Spin dynamics and relaxation in 7.6 nm thin film of La_0.7Sr_0.3MnO₃/SrTiO₃: ac magnetic susceptibility and magnetic viscosity investigations

La_1–xSr_xMnO₃ (LSMO) with the optimum composition of x = 0.3 is a well-known manganite with semi-metallic behavior, high Fermi-level spin polarization, outstanding magneto-resistance, and desirable ferromagnetism with bulk Curie temperature T_C = 370 K well above room temperature.^1–6 These characteristics make it suitable for potential applications in spintronics,^7,8 multiferroic tunnel junctions,⁹ colossal magnetoresistance,¹⁰ magnetic refrigeration,¹¹ and other applications.^3–6,12 Since the use of thin films in modern technologies for device applications is ubiquitous in order to decrease material costs and increase speed of operation of the devices, it is important to understand how the properties of thin films change with change in the thickness of the films. For LSMO thin films, it has been reported that with decrease in the thickness of the films, magnitudes of the spontaneous magnetization M_S,^2,13,14 T_C,^2,13 and magnetocaloric properties¹⁵ are also lowered. Another important issue for the LSMO thin films has been the surface reconstruction at the LSMO-substrate (SrTiO₃) interface containing oxygen vacancies and enhanced concentration of Mn³⁺ ions, which affects the properties of the films both above and below T_C.^5,6,16,17 For example, the presence of non-zero M_S even above $TC$ of the LSMO film has been attributed to these defects in the so-called dead layer.^16,17

In recent papers, we have reported on the magnetic properties of a 7.6 nm thin film of La_0.7Sr_0.3MnO₃ grown on SrTiO₃ (001) substrate using pulsed laser deposition (PLD).¹⁴ These studies showed that despite being high quality, this film is inherently inhomogeneous consisting of a ferromagnetic phase with T_C = 270 K and a magnetically dead layer with thickness of about 1.4 nm inferred from decrease of M_S and $TC$ as compared to those for bulk LSMO.¹³ Also, a blocking type phenomenon usually present in magnetic nanoparticles was also observed with a blocking temperature T_B ∼ 230 K, where the magnetization M measured for the zero-field-cooled (ZFC) case bifurfactes from that measured for the field-cooled (FC) case and T_B decreases with increase in magnetic field H.¹³ Assuming that this T_B ∼ 230 K is due to nanoclusters of spins present in the 1.4 nm dead layer, size of the clusters of about 90 nm is estimated although in Refs. 13 and 14, the cluster size is incorrectly listed as 9 nm due to computational error as explained in a footnote in Ref. 18. In addition, magnetic interaction between the spin clusters and ferromagnetic phase was proposed to lead to the observed inverted hysteresis loops and negative remanent magnetization (NRM) for T_B < T < T_C.¹⁴ This inhomogeneity of the film was found to have detrimental effect on its magnetocaloric properties.¹⁵ 

In this paper, we report results from our detailed magnetization of the spin dynamics and magnetic relaxations of this 7.6 nm LSMO/STO film using temperature dependence of the ac susceptibilities (⁠$χ′$ and $χ′′$⁠) and magnetic viscosity (S). Results from the ac susceptibilities show a broad peak in $χ′$ near 230 K associated with T_B and frequency dependent peaks in $χ′$ and $χ′′$near 270 K associated with T_C. The mean relaxation time $τ$ determined from the Cole–Cole analysis of $χ′$ and $χ′′$ near T_C is shown to fit the Vogel–Fulcher law. The temperature dependence of the magnetic viscosity S determined from M (t) = M (0) – S ln t, with time t up to 2 h shows peak at 230 K above which M (2 h) switches to negative values for T up to T_C. Details of these results and their analysis/discussion are presented in Sec. III.

