Assignment -

This Assignment is on the fundamental of semiconductors (advanced level). Topics are fermi level, trap densities, carrier life time .. etc.

E_g = 1 eV and is temperature independent.

N_c = N_v = 1 x 10¹⁸ cm^-3 at 300K. Nc and Nv have the temperature dependence as regular semiconductors.

m_e^* = 0.2m_o, m_h^* = 0.6m_o, m_o = 9.1 x 10^-31Kg is the free electron mass.

Problem 1 - For the above semiconductor, we introduce acceptors N_a = 6 x 10¹⁷ cm^-3. E_a is 0.1eV above the valence band.

(a) Plot the Fermi level as a function of temperature from 0K to 1000K.

(b) We will introduce donors to keep the Fermi level in the middle of band gap, plot the required donor concentration (in cm^-3) versus donor ionization energy E_d (in eV) at 50K, 300K, and 500K. The donor ionization energy E_d is defined as the energy difference between the donor state and the bottom of the conduction band. The plot should be in semi-log (i.e. log-linear) scale.

Problem 2 - Assume the hypothetical semiconductor is homogeneously doped to n-type with n_o = 1 x 10¹⁷cm^-3 being the electron equilibrium concentration at 300K.

(a) Assume the semiconductor contains a trap concentration N_t = 3 x 10¹⁷cm^-3 and assume the trap is a "donor trap with E_d = 0.4eV" meaning that the trap state is either charge neutral (when capturing an electron or emitting a hole) or charge positive (when capturing a hole or emitting an electron). C_pe = 1 x 10^-10 cm³/s, C_ph = 4 x 10^-10 cm³/s.

Plot the lifetime for the excess carrier as a function of the excess carrier concentration from 5 x 10¹² cm^-3 to 5 x 10¹⁷ cm^-3. Choose the appropriate scale (linear-linear, linear-log, log-linear, or log-log) for the plot to be most informative.

(b) Plot the probability for the donor traps to be occupied versus the excess carrier concentration from 5 x 10¹² cm^-3 to 5 x 10¹⁷cm^-3.

(c) Assume excess holes at a concentration of 1 x 10¹⁶ cm^-3 are injected from the left side of the semiconductor. Plot the hole current density (in Amp/cm²) over the distance (in μm). Assume the electron diffusivity D_e = 20 cm²/s and hole diffusivity D_h = 4 cm²/s.

Problem 3 - Assume an n-type semiconductor of length "L" is non-uniformly doped. L = 1 μm. Because it is an n-type material, you can assume that all the current is due to electron current (the hole current can be neglected). The electron current can be written as

J_n = eD_n dn/dx+ eμ_nnE

The Poisson equation according to the Gauss law is

d²φ/dx² = -e/∈ (N_d⁺(x) - n(x)) since we ignore the hole concentration.

Under zero bias (V = 0), J_n = 0 and the system is at equilibrium.

(a) Calculate the built-in potential at 300K assuming the electron concentration at the two boundaries of the semiconductor is n(x = 0) = 3 x 10¹⁵ cm^-3, n(x = L) = 1 x 10¹⁶ cm^-3,

(b) Derive a single differential equation for n(x). Note that the differential equation should contain n(x) only without other variables such as E or φ. The equation will be a nonlinear differential equation. (Hint: Use Einstein equation D/µ = kT/e ≡ V_T).

(c) Assuming that the solution for (b) is n(x) = 3 x 10¹⁵ + 7 x 10¹⁵ (x/L) cm^-3, find and plot the profile of the ionized donors N_d⁺(x).

(d) Calculate and plot the profile for the charge density ρ(x), E-field E(x), and potential φ(x).