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Model of polarization build-up during corona charging of ferroelectric polymers
Journal of Electrostatics, 30 (1993) 39-46 Elsevier
39
M O D E L OF P O L A R I Z A T I O N BUILD-UP D U R I N G C O R O N A C H A R G I N G OF FERROELECTRIC POLYMERS S.N.Fedosov and A.E.Sergeeva Physics Department, Odessa Technological Institute, ul.Sverdlova 112, 270039 Odessa, Ukraine ABSTRACT The method of a constant charging current is applied to study processes of polarization build-up and space charge accumulation in ferroelectric polymers. A phenomenological model is constructed which adequately explains experimentally observed features. A coercive field and a remanent polarization are evaluated by fitting the experimental data in the model. 1. I N T R O D U C T I O N Ferroelectric polymers attract great interest because of their good piezoelectric and pyroelectric properties which are usually attributed to a high level of remanent polarization. Polarization phenomena in ferroelectric polymers are extensively studied [1-5], but several important features are not consistently accounted for, espacially the non-linear dependence of polarization from the field in the thickness direction [6-8]. Apart from the traditional approach of studying the current response to a step voltage, a new method is developed to investigate transient processes of carrier transport and space charge accumulation in polymers [9,10] the method of a constant charging current. In the present work, the method is applied to study the build-up of polarization in poly(vinylidenefluoride) (PVDF) which is a typical ferroelectric polymer. An important role is assigned in the proposed phenomenological model to injected charge carriers which are supposed to be trapped at crystallite boundaries. The need for a trapped charge to stabilize the polarization in PVDF was already proved [11]. The model explains existence of three stages in the polarization build-up and provides a method for evaluating a coercive field and a remanent polarization in ferroelectric polymers. 2. M O D E L Consider a ferroelectric polymer of thickness x0 equipped with a blocking grounded electrode on one surface (x = x0) while the charge carriers are injected in the bulk at a constant rate through the other surface (x = 0) providing the constant current density i0. The injecting electrode is either a real thin metallic film or a virtual electrode formed by adsorbed and thermalized 0304-3886/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
40 charged panicles. Let a level of the injection be sufficiently high to neglect the intrinsic conductivity that is quite plausible for polymers. T h e experimentally measured variable is the voltage across the sample also called electret poltential. T h e r e is a certain probability for the injected carriers to be trapped considering the heterogeneous structure of the ferroelectric polymer. Besides that, it is reasonable to assume that not all the traps are filled. Considering the ferroelectric nature of the sample, one can present the dielectric displacement as (1)
D ( E ) = e0eE + P ( E ) ,
where e is the relative permittivity which comprises the contributions of all non-ferroelectric polarization processes, e0 the permittivity of vacuum, E the electric field, P the ferroelectric polarization. A simplified d e p e n d e n c e P(E) which is indeed a fraction of the idealised hysteresis loop is given analytically by
P(E) =
0;
E _< Ec ,
e ~ ' (E - Ec ) ;
Ec -< E -< E~,
e0e"
E~ -< E ,
I
(E - Es ) ;
(2)
where e' and e " are the dynamic dielectric permittivities, Ec the coercive field, Es the field corresponding to the end of a plateau in the P(E) curve. It is k n o w n that the switching time in ferroelectric polymers does not exceed the range of milliseconds [12] while the time constants of the voltage and the field are of the o r d e r of minutes. Therefore, the poling process can be considered as quasi-stationary since the polarization P follows the field E without a delay. U n d e r these conditions, the process of charging can be divided into three stages as shown in Figure 1. At the first stage, the field is lower than the coercive value Ec e v e r y w h e r e in the bulk and a strong polarization is not developed. At the second stage, the field near the grounded positive electrode exceeds Ec resulting in formation of two regions (zones). The third zone appears at the onset of the third stage when the field near the positive electrode exceeds the critical value of Es . To obtain analytical expressions for the voltage-time d e p e n d e n c e one should solve the equations for charge transport at every stage of the charging process. T h e r e must be considered that the total charging current remains constant being a sum of the three components. The equations with the initial and boundary conditions are as follows
eoeOE/Ot + OP/0t + e n ~ E = i0,
(3)
e0e0E/0x + 0 P / 0 x = end, (nt >> nc),
(4)
0nt/0t = nc/to,
(5)
41 V(t) = f E(x,t)dx,
(6)
E(0,0) = nc(0,0) = nt(0,O) = E(0,t) = 0, E(xo,tl2)
=
E(Xl2,t)
=
Ec,
E(xo,tz3) = E(x2a,t) = Es, where t is the time, x the coordinate, xo the sample thickness, /~ the mobility of charge carriers, nc and nt the densities of free and trapped carriers, to the capture time constant, t12 and t2s the time at which the first and the second stage are completed, x12 and x2s the coordinates of the time dependent boundaries between the zones.
