## Main

All materials present a positive global capacitance or dielectric constant on account of thermodynamic stability. Nevertheless, local negative capacitance (NC) states can be obtained in various ways2,4,5,6,7,8,9,10,11,12,13. Most interestingly, by placing a ferroelectric in contact with a dielectric or non-ideal electrodes9,14,15, we can prevent it from reaching its ground state (homogenous polarization), forcing it into a configuration of relatively high energy. Such a frustrated ferroelectric will typically display a steady NC response upon application of an electric field2,6,7,9,10. This has been shown in detail for multidomain structures in ferroelectric/dielectric superlattices4,5,11,16,17.

To understand steady-state NC, consider the superlattice in Fig. 1, where ferroelectric (f) and dielectric (d) layers repeat periodically along the stacking direction z. In the absence of free carriers, Maxwell’s first equation dictates D = ρfree = 0, so the z-component of the planar-averaged displacement vector is continuous. We thus have D = Df = Dd, where D is the superlattice displacement while Df and Dd are the layer vectors (z subscript omitted for simplicity). Using the definitions in Fig. 1, this yields

$$D=P+{\epsilon }_{0}{{{{\mathcal{E}}}}}_{{{{\rm{ext}}}}}={P}_{{{{\rm{f}}}}}+{\epsilon }_{0}{{{{\mathcal{E}}}}}_{{{{\rm{f}}}}}={P}_{{{{\rm{d}}}}}+{\epsilon }_{0}{{{{\mathcal{E}}}}}_{{{{\rm{d}}}}},$$
(1)

where ϵ0 is vacuum permittivity, P = L−1(lfPf + ldPd) is the superlattice polarization, $${{{{\mathcal{E}}}}}_{{{{\rm{ext}}}}}$$ is the external electric field along z and the total field in layer i (i = f, d) is

$${{{{\mathcal{E}}}}}_{i}={{{{\mathcal{E}}}}}_{{{{\rm{ext}}}}}+{{{{\mathcal{E}}}}}_{{{{\rm{ind}}}},i}.$$
(2)

Further, as D = Di we have

$${{{{\mathcal{E}}}}}_{{{{\rm{ind}}}},i}={\epsilon }_{0}^{-1}\left(P-{P}_{i}\right),$$
(3)

which shows that induced fields $${{{{\mathcal{E}}}}}_{{{{\rm{ind}}}},i}$$ appear when the local and global polarizations differ. For the f-layer we typically have Pf > P, so that $${{{{\mathcal{E}}}}}_{{{{\rm{ind,f}}}}}$$ opposes Pf; this is the so-called ‘depolarizing field’.

Because of the superlattice periodicity, the total voltage associated to the induced fields is null, implying $${l}_{{{{\rm{d}}}}}{{{{\mathcal{E}}}}}_{{{{\rm{i}}}}{\mathrm{nd,d}}}+{l}_{{{{\rm{f}}}}}{{{{\mathcal{E}}}}}_{{{{\rm{ind,f}}}}}=0$$. Hence, $${{{{\mathcal{E}}}}}_{{{{\rm{e}}}}{\mathrm{xt}}}$$ is the only macroscopic field acting on the system.

To examine the response to a variation of the external field $${\mathrm{d}}{{{{\mathcal{E}}}}}_{{{{\rm{ext}}}}}$$, it is useful to introduce a quantity we call the ‘screening factor’, defined for the f-layer as

$${\varphi }_{{{{\rm{f}}}}}=\frac{\mathrm{d}{{{{\mathcal{E}}}}}_{{{{\rm{ind,f}}}}}}{\mathrm{d}{{{{\mathcal{E}}}}}_{{{{\rm{ext}}}}}}={\epsilon }_{0}^{-1}\frac{\mathrm{d}\left(P-{P}_{{{{\rm{f}}}}}\right)}{\mathrm{d}{{{{\mathcal{E}}}}}_{{{{\rm{ext}}}}}}=\frac{{l}_{{{{\rm{d}}}}}}{L}\left({\chi }_{{{{\rm{d}}}}}^{\prime}-{\chi }_{{{{\rm{f}}}}}^{\prime}\right).$$
(4)

