# Coexistence of Topological and Normal Insulating Phases in Electro-Optically
Tuned InAs/GaSb Bilayer Quantum Wells

M. Meyer<EMAIL_ADDRESS>Julius-Maximilians-Universität
Würzburg, Physikalisches Institut and Würzburg-Dresden Cluster of Excellence
ct.qmat, Lehrstuhl für Technische Physik, Am Hubland, 97074 Würzburg,
Deutschland  T. Fähndrich  Julius-Maximilians-Universität Würzburg,
Physikalisches Institut and Würzburg-Dresden Cluster of Excellence ct.qmat,
Lehrstuhl für Technische Physik, Am Hubland, 97074 Würzburg, Deutschland  S.
Schmid  Julius-Maximilians-Universität Würzburg, Physikalisches Institut and
Würzburg-Dresden Cluster of Excellence ct.qmat, Lehrstuhl für Technische
Physik, Am Hubland, 97074 Würzburg, Deutschland  A. Wolf  Julius-Maximilians-
Universität Würzburg, Physikalisches Institut and Würzburg-Dresden Cluster of
Excellence ct.qmat, Lehrstuhl für Technische Physik, Am Hubland, 97074
Würzburg, Deutschland  S. S. Krishtopenko  Julius-Maximilians-Universität
Würzburg, Physikalisches Institut and Würzburg-Dresden Cluster of Excellence
ct.qmat, Lehrstuhl für Technische Physik, Am Hubland, 97074 Würzburg,
Deutschland  Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de
Montpellier, F-34095 Montpellier, France  B. Jouault  Laboratoire Charles
Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier, F-34095 Montpellier,
France  G. Bastard  Julius-Maximilians-Universität Würzburg, Physikalisches
Institut and Würzburg-Dresden Cluster of Excellence ct.qmat, Lehrstuhl für
Technische Physik, Am Hubland, 97074 Würzburg, Deutschland  Physics
Department, École Normale Supérieure, PSL 24 rue Lhomond, 75005 Paris, France
F. Teppe  Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de
Montpellier, F-34095 Montpellier, France  F. Hartmann  fabian.hartmann@uni-
wuerzburg.de Julius-Maximilians-Universität Würzburg, Physikalisches Institut
and Würzburg-Dresden Cluster of Excellence ct.qmat, Lehrstuhl für Technische
Physik, Am Hubland, 97074 Würzburg, Deutschland  S. Höfling  Julius-
Maximilians-Universität Würzburg, Physikalisches Institut and Würzburg-Dresden
Cluster of Excellence ct.qmat, Lehrstuhl für Technische Physik, Am Hubland,
97074 Würzburg, Deutschland

###### Abstract

We report on the coexistence of both normal and topological insulating phases
in InAs/GaSb bilayer quantum well induced by the built-in electric field tuned
optically and electrically. The emergence of topological and normal insulating
phases is assessed based on the evolution of the charge carrier densities, the
resistivity dependence of the gap via in-plane magnetic fields and the thermal
activation of carriers. For the Hall bar device tuned optically, we observe
the fingerprints associated with the presence of only the topological
insulating phase. For another Hall bar processed identically but with an
additional top gate, the coexistence of normal and topological insulating
phases is found by electrical tuning. Our finding paves the way for utilizing
a new electro-optical tuning scheme to manipulate InAs/GaSb bilayer quantum
wells to obtain trivial-topological insulating interfaces in the bulk rather
than at the physical edge of the device.

###### pacs:

73.21.Fg, 73.43.Lp, 73.61.Ey, 75.30.Ds, 75.70.Tj, 76.60.-k

The discovery of topological materials has opened up new avenues in condensed
matter physics Kane and Mele (2005); Bernevig  _et al._ (2006); König  _et
al._ (2007); Roth  _et al._ (2009) as topological insulators (TIs) are
promising for breakthroughs in fundamental research but also exhibit potential
premise for several device applications such as spintronics and quantum
computing Hasan and Kane (2010). Among the vast amount of topological
materials König  _et al._ (2007); Wu  _et al._ (2018); Zhang  _et al._ (2009);
Vergniory  _et al._ (2019), two-dimensional (2D) TIs based on inverted band
InAs/GaSb quantum wells (QW) heterostructures Liu  _et al._ (2008) are the
most appealing for potential device applications due to the mature growth and
processing technology developed, which is fully compatible with existing Si-
based chips Tournié  _et al._ (2022) combined with the electrical switching
capability of the topological phase transition Liu  _et al._ (2008). The band
inversion in InAs/GaSb QW heterostructures is caused by specific band-edge
alignment of the InAs and GaSb semiconductors arising at their interface. For
thin enough InAs and GaSb layers, the first electron-like (_E_ 1) level lies
above the first hole-like (_H_ 1) level, and the QW has a trivial band
ordering with a normal insulator (NI) phase. As the layer thickness increases,
the _E_ 1 subband becomes lower than the _H_ 1 subband, which leads to an
inverted band structure and the appearance of a 2D TI phase, characterized by
an insulating bulk and spin-polarized gapless helical states at the sample
edges Liu  _et al._ (2008).

