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Polaronic Signatures in Mid-Infrared Spectra: Prediction for LaMnO3 and CaMnO3

Hole-doped LaMnO3 and electron-doped CaMnO3 form self-trapped electronic states. The spectra of these states have been calculated using a two orbital (Mn eg Jahn-Teller) model, from which the non-adiabatic optical conductivity spectra are obtained. In both cases the optical spectrum contains weight in the gap region, whose observation will indicate the self-trapped nature of the carrier states. The predicted spectra are proportional to the concentration of the doped carriers in the dilute regime, with coefficients calculated with no further model parameters.


I. INTRODUCTION
Qualitative discussions of optical absorption from polarons [1] have often been given. For the heavily doped manganites such as La 0.7 Ca 0.3 MnO 3 , polarons form at temperatures T ≥ T c ≈ 250K where ferromagnetic order is lost. Dramatic thermal shifts of optical conductivity [2] agree [3] with the qualitative polaron picture [4]. In the lightly doped end-member compounds LaMnO 3 and CaMnO 3 , polarons must exist at low temperature in the spin and orbitally ordered states. We have previously [5][6][7][8][9] calculated the properties of these polarons in the Mn e g Jahn-Teller (MEJT) [10] model. Here we point out that the polarons have a rich spectrum of local excited states. We provide a quantitative calculation of the corresponding polaron-induced mid-infrared absorption. The transitions between these local levels are broadened by the Franck-Condon effect since each local level prefers its own rearrangement of the lattice. This is a general phenomenon expected in many insulators and may account for mid-infrared absorption in other systems as well [11].
In the usual theory of small polaron absorption [1,12], an electron is trapped at a single site in a non-degenerate orbital. The photon permits a transfer of the electron to a first neighbor site. Direct transfer to the gound state first neighbor polaron (corresponding to dc conductivity) is exponentially suppressed by unfavorable vibrational overlap ("Huang-Rhys") factors. The largest vibrational overlap is for transitions into states whose maximum amplitude overlaps with the zero-point motion of the distorted polaron on the original site. A Gaussian envoelope of Franck-Condon vibrational sidebands is predicted. The cases considered here differ in three ways. (1) Our electron levels are doubly degenerate before distortion. (2) Hund's energy is large which blocks hopping unless the neighbor atom has spin ferromagnetically aligned. (3) Our polarons are not strictly confined to a single site, and consequently have localized excited states in the self-organized potential well. These complications are not all unique to manganites, and should affect polaronic absorption in other materials as well.
Pure LaMnO 3 has a single Mn d electron in the e g dou- . Therefore there is a cooperative Jahn-Teller effect resulting in orbital ordering. The peak [13,14] near 2 eV in σ(ω) is interpreted [14,8] as e g → e g transitions across the Jahn-Teller (JT) gap. Pure CaMnO 3 has no e g states occupied, and σ(ω) peaks at 3.6 eV [14] due to excitations across the gap between occupied (O 2p and Mn t 2g↑ ) states and empty (Mn e g↑ ) states. These materials are insulating antiferromagnets (AF) with the nice property of being continuously dopable. Here we focus on two cases, x = 1 − ǫ (electrondoped CaMnO 3 ) and x = ǫ (hole-doped LaMnO 3 .) Both materials are insulators when pure, and remain so when lightly doped because of polaron formation. We use the model Hamiltonian [6] similar to the one used by Millis [10], except that we have U = ∞ for LaMnO 3 . For CaMnO 3 we include also the t 2g AF exchange term H J and do not need the Hubbard U term for dilute doping. Two e g orbitals per Mn atom are kept to fully represent the symmetries of the orbitals and the crystal, and to drive JT distortions. In the hopping term there is only one parameter, namely, the nearest-neighbor t = (ddσ) integral from Slater-Koster two-center theory.
In the electron-phonon interaction, we include both the oxygen breathing vibrations Q 1 and the local E g -type oxygen motions (Q 2 , Q 3 ) which give the Jahn-Teller distortion.
The indices α and β run though orbitals (ψ 2 , ψ 3 ). H L is the lattice elastic energy of oxygen displacement along Mn-O-Mn bond directions with an Einstein restoring force. Note that the JT modes Q α ( l) and Q α ( l ′ ) are not independent when Mn sites l and l ′ are first neighbors, since the intervening oxygen is shared by both Mn atoms (this is called the "cooperative" phonon effect.) At the LaMnO 3 end of the phase diagram, the dominant term of the Hamiltonian after electron-electron repulsion is H ep . Instead of delocalizing in a conduction band, holes in lightly doped LaMnO 3 lower their energy more by self-trapping with local oxygen rearrangement, forming "anti-JT" small polarons [5]. At the CaMnO 3 end, at the cost of AF exchange energy, spin-flipping in the AF background allows the doped-in electron to lower its energy by delocalization. This spin-polaron effect competes with a localizing JT polaron effect from H ep . The competition determines the size of the spinlattice polaron [6].
