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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