Table Of ContentExistence of an upper limit on the density of excitons in carbon nanotubes by
diffusion-limited exciton-exciton annihilation: Experiment and theory
Yoichi Murakami1,2,∗ and Junichiro Kono1
1Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, USA
2Department of Chemical System Engineering, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
(Dated: January 2, 2009)
Throughaninvestigationofphotoemissionpropertiesofhighly-photoexcitedsingle-walledcarbon
nanotubes,wedemonstrate thatthereis an upperlimit on theachievableexcitonic density. Asthe
9 intensityofopticalexcitation increases, allphotoluminescenceemission peaksarisingfrom different
0 chiralitysingle-walledcarbonnanotubesshowedclearsaturationinintensity. Eachpeakexhibiteda
0 saturationvaluethatwasindependentoftheexcitationwavelength,indicatingthatthereisanupper
2 limit on the excitonic density for each nanotube species. We propose that this saturation behavior
isaresult ofefficient exciton-exciton annihilation through whichexcitonsdecay non-radiatively. In
n
order to explain the experimental results and obtain excitonic densities in the saturation regime,
a
J we have developed a model, taking into account the generation, diffusion-limited exciton-exciton
annihilation,andspontaneousdecaysofone-dimensionalexcitons. Usingthemodel,wewereableto
2
reproducetheexperimentallyobtainedsaturationcurvesundercertainapproximations,fromwhich
] theexcitonicdensitieswereestimated. Thevalidityofthemodelwasconfirmedthroughcomparison
i with Monte Carlo simulations. Finally, we show that the conventional rate equation for exciton-
c
exciton annihilation without taking into account exciton diffusion fails to fit the experimentally
s
- observed saturation behavior, especially at high excitonic densities.
l
r
t PACSnumbers: 78.67.Ch,71.35.-y,78.55.-m
m
.
t
a I. INTRODUCTION of 1 nm, giving rise to large exciton binding energies
m on∼the order of 0.5-1 eV,25,26,27 much larger than those
d- The optical and electronic properties of low- olafrgGearAtshaQnWthRosse(∼of2c0onmjuegVa)t5e,d7 panodlymcoemrsp(arab0l.e4teoVo19r
n dimensional materials have been an important subject and < 0.1 eV28). ∼
o of study in the field of condensed matter physics. In
c particular,one-dimensional(1-D)materialsarepredicted
[
to possess unique properties that are distinctly differ-
1 ent from those at higher dimensions,1,2 primarily due Here, we report results of experimental and theoreti-
v to the enhanced Coulomb interactions among the quan-
calinvestigationsonthepropertiesofphotoluminescence
1 tum confined charge carriers. One common feature
(PL) from excitons in SWNTs through nonlinear pho-
1 of optically-excited low-dimensional systems is the for-
toluminescence excitation (PLE) spectroscopy using in-
2
mation of strongly bound electron-hole (e-h) pairs, or
0 tense optical pulses. From the clear saturation behav-
excitons,3 which dominate interband optical spectra. 1-
. ior observed in the intensities of all the PL features as
1 D semiconductors are expected to show an almost com-
a function of excitation laser intensity as well as the
0
pletesuppressionofopticalabsorptionatthebandedges,
9 complete flattening of the PLE spectra observed at very
with a significantfraction of the total oscillator strength
0 high laser intensities, we show the existence of an upper
taken by the lowest excitonic state.4,5
: limit on the density of excitons that can be accommo-
v
Early reports of lasing from semiconductor quantum dated in SWNTs. Such an upper limit is considered to
i
X wires (QWRs)6,7 invoked much interest in the physics be caused by the diffusive motion of the excitons29,30,31
r of high density 1-D excitons. A number of studies have combined with highly rapid and efficient exciton-exciton
a thus far been performed on such QWR systems during annihilation(EEA)inSWNTs.22 AsdescribedinSection
the last two decades to understand many-body phenom- III, we havedevelopedatheoreticalmodel fordescribing
ena(e.g.,lasing,band-gaprenormalization,biexcitonfor- diffusion-limited EEA processes in 1-D, which enabled
mation, and the Mott transition),8,9,10,11,12,13,14,15 but ustosimulatethePLsaturationcurvesandestimatethe
many aspects are still under debate and not well un- densitiesofexcitonsinSWNTsasafunctionofexcitation
derstood. More recently, studies on high-density e-h intensity.
pairs have been extended to novel 1-D materials such
as conjugated polymers16,17,18,19 and single-walled car-
bon nanotubes (SWNTs).20,21,22 The latter are tubular
materialsmadeofsp2-bondedcarbonatoms,23attracting Aportionofthis workwasdescribedinourearlierlet-
much recent interest from diverse research fields due to ter.32 The purpose of the present paper is to provide a
their unique properties.24 Semiconducting SWNTs are complete description of both experimental and theoreti-
known to have extremely strong quantum confinement cal aspects of this study.
