# Aharonov-Bohm interference and phase-coherent surface-state transport in
topological insulator rings

Gerrit Behner<EMAIL_ADDRESS>Peter Grünberg Institut (PGI-9),
Forschungszentrum Jülich, 52425 Jülich, Germany  JARA-Fundamentals of Future
Information Technology, Jülich-Aachen Research Alliance, Forschungszentrum
Jülich and RWTH Aachen University, Germany  Abdur Rehman Jalil  Peter Grünberg
Institut (PGI-9), Forschungszentrum Jülich, 52425 Jülich, Germany  JARA-
Fundamentals of Future Information Technology, Jülich-Aachen Research
Alliance, Forschungszentrum Jülich and RWTH Aachen University, Germany  Dennis
Heffels  Peter Grünberg Institut (PGI-9), Forschungszentrum Jülich, 52425
Jülich, Germany  JARA-Fundamentals of Future Information Technology, Jülich-
Aachen Research Alliance, Forschungszentrum Jülich and RWTH Aachen University,
Germany  Jonas Kölzer  Peter Grünberg Institut (PGI-9), Forschungszentrum
Jülich, 52425 Jülich, Germany  JARA-Fundamentals of Future Information
Technology, Jülich-Aachen Research Alliance, Forschungszentrum Jülich and RWTH
Aachen University, Germany  Kristof Moors  Peter Grünberg Institut (PGI-9),
Forschungszentrum Jülich, 52425 Jülich, Germany  JARA-Fundamentals of Future
Information Technology, Jülich-Aachen Research Alliance, Forschungszentrum
Jülich and RWTH Aachen University, Germany  Jonas Mertens  Peter Grünberg
Institut (PGI-9), Forschungszentrum Jülich, 52425 Jülich, Germany  JARA-
Fundamentals of Future Information Technology, Jülich-Aachen Research
Alliance, Forschungszentrum Jülich and RWTH Aachen University, Germany  Erik
Zimmermann  Peter Grünberg Institut (PGI-9), Forschungszentrum Jülich, 52425
Jülich, Germany  JARA-Fundamentals of Future Information Technology, Jülich-
Aachen Research Alliance, Forschungszentrum Jülich and RWTH Aachen University,
Germany  Gregor Mussler  Peter Grünberg Institut (PGI-9), Forschungszentrum
Jülich, 52425 Jülich, Germany  JARA-Fundamentals of Future Information
Technology, Jülich-Aachen Research Alliance, Forschungszentrum Jülich and RWTH
Aachen University, Germany  Peter Schüffelgen  Peter Grünberg Institut
(PGI-9), Forschungszentrum Jülich, 52425 Jülich, Germany  JARA-Fundamentals of
Future Information Technology, Jülich-Aachen Research Alliance,
Forschungszentrum Jülich and RWTH Aachen University, Germany  Hans Lüth  Peter
Grünberg Institut (PGI-9), Forschungszentrum Jülich, 52425 Jülich, Germany
JARA-Fundamentals of Future Information Technology, Jülich-Aachen Research
Alliance, Forschungszentrum Jülich and RWTH Aachen University, Germany  Detlev
Grützmacher  Peter Grünberg Institut (PGI-9), Forschungszentrum Jülich, 52425
Jülich, Germany  JARA-Fundamentals of Future Information Technology, Jülich-
Aachen Research Alliance, Forschungszentrum Jülich and RWTH Aachen University,
Germany  Thomas Schäpers<EMAIL_ADDRESS>Peter Grünberg Institut
(PGI-9), Forschungszentrum Jülich, 52425 Jülich, Germany  JARA-Fundamentals of
Future Information Technology, Jülich-Aachen Research Alliance,
Forschungszentrum Jülich and RWTH Aachen University, Germany

###### Abstract

We present low-temperature magnetotransport measurements on selectively-grown
Sb2Te3-based topological insulator ring structures. These topological
insulator ring geometries display clear Aharonov-Bohm oscillations in the
conductance originating from phase-coherent transport around the ring. The
temperature dependence of the oscillation amplitude indicates that the
Aharonov-Bohm oscillations originate from ballistic transport along the ring
arms. The oscillations can therefore be attributed to topological surface
states, which can maintain a quasi-ballistic transport regime in the presence
of disorder. Further insight on the phase coherence is gained by comparing
with similar Aharonov-Bohm-type oscillations in topological insulator
nanoribbons exposed to an axial magnetic field. Here, quasi-ballistic phase-
coherent transport is confirmed for closed-loop topological surface states in
transverse direction enclosing the cross-section of the nanoribbon. In
contrast, the appearance of universal conductance fluctuations indicates
phase-coherent transport in the diffuse regime, which is attributed to bulk
carrier transport. Thus, it appears that even in the presence of diffusive
$p$-type charge carriers in Aharonov-Bohm ring structures, phase-coherent
quasi-ballistic transport of topologically protected surface states is
maintained over long distances.

topological insulators, ring interferometer, Aharonov-Bohm effect, topological
surface states, ballistic transport, universal conductance fluctuations,
phase-coherent transport

##  I Introduction

Phase-coherence has a great impact on transport in mesoscopic systems which
leads to many interesting effects, visible through their quantum mechanical
correction to the conduction as a function of magnetic field or gate voltage
[1, 2]. Typical phenomena associated with phase-coherent transport are weak
(anti-)localization, universal conductance fluctuations (UCFs), or Aharonov-
Bohm (AB) oscillations [1]. Recently, phase-coherent transport has also been
studied in three-dimensional topological insulators (TIs) such as Bi2Te3,
Sb2Te3, Bi2Se3, or their alloys, in which topologically protected spin-
momentum locked surface states are present [3, 4]. Interest in these materials
stems from applications in topoelectronic circuits and topological quantum
computer architectures [5, 6, 7, 8].

