Photoinduced Absorption within Single-Walled Carbon Nanotube Systems

3.6 MB·

by  Livia Noëmi Glanzmann

Photoinduced Absorption within Single-Walled Carbon
Nanotube Systems
LiviaNoëmiGlanzmann,† DuncanJohnMowbray,∗,† DianaGisellFigueroadelValle,‡,¶ FrancescoScotognella,‡,¶
GuglielmoLanzani,‡,¶ andAngelRubio†,§
6 †Nano-BioSpectroscopyGroupandETSFScientificDevelopmentCenter,DepartamentodeFísicadeMateriales,CentrodeFísicade
1 MaterialesCSIC-UPV/EHU-MPCUniversidaddelPaísVascoUPV/EHUandDIPC,E-20018SanSebastián,Spain
0 ‡CenterforNanoScienceandTechnology,IstitutoItalianodiTecnologia,ViaPascoli70/3,I-20133Milano,Italy
2 ¶PolitecnicodiMilano,DipartmentodiFisica,P.zzaL.DaVinci32,I-20133Milano,Italy
§MaxPlanckInstitutefortheStructureandDynamicsofMatter,LuruperChaussee149,D-22761Hamburg,Germany
n
a SupportingInformation
J

9 ABSTRACT: We study the photoabsorption properties of photoactive bulk polymer/
Differential Transmission (arb. units)
2 fullerene/nanotubeheterojunctionsinthenear-infraredregion. Bycombiningpump-probe pump 1 0 -1probe Δ
l-sci] ttwsderpoarietmnhc(sPtipprtCohioosBalcnysMo(ew3pa)-yiphitnhpeaxirnPnoyd3xl(tHli6himn,Ti5oea/)aptPirhaoCnernBend(seMTp(-7D2o/,S,n5D5Ws)-eFdsNTitiyin-TmRlg)elPb(e-APl-dew3)enHpwadeleTelnde)eddasleuancncmdatirdpd[b6aleoet,n6ness.]tin-htWpyaehnefeeouxnfitncuynciblttd-eeiCodstn6hs(1aetS-labWpttuerhetNedysoyTerrinnscyac)amewacnioiictddfhsmiaocnofeptmtthhhhoebyetilorEnaeie1nnsd1--- s=0 s=E111 Wavelength (nm)95010001050 PPBA s=E011 Δ+s 2=Δ1
r
t ducedabsorption(PA)peakisrelatedmainlytothewidthofthephotobleach(PB)peakand 0 Pu1mp-pr2obe d3elay (4ps) 5
m
thechargecarrierdensityoftheSWNTsystem. FormixedSWNTsamples,thePBpeak
. istoobroadtoobservethePApeak,whereaswithinP3HT/PCBM/SWNTblendedsamples PT
t
a P3HTactsasaholeacceptor,narrowingthePBpeakbyexcitondelocalization,whichre-
m vealsaPApeak. OurresultssuggestthatthePApeakoriginatesfromawideningofthe -0.13e -0.10e
- bandgapinthepresenceofexcitedelectronsandholes.Theseresultshaveimportantimpli-
d cationsforthedevelopmentofneworganicphotovoltaicheterojunctionsincludingSWNTs. (6,5) SWNT PCBM
n
o
c
[ 1.INTRODUCTION phonons.25 Thestrongelectron-phononcouplingisevidentinthe

largeRamancross-section,extensivelystudiedwithstandardcon-
1 Single-walled carbon nanotubes (SWNTs),1–3 with their high tinuouswavetechniques. ThemostintenseRamanactivemodes,
v chargecarrierdensities4 andmobilities,5–7 showpromiseaselec- radialbreathingmodesandGmodes, havebeendetectedintran-
0 tron transporting materials for photoactive polymers8 within or- sientcoherentRamanexperiments.26,27
1
ganicphotovoltaicdevices(OPVs). Infact,asignificantenhance- Unfortunately, there are other photoexcited species, resulting
0
0 ment of photocurrent was achieved by inserting SWNTs into the fromhighlynonlinearphenomena,thatmightoccurinSWNTs,be-
0 bulkheterojunction(BHJ)ofanOPV.9–11OPVswithSWNTscan sidessingletexcitons.Forthisreason,theinterpretationoftransient
. be further improved by enriching the content of semiconducting spectraforSWNTsbecomesdifficult, andthisisduetocontribu-
2 tubes,12 thereby reducing electron-hole trapping.13 SWNTs have tions of such species, e.g., triplets,11 biexcitons,28,29 trions,30,31
0
evenbeenuseddirectlyasphotoreceptorswithinBHJs.14,15How- andcharge-carrierphotogeneration.32,33Previoustransientabsorp-
6
1 ever, the power conversion efficiencies (PCEs) of SWNT OPVs tionmeasurements34,35havefoundaphotoinducedabsorption(PA)
: stilllagbehindthebestBHJOPVs.16 TheoptimizationofSWNT peak at ∼950 nm for predominantly (6,5) SWNT samples when
v OPVsrequiresacombinedtheoreticalandexperimentalapproach pumpedinthevisible(VIS)region.ThestructureofthisPAfeature
i
X to understand the photoelectric processes within SWNT compos- provides direct information about the photoexcitation processes
ites.Onlyinsodoingcanwetakefulladvantageoftheoutstanding withintheSWNT.ItisthestructureandoriginofthisPApeak,and
r
a physicalpropertiesofSWNTswithinBJHOPVs. how it depends on the SWNT system, which we will focus upon
TheelementaryphotoexcitationsinSWNTsaresingletexcitons herein.
