Coexistence of Itinerant Electrons and Local Moments in Iron-Based Superconductors

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by  Su-Peng Kou

Coexistence of Itinerant Electrons and Local Moments in Iron-Based Superconductors
Su-Peng Kou1, Tao Li2, and Zheng-Yu Weng3∗
1Department of Physics, Beijing Normal University, Beijing, 100875, China
2Department of Physics, Renmin University of China, Beijing, 100872, China
3Center for Advanced Study, Tsinghua University, Beijing, 100084, China
(Dated: January 15, 2010)
Inviewoftherecentexperimentalfactsintheiron-pnictides,wemakeaproposalthattheitiner-
antelectronsandlocalmomentsaresimultaneouslypresentinsuchmultibandmaterials. Westudy
0
a minimal model composed of coupled itinerant electrons and local moments to illustrate how a
1
consistent explanation of the experimental measurements can be obtained in the leading order ap-
0
proximation. Inthismean-fieldapproach,thespin-density-wave(SDW)orderandsuperconducting
2
pairingoftheitinerantelectronsarenotdirectlydrivenbytheFermisurfacenesting,butaremainly

n inducedbytheircouplingtothelocal moments. Thepresenceofthelocal momentsasindependent
a degreesoffreedomnaturallyprovidesstrongpairingstrengthforsuperconductivityandalsoexplains
J
thenormal-statelinear-temperaturemagneticsusceptibilityabovetheSDWtransitiontemperature.

5 Weshowthatthissimplemodelissupportedbyvariousanomalousmagneticpropertiesandisotope
1 effect which are in quantitativeagreement with experiments.

] PACSnumbers: 74.20.Mn,71.27+a,75.20.Hr
n
o
c
- I. INTRODUCTION anomalous large linear-temperature susceptibility in the
r normal state over a wide temperature regime27. How-
p
ever, how this localized spin picture can be meaning-
u Since the discovery of superconductivity (SC) in iron
s pnictides1–5, with T being quickly raised to 55 K5, in- fully applied to a metallic material (albeit a bad metal
. c in the undoped case of the iron-pnictides) remains un-
t tensive attentions have been focused on possible un-
a clear. Whether the doping effect is similar to that of the
m derlying mechanisms. With the neutron scattering
cuprates as described by a multiband t-J -J like model
measurement6,7 subsequently establishing the fact that 1 2
- is also controversial.
d anSDWorderexistsintheundopedLaOFeAscompound
n below T ≃ 134 K, which has been later generically
SDW
o found in other iron pnictides8–11, the interplay between
c
SC and antiferromagnetism (AF) has become a central
[
issue.
In this paper, we point out that if both itinerant elec-
4
The iron 3d-electrons are believed quite itinerant trons and local moments are allowed to simultaneously
v
1 with their hybridized multi-orbitals forming multiple present in the system, then many basic properties of the
1 Fermi pockets at the Fermi level13. Many theoretical iron pnictides can be naturally accommodated by a sin-
1 efforts12–14,17? –21 are based on itinerant approaches in gle framework in the leading order approximation. To
4 searchingforpossibleSDWandSCmechanismsresponsi- illustrate the point, we study a highly simplified model
1. blefortheironpnictides. Thiskindoftheoryisgenerally with the local moments and itinerant electrons coupled
1 sensitive to the detailed band structure where the Fermi together by a Hund’s rule coupling as schematically il-
8 surfacenesting effectis important. As shownbyarenor- lustrated in Fig. 1. We show that an SDW order of the
0 malizationgroupanalysis21,suchanitinerantmodeldoes itinerant electrons can take place simultaneously with a
v: possess the instabilities towardsthe SDW andSC order- collinearAForderofthelocalmoments,whiletheorder-
i ings. However,howtoreachhigh-Tc intheSCphaseand ing would be absent in either degrees of freedom if they
X
atthe sametime haveaself-consistentdescriptionofthe do not couple, clearly different from the Fermi surface
ar magnetic phase within a unified framework remains a nesting mechanism or the AF ordering in a J1-J2 model.
