Temperature dependent electronic correlation effects in GdNsharma/Defense_Talk.pdf · Temperature dependent electronic correlation effects in GdN Anand Sharma Lehrstuhl Festkorpertheorie,

Temperature dependent electroniccorrelation effects in GdN

Anand Sharma

Lehrstuhl Festkorpertheorie, Institut fur PhysikHumboldt Universitat zu Berlin

December 20, 2007

Anand Sharma Temperature dependent correlation effects in GdN

IntroductionTheoretical evaluation

ab-initio calculationResults

Summary

Outline

1 Introduction

2 Theoretical evaluationGreen function methodMulti- band self- energyPhysical properties

3 ab-initio calculationKinetic energyIntra-atomic exchange

4 Results

5 Summary




Summary

Outline

1 Introduction



4 Results

5 Summary




Summary

Spintronics

It is a technology which exploits the quantum spin as well ascharge states of electrons

Spin- polarized electrontransport.Giant MagnetoResistance(GMR) effect

considered as birth of thistechnology.Prof. A.Fert (France) andProf. P.Grunberg (Germany)shared the 2007 Nobel prizein Physics.




Summary

Spintronics

It is a technology which exploits the quantum spin as well ascharge states of electrons

Spin- polarized electrontransport.Giant MagnetoResistance(GMR) effect

considered as birth of thistechnology.Prof. A.Fert (France) andProf. P.Grunberg (Germany)shared the 2007 Nobel prizein Physics.




Summary

Spintronics

ApplicationsRead heads in magnetic diskrecorders.”Non-volatile” memories.Magnetic tunnel junction.

FutureIn search of materials exploiting100% spin- polarization of chargecarriers. Thus giving a newdirection to the field of electronics.




Summary

Spintronics

ApplicationsRead heads in magnetic diskrecorders.”Non-volatile” memories.Magnetic tunnel junction.

FutureIn search of materials exploiting100% spin- polarization of chargecarriers. Thus giving a newdirection to the field of electronics.




Summary

Real Materials

Examples (of currently studied substances)Composite of thin films : magnetic semiconductors(Europium chalcogenides and Gadolinium pnictides) withtransition metals (Fe) and local moments (Gd).Family of Heusler alloys.Diluted Magnetic Semiconductors (DMS) : transition metaland rare earth doped III-V and II-VI semiconductors.Manganites like A1−xBxMnO3 where A=La,Pr,Nd andB=Sr,Ca,Ba,Pb).




Summary

Real Materials





Summary

Real Materials





Summary

Real Materials





Summary

Kondo Lattice model

Also known as s-d , s-f andferromagnetic Kondo lattice model

DefinitionIt describes a ferromagnetic spin- exchange interactionbetween band of itinerant electrons and a system of localizedspins.




Summary

Many body theoretical model

T αβij : electron hopping matrix

c†iασ, ciασ : electron creation andannihilation operatorsJ : intra- atomic exchangeSz

i ,Sσi = Sx

i + izσSyi : spin operators

z↑ = +1, z↓ = −1

Multi- band Hamiltonian

H = Hkin + Hint

=∑ijαβσ

T αβij c†iασcjβσ −

J2

∑iασ

(zσSzi c†iασciασ + Sσ

i c†iα−σciασ)




Summary

Many body theoretical model

T αβij : electron hopping matrix

c†iασ, ciασ : electron creation andannihilation operatorsJ : intra- atomic exchangeSz

i ,Sσi = Sx

i + izσSyi : spin operators

z↑ = +1, z↓ = −1

Multi- band Hamiltonian

H = Hkin + Hint

=∑ijαβσ

T αβij c†iασcjβσ −

J2

∑iασ

(zσSzi c†iασciασ + Sσ

i c†iα−σciασ)




Summary

Green function methodMulti- band self- energyPhysical properties

Outline

1 Introduction



4 Results

5 Summary




Summary


Analytical approach

Green function :

Gµνlmσ(E) = 〈〈clµσ; c†mνσ〉〉E

Equation of motion :

EGµνlmσ(E) = δlmδµν +

∑jβ

T µβlj Gβν

jmσ(E)− J2

[zσΓµνllmσ(E) + Fµν

llmσ(E)]

