In this paper, we propose new physically based threshold voltage models for short channel Surrounding Gate Silicon Nanowire Transistor with two different geometries. The model explores the impact of various device parameters like silicon film thickness, film height, film width, gate oxide thickness, and drain bias on the threshold voltage behavior of a cylindrical surrounding gate and rectangular surrounding gate nanowire MOSFET. Threshold voltage rolloff and DIBL characteristics of these devices are also studied. Proposed models are clearly validated by comparing the simulations with the TCAD simulation for a wide range of device geometries.
1. Introduction
The requirement for Nano electronic devices to have large computing power with less power consumption and smaller dimensions has lead to device miniaturization. There are a number of drawbacks in the downscaling process of a conventional MOSFET, such as the short channel effects and the gateleakagecurrent. Hence in these conditions, Silicon Nanowire Transistors have garnered a huge attention in the modern semiconductor industry as the alternate option for the conventional MOSFETs due to their highly improved electrical and optical properties
[1]
. Devices based on silicon nanowires are bound to have high electrostatic control over the channel and reduced short channel effects.
Modeling of nanowire field effect transistors have been the subject of several investigation, both analytical and numerical. Analytical models are based on two assumptions, 1) an undoped semiconductor nanowire and 2) Boltzmann statistics
[2

6]
. These assumptions allow, in fact, for a closed form solution of Poisson’s Eqs.
[2
,
3]
, that makes it possible to work out an intrinsic analytical relationship between the gate voltage and surface potential. Alternatively, the assumption is taken of a completely depleted nanowire, consistently with the investigation of Sub threshold Slope (SS) and short channel effects
[7]
. A new study on the gate capacitance of a surrounding gate nanowire transistor studies the acceptor type doping
[8]
such that the carriers are confined only within the inversion layer.
Numerical approaches, instead, account for nearly all of the relevant effects that exert an impact on the device behavior, including motion quantization, sub band splitting, Fermi statistics, quasi ballistic transport, surface and channel orientation, and band structure
[9

11]
. The classical single gate MOSFET is approaching its minimum channel length due to the limits imposed by gate oxide tunneling. For extending the scalability of CMOS technology several non classical MOSFETs have been proposed that are being the subject of intense research.
Fig. 1
shows some of the non planar MOSFET devices with different geometries.
Non Planar MOSFETs with different geometric specifications
The use of several gates has shown good electrostatic control of the channel and, therefore, the possibility of higher reduction in channel length compared to traditional bulk MOSFETs. Structures such as FINFETs, Double Gate, Tri Gate, Surrounding Gate, Omega Gate and GateAllAround MOSFETs are preferred to planar structures as they have a steep sub threshold voltage and reduced leakage currents for very short channel lengths
[12]
. Drain Induced Barrier Lowering (DIBL), threshold voltage rolloff, and offstate leakage current are greatly reduced in these devices
[13
,
14]
.
Surrounding Gate MOSFETs are one of the most promising structures beyond bulk CMOS. Since early investigations on Surrounding Gate / SOI devices
[15]
, most recent experiments have demonstrated GAA Silicon Nanowires with controlled diameters on the order of 36 nm using conventional CMOS technology
[16]
. Theoretically, Surrounding Gate MOSFETs provide better gate electrostatic control capability than planar and double gate devices. We propose to develop a feasible physics based analytical model for the Surrounding Gate Nanowire MOSFET with two different geometries; a Junction Based Cylindrical Surrounding Gate Device and a Rectangular Surrounding Gate Device.
2. Modeling of Cylindrical SG Nanowire MOSFET
A number of analytical models for the Surrounding Gate devices have been proposed in the past by various research scholars. A simple threshold voltage model for a Surrounding Gate MOSFET transistor has been proposed by C.P. Auth and James D. Plummer in 1998
[17]
. This model determines the channel potential using perimeter weighted summation method. Threshold voltage model for a Omega Gate Transistor was proposed by Biswajit Ray and Santanu Mahapathra in 2008
[18]
. This model can be considered as the generalized model for the Surrounding Gate and SemiSurrounding Gate Cylindrical Transistors. TeKuang Chiang proposed a model on surrounding gate MOSFET with localized interface charges with potential approximation using parabolic approximation in 2010
[19]
and using an alternative approach of perimeter weighted summation method in 2011
[20]
. M. Jagadesh Kumar et.al. have proposed a threshold voltage model for the dual material surrounding gate that includes the physical properties of both the materials used in the gate
[21]
. A new threshold voltage model for a short channel Junction Less Cylindrical Surrounding Gate MOSFET has been developed by T.K.Chiang in 2012
[22]
.
