Embed Size (px)
Citation preview
Studying Topological Insulators via
time reversal symmetry breaking and proximity effect
Roni Ilan UC Berkeley
Joel Moore, Jens Bardarson, Jerome Cayssol, Heung-Sun Sim
Topological phases
Insulating phases with metallic surface states.
Properties/existence of surface states protected by bulk properties.
Topological invariants of the bulk are immune to continuous deformations.
Bulk is insulating but non-trivial!
Probes of topological properties
Many features are attributed to surface states as mirrors of topological properties of the bulk
How can we observe this in experiment?
Topological phases and symmetry
Some topological phases require no symmetry:
Quantum Hall effect
Bulk topological invariant
= Chern number
Chiral edge modes protected due to spatial separation
�xy
=
Topological phases and symmetry
Quantum Spin Hall effect (2DTI)
TI surface states are protected by time reversal symmetry
An opportunity to study surface states by breaking that symmetry
Surface states of 2 and 3D TI’s
Figure from Y. Ando J. Phys. Soc. Jpn. 82, 102001 (2013)
2D 3D
Spin-momentum locking
Dirac fermions
Time reversal symmetry
Half integral spin
[H,T ] = 0 | i |T i
hT | i = hT 2 |T i⇤ = �h |T i⇤ = �hT | i
Absence of backscattering
T 2 = �1
h |V |T i = 0
Experimental detection: 2DTI
[Konig et al 07]
CdTe
CdTe
HgTe
I. Trivial II-IV. Non trivial !L=20 microns (L,W) =(1,1) micron (L,W) = (1,0.5) micron
!
Experimental detection: 3DTI
Bi2X3
Figure from Moore and Hasan, Annu. Rev. Condens. Matter Phys. 2011. Experiments by Xia et al. 2008, 2009 and Hsieh et al. 2009
Interesting questions
2D TI: Surface states are 1D.
!
!
Helical states.
Evidence of luttinger liquid physics? Impurity physics?
Exact solutions for non-equilibrium transport?
G = 2e2/h
Interesting questions
3D TI: Surface states are 2D.
!
Dirac Fermions.
Transport? Topological effects?
Majorana fermion physics?
Luttinger liquid physics on the edge of a 2DTI
Does mean edge is non-interacting?
No!
2e2
h
contactcontact
QSHE
[Maslov and Stone 95, Safi and Schulz 95,97; Furusaki and Nagaosa 96; Oreg and Finkel’stein 96; Alekseev et al. 96; Egger and Grabert 98..... Wen 90, Milliken et al 96]
G = ge2
h
contactcontact
QHE G = ⌫e2
h
Need to break time reversal symmetry:
Local Magnetic impurity
Global magnetic field + impurity
[Maciejko et al 09; Tanaka et al 11; Beri and Cooper 12; Soori et al. 12; Timm. 12, Lezmy Oreg and Berkooz, 12]
Local breaking of TR
Luttinger liquid physics on the edge of a 2DTI
To observe interaction effects we must have backscattering.
Goal
To find a controlled way to induce local backscattering
Use the special properties of 2DTI to obtain exact solution for noneq. conductance
designing a tunable impurity
Point contact in the QHE
Point contact in the QSHE
?
magnetic field
Breaks time reversal symmetry
H = �i~vF
�z
@x
+ µB
ge
2~B · ~�
hz
Eg = 2h?
hz
Eg = 2h?
