Chapter 13 Phonons. Contents 13-1

Transcription

1 - Chaptr Phonons Contnts Chaptr Phonons... - Contnts Crystal Vibration On Atom in a Primitiv cll Two Atoms in a Unit cll Spcific Hat Intrnal Enrgy by Modl... - Eampl. Atomic Siz and Spcific Hat Lattic Thrmal Conductivity Klmns-Callaway Modl Umlapp Procsss Callaway Modl Phonon Rlaation Tims Eampl. Lattic Thrmal Conductivity... - Problms Rfrncs Th quantizd nrgy of a lattic vibration is calld a phonon, which is in analogy with a photon of th lctromagntic wav.

2 -. Crystal Vibration.. On Atom in a Primitiv cll Considr th on-dimnsional lastic vibration of a crystal with on atom in a unit cll in Figur.. W want to find th frquncy of an lastic wav in trms of th wavvctor. Thr ar th prcis quation if ach atom wr connctd to its nighbors by prfct springs with spring constant C. Na a n- n n+ n+ n+ n+4 Figur. On-dimnsional monatomic lattic chain modl. a is th distanc btwn atoms (lattic constant). Th atoms as displacd during passag of a longitudinal wav. W assum that th forc at is proportional to th displacmnt as f n C C n n n n (.) Using th Nwton s scond law of motion with an atom of mass m, f n d m dt n (.) Combining ths two, w hav

3 - d dt n m Cn n n (.) Th gnral solution is found to b n A i na t (.4) whr is th angular frquncy of f, whr f is th frquncy. W considr th standing wav for a crystal which has a finit numbr N of atoms. N is larg and w simply connct th two rmot nds of th chain maing a circular chain. Th priodic boundary conditions rquirs that ina (.5) Sinc ina cos na isin na, w hav cosna (.6) which lads to n, whr n =,,..N (.7) a N Not that is changd by a. Considring th first illouin zon, using priodicity w hav a a (.8) Thus, th displacmnt of phonons can always b dscribd by a wavvctor within th rillouin zon. Taing th scond drivativ of Equation (.4) givs

4 -4 d dt n na A i t (.9) Insrting Equations (.4) and (.9) into (.) givs CA i A CA ia ia ina t nat inat ina t inat A A (.0) which lads to C m ia ia (.) Solving for, w hav 4C a sin m (.) Th solution dscribs a wav propagating along th chain with phas vlocity v s, and group vlocity v g. Th frquncy is plottd against th wavvctor in Figur.. Such a curv is nown as a disprsion rlation. Th group vlocity is v g (.) Whn is small compard with /, is linar in. Th phas vlocity is qual to th spd of sound vs as

5 -5 v s (.4) Figur. Phonon disprsion curv of a on-dimnsional monatomic lattic chain for rillouin zon. Th by approimation us a linar rlationship btwn th frquncy and th wavvctor. It can b sn in Figur. that th disprsion curv in th rillouin zon diffrs gratly from disprsion curv for a continuum, which for comparison has bn assumd to hav th sam bhavior as th discrt chain in th long-wavlngth limit (small )... Two Atoms in a Unit cll Considr th on-dimnsional lastic vibration of a crystal with two diffrnt mass atoms in a unit cll, which ar connctd with spring constant C (Figur.).

6 -6 Figur. iatomic linar chain of diffrnt atoms, connctd with spring constant C. For larg M, F n X C X C n n n n (.5) For small m, f n X CX C n n n n (.6) Using th Nwton s scond law of motion, w hav M d X n dt = C( n + n+ X n ) (.7) d dt n m CX n X n n (.8) Th dtrminant should b zro and thn

7 -7 4 Mm C ia ia M m 4C 0 (.9) Solving for and, C mm m M Cm M 8mMC cos a (.0) and C mm m M Cm M 8mMC cos a (.) Whn is small, cos a which lads to 0 and m M C mm (.) Whn is /a at th boundary of th rillouin zon, cos a which lads to C M (.) C m (.4) In Figur.4, th lowr branch has th sam structur as th singl branch w found in th monatomic lattic: vanishs linarly in for small, and th curv bcoms flat at th dgs of th rillouin zon. This branch is nown as th acoustical branch bcaus its disprsion rlation is of th form = v charactristic of sound wavs at small. Th scond branch starts as

