Chng 4: Cc h lng t v cc h thp chiu

Chng 4: Cc h lng t v cc h thp chiu Cc in t thc l cc in t 3 chiu nhng chng c th c to ra x s dng nh chng ch t do chuyn ng theo s chiu t hn. iu ny c th t c bng cch by chng trong mt h th hp gii hn chuyn ng ca chng theo 1 chiu ti cc mc nng lng gin on. Nu s tch ra gia cc mc nng lng ny l ln, cc in t dng nh b ng bng vo trong cc trng thi c bn v khng th c chuyn ng theo chiu ny. Kt qu l mt kh in t 2 chiu (2DEG). Cng mt hiu ng c th t c vi mt h th 2 chiu nhm lm cho cc in t ch chuyn ng t do theo 1 chiu. l mt dy lng t. Phn u ca chng ny lin quan n mt s h th 1 chiu n gin dng by cc in t. Khng th to ra h vung gc su v hn trong thc t nhng tnh n gin ca h ny lm cho n tr thnh mt m hnh c s dng thng xuyn. H c su hu hn cung cp mt s m t tt hn nhiu i vi mt h lng t thc. Cc h parabol c th c nui bng cch thay i thnh phn ca bn dn mt cch lin tc nhng h th ny lin quan nhiu nht n nghin cu t trng. V d cui cng l mt h th tam gic m n c dng nh mt m t th i vi 2DEG to thnh ti mt d

Chng 4: Cc h lng t v cc h thp chiu chuyn tip pha tp. Tip theo ta s xt xem lm th no cc h th ny lm cho cc in t x s dng nh chng l 2 chiu. Cc phn cui lin quan ti s giam cm nhiu hn na cho cc h 1 chiu hoc 0 chiu v mt vi chi tit thay i do cc khi lng hiu dng khc nhau trong cc d cu trc. Ta s xem xt cc h 1, 2 v 3 chiu v iu quan trng l cn c mt k hiu r rng phn nh iu ny. Gi z l hng nui v hng ny vung gc vi cc mt phng ca mt cu trc xp thnh lp. Vect l vect 2 chiu vung gc vi z. Vect l vect 3 chiu.4.1. H vung gc su v hn H vung gc su v hn l v d n gin nht ca mt h lng t. N cng c xem xt chng 1. so snh vi h su hu hn, ta ly gc ta ti chnh gia h sao cho V(z) = 0 trong vng a/2 < z < a/2 v cc ro th cao v hn chng li cc ht by i ra khi vng ny. Ta dng ta z thay cho cho x trong chng ny v n qui c c s dng k hiu hng nui trong cc d cu trc. Cc hm sng l

4.1. H vung gc su v hn

vi nng lng

Cc hm sng l cc hm chn ca z i vi n l (tnh chn l dng (trng thi chn)) v c tnh chn l m (trng thi l) i vi n chn vi trng thi thp nht l trng thi chn. Cc kt qu ca m hnh ny l n gin n mc chng c s dng rng ri bt k bn cht khng thc ca chng.4.2. H vung gc su hu hn Cc h lng t c to ra trong h GaAs AlGaAs l xa vi trng hp l tng ho ca su hu hn. su i vi cc in t c thit lp bi gin on trong CB m n thng c gi di khong 0,3 eV trnh vn ca mt khe vng gin tip trong AlGaAs.Gin on trong

3

4.2. H vung gc su hu hn trong VB thm ch nh hn mc d n c b tr bi khi lng hiu dng ln hn. Cc su ny l kh nh i vi nhiu ng dng nht l nhit phng m n khuyn khch vic s dng cc chuyn tip khc trong c v ln hn. H vung gc vi su c ch ra trn hnh 4.1. H ny cho php i vi gi tr hu hn ca hoc N cn l mt s n gin ho ng k ca mt h thc v th b un cong chng hn nu h khng trung ha in khp ni. Nng lng c o t y h cho php d dng so snh vi cc kt qu i vi h su v hn. Cc trng thi vi b by bn trong h trong khi cc trng thi vi c th lanHnh 4.1. H vung gc su hu hn trongGaAs c su eV v b rnga = 10 nm v n ch ra 3 trng thi lin kt.

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4.2. H vung gc su hu hn . truyn t n Cc trng thi lin kt thng c m t bi nng lng lin kt B ca chng. l nng lng i hi nng in t ln t trng thi lin kt ca n sao cho n c th thot khi h. iu c ngha l Lc u ta xt cc trng thi lin kt. Cc hm sng bn trong h tng t nh cc hm sng i vi h su v hn v c cng i xng chn l (phng trnh (4.1)). Vit c 2 kh nng thnh

i vi a/ 2 < z < a/2 vi Bn ngoi h, tha mn

vi Cc nghim l vi N cn phi nh th chun ho cc hm sng v do , hm m dng trong (4.5) khi z < 0 v hm m m khi z > 0. Do cc hm sng l chn hoc l nn ch cn xt hm m m i vi z > 0 v dng i xng tm i vi z < 0.

5

4.2. H vung gc su hu hn Cc hm sng (4.3) v (4.6) by gi cn c lm khp ti z = a/2. Tnh lin tc ca i hi

Tng t, vic lm khp cc o hm cho

Cn phi thay i iu ny nu cc khi lng hiu dng trong 2 vt liu l khc nhau. Trong (4.7) v (4.8) cha bit cc h s chun ho C v D v nng lng m n xc nh k v C th loi tr C v D bng cch ly (4.8) chia cho (4.7). Khi ,

Thc ra, ta ang lm khp hm n gin N l o hm hm lga m t loi b s chun ha. T (4.9) suy ra

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4.2. H vung gc su hu hn

Nghim ca ca phng trnh siu vit ny khng th thu c mt cch chnh xc nhng l mt bi ton tnh s d dng. C th xt cc nghim t mt th m n c s dng phng on mt gi tr ban u cho php lp li. Nu dng bin s khng th nguyn (4.11) tr thnh

Khi lng ht, su v b rng ca h b st nhp vo trong thng s khng th nguyn Thng s ny xc nh cc gi tr cho php ca v cng n gin hn vic gii phng trnh bng s dng khng th nguyn ny . C 2 v ca (4.12) c v th theo trn hnh 4.2 i vi cc in t trong mt h trong GaAs. N c v

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4.2. H vung gc su hu hn Hnh 4.2. Nghim th ca phng trnh (4.12) i vi mt h vung gc trong GaAsvi su h v b rng a = 10nm. N cho v c 3 trng thi lin kt.

