What is the minimal model of magnetic interactions in Fe-based superconductors?

0.29 MB·

by  J. K. Glasbrenner

What is the minimal model of magnetic interactions in Fe-based superconductors?
J. K. Glasbrenner
National Research Council/Code 6393, Naval Research Laboratory, Washington, DC 20375, USA
J. P. Velev
Department of Physics, Institute for Functional Nanomaterials,
University of Puerto Rico, San Juan, Puerto Rico 00931, USA and
Code 6393, Naval Research Laboratory, Washington, DC 20375, USA
4
1 I. I. Mazin
0 Code 6393, Naval Research Laboratory, Washington, DC 20375, USA
2 (Dated: January 9, 2014)

n
Using noncollinear first-principles calculations we perform a systematic study of the magnetic
a
order in several families of ferropnictides. We find a fairly universal energy dependence on the
J
magnetizationorderinallcases. OurresultsconfirmthatasimpleHeisenbergmodelfailstoaccount
7
fortheenergydependenceofthemagnetizationincoupleofways: firstabiquadratictermispresent

in all cases and second the magnetic moment softens depending on the orientation. We also find
]
n thatholedopingsubstantiallyreducesthebiquadraticcontribution,althoughtheantiferromagnetic
o stripe state remains stable within the whole range of doping concentrations, and thusthereported
c lack of the orthorhombicity in Na-doped BaFe2As2 is probably due to factors other than a sign
- reversal of the biquadratic term. Finally, we discovered that even with the biquadratic term, there
r
isalimit totheaccuracy ofmappingthedensityfunctional theoryenergetics ontoHeisenberg-type
p
u models, independentof the range of the model.
s
.
t
a I. INTRODUCTION
m
-
d Fe-based superconductors are only the second fam-
n ily, after cuprates, of known high-T superconductors
c
o (HTSC). The parent compounds of these materials ex-
c hibitmagneticorderingatlowtemperatures,andintheir
[
phase diagrams the magnetic phase is proximate to the

1 superconductingphase. Thetwoordersareintimatelyre-
v lated as the superconductivity emerges when magnetism
7
is suppressed, for instance, by doping. Therefore, it is
2 generally believed that magnetic fluctuations in these (cid:1836)(cid:2870) (cid:2016)
5
systemsarethelikelydriverofthepairingmechanism.1,2
1 (cid:1836)(cid:2869)
. Magnetic order is also accompanied (with a notable ex-
1
ception discussed later) by a structuralphase transition,
0
and there are compelling arguments that this is also
4
1 drivenby magnetism: (1)density functional calculations FIG.1. (Coloronline)Schematicviewofthetwo-dimensional
: quantitatively reproduce the observedorthorhombic dis- FeplanesintheFe-basedsuperconductors. Thefirst(J1)and
v
tortions, including the amplitude and the counterintu- second(J2)nearest-neighborexchangeinteractionsareshown
i
X itive sign, and also reproduce a qualitatively different insolidanddashedlines. Themomentsonthetwosublattices
r distortionin FeTe;3 (2) the samecalculationsfail to pro- form an antiferromagnetic checkerboard pattern and in our
a duce any distortion in the absence of magnetism. The calculations the moments on one sublattice is rotated by an
angle θ relative to the other. The antiferromagnetic stripe
seemingly counterintuitive factthat the structuralinsta- ◦ ◦
patternscorrespond to θ=0 and θ=180 .
bility sometimes occurs at a temperature slightly above
themagnetictransitionisinfactconsistentwiththiscon-
cept,becauselongrangemagneticorderissufficient,but
notnecessary,forbreakingthe globalC symmetry: itis The local-moment picture has the advantage of being
4
enoughtounequallypopulatemagneticfluctuationswith analytically solvable and simple; Heisenberg-like models
differentk-vectors. Thesecanbedescribedasfluctuating areapopularwaytoapproachthemagnetismofFe-based
domain walls in an itinerant picture4 or as “order-from- HTSCs. This can be considered a reasonable approach
disorder” in the local-moment picture (see Refs. 5–7; for since,withasufficientnumberofparameters,anysortof
a review,see Ref.8). This picture is alsoconsistentwith magneticinteractioncanbemappedontoalocalmoment
observations of fluctuations breaking charge9 and spin10 model. The simplest possible model is a Heisenberg-
C symmetry locally well above the N´eel temperature. type interaction between the first and second nearest
4
2
neighbors.5–7 This model has the desired property that the necessary calculations have been reported at a very
symmetry breaking always occurs above the N´eel tem- early stage, yet were largely overlooked.14 Instead of an
perature, although this splitting diminishes as the mag- unphysical model with only superexchange terms, the
neticinteractionbecomesmorethreedimensional. While same DFT calculations can be mapped with good ac-
themodelreplicatessomephysicalpropertiesofFe-based curacy onto an isotropic (J = J ) Heisenberg model
1a 1b
HTSCs, there is a serious problem that is often over- which includes a biquadratic term (formally the same
looked. The essential physics of this approach can be as found in Ref. 5) with an amplitude of K J ,J .
