Aharonov-Bohm effect and broken valley degeneracy in graphene rings
P. Recher,
1,2
B. Trauzettel,
3,4
A. Rycerz,
5
Ya. M. Blanter,
2
C. W. J. Beenakker,
1
and A. F. Morpurgo
2
1
Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
2
Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
3
Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
4
Institute for Theoretical Physics and Astrophysics, University of Würzburg, D-97074 Würzburg, Germany
5
Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
共Received 14 June 2007; revised manuscript received 27 September 2007; published 5 December 2007
兲
We analyze theoretically the electronic properties of Aharonov-Bohm rings made of graphene. We show that
the combined effect of the ring confinement and applied magnetic flux offers a controllable way to lift the
orbital degeneracy originating from the two valleys, even in the absence of intervalley scattering. The phe-
nomenon has observable consequences on the persistent current circulating around the closed graphene ring, as
well as on the ring conductance. We explicitly confirm this prediction analytically for a circular ring with a
smooth boundary modeled by a space-dependent mass term in the Dirac equation. This model describes rings
with zero or weak intervalley scattering so that the valley isospin is a good quantum number. The tunable
breaking of the valley degeneracy by the flux allows for the controlled manipulation of valley isospins. We
compare our analytical model to another type of ring with strong intervalley scattering. For the latter case, we
study a ring of hexagonal form with lattice-terminated zigzag edges numerically. We find for the hexagonal
ring that the orbital degeneracy can still be controlled via the flux, similar to the ring with the mass
confinement.
DOI: 10.1103/PhysRevB.76.235404 PACS number共s兲: 73.23.⫺b, 81.05.Uw
I. INTRODUCTION
Graphene offers the remarkable possibility to probe pre-
dictions of quantum field theory in condensed matter sys-
tems, as its low-energy spectrum is described by the Dirac-
Weyl Hamiltonian of massless fermions.
1
However, in
graphene, Dirac electrons occur in two degenerate families,
corresponding to the presence of two different valleys in the
band structure—a phenomenon known as “fermion dou-
bling.” This valley degeneracy makes it difficult to observe
the intrinsic physics of a single valley in experiments, be-
cause in many cases, the contribution of one valley to a
measurable quantity is exactly canceled by the contribution
of the second valley. A prominent example that single-valley
physics is interesting is the production of a fictitious mag-
netic field in a single valley by a lattice defect or distortion.
2
The field has the opposite sign in the other valley, so its
effect is hidden when both valleys are equally populated.
Another example is the existence of weak antilocalization in
diffusive graphene, which is destroyed by intervalley
scattering.
3
Therefore, from a fundamental point of view, it is
desirable to find a feasible and controlled way to lift the
valley degeneracy in graphene. From a more practical point
of view, the lifting of the orbital degeneracy is essential for
spin-based quantum computing in graphene quantum dots,
4
which is a promising direction of future research because of
the superior spin coherence properties expected in carbon
structures.
Here, we show that the confinement of electrons in
graphene in an Aharonov-Bohm ring 共see Fig. 1兲 provides a
conceptually simple way to achieve a controlled lifting of the
valley degeneracy. We find that the ring confinement—which
breaks the effective time-reversal symmetry 共TRS兲
in a single valley in the absence of intervalley
scattering
5,6
—leads generically to a lifting of the valley de-
generacy controllable by magnetic flux. To demonstrate this,
we choose an analytical model with a smooth ring boundary
described by a mass term in Sec. II. Such a mass term might
be generated in a real system by the influence of the lattice of
the substrate on the band structure of the sample, see Refs. 8
and 9 for concrete examples of such an effect. However, our
conclusions are not restricted to the mass confinement but
potentially hold for any boundary that preserves the valley
isospin as implied by general symmetry arguments.
6
We
show that the signature of the broken valley degeneracy is
clearly visible in the persistent current and the conductance
through the ring. It is further illustrated how to use the lifting
of the valley degeneracy with flux to manipulate and mea-
sure the valley isospin.
FIG. 1. 共Color online兲 A circular graphene ring of radius a and
width W subjected to a magnetic flux ⌽ threading the ring.
PHYSICAL REVIEW B 76, 235404 共2007兲
1098-0121/2007/76共23兲/235404共6兲 ©2007 The American Physical Society235404-1
In Sec. III, we compare our analytical model for a smooth
boundary to a system where intervalley scattering is strong.
