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Anisotropic temperature dependence of the hall effect in Cu and some CuZn crystals
J. Phys. Chem. Solids
Pergamon Press 1971. Vol. 32, pp. 175-189.
Printed in Great Britain.
ANISOTROPIC TEMPERATURE DEPENDENCE THE HALL EFFECT IN Cu AND SOME Cu-Zn CRYSTALS
OF
C. M. HURl) and J. E. A. ALDERSON National Research Council of Canada, Ottawa, Canada
(Received 4 May 1970) A b s t r a c t - T h e Hall effect in single crystals o f C u (RRR --- 2312) and Cu containing 76, 172 and 1 I00 ppm Zn has been studied in the temperature range 4.2-80°K (i.e. through the low-field/high-field transition). The temperature dependence of the effect is given for H along [111], [I 10], [100], [211l, and other directions which show either a single open orbit or all closed orbits. It is shown how the qualitative features of this dependence can be explained in terms of the known topology of the Fermi surface. Comparable results are also given for polycrystalline samples of Cu, and Cu with 78, and 940 ppm Zn. These indicate that of the singular topological features associated with the open Fermi surface of Cu only orbits of the four-cornered rosette and dog's bone type contribute to the Hall effect in a polycrystal. Some further evidence is given to support the view that the anomalous maximum in the temperature dependence of the effect in a polycrystal, which is centred at about 50*K, arises from the low-field/high-field transition. 1. INTRODUCTION
there are two limiting conditions under which measurements of the galvanomagnetic properties of a metal can be made. In the first case the lifetime of an electron between collisions remains short enough that the topological features of the Fermi surface do not manifest themselves in the electronic motion. This is the low-field case, and in this condition (providing there is sufficient detailed information available about the shape of the Fermi surface) the Hall coefficient (RH) can give information about the anisotropy of the dominant scattering process[l]. In the second case the lifetime of the electron is very long compared with the characteristic time necessary to complete a cyclotron orbit, and the effect of collisions is then a negligible perturbation of the electronic motion compared with the effect of the externally applied fields. This is the high-field condition. In this limit the galvanomagnetic properties are independent of the dominant scattering process[2] and are determined solely by the topological features of the Fermi surface.
THEORETICALLY
In the experimental study of the Hall effect there is no sharp transition between these two conditions. This is because contributions to the Hall field arise from all closed cyclotron orbits in planes perpendicular to the applied magnetic field (H). As the temperature or the impurity concentration is reduced, or the applied field increased, those orbits for which an- passes a certain critical value will pass into the high-field condition. There may be a wide range of to~- values covering a decade or more in which orbits of different effective area pass successively into the high-field condition as the above variables change. In this range the sample is in a mixed high- and low-field state and its condition cannot accurately be descrJbed by either of the above classifications. (to is here the cyclotron frequency corresponding to an applied field H, and z is an average electronic relaxation time for an orbit.) The possibility that a Hall effect experiment may be conducted in a mixed high- and lowfield condition, or that a transition may take place between these conditions during the experiment, complicates considerably the interpretation of the results. For example, we 175
176
C.M.
H U R D and J. E. A. A L D E R S O N
recently showed elsewhere[3] that the field dependence which has been observed for the Hall coefficients of pure, polycrystalline samples of the 1B metals, and which has been attributed to the pure metal[l], is apparently the result of an unrecognized transition between the high- and low-field conditions during the experiment. The presence of trace impurities can reduce the value of ~-7 (where the bar signifies an averaged value over all orbits, such as that obtained from free electron theory) to the point where the transition occurs in the range of H used in the experiment, thereby producing the apparent field dependence of Rn. This field dependence is not present when the impurity concentration is either reduced, so that the high-field condition is maintained throughout the experiment, or is increased so that the low-field condition is maintained throughout [3]*. A similar transition between the high- and low-field conditions must also take place in each of the recent studies of the temperature dependence of RM for high-purity, polycrystalline samples of the IB metals[l, 4-6]. This follows from the fact that samples of comparable purity subjected to comparable magnetic fields are known[3-5, 7] to be in a predominantly high-field condition at 4.2°K, and must therefore pass to the low-field condition when the temperature is increased sufficiently. Not much attention has been paid to the effect of this transition upon the observed temperature dependence of RM. Possibly this is because it involves the problem of how the Hall effect in a polycrystalline sample in the high-field condition should be interpreted. Much is known about the influence of the topology of *It now appears to us that this should have been obvious a priori since a polycrystalline sample gives an average over all situations (a)-(d) of Section I, while the high-field theory shows[2] that of these orientations only (d) gives a Hall voltage which varies non-linearly with H. Since this tends to zero in the high-field limit, the highfield value of R . must be field independent.
the Fermi surface on the Hall effect of each crystallite of a given orientation [2], but how are the contributions from the (presumably) randomly oriented crystallites to be combined to give the macroscopic effect measured in a polycrystalline sample? To find the full analytical solution to this question seems to be a formidable problem [8] and for all practical purposes not much progress has been made with it. The interpretation of high-field results for polycrystalline samples of a metal with an open Fermi surface remains an uncertain procedure [2]. The work described in this paper has been carried out partly in the hope of improving the understanding of this problem (if only from an empirical point of view) and partly to study the effect of electron scattering by different foreign elements upon the Hall effect in a 1B metal. We have measured the Hall effect as a function of temperature for a given direction of applied magnetic field in single cryst.al samples of Cu, and Cu doped with Zn. The results are given in Section 2. We have considered all the principal situations of interest which, in the high-field limit, the topology of the Fermi surface affords:(a) With H along directions of high symmetry, < 111 >, < 110>, < 100>, showing only closed electron orbits or closed hole orbits formed from intersecting open orbits. (b) With H along a direction well away from high symmetry directions, where free electron behaviour might be expected [7]. (c) With H along a symmetry direction showing a single open orbit. (d) With H along a direction, < 211 >, showing two nonintersecting open orbits of different orientations. Apart from the recent results of Saeger and Liick for Cu [5], there appears to have been no other attempt at a systematic study of the temperature dependence of R , in single crystals of the 1B metals, the earlier results for single crystals having usually been obtained at a fixed temperature ( - 4.2°K)[7, 9-14]. Our results are qualitatively in agreement with
ANISOTROPIC TEMPERATURE
DEPENDENCE
those of Saeger and Liick but quantitative comparison is impossible since their sample was not in general oriented exactly to give any of the situations (a)-(d) above. The detailed interpretation is thus difficult since for a small deviation of the field direction from a direction of high symmetry the open orbit contribution influences the temperature dependence of RH in a complicated manner. Orientations of H for which the temperature dependence of RH for Cu is available in the literature are shown in Fig. i.
[Ill]
[ioo1
[I,o1
Fig. 1. Orientations of H for which the temperature dependence.of R , in Cu is available in the literature. Closed symbols represent the present work, and open symbols that of Saeger and Liick [5]. 2. EXPERIMENTAL METHOD AND RESULTS The samples were prepared from Cu of 6N purity of which typical analyses have been given elsewhere [3]. The R R R values (R2~a°K/R4.2*K) of all the samples are given in the captions to Figs. 2 and 3. The single crystal samples were grown by the Bridgman method in high-purity graphite molds from previously prepared polycrystalline alloys of the desired concentration. Specially oriented pure Cu seed crystals were used. The quoted concentration of Zn is the mean of values obtained from off-cuts of each end of the sample, since the method of preparation was naturally conducive to some zone refining during the growth. (The analyses were made by the Analysis Section of N R C C using the method of atomic absorption.) Other details of the sample preparation, the instrumentation and techniques, are identical to those described previously [3, 4]. The temperature dependence of the Hall resistivity Pn (the Hall voltage per unit primary current density) is shown for a pure Cu sample in Fig. 2, and that for three C u - Z n single crystals (with 72, 172 and 1100 ppm. Atomic ppm is implied throughout.) is shown in Fig. 3. These figures also include for comparison the results for polycrystalline samples of Cu, C u + 7 8 ppm Zn and Cu + 940 ppm Zn. The primary current direction is along [IT0] in the single crystals, and the applied magnetic field
J.P C.S. Vol. 32N o. I - - L
OF THE HALL EFFECT
177
(H) is 15.17 kG. The values of ~'~ were calculated from the resistivity of the samples using free electron theory. The effect upon the temperature dependence of p , of additions of Zn is shown individually for each orientation in Fig. 4. The field dependence of the Hall voltage (vn) at 4.2°K was measured both for the polycrystalline samples and for each orientation of the single crystals, vn varies linearly with H for the polycrystalline samples, even though they are in a high-field condition at this temperature (see the discussion in[3]). For the single crystal Cu sample vu follows the expectations of the high-field theory [2]: for sufficiently high fields (above - 5 kG in most cases) it varies linearly with H for the [110], [100], the open square (Fig. 2), and open triangle orientations (i.e. respectively for the situations (c) and (b) of Section 1). For H along [211] Vn decreased roughly as H -t, in agreement with theory. For H along [ 111 ] it could not be said with certainty that vz had attained a linear variation with H at the maximum field available, which presumably indicates that the true high-field condition was not obtained for this orientation. The same applies for all the orientations (except H along [100] and [110])in the Cu + 76 ppm Zn crystal. The presence of the solute makes it impossible to obtain the true high-field condition with the maximum field available. This gives rise to an apparent field dependence of R,, as was discussed in[3], and for this reason we shall throughout discuss the behaviour of On(= R n H ) . 3. H A L L EFFECT IN T H E HIGH-FIELD LIMIT We shall assume that the Fermi surface of Cu is sutficiently well known to need no detailed description [15]. It supports both open and closed cyclotron orbits depending upon the direction of the applied magnetic field H relative to the crystal axes. In the high-field limit we shall differentiate between two cases depending upon whether open orbits in a plane normal to H are (a) impossible, or (b) possible. It is convenient to review briefly some of the results of the theory for these two cases[2, 16]. To simplify the expressions we take H to be applied along the z-axis and E, the external electric field, to be along the x-axis.
(a) There are no open orbits in planes n o r m a l to H The theory in this case is based upon the assumption that the conductivity tensor o-u can be expressed by a series expansion in H tr u = au + cu H - t + bu H-~ + . . . .
