The temperature eﬀect on the carrier concentration can be understood by using equation 1. Temperature enters both the pre-exponential and the exponential term. Increasing temperature has two eﬀects
1. Increases the eﬀective density of states at the band edges (Nc and Nv).
2. Decreases the exponential portion of equation 1, since T is in the denominator.
The exponential terms dominates, as can be seen by a semi log plot of ni vs. inverse temperature, as shown in ﬁgure 2 for Si. The slope of the plot gives the band gap of the semiconductor. Changing the semiconductor material, changes the band gap and also the eﬀective density of states. For diﬀerent semiconductors the plot of ni vs. T is similar with diﬀerent slopes, corresponding to diﬀerent Eg values. Intrinsic carrier concentrations for Ge, Si, and GaAs are compared in ﬁgure 3. The slopes correspond to their respective band gaps, given in table 1. The slope for GaAs is steepest, since it has the highest band gap.
Increase in concentration and/or mobility has the eﬀect of increasing conductivity. While concentration increases with temperature mobility decreases with temperature. But because of the exponential term in equation 3, the dominating term is ni. Consider two temperatures, T1 and T2 with intrinsic carrier concentrations, ni1 and ni2. Using equations 1 and 2 the ratio of the carrier concentration is given by
This leads to an approximately 5 orders of magnitude increase in conductivity. Problem is that almost all devices need to operate at or near room temperature, so that increasing temperature would not be a feasible option. So another method must be found to increase and control the conductivity in Si. This is achieved by doping.