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Exact solutions for space charge transport in dielectrics with non-uniform charge distributions in open circuit
Journal o f Electrostatics, 2 (1976) 187--198 187 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
EXACT SOLUTIONS F O R SPACE C H A R G E T R A N S P O R T IN DIELECTRICS WITH N O N - U N I F O R M C H A R G E DISTRIBUTIONS IN OPEN CIRCUIT*
L. NUNES DE OLIVEIRA and G.F. LEAL FERREIRA Departamento de Ffsica e CiSncia dos Materials, Instituto de Ffsica e Quimica de S~o Carlos, Universidade de S~o Paulo, 13560 --S~o Carlos, SP (Brazil) (Received October 27, 1975, in revised form January 20, 1976)
Summary This paper gives solutions for the space charge motion in dielectrics under open circuit condition in two cases: (a) free space charge with initial density linear in space, and (b) single deep-level trapped charges with initial density linear in space and with the assumption of non-retrapping: An analytical solution is given for the first case; solution of the second case requires numerical computation.
1. Introduction Hitherto, most of the exact solutions for space charge motion have dealt with the free space charge b o x distribution [1,2,3], except the case reported by Van Turnhout [2] for non-uniform charge distribution solved numerically from the beginning. Besides these, there are exact solutions for the emitting electrode [4,5], b u t these, because of the imposed boundary condition {zero electric field at the emitting surface), are useful only under specific conditions. In this article we intend to find solutions for an initially linear space charge distribution, distributed between the electrodes. It will be seen that more complicated relations between charge and space lead to almost prohibitive computational manipulations, even for the simpler boundary condition adaptable to our method of solution -- the method of the characteristics -first introduced in this field b y Many and Rakavy [4]. This method is a general one b u t requires for practical applications the knowledge of the total current. In open circuit, the total current is zero thus introducing a great simplification: (a) for free space charge it is sufficient to obtain the exact solution; (b) considering a single level system of deep traps, a (numerical} solution can be found when retrapping is neglected, i.e., when the density of *Work supported in part by Banco Nacional de Desenvolvimento EconSmico (BNDE), Centro de Aperfei~oamento para Pessoal de Ensino Superior (CAPES), Conselho Nacional de Pesquisa (CNPq) and Fundaq~o de Amparo ~ Pesquisa do Estado de S~o Paulo (FAPESP).
188
trapped charge is given as a function of time. We assume a linear variation of the space density of trapped charge and a zero density of free charge at the beginning of the process. The interesting quantities were obtained by computation (Hewlett--Packard 91000 A calculator); for the voltage across the sample only an approximate result was achieved. In Section 2 we state the equations concerning our problem. The free space charge case is treated in Section 3; Section 4 deals with the trap problem. The results are presented in Section 5. 2. Theory As shown in ref. [4], the general equations for the space charge motion (without diffusion) are: i'c (x', t') = pp '(x', t')E'(x', t') ~x' ic(x',t') = ~
e
~x'
(1)
p'(x',t') + ~-~ p't(x',t')
E ' ( x , t ' ) = p'(x',t') + p't(x',t') - - p ' o - - P ' t o
(2) (3)
where i~
=
p'
_-- free
p E'
= = = = = =
p~ c
p~ p~0
density of conduction current charge density mobility of the free charge the electric field density of trapped charges dielectric permittivity thermal equilibrium density of free charges thermal equilibrium density of trapped charges
One more equation relating the time variation of free and trapped charges at each point x' is needed. Under the assumption of non-retrapping and a single trap level this relation is given by p~t(x',t') = p'to + [p~(x',0) --pto]exp(--t'/T')
(4)
where T' is the trapping time. We will work with a simplified version of eqns. (3) and (4), neglecting both p ~0 and p ~. These are good assumptions for high initial trap concentration and for deep trap level. Our boundary condition is that the density of total current, j'(x', t'), equals zero: ]'(x',t') = Up'(x',t')E'(x',t) + e
