- Email: [email protected]
Effects of warped energy surfaces on the low field hall coefficient of graphite
Solid State Communications, Vol. 26, pp. 333-337. 0 Pergamon Press Ltd. 1978. Printed in Great Britain.
0038-1098/78/05OI-0333
EFFECTS OF WARPED ENERGY SURFACES ON THE LOW FIELD HALL COEFFICIENT
$02.00/O
OF GRAPHITE
R.O. Dillon* and I.L. Spain Laboratory for High Pressure Science, Dept. of Chemical Engineering, University of Maryland, College Park, MD 20742, U.S.A. (Received
3 1 October 1977 by G. Burns)
Unusual field dependence of the low field galvanomagnetic properties of graphite has been attributed to small pockets of mobile minority carriers in the corners of the Brillouin Zone. However, this explanation does not account for the sample or temperature dependence of these phenomena. An alternative exnlanation. based on the trigonal warping of the constant energy surfaces, is presented. minority hole. The trend in R&B) for Kish graphite is generally to more negative values at low field [IO, 1 I]. The mechanism responsible for the unusual field dependence of the low field Hall coefficient must explain the change in the character of the low-field Hall coefficient (a) with temperature for natural crystals and (b) with crystallite size at 77 K for pyrolytic samples. It is postulated here that the trends in the low-field Hall coefficient are not reiated to the minority hole pockets at the corners of the Brillouin Zone. This is suggested in a number of ways. Firstly, the sign of the minority carriers in the Brillouin Zone can only change if the sign of the band overlap parameter y2 changes. Although the dimensions of the pocket are largely determined by the band overlap parameter A, a change of sign of A cannot lead to a change of sign of the carrier type (see [s] for a thorough discussion). It is extremely unlikely that ~2 could change sign with temperature or crystallite size. Secondly, the nature of the pocket can be inferred from the sign of y2. Experiments performed at low temperature (T - 14 K) show that y2 is negative [12-l 51, so that the minority carrier pocket is hole-like. At this temperature, the nature of the low-field Hall coefficient is electron-like for natural, Kish and synthetic crystals (Fig. 1). Thirdly, the Shubnikov-de Haas frequencies for pyrolytic graphite are the same as those for natural crystals [ 161. There is some uncertainty about the interpretation of these data, since two low-frequency signals are seen in some crystals [7, 16, 171 and their origin has not been unambiguously identified. However, it has been further shown from magnetoreflection studies that the band parameters of both pyrolytic and natural crystals are the same within experimental error [ 181. This includes the parameters y2 and A which determine the size and nature of the minority pockets. A fourth piece of information comes from a
UNUSUAL BEHAVIOR in the low-field galvanomagnetic properties of the semimetal graphite, particularly the Hall coefficient, has been generally attributed to the presence of small pockets of mobile minority holes in the corner of the Brillouin Zone. There have been a number of problems associated with this basic premise, and the purpose of the present paper is to suggest a new explanation for these phenomena based on the trigonal warping of the constant energy surfaces. Anomalies in the low field behavior (m < 1, where B is the magnetic field and ii is a measure of the mean carrier mobility) can be seen in an unusual field dependence of the magnetoresistance. More striking effects occur in the Hall coefficient. Soule [I] found that for natural crystals of graphite, the Hall coefficient becomes more positive at 77 K for fields less than about 0.1 T (1 kg) whereas at 4.2 K the trend is reversed (Fig. 1). At 77 K this behavior was attributed to the presence of mobile minority holes [l, 21. Using the accepted dispersion relationship for the electrons and holes in graphite [3], with current estimates for the band overlap parameters (for recent reviews see [4] and [5]) pockets of low-mass, hole-like states occur in the corners of the Brillouin Zone. These pockets have been used to explain [6] Soule’s observation [7] of a de Haas-van Alphen signal of low frequency. For specimens of highly oriented pyrolytic graphite, the trend in the low-field Hall coefficient is consistent with mobile electrons at all temperatures [8] (Fig. 1). However, upon further annealing it was shown [9] that the curve of the Hall coefficient against field at 77 K systematically approached the curve for natural crystals as the crystallite size was increased with heat treatment, so that the low-field behavior changed from that characteristic of a mobile minority electron to that of a mobile * Present address: University New Zealand.
