Complementi di Fisica Lecture 19

Transcription

1 Complement d Fsca Lectue 9 Lvo Lance Unvestà d Teste Teste, 0--04

2 Couse Outlne - Remnde The phscs of semconducto devces: an ntoducton Quantum Mechancs: an ntoducton just a few moe comments on: (3-d) Hdogen atom, angula momentum, spn Sstems wth man patcles: femons and Paul pncple (slghtl moe) Advanced semconducto fundamentals Eneg bands, effectve mass Equlbum Cae Statstcs: denst of states, Fem functon Non-equlbum tanspot of chage caes and the moton of electons n eal cstals Too late fo Smulatons; evstng Shockle; dscussng a devce as an eample Measuements of semconducto popetes L.Lance - Complement d Fsca - Lectue 9

3 Lectue 9 - outlne A ve bef summa on: (3-d) Hdogen atom, angula momentum, spn Sstems wth man patcles: femons and Paul pncple Mult-electon atoms, peodc table, and cstals Back to semconductos: equlbum cae statstcs Themal equlbum and detaled balancng Denst of states Fem pobablt dstbuton functon consequences? see st pat of the couse Boltmann appomaton Numbe of caes at band edges Etc (ntnsc, etnsc semconductos, ) L.Lance - Complement d Fsca - Lectue 9 3

4 3-d wave mechancs (just a hnt ) Hdogen atom Angula momentum Spn

5 Hdogen atom smple : tme-ndependent Schödnge equaton fo the electon: cental Coulomb potental V ( ) q /(4πε 0 ) sphecal coodnates (, θ, φ ) Sepaaton of vaables (3) 3 ntege quantum numbes dentf each soluton n,l,m (, θ, φ ) R nl () Y lm (θ, φ ) ˆ nlm En nlm Lˆ nlm H Eneg ( Boh!) Angula momentum ( l + ) nlm L nlm hm nlm h l ˆ L.Lance - Complement d Fsca - Lectue 9 5

6 Radal pdf Radal pobablt dstbuton functon [ R () ] nl Peaks occu at Boh obts ad L.Lance - Complement d Fsca - Lectue 9 6

7 Sphecal hamoncs Angula Pobablt dstbuton functons Ylm Pobablt of fndng the electon In the sold angle (snθ dθ dφ) L.Lance - Complement d Fsca - Lectue 9 7

8 Angula momentum Also angula momentum s quanted! One can onl measue smultaneousl the magntude squae and one component (the components don t commute!) Catesan and sphecal coodnates: L p Lˆ Lˆ Lˆ + Lˆ h φ Lˆ + Lˆ pˆ h pˆ θ Egenvalues and egenfunctons L Y Lˆ Y lm Lˆ pˆ pˆ Lˆ pˆ + cotθ + θ θ φ sn ( θ, φ) h l( l + ) Ylm( θ, φ) l,, 3,... ( θ, φ) hmy ( θ, φ) l m ntege + l ˆ lm lm pˆ L.Lance - Complement d Fsca - Lectue 9 8

9 Magnetc effects On dmensonal gounds, fo a chaged patcle wth angula momentum we epect a magnetc moment and a contbuton to potental eneg when nteactng wth an etenal B feld: q µ g L U µ B m Zeeman effect (splttng of degeneate levels) and Sten-Gelach epement ( space quantaton : splttng of an atomc beam) Zeeman splttng Sten-Gelach L.Lance - Complement d Fsca - Lectue 9 9

10 Spn Elementa patcles ca also an ntnsc angula momentum ( spn S) besdes the obtal angula momentum (L) The egenstates ae not the sphecal hamoncs: not functons of θ, φ at all! The quantum numbes s, m can be half-ntege The magntude s s specfc and fed fo each elementa patcle, and s called spn Electons have spn s ½, wth two possble egenstates: up and down ˆ 3 S sm h s( s + ) sm s 0,,,,...; m s, s +,..., s Sˆ sm hm sm electons : s χ egenstates and egenvalues : χ + h, egenvalue h, egenvalue L.Lance - Complement d Fsca - Lectue 9 0

