6 Description of Charge Transport in Disordered Organic ...baranovs/publications/2006/Ch_6... · 6.1 Introduction 222 ... ment of organic materials for applications in light-emitting

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6 Description of Charge Transport in Disordered Organic Materials

S.D. Baranovski and O. Rubel

Faculty of Physics and Material Sciences Center, Philipps Universität Marburg, Germany

6.1 Introduction 222 6.2 CharacteristicExperimentalObservationsandthe

ModelforChargeCarrierTransportinRandomOrganicSemiconductors 224

6.3 EnergyRelaxationofChargeCarriersinaGaussianDOS.TransitionfromDispersivetoNondispersiveTransport 228

6.4 TheoreticalTreatmentofChargeCarrierTransportinRandomOrganicSemiconductors 230

6.4.1 Averagingofhoppingrates 230 6.4.2 Percolationapproach 233 6.4.3 TransportenergyforaGaussianDOS 233 6.4.4 Calculationsoftrel andm 235 6.4.5 Saturationeffects 241 6.5 TheoreticalTreatmentofChargeCarrierTransport

inOne-dimensionalDisorderedOrganicSystems 243 6.5.1 Generalanalyticalformulas 245 6.5.2 Driftmobilityintherandom-barriermodel 246 6.5.3 DriftmobilityintheGaussiandisordermodel 248 6.5.4 Mesoscopiceffectsforthedriftmobility 251 6.5.5 Driftmobilityintherandom-energymodelwith

correlateddisorder(CDM) 253 6.5.6 Hoppingin1Dsystems:beyondthe

nearest-neighborapproximation 254 6.6 OntheRelationBetweenCarrierMobilityand

DiffusivityinDisorderedOrganicSystems 255 6.7 OntheDescriptionofCoulombEffectsCausedby

DopinginDisorderedOrganicSemiconductors 258 6.8 ConcludingRemarks 262 References 263

Charge Transport in Disordered Solids with Applications in Electronics EditedbyS.Baranovski.©2006JohnWiley&Sons,Ltd

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222 CHARGETRANSPORTINDISORDEREDSOLIDS

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6.1 INTRODUCTION

In this chapterwe focuson the theoretical description of charge transport in disorderedorganicmaterials.This topic is enormouslybroad since thevarietyof organicmaterialswithdifferentconductingpropertiesisveryrich.Insomeorganicsubstancessuperconduc-tivity has been found with transition temperatures as high as 10K [1]. Some polymericconductorsbasedonpolyacetyleneevidencethevaluesoftheelectricalconductivitieswhicharecomparabletothoseofthebestconductingmetals.Forexample,iniodine-dopedpoly-acetylenepreparedbytheNaarmannmethodavalueoftheconductivityabove105Wcm-1wasobtained[2].Manyorganicmaterialssuchaspolyethylenewithconductivitiesbelow10-8Wcm-1canbeconsideredasgoodinsulators.Itisnotpossible,ofcourse,tocoverthepropertiesofallthesematerialsinonechapter.In[3]somedeviceapplicationsoforganicmaterialsaredescribed.Untilnowelectrophotographicimagerecordinghasbeenthemaintechnique thatexploited theelectricalconductingpropertiesoforganicsolidsonabroadindustrialscale[4].Themaineffortsofresearchersarefocusedpresentlyonthedevelop-ment of organic materials for applications in light-emitting diodes (LEDs) as well as inphotovoltaics.Materialsusedforthesepurposesaremostlyrandomorganic,notablycon-jugated,or/andmolecularlydopedpolymerswithsemiconductingproperties[4,5].There-forewerestrictourdescriptioninthischaptertoconsideringelectricalconductioninsuchorganicsemiconductingmaterials.

Severalhandbookshavebeenpublishedrecently,whicharedevotedtothedescriptionofphysicalpropertiesanddeviceapplicationsoforganicmaterials.Forexample,numerousdetailsofthechemistry,physicsandengineeringofsemiconductingpolymerscanbefoundinrecentbooks[5,6].TotheinterestedreaderwecanalsorecommendthecomprehensivemonographofPopeandSwenberg[7]wherevariouselectronicprocessesinorganiccrystalsandpolymersarebeautifullydescribed.WewilldiscussbelowthechargecarriertransportproblemsraisedinChapterXIVofthatmonograph.

Discussingcharge transportpropertiesoforganicmaterials,oneshouldclearlydistin-guishbetween thepropertiesoforderedsystemssuchasmolecularcrystalsononehandand thoseofessentiallydisorderedsystems likemoleculardopedpolymerson theother.Transport models used to describe electrical conduction in these two distinct classes ofmaterialsareessentiallydifferent.

Inordertodescribegoodconductingpropertiesofsuchorganicmaterialsasmolecularcrystals,oneusuallyemploysratherstandardmethodsofsolid-statephysicsdevelopedforchargetransportincrystallineinorganicsolids.Forapplicationstoorganicmaterialsoneslightlymodifies the standard theoryby taking into account the strongelectron–phononinteraction leading topolaroneffectsandnonlinearexcitationssuchassolitons [1,7].Acompletelydifferentsetofideasisexploitedinordertodescribechargecarriertransportinessentiallydisorderedorganicmaterials,suchasmolecularlydopedpolymers[8],low-molecular-weightglasses[9,10]andconjugatedpolymers[11]. In thesematerialschargetransportisassumedtobeduetothevariable-rangehopping(VRH)ofelectronsorholesviastatesrandomlydistributedinspaceandenergylocalizedstates[1,5,7,8,12,13].Itisthistransportmodethatisdescribedinthischapter.ThequestionmayariseastowhyoneneedsdescribethistransportmodeoncemoreiftheVRHinapplicationtodisorderedorganicsemiconductorshasalreadybeendescribedinnumerousreviewarticles[1,8,12,13]andalsoinrecentbookchapters(forinstance,ChapterXIVin[7]andthechapterofH.Bässler in [4]).Wedo itbecausewefind the treatmentof theVRH inapplication to

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CHARGETRANSPORTINDISORDEREDORGANICMATERIALS 223

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organicdisorderedmaterialsgivenintheaboveliteraturetobeincomplete.Someexamplesoftheincompletenessaregivenbelow.

(i) IthasalreadybecomeatraditiontoclaimthattheGaussianshapeoftheenergydis-tribution of localized states (DOS) assumed for random organic systems preventsanalytical solutionof thehopping transportproblem[7,8,12,13].Weshow, to thecontrary,thatsuchasolutioncanbeeasilyobtainedwithinthestandardsetofideasfortheVRHintroducedin[14].

(ii) IthasalreadybecomeatraditiontoclaimthatthebesttheoreticalapproachtodescribetheVRHinrandomorganicsystemsisbasedonMonteCarlocomputersimulations[1,4,5,7,8,12,13].Weshowbelowthatalthoughcomputersimulationsoftenprovidevaluable informationon transportproperties, the simulation results shouldbe takenwith caution. For instance, some dependences of transport coefficients on materialparametershavebeenconsideredasuniversalsimplybecauseothermaterialparame-terswerenotchangedinthecourseofthesimulations.Furthermorefinitesizeeffectsincomputersimulationshavenotalwaysbeentreatedappropriately.

(iii) Inmostreviewarticlesandbookchapterspublishedsofartheeffectofcarriermobilitydecreasingwithincreasingelectricfieldwasspecifiedasoneofthemostchallengingfortheoreticalexplanation[4,5,7,8,12,13].Wewillshowbelow,forexactlysolvablemodels[15,16]thatthiseffectmightbeanartifactthathasbeencausedbymisinter-pretationofexperimentalresults.

(iv) ThereisnoconsensusbetweenresearchersontheverybasicquestiononwhethertheconventionalEinsteinrelationbetweenthechargecarriermobilityanddiffusivity isvalidforhoppingtransportindisorderedsystems.Weclarifythisproblembelow.

(v) RecentlymanycontradictorypapershavebeenpublishedonthetheoreticaltreatmentofhopingtransportinarandomsystemwithGaussianDOS.Inalldisorderedorganicandinorganicmaterialsaverystrongnonlineardependenceofthecarriermobilitym ontheconcentrationoflocalizedstatesN isobservedexperimentallyinthehoppingregime.Thisstrongdependenceiscausedbytheverystrongexponentialdependenceofthetransitionratesonthedistancesbetweenlocalizedstates.Whileforinorganicmaterialsthedependencem (N)hasbeenwelldescribedtheoreticallywithintheVRHapproach,andhasalreadybecomeasubjectoftextbooks[17],theverysamedepend-encem (N)stilllookspuzzlingformanytheoreticiansworkingwithorganicmaterials.Sometimes it isclaimedthat thedependencem (N) in thehoppingregimeshouldbelinear[18–21],inmarkedcontradictiontoexperimentaldata.Sometimesadependenceoftheformln(m)∝ -g (Na3)-1/3hasbeensuggested[22,23],wherea isthelocaliza-tionlengthandg isanumericalcoefficient.Thespreadofvaluesforg intheliteratureisenormous.Forexample,in[22]g wasestimatedas1.056<g <1.076whilein[23]itwasclaimedtobeintherange1.54<g <1.59,dependingontemperature.Thisdif-ferenceing leadstothedifferenceinm valuesatlowconcentrations(Na3)1/3<0.02ofmore than10ordersofmagnitude!.Since there isnocross-citationbetween[22]and[23],publishedinthesameyear,itisdifficulttoguesswhichofthesecontradic-toryresultstheauthorsconsiderascorrect.Weshowbelowthatneitheroftheseresultsseemcorrect.Furthermore,whileinsomerecentpublicationsitiscorrectlyclaimedthataveragingofhoppingratesleadstotheomissionoftheconcentrationdependence

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ofthecarriermobilitym (N)[23]inotherpublicationsbythesameauthorsitisclaimedthataveragingofhoppingratesiscapableofdescribingtheconcentrationdependenceofthemobility[24,25].Remarkably,whattheauthorscallaveragingofhopingratesin[24,25]isnotwhattheycallaveragingofhoppingratesin[18–21,23].Evenmoreremarkable is the statement of a recent publication [26] that the results of [18–21],claimingthelineardependencem (N),arecorrect.Thereareotherrecentpapersthatleadtothisconclusion[27,28].Thebeliefoftheresearchersworkingwithdisorderedorganicsemiconductorsintheaveragingofhoppingratesissostrongthatsometimestheagreementbetweenexperimentalresultsandthecorrectformulaswithexponentialdependence m (N) are called occasional, because the latter dependence cannot beobtainedbyaveragingofhoppingrates[29].

This unsatisfactory situation with theoretical description of hopping conductivity inorganic disordered solids shows that many researchers working with hopping transportin organic semiconductors are not familiar with the basic fundamental ideas, wellapproved and known in the parallel field of hopping transport in inorganic disorderedsystems.Our aim in this chapter is to provide thedescriptionof these elementary ideasinapplicationtorandomorganicsystems.Forexample,usingtheroutineVRHapproach[30,31]wecalculate thedependencem (N) inorganicdisorderedsolids. Itappears tobein agreement with experimental data. Only in the limit of very dilute systems is thedependenceln(m)∝ -g (Na3)-1/3valid,butwithg 1.73,inagreementwiththeclassicalpercolation result [17] and at variance with the results of [18–21, 23–25, 27, 28].

Inthenextsectionwebrieflydescribethemodelforchargecarriertransportindisor-deredorganicsemiconductorsformulatedandjustifiedinpreviousreviews[1,5,7,8,12,13]. In subsequent sections we describe theoretical methods which provide transparentsolutionsofvarioustransportproblemsintheframeworkofthismodel.

6.2 CHARACTERISTIC EXPERIMENTAL OBSERVATIONS AND THE MODEL FOR CHARGE CARRIER TRANSPORT IN RANDOM ORGANIC SEMICONDUCTORS

Althoughthevarietyofdisorderedorganicsolidsisveryrich,thedetailsofchargetrans-port in most of such materials are common. The canonical examples of disorderedorganic materials with the hopping transport mechanism are the binary systems thatconsistofdopedpolymericmatrixes.Examplesincludepolyvinylcarbazole(PVK)orbis-polycarbonate (Lexan)dopedwitheither strongelectronacceptors suchas, for example,trinitrofluorenoneactingasanelectrontransportingagent,orstrongelectrondonorssuchas, forexample,derivativesof triphenylamineor triphenylmethaneforhole transport.Toavoidthenecessityofspecifyingeachtimewhetherchargetransportiscarriedbyelectronsorholes,inthegeneraldiscussionbelowweusethenotation‘chargecarrier’.Theresultsarevalidforeachtypeofcarriers—electronsorholes.Chargecarriersinrandomorganicmaterialsarebelievedtobehighlylocalized.Localizationcentersaremoleculesormolecu-larsubunits,henceforthcalledsites.

Inordertounderstandthechargetransportmechanisminrandomorganicsolids,letusbrieflyrecallsomeexperimentalobservationsthataredecisiveforformulatingthetransport

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model.Amongthesearethedependencesofthecarrierdriftmobility,diffusivityandcon-ductivityontemperatureT,onthestrengthofappliedelectricfieldF,andonsuchimportantmaterialparametersasthespatialconcentrationoflocalizedstatesN.

(i) Dependence of transport coefficients on the concentration of localized states.Alreadyintheearlystagesofthestudyofmolecularlydopedpolymersitwasestablishedthatthe dependence of the carrier kinetic coefficients on the concentration of localizedstatesisverystrongandthatitisessentiallynonlinear[32].InFigure6.1thedepend-ence of the logarithm of the carrier drift mobility, ln(m), on R = N-1/3 obtained intime-of-flight experiments is shown. This extremely nonlinear dependence at lowconcentrationscanbefittedbytheexpression

µα

∝ -

-

exp2 1 3N

(6.1)

Figure 6.1 Electronandholemobilitydataplottedasafunctionoftheaverageseparationbetweenuncomplexedmoleculestreatedaslocalizedstates(reproducedwithpermissionfrom[32].Copyright1972,AmericanInstituteofPhysics)

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whichGill[32]suggestedinthisform,assumingthata isthelocalizationlengthofchargecarriersinthelocalizedstates.Asshownin[14],suchdependenceoftransportcoefficientsontheconcentrationoflocalizedstatesN ischaracteristicforincoherenthopping transportmechanism inwhich thechargecarrierhopsbetween thenearestavailable localizedstates. Indeed,sucha transportmode isexpectedfor thecaseoflowconcentrationsN,atwhichthemainlimitingfactorforcarriertransitionsbetweenlocalizedstatesisprovidedbytheexponentialdistancedependenceofhoppingprob-abilities (see [14]). At high concentrations, the deviations from the dependencedescribedbyEquation(6.1)wereobservedexperimentally[33],asexpectedforhoppingtransport in theVRHmode thatsucceeds thenearest-neighbor transportmodewithrisingN.Thisresultshouldbeconsideredasthekeyobservationleadingtotheconclu-sionthattransportmechanismindisorderedorganicsolidsistheincoherenthoppingofchargecarriersbetweenspatiallylocalizedstates.

