Coupled Quantum - Scattering Modeling of Thermoelectric Properties of Si/Ge/Si Quantum Well Superlattice

Transcription

1 Proceedgs of IMECE 6 26 ASME Iteratoal Mechacal Egeerg Cogress Chcago, Illos, November 5-1, 26 Coupled Quatum - Scatterg Modelg of Thermoelectrc Propertes of S/Ge/S Quatum Well Superlattce A. Bulusu Graduate Research Assstat Iterdscplary Program Materal Scece Vaderblt Uversty Nashvlle, TN auradha.bulusu@vaderblt.edu D. G. Walker 1 Assstat Professor Departmet of Mechacal Egeerg Vaderblt Uversty Nashvlle, TN greg.walker@vaderblt.edu Cofed structures presumably offer ehaced performace of thermoelectrc devces. 1) Iterfaces ad boudares create scatterg stes for phoos, whch reduces the thermal coductvty. 2) Reduced dmesoalty creases the local desty of states ear the Ferm level, whch creases the Seebeck coeffcet. From these two pheomea, the et effect should be a crease ZT, the performace parameter used to evaluate dfferet materals ad structures. These effects have bee measured ad modeled, but oe of the models attempts to quatfy the electro-phoo coupled effects partcularly the regme where quatum ad scatterg flueces are foud. Usg the o-equlbrum Gree's fucto (NEGF) approach, quatum wells composed of S ad Ge are studed ad the mportat physcs solated. Results show a competg effect betwee the decrease the electrcal coductvty due to scatterg wth the crease electrcal coductvty wth dopg, leadg to 77% decrease the value of the power factor for the case of electro-optcal phoo scatterg. INTRODUCTION The usefuless of thermoelectrc materals s ofte characterzed by the dmesoless thermoelectrc fgure of mert ZT S 2 σt/κ where S s the Seebeck coeffcet, σ s the electrcal coductvty ad κ s the thermal coductvty. Research has bee focused towards developg materals that have a hgh Seebeck coeffcet as well as structures that wll ether have reduced thermal coductvty or ehaced electrcal coductvty resultg a crease ZT [1]. I the past decade, the use of cofed structures such as quatum well ad quatum wre superlattces have gaed atteto as thermoelectrc materals due to a umber of advatages [2, 3, 4]. The reduced dmesoalty of quatum well superlattces creases the local desty of desty of states per ut volume ear the Ferm eergy leadg to a crease the Seebeck coeffcet [5]. The reduced dmesoalty also causes phoo cofemet leadg to reduced thermal coductvty [6, 7]. I addto, the presece of materal terfaces the superlattce promotes phoo scatterg, whch theoretcally reduces the effectve thermal coductvty [2, 3, 4]. Phoo terferece effects caused by phoo-terface scatterg gve rse to badgaps at the terface of the th flms affectg phoo trasport through these structures [8]. Che [9, 1] studed phoo trasport S/Ge superlattces usg the BTE for phoos. Phoo terface scatterg was cluded through a combato of dffuse ad specular scatterg. He foud that the greatest temperature drop occurred at the terfaces rather tha wth the layers due to a combato of dffuse ad elastc scatterg processes thus developg the cocept of phoo egeerg to buld structures havg low thermal coductvty. However, expermetal observatos have ot bee able to acheve the presumed beefts of superlattce thermoelectrc devces despte a theoretcally predcted ad expermetally observed reducto the thermal coductvty of a superlattce compared to ts bulk couterpart [5, 11]. Whle phoo cofemet ad scatterg leads to reduced thermal coductvty, the electrcal coductvty s assumed ot to be sgfcatly affected due to the large semcoductor badgap ad the dsparty betwee the electro ad phoo mea free paths [12]. However, ths assumpto does ot take to cosderato the effects of electro cofemet as well as electro-phoo teracto due to phoo scatterg processes. Hece there s a eed to better uderstad the effect of the other