FESC 2010 1

Spin-spiral calculations of the magnetic properties of Fe- basedsuperconductors

Alexander Yaresko

Max Planck Institute for Solid State Research, Stuttgart, Germany

FESC 2010

May 11, 2010

in collaboration with

FESC 2010 2

Guo-Qiang Liu, Viktor Antonov, and Ole Andersen

MPI FKF

special thanks to:Lilia Boeri, George Jackeli, Wei Ku

for helpful discussions

overview

FESC 2010 3

• band structure, nesting, and susceptibility

• q-dependence of the total energy from spin-spiral calculations

◦ effect of electron and hole doping

◦ applicability of the Heisenberg model

◦ interlayer coupling

• conclusions

new class of unconventional superonductors

FESC 2010 4

H.Luetkens,et al Nature Mat.8, 305 (2009)

0.0 0.2 0.4 0.6 0.8 1.00

30

60

90

120

150

T(K)

x in Ba1-xKxFe2As2

Ts

TC

SC

SDW

H. Chen, et al, EPL 85, 17006 (2008)

• undoped compounds: orthorombic distortions + AFM order

• when doping or pressure suppress magnetic order, superconductivitycomes into play (TC>50K)

• LiFeAs: nonmagnetic with (TC ≈18K) J. Tapp, et al, PRB 78, 060505 (2008)

• electron-phonon coupling seems to be too weak to explain high TC

L. Boeri, et al, PRL 101, 026403 (2008)

it is important to understand magnetic properties!

crystal structure

FESC 2010 5

LaFeAsO

P4/nmm (1111)

La

Fe

As

O

MFe2As2 M=Sr,Ba

I4/mmm (122)

Ba

Fe

As

LiFeAsP4/nmm (111)

Li

Fe

As

• Fe is surrounded by slightly distorted As4 tetrahedra• Fe ions form a square lattice

• FeAs layers are separated by a LaO, M or Li layer(s)

two limits

FESC 2010 6

weak magnetic perturbation

⇓noninteracting susceptibility

χ0(q, ω)

strong magnetic perturbation

⇓self-consistent calculations for

spin spirals

? do the two limits always lead to the same conclusions

? how electron and hole doping affect the magnetic properties

? is stripe AFM order the ground state for doped compounds

? can the j1–j2 Heisenberg model describe the calculated E(q)? similarities and differences between various families compounds

two limits

FESC 2010 6

weak magnetic perturbation

⇓noninteracting susceptibility

χ0(q, ω)

strong magnetic perturbation

⇓self-consistent calculations for

spin spirals

? do the two limits always lead to the same conclusions

? how electron and hole doping affect the magnetic properties

? is stripe AFM order the ground state for doped compounds

? can the j1–j2 Heisenberg model describe the calculated E(q)? similarities and differences between various families compounds

details:• L(S)DA calculations were performed using the LMTO method

• for experimental high-temperature tetragonal crystal structures

orthorhombic distortions were neglected

• electron (δ > 0) or hole (δ < 0) doping per Fe ion was simulated by usingthe virtual crystal approximation

“ t2” bands for BaFe 2As2

FESC 2010 7

-2

-1

0

1

Ene

rgy

(eV

)

T N P T M

xy yz,xz

Γ

M

P

X

N T

T

P

N

three (2×dyz,zx; 1×dxy) hole-like pokets around Γand two electron-like (2×dyz,zx+dxy) pokets around X

and for LiFeAs

FESC 2010 8

xy yz,xz

-2

-1

0

1

Ene

rgy

(eV

)

Γ M X Γ Z

ΓX

M

R Z

A

Γ

M

X

• similar hole-like and electron-like pockets

• the size of the dxy FS is larger than dyz,zx ones ⇒ less effective nesting

• the width of dxy and dyz,zx bands is larger than in BaFe2As2

(dFe-Fe=2.67 A vs. 2.80 A)?

noninteracting susceptibility for LaFeAsO

FESC 2010 9

δ=0

50

60

70

80

90

Re

χ(0)

(arb

.uni

ts)

Γ X_

M_

Γ

0

10

20

30

40

50

Imχ(

ω0)

/ω(a

rb.u

nits

) peak at X due to FS nesting

Γ

M=X_

X M_

J. Dong, et al, EPL 83, 27006

(2008), I. Mazin, et al, PRL 101,

057003 (2008). . .

