Table Of ContentPhonons and their Coupling to Magnons in n = 2
Ruddlesden-Popper Compounds.
A. B. Harris
Department of Physics and Astronomy,
1 University of Pennsylvania, Philadelphia PA 19104
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0 (Dated: January 26, 2011)
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n Abstract
a
J
The number of absorption and Raman optical modes for each Wyckoff orbit in the high-
5
2
temperature tetragonal (I4/mmm) parent lattice of the Ruddlesden-Popper compounds Ca X O ,
3 2 7
]
i with X=Mn or X=Ti is given. We analyze the effect of sequential perturbations which lower the
c
s
- symmetry to Cmcm and Cmc21 and finally include magnetic ordering. We determine the power
l
r
t law behavior (within mean field theory) for the cross section for photon absorption and Raman
m
t. scattering of modes which appear as the symmetry is successively lowered. In the Cmc21 phase we
a
m
give a symmetry analysis to discuss the magnon-phonon coupling which in other systems gives rise
-
d
to “electromagnons.” From our results we suggest several experiments to clarify the phase diagram
n
o
and other properties of these systems.
c
[
2 PACS numbers: 61.50.Ks,61.66.-f,63.20.-e,76.50.+g
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7
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I. INTRODUCTION
The Ruddlesden-Popper (RP) compounds[1] are those of the type A B C , where
n+1 n 3n+1
the valences of the ions are A = +2, B = +4, and C = 2 (usually C is oxygen). At
−
high temperature the crystal structure for many of these compounds is tetragonal, but the
structure can distort in several ways as the temperature passes through structural phase
transitions. In the lowest symmetry phase, these systems are usually ferroelectric and can
also be magnetically ordered, although, in contrast to many magnetoelectrics,[2–7] these two
degrees of freedom do not appear at the same temperature. Nevertheless the RP systems
do have many interesting couplings between their magnetic and dielectric properties. For
example, their spontaneous polarization can be rotated by the application of a magnetic
field and the weak ferromagnetic moment can be rotated by the application of an electric
field.[8,9] The symmetry of the RP’s is very similar to that of the Aurivillius compound
SrBi Ta O analyzed in detail by Perez-Mato et al.[10] and many of our results should also
2 2 9
apply to it. Although there have been several first principles calculations of the properties of
CMO and/or CTO,[8,11,12] the calculations of CF are most relevant to the present paper.
FIG. 1: (Color online) The tetragonal phase of CMO or CTO (space group I4/mmm). The oxygen
ions are thebluespheres,the Caions are theopen green circles, andthe Mn or Tiions (not shown)
are at the centers of the octahedra of oxygen ions. There are two molecules per conventional unit
cell.
−
l
i
B
r
e
y
a
l
i
B
r
e
y
a
Here we will be interested in the two RP compounds Ca Mn O (CMO) and Ca Ti O
3 2 7 3 2 7
2
(CTO), which consist of bilayers of oxygen octahedra enclosing a Mn or Ti ion. It is believed
that the high-temperature phase is tetragonal (I4/mmm)[13,14] and the low-temperature
phase is orthorhombic (Cmc2 )[15] and in Fig. 2 we show the three possible paths that con-
1
nect these structures.[8] First principles calculations[8] indicate that the most likely scenario
involves the appearance, as the temperature is lowered, of the intermediate state Cmcm at a
temperature we denote T . In fact, the Cmcm phase has been observed in the isostructural
>
compounds LaCa Mn O [16] at and below room temperature and Bi Ca Mn O [17] at
2 2 7 0.44 2.56 2 7
room teperature. Accordingly, we will assume that this scenario applies to CMO and CTO
althoughfor these systems no intermediate phase has been observed up to now. However, re-
cent measurements onceramic CMO finda clear pyroelectric signal consistent withthe onset
of ferroelectric order close to T = 280K.[18] This transition is identified as the temperature
>
at which Cmc2 appears. Since this ferroelectric transition seems to be a continuous and
1
well-developed one and since a direct continuous transition between I4/mmm and Cmc2 is
1
inconsistent with Landau theory,[19] the seemingly inescapable conclusion is that the phase
for T slightly greater than T is not I4/mmm, but is some phase which does not allow a
>
spontaneous polarization. Thus the phase at temperature just above T may be the Cmcm
>
phase. In this scenario the Cmc2 phase would appear at a lower temperature T (which
1 <
experimental data implies is quite close to T ). The purpose of the present paper is to
>
discuss the symmetry of the zone center phonons and the Raman scattering cross section
as they are modified by the successive lowering of symmetry as the temperature is reduced
through the structural phase transitions at T and T . Such a symmetry analysis can be
> <
done using standard group theory methods. Here we will supplement that analysis with an
analysis of the phonon cross sections from which we determine their power law dependence
ontheorder parameters (OP’s) which characterize thedistortions fromtetragonalsymmetry
to the lower symmetry structures. This enables us to predict the temperature dependence
of the newly allowed cross sections which appear at the structural phase transitions. We
then give a similar analysis of the magnon-phonon interaction. As a result of this coupling,
the magnon absorption cross section (which usually results from a magnetic dipole allowed
transition) isenhanced by nowbeing electric-dipole allowed. This enhancement hasled some
authors[20–25] to call such magnons “electromagnons.”
