Distribution of charge carrier transport properties in organic semiconductors with Gaussian disorder

0.43 MB·English

by  Jens Lorrmann

DistributionofchargecarriertransportpropertiesinorganicsemiconductorswithGaussian
disorder
Jens Lorrmann,1,∗ Manuel Ruf,1 David Vocke,1 Vladimir Dyakonov,1,2 and Carsten Deibel1,†
1ExperimentalPhysicsVI,JuliusMaximilianUniversityofWu¨rzburg, 97074Wu¨rzburg, Germany
2BavarianCenterforAppliedEnergyResearche.V.(ZAEBayern), 97074Wu¨rzburg, Germany
(Dated:September12,2012)
Thechargecarrierdriftmobilityindisorderedsemiconductorsiscommonlydeterminedfromasingletransit
time graphically extracted from time-of-flight (TOF) photocurrent transients. However, the term transit time
isambiguouslydefinedandfailstodeliveramobilityintermsofastatisticalaverage. Here,weintroducean
advancedcomputationalproceduretoevaluateTOFtransients,whichallowstoextractthewholedistributionof
transittimesandmobilitiesfromthephotocurrenttransient,insteadofasinglevalue. Thismethod,extending
theworkofScottetal.(Phys.Rev.B46,8603),isapplicabletodisorderedsystemswithaGaussiandensityof
states(DOS)anditsaccuracyisvalidatedusingone-dimensionalMonteCarlosimulations.Wedemonstratethe
superiorityofthisnewapproachbycomparingittothecommongeometricalanalysisofhole TOF transients
measured on poly(3-hexyl thiophene-2,5-diyl) (P3HT). The extracted distributions provide access to a very
2
detailedandaccurateanalysisofthechargecarriertransport. Forinstance,notonlythemobilitygivenbythe
1
meantransittime,butalsothemeanmobility—whicharegenerallynotidentical—canbecalculated. Whereas
0
thelatterdeterminesthemacroscopicphotocurrent,theformerisrelevantforanaccuratedeterminationofthe
2
energeticdisorderparameterσwithintheGaussiandisordermodel(GDM).σderivedbyusingthecommon

p geometricalmethodis,asweshow,underestimatedinstead.
e
S
I. INTRODUCTION oftransportpropertiesthroughoutthethin-filmdevice.19–22A
1
1 detailedexperimentalinvestigationoftherelationbetweenthe
Disordered or amorphous semiconductors provide the po- macroscopic behaviour and the microscopic transport statis-

] tential of cheap future electronics with a wide variance of tics, however, requires the extraction of the transport param-
n
n physicalproperties,e.g.,absorptionspectrum. Thisflexibility eter distributions. Furthermore, new conclusions in terms of
- allowstodesignthemforspecificareasofapplication, rang- the energy distribution of the DOS may be drawn from the
s ingfromtransistorsandlightemittingdiodestosunlighthar- shapeofthesedistributions. Suchanapproachshouldsignif-
i
d vestingandconvertingsolarcells. Acommonkeyfeaturede- icantly improve the ability to check the consistency between
t. terminingtheelectricaldeviceperformanceinthesematerials experimentaldataandvarioustheoreticalmodels. Inturn,we
a is the hopping type charge carrier transport. Therefore, a lot consideritasafundamentalsteptowardsamoregeneralfunc-
m
of effort has been put into studies during the last decades to tionaldescriptionofthechargecarriertransportindisordered
- understandthedetailedmechanism,experimentallyandtheo- systems in terms of drift and diffusion. In some cases, such
d
n retically.1–6 descriptionsarealreadyavailable,15,16,18,21,23–26butnotinthe
o Inclassicaldescriptionsthechargetransportisbasedonthe generalformpresentedhere.
c electrostaticforceinduceddriftanddiffusionduetoaspatial First attempts to experimentally extract mobility distribu-
[ chargecarrierdensitygradient.Therelatedparametersarethe tions from time-of-flight (TOF)27–29 measurements or from
2 chargecarriermobilityµandchargecarrierdiffusivityordif- theturn-ondynamicsofpolymerphoto-cellswherepublished
v fusion constant D, respectively, which are connected by the by Scott, etal.30 and Rappaport etal.20,31, respectively. Un-
2 classical Einstein7–Smoluchowski8 relation D/µ = k T/e. fortunately,themethodbyScottetal.wasseldomusedsince
B
2 Here,k istheBoltzmanncoefficient,etheelementarycharge its publication in 1992.32 The same holds for the method
B
9 and T the temperature. This concept successfully describes publishedbyRappaportetal.Moreoverthefullpotentialutil-
1
semiconductordevicescouplingcontinuityandPoissonequa- ising the statistics of the extracted parameter distributions
.
