Evaluation of the constant photocurrent method for determining the energy distribution of localised states in disordered semiconductors

NON-CR iSOUDS

Journal of Non-Crystalline Solids 137&138 (1991) 343-346 North-Holland

EVALUATION OF THE C O N S T A N T P H O T O C U R R E N T METHOD FOR DETERMINING THE E N E R G Y D I S T R I B U T I O N O F L O C A L I S E D S T A T E S IN D I S O R D E R E D S E M I C O N D U C T O R S

J.M.MARSHALL, W.PICKIN and A.R.HEPBURN

Electronic Materials Centre, Dept. of Materials Engineering, University College of Swansea, Singleton Park, Swansea SA2 8PP, U.K. C.MAIN

Department of Electronic and Electrical Engineering, Dundee Institute of Technology, Dundee, U.K. and

R.BRi~GGEMANN

Institut fur Physikalische Elektronik, Universitat Stuttgart, Stuttgart, Germany Numerical modelling is employed to explore the information which can be obtained from studies of the 'constant photocurrent' response for disordered semiconductors. Computer simulation is used to determine the experimental behaviour for a range of N(E) distributions of differing functional form. The results constitute input data to which two different interpretive techniques are applied, allowing an assessment of the viability and limitations of such procedures. 1. INTRODUCTION

2. CONSTANT PHOTOCURRENT METHOD

Many experimental techniques have been suggested

2.1. Experimental and interpretive techniques

for exploring the energy distribution of localised states,

Optical methods for the measurement of N(E) are

N(E), in amorphous semiconductors. However, precisely

attractive, as the relationship between the absorption

because of the disordered nature of the material under

coefficient, c~, and N(E) is relatively simple (at least, if

examination, it is often difficult to assess the accuracy of

the energy dependence of the optical matrix element can

a

be reliably assumed). Measurement of photoconductivity

given procedure.

Moreover,

most

experimental

techniques require the introduction (by design

or

accident) of simplifying assumptions of questionable validity. In some cases (especially in the study of various forms

of

transient

photoconductivity),

numerical

modelling has been very profitably utilised by ourselves and others in assessing the capabilities and limitations of experimental procedures.

as a function of photon energy hv has advantages of simplicity and sensitivity, but suffers from the drawback that the carrier lifetime tends to vary with carrier concentration (and

thus

with

hv).

The

'constant

photocurrent method' (CPM) was proposed to obviate this problem1. For each hv, the light intensity, I(hv), is

For many other forms of

experiment, although the situation presently remains much less clear, numerical assessment is capable of providing insights of equal value. In the present paper,

numerical modelling is

employed to explore the information which can be obtained from the 'constant photocurrent' Characteristics of disordered semiconductors.

adjusted to keep the measured photocurrent constant; implying a constant carrier density and thus a constant lifetime. (x(hv) is then related to 1/I(hv) via a constant of proportionality which may be assessed by a fit to directly measured values of c~ at higher hr. In this study, we have so far evaluated two differing procedures for calculating N(E) from c~(hv). Both

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

344

J.M. Marshall et aL ~Energy distribution of localised states in disordered semiconductors

techniques have been applied to the study of a-Si:H, and

2.2. Computer simulation procedure

thus, for reasons of space, we confine attention to this material

in

the

present

paper.

For

this

case,

photoconduction is assumed to be dominated by the higher-mobility electrons, moving close to the conduction band

mobility

edge,

Ec.

Also,

the

localised

state

distribution is often envisaged in terms of a relatively steep conduction band (CB) tail, a shallower valence band

We are in the process of performing detailed finitedifference-based numerical modelling studies of both CPM and other aspects of the transient and steady state photoconductivity of disordered semiconductors. However, for the present exploratory study, it has been sufficient to ignore complicating factors and to compute the absorption coefficient simply as the convolution:

(VB) tail, and a set of deep-lying 'mid-gap' (MG) states associated with Si dangling bonds. Thus, the dominant

o~(hv) =

W(E,E+hv).N(E).N'(E+hv).f(E) dE Ef -h~,,

transitions are those from (occupied) VB tail and MG states to the CB and its tail (figure 1).

(2)

where N(E) comprises the initial VB and MG states, N'(E+hv) is the final CB states (at energy E+hv-Ec relative to the mobility edge), and W(E,E+hv) the matrix element (taken below to vary as hv -~, unless otherwise stated).

