Effects of interaction and disorder on polarons in colossal resistance manganite Pr_0.68Ca_0.32MnO₃ thin films

According to Firsov et al [18], the temperature dependent resistivity due to TAH of small polarons is given by

$$\begin{eqnarray}&&\rho \left( T \right)=\frac{{\rm h}\cdot {{a}_{0}}}{Z\cdot {{{\rm e}}^{2}}\cdot {\rm x}\cdot (1-x)}\;\frac{{{k}_{{\rm B}}}T}{{{E}_{{\rm Ph}}}}\;{{f}^{-1}}\left( {{\eta }_{2}}(T) \right){{{\rm e}}^{\frac{{{E}_{{\rm A}}}}{{{k}_{{\rm B}}}\cdot T}}}\equiv {{\rho }_{0}}\;{{T}^{{\rm q}}}{{{\rm e}}^{\frac{{{E}_{{\rm A}}}}{{{k}_{{\rm B}}}\cdot T}}}.\end{eqnarray} \tag{ 1 }$$

Here, E_A is the activation barrier, ${{E}_{{\rm Ph}}}=\hbar {{\omega }_{{\rm Ph}}}$ is the phonon energy and a₀ is the hopping distance. Constants have their usual meaning. The pre-factor ρ₀ depends on the parameter η₂, being defined as

$$\begin{eqnarray}&&{{\eta }_{2}}=\frac{{{J}^{2}}}{{{E}_{{\rm Ph}}}\;\sqrt{{{E}_{{\rm A}}}{{k}_{{\rm B}}}T}}\;f\left( {{\eta }_{2}} \right)=\left\{ \begin{array}{ccccccccccccccc} {{{\rm \ \pi\ }}^{3/2}}{{\eta }_{2}},({{\eta }_{2}}\ll 1) \\ 1,({{\eta }_{2}}\gt 1) \\ \end{array} \right.,\end{eqnarray} \tag{ 1a }$$

where η₂ > 1 (q = 1) corresponds to the adiabatic and η₂ ≪⃒ 1 (q = 1.5) to non-adiabatic hopping. The extra term x(1 − x) has been included to take into account on-site Coulomb repulsion [24].

The polaron formation energy E_p can be extracted from optical spectroscopy data (for manganites, e.g., see [3]). Optically induced intra-band transitions which correspond to hopping between adjacent sites give rise to a skewed Gaussian absorption peak approximately centered at photon frequencies ħω = 2E_p. The real part of the optical conductivity is [25, 26]

$$\begin{eqnarray}&&\operatorname{Re}\left( \sigma \left( \omega ,T \right) \right)={{\sigma }_{{\rm dc}}}\left( T \right)\;\frac{{\rm sinh} \left( \frac{\hbar \omega }{2{{E}_{{\rm vib}}}} \right)}{\left( \frac{\hbar \omega }{2{{E}_{{\rm vib}}}} \right)}{{{\rm e}}^{-{{\left( \frac{\hbar \omega }{2{{E}_{{\rm vib}}}} \right)}^{2}}{\rm \ ~\ }\frac{{{E}_{{\rm vib}}}}{2{{E}_{{\rm P}}}}}}.\end{eqnarray} \tag{ 2 }$$

E_vib is the phonon zero-point energy $\hbar {{\omega }_{{\rm Ph}}}/2$ at low temperatures and corresponds to k_BT at high temperatures. Equation (2) requires that the peak width is mainly governed by phonon-induced broadening. In the Firsov–Lang theory this implies that σ_dc is the non-adiabatic hopping conductivity and that the quantity η₃ is small [11].

$$\begin{eqnarray}&&{{\eta }_{3}}=\frac{{{J}^{2}}}{{{E}_{{\rm A}}}\cdot {{k}_{{\rm B}}}\cdot T}\;.\end{eqnarray} \tag{ 2a }$$

Sometimes, the variable range hopping (VRH) model proposed by Mott [27] has been applied to the semiconductor-like resistivity of low bandwidth manganites. However, impurity hopping conductivity is expected at temperatures well below half of the Debye temperature [28], which is of order of Θ_Debye ≈ 320 K [29]. By fitting the VRH model to the experimental data we can exclude the applicability of this model to PCMO in the temperature range from about 100 to 300 K even for as-prepared films with strong lattice disorder. Therefore, we have focused our analysis on the model of TAH.

