Transport in Disordered Systems

by Cristine Villagonzalo

Customarily, the study of solids has largely been confined to crystalline materials. However, the majority of commonly encountered solids are non-crystalline. Also, a real crystalline solid is never absolutely pure. It may contain impurity atoms or may have misplaced or missing ions or atoms. Hence, it is interesting to study solids from a non-crystalline point of view.

A well known model of a disordered system is that of Anderson. It is obtained from a three-dimensional (3D) ideal crystal lattice with the requirement that the potential energy varies randomly from one lattice site to another. For this system at strong enough disorder in the absence of interaction, with no magnetic field present, the electrical conductivity vanishes at the absolute zero temperature. This Anderson metal-insulator transition (MIT) illustrates that the difference between disordered and crystalline metallic materials can be seen dramatically in their transport properties at very low temperatures.

This disorder-driven MIT, as simple as it may appear, has not yet been fully understood. A useful concept is the introduction of a mobility edge, i.e., the value of energy at which the Anderson transition occurs. For states with energies less than or equal to mobility edge, the electrical conductivity is zero, whereas for states with energies larger than the mobility edge the conductivity is found to obey a power law behavior with respect to the energy. However, the exact power law behavior is still being debated. Even more unclear is the behavior of other transport properties as they approach the Anderson transition. Studying the electronic and thermal transport properties of disordered systems, therefore, will give us a better concept on why there are metallic and insulating states of matter and how the transition between these two states occur.

A higher motivation in studying the thermoelectric transport properties is that an alternative, environmental friendly technology for power generation can be found by applying the Seebeck effect. It is the phenomenon that in a long thin bar subjected to a temperature gradient induces an electric field across the bar. If we can only apply the Seebeck effect then we would be able to convert waste heat, such as automobile exhaust and engine heat, into useful power.

Today the Seebeck effect has been applied in devices for measuring temperature. However, its use in power generation has been limited due to inefficiency of devices. Thus, search is on for materials having good thermoelectric properties. One such characteristic is for the material to have a high thermoelectric power. It is the proportionality constant between the temperature gradient and the electric field it induces. What is the thermoelectric power in a disordered system? How does a temperature gradient affect the transport properties in the region near the metal-insulator transition? How the thermoelectric power behaves at low temperature near the Anderson transition has been disputed. Theoretical studies have either claimed that it diverges or it remains a constant as the MIT is approached from the metallic side.

We have studied the temperature dependence of the thermoelectric power in a 3D Anderson model using a linear response formulation. Our results show that thermoelectric power is a constant at the transition and takes on a large value, that is, of the order of a hundred microvolts per Kelvin. However, measurements on doped-semiconductors and on amorphous alloys give values that are at least one order magnitude lower than those predicted in theory. Furthermore, the experiments have shown that the thermoelectric power changes sign. Since the sign of the thermoelectric power depends on the sign of the charge carrier, this then indicates the change in the charge carriers of the system from electrons to holes or vice versa. Why this happens remains a challenging open problem.

Work is in progress for numerically calculating the thermoelectric power using a microscopic approach. Our investigations also include the thermal conductivity and the Lorenz number.



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