sgfcat factor cotrbutg to the thermoelectrc fgure of mert of aoscale devces amely the electrcal coductvty σ. The two fudametal aspects that dfferetate aoscale devces from bulk devces are scatterg effects ad quatum effects. Scatterg aostructures ca sgfcatly affect carrer as well as thermal trasport devces [14]. Ielastc electro-phoo scatterg causes electros to lose or ga eergy wth the lattce whle other scatterg effects such as carrer-carrer, carrer-defect, ad carrer-boudary scatterg ca lead to eergy redstrbuto devces where trasport s ear ballstc. Quatum effects mafest themselves the form of electro tuelg as well as carrer cofemet [14]. These effects usually domate whe the devce legth scales are of the order of the de Brogle wavelegth assocated wth the devce. Quatum cofemet ca lead to reduced desty of 1 Correspodg author 1

2 states avalable for the carrer whle electro tuelg ca have a detrmetal mpact the form of leakage currets very small szed trasstors [15]. However, quatum effects ca be leveraged as a hgh performace alteratve to very-large-scale tegrato (VLSI) [16] the form of resoat tuelg dodes. Shrkg devce dmesos preset a creasg eed for a smple trasport model that ca effectvely couple quatum wth scatterg effects. The eed to corporate scatterg stems from the fact that whle electro-phoo scatterg usually helps restore thermodyamc equlbrum, shrkg devce dmesos may ot esure eough scatterg to restore equlbrum. The smultaeous cosderato of scatterg effects, whch s usually descrbed as partcle behavor, ad quatum effects, whch are wave ature, s extremely dffcult ad computatoally tesve. Classcal trasport models I geeral carrer trasport models ca be classfed to two types. The frst type s the classcal partcle models that are based o fdg the soluto to the Boltzma trasport equato (BT. The BTE based models allow for the cluso of lmted carrer scatterg through a relaxato-tme approxmato treatg the system to be oly slghtly perturbed from equlbrum. Numercal solutos to the BTE are possble usg Mote Carlo techques [17] whch partcle dstrbutos are solved usg stochastc processes. Mazumder et al. [18] descrbes such a approach that cludes phoo dsperso ad varous phoo modes depedetly. For coupled solutos, a electro-thermal model based o momets of the BTE for phoos ad electros was developed to smulate electro-phoo teractos [19]. Ths oequlbrum approach has show that the eergy dstrbutos for optcal ad acoustc phoos dffer sgfcatly suggestg that o-equlbrum behavor s very dfferet from cotuum behavor. Usg a smlar model, Rama [2] showed that oequlbrum sgfcatly affects the locato ad geerato of the hot spot mcroelectroc power devces. To remove the gray assumpto momet-based solutos, a ustructured fte-volume dscrete ordates scheme has bee used by several groups to solve the BTE wth spectral formato [21]. Isotropc scatterg the form of mpurty ad Umklapp scatterg was cosdered ad a favorable match wth exact solutos was foud. Quatum effects the BTE based models are cluded by corporatg a quatum correcto to the trasport model. Commo methods of corporatg quatum effects are the desty gradet formalsm ad the effectve potetal method [22]. The desty gradet formalsm s derved from the equato of moto for the oe partcle Wger fucto where quatum correctos are troduced by expressg the mea potetal eergy as a power seres h. The trasport equato for the Wger dstrbuto fucto ca ow be wrtte the form of a modfed Boltzma Trasport Equato. I the effectve potetal method a spatally-localzed wave packet s used as a represetato of the electro where the sze of the wave packet s defed roughly by the thermal de Brogle wavelegth. The o-local form of the charge dstrbuto troduces a effectve potetal whe the homogeeous potetal s troduced the Hamltoa. The geerato of the effectve