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Observation of interband transitions in Cd3As2
Solid State Commumcations Vol. 4, PP. 65-68, 1966. Pergamon Press Ltd. Printed in Great Britain.
OBSERVATION OF INTERBAND TRANSITIONS IN Cd3As2 E. D. Haidemenakis*~, J. G. Mavroides*, M. S. Dresselhaus* and D. F. Icolesar* Laboratoire de Physique, Ecole Normale Superieure, Paris, France and Lincoln Laboratory~ Massachusetts Institute of Technology, Lexington, Massachusetts (Received 22 October 1965 by E. Burstein; Revised Manuscript Received 15 December 1965) High field oscillatory magnetoreflection measurements performed in crystals of Cd3As2 at liquid helium temperatures have yielded an energy gap Eg = 0.048 eV, and a reduced effective mass m* = 0.092 m0. The absence of a low energy cutoff indicates that no carriers are present in these bands. Nevertheless,1 Hall3 measurements between 4. 2°Kand on these 300°K. samples Theseyield results a carrier suggestconcentration that these carriers, of 3 x 1018/cm which have also been observed in other experiments, must be located elsewhere in the Brillouin zone. The coincidence of the band gap with the plasma frequency indicates that interband effects are important in the interpretation of the magnetoplasma effects. WE HAVE PERFORMED high field magnetoref-
The Cd
lection experiments in Cd3As2. These measurements followed earlier magnetoplasma experiments at low fields, 1 which were motivated by the high mobility of this 2/Vsec material. at room 2, 3 temperaTypical values tare and are 30,10, 000000 at cm 77°K. Previous measurements of the photoconductivity4 as well as of the absorption coefficient5 have been interpreted to yield values of 0. 14 eV and 0. 13 eV respectively, for the optical band gap. An effective mass of (0. 10 ~ 0. 05)mo was deduced from the latter analysis for the carri~rsin this energy band. More recently, Sexer’ has reported an effective density of states mass of (0.050 ± 0.015) m 0 from thermoelectric power measurements. The present measurements provide information about energy bands with extrema located at another point in the Brillouin zone and with semiconducting characteristics. The presence of these interband transitions are expected to be important in the interpretation of5optical and themeasurements such effect. as the 1absorption magnetoplasma
3As2 crystals were grown by crystallization from the vapor phase. The crystal faces were along the direction of growth, and. were prepared by mechanically polishing with 0. 5v diamond ments were taken grit.at Magnetoreflection liquid helium temperatures measureon an undoped sample in the energy range 0.048 < ~w < 0.067 eV, in a Bitter solenoid with the magnetic field and incident light normal to the crystal face. The magneto-optical arrangement which was used has been previously described. 6 Figure 1 shows a typical experimental trace of the variation of the reflectivity with magnetic field at a fixed value of photon energy. Each oscillation in the reflectivity corresponds to an interband transition and is labelled by n = integer. By taking such traces at different values of the photon energy, it was possible to follow each transition and obtain a summary plot, which is shown in Fig. 2. In this analysis only the gross structure was present oscillations as indicated were studied. in the first Finer fIgure by the broken curves but could not be
*Vislting scientists, National Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts. National Magnet Laboratory is supported by- the U. S. Air Force Office of Scientific Research. §Attache de Recherches, Centre National de la Recherche Scientifique. ¶Operated with support from the U. S. Air Force. 65
66
INTERBAND TRANSITIONS IN Cd3As2
SCALE
R (H) R (0) R(0) —
0.10
=
Vol. 4, No. 1
~
>..
