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Band structure enhancement of indirect transitions
Solid State Communications, Vol. 21, pp. 437-439, 1977.
Pergamon Press.
Printed in Great Britain
BAND STRUCTURE ENHANCEMENT OF INDIRECT TRANSITIONS D. Auvergne, P. Merle and H. Mathieu Centre d'Etudes d'Electronique des Solides,* Oniversit6 des Sciences et Techniques du Languedoe, 34060 MontpeUier Cedex, France
(Received 13 September 1976 by M. Balkanski) Indirect and direct transitions in Ga= Inz-xP alloys has been studied in modulation spectroscopy, between 10 and 300 K. Deviation from simple indirect absorption has been studied and shown to be consistant with a model which takes into account band structure enhancement of the absorption coefficient at alloy composition values where direct and indirect edges occur, at nearly the same energy. For the first time we observe structures of Ae~ corresponding to band structure enhanced indirect transitions. WE RECENTLY SHOWED 1 that, in Gax In~_xP alloys, the comparison of modulated reflectivity and transmission spectra gives clear evidence of the participation of L and X conduction band minima, to the indirect process. At 10 K direct determination of r' i c - L i c and Xlc-Llc, crossover compositions (:re = 0.68 and xc = 0.77 respectively) permits to resolve the discrepancies existing in the literature, on this alloy. We present here piezoreflectance studies on this alloy in the composition range corresponding to the I"lc-Lzc crossover. We observe, for the first time, the contribution of resonant indirect transitions to the real part e~ of the dielectric constant. The lowest band edge to indirect zinc blende-type semiconductors corresponds to transitions from the valence-band maximum at r to the lowest conductionband minimum located at k :/: 0. For each scattering process (only one phonon involved), the absorption coefficient relative to indirect transitions from valence states to exciton states can be given by the general expression: 2
[eJ [d] 2.S
1.5
18+50°37-X+( I
InP
I
I
I
I
I
90
I
I
I
GaP
Fig. 1. Composition dependence of the direct (e) and indirect (w: r - X and i : r - L ) gaps. For the indirect edges the energy values correspond to (Bind. X - Eex q-
IhhO.~LA(X)~and (.Eina. L -- Eex q- li~LO(L) ) which are e energles ot the more accurate structures we can obtain in differential spectroscopy. Solid stars correspond to the values obtained from: (a) reference 5; Co) reference 6; (c) reference 7; (d) reference 8; (e) reference 9.
/all2
a "" (SE+fl6op)2(E--Egx+h6op)l/2
1~.36~ -- 0:152X + 0.147X '~J/
(1) l~lsv -~ LIe indirect gap, the LAL and LOL phononassisted transitions are the strongest. It appears that the Pie--Lte crossover occurs in the composition range corresponding to x "" 0.70. Now, Fig. 2 shows the temperature dependence of the thresholds of the direct and indirect edges on Gao.721no.2aPsample. The fundamental edge is still indirect but near 135 K we observe a crossing between the Eo and E~x + h6~LO transitions. That is the resonance of the PlSv -~ Lze indirect transition assisted by emission of the LO phonon. This resonance, which corresponds to the vanishing of the denominator of equation (1), is clearly shown in Fig. 3. This figure
where P and H are respectively the matrix elements for radiative transition and phonon scattering, 8E is the energy difference between the intermediate state and the final (or initial) state and h6op the phonon energy. Since the energy denominator of equation (1) is much smaller for the process involving the lowest conduction band, Fie, structures associated to this intermediate state are expected to be the strongest. Now transitions via I~ze intermediate state involve only phonons of symmetries X1 or LI. So considering * Centre associ6 an CNRS. 437
438
BAND STRUCTURE ENHANCEMENT OF INDIRECT TRANSITIONS
[Ga0'691"031P I
I E [eV]
Vol. 21, No., 5
~
Ga0.721%.28p ] 105~K ~ - ~ / ~
128°K ~ ~ , ~ o \11
_2.2
"',,
L. ~ " .
ELx+fI~L O
E~X"I"f"I~LAI~',,,~
200 K
"',
--m T OK J J,
100
200
I
25°°K
300
I
273°K
I
Fig. 2. Temperature dependence of the thresholds of the direct and indirect edges observed on Gao.~Ino.zsP sample.
