Linearly graded doping drift region: anovel lateral voltage-sustaining layer used for improvement of RESURF LDMOS transistor performances

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1 INSTITUTE OF PHYSICS PUBLISHING Semicond. Sci. Technol. 17 (22) SEMICONDUCTOR SCIENCE AND TECHNOLOGY PII: S (2) Linearly graded doping drift region: anovel lateral voltage-sustaining layer used for improvement of RESURF LDMOS transistor performances Jin He 1,2,XuemeiXi 2,MansunChan 2,Chenming Hu 2, Yingxue Li 1,Zhang Xing 1 and Ru Huang 1 1 Institute of Microelectronics, Peking University, Beijing 1871, People s Republic of China 2 Electronic Research Laboratory of Electronic Engineering and Computer Science Department, University of California, Berkeley, CA 9472, USA jinhe@fermi.eecs.berkeley.edu Received 5 February 22, in final form 17 May 22 Published 18 June 22 Online at stacks.iop.org/sst/17/721 Abstract A linearly graded doping drift region structure, a novel lateral voltage-sustained layer used for improvement of reduced surface field (RESURF) LDMOS transistor performance has been evaluated theoretically, numerically and experimentally in this paper for the first time. Due to the coupling effect of the two-dimensional (2D) electrical field, it is found from the theory developed here that the linearly graded drift region-doped profile can provide a high breakdown voltage while maintaining a high doping dose in the total drift region for minimizing the on-resistance R on.the characteristics of such an LDMOS have been demonstrated by the 2D semiconductor device simulator MEDICI and further verified by our experimental results. We have obtained a reduction of the on-resistance of 5% from 1.3 m cm 2 to5m cm 2 in the on-state, and an increase of the breakdown voltage by a factor of 2.5 from 9 V to 234 V in the off-state, compared to the values for conventional RESURF devices. The experimental results verify the performance improvement predicted by the simulation and theory. (Some figures in this article are in colour only in the electronic version) 1. Introduction High-voltage integrated circuits which typically combine one or more high-voltage transistors on the same chip with lowvoltage circuits are widely used in various electric applications, such as telecommunication, electronic luminescent displays and lamp ballasts, etc. In these circuits, a so-called lateral double-diffused MOS transistor (LDMOS) is the most important conventionally used high-voltage device. In the practical design of the LDMOS transistor, it is a fundamental requirement to minimize the on-resistance while still maintaining a high breakdown voltage. However, these two electrical parameters often conflict with each other in the present day processing technology. For instance, the conventional reduced surface field (RESURF) technique can not only produce a high breakdown voltage but also a high on-resistance [1, 2]. Although many methods, such as the multiple-resistivity drift region technique [3], surface ion implantation in the drift region and use of an SIPOS resistance layer on the oxide layer have been proposed to reduce the on-resistance [4, 5], no considerable improvement has been obtained. In this case, optimization of the device parameters is very important to obtain an acceptable trade-off between the voltage blocking capability and the on-resistance [6 8] /2/7721+8$3. 22 IOP Publishing Ltd Printed in the UK 721

2 JHeet al In the early bulk silicon technology, a lateral nonuniform doping profile was used to avoid low breakdown voltage caused by the radius of curvature of the metallurgical junction [9]. Continuous graded junction terminations [1, 11] were also used for the same purpose. Extending the reduced surface field (RESURF) principle to silicon-on-insulator (SOI) technology, Merchant et al developed a theoretical model for optimizing the breakdown voltage of thin film SOI RESURF LDMOS transistors [12], which predicted that the linear lateral doping profile in the drift region on SOI could be designed to attain maximum breakdown voltage [13, 14]. A computer program was developed to realize this profile and experimental verifications were also performed [15, 16]. However, these results were only available for thin film SOI RESURF LDMOS devices. As is well known, the advantages and applications of thin film SOI technology using the RESURF principle are countered somewhat by high cost of the SOI substrate materials and self-heating of the high-voltage power devices. In this paper, the non-uniform lateral doping profile is employed once again with bulk silicon technology to produce junction isolation RESURF LDMOS devices. Based on modulation of the lateral