Interplay of Coulomb blockade and ferroelectricity in nanosized granular materials

We study electron transport properties of composite ferroelectrics—materials consisting of metallic grains embedded in a ferroelectric matrix. In particular, we calculate the conductivity in a wide range of temperatures and electric fields, showing pronounced hysteretic behavior. In weak fields, electron cotunneling is the main transport mechanism. In this case, we show that the ferroelectric matrix strongly influences the transport properties through two effects: (i) the dependence of the Coulomb gap on the dielectric permittivity of the ferroelectric matrix, which in turn is controlled by temperature and external field, and (ii) the dependence of the tunneling matrix elements on the electric polarization of the ferroelectric matrix, which can be tuned by temperature and applied electric field as well. In the case of strong electric fields, the Coulomb gap is suppressed and only the second mechanism is important. Our results are important for (i) thermometers for precise temperature measurements and (ii) ferrroelectric memristors.

Figure 1

(Top) Sketch of a granular ferroelectric (GFE) material with two metal contacts (source and drain). (Bottom) Sketch of a pair of grains embedded in a FE matrix with radii $R_1$ and $R_2$ and distance $2r$ between them. The vector $\vec{P}$ is the local electric polarization of the FE matrix and the vector ${\vec{E}}_i$ is the internal electric field appearing in the system due to the presence of charged impurities. The vector ${\vec{E}}_e$ is the applied external electric field.

Figure 2

Average dielectric susceptibility $\overline{\chi }$ vs temperature. The solid lines correspond to the longitudinal ${\overline{\chi }}_{\parallel }$, Eq. (B3), and the dash lines correspond to the perpendicular ${\overline{\chi }}_{\perp }$, Eq. (B4), components of susceptibility $\overline{\chi }$. The behavior is shown for both hysteresis branches. Arrows indicate the path around the hysteresis loop for fixed external electric field $E_e=E_i/3$. $T_{\mathrm{C}}$ and $T_{\mathrm{S}}$ are the Curie and the switching temperatures, respectively. The derivative of susceptibility has a jump at the transition point following from the divergence of the microscopic susceptibility ${\chi }_{\vec{n}}$ appearing at the points of polarization switching ($E_s$ or $T_{\mathrm{S}}$). However, in real ferroelectrics, this behavior is absent due to order parameter fluctuations. Therefore the kink in the average susceptibility $\overline{\chi }$ is smeared.

Figure 3

Transport phase diagram of granular ferroelectrics in coordinates of the external electric field $E_e$ vs temperature $T$. $T_{\mathrm{C}}$ is the Curie temperature. AC (shaded region) denotes the activation conductivity, LVRH (cross filled region) and NLVRH (unfilled region) stand for linear and nonlinear electron cotunneling, respectively. MC (colored region) stands for metallic conductivity. The blue line $T_{\mathrm{A}}^{<}$ describes the transition to activation transport; the brown lines $T_{\mathrm{M}}$ describe the transition from activation to metallic regimes, these lines have a physical meaning outside the metallic regime only (colored region is MC). The electric field $E_{\mathrm{D}}$ describes the transition to the metallic regime. The field $E^{*}$ shows the boundary between the linear and nonlinear hopping regimes. This field has physical meaning outside the metallic regimes only, since inside the metallic region the VRH contribution to the conductivity is negligible.

Figure 4

Dielectric permittivity $\epsilon$ (solid line), Eq. (3), and Coulomb gap $E_{\mathrm{C}}$ (dash line) of granular ferroelectrics, Eq. (2), vs temperature at fixed external electric field, $E_e=6\times {10}^6$ V/m. $T_{\mathrm{C}}$ is the Curie temperature.

Figure 5

Conductivity $\sigma$ of GFE, with ${\sigma }_0=2e^2{g}_t^0/a$ being the metallic conductivity in the paraelectric phase, vs temperature at fixed external electric field $E_e=6\times {10}^6$ V/m. Variable range hopping and electron cotunneling, Eq. (7), is the main transport mechanism for temperatures $T<T_{\mathrm{A}}^{<}$. Close to the transition temperature $T_{\mathrm{C}}=400$ K the transport is metallic, Eq. (10). Between these two regions the conductivity has activation behavior, Eq. (11).

Figure 6

Conductivity $\sigma$, with ${\sigma }_0=2e^2{g}_t^0/a$ being the metallic conductivity in the paraelectric phase, vs external electric field $E_e$ at fixed temperature $T=350$ K ($E_{\mathrm{D}}\approx 1.5\times {10}^7$ V/m). The solid and dotted lines correspond to the two hysteresis branches. The arrows show the direction, the hysteresis loop is followed. There are two different situations: (a) the average internal field $E_i=5.7\times {10}^7$ V/m is less than the FE switching field $E_s=6.3\times {10}^7$ V/m and (b) the average internal field $E_i=5.7\times {10}^7$ V/m is larger than the switching field $E_s=3.9\times {10}^7$ V/m. For internal fields $E_i>E_s$, there is no difference between fields $E_{\mathrm{D}}^{<}$ and $E_{\mathrm{D}}^{>}$.

Figure 7

The local permittivity $\epsilon$ of the FE matrix vs local electric field, $E=E_e+E_i$, with $E_e$ and $E_i$ being the external and internal electric fields, respectively. Solid and dash lines correspond to two hysteresis branches. According to the procedure described in the Sec. 2 the dielectric permittivity of the whole GFE is the average of the local permittivity over different internal field orientations. The filled area shows the region of averaging. There are two different situations: (a) the internal field $E_i$ is larger than the switching field $E_s$ and (b) the internal field $E_i$ is less than the switching field $E_s$.

Figure 8

Conductivity of a GFE vs temperature in the metallic regime, Eq. (10), at fixed external electric field $E_e=3\times {10}^7$ V/m. Solid and dash lines correspond to the two hysteresis branches. The inset shows the behavior of the correlation function $C\left(T\right)$ on temperature, Eq. (B6). The conductivity in the metallic regime is controlled by the correlation function.