~
/~'~,,~ 1Ga0.72In0.28PJ
//-E0
135°K~
~
,Rt
--
t?£K
200OK :'O°K
0.52
0.54 i
0.56 |
~,
[~] .
Fig. 3. Temperature evolution of AR/Rspectra obtained on Cao.72Ino.2sPsample, with rough back surface. shows the temperature evolution of the AR/R ~- Ael structures, observed on sample with unpolished back face, in order to prevent back reflection. At low temperature, we observe a structure charac-
0.49 i
~,[~]~ 0.51 a 0.55 i
Fig. 4. Temperature evolution of the AR/R spectra obtained on Gao.69Ino.31Psample. teristic of the excitonic direct gap Eo.a At high temperature an extra oscillation appears. This extra oscillation, observed for the first time, corresponds to a resonance of the indirect transitions produced, in this case, by the energy conservation on the Flc intermediate state which becomes resonant state. This means that the contribution to e ~ of the indirect transition can become comparable to that of the direct gap. This reconance has also been observed at lower temperature on a sample with lower gallium content (x = 0.69). In Fig. 4 we show, from the preceding discussion, that this time, the F - L crossover occurs around 7 0 - 1 0 0 K in agreement with the expected evolution of the transitions energies with composition. We can see in conclusion that the resonant effect observed on GaInP could explain the discrepancies which exist in the literature for the value of the composition where the crossover occurs. It confirms the results published earlier by Pitt et al. 4 on this alloy, and it could be observed on other one like GaA1Sb and GaAsP. These studies allowed us to present a complete description of the conduction band of GaInP and to show, for a fundamental edge that it is possible to observe effects of indirects transitions on the direct reflexion-coefficient.
REFERENCES 1.
i
MATHIEU H., MERLE P. & AUVERGNE D., XIII. Int. Conf. Phys. Semicond. Rome (1976).
2.
ELLIOT R.J., Phys. Rev. 108, 1347 (1957).
3.
CARDONA M., Solid State Phys. Vol. 11, suppl. Academic Press, New York (1969).
,VOW.21, No. 5
439
BAND STRUCTURE ENHANCEMENT OF INDIRECT TRANSITIONS
4.
PITT G.D., VYAS M.K.R. & MABBITT A.V., Solid State Commun. 14, 621 (1974).
5.
ONTON A. & LORENTZ M.R., Proc. Xlnt. Conf. Phys. Semicyond., p. 440. Cambridge (1970).
6.
PITT G.D., J. Phys. C6, 1586 (1973).
7.
WHITE A.M., DEAN P.J., TAYLOR L.L., CLARKE R.C, ASHEN D.J. & MULLIN J.B., J. Phys. C5, 1727 (1972).
.
9.
DEAN P.J., KAMINSKY G. & ZETTERSTRAM R.B., J. Appl. Phys. 38, 3551 (1967). DUNKE W.P., LORENTZ M.R.M. & PETTIT G.D., Phys. Rev. BS, 2978 (1972). On ~ t u d i e l e e t r a n s i t i o n s en s p e c t r o s c o p i e
directes
et indirectes
de l e a l l i a g e
Ca InP
de m o d u l a t i o n . On montre couBent l e c r o i s e m e n t des m i n i ma
F l c e t L l c de l a bande de c o n d u c t i o n e n t r a ~ n e une r~sonance des t r a n s i t i o n s indirectes
e t comment c e t t e
sur l a p a r t l e
r~elle
r ~ s o n a n c e se t r a d u l t par l e a p p a r i t i o n
¢| de l a c o n s t a n t e d i ~ l e c t r i q u e .
de s t r u c t u r e
Pergamon Press.
