Semiconductors, Junction Devices & Quantum Dots

1. Direct vs. Indirect Band Gap Semiconductors

In crystalline semiconductors, the electrical and optical properties are governed by the band gap $E_g$ separating the highest occupied Valence Band Maximum (VBM) from the lowest unoccupied Conduction Band Minimum (CBM):

• Direct Band Gap Semiconductors (e.g., $\text{GaAs}, \text{InP}, \text{GaN}$): Both the conduction band minimum and valence band maximum occur at the same crystal wavevector (typically the zone center $\Gamma$-point, $\vec{k} = 0$). Optical photon transitions ($\Delta \vec{k} \approx 0$) occur directly via vertical electric dipole absorption and rapid radiative electron-hole recombination ($h\nu = E_g$). Essential for efficient optoelectronics: LEDs, laser diodes.
• Indirect Band Gap Semiconductors (e.g., $\text{Si}, \text{Ge}, \text{GaP}$): The conduction band minimum and valence band maximum occur at different wavevectors (e.g., in Silicon, the VBM is at $\Gamma$ while the CBM is near $0.85$ of the zone edge along $[100]$). Conservation of crystal momentum forbids first-order radiative transitions: photon absorption or emission requires simultaneous absorption or emission of a lattice phonon ($\hbar\vec{q}$):
  $$h\nu = E_g \pm \hbar\omega_{\text{phonon}}, \quad \vec{k}_{\text{initial}} + \vec{k}_{\text{photon}} \approx \vec{k}_{\text{final}} \pm \vec{q}$$

  Because second-order phonon-assisted transitions are far less probable, indirect semiconductors exhibit long carrier radiative lifetimes, making them ideal for photovoltaic solar cells and CMOS microelectronics, but inefficient as light emitters.

2. Kane Non-Parabolicity & Anisotropic Effective Masses

Near band extrema, expanding $E(\vec{k})$ to quadratic order yields anisotropic effective mass tensors:

where $m_l^*$ is the longitudinal mass and $m_t^*$ is the transverse mass (e.g., in Silicon, $m_l^* = 0.98 m_0$, $m_t^* = 0.19 m_0$). The density-of-states effective mass is $m_d^* = (m_l^* m_t^{*2})^{1/3}$, while the conductivity effective mass is $m_c^* = 3\left(\frac{1}{m_l^*} + \frac{2}{m_t^*}\right)^{-1}$.

1. Intrinsic Conduction Electron Density $n$ and Hole Density $p$

In a non-degenerate semiconductor ($E_c - E_F \gg k_BT$ and $E_F - E_v \gg k_BT$), the Fermi-Dirac distribution is accurately approximated by the classical Maxwell-Boltzmann tail:

Integrating over the parabolic conduction band density of states yields the thermal equilibrium electron concentration:

where $N_c$ is the effective density of states in the conduction band:

Similarly, the hole concentration in the valence band is:

where $N_v = 2 \left( \frac{2\pi m_h^* k_B T}{h^2} \right)^{3/2}$ is the effective density of states in the valence band.

2. The Law of Mass Action & Intrinsic Carrier Concentration $n_i$

Multiplying $n$ and $p$ eliminates the Fermi energy $E_F$ entirely, establishing the fundamental Law of Mass Action:

where $n_i$ is the intrinsic carrier concentration:

This product $n \cdot p = n_i^2$ holds strictly for any semiconductor in thermal equilibrium, regardless of whether it is intrinsic or heavily doped.

3. The Intrinsic Fermi Level $E_i$

In an undoped intrinsic semiconductor, charge neutrality requires $n = p = n_i$. Equating $n$ and $p$ reveals the position of the intrinsic Fermi level:

Because $\ln(m_h^*/m_e^*) \sim 0$, the intrinsic Fermi level lies virtually at the exact mid-gap ($E_g/2$).

1. Shallow Hydrogenic Donor and Acceptor Impurities

Extrinsic semiconductors are created by intentionally introducing trace dopant atoms into the host lattice:

• $n$-Type Doping (Group V in Group IV, e.g., P, As in Si): Donates an extra valence electron. The extra electron orbits the positively charged donor ion core ($+e$) in a hydrogen-like orbit screened by the large dielectric permittivity of the crystal ($\varepsilon_r \sim 12$):
  $$E_d = \frac{m_e^*}{m_0} \frac{1}{\varepsilon_r^2} E_H = \frac{m_e^*}{m_0} \frac{13.6\text{ eV}}{\varepsilon_r^2} \sim 10 - 50\text{ meV}$$

  The donor Bohr radius is expanded to $r_d = \varepsilon_r \frac{m_0}{m_e^*} a_0 \sim 30 - 80\text{ \AA}$. At room temperature ($k_BT \approx 26\text{ meV}$), virtually all donors are thermally ionized, so $n \approx N_D$.

• $p$-Type Doping (Group III in Group IV, e.g., B, Ga in Si): Lacks one valence electron, creating a bound acceptor state near the valence band edge that accepts an electron, generating mobile positive holes ($p \approx N_A$).

