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Cyclotron resonance studies of polarons and screening effects in GaAs
Physica 117B & 118B (1983) 649-651 North.Holland Publishing Company
CYCLOTRON
649
RESONANCE
G.Lindemann,
STUDIES
OF P O L A R O N S A N D S C R E E N I N G
W.Seidenbusch,
R.Lassnig,
EFFECTS
J.Edlinger,
IN GaAs
E.Gornik
I n s t i t u t fHr E x p e r i m e n t a l p h y s i k Universit~t Innsbruck AUSTRIA P o l a r o n e f f e c t s are s t u d i e d in bulk n - G a A s by c y c l o t r o n r e s o n a n c e m e a s u r e m e n t s u n d e r hot e l e c t r o n c o n d i t i o n s . G o o d a g r e e m e n t b e t w e e n calc u l a t i o n s and e x p e r i m e n t s is obtained. The F r 6 h l i c h p a r a m e t e r d e t e r m i n e d is c l o s e to the l i t e r a t u r e value. An a c c u r a t e t r e a t m e n t of p o l a r o n e f f e c t s is n e c e s s a r y for the d e t e r m i n a t i o n of the b a n d e d g e e f f e c t i v e mass. In G a A s - G a x A l 1 _ x A s h e t e r o j u n c t i o n s the i n f l u e n c e of the free c a r r i e r c o n c e n t r a t i o n on the c y c l o t r o n r e s o n a n c e l i n e w i d t h is i n v e s t i g a t e d and s c r e e n i n g is found to be ineffective. P o l a r o n e f f e c t s are w e a k e r than d e t e r m i n e d in bulk samples.
INTRODUCTION
a) I n v e s t i g a t i o n s
C y c l o t r o n r e s o n a n c e is a u s e f u l tool to i n v e s t i g a t e p o l a r o n s and high f r e q u e n c y t r a n s p o r t e f f e c t s in GaAs. P o l a r o n effects i n f l u e n c e the e f f e c t i v e m a s s of the elet r o n s and t h e r e f o r e the e n e r g y levels in the c o n d u c t i o n band. F r o m a d e t a i l e d study of c y c l o t r o n r e s o n a n c e spectra, o b s e r v e d u n d e r hot e l e c t r o n c o n d i t i o n s , p o l a r o n e f f e c t s a n d n o n p a r a b o l i c i t y can s e p a r a t e l y be determined. In a t w o - d i m e n s i o n a l (2D) e l e c t r o n gas as p r e s e n t at the G a A s - G a x A l 1 _ x A S interface the DC m o b i l i t y can be v a r i e d by i n c r e a s i n g the free c a r r i e r c o n c e n t r a tion. It is t h e r e f o r e i n t e r e s t i n g to study the i n f l u e n c e of the c a r r i e r conc e n t r a t i o n on the c y c l o t r o n r e s o n a n c e l i n e w i d t h in this system. In a d d i t i o n the e l e c t r o n p h o n o n c o u p l i n g is e x p e c t e d to be w e a k e r in the 2D case.
The b u l k samples u s e d w e r e g r o w n by H . B a u s e r ( M a x - P l a n c k - I n s t i t u t , Stuttgart, l i q u i d phase epitaxy) and had a free c a r r i e r c o n c e n t r a t i o n of n = 1 . 1 0 1 3 cm -3 and an i o n i z e d i m p u r i t y c o n c e n t r a t i o n of N = 8 . 1 0 1 3 cm-3. D e t a i l e d c h a r a c t e r i s t i c s can be f o u n d in ref./3/. In both e m i s s i o n and t r a n s m i s s i o n e x p e r i m e n t s the electrons w e r e h e a t e d up by an a p p l i e d electric field E to o c c u p y s e v e r a l L a n d a u levels. T h e c y c l o t r o n r e s o n a n c e s p e c t r u m splits t h e r e f o r e in up to four lines w i t h i n c r e a s i n g e l e c t r i c f i e l d due to transitions between different Landau levels. The s p l i t t i n g of the c y c l o t r o n r e s o n a n c e line r e f l e c t s the e n e r g y d i f f e r e n c e of d i f f e r e n t L a n d a u level t r a n s i t i o n s for a c e r t a i n m a g n e t i c field. In our exper i m e n t s the t r a n s i t i o n e n e r g y is constant, since it is d e t e r m i n e d by the laser f r e q u e n c y or by the n a r r o w b a n d d e t e c t o r line in the e m i s s i o n e x p e r i ments. T h e r e f o r e d i f f e r e n t L a n d a u level t r a n s i t i o n s o c c u r at d i f f e r e n t m a g n e t i c fields. The l o w e r i n g of the e n e r g y of h i g h L a n d a u t r a n s i t i o n s in r e s p e c t to the fundamental transition (between n=0 and n = 1 , n d e n o t e s L a n d a u level indices) can be e x p l a i n e d by two effects: Nonp a r a b o l i