Briefly, this paper is organized as follows. In Sec. II we give a symmetry analysis
of the zone center phonons in the tetragonal (I4/mmm) phase and in Secs. III and IV
3
we give similar analyses for the orthorhombic Cmcm and Cmc2 phases. We discuss the
1
photon absorption and Raman scattering of these modes. For modes which only appear as
a consequence of a phase transition we also determine the power-law dependence of their
optical cross sections on the emergent OP’s. In Sec. V we discuss the coupling of these
modes to magnetic excitations (magnons). Our results are summarized in Sec. VI.
I4/mmm (139)
+ − −
Γ
X X
2 5 3
Cmca (64) F2mm (42) Cmcm (63)
A2 am (36)
1
FIG. 2: The group-subgroup structure of CMO and CTO. We indicate the irreducible representa-
tions (irreps) by which the tetragonal phase is distorted to reach one of the intermediate phases.
The numbering of the space groups follows Ref. 26. The superscripts on the irreps give the parity
with respect to spatial inversion. The irrep Γ− is at zero wave vector and the X irreps are at
5
the wave vectors q = (1/2, 1/2,0). In the final step to reach the A2 am (equivalent to Cmc2 )
1 1
±
structure the other two irreps are introduced simultaneously,[9] so that Cmc2 can be regarded as
1
the result of simultaneous condensation of the three irreps.
II. SYMMETRY ANALYSIS FOR THE TETRAGONAL STRUCTURE
The occupied sites in space group I4/mmm for the n = 2 RP system are given in the
first three columns of Table III, below. The character table for the little group of q = 0 for
the I4/mmm tetragonal structure is given in Table I. For the one dimensional irreducible
representations (irreps) the characters are the matrices. For the two dimensional irreps and
for the irreps for the X points at q = (1/2, 1/2,0) (in rlu’s) the matrices are given in Table
±
II.[28]
4
TABLE I: Character table for the little group of q = 0 for the I4/mmm tetragonal structure. The
operator numbers are those of Ref. 27. The bottom row gives the dipole moment vector P or the
Raman tensor R, where a, b, c, etc. are arbitrary constants. For the two dimensional irreps the
two entries are the values of P and R for each of the two degenerate states. The entry is left blank
for irreps which are optically inactive.
