Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn (2014) SOLID STATE PHYSICS 2 HOURS The pper is divided into 5 questions. Answer compulsory question 1, which is mrked out of 20. Answer ny two out of the optionl questions 2-5, ech of which is mrked out of 15. The brekdown on the right hnd-side of the pper is ment s guide to the mrks tht cn be obtined from ech prt. Plese clerly indicte the question numbers on which you would like to be exmined on the front cover of your nswer book. Cross through ny work tht you do not wish to be exmined. 1 TURN OVER
Question 1 COMPULSORY ) Describe why bnd gps rise for electrons in periodic crystl lttice. [2] b) Show tht the Fermi wvevector k F of conduction electrons in terms of electron density n is given by the expression k F = (3 2 n) 1/3. [2] c) A metl hs simple cubic structure nd lttice constnt of 3 x 10-10 m. Ech tom contributes one electron to conduction. Using the result for the Fermi wvevector from question b), deduce the minimum seprtion in wvevector spce between filled electron stte nd the edge of the first Brillouin zone. [4] d) Wht informtion bout the current crriers in semiconductors is provided by cyclotron resonnce nd Hll mesurements? Describe which physicl prmeters my be mesured to obtin this informtion. [5] e) Explin wht is ment by dimgnetism nd describe the physicl principle which mkes mterils dimgnetic. [2] f) The ion Tm 3+ hs the electronic configurtion 4f 12 5s 2 5p 6. Apply Hund s rules to clculte the quntum numbers J, L nd S. Using the expression for the Lndé g-fctor g = ( 3 + S(S+1) L(L+1) ), 2 2J(J+1) clculte the bsolute mgnitude of the mgnetic dipole moment of the ion in units of the Bohr mgneton. [3] g) Explin why most metls reflect light in the visible region nd re trnsprent to light t higher frequencies. [2] 2 CONTINUED
Question 2 OPTIONAL ) Derive n expression for the effective mss of electrons in terms of their energy ginst wvevector dispersion reltion. [3] b) Using the result of ), nd dispersion curve for electrons cross first Brillouin zone, explin how sttes with pprent negtive mss my rise. [2] c) Explin how the negtive msses of prt b) my be interpreted in terms of prticles with positive mss nd positive chrge. [2] d) An electric field of 10 2 V/m is pplied to metl smple. The resistivity of the smple is 10-8 m nd the electron density is 3 x 10 28 m -3. Clculte the resulting chnge in wvevector of the Fermi sphere of electrons due to the electric field. You my tke the effective mss of the chrge crriers to be equl to the free electron mss. [4] e) Using sketch of Fermi circle in two dimensions, explin which types of scttering process re effective in relxing momentum gined from n pplied electric field. [2] f) Sketch typicl vrition of electron scttering time with temperture in metls, identifying the processes which re likely to dominte t low nd high tempertures. [2] 3 TURN OVER
