HOT ELECTRON POPULATION INVERSION AND BULK NDC IN SEMICONDUCTORS

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HOT ELECTRON POPULATION INVERSION AND BULK NDC IN SEMICONDUCTORS

V. Kozlov

To cite this version:

V. Kozlov. HOT ELECTRON POPULATION INVERSION AND BULK NDC IN SEMICONDUC- TORS. Journal de Physique Colloques, 1981, 42 (C7), pp.C7-413-C7-420. �10.1051/jphyscol:1981751�.

�jpa-00221688�

HOT ELECTRON POPULATION INVERSION AND BULK NDC IN SEMICONDUCTORS

V.A. Kozlov

Institute of Applied Physios, Academy of Sciences of the USSR, Gorky, USSR

Résumé. - L'inversion de population due à la transition radiative entre les sous-bandes de trous lourds et légers donne une conduc- tivité différentielle négative, dans la bande submillimétrique et dans l'infrarouge lointain. Les procédés de création d'une surpo- pulation de trous légers, dans des échantillons de Ge-p pur, à basse température, soumis à de forts champs électriques et magné- tiques sont discutés ici. La surpopulation de la sous-bande de trous légers est due à l'accumulation de trous légers dans des ré- gions spécifiques de l'espace des moments où l'énergie des trous est plus faible que celle du phonon optique. Des résultats obtenus par une simulation de Monte-Carlo sur cette surpopulation sont aus- si présentés ici.

Abstract« - The population inversion of the radiative transition between light and heavy hole subbands results in negative diffe- rential conduvtivity in submillimetre and far infrared waveband.

The means of producing of the liglit hole overpopulation in pure p-Ge samples at low lattice temperatures in strong electric and magnetic fields are discussed in the report. The overpopulation of light hole subband arises due to accumulation of light holes in the specific regions of the momentum space where the energy of holes is smaller than the optical phonon energy. The results of Monte Carlo simulation of the overpopulation are also presen- ted in this report.

Valence band of many semiconductors consists of light ( t ) and heavy ( h ) hole subbands. If sufficient increase in the light hole concentration above its equilibrium value ( m ^ / m j ' is produced, then the population inversion of radiative transition between light and heavy hole subbands may arise (Jig. 1a). The overpopulation may lead to generation and amplification of far infrared and submillimet- re radiation in bulk semiconductors. In the report we describe some ways of producing the overpopulation of light hole subband in crossed electric E and magnetic B fields, when hot holes strongly interact with optical phonons. The calculations and measurements refer to pu- re p-Ge as typical example of semiconductor. The overpopulation oc- curs in pure semiconductors at low lattice temperatures due to accu- mulation of light holes in the spindle region of momentum space. The energy of holes in this region does not exceed optical phonon energy.

Even at low lattice temperatures K hot holes in pure p-Ge strongly interact with optical phonons, so there is a dif- ference in behavior of carriers with energies more or less than opti-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981751

C 7 - 4 1 4 JOURNAL DE PHYSIQUE

c a l phonon energy h O o . I n a c t i v e energy r e g i o n

& > h ~ ) ~ ) h o l e s can emit o p t i c a l phonons spontaneously and t h e c o l l i s i o n frequency i s r a t h e r high: go

10 ''S-' . ^{I n} p a s s i v e energy r e g i o n t h i s s c a t t e r i n g p r o c e s s i s i m p o s s i b l e and t h e c o l l i s i o n frequency i s d e f i n e d by acous- t i c a l o r i m p u r i t y s c a t t e r i n g . The i m p u r i t y s c a t t e r i n g i n pure p-Ge samples i s low and can be n e g l e c t e d i n t h e f a c i l e c a l c u l a t i o n s . There- f o r e t h e c o l l i s i o n frequency i n p a s s i v e r e g i o n i s r a t h e r s m a l l . For spontaneous a c o u s t i c a l phonon emission dA 2 * 1 0 4 0 ~ - '

A t a p p r o p r i a t e v a l u e s of e l e c -

1 t r i c f i e l d i t i s e a s y t o sa- t i s f y s t r e a m i n g c o n d i t i o n s :

To

< f E 4 ' r ~

where tE = mwV0 /e E i s t h e f r e e f l i g h t time through pas- s i v e r e g i o n under t h e i n f l u - ence of e l e c t r i c f i e l d E

m 1s t h e velo-

v, =m$--

c i t y on t h e boundary of pas- s i v e r e g i o n ; T~~ 9;'; ^TA= ' $; .

