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Simulation of relaxation processes in dielectrics
Journal of Electrostatics, 8 (1979) 59--68 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
59
SIMULATION OF RELAXATION PROCESSES IN DIELECTRICS*
ZDZIS~AW S~OWII~SKI Institute of Physics, Technical University of Wrocfaw, 50--370 Wroctaw, Wybrze~e Wyspiaf~skiego 27 (Poland)
Summary The paper proposes a network model for simulating the electrical properties of a dielectric, which enables the relaxation processes in a dielectric to be modelled over a wide range of frequencies. A method for determining the parameters of the model is presented, which is suitable for frequencies from 0 to several MHz. It is also shown how to convert the network model to e and tan5 values.
Introduction In an i n v e s t i g a t i o n o f t h e electrical p r o p e r t i e s o f dielectrics it is i m p o r t a n t t o distinguish b e t w e e n p a r t i c u l a r e f f e c t s relating t o t h e p r o c e s s o f charge t r a n s p o r t ; especially f o r m a t e r i a l s having a c o m p l i c a t e d s t r u c t u r e . T h e p r o p o s e d p h e n o m e n o l o g i c a l m o d e l o f a dielectric enables, at least, partial s e p a r a t i o n o f such effects, and, t o s o m e extent~ a q u a n t i t a t i v e d e s c r i p t i o n o f t h e effects. F u r t h e r m o r e , changes in t h e e x p e r i m e n t a l c o n d i t i o n s e n a b l e t h e i n f l u e n c e o f t h e s e changes o n a p a r t i c u l a r p r o c e s s t o be investigated. D e s c r i p t i o n o f a n e t w o r k m o d e l f o r a dielectric T h e s u b j e c t o f t h e m o d e l l i n g is a c o n d e n s e r w i t h t h e m a t e r i a l u n d e r i n v e s t i g a t i o n as dielectric (see Fig. 1). C r is t h e v a c u u m c a p a c i t y o f t h e c o n -
J t £p
£,-
I
£,
I Itl t
Fig. 1. Network model of a dielectric.
*Paper presented at the Colloquium on Space Charge in Dielectrics, Karpacz, Poland, September 12--17, 1977.
60
denser and R~ the resistivity of the sample for a direct current. The various R , - - C , two-terminal networks model the transient processes of charge transport in the dielectric. To describe precisely all relaxation processes in the dielectric, the R C twoterminal networks (Fig. 2) should comprise a continuum of "relaxation times" r = R C with limits 10;~l. In practice, it is sufficient to replace this continuum by a sequence of discrete R C elements in the range of interest Irma; rma~l. Those processes for which r lies outside this range are included in the elements Co and R~.
I
!
TT1 &
&
Cs
C,-, C.
t
Fig. 2. Model with a finite number of elements. To ensure the required accuracy of the model with a minimum of R C twoterminal networks the values of r should be described by an exp(n~) function. We would like that: Tn -- Tn_ I
=
T.-I
Tn
-- T . _ 2
(1)
Tn-I
and hence AT n Tn
ATn- 1
-
0,
(2)
Tn-I
where Ark = rk - rk-, is the distance between successive values of r. Assuming such a dense distribution, we obtain after integrating eqn. (2): rn/rn-1
= exp a
(3)
Construction of the model of the dielectric requires the determination of Co, R ~ and the sequence {R.(r.) } or {C. (r.) }. This model is correct for a dielectric under the given conditions. Furthermore, it is assumed that the processes to which the dielectric is submitted do not change its electric properties irreversibly. Determination o f the parameters o f the model
The m e t h o d applied to determine parameters of the model consists in investigating its response to a rectangular voltage (Fig. 3).
61
U~tl
IT 12T II4T -U Fig. 3. Square-wave excitation voltage with a period of 2 T.