8 -8 Cm M mm at = 0 and dcrass with incrasing. This branch is nown as th optical branch bcaus th long wavlngth optical mods in ionic crystals can intract with lctromagntic radiation, and ar rsponsibl for much of th charactristic optical bhavior of such crystals. Figur.4 Phonon disprsion curv of a on-dimnsional diatomic lattic chain for rillouin zon. If thr ar p-atoms in a -dimnsional lattic, thr ar p branchs to th disprsion rlation: thr acoustical branchs and ( p-) optical branchs. For ampl on atom in a lattic has acoustical and zro optical branchs. If thr ar two atoms in a lattic, thr will b 6 branchs: acoustical and optical branchs as shown in Figur.5. W hav longitudinal acoustical LA and transvrs acoustical TA mods, and longitudinal optical LO and transvrs optical TO mods. Th group vlocity of th uppr optical mods is small so that thy ar not vry ffctiv in transporting nrgy, but thy may affct hat flow by intracting with th acoustical mods which ar mainly rsponsibl for th thrmal conductivity.

9 -9 Figur.5 Phonon disprsion rlations of a thr-dimnsional diatomic lattic for rillouin zon. Although a crystal may b said to hav thr acoustical branchs, this dos not ncssary man that ths must vrywhr hav diffrnt frquncis. In a cubic structur of PbT, th two transvrs branchs ar dgnrat, which is shown in Figur.6 (a). It should b notd that optical is th trm usd to dscrib all branchs that hav a non-zro frquncy at = 0.

10 -0 (a) (b) Figur.6 (a) Calculatd phonon disprsion of PbT by Zhang t al. (009)[], (b) Th rillouin zon of PbT. Points of high symmtry ar dnotd by, L, X tc.. Spcific Hat Th hat capacity is th ratio of th amount of hat nrgy pr unit tmpratur ris. Th spcific hat is th hat capacity pr unit mass, which is a charactristic of th matrial... Intrnal Enrgy In quantum statistics, th nrgy of a linar lattic vibration is givn by E f o, ph (.5) whr ½ is th zro point nrgy which has no contribution to th hat capacity. is th nrgy of a phonon. o ph f, is th thrmal quilibrium occupancy (digit numbr) of phonons, which is calld th os-einstin distribution function as

11 - f o, ph T (.6) Th intrnal nrgy U, which is th total vibrational nrgy, can b obtaind by summing ovr all normal mods, approimating th lattic as a continuum but with a maimum frquncy. U p ma 0 E g ph d (.7) whr p is pth branch (th ind of phonon branchs) and gph() is th phonon dnsity of stats... by Modl In th by modl, th lattic vibrats as if it wr an lastic continuum, but th vibration frquncis cannot cd a crtain maimum valu, chosn to ma th total numbr of mods qual to th total numbr of classical dgrs of frdom. Th by modl rplacs all branchs of th vibrational spctrum with thr branchs, ach with th sam linar disprsion rlations. Th vlocity of sound is tan as constant for ach branchs, as it would b for a classical lastic continuum. In th by modl th optical mods ar uncrmoniously lumpd into th top of th distribution of acoustic mods, as if thy wr mrly lastic wavs of vry short wavlngth. spit its obvious crudity, th by approimation has th grat advantag of suprm simplicity. If any on paramtr is rquird to masur th nrgy scal of th vibrations of a solid, th by tmpratur is th most appropriat. If any on function is rquird to prsnt th distribution of th lattic frquncy, th phonon dnsity of stats is th simplst. Th disprsion rlation is approimatd, as in Equation (.4), to b v s (.8)

12 - For ach branch, th total numbr of mods (dgr of frdom) N is obtaind by dividing a sphr of -spac by a volum of th smallst wavvctor. N 4 L V 6 (.9) whr V L and v s, w hav N V 6 vs (.0) If thr ar N primitiv clls in a spcimn, th total numbr of acoustic phonon mods is N. by tratd ach mod as a quantizd harmonic oscillator. Th by cutoff frquncy should b N v s 6 V (.) which was also shown in Figur.. Th phonon dnsity of stats (OS) pr ach branch is obtaind by taing drivativ of Equation (.0) with rspct to as g ph (ω) = dn(ω) dω = Vω π v s (.) Substituting Equation (.) into Equation (.), th phonon dnsity of stats is prssd by

