Department of Physics and Astronomy

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Materials theory and spectroscopy

Olle Eriksson

Department of Physics and Astronomy

Uppsala University

(2)

Overview

A general motivation

Theoretical background of DMFT Valence band XPS

XAS of complex oxides

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Collaborators

Abrikosov, Bergman, Bergqvist, Björkman, Burkert, Burlamaqui-Klautau, Chico, Chimata, , Delczeg, Delin, Di Marco, Edström, Etz Grechnev, , Grånäs, Hellsvik, Iusan, Katsnelson, Keshavarz, Kimel, Kirilyuk, Kvashnin, Koumpouras, Lichtenstein, Locht, Mentink, Nordström, Pereiro, Rasing, Rodriques, Russ, Sanyal, Schött,

Skubic, Szilva, Szunyogh, Taroni, Thonig, Thunström,

Wills

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Motivation

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A quantum mechanical problem

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Density functional theory

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Bloch states

?

When I started to think about it, I felt that the main problem was to explain how the electrons could sneak by all the ions in a metal so as to avoid a mean free path of the order of atomic distances. Such a distance was much too short to explain the observed resistances...

To make my life easy, I began by considering wave functions in a one-dimensional periodic potential. By straight Fourier analysis I found to my delight that the wave differed from the plane wave of free electrons only by a periodic modulation. This was so simple that I couldn't think it could be much of a discovery, but when I showed it to Heisenberg he said right away:

'That's it!' Well that wasn't quite it yet, and my calculations were only completed in the

summer when I wrote my thesis on “The Quantum Mechanics of Electrons in Crystal Lattices."

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Bloch states and bands

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Bloch states and bands

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Electrons become magnetic

moments

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Spin moments of Fe-Co alloys

bcc fcc

t per a tom ( µ ) M ag netic momen Β

B

hcp

1 2

Expt

Fe alloy Co

PHYSICAL REVIEW B 59, 419 (1999)

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Ab initio exchange parameters from DFT

E

DOS(E)

1 Stress Tensor

H = E ˆ _band + E _exch (1)

H = ˆ X

i,j,

t _ij c ˆ ^† _i c ˆ _j + X

i

U hˆn ⁱ " ihˆn ⁱ # i (2)

E _band = X

~ k

" ^" _k n ˆ _~ ^"

k + " ^# _k n ˆ _~ ^#

k (3)

E _band =

Z E

_f

+ 1

"⇢(")d" +

Z E

_f

1 "⇢(")d" (4)

U · ⇢(" ^f ) > 1 (5)

hˆn ^" _i i = n

2 + m (6)

hˆn ^# _i i = n

2 m (7)

hˆn ^" _i i = n

2 + m (8)

H = ˆ X

i 6=j

J _ij · (~e ⁱ · ~e ^j ) (9)

H = ˆ X

i 6=j

J _ij · ~ S _i · ~ S _j (10)

~e _i = S ~ _i

| ~ S _i | (11)

~e ⁰ _i = ~e _i + [ ~' _i ⇥ ~e ⁱ ] 1

2 ~e _i · ( ~' ⁱ ) ² (12)

H ˆ ⁰ = X

i 6=j

J _ij · (~e ⁰ i · ~e ⁰ j ) = ˆ H ₀ + ˆ H + ² H ˆ (13)

2 H = ˆ 1 2

X

ij

J _ij | ~' ⁱ ~' _j | ² (14)

U = exp ˆ



i ~' _i · ˆ~/2 (15)

H ˆ ⁰ = ˆ U ^† H ˆ ˆ U = ˆ H ₀ + ˆ H + ² H ˆ (16)

J _ij = 1

4 Tr _!



i (i! _n ) · G ^" _ij (i! _n ) · ^j (i! _n ) · G ^# _ji (i! _n ) (17)

Tr _! = T

X 1

i!

_n

,n= 1

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2 H = ˆ 1 2

X

ij

1 4 Tr _!



i (i! _n ) · G ^" _ij (i! _n ) · ^j (i! _n ) · G ^# _ji (i! _n ) · | ~' ⁱ ~' _j | ² (19) Inter-site Green’s function

Local exchange splitting

Table 1:

State M _S (N i) (µ _B )

FM 1.80

AFM 1.77

H ˆ _ij = hinlm | ˆ H | jn ⁰ l ⁰ m ^{0 0} i (22)

⇠ 2µ ^B (23)

2 H = ˆ 1 2

X

ij

1 4⇡

Z E

_f

1 d" Tr _L



i · G ^" _ij (") · ^j · G ^# _ji (") · | ~' ⁱ ~' _j | ² (24)

J _ij = 1 4⇡

Z E

_f

1 d" Tr _L



i · G ^" _ij (") · ^j · G ^# _ji (") (25)

J _ij = 1

4 Tr _!



⌃ _i (i! _n ) · G ^" _ij (i! _n ) · ⌃ ^j (i! _n ) · G ^# _ji (i! _n ) (26)

Table 1:

State M _S (N i) (µ _B )