The periodic c o m p o n e n t of the response of a single Rn--C, cell equals:
(4) with t = t'-mT
(5)
where t' is the real time and t the reduced time. The current through R~ is given by J~ = U / R ~ ,
(6)
which can be written as (7) R~ 1 + exp ( - ~--~) forT~
= o%
The current through Co is given by Jo = U C o 5 ( t ) .
(8)
In practice, it is not possible to measure the current through Co. The time constant for the charging of Co depends on the electrical properties of the measuring set and imposes a limitation to the determination of the model. For each value of time tk, set by the measuring position for measuring the corresponding current Ih, we may write according to [ 3]
62 with U An
~
-
-
Rn
2
(lO)
l+exp(¼)
We can rewrite eqn. (9) into
exp (_
Alj
n
now
ail •
....
J
alN •
L ak 1
(11)
(12)
akN
where a~n = exp (--tklTn)
(13)
All An values can be determined from:
=
akn
,
(14)
where [akn] -~(L) is the left-hand side reciprocal of the matrix [akn]. To avoid the inconvenient calculations of the left-hand side reciprocal of the generally rectangular matrix for each measurement, proper conditions for t~, rn are chosen to enable normalization of eqn. (14). To start with, we assume k = n. The reciprocals of the left-hand side and right. hand side of the square matrix are then equal and easy to calculate even for large n. The program of the calculations is based on [4]. Next, we multiply both sides of eqn. (14) by the diagonal matrix:
1 + exp(-T/rl) 2 1 + exp(--T/rN-l)
0
2
(15) 0
].
+
exp(-- T/~'~)
2 1 + exp(--T/rN 2
63
we then obtain
J! 1
[DN]
--
I!R,] (16)
[ak.]-I
U
gk
1)RN_~
Normalization Assuming tk = to 10 °"Ik
rn =
(17)
nmax=N,
t0 10 °"1 n
for n ¢
oo
for n = nma
I
=N ,
(18)
and to --- t o ,
(19)
we have I e x p ( - ( 1 0 '' (~-"))) ak,
=
1
for n ¢ N , forn=N.
(2O)
The matrix [a~,] -1 once calculated can be used in the calculation of the model parameters for any ro i.e. it enables coverage of any It0 10°'1; ro 10 °'1Ck-OI range. If in addition, we multiply T by the same n u m b e r as to and r0, [ D , ] will also n o t change. It is n o t possible, considering the technical limitations, to carry out a measurement for one frequency only of the rectangular wave, when r varies by more than 4--5 decades. To obtain results for a wider range of r, some values of T should be used to cover the whole r-band of interest. Omitting c o m p o n e n t Co and replacing the two-terminal networks with long time-constants by R~ we obtain large errors for the values of 1 / R , within the limits of the range considered. To eliminate these errors the succeeding T values should be chosen in such a manner that some values of r (in practice, 0.5--1 decades) are repeated in the neighbouring ranges. The extreme values of elements of the model The components of the network model for r -~ oo can be determined correctly, because components with r > rn can be replaced by R~ for any arbitrary value of TN. For r -- 0, there are difficulties connected with conveying a rectangular wave of short rise-time to the sample and also because of the inertia of the adjustable capacitances of the set. The most precise value of Co can be found by measuring the capacity of the sample with a bridge or
64
resonance m e t h o d at a selected frequency in the proper range I~ 1/T/v; co 1/rll. As will be shown later, the measured value of the capacity equals =
1 + ~2/~
C = [Co; C,;...; CN]
=
(21)
1 1 + (.O2/ ~J N 2 where co is the measuring frequency and co, = I / r , . Hence 1 1 + ~2/~ (22)
[ C 1 , . . . , CN]
C 0 = C-
1
1 + ~2/co~ or
Co
=
C
-
N
1
n ~=1
R,
Tn
1+¢o2T~ .
(23)
Discussion It is convenient to present the described model as a sequence of { I / R . } or {C,} and R ~ , Co, see Figs. 4 and 5. In these figures, maxima associated with several relaxation processes may be noted: the narrow ones are associated with domain polarization, the wider maxima with ion diffusion processes, etc. The shape of the maxima agrees with those obtained by an analysis of the physical ¢
oo.. •
•
•.•°
. . . . .
