Combination of the KKR-method and the DMFT · Ludwig Maximilians-Universit¤at Munchen¤...

Ludwig

Maximilians-

Universitat

Munchen

Combination of the KKR-method and the DMFT1 H. Ebert, J. Minar, A. Perlov, S. Chadov

2 A. Lichtenstein3 M. Katsnelson

4 L. Chioncel5 J. Braun

6 C. de Nadaı, N. B. Brookes

1 University Munchen, Germany2 University Hamburg, Germany

3 University Nijmegen, Netherlands4 University Graz, Austria

5 University Munster, Germany6 European Synchrotron Radiation Facility, Grenoble, France

KKR+DMFT HH – p.1/71

Ludwig

Maximilians-

Universitat

MunchenOutline

Why KKR ?

The Munich SPR-KKR package

The SPR-KKR method

Combination of KKR and DMFT

Results for transition metal systems

Summary


Ludwig

Maximilians-

Universitat

Munchen

Why KKR ?

KKR represents electronic structure in terms ofGreen’s function G+(~r, ~r ′, E)

G+(~r, ~r ′, E) = limε→0

∑

i

φi(~r)φ∗i (~r)

E − Ei + iε

linear response formalism

Dyson equation G = G0 + G∆HG0

CPA alloy theory

description of spectroscopic quantities

central quantity of many-body theories


Ludwig

Maximilians-

Universitat

Munchen

Munich SPR-KKR Package

Systems

Arbitrary ordered/disordered three dimensionallyperiodic systems

Surfaces in cluster or slab approximation

Calculation Mode

Spin-polarised

Scalar- and Fully relativistic

Non-collinear spin configurations

Default: Spin-polarised relativistic Dirac formalism


Ludwig

Maximilians-

Universitat

MunchenFeatures

Electronic Properties

SCF-potential

Dispersion relation

Bloch spectral Function

Density of states

...

Ground State Properties

Spin- and Orbital Moments

Hyperfine Fields

Magnetic Form Factors

...


Ludwig

Maximilians-

Universitat

MunchenFeatures

Response Functions

Spin- and orbital susceptibility

Knight-shift

Field-induced XMCD

Residual resistivity of alloys


Ludwig

Maximilians-

Universitat

Munchen

Spectroscopic Properties

Spectroscopic Properties – including magnetic dichroism

Valence Band Photoemission

Core level Photoemission

Appearance Potential Spectroscopy (non-relat.)

Auger Electron Spectroscopy

X-ray absorption

X-ray emission

X-ray magneto-optics

X-ray scattering

Magnetic Compton scattering

Positron annihilation


Ludwig

Maximilians-

Universitat

Munchen

The Dirac Equation

[~

ic~α·~∇+βmc2+V (~r)+β~σ · ~Beff (~r)︸︷︷︸

Vspin(~r)

]Ψ(~r, E) = EΨ(~r, E)

with an effective magnetic field

~Beff (~r) =∂Exc[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 z′ to have:

Vspin(~r) = βσz′ Beff(r)


Ludwig

Maximilians-

Universitat

Munchen

Atomic sphere approximation – ASA

replace atomic cells by atom centred spheres

keep the volume of the unit cell constant

assume a spherical symmetric potential withinsphere:

V (~r) = V (r) and ~Beff(~r) = ~Beff(r)


Ludwig

Maximilians-

Universitat

Munchen

Single site Dirac equation

restrict potential to a single atomic cellfor atom type t:

V t(~r) = 0 and ~Bt

eff(~r) = 0 for ~r /∈ Ωt

Single site Dirac equation for atom type t

[~

ic~α·~∇+βmc2+V t(~r)+β~σ· ~Bt

eff(~r)

]Ψ(~r, E) = EΨ(~r, E)


Ludwig

Maximilians-

Universitat

Munchen

Ansatz to solve the single site Dirac equation

ψν(~r, E) =∑

Λ

ψΛν(~r, E)

with

ψΛν(~r, E) =

(gΛν(r, E)χΛ(~r)

ifΛν(r, E)χ−Λ(~r)

)

spin-angular functions

χΛ(r) =∑

ms=±1/2

C(l1

2j;µ−ms,ms)Y

µ−ms

l (r)χms

short hand notation Λ = (κ, µ) and −Λ = (−κ, µ)


