Unexpected edge conduction in mercury telluride quantum wells under broken time-reversal symmetry

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Abstract The realization of quantum spin Hall effect in HgTe quantum wells is considered a milestone in the discovery of topological insulators. Quantum spin Hall states are predicted to allow current flow at the edges of an insulating bulk, as demonstrated in various experiments. A key prediction yet to be experimentally verified is the breakdown of the edge conduction under broken time-reversal symmetry. Here we first establish a systematic framework for the magnetic field dependence of electrostatically gated quantum spin Hall devices. We then study edge conduction of an inverted quantum well device under broken time-reversal symmetry using microwave impedance microscopy, and compare our findings to a non-inverted device. At zero magnetic field, only the inverted device shows clear edge conduction in its local conductivity profile, consistent with theory. Surprisingly, the edge conduction persists up to 9 T with little change. This indicates physics beyond simple quantum spin Hall model, including material-specific properties and possibly many-body effects.

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### ARTICLE

Received 3 Feb 2015 | Accepted 20 Apr 2015 | Published 26 May 2015 DOI: 10.1038/ncomms8252 OPEN
## Unexpected edge conduction in mercury telluride quantum wells under broken time-reversal symmetry

Eric Yue Ma 1,2, M. Reyes Calvo 1,2, Jing Wang 1,3, Biao Lian 1,3, Mathias Mu¨hlbauer 1,4, Christoph Bru¨ne 4,
Yong-Tao Cui 1, Keji Lai 1,2,5, Worasom Kundhikanjana 1,2, Yongliang Yang 1, Matthias Baenninger 1,3,
Markus Ko¨nig 1,3, Christopher Ames 4, Hartmut Buhmann 4, Philipp Leubner 4, Laurens W. Molenkamp 4,
Shou-Cheng Zhang 1,3, David Goldhaber-Gordon 1,3, Michael A. Kelly 1 & Zhi-Xun Shen 1,2,3

The realization of quantum spin Hall effect in HgTe quantum wells is considered a milestone

in the discovery of topological insulators. Quantum spin Hall states are predicted to allow

current flow at the edges of an insulating bulk, as demonstrated in various experiments. A key

prediction yet to be experimentally verified is the breakdown of the edge conduction under

broken time-reversal symmetry. Here we first establish a systematic framework for

the magnetic field dependence of electrostatically gated quantum spin Hall devices. We then

study edge conduction of an inverted quantum well device under broken time-reversal

symmetry using microwave impedance microscopy, and compare our findings to a non
inverted device. At zero magnetic field, only the inverted device shows clear edge conduction

in its local conductivity profile, consistent with theory. Surprisingly, the edge conduction

persists up to 9 T with little change. This indicates physics beyond simple quantum spin Hall

model, including material-specific properties and possibly many-body effects.

1 Geballe Laboratory for Advanced Materials, Stanford University, 476 Lomita Mall, Stanford, California 94305, USA. 2 Department of Applied Physics,
Stanford University, 348 Via Pueblo Mall, Stanford, California 94305, USA. 3 Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford,
California 94305, USA. 4 Physikalisches Institut (EP3), Universita¨t Wu¨rzburg, Am Hubland, Wu¨rzburg D-97074, Germany. 5 Department of Physics,
University of Texas at Austin, Austin, Texas 78712, USA. Correspondence and requests for materials should be addressed to Z.-X.S. (email:
[zxshen@stanford.edu).](mailto:zxshen@stanford.edu)

uantum spin Hall (QSH) theory predicts that when the
thickness of an HgTe quantum well (QW) exceeds a
critical value dC, a band inversion happens and the QW
# Q
transitions from a trivial insulator to a QSH insulator with gapless
edge channels protected by time-reversal symmetry (TRS) 1,2 .
While edge conduction in the presence of TRS has been well
established in various transport 3–7 and microscopy studies 8,9, its
predicted breakdown under broken TRS remains unverified 10,11,
except for earlier works showing suppression of conductance
by weak field in devices with strong backscattering at zero field 3,
and a gap opening at the otherwise protected Landau level (LL)
crossing near the field-induced topological phase transition 12,13 .
A systematic magnetic-field-dependent study of QSH edge
conduction combining microscopy and transport techniques is
therefore highly desirable to gain further insight into this key
prediction.

