- Email: [email protected]
Temperature dependence of photoreflectance line shapes in GaAsAlGaAs multiple quantum wells
Superlattices and Microstructures, VoL 3, No. 3, 1987
TEMPERATURE DEPENDENCE IN G a A s / A I G a A s
235
OF P H O T O R E F L E C T A N C E LINE MULTIPLE QUANTUM WELLS
SHAPES
O.J. G l e m b o c k i and B.V. S h a n a b r o o k U.S. N a v a l R e s e a r c h L a b o r a t o r y Washington, DC 20375, U S A Received
August
17,
1986
The t e m p e r a t u r e d e p e n d e n c e of the p h o t o r e f l e c t a n c e (PR) line s h a p e of the e x c i t o n i c t r a n s i t i o n s in GaAs/AIGaAs multiple quantum wells has b e e n studied. At low t e m p e r a t u r e s , near 6K, we f i n d that the PR lines can be d e s c r i b e d by a d i e l e c tric f u n c t i o n of a L o r e n t z i a n o s c i l l a t o r . At h i g h e r t e m p e r a tures, we have o b s e r v e d the evolution of the dielectric function from one w i t h a L o r e n t z i a n a b s o r p t i o n p r o f i l e to one w i t h a G a u s s i a n absorption profile. The dielectric function of excitonic absorptions is e x p e c t e d to e x h i b i t this b e h a v i o r in c a s e s of w e a k e x c i t o n - p h o n o n coupling.
The o p t i c a l m o d u l a t i o n t e c h n i q u e of photoreflectance has provided a simple and a c c u r a t e method for obtaining the energies of c o n f i n e d s t a t e s in such systems as m u l t i p l e q u a n t u m w e l l s (MQW), *,2 modulation doped heterojunctions* and "NIPI" structures. 3 The ability to obtain accurate values for transition e n e r g i e s is made possible by the fact that the p h o t o e x c i t e d c a r r i e r s lead to a changing surface electric field. Therefore, the electroreflectance t h e o r y of A s p n e s can be a p p l i e d to o b t a i n the transition energies. 4 In the case of low e l e c t r i c fields, the f i e l d - i n d u c e d c h a n g e in the d i e l e c t r i c f u n c t i o n is g i v e n by: A~ = AE,
+ i,5£2 - e *e (E - E,
+ it)-"
(I)
where E is the p h o t o n energy, Eg , is the g a p energy, and [ is a L o r e n t z i a n broadening p a r a m e t e r . The c r i t i c a l p o i n t type d e t e r m i n e s n, w h i c h is n=3 for 2D critical points and n=5/2 for 3D c r i t i c a l points. In Eq. (i) , the phase factor G depends u p o n the c r i t i c a l p o i n t type and u p o n f a c t o r s such as nonuniform fields and i n t e r f e r e n c e p h e n o m e n o n . 4 For 2D and 3D c r i t i c a l points, Eq. (i) is r e l a t e d to the third derivative with respect to energy of the unperturbed dielectric function. In the case of MQW, the c o n f i n e m e n t of c a r r i e r s in the well m a t e r i a l leads to strong excitonic resonances in the a b s o r p t i o n profile, an e f f e c t w h i c h persists e v e n at r o o m t e m p e r a t u r e , s We h a v e r e c e n t l y c o m p a r e d PR to p h o t o l u m i n e s c e n c e
0749-6036/87/030235 + 04 $02.00/0
excitation spectroscopy and f o u n d that the t r a n s i t i o n s o b s e r v e d in PR are excitonic up to 250K. 6 For excitons, the e l e c t r i c f i e l d m o d u l a t e d d i e l e c t r i c function is g i v e n by Eq. (i) w i t h n=2. This c o r r e s p o n d s to a first d e r i v a t i v e 4 of w i t h r e s p e c t to e i t h e r the e x c i t o n e n e r g y gap, Eg , or w i t h r e s p e c t to the exciton l i f e t i m e (~i/[). In this case, the p h a s e factor, @, does not d e p e n d u p o n the c r i t ical point type and in the a b s e n c e of nonuniform fields and inteference effects, @ can be u s e d to d e t e r m i n e the relative contributions to A£ from the exciton energy g a p and l i f e t i m e m o d u l a tion m e c h a n i s m s . 6 In addition, in MQW the total integrated intensity of the t r a n s i t i o n can also be m o d u l a t e d by the field, resulting in an n=l c o n t r i b u t i o n from Eq. (i) to the PR signal. 6 The form of the d i e l e c t r i c function u s e d to o b t a i n Eq. (i) a s s u m e s L o r e n t z i a n b r o a d e n i n g . However, it is k n o w n from the theoretical w o r k of T o y o z a w a that in the p r e s e n c e of strong exciton-phonon coupling or w e a k c o u p l i n g at h i g h t e m p e r a tures the form of the a b s o r p t i o n profile can c h a n g e f r o m L o r e n t z i a n to G a u s s i a n . 7 In the case of M Q W there is some experimental evidence of this p h e n o m e n o n in the room temperature experiments of Chemla et al.' who o b s e r v e d that for 100A MQW a Gaussian line shape is r e q u i r e d to d e s c r i b e the a b s o r p t i o n data. Furthermore, r e c e n t PR m e a s u r e m e n t s f o u n d that n=3 (3rd d e r i v a t i v e of 2D c r i t i c a l point) in Eq. (i) fits better to room temperature data than the e x p e c t e d n=2
© 1987 Academic Press Inc. (London) Limited
Superlattices and Microstructures, Vo/. 3, No. 3, 1987
236
llH T = 6K %
I
0.2
I ...~
I !
