Hopping transport of localized p electrons in amorphous carbon films

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Hopping transport of localized p electrons in amorphous carbon films

PHYSICAL REVIEW B VOLUME 39, NUMBER 11 15 APRIL 1989-IHopping transport of localized n. electrons in amorphous carbon filmsK. Shimakawa and K. MiyakeDepartment of Electronics and Computer Engineering, Faculty of Engineering, Gifu University,1-1 Yanagido, Gifu 501-11,Japan(Received 6 September 1988; revised manuscript received 21 November 1988)The temperature dependence of the dc and ac conductivities has been studied in sputtered arnorphouscarbon films. The dc conductivity is proportional to T" with n =15— 17 below room temperature.The mechanism is discussed in terms of the mobility-edge conduction, variable-range hopping,and small-polaron hopping. Quantitatively, these models could not interpret the observed behaviors.Instead, the multiphonon tunneling of localized electrons with weak electron-lattice couplingis suggested to be the dominant transport mechanism. This may be attributed to the ~-bondednature of sputtered amorphous carbon. The ac conductivity is strongly correlated to the dc conductivity,which has never been explained by the current theories based on the pair approximation. Alternatively,the continuous-time random-walk approximation is shown to be a useful approach.I. INTRODUCTIONIt is believed that amorphous carbon (a-C) prepared byevaporation or sputtering is predominantly sp bondedwith an optical gap of 0.4—0.7 eV. ' At a carbon spsite, there are three strong o. bonds and one weak m bondlying normal to the o. bonding plane. The ~ states willform both the valence-'and conduction-band states.The sp sites must be clustered together in aromaticunits. The large size of such clusters could produce localizedstates near the Fermi level E+.The eFect of disorder in a ~-electron system is thereforeof particular interest. The present authors haveshown that the temperature dependence of the dc conductivityin sputtered a-C is empirically described byo. d,=o. oT", with n =15—17, which may suggest that thenonpolaronic multiphonon tunneling of localized m electrons(weak coupling with lattice) dominates chargetransport.Measurement of ac conductivity could provide directinformation about the hopping rate of localized electrons.In the present study, the temperature dependences ofboth dc and ac conductivities are reported. Possibletransport mechanisms, mobility-edge conduction, variable-rangehopping (single-phonon process), and multiphononhopping with strongly (small-polaron) and weaklycoupled states, are discussed. It is suggested that theconventional mobility-edge conduction or small-polaronconduction could not explain the overall features oftransport in a-C. It is also found that the ac conductivityis strongly correlated with the dc conductivity. Thisfeature, which has never been explained by the currenttheory based on the pair approximation, is interpretedwell in terms of the continuous-time random-walk(CTRW) approximation.II. EXPERIMENTThin films of a-C were prepared on Corning 7059 glasssubstrate with a bottom Au electrode by rf (13.56-MHz)sputtering of a 60-mm-diam graphite target (99.999%) inargon gas (0.18 Torr). The forward power applied to thetarget was 100 W and the substrate was held at 20 C.The deposition rate under these conditions is approximately0.6 A/s. Front contacts were made of Au bythermal evaporation. Annealing of samples was done at400 C (1 h) at 3X10 Torr. The ac conductivity wasmeasured using a capacitance bridge (Ando TR-1C).III. EXPERIMENTAL RESULTS AND DISCUSSIONA. dc conductivityThe solid circles in Figs. 1 and 2 show the temperaturedependence of dc conductivity (crd, ) for as-deposited andannealed (400'C, 1 h) films of 0.6 pm thickness, respectively.Annealing results in an increase in hard„as foundby several authors. ' 'The activation energy decreasesgradually as decreasing temperature, suggesting thatthe transport mechanisms are thermal activation tothe mobility edge at higher temperatures and variable-rangehopping at lower temperatures. o. d, formobility-edge conduction is approximately given byeX,Ittoexp( EE/kT), whe—re po is the microscopic electronmobility and X, is the e6'ective density of states forthe conduction band (carrier is assumed to be electron).AE is hard to estimatebecause there is no good straightline in the lno. d, versus 1/T curve even for higher temperature.Taking, for example, DE=0.5 eV for the asdepositedfilm, and assuming the conventional values ofpo-10 cm V 's 'and %,=10' cm, o. d, at 300 K isestimated to be 6X10 Scm ', which is about one orderlarger than the experimentally observed value (seeFig. 