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Insulators
Low-Dimensional Topological Crystalline Insulators Qisheng Wang , Feng Wang , Jie Li , Zhenxing Wang , Xueying Zhan , and Jun He *
Topological crystalline insulators (TCIs) are recently discovered topological phase with robust surface states residing on high-symmetry crystal surfaces. Different from conventional topological insulators (TIs), protection of surface states on TCIs comes from point-group symmetry instead of time-reversal symmetry in TIs. The distinct properties of TCIs make them promising candidates for the use in novel spintronics, low-dissipation quantum computation, tunable pressure sensor, mid-infrared detector, and thermoelectric conversion. However, similar to the situation in TIs, the surface states are always suppressed by bulk carriers, impeding the exploitation of topology-induced quantum phenomenon. One effective way to solve this problem is to grow low-dimensional TCIs which possess large surface-to-volume ratio, and thus profoundly increase the carrier contribution from topological surface states. Indeed, through persistent effort, researchers have obtained unique quantum transport phenomenon, originating from topological surface states, based on controllable growth of low-dimensional TCIs. This article gives a comprehensive review on the recent progress of controllable synthesis and topological surface transport of low-dimensional TCIs. The possible future direction about low-dimensional TCIs is also briefl y discussed at the end of this paper.
1. Introduction ............................................. 2
2. Fundamentals of TCIs ................................ 2
3. Controllable Growth of Low-Dimensional TCIs ............................... 4
4. Surface Electronic Transport of Low-Dimensional TCIs ............................... 7
5. Prospects ............................................... 10
6. Conclusion ............................................. 11
From the Contents
small 2015, DOI: 10.1002/smll.201501381
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DOI: 10.1002/smll.201501381
Q. Wang, F. Wang, J. Li, Prof. Z. Wang, X. Zhan, Prof. J. He National Center for Nanoscience and Technology Beijing 100190 , P. R. China E-mail: hej@nanoctr.cn
1. Introduction
Thermal dissipation, due to the scattering of carriers during
transport process, is a common problem in current silicon-
based electronic devices. The emerging topological insulators
(TIs), a new quantum phase whose surface is conductive but
interior is insulating, open up a hopeful route to solve this
issue. [ 1–3 ] Owing to the relativistic effect (spin–orbit coupling)
and topological protection from time-reversal symmetry,
spin-locked current on the surface/edge of TIs is immune to
any nonmagnetic impurities, which endows them with great
application potential in low-dissipation electronic devices
and quantum information processing. The exciting discovery
and novel properties of TIs motivate scientists to search for
new topological phase classifi ed by other invariants. Topo-
logical crystalline insulators (TCIs), protected by point-group
symmetry, are such kind of topological phase. [ 4,5 ] So far, the
experimentally confi rmed TCIs are SnTe and its related alloy
Pb 1− x Sn x Te(Se) that possess high-symmetry rock-salt crystal
structure. Each high-symmetry surface of TCIs accommo-
dates four Dirac states. Interestingly, through controlling the
crystal symmetry, the topological nature of TCIs can be trans-
formed from nontrivial to trivial phase by strain and electrical
fi eld. Meanwhile, surface states properties of TCIs are highly
tunable by composition and temperature. All above features
indicate TCIs are promising for exploiting tunable electronic
and spintronic devices. However, similar to TIs, surface states
transport of TCIs is usually overwhelmed by bulk carrier. [ 6–8 ]
In order to resolve this problem, researchers dedicated to
growing low-dimensional TCIs since huge surface-to-volume
ratio of low-dimensional TCIs can signifi cantly enhance con-
tribution of carrier transport from surface states. Especially,
as topological nature varies from one high-symmetry facet
to another, controlling the crystal planes orientation of TCIs
nanostructures is crucial for probing unique surface states.
Furthermore, based on low-dimensional TCIs, researchers
successfully observed quantum transport phenomenon from
surface states. It is worth noting that, although it is a short
time after the fi rst theoretical prediction of TCIs by Fu et
al. in 2011, [ 4,5 ] researchers have achieved great success on
growth and surface states transport of low-dimensional
TCIs. This paper comprehensively reviews the recent pro-
gress in both synthesis and surface topological transport of
low-dimensional TCIs. We fi rst introduce the basic principle
of TCIs. Then, we summarize and analyze the newest results
about synthesis and quantum transport of low-dimensional
TCIs. The possible future directions about low-dimensional
TCIs are also proposed in the end.
2. Fundamentals of TCIs
2.1. From TIs to TCIs
One of the big breakthroughs in condensed matter physics is
the discovery of quantum Hall effect (QHE) in the 1980s. [ 9 ]
QHE occurs in 2D electrons system when an intense and per-
pendicular magnetic fi eld is applied to drive the electrons to
circulate in quantized orbits. As a result, the edge of samples
is characterized by dissipationless current fl ows while the
interior becomes inert. QHE is considered to be the fi rst TIs
because Hall conductivity σ xy equals integral multiples ( n )
of quantum conductance e 2 / h and n is the topological invar-
iant. [ 10 ] However, the requirements of low-temperature and
strong magnetic fi eld for generating QHE strictly limit its
practical application in electronic devices.
