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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 Effect
Tejas 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
Current = 1 μA
Magnetic Field = 18 T
Temperature = 1.5 K
• 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”
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”
Spin conductance
Occupations of
Light-Hole (LH) and
Heavy-Hole (HH)
bands
Phenomenology of the Quantum Spin Hall Effect
• Classical spin vs. charge Hall effect
• Charge Hall effect disappears in the presence of time-reversal symmetry
Even under
time reversal
Odd under
time reversal
Constant
• Non-zero spin Hall conductance in the presence of time-reversal symmetry
Even under
time reversal
Even under
time reversal
Constant
• 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
Spinless
Spinless
1D1D
chain
chain
2 =21=
+11+1
Spinful
Spinful
1D1D
chain
chain
4 =42=
+22+2
Impurity
Impurity
QHQH
QSQ
HSH
• “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)
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
• Kramers pairs well defined even when spin is not conserved
• What does inversion symmetry imply?
• 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
4
4
3
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
• 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
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-4
-4
• 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
Band gap of barrier
z
EF
E
Band gap of QW
• L = 600 μm and W = 200 μm
• Gate voltage (VG) used to tune the Fermi level (EF) in HgTe quantum well
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
The QSHE in HgTe Quantum Wells
• The BHZ Model
• Numerical diagonalization?
• Try ansatz
• Writing
• Plugging in explicit expressions and multiplying by Γ5 we get
• Since
The QSHE in HgTe Quantum Wells
• The BHZ Model
• Solutions
• Normalization condition
0.04
0.03
0.02
• Bulk dispersion
0.01
0
-0.01
• Surface dispersion
-0.02
-0.03
where s labels Kramers pairs
-0.04
The QSHE in HgTe Quantum Wells
• The BHZ Model
• Using Landauer-Büttiker formalism for an nterminal device
• For the helical edge channels we expect
• For a 2-point transport measurement between terminals 1 and 4
• A -flux tube threaded into a QS
spin-charge separation. (Qi & Zh
The QSHE in HgTe Quantum Wells
proposal, Ran, Vishwanath, Lee)
• The BHZ Model
• Start from decoupled case
2
H=+e /h
• If the transport is dissipationless where is the
2
resistance coming from?
H=-e /h
• In QSHE don’t we have spin currents of e2/h +
e2/h = 2e2/h and charge currents of e2/h – e2/h =
• Flux threading in quantum Hall
0?
1981)
• 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
Annu. Rev. Condens. Matter Phys. 2011.2:31-53. Downloaded f
by California Institute of Technology on 12/13/13. For
bulk HgTe and its consequences for the HgTe/CdTe QW subband structure. Therefore, one
should first verify whether band inversion in the HgTe/CdTe system exists. A striking manifestation of this is a so-called re-entrant QH effect (26) that has been experimentally observed
(Section 1.3) (see Figure 5). The peculiar band structure ofinverted HgTe/CdTe QWs gives rise
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
a
dQW = 40 Å
b
dQW = 150 Å
0
5
100
40
50
20
E meV–1
E meV–1
0
0
–50
–20
–100
0
5
10
15
B T–1
10
15
B T–1
5
• Only inverted bandstruc- Figure
Bulk Landau levels ( fan diagram ) for an HgTe/CdTe quantum well (QW) in a perpendicular magnetic field B.
Trivial insulator ( d < d ): No level crossing occurs as a function of B, and for a fixed Fermi energy E in the
tures will reenter the quan- (a)
B ¼ 0 gap, the Hall conductance s is always zero. ( b) Quantum spin Hall insulator with d > d : There is a
crossing at some critical field B ¼ B , and for a fixed E in the B ¼ 0 gap, a conduction or valence band
tum Hall states when B field level
Landau level eventually crosses E , giving rise to a re-entrant quantum Hall effect with s
e /h.