1). This suggests that either or both the values of X,and po should be considerably smaller than those of assumedvalues, predicting that the band-edge conductioncannot be dominated by the other transport path. Thermopower(TP) measurement ' could produce informationabout the transport path. ' The small thermopower39 7S78 1989 The American Physical Society


39 HOPPING TRANSPORT OF LOCALIZED m ELECTRONS IN. . . 7579-11107i000IV (X )FIG. 1. Temperature dependence of the dc conductivity forthe as-deposited a-C film. Solid circles are experimental resultsand solid lines are calculated results of the nonadiabatic smallpolaronhopping.(several pV/deg) and the lack of its temperature dependence,which was observed in a-C by Grigorovici et al. ' ,suggest that the transport path lies near the Fermi level(not the mobility edge), although TP measurements havenot been made in the present Alms.Let us check the possibility of transport in localizedtail states pear the mobility edge. The dc conductiv&ty iswritten by'4o d,= e Ig (E)p(E)f(E)[1 f (E— )jdE,where g (E) and p(E) are energy-dependent density of taillocalized states and mobility, and f (E) is the Fermi distributionfunction. As the g (E) and p(E) decrease rapidlywhereas f(E) decreases with decreasing energy (towardthe Fermi energy E~), the conduction path shiftstoward EF as the temperature is lowered, yielding nonactivatedtransport such as is shown in Figs. 1 and 2. However,tail state conduction should also yield large TP andstrong temperature dependence of TP. From the aboveconsideration, it is preferred that the transport path lienear EF in the present a-C.The localized m states may form a continuous distributionacross the band gap and electronic conduction byvariable-range hopping' (lncrd, &x T ' ) is expected tooccur in a-C as in the same manner as tetrahedral a-Geand a-Si. ' It is of interest to examine whether thevariable-range hopping predominates dc conduction.The T '~ plot of dc conductivities for as-deposited (a)and annealed (b) films are shown by open circles in Fig. 3.10—IE1010" 0{} 1210141071000fr (K ')2, 4 2.6 2.8)lT3.0FIG. 2. Temperature dependence of the dc conductivity forthe annealed a-C film. Solid circles are experimental results andsolid lines are calculated results of the nonadiabatic smal1-polaxon hopping.FIG. 3. Temperature dependence of the dc conductivities for(a) as-deposited and {b) annealed a-C films. Temperature scaleis T ' . The fits of variable-range-hopping theory to data athigher temperatures are shown by the solid lines and those atlower temperatures are shown by the dashed lines.


7580 K. SHIMAKAWA AND K. MIYAKE 39The experimental results are not described by a singlestraight line over the measured temperature range. Thefits to the data at higher temperatures are shown by thesolid lines and those at lower temperatures are shown bythe dashed lines. It is hard to distinguish from the figurewhich line (solid or dashed) correctly describes the Tbehavior.The variable-range hopping theory predicts the temperaturedependence of o d, for three-dimensional materialsaslno. , = 3 — BT (2)where 3 and 8 are constants. The theory reproducesmany experimental data, particularly for o.-defect amorphousGe and Si. ' The parameter 8 has been used to estimatethe density of states at the Fermi level N(EF),while some difhculties of this theory have been pointed16, 17tN(EF) here is estimated from 16a 1kB, where a ' isthe effective Bohr radius of localized electrons and k theBoltzmann constant. Assuming a '=10 A, althoughthis value may be smallerthan the practical value for a-Cbecause of the delocalized nature of rr electrons, N(E~)for high-temperature fitting (solid lines) is estimated to be3 X 10' and 7 X 10' cm eV ' for as-deposited (B = 274K'~ ) and annealed (B =224 K'~ ) films, respectively.N(E~) for low-temperature fitting (dashed lines) is estimatedto be 4X10' and 1X10' cm eV ' for asdeposited(B =251 K' ) and annealed (B =204 K'~ )films, respectively. These