challenge. Furthermore, the superconducting pairing of the itiner-
Alternatively local moment descriptions have been ant electrons is also driven by the same coupling with
also promoted22–27 in view of the d-electrons, local the strength reaching strong coupling. Such a picture is
Coulomb and Hund’s rule interactions in the iron pnic- furthersupportedbyaseriesofmagneticpropertiesboth
tides, as opposed to the itinerant RPA-type treatment. below and above the SDW ordering temperature T ,
SDW
Of them the so-calledJ -J model which emphasizes the which show good agreementwith experiments. It is pre-
1 2
As-bridged superexchange couplings22,28,29 between the dicted that in order to consistently account for T ,
SDW
nearest neighboring (NN) and next nearest neighboring magnetization, spin gap, uniform susceptibility, as well
(NNN) local moments of the irons has been used due to as the competition between the AF and SC phases, the
its natural tendency to form the collinear AF order at (high-temperature)normalstateoflocalmomentsshould
low temperature. The local moment approach is espe- be close to a critical regime of quantum magnets, which
cially appealing over the itinerant one in explaining the can be tested by a neutron-scattering experiment.
2
In contrast to the conventional itinerant approach
(a)(cid:13)
where the Coulomb interaction between the itinerant
electrons gets enhanced via the Fermi surface nesting ef-
Q(cid:13) =((cid:13),0)(cid:13) fect, we shall omit such an interaction. Instead, we em-
s(cid:13)
phasize the importance of the coupling between the itin-
erant electrons and some preformed local moments via
the second term H in (1), by a renormalized Hund’s
itinerant bands(cid:13) J0
rule coupling J as follows
0
(b)(cid:13)
local moment(cid:13)
H =−J M ·S (3)
J0 0 i i
Xi
where S is the spin operator for the itinerant electrons
i
andM denotesthelocalmomentatsitei. Inthefollow-
i
0(cid:13) ingweshallfocusontheweakJ0 casewhereMi behaves
like an independent degree of freedom. In the strong
lower Hubbard bands(cid:13) upper Hubbard bands(cid:13) coupling limit of J0, by contrast, Si and Mi should be
lockedtogetherinstronglycorrelatedregimeashasbeen
FIG. 1: (color online) A schematic illustration of itinerant previously discussed in Ref.23.
electron and local moment bands and the profiles of density An essential assumption of this model will be that be-
of states in the present model. (a) A simplified picture of sides the itinerant electrons described above, there are
two itinerant bands: A hole-like band near Γ point and an somed-electrons sitting below the Fermienergythat can
electron-likeonenearM point,whichareseparatedbyamo- also contribute to the low-energy physics by forming ef-
ment Qs = (π,0); (b) The density of states for the itinerant fective local magnetic moments23. Namely, some of the
bands shows an enhancement at chemical potential µ near
d-electron multibands can open up a Mott-Hubbard gap
theoriginwhereboththeholeandelectronpocketscontribute
crossing the Fermi energy, due to the on-site Coulomb
(see(a));Thelocalmomentasanindependentdegreesoffree-
interactionandtheHund’sruleferromagnetic(FM)cou-
dom is contributed by a filled lower Hubbard band, with the
Mott gap crossing the Fermi level such that it does not con- pling, with the filled lower Hubbard bands giving rise
tribute to the low-energy charge dynamics. The Hund’s rule to an effective local moment as illustrated in Fig. 1(b).
coupling between the itinerant electrons at the Fermi level Theselocalizedd-electronsmaybedifferentfromtheitin-
and the local moments will dictate thelow-energy physics. erant electrons as mainly coming from more isolated31
dx2−y2 and dz2 orbitals. In the following we shall simply
assume their existence and explore the consequences of
II. MODEL STUDY
it.