To date, the helical nature of the edge states in InAs/GaSb-based
heterostructures has been experimentally reported in InAs/GaSb bilayer quantum
wells (BQWs) Knez  _et al._ (2011); Du  _et al._ (2015). An inherent property
of InAs/GaSb BQWs is the lack of inversion symmetry in the growth direction,
which directly affects not only the position of the E1 and H1 subbands at the
$\Gamma$ point of the Brillouin zone, but also the opening of an inverted
band-gap at nonzero quasi-momentum wavevector. Moreover, by changing the
strength of the structural asymmetry by applying an external electric field in
the growth direction of the BQW, one can change the band ordering in a
controlled manner. Therefore, by fabricating a dual-gated device from
InAs/GaSb BQWs, it becomes feasible to tune between a normal-insulating (NI)
and a TI regime where the helical edge channels are expected Qu  _et al._
(2015). While the electric-field-induced change of the band ordering is often
seen as a route towards the realization of a topological field effect
transistor Liu  _et al._ (2008), such dual-gating approach also enables
another, not yet anticipated, application in topological devices. By
depositing several top gates, it becomes possible to create regions with both
normal and topological phases in the same sample plane. This should move the
interface between a NI- (or vacuum) and 2D TI-phase, which hosts helical edge
channels, away from the physical edge of the device into the bulk of the
sample.

An improvement of the phase coherence length can then be expected as
backscattering via charged donors (or acceptors) Dietl (2023), caused by
fabrication-related defects, can be significantly reduced. If this strategy is
combined with minimizing the residual bulk conductivity (e.g. by substituting
the GaSb-layer with GaInSb Avogadri  _et al._ (2022)), major obstacles in the
unequivocal observation of the Quantum Spin Hall effect (QSHE) in 2D TIs based
on the InAs/GaSb material system Knez  _et al._ (2010); Nichele  _et al._
(2016); Mittag  _et al._ (2017); Mueller  _et al._ (2017, 2015) could be
overcome. We also emphasize that moving the helical channels away from the
sample edge greatly simplifies the fabrication of topological devices, as it
allows the helical channels to be guided via relatively simple lithography
processes rather than demanding wet chemical etching of nanoscale devices.
However, such dual gating of NI and TI interfaces requires not only spatially
well-defined top-gate electrodes but also spatially well-defined back-gate
electrodes to implement two electric field configurations at a fixed Fermi
energy in a single device. To reduce the process challenges of creating
numerous gate electrodes, an alternative knob for tuning both electric field
and Fermi energy can be used. As shown by Knebl _et al._ Knebl  _et al._
(2018), both the Fermi energy and electric field in InAs/GaSb BQWs can also be
tuned optically utilizing a floating gate at the substrate side and negative
persistent photoconductivity effect at the surface side Schets  _et al._
(1999). It was shown that by illuminating an InAs/GaSb BQW a top- and back-
gating operation can be mimicked, and the Fermi energy can be tuned from an
electron-dominated into a hole-dominated regime through the gap with electron-
hole hybridization.

In this work, we report the simultaneous observation of normal and topological
insulating phases in a single InAs/GaSb BQW due to strong in-plane variations
of the built-in electric field across the QW using both electrical and optical
tuning methods. For an optically tuned Hall bar device, we observe only a 2D
TI phase, whereas for another Hall bar processed identically but with an
additional metallic top gate, the coexistence of a NI and 2D TI phase is
found. The presence of the NI and 2D TI phases can be assessed by the
evolution of the longitudinal resistance with temperature and in-plane
magnetic field Qu  _et al._ (2015).

Figure 1:  (a) Schematic layout of the devices made from the same InAs/GaSb
BQW used for the optical tuning (HB-O, left-hand side) and the electrical
tuning (HB-E, right-hand side). Before the deposition of the top-gate
electrode, the HB-O device was cleaved from the processing piece. In the
middle, an exemplary optical image with a top gate (for HB-E) is displayed.
(b) The longitudinal resistivity $\rho_{xx}$ in both devices as a function of
the illumination time (left panel) and top-gate voltage (right panel). For
HB-E, a unique double-peak structure is observed.

Figure 1(a) shows a schematic layer structure of the investigated InAs/GaSb
BQW and highlights the difference between the fabricated Hall bar (HB)
devices. To compare the transport properties under optical excitation or via a
top gate voltage, the HBs were processed from the same wafer and sample piece.
The device used for the optical excitation experiments (HB-O) was cleaved from
the processing piece just before the metallization of the top gate electrode
for the device HB-E. Further information on the device processing is provided
in the Supplemental Material ref . The top-gate electrode is absent for HB-O
and instead a UV-LED with an energy of $E_{\mathrm{LED}}\approx 5$ eV
($\lambda=248$ nm) at $T=4.2$ K is used. The time scale of the optical tuning
depends on the incident light power Knebl  _et al._ (2018). Thus, for all
measurements, a LED current of $I_{\mathrm{LED}}=2.3$ $\mu$A was used.
Furthermore, an exemplary optical image of a Hall bar with a top gate is
shown. Both HBs have the same width $W=20$ $\mu$m but a different length
$L=40$ $\mu$m and $60$ $\mu$m for HB-O and HB-E, respectively, where $L$ is
the distance between the lateral probes. Due to the different lengths, the
resistivity $\rho$ was used in the following analysis. Also labeled are the
in-plane directions $x$ (current direction) and $y$.

