Negative capacitance from the inductance of ferroelectric switching

Negative capacitance (NC) has been proposed to realize sub-Boltzmann steep-slope transistors in recent years. We provide experimental evidences and theoretical view for ferroelectric NC and inductance induced by polarization switching, based on an as-deposited nanoscale ferroelectric zirconium oxide (ZrO2) layer (nano-f-ZrO2). The experimental results are demonstrated in nano-f-ZrO2, including resistor–inductor–capacitor oscillations, positive reactance in Nyquist impedance plot, enhancement of capacitance, and sub-60 mV/dec subthreshold swing of nanoscale transistors. The theoretical analysis shows that ferroelectric polarization switching yields an effective electromotive force which is similar in behavior to Lenz’s law, leading to inductive and NC responses. Nano-beam electron diffraction reveals ferroelectric multi-domains in nano-f-ZrO2. Under small-signal operation, the switching of net polarization variation in ferroelectric multi-domains contributes to the ferroelectric inductance and NC. Nano-f-ZrO2 provides a pronounced inductance compared to conventional inductors, which would have impacts in a variety of applications including transistors, filters, oscillators, and radio-frequency-integrated circuits. Ferroelectric negative capacitance could be used to overcome the Boltzmann limit for next-generation energy-efficient transistors. This study demonstrates the performance capability of nanothick ZrO2 observing negative capacity and ferroelectric inductance and characterizes the role of multidomains in the observed behaviors.

The ferroelectric NC has been mainly demonstrated with HZO (Hf 0.5 Zr 0.5 O 2 ) in previous reports 9, [18][19][20] , where various crystalline phases exist in HZO. Thus, the post-deposition annealing (PDA) at high temperatures of 500-1000°C is normally required to enrich the ferroelectric orthorhombic phase in HZO [21][22][23] . Furthermore, the PDA temperature down to 450°C has been realized to achieve ferroelectricity in Gd-doped hafnium oxide (HfO 2 ) 23 . The high-temperature PDA, however, is generally detrimental to the semiconductor process integration. Therefore, it is highly favorable to produce ferroelectric thin films without PDA treatments. In fact, the highly crystalline zirconium oxide (ZrO 2 ) with strong ferroelectric responses has been successfully achieved without the need to perform PDA treatments in our previous study 24 .
In this paper, we report experimental observations of the inductance caused by ferroelectric polarization switching under both large-and small-signal operations. Large-signal resistor-inductor-capacitor (RLC) oscillations in time domain and a positive imaginary impedance in the Nyquist plot of metal-ferroelectric-metal (MFM) structures are observed, indicating the presence of significant inductance in the nanoscale ferroelectric ZrO 2 layer (abbreviated as nano-f-ZrO 2 ). We theoretically show that ferroelectric polarization switching produces an "effective ferroelectric-induced electromotive force (emf)," which results in a decrease of the voltage drop across the ferroelectric layer. This effective ferroelectric-induced emf behaves like the emf predicted by Lenz's law, which is opposite to the applied voltage, and so contributes to the inductance and NC effect. The NC induced by ferroelectric polarization switching is also manifested by the enhancement of small-signal capacitance of a paraelectric capacitor as connected in series with a ferroelectric capacitor. The ferroelectric multi-domains in a nanoscale ZrO 2 layer, which are responsible for the small-signal operation of the ferroelectric inductance and NC, are observed via the nano-beam electron diffraction (NBED) characterization. The NC junctionless transistor (NC-JLT) with SS <60 mV/dec operated under a large drain voltage, along with the hysteresis-free operation, is also demonstrated.