I> I-
n3
n2
0.0590 eV
4 0
20
40 60 MAGNETIC FIELD (kG)
T’~’4°K 80
100
FIG. 1 Experimental trace of the magnetic field variation of the reflectivity at ~ w = 0.0590 eV and T 4°K. The identification of the Interband Landau level transitions is indicated by n = Integer. The broken curves indicate possible fine structure superimposed on the principal transitions represented by the solid curve. studied because of the poor signal to noise ratio. However, one of the weaker lines labelled ~ could be followed, despite the poor signal to noise ratio, and its frequency dependence Is Included in the plot of Fig. 2. An analysis of the data shown in Fig. 2 indicates that for a fixed photon energy, the resonant fields for the lines labelled n = 1, 2, 3 are in the ratio of 1/3: 1/5: 1/7. These ratios are consistent with the energy relation for interband transitions between Landau levels for simple parabolic bands without spin, i. e. t~En= Eg + (n + 1/2) ~w~. Furthermore, the n = 1, 2, 3, transitions follow straight lines in accordance
with this simple formula and extrapolate to a band gap of Eg = 0.048 eV. From the slope of these lines, a reduced effective mass m* = 0.092 m0 is obtained. This value for m* is considerably higher than would be expected from comparison with m*/E~ratios for typical semiconductors. It is of int’erest to note that the weak intensity line in Fig. 2 has a resonant field which falls between the n = 1 and n = 2 lines approximately in the ratio 1/3: 1/4: 1/5. These lines could correspond to interband transitions between Landau levels in a simple conduction band and a degenerate valence band having one heavy and one light mass component. Alternatively, the extra structure in Fig. 1
Vol. 4, No. 1
INTERBAND TRANSITIONS IN Cd3As2
67
0.08—
(~
~
~0.O4-
0.02
—
I 0
1
I
20
40
I
I 60
I
I 80
I
I
I
100
MAGNETIC FIELD (kG) FIG. 2 Summary of the interband transitions. The weak transitions are indicated by the triangles while the open circles denote the more intense transitions. The filled circles represent the observed reflectivity minimum between n = 0 and n = 1 and the broken line connects these filled circles. might be due to a larger g-factor which then would produce doublets. On the other hand, the transition labelled n = 0 exhibits a different line shape as can be seen in Fig. 1. The resonant field for this line as well as for the other lines was taken at the maximum reflectivity. From the data in Fig. 2, it would appear that this line ‘may not extrapolate to the above band gap; this deviation may be associated with exciton effects. Such effects have been discussed in connection with the lowest interl~andtransition for some other semiconductors. However, the interpretation of this
lowest transition In Cd3As2 is further complicated by the magnetoplasma effect, since the plasma frequency coincides with the energy gap. 1 Conversely, the interpretation of the znagnetoplasma effect in this material must, include interband effects in addition to the free carrier contribution. This coincidence of wp with Eg is touched upon by Burstein, etal. 8, and discussed in the case of antimony by Dresseihaus and MavroidesP The broken curve in Fig. 2 corresponds to the plot of the sharp minimum which occurs between the n = 0 and the n = 1 transItions, as may ,be seen in Fig. 1. At low photon energies a
68
INTERBAND TRANSITIONS IN Cd3As2
Vol. 4, No. 1
nitude is observed, it must be concluded that the interband transitions occur between semiconductor type bands and that the carriers observed in the Hall effect must be located about another point in the Brlllouin zone.
bending of the broken curve of Fig. 2 is observed. In addition, the shape of the minimum is sensitive to polarization effects, whereas the reflectivity maxima associated with the interband transitions are not significantly affected by the polarization of the light. In these studies only linear polarization was used.
Acknowledgements--We are grateful to Dr. Jean Travernier of the Laboratoire Central des Industries Electriques, in France for graciously supplying us with the crystals. We are also indebted to Dr.A. J. Strauss of Lincoln Laboratory for stimulating discussions and for carrying out the Hall measurements. One of us, E. D. Haidemenakis, gratefully acknowledges the support of the N. A. T. 0. Science Affairs Division. He also wishes to thank Professors Y. Rocard, P. Aigrain and M. Balkanski for their interest in this work, and Professor B. Lax, Mr. K. J. Button, and all the members of the National Magnet Laboratory for their hospitality.
