The tight binding energy dispersion ($E-k$) relation for electrons in a one-dimensional array of atoms having lattice constant $a$ and total length $L$ is $$E = E_0 - \beta - 2\gamma \cos(ka),$$ where $E_0$, $\beta$ and $\gamma$ are constants and $k$ is the wave-vector.
The effective mass ($m^*$) of an electron in a crystal lattice is determined by the curvature of the energy band dispersion relation, $E(k)$. The formula relating effective mass to the second derivative of energy with respect to the wave-vector ($k$) is:
$ \frac{1}{m^*} = \frac{1}{\hbar^2} \frac{d^2E}{dk^2} $This can be rearranged to solve for the effective mass:
$ m^* = \frac{\hbar^2}{\frac{d^2E}{dk^2}} $Deriving Second Energy Derivative
Given the energy dispersion relation:
$ E(k) = E_0 - \beta - 2\gamma \cos(ka) $First, find the first derivative of $E$ with respect to $k$:
$ \frac{dE}{dk} = \frac{d}{dk} (E_0 - \beta - 2\gamma \cos(ka)) $ $ \frac{dE}{dk} = 0 - 0 - 2\gamma (-\sin(ka) \cdot a) $ $ \frac{dE}{dk} = 2\gamma a \sin(ka) $Next, find the second derivative of $E$ with respect to $k$:
$ \frac{d^2E}{dk^2} = \frac{d}{dk} (2\gamma a \sin(ka)) $ $ \frac{d^2E}{dk^2} = 2\gamma a (\cos(ka) \cdot a) $ $ \frac{d^2E}{dk^2} = 2\gamma a^2 \cos(ka) $Calculating Effective Mass
Substitute the second derivative back into the effective mass formula:
$ m^* = \frac{\hbar^2}{2\gamma a^2 \cos(ka)} $Therefore, the effective mass of the electrons in the band is $\frac{\hbar^2}{2\gamma a^2 \cos(ka)}$.
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