For PCMO, a doping level of 0.32 Ca atoms per pseudo-cubic unit cell with a lattice parameter of a₀ = 3.86 Å corresponds to a polaron charge density of about n = 5.6 × 10²¹ cm⁻³. The coordination of the polaron lattice site is Z = 6 and the hopping distance between next-neighbored Mn or O sites equals the lattice constant a₀. A clear identification of the phonon mode relevant for small polaron formation is still missing. There is strong evidence that both the Jahn–Teller phonon mode at E_Ph = 71 meV as well as octahedral bending modes, e.g., at E_Ph = 42 meV can couple to the charge dynamics of Pr_1−xCa_xMnO₃ in a wide temperature range [30–32]. Consequently, quantitative estimates of polaron properties are exemplarily done for both phonon modes.

It is hardly possible to distinguish via dc transport measurements whether hopping of polarons evolves adiabatically or non-adiabatically. For an activation energy E_A of about 140 meV and transfer integral J ranging from 60 to 110 meV (supplemental material (S1)), the parameter η₂ amounts to 0.8–2.8 (1.4–4.8), dependent on whether a Jahn–Teller or a Mn-O-Mn bending mode is selected.

We therefore assume that the hopping in PCMO is close to the adiabatic limit. In this limit, equation (1) gives a room temperature resistivity of ρ(T = 300 K) ≈ 0.06 (0.1) Ω cm, which is close to the range of experimental values (figure 1). The essential features described in the next sections are quite similar in both types of assessment (supplemental material (S3)).

In order to study the effect of lattice disorder, magnetic field and charge ordering on polaron hopping we determine the apparent activation energy

$$\begin{eqnarray}&&{{E}_{{\rm A}}}\left( T \right)\equiv \frac{{\rm D}}{{\rm D}T}\;{\rm ln} \left( \frac{\rho \left( T \right)}{{{T}^{{\rm q}}}} \right).\end{eqnarray} \tag{ 3 }$$

Equivalently, using the obtained apparent activation energy in a 'local' Arrhenius analysis, the intercept of the extrapolated straight line with the 1/T = 0 axis gives the apparent pre-factor ρ₀. E_A(T) = const and ρ₀(T) = const implies the validity of the single polaron model without any corrections.

4.2.1. Hopping mobility in zero-field

Figure 4(a) shows the temperature dependence of the apparent activation energy deduced from the adiabatic limit for samples with different degree of lattice disorder. For as-prepared films deposited at rather low temperatures (IBS-1, T_dep ≈ 973 K) E_A ≈ 145 meV is essentially temperature independent above T = 150 K, i.e., above half of the Debye temperature. Post-annealing causes a slight overall reduction of E_A to 135 meV (IBS-2). At zero magnetic field, the rapid decrease of E_A(T) at temperatures T < 120 K implies enhancement of tunnel-like contributions in the hopping regime [9]. This reduction of the apparent polaron activation barrier, however, cannot give rise to band-like conductivity as observed in the CMR regime at high magnetic fields.

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The most noticeable deviation from the model of TAH of small polarons is a pronounced peak in E_A(T) in the temperature range of T = 120–250 K. It is present (IBS-3 and 4) or evolves after post-annealing (PLD-1), if the substrate temperature during deposition is larger than 973 K. With increasing post-annealing time, the E_A(T) peak shifts to higher temperatures and increases in height. The appearance of the peak is not affected by the method of preparation (IBS or PLD) or the choice of substrate. Note that the samples IBS-3 and IBS-4 were simultaneously deposited in the same run but on different substrates (STO and MgO). Their E_A(T) curves are very similar.

Figure 4(b) reveals the temperature dependent apparent activation energy and pre-factor for a well-annealed sample prepared by PLD. E_A and ρ₀ are essentially constant at sufficiently high temperatures (typically above 220 K). At lower temperatures, the adiabatic E_A significantly increases up to 210 meV and ρ₀ simultaneously decreases by about two orders of magnitude. This peak/dip feature does not depend on whether adiabatic or non-adiabatic hopping is assumed (supplemental material S3).