Under s t r e a m i n g c o n d i t i o n s t h e r e i s a c y c l i c motion of h o l e s w i t h t h e p e r i o d appro- x i m a t e l y equal. t o T E [ l ] .

h o l e almost without c o l l i s i - ons t r a n s i t s through t h e pas- s i v e r e g i o n , p e n e t r a t e s i n t o

( a ) - Valence band of p-Ge; t h e a c t i v e one and r a p i d l y

f l i g h t t r a j e c t o r i e s of

heavy ( l e f t ) and

h t ( r i g h t ) h o l e s back because the 'P- i n v e l o c i t y s p a c e ; ?c) - R e l a t i v e t i c a l phonon emission. I n t h e v o l p e of s p i n d l e and p a s s i v e r e g i o n s , streaming without magnetic

V

^{i s} volume of s p i n d l e r e g i o n ,

i s volwne of p a s s i v e region. f i e l d t h e r e l a t i v e l i g h t h o l e

2

c o n c e n t r a t i o n fle/tlh d r o p s

down t o ( m* /m;)

. T h i s drop a r i s e s as f r e e f l i g h t time f o r l i g h t h o l e s e i s l e s s t h a n t h a t f o r heavy ones.

Thus, f o r t h e i n c r e a s e of l i g h t h o l e c o n c e n t r a t i o n above i t s

e q u i l i b r i u m v a l u e i t i s n e c e s s a r y t o fuLfiZ t h e f o l l o w i n g c o n d i t i o n :

f r e e f l i g h t time f o r l i g h t h o l e s should exceed t h e corresponding va-

l u e f s r heavy-holes. T h i s requirement can

achieved i n s t r o n g c r o s -

s e d E ^and -6 f i e l d s , when f r e e f l i g h t t r a j e c t o r i e s a r e c l o s e d and

l i g h t h o l e s may remain i n t h e p a s s i v e r e g i o n d u r i n g i t s f r e e f l i g h t ,

while heavy h o l e s keep s t r e a m i n g l i k e motion (Fig. l b ) . I n t h i s c a s e

time of heavy h o l e s i s YE,, =

. For s u f f i c i e n t l y s t r o n g e l e c t r i c f i e l d s

r E

- ^{To )} t h e c o n d i t i o n rEh <

i s e a s i l y

f u l f i l l e d ,

F r e e f l i g h t t r a j e c t o r i e s of h o l e s i n c r o s s e d c ^{and 5} f i e l d s i n t h e v e l o c i t y space comprise t h e s e t of circumferences w i t h common c e n t r e VC = C E /B . The p o s i t i o n of t h e c e n t r e of c i r c u m f e r e n c e s i s t h e same f o r l i g h t and heavy h o l e s (Pig. I b ) . The h o l e s move a l o n g c i r c u m f e r e n c e s with c y c l o t r o n f r e q u e n c i e s

= e B / mhl

and

U B e

= e6 / m e t

which a r e d i f f e r e n t f o r heavy and l i g h t h o l e s

(uBt 0 ~ ) h i n p-Ge). If t h e boundary v e l o c i t y of p a s s i v e r e g i o n V. exceeds [4] , t h e n i n t h e p a s s i v e r e g i o n t h e r e e x i s t s a s p i n d l e r e g i o n K of c l o s e d t r a j e c t o r i e s [ 51 , The r a t i o of volumes of s p i n d l e and p a s s i v e r e g i o n s v e r s u s VC is p l o t t e d i n Fig. Ic. The energy of t h e c a r r i e r s moving i n t h e s p i n d l e r e g i o n i s always l e s s t h a n ,

f r e e f l i g h t time of t h e c a r r i e r s i n s p i n d l e r e g i o n i s d e f i n e d by a c o u s t i c a l s c a t t e r i n g and i s approxima- t e l y e q u a l t o TA .

