hocnq-VẬT LÝ BÁN DẪN THẤP CHIỀU-C1.pptx

vt l bn dn thp chiu

VT L BN DN THP CHIUBi ging Cao hc Vt l l thuytTrng i hc S phm H Ni

PGS.TS Nguyn Quang Hc

TI LIU THAM KHOJohn H. Davies, The physics of low-dimensional semiconductors: An introduction, Cambridge university press, 1998.Nguyn Vn Hng, L thuyt cht rn, NXBHQGHN , 1999Nguyn Quang Bu, Nguyn V Nhn v Phm Vn Bn, Vt l bn dn thp chiu, NXBHQGHN, 2007

MC LC M u Chng 1: C s c hc lng t 1.1. C hc sng v phng trnh Schrodinger 1.2. Ht t do 1.3. Ht lin kt: H lng t 1.4. Mt in tch v mt dng 1.5. Ton t v php o 1.6. Tnh cht ton hc ca trng thi ring 1.7. Trng thi m 1.8. Trng thi lp y: Hm lp y Bi tp Chng 2: C s vt l cht rn: in t v phonon trong tinh th 2.1. Cu trc vng mt chiu 2.2. Chuyn ng ca in t trong cc vng

MC LC 2.3. Mt trng thi 2.4. Cu trc vng hai v ba chiu 2.5. Cu trc tinh th ca bn dn thng thng 2.6. Cu trc vng ca bn dn thng thng 2.7. Php o quang ca khe vng 2.8. Phonon Bi tp Chng 3: D cu trc 3.1. Tnh cht chung ca d cu trc 3.2. Phng php nui d cu trc 3.3. K thut vng 3.4. Cu trc xp thnh tng lp: H lng t v ro 3.5. D cu trc pha tp 3.6. Lp bin dng 3.7. D cu trc Si - Ge

MC LC 3.8. Dy v chm 3.9. S giam cm quang 3.10. Php gn ng khi lng hiu dng 3.11. L thuyt khi lng hiu dng trong d cu trc Bi tp Chng 4: H lng t v h thp chiu 4.1. H vung gc su v hn 4.2. H vung gc su hu hn 4.3. H parabol 4.4. H tam gic 4.5. H thp chiu 4.6. S lp y vng con 4.7. H hai v ba chiu 4.8. S giam cm vt ra khi hai chiu 4.9. H lng t trong d cu trc Bi tp

MC LC Chng 5: Vn chuyn xuyn hm 5.1. Bc th 5.2. Ma trn T 5.3. Ma trn T (tip) 5.4. Dng v dn 5.5. Xuyn hm cng hng 5.6. Siu mng v vng mini 5.7. Vn chuyn lin kt vi nhiu knh 5.8. Xuyn hm trong d cu trc Bi tp Chng 6: in trng v t trng 6.1. Phng trnh Schrodinger i vi in trng v t trng 6.2. in trng u

MC LC 6.3. Tenx dn v in tr sut 6.4. T trng u 6.5. T trng trong knh hp 6.6. Hiu ng Hall lng t Bi tp Chng 7: Cc phng php gn ng 7.1. Hnh thc lun ma trn ca c hc lng t 7.2. L thuyt nhiu lon khng ph thuc vo thi gian 7.3. L thuyt k.p 7.4. L thuyt WKB 7.5. Phng php bin phn 7.6. L thuyt nhiu lon suy bin 7.7. L thuyt lin kt cht

MC LC 7.8. L thuyt in t gn t do Bi tp Chng 8: Tc tn x: qui tc vng 8.1. Qui tc vng i veil th tnh 8.2. Tn x tp 8.3. Qui tc vng i vi th dao ng 8.4. Tn x phonon 8.5. S hp th quang 8.6. S hp th gia cc vng 8.7. S hp th trong h lng t 8.8. Gin v nng lng ring Bi tp Chng 9: Kh in t hai chiu

MC LC 9.1. Gin vng ca lp bin iu pha tp 9.2. Cc mu khc vi mu n gin nht 9.3. Cu trc in t ca kh in t hai chiu 9.4. S chn bi kh in t 9.5. S tn x bi tp t xa 9.6. C ch tn x khc Bi tp Chng 10: Tnh cht quang ca h lng t 10.1. L thuyt chung 10.2. Mu Kane v cu trc vng ha tr 10.3. Vng trong h lng t 10.4. S chuyn gia cc vng trong h lng t 10.5. S chuyn gia cc vng con trong h lng t

MC LC 10.6. S khuch i quang v laze 10.7. Exciton Bi tp A1 Bng hng s vt l A2 Tnh cht ca cc bn dn quan trng A3 Tnh cht ca hp kim GaAs-AlAs nhit phng A4 Phng trnh Hermite: Dao ng t iu ho A5 Hm Airy: H tam gic A6 H thc Kramers-Kronig v hm phn ng A6.1. H thc Kramers-Kronig A6.2. Hm phn ng mu

M UCc h thp chiu (low-dimensional system) to ra mt cuc cch mng trong vt l bn dn. Chng da trn c s cng ngh ca cc d cu trc (heterostructure) trong thnh phn ca mt cht bn dn c th thay i trn phm vi nanmt. Chng hn mt lp GaAs b kp gia hai lp AlGaAs tc ng ging nh mt h lng t c bn. Cc mc nng lng c tch rng nu h hp v tt c cc in t c th b by trn mc thp nht. Chuyn ng song song vi cc lp khng b nh hng v cc in t chuyn ng t do theo cc hng ny. Kt qu l mt cht kh in t 2 chiu v cc l trng c th b by theo cng mt cch.Cc thc nghim quang a ra chng c trc tip v dng iu thp chiu ca cc in t v l trng trong mt h lng t. Mt trng thi thay i t dng parabol trn trong h 3 chiu sang dng bc thang (staircase) trong h 2 chiu. iu ny c th hin r rng trong s hp th quang v bc y ca mt trng thi tng cng cc tnh cht quang. iu c ng dng trong cc laze h lng t c dng ngng thp hn dng ngng ca mt thit b 3 chiu.

M UVo cui th k 20, vt l cht rn chuyn hng nghin cu t tinh th khi sang mng mng v cu trc nhiu lp vi nhiu tnh cht vt l mi trong c hiu ng kch thc lng t. Trong cc cu trc c kch thc lng t, cc ht dn b gii hn trong cc vng c kch thc c trng vo c bc sng de Broglie v chu tc ng ca cc qui lut c hc lng t. Chng hn ph nng lng ca cc ht dn tr thnh gin on theo hng ta gii hn. Khi , dng iu ca ht dn tng t nh trong kh 2 chiu. Cu trc vi kh in t 2 chiu c nhiu tnh cht khc thng so vi cu trc ca cc h ht dn 3 chiu. V d nh hiu ng Hall lng t (1980). Cc cu trc tng t ngy cng ph bin trong nhiu linh kin bn dn mi nht l trong quang in t. Hiu ng Gunn (1963) l hin tng dng in chy trong cht bn dn n- GaAs mng t trong mt in p cao th bin thnh cc dao ng vi tn s vi ba. Tn s ny t l nghch vi kch thc vt m cao th t vo. Hiu ng ny c ng dng trong it Gunn lm my pht vi ba.

M UL. Esaki v R.Tsu (1970) to ra siu mng (superlattice) bn dn u tin. l mt cu trc tun hon nhn to gm cc lp xen k ca hai cht bn dn khc nhau c dy lp c nanmt. Siu mng thuc v cu trc nan (nanostructure). S tun hon nhn to l do vng Brillouin b gp li thnh cc vng Brillouin nh hn gi l cc vng mini. Do c th to ra cc cc tiu ca vng dn cao hn vi cc nng lng tha mn cc dao ng Gunn. Cc cu trc nan thu c nh cc cng ngh hin i nh MBE (molecular beam epitaxy), MOCVD (metal-organic chemical vapor deposition),... Cng ngh ny c kh nng to ra cc cu trc vi phn b thnh phn tu v vi chnh xc ti tng lp phn t ring r. Siu mng v h lng t QW (quantum well) l cc cu trc nan phng hay 2 chiu. Cu trc nan 1 chiu gi l dy lng t QW (quantum wire). Cu trc nan khng chiu gi l chm lng t QD (quantum dot).

M USiu mng l nhng cu trc tun hon nhn to ca cc lp vt liu c hng s mng gn bng nhau. Trong siu mng, cc in t chu tc dng ca hai th tun hon l th ca tinh th v th ca siu mng vi chu k ln hn rt nhiu so vi hng s mng. Th ca siu mng c to nn bi s khc bit gia cc mc nng lng ca cc vng dn thuc hai cht bn dn to thnh siu mng. Cc tham s ca siu mng nh chu k siu mng, nng ht ti,... c th iu chnh c nh cng ngh cao. Do , c th iu chnh th siu mng v ph nng lng ca in t nh thay i cc tham s siu mng. iu lm cho siu mng bn dn c nhng c tnh u vit hn so vi cc bn dn thng. Ngy cng c nhiu phng php ch to ra cc cu trc nan khc nhau. Mt trong cc l do lm cho cu trc nan c c bit quan tm hin nay l cc tnh cht in t v dao ng ca chng b bin dng v chng tr thnh thp chiu v i xng thp.