The optical conductivity σ of the PCMO thin film IBS-5 was measured at various photon energies and different temperatures (supplemental material (S4)). Figure 5(a) shows the absorption maximum in the low-energy range at room temperature. This peak was fitted by two skewed Gaussian peaks P1 and P2 (equation (2)). According to the optical studies on LCMO [33] we expect that the peak P1 centered at ħω = 740 meV ≈ 2 E_P is related to photon induced small polaron hopping process between occupied and unoccupied polaron sites. The peak P2 centered at ω = 1474 meV ≈ 4 E_P corresponds to the on-site 'Jahn–Teller' transition between the split Mn3de_g orbitals hybridized to the O2p states.

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Figure 5(b) shows the temperature dependence of the fitting parameters of the hopping-related peak P1, i.e., the polaron formation energy E_P and the vibrational energy E_vib. E_P amounts to about 370 meV at room temperature and increases to about 395 meV at low temperatures. At temperatures below Θ_Debye/2, E_vib ≈ ħ ω_ph/2 ≈ 47 meV is temperature independent. Above the plateau, E_vib increases approximately linear with temperature and the slope corresponds to 1.38 k_B. Note that the fitting parameter σ_dc at room temperature corresponds to a resistivity of about 0.06 Ω cm, which is close to 0.085 Ω cm observed in a dc-measurement of the same film. However, since the parameter η₃ (equation (2a)) amounts to about 1–3, the contribution from electronic dispersion may significantly alter the shape, the maximum peak position as well as the width of the peak [34, 35]. The obtained values for E_P (E_vib) are thus only estimates in the sense of a lower (upper) limit.

4.2.2. Hopping mobility in magnetic fields

Application of magnetic fields strongly affects the polaron mobility and gives rise to an apparent activation energy E_A, which changes with field and temperature. Similar to the zero-field behavior, the obtained features in E_A do not depend on the applied model, i.e., adiabatic or non-adiabatic hopping.

Figure 6(a) compares E_A(T) of samples in different annealing stages at an applied field of B = 9 T and figure 6(b) shows exemplarily the temperature and field dependence of the adiabatic E_A(T) and the apparent pre-factor ρ₀(T) for a moderately annealed sample. The main observations are: (i) samples prepared under conditions, which do not result in a pronounced zero-field peak/dip feature, reveal an apparent activation energy, which monotonously decreases with decreasing temperature (IBS-2 and 3). (ii) The zero-field peak/dip feature at low temperatures is less pronounced in magnetic fields, i.e., the apparent activation energy decreases and the pre-factor increases with increasing magnetic field (figure 6(b)). (iii) The degree of reduction depends on the post-annealing time, i.e., the peak/dip feature is completely suppressed in well-annealed samples (PLD-1 in figure 6(a)).

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Therefore, post-annealing increases the peak/dip feature in E_A(T)/ρ₀(T) in zero field and decreases it in magnetic fields. With increasing post-annealing time the transition temperature T_CMR shifts to higher temperatures (figure 1(b) and table 1). In the crossover regime from hopping to a band-like transport, E_A drastically drops down towards zero and thus becomes meaningless for well-annealed samples (PLD-1 in figure 6(a)). This indicates a crossover from hopping conductivity to band transport with 'metal-like' temperature dependence below T_CMR.

Coherent band-like transport of small polarons evolves, if the energy fluctuations due to scattering at thermal quasiparticles or at lattice disorder are smaller than the renormalized width of the polaron band D_P [11]. Consequently, at sufficiently low temperatures, small polarons behave as heavy particles with effective mass m^* in a small polaronic band, giving rise to a resistivity

$$\begin{eqnarray}&&\rho \left( T \right)=\frac{{{m}^{*}}}{{\rm n}{{{\rm e}}^{2}}}\left( \frac{1}{{{\tau }_{{\rm R}}}}+\frac{1}{{{\tau }_{{\rm QS}}}}+\frac{1}{{{\tau }_{{\rm PS}}}} \right)={{\rho }_{{\rm R}}}+{{\rho }_{{\rm QS}}}(T)+{{\rho }_{{\rm PS}}}\left( T \right),\end{eqnarray} \tag{ 4 }$$

were ρ_R is the residual (temperature-independent) defect contribution, ρ_PS corresponds to phonon scattering and ρ_QS is due to magnon scattering [6] in a magnetically ordered phase.

