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Nonlinear magnetization of bismuth single crystals in high magnetic fields
Physica 79 B (1975) 9 5 - 1 0 1 © North-Holland Publishing Company
L E T T E R TO THE E D I T O R
NONLINEAR MAGNETIZATION OF BISMUTH SINGLE CRYSTALS IN HIGH MAGNETIC FIELDS L. W. ROELAND, G. J. COCK, F. A. MULLER
Natuurkundig Laboratorium van de Universiteit van Amsterdam, Amsterdam, The Netherlands D. SHOENBERG
Cavendish laboratory, Madingley Road, Cambridge, England Received 17 January 1975
The magnetization of bismuth single crystals has been measured in very high magnetic fields up to 340 kG. Along the binary and the bisectrix axes the data (at 1.5 K) show small but significant deviations from linearity, but along the trigonal axis the deviation is much smaller and barely significant.
1. Introduction Recent experiments 1) on the magnetization, M, o f bismuth crystals at fields b e y o n d the last de H a a s - v a n Alphen oscillation (L e. b e y o n d the " q u a n t u m limit") showed that over a considerable field range (roughly 35 to 80 kG) M varied with H as M=-k'H
(1)
+ Mo .
For the field along either the binary or bisectrix axes, k' was roughly 2.5 X 10 -s (i.e. nearly 50% greater than the diamagnetic susceptibility at low fields) and M0 was roughly 0.16 emu cm- 3 for the binary axis and 0.19 emu cm -3 for the bisectrix axis*. Rough measurements 2) for the * EM units are used in this paper for convenience in comparison with theoretical calculations. To convert to SI units note that 1 emu cm -3 -" 103 A m -1 and 1 gauss --" 103/41r A m - 1 .
95
96
L. W. Roeland et al./Nonlinear magnetization of bismuth crystals
field along the trigonal axis (taken to fields only a little above the quantum limit of the holes) suggested that the value of k' did not differ greatly from the initial susceptibility (perhaps 10% greater) and that the intercept M0 was of order 0.10 emu cm -3 , though it might have been zero. Calculations 3) based on the C o h e n - L a x model of the energy levels in a magnetic field provided a qualitative explanation (and in a more recent version4), nearly a quantitative explanation) of this behaviour, but suggested that the linear form of (1) was merely an approximation valid for the limited range of fields covered. For higher fields the calculations suggested that the M - H relation should curve appreciably (becoming less diamagnetic) for the binary and bisectrix axes, but that for the trigonal axis the M - H relationshoutd depart much less from the proportionality characteristic of low fields. These predictions provided the motivation for the present experiments at higher fields in the high field installation s) of the University of Amsterdam, which is particularly suitable for the observation of deviations from linear proportionality between M and H at fields up to about 350 kG.
2. Outline of technique The Amsterdam installation produces a high-field pulse like a "staircase", which steps down from its maximum value (338 kG in these experiments) in 7 equal steps with the field held constant for 50 ms at each step. These periods of constant field allow eddy currents to decay and so permit a precise determination of magnetization at each of the 7 field values. The magnetization is measured by integrating the output voltage of an elaborate pickup coil system 6 ) surrounding the sample, as the field changes from step to step. To allow for the magnetic properties of the material of which the pick-up coil is constructed and for slightly imperfect balancing, it is necessary to subtract from the integrated output voltage the integrated output voltage observed during a second pulse when the sample has been removed. Since, however, the time interval between the two measurements has to be considerable (2 or 3 hours), slight deformations are liable to occur in the coil system, which means that the " e m p t y " integrated voltage output may differ slightly from what it would have been when the sample was present. This can lead to a small error, varying from one experiment to another, but this error should always be strictly proportional to field in any one experiment. Because the 7 consecutive steps of field occur in rapid sequence, this long term deformation effect in the coil system does not, however, upset the relative precision of any
L. W. Roeland et aL /Nonlinear magnetization o f bismuth crystals
97
one series. Thus any deviations from an exactly proportional M - H relation may be determined with a much higher precision than the constant of proportionality itself (i. e. than the mean susceptibility). To allow for the drift in the electronic integrator, which is appreciable even in the short time of the pulse, a computer is used to measure the output signal relative to an interpolated zero line.
