness of 0.243 mm, a section removed for preparing Sample 4, and so on for Sample 5. Samples 2-5 were annealed at the same time at a temperature of 1000°C for 1 hr. Sample 1 was annealed in the same manner except that the temperature control was better. For both anneals the cooling rate did not exceed 3” per min. Data were taken at five temperatures. The samples were in direct contact with the baths used which were liquid He (1.85 and 4*5”K), liquid H, (14.1 and 20el”K) and a high thermal conductivity silicon oil (room temperature). Wires about 0.1 mm in dia. were drawn from material cut from the same plate as the samples. Four such wires were spot welded onto the samples for measuring the Hall effect and magnetoresistance; two on opposite edges for measuring the former and two on the same edge for the latter. The separation of the magnetoresistance contacts was roughly 1 cm except for Sample 1 t = O-072, and Sample 1 t = O+tl where the separation was approximately 7 and 4 mm, respectively. The Hall experimental setup was the standard one; i.e. the electric current direction, magnetic field direction, and a line connecting the two Hall contacts formed 3 mutually perpendicular axes. The sign of the measured potential was kept constant, in spite of the sign reversal for the Hall potential with magnetic field, by introducing a bias voltage into one Hall potential lead. This total potential was measured with the usual potentiometer-photocell amplifier-wall galvanometer system. The incremental method[9] was used to measure the Hall effect and transverse magnetoresistance for fields up to 3.25 w/m”. No contributions to the Hall electric field were detected which were not odd in the magnetic induction. Current densities used
EFFECTS
for the low temperature measurements ranged from approximately 0.8 A/mm2 for the thickest samples to 20 A/mm2 for the thinnest. Limited data were taken with currents smaller by a factor of five for the thickest and thinnest specimens and several intermediate thicknesses. Within experimental error the results were always identical. Sample geometry factors (thickness, width, distance between the resistivity contacts) needed to reduce the data to values of the resistivity and Hall resistivity were measured with a precision micrometer and moving table microscope on Samples 2-5 and, prior to making galvanomagnetic measurements or electropolishing, on Sample 1. For the latter, the ordinary Hall coefficient, Ro, and electrical resistivity, pO, were determined at room temperature using the measured dimensions as well as the galvanomagnetic measurements. Thereafter, as the thickness of Sample 1 was changed by electropolishing, the galvanomagnetic measurements at room temperature were used to determine the dimensions under the assumption that R, and p,, do not vary with thickness at room temperature. Some inhomogeneities occurred in Sample 1 as a result of electropolishing. The regions near the ends of the sample were attacked somewhat more rapidly than the rest of the surface. However, the edges maintained a good geometry throughout the electropolishing processes. Thus this did not affect the region of the Hall and magnetoresistance contacts which were placed centrally on the edges of the sample. The length of Sample 1 varied from 50 to 30 mm during the course of the experiment, while the width varied from 10 to 7 mm. Most of the reduction in length took place in reducing the thickness below 0.100 mm. The only other measureable inhomogeneity was a shallow depression running lengthwise along the middle of each surface of the sample. The depression was about 2 mm wide and deepest for the t = 041 sample, where it was approximately
IN Ni
1297
0.01 mm deep. With the method used here for calculating the effective thicknesses this inhomogeneity does not affect our conclusions regarding the size effect in Ni. 4. RESULTS
Figure 1 shows pH as a function of B for Sample 1, f = 0~190. The results for PH on the other thicknesses, to the scale used in this figure, appear essentially the same. At the I-Ie 2.5
: 0
-
.
-
0
2.0
+
-
20.1 *K 14.1 'K
:
4.15 *K
P
1.85 'K
9
X P l
9 2
1.5
4
d:
P
8 'I e
; 1.0
4 D
0.5
? P
t
0 0
2
1
3
B(weber/m*)
Fig. 1. Hall resistivity vs. magnetic induction at the indicated temperatures for Sample 1 at a thickness of 0.190 mm.
temperatures
PH has a linear dependence on about 1.0 and 3.25 w/m2. The hydrogen data is slightly ‘S’ shaped up to 2.5 w/m2. It is concave upward at low fields, and then turns concave downward at about 1.2 w/m”. Both the 14.1 and 20el”K data are linear between 2.5 and 3.25 wlm2. These details are clearly seen in Fig. 2 where a much larger scale has been achieved by presenting (PH-0’7 lo-‘OB) as a function of B.