aE' ~t'
(x,t') = 0
189
We can simplify our system of eqns. (1--4), adopting the following dimensionless variables (without primes}:
x'
x=--,p d
p' eE' =--,E=--,i Po d-rio
ei' P-Po t' V' = - , t=--, V = pd-po e d2-po
(5)
Here, P0 has any arbitrary value, d is the thickness of the sample (planar symmetry) and V' the value of the potential, with reduced value V; i' represents a quantity with dimension of current density. Then, our system of equations becomes: i c (x, t) = p (x, t ) E ( x , t)
(6)
a a a . . . .a x i c ( x , t ) = -~tt O(x,t) + --~t P t ( x , t )
(7)
a --E(x,t) ax
(8)
= p(x,t) + pt(x,t)
(9)
p t(x, t) = p t ( x , O ) e x p ( - - t / ~ ) a /(t) = p ( x , t ) E ( x , t ) + - ~ t E ( x , t ) = 0
(10)
The method of characteristics [4] consists in using the equations of the characteristics of this system, d (11)
dt x(t) = E(x(t),t)
where x ( t ) defines the flow lines. The time variation of the charge density along a flow line is obtained, after using the system eqns. (6--10), as d - ~ p ( x ( t),t)
apt --
__p2 __ p p t - - -
8t
(12)
where apt/a t is to be found at the m o m e n t a r y position x ( t ) along a given flow line. The time variation of the electric field can also be found. It is given by d
d2x
--E(x(t),t) dt
-
dt 2
- pt(x(t),t)E(x(t),t)
(13)
Here again Pt is to be found at the position x ( t ) along a given flow line. The flow line will be characterized b y its initial position x0, 0 < x0 < 1. A flow line starting at x0, will be at time t at x, which is given by t
x = Xo + . ; E ( x ( t ) , t ) d t 0
(14)
190
3. Free space charge Putting Pt = 0 in eqns. (11), (13) and (12) we obtain the following: dx dt d2x
dt 2 dp dt
-E
(11')
=0
(15)
= _p2
(16)
The integration of these equations yields: p -
po(Xo)
(17)
1 + po(xo)t
(18)
x = xo + E o ( x o ) t
where po(Xo) and E0{x0) give the charge density and the electric field at Xo and at t = 0. Making use of the arbitrariness in the definition of p0, we will write for the initial charge density distribution {19)
p(Xo) = 1 + axo
The origin of the coordinate system will be taken on the position of the zero-field plane, which does not move f o r ] = 0 [6,7]. Accordingly, the electric field is" ax 2
E ( x o ) = x0 + - n °
(20)
2
By eqns. (11) and (15), the electric field keeps its initial value along the flow lines: (21)
E(x(t),t) = E(xo)
In order to find E ( x , t ) we have to eliminate xo from eqn. (21) using eqns. (20) and (18). Alternatively, we could also use directly eqn. (21), substituting it in eqns. (18) and (20) and eliminating Xo in eqn. (20), using eqn. (18). This is believed to be the best way for finding E ( x , t). It results in the second degree equation in E ( x , t), E(x,t) =
l+t+axt+
[l+t 2+2t+2axtl at 2
'h
191
In order to have at t = 0, E ( x , O ) = E(xo), the negative sign must be chosen. So, 1 + t + a x t - - [1 + t 2 + 2t + 2axt] ~
E(x,t) =
at 2
(22)
The charge density can be f o u n d directly from eqn. (22) using Poisson's equation. The interesting quantity to be measured in open circuit is the voltage. It can be found directly by integrating eqn. (22) in x or, in an easier way, by changing the integration variable from x to x0, using eqns. (18) and (20). Thus, l
c
v(t) = f
f
o
b
dx
,:txo
c
f E(x0)[1 + t(1 + ax0)]dx0, b
where b and c are to be found from eqn. (18) respectively with x = 0 and x = 1. After some calculation we find: V(t)
= V ( n ) - - V ( m ) , with
x : [1 a(l + 3t)x a2tx 2 ] Y(x) = ~+t + +- 3 4
atn
= [(1 + t) 2 + 2x'oat] ' / 2 - (1 + t)
atm
= [(1 + t) 2 + 2xo'at] i/~_ (1 + t).
(23)
Here Xo and x~' are the coordinates of the plates with respect to an origin at the zero-field plane. 4. Deep traps without retrapping We assume pt(x,t) = A(x)exp(--t/T),
p(x,O) = 0
So, eqns. (12) and (13) become: do = A ( x ) e x p ( - - t / r ) dt
--p
_ p2 T
)
d E d 2x dx dt - dt 2 - A(x(t))exp(--t]T) dt
(24) (25)
From eqns. (24) and (25) we obtain
dp__p2 +
dt
(_p
+l
)
d2x (26)
192
It is easy to see that eqn. (26) shows that the zero-field plane ~-= 0
does not move [6,7].
