Density of States at the Fermi Energy

In summary, the density of states at the Fermi energy, D(E_F), is given by (3/2)n/E_F. This factor of (3/2) accounts for the volume by considering the total number of available states up to the Fermi surface, N(k), and using the parabolic dispersion relation to find the density of states, D(E).
  • #1
nboogerz
4
0
The density of states at the fermi energy is given by

D(E_F)=(3/2)n/E_F

I understand the density of states is the number of states per energy per unity volume, accounting for n/E_F. I don't understand how the 3/2 multiplying factor accounts for the volume?
 
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  • #2
Dimensionally you are correct. But in this case, unfortunately, you have to perform the detailed calculus steps in order to get that factor. First let us determine the expression for ##n##. In ##\bf k##-space you need to count the total number of occupied states. This can be computed as seen in the steps below [tex] \begin{eqnarray} n&=&2\int_{{\rm FS}}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}} \\
&=& \frac{2}{(2\pi)^{3}}\int_{0}^{k_{F}}dk\int_{0}^{2 \pi}d\phi\int_{0}^{\pi}d\theta\left(k^{2}
\sin(\theta)\right) \\
&=& \frac{2}{(2\pi)^{3}}\left(\int_{0}^{k_{F}}dk\, k^{2}\right)
\left(\int_{0}^{2\pi}d\phi\right)
\left(\int_{0}^{\pi}d\theta\,\sin(\theta)\right) \\
&=& \frac{2}{2\pi^{2}}\int_{0}^{k_{F}}dk\, k^{2} \\
&=& \frac{k_{F}^{3}}{3\pi^{2}}
\end{eqnarray} [/tex] where ##\int_{{\rm FS}}## is an integral from the origin till the (spherical) Fermi Surface (FS). The ##k^{2}
\sin(\theta)## in the second step is simply the Jacobian in spherical coordinates. Now, ##n## is the total number of available (and filled) states for ##k\le k_{F}##. The total number of states available up to some arbitrary ##k## is simply [tex] N(k)=\frac{k^{3}}{3\pi^{2}} [/tex] The density of states (for the isotropic case) is given by [tex] \begin{eqnarray} D(E) &=& \frac{dN(E)}{dE}\\
&=& \frac{dN(k)}{dk}\left(\frac{dE}{dk}\right)^{-1} \end{eqnarray} [/tex] For a parabolic dispersion we have [tex] E=\frac{\hbar^{2}k^{2}}{2m^{*}} [/tex] Therefore, at ##k=k_F## we have [tex] \begin{eqnarray} D(E_{F}) &=& D(E(k_{F}))\\
&=& \frac{m^{*}k_{F}}{\hbar^{2}\pi^{2}}\\
&=& \frac{k_{F}^{3}}{\pi^{2}}\left(\frac{\hbar^{2}k_{F}^{2}}{m^{*}}\right)^{-1}\\
&=& \frac{3}{2}\left(\frac{k_{F}^{3}}{3\pi^{2}}\right)
\left(\frac{\hbar^{2}k_{F}^{2}}{2m^{*}}\right)^{-1} \end{eqnarray} [/tex] From the above expressions you can make the appropriate substitutions [tex] D(E_{F}) = \frac{3}{2}nE_{F}^{-1} [/tex]
 
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1. What is the density of states at the Fermi energy?

The density of states at the Fermi energy is a measure of the number of available energy states per unit volume at the Fermi energy level in a given material. It is an important quantity in condensed matter physics and is typically denoted by the symbol D(EF).

2. How is the density of states at the Fermi energy calculated?

The density of states at the Fermi energy can be calculated using the following formula: D(EF) = (1/2π2)(2m/h2)3/2(EF)1/2, where m is the effective mass of the electrons, h is Planck's constant, and EF is the Fermi energy level.

3. Why is the density of states at the Fermi energy important?

The density of states at the Fermi energy is important because it determines the number of available energy states for electrons at the Fermi energy level. This affects various properties of a material, such as its electrical conductivity and thermal conductivity.

4. How does the density of states at the Fermi energy vary with temperature?

The density of states at the Fermi energy is not affected by temperature in a non-degenerate material (where the number of available energy levels is much greater than the number of electrons). However, in a degenerate material (where the number of available energy levels is similar to the number of electrons), the density of states at the Fermi energy decreases with increasing temperature.

5. Can the density of states at the Fermi energy be directly measured?

No, the density of states at the Fermi energy cannot be directly measured. It is a theoretical quantity that is calculated based on the material's band structure and electron density. However, it can be indirectly measured through experiments such as specific heat and electrical conductivity measurements.

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