Let us begin by discussing *P*-switching in the MFIM stack with soft-DW. The simulated average charge density ((Q)) vs applied voltage (({V}_{app})) characteristics is shown in Fig. 3(a) for ({T}_{fe}) = 5 nm, ({T}_{de}) = 4 nm (Al_{2}O_{3}), (g=1times {10}^{-9},{m}^{3}V/C). Here, (Q) is calculated as the average displacement field at the Metal-DE interface based on following equation.

$$Q=frac{1}{l}[{int }_{0}^{l},{varepsilon }_{0}{varepsilon }_{de}times {E}_{z,de}(x)dx]$$

(1)

where, ({E}_{z,de}) is the z-component of E-field at Metal-DE interface and (l) is the length of the stack. For (|{V}_{app}| < 2,text{V}), a continuous (Q)–({V}_{app}) path exists when the FE is in the MD state and (P)-switching takes place through DW motion (see Fig. 3(b)). If (|{V}_{app}|) is increased above 2 V, MD state (*P*) switches to the poled state (either (Puparrow ) or (Pdownarrow )). Now, with decreasing (|{V}_{app}|), MD state forms from the poled state at a lower (|{V}_{app}|) (0.9 V) and that induces a hysteresis in the (Q)–({V}_{app}) characteristics (discussed in the Supplementary Section). Therefore, for non-hysteretic operation, the MD state needs to be retained by limiting ({V}_{app}). Interestingly, in the MD state, (Q) is higher in the MFIM stack compared to the MIM (Metal-Insulator-Metal) at the same ({V}_{app}) as shown in Fig. 3(a). This implies that the effective capacitance of the MFIM stack is higher than MIM. In a static scenario, such a phenomena is only possible if the FE layer acts as an effective negative capacitor. The (Q-{V}_{fe}^{avg}) characteristics are shown in Fig. 3(a). In the MD state, as the potential drop across the FE layer is non-homogeneous along the x-axis (Fig. 2(b)), the average potential drop across the FE layer (({V}_{fe}^{avg})) is calculated using the following equation.

$${V}_{fe}^{avg}={V}_{app}-frac{1}{l}[{int }_{0}^{l},{V}_{{int}}(x)dx]$$

(2)

Here, ({V}_{int}(x)) is the FE-DE interface potential (Q) is calculated by taking the average displacement field at the FE-DE interface which provides exactly same (Q) as Eq. 1 for the same ({V}_{app}). Figure 3(a) shows that the effective FE capacitance, ({C}_{fe}^{ef{f-awg}}=dQ/d{V}_{fe}^{avg}) is indeed negative while FE is in MD state which implies that the effective average permittivity of the FE layer in the out-of-plane direction, ({varepsilon }_{z,fe}^{eff-avg}) is negative.

The DW-motion induced negative effective permittivity can be described as follows. When ({V}_{app}) = 0 V, the P(downarrow ) and P(uparrow ) domains in FE are equal in size and the local ({E}_{z,fe}) (depolarizing field) is directed opposite to the local (P) (i.e. P(downarrow ) domains exhibit E(uparrow ) and P(uparrow ) domains exhibit E(downarrow )). Note that ({f}_{x,grad}) is non-zero inside of the domain (due to DW diffusion in soft-DW) and that causes the (P) to decrease in magnitude (discussed earlier). Now, with the increase in ({V}_{app}), P(downarrow ) domains grow and P(uparrow ) domains shrink in size, due to positive stiffness of DW motion^{8}. As the DW moves away from P(downarrow ) domain and towards the P(uparrow ) domain, ({f}_{x,grad}) in P(downarrow ) domain decreases and in P(uparrow ) domain increases. Due to this effect as well as increase in ({V}_{app}), the magnitude of local (P) in P(downarrow ) domain increases and in P(uparrow ) domain decreases. Our simulation shows that as a result of this, the depolarizing field (({E}_{z,fe})) in P(downarrow ) domain increases and in P(uparrow ) domain decreases in magnitude. This implies ({f}_{dep}) increases (decreases) in P(downarrow ) (P(uparrow )) domain. The increase (decreases) in ({f}_{dep}) in P(downarrow ) (P(uparrow )) domains is possible as it is accompanied by a decrease (increase) in ({f}_{x,grad}). As the oppositely directed local E-field in FE increases (decreases) with the increase (decrease) in local (P) in both P(downarrow ) and P(uparrow ) domains, the effective local permittivity of the domains (({varepsilon }_{z,fe}^{eff})) become negative. At the same time, in the DW, the asymmetry in (P) distribution (due to unequal P(uparrow ) and P(downarrow ) domain sizes and (P) magnitudes) causes ({F}_{dw}) (comprised of ({f}_{x,grad}) and ({f}_{x,elec})) to decrease compared to the symmetric (P) distribution^{2} (at *V*_{app} = 0 V). Such a decrease in ({F}_{dw}) allows a further increase in average ({E}_{z,fe}^{avg}) (an increase in depolarization energy) in the DW, while the average-(P) (directed opposite to ({E}_{z,fe}^{avg})) in the DW increases (due to unequal (P) magnitudes in P(uparrow ) and P(downarrow ) domain). As a consequence, the permittivity of the DW region also becomes negative. These effective local (and non-homogeneous) negative permittivity (({varepsilon }_{z,fe}^{eff})) of the domain and DW regions give rise to an average effective negative permittivity in the FE layer, i.e. ({varepsilon }_{z,fe}^{eff-avg} < 0).

