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Radiation-induced charge transport in polymer electrets
Radiat. Phys. Chem. Vol. 23, No. 3, pp. 359-362, 1984 Printed in Great Britain.
0146-5724/84 $3.00 + .00 Pergamon Press Ltd.
R A D I A T I O N - I N D U C E D CHARGE TRANSPORT IN POLYMER ELECTRETS K. LABONTE Technical University of Darmstadt, Institute for Electroacoustics, Merckstr. 25, D-6100 Darmstadt, FRG Abstract--Recently, a new physical model has been developed describing the charge dynamics in dielectrics during irradiation. Experimental investigations of the charge transport in polymer electrets were carried out in a modified electron-beam microscope on various materials (FEP, PETP, PVDF). A qualitative comparison of the theoretical results with experimental data shows that, in FEP, electrons are practically immobile, whereas positive charge carders cause a trap-modulated unipolar hole current. For PETP, analogous results are found except that now the mobility of the electrons dominates. In PVDF however, both charge carders must be mobile. INTRODUCTION have suggested a general physical model for the analysis of charge dynamics in dielectrics exposed to ionizing radiation. (t) Since we could show that this model can be successfully applied to electron-beam-irradiated Teflon-FEP foils, we were encouraged to study the numerical results of the theory systematically and to compare them to experimental data for other materials. Some typical results are presented in this paper. RECENTLY,
we
THEORY The basic assumptions of the physical model are given elsewhere. °) Essential features are: (1) The intrinsic conductivity of the sample can be neglected. (2) There exist discrete trap levels for electrons and holes. (3) The charge carrier transport is modulated by four elementary processes: hopping between shallow traps, trapping in deep traps, thermal release from deep traps, and recombination of trapped and free carriers of opposite polarity. (4) The irradiated electrode is ohmic and the nonirradiated electrode is blocking in the direction from the electrode to the sample. This physical model leads to a system of partial differential equations which has been solved with a digital computer by use of finite difference approximations under the following assumptions: (1) The dielectric is irradiated by a nonpenetrating electron beam with a normalized average electronrange r = 0.4. (2) The production rate of electron-hole pairs is assumed to be proportional to the dose-rate profile. The dose-rate and the deposition profiles were taken from the literature.(:)
(3) There is only one deep trap level for electrons and one for holes. (4) Initially, the dielectric is uncharged. (5) The sample is irradiated in one of the two possible open-circuit modes: Mode A: Front electrode floating, rear electrode grounded. Mode B: Front electrode grounded, rear electrode floating. The numerical data included in Figs. 1--4 were calculated for three special cases: (1) Electrons are immobile, holes are mobile. (2) Holes are immobile, electrons are mobile. (3) Both, electrons and holes are mobile. All other parameters are arbitrary, but constant. The theoretical results are given in a normalized form using the following set of equations (physical parameters are primed) (1)
V=v,
ago lad'
t = t" go/E,
eln P+ = -I- It'~ a--go2d'
P = P
r = r'/d, , agod E--~n'
where V is the voltage across the sample, a the irradiated area, go the normalization parameter with dimension 1/~ cm, IB the beam current, d the sample thickness, t the time, E the dielectric constant, r the average range of electrons,/~ ± the mobility of holes ( + ) and electrons ( - ) , and p the spatial excess charge density. Mode
A : Front electrode floating
Figure 1 shows the voltage V ( t ) across the sample as a function of time t in Mode A. Because the dielectric is initially uncharged, the voltage is zero at the onset of irradiation at t = 0. Then the sample is negatively charged by the electron beam and the voltage increases. The initial slope is given by r - 1. 359
360
K. -0.6
LABo~rrE
r=0.4
N -0.5 I..U
o -0.4 O
> -0.3 r'~ I..kl N -0.2 _J ,< -0.1 O Z
0
,,~
Mode A 014
0.2 016 0.'8 NORMALIZED TIME t - - , , -
FIG. 1. Normalized voltage V vs normalized time t with mobilities # _+of holes and electrons as parameter in Mode A, front electrode floating.