Here we use the primed susceptibilities $${\epsilon }_{0}{\chi }_{i}^{\prime}=\mathrm{d}{P}_{i}/\mathrm{d}{{{{\mathcal{E}}}}}_{{{{\rm{ext}}}}}$$, which are all but guaranteed to be positive. (The change in polarization—local or global—will always follow the change in the external field.) The inverse permittivity of the f-layer can then be written as

$${\epsilon }_{{{{\rm{f}}}}}^{-1}=\frac{\mathrm{d}{{{{\mathcal{E}}}}}_{{{{\rm{f}}}}}}{\mathrm{d}D}=\frac{\mathrm{d}{{{{\mathcal{E}}}}}_{{{{\rm{ext}}}}}}{\mathrm{d}D}\left(1+{\varphi }_{{{{\rm{f}}}}}\right)={\epsilon }^{-1}\left(1+{\varphi }_{{{{\rm{f}}}}}\right).$$
(5)

Further, as detailed in Supplementary Note 1, we can derive the voltage response of the dielectric layer $${{{{\mathcal{A}}}}}_{{{{\rm{d}}}}}$$ as

$${{{{\mathcal{A}}}}}_{{{{\rm{d}}}}}=\frac{\mathrm{d}{V}_{{{{\rm{d}}}}}}{\mathrm{d}V}=\frac{{l}_{{{{\rm{d}}}}}}{L}\frac{\mathrm{d}{{{{\mathcal{E}}}}}_{{{{\rm{d}}}}}}{\mathrm{d}{{{{\mathcal{E}}}}}_{{{{\rm{ext}}}}}}={L}^{-1}\left({l}_{{{{\rm{d}}}}}-{l}_{{{{\rm{f}}}}}{\varphi }_{{{{\rm{f}}}}}\right).$$
(6)

Voltage amplification (VA) corresponds to $${{{{\mathcal{A}}}}}_{{{{\rm{d}}}}} > 1$$. This key quantity is fully determined by trivial geometric elements and the screening factor of the f-layer.

We now discuss the dielectric response of a superlattice. Typically the ferroelectric layers will be more responsive than the dielectric ones, so that $${\chi }_{{{{\rm{f}}}}}^{\prime} > {\chi }_{{{{\rm{d}}}}}^{\prime}$$. From equation (3), the induced depolarizing field $$\mathrm{d}{{{{\mathcal{E}}}}}_{{{{\rm{i}}}}{\mathrm{nd,f}}}$$ will oppose $$\mathrm{d}{{{{\mathcal{E}}}}}_{{{{\rm{ext}}}}}$$, and hence φf < 0. One expects the induced field to be smaller in magnitude than the applied one, so that −1 < φf < 0. It follows that $${\epsilon }_{{{{\rm{f}}}}}^{-1} > 0$$ and $${{{{\mathcal{A}}}}}_{{{{\rm{d}}}}} < 1$$, a behaviour we may call normal.

Imagine we make the ferroelectric more responsive, for example by varying its temperature to approach the Curie point. We can eventually reach a situation where the induced $$\mathrm{d}{{{{\mathcal{E}}}}}_{{{{\rm{ind,f}}}}}$$ compensates the applied $$\mathrm{d}{{{{\mathcal{E}}}}}_{{{{\rm{ext}}}}}$$ (φf = −1), and the voltage drops exclusively in the dielectric layers ($${{{{\mathcal{A}}}}}_{{{{\rm{d}}}}}=1$$). The ferroelectric effectively behaves as a metal; we call this ‘perfect screening’.

If we keep softening the f-layer so that $${\chi }_{{{{\rm{f}}}}}^{\prime}\gg {\chi }_{{{{\rm{d}}}}}^{\prime}$$, we access a regime where the ferroelectric ‘over-screens’2: its response is so strong that the induced depolarizing field exceeds the applied one (φf < −1). This yields NC ($${\epsilon }_{{{{\rm{f}}}}}^{-1} < 0$$) and VA in the dielectric ($${{{{\mathcal{A}}}}}_{{{{\rm{d}}}}} > 1$$).

Our formulas show that NC and VA can be obtained from the layer polarizations, readily available from the ‘second-principles’ simulations18,19,20 used to explain NC in PbTiO3/SrTiO3 (PTO/STO) superlattices4,5 (Methods). We now use said methods to monitor the dependence of NC and VA on the design variables offered by these materials (layer thickness, epitaxial strain).

We study PTO/STO superlattices where the PTO and STO layers have a thickness of n and m perovskite cells, respectively, denoted n/m in the following. We consider n and m from 3 to 9, and investigate the response to small fields along z. We also vary the epitaxial strain η between −1% and +3%, choosing the STO substrate as the zero of strain.

