Shih-Ching Lo
a, Yiming Li
b,∗, Shao-Ming Yu
caDepartment of Transportation Technology and Logistics Management, Chung Hua University, Hsinchu 300, Taiwan bDepartment of Communication Engineering, National Chiao Tung University, Hsinchu 300, Taiwan
cDepartment of Computer Science, National Chiao Tung University, Hsinchu 300, Taiwan
Received 18 May 2006; accepted 15 December 2006
Abstract
In this paper, an analytical solution of the Poisson equation for double-gate metal-semiconductor-oxide field effect transistor (MOSFET) is presented, where explicit surface potential is derived so that the whole solution is fully analytical. Based on approximations of potential distribution, our solution scheme successfully takes the effect of doping concentration in each region. It provides an accurate description for partially and fully depleted MOSFET devices in different regions of operation. Comparison with numerical data shows that the solution gives good approximations of potential for MOSFETs under different biases and geometry configurations. The solution can be applied to estimate classical and quantum electron density of nanoscale double-gate MOSFETs.
c
2007 Elsevier Ltd. All rights reserved.
Keywords:Analytical solution; Poisson equation; Surface potential; Double-gate MOSFET
1. Introduction
Advanced integrated circuit (IC) technology imposes new challenges for semiconductor devices [1–13]. Nowadays, double-gate metal-oxide-semiconductor field effect transistors (DG-MOSFET) have attracted much attention due to their improved physical properties compared with the single-gate MOSFETs [1–4,9,12]. One of the important questions is how to analytically calculate the characteristics of a device in an efficient and accurate way, especially for large scale IC simulation. Among the characteristics of a device, the potential dominates the main mechanism of operation of a device. As the potential is obtained, carrier density and transport current can be derived accordingly. Unfortunately, many solution approaches to the nonlinear Poisson equation were based on very strong assumptions and simplifications; therefore, they cannot be applied to simulate DG-MOSFET circuits accurately [3–5].
Therefore, the purpose of this work is to introduce a new one-dimensional (1-D) closed-form analytical approximation of potential distribution in the silicon film of a DG-MOSFET in different regions of operation, such as the depletion, the weak inversion, and the strong inversion. The solution is derived based on the physical mechanism
∗Corresponding address: P.O. BOX 25-178, Hsinchu 300, Taiwan.
E-mail address:[email protected](Y. Li).
0895-7177/$ - see front matter c 2007 Elsevier Ltd. All rights reserved.
Fig. 1. An illustration of the examined DG-MOSFET.
in each region. To make the solution be a fully analytical one, the explicit form of surface potential is also derived by integrating the continuous condition. A comparison with numerical data shows that the solution gives an accurate approximation of potential distribution for a nanoscale DG-MOSFET in all regions of operation. The analytical potential can be incorporated into a quantum correction formula to account for the quantum mechanical effect on the electron density.
The remaining content of this paper is as follows. In Section 2, the nonlinear Poisson equation is presented firstly and an analytical solution is derived under several cases and approximations. Also, the explicit equation of surface potential is derived in this section. Computational concerns are given in Section3, where results of analytical solution are compared with the numerical results. The analytical potential is further incorporated into a quantum correction formula to explore the quantum mechanical effect on the electron density in Section4. Section5draws the conclusions.
2. The nonlinear Poisson equation and analytical solution
The investigated 1-D symmetric DG-MOSFET is illustrated inFig. 1. The Poisson equation governs the operation of semiconductor devices. It comes from Maxwell’s first equation, which in turn is based on Coulomb’s law for electrostatic force of a charge distribution. The Poisson equation is given as
1φ = −q (p − n + Nd−Na) /si, (1)
where1 is the Laplace operator, φ is potential, si is dielectric constant of silicon, q is electron charge, p(x) is
density of hole, n(x) is density of electron, Na is doping concentration of acceptor impurity and Nd is doping
concentration of donor impurity. In the silicon, p(x) = Naand n(x) = n2i/Na, where ni is intrinsic concentration.