3. Results Measurements were made on 5 bismuth crystals grown by the Czochralski method in the form o f cylinders with length roughly equal diameter. The samples were cut to length by spark erosion and then etched to remove surface damage; in two of the crystals the axis of figure was the binary, in two the bisectrix axis and in one the trigonal axis. Typically the volume of a crystal was about 0,2 cm 3 and it was mounted with its axis of figure along the field. All the measurements were made at 1.5 K. In order to show the deviation from iinearity more clearly, each set of 7 points was first fitted by least squares to an expression M = kH
(2)
and this "linearized" value was then subtracted from the actual points to give a set of 7 differences* AM which are plotted against H in figs. 1,2 and 3. It can be seen from figs. 1 and 2 that in spite of a considerable scatter there is significant departure from proportionality in the M - H relation for the binary and bisectrix samples. There does not appear to be any appreciable difference between the two different samples used for each orientation and it may be noted that one of the two bisectrix samples was made from bismuth o f a slightly lower purity. The deviation from linearity is smaller for the trigonal axis (fig. 3) and barely more than experimental error. Part of the scatter below about 100 kG may be due to sampling of the de Haas-van Alphen oscillations in M which should have an amplitude at this orientation of 1 or 2 X 10 -2 emu cm -3 . The de Haas-van Alphen amplitudes of M for the binary and bisectrix directions, for fields above 30 kG should be much smaller * The values of the 7 fields were not always the same from one experiment to another and for this reason, considerably more than 7 field values appear in figs. l, 2, 3 and 4.
98
L. W. Roeland et al./Nonlinear magnetization o f bismuth crystals
(< 3 X 10 -a emu cm -a) and would hardly show on the scale o f figs. 1 and 2. The random scatter is probably mainly caused by the interpolation of the zero line mentioned above. In order to demonstrate this, a number of e m p t y runs from b o t h the binary and the bisectrix series, were subtracted from their mean and analyzed by subtracting a least-squares fit of the form Me = keH, just as with the actual data. The AMe obtained in this way are plotted in fig. 4 and it is clearly seen that they scatter to much the same extent as the data of figs. 1, 2 and 3. If the mean values of k in eq. (2) (which were - 2 . 2 2 -+ 0.06 X 10 -s for the binary axis, - 2 . 2 9 -+ 0.06 × 10 -s for the bisectrix axis and -1.31 -+ 0.05 × 10 -s for the trigonal axis) are added to the points plotted in figs. 1,2 and 3, the observed M - H relations at high fields can be described within experimental error by the empirical formulae
binary 10-~2 / P ,
(3)
M = 0 . 1 1 - 2.66X 10 -s H + 12.4X 10 -12 / ~ ,
(4)
M=0.18-2.58X
10 -s H + 1 0 . 6 X
bisectrix trigonal M=0.23-
1.47X 10 -s H + 2 . 6 X
(5)
10 -12 H 2.
Bi 1.5K Binary axis
20
10
AM
(10.2 e.m.u/cc)
o
go
oo Io
-lC
~
~o
o
-2c ~_ 0
~ _ _
L
100
t
I
200
I
I
300 - - ~ - ~
I
k Gauss
Fig. 1. Plot of the differences ~ M (as explained in the text) against H at 1.5 K, for binary axis along field. The open and full circles distinguish between the two samples.
99
L. W. Roeland et al./Nonlinear magnetization of bismuth crystals
l
Bi 1.5K Bisectrix axis
2C~"
AM 10 (10-2e.m u/cc)
0 oO o
o o
8
°°
-10
8 ° O o I] o
°
~
8 o
~
o
'8°
-20 j
I
100
I
I
I
J
200
300 ~
k Gauss
Fig. 2. As for fig. 1, but for bisectrix axis along field. The full circles are for the sample of less pure material.
l
Bi 1.5K Trigonol axis
20
AM (10_2e,mu/cc ) 10
,
| |
°
!