The studied InAs/GaSb BQW was grown on a (001) GaSb buffer with InAs and GaSb
layer thicknesses of $10.5$ nm each. At these layer thicknesses, in the
absence of a built-in electric field, band structure calculations ref
indicate that the BQW should be a semimetal with a slight overlap of the
conduction and valence band (CB and VB). However, the presence of a built-in
electric field, which is always inherent in real samples, opens a band-gap in
the band dispersion, turning the semimetal into an insulator. How strong such
a built-in electric field needs to be will be addressed later in the
manuscript. If the BQW is turned into a TI, the band structure of the BQW is
inverted and characterized by a camel-back-like band dispersion of the CB and
VB Nichele  _et al._ (2016). Note that this a shape of the CB and VB allows
for the probing the Van Hove singularities (VHS) associated with the band
edges by means of transport measurements Qu  _et al._ (2015); Karalic  _et
al._ (2016); Schmid  _et al._ (2022).

The longitudinal resistivity $\rho_{xx}$ for both devices as a function of the
illumination time ($t$) and the top-gate voltage ($V_{TG}$) are presented in
Fig. 1(b). If not stated otherwise, all measurements were performed at $T=4.2$
K. Under LED illumination, $\rho_{xx}$ of the HB-O device increases from
$\rho_{xx}=1.3$ k$\Omega$ until it reaches its maximum value around $t\approx
30$ s with $\rho_{xx,max}=4.6$ k$\Omega$. Subsequently, the longitudinal
resistivity drops to $\rho_{xx}=3.9$ k$\Omega$ and remains nearly constant for
larger illumination times. As mentioned above, the optical tunability of the
HB-O device is attributed to a negative persistent photoconductivity effect
(NPPC-effect), which in InAs-based QWs is caused by photo-excited electron-
hole pairs in the GaSb cap layer Tong  _et al._ (2017); Gavrilenko  _et al._
(2010); Gauer  _et al._ (1993); Krishtopenko  _et al._ (2019) and an
accumulation of electrons on the GaSb/AlSb superlattice below the bottom
barrier Knebl  _et al._ (2018). Note that the illumination effects are
persistent but can be reverted by switching off the LED and heating up the
sample. More detailed information on the optical tunability of InAs/GaSb BQWs
is given in Ref. Knebl  _et al._ (2018).

In contrast, the longitudinal resistivity of HB-E is tuned with a top gate.
The central resistivity peak at $V_{TG}=V_{peak}$ occurs when the Fermi level
lies inside the band-gap of the InAs/GaSb BQW. For positive values of
$V_{TG}-V_{peak}$, the Fermi level lies in the CB, while decreasing of
$\rho_{xx}$ at negative $V_{TG}-V_{peak}$ is caused by the Fermi level
shifting into the VB. As seen, in the vicinity of its maximum value,
$\rho_{xx}$ shows a unique double-peak structure. Although similar double-peak
features in InAs/GaSb BQWs were also reported in other works Qu  _et al._
(2015); Karalic  _et al._ (2016), their origin still remains unclear as they
were not further investigated. Here, the $\rho_{xx}$ peak values are $5.5$
k$\Omega$ and $5.1$ k$\Omega$ for the left and right peak, respectively. The
dashed lines contrast the range of the optical compared to the electrical
tuning. As can be seen, the full illumination time used for the optical tuning
corresponds to the respective gate voltage range of about
${\Delta}V_{TG}\approx 2.5$ V. Although the optical tuning range of
$\rho_{xx}$ is smaller than for the electrical gating, it is still possible to
tune the Fermi level position from the CB to the VB through the gap region
where the helical edge states appear. However, the maximum resistance value is
also roughly 20% smaller in the latter case than in the HB-E device. In the
following, the reasons for the difference in both devices will be explored.

Figure 2:  (a) Hall resistance traces for different illumination times ranging
from $t=9$ s up to $55$ s. Starting from $t=25$ s, a pronounced nonlinearity
is observed indicating two-carrier transport. (b) Extracted electron (in blue)
and hole (in red) densities and the zero-field resistivity $\rho_{xx}$ (in
black) for different illumination times. After $t=25$ s, an electron and hole
densities can be extracted indicating electron-hole hybridization. (c) Hall
resistance $R_{xy}$ versus magnetic field for $V_{TG}-V_{peak}=+4$ V to $-2$
V. In contrast to HB-O, $R_{xy}$ does not show pronounced nonlinearities. (d)
Electron (in blue) and hole (in red) charge carrier densities and $\rho_{xx}$
at zero magnetic field (in black) as a function of top-gate voltage normalized
to the central peak $V_{peak}$. Within a small region around the gap, both
carrier types are present.

