Theory of high energy optical conductivity and the role of oxygens in manganites

Muhammad Aziz Majidi1,4 , Haibin Su5 , Yuan Ping Feng1 , Michael Rübhausen3,1 , and Andrivo Rusydi1,2,3∗
1
NUSNNI-NanoCore, Department of Physics, Faculty of Science,
National University of Singapore, Singapore 117542, Singapore,
2
Singapore Synchrotron Light Source, National University of Singapore, Singapore 117603, Singapore
3
Institut für Angewandte Physik, Universität Hamburg, Jungiusstrae 11, D-20355 Hamburg, Germany.
Center for Free Electron Laser Science (CFEL), Notkestraße 85, D- 22607 Hamburg, Germany,
4
Departemen Fisika, FMIPA, Universitas Indonesia, Depok 16424, Indonesia and
5
Division of Materials Science, Nanyang Technological University,
arXiv:1101.3059v1 [cond-mat.mtrl-sci] 16 Jan 2011

50 Nanyang Avenue, Singapore 639798, Singapore.

(Dated: July 4, 2018)
Recent experimental study reveals the optical conductivity of La1−x Cax MnO3 over a wide range
of energy and the occurrence of spectral weight transfer as the system transforms from paramag-
netic insulating to ferromagnetic metallic phase [Rusydi et al., Phys. Rev. B 78, 125110 (2008)].
We propose a model and calculation within the Dynamical Mean Field Theory to explain this phe-
nomenon. We find the role of oxygens in mediating the hopping of electrons between manganeses
as the key that determines the structures of the optical conductivity. In addition, by parametrizing
the hopping integrals through magnetization, our result suggests a possible scenario that explains
the occurrence of spectral weight transfer, in which the ferromagnatic ordering increases the rate
of electron transfer from O2p orbitals to upper Mneg orbitals while simultaneously decreasing the
rate of electron transfer from O2p orbitals to lower Mneg orbitals, as temperature is varied across the
ferromagnetic transition. With this scenario, our optical conductivity calculation shows very good
quantitative agreement with the experimental data.

PACS numbers:

Introduction. Manganites have been the subject of ex- ergy phenomena, as their models implicitly assume that
tensive studies since they have exhibited a wealth of fasci- low energy phenomena occuring in these materials are in-
taning phenomena such as the colossal magnetoresistance sensitive to possible high energy excitations. Many such
(CMR), charge-, spin-, and orbital orderings, and transi- models [14–19] typically consider only effective hoppings
tion from paramagnetic insulator to ferromagnetic metal, between Mn sites, while ignoring the electronic states in
as well as multiferroic behavior [1–4]. Upon hole dop- oxygen sites. On the other hand, models that included
ing, the transition from antiferromagnetic insulator to local interactions and hybridization in correlated mate-
ferromagnetic metal has been argued to occur through a rials, might expect pronounced effects at higher energies
mixed-phase process [5]. Whereas for a fixed hole doping that are connected to charge-transfer or Mott-Hubbard
where ferromagnetic order is found, insulator to metal physics [20–23]. Thus, the validity of such theories may
transition simultaneously occurs as temperature is low- have to be tested through experimental studies on the
ered across the ferromagnetic transition [6]. It has been band structures and the optical properties over a wide
generally assumed and experimentally confirmed that the range of energy. In that respect, experimental studies of
magnetic order in these systems is driven by the double- optical conductivity of manganites as functions of tem-
exchange interactions [5, 7–10]. However, explanation on perature and doping in a much wider energy range be-
other phenomena accompanying the ferromagnetic tran- come crucial.
sition seems to be far from complete, and remains as an
A recent study of optical conductivity by Rusydi et al.
open subject.
[24], has revealed for the first time strong temperature
Several theories on the insulator-metal (I-M) transition and doping dependences in La1−x Cax MnO3 for x = 0.3
accompanying the ferromagnetic transition have been and 0.2. The occurrence of spectral weight transfer has
proposed [12, 13]. Although the details of the models and been strikingly found between low (<3eV), medium (3-
scenarios of the I-M transition proposed by these theo- 12eV), and high energies (>12eV) across I-M transition.
ries are quite different, they have similar idea suggesting In fact, as the temperature is decreased, the spectral
that the Jahn-Teller distortion along with the electron- weight transfer appears more noticeably in the medium
phonon interactions stabilize the insulating phase at high and high energy regions than it does in the low energy
temperatures, which is broken by the ferromagnetic order region. Observing how the spectral weight in each re-
below its transition temperatures. These theories, how- gion of energy simultaneously changes as temperature is
ever, have only addressed the static properties or low en- decreased passing the ferromagnetic transition tempera-
ture (TF M ), one may suspect that there is an interplay
between low, medium, and high energy charge transfers
that may drive many phenomena occuring in manganites,
∗ Electronic address: phyandri@nus.edu.sg including the I-M transition. This conjecture is related
 2

