Particle-Hole Transformation in Strongly-Doped Iron-Based Superconductors

0.2 MB·

by  J. P. Rodriguez

epl draft
Particle-Hole Transformation in Strongly-Doped
6
Iron-Based Superconductors
1
0
2

v J.P. Rodriguez1
o
N 1 Department of Physics and Astronomy, California State University, Los Angeles, Cali-
fornia 90032
5
2

]
n
o PACS 74.70.Xa–Pnictides and chalcogenides
c PACS 74.20.Rp–Pairing symmetries (otherthan s-wave)
-
r PACS 74.25.Jb–Electronic structure (photoemission, etc.)
p
u Abstract –An exact particle-hole transformation is discovered in a local-moment description of
s a single layer in an iron-based superconductor. Application of the transformation to a surface
.
t layer of heavily electron-doped FeSe predicts a surface-layer high-temperature superconductor at
a
m strongholedoping. Comparisonwithexistinglow-Tc ironsuperconductorssuggeststhatthecritical
temperature at heavyhole dopingcan be increased by increasing direct ferromagnetic exchange in
-
d between nearest neighbor iron atoms.
n
o
c
[

2
v
Introduction. – The discovery of iron-based superconductors has identified a new
9
route in the search for high critical temperatures [1]. Iron atoms in these materials lie in
7
4 weaklycoupledstacksofsquarelattices[2]. Electronicconductionresideswithinsuchlayers,
7 wherechargecarriersareprimarilyelectrons/holesfromiron3dlevels. Theoptimumcritical
0 temperature in iron-pnictide materials, in particular, coincides with nesting between hole
.
1 FermisurfacespocketsatthecenteroftheBrillouinzoneandelectronFermisurfacepockets
0 atcommensuratespin-densitywave(cSDW)momenta¯h(π/a)xˆand¯h(π/a)yˆ. Here,xˆandyˆ
6
areunitvectorsthatpointalongtheprincipalaxesofthesquarelatticeofironatoms,while
1
a denotes the lattice constant. Strong hole doping can destroy such nesting. In particular,
:
v the electron bands at cSDW momenta rise completely above the Fermi level in the series
i
X of compounds (Ba1−xKx)Fe2As2 at 0.5 < x < 0.7 [3,4]. Angle-resolved photoemission
spectroscopy (ARPES) on the end-member of the series KFe As , with T = 4 K, reveals
r 2 2 c ∼
a only hole Fermi surface pockets at zero two-dimensional (2D) momentum [5].
Strong electron doping can also destroy nesting in iron-based superconductors. ARPES
onamonolayerofFeSe overa dopedSrTiO (STO)substrateandonintercalatedFeSe find
3
onlyelectronFermisurfacepocketsatcSDWmomenta[6–8]. Holebandsatthecenterofthe
Brillouin zone lie buried below the Fermi level. Unlike heavily hole-doped compounds like
KFe As , however, the FeSe surface layer shows high critical temperatures, T 40 - 100
2 2 c
∼
K [9,10]. In addition, ARPES [7,8] and scanning tunneling microscopy (STM) [11,12] on
suchsurfacelayersofFeSe findevidenceforanisotropicgapoverthe electronFermisurface
pockets, with no nodes. Last, a Mott insulator phase is reported nearby at low electron
doping in single-layer FeSe/STO and in voltage-gate tuned thin films of FeSe [13,14]. This
indicates that the limit of strong on-site electron repulsion [15,16] is a valid starting point
p-1
J.P. Rodriguez1
filling, bands Jk <J⊥ Jk >J⊥
1 1 1 1
half filling, none hidden ferromagnet: (π,0,0) hidden N´eel: (π,π/a,π/a)
hole dope, hole bands @ Γ hidden half metal, FS @ Γ nested cSDW metal?
e− dope, e− bands @ M nested cSDW metal? hidden half metal, FS at M
Table 1: Groundstate of two-orbital t-J model (1). Hund coupling is tuned to the QCP at half
filling,whichseparatesacSDWatstrongHundcouplingfromhiddenmagneticorderatweakHund
coupling (ref. [17]). The 3-vector(π,Qx,Qy) describes the hiddenmagnetic order.
to describe superconductivity in heavily electron-doped FeSe.
