Data Provided: A formula sheet and table of physical constants is attached to this paper. SOLID STATE PHYSICS

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Randolf Garrett
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1 Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn (2016) SOLID STATE PHYSICS 2 HOURS Instructions: The pper is divided into 5 questions. Answer question ONE (Compulsory), nd TWO other questions. The totl number of mrks vilble for the exm is 50. Question ONE is mrked out of 20. Questions 2 5 re 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

2 Question 1 COMPULSORY ) Wht re typicl electron densities in metls, semiconductors nd insultors? Explin how electron densities my be controlled in one of the three cses. [3] b) The electron configurtion in potssium is 1s 2, 2s 2, 2p 6, 3s 2, 3p 6, 4s 1. Explin which electrons contribute to conduction. Wht is the role of the electrons in the other shells. [3] c) A metl hs fce centred cubic structure with lttice constnt equl to 3 x m. Ech tom contributes one electron to conduction. Clculte the free electron density in the metl. d) Explin the concept of donors nd cceptors in semiconductors. Why does doping increse significntly the conductivity of semiconductors? [2] [3] e) Sketch grph showing the vrition of the Fermi level s function of temperture in n n-type semiconductor doped with both donors nd cceptors. Explin the temperture behviour of the Fermi level t low nd high tempertures. [3] f) The mgnetistion of prmgnet s function of temperture T is given by M(T) = Ng BJBJ(y), where g is the Lndé fctor, J is the quntum number for the totl ngulr momentum of the toms, B is the Bohr mgneton, N is the number of mgnetic toms per unit volume nd BJ(y) is the Brillouin function with y = g BJB/(kBT), where B is the vlue of mgnetic field. Explin why the mgnetiztion t temperture T cn be relted to the mgnetistion M(0) t temperture T = 0 by M(T)/M(0) = BJ(y). [2] g) Mn 2+ ions hve electronic configurtion 3d 5. Use Hund s rule to clculte the quntum numbers J, L nd S. Determine the mgnitude of the mgnetic moment of the ion in units of the Bohr mgneton. In your nswer you my use the expression for the Lndé g-fctor g = ( 3 + S(S+1) L(L+1) ). 2 2J(J+1) h) Explin why most metls hve silvery (metllic) colour nd some metls, e.g. gold or copper hve yellowish colour. [2] [2] 2 CONTINUED

3 Question 2 OPTIONAL ) Explin the mening of the terms: i) Fermi surfce ii) Bnds nd bnd gps iii) Brillouin zone b) Explin why different bnds in solids hve different widths in energy. How do you expect the widths of the bnds to vry s function of energy, from low energy bnds to high energy bnds? [2] c) 2 Using the expression 2k.G G where k is wvevector nd G reciprocl lttice vector with mgnitude G, using geometricl constructions, derive the shpes of the first nd second Brillouin zones for simple cubic lttice. [3] d) A simple metl hs Fermi wvevector of mgnitude kf in the rnge π/ < k F < 2π/, where is the crystl lttice constnt. Deduce nd sketch which sttes re occupied in the first nd second Brillouin zones. You should use the reduced zone scheme in your [3] nswer. e) An electric field of mgnitude 10 V/m is pplied to metl with electron mobility 10-3 m 2 /Vs. Clculte the resulting displcement of the Fermi surfce. [2] f) Describe mesurement technique which my be used to deduce the vlue of the Fermi energy in metl. [2] 3 TURN OVER

4 Question 3 OPTIONAL ) Indium ntimonide hs dielectric constnt = 17. The donor ionistion energy in this mteril is ED = ev. Clculte the electron effective mss nd the rdius of the ground stte orbit. [3] b) In hevily doped smple of GAs the onset of opticl bsorption occurs t wvelength of 700 nm t 300 K. Clculte the electron Fermi k-vector. The electron (me) nd hole (mh) effective msses in GAs re m0 nd 0.45 m0, respectively, where m0 is the free electron mss. The bnd gp of GAs is 1.43 ev t 300 K. [4] c) Provide n explntion of how the sign nd the effective mss of chrge crriers in semiconductor cn be obtined from cyclotron resonnce mesurements. [3] d) A piece of copper is plced in n externl mgnetic field H, whose mgnitude inside the copper is 10 4 A/m. The mgnetic moment per unit volume in copper is 0.05 A/m. Clculte the mgnetic susceptibility of copper. e) Provide qulittive explntion of the mechnism responsible for spontneous lignment of mgnetic dipole moments in ferromgnetic mteril. The Curie temperture of pure iron is Tc = 1043 K. Assuming tht the mgnetic dipole moment per tom in iron is 2.2 Bohr mgnetons estimte the vlue of the effective internl mgnetic field in iron. [4] 4 CONTINUED

