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The thermoelectric power in chromium and vanadium
J. Phys.
Chem. Solids
Pergamon Press 1963. Vol. 24, pp. 501-506.
THE THERMOELECTRIC
Printed in Great Britain.
POWER
IN CHROMIUM
AND
VANADIUM* A. R. MACKINTOSH
and
L. SILL
Institute for Atomic Research and Department of Physics, Iowa State University, Ames, Iowa (Received
18 December
1962)
Abstract-The thermoelectric power in pure chromium and vanadium has been measured between 4.2 and 340°K and found to be positive in both metals over the whole temperature range. Maxima in the curves of thermopower against temperature at low temperatures are interpreted in terms of a phonon drag component. An anomaly in the results for both monocrystalline and polycrystalline chromium at the Ntel temperature is interpreted in terms of the LIDIARD-OVERHAUSERmodel of conduction electron antiferromagnetism. Hysteresis in the measurements on chromium over the whole temperature range is attributed to the effect of antiferromagnetic domains. An anomaly in the thermopower of vanadium at approximately 217°K is attributed either to a transition to an antiferromagnetic state or to a structural phase transition, possibly involving ordering of impurities. Experiments for distinguishing between these possibilities are suggested.
INTRODUCTION
determined
by the equation
CONSIDERABLE
interest has been evinced recently in the exact nature of the antiferromagnetic phase of chromium and in the possibility of the existence of such a phase in vanadium. The transition to the ordered state in chromium manifests itself through a variety of physical properties,(l-5) and the nature of the magnetic ordering has been explored using neutron diffraction techniques.@-1s) OVERHAUSER has proposed that the antiferromagnetic phase is characterized by the existence of a static spin density wave in which the conduction electron spin density varies sinusoidally through the metal. He has shown that the experimental data can be understood on the basis of three spin density waves, with their wave vectors in the [IOO] directions. The observation by various workers of small anomalies in various properties of vanadium(rs-15) has raised the possibility that a phase with a small amplitude spin density wave might exist in it, and possibly also in other metals. The thermoelectric power in a metal is * Contribution No. 1249. Work was performed Ames Laboratory of the U.S. Atomic Energy mission. A
in the Com-
s=--.-
?ra k2T 3
e
ai0gi -+II as
alogE aa 1EF
(1)
where i is the mean free path of electrons on a Fermi surface of area x, and the derivatives are evaluated at the Fermi surface. It thus provides an excellent tool for the study of small changes in the state of metals, since it is extremely sensitive both to changes in the electronic structure and to the mechanisms which scatter the electrons. The experiments described in this paper were undertaken primarily in order to elucidate further the nature of the magnetic transition in chromium and to gather more evidence about the possibility of a phase transformation in vanadium. THE
EXPERIMENTAL
PROCEDURE
The experimental procedure used was basically the same as that described by BORN et a2.(ls) A temperature difference of about 2°K was maintaind across the specimens by means of a heater soldered to one end, and the mean sample temperature varied by using a heat-leak chamber. The temperature at each end of the sample and its mean
501
A.
502
R.
MACKINTOSH
temperature were measured with copper-constantan thermocouples. The thermoelectric voltage between the specimen and copper leads was measured with a thermofree Rubicon microvolt potentiometer and the absolute thermoelectric power of the specimen determined by standardizing with a piece of pure lead, whose absolute thermopower is known.(r” The relative accuracy of the measurement is estimated to be within i. O-05 pV/“K for the thermoelectric power and within I_ O.OS”K for
and
L.
SILL
20 to 90°K using liquid hydrogen and finally 4.2 to 30°K using liquid helium. Measurements were made on a single crystal and a polycrystal of chromium and a polycrystalline vanadium sample. In each case the specimen was in the form of a rod, approximately 15 mm by 2 mm. The ratio of the resistivity at 300°K to that at 4~2°K was about 150 for the chromium samples and the orientation of the axis of the single crystal was at 10” from the [ill] direction in a (110) plane. The resistivity ratio of the vanadium was
SWGLEi CRYSTAL G~MIUM
0
0
20
40
60
80
lo0
FIG. 1. The thermoelectric
120
140
160 180 200 TEMPERATURE,
power of chromium
the temperature. The systematic errors are more difficult to evaluate, but above 20°K the absolute accuracy of the thermoelectric power measurements is estimated to be within 2 0.1 pV/‘K, and that of the temperature to be within -t_ O.S’K. Below 20°K the results are somewhat less precise because of the difficulty of establishing equilibrium and the lack of reproducibility of the thermocouples. In addition, non-reproducible results were initially obtained for vanadium above 20°K on account of adsorption of helium exchange gas, In order to overcome this difficulty the temperature range was covered in three stages; in order, 70 to 340°K using liquid nitrogen in the bath,
220 ‘K
7.40
280
300
320
340
as a function of temperature.