The boundary v e l o c i t y of p a s s i v e r e g i o n f o r l i g h t h o l e subband

v0, @iG'$Te exceeds t h e corresponding boundary v e l o c i t y

:

'

,

, f o r heavy h o l e subband.

v e l o c i t y V,

s a t i s f i e s t h e c o n d i t i o n s VOh

VC Vat t h e n t h e s p i n d l e r e g i o n

K t ^{i n} t h e l i g h t h o l e subband i s p r e s e n t , but heavy h o l e s p i n d l e r e g i o n i s a b s e n t [6,7] (Fig. I b ) . I n t h i s c a s e t h e h o l e s may be ac- cumulated i n t h e s p i n d l e r e g i o n of l i g h t h o l e s He ( c f . [ 51

be- cause f r e e f l i g h t time of l i g h t h o l e s i n K t i s e q u a l t o TA . ^The

motion of heavy h o l e s l o o k s l i k e streaming, The average f r e e f l i g h t time of heavy h o l e s i s approximately e q u a l t o rEh =

/ e E ^and

i s s m a l l e r t h a n TA .

I f t h e e l e c t r i c f i e l d i s l a r g e enough s o t h a t

t h e n i t i s p o s s i b l e t o e s t i m a t e t h e r a t i o of l i g h t and heavy h o l e concentra- t i o n s ne / h

by t h e f o l l o w i n g formula:

where V : ^{/V: i s} t h e r a t i o of volumes of s p i n d l e and p a s s i v e r e g i o n s i n l i g h t subband, t h i s r a t i o i s about 0.3 a t VC= V,, (Fig, Ic).

A t helium temperature i n pure p-Ge samples t h e r a t i o 2.A > /< reh' i s

approximately e q u a l t o

i n e l e c t r i c f i e l d s f o r which

.These

e s t i m a t i o n s show t h a t t h e accumulation of l i g h t h o l e s i n

r e g i o n

l e a d s t o t h e o v e r p o p u l a t i o n of l i g h t h o l e subband. The r e l a t i v e con-

c e n t r a t i o n of l i g h t h o l e s more t h a n 10 t i m e s exceed e q u i l i b r i u m

v a l u e

me

C7-4 16 JOURNAL DE PHYSIQUE

The overpopulation of l i g h t h o l e subband w a s simulated by Monte Carlo method. The s p h e r i c a l p a r a b o l i c subbands were used i n t h e calcu- l a t i o n s . E f f e c t i v e masses of l i g h t and heavy h o l e s were chosen equal t o me -0,043 m. and

=0.35 m, . ^In t h i s s i m u l a t i o n o p t i - c a l , a c o u s t i c a l and i o n i z e d impurity s c a t t e r i n g w i t h i n t r a b a n d and i n t e r b a n d t r a n s i t i o n s were t a k e n i n t o account. Acoustical s c a t t e r i n g at T = 4K w a s described by z e r o p o i n t approximation, and at T t 77 fi

by e q u i p a s t i t i o n approximation. I n t h e i n t e r m e d i a t e range of l a t t i c e temperatures, where such approximations a r e i n v a l i d s t r i c t c a l c u l a - t i o n s were made by modified s e l f - s c a t t e r i n g procedure [81 which de- s c r i b e d s c a t t e r i n g s t a t i s t i c a l l y without any approximations.

Fig. 2

( a ) - R e l a t i v e c o n c e n t r a t i o n of l i g h t and heavy h o l e s , ne i s c o n c e n t r a t i o n of l i g h t h o l e s , n h , - heavy ones, H

kOe;

( b ) - g a l l and d i s s i p a t i v e c u r r e n t s , b~ i s H a l l c u r r e n t ,

i s d i s s i p a t i v e one, d o h = e ( n h + n ) V o h , H

kOe; ( c ) - R e l a t i o n between n e / n

^and

(d? - Experimental v a l u e s of H a l l and d i s s i p a t i v e c u r r e n t

Simulation r e s u l t s a r e p r e s e n t e d i n Fig. 2a,b,c. The c o n s i d e r a b l e

i n c r e a s e of nt/nh t a k e s p l a c e at VC r VOh , when r e g i o n K h disappe-

ars (Pig. 2a). A t T = 4 K t h e c o n c e n t r a t i o n of l i g h t h o l e s may even

exceed t h a t of heavy ones. Fig. 2b shows t h e dependence of H a l l

and d i s s i p a t i v e i d c u r r e n t s on E f i e l d . The dashed l i n e s show

t h e jH and dd c u r r e n t s f o r t h e s i n g l e band model where only heavy

p o p u l a t i o n of l i g h t h o l e s i s t h e absence of t h e H a l l c u r r e n t drop at

