Band Theory of Ferromagnetism
Angle Resolved Photoelectron
Spectroscopy (ARPES)
Phys. 342/555:Ch12B (2008)
The Band Theory of Ferromagnetism
Model including both band structure and correlation of
same spin electrons. Stoner and Wolfarth
Ansatz:
E↑ (k) = E(k) −
E↓ (k) = E(k) −
Define
n↑ − n↓
R=
N
I(n↑ + n↓)/2N
In↑
Stoner I is a parameter
describing energy
reduction due to
correlation
N
In↓
N
M=
μB N
V
IR
N
~
IR
E↓ (k) = E (k) +
N
R=
μB N n↑ − n↓
V
~
E↑ (k) = E (k) −
~
E (k) = E(k) −
N
I(n↑ + n↓ )
N
Phys. 342/555:Ch12B (2008)
The Band Theory of Ferromagnetism
Calculate R
n↑ − n↓
R=
N
1
1
1
R = ∑ f↑ − f↓ = ∑
N k
N k exp( E˜ (k) − IR /2 − E F /kT) + 1
1
1
− ∑
N k exp( E˜ (k) + IR /2 − E F /kT) + 1
There are certain conditions when R is non-zero even
when B=0, i.e. Ferromagnetism
Expand for small R
Δx
2
Δx
f (x − ) − f (x + ) = − f '(x)Δx − f '''(x)
2
3!
2
Phys. 342/555:Ch12B (2008)
The Band Theory of Ferromagnetism
1
1
1
∂f (k)
∂ 3 f (k)
3
R = ∑ f↑ − f↓ = ∑
IR −
(
IR)
∑
˜
N k
N k ∂E (k)
24N k ∂E˜ (k) 3
Look at f’ and f’’’. First derivative is negative the third
is positive. Ferromagnetism occurs when R>0
For R>0
T=0
I
∂f (k)
−1− ∑
>0
N k ∂E˜ (k)
∂f (k)
I
V
− ∑
=
˜
N k ∂E (k) (2π) 3 N
I
∂f (k)
V
− ∑
=
˜
N k ∂E (k) (2π) 3 N
⎛ ∂f ⎞
∫ dk⎜⎝− ∂E˜ ⎟⎠
Spin
1V
˜
∫ dkδ( E − E F ) = 2 N D(E F )
Phys. 342/555:Ch12B (2008)
The Band Theory of Ferromagnetism
Define
V
˜
D(E F ) =
D(E F )
2N
Stoner Criterion
The Curie Temperature is
the temperature where
Ferromagnetism goes to 0
ID˜ (E F ) > 1
I
∂f (k)
−1− ∑
=0
N k ∂E˜ (k)
Phys. 342/555:Ch12B (2008)
The Band Theory of
Ferromagnetism
Calculated I
Calculated D(E)
Phys. 342/555:Ch12B (2008)
The Band Theory of Ferromagnetism
Consider applying a magnetic Field
1
1
1
R = ∑ f↑ − f↓ = ∑
N k
N k exp( E˜ (k) − IR /2 − μB B − E F /kT)
1
1
− ∑
N k exp( E˜ (k) + IR /2 + μB B − E F /kT)
R = D˜ (E F )(IR + 2μB B)
N
2 N
˜
M = μB R = D(E F )(IM + 2μB B)
V
V
First order when T=0
N D˜ (E F )
M = 2μ
B
V 1− ID˜ (E F )
2
B
χ=
μ 0M
B
χ=
χ Pauli
1− ID˜ (E F )
Phys. 342/555:Ch12B (2008)
The Temperature Behavior of a
Ferromagnetic in the Band Model
A simple model to account for band theory
Consider only d-electrons at Fermi energy
In Ni there are
~0.54 d holes
per atom. So
magnetic
moment is
μBeff~0.54μB
Phys. 342/555:Ch12B (2008)
Band Model of Ferromagnetism
Model, density of spin up and down bands is a delta
function, separated in energy by exchange spliting I.
