a brief introduction to qsh & qah states in condensed
TRANSCRIPT
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A Brief Introduction to QSH & QAH States in Condensed
Matter System
Tongyang Zhao
11/05/2019
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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
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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
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• 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?
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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
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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
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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
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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
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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).
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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).
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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
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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”
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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 #”
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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).
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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).
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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
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Experimental verification for HgTe surface state
Olivier Crauste et al.. arXiv preprint arXiv:1307.2008 (2013).
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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
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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
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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
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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).
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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).
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Realization of QAH state (cont.)
• Cr-doped (BixSb1-x)2Te3 system
Cui-Zu Chang et al, Science 340 (6129), 167-170 (2013).
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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
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Recent progress in QAH system study
• 1. Enhancement of critical temperature
Reis et al., Science 357, 287–290 (2017)
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Recent progress in QAH system study
• 2. Robust magnetization of hard ferromagnetic material
Cui-zu Chang et al., arXiv preprint arXiv:1412.2785 (2015).
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Recent progress in QAH system study
• 3. Candidate for new topological states
Qing Lin He et al., Science 357, 294–299 (2017).
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Recent progress in QAH system study
Qing Lin He et al., Science 357, 294–299 (2017).
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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