Chapter 2
Spinelektronik: Grundlagen und Anwendung spinabhängiger Transportphänomene 1
Winter 05/06
2.0 Scattering of charges (electrons)
In order to establish a steady current under an applied magnetic field, the electrons must be scattered inelastically. Main scattering mechanisms in (non-magnetic) metals
• phonons (temperature dependent)
• defects (temperature independent)
Defects are responsible for residual resistivity at low temperature.
2.1 General Expression for j
Current density
€
r j = − e
4π
3d r
∫ k ν r ( k r ) ⋅ f ( k r ) = σ ⋅ E
introduce expression for
€
f ( r
k ) = f
o( r
k ) − τ ⋅ υ ⋅ eE (− ∂f
o∂E )
€
r j = − ε
4π
3d r k ( υ r ( r
k ) ⋅ f
o(k )) − υ r ( r k ) o υ r r
k ⋅ τ ( r k ) ⋅ e r
E − ∂f
o∂E
∫
∫
first term vanishes → symm. Integration, asymm. Integrand
€
⇒ σ = e
24π
3d r k τ ( r
k ) υ r ( r
k ) o υ r ( r k ) − ∂f
o∂E
∫
Metals: Behavior determined by Fermi function
€
f
o→ changes only within
€
kΤ
→€
∂f
o∂E = −δ(E − E
F)
→ only states at
€
Ε
F contribute to the transportSpinelektronik
Spinelektronik: Grundlagen und Anwendung spinabhängiger Transportphänomene 2
Viewgraph 2 Viewgraph 1
€
σ
= e4
π
3hE=E∫
FdSFτ
(k r F)υ
(k Fυ
r )(orυ
(k F) k F)→ Integration only over Fermi surface
Actual shape of the Fermi surface is important.
For a cubic crystal (e.g. Cu)
€
σ = σ
xx= σ
yy= σ
zz€
σ = e
212π
3h ds
FFS
∫ υ
Fτ ( k r
F)
for a free electron gas this again yields the Drude conductivity
€
σ
met= e
2n ⋅ τ (k
F) m
∗For semiconductors, the situation is more complicated as EF lies within the gap. There are contributions from both electrons (conduction band) and defect electrons – holes (valence band).
Drude term:
€
σ
sc= e
2nτ
Nm
n+ e
2pτ
Pm
pImportant:
€
τ
n( r
k ) ≠ τ
p( r k )!
2.2 Influence of a Magnetic Field
Assuming only weak magnetic fields, the electron in the solid are subject to the Lorentz force
€
F = r −e( r
E + υ × r r B )
Boltzmann equation must include an extra term
Spinelektronik: Grundlagen und Anwendung spinabhängiger Transportphänomene 3
Viewgraph 3
Winter 05/06
€
∂f
∂t
magn= − e
h ( υ × r r B ) ⋅ ∂f
k r∂ r k
Problem:
€
f
k r cannot be simply replaced by€
f
o( r
k )
, as the influence of€
B r
on the equilibrium distribution
€
f
o( r
k )
is zero.→ next order in the expansion is needed.
This yields for the deviation
€
g( r k )
from€
f
o(k )
€
g( r
k ) = −τ [ υ r ∂f
o∂ r r − e h
E r ∂f
o∂ r k − e
h ( υ × r r B ) ∂g
∂ r k ]
despite
€
∂g
∂ r
k << ∂f
o∂ r
k
, the last term is important, because the magnetic field B can cause strong “equivalent electric fields”.in metals E~1 V/m
€
B = 100mT → υ
F⋅ B = 10
6m
s ⋅ 1 Vs
m
−2= 10
6V m
A simple solution for g can only be given for the NFE approximation
€
g = τ e r
E ⋅ υ ⋅ r ∂f
o∂ε
+ e
m
∗( υ × v v B ) ∂g
∂υ ]
assuming
€
B r ⊥ r E
.A solution to this problem
€
g = −eτ − ∂f
o∂E
1
1 + ω
02τ
2[ (E
x− E
yω
0τ )υ
x+ (E
y+ E
xω
oτ )υ
y] Spinelektronik
Spinelektronik: Grundlagen und Anwendung spinabhängiger Transportphänomene 4
with the cyclotron frequency
€
ω
o= eB
m
∗ (valid for electrons,€
−ωo for holes!) This situation describes the Hall effect
€
j
x= M
1E
x− M
2E
y€
r j = σ r E
€
j
y= M
1E
x+ M
1E
y (2 x 2) tensor For metals, one obtains
€
M
1= e
2n m
∗τ (E
F) 1 + ω
02τ
2(E
F)
B=0 → M
1= e
2nτ m
∗
€
M
2= e
2n m
∗τ
2(E
F) ⋅ ω
01+ ω
02τ
2(E
F)
B=0
→ M
2= 0
The Hall field Ey is measured with
€
j
y= 0
€
⇒ E
y= − j
xM
2M
12+ M
22Spinelektronik: Grundlagen und Anwendung spinabhängiger Transportphänomene 5
Viewgraph 4
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The Hall coefficient
€
RH is defined as
€
j
x= 1 R
H⋅ E
yB
€
R
H= − 1 en
€
⇒ Hall effect depends linearly on the magnetic field B and on the electron density. Hall coefficient is field independent.