Figure 1:  Aharonov-Bohm interferometer: (a) Schematics of the selectively-
grown ring structure. (b) Scanning electron micrograph of a selectively-grown
Sb2Te3-based ring device (sample A) with an inner and outer radius of 100 nm
and 150 nm, respectively.

In previous studies, various transport properties of straight three-
dimensional TI-based nanowires and nanoribbons have been investigated
theoretically and studied experimentally in micrometer- and nanometer-sized
systems. The observed effects range from weak antilocalization and conductance
fluctuations to the manifestation of quasi-ballistic transport of
topologically protected surface states, inducing Aharonov-Bohm-type
conductance oscillations when applying a magnetic field along the wire or
ribbon [9, 10, 13, 12, 13, 14, 15, 16].

Three-dimensional TIs often tend to be intrinsically doped due to the
formation of crystal defects during growth, leading to an additional bulk
transport channel as the Fermi level either crosses the conduction or valence
band [17, 18]. With respect to phase-coherent transport, it is a difficult
task to disentangle the contributions from bulk and topologically protected
surface states. On the one hand, the picture regarding UCFs is not so clear.
In nanoribbons of ternary materials with a relatively small bulk contribution,
the UCFs have been attributed to surface states [14], while in Bi2Se3
nanoribbons they have been assigned to bulk carriers [19]. On the other hand,
the AB-type oscillations observed in nanoribbons are generally believed to be
due to phase-coherent loops formed by topologically protected surface states
in the transverse direction around the perimeter of the cross section [9, 10,
13, 12, 13, 20, 15, 16]. From the exponential decrease of the oscillation
amplitude with temperature, it was deduced that the transport is quasi-
ballistic [13, 15, 16]. However, not much information has been available on
phase-coherent transport of topologically protected surface states along the
axial direction of the nanoribbon, i.e., the direction along which the
(surface-state) current flows. To address this issue, we measure planar Sb2Te3
ring-shaped interferometers and investigate the Aharonov-Bohm effect with an
out-of-plane magnetic field. Here, we identify a clear peak in the Fourier
spectrum of the magnetoconductance corresponding to magnetic flux quantum-
periodic oscillations. From the decrease of that peak with temperature, the
corresponding transport regime is identified. The comparison of different ring
sizes as well as a detailed analysis of the spectrum of the UCFs and the
Aharonov-Bohm oscillations in straight nanoribbons allow a comprehensive
investigation of the phase-coherence in the different transport channels. The
interpretation of the transport measurement data is supported by quantum
transport simulations.

##  II Experimental

The Sb2Te3 layer was grown by molecular beam epitaxy employing a selective-
area growth approach [21]. For the substrate preparation, first, 5 nm of a
Si(111) wafer was thermally converted into SiO2. Subsequently, a 20-nm-thick
Si3N4 layer was deposited by low-pressure chemical vapor deposition. The ring
and nanoribbon structures were defined by electron beam lithography. Using
reactive ion etching (CHF3/O2) and hydrofluoric acid wet etching, the Si3N4
and the SiO2 layers were etched, respectively, to reveal the Si(111) surface
locally. The structured Si3N4/SiO2 layers formed the selective-area growth
mask. The standard parameters for selective growth of Sb2Te3, given a
substrate temperature of $300\,^{\circ}$C, a Sb-cell temperature of
$470^{\circ}$C, and a Te-cell temperature of $325\,^{\circ}$C, resulted in a
growth rate of 7 nm/h. The 20-nm-thick TI ring and nanoribbon structures were
grown in the Te-overpressure regime. To prevent oxidation the Sb2Te3 layer was
capped by a 5-nm-thick AlOx layer. From Hall measurements at
$1.5\text{\,}\mathrm{K}$ we determined a hole carrier concentration of
$7.4\times 10^{13}\,\textnormal{cm}^{-2}$ and a mobility of
$152\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{V}\mathrm{s}$ (see
Supplementary Material SI).

The Ohmic contacts composed of a 5-nm-thick Nb layer and a 100-nm-thick Au
layer were deposited on top of the TI layer after removing the AlOx capping in
the contact areas by wet chemical etching and argon sputtering. Rings of two
different sizes were investigated, i.e., samples A and B with an outer radius
of $150\text{\,}\mathrm{n}\mathrm{m}$ and $200\text{\,}\mathrm{n}\mathrm{m}$,
respectively, both with an annulus width of $50\text{\,}\mathrm{n}\mathrm{m}$.
Figure 1 shows a schematic of a selectively-grown ring structure as well as a
scanning electron micrograph of sample A. To further characterize the
properties of the Sb2Te3 layers, a 100-nm-wide nanoribbon (sample C) prepared
in the same run as the ring structures was fabricated. The basic transport
properties were extracted from a 500-nm-wide Hall bar structure (see
Supplementary Material SI).

The measurement of the ring structures were conducted in a 3He-cryostat with a
base temperature of $400\text{\,}\mathrm{m}\mathrm{K}$. The magnetic field is
applied perpendicularly to the substrate plane using a superconducting magnet.
The magnetotransport of the nanoribbon and Hall bar were measured in a
variable temperature insert with a base temperature of
$1.4\text{\,}\mathrm{K}$. The current was measured using a standard two-probe
voltage sensing lock-in setup for all samples.

##  III Results and discussion

###  III.1 Aharonov-Bohm effect in ring structures

Figure 2a shows the normalized magnetoconductance $G/G_{0}$ of a ring
interferometer structure (sample A) with $G_{0}=2e^{2}/h$. The measurement
temperature is varied from $0.4\text{\,}\mathrm{K}$ to
$5.0\text{\,}\mathrm{K}$. The magnetic field is oriented perpendicularly to
the substrate plane so that a magnetic flux is threading the ring aperture.
The corresponding data of the second ring structure (sample B) is presented in
the Supplementary Material SII.

Figure 2: Magnetotransport of ring structures: (a) Normalized
magnetoconductance of sample A at temperatures in the range of
$0.4\text{\,}\mathrm{K}$ to $5.0\text{\,}\mathrm{K}$, with $G_{0}=2e^{2}/h$.
The inset shows a detail of the oscillations at
$800\text{\,}\mathrm{m}\mathrm{K}$ in a smaller magnetic field range. The
measured period $\Delta B$ of $82.5\text{\,}\mathrm{m}\mathrm{T}$, fits
accurately to the peak observed in the FFT of the measured data. The legend
for the different temperatures is given in (b). (b) Fourier spectrum of the
magnetoconductance shown in (a). The expected frequency according to the
minimum and maximum radius as well as the mean radius are indicated by dashed
lines. (c) Integrated amplitude $\mathcal{A}$ of the peak in the FFT at
$12\text{\,}\mathrm{T}^{-1}$ as a function of temperature. The solid line
represents an exponential fit.

The magnetoconductance exhibits several features. The most striking one is a
peak at zero magnetic field, which is due to the weak antilocalization effect.
This peak structure has been observed previously in TI nanoribbon structures
of similar width and is due to electron interference combined with strong
spin-orbit coupling (see Supplementary Material SIII) [14, 15]. Another
feature are pronounced conductance fluctuations with larger amplitude over
larger B-field intervals, in particular at low temperatures. These are caused
by the interference of a limited number of trajectories due to the small
dimensions of the sample [22], which will be discussed in more detail at a
later stage to provide complementary information on phase-coherence. A closer
look at the magnetoconductance reveals that regular oscillations with a
smaller amplitude are superimposed on the conductance fluctuations. A
magnification of a smaller magnetic field region is shown in the inset of Fig.
2a. We found that the oscillation period is about $\Delta
B=$82.5\text{\,}\mathrm{m}\mathrm{T}$$. We attribute these regular features to
the Aharonov-Bohm effect in the ring-shaped conductor [23]. Indeed, the period
$\Delta B$ fits very well to oscillations with a flux period of
$\phi_{0}=\Delta B\cdot A$, with $\phi_{0}=h/e$ the magnetic flux quantum and
$A=\pi r_{\mathrm{mean}}^{2}$ the area of the disc with radius equal to the
mean radius of the ring, $r_{\mathrm{mean}}$ of
$125\text{\,}\mathrm{n}\mathrm{m}$.

The periodic features in the magnetoconductance are analyzed using a fast
Fourier transform (FFT), as shown in Fig. 2b. Note that the Fourier transform
was applied to the original data without any filtering. The FFT shows a
distinct peak at a frequency $f_{B}$ of $12\text{\,}\mathrm{T}^{-1}$, which
corresponds to the expected value for the mean radius of the ring indicated by
the vertical dashed green line in Fig. 2b. In general, the peak lies within
the frequency limits given by the inner ($r_{\mathrm{min}}$) and outer
($r_{\mathrm{max}}$) ring radii, indicating that the trajectories of the
electron partial waves cover the entire ring area. As the temperature
increases, the height of the peak decreases corresponding to a reduction of
the oscillation amplitude. At about $3.0\text{\,}\mathrm{K}$, the peak has
disappeared. In addition to the peak at about $12\text{\,}\mathrm{T}^{-1}$, a
weaker feature is observed at about $25\text{\,}\mathrm{T}^{-1}$, where the
second harmonic is expected.

From the decrease of the integrated peak height $\mathcal{A}(T)$ at
$12\text{\,}\mathrm{T}^{-1}$ in the FFT with increasing temperature, we
estimate the phase-coherence length $l_{\varphi}$. The integration is
performed within a window bounded by the frequencies corresponding to flux
quantum periodicity when considering the inner and outer radius of the ring
(cf. Fig. 2b). We consider an exponential decay $\mathcal{A}\propto\exp(-\pi
r_{\mathrm{mean}}/l_{\varphi}(T))$ [4], with $\pi
r_{\mathrm{mean}}=$393\text{\,}\mathrm{n}\mathrm{m}$$ the length of one of the
ring arms and $\mathcal{A}$ being a measure of the oscillation amplitude.
Indeed, the peak height decay is very well fitted by an exponential decrease
with $l_{\varphi}(T)\sim T^{-1}$, as shown in Fig. 2c. From the fit, we obtain
a phase-coherence length of $l_{\varphi}=$722\text{\,}\mathrm{n}\mathrm{m}$$
at a temperature of $0.4\text{\,}\mathrm{K}$, which is considerably longer
than the length of the ring arm.

Aharonov-Bohm oscillations arise when the phase-coherence length $l_{\varphi}$
is of the order of the length of a ring arm, which is the case for our ring,
as shown above. In principle, in our intrisically doped TI samples, the
oscillations can originate from bulk carriers as well as from charge carriers
in topologically protected surface states. However, the exponential decay of
the FFT amplitude with $l_{\varphi}(T)\sim T^{-1}$ indicates that the
transport is in the quasi-ballistic mesoscopic regime [8]. Hence, we
anticipate that the observed AB oscillations are mainly due to transport of
topologically protected surface states. Indeed, in TI nanoribbons it was
deduced that these surface states with spin-momentum locking have an enhanced
transport mean free path due to strong anisotropic scattering [26, 19]. This
is consistent with our previous findings based on magnetotransport
measurements on Sb2Te3 layers, in which a high-mobility two-dimensional
channel was identified that corresponds to these surface state [1]. Note that
the oscillation ampitude in 2 is small compared to the conductance quantum.
This can be due to many reasons, e.g. resistances in series from the legs and
contacts as well as a suppressed amplitude due to disorder interaction which
can also be seen in the simulations in 3.

Figure 3: Tight-binding simulations of a metallic and topological insulator
ring structure: (a) Schematic of the tight-binding model, with semi-infinite
leads (a few unit cells indicated in red) attached to a (disordered)
scattering region (in blue). (b) The number of transport channels as a
function of energy for a bulk metallic ring and a (bulk-insulating) TI ring.
The vertical dashed lines delineate the energy window over which the
magnetoconductance is averaged. (c),(d) The conductance as a function of the
magnetic flux threading the aperture of (c) a bulk metallic ring and (d) a
(bulk-insulating) TI ring for different values of the disorder strength in the
scattering region. (e),(f) The Fourier transform of the conductance profile in
(c) and (d), respectively.

