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Low field Hall effect in potassium
Solid State Communications,Vol. 16, pp. 1371—1373, 1975.
Pergamon Press.
Printed in Great Britain
LOW FIELD HALL EFFECT IN POTASSIUM C.M. Hurd and J.E.A. Alderson National Research Council of Canada, Ottawa K1A 0R9, Canada (Received 19 February 1975 by M.F. Collins)
The anisotropic electron relaxation time in K caused by the freezing Out of Umklappprocesses on certain areas of the Fermi surface is predicted to give a temperature-dependent component to the low-field Hall effect at low temperatures. A previous attempt to demonstrate this component was dc feated by the unsuitability of the existing data which contain an overwhelming high-field contribution in the temperature range of interest. To provide more suitable data, we have measured the Hall effect of polycrystalline Kas the conditions tend towards the low-field limit. It is shown how the results can be qualitatively interpreted in terms of a competition between this predicted Umklapp component and the high-field tendencies inherent in the galvanomagnetic effect. RECENT calculations13 of the anisotropic electron relaxation time due to Umklapp scattering in K point to certain consequences for the low temperature Hall effect. Briefly, the argument is that in the temperature range where such scattering dominates the ideal resistivity (which is roughly 2—20°K)the probability of an Umklapp electron—phonon scattering event will depend rather strongly upon the electron’s initial state. This is because as the temperature is reduced such scattering will be increasingly restricted to electrons represented on those regions of the Fermi surface which are closest to the Brillouin zone boundaries. (Such regions are centred about the [1101and [111] directions, and it has been estimated2 that at 6°Kthe mean lifetime of an electron in one of the states is only about one third of that in a state close to a [100] direction.) Thus although the metal has an essentially spherical Fermi surface, in the above temperature range the anisotropy of the Umklapp process effectively splits the conduction electrons into two groups which are distinguished by markedly different relaxation times.
mass and anisotropic relaxationtime at every point on the Fermi surface, it should show a temperature dependence in sympathy with that of the anisotropy of the electron—phonon scattering. Calculations show2’3 that the latter peaks at about 6°K.On the lower side of this temperature, where there is a practic. ally exponential elimination of all phonon scattering, this anisotropy falls off sharply, while on the higher side it decreases rather less abruptly to become unity probably at about2 60°K. Hasegawa2 tried to find a corresponding behaviour in the Hall effect, but his approach was defeated by the unsuitability of the few data then available.4’5 As he pointed out, these data (which cover the range 4—295°Kat a fixed field of~1.5 T) in fact apply to the low-field condition only above about 25°K;below that temperature the low-field criterion (o.~r~ 1 in the usual notation) is no longer satisfied by alt the contributing cyclotron orbits. For decreasing ternperature below about 25°K,these data thus contain an increasing high-field component which confounds any comparison with the low-field theory. Ideally, what is therefore required is a set of high-precision low-field Hall effect data obtained for a very pure
Since the low-field Hall effect is determined by a summation involving the electron’s velocity, effective 1371
LOW FIELD HALL EFFECT IN POTASSIUM
1372
Vol. 16, No. 12
1.00 021
IT
00.57
COOT
* 0.11
K
OkoI
5102025
FIG I Temperature dependence of the effective number of electrons per atom n’~obtained from the Hall effect in the applied fields indicated. The experimental uncertainty due to errors in the field setting and in the measurement of the galvanomagnetic voltage is field dependent; it amounts to about ±1 per cent at 0.1 T and reduces uniformly to about ±0.1 per cent at 2.0 T. The text attributes a decrease in n*(T) to the effect of anisotropic r(k) caused by the freezing out of Umklapp processes, while an increase arises from the onset of the high-field condition. sample over the range of say 4—60°K.Experimentally this requirement of course represents incompatible ideals: to produce the low-field effect in a pure sample at low temperatures demands an appropriately small applied field strength, but the continual reduction of this parameter is self-defeating because the imprecision of the datum increases correspondingly. Even with the most sensitive techniques available, such a direct approach to the low-field behaviour at low temperatures rapidly becomes futile. Perhaps the best experimental cornpromise is to study the behaviour of the Hall effect as the low-field condition is approached; the data then inevitably contain high or intermediate-field contributions but these are generally distinguishable so that an inference of the low-field behaviour is possible.6 We have made this approach to the low-field Hall effect in K. Figure 1 shows (for direct comparison with Hasegawa’s predictions2) the temperature dependence of n” (the effective number of conduction electrons per atom) obtained from the Hall effect measured over the range 4—.27°Kand at fixed fields between 0.1— 2.0 T. Thes~data were obtained using two samples which gave essentially identical results prepared from the same starting material and to the ~samedimensions as described previously.4 A third sample cut from the same material (and of thickness 0.058 cm) was used for independent measurements of the resistivity required to calculate the w~,rvalues of Fig. 2. The —
—
~
WCT
FIG. 2. The data of ‘ig. 1 plotted as a function of w~,rcalculated from the measured resistivity using free electron theory. This plot confirms the assignment of an increase in n”(T) to the onset of the high-field condition. (For the symbols’ key see Fig. 1.)
