Testing the sign-changing superconducting gap in iron-based superconductors with quasiparticle interference and neutron scattering.

Abstract

We present a phenomenological calculation of the quasiparticle interference (QPI) pattern and inelastic neutron scattering (INS) spectra in iron-pnictide and layered iron-selenide compounds by using material specific band structure and superconducting (SC) gap properties. As both the QPI and the INS spectra arise due to scattering of the Bogolyubov quasiparticles, they exhibit a one-to-one correspondence of the scattering vectors and the energy scales. We show that these two spectroscopies complement each other in such a way that a comparative study allows one to extract quantitative and unambiguous information about the underlying pairing structure and the phase of the SC gap. Due to the nodeless and isotropic nature of the SC gaps, both the QPI and INS maps are concentrated at only two energies in pnictide (two SC gaps) and one energy in iron-selenide, while the associated scattering vectors q for scattering of sign-changing and same sign of the SC gaps change between these spectroscopies. The results presented, particularly for the newly discovered iron-selenide compounds, can be used to test the nodeless d-wave pairing in this class of high temperature superconductor.

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Testing the sign-changing superconducting gap in iron-based superconductors with quasiparticle
interference and neutron scattering
Tanmoy Das1, and A. V. Balatsky1,2
1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA.
2Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA.
(Dated: October 19, 2011)
We present a phenomenological calculation of the quasiparticle-interference (QPI) pattern and inelastic Neu-
tron scattering (INS) spectra in iron-pnictide and layered iron-selenide compounds by using materials specific
band-structure and superconducting (SC) gap properties. As both the QPI and the INS spectra arise due to
scattering of the Bogolyubov quasiaprticles, they exibit an one-to-one correspondence of the scattering vectors
and the energy scales. We show that these two spectroscopies complement each other in such a way that a com-
parative study allows one to extract the quantitative and unambiguous information about the underlying pairing
structure and the phase of the SC gap. Due to the nodeless and isotropic nature of the SC gaps, both the QPI
and INS maps are concentrated at only two energies in pnictide (two SC gaps) and one energy in iron-selenide,
while the associated scattering vectors q for scattering of sign-changing and same-sign of the SC gaps change
between these spectroscopies. The results presented, particularly for newly iron-selenide compounds, can be
used to test the nodeless d−wave pairing in this class of high temperature superconductors.
PACS numbers: 74.70.Xa,74.55.+v,74.20.Rp,74.25.Jb
I. INTRODUCTION
The most crucial information to unravel the mechanism of
pairing is the structure of the superconducting (SC) gap func-
tion, a measure of the amplitude and the phase of electron
pairs. Although the SC-gap function of conventional phonon-
mediated (attractive pairing interaction) superconductors has
the same sign all over the k-space (s−wave symmetry), that
of spin-fluctuation mediated (repulsivepairing interaction) su-
perconductors is expected to exhibit a sign reversal between
the Fermi momenta connected by the characteristic ‘hot-spot’
wave vector Q of spin-fluctuations.1,2As a consequence of
the sign reversal, nodal planes in which the SC gap van-
ishes should exist in k space. If such nodal planes inter-
sect the Fermi surface (FS), as in d−wave cuprate, nodal
quasiparticle states emerge in the low-energy spectra. How-
ever, if the FS consists of small pockets and do not touch
the nodal line, as s±pairing in iron-pnictide and single
layer iron-chalogenides3or d-wave pairing in double layered
iron-selenide (Fe2Se2),4–7develop despite a sign reversal of
the pairing. Therefore, to precisely establish an unconven-
tional pairing symmetry and separate it from a conventional
s−wave, the relative sign of the SC gap on the FS must be
established.
The magnetic resonance mode develops in the SC state
and was shown theoretically to arise from the sign-flip of the
SC gap at the ‘hot-spot’.6,8–11However, there are other the-
ories which also reproduce the resonance mode without the
requirement of sign change of the SC gap.12–14Alternative
and important new spectroscopy comes from the quasiparti-
cle interference (QPI) pattern, measured by scanning tunnel-
ing microscopy (STM), which in principle visualizes all pos-
sible elastic scattering vectors. It therefore can distinguish
the sign-changing ‘hot-spot’ vectors by studying its evolution
with varying magnetic field.15–18Magnetic field breaks the
time-reversal symmetry of the SC quasiparticles and thus il-
luminate those scattering vectors which scatter quasiparticles
of same pairing phase, thanks to the remarkable properties
of the Bogolyubov coherence factors.19These observations
can be questioned based on at least two arguments: (1) mag-
netic field induced vortex states drastically redistribute the
spectral weight across the ‘bright-spots’ on the constant en-
ergy surface, making this procedure complicated; (2) a mag-
netic impurity always carries a scalar potential which allows
simultaneous scattering of quasiparticles of opposite pairing
phase.17,18We show in this paper that the INS and QPI maps
complement each other and allow determination of the rel-
ative phase. Thus, we propose, a comparative study between
the Inelastic neutron Scattering (INS) spectra and the QPI pat-
tern, taken together, can be a viable tool to quantitatively and
unambiguously determine the relative sign of the SC gap.