Chng 1: C s c hc lng t 1.1. C hc sng v phng trnh Schrodinger C hc sng l hnh thc lun c bn ca l thuyt lng t. N lin quan n hm sng tun theo phng trnh Schrodinger ph thuc vo thi gian Nu th t (1.1) suy ra Cn tun theo phng trnh Schrodinger khng ph thuc vo thi gian

Trong trng hp 3 chiu, thay

1.1. C hc sng v phng trnh Schrodinger

Nghim ca (1.1) l

E l nng lng ca ht. Cc nghim ny m t cc trng thi dng ca ht vi nng lng xc nh. 1.2. Ht t do Khi , V(x) = 0. (1.4) tr thnh

l phng trnh sng chun (Helmholtz). Cc nghim ca (1.6) c th l hm m phc hoc hm lng gic thc sau

Do ,

1.2. Ht t do

Theo c hc c in, ng nng Hai h thc trung tm ca l thuyt lng t c (Einstein), (1.8) (de Broglie). (1.9) H thc tn sc: Vn tc pha Vn tc nhm l vn tc c in.

1.2. Ht t do

B sng (wave jacket) trn hnh 1.1 ch ra ngha ca hai vn tc ny. Sng con (wavelet) bn trong b sng chuyn ng v pha trc vi vn tc pha trong khi hnh bao (envelope) chuyn ng vi vn tc nhm. Hnh 1.1

Nu l s thc. Nu l s o vi Cc hm ny u l thc v u b phn k khi theo t nht mt hng. Do , cc hm ny ch c th c s dng trong mt vng khng gian hn ch.

1.3. Ht lin kt: h lng t

Xt in t chuyn ng trong mt vng khng gian hu hn. N c gi l mt h lng t (quantum well (QW)) hay mt ht trong mt hp (a particle in a box). V d n gin nht l mt h vung gc su v hn v c minh ha trn hnh 1.2. H th c V(x) = 0 khi 0 < x < a v khi hoc . a l b rng hoc na b rng ca h. Bn trong h, ht chuyn ng t do. Nghim ca phng trnh Schrodinger trong h c th l hm m phc, hm lng gic thc khi E > 0 hoc hm m thc, hm hyperbol khi E < 0 ging nh xt phn 1.2. Cc iu kin bin loi b chuyn ng t do. Ngoi h, khng c ht. Do , trong ro. Do hm sng lin tc nn cn c cc iu kin bin T , v t Do , cc hm sng v nng lng c dng


Hnh 1.2. H vung gc su v hn trong GaAs vi b rng a = 10 nm dc theo x trong ch ra 3 mc nng lng v hm sng u tin.

S nguyn n = 1, 2,.... l s lng t k hiu trng thi. S lng t ha l gii hn ln cc nng lng cho php v n xut pht t cc iu kin bin p t ln chuyn ng ca ht. Cc ht chuyn ng t do qua ton b khng gian c cc mc nng lng lin tc. Cc ht b giam cm trong mt vng khng gian c cc mc nng lng gin on. Thng mt ht b lin kt trong mt khong nng lng no v t do trong cc khong nng lng khc vi mt hn hp cc mc nng lng lin tc v gin on. Trng thi thp nht c nng lng iu i lp vi c hc c in m trng thi ng vi nng lng thp nht c ht bt c ch no trn y h vi ng nng bng khng v E = 0. Dng iu nh vy vi


phm nguyn l bt nh trong c hc lng t. Do , thm ch trng thi thp nht c nng lng im khng dng. Cc hm sng trong h th vung gc c tnh cht i xng. Cc hm sng ng vi n l l cc hm chn ca x phn gia h trong khi cc hm sng ng vi n chn l cc hm l ca x. iu ny s r rng hn nu vic nh s trng thi bt u t 0 thay v t 1. S i xng ny ng i vi mi h c th V(x) l mt hm chn ca x. Tnh cht i xng ny l quan trng rt ra cc qui tc lc la cho nhiu qu trnh v d nh s hp th quang. Cc phng php quang cung cp cc k thut trc tip o cc mc nng lng trong mt h lng t. Xt s hp th quang trong mt h lng t. H lng t l mt m hnh nhn to t c ng dng trong th gii thc. Mc d khng th to ra mt h su v hn, hin nay c th nui c cc cu trc gn vi cc h hu hn l tng. H su v hn thng dng nh mt php gn ng do cc kt qu n gin ca n.


Hnh 1.3. S hp th quang trong mt h lng t to bi mt lp GaAs gia cc lp AlGaAs. (a) H th trong vng dn v vng ha tr trong mi vng c hai trng thi lin kt v khe nng lng ca GaAs ln hn nhiu so vi hnh v. (b) Nhng s chuyn gia cc trng thi trong h lng t sinh ra cc vch hp th gia cc khe vng ca h GaAs v ro AlGaAs. Mt d cu trc bao gm mt lp GaAs mng b kp gia cc lp AlGaAs dy cung cp mt h lng t n gin nh trn hnh 1.3(a). AlGaAs thc ra l Cc in t t do c nng lng Cc in t trong bn dn trong vng dn (CB) c th thay i nng lng theo 2 cch. Theo cch th nht, nng lng cn c o t y vng thay v t khng. Theo cch th hai, cc in t x s nh th chng c khi


lng trong khi lng hiu dng trong GaAs. Do , H nhiu lp gm cc lp GaAs v AlGaAs xen k nhau c tc dng ging nh mt h lng t v trong AlGaAs cao hn trong GaAs v chnh lch to ra hng ro giam cm cc in t. in hnh l m n khng ln. Tuy nhin, tag s lm gn ng n bng v cng tm cc mc nng lng trong mt h c b rng a. Nng lng ca cc trng thi lin kt c k hiu l bng

C th o cc mc nng lng ny bng cch chiu nh sng ln mu v xc nh xem nhng tn s no b hp th. Cc bn dn c cc mc nng lng trong cc vng khc. Vng quan trng nht trong s l vng ho tr (VB) m n nm di vng dn. nh vng ho tr ti v vng ny b un cong xung di theo h thc trong cha khi lng hiu dng khc


l trong GaAs). Cc vng dn v vng ho tr c tch ra bi khe vng li l mt h lng t v trong h GaAs mc khc vi mc ca trong cc ro AlGaAs. Nng lng ca cc trng thi lin kt c k hiu bi bng

Mi iu o ngc trong vng ho tr nh ch ra trn hnh 1.3(a). VB y hon ton v CB trng hon ton trong mt bn dn tinh khit nhit khng. S hp th quang chuyn mt in t t VB ln CB. Trong GaAs, iu ny xy ra vi iu kin l l khe vng ca GaAs). Tng t, trong AlGaAs. Qu trnh ny li pha sau mt trng thi trng hay l l trng trong VB. Do , ch s di h dng ch cc thng s ca VB. i vi h GaAs, s hp th quang khng th bt u ti do cc trng thi trong h b lng t ho. Nng lng thp nht m ti xy ra s hp th c cho bi hiu gia nng lng thp nht


trong h trong CB v nng lng thp nht trong h trong VB. S hp th c th xy ra ti nhng nng lng cao hn do vic s dng cc trng thi khc. Cc chuyn tip mnh nht xy ra gia cc trng thi tng ng trong hai vng. Do , c th t V th, s hp th mnh xy ra ti cc tn s c cho bi

Cc nng lng trn ging nh cc nng lng trong mt h lng t m khi lng hiu dng l c cho bi N gi l khi lng hiu dng quang. Nu cc h thc s su v hn, c v s vch vi cc tn s tha mn (1.15). Cc ro trong bn dn l hu hn v s hp th xy ra trong cc ro AlGaAs i vi mi tn s m Ph thu c trn hnh 1.3(b) khi gi thit c 2 trng thi lin kt trong c CB v VB. Khng c


s hp th i vi v c mt vng hp th lin tc i vi Gia 2 tn s ny c 2 vch gin on sinh ra bi cc chuyn tip gia cc trng thi trong h lng t vi cc nng lng tha mn (1.15). C th tnh c b rng h t cc nng lng ca cc vch ny nu bit cc khi lng hiu dng. l mt cch kim tra thy rng cc lp c ch to mt cch chnh xc. Trong thc t, ngi ta thng tin hnh mt th nghim khc mt t gi l s pht quang nh sng PL (photoluminescence). nh sng vi nng lng c chiu vo mu v n kch thch nhiu in t t VB ti CB khp ni. Mt s trong s cc in t ny b by trong h lng t v xy ra iu tng t i vi cc l trng trong VB. Khi , c th c mt in t ri t CB vo mt l trng trong VB v gii phng s khc bit nng lng thnh nh sng. S pht quang ny ngc vi s hp th v c th xy ra ti cng cc nng lng. Thng ch nhn thy cc mc thp nht. Do , ph PL ch ra mt vch ti Mt v d v ph PL c ch ra trn hnh 1.4. V d c 4 h c cc b rng khc nhau m mi mt h ng gp mt nh vo PL.


Hnh 1.4. PL nh mt hm ca bc sng i vi mt v d vi 4 h lng t vi cc b rng khc nhau m cc CB v VB ca chng c ch ra bn phi. Cc ro gia cc h dy hn nhiu so vi hnh v.

