Absence of Weak Localization Effects in Strontium Ferromolybdate

1. Introduction

In a disordered electronic system, the electron motion is diffusive due to scattering at non-uniformities. Weak localization is a physical effect which occurs in disordered electronic systems at very low temperatures. The effect manifests itself as a correction Ds to the conductivity (or correspondingly the resistivity) of a metal or semiconductor arising in the case that the mean free path l is in the order of the wavelength λ_F = 2π/k_F of the carrier wavefunctions, i.e. k_Fl~1, with k_F denoting the Fermi wave vector. The weak localization correction comes from quantum interference of back-scattered electrons. Labeling all the trajectories by the time t it takes for a classical particle to go around a loop, the classical probability dP that the diffusing particle returns into the phase volume dV at a given time t is given by [1] (p. 153)

$$dP=\frac{dV}{{(4\pi Dt)}^{d/2}}$$

(1)

where P is the probability for a quantum mechanical particle to return to the starting point (the wave function of both time-reversed paths will constructively interfere with each other), D= v_F²t/d is the diffusion constant, t is the elastic scattering time and d is the dimension. The relevant phase volume can be estimated as v_F dt (δρδφ)^d−1[2,3], where v_F= (2E_F/m_e)^1/2 is the Fermi velocity, E_F the Fermi energy, m_e the electron mass, δρ and δφ characterize the transverse distance between the electron trajectories 1 and 2 at the intersection point and δφ is the intersection angle between the both trajectories. For the interference between paths 1 and 2 to be effective, the uncertainty relation should hold

$$p_F\delta \rho \delta \varphi =\hslash k_F\delta \rho \delta \varphi =h$$

(2)

where p_F = m_ev_F is the Fermi momentum and h the Planck constant. This leads to a relative change in conductivity of two and three dimensional systems yielding [1] (p. 183):

$$\begin{gathered} \frac{\delta {\sigma }_{2D}}{{\sigma }_0}=-\frac{2}{\pi k_F{l}_e}\ln \frac{l_{\Phi }}{l_e}, \\ \frac{\Delta {\sigma }_{3D}}{{\sigma }_0}=-\frac{3}{2{(k_F{l}_e)}^2} \end{gathered}$$

(3)

where s₀ is the residual low temperature conductivity taken as the Drude conductivity, i.e. s_D = e²n_et_e/m_e, determined by relaxation time t of the dominating charge scattering mechanism, the electron charge e, the electron density n_e, the elastic scattering time τ_e and the electron mass m_e. In eq. (3), the negative sign is due to the fact that the returning trajectory should arrive at the intersection point with the momentum almost opposite to the initial one. This means that interference lowers the conductivity. Note that for small sizes b the volume (Dt)^d/2 should be replaced by (Dt)^d/2b^3-d since the charge carrier has a chance to diffuse repeatedly from one wall to the other and the probability of finding it at any point across films or wires limited in size will be the same [4]. However, we are considering macroscopic sizes rather than sizes of the order of atomic distances. The lower cutoff τ_e is justified with the fact that within a time τ_e no elastic scattering occurs, and therefore there will be no closed trajectories. The upper limit is given by [5]:

$${\tau }_{\Phi }^{-1}\sim {\left(\frac{T}{D^{d/2}{N}_0{a}^{3-d}}\right)}^{2/(4-d)},$$

(4)

where D = (v_F²t_e/d) is the diffusion constant, N₀ is the one-spin density of states, d the characteristic sample dimension. The phase coherence time t_ϕ defines the phase coherence length l_ϕ = (Dt_ϕ)^1/2. This yields conductivity corrections for two and three dimensional systems amounting in terms of quantum conductance e²/h to [1] (p. 282):

$$\begin{gathered} \Delta {\sigma }_{2D}=-\frac{2e^2}{\pi h}\ln \frac{l_{\phi }}{l_e}=-\frac{e^2}{\pi h}\ln {\left(\frac{l_{\phi }}{l_e}\right)}^2, \\ \Delta {\sigma }_{3D}=-\frac{e^2}{\pi h}\left(\frac{1}{l_e}-\frac{1}{l_{\phi }}\right), \end{gathered}$$

(5)

with l_e the mean free path of the electron.

