KaSap Electronic Materials_Chapter_2_problem_solutions

KaSap Electronic Materials_Chapter_2_problem_solutions -...

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Chapter 2 solutions ___________________________________________________________________________________ 2.2 a. Apply the equation for temperature dependence of resistivity, ρ ( T ) = ρ o [1 + α o ( T - T o )]. We have the temperature coefficient of resistivity, α o , at T o where T o is the reference temperature. The two given reference temperatures are 0 ° C or 25 ° C, depending on choice. Taking T o = 0 ° C + 273 = 273 K, ρ (-40 ° C + 273 = 233 K) = ρ o [1 + α o (233 K 273 K)] ρ (25 ° C + 273 = 298 K) = ρ o [1 + α o (298 K 273 K)] Divide the above two equations to eliminate ρ o , ρ (-40 ° C)/ ρ (25 ° C) = [1 + α o (-40 K)] / [1 + α o (25 K)] Next, substitute the given values ρ (25 ° C) = 2.72 × 10 -8 Ω m and α o = 4.29 × 10 -3 K -1 to obtain K)] )(25 K 10 (4.29 + [1 K)] )(-40 K 10 (4.29 + [1 m) 10 (2.72 = C) (-40 1 - 3 - -1 -3 8 - × × Ω × ° ρ = 2.03 × 10 -8 Ω m b. In ρ ( T ) = ρ o [1 + α o ( T T o )] we have α o at T o where T o is the reference temperature, for example, 0 ° C or 25 ° C depending on choice. We will choose T o to be first at 0 ° C = 273 K and then at -40 ° C = 233 K so that ρ (-40 ° C) = ρ (0 ° C)[1 + α o (233K 273K)] and ρ (0 ° C) = ρ (-40 ° C)[1 + α - 40 (273K 233K)] Multiply and simplify the two equations above to obtain [1 + α o (233 K 273 K)][1 + α -40 (273 K 233 K)] = 1 or [1 40 α o ][1 + 40 α -40 ] = 1 Rearranging, α -40 = (1 / [1 40 α o ] 1)(1 / 40) α -40 = α o / [1 40 α o ] i.e. α -40 = (4.29 × 10 -3 K -1 ) / [1 (40 K)(4.29 × 10 -3 K -1 )] = 5.18 × 10 -3 K -1 Alternatively, consider, ρ (25 ° C) = ρ (-40 ° C)[1 + α -40 (298 K 233 K)] so that α -40 = [ ρ (25 ° C) ρ (-40 ° C)] / [ ρ (-40 ° C)(65 K)] α -40 = [2.72 × 10 -8 Ω m 2.03 × 10 -8 Ω m] / [(2.03 × 10 -8 Ω m)(65 K)] α -40 = 5.23 × 10 -3 K -1 c. We know that 1/ ρ = σ = en μ where σ is the electrical conductivity, e is the electron charge, and μ is the electron drift mobility. We also know that μ = e τ / m e , where τ is the mean free time between electron collisions and m e is the electron mass. Therefore, 1/ ρ = e 2 n τ / m e 2.1
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