Vng ho tr khng n gin nh ta gi

thit. Mt m hnh tt hn gi thit rng c hai loi l trng l l trng nng v l trng nh. Do , c th thy 2 h vch ph mc d cc l trng nng ng ch hn nhiu. Cc in t v l trng cn lin kt vi nhau to ra cc exciton tng t nh cc nguyn t hir v iu ny lm thay i cc nng lng mt cht.

1.4. Mt in tch v mt dng

Hm sng tha mn phng trnh Schrodinger. Bnh phng mun hm sng mt xc sut tm thy ht ti x. (1.16) Nu ht c in tch q th mt in tch ca ht (1.17) hoc in tch trong vng dx xung quanh x. (1.18) Nu in t b giam cm trong mt th tch no th in tch tng cng trong th tch ny cn phi l q. Khi , mt in tch ca ht. (1.19) in tch ton phn (1.20) T , (1.21) l iu kin chun ho hm sng v n c ngha l l mt xc sut tm thy ht.


Khng phi mi hm sng u c chun ho nh th. V d hm sng ca in t t do khng th chun ho theo cch v tch phn theo ton b khng gian b phn k. Khi , ta ch c th ni n xc sut tng i. C th thot khi kh khn ny bng cch bt u vi in t trong mt hp ln nhng hu hn v chuyn th tch hp ti v cng cui qu trnh tnh ton. S chun ho em li th nguyn vt l cho hm sng. V d xt mt h su v hn. Cc hm sng v iu kin chun ho hm sng l

Nu thc th hm sng chun ho l (1.23) S chun ho lm cho hm sng c th nguyn l (chiu di) theo mt chiu. iu ny c th dng kim tra.


Mt sng phng trong mt th tch v hn c th c chun ho theo mt cch khc mt cht. y, mt c th t cho mt ht cho. Cn tn ti mt dng J (hoc ch l dng mt chiu) gn vi mt in tch v chng tha mn phng trnh lin tc (1.24) c s bo ton in tch v bo ton ht. Trong trng hp 3 chiu, tr thnh Ta xy dng mt dng t phng trnh Schrodinger ph thuc vo thi gian

Nhn 2 v ca (1.25) vi , ta c


Ly lin hp phc (1.25) v nhn vo v tri vi , ta c

Nu l thc v tnh n th t (1.26) v (1.27) suy ra

Nu s dng

th t (1.28) suy ra


l biu thc ca mt dng biu din qua hm sng. Trong trng hp 3 chiu, tr thnh Mt trng thi dng c hm sng phc th vn mang dng (khng i theo thi gian). Cn mt trng thi dng c hm sng thun thc th khng mang dng. iu ny p dng cho mt ht trong mt hp v cc trng thi lin kt ni chung. Mt s chng chp cc trng thi lin kt cn sinh ra mt dng. Do , hm sng trong c hc lng t ni chung l mt i lng phc thc s. lm v d, xt N m t mt sng phng chuyn ng theo hng +x. Mt in tch ng u qua ton b khng gian v dng ging nh kt qu mong mun l Phng trnh Schrodinger l tuyn tnh. Do , cc hm sng c th c xy dng bng cch chng chp cc nghim c bn. V d nh (1.33) m t mt s chng chp ca cc sng truyn theo cc hng ngc nhau. Biu thc c hc lng t i vi dng cho kt qu mong mun l


Hnh 1.5. Dng c mang bi cc sng phn r lan truyn ngc. (a) Mt ro v cng dy cha mt sng hm m phn r n m n khng mang dng. (b) Mt ro hu hn cha c hai sng hm m tng v gim v n cho dng i qua. Hm sng l phc v do , hnh v ch l mt m t th. Xt hai sng phn r lan truyn ngc

Khng c thnh phn no t n mang dng v n l

thc nhng s chng chp cho

Sng cn phi cha cc thnh phn phn r theo c hai hng vi s khc pha gia chng i vi dng chy. nh hng ny c ch ra trn hnh 1.5 i vi mt sng va chm vi mt ro. Mt sng dao ng chuyn thnh mt sng phn r bn trong mt ro cao. Nu ro v cng dy, n cha mt sng phn r n v khng c dng thc. Mt ro hu hn cho mt dng (nh) i qua v cn phi cha hai sng phn r lan truyn ngc. Sng quay li t u xa ca ro mang thng tin l ro hu hn v c mt dng chy.

1.5. Cc ton t v php o

Xt xem cc i lng vt l c th c suy ra t hm sng nh th no?1.5.1. Cc ton t Theo c hc lng t, cc i lng o c c th c biu din bng cc ton t m chng tc ng ln hm sng mc d hm sng t n khng th o c. Ta , xung lng v nng lng ton phn c biu din bng cc ton t sau y tc dng ln hm sng

C th xy dng cc ton t phc tp hn t cc thnh phn ny. V d nh hm Hamilton cho nng lng ton phn ca ht c in trong loi h bo ton nng lng. N tr thnh ton t Hamilton trong c hc lng t v c cho bi


S cn bng tc ng ca ton t ny vi tc ng ca ton t nng lng cho hay

Ta tr li phng trnh Schrodinger ph thuc thi gian (1.1). By gi c th vit phng trnh Schrodinger khng ph thuc vo thi gian thnh trong E l mt s ch khng phi l ton t. N ging vi phng trnh tr ring ca ma trn v c gi l phng trnh tr ring ca ton t. c gi l hm ring hoc trng thi ring v E l tr ring tng ng. C th biu din mt dng theo ton t xung lng nh sau

iu ny chng t dng lin quan ti vn tc p/m. Tc dng ca ln sng phng cho


(1.43) l mt phng trnh tr ring do tr ring xung lng Theo c hc lng t, ch c nhng tr s kh d ca mt i lng vt l o c l cc tr ring ca ton t tng ng vi n. Nu hm sng l mt hm ring ca ton t ny ging nh trong trng hp ca sng phng v xung lng, i lng o c c mt tr s xc nh. Xt tc dng ca ln mt ht trong mt hp

Cc hm sng ny khng phi l cc hm ring ca v do khng c tr ring xc nh ca xung lng. Cc php o xung lng dn n mt khong gi tr m ta c th m t c im ca n theo mt gi tr trung bnh ( y l khng) v mt s m rng. Hm l hm ring ca Do , tr ring ng nng ca ton t ng nng c tr s xc nh. Nhng vn tng t ny sinh khi ta o ta ca mt ht c xem xt phn sau.


1.5.2. Gi tr k vng Cho hm sng i vi mt ht no . Ta cn xem xt ta trung bnh ca ht v ht nh x nh th no ta . Ta khng th ni ht ti mt im ring no nh trong c hc c in v ta ang s dng mt bc tranh trn c s sng. Mt xc sut tm thy ht l v c th dng n tm tr trung bnh nh sau

trong c s dng k hiu gi tr k vng hay gi tr trung bnh. i vi cc hm sng chun ho,

xem ht c nh x nh th no, ta thng o lch chun sau trong l gi tr k vng ca c cho bi


Ly trng thi thp nht ca mt ht trong mt hp lm v d. Khi ,

Ht chc chn tm c gia h nhng vi s m rng ng k xung quanh . N tng ln i vi nhng trng thi cao hn. C th xem xt tng t i vi xung lng ca ht . Gi tr k vng ca mt i lng vt l o c q no c cho bi

trong l ton t tng ng. V d gi tr k vng ca xung lng l


v c xc nh mt cch tng t. Cc gi tr k vng l cc i lng vt l v do cn phi l cc s thc. N i hi cc i lng vt l c biu din bng cc ton t hecmit. Cc ton t nh th c cc tr ring thc bo m cc php o trn hm sng cho cc gi tr thc. Cc gi tr k vng ca cc trng thi dng l khng i theo thi gian v s ph thuc ca chng vo thi gian bin mt gia v . V d l khng i i vi mi trng thi dng v do ht th hin l dng. Cn mt s chng chp trng thi ht chuyn ng vi ngha l thay i theo thi gian. i vi h mt chiu, c th xy dng mt hm sng chuyn ng t hai trng thi u tin Khi hm sng ny thay i theo thi gian, ta trung bnh tr thnh

Ht dao ng trong h vi tn s gc cho bi s khc bit gia v .


1.5.3. Chuyn ng ca b sng C hc c in da trn khi nim cht im vi ta v xung lng c xc nh chnh xc. iu ny khng ng trong c hc sng. S tng t t nhin l mt b sng. l mt sng b gii hn trong mt vng hu hn bi mt hnh bao. N cng minh ha cc gi tr k vng. Xt mt sng mang phng v bin iu n vi mt hnh bao Gauss ti t = 0

Mt xc sut ca n l mt hm Gauss chun ho vi trung bnh v lch chun d

T , v ti . C th nh x b sng bng cch chn d thch hp.