c i t y and e l e c t r o n L O - p h o n o n c o u p l i n g (polarons). The i n f l u e n c e of n o n p a r a b o l i c i t y i n c r e a s e s slowly w i t h i n c r e a s i n g e l e c t r o n energies. In contrast, the r e s o n a n t part of the e l e c t r o n p h o n o n coupling, o b s e r v e d in our e x p e r i ments, is small for small e n e r g i e s but i n c r e a s e s strongly, w h e n the e n e r g y of an e l e c t r o n b e c o m e s c o m p a r a b l e to the e n e r g y of the g r o u n d state plus the o p t i c a l p h o n o n energy. Due to the d i f f e rent e n e r g y d e p e n d e n c e these two e f f e c t s
EXPERIMENTAL
RESULTS
The e x p e r i m e n t s w e r e p e r f o r m e d in emission as w e l l as in t r a n s m i s s i o n . The e x p e r i m e n t a l s y s t e m for the e m i s s i o n e x p e r i m e n t s is d e s c r i b e d in detail in ref./I/. E m i t t e r and d e t e c t o r are immersed in l i q u i d He and p l a c e d in two ind e p e n d e n t m a g n e t i c fields, w h i c h can be t u n e d separately. T h e n a r r o w - b a n d transition b e t w e e n the i m p u r i t y levels Is2p (m=+1) of s h a l l o w i m p u r i t i e s in a m a g n e t i c f i e l d of a GaAs d e t e c t o r is u s e d for the s p e c t r a l a n a l y s i s of the e m i t t e d radiation. The c y c l o t r o n r e s o n a n c e t r a n s m i s s i o n m e a s u r e m e n t s w e r e p e r f o r m e d u s i n g an o p t i c a l l y p u m p e d far i n f r a r e d l a s e r /2/. The f o l l o w i n g w a v e l e n g t h s w e r e used: 96,52 ~ m , 1 1 8 , 8 3 ~ m , 1 3 3 , 1 2 ~m and 170,97 ~m.
0 378-4363/83/0000-0000/$03.00 © 1983 North-Holland
of bulk n - G a A s
G. Lindemann et al. / Cyclotron resonance studies o f polarons
650
.4
can be separated: I) N o n p a r a b o l i c i t y is d e t e r m i n e d from the f u n d a m e n t a l L a n d a u level t r a n s i t i o n a f t e r c a l c u l a t e d p o l a r o n e f f e c t s have been substrac~e~. O •
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Fig.2" Cyclotron resonance line splitting G
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Fig.l: F u n d a m e n t a l c y c l o t r o n r e s o n a n c e e n e r g y and p o l a r o n e n e r g y c o r r e c t i o n Ep as a f u n c t i o n of the m a g n e t i c field; full symbols: e x p e r i m e n t a l l y o b s e r v e d values; open symbols: e x p e r i m e n t a l v a l u e s a f t e r p o l a r o n c o r r e c t i o n s ; full line: calculation including nonparabolicity; d o t t e d line: p o l a r o n c o r r e c t i o n of Eq.1; d a s h e d line: p o l a r o n c o r r e c t i o n using a v a r i a t i o n a l p o l a r o n t h e o r y /4/. The d a s h e d and the d o t t e d lines in Fig.1 are a c o m p a r i s o n of the p o l a r o n e n e r g y correction Ep c a l c u l a t e d u s i n g the approximation Ep=~ c_~=~
~ . ~a-
(I)
and a c o r r e c t v a r i a t i o n a l p o l a r o n t h e o r y /4/. h ~ is the o b s e r v e d c y c l o t r o n reson a n c e e n e r g y i n c l u d i n g p o l a r o n effects, h~c is the c o r r e c t e d e n e r g y ; e d e n o t e s the F r ~ h l i c h p a r a m e t e r . It is e v i d e n t that the a p p r o x i m a t i o n should not be u s e d for h i g h e r m a g n e t i c fields. T h i s e f f e c t has been u n d e r e s t i m a t e d in m o s t of p r e v i o u s w o r k r e s u l t i n g in a too h i g h b a n d e d g e e f f e c t i v e m a s s m o . The full symbols are the e x p e r i m e n t a l data. T h e s e are c o r r e c t e d for p o l a r o n e f f e c t s (open symbols) a n d f i t t e d u s i n g a t h r e e level m o d e l for n o n p a r a b o l i c i t y /5/. W i t h m~ as a f i t t i n g p a r a m e t e r d e t e r m i n e d to m ~ = 0 , 0 6 5 0 ± 0 , 0 0 0 5 m o . The d i f f e r e n c e to the l i t e r a t u r e v a l u e of m * = 0 , 0 6 6 5 mo is m a i n l y due tO d i f f e r e n t P o l a r o n c o r r e c tions. W i t h o u t a p o l a r o n c o r r e c t i o n (full symbols) w e w o u l d get mo=0, m00661 in v e r y close a g r e e m e n t w i t h the lite rature.
