Chapter 16 Theoretical Model of Thermoelectric Transport Properties

Transcription

1 6- Chapter 6 Theoretcal Model of Thermoelectrc Transport Propertes Contents Chapter 6 Theoretcal Model of Thermoelectrc Transport Propertes 6- Contents 6-6. Introducton 6-6. THORTICAL QUATONS Carrer Transport Propertes Scatterng Mechansms for lectron Relaxaton Tmes Lattce Thermal Conductvty Phonon Relaxaton Tmes Phonon Densty of States and Specfc Heat Dmensonless Fgure of Mert RSULTS AND DISCUSSION lectron or Hole Scatterng Mechansms Transport Propertes Summary 6-55 RFRNCS 6-56 Problems 6-65

2 6- A generc theoretcal model for fve bulk thermoelectrc materals (PbTe, Te3, SnSe, S0.7Ge0.3, and MgS) has been developed here based on the semclasscal model ncorporatng nonparabolcty, two-band Kane model, Hall factor, and the Debye-Callaway model for electrons and phonons. It s used to calculate thermoelectrc transport propertes: the Seebeck coeffcent, electrcal conductvty, electronc and lattce thermal conductvtes at a temperature range from room temperature up to 00 K. The present model dffers from others: Frstly, thorough verfcaton of modfed electron scatterng mechansms by comparson wth reported expermental data. Secondly, extensve verfcaton of the model wth concomtant agreement between calculatons and reported measurements on effectve masses, electron and hole concentratons, Seebeck coeffcent, electrcal conductvty, and electronc and lattce thermal conductvtes. Thrdly, the present model provdes the Ferm energy as a functon of temperature and dopng concentraton. Fourthly, the veloctes of sound are calculated usng the Debye model rather than taken from lterature. After verfcaton of the present model, we were able to examne a recently attracted materal of SnSe, ndcatng a sgnfcant mprovement on the dmensonless fgure of mert. 6. Introducton Thermoelectrc (T) materals drectly convert thermal energy to electrcty or vce versa wthout movng parts, mantenance, and sounds. Htherto, they have been used only n nche applcatons such as space exploraton or electronc temperature control due to ther low thermal effcences.[, ] It has long been thought that f ther effcences are mproved they could be used for waste heat recovery or large scale solar power converson. Snce the early 90s, a great deal of effort on understandng the physcs for T transport propertes recently brought a remarkable mprovement, especally wth nanostructures.[3-5] It s known that the modelng of nanostructures s based on the frame work of bulk structures.[6] Many models and experments are reported n the lterature.[6-7] Semclasscal theores based on the lnearzed oltzmann transport equaton (T) wth the relaxaton tme approxmaton (RTA) have successful agreement wth the experments even wth ther lmtatons (elastc scatterng).[6, 8, 9] A specfc model was typcally developed for a specfc materal, whch leaves t uncertan whether t could also work for other materals. The RTA ncorporates the electron and phonon scatterng mechansms, whch are of great mportance for determnng the T transport propertes.[0] Notably, a varety n the use of electron scatterng mechansms prevals n lterature. Ab nto numercal calculatons lead to understandng electron and phonon band structures, dsperson relatons, densty-of-states effectve masses.[, ] However, they are ntrnscally ncapable of predctng expermental values due to ts nablty to handle temperature-dependent band gaps

3 6-3 and densty-of-states effectve mass[3, 3]. Therefore, the theoretcal model becomes mportant for correctly predctng the T transport propertes. It s nterestng to note that, although there are a great number of research reports n lterature, few complete theoretcal models thorough from electrons to phonons have been reported.[6, 5, 8] Snce electrons and phonons nterrelated, t s mportant to nclude both n the model. In ths work, an attempt s made to develop a generc transport model wthout the fttng parameters. Ths work dscusses the twoband Kane model, bpolar effect, electron and phonon scatterng mechansms, ansotropy of effectve mass (Hall factor), nonparabolcty, electrcal and lattce thermal conductvtes, and specfc heat. Although some materals[6, 4] may have more than two bands, here we delberately choose materals[0, 5-8]for whch two bands suffce. The present model uses the T[9, 30] wth the RTA[9] and the Debye-, Klemens-Callaway expressons[3, 3]. The bnary compounds wth the comprehensve expermental data are PbTe[6], Te3[5], SnSe[8], S0.7Ge0.3[0], and MgS[7]. The organzaton of the chapter s as follows. Frst, n Secton II, a theoretcal model s provded wth the equatons: carrer transport propertes, electron scatterng mechansms, lattce thermal conductvty, phonon relaxaton tmes, and phonon densty of states and specfc heat. In Secton III, the calculated results for fve bnary compounds are dscussed and compared wth recently reported expermental data. Fnally, n Secton IV, the conclusons are gven. The formulaton of ansotropy factor n Hall factor s gven n Supplementary. 6. THORTICAL QUATONS 6.. Carrer Transport Propertes In ths work the nonparabolc two-band Kane model ncludng the Hall factor for the transport propertes s derved. For many semconductors, electrons respond to appled felds wth an effectve mass that depends on the crystallographc orentaton of the feld. Herrng (955 and 956)[8, 33] and Zman (960)[30] observed that most actual band structures have ellpsodal energy surfaces whch requre longtudnal and transverse effectve masses n place of the three prncpal effectve masses. In the Kane model that consders nonparabolcty, the energy dsperson s gven by[, 34] g kl kt ml mt (6.)

4 6-4 where s the carrer s knetc energy, the Planck constant (h/), g the band gap, kl and kt the longtudnal and transverse wavevectors, respectvely, and ml and transverse effectve masses. The densty of states (DOS) s gven by[] m t the longtudnal and g N v, m 3 d, 3 k T g g (6.) where subscrpt denotes for the majorty carrer and for the mnorty carrer (for n-type semconductors the majorty carrers are electrons whle the mnorty carrers are holes). Nv, s the degeneracy of the carrer valleys, m, the DOS effectve mass of sngle valley (see q. (6.9a)), k the oltzmann constant, T the absolute temperature, k T, and the reduced Ferm energy k T g g defned by[, 35] F d F F g the reduced band gap. The Hall coeffcent s R H, n e H, A n e (6.3) where n, s the Hall carrer concentraton, n the carrer concentraton, e the carrer charge H (negatve for electrons and postve for holes), and Hall coeffcent H A the Hall factor[]. Measurements of the R, are the most wdely used method for determnaton of the carrer concentraton n semconductors[35]. The Hall coeffcent negatve. It s mportant to understand that when the Hall factor measured carrer concentraton n becomes the Hall carrer concentraton R, s taken negatve when e s H A s assumed to be unty, the general Ferm ntegral wth three exponents (l, n, and m) and carrer ndex () s[6] n,. For brevty, a H n F m l, 0 f 0, l n g 3 m d g (6.4) Where j, j (6.5)

