1-D matrix method. U 4 transmitted. incident U 2. reflected U 1 U 5 U 3 L 2 L 3 L 4. EE 439 matrix method 1

Transcription

1 -D matrx method We ca expad the smple plae-wave scatterg for -D examples that we ve see to a more versatle matrx approach that ca be used to hadle may terestg -D problems. The basc dea s that we ca break a problem havg a complcated potetal profle to a sequece of costat potetal regos. Sce we already kow the TISE solutos for regos of costat potetal, the problem bols dow to coectg the solutos at each terface. A matrx approach leds tself well to ths type of problem. cdet reflected U 4 trasmtted U U U 3 U 5 L L 3 L EE 439 matrx method

2 Also, we wll see that the method ca be used to fd eergy levels cofg quatum wells. etc E E Fally, t ca be used to obta approxmate solutos to complex potetal profles. EE 439 matrx method

3 r t A C A 3 t r B D B 3 D 3 3 B 4 D C 3 A 4 C 4 I a mult-layer problem, the dffcultes come hadlg the reflectos at all of the terfaces. However, f we ca determe how the plae waves relate from oe sde of a costat potetal rego to the other, cludg the effects of scatterg at the terfaces, the we ca relate the trasmtted ampltude to the cdet ampltude or reflected to cdet) of the overall system. A For each rego, we ll try to wrte a matrx of the form: [M B ] where [M ] s a x matrx descrbg the th rego. EE 439 matrx method 3 C D

4 r t A C A 3 t r B D B 3 D 3 3 B 4 D C 3 A 4 C 4 apple A B [M ] apple C D [M ] apple A3 B 3 [M ][M 3 ] apple C3 D 3 [M ][M 3 ] apple A4 B 4 [M ][M 3 ][M 4 ] apple apple t r [I L][M ][M 3 ][M 4 ][I R ] 0 apple C4 D 4 EE 439 matrx method 4

5 apple r [I L][M ][M 3 ][M 4 ][I R ] apple t 0 apple r apple M M M M apple t 0 trasmsso: reflecto: t M r M M Fdg reflecto ad trasmsso coeffcets s a process of multplyg dvdual layer matrces to obta a overall system matrx. The reflecto ad trasmsso coeffcet ratos are foud from the elemets of the system matrx. EE 439 matrx method 5

6 Ufortuately, our approach wo t be qute as tdy as the prevous slde would mply. Sce the electro waves are scattered at terfaces betwee layers, we eed a matrx to descrbe what happes at each terface. Of course, a terface s ot a property of sgle layer, but depeds o the layer propertes o ether sde of the terface. Also, the waves chage phase as they propagate across regos where E > U, or they grow ad decay expoetally across regos where E < U. We wll eed propagato matrces to descrbe these chages. Itutvely the, each layer leads to two matrces to be cluded the sequece, oe propagato matrx ad oe terface matrx. However, oce we have the form of the propagato ad terface matrces, we ll see that we ca combe them the rght way to obta a smple, oe-matrx descrpto of each layer. However, gettg to ths pot s ot tutve, so we ll take a roud-about approach, but oe that wll hopefully gve a clearer pcture of what we re dog. EE 439 matrx method 6

7 Iterface matrces E Case - E > U o both sde of the terface. U U x We assume the terface occurs at x x. For x < x, the potetal s U, ad for x > x, t s U. Sce E > U ad E > U, we use plaewave solutos o both sdes of the terface. ) ) )] )] )] )] Apply the boudary codtos EE 439 matrx method 7

8 Use the two equatos to wrte A ad B terms of C ad D. I matrx form A I B I I I C D Note that the matrx elemets ths case are all real. EE 439 matrx method 8

9 Case - E > U o the left ad E < U o the rght. As before, we assume the terface occurs at x x. For x < x, the potetal s U, ad for x > x, t s U. For x < x, we eed plae wave solutos ad for x > x, we use growg ad decayg expoetals. ) ) )] )] U E U x )] )] Usg the coecto rules at the terface: EE 439 matrx method 9

10 Solvg for A ad B terms of C ad D: I matrx form A B k k k k C D Note that the matrx elemets ths case are all complex. EE 439 matrx method 0

11 Case 3 - E < U o the left ad E > U o the rght. Same sog, thrd verse. For x < x, the potetal s U, ad for x > x, t s U. For x < x, we eed growg ad decayg expoetals ad for x > x, we use plae wave solutos. ) ) )] )] U E U x )] )] Usg the coecto rules at the terface: EE 439 matrx method

12 Solvg for A ad B terms of C ad D: I matrx form A B k k k k C D The matrx elemets are aga all complex. EE 439 matrx method

13 Case 4 - E < U o the left ad E < U o the rght. U Fal staza. Everythg s growg ad decayg expoetals. U E x ) ) )] )] )] )] Applyg the boudary codtos oe last tme: EE 439 matrx method 3

14 Solvg for A ad B terms of C ad D: I matrx form A B C D The matrx elemets are aga all real. EE 439 matrx method 4

15 Summary: E > U E > U E > U E < U E < U E > U E < U E < U EE 439 I I k k k k I I k k k k k k k k k k k k It s pretty easy to see that k should be replaced wth -α regos where E < U. matrx method 5

16 Propagato matrces As a travelg wave crosses a rego where E > U, t chages phase. For a wave travelg the x drecto, we ca wrte by specto: C A expk L ) E U A k C For a wave travelg the -x drecto B D B exp k L ) x x D Expressed matrx form: apple A B apple P P P P apple C D apple A B apple exp k L ) 0 0 expk L ) apple C D EE 439 matrx method 6

17 For a evaescet wave a rego where E < U, there s o phase chage, but the ampltude must chage expoetally. C A exp L ) D B exp L ) U E A α C B D x x I ths case, the propagato matrx s apple A B apple exp L ) 0 0 exp L ) apple C D EE 439 matrx method 7

18 U 4 cdet trasmtted U U U 3 U 5 reflected L L 3 L 4 Lookg aga at the potetal posed o the frst slde, we see that we eed a whole strg of terface ad propagato matrces. apple r [I ][P ][I 3][P 3 ][I 34][P 4 ][I 45] apple t 0 apple M M M M apple t 0 t M r M M T k 5 k t k 5 k M R k r k M M EE 439 matrx method 8

19 The matrx techque leds tself well to programmg Matlab or some other laguage. However, hadlg the terfaces s a bt uweldy sce the terface matrx volve propertes of two layers. It would be ce f everythg about a gve layer could be cluded oe matrx. Ca ths be doe? Look at the form of a terface matrx. I, 3 k k 4 k k 5 k k k Mathematcally, t ca be splt two: 3 I, p 4 apple k 5 p k ) k ) k The matrx wth k represets the left sde of the terface. The matrx wth k represets the rght sde of the terface. [, ][ ][ ] Alteratvely, the matrx wth k represets the rght ed of the th layer. The matrx wth k represets the left ed of the )st layer. EE 439 matrx method 9 k

20 Look a secto of the sequece of matrces from our orgal problem. [ ] [ ][ ][ ][ ][ ][ ][ ] Splt the terface matrces [ ] [ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ] [ ][ ][ ][ ][ ][ ][ ][ ][ ][ ][ ] where [ ][ ][ ][ ][ ] [ ][ ][ ][ ] The layer matrx [M ] cotas all of the formato about a partcular layer. The parameters for layer show up oly that partcular matrx. Ths makes t easer to specfy ad compute the matrces a program. EE 439 matrx method 0

21 [ ][ ][ ][ ] I the layer, f E > U propagatg wave) [M ] k [M ] k exp k L ) 0 cos k L ) k s k L ) 0 exp k L ) k k k s k L ) cos k L ) If E < U evaescet wave) [M ] [M ] EE 439 exp 0 cosh L ) sh L ) L ) 0 exp L ) sh L ) cosh L ) matrx method

22 Example - tuelg through a square barrer redux) We ve doe ths before ad kow the result. Ths may a good test for our matrx approach. cdet reflected U 0 x 0 x L U U o trasmtted 3 There s a electro cdet from the left rego where U 0), so we eed a left half matrx for rego at x 0. We eed a layer matrx of the E < U varety) for the barrer. Fally, we must have a rght-half matrx for rego 3 at x L. Sce k k 3 k ad rego s characterzed by α, we ca dspese wth the subscrpts. [M] k k cosh L) sh L) sh L) cosh L) k k Now comes tedous algebra to get to the aswer. Note that to the fd the trasmsso probablty, we oly eed M. EE 439 matrx method

23 M cosh L) cosh k k ) sh sh ) EE 439 ) ) sh ) sh sh sh k3 T k M M sh L) ) ) ) ) As we saw earler whe we frst looked at tuelg. sh ) 6E Uo Uo E) exp L) matrx method 3

24 Boud states Ca the matrx method be used to lear somethg about boud states? It requres a slghtly dfferet approach, sce a boud state does ot propagate ad so we wll ot calculate trasmsso or reflecto probabltes. A boud state s characterzed by the requremet that ψ x ± ) 0. Ths requremet meas that, the put ad output regos, the wave fucto must be the form of a decayg expoetal. 0 B [M] C 0 M 0 So the matrx procedure would be to fd the total matrx descrpto for the problem, ad the fds the roots of the M matrx elemet. EE 439 matrx method 4

25 Example - fte heght square well redux) U U o U U o 3 L/ U 0 L/ 0 x To gve a quattatve comparso, use U o ev ad L m. Usg the eve / odd approach wth solvg the trascedetal characterstc equato repeatedly gve four solutos: w.3 E ev w E ev w.608 E 0.60 ev w E ev EE 439 matrx method 5

26 U Uo U Uo L/ ] [ ][ M EE 439 ][ L/ U0 0 [ 3 x ] cos kl) k k cos kl) k s kl) k s kl) cos kl) s kl) matrx method 6

27 To fd the boud states, set M 0 ad fd the roots. cos kl) k k s kl) 0 Of course, k ad α deped o eergy, so we wll be fdg partcular eerges for whch the above equato goes to 0. A easy way to see what s gog o s to make a plot. 6! 5! Just from the plot, we see that the approxmate eerges are: M! 4! 3!!! 0! -! -! 0! 0.! 0.! 0.3! 0.4! 0.5! 0.6! 0.7! 0.8! 0.9!! Eergy ev)! E 0.06 ev E 0.5 ev E ev E ev Wth just a bt of effort, the umbers ca be made more precse. EE 439 matrx method 7