In this work, we combine transport and microwave
impedance microscopy (MIM), a scanning probe technique
that measures local electromagnetic response 14–16, to study two
HgTe/(Hg0.3Cd0.7)Te QW devices, above (7.5 nm) and below
(5.5 nm) the theoretical dC of 6.7 nm. At zero magnetic field, edge
conduction is only observed in the 7.5 nm device, consistent with
QSH theory. In the case of broken TRS by a magnetic field, we
first develop a rigorous treatment for the field dependence of

electrostatically gated HgTe QW devices and show that the widely
accepted picture of re-entrant behaviour in Hall conductance 3
needs to be revisited. We then discuss our field-dependent MIM
and transport data. They agree well with our expectations in
general, but with one surprise: the edge conduction in the 7.5 nm
device survives up to 9 T with no sign of disappearing, well
beyond the expected crossover field of B3.8 T, when the QW
should transition into a trivial insulator. The stark contrast
suggests the necessity of including more physics to understand
QSH in real systems. We discuss the possibility of conduction
through trivial edge states enhanced by spin-orbit coupling and
effects of electron–electron interactions.

Results
5.5 versus 7.5 nm QW in zero magnetic field. We first establish
the influence of the different QW thicknesses in transport measurements. We use a two-step etching process to define mesas
with clean physical edges for both devices (Fig. 1a, Supplementary
Fig. 1 and Supplementary Note 1). The conductive substrate
(n þþ GaAs) 4 mm below the QW is used as the back gate to tune
the QW from p-type through the bulk gap to n-type, as shown in
the two-terminal resistance (R2) taken at 4.5 K and 0 T, of a strip
50-mm long and 1,225-mm wide (Fig. 1c,f). When the Fermi level

5.5 nm QW 7.5 nm QW

a MIM-Im c
8

MIM-Im

MIM-Re

c f

_T_ = 4.5 K
_B_ = 0 T

_x_

–2 0 2 4 6

_V_ g (V)

–2 –1 0 1 2 3

_V_ g (V)

d g

_V_ g (V)

MIM-Im MIM-Re

_V_ g (V)

MIM-Im MIM-Re

2.2

0.4

–1.2

_x_ _x_ _x_ _x_

4.0 V 0.7 V 4.0 V 0.7 V

_V_ g

5.0

2.5

–2.0

1 GHz

Impedance

match

_W_

MIM-Im
MIM-Re

e h

–1.2 V

0.4 V

Topo

0 2 4 6 8

2.2 V

–2.0 V
5.0 V

2.5 V

2 Topo 20

Topo

10 –10 10 –8 10 –6 10 –4

0 2 4 6

_x_ (μm) _x_ (μm)

_�_ xx(Ω –1 )

Figure 1 | Microwave impedance microscopy (MIM) set-up and zero field gate dependence of 5.5 and 7.5 nm devices. (a) Schematic of MIM set-up and
device structure, showing line scan direction x. (b) MIM response as a function of the sheet conductance of the quantum well, obtained by finite element
analysis. The sensitivity window is centred at 2 � 10 � 7 O � 1 and extends roughly 2 orders of magnitude in both directions, as indicated by the two-tone
colour scale. (c) Gate-dependent two-terminal resistance of a 50-mm long stripe on the 5.5 nm device. (d) Typical MIM-Im and MIM-Re single-line scan
versus gate voltage (Vg) across a B5-mm wide mesa at 4.5 K. Physical edges are indicated by grey dashed lines. Two-tone colour scale is used in MIM-Im
to reflect the MIM sensitivity window in b, with neutral grey corresponding to the maximum in MIM-Re. The mesa appears homogeneously insulating with
no edge conduction in the bulk gapped region. The scale bar is 1 mm, for x direction only. (e) MIM-Im line-cuts when the QW is in p-type ( � 2.0 V), gap
(2.5 V) and n-type (5.0 V). Black solid line is the mesa surface topography detail. MIM stays in constant-height mode with the tip B30 nm above the
average mesa surface; when moving into the etched-away region outside the mesa, the tip-sample distance quickly increases to B160 nm and thus the
MIM-Im signal decreases significantly. For the same reason the right part of the mesa has a slightly higher MIM-Im signal due to higher topography. This
crosstalk only contributes a gate-independent modulation to the MIM signal. (f–h) Same measurements for the 7.5 nm device. Note the lower maximum
two-terminal resistance and prominent edge conduction in the gapped region. Crosstalk from topography is again visible as a gate-independent modulation
of MIM signal.

