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

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

TNTL
Journal Club
Phys. Rev. Lett. 108, 187201 (2012)
Yaroslav Tserkovnyak and Daniel Loss

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

Dongwook Go
1


2


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
3


Chiral Electron Mode

Bloch sphere representation
of the spin coherent state
Half integer quantum Hall effect and
anomalous quantum Hall effect
4


,
The problem is analogous to the IQHE
Let
Zero mode solution (n=0) is

with

where
5


A square-integrable solution is
,

→ Chiral Zero Mode

Characteristic width of the chiral mode :
Gap between the chiral zero mode and the gapped state :
6


Contents

• Onsager Reciprocity Principle : DW dynamics induced chiral mode current
7


Electromagnetic Response of TIs
TI surface electron coupled to guage 3-potential

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

Band structure for
the TI surface electron

or, explicitly

and

Bloch sphere representation
of the spin coherent state
8


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
the previous results.

, an additional charge density and current density agrees with

and

10


Contents

• Free energy of the DW coupled with the chiral mode current
11


Emergent Guage Field : MI-TI exchange coupling
TI surface electron at the proximity of MI

where

and

Thus, magnetic texture induces a charge response given by

,
12


Contents

13


Soft Dynamic Coordinates of the DW
Ferromagnetic DW with a perpendicular magnetic anisotropy
e.g. CoFeB alloys

with the boundary condition
Minimizing the free energy gives
,

→ Neel wall
→ Bloch wall

where

The degeneracy with respect to
and
is generally lifted by
spatial pinning fields, and applied fields or spin-orbit interactions.
14


Free Energy from the Equilibrium Chiral Mode Current
,

For the chiral zero mode,
characteristic width :

,

bandwidth

Equilibrium current density of the chiral zero mode is
where
In general,

where

in this case.
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Free Energy from the Equilibrium Chiral Mode Current
※ This step assumes TI electron is unperturbed by ferromagnetic proximity :

Let the free energy associated with the equilibrium chiral mode current
then,
Integration of the above equation leads to

First term : enhances the tendency to form magnetic textures, such as Skyrmion lattices
Second term : out-of-plane anisotropy
16


Free Energy from the Non-Equilibrium Chiral Mode Current

where

parameterizes the Luttinger-liquid strength of the electron-electron forward scattering.
17


Free Energy from the Non-Equilibrium Chiral Mode Current
Let
Similar to the previous case

where
thus

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

18


Contents

19


LLG Equation and the DW Motion
Now the total free energy is

where

and

with

So, the full LLG equation for the dynamics becomes

where

external field

exchange

anisotropy

equilibrium current non-equilibrium current
20


,

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.
21


DW dynamic equations

and

Energy dissipation leads to
(Neel wall)

(Bloch wall)
which corresponds to the lowest magnetostatic energy
22


Contents

23


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

24


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

Reciprocity principle relates DW dynamics-induced charge pumping

25


Contents

26


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
27


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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