X-ray Dichroism and the Pseudogap Phase of Cuprates
S. Di Matteo1, 2 and M. R. Norman3
Laboratori Nazionali di Frascati INFN, via E. Fermi 40, C.P. 13, I-00044 Frascati, Italy
Equipe de physique des surfaces et interfaces, UMR-CNRS 6627 PALMS,
Université de Rennes 1, 35042 Rennes Cedex, France
Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA
(Dated: October 26, 2018)
A recent polarized x-ray absorption experiment on the high temperature cuprate superconduc-
tor Bi2Sr2CaCu2O8+x indicates the presence of broken parity symmetry below the temperature,
T*, where a pseudogap appears in photoemission. We critically analyze the x-ray data, and con-
clude that a parity-breaking signal of the kind suggested is unlikely based on the crystal structures
reported in the literature. Possible other origins of the observed dichroism signal are discussed.
We propose x-ray scattering experiments that can be done in order to determine whether such
alternative interpretations are valid or not.
PACS numbers: 78.70.Ck, 78.70.Dm, 75.25.+z, 74.72.Hs
I. INTRODUCTION
Twenty years since the discovery of high-temperature
cuprate superconductivity, there is still no consensus on
its origin. As the field has evolved, more and more at-
tention has been directed to the pseudogap region of
the phase diagram in underdoped compounds and the
possible relation of this phase to the superconducting
one.1 Time-reversal breaking has been predicted to occur
in this pseudogap phase due to the presence of orbital
currents2 and a subsequent experiment3 using angle-
dependent dichroism in photoemission has claimed to de-
tect this. However, this result has been challenged by
others4 and an independent experimental verification of
this would be highly desirable.
Recently, Kubota et al.5 performed Cu K edge circu-
lar and linear dichroism x-ray experiments for under-
doped Bi2Sr2CaCu2O8+x (Bi2212), claiming no time-
reversal breaking of the kind predicted in Ref. 2 exists,
and that, on the contrary, a parity-breaking signal (but
time-reversal even) is present with the same tempera-
ture dependence as the photoemission dichroism signal,
that was interpreted as x-ray natural circular dichroism
(XNCD) as seen in other materials.6
The aim of the present paper is to critically examine
the conclusions of Kubota et al.5 In particular, we find
that the XNCD signal for the average7 Bi2212 crystal
structure should be zero along all three crystallographic
axes, therefore casting doubt on the original interpreta-
tion of Ref. 5. To look into alternate explanations, we
performed detailed numerical simulations aimed at ex-
plaining the observed signal. At the basis of our study
is the simple observation (see, e.g., Ref. 8) that circu-
lar dichroism in absorption can be generated either by
a non-magnetic effect in the electric dipole-quadrupole
(E1-E2) channel (XNCD, a parity-breaking signal), or
by a magnetic signal in the E1-E1 channel (parity-even).
Alternately, such a signal can be due to contamination
from x-ray linear dichroism (XNLD). We propose x-ray
experiments that could be used to investigate these mat-
ters further.
The structure of the paper is as follows. In Sec. II we
show how symmetry constrains any possible XNCD sig-
nal that would be observed in Bi2212. We also perform
numerical simulations for XNCD assuming an alignment
displaced from the c-axis, using several crystal structure
refinements proposed in the literature. We also calculate
the XNLD signal and comment whether XNLD contam-
ination could be responsible for the observed signal. In
Sec. III we calculate the x-ray magnetic circular dichro-
ism (XMCD) signal at both the Cu K and L edges as-
suming magnetic order on either the copper or oxygen
sites. Finally, in Sec. IV, we draw some general conclu-
sions from our work.
II. NON-MAGNETIC CIRCULAR DICHROISM
IN BI2212
The structure of Bi2212 is strongly layered, with insu-
lating BiO blocks intercalated between superconducting
CuO2 planes. Crystal structure refinements reveal the
presence of an incommensurate modulation whose origin
has been the subject of much debate. The presence of
this modulation has complicated the determination of the
average crystal structure. Two different average struc-
tures have been proposed in the literature for Bi2212:
Bbmb9,10 and Bb2b11,12,13,14, where b is the modulation
direction for the superstructure. We follow the general
convention in the cuprate literature and use a rotated
basis compared to those in the International Tables for
Crystallography15 (respectively, Cccm, No. 66, and Ccc2,
No. 37). In this way, the c-crystallographic direction is
orthogonal to the CuO2 planes.
The Bbmb structure is globally centrosymmetric and,
as such, does not admit a non-zero value for the parity-
odd operator ~L · (ǫ̂∗ × ǫ̂)(~Ω · k̂), whose expectation value
determines the XNCD signal (~L, k̂, ǫ̂ and ~Ω are, re-
spectively, the angular momentum operator, the direc-
tion and polarization of the x-ray beam, and the toroidal
http://arxiv.org/abs/0704.0599v2
momentum operator, see, e.g., Refs. 8,16). Therefore, if
the signal measured by Kubota et al.5 were a true XNCD
signal, this would imply a lower crystal symmetry than
Bbmb. We note that most refinements in the literature
suggest Bbmb,9,10 and this crystal structure is also con-
sistent with recent photoemission data17 that indicate
the presence of both a glide plane and a mirror plane
based on dipole selection rules.
The other average structure that has been suggested
by x-ray and neutron diffraction is Bb2b.11,12,13,14 This
space group is not centrosymmetric and, therefore, a par-
ity breaking signal like that of XNCD is in principle al-
lowed. However, not all wave vector directions are com-
patible with the presence of a XNCD signal, as demon-
strated below by symmetry considerations. In the last
part of Section II, we numerically calculate the XNCD
for a geometrical configuration allowing a signal - like
~k‖(101) - by means of the multiple-scattering subroutine
in the FDMNES program.18
In the context of this program, atomic potentials are
generated using a local density approximation with a
Hedin-Lundqvist form for the exchange-correlation en-
ergy. These potentials are then used in a muffin tin
approximation to calculate the resulting XANES sig-
nal by considering multiple scattering of the photo-
electron about the absorbing site within a one-electron
approximation.19 In the future, it would be desirable
to repeat these calculations by using input from self-
consistent band theory, as has recently been done for the
Bbmb structure in regards to angle resolved photoemis-
sion spectra.20
In the Bb2b setting, Cu ions belong to sites of Wyck-
off multiplicity 8d. These eight equivalent copper sites
can be partitioned in two groups of four sites that are
related by the vector (1/2,0,1/2). Within each group
the four sites are related by the symmetry operations
{Ê, Ĉ2y, m̂x, m̂z}, where Ê is the identity, Ĉ2y is a two-
fold axis around the b crystallographic axis, and m̂x(z)
is a mirror-symmetry plane orthogonal to the a(c)-axis.
The absorption at the Cu K edge, expressed in Mbarn,
can be calculated through the equations:
σ(±) =
j , (1)
j = 4π
2αh̄ω
|〈Ψ(j)n |Ô
(±)|Ψ
2δ[h̄ω − (En − E0)]
The operator Ô(±) ≡ ǫ̂(±) ·~r(1+ i
~k ·~r) in Eq. (2) is the
usual matter-radiation interaction operator expanded up
to dipole (E1) and quadrupole (E2) terms, with the pho-
ton polarization ǫ̂ and the wave vector ~k, where we label
left- and right-handed polarization by the superscript ±.
n ) is the ground (excited) state of the crystal,
and E0 (En) its energy. The sum in Eq. (2) is extended
over all the excited states of the system and h̄ω is the
energy of the incoming photon, with α the fine-structure
9.029.019.008.998.98
Energy (keV)
Pet (XNCD x 1000)
Glady (XNCD x 400)
Kan (XNCD x 100)
FIG. 1: (Color online) XANES signal for ~k‖(001) and XNCD
signal for ~k‖(101) at the Cu K edge for three Bb2b crystal
refinements, with a cluster radius of 4.9 Å. The crystal struc-
tures are Pet for Petricek et al.11, Glady for Gladyshevskii
and Flükiger13 , and Kan for Kan and Moss12. The XNCD
signals have been multiplied by the factors indicated. Each
successive set of curves is displaced by 0.4 Mbarn.
constant. Finally, the index j = 1, ..., 8 indicates the
lattice site of the copper photoabsorbing atom in the
unit cell. Eqs. (1) and (2) are the basis of the numer-
ical calculations of the FDMNES program.18 The eight
contributions can be written as the sum of two equal
parts coming from the two subsets of four ions related by
the (1/2,0,1/2) translation. Within each subset, the four
absorption contributions are related to one another by
the symmetry operations σ2 = Ĉ2yσ1, σ3 = m̂xσ1, and
σ4 = m̂zσ1, implying that the total absorption is:
σ = 2(1 + m̂z)(1 + m̂x)σ1 (3)
The group from σ5 to σ8 is equivalent to the first group of
four modulo a translation (this is the reason for the factor
of two in Eq. (3)). Notice that the symmetry operators in
Eq. (3) are meant to operate just on the electronic part
of the operator Ô in Eq. (2).