,.•
°
•.o•
log tr/v o) ~ o.~ n Fig. 4. S e q u e n c e o f {1/Rn}.
66
.
.a._ . .
. . .
.
.
.
.
.
.
. .
.
:
.,...., ,
. .
.
. .
. . ..**
. .
. .
.
.
l*.
.**
. , ..**
log (r/To)
s
0.1 n
Fig. 5. Sequence of {C,}.
processes, see [l-2]. Values of the model parameters obtained at different temperatures give information about the influence of the temperature on these processes (Fig. 6). But discrimination of processes on the basis of measuring results of only one parameter is not possible because the contribution of some processes into the total effect could be too small (comparable with the experimental error).
-.
.:
/
Fig. 6. Temperature dependence of the model parameters.
The model can be easily converted into e(w) or tan 6 (w ). This enables a comparison of experimental results obtained by applying different methods, and also to complete one experiment with results of the others. Impulse measurements are especially suitable for frequencies from 0 UP to 1 MHz. For higher frequencies - between several MHz and optical frequencies - it is more advisable to apply e and tan 6 measurements. By applying resonance, microwave and optical methods, it is possible to extend the model to very high frequencies and to obtain a comprehensive description of the dielectric being investigated.
66
Conversion of the network model to e and tan 6 values The admittance of the model can be written as:
1
Y=--
RI
1
+...+
1 + ¢o~/¢o2
( "{- jLO
1
1
R.
1 + ¢O2N/¢O2
+
1
--
+
R~
i
i
Co + C 1 1 . { . ( . 0 2 / ( . o 2 "{-..,"{-C N
1+¢°2/¢°~v
)
(24)
Hence
1 1 1 e(¢o) = -:- [Co, C,, . . . , CN] C'p
1 + co2/w~
(25)
1 +
If we replace by 1 the terms of ~ k less than the current frequency, and by zero, Wk larger than the current frequency in the vector [1; 1/(1+~o2/¢o~); ...; 1/(1+co2/co~)] in the range look; ¢ok-l[, we obtain: 1
e(co) = Cp
k
= C,,
(26)
or
1
k
1
The expression obtained for e(co) is a step function n o t very different from the continuous function given by eqn. (25).
67 F r o m the admittance of the model (24) we have -
1
1
1 1 + col/co2
1
RN'Roo
Rl
1
+
co,~lco2 1
tan ~ (co) =
coICo, c , , . . . , ON]
1+
(28)
co2/co~ 1
I + co 2 /co~
Reducing the above according to the same rule as in (25) we obtain in the range Icok, ¢ok+~l I N ~-i- E
R.
tams(co) =
co
1 --
n--k
RN
o + ~C. n=l
Conclusion The experiment which should be carried o u t to obtain the parameters of the model is very simple. It consists in exciting the investigated sample by a rectangular voltage and measuring the values of the current at tk m o m e n t s of time. Having the J(tk ) values, we can calculate from eqn. (16) the terms of the sequence {1]Rn ) or {Cn} = {rn/Rn). The sequence {1]Rn } or {Cn ) form the basis for a discussion of the relaxation processes in the sample. If we also perform a measurement of e for the frequency chosen, we can calculate, using eqns. (25) and (28), e(co) and tan5 (co). The frequency range possible includes very low frequencies which cannot be reached b y bridge or resonance methods.
Acknowledgments The author wishes to thank Professor A. Szaynok for many helpful suggestions.
68 References 1 2 3 4
G. Tomandl, J. Non-Cryst. Sol., 14 (1974) 101. H. Namikawa, J. Non-Cryst. Sol., 18 (1975) 173. Z. S~owib.ski, Raporty Inst. Fiz. PWr, 1976, nr 237 (in Polish). J. StaSko, Biblioteki programbw i podgrogrambw maszyn Odra serii 1300. Skrypt. Wrochw: Polit. Wroct. 1975 (in Polish).
