THE METHOD OF MOMENTS FOR THE ANALYSIS OF TRANSIENT HOT-CARRIER PHENOMENA

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THE METHOD OF MOMENTS FOR THE ANALYSIS OF TRANSIENT HOT-CARRIER PHENOMENA

L. Reggiani, R. Brunetti, C. Jacoboni

To cite this version:

L. Reggiani, R. Brunetti, C. Jacoboni. THE METHOD OF MOMENTS FOR THE ANALYSIS

OF TRANSIENT HOT-CARRIER PHENOMENA. Journal de Physique Colloques, 1981, 42 (C7),

pp.C7-111-C7-116. �10.1051/jphyscol:1981711�. �jpa-00221647�

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10, Tome 42, octobre 1981 page C7-111

THE METHOD OF MOMENTS FOR THE ANALYSIS OF TRANSIENT HOT-CARRIER PHENOMENA

L. Reggiani, R. Brunetti and C. Jacoboni

Gruppo Nazionale di Struttura delta Materia, Istituto di Fisica dell' Universi- ty di Modena, Via Campi 213/A, 41100 Modena, Italy

Résumé.- La méthode des moments est proposée pour analyser le comportement de la vi- tesse de dérive et du coefficient de diffusion en régime non stationnaire . Les ré- sultats obtenus avec la méthode de Monte Carlo dans Si, Ge, CdTe sont interprétés par référence aux collisions des porteurs avec le réseau et à la structure des bandes d'énergie de conduction.

Abstract.- The method of moments is here suggested to analyse the transient behaviour of drift velocity and diffusion coefficient. Monte Carlo results obtained for Si, Ge and CdTe are reported and interpreted with reference to scattering mechanisms and peculiarities of the band structure of these semiconductors.

1. Introduction.- The microscopic definition of the transport coefficients related to the phe nomenological current equation, in one dimension for simplicity,

j ( * , t ) =

^e

[^(x

^J

t)v

⁴

(E) - fyE)^*H*,t)~\ U)

j being the c u r r e n t density , n the c a r r i e r c o n c e n t r a t i o n , v

d

the drift v e l o c i t y , ~DQ the longitu dinal diffusion coefficient, E the e l e c t r i c field and e the unit c h a r g e , i s s t i l l a debated prob_

lem in the study of high-field t r a n s p o r t phenomena / 1 - 3 / . Gantsevich et al /L,/ demonstra_

ted that E q . ( l ) can be derived from the l i n e a r i z e d Boltzmann equation under the assumption t h a t : i) the c o n c e n t r a t i o n g r a d i e n t is small and, ii) the time i s long enough for the momentum d i s t r i b u t i o n function to be the s t a t i o n a r y o n e . Owing to t h e s e l i m i t a t i o n s , any a n a l y s i s of tra_

nsient phenomena will r e q u i r e an "a p r i o r i " o p e r a t i v e definition of the t r a n s p o r t c o e f f i c i e n t s . The aim of this p a p e r is to p r e s e n t the method of moments a s a suitable technique to over_

come the limitations cited a b o v e , and to i l l u s t r a t e some applications t o c a s e s of g e n e r a l inte r e s t .

2 . T h e o r y . - Under t r a n s i e n t conditions and high fields a one-dimensional c u r r e n t equation of the following form is assumed / 5 , 6 / :

! =• g, \ a,w •>• b Om •+. c J<w ( 2 ) J L ax D t J

where the coefficients a, , b and c can depend upon time and bias field. E q . ( 2 ) differs from E q . ( l ) for the p r e s e n c e of a t i m e - d e r i v a t i v e t e r m . The justification and the p h y s i c a l meaning of the different coefficients of E q . ( 2 ) can be d i r e c t l y obtained by using the continuity equation and the moments of the spatial d i s t r i b u t i o n / 6 / , thus obtaining:

cu = A- <*> (3)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981711

C7-112 JOURNAL

DE PHYSIQUE

S i - e l e c t r o n s

Longitudinal e f f e c t i v e mass

7

⁼ 0.916 Transverse e f f e c t i v e mass m t = 0.19 C r y s t a l density

P

= 2.33 gr/cm3 Sound v e l o c i t y vs = 9.04

lo5

cm/sec Non-parabolicity parameter

w

= 0.5 eV -1

Acoustic deformation p o t e n t i a l El = 9.0 eV

where b r a c k e t s indicate ensemble a v e r a g e . Under s t a t i o n a r y conditions ( i . e . in the limit t +

W ) ( Z

r e d u c e s t o the drift velocity;

c

= 0 ,

a s expected b e c a u s e the solution of E q . ( l ) i s symmetric; -b coincides with the longitudinal diffusion coefficient defined by F i c k ' s l a w .

