Chapter Six Free Electron Fermi Gas

Transcription

1 Chapt Six Elcton mi Gas What dtmins if th cystal will b a mtal, an insulato, o a smiconducto? E Band stuctus of solids mpty stats filld stats mpty stats filld stats E g mpty stats filld stats E g Conduction band Valnc band filld / Conduction band mpty patially filld E g < B T E g >> B T Mtal smiconducto Insulato Conduction lctons a availabl Conduction lctons a availabl at high T o by doping No conduction lctons

2 Basic ida : pushing atoms togth to fom a cystal f atoms molculs cystals disct ngy lvls splitting of lvls band of stats Low ngy lvls main disct and localizd on atoms. Co stats High ngy lvls split to fom bands of closly ngy lvls that can xtnd though th cystal valnc and conduction bands

3 lcton modl tat conduction lctons as f paticls Continuum stats dnsity of stats mi statistics occupancy of stats Thmal poptis Thmal ngy, hat capacity, Elctical and thmal tanspots scattings of conduction lcton Magntic fild ffct

4 conduction lctons in th box Not intacting lctons (xcpt w/. walls of th box) K.E. P m ( h) m In ality, intactions of lctons : Ions stady Coulomb intaction (lcton binding) but Scning by co lctons wans th attaction at lag distanc Pauli xclusion pincipl quis that conduction lctons stay away fom co lctons localizd at th atoms. Elctons stong Coulomb pulsion but Coulomb pulsion Pauli xclusion pincipl Elctons tnd to stay apat 4

5 In on dimnsion U m L Bounday condition thfo, ϕ n Asin ( x) n U(x) x L x ϕn () ϕn(l) w/. n lswh Schöding quation nπ L and h d m dx n ( h ) Ψ + U(x) Ψ Ψ n m How to accommodat N lctons on th lin? h ( nπ ) ml Pauli xclusion pincipl + spin dgnacy (two spins p lvl) Stat to fill th lvls fom th bottom (n) and continu to fill high lvls with lctons until all N lctons a accommodatd.,, n, wh n is th valu of n fo th uppmost filld lvl. 5

6 In gnal cass, such as piodic chain Bounday condition Dnsity of stats ϕn (x) ϕn(x + L) On stat vy -intval π/l D()d D( )d D( ) D()d d (L/π ) d / d D() n ± n nπ L (π /L) L π unifom and h /m (L/π ) ml m D( ) h /m πh ml πh ( h) singly spin dnsity of stats in on dimnsion D() -/ 6

7 In th dimnsions, SchÖding quation h m x + y + z Ψ + U(x, y,z) Ψ Ψ Bounday condition : Ψ is piodic in x, y, and z with piod L π 4π π 4π π x, ±, ±,... ;y, ±, ±,... ;z, ±, ± L L L L L On stat vy -volum intval x y z (π/l) L D() (π /L) π D( )d D()4π x 4π y ( π) z d V m d h V ( π) V 4π 4π ( m) h d V ( π) d d / d 4π L,... 7

8 D( ) V m 4π h / x fo spin dgnacy singly spin dnsity of stats in th dimnsions D() / Conduction lctons : f to mov though th cystal Dnsity of conduction lctons n N/V (Tabl ) typically n ~ ~ cm - mostly s obital lctons but also p and d D( ) V m π h / dnsity of stats in th dimnsions 8

9 Diffnc btwn lctons and phonons Elctons Phonons Numb NnV fixd N ~ B T vais w/. T Dgnacy mions Bosons (mi-diac statistics) (Planc distibution) two p obital stat n p mod xcitd Dispsion ω Dnsity of stats D() / D(ω) ω up to ω D Dby Gound stats T, ill ngy lvl fom bottom : p lvl n 4 highst lvl occupid w/. mi ngy Maximum ngy : h /m 9

10 z Stats w/. a occupid mi sph volum in -spac occupid by lctons in th gound stats y mi sufac stats w/. x 4 V N π (π ) # of lctons spin volum of mi sph D() π N V / and h π N m V / typically, ~ -8 cm - ~ V

11 n T

12 T D() N D( )f( )d D( )d f() f() is th pobability that a stat of ngy is occupid f() {,, > mi ngy is impotant bcaus lctonic poptis a dominatd by stats na only B T <<

13 init tmpatus Kintic ngy of lcton incass du to th incas of thmal ngy occupy high ngy lvls What is th pobability of occupancy of an lcton stat w/. ngy at T? Boltzmann facto xp(- / B T)? o phonons (Bosons) Elctons a mions : quantum ffcts such as Pauli xclusion pincipl Standad poblm in statistics (s appndix D) mi-diac distibution f( ) xp [( µ )/ T] B + wh µ is th chmical potntial to consv lcton numb At T µ, whn µ, f() changs discontinuously At finit T, whn µ, f()/ Whn (-µ) >> B T, f() Boltzmann distibution

