field enhanced conduction

8/14/2019 Field Enhanced Conduction

1/54

FIELD ENHANCED CONDUCTION

The dielectric properties which we have discussed so far mainly consider theinfluence of temperature and frequency on & ' and s" and relate the observedvariation to the structure and morphology by invoking the concept of dielectricrelaxation. The magnitude of the macroscopic electric field which we considered was

necessarily low since the voltage applied for measuring the dielectric constant and lossfactor are in the range of a few volts.W e shift our orientation to high electric fields, which implies that the frequency underdiscussion is the power frequency which is 50 Hz or 60 Hz, as the case may be. Sincethe conduction processes are independent of frequency only direct fields are consideredexcept where the discussion demands reference to higher frequencies. Conductioncurrent experiments under high electric fields are usually carried out on thin filmsbecause the voltages required are low and structurally more uniform samples are easilyobtained. In this chapter we describe the various conduction mechanisms and refer toexperimental data where the theories are applied. To limit the scope of considerationphotoelectric conduction is not included.

7.1 S O M E G E N E R A L C O M M E N T SApplication of a reasonably high voltage -500-1000V to a dielectric generates a currentan d let's define the macroscopic conductivity, for limited purposes, using Ohm's law.The dc conductivity is given by the simple expression

C7=AE

TM

Copyright n 2003 by Marcel Dekker,Inc.All Rights Reserved.


2/54

3 3 0 Chapter?where a is the conductivity expressed in (Q m)"1 , A the area in m 2, and E the electricfield in V m"1. The relationship of the conductivity to the dielectric constant has not beentheoretically derived though this relationship has been noted for a long time. Fig.7.1shows a collection of data1 for a range of materials from gases to metals with thedielectric constant varying over four orders of magnitude, and the temperature from 15Kto 3000K. Note the change in resistivity which ranges from 1026 to 1 0 ~ 1 4 Q m . Threelinear relationships arerelationship is given as noticed in barest conformity. For good conductors the

log p + 3 log e' = 7.7For poor conductors, semi-conductors an d insulators the relationship is

(7.2)

2OI2Xo>-"H>(/>UJ(E

T I T A N A T E S

FERRO-ELECTRICS

CA RBO N AT 0CGRAPHITE AT 0C

C OPPER AT 500CSILVER AT 15K

G L Y C E R I N E /AT 800C

Sn-BiTUNGSTEN AT 3500K /

SILVER AT 0CS U P E R C O N D U C T O R S C OPPER AT 15 K

(7.3)

I02 I03 I04DIELECTRIC CONSTANT

Fig. 7.1 Relationship between resistivity and dielectric constant (Saums and Pendleton, 1978,with permission of Haydo n Book Co.)Ferro-electrics fall outside the range by a wide margin. The region separating theinsulators and semi-conductors is said to show "shot-gun" effect. Ceramics have ahigher dielectric constant than that given by equation (7.3) while organic insulators havelower dielectric constant. Gases are asymptotic to the y-axis with very large resistivityand s' is close to one. Ionized gases have resistivity in the semi-conductor region.

TM



3/54

From the definition of complex dielectric constant (ch. 3), we recall the followingrelationships (Table 7.1):

Table 7.1Summ ary of definitions for current in alternating voltageQuantity

Charging current, IcLoss current, ILTotal current, I

Dissipation factor, tan8Power loss, P

FormulaC O C08' Vc o C 0 s " V

c o C0 V(s'2 + e" 2)1/28" / 8'

c oC 0e 'V 2 tan5

Unitsamperesamperesamperes

noneWatts

7.2 MOTION OF CHARGE CARRIERS IN DIELECTRICSMobility of charge carriers in solids is quite small, in contrast to that in gases, because ofthe frequent collision with the atoms of the lattice. The frequent exchange of energydoes not permit the charges to acquire energy rapidly, unlike in gases. The electrons aretrapped and then released from localized centers reducing the drift velocity. Since themobility is defined by W e = jj,e E where W e is the drift velocity, |ne the mobility and E theelectric field, the mobility is also reduced due to trapping. If the mobility is less than~5x lO ~ 4 m 2 / Vs the effective mean free path is shorter than the mean distance betweenatoms in the lattice, which is not possible in principle. In this situation the concept of themean free path cannot hold.Electrons can be injected into a solid by a number of different mechanisms and the driftof these charges constitutes a current. In trap free solids the Ohmic conduction arises asa result of conduction electrons moving in the lattice of conductors and semi-conductors.In the absence of electric field the conduction electrons are scattered freely in a solid dueto their thermal energy. Collision occurs with lattice atoms, crystal imperfections andimpurity atoms, the average velocity of electrons is zero and there is no current. Themean kinetic energy of the electrons will, however, depend on the temperature of thelattice, and the rms speed of the electrons is given by (3kT/m)L2.If an electric field, E, is applied the force on the electron is eE and it is accelerated indirection opposite to the electric field due to its negative charge. There is a net driftvelocity and the current density is given by

TM



4/54

(7.4)awhere N e is the number of electrons, \ \ . the electron mobility, V the voltage and d thethickness.We first consider Ohmic conduction in an insulator that is trap free. The concept ofcollision time, ic, is useful in visualizing the motion of electrons in the solid. It isdefined as the time interval between two successive collisions which is obviously relatedto the mobility according to

j U = eTcm* (7.5)

where m* is the effective mass of the electron which is approximately equal to the freeelectron mass at room temperature.The charge carrier gains energy from the field and loses energy by collision with latticeatoms and molecules. Interaction with other charges, impurities and defects also resultsin loss of energy. The acceleration of charges is given by the relationship, a = F/m* = eE/m* where the effective mass is related to the bandwidth W b. To understand thesignificance of the band width we have to divert our attention briefly to the so-calledDebye characteristic temperature2.In the early experiments of the ninteenth century, Dulong and Petit observed that thespecific heat, Cv, was approximately the same for all materials at room temperature, 25J/mole-K. In other words the amount of heat energy required per molecule to raise thetemperature of a solid is the same regardless of the chemical nature. As an exampleconsider the specific heat of aluminum which is 0.9 J/gm-K. The atomic weight ofaluminum is 26.98 g/mole giving C v = 0.9 x 26.98 = 24 J/mole-K. The specifc heat ofiron is 0.44 J/gm-K and an atomic weight of 55.85 giving Cv = 0.44 x 55.85 = 25J/mole-K. On the basis of the classical statistical ideas, it was shown that Cv = 3 R whereR is the universal gas constant (= 8.4 J/g-K). This law is known as Dulong-Petit law(1819).Subsequent experiments showed that the specific heat varies as the temperature islowered, ranging all the way from zero to 25 J/mole-K, and near absolute zero thespecific heat varies as T3. Debye successfully developed a theory that explains theincrease of C v as T is increased, by taking into account the coupling that exists betweenindividual atoms in a solid instead of assuming that each atom is a independent vibrator,

TM



5/54

as the earlier approaches had done. His theory defined a characteristic temperature foreach material, 0, at which the specific heat is the same. The new relationship is Cv (0) =2.856R. 0 is called the Debye characteristic temperature. For aluminum 0 = 395 K, foriron 0 = 465 K and for silver 0 = 210 K. Debye theory for specific heat employs theBoltzmann equation and is considered to be a classical example of the applicability ofBoltzmann distribution to quantum systems.Returning now to the bandwidth of the solids, W b may be smaller or greater than k0.Wide bandwidth is defined as Wb > k0 in contrast with narrow bandwidth where Wb (e a W^nhkT).For a lattice spacing of 50 nm we get |i> 3.8 Wb/kT.If these conditions are not satisfied then the conventional band theory for the mobilitycan not be applied. The charge carrier then spends more time in localized states than inmotion and we have to invoke the mechanism of hopping or tunneling between localizedstates. Charge carriers in m any molecular crystals show a mobility greater than 5 x 10"4

TM



6/54

m2 V s"1 and varies as T15. This large value of mobility is considered to mean that theband theory of solids is applicable to the ordered crystal and that traps exist within thebulk.The Einstein equation D/|ii - kT/eV where D is the diffusion co-efficient may often beused to obtain an approximate value of the mobility of charge carriers. For most

-2 3polymers a typical value is D = 1 x 10" m s" and substituting k = 1.38 x 10" J/K, e =1.6 x 10"19C and T = 300 K we get |n = 4 x 10"11 m 2 V'V1 which is in the range of valuesgiven in Table 7.2.Table 7.2Mobility of charge carrier in polymers [Seanor, 1972]

polymer Mobility(x 10'8m2Vs'1)Activation energy (eV)

Poly(vinyl chloride)Acrylonitrile vinylpyridinecopolymerPoly-N-inyl carbazolePolyethylenePoly(ethylene terephthalate)Poly(methyl methacrylate)Commercial PMMAPoly-n- butyl-methacrylateLucitePolystyreneButvarVitelpolyisopreneSiliconePoly(vinyl acetate)Below T O

Above TG

7310'3-10"2i o - 3

I x l O '22.5 x 10'73.6 x 10'72.5 x 10'63.5 x 10'91.4 x 10'7

4.85 x I O " 74.0 x I O " 72.0 x I O " 83.0 x I O " 1 02.2 x 10'8

0.4-0.520.24(Tanaka, 1973)0.24(Tanaka, 1973)0.52 0.090.48 0.090.65 0.090.52 0.090.69 0.090.74 0.091.08 0 .1 31.08 0 .1 31.73 0 . 1 70.48 0.091.21 0.09(with permission from North Holland Publishing Co.)

