EFFECT OF LOCALIZED ELECTRONIC STATES ON SIMPLE ELECTRON TRANSFER REACTIONS AT FILM-COVERED ELECTRODES

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EFFECT OF LOCALIZED ELECTRONIC STATES ON

SIMPLE ELECTRON TRANSFER REACTIONS AT

FILM-COVERED ELECTRODES

K. Doblhofer, J. Ulstrup

To cite this version:

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EFFECT OF LOCALIZED ELECTRONIC STATES

ON SIMPLE ELECTRON TRANSFER REACTIONS

AT FILM-COVERED ELECTRODES

K. DOBLHOFER

Fritz-Haber-Institut der Max-Planck-Gesellschaft 1 Berlin 33 Dahlem Faradayweg 4-6, Germany

and J. ULSTRUP

Chemistry Department A, Building 207

The Technical University of Denmark 2800 Lyngby, Denmark

R6sumB. - Nous avons discutk des diffkrents mkcanismes pour le transport klectronique & travers des couches minces d'isolant et de semi-conducteur dans les systemes mktal-film-mktal et mktal- film-klectrolyte. Pour les couches minces

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50 A) le mkcanisme de transport est lik B I'effet tunnel elastique direct de resonance ou inklastique. Au contraire, le transfert klectronique B travers des couches plus kpaisses se fait par une bande de conduction ou par des sauts dus aux phonons A travers des ktats klectroniques localises.

Nous avons aussi prksente des donnees experimentales du transfert klectronique i travers des couches minces de polymeres d'acrylonitrile dkposks sur des substrats d'acier inoxydable et de platine, soit dans des jonctions d'ktats-solides soit dans des systkmes klectrochimiques impliquant des rkactions simples d'oxydo-rkduction. La cornparaison avec les relations prkvisibles entre densitk de courant, surtension et les propriktks des films montre que le mkcanisme de transfert d'klectrons dans ces systemes se comporterait comme un effet tunnel inklastique ou des sauts assistks par des phonons.

Abstract.

-

Various mechanisms for electron transport across thin insulator or semiconductor films in metal-film-metal and metal-film-electrolyte systems are discussed. For thin layers (i. e.

5 50

A) the transport mechanism may thus be direct and resonance elastic tunnelling, and inelastic tunnelling, whereas electron transfer across thicker layers proceeds by band conduction or phonon-assisted hopping via localized electronic states.

Experimental data for electron transfer across thin polyacrylonitrlle films on stainless steel or platinum substrates in both solid-state junctions and electrochemical systems involving simple redox reactions are presented. Comparison with predicted relationships between current density, overvoltage, and film properties shows that the electron transfer mechanism in these systems is likely to be inelastic tunnelling and/or phonon-assisted hopping.

1. Introduction. - Electron transfer (ET) across thin insulating or semiconducting films has been the subject of comprehensive experimental investigations in relation to a variety of different systems such as metal-insulator-metal junctions [l], heterogeneous gas phase chemical reactions [2], passivation [3] and electrocatalytic phenomena [4], and biological mem- branes [5]. Extensive theoretical treatments of metal- insulator-metal junctions have also been provided [I]. For thin films the charge transfer characteristics are thus described in terms of direct [I] and resonance tunnelling [6], and inelastic tunnelling [I], whereas

the ET through thick layers proceeds either by electron or hole conduction in delocalized bands or by phonon- assisted hopping via discrete levels in the band gap.

The nature of the elementary E T steps across the film must be the same in metal-film-metal, metal-film-

electrolyte, and membrane systems. However, the initial and final electronic states in the former systems are both band states, whereas one or both are localized on depolarizer ions in the latter. The three kinds of processes are therefore expected t o display interesting common features. This has recently been exploited in attempts to formulate a theory of thin film ET in electrochemical and membrane systems [7-111. These ap- proaches have rested on both the general theory of charge transfer reactions in condensed phases evaluated over the last decade and a half [12], and the formalism of bulk and heterojunction solid-state conduction [I]. We recapitulate here at first briefly the major results of these theoretical achievements with emphasis o n electrochemical systems and secondly, we discuss some experimental data for electron transport across metal-supported thin polymer layers. These materials

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C5-50 K. DOBLHOFER AND J. ULSTRUP

present a special interest since their ET properties in both solid-state and electrochemical systems can be investigated.