With the Boltzmann relationship, we have p(x) = Naexp(−qφ/Vt) and n (x) = ni2/Na exp(qφ/Vt), where Vt is
thermal voltage, which is considered as a constant. Therefore, the Poisson equation and the boundary conditions can be rewritten as 1φ = q Na si " 1 − exp −φ Vt + ni Na 2 exp φ Vt −1 # , −tsi 2 < x < tsi 2 φ = φs, ∂φ/φx = −Cox(VG−VFB−φs) /si, x = −tsi 2 φ = φs, ∂φ/φx = Cox(VG−VFB−φs) /si, x = tsi 2, (2)
whereφs is surface potential. VG is gate voltage and VFBis flat-band voltage. toxis thickness of oxide, tsiis thickness
of silicon film and Coxis capacitance of oxide. Since the device is symmetric, the problem is rewritten as
1φ = q Na si " 1 − exp −φ Vt + ni Na 2 exp φ Vt −1 # , −tsi 2 < x < 0 φ = φs, ∂φ/φx = −Cox(VG−VFB−φs) /si, x = −tsi 2 φ = φb, ∂φ/φx = 0, x = 0, (3)
where∂φ/∂x = 0 at x = 0 is obtained from the assumption of symmetric applied bias [4] and φb is potential
at the center of the silicon region, i.e., φ (x = 0) = φb. As Eq.(3) is solved, the solution of the remaining part
Fig. 2. Variation of total charge density in silicon as a function of surface potentialφsfor a MOS device. 2.1. Solution of Poisson equation
Multiplying(∂φ/∂x) dx on both sides of the Poisson equation of Eq.(3)and integrating from the surface toward the center of the silicon film, i.e., x = −tsi/2to 0, we have
∂φ ∂x = ± ( 2q Na si ( φ − φb+Vtexp −φb Vt −Vtexp −φ Vt + ni Na 2 φb−φ + Vtexp φ Vt −Vtexp φ b Vt ))1/2 . (4)
Eq.(4)is a nonlinear equation, which involves polynomial and exponential functions. It needs to be simplified so as to derive its solution. According to the operating property of semiconductor devices, the solution can be discussed in two regions: (a) depletion and weak inversion region and (b) strong inversion region, which are illustrated inFig. 2.
Letφf =Vtln(Na/ni). For the case of the depletion and weak inversion, we assume 0 < φ < 2φf andφ Vt,
i.e., Vtexp(−φb/Vt) − Vtexp(−φ/Vt) ≈ 0 and exp φ − 2φf/Vt − exp φb−2φf/Vt ≈ 0. Eq.(2)can be
approximated and rewritten as follows ∂φ ∂x = ± v u u t 2q Na si (φ − φb) " 1 − ni Na 2# . (5)
The positive sign is for 0< x < tsi/2and the negative sign is for −tsi/2< x < 0. Then, another integration of Eq.(5)
yields φ = φb+(φs−φb) 1 − v u u t q Na 2(φs−φb) si " 1 − ni Na 2# x + tsi 2 2 , (6) for −tsi/2< x < 0 and φ = φb+(φs−φb) 1 + v u u t q Na 2(φs−φb) si " 1 − ni Na 2# x − tsi 2 2 , (7)
for 0< x < tsi/2. Next, for the case of the strong inversion, we assume 0< Vt φ − 2φf, i.e., Vtexp(−φb/Vt) −
Vtexp(−φ/Vt) ≈ 0, exp (φ/Vt)−exp (φb/Vt) φb−φ and exp φ − 2φf/Vt−exp φb−2φf/Vt φ−φb.