In the sub threshold operation of a Junction Based device, the channel region is fully depleted by the flat band voltage and the gate bias is used to induce the minority carriers at the semiconductor – insulator interface. Due to the 3D channel potential,
φ
(
r
,
z
,
θ
) is symmetrical in the
θ
direction. Hence the 2D Poisson equation for a Junction Based Cylindrical Surrounding Gate Transistor with a uniform impurity distribution is given by
[22]
.
Junction Based Cylindrical Surrounding Gate (JBCSG) MOSFET.
where
φ
(
r, z
) is the 2D channel potential and
N_{A}
is the channel doping density.
Using the parabolic potential approximation method, the 2D potential vertical to the channel is given by
[22]
,
Certain assumptions are done to determine the channel potential
[22]
. They are as given below:

1) The electrical flux between the silicon film and surrounding gate oxide must be continuous.

2) The electrical field atr= 0 must be zero due to the symmetry of the channel potential along therdirection.
Now the channel potential can be obtained as
Where
V_{fb}
is the flat band voltage,
C_{ox}
is the gate oxide capacitance per unit area, and
t_{si}
is the silicon film thickness.
φ_{s}
(
z
) and
φ_{c}
(
z
) are the surface and centre potential respectively that should satisfy the following Eq.
[22]
.
In case of a Junction Based Transistor, we are concerned about the surface potential and the centre potential has to be eliminated.
Hence by substituting Eqs. (3) and (4) in Eq. (1) we get,
φ_{s}
is the surface potential and
λ
is the scaling length. Their values are given by,
where
t_{ox}
is the oxide film thickness. The general solution of the differential equation in (5) is given by,
The coefficients of a and b can be determined by applying the boundary conditions as
φ_{s}
(
z
= 0) = 0 and
φ_{s}
(
z
=
L_{eff}
) =
V_{ds}
. By applying the above conditions we get,
By solving and rearranging, the values of a and b are obtained as,
The surface potential is given by,
Where,
And here the effective length is given by,
L_{g}
is the gate length while
L_{s}
and
L_{d}
are source and drain depletion widths respectively. Now
L_{d}
, the Debye length, by accounting for the transition regions separating the drift and diffusion regions of the channel, is given by
[23]
.
where
T
is the absolute temperature and
K
is the Boltzmann constant. The Minimum Surface Potential in (8) is now derived from the previous equations as,
The bulk potential
φ_{B}
is now given by
n_{i}
is the intrinsic carrier concentration of silicon.
Setting the Minimum Surface Potential (MSP) to be two times the bulk potential and solving for the gate voltage
V_{gs}
, the threshold voltage for the Junction Based transistor is obtained.
The values of
A_{JB}
,
B_{JB}
and
C_{JB}
and the associated fitting parameters are listed here.
3. Modeling of Rectangular SG Nanowire MOSFET
An analytical threshold voltage model for GAA Nanoscale MOSFETs considering the hot carrier induced interface charges have been proposed by Z.Ghoggali et al., in 2008
[24]
. Quantum confinement and its effects on threshold voltage variations in short channel GAA devices have been studied by Y.S.Wu and Pin Su in 2009
[25]
. A physically based classical model for body potential of a GAA nanowire transistor has been proposed by Biswajit Ray and Santanu Mahapathra in 2008
[26]
. A quasi and analytical model for predicting the potential of a nanowire FET has been proposed by De Michielis in 2010
[27]
. Here, we consider a lightly doped rectangular surrounding gate nanowire MOSFET, bound to have high gate control, in the weak inversion region, where both fixed and mobile charge densities in the channel are negligible. In modeling the threshold voltage, the quantum effects are also considered, as the quantization of the electron energy in ultra thin devices can never be ignored. One important consequence of the quantum mechanical carrier distribution, in accordance with the device behavior, occurs when the device geometries and silicon thickness are varied, so a reliable compact model for the nanowire transistors must also take into account the quantum effects resulting out of these variations.
