Spin-dependent and spin-independent channels of electrical transport in perovskite manganites

Abstract

A model with two channels of electrical transport (TCET) for perovskite manganites is proposed, and it is described by an equivalent device with two current-carrier channels. In one channel, there is a spin-independent resistor (R₃) with an equivalent resistivity of ρ₃. In the other channel, there are two spin-dependent resistors in series (R₁ and R₂) with an equivalent resistivity of ρ₁ + ρ₂. The component ρ₁ includes residual resistivity and the resistivity contributed by crystal-lattice scattering. The other component of the equivalent resistivity, ρ₂, originates from the spin orientations of the itinerant electrons and the local electrons of the outer O 2p and Mn 3d orbits that deviate from the orientation of their ground states when the test temperature is close to the Curie temperature. Using this model, we fitted the experimental curves of the resistivity versus test temperature for single-crystalline La_1−xSr_xMnO₃ (0.00 ≤ x ≤ 0.40) and polycrystalline La_0.6Sr_0.4Mn_1−xFe_xO₃ (0.00 ≤ x ≤ 0.30). In addition, we investigated the effects of the fraction of the antiferromagnetic phase, scattering at the crystallite interfaces, and the crystal-cell constants on the samples' resistivity. The physical mechanism of the TCET model was explained using an O 2p itinerant-electron model, which has been used in other studies to explain the magnetic ordering of several series of spinel ferrites and perovskite manganites.

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1. Introduction

There have been many investigations on the electronic transport and magnetic properties of perovskite oxides^1–12 because of their potential application in both electronic and magnetic information-storage devices. There are several models for the electronic-transport properties of these materials, and spin-dependent transport has been taken into account by most researchers. One of the popular views is that there is polaron hopping below the Curie temperature (T_C) on the basis of the double-exchange interaction.^2,3,13 In this view, the current carriers of perovskite manganites originate from the d electrons of manganese cations. Another view was proposed by Alexandrov et al.,^13,14 who argued that the current carriers of perovskite manganites originate from the p holes of oxygen anions rather than the d electrons of manganese cations, based on the experimental results of electron-energy-loss spectroscopy^15,16 and X-ray absorption spectroscopy.¹⁷ Alexandrov et al.^13,14 therefore established a current-carrier-density collapse (CCDC) model for the electrical-transport mechanism, and many researchers followed with investigations on this CCDC model.^18–21

The presence of p holes of oxygen anions in an oxide can be easily understood as there are O¹⁻ anions in addition to O²⁻ anions. When compared with an O²⁻ anion (2s²2p⁶) with a closed outer electron shell, it is clear that a p hole exists in the outer orbit of an O¹⁻ anion (2s²2p⁵). This fact was further confirmed by theoretical calculations^22,23 and experiments.^24–26

Taking into account that there are both O¹⁻ and O²⁻ anions in oxides,^22–27 our group has proposed an O 2p itinerant-electron model for magnetic oxides (IEO model),^28,29 which is similar to the O 2p hole model proposed by Alexandrov et al.^13,14 Using the IEO model, our group studied the relationship between the magnetic moment and cation distribution of several series of Cr (Ti, Mn)-doped spinel ferrites.^28–36 We also studied the magnetic ordering in perovskite manganites.^37–40

In this study, we used a model of two channels of electrical transport (TCET), which was derived from the IEO model, to fit the temperature dependence of the resistivity for two series of perovskite manganites: single crystallines La_1−xSr_xMnO₃ (0.00 ≤ x ≤ 0.40) reported by Urushibara et al.⁴¹ and polycrystallines La_0.6Sr_0.4Mn_1−xFe_xO₃ (0.00 ≤ x ≤ 0.30) prepared in this work. In addition, we will discuss the effects of the proportion of the antiferromagnetic phase, scattering at crystallite interfaces, and crystal-cell constants on the samples' resistivity.

2. Experiment and results

The polycrystalline perovskites examined in this study, La_0.60Sr_0.40Fe_xMn_1−xO₃ (0 ≤ x ≤ 0.3), were prepared using the sol–gel method:^42,43 (i) stoichiometric amounts of La₂O₃ (purity: 99.99%, fired in air at 1073 K for 3 h before it was used), Sr(NO₃)₂ (purity: 99.5%) powder, and a Mn(NO₃)₂ solution (purity: 50%) were dissolved in a dilute HNO₃ solution at 343 K. Citric acid and ethylene glycol were added to the mixed solution as complexing agents until a completely homogeneous transparent solution was obtained. The molar ratio of citric acid (or ethylene glycol) to the perovskite molecule was 6 : 1. (ii) The mixture was then evaporated at 363 K for 24 h in a water bath until a highly viscous residue was formed. Upon further heating at 373 K for 12 h and then at 473 K for 24 h, a gel developed. The gel was ground. (iii) All ground gel samples of different compositions were further heat treated. The temperature was increased from 473 to 573 K, in increments of 25 K, with 45 min spent at each step. Next, the temperature was increased from 573 to 673 K in increments of 20 K, with 50 min spent at each step. Finally, the temperature was increased from 673 to 773 K in increments of 10 K, with 1 h spent at each step. The samples were finely ground for 30 min to obtain a homogeneous mixture. After that, the samples were calcined at 873 K for 5 h to eliminate the organic material, ground, and calcined again at 1073 K for 10 h to eliminate any residual organic material. Each sample was then divided into two parts to measure their magnetic and electrical properties, and they were subjected to further heat treatment using slightly different methods.

