Exploring the density of states in isotropic parabolic bands
Continuing my journey on quantum mechanics in crystalline materials here, I’ve been fascinated by the relationship between energy and wave vector in the density of states for isotropic parabolic bands. This relationship is pivotal in comprehending the electronic properties of materials, particularly within the field of materials science.
Understanding the density of states in energy, as opposed to the more familiar density in k-space, requires a precise knowledge of the band structure and how energy E relates to the wave vector \mathbf{k}. In isotropic parabolic bands, by definition, the states of an energy E are situated on a spherical surface in the k-space, and the number of states between E and \mathrm{d}E corresponds to the number within a spherical shell.
The energy at the bottom of the band is described by:
E = \frac{\hbar^2 k^2}{2m_{\text{eff}}} + V
where \hbar is the reduced Planck’s constant, k is the wave vector, m_{\text{eff}} is the effective mass, and V is the potential energy at the band edge.
To compute k, I use the formula:
k = \sqrt{\frac{2m_{\text{eff}}}{\hbar^2}(E - V)}
The derivation of the density of states starts by considering a thin spherical shell in k-space, where the radius k and thickness \mathrm{dk} yield a volume \mathrm{dV}. The volume of this shell is approximated by the surface area of the sphere times the thickness of the shell:
\mathrm{dV} = 4\pi k^2 \mathrm{dk}
From this, I consider the expression for the volume of the shell in terms of energy:
\mathrm{dV} = 4\pi \left(\frac{2m_{\text{eff}}}{\hbar^2}\right)(E - V) \frac{1}{2} \sqrt{\frac{2m_{\text{eff}}}{\hbar^2}} \frac{1}{\sqrt{E - V}} \mathrm{dE}
Simplifying this, the formula for the volume in terms of energy becomes:
\mathrm{dV} = 2\pi \left(\frac{2m_{\text{eff}}}{\hbar^2}\right)^{3/2} \sqrt{E - V} \mathrm{dE}
With the volume known, I can now determine the density of states in energy, considering the factor of two for the two spins in each state:
g(E) = \frac{1}{2\pi^2}\left(\frac{2m_{\text{eff}}}{\hbar^2}\right)^{3/2} \sqrt{E - V}
This function of g(E) shows how the density of states scales with the square root of the energy above the potential V, indicative of the quantum mechanical nature of electron behavior in crystalline materials.
Understanding these relations provides deep insights into the physical properties of materials, enabling predictions about their electronic behavior under various conditions. This understanding is essential for advancing technologies in semiconductors and other materials critical to modern electronics and optoelectronics.
For more insights into this topic, you can find the details here.