 Home Mathematics  The following section is not a detailed description of the numerical methods of device physics for the MOSFET but it gives the basic equations that are necessary for a complete classical description of devices with no approximations. It also sets the ground for the description of devices in terms of quantum transport. The numerical solution of these equations is a major subject in itself, but it does not fall within the scope of the present book. There are many books on numerical calculations in general.

So far, any analysis we have performed of semiconductor devices has relied on either simplifying the Poisson and continuity equations (as we did for the PN and Schottky diodes) or on devising a semiempirical model (as we did for the MOSFET). Rigorous classical calculations which do not necessitate approximations exist, but they lead to a system of differential equations rather than one equation which by itself can give the device response. In previous uses of the Poisson equation for the PN and Schottky junctions the depletion layer was assumed to be fully depleted; this is an approximation. In principle, the charge density p is equal to Furthermore, since any device, not just the MOSFET, is made of many materials with different dielectric constants e, the Poisson equation should be written in the more general form where £ is the electric field and p is given by equation 7.21. It should be immediately apparent that the Poisson equation cannot be solved by itself because both £ and the electron and hole densities n and p are unknown. Equation 7.22 can of course be combined with equations for n and p to form a system of equations as already stated. Actually, the Poisson equation is a necessary, indispensable equation in all models we are going to present.

The simplest system of differential equations for an N-channel MOSFET is produced when the Poisson equation is complemented by the continuity equation for electrons at steady state (cf. equations 3.33 and 3.34) where Jn is given by the drift-diffusion model in 3 dimensions and Rn is the recombination rate for electrons. Note that in the channel the electrons are the majority carriers and hence the full Schottky-Reed-Flall theory must in principle be used for Rn. Flowever, for unipolar devices as the MOSFET, i.e. for devices operating with mainly one type of carrier, the recombination rate is small and is many times omitted.

From the discussion in section 7.2 it should be obvious that when the substrate is P-type, the channel is N-type and vice versa. Therefore, the above set of equations is not sufficient for a domain of solution where the character of the semiconductor changes from P-type to N-type. The density of holes p must be included in the system of equations and a third equation becomes necessary. We note, however, that if the domain of solution is restricted to the channel, a solution is possible by the above restricted system of 2 equations with 2 unknowns. A flow chart for the solution of this simple 2x2 system of differential equations is shown below. FLOWCHART 1

A much better model is obtained if all V, n,p are included but the electrons are treated with a higher accuracy than the holes. This can be accomplished by treating the electrons by the Boltzmann equation (cf. equations 4.3-4.7) and the holes by the continuity equation. Thus with f{k,r) being the distribution function for electrons, in the notation used in chapter 4, we have for the Boltzmann and continuity equations, 7.25 and 7.26 respectively where q= -e and To accomplish self-consistency, the distribution function must also appear in the Poisson equation. This is evidently achieved by writing for the electron density n(r) including spin The above system of equations can be solved numerically by discretizing all the unknown functions or by numerically solving the Poisson equation on the one hand and solving the Boltzmann by expanding in the spherical harmonics Ylm (0,cp) . A flow chart for the solution of the set of equations just described, is shown below. FLOWCHART 2 denotes here an initial guess of the non-equilibrium distribution function and not the value at equilibrium.

The next step in the modelling of the MOSFET is the inclusion of the Schrodinger equation. The inclusion of the latter in the modelling of the MOSFET was found necessary because of the continued miniaturization of the MOSFET and the corresponding effect on the charge density in the transistor channel. In particular, along with the shortening of the channel length, there comes a corresponding reduction of the oxide thickness. This in turn increases the electric field perpendicular to the Si/oxide interface with the result that the bending of the conduction band near the interface becomes very steep, see figure 7.11. Then the electrons in the conduction band of Si cannot be considered to be in a conventional

3-dimensional semiconductor, but in a 2-dimensional one, parallel to the oxide/Si interface with the third dimension, the one perpendicular to the interface, being of nanometric length. The wavefunction of the electrons in the channel experience quantization effects in their density of states, as described in chapter 5. Then a 2-dimensional electron gas (2DEG) is said to be created. The situation is depicted in figure 7.11, where we show the potential well perpendicular to the Si/oxide interface and the corresponding wavefunctions for the first 3 bound levels of the well. Although the above described situation constitutes a FIGURE 7.11 Quantization of the electrons in the region below the oxide due to the surface potential in the semiconductor. The integer i stands for valley.

quantum effect, we include it here in the classical simulations section, because the transistor equations used in such models for the current density J are still classical.