with a large binding energy of about 400 meV.17–22 Pump-probe In this work, we employ state-of-the-art pump-probe tran-
spectroscopyisapowerfultoolforstudyingopticalpropertiesand sient absorption spectroscopy to measure the excited state dy-
exciton dynamics over the femtosecond to microsecond time do- namics of a (6,5) and (7,5) SWNT mixture, and combined
main. As an example, the size of the exciton and its diffusion in with poly(3-hexylthiophene-2,5-diyl) (P3HT) and [6,6]-phenyl-
theincoherentregimehavebeenexperimentallyestimatedfor(6,5) C -butyric acid methyl ester (PCBM) in P3HT/PCBM/SWNT
61
SWNTs23 usingthephasespacefillingmodel.24 Also,theenergy blendedsamplesinthenear-infrared(IR).Wethenemploylinear
ofrelaxationfromhigherexcitonstatestothefirstexcitonE11tran- responsetimedependent(TD)densityfunctionaltheory(DFT)in
sitionhasbeenmeasured,yieldingaveryshortdecaytimeof50fs. frequency-reciprocalspacewithintherandomphaseapproximation
This is due to the peculiarity of a one-dimensional system, such (RPA)tomodelthemeasuredtransientspectrafor(6,5)and(7,5)
as a SWNT, and the strong coupling to highly energetic optical SWNTsandblendedP3HT/PCBM/SWNTsamples.
1
Webeginbyprovidingdetailsofthesamplefabricationinsec- ofthesysteminthegroundandexcitedstates.Wemodelthesystem
tion2.1, opticalcharacterizationofthesampleinsection2.2, the intheexcitedstatewithinDFTbyfixingthetotalmagneticmoment
theoretical methods employed in section 2.3, and computational µ,andthroughtheadditionofchargeQtothesystem.Specifically,
parameters used in section 2.4. In section 3.1 we provide a de- we use the singlet (S =0) to model the ground state, the triplet
tailed comparison of the absorbance and differential transmission (S =1) to model a single exciton, the quintet (S =2) to model a
spectraobtainedforourSWNTandblendedP3HT/PCBM/SWNT pairofexcitons,andthequartet(S =3/2)withanadditionalcharge
sampleswiththoseavailableintheliteratureinthenear-IRregion Q=−etomodelanegativetrion,i.e.,apairofexcitedelectronsand
asafunctionofthepump-probedelay.Weshowinsection3.2how asinglehole.
the differential transmission spectra may be interpreted in terms The optical absorption spectra are obtained via linear response
of a difference in absorption spectra. After briefly justifying our TDDFT-RPA,36–40fromtheimaginarypartofthemacroscopicdi-
methodformodelingopticallyexcitedstatesinsection3.3,weem- electricfunction,Im[ε(q,ω)],inthelimit(cid:107)q(cid:107)→0+. Detailsofour
ployTDDFT-RPAcalculationsof(6,5)SWNTs,(7,5)SWNTs,and implementationareprovidedinAppendixA.
acombinedPT/PCBM/(6,5)SWNTsysteminsection3.4tosim- Tomodelanexcitedsingletstateofthesystembasedonafixed
ulate the measured differential transmission spectra, and explain magneticmomentcalculation, we“swap”betweenthespinchan-
the dependence of the observed PA peak on charge carrier den- nels(s∈{↑,↓}or{0,1})theeigenvaluesandeigenfunctionsofthe
sity in section 3.5 and the photobleach (PB) peak width in sec- levelsbeyondhalfthenumberofelectrons,Ne/2. Moreprecisely,
tion 3.6. This is followed by concluding remarks. A derivation wedefine
(cid:40)
oftheTDDFT-RPAformalism,andtheinfluenceoflocalfieldef- s(cid:48)= s+1 mod2 ifn>Ne/2 (1)
fects on the SWNT spectra are provided in Appendix A. Atomic s otherwise
units((cid:125)=me=e=a0=1)havebeenusedthroughoutunlessstated Inthiswayweobtaintheelectronicstructureofasingletexcited
otherwise. statethatisconstrainedtohavethesametotalelectrondensityas
thetripletgroundstate,ρS=0(r)≡ρS=1(r).
ex gs
2.METHODOLOGY Notethat,asweareprimarilyinterestedhereinwiththeabsorp-
tionspectraneartheironsets, localfieldeffectsmaybeneglected
2.1. Sample Fabrication. The glass substrates were pre- withoutimpactingourresults.ThisisdemonstratedinAppendixA,
cleaned with acetone and isopropanol and dried under a flow where we compare ground state TDDFT-RPA spectra for a (6,5)
of dry nitrogen. For preparation of the samples, regular P3HT SWNTwithandwithoutincludinglocalfieldeffects.
poly(3-hexylthiophene-2,5-diyl) and PCBM ([6,6]-phenyl-C61- 2.4. Computational Details. All DFT calculations were per-
butyricacidmethylester)weredissolvedinortho-dichlorobenzene formedwithlocallycenteredatomicorbitals(LCAOs)andthepro-
(ODCB) at a 1:1 ratio. We employed 704148-SWNTs produced jector augmented wave (PAW) implementation within the gpaw
®
usingtheCoMoCAT catalyticchemicalvapordeposition(CVD) code.41–43 We used a double-zeta polarized (DZP) basis set for
method. The SWNTs were also dispersed in ODCB and then representing the density and the wave functions and the PBE ex-
sonicated for 1 hour. No debundling or removal of metallic changecorrelation(xc)-functional.44 Allcalculationsemployeda
SWNTs was performed on the sample, which was most likely room temperature Fermi filling (kBT ≈25 meV), with total ener-
aggregated. Theresultingsolutionwasthenspin-coatedontopof giesextrapolatedtoT →0K,i.e.,excludingtheelectronicentropy
theglasssubstratestoobtaintheSWNTsample. Inthecaseofthe contribution to the free energy −ST. In this was we avoided an
P3HT/PCBM/SWNT sample, the dispersed SWNTs were added unrealisticsmearingoftheexcitedelectronandholeinthetriplet
to the P3HT/PCBM solution at a 1:1:1 ratio and then sonicated calculations.Weincluded2⁄3Nemanybandswithinthecalculations,
for1hour. Thesolutionwasthenspin-coatedontopoftheglass whichhasbeenshowntobesufficienttoconvergethefirstπ→π∗
substratesat1000rpmfor2minutes.Allthematerialswerebought transitionswithingraphene.39
fromSigma-Aldrich. StructuralminimizationwasperformedwithintheAtomicSim-
2.2. Optical Characterization. The ground state absorption ulation Environment (ASE),45 until a maximum force below
spectra were collected with a PerkinElmer spectrophotometer 0.05eV/Åwasobtained.Weemployedmorethan5Åofvacuumto
(Lambda1050WBInGaAs3DWBDetectionModule). Thelaser thecellboundariesorthogonaltothe(6,5)SWNT,(7,5)SWNTand
systememployedforultrafasttransientabsorptionwasbasedona polythiophene(PT),andobtainedoptimizedunitcellsparameters
Ti-Sapphirechirppulseamplifiedsource;withamaximumoutput of 40.92, 44.79, and 7.87 Å along their axes, respectively. Here,
energyofabout800µJ,1kHzrepetitionrate, centralwavelength PTismodeledusingtwothiopheneunitsins-transconfiguration.
