Chng 2: in t v phonon trong tinh th

Chng 2: in t v phonon trong tinh th 2.1. Cu trc vng trong trng hp 1 chiu Th nng trong tinh th thc phc tp hn so vi cc h nghin cu nhng c tnh tun hon. Trong trng hp 1 chiu, n c ngha l trong a l hng s mng hay kch thc ca mi mng n v ca tinh th. Mt v d khng thc ca matt th tun hon c v trn hnh 2.1. Th tun hon c khai trin thnh chui Fourier nh sau

l ln ca cc vect mng o. Tn ny thch hp i vi s chiu cao hn. Hnh 2.1. Mt v d v mt th tun hon m n ch ra hng s mng a. Tnh tun hon ca th ng iu g i vi cc hm sng? Cc hm sng ca mt h v hn vi mt th u l cc sng phng trong t h s chun ho bng 1. Mt ca chng khng i

2.1. Cu trc vng trong trng hp 1 chiu khp ni. N c k vng do th l nh nhau khp ni. Do , khng c l do g mt in t thch im ny hn im khc. Ni cch khc, h bt bin tnh tin. iu kin ny b ni lng khi c mt th tun hon nhng c th xem n l hp l k vng mt thay i theo cng mt cch trong mi mt mng n v. V th, ta hi vng mt l mt hm tun hon ging nh th, ngha l iu ny c th t c nu ta nhn cc sng phng ca khng gian t do vi mt hm tun hon Kt qu c gi l hm Bloch

Mt l tun hon nh i hi mc d cc l khc nhau i vi mi mt gi tr ca k. (2.2) l mt pht biu ca nh l Bloch i vi cc hm sng trong mt tinh th. Mt dng tng ng ca (2.2) l

K hiu k by gi gi l s sng Bloch v v dng (2.3) chng t ka cho s thay i pha ca hm sng gia cc mng n v. k khng phi l mt s sng thng thng vi ngha l xung lng c hc ca ht v s c

2

2.1. Cu trc vng trong trng hp 1 chiu ca c ngha l hm sng cha nhiu xung lng. c gi l xung lng tinh th. Trong nhiu trng hp, n ging vi xung lng thng thng. Gi s k nm trong khong Do , Khi , Hm tun hon theo x vi cng chu k a nh C th ghp chng vi nhau c mt hm tun hon mi

Ta rt gn s sng Bloch v khong m n c gi l min Brilloun th nht. C th lm iu tng t i vi mi gi tr ca k bng cch tr i mt bi s thch hp n ca Ni cch khc, ta thm mt vect mng o thch hp vo k. Khi , c th dng cp k hiu trng thi thay cho k. Hm sng tr thnh

S th nht trong k c th ly bt k gi tr no c gi l s min m rng k hiu hm sng. Cn s th hai trong c s dng

3

2.1. Cu trc vng trong trng hp 1 chiu

Hnh 2.2. Cu trc vng ca mt tinh th 1 chiu trong cc s min (a) m rng, (b) rt gn v (c) lp li. (d) l mt trng thi theo nng lng. Cc ng m ch ra trong mt th tun hon yu vi cc vng c k hiu bi n trong lc cc parabol mnh l i vi cc in t t do.

4

2.1. Cu trc vng trong trng hp 1 chiu v n c gi l s min rt gn. S min lp li l s th ba s dng tnh tun hon theo k lp li tt c cc hm theo mi mt Vic s dng cc s ny m t cc vng nng lng nh trn hnh 2.2. 2.1.1. S to thnh cc khe vng Ta xem xt cc hm sng khi c mt mt th tun hon v by gi xt n cc nng lng. Chng c k hiu vi s sng trong min Brillouin th nht v ch s vng n thnh i vi a s s sng, nng lng i vi cc in t t do. Cc ngoi l l cc bin vng trong , y, thot khi dng parabol ca in t t do tr thnh phng m n gp bin vng. iu ny xy ra c trn v di v li mt khe gia trong tinh th khng c cc trng thi lan truyn. Cc khe vng ny chia thnh cc vng nng lng. Chng l mt c im chung ca mt h tun hon v c th xem xt ngun gc ca chng theo mt vi cch. Cu trc vng thng c v trong s min rt gn. N c dng bng

5

2.1. Cu trc vng trong trng hp 1 chiu cch tnh tin cc phn ca min m rng qua cc vect mng o i vi parabol ca in t t do, phn trung tm vi ng yn. Phn gia v c tnh tin qua chuyn n vo min Brillouin th nht trong lc phn gia v c dch chuyn bi Xt tng t i vi cc phn nng lng cao hn cho cu trc an cho nhau nh ch ra bi ng mnh trn hnh 2.2(b). S gp np ca min ny l do tnh tun hon ca tinh th v lm cho nng lng c v phc tp. Nng lng trong tinh th c nghin cu theo cng mt cch. Cc on vi cc du ngc li ca k c ghp ni cho cc ng cong trn lm y min vi cc khe bin hoc trung tm. Cui cng, cc vng nng lng trong min rt gn c th c lp li cho s min lp li trn hnh 2.2(c). N t c s dng hn nhng lm sng t mt s vn v s vn chuyn bao hm bin vng. Xt ngun gc khe vng. Mt sng tn x t cc nguyn t theo mi hng ging nh lan truyn qua mng tinh th. Trong a s trng hp, cc ng gp t cc nguyn t khc nhau c xu hng trit tiu ln nhau nhng theo cc hng no , chng giao thoa mang tnh xy dng v to ra mt chm

6

2.1. Cu trc vng trong trng hp 1 chiu Hnh 2.3. iu kin i vi phn x Bragg ca mt sng c bc sng ti theo gc ln cc mt ca mt tinh th nm cch nhau mt khong l a.

tn x mnh. l s phn x Bragg do bn cht tun hon ca mng c minh ha trn hnh 2.3. Mt chm ti to ra mt chm tn x t mi mt mt mng. S khc nhau v qung ng gia cc chm ln cn cn phi l mt s nguyn ln bc sng nu chng tng cng ln nhau. T tam gic ch ra rng s khc nhau ny bng Do , u kin ca phn x Bragg l Trong tinh th 1 chiu, cc in t chuyn ng vung gc vi cc mt. Do , (tn x ngc) v iu kin Bragg tr thnh Khi iu kin ny c tha mn, tn x ngc tr nn mnh n mc in t khng th lan truyn qua tinh th v tr thnh mt

7

2.1. Cu trc vng trong trng hp 1 chiu sng ng. S tn x t mng xy ra i vi tt c cc s sng mc d n rt mnh ti iu kin Bragg. Xt mt sng phng vi s sng k v nng lng truyn qua mng. S tn x trn ln cc thnh phn vi s sng i vi mi n. l ni a vo cc vect mng o. Ni chung, cc thnh phn thm vo c nng lng khc nhau. Do , s trn l yu. N tr nn mnh ch khi cc sng c cng nng lng m iu ny i hi cc s sng c cng ln v k c nh: Ta gp li iu kin Bragg cho phn x mnh v vic m ra cc khe vng xy ra bt c khi no iu ny nhn mnh vai tr ca cc vect mng o. S trn ca cc sng phng l c s ca l thuyt in t gn t do. Mt cch tng ng khc xem xt s hnh thnh ca cc khe vng l cn lu rng lun lun c mt cp trng thi suy bin trong trng hp in t t do l m c hai trng thi ny u cho cng mt . thay th, chng c th c kt hp vi nhau cho v m mt ca chng dao ng. Cc dao ng ny l khng quan trng trong

8

2.1. Cu trc vng trong trng hp 1 chiu khng gian t do trong c th khng i nhng tr nn c ngha khi chng chp ln th tun hon nh minh ha trn hnh 2.4. Trong trng hp

Hnh 2.4. Th tun hon v cc mt gn vi 2 hm sng ti bin min th nht

ny, mt cosin to iu kin h tr phn a thch hn ca th tun hon trong khi mt sin to iu kin hng ti cc i hn l cc thung lng. Nh vy, by gi sng cosin c nng lng thp hn v tnh suy bin gia 2 trng thi b ph v sinh ra khe vng. L lun ny ch c tc dng khi mt v th c cng chu k v tng tc mt cch hp tc m n li i hi

