THE UNIVERSITY OF NEW SOUTH WALES. SCHOOL OF PHYSICS FINAL EXAMINAnON JUNE/JULY 2008

Transcription

1 THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF PHYSICS FINAL EXAMINAnON JUNE/JULY 2008 PHYS3080 Solid State Physics Time Allowed - 2 hours Total number ofquestions - 5 Answer ALL questions All questions are NOT of equal value This paper may be retained by the candidate Candidates may not bring their own calculators The following materials will be provided by the Enrolment and Assessment Section: Calculators Answers must be written in ink. Except where they are expressly required, pencils may only be used for drawing, sketching or graphical work

2 Data and Formula Sheet. dq dt Q=-=KAdt dx K =.!..vlc 3 F = q(vxb) I = nave et v=--e me J=aE a = nel-l N A = 6.023x10 26 (kg.mol!1 h = 6.63xI0-34 Js n= 1.05xl0-34 Js... [2e (V V. (») s.] V MOO nhv J=Josm - ot+-sm\oot +uo ' 0 =--=n oo 2e 2e nphonon ~ exp(- edit) V MOO o =-- = nv<1> 2e v = 2~(T)/e Aphonon ~ exp(+ edit) k = (3Jt2 N)I/3, "= 11 2 (3Jt 2 N)2/3 e = 1.6xl0-19 C F <OF ~tlh ~ nl2 v 2m V j=josin(2ev t+6 0 ) (ae) j=josin6, jo= ennskexp(_kd) (de) n. m, h -15 _I oo 2eV - = <1>0 = 2.07xlO VHz v = - =-- 2e 2Jt h

3 Question 1 (22 Marks) In a solid the mean energy per oscillator (or phonon energy) is given by _ 1 no) c = -no) +...,--::--::--=- 2 (e/iw/k B T-1) (i) (ii) (iii) (iv) (v) (vi) Explain briefly what the factors no) and (e/iw/k~t -1) in the second term represent. (4 marks) What type ofquantum statistics do phonons obey and why? (2 marks) Use the expression for E to show that, in thermal equilibrium at temperature T, the energy of a sufficiently long wavelength mode is kbt. (4 marks) Estimate the number ofmodes that will be excited at temperatures T«So' (5 marks) Thus show that for T«So the heat capacity is given approximately by Cv ~ NkB(TjSO)3. (5 marks) Comment briefly (l-2lines) on the agreement between the form of Cv in (vi) and experimental data for Cv at low temperatures. (2 marks)

4 Question 2 (20 Marks) (a) The Fermi-Dirac distribution function 1 f(e) = ---;--(E----;-EJ 1+exp F kbt gives the state occupation probability for electrons in a free electron metal. (i) Define all symbols in this expression. (2 marks) (ii) The total number of occupied electron states, N(E), in the energy range E- E+ de is given by the product ofthe occupation probability f( E) with the density states function g(e). Sketch the form ofthe three quantities N(E), fee), g(e) for a simple free electron metal. Indicate the situation for T = OK and T F»T»OK, where T F is the Fermi temperature. (Put both curves on one plot or use a separate plot for each, as you prefer.) (6 marks) (b) Give a concise explanation ofthe reason the observed electronic (i.e. the conduction electrons) contribution to the heat capacity of a metal is only a small fraction ofthat expected classically. Include a sketch illustrating your answer. (4 marks) (c) The Fermi energy at OK is given by E FO = ~~Jl?nr3.. 2m (i) Calculate EF,o for aluminiuin metal (AI is trivalent with density PAl = 2.70x10 3 kgm -3 and atomic mass kg(kmoler\ give your answer in electron volts. (4 marks) (ii) Determine the Fermi velocity and the de Broglie wavelength of an electron moving in aluminium at the Fermi energy. (2 marks) (d) A particular sample ofaluminium has drift velocity Vd = 2.16 ms-i in an electric field E = 500 Vm- I. Estimate (i) the electron'mobility; (ii) the relaxation (scattering) time. (2 marks)

5 Question 3 (18 Marks) For electrons moving in the energy band ofa one-dimensional chain of atoms we have the two important results do) 1 de. (1) V = - = - - velocity of an electron wavepacket dk h dk and (2) 6E = -ee6x motion ofelectron wavepacket in a uniform, static electric field E. (i) (ii) (iii) (iv) (v) Use results (1) and (2) to show that h dk = -ee; show all your working (4 marks) dt What is the quantity hk? Explain concisely, 3-4 lines only. (2 marks) Take the time derivative of(1) together with the result ofpart (i) above to show that d 2 E m* = 11? -2' (6 marks), / dk What is the quantity m:? Explain briefly, 3-4 lines only. (4 marks) What is the significance of m* for semiconductors and why is m*important e e technologically? (2 marks)..."::'"

6 Question 4 (22 Marks) (a) The diagram at right shows schematically the energy bands of an n-type doped semiconductor. This semiconductor has electron effective mass 0.067m e and dielectric constant Assuming the bound donor electrons to be in Bohr orbits (hydrogenic impurity model), (i) (ii) calculate the first donor ionization energy within the Bohr picture. (4 marks) Given that the semiconductor exhibits a sharp increase in electrical conductivity when illuminated with radiation ofwavelength A:S 860 nmfind E d referenced to the top ofthe valence band. (4 marks) -k E k (b) The valence band shown in the figure above may be taken to be parabolic and is described by E = _10-37 k 2 J near to the band edge. An empty state at k = 10 9 k x m-i provides a mobile hole. For the mobile hole calculate, paying particular attention to the sign, (i) the effective mass, (4 marks) (ii) the energy, (4 marks) (iii) the momentum, (3 marks) (iv) the velocity. (3 marks) t.1 ~,.j

7 Question 5 (18 Marks) Write briefnotes (about 2-3 pages for each, including diagrams and any equations you include, but no more than this) on two only from the list offour topics given below. Use simple diagrams and/or sketch graphs to illustrate your answers where appropriate ensuring that you label these and refer to them in your account. (a) Crystalline solids. Your answer must discuss (but is not limited to) lattice and basis, bravais lattices and symmetry, unit cells, packing fraction. (9 marks) (b) The BCS theory of superconductivity. Your account must include a description ofthe isotope effect, electron pairing and energy considerations, virtual phonon exchange and the superconducting energy gap. (9 marks) (c) Effect of dopant concentration and temperature on the electrical conductivity of semiconductor materials. The relevant sketch graphs must be included in your answer. (9 marks) (d) The Josephson effects. You may use information from the data sheet and refer to the diagram included below. (9 marks) 4,-.. <r:... ' 8 '-' o """"""""" ':... r..._-----" o 1 2 V(mV) 3 4