9 tions9,10 in combination with physical models concerning, was hardly ever discussed. In the case of Scott’s method
0 e.g.,dissociation11,12orrecombination.13,14 this is probably due to the fact that the method as such is
2
However,inthecaseofdisorderedsystemsitisknownthat not suitable for systems exhibiting a Gaussian density of
1
theconceptofmobilitycanbeill-defined15or,inotherwords, states(DOS),foundinmanyorganicsemiconductors,e.g.,the
:
v the transport is determined by a broad distribution of mobil- widelyusedconjugatedpolymerpoly(3-hexylthiophene-2,5-
Xi ities. This fact is due to the random nature of the charge diyl)(P3HT).
carrier transport in an energetically and spatially disordered InthispaperweextendthemethodbyScottetal.withre-
r
a environment. It leads to energetic relaxation of the charge specttoaGaussianDOSandrevealtheadvantagesofthisap-
carrierswithtime,15–18 andtoaninhomogeneousdistribution proachovertheconventionalgeometricaltechniquetoextract
mobilities. The geometrical method determines the charge
carriermobilityfromaparticulartransittime,whichisgraph-
ically extracted from TOF transients. This transit time is
∗[email protected] defined by the intersection of the fits to the pre- and the
†[email protected] post-transit sections of the photo current transient, respec-
2
tively15,33. Henceforth we refer to the methods compared in for purely Gaussian (non-dispersive) transport, together with
thisworkastheScottmethodandthegeometricmethod, re- a decreasing mean drift velocity with time.16,38 As a conse-
spectively. To verify the accuracy of the our new approach quence,thedistributionoftransittimesisbroadenedandthus
and the modifications to Scott’s method we utilise a one- thedropinthephotocurrentattheextractionisblurred.
dimensional Monte Carlo simulation. Generally, the justifi- The TOF transients for holes in P3HT where measured in
cationofanalternative,newapproachisbestdemonstratedin the temperatures range between T =110−300K and at dif-
comparisontoawell-establishedstandardmethod.Hence,the ferentelectricfieldsbetweenF =1.3·107−1.9·108 V.
m
comparisonisdoneherebyevaluatingtheholeTOFtransients
measuredonP3HTwiththeScottandthegeometricmethod,
respectively.Subsequently,weidentifythestatisticalaverages III. EVALUATIONMETHOD
of the transit times and the mobilities from the transit time
andmobilitydistributions,andmatchthemagainstthetransit Before describing the Scott method, it is worth having a
timesandmobilitiesdeterminedfromthegeometricanalysis closer look at the photocurrent decay prior to extraction of
ofphotocurrenttransients. chargecarriers,asitsfunctionalapproximationisanessential
Although convenient to use, it is, however, well known partofthismethod.
that the geometric method extracts single value transit times IncaseofapurelyexponentialDOSthistransientdecaycan
which account for the fastest carriers only.34,35 We demon- be explained within the framework of Scher and Montroll15
stratebelow,thatthecorrespondingmobilitiesarenotrelated yielding a power law for the pre-transit (j (t)∝t−(1−αpre))
0
to the average values and that the relative deviation depends photocurrent. Note that this behaviour in combination with
strongly on temperature and electric field. Furthermore, the the predicted post-transit current j(t)∝t−(1+αpost) builds the
geometric method is more prone to evaluation errors, as, in basisofthegeometricanalysis.33 Incontrasttotheexponen-
thecaseofsystemsexhibitingaGaussianDOS,thefittedre- tialDOS,wheretransientmeasurementsstrictlyfollowthese
gionsjustspanoveraverysmallpartofthecurrenttransient power laws at least “as long as one would be able to mea-
onadouble-logarithmicscaleandtheregions’boundariesare sure”,39 a Gaussian DOS yields a saturation of the transient
judgedbytheeye,whichmayleadtohugeuncertainties. On photocurrent in time, thus it is a power law with a time de-
the other hand, the Scott method, as will be shown below, pendentexponent j (t)∝tα(t). Thetimedependenceofα(t)
0
needsjustonefitspanningovermorethanoneorderofmag- isrelatedtoexperimentalparameters,suchasthetemperature
nitude in time. Novikovetal. explicitly discussed the conse- and the applied electric field, and material properties (ener-
quencesofusingpoorlydefinedtransittimesormobilitiesin getic and spatial disorder).16,38 Unfortunately, a plain func-
Ref. 36 and asked for a safe and reliable procedure for the tional description of α(t) and, thus, the photocurrent decay
analysisof TOF data. Webelievethatthenewapproachpre- j0(t),similartothepowerlawrelationforaexponentialDOS,
sented here fulfils this request, by experimentally providing is not available. However, taking into account that α(t) is a
detailedandcomprehensiveinformation,asitisusuallyonly graduallydecreasingfunctionandthat limα(t)=0,wesug-
knownfromMonteCarlosimulations. t→∞
gestthefollowingempiricalapproximationof j (t):
0
j (t)= j∗t−α(t) (1a)
0 0
II. EXPERIMENTALMETHOD
(cid:16)τ(cid:17)β
α(t)= . (1b)
t
The studied devices were prepared in a diode configura-
tion, with P3HT 4002E (∼94% regioregular; Rieke Metals
j∗isascalingfactordeterminingthephotocurrentatt=1s,
Inc.,withoutfurtherpurification)astheactivematerial,drop 0
while τ is a lifetime and β is a stretching parameter defining
castonITOcoveredglasswithanaluminiumelectrodether-
the dynamics of the decay. We confirm the accuracy of this
mallyevaporatedontop. Thesamplethicknessandexcitation
expressioninSec.Vusingone-dimensionalMonteCarlosim-
intensity were chosen in a way to ensure a photogeneration
ulationsandshowthatallthreeempiricallydefinedfittingpa-
ofchargecarriersconfinedtothefirst10%ofthebulkthick-
rametersexhibitastricttemperaturedependence. Thefidelity
ness. Fortheopticalexcitationweusedthesecondharmonic
ofEqn.(1)canalreadybeseenfromFig.1.
