Topological Insulator and Related Topics (Volume 108) (Semiconductors and Semimetals, Volume 108)

233 Pages·2021·9.486 MB·English

by  Lu Li (editor), Kai Sun (editor)

SERIES EDITORS
CHENNUPATI JAGADISH
Distinguished Professor
Department ofElectronic Materials Engineering
Research Schoolof Physicsand Engineering
Australian National University
Canberra,ACT2601, Australia
ZETIAN MI
Professor
Department ofElectrical Engineering andComputer Science
University ofMichigan
1310Beal Avenue
AnnArbor, MI 48109
United Statesof America
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Contributors
YongP.Chen
DepartmentofPhysicsandAstronomy;BirckNanotechnologyCenter;PurdueQuantum
ScienceandEngineeringInstitute;SchoolofElectricalandComputerEngineering,Purdue
University,WestLafayette,IND;QuantumScienceCenter,OakRidge,TN,UnitedStates;
InstituteofPhysicsandAstronomyandVillumCentersforDiracMaterialsandforHybrid
QuantumMaterials,AarhusUniversity,Aarhus-C,Denmark;WPI-AIMRInternational
ResearchCenterforMaterialsSciences,TohokuUniversity,Sendai,Japan
YulinChen
StateKeyLaboratoryofLowDimensionalQuantumPhysicsandDepartmentofPhysics,
TsinghuaUniversity,Beijing;SchoolofPhysicalScienceandTechnology,ShanghaiTech
University,Shanghai,China;DepartmentofPhysics,UniversityofOxford,Oxford,
UnitedKingdom
TianLiang
StateKeyLaboratoryofLowDimensionalQuantumPhysics,DepartmentofPhysics,
TsinghuaUniversity,Beijing,People’sRepublicofChina;RIKENCenterforEmergent
MatterScience(CEMS),Wako,Japan
Chao-XingLiu
DepartmentofPhysics,ThePennsylvaniaStateUniversity,UniversityPark,PA,
UnitedStates
JaySau
DepartmentofPhysics,CondensedMatterTheoryCenterandTheJointQuantumInstitute,
UniversityofMaryland,CollegePark,MD,UnitedStates
SumantaTewari
DepartmentofPhysicsandAstronomy,ClemsonUniversity,Clemson,SC,UnitedStates
YangXu
BeijingNationalLaboratoryforCondensedMatterPhysicsandInstituteofPhysics,Chinese
AcademyofSciences,Beijing,China
HaifengYang
SchoolofPhysicalScienceandTechnology,ShanghaiTechUniversity,Shanghai,China
LexianYang
StateKeyLaboratoryofLowDimensionalQuantumPhysicsandDepartmentofPhysics,
TsinghuaUniversity,Beijing,China
JiabinYu
CondensedMatterTheoryCenter,DepartmentofPhysics,UniversityofMaryland,College
Park,MD,UnitedStates
vii
Preface
Topological materials are novel quantum materials in which exotic physical
propertiesprohibitedbytraditionaldoctrinesariseduetotheunconventional
topological structures in their quantum wave functions. Although the initial
discoverywasmadeintheearly1980s,mosttopologicalmaterialsarerevealed
quiterecently.Predicted andexperimentally confirmedin2008,topological
insulators establish topologically protected surface states with a series of
remarkable novel physical phenomena, such as spin-momentum locking
andlinear(Dirac)energy-momentum dispersion.Thesuccessleadstoarich
familyoftopologicalmaterials,includingtopologicalsuperconductors,topo-
logical Dirac/Weyl semimetals, topological crystalline insulators, and corre-
lated topological insulators, among others. The rapid progress opens a
pathway for intriguing applications in future electronics, sensing, and
communications.
Thisbookprovidesin-depthreviewsofintriguingtopicsintopological
materials.Ontheexperimentalside,Chapter1reviewsthedetailedphoto-
emissionresultsontheelectronicstructuresofthebulkandsurfacestatesof
the big family of the topological insulators and topological semimetals.
Chapter2 further reviews theuniqueelectrical and thermoelectric transport
propertiesoftopologicalsemimetals.Moreontheapplicationside,Chapter3
reviews the discovery of quantum Hall effects in specially designed and
fabricated topological insulator devices. On the theoretical side, Chapter 4
summarizesthestate-of-the-artprogressintheoreticalandexperimentalstud-
iesoftopological superconductors, Majoranamodes, andtopological qubits.