values, perhaps an upper limit,seem to be unreasonably small. It is expected that thedensity of defect is much higher than that of m-defect(10' —10 cm eV ') a-Ge and a-Si. ' It is hencesuspected that the dc transport cannot be dominated bythe variable-range hopping. This may be attributed toweak localization of n defects (large eff'ective a '): Theprobability of single-phonon hopping decreases with increasingeNonactivated dc transport observed here is similar tothat for the small polaron which is formed in the strongelectron-lattice coupling.' lt is of interest to knowwhether the experimental results can be interpreted interms of the small-polaron hopping. Before proceedingwith the discussion, we summarize the small-polaronhopping theory in which both the acoustic and opticalare taken into consideration.In general, the dc hopping conductivity can be given13phononsad, =n, (eR) I;„/6kT, (3)where n, is the number of carriers, R the hopping distance,and I;„the minimum hopping rate.If the electron-lattice coupling is strong enough, thesmall polaron can be formed in disordered materials. 'Electrons should couple with both the optical and acousticphonons. Following Holstein,'Gorham-Bergeronand Emin presented an exact calculation of the nonadiabaticmultiphonon transition rate for the small polaron:'2J, q1/22(E;~+E )kTXexpX expQ28(E;t'+E )kTEgokTEackTexp2kTwhere J, is an electron transfer integral between sites iand j, and 5 the energy difference between sites i and j.E;,E, Eg, and E~ are defined as follows:EQP4kT2Ebh-.%co"' 2kT2EbE Bc- 4kT X A'cog„AdoE„=— gAM%cogac 1 2kT ac g, acEb tanhX %cog gc 4k Tg, acwhere co, is the mean optical frequency (small dispersionof optical mode is assumed here), co~„ is the acousticphonon frequency at wave vector g, and N is the numberof phonon modes. Two energies, Eb" and Eb', are the polaronicbinding energies related to optical and acousticphonons,respectively.The solid lines, (a), (b), and (c) in Fig. 1, are calculatedresults using Eqs. (3)— (5). The phonon density of stateswas approximated to be g(v) oc v for an acoustic phononwith a cutoff'(Debye) frequency vD and the mean opticalphonon frequency was assumed to be vo=3vD. One ofthe important physical parameters is the Debye frequencywhich determines the shape of curve. The curves,Figs. 1(a), 1(b), and 1(c), are for vD = 2. 5 X 10',5. 5 X 10', and 1 X 10' s ', respectively. The other physicalparameters required for calculation are taken to beEb'=E„'"=0 5 eV, 6=0 015 eV, n, =1X10' cmwhich is quoted from the electron spin resonance (esr),and R = 5 A which may be reasonable for the average siteseparation of small-polaron hopping. Fitting to experimentaldata [curve shown in Fig. 1(b)] produces J;, =1.0eV. It should be noted, however, that estimated J;, . (1.0eV) seems to be very much larger than that ((0.1 eV)predicted for the nonadiabatic treatment of the small polaronBeyond this value (=0. 1 eV), Eq. (4) cannot beapplied to the nonadiabatic small-polaron hopping. Byassuming J,"=0. 1 eV, n, =1X10 cm is required forfitting, which seems to be too large for the carrier density.The solid lines, (a), (b), and (c) in Fig. 2, are calculatedresults for vD= 3.5 X 10'2, 7.0 X 10', and 1.5 X 10' srespectively. Other physical parameters used for the calculationare the same as those for as-deposited film. Fittingto experimental data [curve shown in Fig. 2(b)] producesJ;~=4.5 eV, which is too large for nonadiabatic(4)(5)


39 HOPPING TRANSPORT OF LOCALIZED m ELECTRONS IN. . . 7581hopping of the small polaron. When J; =0. 1 eV is assumed,n, =2 X 10 ' cm is required, which may also betoo large.The above results suggest that the small polaron couldnot dominate dc transport in a-C, although the qualitativenature is well explained by the small-polaron hoppingtheory. The small polaron cannot be formed whenthe extent of localized states is larger than the averagelattice spacing. ' ' The origin of the defect states in a +-electron system should therefore be considered. Robertsonsuggested that the deep localized states originatedfrom large ~-bonded clusters: The wave function is delocalizedacross its cluster and the electronic hoppingshould occur between clusters. The large-radius localizedelectrons could couple most effectively to longwavelength(acoustic) phonons vo which are given by(ao/a ')vii, where ao is the