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Hole mobility in GeTe single crystals
Volume 63A, number 2
PHYSICS LETTERS
31 October 1977
H O L E M O B I L I T Y IN G e T e S I N G L E C R Y S T A L S O. VALASSIADES and N.A. ECONOMOU Department of Physics, University of Thessaloniki, Greece Received 26 April 1977 The mobility of GeTe indicates anisotropic behaviour. The data based on a band structure compatible with the experimental rhombohedral C3v class, show scattering due to antistructure defects in the Te rich region, while scattering from neutral centers in the Ge side. •
5
Germanium telluride, in its low temperature modification, crystallizes into a structure of rhombohedral symmetry of the C5v class. This structure is a subgroup of the arsenic (A7) structure, the difference being that the two sublattices in GeTe consist of inequivalent atoms leading to the absence of the center of symmetry. The degree of distortion from the NaC1 structure, to which the high temperature modification of GeTe belongs, is relatively small, the angle of distortion being 1°45 [11. Germanium telluride belongs to a class of materials with 10 bonding electrons per atom pair. This electronic structure favours the formation of either a semiconductor, with five occupied bands separated from the unoccupied bands by a narrow energy gap, or a semimetal with a small band overlap [2]. This leads to similarities between the cubic small gap materials of the IV-V1 compounds and the group V elements which belong to the A 7 class. Thus the band structures of these classes of materials are similar with the differences caused by the symmetry difference. The link between the Fermi surfaces of these two classes was pointed out by Allgaier [3]. The band structure calculations on GeTe [4, 5] as well as the interpretation of its transport properties are based on the NaC1 type lattice. This simplifies the calculations, but in order to apply it for the interpretation of experimental data at low temperatures where the rhombohedral distortion is present the connection with the A 7 structures should be taken into account. The dominant effect of the rhombohedral distortion is to make the points L and T non equivalent. The carrier pockets are situated near the L and T points in the Brillouin zone. Since GeTe is always a p type semiconductor one should not be concerned with the elec-
tron surface but rather concentrate on the corresponding hole surface. By analogy to the V group materials, the hole pocket could be considered as an ellipsoid of revolution with the axis almost parallel to the trigonal axis. For this case the following equation for the current density (6) applies
Ii :
!
oij [E + (1/ep)I × H ] j ,
(t)
where ff is the electric field strength,/t the magnetic field, e the charge of the electron (taken positive) and p the number of holes near the energy extremum per unit volume of the crystal. The components of the tensor, referred to the principal axes, are given by [7] Oij =
(2)
ep lAi6 ij ,
while the Hall coefficient components rij, k, referring only to holes, will be e p~ r23,1 = P~l/-t3 - - r12,3 = °11033 '
e - (o 11)2 '
(3)
where tai are the components of the hole mobility tensor
/l =
(10 0) #1 , 0 /a3
(4)
along the respective crystal axis. On the basis of the above considerations the conductivity in the low temperature modification of GeTe should be anisotropic, a fact which has already been reported [8,9], the conductivity surface being an ellipsoidal surface of revolution with the axis parallel to the (111). On the other hand the Hall coefficient should be isotropic. 133
Volume 63A, number 2
PHYSICS LETTERS
0/1
140
";. I00 ,-&
o..._ "~.
T=3OO°K
I~
P3 °"'o. \ \
60
13 \ \
40
o
20 48-
Two features are noticeable: (a) the anisotropic values of the mobilities have a different dependence on stoichiometry in the two regions (i.e. the rich in Ge or the rich in Te region), and (b) the anisotropy vanishes at points near the limits of the stability range of GeTe, which according to Kolomoets et al. extend from 49%Te to 54%Te [12]. In analysing the results these two regions are treated separately. On the Te rich side, the data can be fitted into an equation of the form
tl
tZ~ "7
1
\\ xx.~
//(x) = a x ( l - --5 '
~ s'2
54
1,[%Tel Fig. 1. The Hall mobility of germanium telluride at room temperature.
