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Topological Insulators &
Their Spintronics Application
Xue-sen WangNational University of Singapore
Topological Insulator: A piece of material that is an
insulator (or semiconductor) in its bulk, but supports
spin-dependent conductive states at the boundaries
due to spin-orbital coupling
Spintronics in Future Information Technology
(S-Q Shen, AAPPS Bulletin 18(5), 29)
Spintronics: Manipulation of electron spin for information
storage, transmission & processing
Traditional spintronics: e.g. magnetic disk, size limit
Current spintronics: e.g. GMR, TMR, still with a charge
current
Ideal Spintronics: Pure spin current & spin accumulation
controlled by electric field/voltage, dissipationless
Quantum spin Hall effect (QSHE): an attractive option
Hall Effect & Spin Hall Effect (SHE)
(From Y.K. Kato, Sci. Am. 2007(10) 88; also see Hirsch, PRL 83, 1834)
SHE: Separate electrons of different spins without using a magnetic field
Spin current can be generated & controlled
with an electric field or voltage, important to spintronics
)(2 jjjS e
Observations of Spin Hall Effect
SHE
Reverse SHE
(Valenzuela & Tinkham, Nature 442, 176; Kent, Nature 442, 143)
Electronic measurement:
A spin Hall voltage VSH
generated by Reverse-SHE
Spin accumulation at
edges of GaAs stripe
observed by Kerr rotation
microscopy
(Kato et al., Science 306, 1910)
E
Mechanism of SHE: Spin-dependent scattering
Extrinsic mechanism: Scattering by magnetic field or
magnetic impurities
Zeeman energy: BμU
Stern-Gerlach effect: BμF )(
From relativity, an electron
moving in an electric field feels a magnetic field
Intrinsic mechanism:
EβB ceff
sp )( VH RR Intrinsic Rashba spin-orbit coupling:
R: small in light atoms (< 1 meV), significantly enhanced in
narrow-gap semiconductors containing Bi, Hg…
More observable in low-D structures, controllable with electric field
A moving electric field induces a magnetic field:
V: potential gradient in atom, or at a boundary:
1) Edges of a 2D electron gas
2) Surface or Interface, adjustable with bias voltage
Rashba Effect EβB ceff
Boundary states & SHE sp )( VH RR
A stripe of 2DEG
0V 0V
Non-zero and opposite V at two edges (or surfaces) of a 2DEG
channel (or a film) Spin-filtered edge (surface) states
A thin film
V = 0 in bulk region if it has inversion symmetry
Major contribution to V still comes from atomic potential
Edge
Surface
×
Spin-Orbital Coupling in 2DEG or on Surface
(Sinova et al., PRL 92, 126603; Ast et al., PRL 98, 186807)
sp )( VH RR
ARPES of Bi/Ag(111)
σek z// )(RRH or:
k
s
Quantized conductance through a quantum wire or point contact:
Transmitted states (modes) Ntrans, can be changed by gate bias Vg
trans0trans
22 NGNheG
Quantum conductance unit: G0 = 2e2/h = 7.75 S
Quantum Transport in Low-D Systems
Edge states & QHE in 2DEG Channel
Skipping OrbitsTo
Edge States
2DEG in a normal B
Four-terminal Hall resistance:
neh
IVV
IVV
R 12
24
13
2424,13
or: hne
xy
2
Quantum Hall Effect & Quantum Spin Hall Effect in 2D
(Nagaosa, Science 318, 758; Day, Phys. Today 61(1), 19; Kane & Mele, PRL 95, 146802)
E
kValence band
Conduction band
Bulk Insulator with Spin-dependent Kramers-pair gapless edge states
Kramers-pair Edge States: elastic backscattering forbidden by
time reversal symmetry, robust against weak disorder
Graphite
1.42 Å
K
M
Γ
K’ Dirac point
Mass-less (relativistic) fermion
Zero bandgap
Semimetal
Single-layer graphite:
Graphene
3.4 Å vkE
DOS |E|
QSHE in Graphene
(from Kane & Mele, PRL 95, 226801)
Spin-orbit coupling for edge
states: sp )( VVSO
x
y
Spin-filtered edge states: Electrons with
opposite spin propagate in opposite
direction; jS may be non-dissipative
A spin current will flow between leads
attached to the opposite edges:
2/eVI S
Quantized SH conductivity: 2/esxy
Graphene: a 2D Spin Hall insulator
Generate a spin current without dissipation
Spin-filtered edge states in graphene are insensitive to disorder:
Elastic backscattering is prohibited by time reversal
Existence of other spin Hall insulators with stronger SO interaction?
Spin Hall gap in graphene: 2SO ~ 2.4 K
(Kane & Mele, PRL 95, 226801; only ~ 0.01 K in Yao et al, PRB 75, 041401)
Operation temperature not practical!
