Elucidation of the electronic structure of semiconducting single-walled carbon nanotubes by electroabsorption spectroscopy

0.17 MB·English

by  Hongbo Zhao

Elucidation of the electronic structure of semiconducting single-walled carbon
nanotubes by electroabsorption spectroscopy
Hongbo Zhao∗ and Sumit Mazumdar
Department of Physics, University of Arizona, Tucson, Arizona 85721, USA
(Dated: February 6, 2008)
7
We report benchmark calculations of electroabsorption in semiconducting single-walled carbon
0
nanotubesto providemotivation toexperimentalists toperform electroabsorption measurement on
0
these systems. We show that electroabsorption can detect continuum bands in different energy
2
manifolds, even as other nonlinear absorption measurements have failed to detect them. Direct
n determinationofthebindingenergiesofexcitonsinhighermanifoldstherebybecomespossible. We
a
also find that electroabsorption can provide evidence for Fano-type coupling between the second
J
exciton and thelowest continuumband states.
4
2 PACSnumbers: 73.22.-f,78.67.Ch,71.35.-y

]
i
c Semiconducting single-walled carbon nanotubes (S- (EA),whichmeasuresthedifferencebetweentheabsorp-
s SWCNTs) are being intensively investigated because of tion α(ω) with and without an external static electric
-
l their unique properties and broad potential for appli- field,astheidealtechniqueforunderstandingtheoverall
r
t cations [1, 2]. Recent theoretical investigations have energy spectra of S-SWCNTs. EA has provided valu-
m
emphasized the strong role of electron-electron interac- able information on both conventional semiconductors
. tions and the consequent excitonic energy spectra in S- [15] and π-conjugated polymers [16, 17, 18]. The sim-
t
a
SWCNTs [3, 4, 5, 6, 7]. While within one-electron the- ilarity in the energy spectra of π-conjugated polymers
m
orytwo-photonabsorption(TPA)beginsatthesameen- and S-SWCNTs [7, 9] makes EA particularly attrac-
- ergy threshold as the lowest one-photon absorption, ex- tive. EA spectroscopy of S-SWCNTs has already been
d
n citon theories predict a significant energy gap between attempted [19], while continuous wave photomodulation
o the lowest two-photon exciton and the optical exciton spectroscopy has been interpreted as electroabsorption
c [8, 9]. This energy gap gives the lower bound to the causedbylocalelectricfields [20]. EAmeasurementsare
[
binding energy of the lowest exciton, and has been de- currentlydifficultascompleteseparationofsemiconduct-

1 termined experimentally [8, 9, 10, 11]. Exciton theories ingfrommetallicSWCNTshasnotbeenpossibletodate.
v of S-SWCNTs therefore may be considered to have firm Recent advances in the syntheses of chirality enrichedS-
7
footing. SWCNTs [21] strongly suggest that EA measurements
7
will become possible in selectS-SWCNTs in the near fu-
5 We note, however, that existing experiments have fo-
1 cused almost entirely on the lowest exciton and infor- ture. We present here benchmark calculations of EA for
0 mation on the overall energy spectra of S-SWCNTs is several wide nanotubes that give new insights to their
7 severely limited. To begin with, no signature of even electronicstructures,andprovidethe motivationforand
0
guidance to experimental work.
/ the lowest continuum band is obtained from such ex-
t As in our previous work [7, 9] we choose the semiem-
a periments. Equally importantly, finite diameters of S-
m SWCNTsleadtosubbandquantizationandconsequently piricalπ-electronPariser-Parr-Pople(PPP)model[22]as
our field-free Hamiltonian,
- aseriesofenergymanifoldslabeledn=1,2,...,etc.with
d increasing energies (see Fig. 4 in Ref. [9]), and although
n emission studies detect the optical exciton in the n = 2 H0 =− X tij(c†iσcjσ +c†jσciσ)+UXni↑ni↓
o
manifold (Ex2) [12], it has not been possible to deter- hiji,σ i
c (1)
: mine its binding energy. TPA or transient absorption 1
v + XVij(ni−1)(nj −1).