Number Operator Γ+ Γ+ Γ+ Γ+ Γ+ Γ− Γ− Γ− Γ− Γ−
1 2 3 4 5 1 2 3 4 5
1 (xyz) 1 1 1 1 2 1 1 1 1 2
37,40 (yxz),(y,x,z) 1 1 1 1 0 1 1 1 1 0
− − − −
4 (xyz) 1 1 1 1 2 1 1 1 1 2
− −
3,2 (xyz),(xyz) 1 1 1 1 0 1 1 1 1 0
− − − −
39,38 (yxz),(yxz) 1 1 1 1 0 1 1 1 1 0
− − − −
25 (xyz) 1 1 1 1 2 1 1 1 1 2
− − − − −
13,16 (yxz),(yxz) 1 1 1 1 0 1 1 1 1 0
− − − −
28 (xyz) 1 1 1 1 2 1 1 1 1 2
− − − − −
27,26 (xyz),(xyz) 1 1 1 1 0 1 1 1 1 0
− − − −
15,14 (yxz),(yxz) 1 1 1 1 0 1 1 1 1 0
− − − −
P a(ˆiˆi+ˆjˆj) c(ˆiˆi d(ˆiˆj e(ˆikˆ+kˆˆi) gkˆ hˆi
or R +bkˆkˆ ˆjˆj) +ˆjˆi) e(ˆjkˆ+kˆˆj) hˆj
−
We now discuss briefly the introduction of order parameters (OP’s). Starting from the
high-temperature tetragonal, I4/mmm, structure, distortions of the irreps Γ−, X− and X+
5 3 2
are condensed as the temperature is lowered to eventually reach the low-symmetry low-
temperature Cmc2 structure. A tabulation of these symmetry-adapted distortion vectors,
1
Ψ(X), for irrep X is given in Table III. In the dominant component of the distortion of irrep
n
X+ the octahedra are rotated about the crystal c axis like interlocking gears. The rotations
2
in the upper (z = τ) and lower (z = τ) planes of a bilayer are equal, so in the notation
−
of Refs. 29 and 30 these rotations are both denoted (0,0,Θ). The dominant component
of the distortion of irrep X− consists of 1) a rotation about the tetragonal [1,0,0] with the
3
rotation alternating in sign as one moves along the [1,0,0] direction and 2) a rotation about
the tetragonal [0,1,0] direction with the rotation alternating in sign as one moves along the
5
TABLE II: Representation matrices M(G)( ) for the operators which are the generators of the
O O
irrep G. Here r′ = r and the σ’s are the Pauli matrices. For the irreps G at zero wave vector the
O
matrices for the translations T (x+1,y,z), T (x,y+1,z), and T (x+ 1,y+ 1,z 1)
1 ≡ 2 ≡ 3 ≡ 2 2 − 2
are unity. For the irreps at the X wave vectors the matrices are M(T ) = M(T ) = 1 and
1 2
−
M(T ) = σ . The first row and column for the X irreps refer to wave vector q These matrices
3 z 1
−
are related to those of Ref. 19 by a unitary transformation which, for zero wave vector takes
(σ ,σ ,σ )into( σ , σ ,σ )andforthewavevectorXtakes(σ ,σ ,σ )into( σ ,σ , σ ).
x y z x y z x y z z y x
− − − −
= m m
4 d z
O R
r′ = (y,x,z) (y,x,z) (x,y,z)
M(X+1)( ) = σx 1 1
O −
M(X+2)( ) = σx 1 1
O −
M(X+3)( ) = iσy σz 1
O − −
M(X+4)( ) = iσy σz 1
O − −
M(X−1)( )= σx 1 1
O − − −
M(X−2)( )= σx 1 1
O −
M(X−3)( )= iσy σz 1
O − −
M(X−4)( )= iσy σz 1
O
M(Γ−5)( ) = iσy σx 1
O
M(Γ+5)( ) = iσy σx 1
O −
[0,1,0] direction. Since this distortion is an even function of z, we identify this distortion as
(Φ,Φ,0) in the upper layer and ( Φ, Φ,0) in the lower layer.
− −
If we assume that the distortions aresmall, then the crystal structure in any of the phases
can be approximated as a distortion from the tetragonal structure which is written as linear
combination of the distortions of the three irreps listed in Table III:
Φ = Q(X)Ψ(X) ,
m m
m,X
X
where Ψ(X) is the 12 component vector of irrep X as in Table III. Since the notation to
m
distinguish the various irreps which we will encounter is somehwat complicated, we summa-
rize the notation in Table IV. If we regard the basis vectors Ψ(X) as fixed, then application
m
6
(X) (X)
TABLE III: Basis functions Ψ and Ψ for the distortion under irrep X, where X = 5,3,2
1 2
indicates irrepΓ−, X−, andX+, respectively.[28]For each site wegive thethreecomponents of the
5 3 2
vector displacement. We assume normalization so that the sum of the squares of the components
is unity. The values of the structural parameters are given in Ref. 13: ρ = 0.311, ξ = 0.100,
χ = 0.205, and τ = 0.087. Since the a and b sites have the same symmetry we will refer to them
both as a sites.