Based on the Shubnikov-de Haas oscillation measurements, it was found that the
surface-state carrier concentration is about an order of magnitude lower than
the total one. Additional information on the relevant transport regime can be
found in the Supplementary Material SI.

To better understand the impact of elastic scattering due to disorder on the
AB effect of the ring for bulk versus topological surface states, we perform
quantum transport simulations. We make use of the quantum transport simulation
package Kwant [28], the efficient parallel sparse direct solver MUMPS [29],
and the Adaptive package [30] to efficiently sample the parameter space, i.e.,
energy and flux. We employ the same tight-binding modeling approach as in
Refs. [31, 32] and refer to these works for more details. Disorder is
considered by adding a randomly fluctuating on-site energy with a
characteristic disorder strength $S_{\mathrm{dis}}$ in the scattering region
of the tight-binding model. In Fig. 3, the simulation results are summarized.
The assumed sample geometry for the simulation is depicted in Fig. 3a. We
compare the magnetoconductance of a bulk metallic ring with that of a bulk-
insulating TI, where the bulk metallic states are described as a free electron
gas with an effective mass that is appropriate for the bulk states of Sb2Te3
near the Fermi level. The energy window is chosen for both systems such that
they have a comparable magnetoconductance in the clean limit without disorder
(cf. Fig. 3b). As can be seen in Fig. 3c, without any disorder, the metallic
ring displays the most pronounced AB effect, with the appearance of many
higher harmonics (cf. Fig. 3e). When disorder is introduced, however, the
magnetoconductance oscillations are quickly suppressed, as well as the
conductance itself. The bulk metallic states are easily driven into a highly
diffusive regime by disorder, which hinders the transport along the ring and
the corresponding AB signature. For the TI surface states, the behavior is
quite different (cf. Figs. 3d and f). While the AB peak and its harmonics in
the Fourier spectrum are not so pronounced as for the metallic ring in the
clean limit, disorder has a much weaker impact on the conductance and its flux
quantum-periodic oscillations. This reflects the resilience of TI against
elastic backscattering in the presence of disorder, which is also observed in
straight nanoribbons and multi-terminal junctions [13, 33, 14, 15, 32].
Because of their spin-momentum locking properties and being bound to the
surface, a robust (quasi-)ballistic transport regime can be established for TI
surface states even in the presence of relatively strong disorder throughout
the ring geometry.

###  III.2 Conductance fluctuations

In order to gain more information on the phase-coherent transport, the
investigation of Aharonov-Bohm oscillations is followed up by an analysis of
the conductance fluctuations also present in the magnetoconductance shown in
Fig. 2a. The temperature dependence of $l_{\varphi}$ relevant for this
phenomena can be determined from the correlation field $B_{c}$. This quantity
is extracted from the normalized fluctuation patterns in the
magnetoconductance $\delta G/G_{0}$ depicted in Fig. 4a obtained after
subtracting the slowly varying background and filtering out the Aharonov-Bohm
oscillations.

Figure 4: Universal conductance fluctuations: (a) Normalized fluctuation
pattern present in the magnetoconductance measurements of the Sb2Te3 ring
structure presented earlier, with $G_{0}=2e^{2}/h$. The magnetic field is
oriented perpendicular to the ring structure. (b) Phase-coherence length
$l_{\varphi}$ of the ring structure extracted from the correlation field
$B_{c}$ as a function of temperature. The blue line indicates an exponential
decrease of the phase-coherence length following $T^{-0.26}$.

It can clearly be seen that the fluctuation amplitude substantially decreases
with increasing temperature, while the pattern itself is consistent over all
temperatures. The correlation field $B_{c}$ is determined using the
autocorrelation function: $F(\Delta B)=\langle\delta G(B+\Delta B)\delta
G(B)\rangle$ [22]. Here, the full-width half maximum
$F(B_{c})=\frac{1}{2}F(0)$ defines $B_{c}$. In the diffusive regime,
$l_{\varphi}$ can be determined using
$l_{\varphi}\approx\gamma\phi_{0}/B_{c}d$ [35], with $d$ the width of the ring
arms and the width of leads to the ring, which is
$50\text{\,}\mathrm{n}\mathrm{m}$ in our case. For the pre-factor $\gamma$ we
choose $0.42$ [35] for $l_{\varphi}$ larger than the thermal length (see
Supplementary Material SI). The resulting values of $l_{\varphi}$ determined
from the correlation field are shown in Fig. 4b. As indicated by the blue line
the decrease of the phase-coherence length $l_{\varphi}$ with increasing
temperature can be fitted by a dependency of $l_{\varphi}\propto T^{-0.26}$.
The temperature dependence of the sample is slightly lower than the expected
dependence of $T^{-1/3}$ for a quasi one-dimensional system [36]. The maximum
of $l_{\varphi}=$330\text{\,}\mathrm{n}\mathrm{m}$$ is smaller than the
corresponding value determined from the Aharonov-Bohm oscillations. We
attribute this discrepancy to different contributions to the overall phase-
coherent transport. As outlined in the previous section, we concluded that the
Aharonov-Bohm oscillations originate from (quasi-)ballistic transport of
topologically protected surface states. In contrast, conductance fluctuations
by nature only show up in the diffusive transport regime. We can therefore
attribute the appearance of conductance fluctuations to diffusive bulk
transport.