Z t...._a

_=

2.3. Results and Comments In the present restricted space, we can only highlight I////////////~',,

Energy,

lx\\\\',~

some important features of our study. Specifically:

E

FIGURE 1

Optical transitions dominating the C.P.M. response for a-Si:H

(i)

For

a

step-function

CB

edge,

equation

(1)

yields excellent results when applied to a model involving A recent interpretive procedure, by Hata and Wagner 2, utilises the attractively straightforward expression3: N(E).f(E) = [ dot/d(hv) ]hv=Ec-E

an exponential VB tail (as concluded by Hata and WagnerZ). Moreover, the procedure performs equally well

(I)

when a Gaussian MG distribution is added as in figure 2a. (solid line = actual N(E), open circles = computed N(E)).

This makes no specific assumption about the functional form of N(E) for the tail and mid-gap states. However, it is

(ii)

assumed that the CB has a steep tail plus a slowly-varying

extended state energy distribution plus an exponential

or constant N(E) for extended states, and also, less

(characteristic energy 0.025 eV) tail, the computed data in

critically, that the optical transition matrix element is

figure 2b are obtained. Very similar results are obtained

independent of hr.

for a linear CB tail, while other reasonable CB/tail N(E)

f(E) is the occupation function, for

which we here assume zero-temperature values. The second technique, used by Ko6ka and co-workers 4 involves assumptions of an exponential VB tail, a Gaussian distribution of MG states, and a sharp CB edge. The analysis then involves obtaining a best-fit of such a model to the experimental data.

However, if we examine a CB having a linear

distributions also distort the calculated value in the range 0.8-1.4 eV to a similar extent. This effect was not recognised by Hata and Wagner 2, who performed only a restricted evaluation of the effect of CB shape upon the computed N(E), and retained a flat band beyond a limited (-0.2eV) rising edge.

345

J.M. Marshall et al. / Energy disMbution of localised states in disordered semiconductors

"~8-

where • is an average o f N(E) over the range Ea-hv to

(a)

Ec-hv. Such averaging will severely distort the apparent

form of the MG states. Formally, N(E) is more properly ~

4-

related to the second derivative in this energy range,

g22

o

and we have verifed this prediction "

~

1.8

1.6

, , ,, ,, , , , , , , , ; ; ; 1.4

1.2

1.0

0.8

0.6

0.4 0.2

impossible to establish the energy range over which

0.0

Energy (beio,~ CB edge) (eV) "~" 8-

d2[c~(hv)/W(hv)]/d(hv) z is the correct function to compute!

,

(b)

by inspection of

simulation data. In practice however, it is virtually

We are also subjecting the procedure of Kofika et el. 4, to a rigorous examination, the full results of which will be

= 6-

presented elsewhere. One important question which can be

~4-

addressed in this restricted space concerns its sensitivity to z em ~

2

2-

energy distributions of states which do not correspond to

.

0

the assumed exponential VB tail and Gaussian MG form. i

1.8

i

i

i

1.6

i

i

1.4

i

1.2

i

i

I

1.0

i

i

0.8

i

i

0.6

i

i

i

0.4 0.2

0.0

To explore this in our preliminary studies, we have

Energy (below Ct3 edge) (eV)

used equation (2), with W(hv) varying as hv -1 (as assumed

FIGURE 2

Energy distribution of occupied states, computed using equation (1), for a rectangular and a more realistic conduction band/tail Upon inspection, the above behaviour is not hard to understand. Consider a linear CB and tail, each with slope 9~. Neglecting CB states between the Fermi energy(El) and

{i} A distribution of the form shown in figure 2b

(exponential VB and CB tails of characteristic energies Eev=60 and Eec=25 meV respectively, linear VB and CB extended state distributions, Gaussian MG states of f.w.h.m.

the base of the tail (Ea=Ec-AE), equation (2) yields: P Ef cffhv) = ] W(hv,E).N(E).N'(E+hv) dE. UE~ - h ~

by Ko~ka et al.) to compute cffhv) for the following cases.

200 meV centred at Emg=8O0meV below the CB edge). (3)

Since dN'(E+hv)/dhv = X, and N'(Ea) = 0, then if W(hv) is

{ii} Replacement of the CB in (i) with a step-function N(E) (as for figure 2a)

independent of E itself,

{iii} Replacement of the MG distribution in (i) with: O Ef

d[(z(hv)AV(hv)]/dhv = • . . / N(E).dE •-, E a _ h ~

(4)

AS long as the energy Ea-hv lies below the MG states (and Ef above them), their contribution to (4) will be substantially independent of hv, and N(E) as calculated

(a) a rectangular distribution of width 300 meV (b) a distribution in which N(E) decays exponentially (with characteristic energy 40 meV) on either side of its peak {iv} Replacement of the VB tail in (i) with a Gaussian tail (as inferred from time-of-flight experiments~ )

will appear flat. Even if N'(E) ceases to increase at Ec (at a value

Rows {i} to {iv} of the table below summarise the

Nc = )~.AE), then (4) remains unaffected for hv
calculated parameters for these energy distributions, as obtained using the programme developed by Kurnia 6.