a = 10 nm m n cho Cn bc 2 v phi lun lun dng v do ta ch cn xt cc khong ca trong maim mt trong v l dng. i vi n c ngha l trong khi lm y cc khe v n dng i vi trong cc khong C cc cch khc vit li (4.12) v d nh n c th c thu gn thnh

nhng cn thn trng khi s dng la chn chnh xc ca v tri trong mi

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4.2. H vung gc su hu hn mt khong ca Do , ta s dng biu thc ban u. Cc nghim ca (4.14) xy ra khi 2 ng ct nhau. C 3 nghim k hiu l n = 1, n = 2 v n = 3. Mt s kt qu quan trng rt ra t th ny nh sau. (i) ng cong i vi v phi ca (4.12) ct trc ti trong khi cc ng cong ca v ct ti Do , s nghim c cho bi

c lm trn ti s nguyn gn nht. iu ny chng t rng lun lun c t nht mt nghim ca (4.12). Mt h vung gc 1 chiu lun lun c t nht mt trng thi lin kt nhng lm nng hoc thu hp h (mc d trng thi c th b lin kt rt yu). Kt qu ny ng cho mi h 1 chiu ch khng phi ch i vi h vung gc. N cng ng trong trng hp 2 chiu mc d cc h nng c mt trng thi lin kt rt yu. N khng ng i vi cc h 3 chiu m chng cn phi vt qua mt bn knh hoc su ti hn tn ti mt trng thi lin kt. (ii) Xt mt h rt nng vi ch mt trng thi lin kt v nh. Ta dng

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4.2. H vung gc su hu hn php gn ng trong (4.12) m n tr thnh N c th chuyn thnh phng trnh bc 2 vi nghim l

Du c th khng c tnh n do cn phi thc. Vic khai trin cn bc 2 theo nh l nh thc ti bc 2 cho

Do , nng lng l

Nng lng lin kt l N ph thuc vo v do l nh i vi mt h nng. (iii) Nu h l rt su, cc nghim nm trn cc phn dc ng ca cc ng cong tip gip v cc giao im tin n Khi v cc nng lng c cho bi ging nh cc nng lng i vi mt h su v hn (phng trnh (4.2)).

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4.2. H vung gc su hu hn (iv) Tp trung vo mt nghim vi n cho nh mt hm ca Phng trnh (4.15) chng minh rng trng thi lin kt ny xut hin khi v su ca h l Trng thi ch lin kt yu v B l nh khi khng ln hn nhiu so vi n. Hng s suy gim dng hm m trong hm sng (4.5) bn ngoi h l nh v hm sng xuyn qua mt qung ng di vo trong cc ro. Xc sut tm thy ht bn trong h l nh v c o bi

Vic lm cho h su hn na dn n lm tng v trng thi tr nn lin kt tt hn. (v) c im ca cc hm sng thay i khi trng thi tr nn lin kt tt hn. Khi trng thi tr thnh trng thi lin kt trong h v o hm hu nh bng 0 ti thnh h v n c lm khp vi mt sng gim chm trong cc ro. S tng lm tng ng nng ca n cho m ti im , bin hm sng hu nh bng 0 ti cc bin v ui dng hm m ch c mt bin nh.

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4.2. H vung gc su hu hn Hu ht cc kt qu ny p dng cho bt k h th no. Cc trng thi vi khng phi l cc trng thi lin kt. Chng ko di t n theo trc z v tt c cc nng lng u c php vi 2 nghim i vi mi nng lng. N khng c ngha l cc hm sng l cc sng phng n gin vi mt ng u mi ni. Chng b mo khi chng i qua h nht l khi nng lng ca chng khng xa trn iu ny c tnh ton theo cng mt cch nh cc ro vung gc. Mt cch ch ra s mo l vi mt trng thi a phng N c nh ngha bi (1.102) v thin v cc mc nng lng ca h vi mt hm sng ca chng ti z. Hnh 4.3. Mt trng thi a phng gia h vung gc trongGaAs c b rng 10 nm v su 0,3 eV. Kt qu i vi cc in t t do t l vi c ch ra so snh.

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4.2. H vung gc su hu hn N c v th trn hnh 4.3 i vi tm h trn hnh 4.1. Mi mt trong 3 trng thi lin kt ng gp mt hm Trng thi lin kt th hai trong s c mt nt ti z = 0 v do mt trng thi a phng ca n trit tiu ti im ny. V th, n c ch ra nh mt ng t nt. Cc trng thi l t do i vi nhng mt trng thi a phng b mo mnh i vi cc nng lng thp v iu phn nh mt s mo ca cc sng phng bi h. c bit l tng t 0 ging nh khc vi s phn k kiu i vi cc in t t do. S mo ca cc hm sng l quan trng trong cc qu trnh trong mt in t b kch thch t mt trng thi lin kt ti mt trng thi ch ngay trn nh h v dn ti cc nh hng trng thi cui. Mt v d l s hp th quang. Mt trng thi a phng i vi l thp hn mt trng thi a phng i vi cc in t t do do mt phn ca n i vo cc trng thi lin kt. C th thu gn h vung gc ti hm Cng S c th nguyn ca nng lng nhn vi chiu di. s dng kt qu i vi h hu hn, ta ly gii hn trong khi gi l hng s. Khi , v ta c th dng kt qu i vi mt h vung gc (phng

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4.2. H vung gc su hu hn trnh (4.20)). N chng t rng ch c mt trng thi lin kt c nng lng lin kt

4.3. H th parabol N c mt th nng c cho bi v m t mt dao ng t iu ho. Mt thc hnh vt l n gin l mt khi lng trn u ca mt l xo. Trong trng hp ny, z l di t cn bng v K l hng s l xo (o bng hoc Cc dao ng ca mng tinh th (cc phonon) cng c th c m t bi cc th parabol. Mt v d khc l mt vng c mt in tch khng i m nghim ca phng trnh Poisson c dng parabol. Cng c th nui cc h parabol bng cch thay i lin tc thnh phn ca hp kim. Mt t trng cng sinh ra mt th parabol. Mt ht c in c khi lng m chuyn ng trong th (4.23) thc hin dao ng iu ho vi tn s c tnh quan trng l tn s khng ph thuc vo bin (mc d n

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4.3. H th parabol khng phi l duy nht i vi th parabol). Kt qu ny b hn ch trong thc hnh do th (4.23) thng l mt php gn ng ch p dng i vi z nh. Trong c hc lng t, ta cn gii phng trnh Schrodinger khng ph thuc vo thi gian

trong (4.24) c dng loi tr K nhm s dng Bc u tin l loi b v thay chng bng cc s thun ty (cc i lng khng th nguyn) . Bi ton vt l (4.25) bng cch c rt gn v mt bi ton ton hc thun ty. Ta lm iu ny bng cch nh ngha mt thang chiu di v mt thang nng lng nh di y. Vic nhn (4.25) vi cho S hng I trong mc vung c th nguyn ca nghch o bnh phng chiu di trong lc s hng II c th nguyn ca bnh phng chiu di t Do , h s trc cn phi c th nguyn ca nghch o ly tha bc 4 ca chiu di. iu gi cho ta nh ngha thang chiu di loi b

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4.3. H th parabol h s hng s bng cch t T suy ra D dng thot khi cc i lng vt l cn li bng cch nh ngha thang nng lng sau Kt qu l phng trnh Schrodinger khng th nguyn

