Quantum Hall Effect without Applied Magnetic Field Guang-Yu Guo ( ) - PowerPoint PPT Presentation

Quantum Hall Effect without Applied Magnetic Field Guang-Yu Guo ( 郭光宇 ) Physics Department, National Taiwan University, Taiwan (Colloquium Talk in NTU Physics Dept., Oct. 4, 2016)

Plan of this Talk I. Introduction 1. (Integer) quantum Hall effect 2. Spontaneous quantum Hall effects (SQHE) 3. Why search for SQHE in layered 4d and 5d transition metal oxides II. Chern insulator in 4d and 5d transition metal perovskite bilayers 1. 4d and 5d metal perovskite bilayers along [111] direction 2. Magnetic and electronic properties 3. Quantum anomalous Hall phase III. Quantum topological Hall effect in chiral antiferromagnet K 1/2 RhO 2 1. Physical properties of layered oxide K x RhO 2 2. Non-coplanar antiferromagnetic ground state structure 3. Unconventional quantum anomalous Hall phase IV. Conclusions

I. Introduction Hall resistance 1. (Integer) quantum Hall effect 1) Ordinal Hall Effect [Hall 1879] magneto-resistance Lorentz force q  v B Hall resistance   R V / I ( E L ) / ( j W ) L L x x   ( L W / ) xx     R V / I ( E W ) / ( j W ) H H y x xy  (1/ nq B ) Edwin H. Hall (1855-1938) OHE is a widely used characterization tool in material science lab.

2) (Integer) quantum Hall Effect [von Klitzing et al., 1980] In 1980, von Klitzing et al discovered QHE. [PRL 45, 494] [Wei et al., PRL 61 (1988) 129]   (1/ ) i R xy K Klaus von Klitzing (1943-present)   0 von Klitzing constant xx R K = 1 h/e 2 = 25812.807557(18)  Conductance quantum  xx ≈ 0, superconducting states (  xx ≈ ∞ )?  0 = 1/ R K = 1 e 2 /h        0 0 1/ xy xy     1 ρ , σ [ ] ρ ,        0 1/ 0     xy xy         1/ i , 0, they are insulating phases. xy xy 0 xx

[Wei et al., PRL 61 (1988) 129] Formation of discrete Landau levels   2 i e /h xy 2DEG: E j = ħ  c ( j +1/2) E F   0 xx  c = T = 1.5 K, B = 18 T [von Klitzing et al., PRL 45 (1980) 494] Bulk quantum Hall insulating state

Quantization of Hall conductance Thouless et al. topological invariance argument [PRL49, 405 (1982); PRB31, 3372 (1985)] 2 2 e    n , n Chern (TKNN) number xy (Berry curvature) h       1 u u u u         i i n dk dk ( ), k ( ) k i i k i k i k i k   2D x x z z      2 k k k k   BZ i x y y x David Thouless (1934 - ) QH phases are the first discovered topological phases of quantum matter; QH systems are the first topological insulators 2D BZ is a torus. Chern theorem: with broken time-reversal symmetry. Topological invariant         dk dk ( ) k ( ) k dS 2 C . 2D x x z S is Chern number. BZ Q: A nonzero conductance in an insulating system! How can it be possible?

Existence of conducting edge states (modes) (a) Laughlin gauge invariance argument [Laughlin, PRB23 (1981) 5632; Halperin, To do measurements, a finite size sample and hence boundaries must be created. PRB25 (1982) 4802] (b) Bulk-edge correspondence theorem Robert Laughlin (1950 - ) When crossing the boundary between two different Chern insulators, the band gap  E neV would close and open   I x , y   again, i.e., metallic edge 0 I 2 e states exist at the edge   y  n . xy V h whose number is equal x Bending of the LL to the difference in Chern number.

(c) Explicit energy band calculations 2D TB electrons with 2 edges under  [Hatsugai, PRL 71 (1993) 3697] IQHE is an intriguing phenomenon due to the  = p / q = 2/7. occurrence of bulk topological insulating phases with dissipationless conducting edge states in the Hall bars at low temperatures and under strong magnetic field. Hall resistance is so precisely quantized that it can be used to determine the fundamental constants and robust metallic edge state is useful for low-power consuming nanoelectronics and spintronics. Q: High temperature IQHE without applied magnetic field?