In the year of 2006, Zhang et al. [ 1,2 ] theoretically predi-
cated quantum spin Hall effect (QSHE) in 2D HgTe quantum
well, in which strong spin–orbital coupling replaces the role
of external magnetic fi eld to force the current to move in one
direction without back scattering. Such new TIs belong to a
novel topological classifi cation of a Z 2 index. Time-reversal
invariant property protects the surface spin-polarized elec-
tron fl ow from the scattering of nonmagnetic impurities.
This predication was subsequently verifi ed by experiments in
2007. [ 3 ] Channel conductivity σ xx was observed to be quan-
tized to 2 e 2 / h in zero magnetic fi eld, proving the existence of
gapless edges states in CdTe/HgTe/CdTe quantum well. This
encouraging discovery profoundly boosts the research of TIs.
Soon 3D TIs such as Bi 1− x Sb x , Bi 2 Se 3 , Bi 2 Te 3 , and Sb 2 Te 3 with
2D surface gapless states are validated by both theory [ 11 ]
and experiments. [ 12,13 ] It is worth emphasizing that Bi 2 Se 3 ,
Bi 2 Te 3 , and Sb 2 Te 3 are ideal building blocks for the study of
topological surface states when we take account of the fol-
lowing aspects: [ 6,8,14 ] (1) they have simple chemical stoichio-
metric ratio, (2) they are of layered crystal structure that
each covalently bonded quintuple layer interacts with each
other by weak van der Waals forces, enabling the synthesis
of few-layer nanoplates by conventional vapor deposition or
mechanically exfoliated methods, and (3) their surface is ter-
minated by a single Dirac cone with a relatively large bulk
bandgap (≈0.2–0.3 eV) which makes it accessible to surface
states even at room temperature.
Inspired by the TIs, theorists are committing themselves
to fi nd new type of TIs by other symmetry. Fu et al. [ 4 ] fi rst
proposed that insulators with mirror symmetry, namely
TCIs, could also obtain robust surfaces states. They subse-
quently present defi nite materials of TCIs that are SnTe and
its related alloy Pb 1− x Sn x Te(Se). [ 5 ] The theoretical predi-
cation was immediately confi rmed by three groups who
detected the linear Dirac dispersion on mirror-symmetry
surfaces of TCIs by angle-resolved photoemission spectros-
copy (ARPES). [ 15–17 ] The discovery of TCIs considerably
extends the family of TIs. In starkly contrast to TIs, TCIs
show differences in the aspects of (1) they are of highly
symmetry crystal structures, (2) gapless metallic states only
reside on those mirror-symmetry surfaces such as (100),
(111), and (110), (3) Dirac cones on TCIs surface can be
opened up by breaking symmetry, manipulating tempera-
ture, and tailoring compositions. Basic information about
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TIs and TCIs is listed in Table 1 from which we can clearly
see peculiar characteristics of TCIs. Take Pb 1− x Sn x Te as a
representative example, it crystallizes in the form of cubic
crystal structures. And it undergoes a band inversion and
obtains topological protection when Sn content ( x ) reaches
0.38 at 9 K. [ 15 ] Temperature also drives the conversion of
Pb 1− x Sn x Te from topological nontrivial insulator to trivial
insulator when it exceeds the critical temperature ( T c ).
Intriguingly, external physical disturbs such as strain and
electric fi eld can open up the surface gapless Dirac states by
destroying the crystal symmetry. These peculiar characteris-
tics make TCIs a promising materials system in the applica-
tions of tunable spintronic devices.
2.2. Surface Electronics of TCIs
TCIs harbor four Dirac cones on each high-symmetry sur-
face. Figure 1 a presents the electronic structure of (100) sur-
face in bulk SnTe. [ 5 ] Two surface bands with opposite mirror
eigenvalues cross each other and form a Dirac point along
TX . Four Dirac points can be found on the four equivalent
TX as shown in Figure 1 b. The surface Dirac states show a
Lifshitz transition that the Fermi surface fi rst exhibits two
disconnected hole pockets outside X . [ 18 ] And they then close
to each other with the decrease of Fermi energy and touch
each other. A large electron pocket with a hole inside is
formed fi nally (Figure 1 c). Another important feature of TCIs
lies in the tunability of topological surface states through
composition and temperature. [ 17,19–21 ] Figure 1 d shows the
ARPES of Pb 1− x Sn x Se (001) surface at various tempera-
tures in the vicinity of X . It obviously presents that surface
state is gapped above 100 K at which bottom of conduction
band connects to top of valence band with the formation of
a Dirac node. [ 15 ] Composition dependence of topological sur-
face states is also proved by Chen et al. in Pb 1− x Sn x Te (111)
fi lms. [ 22 ] Madhavan et al. [ 23,24 ] further pointed out that the
nontopological regime also host surface states. However, the
weight of Dirac surface states decreases when Sn content ( x )
approaches the trivial phase, imparting the mass to the mass-
less Dirac electrons. Zero-mass Dirac fermions protected by
crystal symmetry were found to coexist with massive Dirac
small 2015, DOI: 10.1002/smll.201501381
Table 1. Fundamental information of TIs and TCIs. Insets: a) quantum Hall effect in 2D electron system with dissipationless edge states. b) Back scattering from nonmagnetic impurities in TIs surface is prohibited. Reproduced with permission. [ 60 ] Copyright 2011, Nature Publishing Group. c) 2D helical surface states of 3D TIs. d) Energy dispersion of spin nondegenerate surface states on 3D TIs. Reproduced with permission. [ 61 ]
Copyright 2013, The Physical Society of Japan. e) Dirac-cone surface states on two different surface planes of (001) [ 16,28 ] and (111). [ 18 ] Reproduced with permission. [ 18 ] Copyright 2013, The American Physical Society.
