Ebert - Psi-k

Ludwig
Maximilians-
Universität
München
The Munich SPRKKR-program package
A spin polarized relativistic
Korringa-Kohn-Rostoker (SPR-KKR) code
for Calculating Solid State Properties
H. Ebert, M. Battocletti, D. Benea, S. Chadov, M. Deng, H. Freyer,
T. Huhne, M. Kardinal, D. Ködderitzsch, M. Kosuth, S. Mankovskyy,
J. Minar, A. Perlov, V. Popescu, Ch. Zecha and many others
Hubert.Ebert@cup.uni-muenchen.de
Universität München, Physikalische Chemie
München, Germany
KKR07 – p.1/100

Ludwig
Maximilians-
Universität
München
Outline
Introduction
Green’s functions
Scattering theory
Angular momentum representation
Calculating the scattering path operator
The impurity problem
Substitutional disordered alloys
Relativistic formulation
Combination of KKR with the DMFT
Applications
KKR07 – p.2/100

Ludwig
Maximilians-
Universität
München
Standard Band Structure Problem
Born-Oppenheimer Approximation
reduction of many-particle problem
to single-particle problem
e.g. via density functional theory (DFT)
[
]
−⃗∇ 2 + V (⃗r)
ψ(⃗r, E) = Eψ(⃗r, E)
periodic potential V (⃗r) = V (⃗r + ⃗R n )
leads to the Bloch theorem
T ⃗ R n
ψ ⃗k (⃗r, E) = e −i⃗ k ⃗R n
ψ ⃗k (⃗r, E)
KKR07 – p.3/100

Ludwig
Maximilians-
Universität
München
Original KKR-method
Replace differential Schrödinger equation
[
]
−⃗∇ 2 + V (⃗r)
ψ ⃗k (⃗r, E) = E ⃗k ψ ⃗k (⃗r, E)
by integral Lippmann-Schwinger equation
ψ ⃗ k (⃗r, E) = ei ⃗k⃗r +
∫
d 3 r ′ G 0 (⃗r, ⃗r ′ , E) V (⃗r ′ ) ψ ⃗ k (⃗r ′ , E)
with free electron Green’s function G 0 (⃗r, ⃗r ′ , E)
Korringa, Physica 13 392 (1947); Kohn, Rostoker, PRB 94 1111 (1954)
KKR07 – p.4/100

Ludwig
Maximilians-
Universität
München
KKR secular equation
Ansatz: Ψ j ⃗ k
(⃗r, E) = ∑ lm Aj ⃗k
lm il R l (r, E) Y lm (ˆr)
Application of variational principle ⇒ secular equation
[t(E j ⃗k )−1 − G 0 ( ⃗ k, E j ⃗k ) ]
A j⃗k = 0
t(E)
G 0 ( ⃗ k, E j ⃗ k
)
t-matrix
structure constants matrix
minimal numerical basis set used
splitting into potential and geometrical part
no standard algebraic eigenvalue problem
KKR07 – p.5/100

Ludwig
Maximilians-
Universität
München
Lippmann-Schwinger equation
H 0 : Hamiltonian for unperturbed reference system
(E − H 0 ) Ψ 0 (⃗r, E) = 0
(E − H 0 ) G 0 (⃗r, ⃗r ′ , E) = δ(⃗r − ⃗r ′ )
G 0 (⃗r, ⃗r ′ , E) solving inhomogeneous equation
Treatment of perturbed system H = H 0 + V (⃗r)
(E − H) Ψ(⃗r, E) = 0
(E − H 0 ) Ψ(⃗r, E) = V (⃗r)Ψ(⃗r, E)
solution supplied by Lippmann-Schwinger equation
∫
Ψ(⃗r, E) = Ψ 0 (⃗r, E) + d 3 r ′ G 0 (⃗r, ⃗r ′ , E) V (⃗r) Ψ(⃗r, E)
KKR07 – p.6/100

Ludwig
Maximilians-
Universität
München
Dyson Equation
H 0 : Hamiltonian for unperturbed system
(E − H 0 )G 0 (⃗r, ⃗r ′ , E) = δ(⃗r − ⃗r ′ )
Treatment of perturbed system
H = H 0 + V (⃗r)
(E − H)G(⃗r, ⃗r ′ , E) = δ(⃗r − ⃗r ′ )
solution supplied by Dyson equation
G(⃗r, ⃗r ′ , E) = G 0 ∫
(⃗r, ⃗r ′ , E)
+ d 3 r ′′ G 0 (⃗r, ⃗r ′′ , E) V (⃗r ′′ ) G(⃗r ′′ , ⃗r ′ , E)
KKR07 – p.7/100

Ludwig
Maximilians-
Universität
München
Spectral representation of G(⃗r, ⃗r ′ , E)
ψ i (⃗r, E i ) complete and orthogonal set of eigenfuctions
(E i − H) ψ i (⃗r, E i ) = 0
(E − H) G(⃗r, ⃗r ′ , E) = δ(⃗r − ⃗r ′ )
G(⃗r, ⃗r ′ , E) for fixed ⃗r ′ and E can be expanded into a
series of ψ i (⃗r, E i ) (Mercer’s theorem)
∑
G + (⃗r, ⃗r ′ , E) = lim
ɛ→0
retarded Green’s function
i
ψ i (⃗r)ψ × i (⃗r ′ )
E − E i + iɛ
poles of G + (⃗r, ⃗r ′ , E) indicate eigenvalues
lim ɛ→0 avoids divergencies
NB: × is in general equivalent to †
KKR07 – p.8/100

Ludwig
Maximilians-
Universität
München
Densities and expectation values
The Dirac identity lim
ɛ→0
I 1
x + iɛ
= −π δ(x) leads to
− 1 π IG+ (⃗r, ⃗r ′ , E) = ∑ i
ψ i (⃗r)ψ × i (⃗r ′ )δ(E − E i )
IG + (⃗r, ⃗r ′ , E) can be interpreted as density matrix
Density of states
n(E) = − 1 π I ∫
V
d 3 r G + (⃗r, ⃗r, E)
Particle density n(⃗r) = − 1 π I ∫ EFdE
G + (⃗r, ⃗r, E)
Expectation value
〈A〉 = − 1 π I ∫ EFdE ∫
V
d 3 r A G + (⃗r, ⃗r, E)
KKR07 – p.9/100

Ludwig
Maximilians-
Universität
München
Analyticity of the Green’s function
Extension of G + (⃗r, ⃗r ′ , E) to complex energies z
with
(z − H) G(⃗r, ⃗r ′ , z) = δ(⃗r − ⃗r ′ )
ψ i (⃗r)ψ × i (⃗r ′ )
G(⃗r, ⃗r ′ , z) = ∑ i
z − E i
implies G(⃗r, ⃗r ′ , E) to be analytical for Iz ≠ 0 and
G(⃗r, ⃗r ′ , z) = G × (⃗r ′ , ⃗r, z ∗ )
Accordingly one has e.g. for the particle density
n(⃗r) = − 1 ∫ EF
π I dE G + (⃗r, ⃗r, E) = − 1 ∫ EF
π I dz G(⃗r, ⃗r, z)
−
∩
KKR07 – p.10/100

Ludwig
Maximilians-
Universität
München
Path integrals in the complex energy plane
particle density n(⃗r)
n(⃗r) = − 1 ∫ EF
π I dE G + (⃗r, ⃗r, E)
−
= − 1 ∫ EF
π I dz G(⃗r, ⃗r, z)
Iz
∩
density of states
n(z) = − 1 π I ∫
d 3 r G(⃗r, ⃗r, z)
for fixed Iz ‖ real E-axis
3
2.5
Im E=0.001 Ry
Im E=0.05 Ry
Im E=0.01 Ry
Im E=0.1 Ry
E 0
E F
Rz
n Cu
(z) (sts./eV)
2
1.5
1
0.5
0
-10 -8 -6 -4 -2 0 2 4
Re z (eV)
NB: In the following: E stands for a complex energy z
KKR07 – p.11/100

Ludwig
Maximilians-
Universität
München
Benefits of Green’s function based methods
Straightforward application to
any type of host/impurity system
surfaces or arbitrary layered systems
disordered systems using appropriate alloy theory
response to an external perturbation
response functions/susceptibilities
transport (Kubo-Greenwood or
Landauer-Büttiker-formalism)
spectroscopy
many-body problems
etc.
It’s worth to work with Green’s functions ...
... but how to calculate them ??? KKR07 – p.12/100