Eq.(4)can be approximated and rewritten as follows ∂φ ∂x = ± s 2q Na si ni Na 2 Vtexp φ Vt −Vtexp φ b Vt . (8)
Also, the positive sign is for 0< x < tsi/2and the negative sign is for −tsi/2 < x < 0. Then, another integration of
Eq.(8)gives φ = φb−Vtln cos s qn2i 2VtsiNa exp φ b 2Vt x 2 , (9)
for −tsi/2 < x < tsi/2. Eqs.(6),(7) and(9)give the general solution of potential under weak inversion and strong
inversion, respectively. However, the surface potential is still unknown. To obtain a completely analytical solution of the Poisson equation, the explicit function of surface potential is needed.
2.2. Explicit expression of surface potential
Derivation of surface potential (φs) is just the same as it is for the potential (φ). Firstly, multiplying (∂φ/∂x) dx on
both sides of the Poisson equation of Eq.(3)from the surface toward the center of the silicon film, i.e., x = −tsi/2to
0, and substituting the boundary into it yields (VG−VFB−φs)2 γ2 = (φs−φb) + Vtexp −φb Vt −Vtexp −φs Vt + ni Na 2 (φb−φs) + Vtexp φ s Vt −Vtexp φ b Vt , (10)
whereγ =√2qεsiNa/Cox andφf =Vtln(Na/ni). To solve φs, another expression relatingφs andφbis required.
The relation is obtained from the condition of full depletion for body charge and the simplification of discretization of Eq.(3) [6–8]. According to the condition of full depletion for body charge Qb = Coxγ
√
φs−φb ≤ q Natsi/2,
φbis given asφs−λtsiCox(VG−VFB−φs) /2εsi−Vxd, whereλ is a fitting parameter and Vxd =q Natsi2/8εsi. The
critical voltage VG =VCat which is changed from partially depleted (PD) to the fully depleted (FD) device needs to
be calculated before deriving the analytical solution. It will be derived from the condition thatφb=0 andφs ≈Vxd.
VC=VFB+Vxd+γ Vxd+Vt exp −Vxd Vt −1 +Vt exp Vxd−2φf Vt −exp −2φf Vt 1/2 . (11) Also, the analytical solution of surface potential is discussed in the (a) depletion and weak inversion and (b) strong inversion regions.
Firstly, surface potential of PD devices is discussed, i.e.,φb=0. For the case of the depletion and weak inversion,
Eq.(10)can be approximated and solved as φp
sd=VG−VFB+γ
2/2 − γqγ2/4 + V
G−VFB+A, (12)
where A = Vt 1 + exp −2φf/Vt and φsdp is the surface potential of a PD device in the depletion and weak
inversion region. According to the studies [7,8], surface potential of the strong inversion region is given by φp si= fφ+Vtln h VG−VFB− fφ 2 /γ2− f φ+1i Vt , (13)
where φsip is the surface potential of a PD device in the strong inversion region. fφ is empirically given as fφ = (2φf +φp
sd −
q (φp
Eq.(10)can be approximated and solved as φf
sd=VG−VFB−γ
p
Vxd, (15)
whereφsdf is the surface potential of a FD device in the depletion and weak inversion region. For the case of the strong inversion, the surface potential is given as
φf
si=2φf +Vtln{[(VG−VFB− fφ)
2/γ2−V
xd]/Vt[1 − exp(−Vxd/Vt)]}, (16)
whereφsif is the surface potential of a FD device in the strong inversion region. Also, the surface potential equation can be linked by the following smooth function, which is given as
φf st=φ f si−Vtln n 1 + exph(φsif −φf sd)/Vtio , (17)
whereφstf is the surface potential of a fully depleted DG-MOSFET in all operating regions. If a device can change the operation mode from PD device to FD device as VG is increased, then the smooth function is employed to link the
operation mode [4]: φm s = φp st 1 + l1exp [(VG−VC) /n1Vt] + φ f st 1 + l1exp [−(VG−VC) /n1Vt], (18) where l1and n1are fitting parameters andφsmis the unified surface potential.