2
II. EXPERIMENT OPA, 654 nm CW laser diode, 658 nm
)
s
nit (a)
A. Experimental methods u
b. (8,3)
r
a
The sample was prepared by ultrasonicating CoMo- (
CAT SWNTs in D2O with 1 wt% sodium cholate for 1 sity (6,5) 1000 1100
hour, followed by ultracentrifugation at 111,000 g for 4 en (7,5) Wavelength (nm)
hours. This centrifugation condition is sufficient to re- nt
move SWNT bundles effectively.33 Only the upper 50 % L I
P
of the supernatant was collected and used for the ex-
periment. The solution was put in a 1-mm-thick quartz
cuvette. The optical density of the sample around the s) (b) (6,5) OPA, 654 nm
E22 resonance was below 0.2, which helped avoid non- unit (8,3) high power
uniform excitation and re-absorption of the emitted PL b. (7,5) #4
r
within the sample. The excitation source was an optical a #3 #5
(
parametric amplifier (OPA), producing 250 fs pulses y #2 #6
at a repetition rate of 1 kHz, tunable in∼the visible and sit low power #1 #7
n
near-infrared ranges, pumped by a chirped-pulse ampli- e
nt
fier (Clark-MXR, Inc., CPA-2010). Optical filters were L I
carefullyselectedandsetinthebeampathtothoroughly P
eliminate any parasitic wavelength components (mostly
950 1000 1050 1100 1150
in the ultravioletandnear-infraredregions)containedin
Wavelength (nm)
the OPA beam. The OPA beam was focused onto the
sample to a spot size of 300-400 µm. Only the central
2 mm portion of the OPA beam profile ( 6 mm) s) (c) (6,5) (7,5)
∼was taken out by using an aperture just before∼the focus unit (8,3) high power#4
to enhance the spot uniformity at the sample. The PL b. #3 #5
r
from the sample was focused onto the monochromator (a #2 #6
entrance and recorded with a liquid-nitrogen-cooled In- y low power #1 #7
GaAs 1-Darraydetector. The obtainedPLspectrawere nsit
e
corrected for the wavelength-dependence of the grating nt
efficiency and detector sensitivity. L I
For the data shown in Fig. 1(c), a different sample as P
wellasadifferentexcitationlightsourcewasusedforver-
950 1000 1050 1100 1150
ifyingtheuniversalityofthephenomenaobserved. Inthis
Wavelength (nm)
case, the sample was a dried film of CoMoCAT SWNTs
embeddedinι-carrageenan,formedbydryingamixedgel
of ι-carrageenan and the centrifuged supernatant of Co- FIG. 1: (color online) Pump-intensity-dependentPL spectra
MoCAT SWNTs on the surface of a sapphire substrate. measuredfor(a,b)thecentrifugedsupernatantofCoMoCAT
Thesapphiresubstrateservedasamechanicalsupportas SWNTs dispersed in D2O and (c) CoMoCAT SWNTs em-
beddedinadriedι-carrageenanfilm: (a)Blacksolid—spec-
well as a heat sink of the film during the measurements.
trum obtained with OPA pulse (654 nm, 29 nJ). Red dotted
Theexcitationlightwas1kHzand 250fsopticalpulses
∼ curve— spectrum obtained with a CW laser diode (658 nm,
with a centralwavelengthof653nm (FWHM = 10 nm),
100µW).Insetshowsthatthetwospectracoincidewhenthe
produced by filtering whitelight pulses generated by fo-
OPApulseenergyisverylow(300pJ).(b)ChangeofPLspec-
cusing the CPA beam onto a sapphire crystal. tra with the pulse energy of OPA beam (654 nm) varied be-
tween1nJand30nJintheorderof#1to#7. Curve#4cor-
responds to the highest fluence (∼1.3 × 1014 photons/cm2).
B. Experimental results (c) Change of PL spectra with the pulse energy of 653 nm
light(FWHM=10nm)variedbetween1nJand20nJinthe
order of #1 to #7.
Figure 1(a) compares two PL spectra. The black solid
curve was obtained using the OPA with a wavelength of
654 nm (or 1.90 eV) and a pulse energy of 29 nJ, while
thereddottedcurvewasobtainedusingaweak(100µW) ure 1(b) shows PL spectra measured with pulse energies
CWlaserwithawavelengthof658nm(or1.88eV).Itis of1nJ(curves#1and#7),4nJ(#2and#6),10nJ(#3
seenthattherelativeintensitiesofdifferentPLpeaksare and#5),and30nJ(#4). The (7,5)peakisdominantat
drastically different between the two curves. The inset low fluences while the (6,5) peak becomes dominant at
confirms that the two spectra coincide accurately when high fluences. It is important to note that the different
the OPA pulse energy was kept very low (300 pJ). Fig- curvesweretakenin the orderof#1 to #7,demonstrat-
3
ing that the observed changes are reproducible and are
(a) (e)
12 -2
notcausedbyanylaser-inducedpermanentchangeinthe 2.2 1.2 × 10 cm (6,5)
sample. Additionally,notethatthePLintensitiestendto (6,5) (8,4)
saturateathighlaserfluences,whiletheirpeakpositions (8,4) 983 nm
2.0 1034 nm
do not change at all. 1125 nm
(7,6) (7,5)
Figure 1(c) shows PL spectra measured for the dried (7,5) (7,6)
1.8
ι-carrageenan film using 653 nm optical pulses with a (8,3) 1.2 × 1012 cm-2
FWHM bandwidth of 10 nm at different pulse energies.
