Structural and magnetotransport studies of magnetic ion doping for monovalent-doped LaMnO₃ manganites

Abstract

In this paper, we report the structural, electrical, and magnetic properties of polycrystalline La_0.85-xSm_xNa_0.15MnO₃ (x = 0.05, 0.1, 0.15) manganites. Rietveld refinement of x-ray data infers that doped manganite compounds possess a rhombohedrally distorted structure (space group $R\overline{3}C$). Both lattice parameter and unit cell volume decrease and a systematic change in both Mn-O-Mn bond angle and tolerance factor is observed with Sm content. Resistivity measurements discern metal-insulator transition (T_P). For x = 0.15 sample, a double metal-insulator transition with a single ferromagnetic transition is depicted. All samples exhibit extrinsic magnetoresistance (MR) effect. A large value of MR of 65% (253 K, 8 T) is associated with grain and grain boundary formation. The highest low-field MR of 23% (12 K, 2 T) and 35.2% (23 K, 2 T) for x = 0.05 and 0.1 is observed. The electronic and magnetic inhomogeneities induced by Sm and nonmagnetic metal Na phases account for MR properties.

APPENDIX

The understanding of the dynamical properties of materials requires the formulation of an effective interionic potential and is of substantial importance, as there is much disagreement as to whether long- or short-range interactions are at the origin of the properties of doped manganites. While developing a specific model, we assume that the change in force constants is small; the short-range interactions are effective up to the second-neighbor ions; and the atoms are on hold by elastic forces in the harmonic approximation without any internal strains within the crystal. The crystal energy for a particular lattice separation (r) as:

$$U\left( r \right) = {U_{\rm{C}}}\left( r \right) + {U_{\rm{R}}}\left( r \right) + {U_{\rm{V}}}\left( r \right)\quad .$$

(A1)

The first term is the Coulomb energy, and follows:

$${U_{\rm{C}}}\left( r \right) = - \sum\limits_{ij} {{{{Z_i}{Z_j}{e^2}} \over {{r_{ij}}}}} = - {{{{{\alpha }}_{\rm{m}}}{Z^2}{e^2}} \over r}\quad .$$

(A2)

Here, α_m as the Madelung constant36 and r_ij being the separation distance between i and j ions.

The short-range overlap repulsive energy is the second term in Eq. (A1) as:

$$\matrix{{{U_{\rm{R}}}\left( r \right) = nb{{{\beta }}_{ij}}\exp \left( {{{{r_i} + {r_j} - {r_{ij}}} \over {{\rho }}}} \right)\quad } \hfill & { + n \prime b{{{\beta }}_{ii}}\exp \left( {{{2{r_i} - k{r_{ij}}} \over {{\rho }}}} \right)} \hfill \cr {} \hfill & { + n\prime b{{{\beta }}_{jj}}\exp \left( {{{2{r_j} - k{r_{ij}}} \over {{\rho }}}} \right),} \hfill \cr }$$

(A3)

following Hafemeister and Flygare.37 The notations r_i (r_j) refer to the ionic radii, k is the structure factor, n (n′) is the number of nearest (next nearest) ions, respectively. Further, the notations b and ρ denote the hardness and range parameters, respectively.

The Pauling coefficients, β_ij are defined in terms of valence [Z_i (Z_j)] and number of the outermost electrons [n_i (n_j)] in the anions (cations), respectively, as:

$${\beta _{ij}} = 1 + \left( {{z_i}/{n_i}} \right) + \left( {{z_j}/{n_j}} \right)\,\,\,\,\,.$$

(A4)

The last term in Eq. (A1) is the vdW energy, denoted as:

$${U_{\rm{V}}}\left( r \right) = - \left( {\sum\limits_{ij} {{{{c_{ij}}} \over {r_{ij}^6}}} + \sum\limits_{ij} {{{{d_{ij}}} \over {r_{ij}^8}}} } \right)\quad ,$$

(A4a)

$$= - \left( {{C \over {{r^6}}} + {D \over {{r^8}}}} \right)\quad ,$$

(A4b)

due to dipole–dipole (d–d) and dipole–quadruple (d–q) interactions. The abbreviations C and D represent the overall vdW coefficients, defined as36:

$$C = {c_{ij}}{S_6}\left( r \right) + {1 \over 2}\left( {{c_{ii}} + {c_{jj}}} \right){S_6}\left( 0 \right)\quad .$$

(A5a)

and

$$D = {d_{ij}}{S_8}\left( r \right) + {1 \over 2}\left( {{d_{ii}} + {d_{jj}}} \right){S_8}\left( 0 \right)\quad ,$$