Thethirdtermin(1)describesthe predominantinter-
A. Minimal model actionbetweenthese localmomentsby aHeisenberg-like
model
Our model Hamiltonian is composed of three terms
H =J M ·M +J M ·M (4)
J2 2 i j 2 i j
H =Hit+HJ0 +HJ2. (1) hiXji∈A hiXji∈B
The first term
where the NNN superexchange coupling J is bridged
2
H = (ε −µ)c† c (2) by the As ions between the diagonal iron sites, with
it k kσ kσ A and B referring to two sublattices of the square Fe
Xk,σ
ion lattice, according to the LDA calculation28,29 and
describes the itinerant electrons forming the hole and analysis22. Note that the NN exchange J , bridged by
1
electron pockets near the Fermi energy as illustrated the Asions,canbe eitherAForFMinnatureandmuch
in Fig. 1(a), which are located at the Γ point and weaker than J2 for the isolated dx2−y2 and dz2 orbitals
the M point, respectively, and separated by momenta due to the symmetry reason22,31,32. Furthermore, the
Q = (π,0) and (0,π) [only the former is shown in Fig. itinerant electrons can effectively induce an additional
s
1(a)] in an extended Brillouin zone (BZ). For simplicity NN FM interactionbetween the localmoments, whichis
we shall assume the symmetric dispersions for the hole assumedtobepredominant(tobeconsistentwiththelat-
andelectronbands,withε =−ε suchthatthehole tice distortioninduced by the SDW orderingobservedin
k k+Qs
and electron pockets are exactly nested at ε = µ = 0, theneutronscatteringmeasurement6,8,seebelow). Thus
k
where µ is the chemical potential. Such nesting of the we shall neglect the effect of J to the leading order ap-
1
Fermi pockets will be lifted as the increase (decrease) of proximation. Of course, one may always add such J
1
µ, which effectively controls the electron (hole) doping termaswellastheCoulombinteractionbetweentheitin-
(the undoped case in iron pnictides may correspond to erantelectrons into the abovehighly simplified model to
some small but finite |µ| here). makeitmorerealistic. Butforthepurposeofidentifying
3
the most essential components and the simplicity of the
0.6
model, we shall focus on the minimal model (1) in the )
B 0.6
following study. g 0.5 |Mtot| J2 "spin-wave" gap
on ( 0.4 Gap/0.4
B. Mean field approximation ati 0.2
z 0.3
ti SDW
e 0.0
1. Effective description of the local moments n 0.2 0.0 0.2 0.4 0.6 0.8
g
Ma 0.1 TSDW T/J2
According to (4), the local moments M will antifer- |m|
i
romagnetically fluctuate in each sublattice of the iron
0.0
square lattice, and thus may be redefined by M ≡ 0.0 0.2 0.4 0.6 0.8 1.0
i
Mpini withpi ≡eiQs·ri withQs =(π,0)and(0,π)such T (J2)
that the unit vector n will fluctuate smoothly in each
i FIG. 2: (color online) The total magnetization and the in-
sublattice. For the iron pnictides, the presence of the
duced moment of itinerant electrons at a fixed chemical po-
Mott-Hubbardgapisnotexpectedtobeverylarge(∼0.6
tentialµ=0.2J2. Inset: thegapsinthe“spinwave”spectrum
eVasindicatedintheopticalexperiment30). Itisenough
Ωq of local moments and the SDW spectrum Ek of itinerant
toprotect thelocalmomentsfromamplitudefluctuations electrons.
overawidetemperatureregimepresumablymuchhigher
thanthe SDWorderingtemperatureT aswellasT .
SDW c
But it also means that in reality the local moment M is 2. Mean-field theory
notquantizedanddescribedby aHeisenberg-likeHamil-
tonian (with S =2 for instance). It is important to note that such fluctuations will
Thus it would be more suitable to use a nonlinear σ- strongly couple to itinerant electrons via H , for the
J0
model33,34 to characterize the low-energy fluctuations of holeFermipocketsaroundtheΓpointandelectronFermi
local moments in replace of (4): pocketsaroundtheM pointareapproximatelyconnected
by the momentum Q at small µ, as shown in Fig. 1(a).
s
LJ2 =a=XA,B(cid:26)21g0 (cid:2)(∂τna)2+c2(∇rna)2+iλa(n2a−1)(cid:3)(cid:27) pInarptiacrlet-ichuollaerp,adirrsivceannbsiymuHltJa0n,etohueslyloccoanldmenosmeeanttas sapned-
cificwavevectorQ ,givingrisetoanAForderatafinite
(5) s
mean-field temperature T , which can be stabilized
withc≃4MJ (withthelatticeconstantoftheFesquare SDW
2
presumably by a weak interlayer coupling.
lattice taking as the unit) and g ≃16J . (Note that n
0 2 a
AssumeanSDWorderparameterfortheitinerantelec-
(a=A,B)heredenotestheunitvectorinagivensublat-
tice such that two separated N´eel orders would emerge, trons
if λ = 0 at T = 0.) Denoting n ≡ hn i, the fluctu-
a 0 a hS i=mp 6=0 (9)
ations of δn ≡ n − n is described by the propagator i i
0
D (q,τ)=−hT δn(q,τ)·δn(−q,0)i with33,34
0 τ with a specific wavevector Q = (π,0) in p ≡ eiQs·ri.
