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 order2. 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

rqrMrqrM

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

0110

,)()(

riqriq

rqi

rqirqirqi

xx eee

eeecrcrM

)cos(sincos rqrqmrqm yx

xm

cos 22yx mm 1221 xm 1221 iimy

)sin(0

0,)()(

22

12

22

12

rq

e

ei

ieecrcrM rqi

rqirqirqi

yy

zrqi

rqirq

irq

i

zz me

eeecrcrM

2211

22

12

22

12

1001

,)()(

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 matrix3. 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

rrrrgrrdrdrVdriqq

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

][)()(|'|)'()('][][

,

XCErrwdrrrrnrndrdrTE

iir )(

)()(][ XCLSDAXC rdrE

)()( rtrrn

][2

trJUEE 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

Ener

gy (R

y/Ce

ll)

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 function2. DFT calculation