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Influence of band inversion upon the electrical properties of PbxSn1−xSe in the low carrier concentration range
Solid State Communications,
‘Vol. 7,
pp.
1777—1779, 1969.
Pergam..~nPress.
Printed in Great Britain
INFLUENCE OF BAND INVERSION UPON THE ELECTRICAL PROPERTIES OF Pb~Sn~..~Se IN THE LOW CARRIER CONCENTRATION RANGE* J.R. Dixon and G.F. Hoff U.S. Naval Ordnance Laboratory, Silver Spring, Maryland 20910 and University of Maryland, College Park, Maryknd 20740 (Received 9 September 1969: In revised form 10 October 1969 by E. Burstein)
Intrinsic-like behavior of the electrical resistivity and Hall coefficient has been observed for3.PbThe 0.77 Sn onset 023 of Se this having behavior a free-hole occurs concentration at the unusually of low 2.3 xtemperature 1O’~cm of 80°K. It is shown that this is an expected consequence of the valence-conduction band inversion phenomenon proposed by Dimmock, Meingailis, and Strauss.
CONSIDERABLE interest has been generated by
studies of ~ Pb~Sn 1_~Se and 5,6 Pba,Sni_xTe having carrier concentrations greater thanshown 3. For such materials it was 1 x 10~ cm that small but distinct breaks occur in the ternperature dependence of the electrical resistivity p at the point of band inversion. In contrast, the Hall coefficient RH was essentially temperature independent at this point. The purpose of the work described here was to study the effects of band inversion upon p and R~at lower carrier concentrations. We believed that the influence of band inversion would be enhanced considerably by decreasing the carrier concentration.
the band-inversion model of Dimmock, Meingailis, and Strauss for the semiconducting alloys Pb~Sn 1_~Se and Pb~Sn1_2,Te.According to the model, the energy gap E9 between conduction and valence bands is a strong function of the alloy fraction x and the sample temperature T.’ ,2 As T is varied for certain alloys, E9 goes to zero and then increases again as the conduction and valence band edges cross and the two bands interchange their roles. The small and controllable band gap involved in this model suggested the possibility of fabricating useful infrared devices (lasers, detectors, and filters) with spectral responses that could be varied over wide ranges. Considerable progress in this direction 3 has already been made. Despite these advances, very little is known about the effects of band inversion on the electrical properties of these alloys. Work already~ done in this area has been limited largely to
Our hypothesis was based upon the elementary reasoning thatfor forthermal low carrier concentrations the effective gap excitation of carriers, Ethermal , is smaller than for high carrier concentrations. This follows directly from the approximate relationship E E E thermal a + F’ and the fact that the Fermi energy EF decreases with decreasing carrier concentration. For sufficiently small carrier concentrations, E — —
This work constitutes a portion of a thesis to be submitted to the University of Maryland by G.F. Hoff in partial fulfillment of the requirements of the Ph.D. degree in physics. *
.
t~erma
.
.
it
is under this condition
approximates E 0, and 1777
1778
INFLUENCE OF BAND INVERSION UPON Pba,Sni~Se 40
0.12 >
80
120
160
200
240 I
Vol.7, No.24
280
PREDICTEDONSETOF
~ 0.0
~i—:
__________________________
Pb
0 77Sn0 17cm3 ~Se p~2.3x10
16.0
—
14.0
_~çscS”°~°~k~ ~ ~
20.0 24.0
—
16.0
HALL COEFFICIENT, RH
~__
12.0—
-
12.0 -
Q~,
8.0 4.0
~
E
‘P
~
10.0
0.0
C
~
8.0
—
-
-4.0
RESISTIVITY.p
~
6.0
-
4.0
-
2.0
—
0.0
0
40
80
I I I 120 160 200 TEMPERATURE (°K)
I 240
I 280
FIG. 1. Intrinsic behavior of the Hall coefficient and electrical resistivity for p-type Pb 0 77Sn023Se having cm ~. The temperature at which intrinsic behavior was predicted is indicated in the upper part of the figure by the intersection of the curves representing Ethermal and 4kT, as discussed in the text.
