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The effect of hot phonons and coupled phonon-plasmon modes on scattering-induced NDR in quantum wells
Solid-State Electronics Vol. 31, No. 3/4, pp. 683-685, 1988 Printed in Great Britain
0038-1101/88 $3.00+0.00 Pergamon Journals Ltd
THE E F F E C T OF HOT PHONONS AND C O U P L E D P H O N O N - P L A S M O N MODES ON S C A T T E R I N G - I N D U C E D NDR IN Q U A N T U M WELLS
B.K. Ridley* and M. A 1 - M u d a r e s * *
**Dept.
*Dept. of Physics, U n i v e r s i t y of Essex, Colchester, E n g l a n d of E l e c t r i c a l E n g i n e e r i n g , U n i v e r s i t y of Glasgow, Glasgow, S c o t l a n d
ABSTRACT We have e x t e n d e d our Monte Carlo s i m u l a t i o n of s c a t t e r i n g - i n d u c e d NDR in A 1 . s G a 2 A s / G a A s q u a n t u m wells by i n c l u d i n g (a) the effect of hot p h o n o n s (b) coupled p h o n o n - p l a s m o n modes (c) degeneracy. Hot p h o n o n s were m o d e l l e d using a p h e n o m e n o l o g i c a l lifetime which we ranged from 3ps to lOps. C o u p l e d modes were m o d e l l e d in the a n t i s c r e e n i n g approximation. B u l k - l i k e modes were assumed in both cases. NDR is q u e n c h e d if the p h o n o n lifetime exceeds 7ps, b u t is little a f f e c t e d if the lifetime is 3ps. The effect of coupled modes is a p p r e c i a b l e at a doping d e n s i t y of iOl8cm -3, v i r t u a l l y e l i m i n a t i n g NDR, b u t at iol7cm -3 the effect is much smaller. Including d e g e n e r a c y has only a small effect on the results. We conclude that NDR is still p o s s i b l e at e l e c t r o n densities around lOl7cm -3 .
KEYWORDS NDR, Q u a n t u m Wells, Monte Carlo, hot phonons,
coupled modes.
INTRODUCTION A n e g a t i v e d i f f e r e n t i a l r e s i s t a n c e (NDR) a s s o c i a t e d with m o d e r a t e e l e c t r o n - e l e c t r o n scattering in the p r e s e n c e of an abrupt t h r e s h o l d for the e m i s s i o n of optical phonons (characteristic of 2D d e n s i t y of states) was p r e d i c t e d some years ago (Ridley, 1982, 1984). Electron-electron s c a t t e r i n g has to be strong enough to r a n d o m i z e energy and m o m e n t u m b e l o w the p h o n o n t h r e s h o l d b u t not so strong as to e s t a b l i s h a M a x w e l l - B o l t z m a n n d i s t r i b u t i o n across the p h o n o n threshold. Monte Carlo simulations of the effect in A I G a A s / G a A s q u a n t u m wells shows that the effect occurs over the range of e l e c t r o n d e n s i t i e s a p p r o x i m a t e l y from lO 17 to lol8cm -3 with a t h r e s h o l d field of 1 . 2 5 k V c m -I (Ai-Mudares and Ridley 1985, 1986). It was, however, evident that at these d e n s i t i e s p h o n o n r e a b s o r p t i o n (hot phonons), p h o n o n - p l a s m o n coupling and s t a t i s t i c a l d e g e n e r a c y w o u l d all affect the NDR (Ridley 1986). It is the p u r p o s e of this paper to report the results i n c l u d i n g these affects in the Monte Carlo simulation. Our Monte Carlo m o d e l has b e e n d e s c r i b e d more fully e l s e w h e r e (Ai-Mudares and R i d l e y 1986). Briefly, it is a m a n y - p a r t i c l e model, now m o d i f i e d to deal with 5Ok electrons and 5Ok v i b r a t i o n a l modes, i n c o r p o r a t i n g a s e l f - c o n s i s t e n t s o l u t i o n of the P e i s s o n and S c h r o