Question

A p-n junction is created by doping the right side of a piece of silicon with 10¹⁴ atoms/cm³ of phosphorus and the left side with 10¹⁸ atoms/cm³ of boron. Assume that the dopants are fully ionized, and assume the junction is at x = 0 with x+ pointed to the right.

a) Plot by hand (roughly to scale) an energy band diagram of the junction and label E_GAP, E_C, E_V, E_F and E_Fi. Using the effective density of states, calculate and label E_F-E_V on the p-side and E_C-E_F on the n-side of the junction. From this plot, calculate and label the built-in voltage V_bi.

b) Calculate V_bi using the intrinsic carrier concentration - does it match the value you calculated in (a)?

c) Plot by hand, as a function of position, the total charge concentration and the concentration of electrons in the conduction band (separate plots, one below the other). In your plots, calculate and label the values of concentration in the p- and n-type regions, and at x = 0 if applicable.

d) Calculate x_n and x_p, the widths of the depletion regions on the n- and p-sides, respectively, and the total depletion width W.

e) Plot the electric field as a function of position, labeling the location and value of maximum electric field and the x-positions of the depletion regions.

f) Plot the voltage as a function of position, labeling the location and value of maximum voltage and the x-positions of the depletion regions

Answer 1

Given that Right side phosphorous (n = type) N_0 = 10^14 cm^-3 Left side N_A = 10^8 cm^-3 V_bi = V_T ln (N_A N_D/n_i^2), n_i = 1.5 times 10^10 = .026 ln (10^18 times 10^14/2.25 times 10^20) = 0.697 volts w = squareroot 2 elementof V_bi/q (1/N_A + 1/N_D) = squareroot 2 times 11.9 times 8.854 times 10^-14/1.6 times 10^-19 times 0.697 (1/10^18 + 1/10^14) total width implies w = 3.03 times 10^-4 cms x_n = w times N_A/N_A + N_D = 3.029 times 10^-4 cms x_p = 3.29 times 10^-8 cms electric field, E_max = -q N_D x_n/elementof = -1.6 times 10^-19 times 10^14 times 3.029 times 10^-4/11.9 times 8.85 times 10^-4 E_max = 4599.7 v/cm E = 1/elementof integral rho v dx