Under t r a n s i e n t conditions, a is by defini tion the time-dependent d r i f t velocity and a value of c different from z e r o , a s indicated by Nag /5/ , weights the contribution of the d r i f t velocity t o the time-derivative of the mean s q u a r e displacement.

Owing to i t s s t a t i o n a r y limiting value, the t e r m

Table 1

-

Physical parameters used i n c a l c u l a t i o n s .

Type o f I n t e r v a l l e y s c a t t e r i n g

1 f 2 f3 gl 92 93

seems t o be a plausible definition f o r a gene_

r a l i z e d diffusion coefficient allowed to depend on time i n a c c o r d with Ref.(3). Consequently,

Eq.(2) should be quite a p p r o p r i a t e t o study t r a n s i e n t phenomena from b a l i s t i c up t o sta- t i o n a r y r e g i m e s i n t e r m s of the time dependell c e of the c o e f f i c i e n t s a , b and c o r of the fi- r s t t h r e e moments a s defined by Eqs.(3-5).

- b + a c = d t

3 . R e s u l t s and d i s c u s s i o n .

-

The dependence with time of the different moments i s calcula-

Ge-holes

Band parameter

I

A

]

= 13.18

1 1

1

B

1

= 8.48

11

I ^{c 1}

^{= 13.14}

Spin-orbit energy A = 0.295 eV Crystal density

P

= 5.32 gr/cm3 Sound v e l o c i t y v, = 3.93

lo5

cmlsec Acoustic deformation p o t e n t i a l

EI)

⁼ 4.6 eV Optical deformation p o t e n t i a l do = 40.3 eV C p t i c a l phonon temperature

@OP = 430 CdTe-electrons

T

- e f f e c t i v e mass

mp

= 0.0963 L - e f f e c t i v e mass mL = 0.5 C r y s t a l density

p

= 6.06 gr/cm3 Optical phonon temperature 8 = 248 K

OP

I n t e r v a l l e y phonon temperature Bint = 220 K S t a t i c d i e l e c t r i c constant

Co

= 10.6 High-frequency d i e l e c t r i c constant E, = 7.13 L -f energy gap EL- 6 r, = 1.5 eV

L 4 i n t e r v a l l e y deformation p o t e n t i a l (DtKILL 10 eVlcm 8 T' -L i n t e r v a l l e y deformation p o t e n t i a l (DtK 10 eV/cm 9

l - L = Phonon

mode

( T A ) (LA) (TO) (TA) (LA) (LO)

ted making u s e of an ensemble Monte C a r l o technique. T h e i n i t i a l conditions of motion a r e taken with a l l c a r r i e r s s t a r t i n g from the same point (x

=

0) and, e n e r g y d i s t r i b u t e d a c c o r d i n g t o an equilibrium maxwellian dis_

tribution f o r Ge and CdJe, a t fixed ene+rgy equal t o (3/2) KT with

k

orthogonal to

E

f o r the c a s e of ~ i : T o exemplify the method, in F i g . 1 the f i r s t t h r e e moments of the d i s t r i b s tion a r e r e p o r t e d f o r t h e c a s e of e l e c t r o n s in C d T e . The time simulation h a s been taken long enough t o match the s t a t i o n a r y conditions. F o r the i n i t i a l conditions h e r e chosen t h e f i r s t m o

Eq. temp.

(K)

220 550 685 140 215 720

-

ment and the sec6nd c e n t r a l moment exhibit in t h e t r a n s i e n t r e g i o n (up t o about 10 p s e c ) a time dependence which l e a d s t o overshoot of the drift velocity a n d , more g e n e r a l l y , to a non-Lange

Coup. const.

(10 e~/cm) 8

0.3 2.0 2.0 0.5 0.8 11.0

vin behaviour of the longitudinal diffusion coefficient. F u r t h e r m o r e , t h e t h i r d c e n t r a l moment evidences a significant time dependence which, i n t e r m s of Eq.

( 2 )

, i s i n t e r p r e t e d

as

a r e 1 5 vant contribution of d r i f t t o s p r e a d i n g through a non-zero value of c .