1 2
described as follows:8 an Fe plane can be viewed as a Moreover noncollinear DFT calculations with o∼ne mag-
bipartite lattice where the only interaction within each netic sublattice rotated with respect to the other, see
sublattice is the second nearest neighbor exchange J , Fig. 1, can only be mapped onto this model. It was
2
while the only interaction between the sublattices is J . then shown that this biquadratic, isotropic Hamiltonian
1
As illustrated in Fig. 1, exchange interaction constant withtemperature-independentparametersisanexcellent
J , if J >J /2, generates a checkerboardantiferromag- model of the magnetic properties of Fe-based HTSCs at
2 2 1
netic (AFM) pattern in each sublattice, while the inter- anytemperature,including the spinwavespectra.15 The
action between the sublattices cancels completely, which model also is consistentwith an orthorhombictransition
holds not only for an arbitrary J , but for any Heisen- occurring above the magnetic one.
1
berg interaction of an arbitrary range. In Ref. 5 it was
These discoveries yielded a much more robust de-
shownthat,afterintegratingoutquantumfluctuations,a
scription of the the magnetic behavior of the Fe-based
nearestneighborbiquadratictermofthe formK(S S )
i j HTSCs, which includes a consistent explanation of the
·
appears in the effective Hamiltonian, with K > 0 and
orthorhombic distortion. Instead of a minuscule “or-
of the order 10−4J. This lifts the infinite degeneracy of
der from disorder” term appearing to drive the physics
the ground state, leaving a double degenerate state of
of these systems, we have a sizable biquadratic term
ferromagnetic stripes running along one of the two crys-
on the mean field level. So what is missing at this
tallographic directions with AFM alternation, matching
point? To date, there is no body of information about
the ground state of the ferropnictides of interest. How-
the biqudratic term. Important questions include: how
everthe smallamplitude ofK isunphysical,makingthis
variable is it from compound to compound? How does
result purely academic and not applicable to any real
it depend on doping? Can it change sign, leading to
material.
a noncollinear ground state while preserving tetragonal
There is another profound problem with the Heisen- symmetry? At the present moment only some answers
berg model. Even though it formally generates the cor- are available. In Ref. 14 only two compounds were stud-
rect ground states for ferropnictides, it fails to explain ied,andtheuncontrollableatomicspheresapproximation
the double stripe structure of FeTe; to do so requires in- was used. While there is no question that the obtained
troducing the third nearestneighborexchange J , which results were qualitatively correct, their quantitative ac-
3
isfoundtobeofthesameorderasJ andJ ,aresultin- curacy remained unclear. Beyond that, in Ref. 16 it was
1 2
consistent with the superexchange picture. Even worse, demonstrated that the biquadratic term depends on the
in order to fit both the ground state and the spin wave details of a material’s band structure. Using accurate
spectra,oneneedstosplitthenearestneighborexchange full potentialDFT calculationswith linear muffin tinor-
into two inequivalent parameters, J and J . The two bitals, it was shown that the biquadratic term is nega-
1a 1b
parametersendupbeingdifferentfromeachother,some- tive in stoichiometric KFe Se , a hypothetical material.
