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 (015) SEMICONDUCTOR PHYSICS AND TECHNOLOGY HOURS The pper hs 3 sections. Section A contins one compulsory question, Sections B nd C ech contin 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 in Section A, one optionl question from Section B nd one optionl question from Section C. 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 SECTION A COMPULSORY QUESTION 1. () (b) (c) (d) (e) (f) (g) At 0C, current of 0 ma flows through p-n junction diode for forwrd bis voltge of 0.5 V. Clculte the current which would flow through the diode if the direction of the pplied voltge ws reversed. [4] A photoconductor is mde from semiconductor with bndgp of 1.5 ev. Sketch how you would expect the externl detector efficiency to vry, s function of the wvelength of the incident light, over the rnge 400 to 1000 nm. Explin the min fetures of your sketch. [4] By considering the current flowing into the bse nd collector in bipolr trnsistor, show tht the current gin (β) is given by 1 where α is the bse trnsport fctor. [] Explin the min differences in the crystl growth conditions required for the growth of semiconductor quntum wells nd quntum dots. [4] Sketch the conduction bnd energy level structure for quntum dot system. Your nswer should include lbelling of the degenercies of the levels involved. [] A smple contins plne of quntum dots t n rel density of m -. The smple is doped t sheet electron density of m -. Deduce the occupncies of the energy levels of the quntum dots t low temperture. [] Explin why quntum dots my hve properties which re desirble for the relistion of high performnce semiconductor lsers. [] CONTINUED

3 SECTION B ANSWER EITHER QUESTION OR 3. () A linerly grded p-n junction hs doping profile given by the formul N (x) = x, where is constnt nd x is the distnce from the centre of the junction. A positive vlue of N (x) corresponds to n-type doping nd negtive vlue to p-type doping. If ech of the dopnt toms is singly ionised (chrge of ±e), show tht the electric field within the depletion region is given by E x e W 4 x where W is the totl width of the depletion region nd is the dielectric permittivity of the semiconductor. [6] (b) Show further tht the built-in voltge Vbi of the junction is given by V bi ew 1 3. [5] (c) Explin why semiconductor junction possesses voltge-dependent cpcitnce nd show tht the depletion cpcitnce per unit re of linerly grded p-n junction is given by 1 e C Vbi V 1 3 where V is the pplied voltge. [3] (d) In linerly grded p-n junction the doping vries from cm -3 n-type to cm -3 p-type over distnce of 3 m. If the semiconductor hs reltive permittivity of 1 nd built-in voltge of 1.5V, clculte (i) the depletion region width; (ii) the mximum electric field in the depletion region; (iii) the cpcitnce per unit re, when there is no pplied voltge. [6] 3 TURN OVER

4 3. () Sketch the energy versus wvevector bnd digrms for (i) direct bnd gp semiconductor; (ii) n indirect bnd gp semiconductor. In ech cse show the mechnisms involved in photon emission. With reference to your digrms, explin why the light production efficiency of n indirect bnd gp semiconductor is generlly significntly less thn tht of direct bnd gp semiconductor. [6] (b) Estimte vlues for the wvenumber of photon emitted by electron-hole recombintion nd for the wvenumber t the Brillouin zone edge. Hence, explin why photon recombintion processes re represented by verticl lines on energy versus wvevector bnd digrms. [] (c) Semiconductor light emitting diodes typiclly hve n externl efficiency tht is significntly less thn the clculted internl efficiency. Briefly discuss three resons for this difference. [3] (d) (e) Explin why these three loss mechnisms tht limit the externl efficiency of light emitting diode re bsent in semiconductor lser. [3] A semiconductor lser diode with cvity length mm nd width 10 µm exhibits the following output power (P) versus current (I) vlues. I (ma) P (mw) By plotting the dt in suitble form determine the threshold current, threshold current density nd externl differentil quntum efficiency. The lser wvelength of the device is 780 nm. [6] 4 CONTINUED

5 SECTION C ANSWER EITHER QUESTION 4 OR 5 4. () Show tht the density of sttes in quntum well is given by * E m D, * where m is the electron effective mss nd the other symbols hve their conventionl mening. Explin which motion gives rise to this density of sttes. [4] (b) A quntum well is doped t sheet electron density of 1 x m -. Clculte the resulting Fermi energy in quntum well with lrge n =1 to n = sub-bnd spcing of 100 mev. The electron effective mss my be tken s 0.07 me. [] (c) (d) (e) (f) (g) If insted the sub-bnd spcing is 0 mev, wht would be the resultnt Fermi energy mesured reltive to the bottom of the lowest sub-bnd? [4] Sketch typicl bsorption spectrum in quntum well, explining which fetures rise from the form of the density of sttes nd which rise from other phenomen. [] An electric field is pplied to quntum well long the quntistion xis. Describe how the bsorption spectrum is likely to be modified by the presence of the electric field. [4] Explin how the effect of electric field s in (e) my led to useful device opertion. [] Describe briefly why the bsorption spectrum for superlttice in n electric field my differ from tht for quntum well. [] 5 TURN OVER

6 5. () A semiconductor microcvity is typiclly composed of three different regions, the distributed Brgg reflector mirrors, the cvity, nd the quntum wells in the cvity. Explin the roles of the three regions. [4] (b) Describe how exciton-polriton modes my rise in microcvity. [4] (c) (d) (e) (f) (g) Explin which physicl properties of polritons enble the occurrence of Bose-Einstein condenstion. [] A quntum well hs width of 10 nm nd fluctution in width of 0.1 nm. Using n infinite well model, estimte the linewidth of typicl photoluminescence line. You my ssume the energy up-shift due to quntum confinement is 60 mev. [] Sketch the form of the trnsmission coefficient s function of energy of n electron which is incident on single brrier tunnelling structure. [] Explin why the trnsmission coefficient differs for double brrier structure. You should include suitble digrm in your nswer. [] Explin how the phenomenon of resonnt tunnelling is importnt for the opertion of quntum cscde lser. Your nswer should focus on the role of the resonnt tunnelling, rther thn the physics underlying the opertion of the whole lser structure. [4] 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