Download - Theory of the space-charge build-up and current transient in a weakly conducting layer with an induced conductivity gradient. Application to photoconduction and thermostimulated current

R. COELHO: Current Transient in a Layer with Conductivity Gradient 563

phys. stat. sol. (a) 31, 563 (1975)

Subject classification: 14.3.4; 16; 22.9

Laboratoire de Gdnie Elec tr ipe de Paris, assoeik a u C.N.M.S., Pontenay-aux-Rosesl)

Theory of the Space-Charge Build-up and Current Transient in a Weakly Conducting Layer

with an Induced Conductivity Gradient

Application to Photoconduction and Thermostimulated Current Analysis

BY R. COELHO

In honour of Prof. Dr. Dr. h c. P. GORLICH’S 70th birthday

The problem is considered of a weakly conducting layer in which the conductivity, initially uniform, becomes a function of depth when the layer is submitted t o a temperature gradient, or irradiated by a partially absorbed exciting radiation. After discussing the steady state E(z, 00) and e(x, a) in the presence of the conductivity gradient, an integrodifferential equation is derived for the transient charge density e(x , t ) . This equation is rewritten in terms of reduced, dimensionless coordinates, and solved by means of an iterative expansion technique involving an approximation which is justified a posteriori. Throughout the paper, extensive use is made of the depth x* where the value of the field is that of the applied field E,,. Neglecting diffusion, the expression is obtained for z*(t) and for the current J ( t ) in the external circuit. J ( t ) is written in terms of o(z*), e(z*) and k*, all these quantities being explicit functions of time. Finally, the form of J ( t ) is discussed for various experi- mental parameters, and with respect to the relevance of this theory t o photoconduction and thermostimulated current analysis.

Sous considkrons le probkme d’une couche faiblement conductrice dans laquellc la con- ductivitk, initialcment uniforme, devient fonction do I’kpaisseur lorsque la conche est soumise A un gradient thermique, ou irradike par un rayonnement partiellement absorb& Aprks discussion de l’ktat permanent E(x, a) e t e(x , M) en presence du gradient de con- ductivitk, on &tablit une equation intkgrodiffkrentielle pour 1’Bvolution de la dcnsitk de charge e(x , t ) . AprAs introduction do coordonnkes rkduites sans dimension, on rksoud 1’6qua- tion au moyen d’un dkveloppement itkratif, en utilisant une approximation qui est justifike ti posteriori. Dans tout le mkmoire, on utilise l’kpaisseur x* oh la valeur du champ est celle du champ appliquk E,. En nkgligeant la diffusion, on obtient l’expression de x*(t) e t du courant J ( t ) dans le circuit exthieur, en fonction de o(z*), e(z*) e t &*, toutes ces quanti- t k s Atant explicitement fonctions de t . Enfin, on analyse la forme de J ( t ) pour differentes valeurs des paramittres expkrimentaux, e t I’utilitk kventuelle de la thkorie dans l’interpr6- tation des rksultats de photoconduction e t du courant stimul6 thermiquement.

1. Introduction Many experimentalists and theoreticians have considered the problems of

transient currents in insulators and semiconductors. However, most of the publi- cations are concerned with the (‘response’’ of a material to a step function of applied voltage [ 1 to 61, or uniform carrier excitation [ 7 ] , under constant applied voltage.

I) 33, Avenue du G6nkral-Leclerc, F 92260, France.

564 R. COELHO

In the present paper, we consider exclusively the phenomena resulting froin a non-uniform carrier excitation in a layer of initially uniform conductivity c, submitted to a constant applied voltage P. Assuming that CJ is a true conductivity independent of the local field E , the current density in the material is uniform and its value is

J = c E .

At time t = 0, origin of the transient, an external perturbation such as the heating of one electrode or the absorption of a radiation (UV or soft X-rays for instance) produces in the sample a transverse conductivity gradient. This conductivity gradient causes a field gradient to appear, and the field gradient, in turn, produces a space-charge distribution according to Poisson’s equation.

The first part of this paper deals with the steady state (t = co). It is a classical, elementary problem, which is summarized here for clarity.