From eqn. (26) we proceed to
r o : [ = p r - In C ( 1 - - p r ) - ~ f The eonstant C is to be found from the initial conditions at t = O: p = 0 dx a n d - - i s the field E(xo) at Xo which labels the flow line. dt Therefore we get
dt
- E(~)
1 -- or
Substituting this in eqn. (14}, we have
texp[rjo~'Pt(1--pr)dt"ldt' x = Xo + E ( x o ) f
Jo
(27)
1 --pr
Hence eqn. (24) becomes exp
-dp -+ P2 = dt
(_p
1) + r exp(--t/r)A
x0
r
p2
t + E(xo) f0
(1 - - p r ) d t "
dt/
1 -- pr (28)
Assuming a linear variation for A ( x ) ,
A ( x ) = 1 + ax
(29)
with the origin, as before, at the zero field plane. Then, eqn. (28) gives __+p2
dt
=
--P +
exp(--t/r)
l+axo+a
t
xf
o
where
we have used eqn. (20).
exp
xo+ 2 J
r
p2dt" 1 -- rp
II dt'
(30)
193 It is possible to perform a c o m p u t e r integration of this equation, obtaining p ( t ) along a flow line and operating at the same time on eqn. (27), to get its
position as a function of time. A result, discussed further in Section 5, is that the density of free charge at the electrodes is close to that at the zero-field plane, po(t). We t o o k this as meaning that the density of free charge is approximately uniform through the sample. With this approximation we obtain V(t), the voltage across the sample, after integrating the system of equations; dV poV+
-dt
-dpo -+p~ dt
=0 (31) =
--P0 +
exp(--t/r)
where eqns. (10) and (24) have been used. 5. Results Figures l(a) and l ( b ) show the plot of the density of free space charge (p) as a function of time for various flow lines, as obtained from numerical integration of eqn. (30). These flow lines are labelled b y the position (x0) which they o c c u p y at t = 0. We assume in both cases p(x,0) = 1 + 0.5x (with the origin of the coordinates at zero-field plane), and in each figure the t w o sets of curves refer to the two chosen values of the parameter T(5 and 10). In Fig. l ( b ) , the zero-field plane is located exactly at the center of the sample, while in Fig. l(a) it is located on one of the electrodes. All lines except the one which refers to zero-field plane (x0 = 0) terminate as the charge front they represent reaches one of the electrodes. We must, therefore, keep track of the position of the front (eqn. (27)} during the numerical integration of eqn. (30}. The end point in each curve gives the density of free charge just in front of the electrode at the corresponding time. It should be noticed that the time of arrival of the various fronts at the electrode is almost independent of r for big values of Ix01 since the density of free charge is small when compared with the trapped charge. We also observe that as t becomes comparable to r, p at x = 0 starts decreasing. This means that the loss of free charge b y electrostatic repulsion (represented by term p 2 in eqn. (29)) is overtaking the effects of the release of trapped charges. Finally, it is apparent from the graphs that the density of free space charge at any point is not t o o different from that at the plane of zero-field and that these differences decrease with increasing t, being smaller for greater r. They also decrease for increasing uniformity in charge distribution (decreasing value of the parameter a, eqn. (28)). For a = 0, the density of free charge would be uniform in space.
194
02
t'-5 .3
5
.2
,0
.4
0.1 ,6
,5 "4 .3
I
TrIO
.I
0
.0
.2
I
i
I
i
I
I
2
3
4
5
--__I
.....
6
]
7
....
t
? 02 T.5 .I .2
-.I
0.I
-.2
.3 -.3
0
r . I0
,o
.I -.I
I
I
I
I
I
I
I
I
2
3
4
5
6
7
t
Fig. 1. T h e c h a r g e d e n s i t y as f u n c t i o n o f t i m e a l o n g a flow line, labelled b y t h e c o o r d i n a t e o f its s t a r t i n g p o i n t , m e a s u r e d f r o m t h e p o s i t i o n o f zero-field plane. I n Fig. l ( a ) t h e zerofield p l a n e c o i n c i d e s w i t h a n e l e c t r o d e , a n d i n Fig. l ( b ) it is at t h e m i d d l e o f t h e sample. T h e t w o sets in e a c h figure refer t o d i f f e r e n t values o f T, T = 5 a n d T = 10.
195
0.5
T -10
0
I
I
I
2
4
6
l
v
0.02
0.01
I
0
2
i
I
4
I
I
6
Z
t
Fig. 2. The electrode voltage as a f u n c t i o n of time. Fig. 2(a) and 2(b) were calculated f r o m the same set o f data leading respectively to Fig. l ( a ) and l ( b ) .