potetal determes the oset of quatzato the system due to the o-local ature of the potetal. The quatum correcto methods have bee foud to gve a excellet match wth the Posso-Schrodger solver for the case of carrer cofemet ad tuelg. More recetly, attempts have bee made to combe quatum correctos wth the Mote Carlo techque [23] whch s a umercal soluto to the BTE. The results have bee foud to match well wth the Schrödger soluto the case of carrer cofemet whle a reasoably close match was observed for the case of tuelg. However, exteso of ths model to 2 ad 3 dmesos remas, computatoally tedous ad dffcult. Quatum trasport models The secod approach to modelg carrer trasport cossts of quatum wave fucto models that volve the soluto to the Schrödger wave equato. These models ca be used to study curret flow over small scales where the trasport ca be ether ballstc or ca volve some type of scatterg. The ma models used to model ballstc trasport are Quatum Trasmttg Boudary Model (QTBM) [24] ad the Quatum Devce Aalyss by Mode Evaluato (QDAM [25]. QTBM volves formulatg the boudary codtos for a gve problem by calculatg the trasmsso ad reflecto coeffcets for a kow boudary potetal. These boudary codtos are the used a dscretzed soluto to obta the wave fucto over the etre problem doma. Whle ths method s sutable for solvg the Schrödger equato for varous boudary potetals, cluso of dsspato due to scatterg becomes very dffcult. QDAME volves dscretely samplg a devce s desty of states usg stadg wave boudary codtos. The stadg waves are decomposed to travelg waves ad jected from the cotacts from whch ther occupaces are assged. All--all, spte of all the progress made small scale devce modelg the fact remas that rgorous scatterg caot be fully cluded most models ad couplg scatterg models to quatum models cotues to rema extremely challegg. The o-equlbrum Gree s fucto formalsm provdes a framework for atural couplg of quatum ad scatterg effects. Ope boudary codtos allow the source ad dra cotacts to be coupled to the devce through smple self-eergy terms that helps elmate workg wth huge matrces that are of the sze of the source ad dra reservor systems ad stead work wth matrces that are the sze of the devce Hamltoa. I addto, the NEGF formalsm allows for the rgorous corporato of both elastc ad elastc scatterg effects usg the cocept of Buttker probes where scatterg s treated as aother cotact, allowg t to be coupled to the devce usg self-eergy terms. We preset a bref syopss of the formalsm the ext secto whle a more thorough ad detaled developmet ca be foud [26] ad [27]. 2

3 THE NEGF FORMALISM Isolated system A solated devce ad ts eergy levels are descrbed usg a Hamltoa H, a Hartree potetal U ad eergy ege-states of the electro, α. ( H U ) ψ ( r) ε ψ ( r). (1) α α α The potetal U s obtaed usg Posso s equato ad accouts for the effect of ay chage the desty matrx o the chael capactace. The chael capactace cossts of a electrostatc capactace that depeds o the delectrc costat ε r ad a quatum capactace whch depeds o the desty of ege states the chael [27]. The electro desty matrx real space s gve by [ ρ ( r, r' )] de f ( E µ ) δ ([ EI H ]). (2) δ(ei-h) s the local desty of states. Usg the stadard expaso for the delta fucto we get δ ( EI H ) 1 1 E I H E I H. (3) 2π From the above equato, the delta fucto ca also be wrtte as δ ( EI H ) G( G ( ; 2π 1 G( E I H. (4) G( s the retarded Gree s fucto whle G (, ts cojugate complex traspose, s called the advaced Gree s fucto. I the tme doma, the Gree s fucto ca be terpreted as the mpulse respose of the Schrödger equato that wll gve us the th compoet of the wave fucto f the system s gve a mpulse exctato at the m th compoet. I the eergy doma the Gree s fucto gves the eergy egevalues for the ege-states that are occuped