d i n g e r equations. Q u a n t i z a t i o n is d e s c r i b e d in the e f f e c t i v e mass f o r m a l i s m and scattering within and b e t w e e n up to i00 subbands can be included. Q u a s i - t w o - d i m e n s i o n a l s c a t t e r i n g rates for all the usual s c a t t e r i n g m e c h a n i s m s is i n c l u d e d a s s u m i n g bulk p h o n o n spectra. E l e c t r o n - e l e c t r o n s c a t t e r i n g is d e s c r i b e d by the bulk m a t r i x e l e m e n t in the Born approximation, w i t h 2D static screening and q u a n t i z a t i o n incorporated. An iterative process is used with e l e c t r o n - e l e c t r o n s c a t t e r i n g to reach c o n v e r g e n c e of the d i s t r i b u t i o n function, and an iterative process is also used to a c c o m m o d a t e the effects of degeneracy. The thickness of the AIGaAs layer is assumed to be equal to the d e p l e t i o n layer thickness, and bulk s c a t t e r i n g p r o c e s s e s are a s s u m e d to prevail. The width of t~e GaAs well is IOOA, b u t at the e l e c t r o n d e n s i t i e s of i n t e r e s t q u a n t i z a t i o n is strongly affected by b a n d bending. In order to d i s t i n g u i s h s c a t t e r i n g - i n d u c e d NDR from NDR via r e a l - s p a c e transfer an 80% A1 c o n c e n t r a t i o n is taken. S i m u l a t i o n s r e p o r t e d here are all for 77k with doping solely in the A I G a A s layer.
HOT P H O N O N S Optical p h o n o n s e m i t t e d by electrons may be r e a b s o r b e d by the electron gas instead of d e c a y i n g into acoustic modes. The lifetime of the latter p r o c e s s in q u a n t u m wells is not known. It is 7ps at low t e m p e r a t u r e in b u l k GaAs (Yon der Kinde and others 1980); Kash and others, 1985) and is expected to be less ( 3ps) at room temperature. P h o n o n r e a b s o r p t i o n is w i d e l y r e g a r d e d as r e s p o n s i b l e for the a n o m a l o u s l y low rate of e n e r g y - r e l a x a t i o n of hot electrons in q u a n t u m wells. (See the reviews of Shah, 1986 and Lyon, 1986.) S u b s t a n t i a l a b s o r p t i o n of optical phonons w o u l d d e s t r o y NDR by p r o v i d i n g a m e c h a n i s m for electrons to jump over the e m i s s i o n t h r e s h o l d b a r r i e r and
683
684
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.
.
.
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6
T h e e f f e c t of h o t p h o n o n s on s c a t t e r i n g i n d u c e d NDR. T h e n u m b e r s on the c u r v e s a r e t h e p h o n o n l i f e t i m e in p i c o s e c o n d s .
~
2.0
I017 cm -3
.
[
I
=
"
N O = 10 18 cell -3
~
.--~
/
-~.
/
V (107cms-']
"~
/ / ~ " ~ ~ ~
ND = lOITcm-t
.~.-~.
_/ I
I
[
I
t
I
1
2
3
4.
5
6
6, ( kVcm-'} Fig.2
T h e e f f e c t of c o u p l e d p h o n o n - p l a s m o n modes. S t a t i s t i c a l d e g e n e r a c y is i n c l u d e d b u t h o t p h o n o n e f f e c t s are not. Dashed lines without coupled modes.
685
reach higher energies. We have m o d e l l e d this p r o c e s s u s i n g a w a v e v e c t o r - i n d e p e n d e n t lifetime, ~, for the polar o p t i c a l phonons ranging from 3ps to lOps. The result is shown in Fig. i. For an A I G a A s d o p i n g d e n s i t y of lOl7cm -3 N D R d i s a p p e a r s w h e n T = lOps and is only m i l d l y a f f e c t e d when T = 3ps. For T = 7ps the NDR is p r e s e n t but weak.