In the case of circular dichroism, the signal is given by
σ = σ+−σ−. If we suppose that no net magnetization is
present in the material (we shall analyze the possibility
of magnetism in Sec. III), then the dichroism is natural,
i.e., necessarily coming from the interference E1-E2 con-
tribution in Eq. (2). In this case the signal is parity-odd,
which implies that m̂z(x) ≡ ÎĈ2z(2x) → −Ĉ2z(2x) (Î is
the inversion operator). Then Eq. (3) becomes
σ = 2(1− Ĉ2z)(1 − Ĉ2x)σ1 (4)
which implies that, of the possible five second-rank ten-
sors involved in XNCD, only the term T
1 − T
−1 sur-
vives. To arrive at this result, we applied the usual op-
erator rules on spherical tensors:21 Ĉ2xT
m = T
−m and
Ĉ2zT
m = (−)
m . This in turn leads to a zero XNCD
along the three crystallographic axes of the Bb2b crys-
tal structure where, e.g., along the c-axis, the signal is
proportional to T
Therefore, even in the Bb2b crystal structure, the
XNCD is exactly zero by symmetry when the wave vec-
tor is directed along the c-axis (i.e., orthogonal to the
CuO2 planes) as in the experiment of Ref. 5. This has
been further checked by numerical calculations with clus-
ter radii up to 6.5 Å, i.e., 93 atoms, centered on the Cu
ion, based on the average crystal structures reported in
Refs. 11,12,13,14. The only possibility to justify theo-
retically the experimental evidence of circular dichroism
of a non-magnetic nature is either by lowering of the or-
thorhombic Bb2b symmetry, a misalignment of the k̂-
direction with respect to the c-axis, or contamination
from linear dichroism. We checked all of these possi-
bilities.
If we take into account the monoclinic supercell pro-
posed in Ref. 13, with space group Cc, this implies a re-
duction of the symmetry operations, with the loss of the
two-fold screw axis. Nonetheless, the glide plane contain-
ing the normal to the CuO2-planes is still present (m̂x
in Eq. 3), which is responsible for the extinction rule of
the quantity 〈Ψn|~L · (ǫ̂
∗ × ǫ̂)(~Ω · k̂)|Ψn〉. Therefore the
XNCD is again identically zero by symmetry, which we
verified by direct numerical simulation of the supercell.
Note that only a reduction to triclinic symmetry would
allow for XNCD in the direction orthogonal to the CuO2
planes.
We also checked for the possibility of misalignment, as
shown in Fig. 1, by a direct calculation with ~k‖(101),
corresponding to a tilting θ ∼ 10o with respect to the
c-axis (note that the a lattice constant is ∼ 5.4 Å, and
the c one ∼ 30.9 Å). We first remark that the calcu-
lated XANES signal compares well to experiment (the
XANES for ~k‖(001) and ~k‖(101) are identical on the
scale of Fig. 1). Despite this, the energy profile of the
XNCD signal is different from the one reported by Kub-
ota et al.5 The main difference is the energy extension of
the calculated dichroic signal, whose oscillations persist,
though with decreased intensity, more than 50 eV above
the edge itself. This characteristic is present for all four
Bb2b refinements we have looked at (and as well for the
monoclinic supercell, which we do not show) and is at
variance with the experimental results of Ref. 5, where
the dichroic signal is confined to a small energy range
around the edge.
Some comments on the calculations are in order. Each
refinement gives rise to a different XNCD signal, and
-0.05
9.029.019.008.998.98
Energy (keV)
 Pet (XNCD x 1000)
 Kan (XNCD x 100)
 Glady (XNCD x 400)
FIG. 2: (Color online) XNCD signal, as in Fig. 1, but for a
CuO5 cluster.
9.029.019.008.998.98
Energy (keV)
R=2.1
R=3.1
R=4.9
R=4.1
R=5.5
R=6.5
XNCD x 1000
FIG. 3: (Color online) XNCD signal at the Cu K edge for
~k‖(101) as a function of the cluster radius (in Å) for the crys-
tal structure of Petricek et al.11. The signal has been mul-
tiplied by 1000. Each successive curve is displaced by 0.06
Mbarn.
their intensities are quite different as well. They are
found to be strongly dependent on the magnitude of the
deviation of the atoms from their Bbmb positions, which
differs significantly among the various Bb2b refinements.
Moreover, the structure of Kan and Moss leads to a mani-
festly different energy profile. This difference is seen even
for CuO4 and CuO5 clusters (the results for a CuO5 clus-
ter are shown in Fig. 2) and has to do with the large de-
parture of this particular structure from the Bbmb one.
Although there has been some criticism in the literature
concerning this particular refinement,10 the point we wish
to make is that each refinement has a different XNCD
signal, showing how sensitive this signal is to the actual
crystal structure. We note that Fig. 1 was done for a
cluster radius of 4.9 Å, i.e., 37 atoms. In Fig. 3, we show
results for the refinement of Petricek et al.11 up to a ra-
dius of 6.5 Å (93 atoms), showing the development of ad-
ditional structure in the energy profile as more and more
atoms are included in the cluster. In the real system, the
effective cluster radius is limited by the photoelectron
escape depth, which is energy dependent.22
We also remark that the magnitude of the XNCD sig-
nal we calculated for a 10 degree misalignment is com-
parable to that measured in Ref. 5. On the other hand,
the signal goes as sin(2θ), where θ measures the displace-
ment from the c-axis. We note that Kubota et al.5 men-
tion that their signal was insensitive to displacements
from the c-axis of 5 degrees, which would argue against
a misalignment given the strong angular dependence we
predict. Moreover, we note that the size of the signal
only depends on the projection of the k vector onto the
a-c plane, i.e., a signal for ~k‖(111) is equivalent to that
for ~k‖(101).
The above leads us to suspect that neither misalign-
ment, nor symmetry reduction, are the basis of the signal
detected in Ref. 5. We now turn to the third possibility
for a non-magnetic signal, that due to intermixing of lin-
ear dichroism. All x-ray beams at a synchrotron have a
linear polarization component (Kubota et al.5 mention
the possibility of up to 5% of linear admixture). The
resulting linear dichroism, which vanishes for a uniaxial
crystal, can swamp the intrinsic XNCD signal for a biax-
ial crystal where the a and b directions are inequivalent
(like Bi2212). This was shown by Goulon et al. for a
KTiOPO4 crystal with the same point group symmetry
(mm2) as Bi2212.23
To analyze this further, we show the linear dichro-
ism (the XANES signal for ~E‖(010) minus the one for
~E‖(100)), calculated for the Bb2b refinement of Glady-
shevskii et al.,14 for various cluster radii, in Fig. 4. Simi-
lar results have been obtained for the other crystal struc-
tures, including the Bbmb refinement of Miles et al.10
The energy profile, with a positive peak followed by a
negative peak, and its location at the absorption edge, is
very reminiscent of the data. Moreover, the size of the
XNLD signal is large, meaning that only a few percent
admixture is necessary to explain the size of the signal
seen in Ref. 5. One issue is that Kubota et al.5 did report
the existence of an XNLD signal, but also claim that it is
temperature independent. This is somewhat puzzling, as
there are significant changes of the lattice constants with
temperature.24 One obvious question would be why such
an XNLD contamination would only appear below T*,
though it should be remarked that there are anomalies
in the superstructure periodicity near T*.24 A definitive
test would be to rotate the sample under the beam, as
any XNLD signal would vary as cos(2φ) where φ is the in-
plane angle relative to the b-axis. Any circular dichroism
(XNCD or XMCD) is instead φ-independent.
9.029.019.008.998.98
Energy (keV)
 R=3.0
 R=4.1
 R=5.1
XANES
XNLD x 50
FIG. 4: (Color online) XNLD signal at the Cu K edge for
~k‖(001) as a function of the cluster radius (in Å) for the crys-
tal structure of Gladyshevskii et al.14. The XNLD signal has
been multiplied by 50. The XANES curve has been displaced
by 0.05 Mbarn.
A final possibility would be a small energy shift be-
tween the left and right circularly polarized beams. Dif-
ferentiating the absorption edge in Fig. 1 would indeed
lead to a signal similar to that seen in Ref. 5 (but with
an enhanced positive peak relative to the negative peak).
But such an energy shift is difficult to imagine with the
particular experimental set up used.
The observed dichroism signal as a function of energy
is also reminiscent of that typically seen for magnetic
circular dichroism: in this case the signal would be of
E1-E1 origin and its main features are indeed expected
to be at the edge itself. In addition, the nature of the
observed signal (a single sharp positive peak followed by
a single sharp negative peak) is also much like a magnetic
signal where the main features are expected to be more
localized in energy starting from the rising edge of the
absorption. Whether this possibility is realistic or not
can only be determined by a quantitative calculation,
which we offer in the next section.