(ii) Dependence of transport coefficients on temperature.Thetemperaturedependenceofthe drift mobility obtained in time-of-flight experiments in random organic solidsusuallytakestheform

µ ∝ -

expT

T0

2

(6.2)

whereT0isaparameter[1,4,5,7,8,12,13].However,inseveralstudiesanArrheniusdependencehasalsobeenreported

µ ∝ -

exp

DkT

(6.3)

whereD is theactivationenergy.Usually the latterdependence inorganicmaterialsisobservedbymeasurementsof theelectricalconductivity infield-effect transistors(see, for instance, [34]). We will show below that both temperature dependencesdescribedbyEquations(6.2)and(6.3)arepredictedtheoreticallyintheframeofthesametransportmodel,dependingonthetotalconcentrationofchargecarriersinthesystem.At lowcarrierdensities thedependencedescribedbyEquation (6.2) ispre-dicted, while at high carrier densities the dependence described by Equation (6.3)shouldbevalid.

(iii) Dependence of transport coefficients on the strength of the applied electric field.Numerous experimental studies have shown that at high electric fields the chargecarrier transportcoefficients increasewith thefieldstrengthF, beingapproximatelyproportionaltoexp[(F/F0)1/2]inabroadrangeoffieldstrengths,withsomeparameterF0[5,7,8,12,13].Morecuriously,ithasbeenreportedthatathightemperaturesthecarrierdriftmobilitymeasuredintime-of-flightexperimentsincreaseswithdecreasingfieldatratherlowfieldstrengthes[5,7,8,12,13].

Whenformulatingtheappropriate transportmechanismforchargecarriers inrandomorganic materials such as molecular glasses and molecular doped polymers, one shouldkeepinmindtheseexperimentalobservations.

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The corresponding model established to treat the charge carrier transport in randomorganicsolidsisasfollows[1,5,7,8,12,13].ItisassumedthatchargecarriersmoveviaincoherenthoppingtransitionsbetweenlocalizedstatesrandomlydistributedinspacewithsomeconcentrationN.Allstatesarepresumedtobelocalized.Theenergiesofchargecar-riersonthesestatesareassumedtohaveaGaussiandistributionsothatthedensityofstates(DOS)takestheform

gNε

σ πεσ

( ) = -

2 2

2

2exp (6.4)

wheres is the energy scale of the distribution and the energye is measured relative tothe center of the DOS. The origin of the energetic disorder is the fluctuation in thelatticepolarizationenergiesandthedistributionofsegmentlengthinthep-ors-bondedmain-chainpolymer[8].TheGaussianshapeoftheDOSwasassumedonthebasisoftheGaussianprofileoftheexcitonicabsorptionbandandbyrecognitionthatthepolarizationenergyisdeterminedbyalargenumberofinternalcoordinates,eachvaryingrandomlybysmallamounts[8].LaterwewillseethattheGaussianshapeoftheDOSaccountsfortheobservedtemperaturedependenceofthekineticcoefficientssuchascarrierdriftmobility,diffusivity,andconductivity.Theenergyscales oftheDOSinmostrandomorganicmate-rials is of the order of ~0.1eV [8]. In the initial model, no correlations between spatialpositionsoflocalizedstatesandtheirenergieswereincluded[8].Forthesakeofsimplicitywewillassumebelowthatthisassumptionvalidunlessthecontraryisspecified.

A tunneling transition rate of a charge carrier from a localized state i to a lower inenergy localized state j dependson the spatial separationrij between the sites i and j asn (rij)=n0exp(-2rij/a),wherea isthelocalizationlengthwhichweassumeequalforsitesi and j.This lengthdetermines the exponential decayof the carrierwavefunction in thelocalizedstates.Thedecaylengthonsinglesiteshasbeenevaluatedinnumerousstudiesoftheconcentration-dependentdriftmobility.Forexample,fortrinitrofluorenoneinPVKtheestimatesa 1.1× 10-8cmanda 1.8× 10-8cmwereobtainedforholesandelec-trons,respectively[32].FordispersionsofN-isopropylcarbazoleinpolycarbonatetheesti-matea 0.62× 10-8cmforholeshasbeenobtained [35].Thepreexponential factorn0depends on the electron interaction mechanism that causes the transition. Usually it isassumed that carrier transitions contributing to charge transport in disordered materialsarecausedbyinteractionswithphonons.Oftenthecoefficientn0issimplyassumedtobeof theorderof thephonon frequency~1013s-1, althoughamore rigorousconsideration isinfactnecessarytodeterminen0.Suchaconsiderationshouldtakeintoaccountthepar-ticular structure of the electron localized states and also the details of the interactionmechanism[36,37].

Whenachargecarrierperformsatransitionupwardinenergyfromalocalizedstatei toahigherinenergylocalizedstatej,thetransitionratealsodependsontheenergydiffer-encebetweenthestates.Thisdifferenceshouldbecompensated,forexample,byabsorptionof a phonon with the corresponding energy [38]. Generally, the transition rate from theoccupiedsitei toanemptysitej canbeexpressedas

v r vr

kTij i j

ij j i j i, , exp expε εα

ε ε ε ε( ) = -

-

- + -

0

2

2 (6.5)

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Ifthesystemisinthermalequilibrium,theoccupationprobabilitiesofsiteswithdifferentenergiesaredeterminedbytheFermistatistics.ThiseffectcanbetakenintoaccountbymodifyingEquation(6.5)andaddingtermswhichaccountfortherelativeenergypositionsofsitesi andj withrespecttotheFermienergyeF.Takingintoaccounttheseoccupationprobabilitiesoneshouldwritethetransitionratebetweensitesi andj intheform[38]

v vr

kTij

ij j F j F j i= -

-

- + - + -

0

2

2exp exp

αε ε ε ε ε ε

(6.6)

With thehelpof theseformulas theproblemof the theoreticaldescriptionofhoppingconductioncanbeeasilyformulated.Onehastocalculatetheconductivitywhichispro-videdbytransitioneventswiththeratesdescribedbyEquations(6.5)or(6.6)inthemani-foldoflocalizedstateswiththeDOSdescribedbyEquation(6.4).

Unfortunately,ithasbecomeacommonbeliefthattheGaussianformoftheDOSpre-ventsclosed-formanalyticalsolutionsofthehoppingtransportproblemsinrandomorganicsystemsandthereforethebestwaytostudytheseproblemsisacomputersimulation[7,8,12,13].Wewilldoourbestinthisreporttoprovetheopposite.However,beforedoingsowe would like to present several important results obtained previously by computersimulations.

6.3 ENERGY RELAXATION OF CHARGE CARRIERS IN A GAUSSIAN DOS. TRANSITION FROM DISPERSIVE TO NONDISPERSIVE TRANSPORT

One of the most remarkable results known for energy relaxation of charge carriers in aGaussianDOSistheexistenceoftheso-calledequilibrationenergy⟨e∞⟩ [7,8,12,13].ThissituationisincontrasttothecaseoftheexponentialDOS,whereintheemptysystematlow temperatures thechargecarrier always relaxesdownward inenergy, asdiscussed in[14].TheenergyrelaxationintheexponentialDOSleadstothedispersivecharacterofthechargetransport,inwhichkineticcoefficientsaretimedependentandthecarriermobilityslowsdown in thecourseof theenergy relaxation,asdescribed in [14]. In theGaussianDOS thechargecarrieronaverage relaxes fromhigh-energystatesdownward inenergyonlyuntilitarrivesattheequilibrationenergy⟨e∞⟩, evenintheemptysystemwithoutanyinteractionsbetween the relaxingcarriers [7,8, 12,13]. In computer simulations [8,12,13],noninteractingcarrierswereinitiallydistributeduniformlyinenergyandtheirfurtherrelaxationviahoppingprocesseswiththeratesdescribedbyEquation(6.5)wastraced.ThetemporalevolutionofapacketofnoninteractingcarriersrelaxingwithinaGaussianDOSisschematicallyshowninFigure6.2.InitiallytheenergydistributionofcarrierscoincideswiththatoftheDOS.Inthecourseoftimethecarrierenergydistributionmovesdownwarduntilitsmaximum⟨e (t)⟩ arrivesattheenergy⟨e∞⟩=limt→∞⟨e (t)⟩=-s2/kT [8,12,13].Thisresultcanbeeasilyobtainedanalytically[39].Inthermalequilibrium

εε ε ε ε

ε ε ε εσ

∞-∞∞

-∞∞

=( ) -( )( ) -( )

= -∫∫

d

d

g kT

g kT kT

exp

exp

2

(6.7)

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Thetimerequiredtoreachthisequilibriumdistribution(calledtherelaxationtime)trel isofkey importance for the analysisof experimental results [40]. Indeedat time scalesshorterthantrel,chargecarriersinitiallyrandomlydistributedoverlocalizedstatesperformadownwardenergyrelaxationduringwhichtransportcoefficients,suchasthecarrierdriftmobility, essentially depend on time, and charge transport is dispersive, as described in[14]. At time scales longer thantrel, the energy distribution of charge carriers stabilizesaround the equilibration energy ⟨e∞⟩, even in a very dilute system with noninteractingcarriers.Insucharegime,transportcoefficientsaretimeindependent.Inotherwords,att trel dispersivetransportissucceededbythenondispersive(Gaussian)transportbehavior.Thisisoneofthemostimportantresultsforchargecarriertransportindisorderedorganicmedia[8].Whileatshorttimesdispersivecurrenttransientswereobservedinsuchmateri-als,atlongtimestransportcharacteristicsarenondispersive,timeindependentandhencetheycanbewellcharacterizedanddescribedincontrasttoanalogousquantitiesininorganicdisorderedmaterialswithexponentialDOS.Inthelattermaterials,asshownin[8],trans-port coefficients in dilute systems are always dispersive (time dependent). Hence, theydependonsuchexperimentalconditionsasthelengthofasampleandthestrengthoftheelectricfield.ThereforethetransportcoefficientsindisorderedmaterialswithexponentialDOScanhardlybecharacterized.On thecontrary, inorganicdisorderedmaterialswithGaussian DOS, transport coefficients do not depend on the experimental conditions att >trel andhencetheycanbewellcharacterized.Ithasbeenestablishedbycomputersimu-lationsthattrel stronglydependsontemperature[8,40],

τ σrel exp∝

B

kT

2

(6.8)

wherenumericalcoefficientB isclosetounity:B 1.1.Ithasalsobeenfoundbycomputersimulationsthatthecarrierdriftmobilitym atlongtimescorrespondingtothenondispersivetransportregimehasthefollowingpeculiartemperaturedependence[7,8,12,13,40]

Figure 6.2 Temporal evolution of the distribution of carrier energies in a Gaussian DOS(s /kT=2.0).⟨e∞⟩ denotesthetheoreticalequilibriumenergydeterminedbyEquation(6.7)(repro-ducedwithpermissionfrom[8].Copyright1993,Wiley-VCH)

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µ σ∝ -

exp

C

kT

2

(6.9)

ForcoefficientC thevalueC=2/3hasbecomeaconventionalone[7,8,12,13,40].Infact,computer simulations [41]give for thiscoefficient thevalueC 0.69,while rathersophisticatedanalyticalcalculations[39,42]predictaclosevalueC 0.64.Equation(6.9)withC=2/3isbelievedtobeuniversal,anditiswidelyusedtodeterminetheenergyscales oftheDOSfromtheexperimentalmeasurementsoftheln(m)versusT -2[43,44].

However,itlooksratherstrangethatthecoefficientsC andB inEquations(6.8)and(6.9)areconsideredastheuniversalones.Infact, it is themainfeatureoftheVRHtransportmodethatthespatialandenergyparametersareinterconnectedinthefinalexpressionsfortransportcoefficients[17].Equations(6.5)and(6.6)showthatboththeenergydifferencebetween localized states and the spatial distance between them determine the hoppingprobability.Hoppingtransportinasystemwithspatiallyandenergeticallydistributedlocal-izedstatesisessentiallyaVRHprocessasdescribedin[14].Insuchaprocess,thetransportpathusedbychargecarriersisdeterminedbybothenergyandspatialvariablesintransitionprobabilities. It means that the temperature dependence of transport coefficients shouldinclude the spatialparameterNa3,while thedependenceof transport coefficientson theconcentrationN of localizedstatesshoulddependon the temperaturenormalizedby theenergyscaleoftheDOS,kT/s.Incomputersimulations[41],aparticularvalueofthespatialparameterwastaken:Na3=0.001andthemagnitudeC 0.69wasobtainedforthispar-ticularvalueofNa3.ThequestionthenarisesonwhetherthisvalueofC isstableagainstvariationsofNa3.

Thereisanotherimportantquestion,alreadyraisedinthescientificliterature[40],thatisrelatedtotheapparentdifferenceinthetemperaturedependencesoftrel andm expressedbythedifferencebetweencoefficientsC andB inEquations(6.8)and(6.9).Therelaxationtimetrel atwhichatransitionfromdispersivetonondispersivetransportshouldtakeplacedependsontheratios /kT morestronglythanonthecarrierdriftmobilitym.Thishasanimportantconsequencethatatime-of-flightsignalproducedbyapacketofchargecarriersdrifting across a sample of some given length must become dispersive above a certaindegreeofdisorder,i.e.,belowacertaintemperatureatotherwiseconstantsystemparame-ters,becauseeventuallytrel willexceedthecarriertransienttime[40].Thetimeevolutionofthetime-of-flightsignalisshowninFigure6.3asafunctionoftheratios /kT.Theearlyanalyticaltheories[39,42]alreadyindicatedthedifferencebetweencoefficientsC andB,althoughtheypredictedC >B incontrasttothesimulationresults.Thereforeabetterana-lytical theory for description of the VRH transport in a system of random sites with aGaussiandistributionisdefinitelydesirable.

6.4 THEORETICAL TREATMENT OF CHARGE CARRIER TRANSPORT IN RANDOM ORGANIC SEMICONDUCTORS

6.4.1 Averaging of hopping rates

Theattempts todevelopamore transparentandusable theory than that in [39,42]wereperformedbyArkhipovandBässler [18–21].The resultobtained for thedependencesof

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the carrier mobility on temperature, T, and on the concentration of localized states, N,reads[18–21]

µ π ασ

σ= -

1 20

5 2

2

1

4

ev N

kTexp (6.10)

Asimilarexpression isprovidedby themodelofRoichmanandco-workers [27,28].This result gives for the temperature dependence a numerical coefficient 1/4 in front of(s /kT)2intheexponent,whichistwiceassmallastheone(1/2)predictedbycomputersimulations[8,40,41]andbypreviousanalytical theories[39,42].Moreimportant, thisexpressionpredictsalineardependenceofthecarriermobilitym onN incontradictiontoexperimentalresults[32].