The sample of a 7.6 nm thin film of La_0.7Sr_0.3MnO₃ was grown on SrTiO₃ (001) substrate with the pulsed laser deposition (PLD) technique using a commercial Neocera PLD system (248 nm KrF excimer laser). Other growth conditions were as follows: substrate temperature = 750 °C; chamber pressure = 100 mTorr O₂ pressure with growth monitored by RHEED analysis; and cool-down to 23 °C in 250 mTorr of O₂ pressure. The lateral dimensions of the film and substrate are: 4.98 × 5.81 mm². The substrate thickness is 0.5 mm. The thickness of the sample was measured by using x-ray reflectivity yielding total thickness of LSMO as 7.6 nm.¹⁴ Other details of the synthesis and structural characteristics of the film are given in our recent paper.¹⁴ Measurements of the ac susceptibilities (using frequencies f_m = 0.1–10 kHz and covering the temperature range of 175 K–310 K) and magnetic viscosity (from 5 K to 320 K) were performed using an ac measurement system attached to the commercial physical property measurement system (PPMS from Quantum Design Inc.). In magnetic measurements, the applied magnetic field was parallel to the plane of the film, thus eliminating the need for corrections due to demagnetization field. The protocol used for magnetic viscosity measurements was as follows: after demagnetizing the magnet coil at 320 K using the oscillating mode to reduce the residual field <2 Oe, the sample was cooled to the measuring temperature (e.g., 5 K) in dc magnetic field H = 50 Oe. After temperature became stable, H was switched to 0 Oe and time dependent magnetization M (t) was measured for time t up to 2 h. The sample was then warmed up to 320 K, the coils were demagnetized again followed by cooling the sample to the next measuring temperature in H = 50 Oe and repeating the M (t) vs t measurements after switching H to zero. From the plot of M (t) vs ln(t), the magnetic viscosity S was determined using the following equation:¹⁹ 

Here, we first focus on how $TC$ is determined from the temperature dependence of magnetization (M vs T) in a ferromagnet, like LSMO, which depends on the applied dc magnetic field H. This information is used later to reconcile the data obtained from the ac susceptibility measurements, although this information is useful on its own since it does not seem to have been discussed in the literature. In Fig. 1(a), the plots of computed dM/dT vs T for H = 50, 100, 200, 500, and 1000 Oe are shown using the M vs T data published earlier.¹³ The minimum in dM/dT vs T is defined as $TC$⁠, which increases with increase in H. Qualitatively, this can be understood since in a ferromagnet, the order parameter is the magnetization (M) that couples directly with H and hence, M is enhanced with increase in H. We have fitted the $TC$ vs H data to the empirical equation

where n is a constant determined from the fit. The fit shown in Fig. 1(b) yields the parameters T_C(0) = 270 K, n = 0.15, and A = 9.74 K/Oeⁿ. A theoretical basis for Eq. (2) is established using Landau theory of phase transitions according to which the free energy F near $TC$ in a magnetic field H can be written in terms of the order parameter M as²⁰ 

Minimizing F with respect to M yields

The coefficient $a(T)=(TC−T)/TC$ is often used and from Takahashi theory for itinerant ferromagnets,²⁰ $b(T)=0$ at $T=TC$⁠. These substitutions yield $H=c(T)M5$ at $TC$⁠, which implies that applied H increases M at $TC$ with M ∼ H^1/5 and so n = 0.2. This is close to n = 0.15 determined from the fit of the data to Eq. (2). The other important information is that for H = 0, T_C(0) = 270 K for this sample, which is used in Sec. III B. This reduced T_C(0) compared to T_C = 370 K for bulk LSMO is believed to be due to nano-size scaling.^13,20,21 This increase of $TC$ with increase in applied field H rules out its interpretation as a spin-glass transition²² or as a blocking temperature^13,23 due to nanoparticles since in both cases, with increase in H, the transition temperature is known to shift to lower temperatures, albeit differently for the two cases.

To measure the ac magnetic susceptibilities $χ′$ and $χ′′$⁠, the coils of the PPMS were first demagnetized at room temperature followed by cooling the sample to 170 K in dc field H = 50 Oe. After temperature became stable, H was switched off to zero and measurements of $χ′$ and $χ′′$ were done with increasing temperatures up to 310 K at different f_m = 0.1, 1, 5, and 10 kHz using H_ac = 10 Oe for two cases of superposed bias dc fields: (i) H = 0 and (ii) H = 100 Oe. The plots of $χ′$ and $χ′′$ vs T from 170 K to 310 K are shown in Fig. 2(a) for the case of H = 0 with the expanded view of the plots near 270 K shown in Fig. 2(b). The temperature range of 170 K–310 K was chosen to include both the blocking temperature T_B = 230 K indicated in earlier studies using H = 50 Oe and the region around $TC$⁠.^13,14 For one frequency, viz., f_m= 1 kHz, the plots of $χ′$ and $χ′′$ vs T for H = 0 and H = 100 Oe are compared in Fig. 3.