III. MAGNETIC DICHROISM IN BI2212
Over the years, there have been several claims of pos-
sible magnetic order in the pseudogap phase of cuprate
superconductors. Recently, a magnetic signal at a (101)
Bragg vector has been observed below T* for several un-
derdoped YBa2Cu3O6+x (YBCO) samples by polarized
neutron diffraction.25 The signal, corresponding to a mo-
ment of order 0.05-0.1 µB, is not simple ferromagnetism
as it was not observed at the (002) Bragg vector. Even
more recently, a Kerr rotation below T* has been de-
tected in underdoped YBCO corresponding to a net fer-
9.029.019.008.998.98
Energy (keV)
 XANES Kan
 XMCD Pet
 XMCD Kan
 XMCD Glady
XMCD x 10000
FIG. 5: (Color online) XMCD for ~k‖(001) at the Cu K edge
with a ferromagnetic moment of 0.1 µB along the c-axis at
each copper site. The cluster radius is 4.1 Å. The XMCD
signal has been multiplied by 10000. The crystal structures
are those of Fig. 1.
romagnetic moment of 10−4 µB.
This motivates us to consider the possibility of a mag-
netic origin for the x-ray circular dichroism detected in
Ref. 5. We restate that in absorption (see, e.g., Ref. 8),
circular dichroism can be generated either by a non-
magnetic effect in the E1-E2 channel (XNCD, a parity-
breaking signal), or by a magnetic signal in the E1-E1
channel (XMCD, parity-even). The first possibility, an-
alyzed in the previous section, does not seem to be com-
patible with the experimental results. In order to analyze
the second possibility, we need to provide the lattice with
a magnetic structure that has a net magnetization (oth-
erwise, the XMCD is zero). In what follows, we shall sup-
pose two magnetic distributions, the first with the mag-
netic moments on the Cu sites, the second on the planar
O sites. The numerical calculations are performed with
the relativistic extension of the multiple-scattering pro-
gram in the FDMNES code,18 and provide results that
are an extension of those of the previous section.
The details of the calculations are as follows: we used
again the average crystal structures discussed in Section
II. For each magnetic configuration we employed clus-
ters with radii ranging from 3.1 Å (a CuO5-cluster), to
4.9 Å (37 atoms) around the Cu photoabsorbing ion. In
the first set of calculations, shown in Figs. 5 and 6, we
built the input potential from a magnetic configuration of
4.55 3d↑ electrons and 4.45 3d↓ electrons (i.e., a moment
of 0.1 µB per copper site). In the second set of calcula-
tions, shown in Fig. 7, we built the input potential from
a magnetic configuration of 2.05 2p↑ electrons and 1.95
2p↓ electrons (i.e., a moment of 0.1 µB per planar oxygen
site). The following results are noteworthy:
a) Differently from the XNCD calculations shown in
-0.05
9.029.019.008.998.98
Energy (keV)
 XMCD Pet
 XMCD Kan
 XMCD Glady
x 10000
FIG. 6: (Color online) XMCD as in Fig. 5, but with a cluster
radius of 3.1 Å, i.e., a CuO5 cluster.
9.029.019.008.998.98
Energy (keV)
 XANES Kan
 XMCD Pet
 XMCD Kan
 XMCD Glady
XMCD x 5000
FIG. 7: (Color online) XMCD for ~k‖(001) at the Cu K edge
with a ferromagnetic moment of 0.1 µB at each planar oxygen
site. The XMCD signal has been multiplied by 5000. The
XANES signal has been displaced by 0.02 Mbarn.
Fig. 1, all crystal structures give basically the same
XMCD spectra. The reason for this behavior may be
related to the fact that XMCD, when x-rays are orthog-
onal to the CuO2 planes, mainly depends on the in-plane
magnetisation density and on the in-plane crystal struc-
ture, which is quite similar for the various refinements.
b) There is a more marked dependence on the cluster
radius compared to the XNCD, as shown by the compar-
ison of Fig. 5 and Fig. 6. The calculations with a radius
bigger than 4.1 Å (6 Cu, 9 O, 4 Sr, and 4 Ca), above the
pre-edge energy, are basically equivalent to those shown
in Fig. 5, with a positive peak at the edge energy, fol-
960950940930
Energy (eV)
 XANES Pet
 XMCD Pet
 XMCD Kan
 XMCD Glady
XMCD x 10
FIG. 8: (Color online) XMCD for ~k‖(001) at the Cu L2,3
edges with a ferromagnetic moment of 0.1 µB along the c-
axis at each copper site. The cluster radius is 4.1 Å. The
XMCD signal has been multiplied by 10.
lowed by a double negative peak, the latter at variance
with the experimental results. On the contrary, the en-
ergy shape obtained for a radius of 3.1 Å is very close to
the experimental one, with a single negative peak after
the sharp positive one, with relatively good agreement
in the energy position and width. We could be tempted
to suppose, therefore, that the virtual photoelectron has
a very small mean free path before decaying and it is
sensitive just to the nearest neighbor oxygens. Indeed
we checked that an identical XMCD profile is obtained
with just the in-plane CuO4-cluster. On the other hand,
the size of the signal we calculate is about an order of
magnitude smaller than that seen in Ref. 5. Since the
XMCD signal is proportional to the moment, then we
would need a moment of ∼1 Bohr magneton per copper
to have a comparable signal. Such a huge moment would
have been observed previously by neutron scattering if
it existed. Of course we cannot exclude that spurious
effects, such as strain fields, could have influenced the
measurement.
c) The energy profile in the case of magnetization at
the oxygen sites is not much different from the cop-
per case, except for the deeper negative peak around E
∼ 8.994 keV, as shown in Fig. 7. Also in this case the dif-
ferent crystal structure refinements give basically equiv-
alent results, as again the CuO2 planes are practically
equivalent in the various cases. Note that the relative
intensity is equivalent to the copper case, as 0.1 Bohr
magnetons per planar oxygen corresponds to 0.2 Bohr
magnetons per CuO2 cell (note we multiply by 5000 in
Fig. 7 as compared to 10000 in Fig. 5).
We also performed simulations for the Cu L2,3 edges
for the magnetic configuration corresponding to Fig. 5,
as shown in Fig. 8, which can be compared with future
experimental investigations in order to confirm whether
or not a net magnetization exists in this compound.
Finally, we remark that the dependence of the XMCD
signal on the tilting angle θ (i.e., the displacement of
the photon wave vector from the c-axis) goes like cos(θ)
and therefore the signal is not very sensitive to small
displacements of 5 degrees, as noted by Kubota et al.5
This different angular dependence from the XNCD signal
suggests a relatively easy way to unravel the question ex-
perimentally: it is sufficient to measure the θ (azimuthal)
dependence of the signal, noting that any XNLD contam-
ination would be tested by the φ (polar) dependence of
the signal.
IV. CONCLUSIONS
In our opinion, the experiments of Kubota et al.5 have
raised more questions than they have answered. Al-
though not treated in our paper, we believe that their
results from the measurement of non-reciprocal linear
dichroism are at this stage not conclusive, as only one di-
rection for the toroidal moment has been investigated, of
the two possible suggested by the orbital current pattern
of Varma.2 The analysis performed in Section II showed,
moreover, that the claimed XNCD signal is probably un-
justified. In fact, even though the space group Bb2b
is non-centrosymmetric, XNCD is absent by symmetry
when the x-ray wave vector is chosen orthogonal to the
CuO2 planes, as in the experimental measurement geom-
etry of Ref. 5. The same extinction rule survives for the
monoclinic supercell structure refined in Ref. 13. More-
over, in both cases, it seems hard to mantain the hy-
pothesis of misalignment, due to the experimental local-
ization of the energy profile around the main absorption
edge, which is absent in the calculations. We also note
the difference of XNCD from the photoemission dichro-
ism results of Kaminski et al.3 A direct comparison is
however not immediate, as the former represents a q̂-
integrated version of the latter (here q̂ is the solid angle in
the space of the photoelectron wave-vector, see, e.g., Ref.
8). A more likely explanation is an XNLD contamination
(Fig. 4), but then the challenge is to understand why such
an effect would only exist below T*. We note that an op-
tics experiment for an optimal doped Bi2212 sample has
seen a change in linear birefrigence below Tc, which was
accompanied by a non-zero circular birefringence.28 In
addition, both Bi222329 and the Fe analogue of Bi221230
exhibit supercells with 222 space groups which would al-
low for dichroism. So, it is conceivable that there is a
subtle structural transition associated with T* which we
suggest could be looked for by diffraction experiments.
A final comment about the physical quantities detected
by x-ray circular dichroism, either natural or magnetic,
is in order. At the K edge of transition metal oxides,
XMCD in the E1-E1 channel, at the excitation energy
E = h̄ω − Eedge, gives information on the expectation
value of L̂ · (ǫ̂∗× ǫ̂) for the excited states at the energy E .