11 Fo eample: Spn: obsevable effects Anomalous Zeeman effect : futhe level splttng n stong B felds Fne Stuctue level splttng due to spn-obt couplng Anomalous Zeeman effect Spn-obt couplng L.Lance - Complement d Fsca - Lectue 9

12 Ths s not the end Hdogen has been a ve nteestng laboato: Odes of magntude of dffeent effects, teated as petubatons, n tems of the a-dmensonal fne stuctue constant α, epessng the stength of the electomagnetc couplng: α e 4πε 0 hc Relatvt, spn-obt Coulomb feld quantaton Electon-poton magnetc moments L.Lance - Complement d Fsca - Lectue 9

13 Man-patcle sstems (just a hnt ) Identcal patcles Bosons and femons Paul Pncple Peodc table

14 Identcal patcles Man-patcle sstems? Let s stat wth two: Wave functon, pobablt dstbuton, hamltonan; S.equaton Fo tme-ndependent potentals: tme-ndep. S.eq. and statona states ( ) ( ) ( ) t V m m H H t d d t t,, ˆ ˆ,,,, h h h + Ψ Ψ Ψ Ψ ( ) ( ) ( ) E V m m e t Et + Ψ,,,, h h h L.Lance - Complement d Fsca - Lectue 9 4

15 Bosons and femons Fo dstngushable patcles (fo nstance, an electon and a poston): patcle s n the (one-patcle) state a ( ) patcle n state b ( ) (, t) ( ) ( ), But: dentcal patcles (fo nstance, two electons) ae tul ndstngushable n quantum mechancs: Thee ae two possble was to constuct the wave-functon: + smmetc : bosons ± ± ( ) A [ ( ) ( ) ( ) ( )], a b b a All patcles wth ntege spn ae bosons All patcles wth half-ntege spn ae femons a b - ant-smmetc : femons L.Lance - Complement d Fsca - Lectue 9 5

16 Femons and Paul pncple Connecton between spn and statstcs (o wave-functon echange smmet) can be poven n elatvstc QM must be taken as an aom n non-elatvstc QM Paul ecluson pncple: Two femons (ant-smmetc w.f.) cannot occup the same a state! Indeed: b ( ) A[ ( ) ( ) ( ) ( )] 0, a a a a It can be shown that: The echange opeato P s a compatble obsevable commutng wth H one can fnd solutons that ae ethe smmetc o antsmmetc Fo dentcal patcles, the wave functon s equed to be smmetc (fo bosons) o ant-smmetc (fo femons) L.Lance - Complement d Fsca - Lectue 9 6

17 Paul Pncple: consequences fo electons Fo electons the total wave-functon (ncludng spn) must be ant-smmetc, and the cannot occup the same state (two pe level allowed, wth opposte spn. The ant-smmet equement allows some wave-functon confguatons, pohbts othes: equvalent to an echange foce Fllng of avalable levels b electons n a bo (neglectng nteactons among electons!): Fem level hghest eneg level occuped at T 0K (see eecses) degeneac pessue : even neglectng electc nteactons between electons, the Paul pncple mples that the closest that two electons can get to each othe s oughl a half a DeBogle wavelength coespondng to the Fem eneg (see eecses) L.Lance - Complement d Fsca - Lectue 9 7

18 Paul pncple: Peodc table of elements Mult-electon atoms ae teated b appomate methods: wave functons ae modfed (and called obtals ), but: the ae labeled b the same quantum numbes n, l, m, and: Obtals ae flled b electons followng the Paul ecluson pncple: two electons cannot have the same quantum numbes (state) L.Lance - Complement d Fsca - Lectue 9 8

19 Peodc table of the elements L.Lance - Complement d Fsca - Lectue 9 9

20 Back to semconductos Themal equlbum Denst of states Fem pobablt dstbuton functon Cae concentatons (Boltmann appo.)

21 Themal equlbum Themal equlbum? Between two bodes o sstems n themal equlbum thee can be no net tansfe of an sot (law of detaled balancng). Themal equlbum: statc, endless, useless wh do we cae? Sstems nea themal equlbum tend to come to equlbum n pedctable was The pedctable behavo of sstems not qute n equlbum allows us to desgn and constuct useful devces! Fom the statstcal pont of vew: Themal equlbum epesents the dstbuton of mamum pobablt, acheved when the detaled balancng between the possble pocesses s eached L.Lance - Complement d Fsca - Lectue 9