Our approach is based on a very simple observation: Both
the INS and the QPI patterns arise from the similar scattering
of the Bogoliubov quasiparticles, forming Cooper pairs (in-
elastic and elastic scatterings, respectively) and thus their ob-
served dispersions, Ω(q)s, must have an one-to-one relation-
shipinthesetwospectroscopies. InINSspectra, ascatteringq
vector will be observed if it connects two quasiparticle states
at kiand kf at which sgn?∆ki
vector will only appear in the QPI spectra if |∆ki| = |∆kf|,
because QPI probes elastic scattering. The comparison be-
tween the INS and the QPI maps can therefore emphasize the
regions of momentum space with opposite sign of the gap
function, connected by scattering momenta. This emphasis
can be further facilitated by studying the magnetic field de-
pendence of the QPI maps. Such a comparative study has
proven to give valuable information about the pairing symme-
try in cuprates,20and heavy-fermions.21
Iron-pnictide:- In the case of iron-pnictide superconduc-
tors, however, the presence of multiple bands at the Fermi
level (EF) and multiple SC gap amplitudes at each bands
make the aforementioned analysis much more complicated
and exotic. In these compounds, the FS consists of discon-
?
?= sgn?∆kf
?and thus its
energy scale will be determined by |∆ki| + |∆kf|. The same
arXiv:1110.3834v1 [cond-mat.supr-con] 17 Oct 2011

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2
nected two concentric hole pockets (namely, α and β pock-
ets) and one electron pocket (γ pocket) centered at Γ and
M points, respectively. Prominent nesting between the hole
pockets and the electron pocket has been shown theoretically
to lead to a s±−pairing in this family of superconductors.3
Within this pairing, all the FS pockets possess nodeless and
isotropic SC gap, but the sign of the gaps is reversed between
the hole pockets and the electron pocket. To complicate the
story, thetwoholepocketsacquireverydifferentmagnitudeof
the SC gaps.11,22,23Therefore, macroscopic geometrical sep-
aration of the two phases, as in the cuprate corner-junction
experiment, can not be performed in these systems to test the
pairing symmetry.24
In our earlier study, we have theoretically shown that be-
cause of the two values of the SC gap in two hole-pockets,
the INS spectra is split into two energy scales.11Furthermore,
as the two hole pockets have very different area as a function
of doping, the corresponding q vectors are also different at
these two resonances. Here, we show that these INS scatter-
ing vectors also appear in two different energy scales of the
QPI patterns. Additional scattering vectors between the states
of same sign of SC gaps, which were prohibited in INS, ap-
pear in QPI. These results are consistent with STM data on
electron doped Ba(Fe1−xCox)2As2as a function of doping.25
Furthermore, the evolution of the QPI pattern as a function of
magnetic field is also consistent with STM studies in single
layer Fe(Se,Te) superconductors.16
Fe2Se2 superconductors:- The recent discovery of high-
Tcsuperconductivity in double layered Fe2Se2based super-
conductors makes the above story even more interesting and
exotic.26–28The FS in this family of superconductors only
hosts electron pockets at M point (no hole-pocket is present
here as in iron-pnictide discussed above). Therefore, the lead-
ing nesting vector occurs between the two electron pockets
which lead to a d-wave pairing (dx2−y2 in 1 Fe unit cell and
dxyin 2 Fe unit cell notation).11Unlike the d-wave pairing in
cuprates, here it gives rise to nodeless and isotropic SC gap
on the FS, in consistent with experiments29–32, but gap func-
tion changes sign between the two electron pockets. We and
others4? ,5have shown earlier that such a d-wave pairing leads
to a magnetic resonance mode at Q which is observed later by
INS measurements.33
Here we show that this INS scattering vector Q will also
show up in the QPI vector at the bias energy equals to the
SC gap magnitude.Furthermore, we phenomenologically
demonstrate that with the application of the magnetic field,
other QPI scattering vectors at which the SC gap do not
change sign can be illuminated. The relative evolution of the
QPI maps as a function of magnetic field will provide a valu-
able test for the relative phase of the SC state on the FS in this
family of superconductors.
