Observability of negative capacitance of a ferroelectric film: Theoretical predictions

We theoretically explore mechanisms that can potentially give rise to the steady-state and transient negative capacitance in a uniaxial ferroelectric film stabilized by a dielectric layer. The analytical expressions for the steady-state capacitance of a single-domain and polydomain states are derived within the Landau-Ginzburg-Devonshire approach and used to study the state stability vs the domain splitting as a function of dielectric layer thickness. Analytical expressions for the critical thickness of the dielectric layer, polarization amplitude, equilibrium domain period, and susceptibility are obtained and corroborated by finite element modeling. We further explore the possible effects of nonlinear screening by two types of screening charges at the ferroelectric-dielectric interface and show that if at least one of the screening charges is very slow, the total polarization dynamics can exhibit complex time- and voltage-dependent behaviors that can be interpreted as an observable negative capacitance. In this setting, the transient negative capacitance effect is accompanied by almost zero dielectric susceptibility in a wide voltage range and low frequencies. These results may help to elucidate the observation of the transient negative capacitance in thin ferroelectric films, and identify materials systems that can give rise to the behavior.

Figure 1

(a) A single-domain state: Free energy dependence on polarization for bulk ferroelectric (left), dielectric layer (center), and the combined single-domain ferroelectric (SDFE)–dead layer (DL) system (right). Polarization (b) and susceptibility (c) hysteresis curves for a bulk FE (left column) and for a FE film with a DL (right column). (d) Typical polarization distribution for polydomain ferroelectric (PDFE), corresponding polarization (e), and susceptibility (f) hysteresis curves. Solid red curves correspond to stable states, while dashed blues ones show the properties of unstable states.

Figure 2

The schematics of the considered systems. (a) A uniaxial ferroelectric film of thickness $h$ and polarization vector $\mathit{P}$ dielectric layer of thickness $d$, respectively, placed between the electrodes. The voltage $V$ is applied to the bottom electrode. (b) A top electrode, a ferroelectric film, a layer of positive and negative surface charges with densities ${\sigma }_p\left(\varphi \right)$ and ${\sigma }_n\left(\varphi \right)$, a dielectric gap of width $d$, and a bottom electrode, which in principle, can provide a direct charge exchange with an ambient media.

Figure 3

The temperature dependence of the SPS layer steady-state capacitance calculated for several values of the film thickness $h=$ 5 nm (red curve), 8 nm (yellow curve), 11 nm (green curve), 14 nm (light blue curve), 17 nm (dark blue curve), and 20 nm (purple curve). The dielectric layer thickness $d=0.2\mathrm{nm}$.

Figure 4

The dependencies of the polarization (a,d), the capacitance (b,e), and the susceptibility (c,f) of the 50 nm thick SPS film on the applied electric field calculated for several temperatures $T=$ 50 K (black curves), 100 K (red curves), 150 K (magenta curves), 200 K (blue curves), 250 K (green curves), and 300 K (purple curves). The bottom row is for a very small external field frequency $\omega {\tau }_{\mathrm{Kh}}={10}^{-4}$, and the top one is for a higher frequency $\omega {\tau }_{\mathrm{Kh}}=5\times {10}^{-2}$. The dielectric layer thickness $d=0.2\mathrm{nm}$; the Khalatnikov time ${\tau }_{\mathrm{Kh}}=\mathrm{\Gamma }/\left|{\alpha }_T{T}_c\right|$.

Figure 5

Polarization distribution in the vertical cross section of the system SPS film/dielectric layer for several values of the dielectric layer thickness $d=0.8\mathrm{nm}$ (a), 0.9 nm (b), 1 nm (c), and 1.2 nm (d). The film thickness $h=50\mathrm{nm}$; $T=300$ K. The equilibrium domain structure is shown.

Figure 6

(a) The dependence of the equilibrium domain period ($2\pi /k_{\mathrm{eq}}$) on the SPS film thickness $h$ calculated near the transition to the paraelectric (PE) phase for different values of the dielectric layer thickness $d$ (shown near the curves). (b) The transition temperature as a function of $d$ calculated for different $h$ values (shown near the curves). (c,d) The PE-FE transition temperatures as a function of $h$ calculated for $d=0.6\mathrm{nm}$ (c) and $d=3\mathrm{nm}$ (d). Solid and dashed curves represent the stability boundaries of the single-domain (SDFE) and polydomain (PDFE) ferroelectric phases, respectively.

Figure 7

The quasistatic dependencies of the polarization (a,c) and the dielectric susceptibility (b,d) of the SPS film on the applied voltage calculated for several values of the dielectric layer thickness $d=0.6\mathrm{nm}$ (a,b) and 1.2 nm (c,d). The voltage amplitude is $V=0.5$ mV (black curves), 1 mV (red curves), 2 mV (green curves), and 3 mV (blue curves) for plots (a,b); $V=10$ mV (black curves), 30 mV (red curves), 40 mV (green curves), and 50 mV (blue curves) for plots (c,d). The film thickness $h=50\mathrm{nm}$, $T=300$ K.

Figure 8

(a,c,e) The quasistatic dependencies of the polarization (black curves), positive (red curves), and negative (blue curves) surface charges, and the total charge at the bottom electrode (green curves). (b,d,f) The quasistatic dependencies of the dielectric susceptibility (black curves) and effective capacitance (green curves). The dependences are calculated for several values of characteristic times ${\tau }_n={\tau }_p=100$ (a,b), ${\tau }_n={\tau }_p=350$ (c,d), and ${\tau }_n={\tau }_p=920$ (e,f) in units of ${\tau }_{\mathrm{Kh}}$. The SPS film thickness $h=100\mathrm{nm}$, dielectric gap thickness $d=1.2\mathrm{nm}$; $T=293$ K. The quasistatic voltage with an amplitude is 2 V and dimensionless very low frequency ${10}^{-3}$ is applied.

Figure 9

(a,c,e) The quasistatic dependencies of the polarization (black curves), positive (red curves), and negative (blue curves) surface charges, and the total charge at the bottom electrode (green curves). (b,d,f) The quasistatic dependencies of the dielectric susceptibility (black curves) and effective capacitance (green curves). The dependences are calculated for several values of characteristic times ${\tau }_n={\tau }_p=10$ (a,b), ${\tau }_n=10,{\tau }_p=100$ (c,d), and ${\tau }_n=10,{\tau }_p={10}^3$ (e,f) in units of ${\tau }_{\mathrm{Kh}}$. The SPS film thickness $h=10\mathrm{nm}$, dielectric gap thickness $d=1.2\mathrm{nm}$, and $T=50$ K. The dimensionless frequency of the quasistatic voltage is ${10}^{-3}$.

Figure 10

(a,b) The dependencies of free energy on the polarization for the different values of applied voltage, −0.8, −0.4, 0, 0.4, and 0.8 V (blue, cyan, black, orange, and red solid curves, respectively). Dashed black curves represent the free energy profile in the absence of free charges and zero voltage. (c,d) The static dependencies of the polarization (black curves), positive (red curves), and negative (blue curves) surface charges, and total charge at the bottom electrode (green curves). Dashed black curves represent the polarization voltage dependence in the absence of screening charges $\sigma$. (e,f) Free energy dependence on the polarization and applied voltage. The dielectric gap thickness $d=1.2\mathrm{nm}$; SPS film thickness $h=10\mathrm{nm}$ and temperature $T=50$ K for plots (a,c,e); $h=100\mathrm{nm}$ and $T=293$ K for plots (b,d,f).