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Quantum magneto-transport in p-type inversion layers of germanium bicrystals
Surface Science 170 (1986) 719-726 North-Holland, Amsterdam
719
Q U A N T U M M A G N E T O - T R A N S P O R T IN p - T Y P E I N V E R S I O N L A Y E R S OF GERMANIUM BICRYSTALS G. L A N D W E H R Physikalisches Institut, Universitg~t W'i~rzburg D-8700 14rfirzburgoFed. Rep. of Germany and S. U C H I D A Department of Applied Physics, University of Tokyo, Tokyo, Japan
Received 29 July 1985; accepted for publication 13 September 1985 A p-type inversion layer adjacent to a grain boundary in medium angle tilt germanium bicrystals behaves fike a two-dimensional disordered system. Superimposed on a random potential is a periodic one, generated by a lattice of edge dislocations. At helium temperatures both resistivity and Hall coefficient increase logarithmically with decreasing temperature. The transverse magneto-resistance shows a logarithmic B dependence in wide ranges of the magnetic field. Many of the experimental data can be explained qualitatively in terms of recently developed many-body theories. 1. Introduction Recently much attention has been paid to the electronic transport in two-dimensional disordered systems. The theory of weak localization was developed b y A b r a h a m s et al. [1]. It predicts a logarithmic increase of the resistance with decreasing temperature and a negative magneto-resistance. The enhancement of the resistance arises from q u a n t u m interference effects, if electrons with opposite m o m e n t u m (cooperons) are scattered by r a n d o m impurities. A transverse magnetic field perpendicular to a 2D layer destroys the interference and gives rise to a negative magneto-resistance. Shortly after the publication of the scaling theory [1], Altshuler et al. [2] and F u k u y a m a [3] showed that a logarithmic divergence of the resistivity can be explained on the basis of m a n y - b o d y effects. Basis of these first order theories is the correction of the self-energy according to C o u l o m b and exchange interaction due to r a n d o m disorder. Altshuler et al. introduced the particle-hole diffusion propagator, which becomes singular for vanishing m o m e n t u m difference. In order to describe the interaction, F u k u y a m a proceeded further [4] and included the cooperon in the evaluation of the self-energy. In the theory of weak localization, the rise in resistance with decreasing
0 0 3 9 - 6 0 2 8 / 8 6 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division) and Y a m a d a Science F o u n d a t i o n
720
G. Landwehr, S. Uchida / Quantum magneto-transport in p-type inversion layers
temperature is a mobility effect and the Hall coefficient should stay constant. The interaction effects modify the density of states and a logarithmic increase of the Hall coefficient with decreasing temperature is predicted. This allows one to distinguish between the two mechanisms. The sign and functional dependence of the magneto-resistance is another possibility to distinguish betwen different models. Not only orbital effects can be produced by a magnetic field, but also spin effects, as shown by Kawabata [5]. The s p i n - Z e e m a n effect results in a positive isotropic magneto resistance. Spin effects can be separated from the orbital ones if the magnetic field is oriented parallel to the 2D layer. Higher order interaction effects have been studied by Fukuyama et al. [6]. At about the same time, Finkelstein [7] developed a set of scaling equations for a system of interacting electrons, taking into account spin relaxation and Zeeman splitting. He arrived essentially at the same conclusions as Fukuyama et al. The most important consequence of Finkelstein's treatment is the prediction that the scaling parameter F should not be constant but depend on both temperature and magnetic field. The various theories of transport in an interacting system have been compared with experiments performed on thin metallic films [8] and on silicon inversion layers in MOSFETs [9]. The available data are not too abundant, however. It would be very desirable to have another system in which the predictions of theory can be checked. It seems that germanium p-type inversion layers, which exist in germanium bicrystals, present such a system. More than 25 years ago, the magneto-transport properties of grain boundaries in Ge were studied by Landwehr and Handler [10]. The data were not compatible with the standard transport theory and dubbed as anomalous at that time. The experiments were discontinued in the sixties and only resumed recently after the new theories, which have been outlined above, were developed. The experiments performed by Uchida, Landwehr et al. [11,12] clearly demonstrated, that p-type inversion layers adjacent to grain boundaries in Ge are indeed a very interesting system to study transport in a 2d disordered system. The system is not a simple one as a consequence of the complicated valence band structure of Ge. Due to the small thickness of the conducting layer of less than 100 ,~, quantum effects, resulting in electrical subbands, are expected. Calculations by Bangert et al. [13] have shown that under the usual circumstances two electric subbands are occupied which can be attributed to light and heavy holes. The impurity potential will not be entirely random, there is a superimposed potential arising from the regular dislocation network. Despite all these complications it has turned out that the magneto-transport properties at helium temperatures show many of the features which are predicted by the interaction theory. There are unexpected results and some of the data obtained in the millikelvin range are not properly understood at present.