Printed in Great Britain
BAND STRUCTURE ENHANCEMENT OF INDIRECT TRANSITIONS D. Auvergne, P. Merle and H. Mathieu Centre d'Etudes d'Electronique des Solides,* Oniversit6 des Sciences et Techniques du Languedoe, 34060 MontpeUier Cedex, France
(Received 13 September 1976 by M. Balkanski) Indirect and direct transitions in Ga= Inz-xP alloys has been studied in modulation spectroscopy, between 10 and 300 K. Deviation from simple indirect absorption has been studied and shown to be consistant with a model which takes into account band structure enhancement of the absorption coefficient at alloy composition values where direct and indirect edges occur, at nearly the same energy. For the first time we observe structures of Ae~ corresponding to band structure enhanced indirect transitions. WE RECENTLY SHOWED 1 that, in Gax In~_xP alloys, the comparison of modulated reflectivity and transmission spectra gives clear evidence of the participation of L and X conduction band minima, to the indirect process. At 10 K direct determination of r' i c - L i c and Xlc-Llc, crossover compositions (:re = 0.68 and xc = 0.77 respectively) permits to resolve the discrepancies existing in the literature, on this alloy. We present here piezoreflectance studies on this alloy in the composition range corresponding to the I"lc-Lzc crossover. We observe, for the first time, the contribution of resonant indirect transitions to the real part e~ of the dielectric constant. The lowest band edge to indirect zinc blende-type semiconductors corresponds to transitions from the valence-band maximum at r to the lowest conductionband minimum located at k :/: 0. For each scattering process (only one phonon involved), the absorption coefficient relative to indirect transitions from valence states to exciton states can be given by the general expression: 2
[eJ [d] 2.S
1.5
18+50°37-X+( I
InP
I
I
I
I
I
90
I
I
I
GaP
Fig. 1. Composition dependence of the direct (e) and indirect (w: r - X and i : r - L ) gaps. For the indirect edges the energy values correspond to (Bind. X - Eex q-
IhhO.~LA(X)~and (.Eina. L -- Eex q- li~LO(L) ) which are e energles ot the more accurate structures we can obtain in differential spectroscopy. Solid stars correspond to the values obtained from: (a) reference 5; Co) reference 6; (c) reference 7; (d) reference 8; (e) reference 9.
/all2
a "" (SE+fl6op)2(E--Egx+h6op)l/2
1~.36~ -- 0:152X + 0.147X '~J/
(1) l~lsv -~ LIe indirect gap, the LAL and LOL phononassisted transitions are the strongest. It appears that the Pie--Lte crossover occurs in the composition range corresponding to x "" 0.70. Now, Fig. 2 shows the temperature dependence of the thresholds of the direct and indirect edges on Gao.721no.2aPsample. The fundamental edge is still indirect but near 135 K we observe a crossing between the Eo and E~x + h6~LO transitions. That is the resonance of the PlSv -~ Lze indirect transition assisted by emission of the LO phonon. This resonance, which corresponds to the vanishing of the denominator of equation (1), is clearly shown in Fig. 3. This figure
where P and H are respectively the matrix elements for radiative transition and phonon scattering, 8E is the energy difference between the intermediate state and the final (or initial) state and h6op the phonon energy. Since the energy denominator of equation (1) is much smaller for the process involving the lowest conduction band, Fie, structures associated to this intermediate state are expected to be the strongest. Now transitions via I~ze intermediate state involve only phonons of symmetries X1 or LI. So considering * Centre associ6 an CNRS. 437
438
BAND STRUCTURE ENHANCEMENT OF INDIRECT TRANSITIONS
[Ga0'691"031P I
I E [eV]
Vol. 21, No., 5
~
Ga0.721%.28p ] 105~K ~ - ~ / ~
128°K ~ ~ , ~ o \11
_2.2
"',,
L. ~ " .
ELx+fI~L O
E~X"I"f"I~LAI~',,,~
200 K
"',
--m T OK J J,
100
200
I
25°°K
300
I
273°K
I
Fig. 2. Temperature dependence of the thresholds of the direct and indirect edges observed on Gao.~Ino.zsP sample.
