Frequency-dependence of conductivity in hopping systems

JOURNAL OF NON-CRYSTALLINESOLIDS 8-10 (1972) 293-315 © North-Holland Publishing Co.

FREQUENCY-DEPENDENCE OF CONDUCTIVITY IN HOPPING

SYSTEMS

A. K. JONSCHER

Department of Physics, Chelsea College, Pulton Place, London S.W.6, England A critical review is given of the experimental evidence relating to the frequency dependence of the electrical conductivity, a (to), in solids in which the flow of current occurs by hopping of localized carriers, whether electrons or ions. It is pointed out that the same dependence is found in semiconducting amorphous systems and in a wide class of insulators, giving a oc con, where 0.5 < n < 1, depending upon temperature. Evidence for a region in which oc to2 is considered to be of dubious reliability. It is pointed out that the conventional theoretical approach to the treatment of the frequency dependence of conductivity leaves certain gaps in our understanding of the physical processes taking place. An alternative approach is developed in which a stochastic sequence of hopping current pulses is Fourier-analyzed to give a frequency dependence of the power-law type found experimentally. The analysis leads to the conclusion that no exponent significantly higher than unity is to be expected - in agreement with all available evidence, and the co2 region is possible as a consequence of the dielectric relaxation in the host matrix in which hopping takes place or of the corresponding hopping process by carriers tightly bound to fixed sites in the matrix. The new theoretical approach enables high-field ac behaviour to be analyzed and experimental results are given to illustrate the discussion. It is also pointed out that the frequency dependence of conductivity is an essential consequence of hopping conduction and not of the disorder of the structure. It is therefore predicted theoretically that similar results should be obtainable in ordered systems in which hopping is known to take place and some experimental results are quoted to support this claim.

I. Introduction The frequency dependence of the electrical conductivity, a(eJ), of n o n metallic solids has been the subject of detailed experimental a n d theoretical investigations by two largely i n d e p e n d e n t a n d separate groups of people w h o m we shall describe as the "dielectrics school" a n d the " s e m i c o n d u c t o r school". The two approaches are rooted in the fact that the electrical conductivity i n a n alternating field m a y be due to two basically different causes: (1) a r o t a t i o n a l m o t i o n o f molecular dipoles with frictional losses, a n d (2) a t r a n s l a t i o n a l m o t i o n of free electric charges. We note that the former m e c h a n i s m c a n n o t give rise to dc conductivity a o, while the latter can. The concept of rotating dipoles has been applied by Debye 1, 2) to explain 293

294

A.K.JONSCHER

the dispersion of ac properties with frequency in liquids containing polar molecules. A single species of non-interacting dipoles would have a complex dielectric susceptibility X* = X' - ix" = B/(1 + icor), (1) where B is a constant and the "relaxation time" ~ is related to the frictional torque restraining the motion of the dipoles. The electrical conductivity becomes

CO2T2 ,, (co) = ,ocoX" (co) =

- -

1 q- 032152"

(2)

Another characteristic parameter frequently employed in the context of dielectrics is the loss tangent,

x"(co) tan6-

e'

e~+X'(co)'

(3)

where ~' and e" are the real and imaginary parts of the complex relative dielectric permittivity of the medium and eoo is the residual value of e' at frequencies which may be considered "infinitely high" for the dipolar mechanism under consideration. It is found that a very wide range of dielectric materials of practical interest possess a loss tangent which is independent of frequency in a wide range of frequencies, coupled with a relatively slight variation of e' z- 6). This implies that X" = constant and hence a (co) oc09. This behaviour is explained in the dielectric context in terms of a "distribution of relaxation times" 7, 8). The alternative approach adopted by the semiconducting school considers the motion of "free" charges in dielectric and semiconducting solids. The charges are considered to move by discontinuous "hopping" jumps between well defined localized sites within the solid, spending most of the time at rest on these sites. This hopping conductivity should be clearly distinguished from "free band" conduction and its frequency dependence may be summarized with reference to fig. la. At sufficiently low frequencies and down to dc there is a constant level go, which may or may not be accessible experimentally within the limitations of lowfield conditions, the only conditions in which conductivity may meaningfully be defined. There follows a region in which conductivity is a monotonically increasing function of frequency with an empirical relation ~r oc co",

(4)

where n takes typically values in the range 0.5 < n < 1. This region is relatively well documented experimentally for a wide range of materials which all have one common characteristic, that conduction in

FREQUENCY-DEPENDENCEOF CONDUCTIVITY

ffo~ O" .log I

IdcI

//

, / x log ~_

~_~

295

I/~ '

""-...