considerably lower, about 30, probably due to the presence of a considerable quantity of adsorbed gases. THE RESULTS
The observed temperature variation of the thermoelectric powers of chromium and vanadium are shown in Figs. 1 and 2. The thermopower of both metals was positive over the whole temperature range. The measurements on both the single crystal and the polycrystal of chromium revealed an abrupt change in the slope of the curve of thermopower against temperature at about 311”K, the
THE
THERMOELECTRIC
POWER
IN
antiferromagnetic NCel temperature. Immediately below this temperature there is a large hump in the curve, and the single crystal results show another, smaller one, with a maximum at about 25°K. No clear evidence for any anomaly was observed in the temperature range near lZO”K, where a further magnetic transition is known to take place,(T) even though a careful search was made in this region in run 2. Considerable hysteresis was observed between runs over the whole range from 4.2”K to 340”K,
CHROMIUM
AND
VANADIUM
503
A small but reproducible anomaly, consisting of an abrupt change in slope of the thermopower curve, was observed in vanadium at about 217°K when the temperature was increased, but not when the temperature was decreased through this value. In addition, a reproducible hysteresis was observed in this region. At low temperature, a rather distinct hump was observed, with a maximum at about 75”K, and the thermoelectric power went abruptly to zero at about 5”K, the superconducting transition temperature.
2.50c2.25
-
z =‘2.00
-
% :I.,,P Yl.50d g,.25I” t-
1.000.75
-
0.50
-
0.25
-
ot_
*
DECIWI
0I
f,
20
40I
60I
60I
100 I
I20 I,,
140
160
160 ],
, 200
TEMPERATUIE,
220
240
,
,
,
260
260
300
Ry?
J
320
lK
FIG. 2. The thermoelectric power of vanadium as
both below and immediately above the NCel temperature. In addition, the thermoelectric power was observed to decrease with temperature immediately above the Ntel temperature in the single crystal while in the polycrystal it increases with temperature. These effects are thought to be genuine, and not due to the experimental technique since measurements on non-magnetic metals, both before and after these experiments using essentially the same technique, yielded completely reproducible results. Similar hysteresis near the NCel temperature in measurements of the electrical resistivity has been reported by ARAJS et aI.
,
a
function of temperature. DISCUSSION
The thermoelectric powers of both chromium and vanadium are positive over the whole temperature range studied and this can be interpreted, according to equation (l), as indicating that the energy surfaces just above the Fermi level have a smaller area than the Fermi surface itself, since ai)& is usually positive. This implies that the Fermi surfaces in these metals are predominantly hole-like. In view of the well known difficulty of interpreting the sign of the thermoelectric power,(m) however, this conclusion should be treated with caution. The low temperature humps in the thermoelectric
504
A. R. MACKINTOSH
of both chromium and vanadium powers can probably be ascribed to phonon-drag. In chromium the humps are not sufficiently distinct to allow a detailed analysis, but the low temperature data for vanadium, excluding the first two points, which, as determined separately by resistivity measurements, are affected by the superconducting transition, give a good fit to the formula S = nT+bT3 (2) where the two terms represent, respectively, the diffusion and phonon-drag contributions. The values of the coefficients are 11= 6.5 x 10-s PV deg-2 b = 24 x 10-a FV deg-4 These can be compared with the values of -9.8 x 10-s PV deg-2 and - 3.5 x 10-4 FV deg-l, respectively, deduced by GOLD and PEARSON@) for pure lead from the results of CHRISTIANet uZ.@l) LIDIARD(“~) first suggested that the antiferromagnetism of chromium is associated with the conduction electrons and showed that the small anomalies in the specific heat and magnetic susceptibility can be understood in this way. OVERHAUSER developed this idea and proposed that the antiferromagnetic phase in chromium is described by three static spin density waves which have wave vectors along the [100] directions and are transversely polarized between the NCel temperature and about 120°K and longitudinally polarized below 120°K. He has shown that this model can account satisfactorily for the change in the physical properties associated with the magnetic transition. The onset of a spin density wave state can also explain the behavior of the thermoelectric power near the magnetic transition. The change in slope of the curve and the rapid increase in magnitude just below the transition is probably caused by the appearance of extra planes of energy discontinuity in the Brillouin zone structure due to the magnetic ordering, together with a rapid variation of magnetic disorder scattering. This is analogous to the behavior of the transport properties of the rare-earth metals near magnetic ordering transitions, which has been discussed by one of us.@) In the case of the rare earths, however, a localized moment is present on the ions above the Neel temperature, whereas
and L.
