On the effect of plasmon-phonon coupling and phonon reabsorption on scattering-induced NDR

Superlattices and Microstructures, Vol. 2, No. 2, 1986

ON M

159

EFFECT OF PLASMON-PHONON COUPLING AND PHONON REABSORPTION ON SCATTERING-INDUCED NDR B.K. Ridley Department of Physics, Colchester,

University England.

of Essex,

(Received 9 December 1985)

Scattering-induced NDR h a s b e e n p r e d i c t e d i n AIGaAs/GaAs q u a n t u m w e l l s n e g l e c t i n g the c o u p l i n g of p l a s m o n s and phonons and phonon r e a b s o r p t i o n . The e f f e c t o f t h e s e on NDR i s d i s c u s s e d a n d i t i s c o n c l u d e d t h e t h e r a n g e of doping densities over which scattering-induced NI)R s h o u l d o c c u r i s reduced. The o p t i m u m d o p i n g d e n s i t y s u g g e s t e d b y t h e d i s c u s s i o n is of order 1017cm-3.

1.

Introduction Scattering-induced negative differential resistance (NDR) 1 , 2 , 3 , 4 , 5 is predicted for a range of carrier concentration in quasi-2D systems as a consequence of the abrupt optical-phonon emission threshold. In the AIGaAs/GaAs s y s t e m Monte C a r l o s i m u l a t l o n 4 , 5 h a s shown t h a t s c a t t e r i n g - i n d u c e d NDR i s t o be e x p e c t e d f o r d o p i n g d e n s i t i e s I 0 1 7 - 1 0 1 8 c m - 3 , and t h a t t h e t h r e s h o l d f i e l d is significantly below that for real-space transfer and intervalley transfer at r o o m - t e m p e r a t u r e and b e l o w f o r w e l l - d e p t h s not too small. A l t h o u g h t h e Monte C a r l o simulation took into account electron-electron scattering it did not treat the coupling of the collective oscillations of the electron gas with the polar-optical phonons. This coupling leads to screening or anti-screening of the electron-phonon interaction and i t introduces plasma oscillations w h i c h may h e an a d d i t i o n a l source of energy relaxation. Another factor of importance in degenerate material is phonon reabsorption. Normally it is assumed that phonons, once emitted, thermalize, but if instead they are reahsorbed with appreciable probability the energy relaxation r a t e w i l l be r e d u c e d . T h e s e e f f e c t s may be e x p e c t e d t o b e significant over the range of carrier densities which is necessary for establishing t h e NDR. I t i s t h e r e f o r e important to establish the role of coupled modes a n d t h e i r i n t e r a c t i o n s with single-particle excitations. This is a difficult problem, but certain general f e a t u r e s c a n be i d e n t i f i e d in a semi-quantitative w a y , and i t l s t h e m a i n p u r p o s e o f t h i s n o t e t o d e s c r i b e how t h i s may b e d o n e . We h o p e t o show t h a t scattering-induced NDR s u r v i v e s t h e c o u p l i n g

0749-6036/86/0201 59 +06 $02.00/0

o f p h o n o n a n d p l a s m a modes a n d p h o n o n reabsorption. The a c c o u n t b e g i n s w i t h a s h o r t description o f t h e NDR m e c h a n i s m . T h i s i s f o l l o w e d b y a b r i e f summary o f t h e s a l i e n t features of coupled modes, in which it is shown how t h e p h o n o n - l i k e c o n t e n t and t h e electron-coupling s t r e n g t h may b e q u a n t i f i e d for long-wavelengths. The e f f e c t on t h e scattering r a t e i s t h e n d i s c u s s e d and conclusions are drawn concerning the scattering-induced N'DR m e c h a n i s m .

2.