2.2. Chuyn ng ca in t trong cc vng Cc in t t do c vn tc nhm ph hp vi vt l c in. Cc kt qu mi tm c i vi mt tinh th.Xt php gn ng

9

2.2. Chuyn ng ca in t trong cc vng cosin n gin cho hnh dng ca vng (b i ch s di n k hiu vng)

N c cng mt dng chung nh vng thp nht n = 1 hoc bt k vng nh s l no khc i vi W dng. Du ca W thay i i vi n chn. Nng lng bt u t 0 v c b rng y l nh ch ra trn hnh 2.5(a). V d ny c cc gi tr gn ng W = 5 eV v a = 0,5 nm i vi mt bn dn thng thng. Khai trin ca cosin i vi k nh cho N l mt parabol v vic so snh n vi dng thng thng ch Hnh 2.5. (a) Nng lng, (b) vn tc v (c) khi lng hiu dng nh mt hm ca s sng i vi mt vng cosin c b rng W = 5 eV trong mt tinh th c hng s mng a = 0,5 nm. ra rng vng cosin c mt khi lng hiu dng Do ,

10

2.2. Chuyn ng ca in t trong cc vng mt vng rng cho mt khi lng hiu dng nh. C th tin hnh mt khai trin tng t gn nh vng xung quanh Vn tc nhm N rt gn thnh i vi k nh m tn ti php gn ng parabol nhng tnh cht chung ca n th v hn. Vn tc gim ti 0 ti c hai u mt ca vng (hnh 2.5(b)). Trn hnh 2.4, cc hm sng tng ng l cc sng ng khng mang dng. C th nh ngha mt khi lng hiu dng trong ton vng ch khng phi ch xung quanh k = 0. i vi cc in t t do, v

Nng lng, vn tc v khi lng hiu dng c v trn hnh 2.5. Khi lng hiu dng ny c ch nht gn cc khe vng m n gn nh khng i. Ti nh vng, khi lng hiu dng m. Do , s tng ca k lm cho v gim. l cc in t t do khc thng. Phng trnh chuyn ng i vi mt ht c in tch q trong in trng v t trng l

11

2.2. Chuyn ng ca in t trong cc vng N c vit trong trng hp 3 chiu. Khi , vn tc nhm c cho bi

Vn tc bng 0 ti c nh v y vng. Mt ht chuyn ng theo mt hng khi n chu tc dng ca mt in trng nu n bt u ti y vng nhng n chuyn ng theo hng ngc li nu n bt u ti nh vng. iu l do khi lng hiu dng m ti nh vng. Tuy nhin, mt cch nhn khc l l dng iu chung l tng (i vi khi lng dng) i vi mt ht c in tch ngc li. Nh vy, cc in t tch in m vi khi lng hiu dng m nh vng c th c m t ging nh cc l trng tch in dng vi khi lng hiu dng dng. Xt mt in t bt u t k = 0 trong mt vng trng v c gia tc bi mt in trng khng i l F (du tr l b i du ca in tch in t). Kt qu ch ra trn hnh 2.6. Phng trnh chuyn ng l m n tng u. Theo hnh 2.5(b), vn tc ca n lc u tng, sau tin ti mt cc i khi s sng i qua v tip theo gim. N gim ti 0

12

2.2. Chuyn ng ca in t trong cc vng khi N tng ng vi khi s dng tnh tun hon ca khng gian k hoc s min lp li. Sau , vn tc tr nn m trc khi gim tr li 0 mt ln na khi k tr li 0 v iu c lp li mt cch tun hon. S tun hon ny c th hin qua cc phng trnh

T suy ra cc in t dao ng c trong khng gian k v khng gian thc. iu ny khc vi in t gia tc u khi khng c mt mt th tun hon. in t khng th x s nh trong trng hp c in do iu ny i hi ng nng ca in t tng khng c gii hn v sung trong mt tinh th c mt b rng hu hn. Hin tng chuyn ng tun hon ny khc vi s gia tc u v c gi l. s dao ng Bloch (hnh 2.6). N c coi l mt ngun bc x vi sng kh d. Tn s khng ph thuc vo b rng vng m ch ph thuc vo s tn hao nng lng bi in trng qua mi mng n v. T cc dao ng Bloch suy ra dng iu ca cc in t trong mt vng y . Cc in t ny khng th tn x bt k ch no trong vng ca chng do tt c cc trng thi u b lp y. Do , chng lm cho cc sng Bloch

13

2.2. Chuyn ng ca in t trong cc vng dao ng v vng nh mt ton b khng mang dng. Ti sao cc in t trong mt vng lp y mt phn mang mt dng trong mt in trng thay v tin hnh cc dao ng Bloch? Cu tr li l cc in t trong mt tinh th thc b tn x (bi cc tp cht, cc phonon hoc cc in t khc) v cc dao ng b gin on lu trc khi hon thnh mt chu trnh. Cc in t li bt u nhng c tn x mt cch lp li che du dao ng. Gn y vng, v phng trnh chuyn ng trong mt in trng c dng

Hnh 2.6. Cc dao ng Bloch nh l mt hm ca thi gian c sinh ra bi in trng trong mt vng cosin c b rng 5 eV.

14

2.2. Chuyn ng ca in t trong cc vng S hng l php gn sung thi gian hi phc p dng cho tn x m n cho m hnh Drude. By gi, vn tc tip cn vi gi tr dng ti nhng thi gian ln trong l linh ng. iu ny c th c vit ging nh kt qu quen thuc i vi dn trong n l mt ht ti. y, l thi gian hi phc xung lng. Cc khe vng cn c gi l l cc khe cm (hay vng cm). Chng b cm nh th no? Mt in trng hoc t trng mnh c th lm cho cc in t i xuyn hm qua khe vng ti vng tip theo. N c gi l s xuyn hm Zener hoc s ph v t v c minh ho trn hnh 2.7 i vi mt in trng. Cc khe ca cc vng c lm nghing do th nng eFx t Hnh 2.7. Cc vng ca h 1 chiu trong mt in trng u c v trong khng gian thc. N ch ra cc dao ng Bloch v s xuyn hm Zener. trng. Mt in t c nng lng khng i b p phi chuyn ng gia cc khe vng vi iu kin l n bn trong vng ca n. iu ny cho php n chuyn ng dc theo chiu di W/ eF ging nh iu xy ra i vi cc dao ng Bloch tm c t (2.14). Tuy nhin, khi n t ti cui khong

15

2.2. Chuyn ng ca in t trong cc vng cho php ca n, c th cc in t i xuyn hm qua khe vo trong vng tip theo thay v phn x t khe. c tnh th tc xuyn hm, ta ly hng ro c b cao v b dy S sng phn r trong khe khi l Xc sut xuyn hm trong mi mt chu trnh dao ng Bloch c chu k

Phng php WKB l php gn ng tt hn. Tc tin ti 0 trong trng yu nhng c th tr nn ng k nu trng rt mnh hoc khe vng nh. Mt ng dng ca kt qu ny l it Zener m n da trn s xuyn hm qua khe vng gy ra s ph v ngc trong cc it p n pha tp mnh (mc d nhiu it Zener bao hm s ph v dn dp). 2.3. Mt trng thi Cu trc vng cng lm thay i mt trng thi. c th c lm gn ng l mt parabol ti c hai cc tr ca vng ging nh cc in t t

16

2.3. Mt trng thi do nhng vi khi lng hiu dng khc. Cc in t t do trong trng hp 1 chiu c mt trng thi ti y vng v do ta hi vng iu ny y ca vng tun hon v c nh. H thc gia n(E) v vn tc l

i vi vng cosin. iu ny chng t nghch o ca cn bc 2 phn k ti nh v y. N p dng cho mi mt vng trong tinh th v kt qu c v trn hnh 2.2 (d). Tch phn ca mt trng thi trong vng cosin (ng vi mt n v chiu di) l

2 tnh n spin v 1/ a chng t c mt trng thi khng gian ng vi mt mng n v c chiu di a. l mt kt qu tng qut. Kt qu trn i vi s trng thi trong mt vng cho php ta d on vt liu 1 chiu l kim loi hay l in mi. Mt s l cc in t ng vi mt nguyn t cho mt vng nh lp y mt na. C cc in t t do ti mc Fermi v n cho mt kim loi. Mt khc, mt s chn cc in t chnh xc