5 6-5 where s the combned carrer relaxaton tme n whch the scatterng rates (where j ndexes the scatterng rates) are added accordng to Matthessen s rule as wll be dscussed n detal n Secton II. The Ferm-Drac dstrbutons f 0, are gven by f 0, e F and f 0, e F g (6.6) Note that for the Ferm-Drac dstrbuton f 0, the reduced Ferm energy F n the frst part f 0, s substtuted by [5], whch fxes the coordnates of the band structures so that the g F majorty carrers are n the postve drecton. The Hall factor A s gven by[] A F, T, nd AK A 3K( K ) (K ) 0 F 0, 0 F, F 0, (6.7) where A K [33, 36, 37]and A [] are the ansotropy factor of the effectve mass and the statstcal factor, respectvely, and K s the rato of the longtudnal to the transverse m l mt component of the effectve mass. Most measurements[5, 8] of carrer concentraton assume the Hall factor of unty. However, n ths study, the effect of the Hall factor s consdered. A s derved from the energy transport n a low magnetc feld (See Supplementary). The Hall carrer concentratons ncludng the Hall factor are then gven by[, 6] n H, N m 3 d, kt 0 0,,, v F T nd A F 3 0, 3 (6.8) where A s the Hall factor as a functon of the Ferm energy F, temperature T, and dopng concentraton n d. The dopng concentraton n d s the absolute dfference between the electron concentraton and hole concentraton. The Ferm energy F s exclusvely determned by solvng q. (6.9)[6], where the Ferm energy s a functon of both the temperature T and dopng concentraton n d. Understandng the varablty of the Ferm energy s of great mportance for the transport propertes. The mportance of the Ferm energy can be seen through qs. (6.4) and (6.6), where the dervatve of the Ferm-Drac dstrbuton functon s non-zero only for energy levels wthn about ± 0. ev at the Ferm energy. Snce the dopng concentraton n d s obtaned from the Hall coeffcent R,, we have H

6 6-6 d H, T, n n, T n n n, F, d H, F d (6.9) The present nonparabolc two-band Kane model provdes expressons for the transport propertes as follows: the electrcal conductvty s gven by[6, 3] N v, e m 3 d, kt 0 3 m c, 3 F, (6.0) where m m d c m m 3 l 3 t m m l t (6.) (6.) where m d s the DOS effectve mass and mc s the conductvty (or nertal) effectve mass whch s usually n the crystallographc drecton[38] (thermoelectrc transport drecton)[6]. For later use, the total DOS effectve mass m s defned by m 3 N v m conductvty s the sum of the conductvtes for the separate bands[39]. d. The electrcal (6.3) Usng the relatonshp of H A [33], ne [40], and n An H [6], the Hall moblty H, for electrons and holes s derved as[] H, A e m c, F 0, 0 0 F0, (6.4) whch s also expressed n terms of the averaged relaxaton tme as[]

7 6-7 c H m e A,, (6.5) where s the averaged relaxaton tme 0 0, 0, 0 F F. Note that the Hall factor s ncluded n q. (6.4). The Seebeck coeffcents, for the majorty and mnorty carrers are obtaned by[, 3] F F F e k, 0,, F g F F e k, 0, (6.6) The total Seebeck coeffcent s gven by[39] (6.7) If we assume a parabolc sngle band model, we could have a smple but conceptual Mott formula[4], or slghtly dfferent verson of the formula[4], whch s n g T k e 3 (6.8) whch s not used n ths work but n nterpretaton of the formula, where the Seebeck coeffcent s proportonal to the carrer densty of states g but nversely proportonal to the carrer concentraton n. The Lorenz number s derved by[6], 0,, 0, F F F F e k L (6.9) The electronc thermal conductvty s gven by[39] T T L T L k elect (6.0)

8 Scatterng Mechansms for lectron Relaxaton Tmes The electron relaxaton tme s the average flght tme of electrons between successve collsons or scatterng events wth the lattce or mpurtes. The relaxaton tme plays the most mportant role n determnng the transport propertes such as the electron moblty, the electrcal conductvtes, thermal conductvty and the Seebeck coeffcent. ardeen and Shockley (950)[43], rooks (95)[44], Howarth and Sondhemer (953)[45], Wlson (953)[9], hrenrech (957, 96)[46, 47], Ravch (97)[48], Nag (980)[49], and Lundstrom (000)[6] studed the fundamental scatterng mechansms by acoustc phonons, polar optcal phonons, and onzed mpurtes. Ravch (97) used the three fundamental scatterng mechansms, derved formulae, and compared wth experments. Hs formulae were based on the frame work of those semclasscal theores shown above. Later, Zayachuk (997)[50] and Frek (00)[5] extended the work of Ravch (97) by addng two scatterng mechansms (nonpolar optcal phonons and short range deformaton potental of vacances) to the three fundamental scatterng mechansms. Thus, fve mechansms have been studed so far. There s a dversty n the use of the scatterng mechansms. For example, lc (006)[3], Huang (008)[5], and Ahmad (00)[53] used the fve mechansms for PbTe and Te3. Ravch (97), Harrs (97)[4], rodo (997)[54], Zhou (00)[6] and ahk (04)[4] used the fundamental three mechansms for PbTe, Te3 and MgS. Vnes (008)[0] used acoustc phonon, polar optcal phonon, and nonpolar optcal phonon mechansms for PbTe. Kolodzejczak (967)[55] used acoustc phonon, polar optcal phonon, and nonpolar optcal phonon mechansms. Amth (965)[56] used polar optcal phonon and onzed mpurtes mechansms for GaAs. Vnng (99)[5] and Mnnch (009)[8] used acoustc phonon and onzed mpurty mechansms for S0.7Ge0.3. Pe (0)[6] used only acoustc phonon mechansm for PbTe. ux (0)[57] showed that acoustc phonon mechansm was domnant for MgS. Chen (04)[58] used only acoustc phonon mechansm for SnSe. There are also many reports that used only the acoustc phonon mechansm for ther analyses[43, 59-6]. Acoustc Phonon Scatterng Longtudnal acoustc phonons may deform the electron band structure leadng to electron scatterng due to the deformaton potental. The man body of the expresson for acoustc phonon scatterng was orgnally provded by ardeen and Shockley (950)[43] and wdely used. Ravch (97)[48] added the effect of nonparabolcty at the energy band edge as a functon of the rato of electron energy to band gap. The electron relaxaton tme for acoustc phonon scatterng modfed[5] s gven by

9 6-9 a a g g a a A, 0, (6.) where 3,, 4, 0 T k m d v d a s a, g a g a K A, 3 8 g a g g a K (6.) where a s the acoustc deformaton potental, s v the velocty of sound (see q. (6.3)), d the mass densty (see q. (6.3c)), and a K the rato of the acoustc deformaton potental for the valence and conducton bands ac av a K [48], whch s often assumed to be unty[5]. Note that the relaxaton tme for acoustc phonon scatterng s a functon of energy 0 where the exponent -½ s known the scatterng parameter. In ths work the velocty of sound s v s derved usng the Debye model rather than takng t from the lterature. Thus we can not only reduce one pece of nput data but also we can see the effect of atomc mass on transport propertes. There can be a margnal dfference between the derved and measured veloctes. The velocty of sound used n ths work s gven by[6] a k v D s 3 6 (6.3) where D s the Debye temperature and the atomc sze a s obtaned by[5] y a y a a A, 3 A A A A d N M a and 3 A d N M a (6.4) where A M, are the atomc masses of component A or, and A d, are the mass denstes of component A or.