3rd stage zonell zonell 'I I ~ ' s ~
~o-~"~
,zo..n..i.
/ !
~
/ '
2rid stage
;/poz qz tio I
M" ~zz I
I ~23
/--Z
[L/~~o~'I ~//~P~°~zl - z~t~t~o x12
corona diso~arge
Xo
---" ~
0
xo
Figure 1. Schematic diagram showing three stagers of the charging. Displacement current can be neglected at the first stage since the field, remaining zero at the injecting electrode, does not vary essentially in the rest of the sample. Therefore, the total current density io has only a conduction component. A polarization component does not exist at this stage because the field is lower than the coercive value Ec and polarization is zero. The sample behaves as a conventional paraelectric. The following expressions are obtained under these assumptions for E(x,t) and V(t) functions at the first stage
42
E(x,t) = 2(ioxt/eoe/,to) 1/2, x -< xiz, t -> 0,
(7)
V(t) = ( 4 / 3 ) (ioxoat/eoe/~to) l/2,
(8)
w h e r e V(t) is calculated according to e q n . ( 6 ) . At the s e c o n d stage, the sample consists of two zones as s h o w n in Figure 1. It is assumed that expression (7) is still valid for the z o n e I. H o w e v e r , o n e c a n n o t l o n g e r neglect the displacement current in the zone II. A n o t h e r a s s u m p t i o n c o n c e r n s the product /~E in e q n . ( 3 ) a n d introduced in o r d e r to o b t a i n the analytical (but u n d o u b t e d l y approximate) solution. T h e value /~E r e p r e s e n t i n g the velocity of charge carriers is assumed constant, since the field E varies o n l y in a rather n a r r o w range between Ec a n d Es in the z o n e II. T h e n , the following intermediate equation can be derived from eqns. (2)- (5) eo(e + e ' ) ( 0 E / 0 t )
(9)
+ votoeo(e + e ' ) ( a Z E / 0 x & ) = io,
Ec -< E _< Es, t12 -< t -< t23, X12 -- X -- x23, E ( X l 2 , t ) = E(xo,tt2) = Ec. T h e final expressisions for E(x,t) of c h a r g i n g are as follows
and V(t) functions d u r i n g the s e c o n d stage
E(x,t) = Ec + [io/eo(~: + e ' ) ] (t - h2 + vototlzs/xo){1 - exp[-(x
- xlz)/voto]}, x12 -< x <_ x23, t12 -< t ~ t23,
V(t) = [ioxotlz/eo(e + e ' ) ] { 4 s h z / 3 t tlz/t) + (t/tl2 + votos/xo + 1 ) ] ( i exp(-
(10)
+ [eo(e + e ' ) E c / i o t i 2 ] ( 1
-
- t12/t - (voto/xo)[1 -
(t - tl/)Xo/votot)])}.