To prepare the samples used to measure the magnetic properties, the powder obtained after grinding was pressed into pellets with a diameter of 13 mm and thickness of about 2 mm under a pressure of 11 ton per cm². The pellets were calcined at 1273 K for 10 h, after which powder samples were obtained by grinding.

To prepare the samples used to measure the electrical-transport properties, the powder obtained after grinding was pressed into pellets under a pressure of 10 ton per cm². The pellet samples were then sintered at 1573 K for 24 h.

Structural characterization was carried out by analyzing X-ray diffraction (XRD) patterns, which were measured using an X-ray diffractometer (X'Pert Pro, PANalytical, The Netherlands) with Cu K_α radiation (λ = 1.5406 Å). The data were collected in the 2θ range of 15° to 120° in increments of 0.0167°. The field and temperature dependence of the specific magnetization, σ(H) and σ(T), were measured using a physical-property measurement system (PPMS, Quantum Design, Inc., USA). Using electrical-transport characterization equipment (Versa Lab, Quantum Design, Inc., USA), the temperature dependence of the electrical resistivity ρ(T) was measured by the conventional four-point probe technique at 50 K.

Fig. 1(a) shows the XRD patterns, which indicate that all the samples had only a single ABO₃ perovskite phase with the space group R3̄c. The XRD patterns were fitted using the Fullprof software (Institut Laue-Langevin, France). The goodness-of-fit factor (s ≤ 1.37) is listed for all samples in Table 1. As an example, the fitted result for the La_0.60Sr_0.40Fe_0.1Mn_0.9O₃ sample is shown in Fig. 1(b). The fitted result; the dependence on the Fe-doping level, x; the lattice parameters, a and c; the crystal-cell volume, v; the Mn–O bond length, d_B–O; and the Mn–O–Mn bond angle, Θ, are listed in Table 1. It can be seen that the values of these parameters slightly varied: the values of a, c, v, and d_B–O increased with increasing x; and the value of Θ decreased with increasing x. The equivalent cubic cell constants (a_e) per formula, calculated using the crystal-cell volume, v, will be used to discuss the samples' resistivity in Section 4.4.

Fig. 1 (a) Powder XRD patterns of La_0.6Sr_0.4Fe_xMn_1−xO₃. (b) Fitted and observed XRD patterns for the sample La_0.6Sr_0.4Fe_0.1Mn_0.9O₃, obtained using the Fullprof software. The crosses represent the observed data, and the curve represents the fitted results; the vertical bars represent the Bragg sites for the space group R3̄c. The difference between the observed and fitted patterns is shown at the bottom.

Table 1 Crystal structural parameters of La_0.6Sr_0.4Fe_xMn_1−xO₃ at room temperature: a and c are the cell constants, v is the cell volume, d_B–O and Θ are the Mn–O distance and Mn–O–Mn bond angle, all which were obtained by fitting the XRD patterns; s is the goodness-of-fit factor. σ_S and μ_obs are the specific saturation magnetization and the average molecular magnetic moment at 10 K, respectively. T_C and T_N are the Curie temperature and the Néel temperature, respectively; T_MI is the temperature at which the resistivity has the maximum value

x	s	a (Å)	c (Å)	v (Å^3)	dB–O (Å)	Θ (°)	σS (A m^2 kg^−1)	μobs (μB)	TC (K)	TN (K)	TMI (K)
0	1.34	5.4861	13.356	348.14	1.9357	172.05	80.55	3.194	364.5	—	292
0.05	1.32	5.4898	13.359	348.69	1.9367	171.66	78.53	3.114	321.8	—	231
0.1	1.30	5.4920	13.362	349.04	1.9374	171.22	78.46	3.112	267.9	—	192
0.15	1.30	5.4936	13.365	349.30	1.9379	170.86	72.77	2.887	187.3	—	133
0.2	1.37	5.4945	13.366	349.44	1.9381	170.67	1.72	0.068	68.9	40.5	—
0.25	1.27	5.4961	13.368	349.69	1.9386	170.35	1.64	0.065	51.3	37.6	—
0.3	1.26	5.4971	13.370	349.88	1.9390	170.16	0.91	0.036	45.5	32.9	—

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Fig. 2(a) shows the magnetic hysteresis loops of the samples measured at 10 K. Using the specific saturation magnetization, σ_S, at 10 K, the average values of the molecular magnetic moment, μ_obs, were obtained. Fig. 2(b) shows the curve of μ_obs versus x. Fig. 2(c) shows the curves of σ versus T under an applied magnetic field of 0.05 T. The values of the Curie temperature, T_C, defined as the temperature at which dσ/dT reaches its minimum value, are shown in Fig. 2(d). In addition, the inset in Fig. 2(c) shows that there was a different Néel temperature, T_N, for each sample with x ≥ 0.20. Below T_N, the value of σ exhibited no significant change; above T_N, the value of σ decreased distinctly. The variation of T_N with x is shown in the inset in Fig. 2(d). The values of μ_obs, T_C, and T_N are listed in Table 1.