ther, with some additional argument, our calculation cap-

tures qualitatively correctly the temperature dependence
of the optical conductivity as the system transforms from
paramagnetic to ferromagnetic phase.
Model. As shown in Fig. 1, we model the crystal struc-
ture of La1−x Cax MnO3 such that each unit cell forms a
cube with lattice constant a set equal to 1, and contains
only one Mn and three O sites, thus ignoring the pres-
ence of La and Ca atoms that we believe not to contribute
much to the structures and temperature dependence of
the optical conductivity. We choose 10 basis orbitals to
construct our Hilbert space, which we order as the fol-
lowing: |Mn eg upper,↑ i, |Mn eg lower,↑ i, |O1 p↑ i, |O2 p↑ i,
FIG. 1: (Color online) Simplified crystal structure of the |O3 p↑ i, |Mn eg upper,↓ i, |Mn eg lower,↓ i, |O1 p↓ i, |O2 p↓ i,
model. The crystal structure is assummed cubic with the unit and |O3 p↓ i. Note that the distinction between | eg upper i
cell containing only one Mn and three O atoms. The O atoms and | eg lower i states is associated with the Jahn-Teller
are labeled 1,2, and 3 to distinguish the p−orbitals belonging splitting. Using this set of bases we propose a Hamilto-
to different O atoms used to construct our Hamiltonian. nian:

1 X † X
to the fact that the hopping of an electron from one Mn H = ηk [H0 (k)]ηk + U nui σ nli σ′
N
site to another Mn site can only occur through an O site. k i,σ,σ′

Considering the difference between the on-site energy

X X
+ Uu nui ↑ nui ↓ + U l n li ↑ n li ↓
of the manganese and that of the oxygen that could be iσ iσ
about 5-8 eV [25], the Mn-O hoppings occur with high X
energy transfer. We hypothesize that if such high en- − JH Si .si . (1)
i
ergy hoppings can mediate a ferromagnetic order, then
other low or high energy phenomena may possibly occur
The first term in the Hamiltonian is the kinetic part,
simulataneously. Thus, the mechanism of I-M transition
in the dc conductivity may not be completely separated whereof ηk† is a row vector whose elements are the cre-
from what appears as the decrease (increase) of the spec- ation operators associated with the 10 basis orbitals, and
tral weight in the medium (high) energy region of the ηk is its hermitian conjugate containing the correspond-
optical conductivity, all of which together may be driven ing destruction operators. Here we consider that each
by the ferromagnetic ordering. Theories based on effec- Mn site contributes 4 eg orbitals (upper and lower each
tive low energy models which only consider Mn sites while of which is with up and down spins), and the three O
ignoring O sites would not be able to address this. sites contribute 6 orbitals (3 from each site each of which
Motivated by the aforementioned conjecture, we de- is with up and down spins). [H0 (k)] is a 10×10 matrix
velop a simple but more general model, in which oxygens in momentum space whose structure is arranged in four
are explicitly incorporated. In this paper, we propose 5×5 blocks corresponding to their spin directions as
our model and calculation of the optical conductivity of  
La1−x Cax MnO3 within the Dynamical Mean Field The- H0 (k)↑ O
[H0 (k)] = , (2)
ory, to explain the experimental results of Ref. [24]. Our O H0 (k)↓
calculated optical conductivity shows that both oxygens
and mangeneses play important roles in forming struc- where O is a zero matrix of size 5×5, and (referring to
tures similar to those of the experimental results. Fur- the choice of coordinates in Fig. 1)

 (1) (1) (1)


EJT 0 tM n−O (1+e−ikx ) tM n−O (1+e−iky ) tM n−O (1+e−ikz )
(2) (2) (2)
tM n−O (1+e−ikx ) tM n−O (1+e−iky ) tM n−O (1+e−ikz )
 (1) 0 −EJT
 
(2)

H0 (k)↑(↓) =  tM n−O (1+e ) tM n−O (1+e )
 ikx ikx
Ep tO−O (1+2e ikx
+2e−iky
) tO−O (1+2e ikx
+2e ) .
−ikz 
 (3)
 t(1)
M n−O (1+e
iky (2)
) tM n−O (1+eiky ) tO−O (1+2e−ikx +2eiky ) Ep tO−O (1+2eiky +2e−ikz ) 
(1) (2)
tM n−O (1+eikz ) tM n−O (1+eikz ) tO−O (1+2e−ikx +2eikz ) tO−O (1+2e−ikx +2eikz ) Ep