Lu l cc s thun ty. Ta k vng kch thc ca cc hm sng s gn ng l v s tch ra gia cc mc nng lng s gn ng l Khng th tm c cc s chnh xc m khng gii cc phng trnh nhng cc c tnh ny l c gi tr. By gi ta gii phng trnh (4.30). Ti ln, s hng c th b qua khi so snh vi v C th loi b s hng ny nu thay

vo (4.30). Mt hm m dng cng s loi b nhng hm sng c th khng c chun ho. Kt qu l phng trnh Hermite i vi sau

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4.3. H th parabol C th gii (4.32) bng cch khai trin thnh mt chui ly tha (Ph lc 4). N c cc nghim a thc ch khi l mt s nguyn chn, ngha l T , N l kt qu i vi cc mc nng lng ca mt dao ng t iu ho trong c hc lng t. Chng cch u nhau mt khong l trn nng lng im khng Lu cc mc nng lng ca dao ng t iu ho thng c tnh t 0 nhng y c tnh t 1 ph hp vi cc h th khc. Cc hm l cc a thc Hermite ngoi mt h s chun ho. Mt s a thc Hermite u tin l

Cc hm sng theo z bao hm c s chun ho l

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4.3. H th parabol

Mt s hm sng thp nht c v th trn hnh 4.4. Chng ch ra s xen k chn l nhn thy c trong cc h vung gc i xng. Hm sng i vi n = 1 l mt hm Gauss n gin c mt xc sut l

lch chun ca mt ny l

Hnh 4.4. H th cc mc nng lngcc cc hm sng ca mt dao ng t iu ho. Th sinh ra bi mt t trng l 1 T tc dng ln cc in t trong GaAs.

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4.3. H th parabol Cc kt qu ny c vai tr rt quan trng i vi mt phm vi rng ca cc bi ton. S cch u ca cc mc nng lng l s tng t ca tn s c in c lp vi bin . N c ngha l bt k b sng no to ra bng cch chng chp cc trng thi khc nhau dao ng vi cng tn s (so snh phng trnh (1.55)). Mt v d v mt h parabol dc nui bng cch thay i thnh phn ca c ch ra trn hnh 4.5. Cc mc nng lng c th c o bi cc chuyn tip quang gia cc trng thi trong cc h trong CB v VB.

Hnh 4.5. (a) Th parabol trong c CB v VB c nui vo trong GaAs bi mt thnh phn thay i t t ca Khe vngc rt gn trong phc ha ny vch l trng nng c ch ra. (b) Squang pht quang trong cc h . parabol.

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4.3. H th parabol Mt qui tc lc la i hi rng c 2 trng thi c cng tnh chn l. Cc chuyn tip c nh s l trong m l trng thi in t, n l trng thi l trng (b i nu n ging nhau v h ch cc l trng nng v l ch cc l trng nh. Thnh phn ca mi mt h thay i t t t GaAs ti qua khong cch 25,5 nm. Khi s dng gi tr c chp nhn hin nay ca ta c cc gi tr cc i

cong ca CB v VB v do cc mc nng lng ph thuc vo v iu ny tng phn vi mt h vung gc m khng c s ph thuc vo v ca cc ro trong m hnh n gin nht l mt h su v hn v s ph thuc l yu i vi cc trng thi su trong mt h hu hn. Khi , th nghim v cc h parabol l mt php th nhy i vi gi tr ca Q.4.4. H th tam gic H tam gic trn hnh 4.6 l mt s m t n gin ca h th ti mt d chuyn tip pha tp. C mt ro cao v hn i vi z < 0 v mt th tuyn tnh i vi z > 0. Vit V(z) theo cch ny l thun tin n m t mt in tch e trong in trng F (tch eF c gi thit l dng).

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4.4. H th tam gic Hnh 4.6. H th tam gictrong ch ra cc mc nng lng v cchm sng. Cc thang i vi cc in t trong GaAs v mt trng l

Lu F c dng i vi in trng khc vi E trnh nhm ln vi nng lng. Ta cn gii phng trnh Schrodinger

vi iu kin bin p t bi ro v hn. Ta li a vo cc bin s khng th nguyn. Thao tc tng t nh i vi dao ng t iu ho ch ra rng cc thang khong cch v nng lng l Phng trnh Schrodinger tr thnh

21

4.4. H th tam gic

N c th c n gin ho hn na bng cch nh ngha mt bin s c lp mi v (4.40) c rt gn thnh phng trnh Stokes hay phng trnh Airy N c xem xt trong Ph lc 5. 2 nghim c lp ca n l cc hm Airy Ai(s) v Bi(s). Ta i hi mt hm sng m n gi c tnh cht khi ging nh khi iu ny c ngha l ta c th loi b Bi(s). Bin cao ti z = 0 i hi C v s cc gi tr m ca s m Ai(s) = 0 c k hiu l hoc loi b du. Do , ta cn bo m rng cc hm sng trit tiu ti z = 0 v cc mc nng lng cho php c cho bi

Mc thp nht c C mt cng thc gn ng c ch l

m n c rt ra t l thuyt WKB. Mc d iu ny l chnh xc nht i

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4.4. H th tam gic vi n ln, n cho v do l kh tt i vi mi n. N cng ch ra rng cc mc nng lng cng gn nhau hn khi n tng do h m rng khi nng lng tng. iu ny tng phn vi h vung gc su v hn c b rng khng i trong cc mc nng lng cng xa nhau hn khi n tng. H parabol cung cp mt trng hp trung gian vi cc mc nng lng tch ra khng i. Cc hm sng khng c chun ho c cho bi

Tt c cc hm sng c cng dng hm v trt n gin dc theo z khi nng lng thay i: cha mt na chu k, cha hai na chu k, v.v... Cc hm sng thiu i xng chn l m ta tm thy n trong cc h xem xt trc y v th tam gic t n l khng i xng theo z. S chun ho c cp trong Ph lc 5.4.5. Cc h thp chiu By gi ta s s dng cc kt qu trn xem lm th no m cc in t 3

23

4.5. Cc h thp chiu chiu c th c to ra x s dng nh chng l thp chiu. im xut pht l phng trnh Schrodinger 3 chiu khng ph thuc vo thi gian Khng c cch gii d dng phng trnh ny nu l mt th nng tng qut nhng mt s dng ca cho php nhng s n gin ho ln. Trong mt cu trc to lp, th nng ch ph thuc vo ta z vung gc vi cc lp. iu ny bao hm cc h lng t c to ra t s xen k cc lp GaAs v AlGaAs v cc in t b by ti mt d chuyn tip pha tp. Nh vy, ch c v (4.44) tr thnh

Th nng khng cho php cc in t chuyn ng t do dc theo x v y. Cc hm sng l cc sng phng nu khng c mt th no c. N gi cho ta th cc sng phng dc theo x v y. Vit hm sng di dng C th thay (4.46) vo (4.45) kim tra xem n c cho mt nghim chnh