of 780 nm and pulse duration of about 180 fs. Excitation pulses WehaveusedPTasasimplifiedmodelforP3HT,thatis,removed
at 590 and 900 nm were generated by noncollinear optical para- thehexylsidechainsofP3HT,sincetheinfluenceoftheP3HTside
metric amplification in a β-barium borate (BBO) crystal, with a chainsonthelevelalignmentandchargetransferisnegligible.40,46
pulse duration of around 100 fs. Pump pulses were focused in a Moreover, asweshallseeinsection3, theinfluenceofP3HT/PT
200 µm diameter spot. Probing was achieved in the visible and ontheIRspectrumoftheSWNTissolelythroughhole-transfer.
nearIRregionbyusingwhitelightgeneratedusingathinsapphire The PT/PCBM/(6,5)-SWNT bulk was modeled by a 39.34×
plate. Chirp-freetransienttransmissionspectrawerecollectedby 40.92Å2layeredstructureoftenthiopheneunitsorthogonaltothe
usingafastopticalmultichannelanalyzer(OMA)withadechirping SWNTaxis,asshowninFigure1(a).Insodoing,thisconfiguration
algorithm. Themeasuredquantityisthedifferentialtransmission, describesthelimitofaminimalSWNT–PToverlap. Todetermine
∆T=T(t)−T(t=0).Oncenormalized,thedifferentialtransmission theimpactofincreasingtheSWNT–PToverlap,wealsoconsidera
∆T/T may be directly compared with the change in absorbance truncatedtenunitPTchainalignedwiththeSWNTandPCBMas
∆Im[ε]=Im[ε(t=0)]−Im[ε(t)]. Theexcitationenergyhasbeen showninFigure1(b).
set to 11 nJ when pumping at 590 nm and then increased to 200 It has previously been shown that changes in the orientations
nJwhenpumpingat900nm,i.e.,selectivelypumpingtheSWNTs. of PCBM next to P3HT only cause energy differences within the
Allmeasurementswereperformedinairatroomtemperature. accuracy of DFT.47 Since the potential energy surface is rather
2.3.TheoreticalMethods. Tomodeldifferentialtransmission flat,47 wehavechosenasmallestC–Cintramoleculardistanceof
measurements,weusethedifferencebetweentheopticalabsorption ∼3.3ÅbetweentherelaxedP3HT,PCBM,andSWNTstructures,
2
Di(cid:1)erential Transmission (arb. units)
1 0 -1
0.20.3 1
(a)
1050
m)
n PB
h (
ngt1000
e
Figure1. SchematicofthePT/PCBM/(6,5)SWNTsystemwith(a)ape- el
v
riodicallyrepeatedPTpolymeralignedorthogonaltotheSWNTand(b)a a
W
finitetenunitPTchainalignedwiththeSWNT.C,H,O,andSatomsare PA
depictedbygray,white,red,andyellowballs,respectively. 950
(b)
andperformedsingle-pointcalculationsfortheresultingconfigura-
tions shown in Figure 1. Although this C–C separation is 0.1 Å 1050
smaller than the interlayer distance of multiwalled carbon nan- m)
otubes(MWNTs)andgraphite,48 ithasbeenchosentoensurean n PB
overlapbetweentheSWNTandPTouterLCAOorbitals. Anin- h (
crease of the PT–SWNT/PCBM distance to ∼3.4 Å changes the ngt1000
e
totalenergybylessthan50meV,i.e.,withintheaccuracyofDFT. el
v
In each case we found the repulsive forces for the close lying C a
atomswere(cid:46)0.1eV/Å. W PA
950
3.RESULTSANDDISCUSSION
3.1. Differential Transmission Measurements. Fig-
0 1 2 3 4 5
ure 2 shows the transient spectra of the SWNT and blended
Pump-probe delay (ps)
P3HT/PCBM/SWNT samples versus probing delay times up to
Figure 2. Normalized differential transmission ∆T/T of (a) SWNT and
5 ps. In both spectra, we find a peak with a maximum around
(b)P3HT/PCBM/SWNTdevicesversuspump-probedelayinpsandprobe
1025 nm. Comparing the transient spectra with the absorbance wavelengthinnm. Pump-probedelaysof0.2,0.3,and1.0ps(spectrain
(black)inFigure3(a),thispeakcanbeassignedtothephotobleach Figure3)aremarkedbyverticallines,whilephotobleach(PB)andphotoab-
(PB) of mainly (6,5) and (7,5) SWNT E transitions.50 For the sorption(PA)peaks(spectrainFigure4)aremarkedbyhorizontallines.