Fig. 2 Magnetic properties of La_0.6Sr_0.4Fe_xMn_1−xO₃: (a) magnetic hysteresis loops at 10 K; (b) magnetic moment, μ_obs, at 10 K as a function of Fe content, x; (c) temperature dependence of the specific magnetization, σ, at an applied magnetic field of 0.05 T; (d) magnetic transition temperature as a function of Fe content, x. The insert of (c) enhanced the variations of the curves. The insert of (d) shows the Néel temperature, T_N, as a function of Fe content, x.

The data points in Fig. 3 show the temperature dependence of the electrical resistivity, ρ, of the samples, and it can be seen that ρ increased with increasing x. In addition, there was a resistivity peak at a characteristic temperature, T_MI, for each sample when x ≤ 0.15, and these characteristic temperatures are listed in Table 1. The curves in Fig. 3 also show the fitted results, which will be discussed in the following sections.

Fig. 3 Temperature dependence of electrical resistivity of the La_0.6Sr_0.4Fe_xMn_1−xO₃ samples. The data points represent the observed values, and the curves represent the fitted results.

The following trends can be observed in Fig. 2 and 3: (i) when x ≤ 0.15, the samples were ferromagnetic conductors. The value of μ_obs decreased slowly from 3.194 μ_B (x = 0.00) to 2.887 μ_B (x = 0.15); the value of T_C decreased rapidly from 364.5 K (x = 0.00) to 187.3 K (x = 0.15); the maximum resistivity increased from 0.0222 Ω cm (x = 0.00) to 4.47 Ω cm (x = 0.15). (ii) When x ≥ 0.20, the samples were semiconductors with both an antiferromagnetic phase and a ferromagnetic phase. The resistivity increased rapidly with decreasing temperature, exceeding 100 Ω cm when the test temperature was below 123.6, 140.2, and 159.3 K, when x = 0.20, 0.25, and 0.30, respectively. In addition, when x = 0.20, 0.25, and 0.30, respectively, the values of T_N of the antiferromagnetic phase were 40.5, 37.6, and 32.9 K; the Curie temperatures of the ferromagnetic phase were 68.9, 51.3, and 45.5 K; and the μ_obs values were 0.068, 0.065, and 0.036 μ_B.

3. Model with two channels of electrical-transport (TCET) for perovskite manganites

In order to explain the electrical-transport properties of perovskite manganites, we propose a model with two channels of electrical transport (TCET), which was derived from the IEO model.^29,37 The IEO model is based on the following three postulates: (i) in a given sublattice, an O 2p electron with a constant spin direction can hop from an O²⁻ anion to an O 2p hole belonging to an adjacent O¹⁻ anion, with the metal cation acting as the intermediary. (ii) The two O 2p electrons in the outer orbit of an O²⁻ anion, which have opposite spin directions, become itinerant electrons of two different sublattices (such as the (A) and [B] sublattices of (A)[B]₂O₄ spinel ferrites, or the A and B sublattices of ABO₃ manganites). (iii) Since an itinerant electron has a constant spin direction in a given sublattice and owing to Hund's rules,⁴⁴ the directions of the magnetic moments of cations with n_d ≤ 4 (where n_d is the number of 3d electrons in 3d transition metal cations), such as Mn³⁺ or divalent and trivalent Cr cations, will couple antiferromagnetically with those of magnetic moments of cations with n_d ≥ 5, such as Mn²⁺ or divalent and trivalent Fe, Co, and Ni cations. This occurs because in the 3d orbits of transition elements, a maximum of five electrons can have their spins aligned in the same direction.

It is well known that there are two sublattices in ABO₃ perovskite manganites: the A–O sublattice and the B–O sublattice. According to the above IEO model, there are O–A–O–A–O and O–B–O–B–O ionic chains, which can be regarded as two current-carrier channels. In La_1−xSr_xMnO₃, the movement of O 2p electrons along the first channel, i.e., the O–A–O–A–O (where La and Sr occupy the A sites) ionic chain, is a spin-independent process because there is no magnetic ordering. The resistance of the first channel is labeled R₃ (with an equivalent resistivity of ρ₃). The movement of O 2p itinerant electrons along the second channel, i.e., the O–Mn–O–Mn–O (where Mn occupies the B site in O–B–O–B–O) ionic chain, is a spin-dependent process below the Curie temperature because there is magnetic ordering. The resistance of the second channel is represented by two resistors in series, R₁ and R₂ (with an equivalent resistivity of ρ₁ + ρ₂), where ρ₁ includes the residual resistivity and the resistivity resulting from scattering by the crystal lattice, and ρ₂ originates from the spin orientations of the itinerant electrons and local electrons of the outer O 2p and Mn 3d orbits that deviate from the orientation of their ground states because of thermal fluctuation. Therefore, the TCET model for a perovskite manganite sample can be represented by the equivalent circuit shown in Fig. 4(a), with the resistance, R, and resistivity, ρ, defined as follows:

Fig. 4 The equivalent circuit of the TCET model of (a) single-crystalline and (b) polycrystalline perovskite manganites. R₁ includes the residual resistance and the resistance resulting from scattering by the crystal lattice. R₂ originates from the spin orientations of the itinerant electrons and local electrons of the outer O 2p and Mn 3d orbitals that deviate from the orientation of their ground states because of thermal fluctuation. R₃ is spin-independent resistance. R₄ originates from the scattering by the crystallite interfaces in the polycrystalline sample.

A series of curves of ρ versus test temperature, T, for single-crystalline La_1−xSr_xMnO₃ (0.00 ≤ x ≤ 0.40) were reported by Urushibara et al.,⁴¹ and they were fitted using the TCET model. As shown in Fig. 5, the fitted curves are very close to the observed results (data points).

Fig. 5 Temperature dependence of electrical resistivity of the single-crystalline La_1−xSr_xMnO₃ samples. The data points represent values observed by Urushibara et al.,⁴¹ and the curves represent the fitted values obtained in this work.

When 0.175 ≤ x ≤ 0.40, we found that the resistivities in eqn (1) can be expressed as follows:

where k_B is the Boltzmann constant. The parameters in eqn (2) were determined by fitting the experimental curves of ρ versus T, and the results are listed in Table 2. For example, the parameters for the sample with x = 0.20 were determined as follows: first, it can be seen that the curve of ρ versus (T₁ + T)³ can be approximated as a straight line at lower temperatures by adjusting the parameter T₁, as shown in Fig. 6(a). We then determined the residual resistivity, ρ₀, and the parameter a₁ from the intercept and slope, respectively, of the straight line in Fig. 6(a). Second, an approximate straight line of ln ρ versus 1/k_BT can be seen at higher temperatures, as shown in Fig. 6(b). We obtained the approximate values of ln a₃ and E₃ from the intercept and slope, respectively, of the straight line in Fig. 6(b). Third, we determined the parameters a₂ and E₂ by fitting the curves of ρ versus T. In this fitting process, the parameters ρ₀, a₁, and T₁ remained constant, but a₃ and E₃ required slight adjustments. The curves of observed ρ, and fitted ρ₁, ρ₂ and ρ₃ versus T are shown in Fig. 6(c). Finally, the dependence of the observed (data points) and fitted (curve) ρ of La_0.8Sr_0.2MnO₃ on the test temperature, T, is shown in Fig. 6(d). It can be seen that the fitted curve is very close to the observed results.

Table 2 Parameters of La_1−xSr_xMnO₃ obtained by fitting the experimental curves of resistivity (ρ) versus test temperature (T) reported by Urushibara et al.⁴¹ The ρ₀ is the residual resistivity in ρ₁; The a₁ and T₁ are the parameters (in the ρ₁) which result from scattering by the crystal lattice; The a₁₁ and E₁ are the parameters in the ρ₁ of the samples with x = 0.10 and 0.15; The a₂ and E₂ are the parameters (in the ρ₂) which originate from the spin orientations of the itinerant electrons and local electrons of the outer O 2p and Mn 3d orbits that deviate from the orientation of their ground states because of thermal fluctuation. The a₃ and E₃ are the parameters (in the ρ₃) which originate from the spin-independent transition

Sr content x	ρ0 (Ω cm)	a1 (Ω cm K^−3)	T1 (K)	a11 (Ω cm)	E1 (eV)	a2 (Ω cm)	E2 (eV)	a3 (Ω cm)	E3 (eV)
0.00	—	—	—	—	—	—	—	0.2440	0.175
0.05	—	—	—	—	—	—	—	0.0220	0.170
0.10	—	—	—	0.009	0.15	4 × 10^5	0.115	0.0050	0.150
0.15	—	—	—	0.0014	0.08	1 × 10^11	0.515	0.0115	0.075
0.175	9.5 × 10^−4	2.4 × 10^10	10	—	—	8 × 10^10	0.650	0.0112	0.0515
0.20	2.3 × 10^−4	1.5 × 10^10	20	—	—	4.5 × 10^10	0.730	0.0095	0.0354
0.30	8.5 × 10^−5	5.5 × 10^−11	30	—	—	9 × 10^9	0.860	0.0084	1 × 10^−4
0.40	7.5 × 10^−5	4.2 × 10^−11	35	—	—	7 × 10^9	0.880	0.0041	5 × 10^−5

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Fig. 6 Fitting process of the temperature (T) dependence of the resistivity (ρ) for single-crystalline La_0.8Sr_0.2MnO₃. The data points in (a, b and d) represent values observed by Urushibara et al.⁴¹ (a) Fitting results of the curve of ρ versus (T₁ + T)³ at lower temperatures. (b) Fitting results of the curve of ln ρ versus 1/k_BT at higher temperatures. (c) Curves of observed ρ (squares),⁴¹ fitted ρ₁ (triangles), fitted ρ₂ (inverted triangles), and fitted ρ₃ (circles) versus T. (d) Dependence of observed (data points⁴¹) and fitted (curve) ρ on T.

