Revista Brasileira de Física, Vol. 6 , Nº 2, 1976
One-Carrier Free Space Charge Motion under Applied Voltage:
The General Solution *
P. C. CAMARGO
Departamento de Ciências Físicas e Matemáticas, Universidade Federal de São Carlos,
São Carlos SP
and
G. F. LEAL FERREIRA
Departamento de Física e Ciência dos Materiais, instituto de Fkica e Química de São Carlos",
Universidade de São Paulo, São Carlos SP
Recebido em 13 de Maio de 1976
E x t e n d i n g the method o f s o l u t i o n r e c e n t l y presented i n t h i s j o u r n a l
,
the system o f p a r t i a l d i f f e r e n t i a l equations d e s c r i b i n g the o n e - c a r r i e r
f r e e space-charge motion under a g i v e n a p p l i e d v o l t a g e
i n t o a system o f two o r d i n a r y d i f f e r e n t i a l
equations.
is
The
transformed
method
a p p l i e d t o f i n d t h e externa1 c u r r e n t observed w i t h s h o r t e d e i e c t r o d e s
is
,
a f t e r space charge l i m i t e d c u r r e n t i n j e c t i o n .
Extendendo-se o método de solução recentemente apresentado nesta Revist a , o sistema de equações d i f e r e n c i a i s p a r c i a i s , que descreve
o
movi-
mento de carga e s p a c i a l monopolar, sob voltagem constante, é
transfor-
mado em um sistema de duas equações d i f e r e n c i a i s o r d i n á r i a s .
O
método
é a p l i c a d o ao c á l c u l o da c o r r e n t e e x t e r n a f o r n e c i d a em c u r t o c i r c u i t o ,
após i n j e ç ã o de carga e s p a c i a l .
1. INTRODUCTION
By o n e - c a r r i e r f ree space charge motion (FSCM) , we mean the time
deve-
lopment o f an excess charge d i s t r i b u t i o n ,
moves
i n which every c a r r i e r
w i t h a v e l o c i t y p r o p o r t i o n a l t o the e l e c t r i c f i e l d a c t i n g upon i t .
* Work supported by FAPESP and CNPq.
**
P o s t a l Address: Caixas P o s t a i s 359-378, 13569
-
São Carlos SP.
As
the e l e c t r o d e s (we are assiiming p l a n a r geometry) may a l s o bear e l e c t r i c
charges, the complete s p e c i f i c a t i o n o f the e l e c t r i c f i e l d needs a r e f e rente
t o them, as open i f unconnected, o r as under an a p p l i e d
i f they are connected through an e.m.f.
voltage
source. They a r e known, respec-
t i v e l y , as charge and c u r r e n t modes.
I n t h i s a r t i c l e w i l l be given a s o l u t i o n o f the FSCM,
in
the
current
mode, assuming t h a t the charge d i s t r i b u t i o n , which completely f i l l s the
dielectric,
i s k n o k ~ ~a,t a g i v e n time, as w e l l as the a p p l i e d
as a f u n c t i o n o f time.
voltage
To achieve t h i s aim, we extended the method
s o l u t i o n r e c e n t l y develo p ed
i
of
f o r the case i n which the charge d i s t r i b u -
t i o n touches o n l y one o f the e l e c t r o d e s .
I t t u r n s o u t t h a t the
system
o f p a r t i a 1 d i f f e r e n t i a l equations d e s c r i b i n g the FSCM, namely the Poisson and c o n t i n u i t y equations, tagether w i t h the c o n s t r a i n t
knowledge o f the v o l t a g e imposes on t h e i r s o l u t i o n s ,
into
a system
is
that
the
transformed
o f two o r d i n a r y d i f f e r e n t i a l equations ( i n Ref. 1
we
had o n l y one) f rom whose s o l u t i o n the q u a n t i t i e s o f i n t e r e s t may be derived.
The method i s a p p l i e d t o f i n d the externa1 c u r r e n t d e l i v e r e d by a
lectric
which was shorted a f t e r being
1 i m i ted c u r r e n t regime.
in
die-
the s t a t i o n a r y space-charge
This case has received some a t t e n t i o n
l i t e r a t u r e ( ~ e f s . 2 , 3 , 4 , 5 ) , s i n c e i t i s c l o s e l y connected
ment. I n the seque1, we wi 1 1 o f t e n quote Ref. 1 ( CALF )
,
with
in
the
experi-
on which most o f
t h i s work i s based.
2. THEORY
We use, throughout t h i s paper, dimensionless v a r i a b l e s as
CALF, p.351.
defined
Thus, we w r i t e t h e Poisson and c o n t i n u i t y equations as
in
where, E i s the e l e c t r i c f i e l d s t r e n g h t , p, the charge d e n s i t y ,
i, the
potential
conduction c u r r e n t , r, t h e depth as measured from the h i g h e r
e l e c t r o d e , and t, the time.
