Detect the orbital distribution of the magnetic fluctuation in the Iron-based superconductors with resonant inelastic X-ray scattering spectroscopy

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by  Da-Wei Yao

Detect the orbital distribution of the magnetic fluctuation in the Iron-based
superconductors with resonant inelastic X-ray scattering spectroscopy
Da-Wei Yao and Tao Li
Department of Physics, Renmin University of China, Beijing 100872, P.R.China
(Dated: January 11, 2016)
Detecting the orbital distribution of the magnetic fluctuation of the Iron-based superconductors
is the key to understand the mechanism for the magnetism,superconductivity and electronic ne-
6 maticity in these multi-orbital systems. In this work, we propose that the resonant inelastic X-ray
1 scattering(RIXS)techniquecanbeusedtoprobeselectivelythemagneticfluctuationondifferentFe
0 3dorbitals. Inparticular,themagneticfluctuationonthethreet2g orbitals, namely3dxz,3dyz and
2 3dxy,can beprobed separately byarranging properlythepolarization of theincidentandoutgoing
photons. Such orbital-resolved information on the magnetic fluctuation is invaluable for the study
n
of the orbital-selective manifestation of electron correlation effect and the entanglement between
a
J the orbital and the magnetic degree of freedom in the electronic nematic phase of the iron-based
superconductors.
8

PACSnumbers:
]
n
o
The multi-orbital nature of the Iron-based supercon- mode in systems deep inside the paramagnetic phase16,
c
- ductors is the most important new feature in these new whichisastrongsignatureforlocalmomentbehavior. It
r
p familyofunconventionalsuperconductors. Evidencesac- isnaturaltoexpecttheelectroninthenarrowestbandto
u cumulatedfromabroadrangeofstudiesindicatethatthe contribute the most to such local magnetic fluctuation.
s orbital degree of freedom is playing an important role in
With these understandings in mind, it is quite em-
.
at the physical processes related to the magnetism, super- barrassing to find there is almost no efficient way to
conductivity,andtheelectronicnematicityofthesystem.
m probe the orbital character of the magnetic fluctuation
For example, the orbital characterof the electron on the
in the Iron-based superconductors. In principle, the or-
-
d Fermi surface may drastically change the nesting condi- bitalcharacterofthemagneticfluctuationcanbeinferred
n tion for itinerant magnetism and may result in a gapless fromthe atomic formfactor inan inelastic neutronscat-
o SDW phase1,2. The pairing potential for BCS supercon-
tering(INS) measurement. However, such a distinction
c ductivity in the system is also subjected to the modu-
is small since the atomic form factor only depends on
[
lation of the orbital charater of the electronic state on the density distribution of the electron and fails to in-
1 the Fermi surface3. More recently, it is found that the corporate the difference in the wave functions of the 3d
v orbitalorderingisintimatelyrelatedtothenematicityin
orbitals. It is the purpose of this paper to propose an
9 the magnetic correlation of the system4–8. Such strong
efficient way to measure the orbital distribution of the
0
entanglement between the orbital and the magnetic de-
magnetic fluctuation in the Iron-based superconductors.
8
gree of freedom may hold the key for the origin of the
1 We propose that the resonant inelastic X-ray scattering
electronic nematicity in the system9,10.
0 technique can be used to probe the orbital character of
. the magnetic fluctuation. In such a probe, it is the dif-
1 At the same time, it has long been debated if an
0 itinerant or a local scenario is more appropriate for ference in the wave functions of the 3d orbitals, rather
6 the description of the magnetism of the Iron-based than difference in the density distribution, is playing the
1 superconductors11–15. Unlike its counterpart in the key role. According to our proposal, we can selectively
: coupletothemagneticfluctuationonagive3dorbitalby
v cuprate superconductors, the magnetism in the Iron-
i based superconductors exhibits strong material depen- properly arranging the polarization of the incident and
X
outgoingphoton.The informationprovidedbysuchmea-
dence and is very susceptible to the change of external
r surement will be invaluable for us to study the interplay
a conditions. Such a variability of magnetism is hard to
between the orbital and magnetic degree of freedom in
understand for a single band system, since the energy
the system. It can also provide key information on the
scale for magnetic correlation is usually the dominating
orbital-selectivemanifestationofthe electroncorrelation
energyscale of the system, but may be consistentwith a
scenarioinvolvingbothitinerantandlocalcomponents12. effect in the Iron-based superconductors.