~ocj uJ

~ogt

~'= ~-/1+~.,:~

Fig. 1. (a) Schematic diagram of the frequency dependence of hopping conductivity, tro is the dc conductivity. (b) The corresponding time-domain graph of the current l(t) in response to unit step-function excitation. (c) The corresponding frequency dependence of the real and imaginary parts of permittivity e* -- e' -- e".

them occurs by hopping 9 - n ) as distinct from free band conduction. The first report is due to Pollak and Geballe 12) who have studied the ac properties of compensated silicon at very low temperatures - a classic example of a hopping system. Since then many more disordered and amorphous or glassy materials have been found to exhibit the same type of behaviour, for example SiOx and A120313) reproduced in fig. 2, the chalcogenide glassy system As2Se314-16) reproduced after Fritzsche 17) in fig. 3, other chalcogenide glasses 18,19), solid polymeric CS 220), anodized ZrO 22t), C r 2 0 322) and glassy V20523) to name but a few of the many published results. The significant points to note about the low-frequency side of these results is that the exponent n in the power-law relation (4) is temperature dependent and varies between zero i.e. dc conduction at high temperatures, through 0.5 at intermediate temperatures to 1 at low temperatures around 100 K or less. No well established case of 1 < n < 2 has come to our notice, neither is there a well documented case of the corresponding tan 6 oc ~0 relation. The second significant point is that there is no well defined activation energy in these results, the same as in the dc conduction through traps 26).

296

A. K. JONSCHER i0-~

10-61 (b)

(a)

10-7

i0-l

lO-e

,.) i0-~ "i" b

10-* _T(°K)

10-~I-- /

~ 10-"

I0 -'°~-*~

/-

'

LO-" i0-q; I0 z

I0 ~

104

Frequenc~ f

(c/s)

I0 s

I0 6

i0 z

105

104

Frequency,f

Ca)

~

(c/s)

I0 ~

I0 6

/C c/s?~ k

10-4 10-1

c" o

SO0

b 10-~

o 7 10-9 b

~..,~(kc/s)

~°o'~'~

--o.-..~ ==o

°"°"°" ---o.... 400

10-,c

II0

,3 I0-,o

0

'~x.

I

5 1031T (°K'l)

~---o-

I0-"

I0

2

'4

t0Vr (OK-b

6

8

Fig. 2. The dependence of the electrical conductivity on frequency and on temperature of evaporated SiOx films (a), and of anodized AI2Oafilms (b). From Argall and Jonscher 13). Some of the published data on the frequency dependence of conductivity in disordered solids show a second region in which conductivity increases quadratically with frequency and sometimes even shows a region of saturation, as for example in polymeric CS 2 2o). This, of course, is exactly the dependence expected on the basis of a simple two-centre hopping mechanism, if the "natural" hopping frequency is 1/z which leads to the same form of frequency dependence as eq. (2), as shown by Pollak and Geballe 12). Returning to fig. 1, we have plotted in diagram c the real and imaginary

297

F R E Q U E N C Y - D E P E N D E N C E OF C O N D U C T I V I T Y

parts of the complex permittivity, e' and e", corresponding to the various parts of the a(co) spectrum. We associate a Debye-type or simple hopping process with the co2 region. Diagram b in fig. 1 represents the corresponding time-domain graph of the current I(t) in response to unit step-function field excitation. ~OCj( h v ) -12

0 ~ i -~

-IO

-8

6

(eV)

-4

-2

o 0~en e, RobeHson

0

//~

i i~kln orld Kolomle~s

2

-- Taylor B~hop & Mdtchelt / /

(~,cm)-~

0 ~0g (n=)

4 ~

-2

-rG

l .................. 0

2

4

kT (50OK) 6

8

IO

gocj (frequency)

12

14

t6

(Hz)

(crn-')

1::

Fig. 3. The ac conductivity of amorphous As2Se3 over a wide frequency range. The right hand scale shows the product of refractive index n and absorption coefficient ~. The dashed line represents an co°.9 dependence. Data above f = 1014 Hz and open circles are by Owen and Robertsonl4), crosses by Ivkin and KolomietslS), and microwave and infrared data are by Taylor et al. la). From Fritzsche17).

The current I(t) is related to the dielectric parameters through the Fourier transforms: oo

eoZ' (co) = [" [ I ( t ) - I~3 cos cot dt,

(5)

o oo

eoZ"(co ) = [" [I (t) - Io~] sin cot dt.

(6)

0

it should be pointed out that since the appearance of the early coz results the conviction has been hardening among researchers that most, if not all cases where they have been reported, are suspect on the grounds of possible spurious effect of even very small series resistances and inductances, which may be due to the leads external to the specimens, or may be caused by

298

A.K.JONSCHER

thin oxide layers on the electrodes giving rise to Maxwell-Wagner effects. It is virtually impossible to make absolutely certain that these spurious influences do not, in fact, dominate the behaviour. A searching analysis of this type of difficulty has been made by Fritzsche in relation to experimental results reported on As2Se 317). Maybe the results on polymeric CS 2 are an exception here, in that the relevant region corresponds to measurements not involving contacts to specimens, i.e. slotted line and X-band waveguide measurements. Although all the data quoted above relate to the ac behaviour of disordered systems, i.e. amorphous or glassy materials, or at least of crystalline materials in which the relevant localized centres are randomly distributed, as is the case with crystalline silicon, the frequency dependence of conductivity is not necessarily confined to disordered solids. There is some evidence that in multi-molecular layers of stearic acid, which represent an ordered system, hopping shows a similar type of frequency dependence, as shown in fig. 4

I... •

)::/t,::

I .!-/:. /

• ~e~,/ee •[ e, 10 -4

10 -3

I ~73K

-

1

10 -2 f

10 -1 Hz

1

10

10 -2

Fig. 4. The frequency dependence of the electrical conductivity of multi-molecular-layer films of stearic acid at various temperatures. Film thickness 250/~. From ref. 24. Each consecutive curve has been displaced upwards by one decade with respect to the previous. one to avoid crowding.

