Negative capacitance field effect transistor (NC-FET) was proposed in 2008 by Salahuddin et al.^[1] to improve the subthreshold slope of the MOSFET. It has been experimentally proven by Asif et al. in Ref. [2]. Although many technologies^{[3, 4]} before it were suggested to improve the subthreshold slope of the MOSFET, they had inherent disadvantages that first had to be overcome. The NC-FET reduces subthreshold swing (SS) below Boltzmann's limit, due to the use of ferroelectric materials. The negative capacitance property of these materials, which actually reduces the SS, is entirely material and process dependent as described in phenomenological Landau-Devonshire theory^[5]. In an NC-FET, any ferroelectric material needs to be integrated directly within the gate stack of a conventional CMOS device and its properties determine the characteristics of the device. These materials are highly process and temperature dependent, and they can switch their properties from one state to another under certain conditions. The coercive field (E_C) and remnant polarization (P₀) of the material and process temperature determine the Landau parameters in Landau-Khalatnikov (LK)^{[6, 7]} equation described in the theory part of the paper.

In this paper, we have obtained an analytical equation to determine the thickness of the ferroelectric material required to get minimum subthreshold swing, under capacitance matching condition. The limit on minimum possible subthreshold swing (S_min) reflects the choice of underlying MOS capacitance (C_MOS) and ferroelectric capacitance (C_FE), which is fundamentally dictated by the need to stabilise the total NC capacitor capacitance. In section 2, we describe the fundamental capacitance constraints that define the minimum subthreshold swing in an NC capacitor. In section 3, we determine the optimal thickness of the ferroelectric material and substrate doping required to observe the NC effect for a minimum subthreshold swing. In section 4, we discussed the dependence of ferroelectric thickness on the coercive field and remnant polarization. Finally, in section 5 we obtain the equation for S_min and corresponding S_min for five different ferroelectric materials.

For the analytical analysis of our model, we have considered a basic MOS capacitor with an extra ferroelectric layer in the gate stack. A schematic of a Metal-Ferroelectric-Metal-Insulator-Substrate (MFMIS) negative capacitance capacitor^[8] and its corresponding capacitance equivalent formed between the ferroelectric and the dielectric layer is shown in Figs. 1(a) and 1(b), respectively. The V_GS is the applied gate voltage and V_GMOS is the internal node voltage formed between the ferroelectric and the dielectric layer. Because our study examines the properties of a ferroelectric layer, we are not going to examine the electrical characteristics of the NC-FET and will instead limit our study to the capacitor only. The subthreshold swing (SS) of a MOSFET device is defined as the change in gate voltage (V_GS) required to change the drain current (I_D) by one order of magnitude^[9]. For the negative capacitance MOS capacitor equivalent shown in Fig. 1(b), SS is given as^[9]

$ {\rm{SS}} \equiv \frac{{\partial {V_{{\rm{GS}}}}}}{{\partial \left( {{\rm{lo}}{{\rm{g}}_{10}}{I_{\rm{D}}}} \right)}} = \frac{{\partial {V_{{\rm{GS}}}}}}{{\partial {V_{{\rm{GMOS}}}}}} \times \frac{{\partial {V_{{\rm{GMOS}}}}}}{{\partial \left( {{\rm{lo}}{{\rm{g}}_{10}}{I_{\rm{D}}}} \right)}} = m \times n. $  (1)

The SS can be reduced by reducing any of the two m or n factors. In the case of the NC capacitor, we are only concerned with the reduction of m to decrease the SS. The factor, (k_B is Boltzmann's constant, T is temperature and q is electron charge) is known as transport factor and is ≈ 60 mV/decade at room temperature.

The unique property of NC effect in an MFMIS structure is to provide an internal voltage (V_GMOS) that is greater than the applied gate voltage (V_GS). From the capacitor divider formed in Fig. 1(b), the partial derivative of the V_GS with respect to V_GMOS, also known as body factor m, can be written as

$ m = \frac{{\partial {V_{{\rm{GS}}}}}}{{\partial {V_{{\rm{GMOS}}}}}} = {1 + \frac{{{C_{{\rm{MOS}}}}}}{{{C_{{\rm{FE}}}}}}} , $  (2)

where is the series sum of substrate capacitance (C_S) and oxide capacitance (C_OX) capacitances.

