Chiral effects in normal and superconducting carbon nanotube-based nanostructures
The novel phenomenon of chiral tunneling in metallic single-wall carbon nanotubes is considered. It is induced by the interplay of electrostatic and pseudomagnetic effects in electron scattering in chiral nanotubes and is characterized by the oscillatory dependence of the electron transmission proba...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Cite this: | Chiral effects in normal and superconducting carbon nanotube-based nanostructures / A.V. Parafilo, I.V. Krive, E.N. Bogachek, U. Landman, R.I. Shekhter, M. Jonson // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1193–1203. — Бібліогр.: 55 назв. — англ. |
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| author | Parafilo, A.V. Krive, I.V. Bogachek, E.N. Landman, U. Shekhter, R.I. Jonson, M. |
| author_facet | Parafilo, A.V. Krive, I.V. Bogachek, E.N. Landman, U. Shekhter, R.I. Jonson, M. |
| citation_txt | Chiral effects in normal and superconducting carbon nanotube-based nanostructures / A.V. Parafilo, I.V. Krive, E.N. Bogachek, U. Landman, R.I. Shekhter, M. Jonson // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1193–1203. — Бібліогр.: 55 назв. — англ. |
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| description | The novel phenomenon of chiral tunneling in metallic single-wall carbon nanotubes is considered. It is induced by the interplay of electrostatic and pseudomagnetic effects in electron scattering in chiral nanotubes and is characterized by the oscillatory dependence of the electron transmission probability on nanotube chiral angle and the strength of the scattering potential. The appearance of a special (Aharonov–Bohm-like) phase in chiral tunneling affects various phase-coherent phenomena in nanostructures. We considered chiral effects in: (i) the persistent current in a circular nanotube, (ii) the Josephson current in a nanotube-based SNS junction, and (iii) resonant electron tunneling through a chiral nanotube-based quantum dot.
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© A.V. Parafilo, I.V. Krive, E.N. Bogachek, U. Landman, R.I. Shekhter, and M. Jonson, 2010
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11, p. 1193–1203
Chiral effects in normal and superconducting carbon
nanotube-based nanostructures
A.V. Parafilo1, I.V. Krive1,2,3, E.N. Bogachek4, U. Landman4,
R.I. Shekhter2, and M. Jonson2,5,6
1B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: krive@ilt.kharkov.ua
2Department of Physics, University of Gothenburg, Göteborg SE-412 96, Sweden
3Physical Department, V.N. Karazin National University, Kharkov 61077, Ukraine
4School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, USA
5School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK
6Division of Quantum Phases and Devices, School of Physics, Konkuk University, Seoul 143-701, Korea
Received May 6, 2010
The novel phenomenon of chiral tunneling in metallic single-wall carbon nanotubes is considered. It is in-
duced by the interplay of electrostatic and pseudomagnetic effects in electron scattering in chiral nanotubes and
is characterized by the oscillatory dependence of the electron transmission probability on nanotube chiral angle
and the strength of the scattering potential. The appearance of a special (Aharonov–Bohm-like) phase in chiral
tunneling affects various phase-coherent phenomena in nanostructures. We considered chiral effects in: (i) the
persistent current in a circular nanotube, (ii) the Josephson current in a nanotube-based SNS junction, and (iii)
resonant electron tunneling through a chiral nanotube-based quantum dot.
PACS: 73.23Ra Persistent currents;
72.10.–d Theory of electronic transport; scattering mechanisms;
74.50.+r Tunneling phenomena; Josephson effects.
Keywords: chiral tunneling, persistent currents, Josephson current, carbon nanotube.
1. Introduction
One of the most spectacular phenomena in physics is
the Aharonov–Bohm (AB) effect predicted in 1959 [1] and
a year later realized in experiment [2] (see also the review
[3] and references therein). This effect is of fundamental
nature, it has a simple theoretical formulation and numer-
ous theoretical and experimental applications. The last
assertion concerns first of all condensed matter physics
where the AB effect is a key idea in a vast amount of theo-
retical and experimental papers.
One of the first among the most significant papers on
the AB effect in condensed matter physics is the work by
I.O. Kulik on non-decaying electric currents in normal
metal systems published in 1970 [4]. In this paper it was
predicted that a perfect (impurity-free) small metallic cy-
linder threaded by magnetic field will support a non-
dissipative (persistent) electric current with the periodicity
of a single-flux quantum 0 = /hc eΦ (see also [5,6]) and
an amplitude (at low temperatures) given by the single-
electron current /Fev L∼ times the number of transverse
channels for a few-channel ring (here Fv is the Fermi ve-
locity and L is the ring circumference). At that time —
15 years before the advent of mesoscopic physics — the
prediction that certain physical characteristics of a real
many body (macroscopic) system (now it is better to say
— mesoscopic) could be sensitive to a single-electron con-
tribution sounded bizarre for many physicists. Although
the fundamental nature of Kulik's prediction was evident
(the paper was published in the most prestigeous physics
journal in the Soviet Union), prospects for the experimen-
tal observation of this effect looked obscure. Nevertheless,
rather soon the prediction was confirmed, at first in indi-
rect experiments [7,8] with massive cylinders where the
A.V. Parafilo, I.V. Krive, E.N. Bogachek, U. Landman, R.I. Shekhter, and M. Jonson
1194 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11
AB persistent currents were induced by electronic states
localized near the surface (whispering gallery states) and
forming effectively a doubly connected (ring) geometry
[9,10]. Later, in the beginning of the 1990's, persistent cur-
rents were measured in a single metallic (gold) ring [11]
(diffusive regime of electron transport [12]) and soon in a
quantum ring formed in 2D electron gas (EG) [13]. In
2DEG electron transport is ballistic and the measurements
[13] were in good agreement with Kulik's prediction [4].
Since that time persistent currents were always a hot to-
pic in condensed matter physics and there is a vast literature
on the problem (see e.g. [14] and the reviews [15–17]). Aha-
ronov–Bohm oscillations were observed not only in metal-
lic rings and cylinders, but in more exotic systems like,
e.g., conducting quasi-1D materials with charge density
wave excitations [18], where the AB effect is induced by
the quantum coherent dynamics of collective modes
[19,20]. Theoretical studies of nontraditional AB effects in
condensed matter range from the calculation of persistent
currents in dielectrics [21] to the study of AB oscillations
induced by superconducting plasmons [22].
Our purpose here is to consider persistent currents in
circular carbon nanotubes and supercurrents in nanotube-
based superconductor/normal metal/superconductor (SNS)
junctions. Both structures have been studied experiment-
tally. The ring-shaped nanotubes (including rings of single-
wall nanotubes) were observed and investigated in
Refs. 23, 24 while measurements of the Josephson current
in a single-wall nanotube-based SNS junction was reported
in Ref. 25.