Galvanomagnetic experiments were carried out on a series of GeTe monocrystals of various stoichiometric compositions, mostly at room temperature, in order to determine the mobility. The measurements were performed on flat (121) circular samples using Wasscher's method [10] which is suitable for anisotropic materials. Details on the application of this method will be published elsewhere [11]. The behaviour is in accordance with the analysis based on the previous considerations. The anisotropy in the conductivity persists up to a temperature coinciding with phase transformation temperature obtained from thermal data. Simultaneous measurements of the Hall coefficient showed that, within the limits of detectability (< 1 ~V), there is no directional anisotropy, indicating that equations (3) hold. At the transformation temperature a sharp discontinuity in both the conductivity and the Hall coefficient is observed, indicating a thermal release of carriers, whereas above the transformation temperature the conductivity becomes isotropic. From the sets of the simultaneous conductivity and Hall coefficient measurements the Hall mobility was calculated by the product ogr. Thus, two mobility components were found;//3 parallel and//1 perpendicular to the c axis. In fig. 1 room temperature mobility data are presented for samples of various stoichiometries. 134
(S)
•
\ 4'9
31 October 1977
where x is the excess mole fraction of Te. Plots of the isothermal behaviour of the mobilities at 300 K on the Te rich side based on the relation (5) (solid lines of fig. 1) indicate that for the mobility//1 perpendicular to the e axis the above relation is completely satisfied, while for the mobility parallel to the c axis//3 there is a deviation for compositions toward the exact stoichiometry. The dependence described by eq. (5) has been attributed by Nordheim [13] and later by Brooks [14] to randomly positioned perturbations due to alloy disorder. This leads to the conclusion that antistructure disorder or interstitials is a dominant factor in GeTe. In such a case the constant A ofeq. (5)depends on the difference of the potentials of the constituents and the carrier energy. That antistructural defects exist in GeTe is a view that has been advanced primarily by Lewis [15] to explain the magnetic susceptibility data. The ratio of the mobility values//1///3 extend from almost 2 at the near stoichiometric region to 1 at the edge of the stability range on the Te rich side. This indicates that the difference in cross section presented by the defects becomes less pronounced as the excess in Te increases. This should be attributed to clustering of vacancies with antistructural defects, since these defects cannot be considered independently, [ 15], the clustering affecting the symmetry of the scattering centers. On the Ge rich side the situation is more complicated. Thus the behaviour of//1 indicates that it is inversely proportional to the number of defects, as expressed by the germanium excess, a fact which is indicative of neutral scattering [16]. On the other hand the behaviour of //3 in this compositional region cannot be attributed to a simple scattering mechanism. Since the temperature where these investigations were performed is relatively high, other scattering modes besides those intro-
Volume 63A, number 2
PHYSICS LETTERS
duced by defects become important. Therefore, a complete analysis should wait until all possible ranges o f temperatures are studied. In conclusion the results presented here indicate that the anisotropic behaviour o f GeTe is a strong factor determining the transport properties and the band structure of the low temperature phase of this material. We would like to thank the National Greek Research Foundation for the partial financial support to one of us
(O.V.).
References [ 1 ] J.N. Bierly, L. Muldawer and O. Beckman, Acta Metall. 11 (1963) 447. [2] M.S. Dresselhaus, Physics of semimetals and narrow gap semiconductors (Texas, 1970) p. 3. [3] R.S. Allgaier and B. Houston, Phys. Rev. B5 (1972) 2186; R.S. Allgaier, in: Proc. Conf. Physics of semimetals and narrow gap semiconductors (Nice-Cardiff, 1973).
31 October 1977
[4] Y.W. Tung and M.L. Cohen, Phys. Rev. 180 (1969) 823. [5] M.L. Cohen and Y.W. Tsang, Physics of Semimetals and Narrow Gaps Semiconductors (Texas, 1970) p. 303; G. Cuicci and G.F. NardeUi, in: Proc. 12 Int. Conf. on Physics of semiconductors, ed. M.H. Pilkahn (Teubner, Stuttgart, 1974) pp 1295. [6] A.H. Wilson, The theory of metals (Cambridge, 1954). [7] R.N. Zitter, Phys. Rev. 127 (1962) 1471. [8] O. Valassiades and N.A. Economou, in: Proc. Conf. Physics of semimetals and narrow gap semiconductors (NiceCardiff, 1973). [9] S.I. Novikova, L.E. Shelimova, N.Kh. Abrikosou and M.A. Korzhuer, Soy. Phys. Solid State 15 (1974) 2267. [10] J.D. Wasscher, Philips Res. Repts. Suppl. no. 8 (1969). [11] O. Valassiades and N.A. Economou, in: Proc. Faculty Math. Phys. Sciences University of Thessaloniki, in press. [12] N.V. Kolomoets, E.Ya. Lev and L.M. Sysoeva, Soviet Phys.-Sol. State 5 (1964) 2101. [13] L. Nordheim, Ann. Physik 9 (1931) 607. [14] H. Brooks and W. Paul, Bull. Am. Phys. Soc., Ser. II, 1 (1956) 48. [15] J.E. Lewis, Phys. Stat. Sol. 38 (1970) 131. [16] C. Erginsov, Phys. Rev. 79 (1950) 1013.
