Negative Capacitance for Electrostatic Supercapacitors

The increasing demand for efficient storage of electrical energy is one of the main challenges in the transformation toward a carbon neutral society. While electrostatic capacitors can achieve much higher power densities compared to other storage technologies like batteries, their energy densities are comparatively low. Here, it is proposed and demonstrated that negative capacitance, which is present in ferroelectric materials, can be used to improve the energy storage of capacitors beyond fundamental limits. While negative capacitance was previously mainly considered for low power electronics, it is shown that ferroelectric/dielectric capacitors using negative capacitance are promising for energy storage applications. Compared to earlier results using (anti)ferroelectric materials for electrostatic energy storage, much higher efficiencies of more than 95% even for ultrahigh energy densities beyond 100 J cm−3 are demonstrated using nonepitaxial thin films suitable for integration on 3D substrates. Stable operation up to 150 °C and 108 charging/discharging cycles is further demonstrated.


Energy Storage in Nonlinear Capacitors
In general, the electrostatic energy density in a dielectric material is given by where E is the electric field and D is the electric displacement field. Note that the relationship between E and D can be nonlinear. Similarly, when describing the total electrostatic The increasing demand for efficient storage of electrical energy is one of the main challenges in the transformation toward a carbon neutral society. While electrostatic capacitors can achieve much higher power densities compared to other storage technologies like batteries, their energy densities are comparatively low. Here, it is proposed and demonstrated that negative capacitance, which is present in ferroelectric materials, can be used to improve the energy storage of capacitors beyond fundamental limits. While negative capacitance was previously mainly considered for low power electronics, it is shown that ferroelectric/dielectric capacitors using negative capacitance are promising for energy storage applications. Compared to earlier results using (anti)ferroelectric materials for electrostatic energy storage, much higher efficiencies of more than 95% even for ultrahigh energy densities beyond 100 J cm −3 are demonstrated using nonepitaxial thin films suitable for integration on 3D substrates. Stable operation up to 150 °C and 10 8 charging/ discharging cycles is further demonstrated.

Introduction
The storage of electrical energy has only been possible since the invention of the capacitor in 1745. [1] When a voltage is applied to a capacitor, energy is stored in the electric field in the dielectric material which separates the two conducting electrodes. The major advantages of the energy storage in capacitors are a high energy storage efficiency, temperature, and cycling stability as well as high power densities. On the other hand, regular dielectric capacitors cannot compete with the orders of magnitude higher energy storage densities of, e.g., batteries or fuel cells. [2] However, so-called supercapacitors, which combine energy stored in a capacitor (neglecting fringing fields), we can write where V is the applied voltage and Q is the charge on the electrodes. Graphically, the stored energy (density) therefore coincides with the area above the Q-V (D-E) curve as shown in Figure 1a for a linear positive capacitor with constant capacitance C  dQ/dV. Since in a linear dielectric D = εE, where ε is the permittivity, it follows that Q = CV and we obtain the well-known expressions w = εE 2 /2 and W = CV 2 /2. In such regular capacitors, the maximum stored energy is limited by the breakdown field strength and permittivity of the dielectric. When thinking about an NC material, the relationship between Q and V (or D and E) cannot be linear for all applied voltages, since such a capacitor would be able to supply infinite amounts of energy. However, materials which show a differential negative permittivity (dD/dE < 0) or capacitance (dQ/dV < 0) can indeed exist if this NC region is bounded by regions of positive capacitance as shown exemplarily in Figure 1b. The simplest way to describe the behavior in Figure 1b is by using a 3rd order polynomial, which is shifted along the Q-axis as described by where a < 0, b > 0 and ΔQ = (−a/b) 1/2 is the remanent charge. Note that the bias point in Figure 1b (red dot), where dQ/dV < 0, would be unstable in an isolated capacitor. [12,[21][22][23] The material would quickly relax to the positive capacitance regions, which would lead to a hysteresis when trying to measure the Q-V curve by forcing a voltage as indicated by the dashed blue arrows. This inherent Q-V hysteresis caused by the unstable NC region is undesirable for energy storage applications, since the area inside the hysteresis loop corresponds to the dissipated energy W loss as shown in Figure 1b. In this case, the energy recovered during discharging of the capacitor (W d ) will be lower than the energy supplied during charging (W c ). Therefore, we can define the hysteresis loss as The average energy densities 〈w loss 〉, 〈w c 〉, and 〈w d 〉 can be obtained by dividing W loss , W c , and W d by the capacitor volume. The energy storage efficiency η of the capacitor is then defined as As can be seen from Equations (4) and (5), if there is no Q-V hysteresis, the energy storage efficiency of a capacitor is η = 1 since W c = W d = W.