For the antiferromagnetic semiconductor samples with x = 0.00 and 0.05, the curves of ρ versus T can only be fitted by the equation The values of the fitted parameters a₃ and E₃ are listed in Table 2.

For the samples with x = 0.10 and 0.15, the curves of ρ versus T can be fitted using eqn (1) and (2), but the expression must substitute for ρ₁ in eqn (2). The values of the fitted parameters a₁₁, E₁, a₂, E₂, a₃, and E₃ are listed in Table 2.

For the polycrystalline samples of La_0.6Sr_0.4Fe_xMn_1−xO₃ (0.00 ≤ x ≤ 0.30) prepared in this work, scattering by the crystallite interfaces affected the sample resistivity, ρ, and the equivalent circuit is shown in Fig. 4(b). We first fitted the observed curves of ρ versus T using eqn (1) and (2), and the effect of scattering at the crystallite interfaces was then analyzed, as discussed in following Section 4.3. The fitted curves of ρ versus T for La_0.6Sr_0.4Fe_xMn_1−xO₃ (0.00 ≤ x ≤ 0.30) are shown in Fig. 3. It can be seen that the fitted curves are very close to the observed results. The values of the fitted parameters are listed in Table 3.

Table 3 Parameters of La_0.6Sr_0.4Fe_xMn_1−xO₃ obtained by fitting the experimental curves of ρ versus T from this work. The ρ₀ is the residual resistivity in ρ₁; The a₁ and T₁ are the parameters (in the ρ₁) which result from scattering by the crystal lattice; The a₂ and E₂ are the parameters (in the ρ₂) which originate from the spin orientations of the itinerant electrons and local electrons of the outer O 2p and Mn 3d orbits that deviate from the orientation of their ground states because of thermal fluctuation. The a₃ and E₃ are the parameters (in the ρ₃) which originate from the spin-independent transition. All these parameters include the effect of the crystallite interfaces

Fe content x	ρ0 (Ω cm)	a1 (Ω cm K^−3)	T1 (K)	a2 (Ω cm)	E2 (eV)	a3 (Ω cm)	E3 (eV)
0	0.0072	2.794 × 10^−10	150	0.05	0.060	0.0044	0.0650
0.05	0.0172	1.050 × 10^−9	107	0.40	0.090	0.0020	0.0865
0.1	0.0737	1.116 × 10^−8	63	25	0.110	0.0019	0.0995
0.15	1.1000	1.159 × 10^−6	20	20 000	0.120	0.0018	0.1070
0.2	—	—	—	—	—	0.0030	0.1140
0.25	—	—	—	—	—	0.0056	0.1200
0.3	—	—	—	—	—	0.0114	0.1245

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4. Discussion

4.1 Spin-dependent and spin-independent electrical transport in perovskite manganites, and their difference from free electron transport in magnetic metals

Fig. 6(c) shows that at lower temperatures, the spin-independent resistivity, ρ₃, of La_0.8Sr_0.2MnO₃ prepared by Urushibara et al. was so high that the electrical transport occurred along the spin-dependent channel, and the value of ρ was close to ρ₁, which increased with increases in the test temperature T owing to the thermal vibration of ions. When T was close to T_C, the transition probability of the itinerant electrons decreased rapidly and ρ₂ increased rapidly because the spin orientations of the itinerant electrons and the electrons in the outer O 2p and Mn 3d orbits deviated from the direction of their ground states. When T was higher than T_C, the spin orientations of the itinerant electrons changed into disorder, and the electrical transport became spin-independent along both O–A–O–A–O and O–B–O–B–O ionic chains.

It should be noted that the resistivity of La_0.6Sr_0.4MnO₃, also prepared by Urushibara et al., was 84 μΩ cm at 2 K,⁴¹ which is far higher than the resistivities of the magnetic metals Fe (8.6 μΩ cm), Ni (6.14 μΩ cm), and Co (5.57 μΩ cm). Therefore, it can be concluded that the spin-dependent transport of itinerant electrons in perovskite manganites below T_C is distinctly different from the spin-independent transport of free electrons in magnetic metals.