Sng mang c xung lng xc nh nhng ta cn trn nhiu sng vi nhau c b sng v do by gi c mt phm vi xung lng theo hm sng. C hai cch rt ra v . Cch th nht l dng nh ngha ca gi tr k vng. Cch th hai l vit hm sng theo xung lng ch khng phi ta . Do mt sng phng c xung lng p xc nh, c th xc nh s phn b ca cc xung lng trong hm sng bng cch phn tch n thnh cc sng phng qua php bin i Fourier. Do , hm sng trong khng gian xung lng lin h vi hm sng trong khng gian thc (khng gian ta ) bi

Cc h s bo m c chun ho ging nh . Khi thc hin php bin i Fourier i vi b sng Gauss (1.56), ta thu c


l mt hm Gauss chun ha vi trung bnh t sng mang v lch chun Tch cc lch chun: l nguyn l bt nh Heisenberg m theo n, khng th o c ta v xung lng ca mt ht ti chnh xc ty . N tri ngc vi bc tranh c in m c ta v xung lng c th xc nh mt cch chnh xc. Cc b sng Gauss em li bt nh cc tiu v theo nguyn l bt nh, V d mt sng phng c xung lng xc nh. Do , nhng n m rng u qua ton b khng gian v dn n


Nguyn l bt nh lm cho trng thi thp nht trong mt h lng t phi c ng nng khc khng. iu khc vi c hc c in m trong ht s ng yn. Nu ht ng yn th xung lng ca ht bng khng. Do , trong lc hu hn v ht mt ch no trong h. Nh vy, m iu vi phm nguyn l bt nh. Cch duy nht trnh iu l ht phi c nng lng im khng trng thi thp nht sao cho c v u khc khng. iu ny c th c s dng c tnh nng lng im khng. V d xt mt h th su v hn c b rng a v ht trong h th. C th coi v do , T ng nng suy ra nng lng ca trng thi c bn l Kt qu chnh xc c thay cho 2. iu ny gii thch s ph thuc ca cc mc nng lng vo b rng a: vic thu hp h lm gim s m rng ca ht trong khng gian thc v do lm tng xung lng v nng lng ca ht. C th m rng nguyn l bt nh tnh nng lng im khng trong bt k h no bng cch bao hm th nng trung bnh.


Xt b sng Gauss. N c bt nh (tch ) cc tiu ti t = 0 nhng iu ny thay i khi b sng thay i theo thi gian. Mt sng phng thay i theo thi gian ging nh vi iu ny p dng cho tng thnh phn Fourier ca b sng. Do , cc hm sng trong khng gian Fourier v khng gian thc i vi t > 0 l


Tha s trc cho mt sng mang vi xung lng chuyn ng vi vn tc pha Hm m u tin trong tch phn ch ra rng b sng xoay quanh v chuyn ng vi vn tc nhm Hm m th hai trong tch phn iu khin b rng b sng v cng b thay i. nh gi tch phn dn n

Xung m rng trong khng gian ging nh n lan truyn. Xung lng gi khng i nu khng c lc tc dng ln ht. Do , tch tng v thng tin v ht b km i theo thi gian. l nh hng in hnh ca s tn sc v d trong thng tin lin lc. S tn sc sinh ra do mt b sng cha mt khong xung lng v cc xung lng ny lan truyn vi cc vn tc khc nhau do b sng c m rng. Cui cng, iu ny ln t b rng ban u. Khong vn tc l v do , trong nhng thi gian ln ta k vng ph hp vi (1.66). Mt xung ngn cha mt khong xung lng rng hn so vi mt xung di v cui cng s tr nn di hn.


1.5.4. Cc tnh cht khc ca cc ton t H thc bt nh c th c ngun gc t tnh cht ca ton t. Ta th o c ta v xung lng ca ht c m t bi b sng. Khi , cn ch n trt t ca cc ton t ta v xung lng. Gi s ta o xung lng ri sau o ta . iu c ngha l cc ton t tc dng ln hm sng l

Trt t ngc li cho

Cc kt qu thu c l khc nhau v trt t ca cc ton t c vai tr quan trng. T trn suy ra Do (1.69) ng i vi mi nn v giao hon vi nhau nu Khi , trt t ca cc ton t khng c vai tr g. Ch c th o 2 i lng vt l mt cch ng thi ti chnh xc ty khi cc ton t m t cc i lng ny giao hon vi nhau.


Cc v d: Mt i lng m ton t m t n giao hon vi ton t Hamilton c gi l hng s chuyn ng (hay tch phn chuyn ng) v gi tr ca n khng thay i theo thi gian. V d nh i vi mt ht t do v do , xung lng ca n gi khng i. Cc hng s chuyn ng thng sinh ra t s i xng no ca h v trong trng hp ny l bt bin tnh tin. Trt t ca cc ton t v d nh v l quan trng v chng khng th o ngc trt t nh l cc s. iu ny cng ng i vi cc ma trn v cc ton t c th c biu din bng cc ma trn. Vic la chn cc ton t ph thuc vo cch m cc hm sng c biu din. Ta rt ra hm sng ca mt b sng trong khng gian xung lng v c th s dng cc ton t tng ng Chng tun theo cng h thc giao hon ging nh trong khng gian ta .

1.6. Tnh cht ton hc ca trng thi ring

Cho cc trng thi ring (hm sng) ca ton t Hamilton vi cc tr ring (nng lng) tng ng l v chun ho mi trng thi sao cho

Khong ly tch phn bao hm vng chuyn ng ca ht. i vi ht trong h vung gc su v hn, Cc trng thi ring vi cc tr ring khc nhau l trc giao, ngha l nu Tch phn tng t tch v hng. i vi h lng t, (1.73) c ngha l nu

Cc trng thi ring khc nhau ng vi cng mt tr ring c gi l suy bin. Khi , c th chn cc trng thi ring sao cho chng trc giao i vi mc d . C th kt hp (1.72), (1.73) c mt h trc chun


Gi thit cc l trc chun ging nh cc sng (1.23) trong h lng t. C th chng minh cc trng thi ring to thnh mt h . iu c ngha l bt k hm sng no tun theo cng cc iu kin bin u c th khai trin thnh mt tng theo cc

T (1.76) suy ra

Ta s thu c hm sng ban u khi thay (1.78) vo (1.76) v c thm iu kin N c gi l h thc ng (h thc ). Khai trin hm sng dng tm s bin i theo thi gian ca mt trng thi bt u ty . Phng php l phn tch theo cc trng thi ring ca phng trnh Schrodinger khng ph thuc thi gian


Mi trng thi ring bin i theo thi gian ging nh v do , trng thi cho bin i ging nh

l cch ta xem xt s bin i ca b sng trong phn 1.5.3.

1.7. Trng thi m

m t mt h cn bit cc nng lng v hm sng ca tt c cc trng thi ca h. iu ny khng th thc hin c ngay c vi cc h n gin nht. i vi nhiu ng dng, ngi ta thng dng thut ng mt trng thi l s trng thi ca h c nng lng nm trong khong t n N ch cho ta bit s phn b nng lng. 1.7.1. Trng hp mt chiu Cc hm sng khng th chun ho theo cch thng thng nu ht chuyn ng qua ton b khng gian. chun ho, ta a ht vo trong mt hp hu hn c chiu di v cho cui kt qu tnh ton. a ht vo trong hp, ta cn chn cc iu kin bin. Hai iu kin bin thng c s dng l (i) cc iu kin bin hp hay c nh trong hm sng trit tiu bin

(ii) cc iu kin bin Born-von Karman hay tun hon trong h c lp li mt cch tun hon vi cng hm sng trong mi mt h. Hm sng ti x = L cn phi lm khp trn vi hm sng ti x = 0 m n i hi

1.7. Trng thi m

Cc iu kin bin c nh ging nh khi ht trong mt hp xt trc y. Cc mc nng lng c cho bi v cc tr cho php ca k l Cc hm sng l cc sng ng. Chng khng mang dng v cc tr cho php ca k u dng. Cc iu kin bin tun hon i hi la chn khc ca k. C th dng cc sng m chy khc vi cc sng sin m chng cn phi tha mn N tha mn iu kin v graien v cc trng thi chun ho l Cc gi tr cho php ca k l Chng gp i so vi iu kin bin c nh. C hai du ca k u c php. C hai trng thi suy bin ti mi mt mc nng lng (tr ti k = 0) vi cc du ngc nhau ca k v vn tc.

1.7. Trng thi m

Liu chng mt trng thi c ph thuc vo iu kin bin la chn hay khng? Kt qu khng nhy vi cc iu kin bin khi Thng xt cc in t t do nh cc sng chy khc vi cc sng ng v do , ni chung ngi ta thng s dng cc iu kin bin tun hon. bin cc gi tr cho php ca k v thnh mt trng thi, ta ly cc gi tr cho php ca k dc theo mt ng thng (h.1.6).

1.7. Trng thi m

Liu chng mt trng thi c ph thuc vo iu kin bin la chn hay khng? Kt qu khng nhy vi cc iu kin bin khi Thng xt cc in t t do nh cc sng chy khc vi cc sng ng v do , ni chung ngi ta thng s dng cc iu kin bin tun hon. bin cc gi tr cho php ca k v thnh mt trng thi, ta ly cc gi tr cho php ca k dc theo mt ng thng (h.1.6). l hnh nh mt chiu n gin ca khng gian k. Cc gi tr cch u nhau mt khong l Chng gn nhau hn khi L tng v c xu hng nho thnh mt min lin tc. S cc gi tr cho php ca k trong vng bng chia cho khong cch gia cc im.

