Analytical Potential Distribution Research Articles

Energy harvesting using a clamped–clamped piezoelectric–flexoelectric beam

In this article, the nonlinear vibration problem of a clamped–clamped monolithic piezoelectric flexoelectric beam is investigated when considering the device as an energy harvester. A scalable model of the beam is derived based on Euler–Bernoulli theory and takes into account geometric nonlinearity resulting from relatively large displacements and restrictive boundary conditions. The Galerkin method is used to discretize the nonlinear reduced-order model which is solved numerically using a continuation technique. The analysis of the dynamic response of the device at the micro and nano-scales demonstrate that such structure can be used to harvest energy despite its symmetry. This result was generally ignored in the literature for clamped–clamped single layered structures. The generated energy was demonstrated to be particularly attributed to the introduction of the longitudinal potential variation and the mid-plane stretching into the beam’s model. It was also found that the transverse piezoelectric coefficient is mainly responsible for the harvested energy. A 3D finite element model is developed providing the validation of the assumed assumptions, including the adopted analytical potential distribution particularly for the piezoelectric case. The nonlinear dynamic behavior of the beam is analyzed by varying the geometric dimensions of the device from the micro to the nano-scale. It is found that a better power density is obtained when larger scaling factors are used. However, a better performance can be achieved for an optimal scale factor at specific narrow frequency bandwidth nearby the fundamental frequency.

Streaming potential analysis and electrokinetic energy conversion efficiency of two immiscible fluids in a nanochannel

In order to conduct extensive investigation of electrokinetic energy conversion in nanofluidic devices, the analysis of streaming potential of pressure driven flow in the two-layer fluidic system through a nanochannel is investigated theoretically. Under the influences of the interfacial electric potential difference and the interface charge density jump, we first obtain the analytical electric potential distribution in the present two-layer fluidic system with consideration of Debye–Hückel linearization assumption. Then the analytical expressions of streaming potential field and flow velocity are derived. Based on the obtained the streaming potential, we finally give the analytical electrokinetic energy conversion efficiency in two-layer fluidic system and discuss the influences of related physical parameters on it. The theoretical result shows that the streaming potential can be viewed as a criterion to estimate the electrokinetic energy conversion efficiency. The electrokinetic energy conversion efficiency can be enhanced by the viscosity ratio and the interfacial slip length, but be restrained by the permittivity ratio and ion friction coefficient. Comparing to the single-layer fluidic system, the electrokinetic energy conversion efficiency can be augmented obviously in the two-layer fluidic system by selecting optimized flow parameters.

The Impact of Tunneling on the Subthreshold Swing in Sub-20 nm Asymmetric Double Gate MOSFETs

<p>This paper analyzes the subthreshold swing in asymmetric double gate MOSFETs with sub-20 nm channel lengths. The analysis of the carrier transport in the subthreshold region of these nano scaled MOSFET includes tunneling as an important additional mechanism to the thermionic emission. It is found that the subthreshold swing is increasing due to tunneling current and that the performance of nano scaled MOSFETs is degraded. The degradation of the subthreshold swing due to tunneling is quantified using analytical potential distribution and Wentzel–Kramers–Brillouin (WKB) approximation in this paper. This analytical approach is verified by two dimensional simulation. It is shown that the degradation of subthreshold swing increases with both reduction of channel length and increase of channel thickness. We also show that the subthreshold swing is increasing in case of different top and bottom gate oxide thicknesses.</p>

비대칭 DGMOSFET의 상하단 산화막 두께비에 따른 터널링 전류 분석

ABSTRACT This paper analyzes the deviation of tunneling current for the ratio of top and bottom gate oxide thickness of short channel asymmetric double gate(DG) MOSFET. The ratio of tunneling current for off current significantly increases if channel length reduces to 5 nm. This short channel effect occurs for asymmetric DGMOSFET having different top and bottom gate oxide structure. The ratio of tunneling current in off current with parameters of channel length and thickness, doping concentration, and top/bottom gate voltages is calculated in this study, and the influence of tunneling current to occur in short channel is investigated. The analytical potential distribution is obtained using Poisson equation and tunneling current using WKB(Wentzel-Kramers-Brillouin). As a result, tunneling current is greatly changed for the ratio of top and bottom gate oxide thickness in short channel asymmetric DGMOSFET, specially according to channel length, channel thickness, doping concentration, and top/bottom gate voltages.

