model /3/ whose band parameters - like for instance the energy gap E ... three-band model formulaeâ. It is the aim of ... certainties are of the order of 1%, except.
K129
Short Notes phys. stat. sol. (b) __ 135, K129 (1986) Subject classification: 71.25; S7.13 C.N.R.S., Service National d e s Champs Intenses, Grenoble 1) P r e s s u r e Dependence of the Electronic Effective M a s s and Effective E-Factor in the Narrow Gaa Semiconductor InSb BY S. HUANT, L. DMOWSK12), M. BAJ3), a n d L . C . BRUNEL
We recently reported a n experimental study of the cyclotron resonance 0+-+1+ (CR) and spin resonance O+-+
0- (SR) in the conduction band of the
narrow-gap semiconductor InSb under hydrostatic p r e s s u r e /I/. We a l s o noticed the remarkable agreement between experimental and theoretical energies of these p r o c e s s e s calculated within a modified /2/ Pidgeon-Brown model /3/ whose band p a r a m e t e r s - like for instance the energy gap E
g ( E = 235.2 mev) o r the spin-orbit energy A ( A = 803 mev) /4/ - w e r e p r e g cisely given i n a recent paper by Goodwin and Seiler /5/. It was a l s o assumed as usual (see for instance / 6 / ) that only E v a r i e s with p r e s s u r e i n a n amount g (chosen in o r d e r to obtain the s a m e agreement under p r e s s u r e as a t z e r o pressure) of 0.14 meV/M??a, which is in very good agreement with experimental measurements /?/.
W e a l s o wrote in the l a s t sent,ence of o u r
paper /1/ that: “Wecan estimate that in the p r e s s u r e range 0 to 1 GPa, the variations of m * and g * can b e calculated with a n accuracy of about 3% using C
C
three-band model formulae”. It is the aim of this short note to briefly p r e c i s e and comment this point. O u r experimental data /1/ allow u s to extract the CR-effective m a s s m*(E) and the SR-effective g-factor &E) f o r one given laser energy E ( a t a given pressure) according, respectively, to fi E =
BCR
1) B. P. 166X, 25. av. d e s Martyrs, 38042 Grenoble Cgdex, France.
2) On leave of absence from the High P r e s s u r e R e s e a r c h Center Unipress, Poland. 3) On leave of absence f r o m the Institute f o r Experimental Physics, University of Warsaw, Poland.
K13 0
physica status solidi (b) 135
In the above expressions, pB is the Bohr-magneton, B
and B denote the SR CR experimentally determined values of the magnetic field where the CR and the
SR, respectively, occur
(see Fig. 2 and 5 of /I/). W e then define the cal-
culated CR-effective m a s s
m:alc
( ECR) and SR-effective g-factor g* calc(ESR) by
h e B ECR
(3)
7
-
m&lc(ECR)
+ '-
where ECR = E ( l )
i
E(O ) and ESR = E(O-) - E(O+) are the calculated CR- and
SR-energies a t the field B, respectively. Note that the quantities appearing in
:I) to (4) a r e only meaningful for the transitions we are interested in. F o r in-
+
+
stance the cyclotron resonance 1 --* 2 would define another CR-effective m a s s because of the strong non-parabolic character of the conduction band. F o r each p r e s s u r e , the comparison between m*and m
*
*
(and g and g calc calc) a t a given resonance energy is a good test of the theory and of our description of the p r e s s u r e effect on the band s t r u c t u r e as well. Instead of correcting the experimental values of m*(E) and $YE) for nonparabolicity of the conduction band in o r d e r to obtain the band-edge values m*
C
and g s w e indeed choose to directly compare m*(E) and gTE) with m:alc(E)
*
(E), respectively, because, within the framework of the theory of calc nearly degenerate bands of /2/ and /3/, t h e r e is no obvious analytical c o r -
and g
respondence between m*(E) and m,* ( o r &E) and g:).