as a f u n c t i o n of t r a n s i t i o n energy. D i f f e r e n t symbols c o r r e s p o n d to d i f f e rent t r a n s i t i o n s . The L a n d a u level indices for the d i f f e r e n t t r a n s i t i o n s are indicated. T r a n s m i s s i o n v a l u e s are char a c t e r i z e d by full symbols, e m i s s i o n v a l u e s by open symbols. A l l s p l i t t i n g s are r e f e r e d to the f u n d a m e n t a l t r a n s i t i on (0-I). D a s h e d curve: s p l i t t i n g b e t w e e n (I-2) and (0-I) due to n o n p a r a b o l i c i t y . Full curves: c a l c u l a t i o n s i n c l u d i n g nonp a r a b o l i c i t y and p o l a r o n effects. The stars are the r e s u l t s for the 2D case. 2) The p o l a r o n e f f e c t s are n o w d e t e r m i n e d from the d i f f e r e n c e s in the m a g n e t i c f i e l d p o s i t i o n of h i g h e r L a n d a u level t r a n s i t i o n s in r e s p e c t to the f u n d a m e n tal transition. T h e s e d i f f e r e n c e s , are p l o t t e d in F i g . 2 as a f u n c t i o n of the t r a n s i t i o n energy for several d i f f e r e n t L a n d a u level t r a n s i t i o n s . T h e d a s h e d c u r v e is the c a l c u l a t e d line s p l i t t i n g for t r a n s i t i o n s b e t w e e n the first (n=1) and the second (n=2) L a n d a u level only due to n o n p a r a b o l i c i t y . The d i f f e r e n c e b e t w e e n this c u r v e and the e x p e r i m e n t a l data is due to p o l a r o n effects. T h e full curves are the c a l c u l a t e d s p l i t t i n g s i n c l u d i n g n o n p a r a b o l i c i t y and p o l a r o n e f f e c t s w h i c h are c a l c u l a t e d u s i n g a v a r i a t i o n a l t h e o r y /4/. The F r ~ h l i c h c o u p l i n g p a r a m e t e r e n e c e s s a r y for the best fit is ~=0,080, w h i c h is in good a g r e e m e n t w i t h the v a l u e found in literature ~=0,068 /6/. The fact that all d i f f e r e n t t r a n s i t i o n s are in g o o d agreem e n t w i t h the c a l c u l a t i o n s u s i n g the same c o u p l i n g c o n s t a n t is a proof, that our e x p e r i m e n t s are c o r r e c t l y i n t e r p r e ted.
b)
2D G a A s - G a x A l 1 _ x A S
heterojunctions
F o r t h e s e i n v e s t i g a t i o n s two MBE g r o w n samples s u p p l i e d by A . C . G o s s a r d (Bell
G. Lindemann et aL / Cyclotron resonance studies o f polarons
Labs.) w e r e used. Sample 1 h a d a DC m o b i lity of ~ = 1 8 0 0 0 0 c m 2 / V s for a c h a n n e l c a r r i e r c o n c e n t r a t i o n of n s = 3 , 6 . 1 0 1 1 c m - 2 at 4,2 K, sample 2 ~ = 1 2 0 0 0 0 c m 2 / V s n s = 4 , 1 . 1 0 1 1 c m - 2 r e s p e c t i v e l y . The DC mob i l i t y can be i n c r e a s e d by i n c r e a s i n g n s t h r o u g h b a n d g a p i l l u m i n a t i o n by a f a c t o r of 2 /7/. In Fig.3 e x p e r i m e n t a l c y c l o t r o n r e s o n a n c e t r a n s m i s s i o n s p e c t r a are shown for two l a s e r w a v e l e n g t h s w i t h o u t c a r r i e r h e a t i n g t h r o u g h an e x t e r n a l e l e c t r i c field. The d i f f e r e n c e b e t w e e n the d a s h e d and full c u r v e s d e m o n s t r a t e s the influence of i n c r e a s i n g n s from 3 , 6 . 1 0 1 1 c m -2 to 6 , 5 . 1 0 1 1 c m - 2 . F r o m the p o s i t i o n of the F e r m i level ( d e t e r m i n e d by SdH o s c i l lations) we a s s i g n the o b s e r v e d line for the lower n s v a l u e to t r a n s i t i o n s b e t w e e n L a n d a u level n=0 to n=1. F o r the h i g h e r n s v a l u e the F e r m i level increases, the line shifts to h i g h e r m a g n e t i c fields and is a s s i g n e d to n=1 to n=2 t r a n s i t i o n s . F r o m F i g . 