FREQUENCY-DEPENDENCE OF CONDUCTIVITY

299

based on measurements by Nathoo 84). Similarly, a strong frequency dependence was reported for crystalline transition metal oxides in which hopping occurs in an ordered array25). We shall return to the significance of these results later in this paper.

2. Previous theories of hopping conductivity It has been recognized by Pollak and Geballe and by everybody else since then that the two-centre hopping is an idealized approximation, that there must be many-centre paths in which a carrier executes a series of hops before coming to a "difficult" hop where it may be held up. In fact, the very existence of a random distribution of hopping sites in a disordered solid must give rise to a wide distribution of hopping "relaxation" times, which must follow an exponential dependence on distance between pairs of hopping centres. An impressive list of papers may be quoted elaborating and refining the fundamental idea of two-centre hopping, including those by PollakZ7-30), Mott31), Austin and Mott3z), Davis and Mott83), predicting a wide variety of frequency dependences from a = c o n s t a n t to a occo2 through a range of powers with exponents between 0 and 1. The starting point of all the treatments mentioned above is the determination, of the time-dependence of the occupanciesf andJj of a pair of centres, labelled, say, i and j, under the action of an alternating electric field of frequency co or alternatively under the action of a step-function field setting in at time t = 0. The quantities J~ and J) are understood to be the time-averaged or "smoothed" occupancies derived through ensemble averaging from the instantaneous actual occupancies, which by the very nature of the stochastic processes involved in hopping can only be either 0 or 1 depending on whether the carrier sits at site i orj. This approach appears to leave a significant gap in our real understanding of the processes in question 34) and we suggest that this situation can be remedied by the alternative approach proposed in the present paper.

3. Frequency dependence of stochastic processes Following the other authors we postulate a stochastic process in which a number of carriers, ionic or electronic, hop randomly in time between localized sites, situated in the matrix. Our treatment of transient or frequency-dependent phenomena differs from the accepted analysis in the transformation from the time-domain to the frequency-domain for stochastic processes.

300

A . K . JONSCHER

Consider first an elementary hopping process consisting of an abrupt transition of a charge q from a site i to a site j in a slab of material of thickness w with metal electrodes on both sides, fig. 5a. The induced charges Q1 and Q2 on the electrodes change as a result of the transition by the amounts AQj = - AQ2 = qrij/w ,

(7)

where rij is the distance between the two sites measured in the direction normal to the electrodes. The presence of a dielectric medium will result in a delayed response in addition to a delta-function, as shown by the schematic diagram 5b, where G(t)=a6(t)+g(t) fort>0, G (t) = 0

for t < 0,

(8)

where 9 (t) is quite generally a decaying function of time, with a characteristic decay time r, due to the finite response time of the dielectric medium.

c~ V," " / / / "

"/////'1

(b)

G(t) ~ ' N

-t Fig. 5. (a) The model of a hopping medium. (b) The time-response G(t) to a hopping event. The strength a of the delta-function is such that oo

f

G ( t ) dt = 1,

(9)

0

and the elementary current due to the hopping event at t = 0 is given by i ( t ) = ( q r i J w ) G (t) = A G (t).

(10)

301

FREQUENCY-DEPENDENCE OF CONDUCTIVITY

The contribution of the elementary hopping current pulse i(t) to the frequency-dependent dielectric parameters Z'(o9) and Z"(o9) arises in accordance with eqs. (5) and (6) through the Fourier transforms oo

c(og) =

zt~

f G(t) cosogtdt,

s(og) =

O

f G(t) sino9tdt,

(11)

0

multiplied by the appropriate amplitude A. We do not know the detailed functional form of these transforms in the absence of detailed assumptions about G(t) but this is not a fundamental limitation. We note that the transforms of the function e x p ( - t / z ) for t > 0 are as given in fig. lc. It is easily shown that the transforms of a large class of decaying time-functions, both monotonic and oscillatory, are given by very similar relations which amount to the properties that c(og) ~ const,

for

co ~ l / r ,

(12)

s(og) ~ ogz

for

co ~ 1/z.

(13)

and

In the absence of external electric fields the hopping events take place randomly all the time between all possible pairs which are accessible on account of their distances. The signal is therefore given, with reference to fig. 6a, by I (t) = ~ A m G (t - tm), (14) ra

where the index m counts the events as seen in the external circuit. With zero external field we must have ~mAm=0, taken over any sufficiently large number of consecutive events. o)

llll [ rl J,I)~lf it. IiIjj III~l I " m

E (t) im

b, °TI IlllU iI I i IllJ I I'i' i[ II IJ~ IIII t ,,F (t)

t=O

t

Fig. 6. Hopping currents as seen in the external circuit. (a) Random sequence in the absence of an electric field. (b) The effect of step-function field setting in at time t = 0. (c) Random sequence (a) subtracted from (b) leaving an ordered sequence.