For this purpose, we study a ring of hexagonal shape with
zigzag edges where intervalley scattering is induced at the
corners of the hexagon. We calculate the spectrum numeri-
cally in a tight-binding approach and find that the orbital
degeneracy can still be tuned by the magnetic flux, similar to
the analytical model. We test this ability against a small dis-
tortion of the sixfold symmetry of the ring and find small
avoided crossings at zero flux.
II. RING WITH SMOOTH BOUNDARY
In this section, we analyze in detail the spectral properties
of a graphene ring subjected to a magnetic flux and its sig-
natures in persistent current and conductance through the
ring assuming a smooth confinement induced by a space-
dependent mass term in the Dirac equation. We also discuss
how to address the valley degree of freedom in such a ring
structure.
A. Spectrum
The graphene ring with valley degree of freedom
=± is
modeled by the Hamiltonian 共ប =c=1兲
H
= H
0
+
V共r兲
z
, 共1兲
where we use the valley isotropic form
7
for H
0
=
v
共p +eA兲·
, with p=−i
/
r,−e being the electron charge,
v
the Fermi velocity, and
i=x,y,z
are the Pauli matrices. The
vector potential is A= 共⌽ / 2
r兲e
, with ⌽ the magnetic flux
threading the ring, see Fig. 1. The term proportional to
z
in
Eq. 共1兲 is a mass term confining the Dirac electrons on the
ring. Introducing polar coordinates, the Hamiltonian H
0
is
written as
H
0
共r,
兲 =−i
v
共cos
x
+ sin
y
兲
r
− i
v
共cos
y
− sin
x
兲
1
r
冉
+ i
⌽
⌽
0
冊
. 共2兲
The angular orbital momentum in the z direction is l
z
=−i
and ⌽
0
=2
/ e. The two valleys
=± decouple, and we can
solve the spectrum for each valley separately, H
=E
.We
note the similarity of H
to a ring with the Rashba
interaction.
10,11
However, an important difference is that for
graphene, the confinement potential acts on the 共pseudo兲spin,
whereas for Rashba interaction, the confinement potential is
spin independent. As a consequence, the confining potential
in Eq. 共1兲 couples the pseudospin components and breaks
effective TRS 共p → −p,
→ −
兲 even in the absence of a
flux ⌽.
5
Since H
commutes with J
z
=l
z
+
1
2
z
, its eigens-
pinors
are eigenstates of J
z
,
共r,
兲 = e
i共m−1/2兲
冉
1
共r兲
2
共r兲e
i
冊
, 共3兲
with eigenvalues m, where m is a half-odd integer, m
=±
1
2
,±
3
2
,.... The radial component
共r兲⬅(
1
共r兲,
2
共r兲)
satisfies H
˜
共r兲
共r兲= E
共r兲 with,
H
˜
共r兲 =−i
v
x
r
+
V共r兲
z
+
v
y
1
r
冉
m
¯
−
1
2
0
0
m
¯
+
1
2
冊
, 共4兲
where we have defined m
¯
=m +共⌽/ ⌽
0
兲.
For V共r兲=0,
1
共r兲 and
2
共r兲 are solutions to Bessel’s dif-
ferential equation of order m
¯
−
1
2
and m
¯
+
1
2
, respectively.
Therefore, the eigenspinor for H
˜
共r兲 with energy E and
V共r兲=0 can be written as
= a
冉
H
m
¯
−1/2
共1兲
共
兲
i sgn共E兲H
m
¯
+1/2
共1兲
共
兲
冊
+ b
冉
H
m
¯
−1/2
共2兲
共
兲
i sgn共E兲H
m
¯
+1/2
共2兲
共
兲
冊
,
共5兲
where H
共1,2兲
共
兲 are Hankel functions of the 共first, second兲
kind and the dimensionless radial coordinate is
=兩E兩r/
v
.
The coefficients a
and b
are determined by the boundary
condition of the ring induced by V共r兲关with V共r兲→ +⬁ out-
side the graphene ring兴. We use the infinite mass boundary
condition
=
共n
⬜
·
兲
, where n
⬜
=共−sin
,cos
兲 at r
=a +
W
2
and with opposite sign at r = a −
W
2
.
