The fraction `(f)` of the number of electrons raised from valence band to conduction band at temperature TK in intrinsic semiconductor is given by........
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To derive the fraction \( f \) of the number of electrons raised from the valence band to the conduction band in an intrinsic semiconductor at temperature \( T_K \), we can follow these steps:
### Step 1: Understand the Energy Bands
In an intrinsic semiconductor, there are two main energy bands: the valence band and the conduction band. The energy gap between these two bands is denoted as \( E_G \).
### Step 2: Identify the Temperature Dependence
At a given temperature \( T_K \), some electrons gain enough thermal energy to jump from the valence band to the conduction band. The fraction of electrons that can make this jump is influenced by the temperature and the energy gap.
### Step 3: Use the Boltzmann Distribution
The fraction \( f \) of electrons that can jump from the valence band to the conduction band can be expressed using the Boltzmann distribution. The formula is given by:
\[
f \propto e^{-\frac{E_G}{kT}}
\]
where:
- \( E_G \) is the energy gap,
- \( k \) is the Boltzmann constant,
- \( T \) is the absolute temperature in Kelvin.
### Step 4: Write the Complete Expression
To express \( f \) in a more complete form, we can include a constant of proportionality \( C \):
\[
f = C \cdot e^{-\frac{E_G}{kT}}
\]
This equation indicates that the fraction of electrons in the conduction band increases with temperature and decreases with a larger energy gap.
### Step 5: Consider the Specifics of the Semiconductor
For specific semiconductors like silicon or germanium, the value of \( E_G \) will be different (approximately 1.1 eV for silicon and 0.66 eV for germanium). However, the general form of the equation remains the same.
### Final Expression
Thus, the fraction \( f \) of the number of electrons raised from the valence band to the conduction band at temperature \( T_K \) in an intrinsic semiconductor is given by:
\[
f = C \cdot e^{-\frac{E_G}{kT_K}}
\]