noninteracting susceptibility for LaFeAsO

FESC 2010 9

δ=0

50

60

70

80

90

Re

χ(0)

(arb

.uni

ts)

Γ X_

M_

Γ

0

10

20

30

40

50

Imχ(

ω0)

/ω(a

rb.u

nits

) peak at X due to FS nesting

Γ

M=X_

X M_

J. Dong, et al, EPL 83, 27006

(2008), I. Mazin, et al, PRL 101,

057003 (2008). . .

for LaFeAsO 1−xFx

FESC 2010 10

δ=0δ=0.1δ=0.2

50

60

70

80

90

Re

χ(0)

(arb

.uni

ts)

Γ X_

M_

Γ

0

10

20

30

40

50

Imχ(

ω0)

/ω(a

rb.u

nits

)

• electron doping sup-

presses the X peak

• maximum shifts to an

incommensurate q

δ=0.2 (x=0.2)

Γ

M=X_

X M_

for LaFeAsO 1−xFx

FESC 2010 10

δ=0δ=0.1δ=0.2

50

60

70

80

90

Re

χ(0)

(arb

.uni

ts)

Γ X_

M_

Γ

0

10

20

30

40

50

Imχ(

ω0)

/ω(a

rb.u

nits

)

• electron doping sup-

presses the X peak

• maximum shifts to an

incommensurate q

• at which FS touch

δ=0.2 (x=0.2)

Γ

M=X_

X M_

. . . BaFe2As2

FESC 2010 11

δ=0

50

60

70

80

Re

χ(0)

(arb

.uni

ts)

Γ X_

M_

Γ

0

10

20

30

Imχ(

ω0)

/ω(a

rb.u

nits

)

Γ

X=X_

M_

P=X_

N M_

. . . Ba1−xKxFe2As2 (x = 2|δ|)

FESC 2010 12

δ=0δ=-0.1δ=-0.2

50

60

70

80

Re

χ(0)

(arb

.uni

ts)

Γ X_

M_

Γ

0

10

20

30

Imχ(

ω0)

/ω(a

rb.u

nits

)

• hole doping also suppresses

the peak at X• the maximum shifts → Γ

The peak of Reχ(q) at X is

suppressed by either electron or

hole doping

. . . LiFeAs

FESC 2010 13

50

60

70

80

Re

χ(0)

(arb

.uni

ts)

Γ X_

M_

Γ

0

10

20

30

Imχ(

ω0)

/ω(a

rb.u

nits

)

Γ

M

X

• nesing is less effective

• χ(q) is lower

• the X peak is split alreadyin undoped LiFeAs

response to finite AFM perturbation

FESC 2010 14

BaFe2As2

AFM || x; FM || y

xyyzxz3z2-r2

x2-y2

MFe/5

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4∆=V -V (eV)

0.0

0.1

0.2

0.3

0.4

M(µ

B)

LiFeAs

AFM || x; FM || y

xyyzxz3z2-1x2-y2

MFe/5

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4∆=V -V (eV)

0.0

0.1

0.2

0.3

0.4

M(µ

B)

induced moment M vs. ∆ = V↑ − V↓

• at small ∆ the moment is induced faster on obitals forming nested FS

BaFe2As2: dMdxy/d∆ = 0.60 ≫ dMd

x2−y2

/d∆ = 0.18

LiFeAs: dMdxy/d∆ = 0.45 ≫ dMd

x2−y2

/d∆ = 0.12• the degeneracy of dzx and dyz states is lifted

• at large ∆ nesting is less important: dMdxy/d∆=dMd

x2−y2

/d∆=0.18

What happens when Fe moments are large?

self-consistent calculations for spin spiral

FESC 2010 15

X→ Γ; q = (q, 0), q=0.875

EH ∼ (j1 + 2j2) cos(πq) + j1

X→ M; q = (1, q), q=0.125

EH ∼ (j1 − 2j2) cos(πq) − j1

EH is a monotonic function of q

self-consistent calculations for spin spiral

FESC 2010 15

X→ Γ; q = (q, 0), q=0.

