Secondary Electron Interference from Trigonal Warping in Clean Carbon Nanotubes
A. Dirnaichner,
1,2
M. del Valle,
2
K. J. G. Götz,
1
F. J. Schupp,
1
N. Paradiso,
1
M. Grifoni,
2
Ch. Strunk,
1
and A. K. Hüttel
1,*
1
Institute for Experimental and Applied Physics, University of Regensburg, 93040 Regensburg, Germany
2
Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany
(Received 15 February 2016; revised manuscript received 6 June 2016; published 11 October 2016)
We investigate Fabry-Perot interference in an ultraclean carbon nanotube resonator. The conductance
shows a clear superstructure superimposed onto conventional Fabry-Perot oscillations. A sliding average
over the fast oscillations reveals a characteristic slow modulation of the conductance as a function of the
gate voltage. We identify the origin of this secondary interference in intervalley and intravalley
backscattering processes which involve wave vectors of different magnitude, reflecting the trigonal
warping of the Dirac cones. As a consequence, the analysis of the secondary interference pattern allows us
to estimate the chiral angle of the carbon nanotube.
DOI: 10.1103/PhysRevLett.117.166804
Clean carbon nanotubes (CNTs) are an excellent material
system to observe Fabry-Perot interference when highly
transparent contacts suppress charging effects [1]. This is
often the case in the hole regime of transport in CNTs [2,3].
So far, experiments mostly concentrated on the effects of
the linear, Dirac-like part of the CNT dispersion relation,
resulting in simple Fabry-Perot (FP) interference [1,46].
Its hallmark is an oscillatory behavior of the differential
conductance GðV
g
;V
b
Þ as a function of both gate voltage
V
g
and bias voltage V
b
, with frequency proportional to
the CNT length [1]. On top of this regular oscillation,
slower modulations are sometimes observed in experiments
[1,5,7]. Such secondary interference has been attributed to
disorder [7,8] or to channel mixing at the CNT-contact
interface [9]. It has been suggested that a slow modulation
can also originate from intrinsic interference effects in
chiral CNTs [10]. In general, being related to a difference
of accumulated phases, secondary interference probes the
nonlinearity of the CNT dispersion relation due to the
trigonal warping and, in turn, the chiral angle [9,10].
In this Letter, we report on the investigation of a peculiar
secondary interference pattern in the hole regime of an
ultraclean CNT. Upon averaging over the fast primary FP
oscillations, the resulting average linear conductance
¯
GðV
g
Þ shows a quasiperiodic slow modulation deep in
the hole regime. We combine detailed tight-binding cal-
culations and fundamental symmetry arguments to identify
the origin of the slow modulation. Our analysis of the gate
voltage dependence of
¯
GðV
g
Þ allows us to estimate the
CNTs chiral angle θ.
We measure the differential conductance of a suspended
CNT attached to 50-nm-thick Pt=Ti leads, separated by a
1.2-μm-wide trench, at T ¼ 15 mK [11]. The fabrication
process is optimized to produce defect-free devices [12].
Figure 1(a) displays the conductance GðV
g
;V
b
Þ of the CNT
device as function of gate voltage V
g
and bias voltage V
b
.
On the electron conduction side (V
g
> 0.35V, see the
Supplemental Material [11]), transport characteristics are
dominated by Coulomb blockade. On the hole side, owing
to the high transparency of the barriers, the CNT behaves as
an electronic 1D waveguide. An oscillatory large conduct-
ance 0.2 G=G
0
1.8 (G
0
¼ e
2
=h) is observed for gate
voltage values 15 V V
g
0 V. The electron wave
vector is affected by both bias and gate voltage, leading
to typical rhombic interference structures in the GðV
g
;V
b
Þ
diagram [1]. A striking feature of our data is the slow
modulation of the conductance pattern as a function of V
g
,
visible as a series of darker and brighter intervals in
Fig. 1(a) alternating on a scale of approximately 2 V.
In Fig. 1(b), we show the differential conductance trace
GðV
g
Þ for V
b
¼ 0. Primarily, we observe a fast oscillation
of the conductance at a frequency f
1
¼ 12.8 V
1
. This
fundamental frequency is directly related to the length of
the cavity via f
1
αeL=πv
F
[1].Forv
F
¼ 8 × 10
5
m=s
[24], we obtain L 1 μm, which is close to the width of
the trench. From the period of the fast oscillation ΔV
g
fast
¼
1=f
1
and the height V
c
of the rhombic pattern in Fig. 1(a),
we extract the gate voltage lever arm α ¼ V
c
=ΔV
fast
g
¼
0.0210 0.0007 [1]. On top of the fast oscillations, the
slow modulation is visible. Figure 1(c) shows the sliding
average
¯
GðV
g
Þ of the conductance as function of V
g
. The
peaks of the average conductance are labeled as
n ¼ 1; ; 6 starting from the band gap. The spacing of
the peak positions E
n
¼ αΔV
g;n
decreases for more neg-
ative gate voltages V
g
.
We perform a discrete Fourier transform (FT) over a
Gaussian window [shaded gray in Fig. 1(b)]. The result is
plotted in log scale in Fig. 1(d) as a function of frequency
and window position. The FT shows regions in V
g
with a
dominant fundamental frequency component f
1
alternating
with regions where the second harmonic with f
2
¼ 2f
1
prevails. These reflect the frequency doubling that is visible
in certain ranges of V
g
in Fig. 1(b). In these regions, the FT
also reveals components from higher harmonic frequencies
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f
n
¼ nf
1
, appearing as horizontal lines in the FT plot. An
analysis of the decay of the higher harmonic amplitudes
in the Fourier transform yields an average length of the
electronic path in the interferometer of 2.7 μm. This length
corresponds to the dwell time of an electron in the device
and provides a lower bound on its phase coherence length
[11]. As shown in Fig. 1(d), the entire spectrum consists
mainly of one fundamental frequency and its harmonics;
i.e., no additional fundamental frequency occurs. Hence,
we can conclude that there are no impurities that subdivide
the CNT into a serial connection of multiple FP interfer-
ometers [8].
The main features observed in the experiment can be
reproduced by a real-space tight-binding calculation, a
description that allows us to realistically include curvature
effects and the spin-orbit interaction in the real-space
Hamiltonian of our system [13,25]. The transport proper-
ties of the CNTs are obtained within the Landauer-Büttiker
approach, using Greens function techniques, very well
suited for transport calculations in the ballistic regime. This
numerical approach can be applied to CNTs with arbitrary
structure. In Figs. 2(a)2(d), our numerical results for the
transmission of four different classes of CNTs are shown.
Strikingly, the slow modulation pattern in the average
transmission
¯
T is observed only for the CNT geometry
in Fig. 2(d), where even in an idealized system the
absence of certain symmetries (discussed below) allows
interferometer channel mixing. As we are going to explain,
the crucial geometrical property determining this secondary
FP pattern is the chiral angle.
Carbon nanotubes can be grouped in four distinct classes
[2628]: armchair, armchairlike, zigzag, and zigzaglike.
The CNT chiral indices (n, m) determine the class: If the
ratio of n m to their greatest common divisor d ¼
gcdðn; mÞ is a multiple of 3, i.e., ðn mÞ=3d Z , the
CNT belongs to the chiral armchairlike class if n m
and is an achiral armchair CNT if n ¼ m. Otherwise, we
are dealing with a zigzaglike CNT unless m ¼ 0, which
characterizes achiral zigzag CNTs.
This classification reflects intrinsic differences in the
CNT band structure, which are of crucial importance to the
transport properties of these systems. In metallic zigzag and
zigzaglike CNTs, the π bands cross at the Dirac points
~
K ¼ðk
¼ 0;k
¼þK
Þ,
~
K
0
¼
~
K [26]. Here, k
and k
are the components of the wave vector parallel and
perpendicular to the CNT axis. In particular, k
is propor-
tional to the crystal angular momentum that stems from the
rotational C
d
symmetry and is opposite in the two valleys.
When considering reflections from the interfaces, this
symmetry only allows for intravalley backscattering.
Thus, each valley constitutes an independent transport
channel, as depicted in Figs. 2(a) and 2(b). In this case,
the FP oscillations are mainly described by the standard
expression for the transmission [14]
0
2
G/G
0
-1
V
b
(
mV)
(b)
1
−14 −12 −10 −8 −6 −2
20
40
60
80
f(1/V)
0
f
1
f
2
f
3
f
4
(d)
gate voltage (V)
1
G/G
0
(a)
1
2
−3
−2
−1
log(|A
FT
|/G
0
)
G
/
G
0
1.4
0.2
(c)
−12 −10 −8 −6 −2−4
6
5
4
3
2
1
−14 0
n=7
1.0
0.6
0
FIG. 1. (a) Differential conductance GðV
g
;V
b
Þ of a clean CNT device in the hole conduction regime, as a function of back gate voltage
V
g
and bias voltage V
b
(G
0
¼ e
2
=h). (b) Zero bias conductance GðV
g
Þ extracted from (a). (c) Average conductance
¯
GðV
g
Þ obtained over
a sliding 0.4-V-wide Gaussian window. A slow modulation is observed. The peak positions are marked with filled circles. The distance
ΔV
g;n
¼ V
g;n
V
g;nþ1
between the nth and the (n 1)th peak decreases with n. (d) Fourier transform of a sliding 0.4-V-wide window in
the signal in (b) as a function of the gate voltage.
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T ðV
g
Þ¼
X
j¼a;b
2jt
1
j
2
jt
2
j
2
1 þjr
1
j
2
jr
2
j
2
2jr
1
r
2
j cos½ϕ
j
ðV
g
Þ
; ð1Þ
where j labels the two independent channels, and the
transmission and reflection amplitudes for the two confin-
ing barriers are given by t
1
, t
2
and r
1
, r
2
, respectively.
The phase accumulated upon one round trip is given by
ϕ
j
ðV
g
Þ¼ðjk
j;l
ðV
g
Þj þ jk
j;r
ðV
g
ÞjÞL, and the wave vector of
the right (left) moving electron k
j;rðlÞ
is linked to V
g
via the
CNT dispersion relation εðk
j;rðlÞ
Þ¼αeV
g
. In zigzag and
zigzaglike CNTs, the accumulated phases are identical for
the two channels since the dispersion in the two valleys is
symmetric, i.e., k
a;r
¼jk
b;l
j and k
b;r
¼jk
a;l
j. According to
Eq. (1), one single FP oscillation occurs when ϕ
j
¼ 2π.
Consequently, the tight-binding model calculations of a
(12,0) CNT in Fig. 2(a) and of a (6,3) CNT in Fig. 2(b)
show featureless single-channel interference patterns with a
fundamental frequency f
1
.
On the other hand, in armchair and armchairlike CNTs,
the bands cross at the Dirac points
~
K ¼ðK
; 0Þ and
~
K
0
¼
~
K; see Figs. 2(c) and 2(d). Two valleys are formed,
which are symmetric with respect to the k
¼ 0 axis and
both characterized by zero crystal angular momentum [26].
Intervalley backscattering is now possible, and the angular
momentum quantum numbers do not provide a means to
distinguish the transport channels.
However, armchair CNTs are invariant under the parity
operation [26], which enables us to identify now two other
independent transport channels a and b. These parity
channels are such that within one pair, backscattering
connects a left mover or right mover in the K valley to
its time-reversal partner in the K
0
valley; see Fig. 2(c).
Equation (1) still describes the FP oscillations but, in
contrast to the zigzaglike class, the two channels accumu-
late different phases ϕ
a
¼ 2k
a;r
L ϕ
b
¼ 2k
b;l
L, owing to
the trigonal warping. In the interference pattern, we thus
expect a beat with a constant average transmission. This
expectation is confirmed by our tight-binding transport
calculations for a (7,7) CNT; see Fig. 2(c).