The orbital angular momentum is either induced from a
spin moment via spin-orbit coupling (as calculated here)
or directly by an orbital current (as in the scenario advo-
cated in Ref. 231). No direct spin information is available
at the K edge and therefore an XMCD measurement is
not directly related to the ground state magnetic moment
as for the L edge. Moreover, the observed states are
those with p-like angular momentum projection on the
photoabsorbing Cu ion, which are extended and there-
fore mainly sensitive to the influence of the oxygen atoms
surrounding the Cu site. In this case, the main contribu-
tion to the XMCD energy profile is expected in an energy
range of 10-20 eV from the main edge to the first shoul-
der in the XANES spectrum, as found in Ref. 5 and in
our own XMCD calculations.
Although the results of Section III are in principle con-
sistent with Ref. 5, the size of the ferromagnetic moment
necessary to get a signal of the magnitude seen by exper-
iment, ∼1 Bohr magneton, is excessive. If such a large
ferromagnetic moment existed, it would have surely been
seen by neutron scattering. From this point of view, ex-
periments performed at the Cu L edge and O K edge
would be desirable as they are more sensitive to the pres-
ence of a magnetic moment.
Finally, we would like to remark that XNCD along
the c-axis is insensitive to orbital currents. These latter,
confined to the CuO2 planes, develop a parity breaking
characterized by a toroidal moment (~Ω) within the CuO2
planes. The XNCD experiment of Ref. 5 would only be
sensitive to the projection of the toroidal moment out of
this plane (i.e., along the direction of the x-ray wavevec-
tor). Therefore, if performed as stated, it cannot tell us
about any possible orbital current order.
To conclude, we believe that the various interpreta-
tions, XNCD, XNLD, or XMCD, have their drawbacks,
and therefore the origin of the experimental signal of
Ref. 5 is still open. In this sense, further experimen-
tal checks of the energy extension of the dichroic signal
would be highly desirable. Based on our results, the most
stringent experimental test on the physical origin of the
signal would come from the measurement of the depen-
dence on the tilting (azimuthal) angle θ, due to the dif-
ferent dependences of XNCD and XMCD, as well as the
dependence on the in-plane (polar) angle φ, which would
test for any possible XNLD contamination.
V. ACKNOWLEDGMENTS
The authors thank John Freeland, Zahir Islam,
Stephan Rosenkranz, Daniel Haskel and Matti Lindroos
for various discussions. This work is supported by the
U.S. DOE, Office of Science, under Contract No. DE-
AC02-06CH11357. SDM would like to thank the kind
hospitality of the ID20 beamline staff at the ESRF.
1 M. R. Norman, D. Pines and C. Kallin, Adv. Phys. 54,
715 (2005).
2 C. M. Varma, Phys. Rev. B 55, 14554 (1997) and Phys.
Rev. Lett. 83, 3538 (1999); M. E. Simon and C. M. Varma,
ibid 89, 247003 (2002).
3 A. Kaminski, S. Rosenkranz, H. Fretwell, J. C. Cam-
puzano, Z. Li, H. Raffy, W. G. Cullen, H. You, C. G. Olson,
C. M. Varma and H. Hoechst, Nature 416, 610 (2002).
4 S. V. Borisenko, A. A. Kordyuk, A. Koitzsch, T. K. Kim,
K. A. Nenkov, M. Knupfer, J. Fink, C. Grazioli, S. Turchini
and H. Berger, Phys. Rev. Lett. 92, 207001 (2004).
5 M. Kubota, K. Ono, Y. Oohara and H. Eisaki, J. Phys.
Soc. Jpn. 75, 053706 (2006).
6 L. Alagna, T. Prosperi, S. Turchini, J. Goulon, A. Ro-
galev, C. Goulon-Ginet, C. R. Natoli, R. D. Peacock and
B. Stewart, Phys. Rev. Lett. 80, 4799 (1998).
7 The average crystal structure is the base orthorhombic unit
cell before the incommensurate superstructure is taken into
account. As the latter involves a translation operator, it
should not affect the symmetry arguments in this paper.
8 S. Di Matteo and C. R. Natoli, J. Synchr. Rad. 9, 9 (2002).
9 A. Yamamoto, M. Onoda, E. Takayama-Muromachi, F.
Izumi, T. Ishigaki and H. Asano, Phys. Rev. B 42, 4228
(1990); D. Grebille, H. Leligny, A. Ruyter, Ph. Labbé and
B. Raveau, Acta Cryst. B52, 628 (1996); N. Jakubowicz,
D. Grebille, M. Hervieu and H. Leligny, Phys. Rev. B 63,
214511 (2001).
10 P. A. Miles, S. J. Kennedy, G. J. McIntyre, G. D. Gu, G.
J. Russell and N. Koshizuka, Physica C 294, 275 (1998).
11 V. Petricek, Y. Gao, P. Lee and P. Coppens, Phys. Rev. B
42, 387 (1990).
12 X. B. Kan and S. C. Moss, Acta Cryst. B48, 122 (1992).
13 R. E. Gladyshevskii and R. Flükiger, Acta Cryst. B52, 38
(1996).
14 R. E. Gladyshevskii, N. Musolino and R. Flükiger, Phys.
Rev. B 70, 184522 (2004). There is no superstructure for
this Pb doped variant, so the crystal structure used is the
actual one, not the average one.
International Tables for Crystallography, 5th ed., ed. T.
Hahn (Kluwer, Dordrecht, 2002).
16 P. Carra and R. Benoist, Phys. Rev. B 62, R7703 (2000).
17 A. Mans, I. Santoso, Y. Huang, W. K. Siu, S. Tavaddod,
V. Apiainen, M. Lindroos, H. Berger, V. N. Strocov, M.
Shi, L. Patthey and M. S. Golden, Phys. Rev. Lett. 96,
107007 (2006).
18 Y. Joly, Phys. Rev. B 63, 125120 (2001).
This program can be downloaded at
http://www-cristallo.grenoble.cnrs.fr/fdmnes.
19 C. R. Natoli, Ch. Brouder, Ph. Sainctavit, J. Goulon, Ch.
Goulon-Ginet and A. Rogalev, Eur. Phys. J. B 4, 1 (1998).
20 V. Arpiainen and M. Lindroos, Phys. Rev. Lett. 97, 037601
(2006).
21 D. A. Varshalovich, A. N. Moskalev and V. K. Kersonskii,
Quantum Theory of Angular Momentum (World Scientific,
Singapore, 1988).
22 The results presented involve a convolution of the calcu-
http://www-cristallo.grenoble.cnrs.fr/fdmnes
lated spectrum with both a core hole (1.9 eV for the Cu K
edge) and a photoelectron lifetime, with the latter having
a strong energy dependence (ranging up to 15 eV with a
midpoint value at 30 eV above the Fermi energy). But the
calculation has a fixed cluster radius.
23 J. Goulon, C. Goulon-Ginet, A. Rogalev, G. Benayoun, C.
Brouder and C. R. Natoli, J. Synchr. Rad. 7, 182 (2000).
In addition, there is also an intrinsic non XNCD contribu-
tion to the circular dichroism for a biaxial crystal, but this
should vanish in Bi2212 for an x-ray beam aligned along
the c-axis, see J. Goulon, C. Goulon-Ginet, A. Rogalev, V.
Gotte, C. Brouder and C. Malgrange, Eur. Phys. J. B 12,
373 (1999).
24 P. A. Miles, S. J. Kennedy, A. R. Anderson, G. D. Gu,
G. J. Russell and N. Koshizuka, Phys. Rev. B 55, 14632
(1997).
25 B. Fauqué, Y. Sidis, V. Hinkov, S. Pailhes, C. T. Lin, X.
Chaud and P. Bourges, Phys. Rev. Lett. 96, 197001 (2006).
26 A. Kapitulnik, unpublished results.
27 It should be remarked, though, that single-particle based
approaches can miss some features of the spectral weight
transfer between the L2 and L3 edges.
28 J. Kobayashi, T. Asahi, M. Sakurai, M. Takahashi, K.
Okubo and Y. Enomoto, Phys. Rev. B 53, 11784 (1996).
29 E. Giannini, N. Clayton, N. Musolino, A. Piriou, R. Glady-
shevskii and R. Flükiger, IEEE Trans. Appl. Supercond.
15, 3102 (2005).
30 Y. Le Page, W. R. McKinnon, J.-M. Tarascon and P. Bar-
boux, Phys. Rev. B 40, 6810 (1989).
31 X-ray dichroism experiments in regards to the orbital cur-
rent scenario of Varma have been discussed by S. Di Matteo
and C. M. Varma, Phys. Rev. B 67, 134502 (2003).

  A recent polarized x-ray absorption experiment on the high temperature
cuprate superconductor Bi2Sr2CaCu2O8 indicates the presence of broken parity
symmetry below the temperature, T*, where a pseudogap appears in photoemission.
