20140328 TNTL journal club axion electrodynamics, TI-FI interface (nomura, nagaosa)

TNTL Journal Club

Dongwook Go

28 March 2014, Journal Club @ Theoretical Nanoscale Transport Lab, POSTECH

Kentaro Nomura and Naoto Nagaosa

Phys. Rev. Lett. 106, 166802 (2011)Surface-Quantized Anomalous Hall Current and the Magnetoelectric Effect in Magnetically Disordered Topological Insulators

Phys. Rev. B 82, 161401(R) (2010)Electric charging of magnetic textures on the surface of a topological insulator

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Topological Insulator

Xia et al., (2008), Hsieh, Xia, Qian, Wray, et al., (2009), and Xia, Qian, Hsieh, Wray, et al., (2009). 2


Hsieh, Xia, Qian, Wray, et al., (2009), Xia, Qian, Hsieh, Shankar, et al., (2009), and Wray et al., (2010).

Topological Insulator

3


Topological Field Theory

Tight-binding + EM field (minimal coupling limit)

Chern-Simon’s Term (3+1 D)

Integrate-outFermionic DOF

X.-L. Qi et al., PRB (2008).

Time Reversal Symmetry

→ 𝑃3 𝑥, 𝑡 = 0 or 1/2 mod 1

Natural unit ( =c=1)

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1. Half-integer quantum Hall effect2. Topological magnetoelectric effect3. Induced magnetic monopole4. Faraday rotation

Consequences from the Chern-Simon’s term

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X.-L. Qi et al., PRB (2008), Science (2009).

1. Half-integer quantum Hall effect2. Topological magnetoelectric effect3. Induced magnetic monopole4. Faraday rotation

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Consequences from the Chern-Simon’s term


Half-Integer Quantum Hall Effect

Surface Chern-Simon’s term

z-integration

𝑛 ∈ ℤ depends on non-topological details of surface

TI

FI𝟎

𝒛

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X.-L. Qi et al., PRB (2008).

Half-Integer Quantum Hall Effect

𝑗𝜇 =𝛿𝑆𝑠𝑢𝑟𝑓

𝛿𝐴𝜇

𝜎𝐻 =1

2𝜋𝑛 +

1

2

𝜎𝐻 =1

4𝜋sgn(𝑚𝑧)

Natural unit ( =c=1)

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e.g. Massive Dirac Fermion

X.-L. Qi et al., PRB (2008).

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Topological Magnetoelectric Effect


Applied Electric Field

Hall Current

Magnetization

Applied Magnetic Field

Induced Electric Field

Charge Polarization(from the Hall current)X.-L. Qi et al., PRB (2008).

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Topological Magnetoelectric Effect


(Gaussian CGS unit)𝜎𝐻 =𝑒2

ℎ𝑐𝑛 +

1

2

From the previous result,

X.-L. Qi et al., PRB (2008).

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Axion Electrodynamics


where , , and

Variation over leads to

X.-L. Qi et al., PRB (2008).

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Axion Electrodynamics


where and

X.-L. Qi et al., PRB (2008).

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Other Examples


Faraday Rotation Inducing Magnetic Monopole


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T-symmetry Broken Surface State


1. External Magnetic Field2. Magnetic Bilayer3. Magnetic Impurity Doping

K. Nomura and N. Nagaosa PRL (2011), PRB Rapid (2010).

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Difficulties in realizing TME

1. Get rid of carriers in the bulk

2. Tune the Fermi energy within the gap of surface Dirac cone

3. Attach the insulating ferromagnetic layer with the magnetization perpendicular to the surface

L.-X. Qi et al., PRB (2008).

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17K. Nomura and N. Nagaosa PRL (2011).


Localization of surface states by magnetic impurities

, where

Impurity-averaged Hamiltonian

Exchange interaction betweenmagnetic impurities and electrons

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Gaussian impurity correlation

Random magnetic impurity

Impurity strength ~ 𝑚 :

K. Nomura and N. Nagaosa PRL (2011).

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Full Hamiltonian

=


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Numerical Simulation Result


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Scaling Analysis


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Moving a Domain Wall

Electric dipole energy

Uniform magnetization configuration can be achieved!

Larger than other (Zeeman, anisotropy …) energy scales.


QH regime

Implication : TME can be realized.

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Summary of the previous work

Role of quenched magnetic disorder?


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Spin textures are charged on TI

K. Nomura and N. Nagaosa PRB Rapid (2010).

Fictitious gauge field Electric field

Magnetic field

Electric charging from spin textures!

QH regime Hall current

Charge pumping

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Spin textures are charged on TI

Vortex Domain wall


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Derivation : functional path integral approach


(1) Fermion Part

where

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(2) Magnetization Part

+ (anisotropy, DMI … )

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(3) Integrate-out Fermionic DOF



where


, (2+1 D CS term)

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where




Electric charging from spin textures!

Electromagnetic Duality

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L.-X. Qi, Science (2009).

,

generalization of dyons

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Vortices


𝑞 ∈ ℤ, vortex charge


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Only vortices with 𝑞 = 1, have finite total charge.

Vortices


Vortex (𝑞 = 1) – anti-vortex (𝑞 = −1) symmetry is broken on TI.


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Domain Walls



Neel wall is charged :

cf. Total charge of Bloch wall =0

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Domain Walls


As DW velocity is given as under a magnetic field,

Electric field induces DW motion

With typical parameters,


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Summary of the previous work


Spin textures are electrically charged on TI, so an electric field can induce motion of spin textures, such as DW.

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The END


20140328 TNTL journal club axion electrodynamics, TI-FI interface (nomura, nagaosa)

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