Modern semiconductor physics research is often carried out at low temperatures where the carrier mobility is high and the thermal broad- ening of quantised energy levels is minimised. 4.2 K, the temperature of liquid helium, is thought of as relatively warm. Temperatures of 300 mK and below are readily accessible with sorption-pumped helium-3 and 3He/4He dilution refrigerators. At such low tempera- tures, the phonons that normally facilitate thermal equilibrium be- tween the charge carriers and the lattice are greatly suppressed. This can lead to the electron system thermally decoupling from the lattice. Conventional thermometers such as ceramic oxide or ruthenium ox- ide resistance thermometers thermally couple to the lattice, meaning that in this regime, they only effectively measure the lattice temper- ature, TL. The effect is further pronounced in low dimensional car- rier systems such as two-dimensional electron gases (2DEGs). This is problematic because the electron temperature, Te, is often of far greater importance than TL in such systems. Recent efforts have been made to develop thermometry techniques that are directly sensitive to Te. These techniques include using the thermopower of quantum point contacts and the Coulomb blockade of quantum dots. However, these techniques have issues with the ease and reliability of fabrication of the thermocouples, and the complexity of their subsequent oper- ation. This thesis presents a direct electron thermometry technique called bar-gate thermometry (BGT) that utilises the diffusion ther- mopower of the 2DEG as a thermocouple. BGT uses simple metal- lic bar-gates, fabricated directly onto the semiconductor wafer, and requires no particularly sophisticated electronic instruments. BGT proves to be a reliable and relatively simple thermometry technique. A device featuring a BGT is presented which allows a determination of the thermal relaxation length, l, of the 2DEG. This comes hand in hand with effective measurements of the inelastic scattering time, τi, and the inelastic scattering exponent, αi, of the system, which are not otherwise easily measured. Finally, an example of how BGTs can be used to measure the thermal conductivity of mesoscopic 2DEGs is given, which is just one possible use case of the technique. An overview of all the necessary and relevant theory needed to under- stand the techniques and results, as well as information on the design and fabrication of the devices used in the experiments is also included.