Journal Club of Topological Materials (2014)

Journal Club of Topological Materials (2014)

If you raised your hand you’re in the wrong place!!

Show of hands, who here is familiar with the concept of topological insulators?

The Quantum Spin Hall EffectTejas Deshpande

Joseph Maciejko, Taylor L. Hughes, and Shou-Cheng Zhang. “The Quantum Spin Hall Effect.” Annual Reviews of Condensed Matter Physics 2, no. 1 (2011): 31-53.

Introduction• Ginzburg-Landau Theory of Phase Transitions

• Classify phases based on which symmetries they break

• Rigorous definition of “symmetry breaking”: ground state does not possess symmetries of the Hamiltonian

• Example: classical Heisenberg model

• Ordered phase characterized by local order parameter

• Phases Defined by Symmetry Breaking• Rotational and Translational: Crystalline Solids (continuous to discrete)

• Spin Rotation Symmetry: Ferromagnets and Antiferromagnets

• U(1) gauge symmetry: Superconductors

Introduction• “Topological” Phases

• Integer Quantum Hall Effect (IQHE) discovered in 1980

• Topological or “global” order parameter Hall conductance quantized in integral units of e2/h

• Fractional Quantum Hall Effect (FQHE) discovered in 1982

• Phase transitions do not involve symmetry breaking

• Experimental implications of “topological order”• Number of edge states equal to topological order parameter (Chern number)

• Edge states robust to all perturbations due to “topological protection”

Current = 1 μAMagnetic Field = 18 TTemperature = 1.5 K

Introduction• Topological Protection

• Current carried only by chiral edge states

• Chiral edge states robust to impurities

• No tunneling between opposite edges

• FQHE• FQHE with (1/m)e2/h (m odd) Hall

conductance gives rise to bosonic quasiparticles

• Example: FQHE with m = 3 has quasiparticles with 3 flux quanta attached

• Chern-Simons theory is the low energy effective field theory

Introduction• Road to Topological Insulators (TIs)

• IQHE without a magnetic field: Haldane model

• Observation of the “spin Hall effect”

Occupations of Light-Hole (LH) and Heavy-Hole (HH) bands

Spin conductance

Phenomenology of the Quantum Spin Hall Effect• Classical spin vs. charge Hall effect

• Charge Hall effect disappears in the presence of time-reversal symmetry

Odd under time reversal

Constant

Even under time reversal

• Non-zero spin Hall conductance in the presence of time-reversal symmetry

Even under time reversal

Constant

Even under time reversal

• Does the quantum version of the spin Hall effect exist?

• Yes! Kane and Mele proposed the quantum spin Hall effect (QSHE) in graphene and postulated the Z2 classification of band insulators

Phenomenology of the Quantum Spin Hall Effect• QSHE as a “topologically” distinct phase

• “Fractionalization” at the boundary

• “Topological” in the sense that the electron degrees of freedom are spatially separated

• Mechanism of spatial separation:

• QHE External magnetic field (time-reversal breaking)

• QSHE intrinsic spin-orbit coupling (time-reversal symmetric)

Spinless 1D chain

HSQHQ

2+2=41+1=2

Spinful 1D chain

Impurity

Spinless 1D chain

HSQHQ

2+2=41+1=2

Spinful 1D chain

Impurity

The QSHE in HgTe Quantum Wells• Review of basic solid state physics

• What does spin-orbit coupling do?

• What does time-reversal symmetry imply?Kramers pair states

• What does inversion symmetry imply?• Kramers pairs well defined even when spin is not conserved

• What do both time-reversal and inversion symmetries imply?

The QSHE in HgTe Quantum Wells• Banstructure of bulk CdTe

• s-like (conduction) band Γ6 and p-like (valence) bands Γ7 and Γ8 with (right) and without (left) turning on spin-orbit interaction

-5

-4

-3

-2

-1

0

1

2

3

4

-5

-4

-3

-2

-1

0

1

2

3

4

• With spin-orbit interaction Γ8 splits into the Light Hole (LH) and Heavy Hole (HH) bands away from the Γ point

• The split-off band Γ7 shifts downward

The QSHE in HgTe Quantum Wells• Banstructure of bulk HgTe

• s-like (conduction) band Γ6 and p-like (valence) bands Γ7 and Γ8 with (right) and without (left) turning on spin-orbit interaction

-4

-3

-2

-1

0

1

2

-4

-3

-2

-1

0

1

2

• The Γ8 splits into LH and HH like CdTe except the LH band is inverted• The ordering of LH band in Γ8 and Γ6 bands are switched

