In many insulators, an added electron or hole self-traps
in a localized state called a small polaron. In the ideal case of
a perfectly periodic insulator, the small polaron forms a very narrow band.
This happens because equivalent small polaron solutions occur at every
lattice site. The hopping integral to go from a small polaron on
site 1 to a small polaron on site 2 has two factors. There is the
purely electronic factor, called *t*, which is the same as it would
be in an undistorted crystal. By itself, this would give a band-width
12t (for a simple-cubic lattice in three dimensions.) There
is also the "Huang-Rhys" factor which accounts for the fact that the distortion
must also move from site 1 to site 2. The zero-temperature Huang-Rhys
factor is the overlap <1|2> of the vibrational wavefunctions for the
ground vibrational state when the electron is on site 1 and when the electron
is on site 2. It is exponentially narrowed, of order exp(-D^{2}/<u^{2}>)
where <u^{2}> is the mean square zero point amplitude of oscillation,
and D is the distortion amplitude. As temperature increases, the
Huang-Rhys factor increases because thermal occupancy of vibrations in
the ground state have greater overlap from one site to the next than do
the ground state vibrations. This factor contains the cross-over
from quantum tunneling to classical thermal activation over the barrier.

The pictures above show schematically how polaron formation and hopping appear in the optical conductivity s(w). In the top part of the left panel, a two-site model is illustrated with no electron present. The two empty levels on adjacent atoms hybridize forming a higher antibonding level |a> and a lower bonding level |b>. When an electron is added, there are two equivalent solutions, |L> and |R>, shown in the middle left and right parts of the left panel. In |L>, the electron is on the left atom, accompanied by a deformation which lowers its energy; in |R>, the electron is on the right atom, with its own deformation. In the models we use, the deformation parameter is an oxygen which lies between the two electron sites. Such a deformation raises the energy of the unoccupied electron site while it lowers the energy of the occupied electron site. The bottom part gives two alternate views of the state |L*> which forms after an external photon has excited the electron out of the bound state on the left atom into the high-lying state on the right atom. The left view, which is purely adiabatic, meaning that the oxygen oscillator has an infinite mass and responds infinitely slowly, the excited state is a single high-energy level. The corresponding s(w) is a delta function at the excitation energy of the static problem. The right view (still in the left panel) is non-adiabatic, meaning that the oxygen mass is not infinite, and the oxygen vibrational frequency is not zero. Then even though the atoms have had no time to move, nevertheless we can regard the state |L*> as a deformed version of the fully relaxed state |R>. The state |L*> can be expanded in terms of vibrational excitations of the state |R>. The biggest terms in the expansion are states with amplitudes of vibration comparable to the distortion magnitudes. However, there is a non-zero contribution to |L*> even from the ground state |R>. The right panel shows the same story for the excitation, in the conventional configuration coordinate form. The initial vibrational wavefunction |L> is shown as a Gaussian. After optical excitation, the electron energy is increased (a vertical transition from the left level shown as the left parabola, to the right level shown as the right parabola.) The vibrational wavefunction has not had time to change and is still the Gaussian centered on the configuration appropriate to a left electron. However, since the electron is on the right, the vibrational wavefunction is a linear combination of the states of the right well, indicated as oscillator levels of the right well. The overlap is greatest with an excited state whose classical vibrational turning point sits right above the origin of the left well. The result for s(w) is a sequence of delta functions, the largest being at the adiabatic excitation energy, but with sidebands (separated by the vibrational quantum hf) all the way down to zero energy (corresponding to dc band conductivity of the polaron.) These are "Franck-Condon" sidebands. Because of the complex and continuous nature of the vibrational spectrum, individual phonon lines are not expected to be resolved, and the vibrational broadening should appear as a Gaussian envelope of the multi-vibrational processes.

When the same ideas are applied to a crystal, the
natural simple assumption is that the dipole transitions <L|p|L*> take
the bound polaron state to continuum states. However, we have found
in various models that other possibilities can alter the picture.
The simplest 3d model we have examined is the "Rice-Sneddon" model for
BaBiO_{3}. The nominal Bi^{4+} ions occupy a simple
cubic lattice. The energy of the Bi *6s* orbital (which is singly
occupied on average) is sensitive to the Bi-O bond lengths. If the
bond lengths increase locally around a Bi, the energy of the *s* orbital
is lowered, encouraging occupancy by two electrons. Changes of bond
lengths occur in equal and opposite pairs because only the oxygen atoms
move. Therefore the neighbors of the Bi site with two electrons are
Bi sites with bonds locally compressed, the *s* orbital raised in
energy, and the electron occupancy depleted. This model describes
the "disproportionation" transition 2Bi^{4+} --> Bi^{3+}+
Bi^{5+}. This transition occurs at high temperature (800K
or higher) and accounts for the insulating behavior of BaBiO_{3}.
Removing one electron leads to partial local relaxation of the stretched
bonds. The hole then is bound at one site, forming a small polaron,
and spending a little time on neighboring sites. Our numerical calculations,
plus exact strong-coupling calculations, show that there are also bound
electron excitations in the vicinity of the hole polaron. In other
words, a potential well self-organizes to bind a hole polaron, and as an
accidental biproduct, creates bound unoccupied electron states as well.
There are six such states, three being dipole active, and in principle
they affect the optical conductivity. However, there are no other
bound hole states besides the one which the hole polaron sits in, so the
optical excitations of the hole are not strongly affected by the electron
bound states. Our paper [1] shows the extra bound electron states
explicitly, but does not attempt a calculation of the polaron-induced mid-infrared
absorption. However, another paper [2] does calculate approximately
the optical absorption of pure BaBiO_{3} which is altered from
that of a band insulator by self-trapping of excitons.

The situation in the antiferromagnetic manganites
[3] LaMnO_{3} and CaMnO_{3} is different and more complex.
The relevant orbitals are now the *E _{g}* doublet (

LaMnO

**References**

1. I. B. Bischofs, V. N. Kostur, and P. B. Allen, “Polaron and Bipolaron Defects in a Charge Density Wave: a Model for Lightly Doped BaBiO3”, cond-mat/0108089 (submitted to Phys. Rev. B).

2. P. B. Allen and I. B. Bischofs, “Self-Trapped Exciton Defects in a Charge Density Wave: Electronic Excitations of BaBiO3”, cond-mat/0108090 (submitted to Phys. Rev. B).

3. Y.-R. Chen, V. Perebeinos and P. B. Allen, "Polaronic
Signatures in Mid-Infrared Spectra: Prediction for LaMnO_{3} and
CaMnO_{3}," (submitted to Phys. Rev. B).

Modified 10 November, 2001

back to P. B. Allen's
homepage