Impact of Hysteresis curve on subthreshold swing in ferroelectric FET

. The changes in Subthreshold Swing (SS) were observed for changes in remanent polarization P r and coercive field E c , which determine the characteristics of the P-E hysteresis curve of ferroelectric in Ferroelectric FET (FeFET). A Metal-Ferroelectric-Metal-Insulator-Semiconductor (MFMIS) structure was used for the junctionless double gate structure. To obtain the SS value, the analytical SS model was used. The ranges of 15 ≤ P r ≤ 30 μC/cm 2 and 0.8 ≤ E c ≤ 1.5 MV/cm , which were reasonable in various experiments and did not generate unstable regions in the relationship of drain current and gate voltage, were considered. As a result, the SS decreased as P r decreased and E c increased due to the capacitance change in the ferroelectric. This phenomenon is because the controllability of channel carriers by the gate voltage increases due to the increasing of change in the ferroelectric voltage for the gate voltage as P r decreases and the memory window increases. Since the SS decreased linearly in the memory window, the SS constantly changed according to the ratio of P r and E c , P r / E c . As the ferroelectric thickness increased, the SS decreased significantly, but the change of SS for the P r / E c was severe. In general, as the channel length decreases, SS increases. However, when the P r / E c decreased to 10 pF/cm , the SS tended to decrease as the channel length decreased. The reason for this can be attributed to the fact that the relative thickness of ferroelectric increases with a small channel length.


INTRODUCTION
High integration is a very necessary factor not only for economic reasons such as reduction in production cost but also for engineering aspects such as applications in the ultra-high frequency region by high-speed switching operations due to reduced transistor sizes.Therefore, reducing the size of the transistor is a key matter, but it becomes an obstacle to high integration because of the increase of short-channel effect that affects the increase in power consumption, such as the increase in subthreshold swing (SS) or the increase in off-current caused by the decrease in transistor size [1 -3].As a result, the decrease in transistor size represents the disadvantage of increased power consumption as well as the advantage of high-speed operation.To solve this problem, the structure of the transistor is being changed three-dimensionally, such as FinFET or GAA FET, and the gate oxide material is made of high-k dielectric stacked with SiO 2 or the hysteresis characteristics of ferroelectric materials are being used [4 -6].Among them, FETs using ferroelectric show an SS of less than 60 mV/dec and are in the spotlight as devices for low power consumption [7 -9].Since the decrease in SS eventually appears as a decrease in parasitic current, the decrease in SS is a very important factor in miniaturizing transistors.However, Boltzmann Tyranny limited the SS value to a minimum value of 60 mV/dec.Even when high-k was used in the already announced junctionless double-gate MOSFET and GAA FET [10 -11], it was confirmed that SS > 60 mv/dec if the negative capacitance effect of ferroelectric was not used.Several papers have already been published showing SS values below 60mV/dec using the Ferroelectric FET (FeFET) to overcome these difficulties [12 -14].
In particular, HfO 2 -based ferroelectric devices are being studied extensively due to their integration, low power consumption, and excellent compatibility with CMOS technology.Therefore, in this paper, the change of SS according to the hysteresis characteristics of HfZrO (HZO) in HfO 2 -based ferroelectric FET (FeFET) is analyzed.The P-E curve, which shows the relationship between ferroelectric polarization and electric field, exhibits hysteresis characteristics and varies greatly depending on the manufacturing process of the ferroelectric material [15 -17].
Since the shape of the P-E curve greatly affects the SS of FeFET, research on the P-E hysteresis curve of ferroelectric is being actively conducted.Das et al. presented the change of composition of Hf and Zr and the transition of the P-E curve according to the heat treatment using Rapid Thermal Annealing (RTA) for the ferroelectric layer having the same composition [18 -19].Chen and Yuan et al. analyzed the effect of NH 3 plasma treatment on reliability during TiN/HZO/TiN formation using the P-E curve, and Dang et al. also studied the endurance of HZO-based ferroelectric capacitor according to the Ti to N ratio during TiN electrode formation [20 -22].The two variables that determine the shape of the P-E curve are the remanent polarization P r and the coercive field E c .As the P r and E c modify the capacitance of the ferroelectric, the two factors have a significant impact on the SS [23 -24].In addition, since P r and E c affect leakage current, wake-up, cycling endurance, and retention of ferroelectric devices, they have a great influence on the electrical characteristics of FeFET using ferroelectric thin film [25 -27].Therefore, in this paper, the effect of changes in P r and E c on the SS of FeFET having a Metal-Ferroelectric-Metal-Insulator-Semiconductor (MFMIS) structure is investigated.A particular focus will be placed on junctionless double-gate FeFETs.Further, the potential of junctionless double-gate FeFETs for integrated circuit applications will be investigated.