2 2
timesevenchangingsign.11Thisimpliesthatnotonlydo The term is also dependent on the size of the local Fe
the exchange constants change qualitatively from com- moment (or, equivalently, the Fe-As/Se bond-length dis-
pound to compound, but that they have a strong, coun- tance), again indicating the necessity for accurate calcu-
terintuitivedependenceontemperature;theinequivalent lations. Finally, the ultimate question that canbe posed
parametersJ andJ ,abovetheN´eeltemperature,be- is whether the total energy is mappable onto the pair
1a 1b
comeequivalentasrequiredbysymmetry,i.e. J =J . interaction at all, linear or quadratic in S S . This is
1a 1b i j
·
This bizarre and inconsistent behavior is the death knell always taken for granted, but there is no a priori reason
forusingthesuperexchangetheorytomodeltheFe-based for that to be true in an itinerant system.
HTSCs,asthereisnoplausiblephysicalmechanismthat
In DFT, the change in energy between different mag-
canexplainthe dramatictemperaturedependence ofthe
neticpatternsaccumulatesviaintegrationovertheentire
superexchange constants.
occupied portion of the Fe band, which extends several
Density functional theory (DFT) calculations, when eV below the Fermi energy. Profound orbital reordering
mapped onto the same Heisenberg model, yield similar inducedby magnetismleadsto the observedstripe order
exchange constants, including the splitting of the near- being lower in energy compared to other patterns,17,18
est neighbor exchange.12,13 This indicates that DFT is andbyextensionaffectstheexchangeconstantsobtained
correct in its description of the Fe-based HTSCs (it can by mapping. These complex changes in the electronic
quantitatively explain the spin wave spectrum, for ex- structureareresponsiblefortheanisotropyintheJ only
−
ample), suggesting that this methodology can be used model and for the large biquadratic term in the J K
−
to resolve the exchange constants conundrum. In fact model,andalsoforthelongerrangeinteractionsinFeTe.
3
This contrasts with the simplistic superexchange model nian,
where both K and J appear only as higher order terms
3
and must be much smaller than J1 and J2. H =J1 Si Sj +J2 Si Sj
· ·
Resolving the incomplete understanding of the bi- Xnn nXnn
quadratictermhasbecomeevenmoreimportantafteran K (S S )2. (1)
i j
intriguingexperimentalreportthattheorthorhombicdis- − ·
Xnn
tortiondisappearsinasmallpartoftheBa1−xNaxFe2As2
phase diagram,while magnetic orderremains.19 The au- We use the experimentally determinedcrystalstructures
thorsfavortheplausibleexplanationthatthebiquadratic whenavailable;forthehypotheticalmaterialKFe Se we
2 2
term changes sign in that region, generating the non- used the lattice constants from Ref. 16. A summary of
collinear structure shown in Fig. 1(c) of Ref. 19. An al- the symmetry groups, lattice constants, and Fe-Fe and
ternative explanation would be that the magnetoelastic Fe-As(Se) bond lengths for all the materials is found in
couplingthatdrivestheorthorhombicdistortionbecomes TableI.TheFeAs(Se)planesarestackedalongthec-axis
smallandthe (still existing)C symmetry breakinggoes separated by a non-magnetic filler plane, except for the
2
undetected. 11 compound FeSe which consists only of FeSe planes.
The Fe layers form a two-dimensional square lattice and
The former explanation, that the biquadratic term
thegroundstatemagneticorderhasbeenconfirmedboth
changes sign upon doping, was supported by the calcu-
lations of Chubukov and Eremin,20 who derived a bi- experimentally21 and theoretically14 to be AFM stripe
order,asschematicallyrepresentedinFig.1(stripeorder
quadratic term in the linear response regime. The prob-
◦
corresponds to θ = 0 or θ = 180 ). In all cases in order
lem with this approach is that it contradicts the DFT
finding3,17,18 that the energy associated with magnetic to accommodate the AFM stripe pattern we double the
cell in the xy-plane (√2 √2).
interactionaccumulatesoveralargeenergywindow,and
×
Inordertostudythebiquadraticcouplingweallowthe
that the local moments of Fe remain large throughout
angleθbetweenthetwoFesublatticestovary. Theangle
the entire phase diagram. On the other hand, knowing
θ, as depicted in Fig. 1, gradually interpolates between
thatmagneticorderinghasastrongeffectonthe density
two equivalent stripes states with q = (1,0) and q =
of states and the orbital composition of the Fe bands, it
(0,1). According to the biquadratic model in Eq. 1, the
is plausible that the sign of the biquadratic interaction
angular energy dependence ∆E(θ)=E(θ) E(0) of the
parameter is not fixed and could change upon doping.
