- Email: [email protected]
Resonant Raman scattering study of localized collective excitations
Super/attices and Microstructures, Vo/. 6, No. 2, 1989
171
RESONANT RAMAN SCATTERING STUDY OP LOCALIZED COLLECTIVE EXCITATIONS D. Gammon and B. V. Shanabrook, Naval Research Laboratory, Washington DC, 20375 USA and D. Musser, Martin-Marietta, Baltimore, MD, 21227 USA (Received
8 August 1988)
The presence of a n e u t r a l d o n o r in a q u a s i - t w o dimensional e l e c t r o n gas of a G a A s / A I G a A s q u a n t u m w e l l causes the c o u p l e d i n t e r s u b b a n d p l a s m o n - L O phonon excitations to split into doublets in the Raman spectrum. The extra peaks are identified as modes which are localized at the neutral donor through the r e s o n a n t dipole field associated w i t h the i n t e r s u b b a n d d o n o r transition. The localized exc i t a t i o n s are treated q u a n t i t a t i v e l y in terms of a dielectric model.
Collective excitations of t h e q u a s i - 2 D e l e c t r o n gas in q u a n t u m w e l l structures (QWS) have drawn considerable i n t e r e s t o v e r the last 10 years. = In particular, the collective excitation of the electron gas in which the associated electric field is normal to the layers, i.e., the intersubband plasmon, is especially familiar. This excitation couples to the polar lattice, producing the coupled intersubband plasmon-LO phonon modes. In the absence of defects these m o d e s travel freely in the p l a n e of the quantum well. In this w o r k we c o n s i d e r the case of a s h a l l o w neutral d o n o r i m m e r s e d in the e l e c t r o n gas. = The q u a s i - 2 D d o n o r has a l a - 2 p z t y p e t r a n s i t i o n in n e a r r e s o n a n c e w i t h the i n t e r s u b b a n d p l a s mon. ~ Thus we interpret the p r o b l e m in terms of a near resonant interaction between the long range e l e c t r i c field of the p l a s m o n - p h o n o n modes and the localized donor transition dipole moment. We find e x p e r i m e n t a l l y that the i n t r o d u c t i o n of the d o n o r c a u s e s e a c h of the c o u p l e d p l a s m o n - p h o n o n p e a k s to s p l i t into a doublet. W i t h t h e h e l p of a simple model we interpret the additional p e a k s as due to l o c a l i z e d m o d e s w h i c h are s h a f t e d up in e n e r g y from the extended modes. LO p h o n o n s l o c a l i z e d or b o u n d to neutral donors have l~n~ been studied in bulk s e m i c o n d u c t o r s ~,~, but this is the f i r s t o b s e r v a t i o n of the q u a s i - 2 D version. P l a s m o n s or c o u p l e d p l a s m o n p h o n o n m o d e s l o c a l i z e d at a n e u t r a l donor have not been observed previously.
0749-6036/89/060171 + 03 SO2.00/0
The d e t a i l s of the r e s o n a n t Raman scattering experiment were described p r e v i o u s l y 2 . The M B E g r o w n s t r u c t u r e consists of alternating layers of GaAs, the center third of which were ~ - d o p ~ d at an estimated density of ixl0 ~° cm -~, and A1 25Ga 75As with the center third a l s o 1 ~ o p e ~ w i t h Si at an e s t i m a t e d 8x10 ~ ° cm -~. B e c a u s e the w e l l w i d t h (L~) was found to vary across the 3 inch wa~er from Lz=182-200A, we were able to g e n e r a t e spectra as a function of well width. The free electron concentration was found from cyclotron resonance measurements to be n=l.5x10 II cm -2. In Fig. 1 we show a partial summary of the data. A more complete description will be published separately 6. The open symbols are the e n e r g i e s m e a s u r e d in the Raman spectra of the single particle excitations which have been discussed previously z, and the filled symbols are the collective excitations. The inset shows the a s s i g n m e n t of the s i n g l e p a r t i c l e transitions. The coupled i n t e r s u b b a n d plasmon-LO phonon peaks are labelled L_ and L+. The peaks a s s o c i a t e d with the d o n o r are labelled with a superscript, d. The lines are the results of a calculation which describes the collective excitations and is based on a mode~ that employs the measured single particle excitation energies. The plasmon energy of a homogeneous gas including the coupling to the polar lattice is c o m m o n l y d e r i v e d by finding the z e r o e s of a q=0 d i e l e c t r i c function. 1 This is equivalent to the solution of c o u p l e d h a r m o n i c oscillators. 7
© 1989 Academic Press Limited
172
Superlattices and Microstructures, Vol. 6, No. 2, 1989
340
,~
300
LO ~L iii:>:i:i::/!:?i:i:iL::iiis:ii?i::~v::i?i?<: i?:: A ii<:i:i s::::iIii:::I>::LLLiLi:: •L:¸ i !! ii!/iLi?ilTiTLii!! L/!!!Lii~ii!i/iii! ~ Lii!~!ii!YL?!i IIY/LI~Ii>~ L Lii!!
