I K J K J. Chapter 1. Problem Solutions. Semiconductor Physics and Devices: Basic Principles, 3 rd edition Chapter 1

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1 Semcouc hyscs evces: sc rcles, r eo her. () cc: 8 cer oms /8 om 6 ce oms ½ oms ol o 4 oms er u cell () cc: 8 cer oms /8 om eclose om om ol o oms er u cell mo: 8 cer oms /8 om 6 ce oms ½ oms 4 eclose oms 4 oms ol o 8 oms er u cell. () 4 G oms er u cell 4 esy esy o G.0 4 As oms er u cell, so h esy o As.0 () 8 Ge oms er u cell 8 esy esy o Ge () Smle cuc lce; r U cell vol r 8 r om er cell, so om vol. () G 4πr 4πr o 00% 8r () ce-ceere cuc lce 4r c h o 5.4% r U cell vol r 6 r her 4 oms er cell, so om vol. G πr r 4 4 π o 00% 6 r oy-ceere cuc lce 4 4r r U cell vol. r o 74% oms er cell, so om vol. G πr 4 r 4 π o 00% o 68% 4r () mo lce 8 oy ol 8r r U cell vol. 8r 8 oms er cell, so om vol. 8 4 r 8 4 π o 00% 8r πr o 4%.4 rom rolem., erce volume o cc oms s 74%; heree er coee s rou, olume 074.

2 Semcouc hyscs evces: sc rcles, r eo her.5 () 54. A rom., 8 r 54. so h r 8. A 8 8 eer o oe slco om o ceer o eres eh r.6 A () umer esy 8 esy ss esy ( A. W. ) 50 ( 809. ) ρ A ρ. rms /.6 () r ( 0. ).04 A A r + r r A so h r A () A-ye; om er u cell esy () esy(a) ye: om er u cell, so esy() A : esy l: esy (sme s ) () : A.W..99 l: A. W So, mss er u cell mss esy s ρ.8 () so h ρ. m / A 4.6 A esy o A esy o () Sme s () Sme merl () Surce esy Sme A oms oms () Sme s () Sme merl.0 () ol esy o Surce esy () Sme s (). Skech. () (),,,, 4 4 o ( ) ( )

3 Semcouc hyscs evces: sc rcles, r eo her. () sce ewee eres (00) les s: 56. A ()sce ewee eres (0) les s: A sce ewee eres () les s: A.4 () Smle cuc: 4.50 A () (00) le, surce esy, om () (0) le, surce esy, om () () le, surce esy, oms 6 () c h () oy-ceere cuc () (00) le, surce esy, Sme s (),(); surce esy () (0) le, surce esy, oms () () le, surce esy, Sme s (),(), surce esy ce ceere cuc () (00) le, surce esy oms () (0) le, surce esy, oms () () le, surce esy, () (00) le o slco smlr o cc, oms surce esy () (0) le, surce esy, 4 oms () le, surce esy, 4 oms r he 4r A () 4 oms olume esy () sce ewee (0) les,

4 Semcouc hyscs evces: sc rcles, r eo her 4.50 A Surce esy oms esy o slco oms 50 4 vlece elecros er om, so esy o vlece elecros 0.8 esy o GAs oms 8 oms A vere o 4 vlece elecros er om, esy o vlece elecros ( 098. ) () rco y weh 50 ( 806. ) () rco y weh 8 0 ( 0. 8) olume esy 0 So A We hve 54. A So () ercee % 0 () ercee % % 00% 6

5 Semcouc hyscs evces: sc rcles, r eo her her. omuer lo. omuer lo. omuer lo.4 π rolem.; hse ω cos λ π ω 0 v ω λ λ π + π rolem.; hse + ω cos λ π + ω 0 v ω λ λ π.5 hc hc hν λ λ Gol: e ( 4. 90) J So λ λ 054. µ m esum: 90. e J So λ λ µ m () lecro: ().. e J 9 m k m/ s 4 h λ λ. A ().. 00 e J m k m/ s λ h λ. A () roo:.. e J 7 9 m k m/ s 4 h λ.0 λ 087. A use Aom: A. W. 8.9 e J m k m/ s 4 h λ. 0 λ 00. A () A 000 k rvel 0 m/s: mv ( 000)( 0 ) 40 4 k m/ s 4 h λ 4 40 λ A 9

6 Semcouc hyscs evces: sc rcles, r eo her.7 k v e v m v v (. ) k m/ s v 4 h λ λ 9. A.8 hc hν λ h e e e m λ e Se λ λ e 0 e hc h h λ m λ m λ e 0 whch yels h λ 00 mc So e J hc hc mc mc λ 00h J 0. ke.9 () mv h m G λ e 80. J 4. 0 e Also mv k m/ s 4 h λ λ 64 A () 4 h λ k m/ s Also v m/ s m 9.0 v / s 4 mv J e hc () hν 0 λ J 5 9 e so.40.4 k 5 () m k m/ s 4 h λ λ 0. A

7 Semcouc hyscs evces: sc rcles, r eo her. 4 h () k m/ s () hc hc c λ h So c( ) J 098. e. 4 () h k m/ s () 6 m J e. () Sme s. (), k m/ s () 6 m J e.4 4 h mv v m 500 v 70 6 m/ s.5 4 h () k m/ s 4 () () s.6 () Ψ (, ) Ψ, Schroer s wve equo, he re soluos o h Ψ, Ψ, + () Ψ (, ) jh m h Ψ (, ) Ψ, + () Ψ (, ) jh m A he wo equos, we o h Ψ(, ) + Ψ (, ) m + Ψ, + Ψ, jh Ψ(, ) + Ψ (, ) whch s Schroer s wve equo. So Ψ(, ) + Ψ (, ) s lso soluo. () Ψ Ψ were soluo o Schroer s wve equo, he we coul wre h Ψ Ψ+ () Ψ Ψ m jh Ψ Ψ whch c e wre s h + + Ψ Ψ Ψ Ψ Ψ Ψ m Ψ Ψ + Ψ Ψ jh Ψ + Ψ v y Ψ Ψ we h () + Ψ Ψ Ψ Ψ + m Ψ Ψ ΨΨ () + Ψ Ψ jh + Ψ Ψ

8 Semcouc hyscs evces: sc rcles, r eo her Sce Ψ s soluo, he h Ψ + () h Ψ j m Ψ Ψ Surc hese ls wo equos, we re le wh h Ψ Ψ Ψ + m Ψ ΨΨ Ψ jh Ψ Sce Ψ s lso soluo, we my wre h Ψ Ψ + () jh m Ψ Ψ Surc hese ls wo equos, we o h Ψ Ψ () 0 m ΨΨ hs equo s o ecessrly vl, whch mes h ΨΨ s, eerl, o soluo o Schroer s wve equo..7 Ψ, As π e jω z + + Ψ, A s π z + A ( ) s π 4π whch yels A A +,, + j, j.8 Ψ, As π e jω z Ψ, A s π + A ( ) s π 4π 0 whch yels A z 0.9 oe h * z 0 Ψ Ψ uco hs ee mlze () o 4 z e 0 z 4 o o o e o 0 o o o e o o 0 o o e e 4 whch yels 09. () o z e o 4 z o 4 o o o e o o o e o J e e whch yels 09. o z e z 0 o o o o e o 0 o o e whch yels J 4 o o 4 o e o o 0 A +,, + j, j

