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Non-linear dielectric losses at high electric fields
Thin Solid Films-Elsevier Sequoia S.A., Lausanne-Printed in Switzerland
$9
Short Communication
Non-linear dielectric losses at high electric fields E. J. LE SUEUR AND A. K. JONSCHER
Chelsea College, Unioersity of London, Pulton Place, London, SW6 5PR (Gt. Britain) (Received June 12, 1972)
The dependence o f the real part o f the dielectric susceptibility Z" on the electric field has been studied extensively in both solids and liquids and departures from the constant low-field value are seen at fields of the order 10 7 V/cm. These departures may be positive or negative and are known as hyperpolarization, their physical causes being the non-linear interaction between the applied field and the molecular dipoles 1-3. By contrast, the imaginary part of susceptibility, the dielectric loss X" has been much less studied and there is little experimental information on this point 4-7 and even less theoretical understanding. Non-linearity of dielectric loss sets in at relatively lower fields, in excess of 104 V/cm and its causes may be sought in the effect of the electric field on the motion of free or bound charges in the material. In the absence of injection, a dielectric does not normally contain any significant densities of free highly mobile carriers. It may contain " f r e e " low mobility charge carriers, whether ions, electrons or polarons which move by some form of hopping mechanism 8' 9 and which exhibit a strong field dependence of the Poole-Frenkel or the Poole type 1°-13. This would be expected to lead to an increase of the dielectric loss with field. Moreover, this mechanism of hopping conduction shows at low fields a characteristic dependence on frequency, o9 of the type 9 Z"(og) oc o9n-1
(1)
implying a dependence of a.c. conductivity tr(og)ocog"
0.5
(2)
often valid over several decades of frequency. In many cases the exponent n is close to unity and the loss is nearly independent of frequency. In sharp contrast with this "free carrier" loss mechanism is the " b o u n d carrier" or Debye loss mechanism in which the loss peak has the characteristic narrow frequency dependence Z" oc
o9,~
1 + 092 z 2
(3)
where z is the relaxation time for the loss mechanism in question. Loss peaks may be superimposed on the broad distribution of loss due to hopping carriers. Thin Solid Films, 12 (1972) $9-S13
S10
SHORT COMMUNICATIONS
A detailed theory of non-linear dielectric losses in high electric fields will be presented elsewhere 14 and only its principal conclusions will be given here. The main object of the present paper is to give some results of detailed measurements of non-linear effects in silicon oxide, SiOx, similar in preparation and properties to material described in earlier work on low-field a.c. properties s, high-field d.c. behaviour is and high-field a.c. behaviour 16. The electrodes were evaporated aluminium. Results are given for two samples; sample A thickness 200 rim, evaporation rate 6.6 nm/sec; sample B thickness 140 nm, rate 3.9 nm/sec. A.c. measurements were made on a resistance-capacitance bridge and balance was normally obtained first at low signal amplitudes. On raising the signal amplitude to levels corresponding to fields in excess of l0 s V/cm in the specimen, an out-of-balance signal was observed which was virtually in phase with the driving field and which contained odd harmonics, Fig. l(a). The fundamental component of the out-of-balance signal could be balanced out completely by a minor adjustment of the capacitance control and a major adjustment of the resistance control, giving a residual signal as shown in Fig. l(c), consisting of the third and higher odd harmonics. The third harmonic is seen to be in phase with the driving signal to within the accuracy of the measurement. By injecting a controlled amount of third harmonic signal it was possible to reveal the residual fifth harmonic in the out-of-balance signal, as shown in Fig. l(d), which again shows the fifth harmonic to be in phase with the driving signal. The amplitude of the in-phase component of the fundamental is given directly by the reading on the resistance arm of the bridge in the balance condition of Fig. l(c). The balance of Fig. l(b) corresponds to the sum of this