What is specific in the transport properties of carbon
nanotube-based mesoscopic structures as compared to the
«ordinary» metallic nanowires? Electron transport in me-
tallic single-wall nanotubes (SWNT) is ballistic and this
property is explained by a specific scattering of charge
carriers by the nanotube defects (see, e.g., [26]). Conduc-
tion electrons in SWNT are Dirac-like particles and their
relativistic spectrum leads to certain peculiarities in elec-
tron scattering. In particular long-range electrostatic poten-
tials in metallic nanotubes do not scatter electrons at all.
This effect is explained by the conservation of helicity for
relativistic particles. In quantum field theory the phenome-
non of particle free penetration through potential barriers is
known as the Klein paradox (see the discussion in Ref. 27).
The specific features of electron scattering in chiral nano-
tubes and their influence on persistent and super-currents
in carbon nanotube-based devices is the goal of the present
paper.
In Sec. 2 we introduce the new concept of chiral tunne-
ling in metallic SWNTs. For a special 2 2× -matrix scatter-
ing potential the transmission and reflection amplitudes are
derived. It is shown that in the local limit the transmission
coefficient ( )D θ is an oscillating function of the chiral
phase 0= cosc Uϕ θ , where 0U is the dimensionless
strength of the scattering potential and θ is the nanotube
chiral angle. Resonant chiral tunneling, ( ) = 0rD θ , occurs
for quantized values of the chiral phase =c nϕ π (where n
is an integer).
In Sec. 3 we evaluate the persistent current in a circular
metallic SWNT in the presence of chiral tunneling. We
show that the chiral phase cϕ plays a crucial role in the
magnetic response of circular carbon nanotubes. In particu-
lar, the parity of the chiral resonance (even or odd n ) de-
termines the character of the magnetic response (paramag-
netic or diamagnetic persistent current). The existance of
non-equilibrium spontaneous persistent currents in an iso-
lated nanotube ring with asymmetric populations of the
Fk± -valleys is briefly discussed.
In the next section we consider the influence of chiral
effects on the supercurrent in a SWNT-based SNS junc-
tion. The equation for the bound state energies (Andreev–
Kulik levels) in the presence of chiral tunneling is derived.
It is shown that for energy independent phase factors (for-
ward and backward scattering phases and the chiral phase)
the spectral equation expressed in terms of scattering data
coincides with the corresponding equation for standard
SNS junction. All information specific to chiral tunneling
is hidden in the oscillatory dependence of the junction
transparency on the chiral angle and chiral phase. In par-
ticular we discuss here the interesting possibility to fabri-
cate highly transparent junction by using high quality car-
bon nanotubes with small chiral angles.
In Sec. 5 the resonant electron transport through a «chi-
ral» quantum dot (QD) (i.e., the QD based on a chiral me-
tallic SWNT) is considered. We show that in the presence
of a chiral scatterer inside the tube the spacings between
the resonant conductance peaks (measured by varying the
gate voltage) strongly depend on the nanotube chiral angle
and the chiral phase (which in principle can be considered
as a controllable parameter). The distribution of the num-
ber of conductance maxima on the level spacing ranges from
δ-function like peaks for armchair nanotubes (equidistant
spectrum of QD energy levels) to a smooth Wigner-Dyson-
like distribution (quasi-random energy spectrum) in chiral
nanotubes in the limit of weak chiral tunneling ( 1D ).
In the Conclusion we summarize the main results and
briefly discuss the influence of electron-electron interac-
tion on chiral tunneling.
2. Chiral tunneling
We evaluate the transmission probability for electron
scattering by special defects (see below) in carbon nano-
tubes. We will assume that the defect potentials are long-
ranged and do not induce inter-valley ( 2 Fk kδ ) electron
scattering. Thus in our model the metallic SWNT Hamilto-
nian is diagonal in the valley index =j ± and takes the
form [28]
ˆ0 exp ( )
= .
ˆexp ( ) 0
x
F
x
i p
H v
i p±
± θ⎛ ⎞
± ⎜ ⎟θ⎝ ⎠∓
(1)
Chiral effects in normal and superconducting carbon nanotube-based nanostructures
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1195
Here Fv is the Fermi velocity, ˆ = ,x xp i− ∂ θ is the chiral
angle of the nanotube ( 0 / 6≤ θ ≤ π ) and the x-axis is di-
rected along the cylinder axis. Notice that we follow [28]
in the definition of chiral angle ( = 0θ for the armchair
nanotube and = / 6θ π for the zigzag nanotube) which is
different from the definition used in Refs. 29,30. The pres-
ence of chiral factors exp( )±θ in the Hamiltonian Eq. (1)
results in special scattering of electrons by a non-diagonal
potential [28] — the effect of which we in what follows
will call chiral tunneling.
The electrostatic (scalar) potential is diagonal in the
pseudospin indices and can not induce electron backscat-
tering in our model Eq. (1) due to the conservation of he-
licity for massless Dirac particles (the Klein paradox). To
get nontrivial scattering of chiral particles we consider the
matrix potential
( ) ( )ˆ ( ) = ,
( ) ( )s
V x V x
V x
V x V x
⎛ ⎞
⎜ ⎟
⎝ ⎠
(2)
which mixes the sublattice components of the electron
wave function. For simplicity we consider all matrix ele-
ments to be real and equal. An effective scattering poten-
tial of the form of Eq. (2) was suggested in Ref. 28 for the
description of electron scattering in metallic carbon nano-
peapods. It is induced by the hybridization of fullerene
molecular orbitals (LUMO) with the conduction electron
states in the nanotube.
To proceed further we will consider the scattering po-
tential Eq. (2) to be «local». However, we can not take the
spatial dependence of ( )sV x to be simply ( )xδ . This is
because a δ -function scattering potential is ill-defined in
the context of the 1D Dirac equation. One has additionally
to define the value of the fermion wave function at the sin-
gular point = 0x (the wave function has a jump at this
point). In order to correctly solve the problem, we at first
consider a rectangular potential of width a and height 0V ,
which allows us to get an analytical solution for the scat-
tering problem. Then we consider the local scatterer limit
by letting 0,a → 0V →∞ while keeping the product
0V a constant.
