Prolog Numerical Modeling in Magnetism

PrologPrologNumerical Modeling in MagnetismNumerical Modeling in MagnetismMacro-Magnetism: Solution of Maxwells Equations –

Engineering of (electro)magnetic devices

Atomic Magnetism: Instrinsic Magnetic Properties

Micromagnetism: Domain Dynamics, Hysteresis

MFM image

Micromagnetic simulation.

Atomic Magnetism-Atomic Magnetism- Modeling Instrinsic Magnetic Properties Modeling Instrinsic Magnetic Properties

Band Models• Spin Polarized First Principle Methods: restricted to simple Magnetic Structures, T=0, no dynamics, no rare earth

elements ... there are attempts to overcome these restrictions

Localized Moment Models Ising-, Heisenberg-, xy-, Standard Model of RE-Magnetism)• Exact Methods: e.g. branch and bound algorithm, transfer

matrix algorithm• Monte Carlo Methods• Selfconsistent Mean Field Method

Atomic Magnetism-Atomic Magnetism- Modeling Instrinsic Magnetic Properties Modeling Instrinsic Magnetic Properties

McPhase 2.4.lnk

Band Models• Spin Polarized First Principle Methods: restricted to simple Magnetic Structures, T=0, no dynamics, no rare earth

elements ... there are attempts to overcome these restrictions

Localized Moment Models Ising-, Heisenberg-, xy-, Standard Model of RE-Magnetism)• Exact Methods: e.g. branch and bound algorithm, transfer

matrix algorithm• Monte Carlo Methods• Selfconsistent Mean Field Method

M. Rotter, Institut für physikalische Chemie, Universität Wien

http://www.ap.univie.ac.at/users/Martin.Rotter/McPhase_-_the_World_of_Rare_Earth_Magnetism.html

Martin Rotter - McPhase Course TU Dresden 2005

McPhase - the World of Rare Earth Magnetism

The Standard Model of RE Magnetism - The Standard Model of RE Magnetism - the Crystal Field Conceptthe Crystal Field Concept

+

+

+

+

+

+

+

+

+

+

Hamiltonian ilm

iml

mlcf OBH

,

)(JE

Q

4f –charge density

McPhase - the World of Rare Earth MagnetismMartin Rotter - McPhase Course TU Dresden 2005

Example: NdCuExample: NdCu2 2

ab

c

Crystal Structure of RCu2

Imma (orthorhombic)

... 9 nonzero CF Parameters9 nonzero CF Parameters

you can use module pointc to calculate CF parameters by the pointcharge model+

++

+

++

+++

+

++


NdCuNdCu2 2 – Crystal Field Excitations– Crystal Field Excitations

orthorhombic, TN=6.5 K, Nd3+: J=9/2, Kramers-ion

Gratz et. al., J. Phys.: Cond. Mat. 3 (1991) 9297

McPhase can• solve CF Model• Calculate Intensities and Energies• Calculate and Plot Charge Density• ...


Make a Crystal Field ModelMake a Crystal Field Modelusing McPhase Module using McPhase Module CfieldCfield

CF Hamiltonian ilm

iml

mlcf OBH

,

)(J

Example files in directory /mcphas/examples/ndcu2b_new/cf

• Edit file Bkq.parameter and enter CF parameters Blm

• Start module cfield - type: cfield –r -B• View output file cfield.out: CF - energies, eigenstates,

transition-matrixelements and corresponding neutron intensities

• Use module convolute to convolute energy vs intensity results with spectrometer resolution function

Module simmannfit can do this again and again for youto fit the result of the calculation to your spectrum by variation of the CF-parameters


Magnetism would be boring Magnetism would be boring without a magnetic fieldwithout a magnetic field

Hamiltonian

i

iBJilm

iml

ml gOBH HJJ

,

)(

Use module cfieldto calculate magnetization

type: cfield –m


Specific HeatSpecific Heat

Use module cpcalc to calculate specific heat

type: cpcalc 5 30 1

Tmin=5 Tmax=30 dT=1


T=10 KT=40 KT=100 K

Use modules chrgplot+javaview to plot 4f charge density

)()(|)(|)(ˆ

,...,06,4,2,0

24

nmT

nmn

mnnnmf ZOecrR Jr


Use modules pointc+chrgplot+javaview

T=2KH=0Same CEF


Module Module mcphasmcphas


Input files for module mcphas: mcphas.j (structure), mcphas.cf (single ion properties), mcphas.tst (table of initial values), mcphas.ini (H,T-range, ...)