( 1)
where a, b and c are constants. In the high-field limit this becomes [2, 16], since we have taken E= = 0 biH-~
oH-=
(2)
where we have simplified the subscripts (bxz =- bt, exu --- c, etc.). It follows that the Hall coefficient (pxulH, where p• is inverse of o'o) tends to a constant value independent of H in the high-field limit, it can be shown
C. M. H U R D and J. E. A. A L D E R S O N
178
tO't" (free electron) 16
250
8 I
I
1.7
0-5
I
I
FIELD DIRECTION
='\
200
[.,1
-2
o-I°-
•
ol ;--"~|
~ \ \
I
]
i<> E
150
\\
To
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.... POLYCRYSTAL o 30"2*FROM ['001
I ~
N
a
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",.~
7"3°FROM[100-111}AXIS
Cu
I00
-
5C
/
/
,____.J 0
I
I
I
20
40
60
80
TEMPERATURE ( ° K ) Fig. 2. Temperature dependence of the Hall resistivity at 15-17 kG for a crystal of Cu with R R R = 2312 with H along the directions indicated in the unit stereographic triangle. Comparable results for a polycrystalline sample are shown by the dashed line.
[16] that this limiting value is given by
R.
H~=
=
l
e(Ne--Nh)
when (3)
where Ne and Nh are respectively the (unequal) effective numbers of electrons and holes per atom. They are thus the volumes in k-space per Brillouin zone enclosed within closed electron and hole orbits, scaled so that the Brillouin zone contains exactly two electrons per atom. Rn can therefore be calculated from (3) when the shape of the Fermi surface is known [7, 17]. The limiting value of RH of (3) will begin to be reached
Ne + N~ tin- ~> Ne--Nh"
(4)
From the results of Table 1 we see that for H along < 100 > and < 110 > t,rr must considerably exceed ~ 2, while for H along < l l l > it must considerably exceed ~12. (b) There are open orbits in planes normal to H [ 16[ We consider the case when open orbits having a single axis direction are present. It can be shown that as H is
A N I S O T R O P I C T E M P E R A T U R E D E P E N D E N C E OF T H E H A L L E F F E C T
179
- (a¢" ( f r e e e l e c t r o n )
0.39
0-37
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=
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FIELD
DIRECTION
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Cu
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+
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0114 CU+ 172ppm Zn
70-
'°t
_ Ca_+ !,oo .
0
.
.
.
p Zn
I
.
20
40 60 TEMPERATURE (*K)
80
Fig. 3. Temperature dependence of the Hall resistivity at 15.17 kG for crystals of C u - Z n with R R R values (in order of decreasing purity) of 544, 263 and 53. The field is applied along the directions indicated in the unit stereographic triangle. Comparable results are shown by the dashed line for polycrystalline samples with 78 and 940 ppm Zn (RRR values are 447 and 80 respectively). increased the conduction by closed orbits tends to zero (except along the direction of H) and in the high-field limit all the components of o'~ (except o-~z) arising from closed orbits are zero. The current flow is then entirely along the open orbit axis direction and is confined in real space to the plane which is normal to the orbit axis direction and which contains the direction of H. The response of the electrons is independent of H, and thus in the high-field limit the problem is reduced to one of quasi-free electrons moving in a plane in real space containing the direction of H and inclined at some angle (8) to the direction of the applied electric field E. From the geometry of
this situation we can write down directly[16] the limiting value of the conductivity tensor for open orbit conduction as d cos z 0 d cos 0 sin 0 o-~ = d cos O sin O d sin~#
H~oo
(5)
where d is a constant which is related to the flux of electrons moving in the open orbit. If the conductivities of (2) and (5) are assumed to be additive, their sum can be inverted to give the full resistivity tensor at large H for mixed open- and closed-
180
C.M. - 2 . 4 0 --
-
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HURD I
I
and J. E. A. A L D E R S O N
-0"80~I• -0.40
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-I.60
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,POLYCRYSTAL
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20 40 60 80 TEMPERATURE (OK)
FIELD DIRECTION
•
[m) -0"8C
I I I l 20 40 6 0 80 TEMPERATURE (<'K)
0
[llol
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-
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Cu
. . . .
Cu + 7 6 ppm
Zn
E] 3 0 . 2 *
FROM
[100]
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Cu + 1 7 2 p p m
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~
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[100-111] AXIS
7.3 °
Fig. 4. T h e data of Figs. 2 and 3 plotted for a fixed orientation as a function of solute concentration. T h e direction o f H is indicated by the symbol, which is maintained t h r o u g h o u t Figs. 2-4. in the case of the polycrystalline results, the dashed line refers to C u + 7 8 p p m Z n and the dash-dot line to Cu + 940 p p m Zn.
Table 1. -R,
Field direction
Ne--Nh
A w a y from high symmetry axes < 100> < 111 > < 110>
!.00 0.51 0.068 0.51
--RH Calculated from c o l u m n 2 X 10-11 m 3 a m p -t s e c -I
(as m e a s u r e d at 4.2°K)
K l a u d e r et al. R R R = 8000
T h i s work R R R = 2312
7.4
7.4
7.4
14.5 109.4 14.5
14.5 83.6 11.9
13.1 24.0* 11.9
*Calculated from the local gradient of the v , vs. H curve.
ANISOTROPIC TEMPERATURE DEPENDENCE OF THE HALL EFFECT orbit conduction. [Compare Lipson [10], whose expression differs in detail from that below apparently because Lipson does not take the adjoint of the cofactor when inverting (2) plus (5).] o--
_.--:---v_dH2 I" sin2 0
/t£~=--tr°'=Bd+c~L--cosOsinO c2 [ b J c z Bd+cZ[Hlc
-- sin 0 cos 0] cos'0 J
(6)
--H/c ] b,/c z J
where B =-- b, sin~O+b2cos20. The first and second tensors represent the open and closed orbit contributions respectively. The element which determines the total transverse field can be written dHz
Po = -sin Ocos O n~® Bd + c 2
cH Bd + c 2"
(7)
Equation (7) shows that for the two cases 0 = 0, rr/2 the open orbit contribution to Pxv is zero. Physically this expresses the fact that when the open orbits conduct exactly along the sample no transverse voltage is produced, and when they conduct exactly across the sample they are not excited. However, the two cases are not exactly equivalent physically, for when 0 = ~r/2 we are left with the closed orbit contribution to Rn represented in (7) but when 0 = 0 the conductivity of the closed orbits in the high-field-limit is negligible compared with I~hatof the open orbits along the length of the sample. Both terms in (7) are then zero; the entire current flows in the open orbits and is unaffected by the applied field H. The Hall effect is then 'shorted-out' by the open orbits which lie along the direction of the primary current. 4. DISCUSSION OF THE RESULTS FOR PURE Cu
Our interpretation of these data is based upon the following argument. In the high-field limit Rn is given by (3). As ~-~ is reduced by some means (in this case it is by increasing temperature) and successive orbits pass into the low-field condition, the sign of the variation of Rn (and hence P n) will also be given by (3). F o r example, if there is a net loss of hole states to the low-field condition as the temperature is raised, then the 'high-field' value of Pn will b e c o m e more positive [through (3)]. This value is not, of course, a true high-field value, but as long as On shows a detectable anisotropy then it can be said that the sample is in a partially high-field condition for that orientation. F r o m the foregoing we can determine the sign of the change in On as particular groups of orbits pass into the low-field condition, but
181
what circumstances determine which orbits are in a high-field condition at a given temperature? It is impossible to give a quantitative reply to this without a full knowledge of ~-~ for each orbit, but our qualitative argument is as follows. As the temperature is raised it is known from other e x p e r i m e n t s [ l ] that (in the usual terminology) electrons in neck states are scattered more than electrons in belly states because the umklapp processes from belly states are 'frozen out' at the lowest temperatures[19]. We therefore expect, all other considerations being equal, that as the temperature is raised those orbits which traverse neck regions of the Fermi surface would suffer the transition to the low field condition before those lying entirely on belly regions. Before we consider the temperature dependence of On in detail, we should comment upon the absolute magnitude obtained for Rn at 4.2°K for those field directions which do not support open orbits. T h e s e directions are with H along [110], [111], [100] and the off-symmetry direction shown by the open triangle in Fig. 2. As is customary [7], we have included [110] in this class although strictly speaking this is not permissible since a non-negligible number of open orbits are excited for this orientation[7, 17, 23] and (3) is no longer valid. I n the high-field condition values of N e - Ni, (and hence Rn) can be calculated for the above orientations from the known Fermi surface. Klauder [17] has made the calculation for Cu using R o a f ' s [ 1 5 ] expression for the Fermi surface. T h e results are shown in Table 1. (More sophisticated expressions for the Fermi surface are n o w a d a y s available [18] but the results from Roaf's formulation are adequate for the following qualitative arguments.) T h e r e is a serious discrepancy between our experimental results and those of Kiauder e t al. [7] only for the [111] orientation. This has two possible causes. On the one hand R n can change f r o m - - 6 t o - - 9 0 x 10-11m s a m p - l s e c -~ when the direction of H is shifted
182
C.M. HURD and J. E. A. ALDERSON
a few degrees off [111], so that a slight misalignment of our sample could account for the discrepancy/13]. On the other hand our sample is considerably less pure than that of Kunzler et al. (values of R R R are given in the Table) and, as the results of Fig. 4 show, On for this orientation depends very strongly upon the impurity concentration. We believe that this second possibility is the most probable since we positioned the sample empirically with great care and are confident that the quoted value is representative of a correctly aligned sample. Furthermore, the arguments concerning purity are supported by the values shown for ~-7. It is clear that when the temperature enters the range in which impurity scattering becomes dominant (i.e. below - 15°K) the ~-~ value does not considerably exceed the value ( - 12) required by (4) for the high-field condition. (a) Orientations showing no open orbits The left-hand side of Fig. 5 shows schematically the orientation of the Fermi surface with respect to the direction of the applied magnetic field for the principal closed orbit directions of Fig. 2. The volume of the Brillouin zone within the shaded portion of the Fermi surface corresponds to the electronic contribution to pH while that outside the unshaded areas and lying between the labelled planes corresponds to the hole contributions. The right-hand side of Fig. 5 shows, again schematically but to the same scale, some principal extremal orbits characteristic of each orientation. The free electron circle is dotted in each case. Figure 2 shows that with increasing temperature above the impurity dominated range each of these orientations shows Pn becoming increasingly positive, indicating a net loss of hole states to the low-field condition. For H along [001] and [110] the hole contributions come respectively from bands of orbitals of the four-cornered rosette and dog's bone type (Fig. 5(b) and (f)). The cyclotron masses of their extremal orbits are
approximately equal ( - 1.3m0[18]) so that for a given H their cyclotron frequencies are approximately equal, and each orbit involves four neck passages per cycle with approximately the same length of circumstance. Furthermore, the electronic contributions to p , for these two orientations are similar; a rough graphical integration shows that the total volume of regions A of Fig. 5(a) amounts to only - 0 . 0 5 electrons/atom (e/a) and is negligible compared with the - 0 . 6 6 e/a contributed by the regions B. The regions A of Fig. 5(e) contribute - 0.76e/a. Thus, insofar as the extremal orbits are representative of the bands of orbits under consideration we expect to find for H along [001] and [110] a very similar temperature dependence and magnitude for pn. This is observed (Fig. 2), and the temperature dependence indicates further that as the temperature increases their rates of loss of holes states are approximately equal. This contrasts with the temperature dependence observed for H along [111]. In this orientation the hole contribution comes from a band of orbits of the six-cornered rosette type of which the extremal is shown in Fig. 5(d). (We referred above to the fact that in the true high-field condition the hole states are very predominant in this orientation-see also Table 1.) The cyclotron mass for this extremal orbit is - 2.4 m0 [18], there are six neck passages per cycle and the length of the circumference is about three times that of the four-cornered rosette of [100]. Thus, for a given increase in temperature, it is reasonable to expect more orbits of the six-cornered rosette type to pass into the low-field condition than either of the types of hole orbits considered for H along [001] and [110]. The temperature dependence shown in Fig. 2 supports this expectation, and it appears from those results that all of the six-cornered rosette type of orbit are in the low-field condition at about 28°K while some of the hole orbits for the other orientations of Fig. 5 continue in the high-field condition up to about
A N I S O T R O P I C T E M P E R A T U R E D E P E N D E N C E OF T H E H A L L E F F E C T
183
[oo0
a
b )iq
C
d
I I e Fig. 5. The left-hand side shows schematically the direction of H for the principal orientations studied which show only closed orbits [but see our remark in the text concerning the orientation (e)]. The orientations of (a), (c) and (e) correspond respectively to the open circle, the closed square and the closed circle symbols of Figs. 2-4. The shaded areas give the electronic contribution to Pn. The right-hand side shows schematically, but to the same scale, some of the principal extremal orbits typical of these orientations. The free electron sphere is shown dotted in each case.