The pulse energies were 1 nJ (curves #1 and #7), 3 nJ 2.2 (b) 1.2 × 1013 cm-2 (f)
(#2 and #6), 10 nJ (#3 and #5), and 20 nJ (#4) mea-
sured in the order of #1 to #7. Figure 1(c) exhibits the
)2.0
samebehaviorasthatshowninFig.1(b),demonstrating V
e
that the observedchanges shownin Fig. 1(b) did not re- y (
sult from any artifacts, e.g., caused by the fluidic nature erg1.8 1.2 × 1013 cm-2
of the sample or by the unnoticed parasitic wavelength n
E
components in the OPA beam. In the following, we use n (c) 13 -2 (g)
the excitationpulsefluence intermsofthe numberofin- atio2.2 4.1 × 10 cm
cidentphotonspercm2 perpulsetoexpresstheintensity cit
of excitation pulses. x2.0
E
Figures 2(a)-2(d) show PLE maps taken with vari-
ous pump fluences. The step size for the pump pho- 1.8
ton energy was 20 meV. The data taken with the low- 4.1 × 1013 cm-2
est fluence (1.2 1012 photons/cm2) [Fig. 2(a)] is es-
× (d) 14 -2 (h)
sentially the same as that taken with low-intensity CW 2.2 1.2 × 10 cm
light. However, as the fluence is increased [Figs. 2(b)-
2(d)], the E22 excitation peaks gradually broaden and 2.0
eventually become completely flat at the highest fluence
(1.2 1014 photons/cm2) — i.e., PL intensities become
× 1.8
independent of the excitation wavelength. The corre-
14 -2
1.2 × 10 cm
sponding PLE spectra are shown in Figs. 2(e)-2(h) for
threePLwavelengthsat983,1034,and1125nm. Again, 950 1000105011001150
Emission Wavelength (nm) Intensity (arb. units)
such changes in the PLE spectra were reproducible over
the fluence range tested here, indicating that no sample
damage was induced. FIG.2: (coloronline) Evolution ofPLEdatawith increasing
In order to obtain PL intensity (I ) versus pump in- pumppulsefluence: (a) 1.2 × 1012,(b) 1.2 ×1013, (c)4.1 ×
tensity(I )relationshipsfordiffePreLntemissionpeaks, 1013, and (d) 1.2 × 1014 photons/cm2. (e-h): PLE spectra
pump
corresponding to (a)-(d) at emission wavelengths of 983 nm
we measured PL spectra at different photon fluences for
(circles), 1034 nm (squares), and 1125 nm (triangles).
various excitation wavelengths. Each PL spectrum was
decomposed and fitted by multiple peaks corresponding
to the SWNT types/chiralities involved in the measured
approximately570,647,and673nm,respectively. Itcan
wavelengthrange. 50%Gaussian+50%Lorentzianline-
be seen that the integrated PL intensity begins to satu-
shape was assumed, and the decomposition was per-
rateatalower(higher)fluencewhenSWNTsareexcited
formedby optimizing the peak-widthsothatthe decom-
resonantly (non-resonantly). Unexpectedly fast satura-
position gives the best fitting for the original PL spec-
tion of the PL from (7,5) with 570 nm excitation (which
trum. The optimum widths at the highest fluence was
isnon-resonant)canbeattributedtoitsproximitytothe
larger by 15 % than those at the lowest fluence in
∼ phonon sideband at 585 nm.34
Fig. 1, and such an increment of the width is consid-
∼
ered to be caused by the enhanced interactions among
excitons or their reduced lifetime in the presence of high
density excitons. Throughout the decomposition analy- C. Data interpretation
sis performed, peak positions of all the PL features and
theratiosamongtheirwidthswerefixedregardlessofthe We interpret these observations as results of very ef-
excitation wavelength and fluence. ficient exciton-exciton annihilation,16,17,18,19,22,35 a non-
Figure3showstheobtainedintegratedPLpeakinten- radiative process that occurs at high exciton densities
sities (I ) plotted against the incident photon fluence where two excitons are spatially close enoughto interact
PL
(I ) for (6,5), (7,5), and (8,3) SWNTs at excitation with each other, resulting in the annihilation of the two
pump
wavelengths of 570, 615, and 658 nm. The resonance excitons and simultaneous creation of an e-h pair in a
wavelengths of these SWNT types at the E levels are higher energy state (either as a bound exciton or an un-
22
4
2.5 the pump wavelength, resulting in the flattening of PLE
(a) spectra. Namely, at very high pump fluences, the PL in-
2.0 tensity is determined not by how efficiently excitons are
createdbut by how many E excitons can be accommo-
11
1.5 (6,5) dated within a particular type of SWNTaswellasbythe
relativeabundance of that type ofSWNT in the sample.