purity of the samples is characterized by the typical residual resistance ratio of 4650 and residual resistivity at I .75°Kof 0.92 n&~2-cm.The galvanornagnetic measurements were made with a conventional cryostat— electromagnet combination using instrumentation described previously.7 The overall sensitivity of the apparatus is 1 nV = 1 mm, while the fixed d.c. primary current in the Hall samples was chosen in the range 5—10 A, depending upon the circumstances. Before discussing the qualitative behaviours shown in Fig. 1, two points concerning these data should be emphasized. First, because it is particularly difficult to determine accurately the thickness of a K sample, we avoided this complication by making measurements relative to the absolute value of the Hall effect observed previously4 at 1.5 T and 20°Kfor material from the same source. Hence the absolute n~values cited in Fig. 1 contain the inaccuracy implicit in that earlier work a point which should not be overlooked in any quantitative use of Fig. 1. Second, measurements made much below about 5°Kshowed exceptionally large scatter and marked irreproducibility between day to day measurements. Although we did not attempt a thorough study of this source of error, it seems possibly connected with the strain suffered by the sample during cooling, and is thus probably dependent upon the rate of cooling to cryogenic temperatures. (Ekin and Maxfield,8 for example, encountered a similar effect in resistivity measurements.) Although we systematically cooled the samples in each run at —
Vol. 16, No. 12
LOW FIELD HALL EFFECT IN POTASSIUM
about the same rate (293 4.2°Kin ~ 30 mm), it was not a sufficient control to produce reproducible results below about 4.7°K.Since data obtained below this temperature do not greatly influence our qualitative arguments, we have omitted them from Fig. 1, but the extra source of error in this range remains unresolved, -+
Our qualitative description of the data shown in Fig. 1 centres about the two competing tendencies arising separately from: (1) the onset of the high-field condition, which increases the value of n~’towards its ultimate limiting value of close to unity, and (2) the effect of the anisotropic r(k) which, for temperatures above 6°Kat least, produces2 the opposite tendency. The temperature dependence of n” thus reflects the competition between these two features. As the ternperature is reduced from about 27°K,and providing the applied field strength is low enough to delay the onset of high-field effects, we see first in n the contribution (2) produced as the Umklapp process are “frozen out” on those parts of the Fermi surface furthest removed from Brillouin zone planes. The anisotropy in r(k) is thereby increased and with it n~is reduced. The effect is of course most noticeable in the lowest applied field strength (0.1 T); the reduction ~f~* is then first evident at about 1 5°Kand becomes increasingly pronounced down to about 8 K where it is swamped by the opposite tendency produced by the onset of the high-field condition. Exactly the same qualitative behaviour is observed in higher fields except that the manifestation *
1373
of high-field behaviour occurs at proportionately higher temperatures (viz, at about 7.5, 9, 13 and 14°Kin fields of 0.1, 0.2,0.5 and 1.0 T, respectively) so that the contribution (2) is increasingly eclipsed. In fact, in a field of 2.0 T the high-field effects, which then first appear at about 20°K,completely obscure any evidence of a decrease in n” produced by (2). This is of course exactly the situation which defeated Hasegawa’s attempt2 to demonstrate in the earlier data4 a component (2) at low temperatures. It is now seen that the component (2) is present in the Hall effect and can be demonstrated at low temperatures in appropriate conditions, but we have still not been able to show its entire temperature dependence predicted by the theory.1’3 In the present experimental circumstances it is not possible to make low-field Hall measurements upon pure K much below 7°K,and so we have been unable to confirm the local maximum predicted to occur in the component (2) at about 6°K.Note that it is wrong to construe that our data obtained at 0.1 T show just this effect, for Fig. 2 shows clearly that at all field strengths studied the upswing in n” occurs at the common value of ~ 1. In other words, it is clear that we have correctly interpreted this upswing as the onset of the high-field condition; it is simply fortuitous that the competition between the components (1) and (2) gives a local maximum at about 6°Kin an applied field of 0.1 T.