비대칭 이중게이트 MOSFET에서 상단과 하단 산화막 두께비가 문턱전압이하 스윙에 미치는 영향

ABSTRACT Asymmetric double gate(DG) MOSFET has the different top and bottom gate oxides thicknesses. It is analyzed the deviation of subthreshold swing(SS) and conduction path for the ratio of top and bottom gate oxide thickness of asymmetric DGMOSFET. SS varied along with conduction path, and conduction path varied with top and bottom gate oxide thickness. The asymmetric DGMOSFET became valuable device to reduce the short channel effects like degradation of SS. SSs were obtained from analytical potential distribution by Poisson’s equation, and it was analyzed how the ratio of top and bottom oxide thickness influenced on conduction path and SS. SSs and conduction path were greatly influenced by the ratio of top and bottom gate oxide thickness. Bottom gate voltage cause significant influence on SS, and SS are changed with a range of 200 mV/dec for      under bottom voltage of 0.7 V. 키워드 : 비대칭 이중게이트, 문턱전압이하 스윙, 전도중심, 상하단 산화막 두께비 Key word : Asymmetric Double Gate, Subthreshold Swing, Conduction Path, Ratio of Top and Bottom Gate Oxide Thickness

비대칭 DGMOSFET의 도핑분포함수에 따른 DIBL

ABSTRACT This paper analyzes the phenomenon of drain induced barrier lowering(DIBL) for doping profiles in channel of asymmetric double gate(DG) MOSFET. The DIBL, the important short channel effect, is described as lowering of source barrier height by drain voltage. The analytical potential distribution is derived from Poisson’s equation to analyze the DIBL, and the DIBL is observed according to the change of doping profile to influence on potential distribution. As a results, the DIBL is significantly influenced by projected range and standard projected deviation, the variables of channel doping profiles. The change of DIBL shows greatly in the range of high doping concentration such as     . The DIBL increases with decrease of channel length and increase of channel thickness, and with increase of bottom gate voltage and top/bottom gate oxide film thickness. 키워드 : 비대칭 이중게이트, 드레인 유도 장벽 감소, 포아송방정식, 채널도핑분포 Key word : asymmetric double gate, drain induced barrier lowering, Poisson equation, channel doping profileJournal of the Korea Institute of Information andCommunication Engineering

10 nm 이하 비대칭 DGMOSFET의 채널도핑농도에 따른 터널링 전류

This paper analyzes the ratio of tunneling current for channel doping concentration of sub-10 nm asymmetric double gate(DG) MOSFET. The ratio of tunneling current for off current in subthreshold region increases in the region of channel length of 10 nm below. Even though asymmetric DGMOSFET is developed to reduce short channel effects, the increase of tunneling current in sub-10 nm is inevitable. As the ratio of tunneling current in off current according to channel doping concentration is calculated in this study, the influence of tunneling current to occur in short channel is investigated. To obtain off current to consist of thermionic emission and tunneling current, the analytical potential distribution is obtained using Poisson equation and tunneling current using WKB(Wentzel-Kramers-Brillouin). As a result, tunneling current is greatly changed for channel doping concentration in sub-10 nm asymmetric DGMOSFET, specially with parameters of channel length, channel thickness, and top/bottom gate oxide thickness and voltage.

도핑분포함수에 따른 비대칭 이중게이트 MOSFET의 문턱전압이동현상

ABSTRACT This paper has analyzed threshold voltage shift for doping profile of asymmetric double gate(DG) MOSFET. Ion implantation is usually used in process of doping for semiconductor device and doping profile becomes Gaussian distribution. Gaussian distribution function is changed for projected range and standard projected deviation, and influenced on transport characteristics. Therefore, doping profile in channel of asymmetric DGMOSFET is affected in threshold voltage. Threshold voltage is minimum gate voltage to operate transistor, and defined as top gate voltage when drain current is  per unit width. The analytical potential distribution of series form is derived from Poisson’s equation to obtain threshold voltage. As a result, threshold voltage is greatly changed by doping profile in high doping range, and the shift of threshold voltage due to projected range and standard projected deviation significantly appears for bottom gate voltage in the region of high doping concentration.