11
blended sample, we have an additional photoinduced absorption
(PA)peakaround960nm. Thesamplesshowaremarkablediffer-
ence in the decay rate of the PB peak. In the case of the SWNT
sample,theE transitionsseemtobecompletelyaccessibleagain pump beams at 570 nm employed in refs 34 (blue) and 35 (ma-
11
aisft4e–r51tpims,ewshloilnegtehre.decayoftheE11excitonintheblendedsample genAtas)swhoewrentuinneFdigtourtehe3(Eb2,c2,dtr)a,ntshietiodniff.erentialtransmissionspec-
The UV/VIS/NIR absorption spectra of the SWNT sample are traforourSWNTsampleisratherinsensitivetowhetherpumping
shownasaninsettoFigure3(a). TheE transitionsof(6,5)and isintheVISorIR.ThissuggeststhatthebroaderPBpeakwithin
22
(7,5)SWNTsareclearlyvisible, withthe E transitionof(11,5) our SWNT sample, shown in Figure 3(a), is related to the sam-
11
SWNTsseenasashoulder. pleitself. IthaspreviouslybeenarguedthatSWNTmixtureshave
BycomparingthetransientspectraoftheSWNTsample(black) an increased exciton transfer and electron-hole trapping between
at delay times of 0.2, 0.3, and 1 ps with the spectra of similar tubes.12,13 Thus,thegreaterPBpeakwidthinourSWNTsample
SWNTsamplesoftheliteratureatthesamedelaytimes,wefound mayberelatedtohavingamixtureofboth(6,5)and(7,5)SWNTs.
a discrepancy within the measured peak width and structure of However,havingaSWNTmixturedoesnotexplaintheoccurrence
theSWNTsamplespectrumandthoseoftheliterature,34,35,49 as ofthePApeakintheblendedP3HT/PCBM/SWNTsample(red).
showninFigure3(b-d). The(6,5)SWNTenrichedsamplesofrefs TakingacloserlookatthetransientspectrumoftheSWNTsam-
34(blue)and 35(magenta)showamuchsmallerpeakwidthand ple in Figure 3(b,c,d), we notice it exhibits a small asymmetry.
anadditionalPApeakbetween940and975nm. ThePApeakin Nevertheless, including a range of ±50 nm from the peak maxi-
thesampleofref49(cyan)isoverlaidbythePBofthe(8,3)SWNT mum,thisasymmetrycannotbecompletelyverified,asitiswithin
(∼966nm)50andthereforenotstronglypronounced. Aswell,the thenoiseofthemeasurement. Thus,thePAappearstocompletely
widthisbroaderduetothemixtureoftubes.Thisisconsistentwith vanishinthespectraoftheSWNTsample.
thedifferentialtransmissionofourSWNTsample. The recovery time dynamics of the PB and PA peaks for the
Todeterminewhetherthepumpenergyhasaninfluenceonthe mixedSWNTsampleandtheP3HT/PCBM/SWNTblendedsample
peakwidth,theSWNTsamplewaspumpedattwodifferentwave- areshowninFigure4.WefindthemixedSWNTsample’srecovery
lengths: 590 nm in the VIS and 900 nm in the IR. These pump timeismuchshorterthanthatoftheblendedP3HT/PCBM/SWNT
energiesarecomparabletothosereportedtheliterature.34,35,49For sample, with the recovery time reported in ref 34 for a predomi-
example,ref 49pumpedintheinfraredregionat930nm,whilethe nantly(6,5)SWNTsampleevenlonger.Ineachcase,thePApeaks
seemtorecoverasfastasthePBpeaksorslightlyslower.
3
(8,3) (6,5) (7,5)
abiexcitation. Inasubsequentpaper,Zhu35suggestedcouplingto
theradialbreathingmodeasareasonforthePApeak.
(a)
Evaluatingtheexperimentaldataofref35showninFigure5,we
e (6 , 5 )( 7 , 5) (11,3)
c
n
a
b
or
s
b
A
500 1000
1 (b) 0.2 ps
s)
nit 0
u
b.
ar
n ( 1 (c) 0.3 ps
o
si
s
mi
s
n
a 0
Tr
al
nti
e 1 (d) 1.0 ps
er Figure5. Normalizedinitialabsorbance(violet)anddifferentialtransmis-
Diff sion∆T/Twithapump-probedelayof1ps(magenta)fromref35,andtheir
difference(green)versusprobewavelengthinnmandenergyineVrelative
totheE11transition. ALorentzianfittothedifferencebetweentheinitial
0 absorbanceanddifferentialtransmissionisalsoshown.Anexcitedelectron
transitionE∗∗modelforthemeasuredabsorbance/differentialtransmission
isshownschematicallyintheupperleft/rightinset.
950 1000 1050
Wavelength (nm)
identifiedthepeakcausingthePApeakbytakingthedifferencebe-
Figure 3. Normalized (a) absorption and (b-d) differential transmis-
sion ∆T/T versus probe wavelength in nm for our SWNT sample tweenthenormalizedabsorbance(violet)andthenormalizedtran-
(black) pumped at 900 nm (solid line) and 590 nm (squares), combined sientspectrum(magenta). Thisresultsinthedifferencespectrum
P3HT/PCBM/SWNTdevice(red)andfromrefs 34(blue), 49(cyan),and oftheprobebeam(green),whenthesystemisfullyexcitedbythe
35(magenta)forpump-probedelaysof(b)0.2ps,(c)0.3ps,and(d)1.0ps. pump beam, that is, all electrons of the E transition are in the
11
WavelengthsoftheE11transitionsfor(8,3),(6,5),and(7,5)SWNTsfrom CBM.Inthiscase,theexcitedsystemgivesrisetoanabsorbance
ref50aremarkedabove.
ofmorethan50%ofthePBpeakbutattheenergyofthePApeak.