μB ⎡
IR
IR ⎤
˜
D(E) =
δ (E − E F − μB B − ) + δ (E − E F + μB B + )⎥
⎢
2
2 ⎦
μB ⎣
eff
Fermi Energy half way between the two δ functions
⎡
⎤
μB eff ⎢
⎥
1
1
R=
−
⎢
⎥
μB ⎢ exp[(−μ B − IR ) /kT] + 1 exp[(μ B + IR ) /kT] + 1⎥
B
B
⎣
⎦
2
2
IμB eff
TC =
μB 4k
Again we want R>0 for Ferromagnetism
Look at B=0 with the abbreviations
μB
˜
R=
R
μ
B
Phys.eff342/555:Ch12B (2008)
Band Model of Ferromagnetism
˜T
⎡
⎤
R
1
1
C
=
Tanh
−
R˜ = ⎢
⎥
˜
˜
exp[2
R
T
exp[−2
R
T
/T]
+
1
/T]
+
1
T
⎣
⎦
C
C
Look
R˜ = 1 forT = 0
R˜ = 0 forT = TC
Tc is the Curie Temperature M(T)=0
When T<<TC
When T~TC
−2TC /T
˜
R = 1− 2e
1/ 2
⎛
⎞
T
R˜ = √ 3⎜1− ⎟
⎝ TC ⎠
Phys. 342/555:Ch12B (2008)
Band Model of Ferromagnetism
Looks pretty good!
Phys. 342/555:Ch12B (2008)
Band Model of Ferromagnetism
Power law wrong
T<<TC Wrong
Problem is low energy
spin excitations
Phys. 342/555:Ch12B (2008)
Band Model of Ferromagnetism
What does this predict above the Curie Temperature?
˜T
⎡
⎤
R
1
1
= Tanh C
R˜ = ⎢
−
⎥
T
⎣exp[−2R˜ T /T] + 1 exp[2 R˜ T /T] + 1⎦
C
C
Only have magnetization when B≠0: expand around
small R and B
μB 1
μB
TC ˜
˜
˜
B0
R=
R=
B0 + R
2k T − TC
2kT
T
C
Curie-Weiss Law
χ=
T − TC
C Curie-Weiss Constant
Phys. 342/555:Ch12B (2008)
Currie-Weiss Law
For T>>TC Currie-Weiss Works
C
χ=
T − TC
For T~TC Currie-Weiss does not works: (T-TC)-4/3
C related to saturation magnetization: Does not work,
gives a C much too small!
We could use Stoner I and definition of Tc to determine Tc
using the measured value of the exchange splitting:
Doesn’t work Tc is much too large!
The failure of this picture is not because we have the band
structure wrong, but because we do not have the low
energy magnetic excitations done properly!
Phys. 342/555:Ch12B (2008)
Calculated Spin Density of State
Half Metallic!!!
Phys. 342/555:Ch12B (2008)
Half Metallic
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
Point Defects
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Phys. 342/555:Ch12B (2008)
ARPES
(angle resolved photoemission spectroscopy)
Momentum conservation
EB
~0
k|| ( f ) = k|| (i ) + k|| ( p )
k|| (i ) = k|| ( f ) =
2 mE k ( f )
sin (ϑ )
Energy conservation
Ek = hν − Φ 0 − EB
Hence, the E B and the k|| (i ) can be obtained.
F. Reinert et. al., New J. Phys. 7, 97(2005).
Phys. 342/555:Ch12B (2008)
New ARPES in SERF
This is being built currently
Scienta R4000
Phys. 342/555:Ch12B (2008)
Phys. 342/555:Ch12B (2008)
Photoemission Matrix Element
Fermi golden rule
Hamiltonian
dω 2π
=
|< Ψf | H'| Ψi >|2 {δ (E f − E i − ω )}
dt
e
e2
2
( A • P + P • A) − eΦ +
|
A
|
H = H0 +
2mc 2
2mc
Pick guage so scaler potential is 0: |A|2 is always small and if we use commutator
[ P, A] = −i ∇ ⋅ A | A |2
dσ
∝|< Ψf | 2 A • P − i ∇ • A | Ψi >|2 {δ (E f − E i − ω )}
dΩ
Second term usually small
A(r,t) = A0 exp(−iωt + iq • r )
Phys. 342/555:Ch12B (2008)
Photoemission Matrix Element: II
dσ
∝|< Ψf | P | Ψi > • A0 |2 {δ (E f − E i − ω )}
dΩ
Use commutation relations of H0 with P and r, yielding
dσ
∝|< Ψf | r | Ψi > • A0 |2 {δ (E f − E i − ω )}
dΩ
dσ
∝|< Ψf | ∇V | Ψi > • A0 |2 {δ (E f − E i − ω )}
dΩ
Single particle
representation
Ψ0N = φ i ΨiN −1 : initial − state
ψ f = uk Φi, j (N −1) : final − state
Hole in ith orbital
Phys. 342/555:Ch12B (2008)
Photoemission Matrix Element: III
dσ
∝|< uk | A • P | φ i >< Φ i, j | Ψi >|2 {δ (E f − E i − ω )}
dΩ
For an independent particle representation,
