Nanotube graphene physics

MASTER’S THESIS
Optical properties of electrochemically doped
single-walled carbon nanotubes
Master’s Educational Program: Photonics and Quantum Materials
Student Daria Satco
signature, name
Research Advisor: Prof. Albert Nasibulin
signature, name, title
Co-Advisor (if applicable): Senior research scientist
Dr. Daria Kopylova
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Moscow 2019
Copyright 2019 Author. All rights reserved.
The author hereby grants to Skoltech permission to reproduce and to distribute publicly paper and electronic copies
of this thesis document in whole and in part in any medium now known or hereafter created.
МАГИСТЕРСКАЯ ДИССЕРТАЦИЯ
Оптические свойства легированных
углеродных нанотрубок
Магистерская образовательная программа: Фотоника и квантовые материалы
Студент: Шатько Дарья
подпись, ФИО
Научный руководитель: Профессор, д.т.н. Насибулин Альберт
подпись, ФИО, должность
Со-руководитель: Старший научный сотрудник,
(при наличии) к.ф.-м.н. Копылова Дарья
подпись, ФИО, должность
Москва, 2019
Авторское право 2019. Все права защищены.
Автор настоящим дает Сколковскому институту науки и технологий
разрешение на воспроизводство и свободное распространение бумажных и электронных копий настоящей
диссертации в целом или частично на любом ныне существующем или созданном в будущем носителе.
Optical properties of electrochemically doped single-walled carbon nanotubes
Daria Satco
Submitted to the Skolkovo Institute of Science and Technology
on 31 May 2019
ABSTRACT
For many years, single-walled carbon nanotubes (SWCNTs) have been an important
platform to study optical properties of one-dimensional (1D) materials, especially due to their
geometry-dependent optical absorption and also due to their potential applications for
optoelectronic devices. Several recent studies suggested convenient approaches to tune SWCNT
thin films transparency and switch their color. One of the methods is electrochemical doping which
allows to tune SWCNTs properties smoothly. The absorption spectrum of SWCNTs in the UVVis-NIR range of light associated with electron transitions between van Hove singularities
strongly depends on Fermi level of tubes. A number of experimental studies have shown the
appearance of new absorption peak in heavily doped SWCNT spectra. Previously, the new
absorption peak in highly doped carbon nanotubes was assigned to a single-particle intersubband
transition within conduction (valence) band. Recently, theory of intersubband plasmons in carbon
nanotubes has been formulated pointing out its diameter and doping level dependence.
Up to now, far too little attention has been paid to the systematic theoretical and
experimental study of intersubband plasmon dispersion in doped SWCNTs. The aim of this
research was to explore the origin of the new peak and to compare theoretical predictions with the
experiment.
The theoretical model was formulated to prove the intersubband plasmonic origin of the
peak and to express its dispersion as a function of the tube diameter and the Fermi energy. It was
demonstrated that the intersubband plasmons are excited due to the absorption of light with linear
polarization perpendicular to the nanotube axis. At each doping level it is possible to identify
dominant intersubband contribution 𝑃"# to the plasmon peak. The calculated plasmon frequency
𝜔% scales with the SWCNT diameter 𝑑' and the Fermi energy 𝐸) as 𝜔% ∝ 𝐸)+.-. / 𝑑'+.1 , which is a
direct consequence of collective intersubband excitations of electrons in the doped SWCNTs. It
was also shown that more than one branch of intersubband plasmons occurs even in one nanotube
chirality. Our mapping of intersubband plasmon frequency may serve as a guide for
experimentalists to search intersubband plasmons in many different SWCNTs.
We designed the experimental setup to dope electrochemically SWCNT thin films. This is
the first time when a systematic measurement of plasmons is conducted for different tube
diameters and film thicknesses. The effectiveness of electrochemical doping was proven both by
optical and electrical measurements. We reached the Fermi level shift up to 1.3 eV with voltages
lower than 3 V. Experimentally measured plasmon demonstrated kink structure as it was expected
from theory. The film structure, such as bundling of tubes and presence of catalytic particles, shift
plasmon frequency and influence the general trend. By changing the tubes diameter and film
structural properties, the frequency of intersubband plasmon can be varied from 0.8 to 1.5 eV,
which is essential for further technical applications.
Research Advisor:
Name: Albert Nasibulin
Degree: Dr. Sc., Ph.D
Title: Professor
Co-advisor:
Name: Daria Kopylova
Degree: Ph.D
Title: Senior Research Scientist
Acknowledgements
I would like to thank my supervisor, Professor Albert Nasibulin, whose expertise and permanent support were driving the project. He was the one who accepted my initiatives and always
came with an advice in the moments of doubt. The whole experimental part of the project was
implemented at the Laboratory of Nanomaterials at Skoltech, led by Albert Nasibulin and would
not be successful without his wise management and inspiring attitude.
Many thanks to all the lab members, who were always open to any discussions and ready to
help with experiment. I am grateful to my co-supervisor, Dr. Daria Kopylova, who introduced me
to the technical part of the project, guided me at the beginning through the each and every step
of experimental work. I highly appreciate fruitful discussions of electrochemical effects with Dr.
Fedor Fedorov, Prof. Tanja Kalio and Prof. Galina Tsirlina. These people opened the door of electrochemical measurements for me. I would like to thank Anton Bubis for his expertise and support
of electrical measurements. Many thanks to Dr. Dmitry Krasnikov and Eldar Khabushev for the
fast and superior synthesis of carbon nanotubes films. I am also thankful to Alexei Tsapenko, who
helped with optical measurements and chemical doping of SWCNT films. I would like to thank
Dr. Iuriy Gladush for fruitful discussions on optical topics and his valuable comments upon the
thesis text.
I acknowledge Skotech for supporting my three months stay at Tohoku University in Japan.
Many thanks to Professor Richiiro Saito and his group for hosting me and leading the theoretical
part of the research. Numerical calculations implemented within the project were based on the
approach of carbon nanotube band structure calculation developed by Saito’s group members. I
am grateful to A.R.T. Nugraha and M. Shoufie Ukhtary for their coaching, teaching and continuous
support during my stay at Japan.
In addition, many thanks to Skoltech professors for sharing their knowledge and experience
with me. This research is inspired by the people and the competing atmosphere. I am grateful to
my colleagues who surrounded me during my Master study, motivating me to be better and more
proficient.
4
Thesis publications
• D. Satco, A. R. T. Nugraha, M. S. Ukhtary, D. Kopylova, A. G. Nasibulin, and R. Saito, “Intersubband plasmon excitations in doped carbon nanotubes”, Physical Review B, 99, 075403.
• D. Satco, D. Kopylova, F. Fedorov, T. Kalio, A. G. Nasibulin, “Intersubband plasmon observation in electrochemically doped carbon nanotube films”(under preparation)
5
Contents
1
Introduction
2
Methods
2.1
2.2
3
10
Computational methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1
Carbon nanotube band structure . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2
Optical matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.3
Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.4
Depolarization effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.5
Dielectric function for SWCNT . . . . . . . . . . . . . . . . . . . . . . . 21
Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1
Sample fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2
Optical measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.3
Optical spectra fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Results and discussions
3.1
3.2
4
7
27
Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1
Absorption of doped SWCNT . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.2
Intersubband plasmon excitation . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.3
Mapping of intersubband plasmon . . . . . . . . . . . . . . . . . . . . . . 34
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.1
Optical absorption of doped SWCNT . . . . . . . . . . . . . . . . . . . . 36
3.2.2
TEM analysis of SWCNT films . . . . . . . . . . . . . . . . . . . . . . . 41
Conclusions
44
6
1
Introduction
For many years, single-walled carbon nanotubes (SWCNTs) have been an important platform
to study optical properties of one-dimensional (1D) materials, especially due to their geometrydependent optical absorption [1–4] and also due to their potential applications for optoelectronic
devices [5–8]. Of the wide interests in the optical properties of SWCNTs, a particular problem
of the doping effects on the absorption of linearly polarized light is worth investigating. So far,
previous studies have confirmed that undoped SWCNTs absorb only light with polarization parallel
to the nanotube axis [9–13], so that when the light polarization is perpendicular to the nanotube axis
the undoped SWCNTs do not show any absorption peak due to the depolarization effect [11, 14].
The optical absorption in the case of parallel polarization can be understood in terms of the Eii
interband excitations from the ith valence to the ith conduction energy subbands, either in singleparticle [2, 9, 11] or excitonic picture [15–18]. On the other hand, much uncertainty still exists
about what happens in the case of doped SWCNTs for the linearly polarized light.
Intersubband plasmons were predicted to exist in semiconductor superlattices with doped quantum wells around the early 1980’s [19–21]. Ripe theory of intersubband plasmons in quantum
wells and its applications for infrared photodetectors was widely discussed in 1990’s [22, 23].
Recently, Delteil et.al [24] proved both theoretically and experimentally that intersubband plasmons in highly doped quantum wells are a consequence of the charge induced coherence, when
single-particle intersubband transitions become transparent and collapse into a resonant collective excitation. The frequency of the plasmonic resonance is different from all the intersubband
transitions, which contribute to it, and easily tunable by doping.
Single-walled carbon nanotubes (SWCNTs) were shown to be a perfect candidate for many
optical and electronic applications including those, where quantum wells were previously used
[25, 26]. Several recent studies suggested convenient approaches to tune SWCNT thin films transparency [27] and switch its color [28]. A large and growing body of literature has investigated
doping induced changes in optical absorption and conductivity of SWCNTs [29–34]. Both chemical [29,30,32] doping and electrochemical gating [31,34] provided with an effective tool of charge
injection to decrease sheet resistance parallel to the increase of film transparency. Thus, SWCNT
thin films were shown to be prospective material for many optoelectronic devices due to the number of various methods to tune their performance. While the increase in film transparency due to
disappearance of characteristical transitions was nicely explained by the Pauli blockade principle,
an unexpected new peak was usually observed at high doping at NIR wavelengths.
For many years the analogy between quantum well and SWCNT intersubband plasmons was
7
neglected. Prior to the experimental study of Igarashi et al. [35], the new absorption peak in highly
doped carbon nanotubes was usually assigned to a single-particle intersubband transition whithin
conduction (valence) band. In 2016, Sasaki et al. [36] emphasized the inability of a single-particle
excitation and suggested the Drude-based theory of intersubband plasmons in carbon nanotubes
pointing out its diameter and doping level dependence. A significant analysis and discussion on
the subject was presented by Yanagi et al. [37], who clearly stated the polarization dependence
as well as doping induced shift of plasmon. Recent advances in numerical study of intersubband
plasmons in chiral SWCNTs [38] have opened new horizons in plasmonic applications of SWCNT
thin films.