COUPLED PHONON-PLASMONS We have not a t t e m p t e d a full d e s c r i p t i o n of the complex i n t e r a c t i o n of electrons with coupled p h o n o n - p l a s m o n m o d e s in our q u a s i - 2 D system. Instead, we have assumed a simple bulk a n t i - s c r e e n i n g approximation. In this we have assumed that dynamic screening is d e s c r i b e d by the bulk L i n d h a r d d i e l e c t r i c function, and we have used this to obtain the d i s p e r s i o n r e l a t i o n of the upper f r e q u e n c y (W+(q)) coupled mode. The lower f r e q u e n c y mode (W_(q)) is assumed to be h e a v i l y damped and hence the i n t e r a c t i o n with this mode is r e g a r d e d as having already b e e n d e s c r i b e d by the e l e c t r o n - e l e c t r o n interaction. This a s s u m p t i o n will fail when the l o w - f r e q u e n c y c o m p o n e n t b e c o m e s p h o n o n - l i k e at high e l e c t r o n densities, but then it is h e a v i l y screened and so the i n t e r a c t i o n b e t w e e n it and an e l e c t r o n can be again ignored. We therefore consider the i n t e r a c t i o n solely w i t h the upper f r e q u e n c y component. This is similar to the a s s u m p t i o n made by Abou E1-Ela and others (1987) in their d i s c u s s i o n of dynamic s c r e e n i n g effects in GaAs except that we include the v a r i a t i o n of f r e q u e n c y with e l e c t r o n density. We refer to this model as the a n t i - s c r e e n i n g a p p r o x i m a t i o n since it d e s c r i b e s an e n h a n c e d i n t e r a c t i o n with the p h o n o n - l i k e mode at d e n s i t i e s up to about ixlOl8cm -3. The effect on NDR is shown in Figk 2 (ignoring hot phonons but including degeneracy). C o u p l i n g b e t w e e n p h o n o n s and p l a s m o n s weakens NDR at lOl7cm -3 and d e s t r o y s it at lOl8cm -3. The cause lies in the increased d i s p e r s i o n i n t r o d u c e d by coupling - the e m i s s i o n threshold is no longer sharp. Moreover, a n t i s c r e e n i n g weakens with i n c r e a s i n g electron temperature. Both of these effects w e a k e n NDR. Thus a l t h o u g h there is an e n h a n c e m e n t of the e l e c t r o n - p h o n o n interaction, that e n h a n c e m e n t is strongly w a v e v e c t o r dependent, and hence also is the frequency shift, and this tends to make the e m i s s i o n t h r e s h o l d diffuse. A diffuse t h r e s h o l d allows the electrons to reach h i g h e r energies, and h o t t e r electrons couple more w e a k l y with the phonons.
DEGENERACY The effect of i n c l u d i n g s t a t i s t i c a l d e g e n e r a c y is to reduce the v e l o c i t y by about 10%. The v e l o c i t y is u n i f o r m l y reduced as a result of fewer electrons being able to transfer to the well. It has little effect on NDR.
SUMMARY AND CONCLUSIONS W h e t h e r NDR survives h o t - p h o n o n and coupled mode effects depends c r u c i a l l y on the p h o n o n lifetime and d o p i n g density. Our analysis p r e s e n t e d here is, of course, far from complete. A l t h o u g h our a s s u m p t i o n s of b u l k formulae for phonons and p l a s m o n s need improving, they are p r o b a b l y not too bad. We have also not yet been able to i n c o r p o r a t e hot phonons, coupled modes, and degeneracy, into one s i m u l a t i o n - s o m e t h i n g that c l e a r l y w o u l d be desirable. Nevertheless, we can say with c o n f i d e n c e that NDR is u n l i k e l y to survive at doping d e n s i t i e s as high as lol8cm -3. It is likely, however, that NDR survives at doping d e n s i t i e s near and p r e f e r a b l y just b e l o w iol7cm -3, p r o v i d e d the p h o n o n lifetime is no higher than 7ps.
ACKNOWLEDGEMENT This work was supported by the U.S. Office of Naval Research.