I
| •
•
•
•
o•
:
~
•
•
•
-10
I 1oo
r
I 200
I
1
300
--~..-
Fig. 3. As for fig. 1 with the trigonal axis along field.
k Gauss
1O0
L.W. Roeland et al./Nonlinear magnetization o f bismuth crystals
In (5) only the points above 100 kG have been used because of the appreciable de Haas-van Alphen oscillations at lower fields. Bearing in mind the appreciable experimental errors it can be seen that there is very satisfactory agreement with the earlier data up to 80 kG. The interesting novel feature of the results is that, as had been suspected, the relation (1) is valid only as a limiting form below about 100 kG for the binary and bisectrix axes. At higher fields the M - H relation becomes appreciably curved for these axes (to about the same extent for both) but there is much less curvature for the trigonal axis. Detailed comparison of these results with a calculation based on the Cohen-Lax nonparabolic band model of the electronic structure of bismuth will be presented elsewhere4), but it may be said here that there is good general agreement between theory and experiment. A particularly satisfying feature of the agreement is that the theory predicts that the binary and bisectrix samples should behave very similarly but quite differently from the trigonal sample, and this is just what the experiments demonstrate. This difference of behaviour between the trigonal sample and the binary and bisectrix samples gives added confidence that the observed curvature at very high fields are genuine rather than any artefact of the experimental technique.
20
_•Ms
10
o
(10- e.rn,uJcc) o ° B
o
oo
~
o
0
o
o
8
o
eo
®
o~
8
o
-10
-20
0
J
I
100
I
I
200
I
I
300 ~
t
kOauss
Fig. 4. As for fig. I but for several e m p t y run values a M e (as explained in the text).
L. W. Roeland et aL /Nonlinear magnetization o f bismuth crystals
101
Acknowledgements This work was made possible by the financial support o f the Stichting voor F u n d a m e n t e e l Onderzoek der Materie (F.O.M.), which in turn is supported by Z.W.O. We also wish to thank Mr. T. E. B r o w n o f the Cavendish Laboratory for growing the crystals.
References 1) N. L. Brignall and D. Shoenberg, J. Phys. C 7 (1974) 1499. 2) N. L. Brignall, Ph.D. Thesis, Cambridge University 1972 and upubllshed data. 3) D. Shoenberg, Proceedings Conference on Semi-metals, Cardiff (Sept. 1973), to be published. 4) J. W. MeClure and D. Shoenberg, to be published. 5) R. Gersdorf, F. A. Muller and L. W. Roeland, Rev. Sci. Inst. 36 (1965) 1100. L. W. Roeland, F. A. Muller and R. Gersdorf, Coll. Int. Cent. Nat. Rech. Sci. no. 166 (1967) 175. 6) R. Gersdorf, F. A. Muller and L. W. Roeland, Coll. Int. Cent. Nat. Reeh. Sci. no. 166 (1967) 185.
L E T T E R TO THE E D I T O R
NONLINEAR MAGNETIZATION OF BISMUTH SINGLE CRYSTALS IN HIGH MAGNETIC FIELDS L. W. ROELAND, G. J. COCK, F. A. MULLER
Natuurkundig Laboratorium van de Universiteit van Amsterdam, Amsterdam, The Netherlands D. SHOENBERG
Cavendish laboratory, Madingley Road, Cambridge, England Received 17 January 1975
The magnetization of bismuth single crystals has been measured in very high magnetic fields up to 340 kG. Along the binary and the bisectrix axes the data (at 1.5 K) show small but significant deviations from linearity, but along the trigonal axis the deviation is much smaller and barely significant.
1. Introduction Recent experiments 1) on the magnetization, M, o f bismuth crystals at fields b e y o n d the last de H a a s - v a n Alphen oscillation (L e. b e y o n d the " q u a n t u m limit") showed that over a considerable field range (roughly 35 to 80 kG) M varied with H as M=-k'H
(1)
+ Mo .
For the field along either the binary or bisectrix axes, k' was roughly 2.5 X 10 -s (i.e. nearly 50% greater than the diamagnetic susceptibility at low fields) and M0 was roughly 0.16 emu cm- 3 for the binary axis and 0.19 emu cm -3 for the bisectrix axis*. Rough measurements 2) for the * EM units are used in this paper for convenience in comparison with theoretical calculations. To convert to SI units note that 1 emu cm -3 -" 103 A m -1 and 1 gauss --" 103/41r A m - 1 .
95
96
L. W. Roeland et al./Nonlinear magnetization of bismuth crystals
field along the trigonal axis (taken to fields only a little above the quantum limit of the holes) suggested that the value of k' did not differ greatly from the initial susceptibility (perhaps 10% greater) and that the intercept M0 was of order 0.10 emu cm -3 , though it might have been zero. Calculations 3) based on the C o h e n - L a x model of the energy levels in a magnetic field provided a qualitative explanation (and in a more recent version4), nearly a quantitative explanation) of this behaviour, but suggested that the linear form of (1) was merely an approximation valid for the limited range of fields covered. For higher fields the calculations suggested that the M - H relation should curve appreciably (becoming less diamagnetic) for the binary and bisectrix axes, but that for the trigonal axis the M - H relationshoutd depart much less from the proportionality characteristic of low fields. These predictions provided the motivation for the present experiments at higher fields in the high field installation s) of the University of Amsterdam, which is particularly suitable for the observation of deviations from linear proportionality between M and H at fields up to about 350 kG.