5 Question 4 OPTIONAL ) Electron (ne) nd hole (np) concentrtions in intrinsic semiconductors re given by the following expressions n e = 2 ( 2πm 3/2 ek B T h 2 ) exp ( E F E G k B T ), n p = 2 ( 2πm 3/2 pk B T h 2 ) exp ( E F k B T ). where me (mp) is the electron (hole) effective mss, EG the bnd gp of the mteril, EF the Fermi level nd T the temperture. Derive n expression for EF s function of EG, me, mp, nd T only. How does the Fermi level chnge with T in the cses of me < mp nd me > mp? [4] b) A smple of silicon contins m -3 donors. Estimte the temperture bove which it begins to show intrinsic behviour. The intrinsic crrier concentrtion in silicon t 300 K is m -3. The vlue of the bnd gp is 1.1 ev. [3] c) A live frog is seen to levitte bove powerful permnent mgnet. Explin the origin of this phenomenon. [2] d) The most importnt contribution to the prmgnetism of CuSO4 comes from Cu 2+ ions for which the mgnetic moment is due to single unpired electrons with zero orbitl ngulr momentum. Write down the probbilities t temperture T tht the moment lies prllel nd ntiprllel to the field. Hence show tht the mgnetiztion for N ions per unit volume in field B is M = N Btnh( BB/(kBT)), where B is the Bohr mgneton. [3] e) Where is the Fermi level locted for the cse of semiconductor with energy gp EG doped with i) donors ii) iii) cceptors both donors nd cceptors, with the concentrtion of donors being lrger thn tht of cceptors? You my ssume tht the temperture is very low such tht kbt << EA, ED, EG, where EA nd ED re binding energies of the cceptors nd donors, respectively. 5 TURN OVER

6 Question 5 OPTIONAL ) A smple of sodium with width of w = 3 mm nd thickness of t = 0.5 mm is plced in perpendiculr mgnetic field of B = 0.1 T. The Hll voltge generted cross the width of the smple is 4.9 nv when current of I = 100 ma is pssed long it. Estimte the electron concentrtion in sodium. [4] b) A smple of germnium is doped with single type of donor, the concentrtion of which is m -3. Estimte the minimum vlue of the mgnetic field required to observe cyclotron resonnce in this smple t 4 K. In your nswer you should tke into ccount tht the collision dimeter of donor, which determines the donor scttering cross-section, is equl to 30 nm nd the effective mss of electrons is me = kg. [5] c) Describe the domin theory of ferromgnetism using references to suitble digrms. Include in your discussion: (i) the reversible nd irreversible motion of mgnetic domins in n pplied mgnetic field; (ii) the mechnisms responsible for the estblishment of stble domin structure. [4] d) Estimte the electron concentrtion in doped semiconductor with bckground dielectric constnt nd effective electron mss 0.02m0 (m0 is the free electron mss), where the plsm ngulr frequency is rd/s. [2] END OF EXAMINATION PAPER 6 CONTINUED

7 PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = C electron mss m e = kg = MeV c 2 proton mss m p = kg = MeV c 2 neutron mss m n = kg = MeV c 2 Plnck s constnt h = J s Dirc s constnt ( = h/2π) = J s Boltzmnn s constnt k B = J K 1 = ev K 1 speed of light in free spce c = m s m s 1 permittivity of free spce ε 0 = F m 1 permebility of free spce µ 0 = 4π 10 7 H m 1 Avogdro s constnt N A = mol 1 gs constnt R = J mol 1 K 1 idel gs volume (STP) V 0 = 22.4 l mol 1 grvittionl constnt G = N m 2 kg 2 Rydberg constnt R = m 1 Rydberg energy of hydrogen R H = 13.6 ev Bohr rdius 0 = m Bohr mgneton µ B = J T 1 fine structure constnt α 1/137 Wien displcement lw constnt b = m K Stefn s constnt σ = W m 2 K 4 rdition density constnt = J m 3 K 4 mss of the Sun M = kg rdius of the Sun R = m luminosity of the Sun L = W mss of the Erth M = kg rdius of the Erth R = m Conversion Fctors 1 u (tomic mss unit) = kg = MeV c 2 1 Å (ngstrom) = m 1 stronomicl unit = m 1 g (grvity) = 9.81 m s 2 1 ev = J 1 prsec = m 1 tmosphere = P 1 yer = s

8 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 x 2 2 +x 2 1 x x 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 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

9 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

10 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

Data Provided: A formula sheet and table of physical constants is attached to this paper. SOLID STATE PHYSICS

Data Provided: A formula sheet and table of physical constants is attached to this paper. SOLID STATE PHYSICS 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