EH ∼ (j1 + 2j2) cos(πq) + j1

X→ M; q = (1, q), q=1

EH ∼ (j1 − 2j2) cos(πq) − j1

EH is a monotonic function of q

spin spirals along Γ–M

FESC 2010 16

Γ→ M; q = (q, q), q=0.125

EH ∼ 2j1 cos(πq) + 2j2 cos2(πq)

a local minimum appears if j2 > j1/2

LaFeAsO

FESC 2010 17

-100

-50

0

E(q

)-E

(0)

(meV

/Fe)

Γ_

X_

M_

Γ_

(0,0) (π,0) (π,π) (0,0)

0.0

0.5

1.0

1.5

MF

e(µ

B)

• the E(q) minimum is at X(stripe AFM order)

• E(q) is reasonably well describedby the j1–j2 Heisenberg model

j1=81 meV, j2=57 meV, j2/j1=0.71

• nonmagnetic solution at small |q|

a

electron doping in LaFeAsO 1−xFx (x = δ)

FESC 2010 18

-100

-50

0

E(q

)-E

(0)

(meV

/Fe)

Γ_

X_

M_

Γ_

δ=0δ=0.1δ=0.2δ=0.3

(0,0) (π,0) (π,π) (0,0)

0.0

0.5

1.0

1.5

MF

e(µ

B)

• deviation from the Heisenberg-likebehaviour increases with doping

• a new minimum at the X–M line

electron doping in LaFeAsO 1−xFx (x = δ)

FESC 2010 18

-100

-50

0

E(q

)-E

(0)

(meV

/Fe)

Γ_

X_

M_

Γ_

δ=0δ=0.1δ=0.2δ=0.3

(0,0) (π,0) (π,π) (0,0)

0.0

0.5

1.0

1.5

MF

e(µ

B)

• deviation from the Heisenberg-likebehaviour increases with doping

• a new minimum at the X–M line

stripe AFM order becomes unstable

already at small δ

band-structure effect?

FESC 2010 19

LaFeAsO; δ=0

dyzdxzdxy

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5Energy (eV)

0

1

2

DO

S(1

/eV

/ato

m)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Ene

rgy

(eV

)

Y Γ X

dyz dyz

LaFeAsO1−xFx; δ=0.1

q=(1,0.0)

-1.0 -0.5 0.0 0.5 1.0 1.5Energy (eV)

0

5

10

15

20

DO

S(s

tate

s/eV

)

a narrow Fe dyz↓ DOS peak just above EF is filled when LaFeAsO is doped

with electrons

band-structure effect?

FESC 2010 19

LaFeAsO; δ=0

dyzdxzdxy

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5Energy (eV)

0

1

2

DO

S(1

/eV

/ato

m)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Ene

rgy

(eV

)

Y Γ X

dyz dyz

LaFeAsO1−xFx; δ=0.1

q=(1,0.0)q=(1,0.05)q=(1,0.10)

-1.0 -0.5 0.0 0.5 1.0 1.5Energy (eV)

0

5

10

15

20

DO

S(s

tate

s/eV

)

a narrow Fe dyz↓ DOS peak just above EF is filled when LaFeAsO is doped

with electrons The peak splits at incommensurate q

BaFe2As2

FESC 2010 20

-50

0

E(q

)-E

(0)

(meV

/Fe)

Γ_

X_

M_

Γ_

(0,0) (π,0) (π,π) (0,0)

0.0

0.5

1.0

1.5

MF

e(µ

B)

• the minimum is at X• E(q) is Heisenberg-like

j1=95 meV; j2=73 meV; j2/j1=0.77

hole doping in Ba 1−xKxFe2As2 (x = 2|δ|)

FESC 2010 21

-50

0

E(q

)-E

(0)

(meV

/Fe)

Γ_

X_

M_

Γ_

δ=0δ=-0.1δ=-0.2δ=-0.3

(0,0) (π,0) (π,π) (0,0)

0.0

0.5

1.0

1.5

MF

e(µ

B)

• the E(q) minimum stays at X up

to x =0.6 (δ = −0.3)

• the stabilization energy

|E(X) − E(0)|decreases with doping

0.0 0.2 0.4 0.6 0.8 1.00

30

60

90

120

150

T(K)

x in Ba1-xKxFe2As2

Ts

TC

SC

SDW

hole doping in Ba 1−xKxFe2As2 (x = 2|δ|)

FESC 2010 21

-50

0

E(q

)-E

(0)

(meV

/Fe)

Γ_

X_

M_

Γ_

δ=0δ=-0.1δ=-0.2δ=-0.3

(0,0) (π,0) (π,π) (0,0)

0.0

0.5

1.0

1.5

MF

e(µ

B)

• the E(q) minimum stays at X up

to x =0.6 (δ = −0.3)

