Transient space charge limited currents including diffusion

Solid State Communications,Vol. 15, pp. 1785—1787, 1974.

Pergamon Press.

Printed in Great Britain

TRANSIENT SPACE CHARGE LIMITED CURRENTS INCLUDING DIFFUSION M. Silver* Physics Department, Birkbeck College, University of London, England (Received 16 April 1974 by C. W. McCombie)

A simplified mathematical treatment of transient space charge limited currents in insulators including diffusion is presented. The results are compared with the original model of Many and Rakavy.1

THE ORIGINAL mathematical treatment of transient space charge limited currents by Many and Rakavy’ and Helfrich and Mark2 made two simplifying assumptions: (1) diffusion was neglected and (2) the electric field at the injecting contact, E(O, t) was set equal to zero. It is clear that when one has a good insulator, there is an excess of charge in the material due to injection from the contact even when no voltage is applied. Depending upon the relative work function between the electrode and the insulator, the magnitude of the charge, and consequently the reverse electric field can be very large. Thus, the field at the cathode is not equal to zero over a wide range of applied voltages. Also since diffusion must be large in order to overcome the drift due to this large reverse field, it is not obvious that diffusion currents can be ignored even when a relatively high voltage is applied.

assumptions is still unresolved. In this paper, I will present a simplified treatment of the problem including diffusion and where E(O, t) is not equal to zero. A reasonable assumption is made on the amount of charge injected from the contact as a function of time. This reduces the second integral of the continuity of current equation to a particularly simple form and the difference between the present and the original Many and Rakavy theoiy is easily seen~ The starting equations for parallel plane geometry are well known but are repeated here for clarity. Assuming a perfect insulator with no trapping: j(x, t) = epn(x, t)E(x, t) eDan(x, t)/ax (1) —

Rosen3 attempted to include diffusion while Shilling and Schachter4 estimated the relative importance of diffusion when E(O, t) =0. Rosental and Lember5 made a computer calculation assuming that the density of carriers at the injecting and collecting electrodes were determined by their thermodynamlu equilibrium values. Unfortunately the voltage range of these calculations was very limited. Baru and Z~flthskli6 calculated transient currents for strongly absorbed light but they ignored diffusion even though E(0, t) *0. Thus, the importance of the two original *

Permanent address: Physics Department, University of North Carolina, Chapel Hifi, N.C. 27514, U.S.A.

öE(x, t)/ôx

=

(4ir/e) en(x, t)

(2)

t)/ax

=

—

ean(x, t)/at

(3)

a/(x,

In what follows, it is assumed that the Debye length, x 2 )~, is very small compared with L, the of the insulator, and that the 0 =thickness (ekT/2irN0e density of carriers at the collectorN(L, t) is zero for times up to the transit time of the injected carriers. Further, only times less than a transit time will be considered here although the results wifi have implications regarding the steady state. When equation (3) is integrated twice over all space, the spatial dependence is removed and only a time dependent equation remains:

1785

1786

TRANSIENT SPACE CHARGE LIMITED CURRENTS

J(t)

(e4u/8irL) {E~(t)

=

—

E~(t)} + 4uNokT/L

(4)

where EL(t) is the field of the collector and E0(t) is the field and N0 is the density of carriers at the inject, ing electrode. N0 is assumed to be constant for all times. It is obvious that equation (4) can be rewritten as:

Vol. 15, No. 11/12

Rakavy’ derived from their approximation. The frac.

(5)

tional critical time for the current to become equal to the Many-Rakavy value is t~,/t0= (x0E0)/(LE~)I. The larger the magnitude of the field ue to the Mott and Gurney cloud, E00 i, the sooner the Many—Rakavy results are obtained. Thus, it was only necessary for them to assume an infinite reservoir and not that E(0, t) = ü in order to achieve their results. Further,

where E~is the reverse field at the injector due to the charge, Q0, in the Mott and Gurney cloud prior to the application the voltage wherenow Q(t) total charge in theofsample. The and problem is is tothe calculate Q(t). This is done by noting that when the system is in equilibrium E(0, t) = E~and no charge is injected. The effect of the applied field is to increase E(0, t) above E~which allows further charge to be injected and Q(t) will increase so long as E(O, t) > E~.Using this notion,

when ay ~ 1, since the results of Many and Rakavy hold after a short time, their calculated after 2 IL3currents dependence ashould transitthen timehold are down also correct. The V to low voltages provided these are large compared with kT/e and xo/L ~ 1. This contradicts the prediction of Adirovich7 who proposed that the V2/L3 law held only if E(0, t) > 0. From the analysis presented here, the Adirovich condition should be relaxed to E(O, t) E~> 0 which gives a much lower critical voltage.