s i
Then a staggered “easy-axis field” from H should be
J0
3g added to the nonlinear σ-model in (5):
D (q,iω )=− 0 (6)
0 n ω2 +Ω2
n q
−J Mm· n . (10)
0 i
where the spin-wave spectrum Xi
TheresultingEuclideanactionisstillquadraticinn and
Ω = c2q2+η2 (7) i
q can be integrated out in a standard way to give rise to
p
the same expressions of (6) and (8), except that now n
and η2 ≡ iλ with the subscription a being dropped, 0
a
is determined by
which is determined by the condition (n )2 =1 as
a
D E
n ≡ hn i
0 a
(n )2−β−1 D (q,iω )=1 (8) = J Mg /η2 m. (11)
0 0 n 0 0
ωnX,q=6 0 (cid:0) (cid:1)
Thus, no matter how weak m is, it can always induce a
with β ≡ 1/k T and ω = 2πnβ. Hence, without cou- collinear AF order of the local moments at the same Q
B n s
pling to the itinerant electrons, n (a = A, B) do not with n 6= 0. In particular, the independent spin wave
a 0
couple to one another, and the local moment M gov- spectrumΩ ofthelocalmomentgainsafinitegapη 6=0,
i q
erned by (5) will intrinsically fluctuate around the two in contrast to the limit of m=0 where n 6=0 can only
0
possible Q ’s. occur at η =0 in a pure J nonlinear σ-model33,34.
s 2
4
Self-consistently, with n0 6= 0 a finite m will always )
e
be induced in the itinerant electrons. At the mean-field F
-0.20
level, it is governedby J2
s/
e
Hit−J0Mn0·Xi piSi, (12) 2 )statB00..1105 1emu/mol)068 lo
leading to the following band folding and reconstruction g 4 4
(( -102 it
′ (−E −µ)α† α +(E −µ)β† β (13) 0u .05 (0
Xk,σh k kσ kσ k kσ kσi 0 100 200 (K)
0.00
where the summation over k is restricted within the re- 0.0 0.2 0.4 0.6 0.8
ducedmagneticBZdefinedbyQs andtheitinerantelec- T (J2)
tronbandsaresplitintoαandβ bands,withtheBogoli-
FIG. 3: (color online) The calculated uniform spin suscepti-
ubov transformation bility χu shows a linear-T behavior above TSDW (marked by
arrow)usingthesamesetofparametersasusedinFig. 2. In-
ckσ = ukαkσ−vkσβkσ, set: The susceptibility χlo contributed by the local moments
c = v σα +u β . (14) andχit fromtheitinerantelectronsareillustratedseparately,
k+Qsσ k kσ k kσ in absolute units comparable to the experimental data with
Here u = [(1−ǫ /E )/2]1/2, v = [(1+ǫ /E )/2]1/2, taking J2 =30 meV.
k k k k k k
and the electron excitation spectrum becomes
as shown in Fig. 2) is close to 0.87 µ for BaFe As 8,11.
B 2 2
E = ǫ2 +∆2 (15) By taking J = 30 meV (incidentally it is compara-
k k SDW 2
q ble to the LDA estimation29 for BaFe As ), one obtains
2 2
in which the SDW gap T ≃ 0.415J = 144 K (compared to the experimen-
SDW 2
tal value 143 K8), and in the inset of Fig. 2, a gap
J M
∆ ≡ 0 |n |. (16) η =0.31J2 =9.3 meV opened up in the spin wave spec-
SDW 0
2 trumofthelocalmomentsisalsofairlyclosetotheexper-
imentalvalue9.8meV11 (asimilarfitforSrFe As 10 can
Finally, the self-consistent mean-field equation reads 2 2
bealsoobtained). Notethatapropercut-offmomentum
|m|= ∆SDW ′ 1 (n −n ), (17) Λ=0.225π is taken in solving (8) and its role in general
kα kβ willbediscussedlatertogetherwiththecomparisonwith
N Xk Ek
other materials.
