FIGURE 22.6 Experimental setup for Hall effect measurements in a long two-dimensional sample. The Hall angl determined by a setting of the rheostat that renders j, =0. Magnetic field B= B(Source: K W Boer, Surveys of Semicond Physics, New York: Chapman Hall, 1990, P 760. With permission.) The curves shown in Fig. 22.5 were obtained for uniform semiconductors under steady-state conditions. Strictly speaking, this is not the case with actual semiconductor devices, where velocity can"overshoot the value shown in Fig. 22.5. This effect is important for Si devices shorter than 0. 1um(0. 25 um for GaAs devices)[ Shur, 1990; Ferry, 1991]. In such extreme cases the drift-diffusion equation(22.9) is no longer adequate, and the analysis is based on the Boltzmann transport equation +vVf +gvF= df (22.10) Here f denotes the distribution function (number of electrons per unit volume of the phase space, i. e, f dn/rd,p), v is electron velocity, p is momentum, and (af/dn)ool is the"collision integral "describing the change of f caused by collision processes described earlier. For the purpose of semiconductor modeling, Eq (22.10) be solved directly using various numerical techniques, including the method of moments(hydrodynamic modeling)or Monte Carlo approach. The drift-diffusion equation(22.9)follows from(22 10)as a special case For even shorter devices quantum effects become important and device modeling may involve quantum [Fery;1991] In a uniform magnetic field electrons move along circular orbits in a plane normal to the magnetic field B with ne angular(cyclotron) frequency @.= gB/m. For a uniform semiconductor the current density satisfies the j=σ(g+R[jB]) (22.11) In the usual weak-field limit o,t < I the Hall coefficient RH =-ring and the Hall factor r depend on the dominating scattering mode. It varies between 3 /8=1. 18(acoustic phonon scattering) and 3157/518=1.93 The Hall coefficient can be measured as rh= V d/I B using the test structure shown in Fig 22. 6. In this expression V, is the Hall voltage corresponding to 1=0 and d denotes the film thickness. e 2000 by CRC Press LLC© 2000 by CRC Press LLC The curves shown in Fig. 22.5 were obtained for uniform semiconductors under steady-state conditions. Strictly speaking, this is not the case with actual semiconductor devices, where velocity can “overshoot” the value shown in Fig. 22.5. This effect is important for Si devices shorter than 0.1mm (0.25 mm for GaAs devices) [Shur, 1990; Ferry, 1991]. In such extreme cases the drift-diffusion equation (22.9) is no longer adequate, and the analysis is based on the Boltzmann transport equation (22.10) Here f denotes the distribution function (number of electrons per unit volume of the phase space, i.e., f = dn/d3 rd 3 p), v is electron velocity, p is momentum, and (¶f /¶t)coll is the “collision integral” describing the change of f caused by collision processes described earlier. For the purpose of semiconductor modeling, Eq. (22.10) can be solved directly using various numerical techniques, including the method of moments (hydrodynamic modeling) or Monte Carlo approach. The drift-diffusion equation (22.9) follows from (22.10) as a special case. For even shorter devices quantum effects become important and device modeling may involve quantum transport theory [Ferry, 1991]. Hall Effect In a uniform magnetic field electrons move along circular orbits in a plane normal to the magnetic field B with the angular (cyclotron) frequency wc = qB/m* n . For a uniform semiconductor the current density satisfies the equation j = s(% + RH[jB]) (22.11) In the usual weak-field limit wct << 1 the Hall coefficient RH = –r/nq and the Hall factor r depend on the dominating scattering mode. It varies between 3p/8 ª 1.18 (acoustic phonon scattering) and 315p/518 ª 1.93 (ionized impurity scattering). The Hall coefficient can be measured as RH = Vy d/IxB using the test structure shown in Fig. 22.6. In this expression Vy is the Hall voltage corresponding to Iy = 0 and d denotes the film thickness. FIGURE 22.6 Experimental setup for Hall effect measurements in a long two-dimensional sample. The Hall angle is determined by a setting of the rheostat that renders jy = 0. Magnetic field B = Bz. (Source: K.W. Böer, Surveys of Semiconductor Physics, New York: Chapman & Hall, 1990, p. 760. With permission.) ¶ ¶ ¶ ¶ f t f q f f t + — + — = Ê Ë Á ˆ ¯ v % p ˜ coll