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Submitted on 1 Jan 1981
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PHONON ASSISTED TRANSITIONS IN THE FREE
CARRIER ABSORPTION OF SEMICONDUCTORS
IN QUANTIZING MAGNETIC FIELDS
H. Spector, T. Rynee
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C6, suppZ6ment au n o 12, Tome 42, d6cembre 1981 page c6-670
PHONON ASSISTED TRANSITIONS IN THE FREE CARRIER ABSORPTION OF
SEMICONDUCTORS IN QUANTIZING MAGNETIC FIELDS
H.N. s p e c t o r s and T.M.
n nee*
*
TRW A b s t r a c t.-
We have i n v e s t i g a t e d t h e e f f e c t of t h e s c a t t e r i n g of f r e e c a r r i e r s i n semicon- d u c t o r s by b o t h a c o u s t i c and o p t i c a l phonons on t h e f r e e c a r r i e r a b s o r p t i o n i n q u a n t i z i n g magnetic f i e l d s . W e f i n d t h a t f o r e l e c t r o m a g n e t i c r a d i a t i o n p o l a r i z e d e i t h e r p a r a l l e l o r p e r p e n d i c u l a r t o t h e inagnetic f i e l d , t h e f r e e c a r r i e r a b s o r p t i o n i s an o s c i l l a t o r y f u n c t i o n of t h e magnetic f i e l d . The o r i g i n of t h i s o s c i l l a t o r y magneto-absorption i s due t o phonon a s s i s t e d t r a n s i t i o n s between t h e v a r i o u s Landau l e v e l s of t h e f r e e c a r r i e r s i n q u a n t i z i n g magnetic f i e l d s . From t h e p e r i o d s of t h e o s c i l l a t i o n s i n t h e f r e e c a r r i e r a b s o r p t i o n , t h e f r e q u e n c i e s of t h o s e phonons which i n t e r a c t most s t r o n g l y w i t h t h e f r e e c a r r i e r s can b e found.The a b s o r p t i o n of e l e c t r o m a g n e t i c r a d i a t i o n by f r e e c a r r i e r s i n i n t r a b a n d t r a n s i t i o n s i n semiconductors can only t a k e p l a c e w i t h t h e simultaneous s c a t t e r i n g of t h e c a r r i e r s by i m p e r f e c t i o n s i n t h e c r y s t a l 1 y 2 . Therefore, t h e dependence of
t h e f r e e c a r r i e r a b s o r p t i o n c o e f f i c i e n t on temperature and wavelength of t h e r a d i a - t i o n f i e l d w i l l depend c r i t i c a l l y on t h e s c a t t e r i n g mechanisms which a r e dominant i n a p a r t i c u l a r semiconductor3. I n s t r o n g magnetic f i e l d s , t h e energy l e v e l s of t h e f r e e c a r r i e r s , which form a continuum i n a p a r t i c u l a r band i n t h e absence of t h e f i e l d , a r e q u a n t i z e d i n t o Landau subbands. Because of t h i s q u a n t i z a t i o n , t h e a b s o r p t i o n c o e f f i c i e n t w i l l become a f u n c t i o n of magnetic f i e l d a s w e l l a s photon frequency and temperature. I n t r a b a n d t r a n s i t i o n s can occur n o t only t o s t a t e s i n t h e same Landau subband b u t a l s o t o s t a t e s i n o t h e r Landau subbands i f t h e energy of t h e photon b e i n g absorbed,
h,
i s g r e a t e r t h a n t h e energy s e p a r a t i o n between subbands, which i s hac, where w i s t h e c y c l o t r o n frequency of t h e c a r r i e r s , f o r p a r a b o l i c energy bands. An o s c i l l a t o r y dependence of t h e f r e e c a r r i e r a b s o r p t i o n w i t h magnetic f i e l d i s p r e d i c t e d 4 where t h e a b s o r p t i o n h a s peaks when = nuc. This o s c i l l a t o r y dependence a r i s e s from o s c i l l a t i o n s i n t h e f r e e c a r r i e r s d e n s i t y of s t a t e s whenever t r a n s i t i o n s can t a k e p l a c e t o a new Landau subband.I n most semiconductors, t h e r e i s a range of temperatures and doping l e v e l s f o r which e i t h e r a c o u s t i c phonons o r o p t i c a l phonons w i l l dominate t h e s c a t t e r i n g of t h e f r e e c a r r i e r s . I n t r a b a n d t r a n s i t i o n s can occur w i t h e i t h e r t h e a b s o r p t i o n o r
% T h i s r e s e a r c h was supported i n p a r t by a g r a n t from t h e NSF M a t e r i a l s Research Laboratory Program, Grant No. DMR 76-24466.
