Physics of Thermal Wave NDE of Semiconductor Materials and Devices
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sample, for example, will be deflected as it passes through the heated region above the surface.7"9 This mirage effect can also be observed within the sample by directing a transmitting probe through the heated region beneath the surface.10 Likewise, using a probe beam directed onto the sample surface one can observe a modulation in reflection, transmission, or scattering.11,12 A related noncontact method is the photothermal measurement of infrared radiation emitted from the material's heated region. 1315 Note that with all these detection methods, thermal wave measurements can be, and most often are, done in air and at room temperature. All the above thermal wave detection schemes involve an average of the thermal wave and, depending on the particular application, offer their own set of advantages and disadvantages. As a surface-sensitive method, we have found modulated reflectance to be the most sensitive, particularly in the study of semiconductors where surface (and interface) characterization is extremely important. Furthermore, it affords the highest spatial resolution attainable and as mentioned above permits measurements on both patterned product wafers and test wafers. The following briefly describes the physics of thermal waves and its applications to semiconductors.
Basic Theory We begin with a qualitative description of what happens when a laser beam is incident on a semiconductor. If the energy per photon £ exceeds the band gap energy Eg, then electrons will be excited from the valence band to an energy E—Eg above the conduction band edge. These photoexcited free carriers, within a fraction of a picosecond, give up this excess energy to the lattice through nonradiative transitions to the unoccupied states near the bottom of the conduction band. This is essentially instantaneous relative to the modulation times used in thermal wave measurements. After a much longer and significant time ranging from submicroseconds to milliseconds, the photoexcited carriers recombine with holes in the valence band, giving up their remaining Eg either to the lattice in an indirect gap semiconductor such as silicon or to an emitted photon (luminescence) in a direct gap semiconductor such as gallium arsenide. Prior to this recombination thus exists a plasma of electrons and holes whose density is governed by diffusion in a manner analogous to the flow of heat from a thermal source. Thus, if an incident laser beam is intensity modulated, one will observe, in addition to the thermal wave, a modulated plasma density whose spatial profile is that of a critically damped wave called a plasma wave.16 The mathematical description of these critical wave phenomena begins with the wave equations for the plasma density N(r,z) and temperature T(r,z): V2N + p2N = -f(i,z)Q/DE V2T + q2T =
(1)
-f(i,z)Q(E-Eg)/KE -NES/KT
- sNEg8(z)/K
(2)
where p is the plasma wave vector defined by p2 = i(a)T + i)IDr = 2i/fij -III2
(3)
and q is the thermal wave vector defined by q2 = icopC/K = 2i'//nr.
(4)
In the above equations, D is the ambipo
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