Question 3 OPTIONAL ) Explin the concept of donors nd cceptors in semiconductors. Give one exmple of donors nd cceptors for silicon. [3] b) A smple of GAs is hevily doped so tht the Fermi level is 300 mev bove the minimum of the conduction bnd. Clculte the wvelength t which the onset of opticl bsorption occurs in this smple t 300 K. The electron (m e ) nd hole (m h ) effective msses in GAs re 0.067 m 0 nd 0.45 m 0, respectively, where m 0 is the free electron mss. The bnd gp of GAs is 1.43 ev t 300 K. [5] c) Explin why the binding energy of Wnnier-Mott exciton is smller thn the binding energy of n electron to donor in the sme mteril. How do the Bohr rdii of the exciton nd the electron on donor compre? [4] d) The mgnetistion of silicon M is 1.5 10-3 A/m when the pplied vlue of the H- field inside the mteril is 500 A/m. Clculte the mgnetic susceptibility of silicon nd the resultnt B-field. [3] 4 CONTINUED
Question 4 OPTIONAL ) Derive the following expression for hole concentrtion n h in the vlence bnd s function of temperture T in n undoped semiconductor: n h = 2 ( 2πm h 3/2 k B T h 2 ) exp ( E F k B T ). You my use the following expressions for the hole density of sttes g h per unit volume nd the electron Fermi-Dirc distribution f(e): g h = 4π(2m h ) 3 2 h 3 E, 1 f(e) = 1 + exp ( E E. F k B T ) Here E F is the Fermi level nd E is the electron energy, which is set to zero for electrons in the vlence bnd with zero momentum. In your derivtion you my ssume tht E E F >>k B T nd use the following integrl x 1/2 e x dx = π/2. [7] b) The intrinsic crrier concentrtion in silicon t 300 K is 2 10 16 m -3. The vlue of the bnd gp in Si is 1.1 ev. Estimte the electron concentrtion t 350 K. [3] c) Briefly explin the two mechnisms by which ferromgnet is mgnetised long the direction of n externl mgnetic field. Wht is the physicl mechnism which is responsible for the spontneous lignment between djcent mgnetic dipole moments in ferromgnet. [3] d) The energy between the electron Lndu levels in semiconductor smple plced in mgnetic field is 2 mev. Clculte the mgnetic field. The electron effective mss is 0.06 m 0 (where m 0 is the free electron mss). [2] 0 5 TURN OVER
Question 5 OPTIONAL ) Derive the following expression for electron cyclotron frequency in semiconductor plced in mgnetic field B: ω c = eb/m. Here m is the electron effective mss nd e is the electron chrge. [2] b) i) Clculte vlues for the Hll coefficient of sodium. Sodium hs body-centred cubic (bcc) structure with unit cell side 0.428 nm. [3] ii) A smple of sodium of thickness 0.5 mm is plced in perpendiculr mgnetic field of 0.1 T. Estimte the Hll voltge generted cross the width of the smple when current of 100 ma is pssed long it. [4] c) i) Explin the physicl bsis of prmgnetism. In your explntion discuss the temperture dependence of mgnetic susceptibility in prmgnetic mterils. [3] ii) A prmgnetic mteril in which L = 0 nd S = 1/2 is plced in mgnetic field B=3 T t temperture of 1K. Using the fct tht the energy U of mgnetic dipole in mgnetic field is given by U =.B = M J g B B, clculte the percentge of the toms tht re in the lowest energy stte. Here g is the Lndé fctor nd B is the Bohr mgneton. [3] END OF EXAMINATION PAPER 6 CONTINUED
PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = 1.60 10 19 C electron mss m e = 9.11 10 31 kg = 0.511 MeV c 2 proton mss m p = 1.673 10 27 kg = 938.3 MeV c 2 neutron mss m n = 1.675 10 27 kg = 939.6 MeV c 2 Plnck s constnt h = 6.63 10 34 J s Dirc s constnt ( = h/2π) = 1.05 10 34 J s Boltzmnn s constnt k B = 1.38 10 23 J K 1 = 8.62 10 5 ev K 1 speed of light in free spce c = 299 792 458 m s 1 3.00 10 8 m s 1 permittivity of free spce ε 0 = 8.85 10 12 F m 1 permebility of free spce µ 0 = 4π 10 7 H m 1 Avogdro s constnt N A = 6.02 10 23 mol 1 gs constnt R = 8.314 J mol 1 K 1 idel gs volume (STP) V 0 = 22.4 l mol 1 grvittionl constnt G = 6.67 10 11 N m 2 kg 2 Rydberg constnt R = 1.10 10 7 m 1 Rydberg energy of hydrogen R H = 13.6 ev Bohr rdius 0 = 0.529 10 10 m Bohr mgneton µ B = 9.27 10 24 J T 1 fine structure constnt α 1/137 Wien displcement lw constnt b = 2.898 10 3 m K Stefn s constnt σ = 5.67 10 8 W m 2 K 4 rdition density constnt = 7.55 10 16 J m 3 K 4 mss of the Sun M = 1.99 10 30 kg rdius of the Sun R = 6.96 10 8 m luminosity of the Sun L = 3.85 10 26 W mss of the Erth M = 6.0 10 24 kg rdius of the Erth R = 6.4 10 6 m Conversion Fctors 1 u (tomic mss unit) = 1.66 10 27 kg = 931.5 MeV c 2 1 Å (ngstrom) = 10 10 m 1 stronomicl unit = 1.50 10 11 m 1 g (grvity) = 9.81 m s 2 1 ev = 1.60 10 19 J 1 prsec = 3.08 10 16 m 1 tmosphere = 1.01 10 5 P 1 yer = 3.16 10 7 s
Polr Coordintes x = r cos θ y = r sin θ da = r dr dθ 2 = 1 ( r ) + 1r 2 r r r 2 θ 2 Sphericl Coordintes Clculus x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dv = r 2 sin θ dr dθ dφ 2 = 1 ( r 2 ) + 1 r 2 r r r 2 sin θ ( sin θ ) + θ θ 1 r 2 sin 2 θ 2 φ 2 f(x) f (x) f(x) f (x) x n nx n 1 tn x sec 2 x e x e x sin ( ) 1 x ln x = log e x 1 x cos 1 ( x sin x cos x tn ( 1 x cos x sin x sinh ( ) 1 x cosh x sinh x cosh ( ) 1 x sinh x cosh x tnh ( ) 1 x ) ) 1 2 x 2 1 2 x 2 2 +x 2 1 x 2 + 2 1 x 2 2 2 x 2 cosec x cosec x cot x uv u v + uv sec x sec x tn x u/v u v uv v 2 Definite Integrls 0 + + x n e x dx = n! (n 0 nd > 0) n+1 π e x2 dx = π x 2 e x2 dx = 1 2 Integrtion by Prts: 3 b u(x) dv(x) dx dx = u(x)v(x) b b du(x) v(x) dx dx
Series Expnsions (x ) Tylor series: f(x) = f() + f () + 1! n Binomil expnsion: (x + y) n = (1 + x) n = 1 + nx + k=0 ( ) n x n k y k k n(n 1) x 2 + ( x < 1) 2! (x )2 f () + 2! nd (x )3 f () + 3! ( ) n n! = k (n k)!k! e x = 1+x+ x2 2! + x3 x3 +, sin x = x 3! 3! + x5 x2 nd cos x = 1 5! 2! + x4 4! ln(1 + x) = log e (1 + x) = x x2 2 + x3 3 n Geometric series: r k = 1 rn+1 1 r k=0 ( x < 1) Stirling s formul: log e N! = N log e N N or ln N! = N ln N N Trigonometry sin( ± b) = sin cos b ± cos sin b cos( ± b) = cos cos b sin sin b tn ± tn b tn( ± b) = 1 tn tn b sin 2 = 2 sin cos cos 2 = cos 2 sin 2 = 2 cos 2 1 = 1 2 sin 2 sin + sin b = 2 sin 1( + b) cos 1 ( b) 2 2 sin sin b = 2 cos 1( + b) sin 1 ( b) 2 2 cos + cos b = 2 cos 1( + b) cos 1 ( b) 2 2 cos cos b = 2 sin 1( + b) sin 1 ( b) 2 2 e iθ = cos θ + i sin θ cos θ = 1 ( e iθ + e iθ) 2 nd sin θ = 1 ( e iθ e iθ) 2i cosh θ = 1 ( e θ + e θ) 2 nd sinh θ = 1 ( e θ e θ) 2 Sphericl geometry: sin sin A = sin b sin B = sin c sin C nd cos = cos b cos c+sin b sin c cos A
Vector Clculus A B = A x B x + A y B y + A z B z = A j B j A B = (A y B z A z B y ) î + (A zb x A x B z ) ĵ + (A xb y A y B x ) ˆk = ɛ ijk A j B k A (B C) = (A C)B (A B)C A (B C) = B (C A) = C (A B) grd φ = φ = j φ = φ x î + φ y ĵ + φ z ˆk div A = A = j A j = A x x + A y y + A z z ) curl A = A = ɛ ijk j A k = ( Az y A y z φ = 2 φ = 2 φ x + 2 φ 2 y + 2 φ 2 z 2 ( φ) = 0 nd ( A) = 0 ( A) = ( A) 2 A ( Ax î + z A ) ( z Ay ĵ + x x A ) x y ˆk