Another kind of plasmon positioned around 6 eV is usually observed in electron energy loss
spectroscopy (EELS) experiments [39, 40], the so-called π plasmon, which is not excited by light
with perpendicular polarization but with parallel polarization [39, 40]. Observations of the πplasmons in SWCNTs [41, 42] or any graphitic materials [43–45], either doped or undoped, are
quite common in the earlier EELS experiments and the peaks are assigned unambiguously. Lin
and Shung in two decades ago theoretically explained the origin of π plasmons in the SWCNTs
as a result of collective interband excitations of the π-band electrons [46, 47]. On the other hand,
the theory for plasmons excited in doped SWCNTs with perpendicularly polarized light is just
available recently by Sasaki et al. [48, 49] and Garcia de Abajo [50], in which they discussed how
the plasmon frequency (ω p ) in a doped SWCNT depends on its diameter (dt ) and Fermi energy
(EF ). However, the dependence of ω p on dt and EF was analyzed within the Drude model, which
is not relevant to intersubband transitions but it deals with intrasubband transitions. In this sense,
there is a necessity to properly describe the intersubband plasmons in the doped SWCNTs for any
SWCNT structure or chirality.
Up to now, far too little attention has been paid to the systematic theoretical and experimental
study of intersubband plasmon dispersion in doped SWCNTs. The project consists of two substantial parts.
Theoretical part of this research is devoted to the calculation of plasmon frequencies for the
doped SWCNTs as a function of diameter and the Fermi energy, considering all SWCNTs with
different chiralities in the range of 0.5 < dt < 2 nm. We further consider optical absorption at the
plasmon frequencies caused by intersubband transitions within the conduction and valence bands,
corresponding to EF > 0 and EF < 0, respectively. We find that the most dominant plasmonic
transition, which we label as Pi j at a certain energy Ei j (following the notation introduced by
Bondarev [51] for the interband plasmon at Eii ), changes with Fermi energy from a Pi j to another
Pi0 j0 . For the smaller (larger) nanotube diameter, we need higher (lower) EF to excite the plasmon.
8
In the experimental part we present optical measurement of intersubband plasmons in electrochemically doped SWCNT thin films with different tube diameters. The objectives of this part are
to determine how plasmon frequency changes with gating potential and whether such factors as
film thickness or structure make the result completely different from the numerical predictions for
an isolated SWCNT.
The rest of this thesis is organized as follows. In Sec. 2, we describe how to calculate the
plasmon frequency for a given SWCNT and how to measure its frequency experimentally. Two
subsections are devoted to numerical (Sec. 2.1) and experimental (Sec. 2.2) approaches. Within
Sec. 2.1 we define the plasmon starting from the definition of the dielectric function and energy
band structure for chiral SWCNT. The complex dielectric function in this work is calculated within
the self-consistent-field approach by considering dipole approximation for optical matrix elements,
from which there exist selection rules for different light polarization. Sec. 3 is devoted to the results and discussion. In Sec. 3.1, we discuss the main results of intersubband plasmon frequencies,
including the opportunity to map them into a unified picture of ω p ∝ (EF0.25 /dt0.7 ). We justify the
fitting by means of graphene plasmon dispersion, considering the model of the rolled graphene
sheet for a SWCNT. In Sec. 3.2, we show the optical measurements of intersubband plasmon in
SWCNT thin films with different tubes diameters. We compare the calculated and measured dispersions and discuss the factors which might influence the difference. Finally, we give conclusions
in Sec. 4.
9
2
Methods
2.1
2.1.1
Computational methods
Carbon nanotube band structure
SWCNTs belong to the group of carbon nanomaterials. Similarly to graphene they consist
of honeycombs with carbon atoms in vertexes and can be imagined as a rolled graphene sheet.
While in graphene all carbon atoms posses sp2 hybridization, carbon nanotubes are characterized
by sp2+δ hybridization due to their curved surface. Moreover, their crystal structure slightly differs
from the perfect graphene cylinder and usually is modeled within density functional theory (DFT)
by total energy minimization. However, DFT calculations are time-consuming and difficult to
generalize. In many applications simpler models give affordable accuracy with nicely explained
effects. Hereafter, we will discuss rolled-graphene model.
The structure of a nanotube is specified by the chiral vector Ch (circumferential direction) and
the translational vector T (nanotube axis direction) (see Figure 1). Ch and T can be expressed by
the real space graphene lattice unit vectors a1 and a2 , given by [52]:
!
!
√
√
3a a
3a a
a1 =
,
; a2 =
,−
2 2
2
2
where |a1 | = |a2 | = a =
(1)
√
3aC−C = 2.49 Å and aC−C is the distance between carbon atoms in
SWCNT. Then the SWCNT lattice vectors are expressed as follows:
Figure 1: The unrolled honeycomb real-space lattice of a nanotube. Pink solid lines restrict the
range of possible chiral angles θ .
10
Ch = na1 + ma2 ;
T = t1 a1 + t2 a2
(2)
2n+m
where n, m are integers and 0 ≤ m ≤ n; t1 = 2m+n
dR , t2 = − dR are as well integers; dR is the greatest
common divisor of (2m + n) and (2n + m). The chiral vector Ch in the basis of two-dimensional
(2D) lattice vectors of graphene a1 , a2 uniquely identifies the SWCNT structure by (n, m), also
known as the chirality.
While Ch and T define the unit cell of (n, m) carbon nanotube in real space, K1 and K2 are the
corresponding reciprocal lattice vectors. Reciprocal space unit vectors are defined by the relation
ai · b j = 2πδi j with δi j = 1 for i = j and 0 overwise. b1 , b2 are given as follows:
2π 2π
2π
2π
b1 = √ ,
; b2 = √ , −
a
3a a
3a
(3)
Then Ch , T and K1 , K2 satisfy the following relations:
Ch · K1 = 2π;
Ch · K2 = 0;
T · K1 = 0
(4)
T · K2 = 2π
(5)
Thus, K1 , K2 are expressed as:
K1 =
1
(−t2 b1 + t1 b2 ),
N
|K1 | =
where T =
2
,
dt
|K2 | =
2(n2 +m2 +nm)
dR
1
(mb1 − nb2 )
N
(6)
2π
,
T
(7)
√
is the length of translational vector T and L = a n2 + m2 + nm is the length of
√
3L
dR
chiral vector Ch . Above we also introduced dt =
N=
K2 =
√
a n2 +m2 +nm
,
π
which is the nanotube diameter and
, which is the number of hexagons in the nanotube unit cell.
The SWCNT first Brillouin zone is given by the line segment with the direction of K2 and the
length of 2π/T . NK1 corresponds to a reciprocal lattice vector of graphite, which means that 2
wave vectors which differ by NK1 are equivalent, whereas µK1 (µ = 0, ..., N − 1) are N discrete
inequivalent k vectors. These N discrete values of k vector result in N one-dimensional energy
bands. Let’s define the wave vector k = (k1 , k2 ) as following:
k=k
Each line of
2π
T
K2
+ µK1 ,
|K2 |
(µ = 1, ..., N and −
π
π
≤k≤ )
T
T
(8)
length in K1 direction is called cutting line and labeled with µ taking values from
1 to N (Figure 2 (a)).
11
Figure 2: (a) Cutting lines of the (10, 5) SWCNT on hexagonal two-dimendional (2D) Brillouin
zone, where N = 70. We also show a closer look at cutting lines around K and K 0 points for (b)
(10, 5) semiconducting SWCNT and (c) (6, 3) metallic SWCNT. In (b) and (c), two approaches
are demonstrated to label the energy bands: with the cutting line index µ (upper) and with optical
transition index i (lower). Intersubband transitions for perpendicular polarization are shown by
arrows.
Hereafter we will discuss the third nearest-neighbor approach for the band structure calculation.
We also assume that only 2pz atomic orbitals which form π molecular orbitals are contributive. In
extended tight-binding model σ orbitals formed by 2s, 2px and 2py orbitals are also included
[53]. However, σ orbitals are almost non-contributive to the optical and transport properties, since
they participate in covalent bonds and are responsible for the formation of crystal lattice. On the
contrary, π orbitals are orthogonal to the carbon-atom plane and uncoupled from σ orbitals. The
energy of π electrons is close to the Fermi level and thus make them most relevant for transport
effects characterization.
The Schrödinger equation for a SWCNT is given by:
H(r)ψks (r) = Eks ψks (r),
(9)
where H(r) is the real-space Hamiltonian, k is the electron wave vector, and s = c (s = v) denotes
the conduction (valence) band. The wave function ψks (r) can be expanded by a linear combination
of the Bloch functions φk` (r) as follows:
ψks (r) =
∑
C`s (k)φk` (r),
(10)
`=A,B
where C`s (k) is the coefficient for the state k. The Bloch function is expressed by
1
φk` (r) = √ ∑ eik·R( j) χ(R( j) − r` − r),
N j
12
(11)
where χ(r) denotes the 2pz atomic orbital, R( j) = j1 a1 + j2 a2 gives the position of the jth unit
cell (with a1 and a2 unit vectors of hexagonal unit cell given by Eq. 1), r` is the position of `th
atom (A or B) in the jth unit cell, and N is the number of unit cells. Substituting Eq. (10) into
Eq. (9) we obtain:
1
√ ∑ C`s (k) ∑ eik·R( j) H(r)χ(R( j) − r` − r)
N `=A,B
j
1
= Eks √ ∑ C`s (k) ∑ eik·R( j) χ(R( j) − r` − r).
N `=A,B
j
(12)
One can rewrite Eq. (12) in a matrix form multiplying χ(R(0) − r`0 − r) to Eq. (12):
∑
C`s (k)Hk`0 ` =
`=A,B
∑
Eks C`s (k)Sk`0 ` ,
(13)
`=A,B
where Hk`0 ` and Sk`0 ` are 2 × 2 Hamiltonian and overlap matrices respectively, defined by:
Hk`0 ` = ∑ eik·(R( j)) H`0 ` ( j),
(14)
Sk`0 ` = ∑ eik·(R( j)) S`0 ` ( j),
(15)
drχ(R(0) − r`0 − r)Hχ(R( j) − r` − r),
(16)
drχ(R(0) − r`0 − r)χ(Rl ( j) − r` − r).
(17)
j
j
and
Z
H`0 ` ( j) =
Z
S`0 ` ( j) =
H`0 ` ( j), S`0 ` ( j) are considered up to the third nearest neighbor sites. The hamiltonian matrix H`0 ` ( j)
has the energies of the corresponding orbitals (π orbitals in our approach) staying at the diagonal
(H`,` = επ ) and transfer integrals (H`0 ,` = t`0 ,` ) as non-diagonal elements. Orbital energies επ ,
transfer integrals t`0 ,` and overlap integrals S`0 ` are adopted from DFT calculations [54]. Inclusion
of the three nearest neighbors gives high accuracy in terms of band structure, because Hk , Sk matrix
elements vanish at longer ranges. Thus, having hamiltonian and overlap matrices, we come to the
generalized problem for eigenvectors and eigenvalues of the form:
Hk Csk = Eks Sk Csk ,
(18)
where Eks = {Ekv , Ekc } gives the energy of valence and conduction subbands for particular SWCNT
and vector Csk = (CAs (k),CBs (k))T gives the coefficients for the wave function represented by
Eq. (10). Within the zone-folding approach Eq. (18) is solved for Hamiltonian of 2D graphene.
We use the wave vector notation of Eq. (8) to the single electron wave function in carbon nanotube
as bra-ket style as |s, µ, ki ≡ ψks (r).