there is no rearrangement of charge so
< Φi, j | Ψi >= δi,0
dσ
∝|< uk | A • P | φ i >|2 {δ (E f − E i − ω )}
dΩ
Symmetry Rules: Very important
Uk always even in plane of emission to have
signal in detector
Phys. 342/555:Ch12B (2008)
Photoexcition in a periodic system
Momentum picked
up by lattice: G
Vertical Excitiation
Phys. 342/555:Ch12B (2008)
Projection of Bulk onto Surface
100 surface of a fcc material:
Top is the Surface Brillouin zone
and bottom shows the bulk Brillouin
zone: This shows the projection of
the bulk onto the surface. The
solid line represents a rod
connection all points in the bulk
zone with the one point in the
surface zone with the same k\\
The perpendicular component of
the momentum of the
photoexcited state inside the
solid is undertermined
experimentally. The value of
k(perpendicular) could be
anywhere on a rod in k-space
with fixed k]]
E i = E kin + eφ − ω
Phys. 342/555:Ch12B (2008)
3-D assuming Free
Electron Bands
Fcc Cu: The vertical axis is normal to the
surface and the horizontal axis is in two different
directions in the bulk. The top shows circles of a
the free electron bands at different energy. The
dashed lines show the k versus E location for
transiton seen with a detector at a fixed angle.
The effects of bands crossing zone boundaries is
shown at the bottom.
Phys. 342/555:Ch12B (2008)
Excitation process in Cu
Normal Emission Cu(100)
The position, number, and
intensity of the peaks
depends upon photon
energy.
This is never a density of
states.
What causes the width??
Phys. 342/555:Ch12B (2008)
Data for Normal Emission Cu(100)
Intensity vs. Photon energy
for energy 0.13 below EF
Data vs. Photon Energy
Dispersion assuming
free electron final
bands
Phys. 342/555:Ch12B (2008)
Al data: procedure
Dashed Free Electron
Solid DFT
Phys. 342/555:Ch12B (2008)
Normal Emission: k\\=0
Surface State
Direct Transitions
Plasmons
Free Electron bands
Real
Data342/555:Ch12B
for normal Emission
Phys.
(2008)
Normal Emission:
This is the kind
of measurement
that should be
made by anyone
wanting to know
the bulk band
structure.
Bottom of band
Phys. 342/555:Ch12B (2008)
X-W band mapping:
fix k|| M in SBZ
Phys. 342/555:Ch12B (2008)
Fix k|| sweep
photon
energy
Phys. 342/555:Ch12B (2008)
Γ−X direction in the Bulk
Gap Wrong
Band Width Wrong
Phys. 342/555:Ch12B (2008)
Γ−X direction in the Bulk
Gap Wrong
Band Width Wrong
Many-body effect
Phys. 342/555:Ch12B (2008)
X-W in the bulk
There is still a problem with the
Gap when compared to DFT
calculations!
Phys. 342/555:Ch12B (2008)
X-W in the bulk
Phys. 342/555:Ch12B (2008)
How did we do?
Phys. 342/555:Ch12B (2008)
Final States
Γ−X
Data
Folded Back
Band Structure
Calculated Band
Structure with dark
lines where there is
amplitude in the
right direction.
Phys. 342/555:Ch12B (2008)
Final States
X-W
Data
Folded Back
Band Structure
Calculated Band
Structure
Phys. 342/555:Ch12B (2008)
Line Width: Imaginary part of
interaction
Phys. 342/555:Ch12B (2008)
Absolute Measurement: 1988
Phys. 342/555:Ch12B (2008)
Band Structure of Na
Phys. 342/555:Ch12B (2008)
Band Structure of Na
Band Width
Wrong!!
Phys. 342/555:Ch12B (2008)
Band Structure of Na
Real part of Self
Energy
Re{Σ(k)} ≡ E(k) − E 0 (k)
Phys. 342/555:Ch12B (2008)
Band Width in Simple Metals
Rs is an inverse
measure of
electron density
Theory Looks
Good, BUT!!!
Phys. 342/555:Ch12B (2008)