Table Of ContentIndirect resonant inelastic X-ray scattering on magnons
Jeroen van den Brink
Institute-Lorentz for Theoretical Physics, Universiteit Leiden,
P.O. Box 9506, 2300 RA Leiden,The Netherlands and
Institute for Molecules and Materials, Radboud Universiteit Nijmegen,
P.O. Box 9010, 6500 GL Nijmegen, The Netherlands
(Dated: February 2, 2008)
7
0 Recentexperimentsshowthatindirect resonantinelastic X-rayscattering (RIXS)isanewprobe
0 ofspin dynamics. HereIderivethecross-section formagnetic RIXSand determinethemomentum
2 dependent four-spin correlation function that it measures. These results show that this technique
n offers information on spin dynamics that is complementary to e.g. neutron scattering. The RIXS
a spectrum of Heisenberg antiferromagnets is calculated. It turns out that only scattering processes
J that involve at least two magnons are allowed. Other selection rules imply that the scattering
2 intensity vanishes for specific transferred momenta q, in particular for q = 0. The calculated
2 spectra agree verywell with theexperimental data.
]
el Introduction. Resonant inelastic X-ray scattering 4p (cid:0)(cid:1) 4p 4p
- (RIXS) is a technique that is rapidly developing due to
r
t the recent increase in brilliance of the new generation 3d 3d 3d
s
synchrotronX-ray sources [1]. As a scattering technique
.
t
a RIXShastwoimportantadvantages. First,itissensitive
m to excitations that are difficult to observe by otherwise, (cid:0)(cid:1) (cid:0)(cid:1) 1s (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) 1s(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) 1s (cid:0)(cid:1) (cid:0)(cid:1)
d- fitoerse[x2a,m3,pl4e,d5i,re6c,t7d].-dSeexccointadt,ioitnpsrinobceuspdrairteecstolyrmboatnhgtahne- qin ω0in qout ω0out
n
energy and momentum dependence of such excitations
o
–unlike for instance optical methods. FIG.1: SchematicrepresentationofthemagneticRIXSscat-
c
teringprocessat atransition metal K-edge. Left: theincom-
[ Experiments are performed at the K-edges of transi- ingphoton(energyωi0n,momentumqin)inducesanelectronic
3 tionmetalions,wherethefrequencyofthe incidenthard transition from a 1s to a 4p level. Middle: exchange inter-
v X-rays is tuned to match the energy of an atomic 1s-4p action between 3d electrons in the presence of the core-hole.
0 transition(5-10 keV). When a photon is absorbedin the Right: de-excitation and outgoing photon (ωo0ut, qout).
4
solida1score-holeiscreatedastheelectronispromoted
1
to the 4p empty states. From the theoretical side it is
0
1 clearthatthescatteringintensityisrelatedtothecharge S(q,ω) vanishes, as well as the longitudinal spin one –
5 dynamics of the system under study [8, 9, 10, 11]. The which is in fact a higher order charge response function.
0 precise nature of this relation was established recently: This implies that we need take the ultra-short lifetime
/
t a systematic expansionin the ultra-shortlife time of the expansion [12, 13] one step further. By doing so I find
a
m core-hole[12,13]showsthattheeffectiveRIXScrosssec- the scattering amplitude for magnetic RIXS, expressed
tion a resonant scattering factor times a combination of in terms of an intrinsic dynamic four-spin correlation
-
d the linear charge response function S(q,ω) and the dy- function of the system that is probed. Moreover I de-
n namic longitudinal spin density correlation function of rive selection rules which are related to the symmetry of
o the d−electrons. the underlying spin Hamiltonian. It turns out that the
c
: A recent breakthrough was achieved by J.P. Hill et firstallowedmagneticscatteringprocessisatwomagnon
v al. [14], who observed that RIXS on the high temper- scattering one. As an example the formalism is used to
i
X ature superconductors La2−xCuxO4 picks up transver- calculate the indirect RIXS spectrum of Heisenberg an-
r sal spin dynamics –magnons. This triggers the question tiferromagnets as function of transferred momentum q.
a what kind of transversal spin correlation function actu- The scattering intensity vanishes at (0,0) and at the an-
ally is measured in magnetic RIXS and how this is dif- tiferromagnetic wavevector q = (π,π). The computed
ferent from other probes of spin dynamics –in particular spectra all agree with the experimental data.