emission of a phonon. When this occurs, the peaks in the free carrier absorption coefficient occur when Qfwo = nu 5, where
wo
is the frequency of the phonon being absorbed (plus sign) or emitted (minus sign). Therefore, from the periods of the oscillation of the free carrier absorption in quantizing magnetic fields, the fre- quencies of those phonons, whether acoustic and optical, which dominate the scat- tering of the free carriers can be determined.The free carrier absorption coefficient is given by
where
wiase
is the transition probability given by second order perturbation theory for absorption or emission of a photon,P
7
,
Y
=2n
&I< flM+l i>1
6(~~-~~~162-6;~)4i
fi is the free carrier distribution function, E is the dielectric constant, n the
number of photons in the radiation field and <flM+li> is the transition matrix ele-
-
merit for the electron-photon interaction with the simultaneous absorption or emis- sion of a phonon. Because the form of the interaction Hamiltonian between the radiation field and the carriers depends upon the polarization of the radiation, we find that the free carrier absorption depends not only on the magnitude of the mag- netic field but the orientation of the field relative to the direction of polari- zation of the radiation. For the radiation polarized along the direction of the magnetic field and for either acoustic or optical phonon scattering via deformation potential scattering, the free carrier absorption coefficient is
JOURNAL DE PHYSIQUE
f o r h e a v i l y doped semiconductors where t h e c a r r i e r s a r e degenerate. Here, n*, aT
*
and n2 a r e t h e l a r g e s t i n t e g e r s s a t i s f y i n g t h e c o n d i t i o n s
-
n * ~
Ef 1n * ~
n = n i + L
,
n1-
-%jrq -
and n2 =$+<--
and t h e r e s t of t h e symbolst
Wa r e a s d e f i n e d i n p r e v i o u s work4 r 6 . The frequency wo i s t h e o p t i c a l phonon f r e - quency when o p t i c a l phonon s c a t t e r i n g o r i n t r a v a l l e y s c a t t e r i n g i s important and is t h e frequency of t h e a c o u s t i c phonon which i n t e r a c t s most s t r o n g l y w i t h t h e f r e e c a r r i e r s when a c o u s t i c phonon s c a t t e r i n is dominant. I n very s t r o n g f i e l d s , t h e s e a r e phonons whose wave v e c t o r q % (mwc
.
For b o t h d e g e n e r a t e and nondegenerateTJg
semiconductors, t h e a b s o r p t i o n c o e f f i c i e n t i s an o s c i l l a t o r y f u n c t i o n of magnetic f i e l d w i t h t h e p e r i o d of o s c i l l a t i o n depending upon t h e frequency o f t h e phonons which i n t e r a c t most s t r o n g l y w i t h t h e f i n e c a r r i e r s . However, t h e d e t a i l e d depen- dence of t h e a b s o r p t i o n on f i e l d and t h e amplitude and p r o d i c i t y of t h e o s c i l l a - t i o n s d i f f e r s between d e g e n e r a t e and nondegenerate m a t e r i a l s .
References
1. H. Meyer, Phys. Rev.
112,
298 (1958).2. R. Rosenberg and M. Lox, Phys. Rev.
112,
843 (1958).3. H.Y. Fan, i n "Semiconductors and Semimetals'", e d i t e d by A. Beer and R.Willardson
(Academic P r e s s , NY, 1967) Vo1.3, p.402.
4. E.R. Generazio and H.N. Spector, Phys. Rev. B 20, 5162 (1979).
