Introduction

Negative capacitance (NC) effects in ferroelectric (FE) films have attracted intensive attention recently in terms of both polarization kinetics modeling and low-power complementary metal-oxide-semiconductor (CMOS) applications. The original idea of NC was proposed from the fact that the Landau-Devonshire theory would possess an intrinsic NC state in a FE capacitor (CAP) with a single domain, namely an S-like polarization-voltage behavior. Although this effect is not observable in a single FE-CAP, it is expected to be stabilized by connecting a linear paraelectric (PE) CAP in series. Furthermore, since it may achieve an internal voltage amplification in gate stacks of field-effect transistors (FETs) by using the FE/PE stacks, it is very attractive that the Boltzmann tyranny (60 mV dec−1 at room temperature) in subthreshold slope (SS) may be surmounted1. Such a device with no hysteresis is often called a NC-FET. In a decade, many groups have focused on studying NC effects in FE/PE stacks and FE/PE gate stack FETs2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24. Concerning NC effects in FE/PE stacks, three kinds of demonstrations have been provided experimentally so far: (i) total capacitance enhancement in the case of no internal metal between FE and PE layers2,3,4, (ii) transient NC effects in AC mode operation5,6,7, and (iii) locally stabilized NC state8. Several models of quasi-static NC associated with domain wall motion in a multiple-domain system have been also proposed9,10,11,12,13. Meanwhile, steep SS values have been demonstrated by incorporating FE/PE gate stacks into FETs with various FE materials14,15,16, various channel materials15,17,18,19 and various FET structures14,20,21,22,23,24.

However, physical understanding of NC effects is still under intensive debate25,26,27,28,29,30,31,32,33,34,35,36. Experimentally observed NC effects are different from each other, and also from the concepts initially proposed. Alternative explanations for the capacitance enhancement and transient NC effects in FE/PE stacks have been also proposed25,26,27,28,29,30,31. For example, a feasibility of capacitance enhancement is explainable from a strong coupling between FE and PE layers25,26,27, while the transient NC is understandable from the viewpoints of overshoot in voltage supply or slower speed of charge compensation relative to polarization switching in FE-CAP28,29,30,31. In fact, it has been argued that NC region of FE material is intrinsically unstable or even impossible32,33. In addition, the SS improvements observed in FETs mostly suffer from critical problems that a large hysteresis is detected, a high voltage is needed and an operation frequency is limited in actual experiments34,35,36.

To sum up, the experimental evidences provided so far are insufficient to conclude the concept of quasi-static NC, and reliable modeling of SS improvement in a FET with FE/PE gate stack is still missing. These should be verified urgently, because they are critical for further advancing the material science as well as electron device performance of FE/PE stacks to elucidate whether the quasi-static NC can be really stabilized or not, and whether the steep SS characteristics so far demonstrated are really promising for low-power CMOS applications or not. A direct way to examine the actual FE effect in FE/PE stack is to investigate the voltage at the internal node, Vint, between FE and PE layers in DC mode, which makes possible of the direct correlation of Vint with SS in FET. In fact, a couple of works on internal potential measurement have been reported, but they have only qualitatively discussed about this issue, resulting that a consistent model could not been provided16,37. We have suspected it might be due to experimental difficulties of measuring the internal potential in FE/PE stack. Therefore, in this work, accurate DC measurements are particularly paid careful attention. The stepwise Vint jumps at the coercive voltage, ±VC, of FE layer in FE/PE stack are demonstrated, and a relationship between Vint jumps and the steep SS in FET with FE/PE gate stack is presented. They are quantitatively understood from the viewpoints of successive polarized domain flipping and depolarization feedbacks from the PE-CAP. The results provide a clear physical insight to understanding the small SS values in FETs with FE/PE gate stacks reported so far.