Ludwig

Maximilians-

Universitat

Munchen

Relativistic quantum numbers

total angular momentum operator ~j = ~l+ 12~σ

~j2χΛ(r) = j(j + 1)χΛ(r)

jzχΛ(r) = µχΛ(r)

total angular momentum quantum number j

j = l± 1/2

magnetic quantum number µ

µ = −j ... + j


Ludwig

Maximilians-

Universitat

Munchen

Relativistic quantum numbers

Spin-orbit operator K

KχΛ(r) = (1 + ~σ ·~l )χΛ(r) = −κχΛ(r)

Spin-obit quantum number κ

κ =

l = j + 1

2 if j = l− 12

−l− 1 = −j − 12

if j = l+ 12

κ −1 +1 −2 +2 −3 +3 −4

j 1/2 1/2 3/2 3/2 5/2 5/2 7/2l 0 1 1 2 2 3 3

symbol s1/2 p1/2 p3/2 d3/2 d5/2 f5/2 f7/2


Ludwig

Maximilians-

Universitat

Munchen

Radial Differential Equations

P ′Λν = −

κ

rPΛν +

[E − V

c2+ 1

]QΛν

+B

c2

∑

Λ′

〈χ−Λ|σz|χ−Λ′〉QΛ′ν

Q′Λν =

κ

rQΛν − [E − V ]PΛν +B

∑

Λ′

〈χΛ|σz|χΛ′〉PΛ′ν

with PΛν(r, E) = r gΛν(r, E)and QΛν(r, E) = cr fΛν(r, E)

The coupling is restricted to µ − µ′ = 0 and l − l′ = 0

−→ 4 coupled functions for |µ| < j; e.g. d3/2,µ — d5/2,µ

−→ 2 coupled functions for |µ| = j; e.g. d5/2,µ=±5/2


Ludwig

Maximilians-

Universitat

Munchen

Single site t-matrix

scattering amplitude f(p)

Ψ(~r, E) = ei~p·~r + f(p)eipr

r

spherical basis

ψΛ(~r, E) = jΛ(~r, E) + ip∑

Λ′

tΛΛ′(E)h+Λ′(~r, E)


Ludwig

Maximilians-

Universitat

Munchen


single site t-matrix tΛΛ′(E)

t(E) =i

2p[a(E)− b(E)] b(E)−1

aΛΛ′(E) = −ipr2[h−Λ , φΛΛ′]r

bΛΛ′(E) = −ipr2[h+Λ , φΛΛ′]r

relativistic Wronskian

[h+Λ , φΛΛ′]r = h+

l cfΛΛ′ −p

1 + E/c2Sκh

+lgΛΛ′


Ludwig

Maximilians-

Universitat

Munchen


spherical spin-dependent potential ( ~M‖z)

s1/2 p1/2 p3/2 d3/2 d5/2

s1/2

d5/2

d3/2

p3/2

p1/2


Ludwig

Maximilians-

Universitat

Munchen

Multiple scattering

c

a

b

scattering T-matrix operator of the crystal T

T =∑

n

tn +∑

n,m

tn G0 tm +

∑

n,m,kn,m6=k

tn G0 tm G0 t

k + · · ·

decomposition into scattering path operator τ

T =∑

n,m

τnm

self-consistent requirement for τnm

in angular momentum representation

τnmΛΛ′ = tnΛΛ′δnm +

∑

Λ′′Λ′′′

tnΛΛ′′

∑

k 6=n

G0 nkΛ′′Λ′′′τ

kmΛ′′′Λ′

τnm = tnδnm + tn∑

k 6=n

G0 nkτkm

formal solution

τ = [t−1 −G]−1


Ludwig

Maximilians-

Universitat

Munchen

Cluster approximation

cut a finite cluster out of the infinite systemcentred on the atom of interest

τ (E) = [ t−1(E)−G(E)︸︷︷︸

real space KKR matrix M(E)

]−1


Ludwig

Maximilians-

Universitat

Munchen

Brillouin zone integration

τnn′

ΛΛ′(E) =1

ΩBZ

∫

ΩBZ

d3k ei~k(~Rn− ~Rn′)

×[t−1(E)−G(~k,E)︸︷︷︸KKR matrix M(~k,E)

]−1ΛΛ′

Symmetry allows to reduce the integration regime ΩBZ

to an irreducible wedge


Ludwig

Maximilians-

Universitat

Munchen

Special point sampling method

τnn,αα(E) =∑

j=1,...,nU

U jτnn,α′α′

0 (E)U j −1

+∑

j=1,...,nA

U jτnn,α′α′T0 (E)U j −1

with τnn,αα0 (E) =

∑~kw~k

ταα(~k, E)