24

4.5. Cc h thp chiu xc i vi x v y v tm phng trnh i vi hm cha bit Phng trnh (4.45) tr thnh

Cc hm m b loi b khi 2 v ca (4.47). iu xc nhn rng d on (4.46) l chnh xc. Ch nhng hm ca z c gi li

Nng lng ca cc sng phng c th c chuyn qua v phi v do ,

Mt s thay th tip theo i vi nng lng

25

4.5. Cc h thp chiu rt gn (4.49) thnh N l phng trnh Schrodinger 1 chiu thun ty theo hng z, cn 2 chiu khc b loi b. By gi gi s rng ta gii c phng trnh ny. N c th l mt h vung gc hoc n c th cn c tnh s. Gi s cc hm sng l vi nng lng T (4.46) v (4.50) suy ra nghim ca bi ton 3 chiu l

3 s lng t v n dng k hiu cc trng thi do chng l 3 chiu khng gian. C th vit gn (4.52) v (4.53) bng cch nh ngha cc vect 2 chiu i vi chuyn ng trong mt phng xy l Do , Cc kt qu trn c minh ha trn hnh 4.7. Bn tri l h th V(z) vi cc

26

4.5. Cc h thp chiu nng lng cho php v cc hm sng ca n. H thc tn sc (4.55) c v th hnh gia. i vi mt gi tr c nh ca n, n n gin l h thc gia nng lng v vect sng ca mt 2DEG t do vi y vng dch chuyn ti H thc i vi mi mt n cho mt parabol gi l vng con (chnh xc l vng con in) bt u ti nng lng khi v th theo Khng c cc vng con i vi mc d cn c cc trng thi cho php y nu khng c s giam cm. i viHnh 1.7. (a) H th vi cc mc nng lng; (b) Nng lng tonphn bao gm ng nng ngang ivi mi vng con v (c) Mt trngthi kiu bc ca mt h gi 2 chiu.V d l mt h vung gc su v hntrong GaAs vi b rng10 nm. ngcong mnh trong (c) l mt trng thi dng parabol i vi cc in t 3 chiu khng b giam cm.

27

4.5. Cc h thp chiu ch c cc trng thi trong vng con thp nht. i vi c cc trng thi trong 2 vng con thp nht vi n = 1 v n = 2. Nng lng b chia ct khc nhau trong 2 vng con. Vng con vi n = 2 c ng nng cao hn theo hng z, c khc vi v do c ng nng v vn tc thp hn theo mt phng ngang. S tch nng lng ny thnh cc thnh phn khc nhau nh th n l mt vect da vo dng n gin ca ton t ng nng trong phng trnh Schrodinger v s khng c duy tr nu ta cn s dng mt hm Hamilton hiu dng phc tp hn. C nhiu vng con hn ti cc nng lng ton phn cao hn. Do , cc in t vi cng nng lng ton phn c th c mt s vect sng ngang khc nhau. iu ny tng t vi bi ton trong l thuyt in t ca mt dn sng vi nhiu mode cho php. Cc vng con lm thay i dng ca mt trng thi n(E). i vi mt vng con cho trc (n c nh), nng lng (4.55) l nng lng ca 2DEG vi y vng ti Do , mt trng thi l mt trng thi ca 2DEG. l mt hm bc c chiu cao bt u ti Mi vng con ng gp mt bc. Do , mt trng thi tng cng n(E) trng ging nh cu

28

4.5. Cc h thp chiu thang vi cc bc nhy ti cc nng lng ca cc vng con. Lu rng n l mt trng thi ng vi mt n v din tch ch khng phi th tch. S hp th quang mt trng thi n(E).4.6. S lp y ca cc vng con Do ta tnh c cc mc nng lng cho php, ta cn phi xem iu g xy ra khi ta lm y h vi cc in t. S cc vng con b lp y ph thuc vo mt in t v nhit . Mt in t. ng vi mt n v din tch c th tm c theo cch thng thng bng cch ly tch phn tch ca mt trng thi n(E) v hm lp y Fermi-Dirac trong l nng lng Fermi:

thun tin, ta chia tch n thnh cc vng con: trong l mt in t trong mt vng 2 chiu bt u ti N c cho bi (1.114) v tr thnh

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4.6. S lp y ca cc vng con N c th c n gin ho trong gii hn nhit cao v nhit thp (khi so vi nh trc y. Gi s rng ta ang gii hn nhit thp m cc in t l suy bin. Khi ,

N c minh ha trn hnh 4.8 i vi 2 gi tr ca Gi tr thp hn di v ch nm trong vng con th nht. C mt s lp y nh c th b qua ca cc vng con th hai v cao hn vi iu kin l Cc in t x s dng nh chng trong mt h 2 chiu vi mt mt trng thi kiu bc n gin. Tt c cc in t b gia trong cng trng thi i vi chuyn ng vung gc vi th giam cm v khng th chuyn ng dc theo z khi n i hi chng thay i trng thiHnh 4.8. S lp y ca mt trng thi kiu bc i vi mt h gi 2 chiu. Ch c mt vng con b lp y nu nng lng Fermi ly gi tr thp hn nhng 2 vng con b lp y ti gi tr cao hn.

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4.6. S lp y ca cc vng con ca chng. Gii hn ny c th t c bng th nghim i vi mt 2DEG. Bn cht 2 chiu l hi tinh t v d dng mt i nu nhit tng ln hoc nu cc in t thu c nng lng t ngun bn ngoi no nh in trng v i vo vng con cao hn. Nng lng Fermi i vo vng con th hai nu mt in t tng qu xa nh trng hp i vi Cc in t ti nng lng Fermi by gi c th mt trong 2 vng con. Cc vng con ny c cc vn tc khc nhau trong mt phng ngang. Mt mc gii hn i vi chuyn ng dc theo z l c th c bi tn x gia v N cho mt du hiu th nghim khi c hn mt vng con b lp y: s tn x gia cc vng con lm cho linh ng gim. H ch l gi 2 chiu mc d n gi xa trng hp gii hn ca chuyn ng t do theo tt c 3 chiu. Mt in t cc i m n c th lp y h trc khi i vo vng con th hai c cho bi S tch gia cc mc nng lng dc theo z s c lm cc i tng mt ny. Mt h hp t c iu cho n khi cc mc nng lng b p ra ngoi nh. Mt tnh hung th v xy ra nu cc mc nng lng c v li i vi

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4.6. S lp y ca cc vng con Hnh 4.9. H gi 2 chiu trong mt h th su hu hn. Cc in t vicng nng lng ton phn c th blin kt bn trong h (A) hoc t do(B).

