and one unit cell contains 8 silicon atoms. The atomic density of silicon is

Transcription

1 Chaptr Vsualzato o th Slo Crystal (a) Plas rr to Fgur - Th 8 orr atoms ar shar by 8 ut lls a thror otrbut atom Smlarly, th 6 a atoms ar ah shar by ut lls a otrbut atoms A, 4 atoms ar loat s th ut ll H, thr ar total 8 slo atoms ah ut ll (b) Th olum o th ut ll s V ut ll A 5 4 m 6 m, a o ut ll otas 8 slo atoms Th atom sty o slo s slo atoms 8 5 (slo atoms) m S Vut ll H, thr ar 5 slo atoms o ub tmtr () I orr to th sty o slo, w to alulat how hay a ual slo atom s 8 g/mol Mass S atom 4 67 g/atom 6 atoms/mol Thror, th sty o slo ( S ) g/m s ρ S S Mass S atom g / m Frm Futo (a) Assum = quato (7), () boms ½ H, th probablty s ½ (b) St = + a = quato (7): () 7 / Th probablty o g ltros stats at + s 7 Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

2 * For Problm Part (b), w aot us approxmatos suh as quatos (7) or (7) s - s thr muh largr tha or muh smallr tha - () () s th probablty o a stat bg ll at, a -() s th probablty o a stat bg mpty at Usg quato (7), w a rwrt th problm as ( ) ( ) / / whr / / / / / / ow, th quato boms / / Ths s tru a oly Solg th quato abo, w (a) Assum = a T > K quato (7) () boms ½ H, th probablty s ½ (b) () s th probablty o a stat bg ll at, a -() s th probablty o a stat bg mpty at Usg quato (7), w a rwrt th problm as ( ) ( ) / / whr Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

3 / / / / / / ow, th quato boms / / Ths s tru a oly Solg th quato abo, w () Th plot o th Frm-Dra strbuto a th Maxwll-Boltzma strbuto s show blow Probablty Frm-Dra Dstrbuto Maxwll-Boltzma Dstrbuto (- )/ Th Boltzma strbuto osrably orstmats th Frm strbuto or small (- )/ I w st (- )/ = A quatos (7) a (7), w ha A A Solg or A, w A A A A l Thror, th Boltzma approxmato s aurat to wth % or (- )/ Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

4 4 (a) Plas rr to th xampl S 7 Th rato o th trog otrato at km abo sa ll to th trog otrato at sa ll s g by km / ) km ( km Sa Ll )/ Sa Ll / ) Sa Ll whr km Th rato s Sa Ll alttu mass o 6 m 866 molul alrato o graty 4 g 98 m s rg ( ) km 4 6 ( 4 56 rg)/( 8 rgk 7 K) ) Sa Ll S trog s lghtr tha oxyg, th pottal rgy r or trog s smallr, a osqutly th xpotal trm or trog s largr tha 5 or oxyg Thror, th trog otrato at km s mor tha 5% o th sa ll otrato (b) W kow that O O ) ) km Sa Ll ) km ) Sa Ll 5,, a 4 ) O ) Sa Ll Sa Ll Th, ) O ) km km ) ) km Sa Ll ) O ) Sa Ll Sa Ll O ) O ) Sa Ll km It s mor -rh tha at sa ll 5 / / / / Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

5 / / 6 (a) 5K () 5 K (b) At K, th probablty o a stat blow th Frm ll bg ll s a a stat abo th Frm ll bg ll s So a total o 7 stats ar ll whh mas thr ar 4 ltros (s ltros a oupy ah stat) th systm Dsty o Stats 7 S th smoutor s assum to b, W ar ask to us quatos (7) a (74) to approxmat th Frm strbuto (Ths mas that th opg otrato s low a s ot wth a w s rom or A lghtly op smoutor s kow as a o-grat smoutor) Th arrr strbuto as a uto o rgy th outo ba s proportoal to Dstrbuto() / /, whr -(- )/ s rom quato (7) Takg th rat wth rspt to a sttg t to zro, w obta Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