−
11, 111, and 1111 families is predicted to be,
As previously discussed, the results of Ref. 16 show that
for the hypothetical KFe Se compound, which can be
2 2 ∆E(θ)=4Ksin2θ, (2)
viewed as a case of extreme hole doping, the biquadratic
term does indeed change sign.
The 122 family belongs to a centered symmetry group,
Our goal in this paper is to make a systematic inves- and therefore rotating θ by 180◦ takes the system from
tigation of the biqudratic interaction in representative one stripe patternto another,inequivalent one. The two
Fe-pnictide and Fe-chalcogenide families to address the differbythestackingorderalongcandtheenergydiffer-
variability of the biquadratic interaction. For this in- enceisproportionaltotheinterplanarexchangeconstant
vestigation, we use the all-electron, full-potential Linear J⊥. Taking this into account,the angular energy depen-
Augmented Plane Waves (FLAPW) method. We find dence for the 122 family is,
that the biquadratic parameter can vary within large
limits, but that it does not change sign in the accessi- θ
ble ranges of doping. We conclude, in particular, that ∆E(θ)122 =4Ksin2θ−16J⊥sin2(cid:18)2(cid:19). (3)
the observed lack of an orthorhombic distortion in the
◦
Na-dopedBaFe As compoundinRef.19islikelydueto This splits the degeneracy of the θ = 0 and θ = 180
2 2
the inaccessibility of the tetragonal symmetry breaking states by 16J⊥.
by the experimental tools used in that work. In order to calculate the angular energy dependence
weperformfullynoncollinearfirst-principlescalculations
using the ELK code.22 ELK implements density func-
tionaltheory(DFT)withinaFLAPWbasissetwithlocal
orbitals. In our calculations we use the Perdew-Burke-
II. METHODS
Ernzerhof(PBE)exchange-correlationfunctional.23ELK
allows constrained magnetic moment calculations where
We investigate the magnetic order in representative the moment direction and/or magnitude can be fixed.
compounds from different families of the iron-based su- To study the effect of hole doping on BaFe As , we use
2 2
perconductors: FeSe for the 11 family; LiFeAs and thevirtualcrystalapproximation(VCA)inthestandard
NaFeAs for the 111 family; BaFe As , BaFeCoAs , way, in which homogeneous doping is achieved by re-
2 2 2
CaFe As , SrFe As , KFe As , and KFe Se for the 122 placing the Ba atoms with fictitious atoms of fractional
2 2 2 2 2 2 2 2
family; and LaFeAsO for the 1111 family. The magnetic nuclear charge between those of Ba and Cs. The density
order is modeled using the J J K model Hamilto- of states (DOS) aroundthe Fermi energy of BaFe As is
1 2 2 2
− −
4
TABLE I. Summary of the crystallographic symmetry groups, lattice structure (lattice constants a and c, and fractional
coordinates z of thenon-magnetic planes), Fe-Fe/Fe-As(Se)bond lengths, thefitted parameters of Eqs. 2 & 3, and theenergy
change dueto thesoftening of the magnetic moment for all thestudied compounds.
Compound Sym. a c zAs,Se (zLi,Na,La) dFe-Fe dFe-As(Se) M(0) K J⊥ E(0)|M(90)−E(0)|M(0)
Group (˚A) (˚A) (frac.) (˚A) (˚A) (µB) (meV) (meV) (meV)
FeSe P4/nmm 3.803 6.084 0.2708 (−) 2.689 2.516 2.72 9.67 − -0.04
LiFeAs P4/nmm 3.793 6.366 0.2365 (0.6541) 2.682 2.421 1.85 9.75 − 4.21
NaFeAs P4/nmm 3.949 7.040 0.2028 (0.6460) 2.793 2.437 2.18 15.43 − 7.30
LaFeAsO P4/nmm 4.037 8.742 0.1513 (0.6415) 2.854 2.413 2.09 13.24 − 4.97
SrFe As I4/mmm 3.930 12.324 0.3604 (−) 2.779 2.390 1.94 10.57 0.79 2.78