FM 1.80

AFM 1.77

H ˆ _ij = hinlm | ˆ H | jn ⁰ l ⁰ m ^{0 0} i (22)

⇠ 2µ ^B (23)

2 H = ˆ 1 2

X

ij

1 4⇡

Z E

_f

1 d" Tr _L



i · G ^" _ij (") · ^j · G ^# _ji (") · | ~' ⁱ ~' _j | ² (24)

J _ij = 1 4⇡

Z E

_f

1 d" Tr _L



i · G ^" _ij (") · ^j · G ^# _ji (") (25)

G _ij (z) =

⌧

i 1

z H ˆ ⌃(z) ˆ j (26)

J _ij = 1

4 Tr _!



⌃ ^s _i (i! _n ) · G ^" _ij (i! _n ) · ⌃ ^s j (i! _n ) · G ^# _ji (i! _n ) (27)

⌃ ^s _i (i! _n ) =

✓

H ˆ _i ^" H ˆ _i ^#

◆ +

✓

⌃ ^" _i (i! _n ) ⌃ ^# _i (i! _n )

◆ (28)

i =

✓

H ˆ _i ^" H ˆ _i ^#

◆ (29)

j i

Lichtenstein et al JMMM 67 65 (1987)

2 H = ˆ 1 2

X

ij

1 4 Tr _!



i (i! _n ) · G ^" _ij (i! _n ) · ^j (i! _n ) · G ^# _ji (i! _n ) · | ~' ⁱ ~' _j | ² (23)

G _ij (z) =

⌧

i ˆ G(z) j =

⌧

i 1

z H ˆ j (24)

m = N ^" N ^# =

Z E

_f

1 ⇢ ^" (")d"

Z E

_f

1 ⇢ ^# (")d" (25)

H ˆ _ij = hinlm | ˆ H | jn ⁰ l ⁰ m ^{0 0} i (26)

Table 1:

State M _S (N i) (µ _B )

FM 1.80

AFM 1.77

⇠ 2µ ^B (27)

2 H = ˆ 1 2

X

ij

1 4⇡

Z _E

_f

1 d" Tr _L



i · G ^" _ij (") · ^j · G ^# _ji (") · | ~' ⁱ ~' _j | ² (28)

J _ij = 1 4⇡

Z E

_f

1 d" Tr _L



i · G ^" _ij (") · ^j · G ^# _ji (") (29)

G _ij (z) =

⌧

i 1

z H ˆ ⌃(z) ˆ j (30)

J _ij = 1

4 Tr _!



⌃ ^s _i (i! _n ) · G ^" _ij (i! _n ) · ⌃ ^s j (i! _n ) · G ^# _ji (i! _n ) (31)

⌃ ^s _i (i! _n ) =

✓

H ˆ _i ^" H ˆ _i ^#

◆ +

✓

⌃ ^" _i (i! _n ) ⌃ ^# _i (i! _n )

◆ (32)

i =

✓

H ˆ _i ^" H ˆ _i ^#

◆ (33)

⌃(~k, i! ˆ _n ) ! ˆ ⌃(i! _n ) (34)

G ˆ ¹ = ˆ G ₀ ¹ ⌃ ˆ (35)

⌃(~k, i! ˆ _n ) ! ˆ ⌃ (36)

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Heisenberg spin Hamiltonian

M vs T for bcc Fe

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Atomistic Landau-Lifshitz equation

Separate fast variables (electrons) and slow

(atomic spins) and the EOM together with Landau- Lifshitz damping term gives:

The effective field B is given by

Precession Damping

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Time scales

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Some examples of J ^ij

Y. Kvashnin et al. PRB 91 125133 (2015)

1.0 1.5 2.0 2.5 3.0 3.5 R_ij / a

-0.1 0.0 0.1 0.2

J

ij

(mRy)

MTHORT

hcp Gd

0.8 1.0 1.2 1.4 1.6 1.8 2.0 R_ij / a

0.00 0.05 0.10 0.15

J

ij

(mRy)

MTHORT

fcc Ni

For surface magnons see: Corina Etz, et al. , ''Atomisic spin-dynamics and surface magnons'' J. Physics. Cond. Matter. 27 , 243202 (2015)

Y. Kvashnin et al. PRB 91 125133 (2015)

I.L.M. Locht et al Phys. Rev. B 94, 085137 (2016)

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All thermal switching

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Atomistic spin-dynamics; foundations and applications

O. Eriksson

Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden

A. Bergman

Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden

L. Bergqvist

Department of Materials and Nano Physics, KTH Royal Institute of Technology, Electrum 229, SE-164 40 Kista, Sweden

J. Hellsvik

Department of Materials and Nano Physics, KTH Royal Institute of Technology, Electrum 229, SE-164 40 Kista, Sweden

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NiO - an example

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Dynamical mean field theory

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Dynamical mean field theory

FICTITIOUS SYSTEM REPRODUCING THE DYNAMICS The Hubbard model is mapped into an Anderson Impurity Model

The mapping is made with the condition of preserving the local

Green’s function and is exact in the limit of infinite nearest neighbors

U-matrix expressed in terms of Slater integrals

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Some equations

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DMFT loop

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Correlated basis

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Exact Diagonalization Solver

Igor Di Marco

Local correlation effects in the electronic structure of Mn doped GaAs with LDA+DMFT Igor Di Marco

The finite size problem can be solved exactly with a direct construction of all the accessible many-body states.