9 Semcouc hyscs evces: sc rcles, r eo her.0 () k ω cos k ω 0 v + ω k v 5. 0 m s / 9. v s 06 / () π π π k λ λ k λ 49. A Also 4 h λ k m/ s hc hν 0 λ J.960 e. ψ() A jk+ ω where k m h e k m ω h ω.80 r / s. h 4 π π m so J e e e 8. 0 e. () h π m π ( J ) So J 06. e J 04. e () hc hc hν λ λ λ λ m λ 59. µ m.4 () he e oel well h π m m h π so π 56

10 Semcouc hyscs evces: sc rcles, r eo her 5. 0 () h π m ( + ) h π ( + ) m π () J ery he (+) se s Joules lrer h 0 mj. Quum eecs woul o e oservle..5 euro : 4 h π π 7 4 m e elecro he sme oel well: π e.6 Schroer s wve equo ψ () m + ( ()) ψ h We kow h ψ() 0 () 0 + so hs reo ψ () m + ψ () 0 h Soluo s o he m ψ() Acos + s 0 m where h oury coos: ψ() 0 +, So, rs moe: ψ () Acos where π so π h m Seco moe: ψ () s where π so hr moe: ψ () Acos where π so ourh moe: ψ 4 () s where 4π so 4 4 π h m 9 π h m 6π h m.7 he - wve equo cres coes, (,y,z) ψ(, y, z) ψ(, y, z) ψ(, y, z) y z m + ψ (, y, z) 0 h Use sero o vrles, so le ψ(, y, z) X()()() Y y Z z Susu o he wve equo, we e X Y Z m YZ + XZ + XY + XYZ 0 y z h m v y XYZ le k, we h o X Y Z () k 0 X Y y Z z We my se 4

11 Semcouc hyscs evces: sc rcles, r eo her X X k so + k X 0 X Soluo s o he m X() Ask + cosk oury coos: X() π X( ) 0 k where,,,... Smlrly, le Y Z k y k z Y y Z z Aly he oury coos, we π y k y, y,,,... π z k z, z,,,... rom quo () ove, we hve k k k + k 0 y z m k + k + k k y z h so h h π m + + y z y z.8 he -mesol e oel well: + ψ, y ψ, y m + ψ(, y) 0 y h e ψ(, y) X()() Y y susu, X Y m Y + X + XY 0 y h ve y XY So X Y m X Y y h e X k X X + k 0 X Soluo s o he m: X Ask + cosk u X( 0) 0 0 So X Ask Also, X( ) 0 k π Where,,,... π So h k We c lso ee Y k y Y y Soluo s o he m Y sk y + cos k y y y u Y( y 0) 0 0 Y( y ) 0 k π y y so h π y k y m k k + 0 y h whch yels h y y + m π J Smlres: eery s quze erece: ow uco o eers.9 () ervo o eery levels ecly he sme s he e. h π () m, h π m 5

12 Semcouc hyscs evces: sc rcles, r eo her () 4 A π e () π e.0 () reo, > 0 ψ () m + ψ () 0 h Geerl m o he soluo s ψ () A e j + e j where m h erm wh rereses ce wve, erm wh A rereses he relece wve. eo, < 0 ψ () m + ψ () 0 h he eerl soluo s o he m ψ () A ej + ej where m h erm volv rereses he rsme wve, he erm volv A rereses he relece wve; u rcle s rsme o reo, wll o e relece so h A 0. ψ () ej ψ () A ej + ej () oury coos: () ψ 0 ψ 0 () () ψ () ψ 0 0 Aly he oury coos o he soluos, we A + A om hese wo equos, we A + he releco coece s * AA * + he rsmsso coece s 4 +. reo, > 0, we hve ψ () A e where m h.4 e. e S m roly comre o 0, ve y ψ () e ψ () 0 () A 9 0 e % () 48 A 9 0 e %. 6 e,. e We hve h U W / 6

13 Semcouc hyscs evces: sc rcles, r eo her G 6 e where m h / ( 6.) 60 S m 0 0 m 9 U W 9 0 e m Assume h quo [.6] s vl: G 6 e () m S m h m o 9 U W 0067 (. ) 9.0 ( ) 60. / m e () m ( 08. ) S 9 0 m o m 9 4 U W / e e 6 e 4 0 0, 0, 0 m m k m h ( 0 ) S m So 6 e U W / eo, 0 ( < 0 ); eo, ( 0 < < ) ; eo, 0 ( > ). () eo ; ψ () A ej + ej (ce) (relece) eo ; ψ () A e + e eo ; ψ () A e j + ej () reo, he erm rereses relece wve. owever, oce rcle s rsme o reo, here wll o e relece wve whch mes h 0. oury coos: 0: ψ ψ A + A + ψ ψ j A j A : ψ ψ A e + e A ej A lso 7

14 Semcouc hyscs evces: sc rcles, r eo her ψ ψ Ae e j A ej rsmsso coece s ee s * AA * AA so rom he oury coos, we w o solve A erms o A. Solv A erms o A, we A ja + e e 4 l s j e + e e j We he h l + 4 e+ e r m * * AA AA e 4 e We hve h sce >>, he wll e lre so h e>> e we c wre AA * l +4 * AA e 4 whch ecomes e r * * AA AA + e 4 Susu he eressos, we m + h m m h h h m J h * m AA e * h m 6 J h * AA G 6 e AA AA * G 6 m AA lly e *.6 eo : 0 ψ m + ψ 0 h ψ A ej + ej (ce wve) (relece wve) m where h eo : ψ m + ψ 0 h ψ A e j + ej (rsme (relece wve) wve) m where h eo : ψ m + ψ 0 h ψ A e j (rsme wve) m where h here s o relece wve reo. 8

15 Semcouc hyscs evces: sc rcles, r eo her he rsmsso coece s ee s * * v AA AA * * v AA AA rom oury coos, solve A erms o A. he oury coos re: 0: ψ ψ A + A + ψ ψ A A : ψ ψ A e j+ e j A ej ψ ψ Aej e j Aej u π ej e j, elm, A, rom he ove equos, we hve () eo : Sce >, we c wre ψ m ψ 0 h eo : 0, so ψ m + ψ 0 h eo : ψ 0 he eerl soluos c e wre, kee m h ψ mus rem e < 0, s ψ e+ ψ A s + cos ψ 0 where m h m h () oury coos: 0: ψ ψ ψ ψ : ψ ψ A A s + cos 0 A A A sce, he G A rom A, we c wre G G whch ves G ur, hs equo c e wre s m h m h hs ls equo s vl oly secc vlues o he ol eery. he eery levels re quze..8 4 me o 4π h o me o 4π h o ( J) ( e ) π 58. ( e ) 9

16 Semcouc hyscs evces: sc rcles, r eo her 58. e. 95e. 5e e 4.9 We hve / ψ 00 π r G e o o * r r r o G 4π ψ ψ 4π e o π 4 r r e o o o he mmum roly () r 0 r 4 + r G r G r e r e S o o o o UW whch ves 0 r + r o o r o s he rus h ves he rees roly..40 ψ 00 s eee o θ φ, so he wve equo shercl coes reuces o mo ψ r + ( 0 () r ) ψ r r r h where () r ψ 00 e r h 4π m r π o o o / r G e o o / r o G o G e o ψ 00 r π r ψ 00 r π o so h r ψ 00 r r 5 / r e G r 5 / G G G G G o o o o G / h G o e o o π o o r r r r e e π o o o o Susu o he wve equo, we hve 5 / r r r r e e r π m r + + h mr where 4 me o 4π h o o h m he ove equo ecomes / π r G r e r r + m o h o π S U W o o o o h m / h + 0 mr o o o o S r G e o o UW o r r o o o o whch ves 0 0, shows h ψ 00 s ee soluo o he wve equo..4 All elemes rom Grou colum o he eroc le. All hve oe vlece elecro he ouer shell. 0 0