G. Landwehr, S. Uchida / Quantum magneto-transport in p-type inversion layers
721
In the recent past, extensive experimental data on Ge bicrystals with different tilt angles have been obtained. To expose and discuss them in detail would take considerable space, which is not available here. Consequently, only the most significant and general features will be discussed subsequently.
2. Experimental Ge bicrystals were grown by the Czochralski technique by tilting (001) seeds along a (100) axis by angles O of 9, 10, 15 and 25 ° . A lattice of edge dislocations with a distance D = a/2 sin(0/2) is supposed to arise at the interface (a = lattice constant). Because edge dislocations in Ge have acceptor character, a p-type inversion layer arises for tilt angles 0 > 7 °. The conductivity is anisotropic [14] depending on whether the current is oriented parallel or perpendicular to the dislocations. For large 0-values the difference almost vanishes. The magneto resistance is also anisotropic, as will be shown subsequently. Actually, the crystallographic properties of grain boundaries are not quite as simple as indicated, but it seems nevertheless justified to use the model as a basis for discussion. The transport experiments were performed in the temperature range from 50 m K to 4.2 K in magnetic fields up to 12.5 T or from 1.5 to 4.2 K in fields up to 23 T at the Max-Planck-Institut fflr Festkbrperforschung, Hochfeldmagnetlabor Grenoble.
3. Subband calculations In order to have a solid basis for the interpretation of the experimental data, subband calculations were performed [13,15]. A one-dimensional Schr~Sdinger equation was solved self-consistently in conjunction with Poisson's equation. The band structure was taken into account by inserting the well-known Luttinger and Kohn 6 x 6 matrix. It turned out that for a typical free hole concentration of 4 × 1012/cm2 two electric subbands are occupied, which can be attributed to light and heavy holes. For this hole concentration about 20% of the carders are predicted to reside in the light hole band. Whereas the light hole band is roughly isotropic, the heavy one is strongly warped. Both subbands are calculated to be strongly non-parabolic. The cyclotron mass of the heavy holes is around 0.4m 0 over a rather wide concentration range. The light hole mass is enhanced significantly over the bulk value, it rises from 0.25m 0 at a total hole concentration of p = 4 x 1012/cm2 to about 0.37m 0 at p = 1013/cm 2. The change with respect to the bulk value arises from the existence of a strong electric field at the interface, which causes considerable band mixing.
722
G. Landwehr, S. Uchida / Quantum magneto-transport in p-type inversion layers
The calculations are in essential agreement with S h u b n i k o v - D e Hass experiments on samples with a tilt angle of 15 ° [15]. Specimens prepared from that particular bicrystal showed an unusually high conductivity. Well developed Shubnikov-De Haas oscillations showed two periods, from which the carrier concentrations of light and heavy holes could be deduced. They were in reasonable agreement with the theoretical predictions. The same was true for the effective mass of the light holes which were derived from the temperature dependence of the amplitude of the quantum oscillations.
4. Experimental results
4.1. Temperature dependence of resistivity and Hall coefficient It was reported earlier already [11] that for bicrystal specimens with tilt angles of 10 and 15 ° both the resistance per unit area and the reduced Hall coefficient increased logarithmically with decreasing temperature. The same behaviour was observed for samples with 25 ° and 9 ° tilt angle. Typical data are shown in fig. 1. The slope of the Hall coefficient curve was between 1.5 and 4 times as large as that of the resistivity. It turns out, however, that the Hall resistance of all specimens studied shows a dependence on the magnetic field, differing from sample to sample. An example is shown in fig. 2. The differences cannot be explained on the basis of a two-carrier model alone. Therefore the dependence of the Hall coefficient on temperature is sensitive to the range of the magnetic field chosen. But nevertheless the obtained data are significant because only the interaction theory predicts a logarithmic increase of the Hall coefficient with reduced temperature, whereas in the weak localization model the Hall coefficient should stay constant. When the resistivity measurements were extended to lower temperatures, it turned out that the logarithmic behaviour did not extend to the lowest temperatures used [12]. In fig. 3 the relative change of resistivity (with respect to its value at 4.2 K) has been plotted as a function of temperature on a logarithmic scale for two samples with 0 = 15 °, which differ in the orientation of the dislocation lines relative to the current direction. A significant difference is observed for the two kinds of specimens. In the lower curve obtained on a specimen with perpendicular dislocations, the resistivity increases logarithmically between 5 and 0.4 K in order to level off then. On the parallel sample the resistivity changes between 5 and l K in a similar way in order to increase substantially around 1 K and in order to level off about at 200 mK. A very similar behaviour was observed for 10 ° samples with perpendicular and parallel dislocations. These data indicate again that the p-type layers studied do not behave entirely as a disordered 2D system; obviously the orientation of dislocations plays a significant role.