(EF) is gate-tuned to the bulk gap, R2 reaches 6 MO (150 MO/&)
for the 5.5 nm QW but only 50 kO for the 7.5 nm QW, consistent
with contribution from edge conduction.

We then use MIM to confirm the edge conduction in real
space. MIM delivers a small excitation (oB0.1 mW) at 1 GHz to
a metallic tip and measures the reflected signal with a spatial
resolution of B150 nm. The two output channels, MIM-Im and
MIM-Re, are proportional to the change of the imaginary and real
parts of the tip-sample admittance 15 . For a buried QW structure,
the MIM-Im signal is an intuitive semi-logarithmic
representation of QW conductivity with a calibrated two-tone
colour scale (Fig. 1b, Supplementary Fig. 2 and Supplementary
Note 2). Repeated MIM single-line scans are performed across a
B5-mm wide mesa while tuning the gate voltage, producing
Fig. 1d,g (for 5.5 and 7.5 nm QW, respectively). When EF is in
valence or conduction band, both the 5.5 and 7.5 nm device
appear homogeneously conductive (sxx4B2 � 10 � 6 O � 1 );
when EF is tuned towards the bulk gap, the 5.5 nm device
becomes homogeneously insulating (sxxoB2 � 10 � 8 O � 1 )
while the 7.5 nm device only becomes insulating in the bulk—
the edge remains conductive (Fig. 1d,e,g,h, also Supplementary
Fig. 3).

The smooth MIM profile of the edge conduction with a fullwidth-half-maximum of B1.5 mm (green trace in Fig. 1h) is not
limited by the spatial resolution. Instead it reveals that local
conductivity gradually decreases by B3 orders of magnitude
going from the edge to the centre of the mesa. While the fullwidth-half-maximum of local conductivity in linear scale
(oB500 nm) is consistent with previous results 7, the origin of
the small MIM signal extending B2 mm into the bulk requires
better understanding 17 .

Expected magnetic field dependence for gated HgTe QW
devices. We now move on to the magnetic field dependence.
Figure 2a,f is the calculated LL fan chart for 5.5nm and 7.5 nm
QWs, respectively. We included an inversion symmetry breaking
term to reproduce the observed avoided crossing of the ‘zero
modes’ at the crossover field of B3.8 T (Fig. 2f) 12,13,18 . Previously
the LL fan charts were directly compared with gate- and fielddependent transport data to demonstrate a ‘re-entrant quantum
Hall effect’ in an inverted device: EF was thought to stay constant
with fixed gate voltage and thus if resides in the gap at zero field
will cross v ¼ þ ( � )1 LL and back to the gap as the field
increases 3 . This picture needs to be revisited, because in
electrostatically gated devices, a constant gate voltage corresponds
to a constant carrier density instead of constant EF; the excess
carriers needed for EF to cross the lowest LLs when starting from
the gap are simply not supplied (Supplementary Note 3).

Figure 2b,g is the calculated bulk density of states (DOS) from
Fig. 2a,f with vertical axis converted to carrier density using a
disorder-broadened LL DOS (Methods): they can now be directly
compared with the gate- and field-dependent transport and MIM
scans. Note that Fig. 2b,g is very similar despite drastic differences
between Fig. 2a,f, because the DOS of a LL is independent of the
LL fan chart and only depends on B (Supplementary Fig. 4). The
difference in field dependence between inverted and non-inverted
QW devices is therefore much more subtle than previously
understood, as demonstrated below.