To include the model for the dynamics of ferroelectric capacitor, the Landau Khalatnikov (LK) equation^{[6, 7]} that describes the correlation between the polarization and the electric field in a single domain ferroelectric material is employed. The voltage drop across the ferroelectric layer is given as

$ E = \frac{{{V_{\rm FE}}\left( Q \right)}}{{{t_{\rm FE}}}} = 2\alpha Q + 4\beta {Q^3} + 6\gamma {Q^5}, $  (3)

where α, β, γ are known as Landau parameters of the material. Furthermore, α < 0, β > 0, γ ≥ 0 defines the negative capacitance property, V_FE is the voltage drop across the ferroelectric film and t_FE is the thickness of the ferroelectric layer. The total charge Q is the same as the total charge in the channel, which is calculated using the depletion width approximation and has already explained in Ref. [10]. The potential balance in the equivalent capacitor network results in applied gate voltage V_GS as the sum of the voltage drop across the ferroelectric capacitor (V_FE) and the internal node voltage V_GMOS is given as

$ {V_{{\rm{GS}}}} = {V_{{\rm{FE}}}} + {V_{{\rm{GMOS}}}}, $  (4)

where and where V_FB is the flatband voltage, V_OX is the voltage across the oxide layer between ferroelectric and silicon substrate and ψ_S is the surface potential or band bending in the semiconductor. The capacitance C_FE is the differential capacitance of ferroelectric layer and is defined . From the LK equation described in Eq. (3), the inverse of C_FE is obtained as

$ C_{{\rm{FE}}}^{ - 1}\left( Q \right) = \left( {2\alpha + 12\beta {Q^2} + 30\gamma {Q^4}} \right) {t_{{\rm{FE}}}}. $  (5)

The total gate capacitance C_T of the NC capacitor is given by

$ C_{\rm{T}}^{ - 1}\left( Q \right) = C_{{\rm{MOS}}}^{ - 1}\left( Q \right) + C_{{\rm{FE}}}^{ - 1}\left( Q \right). $  (6)

The increased gate capacitance due to C_FE of the ferroelectric layer in the gate stack, amplifies the internal node voltage. However, C_T must be positive for all charges for a hysteresis free stable operation of the device^[11–13]. This stability requirement puts a fundamental constraint on C_MOS; i.e.,

$ {C_{{\rm{MOS}}}}{\left( Q \right)^{ - 1}} \geqslant |{C_{{\rm{FE}}}}^{ - 1}\left( Q \right)|, $  (7)

which should hold throughout the NC regime. Eq. (7) puts a minimum limit on the subthreshold swing that can be obtained in an NC-FET. For a given dependence of C_MOS and C_FE on Q, minimum SS can be obtained where the two capacitances match as closely as possible^{[12, 14]}. However, the two cannot be made equal because they have different dependence on the value of Q and it is not possible to obtain the same values for all of the values of Q.

For a fixed C_OX, C_MOS changes with C_S which depend on the channel doping N_A; however, C_S depends on whether the channel is in subthreshold, where , or in inversion, where .

Unlike conventional MOSFETs, where the minimum value of SS is fixed at 60 mV/dec, there is no such single value in NC-FET. The minimum subthreshold swing (S_min) that characterizes the NC-FET is different for different devices. However, the two necessary conditions for minimum subthreshold swing are a) and b) channel should be in subthreshold throughout the NC regime^[15] remain the same.

Mathematically, the condition for being in subthreshold is given by (where is the surface potential, is the built-in potential on p-substrate, N_A is the substrate doping and n_i is the intrinsic carrier concentration), throughout the NC-regime. For uniform substrate doping and the partially depleted case, is related to the depletion charge Q by^[9]

$ {\psi _{\rm{S}}} = \frac{{{Q^2}}}{{2q{\varepsilon _0}{\varepsilon _{\rm{s}}}{N_{\rm A}}}}, $  (8)

where is the permittivity of free space and is the permittivity of silicon substrate. The condition for subthreshold region translates Eq. (8) into , where l < 2.

Condition (7) is general and will work irrespective of values of C_MOS and |C_FE|. However, Eq. (8) changes as we change the FET structure, such as single gate versus multigate^[16] or bulk FET versus depleted FET.