1.0 We also performed optical transmission spectroscopy in
570 nm excitation the E22 range using OPA pulses and found that the ab-
615 nm excitation sorption spectra do not exhibit any change even at high
0.5
654 nm excitation pulse fluences. Thus, nonlinear optical effects such as
phase-space filling in the E range are not playing any
0.0 22
role in the observed PLE broadening/flattening and PL
) 1.0
s (b)
t saturation.
ni
u
.
b
r (7,5) III. THEORY
a
( 0.5
y
t In this section, we develop a theoretical model for ex-
si 570 nm excitation
n plainingtheexperimentalresults,takingintoaccountthe
615 nm excitation
e
generation, diffusion-limited EEA, and spontaneous de-
t 654 nm excitation
n
I cays of 1-D excitons. Under certain approximations, the
0.0
model provides a direct analytical relationship between
(c) the intensity of the excitation light (I ) and that of
pump
the emitted PL (I ) for limiting cases, which allows us
PL
0.4 toestimatethedensityofexcitonsinSWNTsthroughfit-
tingtotheexperimentallyobtainedI vs.I curves.
pump PL
(8,3)
0.2
570 nm excitation A. Model
615 nm excitation
654 nm excitation
Figure 4(a) shows a schematic energy diagram of the
0.0 excitonsinsemiconductingSWNTs under consideration.
0.0 0.2 0.4 0.6 0.8 1.0 1.2
We are interested in calculating the population of exci-
14 2 tons N (indicated by the dotted box) in the lowest en-
Fluence (× 10 photons/cm )
ergy state E . First, excitons are created by optical
11
excitation at an energy around the E level, which is
22
FIG. 3: (color online) Integrated PL intensity versus pump typically in the visible wavelength range. The excita-
fluencefor(6,5),(7,5),and(8,3)SWNTs. Pumpwavelengths tion intensity is denoted by I . As soon as excitons
pump
were 570 nm (circles), 615 nm (squares), and 654 nm (trian- are created, they decay to the E level within a very
11
gles). Theerrorbarsaccountfor ±5%. Thesolid anddashed short time ( 40 fs)36 by transferring their energies to
curvesare fitting by Eq.(15) and Eq.(20), respectively. ∼
the lattice. Recent studies have reported that the ex-
citons created around E levels primarily decay to the
22
E levelwithaprobabilityclosetounity.37,38 Theinflux
11
bound free e-h pair). We assume that the formation of of excitons to the E level is denoted by G . On the
11 in
E excitonsoccursinaveryshorttimescaleafteranop- other hand, the spontaneous decay time τ of the E
11 tot 11
tical excitation around E , because of much faster E - excitonstothegroundstate(G.S.)hasbeenreportedto
22 22
to-E relaxation(e.g., 40fs36)thanthedurationofour be 10–100 ps.39,40,41 Such a spontaneous decay consists
11
∼
OPApulse( 250fs). Thus,excitonsquicklyaccumulate of radiative and non-radiative processes, with respective
in the E st∼ate during and right after photo-creationof rates γ and γ (s−1), where γ + γ γ = τ −1.
11 r nr r nr tot tot
≡
e-h pairs. However, the number of excitons that can be The outflux of excitons from the E level via the spon-
11
accommodated in the E state is limited by EEA. As taneous decay process isdenotedby G . Therefore,the
11 out
the exciton density, n , approaches its maximum value, flux of the PL photons orthe PL intensity (I ) is equal
x PL
EEAbeginstopreventafurtherincreasebyefficientlyre- toηG ,whereη( γ /γ )denotesthebranchingratio
out r tot
≡
moving excitons non-radiatively, which explains the PL for the radiative decay from the E level.
11
saturation behavior. Since EEA serves as a bottleneck As the density of excitons increases, the EEA process
for the exciton density, the PL intensity becomes insen- becomes important. If the e-h pair created in the higher
sitive to whether the excitonswere createdresonantlyor energy state returns back to the E exciton level with
11
non-resonantly around the E level and independent of a probability of λ (0 λ 1), the initial two excitons
22
≤ ≤
5
(a) segments (or excitons) of length l are randomly present
~ E x
22 without overlapping each other, as shown in Fig. 4(b).
I
pump The probability of a new segment (of length l ) to enter
x
the system without overlapping any of the N existing
EEA:
G
in 2ex!"#ex segments, p(N), is given by the product of the following
two probabilities
N Nl
E x
p =1 (1)