SILL
the neutron diffraction results of WILKINSON et a1.(24) and the observation of a nuclear magnetic resonance by BARNES and GRAHAM(~~) strongly indicate that no localized moment is present on the chromium ions above the NCei temperature. It is conceivable that the transition is one in which the metal goes from a state with no localized moment to one in which localized moments are present and are ordered. This would happen, for instance, if the free energy curve for the nonlocalized state cut that for the state with localized moments below the hypothetical ordering temperature of the latter. In this case, however, a sudden change in the electronic structure would be expected and consequently a discontinuity in the thermoelectric power, which is not observed. The ampiitude of the nuclear magnetic resonance, which goes continuously to zero at the transition,@5) and the small specific heat anomaly@) also provide evidence against this possibility. It seems, therefore, that the antiferromagnetism of chromium must be associated with the conduction electrons and that it is adequately represented by a spin density wave model. No evidence for the spin-flip transition at lower temperatures was detected in this work, but this transition does not lead to any new periodicities in the lattice, so the changes in electronic structure and scattering which it causes are probably relatively minor. The large amount of magnetic hysteresis observed both below and above the NCel temperature might tentatively be attributed to antiferromagnetic domain structure. A domain wall in a single crystal is presumably a region over which the phase of the spin density wave varies abruptly. Similar effects in the resistivity of pure chromium have been observed by ARAJS et uZ.tlS) near the Neel point and magnetomechanical damping experiments have been interpreted on the same basis by DE MORTON.@~) The fact that the hysteresis persists above the transition shows that there is a considerable amount of residual short range ordering in the paramagnetic phase. The persistence of order above the NCel temperature was also observed in the neutron diffraction experiments of BACON,(~)although the effect was much greater in a polycrystalline sample than in a single crystal. The small but distinct and reproducible anomaly in the thermoelectric power of vanadium at
THE
THERMOELECTRIC
POWER
IN CHROMIUM
AND
VANADIUM
SO5
about 217°K is also suggestive of some kind of the KNIGHT shift of the Vs1 resonance in pure vanadium on cooling to liquid nitrogen temperaphase change. Small anomalies in the electrical resistivity in this region have been observed by ture through the region of the anomalies, and this LOOMIS and CARLSON and by others. BURGER contrasts strongly with the abrupt vanishing of the Crss resonance at the N6el point in pure chromium. and TAYLOR have observed a very small change in the magnetic susceptibility. LOOMIS and To make this result compatible with the occurrence C~~SON(~~) and BOLEF, DE KXJW and BRANDT(~~) of a magnetic transition in vanadium, it appears necessary to assume that the electrons particihave reported anomalies in the elastic constants pating in the antiferromagnetic ordering have no and thermal expansion coefficients at temperatures S-character so that they do not penetrate the which are extremely sensitive to the impurity content of the vanadium. The two most likely nucleus, and this seems rather unlikely. If vanathen there is particular explanations for these phenomena appear to be dium is antiferromagnetic also. either that there is a transition to a very weakly interest in the fact that it is sup~~ondueting Because of the small number of electrons particiantiferromagnetic phase or that there is a strucpating in the magnetic ordering it is in principle tural phase change, possibly involving ordering energetically possible for a metal to undergo of impurities. In either case the non-appearance successively transitions to an antiferromagnetic of the anomaly when the sample is cooled through state and, at some lower temperature, to a superthe transition is probably due to local supercooling in the poiycrystalline sample. A similar effect has conducting state. There does not yet appear to be sufficient evibeen observed at the ferroelectric transition of dence to distinguish definitely between the SrTiOs by KROGSTADand Moss.