ofapulsedNd:YAG laseremittingatλ=532nm,closetothe
In order to describe the full transient including extraction,
absorption maximum of the polymer,37 together with optical
the integrated number of charge carriers arriving between t
intensityattenuators.
anddt, p (t),issubtractedfrom j (t):
Adjustingthepolarityoftheappliedconstantelectricfield, tr 0
one can select the type of the charge carriers being dragged (cid:18) (cid:90) t (cid:19)
through the bulkand extracted at the counter-electrode. Due j(t)= j0(t) 1− ptr(tˆ)dtˆ . (2)
0
totheextraction,thephotocurrentsteeplydropstowardszero
until no mobile charge carriers are left within the device. Hence, the transit time distribution p (t) can be calculated
tr
However, the photocurrent decreases even before extraction fromtheexperimentalphotocurrent j(t)and j (t)as
0
due to energetic relaxation of the charge carriers within the
DOS.16–18,21 Associated with this relaxation is a dispersion d j(t)
p (t)=− . (3)
of the charge carrier package being stronger than predicted tr dt j (t)
0
3
TABLEI.OverviewoverthefourdifferentmobilitydefinitionsusedthroughoutthisPaper.
Name relationtotransittime physicalcontext Scheme
Scottmethod log(j) 〈t-1〉〈t 〉
(cid:28) (cid:29) j(t) tr tr
1 ensembleaveragemobility;determines j (t)
µm ∝ ttr themacroscopicphotocurrent 0
1 related to average transit time; often ptr(t)
µtr,m ∝ (cid:104)ttr(cid:105) usedinMonteCarlosimulations log(t)
geometricmethod log(j) t t
tr,geo tr,1/2
j(t)
µgeo ∝ttr,1geo triemacehatthewchoicuhntethreeleficrtsrtodceharge carriers jpre
1/2∙j
pre j
1 timeatwhich∼50%ofthechargecar- post
µ ∝
1/2 t rierswereextracted
tr,1/2 log(t)
Six exemplary transit time distributions calculated from Notethatgenerallythemathematicalexpectationofthemo-
measured and simulated TOF transients are plotted in the bility (cid:104)µ(cid:105) is not identical to µtr,m, which is determined by
lower part of Fig. 1. A transit time distribution can (cid:104)t (cid:105), as the former is proportional to the average of the in-
tr
be transformed into a mobility distribution by: pµ(µ) = verse transit time tt−r1, while the latter is determined by the
(cid:0)F·t2/d(cid:1)p (t). averagetransittimet . Consequently,µ andµ mayyield
tr tr m tr,m
The comparison of the Scott and the geometric method is very different values, as yet the mean mobility µm is mathe-
basedonthecomparisonoffourexplicitmobilitydefinitions. maticallylessaffectedbythefractionofslowchargecarriers
Tofacilitatethelaterdiscussionweshortlysummarisethedif- thanµtr,m. Thelattermobilityisoftenfoundinliteratureasan
ferencesbetweenthecomparedmobilities.InTab.Iwedepict output parameter treating the charge transport in disordered
andlisthowtheyareobtained,weassignthemtotheircorre- materials using Monte Carlo21 or Master Equation1,24 simu-
spondingmethodandputanoteontotheirphysicalrelevance. lations. µm, instead, is linked to the average charge carrier
In general the relation between mobility and transit time is: transport and determines the macroscopic current density by
(cid:82)
µ=L/(ttr·F), with the sample thickness L and the applied j=e·n µ·pµ(µ)dµ·F =e·n·µm·F.
electric field F. Thus, the mobility is commonly determined
in TOF transients from a particular transit timettr. The mo-
bilitylabelsandtherelationtothecorrespondingtransittimes IV. SIMULATIONMETHOD
can be found in Tab. I and the transit times are obtained as
follows. In order to verify our approach, a one-dimensional Monte
In the geometrical method ttr,geo is identified from the Carlosimulation42,43wasimplementedtostudychargecarrier
intersection of two power laws fitted to the plateau (jpre) transport by hopping. It considers the capture and the emis-
and the trailing edge (jpost) of the photocurrent transients sionofchargecarriersaccordingtothemultipletrappingand
in double-logarithmic presentation, and ttr,1/2 is the time at release (MTR) model. Before further specifying the simula-
which1/2·jpre= jpost (seelowerschemeinTab.I).Fromthe tion procedure, let us first recall some details of the MTR-
extracted distributions the mathematical expectations can be model developed by Schmidlin etal.44 and Noolandi etal.45
calculatedusingstandardprobabilitytheory,40,41 whichgives Originallythepresenceofextendedstatesinthevalenceand
therelationbetweenthetransittime(cid:104)ttr(cid:105)andthetransittime conduction bands and localised trapping states in the band
distribution ptr(t)as gap were considered. These states are separated by the mo-
bility edge and only charge carriers which are thermally ac-
(cid:90)
(cid:104)t (cid:105)= t p (t)dt . (4) tivated to extended states above the mobility edge contribute
tr tr tr
tochargetransport. Indisorderedorganicsemiconductorsin-
stead,thevalenceandconductionbandsaredescribedinterms
Consequently, the stochastical average of the mobility (cid:104)µ(cid:105),
of a Gaussian distribution of localised states. Transport in
calledµ is
m these localised states occurs via hopping. In such a system,
(cid:90) the transport energy46 is a more relevant concept to describe
µm=(cid:104)µ(cid:105)= µpµ(µ)dµ thehoppingtransportintermsoftheMTR.Thetransporten-
ergyE isaparticulartemperature-dependentenergylevel
L (cid:90) 1 L (cid:28)1(cid:29) trans
= p (t)dt= . (5) within the DOS.17,47 The transport resembles the MTR pro-
tr
F ttr F ttr cessnearandbelowthisenergy,byaquasi-freetransportuntil
4
trappingofchargecarriersaboveE andathermalemission
trans Experiment MC-Simulation
backtoE ofchargecarriersbelow.