Hnh 1.6. Cc gi tr cho php ca k i vi cc iu kin bin tun hon trong mt h c chiu di L c v dc theo mt ng thng nh mt dng n gin ca khng gian k. Cc im ny tnh n cc trng thi khc nhau c sinh ra t chuyn ng trong hp nhng ta cn tnh n chuyn ng bn trong ca ht. V phng

1.7. Trng thi m

din c in,chuyn ng t do c th b tch ra thnh chuyn ng tnh tin v chuyn ng quay xung quanh khi tm. Cc in t cn mang mt xung lng gc gi l spin. Trong c hc lng t, spin c th 2 trng thi l spin ln v spin xung. Mi mt hm sng khng gian c th c gn vi spin ny hoc spin kia. Khi tnh n spin th s trng thi ca in t s gp i. Spin ng gp h s 2 vo mt trng thi. i vi in t trong t trng c mt mmen t gn vi spin m n ng gp mt nng lng l S tch ca vn tc v spin ch duy tr trong c hc lng t phi tng i tnh. Nhng iu kin m trong s tng i tnh c bit l quan trng c th coi nh khng nh hng n cc cht bn dn thng thng nhng cc in t chuyn ng vi vn tc gn vi vn tc nh sng khi chng i gn ht nhn. iu ny dn ti mt nh hng gi l lin kt spin qu o ln nh ca VB v tnh cht ca l trng. C th nh ngha mt mt trng thi trong khng gian k sao cho l s trng thi cho php trong khong t k n v c cho bi

1.7. Trng thi m

Tha s 2 tnh n spin. l mt im. Do , N t l vi th tch (chiu di) ca h. Nh vy, mt trng thi tng gp i nu kch thc ca h tng gp i. Thng dng mt trng thi ng vi mt n v chiu di By gi cn bin n thnh mt trng thi theo nng lng. H.1.7 chi ra cch nhng gi tr cho php ca k cch u nhau chuyn thnh nhng gi tr cho php ca nng lng qua h thc tn sc Cc nng lng ny nm trong mt vng lin tc i vi trong mt h ln. H.v ch ra mt parabol nhng l thuyt xem xt mt h thc tn sc tng qut hn. Mt trng thi gim vi s tng nng lng. Mt khong s sng tng ng vi H.1.7. H thc tn sc khong nng lng i vi cc in t t do

1.7. Trng thi m

S trng thi trong khong ny c th c vit theo hoc theo mt trng thi theo nng lng ng vi mt n v th tch Do ,

H s 2 trong biu thc ca sinh ra do tnh n 2 hng chuyn ng: c mt khong i vi v mt khong khc i vi Nh vy, Nu biu din n qua vn tc nhm v = ta c

(1.88) chng t rng c mt hng s trong trng hp mt chiu m n dn ti mt dn lng t ho. Trong trng hp ca cc in t t do, Mt trng thi phn k ging nh khi l mt tnh cht c trng ca mt chiu.

1.7. Trng thi m

1.7.2. Trng hp ba chiu a cc in t vo trong mt hp c th tch Cc hm sng l cc sng chy theo mi mt hng vi cc iu kin bin tun hon ging nh trong trng hp mt chiu v tch ca chng cho

Ch cc vect 3 chiu. Tch ca 3 sng c vit nh mt sng phng 3 chiu. Cc gi tr cho php ca K theo mi mt trong 3 hng c th kt hp li thnh nhng vect sng 3 chiu

Chng c th c m t bng cc im trong khng gian 3 chiu vi cc trc l m chng to thnh mt li hnh ch nht cch u nhau. Mi mt c s chim th tch Mt ca cc trng thi cho php l trong 2 tnh n spin. Vic chia cho th tch cho mt trng thi trong khng gian ng vi mt n v th tch ca h trong khng gian thc N l mt hng s v c tng qut ho cho trng hp d chiu thnh

1.7. Trng thi m

Cn rt ra mt trng thi nh mt hm ca nng lng. Ch xt cc in t t do v tnh ton phc tp hn i vi mt hm tng qut H.1.8 ch ra 2 qu cu gn gc trong khng gian trong c mt qu cu c bn knh v qu cu kia c bn knh Th tch lp cu gia 2 qu Hnh 1.8. Cch xy dng trong khng gian tnh mt trng thi i vi cc in t t do trong trng hp 3 chiu. Cc lp v c cc bn knh t n tng ng vi cc nng lng t n

cu l S trng thi trong lp cu bng tch ca th tch lp cu vi mt trng thi v kt qu l tng ng vi

1.7. Trng thi m

S trng thi trong lp c cho theo mt trng thi theo nng lng bi T , Do ,

S ph thuc c trng cho trng hp 3 chiu. S d thng ca n y vng yu hn nhiu so vi s ph thuc trong trng hp 1 chiu. Ni chung, mt trng thi ch ra mt c tnh mnh hn y vng h t chiu hn. Cc tnh cht quang nh s hp th b nh hng mnh bi mt trng thi v cc h thp chiu c a thch i vi cc linh kin quang in t do cc mt trng thi ca chng ln hn y vng. Mt trng thi i vi cc in t t do trong cc trng hp 1, 2 v 3 chiu c m t trn hnh 1.9. Trong tt c cc trng hp, mt khi lng nh gn vi mt mt trng thi thp. Mt trng thi i vi mt tinh th 3 chiu l phc tp hn do cc b mt ng nng trong khng gian khng phi l nhng hnh cu. Cc d thng khc ca xut hin bn trong cc vng v cung cp thng tin cho quang

1.7. Trng thi m

Hnh 1.9. Mt trng thi ca cc in t t do trong cc trng hp 1, 2 v 3 chiu

tin cho quang ph. Xt trng hp khi nng lng ch ph thuc vo ln ca m khng ph thuc vo hng ca n. Khi , cc b mt ng nng l mt cu v vic rt ra c tin hnh nh trc tr dng ca V d CB ca GaAs thng c m hnh ho bi biu thc

iu ny l do vng khng c dng parabol i vi cc nng lng cao m

1.7. Trng thi m

1.7.3. nh ngha tng qut ca mt trng thi Xt cc trng thi ca mt h c cc nng lng Khi , mt trng thi theo nng lng c th vit thnh trong l hm delta Dirac. N l mt trng thi tng cng ch khng phi l mt trng thi ng vi mt n v th tch. (1.95) ch c ngha nu ta ly tch phn theo E do s c mt ca cc hm delta. Xt

N c minh ho trn h.1.10. Tch ca bng 1 i vi mi trng thi trong khong v bng 0 i vi mi trng thi nm ngoi khong . Khi ly tng tip theo, n cng thm vo Hnh 1.10. nh ngha hm delta ca mt cho ti tng s cc trng thi gia v trng thi N chnh l iu ta mong mun t mt tch phn ca v (1.95) l mt

1.7. Trng thi m

nh ngha c gi tr ca mt trng thi. xc nhn iu , xt cc in t t do trong trng hp 1 chiu. Khi , c th k hiu cc trng thi theo s sng k ca chng v (1.95) tr thnh

2 tint n spin. Bin tng thnh tch phn khi gi thit mt h ln. Mt trng thi trong khng gian k l Do , tng tr thnh

i bin tch phn t k sang

2 tnh n 2 du ca k di vi mi gi tr ca z. Tch phn l tm thng v ch c ng gp ti z = E khi E > 0 v ta c kt qu ging nh trc

1.7. Trng thi m

1.7.4. Mt trng thi nh x Mt c tnh quan trng trong nh ngha hm delta ca mt trng thi l ch n c th c m rng cho bt k h no. Cc v d xt l bt bin tnh tin vi ng l mt trng thi l nh nhau ti mi im. l mt trng hp c bit. nghin cu h khng c i xng tnh tin, xt cc in t t do b gii hn trong vng x > 0 bi mt thnh khng th xuyn qua ti x = 0. Cc hm sng l sinx v do , chng bng 0 ti x = 0. Nu cc hm sng trit tiu th iu hp l l mt trng thi cng nh vy. C th nh ngha mt trng thi nh x nghin cu trng hp khi ng gp ca mi mt trng thi c xc nh bi mt hm sng ca n ti im kho st. Nh vy, (1.95) tr thnh

T (1.102) suy ra (1.95) khi ly tch phn theo ton b h

Tha s c ngha l mi mt trng thi ng gp vo mt trng

64

1.7. Trng thi m

thi nh x ch trong cc vng c mt cao. iu ny l c ch trong cc h khng ng nht v cha nhiu thng tin hn so vi i vi h 1 chiu vi mt thnh, cc hm sin cho

Mt php ly turn bnh theo x dn ti biu thc ca (1.89) nhng cc dao ng tn ti ti mi khong cch t thnh. Kt qu tng ng i vi mt h 3 chiu b gii hn ti x > 0 l trong N c v trn hnh 1.11 i vi cc in t trong GaAs. Ti x = 20 nm, mt trng thi nh x gn vi cn bc 2 m n duy tr trong mt h v hn nhng n ch ra cc dao ng mnh i vi x nh v trit tiu i vi mi nng lng ti x = 0. C th tng qut hn na mt trng thi nh x ti

N l mt hm ca bin s theo tng hm sng tch ri v c gi l hm ph. N l mt im tip xc t nhin vi cc hm Green trong l thuyt hin i hn.Hm ph cung cp biu din gn v d ca s hp th quang.