In the Lang–Firsov theory, the resistivity of small polarons due to scattering with optical phonons is given by [18]

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{\rho }_{{\rm PS}}}\left( T \right) & = & \frac{{{a}_{0}}\hbar }{2\;Z\;x\;{{e}^{2}}}\left( \frac{{{E}_{{\rm Ph}}}}{{{D}_{{\rm P}}}} \right)\left( \frac{\omega }{\Delta \omega } \right){{\left( \frac{J}{{{E}_{{\rm A}}}} \right)}^{4}}F\left( T \right){{{\rm sinh} }^{-2}}\left( \frac{{{E}_{{\rm Ph}}}}{2\;{{k}_{{\rm B}}}\;T} \right) \\ {} & \equiv & {{\rho }_{{\rm LT}}}\;{{T}^{{\rm q}}}~{{{\rm sinh} }^{-2}}\left( \frac{{{E}_{{\rm Ph}}}}{2\;{{k}_{{\rm B}}}\;T} \right), \\ \end{array}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&F\left( T \right)=\left\{ \begin{array}{ccccccccccccccc} 1\;({{D}_{{\rm P}}}\gt Z\;{{k}_{{\rm B}}}T) \\ \frac{Z\;{{k}_{{\rm B}}}\;T}{{{D}_{{\rm P}}}}\;({{D}_{{\rm P}}}\lt Z~{{k}_{{\rm B}}}T)\; \\ \end{array} \right.\;,\end{eqnarray} \tag{ 5a }$$

where q = 0 or 1, depending whether the polaronic band width D_p is larger or smaller than Z*k_BT. Δω is the dispersion of the involved optical phonons and E_ph denotes a limiting upper frequency of longitudinal polar phonons involved in the scattering process.

In small bandwidth manganites such as PCMO, the FMM like state with coherent polaron transport is not observed at zero magnetic field, i.e., the temperature dependent resistivity increases up to the measurement limit of 10⁵ Ω cm at T = 4.2 K. The transition to the FMM state can be only induced by application of strong magnetic field. The strong drop of the resistivity from above 10⁵ Ω cm to 10⁻³ Ω cm is due to the CMR and appears by the first order meta-magnetic transition [36] from an antiferromagnetic insulating ground state to a ferromagnetic and conducting state. At intermediate temperatures, the meta-magnetic phase transition gives rise to nanoscale phase separation, where the appearance of a conductive phase is embedded in an insulating matrix [37]. The volume fraction of this phase increases with decreasing temperature. Over a broad temperature range below the crossover temperature T_CMR, charge transport thus takes place in an inhomogeneous phase-separated system.

The pronounced hysteresis of the resistivity with respect to field-cooling and field heating in figure 7(a) can be attributed to the nano-scale phase separated state, where the magnetization of the ferromagnetic phase is affected by domain wall pinning at defects. With increasing post-annealing time, i.e., with subsequent defect annihilation, this hysteresis becomes smaller. In addition, T_CMR shifts to higher temperatures.

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In the temperature range, where the volume fraction of the metal-like conducting phase is large and the hysteresis is small, the resistivity is mainly related to the residual contribution and the phonon scattering [6]. In the low-temperature approximation of equation (5) the phonon scattering of small polarons can be written as

$$\begin{eqnarray}&&{{\rho }_{{\rm PS}}}\left( T \right)\approx \rho \left( T \right)-{{\rho }_{{\rm R}}}\approx 4\;{{\rho }_{{\rm LT}}}\;{{T}^{{\rm q}}}\;{{e}^{\frac{-{{E}_{{\rm Ph}}}}{{{k}_{{\rm B}}}\cdot T}}}.\end{eqnarray} \tag{ 5b }$$

5.2.1. Low temperature mobility in magnetic fields

In order to analyze the applicability of equation (5b), we again define the temperature-independent prefactor ρ_LT and the phonon energy E_Ph as apparent quantities, i.e., ρ_LT(T) = const and E_LT(T) = const implies the validity of the model. Figures 7(b) and (c) show the temperature dependence of the apparent phonon energy E_Ph(T) deduced from equation (5b) in the small-band limit (q = 1) for the sample PLD-1 after different post-annealing times t_a (B = 9 T) and at different magnetic fields B (t_a = 20 h). At 10 K, the obtained phonon energy E_Ph ≈ 5 meV is not strongly affected by t_a and B.