302

A.K. JONSCHER

It should be understood that the index m does not count the consecutive hops of any particular carrier, since large numbers of these carriers are hopping "in parallel". In our preliminary analysis we shall make the assumption that the amplitudes A= do not vary significantly and this is how they are represented in fig. 6. The random series represented in fig. 6a gives rise to noise in the system but we do not propose to discuss this problem further at this stage. We now consider the effect of the application of a step function electric field starting at t = 0. We assume that the magnitude of the field is sufficiently small so that only linear effects need be considered, and this means

qErij "~ k T.

(15)

This also implies that the underlying stochastic processes go on with relatively little perturbation which is equivalent to the superposition on the random process of a sequence of "ordered" amplitudes, as shown in figs. 6b and 6c. The sequence of ordered amplitudes of fig. 6c will be denoted by F(t) and in the following analysis we shall be using the index m to count these amplitudes only. The basic features of the sequence F(t) are that the amplitudes are unidirectional, that the times tm are still random but with a tendency to increasing periods between subsequent events until after a sufficiently long time a quasi-steady-state is reached corresponding to dc conductivity, %. This steady-state sequence is also subtracted from the function F(t) which therefore has no hopping events at t ~ m. The initial "rush" of hopping events following t = 0 is due to the fact that most available carriers in the system respond initially to the electric field by hopping preferentially in the direction of the applied force but their movement is gradually impeded by more difficult hops due to longer hopping distances or to "piling up" of other carriers behind one that has been so held up. The central problem of the present paper is to obtain the frequencytransform of the para-random sequence of event F(t) which follows the application of a unit-step field at t = 0. To this end we note that a delayed event G ( t - tin) has Fourier transforms COtm -- S(¢.O) sin tot,,

(16)

s (to; t=) = s (to) cos tot= + c (co) sin tot=.

(17)

C(to; tin) = C(to)

COS

and Hence an event displaced by time t= with respect to t = 0 has Fourier transforms which consist of the Fourier transforms c(to) and s(to) of the nondisplaced event G(t)modulated by the functions sin tot= and cos tot=, as shown schematically in fig. 7. We note that events delayed by t= < z have a

303

FREQUENCY-DEPENDENCE OF CONDUCTIVITY

log c~) (a) log s(-)

G (t) ~x, ~t

(b)

b..

Fig. 7.

"4

Fourier transforms of the undelayed event G(t), (a), and of the delayed event G(t - tin), (b) and (c).

virtually unchanged frequency response,

C(¢D; tm)~C(O~), S(OO; tr,)~S(09), tin
2n/tr,'~1/~.

~3

eoZ'(o~)=f F(t)cosogtdt= .,=,~Amc(og;t,.), o

(19)

304

A.K.JONSCHER ao

e°Z"(09)=f F(t) sin09tdt=,.=l ~ A,.s(09;tm).

(20)

0

The summation will be carried out in the first instance by assuming that all the amplitudes A m are equal to A. We note that for any delay t m there is a range of frequency 09 < 1~tin for which eq. (20) gives the approximate relation: eoZ"(09) ~ A

M

M

~

[ s (09) + 09tmC (09)] ~ A09 ~

m=l

tm,

09 = l i t M . (21)

m=l

This is illustrated in fig. 8a which shows a number of sin 09tin waves corresponding to a range of tm. For any given frequency 09, a value of m = M is found such that t M,~ 1/09. For values of m < M the sine-waves change slowly in the frequency interval up to 09 and may be approximated by eq. (21). For values of m > M, however, the waves oscillate rapidly and average out without contributing to the sum - this explains why it is only necessary to sum the first M waves. To illustrate the principle involved let us compute the conductivity for three simple sequences of firing times, shown in fig. 8b : linear

tm = a m ,

(22)

quadratic

tm= am 2 ,

(23)

exponential

tm = a e x p b m .

(24)

(a) i/t

th,,nat

/'c~

(b)

I

I

m = 1,2,3

t,=am

I I

I

..... I

I

IIIIIIlll

/

L..= a m z

t. =expbm

I[I I

IIII

lx'

I

t "t

"t Fig. 8. (a) Illustrating the summation of sine-waves of varying "periods" 2nitro on the frequency axis. (b) Three simple sequences of firing times- dc or linear: tm ~ am; quadratic: t m = am2; exponential: t m ~ exp bm.