3 it is e v i d e n t that the lines are s h i f t e d from the b u l k v a l u e i n d i c a t e d by an arrow. T h i s is m a i n l y due to an i n c r e a s e d o o n t r i b u t i o n of n o n p a r a b o l i c i t y since the lowest s u b b a n d is l o c a t e d a b o u t 50meV a b o v e the bandedge. The q u e s t i o n a r i s e s w h e t h e r p o l a r o n effects are p r e s e n t in the 2 D - c a s e in the same way as in the bulk. To a n s w e r this q u e s t i o n we have p l o t t e d the d i f f e r e n c e of the m a g n e t i c f i e l d p o s i t i o n s of the two p e a k s in c o m p a r i s o n w i t h the b u l k results in Fig.2 for two w a v e l e n g t h s . It is e v i d e n t that the s p l i t t i n g is c o n s i d e r a b ly s m a l l e r than in the bulk for the same t r a n s i t i o n s . The s m a l l e r s p l i t t i n g can only be e x p l a i n e d by w e a k e r p o l a r o n effects. T h i s is the first e v i d e n c e for a r e d u c t i o n of the e l e c t r o n L O - p h o n o n coupling due to a c o m p l e t e f i l l i n g of the g r o u n d level in the 2 D - e l e c t r o n system. F o r the two s a m p l e s i n v e s t i g a t e d the l i n e w i d t h s d e r i v e d from the s p e c t r a are: sample 1: ~ = 2 , 4 c m -I for 118 ~m ~w = 2 , 1 c m -I for 96 ~m sample 2: ~ 3,0cm -I for 118 ~m ~w = 2 , 4 c m -I for 96 ~m. The l i n e w i d t h d e c r e a s e s w i t h i n c r e a s i n g m a g n e t i c f i e l d by a b o u t 15%. The linew i d t h is a m e a s u r e for the s t r e n g t h of the d o m i n a n t s c a t t e r i n g process. A comp a r i s o n w i t h l i n e w i d t h s t u d i e s on bulk s a m p l e s /3/ e n a b l e s us to d e t e r m i n e an equivalent effective ionized impurity c o n c e n t r a t i o n in the c h a n n e l range. We d e r i v e for sample I N = 5 . 1 0 1 4 c m -3 and for s a m p l e 2 N = 9 . 1 0 1 4 c m -3. T h i s v a l u e s indicate that the l i n e w i d t h is m a i n l y determ i n e d by the r e s i d u a l d o p i n g of the bulk GaAs layer. The o b s e r v e d l i n e w i d t h diff e r e n c e of the two s a m p l e s c o r r e l a t e w i t h the m e a s u r e d DC m o b i l i t y d i f f e r e n cies. T h e i n c r e a s e of the DC m o b i l i t y w i t h i n c r e a s i n g n s is e x p l a i n e d by a s c r e e n i n g of the i o n i z e d i m p u r i t i e s by
651
the a d d i t i o n a l free c a r r i e r s /7/. In the c y c l o t r o n r e s o n a n c e e x p e r i m e n t s no effect of the free c a r r i e r c o n c e n t r a t i o n on the l i n e w i d t h is o b s e r v e d as e v i d e n t from Fig.3. That means, s c r e e n i n g influences d r a s t i c a l l y the DC m o b i l i t y , but does not i n f l u e n c e the c y c l o t r o n reson a n c e linewidth, w h i c h is c o n n e c t e d w i t h the i m a g i n a r y part of the high f r e q u e n c y conductivity. _J oz Z L9 Go
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Fi9.3: L a s e r t r a n s m i s s i o n s p e c t r a for two w a v e l e n g t h s b e f o r e (full curve) and a f t e r (dashed curve) b a n d g a p i l l u m i n a t i on; the f u n d a m e n t a l c y c l o t r o n r e s o n a n c e t r a n s i t i o n for bulk m a t e r i a l are indid a t e d by arrows. ACKNOWLEDGMENTS T h i s w o r k was s p o n s o r e d by the F o n d s zur F 6 r d e r u n g der w i s s e n s c h a f t l i c h e n F o r ~ schung, A U S T R I A (Project S22/05). We w a n t to t h a n k W . Z a w a d z k i for h e l p f u l discussions. REFERENCES /I/ E.Gornik, J.of M a g n e t i s m and M a g n e t i c M a t e r i a l s 11, 39 (1979) /2/ J . H e p p n e r , C . O . W e i s s , U . H U b n e r and G.Schinn, IEEE J.of Q u a n t u m Elect r o n i c s QE-16, 4, 392 (1980) /3/ G . L i n d e m a n n , E.Gornik, R . S c h a w a r z , D . C . T s u i , Inst. Phys. Conf. Ser.56, p.631 (1980) /4/ D . M . L a r s e n in " P o l a r o n s in Ionic C r y s t a l s and P o l a r S e m i c o n d u c t o r s " e d i t e d by J . T . D e v r e e s e , N o r t h H o l l a n d P u b l . C o m p . , L o n d o n 1972 /5/ W . Z a w a d z k i , L e c t u r e N o t e s in P h y s i c s 133, p.85 (1979) /6/ F . C . B r o w n in "Polarons and E x i t o n s " e d i t e d by C . G . K u p e r and G . D . W h i t f i e l d O l i v e r and Boyd, L o n d o n 1962, p.323 /7/ H . L . S t 6 r m e r , A . C . G o s s a r d , W . W i e g m a n n and K . B a l d w i n , Appl. Phys. Lett.39, 912 (1981)