비대칭 DGMOSFET의 채널길이와 두께 비에 따른 문턱전압이하 스윙 분석

This paper has analyzed the variation of subthreshold swing for the ratio of channel length and thickness for asymmetric double gate MOSFET. The asymmetric double gate MOSFET has the advantage that the factors to control the short channel effects increase since top and bottom gate structure can be fabricated differently. The degradation of transport property due to rapid increase of subthreshold swing can be specially reduced in the case of reduction of channel length. However, channel thickness has to be reduced for decrease of channel length from scaling theory. The ratio of channel length vs. thickness becomes the most important factor to determine subthreshold swing. To analyze hermeneutically subthreshold swing, the analytical potential distribution is derived from Poisson’s equation, and conduction path and subthreshold swing are calculated for various channel length and thickness. As a result, we know conduction path and subthreshold swing are changed for the ratio of channel length vs. thickness.

비대칭 이중게이트 MOSFET의 차단전류에 대한 전도중심 의존성 분석

Asymmetric double gate(DG) MOSFET is a novel transistor to be able to reduce the short channel effects. This paper has analyzed a off current for conduction path of asymmetric DGMOSFET. The conduction path is a average distance from top gate the movement of carrier in channel happens, and a factor to change for oxide thickness of asymmetric DGMOSFET to be able to fabricate differently top and bottom gate oxide thickness, and influenced on off current for top gate voltage. As the conduction path is obtained and off current is calculated for top gate voltage, it is analyzed how conduction path influences on off current with parameters of oxide thickness and channel length. The analytical potential distribution of series form is derived from Poisson’s equation to obtain off current. As a result, off current is greatly changed for conduction path, and we know threshold voltage and subthreshold swing are changed for this reasons

10 nm이하 비대칭 이중게이트 MOSFET의 하단 게이트 전압에 따른 터널링 전류 분석

This paper analyzed the deviation of tunneling current for bottom gate voltage of sub-10 nm asymmetric double gate MOSFET. The asymmetric double gate MOSFET among multi gate MOSFET developed to reduce the short channel effects has the advantage to increase the facts to be able to control the channel current, compared with symmetric double gate MOSFET. The increase of off current is, however, inescapable if aymmetric double gate MOSFET has the channel length of sub-10 nm. The influence of tunneling current was investigated in this study as the portion of tunneling current for off current was calculated. The tunneling current was obtained by the WKB(Wentzel-Kramers-Brillouin) approximation and analytical potential distribution derived from Poisson equation. As a results, the tunneling current was greatly influenced by bottom gate voltage in sub-10 nm asymmetric double gate MOSFET. Especially it showed the great deviation for channel length, top and bottom gate oxide thickness, and channel thickness.

이중게이트 MOSFET의 대칭 및 비대칭 산화막 구조에 대한 문턱전압 분석

본 연구에서는 대칭 및 비대칭 산화막 구조를 가진 이중게이트(double gate; DG) MOSFET의 문턱전압 변화에 대하여 분석하였다. 상하단 동일한 산화막 두께을 갖는 대칭 DGMOSFET와 달리 비대칭 DGMOSFET는 상하단 게이트 산화막 두께를 다르게 제작할 수 있다. 그러므로 비대칭 DGMOSFET에서 상단과 하단게이트 산화막 두께의 크기 변화에 따라 대칭 DGMOSFET와 문턱전압을 비교하여 상하단 게이트 산화막 두께의 최적값에 대하여 고찰하고자 한다. 문턱전압을 구하기 위하여 포아송방정식에서 해석학적 전위분포모델을 유도하였으며 도핑분포함수는 가우스분포함수를 사용하였다. 문턱전압 모델을 이용하여 하단게이트 전압, 채널길이 및 채널두께 등에 따라 상하단게이트 산화막 두께가 문턱전압에 미치는 영향을 관찰하였다. 결과적으로 문턱전압은 상하단 게이트 산화막 두께에 따라 크게 변화하였으며 변화하는 경향은 하단게이트 전압, 채널길이 그리고 채널두께에 따라 매우 상이하게 나타나고 있다는 것을 알 수 있었다. This paper has analyzed the change of threshold voltage for oxide structure of symmetric and asymmetric double gate(DG) MOSFET. The asymmetric DGMOSFET can be fabricated with different top and bottom gate oxide thickness, while the symmetric DGMOSFET has the same top and bottom gate oxide thickness. Therefore optimum threshold voltage is considered for top and bottom gate oxide thickness of asymmetric DGMOSFET, compared with the threshold voltage of symmetric DGMOSFET. To obtain the threshold voltage, the analytical potential distribution is derived from Possion's equation, and Gaussian distribution function is used as doping profile. We investigate for bottom gate voltage, channel length and thickness, and doping concentration how top and bottom gate oxide thickness influences on threshold voltage using this threshold voltage model. As a result, threshold voltage is greatly changed for oxide thickness, and we know the changing trend greatly differs with bottom gate voltage, channel length and thickness, and doping concentration.