a carrier concentration of 2.3 x 10
~
that the band inversion phenomenon is expected to have an enhanced influence upon the electrical properties. In particular, we predicted that as a result of this phenomen, intrinsic-like behavior would occur in such material at unusually low temperatures. This is, of course, a simple consequence of the fact that according to the band inversion model, E0 can become very small at such temperatures. The variables involved are illustrated in the upper part of Fig. 1. The solid
line represents the expected variation of E9 and EtherCal with temperature for a Pb077Sn023Se alloy of low carrier concentration. Since the condition for the onset of intrinsic behavior is approximated by E — 4kT (2) thermal
one would expect large thermally-induced changes in RH and p at the temperature for which this relation is first satisfied. This temperature is
Vol.7, No.24
INFLUENCE OF BAND INVERSION UPON Pb~Sn1...~Se
about 90°K, as indicated in Fig. 1 by the intersection of the solid curve with the broken one, the latter representing 4kT. The hypothesis described above is confirmed by our measurements of RH and p, which are presented in the lower part of Fig. 1. These data apply to polycrystalline Pb077 Sn023 Se having a hole concentration of 2.3 x 10 ‘~cm The crystal was prepared by the Bridgman method with an as-grown hole concentration of 3.7 x 10’s cm The final concentration was achieved iso7 inby theanpresence thermal anneal at 550°Cfor 240 hr of a Se deficient ingot of the same material. The good carrier transport properties of this crystal are indicated by the fact that the Hall mobility (RH/p) at 4°Kis 2000cm2/Vsec, a value which we consider to be relatively high for such materials, ~.
~.
1779
dependence of RH in this temperature region. However, we find that all of these features can be well described in terms of a simple parabolic valence-conduction band model incorporating band inversion. Calculations based upon this model indicate that the unusual constancy of RH above the reversal temperature reflects the balance between two opposing processes. The first of these is the increasing thermal energy which increases the number of carriers as the temperature rises. The second is the increasing forbidden energy gap carriers which reduces the number of thermally excited as the temperature is increased. A detailed analysis of the present results and other data is being prepared for publication.
The figure reveals that both RH and P rapidly decrease at about 90°Kand that at higher temperatures RH becomes negative. Such behavior is typical of that found in many other
A knowledge of such large scale variations in Ra and p is important for two reasons. First, the existence of these variations serves as a strong confirmation of the band-inversion model itself. Second, the variations will very likely have an important influence upon the performance of many of the optical devices mentioned above
semiconductors when contributions from intrinsic
when these devices are operated within regions
electron-hole pairs become increasingly important. However, the relatively constant value of RH after it has become negative is atypical. Normally, intrinsic behavior involves a strong temperature
of intrinsic behavior. Acknowledgements We are indebted to Dr. R.S. Allgaier of the U.S. Naval Ordnance Laboratory for many helpful discussions regarding this work. —
REFERENCES 1.
DIMMOCK J.O., MELNGAILIS I. and STRAUSS A.J., Phys. Rev. Lett. 16, 1193 (1966).
2.
STRAUSS A.J., Phys. Rev. 157, 608 (1967).
3.
This work is reviewed by STRAUSS A.J., Trans. AIME, 242, 354 (1968).
4.
HOFF G.F. and DIXON J.R., Bull. Am. phys. Soc. 13, 1378 (1968).
5.
BIS R.F., DIXON J.R. and HOFF G.F., Bull. Am. phys. Soc. 12, 889 (1967).
6. 7.
DIXON J.R. and BIS R.F., phys. Rev. 176, 1942 (1968). CALAWA A.R., HARMAN T.C., FINN M. and YOUTZ P., Trans. AIME, 242, 374 (1968).
-
Intrinsicihnliches Verhalten des elektrischen Widerstandes und des Hall Koeffizienten wurde für Pb 3 hat Dieses Ver077 Sn0•23 Se, beobachtet. welches eine Konzentration halten zetzt freierbei Löcher der ungewöhnlich von 2.3 x 10’~ teifen crn Temperatur von 80°K em. Es wird gezeigt, dass dieses eine erwartete Folge der von Dimmock, Melngailis, und Strauss vorgeschlagenen Valenz-Leitungsband Inversionsersheinung ist.