We are given
and
V ( t ) bei ng the appl i e d v o l tage.
Using the method o f the c h a r a c t e r i s t i c s ,
the f o l l o w i n g
e q u a t i o n s are
ob t a i ned
which may be i n t e g r a t e d t o g i v e the equation o f the f l o w l i n e s
and the charge d e n s i t y a l o n g them,
E(X~)=E(X,O) being a known f u n c t i o n o f t h e p o s i t i o n which may
be
found
frorn p ( x o ) and v(O), and j the externa1 c u r r e n t .
3. THE FUNCTION y(t) AND zít)
We assume t h a t both electsodes are r e c e i v i n g c a r r i e r s ; t h i s
is
surrly
the case, f o r a l l times, i f V(t)=O, and f o r a l i m i t e d range o f time, i f
the v o l t a g e i s n o t so h i g h as t o d r i f t a11 the c a r r i e r s i n a given
rection ( i n t h i s c a s e t h e p r o b l e m w o u l d b e t r e a t e d a s i n R e f . 1).
define, therefore,
f u n c t i o n s y = y ( t ) and z = z ( t ) as g i v i n g
the
coordinates, y and z, o f t h e c a r r i e r s t h a t , a t time t , were
v e l y a t $=O and x=1.
diWe
initial
respecti-
C l e a r l y , we have y(O)=O.and z(0)=1.
Hence, we may o b t a i n the equations (CALF,
p.353)
With the v a r i a b l e s y and z, we may w r i t e , from Eq.(2),
With the h e l p o f these r e l a t i o n s , Eq.(2) may be r e w r i t t e n e i t h e r as
I n Eqs.(5)
at
and (61, ~ ( y and
)
E ( z ) , l i k e E ( x O ) , are t h e e l e c t r i c f i e l d
t=O, expressed as a f u n c t i o n o f t h e v a r i a b l e s y,z and xo.
Charge conservat i o n a1 lows us t o w r i t e
On the o t h e r hand, from Eq. ( I ) , we d e r i v e
234
,
r1
~ ( , t1) =
xp(x,t)dx + V ( t )
.
(8)
We, now, change t h e i n t e g r a t i o n v a r i a b l e , i n E ~ (81,
.
from x t o
g i v e n by Eq. (6a).
x
as
Keeping the time f i x e d , as the i n t e g r a t i o n r e q u i r e s ,
we d i f f e r e n t i a t e Eq. (6a), o b t a i n i n g
which may be i n t e g r a t e d , g i v i n g
wi t h ~ ( r d) e f i n e d as
, obtain
S u b t r a c t i n g Eq.. (5a) f rom (5b), and u s i n g t h e r e s u l t i n Eq. ( l ~ ) we
Using E ~ (.i ' ) , we may w r i t e , f o r E(O,t),
S u b s t i t u t i o n o f Eqs. ( l 2 a ) and (12b), r e s p e c t i v e l y i n t o
Eqs. ( 4 a )
(4b), gives r i s e t o a system o f two o r d i n a r y d i f f e r e n t i a l equations
y,z
and t:
and
in
I t i s desi r a b l e t o add a few words o f comnent about t h i s system.
we note t h a t , f o r +O,
i f E(y)<O,anddz/dt<O,
~ ( t and
)
z(t).
W(O)=O (see E q . ( I I ) ) and W(z)=V(O), so
i f E(z)>O,asrequiredbythedefinitions
s i n g f u n c t i o n o f time, and z a decreasing one.
;
y wi11 be an
We, t h e r e f o r e ,
t h a t , f o r l a r g e times, y and z tend t o the same l i m i t , a
gives d y / d t
z ( t ) i s , however, very de1 i c a t e because,
-
dy/dt>O,
of
Suppose now t h a t v ( ~ ) = o ; t h e r e f o r e , as already said, the
proposed s o l u t i o n w i l l be v a l i d f o r a l l times O
o f y ( t ) and
Fi r s t ,
.
increaexpect
This behavior
f o r y- z
,
Eq.
dz/dt, what would v i o l a t e the c o n d i t i o n t h a t y ( t ) must
an increasing, and z ( t ) a decreasing,
(13)
be
f u n c t i o n o f time.
We have always found t h a t , i n c r e a s i n g accuracy o f the numerical i n t e g r a and
t i o n o f Eqs. ( l 3 ) , a c t u a l l y increases the time f o r which y ( t )
z(t)
behave as expected.