• the stabilization energy

|E(X) − E(0)|decreases with doping

stripe AFM order is more stable

0.0 0.2 0.4 0.6 0.8 1.00

30

60

90

120

150

T(K)

x in Ba1-xKxFe2As2

Ts

TC

SC

SDW

hole vs. electron doping in MFe 2As2 (M=Ba, Sr)

FESC 2010 22

hole doping (δ < 0): M1−xKxFe2As2 (x = 2|δ| )

electron doping (δ > 0): M(Fe1−xCox)2As2 (x = δ)

BaFe2As2 qz=π

-50

0

E(q

)-E

(0)

(meV

/Fe)

Γ_

X_

M_

Γ_

SrFe2As2

δ=-0.1δ=0δ=0.1

Γ_

X_

M_

Γ_

already weak electron doping (δ < 0.1) destabilizes stripe AFM order

magnetic transition in Ba(Fe1−xCox)2As2 is suppressed at x ≈ 0.06J. Chu, et al PRB 79, 014506 (2009)

LiFeAs

FESC 2010 23

-50

0

E(q

)-E

(0)

(meV

/Fe)

Γ_

X_

M_

Γ_

(0,0) (π,0) (π,π) (0,0)

0.0

0.5

1.0

1.5

MF

e(µ

B)

• Fe moment is much smaller

• magnetic solution exists in a

narrow range around X• stabilization energy is only 12

meV/Fe

The loss of kinetic energy is higher

because of compressed lattice?

does the Heisenberg model work for undoped compounds?

FESC 2010 24

q=(π,0):

two decoupled AFM sublattices

each Fe has two FM and two AFM nn

EH = j1[cosα + cos(π − α)] − 2j2

= −2j2 = const.

does the Heisenberg model work for undoped compounds?

FESC 2010 24

q=(π,0):

two decoupled AFM sublattices

each Fe has two FM and two AFM nn

EH = j1[cosα + cos(π − α)] − 2j2

= −2j2 = const.

does the Heisenberg model work for undoped compounds?

FESC 2010 24

q=(π,0):

two decoupled AFM sublattices

each Fe has two FM and two AFM nn

EH = j1[cosα + cos(π − α)] − 2j2

= −2j2 = const.

NO!

FESC 2010 25

La δ=0La δ=0.2

0 45 90 135 180α (deg)

0

10

20

30

40

50

E(α

)-E

(0)

(meV

)

Ba δ=0Ba δ=-0.2

0 45 90 135 180α (deg)

• already LSDA prefers collinear stripe AFM order

• E(α) ∼ sin2 α ∼ −(Si · Sj)2

• energy variation is suppressed by doping, especially in BaFe2As2

orbital degrees of freedom?

FESC 2010 26

variation of orbital occupations as a function of α

yzzxxy

0 45 90 135 180α (deg)

yzzxxy

0 45 90 135 180α (deg)

0.5

0.6

0.7

0.8n

stripe AFM order lifts the degeneracy of dyz and dzx orbitals (δn ∼0.15)

is this the reason for anisotropic j||1

and j⊥1

?

Z. Yin, et al PRL 101, 047001 (2008)

interlayer exchange coupling

FESC 2010 27

stripe AFM order in the ab-plane

Ba

Fe

As

FMc: qz = 0, α = 0AFMc: qz = π, α = 180

BaFe2As2LaFeAsO

0 90 180α (deg)

-4

-3

-2

-1

0

E(α

)-E

(0)

(meV

/Fe)

E(FMc) − E(AFMc) ≈ 4 meV/Fe

strong AFM interlayer coupling inMFe2As2

conclusions

FESC 2010 28

• In the undoped compounds the minimum of the total energy is found at

q=(π,0) corresponding to stripe AFM order

• LaFeAsO1−xFx (δ > 0): the minimum shifts to incommensurate wave

vectors with doping and stripe AFM order becomes unstable

• M1−xKxFe2As2 (δ < 0): stripe AFM order remains stable in a wide range of

K concentrations. However, the stabilization energy decreases with doping• M(Fe1−xCox)2As2 (δ > 0): stripe AFM order is rapidly destabilized when

FeAs layer is doped with electrons

• LiFeAs: the Fe moment and the stabilization energy are much smaller

• strong dependence of the energy on α shows that the magnetic interactions

between Fe spins cannot be described by the simple j1–j2 Heisenbergmodel

• the interlayer coupling in MFe2As2 is much stronger than in LaFeAsO