alexander yaresko- spin-spiral calculations of the magnetic properties of fe-based superconductors
TRANSCRIPT
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
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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
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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
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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