Quantum Hall effect Quantum spin Hall effect
2D TIs 3D TIs 2D TCIs [65] 3D TCIs
Materials 2D electron system CdTe/HgTe/CdTe, AlSb/
InAs/GaSb/AlSb, Bi bilayer
Bi 1− x Sb x , Sb, Bi 2 Se 3 ,
Bi 2 Te 3 , Sb 2 Te 3 ,
Ag 2 Te, Bi 2 (Se 1− x Te x ) 3 ,
(Bi 1− x Sb x ) 2 Te 3
(001) monolayers of rock-
salt IV–VI semiconductors
XY (X = Ge, Sn, Pb, and
Y = S, Se, Te
SnTe, Pb 1– x Sn x Te,
Pb 1− x Sn x Se
Crystal structure – Layered rhombohedral crystal structure except 2D
quantum well
Cubic structure
bandgap – <30 meV except Bi
bilayer (0.1 eV)
≈0.3 eV except Sb
(semimetal)
0.26 eV for
monolayer PbTe
≈0–0.3 eV, invert with
temperature ( T ) and
composition ( x )
Protection TKKN invariant Time-reversal invariant (Z2 topology) Point-group symmetry (Mirror Chern number)
Surface states – Single Dirac cone Four Dirac cones on each surface
Characteristics Requirement of strong
magnetic fi eld
Strong spin–orbital coupling, simple chemical
stoichiometric ratio, large bandgap
Strong spin–orbital coupling, high crystal symmetry,
tunability by stain and composition, large bandgap
Schematic diagram
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electrons due to the symmetry-breaking distortion on the
surface. And the magnitude of the symmetry-breaking dis-
tortion nearly keeps unchanged in the both topological and
nontopological regimes.
Since metallic surface states of TCIs are protected by
crystal symmetries instead of time-reversal symmetries, low-
dissipation logic devices and pressure sensors can be devel-
oped based on topological surface states by breaking the
crystal symmetry. Fu et al. fi rst theoretically demonstrated
topological transistor devices of TCIs thin fi lm grown along
the (001) direction. [ 25,26 ] They proposed the surface states of
an 11-layer SnTe thin fi lm will be gaped when applying a ver-
tical electrical fi led. Figure 1 e shows the surfaces electronic
structures of SnTe (001) surface without application of elec-
tric fi led. After a 0.1 V bias is applied across the thin fi lm, the
surface band opens up (Figure 1 f). They further revealed that
the ferroelectric-type structural distortion opened some or all
of Dirac points, whereas strain moves the Dirac points to the
Brillouin zone. And the perpendicular magnetic fi eld gener-
ates the discrete Landau levels while in-plane magnetic fi eld
causes asymmetry between Dirac points. [ 27 ]
3. Controllable Growth of Low-Dimensional TCIs
Achieving controllable growth is always a very important
step as well as a big challenge for any “new” material. This
is especially true for TCIs in which fragile surface states
may become undetected under the perturbation of defects
and bulk state. [ 17 ] To avoid the disturbance of defects, high
quality TCIs prepared by thermodynamic equilibrium syn-
thesis method, like modifi ed Bridgeman [ 28 ] and self-selecting
vapor growth methods, [ 17 ] are used in the initial experimental
studies. However, as we mentioned before, TCIs are topo-
logical insulators in which the gapless surface states are pro-
tected by mirror symmetry of the crystal. [ 16 ] In another word,
possessing objects with large area of surfaces (surface states)
of specifi c crystalline planes (mirror symmetry) is prerequi-
site if one wants to implement better experimental obser-
vation of possible topology-related phenomena. Taking this
into consideration, synthesizing low-dimensional TCIs is one
of the best choices. After its theoretical prediction in 2011, [ 4 ]
many ways, such as molecular beam epitaxy (MBE) and
vapor deposition method, have been employed to synthesize
low-dimensional TCIs. In the following part, we will give a
review on this rising fi eld. We note that there are still other
ways, [ 29–31 ] such as solution-phase synthesis method, to grow
low-dimensional TCIs. But the products either are too small
for device fabrication or have too bad crystalline quality for
probing surface state. Hence, they will not be discussed in
this section.
3.1. MBE for Thin Film
The fi rst TCI proved by experiment is SnTe. [ 16 ] However,
due to intrinsic Sn vacancies (usually p-type doped state),
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Figure 1. a) Band structure and b) Fermi surface of SnTe (001) surface. c) A set of Fermi surface at different energy with a Lifshit transition. Reproduced with permission. [ 5 ] Copyright 2012, Nature Publishing Group. d) Temperature-dependent ARPES spectra in the vicinity of X . Reproduced with permission. [ 15 ] Copyright 2012, Nature Publishing Group. e) Gapless edge states of an 11-layer SnTe thin fi lm. f) The edge state opens up when applying a perpendicular electrical fi eld. Reproduced with permission. [ 26 ] Copyright 2014, Nature Publishing Group.
