9-JUNCTIONLESS FIELD-EFFECT

SURFACE POTENTIAL MODELING OF JLFETs 335 where 0 is the potential at the center of the silicon film known as the central potential. Now, the electric field at the bottom surface (ES,b) can be obtained by using equation (8.20), boundary condition (2) and x = tSi∕2 in (8.19) as E2 S,b = 2qNDVt Si [ e (S−V) Vt − e (0−V) Vt −S − 0 Vt ] (8.21) Although equation (8.21) appears very simple, it has no closed form analytical solution. Therefore, we use approximations that are valid in different operating regimes of DGJLFET and obtain analytical solutions for equation (8.21). As discussed in Section 3.2, DGJLFETs operate mainly in three regions: (a) accumulation, (b) partial depletion, and (c) full depletion. First, let us consider the case of DGJLFET in accumulation. 8.2.1.A Case A: DGJLFET Operating in the Accumulation Region As discussed in Section 3.2, the electric field along the x-direction ceases to be zero at the flat band condition and begins to increase as the gate voltage is increased forcing the DGJLFETs into the accumulation regime. However, the electric field is still very low in the accumulation regime. The accumulation layer appears only at the surface, whereas the entire silicon film remains neutral. Therefore, the assumption of a constant potential (at least in the neutral regions) in the silicon film is valid. This constant potential can be approximated to be the electron quasi-Fermi potential V [33]. Approximating (x) ≈ V in the silicon film apart from the surface in equation (8.21), we have ES,b ≈ √ 2qNDVt Si [{e (S−V) Vt − 1 } −S − V Vt ] (8.22) Now, close to the flat band conditions, i.e. near accumulation, S > V. Therefore, the ratio e (S−V) Vt − 1 S−V Vt (8.23) is much higher than unity for the accumulation mode and becomes unity only at the flat band condition. Therefore, by neglecting the denominator term of equation (8.23) in equation (8.22), we obtain ES,b ≈ √ 2qNDVt Si [ e (S−V) Vt − 1 ] (8.24)

336 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS M O O + + M 0 y –a a x 2 – tSi Lg 2 tSi FIGURE 8.5 Depletion charges in the DGJLFET in the partially depletion operating mode. Now, using the boundary condition (4), i.e. equation (8.17) in equation (8.24), we have ( VG − VFB − S )2 = Vt [ e (S−V) Vt − 1 ] (8.25) where = ( 2qSiND ) ∕C2 ox is already defined in equation (8.12) Equation (8.25) relates the surface potential to the gate voltage in the accumulation regime. 8.2.1.B. Case B: DGJLFET Operating in the Partial Depletion Region In the partial depletion regime, a part of the silicon film thickness at the center is uncovered and remains neutral while the silicon film close to the surface remains depleted as shown in Fig. 8.5. As done in Section 8.2, we take the depletion approximation to solve the Poisson equation in the partial depletion regime. The charge density can be given as = { 0, 0 ≤ x ≤ a qND, a ≤ x ≤ tSi 2 (8.26) where a = ( tSi∕2 ) − xdep and xdep is the depletion region width close to the bottom gate. Therefore, the Poisson equation gets modified as d2(x) dx2 = { 0, 0 ≤ x ≤ a −qND Si , a ≤ x ≤ tSi 2 (8.27) Integrating equation (8.27) with respect to x and using the boundary condition (1), i.e. equation (8.15), we have E (x) = 0; 0 ≤ x ≤ a (8.28)

SURFACE POTENTIAL MODELING OF JLFETs 337 Integrating equation (8.28) with respect to x and using the boundary condition that (x) = 0 at x=0, we have (x) = 0; 0 ≤ x ≤ a (8.29) Now, integrating equation (8.27) with respect to x once for a ≤ x ≤ 0.5 tSi, we get E (x) = qNDx Si + C1, a ≤ x ≤ tSi 2 (8.30) Now, the electric field diminishes to zero at the boundary of the depletion region, i.e. at x=a, we obtain C1 = −qNDa Si (8.31) Integrating equation (8.30) with respect to x, we get (x) = −qND 2Si ( x2 − 2ax) + C2, a ≤ x ≤ tSi 2 (8.32) Now, the potential should be continuous ((x) = 0) at x=a. Using this boundary condition, we get C2 = 0 − qNDa2 2Si (8.33) The potential distribution in the silicon film for a ≤ x ≤ tSi∕2 can be obtained by using equation (8.33) in equation (8.32) as (x) = 0 − qND(x − a) 2 2Si (8.34) Since the center of the silicon film is neutral and undepleted, the central potential is equal to the electron quasi-Fermi potential 0 = V [33]. Using this, the potential at the bottom surface (x = tSi∕2) can be calculated as S,b = V − qNDx2 dep 2Si (8.35)

338 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS Also, the total charge density in the silicon film can be obtained by applying the Gauss’s law at the Si–SiO2 interface adjacent to the bottom gate as QSC = 2SiES,b = −2Cox(VG − VFB − s) (8.36) For DGJLFETs operating in partial depletion regime, QSC = 2qNDxdep, which relates xdep to the gate voltage as xdep = −Cox ( VG − VFB − s ) qND (8.37) Using the value of xdep in equation (8.35), we obtain a relationship between gate voltage and surface potential as S,b = V − ( VG − VFB − s )2 (8.38) which is same as that obtained for a single-gate SOI JLFET working in the partial depletion regime. Another interesting observation is that using the threshold conditions, i.e. xdep = tSi∕2 and VG = VTh, where VTh is the threshold voltage, we obtain the same result as obtained in Section 3.2.2 as VTh = VFB − qNDt 2 Si 8Si − qNDtSitox 2ox (8.39) 8.2.1.C. Case C: DGJLFET Operating in the Full Depletion (Subthreshold) Region The depletion approximation used earlier is not suitable for subthreshold condition where the depletion regions merge completely and overlap each other. Therefore, the deep depletion approximation, which provides accurate results for the subthreshold full depletion regime, should be used [33]. Under the full depletion operating regime (Fig. 8.6), integrating the Poisson equation (8.3) once, we have E (x) = qNDx Si + C3 (8.40) Now, due to the symmetric structure, E(0)=0, which implies C3 =0. Now, integrating equation (8.40) with respect to x, we have (x) = −qND 2Si x2 + C4 (8.41) At x = 0, (x) = 0. Using this in equation (8.41) yields C4 = 0.

SURFACE POTENTIAL MODELING OF JLFETs 339 M O + + O M 0 y x 2 – tSi Lg 2 tSi QD = qNDtSi FIGURE 8.6 Depletion charges in the DGJLFETs in the full depletion regime. At x = tSi 2 , (x) = S,b. Using this condition, we have a relation between the central potential and the surface potential in the full depletion mode: S,b = 0 − qNDt 2 Si 8Si (8.42) From equation (8.17) and equation (8.21), we have C2 ox( VG − VFB − S,b )2 = −2qNDSiVt [ e (S−V) Vt − e (0−V) Vt −S − 0 Vt ] (8.43) Now, to simplify, we define =0−S Vt , which yields: VG − VFB − S,b = −√ 2qSiNDVt C2 ox √ 1 − [ 1 − e− ] e (0−V) Vt (8.44) The value of can be obtained from (8.42) and used in equation (8.44) to give an analytical relation between the surface potential and the gate voltage. However, equation (8.44) is transcendental and needs to be simplified using mathematical techniques. In the full depletion mode, 0 > S,b. Therefore, is a positive quantity more than unity, which implies 1 − e− < 1 (8.45)

340 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS Expanding the square root on the RHS of equation (8.44) using the binomial theorem, we have VG − VFB − S,b = qSiNDVt 2C2 ox ⎡ ⎢ ⎢ ⎣ 1 − e (0 − V) Vt 2 ⎤ ⎥ ⎥ ⎦ (8.46) Equation (8.46) relates the surface potential to the gate voltage in the subthreshold region. An explicit expression for the surface potential can also be found by rearranging the terms in equation (8.46) as S,b = VG − VTh − qNDt 2 Si 8Si − VtW [ qNDtSi 4CoxVt e (VG − VTh − V) Vt ] (8.47) where W is the Lambert W function [34], which is the inverse of the function z = W (z) eW(z) . The Lambert W function is a well-known function used for circuit analysis of the bipolar junction transistor (BJT) [35]. The proposed model is in well agreement with the TCAD results for various ranges of silicon film doping, silicon film thickness, and gate oxide thickness as shown in Fig. 8.7. In this section, we presented analytical expressions for the surface potential of DGJLFETs utilizing suitable approximations in different operating regions. However, a regionwise approach may cause a convergence problem in circuit simulators. Therefore, a continuous model covering all the operating regions is desired, which will be discussed in the next section. 8.2.2 Parabolic Approximation Technique The parabolic approximation technique or the pseudo-2D method is widely used for surface potential modeling in MOSFETs and tunnel FETs [30, 31, 36–40]. The basis of this method can be understood from Fig. 8.8. The energy band profile (−e) and, hence, the potential variation along any point along the channel length direction (y-axis) is monotonous in the direction along the channel thickness (x-direction). However, the nature of the potential distribution differs as we move along the channel length. Therefore, the potential distribution can be approximated as a quadratic polynomial in x as [36] (x, y) = a0 (y) + a1 (y) x + a2 (y) x2 (8.48) The reason for limiting the order of polynomial to 2 is that utilizing the available boundary conditions for double-gate metal-oxide-semiconductor field-effect transistor (DGMOSFETs), only a quadratic polynomial may be solved. Therefore, higher order terms are ignored [36]. By using such an approximation, the original 2D Poisson equation reduces to a second-order linear differential equation in a single dimension, which may be solved easily.

