Relativistic effects Non-collinear magnetism (WIEN2k...

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


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 …

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


Light Matter



c ≈ ≈ ≈ ≈ 137 au


Speed ofmatter

mass

v = f(mass)

mass = f(v)

Composed of atoms (mass)


Light Matter


Lorentz Factor (measure of the relativistic effects)


c ≈ ≈ ≈ ≈ 137 au


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

( )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


Relativistic increase in the mass of an electron with its velocity (when ve → c)

1) The mass-velocity correction

+Ze



It has no classical relativistic analogueDue to small and irregular motions of an electron about its mean position (Zitterbewegung)

2) The Darwin term

+Ze





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





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


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


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


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

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

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

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)


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 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

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:




2' cme−= εε ( ) ( )22

'

c

rVmrM ee


Then the coupled equations can be written in the form of the radial eq.:

Darwin term

Spin-orbit coupling

( ) ( )11 +=+ llκκNote that:


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



2' cme−= εε ( ) ( )22

'

c

rVmrM ee


( ) ( )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:


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


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:


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

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

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)

Interstitial RegionInterstitial Region


Not relativistic

SOC: Spin orbit coupling

Implementation in WIEN2k


CoreelectronsCore

electrons






Implementation in WIEN2k: core electrons


CoreelectronsCore

electrons




CoreelectronsCore

electrons



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)

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


CoreelectronsCore

electrons




CoreelectronsCore

electrons





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)

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







♦ 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










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 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)

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


The w2web interface is helping you

Non-spin polarized case


The w2web interface is helping you

Spin polarized case

Relativistic effects in the solid: Illustration

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

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-


c = 1/ α α α α ≈≈≈≈ 137 au

AND

Au 1s

(1) Relativistic orbital contraction

[ ]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


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


However, the relativistic contraction of the 6s orbital is large (>20%)

-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


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


Au 6s

(1) Orbital Contraction: Effect on the energy

(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


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


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


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


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


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

-40

-30

-20

-10

0

10

20


2p1/2 2p3/2

3p1/2 3p3/2 4p1/2 4p3/2 5p1/2 5p3/2

( )NRELA

NRELARELA

E

EE −

κκκκ=1 κκκκ=-2


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

(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

+Ze-e

-e

Non-relativistic (NREL)

-e

+Zeff1e

Zeff1 = Z- σ σ σ σ(NREL)



+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



-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)

-40

-30

-20

-10

0

10

20


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)


40.0 0.1 0.2 0.30

1

2

3

4

r2ρρρρ (e/bohr)

r (bohr)


0.40.0 0.1 0.2 0.30

1

2

3

4

r2ρρρρ (e/bohr)

r (bohr)


0.4



Orbitalexpansion


Au 3d Au 5d

-40

-30

-20

-10

0

10

20


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

Atomic spectra of gold

Orbital contraction

SO splitting

SO splitting

Orbital expansion

Ag – Au: the differences (DOS & optical prop.)

Ag Au

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

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

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*


&


(WIEN2k / WIENncm)




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

22

+⋅+⋅++∇−= lBVm

H effBeffe

P

rrrrh σζσµ


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

22

+⋅+⋅++∇−= lBVm

H effBeffe

P

rrrrh σζσµ


Effective electrostatic potential

Effective magnetic field

xcHexteff VVVV ++= xcexteff BBB +=

Exchange-correlation potential

Exchange-correlation field


( ) ...2

22

+⋅+⋅++∇−= lBVm

H effBeffe

P

rrrrh σζσµ


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

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)^


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

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

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

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

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 !!!


♦ 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


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

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.

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)

[email protected]

Thank you for

your attention




Relativistic effects Non-collinear magnetism (WIEN2k...

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