The diagonal elements of H0 (k)↑(↓) represent the local energies, while the off-diagonal elements represent the
 3

hybridizations between orbitals. The first two diagonal following blocks of [G(z)] by a half, while keeping the
elements, i.e. EJT and −EJT , correspond to the Mneg remaining elements unchanged, that is
orbital energies which are split due to the presumedly    
static Jahn-Teller distortion. Each of the remaining three G11 G12 1 G11 G12
⇒ ,
diagonal elements, i.e. Ep , corresponds to the local en- G21 G22 2 G21 G22
(1) (2)
ergy of the O2p orbital. The parameter tMn−O (tMn−O )
   
G16 G17 1 G16 G17
corresponds to hopping between the upper (lower) Mneg ⇒ ,
G26 G27 2 G26 G27
orbital and the nearest O2p orbital. Whereas tO−O cor-    
G61 G62 1 G61 G62
responds to hopping between nearest O2p orbitals. ⇒ ,
G71 G72 2 G71 G72
The second term in Eq. (1) represents the Coulomb    
repulsions between the upper and lower Mneg orbitals in G66 G67 1 G66 G67
⇒ . (6)
a site. The third and forth terms represent the intra- G76 G77 2 G76 G77
orbital Coulomb repulsions. In this work, we take Uu
and Ul to be infinity, forbidding double occupancy in The “mean-field” Green function can then be extracted
each of the lower and upper Mneg orbitals. Finally, the as
fifth term represents the double-exchange magnetic inter-  −1
actions between the local spins of Mn, S, formed by the −1
[G(z)] = [G(z)]eff + [Σ(z)] . (7)
strong Hund’s coupling among three t2g electrons giving
S=3/2, and the itinerant spins of the upper and lower
Mneg electrons, s. Note that we use a well-accepted Next, we construct the local self energy matrix,
general assumption that the on-site Coulomb repulsion [Σnl (z)], corresponding to the second and the fifth terms
in each t2g orbital and the Hund’s coupling among the of the Hamiltonian. Here nl ∈ {0, 1} is the occupation
t2g orbitals are so strong to keep the occupancy of the number of the lower Mneg orbital. The elements of the
three t2g levels fixed at high spin configuration. Thus 10 × 10 matrix [Σnl (z)] are all zero except for the blocks
the charge degrees of freedom of the three t2g electrons  
become frozen, and the remaining degree of freedom to Σ11 Σ16
h i
−J S cos θ+nl U −JH S(cos φ+i sin φ)
= −JH H ,
be considered is the orientation of the collective spin 3/2. Σ61 Σ66 S(cos φ−i sin φ) JH S cos θ+nl U

Method. To solve our model, we use the Dynamical 

Σ22 Σ27
 h i
H S cos θ −JH S(cos φ+i sin φ)
Mean Field Theory [26]. First, we define the Green func- = −JH−J .
Σ72 Σ77 S(cos φ−i sin φ) J H S cos θ
tion of the system, which is a 10 × 10 matrix,
(8)
 −1
[G(k, z)] = [H0 (k)] + [Σ(z)] , (4) The local interacting Green function matrix is then cal-
culated through
with z the frequency variable. Then, we coarse-grain it  −1
over the Brillouin zone as −1
[Gnl (z)] = [G(z)] − [Σnl (z)] , (9)
1 X
[G(z)] = [G(k, z)]. (5) where θ and φ are the corresponding angles representing
N
k
the direction of S in the spherical coordinate.
For each Mn site with a given nl , the probability of
In defining [G(k, z)], all the interaction parts of the
Mn spin S having a direction with angle θ with respect
Hamiltonian (all terms other than the kinetic part), are
to the direction of magnetization (which is defined as the
absorbed into a momentum-independent self energy ma-
z−axis) is given by
trix, [Σ(z)], which will be solved self-consistently. Note
that in this algorithm, we need to go over the self-
consistent loops in both Matsubara (z = iωn + µ) and e−Snl (cos θ)
Pnl (cos θ) = , (10)
real frequency (z = ω + i0+ ). Znl
On taking Uu and Ul to be infinity, to some ap-
where
proximation, we forbid the double occupancies in states
|Mn eg upper,↑ i, |Mn eg lower,↑ i, |Mn eg upper,↓ i, and Z
|Mn eg lower,↓ i by throwing them out of our Hilbert space. Znl = d(cos θ)e−Snl (cos θ) (11)
To do this, according to the structure of Hamiltonian ma-
trix in Eqs. (2) and (3), we multiply the weights of all is the local partition function, and
the diagonal elements with indices 1, 2, 6, and 7, and all
+
the corresponding off-diagonal elements connecting any
X
Snl (cos θ) = − ln det[Gnl (iωn )]e−iωn 0 (12)
pair of them by a half. Thus, after obtaining the ma- n
trix [G(z)] from Eq. (5), the effective [G(z)] (let’s call
it [G(z)]eff ) can be obtained by multiplying each of the is the effective action.
 4