2. Summary of theoretical results.

-

The chemical nature of the systems to be discussed below excludes charge transfer by ionic migration. Our discussion is therefore restricted to electron or hole currents. At sufficiently thin barriers the ET proceeds by tunnelling, whereas carrier mobility mechanisms prevail at thick films. The dominance of a particular ET channel is inherent in the fact that although tunnelling involving metal electronic levels around the Fermi level requires a low activation energy the corresponding tunnelling barrier is high. On the other hand, ET via the conduction band typically requires activation energy but in return proceeds with a probability approaching unity when sufficient energy is acquired. We consider in turn these two main groups of ET mechanisms.

2 . 1 TUNNELLING MECHANISMS. - The analogies and differences between ET in solid-state and electrochemical systems are conveniently summarized by reference to the simplest case of elastic tunnelling through a uniform barrier in which both the electronic energy and momentum component parallel to the junction, kll, are conserved. Thus, within the effective mass approximation the general expression for the current density, jLR, in metal sandwich junctions (from left to right in a vertical junction) is [I]

where e is the carrier charge,

m

its effective mass, & its

energy relative to the bottom of the conduction band of the metal to the left, n,(&) and nR(&) the Fermi functions of the left and right metal, respectively, and D(e, kll, eV) the tunnelling probability for given e, kll, and bias voltage eV. In electrochemical systems the electron is initially (finally) strongly localized on the depolarizer molecule (for an anodic (cathodic) reaction) and also strongly coupled to the solvent phonon modes. The current density (j,,) expression analogous to eq. (1) is then [8]

where @(c) determines the depolarizer concentration, c, in the outer Helmholz layer, q is the overvoltage, and

p

= kg T, where kB is Boltzmann's constant, and T the absolute temperature. D(e, klj, eq) is the probability of electron tunnelling from the localized depola- rizer level to a delocalized metal level of given e and

kll, and EA(&, eq) the activation energy required to reach the saddle point of the reaction hypersurface with respect to the solvent and depolarizer phonon modes. Eqs. (1) and (2) have been the subject of theoretical analysis on the basis of both WKB and transfer Hamiltonian approximations [I, 81. However, lack of barrier homogeneity or energy and momentum conser- vation is expected to open the following additional reaction channels :

2.1 .1 Fluctuating barrier tunnelling. - An amorphous barrier contains exponentially decreasing densities of shallow donor and acceptor states in the band gap [13]. In addition to hop reaction channels (see below) this induces a random variation of the barrier height across the film 1141. For small fluctuation ampli- tudes this effect is manifested by a current density increase at low bias voltage up to an order of magni- tude [14]. This may be of importance for the determi- nation of the upper limit of the barrier width for which tunnelling is the ET mechanism.

2 . 1 .2 Impurity-assisted elastic resonance tunnelling. Barrier inhomogeneity may also be caused by the presence of local impurity centres represented by deep and highly localized states. They can provide new reaction channels in which the electron is transferred via the localized level, enhancing the current relative to direct tunnelling [6, 10, 151.

In the limit of weak coupling between the electron temporarily accommodated in the impurity level and the barrier phonons the two-step ET is still an elastic, although no longer momentum-conserving process [I]. The current density for the elastic resonance ET via a single impurity level, written in a form analogous to eq. (1) is thus, for the solid-state junction [6, 151

where pL(e) and pR(e) are the level densities of the metal to the left and right, respectively, and T(E, eV) the resonance tunnelling matrix element. The analogous expression for the 'electrochemical (anodic) current density is [lo]

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where

I

i

>,

I

d

>,

and

I

f

>

are the electronic wave functions of the initial, resonance, and final state,

A A

respectively,

Vfd

and

Vdi

the coupling operators for the states indicated by the subscripts, and y (+ 0

+

for weak interactions) the broadening of the intermediate state.

For a uniform distribution of the impurity centres across the film the main results pertinent to the systems below are [6, 101 :

(1) In both solid-state and electrochemical systems the resonance tunnelling currents via low-lying (i. e. below the Fermi Ievel of the donor electrode) strongly localized centres can assume up to a few multiples of the value for direct tunnelling provided that the impurity concentration exceeds 1019-1020 ~ m - ~ . The effect is larger for less localized centres and more pronounced the thinner the barrier, the higher the energy of the impurities, and the closer they are located to the barrier centre.

(2) The current-voltage characteristics differ from those predicted for direct tunnelling by having lower logarithmic current derivatives. For electrochemical systems the transfer coefficient is sensitive to the position of the resonance level relative to the metal Fermi level, and its value predicted to change in overvoltage regions where the two levels cross.