Suppose the PA peak is due to absorption by an excited elec-
s)
nit 1 tron,i.e.,anexcitedelectrontransitionE∗∗,asdepictedschemati-
u callyinFigure5. Itshouldbenotedthatthesingleparticlepicture
arb. employed here refers to the system being excited, rather than the
n ( excitation process itself. When the system is initially excited by
o thepump,thedifferenceinfillingfortheE transition,∆f =2,
si 11 11
mis whilethedifferenceinfillingfortheE∗∗transition’slevels∆f∗∗=0.
n Whenthesystemissubsequentlyexcitedbytheprobe, thatis, in
Tra the presence of an exciton, ∆f1(cid:48)1 =0, while ∆f∗(cid:48)∗ =1. Since a
al transition’sintensityisproportionaltothedifferenceinfilling,the
nti 2:1 ratio between PB and PA peaks is already accounted for by
e
er0 ∆f11:∆f∗(cid:48)∗. Thisrequiresthe E11 and E∗∗ transitionstohaveal-
Diff mostthesameoverlaps.
0 2 4 6 8 10
Pump-probe delay (ps) AlthoughthisdoesnotruleoutthepossibilityofanE∗∗transition
being responsible for the PA peaks close to the PB peaks within
Figure4. Normalizeddifferentialtransmission∆T/T asafunctionofthe
theSWNTsystems,itmakesitratherunlikely. Moreimportantly,
pump-probedelayinpsmeasuredatthemaximumofthephotobleach(PB)
it suggests the PA and PB peaks most likely arise from the same
peak (solid lines) and photoabsorption (PA) peak (dashed lines) for our
SWNTsample(black), combinedP3HT/PCBM/SWNTdevice(red), and E11 transition. In fact, in section 3.4 we will show that the E11
for(6,5)SWNTsfromref34. transitionisblueshiftedbyabandgapwideningintheexcitedstate,
potentiallyexplainingtheoriginofthePApeak.
3.3.ModelingOpticallyExcitedStates. Asdiscussedinsec-
tion 2.3, we model the optically excited state using a singlet ex-
The main reason for the extinction of the PA peak is the large cited state whose total electron density is constrained to be that
peakwidth,whichiscrucialforthevisibilityofthePApeak.How- of the triplet ground state. This singlet excited state is a suit-
everthisdoesnotexplainitsorigin. able approximation to the optically excited state if the spin den-
3.2.InterpretingdifferentialtransmissionasaDifference sities associated with the Kohn Sham (KS) valence band maxi-
Spectra. Zhuetal.34 foundaPApeak,andsuggesteditisdueto mum (VBM) and conduction band minimum (CBM) eigenfunc-
4
tions, |ψ |2 and |ψ |2, are only weakly dependent on the of the bulk system (dashed)shows the onset of PT absorbance at
VBMs CBMs
spinchannels.Inotherwords,ifthetwospinchannelsinthetriplet ∼925nm,orthogonaltotheE transition. Intherangeof925to
11
DFTcalculationarebasicallyequivalentuptoaphasefactor,i.e., 1100nm,bothsystems,the(6,5)SWNTandthebulk,overlap.This
|(cid:104)ψSn↑=1|ψSn(cid:48)=↓1(cid:105)|≈δnn(cid:48),theopticallyexcitedstateshouldhaveasimi- suggeststhatneitherPTnorPCBMtransitionsareinvolvedwithin
larelectrondensity.If,moreover,thesingletandtripletgroundstate thisenergyrange,andthereareonly(6,5)SWNTtransitions. The
eigenfunctions are also basically equivalent up to a phase factor, excitedstateabsorbances(green)ofbothsystemsshowtwosignif-
i.e.,|(cid:104)ψSng=s1|ψnS(cid:48)=g0s(cid:105)|≈δnn(cid:48),thissingletexcitedstateshoulddescribe icantchanges: thePBpeakisshiftedtohigherenergyandthereis
theopticallyexcitedstatequitewell. anewtransitionat∼880nm.Thisnewpeakwithratherlowinten-
Wefindthisisindeedthecaseforthe(6,5)and(7,5)SWNTs, sityisabsorbingorthogonaltothetubeaxis,andcanbeassignedto
with the KS eigenfunctions having approximately the same spa- anintratubeinterbandtransitionE∗∗ oftheexcitedelectroninthe
tial distribution in the singlet and triplet ground states. For the CBMtoanenergylevel∼1.1eVabovetheCBM.However, this
blended SWNT/PT/PCBM systems, the KS eigenfunctions have peakvanisheswhenlocalfieldeffectsareincluded,asshowninAp-
similarspatialdistributioninbothspinchannelsforthetripletDFT pendixA,whichmayberelatedtomodelingdebundledSWNTs.51
calculation,butdifferfromthesingletgroundstateKSeigenfunc- The schematic in Figure 6 explains the origin of the E tran-
11
tions.Inparticular,theoccupiedCBMlevelofthetripletDFTcal- sitionshiftsintheexcitedstates. Asdiscussedinsection2.3, we
culationisahybridizationofthefirstthreeCBeigenfunctionsfrom modeled the excited state by computing a triplet state in the first
the singlet ground state calculation. This is indicative of charge step.Thesingletexcitedstateisobtainedbyswappingtheeigenen-
transferintheblendedSWNT/PT/PCBMsystem. ergiesandeigenfunctionsbetweenspinchannelsfortheoriginally
Overall,theseresultsstronglysuggestthatthetotalelectronden- unoccupied states, i.e., n>Ne/2. In this way, the electron and
sity of the triplet DFT calculation should be quite similar to that the hole are arranged in the same spin channel. As a result, the
of the optically excited state. This justifies our use of the singlet bandsincludingtheelectronandholeareshiftedcloserinenergy
excitedstateelectronicstructure,obtainedby"swapping"between (electron-holebinding), whiletheotherchannelwiththeelectron
spinchannelstheCBeigenvaluesandeigenfunctionsofthetriplet inthevalenceband(VB)increasesitsbandgap.Thisisnotseenin
groundstate,tomodeltheopticallyexcitedstate. thetripletcalculationitself. Duetothiswideningofthebandgap,
3.4. TDDFT-RPA Simulated Spectra. The differential trans- thesecondexcitationofanelectronintheVBisathigherenergy.
missionsinthelongtermlimitofthe(6,5)SWNT(black)andbulk Inthechangesoftheabsorbance,i.e.,differentialtransmissionin
PT/PCBM/(6,5)SWNTsystems(red)inFigure6arecalculatedus- thelongtermlimit,theE∗∗ transitionsgiverisetonegativepeaks.
