**2.7 Discussion and Conclusion**

**3.1.1 Experiments on the Critical Behavior of Perovskite Manganites 56**

Perovskite manganites (R_{1−x}A_{x}MnO_{3}) (cf. Sec. 1.6.1) have been investigated ex-
tensively for decades due to their exotic structural, electronic, and magnetic behav-
ior and due to the existence of colossal magnetoresistance (CMR)—the dramatic
reduction of resistivity in a magnetic field[234,104,148,14,197]. In these compounds,
different choice of *R* and *A* have different atomic sizes and therefore produce
varying stresses on the Mn-O-Mn bond. It was shown that the average *R/A* size
in terms of ionic radii, expressed by the tolerance factor *f*^{[148,197]}, affects their
critical behavior. To investigate the PM-FM phase transition, various experiments
have been performed based on the DC magnetization^{[68]}, AC suceptibility^{[162]}, and

3.1 Introduction

specific heat^{[117]}measurements followed by scaling analysis with Arrot plot, mod-
ified Arrot plot (MAP)^{[117,92]}, Kouvel-Fisher (KF) formalism, and critical isotherm
(CI)[155,185,52,162] analysis. These experiments clearly show that a change either in
the doping level (*x*) or in the tolerance factor (*f*) lead to different critical expo-
nents.

In fact, although initial experimental investigations^{[142,68]} suggested inconclu-
sive results as regards the universality classes, a number of subsequent experimen-
tal studies on some samples[117,233,52,163] showed that the critical exponents were
close to those of the tricritical mean-field theory (*β* = ^{1}_{4}, *γ* = 1, and *δ*= 5)^{[94]}.
For instance, in a polycrystalline La_{0.6}Ca_{0.4}MnO_{3}, the specific heat and magne-
tization measurements with MAP and CI analysis^{[117]} yield *α*= 0*.*48±0*.*06, *β* =
0*.*25±0*.*03,*γ*= 1*.*03±0*.*05, and *δ*= 5*.*0±0*.*8. The magnetization data for polycrys-
talline La_{0.1}Nd_{0.6}Sr_{0.3}MnO_{3} yields two sets of results, namely, *β* = 0*.*248±0*.*006,
*γ*= 1*.*066±0*.*002with MAP and*β*= 0*.*257±0*.*005,*γ*= 1*.*12±0*.*03, and*δ*= 5*.*17±0*.*02
with KF method and CI analysis. In polycrystalline Nd_{0.67}Sr_{0.33}MnO_{3}, the mag-
netization data analysed with MAP yields *β* = 0*.*23±0*.*02, *γ* = 1*.*05±0*.*03, and
*δ*= 5*.*13±0*.*04.

Although the above mentioned samples exhibit tricritical mean-field exponents,
there are other compounds[68,92,118,162,220,176,167,185,53,69,154] whose critical expo-
nents deviate from those of the tricritical mean-field values. For instance, in single
crystal La_{0.7}Sr_{0.3}MnO_{3}, the critical exponents obtained via Arrot plot and CI anal-
ysis are *β*= 0*.*37±0*.*04, *γ* = 1*.*22±0*.*03, and *δ* = 4*.*25±0*.*2^{[68]}. In polycrystalline
Pr_{0.5}Sr_{0.5}MnO_{3}, the MAP analysis yields*β*= 0*.*443±0*.*002,*γ*= 1*.*339±0*.*006, while
the KF method as well as CI analysis give*β*= 0*.*448±0*.*009,*γ*= 1*.*334±0*.*010, and
*δ*= 3*.*955±0*.*001^{[185]}.

In a number of studies, a change in the doping level *x* in the same compound

led to different critical exponents[68,92,117,162,233]. For instance, different sets of
critical exponents were obtained for La_{1−x}Ca_{x}MnO_{3} when *x* = 0*.*2^{[92]} and *x*=
0*.*4^{[117]}. Similar behavior was noted for La_{1−x}Sr_{x}MnO_{3} when *x*= 0*.*3^{[68]} and *x*=
0*.*125^{[162]}. This was also observed in Ref.^{[233]} for a different compound, namely,
Nd_{1−x}Sr_{x}MnO_{3} with*x*= 0*.*33and*x*= 0*.*4.

The above mentioned experimental findings clearly indicate that different criti-
cal exponents are obtained with different choices for R and/or A as well as with dif-
ferent levels of doping,*x*. Although in some of these experimental works[68,162,52],
the critical exponents were compared with those of the existing theoretical mod-
els, namely, mean-field, tricritical mean-field, 3D Ising, and 3D Heisenberg, these
existing models were unable to reproduce their widely varying critical exponents.

**3.1.2** **Studies on Microscopic Models**

An essential microscopic mechanism in perovskite manganites is the coupling of
spin and lattice degrees of freedom (cf. Sec.1.6.1). The electron-phonon JT cou-
pling leading to lattice distortions are known to be important as indicated by a
number of theoretical[151,188,148,149,6] and experimental studies[99,200,43,24,26]. A
pronounced variation of electrical resistivity and a large shift of *T*_{c} after isotope
exchange (^{18}O for ^{16}O) indicate a strong spin-lattice coupling^{[257,14]}. The lattice
distortions are shown to be directly related to the imposed strain due to pertur-
bations induced via changes in*R*, *A*, and*x*[148,149,26,200,105]. A quantitative analy-
sis^{[149]}predicted a dramatic sensitivity of material properties to strain, particularly
the shifting of*T*_{c} with strain. Different strain modes are shown to evolve depend-
ing on whether the perturbation is due to the size distribution of *R/A*atoms or to
the change in the doping concentration *x*. Elastic interactions are shown to play
an essential role in the formation of superstructures^{[115,116]} and texturing^{[6]} ob-

3.1 Introduction

served in perovskite manganites. In addition to JT distortions, another essential
microscopic mechanism is the spin-spin DE interaction[255,148,197]. The microscopic
models for the magnetic (and electrical) properties of perovskite manganites are
based on the framework of these two mechanisms which provide a satisfactory ex-
planation for the origin of CMR and the change in resistivity as the system passes
through the PM-FM transition. They also explain satisfactorily the systematic vari-
ation of*T*_{c}with doping[99,148,188]. A number of Monte Carlo (MC) simulations have
been performed[32,158,157,156]on a DE model Hamiltonian^{[32]}

*H*=−*J*^{X}

*i,j*

q1 +**S**_{i}*.***S**_{j} (3.1)

for the investigation of the static critical behavior. These simulations yield *ν* =
0*.*6949±0*.*0038, *β* = 0*.*3535±0*.*0030, *γ* = 1*.*3909±0*.*0030 in Ref.^{[32]}, *β* ≈0*.*365 in
Refs.^{[158,156]}, *β* = 0*.*36±0*.*01 in Ref.^{[157]}, and *ν* = 0*.*686±0*.*010, that are close
to those of the 3D Heisenberg model (with SR interaction). This suggests that
the DE and 3D Heisenberg models^{[36,91]} belong to the same universality class.

Although a few perovskite manganite samples^{[162,167]} are found to have critical
indices near to the 3D Heisenberg model predictions, a vast majority of sam-
ples[142,68,92,117,220,185,233,52], as discussed above, exhibit a widely varying sets of
critical indices including the tricritical mean-field exponents. Thus, although the
DE is widely accepted as one of the key mechanisms for CMR in perovskite man-
ganites, models involving the DE interaction can not capture their widely varying
critical behavior near the PM-FM phase transition.