Proposal of a Negative Capacitance Supercapacitor
In the following we will describe a way to create a supercapacitor using NC, but without reducing the storage efficiency due to hysteresis. If we combine a positive capacitance material (V = Q/C) with an NC material described by Equation (3) in series (see Figure 1c), we obtain the following Q-V relation of the total capacitor Since the combined supercapacitor should ideally have no Q-V hysteresis to achieve η = 1, the condition 1/C > −a must be satisfied, which was originally proposed for hysteresis-free NC transistors by Salahuddin and Datta. [19] An exemplary Q-V curve of such an NC supercapacitor can be seen in the right panel of Figure 1c. Due to the positive curvature of the nonlinear Q-V curve, W can be significantly increased compared to the linear capacitor in Figure 1a, even at identical voltage V and stored charge Q. Using Equations (2) and (6), the energy stored in such a capacitor is given by Adv. Energy Mater. 2019, 9,1901154  Energy stored in a capacitor is given by the area (green) above the Q-V curve. b) Capacitor with negative differential capacitance dQ/dV < 0 due to dD/dE < 0. Exemplary Q-V characteristic with negative capacitance region. In isolation, the NC region is unstable, and hysteresis is observed (dashed blue arrows) leading to an energy loss (orange area). c) Combining NC and positive capacitance to create an NC supercapacitor with improved energy storage (green area) and without hysteresis.
To determine the maximum energy storage enhancement compared to the same capacitor without NC layer, we investigate the case where 1/C = −a, which corresponds to the optimal capacitance matching condition of both layers. [19] After normalizing W to the energy of the capacitor C, at the same voltage, which is W 0 = CV 2 /2, we obtain a maximum energy storage enhancement W/W 0 at the voltage V = V max (see Supporting Information for a detailed derivation). In a first order approximation we can write which surprisingly is independent of b and only depends on the product aC. In the case of perfect capacitance matching (aC = −1), the enhancement is roughly two, which means that the energy stored at identical voltage is doubled. The voltage V max , where this maximum enhancement is observed can be obtained from It should be noted, that both a and 1/C are proportional the thickness of the NC layer and dielectric, respectively. For energy storage applications, the NC supercapacitor should be designed such that it operates around the voltage V max . This enhancement of the stored energy at identical voltage due to positive Q-V curvature is the one of the main advantages of the proposed NC supercapacitor. Furthermore, the overall leakage current through the capacitor will be reduced, and the breakdown voltage will be improved by adding the NC layer. [24] Therefore, it should be possible to apply much higher voltages to the NC supercapacitor compared to the same capacitor without NC layer, leading to significantly higher energy densities. Lastly, as shown above, with the hysteresis-free condition 1/C > −a, the theoretical storage efficiency will be η = 1. Note that when combining two positive capacitance layers, W/W 0 is always smaller than one.

