# Magnetic-field driven evolution of zero-energy mode on Bi islands deposited
on Fe(Te,Se)

Kailun Chen, Chuanhao Wen, Zhiyong Hou, Huan Yang,∗ and Hai-Hu Wen† National
Laboratory of Solid State Microstructures and Department of Physics,
Collaborative Innovation Center of Advanced Microstructures, Nanjing
University, Nanjing 210093, China

###### Abstract

We investigate the magnetic-field dependent evolution of the zero-bias
conductance peaks (ZBCPs) on the nanoscale bismuth islands grown on the
FeTe0.55Se0.45 substrate. The ZBCPs can be observed throughout the entire
region on these islands, and their characteristics align with the signatures
of Majorana zero modes. Remarkably, the evolution of ZBCPs on these islands
exhibits anomalous behavior under varying magnetic fields: The magnitude of
ZBCPs is first enhanced at weak fields lower than 2 T and then suppressed as
the fields further increase. We attribute the non-monotonic evolution of the
ZBCPs to the magnetic-field-enhanced topological edge states on these Bi
islands. Our findings provide valuable insights into the probable origin of
the Majorana zero modes in the Bi-island platform and the magnetic-field
response of topological edge states.

##  I Introduction

Majorana zero modes (MZMs) have attracted intensive interest because of their
potential applications in fault-tolerant topological quantum computing
SCZhangReview ; AndoReview ; NonAbelian , and these modes can be realized in
topological superconductors. One of the effective methods for achieving
topological superconductivity is to induce superconductivity in the
topologically insulating layer through the proximity effect from the adjacent
superconductor LFuProximity . For example, zero-energy modes were observed in
vortex cores of the heterostructures composed by the topological insulator
Bi2Te3 and the conventional superconductor NbSe2 JiaReview ; Jiaprl  or the
iron-based superconductor Fe(Te,Se) ChenMYSA . In addition, MZMs were also
observed at terminals of vortex lines in superconductors with topologically
non-trivial bands, such as some iron-based superconductors FeTeSeMajorana ;
FeTeSeHanaguri ; HuReview ; LiFeOHFeSe ; CaKFe4As4 ; LiFeAs  or other
materials WS2LiW ; CVSWangZY .

On the surface of a topologically non-trivial superconductor, topological edge
states appear at some boundaries SCZhangReview ; Hasan , such as the twin
boundary WangZYScience  or the step edge JiaoLUTe2 . In one-dimensional cases,
topological edge states can exist in a semiconducting or spin-orbit-coupled
nanowire, as well as in a ferromagnetic atom chain neighboring to a
superconductor AliceaReview , where they appear as MZMs. Experimental
observations of these modes are realized at the ends of one-dimensional
semiconducting nanowires SemiChain ; Quantumdot  and magnetic atomic chains
FeChain  grown on the surface of an $s$-wave superconductor. As for two-
dimensional heterostructures, MZMs CrBr3  or Majorana edge modes Feisland ;
PbCoSi  are observed on ferromagnetic islands grown on $s$-wave
superconductors. In addition, a robust zero-energy mode is observed in a
trilayer heterostructure MnTe/Bi2Te3/Fe(Te,Se) ZhangT .

Bismuth is a semimetal with strong spin-orbit coupling, and it is a good
platform for investigating the topological superconductivity or the MZMs when
it is adjacent to a superconductor. The Majorana edge states may exist at the
boundary of the Bi layer with the former mentioned configuration. And the
experimental evidence has been demonstrated at the edges of Bi bilayers
Bibilayer  and Bi films decorated with magnetic iron clusters Edgechannel
grown on the superconducting substrate. In our previous research, robust zero-
energy modes were observed on specific Bi islands deposited on the iron-based
superconductor Fe(Te,Se) Biisland . The zero-energy modes are likely caused by
the interference of two counter-propagating topological edge states at the
boundary of Bi islands. This kind of edge states in a topologically non-
trivial system is protected by the time-reversal symmetry, and is impervious
to impurity scattering in the absence of magnetic fields. Therefore, after
applying varying magnetic fields, the evolution of the edge states is also an
interesting issue that has been scarcely reported in experiments.

In this work, we examined the evolution of the ZBCP magnitude on some Bi
islands grown on the FeTe0.55Se0.45 substrate when applying magnetic fields.
Based on the statistic of the measuring areas, the ZBCPs exist only on some of
the islands with a diameter of 4-8 nm and can be observed in the entire region
of these islands. The characteristics of all the ZBCPs are also consistent
with those of MZMs, indicating a topologically non-trivial origin of the
ZBCPs. Notably, we observed an anomalous, yet general behavior of the ZBCPs
upon varying magnetic fields: The intensity of the ZBCPs on Bi islands is
first enhanced at weak fields lower than 2 T, and subsequently decreases as
the fields further increase. The strengthening of the ZBCPs at weak fields may
be attributed to the magnetic-field tuning on the edge states to the inner
part of the island.