The Schrodinger equation for the direction perpendicular to the oxide surface reads where Xi?j is the (envelope) wavefunction and , are the eigenvalues for the i,h valley in the jth sub-band. The term m, z is the effective mass of the ith valley along the z direction which is perpendicular to the Siloxide interface. Note that we use U for potential energy to avoid confusion with V reserved for potential, i.e. the solution of the Poisson equation. Remember that the relation between the two is U= -eV. The 1-dimensional potential energy Ud(z) can be obtained from the 2-dimensional Poisson equation. The latter can be solved numerically, on a mesh, in a sub-domain called quantum box, as shown in figure 7.11. Equation 7.28 is then solved along any of the lines of the mesh that are perpendicular to the Si / oxide interface by imposing zero boundary conditions on both ends. Along any such line Uw(z) can be extracted from the solution of the Poisson equation.

The electron density n(z) along these perpendicular lines to the interface can be computed from the relation with g2D given by equation 5.12 and/0 is the equilibrium Fermi-Dirac distribution. The above model can be extended by augmenting the domain of solution of the Schrodinger equation to 2 dimensions, namely the length x and depth z of the channel. Then the electron density is likewise given by with gu> given by equation 5.19. The peak electron concentration along the channel thus calculated is shown in figure 7.12a, figure redrawn from . One can see that the electron concentration does not behave in the simple manner of the previous section. The current IDS as a function of Vos and Vcs is shown in figure 7.12b, redrawn again from . Details about the calculation and a flow chart are given in the next chapter on quantum wells, see flow chart 3.

As miniaturization continued according to Moore’s law, the ballistic regime of operation is reached. Present-day FETs already operate in this regime. The Boltzmann equation becomes simplified in the ballistic regime because the collision term is absent from the equation. We will allocate more space to the description and solution of this Boltzmann equation because a) it leads naturally to the quantum description and b) the solution can be inferred from physical intuition and no numerical work is necessary in this case. A lot of the physical intuition actually derives from the Landauer theory of chapter 5. A Landauer FIGURE 7.12a Calculations of the peak electron concentration in the channel by Pirovano et al  using the 2-dimensional Schroedinger equation and a classical transport model. FIGURE 7.12b Calculations by Pirovano et al  of the current ID as a function of gate voltage VG using the 2-dimensional Schroedinger equation and a classical transport model. For comparison a classical calculation where the Schroedinger equation is not used is also shown.

theory of the MOSFET will be given in the next section. The solution of the Boltzmann equation given here is due to M. Lundstrom  and we follow it closely.

The MOSFET channel is considered bounded by two oxide layers, one above it, one below, and connected to two reservoirs, the source and the drain as shown in figure 7.13. Injection of carriers from both the source and drain reservoirs into the channel takes place as the Landauer theory assumes. Let the corresponding Fermi energies be Ещ and £/■>. Obviously FIGURE 7.13 Layout of the nanodevice used for the Boltzmann calculations by Rhew et al .

thermodynamic equilibrium prevails inside the leads-reservoirs. The Boltzmann equation in the steady state of the ballistic regime (i.e. no collisions) takes the form, assuming variation in one dimension only, where vx and Sx are the velocity and electric fields along the channel, i.e. along the x-direc- tion and / denotes the distribution function in the ballistic regime. Note that the Plank constant in the denominator of the second term has been absorbed in the derivative. For simplicity we assume that all the electrons in the channel occupy only the lowest sub-band of the conduction sub-bands shown in figure 7.11.

