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 [69] and the Kondo [1015] 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
+ σk
SO
+ τk
c
,
k
k
k
k
+ τk
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 [69] 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