G. Landwehr, S, Uchida / Quantum magneto-transport in p-type inversion layers
723
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724
G. Landwehr, S. Uchida / Quantum magneto-transportin p-type inversion layers
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In fig. 4 data for the transverse magneto-resistance are shown for a specimen with 0 = 25 ° and perpendicular dislocations. The change in resistivity per square has been plotted as a function of a transverse magnetic field. The angle ~ between magnetic field and the grain boundary normal is the parameter in the figure. If the magnetic field is oriented parallel to the grain boundary (lower curve), the change in resistance between 2 and 10 T is logarithmic, at higher magnetic fields a linear component seems to be superimposed. In the upper curve (BL), R is logarithmic only up to about 5 T and there is a very pronounced linear component at high magnetic fields. It should be noted that the magneto-resistance is quite substantial, it amounts to about 25% at 20 T. A very similar behaviour was observed for samples with 0 = 10 ° and perpendicular dislocations [11,15]. For specimens with parallel dislocations the linear component of the magneto-resistance in magnetic fields perpendicular to the grain boundary is much less pronounced [16]. It has already been reported [12] that in the temperature range down to about 1 K the magneto-resistance in parallel fields scales rather well with B / T which indicates that the effect arises from Zeeman splitting. Orbital effects are not expected because the holes are not free to move perpendicular to the surface. At lower temperatures, however, the scaling is poor, which has to do with the appearance of a negative magneto-resistance. It only shows up with specimens with parallel dislocations (see ref. [12]). For samples in which the
G. Landwehr, S. Uchida / Quantum magneto-transport in p-type inversion layers
725
dislocations are oriented perpendicular to the current, the magneto-resistance is always positive. The experimental data for bicrystals with a tilt angle of 10 ° were the most detailed and therefore thoroughly analysed in terms of the theoretical methods. It turned out that it was possible to obtain reasonable agreement between theory and experiment for not too high magnetic fields for samples with parallel and perpendicular dislocations if higher order effects were incorporated and if an anisotropy factor of about 2.5 was introduced [16]. For specimens with perpendicular dislocations, however, no reasonable set of parameters could be obtained in the high field range due to the strong linear component of the magneto-resistance. This seems to arise form the periodic potential, which is introduced by the edge dislocations. Regularly spaced dislocations are supposed to set up a superlattice potential which interacts with the free holes. A minigap is expected in the energy versus wave vector relation at k vectors roughly twice the Fermi wave vector according to the subband calculations [15]. Because certain approximations had to be made in the calculations, it cannot be excluded that the Fermi surface has contact with the mini-zone boundary. In high magnetic fields open orbits could arise which might be responsible for the observed linear magneto-resistance. Actually one cannot expect that a theory based on random disorder is suitable to explain grain-boundary data in a wide range of magnetic fields. It should be emphasized that in order to interpret the logarithmic increase of magneto-resistance in magnetic fields parallel to the grain boundary, it is necessary to invoke substantial Zeeman splitting. To explain the data, the g-factor for the holes in the 10 ° specimens is required to be about 5 and for the 15 ° specimens it must be assumed to be larger than 10. This, however, is hard to accept. An obvious shortcoming of our comparison between theory and experiment is that we are dealing with light and heavy holes, whereas the theory is based on a one-carrier model. It seems that the existence of two kinds of holes significantly enhances the magneto-resistance. This can also be concluded from experiments on p-type GaAs-(GaA1)As heterojunctions [17]. It would be desirable, if a two-carrier theory of magneto-transport in a disordered two-dimensional system were available. It seems that the neglect of the anisotropy of the Fermi surface would not unduly oversimplify the situation. We conclude this from recent measurements of the magneto resistance of p-channel Si MOSFETs of (111) and (110) orientation [18].