SURFACE POTENTIAL MODELING OF JLFETs 341 (a) 0.0 –0.2 –0.4 Symbol: Sentaurus Line: Model Symbol: Sentaurus Line: Model Symbol: Sentaurus Line: Model –0.6 φs (V) –0.8 (b) 0.0 –0.2 –0.4 –0.6 φs (V) –0.8 (c) 0.0 –0.2 –0.4 –0.6 φs (V) –0.8 –2.0 –1.5 –1.0 –0.5 0.0 VG (V) V = 0 V 0.5 tox = 2 nm, 5 nm, 8 nm tSi = 8 nm, 10 nm, 12 nm ND = 1019 cm–3, 1.1 × 1019 cm–3, 1.2 × 1019 cm–3 ND = 1019 cm–3 tSi = 10 nm V = 0 V ND = 1019 cm–3 tox = 8 nm V = 0 V tSi = 10 nm tox = 8 nm 1.0 1.5 2.0 FIGURE 8.7 Comparison between the surface model obtained in [33] with the simulation data for different (a) silicon film doping, (b) silicon thickness, and (c) gate oxide thickness. Utilizing the boundary conditions (equations 8.15–8.17) and the coordinate system used for the operating regionwise approximation in Section 8.2.1, we will try to solve equation (8.48) and obtain a relationship between the surface potential and the gate voltage, which is continuous and valid for all operating regions of interest [42]. Utilizing the first boundary condition, i.e. the electric field at the center is 0 [E (0, y) = 0] in equation (8.48), we obtain, a1 (y) = 0. Now, equation (8.48) reduces to (x, y) = a0 (y) + a2 (y) x2 (8.49) Using the boundary condition (2) that the potential at the center is 0, i.e. [ (0, y) = 0 ] , we obtain a0 (y) = 0, where 0 is the central potential. Now,

342 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS 0.5 0 –0.5 0 –0.5 –1 –1.5 –2 –1 0 (c) (d) 2 (a) (b) 4 6 Full depletion Flat band Threshold voltage Accumulation mode EC EC EC EC EV EV EV EV EFi EFi EFi EFi EF EF EF EF Location along film thickness (nm) Energy (eV) Energy (eV) 0 –0.5 –1 –1.5 –2 Energy (eV) 0.5 0 –0.5 –1.5 –1 Energy (eV) 8 10 0246 Location along film thickness (nm) 8 10 0 2 4 6 Location along film thickness (nm) 8 10 0246 Location along film thickness (nm) 8 10 φs φ0 FIGURE 8.8 Energy band profiles along the cutline B–B’ (Fig. 8.1) in different operating modes of JLFETs. utilizing the third boundary condition, i.e. the potential at the surface is the surface potential [ ( tSi∕2, y ) = S,b ] , we obtain a2 (y) = ( S,b − 0 ) 4 t 2 Si (8.50) Therefore, the potential distribution can be expressed as (x, y) = 0 + ( S,b − 0 ) 4x2 t 2 Si (8.51) Now, using the boundary condition that the electric displacement vector at the bottom Si–SiO2 interface should be continuous, we have oxEox = SiESi, which can be elaborated using equation (8.17) as − Cox(VG − VFB − S,b) = −Si d (tSi 2 , y ) dx = 4SiΔ tSi (8.52)

SURFACE POTENTIAL MODELING OF JLFETs 343 where Δ = ( 0 − S,b ) . Although equation (8.52) relates the gate voltage to the surface potential, the central potential is an unknown term. Therefore, another relationship between S,b and 0 is required. For this, we can relate the charge density in the bottom half silicon film using Gauss’s law at the bottom Si–SiO2 interface and utilize the symmetric conditions in DGJLFET to find the total charge density as QSC = −2Si d (tSi 2 , y ) dx = 8SiΔ tSi (8.53) Now, the total charge density in the silicon film consists of both mobile electrons and the depletion charge, i.e. = qND [ e (−V) Vt − 1 ] (8.54) Integrating equation (8.54) throughout the silicon film with respect to x, we get the total charge density inside the silicon film as QSc = qNDtSi − qND ∫ tSi∕2 −tSi∕2 e ((x,y)−V) Vt dx (8.55) Replacing (x, y) by equation (8.51), we have QSc = qNDtSi − qNDe (0−V) Vt ∫ tSi∕2 −tSi∕2 e − ( 4Δ t 2 SiVt ) x2 dx (8.56) The integral on the right side of equation (8.56) takes the form of the error function which can be solved using √ c ∫ q p e−Cx2 dx = [ erf ( q √c ) − erf ( p √c )] 2 (8.57) Using the above integral, we obtain total charge density in the silicon film as QSc = qNDtSi ⎡ ⎢ ⎢ ⎣ 1 − e (0−V) Vt 2 √ Vt Δ { erf (√ Δ Vt )}⎤ ⎥ ⎥ ⎦ (8.58) Although equation (8.58) relates the total charge density inside the silicon film to the surface and central potential, the obtained expression is not analytical. However, if we carefully observe, the difference between the surface potential and the central

344 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS 0.2 0.0 –0.2 –0.4 –0.6 –0.8 –6 –4 –2 0 Position in silicon, X (nm) Simulation p+ Poly Gate, VDS = 0 V, NSi = 1 × 1019cm–3, tox = 2 nm VGS = 1 V, 0.75 V, 0.5 V, 0.25 V, 0 V, –0.25 V Model Electric potential (V) 246 FIGURE 8.9 Comparison of the potential in the channel region along the channel thickness for different gate voltages calculated utilizing the analytical model [42] and the simulations. potential Δ is more than the thermal voltage apart from the flat band and weak accumulation conditions [42]. Therefore, the error function term in equation (8.58) tends to unity and equation (8.58) can be simplified into an analytical relation. Now, using equation (8.58), a second relationship between the surface potential and the central potential may be found as 8SiΔ tSi = qNDtSi ⎡ ⎢ ⎢ ⎣ 1 − e (0−V) Vt 2 √ Vt Δ ⎤ ⎥ ⎥ ⎦ (8.59) Equations (8.52) and (8.59) can be solved in a self-consistent manner to obtain the surface potential in terms of the gate voltage. A numerical example calculated using the above equations is shown in Fig. 8.9 and compared with simulation results obtained using a commercial TCAD silvaco Atlas. The model shows a good agreement with the simulations and provides a continuous model for relating surface potential to the applied gate voltage [42]. However, it may be noted that the assumption of a parabolic potential profile is true when the JLFETs operate in the full depletion or partial depletion mode as shown in Fig. 8.8. Close to the flat band condition, in the weak accumulation regime, the potential profile does not resemble a parabola. However, the JLFETs are generally designed to operate in the flat band condition only once the ON-state is reached. Therefore, even the parabolic approximation works well for modeling JLFETs.

SURFACE POTENTIAL MODELING OF JLFETs 345 8.2.3 Initial Guess technique As discussed in Section 8.2.1, although the Poisson equation for JLFETs appears simple, it has got no closed form analytical solution unlike the undoped MOSFETs [43–46]. We may find an analytical relationship between the gate voltage and the surface potential if we solve the Poisson equation for undoped DGMOSFETs and use it as an initial guess [47]. In the next iteration, the Poisson equation for the DGJLFETs may be updated based on the initial guess. The surface potential model can then be developed utilizing the boundary conditions. We begin our analysis from the simple Poisson equation for DGJLFETs, which we formulated as equation (8.14). Now, in an undoped DGMOSFET, the depletion charge density can be neglected. Therefore, by including only the mobile electron charge density in equation (8.14), we obtain the Poisson equation for DGMOSFETs as d2(x) dx2 = − Si = qnie (−V) Vt Si (8.60) where ni is the intrinsic carrier concentration. Note that we ignored the hole contribution as done in Section 8.2.1. Using the similar steps as done in Section 8.2.1, we obtain the expression for the electric field by the following equation: 2 d dx (d2(x) dx2 ) = d(E(x) 2) dx = 2 qni e (−V) Vt Si d dx (8.61) Integrating equation (8.61), we get d dx = √ 2nikT Si e (−V) Vt + C (8.62) where C is the constant of integration. Rearranging equation (8.62), we get d √ 2nikT Si e (−V) Vt + C = dx (8.63) Integrating both sides, we may get an expression for the potential as a function of x. The integral on the LHS can be performed using the substitution method. We may use the integration constant C in two ways, yielding two different expressions. We can rearrange equation (8.63) as shown below: d √2nikT Si √[ e (−V) Vt + C1 ] = dx (8.64)

346 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS where C1 is another constant. If we substitute e(−V)∕2Vt = u and perform the integration of equation (8.64) using the appropriate boundary conditions, we arrive at the standard expressions given in [43]. However, for finding the initial guess, we should not use any boundary condition as the initial guess is only an approximate solution to the Poisson equation and not the final solution. Therefore, we use a different approach without changing the constant C and rearrange equation (8.63) as d 2Vt √ q2ni 2SikT e (−V) Vt + C′ = dx (8.65) √ where Vt = kT∕q and C′ is yet another constant. Now, substituting ( q2ni ) ∕ ( 2SikT) e(−V)∕2Vt = u, we have, ( 2Vtdu) ∕u = d. Substituting this in equation (8.65), we get du u √ u2 + C′ = dx (8.66) Now, using C′ = −( 2∕tSi)2 , where is a constant, and substituting y = u(tSi∕2), we obtain dy y √y2 − 1 = 2 tSi dx (8.67) Integrating both sides, we get, sec−1y = 2x∕tSi + k, where k is a constant. Substituting the value of y and u in integral of equation (8.67), we get (x) = V − 2Vt ln ⎡ ⎢ ⎢ ⎣ tSi 2 √ q2ni 2SikT cos (2x tSi + k )⎤ ⎥ ⎥ ⎦ (8.68) Now, utilizing the boundary condition d∕dx = 0 at x = 0, we get tan (k) = 0 yielding k = 0. Therefore, for an undoped DGMOSFET, we can solve analytically to obtain a closed form solution of the potential as [44, 45] (x) = V − 2Vtln ⎡ ⎢ ⎢ ⎣ tSi 2 √ q2ni 2SikT cos (2x tSi )⎤ ⎥ ⎥ ⎦ (8.69) Now, the Poisson equation for the DGJLFETs can also be written as d2(x) dx2 = − Si = q [ ni e (−V) Vt ] Si − qND Si (8.70)