Ludwig
Maximilians-
Universität
München
Born series
Lippmann-Schwinger equation in operator notation
with G 0 (⃗r, ⃗r ′ , E) = 〈⃗r|G 0 (E)|⃗r ′ 〉
and the energy argument E omitted in the following
|Ψ 1 〉 = |Ψ 0 〉 + G 0 V |Ψ 1 〉
iteration leads to the Born series
|Ψ 1 〉 = |Ψ 0 〉 + G 0 V |Ψ 0 〉 + G 0 V G 0 V |Ψ 0 〉
+G 0 V G 0 V G 0 V |Ψ 0 〉 + ...
solves Lippmann-Schwinger equation iteratively
NB: sometimes used for FP single-site problem
KKR07 – p.13/100

Ludwig
Maximilians-
Universität
München
Transition matrix T
The transition or T-matrix T relates by definition
the unperturbed and perturbed systems via
V |Ψ 1 〉 = T |Ψ 0 〉
inserting repeatedly |Ψ 1 〉 = |Ψ 0 〉 + G 0 V |Ψ 1 〉 gives
T = V + V G 0 V + V G 0 V G o V + · · ·
Iterating the Dyson equation the same way one is led to
V G = T G 0
Lippmann-Schwinger equation |Ψ 1 〉 = |Ψ 0 〉 + G 0 T |Ψ 0 〉
Dyson equation G = G 0 + G 0 T G 0
KKR07 – p.14/100

Ludwig
Maximilians-
Universität
München
Single-site t-matrix t n
Decomposition of the potential V into
non-overlapping potentials V n centered at sites n
V = ∑ n
V n
T-matrix for an isolated potential V n
t n = V n + V n G 0 t n
= ( 1 − V n G 0
) −1 V
n
= V n ( 1 − G 0 V n) −1
single-site t-matrix t n
KKR07 – p.15/100

Ludwig
Maximilians-
Universität
München
Expanding the T -matrix T
Introducing the auxilary quantity Q n
T = V + V G 0 V + V G 0 V G o V + · · ·
= ∑ V n + ∑ V n G 0 V m +
∑
n n, m
= ∑ n
Q n
n, m, k
V n G 0 V m G 0 V k + · · ·
one has
and finally
T = ∑ n
t n +
Q n = t n + t n G 0
∑
∑
n, m
(n≠m)
t n G 0 t m +
m
(m≠n)
∑
Q m
n, m, k
(n≠m, m≠k)
t n G 0 t m G 0 t k + · · ·
KKR07 – p.16/100

Ludwig
Maximilians-
Universität
München
T
= ∑ n
= ∑ n,m
Scattering path operator τ
t n +
∑ n, m
(n≠m)
t n G 0 t m +
∑
n, m, k
(n≠m, m≠k)
[
t n δ nm + t n G 0 t m (1 − δ nm ) +
t n G 0 t m G 0 t k + · · ·
∑
k
(k≠n, k≠m)
t n G 0 t k G 0 t m + · · ·
]
= ∑ n,m
τ nm
scattering path operator τ nm
τ nm = t n δ nm + ∑ k≠n
= t n δ nm + ∑ k≠n
t n G 0 τ km
τ nk G 0 t m
Gyorffy (1971)
KKR07 – p.17/100

Ludwig
Maximilians-
Universität
München
Standard scattering experiment
ψ(⃗r, E) =
e i ⃗k·⃗r
} {{ }
incoming
plane wave
+ f(θ, φ) eikr
}{{} r
outgoing
spherical wave
scattering amplitude f(θ, φ)
KKR07 – p.18/100

Ludwig
Maximilians-
Universität
München
Single-site scattering problem
Standard problem in a scattering experiment
scattering of a plane wave at a target
Lippmann-Schwinger equation
∫
ψ(⃗r, E) = e i⃗k⃗r + d 3 r ′ G 0 (⃗r, ⃗r ′ , E) V n (⃗r) ψ(⃗r, E)
Ω n
∫
= e i⃗k⃗r + d
∫Ω 3 r ′ d 3 r ′′ G 0 (⃗r, ⃗r ′ , E) t n (⃗r ′ , ⃗r ′′ , E) e i⃗ k⃗r ′′
n Ω n
with |⃗k| = √ E
KKR07 – p.19/100

Ludwig
Maximilians-
Universität
München
Single site potential V n (⃗r)
single site potential: V n (⃗r) = f n (⃗r) V (⃗r)
theta or shape function f n (⃗r) for cell n
f n (⃗r) =
{
1 for ⃗r ∈ Ω n
0 for ⃗r /∈ Ω n
expansion into (real) spherical harmonics Y L (ˆr)
V n (⃗r) = ∑ L
V n L (r) Y L(ˆr)
f n (⃗r) = ∑ L
f n L (r) Y L(ˆr)
with cell centered coordinates ⃗r and L = (l, m)
muffin-tin approximation (assumed in the following)
V n (⃗r) =
{
V n (r) for r < r mt
0 else
KKR07 – p.20/100

Ludwig
Maximilians-
Universität
München
Expansion of plane waves
Bauer’s identity
e i⃗ k⃗r = 4π ∑ L
i l j l (kr) Y L (ˆr) Y L (ˆk)
j l (kr) spherical Bessel function with k = √ E
spherical functions: linearly independent solutions to the
free electron problem in spherical coordinates
(Helmholtz equation)
j l (kr)
n l (kr)
h + l (kr)
spherical Bessel function
von Neumann function
Hankel function (1st kind)
with h + l (kr)= j l(kr)+in l (kr)
KKR07 – p.21/100

Ludwig
Maximilians-
Universität
München
Spherical functions j l (kr) and n l (kr)
spherical Bessel functions
j l (kr)
von Neumann functions
n l (kr)
0.4
0.2
l=0
l=1
l=2
0.4
0.2
l=0
l=1
l=2
j l
(kr)
0
n l
(kr)
0
-0.2
-0.2
-0.4
0 5 10 15 20
regular
irregular
kr
(kr) l
(2l+1)!!
asymptotic behavior
0←r
←−
-0.4
0 5 10 15 20
j l (kr)
r→∞
−→
kr
1
kr sin(kr − l π 2 )
− (2l−1)!!
(kr) l+1 n l (kr) − 1
kr cos(kr − l π 2 )
KKR07 – p.22/100

Ludwig
Maximilians-
Universität
München
Free electron Green’s function
G 0 (⃗r, ⃗r ′ , E) =
∫
1
(2π) 3
d 3 k ei⃗ k⃗r e −i⃗ k⃗r ′
E − k 2
= − 1
4π
e −i√ E|⃗r−⃗r ′ |
|⃗r − ⃗r ′ |
using Bauer’s identity
G 0 (⃗r, ⃗r ′ , E) =
⎧
⎪⎨
⎪⎩
−ik ∑ L
−ik ∑ L
j L (⃗r, E)h +×
L (⃗r ′ , E) for r < r ′
h + L (⃗r, E)j× L (⃗r ′ , E) for r > r ′
with j L (⃗r, E) = j l ( √ E r)Y L (ˆr) and L = (l, m)
KKR07 – p.23/100

Ludwig
Maximilians-
Universität
München
Angular momentum expansion of ψ(⃗r, E)
Solving the Schrödinger equation
using the ansatz
[
]
−⃗∇ 2 + V (r)
ψ(⃗r, E) = Eψ(⃗r, E)
ψ(⃗r, E) = ∑ L
R l (r, E)Y L (ˆr)
leads to the radial Schrödinger equation
[
− ∂2 l(l + 1)
+
∂r2 r 2
+ V (r)
for the radial wave function R l (r, E)
]
R l (r, E) = 0
KKR07 – p.24/100

Ludwig
Maximilians-
Universität
München
Partial wave decomposition
Lippmann-Schwinger equation
∫
ψ(⃗r, E) = e i⃗k⃗r + d 3 r ′ G 0 (⃗r, ⃗r ′ , E) V n (⃗r) ψ(⃗r, E)
Ω n
∫
= e i⃗k⃗r + d
∫Ω 3 r ′ d 3 r ′′ G 0 (⃗r, ⃗r ′ , E) t n (⃗r ′ , ⃗r ′′ , E) e i⃗ k⃗r ′′
n Ω n
angular momentum expansion
R L (⃗r, E)=j L (⃗r, E)
∫
−ik d 3 r ′ h + L (⃗r >, E)j × L (⃗r , E)j × L (⃗r

Ludwig
Maximilians-
Universität
München
Asymptotic limit for r > r mt
For r > r mt one has
R L (⃗r, E) = j L (⃗r, E)
−ikh + L (⃗r, E) ∫Ω n
d 3 r ′ j × L (⃗r ′ , E)V (r ′ )R L (⃗r ′ , E)
} {{ }
t LL ′=t l δ LL ′
= j L (⃗r, E)
∫ ∫
−ikh + L (⃗r, E) d 3 r ′ d 3 r ′′ j × L (⃗r ′ , E)t(⃗r ′ , ⃗r ′′ , E)j L (⃗r ′′ , E)
Ω n Ω
} n
{{ }
t LL ′=t l δ LL ′
R L (⃗r, E) = j L (⃗r, E) − ikh + L (⃗r, E)t l(E)
KKR07 – p.26/100