17 Semcouc hyscs evces: sc rcles, r eo her her. o were o crese, he eery woul ecrese he merl woul e o ehve less lke semcouc me lke mel. o were o ecrese, he eery woul crese he merl woul e o ehve me lke sul.. Schroer s wve equo h Ψ(, ) Ψ(, ) + () Ψ(, ) jh m e he soluo e o he m Ψ, u e j k h eo, () 0, so susu he roose soluo o he wve equo, we o h { jku e j k m h + () () U u e j k h W j () j h ue j k h h jk u j k h () e j k h () U u e j k h W + () u e j k h whch ecomes h { e m + jk u + hs equo c he e wre s () + + () ku () jk u u m + u () 0 h Se u () u () reo, hs equo ecomes () () u jk u + k α u () 0 where α m h Q... reo, (). Assume he sme m o he soluo Ψ, u e j k h Susu o Schroer s wve equo, we o h { jk u e j k m h () e j k h () U u e j k h W + () u e j k h () u e j k h + jk u + hs equo c e wre s () + + () ku () jk u u m m () + () 0 u u h h Se u () u () reo, hs equo ecomes u() jk u () + where α m h m k α + u 0 h () Q...

18 Semcouc hyscs evces: sc rcles, r eo her. We hve u () jk u + () k α u () 0 he roose soluo s u () A e j ( α k ) + e j ( α + k ) he rs ervve s u () j ( α k ) Ae j ( α k ) j( α + k) e j( α + k) he seco ervve ecomes u() j( α k) Ae j( α k) + j( α + k) e j( α + k) Susu hese equos o he erel equo, we α k Ae j α k ( α + k) e j( α + k) + jk{ j( α k) A e j( α k) j( α + k) e j( α + k) } k α { Ae j( α k) + e j( α + k) } 0 l k q α Ae j( α k) + m cα + αk + kh + kα + k k αqe j( α + k) om erms, we hve α αk + k k α k We h 0 0 Q... he erel equo u () he roose soluo, he roceure s ecly he sme s ove..4 We hve he soluos u () A e j ( α k ) + e j ( α + k ) 0 < < u ( ) e j ( β k ) + e j ( β + k ) < < 0 he oury coos: u () 0 u () 0 whch yels 0 A+ 0 Also u u 0 0 whch yels ( α k) A ( α + k) ( β k) + ( β + k) 0 he hr oury coo s u () u ( ) whch ves Ae j α k + e j α + k ( β ) e ( β ) e j k + j + k hs ecomes Ae j α k + e j α + k ( β ) e ( β ) 0 e j k j + k he ls oury coo s u u whch ves j( α k) Ae j( α k) j( α + k) e j( α + k) j( β k) e j( β k) j( β + k) e j( β + k) hs ecomes α k Ae j α k ( α + k) e j( α + k) ( β ) e ( β ) ( β ) ( β ) k j k + + k e j + k 0.5 omuer lo.6 omuer lo.7 s α + cosα cos k α e k y, α s + cos cos y oser o hs uco y 4

19 Semcouc hyscs evces: sc rcles, r eo her { s + cos } s y We o ( )() + () s cos y y S S s y UW y s y + y s cos s s y k π, 0,,,... s y 0 So h, eerl, he ( α) α 0 y ( k) k A / m α m m α h k h h k hs mles h α 0 k π k k.8 α ( α) 9 s + cosα cos k α () k π cos k s o: α π : o: α 66. π ( o y rl err) m α h α h m So ( α) α J α 054. e So α π 504. e α 66. π 4.45 e UW y.64 e () k π cos k + s o: α π o: α.54 π e e 4 so 69. e k π cos k s o: α π o: α 44. π 57. e e 6 so 4.6 e () k 4π cos k + s o: α 4 π o: α 4.7 π e e 8 so 4.66 e.9 () 0 < k < π k 0 cos k + y rl err: s o: α 08. π o: α π rom rolem.8, α 054. e 06. e 504. e so e () π < k < π Us resuls o rolem.8 s o: α 66. π o: α π 5

20 Semcouc hyscs evces: sc rcles, r eo her 4.45 e e 4 so 87. e π < k < π s o: α.54 π o: α π e e 6 so 8. e () π < k < 4π s o: α 44. π o: α 4 π e e so 67. e.0 6 s α + cosα cos k α e eery s () k π cos k s o: α π o: α 56. π (y rl err) rom rolem.8, ( α) ( 054. ) e 504. e 660. e so.6 e () k π cos k + s o: α π o: α.4 π e e 4 so.79 e k π cos k s o: α π o: α. π 57. e e 6 so 4. e () k 4π cos k + s o: α 4 π o: α 4.6 π e e 8 so. e. Allowe eery s Use resuls rom rolem.0. () 0 < k < π s o: α π (y rl err) o: α π We hve α 054. e e 504. e so 068. e () π < k < π s o: α 56. π o: α π 660. e e 4 so.6 e π < k < π s o: α.4 π o: α π 6

21 Semcouc hyscs evces: sc rcles, r eo her e e 6 so 4.7 e () π < k < 4π s o: α. π o: α 4 π e e 8 so 7.9 e. 00 ; ( 00) e e e e e e. he eecve mss s ve y * m h k We hve h curve A> curve k k so h * * m curve A < m curve 4.4 he eecve mss hole s ve y * m h k We hve h curve A> curve k k so h * * m curve A < m curve.5 os A, : < 0 k velocy reco; os, : > 0 velocy + reco; os A, ; k < 0 os, ; > 0 k.6 k h m A k 0 A So eve eecve osve eecve k A + k 0 9 m A: ( 007. ) 6. 0 m whch yels so m k curve A; m m o : ( 07. ) 6. 0 m whch yels m k so urve : m m o.7 k h m m mss; mss; 7

22 Semcouc hyscs evces: sc rcles, r eo her * 9 k 0. A 0 m urve A: ( 008. ) 6. 0 m whch yels m m 4.40 k m o urve : ( 04. ) 6. 0 m whch yels m m k m o.8 () hν 9 ( 4. ) 6. 0 ν 4 h ν z () 8 c 0 7 λ m 4 ν 4. 0 λ µ m.9 urve A: ecve mss s cos urve : ecve mss s osve rou k 0, s eve rou k ± π..0 k k ( α) s α k k k + αs αk k So α cos αk k k cos α α k k k We hve m * h k α h m * h α. he -mesol e oel well, () 0 whe 0 < <, 0 < y <, 0 < z <. hs reo, he wve equo s + + ψ, y, z ψ, y, z ψ, y, z y z m + ψ (, y, z) 0 h Use sero o vrles echque, so le ψ(, y, z) X()()() Y y Z z Susu o he wve equo, we hve X Y Z YZ + XZ + XY y z m + 0 XYZ h v y XYZ, we o X Y Z m X Y y Z z h e X X k + k X 0 X he soluo s o he m X() As k + cos k Sce ψ(, y, z) 0 0, he X() 0 0 so h 0. Also, ψ(, y, z) 0, he X() 0 so we mus hve k π, where,,,.. Smlrly, we hve Y Z k y k z Y y Z z rom he oury coos, we 8