VC

VOh w i t h t h e i n c r e a s e of f i e l d E . T h i s behavior of j i s due t o l i g h t holes. The average H a l l v e l o c i t y of l i g h t h o l e s i s al- H most e q u a l t o VC and exceeds t h e H a l l V e l o c i t y of heavy holes. This phenomenon can be used f o r experimental d e t e r m i n a t i o n of n, I hh ^by measuring 1

a t V t Voh . Fig. 20 shows J o i n t r e l a t i o n between

j,, /

^{,,h and 1 nh} . This r e l a t i o n f o l l o w s from t h e r e s u l t s of s i m u l a t i o n s of overpopulation f o r d i f f e r e n t magnetic f i e l d s , lat- t i c e temperatures and c o n c e n t r a t i o n s of i o n i z e d i m p u r i t i e s .

small value of heavy hole H a l l v e l o c i t y at VC

VOh i s r e s p o n s i b l e f o r t h i s r e l a t i o n .

The measured H a l l and d i s s i p a t i v e c u r r e n t s a r e given i n Fig. 2d.

The Measurements were t a k e n i n p-Ge samples w i t h h o l e concentratioris

P

2'

at n i t r o g e n and helium temperatures. The experiments

were made at pulsed regime w i t h low r e p e t i t i o n r a t e . The measuring technique was described elsewhere ( 9 1 . ^{A t T}

^{4 K} i n low e l e c t r i c f i e l d s t h e process of t h e impact i o n i z a t i o n of i m p u r i t i e s t a k e s p l a c e , t h a t i s

t h e value?

j

and

a r e given

h i g h enough f i e l d s when i m p u r i t i e s a r e f u l l y ionized. The comparison of e x p e r i - mentally obtained dependences of l,, w i t h t h e j o i n t r e l a t i o n given i n Fig. 2c shotvs t h a t t h e c o n c e n t r a t i o n of l i g h t h o l e s w a s 4 times h i g h e r t h a n t h e e q u i l i b r i u m value [l01 . Recently Komiyama e t al. [ l l )

( c f .

has measured t h e dependence of t h e H a l l f i e l d E ,,, ^{i n}

p-Ge on t h e magnetic f i e l d 6 . He observed t h e i n c r e a s e i n E, when

K h The accumulation of l i g h t h o l e s l e a d s t o overpopulation of l i g h t and K t r e g i o n s appeared, but t h e d a t a on nI I", were absent.

h o l e subband and t o p o p u l a t i o n i n v e r s i o n of t h e r a d i a t i v e t r a n s i t i o n

f!

h ^{[ 6 ]} . ^{It i s} p o s s i b l e t o make a f a c i l e c a l c u l a t i o n of t h e a b s o r p t i o n c o e f f i c i e n t of f a r i n f r a r e d r a d i a t i o n i n p-Ge under over- p o p u l a t i o n of l i g h t hole subband. In t h e s e c a l c u l a t i o n s f r e e c a r r i e r a b s o r p t i o n of heavy h o l e s was d e s c r i b e d by Drude formula w i t h charac-

- 1

t e r i s t i c s c a t t e r i n g frequency e q u a l t o

h . T r a n s i t i o n l ' h ^pro-

v i d e s t h e n e g a t i v e term of t h e a b s o r p t i o n c o e f f i c i e n t . This term i s p r o p o r t i o n a l t o t h e d i f f e r e n c e between d i s t r i b u t i o n f u n c t i o n s fh- ft .

The r a t i o f t / lh may be e v a l u a t e d i n accordance w i t h d e n s i t y of s t a t e s as fe I ^{f h % (h Iim, 3/3/(nh 1} m h 3 l ? - ~ h e r e f o r e f o r t h e n e g a t i v e a b s o r p t i o n c o e f f i c i e n t it i s necessary t o o b t a i n t h e consi- d e r a b l e i n c r e a s e of _e / _h ^{above i t s} e q u i l i b r i u m value (m, I ^m,,)

.