CYCLOTRON
649
RESONANCE
G.Lindemann,
STUDIES
OF P O L A R O N S A N D S C R E E N I N G
W.Seidenbusch,
R.Lassnig,
EFFECTS
J.Edlinger,
IN GaAs
E.Gornik
I n s t i t u t fHr E x p e r i m e n t a l p h y s i k Universit~t Innsbruck AUSTRIA P o l a r o n e f f e c t s are s t u d i e d in bulk n - G a A s by c y c l o t r o n r e s o n a n c e m e a s u r e m e n t s u n d e r hot e l e c t r o n c o n d i t i o n s . G o o d a g r e e m e n t b e t w e e n calc u l a t i o n s and e x p e r i m e n t s is obtained. The F r 6 h l i c h p a r a m e t e r d e t e r m i n e d is c l o s e to the l i t e r a t u r e value. An a c c u r a t e t r e a t m e n t of p o l a r o n e f f e c t s is n e c e s s a r y for the d e t e r m i n a t i o n of the b a n d e d g e e f f e c t i v e mass. In G a A s - G a x A l 1 _ x A s h e t e r o j u n c t i o n s the i n f l u e n c e of the free c a r r i e r c o n c e n t r a t i o n on the c y c l o t r o n r e s o n a n c e l i n e w i d t h is i n v e s t i g a t e d and s c r e e n i n g is found to be ineffective. P o l a r o n e f f e c t s are w e a k e r than d e t e r m i n e d in bulk samples.
INTRODUCTION
a) I n v e s t i g a t i o n s
C y c l o t r o n r e s o n a n c e is a u s e f u l tool to i n v e s t i g a t e p o l a r o n s and high f r e q u e n c y t r a n s p o r t e f f e c t s in GaAs. P o l a r o n effects i n f l u e n c e the e f f e c t i v e m a s s of the elet r o n s and t h e r e f o r e the e n e r g y levels in the c o n d u c t i o n band. F r o m a d e t a i l e d study of c y c l o t r o n r e s o n a n c e spectra, o b s e r v e d u n d e r hot e l e c t r o n c o n d i t i o n s , p o l a r o n e f f e c t s a n d n o n p a r a b o l i c i t y can s e p a r a t e l y be determined. In a t w o - d i m e n s i o n a l (2D) e l e c t r o n gas as p r e s e n t at the G a A s - G a x A l 1 _ x A S interface the DC m o b i l i t y can be v a r i e d by i n c r e a s i n g the free c a r r i e r c o n c e n t r a tion. It is t h e r e f o r e i n t e r e s t i n g to study the i n f l u e n c e of the c a r r i e r conc e n t r a t i o n on the c y c l o t r o n r e s o n a n c e l i n e w i d t h in this system. In a d d i t i o n the e l e c t r o n p h o n o n c o u p l i n g is e x p e c t e d to be w e a k e r in the 2D case.
The b u l k samples u s e d w e r e g r o w n by H . B a u s e r ( M a x - P l a n c k - I n s t i t u t , Stuttgart, l i q u i d phase epitaxy) and had a free c a r r i e r c o n c e n t r a t i o n of n = 1 . 1 0 1 3 cm -3 and an i o n i z e d i m p u r i t y c o n c e n t r a t i o n of N = 8 . 1 0 1 3 cm-3. D e t a i l e d c h a r a c t e r i s t i c s can be f o u n d in ref./3/. In both e m i s s i o n and t r a n s m i s s i o n e x p e r i m e n t s the electrons w e r e h e a t e d up by an a p p l i e d electric field E to o c c u p y s e v e r a l L a n d a u levels. T h e c y c l o t r o n r e s o n a n c e s p e c t r u m splits t h e r e f o r e in up to four lines w i t h i n c r e a s i n g e l e c t r i c f i e l d due to transitions between different Landau levels. The s p l i t t i n g of the c y c l o t r o n r e s o n a n c e line r e f l e c t s the e n e r g y d i f f e r e n c e of d i f f e r e n t L a n d a u level t r a n s i t i o n s for a c e r t a i n m a g n e t i c field. In our exper i m e n t s the t r a n s i t i o n e n e r g y is constant, since it is d e t e r m i n e d by the laser f r e q u e n c y or by the n a r r o w b a n d d e t e c t o r line in the e m i s s i o n e x p e r i ments. T h e r e f o r e d i f f e r e n t L a n d a u level t r a n s i t i o n s o c c u r at d i f f e r e n t m a g n e t i c fields. The l o w e r i n g of the e n e r g y of h i g h L a n d a u t r a n s i t i o n s in r e s p e c t to the fundamental transition (between n=0 and n = 1 , n d e n o t e s L a n d a u level indices) can be e x p l a i n e d by two effects: Nonp a r a b o l i c i t y and e l e c t r o n L O - p h o n o n c o u p l i n g (polarons). The i n f l u e n c e of n o n p a r a b o l i c i t y i n c r e a s e s slowly w i t h i n c r e a s i n g e l e c t r o n energies. In contrast, the r e s o n a n t part of the e l e c t r o n p h o n o n coupling, o b s e r v e d in our e x p e r i ments, is small for small e n e r g i e s but i n c r e a s e s strongly, w h e n the e n e r g y of an e l e c t r o n b e c o m e s c o m p a r a b l e to the e n e r g y of the g r o u n d state plus the o p t i c a l p h o n o n energy. Due to the d i f f e rent e n e r g y d e p e n d e n c e these two e f f e c t s