Below, we identify a particle-hole transformation of a local-moment description for a
single layer in an iron-based superconductor [15–17]. The t-J model that results includes
the minimum d and d iron orbitals [18,19]. Above half filling (electron doping), mean
xz yz
field and exact calculations based on a hidden half metal state that it harbors predict
electronic structure that is very similar to that shown by high-T surface layers of FeSe
c
[20]. TheexactcalculationsalsopredictisotropicCooperpairsattheelectronFermisurface
pockets,inadditiontoremnantisotropicCooperpairsofoppositesignonburiedholebands.
Application ofthe particle-holetransformationto a surface layerof FeSe predicts a surface-
layer iron-based superconductor that is heavily hole-doped, and that exhibits high T [21].
c
We suggest that heavily hole-doped iron superconductors such as KFe As show relatively
2 2
low critical temperatures because of non-ideal Heisenberg exchange coupling constants.
Local Moment Hamiltonian. – Our starting point is a two-orbital t-J model over
the square lattice, where on-site-orbital Coulomb repulsion is infinitely strong [18,19]:
H = X[−(tα1,βc†i,α,scj,β,s+h.c.)+J1α,βSi,α·Sj,β]+
hi,ji
X [−(tα2,βc†i,α,scj,β,s+h.c.)+J2α,βSi,α·Sj,β]+
hhi,jii
X(J0Si,d−·Si,d++U0′n¯i,d+n¯i,d−+limU0→∞U0ni,α,↑ni,α,↓(cid:1). (1)
i
Above, S is the spin operator that acts on spin s = 1/2 states of d = d and
i,α 0 (x−iy)z
−
d+=d orbitalsαinironatomsatsitesi. Repeatedorbitalandspinindicesin(1)are
(x+iy)z
summed over. Nearest neighbor and next-nearest neighbor Heisenberg exchange across the
links i,j and i,j is controlled by the coupling constants Jα,β and Jα,β, respectively.
h i hh ii 1 2
Hopping of an electron in orbital α to an unoccupied neighboring orbital β is controlled by
thematrixelementstα,β andtα,β. Finally,J isaferromagneticexchangecouplingconstant
1 2 0
that imposes Hund’s Rule. The last term in (1) suppresses double occupancy at a site-
orbital, where n =c† c is the occupation operator for a spin-s electron in orbital
i,α,s i,α,s i,α,s
α atsite i. The next-to-lasttermin (1) measuresthe energy cost, U′ >0, of a pair of holes
0
at an iron site, on the other hand, where n¯ = 1 n counts holes at site-orbitals
i,α −Ps i,α,s
below half filling. Observethatn¯ can be replacedby n¯ , which counts singletpairs at
i,α i,α
−
site-orbitals above half filling. Last, notice that the operation d e±iθd is equivalent
± → ±
to a rotation of the orbitals by an angle θ about the z axis. Spin and occupation operators
remain invariant under it. Magnetism described by the two-orbital t-J model (1) is hence
isotropic, which suppresses orbital order and nematicity [22].
Becausethe spin-1/2moments live onisotropicd orbitals,two isotropicnearestneigh-
±
bor and next-nearest neighbor Heisenberg exchange coupling constants exist:
Jk =Jd±,d± and J⊥ =Jd±,d∓ (n=1,2). (2)
n n n n
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Particle-Hole Transformation in Strongly-Doped Iron-Based Superconductors
(A) HOLE DOPING (B) ELECTRON DOPING
Y Y
G coherent 3d yz G
R incoherent 3d yz R
E E
N Fermi level N
E E
0 π/a 0 π/a
k k
x x
π/a 3d xz π/a
3d yz
y y
k k
0 0
0 π/a 0 π/a
k k
x x
Fig. 1: Electronic structure of half metal states characterized by hidden magnetic order depicted
by insets to Fig. 2. Dispersions in energy are fixed at wavenumberky =0.