The good agreement
between experiment and calculation is very apparent on Fig. 1 (2) which shows m*(E) and mcalc(E) (g(E) and gcalc(E)) a t various p r e s s u r e . Except for the cyclotron resonance data a t a n energy of 21.15 meV - which is close to the phonon energies /1/ - the agreement always l i e s within the 3% mentioned above. The slight deviation between calculated and experimental cyclotron effective m a s s e s just below the reststrahlen band is due to the polaron effect /8/ which is not taken into account in the present calculation. A better agreement could
be obtained by modifying the Goodwin and Seiler' s p a r a m e t e r s /5/,
but no such
attempt was made in our work since these p a r a m e t e r s describe accurately a huge amount of magneto-optical data in a much wider range of energy than in our work. T h i s good agreement allows u s to conclude that the three-band
Short Notes
K131 Fig. 1. CR-effective m a s s at T = 4.2 K for different p r e s s u r e s P: (a) 0, (b) 545 MPa, (c) 1 GPa; The points a r e deduced from the experimental data of our previous paper /l/, except for one m o r e point a t P = 0 with the energy E = 27.85 meV. ExperimentaI uncertainties a r e of the o r d e r of 1%, except for the points above the r e s t s t r a h l (hatched zone) where they are of the o r d e r of 2%. The lines are calculated
formulae for m*and g y which, respectively, read /2/ C
C
give the band edge values with a precision of 3% in all our p r e s s u r e regime. (E = 23.2 eU) is related to the interband-momentum 2 p 2 = 2P m/ti (with m for the free electron mass); P the band-parameter F (F = -0.2) arises from the k. 6 interaction of the r con-
In the above formulae, E
p m a t r i x element P /4/ b y E
6
duction band with the higher bands of the s a m e symmetry, while N1 (N = -0.55)
1
comes from the spin-orbit coupling of the
photon c q y i m e V
-----*
Fig. 2. SR-effective g-factor at T = 4.2 K f o r the p r e s s u r e s P: (a) 0, (b) 300 MPa, (c) 545 MPa, (d) 1 GPa. The points again a r e deduced from the experimental points of /1/ with a n accuracy of 1%and the lines a r e calculated
physica s t a t u s solidi (b) 135
K132
'
, -
Fig. 3. Band edge effective m a s s m z (dashed line) and effective g-factor (full line) computed v e r s u s p r e s s u r e using the three-band formulae (5) and
higher bands which have the m e t r y /2/. /
0
r8sym-
Fig. 3 a l s o shows m:and
computed as a function of p r e s s u r e
,, 05 pressure (bfa)
-;o
up to 1 . 2 GPa. They clearly display the g r e a t influence of the s m a l l energy gap /9/: f o r instance at a p r e s -
s u r e of 1 GPa, the effective m a s s is increased by a l m o s t 5%
and the
g-factor (in absolute value) is decreased by the s a m e quantity, while the energy gap is increased by a l m o s t 60%. p r e s s u r e experiments are thus v e r y useful to strongly modify the electronic p r o p e r t i e s of the narrow-gap material. We are glad to thank helpful discussions with Prof. W. Zawadzki whose comments /9/ on o u r previous paper /1/ Two of us, L . D. and M. B.
, have
stimulated t h i s complementary work.
the p l e a s u r e to acknowledge the financial
support they received from the UniversitC Scientifique et MCdicale d e Grenoble. References
/1/ S. HUANT, L . DMOWSKI, M. BAJ, and L . C . BRUNEL, phys, s t a t . sol. 125, 215 (1984). (b) /2/ M.H. WEILER, R . L . AGGARWAL, a n d B . LAX, Phys. Rev. B 17 , 3269 (1978). M.H. WEILER, in: Semiconductors and Semimetals, Vol. 16, E d . R . K . WILLARDSON and A. B. BEER, Academic P r e s s , Inc., 1981 (p. 119). /3/ C . R . PIDGEON and R . N . BROWN, Phys. Rev. -146, 575 (1966). /4/
1, 249 (1957). E.O. KANE, J. Fhys. Chem. Solids -
/5/ M. W. GOODWIN and D. G. SEILER, Phys. Rev. B 27, 3451 (1983). /6/ G, MARTINEZ, in: Handbook of Semiconductors, Vol. 2 , Ed. M. BALKANSKI, North-Holland Publ. Co., 1980, (p. 181). /7/N.
MENUYK, A.S. PINE, J . A . KAFALAS, a n d A . J . STRAUSS, J. appl.
45, 3477 (1974). Phys. /8/ D. B. MC COMBE and R . KAPLAN, Phys. Rev. L e t t e r s 21, 756 (1968). -
/9/ W. ZAWADZKI, private communication.
(Received F e b r u a r y 20, 1986)