Chapter5discussesthelatestprogressonpseudo-gaugefieldasagenerictool
to characterize various exotic phenomena in topological semimetals.
ix
CHAPTER ONE
Electronic structures
of topological quantum materials
studied by ARPES
Lexian Yanga, Haifeng Yangb, and Yulin Chena,b,c,*
aStateKeyLaboratoryofLowDimensionalQuantumPhysicsandDepartmentofPhysics,TsinghuaUniversity,
Beijing,China
bSchoolofPhysicalScienceandTechnology,ShanghaiTechUniversity,Shanghai,China
cDepartmentofPhysics,UniversityofOxford,Oxford,UnitedKingdom
∗Correspondingauthor:e-mailaddress:[email protected]
Contents
1. IntroductiontoARPES 2
1.1 Basicconcept 2
1.2 Generalprinciple 5
1.3 Experimentalinstrument 10
1.4 ARPESspectrum 15
2. ARPESstudiesontopologicalquantummaterials 16
2.1 Topologicalinsulatingphases 17
2.2 Topologicalsemimetals 22
2.3 Topologicalsuperconductors 32
3. Summaryandperspective 34
References 35
Abbreviations
2D two-dimension(al)
3D three-dimension(al)
ARPES angle-resolvedphotoemissionspectroscopy
MBS Majoranaboundstates
QH quantumHall
QSH quantumspinHall
SOC spin-orbitalcoupling
STM scanningtunnelingmicroscope
TCI topologicalcrystallineinsulator
TCS topologicalchiralsemimetal
TDS topologicalDiracsemimetal
TI topologicalinsulator
TMD transitionmetaldichalcogenide
TNLS topologicalnodallinesemimetal
SemiconductorsandSemimetals,Volume108 Copyright#2021ElsevierInc. 1
ISSN0080-8784 Allrightsreserved.
https://doi.org/10.1016/bs.semsem.2021.07.004
2 LexianYangetal.
TQM(s) topologicalquantummaterial(s)
TRS time-reversalsymmetry
TSC topologicalsuperconductor
TSS topologicalsurfacestates
TWS topologicalWeylsemimetal
UHV ultrahighvacuum
UV ultraviolet
1. Introduction to ARPES
1.1 Basic concept
ARPES is based on the photoelectric effect Heinrich Hertz discovered in
1887 when studying the spark discharge effect to confirm Maxwell’s elec-
tromagnetic theory (Hertz, 1887). He found that the maximum kinetic
energy of photoelectrons is independent on the light intensity but propor-
tionaltothefrequencyoftheincidentlight.Moreover,thelightfrequency
must be higher than a material-dependent threshold value to liberate elec-
tronsfromsolids.Lateron,AlbertEinsteinsuccessfullyresolvedthesemys-
teriesbythesimpleconceptofphotonandwasawardedtheNobelPrizein
1905 (Einstein, 1905). In his theory, the maximum kinetic energy of pho-
toelectrons reads:
Emax ¼ hυ(cid:2)ϕ, (1)
kin
where hυ is the photon energy, ϕ is called the work function of the solid
material (Hu€fner, 2003).
Fromtheperspectiveofelectronicstructure,electronsinsolidsarebound
atthebindingenergyE withrespecttotheFermienergyE (thephotoelec-
B F
tronsatmaximumkineticenergyareexcitedfromE ofthesolids).Thegen-
F
eral relationship between the kinetic energy of photoelectrons and the
binding energy can thus be written as (Fig. 1A):
E ¼ hυ(cid:2)ϕ(cid:2)jE j: (2)
kin B
Therefore, if we can accurately measure the kinetic energy of photoelec-
trons, we can calculate the binding energy of electrons in solids.
Likewise,themomentumofelectronsinsolidscanbededucedfromthe
pffiffiffiffiffiffiffiffiffiffiffiffiffi
momentumofphotoelectrons(jKj ¼ 2mE )withmtheelectronmass,if
kin
the photoemission process respects the momentum conservation law.
Fig. 1 Basic working principle of ARPES. (A) The energetics of photoemission process. (B) Schematic of the emission and collection of
photoelectrons.
4 LexianYangetal.
However, this is not completely true since the translational symmetry per-
pendicular to the sample surface is broken. Fortunately, the translational
symmetryparalleltothesamplesurfaceisstillrespectedthustheparallelelec-
tron momentum of photoelectrons is conserved during photoemission.