11 1
− L
( = + ) which is the probability for the center of the new seg-
G tot r nr
out I =$G ment to land on an unoccupied area [hatched regions in
PL out
Fig. 4(b)], and
G. S.
N
l
x
(b) p2 = 1 (2)
lx (cid:18) − L−Nlx(cid:19)
which is the probability that the occupying length (l )
x
of the new segment (whose center has landed on an un-
L
occupied area) does not interfere with any of the N ex-
isting segments [vertical thick bars in the lower part of
Fig. 4(b)]. Hence, p(N) is written as
N
Nl l
x x
p(N)= 1 1 . (3)
L–N l N l − L − L Nl
x x (cid:18) (cid:19)(cid:18) − x(cid:19)
The expectation value of the increment of N due to the
introduction of a new segment into the system, ∆N
FIG. 4: (color online) (a) Schematic energy diagram of the h iN
(0 < ∆N 1), depends on the type of two-particle
system considered in the model. The dotted box enclosing h iN ≤
the lowest energy level (E11) is the domain of interest where annihilationconsidered. Forthegeneral“ex+ex λex”
→
N excitonsarepopulated. Allthesymbolsaredefinedin the case, ∆N is expressed as
h iN
text. (b) Schematic description of N = 4 excitons randomly
distributed overa SWNTwith a length of L. Thehorizontal h∆NiN =p(N)−(1−λ)(1−p(N)). (4)
arrowwithalengthoflxdenotestheaveragelengthtraversed
byoneexcitonduringitsspontaneousdecaylifetimeτtot. The In the following, the derivation for the case of λ = 1 is
lowerpartistheequivalentoftheupperbutemphasizesthat shownas a specific example,since this caseis considered
thetotallengthoftheunoccupiedregion isL−Nlx whereN torepresentourexperimentalsituation. Thefinalresults
vertical thick bars denote the borders of the areas occupied will be presented for both the λ=1 and λ=0 cases.
bythoseexcitons. TheendsoftheSWNTareassumed tobe
a cyclic boundary.
B. Solutions in limiting cases
are eventually reduced to λ excitons (as an expectation 1. Steady-state limit
value) through such an EEA process. We make the fol-
lowing two assumptions: (1) Once two excitons intersect Here, we consider the steady-state limit, where the
in a SWNT, EEA occurs instantaneously with a prob- numberofexcitonsN inthesystemissteady,correspond-
ability of one, and (2) the positions where excitons are ingto CWexcitation. Inorderto derivethe relationship
created in SWNTs by optical excitations are random. between I and I , we consider the relationship be-
pump PL
Figure 4(b) schematically shows a situation in which tween G and G [see Fig. 4(a)], where both G and
in out in
N excitons exist in a SWNT of length of L. Each exci- G arerates,havingunitsofs−1. WhenN issufficiently
out
ton is considered to “occupy” a characteristic length lx smalland EEAis negligible, Gin =Gout =γtotN. When
in a SWNT. In a static limit, lx should simply be the EEA is present, however, this relationship becomes
exciton size. On the other hand, when exciton diffusion
G γ N
is present, l is assumedto be the averagedistance trav- out tot
x G = = . (5)
in
eledbyanexcitonduringthe spontaneousdecaylifetime ∆N ∆N
h iN h iN
τ and is √Dτ , where D (cm2/s) is the exciton
tot tot
∝ In the case of λ=1, Eq. (4) and Eq. (5) lead to
diffusion constant. Namely, it is assumed that l is de-
x
termined by the diffusion length, and any two excitons γ N
tot
creFaitresdt,wait1h-iDn lsxpaucnedoerfgloenEgEthAL. is considered where N Gin = 1− NLlx 1− lLx 1− NLlx -1 N . (6)
(cid:0) (cid:1)n (cid:0) (cid:1) o
6
Weintroducetwodimensionlessvariablesζ Nl /Land created at t = 0, while the PL intensity I is propor-
x PL
≡
ψ G l /γ L, the former of which is a dimensionless tional to the number of excitons N that survived EEA.
in x tot
≡
exciton population (0 ζ < 1). Using these variables, The relationship between N and N is described by the
0
≤
Eq. (6) can then be rewritten in dimensionless form: differential equation
dN
ζ dN = (11)
ψ = (1 ζ) 1 lx(1 ζ)−1 lLxζ . (7) 0 h∆NiN
− − L − where ∆N is given by Eq. (4). When λ = 1 is as-
n o h iN
sumed, Eq. (11) is rewritten as
Expanding the second factor in the denominator of
Eq. (7) and eliminating higher-orderterms of l /L leads dN 1
x 0
= . (12)
to an equation of only ζ and ψ, expressed as dN 1 Nlx 1 lx 1 Nlx −1 N
− L − L − L
ζ
ψ = (1−ζ)κP∞=0(−κ1!)κ (cid:16)1−ζζ(cid:17)κ. (8) AdAisgmaieninnt,shisoeinnplceresesv(cid:0)IiPovLaurs∝iacabNsle(cid:1)esn(asnζtdea≡IdpyuN(cid:0)-msltpxa/t∝LelNiamn0(cid:1),idt)I,ψPowLe≡=inNct1r0ζoldxa/unLcde.