If the electrons are immobile (/t_ = 0, upper curve), the voltage increases almost linearly with time. But if the electrons are mobile ~ _ = 1, other curves), the voltage increases more slowly, and a steady-state value is reached after some time beyond the present time scale. The mobility of the electrons determines the qualitative behavior of the voltage in this mode. This can be explained considering the build-up of the excess charge density p(x, t) depicted in Figs. 2 and 3. Because of the boundary condition of this mode, the electric field is zero at the floating front electrode. It becomes negative in the bulk of the sample due to the injected electrons. If the electrons are immobile (Fig. 2), the holes generated in the irradiated region are driven in the direction towards the front electrode. They cannot leave the sample since the front electrode is floating and recombine
p(x,t)
with electrons. The nonirradiated rear electrode is blocking and no charge carriers can traverse the nonirradiated region. The injected electrons are collected in the irradiated region. After the onset of the irradiation, the charge centroid moves slowly from the average range of electrons towards the maximum range; therefore, the voltage increases almost linearly. On the other hand, if the electrons are mobile (Fig. 3), they are driven by the negative electric field through the nonirradiated region towards the rear electrode. The injected electrons can leave the sample. After some time, a steady-state charge distribution is reached; the voltage becomes constant.
Mode B: Rear electrode floating The corresponding results for Mode B are depicted in Fig. 4. Again, the normalized voltage across the sample is shown vs the normalized time. In this mode,
p=(x,t) -
3,0
I
~
.
-as
FIG. 3. Normalized charge density p as a function of normalized depth x and normalized time t in a three dimensional plot for mobile electrons/~_ = 1, Mode A.
I A
.0.2 ¸
1
~ -0.15. O a 1.1.1 N
-0.1
~ -0.05 O Z
0.2 0.4 0.6 0.8 NORMALIZED TIME t---,,Fro. 2. Normalized charge density p as a function of normalized depth x and normalized time t in a three dimensional plot for immobile electrons ~_ = 0, Mode A.
FIG. 4. Normalized voltage V vs normalized time t with mobilities # ± of holes and electrons as parameter in Mode B, rear electrode floating.
361
Radiation-induced charge transport in polymer electrets all curves have the initial slope - r and reach different steady-state values. If only charge carriers of one polarity are mobile (two upper curves), the voltage increases monotonously, until it reaches its steadystate value. When electrons and holes are both mobile (lower curve), the voltage may overshoot the steadystate value and approach its final value decreasing from a maximum. Again, the behavior can be explained with the field-driven charge carriers. Because of the boundary condition in Mode B, the electric field vanishes in the nonirradiated region, while it is positive in the irradiated region. Thus the nonirradiated region is blocking whereas the irradiated region is in ohmic contact with the irradiated front electrode. Since the front electrode is grounded, negative charge carriers can leave and positive ones can enter the dielectric through this electrode, both driven by the positive field. The number of beam-generated electron-hole pairs exceeds the number of beam-injected electrons by at least one order of magnitude. Thus, the number of free electrons and holes in the irradiated region is approximately equal. This explains the similar behavior of the two upper curves. If both, positive and negative carriers are mobile, the negative charge density, which is built up by the injected electrons, is simultaneously reduced by an electron current leaving and by a hole current entering the sample. This results in a lower value for the steady-state voltage in the lower curve.
E X P E R I M E N T A L RESULTS Experimental curves are shown for three electret materials, namely Teflon FEP (Dupont), Hostaphan PETP (Kalle), and Solef PVDF (Solvay). FEP and PETP foils were 25 # m and P V D F foils 65 # m thick. Virgin samples were sandwiched between evaporated circular A1 electrodes of 100 nm thickness and 5 cm dia. They were irradiated in a Split-Faraday-Cup arrangement (3,4) with a monoenergetic electron beam of 40 keV beam energy and 20 crn 2 irradiated area. Corresponding to the theoretical part of this paper, the voltage across the sample in the two open-circuit Modes A and B was measured by a high-impedance Keithley electrometer. Mode A : Front electrode floating Typical results of Mode A experiments are shown in Fig. 5. The voltage is plotted as a function of time for all three materials. The beam current is 10 nA. With FEP, the voltage builds up almost linearly. With PETP and PVDF, the increase of the voltage diminishes and a steady-state value is reached beyond the present time scale.