The paper is organized as follows. In Sec. II, we provide
the formalism for both QPI and INS spectra. We extract out
very simple equations for these two spectroscopies which can
be used with experimental inputs to reconcile them. In Sec.
III, we present the results for iron-pnictide, whereas the same
results but for layered iron-selenide are given in Sec. IV. The
magnetic field dependence of the QPI map is computed for
layered iron-selenide system in Sec. IVA. Finally, we con-
clude in Sec. V.
II.FORMALISM
A direct correlation between the INS and the QPI spectra
can only be made within the bare level in which the same
green’s functions are involved. This correspondence there-
fore would at least qualitatively be correct for dressed Greens
functions. In fact, one can again evaluate the Green’s func-
tions for the filled state from ARPES spectral weight A, as
G(k,iωn) = π?A(k,iω?)/(iωn− ω?), which is exact for a
band system. In a multiband superconductor, the Green’s
functions for the normal and anomalous part can be written
in the eigenbasis as
single band case, but averaged over orbital indices in a multi-
Gν(k,iωn) =
α2
ν(k)
iωn− Eν(k)+
?
β2
ν(k)
iωn+ Eν(k),
(1)
Fν(k,iωn) = αν(k)βν(k)
×
1
iωn− Eν(k)−
1
iωn+ Eν(k)
?
, (2)
respectively. n is the Matsubara frequency. Here Eν(k) =
±?ξ2
binding parametrization to the material specific LDA bands,
as discussed later. ∆ν(k) = ∆0
pendent SC gap with ∆0
nitude. g(k) is the structure factor for the pairing symmetry
which is taken to be same for all bands in a given system.
?
herence factors for the quasiparticle states ±Eν(k), respec-
tively.
The above Green’s functions can be projected to the orbital
basis as Gsp(k,iωn) =?φs
orbital.
QPI profile:-STM measures local density of state (LDOS)
of quasiparticles which arrive to the tip after going through
multiple intrinsic elastic scattering (due to magnetic and non-
magnetic scatterer17,18,34) in the system. Such scattering vec-
tors can be visualized by Fourier transforming the LDOS into
the q space to obtain17,18,34–36
?
× Vq
+ Fν(k,iΩn)Fν?(−k − q,−iΩn)?.
Here M is the orbital to band matrix element made
of theeigenvectors as
Mνν?
q)φs
important for shaping the spectral weight distributions for
multiband case. Nevertheless, the relative intensity of the
ν(k) + ∆2
ν(k) is the non-interaction band modeled by tight-
ν(k) is the νthBogolyubov quasiparticle
band. ξ2
νg(k) is the momentum de-
νis the band specific SC gap mag-
αν(k)(βν(k)) =
1
2
?
1 ∓
ξν(k)
Eν(k)
?
are the Bogolyubov co-
ν(k)|Gν(k,iωn)|φp
ν(k)?and same
for F. φs
ν(k) represent the eigenvectors of νthband onto sth
Brstu(q,iΩn) =
k,ν,ν?
Mνν?
rstu(k,q)
?Gν(k,iΩn)Gν?(k + q,iΩn)
(3)
rstu(k,q)
The matrix-element can be
=φr∗
ν(k +
ν?(k)φt∗
ν?(k)φu
ν(k + q).

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scattering vectors is determined by the nesting conditions
and the SC gap amplitude for elastic scattering. In addition,
a scattering matrix-element, C(k,q), appears due to the
coherence factors, αν(k) and βν(k), of the Bogolyubov
quasiparticles which is sensitive to the momentum-dependent
phase of the SC order parameter and the symmetry of the
scattering potential, Vq.15,19Taking these facts into account,
we simplify Eq. 3 to calculate the total QPI map as
?
×δ?ΩQPI− Eν(k)?δ?ΩQPI− Eν(k + q)?.