@~) LOOM& and CARLSON have shown that a hypotheses of a structural or a magnetic transition, brittle-ductile transition occurs in vanadium at although the experiments of Barnes and Graham about 200°K. They conclude that this transition appear to favor the former. It is, of course, possible is due to the interaction of dislocations with that both occur and that they are related. Further minute amounts of impurity and suggest that it experimental data therefore seem to be required may be accentuated by the ordering of impurities. to settIe this point. A careful neutron diffraction They cite asevidencefor this conclusion the internal study at low temperatures might reveal the friction peak at around 200°K and the change in presence of magnetic ordering, although failure to the slope of the lattice parameter vs. temperature discover magnetic reflections need not imply its curve in crystal bar vanadium at about the same absence.(rl) It would be interesting to determine temperature. One would expect that such a whether there is any magnetic field dependence transformation would have a considerable effect of the transport properties near the anomaly, as on the elastic moduli, but it is not so clear that it would be expected if it were magnetic in origin. should affect the transport properties. If, as is Perhaps the most valuable effort, however, would likely, however, it alters the lattice vibration spec- be directed towards the preparation of purer trum, it will affect the electron-lattice scattering vanadium crystals. Many workers have observed probability and hence the transport properties, in that the temperature of the anomalies is very deaddition to any possibIe effect on the electronic pendent on impurity concentration. In particular structure. In particular the thermoelectric power, LOOMIS and CARLSON@ have shown that rewhich is very sensitive to the electronic relaxation moving gaseous impurities lowers the transition time, might be appreciably affected. temperature. It would be of great interest to deterIf, alternatively, there is a small amplitude spin mine whether this process can be continued indensity wave in vanadium at low temperatures, all definitely. of the anomalies can readily be explained in terms of the modification of the electronic structure Acknowledgements-We are very grateful to Professor which this entails. This mechanism would also 0. N. CARLSON for supplying the chromium and vanadium samples used in these experiments. Valuable account for the small anomaly in the magnetic discussions with R. G. BARNES, S. LEGVOLD, J. F. susceptibility.(ls) On the other hand, BARNESand SMITH, A. V. GOLD and D. I. BOLEF are gratefully GRAHAM@~)observed only a very small change in acknowledged.
506
A.
R.
MACKINTOSH
and
L.
SILL
REFERENCES 1. PURSEY H., J. Inst. Met& 86, 362 (1958). 2. DE VRIES G. and RATHENAUG. W., J. Phys. Chem. Solids 2,339 (1957). 3. COLLINGSE. W., HEDGCOCXF. T. and SIDDIQUIA., Phil. Mag. 6,155 (1961). 4. BEAUMONTR. H., CHIHORAH. and MORRISONJ. A., Phil. Mag. 5, 188 (1960). 5. WHITE G. K., .4ustraZ. J. Pkys. 14, 359 (1961). 6. CORLISS L., HASTINGSJ. and WEISS R., Phys. Rev. Letters 3, 211 (1959). 7. BYKOV V. ?;., GOLOVKIN V. S., ACEEV N. V., LEVDIA V. A. and VINOGRADOVS. J., Dokl. akad. nauk. S.S.S.R. 128, 1153 (1959). 8. WILKINSON M. K., WOLLAN E. 0. and KOEHLER W. C., Bull. Amer. phys. Sot. 5,456 (1960). 9. BACOX G. E., Acta Cryst. 14, 823 (1961). IO. SHIUNE G. and TAKEI W. J., J. phys. Sot. Japan (to be published). 11. OVERHAUSER A. W., Phys. Rev. 128, 1437 (1962). 12. LOOMIS B. A. and CARLSON0. N., Reactive Metals (Edited bv CLAUGH W. R.1 Vol. 2. D. 227. Inter.&ence, Gew York (1959).’ ._ 13. BURGERJ. P. and TAYLOR M. A., Phys. Rev. Letters 6, 185 (1961). 14. WHITE G. K. and WOODS S. B., Phil. Trans. Roy. Sot. 251, 273 (1959).