65

1.7. Trng thi m

Hnh 1.11. Mt trng thi nh x nh mt hm ca nng lng v khong cch t mt thnh khng th xuyn qua ti x = 0 trong GaAs.

66

1.8. Trng thi lp y: Hm lp y

Mt trng thi cho ta bit cc mc nng lng ca mt h v by gi ta cn lp y cc mc ny bng cc in t hoc cc ht khc. cn bng, s ht trung bnh lp y mt trng thi ch ph thuc vo nng lng ca chng v c cho bi mt hm lp y ph thuc vo bn cht ca cc ht c lin quan. Cc in t, proton v cc ht khc mang spin bn nguyn v c gi l cc fecmion. Chng tun theo nguyn l loi tr Pauli m theo , khng th c hn 1 fecmion chim gi mt trng thi cho. Trc ht ta xem xt hm lp y i vi cc fecmion do n l quan trng nht trong cc bn dn v sau xem xt cc cht khc. 1.8.1. Hm lp y Fermi-Dirac Nguyn l Pauli i vi cc fecmion gii hn s lp y ca mt trng thi l 0 hoc 1. S lp y trung bnh b chi phi bi hm phn b Fermi-Dirac (hay ngn gn l , l hng s Boltzmann, v l mc Fermi

67


trong cc bn dn v thay i theo nhit . Hm Fermi-Dirac c v th i vi mt s nhit khc nhau trong gi khng i trn hnh 1.12. Hnh 1.12. Hm phn b Fermi-Dirac ti 5 nhit khc nhau vi mc Fermi khng i l meV.

c tnh quern trng u tin ca phn b Fermi-Dirac l ch n ly cc gi tr gia 0 v 1 nh mong mun t nguyn l Pauli. khi Do s lp y ca mt trng thi c th l 0 hoc 1 nn cng c th coi nh xc sut ca trng thi b lp y (hoc chim gi). N l mt hm gim ca nng lng. N cng c ngha l mt trng thi d b lp y hn nu nng lng ca n thp hn. S chuyn t 1 v 0 tr nn r rt hn khi nhit gim v n tr thnh hm bc Heaviside trong gii hn nhit khng

68


Tt c cc trng thi di b lp y hon ton v cc trng thi trn u b trng. l k hiu gi tr ca mc Fermi nhit khng. Nhit Fermi Thc t l gii hn ca ti nhit khng l nh ngha cht ch ca mc Fermi. i lng m ta gi l thc ra l th ho hc. Trong vt l bn dn, c v u gi l mc Fermi. S chuyn t f = 1 ti 0 m rng khi nhit tng vi b rng khong so snh, meV nhit phng (300 K). Mc d nng lng nhit thit lp thang o (hay ln) ca vng chuyn tip, b rng vng ny ln hn mt vi ln. i vi cc nng lng trn nhiu sao cho th l phn b Bolzmann c in v n gi xa s bo ho ca f ti 1. Cc

69


bn dn c in trong gii hn ny. Trong trng hp ca cc fecmion, s lp y trng thi ch c th l 0 hoc 1 v ta c th m t h theo cc l trng ch khng theo cc in t. y, mt l trng c nh ngha n gin nh s khng c mt ca mt in t. Phn b ca cc l trng c cho bi

Lu s o ngc ca cc nng lng. Trong trng hp ny, hm m tr nn ln i vi cc nng lng m v do ,

By gi cc l trng tun theo phn b Boltzmann ging nh trong VB ca mt bn dn c in. 1.8.2. S lp y ca cc trng thi By gi ta c biu thc ca mt trng thi v s fecmion trung bnh lp y mi trng thi Tch ca chng cho mt ca cc

70


trng thi lp y trong h. S in t tng cng c xc nh t tch ny

T = 0, hm Fermi tr thnh hm bc v

cc nhit cao, cn phi thc hin tch phn y (1.111) v trong a s trng hp, iu ny khng th thc hin c v mt gii tch. i vi kh in t 2 chiu (h.1.9(b)), mt trng thi ca n l khng i ti i vi Mt in t ng vi mt n v din tch c cho bi

Gi thit khng i. Hm b ly tch phn t l vi hm Fermi trn h.1.12 m khng i. Mt rt gn thnh (1.112) trong gii hn

71


cho i vi cc nhit T > 0, c th n gin tch phn bng cch i bin v khi ,

Mt tng nh mt hm ca nhit nu gi khng i. iu ny l do ui ca hm Fermi (h.1.12) ngy cng m rng hn ti nhng nng lng cao hn v bt gi nhiu in t hn. iu ny khng cn bng vi s gim ca hm Fermi cc nng lng thp v tch phn b ct ti y vng E = 0. Thng ta khng hi vng mt in t thay i theo nhit . Nh vy, cn phi gim thay v gi khng i. Xt mc Fermi nh mt i lng c th iu chnh gi s in t mong mun trong h. Dng h thc vit li (1.114) theo ch khng theo v t ,

72


T ,

R rng, gim khi T tng v khi Cc hm lp y tng ng c v th trn h.1.13.

H.1.13. Hm phn b Fermi i vi kh in t 2 chiu trong GaAs ti mt khng i l Mc Fermi di chuyn xung di t meV khi nhit tng.

i vi nhit thp phn b gn vi hm bc m n duy tr khi T = 0 v gn vi gii hn ca n. a s cc trng thi lp y hon ton hoc trng ngoi tr cc trng thi trong mt vng xung quanh c b dy c mt vi ln Phn b ny c gi l phn b suy bin. N l mt tnh cht c trng ca mt kim

73


loi thc nh Al hoc Cu nhit phng (thc ra l nhit m ti n duy tr l cht rn). Cho n cc hin tng nng lng thp nh vn chuyn thun tr, ch cc trng thi lp y mt phn gn mi ng vai tr quan trng. Cc trng thi lp y hon ton di nhiu khng th tham gia do bt c phn ng no i hi lm thay i trng thi ca chng chng tt c cc trng thi gn v nng lng cng c lm y. Nhiu i lng nh dn do cha h s m n to ra nh nhn gn mc Fermi. Ta c

Trong gii hn nhit thp, tr thnh hm bc, Mi iu xy ra mc Fermi. l tnh cht in hnh ca kim loi nhng khng phi i vi bn dn c in. Khi tng nhit qu Khi , hm m mu s ca phn b Fermi lun lun ln thm ch ti cc tiu ca n xy ra ti y vng E = 0. Ta lun lun xa hng v ui phn b v c th b + 1 c (1.108). N l gii hn khng suy bin trong phn b Boltzmann c duy tr trn

74


ton vng v Cc bn dn pha tp nh c in thng trong gii hn ny v tt c cc in t u c th ng gp vo cc qu trnh v d nh s dn. Mt kh in t trong kh 2 chiu c dng

trong v gi l mt trng thi hiu dng trong vng dn. Nhiu h in t thp chiu khng trong gii hn suy bin v cng khng trong gii hn khng suy bin. Hai gii hn ny b tch ra th bi nhit Fermi in hnh l K i vi mt kh in t 2 chiu. Do , n suy bin trong hli lng nhng khng suy bin ti nhit phng. 1.8.3. Bn dn pha tp nh c in Cng thc ca mt in t gn vi cng thc dng trong vt l bn dn c in. Ta s a mt sai lch nh vo trong thng k bn dn c in mt phn nhn mnh rng nhiu kt qu c in khng p dng c cho cc h thp chiu. y ch gii hn xt cc h 3 chiu ng nht. Mt in

75


t n c cho bi mt biu thc ging nh (1.118) nhng vi mt trng thi hiu dng c cho bi

Trong cc bn dn c in, ta cn bao hm c CB v VB vi mt trng thi nh trn h.1.14. y CB ti ch khng phi ti E = 0 v n lm thay i biu thc ca mt in t thnh Hnh 1.14. CB v VB trong mt bn dn pha tp nh, N ch ra mc Fermi trong khe vng v cc ht ti trong c 2 vng.

nh VB l ti c tch ra t CB bi khe vng (vng cm) Trong bn dn khng pha tp, VB hon ton y in t. Do , thch hp hn ta dng mt l trng p thay cho mt in t n v dng phn b tng ng (1.110). Tng t,

76


trong l mt trng thi hiu dng trong VB. Cc gi tr ca n v p thng c iu khin bng cch thm mt nng nh ca cc tp cht vo trong bn dn. Cc cht cho (donor) gii phng cc in t vo trong CB trong khi cc cht nhn (acceptor) sinh ra cc l trng trong VB. Ngun khc ca cc ht ti l kch thch nhit ca cc in t t VB ti CB m n b nh hng mnh bi nhit . Thng thng cc ht ti b chi phi bi s pha tp. N cho vt liu khng thun v trong vt liu loi n vi cht cho ng vi mt n v th tch. Tng t, vt liu loi p vi mt cht nhn c C th rt ra mc Fermi t (1.120) hoc (1.121). Trong trng hp ca bn dn khng pha tp