13
Hence, to obtain the energy band structure of carbon nanotubes, we adopt the zone-folding
approximation of graphene with long-range atomic interactions up to the third nearest-neighbor
transfer integrals, or the so-called third nearest-neighbor tight-binding (3rd NNTB) model [55,
56]. Although this approach does not include the curvature effect, the resulting band structure is
sufficiently accurate for SWCNTs with diameter larger than 1 nm [57]. Note that in contrast to the
simplest tight-binding approach, the subbands within the valence and conduction bands in the 3rd
NNTB model are not further symmetric with respect to E = 0. Therefore, the SWCNTs properties
are more sensitive to the doping type (n-type or p-type) as usually observed in experiments.
2.1.2
Optical matrix elements
We consider a SWCNT subjected to perturbation by light whose vector potential, electric field,
and magnetic field are denoted by A, E, and B, respectively. The vector potential of the electric
field of incident light at the position of r and time t is given by:
A(r,t) = A0 n cos(q · r − ωt),
(19)
where A0 , ω, q, and n denote the vector potential amplitude, angular frequency, wave vector in the
direction of propagation, and unit vector of polarization direction, respectively. The magnetic and
electric fields are related with A by E(r,t) = −dA/dt and B(r,t) = ∇ × A, respectively.
The single-particle Hamiltonian in the presence of external electromagnetic field is given by:
H(r,t) = H(r) +
ih̄e
A(r,t) · ∇,
m
(20)
where e > 0 is elemental charge and m is the mass of electron. The optical matrix element is given
by hs2 , µ2 , k2 | ih̄e
m Aq · ∇|s1 , µ1 , k1 i, where Aq is Fourier component of the vector potential A(r,t).
For the light propagating parallel to the nanotube axis (n k T) (Figure 3 (a)), Aq in the jth unit
cell can be expressed as [58]:
||
iq·R( j)
Aq (R( j)) = A0 n|| e
dt
= A0 n|| 1 + iq sin θ j .
2
In the case of perpendicularly polarized light (n ⊥ T) (Figure 3 (b)), Aq is expressed as:
A0
dt
⊥
iq·R( j)
iθ j
−iθ j
Aq (R( j)) = A0 cos θ j n⊥ e
= n⊥ (e + e
) 1 + iq sin θ j ,
2
2
(21)
(22)
where we take the direction of n as n|| = (0, 0, 1) and n⊥ = (1, 0, 0). We also take into account
the fact that qdt is sufficiently small compared with the unity, which means that in both cases
the dominant contribution to matrix element comes from the first term, whereas the second term
including q = |q| can be neglected, which is known as the dipole approximation. Hereafter we will
14
Figure 3: Projections of probe photon wave vector q, electric field E and magnetic field B onto
nanotube surface and cross section for (a) parallel polarization and (b) perpendicular polarization.
Nanotube is considered as a hollow cylinder.
consider only the dominant terms. The optical matrix element corresponding to a transition from
an initial state (s1 , µ1 ) to a final state (s2 , µ2 ) in tight-binding approximation of Eq. (10) has the
following form:
hs2 , µ2 , k2 |Aq · ∇|s1 , µ1 , k1 i =
0
1
Cks22 ∗µ2 `0 Cks11 µ1 ` ∑ e−ik2 ·R( j ) eik1 R( j)
∑
N `,`0 =A,B
j, j0
×h j0 , `0 |Aq (R( j)) · ∇| j, `i,
(23)
where | j, `i = χ(R( j) − r` − r) is the bra-ket form for the atomic orbital introduced in Eq. (11).
For the sake of conciseness we’ll adopt the following notation for the dipole matrix element:
Ms11s2 2 (k1 , k2 ) = hs2 , µ2 , k|n · ∇|s1 , µ1 , ki. Let us now discuss Eq. (23) for the two cases of parµ µ
allel and perpendicular polarization one by one.
15
Perpendicular polarization
The matrix element for perpendicularly polarized light takes the following form:
hs2 , k2 , µ2 |Aq · ∇|s1 , k1 , µ1 i =
0
A0
Cks22 ∗µ2 `0 Cks11 µ1 ` ∑ eik1 ·R( j) e−ik2 ·R( j ) ×
∑
N `,`0 =A,B
j, j0
1
A0
0 0 −iθ j
0 0 iθ j
Cks22 ∗µ2 `0 Cks11 µ1 ` ∑ h j0 , l 0 |∇| j, li · n⊥ ×
h j , l |e
∇| j, li · n⊥ + h j , l |e ∇| j, li · n⊥ =
2
N `,`0∑
=A,B
j , j0
0
A0
1 i(k1 ·R( j)−k2 ·R( j0 )−θ j )
e
+ ei(k1 ·R( j)−k2 ·R( j )+θ j ) =
Cks22 ∗µ2 `0 Cks11 µ1 ` ∑ h j0 , l 0 |∇| j, li · n⊥ ×
∑
2
N `,`0 =A,B
j, j0
0
1 i((k1 −k2 )·R( j)−θ j )
i((k1 −k2 )·R( j)+θ j )
e
+e
e−ik2 ·(R( j )−R( j)) =
2
A0
1 i((k1 −k2 )·R( j)−θ j )
s2 ∗
s1
i((k1 −k2 )·R( j)+θ j )
Ck2 µ2 `0 Ck1 µ1 ` ∑
e
+e
×
=
N `,`0∑
2
j
=A,B
0
∑0 n ⊥ · h j0, l 0|∇| j, lie−ik2·(R( j )−R( j))
j
(24)
Here we define two-dimensional unit vectors originated from carbon nanotube lattice vectors Ch , T
[58]:
Ch
K1
=
,
|Ch | |K1 |
T
K2
eT =
=
.
|T| |K2 |
eC =
(25)
Then vectors k1 , k2 , and R( j) can be presented in the following form:
k1 = µ1 |K1 |eC + k1 eT ,
k2 = µ2 |K1 |eC + k2 eT ,
R( j) =
θj
eC + Rz ( j)eT .
|K1 |
(26)
Using Eq. (26) we simplify the phase in Eq. (24):
(k1 − k2 ) · R( j) ± θ j = (k1 − k2 )Rz ( j) + (µ1 − µ2 ± 1)θ j
16
(27)
Taking the summation on j in Eq. (24) we get δ (k2 − k1 ) and δ (µ2 − µ1 ± 1). Finally the optical
matrix elment takes the following form:
A0
Cks22 ∗µ2 `0 Cks11 µ1 ` ∑ ei(k1 −k2 )Rz ( j) ×
∑
N `,`0 =A,B
j
0
1 i(µ1 −µ2 −1)θ j
e
+ ei(µ1 −µ2 +1)θ j ∑ n⊥ · h j0 , l 0 |∇| j, lie−ik2 ·(R( j )−R( j)) =
2
j0
A0
s2 ∗
s1
i(k1 −k2 )Rz ( j) 1
i(µ1 −µ2 −1)θ j
i(µ1 −µ2 +1)θ j
Ck2 µ2 `0 Ck1 µ1 ` ∑ e
e
+e
×
=
N `,`0∑
2
j
=A,B
hs2 , k2 , µ2 |A(q) · ∇|s1 , k1 , µ1 i =
0
∑0 n⊥ · h j0, l 0|∇|00, lie−ik2·R( j ) =
j
1
s2 ∗
s1
= A0 ∑ Ck2 µ2 `0 Ck1 µ1 ` δ (k1 − k2 ) δ (µ1 − µ2 − 1) + δ (µ1 − µ2 + 1) ×
2
`,`0 =A,B
∑ n ⊥ · h j, l 0|∇|00, lie−ik2·R( j)
(28)
j
Parallel polarization
Applying the same approach to the parallel polarized light matrix element takes, we’ll get the
following:
hs2 , k2 , µ2 |Aq · ∇|s1 , k1 , µ1 i =
=
=
=
0
A0
Cks22 ∗µ2 `0 Cks11 µ1 ` ∑ h j0 , l 0 |∇| j, li · n|| · ei(k1 ·R( j)−k2 ·R( j )) =
∑
N `,`0 =A,B
j, j0
0
A0
Cks22 ∗µ2 `0 Cks11 µ1 ` ∑ h j0 , l 0 |∇| j, li · n|| ei(k1 −k2 )·R( j) e−ik2 ·(R( j )−R( j)) =
∑
N `,`0 =A,B
j , j0
0
A0
Cks22 ∗µ2 `0 Cks11 µ1 ` ∑ ei((k1 −k2 )·R( j)) ∑ n|| · h j0 , l 0 |∇| j, lie−ik2 ·(R( j )−R( j)) =
∑
N `,`0 =A,B
j
j0
= A0
∑
0
`,` =A,B
2.1.3
0
A0
Cks22 ∗µ2 `0 Cks11 µ1 ` ∑ eik1 ·R( j) e−ik2 ·R( j ) h j0 , l 0 |∇| j, lin|| =
∑
N `,`0 =A,B
j, j0
Cks22 ∗µ2 `0 Cks11 µ1 ` · δ (k1 − k2 )δ (µ1 − µ2 ) ∑ n|| · h j, l 0 |∇|00, lie−ik2 ·R( j) (29)
j
Selection rules
According to the Fermi’s golden rule, transition probability is proportional to the square absolute value of dipole matrix element: Pi→ j ∼ |Mi j |2 . Thus, the transition will never be observed if
the matrix element has zero value. The conditions under which matrix elements get non-zero values are called selection rules. The concept of optical selection rules for SWCNTs was originally
discussed by Ajiki and Ando [9], who formulated the optical matrix elements by current-density
operator. They proved that the allowed transitions are always vertical (k1 = k2 ) and the cutting line
index should be conserved for parallel polarization (µ1 = µ2 ). On the other hand, the optical tran17
sition for perpendicular polarization occurs within nearest neighbor cutting lines, µ2 = µ1 ± 1. In
previous sections we already discussed the derivation of optical matrix elements within the dipole
approximation. Let us summarize the result. For parallel polarization, the optical matrix elements
are
Ms11s2 2 (k1 , k2 ) =
µ µ
∑
0
`,` =A,B
Cks22 ∗µ2 `0 Cks11 µ1 ` δ (k1 − k2 )δ (µ1 − µ2 ) ∑ n|| · h j, `0 |∇|0, `ie−ik2 ·R( j) ,
(30)
j
and for perpendicular polarization we obtain
µ µ
Ms11s2 2 (k1 , k2 ) =
∑
`,`0 =A,B
1
Cks22 ∗µ2 `0 Cks11 µ1 ` δ (k1 − k2 )
2
δ (µ1 − µ2 − 1) + δ (µ1 − µ2 + 1) ×
× ∑ n⊥ · h j, `0 |∇|0, `ie−ik2 ·R( j) .
(31)
j
It should be noted that the results of optical selection rules are the same either by considering
dipole approximation or current-density operator [9].
