Gradient voltage amplification effect in FDSOI NCFET with thickness-variable ferroelectric layer

In this paper, a negative capacitance field effect transistor with thickness variable ferroelectric layer (TVFL NCFET) based on the fully depleted silicon on insulator (FDSOI) is proposed. The TVFL NCFET features the linearly increased ferroelectric layer thickness along the channel from source to drain. The gradient voltage amplification effect caused by the TVFL is analyzed according to the proposed capacitance model and simulation. Both of the model and numerical results indicate that the TVFL leads to a gradient increased electrostatic potential distribution along the bottom of the ferroelectric layer. The influences of gradient voltage amplification effect on the transfer characteristics, the output characteristic, the ratio between on-state-current (I ON) and off-state-current (I OFF), the drain induced barrier lowering (DIBL) and the subthreshold swing (SS) are investigated. The results show that the TVFL NCFET achieves the SS of 53.14 mV/dec, which is reduced by 19% when compared to the conventional NCFET. Meanwhile, large ION/IOFF is also realized and up to 1012 at most.


Introduction
The negative capacitance field effect transistor (NCFET) has been widely investigated in recent years because of its ability in improving subthreshold swing (SS) and reducing power dissipation.The operation of negative capacitance introduced by ferroelectric layer and other capacitances in the NCFETs is described by the capacitance matching [1,2].Meanwhile, the ferroelectric layer in NCFET is expected to realize the low SS [3].The ferroelectric materials doped hafnium oxides like Hf x Zr 1-x O 2 (HZO) are wildly used as ferroelectric layer to improve the performance and compatibility [4][5][6][7][8][9].The 2D thin film channel is also considered because it's excellent ability in achieving steep SS and large I ON /I OFF ratio [9][10][11].The special structure such as T-shaped gate NCFET [12], highly-doped double-pocket double gate NCFET [13,14] have been proposed and shown great ability in achieving steep SS [15] introduces a box ferro FDSOI which attaches ferroelectric layer to buried oxide to mitigate the negative differential resistance (NDR) effect.The first principle explanation of the NDR effect is presented, and the element which can influence the characteristic of NCFETs is also investigated in this reference.The thickness of the ferroelectric layer plays a role in the electric characteristic of the NCFET [16] indicates that the thicker ferroelectric layer brings steeper SS but larger hysteresis.
In this paper, a novel negative capacitance field effect transistor with thickness variable ferroelectric layer (TVFL NCFET) is proposed.The TVFL can result in a gradient increased electrostatic potential distribution along the bottom of the ferroelectric layer, named gradient voltage amplification effect.Which influences the device characteristics especial reduces the SS.A capacitance model is established to reveal the mechanism of the gradient voltage amplification effect caused by TVFL.The Sentaurus TCAD tool is used to investigate the gradient voltage amplification effect and its influences on the device characteristics.By calibrating with experiment baseline FDSOI in [17], the simulation method and models are determined, the models used in this work including the effective intrinsic density, doping-dependent mobility, doping-dependent SRH, Auger, high field saturation and FEPolarization.