This brief discussion of mobility may be summarized as follows. If the mobility ofcharge carriers is greater than 5 x I O " 4 m 2 V s"1 and varies as T"n the band theory may beapplied. O therwise we have to invoke the hopping m odel or tunneling between localizedstates as the charge spends m ore time in localized states than in motion. The temperaturedependence of mobility is according to exp (-E^ / kT). If the charge carrier spends moretime at a lattice site than the vibration frequency the lattice will have time to relax an d

TM



7/54

within the vicinity of the charge there will be polarization. The charge is called apolaron and the hopping charge to another site is called the hopping model ofconduction. Methods of obtaining mobilities and their limitations have been commentedupom by K u and Lepins (1987), and, Hilczer and Malecki (1986). Table 7.3 shows thewide range of mobility reported in polyethylene.

Table 7.3Selected Mobility in polyethylene3

Mobility (x 10'8m2Vs"')l . O x 10'7(20C)1 .6xlO" 5(70C)2 .2x lO '4 (90C)500

l O t o l x l O " 5l . O x 10'7

l . O x 10'8(20C)4.2x10'7(50C)2.3xlO '6(70C)

AuthorW intle (1972)1Davies (1972)2Davies(1972)3Tanaka (1973)4Tanaka and Calderwood (1974)5Pelissouet. al. (1988)6Nathet. al. (1990)7Lee et. al. (1997)8Leeet. al. (1997)

Glaram has described trapping of charge carriers in a non-polar polymer4. The chargemoves in the conduction band along a long chain as far as it experiences the electricfield. At a bend or kink if there is no component of the electric field along the chain, thecharge is trapped as it cannot be accelerated in the new direction. The trapping site iseffectively a localized state and the charge stops there, spending a considerable amountof time. Greater energy, which may be available due to thermal fluctuations, is requiredto release the charge out of its potential well into the conduction band again. In thetrapped state there is polarization and therefore some correspondence is expectedbetween conductivity and the dielectric constant as shown inFig. 7.1.

1 H. J. Wintle, J. Appl. Phys., 4 3 (1972 ) 2927).2 Quoted in Tanaka and Calderwood (1974).3 Quoted in Tanaka and Calderwood (1974).4 T. Tanaka, J. Appl. Phys., 4 4 (1973) 2430.5 T. Tanaka and J. H. Calderwood,7 (1974) 12956 S. Pelissou, H. St-Ongeand M . R. Wertheimer, IEEE Trans. Elec. Insu. 23 (1988) 325.7 R. Nath, T. Kaura, M. M . Perlman, IEEE Trans. Elec. Insu. 25 (1990)4 198 S. H. Lee, J. Park, C. R. Lee and K . S. Luh, IEEE Trans. Diel. Elec. Insul., 4 (1997) 4 25

TM



8/54

A brief comment is appropriate here with regard to the low values of mobility shown inTable 7.2.The various energy levels in a dielectric with traps are shown in Fig. 7.2. Forsimplicity only the traps below the conduction band are shown. The conduction bandand the valence band have energy levels Ec and Ev respectively. The Fermi level, EF, liesin the energy gap somewhere in between the conduction band and valence band.Generally speaking the Fermi level is shifted towards the valence band so that EF < V z(E c - Ev). We have already seen that the Fermi level in a m etal lies in the middle of thetwo bands, so that the relation EF = V z (E c - Ev) holds. The trap level assumed to be thesame for all traps is shown by E t and the width of trap levels is AEt = Ec -Et.Using Fermi-Dirac statistics the ratio of the number of free carriers in the conductionband, nc, and in the traps, n, is obtained as [Dissado and Fothergill, 1992]

n (7.8)^ ^where N e f f and N t is the effective number density of states in the conduction band, andthe num ber density of states in the trap level, respectively. The ratio

nt nc +n t ntnc (7.9)

is the fraction of charge carriers that determines the current density. O bviously thecurrent will be higher w ithout traps as the ratio will be unity. This ratio will be referredto in the subsection (7.4.6) on space charge limited current in insulators with traps.Equation (7.8) determines the conductivity in a solid with traps present in the bulk. Thechange in conductivity due to a change in temperature, T, may be attributed to a changein mobility by invoking a thermally activated m obility according to

E -E,\ (7.10)

In an insulator it is obvious that the number of carriers in the conduction band, nc, ismuch lower than those in the traps, n t, and the ratio on the left side of equation (7.8) isin the range of 10"6 to 10"10. The mobility is 'unfairly' blamed for the resultingreduction in the cu rrent and the m obility is called trap limited. We will see later, during

TM



9/54

the discussion of space charge currents that this blame is balanced by crediting mobilityfor an increase in current by calling it field dependent.

CONDUCTION BAND E,

EvVALENCE BAND

Fig. 7.2 A simplified diagram of energy levels with trap energy being closer to conduction level.There is a min im um value for the mobility for conduction to occur according to the bandtheory of solids. Ritsko5 has shown that this minimum mobility is given by

In ea2

where a is the lattice spacing. For a spacing of the order of 1 run the m inimum m obility,according to this expression, is ~10"4 m 2 Vs" 1 which is about 6-10 orders of magnitudehigher than the mobilities shown in Table 7.1. Dissado and Fothergill (1992) attributethis to the fact that transport occurs within interchain of the molecule rather than withinintrachain.The mobilitiy of electrons in polymers is ~ 10"' m 2 V'V1 and at electric fields of 100MV/m the drift velocity is 10"2 m/s. This is several orders of magnitude lower than ther.m.s. speed which is of the order of 103 m/s.

7.3 IONIC CONDUCTIONWhile the above simple picture describes the electronic current in dielectrics, traps anddefects should be taken into account. For example in ionic crystals such as alkali halidesthe crystal lattice is never perfect and there are sites from which an ion is missing. At

TM



10/54

sufficiently high electric fields or high temperatures the vibrational motion of theneighboring ions is sufficiently vigorous to permit an ion to j u mp to the adjacent site.This mechanism constitutes an ionic current.Between adjacent lattice sites in a crystal a potential exists, and let be the barrierheight, usually expressed in electron volts. Even in the absence of an external field therewill be a certain n um ber of j umps per second of the ion, from one site to the next, due tothermal excitation. The average frequency of jum ps vav is given by

= v 0exp --M (7.12)\KlJwhere v is the vibrational frequency in a direction perpendicular to the jump, a thenumber of possible directions of the j u mp and the other symbols have their usualmeaning, v is approximately 1012 Hz and substituting the other constants the pre-exponential factor comes to ~ 1016 Hz. An activation energy of < |) = 0.2 eV gives theaverage j ump frequency of ~ 10'' Hz.In the absence of an external electric field, equal number of jumps occur in everydirection and therefore there will be no current flow. If an external field is applied along^-direction then there will be a shift in the barrier height. The height is lowered in onedirection by an amount eEA, where A , is the distance between the adjacent sites andincreased by the same amount in the opposite direction (Fig. 7.3). The frequency ofj u m p in the +E and -E direction is not equal due to the fact that the barrier potential inone direction is different from that in the opposite direction. The j u m p frequency in thedirection of the electric field is:

= ^ 0exp -T^7 I exp |- I (7-13)V KE av


12/54

leading to(7.19)E

kTThis expression is referred to as field dependent mobility though u.0 is independent ofthe electric field. If the electric field satisfies the condition eE A ,kT (E < 10MV/m)then we can substitute in equation (7.17) the approximation

(7.20)kT kT

and equation (7.17) may be rewritten as

d kT\kT)The current density is equal to

j = NeWd= Nv.} -exp I -1 (7.22)d kT ( kT) V 'where N is the number of charges per unit volume. The current is proportional to theelectric field, or in other words, Ohm's law applies.If the electric field is high then the approximation eE A ,kT is not justified and instrong fields the forward j u m p s are much greater in number than the backward j um psand we can neglect the latter. The curren t then becomes6'7

J =Nev_FA = 70 expf -- -+I (7.23) ( kT 2kTJwhere the pre-exponential is given by

TM



13/54

(7.24)/ z ~ V "In practice it is sufficiently accurate to express the current density using equation (7.17)as

---sinn (7.25)kTj { kT )To demonstrate the relative values of the two terms in brackets in the right side of eq.(7.25) we substitute typical values; < |> = 1.0 eV, T = 300 K, E = 100 MV/m, A = 10 nm , k= 8.617 x 10 " 5 eV/K = 1.381 x 10 ~ 23 J/K. We get


14/54

30 Temperature (C)50 100Nylon 66pi. PVCPVCPolyethylene Ox ideHDPEPETPVFPVDFPPEVA '

Fig. 7.4 Range of temperature for observing ionic conduction in polym ers (Mizutani and Ida,1988, with permission of IEEE).7.4 CHARGE INJECTION INTO DIELECTRICS

Several mechanisms are possible for the injection of charges into a dielectric. If thematerial is very thin (few nm) electrons may tunnel from the negative electrode intounoccupied levels of the dielectric even though the electric field is not too high. Fieldemission and field assisted thermionic emission also inject electrons into the dielectric.We first consider the tunneling phenomenon.

