Data Provided: A formula sheet and table of physical constants is attached to this paper. Linear-linear graph paper is required.

Transcription

1 Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Liner-liner grph pper is required. DEPARTMENT OF PHYSICS AND ASTRONOMY Spring Semester ( ) SEMICONDUCTOR PHYSICS AND TECHNOLOGY 2 HOURS The pper hs compulsory question followed by two sections, ech contining two optionl questions. All questions re mrked out of twenty. The brekdown on the right-hnd side of the pper is ment s guide to the mrks tht cn be obtined from ech prt. Answer THREE questions. You must ttempt the compulsory question nd one optionl question from ech section. 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 COMPULSORY QUESTION 1. () (b) (c) (d) (e) (f) (g) (h) A piece of silicon is doped with rsenic toms. The doping density is cm 3. Clculte the free electron nd hole concentrtions, ssuming the intrinsic crrier concentrtion is cm 3. [2] Sketch schemtic energy bnd digrms for p-n junction under (i) zero pplied bis, (ii) reverse bis voltge, (iii) forwrd bis voltge. Your digrms should show clerly the positions of the conduction nd vlence bnds, the impurity energy levels, the Fermi level, the free crriers nd the depletion region. [3] Explin wht is ment by the Erly effect in bipolr trnsistor nd drw digrm to show how the collector current (I C ) versus collector-emitter voltge (V CE ) chrcteristics re influenced by this effect. [3] Two p-n junctions re found to hve reverse brekdown voltges of 100V nd 5V. Explin, with resons, which junction hs the lower doping level. [2] Describe the physicl principles underlying the technique of photoluminescence spectroscopy. [2] A quntum well structure of 10nm width exhibits photoluminescence linewidth of 1 mev. Estimte the fluctutions in well width which my give rise to this linewidth. You my tke the quntistion energy of the lowest electron sttes to be 50 mev. [4] The linewidths of photoluminescence spectr from lrge numbers of selfssembled quntum dots re typiclly very much lrger thn those for quntum wells. Describe briefly the most probble reson for this. [2] Explin wht fctors re likely to determine the photoluminescence emission energy of quntum well structure. [2] 2 CONTINUED

3 SECTION A ANSWER EITHER QUESTION 2 OR 3 2. () (b) Explin why the externl quntum efficiency of semiconductor lser diodes is higher thn semiconductor light emitting diodes. [4] Drw schemtic digrm showing how the opticl output power from semiconductor lser diode depends upon current flow through the device. Describe briefly the min fetures nd explin how the externl quntum efficiency could be determined from your digrm. [5] (c) By considering lser cvity of length L nd with mirror reflectivity R, show tht the threshold gin is given by g th 1 1 i ln L R where α i is the internl opticl loss. [3] (d) The externl quntum efficiency ( ex ) of series of lser diodes with differing lengths (L) is mesured nd gives the following vlues η ex L (cm) Given tht the externl quntum efficiency of semiconductor lser diode cn be written s i m ex, i m where m is the mirror loss nd i is the internl quntum efficiency, formulte suitble eqution nd plot grph to determine vlues of internl quntum efficiency ( i ) nd internl opticl loss ( i ) of the lser. You my ssume mirror reflectivity R=0.3. [8] 3 TURN OVER

4 3. () (b) (c) (d) Show tht for n idel p-n junction the reltionship between the pplied externl voltge V nd the current I is given by I B I0 exp qv k T 1 where T is the temperture, k B is the Boltzmnn constnt nd I 0 is constnt pre-fctor. [5] Describe two non-rditive recombintion processes tht cn occur inside semiconductors. [3] Write down the definition of rditive efficiency η r in terms of the spontneous emission, non-rditive recombintion nd crrier lekge rtes inside semiconductor. You my ssume tht stimulted emission processes re negligible. [2] If G gen is the rte t which crriers re generted per unit volume in the ctive region of semiconductor device, show tht G gen ii qv where I is the current flowing into the terminls of the device, V is the volume of the device ctive region nd q is the electronic chrge. [2] (e) The electricl to opticl conversion efficiency in light emitting diodes is given by 4n1n2 1 cosc n n where n 1 nd n 2 re the refrctive indices of ir nd the semiconductor, respectively nd θ c is the criticl ngle. Clculte the conversion efficiency of light emitting diode mde from semiconductor with refrctive index 3.5. [2] (f) A light emitting diode operting t wvelength of 730 nm genertes n internl opticl power of 5 mw when driven with current of 10 ma. Derive n eqution for the power generted by spontneous emission recombintion nd hence clculte (i) the spontneous rditive nd (ii) non-rditive recombintion rtes if the device hs η i = 0.6 nd n ctive region volume m 3. You my ssume tht the crrier lekge rte is zero. [6] 4 CONTINUED

5 SECTION B ANSWER EITHER QUESTION 4 OR 5 4. () (b) (c) (d) (e) (f) (g) Sketch the form of the density of sttes s function of energy in three, two, one nd zero dimensionl systems. [2] Explin the potentil dvntges for device opertion of reducing dimensionlity in the sequence from three to zero dimensions. [2] Describe the physicl principles which underlie the technique of modultion doping of semiconductor heterostructures. [4] Sketch the conduction bnd energy profile for modultion doped heterostructure under equilibrium conditions. Discuss the physicl resons for ny key fetures in the sketch. [4] The wide bnd gp region of such heterostructure is doped in lyer of 10nm thickness t doping level of 1 x m -3. Clculte the Fermi energy of the two dimensionl electron gs which my result. You my tke the * 2 density of sttes in two dimensions to be given by DE m, nd the electron effective mss m * to be 0.07 m e. [4] Wht re everydy pplictions of modultion doped devices? In wht frequency rnge do such devices operte? [2] Drw schemtic digrm of the vrition of electron mobility ginst temperture in high qulity modultion doped heterostructure, lbelling ny feture which results specificlly from the modultion doping. [2] 5 TURN OVER

6 5. () (b) (c) (d) (e) (f) (g) Explin why exciton binding energies re enhnced in quntum wells compred to bulk semiconductors. [2] Sketch typicl vrition of exciton binding energy versus well width, from very nrrow (~0.5 nm) to very wide wells (~30 nm), explining ny key fetures in the sketch. [2] Wht re the dvntges for device opertion of the enhnced exciton binding energy in quntum wells reltive to bulk semiconductors? [2] Compre the behviour of bsorption spectr in electric field in isolted quntum wells with tht in superlttice, explining ny key resons for the differences. [4] A superlttice hs 10 periods of quntum wells nd brriers. A voltge of 0.1 V is pplied cross the 10 periods. Clculte the splitting of the vlence bnd to conduction bnd bsorption spectr which will result. [2] Explin the key physicl principles which underlie the opertion of quntum cscde lsers. [6] Explin wht role superlttices ply in the successful opertion of quntum cscde lsers. [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