Univ Toronto, Nov 4, 2009
Topological Insulators
J. G. Checkelsky, Y.S. Hor, D. Qu, Q. Zhang, R. J. Cava, N.P.O.
Princeton University
1.
2.
3.
4.
5.
Introduction
Angle resolved photoemission (Hasan)
Tentative transport signatures
Giant fingerprint signal
Insulator and Superconductor
A new class of insulators
Conventional insulator
Topological insulator
Top
cond. band
cond. band
valence band
Fu, Kane ’06
Zhang et al. ’06
Moore Balents ‘06
Xi, Hughes, Zhang ‘09
Bottom
cond. band
cond. band
valence band
valence band
Surface states
may cross gap
Surface states are helical (spin locked to k)
Large spin-orbit interactn
Surface state has
Dirac dispersion
s
s
crystal
k
ky
s
kx
Protection of helical states
1. Time-reversal invariance prevents gap formation at crossing
cond. band
Under time reversal
(k↑)  (-k↓)
cond. band
?
s
k
valence band
valence band
Violates TRI
Kane, Mele, PRL ‘05
2. Suppression of 2kF scattering
2D Fermi
surface
Spinor product kills matrix element
Large surface conductance?
Helicity and large spin-orbit coupling
• Spin-orbit interaction and surface
E field  effectv B = v  E in rest frame
B
s
• spin locked to B
k
E
• Rashba-like Hamiltonian
H  vF n̂ k s
v
E
s
B
v
k
spin aligned with B in
rest frame of moving electron
Helical, massless Dirac states
with opposite chirality on opp.
surfaces of crystal
Like LH and RH neutrinos in
different universes
A twist of the mass (gap)
Doped polyacetylene (Su, Schrieffer, Heeger ‘79)
H
H
H
e /2
D(x)
Domain wall (soliton)
traps ½ charge
 pv D( x )
H  *

D
(
x
)