Schematic Diagram of a Rectangular Surrounding Gate Nanowire MOSFET
The proposed physically based closed form model holds good for ultra thin and ultra short channel surrounding gate devices and does not employ any unphysical fitting parameter. The compact threshold voltage model is obtained by solving the 3D Poisson equation and 2D Schrodinger equations in the inversion region. These equations are then consistently solved to obtain the potential distribution and inversion charge density.
In the weak inversion regime, we have approximated the Poisson equation as Laplace equation with the inversion charge density neglected, and thus the two equations are decoupled. Assuming a flat potential on the plane perpendicular to the sourcedrain direction in the nanowire, neglecting the charge densities ensure that no self consistency problems arise in the nanowire. The mid gap metals are used for gate, intended to suppress the silicon gate poly depletion induced parasitic capacitances. The 3D Poisson equation is solved to obtain the threshold voltage in the weak inversion region, including the parabolic band approximation. Hence the potential distribution in the insulator and silicon regions can be expressed as
[28]
.
The potential
ψ
in terms of x (width), y (height) and z (length) is to be determined. The boundary conditions defined by the physics of the device
[9]
are given by,
Where, V
_{g}
is the gate voltage,
ψ_{bi}
is the built in potential, L is the channel length, V
_{ds}
is the drain to source voltage which is negligible for low V
_{ds}
. For a rectangular surrounding gate device, we have to find the insulator potential on all sides of the channel under consideration. So the height and width of the channel is also taken into account. The insulator potential is now expressed as,
Here Φ
_{ms}
is the work function difference. By applying the superposition principle, the electrostatic potential can be now written as
Here V
_{g}
(x, y) is the 1D solution of the Poisson equation that satisfies the gate boundary conditions. U
_{L}
satisfies the source boundary condition but it is bound to have a null value on the gate and drain boundaries. Similarly U
_{R}
satisfies the drain boundary condition and it is bound to have a null value on the gate and source boundaries. On further evaluation, the term V
_{g}
+U
_{L}
is found to satisfy the potential equation when U
_{R}
is on null value and in an exact repetition the term V
_{g}
+U
_{R}
satisfies the potential equation when U
_{L}
is on null value. From Eq. (30),
By solving the above equation using LDE method we obtain the value of
ψ
. The Potential equation is deduced after rigorous analytical calculations as,
Where,
Threshold voltage for the undoped body devices is defined as the gate voltage when the integrated charge at the virtual source becomes equal to the critical charge (Q
_{T}
). The first series term in Eq. (41) is enough to find the potential at the virtual source; it is only taken into account for the further calculations. Once the potential distribution at every point of the cross section of the channel is known we calculate the inversion charge density by using surface integral over the surface area of the channel. Hence the inversion charge can be expressed as,
Where,
q
is the elementary charge, U
_{T}
is thermal voltage, and
n_{i}
is the intrinsic carrier concentration. The charge equation can now be approximated as,
Here,
z_{c}
is the virtual source position, which is half of the channel length for low
V_{ds}
. Using the inversion charge we can obtain the classical threshold model as,
As MOSFET devices are further scaled into the deep nanometer regime, it has become necessary to include quantum mechanical effects while modeling their device behavior. The potential distribution obtained in (41) is quasiparabolic in nature. Therefore in this paper, we approximate the actual potential as well as the square well potential since it is difficult to solve the Schrodinger equation to obtain the potential.
In the square potential well approximation, the minima of conduction band energy at the centre position is given by,
Band diagram perpendicular to the gate  square well potential of a Rectangular Gate Silicon Nanowire MOSFET
Using the above value of potential energy in (46), the Schrodinger equation becomes,
Here
and
h
is the planks constant,
ζ
is the wave function and E is the energy of the electron wave. The solution for this equation is obtained by variable separation method as,
The transverse and longitudinal masses and lengths in x and y directions tae different values based on the direction of quantization. The energy reaches its minimum when the masses reach the maximum value (48). In silicon, six energy valleys are found to be present in its band structure (two lower energy valleys, two middle energy valleys, and two higher energy valleys). If the thin film of device has equal height and width, the two lower energy valleys and two middle energy valleys combine together to produce four lower energy valleys and the other two higher energy valleys remain in their own state. Thus the charge per unit length per valley of silicon is expressed as,
Where, N
_{1D}
is the 1D densityofstates and f (E) is the FermiDirac distribution function. E is the energy of the electron wave. The terms i
_{x}
and i
_{y}
are positive natural numbers.
Now, the charge is given by,
Where, m
_{z}
is the mass of the valley which is perpendicular to the direction of quantization. The Fermi energy level is much lower than the conduction band energy in weak inversion region. Hence the charge equation can be approximated using Boltzmann equation as,
Using (48) and (51), the total integrated charge at the virtual source can be obtained as,
Where,
Here the
m_{t}
and
m
_{1}
are the transverse and longitudinal effective masses of the energy valleys of silicon. The lengths
i_{x}
and
i_{y}
carry distinct values contingent on the direction of quantization. Finally, the quantum threshold voltage model becomes,
Where,
The impacts on the threshold voltage due to quantum effects is acquired by using the following equation,
Here, ΔV
_{T}
is the difference between the quantum threshold voltage and the classical threshold voltage.
4. Results and Discussions
The Junction Based Cylindrical Surrounding Gate Transistor is simulated with the following specifications for the device: V
_{ds}
= 0.1 V, N
_{A}
=1x10
^{17}
cm
^{−3}
and N
_{D}
=1x10
^{19}
cm
^{−3}
are the drain voltage, doped ptype silicon film and the high ntype doping densities, respectively. The model simulations are validated by comparing the MATLAB simulation results with the results of the TCAD simulations.
Fig. 5
shows the relation between the threshold voltage rolloff and effective channel length for varying oxide thicknesses of a cylindrical surrounding gate device. Three different oxide thicknesses of 1, 3 and 5 nm are used. The thinnest gate oxide thickness of 1 nm shows the minimum threshold voltage degradation. As the gate oxide thickness increases, the threshold voltage rolloff also increases leading to the subdued performance of the device. The threshold voltage tends to remain constant in a range of 0.4V with the effective channel length of 35nm.
Threshold voltage rolloff versus effective channel length for a Junction Based Cylindrical Surrounding Gate Nanowire Transistor with different oxide thicknesses
Fig. 6
. shows the variation of the threshold voltage rolloff versus effective channel length for different silicon film thicknesses of a cylindrical surrounding gate device. Decreasing the effective channel length shows that the threshold voltage rolloff of the device is highly increased. Three different silicon thicknesses of t
_{si}
= 5, 10, and 15 nm are used in the simulations. Reducing thickness of the silicon film leads to the reduction of threshold voltage. As the thickness is less than 10 nm, the device tends to show abnormal variations in terms of threshold voltage. The simulations are further compared with TCAD simulations and are found to be in excellent coordination.
Threshold voltage rolloff versus effective channel length for a Junction Based Cylindrical Surrounding Gate Nanowire Transistor with different silicon thicknesses
Any transistor whose channel length is reduced to a nanometer range, there enters the problem of short channel effects (SCE). DIBL (Drain Induced Barrier Lowering) is one among these SCE that affects the performance of the transistor.
Fig. 7
. shows the impact of the effective channel length on DIBL for different drain biases of a cylindrical surrounding gate device.
DIBL versus effective channel length for a Junction Based Cylindrical Surrounding Gate Transistor
In
Fig. 7
, the minimum V
_{ds}
is kept as 0.1V and as they are gradually increased to higher values of 0.5, 1.0 and 1.5 V, the DIBL is plotted as the difference between the lower and higher bias values. It shows that an increasing drain bias and decreasing effective channel length will lead to the device suffering high Short Channel Effects.
Fig. 8
. represents the variation of total quantum integrated charge of a rectangular surrounding gate device with the gate voltage at 0.3V for different film widths. Eq. (45) is used to obtain the integrated charge with only one energy level and one series term. It clearly shows that the decrease in the film thickness leads to the increase in the quantum threshold voltage which is actually due to the increase in energy quantization of the transistor.
Variation of quantum integrated charge at virtual source with gate voltage for a Rectangular Surrounding Gate Nanowire Transistor with different film widths
Fig. 9
shows the variation of quantum threshold voltage with width and height of the film for a rectangular surrounding gate device at a channel length of 20 nm. The impact of the device dimensions like film height and width on the quantum threshold voltage of the device is explained. The short channel effects tend to decrease along with the energy quantization and this can be further explained as a result of increase in the effective band gap of silicon due to quantum effects. The effect of confinement, expressed as the difference in the threshold voltage and its variation with the channel length L, of a Rectangular surrounding gate device is illustrated in
Fig. 10
. The most important thing about this gate all around nanowire transistor is that any change in one of the dimensions can be nullified by proper tuning of other dimensions as the transistor is symmetric about its height and width.
Variation of quantum threshold voltage with film height and width for a Rectangular Surrounding Gate Nanowire Transistor (L=20nm)
Variation of threshold voltage with film height for a Rectangular Surrounding Gate Nanowire Transistor (L= 20nm and W=9nm)
Fig. 11
shows the variation of the classical threshold voltage and quantum threshold voltage with the film height of a Rectangular surrounding gate device at a constant width of 9 nm. The value of the classical threshold voltage ranges from 0.27 V to 0.29 V for the corresponding changes in the film height. Similarly the quantum threshold voltage ranges from 0.3 V to 0.31 V. It shows that the device has a highly improved control over the threshold voltage. The process parameters are almost same for the cylindrical gate and surrounding gate devices except for the fact that radius is the main factor in the cylindrical gate device while width plays a very important role in the rectangular gate device.
Variation of quantum and classical threshold voltage with film height for a Rectangular Surrounding Gate Nanowire Transistor. Here L=20nm and W=9nm.
As the thickness of the silicon film increases and the effective channel length decreases, threshold voltage rolloff increases in a cylindrical gate device. Similarly as the film width increases, the threshold voltage decreases in a rectangular gate device. This will lead to simultaneous increase in the off current, resulting in the performance degradation of device. In the cylindrical gate device, increase in gate oxide thickness and reduction in effective channel length leads to higher threshold voltage rolloff. In a rectangular gate device, as the film height increases and the channel length decreases, threshold voltage also decreases but the short channel effects are reduced due to energy quantization. The increase in drain bias and reducing channel length is bound to increase the short channel effects in both geometries. The major difference between the two geometries is that the rectangular gate device is bound to get affected by corner effects much more in comparison with the cylindrical gate device.
5. Conclusion
A new physically based threshold voltage models for a Junction Based Cylindrical Surrounding Gate Nanowire Transistor and Rectangular Gate Surrounding Gate Nanowire Transistor have been developed. The simulations are validated by using the TCAD results. This physics based models can be extended to study the IV characteristics of the devices affording to their simplicity and computational efficiency.
BIO
M. Karthigai Pandian was born in Madurai, India in 1981. He received his Bachelors of Engineering degree from the department of Electronics and Instrumentation from Karunya Institute of Technology, Coimbatore and Masters Degree in Applied Electronics from Anna University, Chennai in 2002 and 2006 respectively. He is currently employed as an Assistant Professor at Pandian Saraswathi Yadav Engineering College, Sivagangai and he is also pursuing Ph.D degree in the Department of Electronics and communication engineering, Thiagarajar College of engineering, Tamilnadu, India under Anna University Chennai. His research interest is in the area of analytical modeling and simulation of Multi gate nanowire transistors.
N. B. Balamurugan received his B.E and M.E degrees, both in electronics and communication engineering from the Thiagarajar College of Engineering (TCE), Tamilnadu, India. He has obtained his Ph.D degree in nanoelectronics at Anna University, India. From 1998 to 2004, he worked as a lecturer in R.V.S.college of engineering and technology, Tamilnadu, India. He is currently working as an Associate Professor in the Department of Electronics and Communication Engineering, Thiagarajar College of Engineering (TCE), Tamilnadu, India. He has published more than 60 research papers as sole or joint author in the field of device modeling and simulation. His research interests include analytical modeling and simulation of semiconductor device structures like Nanoscale SOI MOSFETs, Nanowire Transistors and Tunnel FETs.
Wan Yuting
,
Sha Jian
,
Chen Bo
,
Wang Yanjun
,
Wang Zongli
,
Wang Yewu
2009
“Nanodevices based on Silicon Nanowires”
Recent Patents on Nanotechnology
3
(1)
1 
9
Jimenez D.
,
Saenz J.J.
,
Iniguez B.
,
Sune J.
,
Marsal L.F.
,
Pallares J.
2004
“Modeling of Nanoscale GAA MOSFETs”
IEEE Electron Device Lett.
25
(5)
314 
316
DOI : 10.1109/LED.2004.826526
Jimenez D.
,
Iniguez B.
,
Sune J.
,
Marsal L.F.
,
Pallares J.
2004
“Continuous Analytic IV model for Surrounding Gate MOSFETs”
IEEE Electron Device Lett.
25
(8)
571 
573
DOI : 10.1109/LED.2004.831902
Iniguez B.
,
Jimenez D.
,
Roig J.
,
Hamid H. A.
,
Marsal L. F.
,
Pallares J.
2005
“Explicit Continuous model for Long Channel undoped Surrounding Gate MOSFETs”
IEEE Transactions on Electron Devices
52
(8)
1868 
1873
DOI : 10.1109/TED.2005.852892
Yu B.
,
Lu W.Y.
,
Lu H.
,
Taur Y.
2007
“Analytic Charge Model for Surrounding Gate MOSFETs”
IEEE Transactions on Electron Devices
54
(3)
492 
496
DOI : 10.1109/TED.2006.890264
Borli H.
,
Kolberg S.
,
Fjeldly T.A.
,
Iniguez B.
2008
“Precise Modeling Framework for short channel DoubleGate and GAA MOSFETs”
IEEE Transactions on Electron Devices
55
(10)
2678 
2686
DOI : 10.1109/TED.2008.2003221
Auth C. P.
,
Plummer James D.
1997
“Scaling theory for Cylindrical, Fully depleted, Surrounding Gate MOSFETs”
IEEE Electron Device Lett
18
(2)
74 
76
DOI : 10.1109/55.553049
Ruiz FJG
,
TiendaLuna IM
,
Godoy A.
,
Donetti L.
,
Gamiz F.
2010
“A Model of the Gate Capacitance of Surrounding Gate Transistors: Comparison with Double Gate MOSFETs”
IEEE Transactions on Electron Devices
57
(10)
2477 
2483
DOI : 10.1109/TED.2010.2058630
Gnani E.
,
Gnudi A.
,
Reggiani S.
,
Baccarani G.
2008
“Quasi Ballistic Transport in Nanowire FETs”
IEEE Transactions on Electron Devices
55
(11)
2918 
2930
DOI : 10.1109/TED.2008.2005178
Gnani E.
,
Gnudi A.
,
Luisier M.
,
Baccarani G.
2008
“Band effects on Transport Characteristics of Ultrascaled Nanowire FETs”
IEEE Transactions on Nanotechnology
7
(6)
700 
709
DOI : 10.1109/TNANO.2008.2005777
Poli S.
,
Pala M.
,
Poiroux T.
2009
“Full Quantum Tratment of Remote Coulomb Scattering in Silicon Nanowire FETs”
IEEE Transactions on Electron Devices
56
(6)
1191 
1198
DOI : 10.1109/TED.2009.2019380
Poiroux T.
,
Vinet M.
,
Faynot O.
,
Wildeiz J.
,
Lolivier J.
,
Ernst T.
,
Previtali B.
,
Delonibus S.
2005
“Multiple Gate Devices :Advantages and Challenges”
Microelectronic Engineering
80
378 
385
DOI : 10.1016/j.mee.2005.04.095
Oh S.
,
Monroe D.
,
Hergenrother JM.
