Michael Fuhrer
Director, FLEET
Monash University
24th Canberra International Physics
Summer School – Topological Matter
Lecture 1
Aims of this lecture series
• Introduce electronic transport measurements
• What do we measure?
• How do we interpret the measurements?
• What are (some of) the signatures of topology in electronic transport?
• 2D topological insulators
• Quantum Hall effect and the “edge state picture”
• Quantum spin Hall effect
• Interpreting conductance within Landauer-Buttiker formalism and edge
state picture
• 2D Dirac surface state of 3D TIs
•
•
•
•
•
Graphene as an example of a 2D Dirac system
Conductivity of a disordered 2D Dirac system
Non-linear screening and electron-hole “puddling”
Weak localization and weak anti-localization
Landau levels of the 2D Dirac system
Longitudinal and Hall resistivity
Vx
Vy
B  Bz ẑ :
L
Ix
W
2D Longitudinal resistivity:
2D Hall resistivity:
Ix
E x W Vx
 xx 

jx
L Ix
 xy    yx  
Ey
jx

Vy
Ix
Note:
In 2D, [resistivity] = Ω
Relaxation-time approximation
F  eE  v  B  
dp
dt
ky
δk
Fermi surface
kx
Relaxation-time approximation:
Steady state:
Ffriction
k
 

Fnet  eE  v  B   
dp
d
  k
dt
dt
k dp

0

dt
k

 eE  v  B 

mv

 eE  v  B 
B  Bz ẑ :
e
v x   E x  cv y
m
e
v y   E y  cv x
m
j  nev :
ne 2
jx  cj y 
Ex
m
ne 2
j y  cjx 
Ey
m
Note: v is average velocity
i.e. “drift velocity”
Cyclotron frequency
eB
c 
m
Resistivity Tensor

  m
E  ρj  ρ   2
 ne 
m
 xx  2
ne 
B
 xy 
ne
 xx ,  xy
 1

   c
“Hall effect” – depends only on B and n,
can be used to measure n
 xx
c  1
1 
Resistivity in 2D does not depend on B
 xy
m
ne 2
c 
B
What do we measure at higher magnetic field?
 xx ,  xy
 xy
m
ne2
 xx
c  1
c  1 :
B
Strong deviations from semi-classical expectation
Shubnikov-de Haas oscillations, and eventually quantum Hall effect:

  0
Resistivity: ρ  
h
 2
 e
h 

0
e 2  Conductivity: σ  
 2

e
0 

 h

e 2 
h 

0 

ν is integer, quantized to approx 1 in 109
no materials parameters appear in conductivity, only fundamental constants
2D electrons in a strong magnetic field
 p  eA 2

 U    E

 2m

Choose gauge:
B  Bzˆ : A  Byxˆ
Take U = 0
  p x  eBy 2  p y 2 

   E
2m


e
ik x x
 y
“Landau gauge”
eB
c 
m
 py2 1

2
2
 mc  y  yk     y   E  y 

 2m 2

harmonic oscillator centered on
k x
yk 
eB
1

E  c  n  
2

Landau levels, U = 0
momentum space
E
7  c / 2
5c / 2
3c / 2
 c / 2
k
1

E  c  n  
2

1 dE
vx 
0
 dk x
Energy gaps:
System with filled LL is
insulator
Group velocity is zero
real space
y
+kx states
x
Real space:
  eik x x   y 
k
yk  x
eB
States of greater kx are located at larger +y
-kx states
Add an electric field
U  eE y y
 p y 2 eBy  k x 2


 eE y y    y   E  y 

2m
 2m

  eik x x   y 
 py2 1

2
2
 mc  y  yk   eE y y    y   E  y 

 2m 2

 py2 1

 2 
2eE y 
2
2




 mc  y   2 yk 
y  yk    y   E  y 

2 
2
m
2
m

c 




2
2
p 2 1



eE
eE y  


y
y
2

   y   E  y 
   eE y yk 
 mc  y   yk 
2 
2
m

2
m

 2m 2
c 
c 




harmonic oscillator centered on:
Eigenenergies:
yk 
eE y
mc2

k x eE y

eB mc2
eE y 
 k x
1


E  c  n    eE y 

2 
2

 eB 2mc 
Landau levels, constant E-field (U = -eEyy)
E
eE y 
 k
1


E  c  n    eE y  x 
2 
2

 eB 2mc 
1 dE
vx 

 dk x
kx
vx 
Ey
Bz
All electrons have same velocity, in –v X B direction
Same as classical result for electron in crossed E, B fields:
v
B = Bzz
jx   nev x
eB
n
h
e2
jx  E y
h
E = Eyy
degeneracy of Landau level
Quantised Hall effect!
Topological insulator: carries
current perpendicular to E
Harmonic confining potential
1
U  m02 y 2
2
 p y 2 eBy  k x 2 1

2 2

 m0 y    y   E  y 

2m
2
 2m

  eik x x   y 
Again, complete the square to recast as harmonic oscillator:
2
2
 p 2 1  2 2



 c 
1
y
2
2
0
c

 m 2 yk  mc 0  y   2  yk     y   E  y 
c 0
2
 2m 2
  0   

1
1 02c2 2

E   n  c 0  m 2 yk
2
2
c 0

c20  c2  02
States centred on:
c2
c2 k x
yk  2
2
0
0 eB
Landau levels, parabolic U
E
1
1 02c2 2