To follow the evolution of the charge carrier densities, magneto-transport
measurements for HB-O and HB-E with an out-of-plane magnetic field were
performed. In Fig. 2(a), the Hall measurements for HB-O for different
illumination times $t=9-55$ s and $B_{z}=0-2$ T are presented. For short
illumination times, the Hall resistance ($R_{xy}$) is linear, indicating that
only electrons are present and Quantum Hall plateaus are visible. However, at
larger illumination times, a pronounced nonlinearity is observed, which is
attributed to the presence of two distinct charge carriers. An analysis of
$R_{xy}$ in a magnetic field, performed on the basis of a two-charge-carrier
model Meyer  _et al._ (2021), provides the electron (blue) and hole (red)
densities as a function of the illumination time, summarized in Fig. 2(b)
together with the zero-field resistivity $\rho_{xx}$ (in black). At shorter
illumination times, electrons are the majority charge carriers with a maximum
density of $n_{max}=4.2\cdot 10^{11}$ cm-2 and mobility $\mu_{max}=2.6\cdot
10^{4}$ cm2/V$\cdot$s. Starting from $t=25$ s, both electron and hole
densities are observable. As they coincide, they represent a charge neutrality
point (CNP) in the system. Further increasing t switches the charge-carrier
transport gradually from _n-_ to _p-_ type. Thus, the HB-O device features a
broad range of illumination time with an electron-hole hybridization regime,
which is an inherent property of a 2D TI phase in InAs/GaSb BQWs Qu  _et al._
(2015).

Similar measurements were performed for the HB-E device. Figure 2(c) displays
the Hall resistance as a function of magnetic field $B_{z}=0-2$ T shown for a
chosen range of $V_{TG}-V_{peak}=+4$ V to $-2$ V. In contrast to the HB-O
device, $R_{xy}$ remains linear for almost all gate voltage values except for
$V_{TG}-V_{peak}=0$ and $-1$ V, a weak nonlinearity is observable in $R_{xy}$.
In addition, no quantum Hall plateaus are visible in the low magnetic field
range due to the electron mobility being roughly 50% lower compared to HB-O
device. The results of the analysis of $R_{xy}$ within the two-charge-carrier
model are shown in Fig. 2(d). For positive values of $V_{TG}-V_{peak}$, only
electrons are present with the highest mobility $\mu_{max}=1.6\cdot 10^{5}$
cm2/V$\cdot$s at $n_{max}=2.7\cdot 10^{12}$ cm-2. This mobility value is
roughly a factor 4 lower than the highest values reported for BQWs Nguyen  _et
al._ (2015, 2016). However, it should still be sufficient for observing both
VHS in the CB and VB using solely the top gate if the sample is in the TI
regime. Around the maximum value of $\rho_{xx}$ there is a small range of the
gate voltages $V_{TG}-V_{peak}=0$ to $-1$ V, in which electrons and holes
coexist. However, the range of the electron-hole hybridization does not extend
into the VB as it is expected for a 2D TI phase in InAs/GaSb BQWs Qu  _et al._
(2015). At negative $V_{TG}-V_{peak}$ values, less than $-1$ V, only holes
remain as charge carriers, whose maximum mobility reaches $\mu_{max}=6\cdot
10^{3}$ cm2/V$\cdot$s at $p_{max}=1.2\cdot 10^{12}$cm-2.

Despite the qualitatively similar dependencies of the longitudinal resistivity
$\rho_{xx}$ in the HB-O and HB-E devices, the distinctive ranges for the
electron-hole hybridization may indicate different topological phases of the
InAs/GaSb BQW in both devices Qu  _et al._ (2015). In-plane magnetic field
dependent measurements are a powerful tool to distinguish between a NI and TI
gap in InAs/GaSb BQWs. An in-plane magnetic field $B_{||}$ along the
$x$-direction shifts the bands in the $k_{y}$-direction and vice versa by
${\Delta}k_{x/y}=eB_{y/x}\langle{z}\rangle/\hbar$, where $\langle{z}\rangle$
is the average distance between the electron and hole gases Yang  _et al._
(1997); Choi  _et al._ (1988). This leads to a phase transition into the semi-
metallic phase for a TI sample, whereas a NI sample is almost unaffected Qu
_et al._ (2015); Wu (2017).

Figure 3:  (a,b) $\rho_{xx}$ as a function of illumination time for different
in-plane magnetic fields $B_{x}$ (panel (a)) and $B_{y}$ (panel (b)). For
$B_{x}=4$ T and $B_{y}=2$ T, the resistivity peak vanishes indicating the
closing of the TI gap. (c,d) $\rho_{xx}$ versus $V_{TG}-V_{peak}$ for
different in-plane magnetic fields $B_{x}$ (panel (c)) and $B_{y}$ (panel
(d)). The double-peak feature vanishes at around $B_{x}=4.5$ T and $B_{y}=2.5$
T, and the remaining peak prevails until 10 T for both in-plane magnetic field
orientations. (e) The relative change in the difference in the longitudinal
resistance between the peak and saturated value at the VB
$\Delta\rho_{xx,max}$ as a function of the in-plane magnetic field. For the
HB-O device, $\Delta\rho_{xx,max}$ drops to zero, while it remains close to
unity in the HB-E device.