In this work we calculate the excitation spectrum of the spin-lattice polaron in CaMnO 3 by Born-Oppenheimer approximation, and for LaMnO 3 we calculate the excitation spectrum of the self-trapped hole in a Cho-Toyozawa [15] approximation. Our non-adiabatic solutions allow us to calculate the Franck-Condon broadened optical excitation spectra of these lightly doped materials and predict new spectral features that should be experimentally visible.

II. SPIN-LATTICE POLARON SPECTRUM IN DOPED CAMNO3
Undoped CaMnO 3 has Mn 4+ (t 3 2g ) ions with S = 3/2 on an approximately simple cubic lattice, and bipartite (G type) AF order below T N = 125K. One single dopedin electron will align with the S = 3/2 core spin by Hund coupling. Hopping to first neighbors is then blocked by the AF order. To reduce energy by delocalization, one Mn S = 3/2 atom flips. Then the nearby oxygens distort, to lift the e g degeneracy. This gives a seven-site spinlattice polaron [6]. The parameters were chosen to be |t|/JS 2 = 158, and Γ = g 2 /K|t| = 0.25. Each of the seven Mn atoms provides two e g orbitals to form a 14dimensional Hilbert space in which the doped-in electron is partially delocalized.
The ground state of this polaron is calculated in adiabatic approximation. With the local oxygens distorted in the optimized pattern, the ground state lies in the gap with energy lowered from the e g band by −2.13|t|. The remaining 13 excited states contain three dipole-allowed optical excitations. By re-optimizing the oxygen distortion pattern, these states (with original energies 1.20|t| and 1.93|t| above the e g band) can have their energy lowered to sit below the e g band by −0.67|t|. When the electron is optically excited to one of these states, oxygens do not have time to rearrange to a new optimal pattern, and therefore are left in a combination of vibrationally excited states with respect to the re-optimized oxygen pattern of the electronic excited state. The Franck-Condon principle therefore applies and gives vibrational sidebands. The energy level diagram is shown in Fig. 1.
To examine the optical spectrum of this seven-site polaron surrounded by 36 oxygens, we start from one of the three symmetry-equivalent electronic ground states ψ 0 ( r, u 0 ) (e.g., θ = 0 [6]) with local lattice distortion u 0 = (u 0,1 , u 0,2 , ..., u 0,36 ) which minimize the energy E 0 ( u) of the ground state. For distortions u near u 0 , the vibronic eigenstates associated with the electron ground state are approximated by the Born-Oppenheimer product: where φ i,ni (u i − u 0,i ) is the n i phonon state of the harmonic oscillator centered at u 0,i . The usual approximation is made here that ψ 0 (r, u) can be accurately represented by ψ 0 (r, u 0 ). The true electronic ground state varies slowly with u, and also depends on the direction u − u 0 of deviation from u 0 , but corrections are small at low temperatures where the initial state is not vibrationally excited. Then the adiabatic ground state energy sheet E 0 ( u) depends quadratically on ( u i − u 0,i ) and we treat the oxygens as 36 independent Einstein oscillators, with energies Among the 13 polaronic excited states, there are only three states that couple to the ground state by light. They are |ψ z >≡ (|ψ 3 ,ẑ > −|ψ 3 , −ẑ >)/ √ 2 and two other states which are rotations of |ψ z > tox andŷ directions. Here |ψ 3 ,ẑ > denotes the ψ 3 orbital of the Mn atom sitting immediately above the central Mn. These three excited states have their own optimal lattice distortion patterns u z , u x and u y . Starting from u 0 , these electronic states remain eigenstates along the paths towards u z , u x and u y in the lattice distortion space. Therefore in these chosen directions, the vibronic eigenstates associated with the electronic excited states have the product form, e.g.: Consider the optical excitation from Ψ 0 to Ψ z for example. The T = 0K spectrum is given by The matrix element has both an electronic part and a phonon part. The electronic part involves the momentum matrix element < ψ z |p z |ψ 3 , 0 >, where |ψ 3 , 0 > denotes the ψ 3 orbital of the central Mn atom. This matrix element is < ψ z |p z |ψ where m is the electron mass and a the lattice constant. This result can be derived by looking at FM spin order, and requiring that the corresponding FM Bloch eigenstates | k > of H t (with energy ǫ( k)) should have velocity < k|p z /m| k >= ∂ǫ( k)/∂hk z . The phonon part of the matrix element is |< n ′ , u z |n, u 0 >| 2 , where the initial lattice vibrational state has n vibrational quanta relative to the oxygen equilibrium positions u 0 . The lattice vibrational state is unchanged during the electron excitation, but must be expanded in the new basis (denoted by quanta n ′ ) around the new oxygen equilibrium positions u z . We consider only zero temperature optical conductivity, which restricts the initial vibrational quantum numbers n to be zero. The non-zero overlaps of this initial phonon state with the new phonon basis states of the displaced lattice give a spectrum with a Gaussian profile of delta function peaks of the multi-phonon transitions accompanying the electron exitation. The phonon frequency is taken to bē hω = 0.084 eV. This is a little higher than the stretching frequency reported from Raman data [16], but the discrepancy does not affect our conclusions. The hopping integral t is taken from density-functional calculations [17] to be t = −0.75 eV. The delta functions are broadened into Gaussians of width √ n ′ γ where γ = 0.014eV and n ′ is the total vibrational quantum. The results are shown in Fig. 2.  [6]. There are two other possible ground states with orientations θ = 2π/3, and θ = 4π/3, which permute the directionsx,ŷ andẑ. Therefore we show the average spectrum. The absolute value of σ(ω) is shown for 10% electron doping.