###  III.3 Nanoribbon measurements

So far we could show that Aharonov-Bohm oscillations appear in planar ring
structures. The transport regime was identified to be in the ballistic regime.
However, for the topological surface states of TI nanoribbons, Aharonov-Bohm-
type oscillations are also expected to show up under the application of an
axial magnetic field. In this case, the oscillations originate from the
interference of topologically protected surface states enclosing the magnetic
flux penetrating the cross section of the nanoribbon. We verify this effect as
well by measuring a 100-nm-wide Sb2Te3 nanoribbon from the same growth run as
the planar ring under application of an in-plane field along the nanoribbon
axis. Figure 5a shows the corresponding magnetoconductance measurements at
temperatures in the range of $1.7\text{\,}\mathrm{K}$ to
$20\text{\,}\mathrm{K}$ under application of a magnetic field up to
$\pm$$13\text{\,}\mathrm{T}$.

Figure 5: Magnetotransport in a nanoribbon: (a) Magnetoconductance of a
nanoribbon structure (sample C) under application of an axial in-plane
magnetic field in the range of $1.4\text{\,}\mathrm{K}$ to
$20\text{\,}\mathrm{K}$. The legend can be found in (c). (b) Corresponding
magnetoconductance after subtracting the slowly varying background signal. (c)
Fourier transform of the magnetoconductance data shown in (b) at different
temperatures. (d) Amplitude of the peak in the FFT spectrum as a function of
temperature. The solid line represents the fit.

The data shows a peak at zero magnetic field, which can be attributed to weak
antilocalization [14, 15]. On top of this feature, clear low-frequency
magnetic field-dependent oscillations are observed. To distinguish the
magnetic field-dependent oscillations from the weak antilocalization behaviour
of the nanoribbon, a smooth background has been subtracted by applying a
first-order Savitzky–Golay filter. The corresponding curves are depicted in
Fig. 5b. A clear peak at a frequency of around $0.45\text{\,}\mathrm{T}^{-1}$
can be seen in the Fourier spectrum in Fig. 5c. The cross-sectional area
determined from the frequency is $1.86\times 10^{-15}\,\mathrm{m}^{2}$, which
is matching very well to the cross-sectional area of $2\times
10^{-15}\,\mathrm{m}^{2}$, which is determined from the film thickness of
$20\text{\,}\mathrm{n}\mathrm{m}$ at a nanoribbon width of
$100\text{\,}\mathrm{n}\mathrm{m}$. The temperature dependence of the peak
suggests a strong dependence on the phase-coherence of our carriers. The peak
amplitude should vanish with temperature corresponding to a vanishing phase-
coherence length of a carrier as a function of temperature. Similarly to the
planar ring structures we determined the decay of the Aharonov-Bohm integrated
peak height in the FFT spectrum shown in Fig. 5b with temperature. Once again
the decay follows an exponential dependence according to
$\mathcal{A}\propto\exp(-P/l_{\varphi}(T))$, with $P$ the nanoribbon
perimeter. The temperature dependence indicates that the transport is quasi-
ballistic [13, 15, 16]. From the fit we deduced a phase-coherence length of
$l_{\varphi}(T=2\mathrm{K})=600$ nm at $2\text{\,}\mathrm{K}$. The observed
oscillations can be attributed to phase-coherent oscillations around the
perimeter of the nanoribbon. This effectively proves the existence of surface
states in the investigated material.

##  IV Conclusion

From the temperature dependence of the Aharonov-Bohm oscillation amplitude
Sb2Te3 ring interferometers we found that the phase-coherent transport takes
place in the quasi-ballistic regime. Comparing the quantum transport
simulations of a metallic and topological insulator ring structures, we
conclude that the quasi-ballistic transport is attributable to the
topologically protected surface states. The underlying reason is that these
states are resilient against elastic backscattering in the presence of
disorder, unlike the diffusive bulk states. In addition to the periodic
Aharonov-Bohm oscillations, the magnetoconductance trace also contains
irregular conductance oscillations. Since the appearance of this phenomena
requires transport in the diffusive regime, we conclude that transport in the
bulk channel is responsible in this case. Finally, on straight nanoribbons
fabricated in the same growth run, regular Aharonov-Bohm oscillations are
observed under the application of an axial magnetic field. As for the planar
ring structures a quasi-ballistic regime was identified. Our investigation on
planar ring structures as well as on straight nanowire thus leads us to the
conclusion, that the phase-coherent transport in lateral as well as in
transverse direction is quasi-ballistic. Furthermore, it seems that the
transport in the topological surface states is decoupled from the phase-
coherent diffusive transport in the bulk channel. The present work is an
important milestone to the distinction between quantum transport in
topologically protected surface states and in the bulk channel. Our results
thus help to design future topological devices based on phase-coherent
transport with topological insulator surface states.

## Acknowledgments

We thank Alexander Ziessen and Fabian Hassler for fruitful discussions,
Herbert Kertz for technical assistance, and Florian Lentz and Stefan
Trellenkamp for electron beam lithography. This work was partly funded by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy - Cluster of Excellence Matter and Light for
Quantum Computing (ML4Q) EXC 2004/1 – 390534769. D.H. and K.M. acknowledge the
financial support by the Bavarian Ministry of Economic Affairs, Regional
Development and Energy within Bavaria’s High-Tech Agenda Project ”Bausteine
für das Quantencomputing auf Basis topologischer Materialien mit
experimentellen und theoretischen Ansätzen” (grant allocation no. 07 02/686
58/1/21 1/22 2/23).

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Supplementary Material: Aharonov-Bohm interference and phase-coherent surface-
state transport in topological insulator rings

##  SI Measurements on Hall bar structures

Basic Hall measurements have been conducted on a 500-nm-wide Hall bar
structure in order to obtain the electronic material properties of the Sb2Te3
topological insulator thin film. The Hall bar was selectively grown in the
same growth run as the ring and nanoribbon structures. Supplementary Figure S1
shows an electron beam micrograph of the Hall bar sample.

Supplementary Figure S1: Scanning electron micrograph of the 500-nm-wide Hall
bar structure.

