Quantum Hall Effect

(1)

Guang-Yu Guo (郭光宇)

Physics Department, National Taiwan University, Taiwan

Quantum Hall Effect

without Applied Magnetic Field

(Colloquium Talk in NTU Physics Dept., Oct. 4, 2016)

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I. Introduction

Plan of this Talk

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. Physical properties of layered oxide K_xRhO₂

2. Non-coplanar antiferromagnetic ground state structure 3. Unconventional quantum anomalous Hall phase

IV. Conclusions

III. Quantum topological Hall effect in chiral antiferromagnet K_1/2RhO₂ 1. 4d and 5d metal perovskite bilayers along [111] direction

2. Magnetic and electronic properties 3. Quantum anomalous Hall phase

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1. (Integer) quantum Hall effect

I. Introduction

1) Ordinal Hall Effect [Hall 1879]

Edwin H. Hall (1855-1938)

Lorentz force qv B

/ ( ) / ( )

(1/ )

H H y x xy

R V I E W j W nq B



  



/ ( ) / ( )

^L ^L_xx( / )^x ^x

R V I E L j W

 L W

 



Hall resistance

magneto-resistance

Hall resistance

OHE is a widely used characterization tool in material science lab.

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2) (Integer) quantum Hall Effect [von Klitzing et al., 1980]

Klaus von Klitzing (1943-present)

In 1980, von Klitzing et al discovered QHE.

(1/ )

xy i RK

 



_xx ≈ 0, superconducting states (



_xx ≈ ∞)?

xx 0

  von Klitzing constant R_K = 1 h/e²

= 25812.807557(18)  Conductance quantum

₀ = 1/R_K = 1 e²/h

[PRL 45, 494]

1

0

0 0 1/

, [ ] ,

0 1/ 0

1/ , 0, they are insulating phases.

xy xy

xy xy i xx

 

   

 

   

      

   

ρ σ ρ

[Wei et al., PRL 61 (1988) 129]

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E_F

Formation of discrete Landau levels 2DEG: E_j = ħ



_c(j+1/2)

_c =

e /h2

xy i

 

xx 0

 

Bulk quantum Hall insulating state

T = 1.5 K, B = 18 T

[von Klitzing et al., PRL 45 (1980) 494]

[Wei et al., PRL 61 (1988) 129]

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Quantization of Hall conductance

Thouless et al. topological invariance argument

2

2DBZ

2 , Chern (TKNN) number

( ), 1 ( )

2

xy

i i i i i i

x x z z

i x y y x

n e n h

u u u u

n dk dk i

k k k k





 

     





 ^k  ^k    ^k  ^k   ^k  ^k 

[PRL49, 405 (1982); PRB31, 3372 (1985)]

David Thouless (1934 - )

QH phases are the first

discovered topological phases of quantum matter; QH systems are the first topological insulators with broken time-reversal

symmetry. Topological invariant is Chern number.

(Berry curvature)

2D BZ is a torus. Chern theorem:

2DBZ

( ) ( ) 2 .

x x z S

dk dk    dS  C



^k



^k

Q: A nonzero conductance in an insulating system! How can it be possible?

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(a) Laughlin gauge invariance argument

[Laughlin,

PRB23 (1981) 5632;

Halperin,

PRB25 (1982) 4802]

Existence of conducting edge states (modes)

To do measurements, a finite size sample and hence boundaries must be created.

0 2

,

.

x y

y xy

x

neV I E

I e

V n h



  

 

 

Robert Laughlin (1950 - )

Bending of the LL

(b) Bulk-edge correspondence theorem

When crossing the boundary between two different Chern

insulators, the band gap would close and open again, i.e., metallic edge states exist at the edge whose number is equal to the difference in Chern number.

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(c) Explicit energy band calculations

IQHE is an intriguing phenomenon due to the 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.

 = p/q = 2/7.

2D TB electrons with 2 edges under 

[Hatsugai, PRL 71 (1993) 3697]

Q: High temperature IQHE without applied magnetic field?

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2. Spontaneous quantum Hall effects

1) Anomalous Hall Effect [Hall 1881]

[Zeng et al.