average lattice'spacing.The delocalized electrons across its cluster will behave aspointlike defects with large n '. Hence, a'here shouldbe taken to be cluster size l, which behaves as "defect."Taking l, =60 A, which is somewhat arbitrary but is consistentwith the prediction by Robertson, and vD =390K for graphite, vo = 2 X 10" s'( To = h vo/k = 10 K) isestimated by assuming ao =1.5 A. As the binding energyin Eq. (5) under this situation is approximately given byEb'(ao/l, ), ' the electron-lattice coupling is expected tobe considerably smaller and hence no small polaron canbe formed in a-C. It is also believed that the singlephononhopping process is not expected to occur for sucha small vo. ' ' ' We should next consider the multiphononprocess with weak-coupling states.The configurational coordinate diagram for a weakcoupling is shown in Fig. 4. The energy EM in Fig. 4 is-the measure of coupling strength. The nonradiative'recombination (phonon emission) rate with weak couplingis given by' 'R (6)= C exp( — yp)[1 — exp(h vo/kT)] (6)where C=vo, y=ln(A/EM) — 1, and p =b. /hvo. Themultiphonon hopping processes must involve absorptionand emission of p phonons. The absorption process isproportional to the Bose factor to the pth power as[exp(hvo/kT) — 1] ~. Under the condition of hvo &&kT(high population of phonons), the rate R (b, ) for the emissionwill be the same as that for the absorption and isproportional to (kT/hvo)~. Thus Eq. (6) can be applicableto the present hopping problem. The electron overlappingterm, which may be given as exp( — 2yR '), whereR' is the actual hopping distance and is roughly given byR — 2l„and y is the tunneling parameter, is implicitly includedin a constant C. R is the average center-to-centerdistance between clusters. The parameter y is the measureof coupling strength. Note that Eq. (6) is similar tothe multiphonon jump rate for weak-coupling states derivedby Emin' (see Appendix). od, is obtained fromEqs. (3) and (6). As n, in Eq. (3) must be given byX(EF)kT and R (b, ) here should be I;„,o d, is proportionalto T~. The value of p should be an integral number.However, if 6 or vz distributes around a certainmean value, p wi11 be nonintegral.The same data as Figs. I and 2 are replotted on a logarithmicT scale in Fig. 5. Data, Fig. 5(a) for as-depositedand Fig. 5(b) for annealed, are well fitted by the straightlines for all measured temperatures, showing that the logarithmicT plot is better than a T ' plot. This shows10 "IE-1010QO1 012enej(10 14conf igUration2.0 2.2~og iQ[T (K)]2.4FIG. 4. Configurational coordinate diagram expected for localizedstates in the weak electron-lattice coupling.FICi. 5. Logarithmic temperature dependence of the dc conductivitiesfor (a) as-deposited and (b) annealed a-C films.


7582 K. SHIMAKAWA AND K. MIYAKE 39hard, is proportional to T", suggesting that the multiphononhopping with weak-coupling states dominates electronictransport in a-C. The values of n are 17.4 and 15.1for as-deposited and annealed films, respect;ively.1P8323 KB. ac conductivityBefore evaluating physical parameters n„R, and Iwhich appeared in Eq. (3), we turn to ac conductivitywhich will give information about the hopping rate. Theac conductivity o„for many disordered materials can beempirically described by cr„=Af' at high frequency,where A and s ( & 1.0) are usually temperature-dependentparameters.'This relation can be interpreted eitherby the quantum-mechanical tunneling'(@MT) orcorrelated barrier hopping (CBH). These tuodels arebased on the pair approximation in which the motion ofthe carrier is contained within a pair of sites. If the dcand ac conductivities arise from the same hopping mechanisrn,the pair approximation cannot be applied.' " Asthe dc conductivity for a-C is expected to be dominatedby the hopping process, the continuous-time randornwalk(CTRW) approximation originally developed byScher and Lax may be (he appropriate approach.In the CTRW, ac conductivity is given aso (co)=Eico+—— 1I +lcdwhere K is a constant and (1/(I +ice)) denotes theaverage over jump frequency distribution P(I ). Underthe condition P(I ) cc I ', which is encountered in mostof the disordered materials,'a simple form of ac conductivityis given byl CO'7' ln(1+i co~)where ~ is the rnaximurn hopping time and approximatelyequal to I „ in Eqs. (3) and (6). It is noted that o„ isstrongly correlated to the dc conductivity. Thus the rneasurementof ac conductivity could provide knowledge ofthe hopping rate of localized electrons.Figures 6 and 7 show the ac conductivity (o„)for varioustemperatures in as-deposited and annealed films, respectively.o„ is proportional to f' at higher frequencies,where f is the frequency and s is the temperaturedependentparameter. The observed features are similarto those observed for many amorphous semiconductors.'At higher temperatures and lower frequencies,the magnitude of o.