Introduction to the Quantum Hall Effect and Topological...
Transcript of Introduction to the Quantum Hall Effect and Topological...
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Introduction to the Quantum Hall Effect andTopological Phases
Mark O. Goerbig
Ecole du GDR “Physique Mesoscopique, Cargese, November 2016
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Historical Introduction
What is the common point between
• graphene,
• quantum Hall effects
• and topological insulators?
... and what is it?
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The 1920ies: Band Theory
• quantum treatment of (non-interacting) electrons in aperiodic lattice
• bands = energy of the electrons as a function of aquasi-momentum
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1950-70: Many-Body Theory
• Physical System described by a local order parameter(a) ∆k = 〈ψ†
−k,↑ψ†k,↓〉 (superconductivity)
(b) Mµ(r) =∑
τ,τ ′〈ψ†τ (r)σ
µτ,τ ′ψτ ′(r)〉 (ferromagnetism)
• Ginzburg-Landau theory of second-order phase transitions(1957)
∆ = 0(disordered)
↔ ∆ 6= 0(ordered)
• symmetry breaking(a) broken (gauge) symmetry U(1)(b) broken (rotation) symmetry O(3)
• emergence of (collective) Goldstone modes(a) superfluid mode, with ω ∝ |k|(b) spin waves, with ω ∝ |k|2
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The Revolution(s) of the 1980ies
3 essential discoveries:
• integer quantum Hall effect (1980, v. Klitzing, Dorda,Pepper)
• fractional quantum Hall effect (1982, Tsui, Störmer,Gossard)
• high-temperature superconductivity (1986, Bednorz, Müller)
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Integer Quantum Hall Effect (I)
8 12 160 4Magnetic Field B (T)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ρ xy
(h/e
)2
0
0.5
1.0
1.5
2.0
ρΩ
xx(k
)
2/3 3/5
5/9
6/11
7/15
2/53/74/9
5/11
6/13
7/13
8/15
1 2/3 2/
5/7
4/5
3 4/
Vx
VyIx
4/7
5/34/3
8/57/5
123456
Magnetic Field B[T]
[mesurement by J. Smet et al., MPI-Stuttgart]
QHE = plateau in Hall res. & vanishing long. res.
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Integer Quantum Hall Effect (II)
Quantised Hall resistance at low temperatures
RH =h
e21
n
h/e2: universal constantn: quantum number (topological invariant)
• result independent of geometric and microscopic details
• quantisation of high precision (> 109)
⇒ resistance standard: RK−90 = 25 812, 807Ω
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Fractional Quantum Hall Effect
partially filled Landau level → Coulomb interactions relevant
1983: Laughlin’s N -particle wave function
• no (local) order parameter associated with symmetrybreaking
• no Goldstone modes• quasi-particles with fractional charges and statistics
1990ies : description in terms of topological (Chern-Simons)field theories
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The Physics of the New Millenium
• simulation of condensed-matter models with optical lattices(cold atoms)
• 2004 : physics of graphene (2D graphite)
• 2005-07 : topological insulators
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Graphene – First 2D Crystal
• honeycomb lattice =two triangular (Barvais) lattices
AB
B
B
e3
e1
2e
band structure
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Band Structure and Conduction Properties (Bis)
I II I II
gap
metal (2D)
energy
Fermi
level
momentum
electron metal hole metal
Fermi
levelinsulator (2D)
semimetal (2D) Fermi
level
graphene (undoped)Fermi
level
Fermilevel
energy
density
of states
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Topological Insulators
generic form of a two-band Hamiltonian:
H = ǫ0(q)1+∑
j=x,y,z
ǫj(q)σj
• Haldane (1988): anomalous quantum Hall effect → quantumspin Hall effect (QSHE)
• Kane and Mele (2005): graphene with spin-orbit coupling• Bernevig, Hughes, Zhang (2006): prediction of a QSHE in
HgTe/CdTe quantum wells• König et al. (2007): experimental verification of the QSHE
⇒ 3D topological insulators (mostly based on bismuth):surface states ∼ ultra-relativistic massless electrons
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Outline of the Classes
Mon : Introduction and Landau quantisation
Tue : Issues of the IQHE; Introduction to the Berry phase
Thu : Simple models for topological insulators
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Further Reading
• D. Yoshioka, The Quantum Hall Effect, Springer, Berlin (2002).
