IEEE ELECTRON DEVICE LETTERS, VOL. 38, NO. 8, AUGUST 2017
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A General and Transformable Model Platform for Emerging Multi-Gate MOSFETs Chuyang Hong, Jun Zhou, Jiasheng Huang, Rui Wang, Wenlong Bai, James B. Kuo, Fellow, IEEE , and Yijian Chen, Member, IEEE Abstract — The complete general solution of nonlinear 1-D undoped Poisson’s equation, in both Cartesian and cylindrical coordinates, is derived by employing a special variable transformation method. A general model platform for various types of emerging multi-gate MOSFETs is further constructed and verified with TCAD simulations. It is shown that this model platform is suitable for analyzing a series of emerging devices, such as double-surroundinggate, inner-surrounding-gate, and outer-surrounding-gate nanoshell MOSFETs, all of which require different boundary conditions from the conventional gate-all-around nanowire device. Index Terms — Poisson’s equation, general solution, multi-gate devices.
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ULTI-GATE MOSFETs are widely considered as the viable candidate for both logic and memory device technologies as they gradually evolve into the deep nanometer regime and break the limitation of horizontal scaling [1], [2]. With their significantly improved short-channel control capability, multi-gate device structures include not only the commonly seen gate-all-around (GAA) nanowire MOSFET, but also emerging device schemes such as doublesurrounding-gate (DSG), outer-surrounding-gate (OSG), and inner-surrounding-gate (ISG) nanoshell MOSFETs [3]–[17], as shown in Fig. 1. For example, DSG nanoshell MOSFET was proposed to overcome the severe fabrication challenges of vertical GAA nanowire MOSFET because of its significant reduction of process induced variations and mechanical instability [4]–[9]. OSG/ISG nanoshell structures have also been Manuscript received June 15, 2017; revised June 26, 2017; accepted June 27, 2017. Date of publication June 30, 2017; date of current version July 24, 2017. This work was supported in part by the National Natural Science Foundation of China under Grant 61574002, in part by the Natural Science Foundation of Guangdong Province, China, under Grant 1414050004175, and in part by the Shenzhen City’s Strategic Development Fund for Fundamental Research under Grant JCYJ20150630152545236. The work of Y. Chen was supported by Shandong University under a granted adjunct professorship in its School of Microelectronics. The review of this letter was arranged by Editor A. Ortiz-Conde. (Chuyang Hong and Jun Zhou contributed equally to this work.) (Corresponding author: Yijian Chen.) C. Hong, J. Zhou, J. Huang, R. Wang, and W. Bai are with the School of Electronic and Computer Engineering, Shenzhen Graduate School, Peking University, Shenzhen 518055, China. J. B. Kuo is with the Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan. Y. Chen is with the School of Electronic and Computer Engineering, Shenzhen Graduate School, Peking University, Shenzhen 518055, China, and also with the School of Microelectronics, Shandong University, Jinan 25010, China (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LED.2017.2722227
Fig. 1. Schematic of (a) gate-all-around (GAA) nanowire MOSFET, (b) double-surrounding-gate (DSG) nanoshell MOSFET, (c) inner-surrounding-gate (ISG) nanoshell MOSFET, and (d) outersurrounding-gate (OSG) nanoshell MOSFET.
adopted for 3-D NAND flash devices to enable high density and reliability [10]–[15]. In the past years, compact models for FinFET and GAA nanowire MOSFET have been extensively studied and reported [18]–[22]. Most of these models are developed by solving Poisson’s equation in the corresponding coordinate suitable for only one type of devices (i.e., Cartesian coordinate for FinFET, or cylindrical coordinate for GAA nanowire MOSFET). Hence, an intriguing question is whether a unified model platform capable of connecting the device physics in Cartesian and cylindrical coordinates exists. Moreover, a functional model for DSG, OSG and ISG nanoshell MOSFETs is still missing, although some progress has been made recently [7]–[9], [23]. For example, in [8] and [23], the depletion approximation was assumed, which limits the model’s application within the sub-threshold regime. In [9], the reported particular solution of Poisson’s equation is based on the symmetric boundary conditions, thus only suitable for GAA nanowire MOSFET. In [7], it was found that the integration constants of certain “general” solution cannot yield physically meaningful potential unless defined in the complex domain. On the other hand, our physical intuition gives us a hint that there should exist a generic modeling approach that works for all of GAA nanowire and DSG/OSG/ISG nanoshell MOSFETs as these devices share the same Poisson’s equation in the cylindrical coordinate (despite that their boundary conditions are different). To investigate the above issues, we shall first introduce a novel variable transformation method to correlate the nonlinear 1-D Poisson’s equations in the Cartesian and cylindrical coordinates [24], [25]. Their complete forms of multi-branch general solutions are then derived, which confirms the possibility
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of a transformable model platform for both FinFET and other types of cylindrical MOSFETs. The physical and mathematical issues of co-existence of multi-branch potential distribution within the same device will be addressed. A general model is built based on different types of boundary conditions present in various emerging MOSFETs, and verified with TCAD simulations using the DSG and OSG MOSFETs as examples. The nonlinear 1-D undoped (or lightly doped) Poisson’s equation in the cylindrical coordinate is written as [21]: d 2z 1 dz = δe z , + 2 dr r dr
(1)
where δ = q 2 n i /(kT εsi ) and z(r ) = q [ϕ (r, y) − V (y)]/kT . ϕ (r, y) is the potential, V (y) is the quasi-Fermi potential. Two new variables are introduced as [24]–[26]: x = ln r, u = z + 2ln r.
(2)
Using the above two variables, Eq. (1) can be transformed to: d 2u = δeu . (3) dx2 Note that Eq. (3) is in the same form with the 1-D Poisson’s equation in the Cartesian coordinate. This indicates that 1-D undoped Cartesian and cylindrical Poisson’s equations can be transformed to each other. Integration of Eq. (3) with respect to u yields: 1 du 2 ( ) − eu = C, (4) 2δ d x where C is an integration constant equivalent to the coupling factor α in [27], C1 in [28] and G in [29], while its mathematical meaning here contains less ambiguity. Eq. (4) can be further integrated to: 1 x=± √ du. 2δ(eu + C) The integration result of the above equation depends on the sign of C [30]–[32]. In other words, there exist multiple branches of general solution of Eq. (4):
⎧ ⎪ −Cδ ⎪ ⎪ −2 ln sin + ln (−C) , C < 0, (x + D) ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ (5.1) ⎪ ⎪ ⎨−2 ln (x + D) + ln2 − lnδ, C = 0, u= ⎪ (5.2) ⎪
⎪ ⎪ ⎪ ⎪ Cδ ⎪ ⎪ + ln (C) , C > 0, −2 ln sinh (x + D) ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎩ (5.3) where D is the second integration constant. Eq. (5.1) is the well-known solution for symmetric double-gate MOSFET for which the symmetric boundary condition du/d x|x=0 = 0 is required to ensure C < 0 (e.g., according to Eq. (4)) [33], [34]. For asymmetric double-gate MOSFETs, the symmetric boundary condition is no longer valid. However, if there still exists a zero-field condition du/d x|x=x0 = 0 at certain channel location x 0 , C< 0 remains valid and only solution (5.1) is applicable. Otherwise, Eq. (4) should be rigorously applied to judge the sign of C and all the three branches (Eqs. (5.1)-(5.3))
can possibly be the applicable solution [19], [27]. In principle, we can calculate the critical value of (V g − V )cri at which C= 0 in an asymmetric double-gate MOSFET by substituting Eq. (5.2) into the silicon-oxide interface boundary condition and removing the constant D. When the applied voltageVg −V passes the critical value (V g − V )cri , the solution branch will change from (5.3) to (5.1). It can be readily proven that when C approaches zero, solutions (5.1) and (5.3) both degrade to (5.2). Thus, a smooth transition between neighboring solution branches is evident. In [19] and [35], the transitional branch (5.2) was not reported while (V g − V )cri was estimated by coupling solution (5.1) or (5.3) with the interface boundary condition with C forced to approach zero. These two approaches lead to the same result. The complete general solution of Poisson’s equation in the cylindrical coordinate also contains three branches. By simply substituting Eq. (2) into Eqs. (5.1-5.3), we have:
⎧ ⎪ −Cδ ⎪ 1 ⎪ −2 ln r − 2 ln √ sin (lnr + D) , ⎪ ⎪ −C ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ C < 0, (6.1) ⎪ ⎪ ⎨−2lnr − 2 ln (lnr + D) + ln2 − lnδ, z= ⎪ C= 0, (6.2) ⎪
⎪ ⎪ ⎪ ⎪ Cδ ⎪ ⎪ −2 ln r − 2 ln √1 sinh (lnr + D) , ⎪ ⎪ C 2 ⎪ ⎪ ⎪ ⎩ C > 0. (6.3) For GAA nanowire MOSFET, the symmetric boundary condition dz/dr |r=0 = 0 requires C = 2/δ> 0 such that Eq. (6.3) is the only applicable solution, which degrades to the well-known solution adopted by many GAA MOSFET models [36], [37]: z = A − 2 ln Br 2 + 1 where A = − ln δe−2D /8 and B = −e2D . Nevertheless, to our knowledge, the other two branches (6.1)-(6.2) have never been reported in semiconductor device literature. It should be reminded that for certain emerging cylindrical devices such as DSG, ISG and OSG nanoshell MOSFETs, the symmetric boundary condition mentioned above is invalid since the central point r = 0 is not enclosed within the channel. Consequently, the third branch (6.3) may not be the only applicable solution and the three branches must all be considered [38]. However, some authors (e.g., [7], [9], [16], [39]) seemed to be unware of the three-branch solution structure when modeling cylindrical MOSFETs. In general, if there exists a location r0 where (r dz/dr )|r=r0 + 2 = 0 is satisfied, only solution (6.1) is applicable since C < 0 is required. Considering a commonly seen case wherein there exists a critical point rc in the radial direction such that dz/dr |r=rc = 0 in the channel of a multi-gate MOSFET, Eq. (4) is written as: 2 C = − e zc rc2 , (7) δ where z c is the value of z at r = rc . The existence of zero-field points in the channel is a typical condition for a DSG nanoshell MOSFET when its inner and outer gates are biased at roughly the same voltage. It also holds true for an OSG (or ISG) nanoshell MOSFET because of the required zero-field boundary condition on the outer (or inner) surface (i.e., rc = r1(2) ) [7]. At different channel locations,
HONG et al.: GENERAL AND TRANSFORMABLE MODEL PLATFORM FOR EMERGING MULTI-GATE MOSFETs
Fig. 2. ϕ(rc ), ϕ(r½ ) and ϕ(r¾ ) as functions of V (y ) for a DSG nanoshell MOSFET. The inserted figure shows C as a function of V (y ).
the values of rc and z c may vary and consequently the integration constant (of r ) C is actually a function of the channel location y, with its sign depending on the operating condition and the y location. For a DSG nanoshell MOSFET, the inner and outer siliconoxide interface boundary conditions are given as:
∂ϕ Cox1 Vg − ϕ1 − ϕ S1 = −εSi , (8.1) ∂r r=r1
∂ϕ , (8.2) Cox2 Vg − ϕ2 − ϕ S2 = εSi ∂r r=r2 where the inner and outer capacitance (per unit area) are Cox1 = εox / {r1 ln [r1 / (r1 − tox )]} and Cox2 = εox / [r2 ln (1 + tox /r2 )], respectively. ϕ1 /ϕ2 is the workfunction difference between the inner/outer gate and channel. ϕ S1 and ϕ S2 are the potentials of the inner and outer channel surfaces. By coupling Eqs. (6) with (8), the values of C, D, and rc can be numerically solved. To develop a compact/analytical model for circuit simulation, certain approximations are necessary to simplify the numerical process. This needs to be further studied in the future. Since both charges near the inner and outer surfaces of a DSG nanoshell MOSFET contribute to the drain current, we can calculate the drain current by applying Pao-Sah integral [7], [40] to both surfaces: 2πr1 Vds 2πr2 Vds Ids = μ Q i1 (V ) d V + μ Q i2 (V ) d V L L 0 0 (9) where Q i1 (V ) = −εSi (∂ϕ/∂r |r=r1 ) and Q i2 (V ) = εSi (∂ϕ/∂r |r=r2 ). The boundary between the inner and outer regions (in the channel) is along the zero-field surface defined by Eq. (7). For a given Vg , Q i1 and Q i2 are functions of V according to Eqs. (6) and (8). For an OSG/ISG nanoshell MOSFET, Gauss’s Law requires the zero-field boundary condition on the surface inner/outer and Eq. (8.1) or (8.2) changes to: ∂ϕ ∂r r=r = 0. Sim1(2) ilarly, the drain current of an OSG/ISG nanoshell MOSFET can be calculated by setting Q i1(2) = 0 in Eq. (9). Let us first use a DSG nanoshell MOSFET as an example to demonstrate the functionality of our model. Fig. 2 illustrates the dependence of potentials at r1 , r2 and rc on V (y). For a given Vg , when V (y) is low, ϕ (rc ) is much lower than ϕ (r1 ) and ϕ (r2 ), which indicates the well-known “U”-shape profile of the potential along the radial direction. The difference between ϕ (rc ) and ϕ (r1 ) or ϕ (r2 ) becomes smaller as V (y) increases. When V (y) exceeds a certain value, the curves of ϕ (rc ), ϕ (r1 ) andϕ (r2 ) merge together.
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Fig. 3. Model calculation and TCAD simulation of drain current Id s versus (a) Vd s , (b) Vg for a DSG nanoshell MOSFET.
Fig. 4. (a) ϕ(r½ ) and ϕ(r¾ ) as functions of V (y ) for an OSG nanoshell MOSFET. The inserted figure shows C as a function of V (y ). (b) Model calculation and TCAD simulation of Id s versus Vd s for an OSG nanoshell MOSFET.
This suggests that the potential profile becomes nearly flat along the radial direction r when V (y) is higher. A similar so-called “quasi flat-band condition” can be observed in a double-gate MOSFET [27]. However, unlike a symmetric double-gate MOSFET wherein C can be judged as a negative value, the sign of C for a DSG nanoshell MOSFET remains to be determined from Eq. (4). As can be seen in the inserted (smaller) figure, C can be negative, positive, or zero asV (y) changes from zero to Vds . We further calculate the drain current of a DSG nanoshell MOSFET and compare the calculation results with TCAD simulations [41]. A lower electron mobility of μ = 300cm 2 /V s and a mid-gap work function for the gate material are assumed in this letter. In Fig. 3, we show the drain current versus gate and drain voltages calculated from our model for a DSG nanoshell MOSFET. A comparison with the results of TCAD simulation confirms their excellent agreement. Fig. 4(a) shows ϕ (r1 ) and ϕ (r2 ) as functions of V (y) for an OSG nanoshell MOSFET. Similar to a DSG nanoshell MOSFET, the difference between the potentials at rc (i.e., r1 ) and r2 becomes smaller as V (y) increases. Also, the inserted figure shows that the sign of C changes along the channel according to the value of V (y). This confirms the co-existence of multi-branch potential solutions within the same device. Fig. 4(b) shows the drain current versus drain voltage of an OSG nanoshell MOSFET calculated from our model and with TCAD simulations, which evidently verifies our model accuracy. In conclusion, the general solution of nonlinear 1-D undoped Poisson’s equation is obtained. It is found that Poisson’s equations in Cartesian and cylindrical coordinates can be transformed to each other. Based on a variable transformation method, a general model platform for various types of multi-gate MOSFETs is constructed and verified with TCAD simulations.
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