Thereasonforthesecontradictionsisrathertransparent.Equation(6.10)isobtainedbythe configurational averaging of hopping rates [18–21]. Although this method has beenalready analyzed in textbooks (see, for instance, [17]) andqualified as inappropriate for

Figure 6.3 Time-of-flight signal, parametric in s /kT (sample length 8000 lattice planes,F =6×105Vcm-1)(reproducedwithpermissionfrom[8].Copyright1993,Wiley-VCH)

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treatmentofhoppingtransportindisorderedmaterials,itisrepeatedlyusedbynumerousresearchers.Thisoccursnotonlyinthestudiesofchargetransport,butalsointhestudiesofthehoppingcarrierenergyrelaxation.Forexample,itisoftenclaimedintheliteraturethatexponentialdependenceofthelocaltransitionratesonthedistancesbetweenlocalizedstatesis‘averagedout’inthefinalresultsfortherelaxationrateleadingtothelineardepend-enceoftherelaxationrateontheconcentrationofstatesN (see,forinstance[45]).However,ithadbeenalreadyemphasizedbyAmbegaokaret al.in1971[46]ataveryearlystageofthestudyofhoppingtransportthatanessentialingredientofasuccessfultheoryforhoppingtransportindisorderedmaterialsisthatittakesintoaccountthathoppingisnotdeterminedbytherateof‘average’hops,butratherby therateof thosehops thatare‘mostdifficultbutstillrelevant’:hoppingconductionisinfactapercolationproblem.Unfortunately,thebeliefoftheresearchersworkingwithdisorderedorganicsemiconductorsintheaveragingofhoppingratesissostrongthatsometimestheagreementbetweenexperimentalresultsontheexponentialdependencem (N)withEquation(6.1)arecalledoccasional,becausethelatterequationcannotbeobtainedbyaveragingofhoppingrates[29].Wethereforefinditnecessarytoclarifyoncemoretheinvalidityof theensembleaveragingofhoppingratesforthedescriptionofhoppingtransportindisorderedmaterials,althoughthisinvalidityhasbeenalreadyexplainedintextbooks(see,forinstance,[17]).

Intheapproachbasedontheaveragingofhoppingrates,oneassumesthatcarriermobil-ityisproportionaltotheaveragehoppingrate⟨n⟩ multipliedbythesquaredtypicaldisplace-mentr2ofachargecarrierinsinglehoppingevents:m ∝ r2⟨n⟩.Theshortcomingofsuchatreatmentismostlytransparentinthecaseofhightemperatures,kT >s,atwhichchargetransport takesplacevianearest-neighborhopping.Thelatter is trueunder theconditionNa3<<1necessaryforelectronstatestobelocalized,asassumedinthemodel[7,8,12,13,40].AthighT, theenergy-dependent terms in transitionprobabilitiesdonotplayanessentialroleandthehoppingratesaredeterminedmostlybyspatialseparationsbetweenlocalizedstates:n (r) n0exp(-2r/a).Multiplyingthistransitionprobabilitybytheprob-abilitydensitytofindthenearestneighboratagivendistancer,providedthetotalconcen-trationofsitesisN:j(r)dr=4pr2N exp(-4pr3N/3)dr andintegratingoverdistances,oneobtainsfortheaveragehoppingrate:

v r vr

r N r N v N= -

-

∞

∫ d 00

2 30

324

4

3exp exp

απ π π α (6.11)

Itiseasytounderstandthisresult.Duetotheverystrongdecreaseofthefunctionn (r)=n0 exp(-2r/a) with increasing r at the scale r a and due to the weak dependenceof the functionj(r)= 4pr2N exp(-4pr3N/3)onr atr a , theaveragehopping rate inEquation(6.11)isdeterminedbytransitionswithr a ,sincethemaincontributiontotheintegralcomesfromsuchdistancesr a.Assumingm ∝ r2⟨n⟩ withr a and⟨n⟩ describedbyEquation(6.11)oneobtains[18–21,27,28]m ∝ n0Na5,asexpressedinEquation(6.10).Thisresultishoweverinvalidforadilutesystemoflocalizedstates,forwhichtheconditionNa3<< 1shouldbefulfilledtojustifythelocalization.Theaveragehoppingrateisdeter-mined by transitions over the distances r a with the raten n0. The probability offindingsuchaclosepairofsiteswithr a atthetotalconcentrationN ofsitesispropor-tionaltoNa3.Thisisthereasonwhytheresultfortheaveragehoppingrate⟨n⟩ islinearlyproportional to N. However, the charge carrier cannot move over considerable distancesusingonly transitionswith the lengthr a ina systemwith lowconcentrationof sites

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Na3 << 1. Therefore, the averaging of hopping rates cannot describe the charge carrierkineticcoefficientsinarandomorganicmaterialsandothertheoreticalmethodsshouldbeused.

6.4.2 Percolation approach

Oneofthemostpowerfultheoreticaltoolstodescribechargecarriertransportindisorderedsystems isprovidedby thepercolation theoryasdescribed innumeroushandbooks (see,for instance, [17]).Webrieflydescribed thismethod inChapter2[14].According to thepercolationtheory,onehas toconnectsiteswithfastest transitionrates inorder tofulfilltheconditionthattheaveragenumberZ ofbondspersiteonthetransportpathisequaltotheso-calledpercolationthreshold[17,47,48]:

Z B≡ = ±C 2 7 0 1. . (6.12)

Thismethodhasbeenverysuccessfullyappliedtothetheoreticaldescriptionofhoppingtransport indopedcrystalline semiconductors [17]andalso indisorderedmaterialswithexponential DOS [49]. The treatment of charge transport in disordered systems with aGaussianDOSintheframeworkofthepercolationtheorycanbefoundin[50,51].However,thistheoryisnoteasyforcalculations.ThereforeitisdesirabletohaveamoretransparenttheoreticaldescriptionoftransportphenomenaindisorderedsystemswithaGaussianDOS.Inthenextsectionwepresentsuchanapproachbasedonthewell-approvedconceptofthetransportenergy,thatwassuccessfullyappliedearliertodescribetransportphenomenaininorganicdisorderedsystemswithexponentialDOS.

6.4.3 Transport energy for a Gaussian DOS

Theroutineandsofarwidelyacceptedinterpretationofthetemperaturedependenceofthecarrier mobility described by Equation (6.9) claims that activation of carriers from theenergylevel⟨e∞⟩ determinedbyEquation(6.7)tosometransportlevelisresponsibleforthedriftmobilitym [4,8,12,13,40].Furthermore,ithasbeenclaimed[52]thatEquation(6.9)alongwith(6.7)wouldascertainthetransportleveltobeat⟨e∞⟩ +(Cs)2/kT ≈ -5s2/(9kT).However, in such interpretation one treats the hopping transport as a pure activation ofchargecarriersfromthelevel⟨e∞⟩ overtheactivationbarrierwiththehight(Cs)2/kT neces-sarytoobtainEquation(6.9).Thetransportmechanism,inwhichtransitionprobabilitiesarelimitedsolelybytheenergyactivation,mightbevalidinorderedcrystallinesemicon-ductors.Transportinrandomdisorderedmaterialsis,however,ahoppingprocesslimitednotonlybytheenergyactivation,butalsobythenecessityforchargecarrierstotunnelinspacebetweenthelocalizedstates.Itistheinterplaybetweenspatialandenergy-dependentterms in the transitionprobabilities thatdetermines thehopping transportprocess in theVRHregime.Remarkablyitissometimesclaimed[52]thatindisorderedsystemswithaGaussian DOS there is no transport energy in the VRH sense. We disagree with suchstatements and show below that the transport energy for the Gaussian DOS can becalculated in full analogy with the corresponding derivation for the exponential DOSdescribedin[14],whereitwasshown,following[53–57]thataparticularenergylevelet,

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called by Monroe [53] the transport energy (TE), determines all hopping transportphenomena.

Theconceptof transportenergycanbeeasilyextended todescribehopping transportphenomenainadisorderedsystemwiththeGaussianDOS[58].SincethederivationoftheTEfortheGaussianDOSisabsolutelyanalogoustothatfortheexponentialDOSdescribedin[14],wepresentherethisconceptinacompactedform.TheintroductionoftheTEmakessense only at low temperatures, kT <s, in the VRH regime, since otherwise the trivialnearest-neighborapproximationisvalid(see[14]).Qualitatively, theenergyrelaxationofcarriersinaGaussianDOSisschematicallyshowninFigure6.4.Atlowtemperatures,kT <s,carriersplacedinthehigh-energypartoftheGaussianDOSperformdownwardenergytransitionsuntiltheyreachaparticularenergylevelet calledthetransportenergy.Atet thecharacteroftherelaxationchanges.Afterahoptoastatebelowet,thecarrierhopsupwardinenergytoastateinthevicinityofet.Thishoppingprocessnearandbelowet resemblesamultiple-trappingprocesswhereet playstheroleofthemobilityedge.Inordertocalculateet,oneshouldfindthemaximumofthehoppingtransitionprobabilitywithrespecttotheenergyofthefinalstateex foracarrierplacedinitiallyatasitewithdeepenergyei:

d

d

v i x

x

↑ ( ) =ε εε

, .0 (6.13)

Itiseasytoshow[58]thatthefinalenergyet determinedbyEquation(6.13)doesnotdependontheinitialenergyei andhenceitisuniversalforgivenparameterss /kT andNa3.TheTEinarandomsystemwiththeGaussianDOSisdeterminedas

ε σ α σt = ( )X N kT3, (6.14)

whereX(Na3; kT/s)isthesolutionoftheequation[58]

Figure 6.4 SchematicpictureofcarrierenergyrelaxationintheGaussianDOSviathetransportenergy et (reproduced with permission from [31]. Copyright 2000 by the American PhysicalSociety)

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exp expX

t t N kTX2

22

4 31 2 3 1 3

29 2

-( )

= ( ) -∞

-

∫ d π α σ (6.15)

Ithasbeenshown[58]thatatlowtemperatures,kT <s,themaximumdeterminedbyEquation(6.13)issharpandthereforetheintroductionoftheTEforaGaussianDOSmakessense.AnequationfortheTEinaGaussianDOSliterallycoincidingwithEquation(6.15)waslaterpublishedbyArkhipovet al.[59],whoalsoclaimedthattheconceptoftransportenergyisapplicabletopracticallyallrealisticDOSdistributions.Wecannotagreewiththelatterstatement.Ithasbeenshownin[58]that,forexample,inasystemwiththeDOSinthe form g ε ε ε( ) ∝ -( )exp 0 themaximumdeterminedbyEquation (6.13) is sobroadandthepositionofet issodeepinthetailoftheDOSthatintroductionoftheTEmakesnosense.TheDOSdescribedbythelatterformulaisknowntobevalidforsuchabroadclassofdisorderedsystemsasthemixedcrystalswithcompositionaldisorder[60].Henceit should be considered as a realistic one. Furthermore, sometimes the TE approach isapplied toasystemwithaconstant,energy-independentDOS[61].Wedonot think thatsuchaprocedureismeaningful.IftheenergydependenceoftheDOSisweak,thetransportpathintheequilibriumconditionscorrespondstothevicinityoftheFermilevel.Concomi-tantly, the temperaturedependenceof theconductivityobeys theclassicalMott formula:s (T)∝ (T0/T)1/4,asdescribedin[14].Inthenexttwosubsectionswedescribehowonecanuse theconceptof transportenergy inorder toexplainexperimentallyobserveddepend-ences of the transport coefficients on temperature and on the concentration of localizedstates,N,inasystemwithGaussianDOS.

6.4.4 Calculations of trel and m

Afterhavingunderstood the relaxationkinetics, it is easy to calculatetrel andm.Letusstartwithtemperaturedependencesofthesequantities.Weconsiderthecase⟨e∞⟩ <et <0,whichcorresponds toany reasonablechoiceofmaterialsparametersNa3 andkT/s [58].Aftergenerationofcarriers inanonequilibriumsituation, thecarrierenergydistributionmovesdownwardinenergywithitsmaximum⟨e (t)⟩ logarithmicallydependentontimet [8,40,41].Statesabove⟨e (t)⟩ achievethermalequilibriumwithet attime t,whilestatesbelow⟨e (t)⟩ havenochanceattimet toexchangecarrierswithstatesinthevicinityofet,andhencetheoccupationofthesedeepenergystatesdoesnotcorrespondtotheequilibriumoccupation. The system of noninteracting carriers comes into thermal equilibrium whenthetime-dependentenergy⟨e (t)⟩ achievestheequilibriumlevel⟨e∞⟩ determinedbyEquation(6.7).Thecorrespondingtimeistherelaxationtimetrel [8,40,41].Atthistime,dispersiveconductionisreplacedbytheGaussiantransport[8,40,41].Aslongastherelaxationofcarriersoccursviathermalactivationtothelevelet,therelaxationtimetrel isdeterminedbytheactivatedtransitionsfromtheequilibriumlevel⟨e∞⟩ tothetransportenergyet [31].HenceaccordingtoEquation(6.5),trel isdeterminedbytheexpression[31]

τ εα

ε εrel

t t= ( ) + -

- ∞vr

kT0

1 2exp (6.16)

where

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r gε π ε εε

( ) = ( )

-∞

-

∫4

3

1 3

d (6.17)

From Equations (6.14–6.17) it is apparent that the activation energy of the relaxationtimeinEquation(6.16)dependsonbothparameterss /kT andNa3,andtherefore,generallyspeaking,thisquantitycannotberepresentedbyEquation(6.8)and,ifatall,thecoefficientB shoulddependonthemagnitudeoftheparameterNa3.

Letuscalculatetheexponentinthetemperaturedependenceoftrel inEquation(6.16),ln(treln0),andplotitasafunctionof(s /kT)2forvariousvaluesofNa3.SolvingEquation(6.15) numerically and using Equations (6.7), (6.14–6.17), one obtains results given inFigure6.5.ForNa3=0.001,thedependencedescribedbyEquation(6.8)withB 1.0isingoodagreementwithcomputersimulationsthatgiveB 1.1[8,40].HencetheresultoftheanalyticcalculationsconfirmsEquation(6.8).However,Equation(6.8)canbecon-sideredonlyasanapproximationbecause itoriginates fromratherdifferent temperature

Figure 6.5 TemperaturedependencesoftherelaxationtimefordifferentvaluesofNa3.SolidlinesrepresentthebestfitsintheformofEquation(6.8)fordependencesobtainedbynumericalsolutionofEquations(6.15)and(6.16).ThevaluesoftheparameterB obtainedfromsuchafitarespecifiedin the text (reproduced with permission from [31]. Copyright 2000 by the American PhysicalSociety)

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dependencesofdifferenttermsintheexponentofEquation(6.16).Thispossibilityofeffec-tivedescriptionviaEquation(6.8)isprovidedbythestrongtemperaturedependenceof⟨e∞⟩ givenbyEquation (6.7)while the temperaturedependencesof thequantitieset andr(et)areweakerandtheyalmostcanceleachotherinEquation(6.16).However,atanothervalueof theparameterNa3=0.02, the relaxation timetrel isdescribedbyEquation(6.8)withanothervalueofthecoefficientB 0.9.ThisclearlyshowsthatthetemperaturedependenceinEquation(6.8)cannotbeuniversalwithrespecttotheconcentrationoflocalizedstatesN andthedecayparameterofthecarrierwavefunctiona.

