2.4.7.7 Thermo-electric effects in semiconductors

The dependence of the current in a semiconductor on temperature can be included by generalizing the drift-diffusion equation for the current. The proportionality constant between the current density and the temperature gradient is the product of the conductivity and the thermo-electric power.

The derivation starts by generalizing the diffusion current to include a possible variation of the diffusion constant with position, yielding:

If the semiconductor is non-degenerate the electron density can be related to the effective density of states and the difference between the Fermi energy and the conduction band edge:

yielding:

For the case where the material properties do not change with position, all the spatial variations except for the gradient of the Fermi energy, are caused by a temperature variation. We postulate that the current can be written in the following form:

P is the thermo-electric power in Volt/Kelvin. From both equations one then obtains an expression for the thermo-electric power:

If the temperature dependence of the mobility can be expressed as

The thermo-electric power becomes:

for n-type material and similarly for p-type material:

The Peltier coefficient is related to the thermo-electric power by:

If electrons and holes are present in the semiconductor one has to include the effect of both when calculating the Peltier coefficient by

The resulting Peltier coefficient as a function of temperature for silicon is shown in the figure below:

peltier.xls

Fig. 2.4.7.x Peltier coefficient for p-type (top curve) and n-type (bottom curve) silicon as a function of temperature. The doping density equals 10¹⁴ cm^-3.

The Peltier coefficient is positive for p-type silicon and negative for n-type silicon at low temperature. Al high temperature the semiconductor becomes intrinsic. Given that the mobility of electrons is higher than that of holes, the Peltier coefficient of intrinsic silicon is negative.

The current and heat flow are related to the Fermi energy gradient and the temperature gradient. Using the equations above, the current can be written as:

where s_n is the conductivity of the n-type semiconductor.

The heat flow, H, in units of W/cm² is given by:

This expression obtained by using the Onsager relations and by requiring that the heat flow in the absence of current is given by:

Where k is the thermal conductivity of the material.