Topological Materials

『遠いのは、距離じゃなくて次元なんだよ。』

2D Topological Insulators

CdTe/HgTe/CdTe Quantum Well

[Bernevig2006] [König2007]

• Band inversion found in bulk \(\ce{HgTe}\) as long as the thickness exceeds a certain threshold, while normal ordering in \(\ce{CdTe}\).

• Gap zero at \(\Gamma\) in \(\ce{HgTe}\) opened by quantum confinement effect.

• Conductance verified to be quantized by \(2e^2/h\) under zero magnetic field when the Fermi level is tuned into the band gap.

AlSb/InAs/GaSb/AlSb Quantum Well

[Knez2011] [Knez2012]

From InAs/GaSb/AlSb composite quantum well structure preparation with help of reflectance anisotropy spectroscopy

• It’s a story between the holes of \(\ce{InAs}\) and electrons of \(\ce{GaSb}\).

  • Dashed lines for bands before composition, while solid lines for bands after composition.

  • Dashed blue lines for the highest valence band of \(\ce{GaSb}\).

  • Dashed red lines for the lowest conduction band of \(\ce{InAs}\).

  • After composition, the dashed blue electrons near \(\Gamma\) drop to fill the dash blue holes. Bingo! The solid blue electrons.

  • Now the dashed blue line near \(\Gamma\) become holes — band inversion.

  • As we move away from \(\Gamma\), the energy levels of holes and electrons become close before they intersect. However, this does not render the system conductive due to anticrossing gap — exactly the effect in chemistry that creates covalent bonds.

  • As we move further, the dashed lines go back to the solid lines of their own colors — no more band inversion.

Candidates of 2D Topological Materials

• Bilayer \(\ce{Bi}\) metal.

  • Evidence for the existence of edge states has been found [Yang2012].

• Monolayer \(\ce{Na2IrO3}\).

• Vapor deposition of metal atoms onto graphene.

  • Spin-orbit interaction enhanced.

• Silicene: existence in isolation not proven.

Remark: Why is Two-Dimensional More Preferable?

『… 2次元トポロジカル絶縁体には, 無散逸のエッジ電流や量子スピンホール効果など, 3次元トポロジカル絶縁体にはない興味深い物理がある…』

3D Topological Insulators

Bi_1-xSb_x Alloy

[Fu2007] [Hsieh2008]

• Odd number of band inversion at TRIMs.

• Band gap exists in bulk for

  \[0.09 < x < 0.23.\]

• Surface states too complex.

• High surface carrier concentration and high crystal quality.

Tetradymite

Note

\(c\)-axis corresponds to the \((111)\)-direction of \(\ce{NaCl}\) lattice.

• \(\ce{Bi2Se3}\) (confirmed [Xia2009]), \(\ce{Bi2Te3}\) (confirmed [Chen2009]), and \(\ce{Sb2Te3}\) (confirmed [Jiang2012]) are predicted to be TIs [Zhang2009].

• Tetradymites:

  • A-B-C-A-B-C packing of quintuple layers.

  • Each quintuple layer of the form Se-Bi-Se-Bi-Se.

  • Van der Waals cohesion.

• Only one Dirac cone, around \(\overline{\Gamma}\) of surface BZ.

• Easy fabrication. Surface states are all topological.

• Pure crystal hard to obtain. Observation of surface transport disrupted.

Note

Problem here: bulk conductivity too high.

More Tetradymite Materials

• High resistivity found in \(\ce{Bi2Te_{1.95}Se_{1.05}}\) [Ren2010].

• With SdH and Hall data, it is found that surface states contribute \(6\%\) of the total conductivity while the rest \(94\%\) are from the bulk states.

• See also \(\ce{Bi_{2-x}Sb_{x}Te_{3-y}Se_y}\) [Ren2011].

• \(\ce{Bi_{2-x}Sn_xTe_2Se}\): Fermi level dragged into band gap also by doping [Ren2012]. Surface states contributes up to \(50\%\) of the total conductivity.

BiQ Homologous Series

• Formula \(\ce{(Bi2)_n(Bi2X3)_m}\).

• Structure: packing of multi-layers. Covalent inter-multi-layer while van der Waals intra-multi-layer.

• \(\ce{(Bi2)(Bi2Se_{3-x}S_x)}\) found to be topological semimetal for \(x=0.4\) [Valla2012].

• \(\ce{(Bi2)(Bi2Te3)_2}\), i.e. \(\ce{BiTe}\), confirmed to be topological, yet unknown if it is insulator [Cava2013].

TlBiSe₂

[Yan2010] [Lin2010] [Sato2010] [Kuroda2010] [Chen2010]

• Structure similar to tetradymite.

• Topological phase transition from \(\ce{TlBiS2}\):

  • Topological insulator \(\ce{TlBi(S_{1-x}Se_x)_2}\) for \(x>0.5\) [Xu2011], trivial insulator for \(x<0.5\).

  • Gap found at the Dirac point near \(x=0.5\) [Sato2011] [Souma2012a], of yet unknown origin, which should have been degenerate by Kramers theorem.