trans
2 inc. field inc. temperature
Hence,inoursimulationtherelease,i.e.,theemissionfrom
aEEtttrrraaannpss bsatyacttheh,aertrgrmaenasclfaearrcrsiteivaractmihoaonrvg,eeasncqdauraraibseior-vftreoeetthheuentttrriaalnnisstppooisrrtttreeannpeeprrggeyyd sity j / a.u. 100246 11000 ed tnerruc
apsugeranIenidnmietnieoacnrletlryteo,eafacctlhhhlneecishqmauaorengdseeesllcueaacclrhtlrroioaewdsrsseTtaaOhrneeFds.siiemstueinxlatvtriaioccnitneoidtfy.trtTaohnteshieetinmitllemumdeeai--- current den 10246 Tfiot Fto teraqn.(s1ie)nt 01.1 .u.a / j ytisn
natedelectrodeandresideatthetransportenergyEtrans. Dur- ptr 1 0.01 art
ingthetimeuntilcapture n inc. field inc. temperature n
o s
twhiethchthaergmeocbairlriiteyrµa0t(Fth,Teτ)ctr=aandlsnipsτ(ot0ζarn)tceene∆rxgy=mτco·vµe0(qFu,aTsi)-f·r(Fe6e). nsit time distributi 6420 aappffttttrree((ttrr)) eeqq..((73)) 00000.....86420 noitubirtsid emit ti
Where τ0 =[ν0·exp(−2·γ·ax)]−1 is the trapping rate with tra 10-7 10-6 10-5 10-7 10-5 10-3 rtp
the attempt-to-escape frequency ν0, the inverse localisation time t / s time t / s
length γ and theintermolecular distancea . ζ is auniformly
x
distributed random number between ]0,1], F is the electric
FIG. 1. Upper part: Measured TOF transients of hole currents
field and T the temperature. The charge carrier is then ran-
in P3HT at T = 260K for different applied voltages F = 6.7·
domly trapped into one of the trapping levels Etrap with a 107V, 1.1·108V, 1.5·108V (left). SimulatedTOFtransientsF=
m m m
probability according to the Gaussian energy distribution of 1·107V fordifferenttemperaturesT =300K, 250K, 200K(right).
m
theselevels. Thereleasetimefromatraplevelisgivenby Thepre-transitdecayofthephotocurrent(dottedlineswithmarkers)
(cid:18) (cid:19) isreproducedbythefittoEqn.(1)(lineswithoutmarkers). Lower
E −E −a ·F
τ =τ ·exp trans trap x ·ln(ζ), (7) part: Transit-timedistribution ptr(t)fortheexperimental(left)and
r 0
kB·T thesimulated(right)transientscalculatedfromEqn.(3). Notethat
theextracteddistributions(solidlines,right)matchexactlythedistri-
wherek theBoltzmannconstant. Fromthetemporalevolu-
B butions(dottedlines,right)calculatedbyEqn.(9)fromthetemporal
tion of the charge carriers’ positions in the sample, a mean decreaseoftheamountNofchargecarriersinsidethedevice.
currentcanbecalculatedby
j(t)=e·n ·v(t), (8)
0 Ref.48.WeassumedafixedtransportenergyatE =0eV,
trans
because we found just little impact of a temperature depen-
where e is the elementary charge, n the number of initially
0
dent transport energy. Further input parameters are the in-
generatedcarriersandv(t)thecarriers’meanvelocity.
verselocalisationlengthγandtheintermoleculardistancea
By suitable parameterisation of the model, experimental x
and the charge carrier mobility µ (F,T). To fully calibrate
measurements were reconstructed by simulation, offering a 0
thesimulationwesimultaneouslyfittedasetofmeasurements
deeperinsightintomicroscopicchargetransportphenomena.
at different temperatures and electric fields using the global
Todecreasethenumberfreefittingparameterswefirstlyfixed
minimisation algorithm of differential evolution.49 The used
someofthemtoexperimentallydeterminedvaluesaswellas
parameters are listed in Tab. II, where we explicitly mark a
physically reasonable literature values. The energetic disor-
parameterasfixedorfitted.