Magnetic field dependence of 5.5 nm QW. The field dependence
of the 5.5 nm device agrees well with expectation (Fig. 2a–e).
Gate- and field-dependent two-terminal resistance (Fig. 2d)
shows a simple metal–insulator transition at low densities, with
the insulating region expanding linearly with field. MIM scans of
a 7.2-mm long section of the mesa confirm that the mesa stays

mostly homogeneously conductive or insulating with smooth
transitions (Fig. 2e). The observed behaviour can be well understood by assuming that the mobility edge is at a fixed LL filling
factor v (B þ 0.15/ � 0.2 in this particular device, orange and
blue lines in Fig. 2c–e).

Magnetic field dependence of 7.5 nm QW. The field dependence
of the 7.5 nm device can be best illustrated through comparison
with the 5.5 nm device. We start with expectations from a simple
QSH model. Comparing Fig. 2g,b, we know the bulk should
behave similarly in both devices, insulating between the two
mobility edges (|v|oB0.2) and conductive outside this range.
The major predicted difference lies in the regime with insulating
bulk (Fig. 2h): edge states are expected to exist between 0 and
3.8 T (Region I) due to the anomalous bending of the ‘zero
modes’ near physical edges (Fig. 2f inset), but become progressively weaker due to disorder scattering and evolution of the ‘zero
modes’, until completely disappearing into the bulk at 3.8 T, at
which point the QW becomes a ‘trivial insulator’ (Region II) 19 .
We stress here that the LL filling factor in region I and II is low
enough (|v|oB0.2) that no integer quantum Hall effect should
be present, as opposed to previous expectations 3 (Supplementary
Fig. 5). The gradual disappearance of the edge conduction from
region I to II should be directly observable in real-space images; a
reduced resistance should also be measured in region I due to the
edge conduction, contrasting with a fully insulating behaviour in
region II.

In the measured two-terminal resistance data on the 7.5 nm
device, however, the expected fully insulating region II breaks into
two sub-regions II and II* (Fig. 2i), where II* is not insulating.
The nature of this discrepancy is readily revealed in the MIM
images (Fig. 2j): whereas the bulk has already become insulating
in II*, edge conduction, which is predicted to disappear beyond
the crossover field (3.5 ± 0.5 T as determined experimentally),
persists up to 9 T with no sign of disappearing, therefore making
the QW continue to appear conductive in transport. The edge
conduction gradually disappears towards the p-type side; the QW
thus becomes homogeneously insulating and the resistance goes
up by 2 orders of magnitude in region II of Fig. 2i. The boundary
between II and II* (green dashed line in Fig. 2i) has a similar
slope as the bulk mobility edge (vB þ 0.2, orange dashed line)
but with an offset in density, and therefore extrapolates to a
different ‘charge-neutral point’ at 0 T (green dotted line). The
regions where edge conduction is present (Region I and II*),
although artificially divided, show virtually no discontinuity. In
fact, the edge conduction appears even clearer at finite field
(compare 0 T/1.0 V and 5 T/1.0 V in Fig. 2j), as the bulk becomes
more homogeneously insulating, likely due to the localization
caused by the magnetic field.

A careful analysis of the MIM line profile further demonstrates
that the local edge conductivity dominates electronic transport in
both low- and high-field regimes. Figure 3a,c shows the averaged
line-cuts of the MIM images in the two columns corresponding to
the 3 and 7 T case in Fig. 2j. The extracted edge and bulk MIM
signal is plotted against Vg in Fig. 3b,d, together with the
corresponding R2 data. It is immediately obvious that both the peak
position and magnitude of R2 have a strong correlation with the
edge signal, but not the bulk signal. This excellent agreement
between local signal and transport confirms that the unexpected
edge conductivity is due to edge states that can conduct DC current
across mesoscopic distances, instead of localized states at the edges.

Discussion
These results in HgTe QWs represent an opportunity to learn
physics beyond the simple textbook QSH theory—a rare

5.5 nm QW 7.5 nm QW

a 40

f 40 g

_E_

_x_

_E_

_x_

2 4 6 8

_B_ (T)

10 4

10 3

10 2

1.0

0.5

0.0

–0.5

–1.0

0.4

0.0

–0.4

_x_

DOS (a.u.)

1.0

0.5

0.0

–0.5

–1.0

0.2

0.0

DOS (a.u.)