From the early days of the field, it had been anticipated The RIXS spectroscopy is a two-step process(here we
bysomeauthorsthattheelectroncorrelationondifferent will only consider the direct RIXS process) to probe the
Fe 3d orbitals may be different14,15. Electrons on some electronic excitation in the valence shell using an atomic
orbitals may behave more like localized moment than resonance between a core level and the valence shell as
other electrons. Indeed, recent ARPES measurements an intermediate step17. The incident photon first ex-
have already find large difference in mass renormaliza- citesacoreelectronintothevalenceshellnearanatomic
tion in different 3d orbitals in some systems13. RIXS resonance, leaving a core hole which is later annihilated
measurementsalsofoundevidencefordispersivemagnon by another valence electron accompanied by an emitted
2
photon. As the resonance energy can be quite high, the in which r = ǫ r. Here 3/2,m is the
in(out) in(out) j
· | i
momentumtransferbythescatteredphotoncanbequite mj-th component of the 2p23 multiplet and is given
substantial. RIXS can thus be used to probe excitation by 3/2,m = C m,m , in which
in a large portion of the Brillouin zone. At the same | ji Pm,ms 3/2,mj,m,ms| si
C is the C-G coefficient. m(= 0, 1) and
time, as a result of the strong spin-orbital coupling in 3/2,mj,m,ms ±
m (= , ) denote the eigenvalues of the z-component of
s
the core level, RIXS can also be used to probe the spin ↑ ↓
the orbital and spin angular momentum(l and s ) for
z z
dynamics of the valence electron . In this paper, we will
the 2p core electron.
focus on the Fe L edge resonance which involves the
3 3 Ti can be interpretedas a single particle operator act-
atomic transition between the Fe 2p2 core level and the
ing on the valence electron state on the i-th Fe site. As
Fe 3dorbital. Sucharesonancehasalreadybeenusedto
a result of the spin-orbital coupling in the core level, T
i
probethelocalmagneticfluctuationintheheavilydoped
caninducebothspinconservingandspinfliptransitions.
iron-based superconductors16.
Completing the summation over m , we find T can be
j i
Using the second order perturbation theory, the tran-
written as the following second quantized form
sition amplitude between the valence states induced by
the scattered photon can be expressed in the form17
T = n c† c
i X µ,ν i,µ,σ i,ν,σ
F = f † G(z) g . (1) µ,ν,σ
f,g h |Dout Din| i
+ sz σc† c
Here g and f are the initial and final state of the sys- X µ,ν i,µ,σ i,ν,σ
tem. |Gi(z)=| i |nihn| isthepropagatorfortheinterme- µ,ν,σ
diated state aPftenrzt−hEenfirst photon excitation. n is the + Xs+µ,νc†i,µ,↑ci,ν,↓
| i
intermediate state with a core hole, z = E +¯hω+iΓ . µ,ν
g n
Eg and En are the energies of the ground state and the + Xs−µ,νc†i,µ,↓ci,ν,↑ (5)
intermediate state, Γ is the inverse life time of the in-
n µ,ν
termediatestate n causedbyothernonradiatvieatomic
| i
processes. ¯hω is the energy of the incident photon. Here µ,ν = 1,..,5 are the indices of the five 3d orbitals.
D = im1ω Pieik·riǫ·pi is the transition operator of the We will use the following convention for numbering the
nonmagnetic coupling between the photon and the elec- 3d orbitals, namely 1 = xz , 2 = yz , 3 = xy ,
| i | i | i | i | i | i
tron. k is the wave vector of the incident or outgoing 4 = 3z2 r2 ,and 5 = x2 y2 . σ = , isthespinof
photon. Here ǫ is the polarization vector of the incident |thie va|lenc−e eleictron|. ic | is−theiannih↑ila↓tion operator
i,µ,σ
or outgoing photon. ri and pi are the position and mo- foranspin-σ electrononthe µ-thorbitalofsite i. Inthe
mentum operator of the i-th electron. above equation, the matrix elements are given by
Since the radius of the Fe 3d orbital is much smaller
than the distance between neighboring Fe ions in the 2
n = µr m mr ν
Iron-basedsuperconductors,the dipole approximationis µ,ν 3Xh | in| ih | out| i
applicableandwecanapproximatetheplanewavefactor m
1
eik·ri in the transitionoperator by a constant eik·Ri17, sz = m µr m mr ν
in which R is the position of tDhe atomic site that the µ,ν 3X h | in| ih | out| i
i m
i-th electron resides. With such an approximation, one 1
s+ = A µr m m+1r ν
can show that the RIXS transition amplitude becomes µ,ν 3X mh | in| ih | out| i
m
F = f (ǫ D† )G(z)(ǫ D )g , (2) 1
f,g h | out· kout in· kin | i s−µ,ν = 3XAmhµ|rin|m+1ihm|rout|νi, (6)
in which Dk =Pieik·Riri is the dipole operator at wave m
vector k. In the Fe L edge resonance, the intermediate
state always involves3a core hole in the 2p23 multiplet in which |mi is eigenstate of the lz with eigenvalue m,
A = 2 m(m+1).