SURFACE POTENTIAL MODELING OF JLFETs 347 Therefore, we may use the solution for the first term in the RHS of equation (8.70) as an initial guess to find an approximate solution. Differentiating equation (8.69) twice with respect to x and substituting into equation (8.70), we get d2(x) dx2 = q Si [ 82SikT q2t 2 Si sec2 (2x tSi ) − ND ] (8.71) Now, integrating equation (8.71) once with respect to x, we get E (x) = −d dx = −[ q Si {4SikT q2tSi tan (2x tSi ) − NDx } + C1 ] (8.72) where C1 is the constant of integration. Applying the boundary condition due to symmetry in DGJLFETs that E (x) = 0 at x = 0, we get C1 =0. Integrating equation (8.72) again with respect to x, we obtain (x) = q Si [ −2SikT q2 ln { cos (2x tSi )} − NDx2 2 ] + C2 (8.73) where C2 is another constant, which can be evaluated by using the boundary condition that the potential at the center (x) = 0 at x = 0, yielding C2 = 0. Now, utilizing the boundary condition that the electric displacement vector must be continuous at the bottom Si–SiO2 interface, i.e. equation (8.17), gives the required relationship between the gate voltage and the surface potential in terms of the design parameters silicon film doping (ND), silicon film thickness (tSi), and the integration constant obtained from the solution of the Poisson equation for DGMOSFETs as Cox ( VG − VFB − s ) = [ 4SikT qtSi tan () − qNDtSi 2 ] (8.74) However, we need to find an explicit expression for to relate the gate voltage and the surface potential using equation (8.74). For this, we use the Gauss’s law, which gives the charge density (QSC) inside the silicon film as QSC = 2SiES,b (8.75) where ES,b is the electric field at the bottom Si–SiO2 interface. Now, the charge density can be expressed as the sum of mobile charge density (Qm) and the depletion charge (QD) as QSC = Qm + QD, where Qm = −2 ∫ tSi∕2 0 ni e ((x)−V) Vt dx (8.76) QD = 2NDtSi (8.77)

348 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS Using equations (8.75) –(8.77), we obtain a relationship between the integral constant and the electron quasi-Fermi potential V as V = −Vt ln ⎡ ⎢ ⎢ ⎣ 4SikT tan () tSiq2 ∫ tSi∕2 0 ni e (x) Vt dx ⎤ ⎥ ⎥ ⎦ (8.78) The value of the integral constant can be obtained from equation (8.78) and used in equation (8.74) to yield a surface potential model for DGJLFETs [47]. 8.2.4 Finite Difference Approach The parabolic approximation method, discussed in Section 8.2.2, assumes that the potential profile along the direction of channel thickness (x-axis) at any point along the channel length (y-axis) is parabolic with some arbitrary coefficients, which can be obtained utilizing the boundary conditions. Although this assumption is true when the JLFETs operate in the full depletion or partial depletion mode, close to the flat band condition, in the weak accumulation regime, the potential profile does not resemble a parabola. Hence, the parabolic approximation is not valid in this regime. Also, in the JLFETs, the channel thickness needs to be ultrathin to achieve volume depletion. Therefore, we may follow a discretization step and use the well-known finite difference technique for modeling JLFETs. This discretization step not only relaxes any preassumption of the potential profile but also provides a computationally efficient model [48, 49]. The modeling approach utilizing the finite difference method uses three discrete points as shown in Fig. 8.10: first at the top Si–SiO2 interface, second at the center of the channel region, and third at the bottom Si–SiO2 interface to solve the Poisson 0.5 0 –0.5 –1 0 2 4 6 8 10 Location along film thickness (nm) Energy (eV) Full depletion EV EC EF EFi φs φ0 FIGURE 8.10 The points considered for the solution of the Poisson equation using the finite difference method.

SURFACE POTENTIAL MODELING OF JLFETs 349 equation along the channel thickness. For the case of JLFETs, it is a valid approximation since the points are located only 5 nm apart for a channel thickness of 10 nm, which is typically used while designing JLFETs. As the channel thickness reduces, the discretization step tends to be more accurate. The Poisson equation for the DGJLFETs, i.e. equation (8.70), may be expressed as d2(x) dx2 = − Si = qni [ e (−V) Vt − ND ni ] Si (8.79) Following the same approach as in Sections 8.2.1 and 8.2.3, i.e. multiplying both sides by 2 (d∕dx), we have d dx(d dx )2 = d dx E(x) 2 = 2qni [ e (−V) Vt − ND ni ] Si d dx (8.80) Now, integrating equation (8.80) from the center of the channel with E (0) = 0 and (0) = 0 to the bottom Si–SiO2 surface with E ( tSi∕2 ) = ES,b and ( tSi∕2 ) = S, we have ∫ ES,b 0 d dx E(x) 2 = ∫ S 0 2qni [ e (−V) Vt − ND ni ] Si d dx (8.81) Solving, we get E2 S,b = 2qniVt Si [ e (S − V) Vt − e (0−V) Vt − ND ni ( S − 0 ) Vt ] (8.82) Now, using the finite-difference method along the three points discussed earlier, the first derivative of the potential can be expressed as d(x = 0) dx = ( −tSi 2 ) − (tSi 2 ) tSi =S − S tSi = 0 (8.83) which is also evident from the boundary condition that the electric field at the center is zero due to the symmetric operation in DGJLFET. Now, the second derivative of

350 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS the potential using the finite difference method along the three chosen points can be obtained as d2(x = 0) dx2 = 1 tSi 2 ⎡ ⎢ ⎢ ⎢ ⎣ ( −tSi 2 ) − (0) tSi 2 − (0) − (tSi 2 ) tSi 2 ⎤ ⎥ ⎥ ⎥ ⎦ (8.84) Solving equation (8.84) and using equation (8.79), we get S − 0 = qt2 Si 8Si [ ni e (0−V) Vt − ND ] (8.85) Equation (8.85) relates the surface potential to the central potential without any assumption about the shape of the potential profile. Moreover, ignoring the contribution due to the mobile electrons, it converges to the results obtained using depletion approximation for the JLFETs operating in the full depletion mode in Section 8.2.1. Equation (8.85) alone cannot be used to find a relationship between the gate voltage and the surface potential. We need another expression relating the gate voltage to the central or the surface potential for a self-consistent solution. Following the approach in Section 8.2.1–8.2.3, we use the boundary condition that the electric displacement vector at the bottom Si–SiO2 interface is continuous and the Gauss’s law as SiES,b = QSC 2 = −Cox ( VG − VFB − s ) (8.86) Using equation (8.82) in equation (8.86), we get the following expression relating the surface potential, central potential, and the gate voltage: Cox ( VG − VFB − s ) = − √√√√2qSiniVt [ e (S − V) Vt − e (0 − V) Vt − ND ni ( S − 0 ) Vt ] (8.87) Equations (8.85) and (8.87) can be solved in a self-consistent way to get the surface potential model for DGJLFETs. At this juncture, we point out that equation (8.85) could have been directly derived by using the Maclaurin series expansion of the potential about the center as (x) = (0) + d(x = 0) dx x + d2(x = 0) dx2 x2 + d3(x = 0) dx3 x3 + … (8.88)

CHARGE-BASED MODELING APPROACH 351 Because of the symmetric structure of DGJLFETs, the odd power derivatives would cease to be zero. Putting the values from equation (8.79), we get (x) = 0 + qni [ e (0−V) Vt − ND ni ] Si x2 (8.89) Substituting x = tSi∕2 in equation (8.89), we obtain equation (8.85) without using any sophisticated technique. 8.3 CHARGE-BASED MODELING APPROACH Until now, we have discussed about the surface potential modeling approach for the JLFETs. There is yet another modeling approach, which relates the charges in the semiconductor film directly to the gate voltage, known as the charge-based model. The charge-based modeling approach is more practical compared to the surface potential–based models as it is indeed the charge density, which determines the drain current and the behavior of the FETs [25–29]. However, obtaining an analytical and computationally efficient charge-based model is a challenge. Therefore, charge-based models are not commonly used in circuit simulators. Fortunately, computationally efficient charge-based models have been derived for JLFETs like the Ecole poly- ´ technique fed´ erale de Lausanne (EPFL) charge-based model (EPFL JL-1.0) [25]. We ´ discuss the charge-based modeling approach in detail in this section. Throughout Sections 8.2.1–8.2.4, we have related the charge in the semiconductor to the gate voltage and the surface potential by utilizing the Gauss’s law as equation (8.86). However, we need to eliminate the surface potential term from equation (8.86) to get an explicit relationship between the charge density in the silicon film and the gate voltage to get the charge-based model for DGJLFETs [48]. For this, we proceed by using equation (8.82) in equation (8.86) which relates the charge density to the gate voltage, surface potential, and the surface potential as QSC = 2 sign ( 0 − S ) √√√√2qSiniVt [ e (S − V) Vt − e (0 − V) Vt − ND ni ( S − 0 ) Vt ] = −2Cox ( VG − VFB − s ) (8.90) The signum function has been used in equation (8.90) to determine the exact nature of the channel charge as the operation mode changes from full depletion to accumulation. In the full depletion and partial depletion mode, 0 > S, the total charge density is dominated by the depletion charges and the semiconductor charge density is positive. However above flat band conditions, S > 0, the mobile electrons dominate the total charge density and the net charge density becomes negative. However, at the flat band condition, the net charge density inside the semiconductor is zero and

