Origin of the superconducting state in the collapsed tetragonal phase of KFe2As2

6.9 MB·English

by  Daniel Guterding

Origin of the superconducting state in the collapsed tetragonal phase of KFe As
2 2
Daniel Guterding, Steffen Backes, Harald O. Jeschke, and Roser Valent´ı
∗
Institut fu¨r Theoretische Physik, Goethe-Universita¨t Frankfurt,
Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany
Recently, KFe As was shown to exhibit a structural phase transition from a tetragonal to a
2 2
collapsed tetragonal phase under applied pressure of about 15 GPa. Surprisingly, the collapsed
tetragonal phase hosts a superconducting state with T 12 K, while the tetragonal phase is
c
∼
a T 3.4 K superconductor. We show that the key difference between the previously known
c
≤
non-superconducting collapsed tetragonal phase in AFe As (A= Ba, Ca, Eu, Sr) and the super-
2 2
5 conducting collapsed tetragonal phase in KFe2As2 is the qualitatively distinct electronic structure.
1 While the collapsed phase in the former compounds features only electron pockets at the Brillouin
0 zone boundary and no hole pockets are present in the Brillouin zone center, the collapsed phase in
2 KFe As has almost nested electron and hole pockets. Within a random phase approximation spin
2 2
fluctuation approach we calculate the superconducting order parameter in the collapsed tetragonal
r
p phase. We propose that a Lifshitz transition associated with the structural collapse changes the
A pairing symmetry from d-wave (tetragonal) to s (collapsed tetragonal). Our DFT+DMFT calcu-
±
lationsshowthateffectsofcorrelationsontheelectronicstructureofthecollapsedtetragonalphase

8 are minimal. Finally, we argue that our results are compatible with a change of sign of the Hall
coefficient with pressure as observed experimentally.

]
n
PACSnumbers: 71.15.Mb,71.18.+y,74.20.Pq,74.70.Xa
o
c
-
r The family of AFe As (A= Ba, Ca, Eu, K, Sr) su- tablished by a few experimental and theoretical inves-
2 2
p
perconductors, also called 122 materials, has been inten- tigations based on angle-resolved photoemission spec-
u
sively investigated in the past due to their richness in troscopy, de Haas-van Alphen measurements, and den-
s
. structural, magnetic and superconducting phases upon sity functional theory combined with dynamical mean
t
a doping or application of pressure1–6. One phase whose field theory (DFT+DMFT) calculations that correlation
m
propertieshavebeenrecentlyscrutinizedatlengthisthe effects crucially influence the behavior of this system at
- collapsed tetragonal (CT) phase present in BaFe2As2,
d
CaFe As , EuFe As , and SrFe As under pressure and
n 2 2 2 2 2 2
inCaFe P 7–14. Thestructuralcollapseofthisphasehas
o 2 2
beenshownto beassistedby the formationof As 4p -As
c z
[ 4p bonds between adjacent Fe-As layers giving rise to
z
3 a bonding-antibonding splitting of the As pz bands15.
It has been argued that this phase does not support
v
8 superconductivity due to the absence of hole cylinders
6 at the Brillouin zone center and the corresponding sup-
0 pression of spin fluctuations10,16,17. However, recently
3 Ying et al.18 investigated the hole-doped end member
0
of Ba K Fe As , KFe As , under high pressure and
1 x x 2 2 2 2
1. observe−d a boost of the superconducting critical temper-
0 atureT upto12Kpreciselywhenthesystemundergoes
c
5 astructuralphasetransitiontoaCTphaseatapressure
1
P 15 GPa. These authors attributed this behavior
: c ∼
v to possible correlation effects. Moreover, measurements
i of the Hall coefficient showed a change from positive to
X
negative sign upon pressure, indicating that the effective
r
a nature of charge carriers changes from holes to electrons
with increasing pressure. Similar experiments are also
reported in Ref. 19.