2.1 .3 Inelastic tunnelling. - Both theoretical [16] and experimental evidence [17] suggest that lattice dis- tortions are associated with deep impurity centres leading to excitation of local and collective phonon modes. Since our present discussion deals with ET above the Debye temperature we shall consider modi- fications of the equations above from such << inelasti- city )) effects in the limits of high temperature and

strong phonon coupling. Thus :

(1) The wave functions

I

i

>,

I

d

>,

and

I

f

>

now represent the total system, i. e. both the electron and the barrier phonons.

(2) Eqs. (3) and (4) must be averaged with respect to all initial and summed with respect to all final barrier phonon states.

(3) The electrochemical system contains two phonon systems, i. e. those of the barrier and solvent regions. Eq. (4) can be interpreted as if averaging over the total phonon system has been performed, EA(&, eq) then being the activation energy including contributions from all phonon modes.

The role of the nuclear mode reorganization in the barrier and electrolyte regions can be illustrated with reference to figure 1. Thus, figure l a shows the electronic terms for the solid-state junctions. In the intermediate state the impurity level is occupied, whereas it is empty in the initial and final states. The latter therefore corresponds to the same lattice structure represented by the equilibrium coordinate values q,,. On the other hand, q,, is different in all three states for the electrochemical systems (Fig. lb).

FIG. 1. -Dependence of the total potential energy on nuclear coordinates in the film and electrolyte regions in the initial (i), intermediate (d), and final (f) electronic states, a : Solid-state

junction. b : Electrochemical system, where dl and d l refer to an

intermediate state of low and high energy, respectively.

In summary, we provide the explicit current density expressions. For the solid state junction T(E, eV) in eq. (3) takes the form [la]

where Lfd(&) and Ldi(&) are the electronic matrix elements coupling the impurity level with the donor and acceptor levels, respectively (cf. eqs (3)-(S)), Udi(&) and Ufd(&) the saddle points of the intersection hyper- surfaces indicated by the subscripts, and E:~(&, e V )

the activation energy for the ET from donor to impurity.

For electrochemical systems two cases emerge. If the intermediate level energy is low, T ( E , eq) is formally given by eq. (6) [19], but both ET between metal and intermediate level, and between intermediate and depolarizer levels may now be rate determining. Moreo- ver, levels at the barrier centre no longer necessarily dominate. As a result, different current-voltage characteristics may emerge with transfer coefficients almost anywhere between zero and unity. On the other hand, if the resonance energy is high (Fig. lb), T(E, q ) takes the form [20]

where AU(8) is shown in figure lb, and the activation

energy now corresponds to ET directly between metal and depolarizer. From eqs. (6) and (7) we notice the following :

(1) Intermediate states of high energy provide en- hancement factors relative to direct tunnelling very close to those for elastic resonance tunnelling.

(2) High-energy intermediate states close to the barrier centre give the dominating contribution, since this position ensures optimum values of the electronic matrix elements.

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C5-52 K. DOBLHOFER AND J. ULSTRUP

ment factors may be smaller due to the activation a, <

4.

Moreover, the influence of the film thickness factor. is only reflected in eqs (8) and (9) and not directly (4) ET via low-lying levels inside the film region in the absolute values of the exchange current densi- would display low values of the logarithmic derivative ties, in contrast to tunnelling currents.

and the elecGochemica1 transfer coefficient for current flow in both directions. This is contrary to both resonance tunnelling and inelastic tunnelling through high- energy intermediate levels.

While the various kinds of tunnelling processes pertaining to the scheme given above are well known in solid-state junctions [I], a similar approach to electrochemical systems has only recently been attempted [9]. Thus, the behaviour of simple ET reactions at oxide- covered platinum and iron electrodes, and the reduction of dioxygen at carbon-supported metal phthalocyanines are compatible with direct elastic [8], elastic resonance [lo], and inelastic tunnelling [21], respectively.

2 . 2 MOBILITY MECHANISMS. - We consider here two mechanisms which represent limiting cases for electron transport across films of sufficient thickness that tunnelling can be excluded.