352 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS 4 × 10–2 3 × 10–2 2 × 10–2 1 × 10–2 –1 × 10–2 –2 × 10–2 –3 × 10–2 –4 × 10–2 0.3 0.35 0.4 TSi = 20 nm QSC (C/m2) ND: 1017 cm–3 ND: 1018 cm–3 ND: 5 × 1018 cm–3 ND: 1019 cm–3 0.45 Electric potential at the centre (V) 0.5 0.55 0 φFB φFB φFB FIGURE 8.11 The charge density as a function of the central potential for different doping concentrations obtained by numerical solution of equation (8.79) [48]. the central potential is equal to the surface potential, which may be obtained by using ( d2(x) ) ∕dx2 = 0 in equation (8.79) yielding 0 = S = FB = V + VTln (ND ni ) (8.91) Now, the numerical simulations of equation (8.79) shown in Fig. 8.11 indicate that the central potential remains stuck at the flat band potential obtained by equation (8.91) even when the DGJLFETs are biased in the accumulation regime [48]. This asymptote in the central potential is attributed to the shielding of the electric field by the surface-accumulated electrons. Therefore, to relate the charge density to the gate voltage in the accumulation mode, we may use the value of the central potential obtained from equation (8.91) in equation (8.90) as QSC (accu) ≈ −2 √2qSiniVt √√√√√√√√√√√ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ e (S − V) Vt − ND ni ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ 1 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ − ND ni {( S − V ) Vt − ln (ND ni )} ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (8.92)

CHARGE-BASED MODELING APPROACH 353 We can clearly observe in equation (8.92) that the ratio of the terms denoted by 1 to that of the terms denoted by 2 is larger than unity for the accumulation mode of operation. Therefore, we can omit the term denoted by 2 and approximate equation (8.92) as QSC (accu) ≈ −2 √2qSiniVt √ e (S − V) Vt − ND ni (8.93) Now, using equation (8.93), we obtain a relationship between the surface potential and the charge density as S ≈ V + Vtln ND ni ( 1 + Q2 SC 8SiqNDVt ) (8.94) Utilizing the value of surface potential obtained from equation (8.94) in equation (8.86), we obtain an analytical expression relating the charge density in the accumulation mode of operation to the gate voltage: VG − VFB − V ≈ − QSC 2Cox + Vtln ND ni ( 1 + Q2 SC 8SiqNDVt ) (8.95) In general, the charge density can be expressed as a sum of the mobile charge density and the depletion charge density, i.e. QSC = Qm + QD = Qm + qNDtSi. The calculation of mobile charge density is also essential as it can be directly used to get an analytical expression for the drain current using the Pao–Sah integral, which we discuss in Section 8.4.2. A general approach may also be utilized for finding a charge-based model for JLFET. Equation (8.90) may be reexpressed after defining C = e(0 − V)∕Vt and K = ( qnit 2 Si ) ∕8VtSi as VG − VFB − V = Vt lnC + KVt ( C − ND ni ) − 2 sign ( 0 − S ) Cox × √2qSiniVt √√√√√√√ C ⎡ ⎢ ⎢ ⎢ ⎣ e K ( C−ND ni ) ⏟⏞⏞⏟⏞⏞⏟ 1 −KND ni − 1 ⎤ ⎥ ⎥ ⎥ ⎦ + K (ND ni )2 (8.96) Equation (8.96) may be used to calculate C using standard numerical techniques once the gate voltage and the central potential are known [48]. The mobile charge density may be known once C is obtained from equation (8.96). Combining

354 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS equations (8.85) and (8.86), the generalized charge-based model for DGJLFETs may be obtained as QSC = −2Cox(VG − VFB − V + KVtND ni − KVtC − Vt lnC) (8.97) Although equation (8.97) represents a generalized charge-based model for DGJLFETs, the numerical approach of calculating C should be avoided for compact models used in the circuit simulators. However, using valid approximations, we obtained an analytical relation for charge density in the accumulation regime [48]. We extend this approach to find the charge-based model for DGJLFETs in the depletion mode of operation. In the depletion regime, the central potential is larger than the surface potential and the mobile charge density is lower than the depletion charge density. This implies that the exponential term in equation (8.96) indicated by 1 is less than unity [48]. Assuming C = ND∕ni, which appears like the flat band relationship results in the best possible case for the exponential term being unity. Under this assumption, the charge density can be expressed as QSC (dep) = 2 √2qSiniVt √ KND ni (ND ni − C ) (8.98) From equation (8.98), C may be extracted as C (dep) = ND ni [ 1 − ( QSC qNDtSi )2 ] (8.99) Putting the value of C obtained from equation (8.99) in equation (8.96), we get the charge-based model for the DGJLFETs even in the depletion regime as VG − VFB − V = Vt ln ND ni [ 1 − ( QSC qNDtSi )2 ] − Q2 SC 8qNDSi − QSC 2Cox (8.100) The results obtained by the approximations based on the operating regions shows a good match with the TCAD simulations (Fig. 8.12.), indicating the accuracy of the proposed charge-based model [48]. It is also possible to find an analytical expression for the mobile charge density in the DGJLFETs utilizing the parabolic approximation method [42] used in Section 8.2.2. From equation (8.52), we have Δ = ( Qm + qNDtSi) tSi 8Si (8.101)

DRAIN CURRENT MODELING APPROACH 355 10–1 10–2 10–3 10–4 10–5 10–6 10–7 –3 –2 TSi = 10 nm ND (cm–3): 1018, 1019, 3 × 1019, 5 × 1019 –1 0 VG – Δϕ(V) 1 2 0 5 × 10–2 1 × 10–1 Mobile charge density (abs. val. C/m2Mobile charge density (abs. val. C/m ) 2) 1.5 × 10–1 FIGURE 8.12 Mobile charge density as a function of effective gate voltage, i.e. gate voltage – work function of the gate electrode, for different silicon film doping. The TCAD simulations are shown with dotted lines, whereas results obtained by the model are shown as a solid line [48]. Replacing the value of Δ in equation (8.59) and utilizing equation (8.52), after rearrangement, we obtain a closed form relationship between the mobile charge density and the gate voltage as VG − VTh − V = −′ 2Cox Qm + Vt ln [ −Qm √ qNDtSi + Qm 2SiVtq2N2 DtSi ] (8.102) where ′ = 1 + CoxtSi 4Si and VTh is the threshold voltage given by equation (8.39). 8.4 DRAIN CURRENT MODELING APPROACH As discussed in Section 8.1, the main motivation for developing surface potentialbased or charge-based models is to get more physical insight into the working of the FETs. These models can be used to obtain the drain current of the FETs for analyzing its static behavior such as the transfer characteristics, the output characteristics, subthreshold swing, DIBL, and so on. In this section, we discuss the different approaches for modeling the drain current in DGJLFETs utilizing the insights gained

356 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS from Sections 8.2.1–8.2.4. First, we take the simplest bulk current model, which utilizes Ohm’s law to calculate the drain current in Section 8.4.1. Then, we discuss the classical work of Pao–Sah, which allows us to calculate the drain current accurately considering both drift and diffusion components once the mobile charge density is known. 8.4.1 Bulk Current Model In DGJLFETs (Fig. 8.13), above the threshold voltage, the current flows through the center of the channel. Therefore, we may calculate the drain current by simply using Ohm’s law, i.e. dV = IdR, where dR is the differential channel resistance, which can be expressed as dR =dl A = dy W2 (tSi 2 − xdep) = dy 2qNDnW (tSi 2 − xdep) (8.103) where is the sheet resistivity and is the sheet conductivity given by = qNDn. However, the depletion region width xdep is an unknown in equation (8.103). However, xdep can be easily calculated from our analysis in Section 8.2 utilizing the depletion approximation. By combining equations (8.35) and (8.36), we obtain a quadratic equation in xdep as qNDxdep = Cox ( VG − VFB − V + qNDx2 dep 2Si ) (8.104) VS VG VG VD tox tSi Source Channel Drain x (0,0) y FIGURE 8.13 The DGJLFET with a modified coordinate system used for the bulk current modeling in [50].

