纳米p型SiC-硅橡胶复合材料高压非线性I-V机理研究

1070-9878/10/$25.00 ? 2010 IEEE

Mechanisms Leading to Nonlinear Electrical Response

of a Nano p -SiC/Silicone Rubber Composite

X. Wang, J. K. Nelson, L. S. Schadler

Rensselaer Polytechnic Institute

110 Eighth Street Troy, NY 12180, USA and H. Hillborg,ABB Corporate Research

V?ster?s, Sweden

ABSTRACT

It is we ll known that hopping of charge carrie rs via spatially and e ne rge tically distribute d localize d state s is a primary transport me chanism in many disorde re d se miconductors and polyme ric die le ctrics. In this contribution, the nonline ar I-V physics of a 25vol% 50nm p -SiC/silicone rubbe r composite for high voltage fie ld grading application was inve stigate d, and a composite bulk hopping me chanism propose d. It is hypothe size d that ne are st-ne ighbor hole hopping occurs through thin rubbe ry laye rs be twe e n the SiC particle s, and is the me chanism gove rning the nonlinear electric response of SiC/silicone rubber nanocomposites.

Index Terms —Nonlinear electrical, mechanism, high field, hole hopping transport, nearest-neighbor, thin rubbery layer, field grading, nanocomposite.

1 INTRODUCTION

HIGH-VOLTAG E E-field grading [1-4] is typically

implemented via the use of composites with high dielectric constant or composites that possess desirable high field nonlinear resistive behavior. The latter, high field nonlinear resistive field grading composites [4-10], which serve to avoid excess electrical stress concentrations under both ac and dc applications, are commonly made of an insulating polymer matrix, especially rubber such as ethylene propylene diene monomer (EPDM) or silicone rubber, loaded with conducting carbon black, and/or ceramic semiconductor such as SiC, ZnO, or even an organic semiconductor like polyaniline emeraldine base.

In general, the I-V dependence of nonlinear field grading composites can be described by a simple empirical power-law equation D V I v (?>1).The real physics behind this empirical electrical response, however, has not been rigorously studied. It is commonly speculated that, at loadings below percolation, two competing mechanisms such as tunneling [11-13] or hopping [13-19] across thin polymer matrix layers separating filler particles dictate the composite I-V nonlinearity. Above

percolation when direct filler pathways are formed, the

composite nonlinear behavior may be the result of carrier tunneling or field-assisted thermionic emission across particle contact double Schottky barriers similar to the behavior

governing varistors [20-22]. On the other hand, mechanisms like electrode Schottky injection [23-25], bulk Poole-Frenkel emission [26-29] as well as space charge-limited currents [30-31] could also dominate or contribute to the overall composite I-V nonlinearity.

Among the proposed mechanisms, hopping of charge carriers across spatially, as well as energetically discrete, localized states has been widely reported to be a primary charge transport mechanism in many amorphous semiconductors, dielectrics or composites [32-38]. In such cases, carrier hopping usually takes place between neighboring sites of small separation, called nearest-neighbor hopping,or between sites of close energy level but slightly larger separation, called variable range hopping, which is more favorable at low temperature. In this paper, a detailed set of electrical characterization and data analysis leads to a hypothesis of valence band tail nearest-neighbor hole hopping as the mechanism governing the high field I-V nonlinear behavior in nano p -SiC/silicone rubber composites.

2 EXPERIMENTAL The SiC powder filler used in this study was 50 nm in diameter and of beta phase. Secondary-ion mass

spectroscopy (SIMS) characterization done by the Evans

analytical company determined the SiC powder doping level

Manuscript received on 22 April 2010.

and type. A pure 800-1,200 cSt low viscosity polydimethylsiloxane (trimethylsiloxy terminated, vinylmethylsiloxane (0.8-1.2 mole%)-dimethylsiloxane (99.2-98.8 mole%) copolymer) from Gelest Inc. was used as

the matrix silicone resin.