Acknowledgements Most of the experiments reported here were done at the Max-Planck-Institut fiir FestkSrperforschung, Hochfeldmagnetlabor Grenoble. The experiments in
726
G. Landwehr, S. Uchida / Quantum magneto-transport in p-type inoersion layers
the millikelvin range were performed by Dr. G. Remenyi and the subband c a l c u l a t i o n s b y D r . E. B a n g e r t , U n i v e r s i t a t W i ~ r z b u r g . T h e a u t h o r s w o u l d like t o t h a n k P r o f e s s o r H. F u k u y a m a f o r i n t e r e s t i n g d i s c u s s i o n s . T h e s u p p o r t b y t h e " S t i f t u n g V o l k s w a g e n w e r k " is g r a t e f u l l y a c k n o w l e d g e d .
References [1] E. Abrahams, P.W. Andersen, D.C. Licciardello and T.V. Ramakrishnan. Phys. Rev. Letters 42 (1979) 673. [2] B.L. Altshuler, A.G. Aronov and P.A. Lee, Phys. Rev. B44 (1980) 1288. [3] H. Fukuyama, J. Phys. Soc. Japan 48 (1980) 2169. [4] H. Fukuyama, J. Phys. Soc. Japan 50 (1981) 3407. [5] A. Kawabata, Surface Sci. 113 (1982) 527. [6] H. Fukuyama, Y. Iwasa and H. Yasuhara, J. Phys. Soc. Japan 52 (1983) 16. [7] A.M. Finkelstein, Soviet Phys.-JETP 57 (1983) 97. [8] See, e.g., G.J. Dolan and D.D. Osheroff, Phys. Rev. Letters 43 (1979) 721. [9] M.J. Uren, R.A. Davies, M. Kaveh and M. Pepper, J. Phys. C14 (1981) 5737. [10] G. Landwehr and P. Handler, J. Phys. Chem. Solids 29 (1962) 891. [11] S. Uchida and G. Landwehr, in: Application of High Magnetic Fields in Semiconductor Physics, Lecture Notes in Physics, Vol. 177, Ed. G. Landwehr (Springer, Berlin, 1983) p. 65. [12] G. Remenyi, S. Uchida, G. Landwehr, A. Briggs and E. Bangert, Surface Sci. 142 (1984) 43. [13] E. Bangert, S. Uchida and G. Landwehr, Solid State Commun. 45 (1983) 869. [14] B.M. Vul and E.I. Zavaritskaya, in: Physics of Semiconductors 1978, Inst. Phys. Conf. Ser. 43, Ed. B.L.H. Wilson (Inst. Phys., London-Bristol, 1979) p. 421. [15] G. Landwehr, E. Bangert and S. Uchida, Solid State Electron. 28 (1985) 171. [16] G. Landwehr and S. Uchida, in: Localization and Metal Insulator Transitions, Eds. D. Adler and H. Fritzsche (Plenum, New York, 1985) p. 379. [17] G. Remenyi, W. Heuring and G. Landwehr, to be published. [18] R. Baunach and G. Landwehr, to be published.