REFERENCES 1.
HAYMAN B. and CARBOTTE J.P.,Phys. Rev. B6, 1154 (1972).
2. 3.
HASEGAWA A., J. Phys. F: Metal Phys. 4, 1024 (1974). TROFIMENKOFF P.N., J. Low Temp. Phys. 16,455 (1974).
4.
ALDERSON J.E.A. and FARRELL T.,Phys. Rev. 185, 876 (1969).
5. 6.
SIEBENMAN P.G. and BABISKIN J., Phys. Rev. Lett. 30, 380 (1973). ALDERSON J.E.A. and HURD C.M.,Phys. Rev. B7, 1226 (1973).
7.
ALDERSON .LE.A., FARRELLT. and HURD C.M.,Phys. Rev. 174, 729 (1968).
8.
EKIN J.W. and MAXFIELD B.W.,Phys. Rev. B4, 4215 (1971).
Pergamon Press.
Printed in Great Britain
LOW FIELD HALL EFFECT IN POTASSIUM C.M. Hurd and J.E.A. Alderson National Research Council of Canada, Ottawa K1A 0R9, Canada (Received 19 February 1975 by M.F. Collins)
The anisotropic electron relaxation time in K caused by the freezing Out of Umklappprocesses on certain areas of the Fermi surface is predicted to give a temperature-dependent component to the low-field Hall effect at low temperatures. A previous attempt to demonstrate this component was dc feated by the unsuitability of the existing data which contain an overwhelming high-field contribution in the temperature range of interest. To provide more suitable data, we have measured the Hall effect of polycrystalline Kas the conditions tend towards the low-field limit. It is shown how the results can be qualitatively interpreted in terms of a competition between this predicted Umklapp component and the high-field tendencies inherent in the galvanomagnetic effect. RECENT calculations13 of the anisotropic electron relaxation time due to Umklapp scattering in K point to certain consequences for the low temperature Hall effect. Briefly, the argument is that in the temperature range where such scattering dominates the ideal resistivity (which is roughly 2—20°K)the probability of an Umklapp electron—phonon scattering event will depend rather strongly upon the electron’s initial state. This is because as the temperature is reduced such scattering will be increasingly restricted to electrons represented on those regions of the Fermi surface which are closest to the Brillouin zone boundaries. (Such regions are centred about the [1101and [111] directions, and it has been estimated2 that at 6°Kthe mean lifetime of an electron in one of the states is only about one third of that in a state close to a [100] direction.) Thus although the metal has an essentially spherical Fermi surface, in the above temperature range the anisotropy of the Umklapp process effectively splits the conduction electrons into two groups which are distinguished by markedly different relaxation times.