11 L
¢. 0
400 >l--
DONOR RELATED -0.2 DATA
....
GAUSSIAN
LORENTZIAN
Z "' I-Z m
........
-0.4
e," l:*..
-0.6
I 1.522
I 1.524
I 1.526
I 1.528
ENERGY (eV) Figure I. Comparison of 6K PR data (dotted line) to &a, calculated from dielectric functions having Lorentzian
(solid) a n d profiles.
Gaussian
(dashed)
absorption
(exciton). ~ These facts suggest a break down of Eq. (i) at r o o m t e m p e r a t u r e . In o r d e r to b e t t e r u n d e r s t a n d exciton lines in MQW, we h a v e c o m p a r e d at v a r i o u s t e m peratures, PR l i n e s h a p e s to Aa calculated from excitonic dielectric functions having Gaussian and Lorentzian absorption profiles. Henceforth, we will u s e the terms Lorentzian and Gaussian in referr i n g to t h e s e d i e l e c t r i c functions. The MQW sample u s e d in o u r e x p e r iments had barrier and well widths of 150A and 200A respectively. The total t h i c k n e s s of the Q W m a t e r i a l w a s 2~. In o r d e r to r e d u c e the s u r f a c e f i e l d s in the v i c i n i t y of the q u a n t u m w e l l s , the s a m p l e was c l a d w i t h 3 0 0 0 A of A I G a A s . Measurem e n t s w e r e p e r f o r m e d at v a r i o u s temperat u r e s b e t w e e n 6K a n d 250K. For the p r o b e beam (reflectance light), we used the radiation f r o m a CW d y e l a s e r , p u m p e d by an Ar i o n l a s e r . The modulation beam was c o m p o s e d of the b l u e - g r e e n l i g h t f r o m the Ar i o n l a s e r a n d w a s c h o p p e d at 1 7 0 0 Hz. Neutral density filters were used to r e d u c e the i n t e n s i t i e s of the probe and pump b e a m s . T h i s a p p a r a t u s a l l o w e d us to o b t a i n PR s p e c t r a w i t h a r e s o l u t i o n near 0 . 1 5 meV. The r e s u l t s of our m e a s u r e m e n t s are s h o w n in F i g s . (1)-(3) in w h i c h we plot the data (dotted line) a n d f i t s to the data using Lorentzian (solid line) and Gaussian (dashed line) profiles. The
s p e c t r a l f e a t u r e s of i n t e r e s t to us are the two lowest l y i n g p a r i t y a l l o w e d QW transitions. In o u r notation, the two numbers correspond to the q u a n t u m i n d i c e s of the c o n d u c t i o n and valence subbands, respectively, and the letter denotes w h e t h e r the v a l e n c e s t a t e is of h e a v y or light hole character. The Lorentzian l i n e w a s o b t a i n e d f r o m Eq. (I) w i t h n=2, while the "Gaussian" l i n e is the f i r s t derivative of an excitonic dielectric f u n c t i o n c a l c u l a t e d u s i n g the c o n v o l u t i o n f o r m a l i s m I° a n d G a u s s i a n b r o a d e n i n g . 7 in the l a t e r case, the f u n c t i o n a l f o r m of ~ is the product of a Gaussian with a degenerate hypergeometric f u n c t i o n . ~* In t h i s p a p e r , we have retained only the f i r s t t e r m in the s e r i e s e x p a n s i o n of the degenerate hypergeometric function. This approximation is r e a s o n a b l e for e n e r g i e s w i t h i n 2[ of the e x c i t o n g a p energy. We h a v e a l s o a s s u m e d t h a t n e a r the f u n d a m e n tal g a p of the MQW, the PR signal is d e t e r m i n e d o n l y by AE~ . 12 In Fig. (I) , we s h o w the fit to the d a t a at 6K. E x c e p t in the r e g i o n of the spectrum where donor related absorptions o c c u r , the a g r e e m e n t b e t w e e n the d a t a and the fit to the L o r e n t z i a n is e x c e l l e n t . In c o m p a r i n g the G a u s s i a n and Loren~zian fits, we see t h a t b o t h f u n c t i o n a l forms are g o o d n e a r the transitions energies, b u t t h a t t h e y d i f f e r m o s t in the w i n g s of the c u r v e s . In Fig. (2) , we show the