##  II Experimental methods

The single crystals of FeTe0.55Se0.45 were synthesized by the self-flux method
samplegrowth . The crystals were annealed at $400^{\circ}$C for 20 h in an O2
atmosphere to eliminate the interstitial Fe atoms and then quenched in the
liquid nitrogen. The single crystal was cleaved in an ultrahigh vacuum with a
pressure of about $1\times 10^{-10}$ Torr before the growth of the Bi islands.
High-purity Bi (99.999%) powders were heated to $360^{\circ}$C in the effusion
cell (CreaTec) and then evaporated to the cleaved surface of Fe(Te,Se) at room
temperature by molecular beam epitaxy method. The nanoscale bismuth islands
can be grown on the Fe(Te,Se) substrate. Afterwards the sample was transferred
to the scanning tunneling microscopy/spectroscopy (STM/STS) head which was
kept at a low temperature. The STM/STS measurements were carried out in a
USM-1300 system (Unisoku Co. Ltd.) with an ultrahigh vacuum, low temperature,
and high magnetic field. The tunneling spectra were measured by a lock-in
technique with an amplitude of 0.2 mV and a frequency of 938 Hz. The tips in
the measurements were made by tungsten using the electrochemically etching
method. All measurements were taken at 0.4 K unless in some specified cases.
The magnetic field was applied along the $c$-axis of Fe(Te,Se) substrate or
equivalently perpendicular to the Bi islands.

##  III Results

###  III.1 Bi islands with and without ZBCPs

Figure 1: (a) Topography of an area containing 6 Bi islands grown on the
Fe(Te,Se) substrate (setpoint conditions: $V_{\mathrm{set}}=1$ V,
$I_{\mathrm{set}}=20$ pA). (b) Typical tunneling spectra measured on the
islands and the substrate nearby ($V_{\mathrm{set}}=10$ mV,
$I_{\mathrm{set}}=200$ pA). The island #1 is the only one with ZBCP. (c)
Statistics on the number of islands with and without ZBCPs versus the area of
all islands we have measured.

Figure 1(a) shows a typical topography of the Bi islands grown on the
Fe(Te,Se) substrate. The islands are randomly distributed on the flat surface
of the substrate, with the dimensions of several nanometers. The shapes of
these islands are also irregular, and there are even some wrinkles near the
boundary indicating the lattice distortion there. Although the islands have
different sizes, the height of them is all about 7 Å which is consistent with
the thickness of Bi(110) monolayer islands Biheight . These features of the Bi
islands are consistent with those in our previous work Biisland .

Figure 1(b) shows typical tunneling spectra measured on the six islands in the
field of view of Fig. 1(a). We also present the typical tunneling spectrum
measured on the Fe(Te,Se) substrate in Fig. 1(b), and it shows a fully gapped
feature. The superconducting gap of the Fe(Te,Se) substrate varies from 1.1 to
2.1 meV determined by calculating the energy difference between coherence
peaks ChenMYCdGM . Tunneling spectra measured on Bi islands #2-#6 are similar
to the one measured on the Fe(Te,Se) substrate. In contrast, the tunneling
spectrum measured on island #1 is different, i.e., a ZBCP appears in the
tunneling spectrum measured on this island. It should be noted that the ZBCP
can be observed in the spectra measured on the whole island Biisland . We have
investigated 146 islands with monolayer thickness, and only 23 of them exhibit
ZBCPs. The probability of finding a Bi monolayer island with the ZBCPs is
about 16%. We also note that there are some bilayer Bi islands, but none of
them hold the ZBCPs. The number statistics are also carried out in the region
of monolayer islands, and the result is shown in Fig. 1(c). One can see that
the areas of islands with ZBCPs are mainly distributed from 10 nm2 to 50 nm2,
corresponding to the diameter of about 4-8 nm. When the areas of Bi islands
exceed 50 nm2, no ZBCPs has been observed in these islands.

Figure 2: (a) Topography of a Bi island (#7) ($V_{\mathrm{set}}=1$ V,
$I_{\mathrm{set}}=20$ pA). (b) Typical tunneling spectra measured at marked
positions shown in (a) ($V_{\mathrm{set}}=10$ mV, $I_{\mathrm{set}}=200$ pA).
(c) Line-profile of tunneling spectra taken along the dashed arrow in panel
(a) ($V_{\mathrm{set}}=10$ mV, $I_{\mathrm{set}}=200$ pA).