Results
As-deposited f-ZrO 2 with NC. The as-deposited ZrO 2 was used as the nanoscale ferroelectric layer in this study. Supplementary Figure 1 (Supplementary Information, Note 1) shows the polarization-electric field (P-E) hysteresis loop of the Pt/ ZrO 2 (12 nm)/Pt MFM structure, proving the existence of ferroelectricity of the as-deposited nanoscale ZrO 2 layer. The asdeposited nano-f-ZrO 2 was integrated into the gate stack in the nanoscale Si JLT as schematically shown in Fig. 1a. The top view of the Si channel (~70 nm in length) and contact pads taken by helium ion microscopy is shown in Supplementary Figure 2 ( Supplementary Information, Note 1). Figure 1b, c shows the cross-sectional transmission electron microscope (TEM) and scanning electron microscope (SEM) images of the channel and the gate stack, revealing the Si channel with an Ω-shaped gate on a silicon-on-insulator substrate. The gate structure comprises a HfO 2 (~6 nm) high-K dielectric layer stacked with an as-deposited TiN(~15 nm)/Pt(~70 nm)/f-ZrO 2 (~12 nm)/TiN(~15 nm)/Pt (~70 nm) MFM structure, as shown schematically in Fig. 1a. Figure 1d shows the pronounced ferroelectric characteristics of a separately prepared MFM test structure with the same MFM stacking as shown in Fig. 1a. Note that the TiN layer is adopted here to increase the adhesion of the overlaying Pt layer for improving the yield of fabricated transistors. Figure 1e shows the forward and backward I d (drain current)-V gs (gate voltage) curves of the NC-JLT. The I d -V ds (drain voltage) is provided in Supplementary Figure 3 ( Supplementary Information, Note 3). The hysteresis-free I d -V gs characteristics are observed, which can be deduced from the positive overall capacitance as required by the stability of NC 4,6,[8][9][10][11]13,14,16 . The plot of SS at V ds = 1 V versus V gs is shown in Fig. 1f, indicating SS <60 mV/dec and a minimum SS of 46 mV/dec. (The I d -V gs curve of the JLT without the f-ZrO 2 layer is shown in Supplementary Information, Note 4 for a comparison.) This result demonstrates the sub-60 mV/dec SS in NC transistors operated at a high V ds (1 V), and reveals the NC effect of nano-f-ZrO 2 under the large-signal on-off operation of transistors.
Ferroelectric inductance with large-signal switching operation. Figure 2a, b shows the transient responses of the as-deposited Pt/ ZrO 2 (12 nm)/Pt MFM structure, that is, the time-domain voltage and current waveforms (V FE and I FE ) across nano-f-ZrO 2 under the excitation of a voltage pulse V in = 0.5 and 2 V. In principle, the paraelectric insulator in metal-insulator-metal structures exhibits typical resistor-capacitor (RC) charging and discharging features. However, the non-RC responses, that is, the significant damped oscillations of V FE and I FE are clearly observed in Fig. 2a, b. The overshoot/undershoot of V FE and the damped oscillation of I FE agree well with typical characteristics of the underdamped RLC response 25 . As compared with Fig. 2a (V in = 0.5V), the suppressed oscillations of V FE and I FE in Fig. 2b (V in = 2 V) are ascribed to a decrease of the parallel resistance across nano-f-ZrO 2 due to an increase of the leakage at a higher V in . Therefore, Fig. 2a, b indicate the existence of the inductive component in nano-f-ZrO 2 . In order to confirm the existence of inductance, an impedance analyzer was used to probe the ZrO 2 MFM structure. Figure 2c shows the Nyquist plot of the complex impedance (Z r + jZ i ) of the Pt/ZrO 2 (12 nm)/Pt MFM structure. An inductive loop, characterized by positive imaginary impedance, appears in the Nyquist plot. Therefore, the electrical characteristics of ZrO 2 MFM structure can be described in terms of an equivalent circuit containing inductance [26][27][28][29][30] , confirming the presence of inductance in nano-f-ZrO 2 . Recently, the impedance behaviors of ferroelectric-like materials have been noticed in the metal-halide perovskite solar cells [31][32][33] , and their Nyquist plots are similar to that shown in Fig. 2c. The analysis of impedance spectra of these perovskite solar cells has also revealed that an inductive component is present in the equivalent circuit 31 .
The physical origin of ferroelectric inductance can be derived from Maxwell's equations (Supplementary Information, Notes 6-7). The charge redistribution caused by ferroelectric polarization switching leads to a decrease of the voltage drop across the ferroelectric layer, which is terminologically denoted as the "effective ferroelectric-induced emf". As schematically illustrated in Fig. 2d, the polarity of this effective ferroelectric-induced emf is opposite to the applied voltage, which is similar in behavior to Lenz's law, indicating that the induced emf acts against the applied voltage. Therefore, the effective ferroelectric-induced emf gives rise to the inductive responses of the ferroelectric layer. Accordingly, a simple equivalent RLC circuit of the ferroelectric layer is proposed in Supplementary Figure 9  The inductance density of nano-f-ZrO 2 is approximately on the order of 10 mH/ cm 2 . As compared with conventional inductors, which are of spiral geometries and suffer severely from large occupied areas and low inductance densities (<10 μH/cm 2 ) 34 , nano-f-ZrO 2 offers a much larger inductance density. Because of the presence of the inductance and capacitance in the simple equivalent circuit of the ZrO 2 MFM structure as shown in Supplementary Figure 9, nanof-ZrO 2 can serve as the filters/oscillators/resonators in the applications including radio-frequency (RF) communication and signal-processing systems 25 .