Results

DC measurement of internal potential in FE/PE stack

To accurately estimate Vint in FE/PE stack in DC mode, a PE/PE stack was firstly inspected quantitatively. In an ideal case, total charges at the internal metal, which is electrically floating, have to be conserved as long as no leakage current through both CAPs is assured (Fig. 1a). However, since the DC output impedance at the floating node is infinite in the ideal case, it is substantially impossible to estimate Vint experimentally. In actual cases, a finite resistance of capacitors enables to measure Vint in DC-mode, while a time constant should be considered for a quantitative analysis. More importantly, a small but a finite amount of charge-flow from the internal metal to a measurement system should also be paid attention to, because the measured voltage in itself is significantly affected by the input impedance in the measurement system. Thus, the actual equivalent circuit concerned in this work is shown in Fig. 1b. Based on the formulation shown in Supplementary Note 138 and the measured resistance of PE CAPs (~1012 Ω), a voltage measurement system with the input impedance higher than ~1014 Ω was needed for accurate DC-mode measurements. We used a high input impedance voltmeter equipped with a high precision current preamplifier, which was able to adjust a voltage to suppress a current flow down to zero (sub-fA level). By doing so, the effective input impedance was enhanced up to 1016 Ω. Figure 1c compares Vint in PE/PE stacks measured in the present system and in conventional voltmeter with an input impedance of ~1010 Ω. When the present system was employed, the accurate measurement of Vint was successfully assured for capacitors with down to 1 pF. Namely, it is critically important for characterizing the Vint quantitatively to take the input impedance of the measurement system into account.

Fig. 1: Understanding and investigation of Vint in PE/PE stacks.
figure 1

a Ideal and b actual equivalent circuits in the internal potential measurement in PE/PE stacks. C1/C2 and R1/R2 are the capacitance and insulating resistance of each FE-CAP. V1 is the voltage applied on the top PE-CAP. RZ is the input impedance of Vint measurement system. c Vint, with V = 1 V and C1 = 200 pF, as a function of C2 from 1 pF to 33 nF for two cases using high input impedance system and conventional DC voltmeter. Only high impedance system enables to estimate the internal potential in DC mode measurement.

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In actual measurements of Vint in FE/PE stacks, commercially available PZT films with Pt electrodes were used as the FE-CAPs. The typical charge–voltage (Q-VF), capacitance–voltage (CF-VF) and leakage (I-VF) characteristics are shown in Supplementary Fig. 1. The capacitance at the center of CF-VF characteristics was ~0.25 nF and the leakage current density at 1 V was below ~10−8 A cm−2, which is comparable to the level reported for the state of the art PZT39,40 and corresponds to 5 × 1010 Ω. The equivalent circuit of the Vint measurement in FE/PE system shown in Fig. 2a was assumed, in which impedances in both PE-CAP and measurement system were much higher than that in PZT. In prior to measurements, the FE-CPA was polarized by a negative voltage, and the internal terminal of FE/PE stack was grounded to remove the unknown charges stored. More details in the measurements are described in the method section. Figure 2b and c show Vint and δVint/δV in sweepings of the total voltage, V, in the case with PE capacitance (CP) of 0.5 nF. A big Vint jump is seen in both forward and backward V sweepings, and each of them corresponds to the differential gain of Vint (δVint/δV > 1). It is noted that the big Vint jump is followed by successively oscillating small ones as shown in enlarged part of Fig. 2b. VF calculated by subtracting measured Vint from V is shown as a function of V in Fig. 2d. Note that the VF drops along with the Vint jumps occur very near ±VC in FE-CAP. It directly indicates that the Vint gain is associated with the polarization flip in a FE layer. This fact is critically important from the viewpoint that it is beyond the qualitative observation in the similar measurements reported recently16,37. It is worthy of mentioning that the measured voltage was time dependent and it gradually changed due to the finite resistance in FE and PE layers as mentioned above. Therefore, the absolute value of Vint is not as expected in CP-CF circuit. Both V and Vint are, however, accurately measurable and VF obtained by V-Vint should also be accurate quantitatively, as long as the input impedance of voltmeter is effectively higher than the output impedance in the PE-CAP and FE-CAP and the measurement time is substantially smaller than the time constant of the FE/PE system. This is further confirmed by the results in Supplementary Fig. 2, that the experimental observations are well reproduced by repeating the measurement for three times and the VF positions corresponding to Vint jumps remain to be near ±VC even by changing CP from 15 nF down to 0.1 nF. In addition, a possible origin of the Vint variation in FE/PE system might be the time constant variation originated from the bias dependence of resistance and/or capacitance of FE-CAP. But, such a concern is irrelevant to the present experimental observations, because the results are not affected at all even in the FE/PE stacks with different time constants (with different CP values).

Fig. 2: Vint measurement in a FE/PE stack.
figure 2

a Equivalent circuit of Vint measurement in FE/PE system. RF and RP are the insulating resistance of FE- and PE-CAP respectively. Note that high impedance system is needed to get the accurate Vint. b Vint-V, c δVint/δV-V and d VF-V characteristics during V sweeping of FE/PE system. Vint jump occur along with VF drop at VF = ~± VC, corresponding to δVint/δV > 1, followed by the oscillating small ones.

