Long-range antiferromagnetic interactions in ZnFe O and CdFe OLong-range antiferromagnetic interactions in ZnFe2O4 and CdFe2O4 빇쁒 CoContributors Supports 뫓ꑨ늦띾ꗍ뱂껊ꗍ:

LongLong--range range antiferromagneticantiferromagnetic interactions interactions in ZnFein ZnFe22OO44 and CdFeand CdFe22OO44

鄭靜鄭靜

CoContributors Supports碩士畢業生劉晉生 : Initial studies 國科會大學部學生葉學舫 : Diffraction 國家高速電腦中心

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OutlineOutline

FerritesFerritesDescribing interactionsDescribing interactionsPrevious research worksPrevious research worksWhat and how we can approach these systemsWhat and how we can approach these systemsCalculationalCalculational methodsmethodsQuantitative resultsQuantitative resultsProspectsProspects

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Cubic Cubic SpinelSpinel Ferrite : Ferrite : CompoundsCompounds

MFe2O4 MO‧Fe2O3

FeFe2O4 Fe3O4 magnetite lodestoneCoFe2O4NiFe2O4

ZnFe2O4CdFe2O4

Fe 4s23d6

Co 4s23d7

Ni 4s23d8

Zn 4s23d10

Cd 5s24d10

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Cubic Cubic SpinelSpinel Ferrite : Ferrite : Crystal StructureCrystal Structure

O : FCC

B : Octahedral sites

A : Tetrahedral sites

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Cubic Cubic SpinelSpinel Ferrite : Ferrite : Crystal StructureCrystal Structure

Tetrahedral (A) site

Octahedral (B) site

a

Oxygen FCC

8(MO Fe8(MO Fe22OO33))

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Cubic Cubic SpinelSpinel Ferrite : Ferrite : Magnetic StructureMagnetic Structure

2

3+

3B

+

+B

B site (Fe

B site (

A site5

F

(M )

e 5

)

)µ

µ↑ −

↓ −

normal spinelZnO．Fe2O3CdO．Fe2O3

Fe 4s23d6

3+

3+B

B+

B2+

2

B site (M 2

for

A s

B site (Fe 5 )

ite (Fe

N

+

i )

5 )

µ

µ

µ

↑ −

↓

↑ −

inverse spinelFeO．Fe2O3CoO．Fe2O3NiO．Fe2O3 Ni 4s23d8

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Cubic Cubic SpinelSpinel Ferrite : Ferrite : ZnOZnO．．FeFe22OO33

Antiferromagnetic insulators

Frustrated magnetic systemsFrustrated magnetic systems

8

( )( )

0

0

0

2 for 1define and

for 12

E E EE SE SJ E E

E E JE E J

↑↑ ↑↓↑↑

↑↓ ↑↓ ↑↑

↑↑

↑↓

≡ +↑↑ ↑ ≡ + ↑↓ ↓ ≡ −≡ −

= −⇒ ⇒

= +0 1 2E = E - JS S

Describing interactionsDescribing interactions

0 0

J ferroE NJJ antiferroE NJ> →↑↑↑↑↑ = −

< →↑↓↑↓↑ = +

K K

K K

112 i i

iH J S S += − ∑

9

Describing interactions (continued)Describing interactions (continued)

1 2

1 2

including up to 2nd neighbour interact

(( ) i

)on

E N J JE N J J

↑↑↑↑↑ = − −↑↓↑↓↑ = + −

K K

K K

12 n i i n

n i

H J S S += − ∑ ∑

10

Previous researchPrevious research

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Previous research (Continued)Previous research (Continued)ZnFe2O4

TN=10.5K (Konig et al SSC 1970)θCW ≈ +50K (Schiessl et al PRB 1996)θCW ≈ +120K ≈ + 10TN (Kamazawa et al PRB 2003)~ 4.2µB antiferromagnetic J3 (Yamada et al PRB 2002)ferromagnetic J1 for T>TN

CdFe2O4TN ≈ 13K ??? (Kamazawa et al PRB 2004)θCW ≈ -53KJ1 ≈ 5.9meV > 0 antiferromagnetic J1

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What and how we can approach these systemsWhat and how we can approach these systems

The extent of the FeThe extent of the Fe--Fe interactions Fe interactions JJnnthe nththe nth--neighbourneighbour interactionsinteractionsGenerate different configurationsGenerate different configurationsUsing Using IsingIsing model model –– colinearcolinear approachapproachPerform firstPerform first--principles calculationsprinciples calculationsSurprising results?Surprising results?The corresponding analysesThe corresponding analysesProspectsProspects