NC and ferroelectric inductance. The connection between NC and the effective ferroelectric-induced emf can be understood as follows: where i D is the displacement current through the ferroelectric layer with the capacitance C FE , Δt is the time scale of ferroelectric polarization switching, and ΔV is the change of the voltage drop across the ferroelectric layer caused by polarization switching.
Since the effective ferroelectric-induced emf leads to a decrease of the voltage drop across the ferroelectric layer, that is, ΔV<0, Eq. 1 indicates C FE < 0.
NC observed with small-signal switching operation.  Figure 3b compares the P-E characteristics of the as-deposited paraelectric HfO 2 , paraelectric ZrO 2 (p-ZrO 2 ), and f-ZrO 2 layers. Note that the paraelectric/ferroelectric properties of the as-deposited nanoscale ZrO 2 layer can be controlled by the process conditions of atomic layer deposition (ALD) as described in Materials and methods. Figure 3c shows the small-signal capacitance of the three capacitor structures, in which the dielectric layers are HfO 2 , p-ZrO 2 in series with HfO 2 (denoted as p-ZrO 2 /HfO 2 ), and f-ZrO 2 in series with HfO 2 (denoted as f-ZrO 2 /HfO 2 ), respectively. The capacitance of the p-ZrO 2 /HfO 2 structure is lower than that of the HfO 2 capacitor, as expected from the series connection of two paraelectric capacitors.   However, the capacitance increases (×1.1) as the HfO 2 capacitor is connected in series with the f-ZrO 2 capacitor (the f-ZrO 2 /HfO 2 structure). The capacitance of the f-ZrO 2 layer can be estimated from Fig. 3c to be C FE ¼ À13:70μF=cm 2 ; (Supplementary Information, Note 10). Hence Fig. 3c gives a clear demonstration of the NC effect as a paraelectric capacitor is connected in series with a ferroelectric capacitor 4,6,12 . Figure 3d shows the smallsignal capacitance of the f-ZrO 2 /HfO 2 , HfO 2 , and p-ZrO 2 /HfO 2 capacitor structures as a function of the frequency. The frequency dispersion of the capacitance is not obvious in the f-ZrO 2 /HfO 2 capacitor in the frequency range between 10 kHz and 1 MHz, indicating that the time scale of ferroelectric polarization switching in nano-f-ZrO 2 is <1μs.

Discussion
Note that the capacitance shown in Fig. 3 was obtained based on the small-signal measurement at a 1 V direct-current (DC) bias and a 30 mV alternating-current (AC) perturbation. However, for a layer of ferroelectric single domain, the small-signal modulation hardly causes ferroelectric polarization switching because the amplitude of small signals is much lower than the coercive voltage for polarization switching. The ferroelectric polarization switching under small-signal operation can be understood from ferroelectric multi-domains, as illustrated in Fig. 4a. It has been reported that 180°out-of-phase ferroelectric multi-domains are favorable for the reduction of stray fields in ferroelectric layers 35,36 . (The stability of ferroelectric multi-domains is c Nyquist impedance plot of the as-deposited Pt/ZrO 2 (12 nm)/Pt MFM structure, where Z r and Z i are the real and imaginary parts of the complex impedance. The measurement was carried out by an impedance analyzer in the frequency range from 1 MHz to 10 mHz at a 1 V direct-current (DC) voltage and a 10 mV alternate current (AC) perturbation. The inset shows the Nyquist plot of the resistor-capacitor (RC) dummy cell comprising a series resistor (100 Ω) and an RC element (1 μF) capacitance and 1 MΩ resistance in parallel). The dummy cell was provided by Metrohm AUTOLAB ® for standard test. It can be seen that an RC circuit gives a semi-circle of negative imaginary impedance in the Nyquist plot. As compared with the RC dummy cell, positive imaginary impedance is seen in the lower part of the Nyquist plot, indicating the presence of inductance in the equivalent circuit of nano-f-ZrO 2 . d Schematic diagram of the effective ferroelectric-induced electromotive force (emf) originating from ferroelectric polarization switching. This effective ferroelectric-induced emf behaves like the emf predicted by Lenz's law, which opposes the applied electric field and so contributes to the inductance of the ferroelectric layer discussed in Supplementary Information, Note 11.) Hence, the polarizations in ferroelectric multi-domains are organized with alternate up and down orientations 35,36 . The DC bias generates a net polarization in ferroelectric multi-domains. The