Superlattices and Microstructures, Vol. 3, No. 3, 1987
0.2
I i
¢"
237
/!11H /~ i .~
11 L
"" 'ii 1 /
~
~
.t
0 _u--c='r-______."
",.,.
% . 1 " - _ ~ - -
....
-0.2 >" I--
DATA LORENTZIAN GAUSSIAN
----
Z uJ I-Z
-0.4
n13_
-0.6
:::.
I
I
I
I
1.515
1.520
1.525
1.530
ENERGY (eV) Figure 2. Comparison of 77K PR data (dotted line) to A£z calculated from dielectric functions having Lorentzian
(solid) a n d profiles.
Gaussian
(dashed)
absorption
T=150K (:~ 11 H
I~ 11
0.2
...""
"
r. ~""%'t \'..
!
\',.
t0 .....
>.
.-r
-o.2
I-Z ,,, I-Z m nr" S_
........ DATA LORENTZlAN --GAUSSIAN
-0.4
-O.6
I
i
I
I
I
1.490
1.495
1 .N?O
1.505
1.510
ENERGY (eV) Figure 3. Comparison of 150K FR d a t a (dotted line) to A£z calculated from dielectric functions having Lorentzian
(solid) a n d profiles.
Gaussian
(dashed)
absorption
fits at 77K. It is e v i d e n t t h a t in t h i s case, the L o r e n t z i a n fit is n o t as good as it was at 6K and t h a t it b e g i n s to b r e a k d o w n in the w i n g s of the c u r v e s . A
close examination of Fig. (2) r e v e a l s t h a t the d a t a cannot be satisfactorily represented by either a G a u s s i a n or a Lorentzian profile, but rather by some
238 intermediate form. At 150 K, [Fig. (3)] the t r a n s f o r m a t i o n to a G a u s s i a n dielectric function is nearly complete. We n o t i c e that a l t h o u g h there is some disagreement in the wings, the G a u s s i a n fit is q u i t e good. Clearly, at this temperature, the L o r e n t z i a n fit is i n a d e q u a t e . Our results show that in high q u a l i t y samples, the e x c i t o n i c d i e l e c t r i c function of M Q W has a L o r e n t z i a n p r o f i l e at low t e m p e r a t u r e s and that this p r o f i l e becomes Gaussian with increasing temperature. This is c o n s i s t e n t w i t h the theor e t i c a l w o r k of T o y o z a w a , in w h i c h it was s u g g e s t e d that s t r o n g e x c i t o n - p h o n o n coupiing or w e a k c o u p l i n g at h i g h t e m p e r a tures leads to G a u s s i a n absorption profiles. 7 Furthermore, we can n o w u n d e r s t a n d w h y n=3 in Eq. (i) p r o v i d e s a reas o n a b l e fit to r o o m t e m p e r a t u r e data. 9 By u s i n g n=3, one e s s e n t i a l l y takes another derivative of the L o r e n t z i a n and in the p r o c e s s s h a r p e n s up the line, improving the fit in the w i n g s of the curve. This m i m i c s the first d e r i v a t i v e of an excitonic d i e l e c t r i c f u n c t i o n w i t h a G a u s s i a n absorption profile. The results presented above call attention to the fact that c a u t i o n m u s t be e x c e r c i s e d in p e r f o r m i n g analysis of PR data. W h i l e the g a p e n e r g i e s o b t a i n e d f r o m the Lorentzian and Gaussian fits agree to w i t h i n e x p e r i m e n t a l error, this is not the case w i t h the b r o a d e n i n g parameters, F, w h i c h d i f f e r s by 17% for the two types of a b s