E   n  c 0  n 2 yk
2
2 c 0

n=3
n=2
n=1
kx
y
1 dE k x  02 
 2 
vx 

 dk x
m  c 0 
+kx states have vx > 0
x
-kx states have vx < 0
Thought experiment: hard-walled confining potential
U
E
EF
y
kx
y
E
x
EF
kx
the “edge state picture” of quantum Hall effect

Ey

Ey
E
7
 c
2
5
 c
2
EF
3
 c
2
•
Confining potential pushes up Landau levels
at edges of sample
•
Fermi energy crosses Landau levels
•
Edges are metallic – charge can move in/out
•
1D edge states carry current in E x B direction
1
 c
2
edge
edge
E
B
y
E
EF
EF
kx
kx
What is the conductance of the 1D edge state?
E
EF
kx
1D band (spin non-degenerate)
 2k 2
E
2m *
k  n
E
 dn 
 
dn  dk 
1
2
D E  



dE  dE  v hv


 dk 
EF
True for any E(k)!
 N
...
L
 6  4  2
2 4 6
,
,
,0,
,
,
...
L
L
L
L L L
N
L
k
Conductance Quantization in 1D
lead
1D Channel
lead
E
Apply voltage V,
Calculate current I:
I  nev
ΔE=eV
I
1  dn 

E ev
2  dE 
1 2
I  
2  hvF
k
Left-moving Right-moving
states
states

eV ev F

e2
I V
h
Including spin degeneracy:
quantum of conductance: G0 = 2e2/h = (12.9 kΩ)-1
Conductance Quantization
B.J. van Wees et al., (1988)
Conductance and Hall Effect in Quantum Hall regime
ν = # of edge modes = # of full Landau levels
potential = V
I
ν=2
0
V
I
potential = 0
V
Current = ?
e2
I x  V
h
E
ΔE=eV
V y  V
 xy  
k
Left-moving
states
Right-moving
states
ν modes carry
(e2/h)V each
Vy
Ix

h
e 2
A multi-terminal device
V
e2
I  V
h
V
V
0
Ix
0
0
ΔVx=0
ΔVy=V
Voltage probe has I = 0
potential is same for
incoming/outgoing modes
Potential same here and here
Vy
Vx
 xx 
0
I
V y
h
 xy 
 2
I
e
Is the edge-state picture “real”?
Scanning capacitance microscopy of AlGaAs/GaAs quantum well
in quantum Hall regime
New J. Phys. 14, 083015 (2012)
Current does flow along 1D modes
highly confined to sample edges in
QHE and QSHE
Scanning SQUID image of current in an
HgTe quantum well in the quantum spin Hall effect regime
Nature Materials 12, 787 (2013)
“Bulk-edge correspondence”
Gapless states must exist at interface between different topological phases
 1
 0
(vacuum)
# of chiral modes  
Quantised Hall conductivity is both bulk and edge property!

 0
σ 2
 e

 h
e2 
 
h

0 

e.g. light shining on bulk of sample shows a quantised Faraday
rotation – induced current at angle to polarisation
e.g. transport current is carried along edges of sample
Landau levels, constant E-field (U = -eEyy)
E
eE y 
 k
1


E  c  n    eE y  x 
2 
2

 eB 2mc 
1 dE E y
vx 

 dk x Bz
kx
Same as classical result for electron in crossed E, B fields:
j x  nevx
eB
n
h
e2
jx  E y
h
degeneracy of Landau level
Quantised Hall effect!
Quantum spin Hall effect
The trio of quantum Hall effects
Science 340, 153 (2013)
Quantum Hall and quantum anomalous Hall effects: edge modes propagate in a single direction, so no possibility
of scattering if bulk is sufficiently wide and insulating
Quantum spin Hall effect: edge modes propagate in both directions (with opposite spin).
Scattering is possible with flip of spin.
1D “Contact Resistance”
Drops in average
electrochemical potential occur
at contacts
Dissipation (relaxation) occurs
in contacts
S. Datta, 1995
Landauer Formula
2e 2
G
h
T
(Ti = transmission probability of ith mode)
i
i
1-T
1
0
E
T
E
E=eV
k
k
Resistance of a Barrier
h 1
h
h 1 T
R 2  2  2
2e T 2e
2e T
Inevitable quantized
“contact” resistance
h
Rcontact  2
2e
Resistance of the
barrier
h 1 T
h R
Rbarrier  2
 2
2e T
2e T
Two Barriers - Incoherent Addition
“Incoherent” = ignore interference terms
(classical particles)
T  T1T2  T1T2 R 1R 2  T1T2 R 1 R 2  ...
2
T1T2

1  R 1R 2
h R
Rbarrier  2
2e T
R  R1   R 2 

    
T  T1   T2 
For incoherent transmission Rbarrier adds like Ohm’s law
2
Mean free path
Define a length le of 1D conductor such that T = ½ (and R = ½):
h R
h
Rbarrier le   2  2
2e T 2 e
A conductor of length L has (L/le such sections):
h L
Rbarrier L   2
2e l e
and
h
h L
Rtotal L   Rc  Rbarrier ( L)  2  2
2e
2e l e
1D resistivity:
1D
d
h

Rtotal ( L)  2
dL
2e l e
Note: le called “mean free path”, “elastic length”, “backscattering length”, etc.
Mean free path in QSHE: experiments
HgTe quantum well in the QSHE regime:
le ~ 7 μm
Nature Materials 12, 787 (2013)
QSHE in WTe2:
le ~ 150 nm
Nature Physics 13, 677–682 (2017)
Questions?