hocnq-VẬT LÝ BÁN DẪN THẤP CHIỀU-C7.pptx

Chng 7: Cc phng php gn ng


Ch mt t cc bi ton trong vt l v k thut c th c gii mt cch chnh xc v ngi ta cn phi nh n cc phng php gn ng hay cc phng php s gii phn ln cc bi ton ny. V d xt mt in t trong mt h vung gc m th ca n b nghing i bi tc dng ca mt in trng. Nng lng v hm sng i vi cc trng thi ca n ch thay i t nu trng l nh. Trng thi thp nht tr nn b phn cc trong phn su hn ca h m n gy ra mt s gim bnh phng nng lng ca n. L thuyt nhiu lon cung cp mt khun kh tnh nhng thay i ny v v d ny c tho lun trong phn 7.2.

Cch tip cn ny l c hiu qu nu th c th c chia thnh mt phn ln c th c gii chnh xc v mt nhiu lon nh. Cc phng php khc cn c s dng nu n khng phi nh vy. Phng php WKB c m t trong phn 7.4 c kh nng ng dng cho cc th thay i chm trong khng gian v lin quan cht ch vi c hc c in. Phng php bin phn (phn 7.5) ch cho nng lng ca trng thi c bn nhng c chnh xc khng g snh c v c th bao hm tng tc in t - in t v nhng s phc tp khc.


C nhiu ng dng cho cu trc vng. Phng php trong phn 7.3 cho dng ca cc vng nng lng gn mt khe. Khe l vng quan trng nht trong mt cht bn dn. C 2 phng php chung theo cc quan im tri ngc nhau. Phng php lin kt cht (phn 7.7) da trn c s bc tranh ca cc nguyn t c lp ng thi kt hp vi nhau to thnh vt rn trong cc vng c ngun gc t cc mc nguyn t. Ngc li, phng php in t gn t do trong phn 7.8 gi thit rng vt rn ch gy nhiu yu ti chuyn ng ca cc in t. N ch ra rng bt k th tun hon no mc d l yu cng sinh ra cc khe vng.

Trc khi i vo hin tng vt l, thun tin cn c mt c s ton hc no . Kt qu trc y ca chng ta da trn phng trnh vi phn Schrodinger v cc hm ring nhng nhiu l thuyt nhiu lon thun tin hn trong mt hnh thc lun trn c s cc ma trn khc vi cc ton t vi phn.

7.1. Hnh thc lun ma trn ca c hc lng t

Trong phn 1.5.1. ta lu rng thut ng cc trng thi ring v cc tr ring dng m t cc nghim ca phng trnh Schrodinger l rt tng t vi thut ng dng cho cc phng trnh ma trn. N l n gin lm cho


s kt hp ny cht ch hn v vit li phng trnh Schrodinger vi cc ma trn thay cho cc cc ton t vi phn. N to ra cc i lng gi l cc phn t ma trn c mt trong khp l thuyt nhiu lon.

Tnh cht quan trng nht m php bin i ny da trn n l tnh trc giao c m t trong phn 1.6. N c ngha l c th nh ngha mt tp hp cc hm sao cho

Tch phn trit tiu nu 2 trng thi l khc nhau v bng n v nu 2 trng thi l nh nhau. iu c ngha l cc hm l trc chun (va trc giao va chun ho bng n v). Tch phn ly theo khong quan tm m n c th l hu hn hoc v hn v theo 1, 2 hoc 3 chiu. Vic la chn hm ph thuc vo vng quan tm: cc hm sin v cosin hoc cc hm m phc l mt s la chn r rng i vi mt h 1 chiu tri di t n hoc

cc hm sin trong mt h t 0 n a. Trong 2 v d ny, cc kt qu l quen thuc t l thuyt Fourier. Tuy nhin, vic la chn cc hm rng hn nhiu so vi iu ny v cc nghim ca bt k phng trnh Schrodinger no u c th c lm trc chun.

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Phng trnh Schrodinger c th c vit thnh trong l

mt ton t vi phn m ta tnh n nhiu dng i vi n. Khai trin

theo mt h no ca cc hm

N c vit vi mt tng theo n v iu thch hp cho mt h b gii hn ti mt vng hu hn ca khng gian chng hn nh mt ht trong mt hp vi Tng c th tr thnh mt tch phn i vi mt vng v hn ging nh trong trng hp ca mt php bin i Fourier khi dng

By gi thay biu thc ny vo phng trnh Schrodinger. Kt qu l

Nhn 2 v v bn tri vi ri ly tch phn. N cho

Gi thit rng tng v tch phn hoc cc ton t vi phn c th c t do

sp xp li. Tch phn v phi chnh xc l nh ngha ca s trc chun v

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c rt gn ngay thnh Tt c cc s hng trong tng khi trit tiu tr v do , v phi tr thnh V tri khng n gin ho theo cch nhng ta c th nh ngha mt i lng

m n c gi l yu t ma trn ca gia cc trng thi m v n. T , phng trnh Schrodinger c rt gn thnh

By gi, n l mt phng trnh tr ring i vi ma trn H

H

trong E l tr ring v l vect ring. l s tng ng ma trn ca phng trnh vi phn Schrodinger ban u. N cng c th c vit thnh

(EI H) = 0, trong I l ma trn n v. iu kin phng trnh ny c cc nghim khng tm thng l nh thc ca ma trn EI H phi trit tiu

detEI H = 0. (7.8)

N c gi l phng trnh c trng. Cc cn bc 2 ca n xc nh cc

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nng lng cho php Ma trn n v thng c b i.

N ch ra rng c mt mi quan h cht ch gia s m t ca cc trng thi nh l cc hm trong khng gian v d nh m ta s dng n cho n nay v s m t da trn cc ma trn trong ta khai trin hm sng theo mt h trc chun no v dng cc bin m t trng thi. Phng trnh ma trn khng hon ton n gin nh cc bi ton tr ring thng thng do cc chiu l v hn. Tuy nhin, hu nh tt c cc phng papa gn ng ct xn c s ca cc trng thi dng khai trin hm sng v ma trn tr thnh hu hn.

K hiu i vi cc phn t ma trn l nh vo Dirac. y, l

mt ket k hiu trng thi v bra l lin hp phc ca n. Lu s tit kim k hiu so vi cc dng d hiu hn v d nh dng Dac ch gi n m n gn cho cc trng thi. Thc ra trng thi n c th c biu din thnh v iu ny l khng quan trng do thay th ta c th la chn khai trin trng thi theo mt h no ca cc hm ging nh trong (7.2) v cc thng s khai trin c gi tr nh l mt s m t hm sng thnh K hiu Dirac loi b chi tit khng quan trng ny. Thc t l

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k hiu c mt ngha ton hc su sc hn m n san lp h ngn cch gia cc ma trn v cc phng trnh vi phn.

Mc d bt k mt tp hp trng thi no c th c dng rt gn ton t vi phn ti mt ma trn mt s la chn c bit thun tin l cc hm ring ca (nu ta bit chng). C cc trng thi dng ca phng trnh Schrodinger Khi ,

Dng cui rt ra t tnh trc chun ca cc trng thi v ch ra rng ton t Hamilton c dng cho trong c s ny. Nh vy, mt cch khc m t cc trng thi ring ca bt k ton t no khng ch ton t Hamilton l ni rng cc trng thi ny cho ho ton t.

iu c th xy ra l mt ton t khc cng c dng cho trong h ca cc trng thi lm cho ho Do , cng cc trng thi l cc hm ring ca c hai ton t. iu ny ch c th xy ra nu 2 ton t giao hon vi nhau (phn 1.6). Trong trng hp ny, i lng vt l tng ng vi ton t th

hai c ni l mt hng s (tch phn) ca chuyn ng. Ly cc in t t

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do theo 1 chi lm v d. Cc sng phng ng thi cho ho c ton t Hamilton v ton t xung lng

(phn 1.5). iu ny chng t rng chng c c nng lng v xung lng xc nh.

By gi ta bin ton t (vi phn) Hamilton thnh mt ma trn v phng trnh Schrodinger thnh mt phng trnh ma trn. Cng mt qu trnh c th c lp li trn bt k ton t no. Nh ta thy trong cc phn tip theo, l thuyt nhiu lon bao hm s tch ton t Hamilton thnh mt phn ln

m ta bit lm th no cho ho v mt nhiu lon nh m n i vo cc biu thc i vi nng lng hoc hm sng theo cc phn t ma trn gia cc trng thi ring ca

Cc ton t vi phn tng ng vi cc i lng quan st c cn phi l cc ton t ecmit bo m rng cc tr ring ca chng biu din cc gi tr c th o c l thc (phn 1.5.2). Cc ma trn tng ng cng cn phi l cc ma trn ecmit m n c ngha l M = M hay

Vi chun b s b trn y, by gi ta c th tip cn l thuyt nhiu lon.

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7.2. L thuyt nhiu lon khng ph thuc vo thi gian

Ta bt u vi mt v d quen thuc l mt in t trong mt h lng t. Hnh 7.1(a) ch ra h th hnh ch nht hu hn thng thng m ta nghin cu chng 4. N l tm thng nu su l v hn v th v hn nu n l hu hn. By gi gi s rng mt in trng tc dng ln mu v n lm nghing th nh ch ra trn hnh 7.1(b). Trng thi c bn thay i nh ch ra trn hnh v. Hm sng ca n tr nn bt i xng ta trung bnh chuyn ng cch xa tm h (ly thnh x = 0) hng v pha m nng lng ca n b gim i bi in trng. in trng lm phn cc in t v sinh ra mmen lng cc Ta hi vng lng cc t l vi trng F i vi trng nh v xc nh mt h s phn cc (vi cc

n v ca th tch) bi S dch hm sng lm gim nng lng ca n mt lng l l nng lng ca lng cc cm

ng.

Hnh 7.1. Mt h lng t (a) vi cc

th phng v (b) trong mt in trng

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iu xy ra trong trng hp ny l bi ton c th c gii mt cch chnh xc. Th trn hnh 7.1(b) l tuyn tnh khp ni v do , hm sng bao gm cc hm Airy lm khp ti cc bin ca h. iu ny l khng th v v i hi cc bng ln hoc my tnh. chnh xc nh th c th l khng cn thit v trong phn 7.2.3 ta s thy rng mt nghim chnh xc thm ch c th l khng c mong mun. tm h s phn cc ta c th gi thit rng s thay i tc dng trong th nng l nh v tm mt khai trin ca hm sng v nng lng theo cc ly tha ca in trng tc dng. T tho lun trn suy ra ta cn

(i) s thay i hm sng ti bc nht theo trng hoc

(ii) s thay i hm sng ti bc hai theo nng lng.

Nhng i hi gii hn ny ho ra l rt tng qut v by gi ta s pht trin mt s tnh ton i vi cc i lng ny. Loi l thuyt nhiu lon ny c th tht bi thng l i vi cc l do vt l tt v ta s thy rng n khng cho cu chuyn y thm ch i vi h lng t n gin trong in trng. By gi ta s rt ra nhng kt qu chung i vi l thuyt nhiu lon ny v sau p dng chng cho mu h lng t trong in trng.