inelastic neutron scattering. The aim of this Letter is to Series expansion of the scattering cross section. The
answer that question. probability for X-rays to be scattered from a solid state
Experimentally the magnetic RIXS signal is already system can be enhanced by orders of magnitude when
already in the undoped cuprate La2CuO4 –in fact it is the energy of the incoming photons is in the vicinity of
strongest in this case. The magnetic loss features are at an electronic eigenmode of the system –i.e. in the vicin-
energies well below the charge gap of this magnetic in- ity of a resonance. At a transition metal K-edge a 1s
sulator, at energies where the charge response function electron from the inner atomic core is excited into an
2
empty 4p state, see Fig. 1. In transition metal systems of this scattering amplitude it is essential that we split
the empty 4p states are far (10-20 eV) above the Fermi- up the intermediate state Hamiltonian into two parts:
level,sothattheX-raysdonotcausedirecttransitionsof Hint = H0 + H1, where H0 is the Hamiltonian of the
the 1s electron into the lowest 3d-like conduction bands system without core-hole and H1 the part of the Hamil-
of the system. Still this technique is sensitive to low en- tonian that is active in the presence of a core-hole.
ergyexcitationsofthed−electronsbecausetheCoulomb Spin Hamiltonian withcore-hole. Wewillcalculatethe
potential of 1s core-hole can couple to e.g. very low en- resonantX-raycrosssectioninaMott-Hubbardinsulator
ergy electron-hole excitations when the system is metal- at zero temperature. We assume that this system is de-
lic. Since the charge excitations are caused by the core- scribed a single band Hubbard model at strong coupling
hole, this scattering mechanism is sometimes referred to and at half filling. In this case the electrons are local-
as indirect RIXS. ized and the only low energy degree of freedom is their
InthisLetter,however,wewillconsiderinsulatingsys- spin. It is well known that in a Mott-Hubbard insula-
tems –in particular Mott insulators, where the only re- tor the magnetic exchange integrals are determined by a
maininglow-energydegreesoffreedomarethespinones. virtualhoppingprocessofelectrons. We denotethe hop-
So in order to determine the RIXS scattering amplitude ping amplitudes of the valence electrons by t where i
ij
we need to establish how in this case the core-hole cou- andj denotelatticesiteswithlattice vectorsR andR .
i j
ples to magnetic excitations. Our starting point is the The Coulomb interaction between electrons at the same
Kramers-Heisenberg formula for the resonant scattering site is U, so that in second orderperturbationtheory we
cross section [15, 16] have the exchange interaction J =2t2/U and the spin
ij ij
dynamics is governed by a Heisenberg spin Hamiltonian
d2σ
∝ |A |2 δ(ω−ω ), with of the form
dΩdω(cid:12) fi fi
(cid:12)res Xf
(cid:12)(cid:12) hf|Dˆ|nihn|Dˆ|ii H0 = JijSiSj = Jk Sk·S−k, (3)
Afi = ωres . (1) Xi,j Xk
Xn ωin−En−iΓ
whereJ istheFouriertransformofJ . Itiswellknown
k ij
and f and i denote the final and initial state of the thatforneutronscatteringonsuchaspinsystemthetwo-
system, respectively. The sum is over f is over all fi- spin correlation function e−iωthSα(0)Sα (t)idt is
α q −q
nal states. The momentum and energy of the incom-
measured,wherethesumαPisoRverthethreespincompo-
ing/outgoing photons is q and ω0 and the loss
in/out in/out nents. We will see shortly that magnetic RIXS measures
energy ω = ωo0ut −ωi0n is equal to the energy difference a very different four-spin correlation function.
between the final and initial state ωfi = Ef − Ei. In Intheintermediatestateacore-holeispresent. Weas-
the following we will take the groundstate energy of our sumethecore-holepotentialU tobelocal,i.e. asacting
c
system as reference energy: Ei = 0. In the scattering exclusively on those (valence) electrons that belong to
amplitude Afi the resonantenergy is ωres, n denotes the theatomwiththe core-hole. Whenonsitemacore-hole
intermediatestatesandDˆ isthedipoleoperatorthatde- ispresent,theexchangeinteractionsthatinvolvethespin
scribes the excitation from initial to intermediate state on site m becomes stronger, as the virtual intermediate
and the de-excitation from intermediate to final state. state with two electrons on the site with the core-hole
The dipole operator is given in more detail in for in- are lowered in energy by U . On the other hand the vir-
c
stanceRef.[12]. TheenergyoftheincomingX-rayswith tual state with two holes present on site m is at U +U .