pv


Mobius strip
1. Gap-twist produces domain wall
2. Domain wall traps fractional charge
3. Topological (immune to disorder)
Mobius strip like
Dirac fermions as domain wall excitation
Jackiw Rebbi, PRD ‘76
Goldstone Wilczek, PRL ‘81
Callan Harvey, Nuc Phys B ‘85
Fradkin, Nuc Phys B ‘87
D. Kaplan, Phys Lett B ’92
QFT with background mass-twist field
Dirac modes on domain walls of mass field
p 
m( x)
H 
 m( x)
 p
Y
Y+
m(x)
Chiral zero-energy mode
Domain-wall fermion
x
Callan-Harvey: Domain walls exchange chiral current to solve anomaly problm
Chiral surface states?
Topological
insulator
k
vacuum
z
Fu Kane prediction of topological insulator
Fu, Kane, PRB ‘06
Mass twist
k
z
Bi
Bi1-xSbx
Sb
Mass twist traps surface
Weyl fermions
ARPES confirmation
Hsieh, Hasan, Cava et al.
Nature ‘08
Confirm 5 surf states in BiSb
Angle-resolved photoemission spectroscopy (ARPES)
20 eV photons
+
velocity selector
quasiparticle
peak
m
k||
Intensity
E
E
ARPES of surface states in Bi1-x Sbx
Hsieh, Hasan, Cava et al. Nature 2008
ARPES results on Bi2Se3 (Hasan group)
Xia, Hasan et al. Nature Phys ‘09
Large gap ~ 300meV
As grown, Fermi level in
conduction band
Se defect chemistry difficult to
control for small DOS
Photoemission evidence for Topological Insulators
Why spin polarized?
Rashba term on surface
What prevents a gap?
 Time Reversal Symmetry
Hsieh, Hasan et al., Nature ‘09
What is expected from
transport?
•No 2 kF scattering
•SdH
•Surface QHE (like
graphene except ¼)
•Weak anti-localization
Bi2Se3: Typical Transport
Metallic electron pocket with
mobility ~ 500-5000 cm2/Vs
Carrier density ~ 1019 e-/cm3
Roughly spherical Fermi
surface (period changes by
~ 30%)
Quantum oscillations of Nernst in metallic Bi2Se3
Major problem confrontg transport investigation
As-grown xtals are always excellent conductors,
m lies in conduction band (Se vacancies).
r (1 K) ~ 0.1-0.5 mWcm,
m* ~ 0.2,
n ~ 1 x 1018 cm-3
kF ~ 0.1 Å-1
Fall into the gap
Hor et al., PRB ‘09
Checkelsky et al., arXiv/09
Solution:
Tune m by Ca doping
target
electron
doped
cond. band
m
valence band
Decrease
electron density
hole
doped
Resistivity vs. Temperature : In and out of the gap
Checkelsky et al., arXiv:0909.1840
Onset of nonmetallic behavior ~
130 K
SdH oscillations
seen in both n-type
and p-type
samples
Non-metallic
samples show no
discernable SdH
Magnetoresistance of gapped Bi2Se3
Checkelsky et al., arXiv:0909.1840
Giant, quasi-periodic, retraceable conductance fluctuations
Logarithmic
anomaly
Conductance
fluctuations
Magneto-fingerprints
Fluctuations retraceable
Giant amplitude
(200-500 X too large)
Retraceable
(fingerprints)
Spin degrees
Involved in
fluctuations
Checkelsky et al., arXiv:0909.1840
Angular Dependence of R(H) profile Cont.
For δG, 29% spin
term
For ln H, 39% spin
term (~200 e2/h total)
Theory predicts both
to be ~ 1/2π
(Lee & Ramakrishnan),
(Hikami, Larkin, Nagaoka)
Universal Conductance Fluctuations
Stone, Lee, Fukuyama (PRB 1987)
LT
H
Quantum diffusion
Conductance -- sum over Feynman paths
G   Ai Aj  | Ai |2 + | Ai Aj | e
*
i, j
i
i (i  j )
i, j
Universal conductance fluctuations (UCF)
dG = e2/h
in a coherent volume defined by thermal length LT = hD/kT
At 1 K, LT ~ 1 mm
our xtal
LT
For large samples size L,
“Central-limit theorem”
1/ 2
e 2  LT 
dG   
h L
L = 2 mm
UCF should be unobservable in a 2-mm crystal!
Size Scales
dGmeasured ~ e 2 / h
Taking typical 2D LT = 1 µm at 1 K,
For systems size L > LT, consider
(L/LT)d systems of size LT, UCF
suppressed as
( LT / L)
L  D in
LT  D / k BT
2 d / 2
For AB oscillation, assuming 60
nm rings, N-1/2 ~ 10-8
Quasi-periodic fluctuations vs T
Fluctuation falls off
quickly with temperature
For UCF, expect slow
power law decay ~T-1/4 or
T-1/2
AB, AAS effect
exponential in LT/P
 Doesn’t match!
Non-Metallic Samples in High Field
Fluctuation does not change character
significantly in enhanced field
Next Approach: Micro Samples
Micro Samples Cont.
Y
Y+
x
Sample is gate-able
SdH signal not seen in 10
nm thick metallic sample
Exploring Callan-Harvey
effect in a cleaved crystal
(b)
(a)
(c)
Desperately seeking Majorana bound state
Fu and Kane, PRL 08
Surface topological states
f=0
SC1
Majorana bound state
f=p
SC2
D(x)
Open D at m by
Proximity effect
 1, 2   dx [e
Majorana oper.
 if / 2
1, 2 ( x)c ( x) + e
*
bound state wf
+
if / 2
1, 2 ( x)c( x)]
electron creation oper.
Neutral fermion that is its own anti-particle
Gap “twist” traps
Majorana
Cu Doping: Intercalation between Layers
Intercalation of Cu between
layers
Confirmed by c-axis lattice
parameter increase and STM
data
Crystal quality checked by Xray diffraction and electron
diffraction
Hor et al., arXiv 0909.2890
Diamagnetic Response at low T
Typical M for type II: -1000
A/m
From M(H), κ ~ 50
χ ~ -0.2
Impurity phases not SC
above 1.8 K (Cu2Se,
CuBi3Se5, Cu1.6Bi4.8Se8....)
Small deviations from Se
stoichiometry suppress SC
Cu Doping: Transport Properties
Not complete
resistive
transition
Up to 80%
transition has
been seen
Carrier
density
relatively high
Upper Critical Field HC2
HC2 estimate by
extrapolation
Similar shape for H||ab
HC2 anisotropy moderate
ξc = 52 Å , ξab = 140 Å
Ca Doping: Conclusions
Ca doping can bring samples from ntype to p-type
Non-metallic samples at threshold
between the two reveal new transport
properties
G ~ ln(H) at low H
δG ~ e2/h, quasiperiodic
Hard to fit with mesoscopic interpretation
No LL quantization seen up to
32 T
Metallic nanoscale samples
show no LL
Summary
Doping of Bi2Se3 creates surprising
effects
Ca doping: Quantum Corrections to
Transport become strong
Cu Doping: Superconductivity
Next stage:
1. nm-thick gated, cleaved crystals
2. Proximity effect and Josephson
current expt
END