Solid State Physics Simulations

• Graham Keeler of University of Salford
• Roger Rollins of Ohio University
• Steven Spicklemire of University of Indianapolis
• Ian Johnston, University of Sydney, (idj@suphys.physics.su.oz.au)

• LATCE1D (Wavefunctions for a One-Dimensional Lattice), written by Ian Johnston, and included here, is described under the Quantum Mechanics heading.

• SOLIDLAB (Build Your Own Solid State Devices), written by Steven Spicklemire, is a simulation of a semiconductor device. The device can be "drawn" by the user, and the characteristics of the device adjusted by the user during the simulation. The user can see how charge density, current density and electric potential vary throughout the device during its operation.

• LCAOWORK (Wavefunctions in the LCAO Approximation), written by Steven Spicklemire, is a simulation of the interaction of two dimensional atoms within small atomic clusters. The atoms can be adjusted and moved around while their quantum mechanical wave functions are calculated in real time. The student can investigate the dependence of various properties of these atomic clusters on the properties of individual atoms, and the geometric arrangement of the atoms within the cluster.

• PHONON (Phonons and Density of States), written by Graham Keeler, calculates and displays phonon dispersion curves and the density of states for a number of different three-dimensional cubic crystal structures. The displays of the dispersion curves show realistic curves and allow the user to study the effect of changing the interatomic forces between nearest and further neighbor atoms and, for diatomic crystal structures, changing the ratio of the atomic masses. The density of states calculation shows how the complex shape of real densities of states are built up from simpler distributions for each mode of polarization, and enables the user to match the features of the distribution to corresponding features on the dispersion curves. In order to help with visualization of the crystal lattices involved, the program also shows 3-dimensional projections of the different crystal structures.

• SPHEAT (Calculation of Specific Heat), written by Graham Keeler, calculates and displays the temperature variation of the lattice specific heat for a number of different theoretical models, including the Einstein model and the Debye mode. It also makes the calculation for a computer simulation of a realistic density of states, in which the user can vary the important parameters of the crystal, including those affecting the density of states. The program can display the results for a small region near the origin, and as a T cubed plot to enable the user to investigate the low temperature limit of the specific heat, or in the form of the equivalent Debye Temperature to enhance a study of the deviations from the Debye model. The Schottky specific heat anomaly can also be investigated.

• BANDS (Band Structure), written by Roger Rollins, calculates and displays, for easy comparison, the energy eigenstates and corresponding wavefunctions for an electron in a 1-D symmetric V(x) = V(-x) periodic potential of arbitrary shape and of strength V0. The method used is based on an exact, non-perturbative approach so that the energy dispersion curves and band gaps can be obtained for large V0. Wavefunctions can be displayed, and compared with one another, by clicking the mouse on the desired states on the energy dispersion curve. Changes in band structure can be followed as changes are made in the shape of the potential. The variation of the band gaps with V0 is calculated and compared with the two opposite limits of very weak V0 (perturbation method) and very strong V0 (isolated atom). Even the experienced Condensed Matter Researcher may be surprised by some of the results! Open ended class discussions can result from the interesting physics found in these conceptually simple model calculations.

• PACKET (Electron Wave Packet in a 1-D Lattice), written by Roger Rollins, shows a live animation, calculated in real time, demonstrating how an electron wave packet in a metal or semi-conducting crystal move under the influence of external forces. The time dependent Schrödinger equation is solved in a tight binding approximation, including the external force terms, and the motion of the wave packet is obtained directly. The main objective of the simulation is to show that an electron wave packet formed from states with energies near the top of an energy band are accelerated in a direction opposite to the direction of the external force; they have a negative effective mass! The simulation deals with motion in a one-dimensional lattice but the concepts are applicable to the full three dimensional motion of an electron in a real crystal. Numerical experiments on the motion of the packet reveal interesting physics such as: Periodic motion of a packet under constant applied force (Bloch oscillations), when does the usual semiclassical model fail, what happens to the dynamics of the packet when placed in a superlattice with lattice constant twice that of the original lattice.