2 2
CaFe As I4/mmm 3.896 11.683 0.3665 (−) 2.755 2.376 1.83 9.39 1.33 1.58
2 2
KFe As I4/mmm 3.842 13.860 0.3525 (−) 2.716 2.389 2.46 6.36 0.40 9.48
2 2
KFe Se I4/mmm 3.914 14.037 0.3434 (−) 2.767 2.355 2.47 -3.29 -0.05 0.48
2 2
BaFe As I4/mmm 3.942 13.021 0.3545 (−) 2.791 2.397 1.99 10.84 0.23 3.33
2 2
Ba0.9Na0.1Fe2As2 ” ” ” ” ” ” 1.96 9.85 0.21 4.04
Ba0.8Na0.2Fe2As2 ” ” ” ” ” ” 1.94 8.73 0.22 4.73
Ba0.7Na0.3Fe2As2 ” ” ” ” ” ” 1.93 7.80 0.26 5.30
Ba0.6Na0.4Fe2As2 ” ” ” ” ” ” 1.93 7.16 0.30 7.04
Ba0.5Na0.5Fe2As2 ” ” ” ” ” ” 1.94 5.56 0.24 5.37
Ba0.4Na0.6Fe2As2 ” ” ” ” ” ” 1.96 4.42 0.21 6.15
BaFe0.5Co0.5As2 ” ” ” ” ” ” 1.27 0.94 0.039 −
dominated by Fe and As states, and so using VCA has theexceptionsofKFe Se andBaFe Co As ,followa
2 2 0.5 0.5 2
the primary effect of shifting the Fermi energy and re- similar pattern, with the energy difference between the
◦
movingafractionalnumberofelectronsfromthevalence groundstripestateandtheleastfavorableθ =90 config-
band. uration varying between 30-60 meV/Fe. Our results for
Convergence was checked as a function of the size of LaFeAsO, BaFe As , and KFe Se agree well with pre-
2 2 2 2
thek-pointmesh. DifferentsizeMonkhorst-Packk-point vious calculations.14,16 The biquadratic interaction con-
grids were used for the different families of compounds: stant K is fairly large and positive in most materials
9 9 8 for 11; 9 9 8 for 111; 8 8 9 for 122; (again excepting KFe2Se2 and BaFe0.5Co0.5As2) and of
an×d 6× 6 4 for 1×111×respectively. D×ue t×o the small the same orderas J andJ .13,14 One ofthe factorsthat
1 2
× ×
energy differences the energy convergence criterion was influences K is the Fe-Fe bond length, as K tends to be
set to 10−7 Ha. largerincompoundswhichhaveagreaterFe-Fedistance
such as LaFeAsO, NaFeAs and BaFe As . The inter-
2 2
layer coupling J⊥ in the 122 family is about an order of
magnitude smaller thanK,and the constantvaries from
III. RESULTS AND DISCUSSION
material to material.
We checked the relative energies of the different mag- BaFe0.5Co0.5As2 and KFe2Se2 are exceptions to the
netic orders. In all cases we obtained that in the FM above trends. BaFe0.5Co0.5As2 was calculated using the
configuration the magnetic moment collapses and this VCA and represents electron doping of BaFe2As2. This
configuration is much higher in energy than the AFM doping softens the ground state moment by 36% and
◦ ◦
configurations. The energy of the checkerboard AFM destabilizes the local moment for angles 45 <θ <150 ,
configuration was found to be higher in energy than the in which range it collapses. This level of electron dop-
AFM stripe ground state; hole doping decreased the rel- ing also suppresses the biquadratic and interplanar in-
ative energy difference, although the AFM stripe state teractions, reducing both by an order of magnitude.
remained as the ground state configuration. KFe2Se2, on the other hand, exhibits a negative bi-
quadratic interactionterm in agreementwith the results
The angular dependence of the energy difference
ofRef.16.24WhileitexhibitsanegativeK,bulkKFe Se
∆E(θ) = E(θ) E(0) for different compounds is plot- 2 2
− is a hypothetical material that cannot be stabilized in
ted in Fig. 2(a). It is clear that biquadratic coupling is
experiment,25 so here it just serves as a proof of concept
presentinallofthesecompounds. Wefittedtheseresults
that a negative K is possible.