Here, it is important to note that the appearance of negative effective permittivity is essentially an apparent phenomena of change in long range interaction of (P), its gradient and/or DW energy under DW motion. In particular, the change in (P) in MFIM is not directly driven by the local E-field, rather, the change in (P) is driven by the applied E-field induced domain-wall motion. Therefore, the change in local E-field is the effect of change in (P) and not the opposite. In other words, the depolarizing E-field appears depending on the change in (P) induced by DW motion. Even though, such phenomena leads to a negative effective permittivity of the FE layer, the susceptibility of the FE layer and the whole system (MFIM stack) is positive with respect to the applied E-field. That implies, the change in polarization is always in the direction of the change in applied E-field.

As we have identified that the ({F}_{dw}) plays a crucial role in providing negative ({varepsilon }_{z,fe}^{eff-avg}), therefore, it is intuitive that the NC effect is dependent on its components, i.e. ({f}_{x,grad}) and ({f}_{x,elec}). To investigate such dependency, the average effective NC path in the (Q)–({E}_{z,fe}^{avg}) responses of MFIM stack for different (g) are shown in Fig. 3(c). Here, ({E}_{z,fe}^{avg}) is the z-component of E-field in FE averaged along the length (x-axis), which we calculate as ({E}_{z,fe}^{avg}={V}_{fe}^{avg}/{T}_{fe}). Figure 3(c) shows an increase in the NC effect with an increase in (g). Here, similar to the earlier works^{2,3,4}, an increase in the NC effect implies an increase in (1/|{varepsilon }_{z,fe}^{eff-avg}|=|d{E}_{z,fe}^{avg}/dQ|). As the ({f}_{x,grad}) increases with the increase in (g), a higher energy modulation is achieved by displacing the DW, which further provides a higher increase (or decrease) in ({f}_{dep}) in P(downarrow ) (or P(uparrow )) domains, leading to larger NC effect. Similarly, (dP/dx) increases as the number of domains and the DWs increase with the decrease in ({T}_{fe}) (discussed before). Therefore, ({f}_{x,grad}) increases and provides an increased NC effect with decreasing ({T}_{fe}) (Fig. 3(d)). However, the soft-DW induced NC path does not depend of ({T}_{de}) (Fig. 3(e)). This is because, in the MD state, the average depolarization field (which is zero at *V*_{app} = 0) as well as ({f}_{x,grad}) and ({f}_{x,elec}) are independent of ({T}_{de}) within the limit of soft-DW. Interestingly, the MD-NC path does depend on the relative DE permittivity (({varepsilon }_{de})) as shown in Fig. 3(f). This is because the in-plane E-field, ({E}_{x,fe}) in the DW needs to satisfy the in-plane boundary condition at the FE-DE interface, which is ({E}_{x,fe}={E}_{x,de}), where ({E}_{x,de}) and ({E}_{x,fe}) are the in-plane E-field in DE and FE, respectively. As the ({E}_{x,de}) increases with the decrease in ({varepsilon }_{de}) (considering similar (P) difference between two consecutive domains), therefore, ({E}_{x,fe}) also increases in FE, which further increases the ({f}_{x,elec}) stored in the DW. Therefore, the ({F}_{dw}) increases and hence, NC effect increases with the decrease in ({varepsilon }_{de}) as shown in Fig. 3(f). From this analysis, we can summarize that, (i) an FE material with higher (g), (ii) ({T}_{fe}) scaling and/or (iii) using DE materials with low ({varepsilon }_{de}) are key design knobs to enhance DW-induced NC effect in MFIM stack. Note that in all of the cases discussed above, the MD NC path does not coincide with the Landau path (Fig. 3(c–f)) and the MD NC effect is less ((mathrm{1/|}{varepsilon }_{z,fe}^{eff-avg}|) is less) compared to the NC effect that corresponds to Landau path.

