# Plateau Regions for Zero-Bias Peaks within 5$\%$ of the Quantized
Conductance Value $2e^{2}/h$

Zhaoyu Wang  equal contribution State Key Laboratory of Low Dimensional
Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084,
China  Huading Song  equal contribution<EMAIL_ADDRESS>Beijing Academy of
Quantum Information Sciences, 100193 Beijing, China  Dong Pan  equal
contribution State Key Laboratory of Superlattices and Microstructures,
Institute of Semiconductors, Chinese Academy of Sciences, P. O. Box 912,
Beijing 100083, China  Zitong Zhang  equal contribution State Key Laboratory
of Low Dimensional Quantum Physics, Department of Physics, Tsinghua
University, Beijing 100084, China  Wentao Miao  State Key Laboratory of Low
Dimensional Quantum Physics, Department of Physics, Tsinghua University,
Beijing 100084, China  Ruidong Li  State Key Laboratory of Low Dimensional
Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084,
China  Zhan Cao  Beijing Academy of Quantum Information Sciences, 100193
Beijing, China  Gu Zhang  Beijing Academy of Quantum Information Sciences,
100193 Beijing, China  Lei Liu  State Key Laboratory of Superlattices and
Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, P.
O. Box 912, Beijing 100083, China  Lianjun Wen  State Key Laboratory of
Superlattices and Microstructures, Institute of Semiconductors, Chinese
Academy of Sciences, P. O. Box 912, Beijing 100083, China  Ran Zhuo  State Key
Laboratory of Superlattices and Microstructures, Institute of Semiconductors,
Chinese Academy of Sciences, P. O. Box 912, Beijing 100083, China  Dong E. Liu
State Key Laboratory of Low Dimensional Quantum Physics, Department of
Physics, Tsinghua University, Beijing 100084, China  Beijing Academy of
Quantum Information Sciences, 100193 Beijing, China  Frontier Science Center
for Quantum Information, 100084 Beijing, China  Ke He  State Key Laboratory of
Low Dimensional Quantum Physics, Department of Physics, Tsinghua University,
Beijing 100084, China  Beijing Academy of Quantum Information Sciences, 100193
Beijing, China  Frontier Science Center for Quantum Information, 100084
Beijing, China  Runan Shang  Beijing Academy of Quantum Information Sciences,
100193 Beijing, China  Jianhua Zhao<EMAIL_ADDRESS>State Key Laboratory of
Superlattices and Microstructures, Institute of Semiconductors, Chinese
Academy of Sciences, P. O. Box 912, Beijing 100083, China  Hao Zhang
<EMAIL_ADDRESS>State Key Laboratory of Low Dimensional Quantum
Physics, Department of Physics, Tsinghua University, Beijing 100084, China
Beijing Academy of Quantum Information Sciences, 100193 Beijing, China
Frontier Science Center for Quantum Information, 100084 Beijing, China

###### Abstract

Probing an isolated Majorana zero mode is predicted to reveal a tunneling
conductance quantized at $2e^{2}/h$ at zero temperature. Experimentally, a
zero-bias peak (ZBP) is expected and its height should remain robust against
relevant parameter tuning, forming a quantized plateau. Here, we report the
observation of large ZBPs in a thin InAs-Al hybrid nanowire device. The ZBP
height can stick close to $2e^{2}/h$, mostly within $5\%$ tolerance, by
sweeping gate voltages and magnetic field. We further map out the phase
diagram and identify two plateau regions in the phase space. Despite the
presence of disorder and quantum dots, our result constitutes a step forward
towards establishing Majorana zero modes.

Majorana zero modes (MZMs) [1, 2] have been extensively searched in hybrid
semiconductor-superconductor nanowire devices since the first material
prediction in 2010 [3, 4]. One key prediction, a quantized zero bias peak
(ZBP) in tunneling conductance [5, 6, 7, 8], still remains illusive so far [9,
10, 11, 12, 13, 14, 15, 16]. Moreover, theory developments have proposed the
concept of quasi-MZMs due to smooth potential variation [17, 18, 19, 20, 21,
22, 23] or disorder [24, 25, 26, 27, 28, 29]. These quasi-MZMs, though
topologically trivial, can also lead to quantized ZBPs. The quantization
mechanisms for MZMs and quasi-MZMs are similar: the conductance is solely
contributed by one isolated MZM while the second MZM is decoupled. Recent
experimental progress has reported large ZBPs whose height can reach
$2e^{2}/h$ [13, 14]. However, a single peak at $2e^{2}/h$ is not enough to be
entitled as ‘quantized’. A quantized ZBP requires the peak height sticking to
$2e^{2}/h$ by tuning all relevant experimental knobs: a plateau defined by
parameter sweepings. This plateau phenomenon is still missing in experiments.