Temperature modifies the phase coherence length l_ϕ and, therefore, modifies the weak localization correction. The temperature dependence of l_φ can be described as [6,7]

$$\frac{1}{l_{\varphi }^2(T)}=\frac{1}{l_{\varphi }^2(0)}+A_{ee}{T}^{p_1}+A_{ep}{T}^{p_2},$$

(6)

where l_φ(0) is the zero-temperature phase coherence length and A_eeT^p1 and A_epT^p2 represent the contributions from two different dephasing mechanisms, e.g. electron-electron (ee) and electron-phonon (ep) interactions, respectively. Note that the temperature exponents p₁ and p₂ changes fundamentally with the system dimensionality [8]. If electron-electron interaction is the dominant dephasing mechanism (also denoted as Nyquist dephasing mechanism), this gives rise to τ_ee ∝ T^-p with values of p equal to 0.66, and 1 for d = 1 and d = 2, respectively [1] (chapter 13.6.4), [5]. For d = 3 eq. (4) yields p = 2 while the result in [1] (p. 509) is p = 1.5. The electron-phonon interaction decoherence mechanism would provide a temperature dependence of the dephasing time t_ϕ ∝T^-3 [9]. Typically, one finds τ_φ ≈ τ_ep ∝ T^-p with the exponent of temperature p ≈ 2–4.

In metals and semiconductors at low temperatures, quasielastic (small energy transfer) electron-electron scattering is the dominant dephasing process. Assuming only one dephasing process and accounting that l_φ(0) is in the order of 50 nm to several micrometers, that is, usually l_φ(0) >> l_φ(T) [6,7,10,11,12], a simplified approximation formula is given by [11]:

$$\frac{1}{l_{\varphi }^2(T)}\approx A´T^{p_1},$$

(7)

with p₁ = 1 for quasielastic electron-electron scattering in agreement with experimental results in Bi₂Te₃ thin films [13], 50 nm-thick Cd₃As₂ films [11], Bi₂Te₃ single crystals [7], Mo_xW_1-xTe_2+d ultrathin films [14]. As a result, we obtain for quasi-elastic electron-electron scattering:

$$l_{\phi }^2=l_{\phi }^2(T_0)\cdot \frac{T_0}{T},$$

(8)

Consequently, eq. (5) transforms to

$$\begin{gathered} \begin{array}{l}\Delta {\sigma }_{2D,WL}=-\frac{e^2}{\pi h}\ln \left(\frac{l_{\varphi }^2(T_0)}{l_e^2}\cdot \frac{T_0}{T}\right)=\frac{e^2}{\pi h}\ln \left(\frac{l_e^2}{l_{\varphi }^2(T_0)}\cdot \frac{T}{T_0}\right)=\\ =\frac{e^2}{\pi h}\left[\ln \left(\frac{l_e^2}{l_{\varphi }^2(T_0)}\right)+\ln \left(\frac{T}{T_0}\right)\right]\end{array}, \\ \delta {\sigma }_{3D,WL}=-\frac{e^2}{\pi }\left(\frac{1}{l_e}-\frac{1}{l_{\varphi }}\right)=-\frac{e^2}{\pi \cdot l_e}+\frac{e^2}{\pi \cdot l_{\varphi }(T_0)}{\left(\frac{T}{T_0}\right)}^{1/2} \end{gathered}$$

(9)

Finally, the total conductivity consisting of 2D at high magnetic fields and 3D contributions is written as:

$$\sigma (T)={\sigma }_0+A\cdot \ln \left(\frac{T}{T_0}\right)+B\cdot {\left(\frac{T}{T_0}\right)}^{1/2},$$

(10)

with

$${\sigma }_0={\sigma }_D-\frac{e^2}{\pi hb}\cdot \ln \left(\frac{l_{\varphi }^2(T_0)}{l_e^2}\right)-\frac{e^2}{\pi h\cdot l_e},$$

(11)

and the coefficients:

$$A=\frac{e^2}{\pi hb},B=\frac{e^2}{\pi h\cdot l_{\varphi }(T_0)},$$

(12)

where b is the effective thickness of the 2D layer introduced to convert a 2D conductivity to a 3D one. A minimum of conductivity appears only when the second right-side term in Equation (10) changes sign, i.e.:

$$B=-\frac{2A}{{(T_{\max }/T_0)}^{1/2}}\underset{T_0=T_{\max }}{=}-2A,$$

(13)