[Here we assumed M = 1 for simplicity and took analytical
continuation iΩn → ΩQPI + iη (η is a small broaden-
ing).]The explicit form of the scattering matrix element
is Cνν?(k,q) =
q)]αν(k + q)βν(k + q)?2, where the negative and posi-
time-reversal) and a magnetic (odd under time-reversal)
potential, respectively.15At EF, it even simplifies to
Cνν?(kF,q) =?sgn[∆ν(kF)]∓sgn[∆ν?(kF+q)])2. There-
we can write the conditions to obtain a non-vanishing QPI
vector as
B(q,ΩQPI) ≈ Vq
k,ν,ν?
Cνν?(k,q)
(4)
?sgn[∆ν(k)]αν(k)βν(k) ∓ sgn[∆ν?(k +
tive sign represent scattering through a scalar (even under
fore, focusing on the low-energy region where ΩQPI ≤ ∆,
Ωνν?
QPI(q) = |∆ν(kF)??=??∆ν?(kF+ q)??,
sgn?∆ν(kF)?= sgn?∆ν?(kF+ q)?for mag.imp.
(5)
(6)
(7)
sgn?∆ν(kF)??= sgn?∆ν?(kF+ q)?for scalar imp.,
Note that as magnetic impurity is always associated with
a scalar potential, it, in principle, involves QPI scatterings
which satisfies Eq. 6 as well but its intensity will depend on
the relative strength of the potential.
INS spectra:-The calculation of the INS spectra follows
similarly with the exception that the latter is an inelastic scat-
tering of the quasiparticle spectra. INS probes the imaginary
part of the susceptibility which can be written in the orbital
basis as11,37
χ0rstu(q,iΩm) = −1
×?Gν(k,iωn)Gν?(k + q,iωn+ iΩm)
2
?
k,n,ν,ν?
Mνν?
rstu(k,q)
+ Fν(k,iωn)Fν?(−k − q,−iωn− Ωm)?.
We have shown earlier in Ref. 11 that in many cases the dif-
ference between realistic matrix element and matrix element
assumed to be a smooth function of energy and momentum
and thus is not important in calculating the magnetic scatter-
ing structure. The INS dispersion is mostly governed by the
(8)
locus of the discontinuous jumps in χ0and due to Kramers-
Kroning relationsship, corresponding χ??
same location [within random-phase approximation (RPA),
the peak position shifts slightly to a lower energy]. Similar
to Eq. 4 above, we absorb the INS scattering matrix-element
term in Cνν?(k,q) and performing the Matsubara summation
in Eq. 8, we obtain the total BCS χ0as
?
×δ?ΩINS− Eν(k) − Eν(k + q)?.
Here, the explicit form of C is Cνν?(k,q) = βν(k)αν?(k +
q)?αν(k)βν?(k + q) − βν(k)αν?(k + q)?
This implies that the magnetic structure in BCS χ0below
ΩINS≤ 2∆ is entirely governed by two conditions:6,8–11
0attains a peak at the
χ0(q,ΩINS) ≈
k,ν,ν?
Cνν?(k,q)
(9)
=
?1 −
sgn[∆ν(kF)]sgn[∆ν?(kF+ q)]?at EF.
Ωνν?
sgn?∆ν(kF)??= sgn?∆ν?(kF+ q)?.
INS(q) =??∆ν(kF)??+??∆ν?(kF+ q)??,
(10)
(11)
We will use QPI Eqs. 5-7 and INS Eqs. 10 & 11 to perform
a comparative analysis of the two spectroscopic data in iron-
pnitide and iron-selenide superconductors.
III.RESULTS ON IRON-PNICTIDE
FSs in pnictide:-The low-energy Hamiltonian of Iron-
pnictide system is dominated by five d-orbitals of the Fe
atoms. We take the tight-binding model from Ref.37where
the parameters are fit to the corresponding first-principles dis-
persion. The doping is evaluated within rigid-band shift ap-
proximation. At x = 0.2, the computed FS consists of two
concentric hole pockets at Γ points which are called α- [in-
ner pocket as depicted by red line in Fig. 1(a)] and β-pockets
(blue line) and one electron pocket at M point (green line).
All results in the this paper are presented in the unfolded BZ
coming from 1 Fe unit cell.
