Particles and Quasi-Particles in the Box

All students who study physics have a basic introduction to quantum mechanics in terms of the one-dimensional ‘Particle in the Box’, in which the wavefunctions of a confined particle are found to follow simple rules. These are based around the fact that the wavefunction of the electron must go to zero at the wall of an infinite well, and to a small value if the confinement potential is not infinite. The levels are sine functions for an infinite well, and resemble sine functions as the potential height decreases, the ground state always has zero nodes with successive levels having 1,2,3…

Work in the LCN, in collaboration with the Cavendish Laboratory, Cambridge has shown that this simple picture breaks down due to the repulsion between electrons. The system used for this study was a one-dimensional (1D) quantum wire formed by surface gates which confine the two dimensional electrons within a GaAs-AlGaAs heterostructure. The quantum wire has a specialised characteristic – the transverse motion of the electrons is confined, i.e. energy takes the discrete form. The confinement is so strong that the system is essentially like a particle in the box, which is one-dimensional (1D), with each 1D energy level having a quantised conductance of 2e2/h. As the width of the quantum wire is changed, the conductance changes in steps of 2e2/h as each level passes through the Fermi energy. The experimental device allows observation of the movement of the levels as the wire width is varied.

The mutual interaction between the electrons results in a distortion of the wave function and a change in their energy, which is most pronounced in the ground state. The repulsion varies inversely with the distance between electrons and so has a rough maximum value of 1/L: comparing this with the quantisation energy of 1/L2 we see that as L (the dimension of the box) increases, so the energy of the levels is increasingly determined by the interaction between them, rather than by their spatial confinement. As the interaction increases, the electrons can no longer be described by a single particle treatment but influence each other very strongly and are termed quasi-particles. This can lead to an effective modification of the effective mass, and in some circumstances they behave as if their charge, or mass, is different to the value of a free electron.

The LCN experiment showed that as L increased, so the ground state with zero nodes crosses the first excited state, which now becomes the new ground state. Further increase in width results in the zero node state crossing the higher levels producing a complex pattern of level movement which sometimes appeared as a superposition,(crossing) or else hybridisation, ( anti-crossing).

Lifting the spin degeneracy with a magnetic field increased the complexity and seemed to produce both level superposition and hybridisation due to the successive mixing of parallel and anti-parallel spins. Normally such a hybridisation cannot occur, but it is not clear if this arises from a lack of symmetry in the 1D well. Further experiments on this topic are continuing with a view to using the electron-electron interaction for the controlled manipulation of the wavefunctions.

Reference: Sanjeev Kumar, Kalarikad J. Thomas, Luke W. Smith, Michael Pepper, Graham L. Creeth, Ian Farrer, David Ritchie, Geraint Jones, and Jonathan Griffiths, Many-body effects in a quasi-one-dimensional electron gas. Phys. Rev. B 90, 201304(R) http://journals.aps.org/prb/abstract/10.1103/PhysRevB.90.201304

Figure:

For a box with infinitely high enclosing potential walls these have the form sine nπx/L, where L is the dimension of the box and n is the level index, I,2,3… where the ground state is 1 (Fig. 1). If the walls are not infinitely high or the confining potential is a different shape, such as that for a harmonic oscillator, the wavefunctions are modified but the number of nodes of the wavefunction increases with increasing energy. For the case of infinitely high barriers the energy varies as 1/L2. Consequently increasing L results in a decrease in the energy of the levels and they become closer.

The experiments showed that as the channel is widened so the ground (1) and first excited state (2) moved closer together and eventually crossed, as they did so the conductance jumped  by twice the normal 2e2/h (Fig. 3). After the crossing, 2 now becomes the new ground state, as shown in Fig. 2.

Other contributors: 
Kalarikad J. Thomas; Luke W. Smith; Ian Farrer; David Ritchie; Geraint Jones; Jonathan Griffiths