However,theyareatmuchhigherenergythanthePApeaksmea-
sured experimentally. On the other hand, the shifts ∆ of the E
11
transitions in the excited state cause a reduction of the transient
absorbance peaks at ∼960 nm. The larger shift within the (6,5)
SWNT system even leads to a small negative peak in the change
inabsorption.Overall,the(6,5)SWNTdifferencespectrumagrees
qualitativelywiththatofref 34,showninFigure3(b).
Thisisexactlytheoppositefromwhatweseeintheexperimental
dataobtainedfromtheSWNTandtheblendedsamples.There,we
havenoPAinthetransientspectrumoftheSWNTsample, buta
more pronounced reduction of the peak at the higher energy end
fortheblendedsamplespectrum. Tounderstandtheoriginofthis
discrepancy, we consider the role of charge carrier loading, i.e.,
additionandremovalofelectronsandholes,andpeakbroadening
onthedifferentialtransmissionspectrainthefollowingsections.
3.5.RoleofChargeCarrierLoading. Wecalculatedtheex-
citedelectron(blue)andhole(red)densitiesasthedifferencebe-
tweentheelectrondensityinthegroundstateandtripletconfigura-
tion,asshowninFigure7.
Figure 6. The TDDFT-RPA absorbance Im[ε] of a (6,5) SWNT (solid
lines)andforthecombinedPT/PCBM/(6,5)SWNTsystemdepictedinFig-
ure1(a)(dashedlines)and(b)(diamonds)intheground/excitedstatebythe
pump/probe(violet/green)asdepictedschematicallyintheupperleft/right Figure 7. Electron (blue) and hole (red) densities for a combined
PT/PCBM/(6,5) SWNT system from the DFT total electron density dif-
inset,andchangeinabsorbancebetweenthegroundandexcitedstatefora
(6,5)SWNT(black)andacombinedPT/PCBM/(6,5)SWNTsystem(red) ference between the excited and ground state. The charge transfer upon
excitationof−0.13efromthePTtothe(6,5)SWNTandof−0.10efromPT
versuswavelengthinnmandenergyineVrelativetotheE11transition.To- toPCBMisdepictedschematically. C,H,O,andSatomsaredepictedby
talabsorbance(thicklines)andabsorbanceforlightpolarizedperpendicular
gray,white,red,andyellowballs,respectively.
totheSWNTaxis(thinlines)areshown.
ing the difference between the TDDFT-RPA ground state (violet) InthiswayweincludeallelectronredistributionsintheVBand
andexcitedstate(green)absorbances.Thegroundstateabsorbance CB. The difference in the densities reveals a hole located mainly
5
on P3HT/PT, whereas the electron is predominantly on the (6,5) SWNTsystems.
SWNTandthePCBM.Thisresultsinanetchargetransferof0.23 Unfortunately, this still does not explain, why the blended
electronsfromthePTtoitsneighboringmolecules.Theamountof P3HT/PCBM/SWNT sample has a more pronounced PA peak,
calculated charge transfer within the bulk system is in agreement sinceactuallywewouldexpecttheoppositeconsideringtheamount
with previous DFT results for P3HT/fullerene and P3HT/SWNT ofchargecarriers,thatis,theamountofhole,ontheSWNT.
heterojunctions.52,53 Comparing the calculated change in absorbance for (6,5) and
Thechargetransferwithinthebulksystempartiallyfillsthehole (7,5)SWNTsinFigure8,wefindthe(7,5)SWNTexhibitsaless
intheexcitedstate. ThistransferofchargefromPTintothe(6,5) pronouncedPApeakthanthe(6,5)SWNTforasingleexciton, a
SWNTVBMstabilizesit,makingthebandgapwidenlessthanin negativetrion,andapairofexcitons.Ineachcase,the(7,5)SWNT
theisolated(6,5)SWNTsystem.Eventhoughthetransitionismore excited state calculation gave a smaller gap widening ∆ than the
intense,duetotheadditionalelectrons,thereductionofthePBpeak (6,5)SWNT.Moreimportantly,the(7,5)SWNTPApeakoverlaps
atthehighenergyendislesspronouncedduetothesmallershift∆. withthePBpeakofthe(6,5)SWNT.Thissuggeststhatforamix-
TodeterminehowtheamountofholetransferfromtheSWNT tureof(6,5)and(7,5)SWNTs,aPApeakwillonlybevisiblewhen
and higher loading of charge carriers influences the shape of the excitedelectronsremainonthe(6,5)SWNT.
transient spectrum at the E11 transition of SWNT systems, we 3.6. Influence of PB Peak Width. Itwasalreadyshownthat
haveincreasedtheamountofchargecarriersonthe(6,5)and(7,5) theintensityandvisibilityofaPApeakdecreases,ifthebroaden-
SWNTsfrom0.24and0.22to0.49and0.45e/nm,respectively.In ingistoohigh.29Toseewhathappenswiththewidthandtheshape
Figure8wecomparethechangeinTDDFT-RPAabsorptionIm[ε] ofthepeakfordifferentbroadenings,wemodeledtheSWNTsam-
(fromeq6ofAppendixA)of(6,5)and(7,5)SWNTswith(a)asin- ple spectrum by combining the calculated (6,5) and (7,5) SWNT
gleexciton,(b)anegativetrion,i.e.,apairofexcitedelectronsand transientspectraforvariousbroadeningsΓ.