The isotropy of the d orbitals also implies intra-orbital hopping matrix elements that are
±
isotropic and real: tk =td±,d± for n=1,2. Finally, the reflection properties of the d and
n n xz
d orbitalsalsoimplyrealinter-orbitalhoppingmatrixelementsbetweennearestneighbors
yz
with d-wavesymmetry [19]: t⊥(xˆ)= t⊥(yˆ), where t⊥ =td±,d∓. Inter-orbitalnext-nearest
1 − 1 1 1
neighbor hopping matrix elements td±,d∓ also show d-wave symmetry, but they are pure
2
imaginary. They consequently result in hybridization of the d and d orbital bands.
xz yz
Table 1 summarizes the expected phase diagram of the two-orbital t-J model (1) near a
quantum critical point into hidden magnetic order [17].
Particle-Hole Transformation. – We first define the particle-hole transformation
that is relevant to iron-based high-temperature superconductors in momentum space for
electrons in either the d or d orbitals. The corresponding electron destruction operator
xz yz
reads
1
cs(k0,k)= −1/2XXe−i(k0α+k·ri)ci,α,s, (3)
N
α=0 i
where = 2N denotes the number of sites-orbitals on the square lattice of iron atoms,
Fe
N
and where the indices 0 and 1 denote the d and d+ orbitals α. The quantum numbers
−
k = 0 and π therefore represent the d and the ( i)d orbitals. We then define the
0 xz yz
−
particle-hole transformationby the replacements
c (k ,k) c (k ,k+Q )† and c (k ,k)† c (k ,k+Q ), (4)
s 0 → s 0 k0 s 0 → s 0 k0
whereQ =(π/a)yˆandQ =(π/a)xˆ. Figure1displaystheactionoftheabovetransforma-
0 π
tiononelectronicstructure. Whatthenistheformoftheaboveparticle-holetransformation
in real space for electrons in d orbitals? Comparison of (3) and (4) yields the equivalent
±
particle-hole transformationin real space:
c ( 1)yi/ac† and c† ( 1)yi/ac (5)
i,α,s → − i,pi(α),s i,α,s → − i,pi(α),s
p-3
J.P. Rodriguez1
(A) MOBILE HOLE AT J = 0 (B) MOBILE ELECTRON AT J = 0
0 0
-34 -34
-35 -35
⊥J)1 -36 ||J)1 -36
Y ( Y (
G G
R R
E E
N N
E -37 E -37
S = 1/2, even
SS == 13//22,, oedvedn 3d (x+iy)z
S = 3/2, odd
-38 S = 5/2, even -38 3d (x−iy)z
S = 5/2, odd
S = 7/2, even
S = 7/2, odd incoherent 3d xz hole
coherent 3d xz & yz holes coherent 3d yz electron
-39 incoherent 3d xz & yz electrons -39 incoherent 3d yz hole
(1,1) (0,0) (1,0) (2,0) (2,1) (2,2) (1,1) (0,0) (1,0) (2,0) (2,1) (2,2)
MOMENTUM (π/2a) MOMENTUM (π/2a)
Fig. 2: Exact spectra for t-J model, Eq. (1), over a periodic 4×4 lattice, with parameters (a)
Jk = 0, J⊥ > 0, Jk = 0.3J⊥ = J⊥, tk = −3J⊥, t⊥(xˆ) = −2J⊥, t⊥(yˆ) = +2J⊥, tk = −J⊥,
1 1 2 1 2 1 1 1 1 1 1 2 1
and td±,d∓ = 0 in the mobile-hole case. Model parameters transform to (b) Jk > 0, J⊥ = 0,
2 1 1
Jk = 0.3Jk = J⊥, tk = 2Jk, t⊥(xˆ) = +3Jk, t⊥(yˆ) = −3Jk, tk = −Jk, and td±,d∓ = 0 in the
2 1 2 1 1 1 1 1 1 2 1 2
mobile-electron case.
where p (d ) = d for iron sites i on the A sublattice of the checkerboard, and where
i
± ±
p (d )=d for iron sites i on the B sublattice of the checkerboard.