50°K. Above about 30°K the behaviour of Pn tends to follow that found for the orientations with H along [112] and the off-symmetry directions, and we shall return to this point at the end of this Section. -the variation ofpH with temperature observed for the open triangle orientation of
Fig. 2 (i.e. with H well away from highsymmetry directions) is not monotonic. This is unexpected since it has been suggested [7] that free electron behavior should be observed for such an orientation since only closed electron orbits were thought to be possible in this case. However, Saeger and Liick[13]
184
C.M.
H U R D and J. E. A. A L D E R S O N
following Section) and that as the temperature is increased above the impurity dominated region (i.e. above - 15°K) these orbits, which presumably pass closer to neck regions than electron orbits, are affected to a greater degree by the phonon scattering. At about 30°K they are all in the low-field condition, and pH follows the behaviour found for H along [111], [112] and the open orbit directions. We shall return to this point in the following.
have measured at 4.2°K the complete dependence of RH for Cu upon crystal orientation, and their results show that within the centre of the unit stereographic triangle the value of RH can vary widely and it only has the free electron value along a very narrow contour. Presumably this contour represents those orientations for which the net effect of the contributions to Rn fortuitously gives the free electron value and does not necessarily imply free electron behavior. Our results support this, since our orientation lies on the contour (as the values quoted in Table 1 show), and yet the temperature dependence is typical of at least two major contributors to OH- It appears from our results that there is some contribution from hole orbits for this orientation (the results obtained for the alloys are particularly relevent in this respect. See the
(b) Orientations showing open orbits The left-hand side of Fig. 6 shows schematically the orientation of the Fermi surface with respect to the applied magnetic field for those orientations showing open orbits. The open orbits lie in the unshaded regions A and B, and have axes oriented in the sample as shown in the right-hand side of the figure. For
[ood H
lilZl OPEN ORBITS S
[ilol
[
Ill
!
~
_
OPER
""'~ 0117o1
]
b
a [ooll
[.z)
~PEN ORBITS A
• [lio]
~.~I
[,TI
":" J [5o]
C
d
Fig. 6. The le•hand side shows schematically the direction of H for the principal orientations studied which show open orbits. The orientations of (a) and (c) correspond respectively to the closed triangle and open square symbols of Figs. 2-4. The shaded areas give the electronic contribution to p . and the open areas correspond to the bands of open orbits. The right-hand side shows the orientation of these open orbits with respect to the sample and to the direction of the primary current J.
AN1SOTROPIC TEMPERATURE DEPENDENCE OF THE HALL EFFECT H along [112] and the primary current along [1i0] there are both transverse and longitudinal open orbit directions. As we showed in Section 3, in the high-field limit the transverse orbits will not be excited and the total current flow will be along the longitudinal orbits, the closed orbit conductivity being negligibly small by comparison. The longitudinal open orbits thus act as a short-circuit to the Hall effect since the response of electrons in them is independent of H. We expect therefore On ~ 0 as ~-7 increases, and Fig. 2 shows that this behaviour is observed as the temperature is reduced to the residual resistance region in which region ~-~ is limited by residual impurity scattering and On becomes consequently independent of temperature. As the temperature is increased above this region the mean free path of the electrons in the longitudinal open orbit is continuously reduced by the phonon scattering and their ability to short-circuit and Hall effect decreases corespondingly. Thus On increases with temperature and, as would be expected, is found to vary linearly with the resistivity (i.e. with the electronic mean free path) up to a temperature ( - 2 8 ° K ) above which the mean free path is short enough that the longitudinal open orbits are no longer influential. When the sample is rotated through 5 ° about the primary current axis the situation becomes as depicted in Fig. 6(c) and (d) (shown by open squares in Fig. 2). Only a narrow band of open orbits is possible and, since these are in the transverse direction, they are not excited. For this orientation we have therefore no operative open orbits and no effective closed hole orbits. The temperature dependence of pn is thus expected to be monotonic up to the low-field region, andRn should be given by (3). The first expectation is observed, while the second could only be checked quantitatively if an elaborate programme such as that used by Klauder [17] was available. However, it seems from the value of Rn observed at 4.2°K for this orientation that there must be a fault
185
in the above argument since Rn corresponds to an Ne [of (3)] of 1"86, and it is difficult to understand how any orientation free from open orbit and hole orbit contributions can appear more free electron like than the free electron value for a monovalent metal. (We note that the results of Saeger and LiJck [5] for this orientation confirm our value of Rn.) And yet it seems clear from the results for the alloys, which are discussed in the following Section, that this orientation does not in fact contain any singular high-field topological features, such as open orbits or easily-quenched hole orbits. It is the only orientation among those studied for which there is no significant change in the temperature dependence of On when solute is added (see Fig. 4). Furthermore when the singular low temperature topological features of say the [112] and the open triangle orientations are removed to the low-field condition by the addition of solute, the temperature dependence of On then follows that for the orientation presently under discussion (see Figs. 3 and 4). We can at the present time offer no explanation why this orientation corresponds to an apparently unreasonable value of Ne. We referred above to the observation (Fig. 2) that for temperatures above about 30°K and up to the low-field limit the same temperature dependence of On is observed for all orientations except H along [001] and [110]. This can be understood simply from Figs. 5 and 6. As the dominant topological feature which differentiates the orientations at 4.2°K (e.g. the open orbits, the six-cornered rosette, or the unspecified hole orbits which we believe contribute to the open triangle orientation) is quenched to the low-field condition by increasing temperature, the remaining orbits in the high-field condition will be entirely closed electron orbits. The high-field portion of On will therefore be determined by the volume of the Fermi surface, which is independent of the orientation. Whereas with H along [001] and [110] the singular features in the low temperature
186
C.M.
H U R D and J. E. A. A L D E R S O N
results come respectively from the hole orbits of the four-cornered rosette and dog's bone type. Some of these orbits can persist in the high-field condition up to about 50°K, and it is presumably their presence which is responsible for the anisotropy which is observed (in the approximate range 30-50°K) between these two orientations and all the others we have considered. 5. DISCUSSION OF THE RESULTS FOR THE ALLOYS
With 76 ppm Zn in solid solution the results (Figs. 3 and 4) indicate that the electronic mean free path is sufficiently reduced at 4.2°K that with the fields available the bulk of the orbits which distinguish between the [111], [211], the open square, and the open triangle orientations never achieve the high-field condition. The temperature dependence Of pn for these orientations then tends to the same value (a feature which is even more pronounced with 172 ppm Zn) since, with these distinguishing features removed, the high-field Hall effect is determined solely by electronic orbits and thus by the volume of the Fermi surface. The addition of solute therefore drastically changes the temperature dependence of pn (see Fig. 4) for the [111] orientation, since the bulk of the six-cornered rosette type of orbits are never fully in the high-field condition. Similarly that for the [211] orientation is also greatly changed because the influence of the longitudinal open orbits is reduced and they can no longer effectively short circuit the closed orbit contribution. The temperature dependence of Pn for the open triangle orientation (situation b of Section 1) is changed below about 25°K, indicating that some contribution from hole orbits existed in the pure Cu crystal and has been lost to the lowfield region as solute is added. We pointed out in Section 4 how this is contrary to expectations [7, 13]. Only the temperature dependence of pn for the open square orientation is relatively unaffected by the addition of solute.