Ludwig
Maximilians-
Universität
München
Expression for the single-site t-matrix
Matching at the muffin-tin radius
˜R l (r, E) · N l (E) = R l (r, E) → j l (kr) − ikh + l (kr)t l(E)
d
dr R l(r, E) → d [
]
j l (kr) − ikh + l
dr
(kr)t l(E)
˜R l : unnormalised solution of radial Schrödinger equation
2
R d
(r,E)
0
-2
-4
matching radius r mt
t l (E) = − 1
ik
[j l , ˜R l ] r=rmt
h
h + l , ˜R l
i
r=r mt
-6
-8
R d
(r,E)
j d
(kr) - ik h d
+
(kr) td (E)
Wronskian
[u, v] = u(r) d dr v(r) − v(r) d dr u(r)
0 1 2 3 4
radius (a 0
)
KKR07 – p.27/100

Ludwig
Maximilians-
Universität
München
Phase shift δ l (E)
2
0
2
0
R d
(r,E)
-2
-4
R d
(r,E)
-2
-4
phase shift ∆r = δ d
(E) / k
matching radius r mt
-6
-8
matching radius r mt
0 2 4 6 8 10
R d
(r,E)
A j d
(kr) + B n d
(kr)
-6
-8
R d
(r,E)
j d
(kr) x 5
A j d
(kr) + B n d
(kr)
0 1 2 3 4
radius (a 0
)
radius (a 0
)
R l (r, E)
r=r mt
= A j l (kr) + B n l (kr)
r→∞
−→ A (
kr sin kr − lπ 2
√
A 2 + B 2
r→∞
−→
kr
sin
)
− B kr cos (
kr − lπ 2
(
kr − lπ )
2 + δ l(E)
)
phase shift δ l (E)
KKR07 – p.28/100

Ludwig
Maximilians-
Universität
München
Phase shift δ l (E) and single site t-matrix t l (E)
t l (E) = − 1 k sin ( δ l (E) ) e iδ l(E)
δ l Au
(E)
3.2
2.8
2.4
2
1.6
1.2
0.8
0.4
0
phase shift δ l (E)
-0.4
0 2 4 6 8 10
energy (eV)
s
p
d
- t l Au
(E)
single site t-matrix t l (E)
2.5
2
1.5
1
0.5
0
-0.5
-1
0 2 4 6 8 10
energy (eV)
Im s
Im p
Im d
Re s
Re p
Re d
scalar-relativistic results for Au
KKR07 – p.29/100

Ludwig
Maximilians-
Universität
München
Single site Green’s function G ss (⃗r, ⃗r ′ , E)
free electrons
[E − H 0 ] G 0 (⃗r, ⃗r ′ , E) = δ(⃗r − ⃗r ′ )
single site potential
[E − H] G ss (⃗r, ⃗r ′ , E) = δ(⃗r − ⃗r ′ )
j l (r) ↔ j l (r)
h + l (r) ↔ h+ l (r)
R l (r) ↔ j l (r) − ikh + l (r)t l
H l (r) ↔ h + l (r)
G 0 (⃗r, ⃗r ′ , E) =
−ik ∑ j L (⃗r, E)h +×
L (⃗r ′ , E)
L
G ss (⃗r, ⃗r ′ , E) =
−ik ∑ R L (⃗r, E)H × L (⃗r ′ , E)
L
KKR07 – p.30/100

Ludwig
Maximilians-
Universität
München
Alternative Convention for G ss (⃗r, ⃗r ′ , E)
boundary conditions
at muffin-tin radius r mt
G ss (⃗r, ⃗r ′ , E) = −ik ∑ L
R L (⃗r, E)H × L
(⃗r, E)
r→r mt
R l −→
Z l = R l t −1
l
jl − ikh + l t l
replace R L and H L by Z L and J L
r→r
H
mt
l −→ h
+
l
G ss (⃗r, ⃗r ′ , E) = ∑ L
Z L (⃗r, E)t l (E)Z × L
(⃗r, E)
− ∑ L
Z L (⃗r, E)J × L
(⃗r, E)
r→r
J
mt
l −→
jl
H l = i k
(
Zl − J l t −1 )
l
r→r
Z
mt
l −→
r→r
J
mt
l −→
jl t −1
l
jl
− ikh + l
Bristol-Oak Ridge convention
KKR07 – p.31/100

Ludwig
Maximilians-
Universität
München
Single site density of states n ss (E)
Density of states DOS
n ss (E) = − 1 π I ∫
d 3 r G ss (⃗r, ⃗r ′ , E)
= − 1 ∑
I
5
π
L
⎡
4
∫
⎣t L (E) d 3 r Z × L (⃗r, E)Z 3
L(⃗r, E)
Ω n 2
⎤
∫
− d 3 r J × 1
L (⃗r, E)Z L(⃗r, E) ⎦
0
Ω n
for real energies
= − 1 ∑
∫
I t l (E) d 3 rZ × L
π
(⃗r, E)Z L(⃗r, E)
L
Ω n
= n ss
l
(E)
l-resolved single site
DOS n ss
l
(E)
n Au
(E) (sts./eV)
6
-10 -8 -6 -4 -2 0
energy (eV)
scalar-relativistic results for Au
tot
s
p
d
KKR07 – p.32/100

Ludwig
Maximilians-
Universität
München
Multiple scattering
a
b
c
Solve Dyson equation
for
G = G 0 + G 0 T G 0
T = ∑ n,m
τ nm
τ nm = t n δ nm + ∑ k≠n
t n G 0 τ km
KKR07 – p.33/100

Ludwig
Maximilians-
Universität
München
Reexpansion of the spherical Bessel functions
Starting from e i ⃗k( ⃗R+⃗r) = e i ⃗k ⃗R e i⃗k⃗r and Bauer’s
identity e i⃗k⃗r = 4π ∑ i l j L (r, k 2 ) Y L (ˆk) one finds:
L
j L ( ⃗R + ⃗r, E) = 4π ∑ L ′ L ′′ i l−l′ −l ′′ C LL ′ L ′′ j L ′( ⃗R, E) j L ′′(⃗r, E)
with the Gaunt coefficients
C LL ′ L ′′ = ∫
dˆr Y L (ˆr) Y L ′(ˆr) Y L ′′(ˆr)
KKR07 – p.34/100

Ludwig
Maximilians-
Universität
München
Reexpansion of the Hankel functions
From − 1
4π
e ik |⃗r− ⃗R|
∣
∣⃗r − ⃗R ∣
= −ik ∑ L
the expansion for j l ( ⃗R + ⃗r) one finds
j L ( ⃗R, k 2 ) h +×
L
(⃗r, k2 ) and
h + L ( ⃗R + ⃗r, E) = ∑ L ′ G 0LL ′( ⃗R, E) j L (⃗r, E)
real-space structural Green’s function matrix
or
G 0LL ′( ⃗R, E) = 4π ∑ L ′′ i l−l′ −l ′′ C LL ′ L ′′ h+ L ′′ ( ⃗R, E)
G nm
0LL ′ (E) = 4π ∑ L ′′ i l−l′ −l ′′ C LL ′ L ′′ h+ L ′′ ( ⃗R m − ⃗R n , E)
for ⃗R = ⃗R m − ⃗R n
KKR07 – p.35/100

Ludwig
Maximilians-
Universität
München
Multicenter expansion of G 0 (⃗r, ⃗r ′ , E)
Using the reexpansion of h + L ( ⃗R + ⃗r, E) and
cell-centered coordinates ⃗r n = ⃗r − ⃗R n one gets
⃗r n ⃗R m − ⃗R n
site m ⃗r m
site n
G 0 (⃗r n , ⃗r m ′ , E) =
⎧
⎪⎨ −ik ∑ j L (⃗r n , E) h + L (⃗r n ′ , E) n = m
L
∑
⎪⎩ j L (⃗r n , E) G nm
0 LL
(E) j ′ L ′(⃗r ′ m , E) n ≠ m
LL ′
KKR07 – p.36/100

