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УДК 538.945
Band Structure Modification Due to the Spin-orbit Coupling in the Three-orbital Model for Iron Pnictides
Maxim M. Korshunov*
Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Kirensky Institute of Physics Federal Research Center KSC SB RAS Akademgorodok, Krasnoyarsk, 660036
Russia
Yuliya N. Togushova^
Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 19.11.2017, received in revised form 21.11.2017, accepted 20.04.2018 We study the effect of the spin-orbit coupling on the band structure and the Fermi surface of the three-orbital model within the two-iron Brillouin zone. Due to the presence of two irons in the crystallographi-cally correct unit cell, the spin-orbit coupling can be divided into the intra- and intercell parts with respect to the one-iron unit cell. We show that the intercell part produces the reconstruction of the Fermi surface in the form of the pronounced splitting between the previously degenerate electron (n, n) -pockets along the (0,n) — (n,n) direction. Intracell part shifts the bands around (0, 0) point and removes degeneracy there. There are also some other band shifts but they should not affect the low-energy physics because they occur at energy scales of about 1 eV below the Fermi level.
Keywords: Fe-based superconductors, spin-orbit coupling, band structure, Fermi surface. DOI: 10.17516/1997-1397-2018-11-4-430-437.
Discovery of superconductivity in iron-based systems in 2008 revived the interest in multiband systems [1]. Fermi surface in iron pnictides is formed by at least three bands. According to the density-functional studies (DFT) within LDA (local-density approximation) and GGA (generalized gradient approximation) and ARPES (angle-resolved photo-emission spectroscopy), thee bands originate from the three t2g iron d-orbitals. That is, the hybridization of the xz, yz, and xy orbitals results in two electron-like Fermi surface pockets around X = (n, 0) and Y = (0, n) points and one hole-like pocket around r = (0,0) point in the 1-Fe Brillouin zone. Nesting between two groups of sheets is the driving force for the spin-density wave antiferromagnetic order in the undoped systems [2]. The scattering with the wave vector connecting hole and electron pockets is the most probable candidate for the superconducting pairing in the doped systems. In particular, the spin-fluctuation approach result the extended s-wave gap which changes sign between hole and electron sheets (s± state [3]) as the leading instability [1,4-7].
There are several low-energy models for iron pnictides. Simplest one is the two-orbital model [8] correctly describing the dominant dxz-dyz contribution to the Fermi surface. It has
* mkor@iph.krasn.ru tytogushova@sfu-kras.ru © Siberian Federal University. All rights reserved
many disadvantages including misplacement of the second hole pocket, wrong Fermi velocities as compared with ARPES and DFT, and absence of dxy orbital contribution which is known to appear near or even at the Fermi level (both ARPES and DFT). Here we use the three-orbital model from Ref. [9] that comes from the t2g manifold. In particular, there the dxz and dyz orbitals are hybridized to form two electron Fermi surface sheets around X and Y points and one hole sheet around r point of the 1-Fe Brillouin zone (1-Fe BZ). The dxy orbital is decoupled from the others and form an outer hole pocket around r point.
One of the puzzles in iron-based superconductors is the anisotropy of the spin resonance peak [10]. It was found that transverse and longitudinal components of the spin susceptibility,
Xh__and 2\zz, violate the spin-rotational invariance, (S+S-) = 2 (SzSz). Later have to be
obeyed in the disordered system. One of the solution to the puzzle is the effect of the spinorbit (SO) interaction that can break the spin-rotational invariance like it does in Sr2RuO4 [11]. This approach was used before to study the spin response in model from Ref. [9] within the 1-Fe BZ [12].
There is a complication coming from the As, which forms square lattice planes between the lattice sites of, but also above and below, the square lattice of Fe. This alternating pattern of As makes the correct real space unit cell twice the 1-Fe unit cell. The corresponding 2-Fe BZ is twice as small as the 1-Fe one and called "folded BZ". For the simplest case of single-layer Fe pnictides the folding wave vector is two-dimensional and equal to Qf = (n, n). Most experimental results and DFT band structure are reported in the folded BZ since crystallographically it is the correct one. However, some experiments sensitive to the Fe positions, like neutron scattering on Fe moments, have more meaning in the 1-Fe BZ ("unfolded" zone).