Apart from the unlikely possibility that this results from a fortuitous cancellation of two competing contributions, this result suggests that for this orientation in pure Cu there are no effective open orbits or easily-quenched closed orbits which make a contribution to the Hall effect. We referred to this point in the preceding Section. The temperature dependence of p , for the [001] and [110] orientations with 76 ppm Zn retained its form found in pure Cu, although IOn[ at any temperature is reduced by the addition of solute. This indicates that some of the four cornered rosette and dog's bone types of orbit have passed to the low-field condition, but the persistence of the bulk of these orbits in the high-field condition as solute is added is shown (Figs. 3 and 4) by the clear distinction between the temperature dependence observed for the [001] and [110] orientations compared with that of all the other orientations. Note that the addition of 76 ppm Zn makes relatively no change in the results for the polycrystalline alloys (cf Figs. 2-4). This indicates how little the singular topological features of the high-symmetry orientations contribute to the Hall effect in a polycrystalline sample in the high-field region. For example, although a specific orientation (say [111]) has an enormous OH value in the high-field limit, the occurrence of this exact orientation among the crystallites of a polycrystalline sample is so rare that our results (which we shall presume are typical of a random polycrystal) show that it makes no significant contribution to p,. (Its contribution is in any case offset by that from the [211] orientation and this cancellation effect may also be important in a polycrystal.) It is clear that the bulk of the effect in a polycrystal in the high-field region comes from orbitals typical of the [110], [100] and open square orientations. This can also be seen from the results of Saeger and LiJck [13]. A surface integral of Rn taken over their unit stereographic triangle (i.e. their Fig. 2) should give the value of RH for a random
ANISOTROPIC TEMPERATURE
DEPENDENCE
polycrystalline sample, and it is clear from their figure that the singular orientations of high symmetry would make only a very small contribution to the integral. The lower part of Fig. 3 shows the comparison between the results for a single crystal containing 1100 ppm Zn and a polycrystal containing 940 ppm Zn. On in the single crystal showed no detectable anisotropy at 4.2°K for any orientation, indicating that for this concentration in the fields available none of the orbits lying on singular features of the topology of the Fermi surface were in the high-field condition.
6. CONCLUSIONS
In describing some similar measurements Saeger[20] has presented his data in the form of Kohler plots (i.e. RH vs. p/B, where p/B can be either f(B)T or f(T)B; p is the electrical resistivity). Saeger invokes a somewhat complicated argument in terms of the relative magnitude of different types of scattering to explain his observed deviations from Kohler's rule. The present results could also be presented in this manner, but it seems to us that such plots do not lead to any greater fundamental understanding than say Figs. 2 and 3. For example, our results show that Kohler's rule is obeyed for each orientation of the pure Cu sample up to about 35°K, while those for the Cu + 76 ppm Zn show that it is obeyed to different temperatures along different orientations. All of these plots, both for the pure Cu and the alloys, can be summarized by the remark that empirically Kohler's rule is found to be obeyed for a given orientation only so long as the sample is in a high-field condition. This is just a statement of validity of Kohler's rule; in the high-field condition the galvonamagnetic properties are independent of the nature of the scattering process (under which conditions Kohler's rule is defined) while in the low-field condition any anisotropy of the scattering mechanism becomes significant. Kohler's rule is not valid
OF THE HALL EFFECT
187
under these circumstances, and hence the observed deviations from it. Theoretically the low-field condition for a given orbit is defined by tor,~ 1, which experimentalists frequently interpret as to--7< 1 for a bulk sample, Our results show the unreliability of ~-7 as calculated from free electron theory as a measure of the low-field condition. Even for to--7~ 0.16 there is still a substantial effect due to those orbits which remain in the high-field condition. This work has a bearing upon the local maximum which has been observed in the temperature dependence of Pn for the group IB metals at about 50°K[1, 3, 13]. It is clear that of the singular topological features characteristic of given symmetry directions (e.g. open orbits, six-cornered rosette type of orbit, etc.) only the dog's bone and the fourcornered rosette type of orbit contribute significantly to the Hall effect in a polycrystalline sample. The temperature dependence of their contribution to Pn in a polycrystalline alloy will be of the form found for the [100] and [110] orientations in say the Cu + 76 ppm Zn (Fig. 3); i.e. it is relatively monotonic outside the residual impurity range. The contributions from other orientations, however, will have the typical temperature dependence shown for the open square orientation in Fig. 3. This shows a local maximum at - 45°K as On passes from a value at high temperature, which is determined by the anisotropy of the dominant scattering process, to one (at - 4.2°K) determined solely by the topological features of those orbits still in the high-field condition. Since On in a polycrystal is an average of these two types of contribution it is clear (though we have not proved it) that pn could show a maximum at 50°K. Note that this argument does not require the Fermi surface to be either open or even multiply-connected, since we have shown that the only orientations for which these topological features contribute to the Hall effect in a polycrystal give, for Cu at least, a monotonic variation of On with
188
C.M.
H U R D and . I . E . A .
temperature in the range where the anomalous maximum is observed. Perhaps here lies an explanation for the fact that a somewhat similar anomalous maximum has been observed in numerous metals with very different Fermi surfaces, viz. the alkali metals/21], Cd and Zn [5] and the group-lB metals [4]. It might be said that against the above argument rests the fact that the anomalous maximum exists in single crystal (Fig. 4) and polycrystalline noble metal alloys/3] which, judging from the lack of anisotropy o f p n in the single crystals, are apparently in the low-field condition throughout the experiment (e.g. C u + l l 0 0 ppm Zn in Fig. 4 or/3]). There would appear to be no question in these alloys of a transition between the low- and high-field conditions. An answer to this criticism is that a lack of anisotropy of On for a single crystal does not necessarily imply that all the orbits are in a low-field condition. In the high-field condition the contribution to the Hall effect from orbits lying on a closed electron sheet is proportional to the volume of phase space enclosed by the surface, and is therefore isotropic. Consequently, once the singular features of the Fermi surface in Cu are removed to the lowfield condition either by increasing temperature or through the addition of solute, there will remain no evident anisotropy of o n - b u t the contributing electronic orbits may be still in the high-field condition. (Such is probably the case for the example of the [ 111 ] orientation of Fig. 2 in the range above - 30°K.) An anisotropy in the temperature dependence of pn in a single crystal is a sufficient but not a necessary condition of a prevailing high-field state. It follows from all of this that the results for C u + 1100 ppm Zn can be representative of a sample which is not entirely in the low-field condition, and it remains entirely possible that the anomalous maximum in pn observed in polycrystalline samples arises from the effect of the low-field/high-field transition. Saeger [20] has observed for a crystal of 'pure' Cu
ALDERSON
( R R R -~ 168, indicating a purity less than our
C u + 176 ppm Zn) that the position of the anomalous maximum did not vary when ~-~ was changed by a factor of two. This is cited as evidence against the preceding conclusion. An explanation of this now follows from our results: comparison of Figs. 2 and 3 shows that equivalent or greater changes of co---~ produced by the addition of solute make relatively little change to the temperature dependence in the region of 50°K observed for the various orientations. Thus Saeger's results are in agreement with ours, but they do not preclude, in our opinion, the possibility that the anomalous peak arises from the low-field/high-field transition. Finally, it should be mentioned that the qualitative interpretation proposed in Sections 4 and 5 is conceptually the same as that already suggested by Ashcroft[22] to explain the field dependence of the Hall coefficient of AI and In. In these metals it is thought that the conduction process can be adequately represented by a two-band model formed respectively from the 'heavy' holes of the second Brillouin zone and the 'light' electrons of the third zone. In the data considered by Ashcroft ~ is changed by varying the applied field strength, not the temperature as in our case, and Ashcroft shows that with two-band conduction of the above type the sign change of Rn frequently observed in these metals is brought about by the high-field/low-field transition of the hole surface in the second zone. In the high-field condition this surface supports, of course, only hole-like orbits, but in the low-field condition a representative point on it may have either locally hole or electron-like properties depending upon whether it is close to edges of the intersecting surfaces in the zone. Acknowledgements- I t is a pleasure to thank P. Tymchuk, E. C. Goodhue and J. A. H. Desaulniers of the Analytical Section, and F. Turner of Materials Preparation for the work they have done on our behalf, and to acknowledge the cooperation of L. D. Calvert and his staffof the X-ray Section. We are grateful to I. M. Templeton for commenting upon the manuscript.
ANISOTROPIC TEMPERATURE
DEPENDENCE
REFERENCES
I. D U G D A L E J. S. and F I R T H L. D., J. Phys. C. (Solid State) 2, 1272 (1969). 2. F A W C E T T E.,Adv. Phys. 13, 139 (t964). 3. A L D E R S O N J. E. A., F A R R E L L T. and H U R D C. M., Phys. Rev. B 1, 3904 (1970). 4. A L D E R S O N J. E. A . , F A R R E L L T . and H U R D C. M., Phys. Rev. 174, 729 (1968). 5. S A E G E R K. E. and L U C K R., Phys. Kondens. Mater. 9, 91 (1969). 6. H U P P M A N N W. and S T A N G L E R F., Phys. Status. Solidi 31, KI07 (1969). 7. K L A U D E R J. R., REED W. A., B R E N N E R T G. F. and K U N Z L E R J. E., Phys. Rev. 141, 592 (1966). 8. Z I M A N J. M., Phil. Mag. 3, I 117 (1958). 9. K U N Z L E R J. E. and K L A U D E R J. R., Phil. Mag. 6, 1054(1961). 10. LIPSON S. G., Proc. R. Soc. (London) A293, 275 (I 966). I I. C H A M B E R S R. G., Proc. R. Soc. (London) A238, 344 (1956). 12. A L E K S E E V S K I ! N. E. and G A 1 D U K O V Yu. P.,
OF THE HALL EFFECT
189
Zh. Eksp. teor. Fiz. 42, 69 (1962). [English translation: Soviet Phys. JETP 15, 49 ( 1962)]. 13. S A E G E R K. E. and LLICK R., Phys. Status. Solidi 32, 161 (1969). 14. KOSTER W., H A G M A N N D. and S A E G E R K. E., Z. Metallk. 54, 619 (1963). 15. S C H O E N B E R G D. and R O A F D. J., Phil. Trans. R. Soc. A255, 85 (1962). 16. P I P P A R D A. B., The Dynamics of Conduction Electrons, Blackie and Son Ltd., London (1965); Rep. Prog. Phys. 23, 176 (1960). 17. KLA U DER J. R., Bell Syst. tech. J. 40, 1349 ( 1961 ). 18. H A L S E M. R., in The PhysicsofMetals-I Electrons (Edited by J. M. Ziman) Cambridge University Press, 1969; Thesis, Cambridge University (1967). 19. Z I M A N J. M., Phys. Rev. 121, 1320 (1961). 20. S A E G E R K. E., Phys. Status. Solidi. 28, 589 (1968). 2 I. A L D E R S O N J. E. A. and F A R R E L L T., Phys. Rev. 185, 876 (1969). 22. A S H C R O F T N. W., Phys. Kondens. Mater. 9, 45 (1969). 23. D A T A R S W. R., Phys. Lett. 29A, 700 (1969).