Nowweturntothecalculationofthecarrierdriftmobilitym.Weassumethatthetran-sienttimettr,necessaryforacarriertotravelthroughasampleislongerthantrel,andhencethechargetransporttakesplaceintheequilibriumconditions.Asdescribedabove,everysecondjumpbringsthecarrierupwardinenergytothevicinityofet, beingsucceededbya jump to the spatially nearest site with deeper energy determined solely by the DOS.Therefore, inorder tocalculate thedriftmobility,m , it iscorrect toaveragethe timesofhopping transitions over energy states below et, since only these states are essential forcharge transport in thermal equilibrium [30, 31]. Hops downward in energy from thelevelet occurexponentiallyfasterthanupwardhopstowardset.Therefore,onecanneglectthe former in the calculation of the average time hti. The carrier drift mobility can beevaluatedas

µ ε

e

kT

r

t

2t( )

(6.18)

wherer(et)isdeterminedbyEquations(6.17),(6.14),(6.15)and(6.4).Theaveragehoppingtimehastheform[30]

t g v gr B

kT

t

= ( )

( ) ( )

+-

-∞

--

-∞∫ d d t c1 3

tε ε ε ε εα

ε εε 1

01 2

expεεt

∫ (6.19)

whereBc 2.7 is thepercolationparameter takenfromEquation(6.12).ThisnumericalcoefficientisintroducedintoEquation(6.19)inordertowarranttheexistenceofaninfinitepercolationpathoverthestateswithenergiesbelowet.UsingEquations(6.4),(6.14),(6.15),(6.18) and (6.19), oneobtains for the exponential terms in the expression for the carrierdriftmobilitytherelation

ln expµ ε π απer v

kT BN t t

X20

1

3 224

3t

c

dt( )

= - -( )

-

-∞∫

- -

-1 3

21

2

X

kT kTtσ σ

(6.20)

ItisEquation(6.20)thatdeterminesthedependenceofthecarrierdriftmobilityontheparametersNa3ands /kT.

InFigure6.6thedependenceofthedriftmobilityontemperatureatNa3=0.01isshownfor several values ofs. Equation (6.20) confirms the validity of Equation (6.9), thoughwiththecoefficientC slightlydependentonthevalueoftheparameterNa3.Incomputer

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simulationsthevalueNa3=0.001waschosenandC 0.69wasobtained[41].Equation(6.20)predictsaveryclosevalueC 0.68forNa3=0.001.However,atNa3=0.02weobtainfromEquation(6.20)adifferentvalueC 0.62,whichshowsthatthiscoefficientinEquation(6.9)indeeddependsonthevalueofparameterNa3asexpectedfortheVRHtransportprocess.VerysimilardataforC wereobtainedinseveralotherpapers[61–63].

It is clear from thispicturehowsensitive is themobility to thevalueof temperature.Comparisonofthesedependenceswithexperimentalmeasurementsoftheln(m)versusT -2(someare shown inFigure6.7)provides informationon the energy scales of theDOS(see,forexample,[8,44]).WewouldliketoemphasizehereoncemorethatthechoiceofthenumericalcoefficientC inEquation(6.9)isimportantfortheestimationofs fromthecomparisonwithexperimentaldata.Thisparameter isnotuniversal,beingdependentonthevalueofNa3.

InFigure6.8,thedependenceofthedriftmobilityonNa3isshownforkT/s=0.3.AlsoexperimentaldataofGill[33]areshowninthefigure.Itiswellillustratedthattheslopeof the mobility exponent as a function of (Na3)-1/3 given by the theory described aboveagreeswiththeexperimentaldata.Comparisonbetweenthetheory[30]andexperimentalresults[32,33,35,64]providesanestimateforthedecayparametera ofthecarrierwave-functioninlocalizedstates.Thevaluesbetweena 0.1nmanda 0.3nmareobtained

Figure 6.6 Temperaturedependenceofzero-fieldmobilityinorganicsemiconductorsobtainedviaEquation (6.20) for different disorder energy scales,s (reproduced with permission from [109].Copyright2006bySpringer)

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dependingonthechosenpolymerandthedopingspecies.Atverylowconcentrationofthelocalizedstates,N, theprobabilityofcarrier tunneling in spacedominates the transitionrate inEquation (6.5). In this regimeof thenearest-neighborhopping, theconcentrationdependenceofthedriftmobilityisdescribedbytheexpression[17]

µ γα

∝ -

-

expN 1 3

(6.21)

whereg=1.73± 0.03[17]asexplainedin[14].Equation(6.21)isillustratedbythedashedlineinFigure6.8.

Inseveralrecentpublications,thedependencedescribedbyEquation(6.21)forrandomorganicsolidswasclaimed,butwithdifferentnumericalcoefficientsg [22–25].Wesupposethatthesedeviationsintheobtainedvaluesofthecoefficientg fromthewell-knowntext-bookresultg=1.73± 0.03arecausedby theunjustifiedassumption thatcarriersalwayshop to theneighboringsitewith theoptimalexponent in the transitionratedescribedby

Figure 6.7 Experimentaldatafortemperaturedependenceofzero-fieldmobilityinorganicsemi-conductors:(1)di-p-tolylphenylaminecontaining(DEASP)-traps[110];(2)(BD)-dopedpolycabonate[43];(3)(NTDI)-dopedpoly(styrene)[111];(4)(BD)-dopedTTA/polycarbonate[112](reproducedwithpermissionfrom[109].Copyright2006bySpringer)

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Equation(6.5)[22–25].Suchanassumptionisoftenmadeintheoreticalstudiesofhoppingtransport in organic materials (see, for instance [65]). Expression (6.12) shows howeverthat,whencalculatingtheDC hoppingtransport,oneshouldtakeintoaccountonaveragenot a single neighbor, but at least Bc 2.7 neighbors per site. This average numberBc 2.7isresponsibleforthecorrectvalueofthecoefficientg=1.73± 0.03inEquation(6.21)asdescribedin[14]followingthetextbooks[17].

Wewouldalso like tocommenthereon thecontradictorystatements in the literatureabout the invalidity/validityof theaveragingprocedureforhoppingrates todescribe thehoppingconductivityinrandomsystems.InSection6.4.1,itisshownthatthisprocedureininvalid(seealso[17]).Thishasbeenconfirmedinseveralrecentpublications(see,forinstance,[23]).However, inotherpublicationstheaveragingprocedureforhoppingrateswasclaimedcapableofdescribing thehoppingconductivity [22,24,25,66]. It isworthnotingthattheprocedure,whichtheauthorscalltheaveragingofhoppingratesin[22,24,25,66]isnottheprocedure,whichtheycalltheaveragingofhoppingrates[23]andwhichwasusedin[18–21]anddescribedinSection6.4.1.In[22,24,25,66],theaveragehoppingrateisdefinedviathefollowingchainofarguments.Ononehand,thegeneralexpressionfortheDC conductivityinthefollowingformisused[67]

Figure 6.8 ConcentrationdependenceofthedriftmobilityevaluatedfromEquation(6.20)(solidline)andthedependenceobservedexperimentally[33](squares)forTNF/PEandTNF/PVK.ThedashedlinereferencestheresultofEquation(6.21)withg =1.73forthenearest-neighborhoppingmode(reproducedwithpermissionfrom[30].Copyright2004bytheAmericanPhysicalSociety)

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σ εµ ε εDC d= ( ) ( )∫e n (6.22)

wheree istheelementarycharge,ñ(e)de istheconcentrationofelectronsinthestateswithenergiesbetweene ande +de andm(e)isthemobilityoftheseelectrons.

Ontheotherhand,oneusesthegeneralexpressionfortheDC conductivityintheformsDC=emn,wheren isthetotalconcentrationofthechargecarriers

n n= ( )-∞

∞

∫ dε ε , (6.23)

andn(e)=g(e)f(e)istheproductofthedensityofstatesg(e)andtheFermifunctionf(e)dependentonthepositionoftheFermienergyeF.

Thequantitym obtainedasm=sDC/en iscalledtobeproportionaltothe‘averagehoppingrate’⟨n⟩ :m ∝ (e/kT)⟨r2⟩⟨n⟩,where⟨r⟩ istheaveragelengthforlocalhoppingevents.Inotherwordsthe‘averagehoppingrate’⟨n⟩isdeterminedviathisrelation.AftercalculatingsDC viaEquation (6.22) andn viaEquation (6.23) andusing the expressionm =sDC/en it isclaimedthatm wascalculatedviaaveragingofhoppingrates.Wewouldjustliketoremarkthatthisdefinitionoftheaverageratelooksunjustified.Itisnotthedefinitionoftheaveragehopping rateused in [23]and ithasnothing todowith theaveragingprocedureused in[18–21]leadingtoEquation(6.10),whichisstilloftenclaimedascorrect(see,forinstance,[26–28]).

6.4.5 Saturation effects

Sofarwehavediscussed thedriftmobilityofchargecarriersunder theassumption thattheirconcentrationismuchlessthanthatofthelocalizedstatesintheenergyrangeessentialforhoppingtransport.Insuchacase,onecanassumethatcarriersperformhoppingmotionindependentlyfromeachotherandcalculatetheconductivityasaproduct

σ µDC = e n, (6.24)

wheren istheconcentrationofchargecarriersinthematerialandm istheirdriftmobilitycalculatedviaEquation(6.20)undertheassumptionthatthesystemisemptyandachargecarrierisnotaffectedbythepossibilitythatlocalizedstatescanbeoccupiedbyothercar-riers.Insucharegime,Equations(6.20)and(6.24)describethedependencesofthecon-ductivityontemperatureandonthevalueoftheparameterNa3,whilethedependenceoftheconductivityontheconcentrationofchargecarriersnislinearinaccordwithEquation(6.24). If,however, theconcentrationn ofchargecarriers is increased so that theFermienergyeF inthermalequilibriumorthequasi-Fermienergyunderstationaryexcitationislocatedenergeticallyhigherthantheequilibriumenergy⟨e∞⟩ determinedbyEquation(6.7),oneshoulduseamoresophisticatedtheoryinordertocalculatesDC.

Let us first estimate the critical concentration of charge carriers nc below which theconsiderationbasedon Equations (6.20) and (6.24) is valid. In order to estimatenc, thevalueoftheFermienergyeF shouldfirstbecalculated.ThepositionoftheFermilevelisdeterminedbytheequation

g

kTn

ε εε ε( )

-( )[ ]=

-∞

∞

∫d

1+exp F

(6.25)

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Atlown,forallenergiesinthedomaingivingthemaincontributiontotheintegralinEquation(6.25),theexponentialfunctioninthedenominatoroftheintegralislargecom-paredwithunity (thenondegeneratecase).Thenastraightforwardcalculation [50]givesfortheDOSdeterminedbyEquation(6.4)atlowtemperatures

ε σF -

-

1

2

2

kTkT

N

nln (6.26)

FromtheequationeF (nc)=⟨e∞⟩,where⟨e∞⟩ isdeterminedbyEquation(6.7),oneobtainsfornc theestimate

n NkT

c exp -

1

2

2σ (6.27)

Atn <nc,theFermilevelissituatedbelowtheequilibrationenergy,⟨e∞⟩,andthechargetransport can be described for independent carriers via Equations (6.20) and (6.24). Atn >nc, the theory for charge transport should be essentiallymodified.These argumentscanbeeasilyconvertedinorder toconsider thecaseofconstantconcentrationofchargecarriersandchangingtemperature.ThenatsomeparticulartemperatureTc equationeF (Tc)=⟨e∞⟩ willbefulfilled.AtT >Tc, thecarriermobilityandconductivitycanbedescribedbyEquations(6.20)and(6.24),whileatT <Tc,anessentialmodificationofthetheoryisneeded. Using Equations (6.7) and (6.26) and solving equationeF (Tc) = ⟨e∞⟩ for Tc, oneobtains[50]

Tk N n

c σ

21 2 1 2ln ( ) (6.28)

Inordertodevelopatheoryforhoppingtransportatn >nc (oratT <Tc)oneshouldtakeinto account explicitly the filling probabilities of the localized states by charge carriers.OnepossibilityistosolvethepercolationproblemwithtransitionratesbetweenlocalizedstatesdescribedbyEquation(6.6)thatincludesthevalueoftheFermienergyrelatedviaEquation (6.25) to the concentration of carriers, n [50, 51]. An alternative theoreticalapproachtodescribehoppingconductivityintheGaussianDOS,takingintoaccounttheoccupationoftheessentialfractionoflocalizedstatesbychargecarriers,wasrecentlysug-gestedbySchmechel[62,63].SchmechelextendedtheconceptoftransportenergydescribedinSection6.4.3takingintoaccountthepossibilitythatthelocalizedstatescanbeessentiallyfilledbychargecarriers.AnotherkindofpercolationapproachtotheproblemwassuggestedbyMartenset al.[61].Wewillnotdescribethesetheoriesindetail;theinterestedreadercanfindacomprehensiveanalysisofsomeofthemtheoriesintherecentpaperofCoehoornet al.[68].Wewouldlike,however,toemphasizeoneverypronouncedresultofthosetheo-ries.AssoonastheFermienergydeterminedbyEquation(6.25)liesessentiallyabovetheequilibration energy determined by Equation (6.7), the temperature dependence of theelectricalconductivity isno longerproportional toexp[-(Cs /kT)2]as in thecaseof lowcarrierconcentrationsatwhichEquations(6.9),(6.20)and(6.24)arevalid.Theconductiv-ityinsteadcloselyfollowstheroutineArrheniusbehavior[50,51].Forexample,thepercola-tionapproachpredictsatn >nc thetemperaturedependence[50,51]

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σ σ ε εDC

t F= - -

0 exp

kT (6.29)

wheres0 is thepreexponentialfactor,onlyslightlydependentontemperatureandontheconcentrationofsitesN;thetransportenergyet isdeterminedvia

ε σ β α σt = -

21 2 1 2

3

4

3

lnN

s kT kT (6.30)

wheres isthesolutionoftranscendentalequation

sN

s kT kT= -

ln1 2

3

4

3β α σ (6.31)

withb 0.045.TheFermienergyeF isdeterminedviatheequation[50,51]

ε σ σπ ε

F

F

= -( )

22

1 2 1 21 2lnN

n (6.32)

wheres istheenergyscaleoftheDOSinEquation(6.4).Itisworthnotingthattherearenumerousobservationsofasimpleactivatedtemperature

dependenceof theconductivity indisorderedorganicmaterials [69–72].Particularly theArrhenius temperature dependence of the conductivity is often observed in field-effecttransistors,wherechargecarrierconcentrationisusuallyhigh(see,forinstance,[34]).