We need to average [Gnl (z)] over all possible θ and nl 2

values as Oxygen band Manganese band
Z µ
1.5 3
[G(z)]ave = (1 − hnl i) d(cos θ)P0 (cos θ)[G0 (z)] 1
4 5 6 7 8 9

DOS (eV )
-1
Z Jahn-Teller
Jahn-Teller
+ hnl i d(cos θ)P1 (cos θ)[G1 (z)], (13) 1 DE DE DE
2
Ueff
where hnl i is the average occupation of lower Mneg or-
bital. The new self energy matrix is extracted through Ueff
0.5

[Σ(z)] = [G(z)]−1 − [G(z)]−1

ave . (14)
0
Finally, we feed this new self energy matrix back into the -16 -12 -8 -4 0 4 8 12 16
definition of Green function in Eq. (4), and the iteration Energy (eV)
process continues until [Σ(z)] converges.
After the self-consistency is achieved, we can compute FIG. 2: (Color online) Calculated density of states (DOS).
the density of states as See text for the parameter values used in this figure and the
detailed explanation of the structures of the DOS.
1
DOS(ω) = − ImTr[G(ω + i0+ )] . (15)
π
We can also compute the optical conductivity tensor as three levels by the hybridization between 2p orbitals of
the neighboring oxygen atoms. The structures labeled
πe2 with 4 through 9 result from the eg orbitals of mangene-
 
f (ν, T ) − f (ν + ω, T )
Z
σαβ (ω) = dν × ses. As shown in the figure, there are three mechanisms
h̄ad ω
1 X that split the Mneg states into 6 levels: static Jahn-Teller
Tr[vα (k)][A(k, ν)][vβ (k)][A(k, ν + ω)], (JT) distortion, Coulomb repulsion Ueff ) between lower
N and upper JT-split eg states, the double-exchange (DE)
k
(16) interaction between spins of electrons in the lower and
upper JT-split states and the Mn spins formed by the
where [vλ (k)] = ∂[H0 (k)]/∂kλ is the Cartesian compo- Hund’s coupling among the Mn t2g electrons [28].
nent of the velocity matrix, [A(k, ν)] = [G(k, ω +i0+ )]− Figure 3 shows our calculated optical conductivity
[G(k, ω − i0+ )] /(2πi) the spectral function matrix, and


for T ≈ 194K (> TF M ). The ferromagnetic transi-
f (ν, T ) the Fermi distribution function. Note that the tion temperature for this set of parameters is roughly
dimensional pre-factor πe2 /(h̄ad) [26] with d = 3 is intro- TF M ≈ 160K (based on extrapolation of the mean-field
duced to restore the proper physical unit, since the rest trend). The parameter values for T ≈ 194K are the same
of the expression was derived by setting e = h̄ = a = 1. as those used in Fig. 2. In Figure 3, we demonstrate
In our model, the system is isotropic, and we are only how we tune the the profile of the optical conductivity to
interested in the tranverse components σαα (ω) ≡ σ(ω), achieve the best resemblance with the experimental data
which are equal for all α ∈ {x, y, z}. in Ref. [24]. It is important to note that our model is
Results. Our calculated density of states is shown in not meant to address the dc conductivity, as we already
Fig. 2. The parameter values used for this calculation are anticipate that it cannot form an insulating (or nearly in-
(1) (2)
EJT =0.5eV, Ep =-6.5eV, tMn−O =1.2eV, tMn−O =0.8eV, sulating) phase at T > TF M , possibly due to not incorpo-
tO−O =0.6eV, U =10eV, JH =1.5eV, a=3.9345Å, and T ≈ rating electron-phonon interactions [12, 13, 27]. Rather,
194K (corresponding to β ≡ 1/T = 60 eV−1 ). These our goal is to show how this simple model can capture
parameter values are chosen considering rough estimates qualitatively the general profile of the optical conductiv-
given in other papers [12, 13, 25] and adjusted so as to ity from about 1 eV away from the Drude peak up to 22
give best agreement with the experimental optical con- eV (the energy limit of the experimental data).
ductivity data in Ref. [24]. The DOS is normalized On calculating the optical conductivity from Eq. (16)
such that the integrated area is equal to 8, since in each we introduce an imaginary self energy for the O2p states,
unit cell there are 6 orbitals coming from oxygens and −iΓ, where τ = 1/Γ corresponds to the lifetime of the
effectively 2 from manganeses, considering the restric- O2p states. The red curve in Fig. 3 shows the result if
tion given by relation (6). The chemical potential is we use the self-consistent chemical potential, µ, in Eq.
self-consistently adjusted to satisfy the electron filling of (16). Here, we observe that the resulting profile around
6+(1-x)=6.7, mimicking the situation of La1−x Cax MnO3 the medium energy region (≈ 5-11 eV) does not satis-
for x = 0.3. The structures of the density of states can factorily resembles that of the experimental data in Ref.
be explained as the following. The three peaks labeled [24], since some spectral weight seems to be missing in
1,2,3 result from the fact that there are three oxygen that region. We argue that the reason for this is related
atoms in a unit cell, where the degeneracy is broken into to the fact that our self-consistent chemical potential,
 5

