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# QUANTUM THEORY OF THE MOSFET

We now come to a quantum theory of the MOSFET. As already noted, present-day MOSFETs have channel lengths of the order of tens of nanometers or even smaller, so according to the introductory discussion of chapter 5 a classical particle approach is no longer valid. Furthermore, the ballistic transport, expounded in the previous section, must be a necessary ingredient of such a theory. We emphasize again that the use of the Schrodinger equation, although necessary, is not adequate to characterize a methodology as quantum. It is the use of a quantum transport method to analyze the device that does so. As usual, the effective mass equation will be used. We follow closely the analysis of K. Natori [5], one of the pioneers in the subject. This analysis is not only useful for MOSFETs, but can be applied to many devices, so we will use it in the next chapter for devices made out of semiconductors other than Si.We assume here the coordinate system used in the previous section and throughout this chapter. The device is homogenous in the у direction and of width W. The coordinate x is along the channel length and z is along the channel depth.

For a Si channel in a MOSFET the effective mass equation reads

Since the device is uniform in the у direction, the potential energy along у can be taken as that of a very wide quantum well of width W and of a constant deep potential, as shown in figure 7.18a. The potential energy along x will be of the same form as that used in the solution of the ballistic Boltzmann equation in the previous section. This is shown in figure 7.18b. The maximum at x,op of U along x is shown and this specific point will play a major role in the quantum theory of the MOSFET as it did in the solution of the Boltzmann equation.

FIGURE 7.18 Potential energy distributions of an electron in a MOSFET along the three spatial directions (a) along its width y, (b) along the channel length x, and (c) along its depth z.

Finally, the potential energy along the depth z of the channel will have the usual triangu- lar-like barrier form shown in 7.11 which is narrow enough as to quantize the eigenfunctions in z direction. We reproduce it here in figure 7.18c for completeness.

We assume that the wavefunction can be written in the following form

where W is the width in the у-direction and ny, nz are the quantum numbers in the у and z direction respectively. Note that equation 7.39 cannot be obtained by the method of separation of variables because the triangular barrier shown in figure 7.18c is x-dependent. However since the potential along x is slowly varying compared to z, the Ф,,. (x,z) can be taken as the solutions of the following quasi-l-dimensional equation

where the coordinate x is treated as a parameter. This Schroedinger equation is solved along 1-dimensional cuts perpendicular to the plane of the gate. In 7.40 above the position of the line cut is explicitly shown through the x dependence. Furthermore, the valley index does not appear explicitly. However, the valley index will appear later in the expression for the current to account for the proper counting of the number of electrons emitted from the source and from the drain.

The function X(x) can be obtained from the WKB approximation discussed in chapter 6. From equation 6.17 we get for X(x) for a wave propagating from source to drain

where

and C is a constant. Note that nyn/W acts like a wavevector in the у direction and the eigenvalues in the у and z direction act as an effective potential in the WKB expression. Around the critical maximum xtop equation 7.41 may be approximated by a single plane wave as

where

To calculate the current I we use the Landauer theory of chapter 5 with slight modifications which are necessary here. As usual, we assume that 1) injection from both source and drain takes place so that the net current is the sum of two currents travelling in opposite directions and 2) the electrons injected from the source remain in thermodynamic equilibrium with the reservoir of the source and so do the electrons injected from the drain. On the other hand, we take into account here the different valleys of Si when summing over all the eigenstates normal to the propagation direction and we also take into account that the final state of any electron must be empty. We do not care what happens to the electrons as they cross the channel in each of the two directions. Instead of examining the particular scattering events we simply multiply each beam by the transmission coefficient, which is the same for both directions, to obtain the number of electrons reaching the drain from the source and vice-versa.

The flow of electrons in each direction is given by their velocity times their density times the electron charge (e). The electron density is given by the ID density of states times the respective (source or drain) Fermi-Dirac distribution. If EFS and EFD denote the Fermi levels in the source and drain respectively and the equilibrium Fermi-Dirac distribution is denoted by f(EF,E) (as in chapter 5) we have for the current flowing from source to drain

where the term in parenthesis in 7.43 guarantees that the receiving state is empty and the summation over / indicates summation over valleys. Similarly, for the current injected from the drain and reaching the source we have

Evidently

The 1-dimensional density of states gio given by equation 5.20 is not appropriate here because the present potential U(x,z) does not resemble in any way the confining potential of section 5.2b. The function gw of equations 7.43 and 7.44 can be computed easily: the

number of states per spin per dkx interval is 1/1— I so that the number of states per dkx interval per length is l/n and from the identity

we get

The velocity vx in the effective mass approximation used here is

so that the product of the first two factors in 7.43 and 7.44 gives the constant (тгй)-1. Then equation 7.45 reduces to

We remind the reader that if the Fermi level at the source reservoir is E?s, the Fermi level at the drain reservoir is EFD = EFS-eVDs.