respose to the appled mpulse. The dagoal elemets of the spectral fucto, whch s the dfferece betwee the retarded ad advaced Gree s fucto, represet the occuped local electro desty of states. A( r, r', 2πδ ( EI H ) G( G (. (5) Thus the electro desty matrx for a solated devce ca also be wrtte the form de [ ρ ] f ( E µ )[ A( E )]. (6) 2π Ope system Now cosder the case of a ope system cosstg of a chael coected to source ad dra cotacts. Let µ 1 ad µ 2 be the chemcal potetals of the source ad the dra. The dstrbuto of electros the sem-fte source ad dra s sad to follow the Ferm dstrbuto. For a two dmesoal flm the Ferm fucto s gve by cosderg the electros to have perodc boudary the fte x ad y drectos ad experece cofemet alog the z drecto. f ( ( E )) N l 1 exp 2 D kbt where N mck BT. (7) 2 2πh The dfferece the Ferm levels of the source ad dra causes electros to flow from the source to the dra through the chael. Whe o scatterg s corporated the chael ca be cosdered to be ballstc ature ad s expected to have zero resstace to curret flow. However, expermetal measuremets [28] have show that the maxmum measured coductace of a oe-eergy level chael approaches a lmtg value G 2q 2 / ħ 51.6(KΩ) -1. The reaso for ths lmt to coductace arses from the fact that curret the cotacts s carred by fte trasverse modes whle the umber of avalable modes the chael s lmted. Ths meas that the desty of states the cotacts s spread over a large eergy rage whle the chael desty of states les specfcally betwee µ 1 ad µ 2. Upo couplg the cotact ad the chael, some of the desty of states from the cotact spread to the chael whle the chael loses some of ts desty of states to the cotact. As a result, the couplg causes the desty of states the chael to spread out over a wder rage of eergy levels resultg a reducto the umber of states lyg betwee µ 1 ad µ 2. Cosequetly, the overall effect of the couplg s to broade the rage of eergy levels of the chael leadg to a reducto the umber of states the rage µ 1 to µ 2 that are avalable for electro flow through the chael. I the NEGF formalsm the couplg of the devce to the source ad dra cotacts s descrbed usg self-eergy matrces Σ 1 ad Σ 2. The self-eergy term ca be vewed as a modfcato to the Hamltoa to corporate the boudary codtos. Accordgly, equato 1 ad 4 ca be rewrtte as ( H U Σ Σ ) ψ ( r) ε ψ ( r) (8) 1 2 α α α 1 G ( E I H Σ Σ. (9) 1 2 The self-eergy term orgates from the soluto of the cotact Hamltoa. I ths sem-fte system, whch s coected to the chael, there wll be a cdet wave from the chael as well as a reflected wave from the cotact. The wave fucto at the terface s matched to coserve eergy gvg the boudary codto Σ t exp(ka) (1) where t s a result of the dscretzato ad s gve by 2 h t. (11) * 2 2m a The broadeg of the eergy levels troduced by coectg the devce to the source ad dra cotacts s corporated through the Gamma fuctos Γ 1 ad Γ 2 gve by 3

4 Γ ( Σ Σ ) Γ ( Σ Σ ) The electro desty for the ope system s gve by de [ ] [ (12) ρ G ( ]. (13) 2π G ( represets the electro desty per ut eergy ad s gve by G GΣ G Electro-phoo couplg [ 1 ] 1 [ 2 ] 2 where [ Σ ] Γ f Γ f. (14) A electro a devce ca teract wth ts surroudgs elastcally or elastcally depedg o the ature of surroudgs. Elastc teractos occur whe the surroudgs are rgd leadg to eergy ad mometum coservato. Such teractos are coheret ature. Ielastc teractos occur whe eergy s dsspated through the emsso or absorpto of phoos, photos, etc. Such types of teractos are kow as coheret or phase-breakg processes. Ielastc scatterg s geeral dffcult to model exactly. I ths paper we treat the phase-breakg scatterg to be early elastc,.e. E E ω E - ω. The effect of scatterg s corporated to the coheret model dscussed prevously by treatg t as aother cotact where electros eter ad leave