2.1.4
Depolarization effect
In both optical spectroscopy and EELS, plasmons are observed as prominent peaks in the spectra. These peaks clearly appear in numerically simulated optical absorption, if it is considered to
be proportional to Re(σ /ε) [22], as well as in EELS, which is proportional to the energy lossfunction, Im(−1/ε) [59, 60]. Here σ and ε are, respectively, optical conductivity and dielectric
function as functions of light frequency ω. We will show further that the dielectric function in the
optical absorption accounts for the depolarization effect (a common expression for the absorption
without depolarization is Re(σ )), which means that the screening of the external electrical field is
included in the calculation of optical absorption for both perpendicular and parallel polarizations of
light. Indeed, the depolarization effect is essential for explaining the anisotropy of optical absorption in SWCNTs [14,61,62]. Even though we expect the intersubband single-particle excitations in
doped carbon nanotubes for perpendicular polarization (orange peaks at Figure 4 (a), red arrows at
Figure 4 (b)), they are suppressed and never observed in experiment. The plasmon peaks originate
from zero points of the real part of ε(ω), i.e., Re[ε(ω)] = 0, followed by a relatively small value
of its imaginary part, Im[ε(ω)], in comparison with the maximum of Im[ε(ω)]. Indeed, as we see
from Figure 4 (a) the peaks for A(ω) = Re σ (ω)/ε(ω) and EELS = Im(−1/ε) appear at the
same energy, whereas single-particle excitations are completely transparent, as it was discussed
in [24].
In order to express the absorption of light let us discuss the effective field within the carbon
18
Figure 4: (a) Calculated absorption with (Re σ (ω)/ε(ω) ) and without (Reσ (ω)) depolarization
effect for (16,2) SWCNT. Yellow line corresponds to EELS spectra. (b) Energy bands and DOS
for (16,2) SWCNT. Colored lines correspond to the bands Ei comming from the closest cutting
lines to the K (K’) points. Solid (dotted) lines correspond to K (K’) point. Red arrows correspond
to transitions allowed by selection rules for perpendicularly polarized light.
nanotube. We consider external electrical field as shown at Figure 3:
E(x, y, z,t) = E0 (x, y, z)ei(qy−iωt) .
(32)
As previously, two cases for parallel (E|| ) and perpendicular (E⊥ ) polarization are discussed separately. Let us start with perpendicularly polarized electrical field. In a cylindrical coordinate
system the field is expressed as:

cos θ





E⊥ (r, θ , z,t) = E0 eiqr sin θ −iωt − sin θ  .


0
(33)
imθ , where J (qr) are Bessel’s functions of the
Using the expansion of eiqr sin θ = ∑∞
m
m=−∞ Jm (qr)e
first kind, and fixing r = rt = dt /2 at the tube surface, we come to the following form of the
perpendicular electrical field:
E0
E0 l
Ex (θ , ω) = ∑
Jl+1 (qrt ) + Jl−1 (qrt ) eilθ −iωt = ∑
Jl (qrt )eilθ −iωt = ∑ Exl eilθ −iωt ,
2
r
t
l
l
l
E0
Ey (θ , ω) = ∑
Jl+1 (qrt ) − Jl−1 (qrt ) eilθ −iωt = ∑ Eyl eilθ −iωt ,
(34)
2i
l
l
where Ex,y (θ , ω) are the Fourier series of perpendicular electrical field E⊥ (x, y,t), θ = 2πx/L and
L is the length of chiral vector. For perpendicularly polarized field the current density and charge
19
density can be also expanded into Fourier series:
ρ = ∑ ρ l eilθ −iωt ,
(35)
l
j = ∑ jl eilθ −iωt .
(36)
l
Current density j and charge distribution ρ are related through the continuity equation:
∂ρ
+ ∇ · j = 0.
∂t
(37)
Then putting Eqs. 35, 36 into Eq. 37 we can express Fourier components ρ l through the current
components jxl :
2πl l
j.
Lω x
ρl =
(38)
Induced charge oscillations give rise to the polarization oscillation in circumference direction:
Px = ∑l Pxl eilθ −iωt , where Pxl =
ilρ l
2εs |l|
[14]. Then, for the displacement vector components Dlx we
will get:
l
Dlx = Exl + 4πPxl = εxx
(ω)Exl ,
(39)
where εxx (ω) is the xx component of dielectric permittivity tensor. Next, we’ll adopt the differential
l (ω)E l to obtain the connection between components of dielectric
form of Coulomb law jxl = σxx
x
function and conductivity:
l
εxx
(ω) = 1 +
4π 2 i|l| l
σ (ω).
εs Lω xx
(40)
The l-th component of optical absorption is expressed as:
1 1
A (ω) =
2 2π
l
Z2π
dθ Re[ jxl Exl∗ ] =
0
1 1
2 2π
Z2π
0
l
σxx (ω) l2
Dlx
1
l
l∗
dθ Re σxx (ω) l
E = Re l
D . (41)
2
εxx (ω) x
εxx (ω) x
According to Eq. 41 absorption is proportional to the ratio σ (ω)/ε(ω). As we have seen above,
the presence of ε(ω) in denominator is the direct consequence of electron screening of the external
field. Thus, the effective field felt by electrons is different from the applied field E(r,t). All the
derivations above were applied for perpendicularly polarized field E⊥ , however, all the steps can
be repeated for parallel polarization E|| and the final result will be the similar.
Let us also notice, that similarly to the section 2.1.2 we assume qrt in Eq. 34 being negligibly
small and thus keep only l = ±1 for E⊥ . Total absorption for perpendicular polarization is:
1
−1 (ω) (ω) σxx
σ (ω) 2
1 σxx
⊥
1
−1
+ −1
D2 = ⊥
D ,
A (ω) = A (ω) + A (ω) =
1
2 εxx (ω) εxx (ω)
ε⊥ (ω)
20
(42)
where σ 1 = σ −1 = σ⊥ and ε 1 = ε −1 = ε⊥ in the absence of magnetic flux trough the SWCNT [38].
Following the same approach, we will get expression for the absorption of parallel polarized light:
A|| (ω) =
σ|| (ω) 2
D ,
ε|| (ω)
(43)
where σ|| = σ 0 and ε|| = ε 0 . Here we adopt l = 0, since Ezl ∝ Jl (qrt ). Let us notice, that the values
of l contributive to the total absorption are in a perfect agreement with the selection rules obtained
in section 2.1.3.
2.1.5
Dielectric function for SWCNT
Following the result obtained in the previous section 2.1.4, we use the relation between dielectric permittivity ε(ω) and the optical conductivity σ (ω) as follows:
ε(ω) = 1 + i
4πσ (ω)
,
ωdt εs
(44)
where εs is surrounding dielectric permittivity (εs = 2 for SWCNT film [35]). We calculate ε(ω)
within the self-consistent-field approach in the following form [46, 63]:
2
Z
f [Es1 ,µ1 (k)] − f [Es2 ,µ2 (k)]
dk
8πe2 h̄2
2
µ µ
Ms11s2 2 (k)
ε(ω) = ε0 +
∑
h̄ωAt m
2π
Es2 ,µ2 (k) − Es1 ,µ1 (k) − h̄ω + iΓ
s1 ,s2
µ1 ,µ2BZ
1
×
,
Es2 ,µ2 (k) − Es1 ,µ1 (k)
(45)
where Ms11s2 2 (k) is the optical matrix element, At = π(dt /2)2 is the cross section area of a SWCNT,
µ µ
and f [E(k)] is Fermi-Dirac distribution function (we perform al the calculations at room temperature T = 300 K). The electron wave function |s, µ, ki is related with the subband energy Es,µ (k),
where s = c (s = v) for a conduction (valence) subband and µ is the index for the cutting line, k is
the electron wave vector.
In Eq. (45), Γ is the broadening factor that accounts for relaxation processes in optical transitions resulting in finite lifetime τ of the electron state. Here we simply assume that Γ does not
depend on ω or EF but is constant, Γ = 50 meV [64]. The numerical integration over k is implemented by the left Riemann sums approximation, where the step dk is chosen to reach an accuracy
∆ε/ε = ±0.01, corresponding to dk = Γ/(5h̄vF ), where vF = 106 m/s is the Fermi velocity in
graphene.
Both dielectric function and optical conductivity are obtained by taking summation of different contributions from all possible pairs of (s1 , µ1 ) and (s2 , µ2 ). Although the summation in
Eq. (45) is performed over all the cutting lines in valence and conduction bands, only limited number of subbands gives nonzero contribution. The (s1 , µ1 ) → (s2 , µ2 ) transition is contributive when
21
µ ,µ
Ms11,s2 2 (k) is nonzero (optical selection rules) and the Pauli exclusion principle is satisfied (the
difference of Fermi-Dirac distributions in Eq. (45) is nonzero).
When we discuss the plasma oscillations in the electron gas, all charges are considered equivalent and contributing to the collective motion. However, it is not the case for SWCNTs, in which
the electronic states consist of N subbands in both valence and conduction bands. The calculated plasmonic excitations in nanotubes show that the plasmon peak is dominated by a particular (s1 , µ1 ) → (s2 , µ2 ) transition. With this regard, and also for clarity in presenting our results,
let us introduce a more convenient notation for the plasmonic transition that can be used generally for all (n, m) SWCNTs. Here our target is to assign one-to-one correspondence between the
(s1 , µ1 ) → (s2 , µ2 ) transition and the intersubband transition energy Ei j , similar to the notation
adopted for the interband optical transitions Eii [2, 65]. The case of s1 6= s2 is the interband transition, while the case of s1 = s2 (with µ1 6= µ2 ) is the intersubband transition. The condition of
s1 = s2 means that we consider the intersubband transition within the conduction (or valence) band.
Therefore, instead of using the cutting line index µ, which strongly depends on the SWCNT structure, we will label the cutting line by integers i starting from the cutting line closest to the K point
as shown in Figures 2 (b) and 2 (c) for (10, 5) semiconducting and (6, 3) metallic SWCNTs, respectively. It is possible to analytically obtain the new cutting line indices (optical transition indices)
around the K and K 0 points [66]. Then, the transitions can be enumerated according to the distance
of the corresponding cutting line from the K or K 0 points [Figure 2 (b)], such as E12 , E13 , E24 , E35
and E46 for a semiconducting SWCNT. In the case of metallic SWCNT [Figure 2 (c)], by excluding the trigonal warping effect [2], we can obtain transitions such as E01 , E12 , E23 , and so on, either
going to the right or left direction away from the K (or K 0 ) point.
2.2
2.2.1
Experimental methods
Sample fabrication
Electrochemical gating is performed using a three electrode cell. The electrochemical cell is
a sandwich-like system: three electrodes covered by electrolyte are placed between two quartz
plates (see Figure 5). Working electrode (W) is made from SWCNT thin film fabricated by aerosol
CVD based on ferrocene thermal decomposition using ferrocene as a catalyst precursor [67]. The
film is first collected on membrane nitrocellulose filter (Millipore HAWP) and then placed on
quartz plate by a dry transfer technique. SWCNT films consist of chaotically placed nanotubes,
which are characterized by certain diameter and chirality distribution and usually form bundles.