of Waikato, Hamilton,
333
334
THE LOW FIELD HALL COEFFICIENT
ak -=as
OF GRAPHITE
0
Vol. 26, No. 5
EVAB
(1)
where k is the electron wavevector, V the electron velocity and B the magnetic field, then the velocity is expressed as a Fourier series: V =
L 298
K
-0.2 -
EP-7
[SOULE (1958)]
-.-
SA-26[SPAIN
etol(l967)]
il i i -0.6
2 V(m) eimws m=,
(2)
where o is related to the time, T, for the electron to complete an orbit by w = 2n/T, and m is the index. Using McClure’s formalism for the Boltzmann transport equation, the conductivity components u,, and oXYcan be expressed,
Pa) (3b)
~4.2
K
B (TESLA)
Fig. 1. Curves of the Hall coefficient, RH, vs. magnetic field, B, for purified natural crystals [ 1] and highly oriented pyrolytic graphite [8]. Note the trends in RH at very low field, particularly the reversal of the trend for the natural crystal between 77 and 4 K. comparison of the pressure dependence of the minority carrier de Haas-van Alphen frequency with that of the low field Hall coefficient. The extremal area (S) for the minority carrier pocket depends strongly on pressure [ 191 (d In S/dp - 0.09 kb-r ) while the analysis of the galvanomagnetic data using a three band model [ZO] (majority electrons and holes and a minority carrier) for highly oriented pyrolytic graphite suggested that, if the minority carriers are enclosed in an ellipsoidal pocket, their extremal area changes very little with pressure (d In S/dp 5 0.01 kb-‘). An alternative suggestion is made for the origin of this anomalous low-field Hall coefficient behavior, based on the trigonal warping of the constant energy surfaces in the k,-k, plane (z-direction perpendicular to basal planes). Due to the warping, the path (hodograph) followed by an electron or hole in a magnetic field, along a contour of constant energy, is also trigonally warped in either reciprocal or real space. Calculations have been made to assess the importance of this warping on the Hall coefficient using the formalism developed by McClure [2 11. Using a parameter, s, with the dimensions of time defined by the quasimomentum equation:
where g is the area enclosed by a hodograph, the coefficients a,, k, are related to the Fourier components of the velocity, V,, (which in turn are related to the extent of the warping), 6 denotes the sign of the carrier, r is the relaxation time and w the cyclotron frequency. Full details of the computational method are given in [22]. For a circular cross-section only VI, a,, b, are non-zero. The higher-order components contribute appreciably to a,, and uXYonly at low fields. The conductivity components are related to the Hall coefficient by, (4) Although the warping of energy surfaces enhances u_,., appreciably in the low field (wr < 1) region, the magnetic field dependence of the Hall coefficient is controlled by uXy and attention will be focussed on it. The form of u,,(B) for a particular carrier pocket is controlled by the extent of trigonal warping, which in turn is a function of k, in the Slonczewski-Weiss [3] dispersion relationship (i.e. a, and b, are functions of k,). The hodographs predicted from this relationship for various values of k, are shown in Fig. 2. Most of the electron hodographs consist of small regions of strong positive curvature and larger areas of weak negative curvature (concave as viewed from outside the figure). The hodographs for the hole surface vary from those like (g) (Fig. 2) with stronger negative curvature than the electron hodographs to those like (j) with no negative curvature. The form of u,,(B) has been calculated using a simplified model of trigonally warped energy surfaces and a constant relaxation time. From this, three important field ranges can be identified:
Vol. 26, No. 5
THE LOW FIELD HALL COEFFICIENT
OF GRAPHITE
335
1 kY
Fig. 2. Sketch of the constant energy surfaces at the Fermi energy for various values of @($ = k,c/2, z-component of wavevector and c the lattice constant). Band parameters are for set B of [23].
