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Relativistic effects
&
Non-collinear magnetism
(WIEN2k / WIENncm)
18th WIEN2k WorkshopPennState University – USA – 2011
Xavier RocquefelteInstitut des Matériaux Jean-Rouxel (UMR 6502)
Université de Nantes, FRANCE
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18th WIEN2k WorkshopPennState University – USA – 2011
Talk constructed using the following documents:Slides of:
Robert Laskowski, Stefaan Cottenier, Peter Blaha and Georg Madsen
Books:- WIEN2k userguide, ISBN 3-9501031-1-2- Electronic Structure: Basic Theory and Practical Methods, Richard M. Martin – ISBN 0 521 78285 6- Relativistic Electronic Structure Theory. Part 1. Fundamentals, Peter Schewerdtfeger, ISBN 0 444 51249 7
Notes of:- Pavel Novak (Calculation of spin-orbit coupling)
http://www.wien2k.at/reg_user/textbooks/- Robert Laskowski (Non-collinear magnetic version of WIEN2k package)
web:- http://www2.slac.stanford.edu/vvc/theory/relativity.html- wienlist digest - http://www.wien2k.at/reg_user/index.html- wikipedia …
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Few words about Special Theory of Relativity
Light
Composed of photons (no mass)
Speed of light = constant
c ≈ ≈ ≈ ≈ 137 au
Atomic units:ħ = me = e = 1
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Few words about Special Theory of Relativity
Light Matter
Composed of photons (no mass)
Speed of light = constant
c ≈ ≈ ≈ ≈ 137 au
Atomic units:ħ = me = e = 1
Speed ofmatter
mass
v = f(mass)
mass = f(v)
Composed of atoms (mass)
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Few words about Special Theory of Relativity
Light Matter
Composed of photons (no mass)
Lorentz Factor (measure of the relativistic effects)
Speed of light = constant
c ≈ ≈ ≈ ≈ 137 au
Atomic units:ħ = me = e = 1
Speed ofmatter
mass
v = f(mass)
mass = f(v)
Momentum: p = γγγγmv = Mv
Total energy:
Relativistic mass: M = γγγγm (m: rest mass)1
1
12
≥
−
=
cv
γ
Composed of atoms (mass)
E2 = p2c2 + m2c4
E = γγγγmc2 = Mc2
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( )22.1
58.01
1
1
122
=−
=
−
=
c
ve
γDetails for Au atom:
ccsve 58.0137
79)1( ==
Speed of the 1s electron (Bohr model):
+Zee-
Lorentz factor (γγγγ)
1
2
3
4
5
6
7
8
9
10
0 20 40 60 80 100 120
c ≈≈≈≈ 137 au
H(1s)
Speed (v)
Au(1s)
« Non-relativistic »particle: γγγγ = 1
1s electron of Au atom = relativistic particle
Definition of a relativistic particle (Bohr model)
au1)1(:H =sve
au79)1(:Au =sve
00003.1=γ
22.1=γnZ
ve∝∝∝∝
au1)1(:H =sve
au79)1(:Au =sve
00003.1=γ
22.1=γnZ
ve∝∝∝∝
Me(1s-Au) = 1.22me
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Relativistic effects
Relativistic increase in the mass of an electron with its velocity (when ve → c)
1) The mass-velocity correction
+Ze
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Relativistic increase in the mass of an electron with its velocity (when ve → c)
1) The mass-velocity correction
It has no classical relativistic analogueDue to small and irregular motions of an electron about its mean position (Zitterbewegung)
2) The Darwin term
+Ze
Relativistic effects
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Relativistic increase in the mass of an electron with its velocity (when ve → c)
1) The mass-velocity correction
It has no classical relativistic analogueDue to small and irregular motions of an electron about its mean position (Zitterbewegung)
2) The Darwin term
It is the interaction of the spin magnetic moment (s) of an electron with the magnetic field induced by its own orbital motion (l)
3) The spin-orbit coupling
+Ze
Relativistic effects
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Relativistic increase in the mass of an electron with its velocity (when ve → c)
1) The mass-velocity correction
It has no classical relativistic analogueDue to small and irregular motions of an electron about its mean position (Zitterbewegung)
2) The Darwin term
It is the interaction of the spin magnetic moment (s) of an electron with the magnetic field induced by its own orbital motion (l)
3) The spin-orbit coupling
The change of the electrostatic potential induced by relativity is an indirect effect of the core electrons on the valence electrons
4) Indirect relativistic effect
+Zeffe
Relativistic effects
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One electron radial Schrödinger equation
Ψ=Ψ
+∇−=Ψ εVH S2
2
1 Ψ=Ψ
+∇−=Ψ εV
mH
eS
22
2
h
HARTREE ATOMIC UNITS INTERNATIONAL UNITS
Atomic units:ħ = me = e = 11/(4 πε πε πε πε0) = 1
c = 1/ α α α α ≈≈≈≈ 137 au
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One electron radial Schrödinger equation
Ψ=Ψ
+∇−=Ψ εVH S2
2
1 Ψ=Ψ
+∇−=Ψ εV
mH
eS
22
2
h
r
ZV −=
r
ZeV
0
2
4πε−=
INTERNATIONAL UNITS
In a spherically symmetric potential
Atomic units:ħ = me = e = 11/(4 πε πε πε πε0) = 1
c = 1/ α α α α ≈≈≈≈ 137 au
( ) ( )ϕθ ,,,,, mllnmln YrR=Ψ
( ) ( ) ( )
∂∂+
∂∂
∂∂+
∂∂
∂∂=∇
2
2
2222
22
sin
1sin
sin
11
ϕθθθ
θθ rrrr
rr
HARTREE ATOMIC UNITS
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One electron radial Schrödinger equation
Ψ=Ψ
+∇−=Ψ εVH S2
2
1
( )lnln
e
ln
e
RRr
ll
mV
dr
dRr
dr
d
rm ,,2
2,2
2
2
1
2
1
2ε=
+++
− hh
In a spherically symmetric potential
Ψ=Ψ
+∇−=Ψ εV
mH
eS
22
2
h
r
ZV −=
r
ZeV
0
2
4πε−=
( )lnln
ln RRr
llV
dr
dRr
dr
d
r ,,2,2
2
2
1
2
1 ε=
+++
−
INTERNATIONAL UNITS
( ) ( )ϕθ ,,,,, mllnmln YrR=Ψ
( ) ( ) ( )
∂∂+
∂∂
∂∂+
∂∂
∂∂=∇
2
2
2222
22
sin
1sin
sin
11
ϕθθθ
θθ rrrr
rr
HARTREE ATOMIC UNITS
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Dirac Hamiltonian: a brief description
Dirac relativistic Hamiltonian provides a quantum mechanical description of electrons, consistent with the theory of special relativity.
