Disorder in Topological Insulators
Eugeniu Plamadeala
Mentors: Tami Pereg-Barnea, Doron Bergman, Gil Refael
Department of Physics, Mathematics and Astronomy
California Institute of Technology
May 11, 2010
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
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Background
Phases and Types of Order
Phases
There exist many phases of matter and classifying them involves
understanding their nature.
Landau-Ginzburg theory: phases classified by the symmetries
they break
New types of order: topological and quantum order. Need more
than symmetry to classify
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Disorder in Topological Insulators
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Background
Phases and Types of Order
Topological Order
Cannot be described by symmetry breaking, long-range
correlation or local order-parameters
Instead, characterized by: ground-state degeneracy, quasiparticle
statistics and edge states
If no broken symmetries, and quantum state is gapped, order is
called Topological.
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
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Background
Topologically Ordered Phases
Topological Insulator
Bulk ground state is an insulator
Edge states: decay exponentially into bulk, can move on the edge
Examples: quantum Hall effect, quantum spin Hall effect
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
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Background
Topologically Ordered Phases
The Gap Is Important
An insulator is in the most basic state
Energy gap between bands prevents low energy excitations
Gap alone does not guarantee insulating state
Counter-example: The 2D integer quantum Hall state
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
5 / 25
Background
Topologically Ordered Phases
Z2 Classification of 2D Topological Insulator
Consider the surface of a crystal
Distinctive property: gapless states on boundary, aka edge states
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
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Background
Topologically Ordered Phases
Z2 Classification of 2D Topological Insulator
Consider surface state within bulk energy gap with momentum k
in 2D BZ
Kramer’s Th: TR-invariance =⇒ k, −k are (energy) degenerate
4 special momenta, Γ1−4 , where k and −k coincide
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
7 / 25
Background
Topologically Ordered Phases
Z2 Classification of 2D Topological Insulator
Between any pair Γa,b , SO interactions lifts degeneracy
Two distinct ways to connect these states
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
8 / 25
Background
Topologically Ordered Phases
The Problem
What if the material has disorder? What happens to the ground
states?
The goal was to understand how different types of disorder affect
the phase
Which type of disorder is more important
How much disorder can a TI resist
And what happens to the TI phase when there is too much
disorder
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
9 / 25
Approach
A 2D TI Model: Graphene Sheets
The Model
Two well-known models exhibiting the TI phase
Graphene sheets + strong SO coupling
Mercury telluride (realized experimentally)
We chose to study graphene because it’s simpler, typically studied on
substrates(automatically disordered).
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
10 / 25
Approach
A 2D TI Model: Graphene Sheets
Basics: Honeycomb Lattice
Start with a tight-binding model on a 2D finite honeycomb lattice
wrapped into a cylinder.
Hu = U
X
i
H = Hu + HNN + HSO
X †
=t
ci cj + h.c.
ci† ci HNN
<ij>
HSO = iλ
X
vij ci† cj
ij
where vij = ẑ · (d̂i × d̂j )
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
11 / 25
Approach
A 2D TI Model: Graphene Sheets
Adding Disorder
For every energy term, there’s a type of disorder
On-site potential disorder, U (if U = const, chemical potential
shift)
NN link disorder, δt
NNN link disorder (SO coupling disorder), δλ
Value
disorder
chosen randomly from uniform distribution
Max ofMax
− 2 , 2
ν=3
A2
ν=2
a2
ν=1
ν=0
Eugeniu Plamadeala (Caltech)
a1
Disorder in Topological Insulators
Caltech, 2010
12 / 25
Approach
A 2D TI Model: Graphene Sheets
Extracting σxy
The Topological Insulator state is characterized by quantized Hall
conductivity
I calculate it by creating a small imbalance in the energy landscape.