techniques used to determine the binding energy of the 2
i i6=j
X optical exciton in the n=1 manifold (Ex1) are not use-
r ful for this purpose, as nonlinear absorptions to states Here c† creates a π-electron with spin σ on the ith car-
a iσ
in the n = 2 manifold will be masked by the strong lin- bon atom, hiji implies nearest neighbors, n = c† c
iσ iσ iσ
ear absorption to Ex1 (this is particularly true here as
is the number of π-electrons with spin σ on the atom
the energy of Ex1 is nearly half that of the states in
i, and n = n is the total number of π-electrons
i Pσ iσ
the n = 2 manifold). Interference effects between Ex2
on the atom. The parameter t is the one-electron hop-
ij
and the n = 1 continuum band, suggested from relax-
ping integral,U is the repulsionbetween twoπ-electrons
ation studies of Ex2 [13, 14], are also difficult to verify
occupying the same carbon atom, and V the intersite
ij
directly. Clearly, measurements that can probe much Coulombinteraction. WehavechosenCoulombandhop-
broader energy regions of S-SWCNTs are called for.
ping parameters as in our recent work [7]. In principle,
In the present Letter, we propose electroabsorption we should also include the electron-phonon interactions
2
[6],sinceEAspectraforrealmaterialswillcontainsigna- 10 kV/cm 50 kV/cm 0.15
turesarisingfromphononsidebands[16]. This,however, 0.005 B 0.1
0.05
will make the EA calculations much too complicated. In
0 0
agreement with prior EA experiments on π-conjugated )
s -0.05
pduoleytmoeerxsc[i1t6o]n,soaunrdcaclocnutliantuiounms hbearnedfisnadretshuaffit cEiAentsliygndailfs- b. unit-0.005 A C (a) (b) --00..115
ferent that there can be no confusion in distinguishing ar 0.2 100 kV/cm 200 kV/cm 0.4
(
between exciton sidebands and continuum bands. α 0.1 0.2
∆ 0
We areinterestedinopticalabsorptionspolarizedpar- 0
-0.2
allel to the nanotube axis and consider only the compo- -0.1
-0.4
nent of the static electric field along the same direction. -0.2 (c) (d) -0.6
The overallHamiltonian is written as [17] 1.0 1.5 2.0 1.0 1.5 2.0
E (eV)
H =H0+eFz =H0+µF, (2)
FIG. 1: (Color online) (a) Linear absorption (red), and EA
whereeisthechargeoftheelectron,F thefieldstrength spectrum of the(10,0) S-SWCNTin then=1 energy region
along the nanotube axis (taken to be the z-direction), for F = 10 kV/cm. (b)–(d)EA spectrafor F = 50, 100, and
and µ the transition dipole operator along z. 200 kV/cm,respectively.
The EA is calculated in two steps [17]. We first diag-
onalize H0 in the space of all single excitations from the
Hartree-Fockgroundstate,usingthesingle-configuration
(10,0)S-SWCNT.Figs.1(b)–(d)showtheEAspectrafor
interaction (SCI) approximation [7]. Eigenstates of S-
F =50, 100, and 200 kV/cm, respectively. EA for other
SWCNTs are of even (A ) or odd (B ) parity, and
g u S-SWCNTs,including chiralones,aresimilar. Thethree
dipole matrix elements are nonzero only between states
most important features of the EA spectra are indicated
of opposite parity [23]. We calculate the field-free ab-
inFig.1(a). ThederivativelikefeatureAcorrespondsto
sorption spectra α(ω;0) from the calculated dipole ma-
the redshift of Ex1. From Eq. (3), the redshift (as op-
trix elements between the ground 1A state and ex-
g posedtoablueshift)istheconsequenceoftheexistenceof
cited B states [7, 9]. We now evaluate the matrix
u anA two-photonexcitonthatiscloserinenergytoEx1
g
elements of µ between all excited states of H0, con- than the 1A ground state, and that also has a stronger
g
struct and diagonalize the total Hamiltonian H with
dipole coupling to Ex1 [17]. In analogy to π-conjugated
the eigenstates of H0 as the basis states, and calcu- polymers [17], we have previously referred to the two-
late the new absorption α(ω;F). The EA is given by
photon excitonstate as the mA [9]. It is this state that
g
∆α(ω;F)=α(ω;F)−α(ω;0).