7.4.1 T HE T U N N E L I N G P H E N O M E N O NIn the absence of an electric field there is a certain probability that electron tunnelingtakes plac e in either direction. If a small voltage is applied across a rectangular barrierthere will be a current which may be expressed as9

4ft Sh a / 2 (7.26)

where ( j ) is the barrier height above the Fermi level, m the mass of an electron, s, thewidth of the barrier.For high electric fields, E = V/s and the current is given as

TM



15/54

J=7-7TexP 1/2 ^3/23heE (2m)1 (7.27)

The current is independent of the barrier height at constant electric field. The theory hasbeen further refined to include barriers of arbitrary shapes as rectangular shape is asimplified assumption. An equivalent barrier height is defined as

s

As>!

where As = s2 - Si. The current density is given by

(7.28)

where the barrier width As is expressed in nm , the voltage V in Volts, < j > in eV and thecurrent density in A/m~. Equation (7.28) is independent of temperature and a smallcorrection is applied by using the expression (Dissado and Fothergill, 1992)

where the left side is the ratio of the current density at temperature T to that at absolutezero. At T = 300 K the second term on the right side is

1.5which is negligible with respect to one as far as measurement accuracy is concerned.Marginal dependence of the conduction current on the temperature is often considered asevidence for tunneling mechanism.A few comments with regard to the measurement of current through thin dielectric filmsare appropriate here. It is generally assumed that equation (7.27) is applicable and themeasured currents are used to derive the thickness of the film. The thickness can also be

TM



16/54

derived from the capacitance of the sample. If the thickness is obtained this way andsubstituted in equation (7.27) the calculated currents are lower than measured ones byseveral orders of magnitude. O ne of the reasons for the discrepancy is that the thicknessof the film may not be uniform. The current depends on this quantity to a considerableextent and in reality the current flows through only the least thick part and therefore thecurrent density may be larger.

7.4.2 SCHOTTKY EMISSIONThermionic field emission has been referred to in Ch. 1 and an applied electric fieldassists the high temperature in electron emission by lowering the potential barrier forthermionic emission. The carrier injection from a metal electrode into the bu lkdielectric is governed by the Schottky theory

1 /2*-0EkT

where A, a constant independent of E and P, is given by

(7.30)

(7.31)

and, s0 and ^ are the permittivity of free space and relative permittivity at highfrequencies. Theory [Dissado and Fothergill, 1992] shows that constant A is given by 4ne m k2/h 3 an d substitution of the constants gives a value of -1.2 x 106 A/m 2 K. Theexpression for current (7.28) consists of the product of two exponentials; the firstexponential is the same as in Richardson equation for thermionic emission. The secondexponential represents the decrease of the work function due to the applied field, and alowering of the potential barrier.(7.32)

According to equation (7.30) a plot of In (J /T 2) versus E172 gives a straight line. Theintercept is the pre-exponential and the slope is the term in brackets. In practice thestraight line will only be approximate due to space charge and surface irregularities ofthe cathode, particularly at low electric fields. Lewis10 found that the pre-exponentialterm is six or seven orders of magnitude lower than the theoretical values, possibly due

TM



17/54

to formation of a metal oxide layer of 2 nm thick. The probability of crossing theresulting barrier explains the discrepancy.Miyoshi and Chino11 have measured the conduction current in thin polyethylene films of50-120 nm thickness and their experimental data are shown in fig. 7.5. The measuredresistivity is compared with the calculated currents, both according to Schottky theory,equation (7.30), and the tunneling mechanism, equation (7.27). Better agreement isobtained with Schottky theory, the tunneling mechanism giving higher currents. Lily andMcDowell12 have reported Schottky emission in Mylar.

7.4.3 HOPPING MECHANISMHopping can occur from one trapping site to the other; the mechanism visualized is thatthe electron acquires some energy, but not sufficient, to move over a barrier to the nexthigher energy. Tunneling then takes place, and from the probabilities of tunneling andthermal excitation the conductivity is obtained as:

= A exp B (7.33)'-rinwhere A is a constant of proportionality and % < n < Y z .

7.4.4 POOLE-FRENKEL MECHANISMThe Poole-Frenkel mechanism13 occurs within the bulk of the dielectric where thebarrier between localized states (Fig. 7.6) can be lowered due to the influence of thehigh electric field. For the mechanism to occur the polymer must have a wide band gapand must have donors or acceptors. Because the localized states of the valence band donot overlap those in the conduction band, the donors or acceptors do not gain enoughenergy to move into the conduction band or valence band, unlike in normalsemiconductors. In terms of the energy band the donor levels are several kT below theconduction band and the acceptor levels are several kT above the valence band. Weessentially follow the treatment of Dissado and Fothergill (1992).

TM



18/54

Fig. 7.5 Relationship of specific resistivity and thickness for thin polyethylene films.I=Theoretical Schottky curve (experimental values at IV); II=Theoretical Schottky curve(experimental values at 0.1V); Ill-Theoretical tunneling curve. (Miyoshi and Chino, 1967, withpermission of Jap. Jour. Appl. Phys.)

For the sake of simplicity we assume that the insulator has only donors as the treatmentis similar if acceptors, or both donors and acceptors, are present. As mentioned above,we further assume that there are no thermally generated carriers in the conduction bandand the only carriers that are present are those that have moved from donor states due tohigh electric field strength.Let

N D = Number of donor atoms or molecules per m 3.N 0 = Number of non-donating atoms or molecules per m 3.N c = Number of electrons in the conduction band.Then the relationship

- AT _AT~JV JV (7.34)holds assuming that each atom or molecule donates one electron for conduction. As anelectron is ionized it moves away from the parent atom but a Coloumb force ofattraction exists between the electron and the parent ion given by

TM



19/54

F = (7.35)

where sr is the d ielectric constant and r the distance between charge centers.lattice parameter MfW(a) (a)

f A. conductionbond It1T

valencebond

s I(b) (b)

NIC) LA.LT(c) (C)

Fig. 7.6 Band formulation for (1) ordered and (2) disordered systems, (a) Potential wells; (b)band structure; (c) density of states; F = Fermi level; W = bandwidth; EC = critical energy forband motion; Eg = forbidden energy gap. The shaded areas in 2(b) and 2(c) represent localizedstates. (Seanor, 1972, with perm ission of North Holland Publishing C o., Amsterdam.)

The potential energy associated with this force is-e' (7.36)

Note that V(r) > 0 as r o o and the dimension of the ionized donor assumesimportance. The electric field changes the potential energy to

TM



20/54

2V(r) = -- eEr (7.37)4;r 0 , . r

To find the condition for the potential energy to be maximum we differentiate equation(7.37) and equate it to zero

(7.38)dr 4 f t ( ) r rSolving for r and noting the maximum by rm

( Y ' 2r =\--- (7.39)m I A j ^ '4ns,rE)The change in the maximum height of the barrier with and without E is

/2(7.40)

The interchange of electrons between the donor states and conduction band is indynamical equilibrium and therefore Nc and N0 are constants. The rate of thermalexcitation of electrons is, therefore, equal to the rate of capture of electrons by thedonors.The rate of thermal excitation of electrons to conduction band is given by

(7.41)where V Q is the attempt to escape frequency and < j ) e f f is the reduced barrier height

t0=eD-AVm (7.42)where SD is the barrier height in the absence of the electric field.The rate of capture of electrons by donors =

TM



21/54

Number density of conduction electrons (Nc) x Number density of ionized donors(ND-N0) x drift velocity of electrons (W e) x capture cross section (S). viz.,Rc=Nc(ND-N0-)WeS =NtWeS (7.43)

where W e is the thermal velocity of electrons in the conduction band. Under equilibriumcond itions, we have the rate of excitation equal to the rate of recom bination

(7.44)The ju m p frequency v0 is approximately equal to

"o -N e f f W e S (7.45)where N e f f \ s the effective density of charge carriers in the conduction band. Substitutingequations (7.34 ) and (7.4 5) in (7.4 4 ) gives

= N e ff W e S(N D - A gexp- (7.46)

In dielectrics the number density of conduction electrons is usually quite small whencompared with the number density of donors, and we can make the approximation thatNDN c. Equation (7.46) then gives

(7.47)

Substituting equation (7.42) and (7.40) in (7.47) givesexP| lexP-6 E 1 / 7-(7-48)2 ^ }The conductivity a = N c e |u may now be expressed as

ea = (N effND) e j U Q x p l exp (7.49)V ff } ( 2kT 47T s1'2 kT l V J

TM



22/54

This is the Poole-Frenkel expression from which the current may be calculated for agiven electric field. A plot of loga versus E1 /2 yields a straight line with slope M pf andintercept C p/:3/ 2

NDr (7.50)-t tf\The same plot of log J verses E is also used as for determining whether the Schottkyemission occurs and to resolve between the two mechanisms we have to apply additional

tests as explained below.The conduction mechanism generally does not remain the same as the temperature israised and it is not unusual to find two mechanisms operating simultaneously, thoughone may predominate over the other depending on the experimental conditions. This facthas been demon strated by the recent conduction current measurements in isotactic (iPP)and syndiotactic polypropylene (sPP)14. The current in pristine iPP is observed to belarger than in crystallized iPP possibly because impurities and uncrystallizablecom ponents collect at the bou ndaries of large spherulites. The relatively low breakdownfield (~ 20 MV m"1) of iPP compared with sPP is thought to support for this reasoning.In contrast sPP, which shows smaller spherulites, does not show appreciable dependenceof conductivity on the electric field. The influence of electric field on current in sPP isshown in fig. 7.7.O hmic conduction is observed at field strengths below 10 MV/m, andfor higher fields the current increases faster.Schottky injection mechanism which is cathode dependent may be distinguished fromPoole-Frenkel mechanism which is bulk dependent in the following way. From the slope_ I tryof log I versus E the dielectric constant S T is calculated according to relationship (7.30)and compared with the dielectric constant measured with bridge techniques. Thetheoretical dielectric constant according to Poole-Frenkel mechanism is four times thatdue to Schottky emission. These considerations lead to the conclusion that at lowtemperatures, < 70C, Schottky emission is applicable whereas at higher temperaturesion transport is the most likely m echanism.The hopping distance may be calculated by application of eq. (7.25) and a hoppingdistance of approximately 3.3 nm is obtained in sPP. Hopping distances of 6.5 nm and20 nm have been reported15' 16 in bi-axially oriented and undrawn iPP respectively. Themolecular distance of a repeating unit in PP is 0.65 nm [Foss, 1963] and therefore theionic carriers jump an average distance of five repeating units. The barrier height

TM



23/54

obtained from eq. (7.25) is 0.82 eV for PP which means that the energy is not adequateto facilitate ionic motion below 70C.

