Polarons in Jahn-Teller Crystals: IntrinsicDifference between eg and t2g Electrons
Takada Group
About a decade ago, Tokura's group made aninteresting comparative study of the effective mass m*
between eg and t2g electrons by measuring thelow-temperature electronic specific heat Ce(T) of themanganese oxides La1-xSrxMnO3 as well as thetitanium ones La1-xSrxTiO3 [1]. They have found thatthe t2g (titanium) system has a larger T-linear
coefficient in Ce(T) than the eg (manganese) one bymore than a factor of two, irrespective of x, implyingthat t2g electrons may be intrinsically much heavier
than eg ones.
Since Mn3+ and Ti3+ ions are, respectively,
considered as typical E e and T t Jahn-Teller (JT)centers, we may anticipate that the above experimental
result directly reflects an inherent difference in themathematical structure of JT centers, namely, either
E e or T t. Motivated by this anticipation and as awork subsequent to a previous one of our group for
the E e polarons [2], we have made a comparative
study of the polaronic effective mass m* between theE e and the T t JT polarons. A comparison with theconventional Holstein polarons is also made.
If the electron-phonon (el-ph) coupling is weak, it isstraightforward to determine the ratio of m* to the bareband mass m by resorting to second-order perturbation
calculation, from which we obtain m*/m as 1+2 forboth JT polarons with
the nondimensional el-phcoupling constant in its conventional definition.
Exactly the same mass ratio can be reproduced in theHolstein model, if we redefine in the model in an
appropriate way. Thus we see that there is no essential
difference in the enhancement of m* among those
three polarons in the small- limit.
In the limit of, a polaron will be completely
localized at a single site, implying m*/m = . For a
finite but very large , the localized polaron will begin
to hop between sites, but the hopping in this case is avery rare event. Thus physics connected with such a
hopping can be well captured by just considering a
two-site system. In fact, by implementing the two-site
calculation by exact diagonalization, we can obtain ageneric feature of m*/m in the strong-coupling (and/or antiadiabatic (the electron hopping
integral t much smaller than the local optic phonon
energy 0) region.
The results of the two-site calculation are shown inFig. 1. We note that the Holstein polaron (thedotted-dashed curve) is characterized analytically by
the relation of m/m*=e-2 , while the E e polaron (thedashed curve) by m/m*=( /2)1/2e- [2], indicating themuch lighter effective mass of the latter system.
Physically the mass enhancement is brought about
by the virtual excitation of local phonons in thosemodel Hamiltonians describing the coupling of anelectron with local optic phonons. In the Holsteinmodel no restriction is imposed on exciting multiple
phonons, but in the E e JT model there is a severe
restriction due to the existence of a conserved quantity
intimately related to the SO(2) rotational symmetry in
the pseudospin space representing the twofold
degenerate orbitals.

Fig. 1. Inverse of the mass enhancement factor, m/m*, as afunction of for the T t (solid curve) and the E e (dashedcurve) JT polarons in comparison with the Holstein one(dotted-dashed curve). All the results are obtained in thestrong-coupling and/or antiadiabatic region.

As for the T t model, no corresponding symmetry
exists in the pseudospin space, explaining its
difference in m/m* from the E e model in Fig. 1. Inorder to understand its difference from the Holsteinmodel, however, a more detailed analysis is needed. Itis well known that the adiabatic potential energy
surface for the T t model contains four equivalentwells for sufficiently large , but the wells are notisotropic; the vibrational t-mode splits into an a1-mode
and two e-modes (i.e., ta1 e). This fact indicatesthat the T t model is a combination of the Holstein
and the E e models, explaining the reason why theresult of m/m* in this model (the solid curve in Fig. 1)
comes between those of the Holstein and the E emodels.
Our model is too primitive to make a quantitative
comparison with experiment, but we find that, at leastin the strong-coupling and/or antiadiabatic region, thedifference in the behavior of m* as a function of canbe ascribed to the presence/absence of rotational
symmetry in the pseudospin space representing theinternal mathematical structure of each JT center.
References[1] T. Okuda, A. Asamitsu, Y. Tomioka, T. Kimura, Y.
Taguchi, and Y. Tokura, Phys. Rev. Lett. 81, 3203
(1998).
[2] Y. Takada, Phys. Rev. B 61, 8631 (2000).
[3] Y. Takada and M. Masaki, J. Molecular Structure
838, 207 (2007).
Authors
Y. Takada and M. Masaki