" T h i s

a s s u m p t i o n

has been

made

f o r

s i m p l i c i t y

reasons

At times l o n g e r than 1 0 p s e c a l i n e a r least- - s q u a r e interpolation of the r e s u l t s well r e p r o

Cd T e - e l e c t r o n s

*

;

d u c e s the s t a t i o n a r y values with vd

=

1 . 4 107 - cm/sec and DQ

⁼

8 5 cm2/sec ( s e e the l i n e s in F i g . I ) . T h e same fitting p r o c e d u r e s t i l l yields

-

j; 1

a small negative slope of the t h i r d c e n t r a l mom%

- nt which l e a d s t o c

=

-1.9 cm. However,

0

.' this non-zero value of c should not be s i g n i f i c a a

- because of the l a r g e uncertainty (25% which in_

$ c r e a s e s with time) a s s o c i a t e d with the c a l c u l a t ion of the t h i r d c e n t r a l moment.

Of c o u r s e , quantitative r e s u l t s r e f l e c t the

"

2

X

microscopic p r o p e r t i e s of the material under in-

&

-

o '

vestigation s u c h a s i t s p o l a r o r covalent n a t u r e and i t s band s t r u c t u r e . To p r e s e n t a systematic a n a l y s i s , in t h e following t h e r e s u l t s f o r the ca_

s e of e l e c t r o n s in S i , holes i n Ge and e l e c t r o n s i n CdTe will be briefly s u r v e y e d . The uncerta_

inty of the r e s u l t s i s estimated to be within 5%

f o r the d r i f t velocity and 10% f o r the diffusion coefficient. The microscopic models used f o r

t (P sec)

t h e s e semiconductors have been found t o i n t e r e r e t s a t i s f a c t o r i l y d r i f t , diffusion a n d , f o r the

ca_

s e of Si and G e , n o i s e measurements in a wide

Fig. 1 - F i r s t moment and second and t h i r d c e n t r a l moments

range

of

electric field strengths and temperatK

as a f u n c t i o n of time f o r t h e case of e l e c t r o n s i n CdTe a t

res /7 , 8 , g / , and are reported i n ~ ~ 1, b l ~

t h e f i e l d and temperature reported. Continuous l i n e i s a

l i n e a r least-square i n t e r p o l a t i o n made f o r t h e r e s u l t s a t

~ i - ~ l - ~ ~ ~ i

2

t reports the results ~ ~ . ~ ~ ~ .

times longer than 10 psec.

obtained a t 300 K f o r field s t r e n g t h s of 2 0 and

100 kV/cm along a (100) d i r e c t i o n . At both fie- l d s d r i f t and diffusion exhibit overshoot effects with peaks which a r e more pronounced a t the highest field s t r e n g t h . S i n c e the higher the a c c e l e r a t i o n the f a s t e r the dissipation s t a r t s , the peaks o c c u r a t s h o r t e r times by i n c r e a s i n g the field s t r e n g t h . I t i s quite natural to c o r r e l a t e the overshoot of the diffusion coefficient to the overshoot of the d r i f t velocity. On this correlation i t h a s t o be noted that the peak of the diffg sion o c c u r s somewhat delayed i n time with r e s p e c t t o that of the d r i f t velocity, f u r t h e r m o r e , within the uncertainty of the c a l c u l a t i o n s , t r a n s v e r s e diffusion h a s not evidenced overshoot e E f e c t s . The values of the t r a n s p o r t coefficients in the t r a n s i e n t a r e found t o be p r a c t i c a l l y in_

dependent from the o r i e n t a t i o n of the f i e l d , a r e s u l t which i s consequence of the initial equal population f o r different valleys h e r e assumed. F i n a l l y , the times involved i n the t r a n s i e n t he_

r e studied a r e below about 0 . 5 p s e c which i n t u r n means that detectable effects will r e q u i r e s t r u c t u r e of the o r d e r of 5 0 0 x.

Ge-holes. - F i g . 3 r e p o r t s the r e s u l t s obtained a t 77 K f o r a field s t r e n g t h of 1 0 kV/cm o r i e n t e d along(100) and<111) d i r e c t i o n s . T h e main f e a t u r e s of t h e time dependence of d r i f t and diffusion r e s e m b l e the S i c a s e . H e r e , a s a consequence of the warped s t r u c t u r e of the heavy-hole band, a s t r o n g anisotropy of the t r a n s p o r t coefficients with values l a r g e r in(100) than i n ( l 1 l ) d i r e c t i o n i s evidenced. I n p a r t i c u l a r , overshoot of both d r i f t and diffusion, a s a r e s u l t of a s m a l l e r effective m a s s , i s b e t t e r evidenced along a(100) d i r e c t i o n while along a (Ill) d i r e c t i o n it is smoothed r e m a r k a b l y f o r the d r i f t velocity and s u p p r e s s e d f o r the diffusion.