196 This substantiates both our approximation that pis the same throughout the sample, and the choice of p (0,t) as the representative value of p (x, t) (we have then a better approximation for Fig. l(b) than for Fig. l(a). Figures 2(a) and 2(b) represent the voltage (V) between the electrodes as a function of time, found from numerical integration of the system of eqn. {31), corresponding to the cases studied in Fig. l(a) and l i b ) respectively. Again, we show plots for r = 5 and r = 10. The initial values of V are found from the initial trapped-charge distributions and from the known positions of zero-field planes. We see in both figures that the voltage has a faster decrease for r = 5, since the free charge takes over more rapidly in this case. Figures 3(a) and 3(b) show the dependence of the voltage on time for the case with no traps and with an initial distribution of free charge (p ix,0) = 1 + ax) with zero-field plane at the electrode and in the middle of the sample respectively. We use eqn. (23) with various values of parameter " a " labelling the curves in Fig. 3(a) and 3ib). The rapid decrease of V in Fig. 3(b) clearly indicates that a nearly uniform distribution of charge is quickly achieved. Figure 3(b), of course, is not so informative, since in this case V = 0 would mean no charges between the electrodes. However, comparison of Fig. 2{a) and Fig. 3(a) shows a similarity in the behavior of the voltage for large t. This stems from the fact that for large t the density of free charge is almost uniform. Figure 4 is a plot of p ix, t) analytically determined from eqn. {22) using Poisson's equation, with a = 2. (See eqn. (19) and the zero-field plane at the middle of the sample.) We see that the charge density tends quickly to become uniform, this seeming to be a general trend in non-uniform charge kinetics. The same is also apparent in the case worked out in ref. [2]. Actually this can be deduced by partial differentiation of eqn. (17) with respect to x0: 2 1 ap (xo, t) = ap ( X o , 0 ) [ l + ] aXo aXo p (xo,O)t This equation shows that if two neighboring points, x0 and x0 + Ax0, differ in charge density at t = 0 by Ap0, at time t, their difference Ap in charge density is 1
2
Ap,= AP° II + p?xo,O)t"1 The distance AX between the neighboring points at time t is by eqn. i18) (even with i =/= 0) Ax = hx0(1 +
pixo,O)t)
Therefore, the spatial variation of charge density at time t and the point on the flow line labelled by x0 is
ap (x,t) = ap (xo,O) ax aXo
+ p(xo,O)t
197
V
05 0.4 0.3 02 O,I
0
I
l
I
2
I
3
1
001
0.005'
0
I
2
3
t
Fig. 3. T h e voltage as a f u n c t i o n of t i m e in t h e free space c h a r g e case. T h e value o f " a " labelling t h e curves i n d i c a t e s t h e initial d i s t r i b u t i o n , p = 1 + a x . In Fig. 3(a) t h e zero-field p l a n e is at a n e l e c t r o d e , a n d in Fig. 3 ( b ) it is a t t h e m i d d l e o f t h e sample.
198
o
f
0.0
.
.
- O4,
.
.
- Q2
.
0.0
~ 0 . 5
~
-
02
- -
x
Fig. 4. The charge density p as a function of x and t (origin at zero-fieldplane at the
middle of the sample), for p(x,0) = 1 + 2x. The labels refer to different instants of time.
T h i s explains t h e t r e n d t o u n i f o r m i t y a n d can be successfully t e s t e d f o r t h e p o i n t x = 0 in Fig. 4. Acknowledgments
We are i n d e b t e d to Drs. Sylvio G. Rosa, Jr. a n d B. G r o s s f o r t h e i r assistance in t h e p r e p a r a t i o n o f this article. References 1 2 3 4 5 6 7
H. Wintle, J. Appl. Phys., 42 (1971) 4724. J. van Tumhout, in M.M. Perlman (Ed.), Electrets,The Electrochemical Soc. Inc., p. 230. G.F. Leal Ferreira and B. Gross, Rev. Bras. Ffs., 2 (1972) 205. A. M a n y and G. Rakavy, Phys. Rev., 126 (1962) 1980. LP. Batra, B.H. Schechtman and H. Seki, Phys. Rev. B, 2 (1970) 1592. J. Lindmayer, J. Appl. Phys., 36 (1965) 196. B. Gross and M. Perlman, J. Appl. Phys., 43 (1972) 853. B. Gross, G.F. Leal Ferreira, L. Nunes de Oliveira,G. Dreyfus and J. Lewiner, Phys. Rev. B, 9 (1974) 5318.