Do you really want to see the MF Do you really want to see the MF equations ?equations ?

ij

jii

iBJiilm

iml

ml ijJgOBH JJHJJ )(

2

1)(

,

))((

jjii

ijjijiji

JJJJ

JJJJJJJJ

ij

jii

effiiBJi

ilmi

ml

ml ijJgOBH JJHJJ )(

2

1)(

,

j

jBJi

effi g

ijJJHH

)(

Cfield can calculate

effiTiBJi

effiii g

HJHMM

,


Bulk Properties Calculated by Bulk Properties Calculated by module module mcphasmcphas

Magnetization output file: mcphas.fum


NdCuNdCu22 Specific Heat Specific Heat

output file: mcphas.fum


Crystal field

T

e-

+

+

L0

T<TC(N)

Spontaneous MagnetostrictionSpontaneous Magnetostriction

Microscopic Source of Magneostriction: Strain dependence of magnetic interactions

Exchange

T<TC(N) L=0, L0

„exchange-striction“

.... Symmetry decreases

T


H <0

Crystal Field

e-

+

+

Exchange Striction

H

H>0

Forced MagnetostrictionForced Magnetostriction

L0 L=0, L0


Calculation of MagnetostrictionCalculation of Magnetostriction

lmi

iml

mlcf OBH

,

)()( J ij

jiex ijJH JJ),(2

1

Crystal Field Exchange

k

T k i i

k i i JH J J,

) , (

lm

Tml

ml B

H J O, ) (

+

...)0()0()(

excf HH

excfel HHEH

cEel 2

1mit

}{ / TkH BeTrZ ZTkF B ln 0

F

Output file: mcphas.jj*Output file: mcphas.xyt


NdCuNdCu2 2 Magnetostriction Magnetostriction

H cfH

cfH )0(

Crystal Field

)0( exH

exH

Exchange - Striction


NdCuNdCu22 Magnetic Phase Diagram Magnetic Phase Diagram

F3

AF1

F1

output file: mcphas.xytUse module phased or displaycontour for color plot of phasediagram

ab

c

F1

lines=experiment


output file: mcphas.hkl


Dispersive Magnetic Excitations

ij

jiijJH SS)(2

1

153

T=1.3 KMF - Zeeman Ansatz


... Spinwaves (Magnons)

ij

jiijJH SS)(2

1

Bohn et. al. PRB 22 (1980) 5447

T=1.3 K

153


Spinwaves (Magnons)

ij

jiijJH SS)(2

1

Bohn et. al. PRB 22 (1980) 5447

T=1.3 Ka

153


Module Module Mcdisp Mcdisp – Calculate Magnetic – Calculate Magnetic Excitation Energies and the Neutron Excitation Energies and the Neutron

Scattering Cross Section Scattering Cross Section

),()ˆˆ('

'

2

2

22

κκκ magS

mc

e

k

kN

dEd

d

''/1

12 kTe

S

'

')(

'21

21

21 ),()}({)}({),( ''

dddd

WWiddN

inelmag SeegFgFS dddd

b

κκ BBκ

*)()(2

1)('' ''' zz

iz dddddd

)(

||||)(

)()(1)(),(

,,0

1

00

jiij ij

THTH nniJJjjJJi

J

κκ MF-RPA


Module Module Mcdisp Mcdisp – a novel fast – a novel fast algorithm for magnetic algorithm for magnetic

excitations – Rotter 2005excitations – Rotter 2005

s

ss

s

s

ss

s

s

s

s

s UUUUMM 1**

10 )(

)(),()()(1 00 κκJ

*

'''

''

'''*'''''

'''

),(),(

ss

ssssss UU κ

κTransformation:

''

''''*''