The QSHE in HgTe Quantum Wells• Quantum Well (QW) fabrication

• Molecular Beam Epitaxy (MBE) grown HgTe/CdTe quantum well structure

• Confinement in (say) the z-direction

• Transport in the x-y plane

• L = 600 μm and W = 200 μm

• Gate voltage (VG) used to tune the Fermi level (EF) in HgTe quantum well

E

z

Band gap of QW

Band gap of barrier

EF

The QSHE in HgTe Quantum Wells• Topological phase transition

• QW sub-bands invert for well thickness d > 6.3 nm

• Intersection of the first electron sub-band with hole sub-bands

The QSHE in HgTe Quantum Wells• The Bernevig-Hughes-Zhang Model

• Hamiltonian with QW symmetries

• Components

• Elegant Hamiltonian form

• Break translational symmetry in the y-direction

BHZ Model

• Since

The QSHE in HgTe Quantum Wells• The BHZ Model

• Numerical diagonalization?

• Try ansatz

• Writing

• Plugging in explicit expressions and multiplying by Γ5 we get

The QSHE in HgTe Quantum Wells• The BHZ Model

• Solutions

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04• Normalization condition

• Bulk dispersion

• Surface dispersion

where s labels Kramers pairs

• Using Landauer-Büttiker formalism for an n-terminal device

• For the helical edge channels we expect

• For a 2-point transport measurement between terminals 1 and 4

The QSHE in HgTe Quantum Wells• The BHZ Model

• If the transport is dissipationless where is the resistance coming from?

• In QSHE don’t we have spin currents of e2/h + e2/h = 2e2/h and charge currents of e2/h – e2/h = 0?

• A -flux tube threaded into a QSH insulator inducesspin-charge separation. (Qi & Zhang, see similarproposal, Ran, Vishwanath, Lee)

• Start from decoupled case

• Flux threading in quantum Hall system. (Laughlin PRB

1981)

II. Spin-charge separation in QSH insulators

H=+e2/h

H=-e2/h

(t

Ej

• Answer 1: dissipation comes from the contacts. Note that transport is dissipationless only inside the HgTe QW

• Answer 2: We do measure charge conductance! The existence of helical edge channels is inferred from charge transport measurements

The QSHE in HgTe Quantum Wells• The BHZ Model

• For normal ordering of bands the Landau levels will get further apart as B increases

• For inverted bandstructures Landau levels will cross at a certain B

• Only inverted bandstruc-tures will reenter the quan-tum Hall states when B field increases

The QSHE in HgTe Quantum Wells• The BHZ Model (Section 5).This isuseful to keep in mind,asmanytheoretical works in thefield usethe simple

“spin up/spin down”picturebut reallymean Kramerspartners.This isalso the reason whytheQSH effect is not to be understood as a quantized SH effect: Because spin-orbit couplingdestroys spin conservation, there is no such thing as a quantized SH conductance in the QSHeffect. This is another way to understand why the correct topological invariant for the QSHeffect is Z2 and not Z. Finally, the BHZ Hamiltonian predicts a single helical edge state peredge. This is useful when we compare the theoretical predictions of the BHZ model toexperiment in Section 4.

4. EXPERIMENTSON HgTeQUANTUM WELLSLess than one year after the 2006 theoretical prediction described in Section 3, a team at theUniversityof Wu rzburg led byLaurensW.Molenkamp observed theQSH effect in HgTe/CdTeQWsgrown bymolecularbeamepitaxy(14). In thissection,wereviewthemainresultsof theseexperiments.

4.1. Landau Levelsand Band Inversion in HgTeQuantumWellsAs described in Section 3, the QSH effect relies heavily on the existence of band inversion inbulk HgTe and its consequences for the HgTe/CdTe QW subband structure. Therefore, oneshould first verify whether band inversion in the HgTe/CdTe systemexists. A striking manifes-tation of this is a so-called re-entrant QH effect (26) that has been experimentally observed(Section 1.3) (see Figure5).Thepeculiar band structureofinverted HgTe/CdTeQWsgives rise