The hysteresis of ferroelectric and FeFET
Figure 1 shows the transition of the P-E hysteresis curve of ferroelectric for the change of remanent polarization P r and coercive field E c .As seen from Figure 1, the absolute value of the negative capacitance of a ferroelectric material can be expressed as the rate of change of polarization P for electric field E. Figure 1(a) is a picture of changing E c after fixing P r .As seen in Figure 1(a), as E c decreases, the absolute value of negative capacitance increases.Also, in Figure 1(b), it can be seen that the absolute value of negative capacitance increases when E c is fixed and P r is increased.In this way, when the P r and E c change, it can be found that the value of negative capacitance changes due to the change of the hysteresis curve of the ferroelectric.Figure 2 shows the FeFET with the junctionless double gate of the MFMIS structure used in this paper.The symbols shown in Figure 2 and the range of device parameters used in this paper are shown in Table 1.

P r and E c values used in a simulation
The P r and E c values vary according to various experiments.Figure 3 shows the P r and E c values corresponding to HZO ferroelectric extracted from published papers [18 -21, 28 -30].As seen in Figure 3, it can be found that the P r and E c values are very diverse depending on the experimental conditions or the ratio of Hf and Zr.Therefore, we will observe the change of SS in the range of P r and E c corresponding to the box value indicated by the dotted line where most of the experimental values are gathered in Figure 3.

The Change of SS and I-V Curves for Pr and Ec
Ding's expansion method [31] was used to obtain the potential distribution within the channel for the MFMIS FeFET shown in Figure 2, and the current-voltage characteristics were obtained using the expansion method presented in the previous paper [32].The swing was obtained using dV gs2 /dlog(I ds ) by the definition of SS with drain current I ds and gate voltage V gs2 to find out the change of swing in the subthreshold region.First, after fixing E c = 1 MV/cm in Figure 4, the drain current-gate voltage characteristics and swing obtained with P r as a parameter are shown.As seen in Figure 4(a), as P r increases under the given conditions, the current level decreases and the subthreshold current decreases more rapidly.In addition, it can be observed that the on-current is almost unaffected by P r , but the off-current rapidly decreases as P r increases.Figure 4(b) shows the swing values obtained by calculating the value of dV gs2 /dlog(I ds ) in current-voltage curves of Figure 4(a).It can be observed that the subthreshold swing decreases as the P r value increases in the subthreshold region with a sufficiently small drain current.In particular, it can be observed that the SS is less than 60 mV/dec in the case of P r = 30 μC/cm 2 under the given conditions.Figure 5 shows the drain current-gate voltage characteristics and swing obtained in the same way as Figure 4 with P r fixed at 20 μC/cm 2 and E c as a parameter.Observing Figure 5(a), it can be seen that the current level decreases as E c increases, and in particular, the off-current decreases rapidly.As shown in Figure 5(b), when E c increases, the threshold voltage increases even though the off-current decreases.Note that the off-current and threshold voltage have a large effect on power consumption.Figure 5(b) shows the swing value obtained in the same way as in Figure 4(b).As E c increases, the swing value decreases, and it can be observed that SS is less than 60 mV/dec when E c = 1.2 MV/cm and E c = 1.4 MV/cm in the subthreshold region under the given conditions.As mentioned above, since the SS changed greatly with the change of P r and E c , this paper will focus on the change of SS for P r and E c .
Next, the relationship between SS and ferroelectric capacitance C fe is explained with the P-E relationship in Figure 1.For the FeFET of the MFMIS structure shown in Figure 2, the relationship between SS and the absolute value of ferroelectric capacitance C fe presented by Tu et al. is as shown in Eq. ( 1) [16].
here, C s is the capacitance of the semiconductor, C ox is the capacitance of the oxide used as the insulator layer.C MIS is the capacitance corresponding to the MISFET (Metal Insulator Semiconductor FET) in the MFMIS FET structure and corresponds to the value of (1/C ox +1/C s ) -1 .In this paper, since we want to examine the change of SS for the P-E curve of ferroelectric, we will eventually examine the change of SS for the change of |C fe |.As seen from Eq. ( 1), SS decreases when |C fe | decreases, and SS also increases when |C fe | increases.As described above, since |C fe | is determined according to P r and E c values, the SS is determined according to P r and E c .

