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Spin electronics at the nanoscale
Michel Viret
Service de Physique de l’Etat Condensé
CEA Saclay
France
Principles of spin electronics:
ferromagnetic metalsspin accumulation
Resistivity of homogeneous materials:
AMRDWR
Spin torque in DWs
Reduced dimensions:
Mesoscopic transportAtomic MR
Different DOS for upand down spins :
Spin dependent electrical transport in ferromagnetic metals
s electrons : low density of states + high mobility
d electrons : large density of states + low mobility
Transport is dominated by s electrons scattered into d bandsd bands split by the exchange energy→ diffusion is spin dependent
→ Two current model :Two conduction channels in parallel
with ρ↑≠ ρ↓
Resistivity :
or (with spin-flip) :
ρρρρρρρρ
↑↓↓↑
↓↑↑↓↓↑
++
++=ρ
4
)(
ρρρρ
↓↑
↓↑
+=ρ
E
d bands
s bands
Spin
down
spin
up
↑↑↑↑
↑↑↑↑↓↓↓↓
↓↓↓↓
Current generated spin accumulation at a Ferro (Co) / Normal metal (Cu) interface:
Typically, at 4.2 K :lsf (Co) ≈ 60 nmlsf (Cu) ≈ 500 nm
H (kOe)
80%
I
V
Multilayers
F metal / NM metal(ex : Fe / Cr, Co / Cu, etc )
4 K
Conf igurat ion P
M MNM
-
+
M MNM
+
-
Conf igurat ion AP
r r
r
r
R R R
RR+= r
R- = R
R+ = (r+R)/ 2
R- = (R+r)/ 2
P
PAP
R
RR −=GMR
rrR
RrRP ≈≈≈≈
++++====
4
rRRAP
++++====<
Introduction to spin electronics
Domain wall resistance
Example of FePd:
MFM image 2 x 2 µm2 at zero field, in the virgin state and after saturation. The up domains are black andthe down ones are white
Resistive
measurements : on a nanostructure withstripe width = 300nm
0.5
7.992
8.016
8.040
8.064
8.088 ρ
CPW
ρCIW
ρ (µΩ
cm
)
H
∆R/R = 8 % within the DWs
Why is it so small?
(R. Danneau et al., Phys. Rev. Lett. 88, 157201 (2002))
Spin transfer from the conduction electrons to the DW
Current
direction
Theory
Two kinds of electrons:
• Localised• Conduction electrons
→ s-d Hamiltonian
Action of a current:
Globally, the conduction electrons transfer gµB to the DW
s-d Hamiltonian :
: localised spins
Precession equation :
s : conduction electrons
⇒⇒⇒⇒
Simple model : the particle approach
( )
⇒⇒⇒⇒
In the rotating frame:
Loc
a l M
omen
t
Electron spinφ0
In the frame of the local moment :
In the laboratory frame
Spin evolution during DW crossing
Loc
a l M
omen
t
Electron spinφ0
In the frame of the local moment :
In the laboratory frame
Spin evolution during DW crossing
exw
F
J
v
δφ
η=0
2wex
F
2
w 1
J
v
)p1(
p2
R
R
δ
−=
∆ η
Loc
a l M
omen
t
Electron spinφ 0
In the frame of the local moment direction :
Loc
a l M
omen
t
Electron spinφ0
In the frame of the local moment :
In the laboratory frame
Spin evolution during DW crossing
exw
F
J
v
δφ
η=0
2wex
F
2
w 1
J
v
)p1(
p2
R
R
δ
−=
∆ η
Loc
a l M
omen
t
Electron spinφ 0
In the frame of the local moment direction :
Loc
a l M
omen
t
Electron spinφ0
In the frame of the local moment :
In the laboratory frame
Spin evolution during DW crossing
exw
F
J
v
δφ
η=0
2wex
F
2
w 1
J
v
)p1(
p2
R
R
δ
−=
∆ η
Loc
a l M
omen
t
Electron spinφ 0
In the frame of the local moment direction :
Loc
a l M
omen
t
Electron spinφ0
In the frame of the local moment :
In the laboratory frame
Spin evolution during DW crossing
exw
F
J
v
δφ
η=0
2wex
F
2
w 1
J
v
)p1(
p2
R
R
δ
−=
∆ η
Loc
a l M
omen
t
Electron spinφ 0
In the frame of the local moment direction :
Loc
a l M
omen
t
Electron spinφ0
In the frame of the local moment :
In the laboratory frame