Pergamon Press 1971. Vol. 32, pp. 175-189.
Printed in Great Britain.
ANISOTROPIC TEMPERATURE DEPENDENCE THE HALL EFFECT IN Cu AND SOME Cu-Zn CRYSTALS
OF
C. M. HURl) and J. E. A. ALDERSON National Research Council of Canada, Ottawa, Canada
(Received 4 May 1970) A b s t r a c t - T h e Hall effect in single crystals o f C u (RRR --- 2312) and Cu containing 76, 172 and 1 I00 ppm Zn has been studied in the temperature range 4.2-80°K (i.e. through the low-field/high-field transition). The temperature dependence of the effect is given for H along [111], [I 10], [100], [211l, and other directions which show either a single open orbit or all closed orbits. It is shown how the qualitative features of this dependence can be explained in terms of the known topology of the Fermi surface. Comparable results are also given for polycrystalline samples of Cu, and Cu with 78, and 940 ppm Zn. These indicate that of the singular topological features associated with the open Fermi surface of Cu only orbits of the four-cornered rosette and dog's bone type contribute to the Hall effect in a polycrystal. Some further evidence is given to support the view that the anomalous maximum in the temperature dependence of the effect in a polycrystal, which is centred at about 50*K, arises from the low-field/high-field transition. 1. INTRODUCTION
there are two limiting conditions under which measurements of the galvanomagnetic properties of a metal can be made. In the first case the lifetime of an electron between collisions remains short enough that the topological features of the Fermi surface do not manifest themselves in the electronic motion. This is the low-field case, and in this condition (providing there is sufficient detailed information available about the shape of the Fermi surface) the Hall coefficient (RH) can give information about the anisotropy of the dominant scattering process[l]. In the second case the lifetime of the electron is very long compared with the characteristic time necessary to complete a cyclotron orbit, and the effect of collisions is then a negligible perturbation of the electronic motion compared with the effect of the externally applied fields. This is the high-field condition. In this limit the galvanomagnetic properties are independent of the dominant scattering process[2] and are determined solely by the topological features of the Fermi surface.
THEORETICALLY
In the experimental study of the Hall effect there is no sharp transition between these two conditions. This is because contributions to the Hall field arise from all closed cyclotron orbits in planes perpendicular to the applied magnetic field (H). As the temperature or the impurity concentration is reduced, or the applied field increased, those orbits for which an- passes a certain critical value will pass into the high-field condition. There may be a wide range of to~- values covering a decade or more in which orbits of different effective area pass successively into the high-field condition as the above variables change. In this range the sample is in a mixed high- and low-field state and its condition cannot accurately be descrJbed by either of the above classifications. (to is here the cyclotron frequency corresponding to an applied field H, and z is an average electronic relaxation time for an orbit.) The possibility that a Hall effect experiment may be conducted in a mixed high- and lowfield condition, or that a transition may take place between these conditions during the experiment, complicates considerably the interpretation of the results. For example, we 175
176
C.M.
H U R D and J. E. A. A L D E R S O N
recently showed elsewhere[3] that the field dependence which has been observed for the Hall coefficients of pure, polycrystalline samples of the 1B metals, and which has been attributed to the pure metal[l], is apparently the result of an unrecognized transition between the high- and low-field conditions during the experiment. The presence of trace impurities can reduce the value of ~-7 (where the bar signifies an averaged value over all orbits, such as that obtained from free electron theory) to the point where the transition occurs in the range of H used in the experiment, thereby producing the apparent field dependence of Rn. This field dependence is not present when the impurity concentration is either reduced, so that the high-field condition is maintained throughout the experiment, or is increased so that the low-field condition is maintained throughout [3]*. A similar transition between the high- and low-field conditions must also take place in each of the recent studies of the temperature dependence of RM for high-purity, polycrystalline samples of the IB metals[l, 4-6]. This follows from the fact that samples of comparable purity subjected to comparable magnetic fields are known[3-5, 7] to be in a predominantly high-field condition at 4.2°K, and must therefore pass to the low-field condition when the temperature is increased sufficiently. Not much attention has been paid to the effect of this transition upon the observed temperature dependence of RM. Possibly this is because it involves the problem of how the Hall effect in a polycrystalline sample in the high-field condition should be interpreted. Much is known about the influence of the topology of *It now appears to us that this should have been obvious a priori since a polycrystalline sample gives an average over all situations (a)-(d) of Section I, while the high-field theory shows[2] that of these orientations only (d) gives a Hall voltage which varies non-linearly with H. Since this tends to zero in the high-field limit, the highfield value of R . must be field independent.
the Fermi surface on the Hall effect of each crystallite of a given orientation [2], but how are the contributions from the (presumably) randomly oriented crystallites to be combined to give the macroscopic effect measured in a polycrystalline sample? To find the full analytical solution to this question seems to be a formidable problem [8] and for all practical purposes not much progress has been made with it. The interpretation of high-field results for polycrystalline samples of a metal with an open Fermi surface remains an uncertain procedure [2]. The work described in this paper has been carried out partly in the hope of improving the understanding of this problem (if only from an empirical point of view) and partly to study the effect of electron scattering by different foreign elements upon the Hall effect in a 1B metal. We have measured the Hall effect as a function of temperature for a given direction of applied magnetic field in single cryst.al samples of Cu, and Cu doped with Zn. The results are given in Section 2. We have considered all the principal situations of interest which, in the high-field limit, the topology of the Fermi surface affords:(a) With H along directions of high symmetry, < 111 >, < 110>, < 100>, showing only closed electron orbits or closed hole orbits formed from intersecting open orbits. (b) With H along a direction well away from high symmetry directions, where free electron behaviour might be expected [7]. (c) With H along a symmetry direction showing a single open orbit. (d) With H along a direction, < 211 >, showing two nonintersecting open orbits of different orientations. Apart from the recent results of Saeger and Liick for Cu [5], there appears to have been no other attempt at a systematic study of the temperature dependence of R , in single crystals of the 1B metals, the earlier results for single crystals having usually been obtained at a fixed temperature ( - 4.2°K)[7, 9-14]. Our results are qualitatively in agreement with
ANISOTROPIC TEMPERATURE
DEPENDENCE
those of Saeger and Liick but quantitative comparison is impossible since their sample was not in general oriented exactly to give any of the situations (a)-(d) above. The detailed interpretation is thus difficult since for a small deviation of the field direction from a direction of high symmetry the open orbit contribution influences the temperature dependence of RH in a complicated manner. Orientations of H for which the temperature dependence of RH for Cu is available in the literature are shown in Fig. i.
[Ill]
[ioo1
[I,o1
Fig. 1. Orientations of H for which the temperature dependence.of R , in Cu is available in the literature. Closed symbols represent the present work, and open symbols that of Saeger and Liick [5]. 2. EXPERIMENTAL METHOD AND RESULTS The samples were prepared from Cu of 6N purity of which typical analyses have been given elsewhere [3]. The R R R values (R2~a°K/R4.2*K) of all the samples are given in the captions to Figs. 2 and 3. The single crystal samples were grown by the Bridgman method in high-purity graphite molds from previously prepared polycrystalline alloys of the desired concentration. Specially oriented pure Cu seed crystals were used. The quoted concentration of Zn is the mean of values obtained from off-cuts of each end of the sample, since the method of preparation was naturally conducive to some zone refining during the growth. (The analyses were made by the Analysis Section of N R C C using the method of atomic absorption.) Other details of the sample preparation, the instrumentation and techniques, are identical to those described previously [3, 4]. The temperature dependence of the Hall resistivity Pn (the Hall voltage per unit primary current density) is shown for a pure Cu sample in Fig. 2, and that for three C u - Z n single crystals (with 72, 172 and 1100 ppm. Atomic ppm is implied throughout.) is shown in Fig. 3. These figures also include for comparison the results for polycrystalline samples of Cu, C u + 7 8 ppm Zn and Cu + 940 ppm Zn. The primary current direction is along [IT0] in the single crystals, and the applied magnetic field
J.P C.S. Vol. 32N o. I - - L
OF THE HALL EFFECT
177
(H) is 15.17 kG. The values of ~'~ were calculated from the resistivity of the samples using free electron theory. The effect upon the temperature dependence of p , of additions of Zn is shown individually for each orientation in Fig. 4. The field dependence of the Hall voltage (vn) at 4.2°K was measured both for the polycrystalline samples and for each orientation of the single crystals, vn varies linearly with H for the polycrystalline samples, even though they are in a high-field condition at this temperature (see the discussion in[3]). For the single crystal Cu sample vu follows the expectations of the high-field theory [2]: for sufficiently high fields (above - 5 kG in most cases) it varies linearly with H for the [110], [100], the open square (Fig. 2), and open triangle orientations (i.e. respectively for the situations (c) and (b) of Section 1). For H along [211] Vn decreased roughly as H -t, in agreement with theory. For H along [ 111 ] it could not be said with certainty that vz had attained a linear variation with H at the maximum field available, which presumably indicates that the true high-field condition was not obtained for this orientation. The same applies for all the orientations (except H along [100] and [110])in the Cu + 76 ppm Zn crystal. The presence of the solute makes it impossible to obtain the true high-field condition with the maximum field available. This gives rise to an apparent field dependence of R,, as was discussed in[3], and for this reason we shall throughout discuss the behaviour of On(= R n H ) . 3. H A L L EFFECT IN T H E HIGH-FIELD LIMIT We shall assume that the Fermi surface of Cu is sutficiently well known to need no detailed description [15]. It supports both open and closed cyclotron orbits depending upon the direction of the applied magnetic field H relative to the crystal axes. In the high-field limit we shall differentiate between two cases depending upon whether open orbits in a plane normal to H are (a) impossible, or (b) possible. It is convenient to review briefly some of the results of the theory for these two cases[2, 16]. To simplify the expressions we take H to be applied along the z-axis and E, the external electric field, to be along the x-axis.