−20
0 2 4 6 8

_B_ (T)

−20

−40

−60

–2

II*

1 3 5 7 9 1 I 10 3 1 3 5 7 9

1 3 5 7

–0.2
10 4

10 3

10 2

_B_ (T)

–1

–2

0 2 4 6 8 _R_ 2 (kΩ) 0 2 4 6 8

_R_ 2(kΩ)

_B_ (T)

_R_ 2 (kΩ)

_B_ (T)

_�_ xx(Ω –1 )
<2×10 –9

2×10 –7

>2×10 –5

_B_ (T)

–2

_V_ g (V)

0 1 3 5 7 9

_�_ xx(Ω –1 )

<2×10 –9

2×10 –7

>2×10 –5

_B_ (T)

_V_ g (V)

0 1 3 5 7 9

–1

Figure 2 | Magnetic field dependence of the 5.5 and 7.5 nm devices. (a) Calculated LL fan charts of the 5.5nm device. Inset shows the bending of the
electron-like (blue) and hole-like (red) lowest LLs near a physical edge. Green dashed line is the Fermi level. (b) Bulk DOS versus carrier density and
magnetic field, converted from a. The conversion is necessary because the back gate linearly tunes the carrier density in the QW rather than the relative
position between EF and the LLs. (c) Magnified image of b corresponding to the density range achievable with our back gate. Electron and hole side mobility
edges are indicated. (d) two-terminal resistance versus back gate voltage and magnetic field at 4.5K. For ease of comparison, the colour scale is chosen such
that the corresponding sheet resistance range corresponds to the MIM sensitivity window in zero field limit (sxy ¼ 0). (e) Corresponding real-space MIM-Im
images. The two-tone colour scale reflects the MIM sensitivity window. Inset shows the surface topography details so one can identify topography crosstalk.
The scale bar is 1 mm. The linearly broadened insulating regime in d,e corresponds to mobility edges at fixed filling factors (orange and blue dashed lines, as
also marked in c). (f–j) Same calculations and experimental results for the 7.5nm device. Note the anomalous bending of the ‘zero modes’ in f inset. Because
the estimated disorder broadening (B0.5 meV) and thermal broadening (B0.35 meV) are much smaller than the anti-crossing gap (B4 meV), the bulk is
expected to remain insulating at the crossover field of 3.8 T as shown in h (Supplementary Fig. 4). Note the smaller tuning range in the 7.5 nm device. The
major unexpected feature is the absence of insulating behaviour in region II* in i, which corresponds to edge conduction that survives up to 9 Tas seen in j.
The ‘trivial insulator’ phase only appears near the p-type side. Purple dashed lines in i correspond to vB ± 0.5 (also marked in h), visible in the 7.5nm device
due to higher mobility.

occurrence in the field of topological insulators where so far
theoretical predictions are routinely confirmed. One might
naively attribute the observed edge conduction in moderate to
high fields to trivial edge states, for example due to a
unidirectional band bending (Fermi level pinning) or fringing
field effect from gating 20 . Such trivial edge states, if exist, are often
assumed to be localized and not contributing to transport 21–23 .
But in this particular material, strong spin-orbit coupling may
suppress direct backscattering 24 and thus ‘protect’ the conduction
via trivial edge states, possibly to mesoscopic lengths. In fact,
both scenarios can account for what we see going from the bulk
gap to the n-type side: edge conduction turning on before
bulk conduction. However neither scenario can account for our

observations near the p-type side: Fermi level pinning would
predict that the edge remains insulating when the bulk becomes
conductive; fringing field effect would predict edge conduction
turned on before the bulk becomes conductive; whereas
experimentally the conductivity of edge and bulk evolve
similarly upon entering the p-type side (for example, Vg ¼ 0 to

- 1.5 V in Fig. 3d).
Moreover, the edge conduction, expected and unexpected, only
appears in the 7.5 nm device—suggesting that it cannot be
completely ‘trivial’, but is related to the thickness-induced
band inversion. Additional crucial ingredients may include
many-body effects 25–28 and material specifics of HgTe QW,
including its strong spin-orbit coupling, n–p asymmetry, mercury

7.5 nm QW–3 T 7.5 nm QW–7 T

3.0 V

tip-sample admittance was obtained using COMSOL Multiphysics 4.4 (details see
Supplementary Note 2).