GeBi₂Te₄

• N-type degenerate semiconductor due to defects [Okamoto2012].

Ge-Based Homologous Series

• Formula \(\ce{(GeTe)_n(Bi2Te3)_m}\).

• \(\ce{GeBi_{4-x}Sb_xTe_7}\) confirmed [Muff2013].

Pb-Based Materials

• \(\ce{PbBi2Te4}\) is p-type.

• \(\ce{PbSb2Te4}\) is n-type.

• Dirac fermion in \(\ce{Pb(Bi_{1-x}Sb_x)_2Te4}\) from n-type to p-type as \(x\) increase [Souma2012b].

Pb-Based Homologous Series

• Formula \(\ce{(PbTe)_n(Bi2Te3)_m}\).

• \(\ce{PbBi4Te4}\) confirmed [Eremeev2012].

Natural Superlattice

• Formula \(\ce{(PbSe)_5(Bi2Se3)_{3m}}\) where \(m=1,2\).

• Alternation of \(m\) times of quintuple layers and \(\ce{PbSe}\) layers.

• Dirac cone exists for \(m=2\).

  • Dirac gap opened due to mixture of states on the upper surface and lower surface.

  • Large bulk gap of 0.5eV due to quantum confinement of \(\ce{Bi2Se3}\).

BiTeCl

• Surface states helical despite bulk inversion symmetry broken [Chen2013].

HgTe (Epitaxial)

• Epitaxial growth on \(\ce{CdTe}\) substrate [Brüne2011].

• Band gap opened by broken symmetry.

Sn (Epitaxial)

• Epitaxial growth of \(\alpha\)-\(\ce{Sn}\) on \(\ce{InSb} (001)\) [Barfuss2013] [Ohtsubo2013].

• Helical surface states observed.

Candidates of 3D Topological Materials

• \(\ce{Ag2Te}\), magnetoresistance proportional to magnetic field in a wide range, possibly of topological origin.

• \(\ce{SmB6}\), possibly topological Kondo insulator.

• \(\ce{Bi_{14}Rh3I9}\):

  • Weak topological insulator \((0;001)\) by calculation.

  • Packing of two-dimensional insulators

  • Honeycomb lattice.

  • Surface states are hard to be detecte by ARPES since they are not on the cleavage surface.

Topological Semimetals

Definition of Topological Semimetals

Topological semimetals may refer two three kinds of materials.

• Ordinary semimetals (i.e. those where the top of the valence band is lower than the bottom of the conduction band) with nontrivial \(\mathbb{Z}_2\) index, e.g. \(\ce{Sb}\).

• Zero-gap semiconductors where the degeneracy is protected by crystal symmetries, e.g. \(\ce{HgTe}\), where the gap may be opened by perturbations that breaks the symmetries.

• Weyl semimetals.

Weyl Semimetals

From Topological surface states and Fermi arcs of the noncentrosymmetric Weyl semimetals TaAs, TaP, NbAs, and NbP

• Chirality as a good quantum number.

• Massless Dirac equation: Dirac equation diagonalized into two \(2\times 2\)-blocks of each chirality.

• Inversion symmetry or TRS broken: spin-degeneracy lifted.

• At intersections of conduction bands and valance bands (i.e. Weyl points): Hamiltonian (\(\pm\) depending on the chirality)

  \[H = \pm \hbar v_{\mathrm{F}} \vb*{\sigma}\cdot \vb{k}.\]

• Weyl points exist in pair of opposite chiralities.

• A Weyl point pair is joined by a Dirac arc, projection of which onto the 2D BZ surface gives gapless surface state.

Candidates of Topological Semimetals

• Heusler compounds and half-Heusler compounds: zero band-gap semiconductors by crystal symmetry.

• AFM phase of \(\ce{Y2Ir2O7}\).

• \(\ce{Nd2(Ir_{1-x}Rh_x)_2O7}\): Mott transition.

• Layers of \(\ce{HgTe}/\ce{CdTe}\) with electric field applied.

• MBE growth of \(\ce{Tl-Se-Bi-S}\) multi-layers.

Heusler Compounds as Topological Semimetals

• Topological semimetal \(\ce{LuPtBi}\) and \(\ce{YPtBi}\) confirmed [Liu2016].

  • Dirac point ~0.5eV below \(E_{\mathrm{F}}\).

  • Non-degenerate spin configuration confirmed by CD-ARPES.

Topological Crystalline Insulator

SnTe

[Hsieh2012] [Tanaka2012] [Dziawa2012]

• \(\ce{SnTe}\): Double Dirac cones found around each \(\overline{X}\) point in the surface BZ, each the projection of two \(L\) points.

• \(\ce{PbTe}\): topologically trivial.

• \(\ce{Pb_{1-x}Sn_xTe}\): topological phase transition around \(x=0.25\).

• The two Dirac cones around each \(\overline{X}\) are separated due to level repulsion (or avoided crossing).

SnSe

• \(\ce{Pb_{0.77}Sn_{0.23}Se}\): trivial insulator at RT while topological at \(T\) goes down, where spin-orbit interaction increases due to lattice shrinking.

