spin order in correlated electron systems yukawa institute for theoretical physics zhi li
TRANSCRIPT
Spin order in correlated electron systems
Yukawa Institute for Theoretical Physics
Zhi Li
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
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
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
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
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,)()(
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
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
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)
Summary
• This is a simple introduction about the spin order in correlated electron system.
1. The form of the wave function
2. DFT calculation