c
respecttotheresonantenergyisωin(thisenergycanthus Adding these two effects leads to the following exchange
eitherbenegativeorpositive: ωin =ωi0n−ωres)andEn is interaction between the spins on site m and j of
the energy of intermediate state |ni with respect to the
U
resonance energy. The last important detail is that the Jc =2t2 =(1+η)J (4)
intermediatestateisnotasteadystate. Thehighlyener- mj mjU2−Uc2 mj
getic1score-holequicklydecayse.g. viaAugerprocesses
and the core-hole life-time is very short. This leads to a and η = U2U−c2Uc2. From this we obtain H1, part of the
core-hole energy broadening Γ which is proportional to intermediatestateHamiltonianthatisactiveinthepres-
the inverse core-hole life-time. ence of a core-hole
TocalculateRIXSamplitudes,weproceedbyformally
expandingthescatteringamplitudeinapowerseries[12] H1 =η sms†mJmjSm·Sj, (5)
Xm,j
∞
Afi = ω∆res ∆1lhf|Dˆ(Hint)lDˆ|ii (2) where the operator sm creates a core-hole on site m.
Xl=1 Spin-spin correlation function as measured in RIXS.
where we introduced ∆=ωin−iΓ and the Hamiltonian In order to finally obtain the magnetic cross section, we
in the intermediate state Hint. For a further expansion need to evaluate the operator (Hint)l in equation (2).
3
(0,0) Hamiltonian of the system (Oˆq=0 ∝ H0). Thus |ii and
|fi are eigenstates of this magnetic scattering operator,
which makes inelastic scattering impossible [18]. This is
(π,π) in stark contrast with conventional two-magnon Raman
q scattering in the optical or UV range. That technique
>
-- is also sensitive to a four-spin correlation function [19],
π
( ,0) butaquite differentone. Thisis obviousconsideringthe
factconventionalRamanscatteringisrestrictedtoq=0
–preciselythe momentum transferwhere RIXS vanishes.
(0,0) Thereforealsothese twotechniquesoffercomplementary
0 1 2 3 4 5
information on spin dynamics.
ω
/J
Two-magnon scattering in antiferromagnets. From
(π,π) equation (7) immediately another selection rule follows.
The projection of the total spin on the z-axis, Sz =
tot
(π,0) iSiz commutes with both H0 and Oˆq. Therefore Stzot
(0,0) iPs conservedduring the scattering process,which implies
that creation of a single magnon by the core-hole is not
possible. Butthecreationoftwomagnons(withopposite
z-projections) is allowed and this is therefore the lowest
ordertransversalspinscatteringprocessthatcontributes
toindirectRIXS.Alsofour-magnonscatteringisinprin-
ciple allowed, but of higher order and therefore smaller
and not taken into account in the linear spinwaveanaly-
sis that follows. Note that on the grounds of symmetry
FIG. 2: Left: RIXS spectrum for a nearest neighbor Heisen- it is possible, in principle, to have magnetic scattering
berg antiferromagnet with exchange interaction J as a func- without creating any additional magnons in the scatter-
tion of transferred momentum q for a cut through the ing process. Physically this situation can only arise at
Brioullin zone that is show on theright. finite temperature, when a magnon with momentum k
that is present in the groundstate is scattered to k+q
bythecore-hole. ThisimpliesthatmagneticRIXShasan
This is a non-trivial task. We first expand (Hint)l in
interesting temperature dependence –which is, however,
a series that contains the leading terms to the scat-
beyond our present scope.