(a) There are no open orbits in planes n o r m a l to H The theory in this case is based upon the assumption that the conductivity tensor o-u can be expressed by a series expansion in H tr u = au + cu H - t + bu H-~ + . . . .
( 1)
where a, b and c are constants. In the high-field limit this becomes [2, 16], since we have taken E= = 0 biH-~
oH-=
(2)
where we have simplified the subscripts (bxz =- bt, exu --- c, etc.). It follows that the Hall coefficient (pxulH, where p• is inverse of o'o) tends to a constant value independent of H in the high-field limit, it can be shown
C. M. H U R D and J. E. A. A L D E R S O N
178
tO't" (free electron) 16
250
8 I
I
1.7
0-5
I
I
FIELD DIRECTION
='\
200
[.,1
-2
o-I°-
•
ol ;--"~|
~ \ \
I
]
i<> E
150
\\
To
[.ol
.... POLYCRYSTAL o 30"2*FROM ['001
I ~
N
a
[IOOl
~
",.~
7"3°FROM[100-111}AXIS
Cu
I00
-
5C
/
/
,____.J 0
I
I
I
20
40
60
80
TEMPERATURE ( ° K ) Fig. 2. Temperature dependence of the Hall resistivity at 15-17 kG for a crystal of Cu with R R R = 2312 with H along the directions indicated in the unit stereographic triangle. Comparable results for a polycrystalline sample are shown by the dashed line.
[16] that this limiting value is given by
R.
H~=
=
l
e(Ne--Nh)
when (3)
where Ne and Nh are respectively the (unequal) effective numbers of electrons and holes per atom. They are thus the volumes in k-space per Brillouin zone enclosed within closed electron and hole orbits, scaled so that the Brillouin zone contains exactly two electrons per atom. Rn can therefore be calculated from (3) when the shape of the Fermi surface is known [7, 17]. The limiting value of RH of (3) will begin to be reached
Ne + N~ tin- ~> Ne--Nh"
(4)
From the results of Table 1 we see that for H along < 100 > and < 110 > t,rr must considerably exceed ~ 2, while for H along < l l l > it must considerably exceed ~12. (b) There are open orbits in planes normal to H [ 16[ We consider the case when open orbits having a single axis direction are present. It can be shown that as H is
A N I S O T R O P I C T E M P E R A T U R E D E P E N D E N C E OF T H E H A L L E F F E C T
179
- (a¢" ( f r e e e l e c t r o n )
0.39
0-37
0"26
=
1
I
FIELD
DIRECTION
[ml
"~'~
DOOI
~=~ "~'~ dl'~.~.o.
_ ",-=.
oJE To
~ ,___.I
~__Zl
Cu
0.118
["ol
..... POLYCRYSTAL 0-30-2 ° FROM I100] ~,- 7-3°FROM [IO0-111] AXIS
+
76 ppm Zn
0114 CU+ 172ppm Zn
70-
'°t
_ Ca_+ !,oo .
0
.
.
.
p Zn
I
.
20
40 60 TEMPERATURE (*K)
80
Fig. 3. Temperature dependence of the Hall resistivity at 15.17 kG for crystals of C u - Z n with R R R values (in order of decreasing purity) of 544, 263 and 53. The field is applied along the directions indicated in the unit stereographic triangle. Comparable results are shown by the dashed line for polycrystalline samples with 78 and 940 ppm Zn (RRR values are 447 and 80 respectively). increased the conduction by closed orbits tends to zero (except along the direction of H) and in the high-field limit all the components of o'~ (except o-~z) arising from closed orbits are zero. The current flow is then entirely along the open orbit axis direction and is confined in real space to the plane which is normal to the orbit axis direction and which contains the direction of H. The response of the electrons is independent of H, and thus in the high-field limit the problem is reduced to one of quasi-free electrons moving in a plane in real space containing the direction of H and inclined at some angle (8) to the direction of the applied electric field E. From the geometry of
this situation we can write down directly[16] the limiting value of the conductivity tensor for open orbit conduction as d cos z 0 d cos 0 sin 0 o-~ = d cos O sin O d sin~#
H~oo
(5)
where d is a constant which is related to the flux of electrons moving in the open orbit. If the conductivities of (2) and (5) are assumed to be additive, their sum can be inverted to give the full resistivity tensor at large H for mixed open- and closed-
180
C.M. - 2 . 4 0 --
-
i
HURD I
I
and J. E. A. A L D E R S O N
-0"80~I• -0.40
I
2.00
-1'60 O.Oq
-0.80~/~.
-I.20
-1.00
-0.80 A
-
-0,60
E >
-2.00
-1"201
0
I0
-,.0ol _0.80_,-./ 2"-
-I.60
3:
- I ,20
-0"80 -0"8)
F
0 -I.6C
-I.20
,POLYCRYSTAL
,
20 40 60 80 TEMPERATURE (OK)
FIELD DIRECTION
•
[m) -0"8C
I I I l 20 40 6 0 80 TEMPERATURE (<'K)
0
[llol
[Iool -
-
-
Cu
. . . .
Cu + 7 6 ppm
Zn
E] 3 0 . 2 *
FROM
[100]
......
Cu + 1 7 2 p p m
Zn
~
FROM
[100-111] AXIS
7.3 °
Fig. 4. T h e data of Figs. 2 and 3 plotted for a fixed orientation as a function of solute concentration. T h e direction o f H is indicated by the symbol, which is maintained t h r o u g h o u t Figs. 2-4. in the case of the polycrystalline results, the dashed line refers to C u + 7 8 p p m Z n and the dash-dot line to Cu + 940 p p m Zn.
Table 1. -R,
Field direction
Ne--Nh
A w a y from high symmetry axes < 100> < 111 > < 110>
!.00 0.51 0.068 0.51
--RH Calculated from c o l u m n 2 X 10-11 m 3 a m p -t s e c -I
(as m e a s u r e d at 4.2°K)
K l a u d e r et al. R R R = 8000
T h i s work R R R = 2312
7.4
7.4
7.4
14.5 109.4 14.5
14.5 83.6 11.9
13.1 24.0* 11.9
*Calculated from the local gradient of the v , vs. H curve.
ANISOTROPIC TEMPERATURE DEPENDENCE OF THE HALL EFFECT orbit conduction. [Compare Lipson [10], whose expression differs in detail from that below apparently because Lipson does not take the adjoint of the cofactor when inverting (2) plus (5).] o--
_.--:---v_dH2 I" sin2 0
/t£~=--tr°'=Bd+c~L--cosOsinO c2 [ b J c z Bd+cZ[Hlc
-- sin 0 cos 0] cos'0 J
(6)
--H/c ] b,/c z J
where B =-- b, sin~O+b2cos20. The first and second tensors represent the open and closed orbit contributions respectively. The element which determines the total transverse field can be written dHz
Po = -sin Ocos O n~® Bd + c 2
cH Bd + c 2"
(7)
Equation (7) shows that for the two cases 0 = 0, rr/2 the open orbit contribution to Pxv is zero. Physically this expresses the fact that when the open orbits conduct exactly along the sample no transverse voltage is produced, and when they conduct exactly across the sample they are not excited. However, the two cases are not exactly equivalent physically, for when 0 = ~r/2 we are left with the closed orbit contribution to Rn represented in (7) but when 0 = 0 the conductivity of the closed orbits in the high-field-limit is negligible compared with I~hatof the open orbits along the length of the sample. Both terms in (7) are then zero; the entire current flows in the open orbits and is unaffected by the applied field H. The Hall effect is then 'shorted-out' by the open orbits which lie along the direction of the primary current. 4. DISCUSSION OF THE RESULTS FOR PURE Cu
Our interpretation of these data is based upon the following argument. In the high-field limit Rn is given by (3). As ~-~ is reduced by some means (in this case it is by increasing temperature) and successive orbits pass into the low-field condition, the sign of the variation of Rn (and hence P n) will also be given by (3). F o r example, if there is a net loss of hole states to the low-field condition as the temperature is raised, then the 'high-field' value of Pn will b e c o m e more positive [through (3)]. This value is not, of course, a true high-field value, but as long as On shows a detectable anisotropy then it can be said that the sample is in a partially high-field condition for that orientation. F r o m the foregoing we can determine the sign of the change in On as particular groups of orbits pass into the low-field condition, but
181
what circumstances determine which orbits are in a high-field condition at a given temperature? It is impossible to give a quantitative reply to this without a full knowledge of ~-~ for each orbit, but our qualitative argument is as follows. As the temperature is raised it is known from other e x p e r i m e n t s [ l ] that (in the usual terminology) electrons in neck states are scattered more than electrons in belly states because the umklapp processes from belly states are 'frozen out' at the lowest temperatures[19]. We therefore expect, all other considerations being equal, that as the temperature is raised those orbits which traverse neck regions of the Fermi surface would suffer the transition to the low field condition before those lying entirely on belly regions. Before we consider the temperature dependence of On in detail, we should comment upon the absolute magnitude obtained for Rn at 4.2°K for those field directions which do not support open orbits. T h e s e directions are with H along [110], [111], [100] and the off-symmetry direction shown by the open triangle in Fig. 2. As is customary [7], we have included [110] in this class although strictly speaking this is not permissible since a non-negligible number of open orbits are excited for this orientation[7, 17, 23] and (3) is no longer valid. I n the high-field condition values of N e - Ni, (and hence Rn) can be calculated for the above orientations from the known Fermi surface. Klauder [17] has made the calculation for Cu using R o a f ' s [ 1 5 ] expression for the Fermi surface. T h e results are shown in Table 1. (More sophisticated expressions for the Fermi surface are n o w a d a y s available [18] but the results from Roaf's formulation are adequate for the following qualitative arguments.) T h e r e is a serious discrepancy between our experimental results and those of Kiauder e t al. [7] only for the [111] orientation. This has two possible causes. On the one hand R n can change f r o m - - 6 t o - - 9 0 x 10-11m s a m p - l s e c -~ when the direction of H is shifted
182
C.M. HURD and J. E. A. ALDERSON
a few degrees off [111], so that a slight misalignment of our sample could account for the discrepancy/13]. On the other hand our sample is considerably less pure than that of Kunzler et al. (values of R R R are given in the Table) and, as the results of Fig. 4 show, On for this orientation depends very strongly upon the impurity concentration. We believe that this second possibility is the most probable since we positioned the sample empirically with great care and are confident that the quoted value is representative of a correctly aligned sample. Furthermore, the arguments concerning purity are supported by the values shown for ~-7. It is clear that when the temperature enters the range in which impurity scattering becomes dominant (i.e. below - 15°K) the ~-~ value does not considerably exceed the value ( - 12) required by (4) for the high-field condition. (a) Orientations showing no open orbits The left-hand side of Fig. 5 shows schematically the orientation of the Fermi surface with respect to the direction of the applied magnetic field for the principal closed orbit directions of Fig. 2. The volume of the Brillouin zone within the shaded portion of the Fermi surface corresponds to the electronic contribution to pH while that outside the unshaded areas and lying between the labelled planes corresponds to the hole contributions. The right-hand side of Fig. 5 shows, again schematically but to the same scale, some principal extremal orbits characteristic of each orientation. The free electron circle is dotted in each case. Figure 2 shows that with increasing temperature above the impurity dominated range each of these orientations shows Pn becoming increasingly positive, indicating a net loss of hole states to the low-field condition. For H along [001] and [110] the hole contributions come respectively from bands of orbitals of the four-cornered rosette and dog's bone type (Fig. 5(b) and (f)). The cyclotron masses of their extremal orbits are