Fig. 2 shows the V_FE–Q characteristics of the ferroelectric material. V_FE decreases in the range of –Q₁ < Q < + Q₁ and hence C_FE is also negative between –Q₁ and + Q₁. However, an n-channel NC-FET must be operated in the Q > 0 branch of the NC regime, so that it can be balanced by the negative charges of the channel. Therefore, is defined as the other boundary of the negative capacitance regime. At , Eq. (4) reduces to and at flatband capacitance of the underlying MOS capacitance is given by^[17]

$ \frac{1}{{{C_{{\rm{MOS}}}}}} = \sqrt {\frac{{{v_{\rm{t}}}}}{{q{\varepsilon _0}{\varepsilon _{\rm{s}}}{N_{\rm{A}}}}}} + \frac{1}{{{C_{{\rm{OX}}}}}}, $  (9)

where v_t = K_BT/q is the thermal voltage. Therefore, the condition in Eq. (7) for minimum subthreshold swing and hysteresis free operation of the device can be translated into

$ {t_{{\rm{FE}}}} = - \frac{1}{{2\alpha }}\left[ {\frac{1}{{{C_{{\rm{OX}}}}}} + \sqrt {\frac{{{v_{\rm{t}}}}}{{q{\varepsilon _0}{\varepsilon _{\rm{s}}}{N_{\rm{A}}}}}} } \right], $  (10)

which relates t_FE, and N_A for minimum subthreshold slope. In addition, Q₁ is the maximum charge value and Eq. (8) becomes

$ {N_{\rm{A}}} = \frac{{Q_1^2}}{{2q{\varepsilon _0}{\varepsilon _{\rm{s}}}l{\psi _{\rm{B}}}}}. $  (11)

The final equation for the optimized ferroelectric thickness is

${t_{{\rm{FE}}}} = - \frac{1}{{2\alpha }}\left[ {\frac{1}{{{C_{{\rm{OX}}}}}} + \sqrt {\frac{{2{v_{\rm{t}}}2{\psi _{\rm{B}}}}}{{Q_1^2}}} } \right]. $  (12)

Thus, for a given ferroelectric material characterized by α, β, and γ, SS is minimized for N_A and t_FE given by Eqs. (11) and (12). It is clear from this equation that the thickness of ferroelectric material required depends on the landau parameter α of the material and the capacitance of the oxide between the ferroelectric and the silicon substrate. It is evident that high value of parameter α reduces the optimal ferroelectric thickness for a minimum subthreshold swing.

The anisotropy constants α and β in Landau equation are calculated by fitting the Q–V_FE characteristics to yield the given value of coercive field E_C and the remnant polarization P₀. Hence, and . Consequently, the equation for α and β becomes

$\alpha = \frac{{3\sqrt 3 }}{4}\frac{{{E_{\rm{c}}}}}{{{P_0}}},\;\;\beta = \frac{{3\sqrt 3 }}{8}\frac{{{E_{\rm{c}}}}}{{P_0^3}}. $  (13)

Therefore, the thickness of the ferroelectric film in terms of coercive field and remnant polarization can be obtained as

$ {t_{{\rm{FE}}}} = \frac{2}{{3\sqrt 3 }}\frac{{{P_0}}}{{{E_{\rm{c}}}}}\left( {\frac{1}{{{C_{{\rm{OX}}}}}} + \sqrt {\frac{{2{v_{\rm{t}}}2{\psi _{\rm{B}}}}}{{Q_1^2}}} } \right), $  (14)

the high coercive field and small remnant polarization results in small t_FE.

Ignoring the higher terms of Q in the Eq. (5), the ferroelectric negative capacitance equation reduces to

$ {C_{{\rm{FE}}}} = \frac{1}{{2\alpha {t_{{\rm{FE}}}}}} = - \frac{2}{{3\sqrt 3 }}\frac{{{P_0}}}{{{E_{\rm{C}}}{t_{{\rm{FE}}}}}}. $  (15)

In an NCFET device, the improved I_ON/I_OFF ratio and small subthreshold swing without hysteresis in the transfer characteristics depends on the capacitance matching (|C_FE| > C_MOS) or |C_FE| ≈ C_MOS as already discussed^{[12, 13]}. For the best results, |C_FE| should be as close as possible to C_MOS, or should be higher than the C_MOS. Based on the values of N_A and t_FE obtained for the material PZT, C_MOS and |C_FE| shown in Fig. 3(a), suggesting |C_FE(Q)| ≥ C_MOS(Q) at each Q, as in the numerical simulation for the same material in the MFIS capacitor^[18]. The corresponding V_FE and V_GMOS obtained from numerical simulations of MFMIS capacitor are shown in Fig. 3(b). It is clear that with the increase in Q, V_FE decreases and this in turn amplifies V_GMOS, and reduces SS below Boltzmann's limit of 60 mV/dec.