4.2 Effect of proportion of the antiferromagnetic phase on the resistivity of La_0.6Sr_0.4Fe_xMn_1−xO₃ (0.00 ≤ x ≤ 0.30)

Our previous investigation^37–39 showed that all Mn and Fe ions are trivalent in La_0.6Sr_0.4Fe_xMn_1−xO₃. According to the IEO model, an itinerant electron with up-spin in a sample with no Fe doping moves along the O²⁻–Mn³⁺–O¹⁻–Mn³⁺–O²⁻ ion chain, which always occupies the highest energy level whenever it reaches a Mn³⁺ or O²⁻ ion, as shown in Fig. 7(a). In this transition process, the itinerant electron consumes a small fraction of the energy of the system. When an itinerant electron with up-spin moves along the O²⁻–Fe³⁺–O¹⁻–Mn³⁺–O²⁻ ion chain, it always occupies the highest energy whenever it reaches a Mn³⁺ or O¹⁻ ion, but the lowest energy level of an Fe³⁺ ion, as shown in Fig. 7(b), because the Fe³⁺(3d⁵) cation couples antiferromagnetically with the Mn³⁺(3d⁴) cation (see the IEO model). In this transition process, the itinerant electron must consume more energy of the system, resulting in lower transition probabilities for the itinerant electrons and increased resistivity. Therefore, the resistivity increases with increasing proportion of the antiferromagnetic phase. As shown in Fig. 7(a and b), the spin orientation of the majority of 3d electrons in Mn³⁺ cations is up; the spin orientation of the majority of local 3d electrons in Fe³⁺ cations and local O 2p electrons in O¹⁻ anions is down.

Fig. 7 Schematic diagrams of the transit of itinerant electrons in a spin-dependent process. (a) An itinerant electron with up-spin moves along the O²⁻–Mn³⁺–O¹⁻–Mn³⁺–O²⁻ ion chain. (b) An itinerant electron moves along the O²⁻–Fe³⁺–O¹⁻–Mn³⁺–O²⁻ ion chain. Here, the majority spin orientation of 3d electrons in Mn³⁺ cations is up; the majority spin orientation of 3d electrons in Fe³⁺ cations and O 2p electrons in O¹⁻ anions is down.

The difference between the energy consumed by the itinerant electrons in polycrystalline La_0.6Sr_0.4MnO₃ and La_0.6Sr_0.4Fe_0.1Mn_0.9O₃ was estimated from their Curie temperatures (364.5 and 267.9 K, respectively; see Table 1) to be 8.3 meV.

4.3 Resistivity (ρ₄) originating from scattering at the crystallite interfaces of polycrystalline samples

It is interesting to compare the resistivity of the polycrystalline La_0.6Sr_0.4MnO₃ sample (ρ_P) prepared in this work to that of the single-crystalline La_0.6Sr_0.4MnO₃ sample (ρ_S) reported by Urushibara et al.,⁴¹ as shown in Fig. 8(a). We found that ρ_P is 97 times (at 50 K) and 9 times (at 360 K) the value of ρ_S, which suggests that the resistivity (ρ₄) originating from scattering at the crystallite interfaces of the polycrystalline sample was far higher those originating from lattice thermal vibrating (ρ₁) and spin-dependent scattering (ρ₂). Therefore, for the polycrystalline manganites, the TCET model can be represented by the equivalent circuit shown in Fig. 4(b). Fig. 8(b) shows a comparison of ρ₄ with ρ_P, where ρ₄ was estimated using ρ₄ = ρ_P − ρ_S.

Fig. 8 (a) Temperature dependence of the resistivities, ρ_P and ρ_S,⁴¹ of polycrystalline and single-crystalline La_0.6Sr_0.4MnO₃, and their ratio, ρ_P/ρ_S. (b) Temperature dependence of the observed resistivity, ρ_P, and the estimated resistivity resulting from crystallite interface scattering, ρ₄, of polycrystalline La_0.6Sr_0.4MnO₃, (c) Temperature dependence of ρ_P of La_0.6Sr_0.4Fe_0.3Mn_0.7O₃ and ρ_S of LaMnO₃, and their ratio, ρ_P/ρ_S. (d) Dependence of the equivalent cubic cell constant, a_e, of the single-crystalline La_1−xSr_xMnO₃ samples⁴¹ and the polycrystalline La_0.6Sr_0.4Fe_xMn_1−xO₃ samples on the doping level, x.

According to the report by Urushibara et al.,⁴¹ the temperature T_MI for maximum resistivity is very close to the value of T_C for single-crystalline La_1−xSr_xMnO₃ (0.175 ≤ x ≤ 0.40). However, as shown in Table 2, the values of T_MI were distinctly lower than T_C for the polycrystalline La_0.6Sr_0.4Fe_xMn_1−xO₃ (0.00 ≤ x ≤ 0.15) samples. This was simply due to ρ₄ of the polycrystalline samples. For the La_0.6Sr_0.4MnO₃ sample, the temperature at which ρ₄ reached a maximum was 287 K (see Fig. 8(b)), which is distinctly lower than its T_C (364.5 K).