i
± ∓
Making the replacements(5) in the two-orbitalt-J modelHamiltonian (1) maintains its
form. Nearest neighbor model parameters, however,transform to
J¯k =J⊥ and J¯⊥ =Jk,
1 1 1 1
t¯k = t⊥(xˆ) and t¯⊥(xˆ)= tk, (6)
1 − 1 1 − 1
with t¯⊥(yˆ) = t¯⊥(xˆ). Next-nearest neighbor model parameters tk Jk and J⊥ remain un-
1 − 1 2 2 2
changed, while corresponding inter-orbital hopping matrix elements, which are pure imagi-
nary with d-wave symmetry [19], transform to
t¯d±,d∓ =+td±,d∓ and t¯d±,d∓ = td±,d∓ . (7)
2 |A 2 |A 2 |B − 2 |B
Last, on-site parameters J , U and U′ for ferromagnetic Hund coupling and for Coulomb
0 0 0
repulsion also remain unchanged. Here, the occupation operators n in the divergent
i,α,s
Hubbard term must be replaced by 1 n .
i,α,s
−
Emergent Electron/Hole Bands. – We shall now compare spectra for one mobile
holeandforonemobileelectron,witht-J modelparametersthatarerelatedtoeachotherby
the previous particle-hole transformation. In the hole-doped case, the Heisenberg exchange
coupling constants are set to Jk = 0, J⊥ > 0, and Jk = 0.3J⊥ = J⊥, while the hopping
1 1 2 1 2
matrix elements are set to tk = 3J⊥, t⊥(xˆ) = 2J⊥, t⊥(yˆ) = +2J⊥, tk = J⊥ and
1 − 1 1 − 1 1 1 2 − 1
td±,d∓ = 0. The latter turns off hybridization between the d and d orbital bands. In
2 xz yz
the electron-dopedcase,nearestneighbort-J modelparametersaresetby(6), whileon-site
p-4
Particle-Hole Transformation in Strongly-Doped Iron-Based Superconductors
and next-nearest neighbor model parameters are unchanged. Figure 2 shows exact spectra
for one mobile hole and for one mobile electron roaming over a periodic 4 4 lattice of
×
iron atoms, in the absence of Hund’s Rule, J =0. Details of the numerical calculation are
0
given in the Supplemental Material and in ref. [19]. Notice that all of the states obey the
particle-hole transformation (4). Figure 3 shows the one-hole and one-electron spectra at
Hund coupling J =2.04J(⊥)k. All states again satisfy the particle-hole relationship (4).
− 0 1
The dispersionofthe lowest-energyspin-1/2mobile-holestatesshownbyFig. 2acanbe
understoodatidealhopping, achievedby suppressinginter-orbitalhopping: t⊥ 0. Ahalf
1 →
metal characterized by hidden magnetic order depicted by the inset to Fig. 2a is predicted
in the absence of Hund’s Rule at largeelectronspin s [18,19]. Electronsare spinpolarized
0
per d orbital, where they follow a hole-type energy dispersion relation
±
ε(0)(k)= 2tk(cos k a+cos k a) 2tk(cos k a+cos k a), (8)
e − 1 x y − 2 + −
with k =k k . Two degeneratehole Fermi surface pockets at zero 2D momentare pre-
± x y
±
dictedforsmallconcentrationsofmobileholesperorbital,x,eachwithaFermiwavenumber
k a = (4πx)1/2. The top of the hole-type band (8) lies ǫ = tk +2tk (k a)2 above the
F F | 1 2| F
Fermilevel. Theone-electronpropagatorcanbe calculatedwithinaSchwinger-boson-slave-
fermionmean-fieldapproximationofthetwo-orbitalt-J model(1)forthe abovehiddenhalf
metal[18,19]. Inthelimitnearhalffilling,at t J,itrevealscompositeelectron-spinwave
| |≫
statesatanenergyǫ +h¯ω (k)abovetheFermilevel,whereω (k)isthespin-wavedisper-
F sw sw
sion atlargeelectronspin s [17]. (See Supplemental Materialandref. [21].) The predicted
0
dispersion relation is traced by the dashed line in Fig. 2a. It compares well with the exact
dispersion of the lowest-energy spin-1/2 excitations at non-ideal hopping matrix elements,
in the absence of Hund’s Rule, and it notably shows electron-type dispersionin the vicinity
of cSDW wavenumbers (π/a)xˆ and (π/a)yˆ. We therefore interpret the dispersion of those
spin-1/2groundstates,whichrespectivelyhaveevenandoddparityunderorbitalswapPd,d¯,
as emergent d and d electron bands. Figure 3a indicates that nesting of the coherent
xz yz
holebands(8)withtheemergentelectronbandsbeginsatacriticalvalueofHundcoupling.