17

2.3. Mt trng thi lp y mt s vng v trng phn cn li. Mc Fermi nm trong mt khe vng gia trng thi b chim gi cao nht v trng thi trng thp nht v vt liu l in mi hoc bn dn. Bc tranh ny c th p dng cho nhiu vt liu 1 chiu trong c cc polyme. Mt v d l polyaxetylen N c mt s l cc in t ng vi mt n v v do , n phi l mt kim loi. V ho hc, cacbon ng gp mt in t vo mt lin kt vi hir. Cn ba in t chia s hai lin kt vi cc n v ln cn. Nh vy, m hnh ny d on c 1,5 lin kt gia cc ln cn Thc t l cu hnh ny khng bn. Chui b mo sao cho cc lin kt n v kp xp xen k dc theo chiu di ca xng sng cacbon v cho mt kiu mu ca (-CH=CH-). n v c chiu di gp i v min Brillouin b chia i. Do , by gi c mt s chn cc in t ng vi mt n v v kt qu l mt bn dn. Bin dng gi l s mo Peierls lm gim nng lng ca h v l mt tnh cht chung ca cc kim loi 1 chiu. S mo mng km ph bin hn trong trng hp 3 chiu. Polyaxetylen c th c pha tp v d vi it v n em li dn cao.

18

2.3. Mt trng thi Nhiu vt liu 1 chiu dn tt hin nay u da trn cc polyaxetylen hoc cc tinh th d hng cao ca cc mui hu c. Chng c th c pha tp v c dng ch to cc linh kin trong ng ch l cc it pht quang (LED). By gi ta s m rng nguyn l ny cho cu trc vng trong trng hp nhiu chiu hn. 2.4. Cu trc vng trong cc trng hp 2 chiu v 3 chiu Cc nguyn l cu trc vng trong cc trng hp 2 chiu v 3 chiu ging nh trong trng hp 1 chiu nhng phc tp hn. Mng o by gi l mng 2 chiu hoc 3 chiu v h thc tn sc cn phi c v th nh l mt hm ca vect sng Vect sng ny li c th chuyn v min Brillouin th nht bng cch cng vi cc vect mng o. Xt mt v d 2 chiu. l mt mng hnh ch nht vi khong cch gia cc nt mng l a dc theo x v b dc theo y. Th tun hon V(x,y) c th c khai trin thnh chui Fourier

19

2.4. Cu trc vng trong cc trng hp 2 chiu v 3 chiu

trong l cc vect mng o. Chng c th c sp xp to ra mt mng o trong khng gian o hay khng gian theo cng mt cch nh cc im ca mng thc hay mng trc tip nh ch ra trn hnh 2.8. C hai mng u l hnh ch nht nhng c cc t l hnh dng tri ngc nhau (nu mt mng l di v mng th mng kia l ngn v dy). Hnh 2.8. Cc mng hnh ch nht 2 chiu trong (a) khng gian thc v (b) khng gian o. Th tun hon li sinh ra cc khe vng v chia khng gian thnh cc min Brillouin. S to thnh cc khe l do s trn ca cc hm sng. Mt hm sng vi vect sng thu c cc thnh phn t mi mt h s Fourier ca th. Ni chung, chng c cc nng lng khc nhau v s trn l nh. S trn mnh m n dn ti cc khe vng xy ra khi cc nng lng l nh nhau m n i hi V tri l khong cch ca ti gc trong lc v phi l khong cch ca t

20

2.4. Cu trc vng trong cc trng hp 2 chiu v 3 chiu v do , ng thc ny xc nh mt mt phng chia i on thng t gc ti iu ny c minh ha trn hnh 2.9. Hnh 2.9. S pht sinh ca mt khe vng bng mt vect mng o Khe xut hin trong mt phng m ,

Th tun hon m ra mt khe vng trn mi mt trong cc mt phng ny m chng chia khng gian thnh cc min Brillouin. Mi mt im trong mng o sinh ra mt mt phng v kt qu i vi mng hnh ch nht c ch ra trn hnh 2.10. Min th nht l mt hnh ch nht n gin. Ngoi min ny, khng gian c chia thnh cc mnh vi s phc tp tng ln. Trong s min rt gn, cc mnh ny cn c tnh tin vo min th nht bng cch cng vi mt vect mng o thch hp. Kt qu i vi 4 min u tin c ch ra trn hnh 2.10. Nu th tun hon l yu, mt c tnh cu trc vng c th tm c bng cch qun ngc tr li parabol ca in t t do vo trong min th nht.

21

2.4. Cu trc vng trong cc trng hp 2 chiu v 3 chiu

Hnh 2.10. 4 min Brillouin th nht ca mng hnh ch nht c biu din bi cc vng ti trong mng o. Chng c tnh tin thng qua cc vect mng o xy dng cc hnh ch nht ni tip nh ch ra bn phi.

22

2.5. Cu trc tinh th ca cc cht bn dn thng thng

Cc bn dn thng thng to thnh cc tinh th i xng cao v tnh i xng ca chng l mt thnh phn quan trng trong l thuyt hin i. V d c th ch ra cc qu trnh quang no l c php hoc b cm t ring tnh i xng m khng cn bit chi tit v cu trc vng hoc cc hm sng. L thuyt nhm cung cp khun kh ton hc phn tch i xng.Cc bn dn thng thng c cc tinh th lp phng. Ba mng lp phng c ch ra trn hnh 2.11(a)-(c). Mng lp phng n gin ch c mt nguyn

Hnh 2.11. (a) Mng LPG, (b) mng LPTK, (c) mng LPTD, (d)mng kim cng, (e) mng sunfuakm ZnS. (f) ch ra cao ca cc nguyn t theo n v hng s mngi vi mng ZnS.

t trong mt n v c th tch Mt nguyn t khc c th c thm tm ca mihnh lp phng to ra mng lp phng tm khi (bcc)vi 2 nguyn t hoc vo tm ca mi mt mt to ra mng lp phng tm din (fcc) vi 4

23

2.5. Cu trc tinh th ca cc cht bn dn thng thng nguyn t trong mi mt n v lp phng. thun tin, ngi ta thng s dng mt n v ch c mt nguyn t. Wigner-Seitz (WS) cha tt c cc im gn nguyn t gc hn so vi gn bt k nguyn t no khc. iu ny ging nh cch xy dng min Brillouin th nht trn hnh 2.10. Kt qu i vi mng LPTK c ch ra trn hnh 2.12(a) trong mng bao quanh nguyn t tm mng. Hnh dng l mt khi bt din ct ct (khi lp phng vi cc gc ca n c lm phng). N c gn kt bi cc mt phng na ng gia gc v cc im gn ca mng. 8 mt lc gic sinh ra t cc mt phng na ng ti cc nguyn t cc gc trong lc 6 mt hnh vung nh hn na ng ti cc nguyn t tm ca cc tip theo. Hnh 2.12(b) ch ra cch cc n v khng chc c thc ny lm khp vi nhau lp y khng gian Hnh 2.12. (a) WS ca mt tinh th LPTK v (b) cch xp cht cc WS

24

2.5. Cu trc tinh th ca cc cht bn dn thng thng Cc bn dn thng thng da trn c s cu trc LPTD nhng vi s nguyn t gp i. Cu trc ca cc bn dn nguyn t nh Si c ch ra trn hnh 2.11(d) v c gi l mng kim cng. C thm mt nguyn t c dch chuyn i (1/2, 1/2, 1/2)a so vi mi mt nguyn t trong mng LPTD. Mt cch khc hnh dung n l ta lu n 4 nguyn t trong mng LPTD (ti (0,0,0), (a/2, 0,0), (0,a/2,0), (0,0,a/2)) nm ti cc nh ca mt khi t din. Mt nguyn t khc khi c thm vo tm ca khi t din trong n c th to ra mt lin kt vi mi mt trong 4 nguyn t ti cc nh. Nh vy, mt mng ca s lin kt khi t din c th lp y khng gian. N tng ng vi quan im ha hc l mi mt nguyn t to thnh 4 lin kt vi cc ln cn ca n. n v lp phng ca kim cng cha 8 nguyn t. Cc hp cht nh GaAs c cu trc sunfua km ZnS nh ch ra trn hnh 2.11(e). S sp xp ca cc nguyn t ging nh trong mng kim cng nhng c 2 li xen k: Ga chim cc v tr gc ca mng LPTD trong khi As chim cc v tr ca khi t din to ra mt n v cha 4 nguyn t ca mi mt loi. Chiu di ca mi mt cnh l hng s mng nm