Ψ=Ψ εDH with
E2 = p2c2 + m2c4
VcmpcH eD ++⋅= 2βα rr
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Dirac Hamiltonian: a brief description
Dirac relativistic Hamiltonian provides a quantum mechanical description of electrons, consistent with the theory of special relativity.
=
0
0
k
kk σ
σα
−=
1
1
0
0kβ
=
01
101σ
−=
0
02 i
iσ
−=
10
013σ
(2××××2) Pauli spin matrices
Momentum operator Rest mass
Electrostaticpotential
VcmpcH eD ++⋅= 2βα rr
(2××××2) unit matrices
Ψ=Ψ εDH with
E2 = p2c2 + m2c4
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Dirac equation: HD and ΨΨΨΨ are 4-dimensional
ΦΦΦΦ and χχχχ are time-independent two-component spinors describing the spatial and spin-1/2 degrees of freedom
ΨΨΨΨ is a four-component single-particle wave function that describes spin-1/2 particles.
=
4
3
2
1
4
3
2
1
ψψψψ
ε
ψψψψ
DH
spin up
spin down
Largecomponents (ΦΦΦΦ)
Smallcomponents (χχχχ)
In case of electrons:
ΦΦΦΦ====
χχχχψψψψ
Leads to a set of coupled equations for ΦΦΦΦ and χχχχ:
( ) ( )φεχσ 2cmVpc e−−=⋅ r
( ) ( )χεφσ 2cmVpc e+−=⋅ r
factor ∝ 1/(mec2)
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Dirac equation: HD and ΨΨΨΨ are 4-dimensional
For a free particle (i.e. V = 0):
( )( )
( )( )
0
0ˆˆˆ
0ˆˆˆ
ˆˆˆ0
ˆˆˆ0
4
3
2
1
2
2
2
2
=
ΨΨΨΨ
++−+−−−
+−−−−−−
cmppip
cmpipp
ppipcm
pippcm
ezyz
eyzz
zyze
yxze
εε
εε
↑
0
0
0,2
φ
cme
↓
0
0
0
,2 φcme
Particles: up & down
− ↑
0
0
0
,2
χcme
−
↓χ0
0
0
,2cme
Antiparticles: up & down
Solution in the slow particle limit (p=0)
Non-relativistic limit decouples Ψ1 from Ψ2
and Ψ3 from Ψ4
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Dirac equation: HD and ΨΨΨΨ are 4-dimensional
For a free particle (i.e. V = 0):
( )( )
( )( )
0
0ˆˆˆ
0ˆˆˆ
ˆˆˆ0
ˆˆˆ0
4
3
2
1
2
2
2
2
=
ΨΨΨΨ
++−+−−−
+−−−−−−
cmppip
cmpipp
ppipcm
pippcm
ezyz
eyzz
zyze
yxze
εε
εε
↑
0
0
0,2
φ
cme
↓
0
0
0
,2 φcme
Particles: up & down
− ↑
0
0
0
,2
χcme
−
↓χ0
0
0
,2cme
Antiparticles: up & down
Solution in the slow particle limit (p=0)
Non-relativistic limit decouples Ψ1 from Ψ2
and Ψ3 from Ψ4
For a spherical potential V(r):
( )( )
Υ−Υ
=
Φ=Ψκσnκ
κσnκ
χ rfi
rg gnκ and fnκ are Radial functions
Yκσ are angular-spin functions
2slj +=
( )21+−= jsκ1,1−+=s
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Dirac equation in a spherical potential
The resulting equations for the radial functions (gnκκκκ and fnκκκκ) are simplified if we define:
2' cme−= εε ( ) ( )22
'
c
rVmrM ee
−+= εEnergy: Radially varying mass:
For a spherical potential V(r):
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Dirac equation in a spherical potential
The resulting equations for the radial functions (gnκκκκ and fnκκκκ) are simplified if we define:
2' cme−= εε ( ) ( )22
'
c
rVmrM ee
−+= εEnergy: Radially varying mass:
Then the coupled equations can be written in the form of the radial eq.:
Darwin term
Spin-orbit coupling
( ) ( )11 +=+ llκκNote that:
For a spherical potential V(r):
Mass-velocity effect
( ) ( )κκ
κκ
κ εκnn
e
n
e
ne
n
e
ggrdr
dV
cMdr
dg
dr
dV
cMg
r
ll
MV
dr
dgr
dr
d
rM'
1
44
1
2
1
2 22
2
22
2
2
22
2
2
=+−−
+++
− hhhh
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Dirac equation in a spherical potential
The resulting equations for the radial functions (gnκκκκ and fnκκκκ) are simplified if we define:
2' cme−= εε ( ) ( )22
'
c
rVmrM ee
−+= εEnergy: Radially varying mass:
( ) ( )κκ
κκ
κ εκnn
e
n
e
ne
n
e
ggrdr
dV
cMdr
dg
dr
dV
cMg
r
ll
MV
dr
dgr
dr
d
rM'
1
44
1
2
1
2 22
2
22
2
2
22
2
2
=+−−
+++
− hhhh
( ) ( )κκ
κε nnnk f
rgV
cdr
df 1'
1 −+−=h
Then the coupled equations can be written in the form of the radial eq.:
and Darwin term
Spin-orbit coupling
( ) ( )11 +=+ llκκNote that:
For a spherical potential V(r):
Due to spin-orbit coupling, ΨΨΨΨ is not an eigenfunctionof spin (s) and angular orbital moment (l).