E2 = (vp� hz)2 + h2?
p
E
Rashba spin orbit coupling
H = �i~vF
�z
@x
� i~2 {↵, @x}�y
v ! v↵ =p↵2 + v2
✓ = tan�1(↵/vF )
tunable impurity
↵ = 0 ↵ = 0↵ 6= 0
[Nitta et al 97;Hinz et al 06]
Rashba SO controllable by gate
Eg = 2↵h/vF
H = �i~vF
�z
@x
+ h�z
� i~2{↵, @
x
}�y
(x) =
8><
>:
R
e
ipRx + r
L
e
ipLx
x < 0
a+ +eip+x + a� �e
ip�x 0 < x < d
t
R
e
ipRx
x > d
Non interacting electrons
0 50 100 150 200M/E0
0.0
0.2
0.4
0.6
0.8
1.0
R
E/E0
3
10
15
R ⇠ (h↵/v2F )2
h/(~vF /d)
Interacting edge
H0 = iv( †"@x " � †
#@x #)
Hint = u2 †" "
†# # + u4
h( †
" ")2 + ( †# #)2
i
"= R #= L
HBS = � †L(0) R + h.c
Helicity reduces the number of degrees of freedom by locking the spin to the momentum
Interacting edge
H = v
4⇡g ((@x�R
)2 + (@x
�L
))2
HBS = � cos(�R � �L)HBS = 1
2
Pn �ne
in�R(0)e�in�L(0) + h.c.
1/4 < g < 1 One relevant term
Maps to a spinless Luttinger Liquid
d�d` = (1� n2g)�
[Kane and Fisher 92, Kane and Teo 09, and many others]
G(T ) ⇠ e2
h
⇣TTB
⌘2(1/g�1)TB/T ! 1
TB/T ! 0G(T ) ⇠ e2
h
h1�
�TBT
�2(1�g)i
TB ⇠ �1/(1�g)
Spinless luttinger liquid with impurity
�
e(x, t) = 1p2(�L(x, t) + �R(�x, t))
�
o(x, t) = 1p2(�
L
(x, t)� �
R
(�x, t))
[Fendley, Ludwig, Saleur 95; Fendley, Lesage, Saleur 96]
HSG
=
18⇡g
R10 [⇧
2+ (@
x
�o
)
2] + � cos�o
(0)
Integrable model, can be solved exactly
�R
�L
�
R
(x, t) ⌘ �
o(x+ t)
�
L
(x, t) ⌘ �
o(�x+ t)
Integrability[Ghoshal and Zamolodchikov 94; Fandley Ludwig and Saleur 95; Fendley Lesage and Saleur 96]
Infinite set of conserved quantities with impurity
find “quasiparticle” basis
One body S matrices
Additive energies H = vP
i pi
Particles scatter one-by-one
Two body S matrices in the bulk
Particles scatter elastically off each other
Does not mean scattering is trivial! Permute quantum numbers, phase shifts
-4 -2 2 4
0.5
1.0
1.5
-4 -2 2 4
0.5
1.0
1.5
-4 -2 2 4
0.2
0.4
0.6
0.8
fj =1
1+e�µj/T+✏jTBA equations for ✏j(✓)
HSG
=
18⇡g
R10 [⇧
2+ (@
x
�o
)
2] + � cos�o
(0)
[Fendley Lesage and Saleur 96, Koutouza, Siano and Saleur 01]
EXACT SOLUTION AT g = 1� 1/m
I(V, TB , T ) =T (m� 1)
2
Zd✓
cosh
2[✓ � ln(TB/T )]
ln
✓1 + e(m�1)V/2T�✏+(✓)
1 + e�(m�1)V/2T�✏+(✓)
1 + e�(m�1)V/2T�✏+(1)
1 + e(m�1)V/2T�✏+(1)
◆
V = �✓1� 1
g
◆I(W ) +W
G(V, TB , T ) =dIBS
dV= G(V/2T, TB/T )
G(V/2T, 0) = g
Include contact correction
Conductance vs. temperature
10�4 10�3 10�2 10�1 100 101 102 103
TB/T
0.0
0.2
0.4
0.6
0.8
1.0
G/(
e2 /h)
m
V/T = 1.0
345
g = 1� 1m
G(T ) ⇠ e2
h
⇣TTB
⌘2(1/g�1)
G(T ) ⇠ e2
h
h1�
�TBT
�2(1�g)i
VOLTAGE DEPENDENCE
10�1 100 101 102 103 104
V/T
0.0
0.2
0.4
0.6
0.8
1.0
G/G
1
m
TB/T = 0.5
345