Often the dependence ln(m) ∝ T -2 or ln(s) ∝ T -2 is considered as evidence for aGaussian DOS, while the Arrhenius temperature dependence ln(m) ∝ T -1 or ln(s) ∝ T -1isclaimedtoindicateapureexponentialDOS,whichwasdescribedin[14].Theim-portantconclusionfromtheaboveconsiderationisthepossibilitytoaccountsuccessfullyforbothkindsoftemperaturedependenceofhoppingconductivitydescribedbyEquations(6.2)and(6.3)intheframeworkoftheuniversaltheoreticalmodelbasedontheGaussianDOS.Thetemperaturedependenceoftheconductivityissensitivetotheconcentrationofchargecarriersn.

6.5 THEORETICAL TREATMENT OF CHARGE CARRIER TRANSPORT IN ONE-DIMENSIONAL DISORDERED ORGANIC SYSTEMS

Inthissectionweshowhowthehoppingchargetransportcanbedescribedtheoreticallyinone-dimensional(1D)disorderedorganicsolids.Particularinteresttothistopicistwo-fold. First, 1D disordered organic systems have been intensively studied experimentally,aiming at their applications in various electronic devices. Furthermore, in some vitallyimportantprocessessuchaselectrontransportalongDNAmoleculesorchargetransportin ionicchannels throughcellmembranes,charge transportcanbeeffectively treatedasone-dimensional.Second,inmanycaseshoppingtransportin1Dsystemscanbedescribed

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theoretically with much higher precision than hopping transport in three-dimensionalsystems. Theoretical study of three-dimensional transport is complicated because it issometimesdifficulttofindthespatialstructureofthetransportpath[17].In1Dsystems,thisproblemdoesnotexist.Achargecarrierinsuchsystemsperformsaseriesofsuccessivetransition events between localized states placed on a 1D chain. Provided the rates forhoppingtransitionsbetweenlocalizedstatesandtheDOSfunctionareknown,theproblemof calculating the transport coefficients can be in many cases solved exactly. Using thisadvantageoftheprecisetheoreticaldescription,onecanclarifyvarioustransportphenom-enadiscoveredexperimentally.Amongsuchphenomenawewilldiscussthedependenceofthecarrierkineticcoefficientsontheappliedelectricfieldandinparticulartheratherpuz-zlingobservationofthedriftmobilityincreasingwithdecreasingfieldstrengthatlowfields[4,7,8,12,13,40].

In recentyearsparticular attentionhasbeendevoted to the studyof charge transportprocesses in columnar discotic liquid-crystalline glasses due to their potential technicalapplicationsforelectrophotographyandastransportmaterialsinlight-emittingdiodes[73,74].Dielectricmeasurementshaveclarifiedthatchargetransportinmostofsuchmaterialsis extremely anisotropic [75] so that it is reasonable to describe the transport of chargecarriersasa1Dprocess[76].Indeed,thevaluesofconductivityalongandperpendiculartothediscoticcolumnsdifferinsuchmaterialsbythreeordersofmagnitude[75].Experi-mentallyobserveddependencesoftheconductivityonthefrequency,onthestrengthoftheappliedelectricfield,andontemperatureshowthatanincoherenthoppingprocessis thedominanttransportmechanisminsuchmaterials[44,75,76].Ithasbeenfoundreasonabletodescribe the transportofchargecarriersasahoppingprocessviamoleculesarrangedinspatiallyordered1DchainswithaGaussiandistributionofsiteenergiesdescribedbyEquation(6.4)[44,76,77].

Letusfirstconsider, following[15,76],asimplestcaseallowingonlyhopstonearestneighborsandassumingthat1Dchainsoflocalizedstatesarespatiallyregular.Insuchacase,onecanomitther-dependenceoftransitionprobabilitiesusingtheeffectivepreexpo-nential factorn0 thatcontains the termexp(-2b/a),whereb is thedistancebetween theneighboringsitesontheconductingchain.TheMiller–Abrahamstransitionrateshaveinsuchacasetheform[15,76]

v vkT

ij i jj i j iε ε

ε ε ε ε, exp( ) = -

- + -

0

2 (6.33)

Otherformsoftransitionprobabilitieshavealsobeenconsideredintheliterature.Sekiand Tachiya [78] studied the 1D conduction with transition rates for charged carriersdescribedbytheMarkuslawthattakesintoaccountthepolaroneffect[79].Another,sym-metricformwasalsousedinordertoanalyzeanalyticallythefieldandtemperaturedepend-encesofthehoppingconductivityin1Dsystems[77]:

v vkT

i ii i

±±= -

-

1 0

1

2, exp

ε ε (6.34)

Two distinct models have been discussed in the literature with respect to the space–energy correlation of localized states responsible for charge transport. In the so-calledGaussian disordered model (GDM), which we considered in previous sections of this

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chapter,suchcorrelationsareneglected.Inthealternativecorrelateddisordermodel(CDM),itisassumedthatenergydistributionsofspatiallyclosesitesarecorrelated,forexample,duetodipoleorquadrupoleinteractions[77,80–82].

Muchattentionhasbeenpaidinrecentyearstothedependenceofthecarriermobilityontheelectricfield.Thisdependenceindisorderedorganicmaterialsinabroadrangeofhighfieldstrengthescanbefittedbythefunction µ ∝ ( )exp ,F F0 whereF0isaparameter[8]. It has been shown that in three-dimensional systems such field dependence can beexplainedonlyintheframeworkoftheCDM[77,80–82].Itischallengingtocheckthisresultbytheexactlysolvable1Dmodels.

Thestudyofthetemperaturedependenceofthedriftmobilityin1Dsystemsisalsoofgreatinterest.ResearchersagreethatthisdependenceintheemptysystemcanbedescribedbyEquation(6.9),althoughthereisnoagreementonthemagnitudeofthecoefficientC inthis formula. Ochse et al. [44] used the three-dimensional value C 2/3 for columnarsystemswith1Dtransport,whileBleylet al.[76]obtainedC 0.9intheircomputersimu-lationfor the1Dcasewithasymmetric transitionprobabilities.TheanalyticcalculationsofDunlapet al.[77]forsymmetricprobabilitiespredictC=1inthe1Dcase.Thevalueof C is decisive for characterization of the disorder parameters from comparison withexperimentaldatain1Dsystems.Weshowinthenextsubsectionanalyticalformulasthatallowonetocalculateexactlytransportcoefficientsin1Dsystemsforhoppingtransitionsbetweenthenearestsites.

6.5.1 General analytical formulas

Thesteady-statedriftvelocityofchargecarriersin1Dperiodicsystemswiththenumberofsites in theperiodN andthedistancebetweenthenearestsitesonthechainb canbewrittenexactlyusingthegeneralsolutionfoundbyDerrida[83]

v

Nbvv

vv

k k

k kk

N

k kk

Nk j k j

=-

+

+

+=

-

+-

=

-+ + -

∏

∑

1

1

1

10

1

11

0

11

,

,

,,

vvk j k jj

i

i

N

+ + +==

-

∏∑

, 111

1

(6.35)

ThisformulawasusedbyDunlapet al.[77]tostudythedriftmobilityina1Dsystemwithsymmetrictransitionrates.Thedriftmobilitym isrelatedtothesteady-statevelocityv as

µ υ=F

(6.36)

Howeverintheexperiments,thedriftmobilityisusuallyobtainedbythetime-of-flighttechnique[8].HerechargecarrierspassonlyoncethroughasampleoffinitethicknessanddriftmobilityiscalculatedbydividingthesamplelengthNb bythemeantransittime⟨t0N⟩ andelectricfieldF:

µ = Nb

t N F0

(6.37)

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MurthyandKehr [84,85]havederived theanalyticalexpression for themean transittimeofcarriersthroughagivenchainofN +1sites:

t v vvvN k k

k

N

k kk

Nj j

j jj k

i

i k0 1

1

0

1

11

0

21

11

= ++-

=

-

+-

=

--

+= += +∑ ∑ ∏, ,

,

,11

1N -

∑ (6.38)

Thisformulagivesthetransittimeforachargecarrierthatstartsonsite0andfinishesonsiteN.Thetimeisaveragedovermanytransitsthroughthesamechainwithgivenvaluesoftransitionprobabilitiesnij.

OneshouldbecautiouswiththeapplicationofEquations(6.37)and(6.38)atlowelectricfields. Even without electric field, carriers starting at site 0 will pass through the finitesystemsolelydueto thediffusionprocess.At lowfields,diffusiondominates themotionofcarriersanditwouldbewrongtouseEquation(6.37).Moreappropriateinsuchacaseistheestimateofthemobilityviathediffusionformula

µ = e

kT

N b

t N

2 2

02 (6.39)

whichpresumesthevalidityoftheEinsteinrelationship.NotethatEquations(6.35)and(6.38)arevalidforanytypeofnearest-neighborhopping

ofnoninteracting carriers in1D systems.We show in thenext section the results of theexacttheoryappliedtoamodelsystemrepresentedbya1Dregularchainwithequalsiteenergies separated by barriers with random heights, a so-called random-barrier model(RBM). In the subsequent sectionsweconsidermore realisticmodels,GDMandCDM,withdistributionofsiteenergies.

6.5.2 Drift mobility in the random-barrier model

TransitionprobabilitiesbetweentheneighboringsitesundertheinfluenceofelectricfieldF intheRBMmodelaregivenbyexpressions

v vebF ebF

kTk k

k k, exp+ = - - + -

1 0

2 2

2

D D (6.40)

and

v vebF ebF

kTk k

k k+ = - + + +

1 0

2 2

2, exp

D D (6.41)

whereDk istheenergybarrierbetweensitesk andk +1.Inthelimitoflongchains,N >> 1,calculationofthedriftmobilityviaEquations(6.35)

and(6.36)orEquations(6.37)and(6.38)foranychainwithagivendistributionoftransi-tionratesisequivalenttotheaveragingofEquations(6.35)or(6.38)overallpossibledis-tributionsoftransitionratesdeterminedbydisorder.WeassumetheGaussiandistributionofbarrierheightswiththetypicalscales.Performingtheaveraging,oneobtainsthefol-lowingexpressionforthedriftmobilityintheinfinitechain[15]

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µ σσ

σ= -

-

+

-

v b

F kT

ebF

kT

ebF

kT0

2

11

2 2 8 2exp erf

eerf erf erfσ

σ21

8 8kT

ebF

kT

ebF ebF

- -

-

} ×exp

σσ

σσ

σ

{

+

-

- -

exp1

2 21

8 2

2

kT

ebF

kT

ebF

kTerf

-1

(6.42)

where erf(x) = (2/p) x0dy exp(-y2) is the error function. In Figure 6.9 the calculated

fielddependencesof thedriftmobilityare shown forparametersb=3.6Å,s=50meV,kT= 25meV [15].The solid line shows theexact solution for an infinite chaingivenbyEquation (6.42). All points in the figure correspond to mobilities for finite chains ofN = 500 sites averaged over 1000 different chains. Circles were obtained via Equations(6.37)and(6.38)whilesquarescamefromEquations(6.38)and(6.39).ResultsoftheMonteCarlocomputersimulations[16]areshownbycrossestodemonstratetheexcellentagree-mentof the simulation resultswith theanalytic theory.At lowfields, thedrift approachEquations(6.37and6.38)forfinitesystemsleadstoanincreaseofthecalculatedmobilitywithdecreasingfieldstrength.Similarresultswereobtainedforallconsideredmodelsofdisorderandvariousformsof the transitionprobabilitiesbetweenneighboringsites [15].This result is apparently artificial, reflecting the fact that charge carriers can penetratethroughafinitesystemviadiffusivemotion,evenintheabsenceofelectricfield.ByusingEquation(6.37)oneoverestimatesthemobilityatlowfields.ThesamehappenswhenusingEquation (6.39) at higher fields. Comparison of Equations (6.37) and (6.39) reveals thestrengthof theelectricfieldFc atwhicha transition from thediffusion-controlled to thedrift-controlledtransittakesplace:Fc 2kT/(eNb)[15].

Figure 6.9 Field dependence of the carrier mobility in the RBM with Gaussian distributionof barriers. The solid line represents exact solution for the infinite chain. Data shown by circlesandsquareswerecalculatedviaEquation(6.38)usingdrift(Equation6.37)anddiffusionrelations(Equation6.39),respectively.DatashownbycrosseswereobtainedbyMonteCarlosimulations[16](reproducedwithpermissionfrom[15].Copyright2001bytheAmericanPhysicalSociety)

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Atlowfields,thetransittimedoesnotdependonthestrengthoftheelectricfield,beingdeterminedmostlybydiffusion.UsingEquation(6.37)oneartificiallyobtainsanincreas-ingdriftmobilitywithdecreasingfieldF.Innumerouspublications,experimentalresultswerereportedthatshowanincreaseofthedriftmobilitywithdecreasingfieldatlowfields[11,86,87].Thiswasalwaysobservedathightemperatures,atwhichdiffusioncandomi-nate the motion of charge carriers. Experimental evidence for decreasing mobility withincreasingfieldisusuallyobtainedusingequationssimilartoEquation(6.37),wherethedriftmobility isdeterminedvia themeasured transit timebydividing thesample lengthbythevalueofthistimeandbythevalueofthefieldstrength.WebelievethatadiffusionequationsimilartoEquation(6.39)shouldbeusedatlowfieldsandhightemperatures.Inthethree-dimensionalcase,thisequationshouldbeslightlymodified,thoughtheconclu-sionisthesame:decreasingdriftmobilitywithincreasingfieldstrengthatlowfieldscanbeanartifactcausedbyusingthedriftequationinsteadofthediffusionequationfortheevaluationofthemobilityonthebasisofthemeasuredtransittime.ThisconclusionwasalsosuggestedbyHiraoet al.[88].Inthe1Dcalculations,thisideaisillustratedbyusingEquation (6.39) instead of Equation (6.37) for finite systems at low electric fields. TheresultisshownbysquaresinFigure6.9.Theexcellentagreementofthediffusionequationwiththeexactcalculationfortheinfinitesystematlowfieldsconfirmsourconclusion.Forthechosenparameters,thelineartransportregimewithmobilityindependentoftheelectricfieldisvaliduptoafieldstrengthofapproximately106V/cm,atwhichanarrownonlinearregimestartswiththemobilityincreasingwithelectricfield.Athigherelectricfields,thisregime is replaced by the trivial decrease of the mobility as F-1. At such high electricfields, all energybarriersbetween siteson the chain are eliminatedby thefield and thetransittimebecomesfieldindependent.Equation(6.37)predictsaF-1dependenceinthisregime.