tering cross section in lowest order in ηJ/∆. A con-
We now apply the theory above to two-dimensional
servative estimate gives, using for the copper K-edge
bipartite S = 1/2 antiferromagnets and determine the
Γ ≈ 1.5 eV and for the high temperature superconduc-
two-magnonRIXS spectrum as a function of transferred
tors J ≈125 meV andU /U ≈0.85[6, 17], that atreso-
c
momentum,atzerotemperature. TothisendtheHamil-
nance ηJ/∆ ≈ 0.22. This makes it a suitable expansion
parameter. Re-summing the leading order terms in the tonian H0 and correlation function Oˆq are bosonized
within linear spinwave theory, where S+ →a+,S− →a
seriesgivesinthe endthe magneticscatteringamplitude i i i i
and Sz → 1 −n , with boson creation/annihilation op-
i 2 i
Afi = ωres η hf|Oˆq|ii (6) erators ai/a+i and number operator ni = a+i ai. After
∆ ∆−ω a Boguliobov transformation into the boson operators
α /α+ we have α+ =u a++v a with
where we find that the scattering operator Oˆ to be i i k k k k −k
q
Oˆq = Jk Sk−q·S−k. (7) uk =rJkǫ=0 + 12, vk =sign[Jk]rJkǫ=0 − 21 (8)
Xk k k
so that the magnetic correlation function that is mea- andǫ =2 J2 −J2,thentheHamiltonianreducesto
k k=0 k
sured in RIXS is proportional to the spin correlator H0LSW = pkǫkα+kαk. It is straightforwardto show now
e−iωthOˆq(0)Oˆ−q(t)idt. that withiPn linear spinwave theory the two magnon part
R This expression is surprisingly simple and elegant. It of the magnetic scattering operator is
shows that indeed momentum resolved indirect RIXS
probes a momentum dependent four-spin correlation OˆLSW = (J +J )(u u
q k−q/2 k+q/2 k−q/2 k+q/2
function. Fromexpression(7)itisimmediatelyclearwhy kX>0
experimentally the magnetic RIXS intensity vanishes at + vk−q/2vk+q/2)−(J0+Jq)(uk−q/2vk+q/2
zero transferred momentum, i.e. at q = 0. In that case
the correlation function is nothing but the steady state + vk−q/2uk+q/2) α+k−q/2α+−k−q/2+h.c. (9)
(cid:16) (cid:17)
4
First moment noted, but at other wave-vectors the agreement stands
4
ω/J outevenmore. ThedataonLa2CuO4showforq=(π,0)
3
apeak ataround500meV, preciselywherewe finditon
2
the basis of a nearest neighbor Heisenberg model with
1 J =146meV –avalue alsofoundby the analysisofneu-
(0,0) (π,0) (π,π) (0,0) tron scattering data [20]. Similar agreement is found at
ht J’=J’’=0
eig J’=J’’=J/20 q=(0.6π,0) and q=(0.6π,0.6π).
w
al Conclusions. We determined the momentum depen-
r
ect dent four-spin correlation that is measured in magnetic
p
al s RIXS. On the basis of this the magnetic RIXS spec-
tot(0,0) (π,0) (π,π) (0,0) trum was calculated for Heisenberg antiferromagnets
--> q
with short and longer range couplings. We derive selec-
tionrulesshowthatonlyscatteringprocessesthatinvolve
FIG. 3: Top: first moment of the RIXS spectrum and bot-
at least two magnons are possible and that the scatter-
tom: total spectral weight for a nearest neighbor Heisenberg
antiferromagnet (full line) and a Heisenberg model with in ing intensity vanishes at zeromomentum transferandat
addition a second and third neighbor exchange (J′ and J′′, the antiferromagnetic lattice vector –which are observed
respectively) (dashed line). in experiment. Moreover theory and experiment agree
very well on the two-magnon peak position and spectral
weight throughout the measured Brioullin zone. These
whereqisthetotalmomentumofthetwomagnonexcita- resultsshowthatRIXS isinprinciple a powerfulltoolto
tion. TheresultingRIXSspectrumisshowninFig.2,for obtain new information on spin dynamics –information
a cut through the Brioullin zone indicated by the right- thatis complementaryto whatcanbe obtainedby other
hand side of the figure. There are several remarkable techniques such as neutron, non-resonant X-ray or con-
features in the spectrum. ventional two-magnon Raman scattering.