Figure 2(a) shows the topography of a nanoscale monolayer Bi island numbered
as #7. The ZBCPs can be observed in the tunneling spectra measured on the
island, such as two spectra measured at the edge and the center of the island
shown in Fig. 2(b). Besides, the energies of the coherence peaks of these two
spectra are similar to those obtained from the spectra measured on the
Fe(Te,Se) substrate, which is a demonstration of the proximity-induced
superconductivity on the Bi island. It should be noted that ZBCPs can be
observed in the spectra taken all over the island, and one can obtain the
conclusion from a set of tunneling spectra measured across the island as shown
in Fig 2(c). Obvious in-gap peaks can be seen in all the spectra in this
panel, and the peak positions are fixed near zero energy. These observations
are similar to those in our previous work Biisland .

###  III.2 Magnetic-field dependent evolution of ZBCPs on Bi islands

Figure 3: (a) Tunneling spectra measured at different magnetic fields and at
the location of red dot in Fig. 2(a) ($V_{\mathrm{set}}=10$ mV,
$I_{\mathrm{set}}=200$ pA). (b)-(d) Zero-energy d$I$/d$V$ mappings recorded in
the same region in Fig. 2(a), and they are measured at different fields
($V_{\mathrm{set}}=40$ mV, $I_{\mathrm{set}}=200$ pA). The edge of the island
is marked out by dashed lines. The color bars are the same in these mappings.

The ZBCPs on some Bi islands are weakened by increase of the temperature
Biisland , which can be understood as the temperature suppression to the
superconductivity in the Fe(Te,Se) substrate. Since the upper critical field
of Fe(Te,Se) is extremely high FeSeTeHc2 , it is interesting to investigate
the field-dependent evolution of the ZBCPs on the Bi islands. Figure 3(a)
shows tunneling spectra measured at different fields of 0, 2, and 5 T at the
center of the Bi island #7. Surprisingly, the increment of the magnetic field
does not suppress the ZBCP monotonically, and conversely the peak magnitude
increases at 2 T compared to that obtained at 0 T. At a higher field of 5 T,
the magnitude of the ZBCP is significantly suppressed, but the ZBCP does not
show any splitting or broadening features.

The differential conductance mapping is a useful method to get the information
about the spatial distribution of density of states (DOS) STMReview1 ;
STMReview2 . Figures 3(b)-3(d) show the recorded spatial distributions of
zero-bias differential conductance of the Bi island #7 in the same area but
under different fields. These three mappings are presented in the same color
scale, thus the brightness directly corresponds to the zero-energy DOS. The
zero-bias differential conductance is almost zero on the Fe(Te,Se) substrate,
reflecting the fully gapped feature. The value is finite on the whole island
at 0 T, which corresponds to the robust zero mode on the island. Some weak
ZBCP magnitude may be due to the surface-lattice distortion of the Bi islands.
At the magnetic field of 2 T [Fig. 3(c)], the inner part of the island becomes
notably brighter, suggesting an increment of the ZBCP magnitude. However, at 5
T [Fig. 3(d)], the zero-energy differential conductance becomes much weaker.
These observations are consistent with the ZBCP evolution in the tunneling
spectra at different fields shown in Fig. 3(a).

Figure 4: (a) Tunneling spectra measured on another island #8 at different
magnetic fields ($V_{\mathrm{set}}=10$ mV, $I_{\mathrm{set}}=200$ pA). The
topography of the island is shown in the inset, and the spectra are measured
at the marked position in the island. (b) Magnetic field-dependent evolution
of the ZBCP intensity on several islands with ZBCPs. The values of the zero-
energy differential conductance at different fields are normalized by the
value at zero field.

A control experiment is carried out on the Bi island #8, and Fig. 4(a) shows
the tunneling spectra measured at the same position but under different
magnetic fields. The magnitude of the ZBCP increases when the magnetic field
increases from 0 to 2 T, and then the value decreases as the field further
increases. Similar experiments are also carried on other four islands #9-#12
with ZBCPs, and the field-dependent zero-bias differential conductance is
shown in Fig. 4(b). Here it should be mentioned that these data are recorded
in the Bi islands away from vortex cores in the Fe(Te,Se) layer, otherwise the
tunneling spectra behave very differently from the spectra measured at other
neighbored fields. The curves in Fig. 4(b) share similar features with varying
magnetic fields: the intensity of the ZBCP increases with increasing field and
reaches its maximum at about 2 T, and afterward the ZBCP magnitude decreases
rapidly with the increase of the field. Therefore, the non-monotonic field-
dependent evolution of the ZBCP magnitude is a common property on these Bi
islands grown on Fe(Te,Se).

