Phys 446: Solid Stat Physics / Optical Proprtis Lattic vibrations: Thrmal, acoustic, and optical proprtis Solid Stat Physics Last wk: (Ch. 3) Phonons Today: Einstin and Dby modls for thrmal capacity Thrmal conductivity HW discussion Lctur 5 Fall 007 Lctur 5 Andri Sirnko, NJIT Matrial to b includd in th tst Oct. th 007 Crystallin structurs. 7 crystal systms and 4 Bravais lattics Crystallographic dirctions d hkl and Millr indics h a Dfinition of rciprocal lattic vctors: What is Brillouin zon Bragg formula: d sinθ mλ ; k G k + b n l + c 3 Factors affcting th diffraction amplitud: Atomic scattring factor (form factor): rflcts distribution of lctronic cloud. In cas of sphrical distribution Structur factor F Elastic stiffnss and complianc. Strain and strss: dfinitions and rlation btwn thm in a linar rgim (Hook's law): Elastic wav quation: σ ij C ε j f aj f a r0 f 4πr j a πi( hu + kv + lw ) 0 j n( r) ijkl kl ijkl kl kl kl u Cff u t ρ j ε ij S σ i k r l d 3 r ( k r) sin n( r) k r sound vlocity v 4 dr C ff ρ
Lattic vibrations: acoustic and optical branchs In thr-dimnsional lattic with s atoms pr unit cll thr ar 3s phonon branchs: 3 acoustic, 3s - 3 optical Phonon - th quantum of lattic vibration. Enrgy ħω; momntum ħq Concpt of th phonon dnsity of stats Einstin and Dby modls for lattic hat capacity. Dby tmpratur Low and high tmpraturs limits of Dby and Einstin modls Formula for thrmal conductivity K Cvl 3 B abl to obtain scattring wav vctor or frquncy from gomtry and data for incidnt bam (-rays, nutrons or light) 5 Summary of th Last Lctur Elastic proprtis crystal is considrd as continuous anisotropic mdium Elastic stiffnss and complianc tnsors rlat th strain and th strss in a linar rgion (small displacmnts, harmonic potntial) Hook's law: Elastic wavs σ ij C ε ijkl kl ε ij Sijklσ kl kl kl u Cff u t ρ sound vlocity Modl of on-dimnsional lattic: linar chain of atoms Mor than on atom in a unit cll acoustic and optical branchs All crystal vibrational wavs can b dscribd by wav vctors within th first Brillouin zon in rciprocal spac What do w nd? 3D cas considration Phonons. Dnsity of stats v C ff ρ 6 Vibrations in thr-dimnsional lattic. Phonons Phonon Dnsity of stats Spcific hat (Ch. 3.3-3.9) 7 Thr-dimnsional lattic In simplst D cas with only narst-nighbor intractions w had quation of motion solution u ( q M F ( ) ( ) n C un+ un + C un un (, ) A i n u t t In gnral 3D cas th quations of motion ar: nαi M α t u n α F m, β mβ nα ωt) N unit clls, s atoms in ach 3N s quations Fortunatly, hav 3D priodicity Forcs dpnd only on diffrnc m-n Writ displacmnts as u (, t) M α u αi ( q) i( q r ωt ) n 8
substitut into quation of motion, gt ω u αi ( q) β, j m mβj iq ( rm rn ) Fn αi u M α M β βj Dαi ( q) - dynamical matri βj ω u ( q) + D ( q) u ( q) 0 αi β, j αi βj βj Dt{ ( q) ω } 0 Dα i - disprsion rlation 3s solutions disprsion branchs 3 acoustic, 3s - 3 optical dirction of u dtrmins polarization (longitudinal, transvrs or mid) Can b dgnrat bcaus of symmtry βj ( q) 0 phonon disprsion curvs in G 9 Phonons Quantum mchanics: nrgy lvls of th harmonic oscillator ar quantizd Similarly th nrgy lvls of lattic vibrations ar quantizd. Th quantum of vibration is calld a phonon (in analogy with th photon - th quantum of th lctromagntic wav) Allowd nrgy lvls of th harmonic oscillator: whr n is th quantum numbr A normal vibration mod of frquncy ω is givn by mod is occupid by n phonons of nrgy ħω; momntum p ħq Numbr of phonons is givn by : (T tmpratur) n( ω, T) u A i( q r ωt) Th total vibrational nrgy of th crystal is th sum of th nrgis of th individual phonons: (p dnots particular 0 phonon branch) kt ω