h th su hu hn. Hnh 4.9 ch ra 3 trng thi lin kt trong h. Tt c cc nng lng u c php i vi chuyn ng dc theo z trn nh h. By gi c th c 2 in t vi cng nng lng ton phn b lin kt trong h vi ln (A) hoc t do dc theo z (B). iu ph thuc vo nng lng c chia tch nh th no gia z v mt phng ngang. in t A b lin kt kh khng chc chn v thm ch i vi mt s kin tn x n hi m n bo ton nng lng ton phn ca in t, ngi ta c th ly n t A n B v cho php n thot khi h. N c gi l s chuyn tip khng gian thc bi s tng t vi s chuyn khng gian trong cc vt liu nh GaAs m cc in t tn x t thung lng ti cc thung

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4.6. S lp y ca cc vng con lng X cc in trng ln. Trong c 2 trng hp, s gim linh ng gn vi s chuyn tip c th cho tr khng vi phn m.4.7. Cc h th 2 chiu v 3 chiu Ngi ta s dng nhiu m hnh ca cc h th 2 chiu v 3 chiu. M hnh n gin nht l m rng h vung gc su v hn bng cch cng nhn vi mt th nh vy i vi mi mt chiu. Trong trng hp 2 chiu, n dn ti mt th hnh ch nht i vi mt hp (vi cc cnh c chiu di a v b) trong mt phng xy. Hm sng l mt tch ca cc sng sin theo mi mt chiu v n cho nng lng

Mt h vung gc vi a = b c cc mc nng lng suy bin. S i xng gia x v y i hi v c nhng suy bin tai nn nh Mt h hnh ch nht 3 chiu c th c nghin cu theo cch tng t. Mc d cc kt qu ny l n gin, cc h thc thng c i xng tr hoc cu. Ta s xem xt ngn gn mt s v d. 4.7.1. H tr

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4.7. Cc h th 2 chiu v 3 chiu Ta bt u vi cc in t t do theo 2 chiu vi Hm sng d hiu nht c cc sng phng theo c x v y v n cho

N l mt sng phng chuyn ng theo mt hng c thit lp bi v vi nng lng Trong ta cc n c th c vit thnh trong c o t hng ca N cn l mt sng phng. thay th, ta c th mong mun m t cc sng i ra theo mi hng t mt ngun im hnh tr khc vi cc sng phng. lm iu , ta cn vit li phng trnh Schrodinger i vi cc in t t do theo 2 chiu khi dng cc ta cc

Gc ch xut hin nh mt o hm v do , mt hm sng c th c tch ra di dng l mt nghim. N ging vi mt sng phng theo v iu phn nh i xng quay trong h. Mc d n l mt nghim i vi mi hm sng cn phi n gi. N cn quay li

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4.7. Cc h th 2 chiu v 3 chiu cng mt gi tr nu ta bao quanh gc v thm vo N gii hn s lng t xung lng gc l cc s nguyn N thng c vit l m theo 2 chiu nhng khi n c th b nhm ln vi khi lng. Hm bn knh tha mn phng trnh

Chuyn ng gc li s hng li tm trong th nng m n y cc trng thi ra xa gc khi xung lng gc ca chng tng. Vic thay E bng i vi E > 0 cho

N l phng trnh Bessel i vi cc nghim v l cc hm Bessel bc loi I v loi II. loi II phn k ti gc v do khng th dng c qua ton b khng gian. Cc hm Bessel ny l nhng sng ng nhng c th c kt hp li cho cc sng chy. Bn cht ging sng ca chng l r rng dng tim cn i vi cc bin ln

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4.7. Cc h th 2 chiu v 3 chiu c sin thay cho cosin. Cc sng dao ng nh mong mun v s gim kiu v bin tr thnh v cng m n cn bng vi s tng chu vi khi sng tn ra trong mt phng. Nu E < 0, cc nghim l cc hm Bessel bin dng v Chng ging nh cc hm m thc: tng trong lc gim t mt s phn k gc. Nghim i vi mt h tr vi cc thnh cao v hn c rt ra t cc kt qu ny. H ny c V(r) = 0 i vi r < a v mt ro khng th xuyn qua i vi r > a. Hm sng cn phi trit tiu ti r = a m n i hi Hm Bessel trit tiu ti cc khng c k hiu bi i vi n = 1,2,... Nh vy, cc gi tr cho php ca vect sng ny l v cc hm sng v nng lng l

Trng thi vi nng lng thp nht c xung lng gc bng khng Khai trin tim cn (4.64) ch ra rng N l chnh xc khi nhng khng sai lm thm ch i vi

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4.7. Cc h th 2 chiu v 3 chiu khi so snh vi gn ng tim cnHnh 4.10 ch ra mt th nghim trong cc nguyn st trn b mt ca vng c th nghim vi u ca mt knh hin vi tunnen qut to thnh mt ro baoquanh gi l bi quy lng t. Knh hin vi tunnen qut khi c s dng to nh ca cc trng thi trong bi quy. l mt chng minh ng ch ca s by tr. Cc php o c th c lm khp khi dng m hnh h lng t m ta va xt mc d cc nguyn t st x s theo mt cch phc tp hn so vi mt thnhcng n gin. Hnh 4.10. (a) nh khng gian ca cc trng thi ring ca mt bi quy lng t; (b) Tit din ca mt bi quy c lm khp vi mt kt hp ca cc trng thi i vi mt hp tr.

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4.7. Cc h th 2 chiu v 3 chiu 4.7.2. H parabol 2 chiu C 2 cch tip cn gii bi ton th parabol trong trng hp 2 chiu. Cch I l thm th ny vo phng trnh Schrodinger bn knh (4.62) v tm cc nng lng cho php v cc hm sng. Cc trng thi thu c c cc gi tr hu hn ca xung lng gc v cn nghin cu t trng khi dng chun i xng. Cc mc nng lng ca dao ng t l ging nh trc. Mc thp nht c nng lng im khng v c N trng thi suy bin vi nng lng Thang ca cc mc nng lng cch u nhau c bo ton. l c tnh quan trng nht ca dao ng t iu ho 1 chiu. Cch II l lu rng th c th tch ra v do phng trnh Schrodinger trong cc ta Descartes c th c thu gn thnh cc phng trnh tch ri i vi x v y. Mi mt phng trnh trong s l mt bi ton 1 chiu m ta gii. Nh vy, nng lng ton phn l m n l mt thang sinh ra t Mt nng lng c th c chia tch theo N cch khc nhau gia v