Tch ca (1.120 ) v (1.121) cho

77


trong khe vng Nh vy, tch np khng ph thuc vo s pha tp cng nh v tr ca mc Fermi. Trong trng hp ca vt liu khng pha tp, H thc ny c ngha quan trng trong cc linh kin c in v c gi l phng trnh bn dn. V d nh cc ht ti ch yu trong mt bn dn pha tp n l cc in t vi nng T phng trnh bn dn suy ra mt l trng th yu l Mt hn ch quan trng ca phng trnh bn dn l vic rt ra n ch ng trong gii hn khng suy bin him khi p dng cho cc h thp chiu. N khng p dng c cho cc bn dn khi pha tp mnh. xc nh chnh xc n v p ta cn mt phng trnh khc. Mt mu v m gi trung ho in v do , mt in tch dng tng cng bng mt in tch m tng cng hay Kt hp n vi phng trnh bn dn cho mt phng trnh bc 2 i vi n hoc p theo iu ny ch ng trong mt h ng nht c nng ca tp cht v ht ti khng i trong ton b khng gian. Tuy nhin, cc h thp chiu thng c pha tp mt cch chn lc sinh ra cc ht ti trong mt vng t cc cht cho ch khc trong khng gian. iu kh hn l cn xc nh nng

78


ca cc ht ti trong trng hp ny. Tt c cc kt qu ny i vi s lp y cc trng thi b gii hn cho cc h cn bng. Khng c h cn bng no c duy tr chnh xc khi h b nhiu t cn bng. V d nh trong vng ngho ca mt it pn di tc dng ca th hiu dock v pha trc. Khi , ta cn l thuyt vn chuyn m t h. Gi s ta bit n hoc p. Khi , t (1.120) hoc (1.121) c th rt ra mc Fermi khng phi cn bng. V d

l mt mc Fermi thc v h khng cn bng. Do , n c gi l mc chun Fermi hay imref (Fermi c ngc). Imref ca in t khc vi imref ca l trng. l mt cch pht biu khc ca V d trong mt it p-n di tc dng ca th hiu dch v pha trc. Vic s dng cc imref c th gy nhm ln v n c th em li l cc ht ti c phn b cn bng ngoi tr s thay i mc Fermi. iu ny thng xa vi thc t nhng k hiu thng c s dng.

79


1.8.4. Kh in t Sau khi xem xt gii hn khng suy bin ca thng k bn dn c in, ta s nghin cu trng hp ngc li l kh in t t do suy bin v b mt Fermi ca n. Gi thit h nhit T = 0 v f(E) l mt hm bc ti Ngng c cho ging nh mt nng lng v i vi cc in t t do. Do , tn ti s sng Fermi tng ng vi Trong khng gian 3 chiu, n xc nh mt mt cu Fermi: tt c cc trng thi bn trong b mt ny c lm y ti T = 0 trong khi cc trng thi ngoi b mt ny u b trng. B mt tr thnh ng trn trong trng hp 2 chiu v mt cp im trong trng hp 1 chiu. Nng lng o t y vng c gi l nng lng Fermi v l mt php o mt in t. C th rt ra s sng Fermi khi s dng phn b u ca cc trng thi trong khng gian Trong trng hp 2 chiu, mt trng thi trong khng gian l v din tch hnh trn Fermi l Do , mt in t l

80


Do , N l cch d nht tm v Cn c cc i lng Fermi khc nh vn tc Fermi i vi cc in t t do (vn tc nhm ti ni chung). Khi bit mt v khi lng hiu dng ca kh in t, c th dng cc cng thc ny tm v xc nh xem kh in t l suy bin hay khng suy bin nhit cho. B mt Fermi c dng phc tp hn nu mt tnh i xng cu vi cc dng c bit trong mt s vt liu. N c th c nh ngha nh l b mt ng nng m th tch ca n duy tr mt mt in t chnh xc. i vi cc bn dn thng thng, php gn ng cu ch duy tr trong nhng trng hp c bit nh vng dn ca GaAs. 1.8.5. Cc hm phn b khc C nhng nh lut phn b khc i vi cc ht khng phi l fecmion. Cc ht c in tun theo phn b Boltzmann

81


trong l th ho hc v n thay cho trong phn b Fermi-Dirac. N ging nh gii hn khng suy bin ca phn b Fermi v c s dng rng ri i vi cc bn dn c in. Cc bozon tun theo thng k Bose-Einstein

Cc bozon l nhng ht mang xung lng gc khng hoc nguyn theo n v iu ny khc vi cc fecmion ly cc gi tr bn nguyn. V d v bozon l mt nguyn t . Nhiu bozon khng phi l ht theo ngha thng thng. Cc phonon v photon l nhng bozon quan trng nht. Th ho hc c iu chnh gi cho s ht khng i khi nhit thay i. Cng mt lp lun c th p dng cho nhng khng p dng c cho cc phonon v photon sinh ra hoc mt i mt cch ty . Do , phonon v photon khng c th ho hc. V d s phonon trong mt kiu c tn s c cho bi

82


iu ny l quan trng khi tnh tc tn x ph thuc vo s phonon trong mi mt kiu. N c gii hn n gin (nng lng nhit trn nng lng ca mt phonon) nhit cao. Dng ca phn b Bose khc nhiu so vi phn b Fermi nng lng thp. N ch c nh ngha i vi v tin ti khi (hoc i vi phonon hoc photon). iu c ngha l cc bozon bo ton v d nh tri qua s ngng kt Bose kt thc trng thi c bn khi nhit gim ti khng. Tnh cht ca cc fecmion v bozon l tri ngc nhau trong gii hn ny: cc bozon c xu hng dn vo mt ch v nn nhiu nht vo trng thi thp nht trong khi cc fecmion b p gi tch ri v ch c th c mt ht ng vi mi mt trng thi. Cc hm Fermi, Boltzmann v Bose c v cng nhau so snh trn hnh 1.15. C mt s gii hn khc p t ln gi tr ca cc hm lp y. Th nht l chng ch p dng cho cc h ln hoc cc h nh tip xc vi mt bnh cha c th cung cp cc ht v nng lng. Chng khng p dng c cho cc

83


Hnh 1.15. Cc hm phn b Fermi-Dirac, Boltzmann v Bose-Einstein c v trn cng mt thang o chung theo

in t bn trong mt chm lng t (mt nguyn t nhn to c lp). Mt gii hn th hai l cc ht cn phi khng tng tc vi nhau v ta gi thit rng xc sut mt in t chim gi mt trng thi khng b nh hng bi cc in t chim gi cc trng thi khc. N lun lun l mt php gn ng khi cc in t tch in v y nhau v n khng ng trong mt s trng hp. Mt v d l s lp y ca mt trng thi tp cht trong mt bn dn m n c th by ln ti 2 in t vi cc spin ngc nhau. Do , c 4 trng thi

84


kh d c minh ho trn hnh 1.16. Hnh 1.16. 4 trng thi kh d ca mt tp cht m n c th by ln ti 2 in t.

(i) trng, nng lng 0; (ii) mt in t, spin ln, nng lng ; (iii) mt in t, spin xung, cng nng lng ; (iv) hai in t, mt spin ln v mt spin xung, nng lng Nng lng b sung U trng thi (iv) c gi l nng lng Hubbard U v biu din s y Coulomb gia 2 in t. N c ngha l in t th hai lin kt km cht hn so vi in t th nht. Nng lng lin kt ca in t th hai thng ch vo khong 5% nng lng lin kt ca in t th nht i vi mt cht cho n gin trong mt bn dn. Xc sut tm thy h trng thi vi nng lng E v n in t t l vi S in t trung bnh trn tp cht khi bng

85


kim tra (1.128), ta b qua tng tc U gia cc in t. Khi , vi f l hm Fermi i vi cc in t khng tng tc. Thng gii hn ngc li c duy tr vi th y U ln n mc xc sut 2 in t chim gi tp cht c th b qua. Khi ,

Mc d tp chc chn b chim gi bi mt in t ti mt thi im no , xc sut khng c cho bi hm Fermi thng thng v 2 gi tr kh d ca spin. T (1.129) suy ra s lp y c th c to ra lm ging mt hm Fermi nu s hng c thm vo th ho hc v iu ny thng c s dng trong cc tnh ton bao hm s lp y ca cc tp cht.