Synthesization

Bulk Single Crystal

• Bridgeman method.

  • \(\ce{Bi2Se3}\), \(\ce{Bi2Te3}\), \(\ce{Bi2Te2Se}\).

• Vapor transport method.

  • PVT (physical vapor transport): \(\ce{SnTe}\), \(\ce{(Pb,Sn)Se}\), \(\ce{(Pb,Sn)Te}\).

  • CVT (chemical vapor transport).

Thin Film

• MBE (molecular beam epitaxy) method.

  • \(\ce{Bi_{1-x}Sb_x}\), \(\ce{Bi2Se3}\), \(\ce{Bi2Te3}\), \(\ce{Sb2Te3}\), \(\ce{(Bi,Sb)_2Te3}\).

• CVD (chemical vapor deposition).

  • \(\ce{Bi2Se3}\).

Nano-Ribbon and Nano-Plate

• VLS (vapor liquid solid) method.

Bulk Insulation

• Bulk carrier density too high due to defects.

• Fixed by doping.

• Tunable between p-type and n-type.

  • Enabling p-n junction using surface states.

Glossary

Tetradymite/テトラジマイト/辉碲铋矿

A mineral consisting of bismuth, tellurium and sulfide, \(\ce{Bi2Te2S}\), a.k.a. telluric bismuth.

References

Bernevig2006

Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells

König2007

Quantum Spin Hall Insulator State in HgTe Quantum Wells

Knez2011

Evidence for Helical Edge Modes in Inverted InAs/GaSb Quantum Wells

Knez2012

Andreev Reflection of Helical Edge Modes in InAs/GaSb Quantum Spin Hall Insulator

Yang2012

Spatial and Energy Distribution of Topological Edge States in Single Bi(111) Bilayer

Fu2007

Topological insulators with inversion symmetry

Hsieh2008

A topological Dirac insulator in a quantum spin Hall phase

Zhang2009

Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface

Xia2009

Observation of a large-gap topological-insulator class with a single Dirac cone on the surface

Chen2009

Experimental Realization of a Three-Dimensional Topological Insulator, Bi2Te3

Jiang2012

Landau Quantization and the Thickness Limit of Topological Insulator Thin Films of Sb2Te3

Ren2010

Large bulk resistivity and surface quantum oscillations in the topological insulator Bi2Te2Se

Ren2011

Optimizing Bi2−xSbxTe3−ySey solid solutions to approach the intrinsic topological insulator regime

Ren2012

Fermi level tuning and a large activation gap achieved in the topological insulator Bi2Te2Se by Sn doping

Yan2010

Theoretical prediction of topological insulators in thallium-based III-V-VI2 ternary chalcogenides

Lin2010

Single-Dirac-Cone Topological Surface States in the TlBiSe2 Class of Topological Semiconductors

Sato2010

Direct Evidence for the Dirac-Cone Topological Surface States in the Ternary Chalcogenide TlBiSe2

Kuroda2010

Experimental Realization of a Three-Dimensional Topological Insulator Phase in Ternary Chalcogenide TlBiSe2

Chen2010

Single Dirac Cone Topological Surface State and Unusual Thermoelectric Property of Compounds from a New Topological Insulator Family

Xu2011

Topological Phase Transition and Texture Inversion in a Tunable Topological Insulator

Sato2011

Unexpected mass acquisition of Dirac fermions at the quantum phase transition of a topological insulator

Souma2012a

Spin Polarization of Gapped Dirac Surface States Near the Topological Phase Transition in TlBi(S1−xSex)2

Okamoto2012

Observation of a highly spin-polarized topological surface state in GeBi2Te4

Souma2012b

Topological Surface States in Lead-Based Ternary Telluride Pb(Bi1−xSbx)2Te4

Eremeev2012

Atom-specific spin mapping and buried topological states in a homologous series of topological insulators

Muff2013

Separating the bulk and surface n- to p-type transition in the topological insulator GeBi4−xSbxTe7

Chen2013

Discovery of a single topological Dirac fermion in the strong inversion asymmetric compound BiTeCl

Valla2012

Topological semimetal in a Bi-Bi2Se3 infinitely adaptive superlattice phase

Cava2013

Crystal structure and chemistry of topological insulators

Brüne2011

Quantum Hall Effect from the Topological Surface States of Strained Bulk HgTe

Barfuss2013

Elemental Topological Insulator with Tunable Fermi Level: Strained α-Sn on InSb(001)

Ohtsubo2013

Dirac Cone with Helical Spin Polarization in Ultrathin α-Sn(001) Films

Nakayama2012

Manipulation of Topological States and the Bulk Band Gap Using Natural Heterostructures of a Topological Insulator

Hsieh2012

Topological crystalline insulators in the SnTe material class

Tanaka2012

Experimental realization of a topological crystalline insulator in SnTe

Dziawa2012

Topological crystalline insulator states in Pb1−xSnxSe

Liu2016

Observation of unusual topological surface states in half-Heusler compounds LnPtBi (Ln=Lu, Y)