Scatterina-induced NDR When e l e c t r o n - e l e c t r o n collisions are frequent enough to maintain a Maxwell-Boltzmann distribution for energy E~l~, where ~ is the polar-optical-phonon energy, but not frequent enough to establish a ~ distribution for E~, then there exists the condition for scattering-induced NDR. U n d e r t h e c i r c u m s t a n c e s t h e distribution function exhibits a knee at E--~w, f a l l i n g away r a p i d l y t o w a r d s l a r g e r energies. These conditions can occur in q u a s i - 2 D s y s t e m s as a c o n s e q u e n c e of t h e abrupt phonon-emission threshold associated with the existence of the non-zero density o f s t a t e s i n a 2D s u b - b a n d . If inelastic scattering i s weak b e l o w E--'li~ t h e e f f e c t i s t o c a u s e an a b r u p t s w i t c h i n g - o n o f a s t r o n g energy-relaxation m e c h a n i s m a t E---~m. T h i s acts like a lid preventing the electron gas expanding to high energies. Increasing the electrlc f i e l d h e a t s up t h e g a s b u t , s i n c e higher velocities are discouraged by the phonon-emission lid, the drift velocity can only decrease as the temperature of the gas r i s e s and m o t i o n i s r a n d o m i z e d . T h i s c a n be s i m p l y d e p i c t e d i n a m o d e l which takes the distribution f u n c t i o n for

© 1986 Academic Press Inc. (London) Limited

Superlattices and Microstructures, Vol. 2, No. 2, 1986

160 E < ~ t o be t h e u s u a l t r u n c a t e d harmonic expansion viz. f([)

= fo + f l

cos

spherical

0

13 (1)

and

IL fo = A e x p ( - E / k B T e)

fl

= -e~vrm

~fo

(2)

~0

/

(3)

/

~E 7

where Te is t h e e l e c t r o n temperature, ~ is the electric field, v is the group velocity and rm i s t h e m o m e n t u m - r e l a x a t i o n time. V e r y r o u g h l y , we c a n i m a g i n e t h e phonon-emission lid to cut off the distribution f u n c t i o n s h a r p l y a b o v e E - - ~ and we c a n i g n o r e t h e c o n t r i b u t i o n of electrons a b o v e E---'ti~ t o t h e c u r r e n t . The c u r r e n t is then proportional to fl alone. As t h e electron-temperature rises the gradient of fo f a l l s and, with it, fl hence the a p p e a r a n c e o f NDR. (A f u l l e r discussion is to be found in r e f . ( 2 ) ) .

/ 4-

Fig.

3.

Couvled ~odes The r e q u i r e m e n t o f a h i g h e l e c t r o n density to establish a MB d i s t r i b u t i o n below E = ~ m e a n s t h a t we m u s t t a k e i n t o a c c o u n t t h r e e new f a c t o r s v i z . 1) The e l e c t r o n gas will tend to screen the interation with the polar optical phonon. 2) to the

Emission of plasmons will energy relaxation.

3. Reabsorption of emitted s l o w down e n e r g y r e l a x a t i o n .

contribute

quanta

will

s(~,s)

= o

(4)

The p e r m i t t i v i t y i s t h e sum o f two components - a lattice c o m p o n e n t s L and a

/%.

f ~ J / /

1 Maximum and m i n i u m w a v e v e c t o r s i n GaAs ( a s s u m i n g a p a r a b o l i c c o n d u c t i o n b a n d ) f o r l[~=IwL, t h e I o n g l t n d l n a l o p t i c a l - p h o n o n e n e r g y , and f o r a b s o r p t i o n from t h e F e r m i l e v e l o r e m i s s i o n from EF+I~ L.

free-electron component s e. The w a v e v e c t o r s of interest a r e t h o s e f o r w h i c h e n e r g y and crystal momentum c a n b e c o n s e r v e d i n a n interaction w i t h an e l e c t r o n viz.