Here, we have improved the device fabrication and report ZBPs near $2e^{2}/h$,
forming a plateau defined by sweeping gate voltages and magnetic field ($B$)
in an ultra-thin InAs-Al nanowire device. We quantify the plateau with a
tolerance of $5\%$, a number commonly used in recent literature [26, 27]. In
the end, we discuss their possible connections to MZMs or quasi-MZMs.

Figure 1: (a) Device SEM (false colored). The contacts and gates are Ti/Au
(5/70 nm in thickness). The substrate is p-doped Si covered by 300 nm thick
SiO2, serving as a global back gate (BG). (b) Hard gap tunneling spectroscopy
at 0 T. (c) $B_{\text{z}}$ dependence of the gap. (d) $B$ scan (aligned with
the nanowire) of a ZBP near $2e^{2}/h$. Lower panel, zero-bias line cut. The
pink bar marks the range from 0.95 to 1.05 in the unit of $2e^{2}/h$ for all
figures. (e) ‘Waterfall’ plots of (d). For clarity, every other line cut is
shown and two colors are used. (f) $V_{\text{TG}}$ and $V_{\text{BG}}$ scans
of the ZBP. (g) Gate and $B$ scans at zero bias. The three-color plots mark
red as regions within $\pm 5\%$ of $2e^{2}/h$. (d) (f) and (g) share the same
color bar.

Figure 1a shows the device scanning electron micrograph (SEM). Material and
growth details can be found in Ref [30]. The InAs diameter ($\sim$ 26 nm) is
much thinner than those commonly used ($\sim$ 100 nm), aiming for fewer
subband occupations. Theory has shown that fewer or single subband regime is
preferred for MZM quantization [18, 31, 32]. The device was measured in a
dilution fridge, base temperature $T\sim$ 20 mK. The electron $T$ can be below
40 mK. A total bias voltage together with a lock-in excitation was applied to
contact N1 with the current $I$ and $dI$ drained from S1. A voltage meter
measured $V$ and $dV$ between N2 and S2. This four-terminal set-up can exclude
the contributions of contact resistance as a systematic uncertainty. The
tunnel gate (TG) and back gate (BG) were used while SG was not well functional
and kept grounded. See Fig. S1 for circuit details.

The tunneling spectroscopy at zero $B$ (Fig. 1b) resolves a hard
superconducting gap. The gap size is $\sim$ 0.39 mV due to the thin Al film.
Fig. 1c shows the $B_{\text{z}}$ dependence of the gap. The z axis is 8∘
misaligned with the nanowire. For $B$ along other directions, see Fig. S2. The
gap remains hard below 2 T and gradually becomes soft above 3 T. Hard gap at
high $B$ is necessary for MZM quantization since a soft gap destroys the
quantization by dissipation [33, 34].

In Fig. 1d we tune the device to a gate setting and find a ZBP near $2e^{2}/h$
(see Fig. 1e for the ‘waterfall’ plots). $B$ is aligned with the nanowire
unless specified. Along this direction, the maximum field allowed is 1.87 T
due to the hardware limit. From 1.3 T to 1.85 T, the zero-bias conductance
remains within the $5\%$ tolerance bar, see the pink background (from 0.95 to
1.05). The ZBP width, $\sim$ 0.3 mV, is 20 times larger than the thermal width
(3.5$k_{B}T\sim$ 15 $\mu$eV for $T$ = 50 mK). A wide peak is necessary for MZM
or quasi-MZM quantization to minimize the effect of thermal broadening. This
requires a large tunnel transmission, reflected by the outside gap conductance
in Fig. 1e being $\sim$ 10 times larger than that in Fig. 1c. Large
transmission results in a finite subgap conductance. Unlike a soft gap, this
Andreev reflection induced subgap conductance does not affect the MZM
quantization [34].

We then set $B$ to 1.35 T and scan gate, see Fig. 1f (for ‘waterfall’ plots,
see Fig. S3). The zero-bias conductance remains near $2e^{2}/h$ over a sizable
gate range. Occasionally it slightly deviates from the $5\%$ tolerance bar due
to small oscillations. The oscillations in $V_{BG}$, also observed in our
previous work [14], are possibly due to the formation of an open island as
visualized by the weak Coulomb blockade diamond in the color map. The island
is likely defined between the barrier region and the S2 contact which forms a
weak barrier due to the ultra-thin diameter. The oscillation causes a small
splitting of the ZBP, suggesting the energy also being modified.

