Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a...
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Topological Phases under Strong Magnetic Fields Mark O. Goerbig ITAP, Turunc ¸, July 2013
Transcript of Topological Phases under Strong Magnetic Fields · Topological Insulators generic form of a...
Topological Phases under Strong Magnetic Fields
Mark O. Goerbig
ITAP, Turunc, July 2013
Historical Introduction
What is the common point between
• graphene,
• quantum Hall effects
• and topological insulators?
... and what is it?
The 1920ies: Band Theory
• quantum treatment of (non-interacting) electrons in aperiodic lattice
• bands = energy of the electrons as a function of aquasi-momentum
Chrystal Electrons and Bloch’s Theorem
• discrete translations T = exp(ip ·Rj)represent a symmetry in a Bravais lattice(described by lattice vectors Rj)
• the operator p (generator ofdiscrete translations) plays therole of a momentum(quasi-momentum or latticemoment)
R j
Chrystal Electrons and Bloch’s Theorem
• discrete translations T = exp(ip ·Rj)represent a symmetry in a Bravais lattice(described by lattice vectors Rj)
• the eigenvalues of p are goodquantum numbers: energy bandsǫl(p)
• Bloch functions:
ψ(r) =∑
p
eip·r/~up(r)
Bravais and Arbitrary Lattices
arbitrary lattice=
Bravais lattice+
basis (of N “atoms”)
M. C. Escher (decomposed)
triangular lattice + complicated basis
There are as many electronic bands as atoms per unit cell
Band Structure and Conduction Properties
I II I II
gap
metal (2D)
energy
Fermi
level
momentum
electron metal hole metal
insulator (2D)Fermi
level
semimetal (2D) Fermi
level
Fermilevel
density
of states
energy
1950-70: Many-Body Theory
• Physical System described by an 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
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)
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.
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
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
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
Graphene – First 2D Crystal
• honeycomb lattice =two triangular (Barvais) lattices
AB
B
B
e3
e1
2e
band structure
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
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
Outline of the Classes
Mon 1: Introduction and Landau quantisation
Mon 2: Landau-level degeneracy and disorder/confinementpotential
Tue 1: Issues of the IQHE
Tue 2: Towards the FQHE, Laughlin’s wave function and itsproperties
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
1. Introduction To the Integer Quantum Hall
Effect and Materials
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
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)
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)
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)
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)
Nobel Prize in Physics 2010 : Graphene
Kostya Novoselov Andre Geim
"for groundbreaking experiments regarding the two-dimensionalmaterial graphene"
What is Graphene?
2s
2p 2p 2p 2px y z zsp spsp
Hybridation sp 2
120o
graphene = 2D carbon crystal(honeycomb)
sp hybridisation2
Graphene and its Family
2D
3D 1D 0D
From Graphite to Graphene
(strong) covalent bondsin the planes
(weak) van der Waalsbonds between the planes
How to Make Graphene: Recipe (1)
put thin graphite chip on scotch−tape
How to Make Graphene: Recipe (2)
~10 : fold scotch−tape on graphite chip and undo :
How to Make Graphene: Recipe (3)
2glue (dirty) scotch−tape on substrate (SiO )
How to Make Graphene: Recipe (4)
lift carefully scotch−tape from substrate
How to Make Graphene: Recipe (5)
place substrate under optical microscope
orientationmarks
thick graphite
graphene ?
less thickgraphite
How to Make Graphene: Recipe (6)
zoom in region where there could be graphene
Electronic Mesurement of Graphene
SiO
Si dopé
V
2
g
Novoselov et al., Science 306,p. 666 (2004)
2. Landau Quantisation and Integer
Quantum Hall Effect
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
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
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)]
IQHE – One-Particle Localisation
n
ε
(n+1)
ν
NL
(a)
density of states
RxyxxR
B=n
h/e n2
FE
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
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
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
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
Fractional Quantum Hall Effect beyond Laughlin
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
FQHE series: ν = p/(2sp+ 1) = 1/3, 2/5, 3/7, 4/9, ...
Jain’s Wavefunctions (1989)
Idea: “reinterpretation” of Laughlin’s wavefunction
ψLs (zj) =
∏
i<j(zi − zj)2sχp=1(zj)
∏
i<j(zi − zj)2s: “vortex” factor (2s flux quanta per vortex)
χp=1(zj) =∏
i<j(zi − zj):wavefunction at ν∗ = 1
Jain’s Wavefunctions (1989)
Idea: “reinterpretation” of Laughlin’s wavefunction
ψLs (zj) =
∏
i<j(zi − zj)2sχp=1(zj)
∏
i<j(zi − zj)2s: “vortex” factor (2s flux quanta per vortex)
χp=1(zj) =∏
i<j(zi − zj):wavefunction at ν∗ = 1
Generalisation to integer ν∗ = p
ψJs,p(zj) = PLLL
∏
i<j(zi − zj)2sχp(zj, zj)
χp(zj, zj): wavefunction for p completely filled levelsPLLL: projector on lowest LL (→ analyticity)
Physical Picture: Composite Fermions
CF = electron+”vortex” (carrying 2s flux quanta)with renormalised field coupling eB → (eB)∗
ν = 1/3
pseudo−vortex
electronic filling 1/3theory
CF
1 filled CF level
electron
"free" flux quantum
(with 2 flux quanta)
composite fermion (CF)
Physical Picture: Composite Fermions
CF = electron+”vortex” (carrying 2s flux quanta)with renormalised field coupling eB → (eB)∗
ν = 1/3
ν = 2/5
pseudo−vortex
theory
CF
2 filled CF levels
electron
"free" flux quantum
(with 2 flux quanta)
composite fermion (CF)
electronic filling 1/3 1 filled CF level
At ν = p/(2ps+ 1) ↔ ν∗ = nel/n∗B = p, with n∗
B = (eB)∗/h:FQHE of electrons = IQHE of CFs
Generalisation: FQHE at half-filling
• 1987: Obervation of a FQHE at ν = 5/2, 7/2 (evendenominator)
• 1991: Proposal of a Pfaffian wave function (Moore & Read;Greiter, Wilzcek & Wen)
ψMR(zj) = Pf
(
1
zi − zj
)
∏
i<j
(zi − zj)2
⇒ quasiparticle charge e∗ = e/4 with non-Abelian statistics
• Further generalisations to ν = K/(K + 2): Read & Rezayi
Multicomponent SystemsLa
ndau
leve
ls
|+>
|−>d
ν = 1/2
ν = 1/2 ν = ν + ν = 1
+
+ −− T
: A sublattice : B sublattice
τ
τ
2
3
1
2
e 1e
e
spin + isospin : SU(4)
A physical spin: SU(2)
two−fold valley degeneracy
B bilayer: SU(2) isospin
SU(2) isospin
C graphene (2D graphite)
(doubling of LLs)
exciton