Instead the good quantum numbers are j and κκκκNo approximation have been made
so far
No approximation have been made
so far
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Dirac equation in a spherical potential
Scalar relativistic approximation
Approximation that the spin-orbit term is small ⇒⇒⇒⇒ neglect SOC in radial functions (and treat it by perturbation theory)
nln gg ~→κ nln ff~→κ
( )nl
nl
e
nle
nl
e
gdr
gd
dr
dV
cMg
r
ll
MV
dr
gdr
dr
d
rM~'
~
4~1
2
~1
2 22
2
2
22
2
2
ε=−
+++
− hhh
dr
gd
cMf nl
enl
~
2
~ h=and with the normalization condition: ( ) 1 ~~ 222 =+∫ drrfg nlnl
No SOC ⇒⇒⇒⇒ Approximate radial functions:
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Dirac equation in a spherical potential
Scalar relativistic approximation
Approximation that the spin-orbit term is small ⇒⇒⇒⇒ neglect SOC in radial functions (and treat it by perturbation theory)
nln gg ~→κ nln ff~→κ
( )nl
nl
e
nle
nl
e
gdr
gd
dr
dV
cMg
r
ll
MV
dr
gdr
dr
d
rM~'
~
4~1
2
~1
2 22
2
2
22
2
2
ε=−
+++
− hhh
dr
gd
cMf nl
enl
~
2
~ h=and with the normalization condition: ( ) 1 ~~ 222 =+∫ drrfg nlnl
No SOC ⇒⇒⇒⇒ Approximate radial functions:
The four-component wave function is now written as:
Inclusion of the spin-orbit coupling in “second variation” (on the large component only)( )
( )
Υ−Υ
=
Φ=Ψlmnl
lmnl
χ rfi
rg∼∼∼
∼∼
ψψεψ ~~~SOHH +=
with
=
00
01
4 22
2 l
dr
dV
rcMH
e
SO
rrh σ
Φ∼ is a pure spin state
χ∼ is a mixture of up and down spin states
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Relativistic effects in a solid
For a molecule or a solid:
Relativistic effects originate deep inside the core.
It is then sufficient to solve the relativistic equations in a spherical atomic geometry (inside the atomic spheres of WIEN2k).
Justify an implementation of the relativistic effects only inside the muffin-tin atomic spheres
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SOC: Spin orbit coupling
Implementation in WIEN2k
Atomic sphere (RMT) RegionAtomic sphere (RMT) Region
CoreelectronsCore
electrons
Valence electronsValence electrons
« Fully »relativistic
Spin-compensatedDirac equation
Scalar relativistic(no SOC)
Possibility to add SOC(2nd variational)
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Interstitial RegionInterstitial Region
Valence electronsValence electrons
Not relativistic
SOC: Spin orbit coupling
Implementation in WIEN2k
Atomic sphere (RMT) RegionAtomic sphere (RMT) Region
CoreelectronsCore
electrons
Valence electronsValence electrons
« Fully »relativistic
Spin-compensatedDirac equation
Scalar relativistic(no SOC)
Possibility to add SOC(2nd variational)
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Implementation in WIEN2k: core electrons
Atomic sphere (RMT) RegionAtomic sphere (RMT) Region
CoreelectronsCore
electrons
« Fully »relativistic
Spin-compensatedDirac equation
Atomic sphere (RMT) RegionAtomic sphere (RMT) Region
CoreelectronsCore
electrons
« Fully »relativistic
Spin-compensatedDirac equation
17 0.00 0 1,-1,2 ( n,κ,occup)2,-1,2 ( n,κ,occup)2, 1,2 ( n,κ,occup)2,-2,4 ( n,κ,occup)3,-1,2 ( n,κ,occup)3, 1,2 ( n,κ,occup)3,-2,4 ( n,κ,occup)3, 2,4 ( n,κ,occup)3,-3,6 ( n,κ,occup)4,-1,2 ( n,κ,occup)4, 1,2 ( n,κ,occup)4,-2,4 ( n,κ,occup)4, 2,4 ( n,κ,occup)4,-3,6 ( n,κ,occup)5,-1,2 ( n,κ,occup)4, 3,6 ( n,κ,occup)4,-4,8 ( n,κ,occup)0
17 0.00 0 1,-1,2 ( n,κ,occup)2,-1,2 ( n,κ,occup)2, 1,2 ( n,κ,occup)2,-2,4 ( n,κ,occup)3,-1,2 ( n,κ,occup)3, 1,2 ( n,κ,occup)3,-2,4 ( n,κ,occup)3, 2,4 ( n,κ,occup)3,-3,6 ( n,κ,occup)4,-1,2 ( n,κ,occup)4, 1,2 ( n,κ,occup)4,-2,4 ( n,κ,occup)4, 2,4 ( n,κ,occup)4,-3,6 ( n,κ,occup)5,-1,2 ( n,κ,occup)4, 3,6 ( n,κ,occup)4,-4,8 ( n,κ,occup)0
case.inc for Au atom
s 0 1/2 -1 2
p 1 3/21/2 1 -2 2 4
d 2 5/23/2 2 -3 4 6
f 3 7/25/2 3 -4 6 8
l s=+1
j=l+s/2 κκκκ=-s(j+1/2) occupation
s=-1 s=+1s=-1 s=+1s=-1
For spin-polarized potential,spin up and spin down are calculatedseparately, the density is averagedaccording to the occupation number
specified in case.inc file.