135
PHYSICS LETTERS
31 October 1977
H O L E M O B I L I T Y IN G e T e S I N G L E C R Y S T A L S O. VALASSIADES and N.A. ECONOMOU Department of Physics, University of Thessaloniki, Greece Received 26 April 1977 The mobility of GeTe indicates anisotropic behaviour. The data based on a band structure compatible with the experimental rhombohedral C3v class, show scattering due to antistructure defects in the Te rich region, while scattering from neutral centers in the Ge side. •
5
Germanium telluride, in its low temperature modification, crystallizes into a structure of rhombohedral symmetry of the C5v class. This structure is a subgroup of the arsenic (A7) structure, the difference being that the two sublattices in GeTe consist of inequivalent atoms leading to the absence of the center of symmetry. The degree of distortion from the NaC1 structure, to which the high temperature modification of GeTe belongs, is relatively small, the angle of distortion being 1°45 [11. Germanium telluride belongs to a class of materials with 10 bonding electrons per atom pair. This electronic structure favours the formation of either a semiconductor, with five occupied bands separated from the unoccupied bands by a narrow energy gap, or a semimetal with a small band overlap [2]. This leads to similarities between the cubic small gap materials of the IV-V1 compounds and the group V elements which belong to the A 7 class. Thus the band structures of these classes of materials are similar with the differences caused by the symmetry difference. The link between the Fermi surfaces of these two classes was pointed out by Allgaier [3]. The band structure calculations on GeTe [4, 5] as well as the interpretation of its transport properties are based on the NaC1 type lattice. This simplifies the calculations, but in order to apply it for the interpretation of experimental data at low temperatures where the rhombohedral distortion is present the connection with the A 7 structures should be taken into account. The dominant effect of the rhombohedral distortion is to make the points L and T non equivalent. The carrier pockets are situated near the L and T points in the Brillouin zone. Since GeTe is always a p type semiconductor one should not be concerned with the elec-
tron surface but rather concentrate on the corresponding hole surface. By analogy to the V group materials, the hole pocket could be considered as an ellipsoid of revolution with the axis almost parallel to the trigonal axis. For this case the following equation for the current density (6) applies
Ii :
!
oij [E + (1/ep)I × H ] j ,
(t)
where ff is the electric field strength,/t the magnetic field, e the charge of the electron (taken positive) and p the number of holes near the energy extremum per unit volume of the crystal. The components of the tensor, referred to the principal axes, are given by [7] Oij =
(2)
ep lAi6 ij ,
while the Hall coefficient components rij, k, referring only to holes, will be e p~ r23,1 = P~l/-t3 - - r12,3 = °11033 '
e - (o 11)2 '
(3)
where tai are the components of the hole mobility tensor
/l =
(10 0) #1 , 0 /a3
(4)
along the respective crystal axis. On the basis of the above considerations the conductivity in the low temperature modification of GeTe should be anisotropic, a fact which has already been reported [8,9], the conductivity surface being an ellipsoidal surface of revolution with the axis parallel to the (111). On the other hand the Hall coefficient should be isotropic. 133
Volume 63A, number 2
PHYSICS LETTERS
0/1
140
";. I00 ,-&
o..._ "~.
T=3OO°K
I~
P3 °"'o. \ \
60
13 \ \
40
o
20 48-
Two features are noticeable: (a) the anisotropic values of the mobilities have a different dependence on stoichiometry in the two regions (i.e. the rich in Ge or the rich in Te region), and (b) the anisotropy vanishes at points near the limits of the stability range of GeTe, which according to Kolomoets et al. extend from 49%Te to 54%Te [12]. In analysing the results these two regions are treated separately. On the Te rich side, the data can be fitted into an equation of the form
tl
tZ~ "7
1
\\ xx.~
//(x) = a x ( l - --5 '
~ s'2
54
1,[%Tel Fig. 1. The Hall mobility of germanium telluride at room temperature.