asinglehole,and(c)apairofexcitons.Unsurprisingly,withthead- In Figure 9 the combined (6,5) and (7,5) SWNT differential
ditionofasecondexciton,asshowninFigure8(c),thewideningof transmission spectra for peak widths of 0.05, 0.1, and 0.2 eV for
thebandgap∆issignificantlyincreased,resultinginastrongerPA thegroundstateandexcitedstateabsorbanceareshown. Apeak
peakathigherenergy(∼940nmforthe(6,5)SWNT).Theseresults widthof0.05eVfortheindependenttubeabsorbancesisingood
areconsistentwiththepredominantly(6,5)SWNTmeasurements agreementwiththepeakwidthofthetransientspectrumofref 34
ofref 35aftera1pspump-probedelay,asshowninFigure3(d). (blue). A peak width of 0.05 eV is also consistent with the ab-
There,thePApeakisat∼950nmandevengainshalfoftheinten- sorption FWHM of ref 35, shown in Figures 3(a). However, in
sityofthePBpeak.Additionally,theexistenceofasecondexciton comparison to the spectrum of our SWNT sample, the combined
withina4nmunitcellisquitereasonable,ascomparedtothecal- transientspectrumhasamuchsmallerpeakwidthandisshiftedto
culatedexcitonsizeof∼2nm.23InFigure8(b)oneoftheholesis higherenergy.ThissuggeststhattheSWNTsampleshowsasignif-
filled,e.g.,throughchargetransferfromP3HTtoaSWNT,thePB icantcontributiondueto(7,5)SWNTs. Furthermore,thetransient
peakbecomesmoreasymmetric,andthePApeakbecomeslessin- spectrumofref34showsaclearasymmetryduetothereductionof
tense. Theseresultsaremoreinagreementwiththemeasurements thepeakatthehigherenergyend,consistentwithourTDDFT-RPA
after a 0.2 ps pump-probe delay of ref 34 shown in Figure 3(b). calculations.Thisasymmetrybecomeslesspronouncedthegreater
Overall,thissuggeststhestrengthofthePApeak,andthedegreeto thebroadeningofthetransitions. Apeakwidthof0.2eVresults
whichitisblueshiftedfromthePBpeak∆,maybeusedasqualita- inaspectrumsimilartoourSWNTsamplemeasurements. Thisis
tivemeasuresofthechargetransferandchargecarrierloadwithin alsoconsistentwiththeabsorptionFWHMofourSWNTmixture
Figure8.TDDFT-RPAabsorbanceIm[ε]ofa(6,5)SWNT(solidlines)and(7,5)SWNT(dashed-dottedlines)intheground/excitedstatebythepump/probe
(violet/green)asdepictedschematicallyintheupperleft/rightinset,andchangeinabsorbancebetweenthegroundandexcitedstateversuswavelengthinnm
andenergyineVrelativetotheE11transitionofa(6,5)SWNT.Theexcitedstateismodeledby(a)asingleexciton,(b)anegativetrion,and(c)apairof
excitons.Totalabsorbance(thicklines)andabsorbanceforlightpolarizedperpendiculartotheSWNTaxis(thinlines)areshown.
6
Energy ω − E (eV)
11 probedlowestenergyexcitedstate,theE11 transition. Ourresults
0 -0.1 confirm that the observed PA peak is due to a blue shift ∆ of the
(a) Γ = 0.05 eV E11 transitionafterpumping. ThePApeakisthereforeanartifact
ofexcitationswithinthespinchannel,whichexperienceawiden-
ingofthebandgapafterelectronandholestabilizationwithinthe
otherspinchannel.
The intensity and the visibility of the PA peak depends on the
charge carrier density, but mainly on the peak width. If the peak
widthisgreaterthantheblueshiftofthemainabsorptionpeak,the
s)0 0 transient spectrum will be dominated by the PB peak.29 Further,
n (arb. unit (b) Γ = 0.10 eV Im∆εce [] oTpsaurhomrixspSimlSWeWsaN.t3eNT4ly,T3s54asWmtaimmpelpeessluecgbogrenoextashadtiienbthrsitatsahtatrmhnaitisphxreitesurvrreibeorlouaoftsaeld(yd6mat,5obe)shaoaasrnuvpdritneio(dg7n(a,56pm),5eSi)axWkStsuW,NreNaTopsTf-.
o n
missi orba SarWecNaTussinwgitahnsoimveilralarpeonfertghyepgeaapks.s,Tbuhtemdioffreeriemntpotyrtpaenstloyf,tShWemNiTxs-
s
ns Ab tureenablesanexcitontransferfromthe(6,5)to(7,5)SWNTs.This
al Tra0 0ge in baowthidsehnoirntgenosfththeeebxacnitdongalipf.etAimseaornestuhlet,(6th,5e)dSiffWeNreTnstiaalndtrainnhsmibiists-
Differenti (c) Γ = 0.20 eV Chan csiaolFlnyosrsptteahcbetirlbuizlmeendodfoednthePth3(e6H,(5T6)/,5PS)CWSBWNMTN/ShTWasdNuaTelmtsooasmtthnpeoleh,PoAtlheeptereaaxnkcs.ifteorntiostlhoe-
P3HT.Asaresult,thePApeakintensityandpositioninthediffer-
entialtransmissionspectramaybeusedasaqualitativemeasureof
excitondensityandchargetransferwithinSWNTsystems.
Theseresultsareimportantfortheunderstandingoftheoriginof
thePApeakinpump-probespectroscopyandwillhelptointerpret
theexcitondynamicswithinSWNTsystems.
0 0
A.LCAOTDDFT-RPA
950 1000 1050
Wavelength (nm)
Theopticalabsorptionspectraareobtainedvialinearresponsetime
Figure9. TDDFT-RPAchangeinabsorbancebetweenthegroundandex- dependent(TD)densityfunctionaltheory(DFT)withintherandom
citedstatefora(6,5)SWNT(solidthinlines), (7,5)SWNT(dash-dotted phase approximation (RPA),36–40 from the imaginary part of the
lines), and their normalized sum (solid thick lines) for broadenings Γ of macroscopicdielectricfunction,Im[ε(q,ω)],as(cid:107)q(cid:107)→0+. Ingen-
(a)50meV,(b)100meV,and(c)200meVversuswavelengthinnmand
energyineVrelativetotheE11 transition. Normalizeddifferentialtrans- eral,withinlinearresponseTDDFT-RPA,36–38thedielectricmatrix
mission∆T/T forourSWNTsample(blackdots)andfromref 34(blue) inreciprocalspaceisgivenby
forapump-probedelayof0.2psalsoprovided.