100a dQW =40 Å

EmeV–1

BT–1 BT–1

EmeV–1

b dQW =150 Å

50

–50

–100

–20

20

40

0

0

0 5 10 15 0 5 10 15

Figure5BulkLandau levels ( fan diagram)for an HgTe/CdTequantumwell (QW) in aperpendicular magneticfield B.(a)Trivial insulator ( d< dc):No level crossingoccursasafunctionof B,andforafixed Fermi energy EF in theB ¼0gap, the Hall conductance sxy isalways zero. ( b) Quantumspin Hall insulator with d > dc: There isalevel crossing at somecritical field B ¼Bc, and for afixed EF in theB ¼0gap,aconduction or valencebandLandau level eventuallycrosses EF,giving riseto are-entrant quantumHall effect with sxy e2/h.

www.annualre views.org The Quantum Spin HallB

E

(A)

B

E

(B)

<r>

E

(C) (D)

<r>

E

(E)

<r>

E

(F)

<r>E

iiiii

ivii

case i, σxy=0 case ii, σxy=0

case iii, σxy=e2/h case iv, σxy=–e2/h

εF εF

εFε

• The concept of “helical” edge state states with opposite spin counter-propagate at a given edge

• QH protected by “chiral” edge states; QSH edge states protected due to destructive interference between all possible back-scattering paths

• Clockwise and anticlockwise rot-ation of spin pick up ±π phase leading to destructive inter-ference

Theory of the Helical Edge StateSpinless 1D chain

HSQHQ

2+2=41+1=2

Spinful 1D chain

Impurity

Spinless 1D chain

HSQHQ

2+2=41+1=2

Spinful 1D chain

Impurity

• The physical description of edge state protection works only for single pair of edge states

• With (say) two forward-movers and two back-ward-movers backscattering is possible without spin flip

• Robust or non-dissipative edge transport requires odd number of edge states

Theory of the Helical Edge State

• Only two TR invariant non-chiral interactions can be added

• Combined with Umklapp term we get (opens a gap at kF = π/2)

• We can “bosonize” the Hamiltonian

• The forward scattering term simply renormalizes the parameters K and vF

forward scattering term

Two-particle backscattering or “Umklapp” term

• Boson to fermion field operators

Stability of the Helical Liquid: Disorder and Interactions

• Total Hamiltonian

• RG analysis Umklapp term relevant for K < 1/2 with a gap:• Interactions can spontaneously break time-reversal symmetry• TR odd single-particle backscattering:• Bosonize Nx and Ny . For gu < 0 fixed points at

Umklapp term

• Due to thermal fluctuations TRS is restored for T > 0• For mass order parameter Ny is disordered + TR is preserved with a gap

• For gu < 0, Ny is the (Ising-like) ordered quantity at T = 0

Stability of the Helical Liquid: Disorder and Interactions

• Total Hamiltonian Umklapp term

• Two-particle backscattering due to quenched disorder Gaussian random variables

• The “replica trick” in disordered systems shows disorder relevant for K < 3/8• Nx and Ny show glassy behavior at T = 0 with TRS breaking; TRS again restored

at T > 0• Where would all these interactions come from? locally doped regions? Band

bending?• But edge states are immune to electrostatic potential scattering• Potential inhomogeneities can trap bulk electrons which may then interact with

the edge electrons

K < 1

Stability of the Helical Liquid: Disorder and Interactions

• Static magnetic impurity breaks local TRS and opens a gap• Quantum impurity Kondo

effect:

• Doing the “standard” RG procedure we get flow equations

Stability of the Helical Liquid: Disorder and Interactions

• Static magnetic impurity breaks local TRS and opens a gap• Quantum impurity Kondo

effect1. At high temperature (T)

conductance (G) is log2. For weak Coulomb

interaction (K > 1/4) conductance back to 2e2/h. At intermediate T the G ~ T2(4K-1) due to Umklapp term

3. For strong Coulomb interaction (K < 1/4) G = 0 at T = 0 due to Umklapp. At intermediate T the G ~ T2(1/4K–1) due to tunneling of e/2 charge

Stability of the Helical Liquid: Disorder and Interactions

• Quantized charge at the edge of domain wallo Jackiw-Rebbi (1976)o Su-Schrieffer-Heeger (1979)

• Helical liquid has half DOF as normal liquid e/2 charge at domain walls• Mass term Pauli matrices ∝ external TRS breaking field• Mass term to leading order

• Current due to the mass field

• For m1 = m cos(θ), m2 = m sin(θ), and m3 = 0

• Topological response net charge Q in a region [x1,x2] at time t = difference in θ(x,t) at the boundaries

• Charge pumped in the time interval [t1, t2]

