Semiconductor Mobility Ratio Calculation
The mobility ($\mu$) of charge carriers (holes or electrons) in a semiconductor is determined by their average relaxation time ($\tau$) and effective mass ($m^*$). The relationship is given by the formula:
$ \mu = \frac{q\tau}{m^*} $
where $q$ is the elementary charge.
Given Ratios
- Ratio of effective mass of hole to electron:
$ \frac{m_h^*}{m_e^*} = \frac{2}{11} $ - Ratio of average relaxation time for hole to electron:
$ \frac{\tau_h}{\tau_e} = \frac{1}{2} $
Deriving Mobility Ratio
We need to find the ratio of the mobility of the hole ($\mu_h$) to the mobility of the electron ($\mu_e$):
$ \frac{\mu_h}{\mu_e} = \frac{\frac{q\tau_h}{m_h^*}}{\frac{q\tau_e}{m_e^*}} $
Since the elementary charge $q$ is constant, it cancels out:
$ \frac{\mu_h}{\mu_e} = \frac{\tau_h}{\tau_e} \times \frac{m_e^*}{m_h^*} $
Substituting Values
From the given information, we have:
- $ \frac{\tau_h}{\tau_e} = \frac{1}{2} $
- $ \frac{m_e^*}{m_h^*} = \frac{11}{2} $ (This is the inverse of the given effective mass ratio)
Substitute these values into the mobility ratio equation:
$ \frac{\mu_h}{\mu_e} = \left( \frac{1}{2} \right) \times \left( \frac{11}{2} \right) $
$ \frac{\mu_h}{\mu_e} = \frac{11}{4} $
Therefore, the ratio of the mobility of the hole to electron is 11:4.
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