For increasing t_a and increasing B likewise, the temperature range, where the model can be applied is strongly enhanced. After long-term post-annealing (t_a = 80 h) and at B = 9 T the narrow band model gives an almost constant E_Ph(T) over the largest temperature range up to T = 40 K. In contrast, the large-band model (D_P > Z k_BT) is worse, i.e., the apparent phonon energy significantly changes with temperature (see open symbols in figure 7(b)).

The small-band model can be applied to the low-temperature resistivity and the range of validity is limited by the finite volume fraction of the FMM phase, i.e., the appearance of pronounced phase separation, which is affected by the magnetic field and lattice disorder. However, the low-temperature approximation equation (5b) describes the T-dependence of the resistivity better than the sinh-dependence (equation (5)). This discrepancy cannot be attributed to additional scattering contributions. This observation is supported by recent DMFT calculations for the non-adiabatic case, where it has been concluded that equation (5b) well describes the band-like transport of small polarons over a large temperature range [10].

In contrast to E_Ph, the pre-factor ρ_LT changes by orders of magnitude with respect to post-annealing time and applied magnetic field. In figure 8(a), the temperature averaged value ρ_LT (T = 15–25 K) is plotted versus post-annealing time and magnetic field. It is important to note that the residual resistivity ρ_R reveals the same strong dependence like the pre-factor (inset). This indicates a strong correlation between phonon and defect related scattering, where the changes of resistivity due to post-annealing or magnetic field increase are indistinguishable (figure 8(b)).

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Since the hopping seems to take place near the adiabatic limit, the contribution of the transfer integral to the activation barrier E_A is significant.

$$\begin{eqnarray}&&{{E}_{{\rm A}}}\approx \frac{{{E}_{{\rm P}}}}{2}-J\;.\end{eqnarray} \tag{ 6 }$$

In order to analyze the additional corrections due to magnetic fields, disorder and high carrier concentrations, we include the magnetization dependence of J and the additional contributions due to defects and inter-site Coulomb interaction.

$$\begin{eqnarray}&&{{E}_{{\rm A}}}\approx \frac{{{E}_{{\rm P}}}}{2}-J(M)+{{W}_{{\rm D}}}+{{W}_{{\rm C}}}\;.\end{eqnarray} \tag{ 6a }$$

6.1.1. The Coulomb contribution W_C

In highly doped manganites at x = 0.33, the contribution of the inter-site Coulomb interaction between charge carriers is rather large and can be estimated by [16]

$$\begin{eqnarray}&&{{W}_{{\rm C}}}\approx \frac{x\;{{{\rm e}}^{2}}}{12\;{{\varepsilon }_{0}}{\rm \ ~\ }{{\varepsilon }_{{\rm s}}}\;{{a}_{0}}{\rm \ ~\ }}.\;\end{eqnarray} \tag{ 7 }$$

Here, ε_s is the static dielectric constant. For PCMO, ε_s is in the range of 30 [38] to 57 [39], i.e., W_C is of the order of 40–20 meV for next neighbor interactions.

6.1.2. The magnetic contribution J(M)

In the Pbnm phase, the apparent activation energy in magnetic fields decreases with decreasing temperature. This decrease is governed by the effect of the double exchange interaction on the transfer integral. J is strongly modified by the next neighbor spin alignment θ_ij [1] with

$$\begin{eqnarray}&&J(M)={{J}_{{\bf FM}}}{\bf cos}\left( \frac{{{{\boldsymbol{ \theta }} }_{{\bf ij}}}}{2} \right)\approx \;\frac{{{J}_{{\bf FM}}}}{\sqrt{2}}\sqrt{1+\frac{{{M}^{2}}}{M_{{\bf S}}^{2}}}\equiv {{J}_{{\bf PM}}}\sqrt{1+\frac{{{M}^{2}}}{M_{{\bf S}}^{2}}}\end{eqnarray} \tag{ 8 }$$

M_s denotes the saturation magnetization.