electrical field distribution in the drift region rather than the traditional junction termination shaping, a novel lateral voltage-sustained layer with a linearly graded doping region is proposed for the junction isolation RESURF LDMOS transistor, and an analytical theory is developed based on the two-dimensional Poisson equation. Following this, a RESURF LDMOS transistor with such a linearly graded drift region-doped profile has been simulated by the 2D semiconductor device simulator MEDICI and implemented practically. The numerical analysis and experimental results illustrate that this novel device has an improved trade-off between the breakdown voltage and on-resistance compared to conventional constant drift region doping structures, verifying our theoretical predictions. We believe that use of this novel structure in high voltage and SPIC circuits can significantly improve system performance. 2. Theory Asimplified schematic cross section of the RESURF LDMOS transistor is shown in figure 1. In the present analysis, the epilayer length of the drift region is defined as the n-type region length L between the n+ drain and p+ well. The epilayer thickness is t 1 with a doping concentration profile of N d (x). We assume that the substrate is thick enough to deplete the substrate charge. The substrate doping concentration is N sub, the depletion layer thickness t 2, the thickness of the field oxide layer t f,thedielectric constant ε ox,andthehorizontal and vertical positions relative to the silicon surface are x and y, respectively. The device is biased in the off-state configuration; substrate, source and gate are grounded while the drain is biased to a positive voltage V d. As shown in [14], the ionization path may be taken as a horizontal path along the top surface or a vertical path at the right edge of the depletion region for the RESURF structure. However, as the thick substrate side supports voltage effect in the epilayer RESUF LDMOS structure, an avalanche breakdown may be described by the ionization integral over Source Gate L N (x) d Substrate Drift region Figure 1. Cross-sections of the proposed RESURF LDMOS transistor. t f t 1 t 2 Drain the horizontal path, assuming that the depletion region reaches the drain diffusion. The ionization integral may be written as I[φ f (x)] = L (φ f (x)) dx (1) where φ f (x) is the electrostatic potential along the top surface of the drift region, α[φ f (x)] istheionization rate and E(x) = φ f (x) = dφf(x) is the magnitude of the lateral electric field in dx Vcm 1. An optimum RESURF structure would have the least ionization for a given drift length and drain voltage. The α[φ f (x)] thatminimizes I for a given V d and L is the solution of Euler s equation α φ f (x) d α =. (2) dx φ f (x) In the case of a power law for the effective ionization rate, we have α[φ f (x)] = A e B/φf(x). (3) As shown in [14], equations (2)and(3)implythat φ f (x) = kx. (4) Thus, the optimum RESURF device would have a uniform lateral electric field profile E = Vd = k. Asaresult, we obtain L φ f (x) = V d x. (5) L It is well known that an avalanche breakdown occurs when the ionization integral equals 1. Our treatment is not the same as that in [14], and the well-known Fulop s formula [17] for silicon materials is adopted here α eff = AE 7 (6) with A = cm 1.Theideal breakdown voltage and the critical field are found to be ( L 6 ) 1/7 V b = (7) A [ L 1 ] 1/7 E C =. (8) A The ideal breakdown voltage and critical peak field for the ideal doping profile and the uniform doping profile are plotted in figure 2. Forthesakeofcomparison, the results of the 722

3 Linearly graded doping drift region Critical electric field [ V/cm ] Breakdown voltage [ V ] 4.x x1 5 3.x x1 5 2.x x x x x1 3 1.x1 3 8.x1 2 6.x1 2 4.x1 2 2.x1 2 Optimum doping result obtained by S.Merchant [14] Uniform doping result Optimum doping result obtained by our analysis Drift region length [ µm] 2.x x1 3 Optimum doping result obtained by S.Merchant [14] (a) Optimum doping result obtained by our analysis uniform doping result Drift region length [ µm] (b) Figure 2. (a)dependence of the breakdown voltage on drift region length for the RESURF structure for optimum and uniform doping profiles, respectively. (b)dependence of the critical electric field on drift region length for the RESURF structure for optimum and uniform doping profiles, respectively. optimum doping SOI drift region obtained by Merchant [14] are also plotted in the figure, where there is good accordance between our analysis and their data. Moreover, we see that the optimum doping profile has a significantly improved breakdown voltage compared with the uniform doping profile with good agreement between different treatments. The potential function φ(x, y) inthedrift region of the silicon film must be satisfied by the Poisson equation, yielding d 2 φ(x,y) + d2 φ(x,y) = qn d(x). (9) dx 2 dy 2 As shown infigure 1, the region under investigation is a box drift region. It is possible to derive from the 2D Poisson equation a 1D equation describing the surface potential in the lateral coordinate (x). In this work, a more general analysis integrates the Poisson equation over the y-direction in the drift region tsi 2 φ(x,y) x 2 dy + E y (x, ) E y (x, t 1 ) = qn d(x) t 1. (1) Under the assumption of a 1D electrical field in the SiO 2 material, the continuity of electrical flux along the Si/SiO 2 interface makes the boundary conditions for (9)satisfy E y (x, ) = ε ox φ f (x) V gs (11) t f where φ f (x) φ(x,) is the potential function along the Si/SiO 2 interface, V gs = V gs V FB,f is the effective gate-tosource bias voltage, and V FB,f is the channel flat-band voltage. Because the gate oxide layer and substrate bias equal zero, neglecting the influence of the work function difference between the metal and semiconductor on the electric field, the electric field at the interface can be approximated by E y (x,) ε ox φ f (x). (12) t f Combining (1)with (12), we obtain t1 2 φ(x,y) x 2 dy ε ox φ f (x) E y (x, t 1 ) = qn d(x) t 1. t f (13) According to the RESURF principle, the epilayer should be completely depleted and the depletion approximation suitable for the drift region. To a first-order approximation, the integral term of (13) can be reduced to afunction of the front surface potential by assuming 2 φ(x,y) 2 φ f(x). This x 2 x 2 approximation generally exists for the depleted structure, as shown in [18, 19]. The relationship between φ f (x) and φ b (x) derived by solving the Poisson s equation (9) inthevertical direction through the above approximation is thus φ f (x) = φ b (x) + E y (x, t 1 )t 1 1 [ ] qnd + 2 φ f t 2 2 x 2 1 (14) where φ b (x) φ(x,t 1 )is defined as the electrostatic potential at the metallurgical junction of the vertical n /p junction. From basic p n junction theory, φ b (x) can be simplified to φ b (x) = qn sub t2 2 2ε. (15) Si Putting (14) and (15) into (13) and after further mathematical simplification, (13) istransformed into the 1D differential equation d 2 φ f (x) α dx 2 f φ f (x) = β f (16) where [ a f = 2ε ox + 2 t 1 t f t1 2 and β f = q ( ) ] 2 t2 N d (x) + N sub. t 1 In order to determine the depletion layer thickness on the substrate side, we must solve for the potential distribution of the double-sided p n junction in the vertical direction. Assuming the depletion width along the drift region length direction to be uniform, the general formula of the doublesided junction for the applied drain voltage V d can be given by E max = V d φ b t 1 + qn d(x)t 1 2 = qn subt 2. (17) 723

4 JHeet al Table 1. Device structureparametersused in the numericalsimulation. Parameters Conventional LDMOS Proposed LDMOS Substrate doping concentration N sub (cm 3 ) Drift region-doped concentration N d (cm 3 ) ( x/l) Epitaxial layer thickness (µm) 4 4 Drift region length (µm) Combining (15)with (17), we obtain the quadric equation t 2 = 1+ N d(x) + 2V d t 1 N sub qn sub t (18) It is evident that t 2 tends to zero when the substrate doping concentration N sub tends to a very high value, such as infinity. The above form is only applicable for the condition when the drain voltage makes the depletion width larger than t 1. For RESURF LDMOS devices, this condition is always valid. As aresultequation (16) becomes d 2 [ φ f (x) 2 + 2ε ] ox φ dx 2 f (x) t 1 t f = q t 2 1 N d(x) + N sub [ 1+ N d(x) N sub + 2V d qn sub t 2 1 ] 2 1 (19) Assuming that the RESURF LDMOS structure has an optimum doping profile, from equation (4)andtheuniformity of the electric field we obtain d 2 φ f (x) = de(x) =. (2) dx 2 dx Inserting (2)into(16) gives [ φ f (x) = q ( ) ] 2 t2 t1 2 N d (x) + N t f sub. (21) 2 t 1 t 1 ε ox / + t f Finally, combining (5)and(21), we obtain ( t2 ) 2. (22) N d (x) = 2(t 1 ε ox / + t f )V d qt f t1 2L x N sub t 1 As shown in (22), the ideal drift region doping profile for maximum breakdown voltage is a linearly graded function, with an amplitude related to the drift region length, epitaxial layer thickness, field oxide thickness and substrate doping concentration. Moreover, the optimum doping gradient can be obtained by differentiating (22)with respect to x: G = 2(t 1 ε ox / + t f )V d qt1 2t. (23) fl Based on the above theoretical analysis, we can draw averyusefulconclusion: to obtain the maximum breakdown voltage and thus an improved trade-off, the linearly graded drift region-doping profile is the ideal choice. For such LDMOS transistor structures, the procedure for obtaining the optimum design parameters is as follows. For a given breakdown voltage V d the required drift region length L is calculated by using (7); following the determination of the field oxide thickness t f and the epitaxial thickness t 1,the slope of the linear doping profile G is obtained from (23), thus the maximum doping concentration N d (L) in the drift region can be obtained. Solving the equations (18) and(22) thengivesthesubstrate doping concentration N sub and the required smallest depletion layer thickness t 2 in the substrate. For instance, the drift region length for BV = 25 V is 12 µm; the slope of the linear doping profile is cm 4 from (23)forafield oxide thickness of 45 Åandanepitaxy thickness of 4 µm. The remaining parameters can also be determined following theabove procedure, e.g., the maximum doping concentration in the drift region is found to be cm 3 and the implantation dose about cm 2. The substrate doping concentration is determined to be cm 3 for the substrate. The following sections describe the numerical analysis and experimental verification obtained by the TMA MEDICI simulations and the implementation of such a RESURF LDMOS transistor. 3. Simulation The cross-sectional view of the proposed RESURF LDMOS is already shown in figure 1. Thedevice parameters are listed in table 1 based on the calculation in the above section for a breakdown voltage of 25 V. The linearly graded drift region-doping profile of N d (x) inthe proposed LDMOS is matched by multiple stepwise variation of the uniform doping concentration with different lengths. In practical devices, the linearly graded drift region-doping profile can be created using asingle phosphorus implant through a mask with a series of openings that are smaller near the source and larger near the drain [11]. For the sake of comparison, a conventional RESURF LDMOS structure with a constant drift region doping concentration cm 3 and epilayer thickness of 4 µm issimultaneously considered, with a drift region length of 12 µmand substrate doping concentration of cm 3. The structure parameters used in the simulation are also shown in table 1. The source and drain doping concentrations are cm 3,while the doping profile of the p-base region is agaussian function with a surface doping concentration of cm 3 and junction depth of 3.2 µm. The ratio of lateral/vertical diffusion is assumed to be.8. The other parameters in both structures are the same unless a special note is given. Due to the coupling effect of the 2D electrical field distribution, the RESURF-based LDMOS demonstrates a higher breakdown voltage. However, it is very difficult to give an analytic expression for the 2D field distribution. In this work we use the commercial semiconductor simulation software MEDICI to verify the characteristics of the LDMOS. The high accuracy of the MEDICI simulation has been supported by extensive experimental results, proving that MEDICI can be used for demonstrating device characteristics. Figure 3 describes the simulated off-state potential contours within both device structures at breakdown. Making 724

5 Linearly graded doping drift region (a) Figure 3. Simulated blocking capability of (a)conventional RESURF LDMOS with a constant drift region doping of cm 3 and contours = 1 V/step, and (b)proposed RESURF LDMOS with a drift region doping of ( x/l) cm 3 and contours = 24 V/step. (b) the extracted ionization integral equal to 1, the avalanche breakdown voltages can be readily obtained. As observed in this figure, the breakdown voltage of the conventional RESURF LDMOS is limited to 9 V, which is due to the enhanced surface peak field near the drain edge. When the breakdown voltage for the linearly graded doping-drift region RESURF LDMOS is 234 V, an enhancement factor of 2.5 is obtained because the smooth profile of the surface electrical field significantly decreases the surface peak field near the source and drain edges. It should be noted that the potential contour lines near the drain in thenewstructurearesmoothed away in figure 3(b), while they are crowded at the edge of the drain in figure 3(a) forconventional LDMOS structures. The simulated one-dimensional field distributions in top silicon field oxide interfaces are illustrated in figure 4,where the surface electrical field distribution of the linearly graded drift region-doped LDMOS is more or less uniform in the entire drift region, leading toanenhanced breakdown voltage of 24 V. In contrast, the electrical field appears to be nonuniform in the conventional RESURF LDMOS, as shown in figure 4(b). The peak field reaches a critical electrical field magnitude of Vcm 1 at the drain edge, giving rise to a breakdown voltage of 9 V. Figure 5 demonstrates the results of the simulated potential distributions at breakdown. We can see from figure 5(a) thatitisjust the linearly graded distribution of the surface potential that leads totheuniform field profile in most of the drift region of the linearly graded drift region-doped LDMOS, as shown by our theory. The two peak fields at the edges of the source and drain are strongly related to the effect of the diffusion radius of curvature of the metallurgical junction, as shown in [2]. In contrast, the potential distribution of the conventional RESURF LDMOS shows a large curvature in the whole drift region, as shown in figure 5(b), which leads to a non-uniform electrical field profile Consequently, the breakdown voltage suffers a considerable degradation. The specific on-resistance R on can be extracted from the simulated forward characteristics in the linear region of the I V curves of the LDMOS. We find that the linearly graded 725

6 JHeet al 2.5x x1 5 2 Surface E-field [V/cm] 1.5x1 5 1.x1 5 5.x1 4 Potential [V] Distance along the semiconductor surface [µm] (a) Distance along the semiconductor surface [µm] (a) 2.5x x1 5 8 Surface E-field [V/cm] 1.5x1 5 1.x1 5 5.x1 4 Potential [V] Distance along the semiconductor surface [µm] (b) Distance along the semiconductor surface [µm] (b) Figure 4. Electrical field distribution near the silicon film/field oxide layer interface of (a) linearly graded doping drift region RESURF LDMOS and (b)conventional RESURF LDMOS. Figure 5. Potentialdistributionatthesiliconfilm/field oxidelayer interface of (a) linearly graded doping drift region RESURF LDMOS and (b)conventionalresurf LDMOS. drift region-doped LDMOS allows a significant reduction of R on when compared with the conventional LDMOS, which is just thenatural result of the high doping dose in the drift region. The numerically calculated values for R on are 1.3 m cm 2 and5m cm 2 for the conventional and our proposed LDMOS, respectively, yielding an improvement of 5%. The simulation results for BV and R on as a function of the drift region length are shown in figure 6, where a considerable improvement in both parameters is observed for the proposed LDMOS, promising a significantly improved trade-off for device performance. Breakdown voltage [V] Solid: Proposed LDMOS Hollow: common LDMOS On-resistance [1 µω.cm 2 ] 4. Experimental results The simulation results were verified by experimental realization of the optimal linearly graded drift region-doped RESURF LDMOS. It was found that a continuous lateral linearly graded doping profile can be achieved by using a layer of oxide or photoresist with a sequence of slit openings for masking of the impurity implantation [11]. These slits are continuously smaller toward the p + /n junction from the n side. All design details for implementing the linear doping profile in the device drift region may be found in [15]. A 2 V CMOS compatible LDMOS process was designed using the T-SUPERM-IV and implemented. The silicon material used was (1)-oriented with a substrate doping concentration of cm 3. After epitaxial growth of the drift region of 4 µm thickness with a doping Drift region length [µm] Figure 6. Trade-off between the breakdown voltage and on-resistance R on as a function of the drift region length for linearly graded doping drift region and conventional RESURF LDMOS transistors. concentration of cm 5,the process began by silicon mesa formation using reactive ion etching. Then, a.1 µm thick LTO was deposited, followed by photo-resist deposition on the drift region mask to define the silicon width and slit spacing. Phosphorus was then implanted into the silicon layer to form the drift region, with an implantation energy of 1 kev and a dose of cm 2 for the drain region. After photo-resist striping, a 4 nm thick LTO was deposited to prevent impurity out-diffusion into the ambient during the 726

7 5. Conclusion Linearly graded doping drift region A novel lateral voltage-sustained layer with linearly graded doping drift region has been evaluated and an analytical model developed. It has been shown that an optimal linearly graded drift region-doping profile for the RESURF LDMOS transistor can lead to a uniform distribution of the surface electric field as well as a linearly graded electrostatic potential, resulting in low on-resistance and high breakdown voltage. The significant characteristics of the RESURF LDMOS proposed here have been well demonstrated by the semiconductor device simulator MEDICI and verified by experimental results. Figure 7. Experimental forward conduction characteristics of the linearly graded doping drift region RESURF LDMOS transistor. Acknowledgments The authors would like to thank all referees for their helpful comments and good suggestions. References Figure 8. Experimental breakdown characteristics of the linearly graded doping drift region RESURF LDMOS transistor. drive-in step, which was performed in an inert ambient for 8 min at 12 Ctosmear out the impurity to obtain a linear doping profile along the drift region after the drift region drive-in. The remaining processes, such as gate region defining, p-well implantation and source and drain formation, were quite similar to those of the conventional LDMOS. Finally, a.5 µm thickltowas deposited and the contact holes to gate, source and drain were defined after LTO densification of 95 Cfor3 min. Further processes were similar to those ofcommon CMOS fabrication. Transistors with a 12 µmlong drift region and 4 nm thick field oxide longer were obtained. Figure 7 shows the forward condition characteristics. The transistor has a threshold voltage of 1.5 V and a specification resistance of 5.4 m cm 2. The experimental breakdown characteristics of the transistor with a breakdown voltage of 24 V are also shown infigure 8. Both the above electrical performance parameters agree well with the simulation results of 234 V and 5 m cm 2 obtained in the above section. We note from figure 7 that the drain current shows non-ohmic behaviour to the low drain voltage in the case of the high gate voltage, indicating the gate leakage current. This bad result can be improved by a high quality gate oxide. [1] Apples J A and Vaes H M J 1979 High voltage thin layer devices IEDM 79 Tech. Dig [2] Colak S 1981 Effects of drift region parameters on the static properties of power LDMOST IEEE Trans. Electron Devices [3] Mena J G and Salama C A T 1986 High-voltage multiple-resistivity drift-region LDMOS Solid State Electron [4] Wildi E J et al 1982 Modeling and process implementation of implanted RESURF type devices IEDM 82 Tech. Dig [5] Nakagawa Akio, Yasuhara Norio and Baba Yoshiro 1991 Breakdown voltage enhancement for device on thin silicon layer/silicon dioxide film IEEE Trans. Electron Devices [6] Xing-bi Chen, Song Z Q and Li Z J 1987 Optimization of the drift region of power MOSFET s with lateral structure and deep junctions IEEE Trans. Electron Devices [7] Parpia Z and Salama C A T 199 Optimization of RESURF LDMOS transistors: an analytical approach IEEE Trans. Electron Devices [8] Xing-bi Chen et al 1996 Lateral high-voltage devices using an optimal variation lateral doping Int. J. Electron [9] Tantraporn W and Temple V A K 1987 Multiple-zone single-mask junction termination extension a high-yield near-ideal breakdown voltage technology IEEE Trans. Electron Devices [1] Stengl R and Gosele U 1985 Variation of lateral doping a novel concept to avoid high voltage breakdown of planar junctions IEDM 85 Tech. Dig [11] Schulze H J and Kunhnert R 1989 Realization of a high-voltage planar junction terminations for power devices Solid State Electron [12] Merchant S, Arnold E, Baumgart H, Mukheerjee S, Pein H and Pinker R 1991 Realization of high breakdown voltage (>7 V) in thin SOI devices Proc. 3rd Int. Symp. Power Semiconductor Devices and IC s, ISPSD pp 31 5 [13] Merchant S 1999 Analytical model for the electrical field distribution in SOI RESURF and TMBS structures IEEE Trans. Electron Devices [14] Merchant S, Arnold E, Baumgart H, Egloff R, Letavic T, Mukheerjee S, Pein H and Pinker R 1993 Dependence of breakdown voltage on drift length and buried oxide thickness in SOI RESURF LDMOS transistor Proc. 5th Int. Symp. Power Semiconductor Devices and IC s, ISPSD pp

8 JHeet al [15] Lai T M L, Sin J K O, Wang M, Poon V M C and Ko P K 1995 Implementation of linear doping profile for high-voltage thin-film SOI devices Proc. 7th Int. Symp. PowerSemiconductor Devices and IC s, ISPSD pp [16] Zhang Shengdong, Sin Johnny K O, Lai T M L and Ko P K 1999 Numerical modeling of linear doping profiles for high-voltage thin-film SOI devices IEEE Trans. Electron Devices [17] Fulop W 1967 Calculation of avalanche breakdown voltage of the silicon p n junctions Solid State Electron [18] Chen Shiao-Shien and Kuo James B 1996 Deep sub-micrometer double-gate fully-depleted SOI PMOS devices: a concise short-channel effect threshold voltage model using a quasi-2d approach IEEE Trans. Electron Devices [19] Niu G F, Chen R M M and Ruan G 1996 Comparison and extension of recent surface potential models for fully depleted short-channel SOI MOSFET s IEEE Trans. Electron Devices [2] Sze S M and Gibbons G 1966 Effect of junction curvature on breakdown voltage of abrupt cylindrical and spherical junctions Solid State Electron

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