2.2.1 Band conductivity. - Electron and hole transport across the conduction and valence band, respectively, prevail at thick junctions where a trapfree film possesses a regular structure, and where coupling to the optical phonons of the lattice can be neglected. In relation to the systems to be considered below we notice that for organic molecular crystals the band gaps are wide (> 0.5 eV) and the bands themselves usually narrow ( z k , T). This, and the amorphous nature of

the polymer films makes a band conduction mechanism unlikely for the latter. However, hopping between shallow states close to the mobility edges would be difficult to distinguish from band conduction. Thus, the field dependence of the mobility edge of the amorphous material would closely parallel that of the lower edge of the conduction band in a semiconductor layer of regular structure. The following features of metal-film-electrolyte systems [7] therefore refer to both crystalline and amorphous film layers. The potential distribution in the film can generally be found by solving the Laplace equation in the film and Helmholz double layer regions. For a sufficiently large Debye length in the film region the potential drop is linear, and the film-electrolyte contact potential, qc, related to the metal potential, qM, by the equation [7]

q c = 9M(l

+

&H a)-' (8)

where E~ and d are the dielectric permittivity and thickness, respectively, of the film, and &, and 8 the corres-

ponding quantities for the Helmholz double layer. This gives the following expressions for the transfer coefficients, a, and a,, corresponding to ET via the valence and conduction band, respectively

The term 8, dl&, 6 is of the order of unity, and thus

2.2.2 Hopping mechanisms. - In disordered solid materials the crystalline band structure model is generally believed to be basically preserved, although the density of states remains finite (i. e. 1018-1020 ~ m - ~ eV-I) in the band gap [13]. These low-density states are bound states and provide a new mobility mechanism of low activation energy and free mean paths (i. e. lower than the interatomic spacing) corresponding to electron hops betwee the states. Since the disorder

also causes a spread in energy of the localized states, hopping is generally a thermally activated phonon- assisted process which, in the Iimit of strong phonon coupling, is equivalent to mobility between small or large polaron states and thermal ET in polar solutions. In this sense hopping is of course not restricted to disordered materials.

Provided that impurity band formation can be igno- red, description of electron transport by hopping across thin amorphous films should comprise both the elementary ET step between two local sites and the construction of percolation paths for particularly favorable consecutive or correlated ET acts [13,22-241. We notice at first that the probability of a single ET can be written formally as

where a is a localization parameter,

R

the distance between the two sites, EA an activation energy which

depends on the energy difference between the two sites and the coupling to the lattice phonons, and A a para-

meter which also depends on these quantities. In the limit of weak coupling EA in fact equals the energy

difference between the sites, and the averaged bulk three-dimensional conductivity, o, obtained either by maximizing the exponent of eq. (10) or from percolation theory takes the form [22-241

where NF is the (constant) density of localized states.

For strong phonon coupling or polaron formation a similar treatment can be performed in principle but gives in general a complicated temperature dependence. However, in the high-temperature limit, when multi- phonon processes prevail, an Arrhenius temperature dependence is expected for sufficiently strong phonon interaction (i. e. E,

2

0.2 eV, where E, is the phonon

reorganization energy), whereas a temperature relationship such as eq. (11) is predicted for weaker inter- action (E,

5

0.1 eV).

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given hop before the subsequent hop occurs. ET via several unrelaxed intermediate states are then most conveniently viewed as a single elementary step for which the activation energy is lower and the effective jump distance substantially higher than for hops between neighbouring sites [19, 251. Secondly, in eva- luating the conductivity of the disordered material one should consider networks of traps throughout the layer and connected by conductivities given by the

hopping rates for ET between individual sites [22-241. The measured conductivity is then determined by the critical percolation conductance, i. e. the highest conductance stretching through the r a n d o i network of traps.

Both the correlated hopping and the percolation paths involve a partial delocalization of the electron. Maximum hopping distances determined experimen- tally may therefore be larger than the nearest-neigh- bour distance and for example correspond to hops between percolation islands of higher than bulk

conductivities.

3. Application to polymer-covered electrodes. - We shall now apply the discussion above to some experimental data for electron transport through thin polymer films deposited on metal substrates. Polymeric organic and metal-organic compounds have exhibited powerful catalytic properties-in both ET and decompo- sition reactions [4], and recent strong evidence suggests that the charge transport mechanism in certain disordered thin polymer layers is in fact electron and hole transfer between localized states [26].

The polymer layers were cathodically deposited at 300 OC, by dc glow discharge in acrylonitrile vapour [27], on stainless steel and platinum substrates pre- treated by electropolishing and argon ion bombard- ment. This gives a film probably consisting of polymer fragments of variable size. Metal-supported polymer layers synthesized in this way possess the following properties expected to make a study of their ET features rewarding in relation to both electrocatalysis, electron transport in disordered solids, and to a certain extent, theoretical treatment :

(1) They provide electronically conducting films which are both mechanically stable and resistent to corrosion.