-30O
FEP25 pm / l
PETP25 .urn
-,oo
/ /
/
vs=,o kov
=1
100
It)
200
300
TIME t Is] FIG. 5. Voltage V vs time t for FEP, PETP and PVDF with beam energy VB= 40 keV and beam current I B= 10 nA in Mode A.
If we compare these results with the corresponding theoretical data of Fig. l, we can attach the upper curve (immobile electrons) to FEP. Because of the qualitative agreement between the other curves of Fig. 1 and the curves for PETP and PVDF in Fig. 5, we conclude that in both materials electrons must be mobile. The mobility of the holes cannot be judged from this result. Mode B: Rear electrode floating Figures 6-8 show the corresponding experimental results for Mode B experiments. The parameter is the beam current IB varying from 1 to 100 nA. In Fig. 6 experimental data for FEP are plotted. The voltage increases monotonously and reaches a steady-state value which depends on Ia. While PETP (Fig. 6(5)) shows the same qualitative behavior as FEP, the experimental data for PVDF (Fig. 8) exhibit a max-
-35- 25 ~m FEP /
VB:'0 keV I
t -'o
i i°t
. I
~ -20' -10
o~
~b
2b
3'o TIME
t [s]
4b
s'o
60
=,
FIG. 6. Voltage V vs time t for FEP with beam energy VB= 40 keV and beam current Ia as parameter in Mode B.
362
K. LABONTE
-300 25 N m
-301
PETP
~ 6 5 / J m
PVI)F 1
-250-
~-200> -150-
*
~C
~> -)00-
_
f
.
-50-
c~ 0
tO
20 30 TIME t is]
~0
50
60
o
___.~.,o-s,~ ,b
2'o
3o
I/o
5'0
eo
TIME t [s]
FIG. 7. Voltage Y vs time t for PETP with beam energy Vn = 40 keV and beam current In as parameter in Mode B.
FIG. 8. Voltage V vs time t for PVDF with beam energy Vn = 40 keV and beam current In as parameter in Mode B.
imum value, which is higher and appears earlier for higher beam currents. After this maximum, the voltage decreases. By comparison with the theoretical results in Fig. 4 we conclude that in PVDF both, electrons and holes must be mobile. We can also attach the middle curve of Fig. 4 to PETP, because we already know that the electrons are mobile in this material.
and experiment has already been achieved for FEP; (1> parameter fitting for PETP and PVDF is presently under way.
CONCLUSIONS The mobilities of electrons and holes prove to be important parameters for the numerical results of our physical model. Experimental data for electronbeam-irradiated FEP, PETP, and PVDF foils could be dearly identified with three special cases of the general model: In FEP, the mobility of the electrons is small compared to the mobility of the holes, while the PETP the opposite is true. In PVDF, both charge carriers are mobile and their mobilities are of the same order. Quantitative agreement between theory
Acknowledgements--The author is indebted to Prof. G. M. Scssler, Prof. B. Gross, R. Gerhard-Multhaupt, and A. Berraissoul for many stimulating discussions and for the permission to use their experimental data on PETP and to H. Eisenhauer for sample preparation.Thanks are also due to the Deutsche Forschungsgemeinschaft for financial support. REFERENCES I. K. LABONTE,IEEE Trans. Nucl. Sci, 1982, NS-29, 1650. 2. S. MATSUOKA,H. SUNAGA,R. TANAKA,M. HAGIWARA and K. ARAIO,IEEE Trans. Nucl. Sci. 1976, NS-23, 1447. 3. W. E. SPEAR,Proc. Phys. Soc. Lond. 1955, B,16& 991. 4. B. GRoss, G. M. SKSSLERand J. E. WEST,J. AppL Phys. 1974, 45, 2841. 5. B. GRO~, R. GI~RHARD-MULTHAUPT and A. BES~OUL, Measurement of Radiation-induced Conductivity of PETP. To be published.