13 - g ph N (.) which is a functional distribution of spctrum. Substituting Equations (.5) and (.) into (.7), th intrnal nrgy U is prssd as U p 0 N d T (.4) whr p is th ind of phonon branchs (mods). Th thrmodynamic dfinition of spcific hat is c V du (.5) Vd dt V whr d is th dnsity and, assuming p to b for bul matrials (by modl), th spcific hat is thn c V T N d Vd T T 0 (.6) Th phonon dnsity of stats pr branch pr mod is prssd as g ph, br (.7) which was compard with primnt by Rauh (98) []. Th rgion of lowr frquncis in th primntal curv of Figur.7 shows th mitur of acoustical and optical branchs significantly contribut to th phonon dnsity of stats rsulting in undrstimat of th by modl, which is shown in Figur.6 as pctd. Th rgion of highr frquncis in th primntal curv shows ovrstimat of th by modl. Intrstingly th intgratd dnsity

14 -4 of stats of th simpl by modl approimatly matchs that of th primntal dnsity of stats. Figur.7 Phonon dnsity of stats vrsus frquncy for it [].Th by modl is compard with primnt by Rauh (98)[]. by Tmpratur Th by tmpratur is dfind as (.8) Equivalntly, T T (.9) Spcific Hat

15 -5 From Equation (.6), th spcific hat c V is finally obtaind by c V T 4 N T 9 Vd d (.40) 0 whr T (.4) Equation (.40) is wll nown th by formula which is widly usd in th litratur. Th vlocity of Sound and Atomic Siz Th vlocity of sound is stimatd using Equations (.) and (.8) as v s N 6 V (.4) N/V can b approimatly obtaind assuming a cubic structur. N is th numbr of primitiv clls and V is th crystal volum that is th product of N and volum of th primitiv cll as [4] N V N N (.4) whr is th atomic siz. Not that th by modl assums on atom in a primitiv unit cll, in which is th lattic constant a. Howvr, whn a crystal has mor than on atom ( n a ) in a primitiv cll, δ is not th lattic constant. In that cas, th volum of th atomic siz prssd by is

16 -6 a n a (.44) Hnc, th vlocity of sound in Equation (.4) can b prssd by 6 v s (.45) whr is th volum of th primitiv cll. Whn na =, w hav that a = as by modl assumd. It is sn that th vibration mods ar drivd basd on atoms rathr than lattics. W can also approimatly calculat th siz of th primitiv cll (atomic siz) for a binary compound using th atomic masss and th mass dnsitis by y y A, A M A NAdA and M NAd (.46) M A M y M y (.47) A d A M N A A (.48) whr y is th fraction of componnt A, NA is th Avogadro s numbr, M A, ar th atomic masss of componnt A or, M ar th man atomic masss of componnt A and, and d A, ar th A mass dnsitis of componnt A or.

17 -7 Eampl. Atomic Siz and Spcific Hat PbT (lad tllurid) is a widsprad thrmolctric matrial with th by tmpratur of 6 K. (a) Estimat th atomic siz of th PbT, (b) dtrmin th vlocity of sound (phas vlocity) for th PbT, and (c) dtrmin th spcific hat at room tmpratur. Solution: From Equation (.46) with y = 0.5 for componnt Pb and th priodic tabls in Appndics A- and A-5, th atomic siz is Pb M Pb N A d Pb 07.g g cm / 0 m T M T N AdT 7.6g g cm / 0 m y y Pb T = cm =.79 Å From Equation (.45), th vlocity of sound is v s.80 J K K.790 m.450 m/ s Js From Equation (.47) and (.48), th man atomic mass and dnsity of PbT can b calculatd y 07.g g g M PbT M Pb y MT 4

18 -8 d PbT M PbT 67.4g 8. N 654 A g cm cm From Equation (.40), c V 9 N Vd PbT T T 0 4 d Th by intgrand is obtaind by intrpolation from Tabl C- with T T 4 d 0.0 Th spcific hat at room tmpratur is c V 9.80 J K m g m 00K 6K J g K In mol basis, c V 07.g 7.6g 0 g g mol J 47.5 g K J 49.7 mol K Commnts: Ths valus of th vlocity of sound and th spcific hat ar comparabl to th primntal valus for PbT.