At larger photon energies, it can bc shown that: pEc-h~ d[c~(hv)/W(hv)]/dhv = ~.j N(E)dE

From the results to date, the slope of an exponential VB appears quite well reproduced, although there is some

Ea - h z o

= )~.AE.qb = Nc.qb

(5)

broadening

in {i}, for which case the CB tail is also

346

J.M. Marshall et al. /Energy distribution of localised states in disordered semiconductors gap states, the two procedures evaluated are both able to

Model

C o m p u t e d data (meV)

{i} {ii} {iii(a)} {iii(b) } {iv} {v}

Eev

Emg

Efwhm

69 60 60 62 61 60

830 660 630 660 630 730

328 154 158 163 176 174

reproduce an exponential VB tail. The procedure of Hata and Wagner z appears preferrable, since it does not require the assumption of an exponential N(E), and should thus be capable of reproducing other distributions. However, the degree of distortion associated with the shape of the

exponential. However even where the VB tail is Gaussian {iv}, the apparent fit to an exponential form remains good. In most cases, the computed peak energy of the MG states is displaced towards the CB. Since we have used equal matrix elements for localised and extended states, the effective base of the band should lie within the tail (if present) rather than at the mobility edge. However, a larger

value

is

obtained

in

case

{i},

where

an

exponential CB tail is present, than (for example) in {iv}, where the tail is absent[ Note also the large value of calculated width of the MG distribution. Both are due to the distortion in N(E) over the range 0.8-1.4 eV, associated with the rising density of CB extended states. The flattening in apparent N(E), corresponding to figure 2b, causes the fitting procedure to yield a wider and deeper distribution of MG states. It also broadens the calculated VB tail. This distortion is largely eliminated when the CB extended state distribution becomes constant as in {ii} or

conduction band merits further examination. When mid-gap states are introduced, the effect of conduction band shape upon the Hata/Wagner procedure becomes serious. In practice, the density of CB states will be much closer to figure 2b than figure 2a, and all information on the energy range between Emg and the VB tail may be lost. The technique of Ko~ka and co-workers 4 appears unable to eliminate this limitation. The procedure not only assumes a form for N(E), but appears quite insensitive to deviations from this notional distribution.

Moreover, changes in the shape of the

conduction band and its tail again exert a strong influence. Until the situation has been further explored, caution appears advisable in drawing detailed conclusions from CPM studies. ACKNOWLEDGEMENTS We acknowledge the support provided by ARC grant KN/991/11 and SERC grant G09955 for parts of this work.

{iv } (or even where the exponential tail is retained). For the cases {ii}-{iv}, force fits to a Gaussian MG distribution yield comparable fitting qualities, despite the significant changes in the true form of the mid-gap states. Row

{v}

illustrates

the

effect of

an

incorrect

assumption of W(hv). Here, the input data for case {ii} have been adjusted to an energy-independent value of W, while the fitting procedure

still assumes

the (hv) -1

dependence. Accurate representation of the VB tail is retained, but there is a Strong influence upon both the peak energy and the width of the MG distribution. 4. CONCLUSIONS We have outlined

some results

of

an

ongoing

assessment of the CPM technique. In the absence of mid-

REFERENCES

1. M.Van~ek, J.Kobka, J.Stuchlik and A.~iska, Solid State Communications, 39 (1981) 1199. 2. N.Hata and S.Wagner, Proc. Materials Research Soc 1991 Spring Meeting Symposium A (in press). 3. H.Curtins and M.Favre, Amorphous Silicon and Related Materials, ed. H.Fritzsche (World Scientific, 1988), pp 329-363. 4. J.Ko~ka, U.Van6~ek and A. T~iska, Amorphous Silicon and Related Materials, ed. H.Fritzsche (World Scientific, 1988), pp 329-363. 5. J.M.Marshall, R.A.Street, M.J.Thompson and W.B.Jackson, Phil. Mag. B, 57 (1988) 387. 6. D.Kurnia, Thesis, University of Wales (U.C.Swansea), 1990.