~
/~'~,,~ 1Ga0.72In0.28PJ
//-E0
135°K~
~
,Rt
--
t?£K
200OK :'O°K
0.52
0.54 i
0.56 |
~,
[~] .
Fig. 3. Temperature evolution of AR/Rspectra obtained on Cao.72Ino.2sPsample, with rough back surface. shows the temperature evolution of the AR/R ~- Ael structures, observed on sample with unpolished back face, in order to prevent back reflection. At low temperature, we observe a structure charac-
0.49 i
~,[~]~ 0.51 a 0.55 i
Fig. 4. Temperature evolution of the AR/R spectra obtained on Gao.69Ino.31Psample. teristic of the excitonic direct gap Eo.a At high temperature an extra oscillation appears. This extra oscillation, observed for the first time, corresponds to a resonance of the indirect transitions produced, in this case, by the energy conservation on the Flc intermediate state which becomes resonant state. This means that the contribution to e ~ of the indirect transition can become comparable to that of the direct gap. This reconance has also been observed at lower temperature on a sample with lower gallium content (x = 0.69). In Fig. 4 we show, from the preceding discussion, that this time, the F - L crossover occurs around 7 0 - 1 0 0 K in agreement with the expected evolution of the transitions energies with composition. We can see in conclusion that the resonant effect observed on GaInP could explain the discrepancies which exist in the literature for the value of the composition where the crossover occurs. It confirms the results published earlier by Pitt et al. 4 on this alloy, and it could be observed on other one like GaA1Sb and GaAsP. These studies allowed us to present a complete description of the conduction band of GaInP and to show, for a fundamental edge that it is possible to observe effects of indirects transitions on the direct reflexion-coefficient.
REFERENCES 1.
i
MATHIEU H., MERLE P. & AUVERGNE D., XIII. Int. Conf. Phys. Semicond. Rome (1976).
2.
ELLIOT R.J., Phys. Rev. 108, 1347 (1957).
3.
CARDONA M., Solid State Phys. Vol. 11, suppl. Academic Press, New York (1969).
,VOW.21, No. 5
439
BAND STRUCTURE ENHANCEMENT OF INDIRECT TRANSITIONS
4.
PITT G.D., VYAS M.K.R. & MABBITT A.V., Solid State Commun. 14, 621 (1974).
5.
ONTON A. & LORENTZ M.R., Proc. Xlnt. Conf. Phys. Semicyond., p. 440. Cambridge (1970).
6.
PITT G.D., J. Phys. C6, 1586 (1973).
7.
WHITE A.M., DEAN P.J., TAYLOR L.L., CLARKE R.C, ASHEN D.J. & MULLIN J.B., J. Phys. C5, 1727 (1972).
.
9.
DEAN P.J., KAMINSKY G. & ZETTERSTRAM R.B., J. Appl. Phys. 38, 3551 (1967). DUNKE W.P., LORENTZ M.R.M. & PETTIT G.D., Phys. Rev. BS, 2978 (1972). On ~ t u d i e l e e t r a n s i t i o n s en s p e c t r o s c o p i e
directes
et indirectes
de l e a l l i a g e
Ca InP
de m o d u l a t i o n . On montre couBent l e c r o i s e m e n t des m i n i ma
F l c e t L l c de l a bande de c o n d u c t i o n e n t r a ~ n e une r~sonance des t r a n s i t i o n s indirectes
e t comment c e t t e
sur l a p a r t l e
r~elle
r ~ s o n a n c e se t r a d u l t par l e a p p a r i t i o n
¢| de l a c o n s t a n t e d i ~ l e c t r i q u e .
de s t r u c t u r e