Topological Phases under Strong Magnetic Fields

Topological Phases under Strong Magnetic Fields Mark O. Goerbig ITAP, Turunç, 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 a periodic lattice bands = energy of the electrons as a function of a quasi-momentum

Chrystal Electrons and Bloch s Theorem discrete translations T = exp(iˆp R j ) represent a symmetry in a Bravais lattice (described by lattice vectors R j ) the operator ˆp (generator of discrete translations) plays the role of a momentum (quasi-momentum or lattice moment) Rj

Chrystal Electrons and Bloch s Theorem discrete translations T = exp(iˆp R j ) represent a symmetry in a Bravais lattice (described by lattice vectors R j ) the eigenvalues of ˆp are good quantum numbers: energy bands ǫ l (p) Bloch functions: ψ(r) = p e ip r/ u p (r)

Bravais and Arbitrary Lattices M. C. Escher (decomposed) arbitrary lattice = Bravais lattice + basis (of N atoms ) triangular lattice + complicated basis There are as many electronic bands as atoms per unit cell

Band Structure and Conduction Properties energy energy Fermi metal (2D) level momentum Fermi level density of states electron metal hole metal insulator (2D) Fermi level gap semimetal (2D) I II I II Fermi level

1950-70: Many-Body Theory Physical System described by an order parameter (a) k = ψ k, ψ k, (superconductivity) (b) M µ (r) = τ,τ ψ τ(r)σ µ τ,τ ψ τ (r) (ferromagnetism) Ginzburg-Landau theory of second-order phase transitions (1957) = 0 (disordered) 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) 2.0 I x V y 3.0 V x 2.5 1.5 2.0 ρ xx (k Ω ) 1.0 6 5 4 3 2 3/ 2 1 3/ 4 2/3 3/5 4/7 1/ 2 3/7 4/9 2/5 1.5 1.0 ρ xy (h/e ) 2 5/9 5/11 0.5 5/3 4/3 8/5 7/5 0 0 4 6/11 6/13 5/7 8/15 4/5 7/13 7/15 8 12 16 Magnetic Field B (T) Magnetic Field B[T] 0.5 0.0 [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 R H = h e 2 1 n h/e 2 : universal constant n: quantum number (topological invariant) result independent of geometric and microscopic details quantisation of high precision (> 10 9 ) resistance standard: R K 90 = 25812,807Ω

Fractional Quantum Hall Effect partially filled Landau level Coulomb interactions relevant 1983: Laughlin s N-particle wave function no (local) order parameter associated with symmetry breaking 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 B e 2 A e 3 e 1 B B band structure

Band Structure and Conduction Properties (Bis) energy energy Fermi metal (2D) level momentum Fermi level density of states electron metal hole metal insulator (2D) Fermi level gap semimetal (2D) I II I II Fermi level graphene (undoped) Fermi level

Topological Insulators generic form of a two-band Hamiltonian: H = ǫ 0 (q)½+ ǫ j (q)σ j j=x,y,z Haldane (1988): anomalous quantum Hall effect quantum spin 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/confinement potential Tue 1: Issues of the IQHE Tue 2: Towards the FQHE, Laughlin s wave function and its properties

Further Reading D. Yoshioka, The Quantum Hall Effect, Springer, Berlin (2002). S. M. Girvin, The Quantum Hall Effect: Novel Excitations and Broken Symmetries, Les Houches Summer School 1998 http://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 Effects http://arxiv.org/abs/0909.1998

1. Introduction To the Integer Quantum Hall Effect and Materials

Classical Hall Effect (1879) (b) R H I C1 C6 C2 C5 B C3 + + + + + + 2D electron gas C4 Hall resistance magnetic field B longitudinal resistance Hall resistance Quantum Hall system : 2D electrons in a B-field Drude model (classical stationary equation): Hall resistance: dp dt = e ( E+ p m B ) p τ = 0 R H = B/en el

