Two-dome Superconductivity in FeS Induced by a Lifshitz Transition– Supplemental Material –

Makoto Shimizu,1 Nayuta Takemori,2 Daniel Guterding,3 and Harald O. Jeschke2, ∗

1Department of Physics, Okayama University, Okayama 700-8530, Japan2Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan

3Fachbereich Mathematik, Naturwissenschaften und Datenverarbeitung,Technische Hochschule Mittelhessen, Wilhelm-Leuschner-Straße 13, 61169 Friedberg, Germany

A. Structure

We use the crystal structures for tetragonal FeS(P 4/nmm space group) as given in Ref. [1]. The struc-ture is defined by a and c lattice parameters and the Sheight hS above the iron plane as shown in Figure S1 (a),(b) and (c), respectively. Note that the slightly decreas-ing height of S above the Fe plane as function of pressure(Figure S1 (c)) translates into a slightly increasing S zfractional coordinate. Relaxation of this position usingdensity functional theory within GGA reverses this trendand is thus unreliable for FeS.

Concerning the reliability of the interpolation, we ob-serve that the experimental data points in particular forthe sulphur height and to a lesser extent also for the lat-tice parameters show some scatter. This is, intentionally,not reproduced by our interpolation. For the c lattice pa-rameter, for example, the experimental value at 4.5 GPais 4.81 A while our interpolation is 4.78 A. This is justifiedby the experimental error bars. Concerning the sulphurheight, the overall variation of only 0.06 A in the entirepressure range is rather small. At 4.5 GPa, our interpo-lated values differs by 0.01 A from the experimental datapoint which is only a modest deviation. Determination ofthe internal coordinates is also experimentally somewhatless straight-forward than the dermination of the latticeconstants. Please note that a less monotonous interpo-lation than the one we chose would lead to unphysicallystrong variations in the elastic constants.

B. Origin of the Lifshitz transition

We investigate which structural change in FeS underpressure is most important for the occurrence of the Lif-shitz transition at P = 4.6 GPa. Bond length dFe-S andδ ≡ ](Fe-S-Fe) in tetragonal FeS are related to latticeconstants a and c and sulfur height hS via

dFe-S =

√a2

4+ hS

2

sinδ

2=

a

2√

2dFe-S

=a√

2a2 + 8hS2

(S1)

∗ jeschke@okayama-u.ac.jp

3.5

3.6

3.7

a (Å)interpolationZhang et al.

4.6

4.8

5c (Å)

1.12

1.14

1.16

1.18

0 1 2 3 4 5 6 7 8 9

hS (Å)

P (GPa)

FIG. S1. Experimental crystal structures as measured byZhang et al. [1] (symbols) together with Bezier interpolation(lines). (a), (b) are the tetragonal lattice parameters, and (c)is the height hS of S above the Fe plane.

Figure S2 shows the dFe-S and δ calculated for the inter-polated series of structures as function of pressure. Wenow investigate the sensitivity of the electronic structureand in particular the unoccupied band with Fe 3dz2 or-bital character near the Fermi level to the two relevantstructural parameters separately. We focus on structuresnear P = 4.6 GPa. Upon fixing Fe-S distances dFe-S andFe-S-Fe angles δ to the values indicated by the dashedlines in Figure S2, we calculate a lattice parameter and

2

2.1

2.15

2.2d

Fe−

S (

Å)

72.9

73

73.1

73.2

73.3

73.4

0 1 2 3 4 5 6 7 8 9

∠(F

e−S

−F

e) (

°)

P (GPa)

FIG. S2. Variation of (a) Fe-S distance and (b) Fe-S-Fe an-gle as function of pressure. Dashed lines indicate the bonddistances and angles chosen for the plots in Fig. S3.

