Competing soft dielectric phases and detailed balance in thin film manganites

Using frequency dependent complex capacitance measurements on thin films of the mixed-valence manganite (La${}_{1-y}$Pr${}_y$)${}_{1-x}$Ca${}_x$MnO${}_3$, we identify and resolve the individual dielectric responses of two competing dielectric phases. We characterize their competition over a large temperature range, revealing they are in dynamic competition both spatially and temporally. The phase competition is shown to be governed by the thermodynamic constraints imposed by detailed balance. The consequences of the detailed balance model strongly support the notion of an “electronically soft” material in which continuous conversions between dielectric phases with comparable free energies occur on time scales that are long compared with electron-phonon scattering times.

Figure 1

Resistance and sample geometry. The $ab$ in-plane resistance (measured upon cooling) shows a pronounced peak at the insulator-to-metal transition $T_{IM}\approx 115$ K, but lacks a COI associated anomaly seen in bulk manganites in the temperature range 200–250 K (Refs. 15, 16, 17). Inset: Sample geometry, where the LPCMO film is the bottom electrode of a trilayer capacitor structure.

Figure 2

Circuit models. (a) Circuit model composed of a Maxwell-Wagner circuit with a series resistance ($R_s$) component that accounts for the in-plane voltage drop along the film. In select frequency ranges, this in-plane voltage drop can be shown to be negligible compared to the $c$-axis voltage drops across the capacitors, effectively removing $R_s$ from the circuit (Ref. 18). Additionally, the AlO${}_x$ dc resistance is exceptionally large and can be approximated as infinite. (b) The resulting circuit model for our measurement bandwidth after approximations and limits are applied. See the Appendix for a complete discussion of the relevant time scales. (c) The logarithmic parametric slope, $(C_M^{\prime }/C_M^{\prime \prime })(\partial C_M^{\prime \prime }/\partial C_M^{\prime })$ (green), is shown to differ from the Cole-Cole (Ref. 20) dielectric response, providing a signature of multiple phases. (d) Circuit model incorporating the complex dielectric function of $C_M$, which has multiple components. Three parallel components, $C_{\mathrm{PMI}}$, $C_{\mathrm{COI}}$, and $R_{\mathrm{met}}$, are in series to fractional areas of the AlO${}_x$ layer.

Figure 3

Parallel circuit model describes complex capacitance data. (a) The measured real capacitance (black) is compared with fits to Eq. (3) (green) at $T=200$ K. (b) Measured imaginary capacitance. The individual capacitances of the PMI (red dash) and COI (blue dot) phases are also shown. At low frequencies, the PMI phase dominates both channels, and at high frequencies the COI dominates in the real channel, but not the imaginary channel. This mixing of relaxation components at high frequencies results in the logarithmic parametric-slope behavior seen in Fig. 2c.

Figure 4

Temperature dependence of selected model parameters identifies phases and verifies detailed balance. (a) The temperature dependence of $\beta$ is shown to match the temperature dependence of the correlations of COI phase, where COI nanoclusters are reported in a related material to appear near 280 K with the phase fully developed below 240 K (Ref. 17), identifying the high-frequency relaxation as the COI phase. The broadening of the PMI phase $\alpha$ is shown to be featureless in this temperature range. (b) The temperature dependence of the ratios of the dielectric amplitudes, $r_{\mathrm{amp}}=$ $a_{\mathrm{COI}}/a_{\mathrm{PMI}}({\epsilon }_{\mathrm{COI}}/{\epsilon }_{\mathrm{PMI}})$, and time scales, $r_{\tau }={\tau }_{\mathrm{COI}}/{\tau }_{\mathrm{PMI}}=a_{\mathrm{COI}}/a_{\mathrm{PMI}}$, are compared. As discussed in the text, the strong correlation between the ratios confirms that the phase competition is governed by detailed balance.

Figure 5

Region of identical activation energies provides clue for detailed balance. Arrhenius plots are shown for ${\tau }_{\mathrm{COI}}$ and ${\tau }_{\mathrm{PMI}}$, with $E_A\approx 118$ meV in the linear region for both dielectric phases. The nearly identical $E_A$'s suggest the phases share a single energy barrier [depicted in the middle panel of Fig. 6b]. Inset: The distributions of hopping-rate time scales are shown to be narrow for both phases, suggesting temporally coherent hopping (PMI dotted red, and COI solid blue).

Figure 6

Detailed balance model. (a) An energy level schematic of a three state system describing the polaron hopping process is presented. Polarons of both dielectric phases absorb thermal fluctuations and hop to an excited state at their characteristic rates, ${\gamma }_{\mathrm{PMI}}$ and ${\gamma }_{\mathrm{COI}}$. The excited state is equivalent for both phases, corresponding to an electron surrounded by an undistorted lattice. The lattice can then relax into distortion states that correspond to polarons of either dielectric phase. (b) Schematic depiction of the temperature evolution of detailed balance and the energy barriers separating the PMI and COI dielectric phases.

Figure 7

Thickness dependence of dielectric constants. The dielectric constants, calculated from the dielectric amplitudes and areas of each phase, ${\epsilon }_i=A_i/a_i$ (ignoring the vanishingly small ${\epsilon }_{\infty ,i}$ of each phase), are shown to saturate near their bulk values once substrate strain is relaxed. The agreement with bulk data validates the areas calculated assuming equal lattice relaxation rates, ${\gamma }_E\approx {\gamma }_E^{\prime }$.

Figure 8

Sample geometry. (a) The dimensions and wire connections to our thin film sample are shown. The $ab$ plane resistance measurements (see Fig. 1) are made using a four point Van der Pauw geometry with the voltage and current leads connected at the corners of the film. Capacitance measurements are made between the central circular electrode and the sample edge with average separation of $\approx$2 mm. (b) A cross section shows the thickness of each layer, with the LPCMO contact spanning the width of the LPCMO film. Equipotentials at dc are shown to be concentrated in the plane of the AlO${}_x$ layer.

Figure 9

Measurement circuit and circuit models. (a) The equivalent circuit, comprising the parallel combination of the measured capacitance $C_B$ and resistance $R_B$, as reported by the capacitance bridge in “parallel” mode. (b) Circuit equivalent including the resistance $R_S$ in series with the parallel combination of $R_0$, $C_1$, and R${}_2$. The in-plane voltage drop is proportional to $R_S$ whereas the perpendicular voltage drop is proportional to the complex impedance of the parallel combination of $R_0$, $C_1$, and R${}_2$. (c) With the series resistance $R_s$ ruled negligible, the complex capacitance is decomposed into a modified Maxwell-Wagner circuit of the manganite impedance and AlO${}_x$ capacitance.

Figure 10

Circuit approximations at select time scales. (a) At dc the entire voltage drop is across the AlO${}_x$ capacitor [see Fig. 8b]. (b) When $\omega \approx 1/R_M{C}_{\mathrm{AlO}x}$, the circuit simplifies to a series combination of the AlO${}_x$ capacitance and manganite resistance. (c) When $\omega \ll 1/R_M{C}_M$, this is approximated by a series combination of the AlO${}_x$ capacitance and manganite capacitance. (d) At very high frequencies the constraints of Eq. (A3) fail and the in-plane voltage drop across the series resistance dominates, reducing the circuit to the series resistance.