Higher order Green functions :

Ising function Γµνklmσ(E) = 〈〈Sz

k clµσ; c†mνσ〉〉E

Spin- flip function Fµνklmσ(E) = 〈〈S−σ

k clµ−σ; c†mνσ〉〉E




Summary


Analytical approach

The electronic self- energy can be understood as :

〈〈[clµσ,Hint ]; c†mνσ〉〉 =∑pγ

Σµγlpσ(E)Gγν

pmσ(E)

which gives the required solution :

Gkσ(E) = [(E + i0+)I − ε(k)− Σkσ(E)]−1

ProblemHow to obtain the self- energy ?Is it possible to guess from the limiting case of the model ?




Summary


Analytical approach



Σµγlpσ(E)Gγν

pmσ(E)


Gkσ(E) = [(E + i0+)I − ε(k)− Σkσ(E)]−1





Summary


Analytical approach



Σµγlpσ(E)Gγν

pmσ(E)


Gkσ(E) = [(E + i0+)I − ε(k)− Σkσ(E)]−1





Summary


Limiting case : T = 0 and empty bands

The spin- dependent (k-independent) form of self- energy is :

Σσ(E) = −JSzσ

2I

+J2S(zσ − 1)

4Gσ

(E − JSzσ

2

)[I − J

2Gσ

(E − JSzσ

2

)]−1

where

Gσ

(E − JSzσ

2

)=

1N

∑q

[(E − JSzσ

2

)I − ε(k-q)

]−1

Interpolating self- energy approach for one-band KLM[W.Nolting et al, PRB 64, 155109 (2001)]




Summary


Limiting case : T = 0 and empty bands

The spin- dependent (k-independent) form of self- energy is :

Σσ(E) = −JSzσ

2I

+J2S(zσ − 1)

4Gσ

(E − JSzσ

2

)[I − J

2Gσ

(E − JSzσ

2

)]−1

where

Gσ

(E − JSzσ

2

)=

1N

∑q

[(E − JSzσ

2

)I − ε(k-q)

]−1

Interpolating self- energy approach for one-band KLM[W.Nolting et al, PRB 64, 155109 (2001)]




Summary


Electronic self- energy

The multi- band self- energy ansatz :

Σσ(E) = −J2

Mσ I+J2

4aσGσ

(E−J

2Mσ

) [I − J

2Gσ

(E − J

2Mσ

)]−1

where

Mσ = zσ〈Sz〉; aσ = S(S + 1)−Mσ(Mσ + 1).

and the non- interacting Green function is :

Gσ(E) =1N

∑k

[(E + i0+)I − ε(k)]−1




Summary


If hopping matrix, ε(k), intra- atomic exchange, J and spinquantum number, S are known

Gσ(E) −→ Σσ(E) −→ Gkσ(E)

Quasi- particle spectral density :

Skσ(E) = −1π

ImTr(Gkσ(E))

Quasi- particle density of states :

ρσ(E) =1N

∑k

Skσ(E)




Summary

Kinetic energyIntra-atomic exchange

Outline

1 Introduction



4 Results

5 Summary




Summary


Density Functional Theory (DFT)

Two theorems by Hohenberg and Kohn [PR 136, B864 (1964)].

F [ρ(r)] = T [ρ(r)] +

∫Vext(r)ρ(r)d3r

+12

∫ρ(r)ρ(r′)| r− r′ |

d3rd3r′ + Exc[ρ(r)]

[−~2

2m4+ VKS(r)

]ψi(r) = εiψi(r)

VKS(r)[ρ(r)] = Vext(r) +

∫ρ(r′)

| r− r′ |d3r′ +

δExc[ρ(r)]δρ(r)

ρ(r) =∑

i

f (εi) | ψi(r) |2




Summary




F [ρ(r)] = T [ρ(r)] +

∫Vext(r)ρ(r)d3r

+12

∫ρ(r)ρ(r′)| r− r′ |


[−~2

2m4+ VKS(r)

]ψi(r) = εiψi(r)


∫ρ(r′)

| r− r′ |d3r′ +

δExc[ρ(r)]δρ(r)