The topological insulator material is grown with a thickness of about
$15\text{\,}\mathrm{n}\mathrm{m}$, which is thinner than for the ring and
nanoribbon structures owing to the dependence of the growth rate on the window
area in the growth mask. A standard lock-in amplifier setup and a variable
temperature insert with a base temperature of $1.5\text{\,}\mathrm{K}$ were
used for the measurements. In Supplementary Figure S2(a) and (b) the Hall
resistance as well as the longitudinal magnetoresistance are shown. The system
shows $p$-type behaviour. This is visible from the slope of the Hall
resistance. The Hall resistance increases linearly indicating an effective
single-channel transport. The sample exhibits a monotonic positive
magnetoresistance as well as a weak antilocalization feature manifesting
itself as a cusp-like dip in resistance around zero field [1]. A hole
concentration of
$n=$$7.4\text{\,}\times\mathrm{1}\mathrm{0}^{13}\,\mathrm{c}\mathrm{m}^{-2}$
and a mobility of
$\mu=$$152\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{V}\mathrm{s}$
were obtained from the Hall and longitudinal resistance measurements.

Supplementary Figure S2: (a) Hall resistance as a function of magnetic field
for various temperatures of the Hall bar structure. (b) Corresponding
longitudinal resistance as a function of magnetic field. (c) Measured
magnetoconductance in a magnetic field range of 0 to $0.2\text{\,}\mathrm{T}$
and fit according to the Hikami-Larkin-Nagaoka theory.

Taking the relatively large carrier concentration into account, we assume that
most of the carriers are located in the bulk valence band. Indeed, we
estimated the bulk carrier concentration to be a factor of more than ten
larger than the concentration in the surface states. The dominating bulk
contribution also results in an effective single-channel transport indicated
by the linear increase of the Hall resistance shown in Supplementary Figure
S2(a). Thus, in order to determine the bulk transport parameters we neglect
the contribution of the holes in the surface states. The approximate effect
hole mass $m^{*}=m_{0}$, with $m_{e}$ the free electron mass, was extracted
from the band structure calculation of Zhang et al. [2]. Assuming that
measured mobility is mainly governed by the bulk carriers we obtained an
elastic scattering time of $\tau_{e}=(m^{*}\mu)/e$ of $4.8\times 10^{-14}$ s
which together with an estimated Fermi velocity $v_{\mathrm{F}}$ of $1.3\times
10^{5}$ m/s results in a diffusion constant of
$\mathcal{D}=\frac{1}{3}v_{\mathrm{F}}^{2}\tau_{e}\approx 0.41\times 10^{-3}$
m2/s and an elastic mean free path $l_{e}=v_{\mathrm{F}}\tau_{e}\approx 11$
nm. The bulk Fermi velocity was calculated from
$v_{\mathrm{F}}=(\hbar/m^{*})(3\pi^{2}n_{\mathrm{3D}})^{1/3}$, with
$n_{\mathrm{3D}}$ the three-dimensional hole concentration. Having
$\mathcal{D}$ available the thermal length defined by
$l_{T}=\sqrt{\hbar\mathcal{D}/k_{\mathrm{B}}T}$ can be calculated. At a
temperature of $1\text{\,}\mathrm{K}$ the thermal length is approximately
$56\text{\,}\mathrm{n}\mathrm{m}$. Since the thermal length is in any case
smaller than the phase-coherence length of bulk carriers determined from
conductance fluctuations, we assumed a scaling factor of $\gamma=0.42$ for the
relation between the correlation field with $l_{\varphi}$ in the main
manuscript.

##  SII Aharonov–Bohm oscillations

We also investigate the magnetotransport of the second Aharonov-Bohm ring
(sample B). For comparison, in Supplementary Figures S3(a) and (b) electron
beam micrographs are shown for sample A and B, respectively. The outer radius
of sample B is $200\text{\,}\mathrm{n}\mathrm{m}$, thus
$50\text{\,}\mathrm{n}\mathrm{m}$ larger than the one of sample A.

Supplementary Figure S3: (a) Scanning electron micro-graph of ring sample A in
comparison to sample B which is shown in (b).

In Supplementary Figure S4(a) the normalized magnetoconductance $G/G_{0}$ of
sample B is shown, with $G_{0}=2e^{2}/h$ and the temperatures varied between
$0.4\text{\,}\mathrm{K}$ to $4.0\text{\,}\mathrm{K}$.

Supplementary Figure S4: Magnetotransport of ring structures: (a) Normalized
magnetoconductance of sample B at temperatures in the range of
$0.4\text{\,}\mathrm{K}$ to $4.0\text{\,}\mathrm{K}$, with $G_{0}=2e^{2}/h$.
The inset shows a detail of the oscillations at
$400\text{\,}\mathrm{m}\mathrm{K}$ in a smaller magnetic field range. The
measured period $\Delta B$ of $47\text{\,}\mathrm{m}\mathrm{T}$, fits well to
the peak observed in the FFT of the measured data. The legend for the curves
belonging to different temperatures is given in (b). (b) Fourier spectrum of
the magnetoconductance shown in (a). The frequency spectrum was smoothed with
a moving average. The expected frequency according to the minimum and maximum
radius as well as the mean radius are indicated by dashed lines. (c)
Integrated amplitude $\mathcal{A}$ of the peak in the FFT at
$23\text{\,}\mathrm{T}^{-1}$ as a function of temperature. The solid violet
line represents an exponential fit with $\exp(-aT)$ corresponding to ballistic
transport without thermal broadening. Also shown are fits assuming diffusive
transport, i.e. with $\exp(-aT^{1/2})$, as well as thermal broadening, i.e.
with prefactor $\sqrt{T}$.