6.5.3 Drift mobility in the Gaussian disorder model

In this section we consider a random-energy model with distribution of site energiesdescribedbyEquation(6.4),presumingthattherearenocorrelationsbetweenspatialposi-tionsofchainsitesandtheirenergies.ThismodeliscalledintheliteratureaGaussiandis-ordermodel(GDM).InEquation(6.33),thesiteenergiesekdependonthestrengthoftheelectricfieldF.Theyarerelatedtothezero-fieldsiteenergies,whichweheredenoteasjk,bytheexpressionek =jk - ekbF.InaccordancewithEquation(6.33),theratiooftheforwardnk,k+1andthebackwardhoppingratesnk+1,kforanypairofneighboringsitesis

v

v kTk k

k k

k k+

+

+= -

1

1

1,

,

expε ε

(6.43)

equivalenttoheconditionofthedetailedbalance.InsertingEquation(6.43)into(6.35),oneobtains

υ =- -( )[ ]

+ -( ) -( )+

-

=

-

∑bN ebNF kT

v kTiebF kT

k kk

N

k

1

11

0

1

exp

expexp

, jvv kTk i k i k ik

N

k

N

+ + + +=

-

=

-

-( )∑∑, exp10

1

0

1

j

(6.44)

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TheonlyrandomvariablesinEquation(6.44)arethezero-fieldsiteenergiesjkandtheforwardtransitionratesnk+i,k+i+1,whichinturndependonlyontheneighboringsiteenergiesjk+i andjk+i+1.Sinceinallproductsappearinginthesecondtermofthedenominator,jk isnotcorrelatedwithjk+i ornk+i,k+i+1, averagingoverdisordercanbecarriedout forexp(-jk/kT) and exp(jk+i/kT)n -1

k+i,k+i+1 separately. Straightforward calculations of v and ⟨t0N⟩thenleadviaEquation(6.37)totheexpressionforthedriftmobility[15]

µσ

σ σ= +

+

-

+ -

21

210

2v b

F

ebF

kT

ebF

kT kT

eberf erfexp

FF

kT

ebF

kT

ebF

kT

2

12

σ

σ

+

-

-

exp exp--

-

{

× + -

+ + +

1

12 2

12 2

expebF

kT

kT

ebF

kT

ebFerf erf

σσ

σσσ

}

-1

(6.45)

Atlowelectricfieldsthisexactexpressioncanbeapproximatedby

µ σ

v b

F kT

ebF

kT0

2

21exp exp-

-

(6.46)

whileathighfieldsmn0b/F.Interpolationbetweenbothapproximationsgives[15]

µ

σ

v b

F

kTebF

kT

0

2

2

1

1

2

1+ ( )

-

-

exp

exp (6.47)

Field dependences of the drift mobility in the GDM are shown in Figure 6.10 forb =0.36nm,s =50meV,andkT =25meV.ThesolidlinerepresentstheexactsolutionfortheinfinitechaingivenbyEquation(45),whiledashedlinesshowthelow-andthehigh-field approximations. One can easily check that for these parameters the interpolationformula(6.47)isinexcellentagreementwiththeexactsolution.Circlesandtrianglesinthefigureshowthecalculatedresultsforfinitesystemsof2000sitesaveragedover1000dif-ferentchains,onceusingtheaveragingofinversetransittimes(circles)andonceusingtheaveragingofinversevelocities(triangles).QualitativelytheseresultsinFigure6.10resemblethosefortherandom-barriermodeldescribedintheprevioussection.Inparticular,increaseofthemobilitywithdecreasingelectricfield,showninFigure6.10bycircles,iscausedbytheinvalidityofthedriftapproximationinfinitesystemsatlowfields.

InFigure6.11thefielddependencesofthedriftmobilityareshowninthePoole–Frenkelrepresentation(lnm versus F )fortwodifferenttemperaturesandtwodifferentaveragingprocedures.ThefigureclearlyshowsthatinthechosenmodelthecarriermobilityhardlycanbedescribedbythePoole–Frenkellawlnm∝ F . ThisconclusionisinagreementwiththeresultsofGarsteinandConwell[80],Dunlapet al.[77]andNovikovet al.[81,82].FollowingtheseauthorsweconsiderinSection6.5.5thedriftmobilityinthemodelwithcorrelateddisorder(CDM).Beforedoingso,wefocusinthenextsectiononthetem-peraturedependenceofthedriftmobilityatlowfieldsintheGDM.

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Figure 6.10 Fielddependenceofthecarriermobilityfornearest-neighbourhoppingwithMiller–AbrahamsratesbetweensiteswithGaussianenergydistribution.Solidlineshowstheexactsolutionforaninfinitechain.Dashedlinescorrespondtothelow-fieldandhigh-fieldapproximations.CirclesshowtheaveragedmobilitiescalculatedaccordingtoEquations(6.37)and(6.38).TrianglesrepresenttheresultsobtainedbyaveragingoftheinversemobilitiescalculatedbyEquations(6.36)and(6.44)forthesamechains(reproducedwithpermissionfrom[15].Copyright2001bytheAmericanPhysicalSociety)

Figure 6.11 Poole–Frenkel plots of the carrier mobilities. Circles show the averaged mobilitiescalculatedbyEquations(6.36)and(6.44);squaresshowthecorrespondingresultsobtainedbyaver-agingof the inversemobilitiesgivenbyEquation (6.48).Thenumberofchainswasc =104withN =500siteseach(reproducedwithpermissionfrom[15].Copyright2001bytheAmericanPhysicalSociety)

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6.5.4 Mesoscopic effects for the drift mobility

The low-fieldmobility forafinitechaindescribedbyEquations (6.44)and (6.36) in thelimitofF → 0isgivenbytheequation

µ = -( ) ( )

-+

-

=

-

=

- -

∑∑eb

kTN kT kT vk k k k

k

N

k

N22

11

0

1

0

1

exp exp ,j j11

(6.48)

For the infinite chain with transition rates described by Equation (6.33) the low-fieldmobilityintheGDMisexactlygivenby[15]

µ σ σ= -

+

-v eb

kT kT kT0

2 2 1

12

exp erf (6.49)

ThetemperaturedependenceofthedriftmobilitygivenbyEquation(6.49)isshownbythesolidlineinFigure6.12.Theslopeoflnm versus(s /kT)2inthiscurvediffersslightlyfrom-1duetothetemperaturedependenceofthepreexponentialfactorinEqution(6.49).ThisresultisingoodagreementwiththecomputersimulationsofBleylet al.[76]andwiththeanalyticcalculationsofDunlapet al.[77],althoughthelatteranalyticcalculationswerecarriedout forcorrelatedsystemswithsymmetric transitionratesdescribedbyEquation(6.34).Thisshowsthatin1DsystemsEquation(6.9)withC 1iscorrectandevenstableagainstthechoiceoftheformofthetransitionrates.

Thetemperaturedependencesofthelow-fielddriftmobilityinfinitesystemscalculatedaccordingtoEquation(6.48)arealsoshowninFigure6.12.Theseresultsarereallyremark-

Figure 6.12 Temperaturedependenceofthelow-fieldmobilityfors =50meV.Solidlinerepresentsthe solution for the infinite chain given in Equation (6.49). Circles and squares show the resultsobtainedbytheaveregingofmobilitiesandaveragingofinversemobilitiescalculatedbyEquation(6.48), respectively,withaveragingoverc =103chainsofN =2000sites.Upwardanddownwardtrianglesarethecorrespondingvaluesforc =104andN =200.DashedlineillustratesthetemperaturedependenceofthedriftmobilityintheformofEquation(6.9)withC =1(reproducedwithpermis-sionfrom[15].Copyright2001bytheAmericanPhysicalSociety)

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able.Theyclearlyshowthatthetemperaturedependenceofthemobilityisrelatedtothesizeofthesystem.Thisisamanifestationofthemesoscopiccharacterofthehoppingtransportprobleminfinitesystems,whichismostpronouncedin1Dcases.Theaveragingoftransittimesorinversemobilitiesovervariousfinitechainscorrespondstothecalculationofthemobility ina longsystemconsistingofall thosechainsconnectedsequentiallyoneafteranother.Onthecontrary,theaveragingofthemobilitiesovervariousfinitechainscorre-sponds to thecalculationof thecarriermobility in thesysteminwhich thesechainsarearrangedinparalleltoeachother.Inthelattercase,the‘fast’chainswithuntypicallyshorttransittimesstronglycontributetotheaveragedvalueofthemobilityandtheydominatethetransportofchargecarriers.Thisistheveryessenceofthemesoscopiceffects[89].

Havinginmindanapplicationtodiscoticorganicdisorderedsystemswheremanyquasi-1Dcurrentchannelsareconnected inparallel,oneshouldconclude that the temperaturedependenceoftheelectricalcurrentinsuchsystemsdoeschangewiththethicknessofthesample.Forexample,comparisonbetweenthedataobtainedbytheaveragingofthemobil-ityvaluesforchainswithN =2000sitesandchainswithN =200suggeststhat,forshorterchainsandhenceforthethinnersamples,thetemperaturedependenceofthedriftmobilityshouldbeweakerthanthatforthickersamples.Indeed‘fast’channelsdominatingelectricalconductioninfinitesystemsariseduetothenarrowervariationofsiteenergiesthanintheinfinitesystems.Thiseffectiscausedbystatisticalfluctuationsintheenergydistribution[89].Closersiteenergiesleadtoaweakertemperaturedependence.

Themesoscopiccharacterof the temperaturedependenceof thecarrierdriftmobilitycanalsobeillustratedbythefollowingconsideration.Intheinfinite1Dchainwithhoppingtransitionsonly to thenearest-neighboringsites, thereexistsacharacteristicvalueof thelocalresistancethatdeterminestheoverallchainresistivity[90].Thecorrespondingenergyeopt ofoneoftheinvolvedsitesis,insomerespect,similartothetransportenergyintroducedintwo-andthree-dimensionalsystems.Indeed,letp(e)betheprobabilityforasitetohaveanenergyabovesomevaluee inaGaussianDOS.Thentheproduct

2 12

2 2

2

2- ( )[ ] -

( )

-

ppε ε

ε σ πεσ

d

d exp (6.50)

istheprobabilitydistributionthatoneoftheenergiesofaneighboringpairofsitesisbelowe,whereastheupperenergyliesintheintervalbetweene ande +de.Thisequationgivesthereforethedistributionofthenearest-neighborresistances.Averagingtheresistances,oneeasilyfindsthattheexponentoftheaverageresistivityisdeterminedbytheenergyeopt s2/kT andtheexponentisequalto2-1(s /kT)2+eopt/kT -e2

opt/2s2=(s /kT)2.ThisleadstoasymptotictemperaturedependencedescribedbyEquation(6.9)withC 1.Weseethatthis asymptotic temperature dependence of the drift mobility is determined by the siteswithenergies in thevicinityofeopts2/kT.Thisenergyincreaseswithdecreasingtem-peratureanditmighthappenthatinafinitesystemtherearenositeswithsuchhighenergiesaseopt.Ifso,thetemperaturedependenceoftheresistivityandthatofthecarriermobilityshouldbeweakerthanthedependencedescribedbyEquation(6.9)withC 1.Thiseffecthasrecentlybeenobservedinastraightforwardcomputersimulationofthenearest-neighborcarrierhoppingina1Dchain[91].Studyingafinitesystem,Pasveeret al.[91]obtainedaweakening of the temperature dependence of the carrier drift mobility with decreasingtemperatureinaccordwiththeabovearguments[90].

Thismesoscopiceffectshouldbetakenintoaccountwhenstudyingmaterialpropertiesatdifferentsamples.Fortime-of-flightstudies,researchersprefertodealwiththicksamples

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inordertoobtainlongertransittimesandtodeterminethedriftmobilitywithhigherpreci-sion.However, fordeviceapplications thinsamplesaremostlyused.Oneshouldbewellawarethatsomepropertiesofthethinsamplesusedfordeviceapplicationsmayessentiallydifferfromthoseofthicksamplesusedintheresearch.Itis,forinstance,trueforthetem-peraturedependencesoftheconductivity,asshownabove.

Sofarwehaveconsideredsystemswithoutcorrelationsbetweenspatialpositionsofsitesandtheirenergies.Inthenextsectionweabandonthisassumptionandconsidertheeffectsofsuchcorrelationsonthecarrierdriftmobility.

6.5.5 Drift mobility in the random-energy model with correlated disorder (CDM)

In order to construct a model with correlated disorder, we first generate provisional siteenergiesfk withaGaussiandistribution

g φσ πλ

φλσ

( ) = -

1

2 2

2

2exp (6.51)

wherelisanintegernumber.Thezero-fieldenergyj ofachargecarrieronsitek isthendeterminedbythearithmeticaverageofprovisionalenergiesf ofneighboringsites:

j φk k jj m

m

= +=-∑ (6.52)

Herel =2m +1isthecorrelationlengthinunitsoftheintersiteseparationb.ThisrecipetocreateasystemwithcorrelatedsiteenergiesissimilartothatsuggestedbyGarsteinandConwell[80].TheresultisthesetofsiteenergiesatzerofieldthathaveaGaussiandistributiondescribedbyEquation(6.4).Duetothecorrelation,twositeenergiesjk andjk+iarenotinde-pendentaslongasi ≤l.ThismakestheanalyticalcalculationofthedriftmobilityslightlymoreelaboratethaninthemodelwithoutcorrelationsdescribedinSections6.5.3and6.5.4.Nevertheless,onecanperformtheaveragingoverdisorderstraightforwardlyandobtainthefollowingexactresultforthecarrierdriftmobilityintheinfinite1Dchainintheform[15]

µ

σ λ

λ σ=

-

-

-

-

-

21

1

0

2

12

v b

F

kT

ebF

kT

kT

ebF

kT

exp

exp

+

+

-

-

12

1 2

12

erfλ

σ

λ σ

ebF

kT

ebF

kTexp + -

+

-

121 2

1 2

2

erfσ

λλ

σ

σ λ

kT

ebF

kT

ebF

kTexp

-

+ +

+ -

11

21 2

1 2

exp

exp

ebF

kTkT

ebF

ebF

kT

erfσ

λλ

σ

+ -

-

121 2

1 2 1

erfσ

λλ

σkT

ebF (6.53)

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InFigure6.13thecorrespondingfielddependencesareplottedfordifferentcorrelationlengthsl forfinitechains(symbols)calculatedviaEquation(6.35)and(6.36)andfortheinfinitechaincalculatedviaEquation(6.53).ResultsforfinitechainsofN =103siteswereaveragedoverc =103realizationsofthechain.TheparticularshapeofthefielddependenceofthedriftmobilityintheCDMfortheinfinitesystemisconsideredindetailinthepapersofGarsteinandConwell[80],Dunlapet al.[77]andNovikovet al.[81,82]andwedonotdiscussithere.