approximately equal ( - 1.3m0[18]) so that for a given H their cyclotron frequencies are approximately equal, and each orbit involves four neck passages per cycle with approximately the same length of circumstance. Furthermore, the electronic contributions to p , for these two orientations are similar; a rough graphical integration shows that the total volume of regions A of Fig. 5(a) amounts to only - 0 . 0 5 electrons/atom (e/a) and is negligible compared with the - 0 . 6 6 e/a contributed by the regions B. The regions A of Fig. 5(e) contribute - 0.76e/a. Thus, insofar as the extremal orbits are representative of the bands of orbits under consideration we expect to find for H along [001] and [110] a very similar temperature dependence and magnitude for pn. This is observed (Fig. 2), and the temperature dependence indicates further that as the temperature increases their rates of loss of holes states are approximately equal. This contrasts with the temperature dependence observed for H along [111]. In this orientation the hole contribution comes from a band of orbits of the six-cornered rosette type of which the extremal is shown in Fig. 5(d). (We referred above to the fact that in the true high-field condition the hole states are very predominant in this orientation-see also Table 1.) The cyclotron mass for this extremal orbit is - 2.4 m0 [18], there are six neck passages per cycle and the length of the circumference is about three times that of the four-cornered rosette of [100]. Thus, for a given increase in temperature, it is reasonable to expect more orbits of the six-cornered rosette type to pass into the low-field condition than either of the types of hole orbits considered for H along [001] and [110]. The temperature dependence shown in Fig. 2 supports this expectation, and it appears from those results that all of the six-cornered rosette type of orbit are in the low-field condition at about 28°K while some of the hole orbits for the other orientations of Fig. 5 continue in the high-field condition up to about
A N I S O T R O P I C T E M P E R A T U R E D E P E N D E N C E OF T H E H A L L E F F E C T
183
[oo0
a
b )iq
C
d
I I e Fig. 5. The left-hand side shows schematically the direction of H for the principal orientations studied which show only closed orbits [but see our remark in the text concerning the orientation (e)]. The orientations of (a), (c) and (e) correspond respectively to the open circle, the closed square and the closed circle symbols of Figs. 2-4. The shaded areas give the electronic contribution to Pn. The right-hand side shows schematically, but to the same scale, some of the principal extremal orbits typical of these orientations. The free electron sphere is shown dotted in each case.
50°K. Above about 30°K the behaviour of Pn tends to follow that found for the orientations with H along [112] and the off-symmetry directions, and we shall return to this point at the end of this Section. -the variation ofpH with temperature observed for the open triangle orientation of
Fig. 2 (i.e. with H well away from highsymmetry directions) is not monotonic. This is unexpected since it has been suggested [7] that free electron behavior should be observed for such an orientation since only closed electron orbits were thought to be possible in this case. However, Saeger and Liick[13]
184
C.M.
H U R D and J. E. A. A L D E R S O N
following Section) and that as the temperature is increased above the impurity dominated region (i.e. above - 15°K) these orbits, which presumably pass closer to neck regions than electron orbits, are affected to a greater degree by the phonon scattering. At about 30°K they are all in the low-field condition, and pH follows the behaviour found for H along [111], [112] and the open orbit directions. We shall return to this point in the following.
have measured at 4.2°K the complete dependence of RH for Cu upon crystal orientation, and their results show that within the centre of the unit stereographic triangle the value of RH can vary widely and it only has the free electron value along a very narrow contour. Presumably this contour represents those orientations for which the net effect of the contributions to Rn fortuitously gives the free electron value and does not necessarily imply free electron behavior. Our results support this, since our orientation lies on the contour (as the values quoted in Table 1 show), and yet the temperature dependence is typical of at least two major contributors to OH- It appears from our results that there is some contribution from hole orbits for this orientation (the results obtained for the alloys are particularly relevent in this respect. See the
(b) Orientations showing open orbits The left-hand side of Fig. 6 shows schematically the orientation of the Fermi surface with respect to the applied magnetic field for those orientations showing open orbits. The open orbits lie in the unshaded regions A and B, and have axes oriented in the sample as shown in the right-hand side of the figure. For
[ood H
lilZl OPEN ORBITS S
[ilol
[
Ill
!
~
_
OPER
""'~ 0117o1
]
b
a [ooll
[.z)
~PEN ORBITS A
• [lio]
~.~I
[,TI
":" J [5o]
C
d
Fig. 6. The le•hand side shows schematically the direction of H for the principal orientations studied which show open orbits. The orientations of (a) and (c) correspond respectively to the closed triangle and open square symbols of Figs. 2-4. The shaded areas give the electronic contribution to p . and the open areas correspond to the bands of open orbits. The right-hand side shows the orientation of these open orbits with respect to the sample and to the direction of the primary current J.
AN1SOTROPIC TEMPERATURE DEPENDENCE OF THE HALL EFFECT H along [112] and the primary current along [1i0] there are both transverse and longitudinal open orbit directions. As we showed in Section 3, in the high-field limit the transverse orbits will not be excited and the total current flow will be along the longitudinal orbits, the closed orbit conductivity being negligibly small by comparison. The longitudinal open orbits thus act as a short-circuit to the Hall effect since the response of electrons in them is independent of H. We expect therefore On ~ 0 as ~-7 increases, and Fig. 2 shows that this behaviour is observed as the temperature is reduced to the residual resistance region in which region ~-~ is limited by residual impurity scattering and On becomes consequently independent of temperature. As the temperature is increased above this region the mean free path of the electrons in the longitudinal open orbit is continuously reduced by the phonon scattering and their ability to short-circuit and Hall effect decreases corespondingly. Thus On increases with temperature and, as would be expected, is found to vary linearly with the resistivity (i.e. with the electronic mean free path) up to a temperature ( - 2 8 ° K ) above which the mean free path is short enough that the longitudinal open orbits are no longer influential. When the sample is rotated through 5 ° about the primary current axis the situation becomes as depicted in Fig. 6(c) and (d) (shown by open squares in Fig. 2). Only a narrow band of open orbits is possible and, since these are in the transverse direction, they are not excited. For this orientation we have therefore no operative open orbits and no effective closed hole orbits. The temperature dependence of pn is thus expected to be monotonic up to the low-field region, andRn should be given by (3). The first expectation is observed, while the second could only be checked quantitatively if an elaborate programme such as that used by Klauder [17] was available. However, it seems from the value of Rn observed at 4.2°K for this orientation that there must be a fault
185
in the above argument since Rn corresponds to an Ne [of (3)] of 1"86, and it is difficult to understand how any orientation free from open orbit and hole orbit contributions can appear more free electron like than the free electron value for a monovalent metal. (We note that the results of Saeger and LiJck [5] for this orientation confirm our value of Rn.) And yet it seems clear from the results for the alloys, which are discussed in the following Section, that this orientation does not in fact contain any singular high-field topological features, such as open orbits or easily-quenched hole orbits. It is the only orientation among those studied for which there is no significant change in the temperature dependence of On when solute is added (see Fig. 4). Furthermore when the singular low temperature topological features of say the [112] and the open triangle orientations are removed to the low-field condition by the addition of solute, the temperature dependence of On then follows that for the orientation presently under discussion (see Figs. 3 and 4). We can at the present time offer no explanation why this orientation corresponds to an apparently unreasonable value of Ne. We referred above to the observation (Fig. 2) that for temperatures above about 30°K and up to the low-field limit the same temperature dependence of On is observed for all orientations except H along [001] and [110]. This can be understood simply from Figs. 5 and 6. As the dominant topological feature which differentiates the orientations at 4.2°K (e.g. the open orbits, the six-cornered rosette, or the unspecified hole orbits which we believe contribute to the open triangle orientation) is quenched to the low-field condition by increasing temperature, the remaining orbits in the high-field condition will be entirely closed electron orbits. The high-field portion of On will therefore be determined by the volume of the Fermi surface, which is independent of the orientation. Whereas with H along [001] and [110] the singular features in the low temperature
186
C.M.
H U R D and J. E. A. A L D E R S O N
results come respectively from the hole orbits of the four-cornered rosette and dog's bone type. Some of these orbits can persist in the high-field condition up to about 50°K, and it is presumably their presence which is responsible for the anisotropy which is observed (in the approximate range 30-50°K) between these two orientations and all the others we have considered. 5. DISCUSSION OF THE RESULTS FOR THE ALLOYS
With 76 ppm Zn in solid solution the results (Figs. 3 and 4) indicate that the electronic mean free path is sufficiently reduced at 4.2°K that with the fields available the bulk of the orbits which distinguish between the [111], [211], the open square, and the open triangle orientations never achieve the high-field condition. The temperature dependence Of pn for these orientations then tends to the same value (a feature which is even more pronounced with 172 ppm Zn) since, with these distinguishing features removed, the high-field Hall effect is determined solely by electronic orbits and thus by the volume of the Fermi surface. The addition of solute therefore drastically changes the temperature dependence of pn (see Fig. 4) for the [111] orientation, since the bulk of the six-cornered rosette type of orbits are never fully in the high-field condition. Similarly that for the [211] orientation is also greatly changed because the influence of the longitudinal open orbits is reduced and they can no longer effectively short circuit the closed orbit contribution. The temperature dependence of Pn for the open triangle orientation (situation b of Section 1) is changed below about 25°K, indicating that some contribution from hole orbits existed in the pure Cu crystal and has been lost to the lowfield region as solute is added. We pointed out in Section 4 how this is contrary to expectations [7, 13]. Only the temperature dependence of pn for the open square orientation is relatively unaffected by the addition of solute.