is visible in TPA and transient absorption [8, 9, 10, 11].
The effect of the nonzero field is to mix A and B
g u Feature B in Fig. 1(a) corresponds to the field-induced
states. Within second order perturbation theory appro-
absorption to the mA . Feature C is a dip in the ab-
g
priateforweakfields,Ex1undergoesaStarkenergyshift
sorption due to the B state at the threshold of the n
u
∆EEx1, given by = 1 continuum band (hereafter the nB [9]). The con-
u
|hEx1|µ|jA i|2F2 tinuum band is recognized by its oscillatory nature, and
g
∆EEx1 =X , (3) its appearanceovera broadenergy regionwhere there is
j EEx1−EjAg no linear absorption. EA can therefore give the binding
energy of Ex1 directly, as the energy difference between
where the sum over A states includes the j =1 ground
g
the features C and A.
state. In addition, A excitons that are forbidden for
g
F = 0 become weakly allowed for F 6= 0. This transfer TheamplitudeofthecontinuumbandsignalinFig.1is
ofoscillatorstrengthsbetweenA andB excitonsisalso muchlargerthanthatoftheexcitonatlowfield. Thishas
g u
quadraticinF. Nondegenerateperturbationtheorycan- beenobservedpreviouslyinacrystallinepolydiacetylene
not, however, describe the mixing of A and B states [16]. Finite conjugation lengths prevent the observation
g u
belonging to continuum bands and the EA in these en- of the continuum band EA signal in disordered noncrys-
ergy regions can be only calculated numerically. As we tallineπ-conjugatedpolymers[18],butthissignalwillbe
discuss below, the same is true for Ex2, which is buried observable in S-SWCNTs where the tubes are known to
within the n = 1 continuum. We have ignored the dark be long. Features due to the exciton and the continuum
excitons [6, 7] in our discussion, as they play no role in can be distinguished easily even when electron-phonon
linear or nonlinear absorption. interactions lead to sidebands, from their different field
We first describe the n = 1 energy region separately. dependence[16]. InFig.2(a)weshowthatthecalculated
InFig.1(a)wehaveplottedthecalculatedlinearabsorp- fielddependence ofboth the energyshift ofEx1 andthe
tion along with EA spectrum in the energy range corre- amplitude of EA signal due to Ex1 for (6,5), (7,6), and
spondingtothen=1manifoldforF =10kV/cmforthe (10,0) S-SWCNTs are quadratic in F up to the largest
3
(a) 11110000----7351≈ (6,5) ≈11110000----1357 ∆α (arb. units)--0000....01000055 13110004 00kV/cm (b) s)α (arb. units)0.00244 ((ab))
∆α (arb. units)11110000----6428≈ (7,6) ≈11110000----8642∆E (eV) -00..01125.00 2100.05E (eV1).10 1.15 ∆α (arb. unit--0000....00000242
1100--31 (10,0) 1100--13 b. units) 0.00 units) 00..0024 (c)
10-5 10-5 α (ar-0.02 arb. 0.00
10-7 10-7 ∆-0.04 (c) α (-0.02
∆
106 107 108 10910101011 1.40 1.42 1.44 1.46 1.48 -0.04
F2 (V2/cm2) E (eV) 1.0 1.5 2.0 2.5 3.0 3.5
E (eV)
FIG. 2: (Color online) (a) Field dependenceof ∆EEx1 (open
FIG. 3: Linear absorption (a) and EA spectrum (b) in the
circles), the EA amplitude of Ex1 (solid squares) and nBu
energy region covering both n = 1 and 2 energy manifolds
(solid triangles) for the (6,5), (7,6), and (10,0) S-SWCNTs.