25

2.5. Cu trc tinh th ca cc cht bn dn thng thng trong GaAs 300 K v mt nguyn t tng cng (s nguyn t ng vi mt n v th tch) l S liu i vi cc bn dn thng thng c tng kt trong Phlc 2.Chiu di ca mi mt lin kt l nm. Cc hng trong tinh th lp phng c xc nh bng cch vit cc thnh phn Descartes ca mt vect khng c du phy trong cc ngoc vung. V d trc z c k hiu l [001] v hng z c k hiu l Cc mt c k hiu theo cch tng t bi cc ch s Miller. Ta vit mt vect vung gc vi mt phng trong cc ngoc dn. V d (001) cho mt phng xy. K hiu {001} c ngha l mt phng (001) v tt c cc mt phng c lin quan bi tnh i xng gm cc mt phng (100), v.v... K hiu trn m t cc h ca nhng mt phng cch u nhau. Nu mt mt phng nh th i qua gc, h ca cc mt phng c m t bi cc nghch o ca cc giao im ca mt phng tip theo vi cc trc theo n v hng s mng. V d xt mt phng bn tri trn hnh 2.13(c) ly gc ti gc bn tri y. N ct cc trc ti (a,0,0) v (0,a,0) v do c k hiu l (111). Nu mt phng ct cc trc ti (a/2,0,0),(0,a/2,0),(0,0,a/2) th n c k hiu l (222). c im ny l quan trng i vi cc tinh th khng lp phng.

26

2.5. Cu trc tinh th ca cc cht bn dn thng thng Hnh 2.13. Cc mt phng quan trong trong cu trc ZnS: (a)(001), (b) (110) v (c) (111).

Xt cc mt phng quan trng trong cu trc tinh th. Cc mt phng nh (001) nh trn hnh 2.13(a) vung gc vi cc trc chnh. Mi mt mt phng cha mt loi nguyn t m n xen k trong cc mt phng k tip ca ZnS cch nhau mt khong l a/4. Vic nui tinh th thng c tin hnh bng cch thm vo cc mt phng (001) v mi mt cp bao gm mt lp n. Do , c 2 lp n ng vi mt n v. GaAs thng tch ra dc theo mt phng (110) nh ch ra trn hnh 2.13(b). Cc mt phng ny song song vi trc [001] v ct cc trc [100] v [010] ti gc Chng cha c 2 loi nguyn t. Hnh v ch ra mt mt phng trong. Mt b mt l ra c dng li sau khi tch nhn mt cu trc khc. Mt phng quan trng th ba l (111) m n nghing nh nhau i vi c 3 trc chnh. Mi mt mt phng ch cha mt loi nguyn t ging nh i

27

2.5. Cu trc tinh th ca cc cht bn dn thng thng

vi (100) v mt mt phng ca mi mt loi c ch ra trn hnh 2.13.(c). T cc hnh 2.11(d) v 2.11(e) suy ra hng [111] l hng c cc trong mt bn dn hp cht do cc lin kt u ch theo cng mt hng t Ga n As. Khng c s khc bit ny trong mt nguyn t m do n c i xng cao hn. Xt im ti (1/8, 1/8, 1/8)a m n nm na ng dc theo lin kt t nguyn t ti gc ti nguyn t ti (1/4, 1/4, 1/4a. Mi mt nguyn t trong Si u c th c phn x qua im ny gi cho cu trc khng i. im ny l mt tm i xng. Trong mt hp cht, s phn x hon i 2 loi nguyn t v do n khng phi l mt i xng tinh th.Nhng i xng nh s quay v s phn x c tng kt trong nhm im tinh th. Cc bn dn nguyn t c iHnh 2.14. Mt khi t din c nh hng ch ra mi quan h ca n vi cu trc tinh th ca cc bn dn hp cht.

xng lp phng cao nht l trong k hiu quc t hoc trong k hiu Schonflies. S thiu tm i xng trong cc hp cht rt gn nhm im ca chng thnh hay N c th xem l mt khi t din c hng nh trn

28

2.5. Cu trc tinh th ca cc cht bn dn thng thng hnh 2.14. i xng thp hn ca cc hp cht c mt s h qu quan trng. V d nh chng l cc cht p in. iu c ngha l mt ng sut no sinh ra in trng. S nn cu trc trn hnh 2.13(c) dc theo hng [111] l hng c cc. N s thay i khong cch tng i gia cc mt phng ca Ga v As. Hai loi nguyn t khng c trung ho chnh xc nhng mang c tnh ngc du. Nh vy, mt s dch chuyn tng i ca cc mt phng thit lp mt vect phn cc m n dn ti mt in trng. Khng phi tt c cc ng sut u c tc dng ny. V d s nn dc theo [001] khng to ra s phn cc. i xng rt gn ca cc hp cht cng cho php chng th hin mt s s phi tuyn quang b cm trong cc nguyn t. Bn cht cc ca cc hng v d nh [111] c ngha l cc b mt l ra (111) v c nhng tnh cht khc nhau. Hnh 2.13(c) ch ra rng khong cch gia cc mt phng k tip ca cc nguyn t xen k nhau. S nui tinh th c tin hnh bng cch thm vo cc cp ca nhng mt phng gn nhau hn. Nh vy, nu GaAs c nui trn mt phng (111) th n kt thc trong mt mt phng ca cc nguyn t Ga m n c gi l mt b mt(111)A. Mt khc,

29

2.5. Cu trc tinh th ca cc cht bn dn thng thng s nui trn mt phng kt thc trong mt mt phng ca cc nguyn t As m n c gi l (111)B. S khc bit l quan trng trong qu trnh nui tinh th v s nui hoc khc ha hc thng hot ng trn 2 b mt vi cc tc khc nhau.2.6. Cu trc vng trong cc bn dn thng thng Ta lng ghp cc kt qu trn nghin cu cu trc vng ca cc bn dn 3 chiu. Trc ht ta xem xt mng o v min Brillouin. Lu cc vect l cc i lng 3 chiu. 2.6.1. Min Brillouin Mt mng lp phng n gin vi cc nguyn t cch nhau mt khong a c mng o lp phng n gin c cc nguyn t cch nhau mt khong Vic thm vo cc nguyn t ph to ra mt mng LPTD trong khng gian thc loi b 3/ 4 cc im trong mng o dn ti mt mng LPTK m n v ca n c cnh l Min Brillouin th nht c cho bi mng WS ca mng ny trn hnh 2.12. Khi bt din ct ct ny c lp li trn hnh 2.15(a) trong ch ra k hiu chun ca cc im i

30

2.6. Cu trc vng trong cc bn dn thng thng vi cc im i xng cao. Ni chung, cc ch ci Hi Lp dng k hiu cc im bn trong min vi cc ch ci La M cho b mt. Cc im quan trng nht nh sau: - l gc ca khng gian - l hng v d nh [100] v gp bin min ti X tm ca mt mt vung vi cc ta v d nh (1,0,0). - l hng v d nh [111] vung gc vi cc mt phng xp cht ca cu trc LPTD. N ct tm ca cc mt lc gic ti L vi cc ta v d nh (1/2, 1/2, 1/2). - l hng v d nh [110] v gp bin ti K trung im cnh chung ca 2 hnh lc gic vi cc ta v d nh (3/4,3/4,3/4). - U l trung im cnh chung ca mt hnh lc gic v mt hnh vung. ng S ni t U ti X. - W nm cc nh trong mi mt nh chung cho 2 hnh lc gic v mt hnh vung. Cc im ny cn gii thch cc th chun ca cu trc vng.