Dnsity of stats Considr D longitudinal wavs. Atomic displacmnts ar givn by: u A Boundary conditions: trnal constraints applid to th nds Priodic boundary condition: iql Thn condition on th admissibl valus of q: q π n whr n 0, ±, ±,... L ω rgularly spacd points, spacing π/l dω Numbr of mods in th L dq intrval dq in q-spac : π Numbr of mods in th frquncy rang (ω, ω + dω): L D( ω) dω dq π D(ω) - dnsity of stats dtrmind by disprsion ω ω(q) 3 4 iq dq q Dnsity of stats in 3D cas Now hav Priodic boundary condition: iql iq yl iqzl Plot ths valus in a q-spac, obtain a 3D cubic msh numbr of mods in th sphrical shll btwn th radii q and q + dq: l, m, n - intgrs Fw nots: Equation w obtaind is valid only for an isotropic solid, (vibrational frquncy dos not dpnd on th dirction of q) W hav associatd a singl mod with ach valu of q. This is not quit tru for th 3D cas: for ach q thr ar 3 diffrnt mods, on longitudinal and two transvrs. In th cas of lattic with basis th numbr of mods is 3s, whr s is th numbr of non-quivalnt atoms. Thy hav diffrnt disprsion rlations. This should b takn into account by in p 3s in th dnsity of stats. V L 3 volum of th sampl Dnsity of stats 5 6
Lattic spcific hat (hat capacity) dq Dfind as (pr mol) C If constant volum V dt Th total nrgy of th phonons at tmpratur T in a crystal: E n ω ( q) 0 n q, p qp ω kt p (th zro-point nrgy is chosn as th origin of th nrgy). - Planck distribution Thn rplac th summation ovr q by an intgral ovr frquncy: Thn th lattic hat capacity is: Cntral problm is to find th dnsity of stats 7 Dby modl assums that th acoustic mods giv th dominant contribution to th hat capacity Within th Dby approimation th vlocity of sound is takn a constant indpndnt of polarization (as in a classical lastic continuum) Th disprsion rlation: ω vq, v is th vlocity of sound. In this approimation th dnsity of stats is givn by: Vq Vq V D( ω ω ) π dω dq π v π v Nd to know th limits of intgration ovr ω. Th lowr limit is 0. How about th uppr limit? Assum N unit clls is th crystal, only on atom in pr cll th total numbr of phonon mods is 3N 3 3 Dby 6π v N ω v( 6 n) 3 D π 8 V frquncy 3 Th cutoff wav vctor which corrsponds to this frquncy is Thn th thrmal nrgy is mods of wav vctor largr than q D ar not allowd - numbr of mods with q q D hausts th numbr of dgrs of frdom Th total phonon nrgy is thn whr N is th numbr of atoms in th crystal and D θ D /T To find hat capacity, diffrntiat whr is "3" from? So, whr ħω/k B T and D ħω D /k B T θ D /T Dby tmpratur: In th limit T >>θ D, <<, C v 3Nk B - Dulong-Ptit law 9 0
Opposit limit, T <<θ D : lt th uppr limit in th intgral D Gt Einstin modl Th dnsity of stats is approimatd by a δ-function at som ω E : D(E) Nδ(ω ω E ) whr N is th total numbr of atoms simpl modl for optical phonons within th Dby modl at low tmpraturs C v T 3 Thn th thrmal nrgy is Th hat capacity is thn Th Dby tmpratur is normally dtrmind by fitting primntal data. Curv C v (T/θ) is univrsal it is th sam for diffrnt substancs Th high tmpratur limit is th sam as that for th Dby modl: C v 3Nk B - th Dulong-Ptit law At low tmpraturs C v ~ -ħω/kbt - diffrnt from Dby T 3 law Rason: at low T acoustic phonons ar much mor populatd th Dby modl is much bttr approimation that th Einstin modl Ral dnsity of vibrational stats is much mor complicatd than thos dscribd by th Dby and Einstin modls. This dnsity of stats must b takn into account in ordr to obtain quantitativ dscription of primntal data. Th dnsity of stats for Cu. Th dashd lin is th Dby approimation. Th Einstin approimation would giv a dlta pak at som