With regard to Equation (10), quantum correction to residual conductivity may be written as [15]:

$$\Delta \sigma (T)=A\prime \ln T+B\prime T^{1/2},$$

(14)

Assuming that Mathiessen’s rule holds, and representing resistivity as a sum of elastic and inelastic contributions where the latter increase with increasing temperature due to a power law (e.g. a term CTⁿ), the resistivity data can be be fitted to [16,17]:

$$\rho (T)=\frac{1}{{\sigma }_0+A\prime \cdot \ln T+B\prime \cdot T^{1/2}}+C_n{T}^n,$$

(15)

with n equal to 3/2, 2, 3, depending on the dominant scattering mechanism [18].

Separate conductivity quantum correction terms in the form A´lnT and B´T^1/2 were introduced for La_0.7Ca_0.3MnO₃ by Kumar et al [19]. Here, the first term A´lnT was attributed also to the Kondo effect [20]. In order to account for higher order scattering mechanisms and to extend the analytical description to higher temperatures, an additional term CTⁿ was included. The electron-electron interaction term B´T^1/2 was applied also to La_0.7−xY_xSr_0.3MnO₃ (0 ≤ x ≤ 0.2) ceramics of 1 mm thickness [21], to La_0.7A_0.3MnO₃ (A=Ca,Sr,Ba) ceramics [22], to La_0.6Re_0.1Ca_0.3MnO₃ (Re = Pr, Sm, Gd, Dy) ceramics of 1 mm thickness all three prepared by conventional solid state reaction [23], to SrRuO₃ thin films with a thickness of 6 to 8 nm deposited by pulsed laser deposition onto (100)SrTiO₃ substrates [16], to ultrathin La_0.7Sr_0.3MnO₃ films (3.5 to 40 nm), deposited by molecular beam epitaxy [17], and to metallic SrRuO₃ and LaNiO₃ thin films deposited by pulsed laser deposition with a thickness of 6 and 240 nm, respectively [24].

In the La_0.7Sr_0.3MnO₃ ultrathin film case, a crossover from T^1/2 to a lnT behavior in the low-temperature resistivity dependence with decreasing thicknesses was related to a change in the dimensionality of the system, going from 3D for samples thicker than 20 nm to 2D in the limit of ultrathin samples. The origin of this effect is that in the case of a film thickness larger than the Landau orbit length L_H = (ħ/2eB)^1/2 the system behaves essentially as 3D while in the opposite electron confinement results in a 2D behavior of the system. In the 3D case of metallic and ferromagnetic SrRuO₃ and of metallic and paramagnetic LaNiO₃ epitaxial thin films the term B´T^1/2 was spitted into two terms where in the 3D case first term b₁T^p/2 accounts for the weak localization and the b₂T^1/2 term stands for the renormalized electron-electron interaction quantum corrections [25]. In the 2D case, both these quantum corrections to conductivity have a similar temperature dependence. Here, only a lnT term remains. The quantum correction of conductivity in metallic LaNi_1-xCo_xO₃ (0 ≤ x ≤ 0.75) below 2 K follows a power law BT^m where away from the metal-insulator transition (x ≤ 0.4) m takes a value of m = 0.3…0.4. Such power-law conductivities are seen at the metallic side of the metal-insulator transition also for other ABO₃ oxides [26].

In this work, we attribute the temperature dependence of the SFMO ceramic resistivity in the absence of a magnetic field to the fluctuation induced tunneling model and the decrease of resistivity in the temperature range from 409 K up to 590 K not to localization effects but to adiabatic small polaron hopping. Both fluctuation induced tunneling and adiabatic small polaron hopping do not favor quantum interference. Also, we demonstrate that the resistivity upturn behavior of SFMO cannot be explained by the weak localization effect.