Here we study the role of the SO coupling in the formation of the band structure and the Fermi surface in the 2-Fe BZ within the three-orbital model from Ref. [9].
1. Three-orbital model in 1-Fe BZ
We write the Hamiltonian of the three-orbital model [9] in the following form:
H0 = E ^cL <W, (1)
k,a,l,m
where l and m are orbital indices, ckma is the annihilation operator of a particle with momenta k and a spin a. We choose the following numbering of orbitals, dxy ^ 1, dyz ^ 2, dzx ^ 3. The model consists of the three t2g orbitals and all of them are hybridized. Matrix of one-electron energies and hoppings, £k, is given by
/ £ik 0 0 \
£k = I 0 £2k £4k I , (2)
\ 0 £4k £3k J
where
£ ik = x - M + 2txy (cos kx + cos ky) + 4txy cos kx cos ky,
£2k = eyz — M + 2tx cos kx + 2ty cos ky + +4t' cos kx cos ky + 2t"(cos 2kx + cos 2ky),
£3k = £xz — M + 2ty cos kx + 2tx cos ky + 4t' cos kx cos ky + +2t"(cos 2kx + cos 2ky),
£4k = 4txzyz sin kx/2 sin ky/2. The following set of parameters (in eV) allows to reproduce the topology of the Fermi surface in iron pnictides: chemical potential m = 0, cxy = —0.70, eyz = —0.34, exz = —0.34, txy = 0.18, t'xy = 0.06, tx = 0.26, ty = —0.22, t' = 0.2, t" = —0.07, txzyz = 0.38.
2. Spin-orbit coupling in 1-Fe BZ
Following Ref. [13], we write the spin-orbit coupling terms, HSO = Lf • Sf, in the
f
second-quantized form assuming that three t2g states behave like an 1 = 1 angular momentum representation,
. A
HSO — i-A £lmn cllaCkm<7',
_ (3)
l,m,n k, a, a'
where elmn is the completely antisymmetric tensor, indices {l,m,n} take values {x,y,z} or equivalently {x, y, z} ^ {dyz, dzx, dxy} ^ {2, 3,1}, and &na' are the Pauli spin matrices. Explicit form of the Hamiltonian is the following:
Hso = iA E [°k2ack3 asgn (a) + ick2jcki9sgn (a) + cJk3jcki9 - h.c.
k,a
Let us introduce vector operators in the orbital space, ^ka = (ck1 a ,ck2a , ck3 a). Then, Ho = £ £k#kj, Hso = £ #L3So *ka'.
k,j k,j, a'
J
J
j
Here
-I 0 -i5< '< 9 sgn (a) 5a '<9
£so — ( 6 Sgn(a) 0
-67'<7 -öa'7sgn (a) -
S<'<7sgn (a) I —
0
— ix £67'<7sgn (a) + 67'<7 (êx - iêysgn (a)
2
where we have introduces the following matrices:
(4)
(5)
(6) (7)
000 0 0 1 0-10
iJx, êx
0 0 1 0 0 0 100
— iJy
0 1 0 -1 0 0 0 0 0
(8)
Here, Ji are the generators of the rotation group O(3). It is well known [14] that O(3) transformation of the (x,y,z) vector is equivalent to the SU(2) transformation of the (£^£2) spinor if
1
1
— Ö(£2 - £2), y —-(¿2 + 8), * — É1É2.
2
2i
The total Hamiltonian H = H0 + HSO can be written explicitly as
H
E [*L (■
lr ^ L V
êk + iAêzsgn (a) j #k7 + i(êx - iêysgn (a)) Ska
: #kt (êk + i A ê^j #kt + Ski (êk - i A
. -l2* kt
+ ^L^ki+i-#k4êx^kt+2^ktê y f ki - 2Ski£y Skt
E(
*kt
( Ski )
(9)
where
H
£ k +i2 ê ■ -x A \ i2ê - 2£
-
i—f 2
i-êx + $êy
y
£k - i^£z
(10)
z
y
iJz.
