For the curved surface of the cylinder, the unit vector is which gives Parameterize ,Since we are confined to the first octant The flux through the slant surface is . The top and the bottom caps are in the directions, the contribution from these two give zero by symmetry. There are two more surfaces if we consider the first octant, they are the positive x-z plane and the positive y-z plane., the normal to the former being in the direction of while that for the latter is along directions. The flux from the former is , while that from the latter is . Adding up all the contributions, the total flux from the closed surface is zero. This is consistent with the fact that the divergence of the field is zero.

1. You are given a MOS Capacitor whose gate material has a work function of 4.1eV , oxide (SiO_{2}) thickness is 10nm and whose body consists of p-type Silicon with doping concentration N_{A} = 10 × 10^{15}/cm^{3}. Assume that the dielectric constant of SiO_{2} is 3.9
and that of Si is 11.68. The electron affinity (E_{0} − E_{c}) of Silicon is 4.05eV , and the
band-gap E_{c} −E_{v} of Silicon is 1.1eV (thus, the Fermi level and work function of intrinsic
Silicon is 4.6eV ). The magnitude of the electron charge q is 1.6 × 10^{19} Coulombs. At room temperature, assume that for Silicon, n_{i} = 10^{10}/cm^{3}.

There are six faces. For the face at x=0, since the surface is directed along direction, the surface integral is The face at x=1 gives +1/2. The faces at y=0 and that at z=0 gives zero because the field is proportional to y and z respectively. The contribution to flux from y=1 is 1 and that from z=1 is 3/2. Adding, the flux is 5/2 units. This can also be done by the divergence theorem. Divergence of the field is 2x+3y, so that the volume integral is

2. Plot the transit frequency of NMOS as a function of V_{GS} for the biasing conﬁguration
shown in ﬁgure Fig. 1. Calculate the transit frequency of MOSFET for V_{GS} = 1.5V . Assume W = 1µm, L = 180nm, C_{GS} = 1.2 ƒ F and C_{GD} = 1 ƒ F . Given V_{DS} = 1.8V .

Note: transit frequency is deﬁned as the frequency at which small signal current gain of the device drops to unity while the drain and source are held at ac ground.

Figure .1

The divergence of the given vector field is . Thus, by divergence theorem, the flux is . We can show this result by direct integration. The unit normal on the surface of the sphere is given by , so that the flux is This integral can be conveniently evaluated in a spherical polar coordinates with The surface element on the sphere is . By symmetry, the flux can be seen to be . (Note that we decided to do the integral involving z4 rather than x4 or y4 because the azimuthal integral gives 2p in this case. The integral is easy to perform with the substitution , which gives the flux to be .

3. For an NMOS and PMOS device operating in saturation region sketch W/L versus V_{GS} − V_{TH} if

I_{D} is constant

g_{m} is constant

The divergence of the field is 3. The flux, therefore, is 3 times the volume of the cone which is The flux is thus The direct calculation of the flux involves two surfaces, the slant surface and the cap, as shown in the figure. The cap is in the xy plane and has an outward normal . The flux (because on the cap z=1 and the cap is a disk of unit radius). Thus it remains to be shown that the flux from the slanted surface vanishes. At any height z, the section parallel to the cap is a circle of radius z. Since, the height and the radius of the cap are 1 each, the semi angle of the cone is 45^{0}.

Thus the normal to the slanted surface has a component along the z direction and in the x-y plane. The component in the xy plane can be parameterized by the azimuthal angle and we can write . The area element can be written as , the factor appears because the length element is along the slant. Thus the contribution from the slanted surface is . Using , this integral can be evaluated and shown to be zero.

4.Plot the following parameters as function of body to source voltage of an NMOS device.

Effective channel length of device

g_{mb} of NMOS when NMOS is in saturation and g_{m} is constant

Drain current when gate and drain are biased at ﬁxed voltage

This problem is to be attempted similar to the problem 5 of the tutorial, i.e., by closing the cap and subtracting the contribution due to the cap. The divergence being 3, the flux from the closed cone is 3 times the volume of the cone which gives The contribution from the top face (which is a disk of radius 2 ) is . Thus the net flux is zero. (You can also try to get this result directly as done in problem 4, where we showed that the flux from the curved surface is zero).

5. An NMOS device operating in subthreshold region has a ξ = 1.2 (ξ is the nonideaelity factor of subthreshold conduction). What variation in V_{GS} results in a ten-fold change in ID ? If I_{D} = 8µA, what is g_{m}? Fig.6