Tunneling studies of energy levels and selection rules in low-dimensional structures

Tunneling processes between a multiple-quantum-wire (MQW) system and a two-dimensional electron-gas (2DEG) system, realized in GaAs-${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As-GaAs heterostructures, are investigated. Both the initial and the final states involved in a tunneling transition are quantized in such a way that no free momentum component exists in the direction of the tunneling current. Under these conditions, the tunneling probability and selection rules turn out to depend strongly on the profile of the confining potential and therefore on the shape of the one-dimensional (1D) wave function. In order to illustrate this effect, we performed calculations of the transition probability for a series of 1D quantum-wire potentials of different shapes, using a model based on the transfer Hamiltonian formalism. In the case of a square-well potential, no additional resonance structures (if compared to tunneling between two 2DEG) are to be expected. For a smooth cosine-shaped potential, however, all 1D states give rise to a multitude of resonance structures in the tunneling current. From our experimental results we conclude that in our present MQW system the potential can be best described qualitatively by a harmonic-oscillator-like profile. In addition, we studied both experimentally and theoretically the temperature behavior of the resonance structures caused by the 1D states and obtained a quantization energy of about 4 meV in the quantum wires. If the thermal energy exceeds the 1D subband spacing, significant changes of the 1D potential, caused by the occupation of higher subbands and reduced screening effects, must be taken into account.