Core states: fully occupied →→→→ spin-compensated Dirac
equation (include SOC)
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1s1/2 →→→→2s1/2
2p1/2 →→→→2p3/2 →→→→3s1/2
3p1/2
3p3/2
3d3/2 →→→→3d5/2 →→→→4s1/2
4p1/2
4p3/2
4d3/2
4d5/2
5s1/2
4f5/2 →→→→4f7/2 →→→→
Implementation in WIEN2k: core electrons
Atomic sphere (RMT) RegionAtomic sphere (RMT) Region
CoreelectronsCore
electrons
« Fully »relativistic
Spin-compensatedDirac equation
Atomic sphere (RMT) RegionAtomic sphere (RMT) Region
CoreelectronsCore
electrons
« Fully »relativistic
Spin-compensatedDirac equation
17 0.00 0 1,-1,2 ( n,κ,occup)2,-1,2 ( n,κ,occup)2, 1,2 ( n,κ,occup)2,-2,4 ( n,κ,occup)3,-1,2 ( n,κ,occup)3, 1,2 ( n,κ,occup)3,-2,4 ( n,κ,occup)3, 2,4 ( n,κ,occup)3,-3,6 ( n,κ,occup)4,-1,2 ( n,κ,occup)4, 1,2 ( n,κ,occup)4,-2,4 ( n,κ,occup)4, 2,4 ( n,κ,occup)4,-3,6 ( n,κ,occup)5,-1,2 ( n,κ,occup)4, 3,6 ( n,κ,occup)4,-4,8 ( n,κ,occup)0
17 0.00 0 1,-1,2 ( n,κ,occup)2,-1,2 ( n,κ,occup)2, 1,2 ( n,κ,occup)2,-2,4 ( n,κ,occup)3,-1,2 ( n,κ,occup)3, 1,2 ( n,κ,occup)3,-2,4 ( n,κ,occup)3, 2,4 ( n,κ,occup)3,-3,6 ( n,κ,occup)4,-1,2 ( n,κ,occup)4, 1,2 ( n,κ,occup)4,-2,4 ( n,κ,occup)4, 2,4 ( n,κ,occup)4,-3,6 ( n,κ,occup)5,-1,2 ( n,κ,occup)4, 3,6 ( n,κ,occup)4,-4,8 ( n,κ,occup)0
case.inc for Au atom
s 0 1/2 -1 2
p 1 3/21/2 1 -2 2 4
d 2 5/23/2 2 -3 4 6
f 3 7/25/2 3 -4 6 8
l s=+1
j=l+s/2 κκκκ=-s(j+1/2) occupation
s=-1 s=+1s=-1 s=+1s=-1
For spin-polarized potential,spin up and spin down are calculatedseparately, the density is averagedaccording to the occupation number
specified in case.inc file.
Core states: fully occupied →→→→ spin-compensated Dirac
equation (include SOC)
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Implementation in WIEN2k: valence electrons
Valence electrons INSIDE atomic spheres are treated within scalar relativistic approximation [1] if RELA
is specified in case.struct file (by default).
[1] Koelling and Harmon, J. Phys. C (1977)
TitleF LATTICE,NONEQUIV.ATOMS: 1 225 Fm-3mMODE OF CALC=RELA unit=bohr
7.670000 7.670000 7.670000 90.000000 90.000000 90.000 000
ATOM 1: X=0.00000000 Y=0.00000000 Z=0.00000000MULT= 1 ISPLIT= 2
Au1 NPT= 781 R0=0.00000500 RMT= 2.6000 Z: 79.0LOCAL ROT MATRIX: 1.0000000 0.0000000 0.0000000
0.0000000 1.0000000 0.00000000.0000000 0.0000000 1.0000000
48 NUMBER OF SYMMETRY OPERATIONS
TitleF LATTICE,NONEQUIV.ATOMS: 1 225 Fm-3mMODE OF CALC=RELA unit=bohr
7.670000 7.670000 7.670000 90.000000 90.000000 90.000 000
ATOM 1: X=0.00000000 Y=0.00000000 Z=0.00000000MULT= 1 ISPLIT= 2
Au1 NPT= 781 R0=0.00000500 RMT= 2.6000 Z: 79.0LOCAL ROT MATRIX: 1.0000000 0.0000000 0.0000000
0.0000000 1.0000000 0.00000000.0000000 0.0000000 1.0000000
48 NUMBER OF SYMMETRY OPERATIONS
♦ no κ dependency of the wave function, (n,l,s) are still good quantum numbers
Atomic sphere (RMT) RegionAtomic sphere (RMT) Region
Valence electronsValence electrons
Scalar relativistic(no SOC)
Atomic sphere (RMT) RegionAtomic sphere (RMT) Region
Valence electronsValence electrons
Scalar relativistic(no SOC)
♦ all relativistic effects are included except SOC
♦ small component enters normalization and calculation of charge inside spheres
♦ augmentation with large component only
♦ SOC can be included in « second variation »
Valence electrons in interstitial region are treated classically
Valence electrons in interstitial region are treated classically
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Implementation in WIEN2k: valence electrons
Atomic sphere (RMT) RegionAtomic sphere (RMT) Region
Valence electronsValence electrons
Scalar relativistic(no SOC)
Possibility to add SOC(2nd variational)
Atomic sphere (RMT) RegionAtomic sphere (RMT) Region
Valence electronsValence electrons
Scalar relativistic(no SOC)
Possibility to add SOC(2nd variational)
SOC is added in a second variation (lapwso):
- First diagonalization (lapw1):
- Second diagonalization (lapwso):1111 Ψ=Ψ εH
( ) Ψ=Ψ+ εSOHH1
The second equation is expanded in the basis of first eigenvectors (ΨΨΨΨ1)
( ) ΨΨ=ΨΨΨΨ+∑ jiN
i
iSO
jjij H 11111 εεδ
sum include both up/down spin states
→→→→ N is much smaller than the basis size in lapw1
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Implementation in WIEN2k: valence electrons
Atomic sphere (RMT) RegionAtomic sphere (RMT) Region
Valence electronsValence electrons
Scalar relativistic(no SOC)
Possibility to add SOC(2nd variational)
Atomic sphere (RMT) RegionAtomic sphere (RMT) Region
Valence electronsValence electrons
Scalar relativistic(no SOC)
Possibility to add SOC(2nd variational)
SOC is added in a second variation (lapwso):
- First diagonalization (lapw1):
- Second diagonalization (lapwso):1111 Ψ=Ψ εH
( ) Ψ=Ψ+ εSOHH1
The second equation is expanded in the basis of first eigenvectors (ΨΨΨΨ1)
( ) ΨΨ=ΨΨΨΨ+∑ jiN
i
iSO
jjij H 11111 εεδ
sum include both up/down spin states
→→→→ N is much smaller than the basis size in lapw1
♦ SOC is active only inside atomic spheres, only spherical potential (VMT) is taken into account, in the polarized case spin up and down parts are averaged.