REFERENCES A b o u EI-EIa, F., Riddoch, F.A., Davis, M. and Ridley, B.K. (1986) ICPS S t o c k h o l m (ed. O. Engstrom, W o r l d Scientific) p. 1567. Ai-Mudares, M.A.R. and Ridley, B.K. (1986) J. Phys. C: Solid State Phys. 19 3179 Kash, J.A., Tsang, J.C. and Hvam, J.M. (1985) Phys. Rev. Lett. 54 2151 Lyons, S.A. (1986) J. L u m i n e s c e n c e 35 121 Ridley, B.K. (1982) J. Phys. C: Solid State Phys. 15 5899 Ridley, B.K. (1984) J. Phys. C: Solid State Phys. 17 5357 Ridley, B.K. (1986) S u p e r l a t t i c e s and M i c r o s t r u c t u r e s 2 159. Shah, J. (1986) I . E . E . E . J . Q u a n t u m Electron. QE-22 17~8. Yon der Linde, D., Kuhl, J. and Klingenburg, H. (1980) Phys. Rev. Lett. 44 1505.
0038-1101/88 $3.00+0.00 Pergamon Journals Ltd
THE E F F E C T OF HOT PHONONS AND C O U P L E D P H O N O N - P L A S M O N MODES ON S C A T T E R I N G - I N D U C E D NDR IN Q U A N T U M WELLS
B.K. Ridley* and M. A 1 - M u d a r e s * *
**Dept.
*Dept. of Physics, U n i v e r s i t y of Essex, Colchester, E n g l a n d of E l e c t r i c a l E n g i n e e r i n g , U n i v e r s i t y of Glasgow, Glasgow, S c o t l a n d
ABSTRACT We have e x t e n d e d our Monte Carlo s i m u l a t i o n of s c a t t e r i n g - i n d u c e d NDR in A 1 . s G a 2 A s / G a A s q u a n t u m wells by i n c l u d i n g (a) the effect of hot p h o n o n s (b) coupled p h o n o n - p l a s m o n modes (c) degeneracy. Hot p h o n o n s were m o d e l l e d using a p h e n o m e n o l o g i c a l lifetime which we ranged from 3ps to lOps. C o u p l e d modes were m o d e l l e d in the a n t i s c r e e n i n g approximation. B u l k - l i k e modes were assumed in both cases. NDR is q u e n c h e d if the p h o n o n lifetime exceeds 7ps, b u t is little a f f e c t e d if the lifetime is 3ps. The effect of coupled modes is a p p r e c i a b l e at a doping d e n s i t y of iOl8cm -3, v i r t u a l l y e l i m i n a t i n g NDR, b u t at iol7cm -3 the effect is much smaller. Including d e g e n e r a c y has only a small effect on the results. We conclude that NDR is still p o s s i b l e at e l e c t r o n densities around lOl7cm -3 .
KEYWORDS NDR, Q u a n t u m Wells, Monte Carlo, hot phonons,
coupled modes.
INTRODUCTION A n e g a t i v e d i f f e r e n t i a l r e s i s t a n c e (NDR) a s s o c i a t e d with m o d e r a t e e l e c t r o n - e l e c t r o n scattering in the p r e s e n c e of an abrupt t h r e s h o l d for the e m i s s i o n of optical phonons (characteristic of 2D d e n s i t y of states) was p r e d i c t e d some years ago (Ridley, 1982, 1984). Electron-electron s c a t t e r i n g has to be strong enough to r a n d o m i z e energy and m o m e n t u m b e l o w the p h o n o n t h r e s h o l d b u t not so strong as to e s t a b l i s h a M a x w e l l - B o l t z m a n n d i s t r i b u t i o n across the p h o n o n threshold. Monte Carlo simulations of the effect in A I G a A s / G a A s q u a n t u m wells shows that the effect occurs over the range of e l e c t r o n d e n s i t i e s a p p r o x i m a t e l y from lO 17 to lol8cm -3 with a t h r e s h o l d field of 1 . 