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4.7. Cc h th 2 chiu v 3 chiu iu xc nhn kt qu i vi s suy bin. Cch tip cn ny c th c m rng cho cc th parabol km i xng hn dng vi v cc th parabol 3 chiu c th c gii theo cch tng t. 4.7.3. Th Coulomb 2 chiu Th Coulomb ht c v s cc trng thi lin kt vi nng lng Hm sng thp nht c s suy gim theo kiu hm m n gin Phng trnh Schrodinger bn knh c th c gii theo cng mt cch nh i vi h parabol m ta khng a ra cc chi tit y. Cc thang nng lng v chiu di l nng lng Rydberg v bn knh Bohr c cho bi

i vi hir, Trong mt bn dn, ta thay m bng trong l khi lng hiu dng ca ht ti

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4.7. Cc h th 2 chiu v 3 chiu bao hm v thay bng trong l hng s in mi phng ca vt liu. Cc thay th ny lm thay i nhiu cc thang v cho v i vi cc in t trong GaAs. Cc s ny l quan trng v chng thit lp cc thang t nhin i vi nhiu qu trnh trong cc bn dn. Thc s chng thng c s dng nh l cc thang nng lng v chiu di theo cc n v nguyn t khng th nguyn hoc n v Rydberg n gin cc tnh ton. Trng thi thp nht c nng lng lin kt l m n ln hn 4 ln so vi khng gian 3 chiu tng ng. iu ny c bit quan trng i vi mt exciton, mt in t v mt l trng c lin kt vi nhau bi s ht Coulomb ca chng. 4.7.4. H cu Xut pht im li l li gii cho chuyn ng 3 chiu t do trong cc ta cc. By gi c 2 s lng t xung lng gc l m n cho xung lng gc tng cng v m n cho thnh phn ca n dc theo mt trc ring c chn theo qui c l z. Phn bn knh ca

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4.7. Cc h th 2 chiu v 3 chiu nghim c th c vit thnh trong tun theo phng trnh

N rt gn vi phng trnh Schrodinger 1 chiu thng thng. N bao hm mt th li tm nh trong trng hp 2 chiu v mt th nng V(R) i xng cu c lng vo. Nng lng khng ph thuc vo s lng t m. Nghim nng lng thp nht c i xng cu vi v trong trng hp ny, ta c bi ton 1 chiu quen thuc i vi C mt s khc bit quan trng v cc iu kin bin: hm sng cn phi khng phn k ti gc m n i hi l cng mt iu kin m cc trng thi l trong mt h 1 chiu i xng cn phi tun theo v cm cc nghim chn. Nh vy, trng thi thp nht trong mt h cu su v hn c bn knh a c vi nng lng Mt h hu hn c bn knh a v su c th c gii bng s tng t vi mt h 1 chiu c chiu rng y 2a. Trng thi thp nht trong h

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4.7. Cc h th 2 chiu v 3 chiu cu tng ng vi trng thi th hai trong h 1 chiu c cho bi ng cong trn hnh 4.2 i vi nm gia v Vng i vi b cm v n cho mt trng thi chn. N c h qu quan trng l mt h cu nng khng c trng thi lien kt. N tng phn vi trng hp 1 chiu trong t nht lun lun c mt trng thi lin kt. H cu i hi hay lin kt mt trng thi. Cc c tnh khc ca nghim rt ra t trng hp 1 chiu. 4.7.5. Th Coulomb 3 chiu H th Coulomb 3 chiu c v s trng thi lin kt vi nng lng Trng thi thp nht l mt hm m

Nng lng Rydberg v bn knh Bohr c nh ngha bi (4.66) v (4.67). Cc kt qu ny iu khin s lin kt ca cc in t trn cc cht cho hir (thng thng) v cc exciton 3 chiu.4.8. S giam cm khc ngoi 2 chiu Trong phn 4.5, ta nghin cu cc in t b giam cm trong mt trng

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4.8. S giam cm khc ngoi 2 chiu thi lin kt dc theo z v x s dng nh chng l 2 chiu. C th giam cm chng nhiu hn na v rt gn s chiu hiu dng ca chng xung 1 hoc 0. Nu ta ly th giam cm l mt hm ca cc in t duy tr chuyn ng t do dc theo z v kt qu l mt dy lng t tng t nh mt ng dn sng in t. Phn tch rt ra t phn tch i vi 2DEG. Ta bt u vi phng trnh Schrodinger 2 chiu i vi th giam cm

C th s dng mt trong cc m hnh n gin ho xem xt phn trc i vi (4.70) hoc n c th i hi cc phng php s nhng gi thit rng n c gii. Khi , hm sng v nng lng ton phn c cho bi

Chng l cc tng t ca (4.52) v (4.53) v gii thch ca chng l tng

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4.8. S giam cm khc ngoi 2 chiu t. Mi mt gi tr ca tr thnh y ca mt vng con 1 chiu m mt trng thi ca n thay i theo kiu Mt trng thi ton phn (ng vi mt n v chiu di) l

N c phc ha trn hnh 4.11 vi parabol i vi cc in t 3 chiu so snh. Mt b mo ra khi trng hp t do thm ch nhiu hn trong trng hp 2 chiu. Cc y ca cc vng con tr nn c cc c tnh mnh hn: cc phn k khc vi cc bc m n quan trng trong cc hiu ng quang phn nh mt trng thi.

Hnh 4.11. Mt trng thi ca mt h gi 1 chiu. ng cong c tnh i vi cc in t trong mt h 9 x 11 nm su v hn trong GaAs. Parabol mnh l mt trng thi i vi cc in t 3 chiu khngb giam cm.

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4.8. S giam cm khc ngoi 2 chiu C th tin ti mt giai on xa hn khi giam cm cc in t hoc l trng theo tt c 3 chiu. in hnh l chng b giam cm theo 1 chiu bng cch nui mt h lng t hoc mt d chuyn tip pha tp v sau gii hn ti mt vng nh bng cch khc axit hoc mt th tnh in. Kt qu l mt chm lng t. V c bn, n l mt nguyn t nhn to. Mt trng thi ch l mt h ca cc hm v khng c chuyn ng t do theo bt k chiu no.4.9. Cc h lng t trong cc d cu trc Mc d tt c cc th c m t cc phn trc cn cc d cu trc sinh ra chng, ta nghin cu chng nh cc h th n gin v b qua tt c cc kh khn m t trong phn 3.11. Ta s tip tc gi thit rng y CB l cng mt im trong khng gian k trong tt c cc vt liu bao hm v khng tnh n nhng phc tp ln hn nhiu gn vi cc VB. Vn gi li l cc khi lng hiu dng khc nhau trong cc vt liu. N c 2 nh hng: gii php th giam cm cn phi tnh n cc khi lng hiu dng khc nhau khi lm khp cc hm sng trong cc vt liu khc nhau v s rt gn ca bi ton 3 chiu ban u thnh bi ton 1 chiu hoc 2 chiu tr nn

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4.9. Cc h lng t trong cc d cu trc km kho lo hn mt cht. n gin, ta xt cc in t b lin kt trong mt h lng t ca GaAs b kp gia hai lp AlGaAs. Trc tin ta s nghin cu nh hng ca cc khi lng khc nhau ln bi ton 1 chiu l mt h hu hn vi b rng a v su m ta gii i vi mt cu trc ng nht phn 4.2. Cc s sng bn trong v bn ngoi c cho bi cc biu thc bin dng

y, l khi lng hiu dng trong h vi l y CB; v l cc i lng tng ng trong ro. su h l iu kin lm khp ln hm sng ti mt giao din cn phi thay i nh trong phn 3.11. S lm khp o hm trong (4.8) c thay bi iu kin Nh vy, cc o hm tun theo

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4.9. Cc h lng t trong cc d cu trc v vic chia cho iu kin lm khp khng thay i i vi cho

Bng 4.1. S ph thuc vo khi lng hiu dng trong cc ro ca cc nng lng ca cc trng thi b lin kt trong mt h rng 5 nm v su 1 eV vi khi lng hiu dng bn trong h.

