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Effect of a magnetic field on donor impurity levels in InSb
Pergamon
J. Phys. Chem, So&i.
EFFECT
Press 1956. Vol. 1.
pp. 143-145.
OF A MAGNETIC FIELD ON IMPURITY LEVELS IN InSb ROBERT Westinghouse
W.
KEYES
AND
Research Laboratories,
R.
J.
Pittsburgh
DONOR
SLADEK 3.5, Pennsylvania
(Reckvcd 24 July 1956) Abstract-_-In an attempt to observe the effect described in the preceding paper, we have measured the Hall constant of two specimens of InSb in very strong magnetic fields. The anticipated effect is found to take nlace. although the details are modified by interaction between states on different impurity atoms. In addition;it is found that the effect of the magnetic field on the electron mobility is larger than is expected on the basis of the usual theories of magnetoresistance. INTRODUCTION
the preceding paper (r) a theory of the effect of a strong magnetic field on the energy levels and wave functions of a hydrog~-ale impurity state in a semiconductor is presented. It is concluded that the change in the energy levels of such an impurity state should be observable in n-type InSb as a decrease in the number of electrons in the conduction band in high magnetic fields at low temperatures. In an attempt to observe this effect we have measured the conductivity and the Hall constant of two specimens of InSb as a function of magnetic field strength up to fields of 80 kilogauss at 4*2OK. The simple hydrogenic model of the donor states presented does not apply to InSb of available purity. Even at temperatures far below the Rydberg (9°K) the electrons do not fall from the conduction band into bound states.@) This is apparently a result of overlap of electronic wave functions centered on different impurity atoms.@) In InSb with 2 x 1014 excess donors per cm3 the average distance between excess donor atoms is only three times the reduced Bohr radius. Nevertheless, the possibihty of observing the freeze-out of carriers in a strong magnetid field still exists, because the reduction in the ipatial extent of the wave functions by the magnetic field may be expected to decrease the overlap between adjacent functions. IN
EXPERIMENT Samples of I&b
were provided for us by Dr. S. W. 143
KUIWICK of the Chicago Midway Laboratories. Two specimens were used for the experiment. One had an excess donor concentration of 2 x lo”* crnmsand the other a concentration of 4 x 10lb cm-s. The samples were cut to dimensions O-1 x 0.2 x 1.0 an*, the surfaces were etched, and potential and current leads were attached with In-Sn solder. The high magnetic field facilities of the Cryogenics Branch of the Naval Research Laboratory were made available to us through the courtesy of Dr. R. T. WEBBW. Using these facilities, we have measured the Hall constant and the conductivity of the specimens as a function of magnetic field up to a field of 82 kilogauss at 42°K. The results of this measurement are presented in Figs. 1 and 2 as plots of number of carriers and of mobihty as functions of magnetic field.
DISCUSSION
OF NUMBER
CHANGE
Fig. 1 shows that in the 4~ 1014 sample the number of csrriers begins to decrease sharply when the field is raised above 20 kilogauss, and at the highest field is fifty times smaller than at zero field. This change is far greater than those obtained from the ordinary mechanisms which lead to a field dependent Hall constant(4*5) and we believe that it results from the creation of bound impurity states in the way described above. Fig. 1 also shows that the large decrease in carrier concentration does not occur in the 2 x lo”* sample. This is consistent with our interpretation, since in the 2 x lW* sample fifty times as many electronic wave functions must be placed in a given volume as in the 4 x 1014 sample, and the overlap of the wave functions will persist to much higher
ROBERT
I44
W.
KEYES
AND
R.
J.
SLADEK
DISCUSSION
OF THE
MOBILITY
Fig. 2 shows that in our samples there is a quite large decrease in mobility at high magnetic fields The decrease is much larger than the usual classical theories of mobility@) predict. Further, these theories always predict a saturation of the magnetoresistance effect(*Te) at high values of the dimensionless parameter rB. In our experiment on the 2 x 1016 sample, values of FB as high as 60 were reached, but the mobility does not appear to reach a saturation value. Thus we conclude that the usual theories of magnetoresistance are not adequate to account for our data.
60
40
B
80
I 100
kilogauss
FIG. 1. The number of electrons, calculated from the Hall constant, as a function of magnetic field for two samples of InSb. n-type InSb. 0 N 10’s sample. * N loI* sample . 4~2°K.
magnetic fields. On the other hand, any field dependence of the Hall constant which had its origin in a property of the conduction band states would probably not be greatly different in the two specimens. We are not in a position to make a detailed comparison of our results with the calculated binding energy,@) because the calculation does not take into account the effects of overlap of wave functions on different donors. These effects we have seen to be of great’ importance. Therefore the simple theory cannot be expected to be quantitatively applicable to the results of our experiment.
to21 0
I 20
40
I 60 B
-m 80
I 100
I
kilogauss
FIG. 2. The mobility of the electrons as a function of magnetic field in samples of InSb with excess electron concentrations of 2 x 10” cm+ and 4 x 10“ cmA. n-type InSb. 0 N lo’@ sample. 0 N 1014 sample. 4.2“K.