where k, is the
Fig. 3. Possible variation of the parameter I = u~Jcr!& as a function of or. The case drawn shows a more pronounced dip for electrons than holes, but this behavior may be reversed for certain types of scattering. (1) In the very low field region (wr << 1) carriers in the region of strong curvature dominate in uxY. (2) In the very high field region (or > 1) uxY asymptotes to its classical value so that (u,$),,, = ne. (3) In an intermediate field region (0.1 < wr < 1) carriers in the region of weak negative curvature can predominate in uxY. Depending on the geometry of the hodograph, these carriers can change the sign of a,, so that pockets of carriers normally labelled as electron-like may behave for this field region as hole-like with respect to the Hall parameter, and vice-versa. Carriers in the region of strong curvature dominate uxY at very low fields since only these carriers can turn through appreciable Hall angles. In a field range
(- 0.01 < wr $ 1) within the low field region, carriers at regions of weaker curvature also turn through appreciable Hall angles, and thus become important in uxY. Based on the above, Fig. 3 shows a possible variation of the parameter r for the electron surface in graphite, where r z &,/CT& and t and c indicate conductivities based on trigonal and circular hodographs, respectively, containing equal areas. The important feature of Fig. 3 is that the regions of negative curvature on the hodographs reduce the electronic, negative, contribution to uxY at fields 0.01 5 wr 5 1. For certain fields it is even possible for the electronic contributions to uxy to be positive. The variation of r(w7) will be similar for the hole surface. Due to the varying degree of negative curvature in the hole hodographs, it is reasonable that the dip in the
336
THE LOW FIELD HALL COEFFICIENT
T(WT) curve for the holes is less pronounced than for the electrons. However, for certain types of scattering which give a dependence of relaxation time on position along the hodograph, this may be reversed. A possible reason for the low field anomalies in graphite is now apparent. Since graphite is a nearlycompensated semi-metal, the hole and electron contributions to RH nearly cancel each other out, so that small changes in the contribution from one carrier can produce large fractional changes in RH. The positive rise in RH at low temperatures and fields can be related to the dip in T(wT) for the electrons. The less pronounced dip for the holes is not of sufficient magnitude to compensate for the change due to the electrons. The exact balance between the electron and hole contributions can lead to a wide variety of behavior in these materials in the intermediate field range. For instance. if the scattering is such that the average hole mobility exceeds that of the electron, the contribution to RH from the holes is enhanced and positive-going trends at low field may result. This is exactly the kind of effect seen by Soule at 77 K (Fig. 1). The type of scattering might also contribute to a dependence of relaxation time on the carrier location on the hodograph. In this way the “harmonics” in the expression for uxy [equation (3b)l could be strengthened or weakened. In this way the puzzling changes of the low field Hall constant with temperature and specimen type can be explained. For instance, at low temperature the predominant scattering mechanism in natural single crystal or Kish graphite is believed to be from charged impurities, while in pyrolytic graphite scattering from grain boundaries predotninates. At higher temperature, phonon scattering is increasingly important. At 77 K, mixed scattering is effective, and the change in the
OF GRAPHITE
Vol. 26. No. 5
behavior of Cooper and Young’s pyrolytic specimen with annealing can be related to reduced scattering from imperfections. Since boundary scattering rates are inversely proportional to carrier velocity, and the mean electron velocity is greater than that for the holes, the hole mobility is expected to increase relative to the electron as boundary scattering is reduced. Support for the present suggestion that negative curvature has an important effect on the conductivity can be derived from cyclotron resonance experiments (24, 251. All resonances can be identified as originating from the carrier pocket located near point K in the Brillouin zone [25], now identified as the electron surface [ 121. Strong hole-like resonances occur from harmonic terms resulting from the trigonal warping with energy (3N + 1)ho, where N is a negative integer in this case. In conclusion, it is suggested that the anomalous features of the low-field galvanomagnetic properties of graphite occur as a result of the warping of the constant energy surfaces. Although the present calculations were made for simplified models of the constant energy surfaces and relaxation time, results for an exact calculation will show similar features to those obtained here, but details will differ. Considerable complexity will be introduced by using the Slonczewski-Weiss dispersion relationship 131, a relaxation time dependent on k, and k,, and also by including effects due to carrier nondegeneracy. It is preferable to attempt such a calculation using numerical rather than Fourier summation techniques. Acknowledgements
- This research was funded by a grant from the United States Atomic Energy Commission. Helpful discussions were held with Dr. Robert Allgaier and Prof. J.W. McClure.
REFERENCES 1.
SOULE D.E.,Ph’hys. Rev. 112,698
2.
MCCLURE J.W.,Phys.
3.
SLONCZEWSKI J.C. &WEISS P.R.,Ph,vs. Rev. 109,272
4.
MCCLURE J.W., Proc. Int. Conf: Phys. Semimetals Pergamon, London (1971).
5.