Ferroelectric/Dielectric NC Supercapacitors
As mentioned earlier, a hysteretic Q-V behavior as shown in Figure 1b is well known from ferroelectric materials, which possess a spontaneous electric polarization P S whose direction can be reversed by the application of an electric field E. [25] Due to the inherent P-E hysteresis of ferroelectric materials, a lot of previous work has been focused on nonvolatile memory devices. [26] However, in recent years it was argued that ferroelectric materials could also exhibit an NC without the typical P-E hysteresis, if coupled to a material with positive capacitance. [19] Growing experimental evidence suggests that this is indeed possible. [12][13][14][15][16][17][18]21,22,27] To qualitatively understand how NC in ferroelectric materials can be used to build a supercapacitor, it is useful to examine the basic physical characteristics of dielectric and ferroelectric capacitors. Figure 2a,b shows the free energy-displacement field (F-D), displacement field-electric field (D-E) and capacitance-displacement field (C-D) relationships of a shortcircuited dielectric capacitor and a ferroelectric capacitor. For the latter case, the displacement field D f is approximately identical to the ferroelectric polarization P f . The blue circles indicate the equilibrium state of each capacitor. For the dielectric capacitor in Figure 2a, the free energy F d is identical to the electrostatic energy w d = D d 2 /2ε d , where ε d is the dielectric permittivity. Since ε d is constant, the electric field in the dielectric E d is proportional to the displacement field D d . Therefore, the capacitance of the dielectric capacitor C d is also constant. When no voltage is applied to such a dielectric capacitor, there is no charge Q on the electrodes, because E d = 0 and thus D d = 0. On the other hand, for a shorted ferroelectric capacitor as shown in Figure 2b, the equilibrium state is characterized by a spontaneous polarization ±P S , which correspond to one of the degenerate energy minima in the double well free energy landscape F f . [18] The model for the ferroelectric is based on Landau-Ginzburg-Devonshire theory assuming a homogeneous ferroelectric polarization P f . [28,29] The polarization charges P f are completely compensated by the charges +Q and −Q on the electrodes. In this simple picture of homogeneous P f , the relationship between P f and the electric field in the ferroelectric E f has an "S"-shape. Since the ferroelectric capacitance C f is proportional to the slope dP f /dE f , C f is negative in the region of smaller P f . This NC region of dP f /dE f < 0 coincides with the free energy barrier where d 2 F f /dP f 2 < 0. However, the capacitance C f in the equilibrium state (around P f = ±P S ) is always positive. [12] This can also be seen from the dependency of the inverse capacitance 1/C f on P f , which is shown on the righthand side of Figure 2a.
To create an NC supercapacitor with a ferroelectric/ dielectric stack as shown in Figure 1c, we first need to ensure that the ferroelectric is in one of its spontaneous polarization states when no external voltage V is applied. This can be achieved, e.g., by introducing a sufficiently large fixed charge σ IF at the ferroelectric/dielectric interface as indicated in Figure 2c. [15,18] Such a fixed charge is typically observed at the interface between different dielectrics and can be engineered, e.g., by changing the dielectric material or thickness in a stack consisting of two or more layers. [30,31] Besides this, σ IF might originate from electrons trapped at deep interface states. The charge σ IF plays the same role as the ΔQ introduced in Equation (3) and effectively shifts the Q-V curve upward. See the Supporting Information for a detailed derivation of the theory for the ferroelectric/dielectric NC supercapacitor. In the example given in Figure 2c, σ IF = −|P S |, which results in a complete compensation of P f at the ferroelectric/ dielectric interface if P f = −P S , leading to D d = 0 and E f = 0. If there would be no charge σ IF , the uncompensated charge P f at the ferroelectric/dielectric interface would result in a large depolarization field in the ferroelectric, making the spontaneous polarization state unstable when V = 0. This would be detrimental for the performance of the NC supercapacitor. Notice that in Figure 2c, the total free energy F has no negative curvature and the Q-V curve has a positive slope, which corresponds to a strictly positive total capacitance C = dQ/dV in the entire voltage range. This can be achieved by matching the capacitances of the ferroelectric and the dielectric layers as discussed in Section 3.
By applying a positive voltage V 1 to the NC supercapacitor as shown in Figure 2d, at first P f increases only slightly in the positive capacitance regime of the ferroelectric which leads to a small increase of the stored charge Q and energy W (green area) with increasing voltage. Notice that in this bias region, the electric fields in the ferroelectric E f is still increasing with increasing V. However, by further increasing the voltage toward V max > V 1 as shown in Figure 2e, the ferroelectric enters the NC region of the "S"-shaped P f -E f curve, where the field in the dielectric E d is amplified due to the decreasing field in the ferroelectric E f . In this case, the total differential capacitance will be larger than that of the dielectric alone. [15,16,18] In this region, a small change in external voltage leads to a large increase in Q and thus also in W. By increasing the voltage further beyond V >> V max the storage enhancement W/W 0 will be reduced and the ferroelectric will enter its second positive capacitance region. Therefore, operating the capacitor at V ≈ V max is desirable for energy storage applications.

