Density of States in 2D Free Electron Gas
The density of states, $N(E)$, represents the number of electronic states per unit energy interval. For a free electron gas in $d$ dimensions, the density of states is generally given by the proportionality:
$N(E) \propto E^{(d/2 - 1)}$
In the specific case of two dimensions ($d=2$), the formula becomes:
$N(E) \propto E^{(2/2 - 1)} = E^{(1 - 1)} = E^0$
This implies that $N(E)$ is a constant value, independent of the energy $E$. Therefore, the graph representing the density of states as a function of energy should be a horizontal line.
Graphical Representation Analysis
- Option 1: Shows $N(E)$ decreasing with energy ($N(E) \propto 1/E$ or similar).
- Option 2: Shows $N(E)$ increasing with energy, proportional to $\sqrt{E}$ ($N(E) \propto \sqrt{E}$), typical for 3D systems.
- Option 3: Shows $N(E)$ as a constant, independent of energy ($N(E) = \text{constant}$). This matches our derived relationship for 2D.
- Option 4: Shows $N(E)$ decreasing sharply with energy ($N(E) \propto 1/\sqrt{E}$ or similar).
The graph representing a constant density of states, independent of energy, corresponds to the correct relationship for a 2D free electron gas.
Important Questions from Free Electron Theory Fermi Energy Velocity
- Copper has an electron number density of $8.3 \times 10^{28} \text{ m}^{-3}$. Its Fermi energy in eV (rounded off to one decimal place) is _____
($\hbar = 1.06 \times 10^{-34} \text{ J.s}$, mass of electron $\text{m}_e = 9.10 \times 10^{-31} \text{ kg}$, charge of electron $= 1.60 \times 10^{-19} \text{ C}$) - Consider one mole of a monovalent metal at absolute zero temperature, obeying the free electron model. Its Fermi energy is $E_F$. The energy corresponding to the filling of $\frac{N_A}{2}$ electrons, where $N_A$ is the Avogadro number, is $2^n E_F$. The value of $n$ is
Crystal structures of two metals A and B are two-dimensional square lattices with same lattice constant $a$. Electrons in metals behave as free electrons. The Fermi surfaces corresponding to A and B are shown by solid circles in figures.
The electron concentrations in A and B are $n_A$ and $n_B$, respectively. The value of $(\frac{n_B}{n_A})$ isIf $X$ is the dimensionality of a free electron gas, the energy ($E$) dependence of density of states is given by $E^{½X-Y}$, where $Y$ is ________.
- Potassium metal has electron concentration of $1.4 \times 10^{28}m^{-3}$ and the corresponding density of states at Fermi level is $6.2 \times 10^{46}$ Joule$^{-1} m^{-3}$. If the Pauli paramagnetic susceptibility of Potassium is $n \times 10^{-k}$ in standard scientific form, then the value of $k$ (an integer) is __________ (Magnetic moment of electron is $9.3 \times 10^{-24}$ Joule T$^{-1}$; permeability of free space is $4\pi \times 10^{-7}$ T m A$^{-1}$)
