Thermalization in a system with a continuous spectrum of states

JOURNA

Journal of Non-Crystalline Solids 137&138 (1991) 25-28 North-Holland

THERMALIZATION

IN A SYSTEM

L OF

NON-CRYSTALLINE SOLIDS

WITH A CONTINUOUS

SPECTRUM

OF STATES

Sergey TARASKIN Moscow Engineering Physics Institute,Moscow 115409, U.S.S.R. Thermal relaxation of electronic or atomic systems characterized by a continuous density of states following an abrupt quenching from initial temperature to a lower value is analyzed in the frame of the multiple trapping model. Thermalization is shown to satisfy the dispersive law, but in contrast to the dispersive transport results the dispersion parameter depends on the initial and final temperatures only and is independent of the energy distribution of states. The general approach is applied to the hydrogen subsystem in a-Si:H.

1. I N T R O D U C T I O N

Similar approach is used also to describe thermal

The interest on the problem of thermalization in a system with continuous spectrum of states arises from the recent investigations of hydrogen

relaxation of hydrogen in a-Si:H after abrupt quenching.

But it should be noted that here the situation

thermalization

is rather different from the mentioned above. Really,

processes in amorphous silicon1-5. Experiments have

here the forms of initial and final distribution functions

shown that the high-temperature state can be frozen

are very similar. Both of them describe the thermal

in by quenching to room temperature and its relax-

equilibrium but at different temperatures. Therefore,

ation time is very long. This stretched reequilibration

it may be something doubtful to transfer the results of

seems to be due to the dispersive diffusion of hydro-

dispersive transport analysis to the problem of thermal-

gen atoms characterized by the time dependent dis-

ization following an abrupt quenching from initial equi-

persive diffusion coefficient D ( t ) = D o o ( w t ) - l + % Here

librium state.

D00 is a microscopic diffusion coefficient, w is a hydro-

relaxation of system of carriers with continuous density

gen attempt frequency and a is a dispersion parame-

of states, which follows an abrupt quenching from ini-

In this report we analyse the thermal

ter, which is closely connected with the form of energy

tial temperature To to a lower value T. In the frame of

distribution of states of hydrogen atoms. For the stan-

multiple-trapping model we consider the general situa-

dard exponential tail the dispersion parameter equals

tion which can be applied to electronic subsystem in dis-

with w the energy scale of trapping

ordered semiconductors as well as to atomic subsystem.

a

=

1 - T/w,

This ap-

The general approach is concretesized for the hydrogen

proach seems to be justified when analysing the dis-

sites and T the measurement temperature.

thermalization in a-Si:H. The relaxation is turned out

persive transport problem of non-equilibrium carriers

to follow mainly dispersive law, but now the dispersion

in the system with continuous density of states,such as

parameter does not depend on energy distribution of

the relaxation of excess carriers for the time-of-flight

traps and is sensitive only to the initial and final tem-

problem 6, or for the photoconductivity kinetics after

peratures, i.e. a = T / T o s

pulsed photoexcitation 7. The common feature of these problems is an essentially non-equilibrium form of initial distribution function. This function relaxes towards the equilibrium while the packet of non-equilibrium carriers shifts to the lower energies. The time of life of the carriers in the packet increases with time so that the diffusion coefficient decreases.

2. M O D E L

Let us consider the system of carriers which can occupy the states characterized by the continuous density of states g(c), as in Fig.1. The reference point of energy, E = 0, separates "mobile" (E < 0) and "immobile" (e > 0) states. The functions gl(g) and g2(g) are

0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved.

S. Taraskin I System with a continuous spectrum of states

26

f(e, t) = f0 (c) exp[--r/1 (g, t)

DENSITY OF STATES

-

-

r/2(t)] + e x p [ - r h (e, t)

t

-- r/2(t)] / r/2(e, t) exp[rh (e, t) + 772(t)]dt 0

(4)

>., where ka

Z

¢

t COo

~1(~, t) = ~oot e x p ( - ~ / r ) , ,5o

(5) 0

are the essential dimensionless functions of the problem under consideration. The equation r/l(e,,t ) = I intro-

FIGURE 1

duces a standard demarcation energy e,(t), which sep-

Schematic density of states used in the model

arates the thermalized shallow states (at ~ < s,(t) and rh >> 1) from non-thermalized deep states

related to band-tail like states and deep states, consequently. The transitions between the immobile states

(at e

> e,(t)

and r/1 << 1 )9. The similar equation r/z(t0 ) = I in-

are possible only via mobile ones. Equilibrium distribu-

troduces the effective thermalization time to of the majority of immobile carriers (see below ). The unknown

tion function f(e, T) at temperature T has the standard form:

function n(t) introducing in (5) can be found from the equation (2) which reflects the particle conservation law. The following consideration is based on the approximate

f(c,T) : {1 + exp[(e - ¢(T))/T]} -1

(1)

solution of this equation, which is possible because of specific energy dependence of distribution function (4).