(11)
E q n . ( l l ) is o b t a i n e d from e q n s . ( 6 ) , (7) a n d (10). It can be s h o w n that the c o o r d i n a t e x~2 of a b o r d e r between zones 1 a n d It moves according to the law x12 = xott2/t. This feature is also considered while o b t a i n i n g e q n . ( l l ) . D u r i n g the third stage of the poling process, the sample consists of three zones, as s h o w n in Figure 1. T h e intermediate equation for E(x,t) in the z o n e I l l is similar to e q n . ( 9 ) , but e' is substituted for e". C o n s i d e r i n g c o n s t a n c y of the field at the x23 b o r d e r a n d continuity of the charge at the back electrode (x = x0), o n e can obtain the following expression for the field at the z o n e I l l E(x,t) = Es + [io/eo(e + e ' ) ] { T - [eo(e + e ' ) ( E s - Ec)/io + + T] e x p [ -
(x - x2a)/voto], x23 -
(12)
x <_ xo, t23 -< t,
where T = t23 - t12 + votohzs/xo + (t - t23)(~: + e ' ) / ( e
+ c").
43 The final expression for the voltage across the sample is not given here because it is rather cumbersome. It is clear that the boundary coordinates between the zones x~2 and x2a are time-dependent and their onward movement to the injecting electrode is similar to the propagation of domain walls in conventional ferroelectrics. 3. E X P E R I M E N T A L Experiments were performed on the 25-/~m thick uniaxially stretched PVDF films containing amorphous and crystalline (Form I) phases in nearly equal volume fractions. Aluminium electrode of 0.1 btm thickness was deposited on one surface of the film by a thermal evaporation in vacuum. Then the free surface of the sample was subjected to a negative corona discharge generated by a pointed tungsten electrode stressed at 14-18 kV, with the metallized surface being grounded. A control grid was provided between the corona electrode and the sample to make the charging process controllable. The total current was automatically maintained constant at a level of 60-100 /~A/m ~ while the transient voltage across the sample was measured by the Kelvin method and subsequently recorded. The experimental set-up was similar to that of Giacometti [9]. 4. R E S U L T S AND D I S C U S S I O N Three stages of the charging process are distinctly seen in Figure 2 where the experimental dependence of the voltage on the total trasferred charge is shown. At the first stage, V is proportional to t 1/2 as prescribed by the expression (8) indicating validity of the deep trapping mechanism. The voltage kinetics was calculated with the following values of the parameters: i0 -- 100 /~A/m 2 , x0 = 25 /~m, e = 10, e' = 725, e" = 45, tt2 = 250 s, t23 = 600 s, s = 5.47, y = 13.3, v0to = 6 ~m, /~t0 = 1.3x10 -ll m2/V, Ec = 42 MV/m, Es = 5 1 . 5 MV/m, Ps = 61 m C / m . Most of the parameters were found by the curve fitting method. The value of /~to was obtained from the initial part of the experimental curve shown in Figure 2. The values of tiE and t23 were also taken from the experimental curve in Figure 2. The numerical values of the coercive field and the remanent polarization are in good agreement with the corresponding data reported in other works [13,14]. On the whole, one can conclude that the proposed model correctly describes the main features of the charging process, especially existence of the three stages in the polarization build-up. The model also emphasizes an important role of the injected charge in establishing the stable polarization in ferroelectric polymers. The most suitable sites for the charge trapping are the crystallite boundaries, where the charge carriers, if trapped, neutralize partly the internal depolarization field. Similar processes are known to take place in ferroelectric ceramics. The favourable conditions for the charge trapping are probably created due to the large-scale fluctuations caused by alignment of permanent dipoles in the crystallites.
44
2
i >
0 0
.t 50
£ I00
150
CHARGE DENSITY (i0t), mC/m 2 Figure 2. Kinetics of the electret potential during the constant-current charging. Shaded region - experiment, solid line - calculation according to the model.