5,7
Here, a is the
ring radius and W its width 共see Fig. 1兲. Eliminating the
coefficients a
and b
gives the energy eigenvalue equation
z= z
*
, with,
z =
H
m
¯
−1/2
共1兲
共
2
兲 −
sgn共E兲H
m
¯
+1/2
共1兲
共
2
兲
H
m
¯
−1/2
共1兲
共
1
兲 +
sgn共E兲H
m
¯
+1/2
共1兲
共
1
兲
, 共6兲
which is equivalent to
=
n, with
the phase of z and n an
integer. In Eq. 共6兲, we have abbreviated
1
⬅兩E兩
共
a−
W
2
兲
/
v
and
2
⬅兩E兩
共
a+
W
2
兲
/
v
. To obtain an analytical approximation of
the spectrum, we use the asymptotic form of the Hankel
functions for large
, including corrections up to order
1/
2
.
12
This indeed is the desired limit as
=兩E兩r/
v
⬃兩E兩a /
v
⬀a / W Ⰷ1 when the ring radius is much larger than
its width
13
and leads to the following energy eigenvalue
equation:
兩E兩 =
v
W
冉
n −
sgn共E兲
2
冊
+
1
2
冉
v
a
冊
2
m
¯
2
兩E兩
−
1
2
sgn共E兲
冉
v
a
冊
2
v
W
m
¯
兩E兩
2
. 共7兲
An iteration of Eq. 共7兲 by replacing 兩E兩 on the right-hand side
of the equation by the first 共leading兲 term of 兩E兩 gives the
energy eigenvalues 共neglecting terms of O关共W/ a兲
2
兴兲
E
nm
=±
n
±
n
m
¯
冉
m
¯
⫿
共
n +
1
2
兲
冊
. 共8兲
In Eq. 共8兲,
n
=
v
共
n+
1
2
兲
/ W, n =0,1,2,..., and
n
=共
v
/ a兲
2
/ 2
n
. These energy eigenvalues are plotted as a func-
tion of flux for n= 0 and different values of m 共its half-odd
integer values reflect the
Berry phase of closed loops in
graphene兲 in Fig. 2. Figure 2共a兲 shows the energy levels for
one valley,
= + 1. It is clearly visible that E
共m
¯
兲⫽ E
共−m
¯
兲,
since effective TRS is broken by the confinement. In Fig.
2共b兲, the spectrum of both valleys is shown with full lines
for
= + 1 and dashed lines for
=−1. At ⌽ =0, E
共m兲
RECHER et al. PHYSICAL REVIEW B 76, 235404 共2007兲
235404-2
=E
−
共−m兲 as it should be, since real TRS is present at zero
magnetic field. Crucially, however, at finite ⌽, E
+
⫽ E
−
in
general, showing that the valley degeneracy is indeed lifted
since effective and real TRSs are broken. If n Ⰷ1, the term
⬀
m
¯
in Eq. 共8兲 becomes suppressed and the valley degen-
eracy is restored, correctly predicting that the spectrum is
insensitive to the boundary condition if 2
/ q
n
ⰆW, where
q
n
=
共
n+
1
2
兲
/ W is the transverse wave number. We show next
that a broken valley degeneracy results in observable features
in the persistent current and the conductance through the
ring.
B. Persistent current and conductance
The persistent current in the closed ring is given at zero
temperature by j =−兺
兺
nm
E
nm
/
⌽, where the sum runs
over all occupied states. In Fig. 3, we show the persistent
current as a function of number of electrons on the ring N
共including spin兲 and magnetic flux relative to the half-filled
band. 共We subtract the contribution to the persistent current
that arises from all states with E ⬍0.兲 The persistent current
is periodic in ⌽ with period ⌽
0
. In Fig. 3共a兲, only one valley,
= + 1, is considered and a finite persistent current at ⌽ =0 is
predicted. Therefore, a nonzero persistent current at zero flux
detects valley polarization. In Fig. 3共b兲, we show the case of
equal population of both valleys. Then, the persistent current
as a function of flux is zero at ⌽= 0 but shows a substructure
共kinks at ⌽⫽ 0兲 within one period directly related to the
broken valley degeneracy at finite flux. We note that this
substructure is due to the linear term in m
¯
of the spectrum,
Eq. 共8兲, which is prominent within the first few transverse
modes n which can host many electrons N.
In Fig. 4, we plot the conductance through the ring
weakly coupled to leads as a function of Fermi energy E
F
共or
gate voltage兲 assuming a constant interaction model
14
with
charging energy U.
15
At ⌽= 0, the conductance exhibits a
fourfold symmetry due to spin and valley degeneracies. A
finite flux breaks the valley degeneracy which is observable
via a splitting of the conductance peaks moving with mag-
netic flux, see Fig. 4.
