The electron density of a n-type semiconductor is proportional to :
\(\exp\left(\dfrac{E_F-E_i}{KT}\right)\)
Write the carrier density with the intrinsic level as reference. Measuring the Fermi level from the intrinsic level Ei rather than from the band edge gives the compact and very useful pair
\(n=n_i\exp\left(\dfrac{E_F-E_i}{kT}\right), \qquad p=n_i\exp\left(\dfrac{E_i-E_F}{kT}\right)\)
The electron density therefore varies as \(\exp\left[(E_F-E_i)/kT\right]\), which is option 1.
The sign is the whole question, and it has a clear physical meaning. In n-type material the Fermi level lies above the intrinsic level, so \(E_F-E_i \gt 0\), the exponent is positive, and
\(n \gt n_i\)
exactly as it must be for electrons to be the majority carrier. Option 2 has the sign reversed — that is the expression for the hole density, which in n-type material correctly comes out smaller than ni.
Why options 3 and 4 cannot be right. Both contain only the band gap Eg and no reference to EF at all. A quantity independent of the Fermi level cannot describe a doped semiconductor, because doping acts precisely by moving EF. In fact those two expressions belong to intrinsic material:
\(n_i=\sqrt{N_cN_v}\exp\left(-\dfrac{E_g}{2kT}\right), \qquad n_i^{2}=N_cN_v\exp\left(-\dfrac{E_g}{kT}\right)\)
so option 4 is the intrinsic concentration and option 3 is its square. Neither responds to doping.
The consistency check. Multiplying the two expressions at the top,
\(np=n_i^{2}\exp\left(\dfrac{E_F-E_i}{kT}\right)\exp\left(\dfrac{E_i-E_F}{kT}\right)=n_i^{2}\)
The exponentials cancel and the law of mass action drops straight out — a neat confirmation that the two signs are the right way round.
The practical use of this form. Given a doping level it returns the Fermi position at once: for \(N_D=10^{16}\) and \(n_i=1.5\times10^{10}\),
\(E_F-E_i=kT\ln\dfrac{n}{n_i}=0.0259\ln\left(6.7\times10^{5}\right)=0.35\ \text{eV}\)
Hence, the electron density is proportional to \(\exp\left[(E_F-E_i)/KT\right]\).
Similar Questions
Consider the following statements :
If an electric field is applied to an n-type semiconductor bar, the electrons and holes move in opposite directions due to their opposite charges. The net current is
1. due to both electrons and holes with electrons as majority carriers.
2. the sum of electron and hole currents.
3. the difference between electron and hole current.Which of these statements is/are correct ?
Consider the following :
1. In semiconductors, the mobility of electrons is more than that of holes.
2. In semiconductors, the resistivity increases with the increase in temperature.
3. Metal has positive TCR.
4. In metals, thermal conductivity is inversely proportional to electronic conductivity at a particular temperature.Which of the following statements are correct ?
The Debye length is a characteristic length for semiconductor. Arrange the doping density given below in such a way that Debye length changes from maximum to minimum.
(A) N = 1015 cm–3
(B) N = 1017 cm–3
(C) N = 3 × 1015 cm–3
(D) N = 1016 cm–3
(E) N = 5 × 1016 cm–3Choose the most appropriate answer from the options given below :
Match the following :
List – I List – II a. Laplace equation i. \(q\mu_n\left(\varepsilon\cdot n+\dfrac{kT}{q}\dfrac{\partial n}{\partial x}\right)\) b. Continuity equation ii. \(G_n-\dfrac{n_p-n_{p0}}{\tau_n}+n_p\mu_n\dfrac{\partial\varepsilon}{\partial x}+\mu_n\varepsilon\dfrac{\partial n_p}{\partial x}+D_n\dfrac{\partial^{2}n_p}{\partial x^{2}}\) c. Current density equation iii. \(\dfrac{\partial^{2}\psi}{\partial x^{2}}+\dfrac{\partial^{2}\psi}{\partial y^{2}}+\dfrac{\partial^{2}\psi}{\partial z^{2}}=\dfrac{qN_D}{\epsilon}\) d. Poisson’s equation iv. \(\nabla^{2}\psi=0\) Choose the correct answer from the codes given below:
A conducting line on an IC chip is 2.8 mm long and has a rectangular cross-section of 1 μm x 4 μm.. A current of 5 mA produces a voltage drop of 100 mV across the line. If the electron mobility is 500 cm2/V-s, the electron concentration is
For non-degenerate semiconductors the product of the majority and minority carrier concentration is fixed as
(a) \(ni ^ {2}\) (b) \(N_cN_v\exp\left(\dfrac{Eg}{kT}\right)\) (c) \(N_cN_v\exp\left(-\dfrac{Eg}{kT}\right)\) (d) \(N_v\exp\left(-\dfrac{Eg}{kT}\right)\)
Which of the above are correct ?
The ionised concentration for donors is given by :
Important Questions from Carriers in Semiconductors
The velocity with which electrons are emitted in the photoemission process
The process of adding impurities to a pure semiconductor is called
Mobility and conductivity are related by which of the following equations?
How many electrons are there in the valence shell of a pure semiconductor?
In a pure silicon, what is the time for an electron to drift $1\mu m$ in an electric field of 100 V/cm?
Assume electron mobility of $1350 \text{ cm}^2/V-s$