Applicationoftheparticle-holetransformation(5)yieldsanewhalfmetalstatedepicted
by the inset to Fig. 2b, where the missing spin-1/2 moment in the third row represents a
spin singlet. It is governed by the two-orbital t-J model (1) at electron doping above half-
filling, with a new set of parameters (6) Jk > 0, J⊥ = 0, tk = +2Jk, t⊥(xˆ) = +3Jk,
1 1 1 1 1 1
t⊥(yˆ) = 3Jk, tk = Jk, and td±,d∓ = 0. As tk 0, Schwinger-boson-slave-fermion
1 − 1 2 − 1 2 1 →
mean field theory for the half metal state predicts emergent hole excitations that disperse
according to the dashed lines in Fig. 2b. (See Supplemental Material and ref. [20].) Again,
themean-fieldresultscomparewellwiththeexactones,whileFig. 3bindicatesthatnesting
of the electron bands with the emergent hole bands begins at the prior critical strength in
Hund coupling.
Cooper Pairs with Emergent Sign Changes. – Considernowtwoelectronsabove
half filling that roamovera 4 4 periodic lattice of iron atoms governedby the two-orbital
×
t-J model (1) [20]. Heisenberg exchangeparametersareset to those listed in the captionto
Fig. 2b,butnewhoppingmatrixelementsarechosenthatleavetheelectronmassesm and
x
m per orbital unchanged at cSDW momenta: tk = 2Jk, t⊥(xˆ) = +5Jk, t⊥(yˆ) = 5Jk,
y 1 1 1 1 1 − 1
and tα,β =0. Such model parametersproduce electron-type Fermisurface pockets centered
2
at cSDW momenta in the hidden half metal state within the mean field approximation.
Details of the exact calculation are given in the Supplemental Material and in ref. [21]. In
thecaseofonemobileelectron,andunderthenewsetofhoppingparameters,thelow-energy
spectrum atthe QCP resemblesthat depicted by Fig. 3b under the previous setof hopping
matrix elements, but with emergent hole bands at zero 2D momentum that show small
mass anisotropy[20]. In the case oftwo mobile electrons,the Hund coupling, J , is tuned
0
−
to a putative QCP defined by degeneracy of the spin resonance at cSDW momenta with
p-5
J.P. Rodriguez1
(A) MOBILE HOLE AT QCP (B) MOBILE ELECTRON AT QCP
-55.4 -55.4
-55.6 -55.6
-55.8 -55.8
-56 -56
⊥J)1 ||J)1
Y ( Y (
G -56.2 G -56.2
R R
E E
N N
E E
-56.4 -56.4
S = 1/2, even
-56.6 S = 1/2, odd -56.6
S = 3/2, even
S = 3/2, odd
-56.8 S = 5/2, even -56.8
S = 5/2, odd
(1,1) (0,0) (1,0) (2,0) (2,1) (2,2) (1,1) (0,0) (1,0) (2,0) (2,1) (2,2)
MOMENTUM (π/2a) MOMENTUM (π/2a)
Fig. 3: Exact spectra of two-orbital t-J model at Hundcoupling −J =2.04J(⊥)k. The remaining
0 1
model parameters are given in the caption to Fig. 2.
the lowest-energyspin-1 state at momentum (π/a)(xˆ+yˆ), which is associated with hidden
magneticorder. Thisdefinitionissuggestedbyasemi-classicalanalysisofthecorresponding
Heisenberg model at half filling, which finds a QCP in such case that separates hidden
magneticorderatweakHundcoupling,asdepictedbytheinsettoFig. 2b,fromacSDWat
strong Hund coupling [17]. At zero net 2D momentum, a bound electron-pair groundstate
exists below a continuum of states. It shows S-wave symmetry according to the reflection
parities listed in Table 2. An excited pair state with Dx2−y2 symmetry exists below the
continuum as well.
