Hopping transport of localized p electrons in amorphous carbon films

PHYSICAL REVIEW B VOLUME 39, NUMBER 11 15 APRIL 1989-I**Hopp ing**

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 l**in**es are calculated results **of** the nonadiabatic smallpolaronhopp**in**g.(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** **localized**tail 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** tail**localized** states and mobility, and f (E) is the Fermi distributionfunction. As the g (E) and p(E) decrease rapidlywhereas f(E) decreases with decreas**in**g energy (towardthe Fermi energy E~), the conduction path shiftstoward EF as the temperature is lowered, yield**in**g nonactivated**transport** 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 cont**in**uous distributionacross the band gap and electronic conduction byvariable-range hopp**in**g' (lncrd, &x T ' ) is expected tooccur **in** a-C as **in** the same manner as tetrahedral a-Geand a-Si. ' It is **of** **in**terest to exam**in**e whether thevariable-range hopp**in**g predom**in**ates 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 l**in**es are calculated results **of** the nonadiabatic smal1-polaxon hopp**in**g.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-hopp**in**g theory to data athigher temperatures are shown by the solid l**in**es and those atlower temperatures are shown by the dashed l**in**es.

7580 K. SHIMAKAWA AND K. MIYAKE 39The experimental results are not described by a s**in**glestraight l**in**e over the measured temperature range. Thefits to the data at higher temperatures are shown by thesolid l**in**es and those at lower temperatures are shown bythe dashed l**in**es. It is hard to dist**in**guish from the figurewhich l**in**e (solid or dashed) correctly describes the Tbehavior.The variable-range hopp**in**g 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 **amorphous**Ge 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 po**in**ted16, 17tN(EF) here is estimated from 16a 1kB, where a ' isthe effective Bohr radius **of** **localized** **electrons** and k theBoltzmann constant. Assum**in**g a '=10 A, althoughthis value may be smallerthan the practical value for a-Cbecause **of** the de**localized** nature **of** rr **electrons**, N(E~)for high-temperature fitt**in**g (solid l**in**es) 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 fitt**in**g (dashed l**in**es) 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 dom**in**ated bythe variable-range hopp**in**g. This may be attributed toweak localization **of** n defects (large eff'ective a '): Theprobability **of** s**in**gle-phonon hopp**in**g decreases with **in**creas**in**geNonactivated dc **transport** observed here is similar tothat for the small polaron which is formed **in** the strongelectron-lattice coupl**in**g.' lt is **of** **in**terest to knowwhether the experimental results can be **in**terpreted **in**terms **of** the small-polaron hopp**in**g. Before proceed**in**gwith the discussion, we summarize the small-polaronhopp**in**g theory **in** which both the acoustic and opticalare taken **in**to consideration.In general, the dc hopp**in**g conductivity can be given13phononsad, =n, (eR) I;„/6kT, (3)where n, is the number **of** carriers, R the hopp**in**g distance,and I;„the m**in**imum hopp**in**g rate.If the electron-lattice coupl**in**g is strong enough, thesmall polaron can be formed **in** disordered materials. 'Electrons should couple with both the optical and acousticphonons. Follow**in**g Holste**in**,'Gorham-Bergeronand Em**in** 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 **in**tegral between sites iand j, and 5 the energy difference between sites i and j.E;,E, Eg, and E~ are def**in**ed 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 dispersion**of** optical mode is assumed here), co~„ is the acousticphonon frequency at wave vector g, and N is the number**of** phonon modes. Two energies, Eb" and Eb', are the polaronicb**in**d**in**g energies related to optical and acousticphonons,respectively.The solid l**in**es, (a), (b), and (c) **in** Fig. 1, are calculatedresults us**in**g Eqs. (3)— (5). The phonon density **of** stateswas approximated to be g(v) oc v for an acoustic phononwith a cut**of**f'(Debye) frequency vD and the mean opticalphonon frequency was assumed to be vo=3vD. One **of**the important physical parameters is the Debye frequencywhich determ**in**es 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 sp**in** resonance (esr),and R = 5 A which may be reasonable for the average siteseparation **of** small-polaron hopp**in**g. Fitt**in**g 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 hopp**in**g. Byassum**in**g J,"=0. 1 eV, n, =1X10 cm is required forfitt**in**g, which seems to be too large for the carrier density.The solid l**in**es, (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. Fitt**in**gto 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. . . 