C. Valley qubit
We now turn to the question of how to make use of the
broken valley degeneracy in order to directly address the
valley degree of freedom in graphene experimentally
共valleytronics
16
兲. The valley degree of freedom forms 共in
principle兲 a two-level system that can be represented by an
isospin 兩⫹典 for valley
= + 1 and 兩⫺典 for valley
=−1. We
point out that the graphene ring weakly coupled to current
leads could be used to investigate the relaxation and coher-
ence of such valley isospins.
1 0.5 0 0.5 1
1.57
1.58
1.59
1
.
6
Φ
0
[
[
[v/W]
5
5
E
+
0m
a)
1 0.5 0 0.5
1
1.57
1.58
1.59
1.6
E
±
0m
[v/W]
Φ
0
[
[
b)
FIG. 2. 共Color online兲 Energy spectrum E
0m
共E ⬎0兲 as a function
of magnetic flux ⌽ for various total angular-momentum values m
共blue m⬎0, red m⬍0兲 using Eq. 共8兲 with a/ W= 10: 共a兲 shows a
single valley only
= +1. The dotted lines are the exact numerical
evaluation of E using Eq. 共6兲.In共b兲, we show the full spectrum
including the other valley
=−1 共dashed lines兲. The restoration of
±⌽ symmetry in the combined spectrum of both valleys and the
lifting of the valley degeneracy at finite ⌽ are clearly visible.
0 0.2 0.4 0.6 0.8
0
2
4
6
8
0.15
Φ
0
[
[
j [eλ /π]
0
N=1
N=2
N=3
N=4
a)
0.5 0.25 0 0.25 0.5
0
2
4
6
Φ
0
[
[
j [eλ /π]
0
N=1
N=2
N=3
N=4
b)
FIG. 3. 共Color online兲 Persistent current as a function particle
number N 共including spin兲 and flux ⌽ for n =0 共E⬎0 兲: 共a兲 includes
only valley
= +1, whereas 共b兲 includes both valleys. Curves for
different N are displaced with dashed horizontal lines defining
j = 0 for each curve. The broken valley degeneracy is clearly
visible in 共b兲 via two substructures of length ⌬⌽=2⌽
0
/
and
⌬⌽=共1−共2/
兲兲⌽
0
, whereas 共a兲 predicts a nonzero persistent
current at ⌽= 0 due to effective TRS breaking in a single valley.
AHARONOV-BOHM EFFECT AND BROKEN VALLEY… PHYSICAL REVIEW B 76, 235404 共2007兲
235404-3
Close to a degeneracy point of two levels belonging to
different valleys 共e.g., at ⌽ =0兲, Eq. 共8兲 predicts a valley
splitting of states with fixed m values controllable by flux,
similar to the Zeeman splitting for electron spins in a mag-
netic field. In semiconductor quantum dots, such pairs of
spin-split states can be addressed via electron tunneling from
and/or to leads weakly coupled to the quantum dot and can
be used for readout of single spins
17
or measuring their re-
laxation 共T
1
兲 time.
18
The graphene ring could be used in very
much the same way to measure the intrinsic valley isospin
relaxation time T
1
in graphene as well as the valley isospin
polarization.
In Fig. 5, we show the situation when some small level
mixing leads to avoided crossings of valley-split states near
the degeneracy point ⌽= 0. Such valley mixing naturally ap-
pears through boundary roughness of the ring or atomic de-
fects in the bulk. Using the magnetic flux as a knob, we can
sweep the system from a 兩⫹典 ground state level, filled with
one electron, to a superposition 共兩+ 典+ 兩−典兲/
冑
2 and further to
a 兩⫺典 ground state, see Fig. 5. Such a situation can be used to
produce Rabi oscillations of the valley isospin states by tun-
ing the system fast 共nonadiabatically兲 from 兩⫹典 to the degen-
eracy point where the spin will oscillate between 兩⫹典 and 兩⫺典
in time: cos共⌬t兲兩+ 典− i sin共⌬t兲兩−典, where 2⌬ is the energy
splitting at the degeneracy point.
19
We expect that this qubit is rather robust if intervalley
scattering is weak, since time-reversal symmetry assures the
共approximate兲 degeneracy of states from different valleys at
zero flux ⌽. Indeed, the Hamiltonian 关Eq. 共1兲兴 has the same
spectrum in both valleys at zero flux, independent of the
shape of the mass potential V共x ,y兲.
20
This means that we do
not rely on a special symmetry of the confining potential
共such as the circle discussed here兲. In addition, long-range
disorder will also not lift the valley degeneracy.