10"5

10-67 1(CO

^ 10'8I= 10'9Q IU10-1 0 ,10 -11

107 108Electric field(Vnf1)

10s

Fig. 7.7 Electric field dependence of currentdensity at various temperatures in sPP ( Kim andYoshino, 2000, with permissionof J. Phys. D: Appl. Phys.)

Another example of non-applicability of Poole-Frenkel mechanism in spite of linearrelationship between logc and E 1 /2 in linear low density polyethylene [LLDPE] is shownin fig. (7.8) . From the slopes of the plots the dielectric constant, S o o , is obtained as 12.8which is much higher than the accepted value of 2.3 for PE. A three dimensionalanalysis of the Poole-Frenkel mechanism has been carried out by leda et. al.18 whoobtain a factor of two in the denominator of the second exponential of eq. (7.49).Application of their theory gives 8*, = 3.2 0.3 which is still considered to beunsatisfactory. Schottky theory gives = 0.96 which is obviously wrong. Nath et. al.19suggest an improvement to the space charge limited current which predicts a straightline relationship between plots of log(I/T2)) versus 1/T at constant electric field. Theparameters obtained for LLDPE are: activation energy 0.83 eV, trap separation distance= 2.8 nm, trap concentration = 4 .5 x 1025 m 3.The Poole-Frenkel expression may be simplified by neglecting the density of carriersthat move against the electric field. The current density is then expressed as20

TM



24/54

J = v l / 2 2r, 1 / 2

eE (7.51)

where a is a constant equal to e /(4jr s0 sr) and all other symbols have already beendefined. Substituting the following values a current density of 3.4 xlO" 9 A/m 2 isobtained: 1 0 - 1 _

t o - 1 7

10 -

82.5

100 300 500 T O O

Fig. 7.8 Typical Poole-Frenkel plots (log a vs. E1^) of the steady state conductivity of LLDPE(Nath etal 1990, with permission of IEEE).

E = 4 MV/m, e = 1.6 x 10'19 C, K = 2.5 x 1(T9 m, v-1 x 1013 s'1, Nt = 1019 traps/m3, Nc =/j-j ->10 /m , T = 300K, ( j ) = 1.9 eV, sr - 2.3. This value is in reasonable agreement withmeasured current density.

TM



25/54

7 .4 .5 S P A C E C H A R G E L IM I T E D C U R R E N T ( T R A P F R E E )Charge injected from the electrode moves through the bulk and eventually reaches theopposite electrode. If the rate of injection is equal to the rate of motion charges do notaccumulate in a region close to the interface and the electrodes are called Ohmic. Ohmicconductors are sources of unlimited number of charge carriers. If the mobility is lowwhich is the normal situation with many polymers then the charges are likely toaccumulate in the bulk and the electric field due to the accumulated charge influencesthe conduction current. A linear relationship between current and the electrical fielddoes not apply anymore except at very low electric fields. At higher fields the currentincreases much faster than linearly and it may increase as the square or cube of theelectric field. This mechanism is usually referred to as space charge limited current(SCLC).A study of space charge currents yields considerable information about the chargecarriers. In developing a theory for SCLC we assume that the charge is distributedwithin the polymer uniformly and there is only one type of charge carrier. Inexperiments it is possible to choose electrodes to inject a given type of charges and ifboth charges are injected from the electrodes recombination should be taken intoaccount. With increased electric field the regions of space charge move towards eachother within the bulk and coalesce. The number and type of charge carrier, and itsmobility and thickness of space charge layer influence SCLC.

In gas discharges the mobility of positive ions is lower than that of electrons by 3-4orders of magnitude. However in the solid state with traps this is not always the case.For example Kommandeur and Schneider21 studied the effect of illumination onanthracene an d found that the current was a function of illumination an d also due to thedifferences in carrier mobility, though equal number of holes an d electrons aregenerated. With illumination focused on the positive side the holes move rapidly to thenegative electrode. When the negative side is illuminated electrons drift slowly creatingspace charge within the solid.

Mott and Gurney22 first derived the current due to space charge limited current incrystals assuming that there were no traps. The following assumptions are made:(1) The electrodes are ohmic and electrons are supplied at the rate of their removal. This

means that there is no potential barrier at the electrode-insulator interface. Thisassumption yields the boundary condition

0 (7.52)

TM



26/54

(2) The current is a function of the number an d drift velocity of electrons and notdependent on the position in the sample, z, measured from the cathode. Denoting thenumber of electrons and the electric field at z as n(z)and E(z) respectively, thecurrent density is:

j =n(z)ejuE(x) *f(z) (7.53)(3) Poisson's equation is applicable

d2V en(z)dz2 Af-i OrvO

(4) There is no discontinuity of the electric field within the dielectric, that is

(7.54)

(7.55)The current through the dielectric is composed of three components: Drift, diffusion anddisplacement. Adding these contributions the current density is expressed as:

/ rjjf

J =n(z)e


27/54

l / 2(7-59)

Substituting this equation in (7.55) we getd j j

V= J( -)1/2z1/2 (7.60)

where d is the thickness of the insulator. Integration gives the current density asr2 (7.61)

This equation is similar to Child's law for space charge due to thermionic emission ingases, though the exponent of V is 3/2 for gases. This equation is also referred to asMott and Gurney square law. The current density is inversely dependent on thicknesseven though the electric field is maintained constant.Differentiating equation (7.60) we get

l /2-- z (7.62,Substituting for /from equation (7.61) the electric field at any point within the dielectricis obtained as

(7.63)2d J/2The number density of charge carriers is given by

( 7 - 6 4 )The electric field increases and the number of charge carriers decreases towards thepositive electrode as shown byFig. 7.9.

TM



28/54

Carrier Density n(x)

FieUTStrengthEOc)

DistanceFig. 7.9 Schematic variation of field strength, eq . (7.63) and carrier density, eq . (7.64) acrossthe sample under trap free space charge limited current flow (Seanor 1972). (With permission ofNorth Holland Co.)

At low voltages the current density is given by the Ohmic currentJ = nejuE (7.65)

At higher voltages the current is SCLC limited and given by equation (7.61). The slopeof the plot of J versus V increases from unity to two though the transition is not asabrupt as the theory indicates. The voltage at which transition from Ohmic conduction toSCLC occurs is obtained by equating (7.65) and (7.61). The transition voltage is given asV = Sed2n (7.66)

Experimentally the transition voltage is determined by the departure from linearity of theJ versus V plot. The carrier density is then calculated using equation (7.64) and themobility calculated from equation (7.65). Alternately \i may be obtained using equation(7.61).

TM



29/54

7.4.6 SPAC E CHARG E LIMITED CUR REN T (W ITH TRA PS)We extend the concepts of the previous sub-section to an insulator in which the trapsexist and investigate the nature of space charge limited current. The basic concept is thatif traps are present the SCLC will be reduced by several orders of magnitude. Rose23proposed that neither the space charge density nor the field distribution is altered bytrapping, but the current-voltage relationship should be modified by a trapping parameter0, which is given by

9= H C (7.67)nt+ncand the current density due to SCLC is given by24

(7.68)

for the trap free situation n t = 0and 0 = 1. With traps present the current is always lowerthan that without. The numbers nc and nt are given byE E (7.69)

n t =--(7.70l + exp[(Ef-Et)/kT]where Nv is the density of states in the valence band and Nt the density of traps. Theenergy symbols have already been defined in fig. 7.2.For traps lying below Ef by severalkT, that is , ( E f EJ/kT > 1 (shallow traps) equation (7.70) may be approximated tor tnt=Ntexp\--f --- (7.71)' kTThe number density of occupied traps dominate and we can assume that nt > nc and theratio of nc/nt is given as

TM



30/54

} ( 7 J 2 )Substituting for 0 in equation (7.68) we get the current density as

9 N v V 2 ( Et-E\8 N. d kT (7-73)

This is Child's law for an insulator with traps. Since u , is the mobility of free carriers aneffective mobility may be defined as :

Substituting this equation in (7.73) gives

Note that equations (7.71) - (7.75) are expressed for the case of holes being the carriersand the trap levels are counted from the valence band.Reverting back to the assumption that the free carriers are electrons we recall theassumption that Ec - EF kT, and this condition is called the shallow traps. As thevoltage is increased the current increases rapidly and at sufficiently high voltages thenumber density of injected electrons is approximately equal to the density of traps. Asthe last traps are being filled 9 approaches unity rapidly, the current increases (equation(7.68)), and the Fermi level will rise above the bottom of the trap energy level.The number density of occupied trap sites approaches the number density of trap levels,nt > Nt in equation (7.71), and the condition of all traps filled is called trap filledvoltage l imit (TFVL). W e refer to this voltage as V T F L . The rapid increase of current caneasily be misunderstood as breakdown but this can be verified by lowering the voltageand short circuiting the electrodes for a considerable time, ~2-3 hrs, and in the case ofTFVL the currents are reproduced with sufficient accuracy. Once this condition isreached a further increase in voltage exhibits a current dependence according to V2 aspredicted by Mott an d Gurney, equation (7.61).

