Conductivity of Intrinsic Semiconductor

       It is known that the conductivity of a semiconductor is,

                         = (n μ_n + p μ_p) q

       If the semiconductor is intrinsic, then the concentration of free electrons n and holes p is same at any time and given by n_i  which is called intrinsic concentration.

                       n_i = n = p for intrinsic semiconductor

       Using the expression of conductivity, we get the expression for the conductivity of intrinsic semiconductor denoted as.
                      σ_i = n_i ( μ_n + μ_n ) q                  ..........(1)

Note : Note that the mobilities of electron and hole are different, through the concentration semiconductor n and p are same in intrinsic semiconductor.

1.1 Temperature Dependence of n_i and σ_i.

       The intrinsic concentration depends on temperature, due to thermal generation. As temperature increases, more number of electrons-hole pairs are generated. Hence the free electron concentration n and hole concentration p increases by same amount. Hence intrinsic carrier concentration n_i also increases.

       As temperature increases,

i) Intrinsic carrier concentration n_i increases.

ii) Intrinsic conductivity σ_i increases.

iii) Intrinsic resistivity ρ_i = 1/σ_i decreases

       The temperature dependence of n_i is expressed by mathematical relationship as,

       where            A₀ = Constant independent of temperature

                             T = Absolute temperature expressed in ^oK

                             E_G0 = Forbidden energy gap at absolute zero temerature

                             k = Boltzmann's constant expressed in eV/^oK = 8.620 x 10³ eV/^oK

                             B = √A₀ = Constant independent of temperature

       The equations (2a) and (2b) show that the charge carrier concentration increases as the temperature increases. The effect of temperature on charge carrier concentration is shown in the Fig. 1.

Fig 1. Effect of temperature on n_i

       The values of E_G0 and B for various materials are given in the table 1.

Table 1 Constants of semiconductors

 Note : The concentrations can be expressed in the units /m³ or /cm³ and accordingly B must be used in m^{-3 o}K^-3/2 or cm^{-3 o}K^-3/2.

1.2 Effect of Light on Semiconductor

       The effect of light on a semiconductor is exactly similar to the effect of heat on a semiconductor. Just as thermal energy causes electrons to break their covalent bonds, similarly the light energy also causes electrons to break their covalent bonds. Under the influence of light energy, electron-hole pairs get generated in a semiconductor, increasing its conductivity.

       When not illuminated there are few free electrons in a semiconductor and its resistance is high called dark resistance. As the light is incident on a semiconductor and it is illuminated it imparts light energy to the electrons. The electrons breaking their bonds move from valence band to conduction band and the conduction can take place readily. Thus there is decreases in resistance of a semiconductor. When illumination increases, a semiconductor may behave comparable to a conductor.

       The effect of light on a semiconductor is to cause increase in the conductivity of a semiconductor.

Note : Both heat and light are responsible to generate electron-hole pairs and hence to increase the conductivity of a semiconductor.

1.3 Effect of Temperature on Mobility (μ)

       It is practically observed that over the temperature range of 100 to 400 ^oK, the mobility changes as  T^-m.

       where value of m is ,

Note : The mobility decreases with increase in temperature.

       Similarly the mobility also changes with electric field intensity E. It remains constant for E less than 10³ V/m. For value of e between 10³ to 10⁴ V/m, μ changes as E^-1/2. While for high values of E, μ changes inversely as E.

1.4 Effect of Temperature on Conductivity (σ)

       The conductivity is directly proportional to mobility and concentration of charge carriers. The mobility decreases while the concentration of charge carriers increases with the temperature.

       But change in concentration of charge due to the temperature is more dominating in intrinsic materials than the change in due to temperature.

Note : Thus the conductivity of intrinsic semiconductor increases with increase in temperature.
       The temperature dependence of conductivity (σ) is given by,
                         σ₂ = σ₁{1+ α₁ ΔT}
where                σ₁= Conductivity at temperature T₁.
                         σ₂ = Conductivity at temperature T₂.
                         α₁   = Resistance temperature coefficient at .
                         ΔT = Temperature rise ( T₂ - T₁)

1.5 Effect on Energy Gap (E_G)

       The dependence of energy gap on temperature is given by,
                            E_G(T) = E_G0 - βT
       where            E_G0 = Energy gap at 0 ^oK.
                            β = Constant depending on material
       The value of β = 2.23 x 10^-4 for Ge while 3.6 x 10^-4 for Si.
       While          E_G0 = 1.21 for Si and 0.785 for Ge.
 Note : E_G decreases as temperature increases.

Example : Find the resistivity of an intrinsic silicon at 300 ^oK if intrinsic concentration of silicon at 300 ^oK is 1.5 x 10¹⁰ per cm³ while μ_n = 1300 cm²/V-sec and μ_p = 500 cm²/V-sec. Assume q = 1.6 x 10^-19 C.

Solution : The given values are, n_i = 1.5 x 10/cm³
.^..             n_i = (1.5 x 10¹⁰)/10^-6   /m³  = 1.5 x 10¹⁶  /m³
And         μ_n = 1300 x 10^-4   m²/V-sec,  μ_p = 500 x 10^-4 m²/V-sec
Now       σ_i = ( μ_n + μ_p) q = 1.5 x 10¹⁶ {1300 + 500} x 10^-4 x 1.6 x 10^-19
                   = 0.000432 (Ω-m)^-1                  ..............Conductivity
.^..         ρ  = 1/σ_i = 1/0.000432 = 2314.8148  Ω-m         .........Resistivity

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