EXACT SOLUTIONS F O R SPACE C H A R G E T R A N S P O R T IN DIELECTRICS WITH N O N - U N I F O R M C H A R G E DISTRIBUTIONS IN OPEN CIRCUIT*
L. NUNES DE OLIVEIRA and G.F. LEAL FERREIRA Departamento de Ffsica e CiSncia dos Materials, Instituto de Ffsica e Quimica de S~o Carlos, Universidade de S~o Paulo, 13560 --S~o Carlos, SP (Brazil) (Received October 27, 1975, in revised form January 20, 1976)
Summary This paper gives solutions for the space charge motion in dielectrics under open circuit condition in two cases: (a) free space charge with initial density linear in space, and (b) single deep-level trapped charges with initial density linear in space and with the assumption of non-retrapping: An analytical solution is given for the first case; solution of the second case requires numerical computation.
1. Introduction Hitherto, most of the exact solutions for space charge motion have dealt with the free space charge b o x distribution [1,2,3], except the case reported by Van Turnhout [2] for non-uniform charge distribution solved numerically from the beginning. Besides these, there are exact solutions for the emitting electrode [4,5], b u t these, because of the imposed boundary condition {zero electric field at the emitting surface), are useful only under specific conditions. In this article we intend to find solutions for an initially linear space charge distribution, distributed between the electrodes. It will be seen that more complicated relations between charge and space lead to almost prohibitive computational manipulations, even for the simpler boundary condition adaptable to our method of solution -- the method of the characteristics -first introduced in this field b y Many and Rakavy [4]. This method is a general one b u t requires for practical applications the knowledge of the total current. In open circuit, the total current is zero thus introducing a great simplification: (a) for free space charge it is sufficient to obtain the exact solution; (b) considering a single level system of deep traps, a (numerical} solution can be found when retrapping is neglected, i.e., when the density of *Work supported in part by Banco Nacional de Desenvolvimento EconSmico (BNDE), Centro de Aperfei~oamento para Pessoal de Ensino Superior (CAPES), Conselho Nacional de Pesquisa (CNPq) and Fundaq~o de Amparo ~ Pesquisa do Estado de S~o Paulo (FAPESP).
188
trapped charge is given as a function of time. We assume a linear variation of the space density of trapped charge and a zero density of free charge at the beginning of the process. The interesting quantities were obtained by computation (Hewlett--Packard 91000 A calculator); for the voltage across the sample only an approximate result was achieved. In Section 2 we state the equations concerning our problem. The free space charge case is treated in Section 3; Section 4 deals with the trap problem. The results are presented in Section 5. 2. Theory As shown in ref. [4], the general equations for the space charge motion (without diffusion) are: i'c (x', t') = pp '(x', t')E'(x', t') ~x' ic(x',t') = ~
e
~x'
(1)
p'(x',t') + ~-~ p't(x',t')
E ' ( x , t ' ) = p'(x',t') + p't(x',t') - - p ' o - - P ' t o
(2) (3)
where i~
=
p'
_-- free
p E'
= = = = = =
p~ c
p~ p~0
density of conduction current charge density mobility of the free charge the electric field density of trapped charges dielectric permittivity thermal equilibrium density of free charges thermal equilibrium density of trapped charges
One more equation relating the time variation of free and trapped charges at each point x' is needed. Under the assumption of non-retrapping and a single trap level this relation is given by p~t(x',t') = p'to + [p~(x',0) --pto]exp(--t'/T')
(4)
where T' is the trapping time. We will work with a simplified version of eqns. (3) and (4), neglecting both p ~0 and p ~. These are good assumptions for high initial trap concentration and for deep trap level. Our boundary condition is that the density of total current, j'(x', t'), equals zero: ]'(x',t') = Up'(x',t')E'(x',t) + e