I =c ψ, where c andc are constants. With ζ and
Summing up to κ = 6 in Eq. (8) is sufficient to repro- pump 2 1 2
ψ, Eq. (12) can be written in a dimensionless form:
duce Eq. (7) for l /L < 0.1. Finally, noting that the
x
denominator of Eq. (8) is equal to the Taylor expansion dψ 1
ostfaatenleimxpitonfoerntλia=l f1un(ectxio+n,exthe seoxlu)tiisonexfporrestsheedsatseady- dζ = (1 ζ) 1 lx(1 ζ)−1 lLxζ . (13)
→ − − L −
n o
ζ ζ Furthermore, similar to the steady-state case, an expan-
ψ = exp . (9)
1 ζ 1 ζ sionisperformedonthesecondfactorinthedenominator
− (cid:18) − (cid:19) of Eq. (13) together with elimination of the higher-order
Here,sinceIPL N andIpump Gin,IPL andIpump are terms of lx/L, resulting in a differential equation of only
∝ ∝
proportionalto ζ andψ, respectively,i.e., I =c ζ and ζ and ψ:
PL 1
I =c ψ, where c and c are real constants.
pump 2 1 2 dψ 1 ζ
On the other hand, the solution for the case of λ = 0 = exp . (14)
dζ 1 ζ 1 ζ
(ex+ex 0) is derived through a similar procedure, − (cid:18) − (cid:19)
→
yielding Finally, by integrating Eq. (14) from 0 to ζ, we obtain
the solution for λ=1 (ex+ex ex) as
ζ →
ψ = . (10)
2(1−ζ)exp −1−ζζ −1 ψ = 1e Ei 1 1 ζ −Ei(1) , (15)
(cid:16) (cid:17) (cid:26) (cid:18) − (cid:19) (cid:27)
Equations (9) and (10), valid for CW PL experiments, where
areimplicitequationsrelatingthePLintensity(IPL)and x ey
the pump intensity (I ) in terms of their respective Ei(x)= dy (16)
pump
dimensionlessvariablesζ andψ. Theseequationscontain Z−∞ y
nofittingparametersexceptthetwolinearscalingfactors is the exponential integral.
c1 and c2 and simply become equivalent (ζ = ψ) in the Similarly, the solution for λ = 0 (ex+ex 0) in the
low density limit (ζ 0). instantaneous limit is →
→
ζ 1
ψ = dζ (17)
2. Instantaneous limit Z0 2(1−ζ)exp −1−1ζ −1
(cid:16) (cid:17)
Here,weconsidertheinstantaneouslimitinwhichcre- where the integral has to be solved numerically. Those
ationofalltheexcitonsbyaninfinitesimallyshortoptical dimensionlessequations(15)and(17)becomeequivalent
pulse and their internal relaxation to E level occur in- (ζ =ψ) in the low density limit (ζ 0).
11 →
stantaneouslyatt=0,beforediffusion-limitedEEAand
spontaneousdecayprocessesbegintooccursubsequently.
IV. COMPARISONS AND DISCUSSION
This limit represents a situation where the duration of
the optical pulse and the time required for intrabandre-
laxation are much shorter than the spontaneous decay A. Comparison of model and experiment
time τ , asinthe caseofthe presentexperimentalsitu-
tot
ation. The pump intensity I in this limit is directly To compare with the experimental data shown in
pump
proportional to N , the number of excitons or e-h pairs Fig. 3, we use Eq. (15), which is for the instantaneous
0
7
limit with λ =1. The choice of this equation is because Exct. c1 c2−1 ζ N R
oftheshortdurationoftheopticalpulsesused( 250fs), Type (nm) (×105) (×10−14) (×105 cm−1)
∼
the very fast ( 40 fs36) and efficient37,38 internal decay 570 2.97 7.74 0.764 1.70 10.3
∼
of excitons created at E levels to the lowest E level, (6,5) 615 3.89 1.54 0.566 1.26 2.78
22 11
and the much longer spontaneous decay time from the 654 4.21 1.13 0.505 1.12 2.28
E level to G. S. (10–100 ps).39,40,41 The solid curves 570 1.14 3.18 0.678 1.51 4.77
11
showninFig.3arefitting curvesusingEq.(15), indicat- (7,5) 615 1.34 2.68 0.656 1.46 4.17
ing that the model agrees well with the experimentally 654 1.22 9.16 0.776 1.72 12.0
observed PL saturation curves. 570 0.73 1.54 0.566 1.26 2.78
(8,3) 615 0.72 2.61 0.652 1.45 4.07
Table I is a summary of the fitting analysis performed 654 0.68 5.05 0.729 1.62 7.05
on the data in Fig. 3 using Eq. (15). The first two
columns show the optimum values of the scaling fac- TABLE I: Summary of the optimal parameters to used
tors c and c −1. These are thought to be propor- fit the experimental data, obtained from the analysis with
1 2 Eq.(15). ζ,N,andRarevaluesatthefluenceof1.02×1014
tional to the oscillator strength for the PL emission photons/cm2, corresponding to thelargest fluencein Fig. 3.