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observing surface state of SnTe has been a challenge. [ 16 ] As
a result, synthesizing high-quality SnTe with few defects
is the fi rst task. MBE, by precisely controlling the deposi-
tion rate and compositions, is a promising candidate. In
2013, Yan et al. grow high-quality SnTe thin fi lm on Si(111)
substrates by MBE for the fi rst time. [ 32 ] Relying on the
good controllability of MBE, thickness- and Pb doping-
dependent properties of SnTe thin fi lm are studied. Soon
after, Assaf et al. synthesized SnTe fi lm by MBE on BaF 2
(001) substrates. [ 33 ] They found that high-quality fi lm, then
an increased Hall mobility and decreased carrier concentra-
tion, could be achieved through using higher growth tem-
perature. To take a step further, Taskin et al. grow SnTe fi lm
using Bi 2 Te 3 , which has a lower lattice mismatch with SnTe,
as substrates. [ 34 ] Beyond this binary TCI, its ternary com-
pound Pb 1− x Sn x Te is also an interesting material because
its topological nature can be tailored by Sn content ( x ). [ 16 ]
According to this idea, Yan et al. synthesized Pb 1− x Sn x Te
fi lm with variable Sn content and found that there is a topo-
logical phase transition from trivial to nontrivial state while
increasing Sn content. [ 22 ]
3.2. Vapor Deposition Method
The thickness and composition of fi lms grown by MBE are
precisely controlled. However, it is expensive and time con-
suming. As compared with MBE, vapor deposition method
is easier, cheaper and considered one of the most promising
routes to productively synthesize nanostructured materials.
For TCIs, the fi rst trial is on SnTe. In 2013, through adjusting
the temperature of substrates, Li et al. successfully synthe-
sized SnTe nanomaterials with different morphologies by
chemical vapor deposition (CVD). [ 35 ] Combining with ab
initio calculation, they found that it is the energy difference
of crystalline planes that determines the morphologies (see
Figure 2 a). Specifi cally, when the substrates were placed in
the relative higher temperature (here 645–675 °C), the Te
poor condition, (100) planes have the lowest surface energy
(left side of Figure 2 a). As a consequence, SnTe microcubes
(without Au catalyst, VS growth mode) and nanowires (with
Au catalyst, VLS and VS growth mode) with (100) exposed
planes were obtained. While on the occasion of lower tem-
perature (525–625 °C), the Te in rich condition, (100) and
(111) planes have a similar surface energy (right side of
Figure 2 a). As a result, complicated SnTe microcrystals
(without Au catalyst, VS growth mode) and zigzag nanowires
(with Au catalyst, VLS and VS growth mode) which exposed
both (100) and (111) planes were grown. Figure 2 b,c shows
the respective results. Almost at the same time, by controlling
the experimental conditions in the CVD process, our group
reported the synthesis of high-quality SnTe nanocrystals and
nanowires with highly symmetry facets. [ 36,37 ] Without the uti-
lization of Au catalyst, the surface orientation and crystal
shapes of SnTe micronanocrystals mainly depend on the
growth temperature which decides the surface energy of SnTe
micronanocrystals. However, when using Au nanoparticles as
the catalyst, the 1D anisotropic growth is excited due to the
induction of catalyst. Combining the effect of temperature,
we obtained octahedron-attached SnTe nanowires and trun-
cated octahedron-assisted SnTe nanowires. More recently,
catalyst composition is found to have a strong impact on the
morphologies of SnTe nanostructure. [ 38 ] Zou et al. pointed
out that liquid An–Sn catalyst with high Sn content induced
the thermally dynamic growth of SnTe triangular nanoplates
with dominant {100} surfaces at high growth temperature
region (500 °C). While SnTe nanowires with dominant {100}
side surfaces were formed in the low-temperature region
(450 °C) under the control of solid or quasisolid Au 5 Sn cat-
alysts. In addition to the infl uence from surface energy of
different crystal planes at given conditions, catalyst composi-
tion played the critical role in deciding the morphologies of
SnTe nanostructures by affecting interface lattice match and
chemical potential of Sn content. Compared with 3D bulks,
1D nanowire are favorable for probing surface state of TCIs
due to their high surface-to-volume ratio. [ 39 ] However, for a
better performance, nanoplates with large top and bottom
surfaces are more preferred. Having this idea in mind, Cha
et al. grew SnTe nanoplates via the CVD method recently. [ 40 ]
Figure 2 d–n gives their results. In detail, SnTe nanoplates
with (100) or (111) planes as top and bottom surfaces were
achieved when grown in a relative low temperature (around
300 °C). While in higher temperatures (350–450 °C), nanorib-
bons and nanowires would appear.
In the conventional vapor deposition method, sub-
strates with dangling bonds, like SiO x /Si, are usually used.
