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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
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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)
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 7
Equation of motion
Model
Gradient expansionExact determination
of coefficients
Spin Torque
Landau-Lifshitz-Gilbert
First part of the talk
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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)
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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)
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 10
Kinetic energy and current
total spin density (to calculate STT)
total spin current
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 11
Rashba
1D system
around Fermi vector
Gangadharaiah, Sun, Starykh, PRB 2008
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 12
Resulting Hamiltonian
: direction defined by SOI
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 13
Heisenberg equation of motion
Kac-Moody algebra
just the continuity equation when we sum over p
spin current density
spin density
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 15
Solving the EOM
Ansatz: gradient expansion
for finite SOI: use
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 16
Gradient expansion
arrange EOM for distinct orders of and solve successively
Here: only up to first order:
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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!
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 21
momentary motion determined by both: current and Rashba field
oscillation frequency dependent on Rashba coupling and current density
Steady current
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 22
accompanied by precession
Precession
x-axis → hard axis
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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)
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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:
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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)
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 29
Suppression of Walker breakdown by SOI
Rashba fieldstabilizes DW
Suppressionof WB
M. Stier, M. Creutzburg and M. Thorwart, submitted to PRB (2014)
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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Interacting domain walls: dragging
effective Hamiltonian:
new contribution to STT:
M. Stier and M. Thorwart, in prep. (2014)
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 31
Interacting domain walls: dragging
M. Stier and M. Thorwart, in prep. (2014)
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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)
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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.
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 34
Equation of motion
Model
Gradient expansionExact determination
of coefficients
Spin Torque
Landau-Lifshitz-Gilbert
Second part of the talk
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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)
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 37
Differential equation
Ordinary inhomogeneous differential equation
dependent on magnetic texture
no analytical solution → solve numerically
but try to understand basics first
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 38
broad DWs → neglect any texture
Differential equation – basics, no
algebraic equation
yields common local STT
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 39
steep DWs → derivatives important
Differential equation – basics, no
common parts get unimportant
expect new behavior
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 40
general DWs → approximate solutions
Differential equation – basics, no
eigenvalues
solutions similar to
expect damping and oscillations
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 42
Numerical results
solve ODE
get spin torque
solve LLG
standard parameters
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 44
DW velocity vs DW width
DW velocity saturates at common value for broad DW decreases to zero at low widths
maximum
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 45
Now with Rashba spin-orbit interaction
basically replace
different basis
different coefficient matrix
Rashba field
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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?
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 47
Non-local Rashba field
Rashba field has the approximate shape of the current in n direction
additional oscillations from bp
terms
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
Page 50
Thank you for your attention!
Michael Thorwarthttp://wwwnano.physnet.uni-hamburg.de
Current induced DW motionin Rashba nanowires 02.05.2014
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