Current-induced domain wall motion in Rashba quantum wirestfp1.physik.uni-freiburg.de/Capri14/lectures/Thorwart_Capri2014.pdf · Current-induced domain wall motion in Rashba quantum

Current-induced domain wall motion in Rashba quantum wires

Michael ThorwartI. Institut für Theoretische Physik

Universität Hamburg

Capri Anniversary Workshop & Spring School 2014

Credits toDr. Martin Stier

in collaboration with Reinhold Egger

Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de

Current induced DW motionin Rashba nanowires 02.05.2014

Page 2

Domain walls in possible applications

application: all-electronic racetrack memory

reliable and fast motion of domain walls

motion at low current densities

S. Parkin, IBM



Page 3

Dynamics of a domain wall

Landau-Lifshitz-Gilbert-Equation

magnetic moments n(x,t) within the wire

given DW texture due to intrinsic effective field

additional spin torque by external field and/or spin polarized current



Page 4

Current-induced spin transfer torque

inject spin polarized electrons

move domain wall

interaction

spin torque on domain wall

additional effects may change s



Page 5

Known expressions for spin transfer torque

adiabatic STT

non-adiabatic STT

“adiabatic” Rashba STT

the (non-)adiabatic torques → complex DW motion effects of the Rashba field?

valid for steep DWs? effects of non-local STT and Rashba field?

(Berger 1996)

(Zhang & Li 2004)

(Obata & Tatara 2008)



Page 6

Outline

set up model

analytical result for STT

control DW velocities by Rashba field

steady current vs pulses

chirality switching

interacting DW: drag

effect of temperature

Analytic expressions for STT from DW gradient expansion

generalized Ansatz for STT

steep DW

non-local STT

(similar to RKKY)

DW velocity vs DW width

role of Rashba field

“Exact” calculation of STT and effects of non-local terms



Page 7

Equation of motion

Model

Gradient expansionExact determination

of coefficients

Spin Torque

Landau-Lifshitz-Gilbert

First part of the talk



Page 8

important: → spin density operator for left and right movers

Model

linearize electron dispersion

1D nanowire → “Luttinger” liquid with g=1

defines almost complete physics

Gogolin, Nersesyan, Tsvelik, Bosonization & Strongly Correlated Systems (1998)



Page 9

Model: 1D quantum wire

kinetic part of the Luttinger liquid

Sugawara form (spin part) → only spin operators

Gogolin, Nersesyan, Tsvelik, Bosonization & Strongly Correlated Systems (1998)



Page 10

Kinetic energy and current

total spin density (to calculate STT)

total spin current



Page 11

Rashba

1D system

around Fermi vector

Gangadharaiah, Sun, Starykh, PRB 2008



Page 12

Resulting Hamiltonian

: direction defined by SOI



Page 13

Heisenberg equation of motion

Kac-Moody algebra

just the continuity equation when we sum over p

spin current density

spin density



Page 14

Relaxation: nonadiabatic STT

relaxation time approx → justified microscopically by system-bath approach*

* M. Thorwart and R. Egger, PRB 76, 214418 (2007)

spin torque is dependent on

set reasonable state

spin current density



Page 15

Solving the EOM

Ansatz: gradient expansion

for finite SOI: use



Page 16

Gradient expansion

arrange EOM for distinct orders of and solve successively

Here: only up to first order:



Page 17

Resulting spin transfer torque

adiabatic STT

non-adiabatic STT

field-like STT

nonadiabatic Rashba field

M. Stier, R. Egger and M. Thorwart, PRB 87, 184415 (2013)

“adiabatic” Rashba field



Page 18

Remarks on theory part

straightforward, no explicit consideration of electric field

“common” STT recovered and new terms beyond

extension to higher order terms in gradient expansiondeformation of DW: M. Thorwart and R. Egger, PRB 76, 214418 (2007)

( but we will introduce an “exact” method later )

Related recent works:D.A. Pesin & A.H. MacDonald, PRB 86, 014416 (2012)

E. van der Bijl & R.A. Duine, PRB 86, 094406 (2012)

X. Wang & A. Manchon, PRL 108, 117201 (2012)

… but now check effect of Rashba field on DW motion!