ρ(r) =∑

i

f (εi) | ψi(r) |2




Summary




F [ρ(r)] = T [ρ(r)] +

∫Vext(r)ρ(r)d3r

+12

∫ρ(r)ρ(r′)| r− r′ |


[−~2

2m4+ VKS(r)

]ψi(r) = εiψi(r)


∫ρ(r′)

| r− r′ |d3r′ +

δExc[ρ(r)]δρ(r)

ρ(r) =∑

i

f (εi) | ψi(r) |2




Summary




F [ρ(r)] = T [ρ(r)] +

∫Vext(r)ρ(r)d3r

+12

∫ρ(r)ρ(r′)| r− r′ |


[−~2

2m4+ VKS(r)

]ψi(r) = εiψi(r)


∫ρ(r′)

| r− r′ |d3r′ +

δExc[ρ(r)]δρ(r)

ρ(r) =∑

i

f (εi) | ψi(r) |2




Summary


DFT-LDA and TB-LMTO-ASA

For Vext(r) : Muffin Tin (MT) potential

Vext(r) ≈ VMT = V (r - R) r ≤ R= V0 r > R

In Atomic Sphere Approximation (ASA), the muffin tinspheres are allowed to overlap with each other.The Tight Binding (TB) formalism minimizes this overlap.For Exc[ρ(r)] : Local Density Approximation (LDA)

Exc[ρ(r)] ≈ Ehomxc [ρ(r)]




Summary










Summary










Summary


GdN : experimental details

Electronic ground state is unconfirmed : Half- metal orsemiconductor ?Ferromagnet having Curie temperature ranging from 58 -90 K.4f7- local moments =⇒ S = 7

2NaCl structure with lattice constant, a = 4.99 A along withatomic positions act as input parameters in theTB-LMTO-ASA program :




Summary








Summary








Summary








Summary


T = 0 bandstructure (DFT-LDA)

The hopping matrix, ε(k) : GdN 5d conduction band




Summary


T = 0 density of states (DFT-LDA)




Summary


T = 0 density of states (DFT-LDA)

∆E = 1.237eV= JS

J = 0.35 eV




Summary

Outline

1 Introduction



4 Results

5 Summary




Summary

T = 0 density of states : LDA, ISA

↑ - spectrum is free fromcorrelation (absence of spin-flip processes) but is rigidlyshifted (JS

2 ).compensation of rigid shiftfulfills the exact (T = 0, σ =↑)limiting case and avoids thedouble counting of interaction.↓ - spectrum demonstratescorrelation effects even atT = 0 K.




Summary

Magnon emission :

Magnetic polaron :




Summary

Magnon emission :

Magnetic polaron :




Summary

Quasi- particle bandstructure




Summary





Summary





Summary





Summary

Quasi- particle spectral density




Summary





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Summary

Quasi- particle density of states

Redshift of opticalabsorption edge for anelectronic transition i.e.shift to lower energiesupon cooling fromT=Tc → T=0.




Summary

Red- shift

Theoretical [A.S and W.NoltingJPCM 18, 7337 (2006)] = 0.34 eVTheoretical [W.Lambrecht, PRB62, 13538 (2000)] = 0.30 eVTheoretical [S.Bhattacharjee andS.M.Jaya EPJB 49, 305 (2006)] =0.49 eVExperimental [F.Leuenberger etal PRB 73, 214430 (2006)] = 0.40eV




Summary

Red- shift





Summary

Red- shift





Summary

Red- shift





Summary

Outline

1 Introduction



4 Results

5 Summary




Summary

Summary

Multi- band self- energy ansatz is formulated.The kinetic (hopping) term is in the form of a matrix.The many body theoretical evaluation is combined withab-initio calculation.Quasi- particle spectral densities and density of states arecalculated.Temperature dependent strong electronic correlationeffects are observed.Theoretically calculated redshift energy is in closecomparison with experimentally measured value.




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Documents

Temperature dependent electronic correlation effects in GdNsharma/Defense_Talk.pdf · Temperature dependent electronic correlation effects in GdN Anand Sharma Lehrstuhl Festkorpertheorie,