8 6
0.8 9

DOS (eV )
∗

σ(10 Ω cm )
TFM ~ 260 K

-1
µ µ chemical potential = µ, Γ = 0.1 eV 194 K 5 310 K

-1
0.6
7 ∗ 8 129 K
4
225 K
0.4 chemical potential = µ , Γ = 0.1 eV 111 K 175 K

-1
125 K
∗ 97 K 3
0.2 chemical potential = µ , Γ = 0.6 eV
6 7 I II III

3
σ(10 Ω cm )
2
σ(10 Ω cm )

-1
0
-1

-3 -2 -1 0 1 2 3 1 Experiment of La0.7Ca0.3MnO3
5 Energy (eV) 6 TFM ~ 160 K 0
0 4 8 12 16 20

−1
Γ = 0.6 eV
-1

5 Energy (eV)
4

3
3

4
3
3
2
2 I II III
1 1 Model
0 0
0 2 4 6 8 10 12 14 16 18 20 22 0 4 8 12 16 20
Energy (eV) Energy (eV)

FIG. 3: (Color online) Calculated optical conductivity. Main FIG. 4: (Color online) Spectral weight transfer in the optical
panel: The red and blue curves represent the calculated opti- conductivity. Main panel: Results of the model. Exclud-
cal conductivites for different position of chemical potential, ing the region containing the Drude peak (0-1eV), the energy
µ and µ∗ , respectively, using only a small broadening (Γ=0.1 range is divided into three regions I,II, and III. The black
eV). The black curve represents the result using chemical po- curve represents the optical conductivity in the paramagnetic
tential at µ∗ with a bigger broadening (Γ=0.6eV). Inset: Re- phase, while the red, green, and blue curves correspond suc-
gion in the density of states showing how the position of chem- cessively to lower temperatures in the ferromagnetic phase
ical potential is shifted. See text for detailed explanation. [29]. The borders between regions I-II and II-III, denoted by
the blue vertical dashed lines, are defined such that the curves
are crossing at these energies. Inset: A replot of the corre-
sponding experimental data from Ref. [24] for comparison.
µ, does not lie inside a pseudogap as it probably would
if we incorporate electron-phonon interactions. In this
model, we only have a pseudogap that results from the
double-exchange splitting, where µ falls slightly to the for the experimental data in Ref. [24], except that we
right outside of this pseudogap. To remedy the missing exclude the region around the Drude peak from our dis-
of spectral weight, we slightly shift the position of chemi- cussion, since to obtain the correct values of conductivity
cal potential to the left, i.e. from µ to µ∗ as shown in the in that region requires a more accurate description of the
inset of Fig. 3. Using this new chemical potential, µ∗ , the renormalized band structure around the chemical poten-
resulting optical conductivity, shown in the blue curve, tial. If we decrease temperature from the paramagnetic
resembles the experimental data better. This suggests to ferromagnetic phase while keeping all the parameters
that the true chemical potential may actually lie inside constant, we find no significicant change in the optical
a pseudogap similar to the situation as though it lies at conductivity, thus spectral weight transfer does not oc-
µ∗ . (Note that, as long as considering the optical con- cur in this way. If we inspect how Eq. (16) determines
ductivity region about 1 eV away from the Drude peak, the optical conductivity, we see that the change in op-
choosing µ∗ between 0.1 and 0.9 eV, i.e. around the val- tical conductivity may become more significant if either
ley, lead to similar results.) Although the profile of the the spectral function, [A(k, ν)], or the velocity operator,
blue curve is already better than the red one, it still has [vα (k)], changes significantly while temperature changes.
more pronounced stuctures than the actual experimental Within our model this can only be accomodated if we al-
data does. To further tune the calculated optical conduc- low some parameters to depend on temperature by some
tivity to better resemble the experimental data, we find manner. By comparing the structures of optical con-
that the overly pronounced structures can be broadened ductivity and the corresponding DOS profile, it is clear
by enlarging the O2p imaginary self energy upto Γ = 0.6 that the spectral weight in the medium energy region
eV. The result after the broadening, which is shown in comes mostly from transitions from O2p to lower Mneg
the black curve, looks very similar to the experimental states, while in the high energy region from O2p to up-
results shown in Fig. 2(b) of Ref. [24] (replotted in the per Mneg states. This fact may suggest that the hopping
(1) (2)
inset of Fig. 4). This similarity in both magnitude and parameters tMn−O and tMn−O depend on temperature.
profile of the energy dependence may be a good measure Furthermore, since the spectral weight transfer occurs
of the validity of our model. most notably across and below TF M , the temperature
(1) (2)
Now we discuss how the model captures the spec- dependence of tMn−O and tMn−O may be related to spin
tral weight transfer when temperature is decreased from correlation.
T > TF M to T < TF M . First, we divide the energy The actual interplay resulting in such a temperature
range into three regions: I (low:≈1-3eV), II (medium:≈3- dependence is believed to be very complicated, since
12eV), and III (high:>∼12eV), following the division made it may involve orbital effects on the dynamic electron-
 6

