The fraction `(f)` of the number of electrons raised from valence band to conduction band at temperature TK in intrinsic semiconductor is given by........
The fraction `(f)` of the number of electrons raised from valence band to conduction band at temperature TK in intrinsic semiconductor is given by........
Text Solution
AI Generated Solution
The correct Answer is:
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:
\[
f = C \cdot e^{-\frac{E_G}{kT_K}}
\]
Topper's Solved these Questions
Similar Questions
Explore conceptually related problems
The ratio of number of holes and number of conduction electrons in an intrinsic semiconductor is
The probability of electrons to be found in the conduction band of an intrinsic semiconductor at a finite temperature
The probbility of electrons to be found in the conduction band of an intrinsit semiconductor at a finile temperature
If the energy gap between valence band and conduction band is 5 eV, then it is
If the energy gap between valence band and conduction band is 10 eV, then the material is a/an
The valence band and conduction band of a solid overlap at low temperature .The solid may be
The valence band and conduction band of a solid overlap at low temperature, the solid may be
The conduction band and valency band of a good conductors are
Knowledge Check
The ratio of number of holes and number of conduction electrons in an intrinsic semiconductor is
The ratio of number of holes and number of conduction electrons in an intrinsic semiconductor is
A
One
B
Greater than one
C
Less than one
D
Infinity
The probability of electrons to be found in the conduction band of an intrinsic semiconductor at a finite temperature
The probability of electrons to be found in the conduction band of an intrinsic semiconductor at a finite temperature
A
Decreases exponentially with increasing band gap
B
Increases exponentially with increasing band gap
C
Decreases with increasing temperature
D
Is independent of the temperature and the band gap
The probbility of electrons to be found in the conduction band of an intrinsit semiconductor at a finile temperature
The probbility of electrons to be found in the conduction band of an intrinsit semiconductor at a finile temperature
A
increase exponenially with increases band gap
B
decreases exponenially with increases band gap
C
decreases with increases temperature
D
is independent of the temperature and the band gap