Perhaps the most significant feature of the data in Fig. 2 is the absence of a low energy cutoff for the observation of interband transitions. However, Hall measurements carried out on this sa~n,plel~dicate carrier concentration n <= T< 3 x 10/cm’~in the atemperature range 4. 2°K 300°K. Therefore, if thesc carriers were as10 sociated with the observed ii~terbandtransitions a Fermi energy EF 0.08 eV would be expected assuming m*/ m = 0. 1 and one equivalent band, thus implying a low energy cutoff ~ w> E~÷EF 0. 13 eV. Since no low energy cutoff of this mag-
References 1. HAIDEMENAKIS E. D., Rapport d’Activite du Laboratoire de Physique des Solides de Ia Faculte des Sciences de Paris (1964). 2. ROSENBERGA.J. andHARMANT.C., J. Appl. Phys. 30, 1621 (1959). 3. SEXER N., J. Phys. Radium 22, 807 (1961). 4. MOSST.S., Proc. Phys. Soc. (London) B63, 167 (1950). 5. TURNER W. J., FISCHLER A. S. and REESE W. E., Phys. Rev. 121, 759 (1961). 6. BROWN R. N., MAVROIDES J. G. and LAX B., Phys. Rev. 129, 2055 (1963). 7. ELLIOTT R.J. and LOUDON R., J. Phys. Chem. Solids 8, 282 (1959). 8. BURSTEIN E., STILES P. J., LANGENBERG D. N. and WALLIS R. F., Proc. Inter. ConI. on the Physics of Semiconductors, Exeter (1962). 9. DRESSELHAUS M. S. and MAVROIDES J. G., Internat. CoaL Opt. Prop. Metals and Alloys, Paris (1965).
a
Des mesures de magnétoreflection oscillatoire champs magnetiques dlevds ont été realis~s sur des cristaux de Cd3As 2 la tempertaure de l’hellum liquide. Elles ont donnê une energte de la bande interdite E~= 0.048 eV et une masse réduite effective m* = 0.092 m0. L’absence d’une ~asse energie de “cutoff” indique qu’ll n’existe pas de porteurs dana ces bandes. N~anmoinsdes mesures de l’effet Hall qui oat êté 18/cm3 faites sur entre ces 4.échantillons 2°Ket 300°K. ont determind une concentration de porteurs de 3 x 10 Ces resultats suggèrent que ces porteurs, deja observés dane d’autres experiences, doivent ètre situés ailleurs dans la zone de Brillouin. Le fait que l’energle de la bande interdite coincide avec Ia fréquence de plasma indique les effets de transitions interbandes sont aussi importants que les effets de magnétoplasma. ~.
OBSERVATION OF INTERBAND TRANSITIONS IN Cd3As2 E. D. Haidemenakis*~, J. G. Mavroides*, M. S. Dresselhaus* and D. F. Icolesar* Laboratoire de Physique, Ecole Normale Superieure, Paris, France and Lincoln Laboratory~ Massachusetts Institute of Technology, Lexington, Massachusetts (Received 22 October 1965 by E. Burstein; Revised Manuscript Received 15 December 1965) High field oscillatory magnetoreflection measurements performed in crystals of Cd3As2 at liquid helium temperatures have yielded an energy gap Eg = 0.048 eV, and a reduced effective mass m* = 0.092 m0. The absence of a low energy cutoff indicates that no carriers are present in these bands. Nevertheless,1 Hall3 measurements between 4. 2°Kand on these 300°K. samples Theseyield results a carrier suggestconcentration that these carriers, of 3 x 1018/cm which have also been observed in other experiments, must be located elsewhere in the Brillouin zone. The coincidence of the band gap with the plasma frequency indicates that interband effects are important in the interpretation of the magnetoplasma effects. WE HAVE PERFORMED high field magnetoref-
The Cd
lection experiments in Cd3As2. These measurements followed earlier magnetoplasma experiments at low fields, 1 which were motivated by the high mobility of this 2/Vsec material. at room 2, 3 temperaTypical values tare and are 30,10, 000000 at cm 77°K. Previous measurements of the photoconductivity4 as well as of the absorption coefficient5 have been interpreted to yield values of 0. 14 eV and 0. 13 eV respectively, for the optical band gap. An effective mass of (0. 10 ~ 0. 05)mo was deduced from the latter analysis for the carri~rsin this energy band. More recently, Sexer’ has reported an effective density of states mass of (0.050 ± 0.015) m 0 from thermoelectric power measurements. The present measurements provide information about energy bands with extrema located at another point in the Brillouin zone and with semiconducting characteristics. The presence of these interband transitions are expected to be important in the interpretation of5optical and themeasurements such effect. as the 1absorption magnetoplasma