Ludwig
Maximilians-
Universität
München
Green’s function for many sites
Combining G = G 0 + G 0
∑n,m τ nm G 0
G ss (⃗r, ⃗r ′ , E) = 〈⃗r|G 0 + G 0 t G 0 |⃗r ′ 〉
= ∑ Z L (⃗r, E)t l (E)Z × L
(⃗r ′ , E)
′
L
− ∑ L
Z L (⃗r, E)J × L (⃗r ′ , E)
G 0 (⃗r n , ⃗r m ′ , E) = ∑ LL ′ j L (⃗r n , E)G nm
0 LL ′ (E)j × L ′ (⃗r m ′ , E) (n ≠ m)
leads to
G(⃗r n , ⃗r ′ m , E) = ∑ Z L (⃗r n , E) τ nm
LL
(E) Z × ′ L
(⃗r ′ ′ m , E)
LL ′ ∑
−δ nm Z L (⃗r n , E) J × L (⃗r n ′ , E)
L
KKR07 – p.37/100

Ludwig
Maximilians-
Universität
München
Angular momentum representation for τ nm (E)
τ nm
LL ′ (E) =
∫Ω n
d 3 r n
∫
Ω m
d 3 r ′ m j× L (⃗r n, E)〈⃗r n |τ nm |⃗r ′ m 〉j L ′(⃗r m ′ , E)
corresponding equation of motion
τ nm (E) = t n (E)δ nm + t n (E) ∑ k≠n
G nk
0 (E)τ km (E)
and
[τ nm ] LL ′ = τLL nm
[ ]
′
G
nm
0
= G nm
LL ′ 0 LL ′ (1 − δ nm )
[t n ] LL ′ = t n l δ LL ′
matrix dimension:
∑ lmax
l=0
∑ +l
m=−l 1 = (l max + 1) 2
KKR07 – p.38/100

Ludwig
Maximilians-
Universität
München
Calculating τ nm for finite array of scatterers
τ (E) = t(E) + t(E) G 0
(E) τ (E)
with
[
τ
] nm = τ
nm
[
[ t ] nm = t n δ nm
G 0
] nm
= G
nm
0
(1 − δ nm )
τ (E) =
[
t(E) −1 − G 0
(E)
] −1
matrix dimension: N scatterers × (l max + 1) 2
KKR07 – p.39/100

Ludwig
Maximilians-
Universität
München
Application of real space approach
cluster approximation
cut a finite cluster out of the infinite system
centred on the atom of interest
applied e.g. for EXAFS calculations
free clusters and molecules
KKR07 – p.40/100

Ludwig
Maximilians-
Universität
München
Cluster approximation for τ nn (E)
τ (E) =
Convergence of cluster approximation
[
t(E) −1 − G 0
(E)
] −1
with cluster size
DOS for central atom of bcc-Fe clusters
(E) (sts./eV)
3.2
2.4
1.6
1 / 0 atoms / shells
9 / 1
15 / 2
(E) (sts./eV)
3.2
2.4
1.6
15 / 2
27 / 3
51 / 4
(E) (sts./eV)
3.2
2.4
1.6
51 / 4
59 / 5
k-space
n ↓ Fe
0.8
n ↓ Fe
0.8
n ↓ Fe
0.8
0
3.2
0
3.2
0
3.2
(E) (sts./eV)
2.4
1.6
(E) (sts./eV)
2.4
1.6
(E) (sts./eV)
2.4
1.6
n ↑ Fe
0.8
n ↑ Fe
0.8
n ↑ Fe
0.8
0
-8 -6 -4 -2 0 2
0
-8 -6 -4 -2 0 2
0
-8 -6 -4 -2 0 2
energy (eV)
energy (eV)
energy (eV)
KKR07 – p.41/100

Ludwig
Maximilians-
Universität
München
Calculating τ nm for periodic array of scatterers
Fourier transformation
τ ( ⃗ k, E) = ∑ ⃗R n
τ n0 (E) e i ⃗k ⃗R n
G 0 ( ⃗ k, E) = ∑ ⃗R n
G n0
0 (E) ei ⃗k ⃗R n
τ nm (E) = 1
Ω BZ
∫
= 1
Ω BZ
∫
d 3 k τ (⃗k, E)
Ω BZ
[
Ω BZ
d 3 k
t(E) −1 − G 0 (⃗k, E)
} {{ }
KKR matrix M( ⃗ k,E)
] −1
e −i⃗ k( ⃗R n − ⃗R m )
G 0 (⃗k, E): KKR structure constants matrix
KKR07 – p.42/100

Ludwig
Maximilians-
Universität
München
Brillouin zone integration
τ nm (E) = 1
Ω BZ
∫
Ω BZ
d 3 k
[
t(E) −1 − G 0 (⃗k, E)
} {{ }
KKR matrix M( ⃗ k,E)
] −1
e −i⃗ k( ⃗R n − ⃗R m )
Symmetry allows to reduce the integration regime Ω BZ
to an irreducible wedge
KKR07 – p.43/100

Ludwig
Maximilians-
Universität
München
Special point sampling method
τ nn (E) =
h∑
j=1
U j τ nn j −1
IBZ
(E) U
with
τ nn
IBZ (E) = ∑ ⃗ k
w ⃗k τ (⃗k, E)
U: h symmetry operations
Special point mesh
IBZ: irreducible part of the Brillouin zone
KKR07 – p.44/100

Ludwig
Maximilians-
Universität
München
Substitutional impurities
Single site approximation
ignore the distortion of the surrounding host atoms
Assuming the impurity on site n = 0 one has
t n =
{
t imp for n = 0
t host for n ≠ 0
KKR07 – p.45/100

Ludwig
Maximilians-
Universität
München
Solving the impurity problem
pure host
τ host
=
⎡⎛
⎢⎜
⎣⎝
t host
t host
...
⎞−1 ⎤
− G 0
⎟
⎠
⎥
⎦
−1
impurity system
⎡⎛
τ imp
= ⎢⎜
⎣⎝
t imp
t host
...
⎞−1 ⎤
− G 0
⎟
⎠
⎥
⎦
−1
partition of matrices leads to
τ 00
imp = [
(t imp ) −1 − (t host ) −1 + (τ 00
host )−1] −1
KKR07 – p.46/100

Ludwig
Maximilians-
Universität
München
Substitutional Rh impurity in Au
DOS of bulk fcc-Rh
1
0.8
s
p
d
2.4
2
DOS of Rh in Au
Au
Rh
Rh single site
n(E) (sts./eV)
0.6
0.4
n(E) (sts./eV)
1.6
1.2
0.8
0.2
0.4
0
-10 -8 -6 -4 -2 0 2
energy (eV)
0
-10 -8 -6 -4 -2 0 2
energy (eV)
KKR07 – p.47/100

Ludwig
Maximilians-
Universität
München
Treating substitutional disordered alloys
Electronic structure of a disorderd alloy A x B 1−x
represented by configurational average
for given concentration and eventually short range order
Super cell technique
Single site approximation: ignore short range order
VCA: Virtual Crystal Approximation
ATA: Average t-matrix Approximation
CPA: Coherent Potential Approximation
Cluster approximation: allow short range order
NL-CPA: non-local Coherent Potential Approx.
KKR07 – p.48/100

Ludwig
Maximilians-
Universität
München
Coherent Potential Approximation – CPA
effective CPA medium represents the electronic
structure of an configurationally averaged
substitutionally random alloy A x B 1−x
=
use mean field description
find best possible single-site scheme
Soven, Physical Review 156 809 (1967)
KKR07 – p.49/100

Ludwig
Maximilians-
Universität
München
CPA condition
Embedding of an A- or B-atom into the CPA-medium – in
the average – should not give rise to additional
scattering
x A + x B =
x A τ nn
A
+ x Bτ nn
B
= τ nn
CP A
with the projected scattering path operator τ nn
α
τ nn
α
= τ nn
CP A
[
1 +
(
t −1
α
)
− t−1 CP A
] −1
τ nn
CP A
KKR07 – p.50/100

Ludwig
Maximilians-
Universität
München
DOS for disordered systems
comparison of results for ordered and disordered case
Fe in Fe 3 Pt
Pt in Fe 3 Pt
(E) (sts./eV)
2
1
disordered
ordered
(E) (sts./eV)
2
disordered
ordered
0
2
1
n ↑ Fe
n ↓ Fe
0
2
(E) (sts./eV)
(E) (sts./eV)
1
n ↑ Pt
n ↓ Pt
0
-10 -5 0
energy (eV)
0
-10 -5 0
energy (eV)
KKR07 – p.51/100

Ludwig
Maximilians-
Universität
München
Bloch spectral function A B ( ⃗ k, E)
Bloch spectral function A B ( ⃗ k, E)
Fourier transformed of real-space Green’s function
A B ( ⃗ k, E) = − 1
πN
I
N∑
∫
n,m
Ω
e ı⃗ k( ⃗R n − ⃗R m )
d 3 r
〈
G
interpretation as a ⃗ k-resolved DOS function
n(E) = 1
Ω BZ
∫
(
)〉
⃗r + ⃗R n , ⃗r + ⃗R m , E
Ω BZ
d 3 k A B ( ⃗ k, E)
limit for ordered systems
A B ( ⃗ k, E) = ∑ j
δ(E − E j ⃗ k
)
KKR07 – p.52/100