59
SIMULATION OF RELAXATION PROCESSES IN DIELECTRICS*
ZDZIS~AW S~OWII~SKI Institute of Physics, Technical University of Wrocfaw, 50--370 Wroctaw, Wybrze~e Wyspiaf~skiego 27 (Poland)
Summary The paper proposes a network model for simulating the electrical properties of a dielectric, which enables the relaxation processes in a dielectric to be modelled over a wide range of frequencies. A method for determining the parameters of the model is presented, which is suitable for frequencies from 0 to several MHz. It is also shown how to convert the network model to e and tan5 values.
Introduction In an i n v e s t i g a t i o n o f t h e electrical p r o p e r t i e s o f dielectrics it is i m p o r t a n t t o distinguish b e t w e e n p a r t i c u l a r e f f e c t s relating t o t h e p r o c e s s o f charge t r a n s p o r t ; especially f o r m a t e r i a l s having a c o m p l i c a t e d s t r u c t u r e . T h e p r o p o s e d p h e n o m e n o l o g i c a l m o d e l o f a dielectric enables, at least, partial s e p a r a t i o n o f such effects, and, t o s o m e extent~ a q u a n t i t a t i v e d e s c r i p t i o n o f t h e effects. F u r t h e r m o r e , changes in t h e e x p e r i m e n t a l c o n d i t i o n s e n a b l e t h e i n f l u e n c e o f t h e s e changes o n a p a r t i c u l a r p r o c e s s t o be investigated. D e s c r i p t i o n o f a n e t w o r k m o d e l f o r a dielectric T h e s u b j e c t o f t h e m o d e l l i n g is a c o n d e n s e r w i t h t h e m a t e r i a l u n d e r i n v e s t i g a t i o n as dielectric (see Fig. 1). C r is t h e v a c u u m c a p a c i t y o f t h e c o n -
J t £p
£,-
I
£,
I Itl t
Fig. 1. Network model of a dielectric.
*Paper presented at the Colloquium on Space Charge in Dielectrics, Karpacz, Poland, September 12--17, 1977.
60
denser and R~ the resistivity of the sample for a direct current. The various R , - - C , two-terminal networks model the transient processes of charge transport in the dielectric. To describe precisely all relaxation processes in the dielectric, the R C twoterminal networks (Fig. 2) should comprise a continuum of "relaxation times" r = R C with limits 10;~l. In practice, it is sufficient to replace this continuum by a sequence of discrete R C elements in the range of interest Irma; rma~l. Those processes for which r lies outside this range are included in the elements Co and R~.
I
!
TT1 &
&
Cs
C,-, C.
t
Fig. 2. Model with a finite number of elements. To ensure the required accuracy of the model with a minimum of R C twoterminal networks the values of r should be described by an exp(n~) function. We would like that: Tn -- Tn_ I
=
T.-I
Tn
-- T . _ 2
(1)
Tn-I
and hence AT n Tn
ATn- 1
-
0,
(2)
Tn-I
where Ark = rk - rk-, is the distance between successive values of r. Assuming such a dense distribution, we obtain after integrating eqn. (2): rn/rn-1
= exp a
(3)
Construction of the model of the dielectric requires the determination of Co, R ~ and the sequence {R.(r.) } or {C. (r.) }. This model is correct for a dielectric under the given conditions. Furthermore, it is assumed that the processes to which the dielectric is submitted do not change its electric properties irreversibly. Determination o f the parameters o f the model
The m e t h o d applied to determine parameters of the model consists in investigating its response to a rectangular voltage (Fig. 3).
61
U~tl
IT 12T II4T -U Fig. 3. Square-wave excitation voltage with a period of 2 T.
The periodic c o m p o n e n t of the response of a single Rn--C, cell equals:
(4) with t = t'-mT
(5)
where t' is the real time and t the reduced time. The current through R~ is given by J~ = U / R ~ ,
(6)
which can be written as (7) R~ 1 + exp ( - ~--~) forT~
= o%
The current through Co is given by Jo = U C o 5 ( t ) .