Figure 3(a) and (b) show the evolution of $\rho_{xx}$ versus $B_{x}$ and
$B_{y}$, respectively. Please note that or the highest applied in-plane
magnetic field of 10 T, a small $z$-component of a maximum of 100 mT could be
extracted for both devices, which is negligible. The resistivity peak vanishes
with increasing magnetic fields, indicating the band-gap closing and
transition to a semi-metallic state at $B_{x}=4$ T and $B_{y}=2$ T. We note
that the anisotropy of the evolution of $\rho_{xx}$ with respect to different
orientations of the in-plane magnetic field is similar to the one observed in
Ref. Qu  _et al._ (2015). In contrast, only one of the $\rho_{xx}$ peaks in
the HB-E device vanishes with increasing in-plane magnetic field, as shown in
Fig. 3(c) and (d). This occurs at around $B_{x}=2.5$ T and $B_{y}=4.5$ T,
while the other peak survives up to $B_{x/y}=10$ T – the highest field
available in our experiments. We emphasize that the fields, at which this
resistivity peak disappears, are comparable to those at which the peak
vanishes in the HB-O device. This indicates that the vanishing $\rho_{xx}$
peak in the HB-E device may also be associated with a gap closing of a 2D TI
phase. On the other hand, the other resistivity peak immune to the in-plane
magnetic field, clearly indicates the presence of a NI phase in the InAs/GaSb
BQW of the HB-E device Qu  _et al._ (2015). Thus, the HB-E device features the
fingerprints of both NI and 2D TI phases in the InAs/GaSb BQW. For better
visualization, Fig. 3(e) illustrates the band-gap evolution in both devices by
providing the relative change in the difference in $\rho_{xx}$ between the
maximum value $\rho_{xx,max}$ and saturated value in the VB $\rho_{xx,hole}$
as a function of the in-plane magnetic field:
$\Delta\rho_{xx,max}(B_{||})=\\{\rho_{xx,max}(B_{||})-\rho_{xx,max}(B_{||})\\}/\\{\rho_{xx,max}(0)-\rho_{xx,max}(0)\\}$.
As clearly seen, $\Delta\rho_{xx,max}(B_{||})$ of the HB-O device drops to
zero with increasing in-plane magnetic field, while it remains close to unity
for the HB-E device.

Figure 4:  (a) $\rho_{xx}$ versus illumination time for different temperatures
from $T=1.7$ K up to $17.5$ K for HB-O device. The resistance peak vanishes at
$T=17.5$ K. (b) Arrhenius plot of $\rho_{xx,max}(1/T)$. The fitting at high
temperatures yields a band-gap $E_{gap}=(1.7\pm 0.6)$ meV. (c) $\rho_{xx}$
versus illumination time for different temperatures from $T=1.7$ to $50$ K for
HB-E device. The left-hand peak for the TI phase and the right-hand peak for
the NI phase vanish around $T=15$ K and $50$ K, respectively. (d) Arrhenius
plot of the left-hand (in blue) and right-hand (in black) peak values
$\rho_{xx,max}$. The fit of the high-temperature regime gives two band-gap
values: $E_{gap}(TI)=(0.9\pm 0.7)$ meV and $E_{gap}(NI)=(8.7\pm 1.1)$ meV,
attributed to 2D TI and NI phase, respectively.

To determine the band-gap values associated with each of the longitudinal
resistivity peaks, temperature-dependent measurements have also been performed
for both devices (see Fig. 4). For HB-O device in Fig. 4(a), the peak
corresponding to the gap vanishes around $T=17.5$ K. Figure 4(b) shows the
Arrhenius plot of the peak resistivity in the HB-O device and its high-
temperature fit by the activation contribution $\exp(-E_{gap}/2k_{B}T)$, where
$k_{B}$ is the Boltzmann constant Avogadri  _et al._ (2022); Knebl  _et al._
(2018). The fit gives the band-gap energy, $E_{gap}=(1.7\pm 0.6)$ meV. At low
temperatures, the dependence is weak which might indicate hopping transport
Nichele  _et al._ (2014). The same analysis has been performed for the
temperature dependence of each of the resistivity peaks for the HB-E device
(see Fig. 4(c)) and the Arrhenius plot is depicted in Fig. 4(d). Contrary to
HB-O device, the double peak for HB-E device allows for the extraction of two
band-gap energies. One peak vanishes around $T=15$ K (left-hand peak, TI)
while the second peak persists until $T=50$ K (right-hand peak, NI). Two
different band-gap energies can be determined: $E_{gap}(TI)=(0.9\pm 0.7)$ meV
and $E_{gap}(NI)=(8.7\pm 1.1)$ meV. The former value coincides within the
error bar with the extracted value for HB-O device. Furthermore, the left-hand
side peak shows the same evolution with the in-plane magnetic field as the
$\rho_{xx}$-peak in the HB-O device. Therefore, we attribute the left-hand
peak to the 2D TI phase, as it also occurs at higher VTG-values similar to
Ref. Qu  _et al._ (2015). In contrast, the higher value for the right-hand
peak is attributed to another phase as it significantly exceeds the maximum
gap energy in the TI regime Qu  _et al._ (2015); Karalic  _et al._ (2016);
Yang  _et al._ (1997) and also does not vanish with in-plane magnetic fields.
Hence, the right peak and the associated band-gap is attributed to a NI phase.
We conclude that these measurements provide evidence that a mixed NI/TI phase
is achieved by tuning the BQW solely with a top gate.