We restrict ourselves to low temperatures (formally, 0 K) and work with periodically repeated supercells that are relatively small in plane (8 × 8 perovskite units). This is sufficient to draw conclusions on the behaviour of real materials at ambient conditions.

Let us first recall the main effect epitaxial strain has on PTO/STO superlattices, as obtained from our simulations. Figure 2a shows the ground state of the 6/6 system for η = −1%: it presents stripe domains in the PTO layer, with local polarizations along the out-of-plane (OOP) z direction. This ‘multi-OOP’ state has been thoroughly studied4,21,22,23,24,25,26.

For large enough tensile strains, we find the PTO layer displays a monodomain state with in-plane (IP) polarization (Fig. 2b). This simulated ‘mono-IP’ configuration is characterized by Px = Py. In reality27, one typically observes the so-called a1/a2 multidomain configuration, with local polarizations alternating between Px and Py. Our monodomain result is a consequence of the relatively small size of the simulation supercell.

Finally, Fig. 2c,d shows states we obtain in some superlattices at intermediate η values, where mono-IP and multi-OOP features mix, reminiscent of similar findings in the literature26,27. Thus, apart from some non-essential size effects, our simulations capture the evolution of PTO/STO superlattices with epitaxial strain.

Figure 3a–d shows detailed results for the 3/3 system. At compressive and slightly tensile strains, we get a multi-OOP solution similar to that of Fig. 2a, with Pz ≠ 0 and Px = 0. As η increases, we see a transition to the mono-IP phase with Pz = 0 and Px ≠ 0. This transition is discontinuous, both the multi-OOP and mono-IP states being stable at intermediate strains (grey area in the figure).

The global dielectric susceptibility is shown in Fig. 3b. As we increase η in the multi-OOP state, we induce a maximum of χxx, signalling the occurrence of an IP polar instability. In the mono-IP state, it is χzz that peaks as η decreases, indicating a soft OOP polar mode. The mono-IP state also displays a peak in χxx at η ≈ 0.6%; this feature, associated to STO and not essential here, is discussed in Supplementary Note 2 and Supplementary Fig. 1.

Figure 3c shows the inverse permittivity (green) and screening factor (orange) of the f-layer. For all considered strains we get $${\epsilon }_{{{{\rm{f}}}}}^{-1} < 0$$ and the associated overscreening (φf < −1). Figure 3d shows the corresponding VA in the d layer, which reaches values as high as 12 as the mono-IP state approaches its stability limit. This giant amplification is related to the maximum in χzz (Fig. 3b), in turn connected to the OOP polar instability of the PTO layer. By contrast, the destabilization of the multi-OOP state upon increasing η— which involves a χxx anomaly—does not result in any feature in $${\epsilon }_{{{{\rm{f}}}}}^{-1}$$ or $${{{{\mathcal{A}}}}}_{{{{\rm{d}}}}}$$.

The 9/9 superlattice presents a similar behaviour (Fig. 3e–h), except we find a gradual transformation from multi-OOP to mono-IP, for η between 0.0% and 0.8%, with the occurrence of the mixed state mentioned above (Fig. 2c,d). The small jump in Px around η = 0.9% is related to the occurrence of an IP polarization in the STO layer (not relevant here; Supplementary Note 2 and Supplementary Fig. 2).

The 9/9 superlattice displays its largest NC response in this intermediate region, reaching fivefold amplifications at the transition between the mono-IP and mixed states. Interestingly, the multi-OOP state of the 9/9 superlattices shows a peculiar behaviour: see for example $${{{{\mathcal{A}}}}}_{{{{\rm{d}}}}} < 0$$ at η = −0.5% in Fig. 3h. In this regime, the PTO layer is in a very stable (stiff) multidomain configuration, while the in-plane compression makes STO electrically soft along z. Hence, the roles reverse and the STO layer displays NC. (More in Supplementary Note 3.) A similar behaviour has been predicted for BaTiO3/SrTiO3 superlattices13.

We run the same study for a large collection of superlattices; Fig. 4a–c summarizes our results. We find the transition region between the multi-OOP and mono-IP states becomes wider for thicker PTO, reflecting the fact that broader layers can accommodate more complex dipole orders, such as the one occurring in the mixed state. (This is consistent with recent observations, for example the occurrence of supercrystals in PbTiO3/SrRuO3 superlattices with PTO layers above 15 cells28.) The mixed state is also favoured by thicker STO layers, a subtle effect probably related to the fact that the stray fields are expelled from the STO layer as it thickens.