(A5b)

c_ij and d_ij are the vdW coefficients due to d–d and d–q interactions. We follow the variational method38 to derive c_ij and d_ij as:

$${c_{ij}} = {3 \over 2}{{e\hbar } \over {\sqrt {{m_{\rm{e}}}} }}{{{\alpha }}_i}{{{\alpha }}_j}{\left[ {{{\left( {{{{\alpha }}_i}/{N_i}} \right)}^{1/2}} + {{\left( {{{{\alpha }}_j}/{N_j}} \right)}^{1/2}}} \right]^{ - 1}}.$$

(A6a)

and

$$\eqalign{{d_{ij}} = {{27} \over 8}{{{\hbar ^2}} \over m}{{{\alpha }}_i}{{{\alpha }}_j}{\left[ {{{\left( {{{{\alpha }}_i}/{N_i}} \right)}^{1/2}} + {{\left( {{{{\alpha }}_j}/{N_j}} \right)}^{1/2}}} \right]^2}\;\;{\left[ {\left( {{{{\alpha }}_i}/{N_i}} \right) + {{20} \over 3}{{\left( {{{{\alpha }}_i}{{{\alpha }}_j}/{N_i}{N_j}} \right)}^{1/2}} + \left( {{{{\alpha }}_j}/{N_j}} \right)} \right]^{ - 1}}. \cr}$$

(A6b)

Here, m_e is the electron mass, α_i is the electronic polarizability, and N_i denotes the effective number of electrons of the ith ion. The values of the overall vdW coefficients are obtained using Eqs. (A5) and (A6), and weighted in terms of appropriate lattice sums [S₆(0), S₆(r), S₈(0), and S₈(r)].36 We believe that there is no uncertainty involved in the evaluation of c_ij and d_ij, due to the fact that the excitation energies are ignored in Eqs. (A6a) and (A6b). While applying for manganites herein, the above description, we shall seek the interionic interaction in between a pair such as Mn–O and La/A–O.

It is clear from the above descriptions that the present effective interionic potential contains only two free parameters (b and ρ), which are determined from the equilibrium conditions:

$${\left( {{{{\rm{d}}U} \over {{\rm{d}}r}}} \right)_{r = {r_0}}} = 0\,\,\,\,\,,$$

(A7a)

and Bulk modulus

$${B_{\rm{T}}} = {1 \over {9k{r_0}}}{\left( {{{{{\rm{d}}^2}U} \over {{\rm{d}}{r^2}}}} \right)_{r = {r_0}}}\,\,.$$

(A7b)

While estimating these values, we use the Bulk modulus39 from experiment and r₀ (obtained from Rietveld refinement of XRD data). The model parameters obtained from Eqs. (A7a) and (A7b) have been used to compute the SOECs (C₁₁, C₁₂, and C₄₄) as40,41

$${C_{11}} = {{{e^2}} \over {4r_0^4}}\left[ { - 5.112Z_{\rm{m}}^2 + {A_1} + {{\left( {{A_2} + {B_2}} \right)} \over 2}} \right]\quad ,$$

(A8a)

$${C_{12}} = {{{e^2}} \over {4r_0^4}}\left[ {0.226Z_{\rm{m}}^2 - {B_1} + {{\left( {{A_2} - 5{B_2}} \right)} \over 2}} \right]\quad ,$$

(A8b)

$${C_{44}} = {{{e^2}} \over {4r_0^4}}\left[ {2.556Z_{\rm{m}}^2 + {B_1} + {{\left( {{A_2} + 3{B_2}} \right)} \over 4}} \right]\quad ,$$

(A8c)

where (A₁, B₁) and (A₂, B₂) are the short-range parameters for the nearest and the next nearest neighbors, respectively. These parameters are further defined as

$${A_1} = {{4r_0^3} \over {{e^2}}}{\left[ {{{{{\rm{d}}^2}} \over {{\rm{d}}{r^2}}}{V_{ij}}(r)} \right]_{r = {r_0}}}\,\,,$$

(A9a)

$${A_2} = {{4{{\left( {{r_0}\sqrt 2 } \right)}^3}} \over {{e^2}}}{\left[ {{{{{\rm{d}}^2}} \over {{\rm{d}}{r^2}}}{V_{ii}}(r) + {{{{\rm{d}}^2}} \over {{\rm{d}}{r^2}}}{V_{jj}}(r)} \right]_{r = {r_0}\sqrt 2 }}\,\,,$$

(A9b)

$${B_1} = {{4r_0^3} \over {{e^2}}}{\left[ {{{\rm{d}} \over {{\rm{d}}r}}{V_{ij}}(r)} \right]_{r = {r_0}}}\,\,,$$

(A10a)

$${B_2} = {{4{{\left( {{r_0}\sqrt 2 } \right)}^2}} \over {{e^2}}}{\left[ {{{\rm{d}} \over {{\rm{d}}r}}{V_{ii}}(r) + {{\rm{d}} \over {{\rm{d}}r}}{V_{jj}}(r)} \right]_{r = {r_0}\sqrt 2 }}\,\,\,,$$