The transmission and reflection amplitudes for the rec-
tangular potential is found by matching the plane wave and
evanescent mode solutions of the Dirac equation at the
points = 0,x a . The transmission amplitude, for instance,
takes the form
____________________________________________________
0
2 2 2 2 2
0 0 0 0
exp{ [2 ( )]}( )1( ) = ,
2 [( ) ( ) e (2 e sin )]sin 2 ( ) cosi i
F F F F
i a k q V E
t E
i v V E v q V q v V a a i v V E aθ θ
θ− + −
κ − + κ − + − κ κ − − κ κ
(3)
_______________________________________________
where 2 22
0 0= 2 / ,cos FV EV E vκ θ− + 0= cos / ,Fq V vθ
= / Fk E v .
In the limit we are interested in the Eq. (3) is strongly
simplified. The corresponding transmission and reflection
amplitudes are
0
0 0
0
0 0
cos exp( cos )
= ,
cos cos( cos ) sin( cos )
sin sin( cos )
= .
cos( cos ) sin( cos )
iU
t
U i U
U
r
cos U i U
θ − θ
θ θ + θ
θ θ
−
θ θ + θ
(4)
From Eq. (4) we get expressions for the scattering data
which will be used in what follows (i.e., for the transmis-
sion coefficient *=D tt , and the forward, fδ , and back-
ward, bδ , scattering phases)
2
2 2 2
0 0
cos( ) = ,
( cos ) ( cos )cos cos sin
( ) = 1 ( );
D
U U
R D
θ
θ
θ θ+ θ
θ − θ
(5)
2
0
2 2
0 0
0
0
sin (2 cos ) ( / 2)cos( ) = arctan =
( cos )cos ( cos )cos sin
cos ( ),
tan( cos )
( ) = arctan .
cos
f
b
b
U
U U
U
U
⎡ ⎤θ θ
δ θ ⎢ ⎥
θ θ− θ⎢ ⎥⎣ ⎦
= θ+ δ θ
θ⎡ ⎤δ θ ⎢ ⎥θ⎣ ⎦
(6)
Here 0 = /o FU aV v is the dimensionless strength of the
«local» scattering potential in our model. Notice that the
formula for the transmission coefficient D after a change
of notations coincides with the analogous expression for
the transmission coefficient in graphene [31]. In our case
the chiral angle θ plays the role of the incident angle of a
particle scattered by a rectangular barrier in 2D graphene.
To understand why the quantity 0 cosU θ appears in the
arguments of some trigonometric functions, it is useful to
find the spectrum of the Dirac equation in the constant
matrix potential
*
ˆ = .d o
s
o d
V V
V
V V
⎛ ⎞
⎜ ⎟⎜ ⎟
⎝ ⎠
(7)
From Eqs. (1), (7) we immediately get the spectrum
2 2 2
0 0= [ cos ( )] ( ),sind FE V v p U U± + θ−α + θ−α (8)
where 0 | | /o FU V v≡ and =| | exp( )o oV V iα . We see that
the only effect of the diagonal potential dV is a constant
shift of the energy spectrum. The off-diagonal potential oV
– (i) induces a gap (if θ ≠ α ), which mixes left- and right-
moving components of the wave function (i.e. it leads to
backscattering events), and (ii) plays the role of «vector»
potential by shifting the momentum to cos ( )op U+ θ−α .
Notice that the potential oV has to be odd with respect to
A.V. Parafilo, I.V. Krive, E.N. Bogachek, U. Landman, R.I. Shekhter, and M. Jonson
1196 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11
time reversal ( t t→− ). Both properties are crucial for
chiral tunneling. We will see in the next section that the
quantity 0 cosU θ (we consider real potentials) changes the
Aharonov–Bohm phase in the problem of persistent cur-
rents. It will be convenient for as to call ( )cφ θ =
0( / 2 )cosU= π θ the dimensionless chiral flux.
It is readily seen from Eqs. (5), (6) that for an armchair
( = 0θ ) nanotube (0) = 1D irrespective of the potential
strength, which is a demonstration of the Klein paradox in
non-chiral metallic nanotubes. In addition, ( ) = 1D θ at
0 cos = (U N Nθ π is an integer). The minimal value of the
transmission probability, 2
min = cosD θ , is reached at
0 cos = ( 1/ 2)U Nθ π + . We will refer to these cases as on-
and off-resonance chiral tunneling. The above considered
«quantization conditions» [31] are typical for quantum
resonant transport (see, e.g., [32,33] where an analogous
formula for the transmission coefficient was obtained for
resonant heat transport through a Luttinger liquid constric-
tion). On- and off-resonance conditions for chiral tunneling
are analogous to the corresponding conditions of construc-
tive ( 0/ = NΦ Φ , where Φ is the magnetic flux and
0 = /hc eΦ is the flux quantum) and destructive
0( / = 1/ 2NΦ Φ + ) interference for Aharonov–Bohm in-
terferometer (see e.g., Eq. (4.25) of Ref. 34). It is worth to
note that unlike in other resonant scattering problems, it is
the potential strength (and not the energy of bound states)
that is quantized in our case. The dependence of the trans-
mission coefficient and scattering phases on the chiral an-
gle for different values of potential strength is shown in
Fig. 1.
3. Persistent currents in chiral nanotubes
Several theoretical papers have studied persistent cur-
rents in ring-shaped SWNTs. They mostly deal with im-
purity free nanotubes and the results obtained concern spe-
cific properties of fullerene toroids [35], the differences in
magnetic response for metallic and small gap semiconduct-
ing nanotubes [36], and the influence of electron-electron
interaction on persistent currents in defect-free SWNTs
[37]. The presence of a short-range scatterer which induce
inter-valley electron backscattering ( 2 Fk kδ ) in suffi-
ciently long nanotubes can be described by a Luttinger
liquid model (for a short range electron-electron interac-
tion) or a Wigner crystal model (for an unscreened Cou-
lomb interaction). The evaluation of persistent currents in
these model can be found in [38,39] (see also the review
[16]). In all cases mentioned the nanotube chirality did not
influence the persistent currents at all. In this section we
consider long-range («soft») defects which can induce only
intra-valley electron scattering. For these processes chiral
effects are significant and they will determine the proper-
ties of the persistent current.
The Hamiltonian of the nanotube in our model is
ˆ= ( )j sH H V x+ where H± and ˆ ( )sV x are determined by
Eqs. (1), (2) and we will model the spatial dependence of
the scattering potential by a rectangular barrier in the local
limit (see the previous section). By placing the scatterer at
some specific point ( =x a ) we have two sets of plane-
wave solutions of the Dirac equation, one to the left ( l )
and one to the right ( r ) of the scatterer. For the «+-valley»
they are as follows ( = ,j l r )
exp ( )[ exp ( ) exp ( )]
=
exp [ ( )][ exp ( ) exp ( )]
F j j
j
F j j
ik x A ikx B ikx
i k x A ikx B ikx
+ −⎛ ⎞
Ψ ⎜ ⎟⎜ ⎟− θ − −⎝ ⎠
. (9)
The coefficients ,j jA B are found from two pairs of equa-
tions. The first pair,
Fig. 1. Transmission coefficient (a) and backward scattering phase (b) as a function of chiral angle at different values of potential
strength 0U : solid curve corresponds to 0 = 70U , dashed curve — 0 = 15U .