Our aim is to analyse the relaxation of the energy distrtibution from the initial f ( e , t = O) = f(e, To) to the 3. S O L U T I O N

final f ( e , t ~ oo) = I ( e , T ) . Thermalization is provided by transitions between

At initial interval of times, t << to, the distribution function f(e, t) follows quasi-equilibrium behaviour

immobile and mobile states which can be described by f(e, t) ~ ~

the following set of balance equations:

exp(¢/T)

(6a)

oo

~.at{n(t ) + / f(e, t)fgl(e) + g2(e)lde} = O,

(2)

for shallow states ¢ < ¢,(t) and has essentially nonequilibrium form at e > e,(t),

0

d f ( e , t) -- [1 - f(e, t)]w0 n(@~ _ f(e, t)wo

expC-elg). (a)

Here n(t) is a concentration of mobile carriers, N c is

f ( s , t ) ~_ f0[1 -- ?]l(g,t)] + [1

-

-

f0(e)]r/2(t).

(6b)

At very long times, t >> to, for similar energy regions it is not difficult to obtain the following expressions:

an effective concentration of mobile states and w0 is an escape frequency ( ~ 101° - 1012s -1) proposed to be in-

r12 + rh/t

(7a),

dependent of energy. The relaxing distribution function

f(e, t) can be found from (3) as

f(e > ~., t) _

,h +r12~l/t

(1 - f0)exp(-~2).

(7~)

For the beginning we consider the initial interval of times, t << to. Substituting (6) into the rate equation (2) and using for simplicity the 5-function approxima-

S. Taraskin /System with a continuous spectrum of states tion of g2(e i , g2(e) = NoS(e - co), one can easily find an approximate expression for q2(t) (<< 1) at t << to :

(8)

gl(g,)fo(e.)~gl

~]2(t) ~---gi(e,)(~£2 -[-ND(O )

27

a consequence of similarity between initial, f0@,T0), and final, f ( e , T ) , distribution functions (in contrast to dispersive transport problem, when similar functions strongly differ from each other).

In some aspects it

m a y be interesting to investigate the relaxation of nonwith ND(O) = N0[1 - fo(eo,t = 0)] the concentration

occupied deep states, No(t).

of non-occupied deep states at t = 0 . This formular

pears to follow the stretched exponential-like behaviour:

Their concentration ap-

is obtained in the case of sufficiently smooth density of states as compared to distribution function f ( e , t ) . For

ND(t) ~--ND(O)exp[--r]2(t)]+

t

the exponential tail, gl (e) = ga (0) e x p ( - e / w ) , with w >

No e x p [ - ~ 2 ( t ) ] ~ / e x p [ ~ z ( t ) ] d t

T the functions g l ( e . ) and f0(e.) and typical energy scales 5el and 5e2 are transformed by the following way:

with tie(t) ~_ fo(e.)Sel/&2 ~ (wot) T/T°. It can be seen

g l @ . ) f o ( ¢ . ) ~--gl(O)(wot) -T/w"

(9)

exp[-((To)/To](wot) T/To ,

(~gl -- (~--~-- wl---)-1 + (1].÷

~

-

(14)

0

~)--1

from

(13)

and (14) that the typical relaxation time of

free carriers and non-occupied deep states coincides with time to a n d relaxation is practically finished when the

,

(10)

energy s . ( t ) reaches (0. 3.2 High concentration of deep states

6e2 ~-- w +

(1

--

1) -1

.

(11)

W h e n the concentration

of

deep states is high

enough to control the position of (, i.e. when ND(O)

Derivating (8) or (9) allows to find the time dependence

gl(((To))&z, the statistical shift of chemical potential

of free carriers concentration and therefore answer the

may be essential. In this case there are two time re-

question under consideration.

gions of relaxation.

The characteristic time to, when q2(to) = 1, can be

At t << to the relaxation follows

the law described above (see (8)), but at to << t << t0 the character of relaxation is changed. At these times

estimated from (8) by the following way:

the states between ((To) and ( ( T ) are thermalized and

to ~ wo 1 exp[-( ( To ) / T],

(12)

quasi-equilibrium distribution (11) shifts to the higher energies approaching the equilibrium one. The relax-

showing its correct interpretation as a thermalization

ation is practically finished when the quasi-chemical po-

time of the majority of trapped carriers.

tential, ((t) = T ln[Nc/ n(t)], reaches the position of

Some particular cases following from (8) at different concentrations of deep states are discussed below.

