Problem 4: Read Appendix 2 below (Sec. 1.4.1 of Kasap) and then solve. A metallic back contact is applied to the CdTe solar cell of Problem 1 using a set up similar to that described in Figure 1.74 (b) on the next page. To form the metallic back contact, two evaporation sources are used, Cu and Au. An initial 3 nm layer of Cu is deposited first and then 30 nm of Au is deposited. After these depositions, the sample is annealed at 150°C for about 45 min to diffuse the Cu into the CdTe; the Au layer remains intact at the back of the CdTe solar cell to serve as the final back contact. The following problem describes the deposition of Au onto a Si chip; however, the process is similar for depositing Cu/Au on the back of a CdTe solar cell

Vacuum deposition Consider air as composed of nitrogen molecules N d What is the concentration n (number of molecules per unit volume) of Na molecules at I atm and 27-c? b. Estimate the mean seperation between the N2 molecules. c. Assume each molecule has a finite size that can be represented by a sphere of radius r Also as- sume that e is the mean free path, defined as the mean distance a molecule travels before col- liding with another molecule, as illustrated in Figure 1.74a. If we consider the motion of one N molecule, with all the others stationary, it is apparent that if the path of the crosses the cross- tance between collisions, there must be at least one stationary traveling molecule π(2r)2, there will be a collision. Since t is the mean dis- sectional area S molecule within the volume S s-π (202 (a) A molecule moving with a velocity v travels a mean dislance Any molecule with e between collisions. Since the center in S gets hit. collision cross-sectional area is S Molecule in the volume Se there must be at least one molecule. Consequently, nse)-1. Molecule. Semiconductor Metal film Evaporated metal atoms Hot Vacuum filament Vacuum pump (b) Vacuum deposition of metal electrodes by thermal evaporation. Figure 1.74 is used to deposit the metal electrode. at Bell Telephone Lobs. A vacuum evaporation I SOURCE: Bell Telephone laboratories, courtesy AlP Emilio Segre Visual Archives as shown in Figure 1.74. Since n is the concentration, we must have n(SE)-1 or t m /r4rn). However, this must be corrected for the fact that all the molecules are in motion. which only introduces a numerical factor, so that Assuming a radius r of 0.1 nm, calculate the mean free path of N2 molecules between collisions at 27 C and I atm. d. Assume that an Au film is to be deposited onto the surface of a Si chip to form metallic interconneo- tions between various devices. The deposition process is generally carried out in a vacuum chamber and involves the condensation of Au atoms from the vapor phase onto the chip surface. In one pro- cedure, a gold wire is wrapped around a tungsten filament, which is heated by passing a large current through the filament (analogous to the heating of the filament in a light bulb) as depicted in Fig ure 1,74b. The Au wire melts and wets the filament, but as the temperature of the filament increases the gold evaporates to form a vapor. Au atoms from this vapor then condense onto the chip surface, to solidify and form the metallic connections. Suppose that the source (filument)-to-substrate (chip) distance L is 10 cm. Unless the mean free path of air molecules is much longer than L, collisions between the metal atoms and air molecules will prevent the deposition of the Au onto the chip sur- face. Taking the mean free path é to be 100L, what should be the pressure inside the vacuum system? SIP a (Assume the same r for Au atoms.)

Many commercially sold vacuum systems use units of Torr for pressure; please express your answer to part (d) both in SI units and in Torr. It is helpful to look up and be familiar with the conversion factor. Also Given that the Cu diffuses into the CdTe and has a uniform concentration throughout much of the CdTe thickness after the annealing step at 150°C for 45 min, please suggest why the Cu layer may be needed for a high efficiency CdTe solar cell. (4) e. Notes: Wrapping a Au wire around a filament is quite primitive; most modern systems use a heated “evaporation boat” or a “crucible” into which Au wire or pellets are added. A two-source system is needed to deposit Cu and then Au. Finally, there are two reason for wanting to achieve a low “base pressure” in the vacuum system. The evaporating atoms may be scattered by or react with the surrounding gas molecules, but also the surrounding gas molecules may impinge on the growing film surface and react with it. Cu atoms and surfaces are quite reactive whereas Au atoms are much less so.

Appendix 2 14.1 MEAN KINETIC ENERGY AND TEMPERATURE The kinetic molecular theory of matter is a classical theory that can explain such seem- ingly diverse topics as the pressure of a gas, the heat capacity of metals, the average speed of electrons in a semiconductor, and electrical noise in resistors, among many interesting phenomena. We start with the kinetic molecular theory of gases, which considers a collection of gas molecules in a container and applies the classical equa- tions of motion from elementary mechanics to these molecules. We assume that the collisions between the gas molecules and the walls of the container result in the gas pressure P. Newton’s second law, dp/dt force, where p mv is the momentum, is used to relate the pressure P (force per unit area) to the mean square velocity v2, and the number of molecules per unit volume N/V. The result can be stated simply as Kinetic [1.101 theory for gdses where m is the mass of the gas molecule. Comparing this theoretical derivation with the experimental observation that where NA is Avogadro’s number and R is the gas constant, we can relate the mearn kinetic energy of the molecules to the temperature. Our objective is to derive Equa- tion 1.10; to do so, we make the following assumptions: 1. The molecules are in constant random motion. Since we are considering a large number of molecules, perhaps 1020 m3, there are as many molecules traveling in one direction as in any other direction, so the center of mass of the gas is at rest. l 7 The mathematics and a more rigorous description may be found in the hextbook’s CD,