0 0.1 0.2 0.3 0.4 0.5
0.75
0.80
0.85
0.90
0.95
0 0.1 0.2 0.3 0.4 0.5
–1.5
–1.0
–0.5
0
0.5
1.0
1.5D
�, rad �, rad
�
�
b
(
)
a b
Chiral effects in normal and superconducting carbon nanotube-based nanostructures
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1197
0
( ) = exp 2 ( ),r lx L i x
⎛ ⎞Φ
Ψ + π Ψ⎜ ⎟Φ⎝ ⎠
(10)
represents the Aharonov–Bohm boundary condition ( L is
the ring circumference, 0 = /hc eΦ is the flux quantum,
and we note that in the absence of a scatterer
( ) ( )l rx xΨ ≡ Ψ so that in this case Eq. (10) is the familiar
twisted boundary condition for a particle on a ring threaded
by a magnetic field). The second pair of equations gives
the connection between the amplitudes ( )l aΨ and ( )r aΨ
induced by local potential scattering. It can be represented
in the form (see the previous section)
*
1
= ,
exp ( 2 )
l r
cl r
r
A At t
irB B
t t
⎛ ⎞
⎜ ⎟⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟πφ⎜ ⎟⎝ ⎠⎝ ⎠
⎜ ⎟
⎝ ⎠
(11)
where ,t r are the transmission and reflection amplitudes
defined in Eq. (4) and 0= ( / 2 )cosc Uφ π θ is the chiral
flux. Notice that the matrix  in Eq. (11) is not a transfer
matrix (in particular 0
ˆdet = exp ( 2 cos ) 1A i U θ ≠ for
0 0U ≠ ). In our case the scattering is a two-channel
process (we have an additional spinor index) and the cor-
responding transfer matrix is 4 4× -matrix. It is easy to
check that the Dirac current is conserved in the scattering
process. An analogous set of equations describes the scat-
tering of electrons in the Fk− -valley.
The solvability condition for the above linear equations
results in the spectral equation
0
0
( ) cos 2 cos = cos [ ( )],F bD k L U kL
⎛ ⎞Φ
θ π ± θ −δ θ⎜ ⎟Φ⎝ ⎠
∓ (12)
where ( )D θ , 0( ) = ( ) cosb f Uδ θ δ θ − θ are determined by
Eqs. (5), (6). Here the upper (lower) signs correspond to
the energy spectrum in the Fk+ -valley ( Fk− -valley). The
term Fk L results in a statistical flux («parity effects» [40])
in the persistent current of an isolated ring (where the total
number of particles is fixed). Chiral tunneling introduces
an additional term 0 cosU± θ , which we named the «chiral
phase» cϕ (or chiral flux = / 2c cφ ϕ π ). Notice that par-
ticles in ( ± )-valleys feel chiral fluxes of opposite signs
(the l.h.s. of Eq. (12).
In the limiting case of a local scatterer that we are inter-
ested in, neither the transmission probability nor the scat-
tering phases depend on energy. So the energy spectrum
( = FE v k± ) is
, eff
0
= arccos ( ) cos 2F
n j
v
E D j
L
⎧ ⎡ ⎤⎛ ⎞Φ⎪± θ π − ϕ +⎢ ⎥⎨ ⎜ ⎟Φ⎢ ⎥⎝ ⎠⎪ ⎣ ⎦⎩
( ) 2 ,b n
⎫
+ δ θ + π ⎬
⎭
(13)
where = 0, 1, 2,..., =n j± ± ± and eff 0= cosFk L Uϕ − θ is
the effective dimensionless flux. The evaluation of the per-
sistent current for a ring at given chemical potential μ ,
( ; ) =J c ∂ΩΦ θ −
∂Φ
(14)
(where Ω is the grand canonical thermodynamic poten-
tial) for the spectrum given by Eq. (13) is straightforward.
The result at finite temperatures T is (we consider here
spinless electrons)
eff
0
0
* = ; =1 1 2
eff
0
eff
0
*
sin 2
2=
( ) 2cos
sin arccos ( ) cos 2 cos ( , )
.
sinh
j k
j
TJ I
T
D j
k D j k
Tk
T
∞
± −
⎛ ⎞Φ
π + ϕ⎜ ⎟Φ⎝ ⎠ ×
π ⎛ ⎞Φ
θ − π + ϕ⎜ ⎟Φ⎝ ⎠
⎧ ⎫⎡ ⎤⎛ ⎞Φ⎪ ⎪θ π + ϕ δ μ θ⎨ ⎢ ⎥⎬⎜ ⎟Φ⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭×
⎛ ⎞
⎜ ⎟
⎝ ⎠
∑
(15)
Here 0 *= / , = / , ( , ) = / ( ).F F F bI ev L T v L L vπ δ μ θ μ −δ θ
We see from Eq. (15) that there is a spontaneous persistent
current (i.e., at zero external magnetic flux = 0Φ , see the
discussion in the review [17]) in each valley. However, at
equilibrium and for a ring with a fixed chemical potential,
for which the energy levels in the two valleys are equally
populated, the net persistent current at zero flux vanishes,
( = 0; ) = 0J Φ θ . This conclusion is of course a conse-
quence of the time-reversal invariance of our problem in
the absence of an external magnetic field.
The influence of temperature on the persistent current
in SWNTs is standard — at hight temperatures ( *T T )
the amplitude of Aharonov–Bohm oscillations is exponen-
tially small. The crossover temperature *T is determined
by the level spacing. In what follows we will consider the
low temperature limit *T T and the case of zero chemi-
cal potential, which corresponds to undoped nanotubes.
