Bloch-Zener Oscillations in Graphene and Topological Insulators

Abstract

We show that superlattices based on zero-gap semiconductors such as graphene
and mercury telluride exhibit characteristic Bloch--Zener oscillations that
emerge from the coherent superposition of Bloch oscillations and multiple Zener
tunneling between the electron and hole branch. We demonstrate this mechanism
by means of wave packet dynamics in various spatially periodically modulated
nanoribbons subject to an external bias field. The associated Bloch frequencies
exhibit a peculiar periodic bias dependence which we explain within a two-band
model. Supported by extensive numerical transport calculations, we show that
this effect gives rise to distinct current oscillations observable in the I-V
characteristics of graphene and mercury telluride superlattices.

Full text

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arXiv:1109.5541v1 [cond-mat.mes-hall] 26 Sep 2011
Bloch–Zener Oscillations in Graphene and Topological Insulators
Viktor Krueckl and Klaus Richter
Institut f¨ ur Theoretische Physik, Universit¨ at Regensburg, D-93040 Regensburg, Germany
(Dated: September 27, 2011)
We show that superlattices based on zero-gap semiconductors such as graphene and mercury
telluride exhibit characteristic Bloch–Zener oscillations that emerge from the coherent superposi-
tion of Bloch oscillations and multiple Zener tunneling between the electron and hole branch. We
demonstrate this mechanism by means of wave packet dynamics in various spatially periodically
modulated nanoribbons subject to an external bias field. The associated Bloch frequencies exhibit
a peculiar periodic bias dependence which we explain within a two-band model. Supported by
extensive numerical transport calculations, we show that this effect gives rise to distinct current
oscillations observable in the I-V characteristics of graphene and mercury telluride superlattices.
PACS numbers: 72.80.Vp, 73.21.Cd, 85.35.Ds, 85.75.Mm
Bloch oscillations, the periodic motion of particles in
a superlattice subject to a constant external field, rep-
resent a fundamental phenomenon in transport through
periodic potentials. Predicted already in the early days
of quantum mechanics [1], Bloch oscillations have been
observed in various fields of physics, ranging from ear-
lier experiments in semiconductor superlattices [2] via
cold atoms in optical lattices [3] to classical optical [4]
and acoustic [5] waves.
While many aspects of con-
ventional Bloch oscillations can be explained by a sin-
gle band description, particularly interesting effects arise
in the case of two coupled minibands [6] energetically
seprated from further bands. Then partial Zener tunnel-
ing at avoided crossings of the two minibands can lead to
a coherent superposition of Bloch oscillations [7], i.e. to
a splitting, followed by as subsequent recombination of
a Bloch oscillating wave packet. This gives rise to a va-
riety of Rabi-type interference phenomena, in particular
double-periodic motions coined Bloch–Zener (BZ) oscil-
lations [8]. Signatures of this effect have already been de-
tected in the THz emission of AlGaAs superlattices [9],
and even the population dynamics has been measured
recently for light [10] and atomic matter waves [11] in
especially tailored binary lattices.
However, materials with a linear Dirac spectrum [12]
should naturally provide the effect, since only a small
gap is opened by a spatially periodic modulation al-
lowing for Zener tunneling between electron and hole
states. Such materials are now at hand with the dis-
covery of graphene [13] and the advent of topological
insulators [14] first realized in two-dimensional mercury
teluride (HgTe) heterostructures [15]. Interesting phe-
nomena for graphene superstructures have already been
theoretically predicted like the formation of extra Dirac
cones [16] and the appearance of a negative differential
conductance [17]. Furthermore recent experiments have
realized graphene superlattices with periodicities down
to a few nm [18].
This raises the question for the existence of Bloch
oscillations and their possible peculiarities in graphene
and topological insulator superlattices that we address in
this manuscript [19]. We show that besides conventional
Bloch oscillations, multiple Zener tunneling between the
coupled electron and hole branches leads to distinct BZ
oscillations that appear to be naturally present in su-
perlattices made of systems with Dirac-like dispersion.