the cotact leadg to o et curret at that cotact [26]. However, because the scatterg termal does ot have a defed Ferm level, the scatterg terms Σ s ad Σ s have to be determed through other meas. As such Σ ( r, r'; D G ( r, r'; (15) s o Γ ( D A(, ad Σ s ( D G(. (16) s o D o s defed as the phoo correlato fucto ad s calculated usg the phoo deformato potetal ad the phoo wave vector β. D o ( E ω( β) ) U ( r) U * β β ( r' ) ( r, r', h ω) δ h (17) β o source ad dra Ferm fuctos. Istead, the phoo eergy represets the domat phoo frequecy avalable for scatterg [13]. I the preset model we assume D ADP s depedet of the electro eergy the chael ad correspods to optcal phoo scatterg oly, whch s expected to be the domat electro scatterer [27]. The source, dra ad scatterg self-eergy terms are used to obta the fal Gree s fucto from whch the et curret at each cotact s calculated as a dfferece the flow ad outflow currets usg equato 19. The etre calculato s carred out self-cosstetly wth Posso s equato to accout for the chage chael potetal wth the chage electro desty. q I Trace[ Γ A] f Trace[ ΓG ] (19) h RESULTS Thermoelectrc effects S/Ge/S quatum well structure The devce uder study cossts of a sgle quatum well formed by alteratg layers of slco, germaum ad slco th flms as show Fgure 1. The thckess of each flm s 2D. Electro cofemet the devce occurs due to two reasos. 1) The very small thckess of the flms gves rse to dscretely spaced eergy subbads the cross-plae drecto. 2) For costat dopg levels all three layers, the larger badgap of the slco layers o ether sde of the germaum layer sets up a potetal barrer for the electros germaum thus causg addtoal cofemet of the electros germaum leadg to a 1D potetal well. Electro trasport the devce uder a appled bas s modeled alog the cofed z drecto. The Seebeck coeffcet for the well was studed for varous temperature rages of the source ad dra cotacts whle varyg the dopg the S/Ge/S well. The source temperature was mataed costat at 3K whle the dra cotact was mataed a hgher temperature relatve to the source. The appled bas raged from to.1v. U β ( r) U β exp( β. r) where 2h U β D ADP β (18) ρωω U β s the teracto potetal whle D ADP s the acoustc deformato potetal. ρ s the mass desty ad Ω s the ormalzato volume. Depedg o the system temperature there are a wde rage of phoos wth whch electros scatter. However our model we use a smplfed case of the electros teractg wth phoos of a sgle frequecy a attempt to llustrate the effect of scatterg o electro trasport devces. For example Do.1eV 2 correspods to a phoo eergy of 2meV [27]. Ths phoo eergy s ot related to the temperature of the devce. I fact, the NEGF model does ot clude temperature wth the system as temperature ca be expressed oly for a system that s equlbrum. The oly place where temperature s cluded the model s the Fgure 1. Schematc represetato of the aodevce modeled the smulato. E s the electro eergy the devce whle s s the scatterg self eergy. Fgure 2 shows the curret-voltage characterstcs for a ballstc quatum well doped to a cocetrato of 5x1 18 cm -3. The 4

5 source-dra temperature dfferece rages from K to a maxmum of 3K. The temperature gradet results a dffuso of electros towards the source, opposg the drecto of the bas leadg to egatve curret values. As bas s creased, electros from the source are draw towards the dra ad, at the Seebeck voltage, balace the flow of electros due to the temperature gradet leadg to zero curret values. The electrcal coductvty for the flms s calculated usg the gradet of the lear porto of the curret desty-voltage curve (see Fgure 2). Curret Desty (A/m 2 ) 4.E9 3.E9 2.E9 1.E9.E -1.E9 Nd 5x1 18 cm -3 dt dt1 dt2 dt correspodg to phoo eergy of 6meV (optcal phoo eergy rage) the value of σ dropped by 77% of the ballstc