19 -9 Th calculatd tmpratur-dpndnt spcific hat for MgSi is plottd along with th masurmnts by Grstin (967)[5] in Figur.8. Figur.8 Spcific hat of MgSi []. Th mars ar th masurmnts by Grstin (967) [5] and th solid lin is th calculation by Equation (.40) using th by tmpratur of 47 K. ulong-ptit Law Equation (.4) with thr branchs in a thr-dimnsional crystal can b prssd as U NT d 0 (.49) whr T Whn th tmpratur is much highr than th by tmpratur ( T ), th bract may b pandd as

20 -0 0 d 0 6 0! d ! d 0 d (.50) Thn, th intrnal nrgy U bcoms U N T (.5) Sinc c V du Vd dt V from Equation (.5), th spcific hat at high tmpraturs is N c Vd V (.5) Hr w hav th famous classical ulong-ptit law, showing that th atomic hat of all solids tnds to th sam valu at high tmpraturs. Th spcific hat du to th lattic vibrations is just pr atom. by T Law Whn th tmpratur is much lowr than th by tmpratur ( T ), on can driv c V T (.5) This is nown th by T law which is primntally obsrvd in Figur.8 at vry low tmpraturs.

21 -. Lattic Thrmal Conductivity.. Klmns-Callaway Modl Th hat currnt dnsity (hat flu) is givn by q l T l T l (.54) A tmpratur gradint causs thrmal non-quilibrium. Th hat currnt dnsity is th product of th nrgy of lattic vibrations and th group vlocity as q l = E(ω)v p (.55) Whr p is th ind of phonon branchs, th wavvctor, and v th on-dimnsional nrgy transport vlocity of th phonons (group vlocity). Using Equation (.5), th hat currnt dnsity q l is q l = ħωf ph (ω)v p (.56) This mans that th hat currnt dnsity q l is th product of thr trms: th nrgy of phonon, th nonquilibrium (disturbd) phonon distribution numbr f, and th group vlocity v. Th phonon distribution numbr dparts from quilibrium in th prsnc of a tmpratur gradint, and in th absnc of any intractions it would vary with tim at any point in th crystal. In ordr to obtain ph f, th oltzmann transport quation is usd as givn in Equation (.). ph df dt f t f f r r f t coll (.57)

22 - Undr th condition that trnal forcs and lctric fild do not affct th vibrational motion of th atoms in a crystal, has a valu of zro. Using th rlaation tim approimation in Sction., w hav f ph v c f 0, ph T T (.58) whr c is th combind rlaation tim. Insrting this into Equation (.56) givs q l = ħωτ c v f 0,ph T T p (.59) From Equation (.54), th lattic thrmal conductivity l is prssd as l = ħωτ c v f 0,ph T For cubic and isotropic systms, th avrag vlocity is conductivity is p v v (.60). Th lattic thrmal l = ħωτ cv f 0,ph T p (.6) This can b prssd in th form of intgral using th phonon dnsity of stats as l = p 0 ω ħωτ cv f 0,ph T g ph(ω)dω (.6) which, using Equation (.) pr unit volum, lads to

23 - l = p 0 ω τ c v ( ħω T ) ħω T ( ħω T ) ( ω π v ) dω (.6) It is assumd that thr ar thr acoustic branchs, which ar on longitudinal and two transvrs branchs (p = ). This assumption is latr vrifid numrically by Esfarjani t al. (0) [6]. Using Equation (.), th lattic thrmal conductivity finally givs l T v 0 T c 4 d (.64) whr th phonon vlocity v is vry clos to th vlocity of sound vs and T. This is a wll-nown Klmns-Callaway modl of th lattic thrmal conductivity. In th cas that both and v ar indpndnt of frquncy, comparing Equation (.6) for c V, th lattic thrmal conductivity l is prssd as l cv C V (.65) whr C V is th hat capacity (spcific hat pr volum), v, C V c V d. Using th man fr path (.66) v l C V Although th abov modl is fairly crud, th prssion for th lattic thrmal conductivity is actually a surprisingly good approimation. Vry oftn, th abov quation is usd to stimat th man fr path on th basis of primntal rsults for th othr paramtrs in th quation.

24 -4 Figur.9 Various mchanisms rducing th lattic thrmal conductivity. If th lattic vibrations ar ntirly th normal mods (N-procsss) as shown in th spcific hat, a hat flow will transport without dcaying or scattring, which implis that th lattic thrmal conductivity b infinit as harmonic oscillations. In fact, anharmonicity prsists along with th N- procsss. Th major mchanism of anharmonicity is calld th Umlapp procsss (U-procsss) which was discovrd by Pirls (99). Th U-procsss do not oby th consrvation of momntums whil th N-procsss do oby, causing th scattring of phonons. Th U-procsss incras with incrasing tmpratur. From Equation (.66), sinc th vlocity of sound v is indpndnt of tmpratur, th lattic thrmal conductivity l dpnds on th man fr path and th hat capacity C V. At low tmpraturs, th U-procsss is rducd and th man fr path may rach svral millimtrs, bing comparabl with th dimnsions of spcimns. Th man fr path thus bcoms constant at th grain boundaris. Hnc l dpnds on CV that varis as T which is shown in Figur.9.