3. Computational calibration and numerical results 3.1. Computational calibration
Since the solution is derived under the simplified assumption based on different operating regions, the error between the exact solution and the analytical solution should be subject to further minimization. Therefore, an adjustment term and fitting parameters are considered after deriving the solution of the Poisson equation and the explicit function of surface potential. By comparing Eq. (4)with Eqs.(5) and(8), we know that the slope of Eq.(4) is larger than or equal to the slope of Eqs. (5)and(8)for x > 0. The slope of Eq.(4)is smaller than or equal to the slope of Eqs.
(5)and(8)for x < 0. An adjustment function is introduced to reduce the solution error. According to our analyses, the adjustment function, eE(x) = ax8+bx2, shows a good approximation. a and b are fitting parameters. Then, the modified analytical solution is
e
φ (x) = φ (x) −Ee(x) . (19)
3.2. Results and discussion
In this study, double-gate NMOSFETs with different gate oxide thickness (tox = 2, 3 and 4 nm), silicon film
thickness (tsi = 20, 30 and 40 nm), and doping concentration (Na = 1016, 1017 and 1018 cm−3) are simulated for
different gate applied bias (VG = 0.1, 0.2, 0.4, 0.6, 0.8 and 1.0 V). Under the given scenario, DG-MOSFETs will
be fully depleted and surface potential is estimated by Eq.(16). From the results, the analytical solution of surface potential has an approximation with error, which is smaller than 0.005% in depletion and weak inversion region. In the strong inversion region, a larger error, which is smaller than 0.01%, is obtained. The results, which are given in Table 1, show that the explicit function gives good approximations of surface potential. Because the potential distribution varies widely, the approximated error in the strong inversion region is larger than the approximated error
Table 1
Numerical and analytical surface potentials of the simulated device
Numerical Analytical Difference Error(C)/(A)
(A) (B) (C) = |(B) − (A)| (%)
VG=0.05 V 0.940459 0.9405 0.000041 0.00436
VG=0.1 V 0.953597 0.953588 0.000009 0.00094
VG=0.7 V 1.03872 1.03866 0.00006 0.00578
VG=1.0 V 1.06509 1.06498 0.00011 0.010
Fig. 3. Comparison of the numerical (solid lines) and analytical (symbols) potentials for the DG-MOSFET with respect to different VG, where
tox=2 nm, tsi=20 nm and Na=1017cm−3. Table 2
Relationship among the fitting parameters, Na, tox, tsiand VG
Parameter Na tox tsi VG
a ∝1/Na ∝1/tox ∝tsi ∝VG
b ∝Na ∝1/tox ∝1/tsi ∝VG
in the depletion and weak inversion region. The differentiation of the analytical solution of surface potential is smaller than the actual distribution. With the explicit surface potential, potential distributions are simulated and compared with numerical approximations, which are simulated by an iterative algorithm. The numerical program is calibrated by the measured data and considered as exact.Figs. 3–6illustrate the comparison of numerical and analytical results. Different profiles of devices are compared to show the accuracy of the analytical solution presented in this work.
Fig. 3examines the simulated results of DG-MOSFET (tox =2 nm, tsi =20 nm, and Na =1017 cm−3) under
different gate applied bias (VG = 0.1, 0.2, 0.4, 0.6, 0.8 and 1.0 V). It shows that the analytical results are in good
agreement with the numerical results. The maximal error is about 0.7% and the average error is about 0.4%. Thus, the analytical solution is accurate enough to apply to circuit simulation.Fig. 4illustrates the application of the analytical solution to simulate devices with different oxide thickness for tsi =20 nm and Na =1017cm−3. The error between
the analytical and numerical solution is within 2%. Potential of DG-MOSFETs with different silicon thickness is simulated for tox = 2 nm and Na = 1017 cm−3 numerically and analytically in Fig. 5. We have found that the
error fluctuations are all less than 4%.Fig. 6illustrates the simulated results of DG-MOSFET with different doping concentration for tsi=20 nm and tox =2 nm. The error is within 2%. The relation between parameters and doping
concentration, oxide thickness, channel thickness and gate voltage are given inTable 2. From the figures, the error increases at the top and bottom surface with the gate bias.