The second part deals with the development of the space-charge distribution and the corresponding transient current as can be measured in an external high-impedance circuit. The transverse conductivity gradient is assumed to reach its final value instantaneously, but the validity of the theory is not limited to this case. In fact, the system being linear in V , a progressive transition between the initial state of uniform conductivity and the final state of non-uniform conductivity can be accounted for by the “linear response” theorem.

2. Steady-State Voltage and Space-Charge Distributions

To take a simple practical case, let us assume that the electrodes on the flat sample are a t different temperatures TI a t x = 0 and To (ambient) a t x = L (Fig. 1). Calling K the thermal conductivity of the material, the conservation of the steady-state thermal flux density @ is written

with the boundary conditions of Fig. 1, integration of d T gives

x=o x=L Fig. 1. Configuration showing temperature and conductivity distributions throughout the sample

Current Transient in a Layer with Conductivity Gradient 565

2.1 Potential and field distributions

As long as the conductivity is uniform (t 5 0), the potential is linear

V ( x ) = v (1 - ;) Xow, if the temperature becomes non-uniform, cr, which is temperature dependent, becomes a function of x, and ( 2 ) must be replaced by

L

which expresses the conservation of the steady-state current density. The integral of the numerator represents the resistance per unit surface between x and L, and the denominator represents the total resistance per unit surface. Note that if (TI - To) /To is small compared to unity, the ratio E(L)/E(O) is not very much larger than unity, and this case is, in principle, the only one where the assumption that cr only depends on T is reasonably valid. Nevertheless, it should be noted that the act.ual field dependence of cr has a tendency to flatten the field distribution.

2.2 Temperature dependence of a

The true tenipera,ture variation of cr is that of a thermally activated process, namely

where U is the activation energy (of the order of 1 eV for usual polymers). However, this Arrhenius law yields mathematical difficulties, and can be replaced by

Equation ( 5 ) is a good approximation of (4) around the ambient temperature To, provided that the coefficient k is properly chosen [%JZ) By identification of the logarithmic derivatives of 0 given by (4) and (5 ) , we find for T r 320 K and U = 1 eV,

a ( T ) = uo exp k ( T - To) . (5 )

- 0.125 K-l . k=-- u kBT2

Using (2) and (5), a($) becomes

O(Z) = cro exp k ( T , - To) 1 - - = cr exp ( ;) (-?) *) This approximation is commonly made by electrical engineers.

566 R. CoF,r,m

with 0, = o(T,) and

Using the reduced variables v = zjil and z = L/il, (6) takes the form ~ ( v ) = - - Ole-" g ez-v . Thus, the conductivity distribution is an exponential, as would result froni the absorption of a photoionizing radiation. The integrals in (3) can be evaluated easily with ~ ( x ) given by (6), and the resulting potential distribution is

or, using the reduced variables ez - e" ez - 1

P(,u) = - ~- V .

By definition of E ( x ) , we now have

or e"

ez - 1 E(V) = Eoz __ I

Equation ( 7 ) shows that E is equal t o the average field Eo = V / L for the value x* of x defined by

(8) ez - 1 x* * - A h - - - - - . -

An extensive use of this particular value of x will be made later in the treatment of the transient.

z

2.3 Steady-state space-charge distribution and stored charge

The steady state space-charge density e(x) is given by Poisson's equation

Here & & e" il il e Z - 1 * @(X) = - E(x) = - E,z ___

The total charge stored by unit area is L L

p(x) dx =- E ( z ) dx il "S 0

0

(9)

Finally, V L Q = E -=-&Eo. A i l


It can be shown that the sign of the stored charge is that of the scalar product of the conductivity and potential gradients. Here, both of these gradients are negative, so that the charge is positive.

2.4 Steady-state czirrent

The steady-state current density is uniform through the sample. Therefore, assuming that the diffusion current density is negligible [9 to 121, thc product ~ ( x ) E ( x ) is uniform. This can be checked easily here by multiplying ~ ( x ) , as given by (6), by E ( x ) as given by (7). The results reads

Jo is the current density o&, in the isothermal state. Thus, the presence of the temperature or conductivity gradient multiplies the current density by the factor z (1 - e-z)-1, the value of which is given in Fig. 2 as n function of z , togcther with the initial transient value J ( 0 ) resulting from the theory given below.

3. The Transicnt In this second part, we study the time dependence of all the quantities men-

tioned above between the application of the conductivity gradient (t = 0 + ) and the steady state (t = 03) studied in Section 2.