More attractive materials for Spintronics
Bi, Z = 83; Pb, Z = 82; Hg, Z = 80; strong SOC
Semimetal or narrow-gap semiconductor
Bulk carrier density ~1017 cm-3, low bulk conductivity
Surface carrier density ~1013 cm-2, surface conduction
may be dominant
Small effective mass: m* = 0.002m0, ~ 106 cm2/Vs,
Fermi velocity vF 106 m/s (comparable with
graphene)
2D & 3D Topological insulators possible
HgTe QW, Bi, Bi1-xSbx , Bi2Se3
Inverted Normal
(König et al., Science 318 (2007) 766)
QSH Edge States in HgTe QW
Critical QW thickness 6.3 nm
5.5 nm
7.3 nm
Inverted?
Normal Normal
Lattice Structure of Bi
Rhombohedral lattice
A
B
C
]112[_
]101[_
4.545 Å
3.95 Å
abc
= 57.23˚
Covalent bond
Honeycomb bilayer
Stacking in [111] direction
Bi(111) bilayer: a 2D Spin Hall Insulator
abc
Covalent bond
Honeycomb bilayer
}
[111]
]112[_
]101[_
(Murakami, PRL 97, 236805; Liu et al., PRB 76, 121301)
2D bandgap 0.2 eV
1 Kramers pair of edge states
Spin Hall Conductivity:
474.0~ es
(Koroteev et al., PRL 93, 046403; Liu et al., PRB 76, 121301)
SOC of Bi(111) surface states
Splitting ~ 0.1-0.2 eV
Spin accumulation at edges of Bi(111) bilayer
With SOC
Sz +
Sz -
When EF in middle of Eg
No charge current
MΓ
K
Number of Kramers pairs at each edge/surface must be odd
(non-zero Z2 invariant): Strong Topological Insulator
E
kValence band
Conduction band
Quantum spin Hall Effect in 3D
Kramers pair
2D Edge states
3D Surface states
(Kane & Mele, PRL 95, 146802; Fu & Kane, PRB 76, 045302)
(Weak topological insulator: with even number of edge-state pairs)
(S-c Zhang, APS Physics 1, 6 (2008))
Strong Topological Insulator
Metallic edge/surface states linear in k meet at an
odd number of points in k-space
Robust against perturbation
Lattice Structure of Sb & Bi
Rhombohedral lattice:
A distorted simple cubic (SC) or FCC
lattice
A
B
C
]112[_
]101[_
Sb: 4.31 Å
Bi: 4.545 Å
Sb: 3.76 Å
Bi: 3.95 Å
abc
= 57.1 (Sb) = 57.23° (Bi)
Covalent bond
Honeycomb bilayer
Electronic Structure of Bi & Sb
EF= 26.7 meV
L
13.8 meV
Band overlap 38 meV
Low carrier density (~1017 cm-3)
Small effective mass
High carrier mobility (~ 105 cm2/Vs)
Long F, ~ 120 Å
T
(for Sb: at H, 177 meV
overlap with inverted La)
La
Ls
Semimetal
Energy Bands of Bi1-xSbx
@ x ~ 4%: Dirac Fermions in 3+1 D
kvkvk 22)()(E
TE
x (%)T
H
H
0 4 7 9 17 22
La
LaLa
Ls
LsLs
30 m
eV
Semiconductor or
Topological Insulator
Inversion of L bands
(Hsieh et al., Nature 452, 970; Teo et al., PRB 78, 045426)
Bi1-xSbx: Topological Insulator
m* ~ 0.002me
(x ~ 7-10%)
3D quantum spin Hall phase 2D surface states
2D quantum spin Hall phase 1D edge states
(Hsieh et al., Nature 452, 970 (Suppl. Info.))
