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Studies of two-dimensional electron gases (2DEGs) in semiconductors form an active and productive field of condensed matter physics research. As well as having interesting inherent properties, they are used as the foundation for constructing various nano-scale electronic devices, such as quantum wires and quantum dots. Conventionally, low temperature measurements of 2DEGs are made by cooling the sample to 1.5 K with liquid Helium-4, to 300 mK with liquid Helium-3, or even down to a few mK using a dilution refrigerator. However, at lower temperatures the electron gas becomes increasingly decoupled from the lattice in which it resides. Below ~ 1 K the coupling can be weak enough for the electron gas to be significantly elevated in temperature due to parasitic heating. In this thesis we present the experimental and theoretical investigation of a refrigeration scheme that has the potential to cool 2DEGs below the temperatures currently available. Cooling to ever lower temperatures would be beneficial for studying fragile fractional quantum Hall states, non-Fermi-liquid behaviour in bilayer 2DEGs, or interactions like the Kondo effect that occur between quantum dots and 2DEGs. The scheme we investigate is called the Quantum Dot Refrigerator (or QDR) and is based upon the energy selective transport of electrons through the singleparticle states of quantum dots. By using a pair of dots, both hot electrons and hot holes can be selectively removed from an otherwise electrically isolated 2DEG. The result is a net current that continuously removes heat. This type of refrigerator is best suited to be used in conjunction with a dilution fridge or Helium-3 system to provide a final stage of cooling. The scheme was first investigated theoretically in 1993 by Edwards et al. but, to our knowledge, has never before been demonstrated experimentally. We detail the fabrication and measurement of a QDR device that is designed to cool an isolated 6 µm2 2DEG. In order to interpret the behaviour of the device, a model was developed to take account of electrostatic interactions between the components of the system (the quantum dots and the isolated 2DEG). Electrostatic interactions were found to be significant for our design, but were neglected in previous work. Our model predicts that their presence complicates, but does not invalidate, the principle of operation of a QDR. By comparing measurements of the current through the QDR with predictions of the model, we show that the observed behaviour is consistent with cooling of the isolated 2DEG by up to 100 mK at ambient temperatures around 250 mK. Although these temperatures are well within the reach of conventional refrigeration techniques, the results are a compelling proof-of-concept demonstration that the QDR principle is sound and can achieve significant refrigeration in the right conditions. Finally, we discuss future directions for improving QDR performance and characterisation, and for lowering the achievable base temperature. We also suggest how QDRs could be used to provide cold reservoirs for a nano-scale electronic device, and explore the limitations that would apply to such an experiment.

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