25 -5 At high tmpraturs, C is almost constant and T V. Thn th Umlapp procsss prdominat l so that l T. Phonon Rlaation Tim Th scattring rat is th rciprocal of th rlaation tim. Th scattring may b causd by th coistnc of diffrnt procsss and a rlaation tim can b dfind for ach procss. Th combind rlaation tim c can b calculatd from individual rlaation tim i according to Matthissn s rul (86), (.67) c i i Th Matthissn s rul assums that th scattring mchanisms ar indpndnt of ach othr (Ashcroft and Mrmin, 976)[7]... Umlapp Procsss W considr thr-phonon procsss in a lattic. For ampl, whn th sum of two wavvctors is within th first rillouin zon at low tmpraturs, it is th normal procsss as shown in Figur.0 (a). At high tmpraturs it may b th cas that th sum of th two wavvctors falls outsid of th zon as shown in Figur.0 (b). Howvr, thr is a rul that no wavvctor must b so larg as to li outsid th first rillouin zon. In othr words, th phonon wavlngth cannot b smallr than th lattic constant. In such a cas, th sum of th two wavvctors is forcd to li insid th first rillouin zon rathr than falling outsid th zon, using th rciprocal lattic vctor G and losing th consrvation of momntum. This is th Umlapp procss or anharmonicity. Th wavvctors can b writtn as

26 -6 G (.68) Whn th procss is normal, G = 0. On th othr hand, whn th procss is Umlapp procss, G 0. (a) (b) (c) Figur.0 (a) Normal procss, (b) Unaccptabl procss, and (c) Umlapp procss G, whr G is th rciprocal lattic vctor.

27 -7.. Callaway Modl Th normal procsss ar harmonic oscillations and cannot b prssd as th scattring or th rlaation tim. On th othr hand, normal procss cannot b ignord bcaus thy indirctly caus th scattring. Callaway (959)[8] suggstd an prssion basd on th wor of by (9)[9] and Klmns (955)[0], which is widly usd. Th lattic thrmal conductivity is givn by l (.69) T c s d T v / 0 4 (.70) T U c U T U c s d d T v / 0 4 / 0 4 (.7) whr is th ratio of th Umlapp procsss to th N-procsss, c and U ar th combind and Umlapp rlaation tims, rspctivly. Th scond Callaway trm is larg for a pur crystal but dcras rapidly with addition of dfcts so that it is dominant ovr a wid tmpratur rang for th pur crystal but bcoms almost ngligibl for an impur crystal []. Most dopd crystal compounds that hav intrinsic dfcts ar impur in fact, so that l...4 Phonon Rlaation Tims

28 -8 Etnsiv studis hav bn mad into phonon rlaation tims, including normal-mod procsss (N-procsss), Umlapp procsss, point dfct scattring, lctron-phonon scattring, and boundary scattring. W us hr th prssions suggstd by Stigmir and Abls (964) []. Vining (99) [] and Minnich (009) [4] also usd thos scattring prssions. Scattring by N-Procsss and Umlapp Procsss Th Umlapp procsss ar thr-phonon scattring involving anharmonicity. Th N-procsss do not dirctly caus th scattring but hlp th Umlapp procsss contribut to th scattring. Thrfor, thy can b prssd in trms of th rlaation tim in vn whn is nglctd in Equation (.70). Th Umlapp scattring rat is th invrs of th rlaation tim, which was first studid by Libfrid and Schlömann (954).[5] Latr, a modifid prssion including th ffct of th N-procsss was givn by [,, 6] T U N A 4 M (.7) A whr N A is Avogadro s numbr, th Grünisn anharmonicity paramtr, mass of compounds A and, th man atomic siz (s Equation (.46)), and Th N-procss scattring rat is givn by M A th atomic T. N U (.7) whr is oftn usd for a good approimation. Scattring by Point fcts