SPAIN 1.L.. Chemistry
6.
DRESSELHAUS
7.
SOULE D.E., IBM J, Res. Dev. 8,268
8.
SPAIN I.L., UBBELOHDE A.R. & YOUNG D.A., Phil. Trans. Roy. Sac. (Londonj
9.
COOPERJ.D.,WOOREJ.&YOUNG
(1958).
Rev. 112. 715 (1958).
and Physics ofCarbon
G. & DRESSELHAUS
(1958).
and Narrow Band Gap Semicond.,
Dallas.
1970. p. 127.
(Edited WALKER P.L., Jr. & THROWER P.A.). 8, 1 (1973).
MS., Ph”vs. Rev. 140A. 401 (1965). (1964).
D,A.,Nature225,731
A262.
1128 (1967).
(1970).
10.
KAWAMURA K.. SAT0 N., AOKI T. & TSUZUKU T., J. Phys. Sot. Japan 41.3027
11.
KAWAMURA K., SAITO T.. TSUZUKU T.. J. Phys. Sot. Japan 42. 574 (1977).
12.
SCHROEDER P.R., DRESSELHAUS
(1976).
M.S. & JAVAN A.. Ph?!s. Rev. Lett. 20, 1292 (1968).
Vol. 26, No. 5
THE LOW FIELD HALL COEFFICIENT
OF GRAPHITE
337
13.
WOOLLAM J.A., Phys. Lett. 32A, 115 (1970).
14.
WOOLLAM J.A.,Phys.
15.
SCHROEDER P.R., DRESSELHAUS M.S. & JAVAN A., Pruc. Int. Conf Phys. Semimetals Gap Semicond,, Dallas, 1970, p. 139. Pergamon, London (1971).
16.
WOOLLAM J.A.,Phys.
17.
WILLIAMSON S.J., FONER S. & DRESSELHAUS
18.
TOY W.W., HEWES C.R., DRESSELHAUS
19.
ANDERSON J.R.. O’SULLIVAN W.J., SCHIRBER J.E. & SOULE D.E., Phys. Rev. 164, 1038 (1967).
20.
SPAIN I.L., Proc. ConJ Hec. Density ofstates,
21.
MCCLURE J.W., Phys. Rev. 101, 1642 (1956).
22.
DILLON R.O., Ph.D. thesis, Univ. of Maryland (1974).
23.
DILLON R.O., SPAIN I.L. & MCCLURE J.W.,J. Phys. Chem. Solids 38,635
24.
GALT J.K., YAGER W.A. & DAIL H.W., Jr., Phys. Rev. 103,1586
25.
WILLIAMSON S.J., SURMA M., PRADDAUDE Commun. 4,37 (1966).
Rev. B3, 1148 (1971).
Rev. B4,3393
and Narrow Band
(1971). MS., Phys. Rev. 140A, 1429 (1965); Carbon 4,29 (1966).
M.S., Carbon 11,575
(1973).
p. 717. NBS, Washington,
D.C. (1969).
(1977).
(1956).
H.C., PATTEN R.A. & FURDYNA J.K., Solid State
0038-1098/78/05OI-0333
EFFECTS OF WARPED ENERGY SURFACES ON THE LOW FIELD HALL COEFFICIENT
$02.00/O
OF GRAPHITE
R.O. Dillon* and I.L. Spain Laboratory for High Pressure Science, Dept. of Chemical Engineering, University of Maryland, College Park, MD 20742, U.S.A. (Received
3 1 October 1977 by G. Burns)
Unusual field dependence of the low field galvanomagnetic properties of graphite has been attributed to small pockets of mobile minority carriers in the corners of the Brillouin Zone. However, this explanation does not account for the sample or temperature dependence of these phenomena. An alternative exnlanation. based on the trigonal warping of the constant energy surfaces, is presented. minority hole. The trend in R&B) for Kish graphite is generally to more negative values at low field [IO, 1 I]. The mechanism responsible for the unusual field dependence of the low field Hall coefficient must explain the change in the character of the low-field Hall coefficient (a) with temperature for natural crystals and (b) with crystallite size at 77 K for pyrolytic samples. It is postulated here that the trends in the low-field Hall coefficient are not reiated to the minority hole pockets at the corners of the Brillouin Zone. This is suggested in a number of ways. Firstly, the sign of the minority carriers in the Brillouin Zone can only change if the sign of the band overlap parameter y2 changes. Although the dimensions of the pocket are largely determined by the band overlap parameter A, a change of sign of A cannot lead to a change of sign of the carrier type (see [s] for a thorough discussion). It is extremely unlikely that ~2 could change sign with temperature or crystallite size. Secondly, the nature of the pocket can be inferred from the sign of y2. Experiments performed at low temperature (T - 14 K) show that y2 is negative [12-l 51, so that the minority carrier pocket is hole-like. At this temperature, the nature of the low-field Hall coefficient is electron-like for natural, Kish and synthetic crystals (Fig. 1). Thirdly, the Shubnikov-de Haas frequencies for pyrolytic graphite are the same as those for natural crystals [ 161. There is some uncertainty about the interpretation of these data, since two low-frequency signals are seen in some crystals [7, 16, 171 and their origin has not been unambiguously identified. However, it has been further shown from magnetoreflection studies that the band parameters of both pyrolytic and natural crystals are the same within experimental error [ 181. This includes the parameters y2 and A which determine the size and nature of the minority pockets. A fourth piece of information comes from a