Shubnikov-de Haas Effect (1930) (a) (b) Hall resistance longitudinal resistance Density of states hω C B c magnetic field B oscillations in longitudinal resistance Einstein relations σ 0 n el / µ ρ(ǫ F ) Landau quantisation (into levels ǫ n ) E F Energy σ 0 ρ(ǫ F ) n f(ǫ F ǫ n )

Quantum Hall Effect (QHE) 2.0 I x V y 3.0 V x 2.5 1.5 2.0 ρ xx (k Ω ) 1.0 6 5 4 3 2 3/ 2 1 3/ 4 2/3 3/5 4/7 5/9 1/ 2 3/7 4/9 5/11 2/5 1.5 1.0 ρ xy (h/e ) 2 0.5 5/3 4/3 8/5 7/5 0 0 4 4/5 5/7 6/11 8/15 7/13 6/13 7/15 8 12 16 Magnetic Field B (T) Magnetic Field B[T] 0.5 0.0 QHE = plateau in R H & R L = 0 1980 : Integer quantum Hall effect (IQHE) 1982 : Fractional quantum Hall effect (FQHE)

Metal-Oxide Field-Effect Transistor (MOSFET) (a) EF metal oxide (insulator) semiconductor conduction band acceptor levels valence band I z metal oxide semiconductor V G II E z E1 (b) EF metal VG oxide (insulator) (c) semiconductor metal oxide (insulator) conduction band acceptor levels valence band EF V G E0 2D electrons z conduction band acceptor levels valence band z z usually silicon-based materials (Si/SiO 2 interfaces)

GaAs/AlGaAs Heterostructure (a) AlGaAs GaAs (b) AlGaAs GaAs E F dopants E F dopants 2D electrons z z Impurity levels farther away from 2DEG (as compared to Si/SiO 2 ) enhanced mobility (FQHE)

Nobel Prize in Physics 2010 : Graphene Kostya Novoselov Andre Geim "for groundbreaking experiments regarding the two-dimensional material graphene"

What is Graphene? graphene = 2D carbon crystal (honeycomb) sp 2Hybridation hybridisation sp 2 2s 2p 2p 2p sp sp sp 2p x y z z 120 o

Graphene and its Family 2D 3D 1D 0D

From Graphite to Graphene (strong) covalent bonds in the planes (weak) van der Waals bonds between the planes

How to Make Graphene: Recipe (1) put thin graphite chip on scotch tape

How to Make Graphene: Recipe (2) : fold scotch tape on graphite chip and undo : ~10

How to Make Graphene: Recipe (3) glue (dirty) scotch tape on substrate (SiO 2)

How to Make Graphene: Recipe (4) lift carefully scotch tape from substrate

How to Make Graphene: Recipe (5) orientation marks graphene? thick graphite less thick graphite place substrate under optical microscope

How to Make Graphene: Recipe (6) zoom in region where there could be graphene

Electronic Mesurement of Graphene SiO 2 Si dopé Vg Novoselov et al., Science 306, p. 666 (2004)

2. Landau Quantisation and Integer Quantum Hall Effect

Transition energy (mev) Relative transmission Relative transmission Infrared Transmission Spectroscopy relative transmission Transmission energy [mev] 1.00 0.98 0.96 80 70 60 50 40 30 20 10 L L (A) 0.4 T 1.9 K (B) E L 3 2 L 1 10 20 30 40 50 60 70 80 Energy (mev) 3 L2 D 2 L3 D ( ) ( ) L L L E 1 c ~ 2e B 0 0.0 0.5 1.0 1.5 2.0 Sqrt[B] sqrt(b) L 0 L L L 1 2 3 1 L2 C 2 L1 C ( ) ( ) (C) B Energy [mev] E 1 E 1 L L A B D (D) C 0 L1 B 1 L0 B ( ) ( ) L1 L2 ( A) relative transmission 1.00 0.98 0.96 0.94 10 20 30 40 50 60 70 80 90 0.4T 0.2T transition C 0.3T 0.5T 0.7T 0.92 1 T 0.90 transition B 0.88 2T 4T 0.86 10 20 30 40 50 60 70 80 90 Energy [mev] Energy (mev) Grenoble high field group: Sadowski et al., PRL 97, 266405 (2007) 1.00 0.99 selection rules : λ,n λ,n±1