KFe2As2 has a few distinct features: at ambient pres- FIG.1. (Coloronline)Crystalstructure,schematicFermisur-
sure, the system shows superconductivity at Tc = 3.4 K face (dashed lines) and schematic superconducting gap func-
and follows a V-shaped pressure dependence of T for tion (background color) of KFe As in the one-Fe Brillouin
c 2 2
moderate pressures with a local minimum at a pres- zonebeforeandafterthevolumecollapse. TheLifshitztran-
sure of 1.55 GPa20. The origin of such behavior and sitionassociatedwiththeformationofAs4pz-As4pzbondsin
the nature of the superconducting pairing symmetry theCTphasechangesthesuperconductingpairingsymmetry
are still under debate21–27. However, it has been es- from dxy to s .
±
2
P = 0 GPa28–35. Application of pressure should nev- CaFe As (CT)
(a) 1 2 2
ertheless reduce the relative importance of correlations
with respect to the bandwidth increase. In fact, recent
0.5
DFT+DMFT studies on CaFe As in the high-pressure
2 2
CT phase show that the topology of the Fermi surface is eV] 0
basically unaffected by correlations36,37. One could ar- E [
gue though, that at ambient pressure CaFe2As2 is less -0.5
correlated than KFe As and therefore, in KFe As cor-
2 2 2 2
relation effects may be still significant at finite pressure. -1
KFe As (CT)
In order to resolve these questions, we performed den- (b) 1 2 2
sity functional theory (DFT) as well as DFT+DMFT
calculations for KFe As in the CT phase. Our results 0.5
2 2
show that the origin of superconductivity in the col- V]
lapsedtetragonalphaseinKFe2As2 liesinthequalitative E [e 0
changes in the electronic structure (Lifshitz transition)
-0.5
experienced under compression to a collapsed tetragonal
phaseandcorrelationsplayonlyaminorrole. Whereasin
-1
thetetragonalphaseatP =0GPaKFe2As2featurespre- Γ X M Γ Z
dominantly only hole pockets at the Brillouin zone cen-
dz2 dx2-y2 dxy dxz dyz
ter, at P 15 GPa in the CT phase significant electron
∼
pockets emerge at the Brillouin zone boundary, which
FIG. 2. (Color online) Electronic bandstructure of the col-
together with the hole pockets at the Brillouin zone cen-
lapsed tetragonal phase in (a) CaFe As and (b) KFe As .
2 2 2 2
ter favor a superconducting state with s symmetry, as
The path is chosen in the one-Fe equivalent Brillouin zone.
we show in our calculations of the super±conducting gap The colors indicate the weights of Fe 3d states.
function using the random phase approximation (RPA)
spin fluctuation approach. Moreover, our results in the
tetragonal phase of KFe2As2 at P = 10 GPa suggest conventional I4/mmm unit cell.
a change of pairing symmetry from dxy (tetragonal) to The electronic bandstructure in the collapsed tetrag-
s upon entering the collapsed phase (see Fig. 1). This onal phase of CaFe As and KFe As is shown in Fig.
sc±enario is distinct from the physics of the CT phase in 2. These results alr2ead2y reveal a 2strik2ing difference be-
CaFe2As2, where the hole pockets at the Brillouin zone tween the CT phases of CaFe2As2 and KFe2As2: while
center are absent. For comparison, we will present the theformerdoesnotfeatureholebandscrossingtheFermi
susceptibility of collapsed tetragonal CaFe2As2, which is level at Γ and only one band crossing the Fermi level at
representative for the collapsed phase of AFe2As2 (A= M (π,π,0), the latter does feature hole-pockets at both
Ba, Ca, Eu, Sr). Our findings also suggest an explana- Γ and M in the one-Fe equivalent Brillouin zone. The
tion for the change of sign in the Hall coefficient upon reason for this difference in electronic structure is that
entering the CT phase in KFe2As2. KFe2As2 is strongly hole-doped compared to CaFe2As2.