For not too large magnetizations M < M_s, equations (6a) and (8) imply a linear decrease of E_A with the square of the paramagnetic magnetization. Actually, the temperature dependent apparent activation energy at different magnetic fields can be combined to a single master curve by plotting E_A(T,B) as a function of B²/T² ∝ M² (figure 9). The field-depending upturns in the E_A-M² scaling is observed for data points corresponding to temperatures above the peak in E_A(T) (figure 4). Using J = J_PM ≈ 60−110 meV (supplemental (S1)), equation (8) implies that full spin polarization increases the J by about ΔJ ≈ 25–45 meV. Experimentally, the maximum reduction of the activation energy due to paramagnetic magnetization (at 9 T) amounts to about 27 meV for moderately annealed samples.

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6.1.3. The disorder contribution W_D

In the Pbnm phase the zero-field activation energy ranges from about 130 to 144 meV (see also figure 10(a)). E_A is not strongly affected by the choice of substrate and the method of preparation but commonly decreases with increasing deposition temperatures and post-annealing time. Therefore, the trapping contribution W_D to the activation barrier is of the order of 10–15 meV.

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The spread is larger in strong magnetic fields (113–133 meV at 9 T, see figure 10). In addition, the dependence of the apparent activation energy on the magnetization is more pronounced for samples with a low degree of disorder (compare the moderately annealed samples PLD-1 and PLD-2 in figure 9). This implies that disorder in the octahedral tilt system rather gives rise to local variations in the transfer integral than to trapping.

Summing up all contributions in equation (6) for the charge disordered Pbnm phase of well-annealed samples in zero-field, we have W_C ≈ 30 meV, J ≈ 60–110 meV, W_D = 0 and E_A ≈ 130 meV and thus the polaron formation energy should be of the order of 320–420 meV. Since the formation energy determined from optical conductivity (figure 5(b)) amounts to about E_p ≈ 370 meV, J ⩾ 85 meV seems to be a rather reasonable choice for the transfer integral. According to equation (8), this would imply a reduction of the activation energy in strong fields of about 35 meV, which is very close to the experimentally observed reduction of about 33 meV in long-term annealed samples (sample PLD-1 at t_a = 80 h in figures 4 and 6).

It is instructive to review the deviations from single polaron hopping, i.e., the appearance of temperature and field depending apparent activation energies and pre-factors, with respect to the Meyer–Neldel compensation rule. This rule states that in a thermally activated transport process, a decrease of mobility due to an increase of the activation energy is partly compensated by an increase of the pre-factor [40].

$$\begin{eqnarray}&&{{\rho }_{0}}\propto {{{\rm e}}^{-\frac{{{E}_{{\rm A}}}}{{{k}_{{\rm B}}}\,{{T}_{{\rm MN}}}}}},\end{eqnarray} \tag{ 9 }$$

where T_MN is the Meyer–Neldel temperature.

Generally, the Meyer–Neldel analysis is applied to systems with TAH of charge carriers, where the activation energy and the pre-factor are changed by sample property variations, such as doping or defect concentrations [41]. A publication by Emin [42] on thermally activated polaron hopping states that Meyer–Neldel compensation arises in the adiabatic treatment of small polaron hopping, if the effect of the charge on the barrier at the coincidence positions via a change on the atomic vibrations is taken into account. This can be expressed by an effective contribution to the transfer integral, where J_eff = k_B T_MN.

Near room temperature (280 K–300 K), i.e., in the paramagnetic phase, the apparent quantities depend only weakly on temperature and thus can be represented by field dependent mean values. Figure 10 shows the mean apparent E_A(B) and the pre-factor ρ₀(B) for various samples and annealing states for magnetic fields between 0 and 9 T. The linear dependence in the semi-log plot implies that PCMO follows the Meyer–Neldel rule. The Meyer–Neldel temperature T_MN is not strongly affected by the details of the preparation and amounts to 370 ± 30 K. Since k_B T_MN ≈ 33 meV, the Meyer–Neldel temperature agrees well with the maximum magnetically induced change of the transfer integral of ΔJ ≈ 35 meV as expected for J ≈ 85 meV.