The magnetoconductance show the same kind of features as for sample A, i.e., a
peak at zero resistance due to the weak antilocalization effect as well as
conductance fluctuations. Superimposed, one also finds regular Aharonov-Bohm
oscillations. This becomes clear by having a look on the magnification of a
smaller magnetic field region depicted in Supplementary Figure S4(a) (inset).
From that we extracted an oscillation period of $\Delta
B=$47\text{\,}\mathrm{m}\mathrm{T}$$. The period $\Delta B$ fits very well to
oscillations with a flux period of $\phi_{0}=\Delta B\cdot A$, with
$\phi_{0}=h/e$ the magnetic flux quantum and $A=\pi r_{\mathrm{mean}}^{2}$ the
ring area for the mean radius $r_{\mathrm{mean}}$ of
$175\text{\,}\mathrm{n}\mathrm{m}$.

As for ring sample A, the periodic features in the magnetoconductance are
analyzed by a fast Fourier transform (FFT), as shown in Supplementary Figure
S4(b). In the FFT spectrum we resolve a distinct peak at a frequency $f_{B}$
of about $21.5\text{\,}\mathrm{T}^{-1}$, which fits to the expected value for
the mean radius of the ring indicated by the vertical dashed green line in
Supplementary Figure S4(b). Increasing the temperature causes a decrease of
the peak height. In contrast to sample A no second harmonic feature is
observed in the Fourier spectrum.

The phase-coherence length $l_{\varphi}$ was determined from the decrease of
the integrated peak height $\mathcal{A}(T)$ at $21.5\text{\,}\mathrm{T}^{-1}$
in the Fourier spectrum with temperature. The integration is performed between
frequencies corresponding to the minimum and maximum ring radii (cf.
Supplementary Figure S4(b)). As for sample A, we assumed an exponential decay
$\mathcal{A}\propto\exp(-\pi r_{\mathrm{mean}}/l_{\varphi}(T))$, with $\pi
r_{\mathrm{mean}}=$550\text{\,}\mathrm{n}\mathrm{m}$$ the length of one of the
ring arms. The peak height decay is very well fitted by an exponential
decrease with $l_{\varphi}(T)\sim T^{-1}$, as shown in Supplementary Figure
S4(c). From the fit, we obtain a phase coherence length of
$l_{\varphi}=$859\text{\,}\mathrm{n}\mathrm{m}$$ at a temperature of
$0.4\text{\,}\mathrm{K}$.

Supplementary Figure S4(c) also shows fits for the peak amplitude for
different transport regimes. Generally the temperature dependence of the
Aharonov-Bohm oscillation amplitude can be described by [3, 4]

$\delta
G\propto\left(\frac{E_{\mathrm{Th}}}{k_{\mathrm{B}}T}\right)^{(1/2)}\exp(-\pi
r/l_{\varphi})\,,$ (S1)

where $E_{\mathrm{Th}}$ is the Thouless energy [5], i.e., for diffusive
conductors it is given by $E_{\mathrm{Th}}=\hbar\mathcal{D}/L^{2}$, with
$\mathcal{D}$ the diffusion constant and $L$ the ring circumference. For the
ballistic case the situation for the Thouless energy is more subtle [6]. In
case that the thermal energy $k_{\mathrm{B}}T$ is larger than
$E_{\mathrm{Th}}$ thermal broadening occurs, leading to a pre-factor
$T^{-1/2}$ of the oscillation amplitude. In case that $E_{\mathrm{Th}}$ is
larger than $k_{\mathrm{B}}T$ this factor can be neglected. Furthermore, the
temperature dependence of $l_{\varphi}$ differs for the diffusive and
ballistic case. Ludwig and Mirlin theoretically found that for the diffusive
case the phase-coherence length is proportional to $T^{-1/2}$ [7], whereas in
the ballistic case one expects $l_{\varphi}\sim T^{-1}$ [8]. In diffusive
metallic and semiconducting rings a temperature dependence of $l_{\varphi}\sim
T^{-p}$ with $p$ in between $0.5$ and $0.75$ was observed [3, 9]. Whereas in
clean semiconductor heterostructure rings [10, 11] as well as in clean
graphene ring structures [12] a temperature dependence of $l_{\varphi}\sim
T^{-1}$ was found, indicating a ballistic transport regime [8]. In topological
insulator nanoribbons exposed to a magnetic field along the ribbon axis,
Aharonov-Bohm oscillations are observed due to the presence of tubular
topologically protected surface states. These systems were also found to be in
the ballistic regime [13, 14, 15]. In Supplementary Figure S4(c), fits
assuming different scenarios are shown, i.e., thermal broadening vs. no
thermal broadening and diffusive vs. ballistic transport. Obviously, a good
fit was only obtained for the ballistic case without thermal broadening
implying a large Thouless energy and a long mean free path. The latter can be
explained by the reduced backscattering of carriers in topologically protected
surface states.

##  SIII Weak antilocalization analysis

As mentioned in the previous section, the magnetoresistance shown in
Supplementary Figures S2(b) exhibits a small cusp-like dip at zero field which
can be assigned to the weak antilocalization effect. The weak antilocalization
effect for two-dimensional systems can be described by the the
Hikami–Larkin–Nagaoka (HLN) formula [16]. A fitting to the experimental
magnetoconductance (cf. Supplementary Figure S2(c)) results in a phase-
coherence length of $225\text{\,}\mathrm{n}\mathrm{m}$ with an $\alpha$-factor
of $-0.37$ at base temperature. The $\alpha$ factor is a measure of the number
of transport channels and is close to the value of $-0.5$ corresponding to a
single channel. Our values of $\l_{\varphi}$ and $\alpha$ indicate a strong
bulk conduction of the system, which is typical for this type of samples as
the charge carrier concentration is significantly higher and the mobility is
lower than expected for Dirac surface states. The phase-coherence length
extracted here fits well to the value obtained from the conductance
fluctuations, indicating that both effects are governed by bulk carriers.