where n = 1/ e−β(Ek+µ)+1 and n =
kα kβ
1/ eβ(Ek−µ)+1 . (cid:0) (cid:1)
2. Uniform magnetic susceptibility
(cid:0) (cid:1)
C. Physical properties Using the same set ofparameters,the uniform suscep-
tibility
1. Mean-field results
χ =χ +χ (19)
u lo it
is calculated and presented in Fig. 3, which exhibits
Solving the above mean-field equations, one can de-
a pseudogap behavior below T and a rough linear-
termine T for the collinear AF order and the total SDW
SDW
temperaturedependenceinthenormalstate,mainlydue
magnetization defined by
tothe contributionχ fromthe localmoments asshown
lo
M ≡gµ (Mn +m), (18) in the inset. As indicated in the latter, both the magni-
tot B 0
tude and slope of χ are also quantitatively comparable
lo
(g = 2). The results are shown in Fig. 2. Here we have to the experimental measurements27,39,40. Here χ is
lo
fixed the parameters M = 0.8 and J = J throughout given by34
0 2
the paper, with J tunable. The hole dispersion of the
2 2 −ω2 +c2q2+η2
ikt02in)ewraitnht αele=ctr7oJn2sainsdpkar0a=me0t.e1rπizebdasaesdεokn=the−AαR(kP2E−S χlo = 3χ⊥n20+2β−1ωXn,q(ωn2n+c2q2+η2)2 (20)
measurementfortheso-called“122-type”BaFe As 35–38.
2 2
in units of (gµ )2 where χ =1/g .
It is noted that the above choice of the parameters B ⊥ 0
The contribution from the itinerant electrons is given
is not necessarily optimized. But such a set of param-
by
eters can give rise to a quantitative account of a se-
ries of important experimental results. The calculated β ′
|Mtot| ≃ 0.85 µB at T = 0 (which is independent of J2 χit = 2N k[nkα(1−nkα)+nkβ(1−nkβ)] (21)
X
5
as marked in the left bottom in Fig. 4, where the sign
0.6 1.2 change of the pairing amplitude is also indicated. The

2.0 main panel of Fig. 4 shows that λ is fairly large in a
0.4 0.8 wide regime (it even diverges at µ = 0 due to the ar-
tificial nesting effect of the Fermi pockets). Thus the
1.5 0.2 Mtot/2 0.4
superconducting phase is expected to strongly compete
TSDW/J2
with the SDW state at low doping.
0.0 0.0
1.0 0.8 0.9 1.0 1.1 We further find that M and T are quite flat as
tot SDW
g /g function of µ, insensitive to doping. But the SDW order
0 c
0.5 is very sensitive to the aforementioned momentum cut-
offΛ,whichdecidesthe criticalcouplingconstant33 g =
c
g0/gc=0.84 4πc/3Λinthe nonlinearσ-model(5). Indeed,asgivenin
0.0
the inset of Fig. 4, both M and T are well scaled
tot SDW
0.0 0.1 0.2 0.3 0.4 0.5 togetherandmonotonicallydecreasewiththe increaseof
g /g ∝Λ, whereas λ remains flat until reaching beyond
0 c
g /g = 1 (on the right hand side of the dashed vertical
0 c
FIG. 4: (color online) The pairing strength λ vs. µ, with line). Aquantitativecheckrevealsthatagoodagreement
the relative sign change of the pair amplitude shown at the
with experiments always occurs in the regime that the
left bottom. Inset: Variations of λ, Mtot, and TSDW vs. the
couplingconstantg is notfarfromthe quantumcritical
parameter g0/gc of thenonlinear σ-model (5) (see text). 0
point g . For instance, by increasing Λ, say, to g /g ≃
c 0 c
1.03, |M | reduces to 0.42 µ and T ≃ 0.2J =
tot B SDW 2
139 K by choosing J = 60 meV. Then one can get a
in units of (gµ )2. χ is mostly Pauli-like,except for an 2
B it
good quantitative account for the similar properties of
upturn at low temperature as shown in the inset of Fig.
the so-called “1111” compounds6,9, like LaOFeAs. Note
3, which is due to the enhanced density of states in the
that as compared to the “122” materials the J value
induced SDW state where the SDW gap is near but not 2
here is doubled, which is also comparable to the LDA
right at the Fermi level. It results in the upturn of the
estimation29 for LaOFeAs. Using the BCS formula
total susceptibility at low temperature as shown in Fig.
3.