• S. M. Girvin, The Quantum Hall Effect: Novel Excitations and BrokenSymmetries, Les Houches Summer School 1998http://arxiv.org/abs/cond-mat/9907002
• G. Murthy and R. Shankar, Rev. Mod. Phys. 75, 1101(2003).http://arxiv.org/abs/cond-mat/0205326
• M. O. Goerbig, Quantum Hall Effectshttp://arxiv.org/abs/0909.1998
• B. A. Bernevig, Topological Insulators and TopologicalSuperconductors, Princeton UP (2013).
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1. Introduction To the Integer Quantum Hall
Effect and Materials
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Classical Hall Effect (1879)
B
I
longitudinal Hallresistance resistance
C1
C4
C2 C3
C5C6
2D electron gas_ _ _ _ _ _
++ + + ++
Quantum Hall system :2D electrons in a B-field
Hal
l res
ista
nce
magnetic field B
RH(b)
Hall resistance:
RH = B/enel
Drude model (classical stationary equation):
dp
dt= −e
(
E+p
m×B
)
− p
τ= 0
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Resistivity and Conductivity
σ0Ex = −enel
mpx −
enel
mpy(ωCτ)
σ0Ey =enel
mpx(ωCτ)−
enel
mpy
Ohm’s law : E = ρj with current density j = −enelv = −enelp/m
⇒ resistivity/conductivity tensor
ρ =1
σ0
(
1 ωCτ−ωCτ 1
)
σ = ρ−1 =σ0
1 + (ωCτ)2
(
1 −ωCτωCτ 1
)
Link with mobility µ = eτ/m: ωCτ = µB
Hall resistivity : ρH = ωCτ/σ0 = B/enel
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Shubnikov-de Haas Effect (1930)H
all r
esis
tanc
e
magnetic field B
long
itudi
nal r
esis
tanc
e
Bc
(a)
Den
sity
of s
tate
s
EnergyEF
hωC
(b)
oscillations in longitudinal resistance→ Einstein relations σ0 ∝ ∂nel/∂µ ∝ ρ(ǫF )→ Landau quantisation (into levels ǫn)
σ0 ∝ ρ(ǫF ) ∝∑
n
f(ǫF − ǫn)
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Quantum Hall Effect (QHE)
8 12 160 4Magnetic Field B (T)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ρ xy
(h/e
)2
0
0.5
1.0
1.5
2.0ρ
Ωxx
(k)
2/3 3/5
5/9
6/11
7/15
2/53/74/9
5/11
6/13
7/13
8/15
1 2/3 2/
5/7
4/5
3 4/
Vx
VyIx
4/7
5/34/3
8/57/5
123456
Magnetic Field B[T]
QHE = plateau in RH & RL = 0
1980 : Integer quantum Hall effect (IQHE)1982 : Fractional quantum Hall effect (FQHE)
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Metal-Oxide Field-Effect Transistor (MOSFET)
conductionband
acceptorlevels
valenceband
conductionband
acceptorlevels
valenceband
conductionband
acceptorlevels
valenceband
E
z
FE
z
F
E
z
F
(a)
(b) (c)
VVG
G
metal oxide(insulator)
semiconductor
metal oxide(insulator)
semiconductor metal oxide(insulator)
II
I
VG
z
z
E
E
E
1
0
metaloxide
semiconductor
2D electrons
usually silicon-based materials (Si/SiO2 interfaces)
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GaAs/AlGaAs Heterostructure
dopants
AlGaAs
z
EF
GaAs
dopants
AlGaAs
z
EF
GaAs(a) (b)
2D electrons
Impurity levels farther away from 2DEG (as compared toSi/SiO2)
⇒ enhanced mobility (FQHE)