The three scans in Figs. 1d and 1f were in close measurement sequence without
noticeable charge jumps in between. These three scans simultaneously pass
through one ‘sweet spot’ in the parameter space ($B$ = 1.35 T, $V_{\text{TG}}$
= -9.137 V, $V_{\text{BG}}$ = 6.886 V), and their zero-bias line cuts all
resolve a plateau feature near $2e^{2}/h$. The plateau in $V_{\text{TG}}$
exceeds 100 mV, significantly wider than a fine-tuned sharp crossing. We
therefore identify this combined feature as a ZBP-plateau near $2e^{2}/h$,
mostly within the $5\%$ tolerance bar.

Fig. 1g shows the zero-bias map by scanning both $B$ and gate voltages, also
passing through this ‘sweet spot’. The three-color plots resolve the ‘red
islands’ as the $2e^{2}/h$-regions (within $\pm 5\%$). Occasionally, the
conductance can slightly exceed 1.05, by $\sim 1\%$ (the cyan region). The
‘red island’ is similar to simulations on partially separated MZMs [26]. Based
on the Coulomb blockade diamond size $\sim$ 60 $\mu$V and its period in
$V_{\text{BG}}$ ($\sim 12.7$ mV) in Fig. 1f, we can extract the lever arm
between $V_{\text{BG}}$ and the energy scale possibly related to the electro-
chemical potential. The size of the ‘red island’ in Fig. 1g is $\sim$ 65 mV in
$V_{\text{BG}}$, corresponding to an energy scale of $\sim$ 300 $\mu$eV.

In the MZM or quasi-MZM picture, several factors could cause the deviation of
ZBP from $2e^{2}/h$: 1) soft gap; 2) thermal broadening; 3) residual tunnel
coupling between the second MZM and the probe; 4) coupling between the two
MZMs; 5) multiple subband occupation. The first factor, a soft gap [35], not
only destroys ZBP quantization, but is also detrimental to MZM applications
[36, 37] and should be avoided. This problem has been solved by the
observation of a hard gap within the interested $B$ range (Fig. 1c). The
second factor lowers the ZBP height. Since our ZBP width is mainly tunnel
broadened and $\sim$ 20 times larger than the thermal width, this effect is
also small: thermal averaging at $T$ = 50 mK only causes a height change of
$\sim 1\%$ for such a wide peak. The third and fourth factors are closely
related and can either increase or decrease the height from $2e^{2}/h$,
depending on the coupling details [26]. For the last factor, a second channel
could provide a small additional conductance background, increasing the total
height above $2e^{2}/h$ [8]. Based on its saturation conductance (Fig. S2),
our thin nanowire is likely still not in the single subband regime yet. The
small zero-bias conductance in Figs. 1d-g (at non-ZBP regions), which can be
less than 4$\%$ of $2e^{2}/h$, suggests that the multi-band contribution to
the background conductance (if any) is small.

Figure 2: $V_{\text{TG}}$ = -9.03 V for all panels. (a) Zero-bias map in ($B$,
$V_{\text{BG}}$) space. The right panel is the multi-color plot. (b) $B$ scans
at four $V_{\text{BG}}$ settings, see the dashed lines in (a). The bottom
panel shows the zero-bias line cuts, with corresponding colors. (c)
$V_{\text{BG}}$ scan at 1.2 T (top) and 1.0 T (middle) with zero-bias line
cuts shown in the bottom. (d) Line cuts from (b) at 0 T (dashed) and $\sim$
1.73 T (solid) with corresponding colors. All panels share the same color bar.
Figure 3: (a) $B$ scan of a ZBP near $2e^{2}/h$. Lower panel, zero-bias line
cut. (b) Schematic illustration of the angle $\psi$. $B$ is in-plane. (c)
Angle dependence of the ZBP by fixing the $B$ amplitude at 1.35 T, see the
middle black arrow in (a). Lower panel, zero-bias line cut. (d) Two more
rotation scans at 1.2 T (upper) and 1.499 T (middle), see the other two arrows
in (a). Lower panel, zero-bias line cuts. (e) Line-cuts from (d) (upper). For
clarity, half of the angle range (and every other line cut) is shown. All
panels share the same color bar.