der parameter was set to be σ=69meV in accordance with
Note that from the one-dimensional Monte Carlo simula-
theextractedtemperaturedependenceofthezero-fieldmobil-
tions the transit time distribution p (t) can be calculated di-
ityintermsoftheGaussiandisordermodel.21Theattempt-to- tr
rectly by the monitored gradual decrease of the number of
escapefrequencywaschosenequaltoν =1·1013s−1 from
0 chargecarriersinsidethedeviceN(t). Inthiscase p (t)be-
tr
comes
TABLEII.Simulationparameters.Theparametersweredetermined dN(t)
p (t)=− . (9)
byreconstructionofexperimentalTOFtransients. tr dt
Parameter Value Unit Source
σ 69.0 meV µ0(1/T2)21 V. RESULTS
ν0 1·1013 s−1 Ref.48
Etrans 0.0 eV Approx. A. VerifyingtheinitialphotocurrentapproximationEqn.(1)
γ 3.91·109 m−1 Fit
ax 1.26·10−9 m Fit Atfirstwedemonstratethequalityoftheempiricalexpres-
µ (F,T) (0.8−2.15)·10−8 m2/Vs Fit
0 sionEqn.(1)of j (t). Tothisend,wesimulatedasetoftem-
0
5
perature dependent (T =200−300K) TOF transients using
ourone-dimensionalMonteCarlosimulationandcomparethe u. 10-12 0.20
a.
transit time distributions obtained by the Scott method with / exponentialfit linearfit 0.18
thetemporaldecreaseofchargecarriersinsidethedeviceac- ntj*010-13 0.16
cordingtoEqn.(9).Theparameterisedvaluesusedinthesim- e
ulationarelistedinTab.II.IntheupperrightpartofFig.1the curr 10-14 0.14 opxe
ficatyoifsEpqlont.te(d1)totgoetthheerswimithultahteedfuplrles-itmraunlsaittedphtoratoncsiuernretsn.tTdhee- s,initial 10-15 00..1120 βtnen
approximation of the photocurrent by j0(t) is excellent over /τ 10-16 τfromfit(eq.(1)) 0.08
fourordersofmagnitudeintime. Theequivalenceoftheex- e
tbroatchtetdradnissittritbiumtieondsisitsrisbhuotiwonnsinmtahtechlowpeerrfericgtlhytpwaerltloofvFeirg.th1e: lifetim 10-17 6 8j0*f1ro0m1fi2t(eq1.4(1)1)6 0.060.0β8fr0o.m10fi0t.(1e2q.0(.11)4) 00..0064
wholestudiedtemperaturerangeandverifythevalidityofthe (σ/kT)2 (kT/σ)2
b b
extrapolation of j (t) to post-transit times. The correspond-
0
ingaveragetransittimesofthesimulatedensembleofcharge
FIG.2. Temperaturedependenceoftheempiricalfittingparameters
carriersandthevaluesextractedfromthetransittimedistribu- j∗, τ and β together with exponential and linear fits. log(j∗) and
tionsptr(t)byEqn.(4)areequalwithintherangeofnumerical lo0g(τ)areproportionalto1/T2,whileβscaleslinearlywithT02.
precision(∆(cid:104)t (cid:105)/(cid:104)t (cid:105)<0.5%).
tr tr
FurtherevidenceforEqn.(1)comesfromthefact,thatal-
thoughEqn.(1)isanempiricalapproximationthethreefitting
relaxation. Thus, it is very likely that they are related to the
parametersfollowastricttemperaturedependence,ascanbe
average charge carrier trapping as well as the charge carrier
seeninFig.2. Beforeweratethesedependencies,letusfirst releasetimelimitingrelaxation.2,16
haveadeeperlookatthemathematicalmeaningof j∗, τand
0 Basedontheseresults,weconcludethatEqn.(1)isasuit-
βandtheirrelationtophysicalproperties.
ableapproximationforthephotocurrentdecayandthat,thus,
τdefinesthetemporalcurrentsaturation. Duetothedefini-
the Scott method can be applied to systems with a Gaussian
tionofEqn.(1) j (t)hastwofixedcurrentvaluesindependent
0 DOS.
oftheparameterβ, j (t=1s)= j∗and j (t=τ)= j∗·τ.Con-
0 0 0 0
sequently, as long as the current saturation time τ is shorter
than1s j∗isshouldbeproportionaltotheequilibriummobil-
0 B. Evaluatingholephotocurrentswiththenewapproach
ity. AccordingtoNikitenkoetal.16thisshouldbethecasefor
(σ/(k ·T))2<5.4.Finally,βspecifiesthecurvaturebetween
b We examined TOF transients measured on P3HT utilis-
thesetwofixedpointsintimeatt=τandt=1s—thelarger
ing the presented approach. We compare the outcome with
βthefasterthedecay. Thus,asrelaxationisacceleratedwith
theresultsobtainedbytheconventionalgeometricanalysisin
increasingtemperature,βincreaseswithtemperature.