10 6-0 Polar Optcal Phonon Scatterng In a polar sold when the two atoms n a unt cell are not alke, the longtudnal optcal vbratons produce crystal polarzaton that scatters electrons[46], a mechansm that s dffcult to be expressed n terms of the relaxaton tme[45]. However, at hgh temperatures (T >> D ), the energy change after collson s small compared to the electron energy, whch allows the use of the relaxaton tme.[6, 45] Howarth and Sondhemer (953)[45] and hrenrech (96)[47] derved an expresson for polar optcal phonon scatterng. Ravch (97)[48] used that expresson along wth expermental data ncludng nonparabolcty, the screenng effect, and the concept of the onc charge derved by Callen (949)[63]. The screenng effect usually reduces. There s a problem that ths expresson eventually domnates the other mechansms. It was, brefly speakng, found that, when 8 s added to the magntude of the formula gven by Ravch, t unravels ths problem gvng good agreement wth expermental data, even that of Ravch and others (further dscusson s gven n Secton III). A smlar expresson wth 8 was actually derved n the works of Nag (980)[49] and Lundstrom (000)[6]. The fnal relaxaton tme wth 8 for the polar optcal phonon scatterng s especally of mportance at low carrer concentratons. The electron relaxaton tme for polar optcal phonons s po d g g po G T k m e 0,, 8 (6.5) where 0 and are the statc and hgh frequency permttvtes, g g g po G ln ln, (6.6) 0, r k (6.7)

11 6-, g d T k m k (6.8) 0 0, 3 3, 5 0, d f T k m e r g g d (6.9) where k s the carrer wavevector and r, 0 s the screenng radus of the optcal phonon. Ionzed Impurty Scatterng Ionzed mpurty scatterng becomes most mportant at low temperatures, where phonon effects are small. Conwell and Wesskopf (950)[64], rooks (95)[44], latt (957)[65], and Amth (965)[56] studed onzed mpurty scatterng suggestng the rooks-herrng formula[65] to take account of the screenng effect. Later, Ravch (97)[48], Nag (980)[49], Chattopadhyay (98)[66], Zayachuk (997)[50], Lundstrom (000)[6], Frek (00)[5], lc (006), and Zhou (00)[6] provded a modfed rooks-herrng formula n magntude (a factor of ether or 6 ) and the screenng radus (a factor of ether or /4), whereas the man body of the formula except the magntude and screenng radus s the same as the rooks-herrng formula. It s found n ths work that the orgnal rooks-herrng formula wth magntude factor of and screenng radus factor of /4 ft the expermental data used here (Chattopadhyay)[66]. The onzed mpurty scatterng s fnally gven by I g g d I b b b Ze N T k m ln 3 3 0,, (6.30), kr I b (6.3), g d T k m k (6.3)

12 6-0 0, 3 3, 5, 4 d f T k m e r g g d I (6.33) where I r, s the screenng radus of the onzed mpurtes Lattce Thermal Conductvty The thermal conductvty conssts of two parts: the electronc and lattce thermal conductvtes k e k l k. Snce the equaton of the electronc contrbuton to the thermal conductvty was gven n Secton II A, we focus here on the lattce thermal conductvty. Many efforts has been made to understand the lattce thermal conductvty. Debye (9)[6], Herrng (954)[67], Klemens (955)[68], Zman (956)[69], Callaway (959)[3], Stegmeer and Abeles (964)[70], and Vnng (99)[5] studed the fundamental theores and models. Callaway suggested an expresson based on the work of Debye and Klemens, whch s wdely used.[3, 68] The lattce thermal conductvty l k s gven by k k k l (6.34) T x x c s D dx e e x T k v k k / (6.35) T x x U c U T x x U c s D D dx e e x dx e e x T k v k k / 0 4 / (6.36) where s the rato of the Umklapp processes to the N-processes, c and U are the combned and Umklapp relaxaton tmes (see qs. (6.44) and (6.37) for c and U ), respectvely, and T k x. The N-processes cannot be usually presented as the scatterng processes[7]. The second Callaway term k s large for a pure crystal but decrease rapdly wth addton of defects so that t s domnant over a wde temperature range for the pure crystal but becomes almost

13 6-3 neglgble for an mpure crystal.[7] Most doped crystal compounds that have ntrnsc defects are mpure n fact, so that k l k Phonon Relaxaton Tmes xtensve studes[7-80] have been made nto phonon relaxaton tmes, ncludng normal-mode processes (N-processes), Umklapp processes, pont defects, electron-phonon, and boundary scatterng. We use here the expressons suggested by Stegmeer and Abeles (964).[70] Vnng (99)[5] and Mnnch (009)[8] also used those scatterng expressons. Scatterng by N-Processes and Umklapp Processes The Umklapp processes are 3-phonon scatterng nvolvng anharmoncty. As mentoned n Secton II C, the N-processes do not drectly cause the scatterng but help the Umklapp processes contrbute to the scatterng. Therefore, they can be expressed n terms of the relaxaton tme n k even when k s neglected n q. (6.0a). The Umklapp scatterng rate s the nverse of the relaxaton tme, whch s frst studed by Lebfred and Schlömann (954).[7] Later, a modfed expresson ncludng the effect of the N-processes was gven by[5, 70, 8] 0 6 U N A M Aa T D 3 x (6.37) where N A s Avogadro s number, the Grünesen anharmoncty parameter, mass of compounds A and, a the mean atomc sze (see q. (6.4)), and The N-process scatterng rate s gven by M A the atomc x k T. N U (6.38) where s often used for a good approxmaton. Scatterng by Pont Defects The pont defects may nclude vacances, sotopes, dslocatons, substtute atoms, etc. A defect wth dmensons much smaller than the phonon wavelength can be consdered as a pont defect[7]. The scatterng s then caused by the dfference n mass and the dfference n bondng

14 6-4 between the atoms.[7] The scatterng rate for pont defects s gven by Klemens (960)[75], whch s PD kt V0 3 4v s 4 x 4 (6.39) 3 where V 0 s the atomc volume ( a ) and the mass-fluctuaton-scatterng parameter for a bnary compound s gven by[70] M y y M A a s a M M A M, M A M A y M ( y), d 3 M A N Aa and a a A a (6.40) (6.4) where y s the fracton of component A, d the densty of compound and s the stran parameter.[70] Scatterng by lectron-phonon The scatterng of phonons by electrons wll be actve when the band degeneracy temperature s comparable wth the temperature of the lattce.[69] Zman (956) derved an expresson for the electron-phonon scatterng, whch s P, e x ln e x x rcf 3 6 rc, amd, vs 4 4 d x x rc rc F 6 rc, (6.4) where s the acoustc deformaton potental, m v k T a and x k T. rc, d, s oundary Scatterng The phonon-boundary scatterng rate s assumed ndependent of temperature and frequency. The boundary scatterng rate wth an assumpton of purely dffuse scatterng s gven by[30]

15 6-5 v s L (6.43) where L s the effectve length of the sample. The total scatterng tme can be approxmated by addng the scatterng rates n accordance wth Matthessen s rule,[8] c N U PD P (6.44) 6..5 Phonon Densty of States and Specfc Heat Ths analyss s based on the Debye model, n whch the lattce vbrates as f t were an elastc contnuum, but the vbraton frequences cannot exceed a certan maxmum value, chosen to make the total number of modes equal to the total number of classcal degrees of freedom. The Debye model replaces all branches of the vbratonal spectrum wth three branches, each wth the same lnear dsperson relatons. The velocty of sound s taken as constant for each branches, as t would be for a classcal elastc contnuum. In the Debye model the optcal modes are unceremonously lumped nto the top end of the dstrbuton of acoustc modes, as f they were merely elastc waves of very short wavelength. Despte ts obvous crudty, the Debye approxmaton has the great advantage of supreme smplcty. If any one parameter s requred to measure the energy scale of the vbratons of a sold, the Debye temperature s the most approprate. If any one functon s requred to represent the dstrbuton of the lattce frequency, the phonon densty of states s the smplest.[30, 6] The total number of modes N s found by 3 3 dvdng a sphere n wavevector-space by the volume of the prmtve cell N V v, 6 s where V s the volume of the crystal. The phonon densty of states for each branch s obtaned by 3 takng dervatve of N wth respect to leadng to g V v densty of states per each branch per mode s obtaned by[30, 6] ph s. After all, the phonon g ph 3 3 D (6.45) where s the phonon frequency and D the Debye frequency. Usng D Dk (cutoff frequency), s then obtaned by 3 x kt, and g ph 3 kt T D x, the Debye specfc heat n terms of g ph