D. Valley isospin-orbit coupling
In an open ring geometry with adiabatic contacts to leads,
new interesting coherent rotations of the valley isospin occur
while propagating along the ring. The linear term in m
¯
in Eq.
共8兲 can be thought of as a valley isospin-orbit coupling term,
since the valley isospin
couples to the orbital motion m
¯
.A
general incoming spinor is a superposition of spinors belong-
ing to different valleys. Due to the valley isospin-orbit cou-
pling, the angular momentum m 关determined by the incom-
ing 共continuous兲 energy E and the applied magnetic flux ⌽
via Eq. 共8兲兴 will be different for the two valleys. Conse-
quently, the spinor in Eq. 共3兲 will pick up different phases
exp共im
兲 for the two valleys while propagating along the
ring, thereby rotating the valley isospin in a transport experi-
ment.
III. SPECTRUM FOR A HEXAGONAL RING WITH
ZIGZAG EDGES
Here, we compare our analytical model with the infinite
mass boundary described in Sec. II to a ring with strong
intervalley scattering. We numerically investigate the spec-
trum of a ring of hexagonal form with zigzag edges as shown
in Fig. 6. Electrical conduction through this geometry was
studied in 共Ref. 21.兲
In a zigzag nanoribbon, the valley isospin is a good quan-
tum number, i.e., the zigzag boundary does not mix valleys.
22
Since two neighboring arms of the ring are rotated by 60°
with respect to each other, the roles of the A and B sublat-
tices are interchanged in subsequent segments. Explicitly,
this means that if a zigzag edge is terminated on a A side, it
will be terminated on a B side at a neighboring arm of the
ring. Equivalently, in the reciprocal 共k兲 space, this implies
that equivalent states of subsequent zigzag nanoribbon seg-
ments are lying in opposite valleys. This necessarily induces
intervalley mixing at the corners between two subsequent
zigzag nanoribbon segments. This mixing is very strong in
the lowest mode of the ring, where the direction of motion
and the valley is tightly coupled in each arm of the hexago-
nal ring
16,22
共the zigzag edge is therefore another example
Conductance
Φ=0
Φ = 0.1Φ
0
N=1,2,3,4
N=1,2,3,4
N=1,2
N=1,2
N=3,4
N=3,4
N=5,6
N=5,6
N=7,8
N=7,8
N=9,10
N=9,10
N=11,12
N=11,12
N=5,6,7,8
N=5,6,7,8
N=9,10,11,12
N=9,10,11,12
E –NU
F
FIG. 4. 共Color online兲 Ring conductance assuming a constant
interaction model with charging energy U for the first 12 electrons
in the conduction band 共E ⬎0兲:At⌽ =0 共dashed兲, the conductance
shows a fourfold symmetry as a function of Fermi energy E
F
in the
leads due to spin and valley degeneracies. At finite magnetic flux 共
⌽/ ⌽
0
=0.1, full line兲, the conductance peaks shift due to breaking
of the valley degeneracy. Each peak is labeled with the filling factor
N at this specific resonance.
-0.2 -0.1 0 0.1 0.2
1.571
1.574
E
0m
±
Φ [Φ ]
0
m=-½
m=+½
m=-½
m=+½
[v/W]
|+> + |–>
|+> – |–>
|+>
|+>
|–>
|–>
FIG. 5. 共Color online兲 Valley qubit. The lowest four levels 共for
n= 0 and a / W=10兲 are shown as a function of flux. Blue and red
共dashed兲 lines correspond to valleys
= +1 and
=−1, respectively.
The dotted lines take into account small level mixing leading to
anticrossings. The flux ⌽ is used to switch from 兩⫹典 to the crossing
point with the new eigenstates being superpositions of 兩⫹典 and 兩⫺典
as indicated in the figure.
RECHER et al. PHYSICAL REVIEW B 76, 235404 共2007兲
235404-4
where effective TRS in a single valley is broken兲. An elec-
tron wave, approaching a corner of the hexagon in one val-
ley, is either transmitted into the next arm or reflected back
into the same arm. In both cases, the valley index has to flip.