Edge States (a) n+1 n µ LLs bended upwards at the edges (confinement potential) x (b) n 1 ν = n+1 yn+1 max ν = n 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 000000000000000000000000 1111111111111111111111110000 yn max yn 1 max ν = n 1 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 y y chiral edge states only forward scattering ν= n+1 ν= n ν= n 1

Four-terminal Resistance Measurement R ~ µ µ = 0 L 3 2 µ = µ 2 L 2 3 µ = µ 3 L I 1 4 I 6 µ = µ = µ 6 5 R 5 R ~ µ µ = µ µ H 5 3 R L : hot spots [Klass et al, Z. Phys. B:Cond. Matt. 82, 351 (1991)]

IQHE One-Particle Localisation (a) ε E F n density of states NL (n+1) R xx R xy h/e 2 n ν =n B

IQHE One-Particle Localisation (a) ε E F n (b) ε E F n density of states density of states NL (n+1) R xx R xy R xx R xy h/e 2 n ν =n B B

IQHE One-Particle Localisation (a) ε (b) ε (c) ε E F n E F n E F n localised states extended states density of states density of states density of states NL (n+1) R xx R xy h/e 2 n R xx R xy R h/e 2 n 2 xx h/e (n+1) R xy ν =n B B B

ν IQHE in Graphene Novoselov et al., Nature 438, 197 (2005) Zhang et al., Nature 438, 201 (2005) Density of states Graphene IQHE: R = h/e H 2 ν at ν = 2(2n+1) V g =15V T=30mK 1/ν B=9T T=1.6K Usual IQHE: at ν = 2n (no Zeeman)

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 spin branch 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 2.0 I x V y 3.0 V x 2.5 1.5 2.0 ρ xx (k Ω ) 1.0 6 5 4 3 2 3/ 2 1 3/ 4 2/3 3/5 4/7 5/9 1/ 2 3/7 4/9 5/11 2/5 1.5 1.0 ρ xy (h/e ) 2 0.5 5/3 4/3 8/5 7/5 0 0 4 4/5 5/7 6/11 8/15 7/13 6/13 7/15 8 12 16 Magnetic Field B (T) Magnetic Field B[T] 0.5 0.0 QHE = plateau in R H & R L = 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 ψ L s({z j }) = i<j (z i z j ) 2s χ p=1 ({z j }) i<j (z i z j ) 2s : vortex factor (2s flux quanta per vortex) χ p=1 ({z j }) = i<j (z i z j ): wavefunction at ν = 1 Generalisation to integer ν = p ψ J s,p({z j }) = P LLL i<j (z i z j ) 2s χ p ({z j, z j }) χ p ({z j, z j }): wavefunction for p completely filled levels P LLL : 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 electronic filling 1/3 CF theory 1 filled CF level electron "free" flux quantum pseudo vortex (with 2 flux quanta) composite fermion (CF)

Generalisation: FQHE at half-filling 1987: Obervation of a FQHE at ν = 5/2, 7/2 (even denominator) 1991: Proposal of a Pfaffian wave function (Moore & Read; Greiter, Wilzcek & Wen) ( ) 1 ψ MR ({z j }) = Pf (z i z j ) 2 z i z j quasiparticle charge e = e/4 with non-abelian statistics Further generalisations to ν = K/(K +2): Read & Rezayi i<j

Multicomponent Systems A physical spin: SU(2) (doubling of LLs) B bilayer: SU(2) isospin Landau levels +> > ν = 1/2 + d exciton C graphene (2D graphite) ν = 1/2 ν = ν + ν = 1 T + τ 2 τ 1 e2 1 e 3 e two fold valley degeneracy SU(2) isospin : A sublattice : B sublattice spin + isospin : SU(4)