where a is the thermoelectric power, p is the electrical resistivity, and k is the thermal conductivity. The figure of merit Z is ultimately determined by the band gap Eg and the carrier concentration n and is also a function of temperature, exhibiting a maximum, Zmax, at a unique temperature. Thus, by grading Eg and n along the device, it is possible to create a gradation in Zmax. Since the device experiences a temperature gradient, one can tailor it such that Zmax is achieved for each location in the device. Thus, the tailoring corresponds to choosing materials with the best performance at a particular temperature, and then stacking those materials from one side of the device to the other. The ultimate efficiency of power conversion through thermoelectric techniques depends not only on the difference in temperature from one side of the device to the other but also on the figure of merit achieved at each location. Figure 10.6 shows the concept for a three-layer device in which Bi2Te3, PbTe, and SiGe are used. It has been

FIGURE 10.6 Figure of merit estimation for graded thermoelectric device (solid line) as function of temperature. Estimate is based on figure of merit for three compounds, also shown (dashed line). (Taken from [3].)

suggested that the efficiency of a graded thermoelectric device could be twice that of a homogeneous device. Thermionic conversion refers to the conversion of electrons emitted from a heated material into electrical current. Thermionic devices typically contain several dissimilar material joints, and thus graded materials are used to relieve thermal residual stresses. Fuel Cells

Cathodic materials for solid oxide fuel cells have diverse property requirements: high electrical and ionic conductivity, high catalytic activity for oxygen reduction, chemical compatibility with the electrolyte and interconnects, compatibility of the thermal expansion coefficient with the other components in the fuel cell, stability in air at high temperatures, and the ability to be processed into thin films. By modifying the electrode main structure so that graded multilayer configuration is used, problems of poor adherence related to thermal expansion mismatch have been minimized, and the electrochemically active portion of the interface may be widened, increasing the efficiency of the cell.

10.3.5 Optical Fibers

Graded optical fibers have been used successfully for more than a decade to optimize the multiple transmission of light signals of different wavelength [32]. Figure 10.7 illustrates the benefits of graded index fibers. The multimode graded index design can transmit the widest bandwidth of any of the designs. Common glasses are based on silica, borosilicate, or soda-lime, but multicomponent glasses are also used. Methods to manufacture glass-graded optical fibers include chemical vapor deposition and the double-crucible method. Methods to manufacture polymer-graded optical fibers are based on either vapor-phase diffusion processes or monomer reaction processes [3]. The former involves diffusing a second monomer with a lower index of refraction from the outside to the inside of the polymer fiber, followed by copolymerization. The latter involves a copolymer-ization reaction where the two different monomers used have different reaction rates. By mixing the monomers in a glass tube, and polymerizing by application of ultraviolet (UV) radiation, the rates of polymerization are different in the outside of the tube from the inside, and thus the outside of the tube ends up with a different (lower) index of refraction. Details may be found elsewhere [3].

10.3.6 Electrical and Magnetic Behavior

The concept of using graded band-gap semiconductors has existed for more than 40 years. Nongraded heterojunctions exhibit sharp energy spikes corresponding to sharp interfaces; these sharp spikes may act as charge carrier barriers. The employment of compositional gradations in heterojunctions results in smooth band-gap energy transitions, avoiding spikes. Applications have included bipolar transistors, solar cell structures, and separate confinement heterostructures

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