15. BOLEF D. I., DE KLERK J. and BRANDT G. B., BzdZ. Amer. phys. Sot. 7, 236 (1962). 16. BORN H.-J., LEGVOL& S. and SPEDDING F. H., J. abbl. Phvs. 32. 2543 (1961). 17. CH&TIAN-J. W:, JAN j. P.; PEARSONW. B. and TEMPLETON I. B., Proc. Roy. Sot. A245, 213 (1958). 18. ARAJS S., COLVIN R. V. and MARCINKOWSI~I M. J, J. Less-Common Metals 4. 46 (1962). 19. MACDONALD D. K. C., Thermoelectricity, p. 103. Tohn Wilev. New York (1962). 20. GOLD A. V.-&d PEARSON‘W. B., Canad. J. Phys. 39, 445 (1961). 21. CHRISTIAN J. W., JAN J. P., PEARSONW. B. an TEMPLETON I. M., Phil. Mag. 5, 765 (1960). 22. LIDIARD A. B., Proc. phys. Sot. Lond. A66, 1188 (1953). 23. MACKINTOSH A. R., Phys. Rev. Letters 9, 90 (1962). 24. WILKINSON M. K., WOLLAN E. O., KOEHLER W. C. and CABLE J. W., Phys. Rev. 127, 2080 (1962). 25. BARNES R. G. and GRAHA~~ T. P., Phys. Rev. Letters 8, 248 (1962). 26. DE MORTON M. E., Phil. Mag. 6, 825 (1961). 27. KROGSTADR. S. and Moss R. W., Bull. Amer. ph?s. Sot. 7, 192 (1962).
Chem. Solids
Pergamon Press 1963. Vol. 24, pp. 501-506.
THE THERMOELECTRIC
Printed in Great Britain.
POWER
IN CHROMIUM
AND
VANADIUM* A. R. MACKINTOSH
and
L. SILL
Institute for Atomic Research and Department of Physics, Iowa State University, Ames, Iowa (Received
18 December
1962)
Abstract-The thermoelectric power in pure chromium and vanadium has been measured between 4.2 and 340°K and found to be positive in both metals over the whole temperature range. Maxima in the curves of thermopower against temperature at low temperatures are interpreted in terms of a phonon drag component. An anomaly in the results for both monocrystalline and polycrystalline chromium at the Ntel temperature is interpreted in terms of the LIDIARD-OVERHAUSERmodel of conduction electron antiferromagnetism. Hysteresis in the measurements on chromium over the whole temperature range is attributed to the effect of antiferromagnetic domains. An anomaly in the thermopower of vanadium at approximately 217°K is attributed either to a transition to an antiferromagnetic state or to a structural phase transition, possibly involving ordering of impurities. Experiments for distinguishing between these possibilities are suggested.
INTRODUCTION
determined
by the equation
CONSIDERABLE
interest has been evinced recently in the exact nature of the antiferromagnetic phase of chromium and in the possibility of the existence of such a phase in vanadium. The transition to the ordered state in chromium manifests itself through a variety of physical properties,(l-5) and the nature of the magnetic ordering has been explored using neutron diffraction techniques.@-1s) OVERHAUSER has proposed that the antiferromagnetic phase is characterized by the existence of a static spin density wave in which the conduction electron spin density varies sinusoidally through the metal. He has shown that the experimental data can be understood on the basis of three spin density waves, with their wave vectors in the [IOO] directions. The observation by various workers of small anomalies in various properties of vanadium(rs-15) has raised the possibility that a phase with a small amplitude spin density wave might exist in it, and possibly also in other metals. The thermoelectric power in a metal is * Contribution No. 1249. Work was performed Ames Laboratory of the U.S. Atomic Energy mission. A
in the Com-
s=--.-
?ra k2T 3
e
ai0gi -+II as
alogE aa 1EF
(1)
where i is the mean free path of electrons on a Fermi surface of area x, and the derivatives are evaluated at the Fermi surface. It thus provides an excellent tool for the study of small changes in the state of metals, since it is extremely sensitive both to changes in the electronic structure and to the mechanisms which scatter the electrons. The experiments described in this paper were undertaken primarily in order to elucidate further the nature of the magnetic transition in chromium and to gather more evidence about the possibility of a phase transformation in vanadium. THE
EXPERIMENTAL
PROCEDURE
The experimental procedure used was basically the same as that described by BORN et a2.(ls) A temperature difference of about 2°K was maintaind across the specimens by means of a heater soldered to one end, and the mean sample temperature varied by using a heat-leak chamber. The temperature at each end of the sample and its mean
501
A.