FREQUENCY-DEPENDENCE OF CONDUCTIVITY

305

The first one is the trivial case of dc conductivity without dispersion: M

a = Aao92

~

m = Aao92M (M +

1)/2 ~ A a o 9 2 M 2 / 2 .

m=l

Bearing in mind the fact that co -- 1 ~ a M , this gives the frequency-independent answer one would expect. The quadratic distribution of firing times gives the o90.5 relation frequently found experimentally at relatively higher temperatures, while the exponential distribution of firing times gives the limiting law aocog. It is readily shown that the general power-law distribution k > 1,

t,, = a m k,

(25)

leads to the conductivity dependence a (co) oc o91- 1/k,

(26)

and the exponential distribution given by eq. (33) is seen as the limiting case of k>> 1. We have not succeeded in finding a physically plausible distribution law of firing events which would lead to the power-law dependence of conductivity of the type a(og)ocog" with n > 1. We suggest that the region n > 1 is "forbidden" in our mechanism and we think that this is borne out by experimental data, which show no slopes of the log a versus log co graphs significantly in excess of 1, except for the alleged o92 dependence, to which we shall return in the following section. 4. The stochastic nature of the hopping process

Our hopping system consists of a distribution in energy and in space of localized centres which are partially occupied by carriers - electrons, to be definite. These carriers make transitions between pairs of centres, principally the nearest neighbours, but more distant pairs may also participate 26, 31, as). Taking all possible pairs, i, j of localized sites in the system, we may associate with each pair a transition probability AW~; pij = exp

-

kT

) - 2ctRij

'

(27)

where A ~ j = Wj- W~ is the energy difference between the two sites and R i j is their spatial separation. ~ is the reciprocal decay distance of the wave function associated with site i.

306

A.K.JONSCHER

In view of the fact that both AW~j and Rij fluctuate in fairly wide limits, the transition probabilities Pu form a very wide distribution, with many orders of magnitude of difference between the most likely ones and the least likely ones. A carrier on site i will, on average take a time At u to hop to site j, provided the latter is unoccupied, with

1~Atu = O)dPij,

(28)

where 09d is a suitable "attempt-to-jump" frequency. A small applied field changes the probabilities slightly, making "downfield" transitions more likely and "up-field" transitions less likely and this results in the appearance of ordered transitions which constitute our F(t). I f we fix our attention on any single carrier in the system, this carrier will execute hopping jumps between various pairs of centres, gradually progressing in the direction of the applied field. In its motion it will encounter "easy" hops and more "difficult" ones and it will take longer to overcome the latter than the former. With reference to fig. 9, we may represent the probable motion of a carrier starting from a site i on a position-time diagram. The spacings between the various sites are chosen unequal and the carrier "hesitates" longer before making the more distant hops than before the shorter ones. Also, the effect of the field is less on the shorter hops, since the resulting energy difference is smaller, than for longer hops. The motion distanc~ I

t3"

E tstT-

I÷l 142 1+3-

-

~

p

t 8"

Fig. 9. Schematic representation of the distance-time trajectory of two hopping carriers traversing a series of unequally spaced hopping steps, under the action of an electric field. The numbered transitions correspond to "ordered" jumps in the direction of the field. The corresponding tm's are indicated on the time axis for the first carrier only. The shaded regions represent non-return hops across larger distances. The two carriers may not approach closer than a certain distance because of mutual repulsion.

FREQUENCY-DEPENDENCE OF CONDUCTIVITY

307

of the carrier may now be resolved into a directed component which is marked by transitions rn = 1, 2, 3,..., 8, each of which is defined as taking the carrier a stage further than it had ever been before. The corresponding times of non-random hops tm are marked on the time axis. We note that the shaded transition, corresponding to rn = 8 is a "difficult" one, it takes much longer, on average, for a carrier to traverse it and may define it as being "irreversible" in the sense that the probability of a carrier hopping back against the field is negligible. These transitions determine the dc conductivity, all other transitions giving rise to random motion. A train of hops between any two irreversible transitions may be defined as a "path" of relatively easy hops. These paths form an interconnected network and carriers move in many parallel paths simultaneously and also traverse similar paths sequentially. The latter point is illustrated in fig. 9 which shows a second carrier entering from the left across another irreversible transition and progressing in a partially ordered random motion down the force field. The diagram is intended to indicate, however, the retarding effect of the first carrier on the second - space charge prevents the second carrier from following too closely. Taking the ensemble of all the carriers in all the paths, we see that the initial effect of the application of a step-function field will be to make all carriers move in the direction of the field. As more and more carriers come to the ends of their respective paths, with the resulting delays in making the transitions, the movement slows down, the times tm for the system as a whole lengthen, until in the limit they are determined by the difficult transitions. In a purely qualitative way we would expect these two probability distributions to be as shown in fig. 10a, the transit times within paths being on the whole much shorter, the difficult transitions being much longer, and there being a finite overlap between the two distributions. The first distribution determines the immediate response after the application of the step field, the second distribution determines the long-term response and in the limit the dc conductivity. We have indicated the log-normal character of these distributions. Fig. 10b shows diagrammatically the distribution of the firing times tm on the time axis, with a high density at the beginning immediately after the application of the step field, falling off progressively with time. We may define a "density-of-events" function p(t) which determines the average number of non-random hopping events per unit time. p(t) may be expected to have two components, one due to the transitions within the paths, the other due to inter-path transitions with the limiting value Po~ for very long times defining the dc conductivity, fig. 10c.

308

A, K . J O N S C H E R

PoobGbilityof ~ransitions after At U t I /~

/

transit ions

ithin g~ths

\

m=I

log A

tli

tij

. . . .

tm

d)

Fig. 10. (a) Distribution o f the probability of a transition from i to j occurring after a delay A t , j , (b) Distribution o f firing times tm on the time axis following a step-function field excitation. (c) Density-of-events function p(t) (d) Total number of events m(t) as a continuous function.