Sec.VD.IntheupperleftpartofFig.1threeTOF-transients,
Wefoundthat j∗andthelifetimeτscaleas
0 parametricintheappliedelectricfieldatT =260K,areplot-
j∗∝exp(cid:34)−0.45·(cid:18) σ (cid:19)2(cid:35) tienditiatolgdeetchaeyr wofitthhefietsxpoefrimj0(etn)taalcpcohrodtoincgurtroenEtqjn(t.)(.1)Thtoe ethxe-
0 k ·T cellentagreementbetweenthedataandthe j (t)overalmost
B 0
(cid:34) (cid:18) σ (cid:19)2(cid:35) tswuroedortdeemrspeorfatmuraegsniatnuddeeilnecttirmicefiweladss.achHieovwedevfeor,r iatllismeexa--
τ∝exp −0.67· ,
kB·T pected that at very short times j0(t) is not able to approxi-
mate the photocurrent j(t), since then the current decay on
theonehandinvolvesacontributionofthequicklyextracted
electrons at the illuminated contact and on the other hand
(cid:16) (cid:17)2
respectively. Theexponentβdependson kB·T as is limited by the setup’s RC time. The respective transit
σ
time distributions, plotted in the lower left part of Fig. 1,
(cid:18)k ·T(cid:19)2 werecalculatedfromtheexperimentalcurrentdensity j(t)as
B
β=0.64· −0.031. well as the fitted decay j (t) according to Eqn. (3). These
σ 0
transit time distributions — and consequently the mobil-
ity distributions — are approximately inverse Gaussian51–53
Therefore, for the simulated temperature range in Fig. 2, distributed, in good agreement with distributions obtained
(σ/(kb·T))2≤4 j0∗shouldbeproportionaltotheequilibrium from measurements on poly(2-methoxy,5-(2’-ethyl-hexoxy)-
mobility. And, indeed, theprefactorintheexponentialfunc- p-phenylene vinylene) (MEH-PPV) thin films performed by
tionof j∗isfoundtobe0.45,veryclosetothepredictedvalue Rappaportetal.20,31andondiketopyrrolopyrrole-naphthalene
0
of 0.516,50 in case of a constant transport energy E =0. copolymer(PDPP-TNT)thinfilmtransistorsbyHaetal.54In-
trans
τandβhaveverysimilarprefactors, butinversetemperature terestingtonoteisthatinverseGaussiandistributionsarewell
scaling. However, the physical origin of this scaling might known to describe the “First Passage Time Probability”52,53
be the same, as both parameters define the dynamics of the since Schro¨dinger’s theoretical work about brownian motion
6
pµ 1.2 µtr,m µ 0.8 0.8
bility distribution 1000....0864 1/µ2m µgeo n(t < t)trtr,geo 000...642 TTT === 113390000KKK 000...6422/1,rtrt)t < t(n
o 0.0 0.0
m 0.2
onductivity distribution ·pµµ 00000000........06543210 TTT === 113390000KKK 2-1-1mobility / mVsµ11110000--11--0198 inc. temperatureµgeo µm µtr,m
c
10-11 10-10 10-9 10-8 0 2 4 6 8 10 12 14x103
mobility µ / m2V-1s-1 (field F)1/2 / V1/2m-1/2
FIG.3. Upperpart: Mobilitydistributions pµ(µ)forthreedifferent FIG.4. Upperpart, Leftaxis: Fractionofchargecarriers, which
temperaturesT =130,190,300K(fromlefttoright)atafixedelec- are extracted before the geometric transit timettr,geo, to all charge
tricfieldof. F=1.2·108V. Theverticalstraightlinesillustratethe carriers extracted. A temperature and field dependent deviation is
m
fourdifferentdefinitionsofmobilityµtr,m (dash-double-dotted),µm seenbythestraightlines, whichwereaddedasaguidetotheeye.
(dashed),µgeo(solid)andµ1/2(dash-dotted)forT=130K(left)and Upperpart,Rightaxis: Fractionofchargecarriers,whichareex-
T =300K(right). Lowerpart: Conductivitydistributions pµ(µ)·µ tracted beforettr,1/2, to all charge carriers extracted. Lower part:
correspondingtothedistributionsabove. Temperatureandfielddependenceofdifferentlydefinedchargecar-
riermobilities. Asthegeometrictransittimeonlyaccountsforthe
fastestchargecarriers,themobilityatthegeometrictransittimeµgeo
ofparticlesunderaconstantforcecausingdrift.51Thisproba- liesatsystematicallyhighervalues.Themobilityµtr,mfromthemean
transittimetogetherwithitsstandarddeviationisindicatedbytheer-
bilityisactuallyequivalenttothedistributionoftransittimes rorbars. Themeanmobilityµm isphysicallymostrelevantforthe
considered here if a non-dispersive Gaussian charge carrier descriptionofanexperiment. Allmobilitiesshowsimilarfieldand
transport, which is nothing other than a brown motion with temperaturebehaviour.
drift,isassumed.
Derived mobility distributions p (µ) are displayed in the
µ
upperpartofFig.3forthreedifferenttemperaturesatafixed C. Comparisonofthenewapproachwiththecommon
electricfield. Notethedifferentasymmetricshapeof p (µ)at geometricalmethod
µ
differenttemperatures: Forthehightemperature(T =260K),
the maximum of the distribution is shifted to higher mobili- Next we reveal the advantage and accuracy of the Scott
ties, according to many fast charge carriers, whereas the left method by comparing the results of our data evaluation
edgeisslowlydecreasingtowardsslowermobilities. Forthe on experimental data with results obtained with the well-
intermediatetemperature(T =190K),theshapeisquitesym- established geometric approach. Therefore, we remind the
metric,butbroader,withanadditionallongtailofslowcharge readerofthetwodifferentmobilitydefinitionsforeachofthe
carriers. At low temperature (T =140K) the distribution’s twoapproacheslistedinTab.I.
maximum shifts to lower mobilities and the width is further ThefourdifferentmobilitiesarecomparedinFig.3fortwo
broadened. temperatures.Forbothtemperaturesµ hasthehighestvalue
geo
A more intuitive representation of the extracted distribu- followedbyµmandµtr,misfurthershiftedtolowermobilities.