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Electronic Mesurement of Graphene
SiO
Si dopé
V
2
g
Novoselov et al., Science 306,p. 666 (2004)
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2. Landau Quantisation and Integer
Quantum Hall Effect
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Landau quantisation (reminder of first class)
• 2D electrons in continuum limit (|q|a≪ 1)
H(px, py) p = ~q
• Peierls substitution : p → Π = p+ eA(r) for a/lB ≪ 1
H(px, py) → HB(Πx,Πy)
• Quantum mechanics
[x, px] = [y, py] = i~ → [Πx,Πy] = −i~2
l2B
⇒ Ladder operators [a, a†] = 1
Πx =~√2lB
(
a† + a)
Πy =~√2ilB
(
a† − a)
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Simple Landau levels
Schrödinger fermions :
HB|n〉 = ǫn|n〉, a†a|n〉 = n|n〉, ǫn = ~ωC(n+ 1/2)
Dirac fermions (graphene) :
HB =√2~vFlB
(
0 aa† 0
)
, ψn =
(
unvn
)
, ǫλ=±,n = λ~vFlB
√2n
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Simple Landau levels
Schrödinger fermions :
HB|n〉 = ǫn|n〉, a†a|n〉 = n|n〉, ǫn = ~ωC(n+ 1/2)
Dirac fermions (graphene) :
HB =√2~vFlB
(
0 aa† 0
)
, ψn =
(
unvn
)
, ǫλ=±,n = λ~vFlB
√2n
Eigenstates :
ψn=0 =
(
0|n = 0〉
)
ψλ,n= =1√2
(
|n− 1〉λ|n〉
)
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Infrared Transmission Spectroscopy
10 20 30 40 50 60 70 80
0.96
0.98
1.00
B
E
2L3L
2L
3L
0L
1L
Be2cE1 ~1L
1E
1E
A
B
C
D
B
E
2L3L
2L
3L
0L
1L
Be2cE1 ~1L
1E
1E
A
B
C
D
(D)(C)
(B)
Rel
ativ
e tra
nsm
issi
on
Energy (meV)
(A)
0.4 T1.9 K
0.0 0.5 1.0 1.5 2.00
10
20
30
40
50
60
70
80 )(32 DLL )(23 DLL
)(12 CLL )(21 CLL
)(01 BLL )(10 BLL
)(21 ALL
Tran
sitio
n en
ergy
(meV
)
sqrt(B)
10 20 30 40 50 60 70 80 900.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
Rel
ativ
e tra
nsm
issi
on
Energy (meV)
1 T
0.4T
2T4T
10 20 30 40 50 60 70 80 90
0.99
1.00
0.7T
0.2T
0.3T
0.5T
Grenoble high−field group: Sadowski et al., PRL 97, 266405 (2007)
transition C
transition B
rela
tive
tran
smis
sion
rela
tive
tran
smis
sion
Energy [meV]
Energy [meV]
Tra
nsm
issi
on e
nerg
y [m
eV]
Sqrt[B]
selectionrules :
λ, n→ λ′, n±1
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Edge States
ymaxn+1
ν = n ν =
n−1
yymaxymaxn n−1
n+1
n
n−1
(a)
(b)
y
xν = n+1
µ
LLs bended upwards atthe edges (confinementpotential)
chiral edge states⇒ only forward scattering
ν= n+1 ν= n ν= n−1
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Four-terminal Resistance Measurement
I I
R ~
56
2 3
41
R ~ µ − µ = µ − µ
3µ − µ = 02
5
L
H
µ = µµ = µ2 LL3
µ = µ = µ6 5 R
3 R L
: hot spots [Klass et al, Z. Phys. B:Cond. Matt. 82, 351 (1991)]
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IQHE – One-Particle Localisation
n
ε
(n+1)
ν
NL
(a)
density of states
RxyxxR
B=n
h/e n2
FE
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IQHE – One-Particle Localisation
n
ε
n
ε(b)
(n+1)
ν
NL
(a)
density of statesdensity of states
RxyxxR
B
EF
RxyxxR
B=n