16 6-6 c v 3 N V kt D T e x e x 0 x g ph dx (6.46) or equvalently, c v 9 N V T k 3 D 0 x D T e x 4 e x dx (6.47) where N s the number of modes, V s the volume of a crystal, and atomc volume N V s assumed to be the 3 a for a cubc structure[30, 75]. q. (6.47) s called the Debye formula[30] Dmensonless Fgure of Mert The dmensonless fgure of mert ZT s defned to represent the performance of the thermoelectrc materals by a ZT T k (6.48) where T s the absolute temperature, a the Seebeck coeffcent, the electrcal conductvty, and k the thermal conductvty, whch were defned n the precedng secton. The commercal value s ZT. The hgher the dmensonless fgure of mert the better the performance. 6.3 RSULTS AND DISCUSSION 6.3. lectron or Hole Scatterng Mechansms Moblty for PbTe The three fundamental scatterng mechansms (acoustc phonons, polar optcal phonons, and onzed mpurtes) presented n Secton II are examned here and compared to experments. Snce H, A e mc, (q. (6.5)), the Hall moblty was best known to represent the ndvdual or combned scatterng mechansms. Scatterng rates that are the recprocals of the

17 6-7 relaxaton tmes are assumed to be ndependent each other, so that Matthessen s rule can be appled.[9, 30] Ravch (97) studed the three fundamental scatterng rates, presentng three equatons smlar to qs. (6), (8), and (9). Zayachuk (997)[50] and Frek (00)[5] extended the work of Ravch (97).[3] Fgure 6. (a) depcts the calculated moblty-versus-carrer concentraton for n-type PbTe at room temperature, whch s n agreement wth the experments by Ravch (97)[3] and Pe (0)[6], and also very close to the predctons (not shown n the fgure) by Ravch and Pe. The acoustc phonons appear domnant at hgh concentratons whle both the polar optcal and acoustcal phonons contrbute proportonally to the total moblty at low concentratons. Ionzed mpurtes have a neglgble effect on the total moblty. Accordng to some theoretcal works[44, 56, 64, 66], onzed mpurtes are expected to be mportant at low temperatures. In Fgure 6. (b), both the onzed mpurtes and the acoustc phonons contrbute proportonally to the total moblty at 77 K, whch leads to an antcpaton that the onzed mpurtes wll predomnate the other mobltes f the calculatons and experments are conducted at near zero temperatures. Fgure 6.(c) depcts the Hall moblty versus temperature at the carrer concentraton of about 3 x 0 9 cm -3 showng that the acoustc phonons are domnant over the range of temperatures. In Fgures (a) and (c), two separate experments by Ravch and Pe are n agreement wth the present calculatons. It s seen n general that the present scatterng model fts both the experments and predctons by the theores. And the assumpton on the domnance of the acoustc phonon scatterng for PbTe ensures the valdty at room temperature, whch s the reason why PbTe has been preferably selected for many analyses of transport propertes such as the Psarenko s expresson[83]. (a)

18 6-8 (b) (c) Fgure 6. For n-type PbTe, (a) the Hall moblty at room temperature versus carrer concentraton, (b) the Hall moblty at 77 K versus carrer concentraton, and (c) the Hall moblty versus temperature at a carrer concentraton of about 3 x 0 9 cm -3. Moblty for Te3 A queston arses to how the precedng predctons wll work for other bnary compounds. We are obvously nterested n Te3 whch s one of the most wdely used T materals, known to be predomnated by the acoustc phonons[39, 84]. The predomnance by the acoustc phonons s

19 6-9 clearly seen n Fgure 6. (a) and (b) and ts temperature dependence s n agreement wth experments by Jeon (99).[5] (a) (b) Fgure 6. For p-type Te3, (a) the Hall moblty versus carrer concentraton at room temperature, (b) the Hall moblty versus temperature at a carrer concentraton of. x 0 9 cm - 3. Moblty for S0.7Ge0.3 The next nterestng compound s S0.7Ge0.3. Scatterng s domnated by acoustc phonon and onzed mpurtes because slcon germanum s not a polar materal[8]. Fgure 6.3 (a) and (b) ndcate that onzed mpurtes are domnant at room temperature whle acoustc phonons

20 6-0 gradually ncrease ther contrbuton becomng domnant at 000 K. The effect of polar optcal phonons s mnmal over these ranges of carrer concentratons and temperatures except at very low concentratons. (a) (b) Fgure 6.3 For n-type S0.7Ge0.3 the Hall moblty at room temperature versus carrer concentraton, and (b) the Hall moblty versus temperature at a carrer concentraton of 6.7 x 0 9 cm -3.

21 6- Moblty for MgS ahk (04)[4] demonstrated that the calculated transport propertes for MgS are n good agreement wth the measurements of Tan (005)[7] usng the three fundamental scatterng mechansms whch are the acoustc phonons, polar optcal phonons, and onzed mpurtes. On the other hand, Satyala (0)[85] also showed that ther predctons for MgS are n a good agreement wth the measurements of Tan (008)[86] usng only two mechansms whch are the acoustc phonons and onzed mpurtes. Our calculated results usng the three fundamental mechansms are presented n Fgure 6.4 (a) and (b), where the combnaton of the three mobltes more or less proportonally determnes the total moblty over the wde range of carrer concentraton (Fgure 6.4 (a)). At a hgh concentraton of 4 x 0 9 cm -3 n Fgure 6.4 (b), only two mechansms (acoustc phonons and onzed mpurtes) contrbute to the total moblty. It s also seen that our calculaton s n a good agreement wth the measurements of ux (0).[57] The present work successfully supports the ndvdual work of ahk and Satyala. It may be nterestng to note that n Fgure 6.4 (b) the exact slope of the expermental data cannot be acheved wthout combnng the two ndvdual slopes of the acoustc phonons and onzed mpurtes. (a)

22 6- (b) Fgure 6.4 For n-type MgS, (a) the moblty versus carrer concentraton at room temperature, (b) the moblty versus temperature at a carrer concentraton of 4 x 0 9 cm -3. Moblty for SnSe Tn selende has recently drawn much attenton for possble hgh performance due to the unusually hgh ansotropy reported by Zhao (04)[8]. There has been lttle theoretcal modelng of the electron scatterng mechansms. Chen (04)[58] calculated the transport propertes for SnSe usng only acoustcal phonons. However, we calculated the Hall moblty usng the three fundamental mechansms, shown n Fgure 6.5 (a) and (b). At low carrer concentratons, on the order of 0 7 cm -3, acoustc phonons and polar optcal phonons equally contrbute to the total moblty. In Fgure 6.5 (b), t s seen that the calculated moblty reasonably agrees wth the measurements of Zhao.