The key to understanding these different results obtained for two devices is
implicitly presented in Figs 2(b) and 2(d). As is clear from Fig. 2(b), a 2D
TI phase, whose existence has been unequivocally proven experimentally in the
HB-O device, is characterized by the presence of both electrons and holes
originating from the camelback shape of the bands. Since the electron-like _E_
1 subband and the hole-like _H_ 1 subband are localized in the InAs- and GaSb-
layer, respectively Liu  _et al._ (2008), the presence of electrons and holes
results in a charge carrier separation in the QW, hence the appearance of a
built-in electric field. In real samples, the built-in field is determined not
only by spatially separated electrons and holes, but also by the distribution
of charged donors and acceptors in the layers external to the BQW. However,
just from the difference in the electron and hole concentrations in Figs 2(b)
and 2(d), one can already conclude that the built-in electric field in both
devices should differ dramatically. Since the electric field may yield a
transition between the semimetal, 2D TI and NI phases in InAs/GaSb BQW, the
difference in the fields may explain the observation of different phases in
two devices made from nominally the same QW.

Figure 5:  Band-gap of the InAs/GaSb BQW of the HB-O and HB-E devices as a
function of the effective electric field along the growth direction. Positive
and negative band-gap values correspond to NI and 2D TI phase, respectively.
Negative electric field values correspond to the orientation of the electric
field strength in the direction from the substrate to the surface (see Fig.
1). The insets show a plot of band dispersion for several electric field
values: (A) no electric field, semimetal (SM) phase; (B) $-13.5$ kV/cm, 2D TI
phase with $E_{gap}(TI)=1$ meV and (C) $-70$ kV/cm, NI phase with
$E_{gap}(NI)=8$ meV. For the insets, the separation of the main vertical ticks
is $10$ meV and for the horizontal $0.1$ nm-1.

Since the band-gap in InAs/GaSb BQW depends on the electric field Liu  _et
al._ (2008), one can estimate the strength and orientation of the _effective_
built-in electric field across the QW in each device by using experimental
values obtained from temperature-dependent transport measurements (see Fig.
4). Figure 5 shows the band-gap of the InAs/GaSb BQW of the HB-O and HB-E
devices at T = 4.2 K as a function of the effective electric field across the
QW ref ; Krishtopenko and Teppe (2018). The positive orientation of the
electric field corresponds to the direction from the GaSb cap layer to the
substrate (see a schematic layout in Fig. 1(a)). As seen, the application of a
positive electric field to the InAs/GaSb BQW, which is a semimetal in the
absence of an electric field, only increases the overlap between the CB and
VB. On the contrary, a negative field first opens the band-gap in the
InAs/GaSb BQW, causing a sequential transition from the semimetal into the 2D
TI phase, and then induces a second phase transition from the 2D TI to the NI
phase at high field values. Typical 2D plots of band dispersion for the
semimetal, 2D TI and NI phases tuned by electric field are provided in the
insets of Fig. 5. Without an electric field, the sample should be semi-
metallic (panel (A)). The 2D TI phase with a band-gap of 1 meV (a value that
fits well within the error bars of the experimental values obtained in two
devices at once) is realized at an effective electric field of about $-13.5$
kV/cm (panel (B)). This value seems very reasonable since it is significantly
less than the electric field value corresponding to the maximally asymmetric
profile of the distribution of charge carrier suppliers in ungated InAs/AlSb-
based QWs Gavrilenko  _et al._ (2010); Krishtopenko  _et al._ (2012).

The NI phase with a band-gap of about 8 meV, found in the HB-E device,
corresponds to a much larger _effective_ electric field with a strength of
about $-70$ kV/cm (panel (C)). At this point, it becomes apparent that some
inhomogeneities are involved that cause the two different phases. One
possibility would be inhomogeneities in the sample, such as layer
fluctuations. However, these would also be present in HB-O device, where only
one phase is observable. The second and most likely possibility is that when
tuning with the top gate, there are two different electric field
configurations randomly distributed over the area of the Hall bar. These cause
the occurrence of NI and TI phases over the area of the Hall bar. From a
simple resistor network model (see Supplemental Material ref ) it is possible
to estimate the ratio of the area coverage of both phases for HB-E device.
Using this model, we found that approximately 80% of the Hall bar is covered
by a TI phase and 20% by a NI phase.