31

2.6. Cu trc vng trong cc bn dn thng thng Hnh 2.15. (a) Min Brillouin i vi tinh th LPTD trong ch ra k hiu cho im v hng c bit. Cc ng lin nt trn b mt v cc ng t nt bn trong min. (b) Cu trc vng trong m hnh in t t do trong ch ra nh hng cun parabol ngc tr li vo trong min rt gn. Kh hnh dung hn cu trc vng trong trng hp 3 chiu so vi trng hp 1 chiu. N c ch ra trn hnh 2.15(b) i vi cc in t t do trong mng GaAs. Cch thng thng m t cu trc vng l v th dc theo cc hng la chn trong min Brillouin rt gn. on bn tri ch ra cc vng gia L v theo hng Khi , th bt u theo hng ti bin min ti X. Sau , n i dc theo b mt min ti U. im ny tng ng vi K ging nh m t t ni th quay ngc tr li dc theo ti

32

2.6. Cu trc vng trong cc bn dn thng thng S tng ng ca U v K l khng r rng do hng [111] ch c i xng gp ba ln ch khng phi gp su ln. Thc ra cc im c lin quan bi mt php tnh tin ch khng phi l mt php quay. Xt im U c ta (1, 1/4, 1/4). Vic tr i vect mng o (1,1,1) cho kt qu l (0,-3/4,-3/4) m n l mt im K. Hnh 2.15(b) ch ra nh hng ln s tn sc ca in t t do vic cun ngc parabol qua cc vect mng o vo min Brillouin th nht. Mt s ng cong trng ging nh mt parabol ny ti ny lui nh l trong trng hp 1 chiu (h.2.2). Ni chung, s t do b sung trong trng hp 3 chiu lm cho bc tranh phc tp hn nhiu v nhiu vng b suy bin. Tuy nhin, cc vng ca cc bn dn thng thng khng khc nhiu so vi cc vng ca cc in t t do. 2.6.2. Cc c tnh chung Cc cu trc vng i vi Si, Ge, GaAs v AlAs c ch ra trn hnh 2.16. Chng c xc nh khi s dng mt phng php tng i n gin khng tnh n lin kt spin - qu o ch quan trng i vi nh VB. c tnh chung l mt h ca cc VB m n l y trong mt bn dn thun ty

33

2.6. Cu trc vng trong cc bn dn thng thng Hnh 2.16. Cu trc vng ca 4 cht bn dn l Si, Ge, GaAs v AlAs. Cc tnh ton khng bao hm lin kt spin qu o.

T = 0 c tch ra bi mt khe vng t CB trn. N khng c qui c c ly lm nh VB. C th dng ho hc gii thch dng ca cc VB. Si c 4 in t ho tr trong c 2 in t trn qu o s c lm y v 2 in t trn 3 qu o p m chng c th cha tng cng 6 in t. Cc qu o ny c th lai ho cho 4 qu o m chng th ra ging nh cc chn ti cc gc ca mt khi t din

34

2.6. Cu trc vng trong cc bn dn thng thng v lin kt vi nhau to thnh mng kim cng. Bc tranh ny c th lin quan ti cc vng ca Si bng cch xem VB nh c to ra t cc kt hp lin kt ca cc qu o v CB t cc kt hp phn lin kt. Mt c tnh quan trng l nh VB c i xng ging nh p. y CB c i xng ging nh s trong GaAs nhng bc tranh ca cc qu o khng tt i vi CB do cc hm sng m rng tip qua tinh th v gi li t hn c im nguyn t ca chng. Cc VB ca cc hp cht l phc tp hn mt cht so vi cc VB ca cc nguyn t v i xng thp hn ca chng cho php chuyn t in tch t Ga ti As m n em li c im ion cho lin kt. iu ny lm yu s lai ho v VB tch ra khong 8eV thnh mt mt vng n t cc qu o s di vng ba t 3 qu o p. Xu hng ny c thc hin trong cc hp cht II-VI nh ZnSe m chng c cc lin kt mang tnh ion hn v mt khe rng hn gia cc VB dy hn. V mt nh tnh, cc vng ca 4 vt liu trn l tng t vi nhau v hnh dng nh mong mun t cu trc tinh th chung ca chng v cc v tr gn nhau trong bng tun hon. Tuy nhin, nhng khc bit nh v nng lng

35

2.6. Cu trc vng trong cc bn dn thng thng lm thay i trt t ca cc vng c bit l cc CB nm thp. iu ny nh hng ln n cc tnh cht in t ca 4 cht bn dn ni trn. Ngi ta c bit quan tm n cc min nh ca cc CB v VB m chng to thnh bin ca khe vng. 2.6.3. Vng ho tr (VB) C th thu c VB trn hnh 2.16 t th i vi cc in t t do trn hnh 2.15(b) v vic lm khp cc thang nng lng. Nhnh nh tng gp i v do c 4 nhnh ti VB m chng c th gi 8 in t. Mi mt n v nguyn thy (khng phi lp phng) cha 2 nguyn t vi 8 in t ho tr gia chng c lm y chnh xc VB. C t s khc bit gia cc vt liu ngoi tr min gn nh vng. Ta s a ra mt bc tranh n gin v hnh dng chung ca VB v xem xt chi tit min gn nh. Cc hm sng nh VB c i xng ca cc qu o p. iu khng c ngha l cc hm sng Bloch thc s trng ging cc qu o p ca nguyn t m ch c tnh i xng l quan trng. n gin gi s tinh th c cu trc

36

2.6. Cu trc vng trong cc bn dn thng thng

cu trc lp phng n gin v xt cc qu o trn hnh 2.17(a). Cc hm sng l d hng mnh v xen ph mnh theo hng z. iu lm cho cc in t d dng i t nguyn t ny sang nguyn t khc dc theo z. Do , khi lng hiu dng theo hng ny l nh. S xen ph l yu hn nhiu

Hnh 2.17. Cc VB c xy dng t cc quo p. (a) Mng ca cc qu o . (b) Cutrc vng ch ca cc qu o vng l nhdc theo ti bn phi v vng l nng dctheo (hoc ) ti bn tri. (c) Tng cng cc vng t tt c 3 qu o p v n ch ra mtvng nng suy bin kp v mt vng nhn.

trong mt phng xy. Do , cc in tchuyn ng km t do hn v khi lng hiu dng cao hn. Nh vy, cu trc vng i vi ring qu o l d hng mnh nh trn hnh 2.17(b). 2qu o khc c dng iu theo cch tng t i vi trc cc ca chng vvic thm vo c ba cho cc vng trn hnh 2.17(c) m i xng lp

37

2.6. Cu trc vng trong cc bn dn thng thng phng c khi phc. C mt vng nh n vi nng lng (theo quan im ca cc l trng) tng nhanh theo K v mt vng nng suy bin kp. Bc tranh ny s c tnh ton nh lng bng cch s dng m hnh lin kt cht. N gii thch hnh dng chung ca cc VB. Bc tranh n gin trn khng ng i vi cc nng lng rt gn nh vng. Mt phc ho ca min ny c ch ra trn hnh 2.18(a). Cc vng c k hiu theo s i xng ca chng trong l thuyt nhm. 2 vng gi l Hnh 2.18. (a) nh VB ch ra cc l trng nng v nh cng vi vng tch ra. (b) Cc b mt ng nng hoc cc mt cu lch i vi cc l trng nng v nh trong GaAs.

gp nhau ti nh vi mt vng tch ra th ba ngay pha di. Chng b tch ra bi tch spin qu o N l mt hiu ng tng i tnh.

38

2.6. Cu trc vng trong cc bn dn thng thng tch tng ln theo ly tha 1/ 4 ca s nguyn t ca nguyn t. tch vo khong 0,29 eV i vi Ge v 0,044 eV i vi Si. N ln hn khe vng trong cc trng hp cc tr. Vng tch ra c th c b qua nu tch l ln mc d n khng p dng c cho cc vt liu nh nh Si. 2 vng gp nhau c cc khi lng hiu dng khc nhau i vi nh v c gi l cc l trng nng v nh. Chng hi t i vi ln cho vng nng. Gn s tn sc ca chng c m t rt th bi

trong i vi cc l trng nng hoc i vi cc l trng nh. Php gn ng ny l khng tt mc d n c s dng rng ri do tnh n gin v thun tin ca n. Cc php gn ng tt hn chng t rng cc vng va khng parabol va d hng v iu ph thuc vo hng ca cng nh ln ca n. iu ny c ch ra bi cc mt ng nng trn hnh 2.18(b). Khi lng hiu dng nh hn ca cc l trng nh lm cho b mt ng nng ca chng nm bn trong b mt ng nng i vi cc l trng nng. Khng c b mt ng nng no l mt cu nh d on bi php gn ng n gin (2.20). Cc b mt ny l cc mt cu lch.