The 25vol% 50nm-SiC/silicone rubber which displayed the highest composite I-V nonlinearity in a preliminary study, was selected to study the nonlinear I-V mechanism. The composites were prepared by mixing nano-SiC powder with silicone resin at the appropriate ratio in a Hauschild ?high speed mixer. 1wt.% dicumyl peroxide was employed as the silicone rubber crosslinking agent. Samples were compression moulded at 150 q C/1h +180 q C/1h. Prior to any electrical tests, the cured composite discs (approximately 0.5 mm in thickness) were kept in a vacuum oven at ~80 °C

overnight to remove any dicumyl peroxide residue, and gold electrodes of 2 cm in diameter were then sputter deposited

onto both surfaces of the discs. Robust composite as well as SiC powder filler high field I-V characterization was performed either in a customized guarded cell for composites or a powder measurement cell for powders,

which was connected to a Keithley 6517A electrometer and biased with a Bertan 10 kV voltage source. The measurements were generally undertaken at a range of temperatures at or above ambient temperature. To help identify the underlying conduction mechanism, additional composite I-V

measurements were done at a dry ice temperature (~-40 °C).

Steady-state composite or powder current readings were

typically taken 2 h (composites) or 1 min (powders) after the

step voltage was turned on. Some composite transient currents were also recorded with a L abview 6i program for an

absorption current study.

In addition, dielectric spectroscopy measurements were

undertaken using a Novocontrol Alpha Analyzer. The test ac

voltage was 1 volt RMS, and the typical measurement

conditions were: 1 mHz to 0.1 MHz, 30 °C to 120 °C. Pulsed

electro-acoustic (PEA) measurements [39-40] were conducted to assess the internal space charge distribution in the composite at room temperature. The applied dc field for space charge measurement ranged from 7 to 12 kV/mm. The interrogating pulse amplitude was 500 V, with a pulse width of about 10 ns. 3 RESULTS

3.1 SIC POWDER AND BASE SILICONE RUBBER The composite filler, a nano SiC powder, was characterized by secondary-ion mass spectroscopy and found to be p -type with an Al impurity of ~0.9×1020 atoms/cm 3. Electrical characterization revealed that the nonlinear I-V behavior or current density (J ) as a function of electric field (E ) for the SiC powder compact was slightly temperature dependent, as shown in Figure 1.The powder nonlinearity exponent, D , was found to be approximately 4 at ambient temperature, which is better illustrated in the inset plot of powder resistivity vs. field. From an Arrhenius plot (ln J vs. 1000/T) at constant fields, the average charge carrier activation energy ?E for the nano-SiC

Figure 2 shows the dielectric spectroscopy data for the base

silicone rubber, i.e., neat crosslinked silicone resin. A similar

silicone rubber dielectric spectroscopy result was reported by others [41].The data in the range 0.1-10 Hz is of particular interest. The slope of the log(?”) (the imaginary permittivity) vs. log(frequency) is approximately 1, whereas the real permittivity, ?’, remains almost constant. This is typically a sign of dc conduction [42-44]. It suggests that the silicone rubber is a strongly disordered hopping carrier system in which the charge carriers can hop between localized sites, thereby introducing some dc conduction. 3.2 COMPOSITE DISPERSION & NONLINEAR BEHAVIOR A typical Field Emission Scanning Electron Microscopy dispersion image of the 25vol% 50 nm-SiC/silicone rubber (SIR) composite is shown in Figure 3. The composite dispersion seems uniform although there are signs of some particle agglomeration, and it also appears that most particles are not in direct contact with each other. The composite steady -state nonlinear current density vs.

field and the log(resistivity) vs. log(field) measured at a series of temperatures are shown in Figure 4(a-b). The data indicate that the composite starts to display nonlinearity at fields

beyond 2-4 kV/mm, and the nonlinearity exponent, ?,generally drops as temperature increases. Furthermore, compared to the nano-SiC powder compact, the nonlinear J -E behavior of the composite has larger temperature dependence. Corresponding Arrehnius plots (ln J vs. 1000/T) indicate that the composite carrier activation energy is ~0.4-0.6 eV, which is almost one order of magnitude higher than that of the powder compact. This implies that there are likely no significant direct particle-particle contact pathways in the composite, i.e., the 25vol% nanocomposite is below its geometric percolation and most SiC particles are separated by a thin silicone rubber layer.

