UNIVERSITY COLLEGE LONDON University of London EXAMINATION FOR INTERNAL STUDENTS For the following qualifications.'- B. Sc. M. Sci. Physics 3C25: Solid State Physics COURSE CODE - PHYS3C25 UNIT VALUE 0.50 DATE : 30-APR-02 TIME : 10.00 TIME ALLOWED 2 hours 30 minutes 02-C1088-3-100 2002 University of London TURN OVER
Answer any THREE QUESTIONS The numbers in square brackets in the right-hand margin indicate the provisional allocation of maximum marks per sub-section of a question. Mass of the electron m Planck's constant h elementary charge e one electron volt Boltzmann's constant ks permittivity of vacuum e0 = 9.11x 10-31kg = 27r x 1.05 x 10-~Js = 1.60 x 10-19C = 1.60 10-19J = 1.38 10 -z3 JK -1 = 8.85 x 10-12Fm-1 [Part marks]. (a) Sketch the unit cells of the five two dimensional Bravais lattices, giving the essential geometric characteristics. (b) The analogue in two dimensions of a face centred cubic lattice in three dimensions would be a 'face centred square lattice': a square unit cell with lattice points at the corners and one in the centre. Why is this not considered as a distinct two dimensional Bravals lattice? (c) State the coordination numbers and specify the conventional unit cells of the three dimensional face centred cubic and hexagonal close packed structures. Compare the pattern of stacking of close packed planes in these two structures. (d) A boundary is formed between a simple cubic structure and a face centred cubic structure along their (001) faces, with the [100] direction in one parallel to the [110] direction in the other. The face centred cubic structure may be considered rigid with a lattice parameter of 0.14 nm. The simple cubic structure is compressible and has a lattice parameter of 0.1 nm in the absence of external stress. Determine the compressive strain that the simple cubic structure has to suffer in order to create an exact atomic match at the boundary. [5] [3] [7] [5] PLEASE TURN OVER 2
2. (a) Consider a chain of ions -- -X+Y-X+Y-X+Y-X+Y -..- of infinite length with nearest neighbour separation a. If all ions in the chain interact through a pairwise Coulomb potential, show that the total electrostatic energy per ion is e21n2 E= 4~re0a (b) (c) (You may assume that v, oo t 1~n-1 -In 2). Ions additionally interact through a hard core repulsion. What is the physical origin of this interaction? If we model the effects of hard core repulsion using a nearest neighbour pair potential B/a 12, where B is a constant, show that the equilibrium ionic separation is given by (48~_e0B '~ 1/11 a0 = \ e21n 2 } (d) Starting from the equilibrium position, one ion in the chain is displaced towards a neighbour by a distance u, while all others remain at their equilibrium positions. The change in total potential energy, in the harmonic approximation, is written Ku2/2. If we write the pairwise interaction potential for n th nearest neighbours as,(r), show that the effective spring constant K may be written oo d2 n( )l K= 2 ~ dr 2 n= 1 r=r~ where rn = nao is the equilibrium separation of the n th nearest neighbours. (e) Show that K takes the form [31 [41 K-- 2~eoa03 131n2+2~ ( )", [5} CONTINUED 3
. (a) Explain what is meant by a dislocation, and describe what is represented by the Burgers vector. (b) A dislocation loop and its Burgers vector Ue on a given slip plane in a material. Using diagrams, indicate sections of the loop where the dislocation takes screw and edge character, and describe the local geometry near the dislocation core in each of these sections. (c) What is a mixed dislocation? (d) Describe and explain what can happen when the material in which the loop lies is subjected to a gradually increasing shear stress. What happens when the stress is removed? How does the behaviour depend on the direction of the stress? (e) How would the behaviour of a dislocation under applied stress be affected by impurity atoms dispersed in the crystal? I4] [el [2] I5] [3] PLEASE TURN OVER 4
. (a) The density of electronic states per unit volume for a metal according to the free electron model is given by D~E(E) -- DoE 1/2 where E is the electron energy relative to the bottom of the band and Do is a constant. At T -- 0 K the density of states is fully occupied up to the Fermi energy EF and empty for E > EF. Show that 3Nm DFE( EF) - -~F where Nm is the density of free electrons in the metal. (b) Sketch the Fermi-Dirac function for a temperature T well below the Fermi temperature TF = EF/kB, where ku is Boltzmann's constant. Hence estimate the increase in electronic energy per unit volume associated with an increase in temperature from zero to T. Your calculation need only be approximate. (c) Hence show that the electronic heat capacity per unit volume in a metal is roughly given by [5] where Km is a constant of order unity. (d) We can approximate the number of electrons per,,nit volume promoted from the valence band to the conduction band of an intrinsic semiconductor by the expression Ns = A exp(--ej(2kbt)) where E 9 is the band gap energy and A is a constant. Show that the electronic heat capacity per unit volume in this material is roughly given by C~ = KsNsk, ( E9 ~2 \ kbt ] where Ks is another constant of order unity. (e) Estimate the electronic heat capacities per,,nit volume at T : 293 K for a typical metal (EF ---- 10 ev and Nm : 102s m -3) and a typical intrinsic semiconductor (E 9 : 1 ev and Ns : 1019 m-3). Contrast these values with a typical heat capacity per unit volume due to structural vibrations at this temperature. [31 [41 CONTINUED 5
. (a) Describe the electronic densities of states in an n-type and in a p-type semiconductor. Explain the origin and describe the nature of any states located in the band gap. (b) A junction is formed between n- and p-type semiconductors. Explain what is meant by a depletion zone, and give an account of the events that lead to its formation. (c) The created depletion zone has a width dn on the n-type side of the junction and width dp on the p-type side. Assuming the charge density p in the depletion zone is equal to +end and -en,~ on the n-type and p-type sides, respectively, where Nd and Na are the donor and acceptor concentrations, and zero elsewhere, solve Poisson's equation [6] 15] d2 _ P dx 2 660 where e is the relative permittivity of the semiconductor, to show that the difference in electrostatic potential across the junction is 2eeo (d) When the junction is reverse biassed, A is increased and the depletion zone is extended. Name a device in which this effect is exploited. [7] [2] END OF PAPER