(8)
In practice, it is not possible to measure the current through Co. The time constant for the charging of Co depends on the electrical properties of the measuring set and imposes a limitation to the determination of the model. For each value of time tk, set by the measuring position for measuring the corresponding current Ih, we may write according to [ 3]
62 with U An
~
-
-
Rn
2
(lO)
l+exp(¼)
We can rewrite eqn. (9) into
exp (_
Alj
n
now
ail •
....
J
alN •
L ak 1
(11)
(12)
akN
where a~n = exp (--tklTn)
(13)
All An values can be determined from:
=
akn
,
(14)
where [akn] -~(L) is the left-hand side reciprocal of the matrix [akn]. To avoid the inconvenient calculations of the left-hand side reciprocal of the generally rectangular matrix for each measurement, proper conditions for t~, rn are chosen to enable normalization of eqn. (14). To start with, we assume k = n. The reciprocals of the left-hand side and right. hand side of the square matrix are then equal and easy to calculate even for large n. The program of the calculations is based on [4]. Next, we multiply both sides of eqn. (14) by the diagonal matrix:
1 + exp(-T/rl) 2 1 + exp(--T/rN-l)
0
2
(15) 0
].
+
exp(-- T/~'~)
2 1 + exp(--T/rN 2
63
we then obtain
J! 1
[DN]
--
I!R,] (16)
[ak.]-I
U
gk
1)RN_~
Normalization Assuming tk = to 10 °"Ik
rn =
(17)
nmax=N,
t0 10 °"1 n
for n ¢
oo
for n = nma
I
=N ,
(18)
and to --- t o ,
(19)
we have I e x p ( - ( 1 0 '' (~-"))) ak,
=
1
for n ¢ N , forn=N.
(2O)
The matrix [a~,] -1 once calculated can be used in the calculation of the model parameters for any ro i.e. it enables coverage of any It0 10°'1; ro 10 °'1Ck-OI range. If in addition, we multiply T by the same n u m b e r as to and r0, [ D , ] will also n o t change. It is n o t possible, considering the technical limitations, to carry out a measurement for one frequency only of the rectangular wave, when r varies by more than 4--5 decades. To obtain results for a wider range of r, some values of T should be used to cover the whole r-band of interest. Omitting c o m p o n e n t Co and replacing the two-terminal networks with long time-constants by R~ we obtain large errors for the values of 1 / R , within the limits of the range considered. To eliminate these errors the succeeding T values should be chosen in such a manner that some values of r (in practice, 0.5--1 decades) are repeated in the neighbouring ranges. The extreme values of elements of the model The components of the network model for r -~ oo can be determined correctly, because components with r > rn can be replaced by R~ for any arbitrary value of TN. For r -- 0, there are difficulties connected with conveying a rectangular wave of short rise-time to the sample and also because of the inertia of the adjustable capacitances of the set. The most precise value of Co can be found by measuring the capacity of the sample with a bridge or
64
resonance m e t h o d at a selected frequency in the proper range I~ 1/T/v; co 1/rll. As will be shown later, the measured value of the capacity equals =
1 + ~2/~
C = [Co; C,;...; CN]
=
(21)
1 1 + (.O2/ ~J N 2 where co is the measuring frequency and co, = I / r , . Hence 1 1 + ~2/~ (22)
[ C 1 , . . . , CN]
C 0 = C-
1
1 + ~2/co~ or
Co
=
C
-
N
1
n ~=1
R,
Tn
1+¢o2T~ .
(23)
Discussion It is convenient to present the described model as a sequence of { I / R . } or {C,} and R ~ , Co, see Figs. 4 and 5. In these figures, maxima associated with several relaxation processes may be noted: the narrow ones are associated with domain polarization, the wider maxima with ion diffusion processes, etc. The shape of the maxima agrees with those obtained by an analysis of the physical ¢
oo.. •
•
•.•°
. . . . .