We demonstrate the influence of these tunneling events
on the wave packet motion in biased graphene nanorib-
bons and explain the effect by a two-band model. Subse-
quently, we present how transport through graphene and
mercury teluride is affected by such BZ oscillations and
suggest possible setups for an experimental detection.
We start with the dynamics of a wave packet under-
going Bloch oscillations on a graphene nanoribbon with
a periodic mass potential M(x) = M0sin(2πx/a) and
a linear drift potential V (x) = −eEDx as sketched in
Fig. 1(a). To this end we model the electronic struc-
ture of graphene by a conventional tight-binding Hamil-
tonian [21]
Htb=
?
?i j?,β
t c†
i,−βcj,β+ V c†
i,βci,β+ M β c†
i,βci,β (1)
where ?i j? denotes neighbouring unit cells and β = ±1
the sublattice degree of freedom. Based on the transver-
sal eigenstates of the armchair terminated nanoribbon,
we create an initial wave packet localized in one band
with a Gaussian envelope in longitudinal direction cover-
ing several periods a of the periodic potential. Due to this
extent, the wave packet is also localized in momentum
space with a distinct average momentum in x-direction.
We calculate the time evolution of the wave packet by
a Chebechev propagation algorithm [22] and extract the
center-of-mass (COM) motion. In presence of the drift
field EDthe wave packet starts to accelerate and, given
the periodicity of its average momentum k(t), we get a
sawtooth behavior of k(t) known as Bloch oscillations.
Moreover, as shown in Fig. 1(b), a single trajectory of
the wave packet exhibits a beating pattern on top of reg-
ular Bloch oscillations which suggests that more bands
are involved in the time evolution. This behavior is also

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k a
-1
-0.5
0
0.5
1
E(eV)
V(x)
4.2 4.4 4.6 4.8
0
5
10
15
20
E(meV)
4.24.4
ED(mV/nm)
4.6
4.8
0
5
10
15
E(meV)
510
t(ps)
-30
-15
0
15
30
x(nm)
M(x)
d)
e)
c)
b)
a)
FIG. 1.
nanoribbon. a) Sketch of a Gaussian wave packet in presence
of a periodic mass potential M(x) = M0sin(2πx/a) and drift
potential V (x) = −eEDx. b) COM motion of the wave packet
showing a beating pattern (ED = 4.025 mV /nm, nanoribbon
width W = 10 a0, a = 10√3 a0, M0 = 0.1 t). c) Bandstruc-
ture of the superlattice with small avoided crossing at k = 0.
Thick and dashed lines show the first and second Bloch band
from the metallic armchair mode; gray lines higher modes.
d,e) Frequency spectra E = ?ω from the COM motion of a
wave packet for varying drift potential ED for (d) moderate
(M0 = 0.1t) and (e) stronger (M0 = 0.2t) periodic potential.
Dark colors represent strong intensities.
correspond to {1/2, 1, 3/2} times the conventional Bloch fre-
quency.
(Color online) Bloch oscillations in a graphene
The dashed lines
deducible from the mini-band structure of the superlat-
tice as shown in Fig. 1(c). A state initially starting on
the electronic branch (large bullet in Fig. 1(c)) can tun-
nel into the hole branch through a small avoided crossing
at k(t) = 0. To study this dynamics we perform a fre-
quency analysis of the COM motion for different ED.
The Fourier amplitudes of the dominant frequency con-
tributions are visualized by dark colors in Fig. 1(d,e).
Besides the conventional Bloch frequency (white dashed
line), the resulting spectrum shows a pronounced in-
terweaving pattern around half of this frequency (black
dashed line). A stronger periodic potential, and thereby
an increased gap between electron and hole branch, leads
to a rhombic structure as shown in Fig. 1(e). These peri-
odic features in the frequency spectrum arise from the in-
terplay between Bloch oscillations and splitting the wave
packet into (subsequently interfering) electron and hole
branches at k(t) = 0 (see Fig. 1(c) and Supplemental
Material [23]).