value demostratg the sgfcace of electro-phoo scatterg o the electrcal coductvty. Seebeck Coeffcet (µv/k) Calculated (2A/2A) I-plaeS(2A)/Ge(2A) Bulk S.7Ge.3 alloy S(75A)/Ge(15A) S(5A)/Ge(7A) 1E17 1E18 1E19 1E2 Dopg (cm -3 ) Fgure 3. Seebeck coeffcet vs. dopg for S/Ge/S superlattces. Expermetal data take from [11, 29, 3, 31]. -2.E9 Dra Voltage (V) Fgure 2. Curret desty - voltage characterstcs of a ballstc S(2D)/Ge(2D)/ S(2D) quatum well structure. Chael temperature dfferece rages from K to 3K. Fgure 3 shows the predcted ad measured Seebeck coeffcet values foud the lterature as a fucto of dopg levels for S/Ge/S quatum well superlattces. The agreemet betwee the calculated results ad expermets s remarkable cosderg the lmtatos of the model ad ucertaty of several expermetal parameters that wll be dscussed shortly. Fgure 4 shows the curret-voltage characterstcs of the superlattce wth scatterg. Whle the presece of dopat atoms ca lead to mpurty scatterg, as metoed the troducto, we have cosdered electro eergy-depedet ear-elastc phoo scatterg for ths study. I the scatterg model cosdered the free parameter Do was vared from very low phoo eergy of 2meV to 6meV, whch s the optcal phoo eergy lmt the case of slco, ad the I-V characterstcs were collected. Icluso of electro-phoo scatterg caused a sgfcat drop the curret coducted through the chael dcatg creased resstace to curret flow wth scatterg as see from Fgure 4. Whle the electros do ot lose ay eergy to the lattce, they experece a mometum redstrbuto whch tur creases the resstace to curret flow. Uder such crcumstaces the Seebeck coeffcet does ot chage as electros do ot udergo ay eergy chage. Our curret research efforts are focused towards a full elastc scatterg model ad the results wll be publshed as soo as they become avalable. Fgure 5 shows the electrcal coductvty σ values for coheret ear-elastc scatterg for dopg levels of 5x1 18 cm -3 ad 5x1 19 cm -3. For a Do value of.3 ev 2, A umber of factors eed to be cosdered whle comparg the expermetal data to the results calculated by the NEGF model. 1) The superlattces used the expermets cosst of 3 to 12 alteratg layers of slco ad germaum flms havg thckesses of the order of a few agstroms whle our calculatos usg the NEGF model was doe o a sgle quatum well wth flm thckess of 2D each. Whle there s a paucty of expermetal data for sgle layered S/Ge/S flms the cross-plae drecto, there s a broad varato the reported Seebeck coeffcet ad electrcal coductvty for mult-layered S/Ge/S superlattces wth thckess ad dopg levels. 2) The expermetal data for the Seebeck coeffcet ad electrcal coductvty report a spread the actual dopg values used the th flms. The effect of ths spread o the Seebeck coeffcet s evdet Fgure 3 from the plot for the calculated Seebeck coeffcet vs. dopg. It ca be see that a small chage dopg level leads to a sgfcat chage the S values demostratg the ucertaty of the reported data. 3) The calculated σ values for the ballstc as well as scatterg cases are geeral lower tha the expermetally reported values [31, 32]. It must be oted that the electrcal coductvty values used [31] are a ad-hoc estmate from the -plae electrcal coductvty values. Oe practce to calculate the cross-plae electrcal coductvty s to dvde the measured plae electrcal coductvty by the rato of the -plae/crossplae thermal coductvty [32]. Some expermets also report data measured at hgh temperatures [32] to overcome the effect of the large cotact resstace prevalet aoscale devces. Whle such data s more represetatve of the materal propertes of the superlattce, t s mportat to clude the effect of the cotacts to better uderstad the performace of superlattces as thermoelectrc devces. I the NEGF model, the effect of the cotacts s cluded through the broadeg terms Γ 1 ad Γ 2. 5