2. Methods

First we estimate the coefficient A´ in Equation (15). Taking l_F < 0.46 nm at room temperature [27] and b = 20 nm [17], the ratio of the coefficients A´/B´ will be in the order of 10^-2. Since for arguments larger than one the natural logarithm function will be smaller than the root function, the logarithmic term can be neglected. A further indication of small A´ coefficients arises from the depth of the conductance minimum, i.e. the difference r(0)-r(T_min), which changes with increasing magnetic flux [19]. Values of above 35 mW⋅cm in the absence of a magnetic field down to 2.4 mW⋅cm at 7 T [28] yield coefficients A´ = 10^-2 ⋅(r(0)-r(T_min)) [20,29]. In La_0.7Sr_0.3MnO₃ ultrathin films (3.5 to 40 nm), deposited by molecular beam epitaxy [17], the agreement of fits of conductivity data to equation (15) with only the A´lnT term (B`= 0) was much worse than in the case with only the B´T^1/2 term (A´= 0). Therefore, we neglect in the following the term A´lnT. This corresponds to the common practice of describing similar materials like La_0.7−xY_xSr_0.3MnO₃ (0 ≤ x ≤ 0.2) ceramics of 1 mm thickness prepared by conventional solid state reaction [21], La_0.7A_0.3MnO₃ (A=Ca,Sr,Ba) ceramics prepared by conventional solid state reaction [22], and SrRuO₃ thin films with a thickness of 6 to 8 nm deposited by pulsed laser deposition onto (100)SrTiO₃ substrates [16].

The coefficient B´ is given by [15]:

$$B´=0.0309\frac{e^2}{\hslash }\cdot \sqrt{\frac{k}{\hslash D}}=\frac{2.72149\cdot {\Omega }^{-1}{\mathrm{K}}^{-1/2}{\mathrm{s}}^{-1/2}}{\sqrt{D[{\mathrm{m}}^2/\mathrm{s}]}}.$$

(16)

with ħ = h/2p the Planck constant expressed in J s radian^-1. For a mean free path l_e of 0.46 nm at room temperature [27] and a carrier relaxation time t of 1.6×10^-14s [30], the carrier diffusion constant will be 1.32×10^-5 m² s^-1 yielding a coefficient B´≈ 748 W^-1 m^-1 K^-1/2 in satisfactory agreement with values of B´≈ 360…500 W^-1 m^-1 K^-1/2 in in (Ni_0.5Zr_0.5)_1-xAl_x metallic glasses [31] and with a universal value of B´≈ 600 W^-1 m^-1 K^-1/2 of amorphous and disordered metals [32]

The coefficient C_n was calculated as follows: We start with Drude conductivity:

$${\rho }_D=\frac{\hslash k_F}{n_e{e}^2{l}_e},$$

(17)

where n_e the electron density and e the electron charge. Taking n_e= 1.1×10²⁸ m^-3 [33] we obtain ħk_F = 4.8⋅10^-25 Jsm^-1 and

$${\rho }_D=\frac{1.87\cdot {10}^{-15}\Omega \mathrm{m}}{l_e[\mathrm{m}]},$$

(18)

We assume electron mean free paths l_e = 0.46 nm and 1.11 nm at room temperature and 4 K, respectively, calculated from the ordinary Hall coefficient [27]. This yields r_D = 4.07 µWm at room temperature and r_D = 1.68 µWm at 4 K in satisfactory agreement with experimental data of single crystal SFMO in [34]. Now we assume that in a ferromagnetic state below room temperature magnetic scattering controls electrical transport in SFMO at low temperatures [27]. The mean free path of electron scattered by a spin wave with energy E_s travelling through the bcc I4/mmm lattice in thermal equilibrium at a temperature T, is given by [35]:

$$l_e=\frac{S^2{V}^{1/3}{\theta }^{-7/2}}{\pi \zeta (3/2)(E_{\sigma }/kT)}\approx 0.38\cdot S^{9/2}a\cdot {\left(\frac{J}{kT}\right)}^{5/2},$$

(19)

where S is the effective spin S_eff =(S_FeS_Mo)^1/2 = (1/2⋅5/2)^1/2=1.118, V the unit cell volume, q a dimensionless temperature, z the Rieman Zeta function, a the lattice constant, and J the exchange constant of the 180° Fe-O-Fe interaction amounting to -25 K [36,37]. Finally, we arrive at:

$${\rho }_D=2.413\cdot {10}^{-9}\cdot T^{5/2}\Omega \mathrm{m},$$

(20)

Compared to the experimental value of R_2.5=1.4×10^-11 WmK^-5/2 in a relation r = r₀+R_2.5T^2.5 the calculated value of R_2.5 is overestimated by almost two orders of magnitude in part due to the approximation of the Fermi surface as a sphere and the disregard of additional s-d transitions in transition metals which reduce the mean free path [38]. Also, the value of R_2.5 may be lowered by assuming a higher effective spin.