to Eq. 2 for the 11, 111, and 1111 families and Eq. 3 for
the 122family. Thefittedparametersaresummarizedin The softening of the moments shown in Fig. 2(b) con-
Table I. tributes to ∆E(θ) in a non-trivial way. Modeling the
The angular energy dependence of the materials, with variation of the magnetic moments necessitates the in-
5

70
(a) (c)
60 40
V/Fe) 4500 V/Fe) 30
e e
m m
0) ( 30 0) ( 20
E( 20 E(
) - 10 ) - 10
E( E(
0
FeSe
-10 LiFeAs 0
NaFeAs BaFe2As2
-20 BaFe2As2 Ba0.9Na0.1Fe2As2
0 20 40 60 80 100 120 140 160 180 KFe2As2 -100 20 40 60 80 100 120 140 160 180 Ba0.8Na0.2Fe2As2
(De g) CSraFFee22AAss22 (D eg) BBaa00..76NNaa00..34FFee22AAss22
11..0046 (b) LKaFFee2SAes2O 11..0002 (d) BBaa00..54NNaa00..56FFee22AAss22
BaFe0.5Co0.5As2
1.02 0.98
0)1.00 0)0.96
M( M(
) / 0.98 ) / 0.94
M(0.96 M(0.92
0.94 0.90
0.92 0.88
0.90 0.86
0.88 0.84
0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180
(Deg) (Deg)
FIG. 2. (Color online) The energy E(θ)−E(0) and the normalized moment M(θ)/M(0) as a function of the relative angle θ
ofthetwomagnetic sub-systems. Notethatinall instancesthelinesareaguidetotheeyeanddonot representafit. (a) The
angulardependenceoftheenergy forcompoundsbelongingtothe11, 111, 122, and 1111 families of superconductors. (b)The
angular dependence of the normalized moments belonging to the 11, 111, 122, and 1111 families of superconductors. (c) The
angular dependence of the energy for different doping levels of the 122 compound BaFe As . (d) The angular dependence of
2 2
thenormalized moments for different doping levels of the122 compound BaFe As .
2 2
clusion of J exchange terms in Eqs. 2 & 3, as well as inFig.2(c). AdiscussionofthevalidityofusingtheVCA
2
on-site Hund’s exchange terms such as IM(θ)2 (I is the to address the effect of hole-doping on the biquadratic
Hund’s exchange constant). A simpler way to estimate termisincludedintheAppendix. Asisclearinthefigure,
the importance of the moment softening is to calculate the biquadraticinteractionconstantK stronglydepends
E(0) E(0) inthe collinearAFMstripe state onthedegreeofhole-dopinginthematerial. Goingfrom
M(90) M(0)
| − |
configuration, i.e. the energy change from reducing the an undoped system to x=0.6 results in a 60% decrease
ground state moment to the self-consistent amplitude at in K. Extrapolating the hole doping of BaFe As to the
2 2
◦
θ = 90 , where the softening is greatest. The final col- extreme x=1 case, the biquadraticconstantK however
umn in Table I reports these calculations. For pnictides, does not invert and instead nearly vanishes. Because
the difference is positive and only a few meV in magni- of the similarity of results in other materials, we expect
tude. For chalcogenides, the moment softening is slight that doping via the VCA would yield similar results in
and accordingly the energy contribution from the soft- other materials. Of course, in the case of extreme hole
ening is also small. For FeSe, the moment amplitude doping it is necessary to allow the lattice constant and
◦
grows slightly at θ = 90 , and there is a corresponding atomic positions to relax. Here KFe As is an example
2 2
smallgaininenergy. Inallcases,weseethatmostofthe of extreme hole doping, and if the experimental lattice
energy charge in Fig. 2(a) is driven by the biquadratic constants are used, the biquadratic term is 6.3 meV, in
term, with only a modest contribution coming from the contrastto the vanishing biquadratic term inferred from
moment softening. extrapolating the VCA results discussed above. The Fe-
The angular energy dependence for various levels of Fe and Fe-As(Se) bond lengths, influenced by the hole-
hole-dopinginBa1−xNaxFe2As2 viatheVCAisdepicted doping level, plays a role in determining K. This is in
6
Ref. 26, but is instead due to itinerant effects not cap-
(a) (b) tured by the Heisenberg Hamiltonian.