3As2 crystals were grown by crystallization from the vapor phase. The crystal faces were along the direction of growth, and. were prepared by mechanically polishing with 0. 5v diamond ments were taken grit.at Magnetoreflection liquid helium temperatures measureon an undoped sample in the energy range 0.048 < ~w < 0.067 eV, in a Bitter solenoid with the magnetic field and incident light normal to the crystal face. The magneto-optical arrangement which was used has been previously described. 6 Figure 1 shows a typical experimental trace of the variation of the reflectivity with magnetic field at a fixed value of photon energy. Each oscillation in the reflectivity corresponds to an interband transition and is labelled by n = integer. By taking such traces at different values of the photon energy, it was possible to follow each transition and obtain a summary plot, which is shown in Fig. 2. In this analysis only the gross structure was present oscillations as indicated were studied. in the first Finer fIgure by the broken curves but could not be
*Vislting scientists, National Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts. National Magnet Laboratory is supported by- the U. S. Air Force Office of Scientific Research. §Attache de Recherches, Centre National de la Recherche Scientifique. ¶Operated with support from the U. S. Air Force. 65
66
INTERBAND TRANSITIONS IN Cd3As2
SCALE
R (H) R (0) R(0) —
0.10
=
Vol. 4, No. 1
~
>..
I> I-
n3
n2
0.0590 eV
4 0
20
40 60 MAGNETIC FIELD (kG)
T’~’4°K 80
100
FIG. 1 Experimental trace of the magnetic field variation of the reflectivity at ~ w = 0.0590 eV and T 4°K. The identification of the Interband Landau level transitions is indicated by n = Integer. The broken curves indicate possible fine structure superimposed on the principal transitions represented by the solid curve. studied because of the poor signal to noise ratio. However, one of the weaker lines labelled ~ could be followed, despite the poor signal to noise ratio, and its frequency dependence Is Included in the plot of Fig. 2. An analysis of the data shown in Fig. 2 indicates that for a fixed photon energy, the resonant fields for the lines labelled n = 1, 2, 3 are in the ratio of 1/3: 1/5: 1/7. These ratios are consistent with the energy relation for interband transitions between Landau levels for simple parabolic bands without spin, i. e. t~En= Eg + (n + 1/2) ~w~. Furthermore, the n = 1, 2, 3, transitions follow straight lines in accordance
with this simple formula and extrapolate to a band gap of Eg = 0.048 eV. From the slope of these lines, a reduced effective mass m* = 0.092 m0 is obtained. This value for m* is considerably higher than would be expected from comparison with m*/E~ratios for typical semiconductors. It is of int’erest to note that the weak intensity line in Fig. 2 has a resonant field which falls between the n = 1 and n = 2 lines approximately in the ratio 1/3: 1/4: 1/5. These lines could correspond to interband transitions between Landau levels in a simple conduction band and a degenerate valence band having one heavy and one light mass component. Alternatively, the extra structure in Fig. 1
Vol. 4, No. 1
INTERBAND TRANSITIONS IN Cd3As2
67
0.08—
(~
~
~0.O4-
0.02
—
I 0
1
I
20
40
I
I 60
I
I 80
I
I
I
100
MAGNETIC FIELD (kG) FIG. 2 Summary of the interband transitions. The weak transitions are indicated by the triangles while the open circles denote the more intense transitions. The filled circles represent the observed reflectivity minimum between n = 0 and n = 1 and the broken line connects these filled circles. might be due to a larger g-factor which then would produce doublets. On the other hand, the transition labelled n = 0 exhibits a different line shape as can be seen in Fig. 1. The resonant field for this line as well as for the other lines was taken at the maximum reflectivity. From the data in Fig. 2, it would appear that this line ‘may not extrapolate to the above band gap; this deviation may be associated with exciton effects. Such effects have been discussed in connection with the lowest interl~andtransition for some other semiconductors. However, the interpretation of this
lowest transition In Cd3As2 is further complicated by the magnetoplasma effect, since the plasma frequency coincides with the energy gap. 1 Conversely, the interpretation of the znagnetoplasma effect in this material must, include interband effects in addition to the free carrier contribution. This coincidence of wp with Eg is touched upon by Burstein, etal. 8, and discussed in the case of antimony by Dresseihaus and MavroidesP The broken curve in Fig. 2 corresponds to the plot of the sharp minimum which occurs between the n = 0 and the n = 1 transItions, as may ,be seen in Fig. 1. At low photon energies a
68
INTERBAND TRANSITIONS IN Cd3As2
Vol. 4, No. 1
nitude is observed, it must be concluded that the interband transitions occur between semiconductor type bands and that the carriers observed in the Hall effect must be located about another point in the Brlllouin zone.