o r p t i o n lines. Furthermore, we have seen b e t w e e n 6K and 150K, that the PR lines e x h i b i t s p e c t r a l forms that are intermediate between Gaussian and Lorentzian. In this case, a d e t a i l e d line shape analysis is not possible. Therefore, physical significance should not be a t t a c h e d to a fit w i t h o u t a prior, d e t a i l e d k n o w l e d g e of the line shape. The Gaussian nature of excitonic absorption lines can a r i s e u n d e r a v a r i ety of c o n d i t i o n s . T h e s e include strong e x c i t o n - p h o n o n c o u p l i n g and i n h o m o g e n e o u s perturbations. S i m i l a r e f f e c t s are also present in b u l k s e m i c o n d u c t o r s and suggest the p o s s i b i l i t y that G a u s s i a n b r o a d ening may be i m p o r t a n t in b u l k o p t i c a l p r o p e r t i e s . We s u g g e s t that this p h e n o m e non be e x a m i n e d in r e l a t i o n to i n t e r b a n d t r a n s i t i o n s in these materials. Interesting cases i n c l u d e the E~ and E, + ~ t r a n s i t i o n s in Ge and III-V semiconductors for w h i c h Eq. (i) is k n o w n to provide o n l y an adaquate fit to existing electroreflectance data. .3
Superlattices and Microstructures, Vol. 3, No. 3, 1987 In summary, we have p r e s e n t e d PR data w h i c h shows that the line shape of e x c i t o n s in M Q W can c h a n g e from a L o r e n t zian line at low t e m p e r a t u r e s (~6K) to a Gaussian profile at h i g h e r t e m p e r a t u r e s (>I50K). This c h a n g e in line shape complicates the analysis of PR s p e c t r a at v a r i o u s t e m p e r a t u r e s and s u g g e s t s that a more detailed study of the d i e l e c t r i c f u n c t i o n in M Q W s y s t e m s is required.
Acknowledgement-We wish to thank S. Rudin, D.A. Broido and A. K r i m a n for u s e f u l d i s c u s s i o n s . This work was supported in part by the O f f i c e of N a v a l Research.
REFERENCES i.
2.
3.
4. 5.
6. 7. 8.
9.
i0. II. 12. 13.
O.J. G l e m b o c k i , B.V. Shanabrook, N. Bottka, W.T. Beard and J. Comas, Appl. Phys. Lett. 46, 970 (1985). O.J. G l e m b o c k i , B.V. S h a n a b r o o k and W.T. Beard, in the Proc. of the 2nd International Conf. on Modulated Semiconductor Compounds, 1985 (Kyoto) , to be published in Surf. Sci. (1986). H. Shen, X.C. Shen, P. P a r a y a n t h a l , F.H. Pollak, J.N. Schulman, A.L. Smirl, R.A. McFarlane and I. D ' H a e n e n s , this c o n f e r e n c e . D.E. Aspnes, Surf. S c i e n c e 37, 418 (1973). C. W e i s b u c h , R.C. Miller, R. Dingle, A.C. G o s s a r d and W. W e i g m a n n , Solid State Commun. 37, 219 (1981). B.V. S h a n a b r o o k and O.J. G l e m b o c k i , s u b m i t t e d , P h y s i c a l R e v i e w B. Y. T o y o z a w a , P r o g e s s of Theoretical P h y s i c s 2_O0, 53 (1958). D.S. Chemla, D.A.B. Miller, P.W. Smith, A.C. Gossard and Weigmann, J. Q u a n t Elec. QE2__O, 265 (1984). H. Shen, P. Parayanthal, Fred H. Pollak, M i c h a T o m k i e w i c z , T.J. Drummond and J.N. Schulman, Appl. Phys. Lett. 48, 653 (1986). D.E. A s p n e s and J.E. Rowe, Phys. Rev. B5, 4022 (1972). O.J. G l e m b o c k i and B.V. S h a n a b r o o k , to be p u b l i s h e d . B.O. S e r a p h i n and N. Bottka, Phys. Rev. 145, 628 (1966). F.H. Pollak, p r i v a t e c o m m u n i c a t i o n .