3. Results and discussion

Strontium ferromolybdate Sr₂FeMo6-d (SFMO) is a half-metallic, ferrimagnetic compound with a saturation magnetization of 4 µ_B/f.u [28]. However, SFMO does not exhibit a general metallic conductivity mechanism. In the absence of a magnetic field, the temperature dependence of conductivity of SFMO ceramics [28] is well described by the fluctuation-induced tunneling (FIT) model [39], e.g., by the presence of conducting grains separated by nanosized energy barriers where large thermal voltage fluctuations occur when the capacitance of an intergrain junction is in the order of 0.1 fF. Here, tunneling occurs between large metallic grains via the intergrain junctions with a width w and area A. The FIT model is specified by three parameters [39]: (i) The temperature T₁ characterizing the electrostatic energy of a parabolic potential barrier,

$$k{T}_1=\frac{A\cdot w\cdot {\epsilon }_o{E}_0^2}{2},$$

(21)

where k is the Boltzmann constant and the characteristic field E₀ is determined by:

$$E_0=\frac{4V_0}{e\cdot w},$$

(22)

(ii) the temperature T₀ representing T₁ divided by the tunneling constant,

$$T_0=T_1\cdot {\left(\frac{\pi \chi w}{2}\right)}^{-1},$$

(23)

with the reciprocal localization length of the wave function

$$\chi =\sqrt{\frac{2m^{*}{V}_0}{{\hslash }^2}},$$

(24)

where m*_e is the effective electron mass, and (iii) the residual resistivity r₀. The resulting resistivity of this model is then given by [39]:

$$\rho (T)={\rho }_0\exp \left(\frac{T_1}{T_0+T}\right),$$

(25)

In our case, the model parameters amount to T₀ = 141.1 K, T₁ = 25.6 K and s₀ = 1/r₀ =37.16 S/cm (cf. Figure 1). The FIT model was recently applied to intergrain tunneling in polycrystalline Sr₂CrMoO₆ and Sr₂FeMoO₆ ceramics [40], in half-metallic double-perovskite Sr₂BB’O₆ (BB’– FeMo, FeRe, CrMo, CrW, CrRe) ceramics [41] and in Ba₂FeMoO₆ thin films [42].

One feature attributed to weak localization in SFMO ceramics is the decrease of resistivity in the temperature range between 405 K and 590 K [43]. In the following this resistivity behavior should be considered more in detail. According to [43], the resistivity behavior of vacuum-annealed SFMO ceramics above room temperature is separated into three regions: (i) From 300 K up to the Curie temperature of about 405 K the electrical resistivity increases with temperature and shows metallic behavior, (ii) Above 405 K up to approximately 590 K the resistivity decreases with temperature, (iii) Finally, from 590 K up to 900 K the resistivity increases since the material becomes metallic again [43].

Another report on the electrical resistivity of SFMO indicates metallic behavior up to 420 K, a decrease of resistivity in the temperature range 420–820 K, and the reversion to metallic behavior between 820 and 1120 K [44]. A similar resistivity behavior with a resistivity maximum at about 450 K, far above the Curie temperature of ~330 K [45], was obtained for Ba₂FeMoO_6-d while Ca₂FeMoO_6-d shows solely metallic behavior in the whole temperature range 320-1120 K [44].

A more detailed consideration of the reported resistivity behavior of Sr₂FeMoO_6-d and Ba₂FeMoO_6-d above T_C [43,44], reveals a convincing fit to the adiabatic small polaron hopping model [46] (Figure 2). In the small polaron model, electrical conduction of perovskites at higher temperature, i.e., above a certain transition temperature, occurs by small polarons moving through the lattice by thermally activated jumps between neighboring sites. The transition temperature from small polaron motion in a conduction band to small polaron hopping was estimated to be in the order of θ_D/2 with θ_D the Debye temperature which amounts to 338 K for SFMO [47]. The adiabatic small polaron hopping model yields a resistivity of [46]:

$$\rho ={\rho }_{0}T\exp \left(\frac{E_a}{kT}\right)$$

(26)