U : (anti-)unitary symmetry operations

Special point mesh


Ludwig

Maximilians-

Universitat

Munchen

Symmetry considerations

M‖[001]E C2z C+

4z C−4z I σz S−

4z S+4z

TC2x TC2y TC2a TC2b Tσx Tσy Tσda Tσdb

M‖[111]E C+

31 C−31 I S−

61 S+61

TC2b TC2e TC2f Tσdb Tσde Tσdf


Ludwig

Maximilians-

Universitat

Munchen

Scattering path matrix τ ii

cubic spin-dependent system ( ~M‖z)

s1/2 p1/2 p3/2 d3/2 d5/2

s1/2

d5/2

d3/2

p3/2

p1/2


Ludwig

Maximilians-

Universitat

MunchenElectronic Green’s function

G(~r, ~r ′, E) =∑

ΛΛ′

ZnΛ(~r, E)τnn′

ΛΛ′(E)Zn′×Λ′ (~r ′, E)

−∑

Λ

[Zn

Λ(~r, E)Jn×Λ (~r ′, E)Θ(r′ − r)

+JnΛ(~r, E)Zn×

Λ (~r ′, E)Θ(r − r′)]δnn′

normalisation of wave functions for |~r| ≥ rmt

regular solution

ZΛ(~r, E) =∑

Λ′

jΛ′(p~r)t−1Λ′Λ(E)− iph+

Λ(p~r)

irregular solution

JΛ(~r, E) = jΛ(p~r)KKR+DMFT HH – p.24/71

Ludwig

Maximilians-

Universitat

Munchen

Expectation values

Starting for example from the identity:

=G(E) = −π∑

α

|α〉〈α| δ(E − Eα) ,

for the Green’s function G(E) in operator form one finds:Expectation value of operator A

〈A〉 = −1

π=Trace

∫ EF

dEAG(~r, ~r, E)

Trace-operation:

take the trace with respect to the 4× 4-matrices


Ludwig

Maximilians-

Universitat

Munchen

Expectation values

charge density n(~r)

n(~r) = −1

π=Trace

∫ EF

dE G(~r, ~r, E)

spin magnetization m(~r)

m(~r) = −1

π=Trace

∫ EF

dE βσzG(~r, ~r, E)

spin and orbital magnetic moments µspin and µorb

µspin = −µB

π=Trace

∫ EF

dE

∫

Vd3r βσzG(~r, ~r, E)

µorb = −µB

π=Trace

∫ EF

dE

∫

Vd3r βlzG(~r, ~r, E)


Ludwig

Maximilians-

Universitat

Munchen

Energy paths

EFERMI EFERMIEMAX

EMIN

IGRID = 3IGRID = 5

ImEEMIN

SCF: arc in the complex plane used for integration

DOS: straight path along real axis

XAS: special straight path starting at EF

...KKR+DMFT HH – p.27/71

Ludwig

Maximilians-

Universitat

Munchen

Coherent Potential Approximation – CPA

Ideafind an effective CPA medium that represents theelectronic structure of an configurationally averagedsubstitutionally random alloy AxB1−x

=


Ludwig

Maximilians-

Universitat

MunchenCPA condition

Embedding of an A- or B-atom into the CPA-medium – inthe average – should not give rise to additionalscattering

xA + xB =

xAτnn,A + xBτ

nn,B = τnn,CPA

with the projected scattering path operator τnn,α

τnn,α = τnn,CPA[1 +

(t−1α − t

−1CPA

)τnn,CPA

]−1


Ludwig

Maximilians-

Universitat

Munchen

Single-site problem – reconsidered

[−∇2 + V σ(r)− E]Ψ(~r) +

∫Σσ(~r, ~r ′, E)Ψ(~r ′)d3r ′ = 0

self-energy Σσ(~r, ~r E)

Ansatz:Ψν(~r) =

∑

L

ΨLν(~r)

coupled radial integro-differential equations:[d2

dr2−l(l + 1)

r2− V (r) + E

]ΨLν(r, E) =

∫r′2dr′

[∑

L′′

φl(r)ΣLL′′(E)φl′′(r′)

]ΨLν(r′, E)


Ludwig

Maximilians-

Universitat

Munchen

Single site equation

approximation for the self-energy:∫d3r′Σ(~r, ~r ′, E)ΨL(~r ′, E) ≈

∑

L

ΣL′L(E)ΨL(~r, E)

leads to coupled differential equations:

[d2

dr2−l(l + 1)

r2− V (r) + E

]ΨLν(r, E) =

∑

L′

ΣLL′(E) ΨL′ν(r, E)


Ludwig

Maximilians-

Universitat

MunchenKKR Green’s function

Electronic Green’s function:

G(~r, ~r ′, E) =∑

LL′

ZnL(~r, E)τnn′

LL′ (E)Zn′×L′ (~r ′, E)

−∑

L

[Zn

L(~r, E)Jn×L (~r ′, E)Θ(r′ − r)

+JnL(~r, E)Zn×

L (~r ′, E)Θ(r − r′)]δnn′

Multiple scattering path operator:

τnn′

LL′ (E) =1

ΩBZ

∫

ΩBZ

d3k[t−1(E)−G(~k, E)

]−1

LL′ei~k( ~Rn− ~Rn′)


Ludwig

Maximilians-

Universitat

MunchenGreen’s function matrix

Green’s function matrix GnmLL ′ within KKR-formalism

GnmLL′(E) =

∑

L1,L2

〈φL | ZL1〉τnm

LL′ (E)〈Z×L2| φL′〉

− δnm

∑

L1

〈φL | ZL1(r<, E)J×

L1(r>, E) | φL′〉

GnnLL′ – input for the many-body effective

impurity problem

φL(~r) – basis functionsolution of the Schrödinger equation forspherical LDA non-magnetic potential


Ludwig

Maximilians-

Universitat

MunchenKKR+DMFT scheme I

F

KKRΣG

E

DMFT(FLEX)

VLSDA(~r), ΣDMFT

⇓single-site problem: tLL′(z)

⇓multiple scattering: τLL′(z)

⇓KKR-Green’s function: G(~r, ~r ′, z)

⇓charge ρ(~r)

⇓VLSDA(~r)

⇓G-matrix GLL′

⇓

DMFT-solver⇓

ΣDMFT


Ludwig

Maximilians-

Universitat

Munchen

DMFT-FLEX many-body solver I

Effective impurity problem solved by spin-polarisedT-matrix plus Fluctuation Exchange (SPTF)

Hartree and Fock contribution to Σ:

Σ(TH)12,σ (iω) =

1

β

∑

Ω

∑

34σ ′

〈13|T σσ ′

(iΩ)|24〉 Gσ ′

43 (iΩ− iω)

Σ(TF )12,σ (iω) = −

1

β

∑

Ω

∑

34σ ′

〈14|T σσ(iΩ)|32〉 Gσ34(iΩ− iω)

Spin-fluctuation contribution:

Σ(fl)12,σ(τ ) =

∑

34σ ′

W σσ ′

1342(τ )Gσ ′

34


Ludwig

Maximilians-

Universitat

Munchen

Bloch spectral functions

Spin-resolved results for Ni for ~k ‖ [001]

LSDA DMFT

spin up

spin down


Ludwig

Maximilians-

Universitat

Munchen

Spin resolved DOS of bcc-Fe and fcc-Ni

bcc-Fe fcc-Ni

-10 -8 -6 -4 -2 0 2 4

energy (eV)

0

0.5

1

1.5

n↑ Fe(E

) (s

ts./

eV

)

0

0.5

1

1.5

2n↓ Fe

(E)

(sts

./e

V) LDA

LDA+DMFT

-10 -8 -6 -4 -2 0 2 4

energy (eV)

0

0.5

1

1.5

2

n↑ Ni(E

) (s

ts./

eV

)

0

0.5

1

1.5

2

2.5

n↓ Ni(E

) (s

ts./

eV

) LDALDA+DMFT

Fe: U = 2.0 eV, J = 0.9 eV, T = 400 K

Ni: U = 3.0 eV, J = 0.9 eV, T = 400 K


Ludwig

Maximilians-

Universitat

MunchenKKR+DMFT scheme II

ΣG

KKR

EFDMFT (TMA)

VLSDA(~r), ΣDMFT

⇓single-site problem: tLL′(z)

⇓multiple scattering: τLL′(z)

⇓KKR-Green’s function: G(~r, ~r ′, z)

⇓charge ρ(~r)

⇓VLSDA(~r)

⇓G-matrix GLL′

⇓

DMFT-solver⇓

ΣDMFT


Ludwig

Maximilians-

Universitat

MunchenKKR↔ DMFT interface II

KKR DMFT

Complex energy path Real energy axis

G(z)Pade approximation

−−−−−−−−−−−−−−−−−−−−→ G(E)

Σ(z)Kramers−Kronig transformation←−−−−−−−−−−−−−−−−−−−− Σ(E)


Ludwig

Maximilians-

Universitat

Munchen

DMFT-TMA many-body solver II (III)

Effective impurity problem solved by spin-polarisedT-matrix calculated for the zero temperature