from E levels and the oscillator strength for the op-
11
tical absorption around E levels, respectively, as ex-
22
pected from I c N and N c −1I . The right
three columnsPL(ζ∝, N1, and R) 0ar∝e v2aluespuamtp1.02 1014 -1m) 2.0
photons/cm2 (see Fig. 3). The values of N we×re ob- c
5
tained through the relationship N = ζ/l , where l was 0
x x 1 1.5
assumed to be 45 nm (one half of the average exciton ×
(
excursion range defined in Ref. 31). The exciton den- N
sity in the highly saturated regime ( 1 105 cm−1) is y,
stillmorethanoneordermagnitudes∼mall×erthantheex- sit 1.0 Model, Eq. (15)
pectedMottdensityinSWNTs( 7 106 cm−1,assum- en Monte Carlo
iinngdaetnaielxicnitRonefs.iz3e2.ofR∼in1.5thnemr4ig2h)∼,t-ams×ohsatscboleuemnndidsceunsosteeds ant D 0.5
theratiobetweenthenumberofas-createde-hpairsand ult
s
e
the number of excitons that survived the EEA process R
until their spontaneous decay to G. S., or N /N. The 0.0
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
values ofR showthat approximately90%ofthe initially
Initial Exciton Density, N (× 106 cm-1)
createde-hpairs/excitonsdecaynon-radiativelythrough 0
theEEApathwhentheE levelsareresonantlyexcited
22
at a fluence of 1.02 1014 photons/cm2. FIG. 5: (color online). Comparison between the saturation
×
behavior predicted by Eq. (15) (black solid curve) and the
As expected, the values of c1 are similar within the resultobtainedbyaMonteCarlosimulationoftheEEApro-
same chirality type, regardless of the excitation wave- cess. The abscissa denotes the density of initially created
length. However, for the particular case of (6,5) with excitonswhile theordinate correspondsto theresultant den-
570 nm excitation, where the excitation wavelength ex- sity. The experimental range of N0 in Fig. 3 is N0 ≤2×106
actly coincided with the E resonance peak, the value cm−1.
22
ofc ( 3 105)is appreciablysmallerthanthose atthe
1
othere∼xcit×ationwavelengths( 4 105). Suchasmaller
∼ × B. Comparison with Monte Carlo calculation
c value implies that more excitons decayed to the G. S.
1
throughnon-radiativepathscomparedtotheothercases.
This may be due to an emergence of stronger nonlinear To further confirm the validity of the model, we per-
processes that have not been taken into account in the formed a computational simulation based on the Monte
assumptions, such as the breakdown of the λ = 1 as- Carlomethod. Atthebeginningofthesimulation,aran-
sumption and/orthe appearanceofmore-than-two-body domdistributionofN0 excitonsalongaline iscreatedat
annihilationprocessesbecauseoftheveryhighinitiale-h time t = 0, corresponding to the instantaneous limit.
pair density, achieved with the strong E resonance in For t > 0, each exciton undergoes a random movement
22
this particular case. in each computational step dt by the distance given by
the probability distribution of N¯(0,l2), where N¯(x ,σ2)
0 0
The values of c −1 also show the expected tendency denotesthenormaldistributioncenteredatx withvari-
2 0
toward higher (lower) values when the excitation wave- ance σ and l √Ddt. Upon intersection of any two
0
≡
lengthiscloserto(furtherfrom)theE resonancewave- excitons, the EEAof ex+ex ex takes place. In ad-
22
length. Theslightlyhigherc −1valueforthecaseof(7,5) dition,eachexcitonis{eliminate→dfro}mthe systemwitha
2
with 570 nm excitationis againattributed to its vicinity probability of γ dt (= τ −1dt), corresponding to pos-
tot tot
to the E phonon sideband at 585 nm.34 siblespontaneousdecayduringeachcomputationalstep.
22
∼
8
The total density of excitons that have decayed sponta- saturation with non-resonant excitations, the deviation
neously (N) can thus be calculated for each value of the becomes clearerforthe regime ofstrongerPLsaturation
initial density N . with resonant excitations. Most importantly, however,
0
Figure 5 shows a comparison between the model for the density predicted by Eq. (20) has no upper limit as
the instantaneous limit [Eq. (15)] with l = 45 nm and recognizedby the steadily increasing dashed lines shown
x
theresultoftheMonteCarlosimulation,plottingthere- in Fig. 3 [or by the logarithmic form of Eq. (20)], while
lationship of N and N. The simulation was performed Eq.(15)correctlydemonstratestheexistenceofanupper
0
withD =0.42cm2/sandτ =100ps,43 whichresulted limit,asobservedexperimentallyandintheMonteCarlo
tot
in an average exciton displacement of 45 nm based on simulation.
the same simulation performed without EEA. The com- Another important difference between Eq. (15) and
parisonshowssatisfactoryagreement,indicatingthatthe Eq.(20) is described as follows: The solution of the con-
simpleanalyticalformofEq.(15)welldescribesthephe- ventional rate equation [Eq. (20)] contains two indepen-
nomenon of diffusion-limited EEA, hence validating our dent quantities: η/Γ and N Γ. The former can be used
0
model. to scale I . However, since the parameter N Γ simul-
PL 0
taneously scales I ( N ) and changes the shape
pump 0
∝
of the saturation curve, one essentially cannot estimate
C. Comparison with conventional rate-equation the density ofexcitons fromthe fitting analysisbasedon
solution Eq.(20). On the other hand, since the shape ofEq. (15)
has been uniquely determined by its dimensionless rep-
Conventionally, a rate equation of the form resentation, the only thing one can do is to linearly and
independently changethetwoscalingconstantsc andc
1 2
dN(t)
=Gin(t) γtotN(t) γEEAN(t)2 (18) for IPL and Ipump, respectively, yielding sets of ζ and ψ
dt − − values along the scaled (or fitted) I vs. I curves.