‘Vol. 7,
pp.
1777—1779, 1969.
Pergam..~nPress.
Printed in Great Britain
INFLUENCE OF BAND INVERSION UPON THE ELECTRICAL PROPERTIES OF Pb~Sn~..~Se IN THE LOW CARRIER CONCENTRATION RANGE* J.R. Dixon and G.F. Hoff U.S. Naval Ordnance Laboratory, Silver Spring, Maryland 20910 and University of Maryland, College Park, Maryknd 20740 (Received 9 September 1969: In revised form 10 October 1969 by E. Burstein)
Intrinsic-like behavior of the electrical resistivity and Hall coefficient has been observed for3.PbThe 0.77 Sn onset 023 of Se this having behavior a free-hole occurs concentration at the unusually of low 2.3 xtemperature 1O’~cm of 80°K. It is shown that this is an expected consequence of the valence-conduction band inversion phenomenon proposed by Dimmock, Meingailis, and Strauss.
CONSIDERABLE interest has been generated by
studies of ~ Pb~Sn 1_~Se and 5,6 Pba,Sni_xTe having carrier concentrations greater thanshown 3. For such materials it was 1 x 10~ cm that small but distinct breaks occur in the ternperature dependence of the electrical resistivity p at the point of band inversion. In contrast, the Hall coefficient RH was essentially temperature independent at this point. The purpose of the work described here was to study the effects of band inversion upon p and R~at lower carrier concentrations. We believed that the influence of band inversion would be enhanced considerably by decreasing the carrier concentration.
the band-inversion model of Dimmock, Meingailis, and Strauss for the semiconducting alloys Pb~Sn 1_~Se and Pb~Sn1_2,Te.According to the model, the energy gap E9 between conduction and valence bands is a strong function of the alloy fraction x and the sample temperature T.’ ,2 As T is varied for certain alloys, E9 goes to zero and then increases again as the conduction and valence band edges cross and the two bands interchange their roles. The small and controllable band gap involved in this model suggested the possibility of fabricating useful infrared devices (lasers, detectors, and filters) with spectral responses that could be varied over wide ranges. Considerable progress in this direction 3 has already been made. Despite these advances, very little is known about the effects of band inversion on the electrical properties of these alloys. Work already~ done in this area has been limited largely to
Our hypothesis was based upon the elementary reasoning thatfor forthermal low carrier concentrations the effective gap excitation of carriers, Ethermal , is smaller than for high carrier concentrations. This follows directly from the approximate relationship E E E thermal a + F’ and the fact that the Fermi energy EF decreases with decreasing carrier concentration. For sufficiently small carrier concentrations, E — —
This work constitutes a portion of a thesis to be submitted to the University of Maryland by G.F. Hoff in partial fulfillment of the requirements of the Ph.D. degree in physics. *
.
t~erma
.
.
it
is under this condition
approximates E 0, and 1777
1778
INFLUENCE OF BAND INVERSION UPON Pba,Sni~Se 40
0.12 >
80
120
160
200
240 I
Vol.7, No.24
280
PREDICTEDONSETOF
~ 0.0
~i—:
__________________________
Pb
0 77Sn0 17cm3 ~Se p~2.3x10
16.0
—
14.0
_~çscS”°~°~k~ ~ ~
20.0 24.0
—
16.0
HALL COEFFICIENT, RH
~__
12.0—
-
12.0 -
Q~,
8.0 4.0
~
E
‘P
~
10.0
0.0
C
~
8.0
—
-
-4.0
RESISTIVITY.p
~
6.0
-
4.0
-
2.0
—
0.0
0
40
80
I I I 120 160 200 TEMPERATURE (°K)
I 240
I 280
FIG. 1. Intrinsic behavior of the Hall coefficient and electrical resistivity for p-type Pb 0 77Sn023Se having cm ~. The temperature at which intrinsic behavior was predicted is indicated in the upper part of the figure by the intersection of the curves representing Ethermal and 4kT, as discussed in the text.