Experimental Demonstration
To test our proposed NC supercapacitor design, we fabricated ferroelectric/dielectric capacitors using ferroelectric Hf 0.5 Zr 0. 5 Negative capacitance supercapacitor Realizing NC supercapacitors using a ferroelectric/dielectric stack. a) Free energy F d , displacement field D d , and inverse capacitance C d −1 of a dielectric capacitor with shorted electrodes. b) The same representation for a ferroelectric capacitor with shorted electrodes and spontaneous polarization P f . c) Proposed NC supercapacitor consisting of a ferroelectric/dielectric stack with interfacial charge σ IF , which stabilizes P f at zero voltage. d) NC supercapacitor under application of a positive voltage V 1 . e) NC supercapacitor under application of a higher positive voltage V max > V 1 . Due to the NC in the ferroelectric layer, the stored energy (green area) is significantly enhanced.
because it is a lead-free ferroelectric material with a wide process window and a large electronic band gap, which can be deposited conformally on 3D substrates by atomic layer deposition. Al 2 O 3 and Ta 2 O 5 were selected for their high breakdown field strength, relatively high permittivity and low leakage currents in their amorphous form. The stabilization of the ferroelectric orthorhombic phase in the HZO layers was confirmed by grazing-incidence X-ray diffraction measurements, which can be found in the Supporting Information. Furthermore, the morphology and thickness of the individual layers was investigated by transmission electron microscopy (TEM), which can also be found in the Supporting Information. The TEM results show that the HZO films are polycrystalline with a lateral grain size of 10-20 nm, while the Al 2 O 3 and Ta 2 O 5 layers are amorphous. Figure 3 shows experimental data for a TiN/HZO/Ta 2 O 5 /TiN capacitor using a pulsed electrical measurement technique as reported recently. [18] The layer thicknesses of 11.5 and 13.5 nm for HZO and Ta 2 O 5 , respectively, were determined by TEM. The integrated charges during charging (Q c ) and discharging (Q d ) as well as their difference (Q loss ) are shown as a function of the applied voltage in Figure 3b. By integrating the area above the Q-V curve according to Equation (2), we obtain 〈w loss 〉, 〈w c 〉, 〈w d 〉 and η, which are shown in Figure 3c. When examining η as a function of 〈w d 〉 as shown in Figure 3d, even for ultrahigh energy densities above 100 J cm −3 , we can obtain efficiencies of more than 95%. It should be noted that since 〈w d 〉 is the average energy density of the total capacitor and most of the energy is stored in the dielectric, the actual energy density in the dielectric is much higher (up to ≈200 J cm −3 , see Supporting Information for details). By dividing 〈w d 〉 by the discharging time (≈400 ns) we can calculate a maximum power density of 272.5 MW cm −3 . The integral capacitance C int = Q/V at maximum power density is 1.56 µF cm −2 . In Figure 3e, we compare the discharged energy density per area for the HZO/Ta 2 O 5 supercapacitor (W) and the expected one for the Ta 2 O 5 without HZO (W 0 ) based on the measured permittivity of a TiN/Ta 2 O 5 / TiN capacitor, which is ≈23.5. As can be seen from the energy storage enhancement W/W 0 in Figure 3f, the simple theory for the NC supercapacitor (Equation (S5), Supporting Information) agrees well with the experimental data when using the extracted parameters (a and b) from ref. [18].
An important prerequisite for reliable supercapacitor operation is temperature, cycling, and frequency stability. In Figure 4a, the small-signal capacitance per area and the loss factor are shown as a function of frequency. In the low frequency regime at zero bias, tan(δ) is in the range of 0.6% while the capacitance density is about 1.3 µF cm −2 . The stability with charging/discharging cycles can be inferred from the data in Figure 4b. Up to 10 6 cycles, 〈w d 〉 remains roughly constant at about 38 J cm −3 at an applied maximum voltage of 10.6 V. During further cycling up to 10 8 times, 〈w d 〉 begins to decrease to about 31 J cm −3 , while η remains high at about 99 %. This decrease in 〈w d 〉 might be related to the pinning of ferroelectric domains, which then do not contribute to the NC effect anymore. Furthermore, in Figure 4c, it can be seen that 〈w d 〉 is nearly independent of the measurement frequency across a broad frequency range. While some ferroelectric materials exhibit a strong dependence of their properties on temperature, this does not seem to be the case for the HZO films used here, as shown in Figure 4d, where the high 〈w d 〉 and efficiency are unaffected by an increase in temperature up to 150 °C. [9] These characteristics indicate that reliable supercapacitor operation should be feasible using our proposed capacitor design.