Self-energy Σ includes only Hartree and Fockcontributions:

Σ(TH)12,σ (E) =

∑

34σ ′

∞∫

−∞

〈13|T σσ ′

(E′)|24〉 Gσ ′

43 (E′ − E)dE′

Σ(TF )12,σ (E) = −

∑

34σ ′

∞∫

−∞

〈14|T σσ(E′)|32〉 Gσ34(E

′ − E)dE′


Ludwig

Maximilians-

Universitat

MunchenKKR+DMFT scheme III

EF

KKR (1)

KKR (2) (TMA)DMFT

Σ

G

Σ

KKR(1): ρ(~r), V(~r)

⇓

KKR(2): G(E)

⇓

DMFT-solver

⇓

Σ(E)

⇓ KKT

ΣDMFT(z)


Ludwig

Maximilians-

Universitat

Munchen

Comparison of schemes I-III for bcc Fe

Density of states

-10 -5 0

energy (eV)

0

0.5

1

1.5

n↑ (st

s./eV

)

0

0.5

1

1.5

2

n↓ (st

s./eV

) LSDALSDA+DMFT ILSDA+DMFT IILSDA+DMFT III

µspin µorb

LSDA 2.282 0.054

LSDA+DMFT I 2.296 0.049

LSDA+DMFT II 2.287 0.049

LSDA+DMFT III 2.285 0.049

Self-energy

-12 -8 -4 0 4

energy (eV)

-3

-2

-1

0

Re Σ

↑ t 2g(e

V)

-3

-2

-1

0

1

Re Σ

↓ t 2g(e

V)

-12 -8 -4 0 4

energy (eV)

-3

-2

-1

Im Σ

↑ t 2g(e

V)

-3

-2

-1

0

Im Σ

↓ t 2g(e

V)


Ludwig

Maximilians-

Universitat

Munchen

Comparison of schemes I-III for fcc Ni

Density of states

-10 -5 0

energy (eV)

0

0.5

1

1.5

n↑ (st

s./eV

)

0

0.5

1

1.5

2

n↓ (st

s./eV

) LSDALSDA+DMFT ILSDA+DMFT IILSDA+DMFT III

µspin µorb

LSDA 0.563 0.044

LSDA+DMFT I 0.569 0.040

LSDA+DMFT II 0.631 0.044

LSDA+DMFT III 0.622 0.043

Self-energy

-12 -8 -4 0 4

energy (eV)

-3

-2

-1

0

1

Re Σ

↑ t 2g(e

V)

-3

-2

-1

0

1

2

Re Σ

↓ t 2g(e

V)

-12 -8 -4 0 4

energy (eV)

-5

-4

-3

-2

-1

Im Σ

↑ t 2g(e

V)

-5

-4

-3

-2

-1

0

Im Σ

↓ t 2g(e

V)


Ludwig

Maximilians-

Universitat

Munchen

CPA calculations for FexNi1−x alloy

Spin magnetic moments real part of Σ

0 20 40 60 80Fe content (at. %)

0

0.5

1

1.5

2

2.5

3

µ spin

(µB)

Fe LDA

Ni LDA

total LDA

Fe LDA+DMFT

Ni LDA+DMFT

total LDA+DMFT

-10 -5 0

-0.4

-0.2

0

0.2

0.4

Σ t 2g

↓ (eV

)

FeFe10Ni90

Fe40Ni60

Fe75Ni25

-10 -5 0-2

-1

0

1

2

-10 -5 0energy (eV)

-3

-2

-1

0

Σ t 2g

↑ (eV

)-10 -5 0

energy (eV)

-2

0

2

Fe

Fe

Ni

Ni


Ludwig

Maximilians-

Universitat

Munchen

CPA calculations for NixPd1−x alloy

Spin moment Orbital moment

0 20 40 60 80 100

Ni content (at.%)

0.0

0.3

0.6

0.9

1.2

µ spin

(µB)

total LDAtotal LDA+DMFTNi LDANi LDA+DMFTPd LDAPd LDA+DMFT

0 20 40 60 80 100

Ni content (at.%)

0.00

0.05

0.10

µ orb(µ

B)

total LDAtotal LDA+DMFTNi LDANi LDA+DMFTPd LDAPd LDA+DMFT

relativistic implementation

access to spin-orbit coupling induced properties


Ludwig

Maximilians-

Universitat

Munchen

One-step model of photoemission

!

j1(R, E + ω) = const√E + ω

∫d3r1

∫d3r2

φfinal(r1, R, E+ω)†X~qλ(~r1)=G+(~r1, ~r2, E)X

†~qλ(~r2)φ

final(r2, R, E+ω)