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7.2.1. L thuyt tng qut

tng chung l tch ton t Hamilton ca h m ta mun gii thnh 2 phn Chng l

(i) ca h khng nhiu lon m n l ln v c th gii c chnh xc (ging nh h th trc khi tc dng trng) v

(ii) nhiu lon m n cn phi nh. ngha chnh xc ca nh s tr nn r rng hn di y.

Nh vy, im xut pht l ta bit cc nghim ca phng trnh Schrodinger

v mun tm cc nghim ca phng trnh Schrodinger

tng ca l thuyt nhiu lon l khai trin nng lng v hm sng theo cc ly tha ca th nh Thng vit v do cc ly

tha ca xc nh cc bc ca b. Ta t cui qu trnh. Nng

lng v cc hm sng c cc biu thc

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Ch s di gn cho trng thi v ch s trn gn cho ly tha ca m n l bc b.

Vic thay cc biu thc ny vo phng trnh Schrodinger cho

N cn ng i vi mi gi tr ca m n c ngha l cc h s ca cc ly tha ca c 2 v cn phi nh nhau. i vi cc ly tha 0, 1 v 2, n cho

(7.15) l phng trnh Schrodinger khng nhiu lon v do

iu ny kh r rng v cc c tnh bc 0 ca hm sng v nng lng

chnh l cc c tnh khng nhiu lon.

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Vic thay nhng kt qu ny vo phng trnh bc nht (7.16) v sp xp li phng trnh ny cho

Tip theo, ta cn s dng cc kt qu ca phn trc. Khai trin theo h

ca cc trong lc ch s trn li l bc ca b. Vi khai trin ny, phng trnh bc nht (7.19) tr thnh

Ton t ly trong tng v tri do ta bit rng tc dng ca n ln

n gin l Nh vy, bin mt. l mt kt qu ca khai trin By gi, ta c

Bc tip theo l dng tnh trc chun ca cc trng thi v rt gn

thnh mt phn t ma trn. Nhn (7.22) vi v bn tri v ly tch phn.

bn tri, n chn ra s hng vi k = n do tt c cc s hng khc trit tiu do

tnh trc chun. S khc bit lm cho s hng cn li ny cng trit

tiu. vi v phi c tch phn bng n v v s hng vi th tr

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thnh mt phn t ma trn cho

(7.23) chng t rng s thay i bc nht ca nng lng n gin l gi tr k vng ca th nhiu lon trng thi khng nhiu lon

c cc h s ca hm sng, nhn (7.22) vi trong v bn tri v ly tch phn. Ch k = m cn li v tri v lc ny khng b trit tiu. S hng vi bin mt t v phi do tnh trc chun v n li phn t ma trn Nh vy,

v do Do , s thay i ca hm sng ti bc nht l

Lu rng ta khng th xc nh h s theo cch ny. H s ny ch nh hng ti s chun ho v do , ta s ly

By gi tip tc ti bc 2 tm hiu chnh tip theo vo nng lng. Vi nhng kt qu bit, (7.17) tr thnh

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Khai trin hm sng thnh

m n cho

Ta c th ly bn trong tng v thay n bng nh trc. Tip theo, nhn vo bn tri vi v ly tch phn. V tri trit tiu cn li

H s cha bit b b i. Vic lng vo cc h s khc cho

iu ny c th c n gin ho mt t nu ta nh rng V l mt ma trn ecmit v do , v t s tr thnh hoc

By gi ta c th tp hp cc kt qu trn i vi vic khai trin hm sng

ti bc 1 v nng lng ti bc 2

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Qu trnh c th c tin hnh ti bc cao hn nhng n t khi cn v thay cho iu , ta xem xt cc n ca cc cng thc ny.

(i) By gi ta c th thy ci g c ngha l nh. Nu chui hi t nhanh lc u, cc phn t ma trn cn phi nh hn nhiu so vi cc mu s nng lng. Ni cch khc, s lin kt gia cc trng thi gy ra bi nhiu lon cn phi nh hn s tch gia cc mc nng lng. iu ny hu nh khng gy ngc nhin do cc trng thi ca ton t Hamilton ban u s khng phi l ci g ging nh cc trng thi ca h nhiu lon nu nh iu kin ny khng c p ng.

(ii) Kt qu ca l thuyt nhiu lon khng p dng c nu mt mu s nng lng trit tiu. iu ny xy ra nu trng thi trong ta quan st l suy bin vi trng thi khc v khi cn dng cch tip cn khc (fn 7.6).

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(iii) Nhiu phn t ma trn thng trit tiu do i xng. Cng cc phn t ma trn xut hin trong tnh ton cc tc tn x bi l thuyt nhiu lon ph thuc vo thi gian trong cc gii hn do i xng c bit n nh l cc qui tc la chn. H lng t trong in trng cung cp mt v d n gin. Th ca n l i xng theo x gn im gia ca n v do cc trng thi xen k l i xng v phn i xng (chn hoc l theo x). in trng cho mt nhiu lon v do , cc phn t ma trn ly dng

Lin hp phc c th c b i v cc hm l thc. By gi x l mt hm l v do tch cng cn phi l l nu ton b hm di du tch phn cn phi l chn v cho kt qu khc khng. Nh vy, mt trong cc trng thi cn phi l chn v trng thi khc l l. l mt v d n gin ca qui tc la chn. c bit l cc phn t ma trn cho (n = m) trit tiu.

(iv) S thay i bc 1 ca nng lng c th c du ny hoc du khc. Tuy nhin, n thng trit tiu bi i xng v ta cn phi dng s hng th hai. Trong trng hp ca trng thi thp nht thng c quan tm, tt c cc mu s nng lng u l m. Cc t s d nhin l dng v do

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, s thay i bc 2 i vi nng lng ca trng thi thp nht lun lun l m. Nhiu lon trn cc trng thi khc vo trong hm sng v h c th lun lun s dng s t do ny h thp nng lng ca n.

7.2.2. H lng t trong in trng

By gi ta c th gii bi ton t ra lc m u phn ny l h lng t trong in trng. n gin ly mt h su v hn c b rng a xoay quanh gc. Do , v l cc hm sin v cosin xen k. Ta cn cc phn t ma trn i vi s thay i nng lng ca trng thi thp nht trong nhiu lon do in trng l S i xng ca cc hm sng v nhiu lon c th c s dng ch ra rng cc phn t cho u trit tiu v ch tn ti i vi k chn. Nh vy, in trng lin kt trng thi c bn chn ch vi cc trng thi cao hn c i xng l (k chn) v khng c s thay i nng lng ti bc 1. iu ny l r rng v mt vt l do s thay i nng lng khng ph thuc vo du ca F v do xut hin mt s hng bnh phng l s hng thp nht. Khi xt ti bc 2, ta cn phi nh gi

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trong cc s hng khc 0 c gi li v r rng l nng lng b h thp.

i vi mt lin kt thp hn trn s chuyn dch, tnh ring s hng th nht (k = 1). N cn phn t ma trn lng cc

Mu s nng lng l v do lin kt ca chng ta trn s chuyn dch nng lng l

iu ny c th c din t nh sau

- s thay i nng lng/ s khc bit nng lng

(1/10)(s gim nng lng qua h/ s khc bit nng lng) (7.37)

iu ny ch ra cc nng lng xut hin mt cch t nhin trong bi ton. Nhiu lon l in trng v do , phn t ma trn ca n cn phi t l vi s gim th hiu qua vng tch cc. in t bt u gia v bn trong h v do , ta c th phng on Mu s l s khc bit nng

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lng gia trng thi quan tm v trng thi gn nht lin kt vi n. Mt c tnh khi dng cc gi tr ny khc vi (7.36) ch bi h s 2. Trong a s trng hp, d c tnh cc nng lng tham gia bi l thuyt nhiu lon theo cch ny mc d ngi ta cn cn phi nh gi cc tch phn tm gi tr tuyt i.

S thay i nng lng cng c th c vit theo h s phn cc H

lng t c nhng trong mt cht bn dn c hng s in mi v do

iu thch hp l nh ngha bi Nh vy,

Bn knh Bohr (4.67) c a vo trong biu thc th hai i vi

do n l n v t nhin ca chiu di trong mt cht bn dn v st nhp thun tin tt c cc thng s v d nh khi lng hiu dng.

Ta ch xt nh hng ca trng thi gn nht ln s chuyn dch nng lng v cha xt ng gp t cc trng thi khc. Trong trng hp n gin ny

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xy ra iu l tng theo tt c cc trng thi c th c nh gi mt cch gii tch. N lm thay i h s trc trong (7.36) t 256/243 0,010815

ti ng gp t cc trng thi cao hn l nh v n l chung. S khc bit ca cc nng lng trit nh hng ca cc trng thi cao hn qua mu s nh l y v n cng xy ra trong trng hp ny l cc phn t ma trn gim thnh Nh vy, chui hi

t rt nhanh v ring s hng th nht cung cp mt nh gi rt tt. iu ny c th khng phi l trng hp trong mt h lng t hu hn do cc trng thi cao hn l khng lin kt.

nh hng ca in trng ln s hp th quang trong h lng t c ch ra trn hnh 7.2. Nng lng ca c cc in t v cc l trng c rt gn (theo ngha thch hp) bi in trng. S dch ny ca ng hp th c bit l hiu ng Stark giam cm lng t. N lin quan r rt ti hiu ng Franz Keldysh (phn 6.2.1) m n l s thay i bin hp th i vi cc in t t do khc vi cc in t giam cm. Hiu ng Stark giam cm lng t ho trn vo hiu ng Franz Keldysh khi b rng h tng ln v s tch gia cc trng thi lin kt gim. iu ny cng xy ra khi chiu di in

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Hnh 7.2. (a) Mt h lng t vi cc vng

phng m n ch ra nng lng i

vi s hp th gia cc trng thi lin kt

trong h. (b) Cc vng b nghing bi tc

dng ca in trng m n lm gim nng

lng ca c 2 trng hp lin kt v rt

gn nng lng hp th ti Khe

vng c rt gn nhn cho r.

(4.39) tr nn nh hn so vi b rng h v cung cp s giam cm quan trng hn. Hnh 7.3 ch ra mt php o hiu ng Stark giam cm lng t so vi mt l thuyt tng t khc.

Hiu ng Stark giam cm lng t c th c dng thit k cc dng c

quang in t khi dng s thay i hp th mt cch trc tip hoc s thay

i kt hp ca chit sut theo i hi ca h thc Kramers Kronig. Mt

tnh ton y i hi thnh phn quan trng khc l nh hng ca in

trng ln cc exciton.

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Hnh 7.3. (a) Ph hp th ca mt h

lng t a lp nh l mt hm ca

in trng vung gc. Cc h GaAs

rng 9,5 nm b tch ra bi cc ro 9,8

nm Cc trng l (a)

1,0, (b) 4,7 v (c) 7,3 2 nh

nhn trn mi ng cong l do cc

l trng nng v nh. (b) V tr ca cc

nh nhn trong nng lng nh mt

hm ca in trng. Cc ng l cc

c tnh l thuyt.