'' )()( ssssss UJUL κκ


),()()( '''11

''

*

''''

11''' κκ ss

ss

ssssssss L

all other components of Ψ are zero

with definition:

0 if 1

0 if 1

s

s''''

ssss

with definition: *

''''

11'''' )( sss

ssssss LA κ

Generalized eigenvalue problem (analogue to dynmical matrix in the case of phonons!!)

ttA

Solution gives eigenvalues r and eigenvectors ,...),( 21 tt

(1)


(1) may then be inverted to give the following expression for Ψ

'

*''

1'11 )(),(

rrsrrrrsr

ss κ

r

srsrsr

sss

ss UU '*1

*'1

*

'' ))((),('' κκ

using Diracs formula: )(11

lim0

r

rr

iPi

r

ssssss

ss UU '*1

'111

*

'' ),(),( κκback transformation...

*)()(2

1)('' ''' zz

iz ssssss +calculation of absorptive part...


McDisp - fast algorithm - Cookbook

ss

s UM , ... lize...diagona *

''''1

''*1'''' )( s

ssssssssss UJUA

κ...setup Matrix

,... rttA

...solve generalized EV Problem

r

srsrsr

sss

ss UU '*1

*'1

*

'' ))((),('' κκ

),()ˆˆ('

'

2

2

22

κκκ magS

mc

e

k

kN

dEd

d

''/1

12 kTe

S

'

')(

'21

21

21 ),()}({)}({),( ''

dddd

WWiddN

inelmag SeegFgFS dddd

b

κκ BBκ

)(||||: ,, jiTHTHs nniJJjjJJiMijs 1)

2)

3)

4)

5)

NdCuNdCu22F3

AF1

F1


Diffuse ScatteringDiffuse Scattering


McPhase ModulesMcPhase Modules

Symmetry - CFSymmetry - CFLocal Point Symmetry limits the number of nonzero

Crystal Field Parameters(mind: local symmetry at rare earth position may be lower than lattice symmetry, i.e. The

lattice may be cubic, but the local symmetry tetragonal)

Point Group / Latt. Symmetry

Coordinate Orientation Nonzero Blm

O cubic xyz||abc B40,B44,B60,B64

O cubic z||111 B40,B43,B60,B63,B66

D6h hexagonal xyz||abc B20,B40,B60,B66

D4h tetragonal xyz||abc B20,B40,B44,B60,B64

C3v (no lattice) B20,B40,B43,B60,B63

C2h monoclinic B20,B40,B60,B66,B66s

D3d (quasicubic in dhcp)

xyz||abc B20,B40,B43,B60,B63,B66

D2 orthorh. xyz||abc B20,B22,B40,B42,B44,B60,B62,B64,B66


Example: 2nd order CF terms for point symmetry mm2=C2v

We choose here the basis of Racah instead of Stevensoperators for the Crystal field, because these transformlike the spherical harmonic functions

02

02

12

22

22

~

228

3~

28

3~

OO

PiPO

PiOO

yzxz

xy

C2v 1E 1C2 1σy 1σx

A1 1 1 1 1

B1 1 -1 -1 1

A2 1 1 -1 -1

B2 1 -1 1 -1These operators form a reducable representation T2(G) of the point group

Irr.Repr.

Group elements G

'

'2

2'2 )(

~)()'(

~

m

mmm

m OGTO JJCharacter table of mm2

Group Theory basics taken from: Elliott&Dawber Symmetry in Physics, McMillan Press, 1979


The representation T2(G) can be decomposed into irreducibleRepresentations (i.e. „the Olm can be linear combined to another Basis so that in this basis the representation T2 bas block diagonal form with each block corresponding to a irreducible representation“)

222211112 )( BmAmBmAmGT BABA The m‘s tell, how often a representation occurs. mA1 tells, how often the unit representation occurs in the decomposition, i.e. how many different independent basis vectors span this subspace, i.e. how many independent crystal field parameters will occur.