Band structure and LL fan chart calculation. The critical thickness of 6.7 nm was
determined by an 8-bulk-band model for a fully strained HgTe/(Hg0.3Cd0.7)Te QW.
We employ a 4-2D-band effective k � p Hamiltonian for HgTe QW to calculate
the LL fan chart:

-
hðkÞ 0
H ¼ þ HBIA þ HZ ð1Þ
0 h � ð � kÞ

The kinetic term is a 2 � 2 matrix

6.0

5.0

4.0

3.0

–1.5 V

mðkÞ A kx þ iky
hðkÞ ¼ EðkÞI þ -

- - -
mðkÞ A kx þ iky

-  - ð2Þ

A kx � iky

- mðkÞ - A kx þ iky

kx � iky - mðkÞ

0
0 2 4 6 8 0 2 4 6 8

_x_ (μm) _x_ (μm)

where an expansion up to k 4 is used:

F4k 4 ð Þ ¼ M þ Bk 2 þ G4k 4

1 þ F6k 6 ; m k 1 þ G6

F4k 4
EðkÞ ¼ C þ Dk 2 þ

1 þ F6

ð3Þ
1 þ G6k 6

with the coefficients determined from first principle calculations. The inversion
symmetry breaking BIA term and Zeeman term are:

6.0

5.0

4.0

3.0

Edge
Bulk

0 0 0 - D 1
0 0 D 0 C
0 D 0 0 CA ; HZ ¼ mBB

- D 0 0 0

B
B
@

gE 0 0 0
0 gH 0 0
0 0 - gE 0
0 0 0 - gH

C
C
A

; ð4Þ

B
B
@

Edge
Bulk

HBIA ¼

–1 0 1 2 3 –1 0 1 2 3

_V_ g (V) _V_ g (V)

Figure 3 | Transport and MIM line-cut analysis for the 7.5 nm device at 3
and 7 T. (a) Averaged line-cuts of the centre 3 1 section of the real-space

MIM-Im images in the 3 T column of Fig. 2j, with a 1.0 V offset in MIM
signal between each gate voltage. Gate voltage is from - 1.5 to 3.0 V in
0.5-V steps. (b) Edge and bulk MIM-Im signal (at the red and blue dashed
lines in a) and two-terminal resistance plotted against gate voltage.
(c,d) Same plots for the 7 Tcolumn. Transport is clearly dominated by edge
conduction in the bulk gapped regime in both fields, confirming the
extended nature of the edge conduction observed by MIM.

vacancies 21,29,30 and intrinsic surface effect 22,23 . As it stands now,
our discovery could herald broadly relevant physics or simply
reflect specifics of this material. In either case, with HgTe QWs
being the standard model system for QSH, our results should
shed light on extending existing theory to treat other potential
contributions and understanding QSH in real systems.

Methods
QW growth and device fabrication. Each of the two fully strained HgTe/
(Hg0.3Cd0.7)Te QWs was grown by molecular beam epitaxy on a 4 mm fully
relaxed CdTe buffer atop an n þþ GaAs substrate. The mobility exceeds
200,000 cm 2 V � 1 s � 1 in the n-type regime of the 7.5 nm device. QW thickness was
determined by X-ray reflectometry and magnetotransport measurements. The QW
is buried 50 (26) nm under the top surface in the 7.5 (5.5) nm device. The longrange surface topographic features mostly come from the relaxation of the CdTe
layer instead of the QW or the capping layer. Gate-QW capacitance was extracted
from Hall measurement and was used to convert gate voltage to carrier density and
LL filling factor. A two-step etching process was used to create mesas B130-nm
deep with clean physical edges for transport and MIM scans (Supplementary Fig. 1
and Supplementary Note 1).