The common 1 i m i t value o f y ( t ) and z ( e ) , f o r t*,
i s closely related
the t o t a l charge ~ ( t c) i r c u l a t i n g i n the externa1 c i r c u i t .
to
To see t h i s ,
we d i f f e r e n t i a t e Eq. ( 2 ) w i t h respect t o time:.
The e l e c t r i c f i e l d , a c t i n g a t l a r g e times (t*),
upon
those
carriers
which never reach the e l e c t r o d e s i s zero, t h a t i s , E ( x ( ~ ) , ~ ) = o C
. learly,
xO=a , so t h a t
This
equation provides
a t e s t f o r the
correctness o f our calculation,
s i n c e a comes o u t o f the i n t e g r a t i o n o f Eq. (13), and j ( t ) ,
the
externa?
c u r r e n t , rnay be found a c c o r d i n g t o t h e procedure g i v e n i n the n e x t
Sec-
tion.
4. THE EXTERNAL CURRENT AND THE CHARGE DENSITY
The e x t e r n a l c u r r e n t , j l t ) ,
i s g i v e n by
I n t e g r â t i o n o f t h i s e q u a t i o n i n x , from O t o 1 , u s i n g Poisson's equation,
gives
Using Eqs. (12a) and (12b), we o b t a i n
The charge densi t y , a t t i m e t, rnay be found, f o r any r,, u s i n g
i t s p o s i t i o n , x, i s g i v e n by Eq.(6a)
Therefore,
Eq. ( 3 ) ;
( o r Eq.(6b)).
once y ( t ) and z ( t ) a r e known,
t h e e x t e r n a l c u r r e n t and charge
d e n s i t y can be d i r e c t l y c a l c u l a t e d .
5. APPLICATIONS
I n most cases, we do n o t know the i n i t i a l charge d i s t r i b u t i o n .
ver, t h e t r a p f r e e d i e l e c t r i c i s i n the s t a t i o n a r y space charge
c u r r e n t regime, t h e charge d e n s i t y i s indeed known, and i t w i l l
If,,
howe-
limited
be
used
t o i l l u s t r a t e o u r method.
We suppose ohrnic
contact
a t the i n j e c t i n g e l e c t r o d e
Using t h e same dirnensionless v a r i a b l e s , as i n Ref.5,
d u r i n g t h e charge.
t h e s t a t i o n a r y space
charge d e n s i t y i s p(jc,)=
3/4 2 - ' 1 2 . A t $=O, t h e e l e c t r o d e s a r e s h o r t e d
o
,
and we want t o know t h e c u r r e n t d e l i v e r e d by the system (besides the cap a c i t y c u r r e n t peak, a t t=O), as w e l l as t h e charge d e n s i t y as
tion
a
func-
o f time.
To s t a r t w i t h , Fig.1 e x h i b i t s y and z as f u n c t i o n s o f time.
obtained, by the f i r s t o r d e r Runge-Kutta method, on
calculator.
I t i s seen t h a t , even f o r t=20,
t o t h e i r a s y m p t o t i c value, a.
and z=0.5017.
I t seems s a f e t o p u t a=0.4844
p i s almost u n i f o r m .
were
HP-9810-A
desk
z and y a r e n o t t o o
For i n s t a n c e , f o r t=26, we have
+
close
y=0.4672
0.0005.
F i g u r e 2 shows p as a f u n c t i o n o f x and t, f o r 03<1.4
t=1:4,
a
These
.
We see t h a t , a t
Fig.3 displays the external current
j(t),
j
and t h e charge ~ ( t )(see Eq. ( 1 4 ) ) , as a f u n c t i o n o f t. Observe t h a t
has a very f a s t i n i t i a l decay ( 0 ~ < 1 ) , f o l l o w e d by a r a t h e r
The time t
-
slow
one
1 , m e d i a t i n g these two behaviors, may g i v e a gross
.
indica-
t i o n o f the value o f the m o b i l i t y .
F o l l o w i n g a h i n t taken f r o m e x p e r i m e n t s
t h a t the f u n c t i o n 0.375 exp(-4.182Jt)
i n KCN c r y s t a l s , we have
found
very c l o s e l y represents the e x t e r -
na1 c u r r e n t , i n t h e range O<t<2.
The a s y m p t o t i c value o f ~ ( t ) , t h a t i s , Q(m), was found t o be -0.04426
Coming back t o Eq. (15) , we have
giving
E ( a ) = 0.04398
+
0.00054.
E(x,)=
-1
+
3/2
úc:"
Since a was found
.
and a=O.4844+O.OOO5,
in
a
r a t h e r crude
way, we t h i n k t h a t & ( a ) , as o b t a i n e d by i n t e g r a t i o n , p r o v i d e s a
better
value o f t h e e x t e r n a l charge than t h a t g i v e n by - ~ ( a ) .