p

pAppA c

gm *1

1

1Cp... Number of members of class p g.... Number of group elements χ.... Character of class

m

lmmp a

alGT

)2/sin(

))2/1sin(()(

a... Angle of rotation

Class p a χp

E 0 5

C2 π 1

σy π 1

σx π 1

2)1115(4

11 Am

i.e. We expect 2 independent 2nd order CF parameters

A little group theoretical trick for calculating m


The basis of the 2 A1 representation occuring in the decomposition of T2(G) can be found using the projection operator

G

AA GTGg

P )()(1 211

In order to calculate it, we have to epxlicitely write down the reducable representation T2:

02

02

12

22

22

~

228

3~

28

3~

OO

PiPO

PiOO

yzxz

xy

',2

'

',2

'

'22

'

'2

'

)(

)1()(

)1()(

)(

mmxmm

mmm

ymm

mmm

mm

mmmm

T

T

CT

ET

Jx‘=-Jx, Jy‘=-Jy

Jy‘=-Jy

Jx‘=-Jx

',',''1

' )1()1(4

1mmmm

mmm

mmm

AmmP

02

02

112

1

22

22

22

22

1

~~,0

~8

3~~

2

1~

OOPOP

OOOOP

AA

A

B20 and B22 are nonzero.


Symmetry – Bilinear InteractionSymmetry – Bilinear Interaction

ij

jiex ijJH JJ )(2

1

Anisotropic Interaction (J(ij) is a tensor)

ij

jiex ijJH JJ )(2

1

Isotropic interaction (J(ij) is a scalar)


CeCu2 Structure

Ce

Cu

(quasi)hexagonal types of neighbors

M. Rotter et al., Eur. Phys. J. B 14, 29 (2000)M. Rotter et al., JMMM. 214, 281 (2000)

neighbors related by symmetry must have related interaction constants J(ij)


ij

jiex ijJH JJ )(2

1

Anisotropic Interaction –Symmetry Considerations


ETC...


Example: bc mirror plane

0

1

cccb

bcbb

aa

JJ

JJ

J

SJSJ

S

JJH

0

0

00

100

010

001

'

''2

1

2

11010

JJJ

JJJJ

a

b

Symmetry – Quadrupolar Symmetry – Quadrupolar InteractionInteraction

Isotropic Quadrupolar Interaction

dhcp –lattice: between hexagonal sites

dhcp –lattice: between quasicubic sites

Derivation similar to CF operator using representation T(G)=T2(G)xT2 (G)


Example for quadrupolar Example for quadrupolar interactions: PrCuinteractions: PrCu22

+

+

+

+

+

+

+

+

+

+

+

+

H

M


PrCuPrCu22

+

+

+

+

+

+

+

+

+

+

+

+

ij

jiijQ OOCH )()( 22

22 JJ

Ferroquadrupolarer (Cij>0) Austausch (durch CF-Phonon WW)

www.mcphase.de

Settai et. al. JPSJ 67 (1998) 636 012

TO

022

TO02

2 T

O

GMS

Settai et. al. JPSJ 67 (1998) 636


PrCuPrCu22

ij

jiijQ OOCH )()( 22

22 JJ

Ferroquadrupolar (Cij>0) Interaction


The Model describes well:• the quadrupolar phasen diagram• the magnetisation• the magnetostriction• die temperature dependence of elastic constants

Whats about the Dynamics ?


Orbital Excitations (Orbitonen)Orbital Excitations (Orbitonen)

ij

jiQ OOCH )()( 22

22 JJ

+Antiferroquadrupolar (C<0) Interaction

+

+

+

+

+

+

+

+

+

+

Crystal field ilm

iml

mlcf OBH

,

)(JE

Q

4f – charge density


PrCuPrCu22

+

+

+

+

+

+

+

+

+

+

+

+

ij

jiijQ OOCH )()( 22

22 JJ

Ferroquadrupolar (Cij>0) Interaction (via CF-Phonon coupling)



McPhase: www.mcphase.de Rotter, JMMM 272-276 (2003) 481

Г

00L

Ene

rgy

(meV

)

2.5

01 2

Experiment

Kawarazaki et. al., J. Phys. Cond. Mat. 7 (1995) 4051

PrCuPrCu2 2

Orbital Modes T=5 K, H=0 TOrbital Modes T=5 K, H=0 T

MF-RPA Model

?