4π
εGG(cid:48)(q,ω)=δGG(cid:48)−(cid:107)q+G(cid:107)2χ0GG(cid:48)(q,ω), (2)
whereGandG(cid:48)arereciprocallatticevectors,andχ0 isthenon-
showninFigure3(a). GG(cid:48)
interactingdensity–densityresponsefunction,i.e.,thesusceptibil-
A broader peak width for the SWNT and blended
ity.Thisisgivenby
P3HT/PCBM/SWNT samples compared to those in the litera-
Etuxreci3t4o,n35t,4ra9nmsfiegrhftrboemdtuheet(o6h,5a)vteointghe(6(7,5,5))anSdW(N7,T5)mSaWyNbeTemxpixetcutreed. χGG(cid:48)(q,ω) = Ω1 (cid:88)k (cid:88)nm(cid:88)ss(cid:48) ω+εfnnss(cid:48)(cid:48)kk−−εfmmsskk++qq+iη (3)
tAolroecacduyr,,alea2d0i%ngitmopeulreicfitcraotnio-hnolceantracpapuisnegaondetchreea(s7e,5)inSpWoNwTer. × (cid:104)ψns(cid:48)k|e−i(q+G)·r|ψmsk+q(cid:105)(cid:104)ψns(cid:48)k|ei(q+G(cid:48))·r(cid:48)|ψmsk+q(cid:105)
conversion efficiency of an OPV by more than 30 times due to whereηistheelectronicbroadening,i.e.,twicetheinverselifetime
electron-holetrapping.13 However,thepeakwidthoftheblended Γofthetransitions,Ωisthesupercellvolume, fns(cid:48)k istheFermi-
sampleissignificantlynarrowerthanthatoftheSWNTsystem,as Diracoccupation,εns(cid:48)kistheeigenenergy,andψns(cid:48)kistheKSwave
seeninFigure3(a). Thecalculatedenergydifferencesbetweenthe functionofthenthbandinspinchannels(cid:48)atk-pointk.
excited(triplet)andgroundstatesofthe(6,5)SWNTandthebulk For G=G(cid:48)=0, in the limit (cid:107)q(cid:107)→0+, the matrix elements in
PT/PCBM/SWNTsystemsuggestthatanexcitonwithinthebulkis (eq3)reduceto
mstaobreilisttya,balnedbhyeanbcoeuetxc0i.t0o8nelVif.etAimpeo,tfeonrtitahlesbouulrkcesyfostretmheisineclreecatsroedn (cid:104)ψns(cid:48)k|e−i(q+G)·r|ψmsk+q(cid:105)=−iq·(cid:104)ψnεsn(cid:48)sk(cid:48)|k∇−|ψεmmsskk+q(cid:105). (4)
andholedelocalization.Thismightexplainthesmallerwidthofthe
blendedP3HT/PCBM/SWNTabsorbanceandresultingincreasein Including local field effects, the absorption is obtained from
(eq2)bysolvingtheDysonequation
visibiltyofthePApeak.
1
Im[ε(ω)]= lim . (5)
4.CONCLUSIONS (cid:107)q(cid:107)→0+ε−1(q,ω)
00
UsinglinearresponseTDDFT-RPAcalculationsofthegroundand Neglectinglocalfieldeffects,paralleltoq,i.e.,eˆq,ε(ω)=ε00(ω).
excited states of (6,5) and (7,5) SWNTs, we are able to qualita- Substituting(eqs3and4)into(eq2),andsuppressingk-pointde-
tivelyexplainthemeasureddifferentialtransmissionspectraofthe
7
(cid:4)
pendence,weobtainthesimplifiedform AUTHORINFORMATION
Im[ε(ω)]= 4πη(cid:88)(cid:88) fms−fns(cid:48) (cid:32)eˆq·(cid:104)ψns(cid:48)|∇|ψms(cid:105)(cid:33)2. CorrespondingAuthor
Ω nm ss(cid:48) (ω−εns(cid:48)+εms)2+η2 εns(cid:48)−εms *E-mail:[email protected].
(6) Notes
whereeˆqisaunitvectorparalleltoq.Thematrixelementsin(eq6) Theauthorsdeclarenocompetingfinancialinterest.
maybeexpressedas
(cid:4)
(cid:88) (cid:104) ACKNOWLEDGMENTS
(cid:104)ψns(cid:48)|∇|ψms(cid:105) = c∗νns(cid:48)cµms (cid:104)(cid:101)φν|∇|(cid:101)φµ(cid:105)+
µν We acknowledge financial support from the European Projects
(cid:88) (cid:104) (cid:105) (cid:105)
Pa∗ (cid:104)φa|∇|φa(cid:105)−(cid:104)(cid:101)φa|∇|(cid:101)φa(cid:105) Pa , (7) POCAONTAS (FP7-PEOPLE-2012-ITN No. 316633), DYNamo
ins(cid:48) i j i j jms
(ERC-2010-AdG-267374), OLIMPIA (P7-PEOPLE-212-ITN
aij
No. 316832); Spanish Grants (FIS20113-46159-C3-1-P); Italian
where(cid:101)φν and φν are the pseudo and all-electron locally centered Grants (GGP12033); Grupos Consolidados UPV/EHU del Gob-
atomic orbitals (LCAOs) within the projector augmented wave
ierno Vasco (IT-578-13); and computational time from the BSC
(PAW)formalismwithcoefficientscνns(cid:48) forthenthKSwavefunc- RedEspanoladeSupercomputacion.
tioninspinchannels(cid:48),andPAWprojectors
(cid:4)
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