The most interesting situation is when there is resonant
chiral tunneling ( res= 1, = 0bD δ ). In this case the formula
for the persistent current takes the form
0
res 0
=1
sin 2
8= cos ,
2
F
k
k
ev NJ k n
L k
∞
⎛ ⎞Φ
π⎜ ⎟Φ ⎡ ⎤⎛ ⎞⎝ ⎠ π −⎜ ⎟⎢ ⎥π ⎝ ⎠⎣ ⎦
∑ (16)
where N is the total number of spin-1/2 electrons in the
ring (in the half-filled conduction band) and
0 0= cos /n U θ π . As readily seen from Eq. (16), the cur-
rent at 0Φ ≠ persists even at the Dirac point ( = 0μ ). In
an undoped SWNT ring ( = 0μ ) the total number of par-
ticles with energy 0E ≤ and momentum < <F Fk k k− +
is = 4( 2)(N m m+ is an integer). A degeneracy factor of 4
comes from spin× helicity degeneracy and another factor
A.V. Parafilo, I.V. Krive, E.N. Bogachek, U. Landman, R.I. Shekhter, and M. Jonson
1198 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11
of 2 comes from the double degeneracy of zero-energy
modes at each Dirac point. We see that in the absence of
chiral tunneling ( 0 = 0U ) the persistent current is always
paramagnetic (see, e.g., [37]). Now the response of the ring
to a magnetic field depends on the parity of the chiral re-
sonance: for even 0n the persistent current is paramagnet-
ic, for odd 0n we have a diamagnetic persistent current.
At low temperatures, 0T → , and for = 0μ in the off-
resonance case ( 0 0cos ( 1/ 2)U nθ→ π + ) the persistent
current even for a zigzag nanotube (maximal backscatter-
ing coefficient = 1 / 4R ) is highly non-harmonic (it has a
prominent sawtooth-like shape). Depending on the parity
of 0n and the approach to the off-resonance point, the cur-
rent is either paramagnetic (even 0n , from the «left» of the
off-resonance point) or diamagnetic (even 0n , from the
«right» of the off-resonance point) and vice versa (see
Fig. 2). The change from a para- to a diamagnetic response
is associated with a jump of the backscattering phase
Eq. (6) by =bΔδ π each time one passes through the off-
resonance condition. So the parity of the resonance ( 0n )
determines the type of the response up to the off-resonance
point, where it is «smoothly» changed (see Fig. 2).
In an isolated ring (with a fixed total number of par-
ticles) at = 0T the population of zero-energy modes can
be asymmetric. Then the ring will support a spontaneous
persistent current (the sign of current — clockwise or
counter-clockwise — will be determined by the concrete
choice of zero-mode population by chiral electrons). These
currents are not equilibrium currents [41] in the presence
of even small 2 Fk -backscattering, which tends to sym-
metrize the population of the zero-energy modes.
4. Chiral effects on the Josephson current
We consider the influence of nanotube chirality on the
supercurrent through an SNS junction based on a single-
wall carbon nanotube. The standard approach for describ-
ing S/SWNT/S junctions is to model the normal region as a
Luttinger liquid (see, e.g., the review [42]). In Luttinger
liquids there are specific phenomena (strong enhancement
of backscattering off local impurities, spin-charge and
charge-entropy separation) which are absent in Fermi liq-
uids. The most important property for charge transport is
the strong renormalization of the scattering potential by
electron-electron interactions, which results in a power-low
dependence of the differential conductance on temperature
and bias voltage (the Kane-Fisher effect [54]). Backscatter-
ing processes ( 2 Fk kδ ) mix quasiparticles (electrons
and holes) of different helicities and chiral properties of the
nanotube cease to be relevant. It is known [44] that in a
fully transparent junction (without normal backscattering)
the Josephson current is not renormalized by electron-
electron interactions. In long SNS tunnel junctions the in-
fluence of interactions results mostly in a renormalization
(suppression for repulsive interactions) of the junction
transparency [45] (see also the review [32] and references
therein). In both cases the chirality of the junction does not
influence the supercurrent at all.
There is a certain analogy between supercurrents in
long SNS junctions ( 0d ξ , where d is the length of the
junction and 0 0= /Fvξ Δ is the superconducting cohe-
rence length) and persistent currents in normal-metal bal-
listic rings (see, e.g., [46]). We have seen already that chir-
al tunneling leads to new effects in persistent currents.
What is the effect of chiral tunneling on the Josephson
current through an S/SWNT/S junction?
To calculate the supercurrent in an SNS junction from
the expression
4= eJ ∂Ω
∂ϕ
(17)
(where Ω is the thermodynamic potential, ϕ is the phase
difference and the factor 4 counts spin and pseudospin de-
generacies) we need to know the spectrum of Andreev
bound states in the normal region (here a SWNT contain-
ing a «soft» scatterer). Although we know the scattering
characteristics of our potential from Eqs. (5), (6), we can
not from the very beginning use the known formulae for
the spectrum of Andreev–Kulik levels [47] and the Josep-
son current in an SNS junction in terms of junction trans-
parency. The Andreev scattering in graphene was shown
[48] to exhibit peculiarities (specular Andreev reflection)
as compared to the ordinary SNS junctions. Therefore we
will follow the standard approach and find the spectrum by
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
J/
I 0
–0.06 0.06–0.04 0.04–0.02 0.020
� �/ 0
Fig. 2. Persistent current as a function of magnetic flux for differ-
ent values of the chiral angle near the off-resonance point
0( = ( 1/ 2)c nϕ π + ; = 0.402 rad,θ 0 = 70U results in 0 = 20;n
this integer numerates the 20th off-resonance point counted from
= / 2θ π in ( )D θ dependence): the solid curve corresponds to
= 0.398 radθ , the dashed curve to = 0.418 radθ .
Chiral effects in normal and superconducting carbon nanotube-based nanostructures
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1199
solving the Bogoliubov-de Gennes (BdG) equation with
the order parameter ( )xΔ constant in the superconductors
0( ( ) = ei Lx φΔ Δ for < 0x , 0( ) = ei Rx φΔ Δ for >x d , here
=L Rϕ −ϕ ϕ ) and ( ) = 0xΔ in the normal region. The
scatterer is placed at the point =x l inside the normal re-
gion.
