20140113 TNTL journal club, PRL 108, 187201 (2012)

TNTLJournal ClubPhys. Rev. Lett. 108, 187201 (2012)

Yaroslav Tserkovnyak and Daniel Loss

Thin-Film Magnetization Dynamics on the Surface of a Topological Insulator

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13 Jan. 2014 - TNTL Journal Club – PRL 108, 187201 (2012)

Dongwook Go


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Contents

• TI surface Chiral Electron Mode @ DW proximity

• Electromagnetic Response of TIs / Axion Electrodynamics• Emergent guage field : MI-TI exchange coupling

• Free energy of the DW coupled with the chiral mode

• LLG equation and the DW motion

• Onsager Reciprocity Principle : DW-dynamics-induced chiral mode current


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Chiral Electron Mode


Bloch sphere representation of the spin coherent state

Half integer quantum Hall effect and anomalous quantum Hall effect

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,

The problem is analogous to the IQHE

Let

Zero mode solution (n=0) is

with

where5



A square-integrable solution is

,

→ Chiral Zero Mode

Characteristic width of the chiral mode :

Gap between the chiral zero mode and the gapped state :

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Contents





• Onsager Reciprocity Principle : DW dynamics induced chiral mode current


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Electromagnetic Response of TIs

TI surface electron coupled to guage 3-potential

Long-wavelength, low-frequency charge response (Kubo formula)

or, explicitly

and

Band structure for the TI surface electron

Bloch sphere representation of the spin coherent state

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Axion Electrodynamics @ TI surface

With and ,

Integration of the equation above produces the Chern-Simons action for the electromagnetic field

Electromagnetic Lagrangian is given by , where is the ordinary EM Lagrangian, and

with

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Axion Electrodynamics @ TI surface

This Axion term gives modified Maxwell equations as follows :

In the case of time-independent , an additional charge density and current density agrees with the previous results.

and

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Contents



• Free energy of the DW coupled with the chiral mode current



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Emergent Guage Field : MI-TI exchange coupling

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TI surface electron at the proximity of MI

where and

Thus, magnetic texture induces a charge response given by

,


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Soft Dynamic Coordinates of the DW

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Ferromagnetic DW with a perpendicular magnetic anisotropye.g. CoFeB alloys

with the boundary condition

Minimizing the free energy gives

,

→ Neel wall

→ Bloch wall

The degeneracy with respect to and is generally lifted by spatial pinning fields, and applied fields or spin-orbit interactions.

where


Free Energy from the Equilibrium Chiral Mode Current

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Equilibrium current density of the chiral zero mode is

where

In general,

where in this case.

For the chiral zero mode, ,

characteristic width : , bandwidth


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Let the free energy associated with the equilibrium chiral mode current

then,

Free Energy from the Equilibrium Chiral Mode Current

※ This step assumes TI electron is unperturbed by ferromagnetic proximity :

Integration of the above equation leads to

First term : enhances the tendency to form magnetic textures, such as Skyrmion latticesSecond term : out-of-plane anisotropy


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Free Energy from the Non-Equilibrium Chiral Mode Current

where parameterizes the Luttinger-liquid strength of the electron-electron forward scattering.


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Free Energy from the Non-Equilibrium Chiral Mode Current

Let

where

Similar to the previous case

thus

Meanwhile, non-equilibrium chiral current is given by the Landauer-Buttiker formula,


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LLG Equation and the DW Motion

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Now the total free energy is

where with and

So, the full LLG equation for the dynamics becomes

where

external field exchange anisotropy equilibrium current non-equilibrium current


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,



Substitution of the ansatz leads to

and

where

And there’s no generalized force corresponding to the DW position since there’s no pinning potential.

Energy dissipation

is mediated by the equilibrium chiral zero mode current.

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andDW dynamic equations

(Neel wall)

(Bloch wall)

which corresponds to the lowest magnetostatic energy

Energy dissipation leads to

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Onsager Reciprocity Principle : DW-dynamics induced chiral mode current

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Onsager reciprocity

Onsager Reciprocity Principle : DW-dynamics-induced chiral mode current

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Voltage-induced DW dynamics

Reciprocity principle relates DW dynamics-induced charge pumping

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Summary

• Parity-anomaly chiral electron mode induces DW dynamics.

• DW dynamics is parametrized by soft dynamic coordinates.

• In the absence of external field, the DW switches between two types Neel walls, depending on the sign of the spin torque.

• In the presence of external field, the DW switches between two types of Bloch walls depending on the sign of the applied field.

• Onsager reciprocity principle implies charge pumping due to the DW motion.

• Potential application may be “magnetic lithography” such that the position of a ballistic electron channel is controlled

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References

• Yaroslav Teserkovnyak and Daniel Loss, Phys. Rev. Lett. 108, 187201 (2012)

• M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010)

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20140113 TNTL journal club, PRL 108, 187201 (2012)

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