In accordance with our results of the Boltzmann equation of the previous section the electron beam that will reach the opposite reservoir must have energy Ex > E„z (xlop). We therefore assign a transmission coefficient of T = 1 for these energies and a transmission coefficient T = 0 for the ones below this limit. Obviously, tunnelling through the barrier is neglected. Actually, tunnelling through the barrier is the ultimate limiting process for the good operation of a MOSFET. The summation over ny can now be turned into an integration over the normal energy Ey since the width W is usually large. We can therefore write

The density of states g{'D will also be given by an equation equivalent to 7.46 (with x substituted by y). This will give a term of the form EJ,112. Performing the inner integration (which will give Ely2) we finally get (after changing the variable of integration from E to E / KT (KT is the Boltzmann thermal energy)

where Fy(a) is the Fermi-Dirac integral of order ^ defined by

and as usual Ew = £/;s - eVD. This definition of Fy is equivalent to the one we gave in chapter 2.

The mobile charge Q„ per unit area at the bottleneck x = x,op can be evaluated in the same manner we have evaluated NL and NR of equation 5.67. Both contributions from source and drain must be considered. Hence, we can write summing over all sub-bands and valleys

The current IDS as a function of Q„ is shown in figure 7.19, figure reproduced from Natori (2008) [6]. Two curves are shown, the one labelled as EOSA denotes calculations with an Effective One Sub-Band Approximation, i.e. by considering only the lowest sub-band, the other labelled MSM denotes calculations with many sub-bands. The Ids - Vds characteristics are plotted in figure 7.20 and are compared with the measured characteristics of a 70nm MOSFET. It can be seen that the theory of ballistic transport does not accurately represent the experimental curves. The primary reason for the disagreement is that the approximation of a transmission coefficient equal to one is too extreme for such a gate length of a channel made out of Si. Actually, the result given by equation 7.49 should be considered as the limiting case to which all FETs tend as the channel length L diminishes. We note of

FIGURE 7.19 Current IDS as a function of the mobile charge Qn at x=xtop. EOSA stands for Effective One Sub-band Approximation and MSM denotes a calculation including many sub-bands. The figure is reproduced from Natori (2008) [6].

course that in such a limit the current Ids is independent of L. Present-day MOSFETs are nearly ballistic. It is worthwhile noting that III—V FETs, which we are going to examine in the next section, become ballistic at longer channel length than Si MOSFETs because the mean free path X of the III—V compounds are longer than that of Si, hence the condition LL. We now give a simpler theory of semi-ballistic FETs that is probably more physically appealing. The method is due to Lundstrom [6, 7].

FIGURE 7.20 Comparison of the theoretical (Natori) ID-VD curves with experimental curves of a 70 nm MOSFET. Reproduced again from [6].

Unlike the case of long-channel FETs where the current Ids is extracted by an integration over the channel length (c.f. equations 7.7 to 7.13), in the case of nanoFETs the current Ids is due entirely by what happens at the bottleneck x,op. The electrons that will manage to overcome the barrier will reach the drain. The point x,op is very close to the source so that equation 7.7 at the beginning of the chapter can be rewritten here as

i.e. we have neglected any small effect along the channel due to Vds-

From the discussion so far and the solution of the Boltzmann equation in the previous section we have that in saturation

or

But the density of electrons n(z) at the bottleneck with z being the depth direction of the channel is equal to the sum of the incident and reflected fluxes divided by the velocity v(x,op) at that point.

Hence

where rc;, stands for the reflection coefficient and then Jo becomes by substituting for J5 from 7.53

The integral ofn(z) is however Q„{xlop), so by equation 7.51 we get

and for the current Ids we get by virtue of 7.54

where in equation 7.56 we have used that tch = 1 - rch.

It is now time to draw some conclusions about the differences of long channel FETs and short-nanometric channel FETs. The obvious difference exemplified by the above equation

FIGURE 7.21 The critical point in a long and in a short channel MOSFET. In the long channel MOSFET it is the point of pinch-off and is near the drain, while in a short channel MOSFET it is near the source and is the point where the maximum of the barrier is located.

is that the saturation current Ids is not proportional to (Vgs - Уш )2> but to (Vcs - V,fl). This is a direct consequence, as can be seen by an examination of equations 7.51 to 7.56, of treating current as a wave rather than as a flow of particles. Furthermore, the mechanism of the saturation of the current has changed drastically from a physics point of view. Saturation of the current Ids in the long-channel FET occurs at the pinch-off region which is near the drain side of the channel. The basic mechanism is the saturation of the velocity. On the contrary, saturation of the current in the short-nanometric MOSFET occurs near the source side of the channel and is due to the saturation of the flow over the barrier maximum, i.e. at the point we have called xlop. In figure 7.21 we compare the potentials along the channel for short and long L. The point at which the high field region inside the channel begins to be established is indicative of the mechanism of the current saturation. In the long L MOSFET it is near the drain whereas in the short L MOSFET it is near the source.

We now give some approximate expressions for the transmission coefficient that rely more on physical intuition than a strict quantum mechanical theory. These may be used in equation 7.47 if one wishes to include the transmission coefficient in the Natori formula. We differentiate between the linear and saturation regimes because in each region a different length plays the major role. In the linear regime the transmission coefficient TL(E) can be represented simply by

where 1(E) is the energy dependent mean free path of the channel. Obviously TL(E)^> 1 if L 1(E). On the other hand, a different length scale is important in the saturation regime: it is Xiop, the distance from the source to the bottleneck point along the channel. The reason for this is the following: at saturation there is a large electric field in the channel that is located between xtop and the drain, see figure 7.21. If the electrons manage to overcome the barrier at xlop, the probability of being returned back to the source by back-scattering is almost zero. Therefore, we can write that the saturation regime transmission coefficient TS(E) is

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