bending of the broken curve of Fig. 2 is observed. In addition, the shape of the minimum is sensitive to polarization effects, whereas the reflectivity maxima associated with the interband transitions are not significantly affected by the polarization of the light. In these studies only linear polarization was used.
Acknowledgements--We are grateful to Dr. Jean Travernier of the Laboratoire Central des Industries Electriques, in France for graciously supplying us with the crystals. We are also indebted to Dr.A. J. Strauss of Lincoln Laboratory for stimulating discussions and for carrying out the Hall measurements. One of us, E. D. Haidemenakis, gratefully acknowledges the support of the N. A. T. 0. Science Affairs Division. He also wishes to thank Professors Y. Rocard, P. Aigrain and M. Balkanski for their interest in this work, and Professor B. Lax, Mr. K. J. Button, and all the members of the National Magnet Laboratory for their hospitality.
Perhaps the most significant feature of the data in Fig. 2 is the absence of a low energy cutoff for the observation of interband transitions. However, Hall measurements carried out on this sa~n,plel~dicate carrier concentration n <= T< 3 x 10/cm’~in the atemperature range 4. 2°K 300°K. Therefore, if thesc carriers were as10 sociated with the observed ii~terbandtransitions a Fermi energy EF 0.08 eV would be expected assuming m*/ m = 0. 1 and one equivalent band, thus implying a low energy cutoff ~ w> E~÷EF 0. 13 eV. Since no low energy cutoff of this mag-
References 1. HAIDEMENAKIS E. D., Rapport d’Activite du Laboratoire de Physique des Solides de Ia Faculte des Sciences de Paris (1964). 2. ROSENBERGA.J. andHARMANT.C., J. Appl. Phys. 30, 1621 (1959). 3. SEXER N., J. Phys. Radium 22, 807 (1961). 4. MOSST.S., Proc. Phys. Soc. (London) B63, 167 (1950). 5. TURNER W. J., FISCHLER A. S. and REESE W. E., Phys. Rev. 121, 759 (1961). 6. BROWN R. N., MAVROIDES J. G. and LAX B., Phys. Rev. 129, 2055 (1963). 7. ELLIOTT R.J. and LOUDON R., J. Phys. Chem. Solids 8, 282 (1959). 8. BURSTEIN E., STILES P. J., LANGENBERG D. N. and WALLIS R. F., Proc. Inter. ConI. on the Physics of Semiconductors, Exeter (1962). 9. DRESSELHAUS M. S. and MAVROIDES J. G., Internat. CoaL Opt. Prop. Metals and Alloys, Paris (1965).
a
Des mesures de magnétoreflection oscillatoire champs magnetiques dlevds ont été realis~s sur des cristaux de Cd3As 2 la tempertaure de l’hellum liquide. Elles ont donnê une energte de la bande interdite E~= 0.048 eV et une masse réduite effective m* = 0.092 m0. L’absence d’une ~asse energie de “cutoff” indique qu’ll n’existe pas de porteurs dana ces bandes. N~anmoinsdes mesures de l’effet Hall qui oat êté 18/cm3 faites sur entre ces 4.échantillons 2°Ket 300°K. ont determind une concentration de porteurs de 3 x 10 Ces resultats suggèrent que ces porteurs, deja observés dane d’autres experiences, doivent ètre situés ailleurs dans la zone de Brillouin. Le fait que l’energle de la bande interdite coincide avec Ia fréquence de plasma indique les effets de transitions interbandes sont aussi importants que les effets de magnétoplasma. ~.