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Fig. 1. Non-linear current response under high alternating field; (a) the out-of-balance current, (b) the measurement of peak amplitude, (¢) third aM higher harmonies, (d) fifth harmonic. The driving signal at fundamental frequency is shown for reference. Thin Solid Films, 12 (1972) $9-S13
Sll
SHORT COMMUNICATIONS
fundamental component plus all odd harmonics, i.e. the peak in-phase amplitude of the current as shown in Fig. l(a). Even harmonics, if present at all, are at a level below that of the fifth harmonic. The amplitudes of the fundamental and of the higher harmonics were plotted against field and frequency for both specimens at two different temperatures and are given in Figs. 2-5. Figures 2 and 4 show that the fundamental amplitudes increase with increasing frequency and that they are progressively less non-linear in the field so that departures from d.c. are smaller at high fields than at low fields. The third harmonic amplitude is approximately cubic in the field and the fifth may be seen to be consistent with a fifth-power law, as expected. It was found generally that at higher temperatures the harmonics lie above the dc. level while the opposite is true at lower temperatures. The frequency dependence of the fundamental and higher harmonics is shown in Figs. 3 and 5 which also include the “low-field ” dependence. In Fig. 3, the fundamental and the third harmonic are progressively less frequencyLOG
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Fig. 2. The field dependence of d.c. fundamental, third and fifth harmonics for sample A, at 22 “C, with frequency as parameter, fundamental, -.--3rd, - - - 5th +d.c., 0 90 Hz, 0 470 Hz, W 1.1 kHz, 0 4.5 kHz. Zero to peak.
Thin Solid Films, 12 (1972) S9-S13
3
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Fig. 3. The same data as in Fig. 2 plotted against frequency, with the amplitude of the alternating electric field as parameter. high field, 5.5 ~10~ V/cm, - - - - low field, 5 x10“ V/cm. 0 fundamental at 22”C, 0 fundamental at 53 “C, W 3rd at 22 “C, 0 3rd at 53 “C. Zero to peak.
S 12
SHORT COMMUNICATIONS
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Fig. 4. The held dependence of d.c., fundamental and third harmonic for sample B at 101 and - 5 °C. fundamental, - - . . . . 3rd, +d.c., • 90 Hz, O 1.9 kHz. Zero to peak.
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Fig. 5. The same data as Fig. 4 plotted against frequency, with the amplitude of the alternating field as parameter. ~ high field, 8 x 105 V/cm, - - - - low held, 7.3 x 104 V/cm, • fundamental at - 5 °C, O fundamental at 101 °C, [] 3rd at - 5 °C, [ ] 3rd at 101 °C. Zero to peak.
Thin Solid Films, 12 (1972) $9-S13
SHORT COMMUNICATIONS
S 13
dependent than the low-field current, although in the case of the lowest temperature, - 5 °C in Fig. 5, they are virtually parallel. Starting from the direct current-field characteristic, it is possible to compute the harmonic content of the a.c. response due to the non-linearity of the direct current. This current is by definition frequency-independent. Any excess of the measured signal over this "quasi-d.c." current represents the true non-linear a.c. response of our system. At - 5 °C the d.c. contribution in Fig. 5 is so small that the points plotted represent effectively the a.c. contribution, but the "quasi-d.c." becomes significant in the other results. The effect of subtracting the quasi-d.c, from the fundamental is to make it parallel to the lowfield current-frequency graph. In the case of the third harmonic this cannot be said with any assurance because of the large errors involved, but the trend is in the same direction. We conclude that, subject to the experimental uncertainties mentioned above, the frequency dependence of the true a.c. response of the fundamental and third harmonic is approximately