  Pyrite FeS$_2$ has been investigated for a wide range of applications,
including thermoelectrics due to previous observation of large thermopower at
room-temperature. However, the values of thermopower reported in the literature
is extremely sensitive to the nature of sample -- whether they are natural or
lab grown, bulk crystals or thin films -- and an ambiguity in the magnitude and
sign of thermopower of pure FeS$_2$ exists. Variation in the magnitude of
room-temperature thermopower has also been observed in Co-doped samples.
Therefore, it is of interest to clarify the intrinsic thermopower of this
system that could be measured in more pure samples. In this paper, we
investigate the thermoelectric properties of Co-doped FeS$_2$ using first
principles calculations. We apply three different doping schemes to understand
the effect of electron doping in FeS$_2$, namely explicit Co-substitution,
jellium doping and electron addition within rigid band approximation (RBA)
picture. The calculated thermopower is less than $-50$ $\mu$V/K for all values
of Co doping that we studied, suggesting that this system may not be useful in
thermoelectric applications. Interestingly, we find that RBA substantially
overestimates the magnitude of calculated thermopower compared to the explicit
Co-substitution and jellium doping schemes. The overestimation occurs because
the changes in the electronic structure due to doping-induced structural
modification and charge screening is not taken into account by the rigid shift
of the Fermi level within RBA. RBA is frequently used in first principles
investigations of the thermopower of doped semiconductors, and Co-substituted
FeS$_2$ illustrates a case where it fails.
