Cross-plane Seebeck coefficient and Lorenz number in superlattices

Transcription

1 PHYSICAL REVIEW B 76, Cross-plae Seebeck coeffcet ad Lorez umber superlattces Z. Ba, M. Zebarjad, R. Sgh, Y. Ezzahr, ad A. Shakour Electrcal Egeerg Departmet, Uversty of Calfora, Sata Cruz, Calfora 9564, USA G. Zeg, J-H. Bahk, ad J. E. Bowers Departmet of Electrcal ad Computer Egeerg, Uversty of Calfora, Sata Barbara, Calfora 9316, USA J. M. O. Zde ad A. C. Gossard Materals Departmet, Uversty of Calfora, Sata Barbara, Calfora 9316, USA Receved 9 July 27; publshed 9 November 27 Low dmesoal ad aostructured materals have show great potetal to acheve much hgher thermoelectrc fgure of merts tha ther bulk couterparts. Here, we study the thermoelectrc propertes of superlattces the cross-plae drecto usg the Boltzma trasport equato ad takg to accout multple mbads. Posso equato s solved self-cosstetly to clude the effect of charge trasfer ad bad bedg the potetal profle. The model s verfed wth the expermetal data of cross-plae Seebeck coeffcet for a superlattce structure wth dfferet dopg cocetratos. The smulatos show that thermoelectrc propertes of superlattces are qute dfferet from those of bulk materals because the electroc bad structure s modfed by the perodc potetal. The Lorez umbers of superlattces are surprsgly large at low carrer cocetratos ad devate far away from the Wedema-Fraz law for bulk materals. Uder some codtos, the Lorez umber could be reduced by 5% compared to the bulk value. Most sgfcatly, the Seebeck coeffcet ad the Lorez umber of superlattces do ot chage mootocally wth dopg cocetrato. A oscllatory behavor s observed. The effects of temperature ad well ad barrer thckesses o the cross-plae Seebeck coeffcet ad Lorez umber are also vestgated. DOI: 1.113/PhysRevB PACS umbers: 73.5.Lw, Cd I. INTRODUCTION Thermoelectrc materals have attractg applcatos sold-state refrgerato ad drect thermal-to-electrcal eergy coverso. However, ther use s stll lmted maly due to the poor effcecy. The effcecy ca be smply measured by the dmesoless thermoelectrc fgure of mert ZT=S 2 T/K, where S s the Seebeck coeffcet, s the electrcal coductvty, K s the thermal coductvty, ad T s the absolute temperature. The dffculty fdg a good thermoelectrc materal les the terdepedece of electrcal coductvty, Seebeck coeffcet, ad thermal coductvty. There has bee reewed terest thermoelectrc materals sce t has bee proved theoretcally ad expermetally that thermoelectrc fgure of mert of a gve materal system ca be greatly mproved by usg low dmesoal ad aometer-scale structures. I the cross-plae drecto of heterostructure multlayers ad multple quatum well superlattces, the thermoelectrc power factor ca be creased by usg thermoc emsso ad flterg of hgh eergy electros. 1 6 Aother beeft of such materals s the reducto of thermal coductvty, whch has bee observed expermetally 7 9 ad explaed theoretcally by the ehaced phoo scatterg by heteroterfaces ad aopartcles The thermoelectrc charge trasport multple quatum well superlattces the cross-plae drecto has bee vestgated extesvely based o the thermoc emsso theory. 15 However, there s a heret dffculty keepg the cotuty of both electrcal curret ad Pelter heat flux ths model. Besdes, local desty of states of a average bulk medum were used whch s sutable oly whe the mea free path of charge carrers s less tha oe superlattce perod ad the coherece s lost from elastc scatterg. 16 O the other had, order to utlze the thermal coductvty mma of superlattce materals, the superlattce perod should be aroud 1 m or less, whch s usually less tha the mea free path of electros. Oly a few papers have used the desty of states of superlattces, but the very frst lower mbads were cosdered the calculatos. 16,17 Ths s ot eough for thermoelectrc trasport because the optmal posto of the Ferm eergy s wth k B T from the barrer bad edge ad may hgher mbads should be also cluded especally for tall barrer ad hghly doped superlattces. I a recet paper, multple mbads were cocluded; however, sce oly two perods of superlattces were used, the mbad dsperso relato could ot be costructed ad evetually the varato of group velocty for dfferet states was gored. 18 Furthermore, a fxed barrer heght was assumed the above lteratures, although the bad edge profle ad mbad structure chage wth temperature ad dopg cocetrato due to the charge redstrbuto space. Due to the complete modelg, the electroc thermal coductvtes of superlattces were usually estmated roughly accordg to the Wedema-Fraz law of bulk materals. 9 I ths paper, we wll solve the coupled Schrödger ad Posso equatos self-cosstetly ad calculate the dsperso relato for each mbad. Whe multple mbads are cluded electrothermal trasport calculatos, we fd dfferet behavors of Seebeck coeffcet ad Lorez umber compared to bulk materals whe the temperature chages. The cross-plae Lorez umber of superlattces s surprsgly hgh at low carrer cocetratos, whch dstgushes tself from the Wedema-Fraz law of /27/762/ The Amerca Physcal Socety

