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) Sketch energy bnd digrms for p-n junction under (i) zero pplied bis, (ii) reverse bis, (iii) forwrd bis. 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. [4] At 20 C, current of 30 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. [3] Describe, using pproprite digrms, how the vrition of pplied voltge between two of the terminls controls the current flow into the third terminl of bipolr n-p-n trnsistor. [3] Describe the physicl principles underlying the technique of opticl bsorption spectroscopy, s commonly pplied to quntum well structures. [2] Explin which fctors determine the energy t which the onset of bsorption trnsitions occurs for quntum well. [2] A quntum well of width 10 nm hs infinitely high brriers. Clculte the energy seprtion between the n = 2 nd n = 3 confined electron sttes. The effective mss of the electrons is 0.1 m e. Explin qulittively how the energy seprtion my be modified if the brriers re of finite height. [4] Describe the essentil fetures of crystl growth technique which my be employed to grow high qulity quntum wells. [2] 2 CONTINUED

3 SECTION A ANSWER EITHER QUESTION 2 OR 3 2. () (b) It cn be shown tht light emitting diode (LED) hs spontneous emission intensity spectrum 1 I = A(E E g ) 2 exp ( E k B T ) where A is constnt, E is the photon energy, E g the semiconductor bndgp, k B the Boltzmnn constnt nd T the temperture. Find the photon energy t the mximum of the intensity spectrum. Does room temperture mesurement of this energy provide resonble estimte of the semiconductor bndgp? Justify your nswer. [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 for lsing is given by g th = α i + 1 L ln (1 R ) where α i is the internl opticl loss. [3] (d) (e) An InGAsP Fbry-Perot lser operting t wvelength of 1.3 μm hs cvity length of 300 μm. If one of the lser fcets is coted to produce 90 % reflectivity, wht is the threshold gin for lsing, ssuming α i = 10 cm 1? The refrctive index of InGAsP is 3.9. [3] Assuming tht the refrctive index n depends on the wvelength λ s n = n 0 + dn dλ (λ λ 0) find the seprtion λ between the llowed modes for 300 μm long InGAsP Fbry-Perot lser operting t λ 0 = 1.3 μm. You my ssume tht n 0 = 3.9 nd dn/dλ = 2.5 μm 1. [5] 3 TURN OVER

4 3. () (b) Explin briefly why the depletion cpcitnce of semiconductor p-n junction vries under the ppliction of n externl voltge. [2] An brupt p-n junction is doped with N D donors nd N A cceptors per unit volume. Show tht the width of the depletion region in the n-type mteril (x d ) is given by x d 2 V qn ( N D bi A N A N D ) where ε is the permittivity of the semiconductor, q is the elementry chrge nd V bi is the built-in voltge cross the junction. [6] (c) Using the result from prt (b) show tht the depletion cpcitnce per unit re of n symmetriclly doped p-n junction (N A >> N D ) is given by C q N D 2 V V bi where V is the externlly pplied voltge. [4] (d) Cpcitnce mesurements of n symmetriclly doped p-n junction (N A >> N D ) with circulr cross-section nd dimeter of 0.1mm, give the following vlues s function of pplied voltge. Applied Voltge (V) Cpcitnce (pf) By plotting the dt in suitble form, determine the doping density (N D ) nd the built in voltge (V bi ). The reltive permittivity of the semiconductor is 12. [8] 4 CONTINUED

5 SECTION B ANSWER EITHER QUESTION 4 OR 5 4. () (b) (c) (d) (e) (f) Explin why the ppliction of n electric field to quntum well leds to reduction of trnsition energy. The use of suitble sketch my id in your nswer. [2] Explin how the effects of n electric field on quntum well led to n importnt device ppliction. Briefly describe the ppliction. [3] Give two resons with brief explntion why quntum wells re fvourble for the ppliction of question 4(b) s compred to bulk semiconductors. [3] A potentil difference of 0.1 V is pplied to superlttice of length 100 nm. The superlttice contins 10 wells nd brriers of widths 4 nd 6 nm respectively. Wht splitting between the vlence bnd to conduction trnsition energies is expected to result from the ppliction of the electric field? [3] (i) Derive n expression for the density of sttes of two dimensionl quntum well system. Sketch the vrition of the resulting density of sttes s function of energy. [3] (ii) Tking the quntistion xis of the quntum well s long z, nd the sttes in the plne s (x, y), explin which sttes led to the notble fetures in the density of sttes sketched in question 4e(i). [3] Most modern dy semiconductor lsers contin strined lyer quntum wells. Explin which fetures re importnt in the design of such quntum wells. [3] 5 TURN OVER

6 5. () (b) (c) (d) (e) An electron wve is incident on single potentil brrier. Tking ccount of relevnt quntum mechnicl phenomen, sketch nd explin the form of the trnsmission probbility s function of incident electron energy. How does this probbility differ in quntum mechnicl nd clssicl regimes? [3] The single brrier of question 5() is replced by double brrier structure, surrounding single quntum well. The quntum well contins three confined sttes. Explin nd sketch the expected form of the current which flows through the structure s function of pplied voltge. Explin qulittively why the mgnitude of the pek current my differ strongly between key fetures in your sketch. [4] With reference to the form of the current-voltge chrcteristics of question 5(b), describe one ppliction of such double brrier structure. [2] Explin which properties of the structure determine the emission energy of quntum cscde lser, nd how this differs from the cse of conventionl semiconductor lser. [2] Explin how popultion inversion is chieved in quntum cscde lser. [3] (f) A smple contins lyer of quntum dots of rel density 1x10 15 m -2. There is two dimensionl electron gs in close proximity contining n rel density of electrons of 9x10 15 m -2. By referring to the degenercies of quntum dot electron shells, deduce how mny levels of the quntum dots my be occupied by the electron gs. [3] (g) Explin why quntum dots hve fvourble properties for ppliction s sources of single photons. [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 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