N=5 electrons in K=10 orbitals:

M corresponds to 𝑲 𝑵

Too large for standard computational resources!

Block diagonalization up to 30 bath states!

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The hybridization function

∆ (ω)

_app

(27)

The hybridization function

(28)

Some equations

|111111,11100,001>

|core states, correlates states, bath states> form a basis to diagonalize Hamiltonian

|011111,11101,001>

(29)

Exact Diagonalization Solver

Igor Di Marco

Local correlation effects in the electronic structure of Mn doped GaAs with LDA+DMFT Igor Di Marco

The finite size problem can be solved exactly with a direct construction of all the accessible many-body states.

N=5 electrons in K=10 orbitals:

Once the many-body states have been determined, the one-particle

Green’s function can be obtained through the Lehmann representation

(30)

Valence band spectra

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Paramagnetic NiO

P. Thunström et al. PRL 109 186401 (2012)

Experiment

Theory (DMFT)

DOS

Ni-3d

O-2p

(32)

Valence band of Mn-doped GaAs

LDA+U

Nature Communications 4, 2645 (2013)

(33)

The HIA approximation

I.Locht, Thesis

Self-energy obtained from ME-atomic problem, that is

used in solid state calculation

(34)

Valence band spetra of rare-earths

(Loch et al. PRB 2016)

La Ce Pr

Nd Pm Sm

Eu Gd Tb

Energy (eV) Energy (eV) Energy (eV)

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

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

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

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Valence band spetra of rare-earths

(Loch et al. PRB 2016)

Dy Ho Er

Tm Yb Lu

Energy (eV) Energy (eV)

Energy (eV)

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

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

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

(36)

HIKE on NiO

Panda et al PRB 2016

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XAS

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The XAS process

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Hamiltonians

1

The initial state SIAM Hamiltonian:

1 P.W. Andersson, Phys. Rev. 124, 41 (1961).

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Final state

21-Sep-16 CMD26, Groningen – The Netherlands

barbara.brena@physics.uu.se 2

The final state XAS Hamiltonian accounts in addition for the 2p to 3d excitations (Quanty):

(41)

XAS configurations- charge transfer

2

(42)

Computed L-edge X-Ray Absorption Spectra

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PARAMETERS USED IN THE MLFT CALCULATIONS

U_dd

(eV)

U_pd

(eV)

F⁰

(eV)

F²

(eV)

F⁴

(eV)

∆

(eV)

10D_q (eV)

V_eg

(eV)

V_t2g

(eV)

n_d

NiO 8.0 10.5 7.5 10.1 6.7 4.7 0.65 2.1 1.1 8.2

CoO 7.5 8.4 7.0 10.0 6.7 4.0 0.79 2.6 1.4 7.2

FeO 7.0 9.6 6.5 9.2 6.2 4.0 0.85 2.1 1.2 6.2

MnO 6.5 7.2 6.0 9.0 6.1 8.0 0.86 1.8 0.9 5.1

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CHARGE TRANSFER EFFECTS ON COMPUTED SPECTRA

Importance of including impurity-bath interaction black line: hopping included colored line: no hopping 10 Dq varying from 0.1 to 1.5 eV

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Conclusions

• Presented general motivation to why spectroscopy is needed

• for materials theory

• Presented details around dynamical mean field theory

• Provided examples of XPS and XAS spectra for complex oxides

• and rare-earths

• Pointed to need of help

(46)

Importance of including impurity-bath interaction black line: hopping included colored line: no hopping 10 Dq varying from 0.1 to 1.5 eV

(47)

Importance of including impurity-bath interaction black line: hopping included colored line: no hopping

10 Dq varying from 0.1 to 1.5 eV

Increased hopping rate means higher probability

to have an electron transfer from ligand to impurity

higher charge transfer energy ∆ means that

the mixing of different configurations is reduced

highest ∆

(48)

Final state

The final state XAS Hamiltonian accounts in addition for the 2p to 3d excitations (Quanty):

The XAS intensities are computed within the dipole approximation.

(49)

XAS configurations- charge transfer

(50)

LDA AND FITTED HYBRIDIZATION FUNCTION

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X-ray absorption spectra

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Diana Iu an Materials Theory @UU Sandvik, 2016-01-22

850 855 860 865 870 875

E(eV)

Intensity (arb. units)

Experiment ( digitized) Theory

XAS of NiO

E (eV)

Experiment (digitized) Theory

XAS of FeO

710 720

775 780 785 790 795 800

E (eV)

XAS of CoO

640 650

E (eV)

XAS of MnO

J.Luder et al, (unpublished)..

Department of Physics and Astronomy

Materials theory and spectroscopy

Olle Eriksson