We have used Hartree calculations to investigate the electrical properties of a sub-micron resonant tunnelling diode with an etched tear-shaped cross-section. Non-passivated surface states are shown to produce a deep potential minimum near the centre of the tear drop. In the emitter accumulation layer, the conduction electrons screen this potential minimum and move in an ideal flat-bottomed billiard with soft walls which generate a mixed stable-chaotic classical phase space. In the quantum well, the potential minimum is unscreened and there is a large energy splitting between the ground state and first excited state. The ground state of the quantum well therefore provides an in-built spectroscopic probe of a wide range of quantized emitter states. Our studies highlight the potential of tunnelling spectroscopy as an alternative to parallel transport studies of chaotic billiards which only detect states near the Fermi level.