We critically analyze the x-ray data, and conclude that a parity-breaking
signal of the kind suggested is unlikely based on the crystal structures
reported in the literature. Possible other origins of the observed dichroism
signal are discussed. We propose x-ray scattering experiments that can be done
in order to determine whether such alternative interpretations are valid or
not.

X-ray Dichroism and the Pseudogap Phase of Cuprates
S. Di Matteo1, 2 and M. R. Norman3
Laboratori Nazionali di Frascati INFN, via E. Fermi 40, C.P. 13, I-00044 Frascati, Italy
Equipe de physique des surfaces et interfaces, UMR-CNRS 6627 PALMS,
Université de Rennes 1, 35042 Rennes Cedex, France
Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA
(Dated: October 26, 2018)
A recent polarized x-ray absorption experiment on the high temperature cuprate superconduc-
tor Bi2Sr2CaCu2O8+x indicates the presence of broken parity symmetry below the temperature,
T*, where a pseudogap appears in photoemission. We critically analyze the x-ray data, and con-
clude that a parity-breaking signal of the kind suggested is unlikely based on the crystal structures
reported in the literature. Possible other origins of the observed dichroism signal are discussed.
We propose x-ray scattering experiments that can be done in order to determine whether such
alternative interpretations are valid or not.
PACS numbers: 78.70.Ck, 78.70.Dm, 75.25.+z, 74.72.Hs
I. INTRODUCTION
Twenty years since the discovery of high-temperature
cuprate superconductivity, there is still no consensus on
its origin. As the field has evolved, more and more at-
tention has been directed to the pseudogap region of
the phase diagram in underdoped compounds and the
possible relation of this phase to the superconducting
one.1 Time-reversal breaking has been predicted to occur
in this pseudogap phase due to the presence of orbital
currents2 and a subsequent experiment3 using angle-
dependent dichroism in photoemission has claimed to de-
tect this. However, this result has been challenged by
others4 and an independent experimental verification of
this would be highly desirable.
Recently, Kubota et al.5 performed Cu K edge circu-
lar and linear dichroism x-ray experiments for under-
doped Bi2Sr2CaCu2O8+x (Bi2212), claiming no time-
reversal breaking of the kind predicted in Ref. 2 exists,
and that, on the contrary, a parity-breaking signal (but
time-reversal even) is present with the same tempera-
ture dependence as the photoemission dichroism signal,
that was interpreted as x-ray natural circular dichroism
(XNCD) as seen in other materials.6
The aim of the present paper is to critically examine
the conclusions of Kubota et al.5 In particular, we find
that the XNCD signal for the average7 Bi2212 crystal
structure should be zero along all three crystallographic
axes, therefore casting doubt on the original interpreta-
tion of Ref. 5. To look into alternate explanations, we
performed detailed numerical simulations aimed at ex-
plaining the observed signal. At the basis of our study
is the simple observation (see, e.g., Ref. 8) that circu-
lar dichroism in absorption can be generated either by
a non-magnetic effect in the electric dipole-quadrupole
(E1-E2) channel (XNCD, a parity-breaking signal), or
by a magnetic signal in the E1-E1 channel (parity-even).
Alternately, such a signal can be due to contamination
from x-ray linear dichroism (XNLD). We propose x-ray
experiments that could be used to investigate these mat-
ters further.
The structure of the paper is as follows. In Sec. II we
show how symmetry constrains any possible XNCD sig-
nal that would be observed in Bi2212. We also perform
numerical simulations for XNCD assuming an alignment
displaced from the c-axis, using several crystal structure
refinements proposed in the literature. We also calculate
the XNLD signal and comment whether XNLD contam-
ination could be responsible for the observed signal. In
Sec. III we calculate the x-ray magnetic circular dichro-
ism (XMCD) signal at both the Cu K and L edges as-
suming magnetic order on either the copper or oxygen
sites. Finally, in Sec. IV, we draw some general conclu-
sions from our work.
II. NON-MAGNETIC CIRCULAR DICHROISM
IN BI2212
The structure of Bi2212 is strongly layered, with insu-
lating BiO blocks intercalated between superconducting
CuO2 planes. Crystal structure refinements reveal the
presence of an incommensurate modulation whose origin
has been the subject of much debate. The presence of
this modulation has complicated the determination of the
average crystal structure. Two different average struc-
tures have been proposed in the literature for Bi2212:
Bbmb9,10 and Bb2b11,12,13,14, where b is the modulation
direction for the superstructure. We follow the general
convention in the cuprate literature and use a rotated
basis compared to those in the International Tables for
Crystallography15 (respectively, Cccm, No. 66, and Ccc2,
No. 37). In this way, the c-crystallographic direction is
orthogonal to the CuO2 planes.
The Bbmb structure is globally centrosymmetric and,
as such, does not admit a non-zero value for the parity-
odd operator ~L · (ǫ̂∗ × ǫ̂)(~Ω · k̂), whose expectation value
determines the XNCD signal (~L, k̂, ǫ̂ and ~Ω are, re-
spectively, the angular momentum operator, the direc-
tion and polarization of the x-ray beam, and the toroidal
http://arxiv.org/abs/0704.0599v2
momentum operator, see, e.g., Refs. 8,16). Therefore, if
the signal measured by Kubota et al.5 were a true XNCD
signal, this would imply a lower crystal symmetry than
Bbmb. We note that most refinements in the literature
suggest Bbmb,9,10 and this crystal structure is also con-
sistent with recent photoemission data17 that indicate
the presence of both a glide plane and a mirror plane
based on dipole selection rules.
The other average structure that has been suggested
by x-ray and neutron diffraction is Bb2b.11,12,13,14 This
space group is not centrosymmetric and, therefore, a par-
ity breaking signal like that of XNCD is in principle al-
lowed. However, not all wave vector directions are com-
patible with the presence of a XNCD signal, as demon-
strated below by symmetry considerations. In the last
part of Section II, we numerically calculate the XNCD
for a geometrical configuration allowing a signal - like
~k‖(101) - by means of the multiple-scattering subroutine
in the FDMNES program.18
In the context of this program, atomic potentials are
generated using a local density approximation with a
Hedin-Lundqvist form for the exchange-correlation en-
ergy. These potentials are then used in a muffin tin
approximation to calculate the resulting XANES sig-
nal by considering multiple scattering of the photo-
electron about the absorbing site within a one-electron
approximation.19 In the future, it would be desirable
to repeat these calculations by using input from self-
consistent band theory, as has recently been done for the
Bbmb structure in regards to angle resolved photoemis-
sion spectra.20
In the Bb2b setting, Cu ions belong to sites of Wyck-
off multiplicity 8d. These eight equivalent copper sites
can be partitioned in two groups of four sites that are
related by the vector (1/2,0,1/2). Within each group
the four sites are related by the symmetry operations
{Ê, Ĉ2y, m̂x, m̂z}, where Ê is the identity, Ĉ2y is a two-
fold axis around the b crystallographic axis, and m̂x(z)
is a mirror-symmetry plane orthogonal to the a(c)-axis.
The absorption at the Cu K edge, expressed in Mbarn,
can be calculated through the equations:
σ(±) =
j , (1)
j = 4π
2αh̄ω
|〈Ψ(j)n |Ô
(±)|Ψ
2δ[h̄ω − (En − E0)]
The operator Ô(±) ≡ ǫ̂(±) ·~r(1+ i
~k ·~r) in Eq. (2) is the
usual matter-radiation interaction operator expanded up
to dipole (E1) and quadrupole (E2) terms, with the pho-
ton polarization ǫ̂ and the wave vector ~k, where we label
left- and right-handed polarization by the superscript ±.
n ) is the ground (excited) state of the crystal,
and E0 (En) its energy. The sum in Eq. (2) is extended
over all the excited states of the system and h̄ω is the
energy of the incoming photon, with α the fine-structure
9.029.019.008.998.98
Energy (keV)
Pet (XNCD x 1000)
Glady (XNCD x 400)
Kan (XNCD x 100)
FIG. 1: (Color online) XANES signal for ~k‖(001) and XNCD
signal for ~k‖(101) at the Cu K edge for three Bb2b crystal
refinements, with a cluster radius of 4.9 Å. The crystal struc-
tures are Pet for Petricek et al.11, Glady for Gladyshevskii
and Flükiger13 , and Kan for Kan and Moss12. The XNCD
signals have been multiplied by the factors indicated. Each
successive set of curves is displaced by 0.4 Mbarn.
constant. Finally, the index j = 1, ..., 8 indicates the
lattice site of the copper photoabsorbing atom in the
unit cell. Eqs. (1) and (2) are the basis of the numer-
ical calculations of the FDMNES program.18 The eight
contributions can be written as the sum of two equal
parts coming from the two subsets of four ions related by
the (1/2,0,1/2) translation. Within each subset, the four
absorption contributions are related to one another by
the symmetry operations σ2 = Ĉ2yσ1, σ3 = m̂xσ1, and
σ4 = m̂zσ1, implying that the total absorption is:
σ = 2(1 + m̂z)(1 + m̂x)σ1 (3)
The group from σ5 to σ8 is equivalent to the first group of
four modulo a translation (this is the reason for the factor
of two in Eq. (3)). Notice that the symmetry operators in
Eq. (3) are meant to operate just on the electronic part
of the operator Ô in Eq. (2).