~i~i~ii !~i~i~!~!~iiiiiiiii~i~!~i~i~iiii~iii~!iiiii~i~ii~iiiiiii~i~i~i~iiiii~ii~iiiiii~i~i~i~ii~ii~i~i~i~i~i i~i~ili~il ~i~i~! iiliii ili~i~!i>
260 6--
_.--Tu...-~
L-
L,.I
22o
• .... .:.
". .. • " "
•
Eo,:....i
~
/El
- ' 7 Is '
200
2. Model of a dielectric sphere in the backgound dielectric medium.
*
O1 180
Fig.
--
'
'
190
'
'
180
extended p l a s m o n - p h o n o n modes are found from Zm(~) = 0.I A dielectric sphere has collective excitations determined by %(e)
Fig.
i. Peak energies of single particle (closed symbols) and collective (open symbols) excitations as a function of w e l l w i d t h with n = l . 5 x l 0 II cm -2. The solid and dashed lines are the results of calculations d e s c r i b e d in the t e x t . T h e d o t t e d l i n e is t h e donor-like collective excitation w h i c h is p r e d i c t e d but not observed . The inset shows s c h e m a t i c a l l y the single particle transition assignments.
With a nonhomogeneous s y s t e m the approach is not so straightforward. However, by modeling the donor appropriately w e c a n = a p p l y an effective medium t e c h n i q u e ~, a n d c o n t i n u e with the coupled o s c i l l a t o r s approach. We disc u s s h e r e the l i m i t as the n u m b e r of donors goes to zero, i.e., for a single donor. A m o r e c o m p l e t e t r e a t m e n t will be published separately. 6 To account for the localized dipole moment associated with the donor transition immersed within the background d i e l e c t r i c m e d i u m , we c o n s i d e r a constant dipole moment out to some radius, as, w h i c h s h o u l d be the o r d e r of a Bohr radius (Fig. 2) .~n We then have a d i e l e c t r i c f u n c t i o n w i t h i n the s p h e r e (Ss) w h i c h d i f f e r s from the d i e l e c t r i c function outside the sphere, 8m, by the contribution from the donor, 4~Xd. 8 s = 8m + 4~Xd = ] + 4~(Xph + Xpl + ~d). The s u s c e p t a b i l i t i e s h a v e the f o r m of L o r e n t z oscillators. Note that in the absence of the donor the energies of the
= - 2~(~).
This is equivalent to finding the zeroes of an effective dielectric function, 8ef f = 1 + 4~(Xph + Xpl + ~d/3) = 0. In terms of this effective dielectric f u n c t i o n , we have a d d e d to 8m an additional susceptability d u e to the donor, reduced, however, by a factor of 1/3 due to the spherical boundary. Outside the sphere the field of the dipole moment associated with t~e donor,transit i o n f a l l s off as i/r °, and thus the response of the background medium, i.e., the amplitudes of t h e d o n o r - i n d u c e d plasmon-phonon excitations, w i l l be localized. Far away from the donor, the normal modes will be the free p l a s m o n phonon modes. Although we do not know the microscopic details of the quasi-2D donor in an e l e c t r o n gas, we can use the oscill a t o r s t r e n g t h of the d o n o r , and the radius of the sphere as fitting parameters to calculate the excitation energies within this model. As a rough c h e c k on the r e s u l t i n g v a l u e s of the fitting paramaters, we compare t h e m to t h o s e e x p e c t e d for a s h a l l o w d o n o r in b u l k GaAs, and find r e a s o n a b l e a g r e e ment. The resulting energies are plotted in Fig. i, and show good agreement with the data. In addition to the localized plasmon-phonon modes the model also predicts a collective excitation derived from the donor transition. This m o d e is n o t d e t e c t e d e x p e r i m e n t a l l y , w h i c h is a p p a r e n t l y d u e to it~ m u c h smaller scattering cross section, v In s u m m a r y , we h a v e m e a s u r e d the s p e c t r u m of an e l e c t r o n gas in a GaAs quantum well in t h e p r e s e n c e of a n e u t r a l donor. The extra peaks in the
Superlattices and Microstructures, Vol. 6, No. 2, 1989 s p e c t r u m are u n d e r s t o o d q u a n t i t a t i v e l y in terms of localized collective excitations of the system based on a crude but physically intuitive dielectric model. Acknowledgement - This work is supported in part by the Office of Naval Research.