Galvanomagnetic experiments were carried out on a series of GeTe monocrystals of various stoichiometric compositions, mostly at room temperature, in order to determine the mobility. The measurements were performed on flat (121) circular samples using Wasscher's method [10] which is suitable for anisotropic materials. Details on the application of this method will be published elsewhere [11]. The behaviour is in accordance with the analysis based on the previous considerations. The anisotropy in the conductivity persists up to a temperature coinciding with phase transformation temperature obtained from thermal data. Simultaneous measurements of the Hall coefficient showed that, within the limits of detectability (< 1 ~V), there is no directional anisotropy, indicating that equations (3) hold. At the transformation temperature a sharp discontinuity in both the conductivity and the Hall coefficient is observed, indicating a thermal release of carriers, whereas above the transformation temperature the conductivity becomes isotropic. From the sets of the simultaneous conductivity and Hall coefficient measurements the Hall mobility was calculated by the product ogr. Thus, two mobility components were found;//3 parallel and//1 perpendicular to the c axis. In fig. 1 room temperature mobility data are presented for samples of various stoichiometries. 134
(S)
•
\ 4'9
31 October 1977
where x is the excess mole fraction of Te. Plots of the isothermal behaviour of the mobilities at 300 K on the Te rich side based on the relation (5) (solid lines of fig. 1) indicate that for the mobility//1 perpendicular to the e axis the above relation is completely satisfied, while for the mobility parallel to the c axis//3 there is a deviation for compositions toward the exact stoichiometry. The dependence described by eq. (5) has been attributed by Nordheim [13] and later by Brooks [14] to randomly positioned perturbations due to alloy disorder. This leads to the conclusion that antistructure disorder or interstitials is a dominant factor in GeTe. In such a case the constant A ofeq. (5)depends on the difference of the potentials of the constituents and the carrier energy. That antistructural defects exist in GeTe is a view that has been advanced primarily by Lewis [15] to explain the magnetic susceptibility data. The ratio of the mobility values//1///3 extend from almost 2 at the near stoichiometric region to 1 at the edge of the stability range on the Te rich side. This indicates that the difference in cross section presented by the defects becomes less pronounced as the excess in Te increases. This should be attributed to clustering of vacancies with antistructural defects, since these defects cannot be considered independently, [ 15], the clustering affecting the symmetry of the scattering centers. On the Ge rich side the situation is more complicated. Thus the behaviour of//1 indicates that it is inversely proportional to the number of defects, as expressed by the germanium excess, a fact which is indicative of neutral scattering [16]. On the other hand the behaviour of //3 in this compositional region cannot be attributed to a simple scattering mechanism. Since the temperature where these investigations were performed is relatively high, other scattering modes besides those intro-
Volume 63A, number 2
PHYSICS LETTERS
duced by defects become important. Therefore, a complete analysis should wait until all possible ranges o f temperatures are studied. In conclusion the results presented here indicate that the anisotropic behaviour o f GeTe is a strong factor determining the transport properties and the band structure of the low temperature phase of this material. We would like to thank the National Greek Research Foundation for the partial financial support to one of us
(O.V.).
References [ 1 ] J.N. Bierly, L. Muldawer and O. Beckman, Acta Metall. 11 (1963) 447. [2] M.S. Dresselhaus, Physics of semimetals and narrow gap semiconductors (Texas, 1970) p. 3. [3] R.S. Allgaier and B. Houston, Phys. Rev. B5 (1972) 2186; R.S. Allgaier, in: Proc. Conf. Physics of semimetals and narrow gap semiconductors (Nice-Cardiff, 1973).
31 October 1977
[4] Y.W. Tung and M.L. Cohen, Phys. Rev. 180 (1969) 823. [5] M.L. Cohen and Y.W. Tsang, Physics of Semimetals and Narrow Gaps Semiconductors (Texas, 1970) p. 303; G. Cuicci and G.F. NardeUi, in: Proc. 12 Int. Conf. on Physics of semiconductors, ed. M.H. Pilkahn (Teubner, Stuttgart, 1974) pp 1295. [6] A.H. Wilson, The theory of metals (Cambridge, 1954). [7] R.N. Zitter, Phys. Rev. 127 (1962) 1471. [8] O. Valassiades and N.A. Economou, in: Proc. Conf. Physics of semimetals and narrow gap semiconductors (NiceCardiff, 1973). [9] S.I. Novikova, L.E. Shelimova, N.Kh. Abrikosou and M.A. Korzhuer, Soy. Phys. Solid State 15 (1974) 2267. [10] J.D. Wasscher, Philips Res. Repts. Suppl. no. 8 (1969). [11] O. Valassiades and N.A. Economou, in: Proc. Faculty Math. Phys. Sciences University of Thessaloniki, in press. [12] N.V. Kolomoets, E.Ya. Lev and L.M. Sysoeva, Soviet Phys.-Sol. State 5 (1964) 2101. [13] L. Nordheim, Ann. Physik 9 (1931) 607. [14] H. Brooks and W. Paul, Bull. Am. Phys. Soc., Ser. II, 1 (1956) 48. [15] J.E. Lewis, Phys. Stat. Sol. 38 (1970) 131. [16] C. Erginsov, Phys. Rev. 79 (1950) 1013.
135