6 / / / / Th xpotal trms al out Solg th rmag quato yls / / So, th umbr o arrrs th outo ba paks at +/ Smlarly, th al ba, th arrr strbuto as a uto o rgy s proportoal to Dstrbuto() / /, whr -( -)/ s quato (7) Takg th rat a sttg t to zro, w obta / / / / / / Aga, th xpotal trms al out, a solg th rmag quato yls / / Thror, th umbr o arrrs th al ba paks at -/ 8 S t s g that th smoutor s o-grat (ot haly op), s ot wth a w s rom or W a us quatos (7) a (74) to approxmat th Frm-Dra strbuto (a) Th ltro otrato th outo ba s g by / D () () A CB I orr to smply th tgrato, w mak th ollowg substtutos: x x, x x, a x : rom to ow th quato boms Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

7 x / A x x A / / x x x whr x x x x x x / (Gamma uto) H, th ltro otrato th outo ba s / / A Smlarly, th hol otrato s g by / p D () -() B VB - Aga, w mak th ollowg substtutos to smply th tgrato: x x, x x, a x : rom to ow th quato boms p whr B x x / x B / / x x x x x x x x x / (Gamma uto) Thror, th hol otrato th outo ba s / / p B (b) Th wor Itrs mpls that th ltro otrato a th hol otrato ar qual Thror, p A / / / B / Ths smpls to Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

8 A / / B Solg or yls 5 l 9 V; k 86 V K, T K H, th trs Frm ll ( ) s loat at 9 V blow th m-bagap o th smoutor 9 Th ut stp utos st th tgrato lmts D () s zro or <, a D () s zro or > S t s g that th smoutor s o-grat (ot haly op), s ot wth a w s rom or W a us quatos (7) a (74) to approxmat th Frm-Dra strbuto (a) Th ltro otrato th outo ba s g by / D () () A CB I orr to smply th tgrato, w mak th ollowg substtutos: x x, x x, a x : rom to ow th quato boms x / Ax x A whr x x x / H, th ltro otrato th outo ba s / A Smlarly, th hol otrato s g by / p D () VB -() B - Aga, w mak th ollowg substtutos to smply th tgrato: x x x Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

9 x x, x x, a x : rom to ow th quato boms p whr x B x x x x / x B / Thror, th hol otrato th outo ba s / p B (b) Th wor Itrs mpls that th ltro otrato a th hol otrato ar qual Thror, / / B p A Ths smpls to A / / B I w sol or, w obta 5 l 9 V; k 86 V K, T K H, th trs Frm ll ( ) s loat at 9 V blow th m-bagap o th smoutor x x x (a) Th arrr strbuto as a uto o rgy th outo ba s proportoal to / / Dstrbuto(), whr -(-)/ s rom quato (7) Takg th rat a sttg t to zro, w obta / / / / Th xpotal trms al out Solg th rmag quato yls Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

10 / / H, th umbr o arrrs th outo ba paks at +/ (b) Th ltro otrato th outo ba s g by D () () D () () CB Top o th Couto Ba W assum that th uto () alls o raply suh that Top o th Couto Ba Top o th Couto Ba D () () D () () ow w may hag th uppr lmt o tgrato rom th Top o th Couto Ba to : / A Also, orr to smply th tgrato, w mak th ollowg substtutos: x x, x x, a x : rom to Th quato boms x / A x x A / / x x x whr x x x x / x x (Gamma uto) Thror, th ltro otrato th outo ba s / / A (b) Th rato o th pak ltro otrato at = +(/) to th ltro otrato at = +4 s Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

11 ( 4) ( ) 4 / 5 A A / 4 5 / C / 4 / 5 / 4 5 / 95 6 (4/ 5) 56 Th rato s ry small, a ths rsult justs our assumpto Part (b) () Th kt rgy o a ltro at s qual to - C Th arag kt rgy o ltros s K sum o th kt rgy o all ltros total umbr o ltros D () () CB D () () CB / A / A I orr to smply th tgrato, w mak th ollowg substtutos: x x, x x, a x : rom to ow th quato boms A A whr x x / / x x / / x x A A 5 / / / / x x / x / x x x a / x 5 x x x x x 5/ 4 / x x x x x x / H, th arag kt rgy s (/) (Gamma utos) (Gamma utos) ltro a Hol Cotratos (a) W us quato (8) to alulat th hol otrato: Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