We first discuss the results in Fig. 2(a) where the expanded view of the region near 230 K in $χ′$ vs T plot shows a broad peak whose frequency dependence is difficult to determine because of the broadness of the peak. This broad peak is associated with the T_B of the nanoclusters of about 90 nm width present in the 1.4 nm surface layer of the film as noted earlier¹⁸ and discussed in more detail in Ref. 13 although a computational error in Ref. 13 listed the cluster size as 9 nm instead of the correct 90 nm. This observation of the broad peak near 230 K in the ac data albeit weak is a confirmation of associating the peak with T_B of nanoclusters. The broadness of the peak is indicative of the wide distribution in the size of the nanoclusters. The absence of an associated peak in the $χ′′$ vs T data near 230 K may be due to the often observed fact that in nanoparticles (NPs) and spin glasses, the magnitude of $χ′′$ is usually only about a few % of $χ′$⁠.²⁴ For example, in 11 nm nanoparticles (NPs) of La_0.7Sr_0.3MnO₃ prepared by ball-milling, Phong et al.²⁵ reported a similar broad peak near 230 K with the ratio of the peak magnitude of $(χ′′/χ′)∼0.03$ which they associated with T_B of the NPs. Lu et al.²⁶ have reported magnetic studies of La_0.67Sr_0.33MnO₃ nanofibers with diameter ∼ 27 nm and T_C = 360 K, T_B= 343 K, and ratio of peak magnitude of $(χ′′/χ′)∼0.07$⁠; and Rostamnejadi et al.²⁷ similarly investigated 16 nm size nanoparticles of La_0.67Sr_0.33MnO₃ prepared by the sol gel method with T_B= 290 K and ratio of peak magnitude of $(χ′′/χ′)∼0.05$⁠. These results of $χ′′∼few%χ′$ in NPs of LSMO were discussed by the authors in terms of super-spin-glass (SSG) state and/or strongly interacting magnetic NPs.^24–26 

Next, we discuss peaks in both $χ′$ and $χ′′$ observed near 270 K in Fig. 2(a), expanded views of which are shown in Fig. 2(b). With increase in measuring frequency from 0.1 kHz to 10 kHz, magnitudes of both peak decrease and their positions shifts to higher T. At a particular frequency, the peak positions of $χ′$ and $χ′′$ do not match as expected since $χ′′∼∂(χ′T)/∂T$⁠.²⁸ Our analysis of the data (not shown here) has verified that the peak position of $χ′′$ does match with that of $∂(χ′T)/∂T$⁠. An interesting observation for these peaks is that the ratio of peak magnitude of $(χ′′/χ′)∼0.65$⁠, an order of magnitude larger than that observed for the peaks near T_B in NPs^25–27 of LSMO discussed in the preceding paragraph. In the review of the results by Balanda near magnetic phase transitions,²⁴ the ration $(χ′′/χ′)∼0.1$ was reported.

The effect of applied dc field of H = 100 Oe on the peaks of $χ′$ and $χ′′$ in the 7.6 nm LSMO/STO film near 270 K is shown in Fig. 3. The peak in $χ′$ shifts to higher T by about ΔT ≈ 15 K and its magnitude is reduced by a factor of about 5, whereas the peak in $χ′′$ completely disappears. However, the shift in $χ′$ to higher temperatures in H = 100 Oe is consistent with increase of $TC$ with applied H shown in Fig. 1(b). The observed change in intensity of both $χ′$ and $χ′′$ in H = 100 Oe may be due to the stabilization of the magnetic ordering with applied H, thus making $χ′′=0$⁠. Also, it is noted that the nanoscale thickness of the LSMO film investigated here may be the reason for the observation of the frequency-dependent peaks near $TC$⁠. Further insights into this phenomenon may come from similar ac measurements on films with different thicknesses.