In the case of circular dichroism, the signal is given by
σ = σ+−σ−. If we suppose that no net magnetization is
present in the material (we shall analyze the possibility
of magnetism in Sec. III), then the dichroism is natural,
i.e., necessarily coming from the interference E1-E2 con-
tribution in Eq. (2). In this case the signal is parity-odd,
which implies that m̂z(x) ≡ ÎĈ2z(2x) → −Ĉ2z(2x) (Î is
the inversion operator). Then Eq. (3) becomes
σ = 2(1− Ĉ2z)(1 − Ĉ2x)σ1 (4)
which implies that, of the possible five second-rank ten-
sors involved in XNCD, only the term T
1 − T
−1 sur-
vives. To arrive at this result, we applied the usual op-
erator rules on spherical tensors:21 Ĉ2xT
m = T
−m and
Ĉ2zT
m = (−)
m . This in turn leads to a zero XNCD
along the three crystallographic axes of the Bb2b crys-
tal structure where, e.g., along the c-axis, the signal is
proportional to T
Therefore, even in the Bb2b crystal structure, the
XNCD is exactly zero by symmetry when the wave vec-
tor is directed along the c-axis (i.e., orthogonal to the
CuO2 planes) as in the experiment of Ref. 5. This has
been further checked by numerical calculations with clus-
ter radii up to 6.5 Å, i.e., 93 atoms, centered on the Cu
ion, based on the average crystal structures reported in
Refs. 11,12,13,14. The only possibility to justify theo-
retically the experimental evidence of circular dichroism
of a non-magnetic nature is either by lowering of the or-
thorhombic Bb2b symmetry, a misalignment of the k̂-
direction with respect to the c-axis, or contamination
from linear dichroism. We checked all of these possi-
bilities.
If we take into account the monoclinic supercell pro-
posed in Ref. 13, with space group Cc, this implies a re-
duction of the symmetry operations, with the loss of the
two-fold screw axis. Nonetheless, the glide plane contain-
ing the normal to the CuO2-planes is still present (m̂x
in Eq. 3), which is responsible for the extinction rule of
the quantity 〈Ψn|~L · (ǫ̂
∗ × ǫ̂)(~Ω · k̂)|Ψn〉. Therefore the
XNCD is again identically zero by symmetry, which we
verified by direct numerical simulation of the supercell.
Note that only a reduction to triclinic symmetry would
allow for XNCD in the direction orthogonal to the CuO2
planes.
We also checked for the possibility of misalignment, as
shown in Fig. 1, by a direct calculation with ~k‖(101),
corresponding to a tilting θ ∼ 10o with respect to the
c-axis (note that the a lattice constant is ∼ 5.4 Å, and
the c one ∼ 30.9 Å). We first remark that the calcu-
lated XANES signal compares well to experiment (the
XANES for ~k‖(001) and ~k‖(101) are identical on the
scale of Fig. 1). Despite this, the energy profile of the
XNCD signal is different from the one reported by Kub-
ota et al.5 The main difference is the energy extension of
the calculated dichroic signal, whose oscillations persist,
though with decreased intensity, more than 50 eV above
the edge itself. This characteristic is present for all four
Bb2b refinements we have looked at (and as well for the
monoclinic supercell, which we do not show) and is at
variance with the experimental results of Ref. 5, where
the dichroic signal is confined to a small energy range
around the edge.
Some comments on the calculations are in order. Each
refinement gives rise to a different XNCD signal, and
-0.05
9.029.019.008.998.98
Energy (keV)
 Pet (XNCD x 1000)
 Kan (XNCD x 100)
 Glady (XNCD x 400)
FIG. 2: (Color online) XNCD signal, as in Fig. 1, but for a
CuO5 cluster.
9.029.019.008.998.98
Energy (keV)
R=2.1
R=3.1
R=4.9
R=4.1
R=5.5
R=6.5
XNCD x 1000
FIG. 3: (Color online) XNCD signal at the Cu K edge for
~k‖(101) as a function of the cluster radius (in Å) for the crys-
tal structure of Petricek et al.11. The signal has been mul-
tiplied by 1000. Each successive curve is displaced by 0.06
Mbarn.
their intensities are quite different as well. They are
found to be strongly dependent on the magnitude of the
deviation of the atoms from their Bbmb positions, which
differs significantly among the various Bb2b refinements.
Moreover, the structure of Kan and Moss leads to a mani-
festly different energy profile. This difference is seen even
for CuO4 and CuO5 clusters (the results for a CuO5 clus-
ter are shown in Fig. 2) and has to do with the large de-
parture of this particular structure from the Bbmb one.
Although there has been some criticism in the literature
concerning this particular refinement,10 the point we wish
to make is that each refinement has a different XNCD
signal, showing how sensitive this signal is to the actual
crystal structure. We note that Fig. 1 was done for a
cluster radius of 4.9 Å, i.e., 37 atoms. In Fig. 3, we show
results for the refinement of Petricek et al.11 up to a ra-
dius of 6.5 Å (93 atoms), showing the development of ad-
ditional structure in the energy profile as more and more
atoms are included in the cluster. In the real system, the
effective cluster radius is limited by the photoelectron
escape depth, which is energy dependent.22
We also remark that the magnitude of the XNCD sig-
nal we calculated for a 10 degree misalignment is com-
parable to that measured in Ref. 5. On the other hand,
the signal goes as sin(2θ), where θ measures the displace-
ment from the c-axis. We note that Kubota et al.5 men-
tion that their signal was insensitive to displacements
from the c-axis of 5 degrees, which would argue against
a misalignment given the strong angular dependence we
predict. Moreover, we note that the size of the signal
only depends on the projection of the k vector onto the
a-c plane, i.e., a signal for ~k‖(111) is equivalent to that
for ~k‖(101).
The above leads us to suspect that neither misalign-
ment, nor symmetry reduction, are the basis of the signal
detected in Ref. 5. We now turn to the third possibility
for a non-magnetic signal, that due to intermixing of lin-
ear dichroism. All x-ray beams at a synchrotron have a
linear polarization component (Kubota et al.5 mention
the possibility of up to 5% of linear admixture). The
resulting linear dichroism, which vanishes for a uniaxial
crystal, can swamp the intrinsic XNCD signal for a biax-
ial crystal where the a and b directions are inequivalent
(like Bi2212). This was shown by Goulon et al. for a
KTiOPO4 crystal with the same point group symmetry
(mm2) as Bi2212.23
To analyze this further, we show the linear dichro-
ism (the XANES signal for ~E‖(010) minus the one for
~E‖(100)), calculated for the Bb2b refinement of Glady-
shevskii et al.,14 for various cluster radii, in Fig. 4. Simi-
lar results have been obtained for the other crystal struc-
tures, including the Bbmb refinement of Miles et al.10
The energy profile, with a positive peak followed by a
negative peak, and its location at the absorption edge, is
very reminiscent of the data. Moreover, the size of the
XNLD signal is large, meaning that only a few percent
admixture is necessary to explain the size of the signal
seen in Ref. 5. One issue is that Kubota et al.5 did report
the existence of an XNLD signal, but also claim that it is
temperature independent. This is somewhat puzzling, as
there are significant changes of the lattice constants with
temperature.24 One obvious question would be why such
an XNLD contamination would only appear below T*,
though it should be remarked that there are anomalies
in the superstructure periodicity near T*.24 A definitive
test would be to rotate the sample under the beam, as
any XNLD signal would vary as cos(2φ) where φ is the in-
plane angle relative to the b-axis. Any circular dichroism
(XNCD or XMCD) is instead φ-independent.
9.029.019.008.998.98
Energy (keV)
 R=3.0
 R=4.1
 R=5.1
XANES
XNLD x 50
FIG. 4: (Color online) XNLD signal at the Cu K edge for
~k‖(001) as a function of the cluster radius (in Å) for the crys-
tal structure of Gladyshevskii et al.14. The XNLD signal has
been multiplied by 50. The XANES curve has been displaced
by 0.05 Mbarn.
A final possibility would be a small energy shift be-
tween the left and right circularly polarized beams. Dif-
ferentiating the absorption edge in Fig. 1 would indeed
lead to a signal similar to that seen in Ref. 5 (but with
an enhanced positive peak relative to the negative peak).
But such an energy shift is difficult to imagine with the
particular experimental set up used.