increases
c
F
xy
c
c
F
2
F
xy
www.annualre views.org
The Quantu m Spin Hal l Effect
Theory of the Helical Edge State
Spinless
Spinless
1D1D
chain
chain
• 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 backscattering paths
• Clockwise and anticlockwise rotation of spin pick up ±π phase
leading to destructive interference
2 =21=
+11+1
Spinful
Spinful
1D1D
chain
chain
4 =42=
+22+2
Impurity
Impurity
QHQH
QSQ
HSH
Theory of the Helical Edge State
• The physical description
of edge state protection
works only for single pair
of edge states
• With (say) two forwardmovers and two backward-movers
backscattering is possible
without spin flip
• Robust or non-dissipative
edge transport requires
odd number of edge
states
Stability of the Helical Liquid: Disorder and
Interactions
• Only two TR invariant non-chiral interactions can be added
forward
scattering term
Two-particle
backscattering or
“Umklapp” term
• We can “bosonize” the Hamiltonian
• Boson to fermion field operators
• The forward scattering term simply renormalizes the parameters K and vF
• Combined with Umklapp term we get (opens a gap at kF = π/2)
Stability of the Helical Liquid: Disorder and
Interactions
• Total Hamiltonian
•
•
•
•
Umklapp term
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
• For gu < 0, Ny is the (Ising-like) ordered quantity at T = 0
• Due to thermal fluctuations TRS is restored for T > 0
• For
mass order parameter Ny is disordered + TR is preserved with a gap
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
effect
1. At high temperature (T)
conductance (G) is log
2. 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
Fractional-Charge Effect and Spin-Charge
Separation
• Quantized charge at the edge of domain wall
o 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
spin-charge separation. (Qi & Zh
proposal, Ran, Vishwanath, Lee)
Fractional-Charge
Effect
and inSpin-Charge
•QSH
Startinsulators
from decoupled case
II. Spin-charge
separation
=+e /h
Separation • A -flux tube threaded into a QSH insulator induces =-e /h
H
2
2
H
• Simplified analysis:spin-charge separation. (Qi & Zhang, see similar
(t
proposal, Ran, Vishwanath, Lee)
o Assume Sz is preserved
• Start
from
decoupled case
o QSHE as two
copies
of QHE
2 • Flux threading in quantum Hall
H=+e /h
E
1981)
• Thread a π (units of ℏ = c = e = 1) flux ϕ
j
2
• TRS preserved at ϕ = 0 and π; also, π = –πH=-e /h
• Four possible paths for ϕ↑ and ϕ↓:
• Current density from E||:
• Flux threading in quantum
Hall
system.
(Laughlin PRB
• Net
charge
flow:
1981)
3D Topological Insulators
• Introduction
• 2D topological insulator  1D edge states
• Dirac-like edge state dispersion
(a)
(a)
•Vacuum
What happens in(c)
3D?
up spin
down spin
• Surface dispersion is a Dirac cone, like graphene
(c)
down
spin
up spin
Dirac point
Γ
Bulk
Valence Band
k=0
kx
k
Helical spin
polarization
2D Dirac cone
ky
Surface
Brillouin zone
(b)
Energy
Bulk
Conduction Band
E
(b)
Energy
(d)
up spin
down spin
down spin
1D? Nothing!
(c
Vacuum
spin
• 3D up
topological
insulator  2D surface states
• What
happens in
2D Topological
Insulator
(c
Vacuum
2D Topological Insulator
2D Topological Insulator
(d
Bulk
Conduction Band
down
spin
(d
Bulk
up spinBand
Conduction
Dirac point
down
spin
up spin
Dirac point
Bulk
Valence Band
k=0
k
k=0
k
Bulk
Valence Band
3D Topological Insulators
• 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
• Topological invariant
• Trivial: (–1)ν = +1 and Non-trivial: (–1)ν = –1
2D
2D
• “Dimensional increase” to 3D
• Weak TI: (–1)ν = +1 and Strong TI: (–1)ν = –1
3D
3D
3D Topological Insulators
• Simplified topological invariant expression
• 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
• Recall BHZ model
• Gap closing (phase transition)
• k = (0, 0)  M = 0
• k = (π, 0) and (0, π)  M = 4B
• k = (π, π)  M = 8B
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