Expression for j can be rewritten in a tensor form
€
σ ( B) = e
2nτ m
∗1 1 + ω
02τ
21 +ω
0τ
− ω
0τ 1
€
⇒ σ
xx are independent of B in this model. This is a consequence of the effective single band model
€
σ
xy= −σ
yxSide remark: for higher fields there is a transition to quantum Hall effect.
Viewgraph 6
2.3 Normal Magnetoresistance
Experiment shows that normally
€
σ
xx= f ( B)
!→ single band model is insufficient!
Spinelektronik
Spinelektronik: Grundlagen und Anwendung spinabhängiger Transportphänomene 6
A two band model with
€
m
1∗,€
m
2∗ and€
τ
1,τ
2 yields an expression (e.g., Cu belly and neck, see above)€
σ
xx= (σ
1+ σ
2)
2+ (σ
1ω
2τ
2− σ
2ω
1τ
1)
2σ
1+ σ
2+ σ
1ω
22τ
22+ σ
2ω
12τ
12= σ
xx(B)
with
€
σ
1= ne
2τ
1m
1*€
σ
2= pe
2τ
2m
2*and
€
ω
1(B),ω
2(B)
cyclotron frequencies.This behavior is called normal magnetoresistance.
€
σ
xx( B) > σ
xx(O) ⇒
positive magnetoresistanceDefinition of MR:
€
ρ( B) − ρ(o)
ρ(o) = (ω
cτ )
2
€
= ( eB m τ )
2Spinelektronik: Grundlagen und Anwendung spinabhängiger Transportphänomene 7
Viewgraph 7
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€
= ( R
Hρ
o)
2B
2Normal MR can have large values at low T, if the remaining resistivity is small in very clean metals. Typical values are of the or-
der of a few percent.
In very thin films, behavior may be different. Fuchs- Sondheimer model predicts a negative normal magne- toresitance.
2.4 Fuchs-Sondheimer Model
Present derivation relies on
€
f ≠ f ( r r )
, i.e., distribution function does not depend on space.For confined systems this assumption breaks down. For thin films, for ex- ample,
€
f
k r→ 0
at the surfaces/interfaces.The Fuchs-Sondheimer model rederives the Boltzmann equation for a spatially inhomogeneous situation
€
g = df
odE τ ⋅ υ ⋅ eE + h r k m ⋅ r
∇
rg
Thin film with normal z
€
g = df
0dE τ ⋅ υ ⋅ e ⋅ E + τ h k
zm ⋅ ∂g
∂z
with the solution
Spinelektronik
Spinelektronik: Grundlagen und Anwendung spinabhängiger Transportphänomene 8
Viewgraph 8
€
g(z, k ) = υ
xeE df
0dE 1 + exp − m
∗z h τk
z
with boundary conditions
€
g(z = 0) = g(z = t) = 0
(t film thickness) on gets€
σ = e
24π
3h υ( r k )τ ( r
k ) df
0∫ dE 1 + exp − h mz τk
z
drdk
€
⇒ ρ
ρ
B= 1 + 3λ 8t
€
t >> λ
mean freepath
Sie können auch den anderen Grenzfall noch diskutieren.
Initial motivation for the FS model was to determine
€
λ
experimentally.Can be done by measuring resistivity as function of film thickness.
Why is the MR signal negative in some cases?
Viewgraph 10
The FS model is important in context of the giant magnetoresistance (GMR).
Spinelektronik: Grundlagen und Anwendung spinabhängiger Transportphänomene 9
Viewgraph 9
Spinelektronik
Chapter 2
Winter 05/06
Spinelektronik
• constant current density j
• electrons are inelastically (|k A |≠|k B |) scattered from occupied (A) into unoccupied states (B)
• elastic scattering only expands Fermi sphere
Winter 05/06
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Electron (charge) scattering mechanisms
• scattering at phonons ➙ depends on temperature T
• scattering at defects ➙ residual resistivity at low T natrium
~1/T
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• Fermi surface of Cu based on the 6th band (mainly pd-like states)
• states are different at belly and neck
• normal magnetoresistance
can also be observed in Cu
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Hall effect
• electric field E x displaces Fermi surface
• magnetic field B z rotates Fermi surface
• Hall field E H displaces Fermi surface (j y =0!)
magnetoresistance
= difference in δk
xWinter 05/06
Spinelektronik
• resistivity depends linearly on B
• Hall constant is independent of B σ ( B) = e
2n τ
m
∗1 1 + ω
02τ
21 + ω
0τ
− ω
0τ 1
⎛
⎝ ⎜ ⎞
⎠ ⎟
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Quantum Hall effect
• Initial linear increase is replaced by step- like function in ρ xy
• resonances in ρ xx
(closed orbits of
charge carriers within
the sample)
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• 100 nm Co film ➙ high field MR is positive
• peak at low fields is due to anisotropic MR
j
B
j
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Normal magnetoresistance
• amorphous alloy has higher resistivity due to disorder
• normal MR reaches ~ 1%
ρ(300K)
(µΩ · cm) Δρ/ρ
Ni 10.7 +2.5
Fe 15 +0.8
Co 10.3 +3.0
a-Fe 80 B 20 120 ~0
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Spinelektronik
• FS model considers inhomogeneous distribution function
• resistivity in thin films increases with respect to infinite bulk situation
⇒ ρ
ρ B = 1 +
3 λ
8 t
⎡
⎣ ⎢ ⎤
⎦ ⎥
Winter 05/06