III. POLARONIC SPECTRA IN HOLE DOPED LAMNO3
In LaMnO 3 the electron-phonon term H ep stabilizes an orbitally ordered ground state via a cooperative JT oxygen distortion. We use the same model as in CaMnO 3 , with different filling, and fit it to the JT ground state. We previously used this model to predict the resonant multiphonon Raman cross-section [9] and the angle resolved photoemission spectrum including the Franck-Condon broadening [7]. The photoemission process creates a sudden hole without time for oxygen relaxation. Thus it is in a vibrationally excited state. In the doped compound a hole is initially present in its ground vibrational state in a locally distorted oxygen environment. The zeroth order electronic many-body wave function has a hole h in the orbital at the origin 0 (taken arbitrarily to be an x-oriented orbital) in an otherwise perfectly correlated state with one E g electron per atom.
Here the orbitals ψ X and ψ Y are orthogonal x− and y−oriented orbitals (ψ 2 ± ψ 3 )/ √ 2. At the next level of approximation the state |h0 > is coupled to 18 other states. Six of these states keep the hole at the origin but have an orbital defect, or "orbiton" o on one of the surrounding Mn sites ±l wherel isx,ŷ, orẑ, Six more states have the hole moved to the neighboring Mn site, with no orbital defect, The last six states have the hole on a neighbor and leave a misoriented |Y > orbital at the origin, The Hamiltonian H MEJT becomes a 19 × 19 matrix in this truncated Hilbert space.
In the adiabatic solution of the zeroth order hole ground state |h0 >, the neighboring oxygens in ±x, ±ŷ, ±ẑ directions are distorted to positions given in Table I, row 1, which minimize the polaron energy [5].
With the oxygen pattern frozen in these positions, the diagonal element < h0|H ep |h0 > is set to zero. Nonzero Q 3 distortions on the neighboring sites couple the state |h0 > to the six orbital excitations |h0, o ±l >. Row 2 of Table I gives the diagonal elements < h0, o ± ℓ|H ep |h0, o ±l > and row 3 gives the corresponding offdiagonal elements. The electron hopping term H t couples the state |h0 > to states with the hole moved to the one of the six neighboring sites, both without (Eq. 9) and with (Eq. 10) an orbital defect. The energies of these states are given in Table I rows 4 and 5.
Diagonalization of this 19×19 matrix gives the ground state polaron solution and the excited spectra. Among the 18 excited states, there are 9 odd states which are coupled by light to the even parity ground state with momentum matrix elements < Ψ|p|h0 >= ima/h < Ψ|H t |h0 >, whereas the other 9 even states are optically silent. No new parameter is needed for dipole matrix elements to predict the absolute value of σ(ω). The spectra would consist of 9 delta functions, three for E-field polarized inẑ direction σ ⊥ (ω) and six for σ (ω) shown by vertical lines on Fig. (3, a). The parameters were chosen to be ∆ = 0.95 eV, t = 0.5 eV,hω = 0.075 eV, and Mn-Mn distance a = 4.0Å.
In order to investigate the lattice-coupled electronic excitation, the surrounding oxygens should be allowed to vibrate. Since the frozen oxygen positions are not optimal for the excited hole states, Franck-Condon vibrational sidebands should appear in the optical spectrum. Assuming that the individual vibrational sidebands are not energy-resolved, there will be broadening by an amount roughly equal to the energy gain due to structural relaxation. In order to quantitatively describe the process, a theory beyond the adiabatic approximation is needed.