And vertical nanostructures with a variety of morphologies
have been synthesized. But for 2D nanomaterials parallel
to substrates, it turns out to be diffi cult because, as a result
of the strong bonds, the adatoms become hard to diffuse on
the top of substrates. [ 40 ] Thanks to van der Waals epitaxy
(vdWE) method, in which atomically smooth substrates
(such as mica) are used as substrates, many nanomaterials
with 2D morphologies have been synthesized success-
fully. [ 40–42 ] For example, Li et al. prepared single-crystal
topological insulators (Bi 2 Se 3 and Bi 2 Te 3 ) nanoplates
arrays using mica as substrates. [ 40 ] However, previous works
focused on growing 2D layered materials which have strong
intrinsic driving force of 2D anisotropic growth due to their
planar crystal structure. In the year of 2014, our group fi rst
proposed the growth of 2D nonlayered Te hexagonal nano-
plates by vdWE on mica substrate. [ 43 ] vdWE is advantaged
in that (1) it allows large lattice mismatching between the
target product and substrates, (2) excessive strain in the
heterointerface is relaxed due to weak van der Waals inter-
action in the interface, and (3) van der Waals substrates
facilitates the migration of adatoms and thus promotes the
lateral growth of nanoplates. By performing the synthesis of
SnTe and Pb 1− x Sn x Se nanostructures on mica substrates, we
further put forward the general strategy for van der Waals
epitaxial growth of 2D nonlayered semiconductor. [ 44 ] Two
conditions are required for the growth of 2D nonlayered
materials by vdWE: (1) the nonlayered materials should
have the driving force for the 2D anisotropic growth, which
can be excited by optimizing the experimental conditions in
CVD process, and (2) van der Waals materials such as mica
and BN need to be used as the growth substrates. Figure 3
shows representative examples of van der Waals epitaxial
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2D nonlayered material. [ 45 ] Ultrathin 2D Pb 1− x Sn x Se nano-
plates with thickness ranging from 15 to 50 nm have been
successfully synthesized on the surface of mica. Two growth
conditions, in the CVD process, are thought to be the crit-
ical factors that affect the anisotropic growth of 2D TCIs.
One is the substrates temperature that mainly determines
the chemical activity of different crystal planes. In the case
of our previous work about vdWE of 2D Pb 1− x Sn x Se nano-
plates on mica, {110} surfaces of Pb 1− x Sn x Se showed higher
activation at growth temperature of 550 °C compared
with other facets, which leads to 2D anisotropic growth of
Pb 1− x Sn x Se nanoplates. The other important growth param-
eter is substrate surface chemistry property that infl uences
the nucleation, migration of adatoms, interface stability,
and thus the morphology of fi nal products. Our previous
work showed that, when we replaced layered mica with Si
substrate, Pb 1− x Sn x Se preferred to form microplates rather
than nanoplates under the same experimental parameters
as that on mica. This would be understood by the fact
that (1) dangling bonds on surface of 3D bonded Si cause
strong interaction between substrate and adatoms and thus
increase the migration energy barriers of adatoms, (2) large
lattice mismatch between Si (100) surface and Pb 1− x Sn x Se
(≈10.4%) makes it unstable assuming Pb 1− x Sn x Se epitaxi-
ally grows along surface of Si with planar geometry. Even
anisotropic growth of layered materials such as MoS 2 is
strongly affected by surface electronic properties of growth
substrate. A recent work showed MoS 2 tended to form 1D
nanobelt structure on Si instead of 2D nanoplates on SiO 2 .
The higher surface energy of Si (100 meV Å −1 ) compared
with SiO 2 (2.5 meV Å −1 ) explains the occurrence of this
growth behavior. [ 46 ]
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Figure 2. a) The free energies of the (100), (110), (111): Sn and (111): Te surfaces as functions of the relative Te chemical potential Δ µ Te . Wulff constructions of the thermodynamic equilibrium SnTe crystals under the Te-lean and Te-rich conditions are shown in the inset. Wulff constructions b) and SEM images c) of SnTe nanostructures when Te is not in rich. Wulff constructions d) and SEM images e) of SnTe nanostructures when Te is in rich. The scale bars in bottom of (e) are 200 nm. Reproduced with permission. [ 35 ] Copyright 2013, American Chemical Society. f) CVD growth schematic of SnTe nanoplates with SiO 2 /Si used as substrates. g) SnTe unit cell. h) (100) cubic crystals, grown without Au catalyst. i,j) (100) nanoplates. k) (100) nanoribbon. l–n) Nanowires with <100> growth direction. o) (111) nanoplate and p) (111) nanoribbon. Reproduced with permission. [ 50 ] Copyright 2014, American Chemical Society.
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4. Surface Electronic Transport of Low-Dimensional TCIs
As mentioned before that, because their huge surface-to-
volume ratio notably magnifi es the transport of carriers from
surface sates, low-dimensional TCIs are better candidate for
detecting topological surface states by electrical transport. [ 47 ]
More intriguingly, it is easier to observe quantum interference
phenomenon in TCIs nanostructures, which could be under-
stood by the fact that (1) topological surface states have consid-
erable mobility and (2) physical size of low-dimensional TCIs
is comparable to phase coherent length of surface states. [ 48,49 ]
This section surveys the recent research results about topology-
related electronic transport on low-dimensional TCIs.
4.1. Quantum Coherence Transport of Topological Surface States
It is reasonable to deduce that, compared with bulk coun-
terpart, the wave nature of topological surface states in low-
dimensional TCIs is more remarkable. In 2013, our group fi rst
discovered Aharonov–Bohm (AB) interference of surface
Dirac electrons in SnTe nanowires. [ 36 ] AB interference is the
result of quantum interference of two partial electron waves
along edge of nanowires which encircle certain magnetic fl ux.