Figure 1.The nano-SiC powder current density J vs. electric field E at

different temperatures. The inset plot shows powder resistivities at 25q C.

Theoretically, the average inter-particle silicone rubber layer thickness can be estimated based on the following equation for spherical fillers under ideal uniform dispersion [45-46]:

?

???

o

????

a ??1·¨¨?§ 2343

1f r l S (1) where l is the inter-particle distance, r the particle radius, and f the filler volume fraction. Accordingly, the average inter- particle silicone rubber layer thickness for the 25vol% 50 nm-SiC/silicone rubber is calculated to be approximately 14 nm.

3.3 COMPOSITE HOLE HOPPING NONLINEAR

HYPOTHESIS

3.3.1 HOLE HOPPING HYPOTHESIS

Hole injection and transport is acknowledged as a primary charge carrier process in many amorphous polymeric dielectrics [47-51]. Hole hopping conduction has also been identified as a primary conduction mechanism in some systems, e.g., in the amorphous regions of insulating polyethylene [47]. In this study, a high field nonohmic hole hopping transport at silicone rubber valence band tail hypothesis is schematically shown in Figure 5. The energy levels are based on the electronic band configuration of the components. In this figure, the electrodes are sputtered gold coatings, with a Work Function of ~5 eV. The bottom of the conduction band for silicone rubber is assumed to be located near the vacuum level, which is typical for most organic dielectrics [47, 50-51]. The band gap of ?-SiC is around 2.2 eV [52-54], and the bottom of the conduction band is taken as 2.9eV below the vacuum level [52].

Hole injection and transport is hypothesized to be the dominant carrier process in the schematic based on the following two arguments. First, electrode hole injection is more likely to occur because the Schottky barrier height should be much lower compared to that for electron injection. For instance, if we assume the silicone rubber band gap is 6~8

Figure 2.Real, and imaginary permittivity vs. frequency for base silicone rubber. The data within 0.1-10 Hz is a likely indication of hopping conduction.

1.0

10.0

100.0

1.0E-3

1.0E-1

1.0E+1 1.0E+3

1.0E+5

f, Hz

?'

1.0E-4

1.0E-2

1.0E+0

1.0E+2

1.0E-3

1.0E-1

1.0E+1 1.0E+3

1.0E+5

f, Hz

?''

Figure 3. A Field Emission SEM image of the 25vol% 50nm-SiC/silicone rubber composite showing the state of dispersion.

Figure 4.25vol% 50nm-SiC/silicone rubber composite (a) steady-state current density vs. field at various temperatures, and (b) corresponding log(resistivity) vs. log(field) plots.

h +

hop

hop

hop

e -

2.2 eV

CB bottom 2.9eV 2.2 eV

CB bottom 2.9eV

Figure 5.The speculated hole injection and conduction occurred for the 25vol% 50nm p -SiC/SIR composite. The (nearest-neighbor) hole hopping across thin silicone rubber layer VB tail process is hypothesized to be the governing physics for composite high field I-V nonlinearity. The overall band tilting under the field is removed for clarity.

eV, then the hole injection barrier height is around 1-3 eV, whereas the barrier height for electron injection is about 5 eV. Second, even if there are injected electrons, they will likely recombine with holes in p -SiC, therefore not contributing much to the steady-state current. The overall charge process may occur as follows: holes (h +) are Schottky injected from the anode, which travel through the neighboring thin rubbery layer via hopping across localize states near the silicone rubber valence band (VB) tail, followed by bulk conduction in the SiC VB. After that, it is postulated that holes are further internally emitted from p -SiC VB to the VB of the neighboring silicone rubber layer, followed by hopping transfer through that silicone rubber layer and so on, until eventually recombining with electrons at the cathode.

3.3.2 NEAREST-NEIGHBOR HOPPING (NNH) MECHANISM

Because variable range hopping can be ruled out based on the discussion in section 3.4.3 later on, nearest-neighbor

hopping is the primary mechanism addressed here.