29 -9 Th point dfcts may includ vacancis, isotops, dislocations, substitut atoms, tc. A dfct with dimnsions much smallr than th phonon wavlngth can b considrd as a point dfct []. Th scattring is thn causd by th diffrnc in mass and th diffrnc in bonding btwn th atoms.[] Th scattring rat for point dfcts is givn by Klmns (960)[7], which is P T V0 4v s 4 4 (.74) whr V 0 is th atomic volum ( compound is givn by[] ) and th mass-fluctuation-scattring paramtr for a binary M y y M A s (.75) M M A M, M M y M ( y A A ), and A (.76) whr y is th fraction of componnt A, th atomic siz (s Equation (.46)) and s th strain paramtr []. Elctron-Phonon Scattring Th scattring of phonons by lctrons will b activ whn th band dgnracy tmpratur is comparabl with th tmpratur of th lattic [8]. Ziman (956) drivd an prssion for lctron-phonon scattring, which is

30 -0 EP, i whr ln E rcef 6 Erc, i amd, ivs 4 4 de rc E E rc F 6 Erc, i is th acoustic dformation potntial, E m v T a and T. rc, i d, i s (.77) oundary Scattring Th phonon-boundary scattring rat is assumd indpndnt of tmpratur and frquncy. Th boundary scattring rat with an assumption of purly diffus scattring is givn by[4] v s L (.78) whr L is th ffctiv lngth of th sampl. Th total scattring tim can b approimatd by adding th scattring rats in accordanc with Matthissn s rul in Equation (.67), c N U P EP (.79) In ordr to visualiz th individual ffct of th phonon scattring mchanisms, th 4 wightd phonon rlaation tims of is plottd in Figur. for PbT. It is sn that th Umlapp procsss dominats th phonon rlaation tim at low frquncis and th point dfcts bcom dominant at high frquncis. Th lctron-phonon scattring has a minimal ffct on th rlaation tim.

31 - Figur. Wightd phonon rlaation tim vrsus frquncy for PbT []. Sinc w hav dvlopd th prssions for th diffrnt rlaation tims, w can calculat th lattic thrmal conductivity vrsus tmpratur using of Equation (.64) along with th lctronic thrmal conductivity as shown in Figur. for PbT. It is sn that th lattic and lctronic contributions to th combind thrmal conductivity ar almost qual. Th total (combind) thrmal conductivity shows a good agrmnt with th primntal data by Pi at al. (0)[9].

32 - Figur. Th lctronic, lattic, and combind thrmal conductivitis vrsus tmpratur for PbT. Th lctron doping concntration of cm - was usd []. Eampl. Lattic Thrmal Conductivity PbT (lad tllurid) is a widsprad thrmolctric matrial with by tmpratur 6 K. Assuming that th Umlapp procsss ar th dominant scattring mchanism for PbT, (a) dtrmin th by cutoff frquncy f, not angular frquncy ( f ), (b) dtrmin th lattic thrmal conductivity using th Grünisn paramtr =.5 at room tmpratur, and (c) stimat th phonon man fr path assuming th ulong-ptit law for th hat capacity. Solution: From Equation (.46) with y = 0.5 and th priodic tabls in Appndics A- and A-5, th atomic siz is stimatd as shown Eampl., Pb M Pb N A d Pb 07.g g cm / 0 m T M T N AdT 7.6g g cm / 0 m y y Pb T = cm =.79 Å From Equation (.45), th vlocity of sound is v s.80 J K K.790 m.450 m/ s Js

33 - From Equation (.8), sinc f, th by cutoff frquncy is 0.84 s f =.84 THz From Equation (.76), th man atomic mass is g y M y M M T Pb PbT ) ( From Equation (.7), using =, th Umlapp rlaation tim that is hr th combind on as T M N PbT PbT A U From Equation (.64), th lattic thrmal conductivity is T U s l d T v 0 4 Th quation can b rwrittn as T PbT PbT A s l d T M N T v

34 -4 Th by intgral with T 0. 45can b obtaind by intrpolation from Tabl F-. T 0 d Th lattic thrmal conductivity is finally obtaind as l 4. 0 W mk Sinc 6K T 00K, th hat capacity of th ulong-ptit law from Equation (.5) can b usd as C N.890 V J m K 6 V From Equation (.66), th phonon man fr path is l m 6. 4nm v C s V Commnts: Th valu of 4.0 W/mK for th lattic thrmal conductivity sms high compard to th primntal valu of about. W/mK in Figur.. This valu may b improvd if w us th Matthissn s rul with all th phonon scattring mchanisms. Problms.. riv Equation (.).