3.2 Storage of the space charge

Just before the application of the gradient, (t = 0-), the field is uniform ( E = Eo = Ti/L). As soon as the gradient is applied (t => 0), E(x , t ) =# E0 except for x*, and we intend to study the increment

E(x7 t ) - Eo = E’(x, t ) . (12)

By definition, E’(z, 0) = 0. On the other hand, the constant boundary condition on E(x , t )

L J E(z, t ) dx = Ti 0

implies, using (12), that L J E‘(x, t ) dx = 0 . (14) 0

2 pq 2 6 I

Fig. 2. Curve a : Reduced steady-state current density J(co)/Jo as a function of z = L/A = k(T , - To). Curve b: Initial value of the transient current density J(O)/J,, as a

2

I l l function of z 0 2 4 5 8 1 0

I--)

568 R. COELHO

It results from (14) that E’(x , t ) is a continuous function of x and t which vanishes for x = x*(t). (We have discussed x*(m) in Section 2 . ) Hence, a t any instant t 2 0, one can write

X * L J E’(x, t ) dx = - J E’(x, t ) dx . (15) 0 X*

On the other hand, since E, is constant, Poisson’s equation with E’ takes the form

d I dx & - [E’(x, t ) ] = - e (x , t ) .

The above considerations regarding E’(x, t ) enable us to integrate (16) as

X *

and to write z

E ( x , ~ ) = E , + - [ ~ ( x , t ) d x . 1 ‘5 X*

To carry on the treatment of the transient, we must use the charge conservation equation:

where J is the local current density. Neglecting again the diffusion current, J = u(x) E(x , t ) , so that (19) takes the simple form

ae d - (GE) = - --

a t dx and, coming back to (18) for E(x, t ) , (20) becomes

or

Using now (6) for u, and its space derivative du/dx = - @/A, we can write (22) under the form

3.2 New reduced variables

I n addition to the reduced space variable v = x/A, we shall now introduce reduced variables for time, charge density, and field, t80 simplify the writing.

Current Transient in a Layer wit>h Conductivity Gradient 569

Reduced t ime variable. It is natural to adopt, as reduced time variable, the ratio 6 between the time t and a particular dielectric relaxation time of the material. For optimal simplicity, we choose that at x = 0, where the conductivity is ol. We thus have

to, to, 0 = __ = e z __. & 8

Reduced charge density. The choice of the reduced charge density results from that of the reduced time

Reduced f ie ld . We simply use g' = E'/E,. With the reduced time and charge density defined above, (23) can be written

VJ

2) a4 a e e (24)

3.3 Resolution oj equation (24)

We shall now build up an expansion of q(v, 0 ) of the forin

q(v, e ) = z o n . n

The coefficients a,(v) are calculated by successive iterations [ 131 from the first order term.

At time t = 0+, i.e. 8 = 0 + , there is no charge density, so that q(0) = 0 and ao(v) = 0. Equation (24) thus reduces to aqjae = e-*, and the first-order term in the expansion is

ql(v, 0 ) = e-*8 . (25)

Introducing this approximation of q in the right-hand side of (24), one obtains readily

v* or

Strictly speaking, epv* depends on 0. However, if this quantity is assumed to be a slowly varying function of 0 - as will be confirmed later on in Appendix I - (26) can be integrated to give

This gives the coefficient az(v) = - + (2e-2" - e - u * - v ) . Introducing this new approximation qz in the right-hand side of (24), one obtains a new equation which, under the same assumption as above, yields by integration the third- order term.

570 R. COELHO

Sfter repcating a few times this iteration process, the law of recurrence for a,(u) appears easily, and q(v, 0) can be written as

00 e n + i q(v, 0) = 2 (-l)n [ (n + 1) e-t - ne-"'1 e-nv ( 2 7 )

0 (n + l)! *

It is shown in Appendix I1 that expression (27 ) can be written analytically as

(28) Under this form, we can see tha t the limit of q(v, 0) for 0 -+ 00 is ev-c'. We

are now left with the task of finding eW*(O), a task which will be carried o n in the following section.

q(v, 0) = Be-. exp (-6ec.) + e"-.*[l - (Be-. + 1) cxp ( - 0 e ~ ) l .