A. NNH EQUATION MODIFICATION For classical thermally-activated, field-enhanced “nearest-neighbor” middle band gap or impurity band hopping near the Fermi level,the conductivity is derived as follows [14-19]:

a) It is assumed that the number of electrons per unit volume

within a range kT of the Fermi energy is kTN(E F ), where k is Boltzmann’s constant, and N(E F ), the density of local states at the Fermi level.

b) The hopping rate is approximately given by the phonon-induced hopping probability difference towards and against the electric field, which approximately equals:

)]/exp()/[exp()/exp(kT eRE kT eRE kT E hop ph ' Q (2)

where ?Ph is phonon frequency, ?, the reciprocal of the decay length of the localized wave function, and R ,?E hop are the average hopping distance and the average hopping activation energy respectively. c)Current density J is obtained by multiplying the above two terms with e and R, which gives:

)

/sinh()/exp()(kT eRE kT E E eRkTN J hop Ph F ' Q

(3) In this work, we hypothesized a carrier transport process governed by nearest-neighbor hole hopping across the silicone rubber VB tail. As a result, the above carrier population term

)(F E kTN for middle band gap impurity hopping conduction should be replaced with the hole density near the SiC/silicone rubber interfaces, which conceptually is roughly proportional to exp(-E A /2k T), where E A is the SiC acceptor level, reported to be ~0.25 eV [52]. If we further assume h kT Ph / Q , then equation (3) becomes

)/sinh()/2(exp(kT eRE kT E E h kT eR J A hop ' v (4)

or ))(sinh()(E T D T C J , where C(T) and D(T) are as follows:

)/)2

(exp()(kT E E h kT

eR T C A hop ' v (5a) and kT eR T D /)( (5b) In addition, to achieve a more appropriate comparison of equation (4) with the data, a local E -field instead of nominal E -field is needed. This field can be determined simply by

assuming that the dc voltage drop occurs only at the insulating rubber layers. The local E -field at each silicone rubber layer then approximately follows:

)/1(/)(00l d E l d l E E local | (6) where E 0 is the nominal electric field, l the inter-particle

silicone rubber layer thickness, d the SiC particle diameter (~50 nm). Recall that from equation (1), the average silicone rubber layer thickness l for the studied nanocomposite is about 14 nm. Accordingly, E local is derived to be about 4-5 times higher than the nominal field.

B. NEAREST-NEIGHBOR HOPPING FITTINGS Results suggest that the experimental data fit well with the two parameter hyperbolic sine function characterized for nearest-neighbor hopping at each temperature, as shown in Figure 6(a-e). A further plot of ln(C(T)/T ) vs. 1/T shown in Figure 7(a) exhibits a clear linear trend, from the slope of which, the overall hopping activation energy, ?E hop +E A /2, is estimated to be about 0.53 eV according to equation (5a), which falls in the 0.4-0.6 eV range derived from the composite Arrhenius ln J -1/T fitting. The net hopping activation energy, ?E hop , is accordingly given as 0.53 eV - 0.25/2 eV § 0.41 eV. Figure 7(b) shows the D(T) vs. 1/T fitting, which can be roughly fitted to follow a linear trend that intersects the origin (0,0). The overall average hopping distance, R , is then estimated to be approximately 4.3 nm according to equation (5b). From the same equation, the individual hopping distance, R , at each temperature can also be obtained from each D parameter, as listed in Table 1.

Table 1.Derived individual hopping distance R at each temperature.Temperature, °C 25 60

80

100

120

Hopping distance, nm

~4.8

~3.6 ~4.0 ~3.5 ~4.0

Figure 6.25vol% nano-SiC/SIR composite J-E local two parameter hyperbolic sine hopping fittings (a)-(e).

Fitting parameter C and D at each temperature are shown on each figure.

Figure 7.25vol% nano-SiC/SIR (a) ln(C/T ) vs. 1/T , and (b) D(T) vs. 1/T hopping fittings.

0.0E+00

5.0E ?08

1.0E ?071.5E ?07

2.0E ?072.5E ?07

0.001

0.0020.0030.004

1/T, K

?1

D (T )

-30

-26

-22-18

0.0024

0.0029

0.0034

0.0039

1/T, K

-1

l n C (T )/T

Figure 8.A plot of kT ·ln(J/T 2) vs. E 1/2 to compare the data to the Schottky model for a 25vol% 50nm-SiC/SIR composite.