TM



31/54

Under trap filled conditions the space charge is almost entirely due to the charges in thetraps and the charge density isp =eNt (7.76)

The voltage at which TFVL occurs is given by the Poisson's equation(7.77)dz 0 sr

which has the simple solutioneNd2 (7.78)

The trap density at the trap filled conditions is given by the above expression as

A slightly different result is obtained if the trap density at V T F L is calculated according tod\n(z)dz

(7-80)\dzo

Substituting for n(z) from equation (7.64) we get

This equation differs from (7.79) only in the numerical constant.It is not unusual to find that the distribution of traps is continuous with an exponentialdistribution (Ma, et al., 1999) instead of being discrete, and the number of traps may beexpressed by invoking a characteristic temperature, Tt, according to

TM



32/54

\TNt(E) = exp -- I (7.82)' kT t ( kT tlwhere kT, = E t is the characteristic trap energy. The expression for current in the trapfilled limited region is

( 7 . 8 3 )where He is the electronic mobility. We can summarize the I -V characteristics in thecondensed phase as follows. Four different characteristics are observed depending uponthe mechanism that is operative(Fig. 7.10).

1. Region 1: Ohmic conduction due to thermally generated free carriers. Equation (7.4)applies. Slope of line is +1.2. Region 2: Space charge limited current (SCLC) in trap free materials. Voltage is lessthan V T F L - Equation (7.61) applies. Slope of the line is +2

3. Region 3: Trap filled limited current. Voltage is equal to V T F L . Equation (7.73)applies. Slope of the line is >2.

4. Region 4: Trap filled space charge current. Voltage is greater than V T F L - Equation(7.75) applies. Slope of the line is +2.

An example of current increasing very rapidly in the trap filled region is shown in Fig.7.11, (Ma et. al., 1999) in a 136 nm electroluminiscent polymer with aluminumelectrodes. The initial rise is linear due to Ohmic conduction followed by the trap freespace charge current at higher voltages. Fig. 7.12 shows the conduction currents in thinPVAc films at voltages in the range of 4-300 V25. The four regions described above areclearly seen.The parameters derived at T = 30C are as follows: a = 6.82 x 10"12 Q'1 m'1, 0 = 2.8 x10'2, HO = 2 x 10"13 mW1, He = Ho 6 =5.6 x 1045 mV'V1, nc = 1 x 10 20 m3, nt= 1 x10 21 m3, Ec-E t=0.17eV.

7.5 SPACE CHARGE PHENOMENON IN NON-UNIFORM FIELDSThe mobility of charge carriers is independent of the electric field in the ohmic region ofconduction, and under space charge limited conditions, the mobility becomes fielddependent as given by equation (7.19). A closer examination reveals that the drift

TM



33/54

velocity of charge carriers as a function of the electric field depends upon the electric26field as shown in Fig. 7.13 .

J =s '5[2

5rft

(7.84)0.5 \2-2.245 )

Three regions are identified in the Log W d versus Log E plot: (a) The low field Ohmicregion, (b) the high field space charge region where the electric field is non-linear, and(c) the region of negative resistance. In a concentric spherical geometry with the innerand outer electrodes having radii of T [ and r0 respectively it is assumed that the innerelectrode is at Vi and the outer electrode is at V(ro)=0. Injection occurs from the innersphere and the current density is given as (Zeller, 1987), for r0rt, When r0 is large thecurrent is zero.

D

Fig. 7.10 I-V characteristics for space charge limited currents. A-Ohm's Law, B-Trap limitedSCLC, C-Trap filled limit, D-Trap free SCLC. (Dissado and Fothergill, 1993.)

TM



34/54

DMaetal10 Sp-r-

10

10V ( V )

Fig. 7.11 Experimental (Open Square) and theoretical (full line) I-V characteristics of anAl/polymer/al device, (d=136 nm). The theoretical curve is obtained by using equation (7.4)and (7.83) at high voltages. The active area of the device is 4 mm2 (Ma et. al., 1999, withpermission of J. Phys. D: Appl. Phys.)

Iff4

2xlfr2 4 f I igi 2

V(V)

Fig. 7.12 I-V characteristics on a log-log scale for a typical film 1-2 mm at temperatures shown(Chutia and Barua, 1980, with permission of J. Phys. D: Appl. Phys.)

TM



35/54

difftrtntiolrtiutQitc*oo

Lintorlow field

L O G (FIELD)

Fig. 7.13 The local field versus local drift velocity relation can be divided into three regions.The field, current and charge distributions for three regions are qualitatively different anddiscussed separately (Zeller, 1987, with permission of IEEE).In region II the current increases rapidly with a field dependent mobility and theinfluence of space charge is present. In view of the non-uniformity of the electric field itis difficult to express the mobility as a function of field and only a qualitative descriptionis possible.In region III the I-V characteristics become unstable at a critical field, exhibiting anegative resistance. The current distribution becomes uneven in space leading tofilamentary nature, an d current pulses are observed in the external circuit. Zeller (1987)gives field relationships from which the spatial dependence of charge is calculated forhomogeneous and filamentary space charge. The filamentary region is shown to extendfar deeper in the inter-electrode gap and the volume of charge density is also higher forthe filaments when compared with the homogeneous space charge.

7.6 CONDUCTION IN SELECTED POLYMERSThis section shall present experimental evidence for the applicability of the theoriesdiscussed above in some polymers. Several different schemes are available in general fordetermining the mechanism of conduction as shown in Fig. 7.13 and preferably eachchosen scheme should be supplemented by a second scheme. This procedure also acts as

TM



36/54

a cross check on some conclusions that one is tempted to draw on a limited set of data. Abrief description ofFig. 7.14follows:A. Current measurement involves application of a dc or a time varying voltage. Under dcthe current is measured at constant temperature and this is known as isothermalmeasurements. Current measurement at the same time elapsed after application ofvoltage is called isochronal measurements. Either scheme may be employed dependingupon the data acquisition technique. Absorption and desorption currents have beendiscussed in ch.6 and this chapter concerns transport currents. A study of thermallystimulated processes require measurement of thermally stimulated discharge (TSD) orthermally stimulated polarization (TSP) currents. These techniques and associatedpoling techniques such as thermal, corona, and X-ray poling, are discussed in ch. 10.Conduction currents have been measured for several decades with conflicting results inthe same material due to insufficient control of the experimental conditions or withoutrealizing the importance of the morphology. Even experiments on single crystals oftenlead to different interpretations (Seanor, 1972). For example single crystals ofpolyethylene have demonstrated negative resistance and this observation has beenreinterpreted as due to local heating. Thin films of poly (ethylene terephthalate) exhibitdifferent charging characteristics depending upon the rate of crystallization.Conductivity and activation energy of conductivity have been observed to decrease withincreasing crystallinity in a number of polyesters. However, the more recentinvestigation of Tanaka et. al27 shows that the increased crystallinity of low densitypolyethylene makes electron transport easier and the mean free path longer. The choiceof electrode materials influences the parameters derived from the I-V characteristics. Forexample in light emitting polymers (Ma et. al, 1999) use of gold, palladium or nickelanode instead of copper exhibits the trap filled space charge limited currents.In polypropylene the isotactic configuration has much higher conductivity whencompared with the syndiotactic structure (Kim, 2000). The iPP has much largerspherulites and its higher conductivity is attributed to this factor. In contrast sPP hasmore minute spherulites improving its electrical characteristics. In polyamidesconduction is electronic at low temperatures and protonic at higher temperatures. Inpolyethylene doped with carbonyl sites the electric field distribution is found to beindependent of temperature in the temperature range of 49-82C at constant appliedfield. The same characteristic is also observed in undoped polyethylene28.These brief comments are meant to stress the importance of specifying morphology,crystallinity, moisture content, heat treatment and electrode preparation in conductivitystudies. They also indicate the reasons for the difficulty in obtaining repeatable results

TM



37/54

between different investigations, but also, often under apparently the same conditions inthe same laboratory. A review of pub lished literature on conduction m echanism s inpolym ers can be found in ref. 29'30'31 and we refer to these publications only as required.

A B S O R P T I O N i I E S O R P T 1 O N , I R A N S P O R I

Fig. 7.14 Schemes for measuring current in polymers.

