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The fraction (f) of the number of electr...

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: \[ f = C \cdot e^{-\frac{E_G}{kT_K}} \]
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Knowledge Check

  • 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

    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

    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
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