2 5 k V c m -I (Ai-Mudares and Ridley 1985, 1986). It was, however, evident that at these d e n s i t i e s p h o n o n r e a b s o r p t i o n (hot phonons), p h o n o n - p l a s m o n coupling and s t a t i s t i c a l d e g e n e r a c y w o u l d all affect the NDR (Ridley 1986). It is the p u r p o s e of this paper to report the results i n c l u d i n g these affects in the Monte Carlo simulation. Our Monte Carlo m o d e l has b e e n d e s c r i b e d more fully e l s e w h e r e (Ai-Mudares and R i d l e y 1986). Briefly, it is a m a n y - p a r t i c l e model, now m o d i f i e d to deal with 5Ok electrons and 5Ok v i b r a t i o n a l modes, i n c o r p o r a t i n g a s e l f - c o n s i s t e n t s o l u t i o n of the P e i s s o n and S c h r o d i n g e r equations. Q u a n t i z a t i o n is d e s c r i b e d in the e f f e c t i v e mass f o r m a l i s m and scattering within and b e t w e e n up to i00 subbands can be included. Q u a s i - t w o - d i m e n s i o n a l s c a t t e r i n g rates for all the usual s c a t t e r i n g m e c h a n i s m s is i n c l u d e d a s s u m i n g bulk p h o n o n spectra. E l e c t r o n - e l e c t r o n s c a t t e r i n g is d e s c r i b e d by the bulk m a t r i x e l e m e n t in the Born approximation, w i t h 2D static screening and q u a n t i z a t i o n incorporated. An iterative process is used with e l e c t r o n - e l e c t r o n s c a t t e r i n g to reach c o n v e r g e n c e of the d i s t r i b u t i o n function, and an iterative process is also used to a c c o m m o d a t e the effects of degeneracy. The thickness of the AIGaAs layer is assumed to be equal to the d e p l e t i o n layer thickness, and bulk s c a t t e r i n g p r o c e s s e s are a s s u m e d to prevail. The width of t~e GaAs well is IOOA, b u t at the e l e c t r o n d e n s i t i e s of i n t e r e s t q u a n t i z a t i o n is strongly affected by b a n d bending. In order to d i s t i n g u i s h s c a t t e r i n g - i n d u c e d NDR from NDR via r e a l - s p a c e transfer an 80% A1 c o n c e n t r a t i o n is taken. S i m u l a t i o n s r e p o r t e d here are all for 77k with doping solely in the A I G a A s layer.
HOT P H O N O N S Optical p h o n o n s e m i t t e d by electrons may be r e a b s o r b e d by the electron gas instead of d e c a y i n g into acoustic modes. The lifetime of the latter p r o c e s s in q u a n t u m wells is not known. It is 7ps at low t e m p e r a t u r e in b u l k GaAs (Yon der Kinde and others 1980); Kash and others, 1985) and is expected to be less ( 3ps) at room temperature. P h o n o n r e a b s o r p t i o n is w i d e l y r e g a r d e d as r e s p o n s i b l e for the a n o m a l o u s l y low rate of e n e r g y - r e l a x a t i o n of hot electrons in q u a n t u m wells. (See the reviews of Shah, 1986 and Lyon, 1986.) S u b s t a n t i a l a b s o r p t i o n of optical phonons w o u l d d e s t r o y NDR by p r o v i d i n g a m e c h a n i s m for electrons to jump over the e m i s s i o n t h r e s h o l d b a r r i e r and
683
684
V" (I07cm s" ) 1.5 _
0
2.0
3
ND
~
~~
~I0
1,5-
.
.
.
Fig. i
"/
I
2
I
3
l
I
4 5 [ ( kVcm -i )
6
T h e e f f e c t of h o t p h o n o n s on s c a t t e r i n g i n d u c e d NDR. T h e n u m b e r s on the c u r v e s a r e t h e p h o n o n l i f e t i m e in p i c o s e c o n d s .
~
2.0
I017 cm -3
.
[
I
=
"
N O = 10 18 cell -3
~
.--~
/
-~.
/
V (107cms-']
"~
/ / ~ " ~ ~ ~
ND = lOITcm-t
.~.-~.
_/ I
I
[
I
t
I
1
2
3
4.
5
6
6, ( kVcm-'} Fig.2
T h e e f f e c t of c o u p l e d p h o n o n - p l a s m o n modes. S t a t i s t i c a l d e g e n e r a c y is i n c l u d e d b u t h o t p h o n o n e f f e c t s are not. Dashed lines without coupled modes.
685
reach higher energies. We have m o d e l l e d this p r o c e s s u s i n g a w a v e v e c t o r - i n d e p e n d e n t lifetime, ~, for the polar o p t i c a l phonons ranging from 3ps to lOps. The result is shown in Fig. i. For an A I G a A s d o p i n g d e n s i t y of lOl7cm -3 N D R d i s a p p e a r s w h e n T = lOps and is only m i l d l y a f f e c t e d when T = 3ps. For T = 7ps the NDR is p r e s e n t but weak.
