APCTPAPCTP--POSTECH AMS Workshop 2010POSTECH AMS Workshop 2010

Ferromagnetism of Strongly Ferromagnetism of Strongly CCorrelatedorrelatedEEl il i SS i h Di di h Di dEElectronic lectronic SSystems with Disorder ystems with Disorder

Unjong YuUnjong YuS h lS h l f G l St dif G l St diSchoolSchool of General Studiesof General Studies

Gwangju Institute of Science & Technology (GIST)Gwangju Institute of Science & Technology (GIST)gj f gy ( )gj f gy ( )

CollaboratorsCollaborators

D. Galanakis, J. Moreno, M. Jarrell : Louisiana St. U.

A M Nili P S U i i A.-M. Nili : Penn. State University

K Mikelsons J K Freericks : Georgetown Univ K. Mikelsons, J. K. Freericks : Georgetown Univ.

B. Moritz : Stanford Universityf y

D. Vollhardt : U. Augsburg, Germany

K. Byczuk : U. Warsaw, Poland

R. S. Fishman : Oak Ridge National Laboratory

ContentsContents Introduction Introduction

Dynamical mean-field theory (DMFT)

Dynamical cluster approximation (DCA)

Main body

i i i i A i i Magnetism in the periodic Anderson model with disorder

[U. Yu et al., Phys. Rev. Lett. 100, 246401 (2008)][U. Yu et al., Phys. Rev. Lett. 100, 246401 (2008)]

Magnetism in diluted magnetic semiconductors

[U. Yu et al., Phys. Rev. Lett. 104, 037201 (2010)]

Summary

Strongly Correlated SystemsStrongly Correlated Systems

Kinetic energy Electronic correlation

Disorder

Method of solution :Method of solution : Dynamical Mean-Field Theory (DMFT)& Dynamical Cluster Approximation (DCA)

Real system DMFT

Review : Th. Maier et al. Rev. Mod. Phys. 77, 1027 (2005)

Real system DCA DMFT

Review : Th. Maier et al. Rev. Mod. Phys. 77, 1027 (2005)

Real system DCA DMFA MFAReal system DCA DMFApa

ceMFA

Rea

l sp

R

∑( k) ∑( )∑(ω K) ∑

(k)

∑(ω,k) ∑(ω)∑(ω,K) ∑

entu

m

Mom

e

F ( ) F ( ) F ( ) F ( )M Frequency (ω) Frequency (ω) Frequency (ω) Frequency (ω)

Strongl ti

Thermodynamicli it

Non-local effectscorrelation limit effects

Mean-field theory X ☺ XMean field theory X ☺ XQuantum Monte-Carlo,

☺ X ☺Exact diagonalization ☺ X ☺

D i l fi ld ☺ ☺ XDynamical mean-field approx. ☺ ☺ X

☺ ☺ ☺Dynamical cluster approx. ☺ ☺ ☺

Self-consistency loop of DMFT & DCASelf-consistency loop of DMFT & DCA

IMPURITY SOLVERCLUSTER SOLVERCLUSTER SOLVER

GCC0

GC0

GG

SELF CONSISTENCY

Disorder in DMFT : CPA levelDisorder in DMFT : CPA levelmean-field

A B

mean field

BAC GxGxG )1( Green function : BAC )(

S lf Self-energy : …

Exact at D ∞

Disorder in DCA : higher than CPADisorder in DCA : higher than CPA

N

n

nC

c

nGnPG

cluster in theA of # : )(0

)( Green function :

nNnnN

n

c

cxxCnP

)1()(0

Self energy : Self-energy : …CPA

Magnetism in the periodic Anderson Model

with Disorder

Periodic Anderson modelPeriodic Anderson model Periodic Anderson model

c

icf

if

jicij nncctH

Periodic Anderson modelUf

iif

if

i

iij

fcVnnU H.c.c

ii

Canonical model forf

Canonical model for Heavy fermion systems

Mixed-valence materials

K d i l t Kondo insulators

Local moment magnetsg

Ferromagnetism in the PAMFerromagnetism in the PAMKondo regimeKondo regime

Uf U=1.5n=1 3e

f

chemical potential

n 1.3

ratu

rf

mpe

rTe

m

V

FM in the periodic Anderson modelFM in the periodic Anderson model

FM

A N Tahvildar-Zadeh et al Phys Rev B 55 R3332 (1997)A. N. Tahvildar Zadeh et al. Phys. Rev. B 55, R3332 (1997)U. Yu et al. Phys. Rev. B 78, 205118 (2008)

Disordered PAMDisordered PAM ffccffc fVUH H

iii

i

fi

fi

i

ci

cfi

f

ijji

cij fcVnnUnncctH H.c.