X
-
If one considers only the z-component of the SO coupling, then the expression for H simplifies significantly,
Hz — Ho + HsOz
E *L(-
L- ,t V
êk + i^êzsgn (a)
—ç ( *k, *k; K( £)■<»>
where
Hz
/ êk + i ^ êz 0
V
0
ê k — i — £z k 2
(12)
Therefore, z-component of the SO interaction modifies one-electron energies only, but does it differently for spin-up and spin-down elements of the spin-resolved Hamiltonian matrix H. On the other hand, x- and y-components mix spin-up and spin-down matrix elements and thus effectively increase the dimensionality of the problem by a factor of two.
3. Spin-orbit coupling in 2-FeBZ
While working in the unfolded BZ as before, we are missing important effect of the SO hybridization between orbitals on neighboring irons. After such a hybridization the unfolding is not possible any more. Hereafter we utilize the following conjecture: the structure of the SO coupling between two orbitals on neighboring irons is the same as the structure between two orbitals with the same symmetry on the one iron. Thus, if SO coupling between dxz and dyz
orbitals has the form i^£z, the same form holds for dxz on Fe-1 (some one-iron unit cell) and dyz on Fe-2 (neighboring one-iron unit cell). For convenience, we assign different coupling constant A' to intercell SO interaction; one can always put it to be equal to A. Note that when we use a wording 'intercell SO coupling', it does not mean a long-range SO coupling. The SO interaction is always local but since orbitals of two neighboring irons hybridize directly and through As, the wave functions of electrons on these orbitals can overlap that opens up a possibility for the effectively intercell SO coupling.
Brillouin zone folding, i.e. transition from 1-Fe BZ to 2-Fe BZ, is done in two steps. First, momenta are transformed as (kx + ky)/2 ^ k'x, (kx — ky)/2 ^ k'y. Second, the Hamiltonian matrix is doubled by adding the shifted £k+QF, where QF = (n,n) is the folding wave vector. Thus, the Hamiltonian (1) takes the form
H
o—
k' ,a,l,m
£lm J i
£k' ^k' la ck'ma +
E£lm J c
£k'+QF ck' + QFlack+QFma ■
k' ,a,l,m
(13)
Later we skip prime assuming that all momenta are within the 2-Fe BZ. The Hamiltonian can be written in the matrix form analogous to Eq. (9),
H
E(
kf
k| $kf
k|
H
( ^kf \
$kf V J
where ) corresponds to the first (second) set of iron orbitals.
In case of intracell SO coupling, H has the form which immediately follows from Eq. (10):
o o \
H
/ A
£k +12 £ Z
.A ^ A
1 — £ x--£ y
_2_2_
A A
12 £ x + 2 £ y
• A\z
£ k - 1тг £ z
2
\
" Azz
£ k+QF +12£
•A „x A
1 — £ x--£ y
2 2
1 — £ x +--£ y
22 - A z
£k+QF - 12£ /
(15)
For finite A, the result of diagonalization of Eq. (15) is the same as of Eq. (10). New effects come in once we add the intercell SO coupling:
(
H
SOinter —
^Z 1-£ z
•A' „x A' 1_£ x _ _£ y
\'2
2
1 — £ x +--£ y
2 2
•A' .z -1-£ z
•A' .z 1-£ z
. A' A'
1 — £ '
x__$y
2
A' A' \
1 — £ x +--£ y
2 2 •A' —1 — £ z _2_
The total Hamiltonian is given by the sum of matrices (15) and (16),
(16)
Az
£k+1^£z
A x 2A y 1 _£ x _ _£ y
H
A
A
1 —£ _2_
2
■AT. z 1-£ z
•A' „x A'
1 — £ x--£ y
\ 2 2
1-£x + - £y 2 2
•A' —1 — £ z 2
1 — £ x +--.