B between
1298
A. C. EHRLICH
and D. RIVIER
0.6
%':
0.4
B F: P s I h
0.2
: :
I
0 0
1
2
3
Fig. 2. Hall resistivity minus 0.7. 10-‘“B vs. magnetic induction at the indicated temperatures for Sample 1 at a thickness of 0.190 mm.
On this scale a vertical displacement equal to the dia. of one of the data point circles corresponds to a 0.2 per cent change in oH at 3.25 w/m2. The results for I = 0.041 mm are presented in Fig. 3. Some difference in the shape of the dependence of pH on B but not in its magnitude can be distinguished between Figs. 2 and 3. Figure 4 shows the Hall resis-
1
2
1
2
3
B (weber/m*)
6 (weber/m*)
0
0
’
3
B (weberlm')
Fig. 3. Hall resistivity minus 0.7. 10-u’ vs. magnetic induction at the indicated temperatures for Sample 1 at a thickness of 0.041 mm.
Fig. 4. Hall resistivity minus 0.7. 10-*“B vs. magnetic induction for Samples 2-5 at 4.15”K. The thicknesses of the three thinnest Samples (3-5) were obtained by cold rolling.
tivities for Samples 2-5, all of which were measured at 4.15”K. For Samples 3-5 pH is not a linear function of B. The Hall effect and resistivity results for Sample 1 at all thicknesses and Sample 2 are presented in Table 1. The slope and the intercept have been deduced from a least square fit of the data to equation (5) for the intervals of B where pH is a linear function of B, i.e. 1.0-3.25 w/m2 for the He temperature results, and 2.5-3.25 w/m2 for the H, results. The Sample 1, r = O-041 (deformed) results were obtained on the t = O-041 mm specimen after it had been bent around a cylindrical bar 3 mm in dia. and straightened. The Hall coefficients, Ro, have an absolute error no greater than 2 per cent at He temperatures and 4 per cent at H, temperatures. We estimate the uncertainty in the intercepts as 0.05 X lo-” and 0.25 X lo-” mw/A-s for the He and Hz data respectively. The electrical resistivity measurements have an absolute error of approximately 2-5 per cent with the larger figure applying to the He temperature results on the thickest samples. The transverse magnetoresistivity is the error, for all same, within experimental
EFFECTS
1299
IN Ni
Table 1. The slope (R,,) and intercept (yS) of the linear portion of the Hall resistivity as a function of magnetic induction, and the electrical resistivity (p) at the indicated thicknesses and temperatures Intercept (IO-“mw/A-s)
Resistivity (p$) ( lO-1’%m)
Thickness (mm)
Temperature W)
1.005
1.85 4.15 14.10 20.10
0.700 0.699 0.708 0.781
1.05 1.15 1.89 0.33
3.32 3.79 8.14 13.12
I
0.571
1.85 4.15
0.706 0.704
0.79 0.87
3.27 3.68
1
0.401
1.85 4.15 14.10 20.10
0.689 0.689 0.710 0.775
0.60 0.69 1.09 -0.27
3.11 3.65 7.93 12.99
1
0.190
1.85 4.15 14.10 20.10
0.699 0.700 0.722 0.788
044 0.48 090 -0.52
3.36 3.75 8.03 13.11
1
0.100
1.85 4.15 14.10 20.10
0.692 0.694 0.724 0.795
0.43 0.52 0.88 -0.72
3.90 4.27 9.15 14.38
1
0.072
1.85 4.15 14.10 20.10
1
o*O41
1.85 4.15 14.10 20.10
0.692 0.702 0.736 0.801
0.53 0.58 0.71 -0.89
4.06 4.52 9.06 14.23
1 (deformed)
0.041
1.85 4.15
0.710
0.54
5.00 5.48
2
1.002
1.85 4.15
0.712
1.59
5.70 6.04
Sample 1
SlopeCR,)
(1O-?TP/A-s)
thicknesses of Sample 1 equal to or greater than t = 0.190 mm. The results are given for B perpendicular and B parallel to the plate in Figs. 5 and 6 respectively. In the latter, the abscissa is not the magnetic induction in the sample, but rather the applied induction. Fig. 7 shows the results for the smaller thicknesses of Sample 1 (B perpendicular to the plate) at 1 35°K. The results at 4.15”K are
3.99 4.38 9.19 14.63
essentially the same. H, temperature measurements and the He measurements with B parallel to the plate do not differ from those shown in Figs. 5 and 6. The magnetoresistance for Samples 2-5 at 4+15”K is shown in Fig. 8. The temperature dependence of the resistivity of Sample 1, t = 0.190 is shown in Fig. 9 where (p-p(O))/T vs. T is given. p(O) is the
A. C. EHRLICH
1300
and D. RIVIER
-I
0.8
+
-
0.041 mm(dcformrd)
0
-
0.041 mm
14.1 ‘K
.
