**Strain-polarization coupling in KNO thin films.** A series of KNO thin films with the thicknesses of 66 nm, 110 nm, 176 nm and 330 nm were grown by pulsed laser deposition (PLD) on (001)-oriented SrTiO3 (STO) substrates with SrRuO3 (SRO) as the bottom electrode layer. The X-ray diffraction (XRD) θ-2θ scans, as shown in Fig. 1a, reveal the (001)pc (“pc” refers to the pseudocubic unit cell) epitaxial growth of KNO thin films with different thickness. With the decrease of the film thickness, the peaks of the KNO (001)pc shift to lower 2θ values, indicating the expansion of the out-of-plane lattice parameter ~ c. Accordingly, reciprocal space mappings (RSMs), as displayed in Fig. 1b and Supplementary Fig. S1, show that both NNO (103)pc and (013)pc reflection peaks shift to higher Qx value with the decrease of the film thickness, demonstrating the shrink of in-plane lattice parameters of both a and b. The thickness-dependent lattice parameter variations are shown in Fig. 1c, exhibiting the relaxation process of the compressive strain. The tetragonality (c/a ratio) reaches to 1.044 in 66 nm-thick KNO film, while it gradually decreases to 1.005 (thickness ~ 330 nm) with the increase of film thicknesses.

Next, we further study the strain-polarization coupling in KNO thin films The measurements of polarization-electric field loops (P-E Loops, Fig. 2a) and current-electric field curves (I-E curves, Fig. 2b) were carried out. It shows that the thickest KNO film (~ 330 nm) presents a slim P-E loop and negligible switching current, indicating the weak polarization, whereas the typical hysteresis loops and switching current peaks in 66 nm, 110 nm, and 176 nm KNO thin films demonstrate the enhanced ferroelectricity with increasing compressive strain. The remnant polarization (Pr) is improved with the decreasing thickness from 2.2 µC/cm2 at 330 nm to 20.4 µC/cm2 at 66 nm. There results reveal the strain-polarization coupling effect in the KNO thin film, in agreement with previous experimental and theoretical studies in ferroelectric materials30,34,39. The fatigue endurance and temperature stability of the films were further investigated to evaluate performance of the KNO capacitors. As shown in Fig. 2c, d and Fig. S2, after 108 electric field cycles of bipolar triangular waveform with a maximum electric field of 225 kV/cm and a frequency of 100 kHz, the shape of P-E loops in 176 nm KNO thin films was kept the complete hysteresis, until the occurrence of leakage problem after 109 cycles. The KNO film also shows excellent temperature stability in the range of room temperature to 200°C.

**Observation of tunable negative capacitance via strain engineering.** The measurement of the transient NC effect was performed by monitoring the change of ferroelectric capacitor voltage (VF) and charge (QF) across the circuit in a capacitor-resistor series network (Fig. 3a)17. According to the definition of capacitance, C = dQ/dV, NC effect is manifested as an opposite variation tendency of time-dependent QF and VF. Theoretically, the NC state is unstable since it located at the maximum in energy landscape of ferroelectrics37. However, applying a voltage larger than coercive field can switch the polarization from one stable state to the other, the ferroelectrics would pass through the NC region at the same time. Thus, applying pulse voltage to stabilize the polarization state after switching and connecting a resistor in series to delay the delivery of screening charge is helpful to capture the transient NC effect40.

Figure 3b shows the voltage across the Pt/KNO(~ 66 nm)/SRO capacitor which calculated by Vf = Vs – VR, where Vs is the source voltage with a pulse sequence of -0.75 V→ +0.75 V→ -0.75 V, VR is the voltage across series resistor with R = 1200 Ω measured by oscilloscope. The current flowing through circuit I(t) is calculated using I(t) = VR/R, as presented in Fig. 3c. The total charge across the whole circuit at a given time t, Q(t), is calculated by \(Q\left(t\right)={\int }_{0}^{t}I\left(t\right)dt\). Note that the total charge Q(t) includes charge across ferroelectric capacitor, Qf(t), and the charge contributes from parasitic capacitance Cp of oscilloscope. The Qf(t) can be calculated by \({Q}_{f}\left(t\right)=Q\left(t\right)-{C}_{p}{V}_{f}\left(t\right)\). Here we neglect the Cp because it is usually too small, meaning that Qf(t) is replaced by Q(t), as illustrated in Fig. 3d. It is noted that the VF increases to a high point and then drops down during a short period of time after the VS transition from − 0.75 V to + 0.75 V, as shown in the shadow area of region Ⅰ in Fig. 3b and magnified in Fig. 3e. Both current and charge increase in the same time region, implying an opposite variation tendency between VF and QF, which is negative differential capacitance. Similarly, the increasing voltage and decreasing charge have been observed after VS switching from + 0.75 V to -0.75 V in the shadow area of the region Ⅱ in Fig. 3b and Fig. 3e. The transition of VS induces the polarization switching, meaning that ferroelectrics pass from one stable state through the NC region to the other.