(5) (5) (3) (3) (2) (2)
Site n r Ψ Ψ Ψ Ψ Ψ Ψ
1,n 2,n 1,n 2,n 1,n 2,n
A sites
e 1 (0,0,ρ+1/2) u 0 0 0 u 0 a a 0 a a 0 0 0 0 0 0 0
− − −
e 2 (0,0, ρ+1/2) u 0 0 0 u 0 a a 0 a a 0 0 0 0 0 0 0
− − − −
b 3 (0,0,1/2) v 0 0 0 v 0 b b 0 b b 0 0 0 0 0 0 0
− − −
B sites
e 4 (0,0,ξ) w 0 0 0 w 0 c c 0 c c 0 0 0 0 0 0 0
− − −
e 5 (0,0, ξ) w 0 0 0 w 0 c c 0 c c 0 0 0 0 0 0 0
− − − −
O sites
a 6 (0,0,0) x 0 0 0 x 0 d d 0 d d 0 0 0 0 0 0 0
− − −
e 7 (0,0,χ) y 0 0 0 y 0 e e 0 e e 0 0 0 0 0 0 0
− − −
e 8 (0,0, χ) y 0 0 0 y 0 e e 0 e e 0 0 0 0 0 0 0
− − − −
g 9 (0,1/2,τ) z 0 0 0 z 0 0 0 f 0 0 f h g 0 h g 0
1 2
− − −
g 10 (0,1/2, τ) z 0 0 0 z 0 0 0 f 0 0 f h g 0 h g 0
1 2
− − − − − −
g 11 (1/2,0,τ) z 0 0 0 z 0 0 0 f 0 0 f g h 0 g h 0
2 1
− −
g 12 (1/2,0, τ) z 0 0 0 z 0 0 0 f 0 0 f g h 0 g h 0
2 1
− − −
of an operator in the symmetry group of the tetragonal space group to Φ will induce a
O
transformation of the OP’s as [9]
Q(X) = M(X)( )Q(X) ,
O m k,m O k
k
X
where the representation matrices M(X)( ) for the all the relevant operators can be
O O
constructed from those given for the generators of the irrep X given in Table II. This
equation defines the symmetry of the OP’s.
7
TABLE IV: Notation for order parameters. First column: full notation, subsequent column(s):
abbreviated notation.
For Full Abbrev Abrev’
−
m = 1,2 Q(Γ5) Q−(m) Q (m)
m ≡ 5 ≡ 5
m = 1,2 Q(Γ+5) Q+(m)
m ≡ 5
−
p < 5 Q(Γp) Q−(0)
1 ≡ p
p < 5 Q(Γ+p) Q+(0)
1 ≡ p
− −
[Q(Γ5)+Q(Γ5)]√2 Z− Z
1 2 ≡ ≡
− −
[Q(Γ5) Q(Γ5)]√2 Y− Y
1 − 2 ≡ ≡
[Q(Γ+5)+Q(Γ+5)]/√2 Z+
1 2 ≡
[Q(Γ+5) Q(Γ+5)]/√2 Y+
1 − 2 ≡
−
Q(Γ3) Q−(0) X
1 ≡ 3 ≡
m = 1,2 Q(X2+) Q+(q ) Q (q )
m ≡ 2 m ≡ 2 m
−
m = 1,2 Q(X3 ) Q−(q ) Q (q )
m ≡ 3 m ≡ 3 m
m = 1,2, p = 2 Q(Xp+) Q+(q )
m p m
6 ≡
−
m = 1,2, p = 3 Q(X3 ) Q−(q )
6 m ≡ 3 m
As explained in Ref. 9 these distortions form two families, one of which involves wave
vector q (1/2,1/2,0)andthe other of which involves wave vector q (1/2, 1/2,0).[31]
1 2
≡ ≡ −
The wave vectors (1/2, 1/2,0) correspond to a doubling of the size of the primitive unit
±
− −
cell. The first family is described by the three OP’s Q−(q ), Q+(q ), and Z Q(Γ5)+Q(Γ5),
3 1 2 1 ∝ 1 2
− −
and the second by the OP’s Q−(q ), Q+(q ), and Y Q(Γ5) Q(Γ5). In the absence of any
3 2 2 2 ∝ 1 − 2
external fields, when the sample orders, one expects an equal population of domains of the
two families. However, it is possible to select a single family by invoking the coupling[9]
V = aǫ [Q (q )2 Q (q )2] , (1)
xy k 1 k 2
−
where ǫ is a shear component of strain, and k = 2 or 3. One can utilize this interaction
xy
by cooling the sample through T in the presences of a shear stress conjugate to ǫ . This
> xy
will preferentially select q or q , depending on the sign of the stress and the sign of the
1 2
constant a. It is possible that the selection of the wave vector can also be accomplished by
cooling in an electric or magnetic field.[9]
8
Our aim is describe the evolution of the cross section for a) zero wave vector photon
absorption and b) Raman scattering in terms of the various OP’s.
