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

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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 (015) SOLID STATE PHYSICS HOURS The pper is divided into 5 questions. Answer compulsory question 1, which is mrked out of 0. Answer ny two out of the optionl questions -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

2 Question 1 COMPULSORY ) The expression k.g = G describes the condition for Brgg diffrction in periodic lttice, where k is the electron wvevector nd G is reciprocl lttice vector. Using this expression, deduce the forms of the first nd second Brillouin zones for two dimensionl squre lttice. [3] E F 3 n, m where n is the electron density nd m is the electron mss. [3] b) Show tht the Fermi energy of conduction electrons is given by 3 c) A metl hs body-centred-cubic structure nd lttice constnt of 3.5 x m. Ech tom contributes one electron to conduction. Using the expression for the Fermi energy from prt b), deduce the vlues of the Fermi energy, Fermi temperture nd Fermi velocity. You my tke m to be equl to the free electron mss. [3] d) Sketch grph showing vrition of crrier concentrtion in doped semiconductor s function of temperture. Explin the temperture behviour of the concentrtion t low, intermedite nd high tempertures. [3] e) Where is the Fermi level locted for the cse of semiconductor doped with cceptors? Assume tht the temperture is very low kbt << EA, where EA is the binding energy of the cceptor. [1] f) Explin the physicl bsis of prmgnetism. In your explntion discuss the temperture dependence of mgnetic susceptibility in prmgnetic mterils. [3] g) Co + ions hve electronic configurtion 3d 7. Use Hund s rule to clculte the quntum numbers J, L nd S. Determine the mximum vlue of the mgnetic moment of the ion long n rbitrry xis in spce in the 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) ). [] J(J+1) h) Explin the concept of plsmons in metls. Briefly outline n experiment which enbles the observtion of plsmons in metllic films. [] CONTINUED

3 Question OPTIONAL ) Using tight binding model, deduce tht the number of sttes in bnd is given by N, where N is the number of toms. [] 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. [] c) Describe the physicl bsis of technique which my be used to probe the electron occupncy of the conduction bnd in metl, by probing trnsitions between different bnds. [3] d) Electrons in metl hve mobility 1 m /Vs, crrier density 3 x 10 8 m -3, nd mss equl to the free electron mss. (i) Clculte the conductivity of the metl. [] (ii) An electric field of 10V/m is pplied. Clculte the displcement in k-spce of the Fermi distribution which results. [] (iii) How does this displcement compre to the likely vlue of the Fermi wvevector in the metl. A derivtion of the vlue for the Fermi wvevector is not required. [1] e) Describe two types of scttering mechnism which led to resistnce to electron flow in metls. [] f) Explin why the two mechnisms in e) hve very different temperture dependences. [1] 3 TURN OVER

4 Question 3 OPTIONAL ) Provide qulittive explntion of why the binding energy of n electron to donor, which is of the order of 10 s of mev in the most common semiconductors (Si, Ge or GAs), is much smller thn the binding energy of n electron (13.6 ev) in hydrogen tom. How does the Bohr rdius of n electron to donor in semiconductors compre to the Bohr rdius of n electron in hydrogen tom? [3] b) In hevily doped smple of GAs the onset of opticl bsorption occurs t wvelength of 700 nm t 300 K. Clculte the energy of the Fermi level bove the minimum of the conduction bnd. 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) Explin the concept of Wnnier-Mott exciton in semiconductor mteril. Explin why the pproximtion of the effective mss nd mcroscopic dielectric function is vlid in the clcultion of Wnnier-Mott exciton binding energies using hydrogenic model. [] d) A piece of ferric oxide is plced in n externl mgnetic field H. The resultnt mgnetic field B inside the mteril is 1.58 T nd its mgnetistion M is 1500 A/m. Clculte the mgnetic susceptibility of ferric oxide nd the vlue of the H-field inside the mteril. [3] e) Explin how nd why the phenomenon of impurity compenstion my rise in semiconductor initilly doped only with donors or cceptors. [3] 4 CONTINUED

5 Question 4 OPTIONAL ) Derive the following expression for electron concentrtion ne in the conduction bnd s function of temperture T in n undoped semiconductor: n e = ( πm e 3/ k B T h ) exp ( E F E g k B T ). You my use the following expressions for the electron density of sttes ge per unit volume nd the electron Fermi-Dirc distribution f(e): g e = 4π(m e ) 3 h 3 E E g, 1 f(e) = 1 + exp ( E E. F k B T ) Here m e is the electron effective mss, EF is the Fermi level, Eg is the bnd gp 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 EF >>kbt nd use the following integrl x 1/ e x dx = π/. [6] 0 b) A smple of silicon doped with donors ceses to show intrinsic behviour t tempertures below 360 K. Estimte the donor concentrtion. The intrinsic crrier concentrtion in silicon t 300 K is m -3. The vlue of the bnd gp in Si is 1.1 ev. [3.5] c) Briefly explin the three mechnisms which contribute to the totl energy of ferromgnetic crystl, where stble spontneous domin configurtion is developed. How do these mechnisms compete with ech other? [3] d) Clculte the verge mgnetic moment long the field direction per tom t 300 K when prmgnetic mteril hving only spin (i.e. L = 0) is subjected to uniform H-field of 10 6 A/m. You my use the following formul for mgnetic susceptibility of prmgnet Ng J J 1 0 B 3kT, where g is the Lnde fctor, J is the totl ngulr momentum quntum number, B is the Bohr mgneton. Express your nswer in units of the Bohr mgneton. [.5] 5 TURN OVER

6 Question 5 OPTIONAL ) Sketch schemtic digrm of the experiment enbling the observtion of the Hll effect. Strting from the eqution of motion for electrons in crossed mgnetic (B) nd electric (E) fields show tht the Hll electric field in metl is given by E H = eτb E, where, me, e re the electron scttering time, effective mss nd chrge, m e respectively. [5] b) Explin how the sign of the min chrge crriers in semiconductor mteril cn be obtined from cyclotron resonnce nd Hll experiments. [] c) (i) An electron cyclotron resonnce is observed t frequency f = 100 GHz for semiconductor smple plced in n externl mgnetic field. Find the vlue of the mgnetic field. The effective mss of the electron in the smple is me = 0.07 m0, where m0 is the free electron mss. [.5] (ii) Estimte the minimum electron scttering time required to observe cyclotron resonnce t 00 GHz. [.5] d) Ferromgnetic nickel hs Curie Temperture of 510 K nd mgnetic moment of 0.60 B per ion ( B is the Bohr mgneton). Clculte the vlue of the effective internl mgnetic field tht is responsible for the spontneous mgnetistion of nickel. Comment on the min physicl mechnism responsible for this field. [3] END OF EXAMINATION PAPER 6 CONTINUED

7 PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = C electron mss m e = kg = MeV c proton mss m p = kg = MeV c neutron mss m n = kg = MeV c Plnck s constnt h = J s Dirc s constnt ( = h/π) = 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 =.4 l mol 1 grvittionl constnt G = N m kg 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 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 1 Å (ngstrom) = m 1 stronomicl unit = m 1 g (grvity) = 9.81 m s 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 = 1 ( r ) + 1r r r r θ Sphericl Coordintes Clculus x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dv = r sin θ dr dθ dφ = 1 ( r ) + 1 r r r r sin θ ( sin θ ) + θ θ 1 r sin θ φ f(x) f (x) f(x) f (x) x n nx n 1 tn x sec 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 x 1 x +x 1 x + 1 x x cosec x cosec x cot x uv u v + uv sec x sec x tn x u/v u v uv v Definite Integrls x n e x dx = n! (n 0 nd > 0) n+1 π e x dx = π x e x dx = 1 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 + ( x < 1)! (x ) f () +! nd (x )3 f () + 3! ( ) n n! = k (n k)!k! e x = 1+x+ x! + x3 x3 +, sin x = x 3! 3! + x5 x nd cos x = 1 5!! + x4 4! ln(1 + x) = log e (1 + x) = x x + 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 = sin cos cos = cos sin = cos 1 = 1 sin sin + sin b = sin 1( + b) cos 1 ( b) sin sin b = cos 1( + b) sin 1 ( b) cos + cos b = cos 1( + b) cos 1 ( b) cos cos b = sin 1( + b) sin 1 ( b) e iθ = cos θ + i sin θ cos θ = 1 ( e iθ + e iθ) nd sin θ = 1 ( e iθ e iθ) i cosh θ = 1 ( e θ + e θ) nd sinh θ = 1 ( e θ e θ) 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 = φ = φ x + φ y + φ z ( φ) = 0 nd ( A) = 0 ( A) = ( A) 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