aE' ~t'
(x,t') = 0
189
We can simplify our system of eqns. (1--4), adopting the following dimensionless variables (without primes}:
x'
x=--,p d
p' eE' =--,E=--,i Po d-rio
ei' P-Po t' V' = - , t=--, V = pd-po e d2-po
(5)
Here, P0 has any arbitrary value, d is the thickness of the sample (planar symmetry) and V' the value of the potential, with reduced value V; i' represents a quantity with dimension of current density. Then, our system of equations becomes: i c (x, t) = p (x, t ) E ( x , t)
(6)
a a a . . . .a x i c ( x , t ) = -~tt O(x,t) + --~t P t ( x , t )
(7)
a --E(x,t) ax
(8)
= p(x,t) + pt(x,t)
(9)
p t(x, t) = p t ( x , O ) e x p ( - - t / ~ ) a /(t) = p ( x , t ) E ( x , t ) + - ~ t E ( x , t ) = 0
(10)
The method of characteristics [4] consists in using the equations of the characteristics of this system, d (11)
dt x(t) = E(x(t),t)
where x ( t ) defines the flow lines. The time variation of the charge density along a flow line is obtained, after using the system eqns. (6--10), as d - ~ p ( x ( t),t)
apt --
__p2 __ p p t - - -
8t
(12)
where apt/a t is to be found at the m o m e n t a r y position x ( t ) along a given flow line. The time variation of the electric field can also be found. It is given by d
d2x
--E(x(t),t) dt
-
dt 2
- pt(x(t),t)E(x(t),t)
(13)
Here again Pt is to be found at the position x ( t ) along a given flow line. The flow line will be characterized b y its initial position x0, 0 < x0 < 1. A flow line starting at x0, will be at time t at x, which is given by t
x = Xo + . ; E ( x ( t ) , t ) d t 0
(14)
190
3. Free space charge Putting Pt = 0 in eqns. (11), (13) and (12) we obtain the following: dx dt d2x
dt 2 dp dt
-E
(11')
=0
(15)
= _p2
(16)
The integration of these equations yields: p -
po(Xo)
(17)
1 + po(xo)t
(18)
x = xo + E o ( x o ) t
where po(Xo) and E0{x0) give the charge density and the electric field at Xo and at t = 0. Making use of the arbitrariness in the definition of p0, we will write for the initial charge density distribution {19)
p(Xo) = 1 + axo
The origin of the coordinate system will be taken on the position of the zero-field plane, which does not move f o r ] = 0 [6,7]. Accordingly, the electric field is" ax 2
E ( x o ) = x0 + - n °
(20)
2
By eqns. (11) and (15), the electric field keeps its initial value along the flow lines: (21)
E(x(t),t) = E(xo)
In order to find E ( x , t ) we have to eliminate xo from eqn. (21) using eqns. (20) and (18). Alternatively, we could also use directly eqn. (21), substituting it in eqns. (18) and (20) and eliminating Xo in eqn. (20), using eqn. (18). This is believed to be the best way for finding E ( x , t). It results in the second degree equation in E ( x , t), E(x,t) =
l+t+axt+
[l+t 2+2t+2axtl at 2
'h
191
In order to have at t = 0, E ( x , O ) = E(xo), the negative sign must be chosen. So, 1 + t + a x t - - [1 + t 2 + 2t + 2axt] ~
E(x,t) =
at 2
(22)
The charge density can be f o u n d directly from eqn. (22) using Poisson's equation. The interesting quantity to be measured in open circuit is the voltage. It can be found directly by integrating eqn. (22) in x or, in an easier way, by changing the integration variable from x to x0, using eqns. (18) and (20). Thus, l
c
v(t) = f
f
o
b
dx
,:txo
c
f E(x0)[1 + t(1 + ax0)]dx0, b
where b and c are to be found from eqn. (18) respectively with x = 0 and x = 1. After some calculation we find: V(t)
= V ( n ) - - V ( m ) , with
x : [1 a(l + 3t)x a2tx 2 ] Y(x) = ~+t + +- 3 4
atn
= [(1 + t) 2 + 2x'oat] ' / 2 - (1 + t)
atm
= [(1 + t) 2 + 2xo'at] i/~_ (1 + t).
(23)
Here Xo and x~' are the coordinates of the plates with respect to an origin at the zero-field plane. 4. Deep traps without retrapping We assume pt(x,t) = A(x)exp(--t/T),
p(x,O) = 0
So, eqns. (12) and (13) become: do = A ( x ) e x p ( - - t / r ) dt
--p
_ p2 T
)
d E d 2x dx dt - dt 2 - A(x(t))exp(--t]T) dt
(24) (25)
From eqns. (24) and (25) we obtain
dp__p2 +
dt
(_p
+l
)
d2x (26)
192
It is easy to see that eqn. (26) shows that the zero-field plane ~-= 0
does not move [6,7].