7.6.1 C O N D U C T I O N IN POLYETHYLENEPolyethylene is an extensively investigated material because of its simple structure andtechnological im portance. Selected studies are shown, without claim ing completeness, inTable 7.4.Fig. 7.15 shows the range of condu ctivity observed in PE with some recentdata added. The energy spectrum proposed by Tanaka33 for LDPE is shown in Fig.(7.16). From photon absorption measurements the separation of conduction band fromvalence band is determined as 7.35 eV. Such large band gaps are characteristics ofinsulators and fairly extensive data are available in the literature for other polymers. The

TM



38/54

probability of an electron escaping to the conduction band by thermal excitation at room-124temperature is given by exp(- 7.357 kT) -10" and this never occurs.The Fermi level is closer to the conduction band than to the valence band at 1.6 eV anddonor levels exist 1.95 eV, 2.35 eV and 2.95 eV below the Fermi level. The probabilityof an electron moving to the conduction band due to thermal excitation from the 1.95 eV

O ftlevel at 80C is ~ 10" which must be considered as negligible. Donors at such largeenergy levels are known as deep donors. Impurities, however, may have donors at closerlevels increasing the probability. Electronic traps exist below the conduction band butabove the Fermi level.The conduction band has a tail with no sharp edge, but ending near about 1 eV below theconduction band. Electrons move by hopping from site to site in the tail of theconduction band to cause thermally activated mobility. The activation energy E^ , not tobe confused with donor level, is found to be 0.24 eV and |iio is found to have a value of5 x 10"4 m2V" 1s"1 giving a mobility, at room temperature, that is too high by at least threeorder of magnitude. The difference cannot be ascribed to impurities as they are likely toincrease the discrepancy. Tanaka, et. al.34 have observed that increased crystallinity ofLDPE makes electron transport easier, which may explain the discrepancy.Another factor is that the pulse measurement gives a higher value of the mobility whencompared with the steady state measurements [Tanaka 1973]. Kumar and Perlman(1992) point out the differences that should be taken into account while consideringcondction in LLDPE and HDPE. In LLDPE there is a lowering of the barrier for thePoole-Frenkel mechanism to be effective.In FIDPE a larger boundary concentration results in a greater number of trapped charges.The effective field is lower at the boundaries in the steady state, resulting in negligiblefield lowering. In LDPE mobility in amorphous regions is field independent. In HDPE ahigh degree of branching gives rise to high concentration of defects in the amorphousregions ~ 1026 m"3. These defects become major hopping sites and the mobility is fielddependent.

7.6.2 C O N D U C T I O N IN F L U O R O P O L Y M E R SW e consider poly(tetrafluoroethylene) and also a co-polymer of tetrafluoroethylene andethylene, the latter having a trade name of Tefzel by Dupont Nemours Inc. Both arehigh temperature materials with desirable mechanical and thermal properties.

TM



39/54

-iTable 7.4Electrical conduction in polyethyleneUnits: E = MV m"1, U o - m V ' , N t = m " , s = eV.Unless otherwise stated, current -field measuring technique is employed.

Major Conclusions ReferenceVan Roggen"3^1962).36

Tunneling mechanism in single crystal polyethylen eThermally activated ionic con duction, ca = 0.84 - 1.06 eV,N t = 1 . 0 x 1026Space charge in bulk increases cathode-dielectric potentialbarrier, E = 2.63- 13.2, sa = 1.11 -1.45Technique: Surface charge decay; hopping transportm echanism ; Iodine doping increases j o , significantly. |o,0 = Davies"38 (1972)7x 1 0 ' 4 - 5 . 8 x l 0 2 , N t = 1 . 2 x 1019-9 xlO 27 ' sa = 0.92- 1.2Poole-Frenkel mechanism in LDPE

Lawson , (1965)Taylor and Lewis37(1971)

38

Das-Gupta and39Deep donors at 1.95, 2.35 & 2.95 eV below conductionband, ja ,0 = 5 x 10" 4, E^ = 0.24 , Poole-Frenkel m echanismwith hopping at 25C< T< 80CSpace charge lim ited current possibly w ith Poole-FrenkeleffectTrapping of free charge carriers at 27C< T< 90C, sa =1.0 eVTechniq ue: Absorption current; E > 1 MVm"1 , Chargecarrier hoping process in LDPETechniqu e: Absorption and desorption currents, Spacecharge limited currents; distributed energy of trapsTrapping of injected charge carriers, hopping due to fielddependent m obility, X = 2.2 nm, N t = 9.4 x 1025, ea = 1.15eVSemi-empirical theory in divergent fieldsOhmic conduction below 50C and SCLC above, inLDPE. Removal of low m olecular wt. Species increases a

BarbarezJ5 '(1973)Tanaka4 0(1973)Tanaka &Calderwood4 1 , (1974)Adamec&Calderwood 4 2 (1978)Das-Gupta andBrockley4 3 (1978)Pelissou et. al.(1988)4 4Kum ar and Perlman4 5(1992)Boggs4 6(1995)L ee et. al. (1997)47

(A) PTFECompared to polymers like polyethylene and PVAc the number of investigationspublished on PTFE are relatively few and a brief summary is in order. Lilly and

TM



40/54

AOMcDowell measured conduction currents in 254 urn films at fields in the range of 7 -12 MVm"1, at just three temperatures of 162.5, 181.5 and 200C using electrodes havingan area of 46 x 10"4 m2.The measured current is observed to vary with the field accordingto yE1/2 where y is a geometric constant which is unity for absolutely parallel planeelectrodes without an y fringing effects (Table 7.5).They concluded that the mechanismwas both ionic and electronic, each dominant at different electric field and eachinfluenced by space charge.

Fig. 7.15 Field strength dependence of conductivity in LDPE at about 50C (dashed curves atroom temperature according to: A, Lawson (1965); B, Stetter (1967),; C, Bradwell, et al (1971);D, Taylor and Lewis (1971); E, Das Gupta and Barbarez (1973); F, Cooper, et al (1973); G, Rohland Fischer (1973); H, Beyer and Von O lshausen (1974); I, Garton and Parkmaan (1976); J,Johnsen and Weber (1978); K, Das Gupta and Brockley (1978); L, Adamec and Calderwood(1981); M, Nath and Perlman (1990), N, Lee, et al (1997). (Adamec and Calderwood, 1981; withpermission of Inst. of Phys.), supplemented with the data of Nath and Lee.

TM



41/54

E ,

ffii . o

2.0

Band Tail (Localized States)

33. 0

4. 0

7.35

Conduction Band (Extended States)

Electron Traps(almost- empty)

Fermi Level

Donors(a lmost f i l l e d )

Valence Band

Fig. 7.16 Proposed energy spectrum for polyethylene. It does not claim to be quantitative forthe density of states (Tanaka, 1973, with permission of J. Appl. Phys.)Vollmann and Poll4 9 investigated currents in radio frequency sputtered thin films 100 -500 nm thick deposited by different methods on different electrode materials but theinvestigations were lim ited to room tem perature. O ver the whole range of electric fieldsand film thickness investigated, no dependence of current on the electrode material isobserved. Method of polymerization also does not influence the currents. The results areinterpreted in terms of a modified PooleFrenkel effect for dielectrics w ith high impuritycontent.Fig. 7.17 shows the currents measured by Raju and Sussi50 in PTFE in the temperaturerange of 40 - 200C. For E < 4 MVm"1 and T < 100C O hmic conduction is observed inagreement with Vollmann and Poll (1975) in the same temperature range. For higherelectric field strengths ionic conduction, according to equation (7.25), shows reasonable

TM



42/54

1 /Ofit and the hopping distance between ionic sites may be evaluated. Though logl - Eplots yield straight lines with better fit, the Schottky injection or Poole-Frenkelmechanism is not plausible due to the fact that the dielectric constant evaluatedaccording to equation (7.31) is too high.Muller51 and Lilly, et al. (1968) suggested that the space charge alters the electric fieldwithin the bulk which in turn influences the conduction mechanism. The electric field isexpressed as yE where y, called the field modification factor, is a geometric factor an dE is the geometric electric field. Muller observed that y could be as high as 139. Lillyand McDowell found values less than one in Mylar (PET) and values in the range of 13.4- 25.6 in PTFE. The fit for ionic conduction could be improved by assuming a spacecharge modified field. Summary of results for films of two thickness are given in Table7.5. The hopping distance appears to be a function of temperature and electric fieldwhich has also been observed in other polymers.

(B) TE F Z E L As far as the author is aware there is only one study carried out on conductionmechanism in this high temperature dielectric52. W e recall, from chapter 6, that theabsorption current an d desorption current in Tefzel are not mirror images and thedifference between them is defined as the transport current for which the equations inthis chapter are developed. We also recall that evidence was produced for hetero spacecharge effects at the electrodes resulting in a field intensification.Fig. 7.18 shows the log I - 1/T plots at various electric fields and the slope decreases atlower temperatures. This is considered to be due to increased accumulation of chargesleading to greater field intensification. Tunneling mechanism is unlikely due to the factthat the current is temperature dependent. Ionic conduction, equation (7.25), is alsounlikely because log I - E plots are not straight lines. Schottky theory, equation (7.30),appears to be valid, particularly if space charge effects are taken into account bysubstituting yE in place of E in the exponential term. Whenever Schottky injection isinvoked the Poole-Frenkel mechanism should be verified as both mechanisms predict alog I - E 1/2 linearity. The dielectric constant, E O O , calculated using both theories areshown inTable 7.6.A comparison of the dielectric constant data shown in Fig. 6.22 shows that Schottkymodel applies below 100C and for higher temperatures the Poole-Frenkel mechanism isa reasonable theory. The field modification factor, y, is also in the same range as thoseobtained in polyethylene and polyimide.

TM

http://dk2041_06.pdf/http://dk2041_06.pdf/http://dk2041_06.pdf/


43/54

1 1 1 1 1 1 1 10 25 50 75 100 125 150 175 200 225 250

E( kV/on)

Fig. 7.17 Conduction currents in PTFE at various temperatures (Sussi and Raju, 1990).7.6.3 AROMATIC POLYIMIDE

The aromatic polyimide is used in many electronic components such as integratedcircuits and thin film substrates. Several grades such as device grade, corona resistant,Kapton, and Kapton-H film are available with resistivity varying in the range of 109- 10

17Qm over a temperature of 23 - 350C.