EXPERIMENTAL
RESULTS
The e x p e r i m e n t s w e r e p e r f o r m e d in emission as w e l l as in t r a n s m i s s i o n . The e x p e r i m e n t a l s y s t e m for the e m i s s i o n e x p e r i m e n t s is d e s c r i b e d in detail in ref./I/. E m i t t e r and d e t e c t o r are immersed in l i q u i d He and p l a c e d in two ind e p e n d e n t m a g n e t i c fields, w h i c h can be t u n e d separately. T h e n a r r o w - b a n d transition b e t w e e n the i m p u r i t y levels Is2p (m=+1) of s h a l l o w i m p u r i t i e s in a m a g n e t i c f i e l d of a GaAs d e t e c t o r is u s e d for the s p e c t r a l a n a l y s i s of the e m i t t e d radiation. The c y c l o t r o n r e s o n a n c e t r a n s m i s s i o n m e a s u r e m e n t s w e r e p e r f o r m e d u s i n g an o p t i c a l l y p u m p e d far i n f r a r e d l a s e r /2/. The f o l l o w i n g w a v e l e n g t h s w e r e used: 96,52 ~ m , 1 1 8 , 8 3 ~ m , 1 3 3 , 1 2 ~m and 170,97 ~m.
0 378-4363/83/0000-0000/$03.00 © 1983 North-Holland
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G. Lindemann et al. / Cyclotron resonance studies o f polarons
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can be separated: I) N o n p a r a b o l i c i t y is d e t e r m i n e d from the f u n d a m e n t a l L a n d a u level t r a n s i t i o n a f t e r c a l c u l a t e d p o l a r o n e f f e c t s have been substrac~e~. O •
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Fig.l: F u n d a m e n t a l c y c l o t r o n r e s o n a n c e e n e r g y and p o l a r o n e n e r g y c o r r e c t i o n Ep as a f u n c t i o n of the m a g n e t i c field; full symbols: e x p e r i m e n t a l l y o b s e r v e d values; open symbols: e x p e r i m e n t a l v a l u e s a f t e r p o l a r o n c o r r e c t i o n s ; full line: calculation including nonparabolicity; d o t t e d line: p o l a r o n c o r r e c t i o n of Eq.1; d a s h e d line: p o l a r o n c o r r e c t i o n using a v a r i a t i o n a l p o l a r o n t h e o r y /4/. The d a s h e d and the d o t t e d lines in Fig.1 are a c o m p a r i s o n of the p o l a r o n e n e r g y correction Ep c a l c u l a t e d u s i n g the approximation Ep=~ c_~=~
~ . ~a-
(I)
and a c o r r e c t v a r i a t i o n a l p o l a r o n t h e o r y /4/. h ~ is the o b s e r v e d c y c l o t r o n reson a n c e e n e r g y i n c l u d i n g p o l a r o n effects, h~c is the c o r r e c t e d e n e r g y ; e d e n o t e s the F r ~ h l i c h p a r a m e t e r . It is e v i d e n t that the a p p r o x i m a t i o n should not be u s e d for h i g h e r m a g n e t i c fields. T h i s e f f e c t has been u n d e r e s t i m a t e d in m o s t of p r e v i o u s w o r k r e s u l t i n g in a too h i g h b a n d e d g e e f f e c t i v e m a s s m o . The full symbols are the e x p e r i m e n t a l data. T h e s e are c o r r e c t e d for p o l a r o n e f f e c t s (open symbols) a n d f i t t e d u s i n g a t h r e e level m o d e l for n o n p a r a b o l i c i t y /5/. W i t h m~ as a f i t t i n g p a r a m e t e r d e t e r m i n e d to m ~ = 0 , 0 6 5 0 ± 0 , 0 0 0 5 m o . The d i f f e r e n c e to the l i t e r a t u r e v a l u e of m * = 0 , 0 6 6 5 mo is m a i n l y due tO d i f f e r e n t P o l a r o n c o r r e c tions. W i t h o u t a p o l a r o n c o r r e c t i o n (full symbols) w e w o u l d get mo=0, m00661 in v e r y close a g r e e m e n t w i t h the lite rature.
