Electronics, Power Electronics, Optoelectronics, Microwaves ...

1 -6 Electronics, Power Electronics, Optoelectronics, Microwaves, Electromagnetics, and Radarnot monotonic, which results in negative differential conductivity. Physically, this effect is related to thetransfer of electrons from the conduction band to asecondary valley (see Figure 1.3).The limiting value v s of the drift velocity inastrong electric field is known as the saturation velocity and isusually within the 10 7 to 3·10 7 cm/sec range. As semiconductor device dimensions are scaled down to thesubmicrometer range, v s becomes an important parameter that determines the upper limits of deviceperformance. The curves shown in Figure 1.5 were obtained for uniform semiconductors under steady-stateconditions. Strictly speaking, this is not the case with actual semiconductor devices, where velocity can‘‘overshoot’’ the value shown in Figure1.5. This effect is important for Si devicesshorter than 0.1 m m(0.25 m mfor GaAs devices) (Shur, 1990; Ferry, 1991). In such extreme cases the drift-diffusion equation (1.9) is nolonger adequate, and the analysis is based on the Boltzmann transport equation:q fq t þ v H f þ q E H p f ¼q f q t collð 1 : 10ÞHere, f denotes the distribution function (number of electrons per unit volume of the phase space, i.e.,f ¼ d n /d 3 r d 3 p ), v is electron velocity, p is momentum, and ( q f / q t ) coll is the ‘‘collision integral’’ describing thechange of f caused by collision processes described earlier. For the purpose of semiconductor modeling,Equation (1.10) can be solved directly using various numerical techniques, including the method of moments(hydrodynamic modeling) or Monte-Carlo approach. The drift-diffusion equation (1.9) follows fromEquation (1.10) as aspecial case. For even shorter devices quantum effects become important and devicemodeling may involve quantum transport theory (Ferry, 1991).Hall EffectIn auniform magnetic field electrons movealong circular orbits in aplane normal to the magnetic field B withthe angular (cyclotron) frequency o c ¼ q B = m n* .For auniform semiconductor the current density satisfies theequation:j ¼ s ð E þ R H ½ jB Þð 1 : 11ÞIn the usual weak-field limit o c t ,, 1the Hall coefficient R H ¼r / nq and the Hall factor r depend on thedominating scattering mode. It varies between 3 p /8 < 1.18 (acoustic phonon scattering) and 315p /518 < 1.93(ionized impurity scattering).The Hall coefficient can be measured as R H ¼ V y d / I x B using the test structure shown in Figure 1.6. In thisexpression V y is the Hall voltage corresponding to I y ¼ 0and d denotes the film thickness.Combining the results of the Hall and conductivity measurements one can extract the carrier concentrationtype (the signs of V y are opposite for n-type and p-type semiconductors) and Hall mobility m H ¼ r m :m H ¼R H s ; n ¼r = qR H ð 1 : 12ÞMeasurements of this type are routinely used to extract concentration and mobility indoped semiconductors.The weak-field Hall effect is also used for the purpose of magnetic field measurements.In strong magnetic fields o c t .. 1and on the average an electron completes several circular orbits withoutacollision. Instead of the conventional E b ( k )dependence, the allowed electron energy levels inthe magneticfield are given by ( " ¼ h /2p ; s ¼ 0, 1, 2, ... ):E s ¼ " o c ð s þ 1 = 2 Þþ" 2 k 2 z = 2 m n*ð 1 : 13ÞThe first term in Equation (1.13) describes the so-called Landau levels, while the second corresponds to thekinetic energy ofmotion along the magnetic field B ¼ B z .Inapseudo-two-dimensional system like thechannel of afield-effect transistor the second term in Equation (1.13) does not appear, since the motion of