비대칭 이중게이트 MOSFET의 상하단 산화막 두께비에 따른 전도중심에 대한 문턱전압 의존성

ABSTRACT This paper has analyzed the change of threshold voltage and conduction path for the ratio of top and bottom gate oxide thickness of asymmetric double gate MOSFET. The asymmetric double gate MOSFET has the advantage that the factor to be able to control the current in the subthreshold region increases. The analytical potential distribution is derived from Poisson’s equation to analyze the threshold voltage and conduction path for the ratio of top and bottom gate oxide thickness. The Gaussian distribution function is used as charge distribution. This analytical potential distribution is used to derive off-current and subthreshold swing. By observing the results of threshold voltage and conduction path with parameters of bottom gate voltage, channel length and thickness, projected range and standard projected deviation, the threshold voltage greatly changed for the ratio of top and bottom gate oxide thickness. The threshold voltage changed for the ratio of channel length and thickness, not the absolute values of those, and it increased when conduction path moved toward top gate. The threshold voltage and conduction path changed more greatly for projected range than standard projected deviation.

비대칭 이중게이트 MOSFET의 도핑농도에 대한 문턱전압이동

본 연구에서는 비대칭 이중게이트(double gate; DG) MOSFET의 채널 도핑농도 변화에 따른 문턱전압이동 현상에 대하여 분석하였다. 비대칭 DGMOSFET는 일반적으로 저 농도로 채널을 도핑하여 완전결핍상태로 동작하도록 제작한다. 불순물산란의 감소에 의한 고속 동작이 가능하므로 고주파소자에 응용할 수 있다는 장점이 있다. 미세소자에서 필연적으로 발생하고 있는 단채널 효과 중 문턱전압이동현상이 비대칭 DGMOSFET의 채널도핑농도의 변화에 따라 관찰하고자 한다. 문턱전압을 구하기 위하여 해석학적 전위분포를 포아송방정식으로부터 급수형태로 유도하였다. 채널길이와 두께, 산화막 두께 및 도핑분포함수의 변화 등을 파라미터로 하여 도핑농도에 따라 문턱전압의 이동현상을 관찰하였다. 결과적으로 도핑농도가 증가하면 문턱전압이 증가하였으며 채널길이가 감소하면 문턱전압이 크게 감소하였다. 또한 채널두께와 하단게이트 전압이 감소하면 문턱전압이 크게 증가하는 것을 알 수 있었다. 마지막으로 산화막 두께가 감소하면 문턱전압이 증가하는 것을 알 수 있었다. This paper has analyzed threshold voltage movement for channel doping concentration of asymmetric double gate(DG) MOSFET. The asymmetric DGMOSFET is generally fabricated with low doping channel and fully depleted under operation. Since impurity scattering is lessened, asymmetric DGMOSFET has the adventage that high speed operation is possible. The threshold voltage movement, one of short channel effects necessarily occurred in fine devices, is investigated for the change of channel doping concentration in asymmetric DGMOSFET. The analytical potential distribution of series form is derived from Possion's equation to obtain threshold voltage. The movement of threshold voltage is investigated for channel doping concentration with parameters of channel length, channel thickness, oxide thickness, and doping profiles. As a result, threshold voltage increases with increase of doping concentration, and that decreases with decrease of channel length. Threshold voltage increases with decrease of channel thickness and bottom gate voltage. Lastly threshold voltage increases with decrease of oxide thickness.