The higher values of C_FE due to smaller alpha is due to small E_C and large P₀ for fixed ferroelectric thickness, which results in lower voltage step up . Therefore, the gain reduces and consequently I_ON decreases. Although an increase in P₀ reduces the hysteresis due to positive total gate capacitance, it comes at the cost of reduction in I_ON/I_OFF which renders ineffective the primary advantage of using ferroelectric in gate stack^[19].

Similarly, high values of α implies a proportionally low value of |C_FE| for a fixed t_FE. This can also be understood in terms of very high coercive field and low remnant polarization. The higher values of αresults in the smaller |C_FE| than C_MOS, results in total negative capacitance which appears as the hysteresis in the transfer characteristics of the devices^[20]. Therefore, a high coercive field improves the I_ON/I_OFF ratio; however, hysteresis occurs in the characteristics due to instability.

Consequently, we come to the conclusion that in an NC capacitor using ferroelectric material, the material parameters and the thickness of the material are interdependent to achieving a good result.

Considering Eq. (14), a high P₀ and low E_C results in higher t_FE; whereas, a low P₀ and high E_C results in smaller t_FE. The calculated values of t_FE can be used to stabilise the capacitance and enhance the capacitance matching to the obtain the best possible results with high I_ON and no hysteresis. Therefore, we can opt for either higher polarization and smaller E_C or higher E_C with smaller P₀. However, from the design perspective of the NC-FET as experimentally shown in Ref. [20], the gate stack with ferroelectric material giving high E_C and low P₀ resulted in 45% higher I_ON with the same OFF current. Therefore, we can say that higher alpha and consequently lower t_FE results in better characteristics and are also viable from the point of view of scaling these devices.

Unlike S_min = 60 mV/dec for classical FETs, a single S_min does not define the performance of all NCFETs; instead the stability constrained S_min depends on the choice of material system and channel type. In this section, an explicit analytical equation has been obtained to calculate the minimum possible SS (S_min) under capacitance matching conditions. From the calculated S_min for different materials, we can infer the viability of different materials for the designing of NC-FETs.

For a given ferroelectric material characterised by α, β and γ subthreshold swing is minimized for N_A and t_FE given in Eqs. (11) and (12). The equation for minimum subthreshold swing (S_min) is obtained by substituting the value of C_FE in Eq. (2) and can be written as

$ {S_{{\rm{min}}}} \approx \frac{{2.3{k_{\rm{B}}}T}}{q}\left( {1 + M \times {t_{{\rm{FE}}}}} \right), $  (16)

where is a material dependent constant. The values of calculated optimal thicknesses and corresponding S_min for the five different ferroelectric materials viz: lead zirconium titanate (PZT), strontium barium titanate (SBT), barium titanate (BaTiO₃), hafnium silicate (HfSiO) and organic ferroelectric material P(VDF-TrFE) are given in Table 1. The non-ideal effects (such as domain propagation, grain boundaries and domain nucleation of the ferroelectric material) degrade the subthreshold swing of the realistic device and, therefore, the minimum subthreshold swing (S_min) in Eq. (16) can be obtained when we assume that the all these non-idealities are absent^[21]. The value of S_min for PZT material is 52.8 mV/dec using the above equation, which is consistent with the simulated value of 55 mV/dec for the same material by Ref. [23]. Furthermore, the value of 27 mV/dec for the organic ferroelectric material PVDF is in agreement with low SS values for such materials, as described in Ref. [24].

In this paper, we have developed an analytical equation for the ferroelectric thickness and the minimum subthreshold swing possible in an MFMIS capacitor. This captures the impact of ferroelectric material properties on the Landau coefficients, which in turn determines the thickness of the ferroelectric material required to have better capacitance matching and which reduces the subthreshold swing. Finally, this study calculated the minimum subthreshold swing of an NC capacitor with five different ferroelectric materials, which may help us in the selection of a particular ferroelectric material for an actual NC-FET design.