Some years ago, ~ i n d m a ~ e deduced
r ~ ' ~ an e x p r e s s i o n r e l a t i n g t h e e x t e r n a l
c u r r e n t w i t h t h e motion o f t h e z e r o f i e l d plane, namely,
where r ( t ) g i v e s the p o s i t i o n o f the z e r o f i e l d p l a n e ( t h a t i s ,
t)=0.
We wi 11 discuss some r e s u l t s , a r i s i n g from Eq. ( l 7 ) ,
FSCM scheme.
~(r(t),
within
I n t e g r a t i n g Eq. (17) i n t h e time v a r i a b l e , we have
the
Fig.1
-
y and z as f u n c t i o n s o f t i m e f o r SCLC d i s c h a r g e .
Note t h e weak
convergence t o t h e common l i m i t a.
f
-
-
Fig.3
-
o f time.
-
external current Jít)
(---) external circulating charge Q (t )
The e x t e r n a l c u r r e n t ( f u l l ) and charged (dashed) as a
The approximáte v a l u e o f
&(a)
i s a l s o given.
function
where r, is the initial position of the zero field plane. Now, at r(t),
the charges are not moving, and so we may say that -j(t)dt is the charge
swept out by the moving zero field plane, in a locally stationary charge
conservation, we may further say that -&(t) is equal to the charge lying
between the initial value of r, that is r o , and the initial position, s,,
of those carriers that, at time t, are in r(t). For t*, we have s,+ a ,
and we correspondingly wri te
For a monotonically decreasing initial charge distribution,
Lindmayer
puts
Actually, this relation may be derived from ~~.(19),
noting that, for a
monotonically decreasing ~(x,), j is negative, and by ~~.(17),dr/dt is
positive. This means that a>r, and ~(r,)> ~(a). This allows
us
to
write from Eq.(19)
It is easy, now, to show for FSCM that, along a flow line,
This relation shows that ap/ax cannot change sign during the charge motion, and hence a monotonically decreasing p remains monotonically decreasing as the time goes on. For such charge distributions, ~(1/2,t)is
always greater than zero, and we conclude that a<1/2, because the char-
Fig.2
- Tridimensional
x and t .
At
t=1.4,
p l o t o f t h e charge
d e n s i t y as
the charge d e n s i t y i s almost uniform.
a function o f
ges i n i t i a l l y a t *1/2
Therefore,
a c t u a l l y r e a c h t h e e l e c t r o d e a t x=1.
L i n d m a y e r ' s i n e q u a l i t y , Eq.
(ZO),has
Our r e s u l t s s a t i s f y t h e c o n d i t i o n a<1/2,
which i s stronger than Eq. (20).
We g e t a>0.4838,
been d e r i v e d .
and a l s o t h e i n e q u a l i t y ~ ~ . ( 2 1 )
From E q . (211, we have
w h i c h i s t o be compared w i t h t h e e x t r a p o l a t e d
v a l ue
,
namely, a=0.4844 5 0.0005.
6. CONCLUSION
We deem t h a t o u r method g r e a t l y s i m p l i f i e s
the
s o l u t i o n o f the
p a r t i a 1 d i f f e r e n t i a l e q u a t i o n s as compared w i t h t h e d i r e c t
p e r f o r m e d by J. van ~ u r n h o u t ' ~ i,n a s i m p l e r case ( c h a r g e
one o f t h e e l e c t r o d e s )
.
FSCM
integration
touching only
We a l s o t h i n k t h a t a p p r o x i m a t e s o l u t i o n s o f t h e
p r o b l e m w i l l be welcome inasmuch as t h e e x t e r n a 1 c u r r e n t does n o t depend
on t h e f i n e d e t a i l o f t h e charge d i s t r i b u t i o r i .
I n t h i s r e s p e c t , o u r so-
l u t i o n may be h e l p f u l .
We a r e i n d e b t e d t o L.N.
de O l i v e i r a f o r c a l 1 i n g o u r a t t e n t i o n
to
the
meaning o f t h e Lindmayer theorem as e x p r e s s e d i n S e c t i o n 4
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-
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science Publ.,
New York,
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3. D. J. Gibbons, J. Phys. D, A p p l . Phys.,
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Appl.
I nter-
-
5. A. Many, G. Rakavy, Phys. Rev. 126, 1980 (1962).
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7. J. Lindmayer, J. Appl. Phys.
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5,853
8. B . Gross, ti. M. Perlman, J. Appl. Phys.
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"
(1972).
Exact S o l u t i o n s withNon-
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