In the following we quantitatively explain these charac-
teristic BZ features using a periodically modulated one-
-3-2-10123
k a
-20
-10
0
10
20
ε
4.1 4.2 4.3 4.4
α
0
0.2
0.4
0.6
0.8
ω
ω−
ω+
ω−+ω+
ω−+2ω+
2ω−+ω+
ε−
ε+
2g
a)
b)
e-iξ√
√
1-q
q
∫(ε+−ε−) dk
FIG. 2. (Color online) a) Bandstructure of the Dirac model
Hamiltonian (2) for v = 1, ? = 1, a = 1/10, g = 1/2. b)
Frequency spectrum of the Bloch oscillations for different drift
accelerations α = eED/?. Solid lines show the frequencies
nω++mω−given by Eq. (7), dotted (dashed) lines the strong
(weak) tunneling limit.
dimensional Dirac model Hamiltonian,
H(t) =2?v
asin
?ak(t)
2
?
σz+ g σx. (2)
Here a is the period, v is the Fermi velocity and g the
energy gap between the electron and the hole states. The
resulting bandstructure is given by
ǫ±(t) = ±
?
g2+ 2(?v/a)2[1 − cos(ak)](3)
as shown in Fig. 2(a) [to be compared to full tight-binding
result of Fig. 1(c)]. A drift field EDenters the equations
of motion for the quasi-momentum k(t) as ?∂tk(t) =
eED leading to a time evolution of k(t) = αt linear in
t where α = eED/?. Conventional Bloch oscillations
with frequency ωB = αa arise from the periodicity of
k(t) in momentum space in the interval [−π
phase φ between the two branches accumulated during
one oscillation is given by a free propagation and thus
φ = A(eED)−1≈
area in momentum space as depicted in Fig. 2(a). This
free propagation can be expressed by the matrix
a,π
a]. The
16v
a2αwith A =?π/a
−π/a(ǫ+− ǫ−)dk the
U0=
?eiφ/2
0
0e−iφ/2
?
.(4)
Additional to conventional Bloch oscillations on either
branch, there is a strong periodic tunneling between the
electron and the hole states close to the anti-crossing at
k = 0. There, the Hamiltonian (2) can be linearized
[dashed lines in Fig. 2(a)], leading to a typical Landau–
Zener tunneling problem [24]:
HLZ=
??v αtg
g−?v αt
?
. (5)
Scattering between the different branches is described by
S0=
?e−iξ√q
√1 − q
√1 − q −eiξ√q
?
(6)

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3
with the tunneling rate q = 1 − e−2πδ, δ =
ξ =π
4+arg(1−iδ)+δ(logδ−1) is an additional tunneling
phase. From this we can deduce the scattering matrix
for one Bloch oscillation as S = U0S0. The periodicity of
the scattering eigenstates, given by the argument of their
eigenphases, leads to two new Bloch frequencies
g2
2?2vα, and
ω±=αa
π
arccos?±√q sin(φ/2 − ξ)?. (7)
Unlike standard Bloch oscillations these frequencies do
not simply depend linearly on the drift strength α, but
show a rapid interweaving pattern, as shown in Fig. 2(b),
owing to coherences from combined dynamics on the hole
and electron branch. For strong coupling, the tunneling
rate q → 0 leads to a frequency ω±→ ωB/2 [dotted
line in Fig. 2(b)], since for every Bloch cycle the states
tunnel completely between the two branches in momen-
tum space and hence the complete cycle in position space
is twice as long. In the opposite, weak coupling limit
ω±→ aα[1/2 ± (φ/2 − ξ)/π] mod 1 leading to a rhom-
bic frequency pattern shown as dashed lines in Fig. 2(b).
For intermediate tunneling rates the frequencies show a
smooth transition between these limiting cases and are
in very good agreement with the numerically calculated
spectra of Fig. 1(d,e).