39

2.6. Cu trc vng trong cc bn dn thng thng Cc VB l phc tp v kh xc nh nu i xng lp phng b ph v. S ko cng vt liu hoc s nui h lng t lm tng suy bin gia cc l trng nng v l trng nh ti v cng ph hy s ng hng gn ng. Ni chung, ta chp nhn m hnh n gin nht (2.20) mc d n cn nhiu nhc im. 2.6.4. Vng dn (CB) Cc CB v nh tnh l tng t trong cc bn dn thng thng nhng iu ny khng ng i vi cc CB m nhng dch chuyn nh c lin quan vi nhau lm thay i bn cht ca cc tiu thp nht. Ba trng hp thng thng c tm thy i vi im thp nht trong CB l ti hng ti X (dc theo hoc ti L. 2.6.4.1. Cc tiu trong GaAs y CB nm ti trong GaAs c k hiu l l cng mt im trong khng gian nh l nh VB. Do , GaAs c mt khe trc tip v nh sng c th kch thch cc in t i qua khe vng cc tiu (xem phn 2.7). H thc tn sc c dng parabol nng lng thp

40

2.6. Cu trc vng trong cc bn dn thng thng vi khi lng hiu dng trong GaAs. dc ca n tng nhanh km hn nhng nng lng cao v ngi ta s dng cc php gn ng khc (phng trnh (1.94)). N cng mt s ng hng. Khi lng hiu dng thp dn ti linh ng cao ( v n gii thch vic s dng GaAs trong cc tranzito cao tn. 2.6.4.2. Cc cc tiu X trong Si v AlAs Cc im thp nht trong CB ca Si nm dc theo cc hng v d nh [100] khong 85% ng ti bin min (X). Cc im ny b loi b trong min Brillouin ra xa nh VB ti Do , khe vng l khe khng trc tip (xem phn 2.7). C 6 hng v do c 6 cc tiu tng ng trong CB m mi mt trong s sinh ra mt thung lng. Mt php gn ng parabol li c th c s dng cho mi mt thung lng nhng n khng ng hng v xoay quanh cc tiu ch khng phi i vi thung lng [100],

41

2.6. Cu trc vng trong cc bn dn thng thng S i xng i hi dng iu dc theo cc hng ngang y v z l nh nhau. Do , chng c cng khi lng hiu dng ngang Hng dc x cn phi khc i v ch ra mt khi lng hiu dng dc i vi Si, cc khi lng ny l Do , thung lng l bt ng hng mnh nh thng c minh ha bng mt b mt ng nng. Hnh 2.19(a) ch ra v tr ca 6 thung lng trong min Brillouin vi hnh nh phng i ca mt thung lng trn hnh 2.19(b). Thung lng c dng iu x g v ko di dc theo hng dc m n c khi lng ln. Hnh 2.19. (a) Min Brillouin ca Si vi cc b mt ng nng i vi 6 thung lng X tng ng. (b) Hnh nh phng i ca mt trong cc thung lng m n ch ra cc khi lng dc v khi lng ngang suy bin kp Mt trng thi gi dng parabol nhng cn phi s dng khi lng hiu dng mt trng thi Cng thc thng thng cng cn phi nhn vi suy bin tnh n cc thung lng tng

42

2.6. Cu trc vng trong cc bn dn thng thng ng. suy bin tnh n spin cng c duy tr. Vt liu c tnh d hng cao nu cc in t b giam cm trong mt thung lng n. V d nh cc in t c linh ng cao theo cc hng c khi lng hiu dng nh v c linh ng thp ni c khi lng hiu dng cao. C th dng ng sut ph v i xng lp phng v pht hin mt phn tnh cht ny nhng trong vt liu thng thng, ton b h ca cc thung lng u c ng gp. iu ny dn ti dn ng hng v dn ny ph hp vi i xng lp phng ca tinh th. AlAs cng c cc cc tiu thp nht trong CB ca n ti X. iu ny l khc so vi GaAs. Do , bn cht ca cc tiu thay i ging nh mt hm ca x trong hp kim iu ny dn ti mt s nh hng trong cc d cu trc GaAs v AlAs. 2.6.4.3. Cc cc tiu L trong Ge Trong Ge, cc cc tiu ca CB nm ti cc im L tm ca cc mt lc gic ca min Brillouin dc theo cc hng nh [111]. Cc im trn cc mt i din l tng ng v n cho 4 thung lng tng ng hoc 8 na thung

43

2.6. Cu trc vng trong cc bn dn thng thng lng tng ng. Khe vng li l khe gin tip. Mi mt thung lng c th c m t bi mt mt ng nng nh trn hnh 2.19(b) vi trc di hng dc theo hng [111] ca n. Cc khi lng l bt ng hng hn so vi trong Si vi i vi chuyn ng trong mt phng ngang. 2.6.4.4. Cc cc tiu cao hn Mc d CB ca GaAs c cc tiu thp nht ca n ti , c cc thung lng v tinh ti L v X m chng khng cao hn nhiu. Cc thung lng ti L ch vo khong 0,3 eV trn vi mt tp hp ti X cao hn khong 0,2 eV. Mt cc tiu khc ti X ch cn cao hn 0,4 eV. Cc nng lng ny l nh cc in t nhanh c th i vo cc thung lng L t . Mt trng thi L l cao hn nhiu do khi lng hiu dng, suy bin v linh ng u thp hn. S chuyn khng gian to ra mt vng c linh ng vi phn m trong c trng trng vn tc ca GaAs c khai thc trong hiu ng Gunn. Lu rng nh l Bloch ch p dng cht ch cho mt tinh th v hn.Khng c tinh th thc no l v hn v php gn ng ny c bit ng nghi ng

44

2.6. Cu trc vng trong cc bn dn thng thng gn cc b mt. Cng c th c cc trng thi in t ph gn vi cc b mt khng tun theo nh l Bloch v khng c lin quan ti cu trc vng i vi phn cn li ca tinh th. c bit l chng c th nm tm khe vng. Cc trng thi b mt nh vy thng c th c gn vi nhng xy dng li xy ra trn cc b mt. Cc trng thi b mt c th l do cc khuch tt trong cc trng hp khc v i khi ngun gc ca chng l khng r rng. Chng c th c bit quan trng trong thc t trong ng ch l mt trng thi b mt cao gn tm khe vng trn cc b mt (100) ca GaAs. Tnh nng ca cc linh kin v d nh cc tranzito hiu ng trng b nh hng mnh bi cc trng thi b mt.2.7. Cc php o quang i vi khe vng Cc khe vng c th c o bi cc k thut quang n gin v d nh s hp th quang. Tng tc ca nh sng vi tinh th c th c xem nh mt qu trnh tn x n hi bao hm mt photon v mt in t trong c nng lng v xung lng tinh th cn c bo ton. Xt in t c nng lng v xung lng ban u v nng lng v xung lng cui Mt photon vi vect sng c nng lng v xung lng

45

2.7. Cc php o quang i vi khe vng trong c l vn tc nh sng bn trong vt liu. S bo ton nng lng v xung lng i hi in hnh l s khc bit nng lng i vi khe vng ca mt bn dn v n cho Kch thc min Brillouin vo khong trong nm. Do , Nh vy, Q l nh trn thang ca min Brillouin. V th, c th t v b qua s thay i xung lng trong mt chuyn tip quang. Nhng chuyn tip nh th c gi l nhng chun tip thng ng v chng xut hin nh nhng vch thng ng trn cu trc vng Trong mt bn dn vi mt CB nh GaAs, chuyn tip t nh VB ti y CB l thng ng ti (hnh 2.20(a)). Do , n c th c sinh ra bi nh sng v s hp th quang xy ra ngay khi nng lng photon vt qu khe vng N l ngha ca mt khe trc tip. Qu trnh cng c th xy ra theo chiu ngc li. Trong cc linh kin v d nh mt it c th hiu dch v pha trc, cc in t d gn y CB v l trng gn nh VB c bm vo trong mt bn dn. Cc in t d c xu