5. C O N C L U S I O N It is proved that the three-stage charging of ferroelectric polymers is a result of nucleation and growth of strong polarization. An important role in formation of the polarization plays injection of charge carriers and their subsequent trapping in the bulk. It is possible to obtain the values of the coercive field and the remanent polarization as well as several other parameters by comparing the experimental kinetics of the electret potential during the constant-current charging with the curve based on the phenomenological model. Although the model is approximate, calculations show that the confidence limits of the evaluated fit parameters are within the permissible range of not more than -+20%. Considering the more or less universal approach developed in this work, we believe that its results can be extended to examine the polarization phenomena in other materials such as ferroelectric ceramics and composites formed by ferroelectric inorganic crystals dispersed in a polymer matrix. 6. R E F E R E N C E S
N.Murayama, J.Polym.Sci.: Polym.Phys.Ed., 13 (1975) 929. H.von Seggern and T.T.Wang, J.Appl.Phys., 56 (1984) 2448.
45 3 4 5 6 7 8 9 10 11 12 13 14
W.Eisenmenger and M.Haardt, Sol.St.Comm., 41 (1982) 917. B.Gross, H.von Seggern and R.Gerhard-Multhaupt, J.Phys., D, 18 (1985) 2497. G.M.Sessler and A.Berrassoul, Ferroelectrics, 76 (1987) 489. R.Gerhard-Multhaupt, IEEE Trans.Elec.Insul., 22 (1987) 531. B.Gross, R.Gerhard-Multhaupt, A.Berrassoul and G.M.Sessler, J.Appl.Phys., 62 (1987) 1429. G.M.Sessler (ed.), Electrets, Springer, Berlin, 1988. J.A.Giacometti, J.Phys.: D, 20 (1987) 675. J.A.Giacometti and J.S.C.Campos, Rev.Sci.Instrum., 61 (1990) 1143. M.Womes, E.Bihler and W.Eisenmenger, IEEE Trans.Elec.Insul., 24 (1989) 461. T.Furukawa and G.E.Johnson, Appl.Phys.Lett., 38 (1981) 1027. B.Gross and J.A.Giacometti, Appl.Phys.: A, 37 (1985) 89. S.R.Kurtz and R.S.Hughes, J.Appl.Phys., 54 (1983) 229.
39
M O D E L OF P O L A R I Z A T I O N BUILD-UP D U R I N G C O R O N A C H A R G I N G OF FERROELECTRIC POLYMERS S.N.Fedosov and A.E.Sergeeva Physics Department, Odessa Technological Institute, ul.Sverdlova 112, 270039 Odessa, Ukraine ABSTRACT The method of a constant charging current is applied to study processes of polarization build-up and space charge accumulation in ferroelectric polymers. A phenomenological model is constructed which adequately explains experimentally observed features. A coercive field and a remanent polarization are evaluated by fitting the experimental data in the model. 1. I N T R O D U C T I O N Ferroelectric polymers attract great interest because of their good piezoelectric and pyroelectric properties which are usually attributed to a high level of remanent polarization. Polarization phenomena in ferroelectric polymers are extensively studied [1-5], but several important features are not consistently accounted for, espacially the non-linear dependence of polarization from the field in the thickness direction [6-8]. Apart from the traditional approach of studying the current response to a step voltage, a new method is developed to investigate transient processes of carrier transport and space charge accumulation in polymers [9,10] the method of a constant charging current. In the present work, the method is applied to study the build-up of polarization in poly(vinylidenefluoride) (PVDF) which is a typical ferroelectric polymer. An important role is assigned in the proposed phenomenological model to injected charge carriers which are supposed to be trapped at crystallite boundaries. The need for a trapped charge to stabilize the polarization in PVDF was already proved [11]. The model explains existence of three stages in the polarization build-up and provides a method for evaluating a coercive field and a remanent polarization in ferroelectric polymers. 