3.3 Derivatiotk of eu*(e)

Let us rewrite (17), with the reduced variables, under the form

and introduce in (29) the expression of q given in ( 2 7 ) . This gives TI

V*

or

and the expansion (30) can he rewritten under the compact form

Z' = (1 - ct'-u*) [exp (-6e-.) - 11 . (31) A t this point, a test of the validity of solution (28) can be performed. Using

(29), (24) takes the form

q + 8 ' .

Introducing in ( 3 2 ) the expressions of q and g' given in (28) and (31), re- spectively, (32 ) becomes an identity, showing that (28) is an exact solution of

With the reduced variables, the boundary condition (14) on E' becomes (24).

2

J i5' dv = 0, or 0

I l ( z , 8 ) - z + e+*[eZ - 1 - I z ( z , O ) ] = 0 (33) with

z 1

0 e-z

2 1

Il(z, 0) = J exp (-0e-.) dv = J u-le-eu du

and

I&, 0) = J exp (-v - Oe-.) dv = / u-2e-eu du , 0 e-2

Current Transient in a Layer with Conductivity Gradient 57 1

I , is an exponential integral (tabulated), and I, can be written in ternis of I ,

I,@, 0) = ez exp ( - 0 0 - ~ ) - c-O - e ~ , ( z , 0) . From (33) and the above relation, one obtains

(34) ez - 1 - eZexp (-0e-2) + e-0 + OI,(z, 0)

~~ ~~~~ ev*(O) = ~

2 - Il(Z, 0) The limits of eW*@) for 19 = 0 and 0 --f 00 can bc obtained easily from (34).

a) 0 = 0, Il(z, 0) % J (1 - 0 e c W ) dc = x - 0(1 - e - z ) . 0

The numerator of (34) is equivalent to O x , hence

and

ez - 1 b) 8 + 00, I,@, 0) --f 0 , 0I , ( z , 0) + 0 , liin C”* = ---.

cz - 1

O+CO x and

liin w* = In -. e+m 2

The limit v*(co) given by (38) is the same as found directly in Section2 (equation (8)) for the steady state. Moreover, a comparison of (36) and (38) shows that

v*(O) + V * ( C o ) = 2 .

Consequently, the “starred” plane where the field is equal to the applied field E,, moves from a position x*(O) on the left side of the middle plane to the position %*(a) = L - x*(O), symmetrical of ~ “ ( 0 ) with respect to the middle plane. This is shown in Pig. 3. Finally, by comparing (37) with the asymptotic value of q(v, 0) derived from (as), we find that

This is t,he same as given by (9) in the study of the steady state.

Fig. 3. Field distributions; E,is the applied field; E(x, 8 < 1) is the field distribution shortly ‘after application of the conductivity gradient. E(z , 8 = 00) is the steady-state field distribu-

I

x = o X = L tiou, given here for z = 3

572 R. COELHO

3.5 Total stored charge

Q = J ~ ( x , 00) dx = EE, J p ( ~ , 00) dv . L 2

0 0

As was pointed out before, q(v, 00) = evPv*. Hence

(39)

z Q = &En j ev-w*dv = &Eoe-u*(ez - 1) . (40)

0

Using (37), we obtain finally

Q = &E,,X.

This is again the same as obtained directly by (lo), for the steady state, and the agreement constitutes a further confirmation of the validity of our calculation.

3.6 Transient current

With a reasoning similar to t h a t used by Dreyfus and Lewiner [14], we shall now write the transient current density J ( t ) in the external circuit as a function of x*, as given by (34). J ( t ) can be written as a function of the “local7’ current J ( x , t ) , namely the particle current crossing plane x a t time t , and the local displacement current

Sow, according to (18) X

D(z , t ) = EE, + J e(x , t ) dx . X*

Hence

J ( t ) = J ( x , t ) + at X*

or

(43)

(44)

The conservation equation (19) enables us to write the integral of (45) as the difference J ( x * ) - J ( x , t ) . Consequently, (45) becomes

(46)

(47)

dx * J ( t ) = J(X*) - ~ e(x*) dt

with J ( x * ) = o(z*) E, .