Table 7.5Hopping distance an d field modification factor in PTFE

ThicknessT(C)406080100120140160180200

50 umX(nm)3.423.914.554.091.781.070.680.560.30

y1.001.481.171.030.130.100.020.010.01

130 umA,(nm)3.025.70

9.00, (2.69a)5.57, (1.22a)

0.47a

y1.0

2.552.760.62

Electric field > l O M V m ' 1

TM



44/54

-7

-8

-9g-1060 -11

-12-13-14

200 C

4 5 61/2 s-mr-rrl -.1/2E ,(MV/m)

Fig. 7.18 Variation of transport currents in Tefzel with E1/^ at various constant temperatures(W u and Raju, 1995, with permission of IEEE)Table 7.6

Comparison of S O D derived from Schottky and the Poole-Frenkel model (Wu andRaju, 1995).T(C)50708090100150200

SSch2.342.972.981.480.820.570.65

Y1.070.950.951.341.812.162.03

8p_p9.3911.9111.935.913.272.292.58

In device grade polyimide Smith, et. al.53 observed that the current varied according to t"", with 0.6 < n < 0.8 in the time range of 1 - 16,000 s and temperature range of 23 -125C. In contrast Hanscomb and Calderwood (1973) obtained 0.2 < n < 0.6 at shorttimes in the range 80us < t < 2 ms and temperature range of 150 - 175C.

TM



45/54

The Debye theory of dipolar relaxation assumes that the dipoles are independent androtate like a rigid body leading to the expression "rigid rotator". Smith, et. al. (1987)have developed a molecular dipole theory in which the single-phonon-assisted tunnelingbetween traps occurs at low electric fields (E < 10 MVm" 1). A phonon is a fundamentalconcept of energy equal to the quantum of energy of lattice vibration in a crystal and is

f O 13equal to h v p h 0 n o n - The phonon frequency is between 10 " 10 Hz.The polarization current due to phonon assisted tunneling is derived by Smith et. al.(1987) as

where A is a constant with the dimension of (length)3, Nt the number density of traps andn the index of current decay. For polyimide the index is found to be a function of

temperature according to n(T) = a bT = 1.4 - 0.016 T. By substituting typical valuesfor polyimide, A = 1.7 x 10"23 m 3, X = 10 nm, Nt = 4 x 1023 m"3, the frequency isobtained as 0.016 Hz. Obviously this cannot be phonon frequency and Smith et. al.suggest that polarization can still occur due to a hopping charge carrier or a defect ratherthan phonon assisted, explaining the low frequency.A comparison of results of several publications (Table 7.7) on the transport currentcovers a wide range of different polyimides, experimental conditions, method of samplepreparation, etc.Some general conclusions may be drawn. Generally the low fieldtransport current varies linearly with electric field, with a temperature activation energyin the range of 0.4 eV to about 1.4 eV. The linear regime leads to a dependence of LogI o c E or E 1 /2 at high electric fields. The activation energy is also observed to increase athigher temperatures (fig. 7.19).The decreasing slope at longer times and increasing slope at higher temperatures may beindicative of ionic transport and the question that arises is whether phonon assistedtunneling, which is invoked to explain absorption current, also explains the conductioncurrents. The current density in this case is given by

J = 2 eM T N t v,Q X p - - - i n h (7.86)' p kT \6AkT 2kT

TM



46/54

i54where A is the energy difference between two trapping levels. Swamy, et. al haveshown that the measured current agrees with equation (7.86) confirming the phononmodel both for polarization and transport.

7.6.4 A R O M A T I C P O L Y A M I D EAromatic polyamide has excellent dielectric properties and it is a candidate material forhigh voltage, high temperature applications. It is available as continuous filament yarnand as paper, which is made from a mixture of short length staple fibers and smallfibrous particles called fibrids. However its poor mechanical strength has led to thedevelopment of films that have Young's modulous that is higher than many heatresisting polymers such as PET, PEEK and PI55. The dielectric constant and loss factorof the Polyamide films are shown in Fig. 7-20.The conduction currents in aramid polyamide (paper) has been reported by Raju56 asshown inFig. 7-21. The mechanism of conduction has been shown to be ionic in naturewith an activation energy in the range of 1.2 to 1.6 eV . Conduction occurs as a result ofhopping between sites, the hopping distance being calculated as 4.4-8 nm. Themechanism of conduction in linear polyamides (nylons) has been shown to be due toprotons with an activation energy of 1.29 eV 57.

ire;,. 400 300 200 100 SO 25ITII I I I I.- 8TDA-OQA/MPCAP M Q A - O D A

? O*

1.5 2.0 2.5 3.0J O O O / 7 U ) 3.5Fig. 7.19 Transport current in device grade polyimide: resistivity vs temperature (Ohmicbehavior assum ed for the high temperature data (Sm ith, et. al. 1987, with permission of JournalElectronic. Mat.)

TM



47/54

5 6s

3 -

@ 1 kH z

PET Film/

8

7

6 g5 1.91

50 100 150Temperature (C)

200

Fig. 7.20 Temperature dependence of relative permittivity and dielectric dissipation factor ofaramid film specimens (Yasufuku, 1995, with permission of IEEE).

180 C

-10

-11

ISO"

0 10 20E ( X 1 0 "vm"1'

Fig. 7.21 Isochronal currents as a function of applied electric field at various constanttemperature for 127 mm thick aramid paper (Raju, 1992, with permission of IEEE).

TM



48/54

The energy of a hydrogen bond in polyamide is 0.22 eV and the energy required for self^ "^ ^ ~ " ^ I o o 1 1ionization is 0.75 eV. Assuming a mobility of 10" m V" s" and using the measured

18 " \ __ _conductivity, the concentration of ions is evaluated as 5 x 10 m . Though this is anorder of magnitude lower than that in other polyamides, a justification is provided on thebasis that the aromatic content of aramid paper is high, with 85% of carbon replaced bythe benzene ring.

7.7 NUMERICAL COMPUTATIONAdvent of powerful personal computers has become a new tool to evaluate the I-Vcharacteristics and the temporal variation of space charge within the thickness of thematerial. While great progress has been achieved in simulation methods in gaseousdischarges (ch. 9) the literature on numerical computation in solid dielectrics is rathersparse58'59'60. The current through a dielectric consists of three components; the transportor conduction current, the polarization current and the displacement current. Assumingthat the continuity equations are applicable the time dependent current density is givenby

e0er ~^E(x9f) + jp(x,t) + jc(x,t) (7.87),t)-eD n(x,t) (7.88)ox

(7.89)E(x,t)

kTkT

ju0E(x,t) (7.90)

The symbol (x, t) in these equations means that the denoted parameter is a function ofspace and time co-ordinates, D the diffusion co-efficient, P the polarization, Ps thesteady state polarization, n the number density of charge carriers, N the number densityof dipoles and f o , 0 the permanent dipole moment.These equations are apparently intimidating though they are relatively easy to formulateand program for numerical solution. Some explanations of these equations follow.Equation (7.87) has three terms on the right side. The first term is the displacementcurrent C dV/dt and decreases to a relatively small value at large t. The second term isthe current due to polarization and the last term is the conduction current component.Equation (7.88) is the conduction current where the first term on the right side is the

TM



49/54

transport or drift term and the second term is the diffusion term. The mobility is alwaysassigned a positive sign because the charge carriers, whether electrons (moving againstE) or holes (moving along E) generate current in the same direction. The diffusion termis negative because diffusion occurs from a region of higher density gradient to lowergradient.The polarization current, equation (7.89), is identical to eq. (3.11) and becomes zero inthe steady state. Equation (7.90) is the Langevin function, identical to eq. (2.48). Whenthe parameter uE/kT1 the Langevin function may be approximated to 1/3 (uE/kT)though this is not required in numerical computations.The electric field is computed by applying Poisson's equation at each co-ordinate

-^ [ s 0 r E ( x , t) + P(x, t ) ] = en(x, t) (7.91)dx

The electrical field should satisfy the condition

Ead (7.92)

where d is the thickness of the material and E a the applied electric field. The Schottkyemission, equation (7.30), is assumed and the boundary condition is n(jc, 0) = P(x, 0) =0.The known (experimental) parameters are Ea, T and d . The material parameters are N,J I G and sr. The parameters determined are |u (m 2 V'V1), T (s) and A in the Schottkyequation.The simulation is applied to poly(vinylidene fluoride) (PVDF) assuming the followingparameters: N = 2 x 1028 m'3, fi0 = 7 x 10"30 C m , sr = 2.2, T = 353 K, Ea= 30 MV/m andd = 15 um. A relaxation time of 26 s (mechanical p-relaxation) and a mobility of 3 x 10"12 m 2 V'V1 is obtained. A comparison between experimental an d theoretical currents isshown in fig. 7. 21 . Dipole motion is found to affect E considerably, which in turnaffects j c and J except at low values of T. The J - t profiles depend on the dipolarrelaxation time an d carrier mobility, an d differ from those in non-polar dielectrics.