0,067 0,131 0,504 0,981 0,15 0,108 0,446 0,969

Ta li nh ngha v m n ch ph thuc vo khi lng trong h. Khi , iu kin lm khp l

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4.9. Cc h lng t trong cc d cu trc N c th c gii chnh xc theo cng mt cch nh i vi trng hp ca cc khi lng bng nhau. Li gii th trn hnh 4.2 c tin hnh nh trc ngoi tr iu l ng cong cn bc 2 c t thang bi h s V d gi s rng c th thay i v gi v khng i. S tng lm gim v phi ca (4.79) v do , cc nng lng ca cc trng thi lin kt u gim. N khng gy ra s ngc nhin v ni chung ta mong mun mt khi lng cao hn dn ti cc nng lng thp hn. Tuy nhin, s trng thi lin kt gi khng i do n ph thuc vo ch cha cc tnh cht ca h. lm v d, xt mt h GaAs rng 5 nm b kp gia AlAs m n lm nghi ng gi thuyt cho rng ta ch cn xt thung lng v cc vng c dng parabol. Cc khi lng l vi N l mt v d kh cc oan hn so vi cu trc thng thng hn vi v c chn khuch i nh hng ca cc khi lng khc nhau. H ny c v do c 3 trng thi lin kt vi cc nng lng c a ra trong Bng 4.1. Tt c cc nng lng lm gim nng lng gia trn 50 meV khi tng (nh mong mun). Trng thi nh b

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4.9. Cc h lng t trong cc d cu trc lin kt yu n mc o hm ca n ti bin gn nh bng 0 v s thay i khi lng t nh hng n n. Cc hm sng i vi c v th trn hnh 4.12. Nt trong cc hm sng c a vo bi iu kin lm khp (4.75) l r rng. Mc d hm sng y cn phi trn, cc tnh ton hon chnh xc nhn rng hm bao ch ra dng iu ny (hnh 3.22). Vn th hai l s rt gn phng trnh Schrodinger 3 chiu ban u thnh mt phng trnh 1 chiu. Cc phng trnh Schrodinger trong 2 vt liu l (h), (4.80) ( (ro) (4.81) Hnh 4.12. H vung gc hu hn vi su b rng a = 5 nm dc theo z v cc khi lng hiu dng = 0,067 trong h v trong ro.

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4.9. Cc h lng t trong cc d cu trc Th hiu dng t ch thay i dc theo z ging nh trong phn 4.5 v do ta c th vit hm sng thnh Thay hm sng ny vo 2 phng trnh Schrodinger 3 chiu dn ti mt cp phng trnh 1 chiu

S khc bit nng lng gia 2 vng m n to thnh h by gi ph thuc vo v c cho bi

Hiu chnh l m i vi GaAs-AlGaAs do v do , h th tr nn nng hn khi ng nng ngang tng. Nh vy, nng lng ton phn ca mt in t trong mt trng thi lin kt c cho bi

trong nng lng ca trng thi lin kt cng ph thuc vo k thng qua s thay i su ca h.

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4.9. Cc h lng t trong cc d cu trc Bng 4.2. S ph thuc ca vect sng ngang ca cc nng lng ca cc trng thi lin kt trong mt h rng 5 nm v su 1 eV vi khi lng hiu dng trongh v ngoi h. k (nm) 0,0 0,000 0,000 1,000 0,108 0,446 0,969 0,057 0,5 0,142 0,064 0,921 0,106 0,435 0,919 0,069 1,0 0,570 0,254 0,685 0,096 0,397 - 0,076 Ta ly li h GaAs rng 5 nm b kp gia AlAs lm v d vi cc s sng k = 0, 0,5 v 1,0 Cc ng nng, su hiu dng v nng lng ca cc trng thi lin kt c a ra trong Bng 4.2. su ca h c rt gn nhiu ti s sng cao nht sao cho trng thi th ba khng cn b lin kt na. Nng lng ca trng thi th hai gim i 49 mV so vi ng nng ca n l 570 meV. Ta c th tnh n iu mt cch gn ng bi vic a vo mt khi lng hiu dng khc v d nh l nng lng trn n ti k = 0 tnh n c s tng ng nng ngang v s gim nng lng trng thi lin kt dc theo z. Nh vy,

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4.9. Cc h lng t trong cc d cu trc Cc khi lng hiu dng mi ny ph thuc vo ch s n ca trng thi lin kt v c lp bng. Mt cch vt l hn gii thch ngun gc ca chng l lu rng in t s dng phn thi gian ca n trong ro ch khng phi ngay h v do c xu hng t c mt s c trng ca ro. C th chng minh rng trong l xc sut tm thy in t trong h v l xc sut tm thy in t trong ro. C th tin hnh nghin cu su hn khi cu trc vng trong 2 vt liu trn bn ny hoc bn kia ca mt d chuyn tip l khc nhau mt cch nh tnh. Mt v d l GaAs-AlAs trong CB thp nht l ti trn mt bn ca chuyn tip v ti X bn kia. Khi xut hin s xuyn hm. Cc CB v VB xen ph trong mt chuyn tip loi III ging nh gia InAs v GaSb. iu ny kt thc tho lun v lin kt in t trong cc h lng t. Trong chng 5, ta s xem xt tnh hung ngc li ca cc ro thay cho cc h v s vn chuyn do s xuyn hm ca cc in t.

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Bi tp chng 4 4.1. Tnh s trng thi lin kt v mc nng lng thp nht i vi cc in t v cc l trng nng v nh trong mt h GaAs rng 6 nm b kp gi cc lp nh gi php gn ng l mt h su v hn trong bi ton ny. Tnh li nng lng ca chuyn tip quang trong h 6 nm trong mu m s quang pht quang ca n c ch ra trn hnh 1.4. su hu hn to ra s khc nhau nh th no? N c lm tng s ph hp vi thc nghim hay khng hoc c bt k cc dun hiu sai s no khng (trong m hnh hoc s nui)? Mt s dch nh th no xy ra nu b dy h thng ging mt n lp?4.2. Tnh xc sut tm thy mt in t trng thi lin kt thp nht bn trong mt h rng 4 nm khi dng phng trnh (4.21). N i hi t s ca cc h s D/ C m t s ny c th tnh t (4.7) hoc (4.8) sau khi tm c (v sau l k v Gii thch mt cch nh tnh v s ph thuc ca phn ny vo b rng h.4.3. V th nng lng ca cc trng thi lin kt trong mt h GaAs su 0,3 eV nh mt hm ca b rng h t 0 n 20 nm. 4.4. Lm th no tm cc trng thi lin kt trong mt h khng i xng

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Bi tp chng 4 nh h trn hnh 4.13? Khng cn a ra li gii chi tit. Cn phi loi b tt c nhng s n gin ho do i xng. Hu ht cc h nui trong thc t l i xng.Gi s rng tr nn v cng ln to ra mt tng cng. Chng minh rng bi ton by gi li tr nn n gin khi s dng cc kt qu ca phn 4.2.Hnh 4.13. Mt h th khng i xng vi cc ro c chiu cao bn tri v bn phi.