DRAIN CURRENT MODELING APPROACH 357 Solving equation (8.104), we get a simple expression for the depletion region width as xdep ( VG, V ) =Si Cox ⎡ ⎢ ⎢ ⎣ −1 + √ 1 − 2C2 ox qNDSi ( VG − VFB − V ) ⎤ ⎥ ⎥ ⎦ (8.105) Now, to get a simplified expression for the depletion region width, we expand xdep ( VG, V ) using the Taylor series around the threshold voltage VTh (where xdep = tSi∕2) given by equation (8.39), which yields xdep ( VG, V ) = xdep ( VTh, V ) + dxdep ( VTh, V ) dVG ( VG − VTh) + … (8.106) Ignoring the higher order terms, we obtain xdep ( VG, V ) ≈ − Ceq qND ( VG − VFB − V ) + tSi 2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 − Cox 2 + Cdep Cox + Cdep ⏟⏞⏞⏞⏟⏞⏞⏞⏟ 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (8.107) where Cdep is the half depletion capacitance given by Cdep = ( 2Si) ∕tSi and Ceq is the net series capacitance of Cox and Cdep given as Ceq = CdepCox Cdep +Cox . Equation (8.107) can be further simplified by assuming that the bracketed term designated by 1 is unity for DGJLFETs, which are essentially designed with a ultrathin channel to achieve volume depletion such that tSi∕2 is comparable to the tox [50]. Using this approximation, equation (8.107) simplifies to xdep ( VG, V ) ≈ − Ceq qND ( VG − VFB − V ) (8.108) Using equation (8.108), Ohm’s law expressed as dV = IdR can be integrated utilizing the gradual channel approximation to yield: I ∫ Lg 0 dy = ∫ VDS 0 2qNDnW (tSi 2 + Ceq qND ( VG − VFB − V ) ) dV (8.109) which gives a closed form analytical expression for the drain current in the partial depletion regime as I = 2qNDnW Lg [(tSi 2 + Ceq qND ( VG − VFB) ) VDS − Ceq 2qND V2 DS] (8.110)

358 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS Equation (8.110) indicates that the JLFETs and MOSFETs behave in a similar way with a linear current dependence on VDS in the triode region, where VDS is small. The saturation current can also be found from equation (8.110) by using the saturation voltage given as VDS,sat = VGS − VTh. Once analytical expression for the current in linear and saturation regimes is found, the next step is to find an analytical expression for the drain current in the subthreshold regime to obtain a complete drain current model. Since the current in the subthreshold regime is dictated by the diffusion of mobile carriers from the source to the drain, the drain current in subthreshold may be expressed as the diffusion current [51]: I = qWDn dn (y) dy = qWDn [ n(y = 0) − n(y = Lg) ] Lg (8.111) where n(y=0) and n(y=Lg) are the areal electron density at the source end and the drain end, respectively. Since the electrons at the drain end encounter an additional potential barrier equal to the drain voltage as compared to the electrons at the source, we have, n(y = Lg) = n(y = 0)e −VDS Vt . Using this in equation (8.111), we have I = qWDnn (y = 0) [ 1 − e −VDS Vt ] Lg (8.112) The areal electron density at the source end may be written as n (y = 0) = ND ∫ tSi 0 e (x) Vt dx (8.113) Now, in the coordinate system defined in Fig. 8.13 and used in [50], in the subthreshold mode of operation, using the depletion approximation, the Poisson equation may be integrated with the condition that electric field is zero at x = tsi∕2 to yield d(x) dx = −qND Si ( x − tSi 2 ) (8.114) Integrating equation (8.114) and using the boundary condition that (x = 0) = S, we obtain (x) = −qND 2Si ( x − tSi 2 )2 + qNDt 2 Si 8Si + S (8.115) Also, using the Gauss’s law, we have, 2Cox ( VG − VFB − S ) = QSC = qNDtSi.

DRAIN CURRENT MODELING APPROACH 359 Using equations (8.86) and (8.115) in equation (8.113) and defining ′ = 2ND e ⎡ ⎢ ⎢ ⎢ ⎣ qNDt 2 Si 8Si + VG − VFB + qNDtSi 2Cox ⎤ ⎥ ⎥ ⎥ ⎦ ∕Vt (8.116) and ′′ = qND 2SiVt , (8.117) we get n (y = 0) = ′ ∫ tSi∕2 0 e−′′x2 dx =′ 2 √ ′′ erf (√′′tSi 2 ) (8.118) Although the expression obtained for the areal density at the source end is not analytic, the channel region is highly doped in DGJLFETs, which implies that ′′ is much larger than unity [50]. Since the error function reduces to unity if the argument is more than one, equation (8.118) can be simplified. Substituting simplified equation (8.118) in equation (8.112) after replacing the error function by unity, we get a simplified analytic expression for the subthreshold current as I = qWDn ′ 2 √ ′′ [ 1 − e −VDS Vt ] Lg (8.119) The proposed model shows a good agreement with the TCAD simulations for different silicon film doping, thicknesses, and gate oxide thicknesses [50]. 8.4.2 The Pao–Sah Integral In the conventional MOSFETs, modeling the drain current requires the inversion layer charge density as discussed in the introduction of this chapter. However, calculation of the inversion layer charge density is a tedious task as it does not yield a closed form analytical expression and involves numerical integration which is time consuming and cannot be used in circuit simulators [27, 52]. In their classical work, Pao and Sah derived an exact solution for the inversion layer charge density as a function of the electron quasi-Fermi potential only [53]. Their analysis involved an assumption that the inversion layer charge density is negligible beyond a position x = xf where the band bending approaches the Fermi-potential, i.e. = f . This is a valid assumption as the classical definition of the weak inversion regime also starts once the surface

360 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS potential is equal to the Fermi-potential, i.e. S = f . The inversion layer charge density can be found analytically as Qin (x, V) = q ∫ 0 xc n (x, V) dx = ∫ S f n2 i NA e (x)−V Vt dx d d (8.120) where n(x, V) is the electron concentration at location x and electron quasi-Fermi level (V) in the channel, is the potential, NA is the channel doping, and ni is the intrinsic carrier concentration. Using equation (8.120) to get the inversion layer charge in equation (8.1) gives us an analytical expression for the drain current. The Pao–Sah method is the most comprehensive model and takes both drift and diffusion transport into account. Therefore, it is also considered as the benchmark model for modeling drain current in MOSFETs [27]. The Pao–Sah integral approach for calculating the drain current may be generalized for any FET considering only drift and diffusion transport as I = − W Lg ∫ VD VS QmdV = − W Lg ∫ QD QS Qm dV dQm dQm = W Lg ∫ VD VS (QD − QSC)dV (8.121) where Qm is the mobile charge density. For DGJLFETs, an analytical relationship between the mobile charge density and the gate voltage is presented in Section 8.3 as equation (8.102) [42]. Below the flat band condition, the mobile charge density is less than the depletion charge density. Therefore, equation (8.102) may be simplified as Qm − 2Cox ′ Vt ln [ − Qm √2SiVtqND ] = −2Cox ′ ( VG − VTh − V ) (8.122) From equation (8.122), we may write dV dQm =′ 2Cox − Vt Qm (8.123) Using equation (8.123) in equation (8.121) and solving the integral, we get I = − W Lg [ ′ 4Cox Q2 m − VtQm ]| | | | | QS QD (8.124) where QS and QD are the mobile charge densities at the source and the drain end, respectively, and may be calculated by using V=0 for QS and V=VDS in equation (8.122). Equation (8.124) yields a continuous drain current model for DGJLFETs,

DRAIN CURRENT MODELING APPROACH 361 which is valid in all the operating regimes below the flat band [42]. In the linear region, the mobile charges are dominant at both the source and drain ends. Therefore, the first term of equations (8.122) and (8.124) is dominant and governs the current [42]. Ignoring the second term, we get the following expression for the drain current in the linear region: I = 2CoxW Lg ( VG − VTh − VDS 2 ) VDS (8.125) Similarly, in the saturation regime, although the mobile charges dominate on the source side, the channel region is pinched off on the drain side. Therefore, the first term of equations (8.122) and (8.124) dominates for the source side whereas the second term of these equations dominate and should be used for calculating QD and the drain current [42]. Using this simplification, the drain current is obtained as I = CoxW Lg [( VG − VTh)2 − Vt Cox √2SiVtqND e ( VG − VTh − VDS Vt ) (8.126) In the subthreshold regime, the mobile charges are negligible at both source and drain end. Therefore, the second term of equations (8.122) and (8.124) is dominant and should be used for calculation of both QS and QD as well as the drain current [42]. The simplified expression for the drain current based on this assumption can be given as I =WVt Lg √2SiVtqND e ( VG − VTh Vt ) [ 1 − e −VDS Vt ] (8.127) Therefore, using the mobile charge model developed in Section 8.3, we can simply obtain the drain current model. The model shows a good agreement with the TCAD simulation results [42]. However, this model does not account for the current in the accumulation regime. Therefore, we use the charge-based model developed in Section 8.3 for accumulation and depletion regimes for calculating the drain current [48]. Equation (8.121) can also be expressed as I = W Lg ∫ VD VS ( qNDtSi − QSC) dV = W Lg qNDtSiVDS − W Lg ∫ VD VS QSC dV (8.128) Now, QSC dV in the accumulation mode can be easily found using equation (8.95) as QSC dV (accu) ≈ QSCdQSC 2Cox − Q2 SC dQSC 4qNDSi + Q2 SC 2Vt (8.129)

362 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS The integration of the second term in equation (8.128) in the accumulation regime can then be expressed as ∫ VD VS QSC dV (accu) ≈ 1 4Cox Q2 SC | | | | QD QS − 2Vt [ QSC − √8SiqNDVt tan−1 QSC √8SiqNDVt ] | | | | | | QD QS (8.130) QS and QD may be obtained by putting V=0 and V=VDS in equation (8.95), respectively. Now, when the entire channel is in accumulation, we have, VG − VFB − VDS > Vt ln ( ND∕ni ) which ensures that the channel at the drain end is in accumulation. Once this is satisfied, the channel is bound to be in accumulation mode at the source end [48]. In this regime, the drain current can be expressed as I = W Lg qNDtSi − W Lg ∫ VD VS QSC dV (accu) (8.131) Now, QSC dV in the depletion mode can be obtained from equation (8.100) as QSC dV (dep) ≈ Q2 SCdQSC 4SiqND + QSCdQSC 2Cox + Vt2Q2 SC dQSC ( qNDtSi)2 − Q2 SC (8.132) The integration of the second term in equation (8.128) for the depletion region can be obtained as ∫ VD VS QSC dV (dep) ≈ 1 12SiqND Q3 SC | | | | QD QS + 1 4Cox Q2 SC | | | | QD QS − 2Vt QSC| | QD QS +VtqNDtSi [ ln ( qNDtSi + QSC) ( qNDtSi − QSC) ] (8.133) When the entire channel is depleted, we have, VG − VFB < Vt ln ( ND∕ni) at the source end of the channel, which ensures that the channel is depleted at the source end. If this condition is met, the drain end of the channel region is also depleted [48]. Therefore, in the depletion regime, the drain current can be expressed as I = W Lg qNDtSi − W Lg ∫ VD VS QSC dV (dep) (8.134) However, in the partial depletion mode of operation, the source end of the channel is in the accumulation mode whereas the drain end is in the depletion mode and the conditions VG − VFB > Vt ln ( ND∕ni) and VG − VFB − VDS < Vt ln ( ND∕ni) are satisfied. In this case of a hybrid channel, the integral of the second term is divided into two parts: the accumulated channel at from the source end to the flat band

