Question

Temperature dependence of electron-hole mobility due to lattice scattering varies approximately as :

• A. $T^{-3/2}$
• B. $T^{3/2}$
• C. $T^{-2/3}$
• D. $T^{1/3}$

Hint:

To remember which is which: High temperatures mean violent lattice shaking, which hinders movement (Phonon mobility drops, $T^{-3/2}$). High temperatures mean fast electrons that zip past charges too quickly to be pulled off course (Impurity mobility rises, $T^{3/2}$).

Answer 1

The Correct Option is A

Step 1: Concept:
The mobility of charge carriers (electrons and holes) in a semiconductor is limited by various scattering mechanisms. The question asks for the specific temperature dependence of mobility when limited exclusively by lattice scattering (also known as phonon scattering).

Step 2: Key Formula or Approach:
Carrier mobility ($\mu$) is fundamentally defined by the scattering relaxation time ($\tau$): $\mu = \frac{e\tau}{m^*}$.
There are two primary scattering mechanisms in a moderately doped semiconductor:
1. Lattice (Phonon) Scattering ($\mu_L$): As temperature increases, thermal vibrations of the crystal lattice (phonons) become much more violent, increasing the probability of collision. Thus, mobility decreases as temperature increases.
\[ \mu_L \propto T^{-3/2} \]
2. Ionized Impurity Scattering ($\mu_I$): At higher temperatures, charge carriers move much faster (higher thermal velocity) and spend less time in the vicinity of fixed ionized dopants, making them less easily deflected. Thus, mobility increases as temperature increases.
\[ \mu_I \propto T^{3/2} \]

Step 3: Step-by-step Explanation:

• The question specifically singles out lattice scattering.

• According to semiconductor physics derivations, the scattering rate due to acoustic phonons is proportional to $T^{3/2}$.

• Because mobility is inversely proportional to the scattering rate ($\mu \propto \tau \propto 1/\text{rate}$), the mobility dictated by lattice scattering is proportional to $T^{-3/2}$.

Step 4: Final Answer:
The temperature dependence for lattice scattering varies as $T^{-3/2}$, which matches option (A).