The total number of events up to time tm is, therefore tm

m = [" p(t) dt,

(29)

tt

o

and is represented in fig. 10d as a monotonically increasing function of the time 4,. In this graph both m and t m are represented as continuous rather than discrete variables, but this does not affect the main argument. By differentiating (29) we may express p ( t ) in terms of the tin(m) relation: p (t) = d m / d t m .

(30)

We note that the assumption of an exponentially decaying function p (t) = Po exp ( - c~t),

(31)

gives the relation: tm

=

--

(1/CO log ( 1

--

~m/po).

(32)

FREQUENCY-DEPENDENCE OF CONDUCTIVITY

309

In the present instance the total number of hops is limited and is equal to mmax po/Ot. The logarithmic distribution (32) is linear in m for small m and rapidly rises as m approaches mmax. The former approximation gives tr = constant at frequencies higher than ~, and a region of tr oc092 at lower frequencies. This frequency dependence corresponds exactly to a dipolar mechanism, fig. la, with 1/~ = ~. We note that the same co2 dependence would be obtained for any sequence of firing events finite in number stopping at tmax. This will be recognized as being equivalent to a number of dipoles which are flipped by the applied field and there after cannot produce any further motion. In the hopping context this corresponds to carriers which are tightly bound to definite locations and can only oscillate around them, without being able to contribute to dc conductivity. The assumption made earlier in this paper that all the amplitudes An, are of equal magnitude is clearly not strictly valid and refinements may have to be made later to the theory to allow for a spread of values. However, it would seem very unlikely that this assumption should produce a completely misleading final result. =

5. The effect of temperature In our introductory comments we have already drawn attention to the fact that the exponent n in the aoco9n relation is a function of the temperature, tending towards unity, though not apparently exceeding this value at low temperatures and moving towards the value 0.5 at higher temperatures and eventually giving way to frequency-independent conductivity at high temperatures. Within the framework of the present analysis this gradual change of the exponent becomes understandable as evidence of a changing character of the distribution of firing times t,, rather than simply as a change of the hopping frequencies themselves. This is especially noticeable in the fact that the characteristics at different temperatures actually cross in certain cases, implying that at high frequencies the conductivity at higher temperatures may be actually lower than at lower temperatures, cf. fig. 4.

6. High-field effects The preceding discussion of small-signal transient response of hopping systems was concerned with relatively small perturbations of an equilibrium situation dominated by random transitions between localized centres. The only assumptions that had to be made were concerned with the distribution of transition probabilities between various pairs of centres, according to their spacings and energy differentials. There was no need to make any specific

3]0

A.K. JONSCHER

assumptions about the nature of these centres - to that extent the analysis was completely general. By contrast, a discussion of the effects of a high field on a system containing hopping charges must be concerned with the detailed nature of these charges. We propose to distinguish the following cases: (i) "Dipole-like" charges - ions capable of occupying one of two closely spaced equilibrium positions but not easily removed further away. These are responsible for low-frequency loss peaks in many dielectric materials and their effect is to increase the real part of the permittivity at low frequencies. A high electric field superimposed on small-signal alternating field removes the relevant dispersion as charges becomes pinned on one position by the high field, as shown for example by Argall and Jonscher in SiO la). (ii) Electrons in traps and interstitial ions - in both instances the charge occupies positions which are neutral in the absence of the charge. A more complete discussion of the electron trap hopping mechanism may be found in papers by Jonscher and Hi1126,35-~7). The effect of a high field on trap hopping is to increase the hopping probability in the direction of the field and to reduce it accordingly in the opposite direction. In the case of electrons hopping in trapping levels distributed over a finite range of energy, there may also be a rise of the effective carrier temperature, as discussed by Jonscher 37) who associates an energy relaxation time r~ with this effect. This leads to an enhancement of the mobility through increased transition rates between the higher lying trapping states. (iii) Electrons in donors or ions at lattice sites - in both instances the charge occupies positions which are neutral when occupied by it. In this case the charge is held in position by a Coulombic force with an activation energy A W being required to remove the charge to an infinite distance. In the presence of a sufficiently high electric field the barrier height becomes lowered resulting in an increase of the emission probability by a factor which is given by

K(E)

in the case of relatively distant centres, with the Poole-Frenkel coefficient fl = ( e / ~ ) ½.

(34)

The experimentally determined value of fl may be slightly different for a variety of possible reasons which have been reviewed elsewhere 85). In the case of sufficiently densely spaced donor centres which lead to interacting Coulomb potentials, the barrier lowering is proportional to the field, giving

f elE'~ K(El=exP~kT )

(35)

FREQUENCY-DEPENDENCE OF CONDUCTIVITY

311

where 2/is the average donor spacing. This is the Poole mechanism frequently observed with both ionic and electronic conduction as). (iv) Mixed donor-trap conduction in the case of electronic hopping - first proposed by Jonscher and Ansari ag) for SiO and since developed theoretically by Hill and confirmed in a range of other materials z6, 35). In this mechanism the donors may be regarded as a source of carriers for the trapping system and a high electric field enhances their supply.