−0.2

−0.1

0

0.1

0.2

Γ X M Γ Z R A

(a)

ener

gy (

eV)

−0.2

−0.1

0

0.1

0.2

Γ X M Γ Z R A

(a)

ener

gy (

eV)

dFe−S=2.12 ÅdFe−S=2.14 Å

−0.2

−0.1

0

0.1

0.2

Γ X M Γ Z R A

(b)

ener

gy (

eV)

−0.2

−0.1

0

0.1

0.2

Γ X M Γ Z R A

(b)

ener

gy (

eV)

∠(Fe−S−Fe)=73.0°

∠(Fe−S−Fe)=73.1°

∠(Fe−S−Fe)=73.2°

FIG. S3. Change of band structure of FeS for changes in (a)Fe-S distance and (b) Fe-S-Fe angle.

S z coordinate zS by

a = 2√

2dFe-S sinδ

2

zS =1

c

√dFe-S

2 − a2

4

(S2)

Figure S3 shows the result. A very small change in bandstructure results from the substantial Fe-S bond lengthchange corresponding to a pressure increase of roughly2 GPa (Figure S3 (a)) once the Fe-S-Fe angle is keptconstant; on the other hand, the effect of a pure anglechange as is actually relevant in a 2 GPa window leadsto a substantial change in band structure (Figure S3 (b))even if the Fe-S bond lengths are fixed. Thus, the Lifshitztransition can be considered mainly an effect of the defor-mation of the FeS4 tetrahedron rather than its pressureinduced volume reduction.

C. Spin fluctuation formalism

We follow Graser et al. [2] in considering the multi-orbital Hubbard model

H = H0 + U∑i,l

nil↑nil↓

+U ′

2

∑i,s,p6=s

nisnip −J

2

∑i,s,p6=s

Sis · Sip

+J ′

2

∑i,s,p6=s,σ

c†isσc†isσcipσcipσ

(S3)

where c†isσ (cisσ) are Fermionic creation (annihilation)

operators, Sis is the spin operator, nisσ = c†isσcisσ, Udenotes the intraorbital Coulomb repulsion, U ′ denotesthe interorbital Coulomb repulsion, J denotes the Hund’srule coupling and J ′ denotes the pair-hopping term. Thetight binding part of the Hamiltonian is

H0 = −∑i,j

tspij c†isσcjpσ (S4)

where tij denotes the transfer integral between sites iand j, s and p are the orbital indices, and σ denotesthe spin index. The five band tight binding Hamiltonianis obtained using projective Wannier functions [3] andunfolding [4]. We first calculate the static noninteractingsusceptibility

χpqst (q) =−∑k,l,m

ap∗l (k)atl(k)as∗m (k + q)aqm(k + q)

× nF (El(k))− nF (Em(k + q))

El(k)− Em(k + q)

(S5)

where El(k) is the energy value determined by the bandindex l and the wave vector k, and nF (E) is the Fermi

3

FIG. S4. Pressure dependence of the static noninteracting susceptibility for tetragonal FeS.

distribution function. asm is the matrix element of eigen-vectors resulting from diagonalization of tight-bindingHamiltonian H0. Within the framework of the randomphase approximation (RPA) the charge and spin suscep-tibilities can be calculated from the noninteracting sus-ceptibility

[(χRPAc )pqst

]−1= [χpqst ]

−1+ (Uc)pqst[

(χRPAs )pqst]−1

= [χpqst ]−1 − (Us)

pqst

(S6)

where the nonzero components of the interaction tensorsfor the multi-orbital Hubbard model are given by [2]

(Uc)aaaa = U (Uc)aabb = 2U ′,

(Uc)abab =3

4J − U ′ (Uc)baab = J ′

(Us)aaaa = U (Us)

aabb =

1

2J

(Us)abab =

1

4J + U ′ (Us)

baab = J ′, (S7)

which enables us to calculate the two-electron pairingvertex. For the interaction parameters, we find that thechoice U = 1.90 eV, U ′ = U/2, J = U/4 and J ′ = U/4takes us near the instability for all structures.

Pairing calculations.– The superconducting pairingvertex in the singlet channel is given by

Γpqst (k,k′)

=

[3

2Us χ

RPAs (k− k′)Us +

1

2Us

− 1

2Uc χ

RPAc (k− k′)Uc +

1

2Uc

]tqps

.

(S8)

The vertex in the orbital space description can be pro-jected onto band space using the eigenvector resulting

from diagonalization of the tight-binding Hamiltonian,

Γij(k,k′)

=∑s,t,p,q

at∗i (−k)as∗i (k)Re [Γpqst (k,k′)] apj (k

′)aqj(−k′).

(S9)

Using the vertex Γij(k,k′), we solve the gap equation

−∑j

∮Cj

dk′‖

2π

1

4πvF (k′)[Γij(k,k

′) + Γij(k,−k′)] gj(k′)

= λigi(k)

(S10)

where λi denotes the pairing eigenvalue and gi(k) is thegap function.

D. Noninteracting susceptibility

Figure S4 shows the static noninteracting susceptibil-ities of FeS for the orbitals that have significant weightat the Fermi level, (a) dxy, (b) dyz (which is by sym-metry equivalent to dxz in tetragonal FeS), and (c) dz2 .The latter only acquires features near Γ after the Lifshitztransition because the orbital weight is concentrated onone hole pocket.

E. One-iron Fermi surfaces of FeS

Figure S5 gives the orbital character of FeS Fermi sur-faces at two kz values, kz = 0 and kz = π at ambientpressure and at P = 5.5 GPa, after the Lifshitz transi-tion. At P = 0, only little 3dz2 weight is present in thehole pockets near Γ. The new hole pocket around M hasmostly 3dz2 character. Comparison of the sizes of theelectron pockets around (π, 0, kz) and (0, π, kz) between

4

FIG. S5. Fermi surfaces of FeS at two different pressures with orbital weights. They are calculated from the tight bindingmodels which were unfolded to the one-iron Brillouin zone.

5

kz = 0 and kz = π shows that the P = 5.5 GPa elec-tronic structure is more three-dimensional than the am-bient pressure electronic structure. Three-dimensionalsusceptibility and pairing calculations are essential forproperly describing the pressure dependence of supercon-ductivity in FeS.

F. Off-diagonal components of the spinsusceptibility

χ z2,z

2S yz,yz

χS (

1/e

V)

Γ X M Γ 0

0.2

0.4

0.6

0.8

FIG. S6. Pressure dependence of an off-diagonal element ofthe susceptibility for tetragonal FeS.

Figure S6 shows a relevant off-diagonal component of

the spin susceptibility χSbbaa with a = dz2 and b = dyz.

The even more important off-diagonal component of χSbbaa

with a = dz2 and b = dxy is shown in the main text

(Figure 2 (d)). χSbbaa with a = dz2 and b = dyz has

a weak maximum near q = (π, π), due to some nestingbetween dyz/dxz character hole pockets around Γ and thedz2 character hole pocket around M .

G. Solution of the gap equation with intra-orbitalinteraction only

In Figure S7, we demonstrate the effect of inter-orbitalinteraction terms by determining the solution of thegap equation at U ′ = J = J ′ = 0. The solution atP = 4.7 GPa (and higher pressures) is a simple nodelesssign-changing s wave instead of the more complicated s±solution shown in Figure 3 (c) of the main text.

It arises from the q = (π, 0), (0, π) nesting of thedyz/dxz orbitals. Note also the remaining orbital weightof dxy and dz2 around Γ in Figs. 3(a) and (c) of the maintext, which enables these orbitals to participate in a s±solution.

P = 4.70 GPa (U '=J=J '=0)λ = 0.078

kxky

kz

-π

0

π -π0

π

-π

0

π

P = 4.70 GPa (U '=J=J '=0)λ = 0.078

kxky

kz

-π

0

π -π0

π

-π

0

π

FIG. S7. Hypothetical leading gap function for FeS at P =4.7 GPa for U = 1.9 eV and U ′ = J = J ′ = 0. The three-dimensional Fermi surface is plotted in the one-iron Brillouinzone.

6

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