♦ Eigenstates are not pure spin states, SOC mixes up and down spin states
♦ Off-diagonal term of the spin-density matrix is ignored. It means that in each SCF cycle the magnetization is projected on the chosen direction (from case.inso)
VMT: Muffin-tin potential (spherically symmetric)
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Controlling spin-orbit coupling in WIEN2k
♦ Do a regular scalar-relativistic “scf” calculation
♦ save_lapw
♦ initso_lapw
WFFIL4 1 0 llmax,ipr,kpot-10.0000 1.50000 emin,emax (output energ y window)0. 0. 1. direction of magnetization (lattice vectors)
NX number of atoms for which RLO is addedNX1 -4.97 0.0005 atom number,e-lo,de (case.in1), repeat NX times0 0 0 0 0 number of atoms for wh ich SO is switch off; atoms
WFFIL4 1 0 llmax,ipr,kpot-10.0000 1.50000 emin,emax (output energ y window)0. 0. 1. direction of magnetization (lattice vectors)
NX number of atoms for which RLO is addedNX1 -4.97 0.0005 atom number,e-lo,de (case.in1), repeat NX times0 0 0 0 0 number of atoms for wh ich SO is switch off; atoms
• case.inso:
• case.in1(c):(…)
2 0.30 0.005 CONT 1 0 0.30 0.000 CONT 1
K-VECTORS FROM UNIT:4 -9.0 4.5 65 emin/emax/nband
(…)2 0.30 0.005 CONT 1 0 0.30 0.000 CONT 1
K-VECTORS FROM UNIT:4 -9.0 4.5 65 emin/emax/nband
• symmetso (for spin-polarized calculations only)
♦ run(sp)_lapw -so -so switch specifies that scf cycles will include SOC
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Controlling spin-orbit coupling in WIEN2k
The w2web interface is helping you
Non-spin polarized case
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Controlling spin-orbit coupling in WIEN2k
The w2web interface is helping you
Spin polarized case
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Relativistic effects in the solid: Illustration
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Relativistic effects in the solid: Illustration
♦ Scalar-relativistic (SREL):
- LDA overbinding (2%)- Bulk modulus: 447 GPa
+ spin-orbit coupling (SREL+SO):
- LDA overbinding (1%)- Bulk modulus: 436 GPa
⇒ Exp. Bulk modulus: 462 GPa
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Z
ansr 0
2
)1( = bohr 10 ==αcm
ae
h
bohr 013.079
1)1( ==sr
r2ρρρρ (e/bohr)
0.00r (bohr)
0.01 0.02 0.03 0.04 0.05 0.060
10
20
30
40
50Non relativistic (l=0)
r2ρρρρ (e/bohr)
0.00r (bohr)
0.01 0.02 0.03 0.04 0.05 0.060
10
20
30
40
50Non relativistic (l=0)
r2ρρρρ (e/bohr)
0.00r (bohr)
0.01 0.02 0.03 0.04 0.05 0.060
10
20
30
40
50
Radius of the 1s orbit (Bohr model):
+Zee-
Atomic units:ħ = me = e = 1
c = 1/ α α α α ≈≈≈≈ 137 au
AND
Au 1s
(1) Relativistic orbital contraction
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[ ]γα
00
a
cMRELAa
e
== h
ee mmM 22.1=⋅= γ
Z
ansr 0
2
)1( = bohr 10 ==αmc
ah
bohr 013.079
1)1( ==sr
r2ρρρρ (e/bohr)
0.00r (bohr)
0.01 0.02 0.03 0.04 0.05 0.060
10
20
30
40
50Non relativistic (l=0)Relativistic (κκκκ=-1)
r2ρρρρ (e/bohr)
0.00r (bohr)
0.01 0.02 0.03 0.04 0.05 0.060
10
20
30
40
50Non relativistic (l=0)Relativistic (κκκκ=-1)
r2ρρρρ (e/bohr)
0.00r (bohr)
0.01 0.02 0.03 0.04 0.05 0.060
10
20
30
40
50Non relativistic (l=0)Relativistic (κκκκ=-1)Relativistic (κκκκ=-1)
20% Orbitalcontraction
Radius of the 1s orbit (Bohr model):
+Zee-
AND
In Au atom, the relativistic mass (M) of the 1s electron is 22% larger than
the rest mass (m)
bohr010.022.11
791
)1( 02
===γa
Z
nsr
Au 1s
(1) Relativistic orbital contraction
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0 2 4 60.0
0.1
0.2
0.3
0.4
0.5
r2ρρρρ (e/bohr)
r (bohr)
Non relativistic (l=0)Relativistic (κκκκ=-1)
0 2 4 60.0
0.1
0.2
0.3
0.4
0.5
r2ρρρρ (e/bohr)
r (bohr)
Non relativistic (l=0)Relativistic (κκκκ=-1)Relativistic (κκκκ=-1)
Orbitalcontraction
Au 6s
ns orbitals (with n > 1) contract due to orthogonality to 1s
( )0046.1
096.01
1
1
122
=−
=
−
=
c
ve
γ
cn
Zsve 096.017.13
6
79)6( ====
Direct relativistic effect (mass enhancement) →→→→ contraction of 0.46% only
(1) Relativistic orbital contraction
However, the relativistic contraction of the 6s orbital is large (>20%)
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-40
-30
-20
-10
0
10
20
Relativisticcorrection (%)
1s 2s 4s3s 5s 6s
( )NRELA
NRELARELA
E
EE −
r2ρρρρ (e/bohr)
0.00r (bohr)
0.01 0.02 0.03 0.04 0.05 0.060
10
20
30
40
50
Orbitalcontraction
Non relativistic (l=0)Relativistic (κκκκ=-1)
Au 1s0 2 4 6
0.0
0.1
0.2
0.3
0.4
0.5
r2ρρρρ (e/bohr)
r (bohr)