6 Curret Desty (A/m 2 ) 3.E1 2.5E1 2.E1 1.5E1 1.E1 D D.1 D.5 D.1 D.2 D.3 ca be see that the crease electrcal coductvty wth dopg s more tha compesated by the decrease electrcal coductvty due to electro-phoo scatterg. As a result, the overall power factor value dropped to 23% of the ballstc value for the S(2D)/Ge(2D)/S(2D) quatum well superlattce. 1.6E-4 1.4E-4 1.2E-4 Nd 1e18 Nd 1e19 Nd 1e2 5.E9 1.E-4.E Dra Voltage (V) S 2 σ 8.E-5 6.E-5 Fgure 4. Curret desty vs. voltage characterstcs for coheret ear-elastc scatterg for S/Ge/S superlattce for Nd 5x1 19 cm -3. 4) It s see from our smulatos that the values used for the effectve mass for slco ad germaum were foud to have a sgfcat effect o the predcted electrcal coductvty values. For our calculatos we used the effectve mass correspodg to bulk coductvty values whle the varous expermets were performed o sgle crystalle eptaxal layers deposted o a graded S x Ge 1-x substrate. For flm thckesses of the order of a few aometers as used our calculatos, both slco ad germaum ca be cosdered to be sgle crystals allowg us to use the effectve mass for that partcular oretato. I addto, the graded substrate used the expermets troduces lateral stras the slco ad germaum flms chagg the lattce costat ad the effectve mass. There s however very lmted data avalable o the chage effectve mass wth these straed ad ustraed substrates. Electrcal Coductvty (1/Ohm-m) Do (ev 2 ) Nd 5e18 Nd 5e19 Fgure 5. Electrcal coductvty (sgma) values for coheret ear-elastc scatterg for dopg levels of 5x1 18 cm -3 ad 5x1 19 cm -3. The power factor S 2 σ was calculated usg the coductace values for each level of scatterg alog wth the Seebeck coeffcet obtaed for each dopg level cosdered. Whle the electrcal coductvty creases wth dopg geeral t 4.E-5 2.E-5.E Do (ev) Fgure 5. Chage the power factor S 2 σ wth coheret ear-elastc scatterg for S/Ge/S superlattce for three dfferet dopg levels. CONCLUSIONS The o-equlbrum Gree s fucto formalsm was used to study the coupled effects of quatum cofemet ad scatterg o the thermoelectrc behavor of a sgle S(2D)/Ge(2D)/S(2D) quatum well superlattce. Whle creased cofemet s beleved to decrease the thermal coductvty due to phoo cofemet ad terface scatterg, the perceved gas due to decreased thermal coductvty do ot traslate to a correspodg mprovemet the fgure of mert ZT. Whle electro cofemet s beleved to play a mportat role decreasg the electrcal coductvty electro-phoo scatterg s also beleved to cotrbute sgfcatly to ths decrease despte the ear ballstc ature of electro trasport. Usg the oequlbrum Gree s fucto to successfully couple quatum ad scatterg effects t was see that for phoo eerges the optcal rage creased resstace due to scatterg decreased the overall power factor by 77%. ACKNOWLEDGMENTS We are grateful to Prof. Supryo Datta, Dept. of Electrcal ad Computer Egeerg, Purdue Uversty, for provdg us wth the mauscrpt of hs book o whch much of ths work s based. Ths research was fuded through a Vaderblt Dscover Grat REFERENCES [1] Trtt, T. M., Maha, G., Kaatzds, M. G., Nolas, G. S. ad Madrus, D., 21, Thermoelectrc Materals 2. The Next Geerato Materals for Small-Scale Refrgerato ad Power Geerato, Materal Research Socety,