-0.9

-0.7

-0.5

1.0E+3

1.5E+3

2.0E+3 2.5E+3

3.0E+3

sqrt(E)

K T l n (J /T 2

)

3.4 ALTERNATIVE MECHANISMS

3.4.1 ELECTRODE INJECTION

Electrode charge injection can prevail over bulk conduction mechanism in real measurements. It commonly takes place by field-assisted thermionic emission, i.e., Schottky emission or by simple tunneling injection. Tunneling injection can basically be ruled out in this case because the nano-SiC/SIR J-E data show significant temperature dependence. For electrode

Schottky injection, the J-E relationship obeys the following

equation [23-25]:

equals 1202 mA/mm 2K 2). ?Sch is the Schottky constant, ?0 and ? are the vacuum permittivity and high frequency dielectric constant of the material (SIR in this case).

Figure 8 shows the )/ln(2T J kT ~E 1/2plots for the nano-SiC/SIR at all measured temperatures, which seem to follow

straight lines. From the slope ?Sch ,? for silicone rubber matrix

is calculated to be about 0.22, which is one order of magnitude

lower than what was measured in Figure 2(a) as well as the

commonly reported ? of ~3 [41]. This discrepancy in

dielectric constant rules out electrode Schottky injection as the underlying mechanism leading to composite nonlinearity, and implies that the external carrier supply is abundant.

3.4.2 INTERNAL TUNNELING OVER THIN SIR LAYER Tunneling through a thin polymer layer is an important charge transport mechanism. The general conditions for tunneling to occur are: 1) the local E-field should be high enough, for instance, 100 kV/mm; and 2) the polymer layer should be thin enough, typically within 2~3 nm. In the studied 25vol% 50nm-SiC/SIR system, the composite nonlinear behavior starts at fields as low as ~2 kV/mm, which is far below the 100 kV/mm even after taking into account the estimated 4 ~ 5 × local field enhancement. In the meantime, the average silicone rubber layer thickness is calculated to be ~14 nm. Both of these argue against direct internal tunneling.

Furthermore, since the p -SiC hole density is proportional to exp (-E A /2kT ), the temperature dependence (if tunneling only

were occurring) would have a maximum activation energy of

E A

/2, i.e., ~0.13 eV, which is still considerably lower than the

activation energy range of 0.4-0.6 eV estimated from the composite Arrehnius fittings. 3.4.3. VARIABLE RANGE HOPPING OVER A THIN

SIR LAYER

Hopping through a thin polymer layer usually takes place through either variable range hopping (VRH) or nearest-neighbor hopping (NNH). The temperature dependence of the conductivity for both hopping mechanisms follows [32-33, 36]: ))/exp((00s T T V V (8) For variable range hopping, s is typically ?, which gives the well known three-dimensional Mott’s law, whereas nearest-neighbor hopping is identified by a single thermal activation

process, where s is equal to 1. A typical composite ln(conductivity) as a function of 1/T plot is shown in Figure 9, which includes a data point on dry

ice temperature. The data suggest that a transition in

mechanism occurs below room temperature. Above room

temperature, a single thermal activation process, i.e., nearest-neighbor hopping occurs, whereas variable range hopping may

take place at low temperature.

For quantitative analysis, the composite ln(conductivity) vs. 1/T 0.25was plotted and is shown in the inset plot of Figure 9. According to the literature [36-37], in the event of Mott variable range hopping, the fitting parameters ?0and T 0 are closely related to the density of local states at the Fermi level, N(E F ), and the decay length of the localized wave function, ?-1

, within the material as given in equations (9a-9b):

)

(1.183

0F B E N k T

D (9a)

Nano-SiC/SIR. The inset plot shows the corresponding composite ln(conductivity) vs. 1/T 0.25Mott’s variable range hopping fitting at ambient temperature & above.

ph B F e T k E N Q SD V 2

2

/108)(9 ?

o

??a (9b)

Further knowledge such as the average hopping distance, R , or the hopping activation energy, ?E hop can be obtained from equations (9c-9d).