The NC effect of KNO thin films under different strain states was characterized in the same circuit and series resistor, and the results are shown in Fig. 4 and Fig. S3. The same measured electric field were applied by increasing source voltage proportionally with the thickness. The NC regions in Fig. 4a were framed in dotted square and magnified in Fig. 4b and Fig. 4c. Obviously, the maximum decreasing magnitude of electric field appeared in the 66 nm NNO thin film, which is imparted by the largest compressive strain. Then it decreases with the increase of film thicknesses. Hardly no NC effect expect noise of oscilloscope can be detected in the 330 nm-thick sample. The variation of electric field decreasing magnitude reveals that the NC effect observed here is intrinsic but not arisen from the noise and fluctuation of experimental setup. The ferroelectric polarization, P = QF/A, where A was the electrode area (cm2), is normalizationally plotted as a function of electric field in Fig. 4d. The negative slope shown in shadow areas in P-E curves in the second and fourth quadrants indicate the emergence of negative capacitance in these regions. It is revealed that the KNO thin films with the decrease of thicknesses, i.e., the increase of compressive strain, present increasing negative slope, demonstrating the strain tuning of NC effect.

**Simulations of strain tunable NC effect in KNO.** Next, we further study the effect of epitaxial strain on the NC starting from a simple thermodynamic Landau model41–43 (Details in Supplementary material) to understand the mechanism of strain tuning of NC effect in ferroelectric films. In Landau’s theory, assuming that there is only a uniform polarization along the (101)pc direction in orthorhombic KNO, the ferroelectric free energy can be written as \(G\)*=*\({(\alpha }_{1}^{*}+{\alpha }_{3}^{*}{)P}_{1}^{2}+{(\alpha }_{11}^{*}+{\alpha }_{33}^{*}{+{\alpha }_{13}^{*})P}_{1}^{4}+{(2\alpha }_{111}+2{\alpha }_{112}){P}_{1}^{6}+\left(2{\alpha }_{1111}+{2\alpha }_{1112}+{\alpha }_{1122}\right){P}_{1}^{8}-E{P}_{1}+\frac{{\epsilon }^{2}}{{s}_{11}+{s}_{22}}\), where \({\alpha }_{1}^{*}\), \({\alpha }_{3}^{*}\), \({\alpha }_{13}^{*}\), \({\alpha }_{33}^{*}\), \({\alpha }_{111}\), \({\alpha }_{112}\), \({\alpha }_{1111}\), \({\alpha }_{1112}\), \({\alpha }_{1122}\), \({s}_{11}\), \({s}_{22}\) are the Landau coefficients, E is the applied electric field, \(\epsilon\) is the strain. Below Curie temperature and without strain, \({\alpha }_{1}^{*}+{\alpha }_{3}^{*}\) < 0 and \({\alpha }_{11}^{*}+{\alpha }_{33}^{*}+{\alpha }_{13}^{*}\) > 0, thus the free energy-polarization landscape shows double-well characteristic. However, the \({\alpha }_{1}^{*}+{\alpha }_{3}^{*}\) is changed to positive under tensile strain, the energy landscape accordingly becomes a single potential well. While under compressive strain, the \({\alpha }_{1}^{*}+{\alpha }_{3}^{*}\) increases negatively, thus the both potential wells become deeper. Consequently, the polarization corresponding to the minimum free energy increases and the negative differential region related to NC is enlarged (Fig. 5a). The P-E curves can be obtained by the first partial differentiation of free energy with respect to the polarization. Obviously, the negative slope of the curves increases under compressive strain (Fig. 5b). The second partial differential of free energy to the polarization provides the relationship between the permittivity and polarization. In compressive strain states, permittivity is negative and shows considerable tunability in the region of P1 ≈ 0, which means that NC can be adjusted by different strain states to achieve the capacitance matching with the dielectric layer and maximize voltage amplification.

Our experimental results can be better understood from the time-dependent Landau–Khalatnikov equation \(\rho \frac{d{Q}_{F}}{dt}=-\frac{dU}{d{Q}_{F}}\)44, 45. As shown in Fig. 5c and 5d, the solution of the equation under compressive strain shows a larger voltage drop amplitude and duration, in agreement with our experimental results. Differently from changing the resistor in series to control the voltage reduction magnitude in other works20, we use the same resistance in this study to measure the voltage variation of a series of KNO samples, so the increasing voltage reduction magnitude contributes from the intrinsic properties of KNO rather than measured circuit. The calculations and simulations in this work are based on the mono-domain ferroelectrics, while the free energy landscapes need to be modified due to the multi-domain which ferroelectrics actually tend to exhibit. However, they are usually associated with mono-domain configurations, thus the effect of strain on NC effect in our work can still predict experimental results in multi-domain ferroelectrics qualitatively.