By matching the plane wave solutions of the BdG equa-
tion in the normal region at points = 0, ,x l d we derive the
equation for the bound state energies (Andreev–Kulik
levels [47])
0
2cos 2arccos
F
Ed E
v
⎡ ⎤
− =⎢ ⎥Δ⎣ ⎦
2 (2 )( )cos ( ) cos ,
F
E l dD R
v
⎡ ⎤−
= θ ϕ+ θ ⎢ ⎥
⎣ ⎦
(18)
where ( )R θ and ( )D θ are defined in Eq. (5). The spectral
equation, Eq. (18), is the standard equation for the bound
state energies in an SNINS junction (I denotes the scatterer
inside the normal region). The phase factor which an elec-
tron (hole) picks up in the process of chiral tunneling is
energy independent in our model. The phases acquired by
electrons and holes are opposite in sign and cancel out in
the spectral equation. Besides, contrary to the problem of
persistent currents in a ballistic ring, the effective flux
(both statistical and chiral) eff 0= cosFk L Uϕ − θ does not
enter the spectral equation (18). In the process of Andreev
reflection at an S/N boundary two electrons with small
total momentum ( Fk k ) penetrate into the bulk super-
conductor. It means that an electron in the +kF-valley is
reflected as a hole in the –kF-valley (and vice versa). The
electron and the hole have opposite momenta and opposite
pseudospin (but the same helicity). The two possible he-
licities ( 1± ) result in an additional factor 2 in the defini-
tion of the Josephson current Eq. (17). We have already
seen in the previous section that particles in different val-
leys carry effective fluxes effϕ with opposite signs. These
contributions to the spectral equation, Eq. (17), cancel out.
As a result all information in Eq. (17) specific to SWNTs
is hidden in the transmission probability ( )D θ . In particu-
lar, the nanotube chirality does not influence the Josephson
current at all in the absence of normal scattering ( 0 = 0U ).
What are then the effects of chiral tunneling on the Jo-
sephson current? In chiral nanotubes the junction transpa-
rency is an oscillating function of the strength, 0U , of the
«soft» scattering potential. Therefore one can expect an
anomalous (non-monotonic) behavior of the critical current
as a function of potential strength. For resonant chiral
tunneling, 0 0cos =U nθ π , the junction becomes fully
transparent ( = 1rD ) and the supercurrent through an
SNINS junction coincides with (i) the Josephson current
through a superconducting constriction (for a short junc-
tion 0 ,d ξ 0ξ is the superconducting coherence length)
[49] ( )
max ( ) = (2 / )sin ( / 2)sJ eϕ Δ ϕ (the additional factor 2
in this formula is due to pseudospin degeneracy) or (ii) the
supercurrent through a long 0d ξ transparent junction
( )
max ( ) = 2( / )( / )l
FJ ev dϕ ϕ π , where | |ϕ ≤ π . Notice that
for resonant chiral tunneling the supercurrent does not de-
pend on the position of the scatterer inside the normal re-
gion. For off-resonant tunneling the current in a long junc-
tion does depend on the actual position ( =x l ) of the local
scatterer. However, the effect is numerically small. For the
two limiting cases of a symmetric ( = / 2l d ) and a max-
imally asymmetric ( = 0,l d ) junction the supercurrents are
2
off
22 2
2 sincos( ; ) =
1 ( cos )sin cos
F
s
ev
J
d
θ ϕ
ϕ θ ×
π − θ+ θ ϕ
2 2arccos ( cos ),sin cos× θ + θ ϕ (19)
2
off
22 4 2
2 sincos( ; ) =
(1 )sin cos cos
F
a
ev
J
d
θ ϕ
ϕ θ ×
π + θ − θ ϕ
2
2
coscosarccos .
1 sin
⎛ ⎞θ ϕ
× ⎜ ⎟
+ θ⎝ ⎠
(20)
The behavior of the supercurrents ,s aJ for different chiral
angles is shown in Fig. 3. We see that at given θ and ϕ
the current through a symmetric junction is always larger
(although the effect is numerically small) than the current
through an asymmetric junction.
There are proposals to use the pair of Andreev levels in
a short SNS junction as a qubit («Andreev qubit», see
Ref. 50) and for cooling of nanoelectromechanical devices
[51]. In these proposals the coherent dynamics of the An-
dreev–Kulik levels occur deep inside the gap region
0 0( ,E Δ Δ is the superconducting order parameter).
0 0.5 1.0 1.5 2.0
–2
–1
0
1
2
���
J(
)/
I
�
��
0
Fig. 3. The Josephson current(in units 0 = /FI ev d ) in a long
junction, 0 = 70U ; solid curve corresponds to the chiral reso-
nance ( = 0.158θ rad) and two off-resonance cases 0.276 radθ =
and = 0.507θ rad (symmetric junction); the corresponding cur-
rents in asymmetric junction are represented by the dashed
curves.
A.V. Parafilo, I.V. Krive, E.N. Bogachek, U. Landman, R.I. Shekhter, and M. Jonson
1200 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11
This regime could be reached only for almost fully transpa-
rent junctions and chiral nanotube-based junctions could be
promising candidates for the fabrication of SNS junctions
with high transparency 1 1D − .
In the general case of non-resonant chiral tunneling the
minimum energy separation between the pair of Andreev–
Kulik levels in a short SNS junction is ( )gE θ =
02 1 ( )D= Δ − θ . The minimal junction transparency
2
min = cosD θ is reached at 0 0cos = ( 1/ 2)U nθ π + (chiral
off-resonance). So the gap
0( ) = 2 singE θ Δ θ (21)
could be arbitrary small for nanotubes with small chiral
angles.
5. Resonant tunneling through a chiral quantum dot
Resonant electron tunneling in quantum wires (see, e.g.,
the review [52]) is a coherent quantum mechanical pheno-
menon which is extremely sensitive to the electron energy
spectrum. In transport experiments with quantum dots in
the regime of resonant tunneling (resonant transport spec-
troscopy) one gets valuable information about the electron
energies and electron wave functions in the dot by measur-
ing the position and the shape of resonance conductance
peaks as a function of the gate voltage. Single-wall carbon
nanotubes from the very beginning of their discovery were
considered as promising elements for future nano-
electronics. In particular, carbon nanotube-based single
electron transistors (SET) were fabricated and electron
transport through these molecular devices was studied in a
wide range of temperatures (see, e.g., the review [53] and
references therein). The observation at low temperatures of
Coulomb blockade oscillations and resonant electron
tunneling in long (a few hundreds nanometers ) metallic
SWNTs means that electrons are delocalized along the
whole length of the structure. This fact is usually explained
by the specifics (Klein paradox) of scattering of charge
carriers (massless Dirac electrons) in SWNTs by long-
range tube defects [26]. Chiral tunneling in this sense has
already been indirectly observed in SWNT-based quantum
dots. Here we consider the direct influence of chiral tunne-
ling on the resonant transport properties of quantum dots.
