Drain Current Modeling of Tunnel FET using Simpson’s Rule

Tunnel Field Effect Transistor can be introduced as an emerging alternate to MOSFET which is energy efficient and can be used in low power applications. Due to the challenge involved in integration of band to band tunneling generation rate, the existing drain current models are inaccurate. A compact analytical model for simple tunnel FET and pnpn tunnel FET is proposed which is highly accurate. The numerical integration of tunneling generation rate in the tunneling region is performed using Simpson’s rule. Integration is done using both Simpson’s 1/3 rule and 3/8 rule and the models are validated against numerical device simulations. The models are compared with existing models and it is observed that the proposed models show excellent agreement with device simulations in the entire region of operation with Simpson’s 3/8 rule exhibiting the maximum accuracy.


Introduction
Tunnel Field Effect Transistor (TFET) has been identified as a viable alternate to MOSFET in the nano scale semiconductor device category [1][2][3][4]. Since the device operates on band to band tunneling (BTBT) [5,6] phenomena, it shows better resistance against short channel effects observed in MOSFET. The major advantage of TFET over MOSFET is that it provides a sub threshold slope lesser than the minimum achievable limit of 60 mV/decade for MOSFET. Though the device is ambipolar in nature, the OFF current is comparatively low which makes it popular among memory devices and the device is particularly suitable for low power applications. Developing an accurate and computationally efficient drain current model for TFET becomes important for performing fast and error free circuit simulations. The problem with the TFET device is that it has low Arun A V arunav.aav@gmail.com 1 ON current. The band-to-band tunneling rate has an exponential dependence [7] on the lateral electric field. Due to the low lateral electric field, the planar p-i-n TFET has low ON state current. TFET with n+ pocket between source and channel is generally known as pnpn TFET and exhibits significant improvement in ON state current. This is because the n+ pocket increases the electric field in the lateral direction, modulates the energy band profile, and shortens the tunneling width [8]. The other variants of TFET such as heterojunction TFET [9,10] has low ON current when compared with pnpn TFET due to low lateral electric field. Also, the high vertical electric field in these structures increases the interface generation rate. This deteriorates reliability of the device. Another promising device is nanosheet field effect transistor [11][12][13], but the fabrication complexity involved in these devices are challenging and more investigations should be carried out.
Several numerical models for TFET drain current [9,10,14,15] reported in literature are computationally complex and hence it is inefficient to do circuit simulations using these models. Investigations on various modeling approaches [16][17][18][19][20] point out the need to develop an accurate and computationally efficient analytical model for drain current of TFET. Integration of BTBT generation rate is a major challenge in drain current modeling due to the presence of exponential and polynomial terms. Tangent line approximation [21,22] is one of the preferred method for integration of BTBT generation rate. The tangent line approximation can be used to approximate functional values that deviate slightly from exact values. Another method is to perform integration by approximating the function as exponential and neglecting the polynomial term [18]. This paper reports a compact analytical drain current model for planar TFET which is accurate in the entire operating range. The model is extended to pnpn TFET structure and it shows commendable accuracy. Here, the drain current model is formulated by numerical integration of BTBT generation rate using Simpson's rule. Both 1/3 and 3/8 rule is employed for model derivation and the models are compared for its accuracy. BTBT generation rate is a function of electric field which is derived from surface potential. The model is validated against 2D numerical device simulation using Silvaco Atlas [23]. The device simulation tool is calibrated against published experimental data [19,20,[24][25][26][27][28][29][30][31]. A Cross section of p-channel SOI TFET used for simulation and validation is shown in Fig. 1.
The drain current model published in literature [20], due to its tangent line approximation of the parabolic function has limited accuracy. Simpson's rule approximates BTBT generation rate function as parabolic segments. Hence, the proposed method of drain current formulation demonstrates excellent match with the device simulations.
The paper is organized in the following manner. Section 2 discusses model development followed by model validation in Section 3. Section 4 concludes the work.
Drain current is formulated by the integration of generation rate over the entire volume. An accurate surface

Surface Potential and Electric Field
The surface potential of p-channel TFET with respect to distance along the channel is plotted in Fig. 2. From the plot it is observed that in region R 2 the potential is almost constant and is represented as ψ C . Since ψ C is constant in this region, the electric field in this region becomes negligible.
2D Poisson equation for the surface potential of TFET [20] is given by Applying parabolic approximation [29] of potential in y direction and substituting y=0 in Eq. 1 to obtain surface potential as where L dj is the characteristic length of jth region and in region R 1 , it is given by The gate potential ψ(G) can be expressed as where V FB is the flat band voltage. Applying boundary conditions, the final solution for the surface potential in R 1 [20] is and in R 2 The electric field is given by This field is applied to the tunneling generation rate to find the drain current.