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J4

J4

J4

J3

J3

J3

J4

J

188.16BA368.16AB

68.5AA

127.05AA66.97BA126.97AB126.7BB

numberdn (Å)type

12∑ n i i+n

n , iH J S S= −

J2

J3

J2

J2

J2

J1

J1

J1

J1

J

126.0AA

85.5BA165.5AB

43.7AA63.5BA123.5AB

126.0BB

125.2BB

63.0BBnumberdn (Å)type

IsingIsing Model for cubic Model for cubic spinelspinel ferritesferrites

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Three different BThree different B--site site cationcation distributionsdistributions

the only theoretically considered configuration previously

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Ten considered configurationsTen considered configurations

ib1,ib2,ib3−0+ia1,ia2,ia3+0+

FeZnFeInversespinel

nb1,nb2,nb3−+0na1=na2=na3++0

FeFeZnNormalspinel

configurationB2B1Asite

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The nth The nth neighbourneighbour interactionsinteractions12∑ n i i+n

n , iH J S S= −

-6-6-6-3na

+2+2-2+1nb3-2+2+2-1nb2+2-6+2+1nb1J4J3J2J1

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OutlineOutline


18

ManyMany--electron systemselectron systems

{ }{ }( )

{ }{ }( )

222 2 22 2 1 1

2 2 2 2ii i j ie i j i

total

Z Z eZ ee r Rm m r r r R R R

E r R

α βαα α

α α α βα α β α

α

≠ ≠

− − ∇ + − ∇ + + + Ψ − − −

= Ψ

∑ ∑ ∑ ∑∑ ∑rh h r

r r rr r r

rr

{ }{ }( ) { }{ }( )2 2

2 1 ( , )2 2

Born-Oppenheimer approximation + Frozen-core approximation

i e ion i ei i j ie i j

e V r r E rm r r

R R Rα αα

α−≠

− ∇ + + Ψ = Ψ −

∑ ∑ ∑∑r rr r r

rrh

r

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Density Functional TheoryDensity Functional Theory

The of a many-electron system under the influence of some external potential is a

Hohenberg and Kohn 1

of the electron dens

96

i

4

ty ( )rρ

ground - state energy

unique functional r

3 3 312

( ) ( )[ ( )] [ ( )] ( ) ( )

[ ( )] : kinetic energy + the rest .... [ ( )]

: the exact form is unknown

extr rE r G r d r d r V r r d rr r

G rG r

ρ ρρ ρ ρ

ρρ

′= + +′−∫ ∫

r rr r r r

r r

r

r

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KohnKohn--Sham Equation : 1965Sham Equation : 1965

22 3( ){ ( ) [ ( )]} ( ) ( )

2 | |ext xc i i irV r d r r r r

m r rρ µ ρ φ ε φ

′′− ∇ + + + =

′−∫r

h r r rvr r

( ), ( )

effV r rρr r

1 4 442 4 4 43

[ ]

222

3

0

2

( )

[ ] [ ] [ ] [ ] ( ) ( )

[ ] : exchange-correlation energy functional...

( )0 subject to the condition ( ) 0

( )

the exact form is unknown

s xc s i i imi i

xc

r

G T E T r r

E

E rr d r

r ρ

ρ ρ ρ ρ φ φ ρ φ

ρ

δ ρδρ

δρ

= + = − ∇ =

= =

∑ ∑

∫

h

r

r r

rr

r

21

OutlineOutline


22

Numerical results : relative stability (eV/8formu.)Numerical results : relative stability (eV/8formu.)

3.73.73.73.7/1.53.7/1.43.7/1.53.83.93.83.9µ

0.00.90.44.14.95.21.92.72.24.3∆E

ib3ib2ib1ia3ia2ia1nb3nb2nb1naZn

3.73.73.73.83.93.83.83.93.84.0µ

0.01.91.96.97.78.30.81.71.03.5∆E

ib3ib2ib1ia3ia2ia1nb3nb2nb1naCd

The most stable configuration : inverse The most stable configuration : inverse spinelspinel????