We investigate the spectrum of such a ring numerically in
a tight-binding approach with Hamiltonian
H =
兺
i,j
t
ij
兩i典具j兩 +
兺
i
⑀
i
兩i典具i兩. 共9兲
The hopping element in the presence of a magnetic flux is
t
ij
=−t exp关−i共2
/ ⌽
0
兲兰
r
j
r
i
dr ·A兴, where A is the vector poten-
tial and
⑀
i
=0 are the on-site energies. The vector potential is
chosen as A= 共A
x
,0,0兲, with
A
x
= B关y
a
⌰共y兲 − y
a
⌰共− y兲兴⌰共L
C
/2−兩x兩兲, 共10兲
where y
a
共x兲= min关r
a
,
冑
3共L
C
/ 2−兩x兩兲兴, with L
C
=4r
a
/
冑
3 for
the perfect hexagon 共shown in Fig. 6兲 and L
C
=4r
a
/
冑
3+l for
a hexagon ring with one unit cell added to the top and bot-
tom arms. This represents a uniform magnetic field B inside
the ring hole and zero outside. The spectrum as a function of
magnetic flux ⌽ =2
冑
3r
a
2
B is shown in Fig. 7 in an energy
window which lies well within the lowest mode of a zigzag
nanoribbon of width W= r
b
−r
a
.
23
A band of levels in the
lowest mode is shown in Fig. 7共a兲. Within that mode, the
spectrum follows a clear pattern which is observed for ge-
neric values of r
a
and r
b
. It consists of bands separated by
energy gaps. Each band hosts six levels. The top and bottom
levels are nondegenerate with dE/ d⌽ =0 at zero flux 共corre-
sponding to standing waves兲. The other four levels are two-
fold degenerate at ⌽= 0 with a broken degeneracy at finite
flux. These levels correspond to right- and left-going states in
the ring.
We remark that this level pattern reflects the scattering off
a periodic array of six scatterers subjected to periodic bound-
ary conditions.
24
It is to be noted that the orbital degeneracy
of the hexagonal ring can be tuned by the flux, similar to the
ring with the smooth confinement discussed in Sec. II. If the
sixfold rotational symmetry of the ring is broken, the cross-
ings at zero flux become slightly avoided as is shown in Fig.
7共b兲 共blue circles兲 where we have added one unit cell to two
of the parallel arms of the ring 共this corresponds to a length
change of the arms by about 5%兲. This shows that our results
are also relevant for rings with a slightly reduced symmetry.
Note that the sensitivity of the level crossing at zero flux to
the ring geometry is consistent with strong intervalley scat-
tering where time-reversal symmetry does not protect the
degeneracy at zero flux.
IV. CONCLUSION
We have analyzed the Aharonov-Bohm effect in graphene
rings. We have investigated two different ring systems—a
ring with a smooth boundary 共with zero or weak intervalley
scattering兲 and a hexagonal ring with zigzag edges. For the
ring with a smooth boundary, the combined effect of the
effective time-reversal symmetry 共TRS兲 breaking within a
single valley induced by a smooth boundary and the applied
magnetic flux—breaking the real TRS—gives us at hand a
controllable tool to break the valley degeneracy in such
rings. We have shown that this effect of a broken valley
degeneracy by flux is revealed in the persistent current and in
the ring conductance. This tool could be useful for spin-
based or valley-based quantum computing. The presence of a
degenerate pair of levels from different valleys at zero flux is
FIG. 6. 共Color online兲 A hexagonal ring with zigzag edges of
inner radius r
a
and outer radius r
b
and flux through the hole ⌽.
Here, r
a
=3
冑
3l and r
b
=6
冑
3l, with l the lattice spacing
0.07
0.08
0.09
0.1
-0.5 -0.25 0 0.25 0.5
E[t]
Φ[Φ
0
]
a)
0.03
0.0325
0.035
0.0375
-0.25 0 0.25
E[t]
Φ[Φ
0
]
b)
FIG. 7. 共Color online兲 Plot 共a兲 shows an energy band of the
hexagonal ring 共see Fig. 6兲 with ring dimensions r
a
=7
冑
3l and
r
b
=14
冑
3l in the lowest mode 0⬍E ⱗ 0.34t 共Ref. 23兲. The levels are
grouped into bands containing six levels. The topmost and the low-
est level in each band is nondegenerate, whereas the middle four
levels are twofold degenerate at zero flux. This degeneracy is lifted
by the flux through the ring. In 共b兲, we contrast the perfect crossing
of two levels at zero flux 共red dots兲 with anticrossed levels 共blue
circles兲 induced by the addition of one unit cell to each of the two
parallel arms of the ring 关we have shifted the energy axis for the
asymmetric case 共blue circles兲 by 共+3⫻10
−3
兲t for better
comparison兴.
AHARONOV-BOHM EFFECT AND BROKEN VALLEY… PHYSICAL REVIEW B 76, 235404 共2007兲
235404-5