4.4 Effect of crystal-cell constants on the samples' resistivity

For the antiferromagnetic semiconductor, polycrystalline La_0.6Sr_0.4Fe_0.3Mn_0.7O₃, its resistivity was far lower than that of single-crystalline LaMnO₃, with a ρ_P/ρ_S ratio of 0.51% at 270 K and 0.58% at 300 K, as shown in Fig. 8(c). A comparison of Fig. 8(a) and (c) shows an interesting phenomenon: the change in amplitude of the resistivity from that of the polycrystalline ferromagnetic conductor, La_0.6Sr_0.4MnO₃, to that of the antiferromagnetic semiconductor, La_0.6Sr_0.4Fe_0.3Mn_0.7O₃, is far lower than the change in amplitude of the resistivity from that of the single-crystalline ferromagnetic conductor, La_0.6Sr_0.4MnO₃, to that of the antiferromagnetic semiconductor, LaMnO₃. The primary causes of this phenomenon may be the distinct difference in the crystal-cell constants of the two systems, and the proportion of the antiferromagnetic phase. The equivalent cubic cell constants (a_e) of single-crystalline La_1−xSr_xMnO₃ (ref. 41) and polycrystalline La_0.6Sr_0.4Fe_xMn_1−xO₃ are shown in Fig. 8(d), which were calculated using the average volume per formula of each sample. The samples of La_0.6Sr_0.4Fe_xMn_1−xO₃ (0.0 ≤ x ≤ 0.3) and La_1−xSr_xMnO₃ (0.175 ≤ x ≤ 0.40) exhibited rhombohedral structures, while La_1−xSr_xMnO₃ (0.00 ≤ x ≤ 0.15) was reported to have an orthorhombic structure.⁴¹ The value of a_e of the polycrystalline La_0.6Sr_0.4MnO₃ is very close to that of the single-crystalline La_0.6Sr_0.4MnO₃. However, the value of a_e of polycrystalline La_0.6Sr_0.4Fe_xMn_1−xO₃ prepared in our study increased from 3.872 Å (x = 0.00) to 3.878 Å (x = 0.30), while the value of a_e of single-crystalline La_1−xSr_xMnO₃ was reported to increase from 3.874 Å (x = 0.40) to 3.942 Å (x = 0.00). According to previous investigations,^37,38 there are only Mn³⁺ cations in La_1−xSr_xMnO₃ (0.15 ≤ x ≤ 0.40). Therefore, the increase in resistivity from x = 0.40 to x = 0.15 occurred because of the increase in the crystal-cell constant, while the increase in resistivity from x = 0.15 to x = 0.00 occurred because of increases in both the crystal-cell constant and the proportion of the antiferromagnetic phase.

Obviously, an increase (in increments of 0.01 Å, for example) in a_e may rapidly reduce the transition probability of the itinerant electrons whenever spin-dependent or spin-independent transitions take place, resulting in an increase in the resistivity. This may be the reason the samples of La_0.9Sr_0.1MnO₃ and La_0.85Sr_0.15MnO₃ exhibited high resistivity at low temperatures, as shown in Fig. 5. For the sample of La_0.85Sr_0.15MnO₃, ρ > 500 Ω cm when T < 10 K because of the large crystal lattice constant, a_e, which resulted in a lower spin-dependent transition probability of the itinerant electrons. The value of ρ decreased with increasing T below 202 K because of the thermal energy of the itinerant electrons, which resulted in an increase in their spin-dependent transition probability. In the temperature range of 202–234 K, ρ increased with increasing T because the spin orientation of the itinerant electrons and local electrons of the outer O 2p and Mn 3d orbits deviated from their ground state direction, resulting in a decrease in their spin-dependent transition probability.

4.5 Activation energy (E₃) of spin-independent electron transition

The values of the activation energy, E₃, of the spin-independent electron transition in the polycrystalline La_0.6Sr_0.4Fe_xMn_1−xO₃ (0.00 ≤ x ≤ 0.30) samples ranged from 65.0 to 124.5 meV (see Table 3), which are similar to the values (90.5 to 148.3 meV) of the polaron activation energy, E_p, obtained by Liu et al. with the CCDC model for La_0.7Ca_0.3Ti_xMn_1−xO₃ (0.00 ≤ x ≤ 0.07).¹⁸ The comparable results indicate that the TCET model in this work is similar to the CCDC model.^{13,14,18–20} However, the physical mechanism of the TCEP model, which includes spin-dependent and spin-independent electron transitions, is easier to be understood than that of the CCDC model, which assumes that the current-carrier density changes sharply at the magnetic-transition temperature.

Tables 2 and 3 show that among the ferromagnetic samples, E₃ of polycrystalline La_0.6Sr_0.4MnO₃ (65.0 meV) is far higher than that of single-crystalline La_0.6Sr_0.4MnO₃ (0.05 meV) owing to scattering at the crystallite interfaces in the polycrystalline sample. However, among the antiferromagnetic samples, E₃ of polycrystalline La_0.6Sr_0.4Fe_0.3Mn_0.7O₃ (124.5 meV) is distinctly lower than that of antiferromagnetic LaMnO₃ (175 meV), which may have been caused by the different crystal-cell constants, as discussed in the previous section.