Density functional theory calculations were carried In Fig. 3 we show the Fermi surface in the one-Fe
out using the all-electron full-potential local orbital equivalent Brillouin zone at k = 0. In both cases, the
z
(FPLO)38 code. For the exchange-correlation functional FermisurfaceisdominatedbyFe3d character. The
xz/yz
weusethegeneralizedgradientapproximation(GGA)by hole cylinders in KFe As span the entire k direction of
2 2 z
Perdew, Burke, and Ernzerhof39. All calculations were the Brillouin zone, while only a small three-dimensional
converged on 20 20 20 k-point grids. hole-pocket is present in CaFe As (see Ref.41). For
× × 2 2
The structural parameters for the CT phase of KFe2As2,thehole-pocketsatM(π,π,0)andtheelectron
KFe As were taken from Ref. 18. We used the data pockets at X(π,0,0) areclearly nested, while no nesting
2 2
points at P 21 GPa, deep in the CT phase, where is observed for CaFe2As2. It is important to note here,
a = 3.854 ˚A≈and c = 9.6 ˚A. The fractional arsenic z- that the folding vector in the 122 family of iron-based
position (z = 0.36795) was determined ab-initio via superconductors is (π,π,π), so that the hole-pockets at
As
structuralrelaxationusingtheFPLOcode. Wealsoper- M (π,π,0) will be located at Z (0,0,π) after unfolding
formed calculations for the crystal structure of Ref. 19, the bands to the effective one-Fe picture42.
where a preliminary experimental value for the arsenic After qualitatively identifying the difference between
z-position was given. The electronic structure is very the CT phases of CaFe As and KFe As , we calculate
2 2 2 2
similar to the one reported here. For the CT phase of thenon-interactingstaticsusceptibilitytoverifythatthe
CaFe As we used experimental lattice parameters from better nesting of KFe As generates stronger spin fluc-
2 2 2 2
Ref. 40 (T = 40 K, P 21 GPa) and determined tuations. For that we constructed 16-band tight-binding
≈
the fractional arsenic z-position (z = 0.37045) using models from the DFT results using projective Wannier
As
FPLO.AllFe3dorbitalsaredefinedinacoordinatesys- functionsasimplementedinFPLO43. WekeeptheFe3d
temrotatedby45 aroundthez-axiswithrespecttothe and As 4p states, which corresponds to an energy win-
◦
3
(aπ) CaFe2As2 (CT) (bπ) KFe2As2 (CT) (a) 0.8 CaFe2As2 (CT) (b) KFe2As2 (CT)
k k
y y
V] 0.6
e
1/
) [q 0.4
0 0 χ(
(c) 0.2 (d)
0.4
-π-π dz2 0dx2-y2 kxdxπy -π-π dxz d0yz kx π ) [1/eV]q 00..23
χ(
FIG. 3. (Color online) Fermi surface of the collapsed tetrag-
0.1
onal phase in (a) CaFe2As2 and (b) KFe2As2 at kz =0. The Γ X M Γ Γ X M Γ
fullplotspanstheone-FeequivalentBrillouinzone,whilethe dz2 dx2−y2 dxy dxz dyz
area enclosed by the grey lines is the two-Fe equivalent Bril-
louin zone. The colors indicate the weights of Fe 3d states. FIG. 4. (Color online) Summed static susceptibility (top)
and its diagonal components χaa (bottom) in the eight-band
aa
tight-binding model for [(a) and (c)] CaFe As and [(b) and
2 2
dow from 7 eV to +6 eV. Subsequently, we unfold the (d)]KFe As intheone-FeBrillouinzone. Thecolorsidentify
− 2 2
16-band model using our recently developed glide reflec- the Fe 3d states.
tion unfolding technique42, which produces an effective
eight-band model of the three-dimensional one-Fe Bril-
louin zone. At first glance, the observable static susceptibility
We analyse these eight-band models using the 3D ver- displayed in Fig. 4 is comparable for CaFe2As2 and
sion of random phase approximation (RPA) spin fluc- KFe2As2. A key difference is however revealed upon in-
tuation theory44 with a Hamiltonian H = H + H , vestigation of the largest elements, i.e. the diagonal en-
0 int
where H0 is the eight-band tight-binding Hamiltonian triesχaaaa. TheseshowthatinCaFe2As2thesusceptibility
derived from the ab-initio calculations, while Hint is the has broad plateaus, while in KFe2As2 the susceptibility
Hubbard-Hund interaction. The arsenic states are kept has a strong peak at X (π,0,0) in the one-Fe Brillouin
in the entire calculation, but interactions are considered zone, which corresponds to the usual s pairing scenario
only between Fe 3d states. Further information is given that relies on electron-hole nesting. ±In CaFe2As2 the
in Ref. 41. pairing interaction is highly frustrated because there is
The non-interacting static susceptibility in orbital- no clear peak in favor of one pairing channel.