phonon coupling and spin correlation. In that regard, 0.8

our present model, which is not an ab initio based

Density of States (eV )

T = 194 K, up

-1
model, cannot naturally capture these temperature ef- T = 194 K, down
fects. Thus, to capture the plausible physics within 0.6
T = 97 K, up
our present model, we turn to the phenomenological ap- T = 97 K, down
proach by parametrizing the totally non-trivial temper-
ature effecs on hopping integrals through magnetization. 0.4
TFM ~ 160 K
In the simplest level, we may assume a linear dependence
(1) (2)
of the hopping integrals tMn−O and tMn−O on the mag-
0.2
netization. Hence, we may write

 
(1) (1) M 0
tMn−O (M )= tMn−O (0) 1 + c1 , (17) -16 -12 -8 -4 0 4 8 12 16
Ms Energy (eV)
 
(2) (2) M
tMn−O (M ) = tMn−O (0) 1 + c2 , (18) FIG. 5: (Color online) Spin-dependent density of states. The
Ms
black and the red curves lie on top of each other as the spin-
where M/Ms is the ratio of magnetization to the satu- up and spin-down components of the DOS are identical in the
rated magnetization, and c1 , and c2 are constants. paramagnetic phase. While, the green and the blue curves
(1)
Using relations (17) and (18), taking tMn−O (0)=1.2 look quite distinct as DOS becomes polarized in the ferro-
(2)
magnetic phase.
eV, tMn−O (0)=0.8 eV, c1 ≈ 0.23, and c2 ≈ -0.35, at
T =97K for which M/Ms =0.357, for instance, we obtain
(1) (1)
that tMn−O is enhanced to be ≈ 1.3 eV, while tMn−O is the ingredients incorporated in our model are adequate
suppressed to be ≈ 0.7 eV. The results for four different to explain the occurrence of spectral-weight transfers in
temperatures are shown in Fig. 4. As shown in the main La1−x Cax MnO3 in the energy range up to 22 eV. In that
panel, our calculation shows that the spectral weight si- respect, one may argue, for instance, that the high-spin
multaneously decreases (increases) in the medium (high) state (S = 3/2) of the t2g -electrons may become unstable
energy region of the optical conductivity as the system as the system is optically excited by high-energy photons.
becomes ferromagnetic [30]. Our calculation also pro- Accordingly, transitions from high to low-spin states, or
duces a less noticeable decrease of the spectral weight in excitations of electrons from t2g to eg levels may occur.
the low energy region as observed in the experimental Our present model does not incorporate those possibili-
data (see the inset). In both the main panel and the in- ties. However, our calculations prove that the model is
set, the black curve represents the optical conductivity in capable to obtain the spectral-weight transfers with good
the paramagnetic phase, while the red, green, and blue qualitative agreement with the experimental results, thus
curves correspond successively to lower temperatures in suggesting that such other contributions may be minor
the ferromagnetic phase. If we define the positions of the or irrelevant.
borders between energy regions I-II and II-III such that The inset of Fig. 6(a) is to show that for 0-1 eV re-
all the curves are crossing at these energies, we obtain gion our result does not agree with the experiment, since
that theoretical values of these energies are similar to the it does not capture the insulator-metal transition. As
experimental ones. Note that the temperatures varied in mentioned earlier, we argue that this is due to our model
the theoretical and the experimental results should not not incorporating the dynamic Jahn-Teller phonons and
be compared quantitatively, since the theoretical TF M is their interactions with electrons, which may be responsi-
about 100 K too small compared to the experimental one, ble to form an insulating gap in the paramagnetic phase.
possibly due to neglecting other possible exchange inter- The incorporation of such terms to improve our present
actions in our model. Despite this, we believe that any model is under our on-going study.
improvement of TF M by such additional terms would not Conclusion. In conclusion, we have developed a model
change the physics presented in this paper. To show the to explain the structures and the spectral weight transfer
difference in the density of states between paramagnetic occuring in the optical conductivity of La1−x Cax MnO3
and ferromagnetic phases, we display the spin dependent for x = 0.3. The key that makes our model work in
DOS for T ≈ 194K and T ≈ 97K in Fig. 5. capturing the structures of the optical conductivity at
To demonstrate further how the spectral-weight trans- medium and high energies is the inclusion of O2p orbitals
fers in our model compare with the experimetal results, into the model.
we display the relative spectral-weight changes for differ- Further, by parametrizing the hopping integrals
ent regions of energy in Fig. 6. Comparing results in through magnetization, our model captures the spectral
Fig. 6(a) and (b), it is clear that for every region of en- weight transfer as temperature is decreased across the
ergy, I (low), II (medium), and III (high), (excluding 0-1 ferromagnetic transition temperature. Our calculation
eV), our calculations give exactly same trends as those based on this phenomenological parameters suggests that
shown by the experimental results. These suggest that the ferromagnatic ordering increases the hopping param-
 7