TM



50/54

10'4

ID'5

10'6

1 n 1 ii I r-r-

_,

OI *0 ;

0 _O

: t >- o exp.. o calc.I , , i I i l_l 1 I_L_

Fig. 7.22 The experimental (open circle) and thecalculated J-E characteristics (closed circles) ofPVDF after 4 80s charging at 353 K (K aneko et. al.,1992, with permission of Jpn. J. Appl. Phys.)10e 107 10Fa ( V / m )

,8

Table 7.7Summary of selected conduction studies in aromatic polyimideT (K) E (MV/m) A ( e V ) Proposed mechanism423-623323-398393-453273-473323-543298-62310-500295

2-8 x 10< 0 . 40.4-505-454-401-200-1.5>140

-3 Thermally assisted tunneling9'10Proton conduction11Ionic conduction12Ionic conduction13Schottky injection modified byspace charge layer140.5 eV (298


51/54

7.8 CLOSING REMARKSFor a number of years it has been recognized that conduction currents in a polymer areaffected by the molecular weight and its distribution, and by density, morphology andcrystallinity. In general, electrical conductivity decreases with increasing molecularweight, which in turn is related to a decrease in free volume. Increase in viscosity andintermolecular forces are also associated with the increase in molecular weight andtherefore in resistivity. In general conductivity tends to increase with increasingspherulite density and diameter. A regularity of structure and increasing crystallinity willdecrease in ionic conductivity (Das Gupta, 1997). Mizutani and leda (1984) haveproposed a relationship between fluorination and conductivity; increase of fluorinesubstituent decreases the current. PP, PVF, PE, PVC and PCTFE have no or lowerfluorine content and the conduction currents, due to negative carrier, are higher. On theother hand FEP, ETFE, PTFE, PVDF have lower currents, due to positive carriers.Boggs61 has provided a collection of conduction formulas which are convenient to applyto experimental results, with the electric field and temperature as parameters. The needfor careful and reproducible experimental results cannot be overstressed. Wintle62 (1994)has drawn attention to the need for caution in interpreting conduction data. Theoreticalstudies relating the free volume and conductivity help to understand the role ofmorphology. Further development of models for conduction in cables should be aimedat computation of microscopic parameters in the dielectric so that a comparison with aninitial set of values may help to estimate the deterioration of the dielectric (Boggs,1995).

TM



52/54

7.9 REFERENCES1 H. L. Saums and W. W. Pendleton, "Materials for Electrical Insulating andDielectric Functions", Hayden Book Co., 1978.2 R. Eisberg and R. Resnick, "Quantum Physics of atoms, molecules, solids, Nucleiand Particles", II edition, John Wiley and Sons, New York, 1985, p. 388.3 T. Tanaka and J. H. Calderwood, J. Phys. D., 7 (1974) 5.4 S. H. Glarum, J. Phys. Chem. Solids, 24 (1963) 1963.5 J. J. Ritzko, in Electronic Properties of Polymers, Chapter 2, Ed: J. Mort and G.Pfister, John Wiley and Sons., New York 1982.6 L. A. Dissado and J C. Fothergill, Electrical Degradation and Breakdown in Polymers,Peter Peregrinus Ltd., London, 1992.7 J. J. O'Dwyer, "The Theory of Electrical Conduction and Breakdown in SolidDielectrics", Clarendon Press, Oxford, 1973.8 T. Mizutani and leda, IEEE Trans. Elec. Insul., 21 (1986) 833.9 D. A. Seanor, "Electrical Properties of Polymers", Polymer Science, Vol. 2, Ed: A. D.Jenkins, North Holland Publishing Co., Amsterdam, 1972.C. C. K u and R. Liepins, "Electrical Properties of Polymers", Hanser Publishers,Munich, 1987, p. 213.B. Hilczer and J. Malecki, Electrets, Elsevier, Amsterdam (1986).10 T. J. Lewis, Proc. Phys. Soc. (London), B 67 (1956) pp. 187-200.11 Y. Miyoshi and K. Chino, Japanese Journal of Applied Physics, 6 (1967) 181.12 A. C. Lily, Jr. and J. R. McDowell, J. Appl. Phys., 39 (1968) 141.13J. Frenkel, Phys. Rev., 54 (1938) 647.14 D. W. Kim and K. Yoshino, J. Phys. D., Appl. Phys., 33 (2000) 464.15 P. Karanja and R. Nath, IEEE Trans. Diel. Elec. Insul., 1 (1994) 213.16 R. A. Foss and W. Dannhauser, J. Appl. Poly. Sci., 7 (1963) 1015.17 R. Nath, T. Kaura and M. M. Perlman, IEEE Trans, on Elec. Insul., 25 (1990) 419.18 M . leda, G. Sawa and S. Kato, J. Appl. Phys., 4 2 (1971) 3733.19 R. Nath, T. Kaura and M. M. Perlman, IEEE Trans. Elec. Insul., 25 (1990) 419.20 D. K. Das Gupta and M. K. Barbarez, J. Phys. D., Appl. Phys., 6 (1971) 867.21 J. Kommandeur and W. G. Schneider, J. Chem. Phys., 28 (1958) 582.N. F. Mott and R. W. Gurney, "Electronic Processes in Ionic Crystals", Oxford,Clarendon Press, 1940.23 A. Rose, Phys. Rev., 97 (1955) 1538.24 D. Ma, I. A. Hummelgen, B. Hu and F. E. Kariasz, J. Phys. D: Appl. Phys., 32 (1999)2568.25

26 J. Chutia and K. Barua, J. Phys. D: Appl. Phys., 13 (1980) L9.H. R. Zeller, IEEE Trans, on Elec. Insul., EI-22 (1987) 215.Y. Tanaka, N. Ohnuma,26, pp. 258 - 265, 1991.

97 -r-,Y. Tanaka, N. Ohnuma, K. Katsunami and Y Ohki, IEEE Trans, on Elec. Insul., Vol.

TM

http://dk2041_02.pdf/http://dk2041_02.pdf/http://dk2041_02.pdf/http://dk2041_02.pdf/


53/54

28 M. M. Perlman and A. Kumar, J. Appl. Phys., 72 (1992) 5265.29 (a) H. J. Wintle, Engineering Dielectrics, Vol. IIA, Ed. R. Bartnikas and R. M.Eichhorn,ch. 6(1983)588.(b) H. J. Wintle, IEEE Trans. Diel. Elec. Insul., 6 (1999) 1 .30 M. leda, IEEE Trans. Elec. Insul., 21 (1986) 833.

31 D. K. Das-Gupta, IEEE Trans. Diel. Elec. Insul., 4 (1997) 149.32 V. Adamec and J. H. Calderwood, J. Phys.D: Appl. Phys., 14 (1981) 1487.33 T. Tanaka, J. Appl. Phys., 44 (1973) 2431.34 Y. Tanaka, N. O hnuma, K. Katsunami and Y Ohki, IEEE Trans, on Elec. Insul., 26(1991)258.35 A. van Roggen, Phys. Rev. Lett., 9 (1962) 368.36 W. G. Lawson, Br. J. Appl. Phys., 16 (1965) 1805.37 D. M. Taylor and T. J. Lewis, J. Phys. D: Appl. Phys., 4 (1971) 1346.38 D. K. Davies, J. Phys. D: Appl. Phys., 5 (1972) 162.

39 D. K. Das-Gupta and M. K. Barbarez, J. Phys. D: Appl. Phys., 6 (1973) 867.4 0 T. Tanaka, J. Appl. Phys., 44 (1973) 2430.4 1 T. Tanaka and J. H. Calderwood, J. Phys. D: Appl. Phys., 7 (1974) 1295 .4 2 V. Adamec and J. H. Calderwood, J. Phys. D: Appl. Phys., 14 (1981) 1487.4 3 D. K. Das-Gupta and R. S. Brockley, J. Phys. D: Appl. Phys., 11 (1978) 955-962.4 4 S. Pelissou, H. St. Onge, M. R. Wertheimer, IEEE Trans, on Elec. Insul., EI-23(1988), 325.45 A. Kumar and M. M. Perlman, J. Appl. Phys., 71 (1992) 735.4 6 S. A. Boggs, IEEE Transactions, EI-2 (1995) 97.4 7 S. H. Lee, J. Park, C. R. Lee and K . S. Suh, IEEE Trans. Diel. El. Insul., 4 (1997) 425.4 8 A. C. Lily, Jr. and J. R. McDowell, J. Appl. Phys., 39 (1968) 141-168.4 9 W . Vollmann and H. U. Poll, Thin Solid Films, 26 (1975) 201.50 G. R. Govinda Raju and M. A. Sussi, CEIDP Annual Report, 1990, pp. 28 - 31.51 R. S. Miiller, 34 (1963) 2401.52 Z. L. Wu and G. R. Govinda Raju, IEEE Trans. Elec. Insul., 2 (1995) 475.53 F. W. Smith, H. J. Neuhaus, S. D. Senturia, Z. Fen, D. R. Day and T. J. Lewis, Jour.545556575859

Electronic Materials, 16 (1987) 93.Alagiri Swamy, K. S. Narayan and G. R. Govinda Raju: unpublished.S. Yasufuku, IEEE Electrical Insulation Magazine, 11 (6), (1995) 27.G. R. Govinda Raju, IEEE Trans, on Elec. Insul., 27 (1992) 162.W. O . Baker and W. A. Yager, J. Amer. Chem. Soc., 64 (1942) 2171.H. J. Wintle, J. Appl. Phys., 63 (1988) 1705.K . Kaneko, Y. Suzuoki, T. Mizutani and M. leda, Jpn. J. Appl. Phys., 29 (1990) 1506.60 K . Kaneko, Y. Suzuoki, Y. Yokota, T. Mizutani and M. leda, Jpn. J. Appl. Phys., 31(1992)3615 .61 S. A. Boggs, IEEE Trans. Diel. Elec. Insu., 2 (1995) 97.

TM



54/54

62 H. J. Win tle, IEEE Electrical Insu lation Magazine, 10, NO . 6, (1994) 17.

field enhanced conduction

Documents