2 2
•A' .z —1 — £ z 2
•A' ч 1-£ z
. A' A'
1 — £ '
x__0y
2
A
•A' „x , A' y \
1 — £ x +--£ y
2 2 •A' —1 — £ z _2
A
£ k+Q^ £ z ^ £ x +
*F ' '2 •A' „x A
1-£ x--£ y
2 2
2
£ k+Q
A
в- — 1—£ F 2 )
(17)
Again, if one considers only z-component of the SO coupling, the expression for H simplifies:
\
i * ^.A .z
£ k + z
0
H Hz
0
Az £k — 1 — £ z
1Y£
V
0
•A' .z -1-£z
•A' .z 1-£Z
0
0
•A' .z —1 — £ z _2_
£k+QF + 12£z
£ k+Qi
2 /
(18)
4. Results of the band structure and Fermi surface calculations
For A = 0, the band structure and the Fermi surface are shown in Fig. 1. It is essentially the same as in Ref. [9] but within the folded BZ.
First, we switch on only the intercell SO interaction by setting A = 0 and A' = 100 meV. Corresponding band structure and the Fermi surface are shown in Fig. 2. There is a pronounced splitting between the previously degenerate electron M-pockets along the X — M direction. Now, the band structure can not be unfolded. Also, there is a splitting between bands at M point and along the M — Г direction around —1 eV.
A
0
A
0
0
X=(it,0)
M=(7T,ît) r=(0,0)|X=(n-,0) Y=(0,ir) k
Fig. 1. Band structure and the Fermi surface in the three-orbital model without the SO coupling (A = A' = 0) in the 2-Fe BZ
Now we also switch on the intracell SO interaction by setting A = A' = 100 meV. The result is shown in Fig. 3. Apparently, the intracell SO coupling provides shift of the bands around r point. Apart from that, there is no pronounced difference between Fig. 2 and Fig. 3.
r=(0,0)
X=(ir,0)
M=(îr,ir) r=(0,0)|X=(îr,0) Y=(0,ir) k
■k 0
-■k -w
0
Fig. 2. The same as in Fig. 1 but for the finite intercell SO coupling A' vanishing intracell SO coupling A = 0
100 meV and the
Conclusions
If working in the 2-Fe BZ, the SO coupling can be divided into the intra- and intercell parts, though both being local. We conclude that intercell coupling produces the reconstruction of the Fermi surface around the M point. The new band structure can not be unfolded anymore. Intracell SO coupling removes degeneracy of the bands around r point. There is also a splitting between bands at M point and along the M — r direction around —1 eV. However, these energies are far below the chemical potential and should not affect the low-energy physics.
k 0
0
k
x
k
Fig. 3. The same as in Fig. 1 but for the finite intra- and intercell SO coupling, A = A' = 100 meV
This work was supported in part by the Russian Foundation for Basic Research (grant 16-0200098) and BASIS Foundation for Development of Theoretical Physics and Mathematics.
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Изменение зонной структуры из-за спин-орбитального взаимодействия в трехорбитальной модели пниктидов железа
Максим М. Коршунов
Сибирский федеральный университет Свободный, 79, Красноярск, 660041 Институт физики им. Л. В. Киренского ФИЦ КНЦ СО РАН Академгородок, 50/38, Красноярск, 660036
Россия
Юлия Н. Тогушова
Сибирский федеральный университет Свободный, 79, Красноярск, 660041
Россия
Исследовано влияние спин-орбитального взаимодействия на зонную сруктуру и поверхность Ферми трехорбитальной модели в зоне Бриллюэна двух атомов железа на ячейку. Из-за присутствия двух атомов железа в кристаллографической элементарной ячейке спин-орбитальное взаимодействие можно разделить на внутри- и межъячеечную части по отношению к элементарной ячейке решетки железа. Показано, что межъячеечная часть приводит к реконструкции поверхности Ферми в виде выраженного расщепления между ранее вырожденными электронными карманами в точке (п,п) вдоль направления (0,п) — (п,п). Внутриячеечная часть приводит к сдвигу зон вблизи точки (0, 0) и снимает там вырождение. Также имеют место другие сдвиги зон, но они не должны влиять на низкоэнергетическую физику, поскольку возникают на энергиях порядка 1 eV ниже уровня Ферми.
Ключевые слова: сверхпроводники на основе железа, спин-орбитальное взаимодействие, зонная структура, поверхность Ферми.