We use an approach similar to that applied to describe photoemission in the undoped material [7]. The method was proposed by Cho and Toyazawa [15], who used an infinitely large truncated Hilbert space to diagonalize a 1D polaronic Hamiltonian. For each of the states appearing in Eq. (8), Eq. (9) and Eq. (10), we re-optimize the positions of the 6 oxygens surrounding the hole. The relaxed energies are listed in Table I, row 6, 7, and 8. Notice that for each of the states in Eq. (8), the energy gain from relaxation is ∆, similar to exciton creation in the parent compound [8]. Each of these states, including |h0 >, is now allowed to have arbitrary number of vibrational quanta of the surrounding oxygens around their reoptimized positions for that particular state. This adds additional vibrational multiples nhω to the relaxed energy. In this chosen basis, the H ep + H L term of the Hamiltonian is diagonal. The complete Hamiltonian with the off-diagonal term H t can be diagonalized exactly. The hole ground state is not vibrationally excited at zero temperature, and the electronic part of this ground state has even symmetry which is coupled by light to odd excited states.
The Cho-Toyozawa calculation can be simplified by ignoring the sharing of the oxygens by its two nearest Mn neighbors. This partial removal of the "cooperative phonon" constraints was found in our photoemission calculation [7] to make only a small error. The energies of the odd parity states can be obtained as roots of the following analytic equation: The roots E give energies of the excited states for light polarized in ℓ =x,ŷ,ẑ directions. The parameters a i ℓ , b i ℓ and t i ℓ in Eq. 11 are given in table I. The meaning of parameters a i ℓ is the energy gain due to the structural rearrangement. The parameters b i α are energies of the electronic excited states in vibrational ground state with relaxed oxygen distortions. The hopping matrix elements t i α couple the ground state to the electronically excited states and they are proportional to the dipole matrix elements of the allowed transitions. The parameters ∆ and t used in Eq. 11 are all in units ofhω = 75 meV.
The energy of the ground state is lowered by 0.39 eV due to the hole delocalization on the neighboring sites such that only 83 % of the polaron is present on the central site. This value is very close to the estimation by second order perturbation theory [5], which gives −0.167t/Γ−0.488Γt = 0.41 eV (Γ = ∆/(8t)). The excitation spectrum is measured from the renormalized ground state energy and shown on Fig. 3. It should be noted that the electronically excited states have their maximum weight on the neighboring sites and are not allowed to lower their energies by spreading on the next neighboring sites due to the truncation of the Hilbert space. We have done an estimate of the finite Hilbert space effect by comparing the adiabatic spectra using 19 electronic states on 7 sites with those from the 79 states on 27 sites including the next neighboring atoms. The spectrum in a larger basis has been shifted by approximately 0.3 eV to lower energies. The energy peak positions and relative intensities in the adiabatic solution follow the trend of the non-adiabatic result (see Fig 3.a). We expect an overall shift of the spectrum in Fig. 3 to the lower energy by about 0.3 eV, such that the the spectrum sets in at about 0.7 eV.

IV. CONCLUSION
The particular details for the manganites show that interesting processes occur which were not fully anticipated in the qualitative discussions previously given [1]. Similar processes should be at work in other insulators such as cuprates and nickelates.
New spectral features are predicted to appear in the JT gap of lightly doped LaMnO 3 , and in the insulating gap of lightly doped CaMnO 3 . The excitation spectra are calculated in non-adiabatic Born-Oppenheimer or Cho-Toyozawa approximation. These excitations are dipoleallowed with dipole matrix elements proportional to the hopping parameter t.
In the high T paramagnetic phase, in a mean field approximation, spin disorder has the effect of reducing the parameter t by 30%. This estimate suggests that the spectral weight due to hole excitations in LaMnO 3 , for example, is reduced by 50% above the Néel temperature T N ≈ 140K [18], whereas the orbiton excitation cen-tered at 2 eV [8] is not sensitive to magnetic order and hence is temperature independent. In CaMnO 3 , above T N ≈ 125K [18], spin polarons should persist in disordered and increasingly delocalized form. This will cause the spectrum of Fig. 2 to broaden and shift to lower energy.
Our picture makes the prediction that, at frequencies well below the lowest weak peak in the sub-gap region, there should be a weak Urbach tail with characteristic Urbach temperature dependence. The tail is from the lowest-energy processes, and are weak because of small vibrational overlap. The available experimental data [14] on doped compounds agree with our picture. Further experiments on more lightly-doped materials, and especially transmission experiments through thin single crystals, would give a better test of our theory.