-
0.072 mnl
‘.15*K
A
-
0.100 mm
a
_
20,l’K
0
-
.
-
o
-
1.85.K
.
bulk
0 t. t
mm
6
0
0.6 0
Q75
A
&
. A 0
61
.
^o Q_
v
0
0.4
0
* ,e
^g
0
.
k -
s
0.50
: 0
0.25 0.2
2
1
0
3
B (weberlm2) 0
2
1
3
B (weber/m2)
Fig. 5. Transverse magnetoresistivity (with B perpendicular to the plate) vs. magnetic induction at the indicated temperatures for Sample 1 at all thicknesses greater than or equal to 0.190 mm.
08
o
-
20.1 ‘K
A
-
14.1
‘K
.
-
4.15
‘K
v
-
1.85 ‘K
0.6
Fig. 7. Transverse magnetoresistivity (with B perpendicular to the plate) vs. magnetic induction at l+W’K for the indicated thicknesses of Sample 1,
residual resistivity and has been obtained by extrapolating to 0°K the data between T = 1.75”K and T = 4*15”K. The extrapolation was straightforward since p cc T* in this temperature range. Also shown in Fig. 9 is the He data extrapolated up to H, temperatures as well as the H, data extrapolated down to He temperatures. The results on the thinner samples are essentially the same as the data shown. 5. DISCUSSION OF RESULTS (A) Hall effect and magnetoresistance
0
0
1
2
3
B (wcber/m21
Fig. 6. Transverse magnetoresistivity (with /3 parallel to the plate) vs. applied magnetic induction at the indicated temperatures for Sample I at all thicknesses.
The results for Sample 1 for R, at He temperatures are in excellent agreement with the high field measurements reported by Fawcett and Reed[lO] on a single crystal for a magnetic field direction chosen to give only closed orbits. This is not surprising in view of the fact that in Ni, the contact area of the Fermi surface within the Brillouin zone boundary is small and thus open orbits are found in only a small fraction of magnetic
EFFECTS
IN Ni
1301
‘6
12: 0
0
,-
LOC
v
-
0.100
mm
o
-
0.243
mm
*
-
0.435
mm
+
-
1002
mm
0
.
0
. .
0 A
g
0 0
0
0.75
2 d. Q5C
,-
Q2!
>_ -
2
,’
,/’ _/ ,,,_ 0
,/’
9 0
.
1
2
3
3 0
B (weberlm2)
Fig. 8. Transverse magnetoresistivity (B perpendicular to the plate) vs. magnetic induction at 4.1YK for Samples 2-5. The thicknesses of the three thinnest Samples (3-5) were obtained by cold rolling.