Surface Potential in Source Body and Drain body Depletion Regions
The parabolic approximation adopted in region R 1 is valid for depletion regions. Considering the gate fringing field, boundary conditions relating to continuity of electric field varies in the depletion region. This is due to the modified gate body capacitance generally known as fringing capacitance. The gate body capacitance is modified by applying conformal mapping techniques [32], and is shown in Eq. 8 So the boundary condition changes as Applying this boundary condition in Region R 0 Si (10) where L d0 = πt Si t ox Si 2 ox (11) and where N j is the doping in j th region. Similar equations can be written in drain body depletion region.
In the source body depletion region, the boundary conditions in x direction are where V bi0 is the built-in voltage of the source body region.
where k is the Boltzmann constant, T is the temparature and q is the electron charge. Similarly, boundary conditions in x direction at drain body depletion region are where V bi2 is the built-in voltage of the drain body region.
At the source body and drain body interface, applying boundary continuity of surface potential and electric field displacement the surface potential is obtained as Substituting (10) into (17) and (18) (20) By applying diode approximation, the depletion region lengths are given by C 0 , C 2 , D 0 and D 2 are obtained by solving (19) and (20).

BTBT Generation Rate
Kane's band to band tunneling model [33] is derived with constant electric field applied to time independent Schrodinger equation. Later on Keldysh et al. [34] modified the model. The expression for tunneling per cubic centimeteris given by.
where E is the uniform electric field In both cases, the parameters A and B are linear and exponential parameters respectively and is given by, where m * is the effective mass of the carrier, h is the Planck's constant and E g represents the band gap energy.

Drain Current Model Using Simpson's Rule
The band to band tunneling generation rate in source body depletion region is given by Eq. 23 and is plotted in Fig. 3. The expression contains polynomial as well as exponential terms which limits direct integration. Extensive modeling approximation can be adopted in such cases, by eliminating the polynomial term, if the accuracy is not compromised. Here, if such approximations are made, the drain current model becomes highly inaccurate. So band to band tunneling generation rate is numerically integrated over the entire tunneling volume to obtain the drain current. The numerical method used here is Simpson's rule which is an extension of trapezoidal rule. Numerical integration is performed using both Simpson's 1/3 rule and 3/8 rule.

Model Using Simpson's 1/3 Rule
In this approach, the integrand is obtained by approximating it as second order polynomial. The drain current is given as The tunneling region along the channel is defined from x 0 to x 00 as shown in Fig. 3. The G BTB function is approximated to a second order polynomial and integrated over the tunneling region yields x 00 x 00 x 0 To evaluate the polynomial coefficients a 0 , a 1 and a 2 , choose three points in the x axis of the graph shown in Fig. 3. The three points are x 0 , x 00 and the mid point of x 0 and x 00 . x 00 is the point where BTBT tunneling generation rate approaches zero.
x 0 = L S − L 1 (36) where t inversion is the inversion layer thickness A is the BTBT parameter given by Eq. 23. This analytical model provides a closed form equation for drain current which is suitable for circuit simulations.

Model Using Simpson's 3/8 Rule
Simpson's 3/8 rule for integration is derived by approximating the given function with the third order (cubic) polynomial.
G BTB (x) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 (39) To evaluate the polynomial coefficients a 0 , a 1 , a 2 and a 3 , choose four points in the x axis of the graph shown in Fig. 3. The four points are x 0 , x 01 , x 02 , x 00 . Where x 01 and x 02 are given by Solving for a 0 , a 1 , a 2 and a 3 and substituting in G BTB (x), the drain current computed with Simpson's 3/8 rule is obtained as This analytical model provides a closed form equation for drain current.