V sau ta s xt nhiu ng dng ca l thuyt nhiu lon. By gi cho mt v d khc, s hp th quang trong nhiu h c th c m hnh hoc khi dng mt dao ng t iu ho. Cc s hng phi iu ho c th c a vo mt dao ng t m rng m hnh m n cho

trong B v C l nh v c th c nghin cu khi dng l thuyt nhiu lon.

Cch tip cn ny thng c dng trong l thuyt quang phi tuyn.

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7.2.3. Lu

L thuyt nhiu lon thng c p dng tt v cho nhng li gii hp l i vi nhiu bi ton. Tuy nhin, cn nhn thy rng n c th phm sai lm. Xt mt h lng t trong in trng. By gi xt mt vng rng hn ca cht bn dn (hnh 7.4). in trng lm nghing cc ro sao cho ngi ta c cc ro cao hn nhng ro khc ch c mt b dy hu hn trn nng lng ca trng thi lin kt trong h. Nh vy, mt in t trong trng thi lin kt c th i xuyn hm ging nh trng thi cng hng trong mt cu trc ro kp (phn 5.5). S dch nng lng tnh ton c th hin l khng nhy do cc in t khng cn c lin kt na. Thi gian sng ca mt in t trong h c th l rt lu do ro gi dy min l trng khng qu ln. Cc th nghim quang s khng b nh hng vi iu kin l cc in t v l trng kt hp li trc khi xuyn hm ra khi cc h. Ch trong cc in trng ln nht, bn cht cng hng ca trng thi tr nn quan trng

Hnh 7.4. H lng t sau khi tc dng mt in trng m

n ch ra trng thi lin kt trc by gi c th xuyn

hm ra ngoi nh th no.

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v m rng vch hp th nh trn hnh 7.3(a).

Qu trnh ny trong mt in t c kh nng thot ra khi mt trng thi lin kt trc sau khi tc dng in trng do vic h thp ro c gi l s xuyn hm Fowler Nordheim. Mc d ta c kh nng b qua n trong thc t, l thuyt nhiu lon khng tnh ton c s thay i nh tnh. tm hiu nguyn nhn, ta lm mt nh gi th v tc thot ra ca in t. Xc sut xuyn hm qua ro gn bng trong L l b dy ro v l hng s suy gim. y, ta c th t trong l

su h (thc ra ta o t nng lng ca trng thi lin kt khc y h). Tc suy gim khng phi l hng s nhng ta c th dng gi tr ca n gia ng qua ro nh mt nh gi th. Do , Nhng

gi tr ny cho

N trit tiu khi trng tin ti 0 v c im ton hc quan trng l ch n nh th theo mt cch khng gii tch (T(F) c k d c bn ti F = 0). Nh vy, khng th khai trin T thnh mt chui ly tha ca nhiu lon F

25


m n l gi thuyt thit lp l thuyt nhiu lon ca chng ta (phng trnh (7.12)). Mt l thuyt nh vy do s khng bao gi khm ph c s xuyn hm ra khi h. S xuyn hm Zener (phn 2.2) l mt bi ton tng t.

L thuyt nhiu lon ch c th c s dng tnh cc s i vi mt hiu ng c hiu r. Mc d iu ny hu nh lun lun l hin trng trong thc t, iu c ch l cn ghi nh rng tn ti cc hiu ng khng nhiu lon nh s xuyn hm Fowler Nordheim.

7.3. L thuyt

Ta bit rng phn ln cc qu trnh trong mt cht bn dn xy ra gn nh VB v y CV. Mc d ta dng tt cc parabol cho cc vng trong khu vc ny, cc tnh ton y (phn 2.6) ch ra rng iu ny ch l mt php gn ng th i vi CB v php gn ng km i vi VB, trong cc l trng nng v nh l suy bin ti Ta mong mun c mt s mo t chnh xc hn i vi cc vng trong khu vc quan trng ny m khng cn phi dng n cc phng php s. Phng php v nhng m rng ca n nhm ti mc ch ny.

26

7.3. L thuyt

nh l Bloch pht biu rng hm sng trong mt tinh th c th c vit thnh tch trong l mt hm tun hon (phn 2.1). Trong khi rt ra php gn ng khi lng hiu dng, ta cng tranh lun rng n c th l mt php gn ng tt khi gi thit rng

l hng s trong mt vng nh ca khng gian (phng trnh (3.5)). C th ly n nh mt gi l n c th l d dng hn tm cc nghim gn ng i vi mt hm thay i chm hn so vi

Cho phng trnh Schrodinger i vi cc hm Bloch ca tinh th l

trong ton t xung lng nh thng thng v l khi lng ca in t t do. Thay vo y. Cc o hm ca tc dng ln sng phng n gin cho v sng khi b i. N li mt phng trnh i vi ring phn tun hon nh sau

27

7.3. L thuyt

Gi s rng ta gii n ti v bit tp hp cc hm sng v cc

nng lng L thuyt tng qut cho ta kt qu l chng to thnh mt h hm . Do , ta c th s dng chng nh mt c s trong khai trin cc nghim ti mt gi tr khc no ca m n cho mt phng trnh ma trn. Mt cch la chn khc l ta c th dng l thuyt nhiu lon v ta quan tm ch yu n cc gi tr nh ca Cc s hng ph thuc vo trong (7.42) c coi nh mt nhiu lon xa nghim ti Mt

s hng n gin l s thay i nng lng ca mt in t t do trong lc s hng khc cha ton t m n dn ti tn phng php. Lu rng

l vect sng Bloch m ta ang nghin cu n nh l nhiu lon trong lc

l ton t xung lng. Nh vy, cc s hng theo y v z.

Ta tp trung vo mt vng ring n m ta s gi thit n khng phi l suy bin i vi bt k vng no khc ti D dng chng minh rng cc phn ma trn cho v d nh trit tiu. L thuyt nhiu lon bc 2 cho

28

7.3. L thuyt

Mc d biu thc kh cng knh, n l bnh phng theo v lun lun c th c cho ho cho cc dng ca CB m ta thy trong phn 2.6.2. N rt gn thnh mt v hng trong thung lng ca GaAs vi

Ta c th c tng no v cc hnh dng ca cc vng t dng ca (7.43) thm ch khng cn bit cc hm sng chnh xc. Xt thung lng

trong CB ca GaAs. Cc ng gp ln nht s n t cc vng gn nhau v nng lng. nh ca VB cung cp cc trng thi gn nht ti (mc d cc CB cao nht tip theo cch khng xa). Cc phn t ma trn khng trit tiu do CB l i xng (kiu s), VB l phn i xng (kiu p) v ton t cng

l phn i xng. Mu s nng lng l dng v cc trng thi m ta trn l thp hn v nng lng so vi trng thi gc (khng ging vi trng thi c bn m ta nghin cu trong phn 7.2.2). Nh vy, hiu chnh lm tng nng

lng ca in t nh mt hm ca m n lm gim khi lng hiu dng

29

7.3. L thuyt

ca n.

Ly dc theo x lm mt nh gi th i vi ln ca n. Tng c ly theo cc trng thi ti nh VB m chng sinh ra t 3 orbital p (phn 2.6.3). Phn t ma trn theo qui c c vit thnh

N cha trng thi S ca CB c xy dng ngoi cc orbital s v trng thi X ca VB m n c xy dng ngoi cc orbital Phn t ma trn ny xut hin trong cc tnh cht khc v d nh s hp th quang. 2 phn t ma trn khc v d nh trit tiu bi i xng. (7.43) cho nng lng ca CB l

Khi lng hiu dng c cho bi biu thc trong ngoc v c th c vit thnh trong l khe vng ti v

Cui cng, ta cn nh gi Ton t l mt o hm v do mt cch gn ng n chn ra vect sng ca cc trng thi xung quanh khe

30

7.3. L thuyt

vng. Bin ca min Brillouin (X) l ti v hnh 2.16 ch ra rng nh ca VB b gp li trong mi ln t bin v s sng trong P gn ng l

Hng s mng nm v n cho eV. Thc t l gi tr ny duy tr kh tt i vi cc cht bn dn thng thng (Ph lc 2). Khe vng 1,4 eV i vi GaAs v do ta d on

Chiu hng m cc khi lng nh i vi cc khe hp cng l chnh xc. iu ny cng lm hn ch gi tr ca phng php do mt khi lng hiu dng nh c ngha l ng nng tng nhanh vi k v sm tr nn c th so snh c vi mt khe vng hp. S thay i nng lng do s nhiu lon

khi ln hn so vi s tch ca cc trng thi khng nhiu lon v l thuyt nhiu lon tr nn khng chnh xc. Mt cch tip cn c lin quan theo Kane l tt hn trong cc iu kin ny.

Cng cc phn t ma trn xut hin trong biu thc i vi cc nng lng ca VB nhng nhng s o ngc ca cc mu s nng lng c ngha l chng c nh hng ngc nhau. Lin kt ny vi CB vt qu

m n dn ti y vng ln pha trn v VB b cong xung di cho dng

31

7.3. L thuyt

iu mong mun i vi cc l trng. Ta khng th lm nhiu hn pht biu chung ny do cc VB l suy bin ti v l thuyt khng th c p dng mt cch chnh xc. Ta s quay tr li iu ny vi mu Kane.

Phng php cng c th c dng ti cc im xa Trong trng hp ny, ni chung tn ti mt s tuyn tnh trong m n cho vn tc nhm ca mt in t trong vng cng nh cc s hng bnh phng m ta xem xt.

7.4. L thuyt WKB

Thng thng ngi ta cn gii phng trnh Schrodinger i vi mt h trong th V(x) thay i chm trong khng gian. Mt v d l mt th giam cm cc in t trong mt dy c xc nh bi cc cng tch ri trn b mt ca mt d cu trc hnh 3.17(c)). Cc tnh ton chi tit ch ra rng th ny gn ng l parabol ti cc bin vi mt tit din phng gia nh ch ra trn hnh 7.5 khc vi tit din ngang qua mt bn tm. B rng ca h th ti nng lng Fermi c th l 0,2 m hoc ln hn so vi bc sng Fermi l 0,05 m. V mt c in, mt ht c nng lng E s lin kt v pha sau

32

7.4. L thuyt WKB

v pha trc gia 2 im quay c in v trong Trong

c hc lng t, ht c th xuyn hm vo trong ro v gii hn chuyn ng ca n dn ti s lng t ho nng lng ca n.

V mt c in, ng nng ti mt im x l v thay i chm trong khng gian. Nu n khng thay i g c, nghim c hc lng t ch c th l mt sng phng vi Mt phng on r rng i vi mt h trong thay i chm s l cho s sng thay i vi ng nng nh x Pha ca sng phng cng cn phi thay i t tch n gin kx ti tch phn

cho php s thay i ca k. N l c s ca phng php WKB (Wentzel, Kramers v Brillouin). N cng gn lin vi Jeffreys v Rayleigh v c tn

Hnh 7.5. Phin bn n gin ho ca th giam cm

cc in t trong mt dy lng t (th bn tm)

m n c th c phn tch khi s dng phng

php WKB. Cc im quay c in v

xy ra khi

33

7.4. L thuyt WKB

l php gn ng chun c in trong ti liu khoa hc ca Nga. By gi ta s lm cho nhng tng ny c c s vng chc hn v p dng phng php ny cho mt h tam gic.