Ef
V
Ef +2
Ef
V
Ef -2
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
13 / 25
Approach
A 2D TI Model: Graphene Sheets
Extracting σxy
A system is characterized by the set {N, M, t, λ, U, δλ, δt}
For each complete phase diagram( σxy vs {U, δλ}) compute 30
points
For each point {U, δλ}, compute 15 times, take average
For each computation, diagonalize Hamiltonian, extract I vs V ;
time 10-50 minutes
Total time for one phase diagram: 70-350 computing hours
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
14 / 25
Results
Phase Diagrams
Phase Diagrams
Contour plots: 1 −
1-
Σxy
Σxy,max
σxy
σxy (clean)
vs. U/λ, δλ/λ
for 30 x 33 with Λ = 0.3
1-
5
Σxy
Σxy,max
1-
for 30 x 33 with Λ = 0.4
Σxy
Σxy,max
for 30 x 33 with Λ = 0.5
5
3.5
0.5
4
0.5
3.0
2.5
2
Λ
3
2.0
∆Λ
0.3
Λ
∆Λ
Λ
3 0.2
∆Λ
4
0.3
0.2
0.5
2
1.5
0.1
1.0
1
0.1
1
0.1
0
0.0
0
1
2
3
U
4
5
Λ
Eugeniu Plamadeala (Caltech)
6
0.3
0.2
0.5
0
0
1
2
3
4
5
U
3
U
Λ
Λ
Disorder in Topological Insulators
0
1
2
4
Caltech, 2010
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6
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Results
Phase Diagrams
Phase Diagrams
1-
Σxy
Σxy,max
for 50 x 33 with Λ = 0.2
1-
Σxy
Σxy,max
for 50 x 33 with Λ = 0.3
5
7
6
0.5
4
5
3 0.2
0.3
Λ
∆Λ
Λ
∆Λ
0.5
4
0.2
3
2
0.3
2
1
1
0.1
0.1
0
0
1
2
3
4
5
6
0
7
0
1
2
3
U
U
Λ
Λ
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
4
Caltech, 2010
5
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Results
Phase Diagrams
Phase Diagrams
1-
Σxy
Σxy,max
for 50 x 43 with Λ = 0.2
1-
7
3.0
Λ
0.3
2.0
3
1.5
2
1.0
0.1
1
1
2
3
4
0.1
0.5
0.2
0
0
0.3
2.5 0.2
0.5
∆Λ
Λ
5
∆Λ
for 50 x 43 with Λ = 0.4
3.5
6
4
Σxy
Σxy,max
5
6
7
0.0
0
1
2
3
U
U
Λ
Λ
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
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5
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Results
Phase Diagrams
Phase Diagram Cuts
à
1
ò ò ò
2 Π æ ìà ì
à
ì
à
æ
ì
ì
ò
ò
ò
ò
ò
ò
ò
ò
ò
ò
ò
ò
ò
Lattice parameters
æ æ
æ
ìì à à
æ à à à
à ì ì ìàì ì
à
æ
æ
ì
Σxy
ò
ì
à
0.14
æ
æ
30 x 33 with Λ = 0.2
à
30 x 33 with Λ = 0.3
ì
30 x 33 with Λ = 0.4
æ
æ
à
ì
à
æ
æ
ò
1
2Π
æ
0.
1.6
3.2
4.8
U
6.4
8.
Λ
1
ò
2Π æ
0.14
ò
ò
ò
à ìà æ
ì
ì àì
æ
à
ì
ìà ì
æ àì
ò
ò
ò
ò
ò
ò
ò
ò
ò
Lattice parameters
æ
à
ì
à
æ
æ
30 x 33 with Λ = 0.2
à
30 x 33 with Λ = 0.3
ì
30 x 33 with Λ = 0.4
ì ì àì
æ
Σxy
à
ì
0.1
æ
à
æ
à
æ
à
æ
ò
æ
ì
1
2Π
æ
à
0.
1.2
2.4
3.6
4.8
6.
∆Λ
Eugeniu Plamadeala (Caltech)
Λ
Disorder in Topological Insulators
Caltech, 2010
18 / 25
Results
Phase Diagrams
The above shows that:
Indeed the TI phase is very robust to SO coupling disorder, even
more so to on-site potential disorder
Results are geometry independent, somewhat λ independent
Next natural step: investigate the end phases
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
19 / 25
Results
DOS
Density of States
0.12
0.12
0.10
0.10
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0.00
0.00
-3
-2
-1
0
1
2
30 x 33 ž Λ=0.3, U=0
3
-3
0.12
0.12
0.10
0.10
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0.00
-3
-2 -1
0
1
2
30 x 33 ž Λ=0.3, U=0
Eugeniu Plamadeala (Caltech)
3
0.00
-4
Disorder in Topological Insulators
-2
-1
0
1
2
30 x 33 ž Λ=0.3, U=0
-2
0
2
30 x 33 ž Λ=0.3, U=0
Caltech, 2010
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Results
DOS
Density of States
0.14
0.14
0.12
0.12
0.10
0.10
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0.00
0.00
-3
-2
-1
0
1
2
30 x 33 ž Λ=0.3, ∆Λ=0
3
-3
0.14
0.14
0.12
0.12
0.10
0.10
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0.00
-2 -1
0
1
2
30 x 33 ž Λ=0.3, ∆Λ=0
3
0.00
-3
-2 -1
0
1
2
30 x 33 ž Λ=0.3, ∆Λ=0
Eugeniu Plamadeala (Caltech)
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Disorder in Topological Insulators
-3
-2
-1 0
1
30 x 33 ž ∆Λ=0
2
Caltech, 2010
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Results
DOS
Density of States
0.04
0.03
0.02
0.01
0.00
-4
-2
0
2
4
30 x 33 ž UΛ= 5, ∆ːΛ= 5 vs. UΛ= 0, ∆ːΛ= 5
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
22 / 25
Results
DOS
The DOS suggests a likely scenario:
For moderate U disorder, band insulator
For strong U disorder, Anderson insulator
Moderate U disorder and strong δλ, dominated by δλ
Nothing can be said about strong δλ disorder
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
23 / 25
Results
DOS
Conclusions
Constructed a model of a 2D topological insulator based on
graphene
Understood its robustness with respect to disorder
Gained intuition about disordered phases
To clarify end phases, will look at localization and σxx
Eugeniu Plamadeala (Caltech)
Disorder in Topological Insulators
Caltech, 2010
24 / 25
Results
DOS
Acknowledgements
Many thanks to Tami, Doron and Gil for guidance
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Disorder in Topological Insulators
Caltech, 2010
25 / 25