Device structure and mechanism
The structure of the TVFL NCFET investigated in this paper is shown in figure 1.The TVFL NCFET characterized by a ferroelectric layer with linearly increased thickness along the channel from source to drain.Zirconium doped HfO 2 (Hf 0.5 Zr 0.5 O 2 ) is used as ferroelectric material.The Landau parameters are from [18], where α = −5.810× 10 10 cm F −1 , β = 3.286 × 10 19 cm 5 /(F • C 2 ) and γ = 2.165 × 10 28 cm 9 /(F • C 4 ).The thickness of the ferroelectric layer is denoted as T FS near the source and denoted as T FD near the drain.T FS = T FD means the conventional NCFET that has a uniform thickness of the ferroelectric layer.The TVFL NCFET is based on the fully depleted silicon on insulator (FDSOI) technology with the box thickness of 25 nm and top silicon layer thickness of 4 nm.The doping concentration in the source and drain is 1 × 10 20 cm −3 , and in the channel is 1 × 10 17 cm −3 .The gate length and gate oxide thickness are 20 nm and 0.6 nm, respectively.The overlaps between the gate and the source/drain region are the same value of 1 nm.
The ferroelectric layer is defined by LK model [19].The LK equation is expressed as: where ρ is the viscosity associated with polarization-switching dynamic.The free energy U of the ferroelectric layer is expressed as: In this equation, α, β and γ are ferroelectric material parameters, g is a coupling coefficient for polarization gradient term of the free energy.P is the polarization intensity.E stand for external voltage.Then, combining equations (1) and (2), the expression of E can get as followed: Under single-domain states condition, dP/dt = 0 and g = 0, so equation (3) could be simplified as followed: By giving ferroelectric parameters α, β and γ, the polarization characteristic of the ferroelectric layer could be decided.
The definition of electric field is E = V FE /T FE , where V FE is the voltage drop across the ferroelectric layer.Therefore, V FE can be expressed as: By ignoring the high order terms of P, equation (5) can be simplified as: The polarization intensity is numerically equal to the charge density generated by the polarization on the dielectric surface.In a unit area, the polarization intensity is numerically equal to the charge generated by polarization.So the P in equation ( 6) can be changed to Q [3].The capacitance of ferroelectric layer (C FE ) is defined by C FE = ∂Q FE /∂V FE .By the definition and equation (6), C FE can be approximatively calculated: According to the capacitance model shown in figure 2(a), the voltage amplification factor A V can be expressed as followed: where C mos stands for C OX , C Source and C Drain .
Combining equations ( 7) and (8), the voltage amplification A V is related to ferroelectric layer thickness T FE , express as: Normally, lager T FE induces greater voltage amplification.However, changing gate voltage leads to variation of depletion layer thickness at the top of the channel, resulting in variation in C dep in subthreshold region.In this  investigation, the FDSOI technology makes a simplification which ignores the variation of C mos caused by changing T FD in this equation.
According to equation (9), the thicker T FE , the larger A V , which means the greater voltage amplification.For the TVFL NCFET, the ferroelectric layer thickness is linearly increased from source to drain.Therefore, the voltage along the bottom of the ferroelectric layer has been amplified and gradient increased from source to drain.Which is defined as gradient voltage amplification effect.To describe the gradient voltage amplification effect of the thickness-variable ferroelectric layer, a discrete capacitance model is proposed as shown in figure 2(b).In this model, the ferroelectric capacitance, oxide capacitance and depletion capacitance are separated into infinitesimal capacitance respectively.The larger ferroelectric thickness near drain side produces greater voltage amplification near drain.Therefore, V mos1 > V mos2 > V mos3 > K > V mos,n , and leading to V 1 > V 2 > V 3 > K > V n .These voltage differences create a lateral electric filed while it is caused by gate ferroelectric layer instead of drain voltage.They can bring extra current flow through infinitesimal resistance R 1 , R 2 , R 3 , K ,R n-1 in channel.This operation induces developed SS in subthreshold region and enhanced on-state current.
To verify the gradient voltage amplification effect, the Sentaurus TCAD tool is employed to simulate the TVFL NCFET. Figure 3 shows the electrostatic potential distribution in the ferroelectric layer.The T FS is fixed at 5 nm to clearly observe the negative capacitance effect, while the T FD is varied for 5 nm to 20 nm with equal step, standing for the conventional NCFET and TVFL NCFET with different linearly thickness.The voltage of 1.0 V is applied on the gate.As seen in the figures, the ferroelectric layers show a great ability of voltage amplification, which enlarge the gate voltage significantly.
Figure 3(a) is the electrostatic potential distribution in the ferroelectric layer of the conventional NCFET with T FS = T FD = 5 nm.It reached maximum voltage of 1.13 V at the bottom of the ferroelectric layer.Figures 3(b)-(d) are the electrostatic potential distribution in the thickness variable ferroelectric layer.It is obviously that the TVFL leads to a gradient increased electrostatic potential distribution along the bottom of the ferroelectric layer.The TVFL in figures 3(b)-(d) obtain the maximum voltage of 1.332 V, 1.772 V and 2.647 V at the bottom ferroelectric layer near drain side, respectively.As T FD increases, electrostatic potential gets higher, which indicates that the voltage amplification is stronger, and the gate control ability is better.
Figure 4 shows the electrostatic potential at the bottom of the ferroelectric layer for the TVFL NCFET with different T FD and fixed T FS .It's obviously that the voltage amplification near drain is larger than that near source, and the voltage gradient from source to drain is enlarged with the T FD increasing.It is worth noting that the potential at the source side is also improved with the increasing T FD although the T FS is fixed at 5 nm.This is because the ferroelectric layer thickness varies continuedly.From figure 2(b), it's clear that electrostatic potential on the top surface of channel V 1 , V 2 , V 3 , K, V n is determined by the complex coupling of each capacitance column, indicating that the electrostatic potential on the bottom surface V mos1 , V mos2 , V mos3 , K, V mos,n is relevant.