To gain further insight into the nature of peak near 270 K in the 7.6 nm LSMO/STO film, its frequency dependence is analyzed next. In NPs (nanoparticles) and spin-glasses, the shift of the peak in $χ′′$ vs T with frequency is usually analyzed by the parameter $ϕ=ΔTP/[TPΔlog10fm]$⁠, where $ΔTP$ is the shift in $TP$ with a change in frequency from f_m (1) to f_m (2).^29,30 Using f_m (1) = 0.1 kHz and f_m (2) = 10 kHz, ϕ = 0.002 is determined in this sample. An analysis of $ϕ$ for different nano-particle systems and spin glasses in terms of strength of the interparticle interactions (IPI) has been reviewed recently³¹ with references to particular cases given there. For negligible IPI, $ϕ≥0.13$ for non-interacting NPs, whereas for interacting particles, $0.05<ϕ<0.13$ with $ϕ$ decreasing with increase in IPI, and $ϕ$ < 0.05 for spin-glasses. For the LSMO NPs, the reported $ϕ=0.03$ ²⁵ and 0.011.²⁷ The observed ϕ = 0.002 here is an order of magnitude smaller than these values of $ϕ$ suggesting that the transition near $TC$ is due to even stronger IPI, typical of a normal second order transition but with some modifications due to the nanoscale thickness of the film. More details on this emerge from the Cole–Cole analysis of $χ′$ and $χ′′$ given below.

To gain further information from the ac susceptibility data of Fig. 2, the values of $χ′$ and $χ′′$ were determined at different temperatures in the vicinity of $TC$ for all frequencies using these plots. The Cole–Cole plot is the variation of $χ′′$ vs $χ′$ at various temperatures,^24,32,33 which for the LSMO film are shown in Fig. 4 for temperatures near $TC$⁠. Theoretically, the Cole–Cole equations for the complex susceptibility $χ(ω)=χ′(ω)−iχ′′(ω)$ are given by

Here, $χ0$ and $χS$ are, respectively, the isothermal and adiabatic susceptibilities in the limit of lowest and the highest frequency and usually $χS=0$ is taken since the systems cannot respond to ultra-high frequencies. Also, $α$ represents the distribution of the relaxation time $(τ)$ since for α = 0, Eqs. (5) and (6) reduce to the Debye model representing a single relaxation time. Following the procedure described in Refs. 32 and 33, the Cole–Cole plots shown in Fig. 4 are fits to Eq. (7) derived from Eqs. (5) and (6),

The solid red lines are fits to Eq. (7) by using $χS=0$⁠. The relaxation time $(τ)$ at each temperature was calculated using Eq. (6) by using the magnitudes of $α$ determined from the Cole–Cole fits. For T > 271 K, the magnitude of α < 0.2 practically means a single relaxation time.^24,32

The plot of $τ$ vs T is shown in Fig. 5(b) with the solid line fit to the Vogel–Fulcher law given by the following equation:^30,31,34

where $τ0$ is the characteristic time, $ΔE$ is the energy barrier for (macro) spin reversal, $kB$ is the Boltzmann constant, and $T0$ is the measure of strength of the IPI between magnetic clusters in nanoparticles or spin-glasses.^30,31,34 In order to fit the $τ$ vs T data to Eq. (8), the variation of $ln(τ)$ was plotted as a function of $1/(T−T0)$ by varying $T0$ value to obtain the best straight line [Fig. 5(a)]. The highest $T0$ for which the data could be fit to Eq. (8) is 245 K yielding the following parameters: $τ0=$ 1.2 $×$ 10⁻⁹ s (f₀ = 1/τ₀ = 8.3 × 10⁸ Hz) and $ΔE/kB=270K=TC(0)$⁠. These magnitudes of the parameters are physically reasonable since the activation energy $ΔE/kB$ is equal to the Curie temperature, the magnitudes of $f0=1/τ0$ typically lies in the range of 10⁹–10¹² Hz characteristic of Larmor frequency^16,20 and T₀ = 245 K is in-between T_B and T_C (0). Detailed discussion on the role of the superparamagnetic (SPM) clusters present in the 1.4 nm layer and ferromagnetic (FM) phases present in the rest of the sample in the region of T < T_B < T_C(0) in this sample was given in earlier paper¹⁴ with the additional discussion presented in Sec. III D.