(2) The thickness can be varied, although so far this will also usually result in changes of the electronic density of states.

(3) The density of states can be controlled by dilu- tion in an inert polymer of the hexafluoropropene type.

(4) Both solid-state junctions and electrochemical systems with the same film can be studied.

The polymers differ from conventional polymers prepared from the same monomer [28,29] by extensive unsaturation and ability to absorb up to 20

%

0,

when left in air. This strongly implies that many localized reversible redox centres, e. g. quinoide couples are present in the polymer layer. Several of these groups are subject to both frequency and bond length changes when they participate in ET reactions. The electrons are therefore likely to be coupled to the phonon modes in the film region. The chemical nature of the polymer thus suggests that the electrons are transferred across the film by hops between localized states, assisted by fluctuations in nuclear modes around the trap centres. The conductivities were obtained by measuring the voltage drop across the film during constant dc current flow, and with electric contact via an evaporated gold layer. The film thickness was found by capacity measu- rements, and all values reported are based on a value of cS = 5. cS may in fact be as low as 3 for the polymers investigated here which means that the thickness values reported could be lower by up to 40

%.

This has the important theoretical implication that inelastic tunnelling may also be a possible ET mechanism.

Figure 2 shows typical experimental data for polymer layers in solid-state junctions. The measured vol-

FIG. 2. -Applied dc current and measured voltage drop across the system Au-polymer-stainless steel support. Ambient

temperatures unless other-wise indicated.

tage did not depend on the direction of the net current flow, and for voltages lower than about 0.1 V the voltage drop is proportional to the current. Figure 3 shows that the conductivity follows the T-'I4 law over

several decades. In fact, this relationship is not very sensitive, and for low thickness a T - ' / ~ dependence can also be fitted corresponding to two-dimensional

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C5-54 K. DOBLHOFER AND J. ULSTRUP

FIG. 3.

-

Temperature dependence of the Ohmic conductivity

(o) of polymer film.

FIG. 4.

-

Field dependence of the conductivity at ambient

temperatures,

mental data for the solid-state junction suggest the following :

(1) The thickness values exclude elastic tunnelling mechanisms.

(2) The decrease of the conductivity with increasing film thickness is expected for a hopping or polaron mobility mechanism due to the decreasing critical percolation conductivity, but neither for a band mechanism nor for conduction via shallow states close to the mobility edge.

(3) A Frenkel-Poole conduction mechanism can be

excluded in view of the exponential dependence of the conductivity on the field strength.

(4) Space charge limited currents are not likely to prevail as suggested by the linear voltage dependence of the current.

The experimental evidence is thus in favour of a hopping or a polaron mobility mechanism, possibly involving a single hop only for the thinnest films, i. e. inelastic tunnelling in the sense discussed above.

We have supplemented our investigation by measu- rements of the current-overvoltage characteristics of simple electrochemical reactions proceeding at steel- or platinum-supported layers in contact with aqueous solutions of the depolarizer ions. The depolarizer species were the [Fe(CN),]3-/4- and [CU(NH,),]~+~+ couples and molecular dioxygen. Although these data are of a preliminary nature they nevertheless provide certain evidence for the conductivity mechanism of the polymer layers in addition to that obtained from the solid-state junctions. Thus :

(1) Platinum- and steel-supported layers give similar results showing that polymer properties have in fact been measured.

(2) The catalytic activity for the reduction of 0, in alkaline solution decreases in the order platinum >

polymer > carbon > remanite (steel).

(3) The exchange current density strongly decreases when the density of states is decreased by diluting the acrylonitrile polymer with hexafluoropropene in the ratio 1 : 9.

(4) For polymer thickness in the region 150-800

A

linear Tafel plots of slopes 240-360 mV were obtained for both redox couples and for both cathodic and anodic overvoltages, and the plots crossed at zero overvoltage. Thus, no rectification effects and effects of standard redox potential shift were observed. Together all this leaves hopping or polaron mobility as possible ET mechanisms.

(5) The exchange current density decreases approxi- mately exponentially with increasing layer thickness. This is also expected for a hopping mechanism but not for a band conduction mechanism.

The electrochemical data are thus consistent with those for the solid-state junctions. In addition, we should notice that the values of the Tafel slopes correspond to potential drops between 30 and 50

%

of the total metalelectrolyte potential drop. For a linear potential distribution in the film region this would require large maximum hopping distances in line with the correlated hopping and percolation schemes above.

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