Returning to the experiment from Ref. 19, we can use
ourresultstocommentupontheexperimentaldata. Our
results do not support the proposed noncollinear mag-
netic configuration in Na-doped BaFe As . Why might
2 2
this be? One possibility is that, as discussed in Sec. I,
theremightnotbeareentrantC transitioninNa-doped
4
BaFe As and so it would be unnecessary to argue for
2 2
a change in the magnetic state. The argument for the
reentrantC transitionisbasedonnucleardiffractogram
4
measurementsinwhichanapparentrecombinationofthe
FIG. 3. The two magnetic configurations, in a 4×4 two-
nuclear Bragg peaks at low temperatures is observed.
dimensional cell, thatare degeneratefor anypairwise Hamil-
However the Bragg peaks are quite broad, and so a re-
tonian of arbitrary range. The closed circles correspond to
duction,butnotacompleteremoval,oftheorthorhombic
spin-upmomentsand thecrosses tospin-downmoments. (a)
distortion would also be consistent with the data. An-
Thedoublestripe configuration, which istheground stateof
other possibility is that the reentrant C transition is
FeTe. (b) Thesquare configuration. 4
not accompanied by a noncollinear magnetic configura-
tion. The authors of Ref. 19 put fortha model of the C
4
transition with two magnetic configurations fitting well
line with the results of Ref. 16, where K for KFe Se
2 2
tomeasuredx-raydiffractiondata: thenoncollinearcon-
depended strongly on the internal coordinate z .
Se
figuration presented in the main body of the paper and
Regardingthequestionofwhetheritisvalidtoassume
also a collinear stripe state. The noncollinear model is
that magnetic interactions can be accurately mapped to
preferred as it already has C symmetry. The authors
a pairwise Hamiltonian by just including an arbitrary 4
commentthatalinearcombinationofspindensitywaves
number of terms, one can address this issue by compar-
thatproducestripesalongthe xandydirectionsalsore-
ing the two magnetic patterns shown in Fig. 3. It was
storesC symmetry. Withoutmore information,there is
pointedout26 thattheseconfigurationsaredegenerateon 4
no reason to prefer one magnetic configuration over the
themeanfieldlevelforanyHeisenbergmodelofarbitrary
other.
range. Being collinear, the configurations remain degen-
Therefore, in light of our results, there remains two
erate after the inclusion of the biquadratic interaction
plausible interpretations of Ref. 19. The first is that the
term, which can be lifted by either magnetoelastic cou-
pling or by integrating out fluctuations.26 We calculated orthorhombic distortation of Na-doped BaFe2As2 is re-
duced at lower temperatures, but ultimately retains C
theenergydifferencebetweenthesetwoconfigurationsfor 2
symmetry. Thisistheinterpretationweprefer,asitisthe
FeTe using the experimental high-termpature structure
simpler way in which C symmetry would be preserved.
(tetragonal)andfound that the experimentally observed 2
A higher resolutionmeasurementof the temperature de-
double stripe pattern is lower in energy than the square
pendence of the nuclear Braggpeaks is necessaryto rule
patternby8meV/Fe,asmall,butbynomeansirrelevant
this interpretation out. The second is that the C tran-
ornegligiblenumber. Thedegeneracyisonlyliftedonthe 4
sition does occur, but that the magnetic order remains
level of the 4th order (square) ring exchange, which in
the localized Hubbard limit is of the order of t4/U3, as striped and modulates between x and y oriented stripe
patterns. Additional follow-up studies are necessary to
compared to the nearest-neighbor superexchange terms
which are of the order of t2/U. ultimately resolve this question.
Anotherconsequenceofattemptingtomaptotheclas-
sical Heisenberg J -J -J model is that it does not pre-
1 2 3
dictthe doublestripeorsquarepatterntobetheground IV. CONCLUSIONS
state of FeTe, instead predicting spiral phases for large
J . In Ref. 26 it was pointed out that for some pa- We confirmed that biquadratic coupling is universally
3
rameterrangecollinearstructuresmaybe stabilizedover present in Fe-based superconductors. It is of the same
the spiral ones because of quantum fluctuations. In con- order of magnitude as the superexchange interactions.