for the (6,5) S-SWCNT for F = 50 kV/cm. The arrow indi-
(b) and (c) Field dependence of EA signals for the (6,5) S-
cates Ex2. (c) Superposition of the EAs for the n=1 and 2
SWCNT dueto theEx1and the nBu, respectively.
energymanifolds, calculated separately andindependentlyof
one another, at thesame field.
value of F. EA amplitudes due to the nB , also plotted
u
in the figure, exhibit weaker dependence on F at strong EAcalculationswithvaryingU/t: largerCoulombinter-
fields, in agreementwith that observedin the crystalline actionsimplylargerCoulombcouplingbetweenEx2and
polydiacetylenes[16]. InFigs.2(b)and(c)wehaveplot- the n=1 continuum, andgive largerEA signalfor Ex2.
ted EA signals due to Ex1 and the nBu, respectively, at We have observed these characteristics in our calculated
different fields. EA due to the continuum (but not the EA spectra for all S-SWCNTs.
exciton)exhibits the expectedband-broadening[16]as a Coulombcouplingbetweenadiscretestatewithacon-
function of the field. tinuum leads to the well-known Fano resonance, which
We now discuss EA over the entire energy region, fo- manifests itself as a sharp asymmetric line in the linear
cusing on the n = 2 manifold. In Fig. 3(a) we have absorption [24]. Calculation of the absorption spectrum
shown the calculated linear absorption for the (6,5) S- of the (8,0) S-SWCNT has previously found this cou-
SWCNT, while Fig. 3(b) shows the EA spectrum for pling [4]. In contrast to this standard description of the
F = 50 kV/cm. The EA spectrum is dominated by two Fano effect, the interference effect we observe in the EA
distinctandslowlydecayingoscillatingsignalsduetothe spectrum is a consequence of transition dipole coupling
n=1 and 2 continuum bands. Comparisonof Figs. 3(a) betweenEx2andthen=1continuumstates. Oneinter-
and (b) indicates that Ex2 lies within the n=1 contin- esting consequence of this dipole coupling is that unlike
uum. We also note that the signal due to Ex2 is much Ex1, which undergoes redshift in all cases, Ex2 can be
stronger than that due to Ex1. This is a consequence of eitherredshifted[asobservedinourcalculationsfor(6,4),
the interference between Ex2 and the n =1 continuum, (7,6),and(11,0)S-SWCNTs]orblueshifted[observedfor
as we prove by comparing the true EA of Fig. 3(b) with (6,2), (8,0), and (10,0) S-SWCNTs]. The reason for this
that in Fig. 3(c), where the coupling between Ex2 and isexplainedinTableI,wherewehavelistedforthe(8,0)
the n = 1 continuum states has been artificially elimi- andthe (6,4)S-SWCNTs the dominanttransitiondipole
nated. We calculatedEAfromthen=1manifoldstates couplings between Ex2 andA states along with the en-
g
alonebyremovingalln=2statesfromEq.(2);similarly ergydifferencesbetweenthem. Amongthesestates,only
the calculation of EA from the n = 2 states ignored the oneisfromthe n=2manifold,whichisthemA 2state,
g
n = 1 states. Fig. 3(c) shows the superposition of these the equivalent of the mA state in the n = 2 manifold.
g
two independent EAs. The very small EA signal due to All other states belong to the n = 1 continuum. The
Ex2 in Fig.3(c)is completely obliteratedby the smooth relatively large energy difference between the mA 2 and
g
envelope of the n=1 continuum, clearly indicating that Ex2, comparable to that between the mA and Ex1,
g
the much larger signal in Fig. 3(b) is a consequence of indicates that the energy shift of Ex2 is determined pre-
the coupling between Ex2 and the n = 1 continuum dominantlybythedipole-coupled n=1continuumstates.