The large difference in the zero field mobility of the two samples deserves comment. Both samples are degenerate at 4*2”K, so that the electrons which participate in the conduction process have kinetic energies corresponding to the Fermi energy of the sample in question. If the scattering is byionized impurities, the relaxation time varies approximately as the 312 power of the electronic energy. For comparable concentration of scattering centers, the sample with the higher degeneracy temperature therefore has a higher mobility. Since the
EFFECT
OF A MAGNETIC
FIELD
ON DONOR IMPURITY
Fermi energy varies as the 213 power of the electron concentration, the mobilities are proportional to the concentration. For our samples which have a concentration ratio of 50, the ratio of the mobilities is 30. This suggests that the number of scattering centers is about the same in our two samples, and that the low electron concentration in the 4 x lo’* sample is achieved by compensation of donor and acceptor impurities. CONCLUSIONS 1. We find that the energy difference between a donor impurity level and the bottom of the CORduction band can be increased by the application of a strong magnetic field, and the distribution of electrons between the conduction band and impurity states can thereby be altered. 2. The effect of a strong magnetic field on the mobility is greater than can be accounted for by the usual theories of ma~etoresis~ce. Ackmwledgemmts-We are grateful to Dr. T. T. WEBBJZR and the Cyrogenics Branch of the Naval
K
LEVELS
IN InSb ‘145
Research Laboratory for making available to us the facilities for the experimental work described in this paper. We wish to thank the Chicago Midway Laboratories and Dr. S. W. KURNICKfor supplying the InSb used in these experiments. We are indebted to Dr. E. N. ADAMS,who suggested to us the existence of the effects described herein, and to Dr. Y. YAFET,for discussions of the interpretation. Since this experiment was performed, Dr. H. P. R. FREDERIKSE has observed a similar effect, and we thank Dr. FREDERIKSEfor discussing his results with us.
REFmENc!Es
1. YAFETY., KBYJZS, R. W., and ADAMSE. N. Preceding paper. 2. HROSTOWSKI H. J., MORIN F. J., GEBALLE T. H., and WHEATLEYG. H. Phvs. Reu. 100.1672 119551.
3. Sraa~ F. and TALLEY-R.M. P& Rev. ‘100,‘1638 (1955). 4. SWANSONJ. A. Phys. Rev. !@,1799 (1955). 5. WIL-ON R. K., HARMAN Lx., and BEERA. C. Phys. Rev.. %, 1512 (1954). 6. WILSON A. H. Theory of Metals, Cambridge University Press, Cambridge (1953).
J. Phys. Chem, So&i.
EFFECT
Press 1956. Vol. 1.
pp. 143-145.
OF A MAGNETIC FIELD ON IMPURITY LEVELS IN InSb ROBERT Westinghouse
W.
KEYES
AND
Research Laboratories,
R.
J.
Pittsburgh
DONOR
SLADEK 3.5, Pennsylvania
(Reckvcd 24 July 1956) Abstract-_-In an attempt to observe the effect described in the preceding paper, we have measured the Hall constant of two specimens of InSb in very strong magnetic fields. The anticipated effect is found to take nlace. although the details are modified by interaction between states on different impurity atoms. In addition;it is found that the effect of the magnetic field on the electron mobility is larger than is expected on the basis of the usual theories of magnetoresistance. INTRODUCTION
the preceding paper (r) a theory of the effect of a strong magnetic field on the energy levels and wave functions of a hydrog~-ale impurity state in a semiconductor is presented. It is concluded that the change in the energy levels of such an impurity state should be observable in n-type InSb as a decrease in the number of electrons in the conduction band in high magnetic fields at low temperatures. In an attempt to observe this effect we have measured the conductivity and the Hall constant of two specimens of InSb as a function of magnetic field strength up to fields of 80 kilogauss at 4*2OK. The simple hydrogenic model of the donor states presented does not apply to InSb of available purity. Even at temperatures far below the Rydberg (9°K) the electrons do not fall from the conduction band into bound states.@) This is apparently a result of overlap of electronic wave functions centered on different impurity atoms.@) In InSb with 2 x 1014 excess donors per cm3 the average distance between excess donor atoms is only three times the reduced Bohr radius. Nevertheless, the possibihty of observing the freeze-out of carriers in a strong magnetid field still exists, because the reduction in the ipatial extent of the wave functions by the magnetic field may be expected to decrease the overlap between adjacent functions. IN
EXPERIMENT Samples of I&b
were provided for us by Dr. S. W. 143
KUIWICK of the Chicago Midway Laboratories. Two specimens were used for the experiment. One had an excess donor concentration of 2 x lo”* crnmsand the other a concentration of 4 x 10lb cm-s. The samples were cut to dimensions O-1 x 0.2 x 1.0 an*, the surfaces were etched, and potential and current leads were attached with In-Sn solder. The high magnetic field facilities of the Cryogenics Branch of the Naval Research Laboratory were made available to us through the courtesy of Dr. R. T. WEBBW. Using these facilities, we have measured the Hall constant and the conductivity of the specimens as a function of magnetic field up to a field of 82 kilogauss at 42°K. The results of this measurement are presented in Figs. 1 and 2 as plots of number of carriers and of mobihty as functions of magnetic field.