Reducing the temperature towards T_CO, a pronounced peak (up to 80 meV) in the activation energy appears in well-annealed samples and zero field (figure 4). The peak is completely suppressed in the presence of sufficient strong magnetic fields. Such samples commonly show a CMR transition and, as revealed by TEM analysis, the formation of the CO P2₁mn phase. This implies that the temperature independent part of E_A(T) corresponds to the Pbnm phase and that the peak in E_A is caused by charge ordering.

In the P2₁mn phase, the influence of defects is much stronger. Octahedral disorder reduces the peak in the activation energy and counteracts its reduction in magnetic fields. A high degree of disorder can completely suppress the charge ordering peak. Such samples remain in the Pbnm phase also at low temperatures and only the healing of lattice disorder enables the emergence of the CO phase.

Since charge ordering gives rise to the freezing out of mobile polarons, the peak-dip feature in E_A(T) and ρ₀(T) (figures 4 and 6) might also suggest a Meyer–Neldel compensation. However, the analysis of the peak-dip feature does not reveal a single valued Meyer–Neldel temperature. The charge ordering transition is of first order and undergoes via a two phase area, where CO and disordered phases coexist. The CO phase may have a larger resistivity than the charge disordered phase. The difference cannot be huge, since ρ(T) in single crystals, where the CO transition is most pronounced is only showing a kink with a slight increase in the slope of ρ(T). In addition, in situ TEM studies of electric stimulation of the mixed phase show, that the electric current leads to a motion of CO domains [17]. This clearly shows that the CO areas contribute to the electric transport in the two phase region.

We thus argue that the peak-dip feature in E_A(T) and ρ₀(T) represents the influence of charge order or at least of charge short range order while decreasing the single polaron mobility and increasing the polaron–polaron next neighbor interactions. For moderately annealed samples, the polaron formation energy E_P increases below the charge ordering temperature to a value which is about 25 meV larger than at room temperature (figure 5(b)). According to equation (6a), the increase of E_P due to structural phase transition would be a natural explanation for the increase of E_A, which then also contains contributions from the polaron–polarons Coulomb interactions. An increase of E_A because of structure-induced decrease of the transfer integral seems to be unlikely, because in the P2₁mn phase some of the Mn-O-Mn bonding angles are flattened (157.7°–160.6°) compared to the bonding angle in the Pbnm phase (153°, figure 11(a)). The decrease of ρ₀ may arise from the transition from individual to cooperative hopping of the polarons during ordering. We can assume that having a number of polarons simultaneously at their coincidence points between two sites may change the activation barrier as well as the transfer rates and thus would result in a Meyer–Neldel like behavior as suggested in [42].

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Our study of the insulator–metal transition in PCMO films with different degree of octahedral disorder shows that coherent band-like transport of polarons only evolves at high fields and low temperatures, if the defect density is not too high. The defect-induced suppression of the CMR effect coincides with the suppression of the long-range CO P2₁mn phase formation in zero field. This raises the question whether the coincident suppression just accidentally happens due to the defect-induced variation of the transfer integral J or whether the long range order of small polarons is required for the emergence of the CMR via a fully developed double exchange contribution ΔJ(M) and thus a sufficiently large polaron bandwidth D_p.

As far as we know, no unique determination of the space group of the FMM phase has been published for PCMO. Therefore, we have used the determined P2₁mn structure model for x = 0.33 and the intensity variation of two neutron diffraction peaks published in [43] to model the field-induced changes of the displacement vectors. The results are shown in the supplemental material (S2) and are visualized in figure 11(c). Since the modeling of the magnetic field induced structural changes is based on two reflections only, the suggested displacements of the atom position Δ_i are not unique. However, all explored alternative settings which are in agreement with the intensity variations in the diffraction patterns have in common a smoothening of Mn-O-Mn bond angles. This occurs preferentially at those ferromagnetic zig-zag chains which are preformed in the CE type antiferromagnetic and CO structure (figure 11(b)).