It is noted that the data of $τ$ vs T can also be fit for other T < T₀ = 245 K. For example, for T₀ = 230 K, the fit yields τ₀ = 8 × 10⁻¹² s and $ΔE/kB=630K$⁠, and for T₀ = 0 K representing the thermally activated Néel–Brown relaxation, the fit yields physically unreasonable values of τ₀ = 2.5 × 10⁻⁴⁵ s and $ΔE/kB=25077K$⁠. Thus, with decrease in $T0$⁠, the magnitude of $τ0$ decreases and that of $ΔE/kB$ increases.

The procedure used for measurements of magnetic viscosity S was described under Sec. II. In Fig. 6, we show the plot of M (t) vs ln(t) for t up to 2 h for selected temperatures between 5 K and 305 K. The quantities of interest are M (0), the slope of the plot, which according to Eq. (1) is the magnetic viscosity S and M (2 h), the magnitude of the magnetization at the end of 2 h. Plots of S vs T and M (2 h) vs T are shown in Figs. 7(a) and 7(b), respectively. The noteworthy new result is that S peaks at 230 K above which M (2 h) switches from positive to negative values. With further increase in temperature, the magnitude of M (2 h) decreases, eventually becoming near zero above T_C. Although a peak in S vs T has been reported in magnetic nanoparticles related to the spin-glass ordering of the surface Fe³⁺ spins,³⁵ the accompanying switching of the sign of M (2 h) has not been reported in any other system before.

The unique result of Fig. 7, viz., peak in S accompanied by sign switching of M (2 h), can be understood in terms of the similar temperature dependence of the hysteresis loop parameters reported for this sample in our recent paper.¹⁴ These studies showed that between T_B ∼ 230 K and $TC$⁠, the remanence M_r and coercivity H_C show negative values, and hysteresis loops are inverted. The source of these effects of negative remanent magnetization (NRM) and inverted hysteresis loops (IHL) was suggested to be the negative magneto-static interaction between the FM phase and the SPM phase, both being present in this film for T_B < T < T_C. Details of this model are given by Maity et al.³⁶ and Yan and Xu.³⁷ For the 7.6 nm LSMO film, spins in the SPM phase freeze for T < T_B ∼ 230 K, resulting in a peak in S vs T at 230 K, similar to the observations in ferrihydrite nanoparticles.³⁵ The fit of the relaxation time (determined from the Cole–Cole analysis of $χ′$ and $χ′′$⁠) to the Vogel–Fulcher law [Eq. (7)] yielded T₀ = 245 K with $ΔE/kB=270K$⁠. This coincidence of $T0$ to $TB$ and $ΔE/kB$ to $TC$ suggest a strong magnetic coupling of the SPM phase of the surface layer and the FM phase of the rest of the film.

From the magnetic investigations of the 7.6 nm LSMO film presented here, three major results have emerged. First, the variation of $TC(H)$ with applied dc magnetic field H as $TC(H)−TC(0)∼Hn(n=0.15)$ is shown to agree semi-quantitatively with the predictions of the Landau theory. Second, the Cole–Cole analysis of the ac susceptibilities (⁠$χ′$ and $χ′′$⁠) is used to determine the relaxation time $τ$ whose temperature dependence is shown to fit the Vogel–Fulcher law [Eq. (8)] with $τ0$ = 1.2 × 10⁻⁹ s, $T0$ = 245 K, and $ΔE/kB$ = 270 K. The closeness of $T0$ to $TB$ ∼ 230 K of the SPM phase of the nanoclusters present in the surface layer and $ΔE/kB=TC(0)$ infers strong magnetic coupling of the superparamagnetic and ferromagnetic phases. Third, the temperature variation of magnetic viscosity shows a peak at T_B ≃ 230 K above which M (2 h) becomes negative until the T_C is reached. It will be interesting to know if similar interesting effects are present in other magnetic thin films containing SPM spins in the surface layer, in addition to FM aligned spins in the rest/bulk of the film.

We acknowledge funding support from National Science Foundation (NSF) (No. DMR-1608656) for growth and optimization and U.S. Department of Energy (DOE) (No. DE-SC0016176) for the magnetic characterization of our films. This work has been included in our sample database made possible by NASA WV EPSCoR Grant No. NNX15AK74A.

The data that support the findings of this study are available from the corresponding author upon reasonable request.