From eqn. (26) we proceed to
r o : [ = p r - In C ( 1 - - p r ) - ~ f The eonstant C is to be found from the initial conditions at t = O: p = 0 dx a n d - - i s the field E(xo) at Xo which labels the flow line. dt Therefore we get
dt
- E(~)
1 -- or
Substituting this in eqn. (14}, we have
texp[rjo~'Pt(1--pr)dt"ldt' x = Xo + E ( x o ) f
Jo
(27)
1 --pr
Hence eqn. (24) becomes exp
-dp -+ P2 = dt
(_p
1) + r exp(--t/r)A
x0
r
p2
t + E(xo) f0
(1 - - p r ) d t "
dt/
1 -- pr (28)
Assuming a linear variation for A ( x ) ,
A ( x ) = 1 + ax
(29)
with the origin, as before, at the zero field plane. Then, eqn. (28) gives __+p2
dt
=
--P +
exp(--t/r)
l+axo+a
t
xf
o
where
we have used eqn. (20).
exp
xo+ 2 J
r
p2dt" 1 -- rp
II dt'
(30)
193 It is possible to perform a c o m p u t e r integration of this equation, obtaining p ( t ) along a flow line and operating at the same time on eqn. (27), to get its
position as a function of time. A result, discussed further in Section 5, is that the density of free charge at the electrodes is close to that at the zero-field plane, po(t). We t o o k this as meaning that the density of free charge is approximately uniform through the sample. With this approximation we obtain V(t), the voltage across the sample, after integrating the system of equations; dV poV+
-dt
-dpo -+p~ dt
=0 (31) =
--P0 +
exp(--t/r)
where eqns. (10) and (24) have been used. 5. Results Figures l(a) and l ( b ) show the plot of the density of free space charge (p) as a function of time for various flow lines, as obtained from numerical integration of eqn. (30). These flow lines are labelled b y the position (x0) which they o c c u p y at t = 0. We assume in both cases p(x,0) = 1 + 0.5x (with the origin of the coordinates at zero-field plane), and in each figure the t w o sets of curves refer to the two chosen values of the parameter T(5 and 10). In Fig. l ( b ) , the zero-field plane is located exactly at the center of the sample, while in Fig. l(a) it is located on one of the electrodes. All lines except the one which refers to zero-field plane (x0 = 0) terminate as the charge front they represent reaches one of the electrodes. We must, therefore, keep track of the position of the front (eqn. (27)} during the numerical integration of eqn. (30}. The end point in each curve gives the density of free charge just in front of the electrode at the corresponding time. It should be noticed that the time of arrival of the various fronts at the electrode is almost independent of r for big values of Ix01 since the density of free charge is small when compared with the trapped charge. We also observe that as t becomes comparable to r, p at x = 0 starts decreasing. This means that the loss of free charge b y electrostatic repulsion (represented by term p 2 in eqn. (29)) is overtaking the effects of the release of trapped charges. Finally, it is apparent from the graphs that the density of free space charge at any point is not t o o different from that at the plane of zero-field and that these differences decrease with increasing t, being smaller for greater r. They also decrease for increasing uniformity in charge distribution (decreasing value of the parameter a, eqn. (28)). For a = 0, the density of free charge would be uniform in space.
194
02
t'-5 .3
5
.2
,0
.4
0.1 ,6
,5 "4 .3
I
TrIO
.I
0
.0
.2
I
i
I
i
I
I
2
3
4
5
--__I
.....
6
]
7
....
t
? 02 T.5 .I .2
-.I
0.I
-.2
.3 -.3
0
r . I0
,o
.I -.I
I
I
I
I
I
I
I
I
2
3
4
5
6
7
t
Fig. 1. T h e c h a r g e d e n s i t y as f u n c t i o n o f t i m e a l o n g a flow line, labelled b y t h e c o o r d i n a t e o f its s t a r t i n g p o i n t , m e a s u r e d f r o m t h e p o s i t i o n o f zero-field plane. I n Fig. l ( a ) t h e zerofield p l a n e c o i n c i d e s w i t h a n e l e c t r o d e , a n d i n Fig. l ( b ) it is at t h e m i d d l e o f t h e sample. T h e t w o sets in e a c h figure refer t o d i f f e r e n t values o f T, T = 5 a n d T = 10.
195
0.5
T -10
0
I
I
I
2
4
6
l
v
0.02
0.01
I
0
2
i
I
4
I
I
6
Z
t
Fig. 2. The electrode voltage as a f u n c t i o n of time. Fig. 2(a) and 2(b) were calculated f r o m the same set o f data leading respectively to Fig. l ( a ) and l ( b ) .