T ≃ω exp[−(1+λ)/λ], (23)
Thus, the present theory not only gives rise to the c 0
large, linear-temperature dependent susceptibility above
it explains why T can be much higher in the “1111”
T , but also naturally explains the quick drop of the c
SDW compounds5 as ω ∝J .
susceptibilitybelowT duetothegapopeningandits 0 2
SDW
upturn at even lower temperatures generically found in Furthermore, since ω0 ∝ J2, one can estimate the iso-
the iron pnictides39,40. tope effect of the iron mass based on
J ∝1+ |u|2 (24)
2
3. Pairing strength D E
(usingthefactthattheeffectivehoppingintegral∝1+u
The itinerant electrons can also exchange the quan-
under a relative lattice displacement u) and |u|2 ∝
tum fluctuations of the local moments to form Cooper
D E
pairs,whichresonantlyhopbetweentheelectronandhole 1/(M Θ ) ∝ (M )−1/2 as in the Debye-Waller factor
Fe D Fe
pocketswith largemomentumtransfersaroundQs. The (ΘD istheDebyefrequencyandMFetheironmass),with
pairing amplitude is s-wave within each Fermi pocket,
but has to change sign between the two pockets due to α ≡−dlnT /dlnM ≃0.5[ |u|2 /(1+ |u|2 )].
exchanging the spin fluctuation D [(6)], consistent with SC c Fe
0 D E D E
other approaches15,16,21,25 and measurement37. (25)
In Fig. 4, an effective dimensionless pairing strength According to the inset of Fig. 4, TSDW ∝ J2
λ as function of µ is calculated based on the same set of and thus the magnetic isotope coefficient αSDW ≡
parameters with fixing m = 0,i.e, in the absence of the −dlnTSDW/dlnMFe is related to αSC by
SDW order. Here λ is given by
α ≃α (26)
SDW SC
λ≡2(J M)2N (−1)D (k−k′−Q ,ω =0) (22)
0 F 0 s FS
(cid:10) (cid:11) whichisconsistentwiththerecentexperimentalresult42,
where N = 1/(4πα) denotes the density of states at but the positive α here is in contrast with another
F SC
theFermienergyandtheaverageisovertheFermipock- recentexperiment43 whereanegativeα isobtainedfor
SC
ets. Note thattoproperlyestimate the strengthwe have samples of high-pressure synthesis. Whether the isotope
consideredtwoholepocketsatΓandtwoelectronpocket effectispositiveornegativeneedsafurtherexperimental
at two M’s in consistency with ARPES experiments37, clarification.
6
III. CONCLUSION AND DISCUSSION Thisprovidesafurthersupportforthemechanismofthe
Hund’s rule coupling, with the FM spins at the shorter
NN sites in favor of the kinetic energy of the itinerant
Inthispaper,wehaveproposedaminimalmodelbased
electrons.
on some basic band-structure and experimental facts,
and shown that it can provide a systematic and quan- Therefore,it is the Hund’s rule coupling term (3) that
titative account for a series of anomalous magnetic and makes the present minimal model nontrivial. By lock-
SC properties observed in the iron pnictides. ing the SDW order of the itinerant electrons and local
This is a multiband model, composed of two indepen- moments together at the wavevector Qs, an imperfect
dentanddistinctcomponents,i.e.,theitinerantelectrons “nesting” of the hole and electron pockets by Qs and
andlocalmoments,respectively. Herethechargecarriers the short-rangeAF ordering of the local moments at Qs
of the itinerant bands presumably come from the more are synchronized to optimize the total energy. One finds
extendedd-orbitals,likedxy,dzx,anddyz. Thelocalmo- thatTSDW,themagnitudeofthetotalmagnetizationmo-
ments are contributed by the electrons from some more ment,the smallspingapdueto the out-of-phaserelative
localizedorbitals,say,dx2−y2 anddz2 orbitals,formingat fluctuations of the SDW moments between the itinerant
much higher temperature than that of the SDW order- electronsandlocalmoments,canbequantitativelydeter-
ing observed in the experiment. Such local moments are mined, ingoodagreementwiththe experimentalresults.