2. M O D E L Consider a ferroelectric polymer of thickness x0 equipped with a blocking grounded electrode on one surface (x = x0) while the charge carriers are injected in the bulk at a constant rate through the other surface (x = 0) providing the constant current density i0. The injecting electrode is either a real thin metallic film or a virtual electrode formed by adsorbed and thermalized 0304-3886/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
40 charged panicles. Let a level of the injection be sufficiently high to neglect the intrinsic conductivity that is quite plausible for polymers. T h e experimentally measured variable is the voltage across the sample also called electret poltential. T h e r e is a certain probability for the injected carriers to be trapped considering the heterogeneous structure of the ferroelectric polymer. Besides that, it is reasonable to assume that not all the traps are filled. Considering the ferroelectric nature of the sample, one can present the dielectric displacement as (1)
D ( E ) = e0eE + P ( E ) ,
where e is the relative permittivity which comprises the contributions of all non-ferroelectric polarization processes, e0 the permittivity of vacuum, E the electric field, P the ferroelectric polarization. A simplified d e p e n d e n c e P(E) which is indeed a fraction of the idealised hysteresis loop is given analytically by
P(E) =
0;
E _< Ec ,
e ~ ' (E - Ec ) ;
Ec -< E -< E~,
e0e"
E~ -< E ,
I
(E - Es ) ;
(2)
where e' and e " are the dynamic dielectric permittivities, Ec the coercive field, Es the field corresponding to the end of a plateau in the P(E) curve. It is k n o w n that the switching time in ferroelectric polymers does not exceed the range of milliseconds [12] while the time constants of the voltage and the field are of the o r d e r of minutes. Therefore, the poling process can be considered as quasi-stationary since the polarization P follows the field E without a delay. U n d e r these conditions, the process of charging can be divided into three stages as shown in Figure 1. At the first stage, the field is lower than the coercive value Ec e v e r y w h e r e in the bulk and a strong polarization is not developed. At the second stage, the field near the grounded positive electrode exceeds Ec resulting in formation of two regions (zones). The third zone appears at the onset of the third stage when the field near the positive electrode exceeds the critical value of Es . To obtain analytical expressions for the voltage-time d e p e n d e n c e one should solve the equations for charge transport at every stage of the charging process. T h e r e must be considered that the total charging current remains constant being a sum of the three components. The equations with the initial and boundary conditions are as follows
eoeOE/Ot + OP/0t + e n ~ E = i0,
(3)
e0e0E/0x + 0 P / 0 x = end, (nt >> nc),
(4)
0nt/0t = nc/to,
(5)
41 V(t) = f E(x,t)dx,
(6)
E(0,0) = nc(0,0) = nt(0,O) = E(0,t) = 0, E(xo,tl2)
=
E(Xl2,t)
=
Ec,
E(xo,tz3) = E(x2a,t) = Es, where t is the time, x the coordinate, xo the sample thickness, /~ the mobility of charge carriers, nc and nt the densities of free and trapped carriers, to the capture time constant, t12 and t2s the time at which the first and the second stage are completed, x12 and x2s the coordinates of the time dependent boundaries between the zones.
3rd stage zonell zonell 'I I ~ ' s ~
~o-~"~
,zo..n..i.
/ !