12 p p / 5 5 / m m (b) Plas rr to quatos (9a) a (9b) S - a >> a all th mpurts ar oz, = - a, a p = ( ) /( - a ) () S th Frm ll s loat 6 V abo a losr to, th sampl s -typ I w assum that s loat at th m-bagap (~ 55 V), th - = 9 V : - =55 V : - =9 V : - =6 V Usg quatos (85) a (8), w / m a p / 49 m Thror, th ltro otrato s 4 4 m -, a th hol otrato s 49 5 m - * Thr s aothr way to sol ths problm: / 4 5 m a p / 455 m () I T = 8 K, thr s ough thrmal rgy to r mor ltros rom sloslo bos H, usg quato (8), w rst alulat th trs arrr sty at 8 K: 8 K 8 K g / 6 56 m whr ( T a ( T m 8 K) h m 8 K) h / / 9 T 8 m p / 4 9 K T K / m 45 m 9 m Clarly, at 8K s muh largr tha - a (whh s qual to rom th prous part) H th ltro otrato s, a th hol otrato s p=( ) / Th smoutor s trs at 8K, a s loat ry los to th m-bagap arly Itrs Smoutor Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

13 Applyg quato (8) to ths problm yls / p p p p 77 m a 4 m (a) B s a group III lmt Wh a to S (whh blogs to Group IV), t ats as a aptor proug a larg umbr o hols H, ths boms a P-typ S lm (b) At T = K, s th trs arrr sty s glgbl ompar to th opat otrato, p= a =4 6 m -, a = ( = m - ) /p = 5 m - At T = 6 K, 6 K 6 K g / 5 6 m whr ( T a ( T m 6 K) h m 6 K) h / / 9 T m p / 4 9 K T K / m 94 m 9 m Th trs arrr otrato s o mor glgbl ompar to th opat otrato Thus, w ha p a m 4 m, a m / 4 m 7 m / p Th ltro otrato has ras by may orrs o magtu () At hgh tmpraturs, thr s ough thrmal rgy to r mor ltros rom slo-slo bos, a osqutly, th umbr o trs arrrs rass () Usg quato 88, w alulat th posto o th Frm ll wth rspt to T / pt 4 V, T K l 6 At 6 K, th Frm ll s loat 4 V abo th al ba Iomplt Iozato o Dopats a Frz-out 4 From quato (9), w kow that + a - = p + + S + s muh largr tha a -, all th sampls ar -typ, a + - a - = 5 /m Ths alu s assum to b ostat Usg th quatos (8) a (9b), Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

14 p / xp / CT xp, a g g / whr C s a tmpratur pt ostat Usg th sstty o p by p/t, p / T / CT xp g g / Thror, th largr th rgy gap s th lss sst to tmpratur th morty arrr s For th to o th sstty o p, p / T / p g / / T Th tmpratur sstty o th morty arrr s gratr or largr g 5 (a) Lt us rst osr th as o -typ opg Th opat atoms ar loat at rgy s th bagap, ar th outo ba g Th problm stats that w ar osrg th stuato whh hal th mpurty atoms ar oz, = / I othr wors, th probablty o opat atoms bg oz s ½, or orsly, th probablty that a stat at th oor rgy D s ll s ½ From Problm part (a), w kow that ( D )=/, th D = From quato 85, / W also kow that = D a - D =5V / m 9 T T ) 8 m h K / D / D T ) D / T ) T ) Ths quato a b sol tratly Startg wth a arbtrary guss o K or T, w T orgs to 844 K Smlarly, or boro / / / mp 9 T T ) 4 m h K / a / a T ) a / T ) T ) Startg rom T =K, w T orgs to 677K a Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

15 (b) W wat to T whr s Ths a b wrtt as T T / T g / 9 g / 7 K whr m T ) h m T ) h p / / T K 9 / T K / m m a W to sol th quato tratly, as part (a) or = = 7 m - Startg rom T=K, w gt T=777 K or = For = a, w smply rpla th quato abo wth a Startg rom T =K, w T=65 K () I w assum ull ozato o mpurts at T = K, 6 4 For ars: m, p m 5 5 a m m a For boro: p, () Plas rr to th xampl Sto 8 For ars, 9 4 m l 88V 4 p m For boro, 9 4 m l 4V 5 p m () I as o ars + boro, 5 m 4 a 9 m, a m 5 p, a 9 m 9 4 m l 6V l 9 V 4 p m Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