Insteadwewould like to focusour attentionon theother aspect of thephenomenon,namely,ontheincreasingdifferencebetweentheresultsobtainedbyaveragingofmobilities(opensymbols inFigure6.13)and thoseobtainedbyaveraging the inversemobilitiesortransittimes(filledsymbolsinFigure6.13)withincreasingcorrelationlength.Thistrendisclearlyrelatedtothesmallernumberofhighbarriersinsystemswithlongercorrelationlengths,whichfavorsthemesoscopiceffects.TheincreaseofthecorrelationlengthintheCDMisanalogousto thedecreaseof the totalnumberofsites in thefinitechains in theGDM.

6.5.6 Hopping in 1D systems: beyond the nearest-neighbor approximation

So far we have considered hopping in various 1D systems, taking into account carriertransitionsonlybetween thenearest sites.Thequestion thenarisesofhow transitions tofurtherneighborsonthechaincanmodifytheresultsobtained.TheVRHphilosophytellsusthatthetemperaturedependenceoftheconductivityandthatofthecarrierdriftmobility

Figure 6.13 Influenceofthespace-energycorrelationsonthefielddependenceofthecarriermobil-ityatkT =15meVands =50meV.Thedifferentcorrelationlengthsarel =1(circles),l =3(squares),l = 11 (upward triangles), andl = 101 (downward triangles). Open symbols show the averagedmobilitiescalculatedbyEquation(6.44).Solidsymbolsshowthecorrespondingresultsobtainedbyaveraging of the inverse mobilities. Solid lines show the solutions for the infinite chain given byEquation(6.53)(reproducedwithpermissionfrom[15].Copyright2001bytheAmericanPhysicalSociety)

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shouldbedeterminedatlowtemperaturesbytransitionsofchargecarrierstofurtherneigh-bors than thenearestones.Theeffectof such long-distance tunneling transitionson thetemperaturedependencesofthekineticcoeffcientsin1DsystemswasstudiedanalyticallybyZvyaginet al.[90]andbycomputersimulationsbyKoharyet al.[16]andbyPasveeret al.[91].Zvyaginet al.[90]consideredonlytheGDMmodelandshowedthattransitionsto m neighbors beyond the nearest ones lead to the asymptotic temperature dependencedescribedbyEquation (6.9)with C m m= +( )1 2 . For instance,hopping to thesecondnearestneighborsleadstoEquation(6.9)withC = 3 4. However,thisasymptoticbehaviorwith C m m= +( )1 2 canhardlybeachievedatrealisticconditions.Zvyaginet al.[90]have shown that even at (s /kT)2 100, the coeffcient C in Equation (6.9) is closeto0.8.

Ananalogousresultontheweakeffectofdistanttransitionshasbeenobtainedbycom-putersimulations[16].Koharyet al.[16]studiedbycomputersimulationshoppingtransportin1DGDMandCDMsystemsusingtransitionratesintheformofEquation(6.5)insteadofEquation(6.33)thatwasusedforthenearest-neighborhopping.Inthesesimulationsthelatticeconstantb = 0.36nmwaschosenasknown fordiscotic liquid-crystallineglasses,andthevaluesofparametera inEquation(6.5)werevariedbetween0.1and0.3nm.Thisparameterdeterminesthedecaylengthofthecarrierwavefunctioninthelocalizedstates.Transitions to sixneighbors in eachdirectionwereallowed in the simulationprocedure.Themainresultof thesimulationsis thefollowing.FortheCDM,i.e., forasystemwithcorrelateddisorder,tunnelingtofurthersitesthanthenearestneighborsdoesnotplayanyessential role, while for the GDM, i.e., for uncorrelated systems, these distant hoppingtransitionsslightlyaffectthetransportcoeffcients.Theunimportanceofdistanttransitionsforsystemswithcorrelateddisorderisnotsurprising.Insuchsystems,energiesofneighbor-ingsitesareclosetoeachotherduetocorrelationeffects.Thusthevariable-rangehoppingis not significant for such systems, because the closest energy and space states are thenearestneighbors.Thequantitativeconfirmationof thisqualitativelyclearstatement[16]implies that forsystemswithcorrelateddisorderonecanuseanalytic theoriesdescribedaboveandonecanbesurethattherestrictionoftheanalytictheorywhichconsidersonlythenearest-neighborhoppingisnotessentialfortheresultsobtained.

Forsystemswithuncorrelateddisorder theresultof thesimulationisalsoratheropti-misticwithrespecttothevalidityoftheexactanalyticsolutionsdescribedabove.Evenforaratherhighvalueofthedecayconstanta =0.2nm,whichmakesthedistanthopsfavorable,themagnitudeof thecoeffcientC 0.9inEquation(6.9) isnotmuchdifferent fromitsvalueforthenearest-neighborhopping.Thus,evenforsystemswithuncorrelateddisorder,exactanalyticsolutionsobtainedinprevioussections,takingintoaccountonlythenearest-neighborhopping,givereasonablevaluesfortransportcoeffcients[16].

6.6 ON THE RELATION BETWEEN CARRIER MOBILITY AND DIFFUSIVITY IN DISORDERED ORGANIC SYSTEMS

High interestof researchers incharge transport inorganicdisorderedsystemsmotivatednumerousstudiesoftherelationbetweensuchkineticcoeffcientsasthemobilitym andthediffusion coeffcient D of charge carriers in such systems. However, one can find rathercontradictorystatementsonthissubjectinthescientificliteratureandthereforewewouldliketoclarifythisproblem.

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Borsenbereger et al. [92] studied experimentally the behavior of the ratio eD/m in1,1-bis(di-4-tolylamionophenyl)cyclohexane (TAPC) and found that this ratio increasesapproximatelylinearlywithappliedelectricfieldathighfields.Inthelow-fieldregion,theratioeD/m becomesindependentofthefieldstrengthF.Nevertheless,thelimitingvalueofthis ratiodoesnotagreewith thatgivenby theclassicalEinstein relationship.The latterreads[93]

µ =e

kTD (6.54)

Also Monte Carlo computer simulations were carried out that claimed invalidity ofEquation(6.54)fortherelationbetweenthemobilityanddiffusivityofhoppingcarriersinarandomsystemwithGaussianenergydistributionofsiteenergies[94–96].Theamountofdisorderischaracterizedbytheenergyscales oftheDOSdescribedbyEquation(6.4).Withincreasingdisorderandfield,significantdeviationsfromEinstein’slawwereobtainedinthesimulations.TheseresultscontributedtothegeneralbeliefofmanyresearchersthattheEinsteinlawexpressedbyEquation(6.54)isviolatedinrandommedia.

Discussing thevalidityor invalidityof theEinstein relationship forhoppingelectronsoneshouldclearlydistinguishbetweentheequilibriumandnonequilibriumconditionsononehandandbetweenthedegenerateandnondegeneratesystemsofchargecarriersontheother.Furthermore,oneshoulddistinguishbetweentheregimeoflowelectricfieldswithfield-independenttransportcoeffcientsandtheregimeofhighfields,inwhichthenonlineareffects caused by electric fields become significant. Computer simulations [94–96] haveshownthatathighfieldsinanonlineartransportregimethediffusioncoeffcientD dependsmorestronglyontheelectricfieldthanthecarrierdriftmobilitym.ThereforetheratioeD/m increaseswithincreasingfieldandtherelationdescribedbyEquation(6.54)isviolated.Wewillnotconsiderthisnonlinearregimehereduetotherathercomplicateddefinitionofthediffusionconstantinthecaseofhighelectricfieldswhichcausestronganisotropyinthediffusion process. Interested readers can find the necessary information in the literature[94,95].Insteadwediscussbelowtheregimeoflowelectricfieldswithfield-independenttransportcoeffcientsDandm.

AsshowninSection2.5ofChapter2[14]Einstein’slawcannotbevalidinthestrongnonequilibriumconditionsatlowtemperatures,whentransportprocessesaregovernedbytheenergy-lossdownwardinenergyhoppingrelaxationofchargecarriers.Fortheenergy-losshoppingintheexponentialDOS,arelationbetweenD andm similartothatinEquation(6.54)wasfound,althoughthethermalenergykT inthisrelationisreplacedbytheenergyscaleoftheDOS(seeEquation2.53inChapter2[14]).Itis,unfortunately,notpossibletoobtainsucharelationfortheenergy-losshoppinginaGaussianDOS.TheexponentialDOSconsideredinChapter2[14]representstheonlyexceptionamongpossibleDOSfunctions,forwhichtherelationbetweenm (e)andD(e)isindependentofthelocalizationenergy2 and therefore the relationbetween theeffectiveD andm for thewhole systemofchargecarriersperformingtheenergy-losshoppingcanbeformulatedintheuniversalform(seeEquation2.53inChapter2[14]).ThereforewewillnotconsidernonequilibriumconditionsforhoppingintheGaussianDOSandrestrictourconsiderationheretostudyingthevalidityoftheEisnteinlawintheequilibriumconditions.

Theproblemof therelationbetweenD andm in thermalequilibriumforadisorderedsystemwithGaussianDOShasrecentlybeensolvedbyRoichmanandTessler[97].The

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authorsusedtheformuladerivedforageneralenergydistributionofchargecarriersandageneralDOSfunction[98,99]

µε

=∂∂

eDn

n1

F

(6.55)

where n is the total concentration of charge carriers and eF is the Fermi energy. UsingEquation (6.25),onecan rewrite thisexpression foragivendensityof statesg(e) in theform[97]

µε ε ε ε

ε ε

ε ε=

( ) -( )[ ]+ -( )[ ]{ }

( )+

-∞

∞

∫e

kTD

gkT

kT

g

F

F

d

d

exp

exp

ex

11

1

2

pp ε ε-( )[ ]-∞

∞

∫F kT

(6.56)

InthecasewhentheFermienergyisverylowandthemajorpartofthecarrierenergydistributionisfaraboveeF ,theFermidistributioncanbereplacedbytheBoltzmannfunc-tionand theratioof integrals inEquation(6.56)becomesunity.Thiscorresponds to thenondegenerate energy distribution of charge carriers. In such a case, the generalizedEinstein relationdescribedbyEquation (6.56)approaches theclassical form representedbyEquation (6.54).For theGaussianDOSdescribedbyEquation (6.4) thiscondition isvalidatsmalldisorderparameters fordeepFermienergy.InFigure6.14theratiomkT/eD isplotted forGaussianDOSasa functionof the ratioeF/kT fordifferentvaluesofs /kT (followingRoichmanandTessler [97]).Using for thedisorderparameter realisticvaluess 0.1eV,one comes to the conclusion that at room temperature the classicalEinsteinrelationcanholdonlyatrathersmallconcentrationsofchargecarriers.Computersimula-tionsfortheratiom /D insuchconditionsinGaussianDOSaredescribedin[100].Fora

Figure 6.14 Theratiom /D asafunctionof theFermileveleF/kT fordifferents /kT (reproducedwithpermissionfrom[97],Copyright2002,AmericanInstituteofPhysics)

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degenerate system the generalized Einstein relation has to be calculated in its full formusingEquation(6.56)withg(e)describedbyEquation(6.4)[97].

Aquestioncouldariseastowhysomecomputersimulationsprovideresults that lookcontradictory to the above conclusions. For instance, it has been claimed on the basisof straightforward Monte Carlo computer simulations that, in disordered systems withGaussianDOS,hoppingmobilityanddiffusivitydonotobeytheclassicalEinsteinrelationgiven by Equation (6.54) even in the case of noninteracting carriers when the latter areconsideredas independententities [96].On theotherhand,Equation (6.56) forachargecarrierinanemptysystemshouldcoincidewithEquation(6.54).Thisproblemhasbeensolvedin[100],whereitwasshownthatincomputersimulationsthatdemonstrateapparentdeviationsfromEquation(6.54)thesystemofchargecarrierswasnotinthermalequilib-riumbecauseoftheparticularchoiceofthesimulationparameters.Assoonastheequilib-riumconditionswereestablished, the relationbetweenD andm for independent carriersbecameinagreementwithEquation(6.54)[100].

6.7 ON THE DESCRIPTION OF COULOMB EFFECTS CAUSED BY DOPING IN DISORDERED ORGANIC SEMICONDUCTORS

OneoftheinterestingtopicsinresearchondisorderedmaterialsistheeffectofCoulombpotentialsofchargedspeciesontransportproperties.Thedecisiveroleofsucheffectsforvarious charge transport phenomena has been clarified for hopping transport in dopedcrystallinematerialsandininorganicdisorderedmaterialssuchasamorphoussemiconduc-tors.Thistopichasalreadybecomeasubjectoftextbooks[17,101].Withrespecttodisor-deredorganicmaterialsthesituationisnotasfavorable,althoughCoulombeffectsinsuchmaterials can play even a more pronounced role. The dielectric constant of the organicmatrixisusuallyseveraltimessmallerthanthatininorganicmaterials.ThisshouldmakeCoulombeffectsmoresignificantinorganicmaterials.NotmuchhasbeendoneyetinthestudyofCoulombpotentialsinorganicdisorderedsolidsandonlyseveralinitialtreatmentshave been attempted so far (see, for instance [102]). Unfortunately, the results of thesetreatmentsareincontradictiontotheresultsonCoulombeffectsfromtextbooksdevotedtoinorganicmaterials.Thissituationresemblesthatoftheinitialstudyofhoppingtransportinorganics.Researchersbeganthisstudynottryingtousetheexperiencealreadygatheredinthefieldofinorganicsystems.Thereforewewouldliketoanalyzebrieflytheshortcom-ingsofthesuggestedtreatmentsofCoulombeffectsinorganicmaterialsinordertowarnresearcherswithrespecttopossibledrawbacksinsuchtreatments.

