  The hot carrier solar cell (HCSC) has the potential for converting solar
energy into electrochemical energy with an efficiency of 85.4%. For this, in
addition to an idealized light absorber, the HCSC has to be connected to the
external load by means of the so-called \emph{mono-energetic energy selective
contacts} (ESCs). However, the thermodynamic properties that these types of
contact have to exhibit, such as their electric, thermal conductivity and
Seebeck coefficient, have not been explored. This paper aims to fill this gap.
In this respect, we model electron transport in non-ideal ESCs using the
transport theory proposed by Datta and Landauer which has allowed us to
calculate the value of these parameters as a function of the temperature and
electrochemical potential of operation. Our findings also reveal that, to
preserve the HCSC efficiency above 82%, the ESCs could require in the order of
$3 \times 10^{19}$ cm$^{-3}$ electron states. As the ESCs depart from ideality,
the temperature of the hot carriers at which optimum efficiency is obtained
increases to above 2540 K. The mono-enenergetic selective contact characterized
by the highest energy demands an electric, thermal conductivity and Seebeck
coefficient that, when combined, are characterized by a high thermoelectric
figure of merit $(ZT\approx 8)$. We are not aware of any material exhibiting
this figure of merit which illustrates the difficulty in putting the HCSC
concept into practice. Conversely, our work supports the idea that pursuing
materials capable of transporting electrons ballistically through
mono-energetic electron channels can provide the key for achieving materials
characterized by high $ZT$