~
/ '
2rid stage
;/poz qz tio I
M" ~zz I
I ~23
/--Z
[L/~~o~'I ~//~P~°~zl - z~t~t~o x12
corona diso~arge
Xo
---" ~
0
xo
Figure 1. Schematic diagram showing three stagers of the charging. Displacement current can be neglected at the first stage since the field, remaining zero at the injecting electrode, does not vary essentially in the rest of the sample. Therefore, the total current density io has only a conduction component. A polarization component does not exist at this stage because the field is lower than the coercive value Ec and polarization is zero. The sample behaves as a conventional paraelectric. The following expressions are obtained under these assumptions for E(x,t) and V(t) functions at the first stage
42
E(x,t) = 2(ioxt/eoe/,to) 1/2, x -< xiz, t -> 0,
(7)
V(t) = ( 4 / 3 ) (ioxoat/eoe/~to) l/2,
(8)
w h e r e V(t) is calculated according to e q n . ( 6 ) . At the s e c o n d stage, the sample consists of two zones as s h o w n in Figure 1. It is assumed that expression (7) is still valid for the z o n e I. H o w e v e r , o n e c a n n o t l o n g e r neglect the displacement current in the zone II. A n o t h e r a s s u m p t i o n c o n c e r n s the product /~E in e q n . ( 3 ) a n d introduced in o r d e r to o b t a i n the analytical (but u n d o u b t e d l y approximate) solution. T h e value /~E r e p r e s e n t i n g the velocity of charge carriers is assumed constant, since the field E varies o n l y in a rather n a r r o w range between Ec a n d Es in the z o n e II. T h e n , the following intermediate equation can be derived from eqns. (2)- (5) eo(e + e ' ) ( 0 E / 0 t )
(9)
+ votoeo(e + e ' ) ( a Z E / 0 x & ) = io,
Ec -< E _< Es, t12 -< t -< t23, X12 -- X -- x23, E ( X l 2 , t ) = E(xo,tt2) = Ec. T h e final expressisions for E(x,t) of c h a r g i n g are as follows
and V(t) functions d u r i n g the s e c o n d stage
E(x,t) = Ec + [io/eo(~: + e ' ) ] (t - h2 + vototlzs/xo){1 - exp[-(x
- xlz)/voto]}, x12 -< x <_ x23, t12 -< t ~ t23,
V(t) = [ioxotlz/eo(e + e ' ) ] { 4 s h z / 3 t tlz/t) + (t/tl2 + votos/xo + 1 ) ] ( i exp(-
(10)
+ [eo(e + e ' ) E c / i o t i 2 ] ( 1
-
- t12/t - (voto/xo)[1 -
(t - tl/)Xo/votot)])}.
(11)
E q n . ( l l ) is o b t a i n e d from e q n s . ( 6 ) , (7) a n d (10). It can be s h o w n that the c o o r d i n a t e x~2 of a b o r d e r between zones 1 a n d It moves according to the law x12 = xott2/t. This feature is also considered while o b t a i n i n g e q n . ( l l ) . D u r i n g the third stage of the poling process, the sample consists of three zones, as s h o w n in Figure 1. T h e intermediate equation for E(x,t) in the z o n e I l l is similar to e q n . ( 9 ) , but e' is substituted for e". C o n s i d e r i n g c o n s t a n c y of the field at the x23 b o r d e r a n d continuity of the charge at the back electrode (x = x0), o n e can obtain the following expression for the field at the z o n e I l l E(x,t) = Es + [io/eo(e + e ' ) ] { T - [eo(e + e ' ) ( E s - Ec)/io + + T] e x p [ -
(x - x2a)/voto], x23 -
(12)
x <_ xo, t23 -< t,
where T = t23 - t12 + votohzs/xo + (t - t23)(~: + e ' ) / ( e
+ c").