+_k([l±(~/E)]1/2-1 All of these appear to militate against our NDR m e c h a n i s m : the phonon-emission lid will be weakened by s c r e e n i n g and t h e reabsorption o f p h o n o n s , and i t s e f f e c t w i l l be r e d u c e d by i n e l a s t i c collisions with plasmons with energy less than'~. To assess the importance of these processes for scattering-induced HDR i s n o t e a s y , b e c a u s e it is necessary to consider the whole question of coupled phonon-plasmon modes, a task which has yet to be carried out satisfactorily for bulk material, let alone f o r 2D s t r u c t u r e s . Since progress in this topic is significantly more advanced for bulk material, the subsequent discussion here will be directed to bulk modes. The s p e c t r u m o f c o u p l e d m o d e s i s obtained from the condition that the permittivity vanishes viz.

/

// //

~ q ~ k([l±(~/E)]

1/2+1) (5)

w h e r e k i s t h e w a v e v e c t o r and E i s t h e energy of the electron, and t h e u p p e r s i g n is for absorption, the lower for emission, of a quantum of energy ~. For a degenerate electron g a s , k = k F and E = E F . This case is illustrated in Fig.l f o r GaAs, a s s u m i n g a parabolic conduction band. The m a g n i t u d e o f the wavevector is large compared with those associated with polaritons, but it is tiny in comparison with the extent of the Brillouin zone. As a c o n s e q u e n c e t h e dependence of purely lattice m o d e s on w a v e v e e t o r may b e s a f e l y i g n o r e d . In that c a s e i t i s w e l l known t h a t :

,L o, ,°Fo2 21 L~2-~r2j

(6)

w h e r e ~L, ~T a r e t h e l o n g i t u d i n a l and transverse angular frequencies a n d s~ i s

the

Superlattices and Microstructures, Vol. 2, No. 2, 7986

161

R+

~/~

,"

///

..........

I0"

_ _"..'- I_,..,:_ _ _ .

._

I0't

10"r

Fig.

10's

n Icm-~}

10'+

2 Frequencies and normalized scattering s t r e n g t h R± i n b u l k OaAs.

high-frequency permittivity. If, for simplicity, it is assumed that the plasma frequency is similarly independent of wavevcctor, which it is for small wavevectors, then the contribution to the total permittivity made b y t h e e l e c t r o n s is given by the well-known expression:

agrees with the result o f Kim e t the fractional phonon content of given by

al 7. Thus a mode i s

~2_~p2 S(m) =

(11) 2~2-~p2-tOL2

ee(~) = -

~P.._~. s . ~2

(7)

where

~p2

=

and the

ne 2 __

effective

charge

mesm

given

by

~2(w2-tOT 2 )

(e*) 2 = eL 2 The mode f r e q u e n c i e s , long-wavelength limit, eq.(4) which becomes:

m4-~2(~L2+~p2)

is

(8)

in this are obtained

+ tOT2Wp2 = 0

from

(9)

In general the modes have a mixed phonon-plasmon character. What i s r e l e v a n t here is the strength of the interaction o f a c o u p l e d mode w i t h a n electron. The a p p r o a c h t o t h e electron-polar-optical-mode interaction which uses the concept of the Callen effective charge 6 can be exploited here. It is straightforward to show that if the polarization P associated with the coupled mode i s t a k e n t o b e

(12)

(2~2-~p2-WL2)(WL2-O,,T 2 )

where

eL is

eL 2

=

the

Callen

MVoeo2~L 2

charge, (1

_

Sm

1 _ ) Ss

(13)

and s s is the static permittivity, M the reduced mass of the oscillator. All of this can be subsumed into a normalized scattering strength: oJ~L ( ~ 2 - ~ T 2 ) R(m)

=

(14)

(2~2-~p2-mL2)(mL2-t,~ 2 ) p = e*U

(10)

Vo

where U is the relative displacement, Vo i s t h e v o l u m e o f t h e u n i t c e l l a n d e* i s t h e effective charge, a quantification of phonon content and charge can be derived which

R(m) i s u n i t y f o r p u r e p h o n o n s c a t t e r i n g . Fig.2 shows the frequencies m+ a n d ~_ f o r the two branches of coupled modes along with the normalized scattering strengths for GaAs.