4

/1)(89

??

1ᬬ?

§ F B E TN k R SD (9c)

)

(43

3F hop E N R E S

' (9d)

The result indicates that T 0 and ?0 are around K 9

1067.2u and 1101006.1 u m S respectively. Reasonable calculated values of N(E F ), R and ?E hop are ~1317103.6 u eV cm ,~10.2nm and ~0.35eV at ambient temperature after taking 51 D ?. The phonon frequency ?ph derived from equation (9b) is ~120107.9 u s , which is however significantly higher

than practical phonon frequency of ~1

12106 u s near room temperature. This helps rule out variable range hopping as the primary charge conduction mechanism. 3.4.4 SPACE-CHARGE OR POOLE-FRENKEL EFFECT

A classical space-charge-limited current model [30-31] like the Mott-Gurney law generally predicts a constant low nonlinearity exponent, ?, of close to 2, according to: 2

08

9L

V J PHH (10) which cannot explain the J-E nonlinearity for the studied

nano-SiC/SIR. On the other hand, compared to the hopping mechanism, the Poole-Frenkel effect [26-29] is typically associated with low density yet deep level localized sites or traps (~>1 eV) in dielectrics, which is higher than the estimated composite activation energy of 0.4-0.6 eV. In addition, the dielectric spectroscopy data suggested that the base silicone rubber is likely a hopping carrier system with a high density of local states. Therefore, the Poole-Frenkel mechanism can also be ruled out.

3.5 OTHER RESULTS

3.5.1 COMPOSITE ABSORPTION CURRENT

CHARACTERIZATION

Absorption current characterization [55-57] often provides additional insight for understanding the carrier injection and charge transport in a non-ideal dielectric. Figure 10 shows the nanocomposite room temperature absorption current as a function of time at different fields: from the low field, composite linear I-V region of 2 - 4 kV/mm, to fields of 6 - 8 kV/mm in the nonlinear region (refer to Figure 4b). The result indicates that, in a log-log representation, the low field absorption current follows a linear universal law [55-57], i.e.

n

t T A t I )()(, with n close to unity. By contrast, the current drop slows down and a plateau gradually forms at high

(nonlinear) fields, rendering appreciably higher steady-state currents at high fields. According to Jonscher [57], this high field plateau can be attributed to a carrier hopping mechanism, which supports our hopping hypothesis.

3.5.2 COMPOSITE DIELECTRIC SPECTROSCOPY

Figure 11 shows the nano-SiC/SIR dielectric loss, ?”, as a function of frequency at different temperatures. At low frequency, ?”displays a rapid increase, and the data at each

temperature are essentially parallel, which is likely an

indication of thermally-activated quasi-dc conduction. Based

on the loss frequency shift as a function of 1/T (the insert plot), the activation energy, ?E , is found to be about 0.54 eV, which indeed agrees well with the overall 0.53 eV activation

energy derived from the nearest-neighbor hopping fitting in section 3.3.2.

3.5.3 PULSED-ELECTROACOUSTIC (PEA) STUDY

Pulsed-electroacoustic analysis (PEA) [39-40] was undertaken on the 25vol% nano-SiC/SIR composite. The charge buildup at typical field of 7 kV/mm is shown in Figure 12. The data show no sign of homocharge near the cathode for electrode injection, whereas there exists a uniform positive charge buildup across the composite, which could be argued as

a sign for hole conduction.

from low, linear to high, nonlinear fields.

shows log f vs. 1000/T .