and we will treat the denominator in G(z), z En, as m p −
− In this study, we will focus on the spin excitations in-
a constant. If we neglect the interaction effect in the
ducedby the photonscattering. To simplify the analysis
intermediate state, we can derive an effective transition
of the selection rules in the tetragonal environment, we
operator for the valence electrons by summing over the
expand the 2p core level m in terms of their real com-
intermediate core hole states, which takes the form of
| i
ponents that have the p , p and p character, namely
x y z
T= eiq·RiT . (3) x , y and z , as
X i | i | i | i
i
1
Here q = k k is the momentum transfer from the +1 = (x +iy )
in− out | i −√2 | i | i
scattered photon. The operator T is given by
i
0 = z
| i | i
T = r 3/2,m 3/2,m r , (4) 1
i X in| jih j| out 1 = (x iy ). (7)
mj |− i √2 | i− | i
3
One then find that the matrix element for the spin- orbital rin rout spin direction transition probability
dependent transitions become xz x(z) z(x) y |hxz|x|zi|2
yz y(z) z(y) x |hyz|y|zi|2
sz = i µr x y r ν xy x(y) y(x) z |hxy|x|yi|2
µ,ν 3h | in| ih | out| i
i TABLE I: The selection rules for the magnetic transitions in
− 3hµ|rin|yihx|rout|νi. (8) thet2g subspaceandthecorrespondingpolarization arrange-
ments.
In the same way, one find that
i
sx = µr y z r ν y-directionandthe transitionprobabilityis proportional
µ,ν 3h | in| ih | out| i to xz xz 2. Using cyclic permutation between x, y
i and|hz, w| e|cia|n obtainthe selection rules for other setups,
µr z y r ν
in out
− 3h | | ih | | i which are summarized in table I. If we neglect the lig-
i andeffectoftheFe3dorbitalandapproximatethethree
sy = µr z xr ν
µ,ν 3h | in| ih | out| i t2g orbitals with the correspondingatomic orbitals, then
i thetransitionprobabilityisthesameforallthreesetups.
µr x z r ν . (9)
− 3h | in| ih | out| i Thestrengthofthemagneticfluctuationonthethreet2g
orbitals can then be directly compared with each other
Here sxµ,ν = 12(s+µ,ν +s−µ,ν), sxµ,ν = 21i(s+µ,ν −s−µ,ν). in the RIXS measurement.
Withtheaboveequations,wecanselectivelyexcitethe
Now we discuss the application of the above results
magnetic fluctuation in the orbital space of the 3d elec-
in the study of the Iron-based superconductors. From
trons. Now we apply them to the Iron-based supercon-
the begining of the study ofthe iron-basedsuperconduc-
ductors. We will focus on the collective spin fluctuation
tors, it has been anticipated that a two-component pic-
evolvedfromthespinwaveexcitationoftheorderedstate
turewithbothitinerantandlocalizedmomentummaybe
and assume they have similar orbital characters. In a
more appropriate for the description of their magnetic
multi-orbital system, the magnetic order parameter is in behavior11,12. Especially, a linear temperature depen-
generalamatrixoftheform1,7Mµ,ν =Pσσhc†i,µ,σci,ν,σi. denceoftheuniformsusceptibilityisuniversallyobserved
The eigenvalues and eigenvectors of this matrix provide in all iron-based superconductors18. Such a behavior is
us the ordered moment and the orbital that they reside. hard to be interpreted in a picture with only itinerant
In the Iron-based superconductors, the lattice symme- electron, but may indicating the existence of near criti-
try around the Fe ions is broken to D2 in the magnetic callocalizedmoment11. Thefluctuationofsuchlocalized
ordered state. The symmetry allowed expectation value moment at short length scale should be robust against
forthe magneticorderparameterMµ,ν takesthe general carrier doping. Indeed, recent RIXS measurements does
form of findstrongevidenceforrobustdispersivemagnonexcita-
tionatshortlengthscaleintheheavilydopediron-based
M 0 0 0 0
 01,1 M 0 0 0  superconductors16. Among the five 3dorbitals,it is gen-
2,2
erallybelievedthatthe3d orbitalisthemostcorrelated
M = 0 0 M 0 0 . (10) xy
 0 0 03,3 M M  fromboththeoreticalandexperimentalstudies13,14. Itis
 4,4 4,5 
then quite naturalto expect that the robust highenergy
 0 0 0 M M 
4,5 5,5
spinexcitationintheheavilydopediron-basedsupercon-
Thus the magnetic fluctuation on the t2g and the eg or- ductors should have mainly the 3dxy orbital character.