SC gap properties:-From the shape of the FS pockets, there
are at least four interpocket scattering channels exist in pnic-
tide which span along various high-symmetry q directions
as shown by arrows of different colors in Fig. 1(a). Among
the four vectors, Mazin et al.3have shown theoretically that
the nesting for q1and q2are the strongest which lead to a
sign-changing s±-pairing symmetry in this class of supercon-
ductors. This phase symmetry of the SC gap is consistent
with the spin-fluctuation mechanism of electron-pairing. The
pairing and SC gap amplitudes have three essential properties
which are relevant to our present study: (1) SC gap changes
sign between the electron and hole pockets (black to white
background colors in Fig. 1(a) reflect the s±-pairing symme-
try), (2) SC gap magnitude on all FS pockets is nodeless and

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FIG. 1. (a) Computed FS for iron-pnictide within five band tight-
binding model at a representative hole doping of x = 0.2. The
BZ is chosen for 1 Fe unit cell notation. The black to white back-
ground depicts the s±-pairing symmetry which takes the form of
2cos(kxa)cos(kya) in the 1 Fe unit cell. The arrows give different
interband scattering channels which constitute QPI and INS maps.
(b1) According to Eq. 5, the QPI map at
only reveals the intraband scattering (schematic) within the β− FS
pocket. (b2) All possible interband scatterings from α → γ-FS (q1)
and from γ → γ-FS, (q3,4), but not from β → γ-FS (q2) become
turned on at Ω2
INS, the scattering between α- or β- to γ-FSs is only allowed which
changes sign of the SC gap, according to Eq. 11. q2for β → γ-FS
appears at Ω1
appears at ΩINS = |∆α| + |∆γ|. As |∆α| ≈ |∆γ|, INS spectrum in
(c2) will resemble QPI map in (b2) at zero magnetic field. Note that
in the INS calculation, we have not included the Umklapp scattering
which will symmetrize the INS spectra with respect to the Umklapp
vector Q. This is done to facilitate the direct comparison with QPI
maps which are not calculated using Umklapp scattering to mimic
the experimental procedure.
??∆β
??<
??∆α
??≈
??∆γ
??
QPI. (c1-c2) Due to inelastic scattering process in
INS= |∆β| + |∆γ| in (c1). (c2) q1 for α → γ-FS
isotropic, and (3) evidence from ARPES22, STM25and nu-
merous bulk probes23indicate that,
|∆α| ≈ |∆γ| ≈ 2|∆β|,
(12)
at all dopings and for both electron and hole dopings.
Sketch of QPI and INS maps:- All the aforementioned FS
and SC gap properties lead to a very different QPI and INS
properties in pnictide, than the one obtained in single band
cuprates.15,36
(i) Nodeless and isotropic nature of the SC gaps make
all the QPI and INS maps concentrate at only two energy
scales, insead of a characteristic dispersion seen in nodal and
anisotropic d-wave gap in cuprates. QPI maps will be promi-
nent only at Ω1
QPI=
obeying Eq. 5. While, the INS maps will show up only at
Ω1
Eq. 10.
(ii) At Ω1
QPI=
allowed as
Fig. 1(b1) only shows intra-β-FS scattering which concen-
trate near q = 0. As SC gap does not change sign on the
each pocket, some finite magnetic field will be necessary to
illuminate these small q vectors. Of course, in real material,
the finite scale broadening can introduce some quasiparticle
scates of α, γ bands near ΩQPI =
intensity at q2(at zero magnetic field).
(iii) At Ω2
QPI=
vector q1, intraband scattering vectors q3and q4appear in
Fig. 1(b2). q3and q4become illuminated at zero magnetic
field while q1will gain more intensity with finite magnetic
field, in consistent with the data from Ref. 16.
(iv) At Ω1
INS=
through INS study in Fig. 1(c1). Note that, among the four
maps shown in Fig. 1, this is the only place where q2can be
determined precisely.
(v) Similarly, at Ω1
INS=
observed as sketched in Fig. 1(c2). At zero magnetic field,
this INS spectra will match exactly with QPI pattern shown in
Fig. 1(b2).
(vi) Doping dependence of the q vectors and their energy
scales (not studied here) can also be used to gain confidence
on the FS topology and the location of the sign reversal of the
pairing symmetry. The INS energy scales, Ω1,2
like behavior with doping in accord with the dome like behav-
ior of the SC gaps as calculated in Ref. 11. By implication,
the same doping dependence is expected in Ω1,2
a multi SC gaps pnictide system, such one-to-one correspon-
dence is possible as all gaps show similar dome-like doping
dependence.22,23,25The area of the hole pockets increases with
hole doping; simultaneously the same for the electron pocket
decreases. This doping dependence is reversed for electron
doping. Therefore, all q vectors will characteristically fol-
low the doping dependence of their corresponding connecting
FSs.11Interestingly, near x = 0.15 of electron doped side,
α pocket disappears.11,38Therefore, q1should also disappear
and hence Ω2
are yet to be confirmed experimentally.
