Energy Bands Formation

Formation of Energy Bands

• Bloch functions are defined as $\psi_{n,k}(r) = e^{ik \cdot r}u_{n,k}(r)$, where $u_{n,k}(r)$ is periodic in the direct lattice.
• The periodic property of $u_{n,k}(r)$ is expressed as $u_{n,k}(r+T) = u_{n,k}(r)$.
• Density of states, $D(E)$, is the number of states in the energy range $E$ to $E + dE$ divided by $dE$.
• For a 1D crystal with $N$ unit cells of length $a$, the total length is $L = Na$.
• Applying Born-von Karman boundary conditions, $\psi(x+L) = \psi(x)$.
• From this, $e^{ikL} = 1$, leading to $kL = 2\pi n$, where $n$ is an integer.
• Allowed $k$ values are thus $k = \frac{2\pi}{L}n = \frac{2\pi}{Na}n$.
• The spacing between successive $k$ values is $\Delta k = \frac{2\pi}{L} = \frac{2\pi}{Na}$.
• In the 1st Brillouin Zone (BZ), the number of allowed $k$ values is $\frac{\text{width of the } 1^{st} BZ}{\Delta k} = \frac{\frac{4\pi}{a}}{\frac{2\pi}{Na}} = 2N$.
• Each $k$ value can accommodate 2 electrons due to spin; thus, the 1st band can accommodate $4N$ electrons.
• In 3D, Born-von Karman boundary conditions are: $\psi(r+L_x) = \psi(r)$, $\psi(r+L_y) = \psi(r)$, and $\psi(r+L_z) = \psi(r)$.
• The k-values are given by $k_x = \frac{2\pi}{L_x}n_x$, $k_y = \frac{2\pi}{L_y}n_y$, and $k_z = \frac{2\pi}{L_z}n_z$, where $n_x$, $n_y$, and $n_z$ are integers.
• The volume per allowed $k$ value in $k$ space is $\Delta k_x \Delta k_y \Delta k_z = \frac{(2\pi)^3}{L_x L_y L_z} = \frac{(2\pi)^3}{V}$.
• The number of allowed $k$ values inside a sphere of radius $k$ is $\frac{\text{volume of the sphere}}{\text{volume per } k \text{ value}} = \frac{\frac{4}{3}\pi k^3}{\frac{(2\pi)^3}{V}} = \frac{V k^3}{6\pi^2}$.

Metals, Semiconductors and Insulators

• Metals have a partially filled band.
• Semiconductors have a filled band and a band gap $E_g < 4$ eV.
• Insulators have a filled band and a band gap $E_g > 4$ eV.

Direct and Indirect Band Gaps

• A direct band gap occurs when the minimum of the conduction band and the maximum of the valence band are at the same $k$ value (e.g., GaAs).
• An indirect band gap occurs when the minimum of the conduction band and the maximum of the valence band are at different $k$ values (e.g., Si).

Electronic Effective Mass

• Defined as $m^* = \hbar^2 (\frac{d^2E}{dk^2})^{-1}$.
• $m^* > 0$ indicates electron-like behavior.
• $m^* < 0$ indicates hole-like behavior.

Intrinsic Carrier Concentration

• Given by $n_i = \sqrt{N_c N_v} e^{-E_g/2kT}$.
• $N_c = 2(\frac{m_e^* k_B T}{2\pi \hbar^2})^{3/2}$, where $N_c$ is the effective density of states in the conduction band.
• $N_v = 2(\frac{m_h^* k_B T}{2\pi \hbar^2})^{3/2}$, where $N_v$ is the effective density of states in the valence band.