502
R.
MACKINTOSH
temperature were measured with copper-constantan thermocouples. The thermoelectric voltage between the specimen and copper leads was measured with a thermofree Rubicon microvolt potentiometer and the absolute thermoelectric power of the specimen determined by standardizing with a piece of pure lead, whose absolute thermopower is known.(r” The relative accuracy of the measurement is estimated to be within i. O-05 pV/“K for the thermoelectric power and within I_ O.OS”K for
and
L.
SILL
20 to 90°K using liquid hydrogen and finally 4.2 to 30°K using liquid helium. Measurements were made on a single crystal and a polycrystal of chromium and a polycrystalline vanadium sample. In each case the specimen was in the form of a rod, approximately 15 mm by 2 mm. The ratio of the resistivity at 300°K to that at 4~2°K was about 150 for the chromium samples and the orientation of the axis of the single crystal was at 10” from the [ill] direction in a (110) plane. The resistivity ratio of the vanadium was
SWGLEi CRYSTAL G~MIUM
0
0
20
40
60
80
lo0
FIG. 1. The thermoelectric
120
140
160 180 200 TEMPERATURE,
power of chromium
the temperature. The systematic errors are more difficult to evaluate, but above 20°K the absolute accuracy of the thermoelectric power measurements is estimated to be within 2 0.1 pV/‘K, and that of the temperature to be within -t_ O.S’K. Below 20°K the results are somewhat less precise because of the difficulty of establishing equilibrium and the lack of reproducibility of the thermocouples. In addition, non-reproducible results were initially obtained for vanadium above 20°K on account of adsorption of helium exchange gas, In order to overcome this difficulty the temperature range was covered in three stages; in order, 70 to 340°K using liquid nitrogen in the bath,
220 ‘K
7.40
280
300
320
340
as a function of temperature.
considerably lower, about 30, probably due to the presence of a considerable quantity of adsorbed gases. THE RESULTS
The observed temperature variation of the thermoelectric powers of chromium and vanadium are shown in Figs. 1 and 2. The thermopower of both metals was positive over the whole temperature range. The measurements on both the single crystal and the polycrystal of chromium revealed an abrupt change in the slope of the curve of thermopower against temperature at about 311”K, the
THE
THERMOELECTRIC
POWER
IN
antiferromagnetic NCel temperature. Immediately below this temperature there is a large hump in the curve, and the single crystal results show another, smaller one, with a maximum at about 25°K. No clear evidence for any anomaly was observed in the temperature range near lZO”K, where a further magnetic transition is known to take place,(T) even though a careful search was made in this region in run 2. Considerable hysteresis was observed between runs over the whole range from 4.2”K to 340”K,
CHROMIUM
AND
VANADIUM
503
A small but reproducible anomaly, consisting of an abrupt change in slope of the thermopower curve, was observed in vanadium at about 217°K when the temperature was increased, but not when the temperature was decreased through this value. In addition, a reproducible hysteresis was observed in this region. At low temperature, a rather distinct hump was observed, with a maximum at about 75”K, and the thermoelectric power went abruptly to zero at about 5”K, the superconducting transition temperature.
2.50c2.25
-
z =‘2.00
-
% :I.,,P Yl.50d g,.25I” t-
1.000.75
-
0.50
-
0.25
-
ot_
*
DECIWI
0I
f,
20
40I
60I
60I
100 I
I20 I,,
140
160
160 ],
, 200
TEMPERATUIE,
220
240
,
,
,
260
260
300
Ry?
J
320
lK
FIG. 2. The thermoelectric power of vanadium as
both below and immediately above the NCel temperature. In addition, the thermoelectric power was observed to decrease with temperature immediately above the Ntel temperature in the single crystal while in the polycrystal it increases with temperature. These effects are thought to be genuine, and not due to the experimental technique since measurements on non-magnetic metals, both before and after these experiments using essentially the same technique, yielded completely reproducible results. Similar hysteresis near the NCel temperature in measurements of the electrical resistivity has been reported by ARAJS et aI.