2. Outline of technique The Amsterdam installation produces a high-field pulse like a "staircase", which steps down from its maximum value (338 kG in these experiments) in 7 equal steps with the field held constant for 50 ms at each step. These periods of constant field allow eddy currents to decay and so permit a precise determination of magnetization at each of the 7 field values. The magnetization is measured by integrating the output voltage of an elaborate pickup coil system 6 ) surrounding the sample, as the field changes from step to step. To allow for the magnetic properties of the material of which the pick-up coil is constructed and for slightly imperfect balancing, it is necessary to subtract from the integrated output voltage the integrated output voltage observed during a second pulse when the sample has been removed. Since, however, the time interval between the two measurements has to be considerable (2 or 3 hours), slight deformations are liable to occur in the coil system, which means that the " e m p t y " integrated voltage output may differ slightly from what it would have been when the sample was present. This can lead to a small error, varying from one experiment to another, but this error should always be strictly proportional to field in any one experiment. Because the 7 consecutive steps of field occur in rapid sequence, this long term deformation effect in the coil system does not, however, upset the relative precision of any
L. W. Roeland et aL /Nonlinear magnetization o f bismuth crystals
97
one series. Thus any deviations from an exactly proportional M - H relation may be determined with a much higher precision than the constant of proportionality itself (i. e. than the mean susceptibility). To allow for the drift in the electronic integrator, which is appreciable even in the short time of the pulse, a computer is used to measure the output signal relative to an interpolated zero line.
3. Results Measurements were made on 5 bismuth crystals grown by the Czochralski method in the form o f cylinders with length roughly equal diameter. The samples were cut to length by spark erosion and then etched to remove surface damage; in two of the crystals the axis of figure was the binary, in two the bisectrix axis and in one the trigonal axis. Typically the volume of a crystal was about 0,2 cm 3 and it was mounted with its axis of figure along the field. All the measurements were made at 1.5 K. In order to show the deviation from iinearity more clearly, each set of 7 points was first fitted by least squares to an expression M = kH
(2)
and this "linearized" value was then subtracted from the actual points to give a set of 7 differences* AM which are plotted against H in figs. 1,2 and 3. It can be seen from figs. 1 and 2 that in spite of a considerable scatter there is significant departure from proportionality in the M - H relation for the binary and bisectrix samples. There does not appear to be any appreciable difference between the two different samples used for each orientation and it may be noted that one of the two bisectrix samples was made from bismuth o f a slightly lower purity. The deviation from linearity is smaller for the trigonal axis (fig. 3) and barely more than experimental error. Part of the scatter below about 100 kG may be due to sampling of the de Haas-van Alphen oscillations in M which should have an amplitude at this orientation of 1 or 2 X 10 -2 emu cm -3 . The de Haas-van Alphen amplitudes of M for the binary and bisectrix directions, for fields above 30 kG should be much smaller * The values of the 7 fields were not always the same from one experiment to another and for this reason, considerably more than 7 field values appear in figs. l, 2, 3 and 4.
98
L. W. Roeland et al./Nonlinear magnetization o f bismuth crystals
(< 3 X 10 -a emu cm -a) and would hardly show on the scale o f figs. 1 and 2. The random scatter is probably mainly caused by the interpolation of the zero line mentioned above. In order to demonstrate this, a number of e m p t y runs from b o t h the binary and the bisectrix series, were subtracted from their mean and analyzed by subtracting a least-squares fit of the form Me = keH, just as with the actual data. The AMe obtained in this way are plotted in fig. 4 and it is clearly seen that they scatter to much the same extent as the data of figs. 1, 2 and 3. If the mean values of k in eq. (2) (which were - 2 . 2 2 -+ 0.06 X 10 -s for the binary axis, - 2 . 2 9 -+ 0.06 × 10 -s for the bisectrix axis and -1.31 -+ 0.05 × 10 -s for the trigonal axis) are added to the points plotted in figs. 1,2 and 3, the observed M - H relations at high fields can be described within experimental error by the empirical formulae
binary 10-~2 / P ,
(3)
M = 0 . 1 1 - 2.66X 10 -s H + 12.4X 10 -12 / ~ ,
(4)
M=0.18-2.58X
10 -s H + 1 0 . 6 X
bisectrix trigonal M=0.23-
1.47X 10 -s H + 2 . 6 X
(5)
10 -12 H 2.