tionsisgivenbytheconductivitydistribution pµ(µ)·µplotted µ1/2isalwayssituatedbetweenµmandµtr,m.Forthetwocases
in thelower part of Fig.3. Itshows the contributionof each shown in Fig. 3 the relative error of the geometric mobility
mobility to the total current density j. From Fig. 3 it can to the mean mobility is η=0.10 at T =300K; it strongly
t
be seen that, although an enhanced number of low mobility increasesatT =130Ktoη=0.64.
charge carriers show up in the left edge of the mobility dis- In order to quantify the deviation of the geometric transit
tribution at T =140K, their contribution to the total current time from the physically more relevant transit time distribu-
density j issmallasseenfromtheconductivitydistributionin tion,thenormaliseddensityofchargecarriersextracteduntil
t
thegraphFig.3below. Actuallythemaincontributioncomes the transit time t is studied in the upper part of Fig. 4.
tr,geo
fromchargecarrierswithamobilityaroundµ . Wepointout While the geometric method accounts for the fastest 40% of
m
that,usingthetransportenergydefinitionfromSchmechel,23 chargecarriersatroomtemperatureandhighfields,thisvalue
µ isthemobilityofchargecarriersaroundthetransporten- drops to below 5% at lower temperatures and fields. Long
m
ergy. sinceitisknownthatµ accountsonlyforthefastestcharge
geo
7
carriers34,35 and, thus, itwasoftenrecommendedtouseµ
1/2
instead.35 However, the normalised density of charge carri- -1s σgeo,h = 59 meV
eartsroeoxmtractetemdpuenratitlurtter,1v/a2ryshinogwbseatlwsoeeans5li0g%ht−fie6l0d%d.epAensdtreonncge 2-1mV 10-9 σtr,m,h = 69 meV
negative field dependence is instead detected for lower tem- / µ010-10
ptuerreatduerpees.ndHeonwceevisero,battahinigehdealnedct∼ric60fi%eldosfntheaerclyhanrogetecmarpreierars- mobility 1100--1121 σgeo,l =σ 3tr1,m m,l =e V35 meV
wereextracteduntilttr,1/2. d
As the actual charge transport is determined by the mean o fiel10-13 µ0,geo
mobility, a systematic field and temperature dependent error zer10-14 µ0,tr,m
is made when relying on the geometric method. Although,
20 40 60 80 100
themobilityvaluesdeterminedfromt betterrepresentthe
tr,1/2 (temperature T).2 x10-6 / K-2
meanvalues,forverydispersivetransientswithabadsignal-
to-noise ratio t can not be obtained from experimental
tr,1/2 FIG. 5. Zero field mobilities for geometric µ0,geo and mean tran-
data,whileatransittimedistributioncanstillbeextracted.
sittimemobilitiesµ0,tr,m). Thetemperaturedependenterrorofthe
geometricmobilitycanbeclearlyseenandresultsinanunderestima-
tionoftheextractedenergeticdisorderσoftheDOS.Thetransition
D. Extractinganaccurateenergeticdisorderparameterσfor toaweakertemperaturedependenceatlowtemperaturesismoreat-
holetransportinP3HT tributedtoaviolationofthecommonBoltzmannapproximationthan
tothetransitionfromdispersivetonon-dispersivetransport.
AfurtheradvantageoftheScottmethodisthatbesidesthe
mathematical expectations the mobility and transit time can
beprovidedwiththeirstandarddeviationasameasureofthe
metricalmethod. ForP3HTSepiolidP100andSepiolidP200
transport dispersion. In the lower part of Fig. 4 we compare
(BASF SE) σ=56meV and σ=58meV was obtained, re-
the field dependence of the different mobilities µ , µ and
geo m spectively. However,ourresultssuggestthatthesevaluesare
µ (µ isleftoutforclarity). Additionallyweplottedthe
tr,m 1/2 alsounderestimatedandweaproposeσ≈64meVforthehole
standard deviations of the mean transit time t accounting
tr,m transportinSepiolidP100andσ≈66meVforSepiolidP200.
forthetransittimesshorterthant andforthelongertran-
tr,m
Besidestheimpactofthetemperatureandfielddependent
sit times, respectively. As in Fig. 3, Fig. 4 confirms that the
geometric mobility µgeom always exceeds both µm and µtr,m. error of µ0,geo, its deviation to the mean mobility µm can
Furthermore, thegeometricmobilityµ hardlylieswithin alsobeunderstoodregardingtheenergeticdistributionofthe
geom
chargecarriers. Whilethefastestchargecarriersjusthopvia
theboundsofthemeantransittimemobilities’standarddevi-
statesaroundthetransportenergyandthus“feel”asmalleren-
ation. Only at high electric fields and high temperatures this
ergeticdisorder,themobilitycalculatedfromthedistribution
is the case. It can be nicely seen that the standard deviation
ofallchargecarriersaccountsforallstatesintheDOS.
oftheshorttransittimesinFig.4showsanegligiblefield,but
strong temperature dependence, as in Ref. 21 Yet, the elec- Note that, besides the extracted statistical parameters dis-
tric field and the temperature strongly increase the standard cussed herein, the mobility and transit time distributions in-
deviation for the long transit times. We attribute this strong volvemuchmoreinformationabouttheenergydistributionof
asymmetryofthetransittimedistribution ptr(t)aroundttr,mto the DOS. At temperatures below T ≈190K the slope of the
field assisted diffusion.2,55,56 A detailed investigation of this zero field mobilities in Fig. 5 changes and become smaller.