14 () f(,t) f() T.T T.T T.5T T.5T T.T (A.U.) () whn T<.T, µ, and f(, T)/ whn E whn < µ, f(,t)>/ whn > µ, f(,t)</ 4

15 () Elctons xcitd fom blow to abov as T is incasd ~ B T -df/d T, δ-function T.T T.T T.T T.5T Spad ngy gion incass with incasing tmpatu. 5

16 (4) µµ(t) dcass as T incasd why? D() / non-unifom What dos dtmin µ? Total numb of lctons is consvd N Hnc, d D()f(,T) µ(t) π BT N (m) V d 4π h xp ( -µ)/ B ( T) +.. µ ( ) µ ( ) B T/ B T/ 6

17 (5) Usful xpssion fo D() D( dn V (m) stat ) d 4π h c N d D( ) f( ) TK d c c N N c, D( ), and D ( ) N Total thmal ngy and hat capacity of lctons at T Classical point of viw, U N ( B T/) and C V N ( B /) In ality, much small at oom T Not vy lctons gains ngy B T/ 7

18 U d D( ) f(, T) At gound stat, T D()f(,T).6 U d N 5 N N 5 Avag ngy of ach lcton <>.6 At finit tmpatu (T ), lctons a xcitd to high ngy stats and U(T) incass. D()f(,T) T T f(,t) T T 5 8

19 U(T) U C C C d D( ) f(, T) d ( - ) D( ) f( ) + du dt d ( - ) U(K) + U(T) d ( f(, T) D( ) T - ) D( ) N + ( - f( )) d D( ) f( ) d D( ) f( ) d D( ) In gnal, T/T <., df/dt has non-zo valu within coupls of B T C D( D( D( ) ) B ) d T B ( - ) / d D() is about D( ) in th ngy gim ± f(, T) T x dx x x + ( ) - BT T - dx x x x ( ) + f(,t) T d dt (( ) /( T) ) D( ) B T ( ) xp( ( ) /( BT) ) T ( xp( ( ) /( T) ) + ) x T B xp x wh x + x ( + ) BT B B 9

20 - dx x x π x ( + ) C D( π ) N B B T T T π π T N T B B T lctons contibution to hat capacity U C T T.6N T T In gnal, whn T<<Θ D and T<<T / B C γt + AT sum of lcton and phonon contibutions

21 γ π N T B T - m (mass of lcton) m th, obtaind fom masud γ obsvd, is diffnt fom m. Intaction btwn conduction lctons with piodic potntial of th cystal lattic Band ffctiv mass Intaction btwn conduction lctons with phonons. moving lctons dag naby ions along Intaction btwn conduction lctons with thmslvs. A moving lcton causs an intial action in th suounding lcton gas. o som matials, m th can b m. Havy mions such as CAl, CCu Si, and oth xotic supconductos. Sis in Modn Condnsd Matt Physics Vol ditd by H. Radousy Magntisms in Havy mion systms

22 Tanspot poptis Applying E, T diving fild Elctic cunt dnsity Hat cunt dnsity J σ E J κ U J,J u cunt dnsity + L ( ) T T ( T) + TL T E cofficints σ : lctical conductivity κ : thmal conductivity L T : thmallctic cofficint Consid all physics about cais and scattings coupling both lctic and thmal sponss

23 Elctical conductivity Applying an lctic fild Equation of motion At a constant E, E v d v ( )E m dt Et (t) () h dp dt d h dt Elctic fild acclats lctons incass linaly y E y E < occupid x δ x E shifts mi sph in -spac E Each incass byδ τ h

24 4 Cunt dnsity δ m n δ n m n m δ n m n m δ m n m n v J o o o o o o o o o o h h h h h h h + + unshiftd What limits δ? scattings Elctons can scatt to stats of low ngy and duc cunt. Assum collision tim is τ τ E δ h E m n τ τ E m n J h h m n τ σ o g n n + Thmal quilibium Dviation fom non-quilibium J σe And Ohmic dvics Elctic conductivity lcton modl

25 δ Appoachs to a stady stat valu non-quilibium J nv d τ Eτ n m n τ m In classical pictu, all s cay chag at a constant vlocity v d. E Only lctons na th mi sufac contibut to cunt. δ<< nwly mptid n y nwly filld n f x J v n f v paticipating stats g n ( n f + n ) v Paul Dud ( v ) all at v 5

26 @ Cunt is caid only by a faction of lctons tavling at v. Both nwly filld and nwly mptid stats contibut sam cunt. n f lctons n Copp I σ τ n τ m mσ n hols V V m 9. Coss sctional aa Awt Lngth l g Rρl/A ρ(k).7µωcm n /m v.57 6 m/sc ( 9 ) -8.6 coulomb.7 Ωm.5-4 sc l v τ 4-8 m 4 nm o E volt/cm v d ~.4 m/sc action of stats paticipating δn n δ ~ v v d 6 6