Spin evolution during DW crossing
Loc
a l M
omen
t
Electron spinφ0
In the frame of the local moment :
In the laboratory frame
Spin evolution during DW crossing
Loc
a l M
omen
t
Electron spinφ0
In the frame of the local moment :
In the laboratory frame
Spin evolution during DW crossing
Loc
a l M
omen
t
Electron spinφ0
In the frame of the local moment :
In the laboratory frame
Spin evolution during DW crossing
Loc
a l M
omen
t
Electron spinφ0
In the frame of the local moment :
In the laboratory frame
Spin evolution during DW crossing
For a long wall and
Precession around the effective field :
Rotating frameMagnetisation
Magnetic
moment
Direction of
electron
propagation
Laboratory frame
→ The mistracking angle is small (a few degrees) and theinduced spin scattering is weak
The total moment is conserved →
Reaction on the wall
The torque can be decomposed into a constant and periodic partFor long walls, the periodic part averages to zero and the constant part reads:
distortion (steady state)
P = polarisation, j = current density pressure
• Torques: non-homogeneous within the walls + small ‘pressure’ term
• Importance of the magnetic structure of the DW
• Very thin DWs: Enhanced pressure oscillating with thickness
Conclusions :
Effect of the current : Globally, the conduction electrons transfer gµB to the DW→→→→ Spin torque
Preliminary conclusions
Question: Can we build spin electronics devices with domain walls?
DW: topological soliton which can be moved by a current and which scatters electronsRussel Cowburn (Durham/London) : magnetic logic based on DWsStuart Parkin (IBM San Jose) : registery memory
Problems: DWs are not very resistive and cannot be pushed by small currents…
Solution : 1D structures
Ohm’s law: I = G VG = σ S / L if L >>
• lF Fermi wavelength (quantum)
• ℓ mean free path (elastic)
• Lj coherence length (inelastic)
macroscopicdiffusiveatomic ballistic
Different transport regimes
Magnetoresistance in reduced dimensions: the constriction
+ magnetism :
Spin degeneracy removed by the exchange energy
≠ Fermi wavelength for up and down electrons
→ the size of the constriction at which the
number of transmitted channels changes
is spin dependent
↓↑
↓↑
≠→
−±=
=
++∇−=
NN
kEEm
k
dnk
JrVm
H
perpexFlong
perp
ex
)(2
2.
.)(2
2
2
*,
22
η
η
π
σ
⇒ conductance quantized :
G = ΣΣΣΣTi↑↓↑↓↑↓↑↓G0 (G0 = e2/h = 1/26kOhm)
2D electron gases : σ quantized in units of 2G0 , but large H : quantization in G0
Perfect case: continuous materials of cross section close to λF (ex: 2DEG) : k⊥ quantized
Closer to real life :
Metals : λF ≈ 2Å→ quantization requires atomic contacts !Conducting channels defined by overlap of atomic orbitals→ atomic calculation needed+ In the single atomic regime more than one conduction channels are opened for 3d elements. Ni: 4s2 3d8→ potentially 4↓ + 6↑ channels
Magnetic problems :DWs not infinitely thin: Micromagnetic configuration of the relevant atoms?
Experimental problems : Magnetostriction ??For Ni : λ ≈ -40 ppm → 100 µm wire shrinks by 4 nm !
Introduction of a DW in a constriction (ideal case) :DW width = size of constriction (P. Bruno, PRL83, 2425 (1999), Y. Labaye et al., J.A.P.91, 5341 (2002))
Constricted DW width ≈constriction diameter or length
+ Potential barrier of magnetic (exchange) origin (Imamura et al. PRL 84, 1003 (2000)) : large MR effects because a DW can close the conductance channel.