719
Q U A N T U M M A G N E T O - T R A N S P O R T IN p - T Y P E I N V E R S I O N L A Y E R S OF GERMANIUM BICRYSTALS G. L A N D W E H R Physikalisches Institut, Universitg~t W'i~rzburg D-8700 14rfirzburgoFed. Rep. of Germany and S. U C H I D A Department of Applied Physics, University of Tokyo, Tokyo, Japan
Received 29 July 1985; accepted for publication 13 September 1985 A p-type inversion layer adjacent to a grain boundary in medium angle tilt germanium bicrystals behaves fike a two-dimensional disordered system. Superimposed on a random potential is a periodic one, generated by a lattice of edge dislocations. At helium temperatures both resistivity and Hall coefficient increase logarithmically with decreasing temperature. The transverse magneto-resistance shows a logarithmic B dependence in wide ranges of the magnetic field. Many of the experimental data can be explained qualitatively in terms of recently developed many-body theories. 1. Introduction Recently much attention has been paid to the electronic transport in two-dimensional disordered systems. The theory of weak localization was developed b y A b r a h a m s et al. [1]. It predicts a logarithmic increase of the resistance with decreasing temperature and a negative magneto-resistance. The enhancement of the resistance arises from q u a n t u m interference effects, if electrons with opposite m o m e n t u m (cooperons) are scattered by r a n d o m impurities. A transverse magnetic field perpendicular to a 2D layer destroys the interference and gives rise to a negative magneto-resistance. Shortly after the publication of the scaling theory [1], Altshuler et al. [2] and F u k u y a m a [3] showed that a logarithmic divergence of the resistivity can be explained on the basis of m a n y - b o d y effects. Basis of these first order theories is the correction of the self-energy according to C o u l o m b and exchange interaction due to r a n d o m disorder. Altshuler et al. introduced the particle-hole diffusion propagator, which becomes singular for vanishing m o m e n t u m difference. In order to describe the interaction, F u k u y a m a proceeded further [4] and included the cooperon in the evaluation of the self-energy. In the theory of weak localization, the rise in resistance with decreasing
0 0 3 9 - 6 0 2 8 / 8 6 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division) and Y a m a d a Science F o u n d a t i o n
720
G. Landwehr, S. Uchida / Quantum magneto-transport in p-type inversion layers
temperature is a mobility effect and the Hall coefficient should stay constant. The interaction effects modify the density of states and a logarithmic increase of the Hall coefficient with decreasing temperature is predicted. This allows one to distinguish between the two mechanisms. The sign and functional dependence of the magneto-resistance is another possibility to distinguish betwen different models. Not only orbital effects can be produced by a magnetic field, but also spin effects, as shown by Kawabata [5]. The s p i n - Z e e m a n effect results in a positive isotropic magneto resistance. Spin effects can be separated from the orbital ones if the magnetic field is oriented parallel to the 2D layer. Higher order interaction effects have been studied by Fukuyama et al. [6]. At about the same time, Finkelstein [7] developed a set of scaling equations for a system of interacting electrons, taking into account spin relaxation and Zeeman splitting. He arrived essentially at the same conclusions as Fukuyama et al. The most important consequence of Finkelstein's treatment is the prediction that the scaling parameter F should not be constant but depend on both temperature and magnetic field. The various theories of transport in an interacting system have been compared with experiments performed on thin metallic films [8] and on silicon inversion layers in MOSFETs [9]. The available data are not too abundant, however. It would be very desirable to have another system in which the predictions of theory can be checked. It seems that germanium p-type inversion layers, which exist in germanium bicrystals, present such a system. More than 25 years ago, the magneto-transport properties of grain boundaries in Ge were studied by Landwehr and Handler [10]. The data were not compatible with the standard transport theory and dubbed as anomalous at that time. The experiments were discontinued in the sixties and only resumed recently after the new theories, which have been outlined above, were developed. The experiments performed by Uchida, Landwehr et al. [11,12] clearly demonstrated, that p-type inversion layers adjacent to grain boundaries in Ge are indeed a very interesting system to study transport in a 2d disordered system. The system is not a simple one as a consequence of the complicated valence band structure of Ge. Due to the small thickness of the conducting layer of less than 100 ,~, quantum effects, resulting in electrical subbands, are expected. Calculations by Bangert et al. [13] have shown that under the usual circumstances two electric subbands are occupied which can be attributed to light and heavy holes. The impurity potential will not be entirely random, there is a superimposed potential arising from the regular dislocation network. Despite all these complications it has turned out that the magneto-transport properties at helium temperatures show many of the features which are predicted by the interaction theory. There are unexpected results and some of the data obtained in the millikelvin range are not properly understood at present.
G. Landwehr, S. Uchida / Quantum magneto-transport in p-type inversion layers
721
In the recent past, extensive experimental data on Ge bicrystals with different tilt angles have been obtained. To expose and discuss them in detail would take considerable space, which is not available here. Consequently, only the most significant and general features will be discussed subsequently.