applied small-signal voltage causes elongation or compression of the polarization in each domain, resulting in the switching of the net polarization variation in the ferroelectric layer, as shown schematically in Fig. 4a. Figure 4b shows the bright-field cross-sectional TEM image of the as-deposited Pt/ZrO 2 (12 nm)/Pt MFM structure. Figure 4c shows an NBED pattern that was taken by an electron beam focused mainly into an area enclosed by the yellow box in Fig. 4b. Ferroelectricity has been considered as originating typically from the orthorhombic phase 19 . However, because of the overlap between low-symmetry nanoscale domains, an apparent "average" or "pseudo" cubic symmetry would be observed under the NBED characterization of the nanoscale ZrO 2 layer 37,38 . Since Fig. 4c is in good agreement with the simulated diffraction spots generated from the standard cubic ZrO 2 structure of the P 43m space group 39 , the area enclosed by the yellow box can be regarded as a single-crystal grain of "pseudo" cubic ZrO 2 with the ½011 zone axis along the incident beam or the normal of the specimen surface. The NBED pattern indicates the lattice constant of the nanoscale ZrO 2 layer is 0.50 nm. However, it has been reported that the lattice constants of standard orthorhombic ZrO 2 are 0.526, 0.507, and 0.508 nm along the a, b, and c axes, respectively 19 . Also, the standard cubic ZrO 2 has a lattice constant of 0.51 nm and a non-centrosymmetric structure due tõ 0.03 nm displacements of the oxygen sublattices along <111> directions from Fm 3m 39 . Thus, it can be deduced that the nanoscale ZrO 2 layer is of non-centrosymmetric structure and compressively stressed comparing to either the orthorhombic phase or the "pseudo" cubic of the P 43m space group. These are consistent with the grazing incidence X-ray diffraction (GIXRD) pattern of the as-deposited f-ZrO 2 (12 nm) layer (Supplementary Figure 14 in Supplementary Information, Note 12), which shows a deviation of the X-ray diffraction (XRD) peak from the strainfree cubic or orthorhombic ZrO 2 . Therefore, the results revealed by NBED and GIXRD patterns indicate the presence of the noncentrosymmetric structure and compressive strain in the nanoscale ZrO 2 layer, giving rise to the emergence of ferroelectricity [40][41][42] . Figure 4d shows the corresponding dark-field image taken by the reflection indicated by the red circle in Fig. 4c. A part of the ZrO 2 grain is bright by satisfying Bragg's condition of the 11 1 reflection. Afterwards, the TEM sample was tilted to another angle to get more information. Figure 4e shows another brightfield image. The NBED pattern of the grain indicated by the yellow box in Fig. 4e is shown in Fig. 4f, which also agrees with the simulated diffraction pattern of standard cubic ZrO 2 with the P 43m space group along the ½ 112 zone axis. This NBED pattern indicates that the lattice constant is also 0.50nm. A dark-field image taken by the 1 11 reflection is displayed in Fig. 4g, where almost the grain indicated by the yellow box is bright. It should be noted that the regions with bright and dark contrast appear alternately in the ZrO 2 layer in Fig. 4d, g. The regions with bright contrast can be regarded as single-crystal grains with the same orientation, that is, the ½011 zone axis in Fig. 4d and the ½ 112 in Fig. 4g. The regions with dark contrast are ZrO 2 grains whose orientation is different from the grains with bright contrast. In principle, the polarization directions are different between such single-crystal grains pretended to have the same crystal structure of P 43m in case their zone axes are different along a certain The analysis based on Maxwell's equations shows that the charge redistribution caused by polarization switching contributes to an effective ferroelectric-induced emf, which reduces the voltage drop across the ferroelectric layer similar to the behavior described by Lenz's law. Therefore, the effective ferroelectricinduced emf is responsible for the inductive responses and NC of nano-f-ZrO 2 . Contrary to conventional inductors based on spiral electric coils with large footprints and low inductance densities, the planar ferroelectric inductance provides a much larger inductance and occupies a much smaller area. Therefore, the ferroelectric inductance is beneficial to a broad range of applications such as wireless communications, radar, and RFintegrated circuits.

