Electronic Transport in disordered alloys with
short-range order: microstructural and
concentration dependence
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 ,
A. Marmodoro1 and H. Ebert2
1
University of Warwick, 2 Universitat Munchen,
3
University of Edinburgh
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Electrons and Disorder - An Effective Lattice
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Electrons and Disorder - An Effective Lattice
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Electrons and Disorder - An Effective Lattice
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Electrons and Disorder - An Effective Lattice
P
< Gij >= Gij0 + kl Gik0 Ξkl < Glj >
P
Ḡ (k) = N1 j < Gij > e ik·(Ri −Rj ) = (G 0,−1 (k) − Ξ(k))−1
Ξ(k) is a self energy.
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Cluster Approximation
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Cluster Approximation
GIJγ = [G 0,−1 + Ξ − V γ ]−1
IJ
P
γ
γ P(γ)GIJ = ĜIJ ≈< GIJ >
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Cluster Approximation
GIJγ = [G 0,−1 + Ξ − V γ ]−1
IJ
P
γ
γ P(γ)GIJ = ĜIJ ≈< GIJ >
P R
ĜIJ = Ω1BZ Kn [G 0 (k) − Ξ(Kn ]−1 e iKn ·(RI −RJ ) dkn .
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
DFT for disordered systems with SRO: KKR-NLCPA
Scattering path operator (SPO) for an electron, moving in an effective
medium mimicking the average motion in a disordered system
τ̂ ij = t̂ δij +
X
t̂ (G (Ri − Rk ) + δ Ĝ (Ri − Rk ))τ̂ kj .
k6=i
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
DFT for disordered systems with SRO: KKR-NLCPA
Scattering path operator (SPO) for an electron, moving in an effective
medium mimicking the average motion in a disordered system
τ̂ ij = t̂ δij +
X
t̂ (G (Ri − Rk ) + δ Ĝ (Ri − Rk ))τ̂ kj .
k6=i
Effective medium must be translationally invariant so
1
τ̂ =
ΩBZ
ij
Z
dk[t̂ −1 − G (k) − δ Ĝ (k)]−1 e ik·(Ri −Rj ) .
ΩBZ
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
DFT for disordered systems with SRO: KKR-NLCPA
Scattering path operator (SPO) for an electron, moving in an effective
medium mimicking the average motion in a disordered system
τ̂ ij = t̂ δij +
X
t̂ (G (Ri − Rk ) + δ Ĝ (Ri − Rk ))τ̂ kj .
k6=i
Effective medium must be translationally invariant so
1
τ̂ =
ΩBZ
ij
Z
dk[t̂ −1 − G (k) − δ Ĝ (k)]−1 e ik·(Ri −Rj ) .
ΩBZ
Coarse graining in real and reciprocal space (dynamical cluster
approximation (DCA) (M.Jarrell et al.)
1 X iKn ·(RI −RJ )
= δIJ .
e
Nc
Kn
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
DFT for disordered systems with SRO: KKR-NLCPA
Nc
τ̂ (Kn ) =
ΩBZ
Z
d k̃[t̂ −1 − G (k̃ + Kn ) − δ Ĝ (Kn )]−1 .
Ωt
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
DFT for disordered systems with SRO: KKR-NLCPA
Nc
τ̂ (Kn ) =
ΩBZ
Z
d k̃[t̂ −1 − G (k̃ + Kn ) − δ Ĝ (Kn )]−1 .
Ωt
Multiple scattering between cluster sites I and J
τ̂ IJ =
1 X
τ̂ (Kn )e iKn ·(RI −RJ ) .
Nc
Kn
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
DFT for disordered systems with SRO: KKR-NLCPA
Nc
τ̂ (Kn ) =
ΩBZ
Z
d k̃[t̂ −1 − G (k̃ + Kn ) − δ Ĝ (Kn )]−1 .
Ωt
Multiple scattering between cluster sites I and J
τ̂ IJ =
1 X
τ̂ (Kn )e iKn ·(RI −RJ ) .
Nc
Kn
Finally specifying the SPO for impurity clusters, τγIJ , of atoms
embedded into the NLCPA medium. and taking a weighted average
(including SRO) fixes the NLCPA medium
P
γ
Pγ τγIJ = τ̂ IJ .