Apart from the unlikely possibility that this results from a fortuitous cancellation of two competing contributions, this result suggests that for this orientation in pure Cu there are no effective open orbits or easily-quenched closed orbits which make a contribution to the Hall effect. We referred to this point in the preceding Section. The temperature dependence of p , for the [001] and [110] orientations with 76 ppm Zn retained its form found in pure Cu, although IOn[ at any temperature is reduced by the addition of solute. This indicates that some of the four cornered rosette and dog's bone types of orbit have passed to the low-field condition, but the persistence of the bulk of these orbits in the high-field condition as solute is added is shown (Figs. 3 and 4) by the clear distinction between the temperature dependence observed for the [001] and [110] orientations compared with that of all the other orientations. Note that the addition of 76 ppm Zn makes relatively no change in the results for the polycrystalline alloys (cf Figs. 2-4). This indicates how little the singular topological features of the high-symmetry orientations contribute to the Hall effect in a polycrystalline sample in the high-field region. For example, although a specific orientation (say [111]) has an enormous OH value in the high-field limit, the occurrence of this exact orientation among the crystallites of a polycrystalline sample is so rare that our results (which we shall presume are typical of a random polycrystal) show that it makes no significant contribution to p,. (Its contribution is in any case offset by that from the [211] orientation and this cancellation effect may also be important in a polycrystal.) It is clear that the bulk of the effect in a polycrystal in the high-field region comes from orbitals typical of the [110], [100] and open square orientations. This can also be seen from the results of Saeger and LiJck [13]. A surface integral of Rn taken over their unit stereographic triangle (i.e. their Fig. 2) should give the value of RH for a random
ANISOTROPIC TEMPERATURE
DEPENDENCE
polycrystalline sample, and it is clear from their figure that the singular orientations of high symmetry would make only a very small contribution to the integral. The lower part of Fig. 3 shows the comparison between the results for a single crystal containing 1100 ppm Zn and a polycrystal containing 940 ppm Zn. On in the single crystal showed no detectable anisotropy at 4.2°K for any orientation, indicating that for this concentration in the fields available none of the orbits lying on singular features of the topology of the Fermi surface were in the high-field condition.
6. CONCLUSIONS
In describing some similar measurements Saeger[20] has presented his data in the form of Kohler plots (i.e. RH vs. p/B, where p/B can be either f(B)T or f(T)B; p is the electrical resistivity). Saeger invokes a somewhat complicated argument in terms of the relative magnitude of different types of scattering to explain his observed deviations from Kohler's rule. The present results could also be presented in this manner, but it seems to us that such plots do not lead to any greater fundamental understanding than say Figs. 2 and 3. For example, our results show that Kohler's rule is obeyed for each orientation of the pure Cu sample up to about 35°K, while those for the Cu + 76 ppm Zn show that it is obeyed to different temperatures along different orientations. All of these plots, both for the pure Cu and the alloys, can be summarized by the remark that empirically Kohler's rule is found to be obeyed for a given orientation only so long as the sample is in a high-field condition. This is just a statement of validity of Kohler's rule; in the high-field condition the galvonamagnetic properties are independent of the nature of the scattering process (under which conditions Kohler's rule is defined) while in the low-field condition any anisotropy of the scattering mechanism becomes significant. Kohler's rule is not valid
OF THE HALL EFFECT
187
under these circumstances, and hence the observed deviations from it. Theoretically the low-field condition for a given orbit is defined by tor,~ 1, which experimentalists frequently interpret as to--7< 1 for a bulk sample, Our results show the unreliability of ~-7 as calculated from free electron theory as a measure of the low-field condition. Even for to--7~ 0.16 there is still a substantial effect due to those orbits which remain in the high-field condition. This work has a bearing upon the local maximum which has been observed in the temperature dependence of Pn for the group IB metals at about 50°K[1, 3, 13]. It is clear that of the singular topological features characteristic of given symmetry directions (e.g. open orbits, six-cornered rosette type of orbit, etc.) only the dog's bone and the fourcornered rosette type of orbit contribute significantly to the Hall effect in a polycrystalline sample. The temperature dependence of their contribution to Pn in a polycrystalline alloy will be of the form found for the [100] and [110] orientations in say the Cu + 76 ppm Zn (Fig. 3); i.e. it is relatively monotonic outside the residual impurity range. The contributions from other orientations, however, will have the typical temperature dependence shown for the open square orientation in Fig. 3. This shows a local maximum at - 45°K as On passes from a value at high temperature, which is determined by the anisotropy of the dominant scattering process, to one (at - 4.2°K) determined solely by the topological features of those orbits still in the high-field condition. Since On in a polycrystal is an average of these two types of contribution it is clear (though we have not proved it) that pn could show a maximum at 50°K. Note that this argument does not require the Fermi surface to be either open or even multiply-connected, since we have shown that the only orientations for which these topological features contribute to the Hall effect in a polycrystal give, for Cu at least, a monotonic variation of On with
188
C.M.
H U R D and . I . E . A .
temperature in the range where the anomalous maximum is observed. Perhaps here lies an explanation for the fact that a somewhat similar anomalous maximum has been observed in numerous metals with very different Fermi surfaces, viz. the alkali metals/21], Cd and Zn [5] and the group-lB metals [4]. It might be said that against the above argument rests the fact that the anomalous maximum exists in single crystal (Fig. 4) and polycrystalline noble metal alloys/3] which, judging from the lack of anisotropy o f p n in the single crystals, are apparently in the low-field condition throughout the experiment (e.g. C u + l l 0 0 ppm Zn in Fig. 4 or/3]). There would appear to be no question in these alloys of a transition between the low- and high-field conditions. An answer to this criticism is that a lack of anisotropy of On for a single crystal does not necessarily imply that all the orbits are in a low-field condition. In the high-field condition the contribution to the Hall effect from orbits lying on a closed electron sheet is proportional to the volume of phase space enclosed by the surface, and is therefore isotropic. Consequently, once the singular features of the Fermi surface in Cu are removed to the lowfield condition either by increasing temperature or through the addition of solute, there will remain no evident anisotropy of o n - b u t the contributing electronic orbits may be still in the high-field condition. (Such is probably the case for the example of the [ 111 ] orientation of Fig. 2 in the range above - 30°K.) An anisotropy in the temperature dependence of pn in a single crystal is a sufficient but not a necessary condition of a prevailing high-field state. It follows from all of this that the results for C u + 1100 ppm Zn can be representative of a sample which is not entirely in the low-field condition, and it remains entirely possible that the anomalous maximum in pn observed in polycrystalline samples arises from the effect of the low-field/high-field transition. Saeger [20] has observed for a crystal of 'pure' Cu
ALDERSON
( R R R -~ 168, indicating a purity less than our
C u + 176 ppm Zn) that the position of the anomalous maximum did not vary when ~-~ was changed by a factor of two. This is cited as evidence against the preceding conclusion. An explanation of this now follows from our results: comparison of Figs. 2 and 3 shows that equivalent or greater changes of co---~ produced by the addition of solute make relatively little change to the temperature dependence in the region of 50°K observed for the various orientations. Thus Saeger's results are in agreement with ours, but they do not preclude, in our opinion, the possibility that the anomalous peak arises from the low-field/high-field transition. Finally, it should be mentioned that the qualitative interpretation proposed in Sections 4 and 5 is conceptually the same as that already suggested by Ashcroft[22] to explain the field dependence of the Hall coefficient of AI and In. In these metals it is thought that the conduction process can be adequately represented by a two-band model formed respectively from the 'heavy' holes of the second Brillouin zone and the 'light' electrons of the third zone. In the data considered by Ashcroft ~ is changed by varying the applied field strength, not the temperature as in our case, and Ashcroft shows that with two-band conduction of the above type the sign change of Rn frequently observed in these metals is brought about by the high-field/low-field transition of the hole surface in the second zone. In the high-field condition this surface supports, of course, only hole-like orbits, but in the low-field condition a representative point on it may have either locally hole or electron-like properties depending upon whether it is close to edges of the intersecting surfaces in the zone. Acknowledgements- I t is a pleasure to thank P. Tymchuk, E. C. Goodhue and J. A. H. Desaulniers of the Analytical Section, and F. Turner of Materials Preparation for the work they have done on our behalf, and to acknowledge the cooperation of L. D. Calvert and his staffof the X-ray Section. We are grateful to I. M. Templeton for commenting upon the manuscript.
ANISOTROPIC TEMPERATURE
DEPENDENCE
REFERENCES
I. D U G D A L E J. S. and F I R T H L. D., J. Phys. C. (Solid State) 2, 1272 (1969). 2. F A W C E T T E.,Adv. Phys. 13, 139 (t964). 3. A L D E R S O N J. E. A., F A R R E L L T. and H U R D C. M., Phys. Rev. B 1, 3904 (1970). 4. A L D E R S O N J. E. A . , F A R R E L L T . and H U R D C. M., Phys. Rev. 174, 729 (1968). 5. S A E G E R K. E. and L U C K R., Phys. Kondens. Mater. 9, 91 (1969). 6. H U P P M A N N W. and S T A N G L E R F., Phys. Status. Solidi 31, KI07 (1969). 7. K L A U D E R J. R., REED W. A., B R E N N E R T G. F. and K U N Z L E R J. E., Phys. Rev. 141, 592 (1966). 8. Z I M A N J. M., Phil. Mag. 3, I 117 (1958). 9. K U N Z L E R J. E. and K L A U D E R J. R., Phil. Mag. 6, 1054(1961). 10. LIPSON S. G., Proc. R. Soc. (London) A293, 275 (I 966). I I. C H A M B E R S R. G., Proc. R. Soc. (London) A238, 344 (1956). 12. A L E K S E E V S K I ! N. E. and G A 1 D U K O V Yu. P.,
OF THE HALL EFFECT
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