Drain Current Model for pnpn TFET
In comparison with the p-i-n TFET, the pnpn TFET structure is more promising for low-power circuit design [35]. Using parabolic approximation the surface potential of the device [26] is found out to be where 1/u is the characteristic length and j = 2-4 is applicable for regions II−IV respectively. In region I the potential is given by Expression for x 0 remains the same as in f17Eq. 36 and the value of x 00 [26] changes to Electric field is obtained as the derivative of surface potential and is applied in generation rate. Drain current is modeled using Simpson's 1/3 and 3/8 rule by applying generation rate in Eqs. 35 and 46 respectively.

Model Validation
Even after the availability of an accurate surface potential model, it is difficult to obtain a drain current model by direct integration due to the reason specified in Section 2.
In this paper, a novel method of drain current formulation using a numerical integration method called Simpson's 1/3 rule and Simpson's 3/8 rule is proposed. To evaluate the suitability of the model, the proposed drain current model is compared with the device simulations. While performing the device simulations, the models used are concentration dependent mobility, electric field dependent mobility, Shockley-Read-Hall recombination, Auger recombination, bandgap narrowing and Kane's band-to-band tunneling. The constants in Kane's band-to-band tunneling model is fixed as A Kane = 4 × 10 19 and B Kane = 41 [23] so that they resemble experimental results [28]. The proposed models are also compared with the existing drain current models derived by applying tangent line approximation [20] and by neglecting the polynomial term in the band to band tunneling generation rate [18]. Figures 5 and 6 shows the validation of models with device simulation when V DS is −0.05V and −2V respectively. The proposed models show excellent agreement with the device simulations for the entire range of V GS . Model With Simpson's 3/8 rule (44) is slightly more accurate than that with Simpson's 1/3 rule (35). However, the computational time associated with Simpson's 3/8 rule, because  [20] and device simulation for P channel TFET with V GS = −2V of its third order polynomial approximation is significantly higher than the one associated with Simpson's 1/3 rule which uses only second order polynomial approximation. Figure 7 shows the output characteristics (I D -V DS ) for V GS = −2V. The proposed models has a slight error in the saturation region. This is due to the inaccuracy of surface potential model at high drain voltages.
A short-channel TFET with channel length of 20 nm is also used in the analysis and compared with the models in Fig. 8. The proposed models are in good agreement with the simulation results and hence the model is suitable upto a channel length of 20 nm.
In the existing method [20], drain current is calculated by integration of generation rate using tangent line approximation. Integration is done by dividing the tunneling generation rate shown in Fig. 3 into linear segments and calculating the area under the graph using triangular approximation. On the other hand, Simpson's rule approximates the graph with sequence of quadratic parabolic segments instead of straight lines. This makes the model more accurate which closely portrays the device behavior in the entire operating range.
In the existing model, the tangent line is drawn to the generation rate function y = G btb (x) at a particular point x = a and the value of y is then linearly approximated. Now the linear approximation to G btb (x) is written as If the possible error in x is x e , the possible error in y is given by y e = x e dy dx x=a (53) Fig. 8 Comparison of I D vs V GS given by the proposed model, existing model [20] and device simulation for P channel TFET with V DS = −0.05V with 20nm channel length Here x e depicts the error in surface potential and y e is the total error in drain current. This demonstrates the mismatch of the existing drain current model with the actual device behavior in saturation region as shown in Fig. 7. The model is also validated with different source and drain dopings as shown in Fig. 9. The proposed model is accurate with different doping concentrations. The proposed drain current models for pnpn TFET is validated against numerical device simulations. So far, no published models are there for the drain current of pnpn TFET. Figures 10 and 11 shows the I D vs V GS plot for V DS = 1 V and V DS = 0.2 V respectively. The plots show excellent match between the models and simulation results for different drain to source voltages.

Conclusion
In this paper, compact analytical models for the drain current of a planar TFET and pnpn TFET are reported. Investigations on the drain current modeling approaches indicate the need for an accurate method for integration of tunneling generation rate in the source body junction. The proposed modeling approach is based on integration of the tunneling generation rate by Simpson's rule. Both 1/3 and 3/8 rule are used for numerical integration of tunnelling generation rate function. The band to band tunneling generation function is approximated by sequence of quadratic parabolic segments in both the proposed models, whereas in the existing model with tangent line approximation, it is done by straight line segments. While developing the model the source side depletion region is also taken into account. The results demonstrate excellent agreement of the model with device simulations. The accuracy is proved in both ON state and subthreshold region for different device dimensions.