,
a
function of temperature. DISCUSSION
The thermoelectric powers of both chromium and vanadium are positive over the whole temperature range studied and this can be interpreted, according to equation (l), as indicating that the energy surfaces just above the Fermi level have a smaller area than the Fermi surface itself, since ai)& is usually positive. This implies that the Fermi surfaces in these metals are predominantly hole-like. In view of the well known difficulty of interpreting the sign of the thermoelectric power,(m) however, this conclusion should be treated with caution. The low temperature humps in the thermoelectric
504
A. R. MACKINTOSH
of both chromium and vanadium powers can probably be ascribed to phonon-drag. In chromium the humps are not sufficiently distinct to allow a detailed analysis, but the low temperature data for vanadium, excluding the first two points, which, as determined separately by resistivity measurements, are affected by the superconducting transition, give a good fit to the formula S = nT+bT3 (2) where the two terms represent, respectively, the diffusion and phonon-drag contributions. The values of the coefficients are 11= 6.5 x 10-s PV deg-2 b = 24 x 10-a FV deg-4 These can be compared with the values of -9.8 x 10-s PV deg-2 and - 3.5 x 10-4 FV deg-l, respectively, deduced by GOLD and PEARSON@) for pure lead from the results of CHRISTIANet uZ.@l) LIDIARD(“~) first suggested that the antiferromagnetism of chromium is associated with the conduction electrons and showed that the small anomalies in the specific heat and magnetic susceptibility can be understood in this way. OVERHAUSER developed this idea and proposed that the antiferromagnetic phase in chromium is described by three static spin density waves which have wave vectors along the [100] directions and are transversely polarized between the NCel temperature and about 120°K and longitudinally polarized below 120°K. He has shown that this model can account satisfactorily for the change in the physical properties associated with the magnetic transition. The onset of a spin density wave state can also explain the behavior of the thermoelectric power near the magnetic transition. The change in slope of the curve and the rapid increase in magnitude just below the transition is probably caused by the appearance of extra planes of energy discontinuity in the Brillouin zone structure due to the magnetic ordering, together with a rapid variation of magnetic disorder scattering. This is analogous to the behavior of the transport properties of the rare-earth metals near magnetic ordering transitions, which has been discussed by one of us.@) In the case of the rare earths, however, a localized moment is present on the ions above the Neel temperature, whereas
and L.
SILL
the neutron diffraction results of WILKINSON et a1.(24) and the observation of a nuclear magnetic resonance by BARNES and GRAHAM(~~) strongly indicate that no localized moment is present on the chromium ions above the NCei temperature. It is conceivable that the transition is one in which the metal goes from a state with no localized moment to one in which localized moments are present and are ordered. This would happen, for instance, if the free energy curve for the nonlocalized state cut that for the state with localized moments below the hypothetical ordering temperature of the latter. In this case, however, a sudden change in the electronic structure would be expected and consequently a discontinuity in the thermoelectric power, which is not observed. The ampiitude of the nuclear magnetic resonance, which goes continuously to zero at the transition,@5) and the small specific heat anomaly@) also provide evidence against this possibility. It seems, therefore, that the antiferromagnetism of chromium must be associated with the conduction electrons and that it is adequately represented by a spin density wave model. No evidence for the spin-flip transition at lower temperatures was detected in this work, but this transition does not lead to any new periodicities in the lattice, so the changes in electronic structure and scattering which it causes are probably relatively minor. The large amount of magnetic hysteresis observed both below and above the NCel temperature might tentatively be attributed to antiferromagnetic domain structure. A domain wall in a single crystal is presumably a region over which the phase of the spin density wave varies abruptly. Similar effects in the resistivity of pure chromium have been observed by ARAJS et uZ.tlS) near the Neel point and magnetomechanical damping experiments have been interpreted on the same basis by DE MORTON.@~) The fact that the hysteresis persists above the transition shows that there is a considerable amount of residual short range ordering in the paramagnetic phase. The persistence of order above the NCel temperature was also observed in the neutron diffraction experiments of BACON,(~)although the effect was much greater in a polycrystalline sample than in a single crystal. The small but distinct and reproducible anomaly in the thermoelectric power of vanadium at