In summary, we have investigated NI and 2D TI phases arising in InAs/GaSb BQWs
due to the built-in electric field across the QW, which can be tuned optically
and electrically. For the Hall bar device tuned optically, we have clearly
observed the presence of only a 2D TI phase, while for another Hall bar
processed identically but tuned with an additional top gate, the coexistence
of a NI and 2D TI phase has been found. The differences between the NI and 2D
TI phase have been experimentally discriminated by means of the measured
evolution of longitudinal resistance versus temperature and in-plane magnetic
field. These experimental observations show the possibility of an electro-
optical controlled phase diagram, which results in further device flexibility.
With the finding of a mixed NI/TI phase, it becomes possible to distance the
helical edge channels from the processed physical edge of the sample by
creating an NI/TI interface in the bulk. Therefore, the helical edge channels
can be separated from the trivial edge channels, which would make it more
feasible to observe the Quantum Spin Hall effect in this material system.

###### Acknowledgements.

The work was supported by the Elite Network of Bavaria within the graduate
program “Topological Insulators” and by the French Agence Nationale pour la
Recherche (ANR) with “Equipex+ Hybat” (ANR-21-ESRE-0026) project. We
acknowledge financial support from the DFG through the Würzburg-Dresden
Cluster of Excellence on Complexity and Topology in Quantum Matter – ct.qmat
(EXC 2147, project-id 390858490) and by the Occitanie region through the
program “Quantum Technologies Key Challenge” (TARFEP project).

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Supplemental Material

##  .1 A. Sample growth and processing

The sample was grown by MBE on an undoped (001) GaSb-substrate followed by a
2000 nm thick undoped buffer layer. On top of the buffer is a $10\times
2.5$/$2.5$ nm GaSb/AlSb superlattice (SL). After the SL, the InAs/GaSb bilayer
quantum well (BQW) (10.5/10.5 nm) is sandwiched between a bottom 100 nm thick
AlAs0.08Sb0.92-barrier and the top 50 nm thick AlAs0.08Sb0.92-barrier. The
sample is capped with a $5$ nm thick GaSb-layer. After the growth, a gate
dielectric consisting of $5\times 10$/$10$ nm SiO2/Si3N4 was deposited. After
the deposition, the sample piece was cleaved as shown in Fig. S1. The first
column of Hall bars was used for the fabrication of the HB-O devices tuned
optically with no metallic top gate deposited. For the HB-E device, a $30$ nm
Cr and $100$ nm Au top gate was deposited on top. Therefore, the investigated
devices are nominally identical.

Figure S1:  Fabrication process of the Hall bars for both devices. They were
manufactured on the same wafer piece and only after the gate dielectric was
deposited the piece was cleaved. Afterwards, one column of the Hall bars was
used for the fabrication of the HB-O devices intended for the optical tuning,
while on the other columns, the top-gate electrodes were deposited on top of
the Hall bars (the HB-E device).

##  .2 B. Band structure calculations

Band structure calculations in the main text have been performed by using the
eight-band k$\cdot$p Hamiltonian Krishtopenko  _et al._ (2016), which directly
takes into account the interactions between $\Gamma_{6}$, $\Gamma_{8}$ and
$\Gamma_{7}$ bands in bulk materials. This model describes well the electronic
states in a wide range of narrow-gap semiconductor QWs including InAs/GaSb
BQWs Liu  _et al._ (2008). In the Hamiltonian, we also consider the terms
describing the strain effect arising due to mismatch of lattice constants in
the buffer, QW layers and AlSb barriers. The calculations have been performed
by expanding the eight-component envelope wave functions in the basis set of
plane waves and by numerical solution of the eigenvalue problem. Details of
calculations, the form of the Hamiltonian can be found elsewhere Krishtopenko
_et al._ (2016). Parameters for the bulk materials, and valence band offsets
used in the calculations are taken from Ref. Vurgaftman  _et al._ (2001).
Figure S2 shows the band structure calculations for the sample under study in
the absence of built-in electric field across the QW. The results in the
presence of effective electric field are provided in Fig. 5 of the main text.

Figure S2:  Band structure of the InAs/GaSb BQW studied in this work, based on
an eight-band k$\cdot$p Hamiltonian in the absence of built-in electric field
across the QW. The blue and red curves represent the energy-dispersion of the
electron-like (_E_ 1) and hole-like (_H_ 1) subbands, respectively. The
positive and negative values of quasimomentum k correspond to the [100] and
[110] crystallographic orientations. The horizontal dashed line highlights the
slight overlap between the bottom of the conduction band in the [100]
direction and the top of the valence band in the [110] direction. The absence
of an inversion center in the growth direction of InAs/GaSb BQW leads to the
Kramers spin degeneracy lifting at non-zero wave-vector, known as Rashba spin-
splitting. Figure S3:  Simplified resistor network with $n{\times}n$ building
blocks to model the double-peak shape observed for HB-device.