h/e n2
FE
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IQHE – One-Particle Localisation
n
ε
n
ε
n
ε(b) (c)
(n+1)
ν
NL
(a)
density of states density of states density of states
extended states
localised states
RxyxxR
B
EF
Rxy
B
xx
EF
R
h/e (n+1)
h/e n2
2
RxyxxR
B=n
h/e n2
FE
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IQHE in Graphene Novoselov et al., Nature 438, 197 (2005)
Zhang et al., Nature 438, 201 (2005)
V =15V
Density of states
B=9T
T=30mK
T=1.6K
∼ ν
∼ 1/ν
Graphene IQHE:
R = h/e
at = 2(2n+1)
at = 2n
ν
ν
H ν2
(no Zeeman)
Usual IQHE:
g
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Percolation Model – STS Measurement
2DEG on n-InSb surface Hashimoto et al., PRL 101, 256802 (2008)
(a)-(g) dI/dV for different values of sample potentials (lower spinbranch of LL n = 0)
(i) calculated LDOS for a given disorder potential in LL n = 0
(j) dI/dV in upper spin branch of LL n = 0
![Page 35: Introduction to the Quantum Hall Effect and Topological Phasesmeso2016.sciencesconf.org/data/pages/Lecture_Mark_Cargese_1116.pdfTopological Insulators generic form of a two-band Hamiltonian:](https://reader034.fdocuments.us/reader034/viewer/2022042622/5f8c72d0a1eaae26db1a9fb0/html5/thumbnails/35.jpg)
Towards topological band theory
• Berry-ology :
– Berry connexion : ~An(k) = iψ†n(k)∇kψn(k)
– Berry curvature : ~Bn(k) = ∇k × ~An(k)
– Berry phase : γn =∮
dk · ~An(k)
→ Chern number : Cn = 1
2π
∮
BZdk · ~An(k)
![Page 36: Introduction to the Quantum Hall Effect and Topological Phasesmeso2016.sciencesconf.org/data/pages/Lecture_Mark_Cargese_1116.pdfTopological Insulators generic form of a two-band Hamiltonian:](https://reader034.fdocuments.us/reader034/viewer/2022042622/5f8c72d0a1eaae26db1a9fb0/html5/thumbnails/36.jpg)
Towards topological band theory
• Pseudo-Chern number of a (massive) Dirac point :
C =1
2ξsgn(m)
![Page 37: Introduction to the Quantum Hall Effect and Topological Phasesmeso2016.sciencesconf.org/data/pages/Lecture_Mark_Cargese_1116.pdfTopological Insulators generic form of a two-band Hamiltonian:](https://reader034.fdocuments.us/reader034/viewer/2022042622/5f8c72d0a1eaae26db1a9fb0/html5/thumbnails/37.jpg)
Towards topological band theory
• Pseudo-Chern number of a (massive) Dirac point :
C =1
2ξsgn(m)
• Remarks :– half integer (not a true toplogical invariant) due to
non-compact support k ∈ R2
– each Dirac point contributes ±1/2 to total Chern number→ Dirac points (on a lattice) come in pairs to get integer
Chern numbers ! (fermion doubling)– Haldane model (CK = CK′) : C = CK + CK′ = ±1– Kane-Mele model : C↑ = −C↓
![Page 38: Introduction to the Quantum Hall Effect and Topological Phasesmeso2016.sciencesconf.org/data/pages/Lecture_Mark_Cargese_1116.pdfTopological Insulators generic form of a two-band Hamiltonian:](https://reader034.fdocuments.us/reader034/viewer/2022042622/5f8c72d0a1eaae26db1a9fb0/html5/thumbnails/38.jpg)