,.•
°
•.o•
log tr/v o) ~ o.~ n Fig. 4. S e q u e n c e o f {1/Rn}.
66
.
.a._ . .
. . .
.
.
.
.
.
.
. .
.
:
.,...., ,
. .
.
. .
. . ..**
. .
. .
.
.
l*.
.**
. , ..**
log (r/To)
s
0.1 n
Fig. 5. Sequence of {C,}.
processes, see [l-2]. Values of the model parameters obtained at different temperatures give information about the influence of the temperature on these processes (Fig. 6). But discrimination of processes on the basis of measuring results of only one parameter is not possible because the contribution of some processes into the total effect could be too small (comparable with the experimental error).
-.
.:
/
Fig. 6. Temperature dependence of the model parameters.
The model can be easily converted into e(w) or tan 6 (w ). This enables a comparison of experimental results obtained by applying different methods, and also to complete one experiment with results of the others. Impulse measurements are especially suitable for frequencies from 0 UP to 1 MHz. For higher frequencies - between several MHz and optical frequencies - it is more advisable to apply e and tan 6 measurements. By applying resonance, microwave and optical methods, it is possible to extend the model to very high frequencies and to obtain a comprehensive description of the dielectric being investigated.
66
Conversion of the network model to e and tan 6 values The admittance of the model can be written as:
1
Y=--
RI
1
+...+
1 + ¢o~/¢o2
( "{- jLO
1
1
R.
1 + ¢O2N/¢O2
+
1
--
+
R~
i
i
Co + C 1 1 . { . ( . 0 2 / ( . o 2 "{-..,"{-C N
1+¢°2/¢°~v
)
(24)
Hence
1 1 1 e(¢o) = -:- [Co, C,, . . . , CN] C'p
1 + co2/w~
(25)
1 +
If we replace by 1 the terms of ~ k less than the current frequency, and by zero, Wk larger than the current frequency in the vector [1; 1/(1+~o2/¢o~); ...; 1/(1+co2/co~)] in the range look; ¢ok-l[, we obtain: 1
e(co) = Cp
k
= C,,
(26)
or
1
k
1
The expression obtained for e(co) is a step function n o t very different from the continuous function given by eqn. (25).
67 F r o m the admittance of the model (24) we have -
1
1
1 1 + col/co2
1
RN'Roo
Rl
1
+
co,~lco2 1
tan ~ (co) =
coICo, c , , . . . , ON]
1+
(28)
co2/co~ 1
I + co 2 /co~
Reducing the above according to the same rule as in (25) we obtain in the range Icok, ¢ok+~l I N ~-i- E
R.
tams(co) =
co
1 --
n--k
RN
o + ~C. n=l
Conclusion The experiment which should be carried o u t to obtain the parameters of the model is very simple. It consists in exciting the investigated sample by a rectangular voltage and measuring the values of the current at tk m o m e n t s of time. Having the J(tk ) values, we can calculate from eqn. (16) the terms of the sequence {1]Rn ) or {Cn} = {rn/Rn). The sequence {1]Rn } or {Cn ) form the basis for a discussion of the relaxation processes in the sample. If we also perform a measurement of e for the frequency chosen, we can calculate, using eqns. (25) and (28), e(co) and tan5 (co). The frequency range possible includes very low frequencies which cannot be reached b y bridge or resonance methods.
Acknowledgments The author wishes to thank Professor A. Szaynok for many helpful suggestions.
68 References 1 2 3 4
G. Tomandl, J. Non-Cryst. Sol., 14 (1974) 101. H. Namikawa, J. Non-Cryst. Sol., 18 (1975) 173. Z. S~owib.ski, Raporty Inst. Fiz. PWr, 1976, nr 237 (in Polish). J. StaSko, Biblioteki programbw i podgrogrambw maszyn Odra serii 1300. Skrypt. Wrochw: Polit. Wroct. 1975 (in Polish).