4 DISCUSSION The logic for the composite nonlinearity drop as temperature increases can be understood on the basis of nearest-neighbor hopping conduction as follows. First, in terms of fundamental physics, the composite hopping nonlinearity exponent, ?, basically arises from the hopping

probability difference along or against the field, which is expected to be compromised as T increases. Second, mathematically, the hopping nonlinearity, ?, is determined by

the field dependent term )/sinh(kT eRE , which is approximately equal to 2/)/(kT eRE e at high fields. Accordingly,

D E e kT eRE

J kT eRE v v 2/)sinh()/( (11) .ln ~const E

E kT eR D (12)

Equation (12) suggests that the hopping nonlinearity exponent,

?, should decrease as T increases, and it also implies that, ?, is indeed field dependent. Furthermore, the hypothesis of hole hopping across the

silicone rubber valence band tail is largely based on an ideal component band configuration as well as the p -type nature of the filler SiC. When the composite system is

different, hopping could also occur in the middle of the band gap near the Fermi level, or through the organic conduction band tail for electron hopping if the fillers are

n -type semiconductors, metals or carbon black. Finally, an ionic hopping contribution, which has been claimed for some organic dielectric or dielectric composites, cannot be

rigorous ruled out in this study [35, 38], as it also happens

via hopping transfer. 5 CONCLUSIONS A robust investigation has been performed to study the high field nonlinear I-V mechanism for a 25vol% 50nm p -SiC/silicone rubber field grading composite. Results indicate

that thermally-activated, field-enhanced nonohmic hole nearest-neighbor hopping transport across inter-particle silicone rubber layers is likely the mechanism governing the nonlinear behavior. The average hopping activation energy

and average hopping distance are estimated to be ~0.41 eV and ~4.3 nm respectively, both of which are typical for hopping conduction, and the hopping-based composite nonlinearity exponent, ?, generally decreases as temperature increases.

ACKNOWLEDGEMENT

The authors are grateful for the sponsorship from ABB Corporate Research, Sweden, through the Rensselaer

Nanotechnology Center, and the Nanoscale Science and Engineering Initiative of the National Science Foundation under NSF award number DMR-0642573. We would also

like to express sincere gratitude to Dr. Uno G?fvert and Dr.

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Xiaoping Wang (Non member) was born in

China. He got his B.E. in physical metallurgy

from the Chinese Northeastern University, M.E.

in Materials Science from the Beijing Institute

of Aeronautical Materials. After that, he was

employed for a few years at the Beijing

Institute of Aeronautical Materials. He recently

obtained his Ph.D. in Materials Science and

Engineering from Rensselaer Polytechnic

Institute under the guidance of Dr. L inda

Schadler and Dr. Keith Nelson. He is currently working as a Post Doc at Virginia Polytechnic Institute and State University.

Keith Nelson (F’90) was born in Oldham, UK

and received his B.Sc.(Eng.) and Ph.D.

degrees from the University of L ondon, UK.

He is currently Professor Emeritus at the

Rensselaer Polytechnic Institute [previously

Philip Sporn Chair of Electric Power

Engineering]. Prior to his appointment at

Rensselaer, he was manager of Electric Field

Technology Programs at the General Electric

R & D Center in Schenectady, NY. He has

held numerous IEEE appointments including that of the Presidency of the Dielectrics & Electrical Insulation Society, 1995-6, and is currently an IEEE Director elect. He is a chartered electrical engineer, a Fellow the IET and the recipient of the IEEE Millennium Medal.

He nrik Hillborg (Non member)received his

M.Sc. and Ph.D. degrees in Polymer

Technology from the Royal Institute of

Technology in Stockholm, Sweden, in 1994

and 2001, respectively. Since 1995 he has been

working with polymeric materials in different

HV applications at ABB Corporate Research in

V?ster?s, Sweden. During 2002-2003 he

worked as postdoctoral researcher at the

University of Twente, the Netherlands. His

currently hold positions as Senior Principal

Scientist at ABB and Associate Professor at the Department for Fibre and Polymer Technology, Royal Institute of Technology. His research interests concentrate on silicone rubbers, electric field grading materials and polymer nanocomposites

Linda S. Schadle r (Non member) received

her B.S. From Cornell University and Ph.D.

from the University of Pennsylvania in

Materials Science and Engineering. She is

currently the Associate Dean of Academic

Affairs in the School of Engineering and a

Professor of Materials Science and

Engineering at Rensselaer Polytechnic

Institute. Before coming to Rensselaer in

1996, she was on the faculty at Drexel

University and spent 2 years at IBM's T.J. Watson Research Center. She is a Fellow of ASM International and a former

member of the National Materials Advisory Board.