Bi 1.5K Binary axis
20
10
AM
(10.2 e.m.u/cc)
o
go
oo Io
-lC
~
~o
o
-2c ~_ 0
~ _ _
L
100
t
I
200
I
I
300 - - ~ - ~
I
k Gauss
Fig. 1. Plot of the differences ~ M (as explained in the text) against H at 1.5 K, for binary axis along field. The open and full circles distinguish between the two samples.
99
L. W. Roeland et al./Nonlinear magnetization of bismuth crystals
l
Bi 1.5K Bisectrix axis
2C~"
AM 10 (10-2e.m u/cc)
0 oO o
o o
8
°°
-10
8 ° O o I] o
°
~
8 o
~
o
'8°
-20 j
I
100
I
I
I
J
200
300 ~
k Gauss
Fig. 2. As for fig. 1, but for bisectrix axis along field. The full circles are for the sample of less pure material.
l
Bi 1.5K Trigonol axis
20
AM (10_2e,mu/cc ) 10
,
| |
°
!
I
| •
•
•
•
o•
:
~
•
•
•
-10
I 1oo
r
I 200
I
1
300
--~..-
Fig. 3. As for fig. 1 with the trigonal axis along field.
k Gauss
1O0
L.W. Roeland et al./Nonlinear magnetization o f bismuth crystals
In (5) only the points above 100 kG have been used because of the appreciable de Haas-van Alphen oscillations at lower fields. Bearing in mind the appreciable experimental errors it can be seen that there is very satisfactory agreement with the earlier data up to 80 kG. The interesting novel feature of the results is that, as had been suspected, the relation (1) is valid only as a limiting form below about 100 kG for the binary and bisectrix axes. At higher fields the M - H relation becomes appreciably curved for these axes (to about the same extent for both) but there is much less curvature for the trigonal axis. Detailed comparison of these results with a calculation based on the Cohen-Lax nonparabolic band model of the electronic structure of bismuth will be presented elsewhere4), but it may be said here that there is good general agreement between theory and experiment. A particularly satisfying feature of the agreement is that the theory predicts that the binary and bisectrix samples should behave very similarly but quite differently from the trigonal sample, and this is just what the experiments demonstrate. This difference of behaviour between the trigonal sample and the binary and bisectrix samples gives added confidence that the observed curvature at very high fields are genuine rather than any artefact of the experimental technique.
20
_•Ms
10
o
(10- e.rn,uJcc) o ° B
o
oo
~
o
0
o
o
8
o
eo
®
o~
8
o
-10
-20
0
J
I
100
I
I
200
I
I
300 ~
t
kOauss
Fig. 4. As for fig. I but for several e m p t y run values a M e (as explained in the text).
L. W. Roeland et aL /Nonlinear magnetization o f bismuth crystals
101
Acknowledgements This work was made possible by the financial support o f the Stichting voor F u n d a m e n t e e l Onderzoek der Materie (F.O.M.), which in turn is supported by Z.W.O. We also wish to thank Mr. T. E. B r o w n o f the Cavendish Laboratory for growing the crystals.
References 1) N. L. Brignall and D. Shoenberg, J. Phys. C 7 (1974) 1499. 2) N. L. Brignall, Ph.D. Thesis, Cambridge University 1972 and upubllshed data. 3) D. Shoenberg, Proceedings Conference on Semi-metals, Cardiff (Sept. 1973), to be published. 4) J. W. MeClure and D. Shoenberg, to be published. 5) R. Gersdorf, F. A. Muller and L. W. Roeland, Rev. Sci. Inst. 36 (1965) 1100. L. W. Roeland, F. A. Muller and R. Gersdorf, Coll. Int. Cent. Nat. Rech. Sci. no. 166 (1967) 175. 6) R. Gersdorf, F. A. Muller and L. W. Roeland, Coll. Int. Cent. Nat. Reeh. Sci. no. 166 (1967) 185.