Fractional-Charge Effect and Spin-Charge Separation

• Two magnetic islands trap the electrons between them like a quantum wire between potential barriers

• Conductance oscillations can be observed as in usual Coulomb blockade measurements

• Background charge in the confined region Q (total charge) = Qc (nuclei, etc.) + Qe (lowest subband)

• Flip relative magnetization pump e/2 charge

• Continuous shift of peaks with θ(B)• AC magnetic field

drives current

Fractional-Charge Effect and Spin-Charge Separation

• A -flux tube threaded into a QSH insulator inducesspin-charge separation. (Qi & Zhang, see similarproposal, Ran, Vishwanath, Lee)

• Start from decoupled case

• Flux threading in quantum Hall system. (Laughlin PRB

1981)

II. Spin-charge separation in QSH insulators

H=+e2/h

H=-e2/h

(t

Ej• A -flux tube threaded into a QSH insulator inducesspin-charge separation. (Qi & Zhang, see similarproposal, Ran, Vishwanath, Lee)

• Start from decoupled case

• Flux threading in quantum Hall system. (Laughlin PRB

1981)

II. Spin-charge separation in QSH insulators

H=+e2/h

H=-e2/h

(t

Ej

• Simplified analysis:o Assume Sz is preservedo QSHE as two copies of QHE

• Thread a π (units of = ℏ c = e = 1) flux ϕ• TRS preserved at ϕ = 0 and π; also, π = –π• Four possible paths for ϕ↑ and ϕ↓: • Current density from E||:

• Net charge flow:

Fractional-Charge Effect and Spin-Charge Separation

3D Topological Insulators

E

ky

kx

Γ

Helical spinpolarization

2D Dirac cone

SurfaceBrillouin zone

kEn

ergy

k = 0

BulkConduction Band

BulkValence Band

up spindownspin Dirac point

(a)

(b)

(c)

(d)

Vacuum

upspindownspin

2D Topological Insulator

• Introduction• 2D topological insulator 1D edge states

• Dirac-like edge state dispersion

• What happens in 3D?

• 3D topological insulator 2D surface states

• Surface dispersion is a Dirac cone, like graphene

• What happens in 1D? Nothing!

E

ky

kx

Γ

Helical spinpolarization

2D Dirac cone

SurfaceBrillouin zone

k

Ener

gy

k = 0

BulkConduction Band

BulkValence Band

upspindownspin Dirac point

(a)

(b)

(c)

(d)

Vacuum

up spindownspin

2D Topological Insulator

E

ky

kx

Γ

Helical spinpolarization

2D Dirac cone

SurfaceBrillouin zone

k

Ener

gy

k = 0

BulkConduction Band

BulkValence Band

up spindownspin Dirac point

(a)

(b)

(c)

(d)

Vacuum

up spindownspin

2D Topological Insulator

E

ky

kx

Γ

Helical spinpolarization

2D Dirac cone

SurfaceBrillouin zone

k

Ener

gy

k = 0

BulkConduction Band

BulkValence Band

up spindownspin Dirac point

(a)

(b)

(c)

(d)

Vacuum

upspindownspin

2D Topological Insulator

• Topological band theory

• Difficult to evaluate ℤ2 invariants for a generic band structure

• Consider the matrix

• At the TRIM B(Γi) is antisymmetric; we can define

• Trivial: (–1)ν2D = +1 and Non-trivial: (–1)ν2D = –1

3D Topological Insulators

• Topological invariant

• “Dimensional increase” to 3D

• Weak TI: (–1)ν3D = +1 and Strong TI: (–1)ν3D = –1

• With inversion symmetry rewrite δi as

where ξ2m(Γi) = ±1 is the parity eigenvalue of the 2mth band at Γi) and ξ2m = ξ2m–1 are Kramers pairs

• Gap closing (phase transition)• k = (0, 0) M = 0• k = (π, 0) and (0, π) M = 4B• k = (π, π) M = 8B

3D Topological Insulators• Simplified topological invariant expression

• Recall BHZ model

Conclusion and Outlook• The quantum spin Hall effect (QSHE)

• Phenomenology

• Design of quantum wells in the QSHE regime

• Explicit solution of Bernevig-Hughes-Zhang (BHZ) model

• Experimental verification using transport

• Properties of the “2D topological insulator”• Theory of helical edge states

• Effects of interactions and disorder

• Fractionalization and spin-charge separation

• Introduction to 3D topological insulators• Topological Band Theory (TBT)

• Topological Invariant of the QSHE