7581hopp**in**g **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 dom**in**ate dc **transport** **in** a-C, although the qualitativenature is well expla**in**ed by the small-polaron hopp**in**gtheory. The small polaron cannot be formed whenthe extent **of** **localized** states is larger than the averagelattice spac**in**g. ' ' The orig**in** **of** the defect states **in** a +-electron system should therefore be considered. Robertsonsuggested that the deep **localized** states orig**in**atedfrom large ~-bonded clusters: The wave function is de**localized**across its cluster and the electronic hopp**in**gshould occur between clusters. The large-radius **localized****electrons** could couple most effectively to longwavelength(acoustic) phonons vo which are given by(ao/a ')vii, where ao is the average lattice'spac**in**g.The de**localized** **electrons** across its cluster will behave aspo**in**tlike defects with large n '. Hence, a'here shouldbe taken to be cluster size l, which behaves as "defect."Tak**in**g 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 assum**in**g ao =1.5 A. As the b**in**d**in**g energy**in** Eq. (5) under this situation is approximately given byEb'(ao/l, ), ' the electron-lattice coupl**in**g is expected tobe considerably smaller and hence no small polaron canbe formed **in** a-C. It is also believed that the s**in**glephononhopp**in**g process is not expected to occur for sucha small vo. ' ' ' We should next consider the multiphononprocess with weak-coupl**in**g states.The configurational coord**in**ate diagram for a weakcoupl**in**g is shown **in** Fig. 4. The energy EM **in** Fig. 4 is-the measure **of** coupl**in**g strength. The nonradiative'recomb**in**ation (phonon emission) rate with weak coupl**in**gis 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 hopp**in**g processes must **in**volve 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 hopp**in**g problem. The electron overlapp**in**gterm, which may be given as exp( — 2yR '), whereR' is the actual hopp**in**g distance and is roughly given byR — 2l„and y is the tunnel**in**g parameter, is implicitly **in**cluded**in** a constant C. R is the average center-to-centerdistance between clusters. The parameter y is the measure**of** coupl**in**g strength. Note that Eq. (6) is similar tothe multiphonon jump rate for weak-coupl**in**g states derivedby Em**in**' (see Appendix). od, is obta**in**ed 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 **in**tegral number.However, if 6 or vz distributes around a certa**in**mean value, p wi11 be non**in**tegral.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 straightl**in**es for all measured temperatures, show**in**g 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 coord**in**ate diagram expected for **localized**states **in** the weak electron-lattice coupl**in**g.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", suggest**in**g that the multiphononhopp**in**g with weak-coupl**in**g states dom**in**ates electronic**transport** **in** a-C. The values **of** n are 17.4 and 15.1for as-deposited and annealed **films**, respect;ively.1P8323 KB. ac conductivityBefore evaluat**in**g physical parameters n„R, and Iwhich appeared **in** Eq. (3), we turn to ac conductivitywhich will give **in**formation about the hopp**in**g 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 **in**terpreted eitherby the quantum-mechanical tunnel**in**g'(@MT) orcorrelated barrier hopp**in**g (CBH). These tuodels arebased on the pair approximation **in** which the motion **of**the carrier is conta**in**ed with**in** a pair **of** sites. If the dcand ac conductivities arise from the same hopp**in**g mechanisrn,the pair approximation cannot be applied.' " Asthe dc conductivity for a-C is expected to be dom**in**atedby the hopp**in**g process, the cont**in**uous-time randornwalk(CTRW) approximation orig**in**ally 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** most**of** the disordered materials,'a simple form **of** ac conductivityis given byl CO'7' ln(1+i co~)where ~ is the rnaximurn hopp**in**g time and approximatelyequal to I „ **in** Eqs. (3) and (6). It is noted that o„ isstrongly correlated to the dc conductivity. Thus the rneasurement**of** ac conductivity could provide knowledge **of**the hopp**in**g 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 bedom**in**ated by hopp**in**g **of** **localized** **electrons**. The ac**transport** could arise from the same hopp**in**g mechanism.To analyze ac conduction for this case, the CTRW approximationmust be a proper approach. The solid l**in**es**in** Fig. 8 show the calculated cr„(real part) at 323 K us**in**gEq. (8), where hard, is set to the experimental value (seefrequency-**in**dependent conductivity **in** Fig. 8). The I ~;„shown by the arrow is chosen to obta**in** the best fit, ly**in**gclose 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 **of**temperature 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 hopp**in**grate 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 l**in**es are experimental resultsand calculations, respectively. o. d, **in** Eq. (8) is calculatedby tak**in**g n, =N (E+ )k T= 1 X 10' cm at 300K, which is **in**ferred from the ESR data, andR =n, ' =100 A. The best fit to as-deposited film is obta**in**edfor I;„=R(5)=2X10 [T/(10 K)]' sTak**in**g the same n, and R for annealed film, which mustbe a valid assumption s**in**ce the sp**in** density (10' —10''cm ) is relatively **in**dependent **of** anneal**in**g, the bestE1Q--91Q-Q 1323 K273 K215 K177 K138 K100 K10 1p3f {Hz)FIG. 7. Frequency-dependent conductivity as a function **of**temperature 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 l**in**es) and experimental (solid circles)ac conductivities at 323 K for the as-deposited and annealed**films**. The hopp**in**g rate I;„[Eq.(6)] is shown by the arrow.fit to annealed data is obta**in**ed 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 **of**f on „o(seeFigs. 6 and 7), which has been usually observed **in** many**amorphous** materials around this frequency.'lt issuggested that the **in**crease **in** o. d, upon anneal**in**g couldbe due to an **in**crease **in** multiphonon hopp**in**g rate [notdue to an **in**crease **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 l**in**es 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 solidl**in**es are calculated results.F**in**ally, the physical parameter y which appeared **in**Eq. (6) should be made to check the validity **of** the weakcoupl**in**gmodel. 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 preced**in**g paper **in** which the electronoverlapp**in**g term was taken **in**to account. As theoverlapp**in**g term is implicitly **in**cluded **in** the constant C,we could neglect this term here. The estimated y valuehere may satisfy the weak-coupl**in**g condition: Robertsonand Friedman require the condition **of** EM/hvo&(1.Englman and Jortner, on the other hand, have shownthat the weak-coupl**in**g regime should satisfy the condition**of** 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 yield**in**g 6 =2. 1 for 300 K and 6 =0.7 for1'00 K. Em**in**' required a more severe condition, 6 && 1,for weak-coupl**in**g states. However, as will be shown **in**the Appendix, G =2 would still satisfy the weak-coupl**in**gcondition for relatively large p. Thus we conclude thatthe multiphonon hopp**in**g **of** **localized** **electrons** coulddom**in**ate electron **transport** **in** a-C. The **in**crease **in** o. d,by anneal**in**g cannot be due to an **in**crease **in** the carriernumber n„s**in** ecthe sp**in** density (10' —10' cm ) isrelatively **in**dependent **of** anneal**in**g.'This is due to an**in**crease (one order) **in** the hopp**in**g 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 expla**in**edby mobility-edge conduction or variable-rangehopp**in**g or small-polaron hopp**in**g. This might be attributedto the nature **of** m. **electrons** which are not severely**localized**. The **in**teraction **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 **in**terpreted **in**terms **of** the cont**in**uous-time random-walk approximation.It was shown that the hopp**in**g rate could be estimatedfrom the CTRW approximation.ACKNOWLEDGMENTSThe authors would like to thank Pr**of**essor N. F. Mottfor useful comment on a disordered ~-electron system.We also wish to thank Dr. D. Em**in** for comment on themultiphonon process.APPENDIXThe multiphonon jump rate R (b) **of** **localized** **electrons**coupled to one vibrational mode coo has been derivedby Em**in**R (b. )=Kg [I (z) cos(pP„)],z =(2E„/%coo) A„csch(iricoo/kT),(A 1)where Ic. is a certa**in** characteristic frequency (probably anorder **of** coo), I (z) the modified Bessel function given bym( 2/4)kand Eb the b**in**d**in**g energy (measure **of** coupl**in**gstrength), A„ the lattice relaxation amplitude function,and P„ the lattice relaxation-phase shift.In the weak-coupl**in**gR (5) cc [(4E„/ficoo)(kT/ficoo)]~/p!,(A3)where kT»R~p is assumed. When p is large, the Stirl**in**gformula can be used asWrit**in**gand1/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. Em**in**' requires the condition**of** G = (4EI, /ih'coo)( kT/A'coo) (( 1 for weak coupl**in**g.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. Sol**in**, 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. Adk**in**s, 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-Crystall**in**e 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. Holste**in**, Ann. Phys. (N.Y.) 8, 343 (1959).' D. Em**in**, Adv. Phys. 24, 305 (1975).E. Gorham-Bergeron and D. Em**in**, Phys. Rev. 8 15, 3667(1977).2D. Em**in**, 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. Aust**in** 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).