TEMPERATURE DEPENDENCE IN G a A s / A I G a A s
235
OF P H O T O R E F L E C T A N C E LINE MULTIPLE QUANTUM WELLS
SHAPES
O.J. G l e m b o c k i and B.V. S h a n a b r o o k U.S. N a v a l R e s e a r c h L a b o r a t o r y Washington, DC 20375, U S A Received
August
17,
1986
The t e m p e r a t u r e d e p e n d e n c e of the p h o t o r e f l e c t a n c e (PR) line s h a p e of the e x c i t o n i c t r a n s i t i o n s in GaAs/AIGaAs multiple quantum wells has b e e n studied. At low t e m p e r a t u r e s , near 6K, we f i n d that the PR lines can be d e s c r i b e d by a d i e l e c tric f u n c t i o n of a L o r e n t z i a n o s c i l l a t o r . At h i g h e r t e m p e r a tures, we have o b s e r v e d the evolution of the dielectric function from one w i t h a L o r e n t z i a n a b s o r p t i o n p r o f i l e to one w i t h a G a u s s i a n absorption profile. The dielectric function of excitonic absorptions is e x p e c t e d to e x h i b i t this b e h a v i o r in c a s e s of w e a k e x c i t o n - p h o n o n coupling.
The o p t i c a l m o d u l a t i o n t e c h n i q u e of photoreflectance has provided a simple and a c c u r a t e method for obtaining the energies of c o n f i n e d s t a t e s in such systems as m u l t i p l e q u a n t u m w e l l s (MQW), *,2 modulation doped heterojunctions* and "NIPI" structures. 3 The ability to obtain accurate values for transition e n e r g i e s is made possible by the fact that the p h o t o e x c i t e d c a r r i e r s lead to a changing surface electric field. Therefore, the electroreflectance t h e o r y of A s p n e s can be a p p l i e d to o b t a i n the transition energies. 4 In the case of low e l e c t r i c fields, the f i e l d - i n d u c e d c h a n g e in the d i e l e c t r i c f u n c t i o n is g i v e n by: A~ = AE,
+ i,5£2 - e *e (E - E,
+ it)-"
(I)
where E is the p h o t o n energy, Eg , is the g a p energy, and [ is a L o r e n t z i a n broadening p a r a m e t e r . The c r i t i c a l p o i n t type d e t e r m i n e s n, w h i c h is n=3 for 2D critical points and n=5/2 for 3D c r i t i c a l points. In Eq. (i) , the phase factor G depends u p o n the c r i t i c a l p o i n t type and u p o n f a c t o r s such as nonuniform fields and i n t e r f e r e n c e p h e n o m e n o n . 4 For 2D and 3D c r i t i c a l points, Eq. (i) is r e l a t e d to the third derivative with respect to energy of the unperturbed dielectric function. In the case of MQW, the c o n f i n e m e n t of c a r r i e r s in the well m a t e r i a l leads to strong excitonic resonances in the a b s o r p t i o n profile, an e f f e c t w h i c h persists e v e n at r o o m t e m p e r a t u r e , s We h a v e r e c e n t l y c o m p a r e d PR to p h o t o l u m i n e s c e n c e
0749-6036/87/030235 + 04 $02.00/0
excitation spectroscopy and f o u n d that the t r a n s i t i o n s o b s e r v e d in PR are excitonic up to 250K. 6 For excitons, the e l e c t r i c f i e l d m o d u l a t e d d i e l e c t r i c function is g i v e n by Eq. (i) w i t h n=2. This c o r r e s p o n d s to a first d e r i v a t i v e 4 of w i t h r e s p e c t to e i t h e r the e x c i t o n e n e r g y gap, Eg , or w i t h r e s p e c t to the exciton l i f e t i m e (~i/[). In this case, the p h a s e factor, @, does not d e p e n d u p o n the c r i t ical point type and in the a b s e n c e of nonuniform fields and inteference effects, @ can be u s e d to d e t e r m i n e the relative contributions to A£ from the exciton energy g a p and l i f e t i m e m o d u l a tion m e c h a n i s m s . 6 In addition, in MQW the total integrated intensity of the t r a n s i t i o n can also be m o d u l a t e d by the field, resulting in an n=l c o n t r i b u t i o n from Eq. (i) to the PR signal. 6 The form of the d i e l e c t r i c function u s e d to o b t a i n Eq. (i) a s s u m e s L o r e n t z i a n b r o a d e n i n g . However, it is k n o w n from the theoretical w o r k of T o y o z a w a that in the p r e s e n c e of strong exciton-phonon coupling or w e a k c o u p l i n g at h i g h t e m p e r a tures the form of the a b s o r p t i o n profile can c h a n g e f r o m L o r e n t z i a n to G a u s s i a n . 7 In the case of M Q W there is some experimental evidence of this p h e n o m e n o n in the room temperature experiments of Chemla et al.' who o b s e r v e d that for 100A MQW a Gaussian line shape is r e q u i r e d to d e s c r i b e the a b s o r p t i o n data. Furthermore, r e c e n t PR m e a s u r e m e n t s f o u n d that n=3 (3rd d e r i v a t i v e of 2D c r i t i c a l point) in Eq. (i) fits better to room temperature data than the e x p e c t e d n=2
© 1987 Academic Press Inc. (London) Limited
Superlattices and Microstructures, Vo/. 3, No. 3, 1987
236
llH T = 6K %
I
0.2
I ...~
I !