16 6 (a) I w assum ull ozato o mpurts, th ltro otrato s = 7 m - Th hol otrato s p=( ) /=( m - ) / 7 m - = m - Th Frm ll posto, wth rspt to, s 9 7 / 6l8 m / m V l 5 s loat 5 V blow (b) I orr to hk th ull ozato assumpto wth th alulat Frm ll, w to th prtag o oors oup by ltros D V 7 m D D, a 5 V V 9 m % o D / / 6 S oly % o opats ar ot oz, t s to assum that th mpurts ar ully oz () W assum ull ozato o mpurts, th ltro otrato s = 9 m - Th hol otrato s p=( ) / = ( m - ) / 9 m - = m - Th Frm ll posto, wth rspt to, s 9 9 / 6 l8 m / m V l 7 It s loat 7 V blow Aga, w to th prtag o oors oup by ltros orr to hk th ull ozato assumpto wth th alulat Frm ll D V 9 m D D, a 8 V V 78 m 7% o D / / 6 S 7% o opats ar ot oz, th ull ozato assumpto s ot orrt () For T= K, w to us quato () to th ltro otrato s th tmpratur s xtrmly low Frst, w alulat a at T=K: ( T a ( T m K) h m K) h / / 9 T m p / 4 9 K T K / m 9 m 7 m Th ltro otrato s Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

17 K / 8 84 m D A, th hol otrato s p / whr K K / 75 g m S s xtrmly small, w a assum that all th ltros ar otrbut by oz opats H, D m / m 7 m Th ull ozato assumpto s ot orrt s oly 84-7 % o s oz To loat th Frm ll, D l 48 V - = 5-48 = V Thror, th Frm ll s posto V blow, btw a D 7 (a) W assum ull ozato o mpurts, th ltro otrato s = 6 m - Th hol otrato s p=( ) / = ( m - ) / 6 m - = 4 m - Th Frm ll posto, wth rspt to, s / 6l8 m / m V l It s loat V blow W to th prtag o oors oup by ltros orr to hk th ull ozato assumpto wth th alulat Frm ll D V 6 m D 6 D, a D / m % o 6 V / 6 V S oly % o opats ar ot oz, th ull ozato assumpto s orrt Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458

18 (b) W assum ull ozato o mpurts, th ltro otrato s = 8 m - Th hol otrato s p=( ) / = ( m - ) / 8 m - = m - Th Frm ll posto wth rspt to s 9 8 / 6l8 m / m V l 87 It s loat 87 V blow W to th prtag o oors oup by ltros orr to hk th ull ozato assumpto wth th alulat Frm ll D V 8 m D 7 D, a 7 V V 94 m 9% o D / 7 / 6 S 9% o opats ar ot oz, th ull ozato assumpto s ot aurat but aptabl () W assum ull ozato o mpurts, th ltro otrato s = 9 m - Th hol otrato s p=( ) / = ( m - ) / 9 m - = m - Th Frm ll posto, wth rspt to, s 9 9 / 6l8 m / m V l 7 It s loat 7 V blow Aga, w to th prtag o oors oup by ltros orr to hk th ull ozato assumpto wth th alulat Frm ll D V 9 m D D, a 8 V V 78 m 7% o D / / 6 S 7% o opats ar ot oz, th ull ozato assumpto s ot orrt S s ot ully oz a (oz) << (ot-oz), D / / D Solg th quato abo or yls D l Parso uato, I, Uppr Sal Rr, J All rghts rsr Ths publato s prott by Copyrght a wrtt prmsso shoul b obta rom th publshr pror to ay prohbt rprouto, storag a rtral systm, or trasmsso ay orm or by ay mas, ltro, mhaal, photoopyg, rorg, or lkws For ormato rgarg prmsso(s), wrt to: Rghts a Prmssos Dpartmt, Parso uato, I, Uppr Sal Rr, J 7458