Ludwig
Maximilians-
Universität
München
Bloch spectral function A B ( ⃗ k, E) within CPA
Bloch spectral function A B ( ⃗ k, E)
]
A B ( ⃗ k, E) = − 1 π
[F I Trace c τ CP A ( ⃗ k, E)
with averaged overlap integrals
[ ]
− 1 π I Trace (F c − F cc ) τ nn
CP A
F c = ∑ α
F cc = ∑ α,β
x α F α D α
x α x β F α ˜D α F β D β
F αβ
LL ′ =
∫
Ω
d 3 r Z α×
L (⃗r, E) Zβ L ′ (⃗r, E)
KKR07 – p.53/100

Ludwig
Maximilians-
Universität
München
Bloch spectral function A B ( ⃗ k, E) of Fe 0.2 Ni 0.8
Bloch spectral function A B ( ⃗ k, E) of Fe 0.2 Ni 0.8
for ⃗k along Γ – X
minority spin
majority spin
0.9
B
0.9
B
0.7
0.7
Ε (Ry)
0.5
C
A
Ε (Ry)
0.5
A
0.3
0.3
0.1
0.1
−0.1
−0.1
Γ
k
X
Γ
k
X
KKR07 – p.54/100

Ludwig
Maximilians-
Universität
München
Bloch spectral function A B ( ⃗ k, E) of Fe 0.2 Ni 0.8
Bloch spectral function A B ( ⃗ k, E) of Fe 0.2 Ni 0.8
Fermi surface for ⃗k in Γ-X-W plane
minority spin
majority spin
X
W
X
W
W
W
Γ
X
Γ
X
KKR07 – p.55/100

Ludwig
Maximilians-
Universität
München
The Dirac Equation
[ ]
i c⃗α·⃗∇+βmc 2 + ¯V (⃗r)+β⃗σ · ⃗B eff (⃗r) Ψ(⃗r, E) = E Ψ(⃗r, E)
} {{ }
V spin (⃗r)
with an effective magnetic field
⃗B eff (⃗r) = ∂E xc[n, ⃗m]
∂ ⃗m(⃗r)
that is determined by the spin magnetisation ⃗m(⃗r)
within spin density functional theory (SDFT)
Within an atomic cell one can always choose ẑ ′ to have:
V spin (⃗r) = βσ z ′ B eff (r)
KKR07 – p.56/100

Ludwig
Maximilians-
Universität
München
Ansatz to solve the single site Dirac equation
ψ ν (⃗r, E) = ∑ Λ
ψ Λν (⃗r, E)
with
ψ Λν (⃗r, E) =
(
g Λν (r, E) χ Λ (ˆ⃗r)
if Λν (r, E) χ −Λ (ˆ⃗r)
)
spin-angular functions
χ Λ (ˆr) =
∑
m s =±1/2
C(l 1 2 j; µ − m s, m s ) Y µ−m s
l
(ˆr) χ ms
short hand notation Λ = (κ, µ) and −Λ = (−κ, µ)
KKR07 – p.57/100

Ludwig
Maximilians-
Universität
München
Relativistic quantum numbers
total angular momentum operator ⃗ j = ⃗ l + 1 2 ⃗σ
⃗j 2 χ Λ (ˆr) = j(j + 1)χ Λ (ˆr)
j z χ Λ (ˆr) = µχ Λ (ˆr)
total angular momentum quantum number j
j = l ± 1/2
magnetic quantum number µ
µ = −j ... + j
KKR07 – p.58/100

Ludwig
Maximilians-
Universität
München
Relativistic quantum numbers
Spin-orbit operator ˆK
ˆKχ Λ (ˆr) = (1 + ⃗σ · ⃗l )χ Λ (ˆr) = −κχ Λ (ˆr)
Spin-obit quantum number κ
{
κ =
l = j + 1 2
if j = l − 1 2
−l − 1 = −j − 1 if j = l + 1 2 2
κ −1 +1 −2 +2 −3 +3 −4
j 1/2 1/2 3/2 3/2 5/2 5/2 7/2
l 0 1 1 2 2 3 3
symbol s 1/2 p 1/2 p 3/2 d 3/2 d 5/2 f 5/2 f 7/2
KKR07 – p.59/100

Ludwig
Maximilians-
Universität
München
Radial Differential Equations
P ′ Λν
= − κ r P Λν +
[ ]
E − V
c 2 + 1
+ B c 2 ∑
Q Λν
Λ ′ 〈χ −Λ |σ z |χ −Λ ′〉Q Λ ′ ν
Q ′ Λν
= κ r Q Λν − [E − V ] P Λν + B ∑ Λ ′ 〈χ Λ |σ z |χ Λ ′〉P Λ ′ ν
with P Λν (r, E) = r g Λν (r, E)
and Q Λν (r, E) = cr f Λν (r, E)
The coupling is restricted to µ − µ ′ = 0 and l − l ′ = 0
−→ 4 coupled functions for |µ| < j; e.g. d 3/2,µ — d 5/2,µ
−→ 2 coupled functions for |µ| = j; e.g. d 5/2,µ=±5/2
KKR07 – p.60/100

Ludwig
Maximilians-
Universität
München
Single site t-matrix
single site t-matrix t ΛΛ ′(E)
t(E) = i
2p
[a(E) − b(E)] b(E)−1
a ΛΛ ′(E) = −ipr 2 [h − Λ , φ ΛΛ ′] r
b ΛΛ ′(E) = −ipr 2 [h + Λ , φ ΛΛ ′] r
relativistic Wronskian
[h + Λ , φ ΛΛ ′] r = h + l cf ΛΛ ′ − p
1 + E/c 2 S κh +¯l g ΛΛ ′
KKR07 – p.61/100

Ludwig
Maximilians-
Universität
München
Single site t-matrix
spherical spin-dependent potential ( ⃗M‖ẑ)
s 1/2
s 1/2 p 1/2 p 3/2 d 3/2 d 5/2
p 1/2
p 3/2
d 3/2
d 5/2
KKR07 – p.62/100

Ludwig
Maximilians-
Universität
München
Single site t-matrix – phase shift δ κ
for a spherical potential
Fe in Fe 3 Pt
t κ = − 1 p sin(δ κ) e iδ κ
Pt in Fe 3 Pt
δ κ
(E) (j=l+1/2;|µ|=j)
δ κ
(E) (B xc
=0)
3
2
1
0
-1
2
1
0
s
p
d
s 1/2
p 1/2
p 3/2
d 3/2
d 5/2
-1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
δ κ
(E) (j=l+1/2;|µ|=j)
δ κ
(E) (B xc
=0)
3
2
1
0
2
1
0
s
p
d
s 1/2
p 1/2
p 3/2
d 3/2
d 5/2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
energy (Ry)
energy (Ry)
KKR07 – p.63/100

Ludwig
Maximilians-
Universität
München
KKR+DMFT scheme I
KKR
Σ
G
E
F
DMFT
(FLEX)
V LSDA (⃗r), Σ DMFT
⇓
single-site problem: t LL ′(z)
⇓
multiple scattering: τ LL ′(z)
⇓
KKR-Green’s function: G(⃗r, ⃗r ′ , z)
⇓
charge ρ(⃗r)
⇓
V LSDA (⃗r)
⇓
G-matrix G LL ′
⇓
DMFT-solver
⇓
Σ DMFT
KKR07 – p.64/100

Ludwig
Maximilians-
Universität
München
KKR+DMFT single-site problem
[−∇ 2 + V σ (r) − E]Ψ(⃗r) +
Z
Σ σ (⃗r, ⃗r ′ , E)Ψ(⃗r ′ )d 3 r = 0
For the solution Ψ(⃗r) one starts from the ansatz:
Ψ(⃗r) = X L
Ψ L (⃗r)
» d
2
l(l + 1)
−
dr2 r 2
–
− V (r) + E Ψ L (r, E) =
X
L ′′
Approximation for the self-energy:
X
Z
L
d 3 r ′ φ † L ′ (⃗r)Σ L ′ L(E)φ L (⃗r ′ )Ψ L (⃗r ′ , E) ≈ X L
Z
r ′2 dr ′ Σ LL ′′(E) φ l (r)φ l ′′(r ′ )Ψ L (r ′ , E)
Thus one solves pure differential equation:
» d
2
l(l + 1)
−
dr2 r 2
Σ L ′ L(E)Ψ L (⃗r, E)
–
− V (r) + E Ψ L (r, E) = X Σ LL ′(E) Ψ L ′(r, E)
L ′
KKR07 – p.65/100