196 This substantiates both our approximation that pis the same throughout the sample, and the choice of p (0,t) as the representative value of p (x, t) (we have then a better approximation for Fig. l(b) than for Fig. l(a). Figures 2(a) and 2(b) represent the voltage (V) between the electrodes as a function of time, found from numerical integration of the system of eqn. {31), corresponding to the cases studied in Fig. l(a) and l i b ) respectively. Again, we show plots for r = 5 and r = 10. The initial values of V are found from the initial trapped-charge distributions and from the known positions of zero-field planes. We see in both figures that the voltage has a faster decrease for r = 5, since the free charge takes over more rapidly in this case. Figures 3(a) and 3(b) show the dependence of the voltage on time for the case with no traps and with an initial distribution of free charge (p ix,0) = 1 + ax) with zero-field plane at the electrode and in the middle of the sample respectively. We use eqn. (23) with various values of parameter " a " labelling the curves in Fig. 3(a) and 3ib). The rapid decrease of V in Fig. 3(b) clearly indicates that a nearly uniform distribution of charge is quickly achieved. Figure 3(b), of course, is not so informative, since in this case V = 0 would mean no charges between the electrodes. However, comparison of Fig. 2{a) and Fig. 3(a) shows a similarity in the behavior of the voltage for large t. This stems from the fact that for large t the density of free charge is almost uniform. Figure 4 is a plot of p ix, t) analytically determined from eqn. {22) using Poisson's equation, with a = 2. (See eqn. (19) and the zero-field plane at the middle of the sample.) We see that the charge density tends quickly to become uniform, this seeming to be a general trend in non-uniform charge kinetics. The same is also apparent in the case worked out in ref. [2]. Actually this can be deduced by partial differentiation of eqn. (17) with respect to x0: 2 1 ap (xo, t) = ap ( X o , 0 ) [ l + ] aXo aXo p (xo,O)t This equation shows that if two neighboring points, x0 and x0 + Ax0, differ in charge density at t = 0 by Ap0, at time t, their difference Ap in charge density is 1
2
Ap,= AP° II + p?xo,O)t"1 The distance AX between the neighboring points at time t is by eqn. i18) (even with i =/= 0) Ax = hx0(1 +
pixo,O)t)
Therefore, the spatial variation of charge density at time t and the point on the flow line labelled by x0 is
ap (x,t) = ap (xo,O) ax aXo
+ p(xo,O)t
197
V
05 0.4 0.3 02 O,I
0
I
l
I
2
I
3
1
001
0.005'
0
I
2
3
t
Fig. 3. T h e voltage as a f u n c t i o n of t i m e in t h e free space c h a r g e case. T h e value o f " a " labelling t h e curves i n d i c a t e s t h e initial d i s t r i b u t i o n , p = 1 + a x . In Fig. 3(a) t h e zero-field p l a n e is at a n e l e c t r o d e , a n d in Fig. 3 ( b ) it is a t t h e m i d d l e o f t h e sample.
198
o
f
0.0
.
.
- O4,
.
.
- Q2
.
0.0
~ 0 . 5
~
-
02
- -
x
Fig. 4. The charge density p as a function of x and t (origin at zero-fieldplane at the
middle of the sample), for p(x,0) = 1 + 2x. The labels refer to different instants of time.
T h i s explains t h e t r e n d t o u n i f o r m i t y a n d can be successfully t e s t e d f o r t h e p o i n t x = 0 in Fig. 4. Acknowledgments
We are i n d e b t e d to Drs. Sylvio G. Rosa, Jr. a n d B. G r o s s f o r t h e i r assistance in t h e p r e p a r a t i o n o f this article. References 1 2 3 4 5 6 7
H. Wintle, J. Appl. Phys., 42 (1971) 4724. J. van Tumhout, in M.M. Perlman (Ed.), Electrets,The Electrochemical Soc. Inc., p. 230. G.F. Leal Ferreira and B. Gross, Rev. Bras. Ffs., 2 (1972) 205. A. M a n y and G. Rakavy, Phys. Rev., 126 (1962) 1980. LP. Batra, B.H. Schechtman and H. Seki, Phys. Rev. B, 2 (1970) 1592. J. Lindmayer, J. Appl. Phys., 36 (1965) 196. B. Gross and M. Perlman, J. Appl. Phys., 43 (1972) 853. B. Gross, G.F. Leal Ferreira, L. Nunes de Oliveira,G. Dreyfus and J. Lewiner, Phys. Rev. B, 9 (1974) 5318.