ch3

ch3 - Solutions Chapter 3 3.1 Calculate the speed of an...

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Solutions Chapter 3 3.1. Calculate the speed of an electron in Si with kinetic energy 0.013 eV. Draw an equilibrium energy band diagram for silicon and indicate where this electron will be. Compare your calculated thermal speed to the typical drift velocities cited in the text of 10 4 cm/s. How does it compare to typical saturation velocities? The electron speed is ( 29 ( 29 19 5 * -31 5 7 2 0.013 1.6 10 / 2 1.3 10 / 0.26 9.1 10 100 1.3 10 / 1.3 10 / K ce eV J eV E v m s m kg cm m s cm s m - × = = = × × = × × = × Even with this very small kinetic energy, the electron’s instantaneous speed is still about 1000 times faster than the drift velocity. It is comparable to the drift saturation velocity for silicon of 1 × 10 7 cm/s. E C E V E g =1.12eV E K =0.013eV 3.2. Calculate the resistivity for a uniformly doped silicon sample with 10 17 donors per cubic centimeter. Anderson & Anderson 1 2/15/04 Solutions Chapter 3
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From Equation (3.14), σ = q μ n n + q μ p p The material is n-type, and not degenerately doped, so n 0 =10 17 and ( 29 2 10 3 2 3 3 0 17 3 0 1.08 10 1.2 10 10 i cm n p cm n cm - - - × = = = × The mobilities are μ n =6.3 × 10 2 for majority carriers (electrons) at N D =10 17 and μ p =4.6 × 10 2 for minority holes. ( 29 ( 29 ( 29 ( 29 ( 29 19 17 19 3 1 1.6 10 630 10 1.6 10 460 1.2 10 10 n p q n q p cm σ μ μ - - - = + = × + × × = - The resistivity ρ is 1 0.1 - cm ρ σ = = . Note that the contribution of holes to the conduction is negligible. 3.3. A lightly doped Si sample ( N D = 10 14 cm -3 ) is heated from 300 K to 400 K. Is its resistivity expected to increase or decrease? Explain your answer. Repeat for Si with N D = 10 18 cm -3 . For the lightly doped sample, as the temperature is increased the intrinsic concentration increases. It is still true that D i N n ? , though, and to reasonable approximation n 0 is constant. For this doping level, however, lattice (phonon) scattering predominates and mobility decreases with increasing temperature (Figure 3.8). Thus, the conductivity decreases or the resistivity increases For the heavily doped sample, impurity scattering predominates. As the temperature is raised, the carriers’ thermal energy is increased so the Anderson & Anderson 2 2/15/04 Solutions Chapter 3
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effect of the impurity scattering is reduced, and the mobility should increase. Therefore the resistivity will decrease. 3.4. Germanium is an interesting semiconductor because it has a small band gap ( E g =0.67eV). (In fact, for a while it was not considered to be a semiconductor but was classified as a metal. Now it is a semiconductor again.) As a result, it has a higher intrinsic concentration n i than either silicon or GaAs. Do you expect the conductivity of intrinsic germanium to be less than or greater than that of intrinsic silicon? How about compared to GaAs? Why? Because n i depends exponentially on bandgap, and E g for Ge is appreciably less than that of Si (1.12 eV) or GaAs (1.43 eV), one expects Ge to be more conductive than either Si or GaAs, because it has more carriers available for conduction, and its mobilities are comparable.
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