20 20
eter connecting the O2p orbitals and the upper Mneg or-

∆W/W (%)
80 TFM exp
(a) TFM theor (b)
40
bitals, while simultaneously decreasing the hopping pa-
10
0
10 rameter connecting O2p orbitals and the lower Mneg or-
100 150 200
T(K)
250 300
bitals. Although we have yet to check whether or not
∆W/W(194 K) (%)

∆W/W(310 K) (%)
this scenario works in a more complete model incorporat-
0 0 ing the dynamic electron-phonon coupling, we conjecture
that this may be of important part that contributes to
-10 -10
the mechanism of insulator to metal transition in man-
1-4.6 eV 1-3 eV
ganites.
4.6-11.8 eV 3-12 eV
11.8-22 eV 12-22 eV
-20 0-22 eV -20 0-22 eV

TFM theor TFM exp Overall, our results demonstrate the strength of our
-30
100 120 140 160 180 200
-30
100 150 200 250 300
model that one may have to consider as the minimum
T(K) T(K) model before adding other ingredients in order to prop-
erly explain the insulator-metal transition or other fea-
FIG. 6: (Color online) Relative spectral-weight changes. tures in correlated electron systems such as manganites.
∆W (T )/W from (a) our calculations, and (b) the experimen-
tal results of Ref. [24], for different regions of energy.
Rω ∆W/W
is defined as spectral-weight difference ∆W = ω12 σ(ω, T ) −
Acknowledgement. MAM and AR thank George

Rω
σ(ω, TP M ) dω normalized to W (TP M ) = ω12 σ(ω, TP M )dω,
where in this case TP M is 194 K in (a) and 310 K in (b). Sawatzky and Seiji Yunoki for their valuable comments
Positions of the theoretical and the experimental TF M are in- and suggestions. This work is supported by NRF-CRP
dicated by vertical red dashed lines in each panel. Inset in grant ”Tailoring Oxide Electronics by Atomic Control”
(a) is comparison between ∆W (T )/W from the calculations NRF2008NRF-CRP002-024, NUS YIA, NUS cross fac-
(black filled diamonds) and from the experiments of Ref. [24] ulty grant and FRC. We acknowledge the CSE-NUS com-
(black empty diamonds) for 0-1 eV region. The horizontal puting centre for providing facilities for our numerical
green dashed line in the inset is just to highlight the zero calculations. Work at NTU was supported in part by a
position of ∆W/W . MOE AcRF Tier-1 grant (grant no. M52070060).

[1] S. Jin, T. H. Tiefel, M. McCormack, R. A. Fastnacht, R. (1996).