UNUSUAL BEHAVIOR in the low-field galvanomagnetic properties of the semimetal graphite, particularly the Hall coefficient, has been generally attributed to the presence of small pockets of mobile minority holes in the corner of the Brillouin Zone. There have been a number of problems associated with this basic premise, and the purpose of the present paper is to suggest a new explanation for these phenomena based on the trigonal warping of the constant energy surfaces. Anomalies in the low field behavior (m < 1, where B is the magnetic field and ii is a measure of the mean carrier mobility) can be seen in an unusual field dependence of the magnetoresistance. More striking effects occur in the Hall coefficient. Soule [I] found that for natural crystals of graphite, the Hall coefficient becomes more positive at 77 K for fields less than about 0.1 T (1 kg) whereas at 4.2 K the trend is reversed (Fig. 1). At 77 K this behavior was attributed to the presence of mobile minority holes [l, 21. Using the accepted dispersion relationship for the electrons and holes in graphite [3], with current estimates for the band overlap parameters (for recent reviews see [4] and [5]) pockets of low-mass, hole-like states occur in the corners of the Brillouin Zone. These pockets have been used to explain [6] Soule’s observation [7] of a de Haas-van Alphen signal of low frequency. For specimens of highly oriented pyrolytic graphite, the trend in the low-field Hall coefficient is consistent with mobile electrons at all temperatures [8] (Fig. 1). However, upon further annealing it was shown [9] that the curve of the Hall coefficient against field at 77 K systematically approached the curve for natural crystals as the crystallite size was increased with heat treatment, so that the low-field behavior changed from that characteristic of a mobile minority electron to that of a mobile * Present address: University New Zealand.
of Waikato, Hamilton,
333
334
THE LOW FIELD HALL COEFFICIENT
ak -=as
OF GRAPHITE
0
Vol. 26, No. 5
EVAB
(1)
where k is the electron wavevector, V the electron velocity and B the magnetic field, then the velocity is expressed as a Fourier series: V =
L 298
K
-0.2 -
EP-7
[SOULE (1958)]
-.-
SA-26[SPAIN
etol(l967)]
il i i -0.6
2 V(m) eimws m=,
(2)
where o is related to the time, T, for the electron to complete an orbit by w = 2n/T, and m is the index. Using McClure’s formalism for the Boltzmann transport equation, the conductivity components u,, and oXYcan be expressed,
Pa) (3b)
~4.2
K
B (TESLA)
Fig. 1. Curves of the Hall coefficient, RH, vs. magnetic field, B, for purified natural crystals [ 1] and highly oriented pyrolytic graphite [8]. Note the trends in RH at very low field, particularly the reversal of the trend for the natural crystal between 77 and 4 K. comparison of the pressure dependence of the minority carrier de Haas-van Alphen frequency with that of the low field Hall coefficient. The extremal area (S) for the minority carrier pocket depends strongly on pressure [ 191 (d In S/dp - 0.09 kb-r ) while the analysis of the galvanomagnetic data using a three band model [ZO] (majority electrons and holes and a minority carrier) for highly oriented pyrolytic graphite suggested that, if the minority carriers are enclosed in an ellipsoidal pocket, their extremal area changes very little with pressure (d In S/dp 5 0.01 kb-‘). An alternative suggestion is made for the origin of this anomalous low-field Hall coefficient behavior, based on the trigonal warping of the constant energy surfaces in the k,-k, plane (z-direction perpendicular to basal planes). Due to the warping, the path (hodograph) followed by an electron or hole in a magnetic field, along a contour of constant energy, is also trigonally warped in either reciprocal or real space. Calculations have been made to assess the importance of this warping on the Hall coefficient using the formalism developed by McClure [2 11. Using a parameter, s, with the dimensions of time defined by the quasimomentum equation:
where g is the area enclosed by a hodograph, the coefficients a,, k, are related to the Fourier components of the velocity, V,, (which in turn are related to the extent of the warping), 6 denotes the sign of the carrier, r is the relaxation time and w the cyclotron frequency. Full details of the computational method are given in [22]. For a circular cross-section only VI, a,, b, are non-zero. The higher-order components contribute appreciably to a,, and uXYonly at low fields. The conductivity components are related to the Hall coefficient by, (4) Although the warping of energy surfaces enhances u_,., appreciably in the low field (wr < 1) region, the magnetic field dependence of the Hall coefficient is controlled by uXy and attention will be focussed on it. The form of u,,(B) for a particular carrier pocket is controlled by the extent of trigonal warping, which in turn is a function of k, in the Slonczewski-Weiss [3] dispersion relationship (i.e. a, and b, are functions of k,). The hodographs predicted from this relationship for various values of k, are shown in Fig. 2. Most of the electron hodographs consist of small regions of strong positive curvature and larger areas of weak negative curvature (concave as viewed from outside the figure). The hodographs for the hole surface vary from those like (g) (Fig. 2) with stronger negative curvature than the electron hodographs to those like (j) with no negative curvature. The form of u,,(B) has been calculated using a simplified model of trigonally warped energy surfaces and a constant relaxation time. From this, three important field ranges can be identified:
Vol. 26, No. 5
THE LOW FIELD HALL COEFFICIENT
OF GRAPHITE
335
1 kY
Fig. 2. Sketch of the constant energy surfaces at the Fermi energy for various values of @($ = k,c/2, z-component of wavevector and c the lattice constant). Band parameters are for set B of [23].
where k, is the
Fig. 3. Possible variation of the parameter I = u~Jcr!& as a function of or. The case drawn shows a more pronounced dip for electrons than holes, but this behavior may be reversed for certain types of scattering. (1) In the very low field region (wr << 1) carriers in the region of strong curvature dominate in uxY. (2) In the very high field region (or > 1) uxY asymptotes to its classical value so that (u,$),,, = ne. (3) In an intermediate field region (0.1 < wr < 1) carriers in the region of weak negative curvature can predominate in uxY. Depending on the geometry of the hodograph, these carriers can change the sign of a,, so that pockets of carriers normally labelled as electron-like may behave for this field region as hole-like with respect to the Hall parameter, and vice-versa. Carriers in the region of strong curvature dominate uxY at very low fields since only these carriers can turn through appreciable Hall angles. In a field range
(- 0.01 < wr $ 1) within the low field region, carriers at regions of weaker curvature also turn through appreciable Hall angles, and thus become important in uxY. Based on the above, Fig. 3 shows a possible variation of the parameter r for the electron surface in graphite, where r z &,/CT& and t and c indicate conductivities based on trigonal and circular hodographs, respectively, containing equal areas. The important feature of Fig. 3 is that the regions of negative curvature on the hodographs reduce the electronic, negative, contribution to uxY at fields 0.01 5 wr 5 1. For certain fields it is even possible for the electronic contributions to uxy to be positive. The variation of r(w7) will be similar for the hole surface. Due to the varying degree of negative curvature in the hole hodographs, it is reasonable that the dip in the
336
THE LOW FIELD HALL COEFFICIENT