as a f u n c t i o n of t r a n s i t i o n energy. D i f f e r e n t symbols c o r r e s p o n d to d i f f e rent t r a n s i t i o n s . The L a n d a u level indices for the d i f f e r e n t t r a n s i t i o n s are indicated. T r a n s m i s s i o n v a l u e s are char a c t e r i z e d by full symbols, e m i s s i o n v a l u e s by open symbols. A l l s p l i t t i n g s are r e f e r e d to the f u n d a m e n t a l t r a n s i t i on (0-I). D a s h e d curve: s p l i t t i n g b e t w e e n (I-2) and (0-I) due to n o n p a r a b o l i c i t y . Full curves: c a l c u l a t i o n s i n c l u d i n g nonp a r a b o l i c i t y and p o l a r o n effects. The stars are the r e s u l t s for the 2D case. 2) The p o l a r o n e f f e c t s are n o w d e t e r m i n e d from the d i f f e r e n c e s in the m a g n e t i c f i e l d p o s i t i o n of h i g h e r L a n d a u level t r a n s i t i o n s in r e s p e c t to the f u n d a m e n tal transition. T h e s e d i f f e r e n c e s , are p l o t t e d in F i g . 2 as a f u n c t i o n of the t r a n s i t i o n energy for several d i f f e r e n t L a n d a u level t r a n s i t i o n s . T h e d a s h e d c u r v e is the c a l c u l a t e d line s p l i t t i n g for t r a n s i t i o n s b e t w e e n the first (n=1) and the second (n=2) L a n d a u level only due to n o n p a r a b o l i c i t y . The d i f f e r e n c e b e t w e e n this c u r v e and the e x p e r i m e n t a l data is due to p o l a r o n effects. T h e full curves are the c a l c u l a t e d s p l i t t i n g s i n c l u d i n g n o n p a r a b o l i c i t y and p o l a r o n e f f e c t s w h i c h are c a l c u l a t e d u s i n g a v a r i a t i o n a l t h e o r y /4/. The F r ~ h l i c h c o u p l i n g p a r a m e t e r e n e c e s s a r y for the best fit is ~=0,080, w h i c h is in good a g r e e m e n t w i t h the v a l u e found in literature ~=0,068 /6/. The fact that all d i f f e r e n t t r a n s i t i o n s are in g o o d agreem e n t w i t h the c a l c u l a t i o n s u s i n g the same c o u p l i n g c o n s t a n t is a proof, that our e x p e r i m e n t s are c o r r e c t l y i n t e r p r e ted.
b)
2D G a A s - G a x A l 1 _ x A S
heterojunctions
F o r t h e s e i n v e s t i g a t i o n s two MBE g r o w n samples s u p p l i e d by A . C . G o s s a r d (Bell
G. Lindemann et aL / Cyclotron resonance studies o f polarons
Labs.) w e r e used. Sample 1 h a d a DC m o b i lity of ~ = 1 8 0 0 0 0 c m 2 / V s for a c h a n n e l c a r r i e r c o n c e n t r a t i o n of n s = 3 , 6 . 1 0 1 1 c m - 2 at 4,2 K, sample 2 ~ = 1 2 0 0 0 0 c m 2 / V s n s = 4 , 1 . 1 0 1 1 c m - 2 r e s p e c t i v e l y . The DC mob i l i t y can be i n c r e a s e d by i n c r e a s i n g n s t h r o u g h b a n d g a p i l l u m i n a t i o n by a f a c t o r of 2 /7/. In Fig.3 e x p e r i m e n t a l c y c l o t r o n r e s o n a n c e t r a n s m i s s i o n s p e c t r a are shown for two l a s e r w a v e l e n g t h s w i t h o u t c a r r i e r h e a t i n g t h r o u g h an e x t e r n a l e l e c t r i c field. The d i f f e r e n c e b e t w e e n the d a s h e d and full c u r v e s d e m o n s t r a t e s the influence of i n c r e a s i n g n s from 3 , 6 . 1 0 1 1 c m -2 to 6 , 5 . 1 0 1 1 c m - 2 . F r o m the p o s i t i o n of the F e r m i level ( d e t e r m i n e d by SdH o s c i l lations) we a s s i g n the o b s e r v e d line for the lower n s v a l u e to t r a n s i t i o n s b e t w e e n L a n d a u level n=0 to n=1. F o r the h i g h e r n s v a l u e the F e r m i level increases, the line shifts to h i g h e r m a g n e t i c fields and is a s s i g n e d to n=1 to n=2 t r a n s i t i o n s . F r o m F i g . 