space is defined by Eq. (1), where matrix elements as((cid:126)k) Wehavealsoperformedspin-polarizedcalculationsfor
µ
resulting from the diagonalization of the initial Hamil- KFe2As2 at P 21 GPa in order to confirm the anti-
≈
tonian H connect orbital and band-space denoted by ferromagnetic instability we find in the linear response
0
indices s and µ respectively. The E are the eigenvalues calculations. Out of ferromagnetic, N´eel and stripe an-
µ
of H and f(E) is the Fermi function. tiferromagnetic order only the stripe antiferromagnet is
0
stablewithsmallmomentsof0.07µ onFe,inagreement
B
with our calculations for the susceptibility.
χpstq((cid:126)q)=−N1 asµ((cid:126)k)apµ∗((cid:126)k)aqν((cid:126)k+(cid:126)q)atν∗((cid:126)k+(cid:126)q) The leading superconducting gap function of KFe2As2
(cid:126)k,µ,ν in the CT phase is shown in Fig. 5. As expected from
(cid:80) f(Eν((cid:126)k+q(cid:126)))−f(Eµ((cid:126)k)) our susceptibility calculations, the pairing symmetry is
× Eν((cid:126)k+q(cid:126))−Eµ((cid:126)k) s-wave with a sign-change between electron and hole-
(1)
The observable static susceptibility41 is defined as the pockets. While the superconducting gap is nodeless in
sum over all elements χbb of the full tensor χ((cid:126)q) = the kz = 0 plane, the kz = π plane does show nodes
1 χbb((cid:126)q). aa where the orbital character changes from Fe 3dxz/yz to
2 aa Fe 3d . Note that this k = π structure of the super-
a,b xy z
(cid:80)The effective interaction in the singlet pairing channel conductinggapisexactlythesameasinthewellstudied
is constructed from the static susceptibility tensor χpq LaFeAsO compound44, which shows that the CT phase
st
which measures strength and wave-vector dependence of of KFe As closely resembles usual iron-based supercon-
2 2
spin fluctuations, via the multiorbital RPA procedure. ductorsalthoughitismuchmorethree-dimensionalthan,
Both the original and effective interaction are discussed, e.g. in LaFeAsO.
e.g. in Ref. 45. We have shown previously that our im- Wehavealsocalculatedthesuperconductinggapfunc-
plementation is capable of capturing effects of fine vari- tion for KFe As at P =10 GPa in the tetragonal phase
2 2
ations of shape and orbital character of the Fermi sur- and find d as the leading pairing symmetry41. The
xy
face46. dominantd -solutionobtainedinmodelcalculations
x2 y2
−
4
KFe2As2 (CT) via m∗ =1 ∂Re Σ(ω) .
(aπ) kz = 0 (bπ) kz = π 0.03 TambLlDeA I −displa∂yωs (cid:12)ωth→e0 orbital-resolved mass-
ky ky 0.02 renormalizationsm∗/mLD(cid:12)(cid:12)A forKFe2As2 inthecollapsed
tetragonal phase. The obtained values show that local
0.01 nits] electronic correlations in the CT phases of KFe2As2
0 0 0 arb. u and CaFe2As236,37 are comparable. As in CaFe2As2,
-0.01 ∆ [ the effects of local electronic correlations on the Fermi
surface are negligible (see Ref. 41). The higher T of the
-0.02 c
collapsed phase in absence of strong correlations raises
-π -π -0.03
-π 0 kx π -π 0 kx π the question how important strong correlations are in
general for iron-based superconductivity. This issue
FIG.5. (Coloronline)Leadingsuperconductinggapfunction demands further investigation.