43 The final expression for the voltage across the sample is not given here because it is rather cumbersome. It is clear that the boundary coordinates between the zones x~2 and x2a are time-dependent and their onward movement to the injecting electrode is similar to the propagation of domain walls in conventional ferroelectrics. 3. E X P E R I M E N T A L Experiments were performed on the 25-/~m thick uniaxially stretched PVDF films containing amorphous and crystalline (Form I) phases in nearly equal volume fractions. Aluminium electrode of 0.1 btm thickness was deposited on one surface of the film by a thermal evaporation in vacuum. Then the free surface of the sample was subjected to a negative corona discharge generated by a pointed tungsten electrode stressed at 14-18 kV, with the metallized surface being grounded. A control grid was provided between the corona electrode and the sample to make the charging process controllable. The total current was automatically maintained constant at a level of 60-100 /~A/m ~ while the transient voltage across the sample was measured by the Kelvin method and subsequently recorded. The experimental set-up was similar to that of Giacometti [9]. 4. R E S U L T S AND D I S C U S S I O N Three stages of the charging process are distinctly seen in Figure 2 where the experimental dependence of the voltage on the total trasferred charge is shown. At the first stage, V is proportional to t 1/2 as prescribed by the expression (8) indicating validity of the deep trapping mechanism. The voltage kinetics was calculated with the following values of the parameters: i0 -- 100 /~A/m 2 , x0 = 25 /~m, e = 10, e' = 725, e" = 45, tt2 = 250 s, t23 = 600 s, s = 5.47, y = 13.3, v0to = 6 ~m, /~t0 = 1.3x10 -ll m2/V, Ec = 42 MV/m, Es = 5 1 . 5 MV/m, Ps = 61 m C / m . Most of the parameters were found by the curve fitting method. The value of /~to was obtained from the initial part of the experimental curve shown in Figure 2. The values of tiE and t23 were also taken from the experimental curve in Figure 2. The numerical values of the coercive field and the remanent polarization are in good agreement with the corresponding data reported in other works [13,14]. On the whole, one can conclude that the proposed model correctly describes the main features of the charging process, especially existence of the three stages in the polarization build-up. The model also emphasizes an important role of the injected charge in establishing the stable polarization in ferroelectric polymers. The most suitable sites for the charge trapping are the crystallite boundaries, where the charge carriers, if trapped, neutralize partly the internal depolarization field. Similar processes are known to take place in ferroelectric ceramics. The favourable conditions for the charge trapping are probably created due to the large-scale fluctuations caused by alignment of permanent dipoles in the crystallites.
44
2
i >
0 0
.t 50
£ I00
150
CHARGE DENSITY (i0t), mC/m 2 Figure 2. Kinetics of the electret potential during the constant-current charging. Shaded region - experiment, solid line - calculation according to the model.
5. C O N C L U S I O N It is proved that the three-stage charging of ferroelectric polymers is a result of nucleation and growth of strong polarization. An important role in formation of the polarization plays injection of charge carriers and their subsequent trapping in the bulk. It is possible to obtain the values of the coercive field and the remanent polarization as well as several other parameters by comparing the experimental kinetics of the electret potential during the constant-current charging with the curve based on the phenomenological model. Although the model is approximate, calculations show that the confidence limits of the evaluated fit parameters are within the permissible range of not more than -+20%. Considering the more or less universal approach developed in this work, we believe that its results can be extended to examine the polarization phenomena in other materials such as ferroelectric ceramics and composites formed by ferroelectric inorganic crystals dispersed in a polymer matrix. 6. R E F E R E N C E S
N.Murayama, J.Polym.Sci.: Polym.Phys.Ed., 13 (1975) 929. H.von Seggern and T.T.Wang, J.Appl.Phys., 56 (1984) 2448.
45 3 4 5 6 7 8 9 10 11 12 13 14
W.Eisenmenger and M.Haardt, Sol.St.Comm., 41 (1982) 917. B.Gross, H.von Seggern and R.Gerhard-Multhaupt, J.Phys., D, 18 (1985) 2497. G.M.Sessler and A.Berrassoul, Ferroelectrics, 76 (1987) 489. R.Gerhard-Multhaupt, IEEE Trans.Elec.Insul., 22 (1987) 531. B.Gross, R.Gerhard-Multhaupt, A.Berrassoul and G.M.Sessler, J.Appl.Phys., 62 (1987) 1429. G.M.Sessler (ed.), Electrets, Springer, Berlin, 1988. J.A.Giacometti, J.Phys.: D, 20 (1987) 675. J.A.Giacometti and J.S.C.Campos, Rev.Sci.Instrum., 61 (1990) 1143. M.Womes, E.Bihler and W.Eisenmenger, IEEE Trans.Elec.Insul., 24 (1989) 461. T.Furukawa and G.E.Johnson, Appl.Phys.Lett., 38 (1981) 1027. B.Gross and J.A.Giacometti, Appl.Phys.: A, 37 (1985) 89. S.R.Kurtz and R.S.Hughes, J.Appl.Phys., 54 (1983) 229.