##  .3 C. Resistor network model

To describe the shape of the double peak we have applied a resistor network
model Nichele  _et al._ (2014); Avogadri  _et al._ (2022), which was
simplified as horizontal transport is neglected. A $n{\times}n$ square is used
and the voltage $V$ is applied from top to bottom as shown in Fig. S3. Each of
the building blocks has a resistance labeled from $\rho_{11}$ to $\rho_{nn}$.
Therefore, the resistance for each column are just the resistances of the
building blocks added and as an example for $\rho_{c1}$:

$\rho_{c1}=\rho_{11}+\rho_{21}+\rho_{31}+...+\rho_{n1}.$ (S1)

For the total resistance for the bulk, it follows:

$\rho_{bulk}=\left(\dfrac{1}{\rho_{c1}}+\dfrac{1}{\rho_{c2}}+\dfrac{1}{\rho_{c3}}+...+\dfrac{1}{\rho_{cn}}\right)^{-1}.$
(S2)

One could also take the helical ($\rho_{helical}$) and trivial
($\rho_{trivial}$) edge channels into account. As the helical edge channels
normally only have a length of a few $\mu$m, they do not need to be considered
for the macroscopic devices. For the trivial edge channels in a InAs/GaSb BQW,
they often possess a resistivity of a few k$\Omega$/$\mu$m (e.g. 2
k$\Omega$/$\mu$m Nguyen  _et al._ (2016)). In this model, for $L>10$ $\mu$m,
the trivial edge channels do not change the shape of the double peak anymore
and are therefore negligible as well. Then, the total resistance
$\rho_{total}$ can be described as:

$\rho_{total}=\left(\dfrac{1}{\rho_{bulk}}+\dfrac{1}{\rho_{helical}}+\dfrac{1}{\rho_{trivial}}\right)^{-1}\simeq\rho_{bulk}.$
(S3)

Now, we assume that each building block can either be in a NI or TI phase,
where the complete resistance is labeled as $\rho_{NI}$ or $\rho_{TI}$,
respectively. For each phase, the peak for the gap can be approximated by a
Gaussian function:

$\rho_{NI}=\rho_{max,NI}\times\exp\left(-\dfrac{\left(V-V_{NI}\right)^{2}}{S_{NI}}\right);~{}~{}~{}~{}~{}~{}~{}~{}\rho_{TI}=\rho_{max,TI}\times\exp\left(-\dfrac{\left(V-V_{TI}\right)^{2}}{S_{TI}}\right),$
(S4)

where $\rho_{max,NI}$ and $\rho_{max,TI}$ are the respective maximum
resistance values for the band-gap.

Figure S4:  Experimental data (in black) and modeling of $R_{xx}$ for the HB-E
device. All curves are normalized to the top-gate voltage and resistance value
for the higher peak. The best fit is achieved with an area ratio of
$A_{NI}/A_{TI}=20\%/80\%$ (in red). By increasing $A_{NI}$ to 25% (orange) or
decreasing it to 15% (blue) the fit differs from the experimental data.

Furthermore, $V_{NI}$ and $V_{TI}$ are the voltage positions of the band-gap
and $S_{NI}$ and $S_{TI}$ describe the width of the peaks. For modeling the
actual double peak shape, we chose $\rho_{max,TI}=4$ k$\Omega$ (extracted from
the measurements for HB-O device). As $E_{gap}(NI){\approx}4E_{gap}(TI)$ (see
the main text), we determine $\rho_{NI,max}{\approx}4\rho_{TI,max}=16$
k$\Omega$. Estimating the difference in $V_{TI}$ and $V_{NI}$ as around 0.8 V
(see Fig 1(c) from the main text), one may conclude that $V_{TI}=0$ V and
$V_{NI}=-0.8$ V. The other remaining values can be obtained by a double
Gaussian fit for both peaks of the HB-E device, which gives $S_{NI}=0.95$ and
$S_{TI}=0.42$. With these parameters, the double peak for HB-E device can be
qualitatively modeled by varying the ratio of the area coverage of the NI and
TI phase.

The data for the HB-E device (taken from Fig 1(c) in the main text, in black)
together with a few selected curves for different area ratios from the
resistor network model is presented in Fig. S4. All curves are normalized to
the top-gate voltage and resistance value for the TI peak. For a suitable fit
of the model with the data, roughly 80% of the area of the Hall bar needs to
be covered by a TI phase whereas 20% by a NI phase (red). Decreasing the area
for the NI phase to $A_{NI}=15$% (blue) or increasing it to $A_{NI}=25$%
(orange), leads to a significant deterioration of the agreement between the
model and the experiment. Also, in order to observe a double-peak structure,
the TI and NI phases need to be put in a series connection randomly
distributed over the complete width of the Hall bar. Otherwise, only one peak
is prominent, and the second peak is visible at most as a shoulder. Therefore,
the resistor model allows us to ascertain the ratio of the area coverage
between both phases for HB-E device and to get an understanding of how both
phases need to be arranged to observe the double peak. It indicates that only
the ratio between the areas and the series connection is important, while the
exact arrangement has no significance.

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