the same as the frequency dependence of the low-field current in the linear regime. The adjustments to the capacitance balance that have to be made indicate that the onset of non-linearity is accompanied by a slight increase of capacitance but at present this is close to experimental error. These results lead us to the following general conclusions regarding the non-linear dielectric behaviour of silicon oxide. The non-linearity is due predominantly to the conduction processes, believed to be hopping conduction which is characterized at low fields by a frequency dependence of the form given by eqn. (2). The frequency dependence of the high-field conductance suggests strongly that the high field affects to first order the amplitude of the response but not its time dependence. One of us (E.J.L.S.) wishes to acknowledge the tenure of a SRC Research Studentship. REFERENCES 1 2 3 '4 5 6 7 8 9 10 11 12 13 14 15 16
A . D . Buckingham and B. J. Orr, Quart. Rev., 21 (1967) 195. A . D . Buckingham and B. J. Orr, Trans. Faraday Soc., 65 (1969) 673. C . A . Coulson, A. Maccoll and L. E. Sutton, Trans. Faraday Soc., 48 (1952) 106. N. Klein and N. Levanon, J. AppL Phys., 38 (1967) 3721. F. Argall and A. K. Jonscher, Thin Solid Films, 2 (1968) 185. G. Niivik, Thin Solid Films, 6 (1970) 145. K. Everszumrode and W. Rabus, Thin Solid Films, 9 (1972) 465. N. Mort and E. H. Davis, Electronic Processes in Non-crystalline Materials, Clarendon Press, Oxford, 1971. A . K . Jonscher, J. Non-Cryst. Solids, 8/9 (1972) 293. A . K . Jonscher, J. Vac. Sci. Techn., 8 (1971) 135. R . M . Hill, Phil. Mag., 23 (1971) 59, R . M . Hill, Phil. Mag., 24 (1971) J307. M . H . Nathoo and A. K. Jonscher, J. Phys. C: Solid State Phys. 4 (1971) L301. A . K . Jonscher, J. Electrochem. Soc., to be published. A. Servini and A. K. Jonscher, Thin Solid Films, 3 (1969) 341. A . K . Jonscher and C. K. Loh, J. Phys. C: Solid State Phys., 4 (1971) 1341.
Thin Solid Films, 12 (1972) $9-$13
$9
Short Communication
Non-linear dielectric losses at high electric fields E. J. LE SUEUR AND A. K. JONSCHER
Chelsea College, Unioersity of London, Pulton Place, London, SW6 5PR (Gt. Britain) (Received June 12, 1972)
The dependence o f the real part o f the dielectric susceptibility Z" on the electric field has been studied extensively in both solids and liquids and departures from the constant low-field value are seen at fields of the order 10 7 V/cm. These departures may be positive or negative and are known as hyperpolarization, their physical causes being the non-linear interaction between the applied field and the molecular dipoles 1-3. By contrast, the imaginary part of susceptibility, the dielectric loss X" has been much less studied and there is little experimental information on this point 4-7 and even less theoretical understanding. Non-linearity of dielectric loss sets in at relatively lower fields, in excess of 104 V/cm and its causes may be sought in the effect of the electric field on the motion of free or bound charges in the material. In the absence of injection, a dielectric does not normally contain any significant densities of free highly mobile carriers. It may contain " f r e e " low mobility charge carriers, whether ions, electrons or polarons which move by some form of hopping mechanism 8' 9 and which exhibit a strong field dependence of the Poole-Frenkel or the Poole type 1°-13. This would be expected to lead to an increase of the dielectric loss with field. Moreover, this mechanism of hopping conduction shows at low fields a characteristic dependence on frequency, o9 of the type 9 Z"(og) oc o9n-1
(1)
implying a dependence of a.c. conductivity tr(og)ocog"
0.5
(2)
often valid over several decades of frequency. In many cases the exponent n is close to unity and the loss is nearly independent of frequency. In sharp contrast with this "free carrier" loss mechanism is the " b o u n d carrier" or Debye loss mechanism in which the loss peak has the characteristic narrow frequency dependence Z" oc
o9,~
1 + 092 z 2
(3)