Reference (Ref) and counter (C) electrodes are preliminary deposited onto quartz plate by vacuum
22
Figure 5:
The scheme of electrochemical cell. Working electrode (SWCNT film), pseudo-
reference electrode (silver coating), counter electrode (golden coating) are deposited onto quartz
plate and covered with ionic liquid.
evaporation technique. We use silver film as a pseudo-reference electrode and golden film as a
counter electrode. The contacts are made with conductive copper tape. We use ionic liquid (IL) as
an electrolyte due to its large window of stability [68,69] (within this study imidazolium-based ILs
were used). Since ionic liquids are highly sensitive to water, all the samples are prepared within
the nitrogen inert atmosphere inside the glove-box. The ionic liquid is put on the top of electrodes
and then the sample is covered with the second quartz plate. We encapsulate it with epoxy resin
and then take it out the glove-box to measure spectra.
It is highly important to choose the right amount of ionic liquid, which would be enough to fully
charge the electrical double layer. We used 35 µl of ionic liquid, which is argued as following. The
SWCNT electrode can be imagined as a network of billions of single carbon nanotubes. Thus the
effective area of electrode is quite huge and in fact consists of surfaces of different tubes, which are
separately covered by ionic liquid ions. The effective area of SWCNT film is described by specific
surface area SSA = 1000 m2 /g (the BET surface area analysis was performed with Micromeritics
Gemini 2375). The mass of the film can be calculated if the transparency is given, because for
T = 92% we know m92% = 3.1 µg/cm2 [70]. The films used for the sample fabrication are usually
from 70% to 95% transparent. Let’s consider 70% transparent film. Then the mass is:
m70% = m92%
log T70%
= 13.26 µg/cm2 .
log T92%
(46)
Next we can estimate how large is the real surface area of a SWCNT electrode per unit geometrical
area:
2
2
A70%
e f f = SSA · m70% = 132.6 cm (real)/cm (geom).
23
(47)
Full charge per surface area is given by:
Q = CSWCNT · ∆U = 20 µF/cm2 · 4V = 80 µC/cm2 ,
(48)
where CSWCNT = 20 µF/cm2 is the average capacitance of carbon nanotubes [31] and ∆U is the
potential window usually chosen in our experiment. The amount of electrolyte which is enough to
recharge the real electrode area is:
νreal =
Q
= 8 · 10−10 mol/cm2 ,
F
(49)
where F = 105 C/mol is Faraday constant. The amount of electrolyte which is enought to fully
recharge geometrical electrode area is:
−7
νgeom = νreal · A70%
mol/cm2 (geom).
e f f ≈ 10
(50)
In 1 µl of ionic liquid (DEME BF4 ) is 43·10−7 mol, which is already enough for a 1 cm2 electrode.
In our experiment we use 35 µl of ionic liquid.
2.2.2
Optical measurements
Optical absorbance of unpolarized light is measured with UV-Vis-NIR spectrometer Perkin
Elmer Lambda 1050. Electrochemical doping is controlled with Elins potentiostat-galvanostat P40X with potentials applied in the range [-3..4] V. We apply a constant potential for 50 s prior to
acquiring the spectra until the steady-state current is observed. The experiment is conducted at
room temperature. Transmission electron microscopy (TEM) was performed with FEI Tecnai G2
F20 microscope. SWCNT films for the TEM were deposited directly onto the golden TEM grids.
2.2.3
Optical spectra fitting
To obtain plasmon dispersion from optical absorption spectra one needs to find plasmon peak
position. Optical absorption of SWCNTs is a superposition of single-particle transition peaks
(E11 , E22 , E33 , ... for semiconducting tubes and M11 , M22 , ... for metallic tubes), which strongly
depend on tube diameter and chirality, π-plasmon and background absorption, which comes from
amorphous carbon, catalyst particles, defects and bundles [71, 72]. Following the approaches,
discussed elsewhere [72, 73], we fit background with Ae−bλ , where λ is the light wavelength, A,
b are parameters to fit, and A is known to be sensitive to the density of carbon nanotubes and
film thickness [72]. π-plasmon is fitted with a Lorentzian [74] L(x; A, µ, Γ) =
Γ
A
π (x−µ)2 +Γ2 ,
where
x is photon energy in eV, A, µ and Γ are parameters to fit with the following meanings: A is the
24
magnitude of the peak, µ is its center and Γ is half width at half maximum. Finally, to account
for the contributions from transition peaks, we add Lorentzian contours and then fit the original
spectra by the following combination:
n
F(h̄ω) = Lπ (h̄ω) + Ae−bλ (ω) + ∑ Li (h̄ω)
(51)
i=1
where the number n of additional Lorentzians Li is different for different samples, but usually it
varies from 3 up to 6 (peaks coming from E11 , E22 , M11 , ...). Figure 6 (a) demonstrates an example
of spectra fitting with Eq. 51. E11 , E22 , M11 and E33 excitations are fitted with broad Lorentzian
contours, which is primarily caused by broad diameter distribution within the film (in this case, for
example, diameters vary from 1.3 to 1.5 nm with mean value dt = 1.36 nm). These contours make
a rough approximation of many Eii excitations present in spectra and in general we need them fitted
at this step in order to differentiate Eii contributions from π-plasmon and the background absorption. When the information about π-plasmon and remaining background is obtained, we remove
it from the original spectra for further more accurate analysis of single-particle and plasmonic
excitations.
Figure 6: (a) Optical absorbance of 1.4 nm SWCNT film with T = 90% fitted by Eq. 51 . (b)
Characteristical peaks remained after π-plasmon and background subtraction. Gaussian mixture
fitting done to define chiralities in the sample.
Next step is to obtain the information about possible chiralities present in the samples. For
many applications it is enough to know the mean diameter of the sample. However, in order to be
able to compare plasmon dispersion with theoretical predictions, we intend to know the fractions
of different chiralities. Using the Gaussian Mixture model for the normalized absorption, we fit
25
the remaining peaks with linear combination of Gaussians:
k
−
1
e
GM(h̄ω) = ∑ p j √
2πσ j
j=1
(h̄ω−h̄Ω j )2
2σ 2j
(52)
where p j > 0 is the weight of jth Gaussian, so that ∑kj=1 p j = 1, and h̄Ω j is the position of the
center of the peak and σ j is Gaussian RMS width. The number of Gaussians is additionally varied
to obtain the best fitting score. The expectation-maximization (EM) algorithm is used to fit the
parameters of Gaussian Mixture [75].
After the background-free spectra is fitted with the combination of Gaussians (see Figure 6 (b)),
we still need to find the corresponding chiralities. The information from the Figure 6 (a) is used to
distinguish regions of different Eii excitations. We assume the intersection point of two Lorentzians
as a separating point for Eii and Ei+1,i+1 transitions. Thus, each jth Gaussian is assigned to the
i
ith region: {p j , h̄Ω j , σ j }kj=1 → {pij , h̄Ωij , σ ij }n,k
i, j=1 . We refer to Kataura plot data [76] and look for
chiralities, which have excitation energies close to the fitted peaks (∆Eii < σ ij /2) in all Eii regions.
That means, if we take certain (n, m) from Kataura table and find the corresponding peak in E11
region, then we also check if the corresponding peaks are present in E22 and E33 regions. If the
condition is satisfied, we suppose that this chirality is present in our sample, otherwise we ignore it.
Finally, having Gaussians assigned to the chiralities, we assume that the fraction of the particular
chirality is f(n,m) = p1j , where p1j is the weight of the Gaussian. Within this approach each Gaussian
can be assigned to several chiralities.
In order to calculate plasmon dispersion of a film we should calculate its absorption. Here we
model the film absorption as total absorption of a mixture of nanotubes with different chiralities,
which are present in the film. Taking the chirality distribution from the previous step we calculate
the weighted sum of absorptions:
Asum =
∑
f(n,m) A(n,m)
(53)
(n,m)
where Asum is the absorption of SWCNT mixture, A(n,m) is the absorption of the (n, m) tube present
in the mixture with fraction f(n,m) . Having total absorption corresponding to the mixture of tubes
modeled from experimental data, we look for intersubband plasmon following the approach described in [77].
26
3
Results and discussions
3.1
3.1.1
Numerical results
Absorption of doped SWCNT
Let us firstly discuss the absorption spectra of doped SWCNT for a particular (n, m). In this
section we will discuss (10,5) chiral SWCNT with the diameter d = 1.03 nm and the chiral angle
θ = 19.1o . This is just an example to discuss, in general, the same logic can be applied to any other
chiral SWCNT. In Figure 7 (a) we plot Re(σ /ε) of the (10, 5) SWCNT as a function of photon
energy h̄ω for parallel and perpendicular polarization. Several spectra are plotted for different
Fermi energies EF from −2.5 to 2.5 eV. For |EF | < 0.5 eV, since the first energy subband of
conduction (valence) band is not occupied, we can observe interband transitions of all Eii ’s with
i ∈ {1, 2, 3} for the transitions between the valence and conduction bands. When we increase |EF |
more than 0.5 eV, the Eii peaks start to disappear from E11 to E33 because the ith subband in the
conduction (valence) band begins to be occupied (unoccupied) for i = 1, 2, and so on. The position
of Eii peaks (circles, triangles and diamonds for E11 , E22 and E33 , respectively) is redshifted by
increasing doping and then blueshifted before disappearing. The redshift of Eii occurs because
of the depolarization correction, which decreases with doping, whereas the blueshift attests the
parabolic shape of the subbands. The depolarization correction can be seen as the inclusion of
Coulomb interaction between electrons in the calculation of optical absorption [Re(σ /ε)], since
ε(ω) = 1 + ivq σ (ω)q2 /(e2 ω), where vq = 2πe2 /q is the Coulomb potential and q = 2/dt . Hence
the dielectric function can be expressed as in Eq. (44). If Coulomb interaction is neglected, the
position of Eii absorption peaks is constant by doping, not redshifted. Although we do not include
the excitonic effect for simplicity, the presence of redshift in the Eii peaks in our calculation is
consistent with the previous work by Sasaki and Tokura [49]. It should be noted that, by the
exclusion of excitonic effect, for dt = 2 nm, the deviation of the peak positions (defined as maxima
of Re[σ (ω)]) is still less than 10% in comparison with the exciton Kataura plot [18].