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Correlating internal potential gain with steep SS in FET

A relationship of the internal potential gain with the SS improvement in FET operation is next investigated. A FE-CAP was connected through a cable to a poly-Si gate/SiO2 metal-oxide-semiconductor FET (MOSFET) with a channel width, length, and oxide thickness of 200 μm, 500 μm, and 5 nm, respectively. Accumulated MOS capacitance was ~0.7 nF and gate leakage was ~10−9 A cm−2 at 1 V. The equivalent circuit is shown in Fig. 3a. The source-drain current (IDS) and Vint of the FE-CAP/MOSFET stack were measured separately in gate bias (VGS) sweeping under drain voltage (VDS) of 0.1 V. It is mandatory to eliminate unknown charge effects at the internal node in FE/MOSFET prior to each measurement, because floating charges are very likely to affect the charge dynamics in the domain reversal. This is critical for considering the hysteresis in IDS-VGS characteristics of FE/MOSFET system. Figure 3b shows IDS-VGS curves w/ and w/o connecting the FE-CAP. The SS in the case w/ FE-CAP clearly looks sharper than that in w/o case in a given VGS region. Figure 3c plots VGS dependence of SS. In the case w/ FE CAP, SS is improved down to sub-60 mV dec−1 both in forward and backward sweepings. δVint/δVGS as a function of VGS is shown in Fig. 3d. Note that the improvement factor of SS (a ratio of SS w/o to w/ FE-CAP) is roughly the same as δVint/δVGS in Fig. 2d. These results indicate that the steep SS value below the Boltzmann limit is definitely associated with Vint gain in FE/MOSFET system.

Fig. 3: Correlation of Vint gain with steep SS in FET.
figure 3

a Equivalent circuit for measuring IDS-VGS and Vint-VGS characteristics in FE/MOSFET system. Note that Vint-VGS and IDS-VGS were measured separately for the same system and under the same VDS (0.1 V). b IDS-VGS, c SS-VGS and d δVint/δVGS-VGS characteristics of MOSFET w/ and w/o external FE-CAP during sweepings of VGS. A correlation between sub-60 mV dec−1 SS and internal potential gain is demonstrated.

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Electrostatic understanding of internal potential gains

Next, a possible kinetic origin of stepwise Vint jumps is discussed. The charge dynamics associated with the domain flipping in FE/PE stack is a main focus. We suppose the capacitance coupling circuit in FE/PE stack for characterizing internal charge kinetics in the following.

It is here noted that two kinds of charges are involved in FE-CAP: bound charges (Qbound) and free charges (Qfree), as shown in Fig. 4a, where the positive polarization, P, is defined as the arrow from left to right. If FE-CAP is pre-polarized by a negative voltage as that in the experiments, Qbound = −P initially. According to electrostatics, the relevant system should satisfy the following equations before the polarization switching:

$$V_{\mathrm{F}} + V_{{\mathop{\rm{int}}} } = V$$
(1)
$$V_{{\mathop{\rm{int}}} } = \frac{Q}{{C_{\mathrm{P}}}}$$
(2)
$$Q = Q_{{\mathrm{free}}} + Q_{{\mathrm{bound}}} = C_{\mathrm{F}}V_{\mathrm{F}} - P$$
(3)

in which Q denotes total charges accumulated on each CAP. Assuming CF is constant, Vint and VF are expressed by

$$V_{{\mathrm{int}}} = \frac{{C_{\mathrm{F}}}}{{C_{\mathrm{F}} + C_{\mathrm{P}}}}V - \frac{P}{{C_{\mathrm{F}} + C_{\mathrm{P}}}}$$
(4)
$$V_{\mathrm{F}} = \frac{{C_{\mathrm{P}}}}{{C_{\mathrm{F}} + C_{\mathrm{P}}}}V + \frac{P}{{C_{\mathrm{F}} + C_{\mathrm{P}}}}$$
(5)

Since the bound charges are fixed in the case without polarization flipping, the FE/PE stack is similar to a PE/PE one. Thus,

$$\frac{{\delta V_{{\mathrm{int}}}}}{{\delta V}} = \frac{{C_{\mathrm{F}}}}{{C_{\mathrm{P}} + C_{\mathrm{F}}}} < 1$$
(6)

On the other hand, when the polarization flipping occurs at VF = ~VC (taking the forward sweeping as an example), the polarization, P, should change the direction in Eqs. (3), (4) and (5). Namely, a bunch of charges are transferred from FE-CAP to PE-one. If we assume FE layer is with a single domain,

$$\frac{{\delta V_{{\mathrm{int}}}}}{{\delta V}} = \frac{{C_{\mathrm{F}}}}{{C_{\mathrm{P}} + C_{\mathrm{F}}}} + \frac{{\frac{{2P}}{{\delta V}}}}{{C_{\mathrm{P}} + C_{\mathrm{F}}}}$$
(7)

This indicates that δVint > δV can become to be >1 in case with a relatively small CP. Namely, Vint jump along with VF drop should occur in this case.