4.5. Gii bi ton i vi h th c dng hm vi mt cch trc tip. Hm sng l mt hm m gim khp ni ging nh hm sng bn ngoi mt h hu hn. S gin on ca n theo nghing ti z = 0 cn c lm cn bng bi hm trong th. Ly tch phn phng trnh Schrodinger theo z t ngay di 0 ti ngay trn 0 chng minh rng

(v phi trit tiu do n khng cha cc k d). Chng minh rng n cho ph hp vi (4.22).

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Bi tp chng 4 4.6. Mt t trng B sinh ra mt th parabol vi mt hng s n hi trong e l in tch ca ht, m l khi lng ca n v l tn s xiclotron. Tnh nng lng xiclotron v thang chiu di (cn c gi l di t i vi mt in t trong GaAs trong mt trng 1 T.4.7. Tnh cc nng lng ca cc trng thi lin kt trong mt th c dng parabol i vi x > 0 vi mt tng cng ti x = 0.4.8. Dng h thc bt nh Heisenberg (1.63) c tnh nng lng im khng trong mt h parabol. Nu s m rng hm sng l th nng c th vit thnh v ng nng l Vit nng lng ton phn theo tm gi tr ca m n lm cc tiu v thu c kt qu chnh xc4.9. Mt gi tr th ca in trng giam cm cc in t gn mt d chuyn tip pha tp l Xc nh mt vi mc nng lng u tin trong mt th tam gic vi nghing ny. Cc kt qu c ch hay khng i vi h GaAs-AlGaAs trong ro ti z = 0 ch cao khong 0,3 eV?

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Bi tp chng 4 4.10. Mt chm lng t b giam cm theo 2 chiu bi mt th parabol m n tng ln 50 meV qua mt bn knh l 100 nm. S giam cm l mnh hn nhiu theo chiu th ba v c th gi thit rng tt c cc in t gi trng thi thp nht i vi chiu ny. C bao nhiu trng thi b lp y nu mc Fermi nm cao hn cc tiu 12 meV? Cc kt qu s nh th no nu chm c m hnh ho vi mt tng cng ti bn knh l 50 nm.4.11. Tnh mt trng thi i vi cc in t b by trong h 11 x 9 nm 2 chiu su v cng. Cc nng lng ca cc vng con c cho trong (4.60). So snh n vi kt qu 3 chiu (c chuyn i ti mt ng vi mt n v chiu di).4.12. S thay i su ca mt h th trong mt d cu trc nh mt hm ca vect sng ngang nh hng nh th no n s by ca cc in t trong cc vng con khc nhau nh ch ra trn hnh 4.9 i vi mt cu trc ng nht? Dng cu trc GaAs AlAs minh ha.

56

),(yxr=r()),,(,zyxzrR==rrVED)2.4,....(2,1,222==nanmnpeh)1.4()2(sin2)12(cos2)(=+==lnaznalnaznaznppfCED0V

e3,00=Ve()())4.4)(()()(/2/0222zzVzdzdmeyyy=+-h)6.4.(2/022BVm=-=ekh())5.4(exp)(zDzky=-=z.+=z.0e-=VB.0V

e()0,en0,0Va00q()2/10--Ve.d).()(zSzVd-=()2/10V-e0V>eaVS0=()tzz00cosw=)22.4.(222hmSB=)23.4(2/)(2KzzV=1-Nm0z).1-Jm)24.4.(/0mK=we,z()()[]()())25.4(,)2/1(/2/220222zzzmdzdmeyyw=+-h2/2hm.0w2z0e()[]()()())26.4.(/2//222022zmzzmdzdyeywhh=+-.2z()20/hwm02ez.0e()()())31.4(2/exp2zuzz-=y.~2''yyz2z()())28.4).((/2)(/0222zzzdzdyweyh=+-)29.4.(,/000weeeeh==0e()())30.4.(0)(2)(/222=-+zzzzddyey)27.4.(/,/000wmzzzzh==()zue,z2z())32.4.(0122'''=-+-uuzue)34.4.(128)(,24)(,2)(,1)(332210tttHttHttHtH-=-===()zHn1-()zu()zun()zun12-e.)2/1(0wh0wh())33.4,...(2,1,2/10=-=nnnweh.2/1-=ne)35.4.(2exp!21)(2/10204/102/11-=+zmHzmmnznnnhhhwwwpf),(zV)37.4.(2/2/00zmz==Dwh)36.4.(exp)(202/1021-=hhzmmzwpwf.1AsGaAlxx-AsGaAlxx-10w.14,0,23,0eVEeVEVC=D=D,/gCEEQDD=AsGaAl7,03,0,mnhECEDeFzzV=)(.VEDVED()()[])38.4)(()(/2/222zzeFzdzdmeyy=+-h()00==zy.51-MVm())39.4.(2,203/1203/120eFzmeFmeFz===hhe()().00=-===eyysz.+snc-nc=e3/24123~-ncnp()())40.4.(/22zzdzdyey-=.338,21=cna())42.4,...(2,1,23/12==nmeFcnnhee-=zs)41.4.(/22yysdsd=+z320,21c1f())43.4.()()(0-=-==eeefeFzAizAisAizn2f())(zVRV=r()()[]()())44.4.(2/22RERRVmrrrhyy=+-()())46.4).((expexp),,(zuyikxikzyxyx=y)45.4).(,,(),,()(22222222zyxEzyxzVzyxmyy=+++-h()RVr().zu()()()()()()()()()())47.4.(expexpexpexp222expexp)(222222222222222zuyikxikEzuyikxikzVzmmkmkzuyikxikzVzyxmyxyxyxyx==+-+==+++-hhhh()()())48.4.(2222222222zEuzuzVzmmkmkyx=+-+hhh()()())49.4.(2222222222zumkmkEzuzVzmyx--=+-hhh.ne()zun()()()[]()())50.4.(/2/222zuzuzVdzdme=+-h()()2222/yxkkmE+-=heyxkk,()()()()())53.4.(2/,)52.4(),(expexp),,(222,,yxnyxnnyxnkkkkmkkEzuyikxikzyxyx++==hey()()())55.4.(2/)54.4(),(exp,22,mkkEzurkizrnnnnkrhrrrrr+==ey().,),,(yxkkkyxr==rrne10e