Ludwig

Maximilians-

Universitat

Munchen

Photo current intensity

Imsm′s(E,~k;ω, ~q, λ) ∝

∑

ΛΛ′′

il−l′′C−ms

Λ C−m′

s

Λ′′ Yµ+ms∗l (−k)Y µ′′+m′

s

l′′ (−k)

∑

m m′

ei~k(~Rm−~Rm′)∑

n n′

ei~q(~Rn−~Rn′)

∑

Λ′ Λ′′′

τnmΛ′Λ(E′)τn′m′∗

Λ′′′Λ′′(E′)

∑

Λ1 Λ2

M~qλΛ′Λ1

τnn′

Λ1Λ2(E)M~qλ

Λ2Λ′′′ −∑

Λ1

In ~qλΛ′Λ1Λ′′′

with the matrix elements

M~qλΛΛ′ =

∫d3r[TZΛ(~r, E′)]†X~qλ(~r)ZΛ′(~r, E)


Ludwig

Maximilians-

Universitat

Munchen

Fano effect in VB-photoemission

Spin polarisation of photo electronsdue to spin-orbit coupling

spin resolved angle integratedphotoemission experiment

ESRF

e−

spinsemispheredetector

paramagnetic systems

photoelectrons

circularlypolarized

spin polarized


Ludwig

Maximilians-

Universitat

Munchen

Spin-resolved VB-XPS of Cu, Ag and Au

0

50

100In

tens

ity (

arb.

u.) exp.

theory

Cu (600 eV)

10 8 6 4 2 0

Binding energy (eV)

-5

0

5

Inte

nsity

(ar

b.u.

) exp.theory

0

50

100

Ag (600 eV)

10 8 6 4 2 0

Binding energy (eV)

-10

0

10

0

50

100

Au (600 eV)

10 8 6 4 2 0

Binding energy (eV)

-20

-10

0

10

20

I+

∆I+


Ludwig

Maximilians-

Universitat

Munchen

Perturbational treatment of spin-orbit coupling

non-relativistic spin-resolved Photo currentI(E,~k,ms;ω, ~q, λ) =

∫d3 r

∫d3 r′φ∗

final(~r, E + ω)

× X~q,λIm G(~r, ~r′, E)X∗~q,λφfinal(~r, E + ω)

• final states (time reversed LEED state)

φfinal(~r, E) = 4π∑

L

ilY ∗L (k)tlZl(~r)YL(r)χms

• initial statesIm G(~r, ~r′, E) =

∑

L1,L2

ZL1(~r, E)Im τnn

L1,L2Z†L2

(~r, E)

Include spin-orbit coupling (1st order):

G = G0 + G0HSOCG0KKR+DMFT HH – p.50/71

Ludwig

Maximilians-

Universitat

Munchen

Spin-resolved Photo current

Ims

λ ∝∑

L

[tl(E)]2∑

L1L2

[MλLmsL1ms

Mλ∗LmsL2ms

=τL1L2(E)

+∑

L3L4

ξlmsMλLmsL1ms

MSOCL2msL3ms

Mλ∗LL4

=(τL1L2(E)τL3L4

(E))]

Spin Difference ∆Iλ = I↑λ − I

↓λ (for l = 2 d- electrons)

∆Iλ ∝ ξd

∑

l

Slλ Im[τt2g

(4 τeg + τt2g

)]

Slλ =

+1 d → p λ = +1 (LCP)−1 d → f λ = +1 (LCP)

−1 d → p λ = −1 (RCP)+1 d → f λ = −1 (RCP)


Ludwig

Maximilians-

Universitat

Munchen

Fano effect in VB-photoemission

Spin polarisation of photo electronsdue to spin-orbit coupling


ESRF

Detectorsemisphere

e−

spin

ferromagnetic systems

magnetization in plane

photoelectrons

circularlypolarized

spin polarized


Ludwig

Maximilians-

Universitat

Munchen

Fano effect in VB-XPS of ferromagnets

Photocurrent and spin-difference Ephot =600 eVExperiments - N. Brookes et al., ESRF

Fe Co Ni

0

50

100

Inte

nsity

(ar

b.un

its)

exp.LDA

-10 -8 -6 -4 -2 0 2Binding energy (eV)

-2

0

Spin

-dif

fere

nce

(arb

. uni

ts.)