References i. For a review, see G. Abstreiter, M. Cardona, and A. Pinczuk, in: Vol. 54 of Topics in Applied Physics, Ed. by M. Cardona and G. G u n t h e r o d t (Springer, Berlin 1984) p. 5.
173 2. P r e l i m i n a r y r e s u l t s w e r e p r e s e n t e d in, D. G a m m o n , E. G l a s e r , B.V. Shanabrook, and D. Musser, Surf. Sci. 196, 359 (1988). 3. T . A . Perry, R. M e r l i n , B.V. Shanabrook, and J. Comas, Phys. Rev. Lett. 54, 2623 (1985). 4. P.J. Dean, D.D. M a n c h o n , J.J. Hopfield, Phys. Rev. L e t t . , 25, 1027 (1970); see also the review by M.V. Klein in: Vol. 8 of Topics in Applied P h y s i c s , Ed. M. C a r d o n a (Springer, Berlin 1975) p. 174. 5. A.S. B a r k e r , Jr., Phys. Rev. B ~, 2507 (1973). 6. D. Gammon, B.V. S h a n a b r o o k , and D. Musser, to be published. 7. E. B u r s t e i n , in E l e m e n t a r y E x c i t a tions in Solids, Ed. b y A. A. Maradudin and G. F. Nardelli (Plenum Press, New York 1969) p. 367.
171
RESONANT RAMAN SCATTERING STUDY OP LOCALIZED COLLECTIVE EXCITATIONS D. Gammon and B. V. Shanabrook, Naval Research Laboratory, Washington DC, 20375 USA and D. Musser, Martin-Marietta, Baltimore, MD, 21227 USA (Received
8 August 1988)
The presence of a n e u t r a l d o n o r in a q u a s i - t w o dimensional e l e c t r o n gas of a G a A s / A I G a A s q u a n t u m w e l l causes the c o u p l e d i n t e r s u b b a n d p l a s m o n - L O phonon excitations to split into doublets in the Raman spectrum. The extra peaks are identified as modes which are localized at the neutral donor through the r e s o n a n t dipole field associated w i t h the i n t e r s u b b a n d d o n o r transition. The localized exc i t a t i o n s are treated q u a n t i t a t i v e l y in terms of a dielectric model.
Collective excitations of t h e q u a s i - 2 D e l e c t r o n gas in q u a n t u m w e l l structures (QWS) have drawn considerable i n t e r e s t o v e r the last 10 years. = In particular, the collective excitation of the electron gas in which the associated electric field is normal to the layers, i.e., the intersubband plasmon, is especially familiar. This excitation couples to the polar lattice, producing the coupled intersubband plasmon-LO phonon modes. In the absence of defects these m o d e s travel freely in the p l a n e of the quantum well. In this w o r k we c o n s i d e r the case of a s h a l l o w neutral d o n o r i m m e r s e d in the e l e c t r o n gas. = The q u a s i - 2 D d o n o r has a l a - 2 p z t y p e t r a n s i t i o n in n e a r r e s o n a n c e w i t h the i n t e r s u b b a n d p l a s mon. ~ Thus we interpret the p r o b l e m in terms of a near resonant interaction between the long range e l e c t r i c field of the p l a s m o n - p h o n o n modes and the localized donor transition dipole moment. We find e x p e r i m e n t a l l y that the i n t r o d u c t i o n of the d o n o r c a u s e s e a c h of the c o u p l e d p l a s m o n - p h o n o n p e a k s to s p l i t into a doublet. W i t h t h e h e l p of a simple model we interpret the additional p e a k s as due to l o c a l i z e d m o d e s w h i c h are s h a f t e d up in e n e r g y from the extended modes. LO p h o n o n s l o c a l i z e d or b o u n d to neutral donors have l~n~ been studied in bulk s e m i c o n d u c t o r s ~,~, but this is the f i r s t o b s e r v a t i o n of the q u a s i - 2 D version. P l a s m o n s or c o u p l e d p l a s m o n p h o n o n m o d e s l o c a l i z e d at a n e u t r a l donor have not been observed previously.