7.4.1. L thuyt chung

Ta mun gii phng trnh Schrodinger 1 chiu thng thng

trong thay i chm vi ngha l ta s phi lm chnh xc. L lun s b ca chng ta chng t rng pha hm sng chc chn phi l phn quan trng nht v do , ta vit li Vic thay n vo phng trnh Schrodinger dn ti phng trnh i vi pha nh sau

trong s sng nh x c a vo ph hp vi iu ny l chnh xc v cn phi a vo cc php gn ng t c kt qu tt hn. o hm bc 2 l nh do th v s sng c gi thit thay i

34

7.4. L thuyt WKB

chm. Vic b qua n cho

m n chnh xc l dng pha m ta coi l mt nh . N s l chnh xc nu

Khi dng php gn ng n tr thnh

By gi bc sng l v do bt ng thc i hi rng s thay i ca k ng vi mt bc sng l nh hn nhiu so vi k. l mt nh ngha hp l ca mt h thay i chm. gn cc im quay c in do tip

cn E , k gim ti 0 v bc sng tin ti v do , bt ng thc khng c tha mn. Do , ta cn phi sa cha nghim WKB gn cc im quay.

Trc khi lm iu ny, iu c ch l i mt bc xa hn trong li gii WKB. T (7.46) suy ra

trong php gn ng u tin i vi c dng trong o hm bc

35

7.4. L thuyt WKB

2. Ly cn bc 2 v to ra mt khai trin nh thc cho

m n c th c ly tch phn thu c

Nh vy,

m n l dng m php gn ng WKB thng c trch dn. N c rt ra i vi cc ng nng dng nhng c th c m rng cho cc nng lng m (xuyn hm khc vi cc sng lan truyn) bng cch thay

bng v b i trong hm m i vi cc trng thi lin kt, hm m phc c thay bng hm sin hoc cosin.

Tha s trc gip bo ton dng i vi cc trng thi lan truyn. i vi mt sng phng ch thc

Trong phng php WKB, tha s trc xa b s thay i dng m n gy

36

7.4. L thuyt WKB

ra bi s thay i ca k. N bo m rng mt ht c mt mt thp hn (s dng t thi gian hn) trong cc vng m n chuyn ng nhanh hn. Cc cch m trong n bo ton dng cng lu n nhc im ca php gn ng WKB l n b qua cc phn x. N c gi thit l ht c th chu mt s thay i th thun ty bi s thay i s sng ca n trong khi mt phn (nh) ca sng lun lun b phn x.

Mt cch s dng quan trng ca phng php WKB l nhm c tnh tc xuyn hm qua cc ro. Ta mun xc sut khc vi bin v do , hm sng cn phi c ly bnh phng v do , c tnh WKB l

Cc im v l cc bin ro c xc nh bi l mt

Hnh 7.6. S lm khp cc hm sng WKB

trn pha ny hoc pha kia ca mt im

quay c in

37

7.4. L thuyt WKB

s tng qut ho r rng ca Mt v d ca n s c xem xt trong phn 7.4.3.

Ta cn cp n vn ca cc im quay nu ta mong mun p dng phng php WKB cho cc trng thi lin kt. Xt tnh hung ti c ch ra trn hnh 7.6 trong mt ht c in khng th xuyn qua ro ti bn tri. Phng php WKB khng c gi tr trong vng ny v bc sng tin ti v cc v s thay i ca khng th c coi l nh. Mt cch tip cn th trnh c cc im quay bng cch to ra mt s dch chuyn vo trong mt phng phc. Mt cch khc xem xt n l lu rng th l tuyn tnh i vi mt vng nh xung quanh im quay v ta bit rng cc hm Airy l cc nghim ca phng trnh sng trong mt th nh vy. Do , ta c th tng tng sa cha cc nghim WKB ca chng ta i vi

v ln trn mt hm Airy m n lm y vng gia. Vn c v phc tp nhng cc kt qu l n gin

38

7.4. L thuyt WKB

Cc c im quan trng l tha s 2 v pha Xt phc ha hm sng trn hnh 7.6. Sng dao ng c bin ln hn so vi hm m c ngoi suy tr li v n l tha s 2. Tng t, hm cosin cn phi bt u vi mt dc hng ln trn lm khp vi hm m gim. iu ny c ngha l pha ca cosin cn phi gia v 0.

Hng s trong pha l quan trng xc nh cc trng thi lin kt m n l ng dng chung th hai ca phng php WKB. iu kin bin trong mt h vi cc thnh v cng dc l iu kin hm sng tin ti 0 ti cc bin. Do , mt s chnh xc cc na bc sng cn phi c lm khp gia chng. Nh vy, s thay i pha gia v cn phi l mt bi s ca v

iu kin i vi cc trng thi lng t ha l

Lu rng v u l hm ca nng lng.

39

7.4. L thuyt WKB

Mt khc, xt mt h vi cc thnh mm nh bn tm trn hnh 7.5. Mt trng thi cho php by gi cn phi tun theo iu kin lm khp (7.54) ti mi mt u cui m n i hi pha By gi, iu kin lng t ha l

N t c khi mt thnh l mm v thnh kia l cng. Ni chung, chnh xc tng theo n khi nh hng ca cc im quay trong phng php WKB l km chnh xc nht gim i.

7.4.2. Cc trng thi lin kt trong mt h tam gic

Trc ht, ta s dng phng php WKB nh gi cc nng lng ca cc trng thi lin kt trong mt h tam gic (hnh 4.6). N c i

vi x > 0 v mt thnh cng ti x = 0. Ta gii n mt cch chnh xc trong phn 4.4 nhng cc nng lng ph thuc vo cc s khng ca hm Ai(x) m n cn phi tm bng cch tnh s. Do , ta mong mun c mt php gn ng gii tch n gin . im quay bn tri l l mt bin cng

trong lc bin bn phi l mm v c cho bi hoc

40

7.4. L thuyt WKB

iu kin lng t ha l

, Tch phn l tm thng v ta c

Bng 7.1. So snh cc phng php gn ng khc nhau v cc mc nng lng trong mt th tam gic theo n v ca v cc kt qu chnh xc t hm Airy

n Hm Airy WKB Bin phn Bin phn

(chnh xc) (Fang Howard) (Gauss)

1 2,3381 2,3203 2,4764 2,3448

2 4,0879 4,0818

3 5,5206 5,5172

... ... ...

10 12,8288 12,8281

41

7.4. L thuyt WKB

Mt t gi tr c ch ra trong Bng 7.1. chnh xc tng theo n nh mong mun nhng sai s nh hn 1% thm ch i vi n = 1. Do , n l mt php gn ng kh tt.

Mc d cc kt qu tnh s chng thc ti chnh xc ca php gn ng WKB i vi th ny, ta cn kim tra n bng cch dng bt ng thc (7.48). N i hi By gi,

trong , l thang o chiu di gn vi mt th tam gic (phn 4.4). Nh vy, iu kin phng php WKB c gi tr l Nu ta nh gi k ti im gia ca chuyn ng l bt ng thc tr thnh trong l thang o nng lng ca th tuyn tnh. N c tha mn (t nht vi > thay cho >>) thm ch i

vi trng thi thp nht. Do , ta xc nhn chnh xc ca phng php.

42

7.4. L thuyt WKB

7.4.3. Xuyn hm qua ro Schottky

p dng phng php WKB cho s xuyn hm, xt mt th nh ch ra trn hnh 7.7. N l ro Schottky c to thnh gia n GaAs v kim loi. Cc trng thi ti giao din kim loi bn dn gi mc Fermi ti nng lng di vng dn ca bn dn ti x = 0. Ro eV i vi GaAs v ch thay i nh i vi cc kim loi khc nhau. nh hng ca b mt b mt i i vi x ln v mc Fermi gn bin vng dn trong vt liu pha tp n (nng) trung ho . gia l mt lp ngho vi mt in tch khng gian l nh vo cc cht cho b ion ha. iu ny cung cp mt s chnh lch th cn hi phc t ti gn

Nghim ca phng trnh Poisson vi mt in tch khng i ny ch ra rng vng l parabol, trong b dy d c cho bi

Ro Schottky s c xem xt tip chng 9.

By gi ta s nh gi h s truyn qua ca ro ny. l mt bi ton thc t quan trng v s xuyn hm khng phi l iu mong mun nu kim loi tc ng nh mt cng c cch li tt vi vng hot ng ca dng c.

43

7.4. L thuyt WKB

Mt khc, kim loi c th l mt tip xc thun tr v trong trng hp ny, in tr ro l thp nht c th c.

Xt cc in t c nng lng thp Phng php WKB nh gi xc sut xuyn hm (7.53) v n cho

di suy gim nm trong GaAs. Nh vy, d cn phi rt nh. N i hi s pha tp rt cao nu ro l trong sut va phi. iu ny mt phn gii thch tnh cht thng thay i kh khn ca cc tip xc thun tr trn GaAs.

tnh dng (phn 5.4), ta cn phi tnh phn b trong nng lng ca cc in t ti m mt t trong s c cc nng lng trn Mt s in t c th ng l chuyn qua ro v lm tng dng in t nhit c ln b chi phi bi tha s Boltzmann Thm ch cc in t vi

cc nng lng cao km hn c th xuyn hm qua ro d dng hn so vi

44

7.4. L thuyt WKB

cc in t gn y v do mt nhiu cng sc hn tnh dng ngoi tr cc nhit rt thp v cao.

7.5. Phng php bin phn

Phng php bin phn cung cp mt cch nh gi nng lng i vi trng thi thp nht ca mt h. Mc d iu ny c th xem nh mt nhim v chuyn mn ha, phng php ny l quan trng do chnh xc ca n i vi ng dng ca n cho cc bi ton phc tp v do nhiu phng php s l sn c lm cc tiu mt hm. Ta s chuyn qua mt cch trnh by n gin ca l thuyt chung v sau li p dng n cho mt in t n trong mt h tam gic. N l mt bc chun b cho tnh ton phc tp hn v cc mc nng lng v mt ca mt 2DEG ti mt d chuyn tip cha nhiu in t m mt phng php t hp l cn thit bao hm c phng trnh Schrodinger v phng trnh Poisson.

7.5.1. L thuyt chung

Ta mun tm mc nng lng thp nht ca mt h m n tun theo

45


Nhn 2 v vi v bn tri ri ly tch phn. Khi ,

Nh vy, nng lng ca trng thi ny c th tm c t thng s sau

D nhin, mu s thng l n v do s chun ho. Nguyn l bin phn khng nh rng

trong l bt k hm sng no m n tun theo cc iu kin bin thch hp. Nh vy, ta c th c th cc hm khc nhau lm cho nng lng thp nht c th c v ta bit rng ta khng th n di nng lng thc

Chng minh l n gin. Cho v l cc hm ring thc v tr ring thc ca Hm sng ty c th c khai trin theo di dng

trong l cc h s cha bit. By gi thay n vo biu thc bin phn

46


(7.65). Trc ht, mu s cho

Tch phn l n gin do cc trng thi l trc chun. T s cho

(7.68)

Nh vy, biu thc bin phn tr thnh

Nng lng l nng lng thp nht v do , v thng s tha

mn bt ng thc

47


iu ny hon thnh vic chng minh nguyn l bin phn (7.65).