Results and discussions
The capacitance model and simulation results show that the thickness variable ferroelectric layer results in a gradient voltage amplification on the surface of the channel.In order to reveal the influences of gradient voltage amplification effect on the electrical characteristics of the TVFL NCFET, the output characteristics, transfer characteristics, DIBL effect, I ON /I OFF and sub-threshold swing are discussed.
Figure 5 gives the output characteristics of TVFL NCFET.The V GS varies from 0.3 V to 1.5 V with step of 0.3 V.In figure 5(a), it is found that the TVFL NCFETs induce a steep current increment as V DS gets larger at the liner region.But once the device enters saturation region, the current increment becomes slower and followed by current drop.This effect called negative differential resistance (NDR), gets more significant as T FD increases.This is because the voltage on the bottom surface of ferroelectric layer is influenced by both V GS and V DS through gate-drain coupling.The negative capacitance of ferroelectric layer introduces the negative correlation between V mos and V DS .The increasing V DS reduce the charges under ferroelectric layer and lead to increase of V FE in negative capacitance region, thus result in the reduction of V mos .This means the voltage amplification is impeded, which result in the decrease of I D .Meanwhile, according to equation (5), the ferroelectric layer thickness can be regard as a coefficient which modulate the change in charges.As T FD increases, the effect brought by the decrease in charges is amplified.In this case, the increasing T FD results in more significant NDR.   Figure 6 shows the DIBL effect of TVFL NCFETs depends on changing T FD combined with the conduction band energy diagram.In the energy diagram, the V GS is set at 0.1 V to ensure the device is turned off.The V DS varies from 0.05 V to 0.3 V.It can be found that as V DS increases from 0.05 V to 0.3 V, the potential barrier in the channel region increases.This is a strong evident of reversed DIBL.Meanwhile, compared with the conventional ferroelectric layer, the TVFL brings a higher potential barrier, and the barrier increases more with the increase of V DS .As T FD gets thicker, the raise of conduction band barrier gets larger, proving that TVFL with larger T FD has greater ability in reversing DIBL and inhibit short channel effect.The reversed DIBL ensures the gate controllability at small V GS , which helps to reduce drain leakage current as shown in figure 5(b).It also can be seen in figure 6 that the DIBL is reversed to negative.As T FD increases, the negative DIBL become larger in absolute value.This result confirms the finding in conduction band energy diagram.
Figure 7 shows the on current and off current of TVFL NCFETs.It can be seen that as T FD increases, the on current increases, which means better current driving ability.This is because stronger voltage amplification induced by thicker ferroelectric layer contributes to the forming of conductive channel.It also can be seen that the off current decreases significantly as T FD increasing.It means that the leakage drain current is better restrained, which helps to reduce static power consumption.This can be explained by the reversed DIBL.   Figure 8 shows the transfer characteristics of TVFL NCFETs.The logarithmic axis in I D is set.It can be seen that TVFL NCFETs achieve steep I D -V GS curves.With fixed T FS , the larger T FD corresponds to steeper I D -V GS curve, which means smaller subthreshold swing.In the subthreshold region, the larger gate voltage caused by the larger T FD contributes to the formation of the inversion layer, which promotes the increase of drain current during the transistor opening.In addition, larger T FD forms larger lateral potential difference under the gate stack and on the top of the channel, which also promotes the drain current.Thus, the steeper curve can be obtained and leading to the smaller subthreshold swings.Figure 9 gives the SS depends on T FD with fixed T FS .It's obviously that in TVFL NCFETs, the structure with larger T FD has better performance on reducing SS.This is because larger T FD induces greater voltage amplification on the drain side as explained in equation (9) and realizes better gate control and the lateral electric field which can induce extra current flow.The conventional NCFET with T FS = T FD = 5 nm achieves the SS of 65.56 mV/dec in forward sweep and 65.58 mV/dec in reverse sweep, while the TVFL NCFET with T FS = 5 nm and T FD = 20 nm achieves the SS of 53.14 mV/dec in forward sweep and 53.13 mV/dec in reverse sweep, which provides improvements of 12.42 mV/dec forward and 12.45 mV/dec reverse.
It seems like that thicker T FD means better performance, nothing different from increasing the thickness of whole ferroelectric layer.But actually, something is found when take consideration into the structure with T FS = T FD = 20 nm. Figure 10    22 mV maximum and nearly 0 mV at I D = 10 −7 A, also is negligible.For the NCFET with T FS = T FD = 20 nm, it has hysteresis of 237 mV maximum and 60 mV at I D = 10 −7 A, which brings about severe problem of hysteresis and will lead to serious logic confusion in the circuits.In contrast, the TVFL NCFET with T FS = 5 nm, T FD = 20 nm shows hysteresis by 91% and the hysteresis happens at a lager voltage and away from threshold, which means that the impact to switching operation of the device is likely slighter.Therefore, the thickness of the ferroelectric layer should be optimized and the TVFL is a useful method to restrain hysteresis while reducing the SS.