It was fi rst used to characterize topological surface states of
TIs nanostructures by Peng et al. in 2010. [ 7 ] In our work, we
additionally detected Altshuler–Aronov–Spival (AAS) effect
which arises from quantum interference of two interfering
electron waves encircling the magnetic fl ux once ( Figure 4 a).
Both AB and AAS effects deteriorate with the increase of
temperature because of the disturbing from thermal excita-
tion. We further found SnTe nanowire exhibits pronounced
Shubnikov–de Haas (SdH) oscillations under the exposure
of vertical magnetic fi eld (Figure 4 b). SdH oscillations are
ascribed to the successive emptying of Landau levels with
the increase of the magnetic fi eld. The LL fan diagram gives
the intercept of 0.42, indicating the SdH oscillations are of
2D nature. The mobility of surface states is estimated to be
4866 cm 2 V −1 s −1 which is comparable to that of conventional
TIs. This work is the fi rst evidence of topological surface
states in TCIs nanostructures by magneto-transport. Almost
simultaneously, topological surface states transport was
confi rmed in SnTe thin fi lm by Ando et al. [ 34 ] In this work
p-type SnTe thin fi lm was grown on n-type Bi 2 Te 3 thin fi lm
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Figure 3. a,b) Schematic illustrations of van der Waals epitaxial ultrathin nonlayered materials, c) optical microscope, d–f) SEM, and h,i) AFM images of van der Waals epitaxial ultrathin 2D Pb 1− x Sn x Se nanoplates. g) Histogram of Pb 1− x Sn x Se nanoplates thickness, smooth curve is the Gaussian fi t of the thickness distribution. Reproduced with permission. [ 45 ] Copyright 2013, American Chemical Society.
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by MBE. They observed SdH oscillation of Dirac fermions
residing on the SnTe (111) surfaces due to a downward band
bending on the free SnTe surface (Figure 4 c). Based on SnTe
nanoplates, Cha et al. reported a structural phase transition
of SnTe that it transforms from rock salt at high temperature
to rhombohedral structure at low temperature. [ 50 ] However,
the work has not presented the weak antilocalization (WAL)
effect of topological surface sates, which would be because
the SnTe nanoplates are too thick to enhance transport of
surface states.
Multiple Dirac nodes on each high-symmetry surface
bring more complexity to surface transport of TCIs nano-
structures. In addition to the coupling between bulk and sur-
face states, the hybrid of different Dirac states strongly affects
the numbers of the transport channels. Heiman et al. [ 33 ] care-
fully investigate the valley coupling of degenerate TCIs sur-
face in SnTe thin fi lm. As shown in Figure 4 d, the numbers of
carrier valleys (2 α ) extracted from the HLN model change
with Fermi level ( E F ). At low E F , no bulk sates are involved
in the charge transport and the Fermi surface contains four
small 2015, DOI: 10.1002/smll.201501381
Figure 4. a) AB and AAS interferences of topological surface states in SnTe nanowire. b) SdH oscillation of SnTe nanowire. Reproduced with permission. [ 36 ] Copyright 2013, American Chemical Society. c) Angle-dependent SdH oscillation of SnTe thin fi lm. Reproduced with permission. [ 34 ] Copyright 2013, American Physical Society. d) Numbers of transport channels as the function of Fermi level. e) Plot of phase coherence length versus numbers of transport channels. Reproduced with permission. [ 33 ] Copyright 2014, American Physical Society.
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Dirac surface valleys. 2 α = 8 is expected to obtain in this case.
As the Fermi level increases to the point of Lifshitz transi-
tion ( E F = 0.05 eV), pairs of Dirac surface cones merge into
four pairs of concentric constant energy contours. Below
this point, the inner Dirac cone and the outer Dirac cone
host opposite chirality, which makes the quantum coherent
transport in this case extremely complicated. In addition,
the contribution of bulk sates to transport becomes more
and more important, the trivial bulk states swallow the inner
Dirac cone, thus reducing the 2 α to smaller than 2. As the
E F decrease to 0.2–0.5 eV, surface and bulk states coexist
and a maximum of 2 α = 4 can be obtained. With Fermi level
exceeding 0.5 eV, the 2 α sharply decreases which would be
due to the fact that the shape of the surface valleys changes
again in the deep Fermi level. Figure 4 e further shows that
the numbers of transport channel linearly decrease with the
increase of phase coherence length ( L ). The smaller experi-
mental 2 α means the stronger valley coupling and thus
results in a larger coherence length. The valley coupling may
result from the scattering between top and bottom surfaces
of SnTe thin fi lm. And, given the valley degeneracy of SnTe,
the coupling also likely comes from the scattering between
two Dirac states on the same surface. The coupling between
bulk and surface states has also been observed by Kuroda
et al. [ 51 ] who reported that both the numbers of transport
channel and phase coherence length decrease with increase
of temperature. This is because that thermal excited carriers
take part in the transport and interact with the surface states
at higher temperature. The above two works are crucial for
understating the mechanism of surface states transport.