10�1 100 101 102 103 104
V/T
0.0
0.2
0.4
0.6
0.8
1.0
G/G
1
m
TB/T = 2.0
345
W/O FL contact (shifted down)
With FL contact
Conclusion
A setup for transport experiment on the QSHE edge: controlled TR symmetry breaking
+ controlled SO coupling = “point contact”
Interacting edge theory is integrable model: Exact solutions for conductance with FL leads
From experiment: estimate LL parameter Confirm integrability
Confirm effects of leads
Perfect transmission and Majorana fermions in 3DTI
Detect a perfectly transmitted mode and signature of Majorana fermions
Problem with current setups
Large number of modes and finite chemical potential
Large contribution to transport from bulk states
Dirac fermion on a curved surface- TI nanowires
x
[Ando and Suzuura 2002, Ran, Vishwanath, and Lee 2008, Rosenberg, Guo, and Franz 2010, Ostrovsky, Gornyi, and Mirlin 2010, Bardarson and Moore 2013]
(x, y +W ) = e
i⇡ (x, y)
x
(x, y +W ) = e
i⇡+i2⇡�/�0 (x, y)
� = �0/2odd number of modes
T 2 = �1ST = �S
Perfectly transmitted mode
[Ando and Suzuura 2002, Ran, Vishwanath, and Lee 2008, Rosenberg, Guo, and Franz 2010, Ostrovsky, Gornyi, and Mirlin 2010, Bardarson and Moore 2013]
Dirac fermion on a curved surface- TI nanowires
“Whereas bulk transport tends to obscure the observation of surface state transport in the normal state, the supercurrent is found to be carried mainly by the surface states”
Majorana states �� = ⇡
[Veldhorst et al. Nature Mater. 2012]
[Fu and Kane, PRL 2008]
[ Alicea Rep Prog Phys (2012), Beenakker, Annu. Rev. Cond. Mat. Phys. 2013, Grosfeld and Stern 2011, Potter and Fu 2013, Weider et al 2013]
Proximity effect - Josephson Junctions
3DTI and Majorana fermions
En(�) = �q1� ⌧n sin
2 (�/2)
� = ⇡
⌧ = 1
E = 0
+
Zero energy Majorana state requires a perfectly transmitted mode
L ⌧ ⇠Short JJ [Beenakker 2006, Titov and Beenakker 2006]
Josephson junction on a 3DTI nanowire
L/W ⇠ 1
Hosts a PTM
Hosts Majorana fermions
� = �0/2
�� = ⇡
[Ando and Suzuura 2002, Ran, Vishwanath, and Lee 2008, Rosenberg, Guo, and Franz 2010, Ostrovsky, Gornyi, and Mirlin 2010, Bardarson and Moore 2013, Fu and Kane PRL 2008, Cook and Franz 2011, Cook at al 2012]
Current Phase relation - clean wire
� = �0/2� = 0
µW/~v
102030
0
En(�) = �q1� ⌧n sin
2 (�/2)
I /X
n
@En/@�
⌧n =
k2nk2n cos
2(knL) + (µ/~v)2 sin2(knL) kn =
p(µ/~v)2 � q2n
[Titov and Beenakker 2006]
[RI, Jens Bardarson, H.S. Sim, Joel Moore,2013]
µW/~v
102030
0
[RI, Jens Bardarson, H.S. Sim, Joel Moore,2013]
� = �0/2� = 0
Current Phase relation - clean wire
Parity anomaly
�
p
p̄