where z is the relaxation time for the loss mechanism in question. Loss peaks may be superimposed on the broad distribution of loss due to hopping carriers. Thin Solid Films, 12 (1972) $9-S13
S10
SHORT COMMUNICATIONS
A detailed theory of non-linear dielectric losses in high electric fields will be presented elsewhere 14 and only its principal conclusions will be given here. The main object of the present paper is to give some results of detailed measurements of non-linear effects in silicon oxide, SiOx, similar in preparation and properties to material described in earlier work on low-field a.c. properties s, high-field d.c. behaviour is and high-field a.c. behaviour 16. The electrodes were evaporated aluminium. Results are given for two samples; sample A thickness 200 rim, evaporation rate 6.6 nm/sec; sample B thickness 140 nm, rate 3.9 nm/sec. A.c. measurements were made on a resistance-capacitance bridge and balance was normally obtained first at low signal amplitudes. On raising the signal amplitude to levels corresponding to fields in excess of l0 s V/cm in the specimen, an out-of-balance signal was observed which was virtually in phase with the driving field and which contained odd harmonics, Fig. l(a). The fundamental component of the out-of-balance signal could be balanced out completely by a minor adjustment of the capacitance control and a major adjustment of the resistance control, giving a residual signal as shown in Fig. l(c), consisting of the third and higher odd harmonics. The third harmonic is seen to be in phase with the driving signal to within the accuracy of the measurement. By injecting a controlled amount of third harmonic signal it was possible to reveal the residual fifth harmonic in the out-of-balance signal, as shown in Fig. l(d), which again shows the fifth harmonic to be in phase with the driving signal. The amplitude of the in-phase component of the fundamental is given directly by the reading on the resistance arm of the bridge in the balance condition of Fig. l(c). The balance of Fig. l(b) corresponds to the sum of this
o)
-i
f-l
-
1
iAn"AIAA AA AIANA AX'i °' (U (VVV UVVVIUUVItVIVV
I vlv'tv
T,'W
t
Fig. 1. Non-linear current response under high alternating field; (a) the out-of-balance current, (b) the measurement of peak amplitude, (¢) third aM higher harmonies, (d) fifth harmonic. The driving signal at fundamental frequency is shown for reference. Thin Solid Films, 12 (1972) $9-S13
Sll
SHORT COMMUNICATIONS
fundamental component plus all odd harmonics, i.e. the peak in-phase amplitude of the current as shown in Fig. l(a). Even harmonics, if present at all, are at a level below that of the fifth harmonic. The amplitudes of the fundamental and of the higher harmonics were plotted against field and frequency for both specimens at two different temperatures and are given in Figs. 2-5. Figures 2 and 4 show that the fundamental amplitudes increase with increasing frequency and that they are progressively less non-linear in the field so that departures from d.c. are smaller at high fields than at low fields. The third harmonic amplitude is approximately cubic in the field and the fifth may be seen to be consistent with a fifth-power law, as expected. It was found generally that at higher temperatures the harmonics lie above the dc. level while the opposite is true at lower temperatures. The frequency dependence of the fundamental and higher harmonics is shown in Figs. 3 and 5 which also include the “low-field ” dependence. In Fig. 3, the fundamental and the third harmonic are progressively less frequencyLOG
lo+
LOG.1 (O-PI
II. Al I
S-
6-
?-
I
5
LOG E IV/cm) 2
Fig. 2. The field dependence of d.c. fundamental, third and fifth harmonics for sample A, at 22 “C, with frequency as parameter, fundamental, -.--3rd, - - - 5th +d.c., 0 90 Hz, 0 470 Hz, W 1.1 kHz, 0 4.5 kHz. Zero to peak.
Thin Solid Films, 12 (1972) S9-S13
3
LOGfiHzl
Fig. 3. The same data as in Fig. 2 plotted against frequency, with the amplitude of the alternating electric field as parameter. high field, 5.5 ~10~ V/cm, - - - - low field, 5 x10“ V/cm. 0 fundamental at 22”C, 0 fundamental at 53 “C, W 3rd at 22 “C, 0 3rd at 53 “C. Zero to peak.