Changing the Dielectric and Layer Thicknesses
For the HZO/Ta 2 O 5 capacitor shown in Figure 3, the maximum energy storage enhancement W/W 0 is only ≈1.1 which is far from the theoretical maximum ≈2, since the capacitance matching aC ≈ −0.16 was not ideal. To improve the maximum W/W 0 , we fabricated similar capacitors using Al 2 O 3 instead of Ta 2 O 5 (see the Experimental Section), [16] with the additional advantage that Al 2 O 3 has an even higher breakdown field strength than Ta 2 O 5 . [32][33][34] Furthermore, we varied the thickness of both the Al 2 O 3 and the HZO layer to investigate the effect on the energy storage properties. Figure 5a depicts the η versus 〈w d 〉 curves for an HZO thickness of 7.7 nm and varying Al 2 O 3 thicknesses from 1.5 to 4 nm. It is apparent that an increase in the Al 2 O 3 thickness both increases the maximum obtainable 〈w d 〉 as well as the overall efficiency. The increase in η for thicker Al 2 O 3 seems to be directly related to the decreased probability of electron tunneling through the layer. [15,16] The injection of electrons into the ferroelectric/dielectric interface can lead to hysteretic switching, [15,16,18] which will strongly degrade η. This also explains the even higher observed efficiency for the much thicker Ta 2 O 5 film shown in Figures 3 and 4. The increase of 〈w d 〉 for thicker Al 2 O 3 layers on the other hand can be explained by the increased volume fraction of the dielectric material compared to the combined volume of the ferroelectric and the dielectric. Since the electric field in the dielectric is typically much higher compared to the ferroelectric in the NC region, most of the energy in the capacitor is stored in the dielectric and not in the ferroelectric layer. Therefore, decreasing the volume fraction of the ferroelectric layer should increase the overall energy density (see Supporting Information for a detailed derivation).
A similar behavior can be observed in Figure 5b, where the HZO thickness was changed, but the Al 2 O 3 thickness was fixed at 4 nm. The highest maximum 〈w d 〉 ≈ 121 J cm −3 is achieved for 7.7 nm HZO and 4 nm Al 2 O 3 . This corresponds to a maximum power density of 302.5 MW cm −3 and C int = 2.2 µF cm −2 . However, for the actual w d inside the Al 2 O 3 layer, much higher values can be estimated (see the Supporting Information). While the overall trend of the efficiency is almost identical for both HZO thicknesses (mostly related to the properties of the dielectric), 〈w d 〉 for the thinner HZO seems to be roughly scaled by a factor close to the ratio of the total thicknesses (15.3 nm/11.7 nm ≈ 1.31). Therefore, increasing the ratio of dielectric to ferroelectric thickness without degrading the ferroelectric properties might be useful way to improve the overall 〈w d 〉 of the capacitor further. However, one should keep in mind that this might degrade the matching of the dielectric capacitance to the ferroelectric NC region, thus reducing the voltage gain. Therefore, an optimum in between too thick and too thin dielectric layers can be expected. In the Supporting Information it is shown that for good capacitance matching and a small ratio of ferroelectric to dielectric thickness, dielectrics with higher permittivity, and ferroelectrics with a smaller negative permittivity are preferable. In Figure 5c, the charge-voltage curves for the best layer combination of 4 nm Al 2 O 3 and 7.7 nm HZO can be seen, where a strong increase in the differential capacitance is visible in the NC regime. The corresponding energy storage enhancement in Figure 5d shows a higher maximum of ≈1.5 compared to the HZO/Ta 2 O 5 sample due to the much better capacitance matching (aC ≈ −0.58). Figure 6 shows a comparison of reported highest 〈w d 〉 and corresponding efficiencies η for solid state supercapacitors using nonepitaxial materials. [9,[35][36][37][38][39][40][41][42][43] As can be seen, the first NC supercapacitors demonstrated in this work have roughly twice the maximum 〈w d 〉 while at the same time improving the efficiency Adv. Energy Mater. 