CdTe-electrons. - F i g . 4 r e p o r t s t h e r e s u l t s obtained a t 300 K f o r field s t r e n g t h s of 1 0 . 5

and 40.5 kV/cm, the f o r m e r being below and t h e l a t t e r above the value of threshold f o r negative

JOURNAL DE PHYSIQUE

Ge - holes T 5 7 7 K E = l o kV/cm

-

³

..

0

2

²

differential r e s i s t i v i t y (a 1 5 kV/cm). Below

E

t h r e s h o l d the d r i f t velocity exhibits a small

"

1

12 o v e r s h o o t , on the c o n t r a r y a l a r g e overshoot

b

u

i s found f o r t h e longitudinal diffusion coefficg

> 0

e n t . I n fact, a t fields just below threshold,

-

³⁰

e l e c t r o n s have e n e r g y high enough t o u n d e r t k

-

' 0

ke those flights which, in a b s e n c e of interval_

2

20

l e y s c a t t e r i n g , would produce p o l a r runaway.

N

5 T h e s e "superflights" a r e a t the o r i g i n of a

-

¹⁰

s t r o n g initial elongation in the f o r w a r d d i r e 5

n-

o

tion of the s p a t i a l distribution function /lo/ .

o

0.5 1

T h i s effect p e r s i s t s i n time, s o that values of

t ( P set)

the o r d e r of 1 0 p s e c a r e needed b e f o r e statio- n a r y conditions a r e achieved.

Above t h r e s h o l d , i n analogy with the r e ? ults found f o r GaAs and I n P

/11/

, a l a r g e

Fig. 2

-

D r i f t v e l o c i t y ( a ) and l o n g i t u d i n a l d i f f u s i o n c o ~ c

overshoot of the drift velocity is found. At

f i c i e n t ( b ) a s a f u n c t i o n of time f o r t h e case of e l e c t r o n s in

field the overshoot of the longitudinal dif

S i a t T = 300 K. Dashed and continuous curves r e f e r t o e l e c

fUsivity exhibits a negative peak before

t r i c f i e l d s t r e n g t h s of 20 and 100 kV/cm, r e s p e c t i v e l y .

ching i t s stationary value for times of about

1

p s e c . T h i s peculiar behaviour of DL may be explained in t e r m s of the dominant i n t e r v a l l e y s c a t t e r i n g between t h e s a t e l l i t e s and c e n t r a l

- 3 -

valley which p r i v i l e g e s final s t a t e s i n the

ten_

'0

t r a l valley with l a r g e ?;-wavevector antiparal_

2E

l e l t o 2 /12, 13/ . T h e s e s t a t e s a r e p r i v i l c

0

h

ged i n the s e n s e that a c a r r i e r h a s the p o s s i b i

- 0 l i t y t o perform a long flight b e f o r e suffering

p o a n o t h e r i n t e r v a l l e y s c a t t e r i n g ; under i d e a l c o n ditions t h i s long flight o c c u r s periodically i n

,-. r e a l s p a c e and c a r r y e s the e l e c t r o n t o i t s i n i

(100)

t i a l position within a c y c l e . Consequently,

( 1 1 1 )

1 when t h i s p r o c e s s dominates, the ensemble of

r

_ _ _ _ _ _ _ _ _ _ _ _ _ c a r r i e r s , which me/i.ssurned t o s t a r t a t the sa_

- _{n -}

²⁰

me point, will s p r e a d initially a c c o r d i n g t o i t s

o maxwellian momentum distribution t o come c l c

0 1 2 3 4 5

s e r a t the end of the c y c l e . T h i s "pulsed" cha_

t

( P

s e c )

r a c t e r of the motion of the e l e c t r o n ensemble i s damped f a s t l y by the p r e s e n c e of s h o r t e r f E g h t s and o t h e r s c a t t e r i n g mechanisms but, w c thin the t r a n s i e n t i t i s s t r o n g enough t o produ_

Fig. 3 - D r i f t v e l o c i t y ( a ) and l o n g i t u d i n a l d i f f u s i o n c o e f f i

Ce the negative values of the longitudinal d i f f ~

c i e n t (b) a s a f w c t i o n of time f o r t h e case of holes i n Ge a t

sion a s reported in 4(b).

