FYS-1350 Nanofysiikka TTY / Syksy 2014 Laskuharjoitus 6 Tehtävä 1 (Exercise 6.14). We can look at the band structure of an element to get an idea of how many electrons per atom participate in conduction. We can also determine this number based on the free electron density, the atomic mass, and the mass density of the material. a) Calculate the free electron density of lithium. The Fermi energy is 4.72 eV. 226 ◾ Nanotechnology b) Derive a formula for the number of atoms per unit volume. c) Calculate the number of atoms per unit volume of lithium (535 kg/m3, 6.9 6.22 Determine the temperature at which a cube of gold 3 nm g/mol) using the formula that you derived in part (b). on a side becomes d) Determine of free electrons per lithium atom. quantum confined the (vianumber the metal-to-insulator transition). Gold has a density e) What is the electron configuration for Li? Explain whether your answer to (d) 3 of 19,300 kg/m and its atomic mass is 197 g/mol. sense according to this configuration. The free electron(s) of Li is (are) in 6.23 Why is a makes physical model different from the one used for conductors needed to which sublevel? 6.24 6.25 6.26 6.27 determine the conditions of quantum confinement for semiconductors? What name is given to the distance separating an electron–hole pair in a Tehtävä 2 (Exercise 6.22). semiconductor? Determine temperature a cubedot of gold 3 nm on a side becomes True or false?the The band gap atofwhich a quantum is directly proportional to itsquantum confined (via the metal-to-insulator transition). Gold has a density of 19,300 kg/m3 and size. its atomic mass is 197 g/mol. How must the boundary conditions of a potential energy well change to allow forTehtävä tunneling? 3 (Exercise 6.27). wave function,ψ(x), ψ(x),ofofananelectron electronin in aa potential potential well width, L, TheThe wave function, well of of finite finitedepth depthand and is given width, L, is given by ψ(x) = AeCx for x ≤ 0 ψ(x) = F sin(kx) + G cos(kx) for 0 ≤ x ≤ L ψ(x) = Be−Cx for x ≥ L a. Given F = G F= =5,Gk == 5, 1, kand = 2,Cdetermine A. A. a) Given = 1,Cand = 2, determine b) What is ψ (0)? b. What is ψ (0)? c) Given = ψdetermine (0), determine values andBBwhether whether this c. Given ψ(L) =ψ(L) ψ (0), the the values of of L Land thiswave wavefunction corresponds to the ground energy state. See the attached Figure (below). function corresponds to the ground energy state. See Figure 6.22. d) Given ψ(L) = ψ (0), determine the values of L and B if this wave function corresponds d. Given toψ(L) = ψ lowest (0), determine the values of L and B if this wave function the third energy state. corresponds to the third lowest state. e) Use a spreadsheet program to energy graph the wave function from part (d) over the range −2 ≤ x ≤ (L + 2). Indicate the location and value L along the x-axis. e. Use a spreadsheet program to graph the waveoffunction from part (d) over the range −2 ≤ x ≤ (L + 2). Indicate the location and value of L along the x-axis. 6.28 About what size are the smallest features we can see with a light microscope? 6.29 What variable does the STM attempt to keep constant as it scans? ψ(x) 8 7 the range −2 ≤ x ≤ (L + 2). Indicate the location and value of L along the x-axis. 6.28 About what size are the smallest features we can see with a light microscope? 6.29 What variable does the STM attempt to keep constant as it scans? ψ(x) 8 7 6 5 4 3 2 1 0 –1 FIGURE 6.22 L x Homework 6.27c. Tehtävä Exercise 4. In the book and in the lecture notes, the density of states was given for a 3D system. Reconsider the case but now for a 2D system (quantum well): what is the density of states for a 2D system of this kind? This exercise goes slightly beyond the scope of the material discussed in the book, but please encourage yourself to consider it anyway. It is called research, which often means that one should step out from the comfort zone.

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