2019, 9,1901154   The difference is defined as Q loss = Q c -Q d , as reported in ref. [18]. c) Similarly, the average energy densities during charging (〈w c 〉) and discharging (〈w d 〉) as well as their difference (〈w loss 〉) are shown as a function of V. d) Energy storage efficiency η as a function of 〈w d 〉. e) Comparison of discharged energy densities for the Ta 2 O 5 /HZO stack (W) and the Ta 2 O 5 layer alone (W 0 , extrapolated based on small-signal permittivity). f) Comparison of experimental and theoretical W/W 0 . above 85%. At comparable energy densities as reported in literature (〈w d 〉 ≈ 40-60 J cm −3 ), our NC supercapacitors have much higher efficiencies above 97%. Considering that this is the first experimental demonstration of the NC supercapacitor concept, these values are already very encouraging and there should be plenty of room for further improvement. While other groups have reported capacitors with even higher maximum 〈w d 〉 (up to 307 J cm −3 ), these results were all obtained by using epitaxially fabricated ferroelectrics, [11,[44][45][46][47] which limits these materials to applications in planar capacitors on flat and latticematched substrates. By contrast, by using only polycrystalline [48] and amorphous materials, which can be deposited via atomic layer deposition as shown here, almost arbitrary substrates with complex 3D surfaces can be used to achieve much higher energy densities per projected substrate area as experimentally shown before. [36,43] In this way, the effective energy density per projected substrate area can be easily improved by a factor of 30 or higher, depending on the aspect ratio and density of the 3D structures (e.g., density of capacitor trenches). This makes polycrystalline materials much more promising for applications compared to epitaxial ones. While the capacitors used in this work have a rather small area, other authors have recently demonstrated that much larger energy storage capacitors based on similar HfO 2 based antiferroelectric materials are feasible. [43]

Conclusion
We have proposed that by combining an NC layer (e.g., a ferroelectric) with a regular positive capacitance layer (e.g., a dielectric), it is possible to build an NC supercapacitor, which can store large amounts of electric energy with high efficiency. For an optimal NC supercapacitor design, the capacitances of both layers should be closely matched (aC ≈ -1), and the NC layer should be in a positive capacitance state when no voltage is applied (ΔQ-shift). Furthermore, the capacitor should be operated close to V max , to obtain the highest benefit due to the NC effect.
This new concept has three main advantages: (1) By engineering a nonlinear Q-V curve with positive curvature, the stored energy can be increased compared to a regular capacitor even at identical voltage. (   were used, integration on 3D substrates is promising for high storage density applications. Maximum discharged energy densities above 100 J cm −3 were achieved at efficiencies higher than 95%. Stable operation was shown in a broad frequency and temperature range (up to 150 °C) as well as for up to 10 8 charging/discharging cycles. Lastly, it was shown that increasing the ratio of dielectric to ferroelectric thickness can improve both maximum energy density as well as efficiency, since most of the energy is stored in the dielectric layer. For improving the capacitance matching and reducing the volume fraction of the ferroelectric layer at the same time, dielectric materials with larger permittivity and ferroelectrics with smaller negative permittivity were shown to be advantageous. In conclusion, our results pave the way for next-generation solid state supercapacitor technologies for highly efficient short-term storage of electrical energy.

Experimental Section
Sample Fabrication: Metal-ferroelectric-dielectric-metal capacitors were fabricated on Si (100) substrates with a native SiO 2 layer. Bottom TiN electrodes of 12 nm thickness were reactively sputtered in a BESTEC and an Alliance Concept physical vapor deposition tool at room temperature. Subsequently, Hf 0.5 Zr 0.5 O 2 (HZO) films were grown by atomic layer deposition (ALD) in an Oxford Instruments OpAL ALD tool at 260 °C using the precursors TEMA-Hf, and TEMA-Zr with water as the oxygen source. Alternating ALD cycles of TEMA-Hf and TEMA-Zr were applied to achieve a homogeneous distribution of Hf and Zr in the films. For samples using Al 2 O 3 as the dielectric layer, ALD was carried out directly after HZO deposition without breaking vacuum using TMA Park et al. [9] Hoffmann et al. [35] et al. [36] Ali et al. [37] Lomenzo et al. [38] Kim et al. [39] Yang et al. [40] Chen et al. [41] Kozodaev et al. [42] Kühnel et al. [43] This work Efficiency, η