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Illustration with CuZn
80
ordered
(b)
Density of States (states/atom/Ry)
Density of States (states/atom/Ry)
60
50
40
30
20
10
Cu
Zn
(a)
70
60
50
40
30
20
10
0
0
0.1
0.2
0.3
0.4
0.5
Energy (Ry)
0.6
0.7
0
0
Density of States (states/atom/Ry)
60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
total Nc=2
Cu-Cu
Cu-Zn
Zn-Cu
Zn-Zn
total Nc=16
(b)
50
40
30
20
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
EF
|
0.7
The DOS of the Cu50Zn50 alloy system (centre), ordered Cu-Zn(left) and elements Cu and Zn (right).
2
1
2
2
1
2 1
Julie Staunton
, S. Lowitzer
, P. R.Rev.
TulipB3 ,72,
D. 045101,
Koedderitzsch
, A. Marmodoro
Electronic
Transport
and
H. Ebert
in
disordered
University
alloys withof
short-range
Warwick,
order: m
U
D.A.Rowlands
et al., Phys.
(2005);Phys.
Rev. B 73,
165122,
(2006).
Cu50Zn50 - short-ranged order and cluster defects
Density of States (states/atom/Ry)
60
total Nc=2
Cu-Cu
Cu-Zn
Zn-Cu
Zn-Zn
total Nc=16
(b)
50
40
30
20
10
EF
|
0.7
0
0
60
0.1
0.2
0.4
0.5
0.6
60
total
Cu-Cu
Cu-Zn
Zn-Cu
Zn-Zn
(f) α=-1.0
50
0.3
total
Cu-Cu
Cu-Zn
Zn-Cu
Zn-Zn
(f) α=+1.0
50
40
40
30
30
20
20
10
10
0
0
0.1
0.2
0.3
0.4
Energy (Ry)
0.5
0.6
EF
|
0.7
0
0
0.1
0.2
0.3
0.4
Energy (Ry)
0.5
0.6
EF
|
0.7
The DOS of Cu50Zn50 with short-ranged order (left), random (centre) and short-ranged clustering (right).
2
1
2
2
1
2 1
Julie Staunton
, S. Lowitzer
, P. R.Rev.
TulipB3 ,72,
D. 045101,
Koedderitzsch
, A. Marmodoro
Electronic
Transport
and
H. Ebert
in
disordered
University
alloys withof
short-range
Warwick,
order: m
U
D.A.Rowlands
et al., Phys.
(2005);Phys.
Rev. B 73,
165122,
(2006).
The Conductivity; Kubo-Greenwood linear response
From the symmetric part of the conductivity tensor of form,
C = Tr < O1 G O2 G > , the d.c. conductivity is
σµν
where
=
1
lim [σ̃µν (E + , E + ) − σ̃µν (E − , E + ) −
4 η→0
σ̃µν (E + , E − ) + σ̃µν (E − , E − )]
E + = EF + iη, E − = EF − iη, η → 0.
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
The Conductivity; Kubo-Greenwood linear response
From the symmetric part of the conductivity tensor of form,
C = Tr < O1 G O2 G > , the d.c. conductivity is
σµν
where
=
1
lim [σ̃µν (E + , E + ) − σ̃µν (E − , E + ) −
4 η→0
σ̃µν (E + , E − ) + σ̃µν (E − , E − )]
E + = EF + iη, E − = EF − iη, η → 0.
In the KKR formalism
σ̃µν (z1 , z2 ) = −
4m2 X
< Jµi (z2 , z1 )τ ij (z1 )Jνj (z1 , z2 )τ ji (z2 ) >
πNΩ~3 i,j
where Jµi is a current operator associated with site i .
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
The Conductivity; Kubo-Greenwood linear response
From the symmetric part of the conductivity tensor of form,
C = Tr < O1 G O2 G > , the d.c. conductivity is
σµν
where
=
1
lim [σ̃µν (E + , E + ) − σ̃µν (E − , E + ) −
4 η→0
σ̃µν (E + , E − ) + σ̃µν (E − , E − )]
E + = EF + iη, E − = EF − iη, η → 0.
In the KKR formalism
σ̃µν (z1 , z2 ) = −
4m2 X
< Jµi (z2 , z1 )τ ij (z1 )Jνj (z1 , z2 )τ ji (z2 ) >
πNΩ~3 i,j
where Jµi is a current operator associated with site i .
The NLCPA enables the average to be taken (vertex corrections included).