11 L
¢. 0
400 >l--
DONOR RELATED -0.2 DATA
....
GAUSSIAN
LORENTZIAN
Z "' I-Z m
........
-0.4
e," l:*..
-0.6
I 1.522
I 1.524
I 1.526
I 1.528
ENERGY (eV) Figure I. Comparison of 6K PR data (dotted line) to &a, calculated from dielectric functions having Lorentzian
(solid) a n d profiles.
Gaussian
(dashed)
absorption
(exciton). ~ These facts suggest a break down of Eq. (i) at r o o m t e m p e r a t u r e . In o r d e r to b e t t e r u n d e r s t a n d exciton lines in MQW, we h a v e c o m p a r e d at v a r i o u s t e m peratures, PR l i n e s h a p e s to Aa calculated from excitonic dielectric functions having Gaussian and Lorentzian absorption profiles. Henceforth, we will u s e the terms Lorentzian and Gaussian in referr i n g to t h e s e d i e l e c t r i c functions. The MQW sample u s e d in o u r e x p e r iments had barrier and well widths of 150A and 200A respectively. The total t h i c k n e s s of the Q W m a t e r i a l w a s 2~. In o r d e r to r e d u c e the s u r f a c e f i e l d s in the v i c i n i t y of the q u a n t u m w e l l s , the s a m p l e was c l a d w i t h 3 0 0 0 A of A I G a A s . Measurem e n t s w e r e p e r f o r m e d at v a r i o u s temperat u r e s b e t w e e n 6K a n d 250K. For the p r o b e beam (reflectance light), we used the radiation f r o m a CW d y e l a s e r , p u m p e d by an Ar i o n l a s e r . The modulation beam was c o m p o s e d of the b l u e - g r e e n l i g h t f r o m the Ar i o n l a s e r a n d w a s c h o p p e d at 1 7 0 0 Hz. Neutral density filters were used to r e d u c e the i n t e n s i t i e s of the probe and pump b e a m s . T h i s a p p a r a t u s a l l o w e d us to o b t a i n PR s p e c t r a w i t h a r e s o l u t i o n near 0 . 1 5 meV. The r e s u l t s of our m e a s u r e m e n t s are s h o w n in F i g s . (1)-(3) in w h i c h we plot the data (dotted line) a n d f i t s to the data using Lorentzian (solid line) and Gaussian (dashed line) profiles. The
s p e c t r a l f e a t u r e s of i n t e r e s t to us are the two lowest l y i n g p a r i t y a l l o w e d QW transitions. In o u r notation, the two numbers correspond to the q u a n t u m i n d i c e s of the c o n d u c t i o n and valence subbands, respectively, and the letter denotes w h e t h e r the v a l e n c e s t a t e is of h e a v y or light hole character. The Lorentzian l i n e w a s o b t a i n e d f r o m Eq. (I) w i t h n=2, while the "Gaussian" l i n e is the f i r s t derivative of an excitonic dielectric f u n c t i o n c a l c u l a t e d u s i n g the c o n v o l u t i o n f o r m a l i s m I° a n d G a u s s i a n b r o a d e n i n g . 7 in the l a t e r case, the f u n c t i o n a l f o r m of ~ is the product of a Gaussian with a degenerate hypergeometric f u n c t i o n . ~* In t h i s p a p e r , we have retained only the f i r s t t e r m in the s e r i e s e x p a n s i o n of the degenerate hypergeometric function. This approximation is r e a s o n a b l e for e n e r g i e s w i t h i n 2[ of the e x c i t o n g a p energy. We h a v e a l s o a s s u m e d t h a t n e a r the f u n d a m e n tal g a p of the MQW, the PR signal is d e t e r m i n e d o n l y by AE~ . 12 In Fig. (I) , we s h o w the fit to the d a t a at 6K. E x c e p t in the r e g i o n of the spectrum where donor related absorptions o c c u r , the a g r e e m e n t b e t w e e n the d a t a and the fit to the L o r e n t z i a n is e x c e l l e n t . In c o m p a r i n g the G a u s s i a n and Loren~zian fits, we see t h a t b o t h f u n c t i o n a l forms are g o o d n e a r the transitions energies, b u t t h a t t h e y d i f f e r m o s t in the w i n g s of the c u r v e s . In Fig. (2) , we show the
Superlattices and Microstructures, Vol. 3, No. 3, 1987
0.2
I i
¢"
237
/!11H /~ i .~
11 L
"" 'ii 1 /
~
~
.t
0 _u--c='r-______."