Using an electron analyzer, we can directly record the kinetic energy and
emission angles (θ, φ) of photoelectrons as schematically shown in
Fig. 1B. Consequently, the parallel electron momentum kk in solids can
be calculated according to
(cid:3) (cid:3) (cid:3) (cid:3) pffiffiffiffiffiffiffiffiffiffiffiffiffi
(cid:3)kk(cid:3) ¼ (cid:3)Kk(cid:3) ¼ ħ1 2mEkinsinθðcosφxb+sinφbyÞ: (3)
For the electron momentum in the vertical direction,k , although it is not
z
conserved during the photoemission process, we can approximately deter-
mine it under reasonable assumptions. The most commonly used is the
free-electron final state assumption, based on which the k is deduced as:
z
(cid:4) (cid:5)
ħk2 ħ2 k2k+k2z
E ðkÞ ¼ (cid:2)jE j ¼ (cid:2)jE j, (4)
f 2m 0 2m 0
whereE istheenergyofthevalencebandbottom.NotethatE andE are
0 f 0
with respect to E , while E is with respect to the vacuum energy level.
F kin
Thus E ¼E +ϕ (see Fig. 1A). Using Eqs. (3, 4), we obtain:
f kin
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
jkzj ¼ ħ 2mðEkincos2θ+V0Þ, (5)
whereV ¼jE j+ϕiscalledtheinnerpotential.Underthisassumption,itis
0 0
easy to note that k dispersion can be measured by photon-energy depen-
z
dent ARPES measurements, from which the inner potential can be deter-
minedbymatchingtheperiodicityofthek dispersionwiththatofBrillouin
z
zone (Damascelli et al., 2003;Hu€fner, 2003; Luet al., 2012). The fact that
k depends on θ suggests that the electrons photoemitted by a determinant
z
photonenergyarefromacurvedk sphere.Ithasbeendemonstratedthatthe
z
assumption of free-electron final state is not only suitable for simple metals
but also applicable for complex compounds such as correlated materials.
Photon-energy dependent measurement is particularly useful for the iden-
tification of topological surface states (TSS) of TQMs (Chen, 2012; Chen
et al., 2020; Lv et al., 2019a, 2021; Yang et al., 2018; Zhang et al.,
2020),sincethesurfacestatesshownok dispersion,incontrasttothebulk
z
states that usually show obvious k dispersion.
z
ElectronicstructuresofTQMsstudiedbyARPES 5
1.2 General principle
After phenomenologically describing the energy and momentum conver-
sionprocessinARPESexperiment,wenowdiscussthemicroscopicquan-
tum process of photoemission. The photoemission can be treated as an
optical transition from an N-electron initial state to a final state consisting
of N-1 electrons and a photoelectron. The initial N-electron state is
describedbyamany-bodywavefunction thatsatisfies thesurface boundary
condition. It is one of the eigenstates of the N-electron system. The final
stateisdefinedbyoneoftheeigenstatesoftheionized(N-1)-electronsystem
andthecomponentofthewavefunctionofthephotoelectron(thatisusually
approximated by a plane-wave propagating in vacuum with an amplitude
component in the solid). To calculate the intensity of the photoelectrons,
weneedtoknowthetransitionprobabilityfromtheinitialstatetothefinal
state after photo-excitation, which can be approximately given by Fermi’s
golden rule:
(cid:3)D E(cid:3) (cid:4) (cid:5)
2π(cid:3) (cid:3)
ω ¼ (cid:3) ΨNjH jΨN (cid:3)δ EN (cid:2)EN (cid:2)hυ , (6)
fi ħ f int i f i
where ΨN and ΨN (EN and EN) are wave functions (energies) of the initial
f i f i
and final N-electron systems. H describes the optical perturbation of the
int
system:
e
H ¼ (cid:2) ðA(cid:3)p+p(cid:3)AÞ: (7)
int 2mc
Aandparethevectorpotentialoftheexcitationlightandthemomentumof
theelectron,respectively.Withdipoleapproximation,thevectorpotential
oftheultra-violetlightcanberegardedasconstantattheatomicscale,there-
fore r(cid:3)A50 and Eq. (7) reads:
e e e
H ¼ (cid:2) ðA(cid:3)p+½p,A(cid:4)+p(cid:3)AÞ ¼ (cid:2) A(cid:3)p2iħr(cid:3)A5(cid:2) A(cid:3)p:
int 2mc mc mc
(8)
To accurately describe and calculate the photoemission process, one needs
to treat it as a one-step quantum process. The bulk, surface, and vacuum
informationhavetobeincludedintheHamiltoniandescribingthesystem,
which involvesnot onlythe bulk andsurface statesbutalso theevanescent
statesandsurfaceresonancestates.Suchaone-stepmodelistoocomplicated
tobequantitativelysolved.Instead,aphenomenologicalthree-stepmodelis