We next fix $V_{\text{TG}}$ to a slightly different value and explore the
zero-bias map in ($B$, $V_{\text{BG}}$) space, see Figure 2a. Besides the ‘red
island’ as a ‘zone’, we use several discrete colors to label other ranges
which are ‘boundary-like’. Fig. 2b shows four $B$ scans, see the corresponding
colored dashed lines in Fig. 2a. The first scan is outside the ‘red island’
and resolves no ZBPs. Other scans pass through the ‘red island’ and resolve
ZBPs near $2e^{2}/h$ with different $B$ ranges. The green triangles and stars
mark the on-set and ending $B$ values for ZBPs. The ZBP region defined by the
green dashed lines in Fig. 2a roughly matches the ‘red island’ and can serve
as a $2e^{2}/h$-ZBP phase diagram. Note that not every line cut passing
through the ‘red island’ can resolve a plateau feature. For example, the blue
line in Fig. 2b is more ‘peaked-like’ near $2e^{2}/h$.

Fig. 2c shows $V_{\text{BG}}$ scans of the ZBP at 1.2 T and 1 T, corresponding
to the middle and edge of the ‘red island’, respectively. The zero-bias line
cuts illustrate the evolution from ‘peak-like’ (red) to ‘plateau-like’
(black). Fig. 2d shows the line cuts from the four panels in Fig. 2b at $B$ =
0 T (dashed lines) and $\sim$1.73 T. At zero field, the subgap peaks have
similar energies. At high field (1.73 T), some curves resolve ZBPs near
$2e^{2}/h$ while others resolve split peaks, reflecting the phase diagram
boundaries. For ‘Waterfall’ plots and additional scans see Figs. S3 and S4.

The next experimental knob is $B$ direction. $B$ rotation in Al-based systems
was limited before due to orbital effects: the Al bulk gap is easily
suppressed at low $B$ ($\sim$ 0.3 T) during rotation [10]. Here, the ultra-
thin diameter significantly suppresses the orbital effect and the gap can
survive at high $B$ even if misaligned (see Fig. S2). This advantage enables a
rotation for ZBP [38].

Figure 4: (a) Zero-bias map in ($V_{\text{TG}}$, $V_{\text{BG}}$) space at
1.05 T. Grey regions have no data. (b) Re-plotting (a) using the new axis
$V_{\text{A}}$ and $V_{\text{B}}$ (see labeling). Right panel, three-color
plot. (c) Zero-bias maps at 0.3 T and 1.35 T in three-color plots. (d-e) Three
additional example scans ($V_{\text{BG}}$, $V_{\text{TG}}$, $B$) for ZBPs
within region I. (f-g) $B_{\text{z}}$ and gate scans of a ZBP in region II.
All panels share the same color bar.

Figure 3a starts with a $B$ scan (aligned with the nanowire) of a ZBP near
$2e^{2}/h$ (note the minor charge jump at $\sim$ 1.7 T). The $B$ amplitude is
then fixed at three values, see the arrows. Its direction, defined by the
angle $\psi$ (Fig. 3b), is rotated. $B$ is in-plane during rotation (parallel
to the substrate). Fig. 3c shows the angle dependence at 1.35 T. The ZBP
remains close to $2e^{2}/h$ over an angle range of $\sim$ 50∘. Outside this
range, the ZBP height quickly decreases away from $2e^{2}/h$, accompanied by a
gap closing. Finally, the ZBP vanishes at larger angles. For $\psi\sim$ 170∘,
the zero-bias conductance drops slightly below 0.95$\times 2e^{2}/h$ due to a
small peak-splitting which is separated by a minor charge jump. Fig. 3d shows
the angle dependence at two other $B$s, with line cuts shown in Fig. 3e. For
more rotation scans, see Fig. S5.

For the last experimental check, we fix $B$ (aligned with the nanowire) at
1.05 T, and measure the zero-bias map in ($V_{\text{TG}}$, $V_{\text{BG}}$)
space, see Figure 4a. Due to the cross talks between the two gates, every gate
can tune both the tunnel transmission in the barrier region and the electro-
chemical potential in the proximitized wire. Therefore, the $V_{\text{BG}}$
range is adjusted simultaneously for different $V_{\text{TG}}$ settings to
trace the same feature. To save space from the ‘no-data’ region (grey), we
define new gate voltage axes: $V_{\text{A}}=V_{\text{BG}}$
cos$\phi+V_{\text{TG}}$ sin$\phi$; $V_{\text{B}}=-V_{\text{BG}}$
sin$\phi+V_{\text{TG}}$ cos$\phi$. $\phi$ = 30∘. Fig. 4b re-plots Fig. 4a in
($V_{\text{A}}$, $V_{\text{B}}$) axes. Together with Fig. 4c, the three-color
plots show the gradual evolution of the ‘red island’: its area increases from
none at low $B$ to a sizable zone at high $B$. For a complete evolution, see
Fig. S6. Figs. 4d and 4e show three additional examples of ZBP scans within
this parameter region.