Ludwig
Maximilians-
Universität
München
Comparison of schemes I-III for fcc Ni
n ↓ (sts./eV)
2
1.5
1
0.5
Density of states
LSDA
LSDA+DMFT I
LSDA+DMFT II
LSDA+DMFT III
Im Σ ↓ t 2g
(eV)
0
-1
-2
-3
-4
Self-energy
0
-5
n ↑ (sts./eV)
1.5
1
0.5
Im Σ ↑ t 2g
(eV)
-1
-2
-3
-4
0
-10 -5 0
energy (eV)
-5
-12 -8 -4 0 4
energy (eV)
2
µ spin
LSDA 0.563
LSDA+DMFT I 0.569
LSDA+DMFT II 0.631
LSDA+DMFT III 0.622
Re Σ ↓ t 2g
(eV)
Re Σ ↑ t 2g
(eV)
1
0
-1
-2
-3
1
0
-1
-2
-3
-12 -8 -4 0 4
energy (eV) KKR07 – p.66/100

Ludwig
Maximilians-
Universität
München
CPA calculations for Fe x Ni 1−x alloy
Spin magnetic moments
real part of Σ
Fe
Fe 10
Ni 90
Fe 40
Ni 60
µ spin
(µ B
)
3
2.5
2
1.5
1
0.5
Fe LDA
Ni LDA
total LDA
Fe LDA+DMFT
Ni LDA+DMFT
total LDA+DMFT
0
0 20 40 60 80
Fe content (at. %)
Σ t2g
↓
(eV)
Σ t2g
↑
(eV)
0.4
0.2
0
-0.2
-0.4
0
-1
-2
-3
Fe
-10 -5 0
Fe
-10 -5 0
energy (eV)
2
1
0
-1
-2
2
0
-2
Ni
-10 -5 0
Ni
-10 -5 0
energy (eV)
Fe 75
Ni 25
KKR07 – p.67/100

Ludwig
Maximilians-
Universität
München
Kubo-Greenwood Equation
σ µµ =
NΩ〈 π
〉
∑
〈m|j µ |n〉〈n|j µ |m〉δ(ɛ F − ɛ m )δ(ɛ F − ɛ n )
m,n
〈 〉
...
µ ∈ {x, y, z}
= average over configurations, j µ = −i e m
∂
∂r µ
,
using : ∑ m |m〉〈m|δ(ɛ − ɛ m) = − 1 π lim η→0 IG + (ɛ + iη)
Kubo-Greenwood Equation
〉
σ µµ = −
〈j Tr πNΩ µ IG + (ɛ F + iη) j µ IG + (ɛ F + iη)
KKR07 – p.68/100

Ludwig
Maximilians-
Universität
München
Kubo-Greenwood Equation within KKR
⇒ σ µµ = − 1 (˜σµµ [G + , G + ] + ˜σ µµ [G − , G − ]
4
− ˜σ µµ [G + , G − ] − ˜σ µµ [G − , G + ] )
with ˜σ µµ [G ± , G ± ] = −
πNΩ Tr 〈 j µ G ± j µ G ± 〉
after a lot of mathematics
˜σ µµ = − 4m2
π 3 Ω
⎛
∑
⎜
⎝
α,β
+ ∑ α
∑
(with z 1 , z 2 = lim η→0 (ɛ F ± η))
c α c β ˜J αµ
L 4 ,L 1
(z 2 , z 1 )
L 1 ,L 2
L 3 ,L 4
[
{1 − χω} −1
} {{ }
Vertex correction
〈GjGj〉→〈Gj〉〈Gj〉
]
χ ˜J βµ
L 2 ,L 3
(z 1 , z 2 )
L 1 ,L 2
L 3 ,L 4
⎞
∑
c α ˜J αµ
L 4 ,L 1
(z 2 , z 1 )τ CPA,00
L 1 ,L 2
(z 1 )J αµ
L 2 ,L 3
(z 1 , z 2 )τ CPA,00 ⎟
L 3 ,L 4
(z 2 ) ⎠
L 1 ,L 2
L 3 ,L 4
→ for more details see W. Butler, PRB 31 3260 (1985)
KKR07 – p.69/100

Ludwig
Maximilians-
Universität
München
Residual resistivity of Co x Pt 1−x
40
30
present work
S. U. Jen et al., Phys. Rev. 48 12789 (1993)
ρ (10 -6 Ohm cm)
20
10
0
0 0.2 0.4 0.6 0.8 1
x
KKR07 – p.70/100

Ludwig
Maximilians-
Universität
München
Ruderman-Kittel nuclear spin-spin coupling
coupling of nuclear spins
due to spin-dependent scattering
oscillating spin density
Indirect nuclear spin-spin coupling parameter A mn
A mn ∝ cos(2k F R mn )/R 3 mn
free electron gas (FEG) Ruderman and Kittel (1954)
KKR07 – p.71/100

Ludwig
Maximilians-
Universität
München
Probing of distorted electronic structure
Expectation value for hyperfine interaction
〈H mn
hf
〉 = − 1 ∫ EF
π I Trace dE d
∫Ω 3 r H m hf
(⃗r) G(⃗r, ⃗r, E)
m
− 1 ∫ EF
∫
π I Trace dE d
∫Ω 3 r d 3 r ′
m Ω n
H m hf (⃗r) G(⃗r, ⃗r ′ , E) H n hf (⃗r ′ ) G(⃗r ′ , ⃗r, E)
Probe: nuclear spin ⃗µ m in atomic cell m
expressed by hyperfine operator H m hf (⃗r)
H m hf
(⃗r) short ranged – restricted to atomic cell m
second term: indirect nuclear spin-spin coupling
KKR07 – p.72/100

Ludwig
Maximilians-
Universität
München
Indirect nuclear spin-spin coupling tensor
nuclear spin-spin coupling Hamiltonian
H nuc = − 1 2
∑
n≠m
⃗I n A mn
⃗I m
Indirect nuclear spin-spin coupling tensor
A mn = 1
2π C m C n I
( ∑
Λ
∫ EF
dE
∑
Λ ′
⃗¯M
Λ hf m mn
′′′ Λ
(E)τΛΛ ′ (E)
⊗
∑
Λ ′′′
)
( ∑
Λ ′′ ⃗¯M hf n
Λ ′ Λ ′′ (E) τ nm
Λ ′′ Λ ′′′ (E)
)
KKR07 – p.73/100

Ludwig
Maximilians-
Universität
München
Ruderman-Kittel coupling parameter A RK
mn
Results for Cu
A mn
/ (µ n
/I) 2 (nK)
2.5
0
-2.5
-5
-7.5
FEG
Oja et al.
present
Cu
-10
0.5 1 1.5 2 2.5
R mn
(a)
relativistic
small anisotropy
scalar-relativistic
no anisotropy
Oja et al (1989)
free electron gas
1
R 3 cos(2k F R)
slower decay
out of phase
KKR07 – p.74/100

Ludwig
Maximilians-
Universität
München
R- and Q-parameters for Cu
R-parameter
Q-parameter
[ ∑
m
R =
∑
m
A RK
mn
µ 0 2 γ n ρ
Q =
(
A RK
mn
µ 0 2 γ n ρ
) 2
] 1/2
Theory Expt
Oja Present
−0.37 −0.38 −0.42
±0.05
Theory Expt
Oja Present
+0.101 +0.099 +0.095
±0.003
sign indicates
nuclear antiferromagnet
deduced from
NMR line width
KKR07 – p.75/100

Ludwig
Maximilians-
Universität
München
XMCD Experiment
I 0 I = I
0
exp(− d)
circularly polarised
d
Magnetisation
Magnetic circular dichroism
µ λ q
M
L
3p
3p
3s
2p
2p
2s
3/2
1/2
1/2
3/2
1/2
1/2
λ
Absorption coefficient
depends on relative orientation of
magnetisation and polarisation λ
K
1s
1/2
First XMCD experiments:
G. Schütz et al. (1987), K-edge of Fe
KKR07 – p.76/100

Ludwig
Maximilians-
Universität
München
Calculation of XMCD spectra
Fermi’s golden rule
µ ⃗qλ (ω) ∝
∑
|〈Φ f |X ⃗qλ |Φ i 〉| 2 δ(ω − E f + E i )
i occ
f unocc
∝
∑
i occ
〈Φ i |X × ⃗qλ IG+ (E f ) X ⃗qλ |Φ i 〉θ(E f − E F )
relativistic treatment — X ⃗qλ = − 1 c ⃗ j el · ˆ⃗a λ Ae i⃗q⃗r
X-ray circular magnetic dichroism (XMCD)
∆µ ⃗qλ (ω) = µ ⃗q+ (ω) − µ ⃗q− (ω)
Ebert, RPP 59 1665 (’96)
KKR07 – p.77/100