Our structure model thus implies that the meta-magnetic transition to the ferromagnetic state is accompanied by a preferential increase of the transfer integral at the Mn-O-Mn chains with smoother bond angle and that the formation of a CE type CO state is a precursor of the magnetic field induced FMM state. The overall effect of charge order and magnetic field is an increase of the polaron bandwidth with a strong increase in polaron mobility according to equation (5).

With respect to the influence of disorder and magnetic fields on the band-like transport, it is interesting to consider the stability of the small polaron state. The formation of polarons is commonly described by three coupling parameters α, λ, and δ, which are related to fundamental energy scales: the electron–phonon binding energy g, the energy of the involved phonon mode E_Ph, and the bare electronic bandwidth D [44].

$$\begin{eqnarray}&&\;{\boldsymbol{ \alpha }} \equiv \frac{g}{{{E}_{{\bf ph}}}}=\sqrt{\frac{{{E}_{{\bf P}}}}{{{E}_{{\bf ph}}}}}\;{\boldsymbol{ \lambda }} \equiv \frac{\;{{E}_{{\bf P}}}}{Z~J\;}\;{\boldsymbol{ \delta }} \equiv \frac{{{E}_{{\bf ph}}}}{J}.\end{eqnarray} \tag{ 10 }$$

Because of the dependence on the bare bandwidth D, the coupling constants λ and δ decrease with increasing spin polarization [45] (see also equation (8)).

A small polaron state requires that α and λ exceed critical values of the order of unity. Using E_P ≈ 370 meV, J ≈ 85 meV and E_Ph ≈ 42 meV for the Zener mode (71 meV for the Jahn–Teller mode), we obtain λ ≈ 0.7, α ≈ 3 (2.3) and δ ≈ 0.5 (0.8) for the paramagnetic high-temperature phase. Therefore, the electron–phonon coupling is rather strong but the small polaron state is close to the stability boundary because of the smallness of λ. Charge ordering seems to increase the polaron formation energy and, therefore, stabilize the small polaron state.

A large coupling constant α gives rise to a very small polaron bandwidth. Since the adiabacity parameter δ is of the order of 0.5–1.0, D_P in the FMM phase can be calculated according to [46]

$$\begin{eqnarray}&&{{D}_{{\rm P}}}\approx D\;{{{\rm e}}^{-{{\alpha }^{2}}}}.\end{eqnarray} \tag{ 11 }$$

For a coarse estimate of D_p we have applied the mobility model for small polaron bandwidth (equation (5) with D_P < z k_B T) to the well annealed sample PLD-1 (t_a = 80 h, B = 9 T). Using E_Ph ≈ 4.4 meV, ρ_LT ≈ 1.3 × 10⁻⁶ Ω cm K⁻¹ (figures 7 and 8), J ≈ 85 meV and reasonable values for the phonon dispersion of 1 (10) meV, we obtain D_p ≈ 7.5 (2.4) meV. This corresponds to α ≈ 2.2 (2.5).

The polaronic bandwidth seems to be rather small and comparable to the phonon energy, also in well annealed samples and at high magnetic fields. Since band-like mobility requires D_P > E_ph, reduction of D_p due to spin misalignment and octahedral mistilts would eventually even give rise to the emergence of hopping conductivity, as observed experimentally in as-prepared samples at high fields or in well annealed samples at low fields.

In addition, the polaron mass m_p is expected to increase with decreasing bandwidth. Remarkably, the defect-related residual resistivity and the pre-exponential factor of phonon scattering reveal the same dependence on post-annealing time and magnetic field resulting in a universal dependence of the parameters ρ_LT on the residual resistivity ρ_R (figure 8(b)). This implies, that both field and disorder related changes in the low-temperature resistivity are governed by the change of the scattered particle mass m_p. However, the coupling constant λ is rather small in the paramagnetic phase. The increase of the bandwidth D in the FMM phase could therefore result in a state, where small polarons are not stable, i.e., new quasiparticles are formed. It is worthwhile to note, that the observed correlation between ρ_LT and ρ_R does not depend on the applied model of phonon scattering.