field directions. Therefore only a negligible number of the crystallites in our polycrystal are so oriented as to give rise to open orbits. It seems certain that had we been able to extend our measurements to 8 w/m2, as did Fawcett and Reed, we would have also found the same results as they for this range of magnetic field. What is remarkable in our data is that pH is a linear function of B at fields at low as 1.0 w/m”. Ifere, though condition (4a) may or may not be well satisfied, condition (4b) certainly is not. In fact, between I.0 and 3.25 w/m* (rH/yl)” goes from about 3-16 and 4-20 for 4.15 and 185”K, respectively. Put otherwise, l/y, is not proportional to B, so it is difficult to understand why oH is. Sample 2 has a resistivity 1.6 times that of Sample 1 and thus its linearity is even more difficult to understand. The same general remarks apply to Fawcett and Reed’s measure-
1
2
3
I
I
T(‘K)
Fig. 9. Temperature dependence of electrical resistivity for Sample 1 at a thickness of 0.190 mm. The lower dashed line is the 14-20°K data extrapolated to the O-4°K temperature range and the upper dashed line the 1.84.2”K data extrapolated to the 14-20°K temperature range. The circles and triangles indicate two separate experiments for each temperature range. The arrows show the appropriate scale.
ments, whose pH for their highest and lowest field points seem to fall on the same straight line. Table 1 shows a trend of pHs towards decreasing values for Sample 1 as the thickness is reduced from l-005 to O-190 mm. This is not a true size effect, but can be quantitatively accounted for by our having used the ‘infinite flat plate approximation’ in calculating R, and pH8; i.e. we have assumed that the magnetic induction inside the sample is equal to the applied magnetic induction. For the initial dimensions of Sample 1, 50 X 10 X 1 mm, this is not exact. A Ni ellipsoid of these dimensions would have a B(inside) = B(applied)+ 0.06 w/m” for an applied field sufficiently large to saturate the sample [ 1I].
A. C. EHRLICH
1302
When t = O-2 mm (the other dimensions remaining approximately constant), the approximation is better than the experimental error and no further change in pH8 should occur. This is consistent with the data presented in Table 1. There is, of course, no effect on the calculated value of &. A striking feature of the dependence of pH on B is the appearance of ‘knees’ in the curves at about O-6 w/m2, the saturation magnetization of Ni, for all temperatures measured (see Fig. 2). This is typical of a spontaneous magnetization contribution to the Hall effect. In the case of the H, data, the ‘S’ shaped results in Fig. 2 can be explained semi-quantitatively by combining just such a spontaneous contribution with equation (12a). In [8] it was shown that -yI obeys Kohler’s rule while yH does not (‘Ni III’ of [8] is the same as Sample 1 of this paper). It is consistent with this result to suppose that yIs = 0 or, at least, that y18/yH8 + 1. Then, dropping the terms in equations (12) containing yL8 gives PH
=
YHO + YH@+ Ylon
E
Y”O +
YHo2+ YIOZ p1
=
YLO YZ+
YIOP
1_
2YH0’ ‘yffw+ Y1091
(134
YHsm(B)
?I0 YHon+ YIW
C
YHS yHo2 + yloz [
YHS yz+
.
2Yl”rJf”
yLoz y2- + y102
+YHSS(B).
(13b)
m(B) and s(B) could be evaluated by substituting yH and yL for yHoand yIo since the error introduced this way is 2nd order in yH8. In view of the approximations already made, however, a semi-quantitative discussion of equations (13) is sufficient. For the value of B, Bo, such that yHo = yIo, m(B) = 0. (B,= 1.43 w/m* at 14ml”K and 2.72 w/m* at 20al”K in Sample 1.) When B < B. (B > B,), m(B) > 1 (m(B) < 1). Because it is sharply concave upward at about 0.6 w/m* the H2 temperature data
and D. RIVIER