J(t)

Q(t)

(J.L/2L)[Q(t){EL(t) +~~(t)}+ Q0E00]

=

t

Q

=

0

+ (pQ0/x0)

f {EL(t)—E~ 4irQ(t)/e}dt. (6)

0

—

To further simplify the problem, equations (5) and (6) can be written in terms of normalized units and one obtains a.,~/aT= {n(T)/2}{2.,i(T)

—

~(T)} + a2/2

[5(a)]

and ~(T)

=

a{l + y

T

f

{/j(T) + a

—

~(T)}dT

[6(a)]

0

—

2/L3 When cry c~1, one no longer obtains the V relation because the first term on the right in equation (7) does not decay out in a transit time. The condition that a-y ~ 1 is equivalent to V 2. If (L/xo) one makes the critical voltage 0 V~2(kT/e). = 10(kT/eXL/xo)2, ~-‘

one obtains exactly the upper bound for the V2/L3 law derived by Adirovich.7 This agreement is very reassuring. It is obvious that the current does not saturate unless a = 0 which can only be approached when the drift velocity is comparable to the random velocity.

where E 0 is the applied field and j~= ELIEO, T = t/t0,

~(T) = 4mrQ(t)/eE0, y = L/x0 and a = 4rrQo/eE0. and t0 is the space-charge-free transit time. Equations [5(a)]and [6(a)] can be integrated numerically. However, because Jj(T) is a slow function of time, the main features of the numerical results can be displayed analytically from the first order solution by assuming~(T)is a constant when integrating [6(a)] but leaving in the time dependence in [5(a)]. This procedure gives results within a few per cent of the ones obtained from the computer solution and over a wide range of a and equation Using this approximation live purposes, [5(a)] reduces to for illustra. = af 2/2){l e’2~r7T} () 1e_°~~ + (f1 ~.

—

the key parameters aand ~ determine the time dependence of the current. If cry ~ 1, the first term on the right decays very quickly with time. The second term remains, which is, of course, just the one Many and

Equation (7) has been integrated numerically for y = 100 and a varying from 10-2 to 102. Some of these results are shown in Fig. 1. Since ~yhas been set equal to a constant, these results reflect the current voltage characteristics over a range of four decades in applied field. It is obvious that for a~~ 1 a difference between the present results and those of Many and Rakavy show up only at short time. It is interesting to note that as expected, when a is small, the current at the transit time has risen con-8 siderably above behaviour the value atwhen t = 0. Recently, noted a similar one includedRosental trapping. The trapping has the effect of reducing x 0 and consequently a considerably. Without trapping, a very large initial current was calculated but, because a is so small with trapping, the current rises from is initial value by almost two decades, providing that the trapping

Vol. 15, No. 11/12

TRANSIENT SPACE CHARGE LIMITED CURRENTS

8

—

parameters are not too large. The consistency between Rosental’s results and those proposed here suggest that the simplified model is reasonable and can be adapted

-

to the case of trapping as well.

—

-~

6 ~

—

~

—

1 ~ 21 ~

-\

Finally, one comment should be made and that is that none of the treatments to date include the image potential. It would also seem that an approach similar

k

to that used here might be an appropriate starting point for this problem as well.

\~ ~

~

-

.—~

-

I

G -

002

I

004

.i

006

I 008

Acknowledgement The author gratefully acknowledges financial support by the British Science Research Council and the computational aid given by Mr. Kenneth Gray. —

s-~~----i~

~15

~

-)

/‘

1

—

/ct.05

~

-

05

0

0

~

02

1787

04

I

I

06

0-8

1

(tA) FIG. 1. Normalized current vs normalized time is shown,

for various values of a. The upper curve gives results for T < 0.01 while the lower curve shows times for T> 0.1. In the lower curve, some points are shown for T<0.l for smalla.

REFERENCES 1. 2.

MANY A. and RAKAVY G.,Phys. Rev. 126, 1980 (1962). HELFRICH W.~and MARK P., Z Phys. 166, 370 (1962).

3.

ROSEN G.,Phys. Rev. Lett. 17,692 (1966).

4.

SHILLING R. and SCHACHTER H.,J. Appi. Phys. 38, 841 (1967).

5. 6.

ROSENTAL A. and LEMBER L.,Phys. Status Solidi 39, 19(1970). BARU V.G. and ZEMNITSKII Y.N., Soy. Phys. Semicond. 5, 1862 (1972).

7. 8.

ADIROVICH E.I., Solid State Phys. 2, 1282 (1960). ROSENTAL A., Phys. Lett. 46A, 270 (1973).

On présente une resolution mathématique simplifiée du courant transitoire limité de charge d’espace avec la diffusion et un champ non zero a la cathode. On compare les résultats avec la théone originalè de Many et Rakavy.1