Page 19

Possible effects of a Rashba field

I. Field perpendicular to magnetic moments

field acts as an additional “anisotropy”

could prevent a current-induced Walker breadown

from Handbook of Magnetism and Advanced Magnetic Materials, 2007



Page 20

Possible effects of Rashba field

II. Field parallel to magnetic moments

field can directly move the DW (“field-induced motion”)

If field exceeds a critical value, a field-induced Walker breakdown may occur

standard parameters (inspired by Co nanowires):

We will use this set-up for our numerical calculation

DW is moved simultaneously by the current and the Rashba field



Page 21

momentary motion determined by both: current and Rashba field

oscillation frequency dependent on Rashba coupling and current density

Steady current



Page 22

accompanied by precession

Precession

x-axis → hard axis



Page 23

Current pulse

DW still in motion after current pulse ('drifting')

direction of velocity dependent on amplitude at the end of pulse

drifted distance can be remarkably long



Page 24

contains travelled distance during and after current pulse pulse length determines

velocity can be strongly increased or

decreased

Controlling effective velocity by current pulses



Page 25

Non-linear dependence on current density

oscillation period controlled by current density

the stronger the current, the smaller the period

this determines if the pulse ends at positive or negative amplitude

x 1012



Page 26

Momentary Rashba field

Rashba field sizeable around DW position

Nonadiabatic Rashba field can become large

M. Stier, R. Egger and M. Thorwart, PRB 87, 184415 (2013)



Page 27

Role of temperature

β-parameter

Relaxation rate depends on temperature

System-bath model, weak-coupling regime, Markov (single phonon processes):

M. Stier, G. Gurski and M. Thorwart, in prep (2014)

Co nanowire:

Py nanowire:



Page 28

Switching of DW chirality by SOI

Rashba field perpendicular to mag. moments

Competition between Kand H

R

Rasba field can switch DW chirality

x

z y

M. Stier, M. Creutzburg and M. Thorwart, submitted to PRB (2014)



Page 29

Suppression of Walker breakdown by SOI

Rashba fieldstabilizes DW

Suppressionof WB

M. Stier, M. Creutzburg and M. Thorwart, submitted to PRB (2014)



Page 30

Interacting domain walls: dragging

effective Hamiltonian:

new contribution to STT:

M. Stier and M. Thorwart, in prep. (2014)



Page 31

Interacting domain walls: dragging

M. Stier and M. Thorwart, in prep. (2014)



Page 32

Notes on experiments

Miron et al., Nature Mater. 9, 230 (2010)

Miron et al., Nature Mater. 10, 419 (2011)

Co nanowire in trilayer with structural inversion asymmetry

SIA-mediated Rashba field

see also Jamali et al., PRL 111, 246602 (2013)



Page 33

Summary part I

Rashba field may induce or suppress Walker breakdown

field strength and oscillation period proportional to Rashba coupling and current density

drifting of DW after current pulse in distinct direction

drifted distance important for short pulses

We can control drift direction by pulse length.

We can control DW chirality by SOI + current pulses.

We can drag one DW by a current-driven second DW.



Page 34

Equation of motion

Model

Gradient expansionExact determination

of coefficients

Spin Torque

Landau-Lifshitz-Gilbert

Second part of the talk



Page 35

Back-action of DW on currentHow does the spin current react on the DW?

broad DWs? steep DWs?

possible polarization of current in n direction

allow for space (time) dependence of polarization

P

x x

P



Page 36

Same EOM but different Ansatz, no SOI

complete orthogonal basis in spin space

allow for space-time-dependent coefficients

insert Ansatz into EOM → differential equations for coefficients

M. Stier and M. Thorwart, in prep (2014)



Page 37

Differential equation

Ordinary inhomogeneous differential equation

dependent on magnetic texture

no analytical solution → solve numerically

but try to understand basics first



Page 38

broad DWs → neglect any texture

Differential equation – basics, no

algebraic equation

yields common local STT



Page 39

steep DWs → derivatives important


common parts get unimportant

expect new behavior



Page 40

general DWs → approximate solutions


eigenvalues

solutions similar to

expect damping and oscillations



Page 42

Numerical results

solve ODE

get spin torque

solve LLG

standard parameters



Page 43

normalized directions

non-locality of spin torque and spin current

damping

and oscillations

drastic differences to “common” solution at small DW widths

how does it affect DW velocity?

Numerical results



Page 44

DW velocity vs DW width

DW velocity saturates at common value for broad DW decreases to zero at low widths

maximum



Page 45

Now with Rashba spin-orbit interaction

basically replace

different basis

different coefficient matrix

Rashba field



Page 46

Non-local STT with SOI

spin torque and current are now asymmetric

due to terms such as

which depend on direction

Implication for the Rashba field?



Page 47

Non-local Rashba field

Rashba field has the approximate shape of the current in n direction

additional oscillations from bp

terms



Page 48

DW velocity vs DW width with SOI

as for vanishing SOI: strongly dependent on DW width even for comparably broad DWs difference to local solution



Page 49

Summary second part

non-local spin torque (similar to RKKY)

related to this: non-local Rashba field

significant dependence of DW motion on DW width



Page 50

Thank you for your attention!



Page 51

Current induced motion

from Handbook of Magnetism and Advanced Magnetic Materials, 2007

● low β● high threshold current

● high β● Walker breakdown

● inclusion of Rashba interaction may change this picture



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Current-induced domain wall motion in Rashba quantum wirestfp1.physik.uni-freiburg.de/Capri14/lectures/Thorwart_Capri2014.pdf · Current-induced domain wall motion in Rashba quantum