Ludwig
Maximilians-
Universität
München
Magnetic EXAFS
K-edge of Fe in bcc-Fe
L 3 -edge of Pt in Fe 3 Pt
influence of disorder
µ(E) (arb. units)
100.00
80.00
60.00
40.00
20.00
µ(E) (arb.units)
75.00
65.00
55.00
45.00
35.00
0.04
0 200 400 600
energy (eV)
∆µ(E) (arb. units)
0.02
0.00
−0.02
−0.04
0.0 100.0 200.0 300.0 400.0 500.0
energy (eV)
∆µ(E) (arb.units)
0.20
0.10
0.00
−0.10
−0.20
0 200 400 600
energy (eV)
KKR07 – p.78/100

Ludwig
Maximilians-
Universität
München
XMCD at L 2,3 -edges of Gd, Tb, Dy
1.0
Gd(L 3
)
1.0 Gd(L 2
)
1.0 Tb(L 3
)
1.0 Tb(L 2
)
1.0 Dy (L 3
)
1.0 Dy(L 2
)
µ(Mb)
0.5
Experiment
dipole
quadrupole
dip.+quad.
0.5
0.5
0.5
0.5
0.5
0.0
0.0
0.0
0.0
0.0
0.0
∆µ(Mb)
0.02
0.01
0.00
-0.02
0.02
0.00
0.02
0.00
0.02
0.00
0.00
-0.04
0.00
-0.02
-0.02
-0.02
7240 7260
7920 7940
7510 7520 7530
8240 8260 8280
7785 7800
8580 8600
energy (eV)
energy (eV)
energy (eV)
energy (eV)
energy (eV)
energy (eV)
Experiment: H. Wende et al, (PRB, submitted, 2006)
KKR07 – p.79/100

∆µ = δµ + − δµ −
KKR07 – p.80/100
Ludwig
Maximilians-
Universität
München
Field-induced XMCD
X-ray absorption coefficient:
µ λ (ω) ∝ ∑ i
〈Φ i |X † ⃗qλ G0 (E)X ⃗qλ |Φ i 〉 Θ(E − E F ) .
E = E i + ω
The correction to the absorption coefficient δµ λ due
to the term δG B
µ B λ (ω) = µ λ (ω) + δµ λ (ω) ,
δµ λ (ω) ∝ ∑ i
〈Φ i |X † ⃗qλ δGB (E)X ⃗qλ |Φ i 〉 Θ(E − E F ) .
MCXD signal linear with respect to external field:

Ludwig
Maximilians-
Universität
München
Comparison with experiment
Absorption coefficient µ L2,3 for the L 2,3 -edges of Pd
L 2 -edge
L 3 -edge
µ Pd
(a.u.)
L 2
0.25
0.2
0.15
0.1
theor.
experim. *
µ Pd
(a.u.)
L 3
0.8
0.6
0.4
theor.
experim. *
0.05
0.2
0
0.0006
0
0
∆µ Pd
(a.u.)
L 2
0.0004
0.0002
∆µ Pd
(a.u.)
L 3
-0.0005
-0.001
0
3310 3320 3330 3340 3350 3360 -0.0015
energy (eV)
External magnetic field H = 10 T
3160 3180 3200
energy (eV)
Experiment: A. Rogalev et al.
KKR07 – p.81/100

Ludwig
Maximilians-
Universität
München
Application of the modified sum rules
Spin and orbital Pt susceptibilities in Ag x Pt 1−x
200
spin
60
orbital
χ spin
(10 -6 emu/mol)
150
100
50
direct calc.
sum rule
direct calc., NO enhancement
sum rule, NO enhancement
χ orb
(10 -6 emu/mol)
50
40
30
20
10
direct calc.
sum rule
0
0 20 40 60 80
x Ag
(%)
0
0 20 40 60 80
x Ag
(%)
unique tool to separate spin and orbital
susceptibilities
concentration dependence well described
Stoner enhancement accounted for
KKR07 – p.82/100

Ludwig
Maximilians-
Universität
München
Theory of X-ray magneto-optics
Two possible approaches:
- Atomic scattering amplitude f
- Conductivity tensor σ
Extension of Fresnel’s equations
Complex refractive index n(ω):
n ± (ω) = √ ɛ(ω) = 1 − δ 0 ± δ(ω) + i(β 0 ± β(ω))
ɛ(ω) dielectric tensor
ɛ(ω) = 1 + 4πi
ω σ(ω)
σ(ω) Conductivity tensor – microscopical properties of a medium
Circular dichroism
∆n =
1
2 (n + − n − )
= −∆δ + i∆β
Faraday rotation
Θ F = φ F + iɛ F
= 2πid
c
σ xy
√
1−
4π
ω σ xx
Kerr rotation
Θ K =
σ xy
σ xx
√
1−
4π
ω σ xx
KKR07 – p.83/100

Ludwig
Maximilians-
Universität
München
Calculation of the conductivity tensor
Linear response theory (Kubo formula)
written in the terms of Green‘s function:
σ αβ (ω) = i
π 2 V
∑
i
∫
E F
dE
∫
V
d 3 r
∫
V
d 3 r ′
Trace { 〈φ i | j α (⃗r) ImG + (E)j β (⃗r ′ ) |φ i 〉 }
(E − E i + i0 + )(ω + E i − E + iΓ i )
+ Trace { 〈φ i | j β (⃗r ′ ) ImG + (E)j α (⃗r) |φ i 〉 }
(E − E i + i0 + )(ω + E − E i + iΓ i )
G + : Green’s function representing VB-states
φ i : tightly bounded core states
⃗j = ec⃗α: relativistic current density operator
KKR07 – p.84/100

Ludwig
Maximilians-
Universität
München
Results of calculations
Faraday rotation at the K-edge of fcc-Co
θ F
(10 -3 rad)
2
1.5
1
0.5
0
-0.5
Rotation
θ F
(d=2.5µm)
Experiment
0 10 20 30 40 50
µ K
(arb. units)
2
1
0
-1
-2
Absorption
Experiment
Theory
0 10 20 30 40 50
energy (eV)
energy (eV)
Exp: Siddons et al. PRL 64,1967 (1990)
KKR07 – p.85/100

Ludwig
Maximilians-
Universität
München
L 2,3 -edges of disordered fcc Fe 50 Ni 50
Circular dichroic parts ∆δ and ∆β of the n
∆β
∆δ
8×10 -4
0
-8×10 -4
-2×10 -3
1×10 -3
5×10 -4
0
-5×10 -4
-1×10 -3
∆β
∆δ
2×10 -4
-2×10 -4 0
-4×10 -4
2×10 -4
1×10 -4
0
-1×10 -4
-2×10 -4
-10 0 10 20 30
-10 0 10 20 30
L 2,3 -edges of Fe in Fe 50 Ni 50
Energy [eV]
Energy [eV]
L 2,3 -edges of Ni in Fe 50 Ni 50
Exp: Mertins et al., Nucl. Inst. Methods A (2001)
KKR07 – p.86/100

Ludwig
Maximilians-
Universität
München
Fano effect in VB-photoemission
Spin polarisation of photo electrons
due to spin-orbit coupling
spin resolved angle integrated
photoemission experiment
paramagnetic systems
circularly
polarized
ESRF
spin polarized
photoelectrons
detector
semisphere
spin
e −
KKR07 – p.87/100

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!
§
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¤
¨
¤
Ludwig
Maximilians-
Universität
München
One-step model of photoemission
photo current
j(ˆk, E fin ) ∝ √ E fin
∫
d 3 r ∫ d 3 r ′
φ fin (⃗r, ˆk, E fin ) † X ⃗qλ (⃗r)IG(⃗r, ⃗r ′ , E)X † ⃗qλ (⃗r ′ )φ fin (⃗r ′ , ˆk, E fin )
φ fin (⃗r, ˆk, E) = TR φ LEED (⃗r, ˆk, E)
∫
φ LEED (⃗r, ˆk, E) = e i⃗k⃗r + d 3 r ′ G(⃗r, ⃗r ′ , E) V (⃗r ′ ) φ LEED (⃗r ′ , ˆk, E)
KKR07 – p.88/100