162 Starting with the situation at carrier densities of i017cm -3 and below we observe that the scattering strength of the phonon-like, w+ mode is enhanced with increasing carrier concentration. This is anti-screening, an effect first noted by Ehrenreich $ and by Doniach 9 in 1959. The frequency of this mode remains close to ~L" The plasmon-like, ~_ mode is of low frequency, but the scattering strength is similar to that for ~+. This suggests, at first, that energy relaxation via low-frequency plasmon emission cannot be disregarded, but this would be to ignore the implication of a scattering rate of the same order as the frequency, which is the situation in degenerate GaAs. In this material a scattering strength R(~) of unity corresponds to an interaction rate, F , of order i O 1 3 s , w h i c h i s a b o u t a t e n t h o f ~L" Thus ~L/F is large and the phonon-like mode is relatively long-lived even in degenerate material. This is not the case for the plasmon-like mode f o r w h i c h m - / r < I . Such a mode i s h e a v i l y d a m p e d a n d c a n n o t t r a n s p o r t energy away. Energy transferred from an electron is rapidly transferred back to another electron, so this process is really part of the electron-electron interaction, which is already i n v o k e d t o p r o v i d e t h e MB distribution b e l o w E---h~+. E v e n f o r n
Superlattices and Microstructures, Vo/. 2, No. 2, 1986 low and intermediate concentrations, screened at high concentrations. What these conclusions suggest regarding scattering-induced NDR may be summarized as follows: I) Incorporating the interaction with Landau-damped plasmon-like modes into the overall electron-electron energy and momentum randomizing process, with suitable renormalization, is not likely to alter markedly the estimate of the net electron-electron scattering rate. 2) Incorporating screening effects high densities w i l l r e d u c e t h e maximum density a t w h i c h NDR i s a c h i e v a b l e .

at

3) Incorporating antiscreenin8 effects a t low d e n s i t i e s will tend to emphasise the strength of the phonon-emission lid. Thus, the prediction of scattering-induced NDR i n the AIGaAs/GaAs system 4,5 for 1017cm-3
Phonon Reabsorption There is, however, another effect which has, in fact, been mentioned in our discussion in connection with plasmon-like modes but not yet in connection with phonon-like modes, that is, the rapid reabsorption of quanta emitted in degenerate material. The p h o n o n - e m i s s i o n lid is efficient only if the emitted phonons escape from the electron gas to dissipate their energy elsewhere and thermalize. In strongly degenerate material t h i s may o c c u r too slowly for efficient energy relaxation. Indeed, the anomalously long energy relaxation t i m e s o b s e r v e d i n some experiments IO'll are likely to be caused by such reabsorption. This is an effect which will be weak in non-degenerate and weakly degenerate material, a n d may b e e v e n w e a k e r in isolated quantum wells. Nevertheless, it is a process which is likely to reduce the upper limit of the range of doping densities for scattering-induced NDR. T h i s may b e s e e n f r o m t h e f o l l o w i n g analysis. Let nq be the phonon occupation n u m b e r f o r t h e mode w i t h f r e q u e n c y ~L a n d wavevector ~. The n e t r a t e o f i n c r e a s e arising from the unscreened Frohlich interaction with electrons in a parabolic band is dnq = Pq dt " w h e r e To i s t h e scattering time

(15)

To characteristic constant given

electron by

Superlattices and Microstructures, Vol. 2, No. 2, 1986 1

e2~L1/2m * 1 / 2

~o

(m* = e f f e c t i v e p

= 1 q 2

1

23/2~ 3/2

1

( em

mass),

(16J

¢s )

and Pq i s g i v e n by

'B~L ~,1/2 ~E-~q) ! (nq+l) f

f(E)(1-f(E-+lim))dE

Z1 nq

f(E)(1-f(E÷~)dE

E2 .~2q2

Eq =

__

,

2m*

1 (17)

163 per unit energy interval in the vicinity of i s 1 . 1 4 x 1019 ( e V ) - l c m -3 and t h u s = ~ L N ( ~ L ) ~ 8 . 0 6 • 1017cm-3- T h i s i s a l s o c l o s e t o t h e d e n s i t y of s t a t e s i n a s q u a r e w e l l of w i d t h 100 ~ . Now f o r s c a t t e r i n g - i n d u c e d NDR o n l y a s m a l l f r a c t i o n of t h e t o t a l e l e c t r o n p o p u l a t i o n i s above the emission threshold. Even s o , c o n d i t i o n (21) i s u n l i k e l y t o be s a t i s f i e d for a c a r r i e r d e n s i t y of 1018cm - 3 . It should, h o w e v e r , be s a t i s f i e d for a carrier density a r o u n d 1017cm -3 and b e l o w .