frquncy. Summary In thr-dimnsional lattic with s atoms pr unit cll thr ar 3s phonon branchs: 3 acoustic, 3s - 3 optical Phonon - th quantum of lattic vibration. Enrgy ħω; momntum ħq Dnsity of stats is important charactristic of lattic vibrations; It is rlatd to th disprsion ω ω(q). Vq Simplst cas of isotropic solid, for on branch: D( ω) π dω dq Hat capacity is rlatd to th dnsity of stats. Dby modl good whn acoustic phonon contribution dominats. At low tmpraturs givs C v T 3 Einstin modl - simpl modl for optical phonons (ω(q) is constant) 3 At high T both modls lad to th Dulong-Ptit law: C v 3Nk B Ral dnsity of vibrational stats is mor complicatd 4
Thrmal Conductivity Tmpratur gradint in a matrial hat flow from th hottr to th coolr nd. Hat currnt dnsity j (amount of hat flowing across unit ara pr unit tim) is proportional to th tmpratur gradint (dt/): dt j u K K - thrmal conductivity In mtals th hat is carrid both by lctrons and phonons; lctron contribution is much largr In insulators, thr ar no mobil lctrons hat is transmittd ntirly by phonons Hat transfr by phonons phonon gas: in vry rgion of spac thr ar phonons travling randomly in all dirctions, much lik th molculs in an ordinary gas phonon concntration is largr at th hottr nd thy mov to th coolr nd th advantag of using this gas modl: can apply familiar concpts of th 5 kintic thory of gass Elmntary kintic considrations: if c is th hat capacity of th singl particl, thn moving from rgion with T+ T to a T, particl will giv up nrgy c T T btwn th nds of th fr path lngth l : whr τ is th avrag tim btwn collisions Th nt nrgy flu (n concntration) : j u n v dt for phonons, v is constant. nc C; lvτ j u 3 Cvl dt cτ K Cvl 3 3 n dt dt T l v τ v dt cτ - phonon thrmal conductivity 6 Dpndnc of th thrmal conductivity on tmpratur C v dpndnc on tmpratur has alrady bn discussd Sound vlocity v ssntially insnsitiv to tmpratur Th man fr path l dpnds strongly on tmpratur Thr important mchanisms ar to b considrd: (a) collision of a phonon with othr phonons (b) collision of a phonon with imprfctions in th crystal (c) collision of a phonon with th trnal boundaris of th crystal Th phonon-phonon scattring is du to th anharmonic intraction. If intratomic forcs ar purly harmonic no phonon-phonon intraction. At high tmpratur atomic displacmnts ar larg strongr anharmonism phonon-phonon collisions bcom mor important At high T th man fr path l /T : numbr of phonons n T at high T 7 collision frquncy n l /n Suppos that two phonons of vctors q and q collid, and produc a third phonon of vctor q 3. Momntum consrvation: q 3 q + q q 3 may li insid th Brillouin zon, or not. If it's insid momntum of th systm bfor and aftr collision is th sam. This is a normal procss. It has no ffct at all on thrmal rsistivity, sinc it has no ffct on th flow of th phonon systm as a whol. If q 3 lis outsid th BZ, w rduc it to quivalnt q 4 insid th first BZ: q 3 q 4 + G Momntum consrvation: q + q q 4 + G Th diffrnc in momntum is transfrrd to th cntr of mass of th lattic. This typ of procss is is known as th umklapp procss highly fficint in changing th momntum of th phonon rsponsibl for phonon scattring at high tmpraturs 8