T r a n s i t i o n f,

h ^{i s} n o t resonant so t h e a b s o r p t i o n c o e f f i c i e n t i s

n e g a t i v e i n wide frequency range from 4 0 p t o 500p , but t h e abso-

C7-418 JOURNAL DE PHYSIQUE

l u t e value i s r a t h e r small. The evaluation of absorption c o e f f i c i e n t f o r wavelength 2

p a t the hole concentration p - ^{c K'}

gives the value

3 ₅ 10-'

m' . The d i r e c t numerical c a l - c u l a t i o n of the absorption c o e f f i c i e n t i s of much i n t e r e s t .

Discuss now t h e p o s s i b i l i t y of producing the overpopulation of 1igh-t; hole subband i n an i n t e n s e microwave f i e l d of c i r c u l a x p o l a r i -

?i.e.

3

( a ) - Free f l i g h t tra- j e c t o r i e s of heavy ( l e f t ) and l i g h t ( r i g h t ) holes i n a.c.

f i e l d of c i r c u l a r p o l a r i z a t i o n ,

Kt i s accwnulation r e i o n i n l i g h t hole subband; (be - ^Rela-

t i v e concentration of l i g h t and heavy holes, N I = Q , T=4K ,

(1)

W 8 h = $ D

( 2 ) U =

W b h

2qD

- -

r a t i o n g = ^E, ( i X . ~ 0 s u t - j y 3 s i n w t ) . We consider the

of heavy hole cyclotron resonance

%h , when heavy and l i g h t h o l e s manifest d i f f e r e n t behavior. The motion of heavy holes a t cyclotron resonance looks l i k e s t r e m i n g motion ( c f . 1131

and the accumulation region i n heavy hole subband i s always absent (Fig. 3a). Light h o l e s a r e mag- netized because of Wbe >> and t h e r e i s an accumulation re- gion kp i n t h e l i g h t hole

subband - ( ~ i g . 3 a ) . The cha- r a c t e r i s t i c v e X o c i t y f o r t h e c a r r i e r motion i n c i r c u l a r

F: ^{( a ) -} Average ve-

o c l t i e s of heavy and l i g h t

holes; (b) - Relative and

resistive,components of t h e

c u r r e n t , job= e (n,+n,)v,,

volume of K I r e g i o n depends on VB 1 Vp only, and i s expressed by t h e same formula as i n &.c. f i e l d s (Fig. I c ) , i t i s only n e c e s s a r y t o s u b s t i t u t e VC ^by V, . ^{I n high}

magnetic f i e l d s and at h i g h frequency of t h e pump i t i s i o s s i b l e t o

e

e

e n l a r g e l t h e r a t i o VI / ^V i n comparison w i t h d.c. f i e l d s where

v : / v , 5 0 . 3

^P

Pig. 3 and 4 show t h e r e s u l t s of Monte Carlo s i m u l a t i o n f o r t h e a.c. f o r a.c. f i e l d of c i r c u l a r p o l a r i z a t i o n . The r e l a t i v e c o n c e n t r a t i o n n f i e l d exceeds t h e maximum v a l u e n, 1 n h o b t a i n e d f o r &.c. 11 n ^h

f i e l d at t h e same r a t i o T A I ^{To (Fig.3b).} The average heavy h o l e ve- l o c i t y i s s a t u r a t e d w i t h t h e i n c r e a s e of t h e amplitude of e l e c t r i c f i e l d E, as i n t h e s t r e a m i n g (Pig. 4a). Heavy h o l e s p r o v i d e t h e r e - s i s t i v e component of t h e c u r r e n t (Fig. 4b). The motion of l i g h t ho- l e s i s of r e a c t i v e kind. The average l i g h t h o l e v e l o c i t y i s approxi- mately e q u a l t o VI . Light h o l e s produce a l a r g e r e a c t i v e compo- nent of t h e c u r r e n t Fig. 4b. The r e a c t i v e component may even exceed t h e a c t i v e one a t c y c l o t r o n resonance o f heavy h o l e s because of t h e i n c r e a s e of t h e l i g h t h o l e c o n c e n t r a t i o n . T h i s phenomenon may be used f o r experimental d e t e r m i n a t i o n of t h e o v e r p o p u l a t i o n of t h e l i g h t h o l e subband. The s a t u r a t i o n of t h e r e s i s t i v e component of t h e c u r r e n t w i t h t h e f i e l d i n c r e a s e r e s u l t s i n t h e drop of t h e r e s i s t i v e conducti- v i t y at a p p r o p r i a t e e l e c t r i c f i e l d s .

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