Most importantly, Fig. 4 confirms that the strongest amplifications occur at the stability limit of the mono-IP state. It also shows that the multi-OOP region is comparatively unresponsive. Let us now get some insight into the physical underpinnings of these behaviours.

According to equations (4) and (6), VA is determined by the screening factor of the f-layer, which in turn depends on the difference in dielectric response between layers. For example, for the 3/3 superlattice at η = 0.3% we get $${{{{\mathcal{A}}}}}_{{{{\rm{d}}}}}\approx 12$$, with $${\chi }_{{{{\rm{f}}}}}^{\prime}=765$$ and $${\chi }_{{{{\rm{d}}}}}^{\prime}=721$$. This $${\chi }_{{{{\rm{f}}}}}^{\prime}$$ value may seem small; indeed, the ferroelectric is close to developing an OOP polar instability and one would expect susceptibilities around 10,000 (refs. 29,30). By contrast, the computed $${\chi }_{{{{\rm{d}}}}}^{\prime}$$ is surprisingly large, as our model for STO yields χ = 202 for the pure material. (Our simulated STO is stiff compared with experimental measurements4.)

The reason for these surprising $${\chi }_{i}^{\prime}$$ susceptibilities can be traced back to electrostatics: all layers respond similarly to an external field, to minimize the depolarizing fields. Thus, we expect $${\chi }_{{{{\rm{f}}}}}^{\prime}\gtrsim {\chi }_{{{{\rm{d}}}}}^{\prime}$$. For example, for the 6/6 superlattice at η = −1%, which does not display VA, we obtain $${\chi }_{{{{\rm{f}}}}}^{\prime}=96$$ and $${\chi }_{{{{\rm{d}}}}}^{\prime}=95$$ (Supplementary Fig. 3). Then, when we move to a region of the phase diagram where the f-layer presents an OOP instability, the energy gain associated to the development of dPf overwhelms the cost of creating a depolarizing field. Hence, the difference between $${\chi }_{{{{\rm{f}}}}}^{\prime}$$ and $${\chi }_{{{{\rm{d}}}}}^{\prime}$$ grows a little, sufficient to yield large VA values.

The largest amplifications correspond to the region marking the limit of stability of the mono-IP state. Here the f-layers are in an ‘incipient ferroelectric’ state2,13: they are ready to develop an homogeneous OOP polarization whose occurrence is precluded by the presence of the d-layers. Eventually, as we move towards negative η values, the multi-OOP polar instability freezes in, leading to either a pure multi-OOP state or a mixed state, and hardening the z-polarized ferroelectric soft mode. (This resembles the competition between antipolar and polar orders in antiferroelectrics31,32.) This incipient ferroelectric state corresponds to the idealized picture of monodomain NC2,3; our results predict a realization of this archetype.

As shown in Fig. 5a and previously reported4,5, the NC response of multi-OOP states mainly stems from the strong response (large $$\chi ^{\prime}$$) of the domain walls. By contrast, the NC of the incipient ferroelectric state comes from the whole f-layer (Fig. 5b), which partly explains its superior VA performance.

Our results thus suggest a strategy to obtain large VA: work with electrostatically induced incipient ferroelectric states that will typically occur at the boundary between IP and OOP phases in ferroelectrics with imperfect screening. Phase boundaries akin to the ones discussed here have been found experimentally in PTO/STO superlattices27 and predicted in other ferroelectric/dielectric heterostructures13. More specifically, PTO/STO superlattices grown on DyScO3 substrates display a coexistence of a1/a2 (IP) and vortex (OOP) states at room temperature27. Further, the balance between such phases can be tuned by controlling the layer thickness33, which should allow stabilization of a1/a2 states on the verge of developing an OOP polarization, thus fulfilling the conditions to present strong overscreening in the ferroelectric layer (φf −1). Those are clear candidates to display giant incipient ferroelectric VA as predicted here. Let us stress that, despite their limitations (low temperature, only monodomain IP states), our simulations capture the physics of the IP-to-OOP transition; thus, we expect our conclusions to apply to experimentally relevant situations.