As the MD NC path is dependent on ({T}_{fe}), (g) and ({varepsilon }_{de}), therefore, the charge enhancement characteristics also depend on them as shown in Fig. 4(a–c). For simplicity, we only show the charge response for ({V}_{app} > 0). Now, a relation between the charge response in MFIM (({Q}_{MFIM})) and MIM (({Q}_{MIM})) can be written as the following equation.

$$frac{{dQ}_{MFIM}}{{dQ}_{MIM}}=frac{{dQ}_{MFIM}/{dV}_{app}}{{dQ}_{MIM}/{dV}_{app}}={left(1-frac{{C}_{MIM}times {T}_{fe}}{|{varepsilon }_{z,fe}^{eff-avg}|}right)}^{-1}$$

(3)

Here ({C}_{MIM}) (=({varepsilon }_{0}{varepsilon }_{de}/{T}_{de})) is the MIM capacitance per unit area. Now, considering that an increase(decrease) in charge enhancement implies an increase(decrease) in the right-hand side of the above equation, let us discuss the (g), ({T}_{fe}), and ({varepsilon }_{de}) dependency. The charge enhancement in MFIM increases with the increase in (g) (as (mathrm{1/|}{varepsilon }_{z,fe}^{eff-avg}|) increases) as shown in Fig. 4(a). Now with the increase in ({T}_{fe}), (mathrm{1/|}{varepsilon }_{z,fe}^{eff-avg}|) decreases. However, increase in ({T}_{fe}) dominates over decrease in (mathrm{1/|}{varepsilon }_{z,fe}^{eff-avg}|), when ({T}_{fe}) increases from 5 nm to 10 nm and therefore, charge enhancement shows mild increase (due to counteracting factors) as shown in Fig. 4(b). In contrast, when ({T}_{fe}) increases from 10 nm to 15 nm, the increase in ({T}_{fe}) is almost compensated by decrease in (mathrm{1/|}{varepsilon }_{z,fe}^{eff-avg}|) and hence, charge enhancement shows almost similar characteristics (Fig. 4(b) for *T*_{fe} = 10 nm and 15 nm). Now, with the increase in ({varepsilon }_{de}) (but same ({C}_{MIM}), obtained by increasing ({T}_{de}) proportionately, so the (Q)–({V}_{app}) characteristics of MIM remains same), charge enhancement increases as shown in Fig. 4(c). This is because, (mathrm{1/|}{varepsilon }_{z,fe}^{eff-avg}|) increases with the decrease in ({varepsilon }_{de}).

Based on the above discussion, we emphasize that the negative effective permittivity of the MD state is not an intrinsic material parameter of FE, rather, it depends on the physical thickness of the FE film, its gradient energy coefficient and the permittivity of the underlying DE material. It is important to note that, this conclusion is different than a single domain model^{1}, where the negative permittivity originates from negative slope of the Landau-Khalatnikov (LK) equation and hence, remains constant irrespective of ({T}_{fe}) and ({varepsilon }_{de}). Further, ({varepsilon }_{z,fe}^{eff-avg}) is a non-local quantity and can only describe the average characteristics. Moreover, the value of ({varepsilon }_{z,fe}^{eff-avg}) obtained for a particular ({T}_{fe}) and ({varepsilon }_{de}) can not be used to calculate the average charge response of MFIM/MFIS stack with any other ({T}_{fe}) and ({varepsilon }_{de}) due to the dependency of ({varepsilon }_{z,fe}^{eff-avg}) on these parameters. However, ({varepsilon }_{z,fe}^{eff-avg}) can be used to calculate the average charge response of MFIM/MFIS stack with different ({T}_{de}) within the limit of soft DW formation. Furthermore, the local effective permittivity, ({varepsilon }_{z,fe}^{eff}) of is not an intrinsic material property and hence, spatially varies within the FE layer^{3,4}. As the DW moves with the applied voltage, the local effective permittivity value changes with ({V}_{app}). As, ({varepsilon }_{z,fe}^{eff}) is not a spatially static quantity, therefore, one cannot directly use this in a capacitor equation to analyze the local charge response of the heterostructures (i.e. MFIM and MFIS stack). However, ({varepsilon }_{z,fe}^{eff-avg}) can be used in a capacitor equation to calculate the average charge response of MFIM/MFIS stack (within the limit of MD state) as in Eqs. 3–4. Therefore, the reason for introducing ({varepsilon }_{z,fe}^{eff}) and ({varepsilon }_{z,fe}^{eff-avg}) in this work is to analyze the implication of different parameters and to emphasize that, one should not use this characteristics as an intrinsic property of FE.