protected by a Mott gap which is supposed to cross the Compared to the conventional Fermi surface nesting
Fermi level all the time such that they do not directly mechanismfor the itinerantelectrons,the presentmodel
contributetothe chargedynamics. Finallythe twoinde- naturally predicts the presence of pre-formed magnetic
pendent degrees of freedom, the itinerant electrons and momentsabovetheSDWorderingtemperature. Thecal-
local moments, are coupled together by the local Hund’s culatednormal-statemagneticsusceptibilityaboveTSDW
rule interaction inside each iron. shows a linear-temperature behavior consistent with the
This is a highly simplified model. We have totally experiment in magnitude and slope, under the same set
omitted the Coulomb interaction between the itinerant of parameters, which is difficult to understand by the
electrons such that the SDW ordering of the itinerant itinerant electrons alone. Furthermore, while the Fermi
electronsisnotdrivenbyaconventionalpictureofFermi surface nesting mechanism is difficult to explain why Tc
surface nesting effect. For the local moments from the is so high in the iron pnictides, the computed pairing
dx2−y2 and/or dz2 orbitals, we have only kept the domi- strength between the itinerant electrons via exchanging
nantNNNJ superexchangecoupling,bridgedbytheAs the AF fluctuations of the local moments is found to be
2
ions, and neglected the NN coupling J due to the sym- easily in strong coupling regime in the present work, us-
1
metry reason. So the local moments alone do not form ing the same parameters.
the propercollinearAForderasseeninexperiment,and On the other hand, as compared to a conventionalJ -
1
areeffectivelydescribedbythe familiarNNNAFnonlin- J model where the collinear AF transition can occur
2
ear σ-models at two sublattices, respectively. alone at low temperature, in the present model the itin-
Then we have shown that the magnetic ordering can erantelectronsplayacrucialroleindrivingthemagnetic
besimplyrealizedduetothestrongHund’srulecoupling orderingbycoupling tothe localmoments,andby doing
betweenthe twocomponents. Namely the itinerantelec- so drastically change their own dynamics below TSDW,
trons can form an SDW order simultaneously with the resultinginpeculiargapbehaviorsinstaticanddynamic
collinearAF orderofthe localmomentsatthe samemo- susceptibilitiesaswellasthescatteringratechangeinop-
mentum Q below a mean-field transition temperature tical conductivity, which have been observed experimen-
s
T , with H gaining a mean-field energy. tallyandarehardtounderstandifthemagneticordering
SDW J0
Of course, one may further consider the Coulomb in- is due to the J1-J2 superexchange interaction alone.
teraction in the itinerant bands which may result in an Asamatteroffact,inordertoprovideaconsistentex-
SDW instability without involving the local moments. planation of some generic phenomena found in the iron
Or one can further introduce the NN coupling J for the pnictidesincludingtheSDWandSCorderswithinasin-
1
local moments and obtain a collinear AF state indepen- gleframework,thepresentmodelpredictsthatinthenor-
dent of the itinerant electrons. But the key assumption mal state the local moments should be close to a critical
inthisapproachisthattheseeffectsarenegligible,tothe regime of quantum magnets with the dominant J -type
2
leadingorderapproximation,ascomparedto the Hund’s superexchangeinteraction,whichcanbe critically tested
rule coupling term and thus are omitted for simplicity. by a neutron-scattering experiment. The magnetic state
Onecanalwaysmakethemodelmorerealisticbyadding of the local moments is also expected to play an impor-
more perturbative terms later. As a matter of fact, in tant role in the superconducting state. Here the reason
the experimentally observed lattice distortion6,8 accom- behindtheriseofT bydopingorpressureinexperiment
c
panying the collinear AF order, the NN spins along the maybe notdueto apureincreaseofchargecarriernum-
shorter lattice constant direction are always FM parallel berlikeinthecuprates,butratherduetothesuppression
whiletheantiparallelNNspinsusuallycorrespondtothe ofthecollinearAForderinfavorofsuperconductivity,as
longerlatticeconstant8,incontrasttoaJ -drivenmecha- the result of the accompanyingchange in the ratio g /g
1 0 c
nismwhichwouldpreferanopposite lattice distortion24. nearthequantumcriticalregime33,41,asdiscussedabove.
7
However,howthismechanismcanberealizedmicroscop- Weng,M.W.Wu,T.Xiang,J.Zaanen,G.M.Zhang,and
icallyisbeyondthescopeofthepresentwork,whichmay also thank W. Bao,X.H. Chen, P. C. Dai, H. Ding, D.L.
involvedetailedandrealisticlocalinteractions(including Feng, N.L. Wang, H.H. Wen, and X. J. Zhou for gen-
J ), and will be left for a future study. erously sharing their experimental results. This work is
1
supported by NSFC, NBRPC, and NCET grants.
Acknowledgments
We would like to acknowledge stimulating discussions
with D.H. Lee, Z.Y. Lu, Q.H. Wang, Y.Y. Wang, M.Q.
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