0749-6036/89/060171 + 03 SO2.00/0
The d e t a i l s of the r e s o n a n t Raman scattering experiment were described p r e v i o u s l y 2 . The M B E g r o w n s t r u c t u r e consists of alternating layers of GaAs, the center third of which were ~ - d o p ~ d at an estimated density of ixl0 ~° cm -~, and A1 25Ga 75As with the center third a l s o 1 ~ o p e ~ w i t h Si at an e s t i m a t e d 8x10 ~ ° cm -~. B e c a u s e the w e l l w i d t h (L~) was found to vary across the 3 inch wa~er from Lz=182-200A, we were able to g e n e r a t e spectra as a function of well width. The free electron concentration was found from cyclotron resonance measurements to be n=l.5x10 II cm -2. In Fig. 1 we show a partial summary of the data. A more complete description will be published separately 6. The open symbols are the e n e r g i e s m e a s u r e d in the Raman spectra of the single particle excitations which have been discussed previously z, and the filled symbols are the collective excitations. The inset shows the a s s i g n m e n t of the s i n g l e p a r t i c l e transitions. The coupled i n t e r s u b b a n d plasmon-LO phonon peaks are labelled L_ and L+. The peaks a s s o c i a t e d with the d o n o r are labelled with a superscript, d. The lines are the results of a calculation which describes the collective excitations and is based on a mode~ that employs the measured single particle excitation energies. The plasmon energy of a homogeneous gas including the coupling to the polar lattice is c o m m o n l y d e r i v e d by finding the z e r o e s of a q=0 d i e l e c t r i c function. 1 This is equivalent to the solution of c o u p l e d h a r m o n i c oscillators. 7
© 1989 Academic Press Limited
172
Superlattices and Microstructures, Vol. 6, No. 2, 1989
340
,~
300
LO ~L iii:>:i:i::/!:?i:i:iL::iiis:ii?i::~v::i?i?<: i?:: A ii<:i:i s::::iIii:::I>::LLLiLi:: •L:¸ i !! ii!/iLi?ilTiTLii!! L/!!!Lii~ii!i/iii! ~ Lii!~!ii!YL?!i IIY/LI~Ii>~ L Lii!!
~i~i~ii !~i~i~!~!~iiiiiiiii~i~!~i~i~iiii~iii~!iiiii~i~ii~iiiiiii~i~i~i~iiiii~ii~iiiiii~i~i~i~ii~ii~i~i~i~i~i i~i~ili~il ~i~i~! iiliii ili~i~!i>
260 6--
_.--Tu...-~
L-
L,.I
22o
• .... .:.
". .. • " "
•
Eo,:....i
~
/El
- ' 7 Is '
200
2. Model of a dielectric sphere in the backgound dielectric medium.
*
O1 180
Fig.
--
'
'
190
'
'
180
extended p l a s m o n - p h o n o n modes are found from Zm(~) = 0.I A dielectric sphere has collective excitations determined by %(e)
Fig.
i. Peak energies of single particle (closed symbols) and collective (open symbols) excitations as a function of w e l l w i d t h with n = l . 5 x l 0 II cm -2. The solid and dashed lines are the results of calculations d e s c r i b e d in the t e x t . T h e d o t t e d l i n e is t h e donor-like collective excitation w h i c h is p r e d i c t e d but not observed . The inset shows s c h e m a t i c a l l y the single particle transition assignments.