Fe (600eV)

0

20

40

60

80

100

Inte

nsity

(ar

b. u

nits

)

exp.LDA


-4

-2

0

2

4

Spin

-dif

fere

nce

(arb

. uni

ts)

Co (600eV)

0

50

100

Inte

nsity

(ar

b. u

nits

)

exp.LDA


-4

-2

0

2

Spin

-dif

fere

nce

(arb

. uni

ts)

Ni (600eV)


Ludwig

Maximilians-

Universitat

Munchen

Fano effect in VB-XPS of ferromagnets

Photocurrent and spin-difference Ephot =600 eVExperiments - N. Brookes et al., ESRF

Fe Co Ni

0

50

100

Inte

nsity

(ar

b.un

its)

exp.LDALDA+DMFT


-2

0

Spin

-dif

fere

nce

(arb

. uni

ts.)

Fe (600eV)

0

20

40

60

80

100

Inte

nsity

(ar

b. u

nits

)

exp.LDALDA+DMFT


-4

-2

0

2

4

Spin

-dif

fere

nce

(arb

. uni

ts)

Co (600eV)

0

50

100

Inte

nsity

(ar

b. u

nits

)

exp.LDALDA+DMFT


-4

-2

0

2

Spin

-dif

fere

nce

(arb

. uni

ts)

Ni (600eV)


Ludwig

Maximilians-

Universitat

Munchen

Spin-resolved VB-photoemission


ESRF

e−

spinsemispheredetector

ferromagnetic systems

magnetization in plane

spin polarizedphotoelectrons

circularlypolarized

fundamental spectra: I↑+ I

↑− I

↓+ I

↓−


Ludwig

Maximilians-

Universitat

MunchenVB-XPS for Ni

spin-polarisation (I↑+ + I↑

−)− (I↓+ + I↓

−)

-10 -5 0Binding energy (eV)

0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(a.

u.)

LDA+DMFTLDAexp.


-0.2

-0.1

0

0.1

0.2

LDA+DMFTLDAexp.


Ludwig

Maximilians-

Universitat

MunchenVB-XPS for Ni

spin-orbit (I↑+ + I↓

−)− (I↓+ + I↑

−)


0

0.2

0.4

0.6

0.8

1.0

Inte

nsity

(a.

u.)

LDA+DMFTLDAexp.


-0.1

-0.05

0

0.05LDA+DMFTLDAexp.


Ludwig

Maximilians-

Universitat

Munchen

Bloch spectral functions – ARPES

Results for Ni for ~k ‖ [001]

LSDA DMFT

BSF

ARPES


Ludwig

Maximilians-

Universitat

Munchen

Bloch spectral functions – ARPES

Ni: comparison with experiment for ~k ‖ [001]

LSDA DMFT

Expt: Måtrensson, Nilsson (1984), Bünemann (2002), Himpsel (1979), Moruzzi (1978)

BSF

ARPES


Ludwig

Maximilians-

Universitat

Munchen

Bloch spectral functions

Ni: comparison with experiment (ARPES) for ~k ‖ [001]

LSDA DMFT

Expt: Måtrensson, Nilsson (1984), Bünemann (2002)

spin up

spin down


Ludwig

Maximilians-

Universitat

Munchen

ARPES-spectra for Ni(110)

total photocurrent for off-normal emission hν = 21.2 eV

LSDA

-5 -4 -3 -2 -1 0energy (eV)

0

0.5

1

1.5

2

Inte

nsity

0

5

10

15

20

25

θ

DMFT

-5 -4 -3 -2 -1 0energy (eV)

0

0.5

1

1.5

2

Inte

nsity

0

5

10

15

20

25

θ

FULL ARPES-calculation using bulk potential


Ludwig

Maximilians-

Universitat

Munchen

ARPES-spectra for Ni(110)

photocurrent intensity for hν = 21.2 eVoff-normal emission with θ = 5 o

-1 0energy (eV)

0

0.2

0.4

0.6

0.8In

tens

ityExperimentLSDADMFT

Experiment: Osterwalder, Kreutz, Aebi (1999)


Ludwig

Maximilians-

Universitat

Munchen

Magnetic circular dichroism in XAS

Stöhr, JMMM 200 470 (’99)KKR+DMFT HH – p.63/71

Ludwig

Maximilians-

Universitat

MunchenXMCD sum rules

excitation scheme for left circularly polarized light

1/2 3/21/2-1/2-3/2

-1/2 1/2 3/21/2-1/2-3/2

d3/2

p1/2 3/2p

d5/2

-1/2

s

3/21/2-1/2-3/2-5/2 5/2

1/2

XMCD sum rules∫

(∆µL3− 2∆µL2

) dE = N3Nh

(〈σz〉 + 7〈Tz〉)