Effect of SOC on Bi
bulk band near EF
= 13.7 meV
3D Dirac point at L
(from Teo, Fu & Kane, PRB 78, 045426)
Surface States on Different Bi1-xSbx Surfaces
Surface time-reversal-invariant momentum (TRIM)
enclosed by an odd number of electron or hole pockets
Surface Fermi arc
encloses 1 or 3 Dirac
points on all surfaces
“Strong” Topological Insulator
(111) & (110) surfaces
commonly observable
Bi(111) Surface
(Hofmann, Prog. Surf. Sci. 81, 191; Ast & Hochst, PRL 87, 177602)
ARPES measurement of surface states
EF mapping K
Spin direction
of states at EF
Bi(110) Surfaces
1X
2X
ARPES & computed of surface states
EF mapping &
spin directions
(Hofmann, Prog. Surf. Sci. 81, 191; Pascual et al., PRL 93, 196802)
HOPG or MoS2
cleaved in air, ~ 5
hours degas at 300-
550C in UHV
Sb & Bi from thermal
evaporators
Nearly free-standing
structures grow on
aninert surface
STM imaging at RT
UHV STM system
Bi & Sb Nanostructures Grown on Inert Substrates
3D, 2D & 1D Sb Nanostructures on HOPG
Sb4, F = 4 Å/min, 12 Å
deposited at RT. 3D, 2D & 1D islands formed at early stage
(1000 nm)2
1D, h ~ 23 nm
3D, h ~ 60 nm
2D, h ~ 3.5 nm
(100 nm)2
(10 nm)2
(111)-oriented 2D islands
Lateral period:
a = 4.170.12 Å
Bulk Sb: a = 4.31 Å
1D & 2D Bi Nanostructures on HOPG
(1 m)2
2D islands, height ~ 1 nm
(111) oriented
(0.6 m)2
1D nanobelts
Bi(111) bilayer spacing: 3.95 Å
Bi Nanobelts: (110) oriented
Belt surface with rectangular lattice:
4.34 Å × 4.67 Å
(200 nm)2
(2 m)2
Height ~ 1-10 nm
Width ~ 25-70 nm
Narrow belts on
top of wide belt
(9 nm)2
Narrow belt
h ~ 8 Å
Bulk Bi(110): 4.55 Å × 4.75 Å
Layer spacing: 3.28 Å
Bi(110) nanobelts on Bi/Ag(111)
Aligned Bi Nanobelt on Low-symmetry Surface
Aligned Bi nanobelts on Si(111)-
41:In single-domain terrace
observed in Surface Physics Lab,
Inst. of Physics, CAS, Beijing
Bi wetting layer on
Ag(111): with a 2D
rectangular lattice
(300 nm)2
Dangling bondsInert sidewall
Deposited atoms
Self-Assembly of Sb & Bi Nanobelts
(111) top surface of Bi nanobelt
Growth direction
Removal of dangling bonds on Bi(110) by “puckered-layer” atomic reconfiguration (Nagao et al. PRL 2004)
Transformation of Bi(110) to Bi(111)
(1 m)2
(1 m)2
After 10 min 100C annealh ~ 5 – 9 nm
After 10 min 130C annealh ~ 5 – 10 nm
(H. Zhang et al., Nature Physics 5 (2009) 438)
Topological Insulators at
Room Temperature
Bi2Se3: Eg 0.3 eV
Surface states on (111)
Sb2Te3: Eg 0.1 eV
Magneto-Electric Effects in Topological Insulators
Normal insulator:
223
0
1BE
xdtdS
Additional action term: BExdtdS 3
22
137/1/2 ce where
AAxdtd 3
42
Topological insulator: θ = πNormal insulator: θ = 0;
All time reversal invariant insulators can be divided into two classes:
(Qi, Hughes and Zhang, PRB 78, 195424)
Topological Magneto-Electric (TME) Effect
P3 = θ/2π
(Qi, Hughes and Zhang, PRB 78, 195424; Qi et al, arXiv:0811.1303)
A charge near TI induces an
image magnetic monopole:
g qP
g 32
Summary
Topological insulators possess novel properties with potential
spintronic applications due to QSHE
HgTe QW, Bi(111) monolayer, Bi1-xSbx alloy, Bi2Se3 and Sb2Te3 are
possible topological insulators
Bi(111) bilayer/film similar to graphene/graphite
Ultrathin (2-6 bilayers) Bi(111) and Bi(110) nanobelts can be
obtained on inert substrates (e.g. graphite and MoS2)
Bi & Sb nanostructures can be fabricated at much less demanding
conditions than for graphene. Certain growth controls have been
accomplished
Further Studies
Fabrication of Bi1-xSbx (x ~ 10%) thin films and
nanostructures, effect of inhomogeneity
Electronic & spintronic transport measurements, TME
effect: contact, patterning and processing
Controlled growth of Bi & BiSb structures on Si-based
substrates
Other topological materials, e.g. Bi2Se3, Sb2Te3
Universal Intrinsic Spin Hall Effect in 2DEG
(Sinova et al., PRL 92, 126603)
xE x ˆys ,j
Spin current is
polarized in z direction,
with spin Hall conductivity
ys ,j
8, e
E
jσ
x
yssH
Still need charge current xc,j
p
s
(Raghu et al., PRL 100, 156401)
U
V1
V2
Phases of Honeycomb Lattice with Repulsive Interactions
QSE phase more likely in bilayer
lattice of dipolar atoms with
V2 > U, V1
D
A
C
B
y
x
A: ax = 4.47 Å
Lattice distortion across 90-elbow of Bi nanobelt
B: ax = 4.49 Å
C: ax = 4.73 Å
D: ax = 4.88 Å
Variation of X-period
Reverse variation of Y-period
On bulk Bi(110): 4.55 Å × 4.75 Å
2 bilayer (~ 6.6 Å) Bi(110) growth
On Ag(111) with a Bi wetting layer
Semimetal-to-Semiconductor transition in Bi nanowires
(Lin et al., PRB 62, 4610)
Bi(111) Ultrathin Films
electron
hole
1 bilayer: Semiconducting
With SOC
2 – 3 bilayer films: Semimetallic
(Koroteev et al., PRB 77, 045428)
Bi(110): bilayer pairing
Remove dangling bonds on Bi(011) by
“puckered-layer” pairing reconfiguration
(Nagao et al. PRL 2004) >10% in vacuum
(Koroteev et al., PRB 77, 045428)