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Residual Resistivity of Cux Zn1−x Alloys
20
10
-6
ρ (10 Ohm cm)
15
5
0
0
0.2
0.4
0.6
0.8
1
C
The residual resistivity of the Cux Zn1−x alloy system with random disorder (red lines, crosses), SRO (green,
asterisks) and SR-clustering (blue, boxes). (Results without vertex corrections are also shown).
P.R.Tulip et al., Phys. Rev. B 77, 165116, (2008)
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Residual Resistivity of Agx Pd1−x Alloys
Guenault
total disorder
SRO
clustering
30
-6
ρ (10 Ohm cm)
40
20
10
0
0
0.2
0.4
0.6
0.8
1
X
The residual resistivity of the Agx Pd1−x alloy system with random disorder (blue diamonds), SRO (orange circles)
and SR-clustering (black triangles). The experimental results of Guenault et al. are also shown (full red circles).
P.R.Tulip et al., Phys. Rev. B 77, 165116, (2008)
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
The Slater-Pauling curve and ferromagnetic alloys
Atoms
Energy
Fe
Atoms
Energy
Alloy
Fe
Fe
Ni
Fe, Ni
Antibonding
V
Fe, Ni
Bonding
Antibonding
Fe, V
V
Fe, V
Bonding
Ni
Fe, Ni
Alloy
Fe
Fe
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
The Slater-Pauling curve and ferromagnetic alloys
Atoms
Energy
Fe
Atoms
Energy
Alloy
Alloy
Antibonding
Fe, V
Fe
Fe
Ni
Fe, Ni
Antibonding
V
V
Fe, V
Bonding
Ni
Fe
Fe
ntot(E) (sts./eV)
1.6
1.6
0.8
↓
Fe, Ni
Bonding
ntot(E) (sts./eV)
Fe, Ni
0
↑
0.8
0
-10
-8
-6
-4
-2
0
2
4
energy (eV)
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Residual Resistivity of FeCr, FeV Alloys
The residual resistivity of iron-rich FeCr and FeV alloys. (Experimental data from Nikoleav et al. for FeCr).
14
1000
total conductivity
σ↑
σ↓
10
100
8
6
Fe1-xCrx (Expt.)
Fe1-xCrx (Theory)
Fe1-xVx (Theory)
4
σ (a.u.)
-6
ρ (10 Ohm cm)
12
10
2
0
0
0.02
0.04
0.06
0.08
0.1
x
0.12
0.14
0.16
0.18
0.2
1
0.04
0.06
0.08
0.1
0.12
x
0.14
0.16
0.18
0.2
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Residual Resistivity of FeCr, FeV Alloys
The residual resistivity of iron-rich FeCr and FeV alloys. (Experimental data from Nikoleav et al. for FeCr).
14
1000
total conductivity
σ↑
σ↓
10
100
8
6
σ (a.u.)
-6
ρ (10 Ohm cm)
12
Fe1-xCrx (Expt.)
Fe1-xCrx (Theory)
Fe1-xVx (Theory)
4
10
2
0
0
0.02
0.04
0.06
0.08
0.1
0.12
x
0.14
0.16
0.18
1
0.2
0.04
0.08
0.06
0.1
0.12
x
0.14
0.16
0.18
0.2
14
-6
ρ (10 Ohm cm)
12
10
8
Fe1-xCrx (Expt.)
Fe1-xCrx (CPA)
Fe1-xCrx (NLCPA clustering)
Fe1-xCrx (NLCPA SRO)
Fe1-xCrx (NLCPA disordered)
Fe1-xVx (CPA)
Fe1-xVx (NLCPA SRO)
6
4
2
0
0
0.02
0.04
0.06
0.08
0.1
x
0.12
0.14
0.16
0.18
0.2
S.Lowitzer et al., Phys. Rev. B 79, 115109, (2009).
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
How resistivity can reduce when an alloy is deformed.
For a class of transition metal materials
residual resistivity decreases as the materials
are deformed and SRO is removed.
Typically alloys rich in late transition
metals such as Ni, Pd or Pt,alloyed to a
mid-row element such as Cr, Mo or W.
Simple model of ‘working’ of an alloy is
assumed to reduce the number of unlike nearest
neighbours to an atom on the average.
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
How resistivity can reduce when an alloy is deformed.
For a class of transition metal materials
residual resistivity decreases as the materials
are deformed and SRO is removed.
Typically alloys rich in late transition
metals such as Ni, Pd or Pt,alloyed to a
mid-row element such as Cr, Mo or W.