While the ith subband is being occupied with electrons (or holes), the value of Eii increases
because the single-particle excitations occur only for the restricted k-regions, which are far from
kii [2, 11, 66], where the interband energy distance is larger. When the subband is partially occupied, a new peak for perpendicular polarization appears. We expect that such a peak is related with
intersubband plasmon excitations for several reasons: (1) Re(ε) has a zero point close to the peak
position, (2) the peak position is different from the single-particle intersubband i → j transition,
(3) the peak intensity strongly depends on Fermi energy and continuously increases even when the
27
Figure 7: (a) Doping-induced evolution of optical absorption spectra in a (10,5) SWCNT. Solid
(dotted) lines represent perpendicular (parallel) polarization of light. Circles, triangles, and diamonds are a guide for eyes to trace the E11 , E22 , and E33 transition peaks in the case of parallel
polarization. The absorption peaks in the case of parallel polarization are not due to intersubband
plasmons, while the peaks in the case of perpendicular polarization are caused by intersubband
plasmons, as discussed in the main text. (b) Plasmon frequency as a function of Fermi energy for
the (10, 5) SWCNT. The radius of the circles corresponds to the intensity of the Pi j peak. Note that
the weak peaks for 0.6 < EF < 1.1 eV are not plasmonic, but related to E13 absorption. (c) Density
of states (solid line) and charge density (points) for (10, 5) SWCNT as a function of energy. Dotted
vertical lines indicate the positions of kinks for plasmon frequency. (d) Energy band structure for
the (10, 5) SWCNT. Colored bold lines correspond to the subbands coming from the cutting lines
nearest to the K point. Thin solid lines correspond to the subbands from the other cutting lines in
the presented energy range.
subbands are almost occupied and part of transitions is blocked, and (4) the blueshift with increasing the Fermi energy is opposite to the redshift for the single-particle excitation [35]. For highly
positive doping EF > 1.9 eV, the second smaller peak is observed around 1.4 eV as shown in Fig28
ure 7 (a). This peak is another type of plasmon, which differs from the first one at 1.5 − 1.8 eV by
the dominant contributions (see the more detailed discussion in section 3.1.2). Hereafter, we focus
our attention to the first, main plasmon peak, since this one should easily be observed in experiments. The Fermi-energy dependent optical absorption shown in Figure 7 (a) is consistent with
that previously discussed by Sasaki and Tokura [49] for the armchair (10, 10) and zigzag (16, 0)
SWCNTs. However, the present result shows additional plasmon peaks and different doping-type
dependence (for EF > 0 and EF < 0), which appears by introducing more accurate energy band
calculation.
3.1.2
Intersubband plasmon excitation
In Figure 7 (b) we plot the absorption peak position in the case of perpendicular polarization
for the (10, 5) SWCNT as a function of EF . The intensity of each peak is represented by the
circle diameter. We attribute the peak as the plasmon peak and denote its frequency as ω p when
Re[ε(ω0 )] = 0 and ω0 is close (≤ 20 meV) to ω p . Each point in Figure 7 (b) consists of several
circles which correspond to different contributions from the transition of the cutting line pair i → j
measured from the K point. We denote the dominant i → j contribution as Pi j , where the threshold
for dominant contribution was chosen as 10% of maximum contribution for each peak. Here we
omit the valence and conduction band indices (s1 , s2 ) since the dominant transition is the intersubband transition, s1 = s2 . One can clearly observe the kink shape of the function, as well as the
existence of the second plasmon branch at lower frequencies for EF > 2 eV.
In Figure 7 (c) we display the density of states (DOS) and charge density as a function of
Fermi energy for the (10, 5) nanotube. The charge density for electrons at EF > 0 is given by
ρ(EF ) =
R∞
0
D(E) f (E)dE, where D(E) is the DOS. For holes at EF < 0 we modify the charge
density formula by replacing the distribution function f (E) with 1 − f (E). In Figure 7 (d) we
show energy dispersion Es,µ (k), where the energy subbands are labeled according to the approach
discussed in Sec. 2.1.5. The kink positions for the plasmon energy and the charge density ρ(EF )
are shown to be consistent to each other [see grey dotted lines in Figure 7 (c)]. In the threedimensional (3D) Drude model, the plasmon frequency is known to be proportional to the square
√
root of charge density (ω p3D ∝ ρ ). For carbon nanotubes, the Fermi energy dependence was
√
predicted to be consistent with 2D graphene result (ω p2D ∝ E F ) [50]. However, we see from
Figure 7 (b) and 7 (c) that the plasmon frequency is a function of ρ(EF ), which in case of carbon
√
nanotubes is the sum ∑Eii <EF EF − Eii .
The kink in ρ(EF ) appears when EF passes through the next van Hove singularity (Ei ) as
29
shown in Figure 7 (c), which is followed by the Pauli blockade of the ith subband and change
in the dominant contribution to the plasmon from Pi j to another Pi0 j0 , where i0 > i and j0 > j for
EF > 0 (i0 < i and j0 < j for EF < 0). As seen from Figure 7 (b), the first dominant contribution is
P13 (P31 ), the second dominant contribution after the first kink is P24 (P42 ), the third contribution
after the second kink is P35 (P53 ) for EF > 0 (EF < 0). The plasmon intensity [radius of circle in
Figure 7 (b)] increases with increasing the Fermi energy and inceasing ρ(EF ).
The asymmetry of plasmon peak intensity with respect to the n-type and p-type doping is consistent with asymmetric nature of ρ(EF ) for EF > 0 and EF < 0. The minimum plasmon frequency
as well as the Fermi energy at which the plasmon is excited basically depend on the energy band
structure. For example, in Figure 7 (b), the asymmetry in the values of E13 within valence and
conduction bands influences the starting plasmon frequency (h̄ω p = 1.52 eV for the valence band
and h̄ω p = 1.54 eV for the conduction band). Meanwhile, the number of subbands under or above
the Fermi level within the valence or conduction band is essential for accumulating negative contribution to dielectric function in order to observe Re(ε) = 0. Therefore, the interplay between
the intersubband transitions determines the asymmetric nature of the plasmon peak intensity in
the n-type and p-type doping. Note that at EF = 0 eV both real and imaginary parts of ε(ω) are
positive in the energy range of 0 − 4 eV. In the case of p-doped (10,5) SWCNT, the plasmon starts
to appear at EF = 0.6 eV, after the 1st subband becomes partially unoccupied, in which the condition of Re(ε) = 0 is already satisfied. In the case of n-doping, the first small peak appears at
EF = 1.1 eV. However, since Re(ε) 6= 0, this peak is still not a plasmon, but is a single-particle
intersubband transition 1 → 3. It is observed when the 1st subband is partially occupied and when
the depolarization effect, which was completely suppressing absorption before, is relaxed. The
true plasmon peak appears at EF = 1.1 eV, which corresponds to the 2nd subband partially occupied. Thus, the condition to observe the plasmon in SWCNT for perpendicularly polarized light is
to shift the Fermi level up higher than the bottom of the 2nd subband in conduction band [36, 78],
or down lower than the top of the 1st subband in valence band.
In Figure 8(a), we plot intersubband and interband absorption spectra in case of perpendicular
polarization for (10, 5) SWCNT and EF = 1.5 eV. We define the absorption associated with the
ij
ij
i → j transition as Ai j = Re(σ⊥ /ε⊥ ), where σ⊥ is
ij
σ⊥
16 e2
=
dt h
h̄2
m
2 π/T
Z
−π/T
f (Ei (k)) − f (E j (k))
dk
1
|Mi⊥j (k)|2
·
.
2π
E j (k) − Ei (k) − h̄ω + iΓ E j (k) − Ei (k)
(54)
For EF > 0, when we consider the interband transitions, the ith and the jth subbands come from
the valence and conduction band respectively. On the other hand, for the intersubband transitions,
30
Figure 8: (a), (c) Absorption spectra for a doped (10, 5) SWCNT with EF = 1.5 eV and EF = 2.5
eV respectively. Black bold solid line represents the total absorption Atot , considering both the
intersubband and interband transitions. Colored solid lines correspond to the dominant intersubband contributions. The EELS spectrum is plotted with red dash-dotted line. Colored dashed lines
lines correspond to the interband absorptions with the same transition indices as the intersubband
counterparts. Inset depicts the enlarged region for the interband peaks, which are about one orderof-magnitude smaller than the intersubband peaks. (b), (d) Real (ε1 ) and imaginary (ε2 ) parts of
dielectric function along with conductivity (σ1 and σ2 ) for (10, 5) doped SWCNT with EF = 1.5
eV and EF = 2.5 eV. Solid (dotted) vertical line corresponds to Re(ε) = 0 [max(Atot )].
both subbands lie within the conduction band. The total absorption Atot in Figure 8(a) is contributed
from all the interband and intersubband transitions. We see that the peak position and line shape of
the absorption spectrum are consistent with those of EELS spectrum, which is given by Im(−1/ε).
As we already mentioned above, both optical conductivity and dielectric function are superpositions of contributions (σi j , εi j ) from different transitions between the i → j subbands. To
calculate absorption from the i → j transition Ai j , we take only the corresponding term from the
conductivity σi j , while the dielectric function (ε⊥ ) is calculated for all pairs of interband and intersubband transitions according to Eq. (45). As an example, in the case of EF = 1.5 eV in Figure 8(a)
two main contributions are P13 and P24 . In Figure 8(a), we see the peak value of Ai j for intersubband absorption (solid lines) is one order-of-magnitude larger than that for interband absorption
(dashed lines), which clearly shows that the plasmon has an intersubband nature. One may notice
31
that the same P13 and P24 transitions are dominant for both intersubband and interband absorptions.
However, the contributions have different signs and different order of magnitude.
Although the interband transitions seem to give negligible contribution to the plasmon intensity,
they affect the redshift of the zero point for the dielectric function [49], as shown in Figure 8(b).
In fact, the position of the maximum in absorption spectra (dotted vertical line) and the zero of
Re[ε(ω)] (solid vertical line) are slightly different (by ∼1 meV). This difference comes from
Im[ε(ω)], which decreases in the proximity of Re[ε(ω)] = 0, as well as Im[σ (ω)] [Figure 8(b)].
If the dielectric function is a real function of ω, the zero value would give the exact position of
plasmon, which is not the case for a complex ε(ω). Indeed, for ε = ε1 + iε2 and σ = σ1 + iσ2 , the
absorption and the energy loss-function have the following form:
1
1
i,
Im −
= h
ε
ε2 1 + (ε1 /ε2 )2
σ σ2 + σ1 εε21
i.
Re
= h
2
ε
ε 1 + (ε /ε )
2
1
(55)
(56)
2
The maxima of Im(−1/ε) and Re(σ /ε) appear close to the ε1 = 0, but not exactly at this point.
The shift of the maxima strongly depends on slope of ε2 (ω) near the zero point of ε1 .
Hereafter we will focus on dispersion only for major plasmons, which appear first and remain
dominant in terms of its magnitude. However, for EF > 2.0 eV there exist another plasmon at the
lower frequency as shown in Figure 7(c). In Figure 8 (c) we plot the absorption spectra Atot =
Re(σ⊥ /ε⊥ ), as well as EELS spectra by Im(−1/ε⊥ ) (dash-dotted line), as a function of photon
energy for the (10, 5) SWCNT at EF = 2.5 eV. We can see two prominent peaks at 1.86 eV (peak
1) and 1.4 eV (peak 2), which differ by the dominant contributions [Figure 8 (c)], i.e., P35 (from
A35 ) and P24 (from A24 ), respectively . In particular, for the peak 2, the absorption A35 , which is
dominant for the peak 1, gives the negative contribution. This leads to a different behavior of the
peak 2 as a function of EF .
