2Electron and Hole Concentrations Silicon doped with 1016 cm-3 phosphorus atoms, at room temperature (T = 300 K).n = ND = 1016 cm-3, p = 1020/1016 = 104 cm-3Silicon doped with 1016 cm-3 phosphorus atoms and 1018 cm-3 boron atoms, at room temperature.p = NA - ND = 1018 cm-3, n = 1020/1018 = 102 cm-3For a compensated semiconductor, i.e. one that has dopants of both types, it is the NET dopant concentration that determines the concentration of the majority carrier. Use np = ni2 to calculate concentration of the minority carrier.
3Electron and Hole Concentrations Silicon doped with 1016 cm-3 phosphorus atoms and 1018 cm-3 boron atoms, at T = 1000 KNA = 1018 cm-3, ND = 1016 cm-3ni = 1018 cm-3 at T = 1000 KIf ni is comparable to the net dopant concentration. Then the equations on Slide 17 of Lecture 2 must be used to calculate the carrier concentrations accurately. Note, np = ni2 is true at thermal equilibrium.
9Energy band diagramConsider a Si sample maintained under equilibrium conditions, doped with Phosphorus to a concentration 1017 cm-3. For T = 300K, indicate the values of (Ec – EF) and (EF – Ei) in the energy band diagram.Since ni = 1010 cm-3 and n = ni e(EF-Ei)/kT :EF – Ei = kT(ln107) = 7 ∙ kT(ln10) = 7 ∙ 60 meV = 0.42 eVThe intrinsic Fermi level is located slightly below midgap:Ec – Ei = Ec – [(Ec+Ev)/2 + (kT/2)∙ln(Nv/Nc)]= (Ec – Ev)/2 – (kT/2)∙ln(Nv/Nc) = 0.56 eV eV = eVHence Ec – EF = (Ec – Ei) – (EF – Ei) = – 0.42 = eV
10Energy band diagramFor T = 1200K, indicate the values of (Ec – EF) and (EF – Ei). Remember that Nc and Nv are temperature dependent. Also, EG is dependent on temperature: for silicon, EG = 2.8×10-4(T) for T > 300K.At T = 1200K, the Si band gap EG = 1.2 2.8×10-4 (T) = 0.87 eVThe conduction-band and valence-band effective densities of states Nc and Nv each have T3/2 dependence, so their product has T3 dependence. (The ratio Nv/Nc does not change with temperature, assuming that the carrier effective masses are independent of temperature.) Therefore, when the temperature is increased by a factor of 4 (from 300K to 1200K), NcNv is increased by a factor of 64. Intrinsic Semiconductor: EF = Ei