~L,

(h~L+Eq)2 _ , 4Eq

E1 = _

(hm-Eq) 2 E2

=

5.

4Eq and i n t h i s d e r i v a t i o n o n l y t h e s p h e r i c a l p a r t of e l e c t r o n d i s t r i b u t i o n function, f(E), has been taken. (A s i m i l a r e x p r e s s i o n for the non-degenerate case has been given by C o l l i n s and y u I 2 . ) F o r phonons e m i t t e d n e a r t h r e s h o l d Eq = ~ L " and pq = _ _ 1 2"fi~L

I (nq+l)f ~f(E)(1-f(E-hm))dEL ~L

f f(E)(l-f(E+~))dE o

]"

nq x

(18)

Fo r s c a t t e r i n g - i n d u c e d NDR t o work nq must be l e s s t h a n u n i t y . Ouantifyin 8 the r e l a x a t i o n o f phonons v i a n o n - e l e c t r o n i c p r o c e s s e s by t h e t i m e c o n s t a n t Zq, we o b t a i n at steady state: nq = ~ q P q To

(19)

I f nq i s t o be s m a l l , Pq must be d e t e r m i n e d p r l n c i p a l l y by s p o n t a n e o u s emission. The c o n d i t i o n f o r n e g l e c t i n g phonon r e a b s o r p t i o n t h e n becomes 1

J"

f(E)(l-f(E-'h~))dE

hWL ~m L

<< 2~° ~q

(20)

I n GaAs To = 114 f s and Zq, t h o u g h n o t w e l l - k n o w n , i s t h o u g h t t o be a b o u t 5 p s I I , g i v i n g a f a c t o r of 0 . 0 4 6 on t h e r i g h t . The c o n d i t i o n can be s a t i s f i e d o n l y i f t h e e l e c t r o n g a s above t h e t h r e s h o l d i s non-degenerate. If f(E) falls off rapidly a b o v e . ~ L we can e x p r e s s t h e above c o n d i t i o n i n t e r m s o f t h e d e n s i t y of s t a t e s i n t h e v i c i n i t y o f +lfmL, t h u s : nhm << 2~° ~ .05 l~w ~q

(21)

where nliw i s t h e c a r r i e r d e n s i t y above ~mL" Fo r b u l k GaAs N ( ~ L ) , t h e d e n s i t y o f s t a t e s