Fig. 4: Electrostatic understanding of stepwise Vint jumps in FE/PE stacks.
figure 4

a Schematics of FE/PE system. Two kinds of charges are involved in FE-CAP: bound charges (Qbound) and free charges (Qfree). b Understanding of charge dynamics (circles and dots represent holes and electrons respectively), voltage change and Q-VF characteristics in FE/PE system with successive multiple-domain flips. c A possible case that the biggest Vint jump in the forward V sweeping may overlap partly with that in the backward one, then the subthreshold hysteresis in FE/PE gate stack FET may be much reduced in the overlapping Vint range if the bias dependence of semiconductor is ignored.

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In actual FE films, multiple domains are involved and not all domains flip simultaneously. Here, it is reasonably assumed that each domain flips independently and that it has each VC with a tight distribution in FE film. When a certain domain flips with a relatively small CP, VF should be reduced due to a finite amount of charge transfer from FE-CAP to PE-one. It means that remaining domains cannot flip before further increase in VF to VC. When VF is increased to VC again, another domain flips, followed by the VF drop again. Thus, the initial Vint jump and VF drop followed by oscillating Vint-V and VF-V characteristics near VC are clearly explained. The stepwise Vint jumps are analytically described as

$$\frac{{\delta V_{{\mathrm{int}}}}}{{\delta V}} = \frac{{C_{\mathrm{F}}}}{{C_{\mathrm{F}} + C_{\mathrm{P}}}} + \frac{{\frac{{2P_i}}{{{\updelta }}V}}}{{C_{\mathrm{F}} + C_{\mathrm{P}}}}$$
(8)

in which Pi (i = 1, 2, 3…) is the polarization of each domain in a FE film. Since the VF drop means the depolarization field formation in FE-CAP, each Vint jump can be regarded as a depolarization feedback from PE-CAP associated with each domain flip. Namely, successive multiple-domain flips accompanying the depolarization feedbacks from PE-CAP result in zigzag Q-VF characteristics. These views are schematically shown in Fig. 4b in detail. Note that the resulted zigzag Q-VF characteristics is totally different from the conventional Q-VF curve in a single FE-CAP or from the S-curve expected from the original NC theory1. It is also significantly different from the characteristics expected from the recent models with the multiple-domain system, in which a continuous change of polarization with the help of domain wall motion is assumed9,10,11,12,13.

Hysteresis of the Vint gain is critical for achieving steep SS FETs in advanced CMOS. According to our modeling, nearly hysteresis-free Vint gain is made possible in principle by inserting an appropriate CP in the case of the single domain FE in FE/PE stack. The requirement for this condition is the same as that for stabilizing NC effect originally proposed. This is also partly consistent with a recent report in which nearly single domain PZT film together with strict capacitance matching was employed and a nearly hysteresis-free steep SS was demonstrated23,24. In the case of FE layer containing multiple domains, the hysteresis-free Vint gain is difficult to achieve perfectly. Here, let us focus on the biggest Vint gain. Since the V value and amplitude of this Vint jump are controlled by CP and P1, respectively, it becomes possible that the biggest Vint jump in the forward V sweeping may overlap with that in the backward one by adjusting P1 under an optimum CP, (Fig. 4c). Resultantly, the subthreshold hysteresis in FETs should be substantially reduced in the overlapped range. As a matter of fact, the bias dependence of semiconductor capacitance makes hysteresis-free FET more difficult. Thus, hysteresis is seen in most of the steep-SS FETs reported up to now16,22,23. More detailed discussion of the hysteresis is shown in Supplementary Fig. 3 and Supplementary Note 2.

In addition, it should be mentioned that the domain–domain interaction is needed to take into consideration and the internal field inside the film may be reduced in the first domain switching in the real multiple-domain system. These effects should affect the voltages for successive domain flipping. In fact, our results in Fig. 1d show that the successive domain flips, following the biggest one, occur at VF values slightly smaller than VC.