shown in Fig. 2 appear to have a y$ which is negative, i.e. the opposite sign of yHo. Thus the 2nd term in equation (13a) makes a negative contribution to pH in the range B > B, and a positive contribution when B > Bo. This is just what is needed to explain the S shape of the curves of Fig. 2. Furthermore, a negative yHs subtracted out of the curves in Fig. 4 of [8] would bring them closer to the He curves on the same graph, and thus into better agreement with Kohler’s rule. It is interesting to note also that yHss(B) should be linear in B in the high field range. The He temperature magnetoresistivities given in Fig. 5 have the same form as that of Fawcett and Reed[lO] in the same interval of magnetic induction except that they are slightly larger. This is not surprising in view of the fact that the absolute value of the magnetoresistance depends on the details of the scattering process in the high field range, while its functional dependence does not. (B) Size effect
A crude estimate of the mean free path, 1, of the conduction electrons can be obtained from the resistivity by using a free electron model. Assuming 1 conduction electron per atom, at l%“K Sample 1 has an 1 = 0.02 mm. This is large enough to expect size effect in some of the thinner specimens in Table 1. Since size effect in the resistivity is practically absent in the three thinnest samples, however, the considerable change in resistivity between t = 0.190 and t = 0.100 must be attributed to damage done to the sample in handling. Examination of the data shows no detectable corresponding effect in the Hall resistivity or the magnetoresistance. In the high field limit Sondheimer’s spherical band theory predicts no size effect in the Hall effect. (Note that when 1 = a; i.e. when there is size effect, a/r S- 1 is a necessary condition for having the high field limit.) This will hold also for an arbitrarily shaped Fermi surface provided there are a negligible number of open orbits because size effect
EFFECTS
arises from scattering of the electrons at the surface of the sample and the scattering mechanism has no influence on the Hall effect in the high field limit. This is in agreement with our Sample 1 Hall effect results which show practically no size effect (compare Figs. 2 and 3). In the high field limit the transverse magnetoresistance does depend on the relaxation time and can be expected to show size effect when B is perpendicular to the sample plate. For B parallel to the plate, size effect is suppressed since the electrons will be prevented from making contact with the sample surfaces. Both of these points are supported by the data (see Figs. 5-7). Sondheimer’s model predicts a size effect in the resistivity considerably larger than we have found. An explanation for this may be found in the ferromagnetic character of Ni. Generally, in ferromagnetic polycrystalline materials the resistivity in the absence of an applied field is approximately [ 121
1303
IN Ni
our electropolished Sample 1, these effects are seen to be a result of the cold rolling rather than a true size effect. Cold rolling produces preferentially oriented crystallites in the polycrystal. Metals with open Fermi surfaces have highly anisotropic galvanomagnetic effects, and different amounts of preferential orientation in two otherwise identical specimens can be expected to produce appreciable differences in their galvanomagnetic properties. (C) Resistivity The results of the resistivity vs. temperature measurements can be expressed by the equation p(T) --p(O) = aT+ bT2
(15)
where a = 0;
b = 2.7, x 10-59; 1.75”K < T c 4.15”K
(15a)
a = -4.6 x 10-Sa+; where pl and pll are the resistivities in the absence of an applied field when the magnetization is perpendicular and parallel, respectively, to the current direction. That is, even in demagnetized Ni the electrons are moving through magnetic domains with a magnetic field intensity, M, of about O-6 w/m”. Thus the equivalent of 2/3 of the electrons experience a Lorentz force tending to prevent them from coming into contact with the surfaces of the sample. The results shown in Figs. 4 and 8 of the Hall effect and magnetoresistance, respectively, for the Ni samples prepared by cold rolling are very similar to Berlincourt’s[3] results on Cu samples prepared in the same manner; i.e. the Hall voltage is not linear in B and shows a systematic deviation with sample thickness while the magnetoresistance of the samples differs from each other but not in a systematic way. In the light of the results on
b = 2.6, x lo-“e; 14eO”K c T s 20.1”K.