where E_a is the thermal activation energy. The obtained E_a values are 0.045-0.08 eV for Sr₂FeMoO_6-δ and about 0.13 eV for Ba₂FeMoO_6-δ and are in the order of the values of other perovskites and double perovskites: La_1-xSr_xCo_1-yFe_yO₃ [48], Sr_1.6Sm_0.4MgMoO_6-δ, Sr_1.4Sm_0.6MgMoO_6-δ, and Sr_1.2Sm_0.8MgMoO_6-δ [49], as well as Sr₂Fe_1.5Mo_0.5O_6-d and Sr₂Fe_1.5Mo_0.5-xNb_xO_6-d [50].

Weak localization effects were taken into account in order to explain the presence of minima in the ρ–T curves of perovskite oxides exhibiting metallic conductivity [17,19,25,51,52]. It has been suggested [20,21] that the resistivity minimum and, consequently, the resistivity upturn at lower temperature arises from the competition of two contributions—one, usual, increasing and the other, decreasing with the increase of the temperature. In eq. (15), the corresponding terms are (B´lnT)^-1 and C_nTⁿ, respectively. The resistivity versus temperature plots of SFMO ceramics in a magnetic field [28] are very similar to the ones of La_0.7Ca_0.3MnO₃and La_0.7Sr_0.3MnO₃ thin films [17,19]. Also here, a resistivity minimum appears in polycrystalline SFMO ceramics at low temperature [28] which was explained by weak localization [52]. In this case, a quantum correction terms A_WT^p/2 [17,52] was added to the residual resistivity s₀ and an electron interaction term A_pTⁿ to the resistivity. A similar approach was applied to perovskite ceramics (LaNi_xCo_1-xO₃ and Na_xTa_yW_1-yO₃ [53], La_0.5Pb_0.5MnO₃ and La_0.5Pb_0.5MnO₃ ceramics containing 10 at.% Ag in a dispersed form [15].

To evaluate the origin of the low temperature resistivity minimum in SFMO, we fitted the experimental data [28] to eq. (15) assuming A´= 0 (Figure 3, Table 1). The increase of the s₀(B) values correspond to a negative magnetoresistance obtained in ferrimagnetic SFMO ceramics which arises due to the suppression of spin disorder by the magnetic field [27,28,43]. The fitted s₀ values describe a power law magnetic flux dependence of the magnetoresistance, -MR ∝ B^m, with a power of m = 0.284. This value lies in-between the values of m = 0.5 for metallic behavior and electron-electron interaction in the temperature range between the temperature of minimum resistivity and the Curie temperature and m = 0.1 for semiconducting conductivity behavior, both at high magnetic fluxes [54]. The fitted B´ values are three orders of magnitude lower than the ones calculated above. Consequently, the elastic scattering time should be reduced by six orders of magnitude. This is physically nonsense. Also, the C_nTⁿ term does not correspond to physically meaningful quantities. The values of the exponent n cannot be attributed to a certain electron scattering mechanism. On the other hand, electron scattering should sufficiently change in dependence on the magnetic flux. When assuming n = 2 for electron-electron scattering, and taking the value C_n = 2.16×10^-11 WmK^-2 from [33], the fit becomes of much worse quality (Figure 4). Here, a satisfactory fit occurs only for B = 0. Assuming n = 2.5 and taking the value C_n of eq. (20), the fit is even worse with a no satisfactory fit even for B = 0.

Thus, the only possible conclusion is that the resistivity upturn at low temperatures in Sr₂FeMoO₆ ceramics cannot be modeled by the weak localization correction due to quantum interference. Note that neither fluctuation induced tunneling nor adiabatic small polaron hopping provide favored conditions for quantum interference of back-scattered electrons.

4. Conclusions

We have related the temperature dependence of the SFMO resistivity in the absence of a magnetic field to the fluctuation induced tunneling model. The decrease in resistivity above the resistivity maximum around Curie temperature was attribute to adiabatic small polaron hopping instead to localization effects. Both fluctuation induced tunneling and adiabatic small polaron hopping do not favor quantum interference. This is evidenced by the observation that the resistivity upturn behavior of SFMO cannot be explained by weak localization. Consequently, to the best of our knowledge, there is still no convincing evidence for the presence of weak localization in SFMO.