The single photon absorption process results from the electromagnetic coupling V =
E
p E, where E is the electric field of the incident electromagnetic wave and p is the dipole
− ·
moment operator of the system. The symmetry indicated in Tables I and II indicates that
in the tetragonal phase one has
p = a Q (1) , p = a Q (2) , p = a Q−(0) ,
x x 5 y y 5 z z 3
with a = a . Note: in reality we ought to index the modes by an additional index to
y x
distinguish between different modes of the same symmetry. Thus, for instance,
−
n(Γ )
5
p = a Q (1) , (2)
x k,x 5 k
k=1
X
where n(Γ−) is the number of occurrences of Γ−, which, in this case, is [see Eq. (4), below]
5 5
n(Γ−) = 7. For simplicity we will usually not display this index explicitly, but in all our
5
results sums over such an index (or indices) are implied. We discuss the use of Eq. (2)
in Appendix A. The absorption cross section is proportional to the square of the scalar
product of the polarization vector of the incident photon and the dipole moment vector for
the irrep in question.[32,33] (For the two dimensional irrep one has to sum the squares of
the two scalar products.)
Likewise, the Raman scattering process results from the electromagnetic coupling V =
E
α E E , where α is the polarizability tensor, which depends on the phonon dis-
βγ βγ β γ
−
plaPcements as
∂α
(0) βγ
α = α + Q .
βγ βγ ∂Q τ
τ τ
X
We consider the cross section (or intensity) for a process in which the incident photon has
an electric field vector in the β direction and the scattered photon has an electric field
vector in the γ direction (we call this the β-γ polarization) and a phonon Q is created
τ
(or destroyed). The cross section for this process is proportional to the square of the tensor
product of the Raman tensor ∂α /∂Q with the polarizations of the incoming and outgoing
βγ τ
photons.[32,33] (For the two dimensional irreps one sums these squares over the two states
of the irreps.) For both absorption and scattering we implicitly assume that the photon
wave length is very large in comparison to the size of the unit cell.
9
Nowwedeterminewhich modes(atzerowavevector) areallowedinthetetragonallattice.
Using standard methods we find that the reducible representations Γ corresponding to the
space of zero wave vector displacements of the Wyckoff orbits a, e, and g, are decomposed
into their irreps as
Γ(a) = Γ− +Γ− ,
5 3
Γ(e) = Γ− +Γ− +Γ+ +Γ+ ,
5 3 5 1
Γ(g) = 2Γ− +Γ− +Γ− +2Γ+ +Γ+ +Γ+ , (3)
5 4 3 5 2 1
so that the reducible representation Γ corresponding to the space of thevector displacements
of all 12 ions in the primitive unit cell has the decomposition into irreps as
Γ = 2Γ(a)+3Γ(e)+Γ(g) = 7Γ− +Γ−
5 4
+6Γ− +5Γ+ +4Γ+ +Γ+ . (4)
3 5 1 2
The allowed absorption modes [x and y from Γ−(0) and z from Γ−(0)] are listed in Table V.
5 3
Thus we find that in the tetragonal phase there are 20 absorption modes, 6 from Γ− which
3
have z polarization and 7 pairs of doubly degenerate modes (one with x polarization and
one with y polarization) coming from Γ−. Note that of these 20 absorption modes three are
5
acoustic (zero frequency) and are not observed in IR experiments. There are also 15 Raman
active modes, five nondegenerate modes coming from irreps Γ+ and Γ+ and 5 pairs of doubly
1 2
degenerate modes coming from Γ+. Because the lowest symmetry sites of the space group
5
I4/mmm are not occupied in CMO or CTO, it happens that there are no modes for which
the Raman polarization xy is nonzero. Indeed, this result can be used as a signature of the
tetragonal phase of CTO or CMO.
III. ORTHORHOMBIC SPACE GROUP CMCM (# 63)
A. Analysis based on Cmcm symmetry
We now consider the modifications in the optical spectrum when we condense irrep X−
3
at wave vector q = (1/2,1/2,0) to go into the space group Cmcm. The transformation
1
from tetragonal to orthorhombic coordinates is illustrated in Fig. 3 and is given by
x y 1
t t
X = z , Y = − ,
O t O
2 − 4
10