Additionally, our formulas teach us that $${{{{\mathcal{A}}}}}_{{{{\rm{d}}}}}$$ does not depend on the macroscopic permittivity ϵ−1 (equation (6)), while $${\epsilon }_{{{{\rm{f}}}}}^{-1}$$ does (equation (5)). Hence, one can have behaviours such as that of the 3/3 system at η = −1% (Fig. 3): a very negative $${\epsilon }_{{{{\rm{f}}}}}^{-1}$$ (Fig. 3c) not accompanied by a large $${{{{\mathcal{A}}}}}_{{{{\rm{d}}}}}$$ (Fig. 3d). The reason is that this superlattice presents a small χzz (Fig. 3b), which yields large ϵ−1 and $$| {\epsilon }_{{{{\rm{f}}}}}^{-1}|$$. By the same token, having a globally soft superlattice may result in a modest NC of the f-layer, but this does not necessarily imply a small VA. Hence, for VA purposes, we should not disregard very responsive systems where small values of ϵ−1 or $$| {\epsilon }_{{{{\rm{f}}}}}^{-1}|$$ have been observed4,5,16. Rather, we must focus on the response difference between ferroelectric and dielectric layers, as captured by the screening factor φf.

Finally, let us stress that our conclusions are not restricted to an idealized superlattice. Note that an infinite superlattice is equivalent to a ferroelectric/dielectric bilayer contacted with good electrodes, so there is no net depolarizing field. Further, NC is perfectly compatible with non-ideal electrodes and depolarizing fields7; in fact, imperfect screening is at the origin of the effect and can be engineered to induce it2,6,9. Hence, we expect our conclusions to apply to real systems whenever the development of an homogeneous polar state is precluded, including field-effect transistors featuring a ferroelectric/semiconductor bilayer.

We hope this work will bring an impetus to the study of NC, shifting the focus to the quantification and optimization of voltage amplification.

## Methods

The second-principles simulations are performed using the SCALE-UP package18,19,20 and the same approach as previous studies of PTO/STO superlattices4,25,34. The superlattice models are based on potentials for the pure bulk compounds—fitted to first-principles results18—and adjusted for the superlattices as described in ref. 4.

We study a collection of n/m superlattices with layer thicknesses n, m = {3, 6, 9}. Further, we consider an isotropic epitaxial strain η between −1% and 3%, where the STO square substrate (with lattice constant of 3.901 Å) is taken as the zero of strain. Note that STO is a convenient reference on account of the popularity of this substrate in experimental investigations and the fact that it lies on the verge of the OOP-to-IP transformation. Additionally, the STO substrate closely matches the in-plane lattice constant of PTO in the OOP state.

We work with a simulation supercell that contains 8 × 8 perovskite unit cells in the xy plane (perpendicular to the stacking direction). In the z direction, only one superlattice period is considered. Periodic boundary conditions are assumed.

To find the lowest-energy state of an n/m superlattice at a given η and electric field value, we relax the atomic structure by performing Monte Carlo simulated annealings. During the annealings, all atomic positions and strains are allowed to vary, except for the in-plane strains imposed by the substrate. From the resulting atomic structures, we compute local electric dipoles within a linear approximation (that is, we consider the atomic displacements with respect to the high-symmetry reference structure and multiply them by their corresponding Born charge tensors), as customarily done in second-principles studies4.

To compute responses, a small external field of 0.2 MV cm−1 is considered. We checked that this field is small enough to obtain susceptibilities and the other relevant quantities within a linear approximation.

We should mention that it is possible to study materials under various electric boundary conditions (that is, at constant electric field35 or constant displacement36) directly from first principles. Yet, here we adopt a second-principles approach for the sake of computational feasibility. The smallest case simulated in this work (3/3 superlattice) contains 1,920 atoms; the largest (9/9) involves 5,760. Systems of this size remain all but untreatable with today’s first-principles methods.

Finally, let us note that STO is far from being a passive dielectric layer. Indeed, it features structural instabilities of its own: antiphase rotations of the O6 groups that are reproduced by our second-principles model18 and present in our simulations. Further, the O6 tilts compete with an incipient ferroelectric order37, and said polar order can be stabilized under epitaxial strain38. These effects, and their impact on some of our results, are mentioned in the main text of this article and further addressed in Supplementary Notes 2 and 3. In addition, Supplementary Note 4 and Supplementary Fig. 4 summarize the behaviour of a pure STO film as a function of epitaxial strain, as predicted by our second-principles model.