",.,.
% . 1 " - _ ~ - -
....
-0.2 >" I--
DATA LORENTZIAN GAUSSIAN
----
Z uJ I-Z
-0.4
n13_
-0.6
:::.
I
I
I
I
1.515
1.520
1.525
1.530
ENERGY (eV) Figure 2. Comparison of 77K PR data (dotted line) to A£z calculated from dielectric functions having Lorentzian
(solid) a n d profiles.
Gaussian
(dashed)
absorption
T=150K (:~ 11 H
I~ 11
0.2
...""
"
r. ~""%'t \'..
!
\',.
t0 .....
>.
.-r
-o.2
I-Z ,,, I-Z m nr" S_
........ DATA LORENTZlAN --GAUSSIAN
-0.4
-O.6
I
i
I
I
I
1.490
1.495
1 .N?O
1.505
1.510
ENERGY (eV) Figure 3. Comparison of 150K FR d a t a (dotted line) to A£z calculated from dielectric functions having Lorentzian
(solid) a n d profiles.
Gaussian
(dashed)
absorption
fits at 77K. It is e v i d e n t t h a t in t h i s case, the L o r e n t z i a n fit is n o t as good as it was at 6K and t h a t it b e g i n s to b r e a k d o w n in the w i n g s of the c u r v e s . A
close examination of Fig. (2) r e v e a l s t h a t the d a t a cannot be satisfactorily represented by either a G a u s s i a n or a Lorentzian profile, but rather by some
238 intermediate form. At 150 K, [Fig. (3)] the t r a n s f o r m a t i o n to a G a u s s i a n dielectric function is nearly complete. We n o t i c e that a l t h o u g h there is some disagreement in the wings, the G a u s s i a n fit is q u i t e good. Clearly, at this temperature, the L o r e n t z i a n fit is i n a d e q u a t e . Our results show that in high q u a l i t y samples, the e x c i t o n i c d i e l e c t r i c function of M Q W has a L o r e n t z i a n p r o f i l e at low t e m p e r a t u r e s and that this p r o f i l e becomes Gaussian with increasing temperature. This is c o n s i s t e n t w i t h the theor e t i c a l w o r k of T o y o z a w a , in w h i c h it was s u g g e s t e d that s t r o n g e x c i t o n - p h o n o n coupiing or w e a k c o u p l i n g at h i g h t e m p e r a tures leads to G a u s s i a n absorption profiles. 7 Furthermore, we can n o w u n d e r s t a n d w h y n=3 in Eq. (i) p r o v i d e s a reas o n a b l e fit to r o o m t e m p e r a t u r e data. 9 By u s i n g n=3, one e s s e n t i a l l y takes another derivative of the L o r e n t z i a n and in the p r o c e s s s h a r p e n s up the line, improving the fit in the w i n g s of the curve. This m i m i c s the first d e r i v a t i v e of an excitonic d i e l e c t r i c f u n c t i o n w i t h a G a u s s i a n absorption profile. The results presented above call attention to the fact that c a u t i o n m u s t be e x c e r c i s e d in p e r f o r m i n g analysis of PR data. W h i l e the g a p e n e r g i e s o b t a i n e d f r o m the Lorentzian and Gaussian fits agree to w i t h i n e x p e r i m e n t a l error, this is not the case w i t h the b r o a d e n i n g parameters, F, w h i c h d i f f e r s by 17% for the two types of a b s o r p t i o n lines. Furthermore, we have seen b e t w e e n 6K and 150K, that the PR lines e x h i b i t s p e c t r a l forms that are intermediate between Gaussian and Lorentzian. In this case, a d e t a i l e d line shape analysis is not