First of all the spectral weight vanishes at q = (0,0)
Acknowledgments. I thank John P. Hill and Michel
and q = (π,π). This in agreement with the experimen-
van Veenendaal for stimulating discussions and sharing
tal observations[14]. Fromthe scatteringoperator(9) it
unpublished data. I thank Fiona Forte andLuuk Ament
one sees that this selection rule is due to the antiferro-
for critically reading the manuscript. This work is sup-
magnetic ordering that occurs in the Heisenberg Hamil-
ported by FOM and the Dutch Science Foundation.
tonian. It is actually easy to show that the RIXS inten-
sity always vanishes at (π,π) if this scattering vector is
also a reciprocal lattice vector. This holds for instance
alsofor a HeisenbergHamiltonianwith weaksecondand
thirdneighborexchangeintereactions(J′andJ′′,respec- [1] Forareviewsee: A.KotaniandS.Shin,Rev.Mod.Phys.
tively),whichisillustratedbythecalculationRIXSspec- 73, 203 (2001).
trum for an extended Heisenberg antiferromagnet with [2] M.Z. Hasan et al.,Science 288, 1811 (2000).
J′ = J′′ = J/20, shown in Fig. 3. The longer range [3] M.Z. Hasan et al.,Phys. Rev.Lett 88 177403 (2002).
[4] Y.J. Kim et al.,Phys.Rev. Lett.89, 177003 (2002).
couplings transfer spectral weight to scattering vectors
[5] J.P. Hill et al., Phys.Rev.Lett. 80, 4967 (1998)
around(π,π), butdo notinduce weightatpreciselythat
[6] K. H¨am¨al¨ainen et al.,Phys. Rev.B 61, 1836 (2000).
wavevector.
[7] T. Inami et al.,Pys. Rev.B 67, 45108 (2003).
The other remarkable feature feature of the magnetic [8] E.D. Isaacs et al.,Phys.Rev. Lett.76, 4211 (1996).
RIXS spectrumis its strongdispersion. This is apparent [9] P. Abbamonteet al., Phys.Rev.Lett. 83, 860 (1999).
from Fig. 2 and the upper pannel of Fig. 3, which shows [10] K. Tsutsui, T. Tohyama and S. Maekawa, Phys. Rev.
thefirstmoment(averagepeakposition)ofthespectrum. Lett. 91, 117001 (2003).
[11] G. D¨oring et al.,Phys. Rev.B 70, 085115 (2004).
The calculationsfor the nearestneighborHeisenbergan-
[12] J. van den Brink and M. Veenendaal, Europhysics Let-
tiferromagnet(Fig.3)showthatthe magneticscattering
ters, 73, 121 (2006); cond-mat/0311446.
disperses from about ω ≈ 0 around (0,0) and (π,π) to
[13] L. Ament, F. Forte and J. van den Brink,
ω ≈4J at (π,0) and (π/2,π/2). Longer range couplings cond-mat/0609767 (2006).
tend to reduce the first moment of the RIXS spectrum. [14] J.P.Hill,G.Blumberg,Y.-J.Kim,D.Ellis,S.Wakimoto,
The observed dispersion has a two-fold origin. It is in R.J. Birgeneau, S. Komiya, Y. Ando, D. Casa, T. Gog,
partdue totheq-dependenceofthe two-magnondensity unpublished.
[15] H.A. Kramers and W. Heisenberg, Z. Phys. 31, 681
ofstates(DOS), combinedwith the scatteringmatrix el-
(1925).
ementsthattendtopronouncethelowenergytailsofthe
[16] Note that where possible indices and constants are sup-
two-magnonDOS.
pressed to avoid cluttering in the equations.
The consistency at q = (0,0) and q = (π,π) of the [17] P.Benedetti,J.vandenBrink,E.Pavarini,A.Vigliante,
theoretical results and experimental data was already and P. Wochner, Phys. Rev. B. 63, 60408(R) (2001), R.
5
Caciuffo, L. Paolosini, A. Solier, P. Ghigna, E. Pavarini, residual spin related scattering intensity at q=0.
J. van den Brink and M. Altarelli, Phys. Rev. B 65, [19] See, for example: M.G. Cottam, J. Phys. C 5, 1461
174425 (2002). (1972) and references therein.
[18] This holds for the expansion in lowest order of ηJ/∆; [20] R. Coldea et al..Phys. Rev.Lett. 86, 23, 5377 (2001).
higher order terms might in principle give rise to some
