Conductivity of Intrinsic Semiconductor

It is known that the conductivity of a semiconductor is,
= (n μn + p μp) q
If the semiconductor is intrinsic, then the concentration of free electrons n and holes p is same at any time and given by ni  which is called intrinsic concentration.
ni = n = p for intrinsic semiconductor
Using the expression of conductivity, we get the expression for the conductivity of intrinsic semiconductor denoted as.
σi = ni ( μn + μn ) 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 ni and σi.
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 ni also increases.
As temperature increases,
i) Intrinsic carrier concentration ni increases.
ii) Intrinsic conductivity σi increases.
iii) Intrinsic resistivity ρi = 1/σi decreases
The temperature dependence of ni is expressed by mathematical relationship as,

where            A0 = Constant independent of temperature
T = Absolute temperature expressed in oK
EG0 = 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 ni
The values of EG0 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 oK-3/2 or cm-3 oK-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 (EG)
The dependence of energy gap on temperature is given by,
EG(T) = EG0 - βT
where            EG0 = 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          EG0 = 1.21 for Si and 0.785 for Ge.
Note : EG 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 μn = 1300 cm2/V-sec and μp = 500 cm2/V-sec. Assume q = 1.6 x 10-19 C.
Solution : The given values are, ni = 1.5 x 10/cm3
...             ni = (1.5 x 1010)/10-6   /m3  = 1.5 x 1016  /m3
And         μn = 1300 x 10-4   m2/V-sec,  μp = 500 x 10-4 m2/V-sec
Now       σi = ( μn + μp) q = 1.5 x 1016 {1300 + 500} x 10-4 x 1.6 x 10-19
= 0.000432 (Ω-m)-1                  ..............Conductivity
...         ρ  = 1/σi = 1/0.000432 = 2314.8148  Ω-m         .........Resistivity