MIM measurement and response curve calculation. MIM measurements were
implemented using an Attocube-based scanner stack in a Janis He4 cryostat with
9 T superconducting magnet 15 . Topography was taken with a quartz tuning fork
sensor to which the MIM probe is attached; MIM images were taken separately
with the tip B30 nm above mesa surface in constant-height mode. The efficient
capacitive coupling at microwave frequency enables sub-surface sensing without
inducing interband transitions. The tip-sample interaction is in the near-field limit,
so the spatial resolution is determined by the tip diameter (B150 nm) rather than
the microwave wavelength. Imaginary and real parts of the complex tip-sample
admittance were recorded during scanning to produce the MIM-Im and -Re
channel images. Only sxx contributes to the screening of the microwave electric
field; sxy is not probed here. A small bias is applied to the tip to compensate
the work function difference. The relationship between QW conductivity and

with ! ¼ 0.5 meV.

References
1. Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin hall effect.
Phys. Rev. Lett. 95, 146802 (2005).
2. Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum spin Hall effect and
topological phase transition in HgTe quantum wells. Science 314, 1757–1761
(2006).
3. Ko¨nig, M. et al. Quantum spin Hall insulator state in HgTe quantum wells.
Science 318, 766–770 (2007).
4. Roth, A. et al. Nonlocal transport in the quantum spin Hall state. Science 325,
294–297 (2009).
5. Bru¨ne, C. et al. Evidence for the ballistic intrinsic spin Hall effect in HgTe
nanostructures. Nat. Phys. 6, 448–454 (2010).
6. Bru¨ne, C. et al. Spin polarization of the quantum spin Hall edge states. Nat.
Phys. 8, 486–491 (2012).
7. Hart, S. et al. Induced superconductivity in the quantum spin Hall edge. Nat.
Phys. 10, 638–643 (2014).
8. Nowack, K. C. et al. Imaging currents in HgTe quantum wells in the quantum
spin Hall regime. Nat. Mater. 12, 787–791 (2013).
9. Ko¨nig, M. et al. Spatially resolved study of backscattering in the quantum spin
Hall state. Phys. Rev. X 3, 021003 (2013).
10. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys.
82, 3045–3067 (2010).
11. Qi, X. L. & Zhang, S. C. Topological insulators and superconductors. Rev. Mod.
Phys. 83, 1057–1110 (2011).
12. Orlita, M. et al. Fine structure of zero-mode Landau levels in HgTe/HgxCd1-xTe
quantum wells. Phys. Rev. B 83, 115307 (2011).
13. Zholudev, M. et al. Magnetospectroscopy of two-dimensional HgTe-based
topological insulators around the critical thickness. Phys. Rev. B 86, 205420
(2012).
14. Lai, K. et al. Mesoscopic percolating resistance network in a strained manganite
thin film. Science 329, 190–193 (2010).
15. Kundhikanjana, W., Lai, K., Kelly, M. A. & Shen, Z.-X. Cryogenic microwave
imaging of metal-insulator transition in doped silicon. Rev. Sci. Instrum. 82,
033705 (2011).
16. Lai, K. et al. Imaging of Coulomb-driven quantum Hall edge states. Phys. Rev.
Lett. 107, 176809 (2011).
17. Zhou, B., Lu, H. Z., Chu, R. L., Shen, S. Q. & Niu, Q. Finite size effects on
helical edge states in a quantum spin-Hall system. Phys. Rev. Lett. 101, 246807
(2008).
18. Ko¨nig, M. et al. The quantum spin Hall effect: theory and experiment. J. Phys.
Soc. Jpn 77, 031007 (2008).

where D ¼ 1.5B2 meV depending on the QW thickness. In a magnetic field B, we
calculate the LL fan chart by substituting k-k � eA. In the presence of the
BIA term, there is an anti-crossing of LLs with a gap 2D in an inverted sample.
Calculated LL fan charts are converted to DOS intensity plots using the disorderbroadened LL DOS of