In Figure 8 (d), we plot ε1 = Re(ε), ε2 = Im(ε), σ1 = Re(σ ), and σ2 = Im(σ ) as a function of
photon energy. The condition on plasmon excitation is satisfied at two zero points of the real part
of dielectric function (solid vertical line). The absorption maxima (dotted vertical line) are redshifted regarding to Re(ε) = 0, the shift is larger for peak 2, since ε2 is steeper around ω p2 . Here
we can clearly observe the effect of ε2 on plasmonic spectra: A1 /A2 ∝ ε2 (ω p2 )/ε2 (ω p1 ), where we
denote A1 and A2 as the intensities of plasmon peaks 1 and 2. The presence of the second branch
of intersubband plasmon have not been mentioned any of previous works of SWCNTs. However
in recent years, several ab initio studies show the similar second branch for bilayer graphene,
32
branch1
branch2
p
(eV)
2.5 EF =2.0 eV
2.0
1.5
1.0
0.5
1.0
1.5
Diameter (nm)
2.0
Figure 9: Two plasmon branches in doped SWCNTs. Blue circles correspond to the main branch
discussed, orange diamonds correspond to the second branch plasmon, which appear at higher
doping levels. The size of the marker corresponds to the peak intensity.
nanoribbons, and other 2D materials [79–82]. The intraband nature of the second branch plasmon
in graphene nanoribbons was supposed by Gomez et al. [80], which is consistent with our results.
We plot both plasmon branches for SWCNT in Figure 9 for different chiralities of SWCNT with
dt < 2 nm at EF = 2.0 eV. The lower plasmon peak P24 shows a larger chiral angle dependence
since it comes from the cutting lines pairs closer to the K point than the major plasmon P35 .
Thus the similar spreading character is observed for small-diameter SWCNTs (dt < 1 nm) and the
second branch plasmon for bigger SWCNTs (1 < dt < 2 nm).
33
3.1.3
Mapping of intersubband plasmon
In Figure 10, we plot energy of
intersubband plasmon h̄ω p as a func-
2.5
tion of nanotube diameter dt , where
2.0
0.5 < dt < 2 nm, for five Fermi en-
1.5
ergies from EF = 1 to 2 eV. For
1.0 EF =1.0 eV
2.5
EF = 1 eV plasmons are observed
only in tubes with dt > 1 nm. With
increasing EF , the number of tubes
2.0
which have plasmonic excitations in-
1.5
1.0 EF =1.25 eV
2.5
creases, since Eii < EF (Eii ∝ 1/dt )
smaller dt nanotubes. Plasmon en-
(eV)
ergies h̄ω p , as well as their spread-
p
is satisfied for a large EF even for
P01
P12
P13
P24
P35
P01
P12
P13
P24
P35
P46
P01
P12
P13
P23
P24
P35
P46
P87
P98
2.0
1.5
ing for fixed dt and EF , are increas-
1.0 EF =1.5 eV
2.5
ing with decreasing diameter. This indicates the presence of chirality de-
2.0
pendence, which was neglected in
P01
P12
P13
P23
P24
P35
P46
P57
P65
P68
P76
P78
P87
P98
P1011
P01
P12
P13
P21
P23
P24
P34
P35
P46
P57
P65
P68
P76
P79
P87
P911
P109
P1011
1.5
the previous works [48–50]. We see
smaller diameters and higher Fermi
1.0 EF =1.75 eV
2.5
energies come from the cutting line
2.0
pairs, which are close to the K point.
1.5
Therefore, the family spread due to
1.0 EF =2.0 eV
0.5
1.0
1.5
Diameter (nm)
that the dominant contributions for
the curvature effect is inherited by
plasmon frequency.
2.0
Hereafter, we
focus on the Fermi energy and diameter dependence of plasmon frequency, since this information is useful for most experimental studies like
the Kataura plot for optical absorption
[76, 84] or Raman spectroscopy [85].
Figure 10:
Intersubband plasmon frequencies (major
peak) for SWCNTs of all different chiralities (n, m) with
diameters from 0.5 to 2 nm. Five different Fermi energies
are considered. The dominant contributions are pointed
out for each plasmon by specific marker types and colors
[83].
Chirality dependence of plasmon en-
34
ergy is a challenging point for the present method, since the band structure calculation by adopting
the 3rd NNTB model is not satisfactory to build reliable chiral angle dependence or curvature
effect [86].
We numerically fit the diameter and the Fermi energy dependence with power law, as shown in
Figure 11 (a). The result is:
h̄ω p = (1.49 ± 0.004)
EF0.25±0.003
dt0.69±0.005
eV.
(57)
The dt (in nm) and EF (in eV) dependence in Figure 11 (a) can be understood from the dispersion of plasmon in graphene, which is shown in Figure 11 (b) [87]. The intersubband plasmons
in doped SWCNTs, which are nothing but the azimuthal plasmons [48], can be considered as the
plasmons in the rolled graphene sheet, where we have the oscillations of charge around the nanotube axis. Rolling of graphene into SWCNT results in the quantization of plasmon wave vector
(qp ) following the reciprocal lattice vector K1 [52] in the SWCNT since we consider the transitions of electron between different cutting lines. The magnitude of the reciprocal lattice vector is
inversely proportional to the diameter, i.e., |K1 | = 2/dt , similar to the wave vector of the electron
√
along the circumferential direction (q ∝ dt−1 ). From Figure 11 (b), we can see that the q p dependence does not always hold for plasmon in graphene. The plasmon dispersion becomes almost
linear to q p as it enters the interband single-particle excitation (SPEinter ) regime [87]. At the colored frequency range (1.75 EF < h̄ω p < 2.25 EF ) in Figure 11 (b) we fit the dispersion, where we
get ω p ∝ q0.6986
. Therefore, we expect ω p ∝ dt−0.7 for the plasmon frequency of SWCNT, which
p
confirms our finding in Eq. (57). It is noted that the ω p ∝ q0.7
p of graphene’s plasmon is at relatively higher frequency range compared with the obtained plasmon frequency range for SWCNTs
as shown in Figure 10. This is owing to the fact, that in SWCNT the lower limit of photon energy
for single particle excitation (the dash-dotted line in Figure 11 (b)) would be smaller compared
with the case of graphene due to the possible intersubband excitation of electron within the conduction band of SWCNT. This lowering of energy limit for starting single particle excitation by
intersubband transition (SPEinter ) shifts the “almost” linear dispersion of plasmon in graphene to
lower frequency range, too. Thus the fitting to “the almost linear dispersion” is justified.
The Fermi energy dependence of azimuthal plasmon in SWCNT given by Eq. (57) can be
also understood from the dispersion of plasmon in graphene shown in Figure 11 (b). Since the
dispersion of plasmon in graphene is normalized to the Fermi energy as shown in Figure 11 (b),
we can obtain the following equation:
α
qp
h̄ω p =
EF = (q p h̄vF )α EF1−α ,
kF
35
(58)
Figure 11: (a) Fitting of the intersubband plasmon energy as a function of nanotube diameter dt
and Fermi energy EF . We consider SWCNTs with 1 < dt < 2 nm and only the major plasmon peak.
(b) Fitting of the plasmon dispersion of graphene. We found ω p ∝ q0.6986 within the horizontally
dashed frequency range (1.75 EF < h̄ω < 2.25 EF ) that could be related with the intersubband
plasmon excitations in SWCNTs. The other colored dashed areas correspond to the regime where
the interband and intraband single-particle excitations occur in graphene, denoted by SPEinter and
SPEintra , respectively [87].
where we use linear energy band of graphene, EF = h̄vF kF . Since ω p ∝ q0.6986
, we expect the
p
Fermi energy dependence to be ω p ∝ EF0.3 , which is not exact but close to the obtained power law
in Eq. (57). The difference with the obtained power law comes from the fact that the electron
√
energy bands of SWCNTs are not exactly linear as in graphene. It is noted that if we have the q p
dependence of plasmon frequency in graphene, using Eq. (58), we will have ω p ∝ EF0.5 as expected
in the Drude model [48, 50, 87].
3.2
3.2.1
Experimental results
Optical absorption of doped SWCNT
All the spectra are measured for the SWCNT film immersed into ionic liquid. Thus, the original
spectra includes information both about the film and the electrolyte. We start processing data from
the subtraction of ionic liquid contribution (see Figure 12 (a)). Figure 12 (b) demonstrates the
common spectra obtained during film doping.
Figure 13 [a-e] shows spectral data measured for different SWCNT thin films during electrochemical doping. The evolution of optical absorption under doping is characterized by common
36
Figure 12:
(a) Experimental data preprocessing. Total absorption measured at U = 0 V (blue
dashed line) is the sum of SWCNT and IL absorptions. Then the IL spectra (green solid line) is
subtracted to obtain spectra from SWCNT (black line). (b) The absorbance of electrochemically
doped SWCNT film (d = 1.4 nm).
stages: 1) while absolute value of applied potential U is small, absorption remains unchanged; 2)
starting from some sufficiently high potentials (depends on tube diameter and electrode capacitance, usually at U > 0.4 V) E11 starts to disappear; 3) after E11 transition is blocked, E22 peak
goes down (U > 1 V); 4) intersubband plasmon appears when E22 is almost gone (U > 1.5 V); 5)
plasmon peak intensity increases while M11 transition is partially blocked (U > 2 V). As it was
previously discussed elsewhere [36, 37], plasmon peak appears at high doping levels, usually after
E22 transition is suppressed. In order to define the correct position of plasmon peak, we preprocess
spectra following the steps discussed in methods section 2.2.3. The peak can be detected during
both n- and p-type doping, however, in case of p-doping it appears earlier and more intensively in
terms of magnitude [77] (see Figure 14 (a)). Taking into account the advantage of p-doping for
plasmon detection, all the samples were studied under high positive voltages.
Changes observed in optical spectra give us the evidence of Fermi level shift. Resistance
measurement is an alternative way to prove the effectiveness of doping, the deeper the Fermi
level moves to the valence/conduction band, the lower is the resistance value (see Figure 14 (b)).
However, with higher gating voltage resistance reaches saturation value, which means that at these
voltages SWCNTs show metallic properties.
Figure 15 (a) demonstrates intersubband plasmon frequency as a function of applied potential.
All the samples are described in Table 1. Let us notice that the dispersion for tubes with different
37
Figure 13: Evolution of optical absorption of electrochemically doped SWCNT films: (a) 85%
film with d = 2 nm (E11 is preliminary suppressed by chemical doping with HAuCl4 ); (b) 90%
film with d = 1.4 nm; (c) 75% film with d = 1 nm; (d) 80% film with d = 1 nm; (e) 85% film with
d = 1.4 nm. Different colors of lines correspond to different values of applied potential.
diameters lays in different frequency ranges. For the samples with the same diameter the trend is
similar, but the range of potentials is different. Finally, the growing rate of the plasmon frequency
differs for various diameters. In order to explain Figure 15 (a), let us first discuss in more details
how electrochemical gating works in ionic liquids.
Room temperature ionic liquids (ILs) are salts with melting point lower than 20 ◦ C [88], which
means that they compose entirely of positive and negative ions. Electric double layer (EDL) is
formed on the interface between ionic liquid and electrode and can be modeled as a nanogap
Table 1: Experimental sample description. Each film is characterized by the mean SWCNT diameter d and the film transparency at 550 nm T550 . Two different ionic liquids (IL) were used for
the sample fabrication: DEME BF4 and 1-Butyl-3-Methylimidazolium PF6 . Total capacitance was
estimated from the cyclic voltamograms. A is the geometrical are of the working electrode.