4.2. Surface Transport Modulation
Unique properties of TCIs provide more degree to control
the surface states of low-dimensional TCIs. First, the Sn con-
tent ( x ) of Pb 1− x Sn x Te(Se) has a strong impact on the surface
electronic structure. For Pb 1− x Sn x Te, when Sn content goes
beyond 0.38 at 9 K, it undergoes a topological phase transi-
tion from trivial to nontrivial. Our group fi rst demonstrated
this characteristic in Pb 1− x Sn x Te nanowires by electric trans-
port. [ 52 ] By conducting the magneto-transport of Pb 1− x Sn x Te
nanowires, we observed the weak localization (WL) effect
in PbTe nanowires while WAL effects in Pb 0.5 Sn 0.5 Te and
Pb 0.2 Sn 0.8 Te nanowires. The PbTe nanowire exhibits the
semiconductive behavior of electrical transport. The thermal
activation energy ( E a ), extracted from R eE k T/c B≈ at high
temperature (260–133 K), is about 8.2 meV, where k B is the
Boltzmann constant. The change of magneto-conductance
at 2 K displays a sharp downward cusp near zero magnetic
fi eld, which is a signature of WL effect ( Figure 5 a). How-
ever, the Pb 0.5 Sn 0.5 Te shows metallic transport behavior and
WAL effect that the curve of Δ G versus magnetic fi eld B
shows upward cusp around zero magnetic fi eld (Figure 5 b).
The angle-traced magneto-transport confi rms that the WAL
effects originate from the 2D surface states. The topological
phase transition by composition engineering has also been
realized by Xiu et al. [ 53 ] in Pb 1− x Sn x Se thin fi lms. They also
found that strong electron–electron interaction caused the
large resistance of Pb 0.75 Sn 0.25 Se thin fi lms. Except the com-
position, surface states of TCIs can be modulated by thick-
ness and gate voltage. As the thickness of TCIs thin fi lm
reaches the critical value at which top surface and bottom
surface couple with each other, the resistance of thin fi lm
will dramatically increase due to the absence of topological
surface states in this case. Xiu et al. further pointed out the
critical value for Pb 1− x Sn x Se thin fi lm is 10 nm. The resistance
of 10 nm Pb 1− x Sn x Se thin fi lm is about one order of magni-
tude larger than that of 16 nm Pb 1− x Sn x Se thin fi lm. They
further performed the gate voltage modulation in the surface
states of 16 nm Pb 0.93 Sn 0.07 Se thin fi lms at 2 K. As shown in
Figure 5 c, the p-type Pb 0.93 Sn 0.07 Se thin fi lms shows WAL
effects by applying negative bias. The Berry phase ϕ is related
to the Fermi level by
[1 ( / (2 ))]FEϕ π= − Δ (1)
where Δ is the gap size. The Fermi level moves toward valence
band as the gate voltage decreases from positive value to
negative value. And thus E F becomes larger and Berry
phase approaches π . Interestingly, the competition between
quantum interference ( σ qi ) and the electron–electron interac-
tion ( σ ee ) also leads to the transition from WAL to WL effect.
As shown in Figure 5 d, at low temperature, the σ qi serves as
the dominant role and thus leads to the WL behavior due
to the trivial Berry phase of surface states. However, σ ee
becomes the main factor that affects the conduction correc-
tion at high temperature, giving rise to the WAL effect.
If introducing the exotic elements, TCIs nanostructures
will exhibit new electronic properties. In-doped SnTe has
been confi rmed to topological superconductor which hosts
Majorana fermions. [ 54 ] Majorana fermions are charge-neutral
particles whose antiparticles are themselves. They are attrac-
tive because of their great potential for using as a qubit of
fault-tolerant topological quantum computing. [ 55 ] Super-
conductor Cu x Bi 2 Se 3 was found to present unconventional
superconductivity in its point-contact spectra. Sn 1− x In x Te also
holds signatures of unconventional superconductivity which
are important for fi nding massless Majorana fermions. [ 56 ]
Superconductivity of Sn 1− x In x Te has recently been confi rmed
by the Cha group [ 57 ] and the Ando group [ 58 ] individually by
synthesizing Sn 1− x In x Te nanoplates and fabricating their Hall
devices. Cha et al. further pointed out In doping will pro-
foundly enhance the surface states of Sn 1− x In x Te nanoplates
since In doping deceases the bulk mobility. [ 57 ]
At the end of this section, we want to briefl y discuss the
thermoelectric conversion properties of TCIs. Although it is
still unclear whether topological surface properties of TCIs
have relation with their thermoelectric nature, we noticed
that almost all TCIs as well as TIs are excellent thermoelec-
tricity materials. [ 54 ] In this regard, Zhang et al. fi rst demon-
strated diameter dependence of thermoelectric properties of
individual SnTe nanowires. [ 59 ] They found the thermopower
enhances a magnitude of two orders with the decrease of
diameter from ≈913 to ≈218 nm, while, due to the increasing
boundary scattering and phonon-defect scattering, the
thermal conductivity is notably reduced with the decrease of
small 2015, DOI: 10.1002/smll.201501381
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nanowire diameter. As a result, the simultaneous increase of
thermopower and suppression of thermal conductivity results
in an improved fi gure of merit ZT.
