Shaping electron wave functions in a carbon nanotube with a parallel magnetic ﬁeld

M. Marga´nska,

1

D. R. Schmid,

2

A. Dirnaichner,

2

P. L. Stiller,

2

Ch. Strunk,

2

M. Grifoni,

1

and A. K. H¨uttel

2, ∗

1

Institute for Theoretical Physics, University of Regensburg, 93053 Regensburg, Germany

2

Institute of Experimental and Applied Physics, University of Regensburg, 93053 Regensburg, Germany

(Dated: March 29, 2018)

A magnetic ﬁeld, through its vector potential, usually causes measurable changes in the electron

wave function only in the direction transverse to the ﬁeld. Here we demonstrate experimentally and

theoretically that in carbon nanotube quantum dots, combining cylindrical topology and bipartite

hexagonal lattice, a magnetic ﬁeld along the nanotube axis impacts also the longitudinal proﬁle of

the electronic states. With the high (up to 17 T) magnetic ﬁelds in our experiment they can be

tuned all the way from “half-wave resonator” shape, with nodes at both ends, to “quarter-wave

resonator” shape, with an antinode at one end. This in turn causes a distinct dependence of the

conductance on the magnetic ﬁeld. Our results demonstrate a new strategy for the control of wave

functions using magnetic ﬁelds in quantum systems with nontrivial lattice and topology.

As ﬁrst noticed by Aharonov and Bohm [1], when a

charged quantum particle travels in a region of ﬁnite

electromagnetic potential, its wave function acquires a

phase whose magnitude depends on the travelled path.

For particles with electric charge q moving along a closed

path, the phase shift ϕ

AB

= qΦ

B

/h, known as Aharonov-

Bohm shift, is expressed in terms of the magnetic ﬂux Φ

B

across the enclosed area. Because Φ

B

depends only on

the magnitude of the magnetic ﬁeld component normal

to this area’s surface, the phase is acquired along direc-

tions transverse to the magnetic ﬁeld, see Fig. 1(a). In

mesoscopic rings or tubular structures pierced by a mag-

netic ﬁeld the phase changes the quantization condition

for the tangential part of the electronic wave vector as

k

⊥

→ k

⊥

+ ϕ

AB

/r, where r is the radius of the ring or of

the tubulus, and is at the basis of remarkable quantum

interference phenomena [2]. However, as the perpendicu-

lar components of the magnetic vector potential commute

with the parallel component of the momentum, a parallel

magnetic ﬁeld is not expected to aﬀect the wave function

along the ﬁeld.

Also in carbon nanotubes (CNTs), the electronic wave

function acquires an Aharonov-Bohm phase when a mag-

netic ﬁeld is applied along the nanotube axis [3], see

Fig. 1(a). The phase gives rise to resistance oscillations

in a varying magnetic ﬂux [4]. Since the Aharonov-Bohm

phase changes k

⊥

, it also changes the energy E(k) of an

electronic state, through its dependence on the wave vec-

tor k = (k

k

, k

⊥

(B

k

)). Such a magnetic ﬁeld dependence

of the energies has been observed through beatings in

Fabry-Perot patterns [5], or in the characteristic evolu-

tion of excitation spectra of CNT quantum dots in the

sequential tunneling [6–9] and the Kondo [10–15] regimes.

In this Letter we show that the combination of the

bipartite honeycomb lattice, the cylindrical topology of

the nanotubes, and the conﬁnement in the quantum dot

intertwines the usually separable parallel and transverse

components of the wave function. This leads to an un-

∗

andreas.huettel@ur.de

(a)

K

K'

R

e

70

0 n

m

R

e

S

i

O

2

p

++

S

i

B

||

(b)

B

||

(c)

K'

K

B

||

B

||

(d)

FIG. 1. (a) Electrons circulating in closed orbits acquire

an Aharonov-Bohm phase proportional to the enclosed mag-

netic ﬂux. (b) Schematic of a suspended CNT device with its

embedded quantum dot and a magnetic ﬁeld parallel to the

nanotube. (c) Dirac cones of the graphene dispersion relation.

Blue and red lines indicate the lowermost transverse subbands

forming in a CNT. Spin degeneracy is lifted by the spin-orbit

coupling. Quantized k

k

values due to a ﬁnite CNT length

are marked with dots; B

k

= 0. (d) An axial magnetic ﬁeld

changes k

⊥

via the Aharonov-Bohm eﬀect, eﬀectively shifting

the 1-d subbands through the Dirac cones.

usual tunability of the wave function in the direction par-

allel to the magnetic ﬁeld. Experimentally, it manifests

in a pronounced variation of the conductance with mag-

netic ﬁeld, arising from the changes of the wave function

amplitude near the tunnel contacts.

Similar to graphene, in CNTs the honeycomb lattice

gives rise to two non-equivalent Dirac points K and

K

0

(also known as valleys) with associated Dirac cones.

The valley and spin degrees of freedom characterize the

four lowermost longitudinal CNT subbands, as shown in

Fig. 1(c). Our measurements display i) a conductance

rapidly vanishing in a magnetic ﬁeld for transitions asso-

ciated to the K-valley; ii) an increase and then a decrease

of the conductance for K

0

-valley transitions as the axial

2

ﬁeld is varied from 0 up to 17 T. Indications of simi-

lar behavior can also be found in results on other CNT

quantum dots [7, 9]. To our knowledge, no microscopic

model explaining this observation has yet been proposed.

Our model calculation captures this essential diﬀerence

between the K and K’ valley states.

Dispersion relation of long CNTs— In CNTs the

eigenstates are spinors in the bipartite honeycomb lat-

tice space, solving the Dirac equation, Eq. (2) below.

The resulting dispersion is E(k) = ±~v

F

q

κ

2

k

+ κ

2

⊥

, see

Fig. 1(c), where the κ

⊥/k

= k

⊥/k

− τK

⊥/k

are wave vec-

tors relative to the graphene Dirac points K (τ = 1) and

K

0

= −K (τ = −1).

The cylindrical geometry restricts the allowed values

of the transverse momentum k

⊥

through the boundary

condition Ψ(R + C) = Ψ(R ), with C the wrapping

vector of the CNT, giving rise to transverse subbands,

see Fig. 1(c). Furthermore, curvature causes a chirality-

dependent oﬀset τ ∆k

c

of the Dirac points, opening a

small gap in nominally metallic CNTs with k

⊥

= 0, as

well as a spin-orbit coupling induced shift σk

SO

of the

transverse momentum [16] (σ = ±1 denotes the pro-

jection of the spin along the CNT axis). As shown in

Fig. 1(c), the latter removes spin-degeneracy of the trans-

verse subbands. When an axial magnetic ﬁeld is applied,

the Aharonov-Bohm phase further modiﬁes k

⊥

. The en-

ergy E(k

k

, k

⊥

(B

k

)) of an inﬁnite CNT then follows again

from the Dirac equation under the replacements

k

⊥

→ k

⊥

+

ϕ

AB

r

+ σ∆k

SO

+ τ∆k

c

⊥

,

k

k

→ k

k

+ τ∆k

c

k

, (1)

the addition of a Zeeman term µ

B

σB

k

, and a ﬁeld-

independent energy shift due to the spin-orbit coupling.

In CNT quantum dots with eﬀective lengths of few hun-

dreds of nanometers the longitudinal wave vector be-

comes quantized, which gives rise to discrete bound states

(dots in Fig. 1(c)). The magnetic ﬁeld dependence of

E for two bound states belonging to diﬀerent valleys is

shown in Fig. 1(d) for ﬁxed k

k

. A characteristic evolu-

tion, distinct for the two valleys, is observed.