Fe Co Ni

Ru Rh Pd

S S k t lS. Sakarya et al., arXiv:cond-mat/0609557

Disorder in ε f or εcDisorder in ε or εc

• Disorder in c• Disorder in f

253

3.1fc

totn

25.300 f

01

02

U. Yu et al., Phys. Rev. Lett. 2008

Disorder in ε f or εcDisorder in ε or εc

• Disorder in c• Disorder in fAlloy Kondoinsulatorinsulator

Uf Uf

f

U. Yu et al., Phys. Rev. Lett. 2008

How T is determinedHow Tc is determined Periodic Anderson model

iii

i

fi

fi

i

ci

cfi

f

ijji

cij fcVnnUnncctH H.c.

Periodic Anderson model

iiiij

Kondo lattice model (Schrieffer-Wolff transformation)

iii

ijji

cij SJcctH

eff filling-halfnear ~with 2eff UVJ

jiij SSJH

Heisenberg model (RKKY approximation)

|)(|)(with 2ff rrFJJ

ji

jiij SSJH,

|)(|)(with eff jiFcij rrFJJ

)1(SS Transition temperature (Weiss mean-field approximation)

)()1()()()1()1( 202eff

cc

Fcijc FUSSrFSSJSSJT

)()( ccff FFT )()( ffc FFT

)()( ccffc FFT

)()1()()( 21ccccff

c FxFxFT

6.05.1

VU

Magnetism in the periodic Anderson Model

with Disorder

Large FM phase region away from half-filling

Disorder in conduction band energy may enhance Tc

Kondo-insulator behavior at non-integer filling

Magnetism in diluted magnetic semiconductors

Diluted magnetic semiconductorDiluted magnetic semiconductor

Ferromagnet Diluted magneticsemiconductor Non-magnetg semiconductor g

spin-polarizedconduction electronsMagnetism in DMS conduction electrons

MnMn

Spintronics : Spin-electronicsSpintronics : Spin-electronicsSpintronicsp

Magnetics ElectronicsMagnetics Electronics

Magnetic material Semiconductor

Diluted magnetic semiconductorCPU+memory on one chip

Faster }Smaller computerEnergy-efficient

}( Quantum computer ? )

Ga Mn AsGa1–xMnxAs

Tc

Ga3+1–xMn2+

xAs3–Saturation magnetization

3d5 (spin = 5/2)

S u o g e o(T = 0)

K Y Wang et al (2005)K. Y. Wang et al., (2005)S. J. Potashnik et al., Phys. Rev. B (2002)

Magnetic anisotropy (I)Magnetic anisotropy (I)Compressive strain Tensile strainCompressive strain Tensile strain

G M AGa1-xMnxAs

Substrate

In-plane anisotropy Out-of-plane anisotropy

Magnetic anisotropy (II)Magnetic anisotropy (II)

SQUID

[100] [110][100] [110]

K.-Y. Wang et al., PRL (2005)M. Sawicki et al., PRB (2004)

Magnetic anisotropy (III)Magnetic anisotropy (III)

Resonant nanoelectromechanical systemMagneto-optic image & SQUID

[100][100][110]

[110][100] [110][100]

U. Welp et al., PRL (2003) S. C. Masmanidis et al., PRL (2005)

Magnetic anisotropy in (GaMn)As Tensile strain Out-of-plane anisotropy

Magnetic anisotropy in (GaMn)As(GaMn)As Tensile strain Out of plane anisotropy

Compressive strain In plane anisotropysubstrate

(GaMn)As

Compressive strain In-plane anisotropy(spin reorientation) (GaMn)As

[100]substrate

[110][110]

Anisotropic interaction in Mn-dimersAnisotropic interaction in Mn-dimers

STMSTM topography

Accepter-levelsplittingenergy

D. Kitchen et al., Nature (2006)

Theoretical difficulties in (GaMn)AsTheoretical difficulties in (GaMn)As

Strongly-correlated system

Strongly-disordered system

Non-local effects are important.Non local effects are important.