비대칭 DGMOSFET의 도핑분포함수에 따른 전도중심과 문턱전압이하 스윙의 관계

본 연구에서는 비대칭 이중게이트(double gate; DG) MOSFET의 채널 내 도핑분포함수에 따른 전도중심과 문턱전압이하 스윙의 관계에 대하여 분석하였다. 비대칭 DGMOSFET의 채널크기는 매우 작기 때문에 불순물의 수가 매우 작으므로 고 도핑된 채널의 경우에 대하여 분석하였다. 이를 위하여 포아송방정식에서 해석학적 전위분포모델을 유도하였으며 도핑분포함수는 가우스분포함수를 사용하였다. 해석학적 전위분포모델을 이용하여 전도중심 및 문턱전압이하 스윙모델을 유도하였으며 채널길이 및 채널두께가 변할 때, 도핑분포함수의 변수인 이온주입범위 및 분포편차에 따른 전도중심 및 문턱전압이하 스윙의 변화를 관찰하였다. 결과적으로 전도중심이 상단게이트 단자로 이동할 때, 문턱전압이하 스윙 값은 감소하였으며 단채널 효과에 의하여 채널길이 감소 및 채널두께 증가에 따라 문턱전압이하 스윙 값은 증가하였다. This paper has analyzed the relation of conduction path and subthreshold swing for doping profile in channel of asymmetric double gate(DG) MOSFET. Since the channel size of asymmetric DGMOSFET is greatly small and number of impurity is few, the high doping channel is analyzed. The analytical potential distribution is derived from Possion's equation, and Gaussian distribution function is used as doping profile. The conduction path and subthreshold swing are derived from this analytical potential distribution, and those are investigated for variables of doping profile, projected range and standard projected deviation, according to the change of channel length and thickness. As a result, subthreshold swing is reduced when conduction path is approaching to top gate, and that is increased with a decrease of channel length and a increase of channel thickness due to short channel effects.

DGMOSFET의 도핑분포에 따른 상 · 하단 전류분포 및 차단전류 분석

ABSTRACT This paper has analyzed the change of forward and backward current for channel doping concentration to analyze off-current of double gate(DG) MOSFET. The Gaussian function as channel doping distribution has been used to compare with experimental ones, and the two dimensional analytical potential distribution model derived from Poisson's equation has been used to analyze the off-current. The off-current has been analyzed for the change of projected range and standard projected range of Gaussian function with device parameters such as channel length, channel thickness, gate oxide thickness and channel doping concentration. As a result, this research shows the off-current has greatly influenced on forward and backward current for device parameters, especially for the shape of Gaussian function for channel doping concentration. 키워드 : DGMOSFET, 소자파라미터, 상 · 하단 전류, 차단전류, 포아송방정식 Key word : DGMOSFET, device parameter, forward/backward current, off-current, Poisson equationJournal of the Korea Institute of Information andCommunication Engineering

이차원 전위분포모델을 이용한 이중게이트 MOSFET의 항복전압 분석

본 연구에서는 이중게이트 MOSFET에 대한 항복전압의 변화를 채널도핑 및 소자파라미터에 따라 이차원 전위분포모델을 이용하여 분석한 것이다. 낮은 항복전압은 전력소자동작에 저해가 되고 있으며 소자의 크기가 감소하면서 발생하는 단채널 효과에 의하여 이중게이트 MOSFET의 경우도 심각하게 항복전압이 감소하고 있다. 항복전압분석을 위하여 포아송방정식의 이차원 해석학적 전위분포모델을 이용하여 채널도핑농도와 소자 파라미터인 채널길이, 채널두께, 게이트산화막 두께 등에 대하여 항복전압의 변화를 관찰하였다. 분석결과 항복전압은 채널도핑 농도의 크기뿐만이 아니라 소자크기 파라미터에 대해서 커다란 변화를 보이고 있었으며 특히 채널도핑함수인 가우시안 함수의 형태에 따라서도 큰 변화를 보이고 있다는 것을 관찰할 수 있었다. This paper have analyzed the change of breakdown voltage for channel doping concentration and device parameters of double gate(DG) MOSFET using two dimensional potential model. The low breakdown voltage becomes the obstacle of power device operation, and breakdown voltage decreases seriously by the short channel effects derived from scaled down device in the case of DGMOSFET. The two dimensional analytical potential distribution derived from Poisson's equation have been used to analyze the breakdown voltage for device parameters such as channel length, channel thickness, gate oxide thickness and channel doping concentration. Resultly, we could observe the breakdown voltage has greatly influenced on device dimensional parameters as well as channel doping concentration, especially the shape of Gaussian function used as channel doping concentration.