??∆β
??, no interband elastic scattering is
??and Ω2
QPI=
??+??∆γ
Therefore, the QPI map, in
??∆α
??≈
??∆γ
??,
INS=??∆β
??+??∆γ
??∆β
??and Ω2
??∆α,γ
INS=??∆α
??.
??, according to
??∆β
??<
??∆β
??to visualize weak
??∆α
??≈
??∆γ
??, the interband scattering
??∆β
??+??∆γ
??, q2vector can be observed
??∆α
??+??∆γ
??, q1vector will be
INSobtain dome
QPI. Even in
QPI(at zero magnetic field), Ω2
INS. These results
A.Computed QPI maps of iron-pnictide
We compute the QPI maps for pnictide using Eq. 4 for the
FSs given in Fig. 1(a). All results presented in Fig. 2 scales
linearly with the SC gap magnitude as the relationship given
in Eq. 12 is maintained always. We will not include the scat-

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FIG. 2. (a1) Computed QPI map at Ω1
Fig. 1(b1), at this energy most intensed q vectors intraband scatter-
ing within β−FS which concentrate near q = 0. The weak inten-
sity at q2comes from finite broadening of the quasiparticle states at
γ-pocket which allows some spectral weight of this band to appear
around Ω1
|∆γ|. (b1)-(b2) Corresponding experimental QPI data at these two
energy scales for electron doped pnictide at x = 0.06 taken from
Ref. 25. Experimental data show all four q vectors depicted in Fig. 1
at two energies, although their relative intensities are representative
of our calculations in (a1) and (a2), respectively, due to low experi-
mental resolution. As our phenomenological calculation do not cap-
ture the actual intensity of QPI maps, we normalize all QPI maps to
their maximum.
QPI= |∆β|. As sketched in
QPI= |∆β|. (a2) Same as (a1) but at Ω2
QPI= |∆α| ≈
tering matrix-element C [i.e., Eqs. 6 and 7] in this case to
theoretically study the behavior of all q vectors as a function
of energy. The computed results follow the similar behavior
as sketched in Fig. 1, demonstrating that most of the evolution
of the QPI maps can be understood from the simple energy
and momentum conservation rules derived in Eqs. 5- 7.
All the intraband scattering vectors lie so close to the strong
elastic peak at q = 0 that it is often difficult to distinguish. We
have taken broadening to be η = 1 meV which is sufficient for
the quasiparticle states at ω = |∆γ| to extend up to ω = |∆β|,
allowing some elastic scattering at q2in Fig. 2(a1) (although
it is prohibited in a clean limit). But our small broadening
does not create visible intensity at all other interband vectors,
although the experimental data at the corresponding energy
shows some finite intensity at them, compare Fig. 1(a1) with
corresponding experimental data in Fig. 1(b1).
At Ω2
tors between α to γ FS appear on the QPI map as shown in
Fig. 2(a2). As at Ω1
due to residual broadening of the β states upto γ pocket. The
separation between q1and q2can be studied more clearly in
the overdoped region of hole doped side where the areas of
QPI= |∆α| ≈ |∆γ|, all interband scattering vec-
QPI, q2vector can only show up at Ω2
QPI
FIG. 3. (a) Computed INS map at Ω1
cell notation. The intensity peak shifts away from (π,0), implying
that the corresponding resonance peak is incommensurate due to the
shape of the FSs. (b) Same as (a) but at Ω1
peak is closer to the commensurate vector than q2 in (a) as α-FS
pocket is smaller than the β-one. The q values are strongly depen-
dentontheFSareas, henceondoping. Theresultsareconsistentwith
the calculated spin-excitation dispersion plot presented in Ref. 11 in
2 Fe unit cell. Both the incommensurate and the commensurate reso-
nance peaks are observed in INS experiments.40,41Note that the QPI
maps shown in Figs. 2(a1) and 2(a2) will match exactly with (a) and
(b), respectively, if a strong magnetic field is applied in the former
case to eliminate the scattering of same sign of SC gaps.
INS= |∆β| + |∆γ| in 1 Fe unit
INS= |∆α| + |∆γ|. q1
the α and β bands are distinguishably different.