The value of the parameter a₃ (0.0044 Ω cm, see Table 3) of the polycrystalline conductor, La_0.6Sr_0.4MnO₃, is very close to that of the single-crystalline conductor, La_0.6Sr_0.4MnO₃ (0.0041 Ω cm, see Table 2). However, the value of a₃ (0.0114 Ω cm) of the polycrystalline semiconductor, La_0.6Sr_0.4Fe_0.3Mn_0.7O₃, is far lower than that of the single-crystalline semiconductor, LaMnO₃ (0.2440 Ω cm). This may be attributed to the effect of the crystal-cell constant, as discussed in Section 4.4.

4.6 Regression coefficients of ρ₁ and ρ₂

As shown in eqn (2) and Tables 2 and 3, the spin-dependent resistivities, ρ₁ and ρ₂, are represented by parameters (or regression coefficients) that play interesting roles:

(i) The resistivity ρ₁ in eqn (2) is determined by three parameters: the residual resistivity, ρ₀; the parameters a₁ and T₁ resulting from scattering by the crystal lattice. All three parameters of the polycrystalline La_0.6Sr_0.4Fe_xMn_1−xO₃ samples—ρ₀ = 0.0072 to 1.100 Ω cm, a₁ = 2.794 × 10⁻¹⁰ to 1.159 × 10⁻⁶ (Ω cm) K⁻³, T₁ = 150 to 20 K—are higher than those of the single-crystalline La_1−xSr_xMnO₃ samples—ρ₀ = 0.075 to 0.95 mΩ cm, a₁ = 4.2 × 10⁻¹¹ to 2.4 × 10⁻¹⁰ (Ω cm) K⁻³, and T₁ = 35 to 10 K—because of the effect of crystallite interfaces in the polycrystalline samples.

(ii) The resistivity ρ₂ in eqn (2) is determined by two parameters: the amplitude a₂; the activation energy E₂, which originated from spin-dependent scattering. The value of E₂ of the single-crystalline La_1−xSr_xMnO₃ samples increased from 0.515 eV (x = 0.15) to 0.88 eV (x = 0.40), which was the result of competition between two factors. One factor was the increase in the cant angle between magnetic moments of Mn³⁺ cations from 0 (x = 0.15) to 34.9° (x = 0.40).³⁷ Another factor was the decrease in crystal cell volume with increasing x. The value of E₂ of the polycrystalline La_0.6Sr_0.4Fe_xMn_1−xO₃ samples increased from 0.06 eV (x = 0.00) to 0.12 eV (x = 0.15), which can be attributed to the increase in Fe-doping level because the magnetic moments of Fe³⁺ cations couple antiferromagnetically with those of Mn³⁺ cations.³⁹ The a₂ values of the polycrystalline La_0.6Sr_0.4Fe_xMn_1−xO₃ samples are far lower than those of the single-crystalline La_1−xSr_xMnO₃ samples, possibly because of the intense scattering at the crystallite interfaces to reduce the effect of spin-dependent scattering.

5. Conclusion

In order to explain the dependence of the resistivity (ρ) of perovskite manganites on the test temperature (T), we propose a model with two channels of electrical transport (TCET): a spin-independent channel (with resistivity ρ₃) and a spin-dependent channel (with resistivity ρ₁ + ρ₂). At low temperature, ρ₃ is very high, and the electrical transport occurs along the spin-dependent channel. When T is far lower than the Curie temperature (T_C), ρ is very close to ρ₁, which increases with increasing T owing to the thermal vibration of the crystal lattice. When T is close to T_C, ρ₂ increases rapidly owing to the spin orientations of the itinerant electrons, and the local electrons of the outer O 2p and Mn 3d orbits deviating from the orientation of their ground states. Therefore, the spin-dependent electrical transport in perovskite manganites at temperatures below T_C is distinctly different from the spin-independent electrical transport in metals. When T is higher than T_C, the spin orientations of the itinerant electrons change into disorder, and electrical transport occurs along both channels, with the spin-independent transition, ρ, being close to ρ₃.

Several determining factors of the resistivity have been discussed: (i) the increase in the proportion of the antiferromagnetic phase may result in an increase in ρ. (ii) The resistivity ρ₄ originating from crystallite-interface scattering in the polycrystalline samples is far higher than ρ₁ and ρ₂ at temperatures below T_C. (iii) Increases in the crystal-cell constant, in increments as small as 0.01 Å, may result in rapid increases in ρ, which may be the reason of the high resistivity of La_0.9Sr_0.1MnO₃ and La_0.85Sr_0.15MnO₃ at low temperatures.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (NSF-11174069), the Natural Science Foundation of Hebei Province (A2015205111), the Key item Science Foundation of Hebei Province (Grant No. 16961106D), and the Young scholar Science Foundation of the Education Department of Hebei Province (QN2016015).

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