Sample name
d (nm)
T550 (%)
IL
Ctot (µF)
A (cm2 )
2 nm deme
2.0 ± 0.2
85
DEME BF4
6.49
0.52
1 nm deme
1.0 ± 0.2
80
DEME BF4
186.63
0.80
1 nm bmim
1.0 ± 0.2
75
BMIM PF6
276.43
0.92
1.4 nm bmim [1]
1.4 ± 0.1
85
BMIM PF6
15.13
0.75
1.4 nm bmim [2]
1.4 ± 0.1
90
BMIM PF6
128.72
1.05
38
Figure 14:
(a) Optical absorption spectra with background subtracted for 1 nm SWCNT film.
With arrows we show the change of applied voltage and shift of Fermi level. The applied potential
was varied from -2.2 to 2.3 V with step = 0.2 V. Dots show the plasmon peak position. (b) Sheet
resistance for 1 nm SWCNT film as a function of applied potential.
parallel plate capacitor with CG = εεd0 [89], where ε is the relative permittivity, ε0 is the permittivity
of vacuum, and d is the thickness of Stern layer (d ∼ 3 Å [90]). Due to extremely small distance
d between capacitor plates, EDL can possess high capacitance. Thus, large electric field can be
reached at the interface and essentially modulate Fermi level of the electrode [91–93]:
∆U =
∆EF
+φ
e
(59)
where ∆U is the shift of potential on the working electrode, which creates potential difference φ =
∆Q/CG within EDL, and shifts the Fermi level ∆EF /e = ∆Q/CQ . Here ∆Q is the injected charge
and CQ is the quantum capacitance originally introduced by Luryi [94] to account for finite density
of states in low-dimensional systems. Thus, total capacitance of the system can be described as a
series circuit of two capacitors CG and CQ . The accumulated charge ∆Q at the interface increases
until the dielectric breakdown of the EDL or ionic liquid dissociation. The Fermi level shift can be
expressed as following [95]:
∆EF =
Ctot
∆U
CQ
(60)
where Ctot is the total capacitance calculated from cyclic voltammetry by integration as Ctot =
UR 2
I+ (u)−I− (u)du / 2(U2 −U1 )v . Here, we assume that U1 and U2 are the limits of the potential
U1
range, I+ /I− are forward/backward instantaneous currents, v is potential scan rate. Further we will
also discuss Ce f f = Ctot /Ae f f , which is the capacitance per unit active area (here Ae f f is the active
area of working electrode Eq. 47).
39
Figure 15:
(a) Plasmon peak position as a function of applied potential defined from optical
spectra. (b) Comparison of the measured and calculated dispersions. Note, that we convert the
potential into the Fermi energy empirically, no general expression is available for now.
At small potentials the EDL capacitance can be assumed to be constant and then the quantum
capacitance can be estimated directly from the experiment [96]. However, in our experiments we
reach high voltage values, where the following is true: 1) IL-EDL capacitance CG is not constant,
but strongly depends on the applied voltage [90, 97–99]; 2) the system is not fully charged due to
the presence of faradaic current (see Figure 16).
Let us now discuss the results from Figure 15 (a). According to the samples properties, given
in Table 1, samples with 1 nm tubes have similar effective capacitances (Ce f f ∼ 2.8 µF/cm2 ),
whereas samples with 1.4 nm tubes have one order difference in capacitances (0.3 µF/cm2 (for the
sample [1]) vs. 3.2 µF/cm2 (for the sample [2])). Thus, the difference in the capacitances explains
the fact, that for the same tube diameter we reach sufficient doping level at different potentials.
Considering the energies of Eii transitions, we would also expect that the smaller the diameter is,
the higher the doping level we need to detect the plasmon (since E11 ∼ 1/dt [52]). This is not the
case for the potentials applied, since there is no linear correspondence between ∆U and ∆EF , it is
difficult to predict minimum potential to excite the plasmon in a particular sample.
Figure 15 (b) includes both measured and calculated dispersions. Let us remind that the absorption of a film is calculated as a superposition of absorptions coming from different isolated chiral
nanotubes. We do not include film structure, such as tubes bundling or catalytic particles presence.
We expect that the magnitude of measured plasmon would be different with calculated one due to
the number of effects which are not accounted in calculation. The dispersion trend for 1 and 1.4
nm tubes is consistent with that theoretically obtained, even though excitonic corrections are not
40
Figure 16: Cyclic voltametry for the sample with 1.4 nm tubes with T=90%. Scan rate was set to
50 mV/s.
included in the model and the band structure is considered to be rigid and independent of doping.
Let us also notice that in contrast to the previous models, which predicted plasmon frequency being
√
proportional to EF [36, 50], indeed we observe a kink structure, which comes from the model
discussed above. Interestingly, the dispersion for tubes with d = 1.4 nm comes higher than that for
d = 1 nm (green and blue points respectively on Figure 15 (b)). This behavior contradicts with the
theoretical predictions, which state that for the smaller diameters the plasmon frequency should be
higher. In the next section we will discuss transmission electron microscopy (TEM) results for the
SWCNT films, which shed some light on the possible reasons for this discrepancy.
We do not have a general expression to recalculate the potential on WE into the Fermi energy,
however, by comparing the measured and calculated dispersions we can estimate the range of
Fermi energies we reach. Thus, within our electrochemical cell we are able to shift Fermi level
approximately up to 1.3 eV into the valence zone. This is an impressive result even comparing
with previous works [32].
3.2.2
TEM analysis of SWCNT films
CVD synthesized SWCNT films are characterized by a certain distribution of nanotubes lengths
and diameters, as well as by the diameters and lengths of bundles, which are formed by nanotubes
both at the gaseous stage and at the moment of the deposition on the substrate [100]. The mean
diameter of the tubes in the film is usually estimated from the optical absorption spectra, because
the band structure of nanotubes which gives rise to the characteristic peaks is only slightly modulated due to the bundling effect. The mean diameter and length of the bundles is estimated from
41
Figure 17: TEM images for SWCNT films with several mean tube diameters: (a) d = 2 nm; (b)
d = 1.4 nm; (c) d = 1 nm.
the TEM images.
Figure 17 [a-c] demonstrates the TEM images for the SWCNT films which were prepared at the
same synthesis conditions as those films discussed in Table 1 respectively. We can clearly observe
bundles of tubes, catalyst nanoparticles, amorphous carbon coating. According to the obtained
images, the amount of catalyst nanoparticles and its size is similar for all 3 films, that is why we do
not consider its presence as a factor which can essentially influence the consistence between the
theory and experiment. Since we study intersubband plasmons which are circumferential waves,
we are interested in the mean diameter of bundles in each case [a-c]. We have processed 15 images
from different spots per each film similar to those shown in Figure 17. That gave us a statistic
sample of approximately 200 values of bundle diameters per each film. The SWCNT bundles
are log-normally distributed with diameters similarly to the distribution of the size of particles
agglomerates formed during aerosol synthesis [101]. The geometric mean diameter of bundles is
8.7 nm, 6.2 nm and 5.3 nm for the films with 2 nm, 1.4 nm and 1 nm tubes respectively. However,
different size of bundles is first of all related to the tube diameter. That is why we consider the
number of tubes observed within a bundle nb , which given as: nb = db /dt , where db is the diameter
of the bundle measured from the TEM images. This is not a true number of nanotubes in a bundle,
since we see only one side of it on the picture and do not have the information about interior
structure. Nevertheless, nb correlates with an actual structure of a bundle and the number of tubes
in it.
Figure 18 [a-c] demonstrates the distributions of nb for the samples discussed above. Each
distribution is characterized by µg and σg , the geometric mean and geometric standard deviation,
which are analogs of mean and standard deviation for log-normally distributed data. Now we can
42
Figure 18:
Distribution of number of nanotubes in the bundles for certain SWCNT films: (a)
d = 2 nm; (b) d = 1.4 nm; (c) d = 1 nm. Histogram bars correspond to the number of bundles with
a particular number of tubes in it. Solid line is the log-normal distribution fitted to the data.
compare the bundling degree for each film. Let us notice, that the SWCNT film with d = 1 nm is
different from the samples with d = 2 nm and d = 1.4 nm. The mean number of tubes in bundle
for d = 1 nm is 5.9, whereas for other two it is about 4.4. Such a strong bundling effect might be
the reason for the plasmon frequency for 1 nm tubes showing lower energies than that for 1.4 nm.
More detailed EELS analysis of isolated bundles with different numbers of tubes would be helpful
to check this hypothesis. Meanwhile, we can also notice, that the amount of amorphous carbon
is slightly higher for the SWCNT film with d = 1 nm. Amorphous carbon on the tube surface
changes the effective electric field felt by SWCNT electrons. The actual structure of a bundle
is important both for the external field modulation as well as for the electrochemical processes.
Electrical double layer is formed at the interface of ionic liquid and tube surface when the number
of ions in contact with a tube is enough to reorganize the anions and cations distribution. When
tubes are tightly bundled, anions/cations motion is limited and the doping becomes less effective.
To sum up, TEM analysis of the SWCNT films demonstrated, that the samples under consideration differ not only by the mean tube diameter, but also by the mean diameter of a bundle and
the amount of amorphous carbon in the sample. For now, these are the main differences we can
observe from TEM images, further experimental studies would be preferable to develop the topic.
43
4
Conclusions
We have systematically studied intersubband plasmon excitations in doped SWCNTs both the-
oretically and experimentally. The theoretical model was formulated to express plasmon dispersion
as a function of the tube diameter and the Fermi energy. The intersubband plasmons are excited
due to the absorption of light with linear polarization perpendicular to the nanotube axis. At each
doping level it is possible to identify dominant itersubband contribution Pi j to the plasmon peak.
The calculated plasmon frequency ω p scales with the SWCNT diameter dt and the Fermi energy
EF as ω p ∝ (EF0.25 /dt0.7 ), which is a direct consequence of collective intersubband excitations of
electrons in the doped SWCNTs, but not a result of intraband transitions described by the Drude
model. We also show that more than one branch of intersubband plasmons occurs even in one
nanotube chirality. Our mapping of intersubband plasmon frequency may serve as a guide for
experimentalists to search intersubband plasmons in many different SWCNTs.
This is the first time when a systematic measurement of plasmons is conducted for different
tube diameters and film thicknesses. We designed the experimental setup to dope electrochemically
SWCNT thin films. The effectiveness of electrochemical doping was proven both by optical and
electrical measurements. We reached the Fermi level shift up to 1.3 eV with voltages lower than 3
V. Experimentally measured plasmon demonstrated kink structure as it was expected from theory.
The film structure, such as bundling degree of tubes and presence of amorphous carbon particles,
shift plasmon frequency and influence the general trend. By changing the tubes diameter and film
structural properties, the frequency of intersubband plasmon can be varied from 0.8 to 1.5 eV,
which is essential for further technical applications. In our experiments plasmon peak is broad
(half width at half maximum approx. 0.2 eV) due to the broad distribution of tube chiralities
within the samples, however, it can be decreased if a single chirality is obtained.
44
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