„approaches the dc conductivity o. d, .As discussed in Sec. III A, dc conduction seems to bedominated by hopping of localized electrons. The actransport could arise from the same hopping mechanism.To analyze ac conduction for this case, the CTRW approximationmust be a proper approach. The solid linesin Fig. 8 show the calculated cr„(real part) at 323 K usingEq. (8), where hard, is set to the experimental value (seefrequency-independent conductivity in Fig. 8). The I ~;„shown by the arrow is chosen to obtain the best fit, lyingclose to the onset frequency at which conductivity becomesfrequency dependent. The CTRW calculationsIE295 K~ 10 -10 243 K1p 1 2193 K143 K88KI1Q1Qf {Hz)FIG. 6. Frequency-dependent conductivities as a function oftemperature for the as-deposited a-C film.agree very well with experimental o „at 323 K for bothfilrns. I;„for the annealed film is about one order largerthan that for as-deposited film. It is clear that the hoppingrate can be estimated from ac conductivity measurernent,Temperature dependence of ac conductivities as afunction of frequency for both films is shown in Figs. 9and 10. Solid circles and solid lines are experimental resultsand calculations, respectively. o. d, in Eq. (8) is calculatedby taking n, =N (E+ )k T= 1 X 10' cm at 300K, which is inferred from the ESR data, andR =n, ' =100 A. The best fit to as-deposited film is obtainedfor I;„=R(5)=2X10 [T/(10 K)]' sTaking the same n, and R for annealed film, which mustbe a valid assumption since the spin density (10' —10''cm ) is relatively independent of annealing, the bestE1Q--91Q-Q 1323 K273 K215 K177 K138 K100 K10 1p3f {Hz)FIG. 7. Frequency-dependent conductivity as a function oftemperature for the annealed a-C film.10'105


39 HOPPING TRANSPORT OF LOCALIZED m' ELECTRONS IN. . . 758310323 KIEO10—LAv -810—annealedas depositedpedimentcu [at ion30 kHz10—1011103t (Hz)I10510--103 kHzFIG. 8. Calculated (solid lines) and experimental (solid circles)ac conductivities at 323 K for the as-deposited and annealedfilms. The hopping rate I;„[Eq.(6)] is shown by the arrow.fit to annealed data is obtained for I;„=6X 10 ' [T/(10 K)]' ' s '. The calculated results agreewith experimental data except for 30 kHz. Disagreementbetween calculation and experiment at 30 kHz may bedue to an onset of quadratic dependence off on „o(seeFigs. 6 and 7), which has been usually observed in manyamorphous materials around this frequency.'lt issuggested that the increase in o. d, upon annealing couldbe due to an increase in multiphonon hopping rate [notdue to an increase in N(EF)].IEOcQ10—-101010— gs deposi te30 Hz1000/ T (KFIG. 9. Temperature dependence of the ac conductivity forthe as-deposited a-C film. Solid circles are experimental dataand solid lines are calculated results.HzHzHzQ 1 2 )QQQI T (K )30 Hz3Q HzFIG. 10. Temperature dependence of the ac conductivity forthe annealed film. Solid circles are experimental data and solidlines are calculated results.Finally, the physical parameter y which appeared inEq. (6) should be made to check the validity of the weakcouplingmodel. C exp( — yp) =2X 10 23 s'and6X 10 ' s' for as-deposited and annealed films, respectively,yield the same y=4. 5. This y value is dift'erentfrom y =4. 1 in the preceding paper in which the electronoverlapping term was taken into account. As theoverlapping term is implicitly included in the constant C,we could neglect this term here. The estimated y valuehere may satisfy the weak-coupling condition: Robertsonand Friedman require the condition of EM/hvo&(1.Englman and Jortner, on the other hand, have shownthat the weak-coupling regime should satisfy the conditionof G =(EM/hvo)(kT/hvo) 5 l. y=4. 5 leads tob, /E~=phvo/EM-245, which provides EM/hvo=0 07.with p =17.4 yielding 6 =2. 1 for 300 K and 6 =0.7 for1'00 K. Emin' required a more severe condition, 6 && 1,for weak-coupling states. However, as will be shown inthe Appendix, G =2 would still satisfy the weak-couplingcondition for relatively large p. Thus we conclude thatthe multiphonon hopping of localized electrons coulddominate electron transport in a-C. The increase in o. d,by annealing cannot be due to an increase in the carriernumber n„sin ecthe spin density (10' —10' cm ) isrelatively independent of annealing.'This is due to anincrease (one order) in the hopping rate I;„for annealedfilm.IV. CONCLUSIONSThe temperature dependence of dc and ac conductivitieswere studied in both the as-deposited and annealed


