A Brief Introduction to QSH & QAH States in Condensed

Matter System

Tongyang Zhao

11/05/2019

Outline

• Introduction to quantum Hall family

• Quantum Spin Hall (QSH) state

• First theoretical attempt: Haldane, Kane&Mele

• BHZ model: HgTe/CdTe QW

• Sketchy theoretical structure

• Conditions for existence

• Experimental verification

• Quantum Anomalous Hall (QAH) state

• From QSH to QAH

• Realization: 3D magnetic topological insulator

• Recent outlook

Outline

• Introduction to quantum Hall family

• Quantum Spin Hall (QSH) state

• First theoretical attempt: Haldane, Kane&Mele

• BHZ model: HgTe/CdTe QW

• Sketchy theoretical structure

• Conditions for existence

• Experimental verification

• Quantum Anomalous Hall (QAH) state

• From QSH to QAH

• Realization: 3D magnetic topological insulator

• Recent outlook

• QH: from integer to fraction

Introduction to QH family (cont.)

1975

University of Tokyo

Earliest prediction of

IQHE

1980

Klaus von Klitzing

IQHE in Si-MOSFET

1982

Daniel Tsui et al.Fractional version of

QHE in GaAs

1983

Robert B. Laughlin

Wavefunction for FQHE

Even

denominator?

Introduction to QH family (cont.)

1988→2005

Haldane; Kane C.L. and Mele E.J.

Prediction of QSH state on

honeycomb lattice

2006

B. Andrei Bernevig et al.Model for QSH in HgTe/CdTe QW

• Quantum spin Hall regime

2007

Markus König et al.Experimental verification of

spin-polarized edge states

Introduction to QH family (cont.)

2010

Fang group

Prediction of QAH in TI

2013

Xue group

Observation of QAH in magnetic TI

• Quantum anomalous Hall regime

2013~

Novel properties in

QAH related systems

Introduction to QH family (cont.)

• Hall effect in 2D electron system: quantization

• Quantized Hall conductivity & vanishing longitudinal

resistance

• Landau level formalism

• QSH: non-zero spin current

• Time-reversal-symmetry protected

• Spin-orbit coupling (relativistic correction)

• QAH: ferromagnetic order

• “Half-fold” of QSH

Outline

• Introduction to quantum Hall family

• Quantum Spin Hall (QSH) state

• First theoretical attempt: Haldane, Kane&Mele

• BHZ model: HgTe/CdTe QW

• Sketchy theoretical structure

• Conditions for existence

• Experimental verification

• Quantum Anomalous Hall (QAH) state

• From QSH to QAH

• Realization: 3D topological insulator

• Recent outlook

QSH: Hall state without magnetic field

• Quantum spin Hall (QSH) holds

time reversal symmetry

• No external magnetic field

• Spin-orbit coupling induces an

effective magnetic field for electrons

with different spin

• Helical edge states: backscattering

prohibited

• Dissipationless spin current

• Vanishing net charge currentMarkus König, Steffen Wiedmann, Christoph Brüne, Andreas Roth, Hartmut Buhmann, Laurens W. Molenkamp, Xiao-Liang Qi, Shou-Cheng Zhang, Science 318 (5851), 766-770 (2007).

Earliest theoretical approach

• Haldane’s honeycomb lattice for spinless particles

• Kane & Mele: generalization to ½ -spin electrons, considering SOC

• For 𝜆𝑅 = 0 case, phase determined by ratio between 𝜆𝑆𝑂and 𝜆𝑣

• For 𝜆𝑅 ≠ 0, numerical calculation reveals gapless edge state in QSH regime (Fig.1(a))

𝐻 = 𝑡

𝑖,𝑗

𝑐𝑖† 𝑐𝑗 + 𝑖𝜆𝑆𝑂

𝑖,𝑗

𝑣𝑖𝑗𝑐𝑖†𝑠𝑧𝑐𝑗 + 𝑖𝜆𝑅

𝑖,𝑗

𝑐𝑖† 𝒔 × 𝒅𝑖𝑗 𝑧

𝑐𝑗

+𝜆𝑣

𝑖

𝜉𝑖𝑐𝑖†𝑐𝑖 𝑤ℎ𝑒𝑟𝑒 𝑐𝑖

† = (𝑐𝑖,↑† , 𝑐𝑖,↓

† )

C.L. Kane, E.J. Mele. Phys. Rev. Lett 95 (146802) (2005).