22 Fndng the mamum pobablt Fst pat: specf all possble states (solutons to a wave equaton) and a set of appopate bounda condtons Possble egenstates of the sstem (n ou case E-k plot!) Total ntenal eneg of the sstem Rules about fllng states (n ou case the Paul pncple) Rules about consevaton of patcles Second pat: pocedue to fnd the most lkel dstbuton of patcles among the states, that does not volate an of the ules Fndng a mamum subject to constants ( ules ): Lagange s method of undetemned multples Let s stat wth the denst of states (step ) L.Lance - Complement d Fsca - Lectue 9

23 Step : denst of states

24 Denst of states Denst of states g(e) g(e) numbe of allowed states fo electons n the eneg ange (E, E+dE ), pe unt volume of the cstal Fo a geneal soluton (an E ) we should use the full machne of band theo (possble, but complcated!) but we ae manl nteested n the band edges, nomall populated b caes: much smple! Shotcut: Equvalent poblem: denst of states fo electons n a 3-d bo, povded we fnall modf the soluton, takng nto account the effectve mass m* and the band stuctue Equvalent poblem: electons n a bo L.Lance - Complement d Fsca - Lectue 9 4

25 Electons n a bo Infntel deep 3-d potental well: Tme-ndependent Schödnge equaton Sepaaton of vaables m k E me k k : o 0 h h c b a < < < < < < ( ) ( ) ( ) ( ) 0 substtutng and dvdng b :,, k Fo each of the thee functons: a k k < < constant (smla fo the othe two) L.Lance - Complement d Fsca - Lectue 9 5

26 Eneg egenstates and egenvalues Each soluton s assocated wth a 3-d k-space vecto: L.Lance - Complement d Fsca - Lectue 9 6 ( ) ( ) ( ) ( )... 3,,,,,,, sn sn sn,, ± ± ± + + E n n n k k k k m k E c n k b n k a n k k k k A h π π π One soluton pe cell : denst pe unt volume of k-space: 3 π π π π abc c b a

27 Countng the solutons n E ntevals Onl the st octant n k-space coesponds to ndependent solutons Each state can be occuped b two electons wth opposte spn (Paul pncple) allowed eneg states unt volume n k -space 8 abc 3 π abc 3 4π eneg states wth k < k < k + dk abc 4π 3 4 π k dk k-space volume between two sphees eneg states wth E < E < E + de γ k me h dk m me 3 π h ( E) de abc de g( E) ( E) γ V L.Lance - Complement d Fsca - Lectue 9 7 m h Cstal volume V abc de E Denst of states m me 3 π h

28 denst of states Denst of states, smplfed model (bo wth nfntel deep walls) But: bands? Inteacton wth the cstal peodc potental? No poblem: the aveage effectve mass m* and the cstal wave numbe k descbe the nteactons wth the cstal fo E close to E C : E E C m me π h ( E) 3 g h m m h k n n gc ( E) 3 mn π ( E E ) C E E C Smlal fo holes: g V ( E) m p m p π h ( E E) 3 V E E V L.Lance - Complement d Fsca - Lectue 9 8

29 wth aveage effectve masses The effectve masses appeang n the denst of states fo some useful semconductos (S, Ge, GaAs) ae aveaged ove cstal dectons onl GaAs s appomatel sotopc See R.F.Peet, secton 4.., p.94, fo detals on S and Ge L.Lance - Complement d Fsca - Lectue 9 9

30 Step : pobablt dstbuton functon (Fem-Dac)

31 F.-D. pobablt dstbuton functon Fom themodnamcs: The most lkel macoscopc state s the one coespondng to the lagest numbe W of equvalent mcoscopc states, compatble wth a gven total numbe N of electons and a fed total eneg E TOT W themodnamcal pobablt ; entop ln (W ) W N S avalable states at eneg E (patall) flled b N electons E N W E TOT S N N const. const. ( S N ) S!! N! L.Lance - Complement d Fsca - Lectue 9 3

