Density of states (DOS)

The density of states (DOS) is the number of quantum states per unit energy per unit volume available to electrons (or holes) in a given energy band. It is one of the most fundamental quantities in semiconductor physics, governing carrier concentration, gain spectrum, absorption shape, and the temperature dependence of nearly every device parameter.

Definition. Per unit volume per unit energy:

$$g(E) \;=\; \frac{dN}{dE \cdot V},$$

where $N$ is the total number of states in the band up to energy $E$, and $V$ is the volume.

Bulk (3D) semiconductor DOS. For parabolic bands with effective mass $m^*$:

$$g_{3D}(E) \;=\; \frac{1}{2\pi^2} \left( \frac{2 m^*}{\hbar^2} \right)^{3/2} \sqrt{E - E_c} \quad \text{(electrons)},$$

with similar form for holes. The DOS rises as $\sqrt{E}$ above the band edge.

Effective DOS. Integrating over the Maxwell-Boltzmann tail gives the effective DOS at the band edge:

$$N_c \;=\; 2 \left( \frac{2 \pi m_e^* k_B T}{h^2} \right)^{3/2},$$

with similar formula for $N_v$ (valence band). These quantities appear in the carrier concentration formulas:

$$n \;=\; N_c \exp[-(E_c - E_F)/k_B T]$$

(in the non-degenerate limit).

Standard $N_c$ and $N_v$ values at 300 K:

Material	$N_c$ (cm⁻³)	$N_v$ (cm⁻³)
Si	$2.8 \times 10^{19}$	$1.04 \times 10^{19}$
Ge	$1.0 \times 10^{19}$	$5.0 \times 10^{18}$
GaAs	$4.7 \times 10^{17}$	$7.0 \times 10^{18}$
InP	$5.7 \times 10^{17}$	$1.1 \times 10^{19}$
InGaAs (LM to InP)	$2.1 \times 10^{17}$	$7.7 \times 10^{18}$
GaN	$2.6 \times 10^{18}$	$4.6 \times 10^{19}$

The much smaller $N_c$ in direct-gap III-Vs reflects their light electron effective mass — relevant for high-mobility electron transport and modest degeneracy thresholds.

2D DOS (quantum well). For a quantum well of thickness $L_z$, the DOS becomes a staircase:

$$g_{2D}(E) \;=\; \frac{m^*}{\pi \hbar^2 L_z} \sum_n \Theta(E - E_n) \quad \text{per unit volume},$$

where $E_n = (n \pi \hbar)^2 / (2 m^* L_z^2)$ are the confined subband energies and $\Theta$ is the Heaviside step function. Each subband contributes a constant DOS independent of energy within that subband.

The step heights are:

$$g_{2D,n} \;=\; \frac{m^*}{\pi \hbar^2 L_z}.$$

For GaAs ($m_e^* = 0.067 m_e$): $g_{2D,n} = 2.8 \times 10^{13}$ cm⁻²·eV⁻¹ per subband per area, or $2.8 \times 10^{20}$ cm⁻³·eV⁻¹ if $L_z = 10$ nm.

1D DOS (quantum wire). For a 1D confined structure:

$$g_{1D}(E) \;\propto\; (E - E_n)^{-1/2},$$

with singularities at each 1D subband edge.

0D DOS (quantum dot). Discrete delta-function levels:

$$g_{0D}(E) \;\propto\; \sum_n \delta(E - E_n).$$

A perfect QD has discrete atomic-like states; in practice, inhomogeneous broadening from QD size distribution gives broadened (but still very narrow) DOS.

Why dimensionality matters for lasers. The "joint" DOS (electron and hole) for optical transitions:

Dimensionality	$g_J(E)$ near band edge	Implications
3D bulk	$\propto (E - E_g)^{1/2}$	Soft gain peak, wide spectrum
2D QW	Step function	Sharper peak, better thermal stability
1D wire	$\propto (E - E_{n})^{-1/2}$	Even sharper, predicted lower threshold
0D QD	Delta functions	Narrowest peak, lowest theoretical threshold

The progression toward lower dimensionality was a major theme in semiconductor laser research from the 1970s through the 2000s. QW lasers replaced bulk lasers in the 1980s; QD lasers (using self-assembled InAs/GaAs QDs) achieve very narrow gain peaks and are commercialized in some applications.

Joint density of states. For optical absorption (or stimulated emission), the relevant quantity combines conduction and valence band DOS at conserved momentum:

$$g_J(h\nu) \;=\; \frac{1}{2\pi^2}\left(\frac{2 m_r^*}{\hbar^2}\right)^{3/2} \sqrt{h\nu - E_g},$$

where $m_r^* = m_e^* m_h^* / (m_e^* + m_h^*)$ is the reduced effective mass. The absorption coefficient is:

$$\alpha(h\nu) \;\propto\; g_J(h\nu) \cdot |M|^2,$$

with $|M|^2$ the optical matrix element.

DOS in heavily doped semiconductors. At high doping ($N_D > N_c$), the Fermi level enters the conduction band, the semiconductor becomes degenerate, and band-filling becomes important. The Burstein-Moss shift moves the apparent optical band edge to higher energy:

$$\Delta E_{BM} \;=\; \frac{\hbar^2 (3\pi^2 n)^{2/3}}{2 m^*}.$$

For GaAs at $n = 2 \times 10^{19}$ cm⁻³: $\Delta E_{BM} \approx 100$ meV — visible blue-shift in absorption.

DOS-based formulas for carrier statistics.

$$n \;=\; \int_{E_c}^\infty g(E) f(E) \, dE,$$

where $f(E)$ is the Fermi-Dirac distribution. For non-degenerate semiconductors, the Maxwell-Boltzmann approximation gives the simple $n = N_c \exp[-(E_c - E_F)/k_B T]$ form. For degenerate semiconductors (laser active regions, modulation-doped 2DEGs), the full Fermi-Dirac integral is needed:

$$n \;=\; N_c F_{1/2}(\eta), \quad \eta = (E_F - E_c)/k_B T,$$

with $F_{1/2}$ the Fermi-Dirac integral of order 1/2.

DOS in van Hove singularities. Critical points in the band structure (saddle points, band extrema) produce singularities in the DOS:

Critical point	DOS behavior
3D minimum or maximum	$\sqrt{E - E_c}$ or $\sqrt{E_v - E}$
3D saddle point	Kink with logarithmic divergence in derivative
2D minimum or maximum	Step
2D saddle point	Logarithmic divergence
1D minimum or maximum	$1/\sqrt{E - E_c}$

These van Hove singularities are visible in measured photoluminescence and absorption spectra and provide direct evidence for crystallographic band structure.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 16; Coldren, Corzine & Mašanović, Diode Lasers and PICs (2nd ed., 2012), Ch. 4 for laser-relevant DOS treatment; Yu & Cardona, Fundamentals of Semiconductors (4th ed., 2010), Ch. 6 — canonical band-structure and DOS analysis; Sze & Ng, Physics of Semiconductor Devices (3rd ed., 2007), Ch. 1.