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 $m^{\ast }$:

$$g_{3D}(E)=\frac{1}{2{\pi }^2}{\left(\frac{2m^{\ast }}{{\mathrm{\hslash }}^2}\right)}^{3\mathrm{/}2}\sqrt{E-E_c},$$

for electrons, with a 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:

$$N_c=2{\left(\frac{2\pi m_e^{\ast }{k}_{B}T}{h^2}\right)}^{3\mathrm{/}2},$$

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

$$n=N_c\exp [-(E_c-E_F)\mathrm{/}{k}_{B}T]$$

(in the non-degenerate limit).

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

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:

$$g_{2D}(E)=\frac{m^{\ast }}{\pi {\mathrm{\hslash }}^2{L}_z}\sum _n\mathrm{\Theta }(E-E_n)$$

per unit volume, where $E_n=(n\pi \mathrm{\hslash }{)}^2\mathrm{/}(2m^{\ast }{L}_z^2)$ are the confined subband energies and $\mathrm{\Theta }$ is the Heaviside step function. Each subband contributes a constant DOS independent of energy within that subband.

The step heights are:

$$g_{2D,n}=\frac{m^{\ast }}{\pi {\mathrm{\hslash }}^2{L}_z}\mathrm{.}$$

For GaAs ($m_e^{\ast }=0.067m_e$): $g_{2D,n}=2.8\times {10}^{13}$ cm⁻²·eV⁻¹ per subband per area, or $2.8\times {10}^{19}$ cm⁻³·eV⁻¹ if $L_z=10$ nm.

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

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

with singularities at each 1D subband edge.

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

$$g_{0D}(E)\propto \sum _n\delta (E-E_n)\mathrm{.}$$

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:

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:

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

where $m_r^{\ast }=m_e^{\ast }{m}_h^{\ast }\mathrm{/}(m_e^{\ast }+m_h^{\ast })$ is the reduced effective mass. The absorption coefficient is:

$$\alpha (h\nu )\propto g_J(h\nu )\cdot \mathrm{\mid }M{\mathrm{\mid }}^2,$$

with $\mathrm{\mid }M{\mathrm{\mid }}^2$ 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:

$$\mathrm{\Delta }{E}_{BM}=\frac{{\mathrm{\hslash }}^2(3{\pi }^{2}n{)}^{2\mathrm{/}3}}{2m_e^{\ast }}\mathrm{.}$$

For GaAs at $n=2\times {10}^{19}$ cm⁻³ this parabolic-band estimate gives $\mathrm{\Delta }{E}_{BM}\approx 0.4$ eV, the rise of the Fermi level above the band edge. A vertical transition at the Fermi wavevector also starts below the valence-band maximum, which multiplies the shift by $(1+m_e^{\ast }\mathrm{/}{m}_h^{\ast })$, about 13 % more for heavy holes; conduction-band nonparabolicity and bandgap renormalization make the measured blue-shift smaller.

DOS-based formulas for carrier statistics.

$$n={\int }_{E_c}^{\mathrm{\infty }}g(E)f(E)dE,$$

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

$$n=N_c{F}_{1\mathrm{/}2}(\eta ),\eta =(E_F-E_c)\mathrm{/}{k}_{B}T,$$

with $F_{1\mathrm{/}2}$ 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:

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.