(s ) of the eight-band model in the one-Fe Brillouin zone of Finally, the change of dominant charge carriers from
±
KFe2As2 in the CT phase at (a) kz =0 and (b) kz =π. hole to electron-like states measured in the Hall-
coefficient under pressure18 is naturally explained from
our calculated Fermi surfaces. While KFe As is known
2 2
TABLE I. Mass renormalizations m∗/mLDA of the Fe 3d or- toshowonlyhole-pocketsatzeropressure,theCTphase
bitalsinthecollapsedtetragonalphaseofKFe As calculated features also large electron pockets. On a small fraction
2 2
with the LDA+DMFT method. of these electron pockets, the dominating orbital char-
acter is Fe 3d (Fig. 3). It was shown in Ref. 53
d d d d xy
z2 x2 y2 xy xz/yz
1.318 1.3−09 1.319 1.445 that quasiparticle lifetimes on the Fermi surface can be
very anisotropic and long-lived states are favored where
marginal orbital characters appear. As Fe 3d charac-
xy
ter is only present on the electron pockets, these states
basedonrigidbandshifts22,24isalsopresentinourcalcu- contribute significantly to transport and are responsible
lation,butasasub-leadingsolution. Ourresultsstrongly for the negative sign of the Hall coefficient.
suggest that the Lifshitz transition, which occurs upon In summary, we have shown that the electronic struc-
enteringthecollapsedtetragonalphase,changesthesym- ture of the collapsed tetragonal phase of KFe As qual-
2 2
metry of the superconducting gap function from d-wave itatively differs from that of other known collapsed ma-
(tetragonal) to s-wave (CT) (see Fig. 1). The possi- terials. Upon entering the CT phase, the Fermi surface
ble simultaneous change of pairing symmetry, density of of KFe As undergoes a Lifshitz transition with electron
2 2
statesandT potentiallyopensupdifferentroutestoun- pocketsappearingattheBrillouinzoneboundary, which
c
derstanding their quantitative connection. arenestedwiththeholepocketsattheBrillouinzonecen-
In order to estimate the strength of local electronic ter. Thus, the spin fluctuations in collapsed tetragonal
correlations in collapsed tetragonal KFe2As2, we per- KFe2As2 resemble those of other iron-based supercon-
formedfullychargeself-consistentDFT+DMFTcalcula- ductors in non-collapsed phases and the superconduct-
tions. WeusedthesamemethodasdescribedinRef.35. ing gap function assumes the well-known s symmetry.
The DFT calculation was performed by the WIEN2k47 This is in contrast to other known materia±ls in the CT
implementation of the full-potential linear augmented phase, like CaFe2As2, where hole pockets at the Bril-
plane wave (FLAPW) method in the local density ap- louin zone center are absent and no superconductivity is
proximation (LDA) with 726 k-points in the irreducible favored. BasedonourLDA+DMFTcalculations,theCT
Brillouin zone. We checked that the results of FPLO phaseofKFe2As2 issignificantlylesscorrelatedthanthe
and WIEN2k agree on the DFT level. The Bloch wave tetragonal phase, and mass enhancements are compara-
functions are projected to the localized Fe 3d orbitals as ble to the CT phase of CaFe2As2. Finally, we suggest
describedinRefs.48and49. Theenergywindowforpro- that the change of dominant charge carriers from hole to
jection was chosen from 7 to +13 eV, with the lower electron-likecanbeexplainedfromanisotropicquasipar-
boundary lying in a gap i−n the density of states. For the ticle lifetimes.
solution of the DMFT impurity problem the continuous-
time quantum Monte Carlo method in the hybridization
expansion50asimplementedintheALPS51,52projectwas ACKNOWLEDGMENTS
employed (see Ref. 41 for more details). The mass-
renormalizationsaredirectlycalculatedfromtheanalyti- We thank the Deutsche Forschungsgemeinschaft for fi-
callycontinuedrealpartoftheimpurityself-energyΣ(ω) nancial support through Grant No. SPP 1458.