32 Mamaton pocedue Take the logathm, use Stlng s appomaton, and set the dffeental to eo (S ae constant, N vaable): d lnw ln lnw ( lnw ) ln ( ln S! ln( S N )! ln N!) ( lage) [ S ln S S ( S N ) ln( S N ) + ( S N ) N ln N + N ] [ S ln S ( S N ) ln( S N ) N ln N ] ( lnw ) N [ ln( S N ) + ln N ] ln ( S N ) dn 0 dn dn L.Lance - Complement d Fsca - Lectue 9 3

33 Constants: Lagange multples ( ) ( ) ln 0. ln TOT E dn E E N dn N N dn N S W d Intoducng the undetemned Lagange multples -α and -β : L.Lance - Complement d Fsca - Lectue 9 33 ( ) [ ] ( ) ( ) ( ) E E E e E f e E f e N S E N S dn E N S β α β α β α β α β α ln 0 ln Fo closel spaced levels, E E

34 Phscal meanng of α and β α and β: fom themo-dnamcal aguments fo femons, Fem dstbuton: k B ev/k β f k T B EF α k T B ( E) ( E E ) k T + e F B T absolute tempeatue E F electochemcal potental o Fem eneg L.Lance - Complement d Fsca - Lectue 9 34

35 Femons, bosons and classcal lmt Femons: Fem-Dac dstbuton (at most one femon pe state): f FD ( E) ( E E ) k T e F B + Bosons: Bose-Ensten dstbuton (an numbe of bosons pe state) f BE ( E) ( E E ) k T e B B Classcal: Mawell-Boltmann (good lmt of quantum statstcs when: few patcles / hgh tempeatue, small fllng pobablt pe state) f MB ( E) E k T e B L.Lance - Complement d Fsca - Lectue 9 35

36 Numbe of caes at band edges (Boltmann appomaton) Fom hee: We have now undestood all the ngedents used n Lectue_3 to obtan cae concentatons n ntnsc and etnsc semconductos at equlbum allowed and fobdden eneg bands denst of avalable states Fem pobablt denst functon In patcula, emembe that the Boltmann appomaton to the Fem functon nea band edges fo non-degeneate semconductos allowed us to compute concentatons eplctl! Go back to Lectue 3 and appecate the consequences L.Lance - Complement d Fsca - Lectue 9 36

37 Lectue 9 - summa We had a quck look at 3-d wave mechancs, ncludng angula momentum and spn. Man-patcle sstems bought us to consde also dentcal patcles and the wave-functons, that must have a defnte echange (ant)-smmet. Electons ae femons and ae descbed b ant-smmetc (oveall, ncludng spn) wave functons, wth nteestng consequences ( echange foces, degeneac pessue, Fem eneg ). Back to semconductos, we consdeed equlbum statstcs and obtaned both the denst of states and the Fem-Dac pobablt dstbuton functons, essental ngedents to pedct equlbum cae concentatons. Net step: e-consde non-equlbum tanspot of chage caes (dft, dffuson; geneaton/ecombnaton) and ts eplanaton L.Lance - Complement d Fsca - Lectue 9 37

38 Lectue 9 - eecses Eecse 9.: Consde a smplfed model of a conducto wth non-nteactng conducton electons n a 3-d nfnte well. Fnd the Fem eneg and the aveage nte-electon spacng. Appl the esults to the case of alumnum (A7), assumng: denst ρ.70 3 kg/m 3, and thee fee atoms pe electon (hnt: see Bensten, pa.0-5 and eample 0-5). Eecse 9.: Wte down the esults of ths lectue on the denst of states fo the conducton and valence bands and on the Fem pobablt denst functon. Compae them wth those used n pevous lectues to compute the concentaton of caes n semconductos at a gven tempeatue. OK? Eplan the eason fo ntoducng the effectve mass n the denst of states as obtaned fom the nfnte well bo model L.Lance - Complement d Fsca - Lectue 9 38