phenomenawillbepublishedelsewhere. The slope of the geometric values is also lower and the rela-
Finally, in terms of the framework of the GDM,21 we can tivedifferenceis11%. Albeittheevaluatedvaluesofσ =
geo,l
useoftheextractedtemperatureandfielddependentTOFmo- 31meV and σm,l = 35meV are not related to the energetic
bilities to gain the characteristic energetic disorder σ of the disorder σ of the DOS and are just given to number the dif-
DOS.Ba¨ssler etal.usedthemobilitygivenbythemeantran- ferent slopes. Furthermore, we do not necessarily relate this
sit time µ for the derivation of their famous parametric toatransitionfromdispersivetonon-dispersivetransport, as
tr,m
GDMmobilityequation[Eq.(15)inRef.21]. Consequently, commonlysuggested. Firstofall,wedonotseeasignificant
asournewapproachallowsfortheextractionofthisquantity, changeinthephotocurrentshapearoundT ≈190Kandsec-
wecancalculateanaccurateσfortheholetransportinP3HT. ondlyweobservetheverysametemperaturebehaviourofthe
To this end, we demonstrate in Fig. 5 the different scaling mobilityfromsteady-statefieldeffectmeasurements58 onthe
of the zero field geometric mobility µ and the zero field samematerialsystem. As,inthecaseoftheScottmethod,we
0,geo
mean transit time mobility µ (t ) with temperature. The canruleoutasystematicalerrorduetotheevaluationproce-
0 tr,m
extracted energetic disorder σ =59meV is significantly dureasthenatureofthisbehaviour,weattributethismoretoa
geo,h
lower by about 14% for the geometric mobilities compared violationofthecommonBoltzmannapproximationutilisedin
to σ =69meV calculated from µ . The found system- theGDM,whichpredictstheconstantslopeofln(µ)vs1/T2.
m,h tr,m
atic error of the geometric method leads, thus, to a wrong Instead, at low temperatures the full Fermi-statistics need to
estimate of the energetic disorder parameter σ. Comparable be taken into account considering the charge carrier density
valuesforσ werefoundbyMaueretal.57 usingthegeo- effectsonthemobility.23,25
geo,h
8
VI. CONCLUSION information, as it is usually only known from Monte
Carlosimulations
We have introduced a new approach to derive the charge
• the extracted distributions provide detailed informa-
carrier mobility and transit time distribution in energetically
tion about the transporting medium, such as, e.g., a
disordered semiconductors exhibiting a Gaussian DOS from
statistical mean mobility determining the macroscopic
photocurrenttransients.
photocurrentorthedispersionofmobilities,exhibiting
Itisknownthattheinitialphotocurrentdecayinthesesys-
muchpotentialforfurthercarefulstudiesandinputfor
tems cannot be described by a plain analytical relation,16,38
theoreticalconsiderations
such as the power law decay in systems with an exponential
distribution of localised states within the band gap.15 How-
• less error prone, as the pre-transit fit spans over more
ever, a prerequisite of this new approach, which is based on
than one order of magnitude in time and the extracted
theearlierworkofScottetal.,30 isthefunctionaldescription
stochasticalvaluesarenumericallyaveraged
ofthephotocurrentdecayinthepre-transitregimeandtheex-
trapolation into the post-transit regime. Here, we proposed Furthermore, we found that the geometric method clearly
anempiricalapproximation[Eqn.(1)]forthepre-transitpho- overestimates the mobility defined in terms of a mean drift,
tocurrent, which accurately fits our experimental data mea- whichistheexpectationvalueµ ofthemobilitydistribution.
m
suredonP3HT.Thevalidityofthisfunctionalapproximation Thisdeviationbecomesevenmoreprominentatlowerelectric
and the reliability of its extrapolation were shown by one- fields and temperatures. The analysis of hole photocurrents
dimensional Monte Carlo simulations. The transit time dis- with the new approach revealed an accurate energetic disor-
tributions, which we either determined by evaluating simu- derparameterσ =69meVforP3HT.Thisvaluewascalcu-
h
latedphotocurrentswiththenewmethodordirectlyfromthe lated from the mobility given by the mean transit time µ ,
tr,m
temporaldecreaseoftheamountofchargecarriersinsidethe asthisistherelevantparameterconsideredintheframework
simulateddevice,matchexactly. Furthermore,theparameters oftheGDM.21Byusingthelessaccurategeometricalmethod
ofEqn.(1),althoughempirical,werefoundtofollowaphys- we determined a ∼14% smaller value of σ =59 meV
h,geo
icallyreasonabletemperaturedependence. instead.
By applying this method to hole TOF transients measured
onP3HToverawidetemperatureandelectricfieldrangewe
ACKNOWLEDGMENTS
demonstrated its convenient usage and the superiority com-
paredtothecommonlyusedgeometricmethod. Incontrastto
ThecurrentworkissupportedbytheDeutscheForschungs-
thelatterthenewapproachhasthefollowingadvantages:
gemeinschaft(DFG),projectEiNDORSE(DE830/9-1). C.D.
• abletoextractparameterdistributionsinsteadofasin- gratefullyacknowledgesthesupportoftheBavarianAcademy
gle value only, providing detailed and comprehensive ofSciencesandHumanities.
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