In armchairlike CNTs, the parity symmetry is absent, and
hence backscattering from branch a to branch b in the same
valley is also possible. The interference pattern displays
<>
a
k
a,l
0 0
k
||
k
||
k
a,r
T
4
0
(eV)
aa bb
0
T
4
0
(eV)
b aa
0
0
a
b
k
||
k
||
T
(eV)
4
0
-0.16 -0.10
aa bb
armchair
0
<>
T
4
0
(eV)
aab
k
a,r
k
a,l
k
||
k
||
(10,4)
-0.22
(6,3)
-0.16
-0.10
-0.22
(12,0)
(7,7)
-0.16 -0.10-0.22
-0.16 -0.10-0.22
zig-zag
zig-zag-like
armchair-like
E
n
<>
b
b
k
b,r
k
a,l
k
b,r
k
b,l
k
b,l
k
b,r
k
a,r
k
b,l
k
b,r
k
a,l
k
a,r
k
b,l
(a)
(b)
(c)
(d)
FIG. 2. Graphene dispersion relation εðkÞ (contour plots in the top left panel of each subfigure) in the vicinity of a Dirac point and
simplified lowest 1D subbands (line plots, top right) [25]. The solid red line in the contour plot marks the direction of k
. The chiral
angle θ is measured with respect to the direction of the zigzag CNT (dashed line). The bottom panels show exemplary transmission
patterns obtained by numerical tight-binding calculations. The green line represents the sliding average
¯
T of the transmission signal.
(a) Zigzag: Dispersion relations at the two Dirac points K
(green) and K
(red) are identical and symmetric with respect to the k
¼ 0
axis. The transmission curve of a (12,0) CNT shows a simple, single-channel FP inter ference pattern. (b) Zigzaglike: Right- and left-
moving branches within each valley exhibit different wave vectors k
j;r=l
at finite energy. No intervalley scattering is possible in (a) and
(b); see the text. A single-channel-like transmission pattern can be observed for the (6,3) CNT (bottom left). (c) Armchair: Parity
symmetry forbids scattering between a and b branches. At finite energy, the two Kramers channels a and b have different wave vectors
associated with the right- and left-moving states and a beat in the interference pattern is observed in the tight-binding calculation for the
(7,7) CNT. (d) Armchairlike: In the armchairlike CNTs, the parity symmetry is broken and interchannel scattering is enabled (see the
text). A slow modulation of the transmission pattern can be observed in the average transmission
¯
T of a (10,4) CNT (bottom right).
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secondary interference with slow oscillations of the average
transmission. The occurrence of the slow modulation can be
understood from the mode mixing within a simplified model;
see the Supplemental Material [11].Thisobservationis
confirmed by the tight-binding modeling of a (10,4) CNT in
Fig. 2(d). Our calculation clearly demonstrates that valley
mixing effects can occur also in clean CNTs [29] and cannot
be taken as an indicator of disorder.
In a realistic experiment, the coupling between the CNTand
the metallic contacts differs between CNT top and bottom parts
and depends on the fabrications details. In the Supplemental
Material, we have inv estigated the effects of an extrinsic top
versus bottom symmetry breaking at the contacts in zigzaglike
CNTs. It induces a breaking of the rotational C
d
symmetry
and hence allows for transport channel mixing. T ight-b inding
calculations conf irm that then a slow modulation of
¯
G
analogous to the armchairlike case emerges [11].