Ithasbeenestablishedexperimentallythatthedopingeffciencyofdisorderedmaterialsismuchlowerthanthatofcrystallinesemiconductors.Forexample,inamorphousinorganicsemiconductors, such as hydrogenated amorphous silicon, a-Si:H, the concentration ofimpuritiesasdeducedfromexperiments involvingelectronicstates isconsiderably lowerthanthatdeterminedfromthestudyoflocalbondingconfigurationsbyextendedX-rayfinestructureornuclearmagneticresonance[103].Alsoindisorderedorganicmaterialsithasbeenclaimedthatatlowdopinglevels,electrochemicaldopingismuchlesseffcientthanthe field-effect doping in which the same amount of charge carriers is injected into thesystemwithoutinducingchemicallyforeignimpurities[104,105].Moreover,atlowdopant

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concentrations,chemicaldopingcanevenleadtodecreasingcarriermobility[106].Athighdopinglevelsthemobilitysteeplyincreaseswithdopantconcentration[104–106].Qualita-tively this resultwas interpretedbyassuming that inchemicallydopedmaterialschargecarriers can be trapped by Coulomb potentials of ionized dopant species at low dopinglevels[105].Concomitantly,thecarriermobilitycouldbemuchsmallerthanthefield-effectmobilitymeasuredwithout introducingchargeddopants into thesample.Athighdopinglevels in theelectrochemicalprocess theenergy landscapemightbecomemoreuniform,leadingtotheincreaseofthecarriermobility[105].

Twoidenticalattempts[102,107]wererecentlyperformedinordertoputthisargumentontoaquantitativetheoreticalbasis.Webrieflydescribetheseattemptsandshowthattheset of equations suggested in such an approach is irrelevant for the description of theproblem under study since neither charge neutrality nor screening effects were properlytakenintoaccount.

TheauthorsconsideredanarrayoflocalizedstatesofthehostsystemwithsomehighconcentrationNt~1021cm-3andanarrayofdopantatomswithmuchlowerconcentrationNd~1018cm-3.Tobespecificwewillconsider thecaseofdonorsasdopantspecies.Thecaseofacceptorscanbe treated inananalogousway.Donorsaresupposed togive theirelectronsintolocalizedstatesofthehostsystemandtobecomepositivelycharged.Arkhipovet al. [102,107]considereda localizedstateandestimated theCoulombenergyshiftofthisstateduetothepresenceofthechargeddonorswithconcentrationNd.Theyconsideredthe contribution to the Coulomb potential from only the nearest donor. The probabilitydensity,w(r),ofhavinganearestdonoratadistancertoachosenlocalizedstateisdeter-minedbythePoissondistribution

w r r N N r( ) = -

4

4

32 3π π

d dexp (6.57)

The energy of the localized state under consideration is shifted downward by theCoulombpotentialofthenearestdonor.Lettheenergyofthislocalizedstateintheabsenceofdonorsbee.In[102,107]itisarguedthatinthedopedsystemtheenergyofthisstatebecomesequaltoE =e +Ec,where

Ee

rc = -

2

κ (6.58)

istheenergyshiftduetotheCoulombpotentialofthenearestdonor.InEquation(6.58)e istheelectronchargeandkisthedielectricconstantofthehostmaterial.AweakdopingwasconsideredwhentheconcentrationofdopantionsNdremainsmuchsmallerthanthetotaldensityofintrinsichoppingsitesNt.Theauthorsclaimthat,undertheseconditions,theenergyofalmosteverylocalizedstatewillbeessentiallyaffectedonlybythenearestdopantion.InsuchacaseonecaneasilyfindthedistributionG(E)ofsiteenergies,E =e +Ecfromthegivendensityofstates(DOS)forintrinsicenergiesg(e)andfromEquations(6.57)and(6.58)forthedistributionofCoulombenergiesEc.Duetotheeffectoftheposi-tivelychargeddonors,thedistributionG(E)appearsshiftedtolowerenergiesascomparedwiththedistributionofintrinsicenergiesg(e).Hencetheextraelectronssuppliedintothesystembychemicaldopingarenotforcedtofilltheintrinsicdensityofstates(thuspushing

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theFermilevelupwards),butrathertheyfillthedensityofstatesalreadyshiftedtowardslowerenergiesbyCoulombpotentialsofthepositivelychargedonors[102,107].Thereforethe increaseof theFermi levelwithdopingoccursmuchmoreslowly thanwouldbe thecaseintheintrinsicDOSg(e)filledbythesameamountofelectrons.Afterhavingcalcu-latedG(E),Arkhipovet al. [102]studiedthehoppingtransportofelectrons, treatingthesystemasspatiallyhomogeneouswithDOSG(E).Wehavetopointoutthedeficiencyofsuchtreatment.

Inthepicturedescribedabove,oneassumesthatonlyasingledonor,namely,thenearestone,causestheCoulombshiftoftheenergyofalocalizedstateinthehostmaterial[102,107].In[102,107]itisclaimedthatthisassumptionisjustifiedbytheinequalityNd << Nt.MoreovertheauthorsclaimthatthisassumptionunderconditionNd << Nt isobvious[102].Wearguehoweverthatthisassumptionisneitherobvious,norcorrect.Ofcourse,thecon-tributionofdonorstotheCoulombpotentialonalocalizedsitedecreaseswiththedistancefromthesiteasr-1.However,thenumberofdonorsinasphereofradiusr aroundthechosensite increases as r3. Therefore, the contribution to the Coulomb potential of a chosenhoppingsitefrommoredistantdonorsthanthenearestoneincreasesproportionaltor2.Inthe absence of screening, as considered in [107], distant donors contribute more to theCoulombenergyshiftsonlocalizedstatesthanthenearestones.Theconditionexpressedvia inequalityNd << Nt is irrelevant for thisconclusion.Furthermore,oneshould realizethat,inthepicturesuggestedin[102,107],theenergydivergesifoneconsidersonlydonorsaschargedcentersandtakesintoaccounttheCoulombcontributionstotheenergyofanintrinsicsitefrommoreandmoredistantdonors.Thisisatrivialresultdiscussedinseveraltextbooks (see, for example, [17,101]). Inorder to avoid thedivergenceof theCoulombenergy,oneshouldnotrestricttheconsiderationtochargesofonlyagivenpolarityasdonein[102,107].Instead,oneshouldconsiderbothkindsofcharges—positiveandnegative—keepingthesystemelectricallyneutral.Herewithwecometotheimportantthoughtrivialquestion:areelectronselectricallycharged?Theanswertothisquestionisdefinitely‘yes’.Adonorbecomeschargedpositivelyonlybecauseitcangetridofavalenceelectron.In[102,107],electronsbroughtintothesystembydonorsweregivenjustapassiveroletofillthedensityof states shifted to lowerenergiesbypositivelychargeddonors.One should,however,takeintoaccountthatelectronsarealsochargedwithoppositepolaritytothatofthedonors.TheconcentrationofthenegativelychargedextraelectronsintroducedintothesystembydonorsisequaltothatofchargeddonorsNd.Theseelectronsarethecausefortheeffectontheenergiesofintrinsicsites,whichisexactlyoppositetothatofpositivelychargeddonorsexclusivelyconsidered in [102,107].Beingnegativelycharged,electronsshifttheenergiesofintrinsichoppingsitesupward.Theauthorsof[102,107]deliberatelytookintoaccountonlypositivecharges.Ifonewoulddothesame,takingintoaccountonlynegativechargesofextraelectrons,onewouldcometotheconclusionexactlyoppositetothatin[102,107],namelytotheconclusionthatDOSfunctionwouldbeshiftedupwardinenergywithrespecttothatinanundopedsample.Ofcourse,noneofsuchdeliberatecon-siderationscanbecorrect.Oneshouldconsideranelectricallyneutralsystem,takingintoaccountbothpositivelychargeddonors andnegativelychargedelectronsasdescribed intextbooks[17,101].Thecrucialpointinsuchtreatmentsisthequestionofthespatialdis-tributionofcharges.Donorsareassumedtobedistributedrandomlyinspace.Whataboutelectrons?

Letus,following[102,107],consideralightlydopedsamplewhenconditionNd << Nt isfulfilled.ThisassumptionisplausiblesinceestimatesgiveNd ~1018cm-3andNt ~ 1021cm-3

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[102,107].ConditionNd << Nt cannotjustifythatonlythenearestdonortoalocalizedstateshouldbetakenintoaccount,thoughthisstronginequalitywillhelpustoanswertheques-tionofwhereelectronsaresituated.ACoulombenergyshiftofanintrinsiclocalizedstatefromthenearestdonorisdescribedbyEquations(6.57)and(6.58).Itisdeterminedbythetypicaldistancebetweendonorsrd Nd

-1/3.InsertingsuchestimateinEquation(6.57),andtakingk=3asknownfororganicsemiconductors,oneobtainsthetypicalCoulombcon-tributiontotheenergyofalocalizedstatefromthenearestdonoroftheorderEc ~ 0.08eVforNd ~ 1018cm-3 [102,107].Since thewidthof theenergydistributiong(e)of intrinsichoppingsitesisoftheorderof~0.1eV,theeffectofCoulombcentresonintrinsiclocalizedstatesseemsessential.Weclaimhoweverthatintheabovepictureoneshouldconsidernotalocalizedstateandthenearestdonortoit,butratherthelocalizedstatewhichisthenearesttoadonor.While the typicaldistancebetweena localizedstateandthenearestdonor isdeterminedbythedistanceoftheorderrd Nd

-1/3, thedistancebetweenadonorandthenearestlocalizedstateisdeterminedbythedistancertN t

-1/3,whichismuchsmallerthanrd. Inserting such a value for rt into Equation (6.58) and taking for the concentration oflocalizedstatesthemagnitudeNt ~ 1021cm-3,assuggestedin[102,107],wefindthattheCoulombenergyshiftofthelocalizedstatenearesttoadonorEcisabout0.8eV.ThisvalueisbyanorderofmagnitudelargerthantheshiftEc ~ 0.08eVcausedonalocalizedstatebythedonornearesttoit.Inthegroundstateofthesystem,electronswouldtendtooccupydeeper energy levels and therefore theywill be situatedon the intrinsic sites,which areclosesttodopantions(donors).Thereforeoneshouldconcludethatallchargesbroughtintothesystembydonorsaregathered intodipoles formedbypositivelychargeddonorsandthenegativelycharged intrinsiccentersnearest to them.Onecouldsuppose that in [102,107]theeffectwithtypicalscaleofabout0.1eVwasconsidered,neglectingtheeffectwithtypicalscaleofabout1eV.Thesituationis,however,worse.Wehavejustseenthatduetotheeffectofthetypicalscaleof~1eV,theeffectofthescale~0.1eVconsideredin[102,107]doesnotexist,sincenotthepointcharges,butratherveryshortdipolesaffectlocalizedstatesintheintrinsicmaterial.Theextraelectronsbroughtbydonorsaretrappedintostateswithverydeepenergies( -1eVatNt =1021cm-3)whicharethenearesttodonors.Thismightbethereasonwhynoeffectofincreasingconductivityhasbeenobservedexperimen-tallyatlowdopantconcentrationswhenconditionNd <<Nt isfulfilled.TheeffectofheavydopingwhenthestronginequalityNd << Nt breaksdownneedsaspecialtreatment,whichisbeyondourscope.LetusinsteadestimatetheeffectofCoulombpotentialsonagivenintrinsiclocalizedstateinalightlydopedmaterial.

Following [102,107]weconsider agiven localized site in the intrinsicmaterial.Thedistancerd Nd

-1/3fromthislocalizedsitetothenearestdipoleisdeterminedbythecon-centrationofdipoles,whichisequaltotheconcentrationofdonorsNd.Thelengthofthedipoleisdeterminedbytheconcentrationoflocalizedstatesrt N t

-1/3.Thecontributionofsuchadipoletotheenergyofagivenlocalizedstateisoftheorder

Ee

r

r

rc -

2

κ d

t

d

(6.59)

ThisequationreplacestheestimategivenbyEquations(6.57)and(6.58).TheresultofEquation(6.59)fortheenergyshiftEc on‘alocalizedstate’isbyanorderofmagnitudesmallerthanthatobtainedin[102,107],sinceinEquation(6.59)theCoulombcontributionofapointchargeatadistancerd ismultipliedbyasmallfactorrt/rd,whichisequalto0.1

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fortheratioNd/Nt =10-3chosenin[102,107].Thiscontributionislessthan0.001eVanditcanbeneglected.

One should emphasize that the study of the effects of Coulomb potentials on chargetransportindisorderedorganicmaterialsisstillinitsinitialphaseandmorestudyisneededto clarify the role of these effects. In particular, the calculation of transport propertiesaffectedbychargedistributionathighdopinglevelsisachallengingtheoreticalproblem,whichstillawaitsitssolution.

6.8 CONCLUDING REMARKS

Inthischapterwehavepresentedseveralbasicconceptsdevelopedfordescriptionofchargecarriertransportinorganicdisorderedsemiconductors,suchasmolecularlydoped,conju-gatedpolymers,andorganicglasses.Theseconceptsare,toagreatextent,analogoustothetheoreticalconceptsdevelopedearlierfordescriptionofchargetransportininorganicdis-orderedmaterialssuchasamorphousandmicrocrystallinesemiconductors.Therefore,wehavetriedtokeepthepresentationoftheseideasparalleltothatinChapter2ofthisbook[14].However,contrarytoChapter2,twoimportanttopicswerenotconsideredhere—thethermallystimulatedcurrentsandthenonlineartransporteffectsinhighelectricfields.Thedescriptionofthermallystimulatedcurrentsinorganicdisorderedmaterialscanbefoundin theworkofSchmechelandvonSeggern [108].Thedescriptionof thenonlinearfieldeffects in such materials can be found in [77, 80–82]. Furthermore, we focused in thischapter only on the description of the motion of charge carriers through the disorderedmaterial.Suchimportant topicsas the injectionofchargecarriersfromthecontacts intothesystemaswellasthedescriptionofthespace-charge-limitedcurrentsremainedbeyondour scope. Readers interested in these topics can find comprehensive descriptions, forexample,intherecentreviewarticleofH.Bäassler[4]andinChapter7ofthisbook[3].

Acomparisonbetweentheresultsof thischapterandthosedescribedinChapter2ofthisbookshow the roleof theDOSfunctionon the transportphenomena.Although thetransportconceptsusedinthesetwochaptersareverysimilartoeachother,someresultsforGaussianDOSdifferessentiallyfromthosefortheexponentialDOS.ThereforewehaveconsideredthechargetransporteffectsfortheGaussianDOSandfortheexponentialDOSintwoseparatechapters.

Acknowledgements

Theauthorsareindebtedtonumerouscolleaguesforstimulatingandenlighteningdiscus-sions.Inparticular,wewouldliketoexpressourgratitudetoIgorZvyagin(MoscowStateUniversity)forclarifyingtoustheroleofdistanthopsin1DsystemsandtoPeterThomas(PhilippsUniversityMarburg)forclosecollaborationontopicsdiscussedin thischapter.FinancialsupportoftheDeutscheForschungsgemeinschaft,oftheFondsderChemischenIndustrie,oftheOptodynamicCentreatthePhilippsUniversityMarburg,oftheEuropeanCommunity [IP ‘FULLSPECTRUM’ (Ref. N: SES6-CT-2003-502620)] and that of theEuropeanGraduateCollege‘Electron-ElectronInteractionsinSolids’Marburg–Budapestisgratefullyacknowledged.

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