Effective Mass Calculation in 1D Lattice
The effective mass ($m^*$) describes how an electron responds to forces within a crystal lattice. It is defined using the curvature of the energy band ($E(k)$) as follows:
$m^* = \frac{\hbar^2}{d^2E/dk^2}$
Here, $\hbar$ is the reduced Planck constant, and $E(k)$ is the energy dispersion relation.
Deriving the Second Energy Derivative
The given energy dispersion is:
$E(k) = E_0 - \frac{1}{2} W \cos ka$
First, calculate the first derivative of $E(k)$ with respect to the wavevector $k$:
$\frac{dE}{dk} = \frac{d}{dk} \left( E_0 - \frac{1}{2} W \cos ka \right) = \frac{1}{2} Wa \sin ka$
Next, calculate the second derivative:
$\frac{d^2E}{dk^2} = \frac{d}{dk} \left( \frac{1}{2} Wa \sin ka \right) = \frac{1}{2} Wa^2 \cos ka$
Finding Effective Mass at the Band Bottom
The bottom of the energy band corresponds to the minimum energy value. For the given dispersion relation, this occurs when $\cos ka$ is maximum, which is 1 (e.g., at $k=0$).
Evaluate the second derivative at the bottom of the band ($\cos ka = 1$):
$\frac{d^2E}{dk^2}\bigg|_{\text{band bottom}} = \frac{1}{2} Wa^2 (1) = \frac{1}{2} Wa^2$
Substitute this curvature into the effective mass formula:
$m^* = \frac{\hbar^2}{\frac{1}{2} Wa^2} = \frac{2\hbar^2}{Wa^2}$
This derived effective mass matches the first option.
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?- Consider a one-dimensional non-magnetic crystal with one atom per unit cell. Assume that the valence electrons (i) do not interact with each other and (ii) interact weakly with the ions. If $n$ is the number of valence electrons per unit cell, then at 0 K,