possible. Therefore, physical significance should not be a t t a c h e d to a fit w i t h o u t a prior, d e t a i l e d k n o w l e d g e of the line shape. The Gaussian nature of excitonic absorption lines can a r i s e u n d e r a v a r i ety of c o n d i t i o n s . T h e s e include strong e x c i t o n - p h o n o n c o u p l i n g and i n h o m o g e n e o u s perturbations. S i m i l a r e f f e c t s are also present in b u l k s e m i c o n d u c t o r s and suggest the p o s s i b i l i t y that G a u s s i a n b r o a d ening may be i m p o r t a n t in b u l k o p t i c a l p r o p e r t i e s . We s u g g e s t that this p h e n o m e non be e x a m i n e d in r e l a t i o n to i n t e r b a n d t r a n s i t i o n s in these materials. Interesting cases i n c l u d e the E~ and E, + ~ t r a n s i t i o n s in Ge and III-V semiconductors for w h i c h Eq. (i) is k n o w n to provide o n l y an adaquate fit to existing electroreflectance data. .3
Superlattices and Microstructures, Vol. 3, No. 3, 1987 In summary, we have p r e s e n t e d PR data w h i c h shows that the line shape of e x c i t o n s in M Q W can c h a n g e from a L o r e n t zian line at low t e m p e r a t u r e s (~6K) to a Gaussian profile at h i g h e r t e m p e r a t u r e s (>I50K). This c h a n g e in line shape complicates the analysis of PR s p e c t r a at v a r i o u s t e m p e r a t u r e s and s u g g e s t s that a more detailed study of the d i e l e c t r i c f u n c t i o n in M Q W s y s t e m s is required.
Acknowledgement-We wish to thank S. Rudin, D.A. Broido and A. K r i m a n for u s e f u l d i s c u s s i o n s . This work was supported in part by the O f f i c e of N a v a l Research.
REFERENCES i.
2.
3.
4. 5.
6. 7. 8.
9.
i0. II. 12. 13.
O.J. G l e m b o c k i , B.V. Shanabrook, N. Bottka, W.T. Beard and J. Comas, Appl. Phys. Lett. 46, 970 (1985). O.J. G l e m b o c k i , B.V. S h a n a b r o o k and W.T. Beard, in the Proc. of the 2nd International Conf. on Modulated Semiconductor Compounds, 1985 (Kyoto) , to be published in Surf. Sci. (1986). H. Shen, X.C. Shen, P. P a r a y a n t h a l , F.H. Pollak, J.N. Schulman, A.L. Smirl, R.A. McFarlane and I. D ' H a e n e n s , this c o n f e r e n c e . D.E. Aspnes, Surf. S c i e n c e 37, 418 (1973). C. W e i s b u c h , R.C. Miller, R. Dingle, A.C. G o s s a r d and W. W e i g m a n n , Solid State Commun. 37, 219 (1981). B.V. S h a n a b r o o k and O.J. G l e m b o c k i , s u b m i t t e d , P h y s i c a l R e v i e w B. Y. T o y o z a w a , P r o g e s s of Theoretical P h y s i c s 2_O0, 53 (1958). D.S. Chemla, D.A.B. Miller, P.W. Smith, A.C. Gossard and Weigmann, J. Q u a n t Elec. QE2__O, 265 (1984). H. Shen, P. Parayanthal, Fred H. Pollak, M i c h a T o m k i e w i c z , T.J. Drummond and J.N. Schulman, Appl. Phys. Lett. 48, 653 (1986). D.E. A s p n e s and J.E. Rowe, Phys. Rev. B5, 4022 (1972). O.J. G l e m b o c k i and B.V. S h a n a b r o o k , to be p u b l i s h e d . B.O. S e r a p h i n and N. Bottka, Phys. Rev. 145, 628 (1966). F.H. Pollak, p r i v a t e c o m m u n i c a t i o n .