We will model a «chiral quantum dot» by a finite length
( L ) metallic chiral SWNT , Eq. (1), with the «soft» local
scatterer, Eq. (4) placed at a distance l from the left end of
the nanotube. The nanotube is connected to the leads by
normal tunnel barriers which results in a finite (small)
width Γ of the electron energy levels (we will assume the
widths to be energy independent). The electron energy
spectrum, to lowest order on Γ , can be found by assuming
the end-barriers to be infinite. The corresponding boundary
conditions can be formulated as the absence of any elec-
tron (Dirac) current through the ends of the nanotube,
( = 0, ) = 0Dj x L . Since scattering at the ends connects
electrons in the Fk+ and Fk− valleys the current should
be expressed in terms of the 4-spinors = ( , )T T T
+ −Ψ ψ ψ ,
where T denotes transposition. For our Hamiltonian, Eq.
(1), the current looks like
†
0 e 0 0
e 0 0 0
( ) = ( ) ( ) .
0 0 0 e
0 0 e 0
i
i
D F i
i
j x v x x
θ
− θ
− θ
θ
⎛ ⎞
⎜ ⎟
⎜ ⎟
Ψ Ψ⎜ ⎟
⎜ ⎟−
⎜ ⎟⎜ ⎟−⎝ ⎠
(22)
The physically evident solution of the above discussed
boundary conditions is the scattering at the boundaries,
when a left-moving fermion in the «+»-valley is trans-
formed into a right-moving fermion in the «–»-valley and
all analogous processes ( L R± ↔ ∓ ). In a general case this
scattering is accompanied by an energy-independent phase
shift.
By matching the plane wave solutions of the Dirac equ-
ation at the points = 0, ,x l L using our boundary condi-
tions and the matrix Eq,(11) for «+»- and «–»-valleys
(θ→ −θ ) we derive the following spectral equation
cos [2 ( )] = ( )cos [2 ( 2 )]kL R k L l−ζ θ θ − +
0( ) cos (2 2 cos ),FD k L U+ θ − θ (23)
where the total scattering phase ( )ζ θ is
0
2 2 2
0 0
sin(2 cos )cos
( ) = arctan
( cos ) ( cos )cos cos sin
U
U U
⎡ ⎤θ θ
ζ θ =⎢ ⎥
θ θ− θ⎢ ⎥⎣ ⎦
02[ ( ) cos ] .f U= δ θ − θ (24)
Notice that Eq. (23) coincides with Eq. (18) of Ref. 54.
However, the expressions for the scattering data in our
case are different (the assumption for the scattering poten-
tial for which the transmission and reflection amplitudes
were obtained in [54] is not satisfied in our model). The
derived spectral equation has a simple physical interpreta-
tion. The phase terms in Eq. (23) depend on: (i) the «quan-
tization length» 2L (at the boundaries inter-valley electron
scattering occurs and one needs a 2L-path to form a closed
trajectory), (ii) the double forward scattering phase
2 ( )fδ θ which the particle picks up by passing the scatter-
ing potential twice, and (iii) the double chiral phase
02 cos = 2 cU θ ϕ , which is added to the geometrical phase
2 Fk L .
Since the momentum Fk is defined for an undoped na-
notube it can be expressed through the total number of
particles in a half-filled conduction band as = / 2Fk L Nπ ,
where = 4( 2),N m + m is an integer; see the discussion in
Sec. 3). We see that the phase associated with the terms
Fk L does not influence the spectral properties of chiral
quantum dots. For armchair nanotubes ( (0) = 0R and
0(0) = 2f U nδ + π on module 2π ) the spectral equation
reads 0 0cos (2 2 ) = cos 2kL U U− ± , which results in two
Chiral effects in normal and superconducting carbon nanotube-based nanostructures
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1201
sets of equidistant energy levels ( = /FE v LΔ π ) which
are shifted relatively each other by (0) /f Fv Lδ . Notice
that for fδ π the level spacing is approximately halved
(this halved level spacing is usually considered as an aver-
aged level spacing in SWNT). In the case of chiral reso-
nant tunneling the electron energy spectrum in a chiral
nanotube does not depend of the position of the chiral scat-
terer (as for armchair nanotubes). It is still equidistant with
energy spacing /Fv Lπ . In the general case of non-
resonant scattering the energy spectrum is quasi-random
for irrational values /l L . The distribution of the number
of energy levels on the level spacing (normalized by
= /FE v LΔ π ) for a given potential strength and different
chiral angles is shown in Fig. 4. The levels, which differ in
energies by less then the level width Γ , are considered as
«degenerate». We see that two δ -function-like peaks for
= 0θ are developed for sufficiently strong backscattering
( 1D ) to the distribution which resembles Wigner-
Dyson distribution. Although in our model (real potentials
,d oV V of equal strengths) small transmission coefficients
correspond to nonphysical chiral angles ( θ close to / 2π )
small transparencies in chiral tunneling could be realized
in the general case of nonsymmetric potentials.
In the regime of resonant electron tunneling through
quantum dots the distribution of spacings of the peak
(maximum) conductances as a function of gate voltages is
determined by the distribution of the level spacings for
electron energy spectrum. We showed that the mechanism
of chiral tunneling is sensitive to the chirality of nanotubes.
Therefore this phenomenon could be used for determina-
tion of nanotube chiral angle in resonant transport spec-
troscopy.
6. Conclusion
In this paper we have introduced the new concept of
chiral tunneling in metallic single-wall carbon nanotubes.
There are significant differences between the Klein (or
chiral) tunneling of massless 2D Dirac-like particles in
graphene (well studied in recent years; see the review [27])
and the chiral tunneling of 1D massless fermions in
SWNTs. In the 2D scattering problem in graphene even a
scalar electromagnetic potential can backscatter massless
Dirac electrons if the incident angle of scattering particle is
not close to zero (for normal incidence the transmission
Fig. 4. The distribution of the number of energy levels on the level spacing (normalized by = /FE v LΔ π ) for 0 = 20U and different
chiral angles: to = 0.49θ rad ( = 0.8)D (a), = 0.52θ rad ( = 0.75)D (b), = 0.955θ rad ( = 0.4D ) (c), / 2θ ≈ π ( = 0.06)D (d). For
the armchair nanotube we have two sets of equidistant energy levels: / = 0.13E Eδ Δ and 0.36. The distribution is shown for
/ = 1 / 2l L π , = 0.002Γ .