PRL 96 (2006) 2010]

Mn₅Ge₃

0

H R B R MS

  

Spin current

2) Spin Hall Effect

[Dyakonov & Perel, JETP 1971]

Relativistic spin-orbit coupling

Spin current Charge current

(Mott or skew scattering)

' v (E p),

B E

c mc

 ^

  

  

 



2 2

' 1 ( ( ) )

SO 2

H B s V p

 m c

         r  

[Jackson’s textbook]

2

2 2 2 2 3

1 ( ) ( )

2 2

SO

dV r Ze

H s p s L

m c dr r m c r

      

 

  

(Hall effect with applied magnetic field)

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3) Quantum spin Hall effect and topological insulators (a) Intrinsic spin Hall effect



   

 ² ₁ ₂ ² ₂ ²

0 ) 2 ( )

2 ( 5

2 k k S

H m     

Luttinger model

(hole)

i i

l il i

i

e E k

k m F

X k





 











 0 e

E k k

e m

X k  





   ₃ 



Equation of motion

Anomalous velocity

n_h = 10¹⁹ cm^-3, μ= 50 cm /V·s, σ= eμn_h = 80 Ω^-1cm^-1; σ_s= 80 Ω^-1cm^-1

[Science 301, 1348 (2003)]

p-type zincblende semiconductors

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[Kato et al., Science 306, 1910 (2004)]

First observation of the SHE in n-type 3D GaAs and InGaAs thin films

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(b) Quantum spin Hall effect and 2D topological insulators Kane-Mele SOC Hamiltonian for graphene

† †

KM

,

z

i j i z ij j

i j ij

H t c c i



c s v c

   









[Kane & Mele, PRL 95 (2005) 146801; 95 (2005) 226801]

SOC in graphene is too small (<0.01 meV) to make QSHE observable!

2

1 ( 1) , 1 ( 1) 0.

2

s

xy xy

e e

  h

         

Based on Haldane honeycomb model for QHE without Landau levels [PRL 1998].

1 2

ij

 

 v d d

d d

[Chen, Xiao, Chiou, Guo, PRB84 (2011) 165453]

y

A x

B

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Quantum spin Hall effect in semiconductor quantum wells

[Bernevig, Hughes, Zhang, Science 314, 1757 (2006)]

Quantum spin Hall effect in 2D topological

insulator HgTe quantum well [Koenig et al.,

Science 318, 766 (2007)]

Observation of QSHE in quantum wells

[Du et al.,PRL 114 (2015)

096802]

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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 - ) Duncan Haldane (1951 - ) Charles Kane (1963 - )

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4) Quantum anomalous Hall effect (QHE without applied magnetic field)

topological insulator quantum Hall insulator Chern insulator

yes yes

Holy trinity?

???

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[PRL 61 (1988) 2015]

2 1

2 2

[ / 1/ 3]

If / 3 3 sin ,

xy / . t t

M t

ne h









† † †

1 2

, ,

exp( ^z )

H i j ij i j i i i

i j i j i

H t c c t iv  c c M  c c

   









 



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

1 2

vij  

 d d d d

Phase diagram of Haldane model

N.B. Kane-Mele model is two copies of Haldane model with M = 0,  =3/2 and _SO = t₂.

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QAHE in real systems: Magnetic impurity-doped topological insulator films Theoretical proposal:

Bi₂Te₃, Bi₂Se₃ or Sb₂Te₃ films doped with Cr or Fe

[Yu et al., Science 329, 61 (2010)]

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First observation on QAHE in Cr_0.15(Bi_0.1Sb_0.9)_1.85Te₃ (5 QLs) thin films

[Science 340, 167 (2013)]

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[Science 340, 167 (2013)]

QAHE in Cr_0.15(Bi_0.1Sb_0.9)_1.85Te₃ thin films

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 cm²/Vs).

Qikun Xue (1963 - )

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/SrTiO₃.

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2) Layered 4d and 5d transition metal oxides 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?).

3. Why search for SQHE in layered 4d and 5d transition metal oxides

1) Transition metal oxides

A fascinating family of solid state systems:

high T_c superconductivity: YBa₂Cu₃O_6.9

colossal magnetoresistance: La_2/3Ca_1/3MnO₃ half-metallicity for spintronics: Sr₂FeMoO₆ ferroelectricity: BaTiO₃

charge-orbital ordering: Fe₃O₄

Charge-orbital ordering in Fe₃O₄

[Jeng, Guo, Huang, PRL 93 (2004) 156403;

Huang et al., PRL 96 (2006) 096401]

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Perovskite (AB’O₃)_N/(ABO₃)₂ bilayer candidates for a topological insulator

II. Chern insulator in 4d and 5d transition metal perovskite bilayers

Conf bulk

LaReO₃ LaRuO₃ SrRhO₃ SrIrO₃ LaOsO₃ LaAgO₃ LaAuO₃

t_2g⁴ t_2g⁵ t_2g⁵ t_2g⁵ t_2g⁵ e_g² e_g²

‐‐

metallic metallic metallic

‐‐

metallic

‐‐

List of ABO₃ candidates.