QSH theory for II-IV QW: BHZ model

• Bernevig-Hughes-Zhang (2006)

• 4×4 matrix describing 2 bands & 2 spin

• 𝐻 𝑘 =ℎ(𝑘) 00 ℎ∗(−𝑘)

, where

ℎ 𝑘 = Ԧ𝑑 𝑘 ∙ Ԧ𝜎, 𝑑1 𝑘 = 𝐴𝑘𝑥, 𝑑2 𝑘 = −𝐴𝑘𝑦, 𝑑3 𝑘 = 𝑀 − 𝐵(𝑘𝑥2 + 𝑘𝑦

2)

(symmetry-considered tight binding expansion around Γ point)

• Band inversion: when 𝑀/𝐵 > 0 (opposite spin configuration)

• Each spin branch induces conducting edge, net current is zero

Theoretical model for conductivity

• Consider the most generic 2-band model

• (Ignore the spin degree of freedom first)

• ℎ 𝑘 = 휀 𝑘 + Ԧ𝑑 𝑘 ∙ 𝜎 =휀 𝑘 + 𝑑𝑧 𝑘 𝑑𝑥 𝑘 − 𝑖𝑑𝑦(𝑘)

𝑑𝑥 𝑘 + 𝑖𝑑𝑦(𝑘) 휀 𝑘 − 𝑑𝑧(𝑘)

• Hall conductivity is given by the Chern number of the mapping

𝑇2 → 𝑆2, 𝑘 → መ𝑑, defined as

• 𝜔 =1

8𝜋2𝐹𝐵𝑍 d𝑘𝑥d𝑘𝑦

መ𝑑 ∙ 𝜕𝑥 መ𝑑 × 𝜕𝑦 መ𝑑

• Robust under small perturbation, “topological invariant”

Theoretical model for conductivity (cont.)

• Using Kubo formula to calculate the Hall conductivity

• 𝜎𝑥𝑦 = lim𝜔→0

𝑖

𝜔𝑄𝑥𝑦(𝜔 + 𝑖𝛿)

• 𝑄𝑥𝑦 𝑖𝜈𝑚 =1

Ω𝛽σ𝒌,𝑛 𝑡𝑟 [𝐽𝑥 𝒌 𝐺 𝒌, 𝑖 𝜔𝑛 + 𝜈𝑚 𝐽𝑦 𝒌 𝐺(𝒌, 𝑖𝜔𝑛)]

• 𝐽𝑖 𝒌 =𝜕𝐻 𝒌

𝜕𝑘𝑖=

𝜕𝜀 𝒌

𝜕𝑘𝑖+

𝜕𝑑𝑗 𝒌

𝜕𝑘𝑖𝜎𝑗

• 𝐺 𝒌, 𝑖𝜔 = 𝑖𝜔 − 𝐻 𝒌−1

• The final result is

• 𝜎𝑥𝑦 = −1

2Ωσ𝑘

𝜕 𝑑𝛼 𝑘

𝜕𝑘𝑥

𝜕 𝑑𝛽 𝑘

𝜕𝑘𝑦መ𝑑𝛾 𝑘 𝜖𝛼𝛽𝛾.

The summation over k become

integral under continuum limit:

𝜎𝑥𝑦 = −𝑒2

ℎ

1

8𝜋2න𝐹𝐵𝑍

d𝑘𝑥d𝑘𝑦 መ𝑑 ∙ 𝜕𝑥 መ𝑑 × 𝜕𝑦 መ𝑑

“Chern #/TKNN #”

Conditions for QSH state to exist?

• Minimal two-band model• BHZ model: inverted band structure• HgTe/CdTe quantum well

• Criteria: band inversion around 𝑘 = 0• Often induced by strong SOC• Always simultaneous with helical edge

state

Markus König, Steffen Wiedmann, Christoph Brüne, Andreas Roth, Hartmut Buhmann, Laurens W. Molenkamp, Xiao-Liang Qi, Shou-Cheng Zhang, Science 318 (5851), 766-770 (2007).