27 Elcton scatting pocsss σ n τ m Conductivity σ is limitd by scattings (τ, l) fo a pfct cystal, no scatting σ Scatting mchanisms ρ(t) IV III II I Rgim I Lag -ph scattings ρ(t) T Rgim II Small -ph scattings ρ(t) T 5 ρ o Rgim III - scattings ρ(t) T lcton modl T Rgim IV impuity scattings ρ(t) T ~ ρ o 7

28 Rgim I Lag angl Inlastic scatting ( lcton-phonon at high T ) Scatting at # of phonon ( T ) q τ T - Thfo, σ T - ' ρ T This nglct umlapp pocss which givs a diffnt sult. (xponntial as fo in th insulato) Umlapp should dominats in an intmdiat ang of tmpatu. Rgim II Small angl Inlastic scatting ( lcton-phonon at low T ) ' q E ± q E ± hω q Scatting at τ - # of phonon ( T ) Dby Effctivnss facto of collision 8 q << T

29 Rgim IV Elastic scatting (impuitis, boundais, dfcts, ) Impuity concntation tc.. dtmin τ, l ' Engy is consvd E E τ constant constant l v τ constant, indpndnt of T Thfo, σ(t) σ o, ρ(t) ρ o Rgim III Elcton-lcton scatting ' E + + E ' E ' + τ - T possibl stats ' + ' E ' σ(t) T -, ρ(t) T 9

30 Two additional uls : () Multipl scatting mchanisms τ σ ρ τ ρ ph σ ph ph ρ τ σ ρ τ σ impuity impuity impuity Matthissn s ul not xact but ptty good () Rsidual sistanc atio R(K) RRR R(K) ρ(k) ρ o Phonon dominats Impuity dominats RRR, pfct cystal In gnal, RRR ~ to 4 (pu mtal)

31 Expimntal vidncs fo Matthisn s Rul Th diffnt sampls w/. diffnt dfct concntations. McDonald and Mndlssohn (95). J. Lind, Ann. Phys. 5, 5 (9).

32 Motion in magntic filds d Elctic fild E qe h chang magnitud of dt qh d Magntic fild B qv B B h chang diction of m dt Exampl : BBẑ Lontz foc motion diction d dt d dt d dt x y z qb m qb m y x d dt d dt d dt y z x qb m qb m x y x y z solutions (t) (t) (t) A cos(ω A sin(ω C c c t) t) Hlical cicula motion B ω c qb/m cycloton fquncy

33 Cicula motion in both al and - spacs in f lcton modl constant z B y Elcton at movs in obits along th mi sufac sph. Tu fo all mi sufacs, not only fo f lctons. x o tanspot poptis, impotant facto is ω c τ, phas chang of lcton btwn two succssiv scatting vnts.

34 Hall Effct Magntic fild B Bẑ J jxˆ nv d xˆ z y cunt dnsity x z y x Tansvs B E E B -, v d E jb n ŷ B -, v d v qe + qv B jbr H E + ŷ jb ( ŷ) n In gnal, R H E j x y B n Hall cofficint Hall ffct vals dnsity and sign of chag cais. ρ H E j x y V I y Hall sistivity [ Ωm ] ( thicnss) 4

35 Mtal valnc tho xp R H /RH Li.8 Na. K. Alali mtals : OK Rb. Cs.9 Cu.4 Nobl mtals : Ag. Au.5 Cd -. Zn -.8 Al -. numically incoct B -. High-valnt mtals : on hol wong sign 5

36 Thmal conductivity T H j U Th flux of th thmal ngy κ j E T j U κ T L dt dx U κ : thmal conductivity cofficint T Elctic cunt dnsity Hat cunt dnsity J J σ E + LT TL E + U T th ngy tansmittd acoss unit aa p unit tim ( T) κ( T) thmal lctic cunt dnsity In a opn-cicuit hat masumnt, L T J E T σ L T J u TLT T κ T σ TL T κ T σ κ * ju T κ J TL σ T In fact, th nd tm, L T, is vy small in most mtals and smiconductos. Hnc, κ * κ 6

37 Hat cunt fom phonon pvious chapt κ Cv l C g π mv n B Cv Apply to f lctons g BT Ratio of Thmal to Elctical Conductivity κ σ π n BTτ / m n τ/m π B τ T κ π n m B Tτ In pu mtal, th lctonic contibution is dominant at all Ts. In impu mtals o disodd matials, τ is ducd by collisions with impuitis, and th phonon contibution may b compaabl with th lctonic contibution. LT Lonz numb L π B Widmann-anz law L th.45-8 Watt-Ω/K 7

38 A tmpatu-indpndnt Lonz numb dpnds on th laxation pocsss fo lctical and thmal conductivity bing th sam which is not tu at all tmpatus. 8