COUPLED PHONON-PLASMONS We have not a t t e m p t e d a full d e s c r i p t i o n of the complex i n t e r a c t i o n of electrons with coupled p h o n o n - p l a s m o n m o d e s in our q u a s i - 2 D system. Instead, we have assumed a simple bulk a n t i - s c r e e n i n g approximation. In this we have assumed that dynamic screening is d e s c r i b e d by the bulk L i n d h a r d d i e l e c t r i c function, and we have used this to obtain the d i s p e r s i o n r e l a t i o n of the upper f r e q u e n c y (W+(q)) coupled mode. The lower f r e q u e n c y mode (W_(q)) is assumed to be h e a v i l y damped and hence the i n t e r a c t i o n with this mode is r e g a r d e d as having already b e e n d e s c r i b e d by the e l e c t r o n - e l e c t r o n interaction. This a s s u m p t i o n will fail when the l o w - f r e q u e n c y c o m p o n e n t b e c o m e s p h o n o n - l i k e at high e l e c t r o n densities, but then it is h e a v i l y screened and so the i n t e r a c t i o n b e t w e e n it and an e l e c t r o n can be again ignored. We therefore consider the i n t e r a c t i o n solely w i t h the upper f r e q u e n c y component. This is similar to the a s s u m p t i o n made by Abou E1-Ela and others (1987) in their d i s c u s s i o n of dynamic s c r e e n i n g effects in GaAs except that we include the v a r i a t i o n of f r e q u e n c y with e l e c t r o n density. We refer to this model as the a n t i - s c r e e n i n g a p p r o x i m a t i o n since it d e s c r i b e s an e n h a n c e d i n t e r a c t i o n with the p h o n o n - l i k e mode at d e n s i t i e s up to about ixlOl8cm -3. The effect on NDR is shown in Figk 2 (ignoring hot phonons but including degeneracy). C o u p l i n g b e t w e e n p h o n o n s and p l a s m o n s weakens NDR at lOl7cm -3 and d e s t r o y s it at lOl8cm -3. The cause lies in the increased d i s p e r s i o n i n t r o d u c e d by coupling - the e m i s s i o n threshold is no longer sharp. Moreover, a n t i s c r e e n i n g weakens with i n c r e a s i n g electron temperature. Both of these effects w e a k e n NDR. Thus a l t h o u g h there is an e n h a n c e m e n t of the e l e c t r o n - p h o n o n interaction, that e n h a n c e m e n t is strongly w a v e v e c t o r dependent, and hence also is the frequency shift, and this tends to make the e m i s s i o n t h r e s h o l d diffuse. A diffuse t h r e s h o l d allows the electrons to reach h i g h e r energies, and h o t t e r electrons couple more w e a k l y with the phonons.
DEGENERACY The effect of i n c l u d i n g s t a t i s t i c a l d e g e n e r a c y is to reduce the v e l o c i t y by about 10%. The v e l o c i t y is u n i f o r m l y reduced as a result of fewer electrons being able to transfer to the well. It has little effect on NDR.
SUMMARY AND CONCLUSIONS W h e t h e r NDR survives h o t - p h o n o n and coupled mode effects depends c r u c i a l l y on the p h o n o n lifetime and d o p i n g density. Our analysis p r e s e n t e d here is, of course, far from complete. A l t h o u g h our a s s u m p t i o n s of b u l k formulae for phonons and p l a s m o n s need improving, they are p r o b a b l y not too bad. We have also not yet been able to i n c o r p o r a t e hot phonons, coupled modes, and degeneracy, into one s i m u l a t i o n - s o m e t h i n g that c l e a r l y w o u l d be desirable. Nevertheless, we can say with c o n f i d e n c e that NDR is u n l i k e l y to survive at doping d e n s i t i e s as high as lol8cm -3. It is likely, however, that NDR survives at doping d e n s i t i e s near and p r e f e r a b l y just b e l o w iol7cm -3, p r o v i d e d the p h o n o n lifetime is no higher than 7ps.
ACKNOWLEDGEMENT This work was supported by the U.S. Office of Naval Research.
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