trast, our calculations clearly indicate that the sizeable In the studied materials, the biquadratic term is mod-
biqudratic interactions, present in the parent Fe-based estly affected by the softening of the magnetic moment,
HTSC compounds, completely exclude spiral phases al- is influenced by the Fe-Fe and Fe-As bond lengths, and
ready on the mean field level. Moreover, DFT calcula- dependsonthe doping,whichunderlinesthe biquadratic
tions strongly favor double stripes over squares, despite term’s itinerant origin. We find that even in the case of
thefactthatfluctuationsworkintheoppositeway.26Our extremeholedoping,noexperimentallyrealizedmaterial
calculationsstronglysuggestthatthefactthattheexper- exhibits a change of sign in the biquadratic term, so the
imental structure in FeTe appears to be double stripe is collinearAFMstripestateisenergeticallypreferredinall
not related to magnetoelastic coupling, as suggested in instances. Therefore,theapparentexperimentalobserva-
7
tion of a reentrant C transition in Na-doped BaFe As taken from experiment, and calculated ∆E(π/2). For
4 2 2
is likely to be an artifact due to the inaccessibility of Ba Na Fe As we obtained ∆E(π/2) = 11.4 meV,
0.5 0.5 2 2
measuring the C symmetry at low temperatures with which is about a factor of two smaller than the VCA
2
the experimental tools. x=0.5 result of 21.5 meV.
Our results show that in the realm of Fe-based super- To see how structural deformations affect ∆E(θ),
conductors the na¨ıve Heisenberg model is a rather poor we relaxed the structures in the pseudopotential-based
approximation. The biquadratic exchange plays an es- software suite VASP.27,28 In VASP we used pro-
sential role and cannot be neglected in any model calcu- jector augmented wave (PAW) pseudopotentials29,30
lationdescribingthesecompounds. Inadditionthereare and the Perdew-Burke-Ernzerhof generalized gradient
deviations, as observed for FeTe, from the general pair- approximation23 to DFT. Our BaFe As calculations in
2 2
wiseinteractionmodelforlinearandbiquadratictermsof the main text took the internal parameter z from ex-
As
an arbitrary range. In FeTe it is these terms that stabi- periment. We want to make a proper comparison be-
lize the experimentallyobserveddoublestripe inthe cal- tween VCA and a relaxed structure with Na substitu-
culations, and not the magnetoelastic coupling, as con- tions, so in order to do this we relaxed the internal
jectured before.26 It remains to be seen whether these parameter z for both undoped BaFe As and doped
As 2 2
interestingfeaturesarespecific to the parentcompounds Ba Na Fe As inVASPandimportedthecoordinates
0.5 0.5 2 2
ofFe-basedHTSCsoraremorecommonthanpreviously into ELK. We then calculated ∆E(π/2) for BaFe As
2 2
expected. Further calculations and studies should aid in in the VCA with x = 0.5 using the relaxed atomic po-
answering this question. sitions found for undoped BaFe As , and then calcu-
2 2
lated ∆E(π/2) for Ba Na Fe As using both the re-
0.5 0.5 2 2
laxed structure for undoped BaFe As and the relaxed
2 2
ACKNOWLEDGMENTS structure for Ba Na Fe As . The VCA result with
0.5 0.5 2 2
the undoped structure is ∆E(π/2) = 10.3 meV. The
I.I.M. acknowledges Funding from the Office of Naval Ba0.5Na0.5Fe2As2resultusingtherelaxedundopedstruc-
Research (ONR) through the Naval Research Labora- ture is ∆E(π/2) = 3.28 meV, and the result for using
tory’s Basic Research Program. J.V. acknowledges the therelaxedstructureforBa0.5Na0.5Fe2As2is∆E(π/2)=
support of the Office of Naval ResearchSummer Faculty 7.58 meV. IfthecellvolumeofBa0.5Na0.5Fe2As2 isfixed
ResearchProgram. J.G.acknowledgesthesupportofthe and both the internal parameter z and a and c pa-
As
NRC programat NRL. rameters are relaxed and then imported into ELK, then
∆E(π/2)=3.08 meV for Ba Na Fe As .
0.5 0.5 2 2
Overall the VCA overestimates ∆E(θ), but this does
Appendix A: Validity of the virtual crystal not affect the qualitative behavior, i.e., the biquadratic
approximation (VCA) termK doesnotchangesign. Furthermore,usingrelaxed
structures illustrates the sensitivity of the biquadratic
We confirmed the validity of the VCA by doing the interaction to the distance between the Fe and As(Se)
following test calculations using BaFe As . To be- planes,butthesesubtlechangesdonotmateriallychange
2 2
gin, we directly substituted a Na atom for a Ba atom the overall trends. We conclude that the VCA is an ap-
(the two Ba sites are equivalent via symmetry), keep- propriate method for investigating whether doping can
ing the lattice and internal parameters set to the values affect the qualitative behavior of the biquadratic term.
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