Break junction technique :Samples : ferromagnetic bridges suspended with pads of different shapes made in polycrystalline Ni, Co, Fe :
MR in ferromagnetic atomic contacts
Micromagnetics:
Program OOMMF (NIST)
Experimental setup
Bending system in a cryostatPulling ≈ 1 nm/turn
H. Ohnishi, Y. Kondo and K. Takayanagi,
Nature 295, 780 (1998)
TEM pictures while pulling an Au tip from a Au film :
Measurements : breaking
01234567
89
101112131415
-1 -0.8 -0.6 -0.4 -0.2 0 0.2
d (nm)
(e2/h)
Conductance steps : atomic reorganization (+ quantization)
Tunnelling : R = R0 exp(x/x0) , x0=0.045 nm
Tunneling regime - H=0.4T perp
3
4
5
6
7
8
9
10
11
0 0.01 0.02 0.03 0.04 0.05gap (nm)
ln(R)
Two types of measurements
R(θ) → AMR:
0
0.2
0.4
0.6
0.8
1
1.2
-90 -60 -30 0 30 60 90 120 150 180 210 240 270
Angle (degree)
Resistance
R(Η) → ‘DWR’ :
- 9 0 - 6 0 -3 0 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0
6 9 8
7 0 0
7 0 2
7 0 4
7 0 6
A n g l e ( d e g r e e )
Atomic contact:
∆∆∆∆R/Rmin= 75%
9 0 0 0
1 0 0 0 0
1 1 0 0 0
1 2 0 0 0
1 3 0 0 0
1 4 0 0 0
1 5 0 0 0
1 6 0 0 0
1 7 0 0 0
1 8 0 0 0
Atomic contact:
∆∆∆∆R/Rmin= 21%
7 2 0 0
7 4 0 0
7 6 0 0
7 8 0 0
8 0 0 0
8 2 0 0
8 4 0 0
8 6 0 0
8 8 0 0
R (
Oh
m)
Nanostructure:
∆∆∆∆R/Rmin= 1.1%
Atomic contact regime (Fe, 4.2K)
R(θθθθ) curves :
• Significant effect• Clear departure from cos2(θ)
7000
7200
7400
7600
7800
8000
8200
8400
8600
8800
-100 -50 0 50 100 150 200
angle (°)
R (Ohm)
2T, Vdc=0mV
3e2/h
3.5e2/h
→ Channels closing?
Parallel with evolution of conductance with stretching
Tentative explanation of the AMR effects :
Orbitals overlap responsible for the
opened channels
Distortion with field because of spin-orbit
coupling?
+ enhanced effects in reduced dimensions
Ab-initio calculations
Pseudo potential plane wave method + spin-orbit coupling
Fe atomic chain: magnetisation parallel and perpendicular
Tight binding calculations
fcc (111)
bcc (001)
The exact geometry of the contact is
important, especially the coordination
number of the central atom.
Tunnelling regime, Fe (4.2K) :
→AMR still large in tunnelling
Measurements quite ‘erratic’ because different configurations can give the same resistance
Tunnelling defined by the overlap of evanescent orbitals→ Spin-orbit coupling of the same nature but on the evanescent orbitals
Summary of AMR measurements in Fe
6.2
6.25
6.3
6.35
6.4
6.45
6.5
6.55
6.6
-60 -40 -20 0 20 40 60 80 100 120 140 160 180 200Angle(degrees)
R(kohms)
6
6.2
6.4
6.6
6.8
7
7.2
7.4
7.6
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
H(T)
R(kohms)
180°
90°
90°<--180°
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7
7.1
-2.5 -1.5 -0.5 0.5 1.5 2.5H(T)
R(kohms)
← Fe (4.2K)‘Domain wall’ effects
12%
27
28
29
30
31
32
33
34
35
36
37
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
H(T)
R(kohms)
180°
90°
Co (4.2K)↓
DWR ≈ 10% – 20%
21%
E-field efects
3700
3800
3900
4000
4100
4200
4300
4400
4500
4600
4700
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
H (T)
R (Ohm)
Vdc=100mV
Vdc=50mV
Vdc=0
field at 0°
Electric field effects, Fe (4.2K):
Effect of the electric field:
Evidence for spin torque?
At 50mV, j ≈≈≈≈ 5 108 A/cm2
differential conductance for two different magnetic configurations
8000
9000
10000
11000
12000
13000
14000
15000
16000
17000
-0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
Vbias (V)
dI/dV (Ohm)
H=0
H=1T
Conclusions for the MR at the atomic scale
conductance depends on orbital overlapS-O coupling = atomic AMR effectIn reduced dimensions: large S + large O !
Contact:
AAMR : 50 %
DWR : 20%
AAMR>DWR
Tunnelling
TAMR : 100 %
TMR : 35%
TAMR>TMR
tunnelling AMR : same with evanescent orbitals
See: M. Viret et al., Eur. Phys. J. B 51, 1 (2006)
Conclusions
Can DWs be used for spin electronics?
In the bulk :
DW resistance too small + current induced pressure too weak
In atomic constrictions:
Importance of orbital effects (AMR)
DW Resistance enhanced (20%)
Pushing with a current impossible (destruction of the contacts)
The ideal : 1D atomic chains…