35 -5.. riv Equations (.) and (.) with a brif planation... riv Equation (.40)..4. Estimat th atomic siz for a facd cntrd cubic structur of MgSi using both mthods: a) lattic constant = 6.8 Å, and (b) atomic masss and mass dnsitis for th binary compound (y = 0.)..5. Plot th thortical phonon dnsity of stats vrsus frquncy as shown Figur.7 for it with th by tmpratur of 65 K and th fraction of componnts and th mass dnsitis..6. MgSi (magnsium silicid) is a widsprad thrmolctric matrial with th by tmpratur of 47 K. (a) Estimat th atomic siz, (b) dtrmin th vlocity of sound (phas vlocity), and (c) dtrmin th spcific hat for MgSi..7. Provid th thortical curv for th spcific hat of MgSi with th by tmpratur of 47 K as shown in Figur riv th quation for th classical ulong-ptit law..9. riv Equation (.64) and (.66)..0. MgSi (magnsium silicid) is a widsprad thrmolctric matrial with th by tmpratur of 47 K. Assuming that th Umlapp procsss ar a dominant scattring mchanism, (a) dtrmin th by cutoff frquncy f, not angular frquncy ( f ), (b) dtrmin th lattic thrmal conductivity using Grünisn paramtr =.67 at room

36 -6 tmpratur, and (c) stimat th phonon man fr path assuming th ulong-ptit law for th hat capacity. Rfrncs. Zhang, Y., t al., Thrmodynamic proprtis of PbT, PbS, and PbS: First-principls study. Physical Rviw, ().. Rauh, H., t al., Gnralisd phonon dnsity of stats of th layr compounds is it SbT and i(t05s05). J. Phys. C: Solid Stat Phys., 98. 4: p L, H., A Thortical Modl of Thrmolctric Transport Proprtis for Elctrons and Phonons. Journal of Elctronic Matrials, (): p Ziman, J.M., Elctrons and phonons. 960, London: Oford Univrsity Prss. 5. Grstin,.C., Thrmal Study of Groups II IV Smiconductors. Lattic Hat Capacitis and Fr Enrgis of Formation. Hat Capacity of MgSi from 5 00 K. Th Journal of Chmical Physics, (6): p Esfarjani, K., G. Chn, and H.T. Stos, Hat transport in silicon from first-principls calculations. Physical Rviw, 0. 84(8). 7. Ashcroft, N.W. and N.. Mrmin, Solid stat physics. 976, Nw Yor: Holt, Rinhart and Winston. 8. Callaway, J., Modl for Lattic Thrmal Conductivity at Low Tmpraturs. Physical Rviw, 959. (4): p by, P., Th thory of spcific hats. Ann. Phys. Lpz. (4), 9. 9: p Klmns, P.G., Th scattring of low-frquncy lattic wavsby static imprfctions. Proc. Phys. Soc., (): p rman, R., Thrmal conduction in solids. 976, Oford: Clarndon Prss.. Stigmir, E. and. Abls, Scattring of Phonons by Elctrons in Grmanium-Silicon Alloys. Physical Rviw, (4A): p. A49-A55.. Vining, C.., A modl for th high-tmpratur transport proprtis of havily dopd n- typ silicon-grmanium alloys. Journal of Applid Physics, (): p.. 4. Minnich, A., t al., Modling study of thrmolctric SiG nanocomposits. Physical Rviw, (5). 5. Libfrid, G. and E. Schlomann, Warmlitung in ltrisch isolirndn Kristalln. Nachrichtn dr Aadmi dr Wissnschaftn in Gottingn, lia, Mathmatisch- Physialisch Klass, 954: p Klmns, P.G., Thrmal conductivity and lattic vibrational mods. Solid StatPhysics Chaptr , Nw Yor: Acadmic Prss Inc. 7. Klmns, P., Thrmal Rsistanc du to Point fcts at High Tmpraturs. Physical Rviw, (): p

37 -7 8. Ziman, J.M., Th ffct of fr lctrons in lattic conduction. Th Philosophical Magazin, 956. (): p Pi, Y., t al., Low ffctiv mass lading to high thrmolctric prformanc. Enrgy & Environmntal Scinc, 0. 5(7): p. 796.