(15b)
p (0) has been determined by assuming equations (15) and (15a) valid down to T = 0°K. Magnon scattering has been shown to give a quadratic dependence of the resistivity on temperature[ 131 and this is the most straightforward explanation of the observed T2 behavior. It should be noted, however, that s-d electron-electron scattering can also give rise to a T2 term in the resistivity as has been shown by Baber[l4] and more recently discussed by Appel[ 151. There are two problems raised by the resistivity data. The first is that previous low temperature measurements in Ni of p vs. T by Kondorski, Galkina and Tchernikova[ 161 and White and Woods[ 171 have also given a quadratic dependence but with b = 4-O X low5
1304
A. C. EHRLICH
for the former and 1*6X 10m5@-cm/?? for the latter. These results show no correlation with the purity of the specimen, since the residual resistivity of the specimen in [16] is about ten times that in [ 173, which in turn is about ten times that reported here. The differences are much too large to be attributed to combined experimental error or to ordinary divergences from Matthiesen’s ruleflS]. Second, the coefficients in (15b) represent an empirical fit to the data between 14.1 and 2O.l”K and are probably not valid very far beyond this range. It is gratifying that ‘b’ in (15a) and (1 Sb) are almost identical, but somewhat anomalous that the temperature dependence of the resistivity is weaker in the hydrogen temperature range than at He temperatures. (For continuity of the resistivity it must be weaker still somewhere between 4.15 and 14.1”K). While these two probtems are not easily explained, it should be noted that the temperature dependence of a ferromagnetic metal is related to its magnetoresistance, and if the latter is anomalous, the former may be expected to be also. (The magnetoresistance of Ni as a function of temperature is anomalous in its deviation from Kohler’s rule [S, 191). The temperature dependence of the resistivity of a non-magnetic metal in a constant magnetic field will be, in general, different than in the absence of a magnetic field. In a ferromagnetic metal the existence of the domain magnetization means that even without an applied field, the resistivity measured is a sort of magnetoresistance (see equation (14)). As Blatt[20] has pointed out, the magnetoresistance is more sensitive to the electronic relaxation time, T, than is the resistivity since the latter depends on an average of 7 over the Fermi surface, while the former involves power and derivatives of 7.
and D. RIVIER
It is to be expected, then, that the resistivity will be more complicated in ferromagnetic than ordinary metals. Acknowled~e~enis-We wish to thank the members of the Institut de Physique Experimentale de I’Universite de Lausanne for their most kind cooperation during the course of this work. We are particularly grateful to Dr. R. Huguenin for many stimulating discussions and useful suggestions, We are also indebted to Miss G. Wendt for her help with the numerical calculations. Finally, we wish to thank the Swiss Fonds National de la Recherche Scientifique for supporting this work and for the grant that one of us (A. C. Ehrlich) has received for his stay at this laboratory. REFERENCES E. H.,Adv. Phys. 1, I(1952). 1. SONDHEIMER B. N., JEEP 2. See, for example, ALEKSANDROV 16, 871 (1963); ibid., p. 286; COOPER J. N., COTTI P. and RASMUSSEN F. B., Phys. Leti. 19, 560 (1965): FORSVOLL K, and HOLWECH I., Phil. Mug. 9, 435 (1964); ibid., 10, 1961 (1964); HOLWECH I., Phil. Mug. 12,117 (1965). T. G., Phys. Rev. 112,381 (1958). 3. BERLINCOURT 4. SONDHEIMER E. H..Phvs. Rev. 80.401(1950). 5. For references, see SOFFER S. B.,J. appi.‘Phys., In press. N. H., HAMBURG R. E. and 6. ZEBOUNl MACKEY H. J.,Phys. Rev. Lett. 11,260 (1963). 7. LIFSHITZ 1. M., AZBEL M. 1. and KAGANOV M. l.,J. exp. them. Phys.4,41 (1957). R. and RIVIER 8. EHRLICH A. C., HUGUENIN D., J. Phys. Chem. Solids 28,253 (I 967). 9. FONER S.. Phvs. Rev. 101, 1648 ( 1956). IO. FAWCET’? E: and REED W. A., Phys. Rev. 131, 2463 (I 963). II. OSBORN J. A., Phys. Rev. 67,35 1 (1945). 12. VAN ELST H. C., Physic-a 25.708 (1959). 13. For references, see GOODfNGS D. A., Phys. Rev. 132,542 (1963). AI58 383 (1937). 14. BABER W. G.,Proc.R.Soc. 15. APPEL J., Phil. Mag.8, 1071 (1963). 0. S. and E. I., GALKINA 16. KONDORSKI TCHERNIKOVA L. A., J. appL Phys. 29, 243 (1958). 17. WHITE G. K. and WOODS S. B., Phil. Trans. R. Sot. A251.273 (1959). 18. WILSON A. E., The Theory of Metals. p. 310. Cambridge University Press, Cambridge (1954). 19. See also MARCUS S. N. and LANGENBERG D. N.,J.uppf.Phys.34, I367(1963). 20. BLATT F. J., Solid State Physics (Edited by F. Seitz and D. Turnbull) Vol. 4, p. 199. Academic Press, New York (I 957).