7. High-field frequency response Since in all cases of hopping conduction we are concerned with an increase of the transition probabilities in the presence of an electric field, we would expect a rise of the level of dc conduction which by that time is non-linear in the applied field. Qualitatively, the effect may be said to be equivalent to a compression of the time axis in fig. 10b and this means an increase of the minimum frequency from which ac dispersion becomes noticeable. To that extent there is no difference between the effect of a high field and that of raising the temperature, which also produces an enhancement of transition probabilities. On a more quantitative basis, we expect the least probable transitions to be affected most, which may result in a significant change of the hopping sequences t,, or of the function p(t) in the presence of a high field. This would lead to a change of the exponent of the Iocco" dependence - we are no longer justified in referring to conductivity in the non-linear regime. With reference to fig. 10 we may postulate that, in the limit of high fields, the probability distribution of transitions between paths gradually merges with the distribution of transitions within paths. This is consistent with the view that all transitions become relatively easier and very difficult transitions no longer play the crucial role they used to have at low fields. Nevertheless, there is no question of removing completely the frequency dependence since there is still a significant spread of transition probabilities within the system. High-field large-amplitude ac measurements on evaporated SiOx (silicon monoxide) were recently reported by Jonscher and Loh4°). The amplitude of their field signal was well within the range at which Poole-Frenkel effects are seen in the dc case. Fig. 11a shows the results of their measurements. We note that, in comparison with the low-field conductivity, the current at high alternating fields is much less field-dependent than at low fields. This is borne out by fig. I lb in which the ac component of current divided by the field, [/(co) - Iac]/E is plotted against E. It is seen that at higher frequencies the ac component of current stays linear to higher fields than at low frequencies. Qualitatively this may be explained in terms of the notion that the

312

A. K. JONSCHER

Ca)

(b) IO-B

1 0 -3 _

_

~ 10 9 o

~ 1 0 -~

w

"~ 10 ~o

/ 8

?

102

I

I 0 -7

/

1 0 -11 m

400

600

600 1000 E1/~(V. cm-1) ~

104

105

106 E ( V . c m ~)

Fig. 11. (a) The high-field ac current 1(o9) in SiOx after Jonscher and Loh40), plotted in Poole-Frenkel manner as log I versus E t with frequency as parameter; T = 300 K. (b) The data from (a) plotted as [1(o2)- I ~ t i l E against E, showing increasing linearity of high-field current with increasing frequency; from Loh 41). effect of field is most pronounced on the most difficult hopping transitions and that these are in any case less relevant at higher frequencies. Another way of looking at the results of fig. 1 la is to take the frequency dependence of the total current at fixed values of the field amplitude. Here we note that at E=4002 V/cm the current depends on frequency as Iocco °'s, while at 9002 V/cm this becomes/oCtO 0'33. This represents a similar trend to that expected of increasing temperature.

8. Frequency-dependence in ordered systems We are now in the position to comment on the statement made earlier in this paper that the frequency-dependence ought to be observed in all hopping systems, regardless of whether their structure is ordered or disordered. The justification for this lies in the fact that even in a perfectly ordered system in which all hops had equal probability distribution, the actual delay times At u would vary over a certain range which could be considerable. This alone, coupled with the repulsion between closely spaced carriers, would ensure that the initial response to a step function field would be stronger than the steady state, which would be determined by the slowest hops. With reference to fig. 10 we may say that the probability distribution

FREQUENCY-DEPENDENCE OF CONDUCTIVITY

313

in an ordered solid would consist of only one branch, the "intra-path", since there are no "inter-path" hops in an ordered solid. However, just as in the case of high temperatures and high fields, the merging of the two branches does not remove the frequency dependence. 9. Conclusions The well-known experimental fact that the electrical conductivity of a wide range of semiconducting and insulating materials shows a frequency dependence of the form a oco)" has led us to a general formulation of hopping transport in terms of frequency transforms of a series of stochastic events arising from discrete jumps by hopping charges. A very general model is considered in which the hopping charges may be either electrons or ions and they may hop between states that are either neutral when occupied, i.e. donor-like, or neutral when empty, i.e. trap-like. By contrast with the hitherto accepted approach, in which the occupancy of a site is expressed as a continuous function of time, subject to fluctuations due to externally impressed fields, we obtain our frequency response in terms of Fourier transforms of a sequence of discrete events following in time according to certain definite laws. Using an approximate method of summation of a large number of frequency responses we have shown that it is possible to "generate" a range of power-law relations with exponents n ranging from zero, corresponding to dc conductivity, to unity. We suggest that it is not possible to devise a physically plausible sequence of hopping jumps that could give a dependence with the exponent higher than unity, which appears to be in agreement with experimental facts. We point out that the experimentally observed e) 2 dependence needs to be treated with caution on account of serious experimental difficulties which may produce spurious results. However, genuine 092 dependence saturating at very high frequencies is admitted by our analysis in two situations: as a lattice relaxation of the Debye type with a single relaxation time, or in response to a rapid sequence of a finite number of hops truncated abruptly in time. The latter is recognized as the hopping equivalent of the dipole rotation and neither can contribute to dc conduction. The present treatment does not attempt to calculate the hopping sequences for any particular mechanisms of hopping. It sets out to provide a general framework in which subsequent detailed studies may be carried out, especially with regard to the dependence of the exponent n on temperature and on the amplitude of the electric field. We quote some experimental evidence for these phenomena, although it is clear that as yet there is very little published information on these aspects of the ac conduction mechanisms. We note