46

2.7. Cc php o quang i vi khe vng hng kt hp li bng cch ri vo cc l trng hi phc s cn bng. Nng lng c gii phng ra trong qu trnh ny v c th b tn hao cho mng thnh cc phonon thng nh mt tp cht hoc pht ra thnh mt photon. Qu trnh sau (gi l s ti kt hp phng x) chi phi trong cc bn dn vi cc khe trc tip. Do , chng l nhng ngun nh sng c hiu qu. Hu ht cc linh kin quang in t trong c bit l cc laze da trn c s ca cc vt liu nh GaAs vi mt khe trc tip. Trong mt vt liu vi mt khe gin tip nh AlAs, Si hoc Ge, cc cc tr ca cc VB v CB khng xy ra ti cng mt (hnh 2.20(b)). Mt s Hnh 2.20. S hp th quang qua khe vng trong cc loi bn dn khc nhau. (a) S hp th qua khe vng trc tip ti (b) S hp th qua khe vng gin tip b cm nhng cc chuyn tip thng ng xy ra vi mi (c) Chuyn tip qua mt khe vng gin tip vi s hp th c phonon v photon.

47

2.7. Cc php o quang i vi khe vng chuyn tip gia cc im ny i hi mt s thay i ln ca xung lng cng nh nng lng v khng th xy ra vi ring mt photon. Nhng chuyn tip thng ng c th xy ra ti mi im trong min Brillouin nhng nng lng thp nht ca cc chuyn tip quang ny ln hn khe vng cc tiu. Cc chuyn tip quang qua cc khe gin tip c th xy ra nu c mt qu trnh khc cung cp s thay i xung lng. Cc phonon c th lm iu ny v kch thch hoc ng gp mt lng nng lng nh vo chuyn tip nh ch ra trn hnh 2.20(c). Cc tp cht cng c th c bao hm. Cc qu trnh ny sinh ra mt ui trong s hp th quang m n bt u xung quanh nng lng ca khe gin tip . S ti kt hp phng x c th xy ra bng cng cc con ng nhng km hiu qu hn nhiu so vi trong cc vt liu vi khe trc tip. Nh vy, Si, Ge v cc hp kim ca chng ni chung khng phi l cc b pht nh sng c hiu qu. Tuy nhin, mt s it pht quang (LED) s dng GaP vi mt khe gin tip v khai thc s ti kt hp phng x nh cc tp cht pht sng. 2.8. Cc phonon

48

2.8. Cc phonon Cc phonon l cc lng t ca cc dao ng ca cc ion mng ra khi cc v tr ng vi nng lng cc tiu ca chng. Tnh tun hon ca mng c ngha l cc phonon c cu trc vng theo cng mt cch ging nh cc in t. Cu trc vng ca cc phonon l phc tp hn so vi cu trc vng ca cc in t v cc phonon b phn cc. Xt cc dao ng c bc sng rt ln v ln hn nhiu so vi hng s mng. Chng c th c xem nh cc sng m c in trong c 3 sng lan truyn theo mt hng cho. Nu ta chn mt hng n gin v d nh hng [100] trong mt vt liu lp phng th mt mode l dc v 2 mode khc l ngang. Chng c minh ha i vi tinh th 1 chiu trn hnh 2.21. Mode dc ging nh mt sng m trong khng kh: cc ion dao ng ti v lui theo cng hng ca ging nh mt sng chy v do , dch chuyn N cho cc min nn v dn xen k nhau. Cc ion dao ng trong mt phng vung gc vi hng lan truyn trong cc mode ngang vi ging nh sng in t trong khng gian t do. V cc phng din khc, cc phonon n gin hn so vi cc in t. V d nh s vng c cho bi s nguyn t ng vi mt n v nhn vi 3 i

49

2.8. Cc phonon Hnh 2.21. (a) Mode dc v (b) mode ngang c cng bc sng 8a trong tinh th n nguyn t 1 chiu. Hnh v ch ra v tr ca cc nguyn t cn bng v v tr ca chng khi b lch theo vect vi cc phn cc. iu ny khc vi s vng v hn i vi cc in t. Ta s xem xt mt m hnh 1 chiu n gin m n th hin hu ht cc c tnh chnh ca cc phonon trc khi nghin cu cc phonon trong mt bn dn. 2.8.1. Cc phonon trong trng hp 1 chiu M hnh n gin nht l mt hng 1 chiu ca cc nguyn t c khi lng m dc theo x c tch ra bi khong cch a khng chuyn ng nh ch ra trn hnh 2.22. Cc nguyn t c c ni vi nhau bi cc l xo c hng s n hi K . Dao ng b gii hn theo hng x v do ch c cc mode dc. Cho dch chuyn ca nguyn t j l Theo nh lut Newton, trong cc s hng trong cc ngoc n l cc gim chiu di ca cc

50

2.8. Cc phonon Hnh 2.22. Mt hng 1 chiu ca cc l xo v cc khi lng c nh du bi j cho php dao ng dc qua cc dch chuyn lin kt gia cc nguyn t j v Ta k vng cc nghim l cc sng v c th th nghim trong i vi nguyn t j. K hiu phc n gin ho thao tc nhng khng phi l c bn ging nh trong phng trnh Schrodinger. Vic thay n vo phng trnh chuyn ng cho

S ph thuc vo c v tr j v thi gian t b trit tiu ch ra rng nghim ny l thch hp nu

C th thu c kt qu ny bng cch s dng nh l Bloch di dng H thc tn sc l

51

2.8. Cc phonon S sng b gii hn trong min Brillouin th nht ging nh i vi cc in t. S tn sc c biu din trn hnh 2.23(a). H thc ny l tuyn tnh i vi q nh ngc vi h thc dng parabol i vi cc in t v c th vit trong vn tc m Cc nguyn t ln cn c cc dch chuyn tng t trong vng q nh v bc sng di ny nh ch ra trn hnh.2.21(a). dc ca gim khi q tng v tr nn Hnh 2.23. H thc tn sc i vi tinh th 1 chiu vi cc dao ng dc. (a) Tinh th n nguyn t vi cc nguyn t c khi lng m. (b) Chui lng nguyn t vi cc khi lng m v M = 2m v n ch ra nhnh m (ng lin nt) v nhnh quang (ng t nt). di l cc cu hnh nguyn t. C 2 chui c cng hng s mng a v cc l xo c cng hng s n hi K.

52

2.8. Cc phonon phng ti bin min m vn tc nhm ca cc sng gim ti 0 ging nh i vi cc in t. Cc nguyn t cc ln cn cng dao ng ngc pha ti im ny. Cc l xo xen k nhau b nn v dn v n cho tn s dao ng cc i. 2.8.2. Bin dao ng Cn xc nh gi tr ca bin dao ng hon thnh vic phn tch. Do cc phng trnh chuyn ng l tuyn tnh theo di nn bin l ty trong c hc c in. Tuy nhin, trong c hc lng t cc dao ng b lng t ha thnh cc phonon. iu ny c th c thc hin bi mt qui trnh chun trong chuyn ng ca cc nguyn t c phn gii thnh cc mode thng thng. Mi mt mode ging nh mt dao ng t iu ho. Bin c th tm c i vi mt phonon n gin bng cch tnh nng lng ton phn ca dao ng v cn bng n vi gi tr lng t ha i vi tinh th n nguyn t, vn tc ca nguyn t j c cho bi

Do , ng nng l v gi tr trung bnh

53

2.8. Cc phonon ca n l N bng th nng trung bnh i vi dao ng iu ho. Do , nng lng ton phn trung bnh ng vi mt nguyn t cng bng S nguyn t N trong th tch (thc ra l chiu di) c mt khi lng v khi lng ca mi mt nguyn t l m c xc nh bi Khi cn bng nng lng ton phn vi lng t nng lng, ta c T suy ra

i vi cc vect sng nh, v Cui cng, chuyn ng ca mt nguyn t c xc nh bi dch chuyn

Lu s c mt ca th tch ca tinh th. Mt mu ln hn th hin c cc phonon yu hn nhng iu ny b loi tr bi s tng ca mt trng thi ton phn. b trit tiu t cc kt qu vt l v d nh cc tc tn x. 2.8.3. Chui lng nguyn t