S 12
SHORT COMMUNICATIONS
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Fig. 4. The held dependence of d.c., fundamental and third harmonic for sample B at 101 and - 5 °C. fundamental, - - . . . . 3rd, +d.c., • 90 Hz, O 1.9 kHz. Zero to peak.
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Fig. 5. The same data as Fig. 4 plotted against frequency, with the amplitude of the alternating field as parameter. ~ high field, 8 x 105 V/cm, - - - - low held, 7.3 x 104 V/cm, • fundamental at - 5 °C, O fundamental at 101 °C, [] 3rd at - 5 °C, [ ] 3rd at 101 °C. Zero to peak.
Thin Solid Films, 12 (1972) $9-S13
SHORT COMMUNICATIONS
S 13
dependent than the low-field current, although in the case of the lowest temperature, - 5 °C in Fig. 5, they are virtually parallel. Starting from the direct current-field characteristic, it is possible to compute the harmonic content of the a.c. response due to the non-linearity of the direct current. This current is by definition frequency-independent. Any excess of the measured signal over this "quasi-d.c." current represents the true non-linear a.c. response of our system. At - 5 °C the d.c. contribution in Fig. 5 is so small that the points plotted represent effectively the a.c. contribution, but the "quasi-d.c." becomes significant in the other results. The effect of subtracting the quasi-d.c, from the fundamental is to make it parallel to the lowfield current-frequency graph. In the case of the third harmonic this cannot be said with any assurance because of the large errors involved, but the trend is in the same direction. We conclude that, subject to the experimental uncertainties mentioned above, the frequency dependence of the true a.c. response of the fundamental and third harmonic is approximately the same as the frequency dependence of the low-field current in the linear regime. The adjustments to the capacitance balance that have to be made indicate that the onset of non-linearity is accompanied by a slight increase of capacitance but at present this is close to experimental error. These results lead us to the following general conclusions regarding the non-linear dielectric behaviour of silicon oxide. The non-linearity is due predominantly to the conduction processes, believed to be hopping conduction which is characterized at low fields by a frequency dependence of the form given by eqn. (2). The frequency dependence of the high-field conductance suggests strongly that the high field affects to first order the amplitude of the response but not its time dependence. One of us (E.J.L.S.) wishes to acknowledge the tenure of a SRC Research Studentship. REFERENCES 1 2 3 '4 5 6 7 8 9 10 11 12 13 14 15 16
A . D . Buckingham and B. J. Orr, Quart. Rev., 21 (1967) 195. A . D . Buckingham and B. J. Orr, Trans. Faraday Soc., 65 (1969) 673. C . A . Coulson, A. Maccoll and L. E. Sutton, Trans. Faraday Soc., 48 (1952) 106. N. Klein and N. Levanon, J. AppL Phys., 38 (1967) 3721. F. Argall and A. K. Jonscher, Thin Solid Films, 2 (1968) 185. G. Niivik, Thin Solid Films, 6 (1970) 145. K. Everszumrode and W. Rabus, Thin Solid Films, 9 (1972) 465. N. Mort and E. H. Davis, Electronic Processes in Non-crystalline Materials, Clarendon Press, Oxford, 1971. A . K . Jonscher, J. Non-Cryst. Solids, 8/9 (1972) 293. A . K . Jonscher, J. Vac. Sci. Techn., 8 (1971) 135. R . M . Hill, Phil. Mag., 23 (1971) 59, R . M . Hill, Phil. Mag., 24 (1971) J307. M . H . Nathoo and A. K. Jonscher, J. Phys. C: Solid State Phys. 4 (1971) L301. A . K . Jonscher, J. Electrochem. Soc., to be published. A. Servini and A. K. Jonscher, Thin Solid Films, 3 (1969) 341. A . K . Jonscher and C. K. Loh, J. Phys. C: Solid State Phys., 4 (1971) 1341.
Thin Solid Films, 12 (1972) $9-$13