Ramesh, and L. H. Chen, Science 264, 413 (1994). [13] T. V. Ramakrishnan, H. R. Krishnamurthy, S. R. Has-
[2] For a general review on the structure and transport in san, and G. Venketeswara Pai, Phys. Rev. Lett. 92,
manganites, see M. B. Salamon and M. Jaime, Rev. Mod. 157203 (2004)
Phys. 73, 583 (2001) and references therein. [14] A. J. Millis, Boris I. Shraiman, and R. Mueller, Phys.
[3] S.-W. Cheong and M. Mostovoy, Nat. Mater. 6, 13 Rev. Lett. 77, 175 (1996).
(2007). [15] O. Cépas, H. R. Krishnamurthy, and T. V. Ramakrish-
[4] E. Saitoh, S. Okamoto, K. T. Takahashi, K. Tobe, K. nan, Phys. Rev. Lett. 94, 247207 (2005)
Yamamoto, T. Kimura, S. Ishihara, S. Maekawa and Y. [16] Yu-Li Lee and Yu-Wen Lee, Phys. Rev. B 75, 064411
Tokura, Nature (London) 410, 180 (2001). (2007).
[5] A. Moreo, S. Yunoki, and E. Dagotto, Science 283, 2034 [17] M. Stier and W. Nolting, Phys. Rev. B 75, 144409
(1999). (2007).
[6] A. Nucara, A. Perucchi, P. Calvani, T. Avelage, and D. [18] Rong Yu, Shuai Dong, Cengiz Şen, Gonzalo Alvarez, and
Emin, Phys. Rev. B 68 174432 (2003). Elbio Dagotto, Phys. Rev. B 77, 214434 (2008).
[7] P. W. Anderson and H. Hasegawa, Phys. Rev. 100, 675 [19] Chungwei Lin and Andrew J. Millis, Phys. Rev. B 78,
(1955). 174419 (2008).
[8] P. -G. de Gennes, Phys. Rev. 118, 141 (1960). [20] J. Zaanen, G. A. Sawatzky, and J. W. Allen, Phys. Rev.
[9] M. Quijada, J. Černe, J. R. Simpson, H. D. Drew, K. Lett. 55, 418 (1985).
H. Ahn, A. J. Millis, R. Shreekala, R. Ramesh, M. Ra- [21] M. B. J. Meinders, H. Eskes, and G. A. Sawatzky, Phys.
jeswari, and T. Venkatesan, Phys. Rev. B 58, 16093 Rev. B 48, 3916 (1993).
(1998). [22] W. G. Yin, D. Volja, and W. Ku, Phys. Rev. Lett. 96,
[10] V. A. Amelitchev, B. Güttler, O. Yu. Gorbenko, A. R. 116405 (2006).
Kaul, A. A. Bosak, and A. Yu. Ganin, Phys. Rev. B 63, [23] Philip Phillips, Rev. Mod. Phys. 82, 1719 (2010).
104430 (2001). [24] A. Rusydi, R. Rauer, G. Neuber, M. Bastjan, I. Mahns,
[11] Pengcheng Dai, J. A. Fernandez-Baca, E. W. Plummer, S. Müller, P. Saichu, B. Schulz, S. G. Singer, A. I. Licht-
Y. Tomioka, and Y. Tokura, Phys. Rev. B 64, 224429 enstein, D. Qi, X. Gao, X. Yu, A. T. S. Wee, G. Stry-
(2001). ganyuk, K. Dörr, G. A. Sawatzky, S. L. Cooper, and M.
[12] A. J. Millis, R. Mueller, and Boris I. Shraiman, Phys. Rübhausen, Phys. Rev. B 78, 125110 (2008).
Rev. B 54, 5389 (1996) and Phys. Rev. B 54, 5405 [25] W.E. Picket and J.D. Singh, Phys. Rev. B 53, 1146
 8

(1996).
[26] Antoine Georges, Gabriel Kotliar, Werner Krauth, and
Marcelo J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996).
[27] Claude Ederer, Chungwei Lin, and Andrew J. Millis,
Phys. Rev. B 76, 155105 (2007).
[28] If the double-exchange (DE) interactions were not there,
each of the pairs of 4-5, 6-7, and 8-9 would merge into
one. The 4-5 pair corresponds to the lower part of the
JT-split states, while the 6-7 and 8-9 pairs correspond to
the upper part. The 6-7 and 8-9 pairs are further split by
the Coulomb repulsion (U) between lower and upper JT-
split states. Namely, when the lower JT-split level (4-5)
is unoccupied, the upper level is given by 6-7. Whereas,
if 4-5 is occupied, the corresponding upper level is given
by 8-9.
[29] If one shifts µ∗ but to remain around the valley, or
changes the value of Γ from 1 to 6 eV, the effect of
weight transfer is still captured with the crossover be-
tween regions of medium and high energy being relatively
unchanged (within 1 eV), while the amount of weight
transferred may change by not more than one hundred
percent.
[30] Since the self-consistently computed chemical potentials
(µ) we found for the PM and FM phases are almost the
same, we choose to define µ∗ in PM and FM phases by
shifting their corresponding µ to the left by the same
amount. This way, µ∗ for PM and FM phases are pretty
much the same.