0.10 0.15 0.20 0.25 0.30 0.35 0.40
0
50
100
150
200
250
300
N N
NN
a
c
b
d
0.150 0.200 0.250 0.300
0
50
100
150
200
0.200 0.250 0.300 0.350
20
40
60
80
100
120
140
160
0.10 0.20 0.30 0.400
20
40
60
80
100
120
140
160
� E/ E
� E/ E
� E/ E
� E/ E
A.V. Parafilo, I.V. Krive, E.N. Bogachek, U. Landman, R.I. Shekhter, and M. Jonson
1202 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11
coefficient is always 1D = ). In a SWNT the scattering is a
1D problem and one can expect finite-probability reflec-
tion of massless Dirac particles only in chiral nanotubes
where spiral-like electron «motion» along the cylinder axis
mimics some features of 2D scattering. Besides, in non-
trivial scattering ( 0R ≠ ) the scattering potential can not be
purely electrostatic («diagonal» in our representation of
Dirac matrices). The presence of non-diagonal compo-
nents, which are induced by a pseudovector potential
(pseudomagnetic effects), are crucial in chiral tunneling in
nanotubes. In our consideration we use a phenomenologi-
cal approach to the problem and postulate the form of ma-
trix scattering potential Eq. (2) in order to study the general
properties of chiral tunneling. We know at least one exam-
ple — the effective scattering potential induced by fulle-
rene molecules in nano-peapods , when such a matrix po-
tential was derived microscopically [28]. Notice that
magnetic potentials in nanostructures are as a rule long-
range and they are consistent with our assumption of a
«smooth» scattering potential.
We showed that in chiral tunneling not only is a (small)
reflection probability an important physical characteristic
but the chiral phase, 0= cosc Uϕ θ , the quantity associated
with the effective vector potential experienced by the par-
ticle in the process of tunneling, plays a significant role as
well. It is worth to note that the forward and backward
scattering phases in chiral tunneling are related by the sim-
ple expression =f b cδ −δ ϕ . The quantized chiral phase
determines the conditions for resonant ( = ,c n nϕ π is the
integer, = 1rD ) and off-resonant ( = ( 1/ 2),c nϕ π + 2= cosoD θ ) chiral tunneling.
The chiral phase is added to the magnetic flux in the
problem of Aharonov–Bohm oscillations and its appear-
ance can result in a spontaneous persistent current in a ring
with an asymmetric population of zero-energy modes.
Since the particles with opposite helicities acquire chiral
phases of opposite signs, the chiral phases are cancelled in
the Josephson current problem when a pair of electrons
( ,F Fk k− + ) tunnel to the bulk superconductor. We demon-
strated the nontrivial role the chiral phase plays in various
phase coherent phenomena in nanostructures.
The last question we would like to discuss here is the
influence of electron-electron interactions on chiral tunne-
ling. We will assume that the interaction is not strong, oth-
erwise the Luttinger liquid effects which strongly enhance
2 Fk -backscattering violate our assumption of a smooth
diagonal scattering potential. It is physically evident that
for the conditions of resonant chiral tunneling ( = 1rD )
electron-electron interactions do not renormalize chiral
scattering potentials at all. Off-resonance there is finite
backscattering and one could expect its enhancement by
(repulsive) interaction effects. This problem needs special
consideration. We briefly mention here that in the limit of
weak interactions, when the renormalization is induced by
electron scattering on Friedel oscillations [55], the naive
mean-field approximation of the interaction potential does
not lead to logarithmic infra-red singularities in the back-
scattering amplitudes.
We gratefully acknowledge discussions with L. Gore-
lik, A. Kadigrobov, S. Kulinich and V. Shumeiko. AVP
thanks V.Yu. Monarkha for help with numerical calcula-
tions. This work was supported in parts by the Swedish
VR, the European Commission (FP7-ICT-2007-C; proj No
225955 STELE), and the Korean WCU programme funded
by MEST through KOSEF (R31-2008-000-10057-0). The
research of ENB and UL was supported by the US De-
partment of Energy under Grant No. FG05-86ER45234.
IVK acknowledges the Department of Physics at the Uni-
versity of Gothenburg for hospitality.
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|
| id | nasplib_isofts_kiev_ua-123456789-117521 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T16:46:33Z |
| publishDate | 2010 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Parafilo, A.V. Krive, I.V. Bogachek, E.N. Landman, U. Shekhter, R.I. Jonson, M. 2017-05-24T05:17:48Z 2017-05-24T05:17:48Z 2010 Chiral effects in normal and superconducting carbon nanotube-based nanostructures / A.V. Parafilo, I.V. Krive, E.N. Bogachek, U. Landman, R.I. Shekhter, M. Jonson // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1193–1203. — Бібліогр.: 55 назв. — англ. 0132-6414 PACS: 73.23Ra, 72.10.–d, 74.50.+r https://nasplib.isofts.kiev.ua/handle/123456789/117521 The novel phenomenon of chiral tunneling in metallic single-wall carbon nanotubes is considered. It is induced by the interplay of electrostatic and pseudomagnetic effects in electron scattering in chiral nanotubes and is characterized by the oscillatory dependence of the electron transmission probability on nanotube chiral angle and the strength of the scattering potential. The appearance of a special (Aharonov–Bohm-like) phase in chiral tunneling affects various phase-coherent phenomena in nanostructures. We considered chiral effects in: (i) the persistent current in a circular nanotube, (ii) the Josephson current in a nanotube-based SNS junction, and (iii) resonant electron tunneling through a chiral nanotube-based quantum dot. We gratefully acknowledge discussions with L. Gorelik, A. Kadigrobov, S. Kulinich and V. Shumeiko. AVP thanks V.Yu. Monarkha for help with numerical calculations. This work was supported in parts by the Swedish VR, the European Commission (FP7-ICT-2007-C; proj No 225955 STELE), and the Korean WCU programme funded by MEST through KOSEF (R31-2008-000-10057-0). The research of ENB and UL was supported by the US Department of Energy under Grant No. FG05-86ER45234. IVK acknowledges the Department of Physics at the University of Gothenburg for hospitality. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Quantum coherent effects in superconductors and normal metals Chiral effects in normal and superconducting carbon nanotube-based nanostructures Article published earlier |
| spellingShingle | Chiral effects in normal and superconducting carbon nanotube-based nanostructures Parafilo, A.V. Krive, I.V. Bogachek, E.N. Landman, U. Shekhter, R.I. Jonson, M. Quantum coherent effects in superconductors and normal metals |
| title | Chiral effects in normal and superconducting carbon nanotube-based nanostructures |
| title_full | Chiral effects in normal and superconducting carbon nanotube-based nanostructures |
| title_fullStr | Chiral effects in normal and superconducting carbon nanotube-based nanostructures |
| title_full_unstemmed | Chiral effects in normal and superconducting carbon nanotube-based nanostructures |
| title_short | Chiral effects in normal and superconducting carbon nanotube-based nanostructures |
| title_sort | chiral effects in normal and superconducting carbon nanotube-based nanostructures |
| topic | Quantum coherent effects in superconductors and normal metals |
| topic_facet | Quantum coherent effects in superconductors and normal metals |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/117521 |
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