Phng php l tng qut hn mt cht so vi iu do n c th c dng tm nng lng thp nht vi i xng cho trc ch khng ch trng thp nht chung. V d nh n c th dng tm cc trng thi chn thp nht v cc trng thi l thp nht trong h th i xng bng cch bo m rng cc hm th c i xng chnh xc.

T s ca biu thc (7.69) ch ra mt l do ti sao phng php bin phn l ng n. Cc h s u c bnh phng v do cc sai s bc 1 ca hm sng c trn vo trong cc hm sng khc vi ch dn ti cc sai s bc 2 ca nng lng. Nh vy, mt nh gi hp l i vi nng lng c th c rt ra t mt cch th hm sng kh km nhng phng php bin phn s tt hn nhiu nu hm sng th c la chn tt. Trong thc t c 2 cch tip cn thng c s dng. Mt cch tip cn d on dng ca hm

48


sng nhng vi mt s nh cc tham s c th iu chnh m chng khi c la chn lm cc tiu nng lng. Ta s dng phng php ny trong phn tip theo i vi h tam gic. Mt cch tip cn khc l vit

thnh mt tng theo mt s hu hn N ca cc hm sng

Nguyn l bin phn tr thnh

Lu rng cc hm cn phi khng trc chun. Thng l tt hn khi chn cc hm m cc phn t ma trn ca chng d dng tnh c. Cc phng php s c hiu qu l sn c lm cc tiu cc biu thc nh th v s hm c th c tng ln cho n khi t c chnh xc.

7.5.2. Trng thi lin kt trong mt h tam gic

lm v d m n c th c a ra mt cch gii tch, ta s li nh gi nng lng ca trng thi thp nht trong mt h tam gic. Bc quan trng l la chn mt hm sng thch hp. Ta bit rng n s trit tiu ti x = 0 v

49


gim gn ging nh mt hm m khi Mt la chn n gin tha mn c 2 tiu chun l

N l hm sng Fang Howard trong l thuyt ca 2 DEG khi cc tc gi ny ln u tin p dng n cho lp o trong mt MOSFET silic. Thng s b l cha bit. Khc vi vic d on mt gi tr ti lc xut pht, ta s n t do v thu c mt nh gi i vi nng lng m n s l mt hm ca b. Khi , ta c th tm cc tiu ca hm ny v c c nh gi tt nht c th cho dng ca hm sng ny.

Mu s ca nguyn l bin phn (7.65) i hi tch phn

T s phc tp hn nhng tt c cc tch phn rt gn thnh cc giai tha

50


Vic chia 2 biu thc ny ch ra rng

nh gi tt nht ca l cc tiu ca biu thc bn phi (i vi b > 0) v kt qu tnh ton cho

Tha s trc l 2,4764 khng xa vi kt qu chnh xc l 2,3381 mc d y, phng php WKB dng tt hn. Cc kt qu khi dng cc phng php khc c so snh vi nhau trong Bng 7.1. Mt nh gi chnh xc hn c th thu c vi nhiu thng s hn hoc bng cch chn dng hm tt hn.

7.6. L thuyt nhiu lon suy bin

L thuyt nhiu lon trong phn 7.2 tht bi nu trng thi kho st xy ra s suy bin do cc h s trn cc hm sng khng suy bin (7.32) cha s khc bit nng lng trong mu s ca chng v tr thnh v hn. Cn phi c mt l thuyt nhiu lon khc. S trn khc nhau gi rng nh hng mnh nht ca mt nhiu lon s l lm trn cc trng thi suy bin. Nh vy, ta s

51


bt u bng cch tp trung s ch ti chng v tnh n nh hng nh hn ca cc trng thi khng suy bin sau nu cn. Bng cch gii hn s ch ti cc trng thi suy bin v b qua cc trng thi khc, ta rt gn dng ma trn v hn ca phng trnh Schrodinger v mt ma trn rt nh c cc kch thc c cho bi s trng thi suy bin. Thng n c th c gii mt cch chnh xc.

Mt quan im khc dn ti cng mt kt lun. lm v d v mt h vi cc trng thi suy bin, xt mt ht 2 chiu trong mt hp hay chm lng t hnh vung vi chiu di cnh l a trong mt phng xy. N xoay quanh gc vi mt th v hn bn ngoi. Cc nng lng ca chm khng nhiu lon l

vi Trng thi thp nht l trng thi khng suy bin nhng mc tip theo l suy bin kp vi cc hm sng

52


Mt ca 2 hm sng ny c phc ha trn Hnh 7.8 (a) v (b). Mc d n l mt cch la chn r rng i cc trng thi ring ca nng lng ny, n khng phi l s la chn duy nht. Do cc trng thi l suy bin, bt k s kt hp tuyn tnh no ca chng cng s l mt nghim ca phng trnh Schrodinger vi cng mt nng lng. Mt cp n gin khc c cho bi m mt ca n c v th trn hnh 7.8 (c) v (d). Khng c l do u tin bt k la chn ring no trong chm lng t khng nhiu lon.

Vic t do la chn cc trng thi nh

th trit tiu nu ta thm vo mt nhiu

lon m n ph v i xng vung gc

Hnh 7.8. 2 la chn i vi mt ca cp hm

sng suy bin thp nht trong mt chm lng t

vung gc. (a) v (b) l cc la chn ban u

v trong lc (c) v (d) l cc kt hp

53


theo mt cch thch hp. Xt vi K > 0. N h thp nng lng trong cc mt phn t bn phi pha trn v bn tri pha di ca hnh vung v lm tng nng lng trong hai mt phn t khc. Mt th nh vy c th c sinh ra bi mt th hiu dch dng trn cc cng gn cc gc nh bn phi v y bn tri hoc bi ng sut n hi. Cc sai lch trong ch to dn ti mt chm hnh thang s c nh hng tng t. By gi vn l ta chn cc trng thi no. Cc nng lng ca la chn ban u v

khng thay i bi nhiu lon do chng c phn b nh nhau theo nh v y, bn tri v bn phi. Tuy nhin, cc kt hp tuyn tnh ca chng b nh hng. C th l c tp hp pha trn bn phi v pha di bn tri v do , nng lng ca n gim i do nhiu lon trong lc nng lng ca tng ln. S suy bin b ph v do i xng gim v iu quan trng l la chn cc trng thi tng ng vi i xng ca nhiu lon.

Mc d cu tr li l r rng trong trng hp ny, ta c th xc nhn n bng cch dng i s. Mc ch l gii phng trnh Schrodinger mt

cch chnh xc khi bao hm hm Hamilton ban u v nhiu lon

54


nhng gii hn s ch ti cc trng thi suy bin m y ta ch c 2 trng thi nh vy. t rt gn cc ch s di. Ta to ra mt hm sng di dng m cc h s ca n c th c vit nh l mt vect m n tun theo phng trnh Schrodinger ma trn H Ma trn H cha cc phn t ma trn ca ton t Hamilton y gia 2 trng thi kho st. Phn t u tin ca n l

xut hin do l mt trng thi ring ca v phn t ma trn ca

trit tiu v i xng. Tip theo, ta cn

S hng t trit tiu do A v B l cc trng thi ring v trc giao trong lc phn t ma trn ca bao gm cc tch phn nh (7.35). Hai phn t khc c rt ra mt cch tng t. Phng trnh Schrodinger ma trn (7.6)

tr thnh

55


v iu kin (7.8) i vi cc nghim l

Nh vy, Vect ring tng ng vi l (1,1) nh ta mong mun v tng t, (1, - 1) tng ng vi Chng l cc trng thi c phc tho trn Hnh 7.8(c) v (d). S thay i nng lng l tuyn tnh theo nhiu lon K hoc mc d gi tr k vng trit siu trong c 2 trng thi ban u v ta tm c mt s thay i bnh phng nu cc trng thi khng b suy bin.

Khng c nhiu ng dng trc tip ca l thuyt nhiu lon suy bin cho cc h thp chiu. Tuy nhin, nguyn l chnh c s dng rng ri l ngi ta c th gii hn phng trnh Schrodinger cho mt s nh cc trng thi nm gn nhau v nng lng v gii phng trnh hn ch ny mt cch chnh xc. Thng th cc trng thi nm xa hn v nng lng hon ton b b qua nh ta va tin hnh. i khi cn phi bao hm chng bng cch st nhp cc k thut ca l thuyt nhiu lon suy bin v l thuyt nhiu lon khng suy bin v mt cch tip cn h thng l cch tip cn ca Lowdin.

56


Hai ng dng quan trng ca cc tng ny l cho cu trc vng trong cc cht rn. l cc cu trc vng theo m hnh lin kt cht v m hnh in t gn t do.

7.7. Cu trc vng: Lin kt cht

Ta xem xt ton din nghim ca phng trnh Schrodinger i vi mt h th hu hn n trong phn 4.2 v dng cc ma trn T tm cu trc in t ca mt siu mng trong phn 5.6. Mc d li gii ca siu mng l chnh xc, n khng th c m rng mt cch n gin cho cc th khc v cng khng m rng c cho hn 1 chiu. M hnh lin kt cht l mt bc tranh gn ng ca cu trc vng da trn tng xut pht vi cc mc nng lng ca cc nguyn t v em chng li ngy cng gn nhau to thnh mt tinh th. N b sung cho m hnh in t gn t do m n xut pht quan im ngc li v s c m t trong phn tip theo. Trc ht, ta s gii bi ton ca 2 nguyn t c em li gn nhau v sau l bi ton ca tinh th. Cc nguyn t c th thc s l cc nguyn t nhng ta s nghin cu cc h th trong mt d cu trc.

Ta bt u vi mt th n (hoc nguyn t) xoay quanh gc v chn im

57


7.7.1. Hai h: phn t lng nguyn t

Tip theo xt bi ton ca 2 h nh ch ra trn hnh 7.9. Mt h xung quanh

v h kia xung quanh Ton t Hamilton c th c vit thnh

trong l cc h th bn tri v bn phi. D nhin, n cn c th c gii mt cch gii tch i vi cc h vung gc

Hnh 7.9. 2 h th tng t nh mt

phn t lng nguyn n gin. (a)

Cc hm sng v mc nng lng ca

cc h ring; (b) Cc hm sng v mc

nng lng ca cc trng thi chn v

l ca cc h lin kt m n ch ra s

tch gn bng cc phn t ma

trn m chng tng ti (c) trng tinh

th - c ( ng m nt lu rng

hm sng c bnh phng); (d) s

khng trc giao v (e) s chuyn t.

58


bng cch lm khp tng minh hm sng qua tt c cc bc nhy th hoc bng cch dng cc ma trn T. Tuy nhin, n xem nh mt d on hp l l hm sng ca 2 trng thi thp nht ca h kp s bao gm gn nh ton b mt hn hp ca cc trng thi thp nht trong mi mt trong 2 h ring bit

v Theo l thuyt nhiu lon suy bin, ta s gii hn s ch ti 2 trng thi ny v b qua phn cn li. Cc chi tit khng hon ton ging nh l thuyt nhiu lon suy bin hon ton v v l cc nghim ca cc phng trnh Schrodinger khc nhau nh sau

Cc hm sng v do khng trc giao nh hnh 7.9. Tuy nhin, tng chnh xc l ging nhau. l vit hm sng mong mun nh l

mt tng ca cc hm vi cc h s cn phi tm: trong

n chy qua L v R.

