spin order in correlated electron systems yukawa institute for theoretical physics zhi li
TRANSCRIPT
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Spin order in correlated electron systems
Yukawa Institute for Theoretical Physics
Zhi Li
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Outline1. Variety of spin order in condensed matter
1.1 Collinear spin order
1.2 Non-collinear spin order
2. Theory and calculation
2.1 The form of wave function
2.2 Introduction about DFT calculation
2.3 Helical spin order in BaFeO3
3. Summary
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1. Variety of spin order
1.1 Collinear spin order
Ferromagnetism
Such as, Fe
Anti-ferromagnetism
Such as, NiO
Ferrimagnetism
Such as, Fe3O4,
yttrium iron garnet
Spin density wave state in Cr
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1. Variety of spin order
1.2 Non-collinear spin order
Example: Spiral spin (fcc Fe)
The position dependent spin moment could be written as
Vector q is the wave vector of the spiral spin order
)()(
)sin()(
)cos()(
rmrM
rqrM
rqrM
z
y
x
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2.1 The form of wave function If the electronic system is spin ordered, the form of the wave
function has a constraint.
A concrete example: Spiral spin
n, l : lattice and orbital index
Basis function could be plane wave, or Wannier function, or atomic orbital wave function.
nlnlaR
2
1
2
2
0
0)()()(
rqi
rqi
e
errUrc
nla
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2.1 The form of wave function
• Check the spin moment in this from of wave function:
1221
22
12
22
12
01
10,)()(
riqriq
rqi
rqirq
irq
i
xx ee
e
eeecrcrM
)cos(sincos rqrqmrqm yx
xm
cos22yx mm 1221 xm 1221 iimy
)sin(0
0,)()(
22
12
22
12
rq
e
ei
ieecrcrM rq
i
rqirq
irq
i
yy
zrqi
rqirq
irq
i
zz m
e
eeecrcrM
2211
22
12
22
12
10
01,)()(
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2.1 The form of wave function
• Many-body Hamiltonian:• Total energy
note:
1. The wave vector q is coupled to the momentum
2. Generally speaking, the interacting term should be in form of matrix
3. It is almost impossible to get the wave vector q by analytical method
)()()'()'()'()'()'()()('
))()(()(2
)()( 22
rrUrrUrrgrUrrUrdrdr
rrUrVm
rUrdrE
)()'()'()'()('
4
4)(2
22
22
2
12
11
2
2
rrrrgrrdrdrVdriq
q
iqq
rdrm
E
ip
ij
jiii rrgrVm
H )()](2
[ 22
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2.2 Introduction about DFT Calculation
• The first-principles calculation based on density functional theory (DFT)
The total energy of system is in the functional of density matrix expressed in single electronic state
Single electron equation
][)()(|'|
)'()('][][
,
XCErrwdrrr
rnrndrdrTE
iir )(
)()(][ XCLSDAXC rdrE
)()( rtrrn
][2
trJU
EE DFTUDFT
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2.3 Helical spin order in BaFeO3
Wave vector dependent total energy
0.0 0.1 0.2 0.3 0.4 0.5
-0.00015
-0.00010
-0.00005
0.00000
0.00005
0.00010
0.00015
0.00020
0.00025
Ene
rgy
(Ry/
Cel
l)
q
U-0 U-0.1Ry U-0.3Ry U-0.3Ry-G U-0.4Ry U-0.5RY-G U-0.5Ry
G: q(1,1,1)
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Summary
• This is a simple introduction about the spin order in correlated electron system.
1. The form of the wave function
2. DFT calculation