Secondary Electron Interference from Trigonal Warping in Clean Carbon Nanotubes

A. Dirnaichner,

1,2

M. del Valle,

K. J. G. Götz,

F. J. Schupp,

N. Paradiso,

M. Grifoni,

Ch. Strunk,

and A. K. Hüttel

1,*

Institute for Experimental and Applied Physics, University of Regensburg, 93040 Regensburg, Germany

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 [1,4–6].

Its hallmark is an oscillatory behavior of the differential

conductance GðV

Þ as a function of both gate voltage

and bias voltage V

, 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

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

Þ allows us to estimate the

CNT’s 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

Þ of the CNT

device as function of gate voltage V

and bias voltage V

On the electron conduction side (V

> 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

≲ 1.8 (G

¼ e

=h) is observed for gate

voltage values −15 V ≲ V

≲ 0 V. The electron wave

vector is affected by both bias and gate voltage, leading

to typical rhombic interference structures in the GðV

diagram [1]. A striking feature of our data is the slow

modulation of the conductance pattern as a function of V

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

Þ for V

¼ 0. Primarily, we observe a fast oscillation

of the conductance at a frequency f

¼ 12.8 V

−1

. This

fundamental frequency is directly related to the length of

the cavity via f

≃ αeL=πℏv

[1].Forv

¼ 8 × 10

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

fast

1=f

and the height V

of the rhombic pattern in Fig. 1(a),

we extract the gate voltage lever arm α ¼ V

=ΔV

fast

0.0210  0.0007 [1]. On top of the fast oscillations, the

slow modulation is visible. Figure 1(c) shows the sliding

average

GðV

Þ of the conductance as function of V

. 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

¼ αΔV

g;n

decreases for more neg-

ative gate voltages V

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

with a

dominant fundamental frequency component f

alternating

with regions where the second harmonic with f

¼ 2f

prevails. These reflect the frequency doubling that is visible

in certain ranges of V

in Fig. 1(b). In these regions, the FT

also reveals components from higher harmonic frequencies

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¼ nf

, 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 Green’s 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

[26–28]: 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,

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

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]

G/G

-1

(

mV)

(b)

−14 −12 −10 −8 −6 −2

f(1/V)

(d)

gate voltage (V)

G/G

(a)

−3

−2

−1

log(|A

|/G

)

1.4

0.2

(c)

−12 −10 −8 −6 −2−4

−14 0

n=7

1.0

0.6

FIG. 1. (a) Differential conductance GðV

Þ of a clean CNT device in the hole conduction regime, as a function of back gate voltage

and bias voltage V

¼ e

=h). (b) Zero bias conductance GðV

Þ extracted from (a). (c) Average conductance

GðV

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

Þ¼

j¼a;b

2jt

1 þjr

− 2jr

∥r

j cos½ϕ

ðV

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

, t

and r

, r

, respectively.

The phase accumulated upon one round trip is given by

ðV

Þ¼ðjk

j;l

ðV

Þj þ jk

j;r

ðV

ÞjÞL, and the wave vector of

the right (left) moving electron k

j;rðlÞ

is linked to V

via the

CNT dispersion relation εðk

j;rðlÞ

Þ¼αeV

. 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

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

On the other hand, in armchair and armchairlike CNTs,

the bands cross at the Dirac points

K ¼ðK

∥

; 0Þ and

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

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

¼ 2k

a;r

L ≠ ϕ

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,l

0 0

a,r

(eV)

aa bb

(eV)

b aa

(eV)

-0.16 -0.10

aa bb

armchair

(eV)

aab

a,r

a,l

(10,4)

-0.22

(6,3)

-0.16

-0.10

-0.22

(12,0)

(7,7)

-0.16 -0.10-0.22

zig-zag

zig-zag-like

armchair-like

b,r

a,l

b,r

b,l

b,r

a,r

b,l

b,r

a,l

a,r

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

symmetry

and hence allows for transport channel mixing. T ight-b inding

calculations conf irm that then a slow modulation of

analogous to the armchairlike case emerges [11].

For a quantitative analysis, we extract the peak positions

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

ðEÞ¼2ðκ

− κ

ÞL. Here, the κ

≥ κ

≥ 0

are the longitudinal wave vectors measured from the same

Dirac point. In an armchairlike CNT with chiral angle θ,

a;l

¼ −K

∥

− κ

, k

b;r

¼ −K

∥

þ κ

, k

b;l

¼ K

∥

− κ

, and

a;r

¼ K

∥

þ κ

; see Fig. 2(d). For a zigzaglike CNT,

the κ

>=<

are in analogy given by κ

¼ k

a;r

¼jk

b;l

j and

¼jk

a;l

j¼k

b;r

; see Fig. 2(b). In either case, a peak

occurs when

Δϕ

ðEÞ¼2πn: ð2Þ

This result is validated by tight-binding calculations [11].

The phase difference Δϕ

ðEÞ is computed numerically

from the tight-binding dispersion relation ε

ðk

j;i

Þ [15].Itis

shown for different chiral angles θ in Fig. 3. The slope of

Δϕ

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

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

0.31V

gap

ðV

ÞdV

, where α

gap

ðV

Þ is the lever

arm in the gap region. α

gap

ðV

Þ increases in the vicinity

of the band gap starting at V

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

þ ΔE

gap

; 2πnÞ and are

compared to Δϕ

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

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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