2. Experimental Ge bicrystals were grown by the Czochralski technique by tilting (001) seeds along a (100) axis by angles O of 9, 10, 15 and 25 ° . A lattice of edge dislocations with a distance D = a/2 sin(0/2) is supposed to arise at the interface (a = lattice constant). Because edge dislocations in Ge have acceptor character, a p-type inversion layer arises for tilt angles 0 > 7 °. The conductivity is anisotropic [14] depending on whether the current is oriented parallel or perpendicular to the dislocations. For large 0-values the difference almost vanishes. The magneto resistance is also anisotropic, as will be shown subsequently. Actually, the crystallographic properties of grain boundaries are not quite as simple as indicated, but it seems nevertheless justified to use the model as a basis for discussion. The transport experiments were performed in the temperature range from 50 m K to 4.2 K in magnetic fields up to 12.5 T or from 1.5 to 4.2 K in fields up to 23 T at the Max-Planck-Institut fflr Festkbrperforschung, Hochfeldmagnetlabor Grenoble.
3. Subband calculations In order to have a solid basis for the interpretation of the experimental data, subband calculations were performed [13,15]. A one-dimensional Schr~Sdinger equation was solved self-consistently in conjunction with Poisson's equation. The band structure was taken into account by inserting the well-known Luttinger and Kohn 6 x 6 matrix. It turned out that for a typical free hole concentration of 4 × 1012/cm2 two electric subbands are occupied, which can be attributed to light and heavy holes. For this hole concentration about 20% of the carders are predicted to reside in the light hole band. Whereas the light hole band is roughly isotropic, the heavy one is strongly warped. Both subbands are calculated to be strongly non-parabolic. The cyclotron mass of the heavy holes is around 0.4m 0 over a rather wide concentration range. The light hole mass is enhanced significantly over the bulk value, it rises from 0.25m 0 at a total hole concentration of p = 4 x 1012/cm2 to about 0.37m 0 at p = 1013/cm 2. The change with respect to the bulk value arises from the existence of a strong electric field at the interface, which causes considerable band mixing.
722
G. Landwehr, S. Uchida / Quantum magneto-transport in p-type inversion layers
The calculations are in essential agreement with S h u b n i k o v - D e Hass experiments on samples with a tilt angle of 15 ° [15]. Specimens prepared from that particular bicrystal showed an unusually high conductivity. Well developed Shubnikov-De Haas oscillations showed two periods, from which the carrier concentrations of light and heavy holes could be deduced. They were in reasonable agreement with the theoretical predictions. The same was true for the effective mass of the light holes which were derived from the temperature dependence of the amplitude of the quantum oscillations.
4. Experimental results
4.1. Temperature dependence of resistivity and Hall coefficient It was reported earlier already [11] that for bicrystal specimens with tilt angles of 10 and 15 ° both the resistance per unit area and the reduced Hall coefficient increased logarithmically with decreasing temperature. The same behaviour was observed for samples with 25 ° and 9 ° tilt angle. Typical data are shown in fig. 1. The slope of the Hall coefficient curve was between 1.5 and 4 times as large as that of the resistivity. It turns out, however, that the Hall resistance of all specimens studied shows a dependence on the magnetic field, differing from sample to sample. An example is shown in fig. 2. The differences cannot be explained on the basis of a two-carrier model alone. Therefore the dependence of the Hall coefficient on temperature is sensitive to the range of the magnetic field chosen. But nevertheless the obtained data are significant because only the interaction theory predicts a logarithmic increase of the Hall coefficient with reduced temperature, whereas in the weak localization model the Hall coefficient should stay constant. When the resistivity measurements were extended to lower temperatures, it turned out that the logarithmic behaviour did not extend to the lowest temperatures used [12]. In fig. 3 the relative change of resistivity (with respect to its value at 4.2 K) has been plotted as a function of temperature on a logarithmic scale for two samples with 0 = 15 °, which differ in the orientation of the dislocation lines relative to the current direction. A significant difference is observed for the two kinds of specimens. In the lower curve obtained on a specimen with perpendicular dislocations, the resistivity increases logarithmically between 5 and 0.4 K in order to level off then. On the parallel sample the resistivity changes between 5 and l K in a similar way in order to increase substantially around 1 K and in order to level off about at 200 mK. A very similar behaviour was observed for 10 ° samples with perpendicular and parallel dislocations. These data indicate again that the p-type layers studied do not behave entirely as a disordered 2D system; obviously the orientation of dislocations plays a significant role.