[Interpretation von Kawarazaki et. al., J. Phys. Cond. Mat. 7 (1995) 4051]

Könnte nicht auch die Austauschwechselwirkungzu der beobachteten Dispersion führen ?

PrCuPrCu22

Г

00L

Ene

rgy

(meV

)

2.5

Nur Quadrupolaustausch

01 2

Magnetic ExcitationsRotter et. al., Europ. Phys. J. B 14 (2000) 29

NdCu2NdCu2


PrCuPrCu22

Nur Quadrupolaustausch

00L

Ene

rgy

(meV

)

2.5

+ magnetic Interactions

ij

jiex ijJH SS)(2

1

01 2

ij

jiQ OOCH )()( 22

22 JJ

Г

00L

Ene

rgy

(meV

)

2.5

Quadruplar Interaction only

01 2

nurmagnetischer

Austausch


Messung

PrCuPrCu2 2 Orbital modes in Magnetic field Orbital modes in Magnetic field

T=2 K, H||a T=2 K, H||a

McPhase: www.mcphase.de Rotter, JMMM 272-276 (2003) 481

IN12(ILL) März 2004(15 Tesla cryomagnet)

Rechnung


The model describes well:• macroscopic properties and quadrupolar Phase diagram• Magnitude of dispersion of orbital modes

Quadrupolar Effects Neutrons can be scattered by 4f - Orbitons – Orbiton spectroscopy:

- Determination of multipolar Interactions- Modeling of GMS

ij

jiijQ OOCH )()( 22

22 JJ

Crystal field + Ferroquadrupolar (Cij>0) InteractionsSettai et. al. JPSJ 67 (1998) 636

PrCu2


How to start – the story of NdCuHow to start – the story of NdCu22 Suszeptibility: 1/χ(T) at high T

... Crystal Field Parameters B20, B2

2

Specific Heat Cp ... first info about CF levels Magnetisation || a,b,c on single crystals in the paramagnetic state, ... ground state matrix elements Neutron TOF spectroscopy – CF levels

... All Crystal Field Parameters Blm

Thermal expansion in paramagnetic state – CF influence

... Magnetoelastic parameters (dBlm/dε)

Neutron diffraction: magnetic structure in fields || easy axis ... phase diagram H||b - model

... Jbb Neutron spectroscopy on single crystals in H||b=3T

... Anisotropy of Jij - determination of Jaa=Jcc

Magnetostriction ... Confirmation of phase diagram models H||a,b,c, dJ(ij)/dε


The story of NdCuThe story of NdCu22

Inverse suszeptibility at high T

... B20=0.8 K, B2

2=1.1 K Hashimoto, Journal of Science of the

Hiroshima University A43, 157 (1979)

)(10

)32)(12()0()1(

3

15

)32)(12()0()1(

3

1

)(10

)32)(12()0()1(

3

1

22

02

02

22

02

BBJJ

qJJJk

BJJ

qJJJk

BBJJ

qJJJk

c

b

a

Θabc



Specific haet Cp and entropy – first info about levels

Rln2



How to start analysis – the story of How to start analysis – the story of NdCuNdCu22

Magnetization: Kramers ground state doublet |+-> matrix elements

Module cfield can also calculate magnetization using a full set of CF parameters

P. Svoboda et al. JMMM 104 (1992) 1329

5.1/

8.2/

1.2/

,,

))2/()(tanh(

xJc

zJb

yJa

BB

Jgg

Jgg

Jgg

cba

kTMHggM



Neutron TOF spectroscopy – CF levels

... Blm


B20=1.35 K

B22=1.56 K

B40=0.0223 K

B42=0.0101 K

B44=0.0196 K

B60=4.89x10-4 K

B62=1.35x10-4 K

B64=4.89x10-4 K

B66=4.25 x10-3 K


The story of NdCuThe story of NdCu22 Thermal expansion – cf influence

... Magnetoelastic parameters (A=dB20/dε, B=dB2

2/dε)