The observed dichroism signal as a function of energy
is also reminiscent of that typically seen for magnetic
circular dichroism: in this case the signal would be of
E1-E1 origin and its main features are indeed expected
to be at the edge itself. In addition, the nature of the
observed signal (a single sharp positive peak followed by
a single sharp negative peak) is also much like a magnetic
signal where the main features are expected to be more
localized in energy starting from the rising edge of the
absorption. Whether this possibility is realistic or not
can only be determined by a quantitative calculation,
which we offer in the next section.
III. MAGNETIC DICHROISM IN BI2212
Over the years, there have been several claims of pos-
sible magnetic order in the pseudogap phase of cuprate
superconductors. Recently, a magnetic signal at a (101)
Bragg vector has been observed below T* for several un-
derdoped YBa2Cu3O6+x (YBCO) samples by polarized
neutron diffraction.25 The signal, corresponding to a mo-
ment of order 0.05-0.1 µB, is not simple ferromagnetism
as it was not observed at the (002) Bragg vector. Even
more recently, a Kerr rotation below T* has been de-
tected in underdoped YBCO corresponding to a net fer-
9.029.019.008.998.98
Energy (keV)
 XANES Kan
 XMCD Pet
 XMCD Kan
 XMCD Glady
XMCD x 10000
FIG. 5: (Color online) XMCD for ~k‖(001) at the Cu K edge
with a ferromagnetic moment of 0.1 µB along the c-axis at
each copper site. The cluster radius is 4.1 Å. The XMCD
signal has been multiplied by 10000. The crystal structures
are those of Fig. 1.
romagnetic moment of 10−4 µB.
This motivates us to consider the possibility of a mag-
netic origin for the x-ray circular dichroism detected in
Ref. 5. We restate that in absorption (see, e.g., Ref. 8),
circular dichroism can be generated either by a non-
magnetic effect in the E1-E2 channel (XNCD, a parity-
breaking signal), or by a magnetic signal in the E1-E1
channel (XMCD, parity-even). The first possibility, an-
alyzed in the previous section, does not seem to be com-
patible with the experimental results. In order to analyze
the second possibility, we need to provide the lattice with
a magnetic structure that has a net magnetization (oth-
erwise, the XMCD is zero). In what follows, we shall sup-
pose two magnetic distributions, the first with the mag-
netic moments on the Cu sites, the second on the planar
O sites. The numerical calculations are performed with
the relativistic extension of the multiple-scattering pro-
gram in the FDMNES code,18 and provide results that
are an extension of those of the previous section.
The details of the calculations are as follows: we used
again the average crystal structures discussed in Section
II. For each magnetic configuration we employed clus-
ters with radii ranging from 3.1 Å (a CuO5-cluster), to
4.9 Å (37 atoms) around the Cu photoabsorbing ion. In
the first set of calculations, shown in Figs. 5 and 6, we
built the input potential from a magnetic configuration of
4.55 3d↑ electrons and 4.45 3d↓ electrons (i.e., a moment
of 0.1 µB per copper site). In the second set of calcula-
tions, shown in Fig. 7, we built the input potential from
a magnetic configuration of 2.05 2p↑ electrons and 1.95
2p↓ electrons (i.e., a moment of 0.1 µB per planar oxygen
site). The following results are noteworthy:
a) Differently from the XNCD calculations shown in
-0.05
9.029.019.008.998.98
Energy (keV)
 XMCD Pet
 XMCD Kan
 XMCD Glady
x 10000
FIG. 6: (Color online) XMCD as in Fig. 5, but with a cluster
radius of 3.1 Å, i.e., a CuO5 cluster.
9.029.019.008.998.98
Energy (keV)
 XANES Kan
 XMCD Pet
 XMCD Kan
 XMCD Glady
XMCD x 5000
FIG. 7: (Color online) XMCD for ~k‖(001) at the Cu K edge
with a ferromagnetic moment of 0.1 µB at each planar oxygen
site. The XMCD signal has been multiplied by 5000. The
XANES signal has been displaced by 0.02 Mbarn.
Fig. 1, all crystal structures give basically the same
XMCD spectra. The reason for this behavior may be
related to the fact that XMCD, when x-rays are orthog-
onal to the CuO2 planes, mainly depends on the in-plane
magnetisation density and on the in-plane crystal struc-
ture, which is quite similar for the various refinements.
b) There is a more marked dependence on the cluster
radius compared to the XNCD, as shown by the compar-
ison of Fig. 5 and Fig. 6. The calculations with a radius
bigger than 4.1 Å (6 Cu, 9 O, 4 Sr, and 4 Ca), above the
pre-edge energy, are basically equivalent to those shown
in Fig. 5, with a positive peak at the edge energy, fol-
960950940930
Energy (eV)
 XANES Pet
 XMCD Pet
 XMCD Kan
 XMCD Glady
XMCD x 10
FIG. 8: (Color online) XMCD for ~k‖(001) at the Cu L2,3
edges with a ferromagnetic moment of 0.1 µB along the c-
axis at each copper site. The cluster radius is 4.1 Å. The
XMCD signal has been multiplied by 10.
lowed by a double negative peak, the latter at variance
with the experimental results. On the contrary, the en-
ergy shape obtained for a radius of 3.1 Å is very close to
the experimental one, with a single negative peak after
the sharp positive one, with relatively good agreement
in the energy position and width. We could be tempted
to suppose, therefore, that the virtual photoelectron has
a very small mean free path before decaying and it is
sensitive just to the nearest neighbor oxygens. Indeed
we checked that an identical XMCD profile is obtained
with just the in-plane CuO4-cluster. On the other hand,
the size of the signal we calculate is about an order of
magnitude smaller than that seen in Ref. 5. Since the
XMCD signal is proportional to the moment, then we
would need a moment of ∼1 Bohr magneton per copper
to have a comparable signal. Such a huge moment would
have been observed previously by neutron scattering if
it existed. Of course we cannot exclude that spurious
effects, such as strain fields, could have influenced the
measurement.
c) The energy profile in the case of magnetization at
the oxygen sites is not much different from the cop-
per case, except for the deeper negative peak around E
∼ 8.994 keV, as shown in Fig. 7. Also in this case the dif-
ferent crystal structure refinements give basically equiv-
alent results, as again the CuO2 planes are practically
equivalent in the various cases. Note that the relative
intensity is equivalent to the copper case, as 0.1 Bohr
magnetons per planar oxygen corresponds to 0.2 Bohr
magnetons per CuO2 cell (note we multiply by 5000 in
Fig. 7 as compared to 10000 in Fig. 5).
We also performed simulations for the Cu L2,3 edges
for the magnetic configuration corresponding to Fig. 5,
as shown in Fig. 8, which can be compared with future
experimental investigations in order to confirm whether
or not a net magnetization exists in this compound.
Finally, we remark that the dependence of the XMCD
signal on the tilting angle θ (i.e., the displacement of
the photon wave vector from the c-axis) goes like cos(θ)
and therefore the signal is not very sensitive to small
displacements of 5 degrees, as noted by Kubota et al.5
This different angular dependence from the XNCD signal
suggests a relatively easy way to unravel the question ex-
perimentally: it is sufficient to measure the θ (azimuthal)
dependence of the signal, noting that any XNLD contam-
ination would be tested by the φ (polar) dependence of
the signal.
IV. CONCLUSIONS
In our opinion, the experiments of Kubota et al.5 have
raised more questions than they have answered. Al-
though not treated in our paper, we believe that their
results from the measurement of non-reciprocal linear
dichroism are at this stage not conclusive, as only one di-
rection for the toroidal moment has been investigated, of
the two possible suggested by the orbital current pattern
of Varma.2 The analysis performed in Section II showed,
moreover, that the claimed XNCD signal is probably un-
justified. In fact, even though the space group Bb2b
is non-centrosymmetric, XNCD is absent by symmetry
when the x-ray wave vector is chosen orthogonal to the
CuO2 planes, as in the experimental measurement geom-
etry of Ref. 5. The same extinction rule survives for the
monoclinic supercell structure refined in Ref. 13. More-
over, in both cases, it seems hard to mantain the hy-
pothesis of misalignment, due to the experimental local-
ization of the energy profile around the main absorption
edge, which is absent in the calculations. We also note
the difference of XNCD from the photoemission dichro-
ism results of Kaminski et al.3 A direct comparison is
however not immediate, as the former represents a q̂-
integrated version of the latter (here q̂ is the solid angle in
the space of the photoelectron wave-vector, see, e.g., Ref.
8). A more likely explanation is an XNLD contamination
(Fig. 4), but then the challenge is to understand why such
an effect would only exist below T*. We note that an op-
tics experiment for an optimal doped Bi2212 sample has
seen a change in linear birefrigence below Tc, which was
accompanied by a non-zero circular birefringence.28 In
addition, both Bi222329 and the Fe analogue of Bi221230
exhibit supercells with 222 space groups which would al-
low for dichroism. So, it is conceivable that there is a
subtle structural transition associated with T* which we
suggest could be looked for by diffraction experiments.
