The band gaps of diamond and silicon are 5.42 eV and 1.1 eV respectively. Estimate the temperature at which diamond has the same conductivity as Si at 29°C. b) Show that the product of electron and hole concentrations in a semiconductor is constant at a given temperature. How is the energy gap determined from the measurement of electrical conductivity of a semiconductor? c) Obtain an expression for the conductivity of an intrinsic semiconductor. How does it vary with temperature?
The Correct Answer and Explanation is:
a) Temperature for Diamond to Have Same Conductivity as Silicon at 29°C
The electrical conductivity (σ\sigmaσ) of a semiconductor is given by the equation:σ=q⋅(n⋅μe+p⋅μh)\sigma = q \cdot (n \cdot \mu_e + p \cdot \mu_h)σ=q⋅(n⋅μe+p⋅μh)
Where:
- qqq is the charge of an electron,
- nnn is the concentration of electrons in the conduction band,
- ppp is the concentration of holes,
- μe\mu_eμe and μh\mu_hμh are the mobilities of electrons and holes.
For an intrinsic semiconductor, n=pn = pn=p, so:σ=q⋅n⋅(μe+μh)\sigma = q \cdot n \cdot (\mu_e + \mu_h)σ=q⋅n⋅(μe+μh)
The electron concentration nnn in an intrinsic semiconductor is given by:n=Nc⋅e−Eg2kTn = N_c \cdot e^{-\frac{E_g}{2kT}}n=Nc⋅e−2kTEg
Where:
- NcN_cNc is the effective density of states,
- EgE_gEg is the energy band gap,
- kkk is the Boltzmann constant,
- TTT is the absolute temperature in Kelvin.
At a given temperature T1T_1T1, the conductivity of silicon (σSi\sigma_{Si}σSi) can be written as:σSi=q⋅nSi⋅(μe+μh)Si\sigma_{Si} = q \cdot n_{Si} \cdot (\mu_e + \mu_h)_{Si}σSi=q⋅nSi⋅(μe+μh)Si
Similarly, for diamond at a different temperature T2T_2T2, its conductivity is given by:σdiamond=q⋅ndiamond⋅(μe+μh)diamond\sigma_{diamond} = q \cdot n_{diamond} \cdot (\mu_e + \mu_h)_{diamond}σdiamond=q⋅ndiamond⋅(μe+μh)diamond
To find the temperature T2T_2T2 at which the conductivity of diamond equals that of silicon, we can equate the two conductivities:ndiamond(T2)⋅(μe+μh)diamond=nSi(T1)⋅(μe+μh)Sin_{diamond}(T_2) \cdot (\mu_e + \mu_h)_{diamond} = n_{Si}(T_1) \cdot (\mu_e + \mu_h)_{Si}ndiamond(T2)⋅(μe+μh)diamond=nSi(T1)⋅(μe+μh)Si
Since both nnn and the mobilities depend on temperature, we can substitute their expressions to solve for T2T_2T2, the temperature at which diamond achieves the same conductivity as silicon at 29°C.
Given that the energy gaps of diamond and silicon are 5.42 eV and 1.1 eV respectively, we can solve this by comparing the exponential dependence of the electron concentration on temperature.
b) Product of Electron and Hole Concentrations
In an intrinsic semiconductor, the product of the electron concentration (nnn) and hole concentration (ppp) is constant at a given temperature. This relationship arises from the principle of charge neutrality and the recombination process in semiconductors.
The intrinsic carrier concentration is governed by the equation:n⋅p=ni2n \cdot p = n_i^2n⋅p=ni2
Where:
- nin_ini is the intrinsic carrier concentration,
- nnn is the electron concentration in the conduction band,
- ppp is the hole concentration in the valence band.
This product is constant because the number of electrons that enter the conduction band equals the number of holes left in the valence band. The intrinsic carrier concentration nin_ini is a function of temperature and the energy band gap (EgE_gEg):ni=Nc⋅Nv⋅e−Eg2kTn_i = \sqrt{N_c \cdot N_v} \cdot e^{-\frac{E_g}{2kT}}ni=Nc⋅Nv⋅e−2kTEg
Where:
- NcN_cNc and NvN_vNv are the effective densities of states in the conduction and valence bands.
c) Expression for Conductivity of an Intrinsic Semiconductor
The conductivity (σ\sigmaσ) of an intrinsic semiconductor can be expressed as:σ=q⋅ni⋅(μe+μh)\sigma = q \cdot n_i \cdot (\mu_e + \mu_h)σ=q⋅ni⋅(μe+μh)
Where:
- qqq is the charge of an electron,
- nin_ini is the intrinsic carrier concentration,
- μe\mu_eμe and μh\mu_hμh are the mobilities of electrons and holes.
The intrinsic carrier concentration is given by:ni=Nc⋅Nv⋅e−Eg2kTn_i = \sqrt{N_c \cdot N_v} \cdot e^{-\frac{E_g}{2kT}}ni=Nc⋅Nv⋅e−2kTEg
Substituting this into the conductivity equation:σ=q⋅Nc⋅Nv⋅e−Eg2kT⋅(μe+μh)\sigma = q \cdot \sqrt{N_c \cdot N_v} \cdot e^{-\frac{E_g}{2kT}} \cdot (\mu_e + \mu_h)σ=q⋅Nc⋅Nv⋅e−2kTEg⋅(μe+μh)
Temperature Dependence of Conductivity
The conductivity of an intrinsic semiconductor increases with temperature because the intrinsic carrier concentration nin_ini increases exponentially with temperature due to the dependence on e−Eg2kTe^{-\frac{E_g}{2kT}}e−2kTEg. As temperature rises, more electrons gain enough energy to jump from the valence band to the conduction band, increasing both the electron and hole concentrations.
Thus, conductivity increases with temperature in intrinsic semiconductors.
