Test02_solution - Test 2 Solution Problem 1 Admixing Al to...

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Test # 2 Solution Problem # 1 Admixing Al to Ag creates an Ag 1- x Al x alloy (in which Al atoms substitute for Ag atoms) and leads to an increase of the Fermi energy. Using a free electron model and assuming that Al is trivalent while Ag is monovalent calculate the relative concentration, x , of Al atoms which is required to increase the Fermi energy by 1eV. Take into account that the Ag crystal has electron concentration n = 5.85 10 22 cm -3 . Since Al is trivalent and Ag is monovalent, the electron concentration in Ag 1- x Al x alloy is 1 (1 ) 3 xx Ag Al nn x =−+ n x , where n is the electron concentration in Ag. The Fermi energy in Ag is determined by () 2 2/3 2 3 5.49 2 Ag F En m π == = e V . The Fermi energy of the Ag 1- x Al x alloy is therefore 1 6.49 Ag Al F E eV = . Taking into account that 1 1 2 2 3 2 Ag Al FA g m = = A l , we find 1 1 3/2 22 3 22 21 7.52 10 3 Ag Al F Ag Al mE nc m ⎛⎞ ⎜⎟ ⎝⎠ = . Finally, we have 1 0.14 2 Ag Al x n =≈ .
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Problem # 2 Suppose that the electronic structure of a solid that represents a two-dimensional square lattice of divalent atoms can be described using a nearly free electron model with a weak crystal potential such that 2 2 2 U ma π ⎛⎞ ⎜⎟ ⎝⎠ G = ± , where a is the lattice parameter. Is this solid a metal, an insulator or a semiconductor? Explain why. Since the number of electrons per atom is even, the solid can be either an insulator (semiconductor) or a metal (semimetal). This depends on whether or not the bands overlap in energy. If the bands overlap in energy, then instead of one filled band giving an insulator (semiconductor), we can have two partly filled bands giving a metal (semimetal, if bands just touch).
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This note was uploaded on 09/21/2011 for the course PHYSICS 101 taught by Professor Wormer during the Spring '08 term at Aarhus Universitet.

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Test02_solution - Test 2 Solution Problem 1 Admixing Al to...

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