a carrier concentration of 2.3 x 10
~
that the band inversion phenomenon is expected to have an enhanced influence upon the electrical properties. In particular, we predicted that as a result of this phenomen, intrinsic-like behavior would occur in such material at unusually low temperatures. This is, of course, a simple consequence of the fact that according to the band inversion model, E0 can become very small at such temperatures. The variables involved are illustrated in the upper part of Fig. 1. The solid
line represents the expected variation of E9 and EtherCal with temperature for a Pb077Sn023Se alloy of low carrier concentration. Since the condition for the onset of intrinsic behavior is approximated by E — 4kT (2) thermal
one would expect large thermally-induced changes in RH and p at the temperature for which this relation is first satisfied. This temperature is
Vol.7, No.24
INFLUENCE OF BAND INVERSION UPON Pb~Sn1...~Se
about 90°K, as indicated in Fig. 1 by the intersection of the solid curve with the broken one, the latter representing 4kT. The hypothesis described above is confirmed by our measurements of RH and p, which are presented in the lower part of Fig. 1. These data apply to polycrystalline Pb077 Sn023 Se having a hole concentration of 2.3 x 10 ‘~cm The crystal was prepared by the Bridgman method with an as-grown hole concentration of 3.7 x 10’s cm The final concentration was achieved iso7 inby theanpresence thermal anneal at 550°Cfor 240 hr of a Se deficient ingot of the same material. The good carrier transport properties of this crystal are indicated by the fact that the Hall mobility (RH/p) at 4°Kis 2000cm2/Vsec, a value which we consider to be relatively high for such materials, ~.
~.
1779
dependence of RH in this temperature region. However, we find that all of these features can be well described in terms of a simple parabolic valence-conduction band model incorporating band inversion. Calculations based upon this model indicate that the unusual constancy of RH above the reversal temperature reflects the balance between two opposing processes. The first of these is the increasing thermal energy which increases the number of carriers as the temperature rises. The second is the increasing forbidden energy gap carriers which reduces the number of thermally excited as the temperature is increased. A detailed analysis of the present results and other data is being prepared for publication.
The figure reveals that both RH and P rapidly decrease at about 90°Kand that at higher temperatures RH becomes negative. Such behavior is typical of that found in many other
A knowledge of such large scale variations in Ra and p is important for two reasons. First, the existence of these variations serves as a strong confirmation of the band-inversion model itself. Second, the variations will very likely have an important influence upon the performance of many of the optical devices mentioned above
semiconductors when contributions from intrinsic
when these devices are operated within regions
electron-hole pairs become increasingly important. However, the relatively constant value of RH after it has become negative is atypical. Normally, intrinsic behavior involves a strong temperature
of intrinsic behavior. Acknowledgements We are indebted to Dr. R.S. Allgaier of the U.S. Naval Ordnance Laboratory for many helpful discussions regarding this work. —
REFERENCES 1.
DIMMOCK J.O., MELNGAILIS I. and STRAUSS A.J., Phys. Rev. Lett. 16, 1193 (1966).
2.
STRAUSS A.J., Phys. Rev. 157, 608 (1967).
3.
This work is reviewed by STRAUSS A.J., Trans. AIME, 242, 354 (1968).
4.
HOFF G.F. and DIXON J.R., Bull. Am. phys. Soc. 13, 1378 (1968).
5.
BIS R.F., DIXON J.R. and HOFF G.F., Bull. Am. phys. Soc. 12, 889 (1967).
6. 7.
DIXON J.R. and BIS R.F., phys. Rev. 176, 1942 (1968). CALAWA A.R., HARMAN T.C., FINN M. and YOUTZ P., Trans. AIME, 242, 374 (1968).
-
Intrinsicihnliches Verhalten des elektrischen Widerstandes und des Hall Koeffizienten wurde für Pb 3 hat Dieses Ver077 Sn0•23 Se, beobachtet. welches eine Konzentration halten zetzt freierbei Löcher der ungewöhnlich von 2.3 x 10’~ teifen crn Temperatur von 80°K em. Es wird gezeigt, dass dieses eine erwartete Folge der von Dimmock, Melngailis, und Strauss vorgeschlagenen Valenz-Leitungsband Inversionsersheinung ist.