0146-5724/84 $3.00 + .00 Pergamon Press Ltd.
R A D I A T I O N - I N D U C E D CHARGE TRANSPORT IN POLYMER ELECTRETS K. LABONTE Technical University of Darmstadt, Institute for Electroacoustics, Merckstr. 25, D-6100 Darmstadt, FRG Abstract--Recently, a new physical model has been developed describing the charge dynamics in dielectrics during irradiation. Experimental investigations of the charge transport in polymer electrets were carried out in a modified electron-beam microscope on various materials (FEP, PETP, PVDF). A qualitative comparison of the theoretical results with experimental data shows that, in FEP, electrons are practically immobile, whereas positive charge carders cause a trap-modulated unipolar hole current. For PETP, analogous results are found except that now the mobility of the electrons dominates. In PVDF however, both charge carders must be mobile. INTRODUCTION have suggested a general physical model for the analysis of charge dynamics in dielectrics exposed to ionizing radiation. (t) Since we could show that this model can be successfully applied to electron-beam-irradiated Teflon-FEP foils, we were encouraged to study the numerical results of the theory systematically and to compare them to experimental data for other materials. Some typical results are presented in this paper. RECENTLY,
we
THEORY The basic assumptions of the physical model are given elsewhere. °) Essential features are: (1) The intrinsic conductivity of the sample can be neglected. (2) There exist discrete trap levels for electrons and holes. (3) The charge carrier transport is modulated by four elementary processes: hopping between shallow traps, trapping in deep traps, thermal release from deep traps, and recombination of trapped and free carriers of opposite polarity. (4) The irradiated electrode is ohmic and the nonirradiated electrode is blocking in the direction from the electrode to the sample. This physical model leads to a system of partial differential equations which has been solved with a digital computer by use of finite difference approximations under the following assumptions: (1) The dielectric is irradiated by a nonpenetrating electron beam with a normalized average electronrange r = 0.4. (2) The production rate of electron-hole pairs is assumed to be proportional to the dose-rate profile. The dose-rate and the deposition profiles were taken from the literature.(:)
(3) There is only one deep trap level for electrons and one for holes. (4) Initially, the dielectric is uncharged. (5) The sample is irradiated in one of the two possible open-circuit modes: Mode A: Front electrode floating, rear electrode grounded. Mode B: Front electrode grounded, rear electrode floating. The numerical data included in Figs. 1--4 were calculated for three special cases: (1) Electrons are immobile, holes are mobile. (2) Holes are immobile, electrons are mobile. (3) Both, electrons and holes are mobile. All other parameters are arbitrary, but constant. The theoretical results are given in a normalized form using the following set of equations (physical parameters are primed) (1)
V=v,
ago lad'
t = t" go/E,
eln P+ = -I- It'~ a--go2d'
P = P
r = r'/d, , agod E--~n'
where V is the voltage across the sample, a the irradiated area, go the normalization parameter with dimension 1/~ cm, IB the beam current, d the sample thickness, t the time, E the dielectric constant, r the average range of electrons,/~ ± the mobility of holes ( + ) and electrons ( - ) , and p the spatial excess charge density. Mode
A : Front electrode floating
Figure 1 shows the voltage V ( t ) across the sample as a function of time t in Mode A. Because the dielectric is initially uncharged, the voltage is zero at the onset of irradiation at t = 0. Then the sample is negatively charged by the electron beam and the voltage increases. The initial slope is given by r - 1. 359
360
K. -0.6
LABo~rrE
r=0.4
N -0.5 I..U
o -0.4 O
> -0.3 r'~ I..kl N -0.2 _J ,< -0.1 O Z
0
,,~
Mode A 014
0.2 016 0.'8 NORMALIZED TIME t - - , , -
FIG. 1. Normalized voltage V vs normalized time t with mobilities # _+of holes and electrons as parameter in Mode A, front electrode floating.