Subthreshold swing of FeFET
First, to obtain the SS for the changes in P r and E c , an analytical SS model derived from Ding's extension method [31] was used, as in Eq. ( 2), which was already verified in a published paper [31].Ay nx ox is the oxide film capacitance, si  is the dielectric constant of silicon, and Q is the ferroelectric charge, as described in the previous paper [32].The value of the SS obtained using the analytical subthreshold swing model is shown in Figure 6 as a contour plot for P r and E c .As described above, the SS increases as P r increases, and decreases as E c increases.Of course, both P r and E c have a non-monotonic relationship with the SS due to the polarization effect [33], and it can be seen that SS decreases when E c increases in the region where P r is 15 μC/cm 2 or more.In particular, the SS values of 60 mV/dec or less were shown in almost half of the simulation area.
Observing Figure 6, it can be seen that P r and E c must increase simultaneously to maintain a constant SS value.In addition, it can be observed that the SS is almost similar when the ratio of P r to E c is the same.This is because the capacitance C fe of ferroelectric is proportional to |dP/dE| as described in Figure 1, and eventually the SS will change according to C fe in Eq. ( 1) if all conditions are the same.
To examine this relationship in more detail, the relationship between the memory window expressed as 2t f E c and the SS is shown in Figure 7 with P r as a parameter.The simulation ranges are 0 ≤ t f ≤ 10 nm, 15 ≤ P r ≤ 30 μC/cm 2 and 0.8 ≤ E c ≤ 1.5 MV/cm, and the given conditions are indicated in the figure.As shown in Figure 7, the SS decreases as the memory window increases, and the degree of decrease reduces as P r increases.That is, when P r increases, the SS increases, but it can be seen that the dependence on the memory window decreases.As mentioned in Figure 6, if the ratio of memory window and P r is the same, the SS shows almost the same value.That is, when the memory window is 1.5 V at P r = 15 μC/cm 2 , the SS value of about 61 mV/dec is shown, and it can be observed that the same SS value is shown even when the memory window is 2 V and 3 V at P r = 20 μC/cm 2 and P r = 30 μC/cm 2 , respectively.As seen in Figures.6 and 7, it can be seen that the SS decreases as P r decreases and the memory window increases.The decrease in SS means that the ability to control the channel carrier by the gate voltage increases.To examine these matters in greater detail, the change of the voltage V gs1 of METAL1 and the voltage V f of ferroelectric according to the voltage applied at METAL2 in Figure 2 is shown for the remanent polarization with the memory window as a parameter in Figure 8.A large dV gs1 /dV gs2 means that a small change in V gs2 causes a large change in V gs1 .Here, since V gs1 directly affects the drain current, the drain current will also change significantly and the SS will decrease accordingly when dV gs1 /dV gs2 increases.In Figure 8, it can be seen that the smaller P r and the larger memory window increase dV gs1 /dV gs2 , which is also explained in Figure 7. Figure 8 shows how the voltage applied to the ferroelectric changes according to V gs2 .In Figure 2, since the voltage across METAL2 is V gs2 =V gs1 +V f , it can be observed that the relationship is like dV f /dV gs2 =1-dV gs1 /dV gs2 .Therefore, it shows the opposite relationship with dV gs1 /dV gs2 in Figure 8.As P r decreases and the memory window increases, it can be seen that the voltage V f across the ferroelectric greatly varies according to V gs2 .If V f greatly varies according to V gs2 , it means that the drain current can be greatly changed according to V gs2 , that is, the controllability of carriers in the channel by V gs2 is improved, so the SS will decrease.Comparing Figure 7 and Figure 8, it can be seen that the change in P r and memory window is an important factor in determining the degree of change in drain current by V gs2 .
As mentioned in Figure 6 and Figure 7, if the ratio of P r and E c is the same, the SS is almost the same.The characteristic of HZO film is that it shows excellent ferroelectricity even at several nm [34 -35].Therefore, we tried to observe the change of SS according to the ratio of P r and E c using the ferroelectric thickness t f as a parameter, and the result is shown in Figure 9.The SS were calculated for the range of the P r and E c shown in Table 1, and the given conditions are indicated in Figure 9.As shown in Figure 9, the SS decreases significantly when the ferroelectric thickness becomes thicker.However, as the ferroelectric thickness becomes thinner, the SS increases and the change of the SS for the change of the ratio of remanent polarization P r and coercive field E c also decreases.That is, it can be observed that the SS is hardly affected by the ratio of remanent polarization and coercive field when the ferroelectric thickness becomes smaller.This is because the effect of ferroelectricity decreases as the thickness decreases.Looking at Figure 9, it can be seen that the SS is saturated when the ratio of remanent polarization and coercive field is above the critical value.Also, as the ratio of remanent polarization and coercive field decreases, the SS changes greatly according to ferroelectric thickness.However, as the ratio of remanent polarization and coercive field increases, it can be observed that the variation of SS according to ferroelectric thickness decreases.To find out how the P r /E c of ferroelectric affects the current-voltage characteristics according to the channel length of FeFET, the relationship between drain current and gate voltage is shown in Figure 10 when the P r /E c is 10 pF/cm and 30 pF/cm.As described above, it can be observed in Figure 10 that the SS increases as P r /E c increases.In addition, it can be seen that the SS increases rapidly when the channel length decreases with P r /E c being as large as 30 pF/cm.However, when P r /E c is relatively small, it can be observed that the SS decreases further as the channel length decreases.As seen in Figure 1, a small P r /E c means a small C fe , and at this time, it can be observed that the change of the P-E curve becomes more severe.That is, strong ferroelectricity occurs and SS decreases.In particular, as the channel length becomes shorter, the ferroelectric thickness increases relatively, so the ferroelectricity also increases and the SS will further decrease.Note that SS > 60 mV/dec when P r /E c = 30 pF/cm and SS < 60 mV/dec when P r /E c = 10 pF/cm.In addition, as P r /E c decreases, an unstable region appears as shown in the red dotted line in Figure 10, so in this paper, simulations were performed in the range of P r /E c > 10 pF/cm.Based on the characteristics of Figure 10, the change of SS according to P r /E c is shown in Figure 11 with the channel length as a parameter.As mentioned in Figure 10, when P r /E c decreases, it can be seen that SS decreases as C fe decreases.The degree of change of SS according to P r /E c increased as the channel length decreased, and as the channel length increased, the change rate of SS according to P r /E c also decreased due to the relative decrease in ferroelectric thickness.In addition, as P r /E c increases, the change of SS according to the channel length increases, and as P r /E c decreases, the change of SS according to the channel length decreases.In particular, for the reason explained in Figure 10, it can be seen that SS decreases as the channel length decreases with P r /E c being around 10 pF/cm.The ranges of P r and E c indicated in Table 1 were limited to ranges where unstable regions did not appear in the relationship between drain current and gate voltage, if possible.The curvature of the hysteresis curve increases when P r becomes smaller than E c and P r /E c becomes smaller.An unstable region begins to appear in the current-voltage characteristic as shown in Figure 10, and SS rapidly decreases.Figures 9 and 11 show the relationship between P r /E c , channel length, and ferroelectric thickness to maintain the SS value below 60 mV/dec.To find the range of these parameters to satisfy SS < 60 mV/dec under the given conditions, Figure 12 shows a contour graph for channel length and P r /E c using ferroelectric thickness as a parameter.In Figure 12, the upper area of each curve indicated by a red arrow represents the SS < 60 mV/dec area.It can be seen that the SS < 60 mV/dec area increases as the ferroelectric thickness increases.In particular, it can be seen that the P r /E c value must also decrease as the channel length decreases to maintain SS < 60 mV/dec.As ferroelectric thickness decreases, the negative capacitance effect of ferroelectric also decreases, so the SS<60 mV/dec area also decreases.In this way, it was found that by adjusting the values of P r /E c , channel length, and ferroelectric thickness, the SS < 60 mV/dec value could be maintained.