K. SHIMAKAWA AND K. MIYAKE 39a-C films. The dc conductivity is proportional to T",with n =15— 17 over a wide temperature range. Thisnonactivated temperature dependence cannot be explainedby mobility-edge conduction or variable-rangehopping or small-polaron hopping. This might be attributedto the nature of m. electrons which are not severelylocalized. The interaction of a m. electron with the latticecan be weak and the nonpolaronic multiphonon processis expected to occur. The ac conductivity is stronglycorrelated to the dc conductivity and was interpreted interms of the continuous-time random-walk approximation.It was shown that the hopping rate could be estimatedfrom the CTRW approximation.ACKNOWLEDGMENTSThe authors would like to thank Professor N. F. Mottfor useful comment on a disordered ~-electron system.We also wish to thank Dr. D. Emin for comment on themultiphonon process.APPENDIXThe multiphonon jump rate R (b) of localized electronscoupled to one vibrational mode coo has been derivedby EminR (b. )=Kg [I (z) cos(pP„)],z =(2E„/%coo) A„csch(iricoo/kT),(A 1)where Ic. is a certain characteristic frequency (probably anorder of coo), I (z) the modified Bessel function given bym( 2/4)kand Eb the binding energy (measure of couplingstrength), A„ the lattice relaxation amplitude function,and P„ the lattice relaxation-phase shift.In the weak-couplingR (5) cc [(4E„/ficoo)(kT/ficoo)]~/p!,(A3)where kT»R~p is assumed. When p is large, the Stirlingformula can be used asWritingand1/p! = p ~exp(p) .21Tp4EbScop= exp ln&~olimit (z =0), R (6) is given by4Eb&~o— (&/A~o)p ~=(b /%coo)' =exp — ln~o &~oEq. (A3) is rewrittenasR (b, ) ~ exp( yp)(kT/hc—oo) (A4)where y =In(b. /4Eb ) — 1. Emin' requires the conditionof G = (4EI, /ih'coo)( kT/A'coo) (( 1 for weak coupling.However, when p is large (around 15), the smallargumentapproximation [Eq. (A3)] can still be valid evenfor G =2 [see Eq. (A2)].J.J. Hauser, J. Non-Cryst. Solids 23, 21 (1977).N. Wada, P. J. Gaczi, and R. A. Solin, J. Non-Cryst. Solids 35,543 (1980).J. Robertson, Adv. Phys. 35, 317 (1986).4J. Robertson and E. P. O'Reilly, Phys. Rev. 8 35, 2946 (1987).5J. Robertson, Philos. Mag. Lett. 57, 143 (1988).K. Shimakawa and K. Miyake, Phys. Rev. Lett. 61, 994 (1988).7M. Pollak and T. H. Geballe, Phys. Rev. 122, 1742 {1961).H. Scher and M. Lax, Phys. Rev. 8 7, 4491 (1973).9J. C. Dyre, Phys. Lett. 108A, 457 (1985).C. J. Adkins, S. M. Freake, and E. M. Hamilton, Philos. Mag.22, 183 {1970).M. Morgan, Thiri Solid Films 7, 313 (1971).G. Grigorovici, A. Devenyi, A. Gheorghiu, and A. Belu, J.Non-Cryst. Solids 8-10, 793 (1972).N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials, 2nd ed. (Oxford University Press, Oxford,1979), p. 65.H. Fritzsche, in Amorphous and Liquid Semiconductors, editedby J. Tauc (Plenum, London, 1974), p. 221.N. F. Mott, Philos. Mag. 19, 835 (1969).M. H. A. Pramanik and O. Islam, J. Non-Cryst. Solids 45, 325(1981).' M. Ortuno and M. Pollak, J. Non-Cryst. Solids 59460, 53(1983).~ST. Holstein, Ann. Phys. (N.Y.) 8, 343 (1959).' D. Emin, Adv. Phys. 24, 305 (1975).E. Gorham-Bergeron and D. Emin, Phys. Rev. 8 15, 3667(1977).2D. Emin, Phys. Rev. Lett. 32, 303 (1974).N. Robertson and L. Friedman, Philos. Mag. 33, 753 (1976);36, 1013 (1977).A. R. Long, Adv. Phys. 31, 553 (1982).~4S. R. Elliott, Adv. Phys. 31, 135 {1987).I. G. Austin and N. F. Mott, Adv. Phys. 18, 41 (1969).M. Pollak, Philos. Mag. 23, 519 (1971).G. E. Pike, Phys. Rev. 8 6, 1572 (1972).8T. Odagaki and M. Lax, Phys. Rev. 8 24, 5284 (1981).S. Orzeszko, W. Bala, K. Fabisiak, and F. Rozploch, Phys.Status Solidi A 81, 579 {1984).3 R. Englman and J. Jortner, Mol. Phys. 18, 145 (1970).

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