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Origin of the superconducting state in the collapsed tetragonal phase of KFe As :
2 2
Supplemental Information
Daniel Guterding, Steffen Backes, Harald O. Jeschke, and Roser Valent´ı
∗
Institut fu¨r Theoretische Physik, Goethe-Universita¨t Frankfurt,
Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany
I. DETAILS OF THE SUSCEPTIBILITY AND
PAIRING CALCULATION
For calculating the susceptibility we used 30 30 10
k-pointgridsandaninversetemperatureofβ =×40eV×−1.
To calculate the pairing interaction, the susceptibility is
needed at k-vectors that do in general not lie on a grid.
Those susceptibility values are obtained using trilinear
interpolation of the gridded data.
The pairing interaction is constructed using 800
∼
points on the three-dimensional Fermi surface. Con-
sequently, the solution of the gap equation is avail-
able only on those points scattered in three-dimensional
space. In order to obtain a graphical represen-
tation of the gap function on two-dimensional cuts
through the Brillouin zone, we used multiscale radial
basis function interpolation as implemented in ALGLIB
(http://www.alglib.net).
The Hubbard-Hund interaction H includes the on-
int
siteintra(inter)orbitalCoulombinteractionU (U ), the
0
Hund’s rule coupling J and the pair hopping energy J .
0
We assume spin rotation-invariant interaction parame-
ters U =2.4 eV, U =U/2, and J =J =U/4. Because
0 0
ofthelargebandwidthinthecollapsedtetragonalphase,
these comparatively large values are necessary to bring
the system close to the RPA instability. Note however,
that the symmetry of the superconducting gap in this
system does not change, even if significantly reduced pa-
rameter values are considered.
The parameter values used in the RPA method are
renormalized with respect to those in the DFT+DMFT
method,becausetheelectronicself-energyisneglectedin
the usual RPA scheme1,2.
II. THREE DIMENSIONAL FERMI SURFACE
FIG. 1. Three-dimensional Fermi surface of (a) collapsed
tetragonal CaFe As and (b) collapsed tetragonal KFe As
2 2 2 2
We extracted the three-dimensional Fermi surface of (both at P = 21 GPa) shown in the two-Fe equivalent Bril-
collapsedtetragonalCaFe As andKFe As fromFPLO louin zone. The Γ-point is located in the center of the dis-
2 2 2 2
played volume.
using 50 50 25 k-point grids in the two-Fe equiva-
× ×
lent Brillouin zone (Fig. 1). For CaFe As only electron
2 2
pockets in the Brillouin zone corners are observed. The
three-dimensionalholepocketthatarisesfromthebands III. DFT BANDSTRUCTURE AND FERMI
atM(π,π,0)issosmallthatitisnotdetectedherewith SURFACE OF THE NON-COLLAPSED PHASE
UNDER PRESSURE
the given k-resolution. In KFe As one of the hole cylin-
2 2
ders is highly dispersive, while the central hole cylinder
andtheelectronpocketslocatedinthecornersarenested ToverifythatelectronpocketsinKFe As donotgrow
2 2
throughout the entire Brillouin zone. continouslywithappliedpressure,butratherappearasa
2
KFe As (TET, P = 10 GPa) CaFe As (CT, P = 0.53 GPa)
(a) 1 2 2 (a) 1 2 2
0.5 0.5
V] V]
e 0 e 0
E [ E [
-0.5 -0.5
-1 -1
Γ X M Γ Z Γ X M Γ Z
dz2 dx2-y2 dxy dxz dyz dz2 dx2-y2 dxy dxz dyz
(b) π (b) π
k k
y y
0 0
-π -π
-π 0 π -π 0 π
k k
x x
FIG. 2. (a) Electronic bandstructure in the one-Fe equiva- FIG.3. (a)Electronicbandstructureintheone-Feequivalent
lent Brillouin zone and (b) Fermi surface of the P 10 GPa Brillouinzoneand(b)FermisurfaceoftheP =0.53GPacol-
∼
non-collapsedtetragonal(TET)phaseofKFe As atk =0. lapsedtetragonal(CT)phaseofCaFe As atk =0. Thefull
2 2 z 2 2 z
ThefullplotoftheFermisurfacespanstheone-Feequivalent plotoftheFermisurfacespanstheone-FeequivalentBrillouin
Brillouin zone, while the area enclosed by the grey lines is zone, while the area enclosed by the grey lines is the two-Fe
the two-Fe equivalent Brillouin zone. The colors indicate the equivalent Brillouin zone. The colors indicate the weights of
weights of Fe 3d states. Fe 3d states.