A final comment about the physical quantities detected
by x-ray circular dichroism, either natural or magnetic,
is in order. At the K edge of transition metal oxides,
XMCD in the E1-E1 channel, at the excitation energy
E = h̄ω − Eedge, gives information on the expectation
value of L̂ · (ǫ̂∗× ǫ̂) for the excited states at the energy E .
The orbital angular momentum is either induced from a
spin moment via spin-orbit coupling (as calculated here)
or directly by an orbital current (as in the scenario advo-
cated in Ref. 231). No direct spin information is available
at the K edge and therefore an XMCD measurement is
not directly related to the ground state magnetic moment
as for the L edge. Moreover, the observed states are
those with p-like angular momentum projection on the
photoabsorbing Cu ion, which are extended and there-
fore mainly sensitive to the influence of the oxygen atoms
surrounding the Cu site. In this case, the main contribu-
tion to the XMCD energy profile is expected in an energy
range of 10-20 eV from the main edge to the first shoul-
der in the XANES spectrum, as found in Ref. 5 and in
our own XMCD calculations.
Although the results of Section III are in principle con-
sistent with Ref. 5, the size of the ferromagnetic moment
necessary to get a signal of the magnitude seen by exper-
iment, ∼1 Bohr magneton, is excessive. If such a large
ferromagnetic moment existed, it would have surely been
seen by neutron scattering. From this point of view, ex-
periments performed at the Cu L edge and O K edge
would be desirable as they are more sensitive to the pres-
ence of a magnetic moment.
Finally, we would like to remark that XNCD along
the c-axis is insensitive to orbital currents. These latter,
confined to the CuO2 planes, develop a parity breaking
characterized by a toroidal moment (~Ω) within the CuO2
planes. The XNCD experiment of Ref. 5 would only be
sensitive to the projection of the toroidal moment out of
this plane (i.e., along the direction of the x-ray wavevec-
tor). Therefore, if performed as stated, it cannot tell us
about any possible orbital current order.
To conclude, we believe that the various interpreta-
tions, XNCD, XNLD, or XMCD, have their drawbacks,
and therefore the origin of the experimental signal of
Ref. 5 is still open. In this sense, further experimen-
tal checks of the energy extension of the dichroic signal
would be highly desirable. Based on our results, the most
stringent experimental test on the physical origin of the
signal would come from the measurement of the depen-
dence on the tilting (azimuthal) angle θ, due to the dif-
ferent dependences of XNCD and XMCD, as well as the
dependence on the in-plane (polar) angle φ, which would
test for any possible XNLD contamination.
V. ACKNOWLEDGMENTS
The authors thank John Freeland, Zahir Islam,
Stephan Rosenkranz, Daniel Haskel and Matti Lindroos
for various discussions. This work is supported by the
U.S. DOE, Office of Science, under Contract No. DE-
AC02-06CH11357. SDM would like to thank the kind
hospitality of the ID20 beamline staff at the ESRF.
1 M. R. Norman, D. Pines and C. Kallin, Adv. Phys. 54,
715 (2005).
2 C. M. Varma, Phys. Rev. B 55, 14554 (1997) and Phys.
Rev. Lett. 83, 3538 (1999); M. E. Simon and C. M. Varma,
ibid 89, 247003 (2002).
3 A. Kaminski, S. Rosenkranz, H. Fretwell, J. C. Cam-
puzano, Z. Li, H. Raffy, W. G. Cullen, H. You, C. G. Olson,
C. M. Varma and H. Hoechst, Nature 416, 610 (2002).
4 S. V. Borisenko, A. A. Kordyuk, A. Koitzsch, T. K. Kim,
K. A. Nenkov, M. Knupfer, J. Fink, C. Grazioli, S. Turchini
and H. Berger, Phys. Rev. Lett. 92, 207001 (2004).
5 M. Kubota, K. Ono, Y. Oohara and H. Eisaki, J. Phys.
Soc. Jpn. 75, 053706 (2006).
6 L. Alagna, T. Prosperi, S. Turchini, J. Goulon, A. Ro-
galev, C. Goulon-Ginet, C. R. Natoli, R. D. Peacock and
B. Stewart, Phys. Rev. Lett. 80, 4799 (1998).
7 The average crystal structure is the base orthorhombic unit
cell before the incommensurate superstructure is taken into
account. As the latter involves a translation operator, it
should not affect the symmetry arguments in this paper.
8 S. Di Matteo and C. R. Natoli, J. Synchr. Rad. 9, 9 (2002).
9 A. Yamamoto, M. Onoda, E. Takayama-Muromachi, F.
Izumi, T. Ishigaki and H. Asano, Phys. Rev. B 42, 4228
(1990); D. Grebille, H. Leligny, A. Ruyter, Ph. Labbé and
B. Raveau, Acta Cryst. B52, 628 (1996); N. Jakubowicz,
D. Grebille, M. Hervieu and H. Leligny, Phys. Rev. B 63,
214511 (2001).
10 P. A. Miles, S. J. Kennedy, G. J. McIntyre, G. D. Gu, G.
J. Russell and N. Koshizuka, Physica C 294, 275 (1998).
11 V. Petricek, Y. Gao, P. Lee and P. Coppens, Phys. Rev. B
42, 387 (1990).
12 X. B. Kan and S. C. Moss, Acta Cryst. B48, 122 (1992).
13 R. E. Gladyshevskii and R. Flükiger, Acta Cryst. B52, 38
(1996).
14 R. E. Gladyshevskii, N. Musolino and R. Flükiger, Phys.
Rev. B 70, 184522 (2004). There is no superstructure for
this Pb doped variant, so the crystal structure used is the
actual one, not the average one.
International Tables for Crystallography, 5th ed., ed. T.
Hahn (Kluwer, Dordrecht, 2002).
16 P. Carra and R. Benoist, Phys. Rev. B 62, R7703 (2000).
17 A. Mans, I. Santoso, Y. Huang, W. K. Siu, S. Tavaddod,
V. Apiainen, M. Lindroos, H. Berger, V. N. Strocov, M.
Shi, L. Patthey and M. S. Golden, Phys. Rev. Lett. 96,
107007 (2006).
18 Y. Joly, Phys. Rev. B 63, 125120 (2001).
This program can be downloaded at
http://www-cristallo.grenoble.cnrs.fr/fdmnes.
19 C. R. Natoli, Ch. Brouder, Ph. Sainctavit, J. Goulon, Ch.
Goulon-Ginet and A. Rogalev, Eur. Phys. J. B 4, 1 (1998).
20 V. Arpiainen and M. Lindroos, Phys. Rev. Lett. 97, 037601
(2006).
21 D. A. Varshalovich, A. N. Moskalev and V. K. Kersonskii,
Quantum Theory of Angular Momentum (World Scientific,
Singapore, 1988).
22 The results presented involve a convolution of the calcu-
http://www-cristallo.grenoble.cnrs.fr/fdmnes
lated spectrum with both a core hole (1.9 eV for the Cu K
edge) and a photoelectron lifetime, with the latter having
a strong energy dependence (ranging up to 15 eV with a
midpoint value at 30 eV above the Fermi energy). But the
calculation has a fixed cluster radius.
23 J. Goulon, C. Goulon-Ginet, A. Rogalev, G. Benayoun, C.
Brouder and C. R. Natoli, J. Synchr. Rad. 7, 182 (2000).
In addition, there is also an intrinsic non XNCD contribu-
tion to the circular dichroism for a biaxial crystal, but this
should vanish in Bi2212 for an x-ray beam aligned along
the c-axis, see J. Goulon, C. Goulon-Ginet, A. Rogalev, V.
Gotte, C. Brouder and C. Malgrange, Eur. Phys. J. B 12,
373 (1999).
24 P. A. Miles, S. J. Kennedy, A. R. Anderson, G. D. Gu,
G. J. Russell and N. Koshizuka, Phys. Rev. B 55, 14632
(1997).
25 B. Fauqué, Y. Sidis, V. Hinkov, S. Pailhes, C. T. Lin, X.
Chaud and P. Bourges, Phys. Rev. Lett. 96, 197001 (2006).
26 A. Kapitulnik, unpublished results.
27 It should be remarked, though, that single-particle based
approaches can miss some features of the spectral weight
transfer between the L2 and L3 edges.
28 J. Kobayashi, T. Asahi, M. Sakurai, M. Takahashi, K.
Okubo and Y. Enomoto, Phys. Rev. B 53, 11784 (1996).
29 E. Giannini, N. Clayton, N. Musolino, A. Piriou, R. Glady-
shevskii and R. Flükiger, IEEE Trans. Appl. Supercond.
15, 3102 (2005).
30 Y. Le Page, W. R. McKinnon, J.-M. Tarascon and P. Bar-
boux, Phys. Rev. B 40, 6810 (1989).
31 X-ray dichroism experiments in regards to the orbital cur-
rent scenario of Varma have been discussed by S. Di Matteo
and C. M. Varma, Phys. Rev. B 67, 134502 (2003).