states. We have further verified this by performing the States below and above Ex2 contribute to blue and red
4
tions showing that EA measurements in S-SWCNTs can
TABLEI:DominanttransitiondipolecouplingsbetweenEx2
provide valuable information on their electronic struc-
andAg states,aswellasthecorrespondingenergydifferences.
tures that are difficult to obtain from other measure-
The mAg2 state is labeled with an asterisk (*).
ments. In particular, EA spectroscopy can detect both
(n,m) hEx2|µ|jAgi/hEx2|µ|1Agi EEx2−EjAg (eV) n = 1 and 2 continuum bands. EA due to continuum
∗
(8,0) 8.31 −0.423
bands can be easily differentiated from those due to ex-
8.01 0.057
citons. Precise estimates of the binding energies of both
6.22 −0.103
Ex1 and Ex2 can therefore be obtained. Ex1 is pre-
1.22 0.073
dicted to have a redshift, which would provide indirect
(6,4) 15.3 0.049
11.5 −0.051 evidence for the mAg state detected in complementary
11.1 0.081 nonlinear absorption measurements [8, 9, 10, 11]. High
9.64 −0.070 resolution will also allow direct detection of the mA .
g
9.45 −0.080 We find strong evidence for Fano-type coupling between
6.62 −0.025
Ex2 and n = 1 continuum states that has previously
∗
6.45 −0.402
been suggested from the experimental observation of ul-
trafastnonradiativerelaxationof Ex2 [13, 14]. EAmea-
surements can provide a more direct evidence for this.
shifts, respectively, and the energy differences in Table I Syntheses of chirality enriched S-SWCNTs are allowing
rationalizeblue(red)shiftinthe(8,0)((6,4))S-SWCNT. a variety of sophisticated spectroscopic measurements.
The magnitude of the energy shift of Ex2 in all cases is The results reported here provide strong motivation for
smallerthanthatofEx1becauseofpartialcancelations, EA spectroscopy of S-SWCNTs. It is tempting to ex-
even as the amplitude of the EA signal of Ex2 is larger. tend the current EA calculations to the n = 3 energy
Correlated SCI eigenstates of the Hamiltonian (1) region. Although the n = 3 exciton is identifiable from
are superpositions of band-to-band excitations from the linear absorption calculations, our calculations indicate
Hartree-Fockgroundstate. Furthermore,withinthenon- that its coupling to the n = 2 continuum states is even
interactingtight-bindingmodelaswellaswithinHartree- stronger than that of Ex2 with the n = 1 continuum.
Fock theory, matrix elements of the component of the The analysis of excitons and continuum bands become
transition dipole moment along the nanotube axis are rather complicated at these high energies. Work is cur-
nonzero only for “symmetric” excitations, viz., from the rently in progress along this direction.
highestvalencebandtothelowestconductionband,from
This work was supported by NSF-DMR-0406604.
thesecondhighestvalencebandtothesecondlowestcon-
ductionband,etc. Hencethe strongdipole couplingsbe-
tween Ex2 and proximate n = 1 continuum eigenstates
of A symmetry necessarily implies that Ex2 eigenstate
g
contains basis vector components belonging to both n=1 ∗ Current address: Department of Physics, University of
and n = 2 manifolds. This is precisely the signature of Hong Kong, Hong Kong, China
Fanocoupling. TableII showsthe relativeweightsofthe [1] P. L. McEuen, M. S. Fuhrer, and H. Park, IEEE Trans.
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[3] T. Ando, J. Phys. Soc. Jpn. 66, 1066 (1997); ibid, 73,
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3351 (2004).
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[14] Z. Zhu et al., preprint(2006).
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