r2ρρρρ (e/bohr)
r ( )
Orbitalcontraction
Non relativistic (l=0)Relativistic (κκκκ=-1)
Au 6s
(1) Orbital Contraction: Effect on the energy
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(2) Spin-Orbit splitting of p states
1.0
r2ρρρρ (e/bohr)
r (bohr)
0.0 0.5 1.5 2.0
r2ρρρρ (e/bohr)
r (bohr)
2.5
Non relativistic (l=1)Non relativistic (l=1)
Au 5p
0.0
0.1
0.2
0.3
0.5
0.7
0.6
0.4
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(2) Spin-Orbit splitting of p states
l=1
E
j=3/2 (κκκκ=-2)j=1+
1/2=
3/2
orbital moment
spin
+e-e
j=1+1/2=3/2
♦ Spin-orbit splitting of l-quantum number
r2ρρρρ (e/bohr)r2ρρρρ (e/bohr)Non relativistic (l=1)Relativistic (κκκκ=-2)Non relativistic (l=1)Relativistic (κκκκ=-2)
Au 5p
♦ p3/2 (κκκκ=-2): nearly same behavior than non-relativistic p-state
1.0
r (bohr)
0.0 0.5 1.5 2.0
r (bohr)
2.50.0
0.1
0.2
0.3
0.5
0.7
0.6
0.4
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(2) Spin-Orbit splitting of p states
l=1
E
j=1/2 (κκκκ=1)j=1-1/2=1/2
spin
orbital moment
+e -e
j=1-1/2=1/2
♦ Spin-orbit splitting of l-quantum number
r2ρρρρ (e/bohr)r2ρρρρ (e/bohr)Non relativistic (l=1)
Relativistic (κκκκ=1)
Non relativistic (l=1)
Relativistic (κκκκ=1)
Au 5p
♦ p1/2 (κκκκ=1): markedly different behavior than non-relativistic p-state
gκκκκ=1 is non-zero at nucleus
1.0
r (bohr)
0.0 0.5 1.5 2.0
r (bohr)
2.50.0
0.1
0.2
0.3
0.5
0.7
0.6
0.4
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(2) Spin-Orbit splitting of p states
l=1
E
j=3/2
j=1/2 (κκκκ=1)
(κκκκ=-2)j=1+
1/2=
3/2
j=1-1/2=1/2
orbital moment
spin
spin
orbital moment
+e-e
+e -e
j=1+1/2=3/2 j=1-1/2=1/2
Ej=3/2 ≠ ≠ ≠ ≠ Ej=1/2
♦ Spin-orbit splitting of l-quantum number
r2ρρρρ (e/bohr)r2ρρρρ (e/bohr)Non relativistic (l=1)Relativistic (κκκκ=-2)Relativistic (κκκκ=1)
Non relativistic (l=1)Relativistic (κκκκ=-2)Relativistic (κκκκ=1)
Au 5p
♦ p1/2 (κκκκ=1): markedly different behavior than non-relativistic p-state
gκκκκ=1 is non-zero at nucleus
1.0
r (bohr)
0.0 0.5 1.5 2.0
r (bohr)
2.50.0
0.1
0.2
0.3
0.5
0.7
0.6
0.4
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-40
-30
-20
-10
0
10
20
Relativisticcorrection (%)
2p1/2 2p3/2
3p1/2 3p3/2 4p1/2 4p3/2 5p1/2 5p3/2
( )NRELA
NRELARELA
E
EE −
κκκκ=1 κκκκ=-2
(2) Spin-Orbit splitting of p states
Scalar-relativistic p-orbital is similar to p3/2 wave function, but ΨΨΨΨdoes not contain p1/2 radial basis function
r2ρρρρ (e/bohr)r2ρρρρ (e/bohr)Non relativistic (l=1)Relativistic (κκκκ=-2)Relativistic (κκκκ=1)
Non relativistic (l=1)Relativistic (κκκκ=-2)Relativistic (κκκκ=1)
Au 5p
1.0
r (bohr)
0.0 0.5 1.5 2.0
r (bohr)
2.50.0
0.1
0.2
0.3
0.5
0.7
0.6
0.4
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(3) Orbital expansion: Au(d) states
Higher l-quantum number states expand due to better shielding of nucleus charge from contracted s-states
-e
+Ze-e
-e
Non-relativistic (NREL)
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(3) Orbital expansion: Au(d) states
Higher l-quantum number states expand due to better shielding of nucleus charge from contracted s-states
-e
+Ze-e
-e
Non-relativistic (NREL)
-e
+Zeff1e
Zeff1 = Z- σ σ σ σ(NREL)
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(3) Orbital expansion: Au(d) states
Higher l-quantum number states expand due to better shielding of nucleus charge from contracted s-states
+Ze-e
-e
+Ze-e
-e
-e
+Zeff1e +Zeff2e
Zeff2 = Z- σ σ σ σ(REL)
Non-relativistic (NREL) Relativistic (REL)
Zeff1 = Z- σ σ σ σ(NREL) Zeff1 > Zeff2
-e-e
-e
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(3) Orbital expansion: Au(d) states
Higher l-quantum number states expand due to better shielding of nucleus charge from contracted s-states
-e
+Ze-e
-e
-e
+Ze-e
-e
-e
+Zeff1e +Zeff2e
-e
Zeff2 = Z- σ σ σ σ(REL)
Non-relativistic (NREL) Relativistic (REL)
Indirect relativistic effect
Zeff1 > Zeff2Zeff1 = Z- σ σ σ σ(NREL)
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-40
-30
-20
-10
0
10
20
Relativisticcorrection (%)
3d3/2 3d5/2 4d3/2 4d5/2
5d3/2 5d5/2
4f5/2 4f7/2( )
NRELA
NRELARELA
E
EE −
0 1 2 30.0
0.1
0.2
0.3
0.4
r2ρρρρ (e/bohr)
r (bohr)
Non relativistic (l=2)Relativistic (κκκκ=2)Relativistic (κκκκ=-3)
40 1 2 30.0
0.1
0.2
0.3
0.4
r2ρρρρ (e/bohr)
r (bohr)
Non relativistic (l=2)Relativistic (κκκκ=2)Relativistic (κκκκ=-3)
40.0 0.1 0.2 0.30
1
2
3
4
r2ρρρρ (e/bohr)
r (bohr)
Non relativistic (l=2)Relativistic (κκκκ=2)Relativistic (κκκκ=-3)
0.40.0 0.1 0.2 0.30