23 6-3 (a) (b) Fgure 6.5 For p-type SnSe, (a) The moblty versus carrer concentraton at room temperature, (b) moblty versus temperature at a carrer concentraton of 3.3 x 0 7 cm -3. We have seen that the present calculatons for Hall moblty are bascally n agreement wth the theoretcal predctons and wth expermental data for fve bnary compounds whch supports the present model wth the semclasscal theores for the scatterng mechansms or equvalently the carrer relaxaton tmes. Table 6. Input data for fve bnary compounds used. Symbol Descrpton n-type PbTe (A) p-type Te3 (A) p-type SnSe (A) n-type S0.7Ge0.3 (A) n-type MgS (A) Input data MA (g) Atomc mass A 07. a a 8.7 a a 4.3 a M (g) Atomc mass 7.6 a 7.6 a a 7.6 a 8.08 a da (g/cm 3 ) Mass densty A.34 a 9.8 a 5.76 a.33 a.74 a d (g/cm 3 ) Mass densty 6.5 a 6.5 a 4.8 a 5.3 a.33 a

24 6-4 g (ev) andgap 0.8 b 0.3 j 0.86 m.05 q 0.77 v g(t) (ev) Dfferental T T T / T bandgap (T-93K) (450K+T) Nv # of degeneracy 4 c k n 6 q 4 w (valleys) mh*/mo Total DOS 0. c 0.6 k 0.65 o x ffectve mass me*/mo Total DOS 0. c 0.6 k 8.5 o x ffectve mass (T/300K) / (T/300K) 0.7 εs/ ε0 Statc delectrc 400 c 90 d 6 d 7.4 r 0 y constant ε / ε0 Hgh-frequency 3.6 c 85 d 7 d 3.3 y delectrc constant θd (K) Debye temperature 36 d 65 d 55 m,p 58 s 47 z,y Фac (ev) Acoustc.4 e s 7.0 deformaton potental Ka Ф av/ Ф ac f Z Vacancy charge 0.4 e 0. l u 0.05 K m l*/ mt* 3.6 g t 3.6 є Stran parameter 39 h s 7.0 ϒ Grünesen parameter.5 d.79 d. m.5 s.67 z Calculated from Input data da (g/cm 3 ) Densty of compound MA (g) Atomc mass of compound vs (cm/s) Velocty of sound (ref.) ( ) d ( ) d ( ) m ( ) aa ( ) bb Ae or Ah Hall factor Cv_DP (J/mol K) Dulong-Pett lmt a Kttel (005)[87], b Ravch (970)[], c Ravch (97)[3], d Madelung (983)[88], e Zayachuk (997)[50], f Frek (00)[5], g Pe (0)[6], h Abeles (963)[89], Lyden (964)[90], j Goldsmd (964)[39], k Kohler (976)[9], l Huang and Kavany (008)[5], m Zhao (04)[8], n Chen (04)[58], o Sh and Koupaks (05)[3], p He (03)[9], q Slack (99)[93], r Vnng (99)[5], s Stegmeer (964)[70], t Fschett (993)[94], u Pankratov (988)[95], v Koeng (96), w oulet (0) and ahk (04), x Morrs (958) and ahk (04), y Satyala (0), z Wang (00), aa Mnnch (009)[8], bb ahk (04)[4] 6.3. Transport Propertes Wth the carrer relaxaton tmes computed n the precedng secton, the present model s now able to calculate thermoelectrc (T) transport propertes usng the equatons shown n Secton II

25 6-5 A and the nput data lsted n Table 6.. There are always small dsagreements between models and measurements. The dsagreements may be here managed by varyng the deformaton potental for electrons and the Grünesen parameter for phonons. Transport Propertes for n-type PbTe The sngle crystals of lead tellurde have been well known snce the early 950s for thermoelectrc generators at a md-temperature range from 500 K 800K.[3, 59, 96-00] The effectve mass for n-type PbTe was measured by Ravch (97)[3] as 0. mo (where mo s the free electron mass) and also predcted wth the ab nto numercal calculatons by Vnes (008)[0] to be 0.3 mo. The temperature-dependent total DOS effectve mass and the degeneracy of band valleys used are 0. (T/300K) / and 4 from Table 6., respectvely, whch are n good agreement wth measurement and ab nto calculatons.[48, 90] The frst calculaton s to determne the Ferm energy usng q. (6.8), whch s a functon of dopng concentraton and temperature. The Ferm energy versus temperature curves at two dopng concentratons of.8 and cm -3 are plotted along wth the conducton and valence band edges n Fgure 6.6 (a), where the valence band edge frst decreases wth ncreasng temperature, then beng constant from 450 K[0]. The band gap s about 0.35 ev here. The Ferm energy along wth the dopng concentraton s shown n Fgure 6.6 (b), where the Ferm energy rses above the conducton band edge from the dopng concentraton of about 0 9 cm -3. The electron concentraton n Fgure 6.6 (c) s the same as the dopng concentraton over three temperatures and the dopng concentraton s not affected at all by the bpolar carrers (holes). Note that the carrer concentraton could dffer from the dopng concentraton f there s a suffcent bpolar effect. Fgures 6 (a), (b), and (c) are rarely reported n lterature, but t s mportant to understand the varablty of the Ferm energy for studyng the transport propertes. Fgure 6.6 (d) shows the Seebeck coeffcent versus carrer (actually dopng) concentraton at room temperature along wth the measurements by Pe (0)(n-type)[6] and Crocker and Rogers (967)(p-type)[0]. Ths also shows good agreement wth Psarenko relaton (not shown here)[83]. The Seebeck coeffcent for the lowest and hghest carrer concentratons are related to the Ferm energy n Fgure 6.6 (b)[0].

26 6-6 (a) (b)

27 6-7 (c) (d) Fgure 6.6 For n-type PbTe, (a) Ferm energy versus temperature, (b) Ferm energy versus dopng concentraton, (c) Carrer concentraton versus temperature, and (d) Seebeck coeffcent versus dopng concentraton at room temperature along wth measurements by Pe (0)(ntype)[6] and Crocker and Rogers (967)(p-type)[0]. The T transport propertes are now calculated for two slghtly dfferent dopng concentratons, whch are shown n Fgure 6.7 (a), (b), and (c). The calculatons for the Seebeck coeffcent, electrcal conductvty, and thermal conductvty are n excellent agreement wth the measurements by Pe (0), precsely agreeng even wth the slght dfference of the carrer concentratons. Fgure 6.7 (d) shows the electronc and lattce components of the thermal conductvty. The measurements of thermal conductvty can knowngly be only conducted for

28 6-8 the total thermal conductvty so the lattce component s obtaned by subtractng the calculated electronc component from the measurements. It s noted that the lattce component s one approach for mprovement of the fgure of mert. The other approach s to look at electronc propertes. The superb agreement between the calculatons and measurements tells us that the present transport model comples well wth the subtle physcs of electrons and phonons ncludng the scatterng, partcularly wth the T under the RTA and the Debye model. Then, f the present model s so precse, two questons mmedately arse: ) How well the smply calculated electron densty of states (DOS) compares to sophstcated ab nto numercal calculatons, and ) how the smple Debye model enables to precsely predct the complex lattce thermal conductvty. It has been challengng to obtan the electron DOS even wth ab nto numercal calculatons[03, 04] or to obtan phonon DOS that nvolves acoustcal and optcal branches where each has one longtudnal and two transverse drectons[05-08]. (a)