Conclusion Our d i s c u s s i o n ha s b e e n b a s e d on b u l k excitations, but although there are w e l l - k n o w n d i f f e r e n c e s b e t w e e n b u l k and 2D situations, these differences are unlikely to cause major concern. The r e p l a c e m e n t of b u l k modes by s l a b modes and 3D p l a s m o n s by 2D p l a s m o n s i s n o t e x p e c t e d t o i n t r o d u c e large effects in the wavevector range of interest. P l a s m a - w a v e d i s p e r s i o n and damping s h o u l d , o f c o u r s e , be t r e a t e d more fully, but the qualitative conclusions of o u r d i s c u s s i o n o u g h t n o t t o be s e r i o u s l y affected. As f a r as t h e c o u p l i n g o f p b o n o n s and plasmons is concerned this appears not to a f f e c t i n any s e r i o u s q u a n t i t a t i v e way t h e p r e d i c t i o n o f s c a t t e r i n g - i n d u c e d NDR i n t h e AIGaAs/GaAs s y s t e m f o r d o p i n g d e n s i t i e s 10171018cm - 3 ) s t r o n g electron-electron scattering already d e s t r o y s NDR, and so t h e e f f e c t i s t o make more a b r u p t t h e c u t - o f f o f t h e NDR me c ha ni s m a r o u n d 1018cm-3. A more a b r u p t c u t - o f f i s g o i n g t o be a s s i s t e d by t h e a n t i s c r e e n l n g t o screening transition which takes place in t h e v i c i n i t y of 101gem - 3 . Phonou r e a b s o r p t i o n ( e q u i v a l e n t t o h o t phonons) w i l l tend to p e r f o r a t e the phonon-emission lid at high carrier densities, as r e a b s o r p t i o n b e f o r e t b e r m a l i z a t i o n becomes more p r o b a b l e . This process will tend to lower the upper limit of c a r r i e r c o n c e n t r a t i o n b e l o w 1018cm - 3 . On the o t h e r band, the h i g h e l e c t r o n t e m p e r a t u r e and t h e e a s e of e s c a p e of phonons from a s i n g l e t h i n l a y e r w i l l b o t h a c t t o r e d u c e t h e p r o b a b i l i t y of reabsorptiou. None of t h e s e d e l e t e r i o u s e f f c c t s d i s c u s s e d above a p p e a r l i k e l y t o have much i m p a c t when t h e d o p i n g d e n s i t y i s as low a s lO17cm - 3 - a n t i s c r e e u i n g i s weak, t h e l o w - f r e q u e n c y p l a s m o n s a r e h e a v i l y damped, t h e h o t - e l e c t r o n g a s i s n o n - d e g e n e r a t e and phonon r e a b s o r p t i o n s h o u l d be weak. I t may be c o n c l u d e d t h e r e f o r e t h a t s c a t t e r i n g - i n d u c e d NDR s h o u l d be o b s e r v a b l e a t and a r o u n d t h i s d o p i n g d e n s i t y .

Superlattices and Microstructures, Vol. 2, No. 2, 1986

164 Acknowledzement The a u t h o r i s g r a t e f u l to F. R i d d o c h and M. B a b i k e r f o r many u s e f u l d i s c u s s i o n s . T h i s work was funded by t h e O f f i c e of Naval Research. References 1. B.K. R i d l e y 1982 J . P h y s . C: S t a t e P h y s . 1._.555899.

Solid

2.

Solid

3.

4.

5.

B.K. R i d l e y 1984 J . P h y s . C: S t a t e P h y s . 1._7_75357

B.K. R i d l e y 1985 P r o c . 1 7 t h I n t . Conf. on t h e P h y s i c s o f S e m i c o n d u c t o r s 401 ( e d . J . D . Chadi and W.A. H a r r i s o n , S p r i n g e r - V e r l a g , N.Y.) M.A.R. A1-Mudares and B.K. R i d l e y 1985 P r o c . Conf. Hot E l e c t r o n s I n n s b r u c k ( t o be p u b l i s h e d ) M.A.R. A I - M u d a r e s and B.K. R i d l e y 1986 J . P h y s . C: S o l i d S t s t e P h y s . ( t o be published)

6.

H.B. C a l l e n

7.

M.E. Kim, A. Das and S.D. P h y s . Rev. B18 6890

8.

H. E h r e n r e i c h 1959 S o l i d s 8 130

9.

S. D o n i a c h 849

10. J .

1949

1959

P h y s . Rev. 7_66 1394

J.

Proc.

Senturia

1978

P h y s . Chem.

Phys.

Soc. 7-3

Shah, A. P i n c z u k , A.C. G o s s a r d and W. Wiegmann 1985 P h y s . Rev. L e t t .

54 2045 11. J . F . Ryan, R.A. T a y l o r , A . J . T u r b e r f i e l d , A. Macid, J.M. W o r l o c k , A.C. G o s s a r d and W. Wiesmann 1984 P h y s . Rev. L e t t .

53 1841 12. C.L. C o l l i n s and P.Y. Yu Rev. B3___O04501

1984

Phys.