IV. BANDSTRUCTURE, FERMI SURFACE
AND SUSCEPTIBILITY AT LOW PRESSURES
result of the structural collapse, we have also prepared a
In the main paper we used crystal structures at P =
crystalstructureatP 10GPa,usingthedatafromRef.
21 GPa for both materials of interest to ensure that
3. The lattice param∼eters were chosen as a = 3.663 ˚A
results are comparable. Here we show the electronic
and c=13.0 ˚A. The fractional arsenic z-position (z =
As bandstructure, Fermi surface and static susceptibility of
0.35750) was again determined ab-initio via structural
CaFe As at low pressure (P = 0.53 GPa), right after
2 2
relaxation using the FPLO code.
the structural collapse. We used the experimental struc-
tural parameters from Ref. 5, which are the same as in
NoqualitativechangesinthebandstructureandFermi ourpreviousGGA+DMFTstudy6. Thedifferencescom-
surface (Fig. 2) are observed compared to the ambient pared to the high pressure results in the main text are
pressure structure4. Most importantly, electron pock- purely quantitative.
ets which are nested with the hole pockets in the Bril- The electronic bandstructure and Fermi surface are
louin zone center are not present, even at high pressures shown in Fig. 3. The static susceptibility is shown in
slightly below the transition from the tetragonal to the Fig. 4. The overall enhancement of the susceptibility
collapsedtetragonalphase. Weconcludethattheappear- comparedtothefiguresinthemaintextisaconsequence
ance of electron pockets is directly linked to the struc- ofthesmallerbandwidthatlowpressure. Theparticular
tural collapse. enhancementofthepeakbetweenXandMinthed di-
xz
3
CaFe As (CT) a)
(a) 0.8 2 2 12
DMFT
KFe2As2 DOS DFT
10
V] 0.6
1/e 8
χq() [ 0.4 ωA() 6
4
0.2
(b)
0.4 2
0
V] 0.3 -4 -3 -2 -1 0 1 2 3
e ω [eV]
1/ b) c)
χq() [ 0.2 2 .25 Fe 3dz2 DMDFFTT 2 .25 Fe 3dx2-y2 DMDFFTT
1.5 1.5
0.1 ω)
Γ X M Γ A( 1 1
dz2 dx2−y2 dxy dxz dyz
0.5 0.5
FIG. 4. Summed static susceptibility (a) and its diagonal 0 0
-4 -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3
components χaaaa (b) in the eight-band tight-binding model d) e)
for CaFe2As2 (CT, P = 0.53 GPa) in the one-Fe Brillouin 2.5 Fe 3dxy DMFT 2.5 Fe 3dxz/yz DMFT
zone. The colors identify the Fe 3d states. 2 DFT 2 DFT
ω) 1.5 1.5
A( 1 1
agonal element of the susceptibility has its origin in the
0.5 0.5
hole band crossing the Fermi level at the M point, which
in turn contributes a small three-dimensional pocket. 0 0
-4 -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3
ω [eV] ω [eV]
V. DETAILS OF THE LDA+DMFT FIG.5. Thek-integratedspectralfunction(DOS)ofKFe2As2
CALCULATION at P = 21 GPa as obtained within DFT compared to
LDA+DMFT. The upper panel (a) shows the total DOS,
with the DFT result (black line) and the renormalized
For our DMFT calculations we used a temperature of
LDA+DMFT result (red line). The lower four panels show
290 K, i.e. an inverse temperature β = 40 eV−1. The the partial DOS of the Fe 3d orbitals: (b) 3d , (c) 3d ,
z2 x2 y2
interaction parameters are defined in terms of Slater in- (d) 3d and (e) 3d orbital. −
xy xz/yz
tegrals7 with an on-site Coulomb interaction U = 4 eV
and Hund’s rule coupling J = 0.8 eV. As the double
counting correction we used the fully localized limit8,9 of this object are then integrated over all bands v and
(FLL)scheme. Beforethecontinuationoftheimaginary- k-points k to obtain the total density of states within
frequency self-energy to the real axis by stochastic ana- LDA+DMFTshowninFig. 5(a). Additionalprojection
lytic continuation10, the calculation was converged with onto the Fe 3d orbitals (denoted by index m) by
2 107 Monte-Carlosweepsforthesolutionoftheimpu-
×
rity model. A (k,ω)=
FromtheLDA+DMFTcalculationweobtainthespec- mm0
tral function on real frequencies from the analytically 1 1
continued self-energy as −π="Xvv0 Pmv(k)(ω+µ)δvv0 −(cid:15)vv0(k)−Σvv0(k,ω)Pm†0v0(k)#,
(3)
1 1
A (k,ω)= ,
vv0 −π= (ω+µ)δ (cid:15) (k) Σ (k,ω)
(cid:20) vv0 − vv0 − vv0 (cid:21) is then used in the same fashion to obtain the orbital
(1)
resolved density of states shown in Fig. 5 (b)-(e).
whereµisthechemicalpotential, (cid:15) (k)=(cid:15) (k)δ are
vv0 v vv0 The k-resolved spectral function shown in Fig. 6 was
theeigenenergiesoftheLDAHamiltonianandΣ (k,ω)
vv0 obtained in the same manner, but only integrating over
is the impurity self-energy upfolded to Bloch space by
all bands v in order to keep the k-dependence. The or-
projectors P (k)(see Ref. 11 and 12)
mv bital resolved Fermi surfaces shown in Fig. 7 and Fig. 8
wereobtainedbyevaluatingA (ω)attheFermienergy
Σ (k,ω)= P (k)(Σ (ω) Σ )P (k), (2) mm
vv0 m†v mm − DC mv0 (ω = ωF) on a 200 200 k-grid in the two-Fe Brillouin
m ×
X zone.
with the double-counting correction term Σ taken in We observe moderate effects of renormalization and
DC
the fully-localized limit (FLL)8,9. The diagonal entries broadening in the density of states (see Fig. 5) with
4
no significant change at the Fermi level. This is con-
firmed by the k-resolved spectral function (see Fig. 6).
UsingtheFLLdouble-countingcorrection,wedonotob-
serveanytopologicalchangesintheFermisurface,which
closelyresemblestheDFTresult(seeFig. 7andFig. 8).
However, we observe a closing of the pockets at Γ when
the nominal double-counting correction13 is considered.
This indicates that the LDA+DMFT results slightly de-
pend on the double-counting. Note that these changes
donotaffectourconclusionsregardingthesuperconduct-
ing pairing symmetry since only a small fraction of the
Brillouin zone around k = 0 is affected. As the rel-
z
evant electron-hole nesting takes place around k = π,
z
our superconducting pairing calculations based on the
DFTbandstructureshouldberobustwithrespecttothe
effects of correlations.
FIG. 6. The k-resolved spectral function of KFe As at
2 2
P =21 GPa as obtained within LDA+DMFT. The red lines Note that the Fermi surface plots (Fig. 7 and Fig. 8)
showtheDFTbandstructureforcomparison,rescaledbythe havetoberotatedby45◦intheplaneinordertocompare
average mass enhancement of 1.37. The labels on the x-axis with the Fermi surface plots in the original paper. In
correspond to high-symmetry points in the two-Fe Brillouin 122 compounds the Z-point of the two-Fe Brillouin zone
zone. corresponds to the M-point of the one-Fe Brillouin zone.
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