86

Bi tp chng 1

1.1. Dng m hnh h th su v hn c tnh mt vi mc nng lng u tin i vi mt in t trong GaAs trong cc h c b rng 10 nm v 4 nm. Khi lng m cn c thay bng trong l khi lng hiu dng i vi cc in t y CB ca GaAs. Kt qu th no i vi cc h thc trong GaAs/ AlGaAs c su khong 0,3 eV? Lp li tnh ton i vi cc l trng trong h th c su khong 0,2 eV. Cc l trng gm cc l trng nng v l trng nh vi cc khi lng hiu dng tng ng l 1.2. Hnh 1.4 ch ra mt s s o thc nghim ca cc mc nng lng trong cc h lng t thay i b dy. Mu c mt s h lng t ca GaAs c b dy a thay i trong mt lp mng ca Al0,35Ga0,65As. Cc vt ch ra cng quang pht quang (PL). Nng lng ca cc photon pht ra l do s khc bit nng lng ca in t v l trng mt trong cc h. Cc h hp hn c cc trng thi giam cm vi nng lng cao hn v do pht ra nh sng vi tn s cao hn (bc sng ngn hn). iu to ra h 4 vch c ch ra trn vt PL. Nu cho b dy ca 4 h, m hnh ht trong hp d on nng lng PL nh th no? Gi thit cc in t v l trng c khi

87

Bi tp chng 1

lng hiu dng Ci g gy ra nh nhn? 1.3. Cc mt dng v in tch rt ra trong phn 1.4 c bo m bo ton in tch vi iu kin l l thc. Chng minh rng cc ht s c sinh ra hoc b hp th nu V l phc. 1.4. Xy dng hm sng m t cc in t chuyn ng theo hng x vi vn tc v mang mt mt dng l 1.5. Rt ra mt dng i vi cc sng chy v phn r lan truyn ngc (cc phng trnh (1.34) v (1.36)). 1.6. Chng minh rng i vi in t trng thi n trong mt h vung gc su v hn c b rng a,

V phng din c in, ht chc chn tm c bt c ch no trong h (tr khi ht dng li). Chng minh khi v lu l kt qu c hc lng t tip cn vi gi tr ny i vi cc nng lng ln (n ln). 1.7. Xt s truyn ca cc b sng vi cc b rng ban u d = 25 nm v d = = 50 nm trong GaAs Phc ha i vi 2 b sng.

88

Bi tp chng 1

Mt bao lu chng c chiu di gp i? 1.8. S dng chng minh i vi dng chung ca ton t Hamilton chng minh 1.9. Mmen xung lng qu o trong c hc c in c xc nh bi Tng t, trong c hc lng t, ton t v.v... Chng minh rng cc thnh phn khc nhau ca khng giao hon vi nhau, c th l Mmen xung lng ton phn c nh ngha bi Chng minh rng Do , c th xc nh ng thi mt thnh phn ca mmen xung lng ton phn v bnh phng mmen xung lng ton phn. 1.10. Ba trng thi thp nht trong mt th parabol c th vit di dng trong l mt a thc Hermite bc n. V d nh Chng minh rng cc trng thi trc giao vi nhau. 1.11. Hm sng thng c s dng nh mt php gn ng i vi trng thi ring thp nht trong th tam gic i vi

89

Bi tp chng 1

i khi cn s dng mt php gn ng cho trng thi th hai. Hy vit n di dng v chn c n trc giao vi trng thi thp nht. 1.12. Chng minh rng mt trng thi i vi cc in t t do trong trng hp 2 chiu (trng hp n gin nht) bng

1.13. Rt ra biu thc (1.119) i vi mt trng thi hiu dng i vi mt h 3 chiu. Cho

1.14. T phng trnh bn dn v phng trnh trung ha chng minh

i vi mt bn dn c in. Tm kt qu tng ng i vi p. 1.15. Chng minh rng s sng Fermi trong trng hp 3 chiu v rt ra kt qu tng ng trong trng hp 1 chiu.

90

),(txy)1.1).(,(),()(),(2222txtitxxVtxxmyyy=+-hh)()(),(tTxtxjy=)2.1.()()(1)()()(2)(1222constEttTitTxxVxxmx===+-hhjjj())3.1.(,expexp)(wwhh=--EtiiEttT)(xj)4.1).(()()()(2222xExxVxxmjjj=+-h.222222222zyxx++=D=)5.1.(exp)(),(-=hiEtxtxjy)6.1).(()(2222xExxmjj=-h()().cos;sin;exp;exp)(kxkxikxikxx-+=j)7.1).((2022kmkEe=h.22kpmpEh==nwhE==hlhkp==hmk22h=w)10.1(,22mpmkkvph===hw)11.1(,clgvmpmkdkdv====hwclvkmkE=0222hkE

e).2/()(0220mkkh=e.7,03,0AsGaAlCE,0emm),2/()(022hVVmmkEkh+=e067,0=em).2/()(022eCCmmkEkh+=eCEDeVEC3,02,0-Den)13.1.(2202220ammnEeeGaAsCnepeh+VE5,0(=hhmmGaAsgE=whGaAsgGaAsgEE(>wh.VCgEEE-=VEVE11VCee-hn)14.1.(2202220ammnEhhGaAsVnhpeh-AlGaAsgE>whehm.AlGaAsgE>wh)15.1.(11220222++=heGaAsgmmamnEph1Ve1Ce.nnnhe===--+=-=202222022222ammnEammnEhGaAsVeGaAsCVnCnnppeewhhh./1/1/1heehmmm+=.1whGaAsgE

whAlGaAsgE>wh),(txy2),(txy2),(txqydxtxq2),(y==)(),(2xtxqry.),()(2qdxtxqdxx==yr.1),(2=dxtxy2),(txy..sin2)(=axnaxnpj)22.1.(2.sin1,.sin)(0222===nnnnAadxaxnAaxnAxppjnA2/1-22)(Axk=j)25.1.(),(),(),(),(2222ttxitxtxVxtxm=+-yyyhhikxkAex=)(j0=+txJrrxJ/.Jdivr)26.1.(2**22*2tiVxm=+-yyyyyyhh*y)29.1(22***+=xxxxxyyyyyy),(txV2**yyyyytititi=+hhhy())31.1.(22**tqtxximqx==--ryyyyyh)27.1.(2***2*22tiVxm-=+-yyyyyyhh)28.1.(222*222*2yyyyytixxm=--hh)30.1.(22**2yyyyytixxxm=--hh)32.1.(2),(**-=xximqtxJyyyyhx/y.y()[].exp),(tkxiAtxwy-=2Aq=rvmpAmkqJrr===)/()/(2h.nevJ=()()[]()tiikxAikxAtxwy--+=-+expexpexp),(()())34.1.(/22-+-=AAmqkJh()()[]())35.1.(expexpexp),(tixBxBtxwkky--+=-+()())36.1.(Im2***-+-+-+=-=BBmqBBBBimqJkkhhxek)39.1.()38.1(,)37.1(,tiEExippxxx=-==h)h))()()40.1).(2,222xVxmpxHH+-==h)))))()2/(2xVmpH+=H)),(txy)41.1).(,(),()(2222txtitxxVxmyy=+-hh()())43.1.(jjkkAeAexipikxikxhhh)==-=y),()(XExHjj=)yyEH))=)42.1.(2),(**+=yyyympmpqtxJ))p)()ikxAxexp)(=j.kph=)2/(2mpT))=)44.1.(.cos.sin)(axnAaniaxnAxixpnnnpppfhh)-=-=)(xnf.2p)=)45.1(,),()(dxtxxPtx==)48.1.(),(),(),(2*22dxtxxtxdxtxPxxyy...),(txy2),(),(txtxPy)47.1(,22xxx-=D==)46.1(.),(),(),()(*2dxtxxtxdxtxxtxyyy2x2x==aadxaxxax02)49.1(,2.sin2p-==)53.1.(),(),(),(),(**dxxtxitxdxtxptxpyyyyh))51.1.(18,021121)50.1(2131.sin22220222aaaadxaxxaxa-=D-==ppp)52.1(,),(),(*dxtxqtxqyy=)q)())55.1.(cos9322)(12221--=htAaAatxeep)54.1).(()()0,(2211xAxAtxffy+==pD2p*yyx1e2e()())57.1.(4exp21)0,(2202/122--==dxxdtxpy0xx=()())56.1.(2expexp21)0,(22004/12--==dxxxipdtxhpy0=t()h/exp0xipdx=D0xpf)59.1.(2exp),(),()58.1(,2exp),(),(-=+=hhhhpyfpfydxiiptxtpdpiiptptx),(tpf()h/expipxpD0php2y),(txy()[]()()()()[]()())61.1.(2/2exp2/21)0,()60.1(,2/4expexp2/21)0,(2202/122220004/12--==----==dppdtpdppxppidtphhhhhpfpf0pp=)62.1.(22.hh==DDddpx().2/dph=D.=Dx)63.1.(2/hDDpx0=Dp().)/2(2/22amh()mp2/2Dp()xpDD2/hxD0=DDpxpDxD4/axD./2ah=()[]()()())64.1(,2exp2/4expexp2/21),(2220004/12-----=mtipdppxppiatphhhhpfpxDD()tiw-exp()h/expipx()[]()()()()())65.1.(2212/4exp2/exp2/exp2/21),(2220000004/12hhhhhhppydpmdtidppmtpxxppimtpxipdtx+------=+-.2/2mp=wh0pmtpx/00+.2/0mpvph=./0mpvg=)66.1.(2)(22+=DmdtdtxhpxDDmp/D())2/(/mdtmtpxh=DD)68.1.(+-=-=yyyyxxixxixphh))[])70.1.(,h))))))ixppxpx=-=)67.1.(xxixixpx-=-=yyyhh)))69.1.(yyyh))))ixppx=-yA)B)[].0,=BA)))71.1.(,pixpp==h))[][].0,,,==xypyipy))h))H)[].0,=Hp))p)x)[]h))ipx=,=0)()(*dxxxnmff.nmee=)75.1()(0)(1)()(,*===nmnmdxxxnmnmdff0.sin.sin20=axnaxmaapp=)72.1.(1)(2dxxnf)73.1.(nmeene)(xnf.0ax

>-FE().,,TEEfF()TkEEBF-

FE0

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