∫(∆µL3

+ ∆µL2) dE = N

2Nh〈lz〉

Thole, PRL 68 1943 (’92); Carra, PRL 70 694 (’93)KKR+DMFT HH – p.64/71

Ludwig

Maximilians-

Universitat

Munchen

Calculation of XMCD spectra

Fermi’s golden rule

µ~qλ(ω) ∝∑

i occf unocc

|〈Φf |X~qλ|Φi〉|2δ(~ω − Ef + Ei)

∝∑

i occ

〈Φi|X×~qλ =G+(Ef )X~qλ|Φi〉θ(Ef − EF)

relativistic treatment — X~qλ = −1c~jel · ~aλAei~q~r

X-ray circular magnetic dichroism (XMCD)

∆µ~qλ(ω) = µ~q+(ω) − µ~q−(ω)

Ebert, RPP 59 1665 (’96)KKR+DMFT HH – p.65/71

Ludwig

Maximilians-

Universitat

Munchen

X-ray absorption for Fe

X-ray absorption coefficientL2,3-edge of Fe in bcc-Fe

µL2,3for unpolarisedradiation

magentic circular dichroism∆µL2,3

= µ+ − µ−

0 5 10 15 20 25

energy (eV)

0

4

8

12

µFe L 2,3 (

Mb

) LDA

LDA+DMFTExperiment Scherz et al.

0 5 10 15 20 25

energy (eV)

-3

-2

-1

0

1

2

3

∆µFe L 2,

3 (M

b)


Ludwig

Maximilians-

Universitat

Munchen

X-ray absorption for Ni

X-ray absorption coefficientL2,3-edge of Ni in fcc-Ni

µL2,3for unpolarisedradiation

magentic circular dichroism∆µL2,3

= µ+ − µ−

0 5 10 15 20 25

energy (eV)

01234567

µNi

L 2,3 (

Mb

) LDA

LDA+DMFTExp. Scherz et al.

0 5 10 15 20 25

energy (eV)

-2

-1.5

-1

-0.5

0

0.5

∆µN

iL 2,

3 (M

b)


Ludwig

Maximilians-

Universitat

Munchen

Summary

Combination of KKR+DMFT

⇒ alloys⇒ inhomogeneous systems⇒ spectroscopy

Fano-effect in angle-integrated PES of ferromagnetsFe, Co and Ni

⇒ DMFT improves agreement with experiment

Angle-resolved PES

⇒ first promising results obtained


Ludwig

Maximilians-

Universitat

Munchen

Magneto-optical properties

Kubo linear response formalism

σαβ(ω) = −1

ωV

∫ 0

−∞dτe−i(ω+iη)τ

⟨[Jβ(τ ), Jα(0)]

⟩

Absorptive part of the optical conductivity

σabsαβ (ω) =

1

πωV

∫ µ

µ−ωdω′ tr

[Jα=G(ω′)Jβ=G(ω′ + ω)

]

Green’s function matrix include correlation effects

[H0 + Σ(E)− E]G(E) = I


Ludwig

Maximilians-

Universitat

Munchen

Magneto-optical properties of Fe

Optical conductivity and Kerr rotation spectra for bcc Fe

Fe

LDAU=1.5eVU=2.eVU=3eVExp(x1.7)

0 1 2 3 4 5 6 7Energy (eV)

20

40

60

80

σ1 xx(ω

),10

-14 s

-1

0 1 2 3 4 5 6 7Energy (eV)

0

5

ωσ2 xy

(ω),

10-2

9 s-2

Exp x 0.8

0 1 2 3 4 5 6Energy (eV)

-0.6

-0.4

-0.2

0.0

Ker

rro

tatio

n,de

g


Ludwig

Maximilians-

Universitat

Munchen

Magneto-optical properties of Ni

Optical conductivity and Kerr rotation spectra for fcc Ni

LDAU=1.5eVU=2eVU=3eVExp

0 1 2 3 4 5 6 7Energy (eV)

20

40

60

σ1 xx(ω

),10

-14 s

-1

0 1 2 3 4 5 6 7Energy (eV)

-2

0

2

ωσ2 xy

(ω),

10-2

9 s-2

0 1 2 3 4 5 6Energy (eV)

-0.2

0.0

0.2

Ker

rro

tatio

n,de

g


Combination of the KKR-method and the DMFT · Ludwig Maximilians-Universit¤at Munchen¤...

Documents

Transcript of Combination of the KKR-method and the DMFT · Ludwig Maximilians-Universit¤at Munchen¤...