In the following, we consider charge transport through
graphene nanoribbon superlattices and demonstrate that
BZ oscillations lead to clear-cut features in the I-V char-
acteristics. To this end we model a graphene nanoribbon
of width W and length L by the tight-binding Hamil-
toninan of Eq. (1) now with a periodic electrostatic
potential V0sin(2πx/a) leading to a superlattice mini-
bandstructure as shown in the inset of Fig. 3(a). A small
constant mass term M(x) = M0 is additionally consid-
ered which opens up a gap commonly present in experi-
ments on graphene nanoribbons [25]. We assume a linear
potential drop eVSDx/L due to the source-drain voltage
VSDbetween the graphene leads at x = ±L/2. The cur-
rent is calculated by means of the Landauer-B¨ uttiker for-
malism [26],
I(VSD) =2e
h
?∞
−∞
T(E,VSD)[f+(E) − f−(E)]dE,(8)
with f±(E) = {1 + exp[(E ∓ VSD/2)/kBT]}−1.
As shown in Fig. 3, the current through the nanoribbon
is governed by a conventional increase with the bias win-
dow for small VSD, followed by a region of negative dif-
ferential conductance typical for superlattices. At higher
bias, VSD> 0.3V, we observe the emergence of distinct
current oscillations that get more pronounced with in-
creasing gap size, see Fig 3(b). Due to the bias poten-
tial the particles traversing the superlattice must change
their electron-hole character. However, states perform-
ing BZ oscillations exhibit transitions between the two
carrier types only for certain VSD= eEDL if the phase
-3 -2-10123
k a
-1
-0.5
0
0.5
1
E(eV)
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
0 0.1 0.2 0.30.40.5
0
2
4
6
8
10
I(µA)
EF= 0 meV
EF= 30 meV
EF= 60 meV
↑↑↑↑↑↑↑↑
0.4
↑↑↑↑↑↑↑↑↑
0 0.10.20.3
0.5
VSD(V)
0
0.1
0.2
I(µA)
1K
20 K
-0.1-0.050 0.05 0.1
E(eV)
0.34
0.35
0.36
0.37
VSD(V)
a)
b)
FIG. 3.
graphene nanoribbon superlattices (L = 3000√3a0, W =
10a0, a = 30√3a0, V0 = 500 meV) for (a) different Fermi
energies (M0 = 20meV, T = 20K) and (b) different tem-
peratures (M0 = 50meV, EF = 0) showing signatures of
Bloch–Zener oscillations at higher bias.
pected peak positions from phase condition (9). Upper inset:
Bandstructure (for M0 = 20meV), lower inset: Transmission
map T(E,VSD) used in Eq. (8) to get the current of panel (b);
dark colors represent high transmissions.
(Color online) Current-voltage characteristics for
Arrows mark ex-
[see Eq. (4)] fulfills
φ =AL
VSD
= 2(nπ + ξ) + π(n ∈ N) (9)
as shown in the Supplemental Material [23]. In conse-
quence the current is strongly enhanced whenever φ ful-
fills this condition. As shown in Fig. 3(a,b) the current
peaks calculated by Eq. (8) perfectly coincide with the
expected voltages (marked by vertical arrows) deduced
by extracting the area A in momentum space from the
minibandes around the Fermi energy shown as shaded
area in the inset of Fig. 3(a). Vice versa, the experimen-
tal observation of BZ peaks in the I-V characteristics
would allow for ‘measuring’ the miniband structure.
A closer look at the transmission values T(E,VSD)
[see inset Fig. 3(b)] reveals a rhombic structure which
features pronounced transmission maxima piled up for
the particular values of VSD(dashed lines) in accordance
with Eq. (9). Since these maxima are present for various
energies in the conductance window, the resulting cur-
rent is fairly independent of the exact Fermi energy [see
Fig. 3(a)] and temperature [see Fig. 3(b)].