##  IV Discussion

As presented above, we have investigated the magnetic-field dependence of the
ZBCP magnitude in some monolayer Bi islands grown on the superconducting
FeTe0.55Se0.45 substrate. On these specific islands with the size of 4-8 nm,
the zero-energy peak is the common feature of the spectra measured throughout
the entire island. In the process of applying magnetic fields, the ZBCPs are
fixed at zero energy and do not split. Additionally, the width of the ZBCPs
does not broaden. These features exclude the trivial origin of the ZBCPs
caused by Yu-Shiba-Rusinov states or the Andreev bound states ChenXY , and are
consistent with the characteristic of MZMs as reported previously Jiaprl ; IFI
; ChenXY ; Adatom . Thus, the ZBCPs on the Bi islands probably have a
topologically non-trivial origin. As discussed before ChenXY , the ZBCPs may
be due to the magnetic moment just below the particular Bi island. The
magnetic moment can be induced by an interstitial iron atom, and it leads to
the time-reversal symmetry breaking AliceaReview . However, in the present
work, the ZBCP magnitude increases with the increase of magnetic fields and
reaches its maximum at about 2 T; afterwards, the magnitude decreases with
further increase of the fields. This is very different from the situation of
excess iron impurities, and the ZBCP magnitude is robust at a field as high as
8 T IFI . In addition, the effective range by the excess iron atom is very
limited with a radius of about 1 nm IFI  which is much smaller than the size
of the Bi island. Therefore, the excess iron atom as the origin of the ZBCP
may be excluded, and the ZBCP is likely caused by the topological
superconductivity induced on the Bi island with a strong spin-orbital
coupling.

From our previous study ChenXY , the zero-energy states observed on these Bi
islands may be caused by two counter-propagating topological edge states TSC3D
emerging at the edge of the islands. The edge states usually behave as the
spatial oscillation as a form of the Bessel function, and the real-space
period approximately equals to $\pi/k_{\mathrm{F}}$ HaoN1 . Since the Fermi
vector $k_{\mathrm{F}}$ is very small in Bi, the real-space period can be the
scale of several nanometers. When the size of the island is suitable, the pair
of the edge states may form an interfered resonant state which behaves as the
ZBCP. Based on this picture, the applied magnetic fields can tune the real-
space period of the oscillation as well as the decaying parameter of the edge
state to the inner part of the island HaoN1 , which may help to increase the
ZBCP magnitude at fields lower than 2 T. In comparison, the ZBCP magnitude
decreases monotonically with increase of the magnetic field in the situation
of the iron impurity IFI ; HaoN2 , the magnetic monolayer film in the trilayer
heterostructure HaoN1  and the vortex core in Fe(Te,Se) ChenXY .

Another possibility of the non-monotonic evolution of the ZBCP magnitude is
from the vortex core in the Fe(Te,Se) substrate. Although our tunneling
spectra are recorded when the Bi island is away from the vortex core in
Fe(Te,Se), there are also some vortex cores nearby. One of them are shown in
the upper right corner of Fig. 3(d). The superconducting current surrounding
the vortex cores may pass through the Fe(Te,Se) underneath, which may probably
enhance the edge states on the Bi island and induce an increment of the ZBCP
magnitude. At higher magnetic fields, the proximity-induced superconductivity
is strongly suppressed by the fields, and the ZBCP magnitude decreases.
Clearly, further theoretical consideration is highly desired to understand the
non-monotonic relationship between the ZBCP magnitude and the magnetic field
quantitatively, as well as the reason why ZBCPs only appear on specific Bi
islands with particular sizes/shapes.

##  V Conclusion

In summary, we have observed the zero energy modes on certain Bi islands with
the diameter of 4-8 nm deposited on the FeTe0.55Se0.45 substrate. These zero
energy modes may be MZMs and can be found throughout the entire region on
these islands. Further measurements on these islands under varying magnetic
fields reveal an unusual behavior of the MZM magnitude. Specifically, the
magnitude is initially enhanced at weak fields lower than 2 T, but then
becomes suppressed as the field strength increases. Concerning the probable
origin of the zero-energy modes we discussed above, the anomalous evolution is
probably caused by the magnetic-field tuning on the edge states or influenced
by the superconducting current surrounding the vortex cores on the Fe(Te,Se)
substrate. Our findings will stimulate theoretical efforts for understanding
the mechanism of the bismuth topological systems and shed new light on the
topological property of Majorana zero modes under varying magnetic fields.

###### Acknowledgements.

We appreciate very useful discussions with Jörg Schmalian, Ning Hao, and
Haijun Zhang. The work was supported by the National Natural Science
Foundation of China (Grants No. 11974171, No. 12061131001, and No. 11927809)
and the National Key R&D Program of China (Grant No. 2022YFA1403201).

∗<EMAIL_ADDRESS>†<EMAIL_ADDRESS>
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