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(-D²/<u²>) where <u²> 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 BaBiO₃.  The nominal Bi⁴⁺ 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 2Bi⁴⁺ --> Bi³⁺+ Bi⁵⁺.  This transition occurs at high temperature (800K or higher) and accounts for the insulating behavior of  BaBiO₃.  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 BaBiO₃ which is altered from that of a band insulator by self-trapping of excitons.

    The situation in the antiferromagnetic manganites [3] LaMnO₃ and CaMnO₃ is different and more complex.  The relevant orbitals are now the E_g doublet (x²-y² and 3z²-r²) which is occupied by one electron in LaMnO₃ and unoccupied in CaMnO₃.  Hopping onto neighboring sites can be prevented if the spin 3/2 Mn T_2g "core" is misaligned.  CaMnO₃ has simple antiferromagnetic "G" structure with each up spin on one sublattice, each down spin on another, and each first neighbor oppositely aligned.  An added electron in  creates a spin-flip on one site, plus local oxygen bond distortions.  The electron is confined by more distant spin misalignment, and has 7 sites (with 14 orbitals) to visit.  This 14-dimensional space has 14 local eigenstates, the ground state (spin-lattice polaron) and 13 excited states.  Some of these excitations are lower in energy than the energy of a distant Mn site, and three have T_1u symmetry and are dipole active.   Therefore the polaron-induced optical activity s(w) in CaMnO₃ is is predicted to be fairly simple, but different from the prototype polaron model mentioned at the beginning.  Unlike the simple prototype, there is little chance of any w-->0 activity since the polaron is prevented by spin conservation from moving to its first neighbor site.  But similar to the simple prototype, the optically generated local excitation is a superposition of vibrational excitations of a relaxed excited state, so Franck-Condon broadening is expected.
LaMnO₃ is an interesting intermediate case.  The ground state of the pure material is anisotropic.  A cooperative Jahn-Teller distortion sets in around 750K which lowers the symmetry from cubic by creating x-y layers where E_g orbitals are directed alternately toward x and y.  These layers have ferromagnetic spin orientation, but the adjacent layers couple spins antiferromagnetically.  When hole-doped, a small polaron forms.  The potential well created by oxygen distortion is strong enough to bind additional (empty) hole states.  These alter the shape of the predicted s(w) spectrum.  In an untwinned single crystal, the spectrum will be anisotropic.  For both LaMnO₃ and CaMnO₃, our paper [3] makes detailed predictions of  s(w) from the two-orbital model with Jahn-Teller and magnetic terms in the Hamiltonian.

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 LaMnO₃ and CaMnO₃," (submitted to Phys. Rev. B).

Modified 10 November, 2001
back to P. B. Allen's homepage