DRAIN CURRENT MODELING APPROACH 363 position in the channel region (where QSC =0) and the depleted region from the flat band position to the drain end as given below: Ihybrid = W Lg qNDtSi − W Lg ∫ VFB VS QSC dV (accu) − W Lg ∫ VD VFB QSC dV (dep) (8.135) Therefore, the drain current in all the operating regimes can be calculated using the charge-based model obtained in Section 8.3 [48]. Now, we discuss the methodology to obtain the drain current model from the surface potential modeling approach using the Pao–Sah integral. For this, we use a modified version of the Pao–Sah integral as I = − W Lg ∫ S ( Lg ) S(0) Qm dV dS dS = − W Lg ∫ S ( Lg ) S(0) ( QSC − QD ) dV dS dS (8.136) Now, utilizing the relationship between the surface potential and the electron quasi-Fermi potential derived in Section 8.2.1 using the regional approximation approach, we may develop a drain current model for the DGJLFETs [41]. Using equation (8.86) for QSC, equation (8.136) may be expressed as I = W Lg ∫ S ( Lg ) S(0) [ 2Cox ( VG − VFB − S ) + qNDtSi] dV dS dS (8.137) For VGS > VFB + VDS, the mobile electrons accumulate in the entire channel region and the relationship between the surface potential and the electron quasi-Fermi potential given by equation (8.25) should be differentiated to obtain dV∕dS as dV dS = 1 + 2Vt ( VG − VFB − S ) ( VG − VFB − S )2 + Vt (8.138) Using this value of dV∕dS in equation (8.137) and integrating, we obtain the drain current for the DGJLFETs in the accumulation region as Iaccu = W Lg [ qNDtSiV − 4CoxVt(VG − VFB − S) − Cox(VG − VFB − S) 2 +4CoxVt √Vttan−1 (VG − VFB − S) √Vt ] | | | | S(Lg) S(0) (8.139)

364 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS where S (0) and S ( Lg ) are the surface potential values at the source end and drain end, which may be calculated by using V=0 and V= VDS in equation (8.25), respectively. Similarly, for the partial depletion regime (VTh < VGS < VFB), differentiating equation (8.38), we get, dV dS = 1 − 2 ( VG − VFB − S ) (8.140) Utilizing this relationship in equation (8.137) and evaluating the integral, we obtain the drain current in the partially depletion mode as Idep = W Lg [ qNDtSiV − Cox( VG − VFB − S )2 + 4 3 ( VG − VFB − S )3 ]| | | | S ( Lg ) S (0) (8.141) Again, S (0) and S ( Lg ) are the surface potential values at the source end and drain end which may be calculated by using V=0 and V=VDS in equation (8.38), respectively. Now, for VFB < VGS < VFB + VDS, the channel is accumulated near the source end while it is depleted near the drain end. Therefore, for this case of hybrid channel, the integral must be separated for the accumulated channel from source end to the flat band point in the channel, which follows equation (8.25), and for the depleted channel until the drain end, which follows equation (8.38), as Ihybrid = W Lg [qNDtSiV + 2Cox ∫ VG−VFB 0 ( VG − VFB − S ) dV| | | (accu) +2Cox ∫ VDS VG−VFB ( VG − VFB − S ) dV| | | (dep) (8.142) For VGS < VTh, the entire channel is depleted. Using equation (8.47) for the fulldepletion mode, we obtain the drain current as Isub = W Lg Vt2Cox ∫ VDS 0 W [ qNDtSi 4Cox e VG − VTh − V Vt ] dV (8.143) The subthreshold current obtained by equation (8.143) saturates when the operating region changes to the partial depletion mode [54]. At the point of saturation, to improve the accuracy in the partial depletion regime, a simple interpolation may be used to find the total drain current as I = [ I 2 sub + I 2 dep]1∕2 (8.144)

MODELING SHORT-CHANNEL JLFETs 365 30 25 20 15 Symbol: Sentaurus Line: Model –2.0 102 10–1 10–4 10–7 10–10 10–13 –1.5 –1.0 –0.5 0.0 0.5 1.0 1.5 2.0 tSi = 8 nm, 10 nm, 12 nm tox = 8 nm ND = 1019 cm–3 VDS = 0.1 V VG (V) L = 1 μm 10 5 0 IDS (μA) IDS (μA) FIGURE 8.14 Transfer characteristics of the DGJLFETs for different silicon film thickness using the interpolation technique [41]. As can be observed from Fig. 8.14, the drain current model obtained using the interpolation technique shows a close agreement with the TCAD simulations [41]. 8.5 MODELING SHORT-CHANNEL JLFETs In Sections 8.1–8.4, we have discussed the different approaches used for modeling the surface potential, the charge, and the drain current in DGJLFETs. However, the approaches discussed in these sections are accurate only for the long-channel JLFETs where there is a negligible impact of the drain voltage on the surface potential at the source–channel interface as the lateral electric field is low. In this case only, we may separately solve the Poisson equation in the direction along the channel thickness and use it to find the drain current using the Pao–Sah integral or numerical integration. However, in the short-channel JLFETs, due to the proximity of the source–channel interface to the drain, the interaction between the surface potential and the drain electric field is inevitable. This interaction leads to a perturbation in the surface potential at the source–channel interface giving rise to short-channel effects such as draininduced barrier lowering (DIBL), and so on. Therefore, the 2D Poisson equation must be solved for modeling the short-channel JLFETs. One of the simplest techniques to solve the 2D Poisson equation and to model the threshold voltage and the shortchannel effects such as threshold voltage roll-off and DIBL in JLFETs is discussed in the next section.

366 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS 8.5.1 Quasi-2D Scaling Equation The quasi-2D scaling equation simplifies the 2D Poisson equation into an ordinary second-order differential equation, which may be solved analytically. Before discussing the quasi-2D approach, we would like to discuss about the concept of scaling factor and natural length [55]. In the subthreshold regime of operation, the contribution to the total charge density due to the mobile carriers may be ignored and the 2D Poisson equation for MOSFETs may be expressed as d2(x, y) dx2 + d2(x, y) dy2 = qNA Si (8.145) Assuming a parabolic potential approximation along the x-direction as presented in Section 8.2.2, the 2D potential may be expressed as equation (8.48): (x, y) = a0 (y) + a1 (y) x + a2 (y) x2 Utilizing the boundary condition that the electric field is zero at the center and the potential at x=0 is the central potential, equation (8.48) simplifies to equation (8.49): (x, y) = 0 (y) + a2 (y) x2 Now, from the continuity of the electric displacement vector at the bottom Si–SiO2 interface, we get a2 −Cox ( VG − VFB − S,b ) = −Si d (tSi 2 , y ) dx = −SitSia2 (y) (8.146) Putting the value of a2 in equation (8.49), we obtain a relation for the 2D potential as (x, y) = 0 (y) + Cox SitSi (VG − VFB − S (y))y2 (8.147) From equation (8.147), a relation between the surface potential and the central potential can be obtained as (tSi 2 , y ) = S (y) = 0 (y) + CoxtSi 4Si (VG − VFB − S(y)) (8.148) Rearranging equation (8.148), we get a relationship between the surface potential and the central potential as S (y) = 0 (y) + CoxtSi 4Si ( VG − VFB) 1 + CoxtSi 4Si (8.149)

MODELING SHORT-CHANNEL JLFETs 367 Now, putting equation (8.147) into equation (8.145) after utilizing the value of central potential obtained from (8.149), we get d2S (y) dy2 − 4Cox SitSi ( S (y) − VG − VFB + qNAtSi 4Cox ) = 0 (8.150) Now, defining 1∕2 1 = (4Cox)∕SitSi, ∅S = VG − 1, and 1 = VFB − (qNAtSi)∕ 4Cox, where 1 has the dimensions of length and is referred to as the natural length and ∅S is defined as the long-channel surface potential, we get a simplified version of equation (8.150) as a simple second-order differential equation: d2S (y) dy2 − 1 2 1 ( S (y) − ∅S ) = 0 (8.151) This simplified differential equation has the solution of the form: S (y) = ∅S + a e y 1 + b e− y 1 (8.152) It may be noted from equation (8.152) that 1 represents the spread in the potential profile along the lateral direction and depends upon the channel thickness and the gate oxide thickness [55]. A smaller channel or gate oxide thickness improves the effective gate control, leading to a reduced natural length and a lower influence of the drain region on the electrostatics of the channel region. Therefore, the natural length is also an indicator of the influence of the drain electric field on the channel region. The gate length should be much larger than the natural length for mitigating the short-channel effects. According to the quasi-2D scaling theory [55], the FETs should be designed to have a large value of the scaling factor given as = Lg∕2 to suppress the shortchannel effects. Now, for the short-channel DGJLFETs (Fig. 8.15), the Poisson equation is modified as equation (8.2): d2(x, y) dx2 + d2(x, y) dy2 = −qND Si Now, following the approach similar to the DGMOSFETs and utilizing the parabolic potential approximation and the boundary conditions, we obtain (x, y) = 0 (y) + Cox SitSi ( VG − VFB − S (y) ) y2 (8.153) Now, we develop the scaling equations for the JLFETs considering the central potential since the current flows at the center at the verge of threshold conditions in a DGJLFET as discussed in Section 3.2.2 [56]. Therefore, replacing S (y) in equation