1
2
3
4
r2ρρρρ (e/bohr)
r (bohr)
Non relativistic (l=2)Relativistic (κκκκ=2)Relativistic (κκκκ=-3)
0.4
κκκκ=2 κκκκ=-3
κκκκ=3 κκκκ=-4
Orbitalexpansion
(3) Orbital expansion: Au(d) states
Au 3d Au 5d
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-40
-30
-20
-10
0
10
20
Relativisticcorrection (%)
1s 2s 4s
2p1/2 2p3/2
3d3/2 3d5/2
3s
3p1/2 3p3/2
5s 6s
4p1/2 4p3/2 5p1/2 5p3/2
4d3/2 4d5/2
5d3/2 5d5/2
4f5/2 4f7/2
( )NRELA
NRELARELA
E
EE −
Relativistic effects on the Au energy levels
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Atomic spectra of gold
Orbital contraction
SO splitting
SO splitting
Orbital expansion
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Ag – Au: the differences (DOS & optical prop.)
Ag Au
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Relativistic semicore states: p1/2 orbitals
Electronic structure of fcc Th, SOC with 6p1/2 local orbital
J.Kuneš, P.Novak, R.Schmid, P.Blaha, K.Schwarz, Phys.Rev.B. 64, 153102 (2001)
6p1/2
6p1/2
6p3/2
6p3/2
Energy vs. basis size DOS with and without p1/2
p1/2 included
p1/2 not included
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SOC in magnetic systems
SOC couples magnetic moment to the lattice
Symmetry operations acts in real and spin space
♦direction of the exchange field matters (input in case.inso)
♦number of symmetry operations may be reduced (reflections act differently on spins than on positions)
♦time inversion is not symmetry operation (do not add an inversion for k-list)
♦initso_lapw (must be executed) detects new symmetry setting
[100] [010] [001] [110]
1
mx
my
2z
A A A A
A B B -
B A B -
B B A B
Direction of magnetization
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Relativity in WIEN2k: Summary
WIEN2k offers several levels of treating relativity:
♦♦♦♦non-relativistic: select NREL in case.struct (not recommended)
♦♦♦♦standard: fully-relativistic core, scalar-relativistic valence
mass-velocity and Darwin s-shift, no spin-orbit interaction
♦♦♦♦”fully”-relativistic:
adding SO in “second variation” (using previous eigenstates as basis)
adding p1/2 LOs to increase accuracy (caution!!!)
x lapw1 (increase E-max for more eigenvalues, to have
x lapwso basis for lapwso)
x lapw2 –so -c SO ALWAYS needs complex lapw2 version
♦♦♦♦Non-magnetic systems:
SO does NOT reduce symmetry. initso_lapw just generates case.inso and case.in2c.
♦♦♦♦Magnetic systems:
symmetso dedects proper symmetry and rewrites case.struct/in*/clm*
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Relativistic effects
&
Non-collinear magnetism
(WIEN2k / WIENncm)
18th WIEN2k WorkshopPennState University – USA – 2011
Xavier RocquefelteInstitut des Matériaux Jean-Rouxel (UMR 6502)
Université de Nantes, FRANCE
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Pauli Hamiltonian for magnetic systems
( ) ...2
22
+⋅+⋅++∇−= lBVm
H effBeffe
P
rrrrh σζσµ
2x2 matrix in spin space, due to Pauli spin operators
=
01
101σ
−=
0
02 i
iσ
−=
10
013σ
(2××××2) Pauli spin matrices
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Pauli Hamiltonian for magnetic systems
( ) ...2
22
+⋅+⋅++∇−= lBVm
H effBeffe
P
rrrrh σζσµ
2x2 matrix in spin space, due to Pauli spin operators
Wave function is a 2-component vector (spinor) – It corresponds to the large components of the dirac wave function (small components are neglected)
ΨΨ
=
ΨΨ
2
1
2
1 εPHspin up
spin down
=
01
101σ
−=
0
02 i
iσ
−=
10
013σ
(2××××2) Pauli spin matrices
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Pauli Hamiltonian for magnetic systems
( ) ...2
22
+⋅+⋅++∇−= lBVm
H effBeffe
P
rrrrh σζσµ
2x2 matrix in spin space, due to Pauli spin operators
Effective electrostatic potential
Effective magnetic field
xcHexteff VVVV ++= xcexteff BBB +=
Exchange-correlation potential
Exchange-correlation field
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Pauli Hamiltonian for magnetic systems
( ) ...2
22
+⋅+⋅++∇−= lBVm
H effBeffe
P
rrrrh σζσµ
2x2 matrix in spin space, due to Pauli spin operators
Effective electrostatic potential
Effective magnetic field
Spin-orbit coupling
dr
dV
rcM e
1
2 22
2h=ζ
xcHexteff VVVV ++= xcexteff BBB +=
Exchange-correlation potential
Exchange-correlation field
Many-body effects which are defined within DFT LDA or GGA
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Exchange and correlation
From DFT exchange correlation energy:
( )( ) ( ) ( )[ ] 3 , , drmrrmrE homxcxc
rr ρερρ ∫=
Local function of the electronic density (ρρρρ) and the magnetic moment (m)
Definition of Vxc and Bxc (functional derivatives):
( )ρρ
∂∂= mE
V xcxc
r, ( )
m
mEB xc
xc r
rr
∂∂= ,ρ
LDA expression for Vxc and Bxc:
( ) ( )ρρερρε
∂∂+= m
mVhomxchom
xcxc
rr ,
, ( )m
m
mB
homxc
xc ˆ,
∂∂=
rr ρερ
Bxc is parallel to the magnetization density vector (m)^
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Non-collinear magnetism
Direction of magnetization vary in space, thus spin-orbit term is present
( ) ...2
22
+⋅+⋅++∇−= lBVm
H effBeffe
P
rrrrh σζσµ
( )
( )εψψ
µµ
µµ=
+−+∇−+
−+++∇−
...2
...2
22
22
zBeffe
yxB
yxBzBeffe
BVm
iBB
iBBBVm
h
h
♦♦♦♦ Non-collinear magnetic moments
=
2
1
ψψ
ψ ΨΨΨΨ1 and ΨΨΨΨ2 are non-zero
♦♦♦♦ Solutions are non-pure spinors
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Collinear magnetism
Magnetization in z-direction / spin-orbit is not present
( ) ...2
22
+⋅+⋅++∇−= lBVm
H effBeffe
P
rrrrh σζσµ
εψψµ
µ=
+−+∇−
+++∇−
...2
0
0...2
22
22
zBeffe
zBeffe
BVm
BVm
h
h
♦♦♦♦ Collinear magnetic moments
♦♦♦♦ Solutions are pure spinors
=↑ 0
1ψψ
=↓
2
0
ψψ
↓↑ ≠ εε ♦♦♦♦ Non-degenerate energies
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Non-magnetic calculation
No magnetization present, Bx = By = Bz = 0 and no spin-orbit coupling
( ) ...2
22
+⋅+⋅++∇−= lBVm
H effBeffe
P
rrrrh σζσµ
εψψ =
+∇−
+∇−
effe
effe
Vm
Vm
22
22
20
02
h
h
=↑ 0
ψψ
=↓ ψ
ψ0
↓↑ = εε
♦♦♦♦ Solutions are pure spinors
♦♦♦♦ Degenerate spin solutions
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Magnetism and WIEN2k
Wien2k can only handle collinear or non-magnetic cases
run_lapw script:
x lapw0x lapw1x lapw2x lcorex mixer
non-magnetic case
m = n↑↑↑↑ – n↓↓↓↓ = 0
run_lapw script:
x lapw0x lapw1 –upx lapw1 -dnx lapw2 –upx lapw2 -dnx lcore –upx lcore -dnx mixer
magnetic case
m = n↑↑↑↑ – n↓↓↓↓ ≠≠≠≠ 0DOS
EF
DOS
EF
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Magnetism and WIEN2k
Spin-polarized calculations
♦♦♦♦ runsp_lapw script (unconstrained magnetic calc.)
♦♦♦♦ runfsm_lapw -m value (constrained moment calc.)
♦♦♦♦ runafm_lapw (constrained anti-ferromagnetic calculation)
♦♦♦♦ spin-orbit coupling can be included in second variational step
♦♦♦♦ never mix polarized and non-polarized calculations in one case directory !!!
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Non-collinear magnetism
♦ code based on Wien2k (available for Wien2k users)
In case of non-collinear spin arrangements WIENncm (WIEN2k clone) has to be used:
♦ structure and usage philosophy similar to Wien2k
♦ independent source tree, independent installation
WIENncm properties:
♦ real and spin symmetry (simplifies SCF, less k-points)
♦ constrained or unconstrained calculations (optimizes magnetic moments)
♦ SOC in first variational step, LDA+U
♦ Spin spirals
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Non-collinear magnetism
For non-collinear magnetic systems, both spin channels have to be considered simultaneously
runncm_lapw script:
xncm lapw0xncm lapw1xncm lapw2xncm lcorexncm mixer
Relation between spin density matrix and magnetization
mz = n↑↑↑↑↑↑↑↑ – n↓↓↓↓↓↓↓↓ ≠≠≠≠ 0
mx = ½(n↑↑↑↑↓↓↓↓ + n↓↓↓↓↑↑↑↑) ≠≠≠≠ 0
my = i½(n↑↑↑↑↓↓↓↓ - n↓↓↓↓↑↑↑↑) ≠≠≠≠ 0
DOS
EF
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WienNCM: Spin spirals
Transverse spin wave
qRrr
⋅=α
α α α α
R( ) ( )( )θθ cos , sinsin , cos nnn RqRqmm
rrrrr ⋅⋅=
♦ spin-spiral is defined by a vector q given in reciprocal space and an angle θbetween magnetic moment and rotation axis.
♦ Rotation axis is arbitrary
⇒⇒⇒⇒ Translational symmetry is lost !
⇒ But WIENncm is using the generalized Bloch theorem. The calculation of spinwaves only requires one unit cell for even incommensurate modulation q vector.
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WienNCM: Usage
1. Generate the atomic and magnetic structures
2. Run initncm (initialization script)
3. Run the NCM calculation:
♦ Create atomic structure
♦ Create magnetic structure
See utility programs: ncmsymmetry, polarangles, …
♦ xncm (WIENncm version of x script)
♦ runncm (WIENncm version of run script)
More information on the manual (Robert Laskowski)
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Thank you for
your attention
18th WIEN2k WorkshopPennState University – USA – 2011
Xavier RocquefelteInstitut des Matériaux Jean-Rouxel (UMR 6502)
Université de Nantes, FRANCE