Phng trnh Schrodinger gii hn l

Theo nh thng thng, nhn c 2 v v bn tri vi v ly tch phn

59


c cc ma trn

m cc phn t ca n l

Mt c im mi l s xut hin ca ma trn S. N khng ch l ma trn n v v cc trng thi l khng trc giao. Phng trnh ma trn cn gii by gi l H = E S l mt bi ton tr ring m rng.

Ta cn cc phn t ma trn tip tc. Cc thnh phn ca n c ch ra trn Hnh 7.9 (c) (e). Ta bt u vi c xc nh bi

S hng vi n gin cho khi s dng phng trnh Schrodinger (7.83) cho S hng cn li cho gi tr k vng ca th thm vo i

vi hm sng v c gi l trng tinh th. K hiu n l c lu n l m v cc h th l ht nhau. Phn t cho khc l cng th. 2 s hng khng thuc ng cho cng bng nhau

60


Vic thiu tnh trc giao sinh ra s hng s m n trit tiu theo cch khc. S hng khc l t l s hng quan trng nht. N c gi l tch phn chuyn hay tch phn xuyn hm hay tch phn ph. N cha tch ca 2 hm sng v mt trong cc th v chuyn mt in t t h th ny sang h th khc. Gii thch ny s tr nn r rng hn khi ta rt ra qui tc vng trong chng tip theo. y, n cng l m mc d iu ny ph thuc vo cc hm sng. bn phi ca phng trnh, cc phn t cho ca S l n v do s chun ho hm sng v cc phn t khng thuc ng cho u bng s. Nh vy,

H = S =

Cc nng lng c cho bi phng trnh th k

N c cc nghim

61


Php gn ng b qua tt c cc trng thi cao hn s ch tt nu s ph gia cc h l yu. Trong trng hp ny, h s khng trc giao s 0) l m n l s kt hp chn

tng ng vi s kt hp l Trong phn t, chng l cc qu o phn t lin kt v phn lin kt nh ch ra trn hnh 7.9(b). Trng thi vi nng lng cao hn c mt nt trong hm sng gia cc nguyn t m n lm tng ng nng ca n.

S tch lun lun xy ra nhng s xuyn hm l yu. Mt trng thi lng t c th lun lun lm gim nng lng ca n bng cch m rng qua mt vng khng gian ln hn hoc ta ni rng bng cch s dng 2 h khc vi mt h. Nu mt in t lc u mt trong cc h, n s dao ng gia

chng vi mt tn s gc l (phn 1.5). Chu k dao

62


ng s thay i rt di nu t l nh v kh ni rng cc h b lin kt.

7.7.2. Dy h: cht rn lin kt cht

Ta xt bi ton 2 h. N ch l mt bc nh gii bi ton ca mt s v hn cc h c sp xp to thnh mt siu mng hoc tinh th mt chiu. Vic dn xp mt cp h lm cho mc nng lng b ch ra thnh 2 mc tch ra bi 2t. S ch ra tng khi s tch gim v s ph t tng. S dn xp N h lm cho mc nng lng chung ca chng b chia tch thnh N gi tr v chng st nhp thnh mt vng lin tc khi N l m hnh lin kt cht ca mt cht rn.

Ton t Hamilton l

trong l th ca h (hoc ion) n. Obitan gn vi h n tun theo phng trnh Cc obitan l nh nhau ngoi tr v tr ca chng v do , trong l v tr ca h n. Ta vit li hm sng ca tinh th thnh mt tng khi b qua cc ng gp t cc obitan khc. R rng l ta s kt thc vi cng mt phng trnh l ging nh i vi 2 h ngoi tr iu l hng ca ma trn l

63


v cng khc vi 2. N l s dch chuyn v thc ra l ta bit cc hm sng t nh l Bloch (2.2) m theo , cc h s n gin l cc h s pha. Nh vy,

v dng m ca phng trnh tr thnh

N cn phi l nh nhau i vi tng v tr v do cui cng m cn b i.

Cc phn t ma trn ca ton t Hamilton l

S hng l = n trong tng theo cc th c tch ra sao cho ta c th dng

Cc phn t ma trn cn li l phc tp hn so vi cc phn t ma trn i vi cp h do chng c th bao hm 3 v tr l m v n i

vi 2 obitan v l i vi h. Phn ln chng l rt nh v c th b qua. Ta s

64


thc hin php gn dng n gin nht v gi thit rng 3 ch s cn phi c gii hn ti ch 2 v tr st lin nhau m n dn ti cng cc phn t ma trn nh i vi cp h. Phn t cho tr thnh

y, c li l trng tinh th. l gi tr k vng ca th t h trong

obitan trn v tr m. Cc phn t khng thuc ng cho cn st li l

Xt H l ch c th l m hoc m + 1 nhng l = m + 1 cng

c rt ra khi tng cho N ch li l = m m n cho

s li l tch phn khng trc giao i vi cc hm sng trn cc v tr st lin nhau v t l tch phn chuyn.

By gi ta c th hon thnh cc tng i hi trong (7.94). V tri tr thnh

65


Tng t, tng v phi cho

V tr m by gi c th b i c 2 v nh khng nh t trc. Khi , nng lng c cho bi

Cc s hng khng trc giao v trng tinh th thng b i nh trong biu thc cui cng. By gi ta rt ra php gn ng cosin i vi mt vng hp (t nh) m ta cng dng n mt vi ln v d trong phn 2.2. Tch phn chuyn cho vng c rng 4t. Vng ln ngc nu t < 0 m n l trng hp i vi cc trng thi trong h vi n = 2 khc vi n = 1.

Phng php c th d dng c m rng cho 2 hoc 3 chiu v cc h s pha trong (7.98) tr thnh c ly tng theo cc ln cn gn nht V d nh mt mng vung gc 2 chiu c

B rng vng l 4dt trong d chiu. Ta cng c th vt ra ngoi cc ln cn

66


gn nht a cc thnh phn Fourier cao hn vo trong

M hnh lin kt cht chng t rng b rng ca cc vng ph thuc vo cng xuyn hm t h (hoc nguyn t) ny sang h khc. Cc in t lin kt cht vi cc nguyn t ca chng gy ra cc vng hp trong khi cc in t lin kt lng gy ra cc vng rng. Bc tranh nh lng m ta ph v n nu cc nguyn t qu gn nhau do cc vng m rng nhiu n mc chng xen ph ln nhau v cc vng khng th c xem xt mt cch tch ri. Trong thc t, bc tranh lin kt cht l km thm ch i vi cc vng ho tr pha trn ca cc cht bn dn tr khi cc ln cn gn nht v xa hn na c bao hm. Tuy nhin, n thng c s dng nh mt cch tham s ho cc vng. Ngi ta c th biu din cc vng lin kt cht nh mt hm ca cc tch phn xuyn hm gia cc ln cn gn nht, nhng ln cn gn nht tip theo v c th xa hn na v dng cc tch phn ny nh l cc tham s c th iu chnh c lm khp cc vng vi cc kt qu ca mt tnh ton hon chnh hn hoc vi thc nghim. N cho mt dng phim hm n gin i vi m n khi c th dng tnh cc i lng khc nh phn ng quang.

67

7.8. Cu trc vng: Cc in t gn t do

M hnh lin kt cht ch ra cu trc vng pht trin nh th no khi cc nguyn t t xa nhau c mang li gn nhau to thnh mt tinh th dn cch cht. M hnh in t gn t do nh tn gi ca n ng theo quan im ngc li: ta bt u vi cc in t t do, thm vo mt th tun hon yu v xem lm th no iu ny dn ti s to thnh cc vng v cc khe nng lng. N gn vi vic m t cu trc vng trong Chng 2 v cch tip cn in t gn t do l mt c s nh lng tt hn m t cu trc vng ca cc cht bn dn thng thng.

Xt mt tinh th 1 chiu c chu k a v chiu di L m v sau ta s cho n bng v cng. H khng nhiu lon ca cc in t t do c nng lng

v cc hm sng By gi

thm vo mt th tun hon ca tinh th nh l mt nhiu lon. Tnh tun hon ca th trong tinh th c ngha l n c th c khai trin nh mt chui Fourier

trong l cc vect mng o. Th V(x) l thc m n ng

68


Khi th c gi thit l yu, ta c th nh gi nh hng ca n ln cc nng lng khi dng l thuyt nhiu lon thng thng phn 7.2. Nh vy,

Cc phn t ma trn l

Tch phn trit tiu tr khi s sng tng cng bng 0 m n i hi

v trong trng hp ny, tch phn loi b h s 1/ L. Cc trng thi bao hm c minh ha trn hnh 7.10 (a). Mt trng hp ring l m n ch

ra rng i vi mi k. By gi chnh l th trung bnh v dch chuyn u tt c cc trng thi v do , ta s b n i. Khai trin nhiu lon ca chng ta tr thnh

69


Hnh 7.10. Cc trng thi trn vo trong

k bi th tun hon ca mt tinh th 1

chiu. L thuyt nhiu lon khng suy

bin l tt trong (a) m tt c cc

trng thi c cc nng lng khc nhau

nhng tht bi trong (b) m mt

trong cc trng thi tr thnh suy bin

v dn ti s to thnh khe vng.

v hm sng tng ng l

Hm bn trong cc ngoc vung l mt chui Fourier ging nh trong (7.103) v do tun hon. V th, ta chng minh nh l Bloch dng (2.2) m

n pht biu rng hm sng trong mt tinh th c th c vit di dng

mt sng phng nhn vi mt hm vi chu k ca mng. Hm sng cng tho

70


mn kim tra r rng l n tr li mt sng phng n gin trong mt mng rng trong V(x) = 0.

Ta lun lun cn phi kim tra cc mu s nng lng khi dng l thuyt nhiu lon. Khng dng c l thuyt nhiu lon khi 2 nng lng tr nn bng nhau , ngha l S suy bin duy nht trong trng hp 1 chiu l v do , mu s trit tiu khi

hay

L thuyt nhiu lon b ph v ti nhng im ny trong khng gian k mc d th tun hon yu (Hnh 7.10(b)). Tht bi l do i xng mng nh tho lun trong phn 2.1.1.

Cc gi tr ny ca k r rng l ng quan tm v l thuyt tht bi ti . Xt nh hng ca mt thnh phn Fourier ring Do tht bi l kt qu ca s suy bin gia cc trng thi vi cc s sng k v mt cch tip cn r rng l gii phng trnh Schrodinger chnh xc i vi ring 2 trng thi ny (nghin cu cc trng thi khc vi l thuyt nhiu lon nu mun).