Conclusion
In this paper, the TVFL NCFET is proposed and investigated by theoretical model and TCAD simulations.It is found that the TVFL forms the gradient electrostatic potential distribution and has great ability on voltage amplification.Compared to the conventional NCFETs with T FE = 5 nm, the maximum electrostatic potential on the bottom surface of ferroelectric layer in TVFL NCFETs is increased by 98.5% at most.The TVFL NCFETs can achieve I ON /I OFF ratio of 2.12 × 10 12 , which is over 10 2 larger than conventional NCFETs with T FE = 5 nm.The SS of TVFL NCFETs is also reduced by 19% at most.Meanwhile, the simulation results show that compared to the conventional NCFETs with T FE = 20 nm, the TVFL NCFETs reduce hysteresis by 91% at least and can achieve non-hysteresis operation with larger performance loss in SS.

Figure 2 .
Figure 2. The capacitance model of NCFETs.(a) is the conventional capacitance model of NCFETs, which regards ferroelectric capacitance as a whole.(b) is the novel capacitance model of the proposed TVFL NCFETs, which separates ferroelectric capacitance into infinitesimal capacitance.

Figure 4 .
Figure 4.The electrostatic potential at the bottom of the ferroelectric layer.

Figure 5 (
b) zoom in the region of V GS = 0.3 V in figure5(a).The NCFETs are turned off at this V GS .It's obviously that larger T FD results in smaller I D .It means that the TVFL structure with thicker T FD can prevent device from punch through at small V GS .This phenomenon is reflected as negative (or reversed) DIBL.

Figure 5 .
Figure 5.The output characteristics of the TVFL NCFET that (a) V GS = 0.3 V, 0.6 V, 0.9 V, 1.2 V, 1.5 V respectively.(b) zoom in the region of V GS = 0.3 V.

Figure 6 .
Figure 6.The negative DIBL effect in TVFL NCFET.The conduction band energy diagram indicates the rising barrier when V G = 0.1 V and the device is turned off.