Th scond mchanism - phonon scattring rsults from dfcts. Impuritis and dfcts scattr phonons bcaus thy partially dstroy th priodicity of th crystal. At vry low T, both phonon-phonon and phonon-dfct collisions bcom inffctiv: thr ar only a fw phonons prsnt, th phonons ar long-wavlngth ons not ffctivly scattrd by dfcts, which ar much smallr in siz In th low-tmpratur rgion, th primary scattring mchanism is th trnal boundary of th spcimn - so-calld siz or gomtrical ffcts. Bcoms ffctiv bcaus th phonon wavlngths ar vry long - comparabl to th siz of th sampl L. Th man fr path hr is l ~ L indpndnt of tmpratur. low T: K ~ T 3 du to C v high T: K ~ /T du to l Thrmal conductivity of NaF 9 (highly purifid) Anharmonism So far, lattic vibrations wr considrd in harmonic approimation. Som consquncs: Phonons do not intract; no dcay No thrmal pansion Elastic constants ar indpndnt of prssur and tmpratur Hat capacity is constant at high T (T>>θ D ). Anharmonic trms in potntial nrgy: U() c g 3 f 4 - displacmnt from quilibrium sparation at T 0 30 Calculat avrag displacmnt using Boltzmann distribution function: U ( ) kbt U ( ) kbt If anharmonic trms ar small, can us Taylor pansion for ponnt c k Thrmal pansion 4 5 T g f B 3 g + + kbt kbt π 5 4 c c kbt π c thrmal pansion Origin of thrmal pansion asymmtric potntial ( k T ) ( k T ) B B 3 3g k 4c B 3 T Tchniqus for probing lattic vibrations Inlastic X-ray scattring Nutron scattring Infrard spctroscopy Brillouin and Raman scattring k θ k 0 q masuring ω - ω 0 Inlastic X-ray scattring k k 0 ± q ω ω0 ± Ω( q) ω0 q k0 sinθ n sinθ c assumd Ω(q) <<ω 0 - tru for -rays: ħω < 00 mv; ħω 0 ~0 4 V n in of rfraction and θ sin on can dtrmin disprsion Ω(q) main disadvantag difficult to masur ω - ω 0 accuratly This difficulty can b ovrcom by us of nutron scattring Enrgy of "thrmal" nutrons is comparabl with ħω (80 mv for 3 λ Å)
Brillouin and Raman spctroscopy Inlastic light scattring mdiatd by th lctronic polarizability of th mdium a matrial or a molcul scattrs irradiant light from a sourc Most of th scattrd light is at th sam wavlngth as th lasr sourc (lastic, or Railigh scattring) but a small amount of light is scattrd at diffrnt wavlngths (inlastic, or Raman scattring) α β ћω i 0 ћω ћω s Stoks Railigh β α ћω i 0 ћω ћω s Anti- Stoks I Stoks Raman Scattring ω i - Ω(q) ω i Elastic (Railigh) Scattring ω Anti-Stoks Raman Scattring ω i + Ω(q) Analysis of scattrd light nrgy, polarization, rlativ intnsity provids information on lattic vibrations or othr citations 33 Raman scattring in crystallin solids Not vry crystal lattic vibration can b probd by Raman scattring. Thr ar crtain Slction ruls:. Enrgy consrvation: ω i ω ± Ω; s. Momntum consrvation: 4πn k i k s ± q 0 q k 0 q λ λ i ~ 5000 Å, a 0 ~ 4-5 Å λ phonon >> a 0 only small wavvctor (cloz to BZ cntr) phonons ar sn in th st ordr (singl phonon) Raman spctra of bulk crystals 3. Slction ruls dtrmind by crystal symmtry i k s q 0 k i k s q k k i k s ki 34 q Phonon thrmal conductivity Summary K Cvl 3 Mchanisms of phonon scattring affcting thrmal rsistivity: umklapp procsss of phonon-phonons collision important at high T collision of a phonon with dfcts and impuritis in th crystal collision of a phonon with th trnal boundaris of th crystal important at low T Anharmonism of potntial nrgy is rsponsibl for such ffcts as: phonon-phonon intraction thrmal pansion Tchniqus for probing lattic vibrations: Inlastic X-ray scattring, Nutron scattring, Infrard spctroscopy, Brillouin and Raman scattring Elctron nrgy loss spctroscopy 35 SOME USEFUL SLIDES FROM Physics-I 36
Avogadro s numbr and Idal Gass Th Kintic Thory of Gass; Mol Idal Gass Avogadro's Numbr Idal Gas at Constant Tmpratur R is a gas constant 37 M is molar mass; m is molcular mass 38 Prssur, Tmpratur, and Spd of molculs Th Distribution of Molcular Spds mn A is th molar mass M Root-man-squar spd 39 40
Viw at th molcular thory of an idal Gas Hat Capacity Spcific Hat 4 4 Hat Transfr Mchanisms Convction Radiation Conduction QkA( T/L)t, Whr k is th thrmal conductivity 43