E. Gratz et al., J. Phys.: Condens. Matter 5, 567 (1993)

Neutron diffraction+ magnetization: magstruc, phasediag H||b-> model

... Jbb


f(B) [arb.units] T=0K

B

F1

F2

F3

BcAF1 F3

Bc1

Bc2

Bc3

AF1

n(k)=sum of Jbb(ij) with ij being of bc plane k

M. Loewenhaupt et al., Z. Phys. B: Condens. Matter 101, 499 (1996)


NdCuNdCu22 Magnetic Phase Diagram Magnetic Phase Diagram

F3

AF1

F1

output file: mcphas.xytUse module phased or displaycontour for color plot of phasediagram

ab

c

F1

lines=experiment



Neutron spectroscopy on single crystals in H||b=3T

... Anisotropy of J(ij) - determination of Jaa=Jcc

F3

M. Rotter et al., Eur. Phys. J. B 14, 29 (2000)

Jaa=Jcc(R)

NdCuNdCu22F3

AF1

F1

M. Rotter, et al. Applied Phys. A 74 (2002) s751


Magnetostriction ... Confirmation of phasediagram model for H||a,b,c, and determination of dJ(ij)/dε

M. Rotter, et al. J. of Appl. Physics 91 10(2002) 8885


„„The Standard Model of Rare Earth Magnetism has been well The Standard Model of Rare Earth Magnetism has been well established and can describe the magnetic properties of Rare earth established and can describe the magnetic properties of Rare earth

compounds. There is no need for a program like McPhase.“compounds. There is no need for a program like McPhase.“

Nonsense !• In very few RE systems a large number of results of the SM have been

compared to experimental data: e.g. the full magneto-striction tensor has been analysed only in 1 case (NdCu2)

• Quadrupolar Excitations have not been compared to the SM

• There is a number of wrong predictions of the SM: e.g. -magnetoelastic paradoxon in L=0 AF-systems-extra magnetic modes or no modes (CeCu2, CeNi9Ge4,

Nd2CuO4),-wrong saturation moments, e.g. in Eu-Skutterudite- ...


Exchange-Striction

Orthorhombic Distortion

Standard Model of RE Mag... McPhase SimulationHH JJJJ

JJJJ

JJJJ

,)100(,)010(

)100()010(

)100()010(

~

))((

))((

TiiTiibbaa

iiiibbaa

iiiiibbaa

elex

B

A

EHH

?

The magnetoelasticThe magnetoelastic Paradoxon Paradoxon

for L=0for L=0demonstrateddemonstrated at at GdNi2B2C


McPhase - the World of Rare Earth MagnetismMcPhase is a program package for the calculation of

magnetic properties of rare earth based systems. Magnetization Magnetic Phasediagrams

Magnetic Structures Elastic/Inelastic/Diffuse Neutron Scattering

Cross Section


and much more....

Magnetostriction

Crystal Field/Magnetic/Orbital Excitations


McPhase runs on Linux and Windows and is available as freeware.

McPhase is being developed by M. Rotter, Institut für Physikalische Chemie, Universität Wien, Austria

M. Doerr, R. Schedler, Institut für Festkörperphysik, Technische Universität Dresden, Germany

P. Fabi né Hoffmann, Forschungszentrum Jülich, Germany S. Rotter, Wien, Austria

M.Banks, Max Planck Institute Stuttgart, Germany

Important Publications referencing McPhase: • M. Rotter, S. Kramp, M. Loewenhaupt, E. Gratz, W. Schmidt, N. M. Pyka, B. Hennion, R. v.d.Kamp Magnetic Excitations in the antiferromagnetic phase of NdCu2 Appl. Phys. A74 (2002) S751

• M. Rotter, M. Doerr, M. Loewenhaupt, P. Svoboda, Modeling Magnetostriction in RCu2 Compounds using McPhase J. of Applied Physics 91 (2002) 8885

• M. Rotter Using McPhase to calculate Magnetic Phase Diagrams of Rare Earth Compounds J. Magn. Magn. Mat. 272-276 (2004) 481

Epilog Epilog

http://www.ap.univie.ac.at/users/Martin.Rotter/

Prolog Numerical Modeling in Magnetism

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Transcript of Prolog Numerical Modeling in Magnetism