�† = �
�1
�2 c† = �1 � i�2c = �1 + i�2
E(�) = ± cos(�/2)
� = �0/2� = 0
µW/~v
102030
0
Parity anomaly
[RI, Jens Bardarson, H.S. Sim, Joel Moore,2013]
�† = �
�1
�2 c† = �1 � i�2c = �1 + i�2
Current Phase relation - clean wire
�
p
p̄
E(�) = ± cos(�/2)
Current Phase relation - disorder
� = �0/2 µW/~v = 20
hV (r)V (r0)i = g~v
2⇡⇠2De|r�r0|/2⇠2D [Bardarson et al. PRL 2007]
[RI, Jens Bardarson, H.S. Sim, Joel Moore,2013]
Critical current - disorder
0.0 0.5 1.00.0
0.5
1.0I c
[e�
/h̄]
g
0.20.51.02.00.2 0.8
0.0
0.5
µ⇠D/~v = 0.5
[RI, Jens Bardarson, H.S. Sim, Joel Moore,2013]
µ = 0
�/�0
0.0 0.5 1.0�/�0
0.0
0.5
1.0I c
[e�
/h̄]
µ⇠D/h̄v = 2.0
1.5
1.0
0.5
0.20
g = 2
[RI, Jens Bardarson, H.S. Sim, Joel Moore,2013]
Critical current - disorder
Conclusion
Josephson junctions on 3DTI nanowires are a great place to look for signatures of surface
physics:
PTM , Majorana fermions
At finite chemical potential, enhanced by disorder (and at finite temperature)
Not constrained by parity conservation
Thanks
• Paul Fendley (UVA)
• David Goldhaber Gordon Group (Stanford)
• Kam Moler Group (Stanford)
• Fernando De Juan (Berkeley)
E(�)
�
p
p̄
Ic(�)
1/2
e�/2~
�/�0
e�/2~(1� (T/�)2)
T ⇡ 30mK
� ⇡ 1K
Effects of temperature
Some numbers
B = 10mT
vF = 5.5⇥ 105m/s
Eg ⇡ 100� 300mK
E = ±q
(v↵p+ vFh/v↵)2 + ↵20h
2/v2↵
↵0 ⇡ 5⇥ 104m/s
Temperature in experiment ~30mK (taken from Konig et al)
Non interacting electrons
H = �i~vF
�z
@x
� i~2 {↵, @x}�y
R
(x) = (v
2F
+ ↵
2)
�1/4e
i�(x)
✓cos ✓
i sin ✓
◆
L
(x) = (v
2F
+ ↵
2)
�1/4e
�i�(x)
✓i sin ✓
cos ✓
◆
�(x) =
Zx
0
Epv
2F
+ ↵(x)2.
2✓(x) = tan�1(↵/vF )
(x�0 ) =
✓vF
v↵
◆1/2
e
i✓�x
(x+0 )
(x1) = T
x1,x0 (x0) Tx1,x0 = P
x
ei
Rx1x0
dx
(vF
�
z
+↵�
y
)
v
2F
+↵
2 [E+ i
2 (@x
↵)�y
]
Non interacting electrons
(x�0 ) =
✓vF
v↵
◆1/2
e
i✓�x
(x+0 )
(x1) = T
x1,x0 (x0) Tx1,x0 = P
x
ei
Rx1x0
dx
(vF
�
z
+↵�
y
)
v
2F
+↵
2 [E+h�
z
+ i
2 (@x
↵)�y
]
H = �i~vF
�z
@x
+ h�z
� i~2{↵, @
x
}�y
Experimental feasibility
Short junction limit L ⌧ ⇠
L/W ⇠ 1Mode suppression
Flux through wire � > �0/2
Distance from Dirac point µ⇠D/~v
B ⇡ 0.2T
Disorder strength
g ⇡ 1 n ⇠ 8⇥ 1010 cm�2
[Sacepe et al. Nature Commun 2011, Kong et al. Nano Letters 2010, Beidenkophf et al. 2011, ,Williams et al. PRL 2012, Tian et al. Sci. Rep. 2012, Veldhorst et al Nature Mater. 2012, Kim et al Nature Phys. 2012, Cho et al. Nature Commun 2013...]
W ⇠ 400nm
g
⇠D ⇠ 10nm
What’s the big deal about Majorana fermions?
�† = � �1
�2 c† = �1 � i�2c = �1 + i�2
Ground state degeneracy
Topological protection
Potential for the perfect qubit
Non Abelian statistics