5. Prospects
Despite great progress in synthesis and surface transport
of TCIs nanostructures, the depth and width of research on
low-dimensional TCIs are far less than that on conventional
TIs nanostructures. Previous research on conventional TIs
nanostructures supplies a paradigm for us to exploit novel
quantum behavior and potential applications of TCIs nano-
structures. Like TIs, high-speed and low-dissipation elec-
tronic devices are thought to be one of the most promising
applications for TCIs nanostructures. [ 60 ] Considering the
ultrahigh surface mobility of TCIs, [ 34,36 ] ultrafast logic devices
with low-thermal dissipation would be developed based on
top gate and back gate modulation.
What is more, since crystal symmetry warrants the sur-
face states of TCIs, new type of logic device can be realized
by applying vertical electrical fi eld in a dual-gated fi eld-
effect transistor. The strong vertical electrical fi eld breaks
the crystal symmetry and thus opens up the surface Dirac
cones. Furthermore, since crystal symmetry of TCIs can also
be destroyed by strain, highly sensitive pressure sensors are
expected to achieve in TCIs nanostructures. However, the
above two applications rely on ultrathin TCIs nanoplates
with high quality, which is an important opportunity as well
as a big challenge for materials scientist.
In order to realize surface states-based electronic devices,
one must lower the bulk carrier density. One way is to syn-
thesize the low-dimensional TCIs. [ 61 ] Among various nano-
structures, due to the dominant top and bottom surfaces, 2D
nanoarchitectures with distinct mirror-symmetry facets are the
best construction for exploring surface states. However, TCIs
are different from conventional TIs with layered structures
and time-reversal symmetry. TCIs are cubic crystal structure
and the surface states only reside on high symmetry surfaces.
Few-layer conventional TIs such as Bi 2 Se 3 and Bi 2 Te 3 have
been successfully grown by CVD due to its strong intrinsic
driving force for 2D anisotropic growth. However, it is diffi -
cult to do this for TCIs as a result of its nonlayered crystal
structures. Meanwhile, one needs to tailor the surface of TCIs
nanostructures to high symmetry crystal planes. Although our
group have grown Pb 1− x Sn x Se nanoplates with distinct (100)
surfaces by vdWE, ultrathin SnTe and Pb 1− x Sn x Te nanoplates
are not yet reported. Meanwhile, formation of impurities in
the TCIs nanostructures is inevitable during the growth and
device fabrication process. For example, Pb 1− x Sn x Se thin fi lm
and nanoplates are usually p-type doping caused by cation
vacancies. Cations compensation may minimize the density
of cation vacancies. One can further tune the mole ratio of
Pb to Sn in ternary TCIs such as Pb 1− x Sn x Se(Te). Similar to
the work on (Bi − Sb 1− x ) 2 Te 3 nanoplates, this way can push the
Fermi level to the middle of bandgap and thus profoundly
decrease the bulk states. Hybrid structure based on TCIs is
also attractive for chemist and material scientist. [ 62,63 ] Inter-
facing TCIs nanostructures with superconductor, ferromag-
netic insulator, and insulator will bring exotic properties. [ 64 ]
Figure 5. a) Dominated WL effect of PbTe nanowire at 2 K, inset is the SEM image of four-terminal device with scale bar of 2 µm. b) Temperature-dependent magnetoconductance of Pb 0.5 Sn 0.5 Te nanowire. Reproduced with permission. [ 52 ] Copyright 2015, American Chemical Society. c) Gate voltage-modulated surface transport of 16 nm Pb 0.93 Sn 0.07 Se thin fi lm at 2 K. d) Change of magnetoconductance of 16 nm Pb 0.93 Sn 0.07 Se thin fi lm at various temperatures. Reproduced with permission. [ 53 ] Copyright 2015, American Chemical Society.
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For example, interface between superconductor and TCIs is
predicted to generate Majorana fermions.
6. Conclusion
TCIs have been the star materials in condensed matter
physics. The marvelous topological surface states, guaran-
teed by crystal symmetry, render scientists an opportunity
for developing fundamental physics as well as low-dissipa-
tion electronic devices. Compared with the bulk counterpart,
low-dimensional TCIs, due to their large surface to volume
ratio, are more ideal system for exploiting surface states.
This paper comprehensively summarizes the recent progress
in controllable growth of low-dimensional TCIs. The device
applications based on TCIs nanostructures are also carefully
reviewed. Although this paper covers only the very tip of
the iceberg, the intriguing properties of TCIs will excite the
interest of a broad research community. It is worth noting,
compared with TIs, much work on TCIs nanostructures has
been left to do. For example, it has not yet elucidated that
how strain controls surface states transport. The unique elec-
tronic properties of TCIs nanostructures such as multiple
surface states have not been very well documented either.
We believe low-dimensional TCIs will bring us more exciting
breakthroughs in the near future.
Acknowledgements
This work at National Center for Nanoscience and Technology was supported by 973 Program of the Ministry of Science and Tech-nology of China (No. 2012CB934103), the 100-Talents Program of the Chinese Academy of Sciences (No. Y1172911ZX), the National Natural Science Foundation of China (No. 21373065 and No. 61474033), and Beijing Natural Science Foundation (No. 2144059).
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Received: May 15, 2015 Revised: June 16, 2015 Published online: