Definition of the effective masses along the radial and axial directions in cylindrical nanowire (
1. Introduction
The phenomenal success of CMOS technology, and, then the progress of the information technology, can be attributed without any doubt to the scaling of the MOS transistor, which has been pushed during more than thirty years to increasingly levels of integration and performances. Then, MOSFETs have been fabricated always smaller, denser, faster and cheaper in order to provide ever more powerful products for digital electronics. Recently, the scaling rate has accelerated, and the MOSFET gate length is now less than 40 nm, with devices entering into the nanometer world [1][2]. The socalled “bulk” MOSFET is the basic and historical keydevice of microelectronics: its dimensions have been reduced more than ~10^{3} times during the three past decades. However, the bulk MOSFET scaling has recently encountered significant limitations, mainly related to the gate oxide (SiO_{2}) leakage currents [3][4], the large increase of parasitic short channel effects and the dramatic mobility reduction [5][6] due to highly doped Silicon substrates precisely used to reduce these short channel effects. Technological solutions have been proposed in order to continue to use the “bulk solution” until the 45 nm ITRS node. Most of these solutions envisage the introduction of highpermittivity gate dielectric stacks (to reduce the gate leakage, [4], [7][8]), midgap metal gate (to suppress the Silicon gate polydepletioninduced parasitic capacitances) and strained Silicon channel (to increase carrier mobility, [9]). However, in parallel to these efforts, alternative solutions to replace the conventional bulk MOSFET architecture have been proposed and studied in the recent literature. These options are numerous and can be classified in general according to three main directions: (i) the use of new materials in the continuity of the “bulk solution”, allowing increasingly MOSFET performances due to their dielectric properties (permittivity), electrostatic immunity (SOI materials), mechanical (strain), or transport (mobility) properties; (ii) the complete change of the device architecture (e.g. multiplegate devices, Silicon nanowires MOSFET) allowing better electrostatic control, and, as a result, intrinsic channels with higher mobilities and currents; (iii) the exploitation of certain new physical phenomena that appear at the nanometer scale, such as quantum transport, substrate orientation or modification of the material band structure in devices/wires with nanometer dimensions [2], [10].
Nanowire MOSFETs with completely surrounding gate are presently considered as one of the possible solutions for replacing bulk (singlegate) devices and continuing MOSFET scaling in the nanometer scale [11][12]. The main advantage of this architecture is to offer a reinforced electrostatic coupling between the conduction channel and the gate electrode, which considerably reduces shortchannel effects (SCE) compared to conventional bulk devices [11][13]. Then, the constraints on channel doping levels can be relaxed and nanowire MOSFET devices can be designed with intrinsic (pure Si) channels. This offers considerable advantages, especially in terms of mobility and elimination of doping fluctuations. The intrinsic channel can also be beneficial with regard to sourcetodrain transport due to the high probability of ballistic transport.
Nanowire MOSFETs are generally designed with very thin silicon films in order to reinforce the electrostatic control of the gate over the channel. The ultrathin silicon film creates a sufficiently narrow rectangular potential well for inducing the quantization of carrier energy. Carriers are then confined in a rectangular quantum well having feature size close to the electron wavelength. This gives rise to a splitting of the energy levels into subbands (twodimensional (2D) densityofstates) [14][15], such that the lowest of the allowed energy levels for electrons (resp. for holes) in the well does not coincide with the bottom of the conduction band (resp. the top of the valence band). In nanowire devices the carrier energy is quantified in the two directions perpendicular to the sourcetodrain axis (leading to onedimensional (1D) densityofstates). The total density of states in a 1D system is less than that in a threedimensional (3D) (or classical) system, especially for low energies. Carriers occupying the lowest energy levels behave like quantized carriers while those lying at higher energies, which are not as tightly confined in the potential well, can behave like classical (3D) particles with three degrees of freedom. As the surface electric field increases, the system becomes more quantized and more and more carriers become confined in the potential well. The quantum mechanical confinement considerably modifies the carrier distribution in the channel: the maximum of the inversion charge is shifted away from the interface into the Silicon film. Because of the smaller density of states in the 1D system, the total population of carriers is smaller for the same Fermi level than in the corresponding 3D (or classical) case. This phenomenon affect the net sheet charge of carriers in the inversion layer, thus requiring a larger gate voltage in order to populate a 1D inversion layer to have the same number of carriers as the corresponding 3D system. This leads to an increase of the threshold voltage of the MOSFET, which is an important issue, especially as the power supply voltages drop to lower levels. The gate capacitance and carrier mobility are also modified by quantum effects. These considerations indicate that the wave nature of carriers can no longer be neglected in very thin nanowire MOSFETs and has to be considered in simulation studies.
Modeling and simulation of nanowire MOSFETs devices are currently experiencing a growing interest due to unique capabilities: (i) simulation provides useful insights into device operation since all internal physical quantities that cannot be measured on real devices are available as outputs in simulation; (ii) the predictive capability of simulation studies makes possible the reduction of systematical experimental investigation of these new ultrascaled devices; (iii) simulation offers the possibility to test hypothetical devices which have not yet been manufactured. Since computers are today considerably cheaper resources, simulation is becoming an indispensable tool for the device engineer, not only for the device optimization, but also for specific studies such as the investigation of new physical phenomena specific to the ultrashort channels (quantum confinement of carriers or shortchannel electrostatic effects).
This chapter presents a simulation study of electrostatics and electronic transport in radiallysymmetric nanowire MOSFETs by quantum driftdiffusion numerical simulation. In this chapter, we will investigate the operation of radiallysymmetric nanowires using a 2D/3D Schrödinger/Poisson solver, called Cylmos. Various methods have been suggested to model quantum confinement effects. Among the approaches that are compatible with classical device simulators based on the driftdiffusion approach, the physically most accurate method is to include the Schrödinger equation into the selfconsistent computation of the device characteristics [15]. Then, the solver described in this work provides selfconsistent solutions of the 2D Schrödinger equation and the 3D Poisson equation in cylindrical coordinates [16][18], coupled with the driftdiffusion transport equation; this approach is commonly called in literature “quantum driftdiffusion”. Simulated drain current versus gate voltage characteristics have been compared to data obtained from simulation with commercial code with an excellent agreement. Cylmos also provides a lot of additional information and valuable physical insights (such as the 3D profile of electrostatic potential, classical and quantum carrier densities in the channel, the energy levels and total inversion charge) used to investigate the influence of shortchannel and quantummechanical effects. Finally, the quantum driftdiffusion code will be used to analyze electrical parameters such as threshold voltage, draininduced barrier lowering (DIBL) effect and subthreshold swing in radiallysymmetric nanowire MOSFETs. The difference between classical (i.e., without quantum confinement) and quantum threshold voltage, as function of different film silicon thicknesses in nanowire MOSFETs will be also discussed.
The chapter is organized as follows. In section 2 we describe in detail the theoretical background of our approach and the simulation code. After the description of the simulated devices, the Schrödinger, Poisson and current continuity solvers are presented. The complete discretization of these equations and the calculation algorithm are explicitly presented. In Section 3 we compare Cylmos to results obtained using a commercial simulator, particularly in terms of output parameters of the Schrödinger module and drain current. Finally, after this validation step, we use Cylmos to analyze and discuss the operation and performance of circular nanowirebased MOSFETs, including the offstate current, threshold voltage and short channel effects.
2. Theoretical background and description of the simulation code
Cylmos is based on the numerical solving of the PoissonSchrödinger system coupled with the driftdiffusion equation. The Poisson equation is solved on the entire 3D structure. To solve the Schrödinger equation, the device is divided into parallel vertical slices (yz plane) (one slice per mesh point in the x direction). The 2D Schrödinger equation is solved in each slice to obtain the wave functions, the quantum energy levels and the charge density. The solution of the Schrödinger equation in 2D Cartesian coordinates for the nanowires with square or circular section is straightforward to implement numerically, but it requires very extensive computation time [19][22]. To simplify the calculations, in radiallysymmetric nanowires it is possible to reduce the size of the two Poisson and Schrödinger equations. By expressing these equations in cylindrical coordinates and using the property of cylindrical symmetry of the structure (which implies a symmetry of the potential and wave function), the Poisson equation becomes a 2 dimensional equation and the Schrödinger equation becomes one dimensional. The Cartesian coordinates (x, y, z) are converted in cylindrical coordinates (x, r, θ), as shown in Fig.1, and considering these later, the structure is symmetrical with respect to the coordinate θ. Therefore, the coordinate θ will be ignored and the Poisson equation is solved in two dimensions, r and x, on the 2D mesh shown in Fig. 2(a). In the same way, the circular symmetry allows us to simplify the Schrödinger equation, which will be solved on a 1D mesh along the radial direction r, as shown in Fig. 2(b). The Schrödinger equation will be solved along vertical cutlines, perpendicular to the x axis, in each meshpoint i of the xaxis. The advantage of this transformation is that, contrary to the 3DPoisson/2DSchrödinger equations, the 2DPoisson/1DSchrödinger system is numerically less CPU consuming. Nevertheless, this system of equations is more difficult to implement due to the form of the Laplacian operator [16]. In the following, we detail the equations for a nchannel fully depleted nanowire MOSFET, but similar equations can be derived for pchannel structures.
The physical description of electrostatics of the nanowire device requires solving the coupled Poisson equation with the Schrödinger equation. The general Poisson equation is given by:
where
where ℏ is the reduced Planck constant,
where
where
where
Poisson, Schrödinger and driftdiffusion equations are solved using a finite difference scheme with a uniform mesh on a domain including the channel, the source and drain regions, the gateoxide layer and the gate electrodes (Fig. 2(a)). The electric field penetration in the source/drain and electron wave function penetration in the gateoxide can be thus taken into account.
The general flowchart of our code is presented in Fig. 3. In a first step, the quantized energy levels as well as their associated electron wave functions and populations can be obtained from the selfconsistent solving of the Schrödinger and Poisson equations. The next step concerns the continuity equation which is numerically solved to ensure a constant drain current along the device channel. The continuity equation gives the numerical solution of the quasiFermi level, used further in the calculation of the charge density in the SchrödingerPoisson system. After reaching the convergence, the drain current density is finally evaluated from the driftdiffusion formalism. In the following, we will explain in detail the three modules corresponding to the solving procedures for the Poisson, Schrödinger and current continuity equations. The discretization of theses equations and the simplifying assumptions used in the calculations, assumptions mainly related to the cylindrical symmetry of the structure, will be systematically explained and justified.
2.1. Description of the simulated devices
The description of the 3D architecture of the radiallysymmetric nanowire MOSFETs considered in the simulation is presented in Fig. 1. The definition of the main geometrical parameters of this structure is also shown in Fig. 1. The structure is symmetric with intrinsic thin silicon film and highly doped source and drain (N_{SD} = 3×10^{20} cm^{3}). A midgap metal gates (Φ_{m}=4.61 eV) and a 1.0nmthick gateoxide have been also considered. The source is grounded, and the gate and the drain biased at V_{G} and V_{D}, respectively. In order to investigate the influence of shortchannel effects and quantum mechanical confinement, different gate lengths and nanowire diameters will be simulated in section 4.
2.2. Poisson solver
For a given channel geometry and bias conditions, the 2D Poisson equation is solved (see flowchart in Fig. 3) considering a net charge density given by [23]:
where
The Poisson equations expressed in cylindrical coordinates (x, r, θ) and ignoring the dependence in θ due to the circular symmetry of the electrostatic potential can be written under the form:
This equation is discretized using a threepoint finite difference scheme applied to the grid shown in Fig. 2, as follows:
where
where the index
2.3. Schrödinger solver
The Schrödinger equation in cylindrical coordinates (x, r, θ) is given by:
where
Due to the cylindrical symmetry, we may consider wave functions of the form:
where
where the factor
Using relation (13), equation (12) becomes:
We recall here that the Schrödinger equation is solved along vertical cutlines (1D mesh along the radial direction), perpendicular to the x axis, in each meshpoint
As the Poisson equation, Schrödinger equation (14) is discretized using finite differences scheme to obtain a standard matrix eigenvalue problem:
where H is the general Hamiltonian matrix. In practice, equation (16) is solved for each value of the angular momentum quantum number m separately. In the following, we will describe in detail the Hamiltonian H for a given m value.
To simplify the equations, we consider firstly
where the effective mass has been considered independent on r (since we simulate here a fully silicon nanowire). In
where
We explain below how the effective masses are calculated in the cylindrical silicon nanowire considered here.The conduction band of silicon has 6 equivalent minima. The variation of the energy of the conduction band near the minimum is not isotropic and constant energy surfaces are ellipsoids around each of the axes (Fig. 4). In each minimum, the electrons propagating along the axis of the ellipsoid are characterized by a longitudinal effective mass,
In order to exploit the cylindrical symmetry, we use an approximate isotropic effective mass in the radial direction. Because of this assumption, now there are only two different types of conduction band valleys: the valley along the x direction which is doubly degenerate and the valley along the radial (r) direction whose degeneracy is equal to 4. The effective masses (m_{r}, m_{x}) along radial and axial directions for the two types of valleys are given in Table 1.




12  2 


36  4 

It is important to note that in addition to the solving in one dimension of the Schrödinger equation, which drastically reduces the computation time compared to a 2D solving, the computation time is further reduced because of the smaller number of valleys taken into account in the calculations compared to the case where an anisotropic effective mass is considered in the radial direction.The choice of an approximate isotropic effective mass for rdirection is justified by the fact that the eigenvectors and eigenenergies resulting from the cylindrical solver are very similar to those of an exact 2D solver taking into account the anisotropy of the effective mass [24]. In the following, the effective mass
Let’s go back now to the solving the Schrödinger equation. Using the discrete form given by eq. (17), the Schrödinger equation for
where H^{0}(0 stands for
where symbols are given by the following equations:
The matrix H_{0} defined in (20) is a nonsymmetric tridiagonal matrix and the computation of eigenvalues and eigenvectors of such matrix is numerically not optimized. However, using the method proposed in [16] by Zervos and Feiner, we can convert
where
Equation (24) can be rewritten in a discretized form as:
Combining equations (25) and (26) we obtain the following relation for matrix
Then we can obtain the elements of the matrix
Matrix
In equation (29)
Multiplying eq. (29) from the left by
We define:
and
Then, eq. (29) becomes:
which is the new eigenvalue problem. The new Hamiltonian
So far, we explained the resolution of the Schrödinger equation for
In this equation, the Hamiltonian
Equation (36) is a new eigenvalue problem to be solved separately for each
The procedure described before for the solving of the Schrödinger equation is performed for both
The 1D electron gas density is given by:
where
where
where
As we said before, the 1D Schrödinger equation is solved (using the procedure outlined above) in a vertical cutline in each mesh point of the x axis (which means that it will be solved
2.4. Current continuity module
As we explained at the beginning of Section 2, the solution of the SchrödingerPoisson system of equations is coupled with the current continuity equation given by relation (4). In eq. (4), the current density is defined by a model of charge transport, usually obtained in a semiclassical approximation or simplifying the Boltzmann transport equation (ETB). The simplifying assumptions made on this equation lead to several different semiclassical models such as the driftdiffusion model or the hydrodynamic model. Nevertheless, whatever the model used for the calculation of the carrier density (driftdiffusion, hydrodynamic, etc), the condition of current continuity should be guaranteed. Indeed, the continuity equations describe the evolution of carriers in the silicon nanowire film (sourcechanneldrain) in order to maintain a constant current along the film in which there is no charge accumulation.
In this work we consider the driftdiffusion model for which the current density is given by equation (5). The charge transport in the nanowire MOSFET simulated here involves only the minority carrier conduction. Therefore, the current density used herein concerns uniquely the electrons (for an nchannel transistor, NMOSFET), but a similar equation can be used for holes in the case of a pchannel transistor (PMOSFET) and the procedure of solving the current continuity equation will be rigorously identical with that shown below for electrons. In addition, to simplify the equations, we consider a stationary regime of operation and we assume that no process of generation or recombination of carriers occurs in our simulations. With these assumptions, equation (4) becomes:
The current density of electrons can be rewritten as a function of the quasiFermi level of electrons in the silicon nanowire,
where the electron diffusion coefficient,
In this work, the electron mobility
Using the properties of vector analysis and after various algebraic manipulations, equation (44) becomes:
where
where
In cylindrical coordinates,
and G is written as:
Equation (46) is discretized using a threepoint finite difference scheme applied to the mesh shown in Fig. 2, as follows:
where Σ_{i,j} is given by equation (11) and G_{i,j} and C_{i,j} are given by:
The quasiFermi level calculated by eq. (50) is then injected into the system of equations PoissonSchrödinger, where it is used for the calculation of the carrier density. This density is then used to calculate a new potential, which is then reinserted into the module solving the continuity equation that gives a new quasiFermi level. In its turn, this new quasiFermi level will be injected into the SchrödingerPoisson and so on. The loop will stop when the convergence criterion is reached. The final values of the electron density,
2.5. Drain current calculation
The drain current density (per unit area) in the driftdiffusion formalism is finally evaluated from equation (42) which can be written as:
The total current drain as a function of the applied voltages is then calculated by summing the contribution of each current line in the silicon film.
Finally, the code allows us to store the main internal keyparameters such as the carrier density, quantum energy levels, wave functions, potential, quasiFermi level, etc. Some of these parameters will be used for the code validation and for the analysis of the device performances in terms of threshold voltage or shortchannel effects.
3. Simulation code validation
Before using the code for analyzing the operation of nanowire MOSFETs, we compared the results of the SchrödingerPoisson module to the same data obtained from a commercial simulator (Silvaco, [24]) which can simulate cylindrical nanowires. We therefore compare in the following the results issued from Cylmos and Silvacoin terms of quantum energy levels, wave functions and drain current. Each time, we will obtain a very good agreement between the two data sets, which demonstrates the validity of our code.
3.1. Quantum energy levels and wave functions
We begin this comparison by looking in detail the internal parameters of the Schrödinger module, namely the quantum energy levels and associated wave functions. We distinguish here between the two radial effective masses,
Table 2 shows the comparison between the energy levels,











0.0406  0.0406  0.094  0.094  0.111  0.158  0.158  0.194  0.194 

0.0409  0.0409  0.094  0.094  0.112  0.1601  0.1601  0.199  0.199 
The wave functions corresponding to the energy levels reported in Table 2 are plotted in Fig. 5 as a function of the position on the radial direction (for
The same analysis can be conducted for the energy levels and wave functions calculated for the second radial effective mass,







0.064  0.064  0.148  0.148  0.175 

0.067  0.067  0.154  0.154  0.183 
In this analysis we studied the energy levels
Another remark concerns the variation of the wave functions with the radial position of the axis. For
3.2. Drain current
We also compare our code Cylmos with Silvaco in terms of drain current. Unfortunately, Silvaco does not provide the drain current, because the coupling between the SchrödingerPoisson system of equations and the continuity equation is not yet implemented. Nevertheless, Silvaco couples the Poisson equation directly to the continuity equation (i.e. the Schrödinger equation is ignored), which makes possible the calculation of the drain current in the socalled "classical" case (this means that the quantum confinement effects are not taken into account in this approach). To facilitate comparison with Silvaco, we have implemented the same procedure in Cylmos. Thus, in our code it is possible to calculate, in addition to the "quantum" drain current (issued from the selfconsistently solving of the Poisson, Schrödinger and continuity equations), the "classical" drain current (by directly coupling the Poisson equation to the continuity equation). In this “classical” case, the electron density is reevaluated using the wellknown FermiDirac or Boltzmann statistics:
where
The solving procedure is the same as explained in Section 2. The discretized Poisson equation (eq. (8)) is solved taking into account this time no more the electron density issued from the Schrödinger equation, but the "classical" electron density given by eq. (56). The potential is then calculated and is afterward injected directly into the module solving the continuity equation. The output of this module is the new quasiFermi level which is subsequently used in eq. (56) to calculate a new carrier density to be used in the Poisson module. In this way, a new potential is found and will be injected into the continuity equation and so on. The loop stops when the convergence criterion is reached.
Figure 8 shows the drain current, simulated in this way, for a long channel MOSFET (L=40 nm) with a circular nanowire with a diameter D=3 nm. The drain current is plotted as a function of the gate voltage for V_{D}=50 mV. This figure illustrates that a good agreement is obtained between the results of Cylmos (“classical” drain current) and those provided by Silvaco.
We also tested a shortchannel transistor, for which the gate length is equal to the diameter of the nanowire (L=D=8 nm). This transistor is probably provided with quite important shortchannel electrostatic effects and it is necessary to verify that our code correctly reproduces these effects. In addition, the shortchannel effects are considerably enhanced when the drain voltage is high, therefore it is important to test the current characteristics for several V_{D}. Figure 9 shows such a verification on both characteristics plotted in linear scale (for a better view on the regime above the threshold) and in logarithmic scale (to visualize the subthreshold regime). These figures show once again an excellent agreement between Cylmos and Silvaco, which means that the electrostatic effects are properly taken into account in Cylmos.
4. Simulation results
After validating Cylmos by comparison with a commercial simulator, we use in the following our code to analyze the operation and performances of circular nanowirebased MOSFETs subjected to quantum confinement. We will particularly focus on the variation of the drain current as a function of the nanowire diameter and of the gate length and we will discuss the device performances in terms of offstate current, threshold voltage and parasitic shortchannel effects. These key parameters will also be compared between the classical and the quantummechanical case.
4.1. Drain current characteristics
We start with the variation of the drain current as a function of the nanowire diameter. Figure 10(a) plots the drain current characteristics as a function of the gate voltage, as calculated by Cylmos, for two diameters D=5 nm and D=8 nm. Data obtained in both classical and quantum cases are shown in this figure. A first remark concerns the offstate current (i.e., value of the drain current for V_{G}=0 V): for D=5 nm this current is lower than for D=8 nm, whatever the computation method, classical or quantum. Similarly, the current in the subthreshold regime is lower for the smallest diameter. This is due to the socalled “volume inversion” phenomenon which takes place in all multiplegate devices in the subthreshold regime [28][29]. Thus, we logically find this phenomenon here in the circular nanowire, for which the gate completely surrounds the silicon film. Because of the volume inversion, the subthreshold current is directly proportional to the nanowire diameter, which implies that the offstate current decreases when the nanowire is thinned.
A second remark concerns the reduction of the drain current in the quantum case compared to the classical case, for both nanowire diameters considered here. As we explained earlier in this chapter, the existence of a strong quantum confinement of carriers leads to the reduction of the charge inversion in the channel (compared to the classic case where this effect is not taken into account). Therefore, for the same gate voltage V_{G}, the quantum drain current is necessarily smaller than the classical drain current, and this for all nanowire diameters. This behavior is similar to that of all other multiplegate devices where strong quantum confinement effects (1D or 2D) occur [30], mainly due to the geometrical construction of these devices.
Another point concerns the difference between the current calculated in the classical case and the current calculated in the quantum case. This difference is larger for D=5 nm than for D=8 nm, as can be seen in Fig. 10(a), particularly in the subthreshold regime. The explanation comes from the large variation of quantum confinement effects with the nanowire diameter. When the nanowire is thinned, the carrier confinement is more pronounced (which also implies an increase of the allowed energy levels) and the inversion charge density in the channel reduces. This increases strongly the threshold voltage and leads to a significant decrease of the drain current.
Finally, Fig. 10(b) shows the variation of the drain current as a function of the gate length for D=5 nm. In this figure we can observe that when the gate length is reduced (for the same nanowire diameter) the offstate current strongly increases and the subthreshold slope is much higher than in the case of a long channel (L=40 nm), reflecting a significant degradation of the device performances. This is due to the huge increase of shortchannel effects, because in this ultrashort channel having a gate length equal to the nanowire diameter, the gate control on the electrostatic potential of the channel is much less effective. Thus, parasitic electrostatic effects dramatically increase and lead to this large increase in offstate current and subthreshold slope. This behavior appears for both classical or quantum cases.
4.2. Threshold voltage and short channel effects
For a more detailed analysis of these parasitic shortchannel effects, we will now focus on the variations of the threshold voltage as a function of the gate length and of the nanowire diameter. The threshold voltage is extracted using the constant current method [31] from the drain current characteristics as a function of the gate voltage. In Fig. 11 we plot the variation of this threshold voltage as function of the nanowire diameter in the case of a longchannel transistor and in the case of a transistor with a channel length equal to the nanowire diameter (L=D). The results reported in this figure come from a quantummechanical calculation. We note that the threshold voltage of the longchannel transistor is higher than that of the shortchannel transistor with L=D. This is an expected result because the short channel effects that occur in the transistor with L=D greatly reduce the threshold voltage. One can also note that when the diameter of the nanowire is reduced, the threshold voltage remains constant over a certain range of values and next begins to increase with decreasing the nanowire diameter (in both cases, long channel and short channel with L=D). This increase is due, of course, to the fact that the quantum confinement of carriers is more significant when the diameter is reduced, which induces an increase in the threshold voltage (as explained earlier). This increase is higher in the case of long channel than in the case of a shortchannel. This indicates that the strong shortchannel effects that exist in the channel with L=D affect the quantum confinement, which becomes weaker than in a longchannel transistor.
Another way to highlight this phenomenon is to calculate the impact of short channel effects, reflected in the SCE metric, which is equal to the difference between the threshold voltage of the shortchannel transistor with L=D and the threshold voltage of the longchannel transistor (for a given diameter D). SCE is then plotted as a function of the nanowire diameter in Fig. 12. This figure shows that, after an increase and a maximum at D=8 nm, the SCE starts to decrease as the diameter reduces below 8 nm. This behavior is explained by the fact that the strong quantum confinement that occurs in these ultrafine nanowires reduces shortchannel effects, which will probably lead to increased performance of these ultrashort and ultrathin devices.
5. Conclusion
In conclusion, we described in this chapter a numerical solver, called Cylmos, that provides selfconsistent solving of the 2D Schrödinger equation and the 3D Poisson equation in cylindrical coordinates, coupled with the driftdiffusion transport equation. We presented in details the discretization of these equations and the overall solving algorithm used to obtain the final drain current of the device. We also paid a particular attention to justify the approximations and simplifications used in the solving of these equations in cylindrical coordinates. A meticulous validation step was performed by comparing the results of our code with those issued from a commercial code. We demonstrated that our code properly takes into account particular phenomena specific to MOSFET devices based on circular nanowires with surrounding gate, such as volume inversion, short channel and quantummechanical effects. Simulated drain current versus gate voltage characteristics have also been successfully compared to data obtained from simulation with this commercial code. Our code was finally used to analyze electrical parameters such as drain current, offstate current, threshold voltage and shortchannel effects in ultrashort radiallysymmetric nanowire MOSFETs. We have shown that quantummechanical confinement is very important in circular nanowires, but the presence of this phenomenon reduces the impact of parasitic shortchannel effects. This is a very encouraging result for the operation of future integrated circuits based on ultrashort and ultrathin circular nanowires.
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