The whole frequency spectrum of the BZ oscillations
can be monitored, if the current is measured in presence
of a tunable laser field. We present corresponding Flo-
quet transport calculations for graphene nanoribbons in

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k a
-20
-10
0
10
20
E(meV)
-303
k a
-20
-10
0
10
20
E(meV)
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓
↓ ↓ ↓
0
5
10
15
20
VSD(mV)
0
10
20
30
40
50
I(nA)
EF= 7.5 meV
EF= 8.0 meV
EF= 8.5 meV
2
3
4
56
ED(mV/nm)
0
0.5
1
1.5
2
E(meV)
a)b)
c)
FIG. 4. (Color online) Bloch and Bloch–Zener oscillations in
spatially modulated two-dimensional HgTe nanoribbons. a)
Bandstructure for a HgTe nanoribbon with periodically mod-
ulated width W(x) ranging from W0 = 300 nm to W1 = 50 nm
and periodicity a = 200 nm [23]. b) Bloch frequency spec-
trum E = ?ω of the wave packet COM motion as a function
of drift field ED. c) I-VSD characteristics of a nanoribbon
with constant width W = 150 nm and electrostatic mod-
ulation V (x) = V0sin(2πx/a). Small vertical arrows mark
expected maxima from Eq. (9). Inset: corresponding mini-
bandstructure.
the Supplemental Material [23] which show a strongly in-
creased transmission when laser and BZ frequencies are
in resonance.
A different setup featuring BZ oscillations can be cre-
ated from a strip etched out of the two-dimensional topo-
logical insulator based on mercury teluride (HgTe) [15].
Periodically modulating the width of this strip leads to a
cyclic change in the mass of an effective one-dimensional
model [27]. The numerically obtained resulting miniband
structure, shown in Fig. 4(a), exhibits various Landau-
Zener anticrossings within the bulk bandgap of HgTe
which suggest BZ oscillations. In order to study the elec-
tron dynamics we calculate the COM motion of Gaus-
sian shaped edge-state wave packets. Initially, the wave
packet is localized on one edge and the direction of mo-
tion is determined by its spin. The array of multiple
constrictions allows for tunneling to the opposite edge,
leading to an inversion of the motion and Bloch oscil-
lations in the COM motion.
the resulting frequency spectrum features the expected
rhombic pattern inbetween the frequencies of the conven-
tional Bloch oscillations (white dashed lines). Compared
to the graphene system [see Fig. 1(e)] we observe more
complicated, superimposed structures due to the whole
sequence of multiple anticrossings in the band structure.
As shown in Fig. 4(b),
As for graphene we further study the transport prop-
erties of HgTe strips of constant width and a periodically
modulated electrostatic potential resulting in a super-
cell bandstructure shown in the inset of Fig. 4(c). We
chose the Fermi energy close to the band crossing of the
topological edge states and calculate the current using
Eq. (8). Besides a strong negative-differential conduc-
tance at lower bias we get BZ oscillations for VSD> 9 mV
as shown in Fig. 4(c). Similar to the calculations for the
graphene superlattice the oscillations are independent of
the exact choice of the Fermi level. The peak positions
are in good accordance with the expected series of drift
voltages, Eq. (9), marked by arrows in Fig. 4(c) where
A is extracted from the bands around the Fermi energy
shown as shaded area in the inset.
In this manuscript we show that Bloch–Zener oscilla-
tions appear naturally in superlattices made of materi-
als with a Dirac-like spectrum highlighting interference
between electron and hole states.
of those oscillations are explained by a one-dimensional
model Hamiltonian and numerically confirmed for real-
istic setups by means of wave packet simulations and
transport calculations. Furthermore we suggest trans-
port measurements through graphene nanoribbons and
HgTe strips as promising experimental setups that fea-
ture Bloch–Zener oscillations leading to sequences of pro-
nounced current peaks.
This work is supported by Deutsche Forschungsge-
meinschaft (GRK 1570 and joined DFG-JST Forscher-
gruppe Topological Electronics).
mann, F. Tkatschenko and D. Ryndyk for useful con-
versations.
The characteristics
We thank T. Hart-
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