7 [2] Hcks, L.D. ad Dresselhaus, M. S., 1993, Effect of Quatum Well Structures o the Thermoelectrc Fgure of Mert, Physcal Revew B, 47, 12, pp [3] Whtlow, L. W. ad Hrao, T., 1995, Superlattce Applcatos to Thermoelectrcty, Joural of Appled Physcs, 78, 9, pp [4] Hcks, L. D., Harma, T. C. ad Dresselhaus, M. S., 1993, Use of Quatum-well Superlattces to Obta a Hgh Fgure of Mert from No-covetoal Thermoelectrc Materals, Appled Physcs Letters, 63, 23, pp [5] Trtt, T. M., 21, Recet Treds Thermoelectrc Materals Research III, Semcoductors ad Semmetals, 71, Academc Press, Lodo. [6] Balad, A ad Wag, K. L., 1998, Sgfcat decrease of the lattce thermal coductvty due to phoo cofemet a free-stadg semcoductor quatum well, Physcal Revew B, 58, 3, pp [7] Balad, A ad Wag, K. L., 1998, Effect of phoo cofemet o the thermoelectrc fgure of mert of quatum wells, Joural of Appled Physcs, 84, 11, pp [8] Smk, M. V, ad Maha, G. D., 2, Mmum Thermal Coductvty of Superlattces, Physcal Revew Letters, 84, 5, pp [9] Che, G., 1998, Thermal Coductvty ad Ballstc- Phoo Trasport the Cross-Plae Drecto of Superlattce, Physcal Revew B, 57, 23, pp [1] Che, G., 1999, Phoo Wave Heat Coducto Th Flms ad Superlattces, Joural of Heat Trasfer, 121, pp [11] Koga, T., Cro, S. B., Dresselhaus, M. S., Lu, J. L. ad Wag, K. L., 2, Expermetal proof-of-prcple Ivestgato of ehaced Z 3D T 1 oreted S/Ge superlattces, Appled Physcs Letters, 77, 1, pp [12] Dresselhaus, M. S., 23, Naostructures ad eergy coverso, Proceedgs of 23 Rohsehow Symposum o Future Treds of Heat Trasfer. [13] Ludstrom, M. S., 2 Fudametals of Carrer Trasport, Cambrdge Uversty Press, New York. [14] Packa, P. A., 1999, Pushg the Lmts, Scece, 285, pp [15] Rahma, A., Ghosh, A ad Ludstrom, M., 23, Assessmet of Ge -MOSFETs by Quatum Smulato, Techcal Dgest of Iteratoal Electroc Devces Meetg, pp [16] Mazumder, P., Kulkar, S., Bhattacharya, M., Su, J. P. ad Haddad, G.I., 1998, Dgtal Crcut Applcatos of Resoat Tuelg Devces, Proceedgs of the IEEE, 86, 4, pp [17] Lugl, P. 199, The Mote Carlo Method for Semcoductor Devce ad Process Modelg. IEEE trasactos o computer-aded desg of tegrated crcuts ad systems, 9, 11, pp [18] Mazumder, S ad Majumdar, A., 21, Mote Carlo Study of Phoo Trasport Sold Th Flms cludg Dsperso ad Polarzato, Joural of Heat Trasfer, 123, pp [19] La, J ad Majumdar, A., 1996, Cocurret Thermal ad Electrcal Modelg of Sub-mcrometer Slco Devces, Joural of Appled Physcs, 79, 9, pp [2] Rama, A., Walker, D. G. ad Fsher, T. S., 23, Smulato of Noequlbrum Thermal Effects Power LDMOS Trasstors, Sold-State Electrocs, 47, 8, pp [21] Murthy, J. Y. ad Mathur, S. R., 22, Computato of Sub-mcro Thermal Trasport Usg a Ustructured Fte Volume Method, Joural of Heat Trasfer, 124, 6, pp [22] Aseov, A., Watlg, J. R., Brow, A. R. ad Ferry, D. K., 22, The Use of Quatum Potetals for Cofemet ad Tuelg Semcoductor Devces, Joural of Computatoal Electrocs, 1, pp [23] Tag, T. W. ad Wu, B., 24, Quatum Correcto for the Mote Carlo Smulato va the Effectve Coducto-bad Edge Equato, Semcoductor Scece ad Techology, 19, pp [24] Let, C., S. ad Krker, D., J., 199, The Quatum Trasmttg Boudary Method, Joural of Appled Physcs, 67, 1, pp [25] Laux, S., E., Kumar, A. ad Fschett, M., V., 22, Ballstc FET Modelg Usg QDAME: Quatum Devce Aalyss by Modal Evaluato, IEEE Trasactos o Naotechology, 1, 4, pp [26] Datta, S., 2, Naoscale Devce Smulato: The Gree's Fucto Formalsm, Superlattces ad Mcrostructures, 28, pp [27] Datta, S., 25, Quatum Trasport: Atom to Trasstor, Cambrdge Uversty Press, New York. [28] Szafer, A. ad Stoe, A. D., 1989, Theory of Quatum Coducto through a Costrcto, Physcal Revew Letters, 62, pp [29] Dsmukes, J. P., Ekstrom, L., Stegmeer, E. F., Kudma, I. ad beers, D. S., 1964, Thermal ad Electrcal Propertes of Hghly Doped Ge-S Alloys upto 13 K, Joural of Appled Physcs, 1, 35, pp [3] Yag, B., Lu, J. L., Wag, K. L. ad Che, G., 22, Smultaeous measuremets of Seebeck coeffcet ad thermal coductvty across superlattce, Appled Physcs Letters, 8, 1, pp [31] Yag, B., Lu, J., Wag, K. ad Che, G., 21, Characterzato of Cross-Plae Thermoelectrc Propertes of S/Ge Superlattces, Proceedgs of 2 th Iteratoal Coferece o Thermoelectrcs. [32] Yag, B., Lu, W. L., Lu, J. L., Wag, K. L. ad Che, G., 22, Measuremets of Asotropc Thermoelectrc Propertes Superlattces, Appled Physcs Letters, 81, 19, pp