Conling back to the reduced variables, (46), combined with (47), becomes


Fig. 4. Reduced transient current density J(0 ) /Jo for various values of z. (Note that the reduced time 8

contains z : 0 = eZ(uO/E) t )

0 2 4 6 8 7 0 8-

Using (28) for q(w*), noting that E/T = crl = ooeZ and writing dv*/d0 as e-** (d/de) (eu*), we can rewrite (48) under the forin

J(f3) = Joez-o* 1 - [I - exp (-@e-’*)] (49)

By using in (49) the expression of ev*@) which was found in (34), we can obtain the analytical expression for the transient current J ( 0 ) . Although this expression is difficult to handle, i t has been used to carry out the numerical calculations of the curves given in Fig. 4, displaying J ( 0 ) for various values of z .

It is shown in Appendix I11 that the first-order expansion of J ( @ ) , for small 0, is

Equation (50) shows that, right after the application of the conductivity gradient (t = O+),

ez - 1 J ( 0 ) = ~- Jo ,

z whereas in the steady state (t = a), (11) gives

z J ( m ) = JO 1 - ecZ

Equation (50), also giving the initial slopes (dJ/d0), of the curves J ( 0 ) as a function of z , has been used for drawing the curves of Fig. 4.

It should be kept in mind, however, that the above results are strictly valid only if the motion of the charge carriers is not limited by their mobility. As a matter of fact, we have seen that for small 0, q(v, 0) e-”8. In other words, for 0 < 1, q has a small, but non-zero value for any v. This does not agree with the fact that the carriers cannot move with a velocity higher than their “drift velocity” pE. Consequently, the above theory is only valid for times t not too small with respect to the dielectric relaxation time in the material, i.e. for a reduced time 0 not too small with respect to unity. A correct treatment for small 0 should integrate through the sample thickness the response of a step excitation in the elemental thickness dx, as given by the theory of the transient space- charge-limited current [ 151.

4. Conclusions We have considered the problem of a layer of quasi-insulating material, of

which the conductivity depends exponentially on the depth. This is, for instance, the case of a layer with a transverse temperature gradient, or that of a plate irradiated by a partially absorbed radiation.

5 74 R. COELHO

After solving the elementary problem of the steady-state field and space- charge distributions due to the presence of the conductivity gradient, we have written the integrodifferential equation for the transient between the initial state with uniform conductivity and the final state with conductivity gradient. Using an iterative expansion based on an assumption which is justified a- posteriori, we have given an analytic solution which fulfills all the boundary conditions, and we have derived expressions for the transient current in the external circuit. This current jumps to a rather high value J(O), then slowly decays to J(o0) . I ts sign is that of V V . go, which is confirmed by many observa- tions of photocurrent transients with soft X-rays. To illustrate the temperature gradient situation, if one face of a polyethylene layer is heated abruptly to only 65 "C, the transient current jumps to 25J0, then slowly decays to a steady value J ( m ) somewhat less than &To. This confirms the necessity of taking an extreme care, for instance in therniostimulated current experiments, to maintain both electrodes at the same temperature, otherwise the measured current, partly due to the presence of a temperature gradient, may bc misinterpreted.

As was stated above, our treatment should be improved for reduced times 0 small compared to unity. Several extra refinements might also be attempted. On the one hand, account could be taken of a field-dependent conductivity, but only at the expense of the linearity, by which the response of the system to any time-dependent conductivity gradient can be deduced from the response to a step function conductivity gradient.

On the other hand, the problem can be solved in a similar way for another geometry, in particular for the cylindrical structure of a coaxial cable.

A rlmowledgements

We are grateful to Mr. Silovitch, of thc Computation Center of the Ecole Supkrieure d'Electricite for pcrforming the numerical calculations used for drawing the ciirves of Fig. 4.

Appendix I

Time variation of x*

Wehaveseenfrom (36) and (38) thatx*=Au*variesfromx*(O) =illnz/(l-e-z) t o x*(co) = L - x * ( O ) . It is intuitively obvious that the velocity dx*/dt decays from t = O+ to t = 00. Hence, if we obtain the time variation of x* at the start of the transient (t or 0 z 0), we can be sure that the time variation of x* at any later moment will be slower.

In order to assess how fast x* moves initially, we have established, by means of a few algebraic manipulations not reproduced here, the first-order Taylor expansion of as given by (34), and deduced that of v*(O). We find

[1 + W z ) + . . .I 3 ,W*(O) = ev*(@) w-1)

where ea*(@) is given by ( 3 5 ) , and

2 f ( 2 ) = 1 [ 1 - 7 + (1 + f ) e-z] .