The order parameter for superconductivity is the defined as
iF(k ,k)= Ψ c˜ (k ,k)c˜ (k , k)Ψ (9)
0 Mott ↑ 0 ↓ 0 Cooper
h | − | i
times √2, where Ψ is the groundstate of the electron pair, and where Ψ is the
Cooper Mott
| i h |
groundstate of the Mott insulator at half filling. Above, the tilde notation signals the limit
U . Figure 4b depicts (9) using exact groundstates Ψ and Ψ on a 4 4
0 Mott Cooper
→ ∞ h | | i ×
periodiclatticeofironatomsattheputativeQCP.Inparticular,theHundcouplingistuned
Table 2: Reflection parities, orbital-swap parity, and spin of low-energy pair states with zero net
momentum in order of increasing energy. The operator Rx′z, for example, denotes a reflection
about the x′-z plane, where x′ is a diagonal axis. The hidden spinwave is exceptional in the case
of electron doping, where it carries net momentum (π/a)(xˆ+yˆ).
no. Pair State Rxz , Ryz Rx′z , Ry′z Pd,d¯ spin
0 S + + + 0
1 Dx2−y2 + + 0
−
2 hidden spinwave 1
− − −
p-6
Particle-Hole Transformation in Strongly-Doped Iron-Based Superconductors
(A) TWO MOBILE HOLES (B) TWO MOBILE ELECTRONS
3d xz 3d xz
0.5 0.5
)
a
π/
2
(
y
k 0 0
0 0.5 0 0.5
3d yz 3d yz
0.5 0.5
)
a
π/
2
(
y
k 0 0
0 0.5 0 0.5
k (2π/a) k (2π/a)
x x
Fig. 4: Complex order parameter for superconductivity,Eq. 9, with Heisenberg exchangecoupling
constants set in thecaption toFig. 2, and with hoppingmatrix elements (a) tk =−5J⊥, t⊥(xˆ)=
1 1 1
−2J⊥, t⊥(yˆ) = +2J⊥, and tα,β = 0 for two mobile holes. Nearest neighbor hopping matrix
1 1 1 2
elements transform to (b) tk = 2Jk, t⊥(xˆ) = +5Jk and t⊥(yˆ) = −5Jk for two mobile electrons.
1 1 1 1 1 1
Also, inter-orbital on-site repulsion is set to U′ = 1J +1000J(⊥)k, while the Hund coupling
0 4 0 1
constantissetto−J =2.25J(⊥)k. Heisenberg-exchangepairsintheHamiltonian (1)arereplaced
0 1
with 1/2 thecorresponding spin-exchangeoperators to reducefinite-size effects.
so that the groundstate spin-1 states at cSDW momenta, which have even parity under
orbital swap, Pd,d¯, become degenerate with the groundstate spin-1 state at wavenumber
(π/a)(xˆ+yˆ),whichhasoddparityunderorbitalswap. Thecouplingconstants,respectively,
are J = 1.35Jk and J = 2.25Jk at half filling and for two mobile electrons. Notice
− 0 1 − 0 1
that the order parameter displayed by Fig. 4b is isotropic, but that it alternates in sign
between the emergent hole bands at zero 2D momentum and the electron bands at cSDW
momenta [20].
Figure 4a shows the particle-hole conjugate of the order parameter(9) for superconduc-
tivity in the two-orbital t-J model with two-mobile holes that roam over a 4 4 periodic
×
lattice, under the transformation (6)-(7) in parameter space [21]. Notice that it is related
to Fig. 4b by the particle-hole transformation (4). In conclusion, both the electron pair
and the conjugate hole pair display an S+− order parameter for superconductivity, with
remnant pairing on the emergent band of opposite sign. This result is similar to a recent
proposal for S+− pairing in heavily hole-doped iron superconductors that is based on a
phenomenological attractive pairing interaction [23].