The experimental data in Fig. 1(b2) also shows all the cal-
culated q vectors. Subtle discrepancies in the relative inten-
sities of each q vector is expected, because we have not in-
cluded any matrix-element M, and the scattering coherence
factor C in this calculation. Furthermore, the magnitude of
each q vectors do not match quantitatively with our calcula-
tion as the calculation is done for hole doping while the exper-
imental data are available only for electron doping, although
in both cases all three FS pockets are present.
Once magnetic field is applied, the relative intensity of q1
with respect to that of q3,4evolves with the strength of the
field, it has been observed in Fe(Se,Te) compounds, in accord
with our calculatoins.16–18
The proximity of q4to the reciprocal vector (2π,0) and
its equivalent directions, which has been observed both in
pnictide25as well as in iron-chalcogenide16, has been ar-
gued theoretically to arise from Bragg peak, instead of QPI
scattering.39Comparing the evolution of the intensity of q4at
two energies Ω1
can deduce that it is not associated with Bragg peak as the
latter does not have any energy dependence while q4has.
QPIin Fig. 1(a1) and Ω2
QPIin Fig. 1(a2), we
B. Computed INS maps of iron-pnictide
The INS spectra, a direct measure of χ??is calculated using
Eq. 9 and the results are shown in Figs. 3(a) and 3(b) at two
energies where INS spectra is finite. The present phenomeno-
logical approach does not include the overlap matrix-element
M and the RPA correction. We have shown earlier in Ref. 11

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6
that they do not change the essential features of the INS for
these systems.
q2and q1appear at Ω1
pected, because they involve sign change of the SC gaps be-
tween the initial and final states. q2being smaller than q1,
will lead to an incommensurate resonance while the latter
is close to the commensurate one (both are doping depen-
dence as described above). Both the commensurate40and
the incommensurate41resonances have been detected by INS
measurements, although the simultaneous presence of the two
modes is yet to be detected in future measurements with better
experimental resolution.
As mentioned earlier, the INS maps will correspond to the
QPI maps if the latter is performed at zero magnetic field.
INSand Ω2
INS, respectively as ex-
IV.LAYERED IRON-SELENIDE
FS properties:-We turn next to the layered AyFe2−xSe2
based systems. These materials host only electron pockets
at M point with no hole pocket.42To model such electronic
structure, we employ two band tight-binding calculation of
the t2gorbitals of Fe d bands and the parameters are obtained
by fitting to the material specific LDA dispersion.6The result-
ing FSs are shown by green lines in Fig. 4(a), which match
well with ARPES FS.42
SC gap properties:-The absence of hole pocket results in
strong nesting between the electron pockets along q3. We
have shown earlier that such nesting lead to nodeless and
isotropicdx2−y2−pairingsymmetry[in2Feunitcellthepair-
ing symmetry becomes dxy7] and a spin resonance near the
commensurate vector q3.6The spin-resonance has recently
been found experimentally in this class of material by Park
et al.33.
QPI and INS spectra:-Due to one SC gap and one FS (con-
centric electron pockets), both the QPI and INS maps appear
only at one energy scale. The QPI map in this material as
shown in Fig. 4(b1) resembles the QPI map for pnictide in
Fig. 2(b2), with the exception that the electron-hole scatter-
ing q2is absent here. On the other hand, the INS spectra
[Fig. 4(c)] is rotated by 45oin comparison with the same plot
of pnictide in Fig. 3(b) as the ‘hot-spot’ for sign-reversal SC
gap is now aligned along q3. It is interesting to note that al-
though q3scattering is also present in pnictide as seen in QPI
maps, but the leading nesting shifts along q1, giving a very
different pairing symmetry. This observation leads us to con-
clude that subtle differences between the nesting and the scat-
tering which can carry rich physical insight can be untangled
more clearly by comparing the QPI with INS spectra.