4

Current, Transient in a Layer with Conductivity Gradient 575

Equation (AII) can also be written as

with

cp(z) behaves like 2/12 for z Q 1 ((TI - To)/To Q l), then reaches a maximum cpmax = 0.0943 for z z 5.

In other words, 19q(z) is always smaller than 0.18, whatever z may be. This justifies the hypothesis made to integrate (26).

Appendix I1

Analytic devivation of q(v, 0)

Equation (27) can be written

But Xn

ecX = 2 (-l)n ;! 00

0

helice

and 5

CQ zn+2 xe-x dx = 1 - (x + 1) e c X = 2 (-l)n (AII2) s 0 (n + 2) n!'

0

Therefore

[l - (x + 1) e c X ] 1 -- - - X

and consequently

y(v, 6 ) = 0ecv exp (-Oe-*) + e*-"*[l - (ceca + 1) exp (-8e-.)] . (AII3)

Prom this, we can see that

lim p(v, 0 ) = ev-v* . o+m

37 physica (a) 31/2

576

Appendix I11

Berivation of - (en*) and expansion of J(0) a a0

The derivation is easier if (34) is written under the form

[ z - I l ( z , 8)] ew* = ez - 1 - ez exp (-8e-.) + ePe + 8I,(z, 8 ) . (AIII1)

Taking now the derivatives of both sides of (AIII1) with respect to 8, we obtain

with

(AIII2)

(A I11 3)

Introducing (AIIIB) in (AIIIB), we obtain

Together with (49), the relation (AIII4) giving djd8 (ev') gives, in principle, the analytic form of J (8 ) . This form is heavy to handle, but it is relatively easy to find the first-order expansion of J ( 6 ) .

First of all, we can use (AI1) of Appendix I to derive

(A I11 5 )

Then, we can expand J ( 0 ) given by (4S), using the fact that, to the first order in 8,

(AIIIB) Using (AIII5) in this expansion, we obtain

1 - exp (-8e-o*) N Be+'*.

J ( 0 ) = ollEoe-ff* [ l - O f @ ) + . . .] . (A I11 7)

The factor e-"* is deduced from ev' given by (AIl), and 0, = ooez. Consequently

or

(AIIIS) ez - 1 J ( 8 ) = Jo ~ [ i - zej(4 + . . . I .

Z

From (AIIIS), we derive, in particular, the value of J for 8 = O+ :

e' - 1 J(0 ) = J, ___

Z

and its time derivative:


References [l] J. CURIE, Ann. Chim. Phys. 18, 203 (1889). [ 2 ] E. R. \-ON SCHWEIDLER, Ann. Phys. (Germany) 14, 711 (1907). [3] J. LINDBIAYER, J. app. Phys. 36, 196 (1965). 141 R. GOFFAUX, Rev. General Electricit6 74, 125 (1965). [5] J. H. CALDERWOOD and B. K. P. SCAIFE, Phil. Trans. Roy. So?. 369, 217 (1970). [6] H. J. WI~YTLE, J. appl. Phys. 42,4724 (1971); J. non-cryst. Solids 15,471 (1974). [7J E. S. RITTNER, Photoconductivity Conference, John Wiley Xew, York 1956 (p. 215). [8] See for instance, R. TELLIER, L. CONSTANTIN, and J. M. BRENAC, CIGRE paper No 212,

[9] N. F. MOTT and R. W. GURNEY, Electronic Processes in Ionic Crystals, Oxford Uni- (1958).

versity, 1940 (p. 172). [lo] S. M. SKINNER, J. appl. Phys. 26, 498 (1955); J. appl. Phys. 26, 509 (1955). [ll] It. B. SCHILLING and H. SCHACHTER, J. appl. Phys. 38, 841 (1967). [12] C. ~ILTCKENFOSS, Phys. Fluids 11, 2389 (1968). [13] H. LAU, Arch. Elektrotech. 53, 17 (1970). [14] G. DREYFUS and J. LEWINER, Phys. Rev. B 8, 3032 (1973). [15] A. MAXY and G. RAKAVY, Phys. Rev. 126, 1980 (1962).

(Received Ju ly 29, 1975)

37'