T(WT) curve for the holes is less pronounced than for the electrons. However, for certain types of scattering which give a dependence of relaxation time on position along the hodograph, this may be reversed. A possible reason for the low field anomalies in graphite is now apparent. Since graphite is a nearlycompensated semi-metal, the hole and electron contributions to RH nearly cancel each other out, so that small changes in the contribution from one carrier can produce large fractional changes in RH. The positive rise in RH at low temperatures and fields can be related to the dip in T(wT) for the electrons. The less pronounced dip for the holes is not of sufficient magnitude to compensate for the change due to the electrons. The exact balance between the electron and hole contributions can lead to a wide variety of behavior in these materials in the intermediate field range. For instance. if the scattering is such that the average hole mobility exceeds that of the electron, the contribution to RH from the holes is enhanced and positive-going trends at low field may result. This is exactly the kind of effect seen by Soule at 77 K (Fig. 1). The type of scattering might also contribute to a dependence of relaxation time on the carrier location on the hodograph. In this way the “harmonics” in the expression for uxy [equation (3b)l could be strengthened or weakened. In this way the puzzling changes of the low field Hall constant with temperature and specimen type can be explained. For instance, at low temperature the predominant scattering mechanism in natural single crystal or Kish graphite is believed to be from charged impurities, while in pyrolytic graphite scattering from grain boundaries predotninates. At higher temperature, phonon scattering is increasingly important. At 77 K, mixed scattering is effective, and the change in the
OF GRAPHITE
Vol. 26. No. 5
behavior of Cooper and Young’s pyrolytic specimen with annealing can be related to reduced scattering from imperfections. Since boundary scattering rates are inversely proportional to carrier velocity, and the mean electron velocity is greater than that for the holes, the hole mobility is expected to increase relative to the electron as boundary scattering is reduced. Support for the present suggestion that negative curvature has an important effect on the conductivity can be derived from cyclotron resonance experiments (24, 251. All resonances can be identified as originating from the carrier pocket located near point K in the Brillouin zone [25], now identified as the electron surface [ 121. Strong hole-like resonances occur from harmonic terms resulting from the trigonal warping with energy (3N + 1)ho, where N is a negative integer in this case. In conclusion, it is suggested that the anomalous features of the low-field galvanomagnetic properties of graphite occur as a result of the warping of the constant energy surfaces. Although the present calculations were made for simplified models of the constant energy surfaces and relaxation time, results for an exact calculation will show similar features to those obtained here, but details will differ. Considerable complexity will be introduced by using the Slonczewski-Weiss dispersion relationship 131, a relaxation time dependent on k, and k,, and also by including effects due to carrier nondegeneracy. It is preferable to attempt such a calculation using numerical rather than Fourier summation techniques. Acknowledgements
- This research was funded by a grant from the United States Atomic Energy Commission. Helpful discussions were held with Dr. Robert Allgaier and Prof. J.W. McClure.
REFERENCES 1.
SOULE D.E.,Ph’hys. Rev. 112,698
2.
MCCLURE J.W.,Phys.
3.
SLONCZEWSKI J.C. &WEISS P.R.,Ph,vs. Rev. 109,272
4.
MCCLURE J.W., Proc. Int. Conf: Phys. Semimetals Pergamon, London (1971).
5.
SPAIN 1.L.. Chemistry
6.
DRESSELHAUS
7.
SOULE D.E., IBM J, Res. Dev. 8,268
8.
SPAIN I.L., UBBELOHDE A.R. & YOUNG D.A., Phil. Trans. Roy. Sac. (Londonj
9.
COOPERJ.D.,WOOREJ.&YOUNG
(1958).
Rev. 112. 715 (1958).
and Physics ofCarbon
G. & DRESSELHAUS
(1958).
and Narrow Band Gap Semicond.,
Dallas.
1970. p. 127.
(Edited WALKER P.L., Jr. & THROWER P.A.). 8, 1 (1973).
MS., Ph”vs. Rev. 140A. 401 (1965). (1964).
D,A.,Nature225,731
A262.
1128 (1967).
(1970).
10.
KAWAMURA K.. SAT0 N., AOKI T. & TSUZUKU T., J. Phys. Sot. Japan 41.3027
11.
KAWAMURA K., SAITO T.. TSUZUKU T.. J. Phys. Sot. Japan 42. 574 (1977).
12.
SCHROEDER P.R., DRESSELHAUS
(1976).
M.S. & JAVAN A.. Ph?!s. Rev. Lett. 20, 1292 (1968).
Vol. 26, No. 5
THE LOW FIELD HALL COEFFICIENT
OF GRAPHITE
337
13.
WOOLLAM J.A., Phys. Lett. 32A, 115 (1970).
14.
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