DISCUSSION
OF NUMBER
CHANGE
Fig. 1 shows that in the 4~ 1014 sample the number of csrriers begins to decrease sharply when the field is raised above 20 kilogauss, and at the highest field is fifty times smaller than at zero field. This change is far greater than those obtained from the ordinary mechanisms which lead to a field dependent Hall constant(4*5) and we believe that it results from the creation of bound impurity states in the way described above. Fig. 1 also shows that the large decrease in carrier concentration does not occur in the 2 x lo”* sample. This is consistent with our interpretation, since in the 2 x lW* sample fifty times as many electronic wave functions must be placed in a given volume as in the 4 x 1014 sample, and the overlap of the wave functions will persist to much higher
ROBERT
I44
W.
KEYES
AND
R.
J.
SLADEK
DISCUSSION
OF THE
MOBILITY
Fig. 2 shows that in our samples there is a quite large decrease in mobility at high magnetic fields The decrease is much larger than the usual classical theories of mobility@) predict. Further, these theories always predict a saturation of the magnetoresistance effect(*Te) at high values of the dimensionless parameter rB. In our experiment on the 2 x 1016 sample, values of FB as high as 60 were reached, but the mobility does not appear to reach a saturation value. Thus we conclude that the usual theories of magnetoresistance are not adequate to account for our data.
60
40
B
80
I 100
kilogauss
FIG. 1. The number of electrons, calculated from the Hall constant, as a function of magnetic field for two samples of InSb. n-type InSb. 0 N 10’s sample. * N loI* sample . 4~2°K.
magnetic fields. On the other hand, any field dependence of the Hall constant which had its origin in a property of the conduction band states would probably not be greatly different in the two specimens. We are not in a position to make a detailed comparison of our results with the calculated binding energy,@) because the calculation does not take into account the effects of overlap of wave functions on different donors. These effects we have seen to be of great’ importance. Therefore the simple theory cannot be expected to be quantitatively applicable to the results of our experiment.
to21 0
I 20
40
I 60 B
-m 80
I 100
I
kilogauss
FIG. 2. The mobility of the electrons as a function of magnetic field in samples of InSb with excess electron concentrations of 2 x 10” cm+ and 4 x 10“ cmA. n-type InSb. 0 N lo’@ sample. 0 N 1014 sample. 4.2“K.
The large difference in the zero field mobility of the two samples deserves comment. Both samples are degenerate at 4*2”K, so that the electrons which participate in the conduction process have kinetic energies corresponding to the Fermi energy of the sample in question. If the scattering is byionized impurities, the relaxation time varies approximately as the 312 power of the electronic energy. For comparable concentration of scattering centers, the sample with the higher degeneracy temperature therefore has a higher mobility. Since the
EFFECT
OF A MAGNETIC
FIELD
ON DONOR IMPURITY
Fermi energy varies as the 213 power of the electron concentration, the mobilities are proportional to the concentration. For our samples which have a concentration ratio of 50, the ratio of the mobilities is 30. This suggests that the number of scattering centers is about the same in our two samples, and that the low electron concentration in the 4 x lo’* sample is achieved by compensation of donor and acceptor impurities. CONCLUSIONS 1. We find that the energy difference between a donor impurity level and the bottom of the CORduction band can be increased by the application of a strong magnetic field, and the distribution of electrons between the conduction band and impurity states can thereby be altered. 2. The effect of a strong magnetic field on the mobility is greater than can be accounted for by the usual theories of ma~etoresis~ce. Ackmwledgemmts-We are grateful to Dr. T. T. WEBBJZR and the Cyrogenics Branch of the Naval
K
LEVELS
IN InSb ‘145
Research Laboratory for making available to us the facilities for the experimental work described in this paper. We wish to thank the Chicago Midway Laboratories and Dr. S. W. KURNICKfor supplying the InSb used in these experiments. We are indebted to Dr. E. N. ADAMS,who suggested to us the existence of the effects described herein, and to Dr. Y. YAFET,for discussions of the interpretation. Since this experiment was performed, Dr. H. P. R. FREDERIKSE has observed a similar effect, and we thank Dr. FREDERIKSEfor discussing his results with us.
REFmENc!Es
1. YAFETY., KBYJZS, R. W., and ADAMSE. N. Preceding paper. 2. HROSTOWSKI H. J., MORIN F. J., GEBALLE T. H., and WHEATLEYG. H. Phvs. Reu. 100.1672 119551.
3. Sraa~ F. and TALLEY-R.M. P& Rev. ‘100,‘1638 (1955). 4. SWANSONJ. A. Phys. Rev. !@,1799 (1955). 5. WIL-ON R. K., HARMAN Lx., and BEERA. C. Phys. Rev.. %, 1512 (1954). 6. WILSON A. H. Theory of Metals, Cambridge University Press, Cambridge (1953).