Large spin orbit coupling Large spin-orbit coupling

Method of solution: Dynamical cluster approximation

+ k th d+ kp method

Model for (GaAs)MnModel for (GaAs)Mn

non-magnetic ion (Ga)

ti i (M )Carrier spin

i i ( )

magnetic ion (Mn)

SJHH

Localized spin (Mn)

I

IIcpk SJHH

Mn site

Ga1-xMnxAs Exchange interaction (1 eV)Spin-orbit coupling& strain effects

Determination of J valueDetermination of Jc value

Resonant tunneling spec. 0.6 eV [1]Photoemission spec. 1.2 eV [2]Magneto-transport 1.5 eV [3]

0 9 eV [4]IR spectroscopy

0.9 eV [4]1 eV [5]

Optical transmission 1.0 eV [6]

[1] H Oh t l APL 73 363 (1998)[1] H. Ohno et al., APL 73, 363 (1998).[2] J. Okabayashi et al., PRB 58, R4211 (1998).[3] T. Omiya et al., Physica E 7, 976 (2000).

Jc = 1 eV[ ] y , y , ( )[4] M. Linnarsson et al., PRB 55, 6938 (1997).[5] E. J. Singley et al., PRB 68, 165204 (2003).[6] J. Szczytko et al., PRB 64, 075306 (2001).

Results: Magnetization and TResults: Magnetization and Tc

Mn concentration : x = 5 % hole concentration : p = 2.5%

Mn concentration : x = 5 %

Experiments

Results: Magnetic anisotropyResults: Magnetic anisotropyCompressive strain (GaMn)Asz Compressive strain(─0.2%) substrate

(GaMn)Aszθ M

x[110]Φy

x

Tensile strain(+0 2%)

(GaMn)As

(+0.2%) substrate

Results: Magnetic anisotropyResults: Magnetic anisotropy[110]

C i t i[100]

(GaMn)As

Compressive strain

substrateCalculation (DCA)

zθ Mθ

Φ

Mx

Experiments [110]Φy

Monomer Dimer TrimerMonomer Dimer Trimer

…

GGaMn

Cluster anisotropyCluster anisotropy

MMdimer

R

G. Zarand & B. Janko, PRL 2002

Magnetization of dimersMagnetization of dimers

NN-dimers

T = 23.2 K

NNN-dimers

dimer configurations

Cluster anisotropy is valid only for nearest neighbor Mn-pairs.

Magnetic frustrationMagnetic frustration

GaMnMn

t tM

Magnetic frustration

d t ti ti ti

totM

reduces saturation magnetization

reduces Tc

Cluster anisotropyCluster anisotropy

Φ=0°

Φ=45°

Spin-reorientationSpin-reorientationCl t i tCluster anisotropy

Crystal anisotropy

SummarySummary

Magnetism in diluted magnetic semiconductors

D i l l t i ti d i t l Dynamical cluster approximation reproduces experimental

values of Tc, saturation magnetization, magnetic anisotropy, c, g , g py,

spin re-orientation of diluted magnetic semiconductors.

Outlook

Tight-binding + DMFT

Tight-binding + DCA

Tight-binding + DCA in nano-structures

How to select clustersAppendix

How to select clusters

123 1

2 3

3 1 3 21 1

4444

3 1 3 2

4 2 4

1

34

2 1

3

Imperfection=1

Infinite size lattice Cluster 1 Cluster 2

p

Nearest Neighbor 4 2 3Next-Nearest-Neighbor 4 1 03rd–Nearest-Neighbor 4 0 0

How to select clustersHow to select clusters

Strategy to select clusters1. Smaller imperfection

2. Squareness near 1

(Squareness : )2 ddll(Squareness : )2121 2 ddll

l l 1d1l 2l

d

1d

2dD.D. Betts et al. Can. J. Phys. 74, 54 (1996)

Example: AFM Heisenberg model in 2D square lattice(Lanczos exact-diagonalization)

Dagotto & Moreo PRB (1988)

Dagotto & Moreog &Barnes & SwansonReger & Young

Liang

Trivedi & CeperleyGross et al.Tao

Results from “good” clusters

TaoOtimaa et al.

Results from good clusters