A. Magnetic field dependent QPI maps
As mentioned before, another way to distinguish the scat-
tering and nesting vector in QPI map is to study its mag-
netic field dependence of it at fixed energy. At zero magnetic
field, the scattering matrix element C only allows scattering
of quasiparticle states of opposite phase of the SC gap, see
FIG. 4. (a) Computed FS for layered iron-selenide KxFe2Se2system
within two band tight-binding model at a representative in 1 Fe unit
cell notation. The black to white background depicts the dx2−y2-
pairing symmetry which takes the form of cos(kxa) − cos(kya) in
1 Fe unit cell and dxy-wave in 2 Fe unit cell (not shown).6,7The ar-
rows give two interband scattering channels that survive in this class
of material, compared to iron-pnictide in Fig. 1(a). (b1) Sketch of the
QPImapaccordingtoEq.5(theassociatedscatteringmatrix-element
which determines the sign of the SC state of the initial and final states
is ignored here). (b2) Computed QPI map shows all q-vectors pre-
dicted in (b1). The small differences in the magnitude of q-vectors
from the two concentric electron pocket is not distinguishable due to
finite broadening. (c1) In the INS spectra, q4does not show up as it
connects the quasiparticle state of same sign of SC gap. (c2) Corre-
spondingcomputationalresultofINSspectraatΩINS = 2ΩQPI. The
resonance spectra is commensurate at (π,π), while the inclusion of
orbital matrix-element M shifts it to a slightly incommensurate one
at a critical value of the interaction U.6Our prediction agrees with
recent INS measurements in this class of materials.33
Eq. 6. Therefore, the resulting QPI map at ΩQPIwill resem-
bles the INS spectra at ΩINS= 2ΩQPI, although the q = 0
elastic peak will be difficult to remove in the former case.
At finite magnetic field, the other scattering channel, q4, at
whichthescatteringfromstatesofsamesignoftheSCgapbe-
comes turned on as shown in Fig. 5(b). In practice, a magnetic
impurity is always associated with a scalar component (not

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7
FIG. 5. (a) Computed result of QPI pattern in KxFe2Se2using Eqs. 5
and 6 which mimics the zero magnetic field (B) condition at which
those QPI vectors become illuminated which correspond to the scat-
tering of states having opposite sign of the SC gaps.15(b) Same as
(a) but using Eqs. 5 and 7. This means, scatterings of same sign of
the quasiparticle states are only included which mimic a finite ap-
plied magnetic field condition. In principle, a magnetic impurity has
a scalar component which will allow some scattering for q3vector as
well but relatively weak in intensity. (c) Differences between (a) and
(b). The positions of maxima and minima corresponds to q3and q2,
respectively. an unambiguous method to find out which scattering
channel involve sign change of the SC gap, a test which will confirm
the presence of d-wave gap in these materials.
consideredinthepresentcalculationforsimplicity), therefore,
q3should also be present. The relative intensity can be moni-
tored by tuning the strength of the magnetic field.15,16On the
otherhand, the difference between QPI maps in Fig. 5(a) and
5(b), as shown in Fig. 5(c), can also be used to identify the
sign-changing ‘hot-spot’ vector. The positions of maxima and
minima corresponds to q3and q2, respectively. The experi-
mental detection of the maxima and minima can be tested to
confirm the presence of d-wave gap in these materials.
V.CONCLUSIONS
We have developed a phenomenologically that allows one
to connect the INS and QPI maps. We point that both INS
and QPI maps arise due to the quasiparticle scattering of the
Bogoliubov quasiparticles involved and bear a one-to-one cor-
respondenceasafunctionofscatteringvectorandenergy. INS
scattering map deals with inelastic scattering whereas QPI is
generated by elastic scattering of same quasiparticles. In fact,
inelastic and elastic scattering complements these two spec-
troscopies to quantitatively and unambiguously identify the
nature of pairing symmetry in unconventional superconduc-
tors. By applying our model in iron-pnictide and layered iron
selenide compounds, we show that the QPI maps at zero mag-
netic field corresponds exactly to the INS spectra at their rep-
resentative energy (the q = 0 elastic scattering and other spu-
rious effects in QPI maps can be ignored because of “contam-
ination from Bragg peaks”). We also show that upon applying
magnetic field , the QPI scattering at same sign of the SC gaps
can be illuminated in layered iron-selenides and produce de-
tectable changes in QPI. We point that evolution of the QPI
maps can be implemented experimentally to test the possible
nodeless d-wave pairing in this class of materials.
ACKNOWLEDGMENTS
We are grateful to Y. K. Bang, T. Hanaguri, H. Takagi, J.-
X. Zhu, H. H. Wen, H. Ding, N. C. Yeh for useful discussions.
This work was supported by This work was supported by the
U.S. DOE at Los Alamos National Laboratory under contract
No. DE-AC52-06NA25396 and the Office of Science (BES)
and benefited from the NERSC computing allocations.
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