B’ = Al or Ti.

[Xiao et al., NC 2 (2011) 596]

1. 4d and 5d metal perovskite bilayers along [111] direction

Design principle: Start with a band structure having ‘Dirac points’ without SOC, and then examine whether a gap opened at those points with the SOC turned on.

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[Xiao et al., NC 2 (2011) 596]

t_2gmodel: /t=5 with /t=1(red) & =0(green)

t_2gmodel: /t=0.5, /t=1.5 (red) & /t=1.5, =0(green)

e_gmodel: /t=0.2 (red) & /t=0(green)

LaAlO₃/LaReO₃ LaAlO₃/LaOsO₃

SrTiO₃/SrRhO₃ SrTiO₃/SrIrO₃

LaAlO₃/LaAgO₃ LaAlO₃/LaAuO₃

LaAlO₃/LaAuO₃/LaScO₃ LaAlO₃/LaAuO₃/YAlO₃

TI

TI TI

NI TI

Electronic band structure and topology

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[Chandra & Guo, arXiv: 1609.07383 (2016)]

2. Magnetic and electronic properties

ABO₃ Conf Bulk Superlat

LaRuO₃ LaAgO₃ LaReO₃ LaOsO₃ LaAuO₃ SrRhO₃ SrAgO₃ SrOsO₃ SrIrO₃

d⁵ (t_2g⁵) d⁸(e_g²) d⁴(t_2g⁴)

d⁵(t_2g⁵) d⁸(e_g²) d⁵(t_2g⁵)

d⁷(e_g¹) d⁴(t_2g⁴) d⁵(t_2g⁵)

metallic metallic metallic*

metallic*

metallic metallic metallic*

metallic metallic

yes [6]

Yes [10]

(ABO₃)₂/(ABO₃)₁₀ superlattices

*

ABO₃ = LaAlO₃ or SrTiO₃

z-AF i-AF

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Mean-field estimation

Physical properties of the magnetic perovskite bilayers

0 ij i j

i j

E E J  



 





Heisenberg model

0

1

B C 3 j

j

k T 



J

*Large exchange couplings, thus high Curie temperatures;

*(LaOsO₃)₂/(LAO)₁₀ is an insulator, others, metallic.

*(LaRuO₃)₂/(LAO)₁₀ and (SrRhO₃)₂/(STO)₁₀, half-metallic;

*Large anomalous Hall conductivities.

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Band structure of the magnetic perovskite bilayers

(LaOsO₃)₂/(LAO)₁₀ is an insulator with a gap of 38 meV, (SrIrO₃)₂/(STO)₁₀ is a semimetal and the rest are metallic.

(LaRuO₃)₂/(LAO)₁₀ and (SrRhO₃)₂/(STO)₁₀ are half-metallic.

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Quantum confinement of conduction electrons

Both charge and spin densities are confined within the (LaRuO₃)₂

bilayer in the central part of the superlattice.

Although many 3D bulk magnetic materials have been predicted to be half- metallic, quasi-2D fully spin-polarized electron gas systems have been rare.

(LaRuO₃)₂/(LAO)₁₀ (111)

charge density

spin density

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3. Quantum anomalous Hall phase

(LaOsO₃)₂/(LAO)₁₀ is a spin-polarized quantum anomalous Hall (Chern) insulator (Chern number = 2) with the spin-polarized edge current tunable by applied magnetic field.

2 3

3 2

' '

2 Im | | ' ' | | ( ( )) ( ), ( )

(2 ) ( )

x y

z z

AH n n n

n n n n n

n v n n v n

e d k f

 

 _  

     

 





k k

k k k k

k k k



For a 3D Chern insulator,

2

AH c

n e

  hc ^{, n}_c is an integer (Chern number)

is calculated using the maximally localized Wannier functions fitted to GGA band structure.

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1. Physical properties of layered oxide K

_x

RhO

₂

III. Quantum topological Hall effect in K

_1/2

RhO

₂

1) Crystal structure:

2) Interesting properties:

Layered hexagonal -Na_xCoO₂-type structure (P6₃/mmc; No. 194)

with two CdI₂-type (1T) RhO₂ layers stacked along c-axis [2f.u./cell].

It is isostructural and also

isoelectronic to thermoelectric and superconducting material Na_xCoO₂.

It shows significant

thermopower and Seebeck coefficient, and is also

expected to become

superconducting at low temperatures.

[Shibasaki et al., JPCM 22 (2010) 115603]

RhO₂ RhO₂ RhO₂

K K

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1) Energetics of various magnetic structures in K_0.5RhO₂

Ground state: all-in (all-out) non-coplanar antiferromagnetic structure.