(A10b)

where V_ij(r) is the short-range potentials between the ions, which follow

$${V_{ij}}(r) = b{{{\beta }}_{ij}}\exp \left( {{{{r_i} + {r_j} - {r_{ij}}} \over {{\rho }}}} \right) + {c_{ij}}r_{ij}^{ - 6} + {{\rm{d}}_{ij}}r_{ij}^{ - 8}\,\,.$$

(A11)

The elastic force constant κ is derived at the equilibrium interionic distance r₀ following

$${{\kappa }} = {{{r_0}} \over 2}{\left[ {{{{\pi }}^2}\left( {{C_{11}} - {C_{12}}} \right)\left( {{C_{11}} + {C_{12}} + 2{C_{44}}} \right)\left( {{C_{44}}} \right)} \right]^{{1 \over 3}}}\quad .$$

(A12)

We have thus estimated the elastic force constants for a pair such as Mn–O and La/A–O and have the total force constants of the A-doped LaMnO₃. This continuum model thus takes care of the clear physical binding in doped manganites incorporating vdW interactions. However, the true potential must recognize the correct charge distribution and the relative orientations of the interacting atoms in manganites which is a complicated task.

We shall now estimate the acoustic Debye branch characterized by the Debye temperature θ_D and an optical peak defined by the Einstein temperature θ_E. The Debye frequency is characterized as a cut off frequency at the Brillouin zone boundary. It can be expressed in terms of effective value of ionic mass and elastic force constant for crystal lattices with two different kinds of atoms such as Mn–O and La/A–O, which we deal with. The acoustic-mode and optical-mode frequencies are estimated in an ionic model using a value of effective ion charge Ze = −2e.

We choose an acoustic mass M = (2M₊ + M₋) [Mn (O) which is symbolized by M₊(M₋)], κ* = 2κ for each directional oscillation mode to get the acoustic phonon frequency as

$${{{\omega }}_{\rm{D}}} = \sqrt {{{2{{{\kappa }}^*}} \over M}} \quad .$$

(A13)

Furthermore, when the phonons belong to optic modes, their frequency is determined by the reduced mass as μ⁻¹ = M(A)⁻¹ + M(B)⁻¹ where A is the anion (La/A, Mn) and B is the cation (O)

$${{\omega }}_{{\rm{LO}}}^2 = {{{{\kappa }} + {{\eta }}} \over {{\mu }}}\quad ,$$

(A14)

$${{\omega }}_{{\rm{TO}}}^2 = \frac{{{{\kappa }} - {{\eta }}}}{{{\mu }}}\quad ,$$

(A15)

where η is the force constant as

$${{\eta }} = {{8{{\pi }}} \over 3}{{{{\left( {Ze} \right)}^2}} \over {{\Omega }}}\quad .$$

(A16)

ω_LO (ω_TO) symbolized for the longitudinal (transverse) optical phonon frequency and Ω for the volume of the unit cell. We must mention that these computed values for La_1−xA_xMnO₃ with 0.25 ≥ x ≥ 0.35 could not be compared due to the lack of the experimental data. However, the computed values of Debye and Einstein temperature for doped La_0.7Ba_0.3 MnO₃ manganites are consistent with previous specific heat and Raman spectroscopic measurements.

It is true that the two-orbital model based on Wannier functions predicts the electronic states such as charge ordering in manganites. As mentioned, the Wannier function approach of the electronic problem is useful for the description of electron dynamics following a semiclassical theory as well as the magnetic interactions in solids.42 In the present investigations, we do not intend to discuss the electron dynamics as well the magnetic interactions, but focused on determining the acoustic (optical) phonon frequency to estimate the electron–phonon contribution of resistivity in the FMM phase.

Earlier, Millis43 determines the elastic parameters using a mean field approximation with emphasis on Mn–O bond lengths to evaluate the Mn–Mn and Mn–O force constants for the lattice distortions. However, we have considered both Mn–O and La/A–O bond lengths to obtain the Mn–O, La/A–O force constant, and total force constants for strong electron–phonon interaction. The formulated EIoIP includes the long-range Coulomb, vdW interaction, and the short-range repulsive interaction up to second-neighbor ions within the Hafemeister and Flygare approach. The interionic interaction in between a pair such as Mn–O and La/A–O enables us to find the total force constant with consistent Debye and Einstein temperatures.

The developed EIoIP thus takes care of the interactions in between a pair such as Mn–O and La/A–O. The interactions thus are attractive Coulomb, and vdW as well as short-range overlap repulsive interaction following Hafemeister and Flygare type potential. The advantage of using this potential is that it takes care of number of nearest (next nearest) ions, the valence, and number of the outermost electrons in the anions (cations), respectively. Thus, it takes care of the structural parameters that yield an approximately correct description of the interactions between a pair such as Mn–O and La/A–O. Henceforth, we are able to estimate the acoustic and optical phonon frequency consistent with the specific heat and Raman measurements to estimate the electron–phonon contribution of resistivity.