Methods
The MFM structure for the measurements of the P-E hysteresis loop, the time-domain voltage and current responses, and the Nyquist impedance plot. Figure 5 shows the schematic ZrO 2 MFM structure. The bottom Pt electrode was deposited by an electron beam evaporator on a Si substrate. Then a 12-nm f-ZrO 2 layer was deposited on the Pt electrode at 300°C by the plasma ALD (Ultratech, Fiji). The tetrakis(dimethylamino)zirconium (TDMAZr, Zr[N(CH 3 ) 2 ] 4 ) and O 2 plasma were used as the precursor and reactant for zirconium and oxygen, respectively. Afterwards, the top Pt electrode was deposited on ZrO 2 by an RF sputtering.
Series connection of a paraelectric HfO 2 capacitor with a ZrO 2 capacitor for the enhancement of small-signal capacitance. Figure 3a shows the schematic diagram of the capacitor structures under test. All the top, internal, and bottom Pt electrodes were prepared by the RF sputtering. The paraelectric HfO 2 (~12 nm) and f-ZrO 2 (~12 nm) layers were deposited at 300°C by the plasma ALD, while the p-ZrO 2 layer was deposited at a lower temperature of 250°C in the same ALD chamber operated in thermal mode. The precursors for hafnium and zirconium were tetrakis(dimethylamino)hafnium (TDMAHf, Hf[N(CH 3 ) 2 ] 4 ) and TDMAZr, and the oxygen sources were O 2 plasma and H 2 O vapor for the plasma and thermal ALD, respectively. The top and internal Pt electrodes with a radius of 70μm were defined by the photolithography and lift-off processes. The top and internal Pt electrodes were used as the hard masks for the subsequent removal of HfO 2 and ZrO 2 using the CHF 3 dry etching.
Fabrication of NC Si JLTs. Figure 1a shows the schematic structures of the NC Si JLT and the gate stack. The substrate is the silicon-on-insulator wafer with a 150-nm layer of buried oxide and a 70-nm top Si layer. First, the top Si layer was implanted with As + for the formation of a heavily doped n + Si channel with a doping concentration of~5 × 10 18 ions/cm 3 . Subsequently, the Si channel was thinned down to~20nm by multiple dry oxidation and wet buffered oxide etch. Next, the electron beam lithography and the CF 4 dry etching were carried out to define the width and the length of the Si channel. The Ω-shaped gate was then formed by the following dry oxidation and wet buffered oxide etch. The thickness and the shape of the resulting Si channel were confirmed by the cross-sectional TEM image as shown in Fig. 1b were prepared by the RF sputtering. Afterwards, the thermal evaporation, lift-off process, and rapid thermal annealing at 400°C were utilized to form the NiSi contacts at the source and the drain.
Materials characterization. The cross-sectional TEM sample was prepared by the focused ion beam (FIB, Helios NanoLab 600i, FEI) system. Structural analysis of the ferroelectric layer was performed by the TEM (Tecnai G2 F20, FEI) with an acceleration voltage of 200kV. The bright/dark-field images and the NBED patterns were produced by the TEM with different focal planes. The NBED patterns were examined by the DigitalMicrograph (Gatan) software. The cross-sectional image of the gate stack was acquired by the SEM equipped in the FIB system. The crystalline phase of the ferroelectric ZrO 2 layer was identified using the GIXRD (TTRAX III, Rigaku).
Electrical characterization. The Keysight B1500A semiconductor parameter analyzer in the C p -R p mode was used to measure the electrical characteristics of the negative-capacitance Si JLTs and the small-signal capacitance of the f-ZrO 2 /HfO 2 , p-ZrO 2 /HfO 2 , and HfO 2 capacitors. The voltage and current responses in the time domain was probed by the Keysight B1500A equipped with the waveform generator/fast measurement unit module. The measurements were calibrated using the built-in "Module Self Calibration" in the Keysight B1500A analyzer. The ZrO 2 MFM structure was tested by the impedance analyzer (Autolab PGSTAT302N) to obtain the Nyquist plot. For the ferroelectric characterization, the P-E hysteresis curves were acquired using the ferroelectric analyzer (TF2000, aixACCT) operated at 2000 Hz.

Data availability
The data sets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.