Magnetospectrum of a CNT quantum dot— The dif-

ferential conductance of a quantum dot as function of

applied bias voltage V

bias

and gate voltage V

gate

gives

access to its excitation spectrum [16]. Fig. 1(b) shows

a schematic of our device: a suspended CNT grown in

situ over rhenium leads [17, 18]. By tuning the back gate

voltage we can explore both the hole conduction regime

and the electron conduction regime, the latter displaying

clear Coulomb oscillations near the band gap (see also

Sect. VI of the Supplement [19]). This way it was possi-

ble to clearly identify the gate voltage region correspond-

ing to 0 ≤ N ≤ 1 trapped conduction band electrons.

Figure 2(a) shows the stability diagram of the CNT

in this gate voltage region. The resonance lines corre-

spond to the single particle energies of the lowest dis-

crete states of the quantum dot. Two closely spaced sets

12

0.67

0.68

-4

0

-8

N=0

N=1

α

β

4

8

12

0

4

8

0

B

‖

(T)

V

bias

(mV)

4

8

12

0

4

0

B

‖

(T)

β

α

V

gate

(V)

dI/dV

(a.u.)

dI/dV

(10

-5

e

2

/h)

1

0.1

5

1

0.1

0.01

dI/dV

(10

-4

e

2

/h)

0

4

8

0

B

‖

(T)

dI/dV

(10

-5

e

2

/h)

-1

(a)

α

β

K'

K

1

0.1

5

1

0.1

5

K

K'

(b)

(d)

(c)

V

bias

K'

K

V

bias

(mV)

V

bias

(mV)

FIG. 2. (a) Zero magnetic ﬁeld diﬀerential conductance

dI/dV

bias

of a CNT quantum dot with 0 ≤ N ≤ 1 conduction

band electrons. Two pairs α and β of conductance lines, all

four representing Kramers doublets, are visible. (b) Diﬀer-

ential conductance for constant V

gate

= 0.675 V and varying

magnetic ﬁeld |B

k

| ≤ 1.5 T. The Kramers doublets split at

ﬁnite ﬁeld into four states for each α and β. Spin and valley

of the α states for B

k

0.5 T are indicated. (c) Diﬀerential

conductance at the same V

gate

, now for B

k

up to 17 T. The

four visible lines correspond to K

0

states in shells α and β;

the K lines fade out fast. (d) Calculated conductance, us-

ing the reduced density matrix technique and assuming ﬁeld-

independent tunneling coupling of all states to the leads. In

contrast to the measurement, both K and K

0

valley lines

clearly persist at high magnetic ﬁeld.

of two Kramers doublets each are revealed, denoted α

and β. By ﬁxing V

gate

and sweeping a magnetic ﬁeld,

the evolution of the states in the ﬁeld can be recorded,

as shown in Figs. 2(b,c). The Kramers degeneracy is then

lifted, revealing four states in each set α and β.

Low ﬁeld spectra similar to Fig. 2(b) have been re-

ported by several groups [6–9] and are now well under-

stood. A quantitative ﬁt can be obtained by a model

Hamiltonian for a single longitudinal mode, including

valley mixing due to disorder or backscattering at the

contact (see [8] and Sec. VII of the Supplement). For

B

k

> 0.5 T, valley mixing is not relevant, and the Dirac

equation, Eq. (2) below, with curvature-induced shifts,

fully accounts for the evolution of the spectral lines. A

valley and spin degree of freedom can be assigned to each

excitation line at higher ﬁelds, as shown in Fig. 2(b).

We have traced the single particle quantum states from

Fig. 2(b) up to a high magnetic ﬁeld of B

k

= 17 T. As

visible in Fig. 2(b) and (c), the four K lines evolve up-

wards in energy. They are comparatively weak, fading

out already at ﬁelds below 1 T. In contrast, the four K

0

conductance lines evolve initially downwards in energy,

gaining in strength, but then turn upwards above 6 T