Experimental realization of QSH state

• HgTe/CdTe quantum well• Critical thickness: 𝑑𝑐 = 6.3nm

• Helical edge states emerge

B. A. Bernevig, T. L. Hughes, S. C. Zhang, Science 314 (5806), 1757-1761 (2006).

Experimental realization of QSH state (cont.)

Hall resistance measurement for inverted device

-1.4V, n-type

-1.9V, p-type

Markus König, Steffen Wiedmann, Christoph Brüne, Andreas Roth, Hartmut Buhmann, Laurens W. Molenkamp, Xiao-Liang Qi, Shou-Cheng Zhang, Science 318 (5851), 766-770 (2007).

Longitudinal resistance measurement

Experimental verification for HgTe surface state

Olivier Crauste et al.. arXiv preprint arXiv:1307.2008 (2013).

Outline

• Introduction to quantum Hall family

• Quantum Spin Hall (QSH) state

• First theoretical attempt: Haldane, Kane&Mele

• BHZ model: HgTe/CdTe QW

• Sketchy theoretical structure

• Conditions for existence

• Experimental verification

• Quantum Anomalous Hall (QAH) state

• From QSH to QAH

• Realization: 3D topological insulator

• Recent outlook

QSH to QAH: spin-splitting process

• Anomalous Hall state: ferromagnetism induced Hall effect• Additional magnetic effect due to spontaneous magnetization

• Multiple mechanism: intrinsic, skew-scattering, side-jump

• Quantum anomalous Hall (QAH)• Conductivity plateau in ferromagnetic system

• Extreme case: nonzero Hall plateau with zero external magnetic field

• Intrinsic mechanism induced effect

Naoto Nagaosa, Jairo Sinova, Shigeki Onoda, A. H. MacDonald, and N. P. Ong

Rev. Mod. Phys. 82, 1539 – Published 13 May 2010

Realization of QAH state

• From QSH to QAH• QSH state provides a recipe to find QAH insulator

• Destructing band inversion for one certain spin branch

• Criteria• Inverted band structure

• Ferromagnetism in insulator

• HgTe QW?• Possibility lies in B-dependence of longitudinal R

• Difficulty: magnetic doping mechanism

• Absence of spontaneous magnetization

Realization of QAH state (cont.)

• Potential candidate: magnetic topological

insulator

• Bi2Te3, Bi2Se3, Sb2Te3 family

• “Oscillate” between conventional insulator and TI

when varying thickness

• QSH state within certain thickness region

• Interaction with magnetic dopant (Cr, V) :

ferromagnetic order, Van Vleck mechanism

C-X. Liu, S-C. Zhang, X-L. Qi, arXiv preprint arXiv:1508.07106 (2015).

Realization of QAH state (cont.)

Rui Yu, Wei Zhang, Hai-Jun Zhang, Shou-Cheng Zhang, Xi Dai, Zhong Fang.

Science 329 (5987), 61-64 (2010).

Realization of QAH state (cont.)

• Cr-doped (BixSb1-x)2Te3 system

Cui-Zu Chang et al, Science 340 (6129), 167-170 (2013).

Recent progress in QAH system study

• Enhancement of critical temperature• 30mK (2013) → 2K (2015) → ~room temperature (?) (2017)

• Robust magnetization of hard ferromagnetic material• Cr→V

• Candidate for new topological states• Interaction with topological superconductor: realization of Majorana

fermion

• Towards high temperature non-dissipative, low power consumption electronic devices

Recent progress in QAH system study

• 1. Enhancement of critical temperature

Reis et al., Science 357, 287–290 (2017)

Recent progress in QAH system study

• 2. Robust magnetization of hard ferromagnetic material

Cui-zu Chang et al., arXiv preprint arXiv:1412.2785 (2015).

Recent progress in QAH system study

• 3. Candidate for new topological states

Qing Lin He et al., Science 357, 294–299 (2017).

Recent progress in QAH system study

Qing Lin He et al., Science 357, 294–299 (2017).

Conclusion

QSH

Time reversal symmetry with SOC

Inverted band structure

Candidate for realization of QAH state

QAH

Half-fold of QSH insulator

Ferromagnetic topological insulator

Electronics: dissipationless device

Physics: breeding ground for novel topological states