DOSðE; ELLÞ ¼ eB

1
p

G=2
ð5Þ
ðE � ELLÞ 2 þ ðG=2Þ 2

19. Scharf, B., Matos-abiague, A. & Fabian, J. Magnetic properties of HgTe
quantum wells. Phys. Rev. B 86, 075418 (2012).
20. Vera-Marun, I. J. et al. Quantum Hall transport as a probe of capacitance
profile at graphene edges. Appl. Phys. Lett. 102, 013106 (2013).
21. Spicer, W. E. et al. Band gap variation and lattice, surface, and interface
‘instabilities’ in Hg1 � xCdxTe and related compounds. J. Vac. Sci. Technol. A 1,
1735 (1983).
22. Bryant, G. W. Surface-enhanced alloy effects on Hg1-xCdxTe(100) surface states
and the site-dependent coherent-potential approximation. Phys. Rev. Lett. 55,
1786–1789 (1985).
23. Schick, J. T., Bose, S. M. & Chen, A. B. Electronic structures of HgTe and CdTe
surfaces and HgTe/CdTe interfaces. Phys. Rev. B 40, 7825–7830 (1989).
24. Pascual, J. et al. Role of Spin in Quasiparticle Interference. Phys. Rev. Lett. 93,
196802 (2004).
25. Maciejko, J. et al. Kondo effect in the helical edge liquid of the quantum spin
Hall state. Phys. Rev. Lett. 102, 256803 (2009).
26. Lee, D.-H. Effects of interaction on quantum spin Hall insulators. Phys. Rev.
Lett. 107, 166806 (2011).
27. Pikulin, D. I. & Hyart, T. Interplay of exciton condensation and the quantum
spin Hall effect in InAs/GaSb bilayers. Phys. Rev. Lett. 112, 176403 (2014).
28. Du, L., Knez, I., Sullivan, G. & Du, R.-R. Robust Helical Edge Transport in
Gated InAs/GaSb Bilayers. Phys. Rev. Lett. 114, 096802 (2015).
29. Swarts, C. A., Daw, M. S. & McGill, T. C. Bulk vacancies in CdxHg1 � xTe.
J. Vac. Sci. Technol. 21, 198 (1982).
30. Bryant, G. W. Effects of cation vacancies and diffusion on the surface states of
Hg1-xCdxTe. Phys. Rev. B 36, 6465–6471 (1987).

Acknowledgements
We thank Chao-Xing Liu for helpful discussions and A. Budewitz for help in transport
characterizations. This work was supported by Defense Advanced Research Projects
Agency Microsystems Technology Office, MesoDynamic Architecture Program (MESO)
through contract number N66001-11-1-4105. The work at Stanford was also supported
by Center for Probing the Nanoscale, an NSF NSEC under grant PHY-0830228 to Z.-X.S.
and D.G.-G. to partially support Y.Y., NSF grant DMR1305731 to partially support W.K.
and Y.-T.C., the Gordon and Betty Moore Foundation through Grant GBMF3133 and
Foundation’s EPiQS Initiative through Grant GBMF4546 to Z.-X.S. to partially support

Y.Y. and Y.-T.C., the European Union under the project FP7-PEOPLE-2010-274769 to
M.R.C. and the Department of Energy, Office of Basic Energy Sciences, Division of
Materials Sciences and Engineering, under contract DE-AC02-76SF00515 to S.-C.Z. to
support J.W. for theoretical calculations and to D.G.-G. for device fabrication and
characterization. The work in Wu¨rzburg was also supported by the German Research
Foundation (DFG grant HA5893/4-1 within SPP 1666 and the Leibniz Program), the EU
ERC-AG program (Project 3-TOP) and the Elitenetzwerk Bayern program ‘Topologische
Isolatoren’.

Author contributions
E.Y.M. did MIM and transport measurements. M.R.C. and M.M. developed the fabrication process and fabricated the devices. C.A., P.L. and M.R.C. characterized the
material. C.A. and P.L. grew the QW wafers. J.W. and B.L. did the theoretical calculations. M.B. fabricated devices used for preliminary measurements. Data analysis, interpretation and manuscript preparation were done by all authors.

Additional information
Supplementary Information accompanies this paper at [http://www.nature.com/](http://www.nature.com/naturecommunications)
[naturecommunications](http://www.nature.com/naturecommunications)

Competing financial interests: M.A.K and Z.X.S are co-founders of PrimeNano Inc.
which licensed the sMIM technology from Stanford for commercial instrument.

Reprints and permission information is available online at [http://npg.nature.com/](http://npg.nature.com/reprintsandpermissions/)
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How to cite this article: Ma, E. Y. et al. Unexpected edge conduction in mercury
telluride quantum wells under broken time-reversal symmetry. Nat. Commun. 6:7252
doi: 10.1038/ncomms8252 (2015).

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