CONCLUSIONS
The change of SS for the junctionless double gate FeFET with MFMIS structure using ferroelectric was investigated for the change of P r and E c of ferroelectric.The P r and E c were used in the range where the unstable region does not appear in the relationship between the drain current and the gate voltage.However, it was found that the SS significantly decreased when the unstable area appeared.The change of P r and E c directly affected the FeFET by eventually changing the capacitance of the ferroelectric.In particular, the SS decreased as P r decreased and E c increased.As the memory window increased, SS decreased, and in the change of SS according to P r and memory window, it was found that SS constantly changed according to the ratio of P r and E c .In conclusion, SS decreased as P r /E c decreased, and increased when P r /E c increased.The decrease in P r /E c eventually increases the ferroelectricity as the C fe decreases in the P-E hysteresis curve, and the change in the voltage applied to the ferroelectric by the gate voltage applied to METAL2, that is, the absolute value of dV f /dV gs2 increases.As a result, the SS decreased as the carrier controllability by gate voltage was improved.The smaller the channel length, the more severe the change of SS for the P r /E c .When the P r /E c ratio is relatively low, at around 10 pF/cm, the shorter the channel length, the smaller the SS.It is judged that the unstable region suddenly appears under these conditions and the SS rapidly decreases.In the future, short channel effects such as the shift of threshold voltage of FeFET and drain-induced barrier lowering (DIBL) for changes in P r and E c should be further analyzed.