mass and anisotropic relaxationtime at every point on the Fermi surface, it should show a temperature dependence in sympathy with that of the anisotropy of the electron—phonon scattering. Calculations show2’3 that the latter peaks at about 6°K.On the lower side of this temperature, where there is a practic. ally exponential elimination of all phonon scattering, this anisotropy falls off sharply, while on the higher side it decreases rather less abruptly to become unity probably at about2 60°K. Hasegawa2 tried to find a corresponding behaviour in the Hall effect, but his approach was defeated by the unsuitability of the few data then available.4’5 As he pointed out, these data (which cover the range 4—295°Kat a fixed field of~1.5 T) in fact apply to the low-field condition only above about 25°K;below that temperature the low-field criterion (o.~r~ 1 in the usual notation) is no longer satisfied by alt the contributing cyclotron orbits. For decreasing ternperature below about 25°K,these data thus contain an increasing high-field component which confounds any comparison with the low-field theory. Ideally, what is therefore required is a set of high-precision low-field Hall effect data obtained for a very pure
Since the low-field Hall effect is determined by a summation involving the electron’s velocity, effective 1371
LOW FIELD HALL EFFECT IN POTASSIUM
1372
Vol. 16, No. 12
1.00 021
IT
00.57
COOT
* 0.11
K
OkoI
5102025
FIG I Temperature dependence of the effective number of electrons per atom n’~obtained from the Hall effect in the applied fields indicated. The experimental uncertainty due to errors in the field setting and in the measurement of the galvanomagnetic voltage is field dependent; it amounts to about ±1 per cent at 0.1 T and reduces uniformly to about ±0.1 per cent at 2.0 T. The text attributes a decrease in n*(T) to the effect of anisotropic r(k) caused by the freezing out of Umklapp processes, while an increase arises from the onset of the high-field condition. sample over the range of say 4—60°K.Experimentally this requirement of course represents incompatible ideals: to produce the low-field effect in a pure sample at low temperatures demands an appropriately small applied field strength, but the continual reduction of this parameter is self-defeating because the imprecision of the datum increases correspondingly. Even with the most sensitive techniques available, such a direct approach to the low-field behaviour at low temperatures rapidly becomes futile. Perhaps the best experimental cornpromise is to study the behaviour of the Hall effect as the low-field condition is approached; the data then inevitably contain high or intermediate-field contributions but these are generally distinguishable so that an inference of the low-field behaviour is possible.6 We have made this approach to the low-field Hall effect in K. Figure 1 shows (for direct comparison with Hasegawa’s predictions2) the temperature dependence of n” (the effective number of conduction electrons per atom) obtained from the Hall effect measured over the range 4—.27°Kand at fixed fields between 0.1— 2.0 T. Thes~data were obtained using two samples which gave essentially identical results prepared from the same starting material and to the ~samedimensions as described previously.4 A third sample cut from the same material (and of thickness 0.058 cm) was used for independent measurements of the resistivity required to calculate the w~,rvalues of Fig. 2. The —
—
~
WCT
FIG. 2. The data of ‘ig. 1 plotted as a function of w~,rcalculated from the measured resistivity using free electron theory. This plot confirms the assignment of an increase in n”(T) to the onset of the high-field condition. (For the symbols’ key see Fig. 1.)