71


iu ny c gi tr i vi k gn vi Nh vy,

v cc thnh phn Fourier c lin quan ca th ch gm v

Phng trnh Schrodinger tr thnh ma trn 2 x 2

Cc tr ring ca n tha mn

T suy ra

N c v th trn hnh 7.11. i vi k xa trong

n tr thnh

72


Hnh 7.11. Hnh nh m rng ca E(k) gn mt khe nng lng (ng

mnh) ti theo l thuyt in t gn t do. ng m

nt l nng lng ca in t t do.

hoc cng nh th i vi k v thay i ln nhau. l hiu chnh chnh t khai trin nhiu lon thng thng (7.106). Gn ti

v

c bit l v do ta thy rng mt khe c b rng m ra ti

l kt qu quan trng nht ca l thuyt in t gn t do. N chng t rng cc khe vng ti mt na ca cc vect mng o trong cc trng

thi k v tr thnh suy bin v ch ra rng b rng khe l gp i thnh phn Fourier

73


tng ng ca th mng. Phng php c th c m rng cho 2 hoc 3 chiu vi cc kt qu m t trong phn 2.4. Mt ln na cc khe vng nh vo thnh phn Fourier xy ra s suy bin m n by gi xc nh mt mt phng vung gc vi vect ni vi gc v chia i n. Hn 2 sng c th tr nn suy bin v s c gi khi tnh khe trong cc gc ca min Brillouin.

Gi s rng ta mun p dng l thuyt in t gn t do cho mt cht bn dn nh silic. C 4 in t ha tr v do chng nhn thy mt li ion c in tch + 4. Th Coulomb t li ny gn bng 10 eV trong n v v khng b che chn bi h s thng thng do n l th c nhn thy bi chnh cc in t ha tr. R rng l th tun hon l ln v do l thuyt in t gn t do l v dng. Ho ra th thc c th c thay bng mt gi th yu hn nhiu m n c chn sao cho n lm tn x cc in t ho tr chnh xc theo cng mt cch. Khi , cu trc vng c th c xc nh khi dng cc gi th yu hn ny. L thuyt l tinh t v cc gi th c th c tnh nh ab initio. Trong phng php gi th kinh nghim n gin hn, mt s

nh cc thnh phn Fourier c iu chnh sao cho mt t c tnh ti hn

74


ca cu trc vng thu c ph hp vi thc nghim. Ch 3 thnh phn Fourier cn c s ph hp tt i vi Si m n ch ra rng php gn ng in t gn t do vi cc gi th l ph hp ng k i vi cc cht bn dn thng thng.

Bi tp chng 7

7.1. Mt in t trng thi thp nht ca h lng t GaAs nh trn hnh 7.12 (a). B rng tng cng l 15 nm v on chnh gia rng 5 nm su hn100eV

Hnh 7.12. Cc h lng t to bc vi (a) mt h su

hn v (b) mt ro.

so vi phn cn li. nh gi nng lng ca trng thi thp nht vi gi thit rng h su v hn. Lm th no th ca h c chia ra gia v

v n c mt nh hng quan trng hay khng? Lm th no nghin cu mt h tng t vi mt ro khc vi mt h su hn gia (hnh 7.12(b)) v khi no th vic dng l thuyt nhiu lon l thch hp?

75

Bi tp chng 7

7.2. Chng minh rng h s phn cc ca trng thi thp nht trong mt h lng t b lm nghing bi mt in trng (phng trnh (7.38)) cng c th tm c bng cch xem xt s thay i ca cc hm sng v mmen lng cc e(x) thu c.

7.3. Mc d phn ln cc h lng t c mt y phng, cc bin dng khc cng c th c pht trin. Ba v d c ch ra trn hnh 7.13. Hiu ng giam cm lng t Stark thay i nh th no trong cc h ny? c bit l s thay i nng lng c cn l bnh phng ca cng trng hay khng?

7.4. iu g xy ra i vi nng lng ca trng thi th hai trong mt h su v hn khi n chu tc dng ca mt in trng?

Hnh 7.13. Cc bin dng bin dng

quan st thy hiu ng giam cm lng

t Stark. (a) Trng gn lin; (b) khe

vng thon nh dn; (c) h parabol.

76

Bi tp chng 7

7.5. Mt bi ton trong Chng 4 lin quan ti cc mc nng lng trong mt th parabol pht trin vo trong nh gi sai s ca gi thit l th parabol tip tc ln pha trn mi mi khc vi chuyn thnh mt on bng trong cc ro bn ngoi h.

7.6. Xc nh s thay i nng lng ca trng thi thp nht trong mt h lng t su v hn nh vo mt t trng nh (hnh 6.13(a)). Ly vect sng ngang bng khng.

7.7. Xc nh nh hng ca mt in trng trong mt phng xy ln mc nng lng thp nht ca mt chm lng t c l tng ho nh mt h 2 chiu su v cng Kt qu c ph thuc vo hng ca trng trong mt phng hay khng?

7.8. Chng minh rng phng php WKB cho cc gi tr chnh xc ca cc nng lng ca cc trng thi lin kt trong mt th parabol.

7.9. Dng phng php WKB nh gi nng lng ca trng thi thp nht trong mt h th tam gic i xng vi Kt qu chnh xc c th tm c khi dng cc hm Airy.

77

Bi tp chng 7

7.10. Mt cng tch ri trn mt d cu trc GaAs sinh ra mt th parabol trong mt 2DEG m cc mc nng lng ca n c tch ra bi meV khi

c mt mt t in t. Khi nhiu in t hn i vo, parabol bin thnh bn tm vi mt vng phng c b rng W = 50 nm (hnh 7.5). Gi thit rng cc phn parabol gi cng cong. Dng phng php WKB chng minh rng cc mc nng lng mi c cho bi cc nghim ca

trong

7.11. Cn gi tr bng bao nhiu ca i vi dng xuyn hm chi phi dng in t nhit trong mt ro Schottky trn GaAs nhit phng? Ch xt cc s hng hm m. S phn bit l quan trng v mt ro b chi phi bi s xuyn hm c c tnh I(V) gn thun tr trong khi dng in t nhit cho mt it.

7.12. Dng phng php WKB nh gi tc thot ra khi mt h lng t trong mt in trng nh vo s xuyn hm Fowler Nordheim (phn

78

Bi tp chng 7

7.2.3). C th tc dng mt trng ln nh th no trc khi s thot ca cc in t hoc cc l trng bi s xuyn hm gii hn thi gian sng ca mt exciton (gn 1 ps t tn x phonon) ? nh gi ny c thch hp vi thc nghim t hnh 7.3 hay khng?

7.13. nh gi h s truyn qua qua mt ro parabol i

vi E < 0 khi dng phng php WKB. So snh php gn ng ny vi kt qu chnh xc

7.14. Chng minh tnh ton xuyn hm Zener trong phn 2.2. Phng trnh (2.16) c rt ra vi gi thit rng mt ro hnh ch nht gia cc vng. Thc t hn l th tuyn tnh t in trng cho gn bin vng ly ti x = 0. Khi , cho mt ro i xng m n quay v khng ti pha xa ca khe trong khng gian thc Dng

phng php WKB nh gi tc xuyn hm qua ro ny.

7.15. nh gi nng lng trng thi thp nht trong mt h th tam gic i xng vi khi dng phng php bin phn. Mt hm Gauss

l mt d on r rng i vi hm sng nhng c

79

Bi tp chng 7

th th cc hm khc chng hn nh

7.16. Nhiu lon tc dng vo chm lng t vung gc trong phn 7.6 m t mt im yn nga vi mt nh hng ring. Cc kt qu ph thuc nh th no vo s nh hng ca im yn ng? V mt nh tnh, ci g l nh hng ca cng nhiu lon ln cc trng thi tng t trong mt chm lng t trn?

7.17. M rng tnh ton ca phn 7.7.1 cho 2 h c su khc nhau vi cc tr ring khi c lp. B qua cc s hng khng trc giao v trng tinh th. Cc mc ca cc h c lp giao nhau khi i du nhng chng minh rng cc h lin kt khng ct nhau nh c phc ha trn hnh 7.14. Tuy nhin, lin kt l yu.

Hnh 7.14. S khng ct nhau ca 2 h

c lin kt bi t (nhng ng mnh)

khi su tng i ca chng

thay i . Cc ng m nt ch ra cc

tr ring ca cc h khng lin kt.

80

Bi tp chng 7

7.18. Chng minh rng cc kt qu ca tnh ton trc ph hp vi l thuyt nhiu lon khng suy bin khi

7.19. Trong mt siu mng ca InAs v GaSb, nh vng ha tr trong GaSb nm trn y vng dn trong InAs bi meV (hnh 3.5). Xt nh hng ca n ln cu trc vng i vi k trong mt phng ca cc h (vung gc vi hng nui). Nu cc h khng lin kt, ta c

trong khe biu kin

l m v do , cc vng xen ph ln nhau. nh gi nh hng ca lin kt gia cc vng vi gi thit rng n c th c lp m hnh bi mt s hng khng i bng cch vit ra v gii mt ma trn ton t Hamilton n gin. Phc ha cc vng v chng minh rng s khng ct nhau hi phc mt khe nng lng dng. Nng lng im khng trong cc h lng t thc ra c thm vo v

7.20. Xt mt h rng 5 nm v su 0,3 eV trong GaAs. Trng thi thp nht

trong h ny c nng lng lin kt l 0,21 eV v do bn

ngoi h. nh gi s tch khi 2 h nh th c tch ra bi mt ro 5 nm

81

Bi tp chng 7

(ly khi lng hiu dng l 0,067 mi ni). Khng nhm mc ch t chnh xc ln nhng thc hin cc php gn ng mnh c mt biu thc n gin i vi t m n cho bit bc ln ca s tch.

7.21. M rng bi tp trc nh gi v tr v b rng ca vng thp nht trong mt siu mng ca cc h 5 nm v cc ro 5 nm c cao 0,3 eV xen k nhau trong GaAs. Mt tnh s t cc bin ca cc vng ti 0,215 eV v 0,205 eV o t cc nh ca cc ro.

7.22. Tnh khi lng hiu dng gn mt khe vng t (7.113). Khi lng v khe c lin quan vi nhau nh th no v cc kt qu ny c ph hp vi cc d on ca phng php hay khng?

7.23. M hnh in t gn t do d on tt nh th no v cu trc vng ca m hnh Kronig Penney v th trn hnh 5.18.

7.24. nh gi b rng khe gn gc ca min Brillouin th I i vi mt tinh th 2 chiu hnh vung. So snh n nh th no vi khe ti im gia ca mt mt?

82

p

k

r

r

-

-

)

1

.

7

.(

)

(

)

(

*

mn

n

m

dx

x

x

d

f

f

=

*

m

f

=

n

n

m

n

n

n

m

n

a

E

H

a

)

4

.

7

.(

*

*

f

f

f

f

)

n

f

y

H

)

,

y

y

E

H

=

)

)

3

.

7

(

.

=

=

n

n

n

n

n

n

n

n

n

a

E

H

a

a

H

f

f

f

)

)

(

)

.

exp

)

(

ikx

x

k

=

f

(

)

(

)

.

/

sin

/

2

)

(

2

/

1

a

x

n

a

x

n

p

f

=

)

2

.

7

.(

n

n

n

a

f

y

=

=

n

m

n

mn

Ea

a

H

)

6

.

7

.(

H

)

)

5

.

7

(

*

n

m

mn

H

n

H

m

H

f

f

)

)

=

=

.

m

Ea

m

n

=

.

mn

d

a

r

a

r

)

7

.