Difference between present and original views on NC effect

The intrinsic difference between our view and the original proposal on NC effects in FE-based FETs is next discussed. To be fair, only the case of single domain is considered. The Vint gain is actually obtained in both cases. In the present view, it is associated with the domain flip, on the assumption that an intermediate state between two polarized states in the domain is not stable quasi-statically (Fig. 5a). This requires that the gate voltage should be high enough to enable VF > VC. While in the original NC proposal, the intermediate state with a small polarization value can be stabilized with a small CP, and the Vint gain is obtainable even at VF = 0 (Fig. 5b). It remains still unclarified whether it is possible to access to the intermediate polarized state quasi-statically or not. It is, however, strongly inferred that the continuous change of polarization in a domain is very improbable from the present experimental results. This is also consistent with the fact that no observation of the SS improvement under a small gate voltage swing has been reported so far.

Fig. 5: Comparison of present view with initial NC proposal.
figure 5

a In present view, Vint gain is associated with polarization flip, and the intermediate state between two polarization states is unstable. Thus, VF > VC is required to get Vint gain. b In the original NC proposal, the intermediate state can be stabilized by changing the polarization value. Then even VF = 0 enables a Vint gain directly.

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Finally, it is worthy of mentioning that the polarization switching kinetics depends on a specific model such as Kolmogorov-Avrami-Ishibashi or nucleation-limited switching models31,41, while specific switching kinetics does not necessarily lead to the total frustration of the Vint enhancement effect12. Namely, when the depolarization field is formed due to the bound charge movement, the Vint gain can be obtained in any switching kinetics cases. Meanwhile, the gain value, shape and time dependence of Vint, might be affected. But such consideration is beyond the scope of this paper in which the DC-mode polarization switching rather than high-speed switching is focused. When the polarization switching speed is concerned in device applications, the specific domain switching kinetics should be taken into account for the NC effect analysis. In addition, in the case of multiple-domain system, there will be a difference between the cases w/ and w/o internal electrode. In the latter case, the charge flow at FE/PE interface, the local effect of domain switching and the coupling effects between FE and PE layer should be taken into consideration11,25,26,27. It is very interesting to further study these issues.

Discussion

This work demonstrates stepwise internal potential jumps associated with successive domain flips in multiple-domain FE/PE stack, and a direct relationship of this effect with steep SS in a SiO2 MOSFET connected with a FE capacitor. Each domain flip provides an internal potential gain in FE/PE stack with a suitable PE capacitance as originally proposed NC effect, while a stable intermediate state between two polarization states is very unlikely. Thus, the power supply voltage surmounting the coercive voltage on FE layer is required, which is not advantageous for low-power device applications. The nearly hysteresis-free steep SS FET may be possible with a single-domain FE layer by tuning PE capacitance, while that with a multiple-domain FE layer is very challenging. It is likely that most of small SS values so far reported in literatures are explainable by the present model.

Methods

Measurement circuits of FE/PE and FE/MOSFET systems

PZT-CAPs with Pt electrodes available from Radiant Technologies Inc. and conventional PE-CAPs were used. FE and PE CAPs were connected in series by a coaxial cable. A DC-voltage measurement system (Keithley 6430) equipped with a remote sub-fA pre-amplifier was connected to the internal metal node between FE and PE CAPs to measure the internal potential. This enabled us to effectively increase the input impedance of the measurement system up to 1016 Ω. For FE/MOSFET system, poly-Si gate/SiO2/Si MOSFET was connected to PZT-CAP in series. The FE/MOSFET system was connected to a semiconductor parameter analyzer (Keysight B1500) for IDS-VGS characterization, while the high impedance system was connected to the internal metal between FE and MOSFET for internal potential measurement. Both experiments were carried out separately under the same VDS.

Electrical measurement

Before measurement, each terminal in FE/PE and FE/MOSFET circuits was grounded to remove unknown charges left inside. Then the internal potential, Vint, was measured by sweeping the total voltage, V, with a step of 50 mV. In each step, a waiting-time of 0.1 s was set for stabilizing the voltage and a total time of 0.5 s was taken for each point measurement. For the FE/MOSFET system, in addition to Vint measurement, FET characteristics were measured with the source grounded under VDS = 0.1 V. The sweeping step of VGS was 20 mV and the waiting-time for each step measurement was 0.1 s.