( ( T ) at t ~_ {0. The approximate analytical expression describing the time dependence of free carriers concentration t << to is given by the following expression:

3.1. Small concentration of deep states W h e n the concentration of deep states is sufficient-

n(t) ~_ n(to)

ly low, so that -No(0) <~ g((0)~e2, and these states don't

(

l + ~~ (wt 0o) w( t

" - to)

)1

(15)

influence the position of chemical potential, ( ( T ) ((To) = (o (the statistical shift is neglidgible), it follows from (8) that:

while No(t) ~ (wot) - ~ / T being due to the strong modification of initial distribution function at ((to) < c < ((T).

~ ( t ) ~_ ~

T~I

, . ,,-~+r/ro

(13)

Here the dispersion parameter does not depend on the form of energy distribution of traps in the tail. It is

4. A P P L I C A T I O N

to a - S i : H

Possible application of this model to a-Si:H is based on the m o d e r n concepts concerning the properties of

S. Taraskin / System with a continuous spectrum of states

28

the hydrogen subsystem in a-Si:H 1-5. So, the hydrogen

initial energy distribution to the final one following an

atoms can occupy different energy states and play the

abrupt lowering of temperature.

role of carriers discussed above. The mobile states are

model differ essentially from the results of standard ap-

related to the interstitial positions of hydrogen atoms,

proach basing on the analysis of dispersive transport

the band-tail states - to the bond-centred positions 1°

problem. Application of the general approach to the

and deep states - to the monohydride bonds. Then the

thermal relaxation of hydrogen in a-Si:H gives a reson-

non-occupied deep states describe the silicon dangling

able agreement with experimental data.

bonds (DB's).

The results of the

Experimental studies show (see e.g. 4)

that an abrupt quenching leads to a very long relaxation of DB's concentration. The relaxation time of DB's concentration is found to be thermally activated with an energy of about 1.5 eV 4. Comparing this value with relation (12) one can estimate the position of ((To) being also of about the same value. The slow temperature dependence of DB's number is probably related to the

ACKNOWLEDGEMENTS

I am very grateful to M.Klinger and M.Stutzmann for helpful discussions of this problem. REFERENCES 1.

J.Kakalios, R.A.Street and W.B.Jackson, Phys. Rev.Lett. 59 (1987) 1037.

2.

R.A.Street, J.Kakalios et al.,Phys.Rev. B35 (1987) 1316.

3.

R.A.Street, M.Hack and W.B.Jackson, Phys.Rev. B37 (1988) 4209.

4.

R.A.Street and K.Winer, Phys.Rev. B40 (1989) 6236.

5.

W.B.Jackson, J.M.Marshall and M.D.Moer, Phys. Rev. B39 (1989) 1164.

6.

J.Orenstein and M.Kastner, Phys.Rev.Lett. 46

close position of Co to ~, and may be to the dispersion of DB~s energies giving rise to thermally-induced pardmagnetism at sufficiently steep decrease of g2(e) 11. The stretched exponential relaxation of DB's concentration found experimentally (see e.g. 4) qualititavely also agrees with results presented above, when ND(O) ~ gl ((o)&2. But it should be noted that stretched exponent with dispersion parameter c~ = T/To (To is an initial temperature ) describes the relaxation only at t << to (see (14)). Unfortunately, the difference between initial and final

(1981) 1421.

values of DB's concentration is not very large (ND(t --*

~)/ND(O) ~-- 2 -- 3) so that the DB's concentration at t << to is very close to the initial one, ND(O), and is un-

7.

H.Scher and E.W.Montroll, Phys.Rev. B12 (1975) 2455.

likely to be measured experimentally. At t ~ to, when

8.

S.N.Taraskin and M.I.Klinger, Pi~ 'rna Zh. Tekh.Fiz. 16 (1990) I0 (in Russian).

9.

V.I.Arkhipov and A.I.Rudenko, Phil.Ma 9. B45

A N D ~ No(O), the second integral term in (14) becomes essential and effectively gives rise to decreasing of the dispersion parameter, for instance, from a ~ 0.8 (at To = 600K and T = 500K ) to experimentally found value of a ~ 0.64.

(1982) 189. 10. Van de Walle, P.J.H.Denteneer et al., Phys.Rev. B39 (1989) 10791. 11. M.I.Klinger, V.G.Kudryavtsev et al., Phys.Rev. B40 (1989) 6311.

5. C O N C L U S I O N S

In summary, we would like to point out that the model presented above describes the relaxation of the