If the electrons are immobile (/t_ = 0, upper curve), the voltage increases almost linearly with time. But if the electrons are mobile ~ _ = 1, other curves), the voltage increases more slowly, and a steady-state value is reached after some time beyond the present time scale. The mobility of the electrons determines the qualitative behavior of the voltage in this mode. This can be explained considering the build-up of the excess charge density p(x, t) depicted in Figs. 2 and 3. Because of the boundary condition of this mode, the electric field is zero at the floating front electrode. It becomes negative in the bulk of the sample due to the injected electrons. If the electrons are immobile (Fig. 2), the holes generated in the irradiated region are driven in the direction towards the front electrode. They cannot leave the sample since the front electrode is floating and recombine
p(x,t)
with electrons. The nonirradiated rear electrode is blocking and no charge carriers can traverse the nonirradiated region. The injected electrons are collected in the irradiated region. After the onset of the irradiation, the charge centroid moves slowly from the average range of electrons towards the maximum range; therefore, the voltage increases almost linearly. On the other hand, if the electrons are mobile (Fig. 3), they are driven by the negative electric field through the nonirradiated region towards the rear electrode. The injected electrons can leave the sample. After some time, a steady-state charge distribution is reached; the voltage becomes constant.
Mode B: Rear electrode floating The corresponding results for Mode B are depicted in Fig. 4. Again, the normalized voltage across the sample is shown vs the normalized time. In this mode,
p=(x,t) -
3,0
I
~
.
-as
FIG. 3. Normalized charge density p as a function of normalized depth x and normalized time t in a three dimensional plot for mobile electrons/~_ = 1, Mode A.
I A
.0.2 ¸
1
~ -0.15. O a 1.1.1 N
-0.1
~ -0.05 O Z
0.2 0.4 0.6 0.8 NORMALIZED TIME t---,,Fro. 2. Normalized charge density p as a function of normalized depth x and normalized time t in a three dimensional plot for immobile electrons ~_ = 0, Mode A.
FIG. 4. Normalized voltage V vs normalized time t with mobilities # ± of holes and electrons as parameter in Mode B, rear electrode floating.
361
Radiation-induced charge transport in polymer electrets all curves have the initial slope - r and reach different steady-state values. If only charge carriers of one polarity are mobile (two upper curves), the voltage increases monotonously, until it reaches its steadystate value. When electrons and holes are both mobile (lower curve), the voltage may overshoot the steadystate value and approach its final value decreasing from a maximum. Again, the behavior can be explained with the field-driven charge carriers. Because of the boundary condition in Mode B, the electric field vanishes in the nonirradiated region, while it is positive in the irradiated region. Thus the nonirradiated region is blocking whereas the irradiated region is in ohmic contact with the irradiated front electrode. Since the front electrode is grounded, negative charge carriers can leave and positive ones can enter the dielectric through this electrode, both driven by the positive field. The number of beam-generated electron-hole pairs exceeds the number of beam-injected electrons by at least one order of magnitude. Thus, the number of free electrons and holes in the irradiated region is approximately equal. This explains the similar behavior of the two upper curves. If both, positive and negative carriers are mobile, the negative charge density, which is built up by the injected electrons, is simultaneously reduced by an electron current leaving and by a hole current entering the sample. This results in a lower value for the steady-state voltage in the lower curve.
E X P E R I M E N T A L RESULTS Experimental curves are shown for three electret materials, namely Teflon FEP (Dupont), Hostaphan PETP (Kalle), and Solef PVDF (Solvay). FEP and PETP foils were 25 # m and P V D F foils 65 # m thick. Virgin samples were sandwiched between evaporated circular A1 electrodes of 100 nm thickness and 5 cm dia. They were irradiated in a Split-Faraday-Cup arrangement (3,4) with a monoenergetic electron beam of 40 keV beam energy and 20 crn 2 irradiated area. Corresponding to the theoretical part of this paper, the voltage across the sample in the two open-circuit Modes A and B was measured by a high-impedance Keithley electrometer. Mode A : Front electrode floating Typical results of Mode A experiments are shown in Fig. 5. The voltage is plotted as a function of time for all three materials. The beam current is 10 nA. With FEP, the voltage builds up almost linearly. With PETP and PVDF, the increase of the voltage diminishes and a steady-state value is reached beyond the present time scale.