For a quantitative analysis, we extract the peak positions
E
n
¼ αV
g;n
of the slow modulation of the average con-
ductance
¯
G [green dots in Fig. 1(c)] and compare these
values to theoretical predictions. A simple model (see the
Supplemental Material [11]) shows that the slow modula-
tion is governed by the phase difference between Kramers
channels Δϕ
θ
ðEÞ¼2ðκ
θ
>
κ
θ
<
ÞL. Here, the κ
θ
>
κ
θ
<
0
are the longitudinal wave vectors measured from the same
Dirac point. In an armchairlike CNT with chiral angle θ,
k
θ
a;l
¼ K
κ
θ
<
, k
θ
b;r
¼ K
þ κ
θ
>
, k
θ
b;l
¼ K
κ
θ
>
, and
k
θ
a;r
¼ K
þ κ
θ
<
; see Fig. 2(d). For a zigzaglike CNT,
the κ
θ
>=<
are in analogy given by κ
θ
>
¼ k
θ
a;r
¼jk
θ
b;l
j and
κ
θ
<
¼jk
θ
a;l
k
θ
b;r
; see Fig. 2(b). In either case, a peak
occurs when
Δϕ
θ
ðEÞ¼2πn: ð2Þ
This result is validated by tight-binding calculations [11].
The phase difference Δϕ
θ
ðEÞ is computed numerically
from the tight-binding dispersion relation ε
θ
ðk
θ
j;i
Þ [15].Itis
shown for different chiral angles θ in Fig. 3. The slope of
Δϕ
θ
ðEÞ is monotonically increasing with θ and is 0 for the
zigzag case (Fig. 3, left inset) and maximal for the armchair
case (right inset). In the model calculation, the energy is
measured from the Dirac point. In the experiment, however,
the center of the gap is located at V
g
¼ 0.31 V. Hence, to
check whether the experimental peak positions are deter-
mined by Eq. (2), one has to account for an energy shift
ΔE
gap
¼
R
0.31V
0V
α
gap
ðV
g
ÞdV
g
, where α
gap
ðV
g
Þ is the lever
arm in the gap region. α
gap
ðV
g
Þ increases in the vicinity
of the band gap starting at V
g
¼ 0.15 V until it reaches
0.68 0.03 within the band gap [30]. The dots in Fig. 3 are
thus given by the coordinates ðE
n
þ ΔE
gap
; 2πnÞ and are
compared to Δϕ
θ
ðEÞ. The chiral angle θ can thus be used as
a fit parameter. The error bars indicate the experimental
uncertainty for ΔE
gap
, which we are only able to restrict
to a range 55 meV < ΔE
gap
< 60 meV, and α [11]. The fit
provides an estimation of 22° θ < 30° for the chiral
angle; see Fig. 3 (gray shaded area).
At high energies, the chiral angle θ and the trigonal
warping of the graphene dispersion relation alone determine
slope and curvature of the 1D subbands, and thereby the
accumulated phase difference between the Kramers channels
[11]. In a realistic experiment, there is likely an extrinsic
symmetry breaking at the contacts. Thus, channel mixing is
expected for both zigzaglike and armchairlike CNTs and in
either case allows for ev aluation of the chiral angle when
several periods of the slow modulation are recorded.
In conclusion, the secondary Fabry-Perot interference
provides a robust tool to estimate the chiral angle, a key
characteristic that is crucial for understanding carbon
nanotube properties such as the spin-orbit coupling
[31,32] or the KK
0
mixing [27]. In contrast to other
methods like, e.g., Raman spectroscopy or scanning probe
microscopy, which are difficult to combine with transport
spectroscopy, our analysis can be easily integrated with
measurements in the few-electron or in the Kondo regime.
The authors acknowledge financial support by the
Deutsche Forschungsgemeinschaft (Emmy Noether Grant
Hu 1808/1, GRK 1570, SFB 689).
*
andreas.huettel@ur.de
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clarity. Full calculations are shown in Fig. S4 of the
Supplemental Material [11].
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be explained from the CNT curvature. A curvature induced
gap of < 20 meV is estimated for a CNT with θ > 22°. Note
that for our analysis, the nature of the small band gap is not
crucial. We focus on energies ε larger than 90 meV where
the effect of the finite band gap on the dispersion is
negligible.
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PRL 117, 166804 (2016)
PHYSICAL REVIEW LETTERS
week ending
14 OCTOBER 2016
166804-5