7

(

,

a

E

a

r

r

=

).

(

x

n

f

:

)

(

x

n

f

n

n

n

H

m

)

)

(

x

y

.

n

E

)

(

x

n

f

.

H

)

n

f

=

=

=

)

9

.

7

.(

*

*

mn

n

n

n

m

n

m

mn

H

H

d

e

f

e

f

f

f

)

.

n

n

n

H

f

e

f

=

)

H

)

,

mn

H

H

)

.

*

nm

mn

M

M

=

+

+

V

)

0

H

)

H

)

x

i

-

/

h

(

)

(

)

2

2

2

/

2

/

x

m

-

h

(

)

ikx

exp

.

0

H

)

(

)

.

2

/

1

2

0

F

a

e

-

.

0

F

p

a

e

=

a

x

e

p

.

-

=

x

a

1

=

l

)

10

.

7

(

0

n

n

n

H

f

e

f

=

)

l

V

H

H

)

)

)

l

+

=

0

V

)

0

H

)

.

0

V

H

H

)

)

)

+

=

H

)

.

V

)

(

)

)

11

.

7

.(

0

n

n

n

n

E

V

H

H

y

y

y

=

+

=

)

)

)

l

(

)

(

)

(

)

(

)

)

14

.

7

.(

...

...

...

)

1

(

)

0

(

)

1

(

())

)

1

(

)

0

(

0

+

+

+

+

=

+

+

+

n

n

n

n

n

n

E

E

V

H

ly

y

l

ly

y

l

)

)

)

13

.

7

....(

)

12

.

7

(

...,

)

2

(

2

)

1

(

)

0

(

)

2

(

2

)

1

(

)

0

(

+

+

+

=

+

+

+

=

n

n

n

n

n

n

n

n

E

E

E

E

y

l

ly

y

y

l

l

)

18

.

7

.(

,

())

())

n

n

n

n

E

e

f

y

=

=

)

17

.

7

.(

)

16

.

7

(

,

)

15

.

7

(

,

)

2

(

)

0

(

)

1

(

)

1

(

)

0

(

)

2

(

)

2

(

0

)

1

(

)

1

(

)

0

(

)

0

(

)

1

(

)

1

(

0

)

0

(

)

0

(

)

0

(

)

0

(

0

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

E

E

E

H

V

E

E

H

V

E

H

y

y

y

y

y

y

y

y

y

y

y

+

+

=

+

+

=

+

=

)

)

)

)

)

0

H

)

.

0

k

k

k

H

f

e

f

=

)

k

f

0

H

)

n

k

e

e

-

*

n

f

V

)

n

f

)

1

(

n

E

n

f

(

)

-

=

-

k

n

n

n

k

n

k

nk

V

E

a

)

22

.

7

.(

)

1

(

)

1

(

f

f

f

e

e

)

.

)

1

(

n

y

(

)

(

)

)

21

.

7

.(

)

1

(

)

1

(

0

n

n

k

nk

n

k

V

E

a

H

f

f

e

)

)

-

=

-

n

f

)

1

(

n

y

(

)

(

)

)

19

.

7

.(

())

)

1

(

0

n

n

n

n

V

E

H

f

y

e

)

)

-

=

-

)

1

(

n

E

(

)

(

)

)

26

.

7

.(

)

2

(

)

1

(

)

2

(

0

n

k

n

k

nk

nn

n

n

E

a

V

V

H

f

f

y

e

+

-

=

-

)

)

)

1

(

nn

a

-

+

=

n

k

k

k

k

n

kn

n

nm

n

V

a

,

)

1

(

)

1

(

)

25

.

7

.(

f

e

e

f

y

.

0

)

1

(

=

nn

a

(

)

.

/

)

1

(

m

n

mn

nm

V

a

e

e

-

=

n

m

*

m

f

.

n

f

V

)

=

)

23

.

7

.(

*

)

1

(

nn

n

n

n

V

V

E

f

f

)

(

)

-

-

=

-

)

24

.

7

(

*

)

1

(

mn

n

m

n

m

nm

V

V

a

f

f

e

e

)

.

mn

V

2

nk

V

*

kn

nk

V

V

=

)

30

.

7

.(

,

,

)

1

(

)

1

(

-

=

=

n

k

k

n

k

k

k

n

kn

nk

nk

nk

n

V

V

V

a

E

e

e

)

1

(

nk

a

.

2

kn

V

)

1

(

nn

a

-

=

-

=

k

nn

nn

nk

nk

nn

nn

k

n

k

nk

n

a

V

V

a

a

V

V

a

E

)

29

.

7

.(

)

1

(

)

1

(

)

1

(

*

)

1

(

)

2

(

f

f

)

*

n

f

k

e

0

H

)

(

)

)

28

.

7

.(

)

(

)

2

(

)

1

(

)

2

(

0

n

n

k

k

nk

k

nn

k

nk

n

E

a

V

V

a

H

f

f

f

e

+

-

=

-

)

)

)

27

.

7

(

)

2

(

)

2

(

=

k

k

nk

n

a

f

y

)

32

.

7

....(

)

31

.

7

(

...,

,

,

2

+

-

+

=

+

-

+

+

=

k

n

k

k

k

n

kn

n

n

n

k

k

k

n

kn

nn

n

n

V

V

V

E

f

e

e

f

y

e

e

e

k

n

e

e

-

nn

V

n

m

f

f

=

)

33

.

7

.(

)

(

)

(

*

dx

x

x

x

eF

V

n

m

mn

f

f

eFx

x

V

=

)

(

kn

V

=

-

-

=

D

1

1

2

2

1

,

2

1

)

34

.

7

(

,

k

k

k

V

E

e

e

1

k

V

kk

V

.

eFx

V

=

)

1

k

V

n

f

(

)

(

)

2

2

/

2

/

a

n

m

n

p

e

h

=

.

4

/

21

eFa

V

2

(

)

)

36

.

7

.(

243

256

1

2

4

1

e

p

eFa

E

>

D

-

1

1

2

3

e

e

e

=

-

(

)

(

)

-

=

=

2

/

2

/

2

21

)

35

.

7

.(

9

16

cos

2

sin

2

a

a

eFa

dx

a

x

eFx

a

x

a

V

p

p

p

a

B

a

)

38

.

7

.(

243

4096

243

512

4

5

1

2

0

4

2

B

b

a

a

e

p

e

a

e

e

p

a

=

=

(

)

.

2

/

1

2

0

1

F

E

B

a

e

e

-

=

D

a

b

e

.

a

.

/

1

3

k

2

/

1

k

(

)

.

010829

,

0

48

/

15

2

2

-

p

p

4

p

.

QCSE

E

QW

E

)

39

.

7

(

...,

2

/

)

(

4

3

2

+

+

+

=

Cx

Bx

Kx

x

V

.

1

-

MVm

As

Ga

Al

68

,

0

32

,

0

(

)

)

40

.

7

.(

4

exp

2

/

1

3

0

-

h

eF

mV

T

.

2

/

)

2

/(

0

2

2

V

m

k

h

k

0

V

(

)

,

/

0

eF

V

L

k

(

)

,

2

exp

L

k

-

.

G

.

.

p

k

r

r

p

k

r

r

.

K

r

h

p

)

r

0

m

(

)

(

)

(

)

R

K

i

R

u

R

K

n

K

n

r

r

r

r

r

r

.

exp

=

f

(

)

(

)

(

)

(

)

)

42

.

7

(

,

2

.

2

0

2

2

0

0

2

R

u

K

R

u

m

K

p

K

m

R

V

m

p

K

n

n

K

n

per

r

r

r

h

)

r

r

h

r

)

r

r

r

e

=

+

+

+

-

=

h

)

r

i

p

(

)

(

)

(

)

(

)

)

41

.

7

(

,

2

0

2

R

K

R

R

V

m

p

K

n

n

K

n

per

r

r

r

r

)

r

r

r

f

e

f

=

+

K

r

(

)

R

u

K

n

r

r

(

)

(

)

(

)

,

.

exp

R

K

i

R

u

R

K

n

K

n

r

r

r

r

r

r

=

f

(

)

R

u

K

n

r

r

(

)

.

R

K

n

r

r

f

0

.

0

r

)

r

r

r

n

p

K

n

.

0

r

r

=

K

p

)

r

(

)

+

-

=

x

i

k

p

K

x

/

.

h

)

r

r

K

r

p

K

)

r

r

.

.

K

r

(

)

.

0

r

n

e

0

r

r

=

K

(

)

R

u

n

r

r

0

.

0

r

r

=

K

G

(

)

.

2

/

0

2

2

e

m

m

K

h

(

)

+

C

C

E

K

r

e

G

(

)

(

)

(

)

(

)

)

43

.

7

.(

0

0

0

.

0

2

0

,

2

0

2

0

2

2

-

+

+

n

m

m

m

n

n

n

n

p

K

m

m

m

K

K

r

r

r

)

r

r

r

h

h

r

r

e

e

e

e

p

)

r

G

K

r

Y

p

S

x

)

.

/

2

2

2

0

h

P

m

=

P

E

p

)

r

.

P

E

g

E

,

/

1

/

1

g

P

e

E

E

m

+

(

)

(

)

)

44

.

7

.(

2

1

2

/

2

2

2

0

0

2

2

2

0

2

0

2

0

2

2

+

+

=

-

+

+

g

C

V

C

C

C

E

P

m

m

K

E

E

E

P

im

K

m

m

K

E

K

h

h

h

h

h

r

e

.

x

p

(

)

P

im

X

p

S

x

h

)

/

0

=

a

k

x

/

p

=

(

)

0

2

2

2

/

m

K

h

.

061

,

0

e

m

g

E

22

P

E

5

,

0

a

.

/

2

a

p

K

r

.

G

p

k

r

r

.

G

[

]

.

/

)

(

2

)

(

h

x

V

E

m

x

k

-

=

.

)

(

E

x

V

=

R

x

L

x

dx

x

k

)

(

)

(

x

V

(

)

.

/

2

h

V

E

m

k

-

=

(

)

ikx

exp

)

(

x

V

E

-

)

(

'

'

x

c

).

(

x

V

)

(

x

k

[

]

[

]

)

46

.

7

(

),

(

)

(

2

)

(

)

(

2

2

'

'

2

'

x

k

x

V

E

m

x

i

x

-

=

-

h

c

c

(

)

x

c

[

]

.

)

(

exp

)

(

x

i

x

c

y

=

)

(

x

V

)

45

.

7

(

),

(

)

(

)

(

2

2

2

2

x

E

x

x

V

dx

d

m

y

y

=

+

-

h

(

)

=

)

47

.

7

(

,

)

(

'

'

dx

x

k

x

c

c

[

]

)

49

.

7

(

),

(

)

(

)

(

)

(

)

(

'

2

'

'

2

2

'

x

ik

x

k

x

i

x

k

x

+

=

c

c

)

(

x

V

k

/

2

p

)

48

.

7

.(

1

,

2

k

dx

dk

k

k

hocnq-VẬT LÝ BÁN DẪN THẤP CHIỀU-C7.pptx

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Transcript of hocnq-VẬT LÝ BÁN DẪN THẤP CHIỀU-C7.pptx