PL pump
hasbeenusedtoexplainEEAprocessesobservedforvari-
ousmaterialsfrom1-Dto3-D.17,18,19,22,44,45,46Theterms
on the right hand side, from left to right, represent the V. SUMMARY
rate of excitons entering the system (or the E level in
11
the present case), the rate of excitons spontaneously de- Wehaveinvestigatedphotoemissionpropertiesofhigh-
cayingfromthesystemtotheG.S.(atγtot),andtherate density 1-D excitons in single-walled carbon nanotubes.
ofexcitonsleavingthesystembecauseofEEA(atγEEA), As the excitation intensity increases, all photolumines-
respectively. Here,theprobabilityoffindingtwoexcitons cence emission peaks arising from different chiralities
at the same position (or the EEA rate) is assumed to be showedclearsaturationinintensity. Eachpeakexhibited
proportional to N2, and hence, exciton diffusion as well asaturationvaluethatwasindependentoftheexcitation
as the finite size (i.e., length) that each 1-D exciton oc- wavelength,indicatingthatthereisanupperlimitonthe
cupies are not taken into account in Eq. (18). exciton density for each nanotube species. We interpret
Equation (18) is solved for the pulse-wise creation of theseresultsintermsofdiffusion-limitedexciton-exciton
N0 excitons at t=0 [i.e., Gin(t)=δ(N0)] in order to be annihilation processes through which high-density exci-
comparedwiththeproposedmodelfortheinstantaneous tons decay non-radiatively.
limit [Eq. (15)] as well as with the experimental results. To quantitatively understand the saturation behavior
With this initial condition, Eq. (18) is readily solved to observedintheexperiment,wehavedevelopedatheoret-
be icalmodel, takinginto accountthe generation,diffusion-
limitedexciton-excitonannihilation,andspontaneousde-
1 γ
EEA
N(t)= Γ . (19) cays of 1-D excitons. The saturation curve predicted by
N10 +Γ exp(γtott)−Γ (cid:20) ≡ γtot (cid:21) themodelunderappropriateapproximationsreproduced
(cid:16) (cid:17) the experimental saturation curves well, and the fitting
The total number of photons emitted from the sample, analysisallowedustoestimatethedensityofexcitonsfor
IPL, is obtained by integrating Eq.(19) from t=0 to a givendiffusionconstant. We alsocomparedour results
∞
as with Monte Carlo calculations, confirming the validity
∞ −1 of our model. Additionally, we examined the saturation
η 1
I =η γ N(t)dt= ln 1 1+ behavior predicted by the solution of the conventional
PL tot
Z0 −Γ ( −(cid:18) N0Γ(cid:19) ) exciton-exciton annihilation rate equation that does not
(20) take into account diffusion and showed that the solu-
where η = γ /γ as before. Equation (20) describes tion qualitatively failed to fit the experimental data at
r tot
the relationship between I and N in terms of two ad- high exciton densities, showing the inappropriateness of
PL 0
justable components η/Γ and N Γ. its use for excitons in carbon nanotubes. The approach
0
The dashed lines drawn in Fig. 3 are the optimum fits presented in this paper should have wide applicability
by Eq. (20) to the experimental data. While Eq. (20) for predicting the intensities of photoluminescence from
reproducesthebehaviorwellfortheregimeofweakerPL single-walled carbon nanotubes under various excitation
9
conditions. thetemperatureaswellasapplyinganexternalmagnetic
Animportantconclusiondrawnfromthecurrentstudy field.47
is thatalargedensity ofelectron-holepairsis difficult to
achievein single-walledcarbonnanotubes. This imposes
a serious challenge for making active optoelectronic de-
vices basedonsemiconducting single-walledcarbonnan-
Acknowledgments
otubes — 1-D semiconductor lasers,in particular,which
would require densities comparable to, or higher than,
the Mott density. The existence of an upper limit on The authorsthank Ajit Srivastavafor valuable discus-
the density of excitons would also prevent fundamental sions and Erik Einarsson for emendation. We thank the
studies of bosonic characters of 1-D excitons expected Robert A. Welch Foundation (C-1509) and NSF (DMR-
at quantum-degenerate densities. Hence, novel methods 0325474 and OISE-0530220) for support. One of us
are needed for minimizing non-radiative decay processes (Y. M.) was financially supported by Grants-in-Aid for
such as exciton-excitonannihilation in single-walledcar- Scientific Research (#18-09883) from the Japan Society
bon nanotubes, which would require ways to control the for the Promotion of Science (JSPS). Y. M. thanks Tat-
dynamic parametersof excitons such as the spontaneous suya Okubo and Shigeo Maruyama for support to his
decay rate and diffusion constant through, e.g., varying fulfillment of the JSPS research program.
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