29 6-9 (b) (c)

30 6-30 (d) Fgure 6.7 For n-type PbTe, (a) Seebeck coeffcent, (b) electrcal conductvty, (c) thermal conductvty and (d) electrcal and lattce contrbutons to total thermal conductvty. To examne the frst queston, the calculated electron DOS usng q. (6.) are compared (Fgure 6.8 (a)) to the ab nto calculatons of Martnez (975)[03] and lc (004)[09]. Consderng that the effectve range of the dervatve of the Ferm-Drac dstrbuton functon of q. (6.4) on the transport propertes s wthn about ± 0. ev at the Ferm energy (here the conducton band edge at 0.3 ev), Fgure 6.8 (a) s expanded as shown n Fgure 6.8 (b), where the nonparabolc Kane model and parabolc model are shown along wth the ab nto calculatons. We can see that the Kane model closely agrees wth ab nto calculatons whle the parabolc model s clearly defcent n that range. Ths s the reason why the calculated nonparabolc DOS predcts well n comparson wth the sophstcated ab nto calculatons.

31 6-3 (a) (b) Fgure 6.8 For n-type PbTe, (a) Calculated electron DOS versus energy of electrons, and (b) close-up of Fgure 6.8 (a) at the conducton band edge. The sold lne s the calculated DOS and dashed lne s the ab nto numercal computatons by Martnez (975)[0] and lc (004)[09]. To examne the second queston of how the smple Debye model so precsely predcts the lattce thermal conductvty, the calculated phonon DOS usng q. (6.45) s compared to the ab nto calculatons by Zhang (009)[05] (Fgure 6.9 (a)). The ab nto calculatons nvolve many detaled frequency-dependent features wth the sx branches mentoned earler.[05, ] The ntegral of the phonon DOS curve drectly contrbutes to the thermal conductvty by knowng that k l 3Cvl, where C s the heat capacty proportonal to the ntegral of the phonon DOS, v the mean phonon velocty, and l the phonon mean free path (l = v).[9] The present Debye model[6] smply lumps the phonon DOS onto the top of the dstrbuton of acoustc modes

32 6-3 assumng elastc waves. However, the cutoff frequency (whch s attaned from the Debye temperature) s best determned from experment (specfc heat), whch s shown n Fgure 6.9 (a). Ths gves rse to the lumped amount of the ntegral of the present phonon DOS curve, beng almost the same as the ntegral n the ab nto calculatons by Zhang. Ths s the reason why for the second queston. The contrbutons of the ndvdual phonon relaxaton tmes to the total relaxaton tme are llustrated n Fgure 6.9 (b), where the Umklapp processes are domnant up to about THz and then the pont defects become domnant at hgher frequences than THz. The electron-phonon scatterng s neglgble except at the very low frequences. It s nterestng to note that the cutoff frequency lmt n Fgure 6.9 (a) s approxmately n agreement wth the frequency of the relaxaton tme n Fgure 6.9 (b). The weghted phonon relaxaton tme s 4 x x calculated by x e e. (a) (b)

33 6-33 Fgure 6.9 For n-type PbTe, (a) calculated phonon DOS versus frequency [05], and (b) weghted phonon relaxaton tme versus frequency. Fnally, the temperature dependent specfc heat usng q. (6.46) s plotted n Fgure 6.0, where the Debye temperature s a key parameter to ft the measurements. The Dulong-Pett lmt c _ 3k a 3 v DP (see q. (6.4) for a) s calculated to be.8 cal/mol K, whch s n agreement wth the experments by Parknson (954)[96] as shown. The Debye temperature s the prmary parameter n the velocty of sound, acoustc phonon scatterng, relaxaton tme, electrcal conductvty and consequently specfc heat. Usually the Debye temperature s determned from measurement of specfc heat at lower temperatures[96, ]. The superb agreement so far between ths model and the measurements ndcates that not only the model s correct but also the Debye temperature shown n Table 6. s correct. Ths procedure was actually used to newly determne the Debye temperature for SnSe later.[6] Fgure 6.0 For n-type PbTe, specfc heat versus temperature compared wth the measurements by Parknson and Quarrngton (954).[96] The calculated Dulong-Pett lmt s.8 cal/mol K whch s n agreement wth the measurements. Transport Propertes for p-type Te3 Sngle crystals of bsmuth tellurde have been wdely used snce the 950s for both thermoelectrc generators and coolers at room temperature.[39, 00, 3-6] The calculated Ferm energy for p-type Te3 s plotted n Fgure 6. (a) along wth the valence and conducton band edges, ndcatng that the Ferm energy appears nondegenerate (located wthn the band gap beyond ± 3 kt of the conducton and valence band edges). The reason for the

34 6-34 levelng off of the Ferm energy n the fgure s due to the bpolar effect, whch s shown n Fgure 6. (c). For Fgure 6. (b), ncreasng the Ferm energy above the valence band edge by ncreasng the dopng concentraton typcally decreases the Seebeck coeffcent and ncreases both the electrcal conductvty and thermal conductvty (we wll demonstrate ths n Fgure 6.6 (a)). It s not clear whether t mproves or not untl ZT s plotted as a functon of carrer concentraton snce an optmum usually exsts at a dopng concentraton near0 9 cm -3. The electron and hole concentratons were measured by Jeon (99)[5] and compared wth the calculated values usng q. (6.8) as shown n Fgure 6. (c). We could ft the measurements by varyng ether the electron or hole DOS effectve mass. The most challengng work was that the total DOS effectve mass of 0.6mo n Table 6. entered to make ft to the expermental carrer concentraton gves no further adjustment later on the transport propertes such as the Seebeck coeffcent, electrcal conductvty, etc. It s nterestng to note that usng both the nonparabolc Kane model and the Hall factor always gve better results than wthout ether of them. In the same lne of dscusson, for n-type Te3 we could not acheve concomtantly the good comparson shown n Fgure 6.(c) and Fgure 6. (a), (b), and(c) wthout the nonparabolcty and ansotropy (Hall factor). (a)

35 6-35 (b) (c) Fgure 6. For p-type Te3 at dopng concentraton of. 0 9 cm -3, (a) Ferm energy versus temperature, (b) Ferm energy versus dopng concentraton, (c) Carrer concentraton versus temperature compared to experment by Jeon (99). For electronc transport propertes, Huang (008)[5] and Zhou (00)[6] reported computatons usng the nonparabolc Kane model. For lattce thermal conductvty, Huang used ab nto calculatons whle Zhou used the effectve medum approxmaton (MA)[7]. Park (00)[7] attempted to predct the Seebeck coeffcent usng ab nto calculatons. Consderng the wde spread use of bsmuth tellurde (bulk), t s surprsng to fnd that not many theoretcal models have been reported n lterature. As shown n Fgure 6. (a), (b), and (c), we calculated the transport propertes usng the present model, and obtaned agreement wth the measurements by Jeon (99)[5] and Goldsmd (958)[74]. Snce we used the total DOS effectve mass of 0.6 mo,