2. Drift and Diffusion Transport

In the presence of an electric field $\vec{E}$ and carrier concentration gradients $\nabla n$ and $\nabla p$, total electron and hole current densities comprise both drift and diffusion components:

where $\mu_n, \mu_p$ are carrier mobilities and $D_n, D_p$ are diffusion coefficients. In thermal equilibrium ($\vec{J} = 0$), the drift and diffusion currents balance identically, deriving the universal Einstein Relation:

The total electrical conductivity is given by $\sigma = e (n \mu_n + p \mu_p)$.

1. Electrostatics of the Equilibrium p-n Junction

When $p$-type and $n$-type semiconductor regions are brought into intimate metallurgical contact, the steep concentration gradient drives diffusion of majority electrons from $n$ to $p$ and holes from $p$ to $n$. This recombination uncovers fixed, ionized donor cores ($N_D^+$) on the $n$-side and ionized acceptor cores ($N_A^-$) on the $p$-side, creating a space-charge depletion region of width $W = x_p + x_n$.

The resulting internal electric field opposes further diffusion until equilibrium is established with a flat Fermi level ($dE_F/dx = 0$). The total electrostatic potential barrier across the junction is the built-in potential $V_{bi}$:

Solving Poisson's equation $\frac{d^2 V}{dx^2} = - \frac{\rho(x)}{\varepsilon_s}$ under the depletion approximation yields the total depletion layer width under applied voltage $V_a$:

2. The Shockley Ideal Diode Equation

Applying a forward bias voltage ($V_a > 0$) lowers the potential barrier to $V_{bi} - V_a$, exponentially increasing minority carrier injection across the depletion region edges. Integrating minority carrier diffusion currents yields the celebrated Shockley Diode Equation:

where $I_0 = e A \left( \frac{D_n n_{p0}}{L_n} + \frac{D_p p_{n0}}{L_p} \right)$ is the reverse saturation current.

3. Optoelectronic Devices

• Light-Emitting Diodes (LEDs): Under forward bias, electrons and holes are injected across the junction and recombine radiatively, emitting photons with peak energy $h\nu \approx E_g$ ($\lambda \approx hc/E_g$).
• Photovoltaic Solar Cells: Photons with $h\nu > E_g$ generate electron-hole pairs within and near the depletion region. The built-in electric field sweeps electrons to the $n$-side and holes to the $p$-side, producing a photocurrent $I_{sc}$ and open-circuit photovoltage $V_{oc}$.

1. Quantum Confinement & Dimensionality Hierarchy

When the spatial dimensions of a semiconductor crystal are reduced below the exciton Bohr radius $a_B = \varepsilon_r \frac{m_0}{\mu} a_0 \sim 2 - 10\text{ nm}$, continuous electronic energy bands quantize into discrete sub-bands due to quantum confinement:

1. 3D (Bulk): Confinement in $0$ dimensions; $g(E) \propto \sqrt{E}$.
2. 2D (Quantum Well): Confinement along $1$ dimension ($L_z \sim \text{nm}$), free in $x, y$. Energy spectrum: $E(k_x, k_y, n_z) = E_{n_z} + \frac{\hbar^2(k_x^2 + k_y^2)}{2m^*}$. Density of states consists of discrete staircase steps: $g_{\text{2D}}(E) = \frac{m^*}{\pi\hbar^2} \sum_{n} \Theta(E - E_n)$.
3. 1D (Quantum Wire): Confinement along $2$ dimensions ($L_x, L_y \sim \text{nm}$), free along $z$. Energy spectrum: $E(k_z, n_x, n_y) = E_{n_x, n_y} + \frac{\hbar^2 k_z^2}{2m^*}$. Density of states exhibits sharp $E^{-1/2}$ Van Hove singularities: $g_{\text{1D}}(E) \propto \sum_n (E - E_n)^{-1/2}$.
4. 0D (Quantum Dot): Confinement along all $3$ spatial dimensions ($L_x, L_y, L_z \sim \text{nm}$). Completely discrete "artificial atom" spectrum: $E_{n_x, n_y, n_z} = \frac{\pi^2\hbar^2}{2m^*} \left(\frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2}\right)$. Density of states is a collection of Dirac delta-functions: $g_{\text{0D}}(E) = 2 \sum_i \delta(E - E_i)$.

2. Size-Dependent Bandgap Tuning in Quantum Dots (Brus Formula)

For a spherical semiconductor nanocrystal (quantum dot) of radius $R$, Louis Brus (1984) derived the quantum confinement energy shift using the effective mass approximation:

where $\frac{1}{\mu} = \frac{1}{m_e^*} + \frac{1}{m_h^*}$ is the exciton reduced mass.

• The first correction term ($+ \hbar^2\pi^2 / 2\mu R^2 \propto 1/R^2$) represents kinetic quantum confinement energy, pushing the effective band gap upward.
• The second correction term ($- 1.786 e^2 / 4\pi\varepsilon_0 \varepsilon_r R \propto -1/R$) represents screened Coulomb attraction between electron and hole.

By simply tuning the nanocrystal radius $R$ from $1.5\text{ nm}$ to $6.0\text{ nm}$, the emission color of cadmium selenide ($\text{CdSe}$) quantum dots can be tuned continuously across the entire visible spectrum from vibrant blue to deep red.