THE
THERMOELECTRIC
POWER
IN CHROMIUM
AND
VANADIUM
SO5
about 217°K is also suggestive of some kind of the KNIGHT shift of the Vs1 resonance in pure vanadium on cooling to liquid nitrogen temperaphase change. Small anomalies in the electrical resistivity in this region have been observed by ture through the region of the anomalies, and this LOOMIS and CARLSON and by others. BURGER contrasts strongly with the abrupt vanishing of the Crss resonance at the N6el point in pure chromium. and TAYLOR have observed a very small change in the magnetic susceptibility. LOOMIS and To make this result compatible with the occurrence C~~SON(~~) and BOLEF, DE KXJW and BRANDT(~~) of a magnetic transition in vanadium, it appears necessary to assume that the electrons particihave reported anomalies in the elastic constants pating in the antiferromagnetic ordering have no and thermal expansion coefficients at temperatures S-character so that they do not penetrate the which are extremely sensitive to the impurity content of the vanadium. The two most likely nucleus, and this seems rather unlikely. If vanathen there is particular explanations for these phenomena appear to be dium is antiferromagnetic also. either that there is a transition to a very weakly interest in the fact that it is sup~~ondueting Because of the small number of electrons particiantiferromagnetic phase or that there is a strucpating in the magnetic ordering it is in principle tural phase change, possibly involving ordering energetically possible for a metal to undergo of impurities. In either case the non-appearance successively transitions to an antiferromagnetic of the anomaly when the sample is cooled through state and, at some lower temperature, to a superthe transition is probably due to local supercooling in the poiycrystalline sample. A similar effect has conducting state. There does not yet appear to be sufficient evibeen observed at the ferroelectric transition of dence to distinguish definitely between the SrTiOs by KROGSTADand Moss.@~) LOOM& and CARLSON have shown that a hypotheses of a structural or a magnetic transition, brittle-ductile transition occurs in vanadium at although the experiments of Barnes and Graham about 200°K. They conclude that this transition appear to favor the former. It is, of course, possible is due to the interaction of dislocations with that both occur and that they are related. Further minute amounts of impurity and suggest that it experimental data therefore seem to be required may be accentuated by the ordering of impurities. to settIe this point. A careful neutron diffraction They cite asevidencefor this conclusion the internal study at low temperatures might reveal the friction peak at around 200°K and the change in presence of magnetic ordering, although failure to the slope of the lattice parameter vs. temperature discover magnetic reflections need not imply its curve in crystal bar vanadium at about the same absence.(rl) It would be interesting to determine temperature. One would expect that such a whether there is any magnetic field dependence transformation would have a considerable effect of the transport properties near the anomaly, as on the elastic moduli, but it is not so clear that it would be expected if it were magnetic in origin. should affect the transport properties. If, as is Perhaps the most valuable effort, however, would likely, however, it alters the lattice vibration spec- be directed towards the preparation of purer trum, it will affect the electron-lattice scattering vanadium crystals. Many workers have observed probability and hence the transport properties, in that the temperature of the anomalies is very deaddition to any possibIe effect on the electronic pendent on impurity concentration. In particular structure. In particular the thermoelectric power, LOOMIS and CARLSON@ have shown that rewhich is very sensitive to the electronic relaxation moving gaseous impurities lowers the transition time, might be appreciably affected. temperature. It would be of great interest to deterIf, alternatively, there is a small amplitude spin mine whether this process can be continued indensity wave in vanadium at low temperatures, all definitely. of the anomalies can readily be explained in terms of the modification of the electronic structure Acknowledgements-We are very grateful to Professor which this entails. This mechanism would also 0. N. CARLSON for supplying the chromium and vanadium samples used in these experiments. Valuable account for the small anomaly in the magnetic discussions with R. G. BARNES, S. LEGVOLD, J. F. susceptibility.(ls) On the other hand, BARNESand SMITH, A. V. GOLD and D. I. BOLEF are gratefully GRAHAM@~)observed only a very small change in acknowledged.
506
A.
R.
MACKINTOSH
and
L.
SILL
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