368 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS VS VG VG VD tox Lg tSi x Channel (0,0) y FIGURE 8.15 Three-dimensional view of the short-channel DGJLFET used for obtaining the quasi-2D scaling equation in [56]. For simplicity, the source/drain thickness is assumed to be zero and contacts are abutted on the sides. (8.153) utilizing the expression relating the central potential and surface potential from equation (8.149), we have (x, y) = 0 (y) + Cox SitSi ⎛ ⎜ ⎜ ⎝ VG − VFB − 0 (y) + CoxtSi 4Si ( VG − VFB) 1 + CoxtSi 4Si ⎞ ⎟ ⎟ ⎠ y2 (8.154) Utilizing equation (8.154) in equation (8.2), we get d20 (y) dy2 − 8Cox 4SitSi + Coxt 2 Si ( 0 (y) − VG − VFB + qNDtSi 2Cox + qNDt 2 Si 8Si ) = 0 (8.155) Now, defining 1∕2 2 = 8Cox∕(4SitSi + Coxt 2 Si), ∅0 = VG − 2, and 2 = VFB − ( qNDtSi) ∕2Cox − ( qNDt 2 Si ) ∕8Si, where 2 is the natural length for DGJLFETs and ∅0 is the long-channel central potential, we get a simplified second-order differential equation even for the DGJLFETs as d20 (y) dy2 − 1 2 2 ( 0 (y) − ∅0 ) = 0 (8.156) This simplified differential equation has the solution of the form: 0 (y) = ∅0 + a1e y 2 + b1 e − y 2 (8.157)

MODELING SHORT-CHANNEL JLFETs 369 where a1 and b1 are constants. These constants may be found by utilizing the boundary condition along the y-direction, which are 0 (y = 0) = VS = 0 (8.158) 0 ( y = Lg ) = VD = VDS (8.159) Using the boundary conditions given by equations (8.158) and (8.159), we get a1 = 1VG + (8.160) b1 = VG + (8.161) where 1 = e −Lg − 1 2 sinh (Lg ) (8.162) = VDS − 2 ( e −Lg − 1 ) 2 sinh (Lg ) (8.163) = 1 − e Lg 2 sinh (Lg ) (8.164) = −VDS + 2 ( e Lg − 1 ) 2 sinh (Lg ) (8.165) Once the constants a1 and b1 are known, the next step is to find an analytical relation for the threshold voltage of DGJLFETs. The location of the minimum central potential can be obtained by differentiating equation (8.157) and using d0 (y) ∕dy = 0, which yields that the minimum central potential exists at a location: ymin = 2 ln (b1 a1 ) (8.166) The value of the minimum central potential is obtained by putting equation (8.166) in equation (8.157) as 0 ( ymin) = √a1b1 + ∅0 (8.167) Now, at threshold voltage, the neutral channel begins to be uncovered at the center. Therefore, the minimum central potential may be assumed to be zero when the gate

370 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS voltage equals the threshold voltage VTh[56]. Using this, we find the threshold voltage expression for the short-channel DGJLFETs as VTh = 2 ( + ) + 2 + √( 2 ( + ) + 2 )2 − (1 − 4) ( 2 − 4) (1 − 4) (8.168) Equation (8.168) presents a closed form analytical solution to the threshold voltage of short-channel DGJLFETs. It may be noted that the constants and cease to be 0 for long-channel lengths and the threshold voltage approaches 2, which is the threshold voltage expression obtained for the long-channel DGJLFETs in Section 8.2.1. 8.5.2 Scaling Implication for MOSFETs Using Quasi-2D Approach Similar to the DGJLFETs, the threshold voltage of the DGMOSFETs can be obtained using the quasi-2D approach. Since the current flows at the surface in the MOSFETs, the surface potential is considered for DGMOSFETs, which is given as equation (8.152): S (y) = ∅S + a e y 1 + b e− y 1 The boundary conditions in the y-direction in the case for MOSFETs are different from that of JLFETs as given below: S (y = 0) = Vbi (8.169) S ( y = Leff) = Vbi + VDS (8.170) where Vbi is the built-in potential at the source–channel and the channel-drain interface given by Vbi = Vt ln ( NS∕D∕NA ) , where NS/D is the doping of the source and drain regions and NA is the channel doping. It may be noted that for the DGMOSFET, the gate length Lg has been replaced by the effective channel length Leff which is given by Leff = Lg − LS − LD + 2Ld and takes into account the encroachment of the source/drain depletion regions denoted by LS and LD, respectively, in the channel length defined as LS = 2 ( Vbi − S ( ymin)) dS (y = 0) dy (8.171) LD = 2 ( VDS + Vbi − S ( ymin)) dS ( y = Lg ) dy (8.172)

MODELING SHORT-CHANNEL JLFETs 371 and also the contribution of the Debye length, which accounts for the transition between the drift and diffusion regions in the channel given as Ld = √ 2SikT q2NA (8.173) After obtaining the constants a and b using the boundary conditions, the minimum surface potential can be obtained as: S ( ymin) = √ ab + ∅S (8.174) Now, for DGMOSFETs, at the threshold voltage, the minimum surface potential should be equal to twice the Fermi-potential, i.e. S ( ymin) = 2f = 2Vt ln ( NA∕ni ) . The threshold voltage for DGMOSFET may be obtained in this manner. Now, we can draw few conclusions right away. The effective channel length is larger in DGJLFETs as compared to the DGMOSFET. Therefore, JLFETs are expected to perform better as compared to the MOSFETs with respect to shortchannel effects such as threshold voltage roll-off (Fig. 8.16) and DIBL (Fig. 8.17). The DIBL has been defined as the difference in the threshold voltage at a drain voltage of 0.1 V and a higher drain voltage specified in Fig. 8.17. Also, it may be concluded that a lower gate oxide thickness and a smaller channel thickness leads to reduced short-channel effects and facilitates the scaling of the JLFETs [56]. 0 –0.4 –0.8 10 20 tsi = 5 nm tsi = 10 nm tsi = 15 nm tox = 3 nm Na,JB = 1 × 1018 cm–3 Nd,JB = 1 × 1020 cm–3 Nd,JL = 1 × 1020 cm–3 Vds = 0.1 V 30 Effective channel length, Leff (nm) Threshold voltage roll-off (V) 40 Solid Line : DGJLFETs Model Dot Line : DGMOSFETs Model Symbol : DESSIS FIGURE 8.16 Threshold voltage roll-off for the DGJLFETs and the DGMOSFETs for different silicon film thicknesses [56].

372 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS 1.6 1.2 0.8 0.4 0 10 20 tsi = 10 nm tox = 3 nm Na,JL = 1 × 1018 cm–3 Nd,JL = 1 × 1020 cm–3 Nd,JL = 1 × 1020 cm–3 Vds1 = 0.1 V Vds2 = 0.5 V Vds2 = 1.5 V Vds2 = 1.0 V 30 Effective channel length, Leff (nm) DIBL (V) 40 Solid line : DGJLFETs Model Dot line : DGMOSFETs Model Symbol : DESSIS FIGURE 8.17 Drain-induced barrier lowering for the DGJLFETs and the DGMOSFETs for different drain voltage [56]. 8.6 MODELING QUANTUM CONFINEMENT As discussed in Section 5.4.2, as the channel thickness approaches sub-10 nm regime, the quantum confinement effects become dominant and alter the carrier concentration and the effective band gap due to significant discretization of energy bands [57–62]. For effectively capturing the impact of quantum confinement effects, each energy subband must be analyzed. This requires self-consistent solution of the Schrodinger ¨ equation and the Poisson equation. The Schrodinger equation yields the energy of ¨ the different subbands using the effective mass approximation. For instance, the Schrodinger equation for the ¨ ith wave function (i) and the corresponding subband energy level Ei can be stated as [ − ℏ2 2m∗ ∇2 − q ] i = Eii (8.175) where ℏ is the reduced Planck’s constant, m* is the effective mass, and is the electrostatic potential, which can be obtained by solving the Poisson equation (equation 8.2): ∇2 = − Si However, self-consistent solution of the Schrodinger and Poisson equations cannot ¨ be obtained analytically.