T = 77 K and E = 1 0 kV/cm. Continuous and dashed curves r e f e r

The long times involved in the transient at

t o f i e l d o r i e n t e d a l o n g c O O ) and ( l l l > d i r e c t i o n s , r e s p e c t i

near and the high

vely.

the d r i f t velocity overshoot a t f i e l d s well

above t h r e s h o l d mean that detectable e f f ~ c t s a r e possible i n s t r u c t u r e within 0.1 - - 1 um.

/

4. Conclusions. - The study of the t r a n sient behaviour of drift and diffusion a t high fields i s c a r r i e d out in t e r m s of the time a n a l y s i s of the f i r s t moment and the second and t h i r d c e n t r a l moments of the spatial distribution. I n this way a gene_

r a l i z e d longitudinal diffusion coefficient at s h o r t times i s defined through Eq.(4).

F o r the initial conditions h e r e assumed, the drift velocity and t h e longitudinal dif fusion coefficient a r e found t o exhibit ov_

e r s h o o t effects. By i n c r e a s i n g the e l e ~ t r i c field s t r e n g t h overshoot i s favoured and i t s peak shifts at s h o r t e r times. The warped shape of the valence band i s found to b e a t t h e origin of s t r o n g anisotropy ef

fects in both drift and diffusion, higher values of these quantities being a s s o c i a i ed with the direction f o r which the e f f e c tive mass of c a r r i e r s i s smaller. I n C d T e , f o r a field strength just below t h r e s h o l d , when polar optical s c a t t e r i n g i s dominant o v e r o t h e r s c a t t e r i n g m e c h c nisms overshoot effect of longitudinal dif_

fusivity is quite l a r g e r and p e r s i s t s up t o times of the o r d e r of 1 0 psec. F o r f i

Fig. 4

-

D r i f t v e l o c i t y (a) and l o n g i t u d i n a l d i f f u s i o n c o e f f i c i e n t

eld s t r e n g t h s above threshold, the Ion@

(b) as a function of time f o r the case of electrons in CdTe a t T =

tudinal diffusivity can assume negative

= 300 K. Dashed and continuous curves r e f e r t o f i e l d s t r e n g t h s of

values which a r e a s s o c i a t e d with long fli-

10.5 and 40.5 kV/cm, respectively.

ghts o f 2 a r r i e r s s t a r t i n g a t l a r g e and ne_

gative k-values i n the c e n t r a l valley.

According to previous r e s u l t s of Ref .(8) drift velocity i s found t o exhibit the highest peak of i t s overshoot f o r fields above t h r e s h o l d .

Acknowledgments. - We a r e grateful t o the Computer C e n t e r of the Modena University f o r providing computer facilities.

R e f e r e n c e s .

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/3/ D . K. F e r r y and J . R. B a r k e r , J . Appl. P h y s . 5 2 , 818 (1981) and r e f e r e n c e s t h e r e i n /4/ S .V. Gantsevich, V .L. Gurevich and R. Katilius, P h y s . Cond. Matter, 2 , 165 (1974) /5/ B . R . Nag, P h y s . Rev. B x , 3031 (1975)

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/7/ V. B o r s a r i and C . Jacoboni, P h y s . S t a t . S o l . , 0 5 4 , 649 (1972) /8/ L. Reggiani, J . ^Phys . S o c . Japan 9 , (1980) Suppl. A p. 317

/ 9 / R . B r u n e t t i , C. Jacoboni, F . Nava, L . Reggiani, G . Bosman and R . J . J . Z i j l s t r a , t o be publ. in J . Appl. P h y s .

/lo/ A. Alberigi- Q u a r a n t a ,

V .

B o r s a r i , C . Jacoboni and G. Zanarini , Appl. P h y s . Lett.

C7-116 JOURNAL DE PHYSIQUE

22 , 103 (1973)

/11/

^{J .}

Maloney and J . F r e y , J . Appl. P h y s . , ,8, 781 (1977) /12/ W . F a w c e t t a n d H.D. R e e s , P h y s . Lett. A x , 578 (1969)

/13/ G . H i l l , P.N. Robson and W. F a w c e t t , J . Appl. P h y s . 50 , 356 (1979).