## References

  * Weyrich  _et al._ [2017] C. Weyrich, T. Merzenich, J. Kampmeier, I. E. Batov, G. Mussler, J. Schubert, D. Grützmacher, and T. Schäpers, Applied Physics Letters 110, 092104 (2017).
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# Planar graphs as distinguished graph of Morse flows on the 2-disk

Oleksandr Pryshliak

###### Abstract

To investigate the topological structure of Morse flows on the 2-disk we use
the planar graphs as destinguished graph of the flow. We assume, that the flow
is transversal to the boundary of the 2-disk. We give a list of all planar
graph with at least 3 edges and describe all planar graphs with 4 edges. We
use a list of spherical graph with at least 4 edges.

Key words and phrases. Morse flow, planar graphs, spherical graph, topological
invariant.

### Introduction

The topic of this paper refers to applications of topological graph theory to
the classification of Morse flows on surfaces. It uses both topological and
algebraic methods when working with embeddings of graphs on the surface. The
main construction, which dates back to the works of Peixoto, consists in the
construction of a discriminant graph, the vertices of which are sources, and
the edges of which are one-dimensional flowing manifolds.

Classic papers on the topological theory of graphs are books [56, 57, 58, 59,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 73, 74, 76].

Embedded graphs as topological invariants of flows on closed surfaces were
constructed in [7, 19, 23, 24, 25, 43, 1, 32, 35, 48, 33, 34, 41, 31], and on
surfaces with a boundary y [5, 8, 21, 20, 35, 32, 28, 40, 25, 41, 44]. For a
3-manifold, a Hegaard diagram is an embedded 4-valent graph in a surface [49,
51, 39, 27, 29, 13, 4, 3].

Morse streams are gradient streams for Morse functions. If we fix the value of
the function at special points, then the structure of the flow determines the
structure of the function [22, 47].

Topological invariants, as Reeb graphs, of functions on oriented surfaces were
constructed in [18] and [45] and in [22] for unoriented surfaces, and in [10,
16, 17, 30] for of surfaces with a boundary, in [38] for non-compact surfaces.

Embedded graphs as topological invariants of smooth functions were also
studied in papers [6, 9, 17, 16, 38, 37, 42, 22, 27, 55, 50, 2, 46], for
manifolds with a boundary in papers [14, 17, 15], and on 3- and 4-dimensional
manifolds in [36, 26].

To get acquainted with the topological theory of functions and dynamical
systems, we recommend [30, 52, 53, 54, 55].

The purpose of this paper is to describe all possible structures of connected
graphs on a plane with no more than 4 edges.

##  1 Spherical graphs

All graphs in this and the next section are connected. We use spherical graphs
to describe all planar graphs. To specify a planar graph, one of the faces
must be selected in the spherical graph, which will be external on the planar
graph. By projecting a graph from any point of the selected graph by spherical
projection onto the plane, we get a flat graph. We will use the list and
numbering of spherical graphs given in the work [7].

###  1.1 Spherical graphs with one edge

Only two such graphs are possible: a loop (one vertex) and a segment (two
vertices) (1).

Figure 1: spherical graphs with one edge

###  1.2 Spherical graphs with two edges

There are 4 spherical graphs with two edges. They are shown in fig. 2.

Figure 2: spherical graphs with two edges

###  1.3 Spherical graphs with three edges

All spherical graphs with three edges are shown in fig. 3.

Figure 3: spherical graphs with three edges

###  1.4 Spherical graphs with 4 edges

Spherical graphs with 4 edges are shown in Fig. 4.

Figure 4: spherical graphs with 4 edges

So, there are a total of 26 non-isomorphic spherical graphs with 4 edges. We
took this list and numbering from the work [7].

##  2 Planar graphs

The graphs depicted in the previous section are flat (they have a fixed outer
edge). The process of changing the outer face can be described as follows: the
edge between the layer and the new outer face is torn and connected in a new
way, that is, it moves through the point at infinity (the pole of the
stereographic projection). At the same time, this process can be repeated
several times and with different edges lying within the boundary of the outer
face. After obtaining planar graphs in this way, they should be checked for
isomorphism among themselves.

###  2.1 Planar graphs with one edge

As for the sphere, only two such graphs are possible: a loop (one vertex) and
a segment (two vertices) (1).

If we apply the process of overturning due to infinity to the loop, we get the
same graph. This process cannot be applied to a segment.

###  2.2 Planar graphs with two edges

There are 6 planar graphs with two edges. They are shown in fig. 5. Here,
graph 2a is obtained from graph 2 by looping. Similarly, graph 4a is derived
from graph 4.

Figure 5: flat graphs with two edges

###  2.3 Planar graphs with three edges

All planar graphs with three edges are shown in Fig. 7. Graphs obtained from
each other by flipping edges through infinity have the same numbers but
different letters. For example, columns 10a and 10b can be obtained from
column 10.

Figure 6: planar graphs with 3 edges (part 1)

Figure 7: planar graphs with 3 edges (part 2)

So, there are a total of 25 non-isomorphic planar graphs with 3 edges

###  2.4 Planar graphs with 4 edges

To list all planar graphs with 4 edges, we will use the list of spherical
graphs with 4 edges shown in Fig. 4.

In this figure, the following graphs have a single face (or two faces for
which there is an isomorphism that translates one into the other): 1), 2), 3)
4).

Graphs with two asymmetric faces each: 5), 6), 7), 8), 9), 10), 11), 12), 13),
15), 16), 18), 19), 20), 21 ).

Graphs each have three asymmetric faces: 14), 17), 22), 23), 24), 25), 26).

Therefore, there are $4+2\times 15+3\times 7=55$ non-isomorphic planar graphs
with 4 edges.

## Conclusion

In this paper, we have constructed a complete list of planar graphs with no
more than 4 edges. During the construction, we used spherical graphs, fixed
the projection face in them, and checked the obtained projections for
isomorphism as planar graphs.

The results of the work can be applied in many fields, for example, such
graphs are distinguished graphs of gradient vector fields on the 2-disk. So,
there two such flow structures with one saddle, 6 structures with two saddles,
25 structures with tree saddles and 55 structures with four saddles.

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