purity of the samples is characterized by the typical residual resistance ratio of 4650 and residual resistivity at I .75°Kof 0.92 n&~2-cm.The galvanornagnetic measurements were made with a conventional cryostat— electromagnet combination using instrumentation described previously.7 The overall sensitivity of the apparatus is 1 nV = 1 mm, while the fixed d.c. primary current in the Hall samples was chosen in the range 5—10 A, depending upon the circumstances. Before discussing the qualitative behaviours shown in Fig. 1, two points concerning these data should be emphasized. First, because it is particularly difficult to determine accurately the thickness of a K sample, we avoided this complication by making measurements relative to the absolute value of the Hall effect observed previously4 at 1.5 T and 20°Kfor material from the same source. Hence the absolute n~values cited in Fig. 1 contain the inaccuracy implicit in that earlier work a point which should not be overlooked in any quantitative use of Fig. 1. Second, measurements made much below about 5°Kshowed exceptionally large scatter and marked irreproducibility between day to day measurements. Although we did not attempt a thorough study of this source of error, it seems possibly connected with the strain suffered by the sample during cooling, and is thus probably dependent upon the rate of cooling to cryogenic temperatures. (Ekin and Maxfield,8 for example, encountered a similar effect in resistivity measurements.) Although we systematically cooled the samples in each run at —
Vol. 16, No. 12
LOW FIELD HALL EFFECT IN POTASSIUM
about the same rate (293 4.2°Kin ~ 30 mm), it was not a sufficient control to produce reproducible results below about 4.7°K.Since data obtained below this temperature do not greatly influence our qualitative arguments, we have omitted them from Fig. 1, but the extra source of error in this range remains unresolved, -+
Our qualitative description of the data shown in Fig. 1 centres about the two competing tendencies arising separately from: (1) the onset of the high-field condition, which increases the value of n~’towards its ultimate limiting value of close to unity, and (2) the effect of the anisotropic r(k) which, for temperatures above 6°Kat least, produces2 the opposite tendency. The temperature dependence of n” thus reflects the competition between these two features. As the ternperature is reduced from about 27°K,and providing the applied field strength is low enough to delay the onset of high-field effects, we see first in n the contribution (2) produced as the Umklapp process are “frozen out” on those parts of the Fermi surface furthest removed from Brillouin zone planes. The anisotropy in r(k) is thereby increased and with it n~is reduced. The effect is of course most noticeable in the lowest applied field strength (0.1 T); the reduction ~f~* is then first evident at about 1 5°Kand becomes increasingly pronounced down to about 8 K where it is swamped by the opposite tendency produced by the onset of the high-field condition. Exactly the same qualitative behaviour is observed in higher fields except that the manifestation *
1373
of high-field behaviour occurs at proportionately higher temperatures (viz, at about 7.5, 9, 13 and 14°Kin fields of 0.1, 0.2,0.5 and 1.0 T, respectively) so that the contribution (2) is increasingly eclipsed. In fact, in a field of 2.0 T the high-field effects, which then first appear at about 20°K,completely obscure any evidence of a decrease in n” produced by (2). This is of course exactly the situation which defeated Hasegawa’s attempt2 to demonstrate in the earlier data4 a component (2) at low temperatures. It is now seen that the component (2) is present in the Hall effect and can be demonstrated at low temperatures in appropriate conditions, but we have still not been able to show its entire temperature dependence predicted by the theory.1’3 In the present experimental circumstances it is not possible to make low-field Hall measurements upon pure K much below 7°K,and so we have been unable to confirm the local maximum predicted to occur in the component (2) at about 6°K.Note that it is wrong to construe that our data obtained at 0.1 T show just this effect, for Fig. 2 shows clearly that at all field strengths studied the upswing in n” occurs at the common value of ~ 1. In other words, it is clear that we have correctly interpreted this upswing as the onset of the high-field condition; it is simply fortuitous that the competition between the components (1) and (2) gives a local maximum at about 6°Kin an applied field of 0.1 T.
REFERENCES 1.
HAYMAN B. and CARBOTTE J.P.,Phys. Rev. B6, 1154 (1972).
2. 3.
HASEGAWA A., J. Phys. F: Metal Phys. 4, 1024 (1974). TROFIMENKOFF P.N., J. Low Temp. Phys. 16,455 (1974).
4.
ALDERSON J.E.A. and FARRELL T.,Phys. Rev. 185, 876 (1969).
5. 6.
SIEBENMAN P.G. and BABISKIN J., Phys. Rev. Lett. 30, 380 (1973). ALDERSON J.E.A. and HURD C.M.,Phys. Rev. B7, 1226 (1973).
7.
ALDERSON .LE.A., FARRELLT. and HURD C.M.,Phys. Rev. 174, 729 (1968).
8.
EKIN J.W. and MAXFIELD B.W.,Phys. Rev. B4, 4215 (1971).