According to the results, the approximations of the analytical solution also present good agreement with the numerical results. The maximal error is about 4% and the average error is about 1%. If we go a step further to examine the simulated data, the maximal error always occurs near the surface of the channel in all cases. From Eq.(4), the first derivative of potential varies largely near the surface of the channel. However, the approximation, which is determined
Fig. 4. Comparison of the numerical and analytical potentials for the DG-MOSFET with respect to different tox, where tsi = 20 nm and
Na=1017cm−3. The left figure is with VG=0.1 V and the right one is with VG=1.0 V.
Fig. 5. Comparison of the numerical and analytical potentials for the DG-MOSFET with respect to different tsi, where tox = 2 nm and Na=1017cm−3. The left figure is with VG=0.1 V and the right one is with VG=1.0 V.
Fig. 6. Comparison of the numerical and analytical potentials for the DG-MOSFET with respect to different Na, where tsi=20 nm and tox=2 nm.
The left figure is with VG=0.1 V and the right one is with VG=1.0 V.
by Eqs.(5)and(8), is smaller than that. Fortunately, the error is small enough to be neglected without influencing the accuracy of circuit simulation and estimating electron density. In addition, the solution is accurate enough to describe the potential distribution of nanoscale DG-MOSFETs. It can be a helpful equation while simulating the characteristics of DG-MOSFETs so as to design advanced devices.
Fig. 7. The classical (left figure) and quantum corrected (right figure) electron densities using the derived potential solution. Comparison of the numerical and analytical results for the DG-MOSFET with respect to different VG, where Na=1017cm−3, tox=2 nm and tsi=20 nm.
Fig. 8. The classical (left figure) and quantum corrected (right figure) electron density using the derived potential solution. Comparison of the numerical and analytical results for the DG-MOSFET with respect to different VG, where Na=1017cm−3, tox=2 nm and tsi=5 nm. 4. Application to quantum correction of electron density
As mentioned above, the analytical solution presents accurate results and time-efficient solving methods for the potential of DG-MOSFETs. Since classical potential of a device is obtained, with Boltzmann statistics electron density can be calculated by the following formula
nCL=niexp(q (φ − φn) /kT ) . (20)
In Eq.(20), nCLis classical electron density,φnis quasi-Fermi potential of electron and k is Boltzmann’s constant.
Furthermore, the electron density with quantum effect is obtained by coupling the quantum correction model [9–13] with the classical electron density. The unified quantum correction model is given as
nQM=a0nCL " 1 − exp −a1ξ2 1 − 1 2 ξ ξ0 2! −a2ξ3 !# , (21)
where nQM is quantum electron density, a0, a1 and a2 are fitting parameters, ξ0 = Tsi/2λth, ξ = x/λth and
λth = h¯/2 × 9.11 × 10−31kT. ¯h is the reduced Planck constant. Classical and quantum electron density of a
DG-MOSFET with tox = 2 nm, tsi = 20 nm and Na = 1017 cm−3 are shown in Fig. 7. Examining the application
of the analytical solution to nanoscale devices, we simulate a tsi = 5 nm, tox = 2 nm, Na = 1017 cm−3
DG-MOSFET.Fig. 8shows the classical and quantum electron density. The analytical results show good agreement with the numerical results [9–13]. Therefore, the analytical solution of the Poisson equation provides a way to simulate nanoscale DG-MOSFET.
correction model for the numerical simulation of nanoscale DG-MOSFETs. We are currently deriving the solution for the case of asymmetric DG-MOSFETs.
Acknowledgments
This work was supported in part by the National Science Council of TAIWAN under Contract NSC-95-2221-E-009-336, Contract NSC-95-2752-E-009-003-PAE, Contract NSC-96-2415-H-216-001, by the MoE ATU Program, Taiwan under a 2006–2007 grant, and by the Taiwan semiconductor manufacturing company under a 2006–2007 grant.
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