-30O
FEP25 pm / l
PETP25 .urn
-,oo
/ /
/
vs=,o kov
=1
100
It)
200
300
TIME t Is] FIG. 5. Voltage V vs time t for FEP, PETP and PVDF with beam energy VB= 40 keV and beam current I B= 10 nA in Mode A.
If we compare these results with the corresponding theoretical data of Fig. l, we can attach the upper curve (immobile electrons) to FEP. Because of the qualitative agreement between the other curves of Fig. 1 and the curves for PETP and PVDF in Fig. 5, we conclude that in both materials electrons must be mobile. The mobility of the holes cannot be judged from this result. Mode B: Rear electrode floating Figures 6-8 show the corresponding experimental results for Mode B experiments. The parameter is the beam current IB varying from 1 to 100 nA. In Fig. 6 experimental data for FEP are plotted. The voltage increases monotonously and reaches a steady-state value which depends on Ia. While PETP (Fig. 6(5)) shows the same qualitative behavior as FEP, the experimental data for PVDF (Fig. 8) exhibit a max-
-35- 25 ~m FEP /
VB:'0 keV I
t -'o
i i°t
. I
~ -20' -10
o~
~b
2b
3'o TIME
t [s]
4b
s'o
60
=,
FIG. 6. Voltage V vs time t for FEP with beam energy VB= 40 keV and beam current Ia as parameter in Mode B.
362
K. LABONTE
-300 25 N m
-301
PETP
~ 6 5 / J m
PVI)F 1
-250-
~-200> -150-
*
~C
~> -)00-
_
f
.
-50-
c~ 0
tO
20 30 TIME t is]
~0
50
60
o
___.~.,o-s,~ ,b
2'o
3o
I/o
5'0
eo
TIME t [s]
FIG. 7. Voltage Y vs time t for PETP with beam energy Vn = 40 keV and beam current In as parameter in Mode B.
FIG. 8. Voltage V vs time t for PVDF with beam energy Vn = 40 keV and beam current In as parameter in Mode B.
imum value, which is higher and appears earlier for higher beam currents. After this maximum, the voltage decreases. By comparison with the theoretical results in Fig. 4 we conclude that in PVDF both, electrons and holes must be mobile. We can also attach the middle curve of Fig. 4 to PETP, because we already know that the electrons are mobile in this material.
and experiment has already been achieved for FEP; (1> parameter fitting for PETP and PVDF is presently under way.
CONCLUSIONS The mobilities of electrons and holes prove to be important parameters for the numerical results of our physical model. Experimental data for electronbeam-irradiated FEP, PETP, and PVDF foils could be dearly identified with three special cases of the general model: In FEP, the mobility of the electrons is small compared to the mobility of the holes, while the PETP the opposite is true. In PVDF, both charge carriers are mobile and their mobilities are of the same order. Quantitative agreement between theory
Acknowledgements--The author is indebted to Prof. G. M. Scssler, Prof. B. Gross, R. Gerhard-Multhaupt, and A. Berraissoul for many stimulating discussions and for the permission to use their experimental data on PETP and to H. Eisenhauer for sample preparation.Thanks are also due to the Deutsche Forschungsgemeinschaft for financial support. REFERENCES I. K. LABONTE,IEEE Trans. Nucl. Sci, 1982, NS-29, 1650. 2. S. MATSUOKA,H. SUNAGA,R. TANAKA,M. HAGIWARA and K. ARAIO,IEEE Trans. Nucl. Sci. 1976, NS-23, 1447. 3. W. E. SPEAR,Proc. Phys. Soc. Lond. 1955, B,16& 991. 4. B. GRoss, G. M. SKSSLERand J. E. WEST,J. AppL Phys. 1974, 45, 2841. 5. B. GRO~, R. GI~RHARD-MULTHAUPT and A. BES~OUL, Measurement of Radiation-induced Conductivity of PETP. To be published.