Nano Electronics (EC-702 (C)) - Important Questions

Derive the electronic density of states for a two-dimensional electron gas in a quantum well and show that the 2D density of states per unit area per unit energy is constant. Present the final result in the form $$g_{2D}(E)=\left(\frac{m^{*}}{\pi\hbar^{2}}\right)$$ and clearly state all assumptions used in the derivation.

Obtain the quantized energy levels for electrons in an infinite potential quantum well of width $L$. Express the $n^{\text{th}}$ energy level $E_{n}$ in terms of $\hbar$, $m^{*}$ and $L$.

Derive expressions for the electronic density of states for (a) a one-dimensional quantum wire and (b) a zero-dimensional quantum dot. Discuss the energy dependence of $g_{1D}(E)$ and $g_{0D}(E)$ and their physical significance.

Explain how band structure of a semiconductor is modified at the nanoscale. Discuss the concepts of effective mass and bandgap widening or narrowing due to quantum confinement, and indicate how these changes affect carrier transport.

Derive the expression for conductance quantization in a ballistic one-dimensional channel and show that the conductance per transport mode is $$G_{0}=\left(\frac{2e^{2}}{h}\right)\,. $$ Explain the physical origin of the factor of 2 and state the assumptions underlying the Landauer approach.

Describe qualitatively how minibands form in a semiconductor superlattice. For a simple Kronig–Penney type one-dimensional superlattice model, state the condition that leads to allowed minibands and forbidden gaps and explain the role of barrier width and well width on the miniband width.

Discuss the effects of heavy doping on semiconductor band structure and carrier statistics. Include in your answer the concepts of bandgap narrowing, Fermi level shift, and degeneracy of carriers, and describe how these influence conductivity.

Explain the operating principle of a resonant tunneling diode formed using a double-barrier quantum well. Discuss the conditions for resonant tunneling, sketch the expected I–V characteristic showing negative differential resistance, and relate the peaks to quantized energy levels in the well.

For a single quantum well with two occupied subbands having energies $E_{1}$ and $E_{2}$, derive an expression for the total sheet carrier concentration $n_{s}$ at temperature $T$ using Fermi–Dirac statistics. Express your result in terms of the 2D density of states $g_{2D}$, subband energies and the Fermi energy $E_{F}$.