3 it is e v i d e n t that the lines are s h i f t e d from the b u l k v a l u e i n d i c a t e d by an arrow. T h i s is m a i n l y due to an i n c r e a s e d o o n t r i b u t i o n of n o n p a r a b o l i c i t y since the lowest s u b b a n d is l o c a t e d a b o u t 50meV a b o v e the bandedge. The q u e s t i o n a r i s e s w h e t h e r p o l a r o n effects are p r e s e n t in the 2 D - c a s e in the same way as in the bulk. To a n s w e r this q u e s t i o n we have p l o t t e d the d i f f e r e n c e of the m a g n e t i c f i e l d p o s i t i o n s of the two p e a k s in c o m p a r i s o n w i t h the b u l k results in Fig.2 for two w a v e l e n g t h s . It is e v i d e n t that the s p l i t t i n g is c o n s i d e r a b ly s m a l l e r than in the bulk for the same t r a n s i t i o n s . The s m a l l e r s p l i t t i n g can only be e x p l a i n e d by w e a k e r p o l a r o n effects. T h i s is the first e v i d e n c e for a r e d u c t i o n of the e l e c t r o n L O - p h o n o n coupling due to a c o m p l e t e f i l l i n g of the g r o u n d level in the 2 D - e l e c t r o n system. F o r the two s a m p l e s i n v e s t i g a t e d the l i n e w i d t h s d e r i v e d from the s p e c t r a are: sample 1: ~ = 2 , 4 c m -I for 118 ~m ~w = 2 , 1 c m -I for 96 ~m sample 2: ~ 3,0cm -I for 118 ~m ~w = 2 , 4 c m -I for 96 ~m. The l i n e w i d t h d e c r e a s e s w i t h i n c r e a s i n g m a g n e t i c f i e l d by a b o u t 15%. The linew i d t h is a m e a s u r e for the s t r e n g t h of the d o m i n a n t s c a t t e r i n g process. A comp a r i s o n w i t h l i n e w i d t h s t u d i e s on bulk s a m p l e s /3/ e n a b l e s us to d e t e r m i n e an equivalent effective ionized impurity c o n c e n t r a t i o n in the c h a n n e l range. We d e r i v e for sample I N = 5 . 1 0 1 4 c m -3 and for s a m p l e 2 N = 9 . 1 0 1 4 c m -3. T h i s v a l u e s indicate that the l i n e w i d t h is m a i n l y determ i n e d by the r e s i d u a l d o p i n g of the bulk GaAs layer. The o b s e r v e d l i n e w i d t h diff e r e n c e of the two s a m p l e s c o r r e l a t e w i t h the m e a s u r e d DC m o b i l i t y d i f f e r e n cies. T h e i n c r e a s e of the DC m o b i l i t y w i t h i n c r e a s i n g n s is e x p l a i n e d by a s c r e e n i n g of the i o n i z e d i m p u r i t i e s by
651
the a d d i t i o n a l free c a r r i e r s /7/. In the c y c l o t r o n r e s o n a n c e e x p e r i m e n t s no effect of the free c a r r i e r c o n c e n t r a t i o n on the l i n e w i d t h is o b s e r v e d as e v i d e n t from Fig.3. That means, s c r e e n i n g influences d r a s t i c a l l y the DC m o b i l i t y , but does not i n f l u e n c e the c y c l o t r o n reson a n c e linewidth, w h i c h is c o n n e c t e d w i t h the i m a g i n a r y part of the high f r e q u e n c y conductivity. _J oz Z L9 Go
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Fi9.3: L a s e r t r a n s m i s s i o n s p e c t r a for two w a v e l e n g t h s b e f o r e (full curve) and a f t e r (dashed curve) b a n d g a p i l l u m i n a t i on; the f u n d a m e n t a l c y c l o t r o n r e s o n a n c e t r a n s i t i o n for bulk m a t e r i a l are indid a t e d by arrows. ACKNOWLEDGMENTS T h i s w o r k was s p o n s o r e d by the F o n d s zur F 6 r d e r u n g der w i s s e n s c h a f t l i c h e n F o r ~ schung, A U S T R I A (Project S22/05). We w a n t to t h a n k W . Z a w a d z k i for h e l p f u l discussions. REFERENCES /I/ E.Gornik, J.of M a g n e t i s m and M a g n e t i c M a t e r i a l s 11, 39 (1979) /2/ J . H e p p n e r , C . O . W e i s s , U . H U b n e r and G.Schinn, IEEE J.of Q u a n t u m Elect r o n i c s QE-16, 4, 392 (1980) /3/ G . L i n d e m a n n , E.Gornik, R . S c h a w a r z , D . C . T s u i , Inst. Phys. Conf. Ser.56, p.631 (1980) /4/ D . M . L a r s e n in " P o l a r o n s in Ionic C r y s t a l s and P o l a r S e m i c o n d u c t o r s " e d i t e d by J . T . D e v r e e s e , N o r t h H o l l a n d P u b l . C o m p . , L o n d o n 1972 /5/ W . Z a w a d z k i , L e c t u r e N o t e s in P h y s i c s 133, p.85 (1979) /6/ F . C . B r o w n in "Polarons and E x i t o n s " e d i t e d by C . G . K u p e r and G . D . W h i t f i e l d O l i v e r and Boyd, L o n d o n 1962, p.323 /7/ H . L . S t 6 r m e r , A . C . G o s s a r d , W . W i e g m a n n and K . B a l d w i n , Appl. Phys. Lett.39, 912 (1981)