Discussion and Conclusions. – The particle-hole transformation (4)/(5) can be
applied to strongly electron-doped iron-based superconductors ultimately to predict new
iron-basedsuperconductorsthatarestronglydopedbyholes. Heavilyelectron-dopedsurface
layersof FeSe showrecordcriticaltemperaturesas highas T =100K [10]. ARPESreveals
c ∼
twoelectronpocketsatthe cornerofthe two-ironBrillouinzonethatcrossanddonotshow
levelrepulsion[7]. TheauthorhasrecentlyproposedthattheCooperpairsinthesesystems
p-7
J.P. Rodriguez1
areisotropic,butthattheychangesignaccordingtoFig. 4b[20]. Hedescribestheelectronic
structureatthesurfacelayerofFeSewiththetwo-orbitalt-J model(1)atsub-criticalHund
coupling, with hopping matrix elements and Heisenberg exchange coupling constants that
favorthehalfmetalstateshownintheinsettoFig. 2b. Inparticular,Schwinger-boson-slave-
fermion meanfield theory and the exact results displayedby Fig. 3b predictelectron Fermi
surface pockets centered at the two distinct cSDW momenta. This suggests that the Fermi
surfacepockets arerobustin the presence ofhybridizationbetweenthe d andd orbitals
xz yz
throughthe addition ofinter-orbitalhopping to next-nearestneighborsoverequivalentiron
atoms: td±,d∓ =td±,d∓ . Thelatterarepureimaginaryandshowd-wavesymmetry[19].
2 |A 2 |B
Applicationoftheparticle-holetransformation(5)tothetwo-orbitalt-J modelforasur-
facelayerofFeSeimplies aheavilyhole-dopedsurfacelayerthatshowshole-typeFermisur-
face pocketsatthe centerofthe Brillouinzone(Fig. 1a)andhigh-T superconductivity. By
c
(7),thedifferencebetweentheAandBironsitesismaximalpernext-nearestneighborinter-
orbitalhoppingparameters,whicharepureimaginary: t¯d±,d∓ = t¯d±,d∓ . Iron-pnictide
2 |B − 2 |A
andiron-chalcogenidelayerssatisfythe(d-wave)identitiestd±,d∓(xˆ′) = td±,d∓(yˆ′) and
2 |A − 2 |B
td±,d∓(yˆ′) = td±,d∓(xˆ′) , however, which remain true in the presence of a substrate.
2 |A − 2 |B
Here,xˆ′ andyˆ′ denotetheiron-to-pnictide/chalcogenidedirections. Theparticle-holetrans-
formation, t¯d±,d∓, of the original hopping matrix elements does not satisfy these identities,
2
and it is therefore not valid.
We can, however, consider the action of the particle-hole transformation in the absence
ofhybridization: td±,d∓ =0. Bytable1andby(6),Heisenbergexchangecouplingconstants
2
satisfy J¯k < J¯⊥ after the electron-hole transformation (5). Non-ideal Heisenberg exchange
1 1
coupling constants J¯k > J¯⊥ at hole doping may result in a smaller critical temperature,
1 1
whichcouldaccountforthe relativelylowT ofthe heavilyhole-dopedcompoundKFe As .
c 2 2
Compression of iron atoms on a surface layer can result in direct ferromagnetic exchange,
which in turn can help achieve the ideal inequality J¯k <J¯⊥ at hole doping [19]. The hole-
1 1
dopedseriesBa K Fe As showsaLifshitztransitionatholedopingbeyondtheoptimum
1−x x 2 2
critical temperature, where the electron bands at cSDW momenta rise entirely above the
Fermilevel[4]followingFig. 1a[21]. Thepreviousobservationsuggeststhattheapplication
of bi-axial pressure on Ba K Fe As will increase its critical temperature in the vicinity
1−x x 2 2
of the Lifshitz transition. (Cf. refs. [24] and [25].)
∗∗∗
The author thanks Miguel Araujo for correspondence. He also thanks Brent Andersen,
Richard Roberts and Timothy Sell for technical help with the use of the virtual shared-
memory cluster (Lancer) at the AFRL DoD Supercomputing Resource Center. This work
was supported in part by the US Air Force Office of Scientific Research under grant no.
FA9550-13-1-0118and by the National Science Foundation under PREM grant no. DMR-
1523588.
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