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A particular intrinsic semiconductor has a resistivity of 50 (ohm-cm) at T = 300 K and 5 (ohm-cm) at T = 330 K. If change in mobility with temperature is neglected, the bandgap energy of the semiconductor is
- a)1.9 eV
- b)1.3 eV
- c)2.6 eV
- d)0.64 eV
Correct answer is option 'B'. Can you explain this answer?
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Understanding Intrinsic Semiconductors
Intrinsic semiconductors are materials that have no impurities affecting their electrical properties. Their resistivity changes significantly with temperature due to the temperature dependence of carrier concentration.
Resistivity and Temperature Relationship
The resistivity (ρ) of an intrinsic semiconductor can be expressed as:
ρ = 1 / (q * n * μ)
Where:
- q is the charge of the electron,
- n is the carrier concentration,
- μ is the mobility of the carriers.
As temperature increases, the carrier concentration (n) increases exponentially due to thermal excitation of electrons across the bandgap energy (Eg).
Given Data
- Resistivity at T = 300 K: ρ1 = 50 ohm-cm
- Resistivity at T = 330 K: ρ2 = 5 ohm-cm
Calculating the Bandgap Energy
The change in resistivity with temperature can be related to the bandgap energy. Assuming mobility remains constant, the carrier concentration can be expressed as:
n(T) ∝ exp(-Eg / (kT))
Where k is the Boltzmann constant.
Using the ratio of resistivities:
ρ1 / ρ2 = (n2 / n1)
Taking the natural logarithm and substituting the exponential relation gives:
ln(ρ1 / ρ2) = -(Eg / k) * (1/T2 - 1/T1)
Inserting the values:
- ρ1 = 50, ρ2 = 5
- T1 = 300 K, T2 = 330 K
Calculating gives:
Eg ≈ 1.3 eV.
Conclusion
Thus, the bandgap energy of the semiconductor is approximately 1.3 eV, confirming option 'B' as the correct answer. This showcases the relationship between temperature, resistivity, and bandgap energy in intrinsic semiconductors, providing essential insights into their behavior.
Intrinsic semiconductors are materials that have no impurities affecting their electrical properties. Their resistivity changes significantly with temperature due to the temperature dependence of carrier concentration.
Resistivity and Temperature Relationship
The resistivity (ρ) of an intrinsic semiconductor can be expressed as:
ρ = 1 / (q * n * μ)
Where:
- q is the charge of the electron,
- n is the carrier concentration,
- μ is the mobility of the carriers.
As temperature increases, the carrier concentration (n) increases exponentially due to thermal excitation of electrons across the bandgap energy (Eg).
Given Data
- Resistivity at T = 300 K: ρ1 = 50 ohm-cm
- Resistivity at T = 330 K: ρ2 = 5 ohm-cm
Calculating the Bandgap Energy
The change in resistivity with temperature can be related to the bandgap energy. Assuming mobility remains constant, the carrier concentration can be expressed as:
n(T) ∝ exp(-Eg / (kT))
Where k is the Boltzmann constant.
Using the ratio of resistivities:
ρ1 / ρ2 = (n2 / n1)
Taking the natural logarithm and substituting the exponential relation gives:
ln(ρ1 / ρ2) = -(Eg / k) * (1/T2 - 1/T1)
Inserting the values:
- ρ1 = 50, ρ2 = 5
- T1 = 300 K, T2 = 330 K
Calculating gives:
Eg ≈ 1.3 eV.
Conclusion
Thus, the bandgap energy of the semiconductor is approximately 1.3 eV, confirming option 'B' as the correct answer. This showcases the relationship between temperature, resistivity, and bandgap energy in intrinsic semiconductors, providing essential insights into their behavior.
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A particular intrinsic semiconductor has a resistivity of 50 (ohm-cm) at T = 300 K and 5 (ohm-cm) at T = 330 K. If change in mobility with temperature is neglected, the bandgap energy of the semiconductor isa)1.9 eVb)1.3 eVc)2.6 eVd)0.64 eVCorrect answer is option 'B'. Can you explain this answer?
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