36 6-36 the DOS effectve mass of a sngle valley usng the degeneracy (multvalley) of valleys s 0. mo by usng m 3 N v m d, whereby 0. mo used n ths work s close to 0.06 mo measured by Kohler (976)[6] and also 0. mo calculated by ab nto method by Km (005)[]. Most works[39, 9, 8-0] used the degeneracy of 6 valleys except a few ab nto calculatons used the degeneracy of valleys[,, 5]. Decreasng lattce thermal conductvty s mportant for seekng a hgh fgure of mert.[] Here we present the lattce thermal conductvty n Fgure 6. (c), where the lattce thermal conductvty s domnant up to 300 K whle the electronc thermal conductvty proportonally becomes sgnfcant and domnant at hgher temperatures, obvously, due to the bpolar effect n Fgure 6. (c). For ths materal, the bpolar effect negatvely acts on the ZT, whch leads to room for mprovement f the bpolar effect s delayed or elmnated by a dopng technque. The lattce thermal conductvty s manly determned by the Umklapp processes where the Grünesen parameter s the mportant factor as shown n q. (6.37). Therefore, we make a fne adjustment of the lattce thermal conductvty curve only n magntude by adjustng the Grünesen parameter to be.79. Ths s also n agreement wth the expermental value gven by Madelung (983)[88]. (a)

37 6-37 (b) (c) Fgure 6. For p-type Te3 at a dopng concentraton of. 0 9 cm -3, (a) Seebeck coeffcent, (b) electrcal conductvty, and (c) electronc, lattce, and total thermal conductvtes. The electron and phonon DOS along wth the energy and frequency, respectvely, are presented n Fgure 6.3 (a) and (b), where the nonparabolcty and the ntegrals between ths model and other work are seen smlar to those n case of lead tellurde. In Fgure 6.3 (b), the calculated phonon DOS s here compared to the real data measured by Rauh (98)[] not ab nto calculatons as before, whch verfes the Debye model and accordngly the lattce thermal conductvty. The specfc heat was calculated usng q. (6.46) and s shown n Fgure 6.3 (c) where the model show a good agreement wth measurements by essas (0).[3] The Dulong-Pett lmt calculated s 4.7 J/mol K whch s n agreement wth the measurements. It

38 6-38 s seen that the specfc heat at low temperatures seems to follow T 3 law[30, 87]. These features support the present model. (a) (b)

39 6-39 (c) Fgure 6.3 For p-type Te3 at a dopng concentraton of. 0 9 cm -3, (a) electron densty of states versus energy compared to ab nto calculatons by Youn and Freeman (00)[], (b) phonon DOS versus frequency compared to experment by Rauh (98) []and (c) specfc heat versus temperature compared to experment by essas (0)[3], where Dulong-Pett lmt calculated s 4.7 J/mol K. Transport Propertes for p-type SnSe Sngle crystals of tn selende have been used snce the 950s n applcatons such as solar cells, lasers, detectors, phase-change memory alloys, etc. Not much attenton has been pad to thermoelectrc applcatons untl the recent tmes. Therefore, theoretcal modelng s rarely found n lterature, Wasscher (963)[4] analytcally studed the maxmum dmensonless fgure of mert ZT Chen (04)[58] dscussed a nave theoretcal model assumng the parabolc one-band model wth sphercal Ferm surface and acoustc phonon scatterng for the Seebeck coeffcent and Hall moblty. Adouby (998)[5] conducted experments on crystal structures. Narya (009)[6], ank (04)[7], Sass (04)[8] Zhao (04)[8] and Han (05)[9] conducted measurements for the T transport propertes. To the author s knowledge, Chen (04) s the only work on theoretcal modelng although there are many studes on the band structures and the lattce thermal conductvty usng the ab nto calculatons[3, 8, 30-35]. An attempt s made here to predct the T transport propertes for p-type SnSe usng the present model. The nput data are shown n Table 6., where the values gven from the lterature have the references but the values wthout references are estmatons n ths work. The majorty carrer of p-type SnSe s holes. Snce we could not fnd measured total DOS effectve masses from lterature, we had to estmate them by fttng the measurements of concentraton by Zhao (04)[8] as shown n Fgure 6.4 (a). We obtaned 0.65 mo for holes and 9.5 mo for electrons

40 6-40 for the best ft ndcatng the unusually hgh ansotropy (for whch the Hall factor appears the smallest among other materals shown n Table 6.), even showng the calculatons slghtly off from the measurements at the hgher temperatures startng at 700 K. Snce the present model uses the total DOS effectve masses as shown above, the sngle valley DOS effectve masses usng the degeneracy of valleys gve 0.4 mo for holes and 5.9 mo for electrons, whch are comparable to 0.34 mo and 3.0 mo, respectvely, calculated by ab nto method by Sh and Koupaks (05)[3]. However, we were then able to calculate the Ferm energy as a functon of temperature and dopng concentraton. The Ferm energy was plotted n Fgure 6.4 (b) along wth the valence and conducton band edges, where smlar ab nto calculatons by Sh and Koupaks (05)[3] are supermposed for comparson. It should be noted that the temperaturedependent band gap ( T) ev n Table 6. s n agreement wth the 0.86 ev measured by Zhao (04). The temperature-dependent band gap s however forced to ft the expermental transport propertes. A strong bpolar effect at around 600 K s seen n the Ferm energy for both the present and ab nto calculatons. Note that the measurements by Zhao (04) were performed at a low dopng concentraton of cm -3 probably for a crystallographc reason. Unusual dynamcs of the Ferm energy wth respect to the dopng concentraton s shown n Fgure 6.4 (c) correspondng to the bpolar effect. The calculated Seebeck coeffcent n Fgure 6.4 (d) was then compared to the smlar measurements conducted by Chen (04)[58] for the applcablty of the present model as a functon of dopng concentraton at two dfferent temperatures of 300 K and 750 K, showng good agreement between the model and the measurements. (a)

41 6-4 (b) (c)

42 6-4 (d) Fgure 6.4 For p-type SnSe at a dopng concentraton of cm -3, (a) carrer concentraton versus temperature compared to experments by Zhao (04)[8], (b) Ferm energy versus temperature compared to the ab nto calculatons by Sh and Koupaks (05)[3], (c) Ferm energy versus dopng concentraton, (d) Seebeck coeffcent versus dopng concentraton compared to experments by Chen (04)[58]. Snce the DOS effectve masses for holes and electrons are forced to ft the measured temperature-dependent carrer concentratons n Fgure 6.4 (a), there s no way to adjust the calculated Seebeck coeffcent f t s n dsagreement wth the experments (because the effect of carrer scatterng on the Seebeck coeffcent s very small). However, the calculated Seebeck coeffcent n Fgure 6.5 (a) s n agreement wth experments by Zhao, whch supports the present model. The electrcal conductvty s a strong functon of the carrer scatterng, dependng on a combnaton of the scatterng mechansms or the relaxaton tmes. As the detaled dscusson showed n the earler secton, the three fundamental scatterng mechansms properly act on resultng n a good agreement wth the measurements as shown n Fgure 6.5 (b). As for expected n Fgure 6.4 (a), the strong bpolar effect produce many holes addng to the majorty holes, rapdly ncreasng the hole concentraton as well as the electron concentraton, correspondngly ncreasng the electrcal conductvty as shown n Fgure 6.5 (b). Accordng to Zhao (04), a phase transton from low-temperature Pnma to hgh-temperature Cmcm occurs at 750 K, where the present model cannot take nto account the phase transton whch causes the calculatons apart from the measurements at the hgher temperatures, whch s shown n Fgure 6.5 (b). The most spectacular feature s the ultralow thermal conductvty. The electronc and lattce contrbutons to the thermal conductvty are presented as shown n Fgure 6.5 (c), where the lattce thermal conductvty s at glace domnant over the range of temperatures except