54

2.8. Cc phonon Nhiu tinh th trong bao hm cc bn dn thng thng c hn mt nguyn t ng vi mt mng n v m n a vo mt c tnh khc. Xt chui trn hnh 2.23(b). N cc khi lng nng M v khi lng nh m xen k nhau dc theo chiu di ca n v n cho 2 nguyn t ng vi mt n v. Cc hng s mng v hng s l xo gi cc gi tr a v K nh trc. H thc tn sc l

By gi c 2 vng hay 2 nhnh cho h thc tn sc. Nhnh thp hn hay nhnh m trng ging nh trng hp n nguyn t ngoi tr iu l s tng khi lng lm gim tn s. Tuy nhin, nhnh trn hay nhnh quang th khc hn. N c mt cc i ti q = 0 v nghing xung di t t ti bin min m c mt khe gia cc nhnh m v nhnh quang. Cc bin dao ng ca 2 nguyn t trong mi mt n v ca tinh th lng nguyn c th tnh nh i vi chui n nguyn t. Cc bn cht ca cc dao ng gn q = 0 trong 2 nhnh l tng phn nhau trn hnh 2.24. Cc dch chuyn trong nhnh m ging nh cc sng m nh ng t tn gi. 2 nguyn t trong mi mt n v chuyn ng theo

55

2.8. Cc phonon cng hng vi gn nh cng khong cch v qua mt vng nh m n th hin nh th ton b tinh th b nn hoc dn n gin. Ta s dng bc tranh ny khi tnh tng tc gia cc phonon v in t. S mo ca cc lin kt gia cc nguyn t tr nn nh hn i vi mt bin dao ng cho ging nh khi th Cc nguyn t nng v nh chuyn ng theo cc hng ngc nhau trong nhnh quang ging nh trn hnh 2.24(b). iu ny ging vi nh vng i Hnh 2.24. Chuyn ng ca cc nguyn t trong (a) mode m v (b) mode quang vi bc sng rt di i vi mt chui 1 chiu ca cc nguyn t nng v nh xen k nhau vi chui n nguyn t. Li c s mo cc i ca tng l xo v do , tn s ly gi tr cc i ca n. Trong cc hp cht c s chuyn in tch gia 2 loi. dch chuyn tng i ca 2 nguyn t ging nh trong mode quang thit lp mt lng cc

56

2.8. Cc phonon in N li to ra mt trng vect phn cc v mt in trng c th tng tc vi cc sng in t m di y c tn l quang. Cc ht tch in nh cc in t cng cm nhn c in trng v cc phonon quang dc c cc lm tn x cc in t mt cch nhanh chng. Bn cht c cc ny khng thy c trong cc nguyn t do tt c cc nguyn t l nh nhau v khng c s chuyn in tch. Ch c mt loi nguyn t chuyn ng trong mi mt mode ti bin min v cc loi khc ng yn. V d nh ch cc nguyn t nh hn dao ng theo kiu dao ng trn vi cc k st ngc pha. Nh vy, ch c m xut hin trong v biu thc ging nh i vi tn s cao nht trong chui n nguyn t ngoi tr iu l gi tr hiu dng ca K b chia i do by gi c 2 l xo tch ra mi mt nguyn t nh. Dng iu ny tng t nh dng iu ca cc in t ti cc bin min nh ch ra trn hnh 2.4. 2.8.4. Cc phonon trong trng hp 3 chiu Cu trc vng i vi cc phonon c th c v th theo cng mt cch chnh xc nh i vi cc in t. Mt phn nh s tn sc c biu din trn hnh 2.25 i vi Si v GaAs. Cc bn dn c 2 nguyn t trong mt

57

2.8. Cc phonon n v nguyn thy. Do , cu trc chung gn vi cu trc i vi chui lng nguyn t trn hnh 2.23(b). Cc tn s l cao hn i vi Si phn ln Hnh 2.25. Phc ha h thc tn sc i vi cc phonon trong (a) Si v (b) GaAs t n X. Cc nhnh ngang l suy bin kp dc theo hng ny. l do khi lng nh hn ca n. Khi lng ny l 28 so vi khi lng trung bnh l 72 i vi GaAs. S khc bit chnh so vi trng hp 1 chiu l do 3 s phn cc kh d. 2 mode ngang l suy bin dc theo [100] nhng ni chung c 3 nhnh m v 3 nhnh quang . C 2 mode m tr nn tuyn tnh i vi q nh. Cn mode dc c vn tc ln hn. Cc phonon quang ngang (TO) v quang dc (LO) trong cc bn dn c cc

58

2.8. Cc phonon c cc nng lng khc nhau t ti S tch ny l do in trng c thit lp bi cc phonon LO v c cho bi h thc Lyddane-Sachs-Teller. Nng lng ca cc phonon LO l 36 meV trong GaAs. Do , cc phonon LO c th b b qua cc nhit thp ( di khong 77 K) nhng chng l mt co ch quan trng i vi s chuyn nng lng gia cc in t v cc phonon ti nhng nhit cao hn. Khi lng nh hn ca AlAs lm tng nng lng ny ti khong 50 meV nhng nng lng ca cc phonon LO trong khng thay i trn t ch ny sang ch khc nh mt hm ca x.

Bi tp chng 2 2.1. Chng minh rng hai pht biu (2.2) v (2.3) ca nh l Bloch l tng ng vi nhau. 2.2. H.2.5 c v cho mt vng in hnh trong mt bn dn. Lp li n cho mt siu mng c a = 10 nm v W = 50 meV. Khi lng hiu dng v vn tc cc i c th c cc gi tr no?

59

Bi tp chng 2 2.3. Nhiu bn dn c vn tc bo ho i vi cc in t vo khong Cn phi kho st bao nhiu min Brillouin tin ti vn tc ny? Vn tc bo ho t c trong trng khong Khi , ci g gn sung l thi gian hi phc? 2.4. Tnh tn s ca dao ng Bloch trong mt in trng ln n mc vt liu cn chu ng c trc khi b ph v. S dng cc thng s trn hnh 2.5. So snh n vi thi gian sng thng thng gia cc s kin tn x (chng hn tnh t linh ng ). By gi lp li n cho mt siu mng. H no l ng c vin tt hn quan st dao ng Bloch. 2.5. Tnh xc sut ca s xuyn hm Zener i vi cc h c coi nh cc dao ng t Bloch trong bi ton trc. Gi thit rng khe rng khong 1 eV i vi tinh th v c cng chiu rng nh cc vng i vi siu mng. C hay khng mt ca s ca nhng in trng ln m c th quan st dao ng Bloch trc khi s xuyn hm Zener chi phi? 2.6. Lp li i vi mt mng hnh vung phc ha ca 4 min Brillouin u tin i vi mng hnh ch nht trn h.2.10.

60

Bi tp chng 2 2.7. Xc nh mt vi vng nng lng u tin i vi cc in t t do trong mng hnh vung. Chng c v dc theo cc hng i xng cao trn hnh 2.26. Hnh 2.26. (a) Min Brillouin v (b) cc vng nng lng i vi cc in t t do trong mng hnh vung.

2.8. Chng minh rng min Brillouin th nht ca tinh th LPTD cha mt trng thi ng vi mt spin ng vi mt nguyn t ca mng. 2.9. Chng minh rng gc gia 2 lin kt bt k gia cc nguyn t ln cn trong mng kim cng bng 2.10. Chui lng nguyn tr thnh chui n nguyn trong gii hn Mm. iu ny c phn nh nh th no trong h thc tn sc?

61

),()(xVaxV=+()-=-===nnnnnxiGVainxVxV)1.2.(exp2exp)(panGn/2p=()ikxxnexp)(=f1)(2=xnf())2.2).(()(,exp)()(xuaxuikxxuxkkkk=+=y2)(xy())3.2).((exp)(xikaaxkkyy=+2)(xuk.)()(22xaxyy=+).(xuk)(xukkh()()())4.2.(exp/2exp)(exp)()('xikaixxuikxxuxkkkpy==aka//'pp