Conflicts of interest.
The author declares no conflict of interest.

Figure 1 .
Figure 1.The change of the P-E hysteresis curve and the change of the corresponding capacitance for the change of P r and E c .

Figure 2 .
Figure 2. Schematic overview of a junctionless double gate FeFET with MFMIS structure.

Figure 3 .
Figure 3. Material parameters of ferroelectric HZO thin films published in previous papers.

Figure 4 .
Figure 4. (a) The relationship of drain current vs. gate voltage (V gs2 in Figure 2), and (b) swings defined by dV gs2 /dlog(I ds ) with the remanent polarization as a parameter under the conditions shown in the figure.

Figure 5 .
Figure 5. (a) The relationship of drain current vs. gate voltage (V gs2 in Figure 2), (b) swings defined by dV gs2 /dlog(I ds ) with the coercive field as a parameter under the conditions shown in the figure.

Figure 6 .
Figure 6.Contour plot for remanant polarization and coercive field under the conditions presented in the figure.

Figure 7 .
Figure 7. Relationships of subthreshold swing and memory window (= 2t f Ec) with a remanent polarization as a parameter under the conditions given in the figure.

Figure 8 .
Figure 8. Changes in voltage V gs1 across METAL1 and voltage V f across ferroelectric for changes in voltage V gs2 across METAL2 with the remanent polarization and memory window as parameters under the conditions given in the figure.

Figure 9 .
Figure 9. Subthreshold swings for the ratio of remanent polarization and coercive field with a ferroelectric thickness as a parameter under the conditions given in the figure.

Figure 10 .
Figure 10.Relation of drain current and gate voltage with the ratio of remanent polarization P r and coercive field E c and channel length as parameters under the conditions given in the figure.

Figure 11 .
Figure 11.Subthreshold swings for the ratio of the remanent polarization P r and coercive field E c with a channel length as a parameter under the conditions given in the figure.

Figure 12 .
Figure 12.Contour plot for the channel length and P r /E c with the ferroelectric thickness as a parameter under the conditions given in the figure.

Table 1 .
Device parameters for this analytical threshold voltage model.