23 Semcouc hyscs evces: sc rcles, r eo her k π k π y y z z where y,,,... z,,,... rom he wve equo, we hve m k k k + 0 y z h he eery c he e wre s h m + + π y z. he ol umer o quum ses he - mesol oel well s ve ( k-sce) y ( kk k ) k π π where m k h We c he wre k m h k he erel, we o k m m h h Susu hese eressos o he esy o ses uco, we o ( m m ) π π h h o h h h m hs esy o ses uco c e smle wre s ( ) 4π ( m ) / h v y wll yel he esy o ses, so h. / 4π m h / * 4π m ( ) h.4 / * + k 4π m h z * / + m k / h 4π * / 4πm ( k) h / / 4π / m / * 4π m ( ) h.5 () / * 4π m h z k * / m / h k 4π * / 4πm / ( k) h / 4π (. ) / m / * 4π m ( ) h / 4π / m J 9

24 Semcouc hyscs evces: sc rcles, r eo her ( ) 7.60 e e e e e / 7 0. e * 4π m () ( ) h / 4π ( 056. ) / m J ( ).850 e ( ) 005. e 00. e 05. e 00. e.6.7 omuer lo / * m * / m.8! 0!!! 8! 0 8! ( 0)( 9)( 8! ) ( 0)( 9) ( 8! )(! ) ()() e m / * * m 45 ().0 + e () + ( ) 069. k e k + e + e k () k, ( ) ( ) 069. () 5 k, ( ) k, ( ) ( ) + e k ( ) + e k + e () + e 5 () + e 0 () k, () 5 k, ( ) k, ( ) () + e + k k. () 00 k e + e k e k 0

25 Semcouc hyscs evces: sc rcles, r eo her + ( ) k + k + ( ) k + k () 400 k ( ) k + k + ( ) k + k ( ) h 4 π π 0 m J e e, e. 5 As s romo > 0, ssume he roly o 5 se e occue s he sme s he roly o 4 se e emy e + e k k e + e k k e () -mesol e oel well, h π y z + + m y z π e y z 5 elecros, eery se creso o cos oh elecro y z emy se, so ( 076) e () elecros, eery se creso o cos oh y z elecro emy se, so e.5 he roly o se + e occue s + e + e k k he roly o se e emy s + e k e k + e + e k k + + e k ece, we hve h Q...

26 Semcouc hyscs evces: sc rcles, r eo her.6 () A eery, we w e + e k k + e k hs eresso c e wre s + e k 00. e k e k + kl ( 00) + 4.6k () A + 4.6k, 4.6k + e + e k k whch yels () 65. e, 00, A 650. e () + e % k k e ( ) 4.5% + e e k e 00. k 099. whch c e wre s e 99 k l( 99) k k l( 99) So () ( ) e %. () A 000 k e ( ) e % ( ) e % () A.9,, ( ) + e k ll emerures. e k

27 Semcouc hyscs evces: sc rcles, r eo her 00. e , e ( ) e e e 08. ( ) () e A, ( ) 07. e e k so A, e so.40 () A, A k e, he e k e e, So. e e k ( ) () e, A, ( ) 0. e e k A, e k ( ) + e k so ( ) + e k e e k k e ( ) k k + e k () 0, < e( ) 0 0 > e( + ) + 0 A.4 () A m, G + e + e S:. e, ( ) + e Ge: 066. e, k (. ) k

28 Semcouc hyscs evces: sc rcles, r eo her ( ) + e GAs: 4. e, ( ) + e (. ) (. ) 4. 0 () Us resuls o rolem.5, he swers o r () re ecly he sme s hose ve r ()..4 6 ( ) e k e 0 6 k e 0 l 0 k k 055. k 46 l A, 005. So e k l ( 9) k y symmery,, 005., So k l ( 9 ) l ( 9) k () A 00, k e l e () A 500, k e l e 4

29 Semcouc hyscs evces: sc rcles, r eo her 4 her 4 4. e k ( ) k e () Slco () Germum ( ) e k e.90 k y rl err 4. omuer lo GAs e ( ) G e k So e k k A 00 k e A 00 k e G. k e e ( 9.9) whch yels 5. e 00, ( 00) e () k e e k k e e e k, o he mmum vlue / e k 0 k / e k hs yels / k / k he mmum vlue occurs k + () e k Q e e k k e 7

30 Semcouc hyscs evces: sc rcles, r eo her 4 e k o he mmum vlue e k Sme s r (). mum occurs k k 4.6 e k e k where k + 4k + 4k e k k 0 e 4 e omuer lo 4.8 ( A) ( ) ( A) ( ) A k k A e k e e e A k e omuer lo ( A) ( ) e ( ) 4.0 * m m kl G * 4 m J * * Slco: m 056. m, m 08. m 008. e m * * Germum: m 07. m, m 055. m e m * * Gllum Arsee: m 0. 48m, m m e m 4. () m kl 4 () G m m * l e m * J m l e m 4. m k G l k l k 9.80 G 8

31 Semcouc hyscs evces: sc rcles, r eo her 4 ( ) k e omuer lo ( e) e ( ) cos, z z + e k e k z m e η so h k η k We c wre so h e e e k k he erl c he e wre s η k e whch ecomes k 4.5 e η η e( ) k 0 k z e z z z + e k e k Q e η so h k η k We c wre + e k z e k e k z ( k) η e ( η) ( k) η 0 We h η e ηe( η) η ( η ) + 0 z0 So ( k) e k 4.6 r m We hve r m * * Germum, 6, m 0. 55m r r ( 6) (. ). so r 5. 4 A he ozo eery c e wre s m m * G S (. 6 ) e 9

32 Semcouc hyscs evces: sc rcles, r eo her e r m We hve r m * * GAs,., m m r r ( ) (. ). r 04 A he ozo eery s 4.8 () m m J * G S (. 6) e (. 6 ) , > -ye () 4.9 so G kl G l e k e e Assum e k e e () 400 k e / e k e Also / e e () G kl G l e e e

33 Semcouc hyscs evces: sc rcles, r eo her 4 4. k e G kl l 04. e e So e k.8 0 G 9 e () e k e () rom rolem 4., k e 00 G kl 6. 0 G l e rom () rom () 4. () e k e () rom rolem 4., , k e G kl. 0 G l e rom () rom () slco, 00, η k We c wre η 060. / 4

34 Semcouc hyscs evces: sc rcles, r eo her 4 9 η / π π Slco, 00, 50 9 We hve η / π η / π whch ves η 58. / η. k e 4.6 he elecro cocero ( ) ( ) he olzm romo les so * / 4π m e h k * / 4πm e h k k k ee k e ( ) o mmum, se () e k / / 0 e + e / 0 e ( ) whch yels + k k he hole cocero ( ) ( ) rom he e, us he well-olzm romo, we c wre * / 4πm h e k k e k k * / 4π m e h k ee k ( ) e ( ) o he mmum o ( ), se 0. Us he resuls rom ove, we he mmum k 4.7 () Slco: We hve e k We c wre e, k e e ( 4.77) We lso hve 4

35 Semcouc hyscs evces: sc rcles, r eo her 4 e.40 k.950 A, we c wre () k, e e e( 4.77) () GAs: Assume e e e (. 4) Assume e e e ( 4.) 4.8 omuer lo 4.9 () Ge: he o level + e k e A ( ) + e k + k e

36 Semcouc hyscs evces: sc rcles, r eo her 4 4. () 0 5 () () 400 k e e Also (e) 500 k e e Also () 0 5 () () k e e (e) k e e

37 Semcouc hyscs evces: sc rcles, r eo her 4.80 Also () > -ye () S:.50 0 Ge: G GAs: A e () () 7. 0 > -ye ol oze mury cocero > -ye

38 Semcouc hyscs evces: sc rcles, r eo her k e e \ A omuer lo 4.8 omuer lo 4.9 omuer lo ye, so mjy crrer elecros 4.4 () > -ye () > -ye () < -ye () elecros: my crrer holes: mjy crrer so Acce mury cocero, 50 5 o mury cocero 4.4 G kl Germum: k( e )

39 Semcouc hyscs evces: sc rcles, r eo her 4 e + + ( ) y rl err 4.46 omuer lo e G kl Germum, e so whch yels We hve e k so 4.47 omuer lo 4.48 () m kl 4 G m m * * J ( 0059) ( 0) e m () mury oms o e e so e m 045. () -ye, so cce mures () e e k 5 0 e k so e so 4.50 e () k e

40 Semcouc hyscs evces: sc rcles, r eo her e e 5 k ( ) l e () e e so h Acce mures o e e () k 0 G l l e () kl e (), 0 5 () G kl l 045. e G so <, os mus e e G () kl () () (e) l G e G kl l G 047. e k e,.80 G kl l.80 G 09. e k e, G kl l G e 48

41 Semcouc hyscs evces: sc rcles, r eo her () k G l () () l G e k G l l G 058. e k e, l G e (e) k e,.80 G kl ye l.80 G 056. e G kl l G 094. e 49

42 Semcouc hyscs evces: sc rcles, r eo her 4 (e le lk) 50

43 Semcouc hyscs evces: sc rcles, r eo her 5 her 5 5. () () J eµ Ε GAs oe 0 6, µ 7500 / s 9 6 J 6. 0 ( 7500) 0 ( 0) () () J 0 A/ 0 6, () GAs oe 0 6, µ 0 / s J eµ Ε 6 0 ( 0) 0 ( 0) 9 6. J 4.96 A/ 5. () 0 ( 0. ) 00 Ω () σ σa A 0 σ ( 00) 0 σ 00(. Ω ) σ eµ ( 50) 4.60 () σ eµ ( 480) oe: he o coceros oe, he ssume moly vlues re vl. 5. ρ () σ eµ A σa 50 6, µ 00 / s ( 00) 50 ( 00) Ω ma () hs cse 6. 0 Ω ma 6. 0 Ε 5 (), Ε 50 / 00. A v µ Ε ( 00)( 50 ) v / s 5 (), Ε 500 / 00. A v v / s 5

44 Semcouc hyscs evces: sc rcles, r eo her () GAs: ρ kω A 0 σa σ eµ 0 7, µ 0 / s σ Ω So σa ( 500)(. 6) µ m () Slco 0 7, µ 0 / s σ Ω So σa ( 500)( 4.96) µ m 5.5 () Ε / v () v v 0 µ Ε µ Ε µ / s µ Ε ( 800)( ) v.4 s 0 / 5.6 () Slco: Ε k /, v. 06 / s s 6 v. 0 GAs, v 7.5 s 06 / s 6 v 7.50 () Slco: Ε50 k /, v 9.5 s 06 / s 6 v 9.50 GAs, v 7 s 06 / s 6 v rsc semcouc, σ e µ µ + () 0 4, µ 50 / s, µ 480 / s 9 0 σ ( ) σ ( Ω ) () 0 8, µ 00 / s, µ 0 / s 9 0 σ ( ) σ Ω 5.8 () GAs σ eµ 5 µ rom ure 5., us rl err, we 7. 0, µ 40 / s 54

45 Semcouc hyscs evces: sc rcles, r eo her () Slco: σ eµ ρ e 9 ρµ () oe: he o coceros oe r (), he ssume moly vlues re vl. 5.9 σ e µ + µ ( ) e k kl l. e ( 500) 0 e ( ) σ ( ) so σ Ω () () Slco: σ e µ + µ σ ( ) σ ( Ω ) () Ge: 9 σ ( ) σ. 0 ( Ω ) () GAs: 9 6 σ ( ) σ ( Ω ) () σa 4 () Ω 4 () Ω 4 () Ω 5. () ρ 5 eµ Assume µ 50 / s ( 50)( 5) () rom ure 5., 75,

46 Semcouc hyscs evces: sc rcles, r eo her 5 µ 500 / s 5, 0 5 µ 700 / s Assum over he emerure re, 00, ρ ( 500) 9.60 ρ.7 Ω 400, ρ ( 700) 9.60 ρ 9.64 Ω 5. omuer lo 5. () Ε 0 / v µ Ε v 50 0 v / s so * m v ( 08. ) J e () Εk /, v / s 4 ( 08. ) () J e e k 9 9 e >> J σ Ε eµ Ε ( 000) 0 ( 00) J 60. A/ () A 5% crese s ue o 5% crese elecro cocero. So We c wre so whch yels e e 00 k y rl err, we 456 k 5.5 () σ eµ + eµ eµ σ + eµ o he mmum coucvy, σ eµ ( ) 0 + eµ whch yels G / µ (Aswer o r ()) µ Susu o he coucvy eresso eµ σ σ + eµ µ µ m / µ µ whch smles o σ e µ µ m he rsc coucvy s ee s / 56

47 Semcouc hyscs evces: sc rcles, r eo her 5 σ σ e µ + µ e µ + µ he mmum coucvy c he e wre s σ µ µ σ m µ + µ () A () A 00, µ ( 00)( 87. ) µ 88 / s 400, µ ( 00)( 0. 65) µ 844 / s 5.6 σ eµ ρ e ρ ρ e 00. e k k k k , k k ( ) l ( 0). e µ µ µ µ µ 6 / s k k 5.8 µ / + / µ µ µ µ 67 / s 5.0 omuer lo 5. omuer lo 5. J e e () ( 09. )( 000. ) 4 50 () ( 5) whch yels 5. () J e e G J A A/ () AJ ma 57

48 Semcouc hyscs evces: sc rcles, r eo her so J e e J 5 / s e e 6 0 e J 6 A/ cos ll hree os 5.6 J ( 0) e ( 0) 0 e 4 50 J ( 0). A/ J ( 0) e ( 5) 50 e 0 J ( 0) A/ J J ( 0) + J ( 0). + J 5. A/ J e e 0 5 e.5 sce s µm, so J e 0 e ( 48) 0 e e A/.5 + J J eµ Ε + e e 8 + Ε e 8. e Ε. e 8 8. e e 8 + Ε e Ε 5.9 J J + J, r, () J e, () 0 5 e where µ m so 5 J e e, 0 58

49 Semcouc hyscs evces: sc rcles, r eo her ( ) 0 J e, A+ e. 0 7 e J + 6. e A/, hs equo s vl ll, so () A J J J, r, A Also J e r, e J eµ Ε 0 7. r, e ( 000) 0 Ε whch yels e A 0, eµ () 0 Ε 50 whch yels so h ( 8000)( )( A+ ) Ε e / whch yels () J e () + e () µ Ε () e () 5 5 µ 8000 / s so h A 0, () ( 0 059)( 8000) 07. / s r () ( 8000)( ) ( ) A 50 µ m, 9 () ( 07) ( 50) e 5. whch yels 4 7 () () +. 0 Soluo s o he m A 50 µ m, J eµ ( 50) Ε r () 9 5 A+ e. 60 ( 8000) ( ) so h () e J ( 50) 94.9 A/ r J ( 50) Susu o he erel equo, we hve J ( 50) 5. A/ 59

50 Semcouc hyscs evces: sc rcles, r eo her 5 5. e k () +, 04. so h So e G k () J e G e e k k Assume 00, k e, J e G G J e () A 0, J.950 A/ () A 5 µ m, J 7. A/ 5. () J eµ Ε + e Ε + G where 00 0 We Ε 6. Ε Ε Solv he elecrc el, we Ε () J 0 A/ 0. 6 Ε Ε 5. () J eµ Ε + e e o e ( α), J 0 0 µ e α Ε + α e α o o 0 Ε + α Sce So µ k µ e Ε α k e () /α z Ε 0 k α e α / k α z e 0 α k so h e 5.4 rom mle ( ) Ε 0 ( ) z Ε ( ) z 60

51 Semcouc hyscs evces: sc rcles, r eo her l 0 0 ( ) l ( 0. ) l ( ).7 m 5.5 rom quo [5.40] () () 4 0 k Ε e () 000 ( ) () () () 0 Soluo s o he m () A e( α ) () Aα e ( α ) Susu o he erel equo Aαe α Ae α 0 whch yels α A 0, he cul vlue o () 0 s rrry. 5.6 () J J + J 0 r J e e () e ( ) e o We hve k µ ( 6000)( ) / s e J ( 55. 4) 5 0 e J e A/ 0 () 0 J + J r J eµ Ε r e Ε 48Ε e We hve J J r so 48Ε e e whch yels 5.7 omuer lo Ε.580 / 5.8 k () µ ( 95)( ) e so. 96 / s () 8. / s 8. µ µ 09 / s We hve 0 0 m, 4 5 W 0 0 m, 0 0 m () We hve 0 0 m 6 z e.9 m, ma 0 A

52 Semcouc hyscs evces: sc rcles, r eo her 5 () 5.40 () ().90 W / z e 05. m W 0 µ / e W ( 0. ) 0 50 µ 0. 5 m / s 5 / s 5.4 () osve -ye () z z e e m e W µ ( 5) 0 0 µ m / s 87 / s 5.4 () W m () eve -ye z e m 4.90 () µ e W (. 5) µ 00. m / s 00 / s 5.4 () eve -ye z () e µ µ 88 / s e W 9 4 () σ eµ. 60 ( 88) ( ) ρ ρ 088(. Ω ) 6

53 Semcouc hyscs evces: sc rcles, r eo her 6 her ye semcouc, low-jeco so h δ 50 6 τ s 6. () τ s () δ 0 7 τ s so s 6. () ecomo res re equl τ τ So τ 00 + τ s 4 () Geero e ecomo e So 4.50 G G s 6 00 G s hc () hν 0 λ J hs s he eery o hoo. W J / s hoos/s + olume ()( 0. ) eh rs/ s () 9 6 δ δ τ δ δ We hve + + τ J eµ Ε e he hole rcle curre esy s J + µ Ε ( + e) + µ ( Ε ) We c wre ( Ε) Ε + Ε so 65

54 Semcouc hyscs evces: sc rcles, r eo her 6 + µ ( Ε + Ε) ( + ) µ Ε Ε + + τ We c he wre µ ( Ε + Ε ) + τ 6.6 rom quo [6.8] + + τ sey-se, oe-mesol cse, s 6.7 rom quo [6.8], s 6.8 We hve he couy equos () ( δ ) µ Ε ( δ) + Ε + ( δ ) τ () ( δ ) + µ Ε ( δ) + Ε + ( δ) τ y chre eurly δ δ δ δ δ ( δ) ( δ) ( δ) ( δ ) Also, τ τ we c wre () ( δ ) µ Ε ( δ) + Ε + δ () δ + µ Ε δ + Ε + δ ully quo () y µ quo () y µ, he he wo equos. We µ + µ ( δ) + µµ ( ) Ε ( δ) + µ + µ µ + µ ve y µ + µ, he µ + µ ( δ) µ + µ J µµ ( ) + Ε δ µ + µ ee µ + µ µ + µ µµ µ µ + µ + δ ( ) + + δ we hve + + ( δ) ( δ ) µ Ε ( δ) ( ) Q... 66

55 Semcouc hyscs evces: sc rcles, r eo her Ge: 00, Also We hve µ 900, µ 900 0, 49. ( + ) ( 0) ( 49.) / s Also µµ ( ) µ µ + µ ( 900) ( 900) 6. 0 µ 868 / s τ τ τ 4 µ s whch yels τ 54 µ s 6.0 σ eµ + eµ Wh ecess crrers rese +δ +δ -ye semcouc, we c wre δ δ δ σ eµ + δ + eµ + δ σ eµ eµ e µ µ δ so σ e µ µ δ sey-se, δ τ So h σ e µ + µ τ 6. -ye, so h my crrers re holes. Um eero hrouhou he smle mes we hve δ ( δ) τ omoeeous soluo s o he m ( δ) A G e τ J he rculr soluo s ( δ) τ so h he ol soluo s ( δ) τ A G + e τ J A 0, δ 0 so h 0 τ + A A τ τ J e e e eµ e µ µ δ ( 000) ( ) 50 0 e τ J δ τ e he coucvy s σ µ µ µ µ δ so σ

56 Semcouc hyscs evces: sc rcles, r eo her 6 σ e where τ 0 7 s τ J 6. -ye GAs: σ eµ + µ ( δ) sey-se, δ τ. 9 7 σ. 60 ( ) 0 0 σ 057(. Ω ) he sey-se ecess crrer recomo re 0 s 6. < 0, sey-se, so 7 δ() 0 τ 50 0 () δ σ eµ + e µ + µ ( δ) 0, δ δ() 0 e τ 9 6 σ ( ) 5. 0 e τ σ e τ We hve h Aσ AJ AσΕ so 4 0 () e τ ( 00. ) e τ ma where τ 0 7 s 6.4 () -ye GAs, + + δ ( δ ) ( δ ) µ Ε ( δ ) τ Um eero re, so h ( δ) ( δ) 0, he δ ( δ) τ he soluo s o he m δ τ e τ δ e τ τ () mum vlue sey-se, So ( δ) ( δ) τ τ s τ 0 4 eerme whch () δ ( 075. ) 0 4 We hve e τ whch yels τ l 9. µ s 075. () δ We τ l 069. µ s 05. () δ We τ l 088. µ s ()

57 Semcouc hyscs evces: sc rcles, r eo her 6.50 τ τ 0 τ.50 7 s 4 δ 0 7 τ s ecomo re creses y he c () rom r (), τ.50 7 s 6.6 Slco, -ye s δ τ τ e δ 0 e τ e τ A 0 7 s, e δ ( ) δ 7 > 0 7 s, 7 0 δ 6. 0 e τ where τ s () 0 < < 0 6 s δ τ e τ e τ 4 δ 0 e τ where τ 0 6 s A 0 6 s δ µ s 0 4 e δ µ s > 0 6 s 6 δ e τ () () A 0, δ 0 () A 0 6 s, δ () A, δ ye, my crrers re elecros sey-se, ( δ) 0, he () ( δ ) δ 0 τ ( δ) δ 0 Soluo s o he m δ Ae + e + u δ 0 s so h 0. A 0, δ 0 δ 0 e k τ, where µ e / s µ m. Q 69

58 Semcouc hyscs evces: sc rcles, r eo her 6 () J e ( δ ) e 0 e e J.6 e ma/ 6.9 () -ye slco, () cess my crrer cocero δ A 0, 0 so h δ () he oe-mesol cse, ( δ ) δ 0 τ ( δ) δ 0 where τ he eerl soluo s o he m δ Ae + e+, δ rems e, so h 0. he soluo s δ e 6.0 -ye so elecros re he my crrers + + δ δ ( δ ) µ Ε ( δ ) τ ( δ) sey se, 0, Ε 0, so we hve ( δ ) δ δ δ 0 τ where τ he soluo s o he m δ Ae + e + 0 >0, 0 he ecess cocero δ mus rem e, so h 0. A 0, δ () 0 0 5, so he soluo s δ 0 e 5 We hve h µ 050 / s, he k µ / s e τ µ m () lecro uso curre esy 0 J e ( δ ) 0 e 0 e e J 094. A/ Sce δ δ, ecess holes use he sme re s ecess elecros, he J ( 0) A/ () A, J e 5 ( δ ) e0 ( ) e e J 044. A/ J A/ 6. -ye, so we hve ( δ ) ( δ) δ µ Ε τ Assume he soluo s o he m δ Ae s 0 70

59 Semcouc hyscs evces: sc rcles, r eo her 6 ( δ) As e ( s), ( δ ) As e ( s) Susu o he erel equo Ae( s) Ase( s) µ Ε Ase( s) 0 τ s µ Ε s 0 τ v y µ s Ε s 0 he soluo s s µ µ 4 s Ε ± Ε + hs c e rewre s J µ Ε µ Ε s ± + We my ee µ Ε β s β ± + β er h δ 0 > 0, use he mus s > 0 he lus s < 0. he soluo s δ() > 0 δ() Aes < 0 + where s ± + β β ± 6. omuer lo J 6. () rom quo [6.55], ( δ ) ( δ) δ + µ Ε τ ( δ ) µ ( δ ) + Ε δ 0 We hve h k µ e so we c ee 0 µ Ε Ε k e we c wre ( δ) ( δ) δ + 0 Soluo wll e o he m δ δ() 0 e ( α) where α > 0 ( δ) ( δ) αδ ( ) α ( δ) Susu o he erel equo, we hve δ α ( δ) + α ( δ) 0 α α 0 whch yels α S oe h Ε () + + U W 0,, he α τ where µ e / s m µ Ε /, he 7

60 Semcouc hyscs evces: sc rcles, r eo her 6 k e Ε α ce o he elecros ue o he elecrc el s he eve -reco. heree, he eecve uso o he elecros s reuce he cocero ros o ser wh he le elecrc el ye so he my crrers re elecros, he + + δ ( δ ) ( δ ) µ Ε ( δ ) τ Um llumo mes h δ δ 0. τ, we re le wh ( δ) whch ves δ + < 0, δ 0 whch mes h 0. δ G 0 >, 0 so we hve ( δ ) 0 r δ G (o recomo) 6.5 -ye so my crrers re holes, he + δ δ ( δ ) µ Ε ( δ ) τ ( δ) We hve τ, Ε 0, 0 (sey se). we hve ( δ ) ( δ) 0 + < < +, G cos. ( δ) G + G δ + + < <, 0 so we hve ( ) so h δ δ 0 δ + 4 < <, 0 so h ( δ) ( δ) 0,, 5 δ he oury coos re () δ 0 + ; () δ 0 ; () δ couous + ; (4) δ couous ; he lu mus e couous so h ( δ) ( δ) (5) couous + ; (6) couous. Aly hese oury coos, we G δ 5 < < + G δ G δ ( ) < < ( + ) < < µ 875 / s Ε µ Ε ( ) whch ves / s rom he se relo, k µ e

61 Semcouc hyscs evces: sc rcles, r eo her / Assume h, 4π e 4 s he soluo o he erel equo G o rove: we c wre / G 4π e 4 4 / G 4π e 4 4 / + G 4π e 4 4 Also / 4π e 4 4 G / + / G 4π e 4 Susu he eressos o he erel equo, we 0 0, Q omuer lo 6.9 -ye δ δ τ We hve kl G δ G l e 4 k 6.0 () -ye + l G δ G l e G kl 5 50 G l e () δ δ k + l G δ G l e k + l G δ G l e 7

62 Semcouc hyscs evces: sc rcles, r eo her ye GAs; We hve δ δ ( 0. ) 50 5 () + kl G δ G l e We hve G kl 6 50 G l e so e () + kl G δ 5 50 G l e 6. Qus-erm level my crrer elecros k We hve 4 δ G 0 µ m 50 + l G δ elec he my crrer elecro cocero kl We 4 () 0 6 m Q 50 µ 8. 0 ( µ m) e Qus-erm level holes: we hve + kl G δ We hve 0 6, δ δ We µ m 6. () We c wre kl ( e) 0 50 G kl G δ so h ( ) + δ G k l k l + δ kl k ( 00. ) + δ e ( 00. )

63 Semcouc hyscs evces: sc rcles, r eo her 6 δ 000. low-jeco, so h δ 50 () k G l δ 50 G l e 6.4 omuer lo 6.5 omuer lo 6.6 () τ τ τ + + τ τ + τ () We h ee he e eero re s + + where sce hese re he herml equlrum eero recomo res. 0, he τ + τ +. hus eve τ + τ recomo re mles e osve eero re. so h 6.7 We hve h τ + + τ + +δ +δ, he + δ + δ τ + δ+ + τ + δ+ + δ + + ( δ ) τ δ τ δ <<, we c elec he ( δ) ; lso δ δ + τ + + τ + () -ye, >>, >> s δ τ () rsc, δ τ + τ 7 7 δ τ + τ s δ ye, >>, >> 7 δ τ 50 0 s

64 Semcouc hyscs evces: sc rcles, r eo her () rom quo [6.56], ( δ ) δ 0 + τ Soluo s o he m δ τ + Ae + e + A, δ τ, so h 0, δ τ + A e We hve ( δ ) s 0 ( δ ) 0 We c wre ( δ) A 0 ( δ) 0 τ + A A s τ + A Solv A we s A τ + s he ecess cocero s he δ τ where s + s e J 7 τ J 7 s δ 0 0 e s s 4 δ 0 e s () s 0, δ () s 000 / s, δ e J J Q 4 () s, δ 0 e () () s 0, δ () () J () s 000 / s, δ () s, δ() τ () 7 5 A , τ r δ τ 0 5 > 0 ( δ ) δ ( δ) δ 0 0 τ Soluo s o he m δ Ae + e+ A 0, δ δ A+ A W, δ 0 Ae W + e + W Solv hese wo equos, we W A δ e + ew δ ew Susu o he eerl soluo, we δ δ e+ W ew ke + ( W ) e ( W ) δ sh ( W ) δ sh W where δ µ m () τ, we hve ( δ) 0 so he soluo s o he m δ + 76

65 Semcouc hyscs evces: sc rcles, r eo her 6 Aly he oury coos, we δ δ W 6.40 τ, we hve ( δ) δ 0 so h A δ A + A W δ s ( ) W δ W A saw ( + ) whch yels A + sw s A 0, he lu o ecess holes s 0 9 δ 0 A so h A sw 0 +W s s he soluo s ow 8 0 δ 0 W + s () s, 8 4 δ 0 00 () s 0 / s δ W < < 0, ( δ ) G 0 + so h ( δ) G + G δ < < W, ( δ) ( δ) 0, so, δ + 4 he oury coos re: () s 0 W, so h ( δ ) W 0 () s + W, so h δ( W) 0 () δ couous 0 ( δ) (4) couous 0 Aly he oury coos, we GW GW, + 4, W < < 0 G δ W + W 0 < < + W GW δ ( W ) 6.4 omuer lo 77

66 Semcouc hyscs evces: sc rcles, r eo her 6 (e le lk) 78

67 Semcouc hyscs evces: sc rcles, r eo her 7 her 7 7. G l where We () 0 5 () ( ) v () () ( ) v S: Ge:.40 GAs: G l () 4 7 0, 0 S: 065., Ge: 05., GAs: 0. () , 50 S: 0778., Ge: 096., GAs: , 0 S: 084., Ge: 04., GAs: omuer lo 7.4 omuer lo 7.5 () -se: G kl -se: 5 50 G l e G kl 7 0 G l e () G l ( 0059). l

68 Semcouc hyscs evces: sc rcles, r eo her 7 () G e + 4 (. ). (. ) µ m / µ m We hve e Ε m Ε m / () -se 6 0 G l e -se 6 0 G l e () Q Q / / () G l ( 0059). l µ m y symmery 054. µ m e Ε m Ε m / e () k e (-reo) e k e (-reo) G l Q / 84

69 Semcouc hyscs evces: sc rcles, r eo her 7 ( 0059). l () GAs: 0., W Also 05. G l ( ) l G e () e (. ) G 4 e µ m / / () µ m (e) e Ε m e 7.9 () () Ε m / ( 0059). l G e + 4 (. ). (. ) µ m G e + 4 / (. ). (. ) µ m e Ε m / Ε m / Q Q / / 85

70 Semcouc hyscs evces: sc rcles, r eo her G l e k We c wre 00 l l l l l + 00 k + k l l Q Q l l l y rl err () G l ( 0059). l () % che, ssume h he che s ue o, where he mj eeece o emerure s ve y e k l l l l l l l l k l l k l 9 9 U l k W l 9 9 U l W 7 7 l / l k k We c wre k k so h k ( 0059). 00 We he 0.4 k 86

71 Semcouc hyscs evces: sc rcles, r eo her 7 7. () 0 6, G kl 6 0 G l e 0 5, 5 0 G l e () () G l ( 0059). l Q Q / / () e Ε m ( 7. ) Ε m / 7.4 Assume Slco, so k / e / /. G () 80 4, µ m ().0 6, µ m 80 7, µ m () () Also () 096. µ m () 078. µ m µ m () G Q 7 80 / 87

72 Semcouc hyscs evces: sc rcles, r eo her 7 () omuer lo 7.6 () () W G l ( 0059). l 067. S / + + e W Q U W / 0, W , W Ε m W 0, Ε m / 8, Ε / m / () Also G / e µ m e + G / µ m Also W + W 00. µ m. Ε m + ( 5856) W () Ε m / A W Q Q / / 7.7 () G l ( 0059). l () G l ( 0059). l

73 Semcouc hyscs evces: sc rcles, r eo her 7 We c wre e e () W G / e µ m Q / / e Q / 7.9 () elec che S >> 44. so 4.4% che. U W / / () k G l k l k l k + G k l l + So we c wre hs s k l so kl ( ) l 7.95 m 7.0 () W A W W A W We / + + A A e A / + + e + + A A A G 8 5 G 6 / l A l So we W( A) W( ) W A W. / 89

74 Semcouc hyscs evces: sc rcles, r eo her 7 () Ε A Ε j j ( A) ( ) + A W( A) + W Ε A Ε j W W A A + + / A + + A A / G G + G + A A A + 5 G 8 6 / ( A) ( ) j () ( 0059). l so / + Ε m e Q / () l Q Q whch yels l whch yels () We hve () 0 j ( 0) j () 0 j ( 0) j Q Q / / + / 90

75 Semcouc hyscs evces: sc rcles, r eo her 7 0, we (. ) () 0W l so l We c he wre e ( ) ( ) l / j + / j + So 0. + () / 7.4 / e + + ( 0059). l >>, we hve /. 60 ( 7. ) /. +, / 0, / A 60 4, he, 4. 0, 66. he reso requecy s ve y π so h, 67. z 0,.6 z 7.5 e Ε m + juco, so h / + e / e Ε m + Assum h <<, he 9

76 Semcouc hyscs evces: sc rcles, r eo her 7 Ε m e 6. 0 ( 0) W 0 + whch yels 9 We c wre G l.. 9 ( ) l We lso hve / j 4 A so / 9 e Whch ecomes ( 7. ) , he y ero we 7.7 () G l ( 0059). l () Also G e + 4 (. ). (. ) G e + / (. ). (. ) µ m, we hve / whch ecomes We Q Q / / Q / 7.8 A + juco wh 0 4, () A oe-se juco ssume >>, he so e / 9

77 Semcouc hyscs evces: sc rcles, r eo her 7 7 ( ) whch yels 9 () G so G µ m Ε m + ( 9) W () Ε m / ( 0059) l A A / e ( 7) Q / , 4., 05. 6, 089. We c wre A + + e / he + juco + A e so h A e We hve 0, , 660., 6, We Ae G ( 7. ) J G 6 so h , srh le y m m 6 A 0, , 0, whch yels

78 Semcouc hyscs evces: sc rcles, r eo her () l ( 7) , 69., 74. 6, / / 5 A 0 () oe-se juco / 7 e + where s he o cocero he low-oe reo. We hve ( 7) () whch yels () G l where s he o cocero he hh-oe reo. So Q / l whch yels 7. omuer lo () G l -reo Ε ρ( ) e Ε e + 0 We hve e Ε0 < < 0 Ε e + -reo, 0 < < Ε ρ( ) e e Ε + -reo, < < Ε ρ( ) e e Ε + We hve Ε 0, he e so h < < Ε e We lso hve 94

79 Semcouc hyscs evces: sc rcles, r eo her 7 Ε Ε e e + e, 0 < <, e Ε e 7. φ() ρ() () () Ε < < µ m, ρ() + e So Ε e e Ε + A µ m, Ε 0 So 0 e e + e Ε + Ε A 0, Ε() 0 µ m, so e Ε () whch yels ( 7. ) Ε() / ue o oel erece s e φ Ε + z e + + e φ 0, he e e 0 + we c wre z e φ + A µ m φ (. ) φ 86. oel erece cross he rsc reo 4 4 φ Ε() φ 55. y symmery, oel erece cross he - reo sce chre reo s lso 86.. he ol reverse-s vole s he ( 86. ) () he lerly re juco, ρ() e, Ε ρ( ) e Ε z e e + A +, Ε0 So e 0 + e G e Ε () e φ() z Ε + Se φ 0, he 0 e e + + e e φ() G + 95

80 Semcouc hyscs evces: sc rcles, r eo her We hve h e + he whch yels / ( 7. ) ( )

81 Semcouc hyscs evces: sc rcles, r eo her 8 her 8 8. he wr s e e S k e e k e S e k S e k () () l G k e J e m 60 m m 0m 8. e e S k we c wre hs s e + e k S so h k l + e S reverse s, s eve, so 090., we hve S l m 8. omuer lo 8.4 he cross-secol re s 00 A 50 J 0 We hve 4 J J e S J S e so h J.50 0 A/ S We c wre J e + S Q \ τ τ We w τ τ τ whch yels 4.4 J S ( 4.4) We