이중게이트 MOSFET의 스켈링 이론에 대한 문턱전압이하 스윙분석

본 연구에서는 이중게이트 MOSFET에서 스켈링 이론에 대한 문턱전압이하 스윙을 분석하였다. 포아송방정식의 해석학적 전위분포를 구하기 위하여 가우스 전하분포를 이용하였다. 문턱전압이하 스윙의 저하와 같은 단채널 효과를 분석하기 위하여 스켈링이론이 사용되었으며 이중게이트 MOSFET의 특성상 두 개의 게이트 효과를 포함하기 위하여 일반적인 스켈링 이론을 수정하였다. 게이트길이에 대한 스켈링인자가 일반적인 스켈링인자의 1/2일 때 문턱전압이하 스윙의 저하현상이 매우 빠르게 감소하였으며 가우스함수의 이온주입범위 및 분포편차도 문턱전압이하 스윙에 영향을 미치는 것을 알았다. This study has presented the analysis of subthreshold swings based on scaling theory for double gate MOSFET. To solve the analytical potential distribution of Poisson's equation, we use Gaussian function to charge distribution. The scaling theory has been used to analyze short channel effect such as subthreshold swing degradation. These scaling factors for gate length, oxide thickness and channel thickness has been modified with the general scaling theory to include effects of double gates. We know subthreshold swing degradation is rapidly reduced when scaling factor of gate length is half of general scaling factor, and parameters such as projected range and standard projected deviation have greatly influenced on subthreshold swings.

문턱전압이하 영역에서 이중게이트 MOSFET의 스켈링 이론과 단채널효과의 관계

본 연구에서는 문턱전압이하 영역에서 이중게이트 MOSFET에서 스켈링 이론이 단채널효과에 미치는 영향을 관찰하였다. 기존 MOSFET의 경우 출력특성을 일정하게 유지하기 위하여 스켈링 이론을 적용하여 전류 및 스위칭 주파수를 해석하였다. 이중게이트 MOSFET에서 단채널효과에 대한 스켈링 이론의 적용 결과를 분석하기 위하여 문턱전압, 드레인유기장벽감소 및 문턱전압이하 스윙 등을 스켈링 인자에 따라 관찰하였다. 이를 위하여 이미 검증된 포아송방정식의 해석학적 전위분포를 이용하였다. 분석결과 단채널효과 중 문턱전압이 스켈링 인자에 가장 큰 영향을 받는다는 것을 관찰하였다. 특히 채널길이에 스켈링 이론을 적용할 때 가중치를 이용한 변형된 스켈링 이론을 적용함으로써 이중게이트 MOSFET에 가장 타당한 스켈링 이론에 대하여 설명하였다. This paper has presented the influence of scaling theory on short channel effects of double gate(DG) MOSFET in subthreshold region. In the case of conventional MOSFET, to preserve constantly output characteristics,current and switching frequency have been analyzed based on scaling theory. To analyze the results of application of scaling theory for short channel effects of DGMOSFET, the changes of threshold voltage, drain induced barrier height and subthreshold swing have been observed according to scaling factor. The analytical potential distribution of Poisson equation already verified has been used. As a result, it has been observed that threshold voltage among short channel effects is grealty changed according to scaling factor. The best scaling theory for DGMOSFET has been explained as using modified scaling theory, applying weighting factor reflected the influence of two gates when scaling theory has been applied for channel length.

The Analysis of Breakdown Voltage for the Double-gate MOSFET Using the Gaussian Doping Distribution

This study has presented the analysis of breakdown voltage for a double-gate metal-oxide semiconductor field-effect transistor (MOSFET) based on the doping distribution of the Gaussian function. The double-gate MOSFET is a next generation transistor that shrinks the short channel effects of the nano-scaled CMOSFET. The degradation of breakdown voltage is a highly important short channel effect with threshold voltage roll-off and an increase in subthreshold swings. The analytical potential distribution derived from Poisson's equation and the Fulop's avalanche breakdown condition have been used to calculate the breakdown voltage of a double-gate MOSFET for the shape of the Gaussian doping distribution. This analytical potential model is in good agreement with the numerical model. Using this model, the breakdown voltage has been analyzed for channel length and doping concentration with parameters such as projected range and standard projected deviation of Gaussian function. As a result, since the breakdown voltage is greatly changed for the shape of the Gaussian function, the channel doping distribution of a double-gate MOSFET has to be carefully designed.