Non-local transport in QSHE (I)
CdTe/HgTe quantum wells [Roth et al., Science 2009]
I1
2 3
4
56
V
I1
2 3
4
56
V
-0.5 0.0 0.5 1.0 1.5 2.00
5
10
15
20
25
30
35
40
R (
kΩ)
V* (V)
(1 x 0.5) µm2
(2 x 1) µm2
R14,14=3/2 h/e2
R14,23=1/2 h/e2
![Page 39: Introduction to the Quantum Hall Effect and Topological Phasesmeso2016.sciencesconf.org/data/pages/Lecture_Mark_Cargese_1116.pdfTopological Insulators generic form of a two-band Hamiltonian:](https://reader034.fdocuments.us/reader034/viewer/2022042622/5f8c72d0a1eaae26db1a9fb0/html5/thumbnails/39.jpg)
Non-local transport in QSHE l’EHQS (II)
CdTe/HgTe quantum wells [Roth et al., Science 2009]
Fig. 4
0.0 0.5 1.0 1.5 2.00
5
10
15
20
25
R (
kΩ)
V* (V)
I: 1-4
V: 2-3
1
3
2
4
R14,23=1/4 h/e2
R14,14=3/4 h/e2
![Page 40: Introduction to the Quantum Hall Effect and Topological Phasesmeso2016.sciencesconf.org/data/pages/Lecture_Mark_Cargese_1116.pdfTopological Insulators generic form of a two-band Hamiltonian:](https://reader034.fdocuments.us/reader034/viewer/2022042622/5f8c72d0a1eaae26db1a9fb0/html5/thumbnails/40.jpg)
3D topological insulators (I)
Frist generation based on Bi1−xSbx alloys [Hasan and Kane, RMP 2010]
→ band inversion above critical Sb concentration xc ≃ 0.04
![Page 41: Introduction to the Quantum Hall Effect and Topological Phasesmeso2016.sciencesconf.org/data/pages/Lecture_Mark_Cargese_1116.pdfTopological Insulators generic form of a two-band Hamiltonian:](https://reader034.fdocuments.us/reader034/viewer/2022042622/5f8c72d0a1eaae26db1a9fb0/html5/thumbnails/41.jpg)
3D topological insulators (II)
• closing of (∼ mass) gap at band inversion
⇒ Dirac fermions at surface of 3D toplogical insulator (∼ edgestates in 2D) :
Hsurface = vp · σ
p : momentum in surfaceσ : characterises true spin
⇒ single Dirac point (contrary to graphene with 4)
![Page 42: Introduction to the Quantum Hall Effect and Topological Phasesmeso2016.sciencesconf.org/data/pages/Lecture_Mark_Cargese_1116.pdfTopological Insulators generic form of a two-band Hamiltonian:](https://reader034.fdocuments.us/reader034/viewer/2022042622/5f8c72d0a1eaae26db1a9fb0/html5/thumbnails/42.jpg)
3D topological insulators (III)
2nd generation : Bi2Se3, Bi2Te2, Sb2Te3 [Zhang et al., 2009]
(ab initio calculations)
![Page 43: Introduction to the Quantum Hall Effect and Topological Phasesmeso2016.sciencesconf.org/data/pages/Lecture_Mark_Cargese_1116.pdfTopological Insulators generic form of a two-band Hamiltonian:](https://reader034.fdocuments.us/reader034/viewer/2022042622/5f8c72d0a1eaae26db1a9fb0/html5/thumbnails/43.jpg)
3D topological insulators (IV)
ARPES measurements of de Dirac fermions at a Bi2Se3 surface[Hsieh et al., 2009]
→ change of Fermi level by chemical doping (absorption of NO2)