Ludwig
Maximilians-
Universität
München
Photo current intensity within KKR
I ms m ′ s (E, ⃗ k; ω, ⃗q, λ)∝ ∑ Λ Λ ′′ i l−l′′ C −m s
Λ
∑
with the matrix elements
M ⃗qλ
ΛΛ ′ =
C −m′ s
Λ ′′
Y µ+m s∗
l
e i⃗ k( ⃗R m − ⃗R ∑ m ′)
e i⃗q( ⃗R n − ⃗R n ′)
m m ′ n n
∑
′ τΛ nm
′ Λ (E′ )τ n′ m ′ ∗
Λ ′′′ Λ ′′ (E ′ )
Λ ′ Λ
[ ′′′ ∑
M ⃗qλ
Λ ′ Λ 1
τΛ nn′
1 Λ 2
(E) ¯M ⃗qλ
Λ 2 Λ ′′′
Λ 1 Λ 2
∫
− δ n n ′
(−ˆk)Y µ′′ +m ′ s
l ′′ (−ˆk)
∑
]
I n ⃗qλ
Λ ′ Λ 1 Λ ′′′
Λ 1
d 3 r[T Z Λ (⃗r, E ′ )] † X ⃗qλ (⃗r)Z Λ ′(⃗r, E)
KKR07 – p.89/100

Ludwig
Maximilians-
Universität
München
Spin-resolved VB-XPS of Cu, Ag and Au
Cu (600 eV)
Ag (600 eV)
Au (600 eV)
I +
Intensity (arb.u.)
100
50
exp.
theory
100
50
100
50
∆I +
Intensity (arb.u.)
0
5
0
-5
exp.
theory
0
10
0
-10
0
20
10
0
-10
10 8 6 4 2 0
10 8 6 4 2 0
-20
10 8 6 4 2 0
Binding energy (eV)
Binding energy (eV)
Binding energy (eV)
KKR07 – p.90/100

Ludwig
Maximilians-
Universität
München
Perturbational treatment of spin-orbit coupling
non-relativistic spin-resolved photo-current
I(E, ⃗k, m s ; ω, ⃗q, λ) =
∫
d 3 r
• final states (time reversed LEED state)
∫
d 3 r ′ φ ∗ final
(⃗r, E + ω)
× X ⃗q,λ Im G(⃗r, ⃗r ′ , E)X ∗ ⃗q,λ φ final(⃗r, E + ω)
φ final (⃗r, E) = 4π ∑ L
i l Y ∗ L (̂k)t l Z l (⃗r)Y L (̂r)χ ms
• initial states
Im G(⃗r, ⃗r ′ , E) = ∑
Z L1 (⃗r, E)Im τL nn
1 ,L 2
Z † L 2
(⃗r, E)
L 1 ,L 2
Include spin-orbit coupling (1st order):
G = G 0 + G 0 Ĥ SOC G 0
KKR07 – p.91/100

Ludwig
Maximilians-
Universität
München
Spin-resolved Photo current
I m s
λ
∝ ∑ L
[t l (E)] 2 ∑ L 1 L 2
[M λ Lm s L 1 m s
M λ∗
Lm s L 2 m s
Iτ L1 L 2
(E)
+ ∑ L 3 L 4
ξ lms M λ Lm s L 1 m s
M SOC
L 2 m s L 3 m s
M λ∗
LL 4
I(τ L1 L 2
(E)τ L3 L 4
(E))]
Spin Difference ∆I λ = I ↑ λ − I↓ λ
S lλ =
(for l = 2 d- electrons)
∆I λ ∝
∑
ξ d S lλ Im [ ( )]
τ t2g 4 τeg + τ t2g
⎧
l
⎪⎨
⎪⎩
+1 d → p λ = +1 (LCP)
−1 d → f λ = +1 (LCP)
−1 d → p λ = −1 (RCP)
+1 d → f λ = −1 (RCP)
KKR07 – p.92/100

Ludwig
Maximilians-
Universität
München
Fano effect in VB-photoemission
Spin polarisation of photo electrons
due to spin-orbit coupling
spin resolved angle integrated
photoemission experiment
ferromagnetic systems
circularly
polarized
ESRF
spin polarized
photoelectrons
Detector
semisphere
spin
e −
magnetization in plane
KKR07 – p.93/100

Ludwig
Maximilians-
Universität
München
Fano effect in VB-XPS of ferromagnets
Photocurrent and spin-difference
Minár et al., PRL , 95, 166401 (2005)
Intensity (arb.units)
100
50
E phot = 600 eV
Fe Co Ni
exp.
LDA
LDA+DMFT
Intensity (arb. units)
100
80
60
40
20
exp.
LDA
LDA+DMFT
Intensity (arb. units)
100
50
exp.
LDA
LDA+DMFT
Spin-difference (arb. units.)
0
0
Fe (600eV)
-2
-10 -8 -6 -4 -2 0 2
Binding energy (eV)
Spin-difference (arb. units)
0
4
2
0
-2
-4
Co (600eV)
-10 -8 -6 -4 -2 0 2
Binding energy (eV)
Spin-difference (arb. units)
0
2
0
-2
Ni (600eV)
-4
-10 -8 -6 -4 -2 0 2
Binding energy (eV)
KKR07 – p.94/100

Ludwig
Maximilians-
Universität
München
Magnetic Compton Scattering
Standard expression for magnetic Compton scattering:
( )
∂ 2 σ
∂Ω∂p z
↑
−
( )
∂ 2 σ
∂Ω∂p z
↓
= P c r 2 o Ψ 2(⃗S)J − (p z )
Geometry factor:
Ψ 2 = ±σ ( k o cos α cos φ − k ′ cos(α − φ) ) (cosφ − 1) c
m o c 2
Magnetic Compton profile:
J − (p z ) = ∫ ∫ ( n ↑ (⃗p) − n ↓ (⃗p) ) dp x dp y
KKR07 – p.95/100

Ludwig
Maximilians-
Universität
München
Multiple Scattering Formulation
Green’s function in momentum representation
G ms m ′ (⃗p, ⃗p ′ , E) = 1
s
NΩ
∫
∫
d 3 r d 3 r ′ ×
Density of spin-projected momentum
×Φ × ⃗pm s
(⃗r)G + (⃗r, ⃗r ′ , E)Φ ⃗p ′ m ′ (⃗r ′ )
s
= G ms (⃗p, ⃗p ′ , E)δ ms m ′ δ s ⃗p⃗p ′
n m s
(⃗p) = − 1 π
∫ EF
0
ImG ms (E, ⃗p, ⃗p)dE
Magnetic Compton profile
J − (p z ) =
∫ ∫ (
n ↑ − n ↓) dp x dp y
KKR07 – p.96/100

Ludwig
Maximilians-
Universität
München
Application to transition metals
MCP (arbitrary units)
0.2
0.15
0.1
0.05
Magnetic Compton
profile for Ni[001]:
experimental
FLAPW 0.43 broad
KKR 0.43 broad
Anisotropic profiles:
disordered alloy Ni 75 Co 25
∆J 110-001
(arb. units)
0.05
0
-0.05
KKR Bansil et.al
experimental
KKR 0.4 broad
0
0 1 2 3 4 5
momentum p z
(a.u)
Experiment: Dixon et. al (1991)
Measurements on the high-energy X-Ray
beam line(ID15) at the ESRF.
The experimental profile normalised to a
spin moment µ spin =0.58µ B . Momentum
resolution of 0.43 a.u.
0 1 2 3 4 5
momentum p z
(a.u)
Experiment: Bansil et. al (1998)
Momentum resolution of 0.40 a.u.
KKR07 – p.97/100

Ludwig
Maximilians-
Universität
München
Application to UFe 2
Comparison
with experiment
Decomposition
MCP (arbitrary units)
0.2
0.1
0
U (KKR)
Fe (KKR)
experiment
interf (KKR)
total (KKR)
MCP (arbitrary units)
0.2
0.1
0
U (KKR)
U 5f (Lawson)
Fe (KKR)
Fe 3d (Lawson)
interf (KKR)
spd (Lawson)
-0.1
0 1 2 3 4 5
momentum p z
(atomic units)
-0.1
0 1 2 3 4 5
momentum p z
(atomic units)
appreciable interference term
component-resolved contributions do not scale with partial moments
KKR07 – p.98/100

Ludwig
Maximilians-
Universität
München
I didn’t talk about
Friedel’s sum rule
Lloyd’s formula
full potential calculations
calculation of forces
tight-binding (TB) formulation
layer KKR/LEED formalism
non-local CPA
non-collinear magnetism
treatment of interstitials
...
KKR07 – p.99/100

Ludwig
Maximilians-
Universität
München
The story goes on ....
Seattle
Champaign
OakRidge
Warwick
Daresbury
Bristol
München
Wien
Jülich
Halle
Budapest
Messina
Osaka
... KKR forges all over the world
KKR07 – p.100/100