2. Non-coplanar antiferromagnetic ground state structure

Possible metastable magnetic structures

nc-AFM

FM S-AFM z-AFM

t-AFM 3:1-FiM 90-c-AFM

3i-1o-nc-FiM 2i-2o-nc-AFM

90-nc-FiM 90-nc-AFM

[Zhou et al., PRL 116, 256601 (2016)]

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Total energy (E^tot) (meV/f.u.), total spin moment (m_s^tot) (_B/f.u.), Rh atomic spin moment (m_s^Rh) (_B/f.u.) and band gap (E_g), from GGA+U calculations. [VASP-PAW method, GGA+U_eff (Rh) = 2 eV]

0 ij i j

i j

H E J  



 





Heisenberg model

Exchange coupling

J₁= 4.4, J₂= -3.6 meV

Neel temperature T_N = ~20 K

[Zhou et al., PRL 116, 256601 (2016)]

[Henze et al., APA 75, 25 (2002)]

K_0.5RhO₂

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2) Band structure of non-coplanar antiferromagnetic structure

In (a), blue solid lines from GGA+U and red dotted lines from MLWFs interpolations.

An insulator (E_g =0.22eV)

(a) (b)

Is it a topologically trivial or nontrivial insulator?

Crystal field splitting of Rh t_2g orbitals in K_1/2RhO₂ with Rh a_1g is ¾ filled.

8(K_1/2RhO₂)

[Zhou et al., PRL 116, 256601 (2016)]

(32)

3. Unconventional quantum anomalous Hall phase

2 3

3

' ' 2

( ( )) ( ) (2 )

2 Im | | ' ' | |

( ) ( )

z

AH n n

n

x y

z n

n n n n

e d k f

n v n n v n

 



 



  

  



 



k k

k k k k

k



Anomalous Hall conductivity

For a 3D Chern insulator,

2

AH c

n e

  hc

n_c is an integer (Chern number)

Thus, nc-AFM state is a QAH phase with n_c = 2.

1) A Chern insulator

[Zhou et al., PRL 116, 256601 (2016)]

(33)

2) Edge states

Bulk-edge correspondence theorem is fulfilled.

[Zhou et al., PRL 116, 256601 (2016)]

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3) Nature of the quantum anomalous Hall phase

Spin chirality, Berry phase and topological Hall effect

[Taguchi et al., Science 291, 2573 (2001)]

, ,

( ).

i j k

i j k

Spin chirality



^s  ^s ^s s₃

s₂ s₁



solid angle ,

Berry phase  / 2



  Nd₂Mo₂O₇

Nd₂Mo₂O₇

Topological Hall effect: Anomalous Hall effect purely due to Berry phase produced by spin-chiraty in the noncoplanar magnetic strucure.

(35)

A conventional QAH phase is caused by the presence of FM and SOC!

Here, m_s^tot = 0 and no SOC; thus QAH phase is unconventional.

AHC is due to nonzero scalar spin chirality in nc-AFM structure,

, ,

( )

i j k

i j k

 



^s  ^s ^s So it is the quantum topological Hall effect

due to the topologically nontrivial chiral magnetic structure!

[Zhou et al., PRL 116, 256601 (2016)]

2 AH

Total solid angle 4 , Berry phase / 2,

Chern number ( / 2 ) 2 2, AHC =2e /h.

nc





 



 

 

  

(36)

4) Effects of spin-orbit coupling

[Zhou et al., PRL 116, 256601 (2016)]

The nc-AFM structure remains the lowest energy one and it is still a QAH insulator with E_g = 0.16 eV and m_s^tot = 0.08 _B/f.u.

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IV. Conclusions

1. Layered 4d and 5d transition metal oxides are good candidates for

Chern insulators, because 4d and 5d transition metal oxides have stronger SOC but moderate/weak correlation, quite unlike 3d transition metal

oxides where correlation is strong and often leads to Mott insulators.

2. Based on first-principles density functional calculations, we predict

that the high temperature QAH phases would exist in two kinds of 4d and 5d transition metal oxides, ferromagnetic /(LaOsO₃)₂/(LAO)₁₀perovskite [111] superlattice and layered chiral antiferromagnetic K_1/2RhO₂.

3. Further theoretical analysis reveals that the QAH phases in these oxide systems result from two distinctly different mechanisms, namely,

conventional one of the presence of ferromagnetism and SOC, and

unconventional one due to the topologically nontrivial magnetic structure (i.e., exotic quantum topological Hall effect).

Quantum Hall Effect

Physics Department, National Taiwan University, Taiwan