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Numerical results : relative stability (Continued)Numerical results : relative stability (Continued)

ia1 ia2 ia3

4.14.14.14.24.24.24.2µ

1.682.682.18.00.14.18.54∆E

ib3ib2ib1nb3nb2nb1naZn

4.14.14.14.24.24.24.2µ

2.814.414.94.00.27.15.76∆E

ib3ib2ib1ia3ia2ia1nb3nb2nb1naCd

Including the onIncluding the on--site Coulomb interactionsite Coulomb interaction

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Numerical results : Numerical results : JJnn ((meVmeV))

-12.4-2.6-39.7PBE+U-12.3-2.7-39.6PBE+U-12.2-3.2-43.7PBE+U-23.6-19.0-220.4PBE4J34J24J1Zn

-6-6-6-3na+2+2-2+1nb3-2+2+2-1nb2+2-6+2+1nb1J4J3J2J1

JJ11*=(nb1*=(nb1--nb2)/2nb2)/2JJ22*=(nb1*=(nb1--nb3)/4nb3)/4

JJ11<0 J<0 J22<0 J<0 J33<0<0

|J|J22|<<|J|<<|J33||

+22.2+9.8+22.0+9.7+21.1+5.0+28.1-126.24J2*4J1*

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Numerical results : Numerical results : JJnn ((meVmeV))

-10.6-2.2-71.1PBE+U-21.8-18.7-275.9PBE4J34J24J1Cd

+18.9-28.9+24.8-188.84J2*4J1*

-12.3-2.7-39.6PBE+U-23.6-19.0-220.4PBE4J34J24J1Zn

+22.0+9.7+28.1-126.24J2*4J1*

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Numerical results : DOS Numerical results : DOS -- effect of Ueffect of U

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-8 -6 -4 -2 0 2

E-EF (eV)

N(E

)ZFO_nb3_GGA

Zn_AFe_B1Fe_B2O

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-8 -6 -4 -2 0 2

E-EF (eV)

N(E

)ZFO_nb3_GGA+U

Zn_AFe_B1Fe_B2O


29

-5 0-80

-40

0

40

80CFO_nb3

Zn_AFe_B1Fe_B2O

30

-5 0-80

-40

0

40

80CFO_nb3u

Zn_AFe_B1Fe_B2O

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-5 0-80

-40

0

40

80CFO_nb3u

Zn_AFe_B1Fe_B2O

-5 0-80

-40

0

40

80CFO_nb3

Zn_AFe_B1Fe_B2O


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Prospects:Prospects:

Diffraction Diffraction –– Low and High TLow and High TJJ11 FeFe--OO--Fe angleFe angleEffect of U Effect of U ---- JJnnDOS DOS –– difference in ZFO and CFOdifference in ZFO and CFOJJnn for AB interactionsfor AB interactions

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Phase diagram for the 3D electron gasPhase diagram for the 3D electron gas

D. CeperleyNature397_386 (1999)

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Magnetic properties of BN sheet with defectsMagnetic properties of BN sheet with defects

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Phase diagram for the 3D electron gasPhase diagram for the 3D electron gas

Thank you for your attention!Thank you for your attention!

That is all I have to say about That is all I have to say about ZnFeZnFe22OO44 and CdFeand CdFe22OO44

for the momentfor the moment

37

38

nn

nnn(111)

nn

39-5 0-80

-40

0

40

80ZFO_nb1

Zn_AFe_B1Fe_B2O

-5 0-80

-40

0

40

80ZFO_nb3

Zn_AFe_B1Fe_B2O

40-5 0-80

-40

0

40

80ZFO_ib3

Zn_B1Fe_AFe_B2O

-5 0-80

-40

0

40

80ZFO_nb3

Zn_AFe_B1Fe_B2O

41

nb1 nb2 nb3 ib1 ib3nb1 nb2 nb3 ib1 ib3

0.0

2.0

4.0

6.0PBEPBE+U(2.5)PBE+U(4.5)PW91PW91+U(4.5)


0.0

2.0

4.0

6.0 LDALDA+U(2.5)LDA+U(4.5)PW91PW91+U(4.5)


0.0

2.0

4.0

6.0 CFOPBEPBE+U(2.5)PBE+U(4.5)PW91PW91+U(4.5)

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Cubic Cubic SpinelSpinel Ferrite : Ferrite : Magnetic StructureMagnetic Structure

B : Octahedral sites

A : Tetrahedral sites

Documents

Long-range antiferromagnetic interactions in ZnFe O and CdFe OLong-range antiferromagnetic interactions in ZnFe2O4 and CdFe2O4 빇쁒 CoContributors Supports 뫓ꑨ늦띾ꗍ뱂껊ꗍ: