Chng 8: Cc tc tn x: qui tc vng

Chng 8: Cc tc tn x: qui tc vng Qui tc vng ca Fermi l mt trong cc cng c quan trng nht ca c hc lng t. N cho cng thc tng qut ca cc tc chuyn tip. l cc tc m ti cc ht b tn x t trng thi ny sang trng thi khc bi mt nhiu lon. Tn x l trong ngoc kp v n l mt khi nim tng qut hn nhiu so vi iu m ngi ta c th d on. Mt v d r rng c cung cp bi cc tp cht trong mt tinh th. Chng lm tn x mt in t t trng thi Bloch ny sang trng thi Bloch khc. Chng lm thay i xung lng nhng khng lm thay i nng lng ca in t. Tng t, cc phonon (cc dao ng mng) cng lm tn x cc in t nhng trong trng hp ny, cc tn x lm thay i c nng lng v xung lng ca cc in t. Mt v d km r rng hn l s hp th nh sng m n c th c xem nh mt qu trnh tn x trong , mt in t va chm vi mt phonon. Qu trnh ngc li cng xy ra trong mt in t mt nng lng cho mt photon v lm tng s pht x kch thch v t pht. Nh vy, tn x l mt khi nim chung ng ch . Cc v d xut rng c 2 lp rng ca cc qu trnh tn x m ta nghin cu nh sau

Chng 8: Cc tc tn x: qui tc vng (i) Cc th khng i theo thi gian nh cc tp cht trong tinh th m chng khng lm thay i nng lng ca ht b tn x. (ii) Cc th thay i iu ho theo thi gian nh l nh cc phonon v cc photon m chng lm thay i nng lng ca ht mt lng l S thay i nng lng d nhin c d on nhng khng c xem xt y. By gi ta s pht trin l thuyt cho cho 2 trng hp ny vi tn x n hi ca cc in t t cc tp cht lm v d ca th khng i v tn x bi cc phonon lm v d u tin ca mt th iu ha. Cui cng, ta s tnh quang dn mc d m t y v cc hin tng quang trong cc h thp chiu c a ra trong Chng 10.8.1. Qui tc vng i vi cc th tnh Trc tin, ta nghin cu mt nhiu lon khng i v d nh th t cc tp cht trong mt tinh th. Mt cu hi r rng l ti sao ngi ta khng s dng ngay l thuyt nhiu lon khng ph thuc vo thi gian m ta cng pht trin n trong phn 7.2 m n cho ta cc tr ring ca h vi cc tp cht. Cu tr li l c 2 cch tip cn u c gi tr nhng c nhng ng dng khc nhau

8.1. Qui tc vng i vi cc th tnh Gi s rng ta tm c cc tr ring chnh xc ca mt h cha mt s phn b ngu nhin ca cc tp cht. Cc tr ring ny l cc hn hp cc k phc tp ca cc trng thi ca h khng nhiu lon m chng lm cho cc tnh ton tr nn cng knh. Hn na, cc trng thi l khc nhau t mu ny sang mu khc do cc tp cht nhng v tr khc nhau. l mt cch l tng tip cn cc h rt nh trong ta ta k vng thy cc kt qu ring cho tng mu. Cc php o v d nh t tr cung cp mt du hiu ring c trng cho s phn b ca cc tp cht. N xc nh ch mesoscopic. N b chi phi bi s giao thoa gia cc sng in t v do i hi cc mu phi nh hn so vi khong cch m qua pha ca mt in t b ph v bi cc va chm vi cc phonon hoc cc in t khc. in hnh n c ngha l cc cu trc di micromet ti cc nhit hli (di 4 K). Tuy nhin, trong ch v m, ta k vng cc i lng nh dn l ging nhau i vi cc mu khc nhau. Thng tin cha trong cc trng thi ring bit c ly trung bnh mt cch lin tc (v n cng khng th qun l c). Qui tc vng ca Fermi a ra mt trin vng khc trong gii hn ny. Ta tip tc dng trng thi ring ca h sch (vt liu thun ty), cc sng

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8.1. Qui tc vng i vi cc th tnh Bloch i vi mt tinh th hoc cc sng phng i vi cc in t t do. Chng khng cn l cc trng thi ring thc s ca h vi cc tp cht v do , mt in t bt u mt trng thi ring s khng gi mi mi trng thi ny. thay th, cc trng thi khc s trn vo ging nh in t lan truyn v pha trc theo thi gian. Xc sut tm thy in t mt trong cc trng thi khc ny tng tuyn tnh vi thi gian v tc l tc chuyn tip hay tn x mong mun. Mt cch hnh thc hn, ta li gi thit rng ton t Hamilton c th c chia thnh mt phn khng nhiu lon ln m n khng i theo thi gian v c th c gii chnh xc v mt nhiu lon nh m n bt u tc dng ti t = 0. Cc trng thi ring ca l vi nng lng Trc khi tc dng nhiu lon, in t trng thi ban u i m hm sng ph thuc vo thi gian ca n l i vi t > 0, ta cn nghim ca vi iu kin bin l ti t = 0. Theo nh thng thng, phng php

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8.1. Qui tc vng i vi cc th tnh khai trin nghim chnh xc theo cc nghim ca bi ton khng nhiu lon. S khc bit l ch cc thng s khai trin by gi ph thuc vo thi gian. Nh vy,

trong l bin xc sut tm thy ht trng thi j thi im t vi gi tr ban u Thay khai trin (8.3) vo phng trnh Schrodinger (8.2), ta c

hay

S hng u tin mi v b loi b do l nghim ca phng trnh Schrodinger ph thuc vo thi gian vi Dng biu thc tng minh ca ta c

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8.1. Qui tc vng i vi cc th tnh

Nhiu s hng ny c th b loi b theo cch thng thng khi ly cc phn t ma trn. Nhn 2 v vi (i vi trng thi cui) v ly tch phn theo khng gian. Tt c cc s hng v tri trit tiu tr j = f cho

Nh vy, trong l s khc nhau v nng lng gia cc trng thi f v j. (8.8) tng ng vi phng trnh Schrodinger ban u v duy tr chnh xc. By gi l lc thc hin cc php gn ng. Php gn ng bc 0 l b qua hon ton v do , Dng n v phi ca (8.8) c

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8.1. Qui tc vng i vi cc th tnh kt qu bc 1

N c th c ly tch phn i vi v lu ti t = 0. T ,

l biu thc tng qut trong qui tc vng ca Fermi i vi bin xc sut ca trng thi f ti thi im t. Php gn ng gi thit rng tc chuyn tip l nh sao cho trng thi ban u lun lun c th ly nh l trng thi gn y v cc trng thi cui lun lun c th ly nh l cc trng thi gn trng. Ta cn khng c to ra bt c gi thuyt no v s ph thuc ca nhiu lon vo thi gian. Cho n l hng s. Trong trng hp ny, c th a ra khi tch phn trong (8.10) c

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8.1. Qui tc vng i vi cc th tnh Xc sut tm thy in t trng thi cui l

vi l mt kt qu l th c v th trn hnh 8.1 nh l mt hm ca chnh lch nng lng Xc sut tm thy in t trong bt k trng thi cui ring no dao ng nh mt hm ca thi gian vi bin khng i. Hm sinc tr nn hp hn v nng lng ging nh 1/ t trong khi h s pha di lm cho cao ca n tng ging nh Nh vy, n tr nn cao v cng v hp khi m n chnh l Hnh 8.1. Xc sut tm thy ht mt trng thi cui nh mt hm ca chnh lch nng lng ti 3 thi im theo t s 1 : 2 : 4. ng t nt l ng bao ca hm cc thi im khc nhau.

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8.1. Qui tc vng i vi cc th tnh tnh cht ca mt hm Tch phn chun ch ra rng

m n c ngha l

khi Nh vy, xc sut tm thy in t mt trng thi cui vi nng lng ti cc thi gian ln ging nh

N tng tuyn tnh vi thi gian v do c mt tc chuyn v khng i t trng thi i ti trng thi f c cho bi

l qui tc vng ca Fermi. N ch ra rng nng lng ca trng thi cui cn phi c cng nng lng vi trng thi u nh k vng t s bo ton nng lng mc d n ch duy tr chnh xc trong gii hn ca nhng thi gian ln

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8.1. Qui tc vng i vi cc th tnh nhng thi gian nh. hm c b rng nng lng l v do , nng lng cn khng c bo ton chnh xc. B rng ny c th l quan trng trong cc h vi tn x mnh do in t c th khng sng lu trng thi cui ca n trc khi n li b tn x v khng th ly gii hn Trong nhng trng hp nh th, c th c thay bi mt hm A(E) ca mt n v din tch v b rng trong l mt php o thi gian sng gia cc s kin tn x. nh hng ny c gi l s m rng va chm. Ngi ta thng s dng mt dng khc ca qui tc vng khng c hm Thay cho vic xem xt tc tn x t mt trng thi ban u ring bit, ta c th ly tng phng trnh (8.17) thu c tc t bt k trng thi ban u no ti cng trng thi cui. N cho Hm p t vic ly tng cho cc trng thi c nng lng rt gn vi Vi iu kin l phn t ma trn l tng t i vi tt c cc trng thi ny, n c th b y ra ngoi tng m n tr thnh N

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8.1. Qui tc vng i vi cc th tnh trong ta dng nh ngha (1.95) i vi mt trng thi. Nh vy, tc chuyn v l N Dng ny ca qui tc vng Fermi hon ton tng ng vi dng trc do hm ch c ngha bn trong mt tch phn. Cch rt ra n nhn mnh mt im quan trng l cn phi c mt s lin tc ca cc trng thi u v cui i vi qui tc vng Fermi c s dng hoc mt trng thi khng th c nh ngha v hm khng c ngha. By gi ta p dng cng thc ny cho tn x ca cc in t bi cc tp cht. 8.2. Tn x tp cht Tn x tp cht gii hn linh ng ca cc in t nhit thp khi c mt mt t phonon. Bn cht ca th thay i rng. Cc tp cht tch in v d nh cc cht cho v cht nhn ion ha c th Coulomb phm vi xa trong khi cc tp cht trung ha c cc th giam cm phm vi gn phc tp. 2 trng hp ny c nhng nh hng khc nhau ln cc tc tn x tng cng. Tn x hp kim nh vo s sp xp ngu nhin ca Al v Ga trong (Al, Ga)As v tn x giao din nhm c th c nghin cu theo mt cch tng t. Ta s kim tra chi tit cc in t trong mt 2DEG Chng 9. y, ta

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8.2. Tn x tp cht pht trin l thuyt chung. Xt cc in t t do trong trng hp 2 chiu. Cc vect nh c s dng cho v tr theo k hiu chun ca chng ta. t h vo trong mt hp c din tch hu hn A. Cc kt qu vt l khng ph thuc vo gi tr ca A. nh ngha ca cc trng thi u v cui l cc sng phng

vi xung lng b sung sau s kin tn x. Nhiu lon n gin l th nng b sung t tp cht v do , phn t ma trn l

trong l bin i Fourier 2 chiu ca th tn x. Khi lng phn t ma trn ny vo trong qui tc vng (8.17), tc tn x t ti l

Kt qu n gin l tc tn x t l vi bnh phng mun ca bin i Fourier ca th tn x c gi l php gn ng Born. N c s dng

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8.2. Tn x tp cht rng ri v cc iu kin chi tit i vi gi tr ca n c cho trong cc sch v l thuyt tn x. Mt c tnh l l h s trong (8.23) m n ng rng nh hng ca tp cht gim i khi h tr nn ln hn. iu ny khng gy ra ngc nhin do mt tp cht n tr nn d thy hn trong mt h ln hn nhng cc h qu vt l ca s tn x v d nh linh ng khng ph thuc vo kch thc ca h. Xt tc tn x tng cng i vi mt in t. l tc tn x t ti bt k trng thi cui no. N c k hiu l v c hiu nh l mt hm ca C nhiu thi gian sng khc nhau c th c nh ngha i vi mt in t. l thi gian sng n ht (chng li tn x tp cht) hay thi gian sng lng t. Gi thit rng trng thi cui c bo m l trng v do ta khng cn phi lo ngi v cc h s lp y (Fermi). Khi , tc tn x tng cng nh vo mt tp cht n c cho bi vic ly tng tc (8.23) t php gn ng Born theo tt c cc vect sng

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8.2. Tn x tp cht Tng theo c th chuyn i thnh mt tch phn nh trong phn 1.7 khi s dng

y l mt h s A do mt ca cc trng thi cui t l vi din tch ca h. Khng c h s 2 i vi spin do th c gi thit khng lm o hng spin ca in t v do khng c s la chn spin trng thi cui. Tip theo, khng c mt tp cht n no trong bt k mu no ca vt liu thc v c mt s tp cht rt ln. Nu mt trung bnh ca chng l ng vi mt n v din tch, tng s tp cht trong mu c din tch A ca chng ta l Khng d kt hp tn x nh vo nhiu tp cht do cc in t l cc sng v c s giao thoa gia cc sng b tn x bi cc tp cht ln cn. Mt s giao thoa nh th l c bit quan trng trong ch mesoscopic v ph thuc vo cu hnh chnh xc ca cc tp cht. N c bit mnh trong trng hp 1 chiu trong s xuyn hm cng hng cung cp mt v d mnh v s giao thoa (phn 5.5). Thng cho php b qua s giao thoa trong cc mu ln nhit cao v

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8.2. Tn x tp cht gi thit rng s tn x nh vo tng tp cht l c lp vi cc tn x khc. Tc tn x tng cng khi c cho bi vic nhn tc tn x ca mt tp cht n vi

Nhng h s A b trit tiu li cng thc chun i vi thi gian sng n ht ca mt in t. Mt c im quan trng l thi gian sng n ht khng phi l thi gian xut hin trong dn hoc linh ng. Cc i lng ny cha thi gian sng vn chuyn v d nh Hnh 8.2 ch ra hnh hc ca qu trnh tn x v vect sng lin quan nh th no vi gc m qua n cc in t b tn x. S bo ton nng lng i hi rng cc vect sng trc v sau va chm c cng ln v do cc vect nm trn mt ng trn trong khng gian T lng gic suy ra

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8.2. Tn x tp cht Hnh 8.2. Mi quan h trong khng gian gia cc vect sng v gc m qua n mt in t btn x bi mt tp cht.

Thi gian sng n ht trong (8.26) cha mt tng theo tt c cc qu trnh tn x c trng s nh nhau. N c ngha l tn x gc nh m trong l nh tnh nhiu n mc tin a cc s kin tn x ngc trong v hng ca in t b o ngc. Tuy nhin, tn x ngc c mt nh hng ln hn nhiu ln dng so vi tn x gc nh. Nu cho thnh phn chuyn ng ca in t song song vi hng ban u ca n t l vi ngi ta c th d on rng hiu qu ca mt s kin tn x ph thuc vo s thay i trong cosin ny theo Mt tnh ton hon chnh xc nhn n. (8.27) ch ra rng v do , tc tn x vn chuyn c cho bi

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8.2. Tn x tp cht H s thm vo u tin tn x qua cc gc ln. Tn x c coi l ng hng nu khng ph thuc vo Khi , trng s gc khng c nh hng v l c trng ca cc th phm vi gn. S ngc li thng l trng hp i vi cc tp cht tch in trong cc h thp chiu. y, gim nhanh khi tng v c th c mt s khc bit mt bc ln gia v Cc biu thc ny i vi cc tc tn x tng cng l tng qut. Chng c th thng c n gin ho do phn ln cc th c i xng trn tr tn x t cc lng cc tch in. Ta bt u t (8.28) i vi thi gian sng vn chuyn. thun tin, ta vit li tch phn tm thi theo cc vect sng ca trng thi cui N cho

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8.2. Tn x tp cht Trc ca cc ta cc i vi c ly dc theo v do , l gc tn x c nh ngha t trc. Hm i hi ln ca cc vect sng cn phi bng nhau bo ton nng lng nhng ta cn nghin cu bin s ca hm mt cch ng n. Qui tc chung l ta chia cho o hm ca hm trong hm v y l Thc ra, ta dang i bin s tch phn ti nng lng ca trng thi cui l Cui cng, ta cn ln ca vect sng bn trong th v h s trng s. Cng thc cosin cho

Hm p v do n thu gn ti biu thc thng thng l Nh vy, tch phn theo trong (8.29) dn ti

Tch phn theo c rt gn ti khong v c gp i b tr. Tc c th c vit theo cch khc theo q thnh

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8.2. Tn x tp cht Lu rng q khng chy ti v cc. Hnh 8.2 ch ra rng gi tr ln nht ca q tha mn s bo ton nng lng l 2k m n tng ng vi tn x ngc. H s gc c th c b i t cc cng thc ny cho tc n ht. S dn xy ra ti nng lng Fermi trong mt kim loi v do , ta c th n gin t trong cc cng thc ny. Cng mt kt qu a ra i vi cc cht bn dn nu chng l suy bin v d nh mt 2DEG nhit thp nhng mt php ly trung bnh theo vng tch cc ca k cn phi c thc hin nu n khng ng. S tn x cng c th c m t theo cc tit din ngang (cc din tch trong trng hp 3 chiu hoc cc chiu di trong trng hp 2 chiu). Gn vi thi gian sng n ht l qung ng t do trung bnh trong Gi s mi mt tp cht c th c biu din nh mt vch c chiu di vung gc vi vn tc ca in t. Mt in t s va chm vi bt k tp cht no trong mt din tch l trong lc i c mt khong cch l Theo nh ngha ca qung ng t do trung bnh, chnh xc c trung bnh mt tp cht trong din tch ny v do , v

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8.2. Tn x tp cht Vic so snh vi (8.31) sau khi b h s trng s i vi tc vn chuyn ch ra rng

Tit din ngang tng cng thay th c th c vit nh mt tch phn theo tit din vi phn nh sau

Mt tit din vn chuyn cng c th c nh ngha bng cch a h s thng thng vo trong tch phn trong (8.34). Cc tit din l hp dn v mt vt l do ln ca chng cho kch thc biu kin ca mt i tng tn x (n c th khc vi kch thc vt l ca n). Bc tip theo l tnh bin i Fourier i hi i vi tc tn x 8.2.1. Tn x bi mt tp cht phm vi gn Mt v d n gin ca tn x tp cht 2 chiu c cung cp bi mt ro hnh trn c bn knh a. l mt th phm vi gn v c th c s dng nh mt m hnh n gin ca mt tp cht trung ho v d nh mt nguyn

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8.2. Tn x tp cht t Al b khuch tn t mt ro vo trong mt h GaAs. Trng hp tn x ngc li bi th phm vi xa ca mt tp cht ion ha t xa s c nghin cu trong Chng 9. C 2 trng hp u c i xng quay. Th c nh ngha bi

Bin i Fourier ca n l

trong l gc gia v Tch phn theo cho mt hm Bessel. l mt c im chung ca bin i Fourier 2 chiu. T ,

Trong trng hp ring ca mt ro hnh trn, n tr thnh

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8.2. Tn x tp cht trong phng trnh (9.1.30) t Abramowitz v Stogun (1972) c s dng cho tch phn m n em li hm Bessel khc. Tc tn x t l vi bnh phng ca (*.38) v c v th trn hnh 8.3. N ch ra cc dao ng gim sinh ra t cnh sc nt ca th v khng c t mt dng thay i trn hn. Hm kh ging sincx v c mt gi tr gii hn l 1 khi Nh vy, i vi q nh v tn x tr thnh ng hng. Kt qu c s dng rng ri ny c th nhn thy t (8.36) m n cho l mt tch phn theo th. Mt s th c phm vi qu xa tn ti tch phn v khng ch ra gii hn ny. Mt v d quan trng l tn x Rutherford t mt th

Hnh 8.3. Tc tn x ca mt ro hnh trn c bn knh a nh mt hm ca s sng tn x q.

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8.2. Tn x tp cht Coulomb khng b chn. Tit din tng cng ti k nh rt gn thnh iu ny rt khc vi kch thc 2a ca vt chng ngi v phn k khi Hnh 8.3 ch ra rng tn x dc hn xung i vi ca gi tr ln ca q (gn ng i vi l mt mi quan h chung gia kch thc ca chng ngi vt v s sng cc i i vi tn x. Gc tn x c cho bi (8.27) v tr thnh i vi cc gc nh. Gc tn x cc i dn bng m n gim khi nng lng ca in t ti tng. Mt tp cht trung ho trong mt cht bn dn c cc kch thc nguyn t v do , tn x gn nh khng ph thuc vo q i vi Tn x t cc tp cht trung ho do c th c nghin cu nh l ng hng trong mt 2DEG m , Mt c im ca tc tn x trong php gn ng Born l n khng ph thuc vo du ca th. Mt h c su tn x theo cng mt cch nh mt ro c cao Kt qu ny khng ng i vi ln v php gn ng Born tr nn khng chnh xc. V d nh ro hnh trn tr nn khng

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8.2. Tn x tp cht th xuyn qua c khi cao ca n ln hn nhiu so vi nng lng ca in t ti v tn x tr nn khng ph thuc vo Cc phng php chnh xc hn nh cc dch pha cn c s dng trong trng hp ny v tn x t mt chng ngi vt hnh trn khng xuyn qua c c th c gii mt cch chnh xc. Mt l thuyt gn ng cng ch ra rng c nhng lin kt trn cho tit din v do khng th thit k cc vt lm tn x vi cng ty . V d nh tn x ng hng khng th cho mt tit din ln hn 4/ k. Cc bi ton xa hn sinh ra vi mt th ht c bit l khi c mt trng thi lin kt gn nh h. N gy ra lo ngi do tt c cc h 2 chiu u c cc trng thi lin kt. Cc tnh ton chnh xc ch ra rng php gn ng Born c th chp nhn c trong mt phm vi rng ca cc iu kin. Nu bn knh a ca ro trn dn ti khng th ta thu c th vi mt hm 2 chiu. Bin i Fourier l mt hng s v tn x l ng hng. Cc tnh ton thng c n gin ha i nhiu trong gii hn ny v d nh cc tit din n ht v vn chuyn l nh nhau v do , n c dng rng ri trong l thuyt. Tn x trong mt 2DEG b chi phi bi cc tp cht ion ha t xa m tit din ca n xa s ng hng nhng

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8.2. Tn x tp cht n dng nh khng lm gim bt vic s dng cc th hm nh mt m hnh.8.3. Qui tc vng i vi cc th dao ng Loi nhiu lon th hai thay i mt cch iu ha theo thi gian

H s 2 c bao hm n gin ha cc biu thc v sau nhng d dng mt i. Nhiu lon cha cc ton t tc dng ln hm sng nhng khng ph thuc vo thi gian. Cc phn t ma trn trong (8.8) cng c dng v tch phn bc 1 (8.10) i vi cc h s tr thnh

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8.3. Qui tc vng i vi cc th dao ng Xc sut tm thy ht trng thi cui do bng

S hng u tin trong ngoc ging nh s hng i vi nhiu lon tnh (8.12) ngoi tr iu l n xoay quanh khc vi khng hoc Nh vy, trng thi cui c nng lng hp th t nhiu lon. Tng t, s hng th hai cho l mt s mt nng lng cho nhiu lon. S hng th ba cha s giao thoa ca 2 s kin ny v c th b qua i vi m n thng l hin trng. Trong gii hn ny, s tch ra ca 2 hm l s ln hn nhiu so vi b rng ca chng ta l v do , chng c th c nghin cu mt cch c lp. N c ngha l 2 thnh phn hm m ca tch ra. Vic rt ra c thc hin chnh xc ging nh i vi trng hp tnh v kt qu cui cng i vi tc chuyn v c sinh ra bi l

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8.3. Qui tc vng i vi cc th dao ng l qui tc vng ca Fermi i vi mt nhiu lon iu ho v n ch ra rng thnh phn dn ti s hp th nng lng bi in t. N c th c vit cch khc theo mt trng thi nh sau N Cc kt qu i vi phn khc ca cosin l l ging nhau ngoi tr iu l m n tng ng vi s pht ra nng lng t in t. Hai ng dng quan trng ca qui tc vng l cho s tn x bi cc phonon (cc dao ng mng) v cc photon (nh sng) m chng s c xem xt trong cc phn tip theo. Ta cn phi b la mt cht v c photon v phonon t chng b lng t ho trong khi ta nghin cu th iu ho theo kiu c in. Trong trng hp ca cc phonon, ta s tnh tc tn x nh vo phonon n v nhn kt qu vi phn b Bose Einstein. i vi s hp th quang, ta s b qua hon ton vn ny v nghin cu nh sng nh l mt sng c in.

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8.4. Tn x phonon Cc phonon c m t trong phn 2.8 v l cc lng t ca cc dao ng mng trong mt vt rn. Chng l c ch tn x ni tri i vi cc in t nhit phng. C nhiu cch trong cc in t v phonon tng tc vi nhau mc d tt c u sinh ra t cng mt hiu ng c bn: chuyn ng ca cc ion nh vo phonon sinh ra in trng v t trng, cc trng ny nh hng n chuyn ng ca cc in t v lm tn x chng. Ta s xt 2 trng hp quan trng v ngc nhau l s lin kt bin dng cho cc phonon LA (m dc) v lin kt cc cho cc phonon LO (quang dc). 8.4.1. Cc phonon m dc v th bin dng Cc phonon m dc ging nh cc sng m i vi cc bc sng di (q nh). Lin kt n gin nht vi cc in t i vi cc phonon nh vy l qua th bin dng. Phonon nn v gin cc vng xen k ca vt rn. Mt s nn hoc gin u ca tinh th lm cho bin ca tng vng nng lng in t chuyn ng ln hoc xung t l vi bin dng. Hng s t l c gi l th bin dng Cn mt biu thc phc tp hn nu cc vng l suy bin hoc thiu i xng cu. Bin dng dc c nh ngha nh l s tng

28

8.4. Tn x phonon rt nh i vi chiu di ca vt v i vi mt mng n v ca chui (Hnh 2.2(a)), n c cho bi i vi cc bc sng di, ta c th nghin cu chui 1 chiu ca cc nguyn t nh l mt mi trng lin tc v bin dng tr nn mt o hm. di ca mt nguyn t ti z c cho bi (2.30) l v do , bin dng dc l

Th nng c th c tnh nh th bin dng l ng u trong mi mt vng vi iu kin l bc sng l di. N cho

trong bin i vi mt phonon n c a vo t (2.30). Do i vi cc phonon m bc sng di, cui cng n tr thnh

N l dng ca th nhiu lon gy ra bi phonon c s dng trong qui tc vng ca Fermi. Cn mt thnh phn na l s phonon c cc tc tn x tng cng. S phonon chim gi mt mode c vect sng cn bng c cho bi phn

29

8.4. Tn x phonon b Bose Einstein (1.127)

Tc i vi mt phonon n cn c nhn vi s phonon trong mt mode tnh tc tn x tng cng nh vo s hp th. H s i vi s pht ra cc phonon l 1 m t s pht t pht trong lc m t s pht kch thch. S pht t pht thay th c th c coi nh s pht kch thch bi cc thng ging chn khng. S khng i xng ny gia tc hp th v tc pht x l rt quan trng bo m rng cc in t v phonon gia cn bng vi nhau. Tn x trong cc h thp chiu l phc tp do cc phonon thng gi bn cht 3 chiu ca chng k c cc in t b giam cm. Do , trc tin ta s xem xt tng tc gia cc in t 3 chiu v cc phonon v nghin cu 2DEG sau phn 9.6.3. t cc in t vo trong mt hp hu hn c th tch trong cc hm sng l cc sng phng tn x t n Ta bt u vi s hng u tin trong (8.47). N ph thuc vo thi gian ging nh v lm cho in t hp th nng lng t phonon. iu kin bo ton nng lng trong hm do l Phn t

30

8.4. Tn x phonon ma trn l

Tch phn cho m n loi b t s chun ho cc hm sng nu cc vect sng c tng bng v khng nu khc. N i hi l kt qu mong mun i vi s bo ton xung lng. Ta cng cn phi bao hm mt h s tnh n s phonon i vi s hp th. Nh vy, tc tn x t n gy ra bi s hp th ca cc phonon m dc l

H thc c cng c bi hm i vi cc vect sng c s dng thay th bn trong hm i vi nng lng. in t c c vect sng v nng lng t phonon. Na th hai ca nhiu lon (8.47) lm cho in t mt c xung lng v nng lng cho phonon. Nhng khc bit ch l cc du trong cc hm v trong h s chim gi phonon nhm bao hm s pht x t pht cng nh s pht x kch thch. Nh vy,

31

8.4. Tn x phonon

S thay i nng lng khng cho php ta vit mt cng thc n gin i vi ng gp ca cc phonon vo tc tn x vn chuyn nh ta lm i vi cc tp cht. Tuy nhin, ta c th xc nh xem liu chng cc phonon m c kch thch mt nng lng quan trng hay khng? Xt mt 2DEG c mt m i vi n, Tn x ngc cung cp phonon vi vect sng ln nht l m nng lng ca n l Vn tc m gn bng v do , nng lng l Kt lun rt ra l cc phonon m khng kch thch nhiu nng lng v chng thng c nghin cu trong php gn ng chun n hi trong b qua nng lng. Cng mt kt lun rt ra i vi cc in t khng suy bin nhit phng. Dng lm mt v d n gin, ta c th tnh tc tn x tng cng i vi mt in t trong mt cht kh 3 chiu khi dng php gn ng chun n hi. iu kin bo ton nng lng tr thnh ging nh

32

8.4. Tn x phonon i vi cc tp cht. S phonon trong cc mode c lin quan cng ln v do , ta c th dng gii hn khng suy bin ca phn b Bose l

Cc php gn ng ny c ngha l cc tc i vi s pht x v s hp th gn bng nhau v cng cho tc tn x tng cng t n

Sau tt c nhng s n gin ho, n ging nh tc tn x (8.23) i vi cc tp cht. Thc ra l n l trng hp c bit n gin ca tn x ng hng do b trit tiu t phn t ma trn. Vic ly tng theo mi cho tc tn x tng cng i vi mt in t vi vect sng

Tng cho mt trng thi ging nh trong (1.97) ngoi tr iu l khng

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8.4. Tn x phonon c tng theo spin. Ta c th tin n kt qu ny trc tip hn khi dng dng mt trng thi ca qui tc vng Fermi. Nh vy, tc tn x rt gn thnh

Cc thi gian sng n ht v vn chuyn l nh nhau i vi tn x ng hng v do khng cn ch s di Mt nng lng thng thng trong mt kh in t 3 chiu khng suy bin l Mt trng thi v do , ta tin n kt qu bit l

C nhng c ch khc m nh , cc in t v phonon LA tng tc vi nhau. V d nh chuyn ng ca cc li ion sinh ra mt t trng. Quan trng hn l bin dng c th pht sinh mt th p in trong cc cht bn dn hp cht v ngi ta tranh ci v tm quan trng tng i ca th bin dng v lin kt p in trong GaAs. 8.4.2. Cc phonon quang dc v lin kt cc Cc phonon quang tn ti trong cc tinh th c hn mt nguyn t trong mt

34

8.4. Tn x phonon mng n v (phn 2.8.3). Chng cung cp mt s i lp vi cc phonon m do nng lng ca chng l ln (36 meV ti Q = 0 trong GaAs). Chng cng minh ha mt cch lin kt khc hn i vi cc in t. Tn quang sinh ra do 2 nguyn t trong mt mng n v chuyn ng theo cc hng ngc nhau (hnh 2.24(b)). N thit lp mt in trng nu nguyn t mang mt in tch v ta s ch xt trng hp cc ny m n bao gm cc cht bn dn III V. Do dch chuyn tng i ca 2 nguyn t trong mt mng n v l ta c lin quan i vi mt phonon quang, ta k hiu n l

H thc tn sc (hnh 2.23) gn nh phng gn v do , ta c th t l mt hng s. H s pha trc i vi mt phonon n c th tm c theo cng mt cch nh i vi phonon m trong phn 2.8.1 vi kt qu l

trong l s mng n v (cc cp ion) ng vi mt n v th tch.

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8.4. Tn x phonon N tng t nh kt qu (2.29) i vi cc phonon m. S khc bit chnh l s xut hin ca khi lng rt gn c nh ngha bi

trong m v M l 2 khi lng ring. 2 khi lng ion l rt gn nhau trong GaAs v do , Cho 2 ion mang cc in tch ngc nhau l Cc vt liu nh GaAs ch l cc vt liu ion yu v do , in tch hiu dng Khi tng cp ion dao ng, n thit lp mt lng cc in iu ny li sinh ra mt trng phn cc Khng c in tch t do c a vo v do , vect in dch v h thc chung rt gn thnh in th l tch phn ca E m n a vo h s 1/ Q. Th nng i vi mt in t cui cng c cho bi

Tc tn x khi c th c tnh ging nh i vi cc phonon m v

36

8.4. Tn x phonon tc hp th ca cc phonon quang l

trong l s phonon LO chim gi c cho bi phn b Bose Einstein. C 2 s khc bit quan trng t kt qu i vi cc phonon m. Trc ht l h s N lm cho tc tn x ca cc phonon quang cc gim nhanh nh mt hm ca Q. iu ny ngc vi cc phonon m trong s lin kt tng ging nh Q. Th hai l nng lng trao i vi phonon l ln. N c ngha l s hp th ca cc phonon quang ch quan trng cc nhit cao (in hnh l trn 77 K) v chng b loi tr di . Tuy nhin, s pht x t pht c th xy ra cc nhit thp hn nu in t c nng lng v cc phonon quang lm lnh cc in t nng mt cch c hiu qu. Tc tn x nh l mt hm ca nng lng ca mt in t tng nhanh ti ngng m s pht x t pht c th xy ra nhng khi nghing chm v h s lm cho lin kt tr nn yu hn. Mt nh hng quan trng khc ca cc phonon LO l lm ion ho cc

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8.4. Tn x phonon exciton ti nhit phng m ta s xem xt n trong phn 10.7. Mc d in tch hiu dng xem nh mt h s r rng bao hm trong tc tn x, n thng c biu din theo hng s in mi ti cc tn s thp v cao

Cc in t ch c th phn ng vi mt in trng ti cc tn s cao hn nhiu so vi do cc ion l qu nng m n cho mt hng s in mi Ti cc tn s nh hn cc ion cng c th phn ng vi mt trng tc dng v do tch in, chng to ra mt ng gp b sung vo s chn. V th, hng s in mi tr nn ln hn. i vi GaAs, H s pha trc ca cc ngoc vung trong (8.59) gn vi 3 i vi cc cht bn dn III V v do , ta tm c trong GaAs. Cng cc hng s in mi cng xut hin trong h thc Lyddane Sacks Teller gia cc tn s ca cc phonon quang dc v ngang

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8.4. Tn x phonon Cc tn s l khc nhau do in trng c thit lp bi cc dao ng dc cho mt lc hi phc b sung m n lm tng tn s ca chng. Cc dao ng ngang thun ty khng sinh ra cc lng cc v do khng c nh hng ny. Ging nh trong trng hp ca cc phonon m, c nhng c ch khc m nh , cc phonon quang c th tng tc vi cc in t. Mt th bin dng li c th c nh ngha v n l quan trng trong cc cht bn dn thng thng m trong khng c lin kt cc. Trc khi qua phn tn x phonon, ta s a ra mt cch nhn gn gi hn v cc phn t ma trn v vic nghin cu cc hm sng ca chng ta kh l n gin. Ta ch bao hm mt sng phng theo l thuyt khi lng hiu dng nhng b qua hm Bloch m n cn c nhn vi hm sng phng. Tch phn trong (8.49) cn c thay bi

39

8.4. Tn x phonon Cc hm Bloch c tnh tun hon ca mng tinh th v do c khai trin v phi thnh chui Fourier vi cc vect sng ca mng o. S hng chung gi li php ly tch phn nu Nh vy, v cn khng c nh nhau nhng c th khc nhau bi mt vect mng o. l tn x umklapp. N c bit quan trng i vi tn x gia cc thung lng trong Si, trong con ng ngn nht gia 2 thung lng X c th bao gm mt vect mng o. Tn x umklapp c ch ra mt cch t nhin trong s min lp li ca cu trc vng. Mt nghin cu y v tn x in t - phonon cn phi vt ra khi tho lun ngn gn ny. Tng tc b chn bi cc in t hoc cc l trng nu c mt kh m c ca chng v iu ny c th c mt nh hng r rt ln s lin kt. Tn x ca cc l trng kh tnh do bn cht phc tp ca cc vng ha tr. Tn x gia cc thung lng cng cn c bao hm trong mt vt liu vi cc thung lng suy bin trong vng dn nh Si v n c th l tn x umklapp nh va tho lun. Tn x t ti cc thung lng cao hn (X v L) cng tr nn quan trng i vi cc in t nng trong GaAs, trong n tng ti hiu ng Gunn (phn 2.6.4.3).

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8.5. S hp th quang Cc tnh cht quang ca cc h thp chiu cung cp cc ng dng ph bin nht ca chng v c xem xt trong Chng 10. y, ta s dng qui tc vng rt ra mt cng thc chung cho s hp th quang v p dng n cho 2 v d i lp nhau l cc chuyn tip t vng ho tr ti vng dn trong c mt khong hp th lin tc v s hp th gia cc mc ca mt h lng t trong cho php cc tn s ring r. Ta i mt vi cng mt bi ton nh trong tn x in t - phonon. l bi ton trong cc photon cng nh cc in t b lng t ho. y ta s b qua s lng t ho ca trng photon v nghin cu n mt cch n gin nh l mt sng in t c in. iu ny l tha ng i vi cc php o chuyn tip ca hp th quang nhng cn phi c chng minh i vi cc laze trong cn phi c mt nghin cu chnh xc v s pht x t pht v kch thch. Cc tnh ton cng c gii hn cho cc hiu ng tuyn tnh v khng m t cc hin tng nh s nhn i tn s nhn thy vi nh sng c cng rt ln. r rng, cc cng thc s c vit nh th chng m t mt mi trng ng nht hu hn. R rng l iu khng ng i vi mt d cu trc v ngi ta cn tnh n cc trng in t mt cch thch hp bng cch p

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8.5. S hp th quang dng cc iu kin lm khp ti cc giao din. Ta s gi thit rng iu ny c tin hnh khi s dng cc nh lut in t c in. Trong thc t, bc ny c th xa vi thng thng thm ch i vi cc cu trc chung nh mt ng dn sng rib. 8.5.1. Cc phng trnh v m Trong in t hc c in, phn ng vi mt in trng c m t bi c dng v h s phn cc m n thng c st nhp vo trong dich (in dch) Cho in trng l mt thnh phn Fourier n vi tn s gc v vect sng c cho bi phn thc ca Lu v cc du trong trt t y. Trong gio trnh ny dng qui c c hc lng t i vi s ph thuc vo thi gian l nhng k hiu chun trong k thut in l S khc nhau ny v du l khp ni. Nhng ngi dng thng nh ngha bo m rng du hiu c ngha vt l ca biu th s tiu tn l khng thay i. Mt qui tc chung l i i thnh j khp ni.

42

8.5. S hp th quang Phng trnh Maxwell th t l trong dng th hai rt ra t s ph thuc iu ho vo thi gian. Cc trng v c cho bi h thc trong l hm in mi (hng s in mi tng i) v l dn. C 2 u l hm ca tn s v vect sng. Chng l cc tenx hng 2 khc vi cc v hng nhng ta s gi thit rng vt liu l ng hng sao cho s phc tp ny by gi c th b i. Mc d s n gin ho ny c duy tr cho cc cht bn dn lp phng, n khng ng i vi cc d cu trc nh cc h lng t trong ta s thy rng phn ng vi mt in trng vung gc vi h c th khc hn vi phn ng vi mt trng song song. Khng thun tin cho vic s dng 2 i lng v do , chng thng c kt hp vo trong hm in mi phc hoc dn phc Khi , phng trnh Maxwell th t c th c vit di 2 dng

Dng vi dn c 2 s hng do khng gian t do c Hm in mi v dn phc lin h vi nhau bi

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8.5. S hp th quang Chng hon ton tng ng vi nhau v ngi ta dng biu thc c xem l thch hp hn. Cc phn thc v phn o ca chng c k hiu l Hm in mi v dn m ta a vo lc u chnh l cc phn thc ca cc i lng phc tng ng. C mt im phn bit vt l quan trng gia cc phn thc v phn o: v m t s dn m n l mt qu trnh tiu tn trong khi phn khc m t s phn cc m n khng hp th nng lng. N c ngha l ta thng k vng v mt du m ng rng nng lng c gii phng khi h b kch thch. Mt cch khc xem xt iu ny l cn lu rng cho mt dng trong pha vi in trng v do cho mt s tiu tn trong khi cho mt dng trong php cu phng m n gn vi mt s chuyn dao ng ca nng lng nhng khng c dng chy v trung bnh. Ch s khc x (chit sut) phc thng dng cho cc tnh cht quang c nh ngha bi N lin h s sng v tn s ca mt sng nh sng qua Nh vy, tc ng n

44

8.5. S hp th quang bc sng v vn tc trong khi cho tc m ti sng suy gim theo khong cch. Cc phn thc v o lin h vi nhau bi S lm yu i nhanh chng ln) c th sinh ra t mt m ging nh trong mt plasma di tn s plasma ca n v khng nht thit t s hp th ln). Ni cch khc, nng lng c th b phn x khc vi hp th. Ngi ta c th d on rng l cc hm c lp do ngi ta m t s phn cc v s tiu tn khc m chng l cc qu trnh vt l khc bit biu kin. iu ny ha ra l hon ton sai lm: hon ton xc nh nu bit i vi mi tn s v ngc li. Kt qu ng ch ny c gi l h thc Kramers Kronig v c m t trong Ph lc 6. Ta s s dng n trong phn 10.1 rt ra phn o ca t phn thc d tnh hn ca n. 8.5.2. Tng tc in t - phonon Phng php tnh ton l tnh nng lng c hp th t mt sng phng ca nh sng theo 2 cch l dng qui tc vng Fermi v dng in t hc c

45

8.5. S hp th quang in ri cn bng 2 kt qu ny rt ra phn thc ca dn. Xt mt sng nh sng vi mt in trng Gi thit rng sng khng suy gim khi n i qua mu. Cng c mt t trng nhng nh hng ca n thng nh hn nhiu v ta s b qua n. Vect n v cho s phn cc ca in trng m n cn phi nm trong mt phng vung gc vi hng truyn v do Sng ny l sng phng b phn cc nhng mt phc c th c s dng m t cc sng tng qut hn. V mt c in, in trng sinh ra mt dng m thnh phn cng pha ca n l m n tiu tn nng lng tc ng vi mt n v th tch. Khi ly trung bnh theo thi gian, cng sut tng cng b tiu tn trong th tch l By gi ta s rt ra mt biu thc c hc lng t khi dng qui tc vng Fermi. Ton t Hamilton cha cc th v hng v th vect khc vi cc trng mt cch trc tip (phn 6.1). y, ta dng m n i hi mt th vect Mt th v hng s cho loi trng sai lm. Xt mt thnh phn Fourier n

46

8.5. S hp th quang N cho c im quan trng l ch in trng cng hng vi hng ca di chuyn. N c gi l trng dc ngc li vi bn cht ngang ca mt sng in t. Phng trnh Schrodinger i vi mt in t (c in tch e) l

trong l th ca tinh th hoc d cu trc nghin cu. Nhiu lon nh vo sng nh sng l s khc bit gia ton t Hamilton v ton t Hamilton khng c m n cho Lu rng trt t ca v c tha nhn v l mt ton t v khng th chuyn i ch ny ch khc ging nh mt s n gin. S hng vi l bc 2 theo in trng v s c b i. S hng ny trit tiu trong php gn ng lng cc in. S hng khi tc dng ln hm sng cho , ta s dng qui tc o hm tch ca mt vect v mt v hng. By gi i vi sng ngang ca chng ta. Do , ch cn gi li s hng

47

8.5. S hp th quang gn vi v nhiu lon tng cng l Dng tng tc ny c v l nhng lu rng ton t dng (1.42) c v ging vi Nh vy, nhiu lon ca chng ta c th vit th di dng m n c s dng trong in ng lc hc c in v lin quan cht ch vi biu thc n gin nht l Khai trin nhiu lon cho N rt gn vi dng (8.47) ca nhiu lon in t - phonon. S hng u tin biu din s hp th ca mt photon bi in t m n lm tng nng lng ca in t ln mt lng l v tng xung lng ca n ln mt lng l Tng t, s hng th hai m t s pht x mt photon v do , in t mt c nng lng v xung lng. Trong cc hin tng quang, cc cng thc thng c th c n gin ha do photon mang xung lng khng ng k v sinh ra cc chuyn tip trong cc vng (phn 2.7). Cc th nghim quang thng c tin hnh vi nh sng nhn thy hoc cc bc sng gn . l cc bc sng c hng trm nanmt. N di hn nhiu so vi mt mng n v trong mt tinh th bn dn hoc b rng ca mt h lng t m chng thit lp thang o ca cc hm sng in t. Nh vy, ta thng c th b qua xung lng ca photon

48

8.5. S hp th quang v t trong nhiu lon khi nghin cu in trng khng i qua cc trng thi in t. l php gn ng lng cc in. N ngc vi php gn ng chun n hi m ta thc hin i vi cc phonon m. By gi ta c th t nhiu lon vo trong qui tc vng Fermi. Tc chuyn tip t trng thi i ti trng thi f nh s hp th mt photon l

Phn t ma trn trng phc tp mt cht. Cn

V d nh nu in trng b phn cc thun ty dc theo z, Cng sut b hp th bi chuyn tip ny l tch ca tc chuyn tip v nng lng ca tng photon Bc tip theo l ly tng theo tt c cc trng thi u v cui kh d c c cng sut tng cng. Ta cn phi bao

49

8.5. S hp th quang hm cc h s Fermi bo m rng trng thi u c lm y v bo m rng trng thi cui l trng nu khng s chuyn tip s khng th xy ra. Nh vy, h hp th cng sut tc

H s 2 trc tng l tnh n spin. Spin cn phi ch c tnh n mt ln v spin khng thay i bi chuyn tip quang. H s ny lun lun l mt iu kh chu v kh bit v tr tt nht cho n v cc qui c thay i. H cng pht ra nng lng tc nh vo s hng trong nhiu lon. N ging nh tr s i du ca

i s ca hm c th c hi phc ti dng trng bng cch hon i cc k hiu i v j m chng ch l cc ch s tng gi. Cc phn t ma trn khng b nh hng nhng cc hm Fermi thay i thnh

50

8.5. S hp th quang Cc h s ny v du hiu chung by gi ch l s khc nhau gia v v ng thi chng cho

Nh vy, tc hp th nng lng tng cng l

Vic so snh vi kt qu c in (8.65) ch ra rng

l cng thc tng qut i vi phn thc ca dn m n c th c chuyn i thnh phn o ca hm in mi khi dng (8.64). H s 2 c vit trc tng lu rng n sinh ra t vic ly tng theo cc spin. Mc d th tch ca h xut hin mu s, n cn phi c loi b sau khi ly tng v n nh th i vi tn x tp cht. Khi lng trong (8.76) l N l khi lng i vi cc in t t do khc vi mt khi lng hiu dng.

51

8.5. S hp th quang iu ny l v n rt ra t s hng ng nng ca phng trnh Schrodinger ban u khc vi mt php gn ng khi lng hiu dng. Ta s rt ra cng thc tng ng i vi hoc trong phn 10.1. Trc y ta tranh lun rng thng thng ta k vng Cc h s nh hng n du l v cc hm lp y. Hm p (nu v mt h cn bng c N bo m rng v h s hp th nng lng t nh sng. D dng chng minh rng l mt hm chn ca v do lun lun dng cn bng. Trong mt laze, ngi ta b tr s tp trung ca mt s trng thi b o ngc sao cho v trong trng hp ny, h c th gii phng nng lng khuch i mt chm nh sng ti (phn 10.6).8.6. S hp th gia cc vng Trong phn 2.7 ta thy rng s hp th quang l mt cch thun tin o cc khe vng trc tip v by gi ta c th tnh n i vi cc CB v VB n gin c ch ra trn hnh 8.4. Gi thit rng vt liu l 3 chiu, khng pha tp v nhit khng cho VB l y v CB l trng. Cc tng theo cc

52

8.6. S hp th gia cc vng Hnh 8.4. (a) S hp th t VB tiCB trong mt cht bn dn vi mtkhe vng trc tip. (b) S hp th ca cc mu 2 chiu v 3 chiu ca GaAs nhit phng. Th tch hotng ca mu 2 chiu (h lng t a lp) ch bng 1/ 2 th tch hotng ca mu 3 chiu.

trng thi i v j trong (8.76) mi ci tr thnh mt tng theo cc vng vi tng khc theo vect sng Bloch trong tng vng. Trong v d ca chng ta, ta c th gi thit rng trng thi u l trong VB, v trng thi cui l trong CB, v cc h s lp y cho 1 trong trng hp ny nhng trit tiu nu khng phi nh th. Nh vy,

Trc tin, xt phn t ma trn. Cc trng thi Bloch c th c vit di

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8.6. S hp th gia cc vng dng trong l tun hon t mt mng n v ca tinh th ti mng n v tip theo. H s chun ha cc trng thi theo mt cch sao cho trong gii hn ca cc in t t do. Ton t ly o hm bc 1 v do , qui tc nhn cho

Nh vy, phn t ma trn l

S hng th nht trong ngoc vung trit tiu sau khi ly tch phn do n rt gn thnh tch ca 2 trng thi trc giao. S hng cn li cng trit tiu tr do cc sng phng trong cc hm sng. Cch chng minh l dc theo cc ng nh tho lun tn x umklapp trc y. By gi, ta c

54

8.6. S hp th gia cc vng Ta li c th s dng thc t l l tun hon gia cc mng n v rt gn tch phn thnh th tch ca mt mng n v khc vi th tch tinh th

Biu thc ca dn by gi tr thnh

l kt qu chung i vi trong mt tinh th ngoi tr iu l ta b i cc hm lp y v cc tng theo cc vng m chng c th d dng c phc hi. . Mt vi ln ta gi thit rng phn tun hon ca cc hm Bloch ch ph thuc yu vo gn bin ca mt vng (xem (3.5)) v t Phn t ma trn rt gn thnh nu ta lp li gi thit ny y v c th c rt ra khi tng. T ,

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8.6. S hp th gia cc vng Tng trong cc ngoc xc nh mt mt trng thi ng vi mt n v th tch (phn 1.7.3) c gi l mt trng thi lin kt quang N bao gm cc nng lng ca 2 vng nhng gn nh ging nhau v dng vi kt qu thng thng v cha mt h s n l 2 i vi spin. By gi, dn c th c vit dng n gin

Phn t ma trn cng ging nh trong l thuyt (phn 7.3) v ta c th vit Cng hp th lin quan cht ch vi cc i lng khc ph thuc vo P nh cc khi lng hiu dng v c th c vit li theo thay i t trong s cc cht bn dn thng thng. Cc sai khc sinh ra t mt trng thi. Mt trng thi lin kt quang trng ging nhiu vi mt mt trng thi thng thng v tnh cht ca n chi phi Gi s rng 2 vng l parabol v i xng cu trong vng kho st. l mt php gn ng c th chp nhn gn trong GaAs vi

56

8.6. S hp th gia cc vng

Khi , s khc nhau ca cc nng lng xut hin trong nh ngha ca l

Do , s l mt trng thi quen thuc i vi cc in t t do nhng vi y vng c dch ti khe vng v vi khi lng hiu dng lin kt quang c cho bi khi lng rt gn Trong trng hp 3 chiu, mt trng thi tng t 0 ging nh v (8.85) ch ra rng dn quang ngay trn b (bin) hp th ti s cho cng mt kt qu. S liu i vi GaAs c ch ra trn hnh 8.4(b). Dng ca ng cong hp th khng ph hp tt vi cc d on ca chng ta. Vn ny gy ra bi exciton v s c cp trong phn 10.7. Kt lun ny c th c m rng cho cc h thp chiu nu ta khng quan tm qu nhiu n phn t ma trn v s phn cc nh sng (Chng 10). S

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8.6. S hp th gia cc vng thay i ch yu l trong m n c dng iu ging nh mt trng thi i vi cc in t t do (hnh 1.9). Nh vy, b hp th trong mt h lng t c c trng kiu bc v mt dy ch ra mt s phn k kiu Cc kt qu ny li khng chnh xc v nh lng nhng xu hng l cc k quan trng: b hp th quang tr thnh mt c tnh mnh hn khi s chiu gim. B hp th cng dch ti nng lng cao hn bao hm nng lng ca in t v l trng trn y h lng t. Cc mc nng lng l gin on trong mt chm lng t v s hp th phn nh iu ny m n l mt chui cc vch khc vi mt vng. 8.7. S hp th trong mt h lng t lm mt v d i lp, xt s hp th gia cc trng thi lin kt trong mt h lng t c t dc theo z nh ch ra trn hnh 8.5. H ny c nhng vo trong CB ca mt d cu trc v ta phi dng cc hm sng khi lng hiu dng. Ta s nghin cu s phc tp ny trong phn 10.5 v n khng c nh hng nh tnh ln tnh cht. T phn 4.5 ta bit rng cc trng thi c phn tch thnh mt tch ca mt trng thi lin kt theo z v mt sng phng ngang

58

8.7. S hp th trong mt h lng t trong v A l dn tch ca h lng t. Cho L l b dy ca ton b mu (khng ch h) dc theo z sao cho th tch Mi trng thi i i vi chuyn ng dc theo z sinh ra mt vng con ca cc nng lng. Xt phn t ma trn gia 2 trng thi nh th l Dng ca nHnh 8.5. S hp th bi cc chuyn tip gia cc trng thi trong mt h lng t. (a) Cc hm sng dc theo z vi cc mc nng lng. Cc b dyca cc hnh mi tn l cc ch dn thi vi cc cng dao ng t ca cc chuyn tip vi cc ng t nt biu din cc chuyn tip cm. (b) Cutrc vng trong mt phng ngangm n ch ra bn cht thng ng ca cc chuyn tip c php m chng cn phi i t cc trng thi lp y ti cc trng thi trng.

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8.7. S hp th trong mt h lng t ph thuc mnh vo s phn cc nh sng Trc tin gi s rng sao cho in trng song song vi x trong mt phng ca h lng t hoc vung gc vi n (xem hnh 10.6 i vi cc v d ca s phn cc v s lan truyn). Trong trng hp ny, v

do ton t xung lng nh hng ti sng phng trong Khi ,

Tch phn c ly theo tch ca 2 trng thi v trit tiu do tnh trc giao. Khng c nh sng b hp th vi s phn cc ny v cng mt kt qu r rng c duy tr i vi dc theo y. Nh vy, nh sng lan truyn vung gc vi cc lp mt s nh hng thun tin cho cc th nghim (mc d khng phi i vi cc ng dn sng) khng th b hp th trong cc chuyn tip ny. Cc kt qu l khc i vi Khi , in trng vung gc vi h lng t m n i hi nh sng lan truyn trong mt phng ca h. By gi, m n ch nh hng n hm sng ca trng thi lin

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8.7. S hp th trong mt h lng t kt. Nh vy, Tch phn theo cho A nu v bng 0 nu v do , mt vect sng 2 chiu c bo ton. Nh vy, cc chuyn tip quang li l thng ng trong nh ch ra trn hnh 8.5. Phn t ma trn cn li c th c vit gn l Vic thay phn t ma trn ny v cc nng lng t (8.88) vo trong cng thc chung (8.76) cho

N c th c rt gn thnh mt cng thc n gin theo mt s bc sau (i) Vect sng b loi b t hm do v gi thit mt h l tng trong khi lng l nh nhau cho mi mt vng con. n gin cc cng thc t v do , hm tr thnh

(ii) Vect sng by gi ch xut hin trong cc tng theo cc hm lp y. Mi mt tng c dng tnh n mt in t tng cng vi c 2 spin chim gi vng con v c th c k hiu l

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8.7. S hp th trong mt h lng t (iii) Ly tn s t h s pha trc bn trong tng. Hm c ngha l mi mt s hng ch ng gp khi v do , c th c thay bng m khng thay i gi tr. (iv) Thay cho cc phn t ma trn, ta a vo cc cng dao ng t khng c th nguyn c nh ngha bi

Dng th hai ca cng dao ng t vi mt phn t ma trn ca z khc vi c rt ra t th thut vi cc ton t. Cng dao ng t cho b loi tr tt nht t vic ly tng bo m rng ta b i cc chuyn tip t mt trng thi ti chnh n. Sau khi thc hin cc thao tc trn, ta c cng thc n gin sau

S hp th ch c nhn thy ti cc tn s tng ng vi s tch ca cc trng thi lin kt trong h. Nh vy, c nhng vch gin on mc d ph

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8.7. S hp th trong mt h lng t lin tc ca cc trng thi l sn c do gii hn ti cc chuyn tip thng ng. Cc vch s c m rng bi bt k s khc nhau no v khi lng hiu dng gia cc vng con (phn 4.9) v cc chuyn tip vo trong mi trng lin tc trn h lng t xy ra cc nng lng cao. C mt h s chung l 1/ L do mu ch cha mt h lng t m nh hng ca n gim i khi mu tr nn dy hn. Cc h lng t a lp c s dng tng cng s hp th trong thc t v chiu di L c thay bng chu k cu trc. Mt mu th nghim ca s hp th gia cc vng con trong mt h lng t a lp c ch ra trn hnh 8.6. Cng ca mi mt chuyn tip ph thuc vo cc hm sng qua cng dao ng t v s lp y ca cc vng con. Cc cng c th c thao tc qua bng cch thay i hnh dng ca h lng t hoc bng cch thay i nhng s lp y qua pha tp, a vo cc ht ti, bm hoc n gin l mt s thay i nhit . Mt s o ngc b chim c th lm cho trong mt phm vi tn s no . Tuy nhin, c mt kt qu ng ch l cng sut hp th tng cng ca mt mu c nh ngha bi tch phn ca theo tt c cc tn s ch ph thuc vo

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8.7. S hp th trong mt h lng t Hnh 8.6. Php o hp th gia cc vng con trong mt h lng t nhit phng khi dng cc lp vi 50 hlng t c b rng 6,5nm i (a) v 8,2 nm i vi (b)v (c). Mi mt h cha in t. Cc nh nh hn bn phi l nh vo cc phonon.

mt in t tng cng ca n

l mt v d ca qui tc cng v rt ra t qui tc cng f Thomas Reiche Rahn v cc cng dao ng t

Nh vy, mt s thay i ca ti mt tn s cn phi c b tr bi nhng thay i ngc li ni khc. C nhiu p t nh th ln m n s c xem xt chi tit hn phn 10.1.3. Cc cng dao ng t c s dng chung trong cc biu thc i vi

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8.7. S hp th trong mt h lng t cc tnh cht quang v cng thc chung (8.76) c th c vit li thnh

N c th c biu din chn la thnh mt tng theo cc chuyn tip khc vi cc trng thi u v cui (phn 10.1.2). Cng thc (8.83) i vi s hp th gia cc vng cng c th c vit theo cch ny mc d k hiu c l l km thch hp hn i vi mt khong nng lng lin tc. Cc kt qu khc nh khai trin ca cc vng nng lng (xem (7.43)) cng c th c biu din theo cc cng dao dao ng t. Ngi ta c th lo ngi v kh nng ng dng ca qui tc vng trong trng hp ca cc nng lng gin on do n ch c gi tr nu xem xt mt min lin tc ca cc nng lng. Trng bc x thng cung cp mt mi trng lin tc. Mt loi tr l khi trng in t b by trong mt vi hc hoc mt cu trc tun hon m c hai sinh ra cc khe trong ph ca chng. Khi , c th khng c mi trng lin tc ca cc nng lng, qui tc vng tht bi v in ng lc hc ch ra cc hin tng vt l mi th v vi cc ng dng th cho cc thit b quang in t.

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8.7. S hp th trong mt h lng t Khi quay tr li h lng t, ta pht hin thy rng s hp th quang ch xy ra gia cc vng con khc nhau v do ti cc nng lng cho bi s khc nhau gia cc trng thi lin kt trong h. nh sng cn c mt thnh phn in trng dc theo z. Nhim v cn li l cn nh gi phn t ma trn

Mt kt qu quan trng c rt ra t s i xng ca h lng t. Ta bit rng cc hm sng trong mt h i xng v d nh h trn hnh 8.5, trong hoc chn hoc l theo z. o hm lm thay i tnh chn l v phn t ma trn s ch khc 0 nu mt trng thi l chn v trng thi khc l l. l qui tc lc la m n chi phi cc chuyn tip c th quan st thy trong s hp th quang v chnh xc l cng kt qu m ta tm c i vi h s phn cc ca mt h lng t trong in trng (phn 7.2.2). Nh vy, s hp th l c php t trng thi thp nht (n = 1) ti n = 2, 4, ... nhng khng ti cc gi tr l ca n. Kt qu ny p dng cho bt k h i xng no v c th b tht bi do s nui c mt h khng i xng. kt lun, ta s tnh cng dao ng t trong chuyn tip t n = 1 ti n = 2 trong mt h su v hn ti 0 < z < a. Ta cn phn t ma trn

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8.7. S hp th trong mt h lng t

Mt cch la chn khc l t (E1.4) hoc (8.93)

Dng sau l phn t ma trn lng cc v c trch dn ph bin hn. N ph thuc vo b rng h v tr thnh ln trong mt h rng. Mt hi vng r rng l lng cc khng l ny c ngha l s hp th s xy ra mnh. Cng dao ng t l

m n c th c lng vo trong (8.94) tnh Lu l cng dao ng t khng ph thuc vo a v khng tng i vi mt h rng. Tuy nhin, ta cng c th thy rng n l mt chuyn tip mnh do cng dao ng t ca n l 0,96 khai thc ht phn ln qui tc cng f v cc chuyn tip khc t trng thi 1 cn phi yu hn nhiu. Nh vy, chuyn tip thp nht trong mt h lng t c th l mt cch c hiu qu hp th nh

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8.7. S hp th trong mt h lng t sng. Trong thc t, c nhiu chng ngi cn vt qua v d nh s tn x t cc phonon v s thot ra khi cc h v mt l thuyt hon chnh khng th b qua cc exciton. By gi ta thy 2 v d ngc nhau ca s hp th quang trong mt v d dn ti mt vng hp th lin tc v mt v d cho cc nng lng gin on phn nh cc nng lng lin kt trong mt h. Cc c im ny c kt hp trong cc chuyn tip trong mt h lng t t VB ti CB. c th l trng hp quan trng nht trong thc t v n s c xem xt trong Chng 10. 8.8. Cc gin v nng lng ring Trc khi ri b qui tc vng Fermi, ta s xem xt cc gin c lin quan. iu ny r rng khng phi l mt nghin cu l thuyt nhiu lon chnh tc v cc hm Green m ch l mt minh ha cho mi lin h gia qui tc vng Fermi v cc gin Feynman n gin. N cng ch ra 2 loi l thuyt nhiu lon m ta pht trin trong 2 chng ny lin h vi nhau nh th no. c bit l ta s thy rng cc kt qu ca chng ta t l thuyt nhiu lon c th c kt hp cho (gn nh) php gn ng Born cho nng lng

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8.8. Cc gin v nng lng ring ring m n c th l i lng c tnh nhiu nht i vi cc hm Green. Phn ny khng phi l c bn i vi cc chng cn li. Ta rt ra 2 dng khc nhau b ngoi ca l thuyt nhiu lon i vi cc th tnh trong 2 chng va qua. l l thuyt khng ph thuc vo thi gian trong phn 7.2 v qui tc vng Fermi trong phn 8.1. Ci th nht cho s thay i nng lng v hm sng ca mt trng thi ring trong khi ci th hai cho thi gian sng (tc tn x) i vi cc trng thi u. Mc d cc qu trnh ny xut hin khc hn nhau, thc ra chng c lin quan cht ch vi nhau. Xt nh hng ca mt th tp cht m bin i Fourier ca n l ln cc sng phng c nng lng Trong trng hp ny, cc phng trnh (7.31) i vi s thay i nng lng v (8.24) i vi tc tn x tng cng (hoc nghch o ca thi gian sng n ht) tr thnh

69

8.8. Cc gin v nng lng ring Tc tn x c chuyn i thnh nng lng bng cch nhn vi 2 cong thc l tng t vi nhau v c th c kt hp cho mt nng lng phc n c nh ngha bi {P

K hiu c ngha l mt i lng dng v cng b. Dng th hai trong phn chnh ca i lng nghch o v hm c kt hp l hon ton tng ng v ngha vi dng th nht. Cng mt th thut s c s dng i vi hm in mi phc v c gii thch trong phn 10.1.2. K hiu P ch yu lu b qua s hng vi m n dn ti mt mu s trit tiu. Thc ra n c ngha l phn chnh Cauchy ca mt tch phn theo nng lng. By gi ta rt gn vi nng lng ring (tr). l mt i lng trung tm trong l thuyt trng m ta tnh n trong php gn ng Born. Chnh xc hn, nng lng ring c v E nh l cc bin s c lp

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8.8. Cc gin v nng lng ring

Ta nh gi n ti nng lng khng nhiu lon ca trng thi k bng cch t Ta cng bit rng phn thc ca nng lng ring dch nng lng ca trng thi. Cn phn o ca nng lng ring l g? t nng lng phc vo s ph thuc thi gian chun ta c

Phn o lm cho hm sng gim i nh mt hm ca thi gian v mt c dng iu ging nh vi thi gian sng Nh vy, mt phn hm sng trng thi ban u gim v xc sut tn x vo trong trng thi khc tng m n chnh xc l ci ta tnh ton trong qui tc vng Fermi. Phn o ca nng lng ring do cho tc ca cc chuyn tip thc ra khi trng thi ban u m n c vect sng trong v d tn x tp cht

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8.8. Cc gin v nng lng ring ca chng ta. N l t nhin gii thch phn thc ca m n dch nng lng theo cng mt cch v ni rng n sinh ra t cc chuyn tip o. Bc tranh y l in t to ra mt chuyn tip t trng thi ti nhng do n c nng lng sai i vi trng thi ny, n tr li sau mt thi gian hu hn l Nng lng gn hn c bo ton. in t c th gi trng thi khc cng lu th ng gp vo s thay i nng lng v hm sng cng ln. By gi ta c mt s m t thng nht i vi nng lng ring theo cc chuyn tip ti cc trng thi khc bi c 2 qu trnh thc v qu trnh o. Bc tranh ny c chuyn trc tip vo trong cc gin . Hnh 8.7(a) l mt biu din r rng v tn x ca mt din t bi mt tp cht. in t c ch ra nh mt ng thng. N i vo vi vect sng v nng lng tng tc ti nh vi tp cht c cng v i ra vi vect sng S bo ton xung lng c nh vect sng ca in t i ra nhng nng lng ca n khc vi gi tr ban u trong cc chuyn tip o.Tp cht c ch ra nh mt du cho v c tch ra t mt nh bi ng t nt nhn r hn.

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8.8. Cc gin v nng lng ring

Hnh 8.7. Cc gin minh ha mt in t tn x t mt tp cht hoc mt phonon . (a) nh n trong mt in t i vo vi vect sng v i ra vi vect sng Phn t ma trn i vi tng tc l (b) S kt hp ca 2 nh nh vy cho mt nng lng ring. (c) Nng lng ring i vi tn x in t - phonon. Phonon c biu din bng mt ng ln sng. By gi, c 2 l thuyt nhiu lon bc 2 v qui tc vng v do nng lng ring u cha 2 phn t ma trn ging nh v ngi ta c th t hi ci g xy ra i vi phn t ma trn th hai. Cu tr li l gin ny ch cha mt na tc dng. Cc phn t ma trn l ecmit v do lin hp phc c th c vit li thnh m n tng ng vi qu trnh vi cc trng thi u v cui o ngc. i vi tp cht, n c ngha l Vic t 2 qu trnh cng nhau cho gin trn hnh 8.7(b). 2 nh l lin kt bi cc ng t nt ti cng tp cht lu rng n l cng th tc dng 2 ln. N chnh xc l dng ca mt gin nng lng ring

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8.8. Cc gin v nng lng ring trong l thuyt nhiu ht nu cc ng trn cc u cui tng ng vi trng thi ban u b b i. Ta cng c th vit li biu thc nng lng ring phn nh 2 s kin ring r ny nh sau

in t tn x khi t c vect sng lan truyn trong mt thi gian no vi vct sng v cui cng li mt tr li trng thi ban u. Thc ra thng s gia (8.106) c gi l hm truyn n ht (tr) hay hm Green. N l mt ch tt dng li. Ai quan tm n n c th c cc sch v l thuyt nhiu ht. Cc phonon c th c hnh dung theo cch tng t (hnh 8.7(c)). Cc phonon c ch ra nh mt ng ln sng ni 2 nh. Trong trng hp ny, c nng lng v xung lng c chuyn giao ti cc nh v phonon t n c m t bi mt hm Green m n dn ti cc s phonon cho s pht x v hp th. Cc tnh ton khng i xa hn gin trn hnh 8.7 m n c gi l php gn ng cho cc phonon cng nh cc tp cht nhng khng phi tt c cc

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8.8. Cc gin v nng lng ring tnh cht vt l u rt gn thnh mt ci g n gin nh th. V d nh thi gian sng n ht ca mt in t c th tm c bng cch tnh cc gin trn hnh 8.7 nhng thi gian sng vn chuyn th kh hn nhiu.Bi tp chng 88.1. Mt in t trng thi thp nht ca mt h lng t GaAs c b rng 15 nm m n c th c nghin cu nh mt h 1 chiu su v hn. t nhin on gia rng 5 nm tr nn su hn 100 meV. Dng qui tc vng rt ra mt biu thc i vi tc ti in t tn x vo trong cc trng thi khc v nh gi tc i vi s chuyn tip cho php thp nht. Kt qu s khc nh th no nu on gia nng ln thay cho h xung v s dng l thuyt nhiu lon c thch hp hay khng? Lp li tnh ton cc tc tn x khi gi s rng h y phng lc u t nhin b lm nghing bi mt in trng Tnh ton ny lin quan nh th no vi tnh ton trong phn 7.2.2 i vi cc mc nng lng ca mt h lng t trong mt in trng.8.2. Rt ra tc tn x i vi s vn chuyn trong mt h 3 chiu vi cc th i xng cu

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Bi tp chng 8

Phng php tng t vi phng php dng i vi cc phng trnh (8.31) v (8.32).8.3. Ro hnh trn cung cp mt m hnh th nh gi tn x hp kim i vi mt 2DEG trong AlGaAs. Chi mt phng thnh cc mng n v. Cc mng n v cha GaAs c th khng trong khi cc mng n v i vi AlAs c th 1 eV m n gn ng l s khc bit nng lng ca thung lng trong 2 vt liu. Lm gn ng mng n v bng mt ng trn v tnh tc tn x v linh ng trong Ch mt phn nh hm sng xuyn vo trong AlGaAs trong mt d cu trc GaAs AlGaAs thng thng sao cho nh hng nh hn nhiu so vi nh hng trong tnh ton ny.Mt php gn ng tt c th dng o cc nng lng t th nng trung bnh khc vi th nng ca GaAs. Trng hp ny, c cc mng

76

Bi tp chng 8 GaAs v cc mng AlAs u ng gp vo tn x. iu ny to ra s khc nhau l bao nhiu? Mt l thuyt tt cn khng ph thuc vo cc la chn ty nh vy v cc phng php thit lp nh php gn ng th kt hp (CPA) l tt hn nhiu.8.4. Xt tn x Rutherford 2 chiu t mt th Coulomb khng b chn trong mt phng ca mt 2DEG. Chng minh rng tit din vi phn trong php gn ng Born l

trong l bn knh Bohr m n tnh n khi lng hiu dng v hng s in mi (xem (4.67)). Cn n tch phn

Tn ti tit din n ht hay tit din vn chuyn? Bi ton c th c gii chnh xc trong c c hc c in v c hc lng t. iu ny chng t rng php gn ng Born b qua h s Do , php

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Bi tp chng 8 gn ng Born l tin cy vi iu kin l iu ny c ng hay khng i vi mt 2DEG in hnh vi in t trong GaAs? Trong trng hp 3 chiu, php gn ng Born l ging vi kt qu chnh xc trong c c hc lng t v c hc c in.8.5. Gi s rng phm vi xa ca th Coulomb b cn tr bi mt h s hm m m n cho Tnh tc tn x trong trng hp 2 chiu trong cn dng tch phn

Chng minh rng kt qu c nhng c im chung ging nh ro trn nhng khng c cc dao ng. Lu rng n khng phi l dng chnh xc ca mt th Coulomb chn trong mt h 2 chiu m ta s tnh n trong phn 9.4. 8.6. Mt s bi ton ph thuc vo thi gian l n gin gii m khng cn n l thuyt nhiu lon. y l mt s thch ng i vi mt h thp chiu. Gi s rng ta c mt chm lng t tnh vi cc trng thi ring

78

Bi tp chng 8 c nng lng Khi , chm b iu khin bi mt trng c tn s v tuyn m n thm vo mt th nng trong l khng i trong khng gian. Chng minh rng cc trng thi ring ph thuc vo thi gian tr thnh

Dng vit li biu thc trn thnh

Nh vy, mi mt trng thi pht trin cc vng bn ti cc tn s b dch i vi cc cng H ca cc nng lng ny c th cho php cc chuyn tip b sung nu chm c lin kt vi my d khc no i vi cu trc in t ca n.8.7. Xt mt kh in t khng suy bin v so snh nng lng nhit v nng lng trao i vi mt phonon m trong mt s kin tn x in hnh. Php gn ng chun n hi v tc tn x gn lin c gi tr khi no?8.8. So snh cc d on ca (*.54) i vi linh ng b gii hn bi cc

79

Bi tp chng 8 phonon m vi s liu i vi cc cht bn dn nhit phng (Ph lc 2). Tn x nhit phng trong GaAs b chi phi bi cc phonon LO cc v do , kt qu ny l khng ng dng c nhng n c tc dng i vi Si. Ly 8.9. nh gi tc tn x nh vo s hp th ca cc phonon LO cc trong GaAs. Gi thit rng trng thi ban u ca in t c nng lng rt thp sao cho trng thi cui n c nng lng ch ngay trn Trong trng hp ny, s thay i vect sng trong Chng minh rng tc tn x gn ng bng

trit tiu t tt c tr hm v do , tn x l ng hng. N cn phi c ly tng theo mi phonon m n cho mt trng thi v rt gn thnh

linh ng thu c ph hp vi thc nghim i vi GaAs nh th no?

80

Bi tp chng 88.10. Vic rt ra s hp th quang ca chng ta nhm nghin cu sng in t v phng din c in m n khng phi lun lun thch hp. thay th, dn c th c vit vi mt s photon khc vi trong s tng t vi trng hp phonon. Mt th thut tng t c th c s dng tnh dn nh vo mt photon n v khi nhn vi i vi s hp th v i vi s pht x. Ta cn tm i vi mt photon n m n c th c thc hin nh i vi phonon trong phn 2.8.1. Lu rng cng c mt t trng m n mang nng lng bng nhau. Chng minh rng v vit cng thc i vi theo cc s photon. Cn thn trng vi kt qu ny m n cn phi c ly tng theo tt c cc mode in t vi cng tn s nhng cc hng lan truyn khc nhau. C 2 s phn cc cng cn c bao hm.8.11. Tnh bin thin Burstein trong b hp th ca mt bn dn trc tip vi cc vng parabol nh vo pha tp nng (loi p) ti mt nhit thp sao cho cc l trng l suy bin. B qua cc exciton. Lu rng dch khng n gin l nng lng Fermi ca cc l trng v bao hm khi lng ca c CB v VB. iu g xy ra i vi hnh dng ca b hp th?

81

Bi tp chng 88.12. Chng minh phng trnh (8.95) t qui tc cng f. N d dng hn nu ta lu rng l mt hm chn ca v m rng tch phn ti khong t ti C th dng h thc8.13. Chng minh rng 2 dng ca cng dao ng t trong (8.93) l tng ng vi nhau. Trc ht chng minh rng giao hon t vi iu kin l ton t xung lng trong ch xut hin trong Cn nh gi m n c th c tin hnh n gin nht bng cch cng v tr i H thc ton t ny c th c p dng cho cc phn t ma trn v do , Khai trin giao hon t ny v dng tnh cht l v l cc trng thi ring ca ton t Hamilton chng minh rng

8.14. Nu bn gii c bi ton trc, bn s khng kh chng minh qui tc cng f. Phng php l cn nh gi theo 2 cch gi tr k vng ca mt giao hon t kp trng thi i c dng Trc ht dng kt qu trc i vi v giao hon n vi z thu

82

Bi tp chng 8 c N ch l mt hng s v tch phn cn li l n v do s chun ho. Trong nh gi th hai, nhn ngoi giao hon t c c 4 s hng. Lng cc h y ca cc trng thi ring nng lng gia cc ton t khi dng h thc kn (1.79)

v ly tch phn theo bin ph Cc phn t ma trn ca ton t Hamilton l tm thng gia cc trng thi ring nng lng v cc phn t ma trn khng tm thng l cc phn t ma trn ca z. Chng minh rng gi tr k vng c rt gn thnh Cn bng 2 biu thc cho qui tc cng f.8.15. H su v hn l mt h n gin sao cho c th xc minh qui tc cng f mt cch trc tip. Th iu ny i vi i = 1. Cn s dng

Ta cng c th xem xt cc chuyn tip bt u t i = 2. Cng dao ng

83

Bi tp chng 8 t thp nht l m, tng theo cn cn phi cho n v 8.16. Xc nh qui tc lc la i vi cc chuyn tip t trng thi thp nht trong mt h parabol. Tnh cng dao ng t ca chuyn tip cho php u tin v t qui tc cng f chng t rng tt c cc chuyn tip khc cn phi trit tiu.8.17. nh sng vi ti GaAs khng pha tp. Cng (t l vi bnh phng bin ) gim ging nh Rt ra h thc v chng minh rng chiu di gim Gi thit rng cc l trng l nng v dng s liu trong Ph lc 2 vi 8.18. S hp th gia cc vng con trong mt h lng t (xem (8.95)) thay i nh th no theo mt in t ti nhit thp khi c hn mt vng con b chim gi? Cc kt qu tha mn qui tc (8.95) i vi s hp th c ly tch phn theo tt c cc tn s nh th no?

84

.qwhtqwcosiF=y[][])2.8(/)()()()(0ttittVHtH=+=yyyh)))jf)(tV)0H)H)0H)())1.8.(/exp)()(htittiiiefy-=F=.je),(tjF.0H).)0(ijjtad==)(tajF=jjjttat)3.8(),()()(y)(tjFF+F==F+FjjjjjjjjjjjjtdttdaitttaittVtatHta)5.8).(()()()()()()()()(0hh))[]()F=F+jjjjjjttatittatVH)4.8)(()(/)()()(0h)).)(ijjtad=V)jffjeee-==jfjfjjftitVtaidttda)8.8(,exp)()(1)(hhe)7.8).((exp)()(exp)(exp)(*tVtitatVtitatidttdaifjjjjjfjjjff-==-=-h)hhheffee*ff-=-jjjjjjjjtitVtatidttdai)6.8.(exp)()(exp)(h)hhefef()())11.8.(2/2/sin2exp1/exp)(fififfififififtVtiitiVtaeeeeehhh-=--=fiV)(tV)())10.8.(exp1)(0'''=tfififdttitVitahhe0=faif)9.8.(exp)(1)(=hhtitVidttdafifife2sinc321,,tttfiet.2t.fie.sinsinqqq=c())12.8(2sin2/2/sin)(2222222==hhhtctVtVtafifififififeee.t())15.8(22sin22fifittctedpehh-=)14.8(22sin22tdtctfifihhpee-=)13.8(sin2pxdxc.d()())17.8.(/22iffifiVWeedp-=h()())16.8.(/2~)(22tVtafififedphfe()()-iiffiV)18.8.(/22eedph()=-iifeed.fed.d"dt,/th()Ed.tt/h())19.8(,fedd())20.8.(,iffeee=()2/2fifiVWhp=)(rVrqrh()()[])21.8(.exp,.exp2/12/1rqkiArkiAfirrrrr+==--ffrr()()()()[])23.8.(~/222,kqkqVAWkqkrrrrhrrreedp-+=-+qkrr+kr()qVr~()()()())22.8(,~12121*---+--=====qVArderVArderVeAVVrqirkirqkiiffirrrrr)rrrrrrrff.krit/1kr2/1A())24.8.(/1,1+=qkqktapiWrrrrtitqr)2(Dtapn())25.8.(222qdAqrrp.)2()2(DtapDtapAnN=.kr()kqkrrr=+it[]()()()()[]()()()[]())26.8.(2~2~2121222)2(2222)2(-+==-+=peedpeedpptqdkqkqVnqdkqkqVAAAnDtapDtapirsrrrhrsrrrh)2(DtapNqqr./metrtm=trt222/cos1kq=-q()()()[]())28.8.(2~22122222)2(peedptqdkqkqVkqnDtaptrrsrrrh-+=.cos1q-,cosqqqrkrpq=q())27.8.(2/sin2qkq=itqr()qVr~.itrtt=.qr()qVr~()()()[]()())29.8.(22~222~221222'22'022''')2(2'2'2'22')2(---==---=-mkmkkkVkkkdkkdnkdkkkkVkkknDtapDtaptrhhrrrrhrrrrrrrhdqppeedptpp.'qkkrrr+=.trtkrkk=')30.8.(cos2'22'2'qkkkkkk-+=-rr()p,0q())31.8.(cos12sin2~1023)2(qqqptpdkVmnDtaptr-=h()())32.8.(2/1~212220233)2(kqdqqqVkmnkDtaptr-=hpt'kr.2/2'2mkh./'2mkh()kk='dq'k().2/sin2'qkkk==-qrris./mkvh=iivlt=Fkk=222/cos1kq=-q1)2(=iiDtaplns.iliils()qsis=pqqps0242)33.8.(2sin2~dkVkmih./)2(iDtapiknmtsh=().~2qVqcos1-trs()())34.8.(2sin2~2,242==-qpqsqqssppkVkmdih.rrqrq()())36.8(,)(~020cos2--==pqqiqrrqiedrdrrVrderVqVrrr())35.8()(0),(0>>tw.0weeh-=if0wh.0weh+=itie0w0weeh-=if()())43.8.(/202weedphh--=iffifiVWtieV0w)())44.8.(,0weeeh+=iff()2/2fifiVWhp=tie0w-.X()())46.8(,sin/2sin)()(0tqzqtqzqUzezUqqqwrww-XW-=-X-=X=h())45.8.(sin/)(0tqzUzuzeqw--==()tqzUzuqw-=cos)(0()./1auujj--Qr()())47.8.(2/,tiiqztiiqzsqqeeeevqitzVwwr---XW=h)qvsq=w0U()().'QKKrhrrwee+=)48.8.(1exp1--=TkNBQQrrhwd()tiQrw-exp.'KrKrWQNr.1+QNrQNrQr)49.8.(123...''RdeeevQiVRKiRQiKKisKKrhrrrrrrrrW-+WXW=rd'KrdQKKrrr+='1+QNrdqvsQhhr=w()()[])50.8.(22,2''QQKKsQKKKQKvQNWrrrrrrrhrrrhhweeddrp--+XW=++'KrKrQNr.'QKKrrr+=0rW/1W()()()[])51.8.(212,2''QQKKsQKKKQKvQNWrrrrrrrhrrrhhweeddrp+--XW+=--15-kms.2FkwhFkq2=.14,0,101-nmkmeVEFF215210.3-=mnD()()KQKrrree=+.9,02meVkvFsh()()[])53.8.(12122-+WX=KQKvTkQsBrrrhreedrptQKrr+Kr.1/1>>+QBQQTkNNrrrhwKrQrQr()()[]()()[])52.8.(12222222,KQKvTkKQKvQQvTkWsBssBKQKrrrhrrrhhhrreedrpeedrp-+XW==-+XW=+.2/3-TmTEEn)(()().2/3TkKB=re.t()[])54.8.(122sBvKTnkrepthrX=cellsn)55.8(,220LOcellsLOcellsnNUmwmwW==hh0ULOQww=r0rr=Q()..cos)(0tRQUtuQjrrrw-=).()(tenQtpeff=()).(tpntPcells=.1

s2e1s.~,~2121ssseeeiir+=+=()())64.8.(~/1~,1~~00sweeewesiirr+=--=rn().//~cincnkrrrwkw+==.2rrnk=s~=-=2221,ekerrnrk)(1we)(2we()()wewe21,2(e1erk(erEJrr.EJrr1s=()()[]tRQitRwf-rrrexp,.Qerr^er()().cos2,tRQetREw-=rrrrr1s()()().sin/2,0tRQeEtRAww-=rrrrtAE-=/rr)65.8.(2201EPW=sWArApr)r.2Arp)r()()()[])68.8.(.Y+Y-=Y-=AAIAiAprrhrhr)ry()())67.8.(..2/20AeAppAmeVrr)r)rr)++=ArcrystalV())66.8(,202tiVmAepcrystalY=Y++hr)rQr()[].exptRQiQigradEwf--=rrrr0=ArY.QrhYpA)rrwh()()()[]())69.8.(.,00peeeimeEtRVtRQitRQi)rrr)rrrrwww----=..JErrJArr.()./0pme)r()../0pAmeV)rr)=.wh()./zi-=h()==pee)rrr.,1,0,0)71.8.(.++-=zeyexeipezyxh)rr())70.8.(.22200wdwph)rrh--=ijfiEEipejmeEV0rr=Qw+Pd+Ptiew-P()()[]())72.8.(1.222,200wdwwph)rrhh---=+ijjijiEEEfEfipejmeEP()jEf-1()iEf()()[]())73.8.(1.222,200wdwwph)rrhh+---=-ijjijiEEEfEfipejmeEP()()[].1ijEfEf--P+P.0mW()()[]()()[]()())74.8.(11jiijjiEfEfEfEfEfEf-=---()()[]())75.8.(.222,200wdwwph)rrhh---=ijjijiEEEfEfipejmeEP()()()[]())76.8.(.22,2021wdwpwsh)rr---W=ijjijiEEEfEfipejme1s01>s()iEfw()>jEfdw.01>s2s1e()().jiEfEf>)0>wijEE>Kr'Kcjr=Kvir=()()()[])77.8.(.2'2',2021'wdwpwshrrr)rrrrr--W=KEKEKvpeKcmeVCKK()()()()[])80.8.(1.3*'''RdRupRuKeRueKpeKcKKRKKiKcrr)rrrhrrr)rrrrrrrrrnnn+W=-()()()[])79.8.(2/1RupRuKeRpKKRKiKr)rrrhr)rrrrrrnnn+W=Y-()())78.8(,2/1RueRKRKiKrrrrrrnn-W=Y-=h)rip()1=RuKrrn2/1-W()RuKrrnKKrr='()()()()())81.8.(.1,.3*,''RdRupeRuKpKpKpeKcKKcccKKrr)rrrrrr)rrrrrrrnnndnW==()0rncp.0rrnKnuuKr1s()Rur()()()())82.8.(.13*RdRupeRuKpKcellsKccellscrr)rrrrrrnnW=()()()()[])83.8.(222021weedwpwsnhrrrr--W=KKKpmeVCKc()()()()[])84.8.(222021--W=KVCcKKKpmerhrrrweedwpwsn1s()()./00Pimpchr=npkrr.().Enopt()()())85.8.(22021wwpwsnhroptcnKpme=G).(1wsPE./1/1hemm+==ehm/1VCgEEE-=()Enopt()()())87.8.(112022++-=-heVCVCmmmKEEKKhrree()Enopt()())86.8.(2,2022022eCChVVmmKEKmmKEKhrhr+=-=eegE=wh()2/1gEE-().2/1--gEwh()Enopt..'kipekjr)rrr.AL=W),(yxr=r()()()())88.8(,2/,exp222/1mkkErkizARiiikihrrrrr+==Y-efkrer)90.8.(0.''==kikjkkipekjxrrhr)rrr().RkirrYxipe-=/.h)rr)0,0,1(=er.er()()())89.8(.RkRpekixkirhr)rrrrY=Yzipe-=/.h)rr).1,0,0(=erijjieew-=hkkrr'()()[]()[]{}()()[])92.8.(22,,2021wdwpwshrrrr)r---W=kEkEkEfkEfipjmeijjizkji.jn()()[]kjkEfArr/2kr()()./1jiwwd-hdrr())91.8).(()(.'*31'zpezrddzAkipekjizrkkijff)rr)rrrrrr--=()()ijijkEkEee-=-rrdkr.ipjz)krkkrr='jiwwjiww=dw()()()--=ijjijijijinnfmLe,21)94.8.(2wwdpwszp))93.8.(2222izjmipjmfjizjijih)hww==()ws1()01

jif()wer~1s=ijijif,)96.8.(121510.4-m()==jjDDnnmLned)95.8.(,220221pwwspkrr.()()()[]()--W=jijijijiEfEffme,21)97.8.(22wwdpws()zVzV=-)(())98.8.()(/)(*dzzdzdziipjijzff-=h).1s())101.8(96,0272561222212221=-=peepzfh)100.8.(9/16122paz-=)91.8.(38sin2sin2120aidzazdzdazaipazhh)-=-=pp()qVr~()()()()()()()()()[])102.8.(~2,~~20,2qkkqVkqkkqVkkqiqqrrrrhrrrrrrrrrrr+-=G+-=DSeedpteee().kre()()()+++-=qiqkkqVrrrrr)103.8.(0~2ee+0()()()()[]=+--+-qkkiqkkrrrhrrreedee1()=G-S=SqqVirr2~21~()krS.hGkr0rr=qd().kEre=()()())104.8.(0~,2+++-=qiqkEqVEkrrrrrekr./G=hit()h/exptG-2)(tY()()[]()()[]())105.8.(exp~expexp)(G-S+-=S+-Yhrhrrhrrtkkkikkitee(),/exphiEt-(),krekr()()[]./qkkrrrh+-eeqkrr+krS().qkrr+()qVr~.qkrr+kr().~qVr-=()=qVr*~iffiVV=*()2~qVr().~qVrqrqkrr+,qr()()()())106.8.(~01~,qViqkEqVEkqrrrrrr+++--=Se.11-=MVmFtrtit.7,03,0AsGaAlG()()-===pqqqqppt023)3(320233)3()2.8.(sincos12sin2~2)1.8(~41EdKVmKnEdQQQVVmnDtapKDtaptrhh()()./tanh/kakaBBpp=00.1)(dxxJBa()()())3.8(,2/sin222EkakBqpqs=()rebepe024/-=()=rVnf()().2/12200--+=babadxxJex()().4/02rbererVlepe--=21510.3-m./1Bak>>()./002whVJk0whk())5.8.(exp)(000EtkiVJtnkknn+-=F-=hhhwewf()()()-==kkikzJizqqexpsinexp)4.8.(sinexpexp)(000EtVititnnn--=Fwwefhh0V,cos00tVw.neQr()()[])6.8.()0(1)(122202,EKQKKeNWLOLOLOLOKQKweedeeewphrrrhhrrr--+-W=++.2/22LOLOmKwhh=,LOKQ.LOwh.10.4,1;10;3,03122-==X=JmveVmser()())7.8.(011102ENKeLOLOtr-=eepethd()2/10102/W=eewhE0E1+wNwN20E()ws1[]()zpmiHz)h)/,=.2/2mp)H)[]Hz),ij())8.8.(Eizjimipjijzh)ee-=[][])9.8.(,,EizHzi).jiijff-=.-w()ws1()[].,/iHzjimipjz)h)=.zzpzp))[]2,zpz)()()())10.8(11'*'Ejjzzzzjjjj===-ffdii./2mh())12.8.(2561421322Ennnp=-=())11.8.(22Eizjjij-ee'z.5,3@rn.05,0eV++=-gEmwmah,11rnc01/es===cr/2wka,96,02112--=ff().expza-gE>wh2jf