Furthermore, the electrons, see figure 7.13, are assumed to be free to move in the (x,y) plane with a single effective mass, that of the transverse effective mass m, of equation 2.46. The momentum vector p spans the (x,y) plane, i.e. p = (px,py) and the energy E(x,p) is in an effective mass approximation where Ec(x) is the lowest conduction sub-band minimum. This quantity is obtained by solving the Poisson equation in the (x,z) plane self-consistently with the 1-dimensional Schroedinger equation, along 2 as explained in previous paragraphs. Then the electric field is The boundary conditions for the Boltzmann equation can be specified by assuming that injection into the channel from both source and drain occurs according to Fermi-Dirac statistics. Since both the equilibrium and non-equilibrium distribution functions will appear in our formulae, we retain the notation of chapter 4 on the Boltzmann equation and denote the equilibrium Fermi-Dirac distribution by f0 and by / the non-equilibrium one. Furthermore we retain the notation of denoting the Fermi levels of the source and drain by Epi and EFr (left, right) respectively For the source where injection (at x = 0) necessitates px> 0 we have and for the drain where injection (at x = L) into the channel necessitates px < 0 The occupation ofthe(x,p) states in the channel can now be inferred by physical intuition, as mentioned earlier. The main idea is that the injected electrons in the channel remain in equilibrium with the reservoir from which they were injected depending on their momentum and position in the channel. To expound this idea, we draw Ec(x), the lowest conduction sub-band profile, together with the respective Fermi levels in figure 7.14. Ec(x) shows a maximum £max at x,op near the source. We also show in this figure the parabolic band structure for the electrons at an arbitrary point in the channel. The dependence of £(x,p) on p may look simple, but it is good enough for our arguments.

Consider the electrons ejected from the source. Those which will climb over the barrier, i.e. those with pK > pmax where are shown with a solid line on the band structure in figure 7.14. All the other electrons injected from the source with px < pmax will be reflected back. Consider now the electrons injected from the drain. These will have initially a negative px. The ones whose energy is below the barrier, i.e. those with |px | < pmax will travel at most up to xtop and will then be reflected back to the drain. These states are marked with a broken curve on the band FIGURE 7.14 Variation of the conduction band minimum Ec (x) as a function of the distance along the channel of the nanodevice in figure 7.13. Also shown at an arbitrary point along the channel is the dispersion relation E-k. Electrons from the source which go over the barrier are marked with a solid curve on the dispersion relation, electrons from the drain which also go over the barrier are marked with a dotted curve and those from the drain or source which return back with a broken line.

structure diagram. Finally, the electrons from the drain emitted with negative momentum and energy greater than £max will also climb the barrier and are marked with a dotted curve in figure 7.14. The ballistic solution of the Boltzmann equation can now be written down by putting the electrons in two categories: those above the barrier and those below it.

For px> pxmax and for рхХпшх From the above discussion it should be evident that the behaviour of /(x,p) at the top of the barrier x = xlop is of paramount importance. Figure 7.15 shows cross-sections of f(vxyvy = 0) for several drain voltages, where v denotes velocity, at x = x,op. It shows clearly that as Vos is increased, an asymmetry in the occupation of the velocities in the x-direction appears, hence current begins to flow in the channel. The zero current in the case of Vds = 0 is a result of two opposite and equal currents. The voltage VDS spoils this symmetry. Saturation at VDS = 0.6 V is also observed. 2-dimensional diagrams of this situation are shown in figure 7.16.

The evolution of f(vx,vy = 0) along the channel is shown in figure 7.17 for Vds = 0.6 V, i.e. at saturation. At x = 2.5nm, i.e. well inside the source, the distribution is very similar to either that of figure 7.15 or to a cross-section of figure 7.16 at vy = 0. As the wavepackets or ballistic electrons travel along the channel, they gain energy from the field and hence their peak velocity moves to higher velocities. Also, their distribution becomes narrower. As we approach the drain, we find the drain-injected electrons which have a symmetric (with respect to positive and negative v*) distribution since they are reflected back well before reaching x,op. FIGURE 7.15 The 1-dimensional cross-sections of f(vx, vy=0) for several VDS at the top of the barrier xtop for VG=0.6V. Note that saturation begins at VDS=0.2V and is complete at VDS=0.6V. Figure has been reproduced from reference . FIGURE 7.16 The 2-dimensional plots of f (vx, vy). Note that the distribution ofvy retains the equilibrium shape. Figure has been reproduced from reference . FIGURE 7.17 Evolution of f (vx,vy=0) along the channel at saturation (VDS=0.6V). See text for comments. Figure has been reproduced from reference .

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