Simple model of ‘working’ of an alloy is
assumed to reduce the number of unlike nearest
neighbours to an atom on the average.
For late-row TM, EF top of d-bands,
DOS large, d-orbitals pointing between
nearest neighbours.
’Working’ an late TM-rich alloy reduces
no. of unlike nearest neighbours
- electron hopping between neighbouring
late TM atoms enhanced at EF .
Larger Fermi velocity, hence reduced resistivity.
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Residual Resistivity of NiMo, NiCr,PdW
90
80
-6
ρ (10 Ohm cm)
70
60
50
40
CPA
total disorder
SRO
clustering
30
20
10
0
Ni0.8Cr0.2
Ni0.8Mo0.2
Pd0.8W0.2
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Residual Resistivity of NiMo, NiCr,PdW
90
80
-6
ρ (10 Ohm cm)
70
60
50
40
CPA
total disorder
SRO
clustering
30
20
10
0
Ni0.8Mo0.2
Pd0.8W0.2
σNi-Ni
σMo-Mo
σNi-Mo
σNi-Ni s-p-s
σNi-Ni s-p-d
σNi-Ni p-d-p
σNi-Ni p-d-f
σNi-Ni d-f-d
20
15
σ (a.u.)
Ni0.8Cr0.2
10
5
0
SRO
total disorder
clustering
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Residual Resistivity of NiMo, NiCr,PdW
90
80
-6
ρ (10 Ohm cm)
70
60
50
40
CPA
total disorder
SRO
clustering
30
20
10
0
Ni0.8Mo0.2
Pd0.8W0.2
σNi-Ni
σMo-Mo
σNi-Mo
σNi-Ni s-p-s
σNi-Ni s-p-d
σNi-Ni p-d-p
σNi-Ni p-d-f
σNi-Ni d-f-d
20
15
σ (a.u.)
Ni0.8Cr0.2
10
5
0
SRO
total disorder
clustering
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Summary
DFT for disordered systems with SRO,SR-clustering.
Kubo-Greenwood linear response theory for electronic transport.
Effect of composition and number of like nearest neighbours on residual
resistivity.
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Summary
DFT for disordered systems with SRO,SR-clustering.
Kubo-Greenwood linear response theory for electronic transport.
Effect of composition and number of like nearest neighbours on residual
resistivity.
Cux Zn1−x , Agx Pd1−x . SRO reduces resistivity,ρ (EF away from
d-bands).
Ferromagnetic alloys, FeCr, FeV. ρ insensitive to dopant
concentration, SRO. Link to Slater-Pauling curve.
Late row TM-rich alloys, (e.g.NiMo, NiCr,PdW,PdAg) - ρ can
reduce when alloy is deformed.
2
2 1
Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
Electronic 1Transport
and H. Ebert
in disordered
University
alloys withof
short-range
Warwick,
order: m
U
Summary
DFT for disordered systems with SRO,SR-clustering.
Kubo-Greenwood linear response theory for electronic transport.
Effect of composition and number of like nearest neighbours on residual
resistivity.
Cux Zn1−x , Agx Pd1−x . SRO reduces resistivity,ρ (EF away from
d-bands).
Ferromagnetic alloys, FeCr, FeV. ρ insensitive to dopant
concentration, SRO. Link to Slater-Pauling curve.
Late row TM-rich alloys, (e.g.NiMo, NiCr,PdW,PdAg) - ρ can
reduce when alloy is deformed.
References
Poster P516 - Alberto Marmodoro et al.
KKR-NLCPA - D. A. Rowlands, J. B. Staunton and B. L. Györffy, Phys. Rev. B 67, 115109
(2003).
Electronic structure and SRO - D. A. Rowlands et al., Phys. Rev. B 72, 045101 (2005);
D. A. Biava et al., Phys. Rev. B 72, 113105 (2005).
DFT, disorder and SRO -D. A. Rowlands et al., Phys. Rev. B 73 165122 (2006).
Conductivity - P.R.Tulip et al. Phys. Rev. B 77, 165116, (2008).
FeCr,FeV - S. Lowitzer et al., Phys. Rev. B 79, 115109, (2009).
K-state alloys - S. Lowitzer et al., submitted to EPL, (2010).
NLCPA review -D. A. Rowlands, Rep.Prog.Phys. 72, 086501, (2009).
2
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Julie Staunton1 , S. Lowitzer2 , P. R. Tulip3 , D. Koedderitzsch2 , A. Marmodoro
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