[Zeng et al. Mn 5 Ge 3 2. Spontaneous quantum Hall effects PRL 96 (2006) 2010] 1) Anomalous Hall Effect [Hall 1881] (Hall effect with applied magnetic field)    R B R M H 0 S Spin current Charge current 2) Spin Hall Effect [Dyakonov & Perel, JETP 1971] Relativistic spin-orbit coupling [Jackson’s textbook]       v E p Spin current     B ' E  ( ), c mc   1            H B ' s ( V ( ) r p ) SO 2 2 2 m c   2 1  dV r  Ze         H s ( p ) ( s L ) SO 2 2 2 2 3 2 m c dr r 2 m c r (Mott or skew scattering)

3) Quantum spin Hall effect and topological insulators [Science 301, (a) Intrinsic spin Hall effect 1348 (2003)] p-type zincblende   semiconductors 2    5     2    2 Luttinger model H ( ) k 2 ( k S )   0 1 2 2 2 m  2   k     Equation of motion X i F k i il l m  e   e 0  (hole) k E i i        k e  Anomalous velocity    X k E 3 m  k  n h = 10 19 cm -3 , μ = 50 cm /V·s, σ = e μ n h = 80 Ω -1 cm -1 ; σ s = 80 Ω -1 cm -1

First observation of the SHE in n-type 3D GaAs and InGaAs thin films [Kato et al. , Science 306, 1910 (2004)]

(b) Quantum spin Hall effect and 2D topological insulators Kane-Mele SOC Hamiltonian for graphene Based on Haldane honeycomb model for QHE without Landau [Kane & Mele, PRL 95 (2005) 146801; 95 (2005) 226801]   levels [PRL 1998].    † † z H t c c i c s v c KM i j i z ij j 2 e e     i j , ij            s 1 ( 1) , 1 ( 1) 0.  d d xy  xy 2 h  v 1 2 ij  d d 1 2 y x B A SOC in graphene is too small (<0.01 meV) [Chen, Xiao, Chiou, Guo, PRB84 (2011) 165453] to make QSHE observable!

Quantum spin Hall effect in semiconductor quantum wells Observation of QSHE in quantum wells Quantum spin Hall effect in 2D topological [Koenig et al., insulator HgTe quantum well Science 318, [Bernevig, Hughes, Zhang, 766 (2007)] Science 314, 1757 (2006)] [Du et al.,PRL 114 (2015) 096802]

For their pioneering works on topological insulators and quantum spin Hall effect, three theoretical condensed-matter physicists won the 2012 Dirac medal and prize (ICTP in Trieste, Italy) Shoucheng Zhang (1963 - ) Charles Kane (1963 - ) Duncan Haldane (1951 - )

4) Quantum anomalous Hall effect (QHE without applied magnetic field) Holy trinity? ??? yes yes quantum Hall insulator Chern insulator topological insulator

[PRL 61 (1988) 2015] Haldane’s 2D honeycomb lattice model (graphene) for spinless electrons Areas a and b are threaded by fluxes  a and  b = -  a . Area c has no flux.  = 2  (2  a +  b )/  0 .  i =  1. A B          † z † † H t c c t exp( iv ) c c M c c H 1 i j 2 ij i j i i i     i j , i j , i  [ t / t 1/ 3] 2 1    d d If M t / 3 3 sin ,  2 v 1 2 ij    2 d d ne / . h xy 1 2 N.B. Kane-Mele model is two copies of Haldane model with M = 0,  =3  /2 Phase diagram of Haldane model and  SO = t 2 .

QAHE in real systems: Magnetic impurity-doped topological insulator films Theoretical proposal: Bi 2 Te 3 , Bi 2 Se 3 or Sb 2 Te 3 films doped with Cr or Fe [Yu et al., Science 329, 61 (2010)]

First observation on QAHE in Cr 0.15 (Bi 0.1 Sb 0.9 ) 1.85 Te 3 (5 QLs) thin films [Science 340, 167 (2013)]

QAHE in Cr 0.15 (Bi 0.1 Sb 0.9 ) 1.85 Te 3 thin films [Science 340, 167 (2013)] Xue just won the first Future Science Prize (“China’s Nobel Prize”, US$ 1 million) for his team’s observation of the QAHE and also superconductivity in FeSe monolayer/SrTiO 3 . Qikun Xue (1963 - ) Remaining issues: QAHE below 30 mK due to (a) Small band gap (~10 meV); (b) Weak exchange coupling T c = ~15 K (a) Low mobility (760 cm 2 /Vs).

3. Why search for SQHE in layered 4d Charge-orbital ordering in Fe 3 O 4 and 5d transition metal oxides 1) Transition metal oxides A fascinating family of solid state systems: high T c superconductivity: YBa 2 Cu 3 O 6.9 colossal magnetoresistance: La 2/3 Ca 1/3 MnO 3 half-metallicity for spintronics: Sr 2 FeMoO 6 ferroelectricity: BaTiO 3 charge-orbital ordering: Fe 3 O 4 2) Layered 4d and 5d transition metal oxides [Jeng, Guo, Huang, PRL 93 (2004) 156403; Huang et al., PRL 96 (2006) 096401] as Chern insulator candidates Electron correlations in 3d transition metal oxides are strong, which is challenging to describe, and make them become Mott (trivial) insulators. So far, many-body theory appears unnecessary for TI research. Layered 4d and 5d transition metal oxides have stronger SOC (larger band gaps?), moderate/weak correlation (easier to study?) and intrinsic itinerant magnetism (higher mobility?).