G. Landwehr, S, Uchida / Quantum magneto-transport in p-type inversion layers
723
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724
G. Landwehr, S. Uchida / Quantum magneto-transportin p-type inversion layers
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1'5
2'0
Fig. 4. Transverse magnetic resistance (I_L dislocations) of a bicrystal with 0 = 25°. 4~: angle between magnetic field and grain boundary normal. 4.2. Magneto resistance
In fig. 4 data for the transverse magneto-resistance are shown for a specimen with 0 = 25 ° and perpendicular dislocations. The change in resistivity per square has been plotted as a function of a transverse magnetic field. The angle ~ between magnetic field and the grain boundary normal is the parameter in the figure. If the magnetic field is oriented parallel to the grain boundary (lower curve), the change in resistance between 2 and 10 T is logarithmic, at higher magnetic fields a linear component seems to be superimposed. In the upper curve (BL), R is logarithmic only up to about 5 T and there is a very pronounced linear component at high magnetic fields. It should be noted that the magneto-resistance is quite substantial, it amounts to about 25% at 20 T. A very similar behaviour was observed for samples with 0 = 10 ° and perpendicular dislocations [11,15]. For specimens with parallel dislocations the linear component of the magneto-resistance in magnetic fields perpendicular to the grain boundary is much less pronounced [16]. It has already been reported [12] that in the temperature range down to about 1 K the magneto-resistance in parallel fields scales rather well with B / T which indicates that the effect arises from Zeeman splitting. Orbital effects are not expected because the holes are not free to move perpendicular to the surface. At lower temperatures, however, the scaling is poor, which has to do with the appearance of a negative magneto-resistance. It only shows up with specimens with parallel dislocations (see ref. [12]). For samples in which the
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dislocations are oriented perpendicular to the current, the magneto-resistance is always positive. The experimental data for bicrystals with a tilt angle of 10 ° were the most detailed and therefore thoroughly analysed in terms of the theoretical methods. It turned out that it was possible to obtain reasonable agreement between theory and experiment for not too high magnetic fields for samples with parallel and perpendicular dislocations if higher order effects were incorporated and if an anisotropy factor of about 2.5 was introduced [16]. For specimens with perpendicular dislocations, however, no reasonable set of parameters could be obtained in the high field range due to the strong linear component of the magneto-resistance. This seems to arise form the periodic potential, which is introduced by the edge dislocations. Regularly spaced dislocations are supposed to set up a superlattice potential which interacts with the free holes. A minigap is expected in the energy versus wave vector relation at k vectors roughly twice the Fermi wave vector according to the subband calculations [15]. Because certain approximations had to be made in the calculations, it cannot be excluded that the Fermi surface has contact with the mini-zone boundary. In high magnetic fields open orbits could arise which might be responsible for the observed linear magneto-resistance. Actually one cannot expect that a theory based on random disorder is suitable to explain grain-boundary data in a wide range of magnetic fields. It should be emphasized that in order to interpret the logarithmic increase of magneto-resistance in magnetic fields parallel to the grain boundary, it is necessary to invoke substantial Zeeman splitting. To explain the data, the g-factor for the holes in the 10 ° specimens is required to be about 5 and for the 15 ° specimens it must be assumed to be larger than 10. This, however, is hard to accept. An obvious shortcoming of our comparison between theory and experiment is that we are dealing with light and heavy holes, whereas the theory is based on a one-carrier model. It seems that the existence of two kinds of holes significantly enhances the magneto-resistance. This can also be concluded from experiments on p-type GaAs-(GaA1)As heterojunctions [17]. It would be desirable, if a two-carrier theory of magneto-transport in a disordered two-dimensional system were available. It seems that the neglect of the anisotropy of the Fermi surface would not unduly oversimplify the situation. We conclude this from recent measurements of the magneto resistance of p-channel Si MOSFETs of (111) and (110) orientation [18].
Acknowledgements Most of the experiments reported here were done at the Max-Planck-Institut fiir FestkSrperforschung, Hochfeldmagnetlabor Grenoble. The experiments in
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the millikelvin range were performed by Dr. G. Remenyi and the subband c a l c u l a t i o n s b y D r . E. B a n g e r t , U n i v e r s i t a t W i ~ r z b u r g . T h e a u t h o r s w o u l d like t o t h a n k P r o f e s s o r H. F u k u y a m a f o r i n t e r e s t i n g d i s c u s s i o n s . T h e s u p p o r t b y t h e " S t i f t u n g V o l k s w a g e n w e r k " is g r a t e f u l l y a c k n o w l e d g e d .
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