So far, all the ZBPs (from Fig. 1d to Fig. 4e) are within one single region
(region I) in the multi-dimensional parameter space ($B$, $V_{\text{TG}}$,
$V_{\text{BG}}$), see Fig. S7 for additional scans in this region. The
measurement in region I lasts for three weeks with several charge jumps in
between. We note the effects of jumps and hysteresis mainly shift the gate
voltages, maximally by 300 mV. The main plateau features still remain after
the shift with minor variations.

Fig. 4f and 4g demonstrate another plateau region (region II) at a very
different ($B$, $V_{\text{TG}}$, $V_{\text{BG}}$) setting from region I. To
access higher $B$s, we apply $B$ along the fridge z axis ($B_{\text{z}}$). The
small misalignment (8∘) between $B_{\text{z}}$ and the nanowire should not
bring a big difference based on the rotation experiment in Fig. 3. In Fig. 4f,
the zero-bias conductance can stick close to $2e^{2}/h$ from 2 T to 3.28 T, a
$B$-range larger than 1.2 Tesla (with a tolerance of 5$\%$). Outside this
range, the ZBP height drops continuously from $2e^{2}/h$ with a faster
decreasing rate, possibly due to the combined effects of gap closing and
softening for $B$ larger than 3 T. The fridge base $T$ for Fig. 4f is slightly
higher (can reach $\sim$ 30 mK) which may also play a role. Even within the
plateau region, there is still a decreasing trend with a much smaller slope,
possibly due to those mechanisms above.

The overall conductance (barrier transmission) in region II is higher than
that in region I, leading to a larger subgap conductance. Fig. 4g illustrates
the interaction between the zero-energy state and a quantum dot level,
possibly formed near the barrier. Tuning the dot level towards zero energy
causes the splitting of the ZBP, similar to the behavior of partially
separated MZMs [39, 40]. We translate the plateau width in $V_{\text{BG}}$ to
an energy scale of $\sim$ 260 $\mu$eV based on our extracted lever arm. For
additional scans of region II, see Fig. S8. Though the plateau regions (I and
II) in gate voltage space is small, only 50 - 100 mV. Its corresponding energy
scale, $\sim$ 260 - 300 $\mu$eV, is much larger than the thermal energy.

We note that this work addresses the question ‘whether ZBP-plateaus near
$2e^{2}/h$ exist by sweeping all relevant experimental parameters’. It,
however, cannot answer the question ‘whether similar plateaus (for sweeping
all parameters) also exist at non-quantized values’. That would require
exhausting the entire parameter space in more devices. Based on the collected
data so far, we have not observed similar plateaus at other values yet. See
Fig. S9 for additional ZBP data due to quantum dots and disorder in regions
beyond I and II. Occasionally, we could find ‘plateau-like’ features above
$2e^{2}/h$ for one-parameter sweeping (but not all). We notice that for
multiple subband occupation, even MZM could lead to some ‘plateau-like’
features above $2e^{2}/h$ in the open barrier regime with slight disorder [8].
Finally, we compare some scans in Fig. S10 to illustrate the interplay between
the barrier transmission and the ZBP height.

To summarize, we have observed ZBPs near $2e^{2}/h$ which form a plateau
mostly within $5\%$ tolerance by sweeping gate voltages and magnetic field.
Our result is qualitatively consistent with the (multi-subband) MZM as well as
quasi-MZM theories: the first is topological while the second is trivial and
caused by non-uniform potential or disorder. The quasi-MZM scenario is
probably more likely. This work is from a single device whose quality is an
improvement compared to our previous one [14] but still not in the ballistic
regime yet due to the obvious presence of quantum dots and disorder. Besides
the ‘end-to-end’ correlation experiment to reveal gap-closing-reopening [41,
42, 43], future devices on ‘single-terminal’ experiments could aim at 1)
looking for zero-bias peak-to-dip transition near $2e^{2}/h$ with better
tolerance [16, 19, 14]; 2) using a dissipative probe to reveal quantized ZBPs
and suppress others [44, 45, 46, 47, 48]; 3) exploring potentially better
material systems [49, 50].

##  I Acknowledgment

We thank Leo Kouwenhoven for comments. Raw data is available at
https://doi.org/10.5281/zenodo.6546974. This work is supported by Tsinghua
University Initiative Scientific Research Program, Alibaba Innovative Research
Program, National Natural Science Foundation of China (Grant Nos. 12104053,
92065106, 92065206, 61974138, 12004040, 11974198). D.P. also acknowledges the
support from Youth Innovation Promotion Association, Chinese Academy of
Sciences (Nos. 2017156 and Y2021043).

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