Consider a one-dimensional lattice with a weak periodic potential $U(x) = U_0 \cos( \frac{2\pi x}{a})$. The gap at the edge of the Brillouin zone $(k = \frac{\pi}{a})$ is:
Analyzing the Energy Gap
The energy gap in a crystal lattice arises due to the periodic potential. For a one-dimensional lattice with potential $U(x)$, the energy gap at the Brillouin zone boundary $k = \frac{\pi}{a}$ is related to the Fourier components of the potential.
The given potential is $U(x) = U_0 \cos\left( \frac{2\pi x}{a} \right)$.
Calculating the Gap Size
The size of the energy gap ($\Delta E$) at the edge of the Brillouin zone ($k = \frac{\pi}{a}$) is given by twice the magnitude of the corresponding Fourier coefficient of the potential.
- First, express the potential $U(x)$ using Euler's formula: $ \cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2} $ So, $ U(x) = U_0 \left( \frac{e^{i\frac{2\pi x}{a}} + e^{-i\frac{2\pi x}{a}}}{2} \right) = \frac{U_0}{2} e^{i\frac{2\pi x}{a}} + \frac{U_0}{2} e^{-i\frac{2\pi x}{a}} $
- This potential has Fourier components $V_G = \frac{U_0}{2}$ corresponding to the reciprocal lattice vector $G = \frac{2\pi}{a}$.
- The energy gap occurs at $k = \pm \frac{G}{2}$. In this case, $k = \pm \frac{1}{2} \left( \frac{2\pi}{a} \right) = \pm \frac{\pi}{a}$, which matches the edge of the Brillouin zone specified in the question.
- The magnitude of the energy gap at this boundary is given by the formula: $ \Delta E = 2 |V_G| $
- Substituting the value of $V_G$: $ \Delta E = 2 \left| \frac{U_0}{2} \right| = U_0 $
Therefore, the gap at the edge of the Brillouin zone is $U_0$.
Important Questions from Band Theory Effective Mass Holes
- For the energy dispersion of an electron in a one-dimensional solid $E(k) = E_0 - 2\gamma \cos(ka)$, the ratio of the effective mass of the electron in the solid to the free electron mass ($m_e$) at $k = 0$ is $R_0$. Taking $\gamma = 0.5 \text{ eV}$ and $a = 0.5 \text{ nm}$, the value of $R_0$ (rounded off to two decimal place) is _____
($\hbar = 1.054 \times 10^{-34} \text{ J.s}$, $m_e = 9.1 \times 10^{-31} \text{ kg}$, electron charge $= 1.6 \times 10^{-19} \text{ C}$) The dispersion ($E(k)$) of the conduction band (CB) and valence band (VB) for a semiconductor are shown schematically in the figure. Considering the possibility of an electron making a transition from the bottom of the CB to the top of the VB, which of the following options is/are correct?
For nonrelativistic electrons in a solid, different energy dispersion relations (with effective masses $m_a^*$, $m_b^*$, and $m_c^*$) are schematically shown in the plots. Which one of the following options is CORRECT?
The temperature dependence of the electrical conductivity ($\sigma$) of three intrinsic semiconductors A, B and C is shown in figure.
Let $E_A$, $E_B$ and $E_C$ be the bandgaps of A, B and C, respectively. Which one of the following relations is correct?- The energy dispersion for electrons in one dimensional lattice with lattice parameter $a$ is given by $E(k) = E_0 - \frac{1}{2} W \cos ka$, where $W$ and $E_0$ are constants. The effective mass of the electron near the bottom of the band is