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that both increasing temperature and increasing amplitude of the electric field produce an increase of the dc conduction and a decrease of the exponent n in the frequency dependence. We also note that the current at higher frequencies is less nonlinear in the field amplitude than at lower frequencies and at dc. These observations are in qualitative agreement with our theoretical model. We suggest that our method of analysis gives a better insight into the ac conduction mechanisms than has been possible hitherto. It leads, moreover, to the prediction that the frequency dependence is not restricted to disordered solids, with which it has been generally associated, but is a characteristic feature of hopping conduction, regardless of whether this takes place in disordered or in ordered systems. Experimental evidence is quoted which may be interpreted as giving support to this statement.

Acknowledgements The author wishes to acknowledge stimulating discussions with Mr. C. G. Garton, Professors H. Fritzsche and M. Pollak, and with his colleagues at Chelsea College, especially Dr. R. Coates, Dr. R. M. Hill, Mr. E. Le Sueur, Dr. C-K. Loh, Dr. M. Morgan and Mr. M. H. Nathoo. This work and its associated experimental activity were partly supported by research grants from the Post Office Research Station and the U.K. Science Research Council.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)

P. Debye, Polar Molecules (Chemical Catalog Co., New York, 1929). V. Daniel, Dielectric Relaxation (Academic Press, New York, 1967). P. J. Harrop and D. S. Campbell, Thin Solid Films, 2 (1968) 273. P. J. Harrop, G. C. Wood and C. Pearson, Thin Solid Films 2 (1968) 457. M. Gevers, Philips Res. Rept. 1 (1945/46) 279. M. Gevers, Philips Res. Rept. 1 (1945/46) 447. M. Gevers, Philips Res. Rept. 1 (1945/46) 298. C. G. Garton, Discussions Faraday Soc. 42 (1946) 57. H. Fritzsche and M. Cuevas, Phys. Rev. 119 (1960) 1238. A. Miller and E. Abraham, Phys. Rev. 120 (1960) 745. N. F. Mott and W. D. Twose, Advan. Phys. 10 (1961) 107. M. Pollak, and T. H. Geballe, Phys. Rev. 122 (1961) 1742. F. Argall and A. K. Jonscher, Thin Solid Films, 2 (1968) 185. A. E. Owen and J. M. Robertson, J. Non-Crystalline Solids 2 (1970) 40. E. B. Ivkin and B. T. Kolomiets, J. Non-Crystalline Solids 3 (1971) 41. P. C. Taylor, S. C. Bishop and D. L. Mitchell, in: Proc. Tenth lntern Conf. on the Physics o f Semiconductors, Cambridge, Mass., 1970. 17) H. Fritzsche, J. Non-Crystalline Solids 6 (1971) 49. 18) H. K. Rockstad, J. Non-Crystalline Solids 2 (1970) 192. 19) A. I. Lakatos and M. Abkovitz, Phys. Rev. B 3 (1971) 1791. 20) W. S. Chan and A. K. Jonscher, Phys. Status Solidi 32 (1969) 749.

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W. S. Chan and C. K. Loh, Thin Solid Films 6 (1970) 91. D. F. Barbe and D. S. Herman, J. Appl. Phys. 41 (1970) 3116. C. S. Linsley, A. E. Owen and F. M. Hayatee, J. Non-Crystalline Solids 4 (1970) 208, M. H. Nathoo and A. K. Jonscber, J. Phys. C 4 (1971) L 301. S. Kobashima and T. Kawakubo, J. Phys. Soc. Japan 24 (1968) 493. R. M. Hill, Phil. Mag. 24 (1971) 1307. M. Pollak, Phys. Rev. 133 (1964) A564. M. Pollak, Phys. Rev. 138 (1965) A1822. M. Pollak, Discussions Faraday Soc. No. 50 (1970) 12. M. Pollak, Phil. Mag. 23 (1971) 519. N. F. Mott, Phil. Mag. 19 (1969) 835. I. G. Austin and N. F. Mott, Advan. Phys. 18 (1969) 41. E. A. Davis and N. F. Mott, Phil. Mag. 19 (1971) 903. H. Fritzsche, to be published. A. K. Jonscher and R. M. Hill, to be published. A. K. Jonscher, J. Vacuum Sci. Technol. 8 (1971) 135. A. K. Jonscher, J. Phys. C 4 (1971) 1331. R. M. Hill, Phil. Mag. 23 (1971) 59. A. K. Jonscher and A. A. Ansari, Phil. Mag. 23 (1971) 181,205. A. K. Jonscher and C. K. Lob, J. Phys. C 4 (1971) 1341. C. K. Loh, Ph.D. Thesis, Univ. of London, 1971.