Figure 7
Figure6shows the DIBL effect of TVFL NCFETs depends on changing T FD combined with the conduction band energy diagram.In the energy diagram, the V GS is set at 0.1 V to ensure the device is turned off.The V DS varies from 0.05 V to 0.3 V.It can be found that as V DS increases from 0.05 V to 0.3 V, the potential barrier in the channel region increases.This is a strong evident of reversed DIBL.Meanwhile, compared with the conventional ferroelectric layer, the TVFL brings a higher potential barrier, and the barrier increases more with the increase of V DS .As T FD gets thicker, the raise of conduction band barrier gets larger, proving that TVFL with larger T FD has greater ability in reversing DIBL and inhibit short channel effect.The reversed DIBL ensures the gate controllability at small V GS , which helps to reduce drain leakage current as shown in figure5(b).It also can be seen in figure6that the DIBL is reversed to negative.As T FD increases, the negative DIBL become larger in absolute value.This result confirms the finding in conduction band energy diagram.Figure7shows the on current and off current of TVFL NCFETs.It can be seen that as T FD increases, the on current increases, which means better current driving ability.This is because stronger voltage amplification induced by thicker ferroelectric layer contributes to the forming of conductive channel.It also can be seen that the off current decreases significantly as T FD increasing.It means that the leakage drain current is better restrained, which helps to reduce static power consumption.This can be explained by the reversed DIBL. Figure 7 also gives the I ON /I OFF ratio of TVFL NCFETs with different T FD and fixed T FS .The logarithmic axis in I ON /I OFF ratio is set.The TVFL NCFETs show great ability in increasing I ON /I OFF ratio.With the same T FS , I ON /I OFF ratio increases as T FD gets larger.These results indicate that by choosing a proper V DS , the property of drain current can be improved greatly by the TVFL NCFETs.

Figure 8 .
Figure 8.The transfer characteristics of TVFL NCFETs with different T FD and fixed T FS .

Figure 7 .
Figure 7.The I ON and I OFF of TVFL NCFETs with different T FD and fixed T FS .
Figure8shows the transfer characteristics of TVFL NCFETs.The logarithmic axis in I D is set.It can be seen that TVFL NCFETs achieve steep I D -V GS curves.With fixed T FS , the larger T FD corresponds to steeper I D -V GS curve, which means smaller subthreshold swing.In the subthreshold region, the larger gate voltage caused by the larger T FD contributes to the formation of the inversion layer, which promotes the increase of drain current during the transistor opening.In addition, larger T FD forms larger lateral potential difference under the gate stack and on the top of the channel, which also promotes the drain current.Thus, the steeper curve can be obtained and leading to the smaller subthreshold swings.Figure9gives the SS depends on T FD with fixed T FS .It's obviously that in TVFL NCFETs, the structure with larger T FD has better performance on reducing SS.This is because larger T FD induces greater voltage amplification on the drain side as explained in equation (9) and realizes better gate control and the lateral electric field which can induce extra current flow.The conventional NCFET with T FS = T FD = 5 nm achieves the SS of 65.56 mV/dec in forward sweep and 65.58 mV/dec in reverse sweep, while the TVFL NCFET with T FS = 5 nm and T FD = 20 nm achieves the SS of 53.14 mV/dec in forward sweep and 53.13 mV/dec in reverse sweep, which provides improvements of 12.42 mV/dec forward and 12.45 mV/dec reverse.It seems like that thicker T FD means better performance, nothing different from increasing the thickness of whole ferroelectric layer.But actually, something is found when take consideration into the structure with T FS = T FD = 20 nm.Figure10gives the transport characteristics of NCFETs with different T FS and T FD = 5 nm.Both forward sweeping and reverse sweeping are done.The NCFET with T FS = 5 nm, T FD = 5 nm shows almost no hysteresis, but poor performances.The TVFL NCFET with T FS = 5 nm, T FD = 20 nm shows hysteresis of

Figure 10 .
Figure 10.The transfer characteristics of NCFETs with T FS = T FD = 5 nm, T FS = 5 nm, T FD = 20 nm and T FS = T FD = 20 nm.Both sweeping forward and reversely is included.

Figure 9 .
Figure 9.The SS of TVFL NCFETs with different T FD and fixed T FS in both forward and reverse sweeping condition.