MODELING QUANTUM CONFINEMENT 373 In case of MOSFETs, in the strong inversion regime, the inversion carriers at the Si–SiO2 lie in a triangular potential well formed by the gate oxide potential barrier on one side and the heavily bent conduction band acting as a barrier on the other side. Because of a significant quantization in the direction along the channel thickness, i.e. x-direction, the inversion carriers must be treated as a 2D gas which is free to move in the direction along the channel length (y-direction) or the direction along the channel width (z-direction) [57]. Therefore, the quantum confined electrons should be treated with discrete energy subbands with 2D density of states formalism rather than with the conventional continuous 3D density of state formalism taught in the freshman semiconductor physics books and courses. First, we discuss a method to find the carrier density in a particular energy subband for the 2D electron gas, which is essentially the behavior of the quantized electrons. The number of electronic states at energy can be simply thought of as the number of possible momentum values that an electron at that energy level may possess. According to the Heisenberg principle, there is only one electronic state in a phase space of volume ( ΔyΔpy ) (ΔzΔpz ) = h2, where py and pzare the electron momentum along the y-direction and z-direction, respectively, and h is the Planck’s constant. Now, the number of electronic states per unit area (density of states (N(E)) in an energy subband at an energy level between E and E + dE can be expressed as N (E) dE = 2gdpydpz h2 (8.176) where g is the degeneracy of the energy subband, dpydpz is the area in the momentum space where the electron lies within the energy level E and E + dE. Now, using the effective mass approximation, the energy–momentum relationship at the bottom of this energy subband with a ground state energy level E0 can be represented as E − E0 = p2 y 2my + p2 z 2mz (8.177) where E – E0 is the kinetic energy and my and mz are the effective mass of the electrons in the y- and z-directions, respectively. Equation (8.177) represents an ellipse in the momentum space whose area can be given as 2 √mymz ( E − E0 ) . Therefore, the area dpydpz in the momentum space where the electron lies within the energy level E and E + dE can also be written as 2 √mymz dE. Using this expression in equation (8.176), the 2D density of states in the subband can be expressed as N (E) dE = 4g √mymz dE h2 (8.178) Now, the number of electrons in this energy subband can be obtained by multiplying the density of states at this energy level by the probability of finding electrons at

374 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS this energy, which is given by the Fermi–Dirac distribution. Therefore, the electron density in this particular subband is given as n = ∫ ∞ E0 N (E) f (E) dE = 4gkT √mymz h2 ln ( 1 + e Ef −E0 kT ) (8.179) The total electron density in the conduction band may be obtained by summing the contributions of all the energy subbands. For this, we need to find the location of the energy subbands, which cannot be obtained analytically owing to the self-consistent solution of the Schrodinger and Poisson equations [57]. ¨ However, in the subthreshold regime of operation, the mobile charge density is negligible and the energy band profile is governed by the depletion charges. Therefore, in this regime, the Schrodinger equation can be decoupled from the Poisson ¨ equation and solved independently to find the location of the different energy subbands [57]. Let us consider the case of the DGJLFETs. The solution of the Poisson equation for the DGJLFETs, as discussed in Section 8.4.1, can be simplified utilizing equations (8.86) and (8.115) as (x) = VG − VTh − qND 2Si x2 (8.180) where VTh is the threshold voltage given by equation (8.39). Now, we also solve the Schrodinger equation independently. Numerical simulations utilizing TCAD ATLAS ¨ reveal that for large channel thickness and channel doping, the energy subbands lie enclosed within the boundaries of the bent classical conduction band similar to the DGMOSFETs as shown in Fig. 8.18. However, for the case of ultrathin channel, the energy subbands lie within the potential well dictated by the potential barriers of the gate oxide and the potential at the bottom is governed by the bent classical conduction band as shown in Fig. 8.19. Therefore, for the two extreme cases, the energy subbands show different behavior and boundary conditions [62]. The first case of heavy channel doping and large channel thickness with energy bands confined within a parabolic barrier is similar to the standard quantum harmonic oscillator problem with a potential energy given by VQM = q2ND 2Si x2 (8.181) Utilizing the effective mass approximation, the solution of this standard problem can be obtained as Ek,n (QHO) = ( n + 1 2 ) h 2 √ q2ND m∗ kSi (8.182) where Ek,n is the quantized energy of the kth energy subband at the nth orbital, n is the principal quantum number, and m∗ k is the effective mass in the kth energy subband.

MODELING QUANTUM CONFINEMENT 375 1013 1012 1011 VG = 0 V Ek,... Ec 1010 109 108 107 106 105 104 103 102 Classical Quantum –6 –4 –2 Position in the channel (nm) Electron concentration (cm–3) 0246 101 100 FIGURE 8.18 Comparison of the quantum and classical electron concentration along the channel thickness in DGMOSFETs for a channel thickness of 10 nm [62]. 1013 1012 1011 VG = 0 V Ek,n Ec 1010 109 108 107 106 105 104 103 102 Classical Quantum –6 –4 –5 –2 –1 –3 Position in the channel (nm) Electron concentration (cm–3) 0 2 1 4 3 6 5 101 100 FIGURE 8.19 Comparison of the quantum and classical electron concentration along cutline B-B’ (Fig. 8.1) in DGJLFET for a channel thickness of 10 nm [62].

376 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS An important point to note from equation (8.182) is that the location of the energy subband is independent of the thickness of the channel [62]. Now, the case with small-channel thickness resembles the standard quantum well surrounded by infinite potential barriers. The solution for this standard problem is Ek,n (QW) = h2n2 8m∗ k t 2 Si (8.183) However, the potential well at the bottom consists of the bent conduction band and does not have a flat surface in DGJLFETs unlike DGMOSFETs [57–61]. This bottom potential energy is given by equation (8.181) and its contribution to the location of the energy subbands must be included for accuracy. The time-independent perturbation theory enables us to predict the changes in the energy Eigen values by treating the small perturbation energy as an additional Hamiltonian [61]. Therefore, the perturbation potential energy may be expressed as a Hamiltonian (H) as H = q2ND 2Si x2 (8.184) According to the first-order time-independent perturbation theory, the expectation value of the Hamiltonian for a sinusoidal wave function w(x), where w(x) = √ 2 tSi sin [ n tSi ( x + tSi 2 )] (8.185) yields an energy correction term for the energy Eigen values of the energy subbands as E = ⟨w(x) |H| w(x)⟩ (8.186) Solving for the expectation value of energy Eigen values, we get E = q2ND SitSi ∫ tSi∕2 −tSi∕2 sin2 [ n tSi ( x + tSi 2 )] x2dx = q2NDt 2 Si 24Si ( 1 − 6 ( 1 n )2 ) (8.187) Adding the energy correction term obtained from equation (8.187) to equation (8.183), we get an accurate expression for the energy Eigen values representing the different energy subbands as Ek,n (QW) = h2n2 8m∗ k t 2 Si + q2NDt 2 Si 24Si ( 1 − 6 ( 1 n )2 ) (8.188) The location of the lowest energy subband has been compared for the DGJLFETs and the DGMOSFETs for different channel thicknesses in Fig. 8.20. The energy of

MODELING QUANTUM CONFINEMENT 377 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0 2 4 6 8 10 Silicon thickness (nm) Lowest subband energy (eV) JL QW+PP QW+PP Simulation JL Ek=1 IM: Undoped tox = 2 nm JL Ek=2 IM Ek=1 IM Ek=2 JL QHO IM [39] Model QHO 12 14 16 18 20 JL: Nsi = 1 × 1019 cm–3 tox = 2 nm FIGURE 8.20 Energy of the lowest subband for different silicon thicknesses for DGJLFETs [62] and DGMOSFETs [58]. the first subband is higher for the DGJLFETs as compared to the DGMOSFETs for tSi >4 nm and becomes nearly equal as the channel thickness reduces below 4 nm. Also, it can be observed that the energy Eigen values given by the solution for quantum well (with the perturbation potential correction term included) approaches the value obtained from the solution of the quantum harmonic oscillator for a channel thickness of ∼8 nm [62]. Moreover, at this channel thickness, the Eigen value obtained from the solution of the quantum well problem is minimum. Therefore, the channel thickness at which the transition from the quantum well problem to the quantum harmonic oscillator problem takes place can be obtained by finding the minima of equation (8.188). Differentiating equation (8.188) and equating it to zero, we get tSi (transition) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ( h2n2 m∗ k ) q2ND 3Si ( 1 − 6 ( 1 n )2 ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 1 4 (8.189) Since the potential distribution inside the channel in JLFETs is a strong function of both channel thickness and doping, the channel doping at which the transition takes place can also obtained at a particular channel thickness and quantum number (n) using equation (8.189).

378 MODELING JUNCTIONLESS FIELD-EFFECT TRANSISTORS Now, once the location of the energy subbands have been obtained, the next step is to find the shift in the threshold voltage (ΔVTh), which can be expressed as ΔVTh = Vt ln (nClassical nquantum ) (8.190) where nclassical and nquantum are the classical and quantum electron density in the conduction band. The classical electron density was obtained as equation (8.58) in Section 8.4.1. The electron density in a particular energy subband is given by equation (8.179). Now, summing over all the discrete energy subbands over different quantum numbers, we can obtain the quantum electron density as nquantum = 4kT h2 ∑ k ∑ n gkm∗ k ln ( 1 + e Ef −Ec0−Ek,n kT ) (8.191) where gk is the degeneracy of the valley, Ec0 is the conduction band energy at the center of the channel and Ek,n is the energy of the subband. The minima of the conduction band may be obtained as [61] Ec0 = Eg 2 + Vt 2 ln (Nc Nv ) − q ( VG − VTh − ∅B ) + Ef (8.192) where Eg is the band gap, NC and NV are the density of states in the conduction and valence bands, respectively, Ef is the Fermi level, and ∅B = −Vt ln ( ND∕ni ) , where ni is the intrinsic carrier concentration. Now, log(1 + ax) may be simplified as ax when ax<1. In the subthreshold regime, e(Ef − EC0 − Ek,n)∕kT < 1. Therefore, the logarithmic term can be simplified and equation (8.191) may be rewritten as nquantum = 4kT h2 ∑ k ∑ n gkm∗ k e Ef −EC0−Ek,n kT = 4kTND NCh2 e VG−VTh Vt [ 2m∗ d,1 ∑ n e −E1,n kT + 4m∗ d,2 ∑ n e −E2,n kT ] (8.193) where m∗ d,1 and m∗ d,2 are the effective masses of the electrons in the conduction band valleys with degeneracy 2 and 4, respectively. The shift in the threshold voltage can be obtained by putting the value obtained from equations (8.193) and (8.58) in equation (8.190) as ΔVTh = Vt ln ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 2 √ erf ( √tSi 2 ) 4kTND NCh2 e VG−VTh Vt [ 2m∗ d,1 ∑ n e −E1,n kT + 4m∗ d,2 ∑ n e −E2,n kT ] ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ (8.194)

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