2000
“Analytic Description of Short channel effects in Fully Depleted Double Gate and Cylindrical Surrounding Gate MOSFETs”
IEEE Transactions on Electron Devices
21
(9)
445 
447
DOI : 10.1109/55.863106
Jimenez D.
,
Iniguez B.
,
Sune J.
,
Saenz J. J.
2004
“Analog Performance of the Nanoscale Double Gate MOSFET near the Ultimate Scaling Limits”
J. App. Physics.
96
(9)
5271 
5276
DOI : 10.1063/1.1778485
Collinge J.P.
,
Gao M.H.
,
Romano A.
,
Maes H.
,
Claeys C.
1990
“SiliconOnInsulator GateAllAround Device”
IEDM Tech. dig.
595 
599
N. Singh
,
F.Y. Lim
,
W.W. Fang
2006
“Ultra Narrow Silicon Nanowire GAA CMOS Devices: Impact of Diameter, Channel Orientation and Low Temperature on Device Performance”
IEDM Tech. dig.
547 
550
Auth Christoper. P.
,
Plummer James D.
1998
“A Simple Model for Threshold Voltage of Surrounding Gate MOSFETs”
IEEE Transactions on Electron Devices
45
(11)
2381 
2383
DOI : 10.1109/16.726665
Ray Biswajit
,
Mahapatra Santanu
“A New Threshold Voltage Model for Omega Gate Cylindrical Nanowire Transistor”
21st International Conference on VLSI Design, VLSID 2008
447 
452
Chiang TeKuang
2010
“A Compact Analytical Threshold Voltage Model for SurroundingGate MOSFETs with Interface Trapped Charges”
IEEE Electron Device Letters
31
(8)
788 
790
DOI : 10.1109/LED.2010.2051317
Chiang TeKuang
2011
“A Compact Model for Threshold Voltage of SurroundingGate MOSFETs with Localized Interface Trapped Charges”
IEEE Transactions on Electron Devices
58
(2)
567 
571
DOI : 10.1109/TED.2010.2092777
Jagadesh Kumar M.
,
Orouji Ali A.
,
Dhakad Harshit
2006
“A New DualMaterial SG Nanoscale MOSFET: Analytical ThresholdVoltage Model”
IEEE Transactions on Electron Devices
53
(4)
920 
923
DOI : 10.1109/TED.2006.870422
Chiang TeKuang
2012
“A New Quasi2D Threshold Voltage Model for ShortChannel Junctionless Cylindrical Surrounding Gate (JLCSG) MOSFETs”
IEEE Transactions on Electron Devices
59
(11)
3127 
3129
DOI : 10.1109/TED.2012.2212904
Chiang TeKuang
2012
“A New Quasi2D Threshold Voltage Model for ShortChannel Junctionless Double Gate MOSFETs”
IEEE Transactions on Electron Devices
59
(9)
2284 
2289
DOI : 10.1109/TED.2012.2202119
Ghoggali Z.
,
Djeffal F.
,
Abdi M.A.
2008
“An Analytical Threshold Voltage Model for Nannoscale GAA MOSFETs including effects of Hot Carrier Induced Interface Charges”
Design and Test Workshop, IDT 2008
93 
97
Wu Yusheng
,
Su Pin
2009
“Quantum confinement Effects in short channel GAA MOSFETs and its impact on the sensitivity of Threshold Voltage to Process Variations”
Proceedings of IEEE International Conference on SOI
1 
2
Ray B
,
Mahapatra Santanu
2008
“Modeing and Analysis of Body Potential of Cylindrical GAA Nanowire Transistor”
IEEE Transactions on Electron Devices
55
(9)
2409 
2416
DOI : 10.1109/TED.2008.927669
De Mechielis L.
,
Selmi L.
,
Lonescu AM.
2010
“A Quasi Analytical model for Nanowire FETs with Arbitrary Polygonal Cross Section”
Solid State Electronics
54
(9)
Rakesh Kumar P.
,
Mahapthra S.
2011
“Quantum threshold voltage modeling of short channel quad gate silicon nanowire transistor”
IEEE Transactions on Nanotechnology
10
(1)