43 6-43 at the very hgh temperatures, n whch the domnance of the lattce thermal conductvty s n good agreement wth Zhao (04) attrbutng to the low carrer concentraton used. It s nterestng to note that the Grünesen parameter obtaned here s. that s the hghest among others n ths work and n agreement wth an average value. of ab nto calculatons by Zhao (04). The Grünesen parameter characterzes crystal volume change due to phonon frequency, whch affects, n turn, the band structure, the carrer scatterng, and fnally the phonon scatterng. The ZT at the dopng concentraton of cm -3 as a functon of temperature are presented n Fgure 6.5 (d) whch s generally n agreement wth the measurements, where ZT s a very low value of about 0. at room temperature whle ZT s an unprecedented hgh value of about.6 at 93 K by measurement (about. n ths work). (a) (b)

44 6-44 (c) (d) Fgure 6.5 For p-type SnSe at a dopng concentraton of cm -3, (a) Seebeck coeffcent, (b) electrcal conductvty, (c) thermal conductvty, and (d) fgure of mert versus temperature. The model was compared to the experments by Zhao (04).[8] The dynamcs of transport propertes as a functon of carrer (actually dopng) concentraton are presented n Fgure 6.6 (a). It s seen that, wth ncreasng the carrer concentraton, the Seebeck coeffcent decreases whle both the electrcal and thermal conductvtes ncrease. The summary result s convenently presented by the dmensonless fgure of mert [] as shown n Fgure 6.6 (b). It s qute nterestng to see that the calculated ZT values along wth three dfferent temperatures dramatcally ncreases as the concentraton ncreases from the measured

45 6-45 concentraton of cm -3 to the calculated optmum concentraton of about cm -3, where the ZT values at 900 K, 600 K and 300K reveal about 3%, 660%, and 60% ncreases, respectvely, compared to the values at the low carrer concentraton. The predcted ZT value at the md-temperature ranges of about 600 K s about.5 whch would be also very useful n some applcatons. It s hoped that the calculated maxmum ZTs at the optmum concentratons be expermentally examned n the future. (a) (b) Fgure 6.6 For p-type SnSe, (a) electronc propertes versus carrer concentraton, and (b) the dmensonless fgure of mert versus dopng concentraton.

46 6-46 The electron DOS for p-type SnSe s presented n Fgure 6.7 (a), showng that the nonparabolc Kane model fts well the ab nto calculatons by He (03) for the effectve range of energy (± 0. ev) at the Ferm energy. The specfc heat s calculated and compared wth the measurements by Zhao (04), where the Debye temperature D used s 55 K that agrees wth nether a calculated value of 65 K by Zhao (04)[8] nor a calculated value of 5 K by He (03)[9]. The value of 55 K was determned by fttng the expermental data for both the transport propertes and the specfc heat. (a) (b) Fgure 6.7 For p-type SnSe at dopng concentraton of. 0 9 cm -3, (a) electron DOS versus energy compared to ab nto calculatons by He (03)[9], and (b) specfc heat versus temperature compared to experments by Zhao (04)[8].

47 6-47 Transport Propertes for n-type S0.7Ge0.3 The sngle slcon-germanum crystals as a possble materal for thermoelectrc generators were taken as early as 957 by Ioffe[83]. The crystals or alloys were consdered to be a good canddate at hgh temperatures around 000 K for energy converson to electrcty n space wth radosotope thermoelectrc generators (RTG). Dsmukes (964)[0], Abeles (96)[77] and Ros (968)[] studed expermentally whle Abeles (963)[89], Parrott (963)[36], Stegmeer (964)[70], Amth (965)[37], and Gaur (966)[38] studed theoretcally. Recently, Slack (99)[93], Vnng (99)[5] and Mnnch (009)[8] studed the theoretcal modelng. Snce the dscussons here are smlar to those n the earler sectons, we here just present the calculated results for nformaton. Fgure 6.8 (a), (b), and (c) present the carrer concentraton, the dopng concentraton, and Ferm energy, respectvely. And the transport propertes versus temperature along wth the experments as shown n Fgure 6.9 (a), (b) and (c), where the two scatterng mechansms (the acoustc phonons and onzed mpurtes) domnantly determne the total scatterng rate, or the relaxaton tme. The specfc heat versus temperature s plotted n Fgure 6.9 (d), where the Dulong-Pett lmt s calculated to be 4.9 J/mol K. (a)

48 6-48 (b) (c) Fgure 6.8 For n-type S0.7Ge0.3, (a) carrer concentraton versus temperature, (b) Ferm energy versus dopng concentraton, and (c) Ferm energy versus temperature.

49 6-49 (a) (b)

50 6-50 (c) (d) Fgure 6.9 For n-type S0.7Ge0.3 at two carrer concentratons of cm -3 and cm -3, (a) Seebeck coeffcent versus temperature compared to experments by Dsmukes (964)[0], (b) electrcal conductvty versus temperature compared to the experments, (c) thermal conductvty versus temperature compared to the experments, and (d) specfc heat versus temperature compared to experments by Ros (968).[] Transport Propertes for n-type MgS MgS sngle crystals have been well known snce the 950s as thermoelectrc materal whch offers nontoxcty (not lke Pb), cost effectveness (not lke Te), and abundance (not lke Te, rare earths, etc.) at a md-temperature range 500 K to 900 K. Many efforts have been drawn to

51 6-5 electronc structures[39-4], thermal conductvty[43], experments[7, 44], ab nto calculatons[04, 45, 46], nanostructures[47, 48], and modelng of transport propertes[6, 4, 85]. The carrer band structures appear somewhat complex, nvolvng ndrect band gaps and multband. The conducton band edge conssts of two nondegenerate bands separated by a dstance of 0.4 ev. The valence band edge also conssts of two degenerate bands, lght-hole and heavy-hole, wth no separaton.[4, 04, 49] Furthermore, each band has two valleys (or pockets). In the present model, the two conducton band edges are lumped nto one conducton band but ncreasng the valleys to match the total number. The two valence band edges are lumped nto one valence band but ncreasng the number of valleys to match the total number. Thus, the present model assumes only two bands, one conducton band and one valence band each wth four valleys, as shown n Table 6.. In fact ths model allows a ft to expermental data of transport propertes smply by varyng the total DOS effectve masses. We obtaned.0 mo for electrons and.0 mo for holes and recalculated the sngle valley DOS effectve masses usng the degeneracy of 4 valleys, leadng to 0.4 mo and 0.8 mo, whch are unexpectedly n excellent agreement wth the 0.46 mo and 0.87 mo measured by Morrs (958)[50]. The dscusson of the results are smlarly explaned n the earler sectons, so here we just present the results for nformaton. The carrer concentraton and Ferm energy are plotted n Fgure 6.0 (a), (b), and (c). The transport propertes for MgS are presented n Fgure 6. (a), (b), and (c). The specfc heat s shown n Fgure 6. (d). All the calculatons are n good agreement wth the measurements. (a)

52 6-5 (b) (c) Fgure 6.0 For n-type MgS, (a) carrer concentraton versus temperature, (b) Ferm energy versus dopng concentraton, and (c) Ferm energy versus temperature.