With a nonhomogeneous s y s t e m the approach is not so straightforward. However, by modeling the donor appropriately w e c a n = a p p l y an effective medium t e c h n i q u e ~, a n d c o n t i n u e with the coupled o s c i l l a t o r s approach. We disc u s s h e r e the l i m i t as the n u m b e r of donors goes to zero, i.e., for a single donor. A m o r e c o m p l e t e t r e a t m e n t will be published separately. 6 To account for the localized dipole moment associated with the donor transition immersed within the background d i e l e c t r i c m e d i u m , we c o n s i d e r a constant dipole moment out to some radius, as, w h i c h s h o u l d be the o r d e r of a Bohr radius (Fig. 2) .~n We then have a d i e l e c t r i c f u n c t i o n w i t h i n the s p h e r e (Ss) w h i c h d i f f e r s from the d i e l e c t r i c function outside the sphere, 8m, by the contribution from the donor, 4~Xd. 8 s = 8m + 4~Xd = ] + 4~(Xph + Xpl + ~d). The s u s c e p t a b i l i t i e s h a v e the f o r m of L o r e n t z oscillators. Note that in the absence of the donor the energies of the
= - 2~(~).
This is equivalent to finding the zeroes of an effective dielectric function, 8ef f = 1 + 4~(Xph + Xpl + ~d/3) = 0. In terms of this effective dielectric f u n c t i o n , we have a d d e d to 8m an additional susceptability d u e to the donor, reduced, however, by a factor of 1/3 due to the spherical boundary. Outside the sphere the field of the dipole moment associated with t~e donor,transit i o n f a l l s off as i/r °, and thus the response of the background medium, i.e., the amplitudes of t h e d o n o r - i n d u c e d plasmon-phonon excitations, w i l l be localized. Far away from the donor, the normal modes will be the free p l a s m o n phonon modes. Although we do not know the microscopic details of the quasi-2D donor in an e l e c t r o n gas, we can use the oscill a t o r s t r e n g t h of the d o n o r , and the radius of the sphere as fitting parameters to calculate the excitation energies within this model. As a rough c h e c k on the r e s u l t i n g v a l u e s of the fitting paramaters, we compare t h e m to t h o s e e x p e c t e d for a s h a l l o w d o n o r in b u l k GaAs, and find r e a s o n a b l e a g r e e ment. The resulting energies are plotted in Fig. i, and show good agreement with the data. In addition to the localized plasmon-phonon modes the model also predicts a collective excitation derived from the donor transition. This m o d e is n o t d e t e c t e d e x p e r i m e n t a l l y , w h i c h is a p p a r e n t l y d u e to it~ m u c h smaller scattering cross section, v In s u m m a r y , we h a v e m e a s u r e d the s p e c t r u m of an e l e c t r o n gas in a GaAs quantum well in t h e p r e s e n c e of a n e u t r a l donor. The extra peaks in the
Superlattices and Microstructures, Vol. 6, No. 2, 1989 s p e c t r u m are u n d e r s t o o d q u a n t i t a t i v e l y in terms of localized collective excitations of the system based on a crude but physically intuitive dielectric model. Acknowledgement - This work is supported in part by the Office of Naval Research.
References i. For a review, see G. Abstreiter, M. Cardona, and A. Pinczuk, in: Vol. 54 of Topics in Applied Physics, Ed. by M. Cardona and G. G u n t h e r o d t (Springer, Berlin 1984) p. 5.
173 2. P r e l i m i n a r y r e s u l t s w e r e p r e s e n t e d in, D. G a m m o n , E. G l a s e r , B.V. Shanabrook, and D. Musser, Surf. Sci. 196, 359 (1988). 3. T . A . Perry, R. M e r l i n , B.V. Shanabrook, and J. Comas, Phys. Rev. Lett. 54, 2623 (1985). 4. P.J. Dean, D.D. M a n c h o n , J.J. Hopfield, Phys. Rev. L e t t . , 25, 1027 (1970); see also the review by M.V. Klein in: Vol. 8 of Topics in Applied P h y s i c s , Ed. M. C a r d o n a (Springer, Berlin 1975) p. 174. 5. A.S. B a r k e r , Jr., Phys. Rev. B ~, 2507 (1973). 6. D. Gammon, B.V. S h a n a b r o o k , and D. Musser, to be published. 7. E. B u r s t e i n , in E l e m e n t a r y E x c i t a tions in Solids, Ed. b y A. A. Maradudin and G. F. Nardelli (Plenum Press, New York 1969) p. 367.