bital are decoupled at low energy. It has been found Following our proposal, we should polarize the incident
that in the Iron-based superconductors, the two e or- photon in the x(y) and the outgoing photon in the y(x)
g
bital (3d3z2−r2 and 3dx2−y2) have the least correlation direction to detect such local magnetic fluctuation. We
effect among the five 3d orbitals and contribute little to note that with the polarization of the photon fixed we
the collectivespinfluctuationatlowenergy5,14. Wethus can still rotate the wave vector of the photon in a plane
neglect the e orbitals in the following discussion and perpendicular to the polarization vector. As a result,
g
concentrate on the magnetic fluctuation in the t sub- we can scan the momentum of the magnetic fluctuation
2g
space. continuously in the Brillouin zone.
Thethree2porbitals x ,y and z andthethreecom- Another possibility offeredby our proposalis to study
| i| i | i
ponents of the position vector x,y and z each forms a the interplay between the orbital ordering and the mag-
distinct one dimensionalrepresentationof the D group. netic nematicity of the iron-based superconductors. In
2
The same is true for the three t orbital xz ,yz and the presence of orbitalordering,the degeneracybetween
2g
| i| i
xy . Withthese symmetry properties,it is easytoshow the 3d and the 3d orbital is lifted and the two or-
xz yz
| i
that to excite the magnetic fluctuation on the xz or- bitals will contribute differently to the magnetism of the
bital, we should polarize the incident photon in t|heix(z) system5–8. Since the nesting vector for magnetic excita-
-directionandthe outgoingphotoninthe z(x)-direction. tion in the 3d and 3d -dominant band is different(by
xz yz
The spin fluctuation excited in this setup is along the a π/2 rotation along the z-axis), the orbital order in the
4
iron-basedsuperconductors is entangled with their mag- detect the orbital character of magnetic fluctuations in
netic nematicity. In the magnetic ordered state, the or- the iron-based superconductors. We find the contribu-
bitalordercanliftthedegeneracybetweenthe(π,0)and tion from the three t orbitals in the d-shell, namely
2g
the (0,π) ordering pattern by enhancing(reducing) the 3d , 3d and 3d can be selectively probed by polar-
xz yz xy
condensation energy of the preferred(disfavored) mag- izing the incident and outgoing photons in the xz, yz
netic ordering pattern7. In the paramagnetic phase, the and xy plane respectively. We show such orbital-specific
orbitalorderingcaninducelargedisproportionalityinthe informationcanbeveryusefulinthestudyoftheorbital-
strength of spin fluctuation at (π,0) and (0,π) through selective Mottness and the interplay between the orbital
the same mechanism8, which has been observed by re- ordering and the magnetic nematicity of the iron-based
cent neutron scattering measurement19. On the other superconductors. Such a technique can also be used in
hand, the entanglement with the magnetic nematicity other transition oxides material with more than one ac-
can also be a driving force for the orbital ordering in tive orbitals. In our derivation of the RIXS transition
the iron-based superconductors.7 To test the validity of probability, we have made two key approximations. The
such a picture, we can compare the orbital character of first is the negligence of the intermediate state interac-
the magnetic fluctuation at (π,0) and (0,π) in the elec- tion effect. The second is the dipole approximation. We
tronic nematic phase of the iron-based superconductors. believe the qualitative feature of our conclusion will not
To single out the contribution from the 3d and 3d be changed by more accurate treatment, which will be
xz yz
orbital, we should polarize the photon in the x(z) and left for further investigations.
z(x) direction in the first case and in the y(z) and z(y)
direction in the second case. This work is supported by NSFC Grant No. 11034012
In summary, we propose that RIXS can be used to and Research Funds of Renmin University of China.
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