2 BIAN et al. bulk materals. The oscllatory behavor of the Seebeck coeffcet ad the Lorez umber as fuctos of dopg cocetrato ad factors, whch are affectg the oscllatos, are dscussed. II. MODELING We calculated the electroc bad structure of superlattces usg the evelope-fucto method ad the Krog- Peey model oe-dmesoal crystal. 19,2 Oe superlattce perod d s segmeted to small sze elemets. I each elemet, the potetal s assumed to be costat ad the wave fucto soluto of the Schrödger equato ca be wrtte as E Vz: z = A exp z + B exp z, E Vz: z = A s z + B cos z, 1 where Vz s the potetal ad E s the egeeergy of electro the cross-plae drecto. For parabolc bad, the wave umbers are 2 = 2m* 2 Vz E, 2 = 2m* 2 E Vz. 2 If the ketc eergy E k =E Vz s much larger tha zero, the oparabolcty of bad dsperso ca be cluded by replacg the effectve mass Eq. 2 wth m * 1+E k, where s the oparabolcty factor. The wave fuctos adjacet elemets are coected at the terface by the cotuty of wave fuctos ad currets. Accordg to Bloch s theorem for perodc structures, wave fuctos at the start ad at the ed of oe perod are dfferet oly by a phase coeffcet z + d = zexpq d, 3 where q s the Bloch wave vector satsfyg the perodc boudary codto for N perods of the superlattces: q Nd = 2, 4 where s a teger. For each quatzed q the frst Brllou zoe, the electro eergy E alog the cross-plae drecto s vared to fd the egeeergy E ad egefucto each mbad whch satsfy Eqs Now, we revse the dces of eergy ad wave fuctos to reflect the mbad level ad the Bloch wave vector. The Ferm eergy ad charge dstrbuto are foud through charge eutralty equato,.e., the total surface dopg desty the x-y plae s equal to the tegral of the charge dstrbuto alog the z drecto, whch s descrbed by z = m* 2 z 2 f E + E 1+2E de, 5 where E s the ketc eergy the plae ad f s the Ferm-Drac dstrbuto. For parabolc bad =, the tegral the plae has a aalytcal soluto ad the charge dstrbuto s reduced to z = PHYSICAL REVIEW B 76, m * k B T 2 l1 + exp E f E z 2. 6 k B T Subsequetly, the bad bedg ca be calculated from the Posso equato usg above charge dstrbuto: d dzz dzz d = ez N + d z, 7 where e s the electro charge, ad N + d z s the cocetrato of ozed dopat. Here, for type materals, the cotrbuto of holes s eglected. The bad bedg s the used to update the total potetal profle Vz = ez + E c z, 8 where E c s the coducto bad offset. The ew potetal profle chages the soluto of the mbad structure accordg to Eqs Thus, teratos are performed utl both the potetal ad mbad structures coverge. After solvg the mbad structures, the group velocty of each egestate alog the cross-plae drecto ca be calculated from the dsperso curves: E = 1 9 q The electrcal coductvty, the Seebeck coeffcet S, ad the electro cotrbuto to thermal coductvty K e of bulk semcoductor materals ca be derved from the Boltzma trasport equato wth the relaxato tme approxmato. They are all tegral fuctos of the dfferetal coductvty 21 d E = e 2 E 2 z EE f E, 1 E where s the electro mometum relaxato tme ad s the desty of states of coducto bad. By decouplg the desty of states of -plae ad cross-plae drectos, the above equato ca be modfed to the oe applcable to asotropc superlattces: E = e 2 m* 21+2E E + E 2 E f E + E. 11 E + E The cotrbutos of all the electro states to, S, ad K e are couted by cotuous tegrals the -plae drecto ad summatos of dscrete states of multple mbads the cross-plae drecto: = de, S = 1 E + E E f de et 12 de,

3 CROSS-PLANE SEEBECK COEFFICIENT AND LORENZ PHYSICAL REVIEW B 76, K e = 1 et 2 E + E 2 de.4.2 E + E de 2 de. 14 For -type materals, the Seebeck coeffcet s usually wth a egatve sg because the electro charge s egatve ad the average eergy of electros cotrbutg to charge trasport s larger tha the Ferm eergy. The sg of Seebeck coeffcet mght chage whe the Ferm level moves across submbads of low dmesoal materals due to the oscllato of desty of states. I the followg calculatos, we do ot see ay postve sg of Seebeck coeffcet because the quatum cofemet ad eergy quatzato are oly the cross-plae drecto. For coveece, the term Seebeck coeffcet deotes ts ampltude the followg text ad ts sg s take as egatve by default. Accordg to the Wedema-Fraz law, the rato of electroc thermal coductvty to electrcal coductvty of a metal s the product of the temperature ad a costat 2 /3k B /e 2 called Lorez umber. The utless Lorez umber ca be defed as L = e k B2 K e T, 15 whch s reduced to 2 /3 for metal. Smlarly, the utless Lorez umber ca be calculated from Boltzma trasport ad electroc bad structures of semcoductor materals. It ca be show that for bulk semcoductors, the utless Lorez umber L depeds o the major mechasm of electro or hole scatterg ad chages slghtly aroud 3 wth the carrer desty whe polar optcal phoo scatterg ad mpurty scatterg domate. 21 The utless Lorez umber wll be vestgated the followg secto. III. RESULTS E (ev) (a) E (ev) (b) Ec Ef Ev Posto (m) Wavevector q (/m) x1 8 FIG. 1. Color ole a The bad profle of oe superlattce perod. b The dsperso curves of mbads. IGaAs/ IGaAlAs superlattces lattce matched to IP substrate ad embedded wth ErAs semmetallc aopartcles were grow wth the molecular beam eptaxy method. Ths s a flexble materal system because the well-barrer coducto bad offset ca be adjusted by chagg the relatve composto of Ga ad Al the barrer. Whe the Ga:Al rato s 6%:4%, the coducto bad offset s about.2 ev. The measuremets showed creased Seebeck coeffcet ad reduced thermal coductvty. 5,6,9 I the prevous work, there were otceable dscrepacy betwee modelg ad expermet ad the electro cotrbuto to the total thermal coductvty was estmated accordg to the Wedema-Fraz law of bulk materals. I the followg, we focus o the calculato of thermoelectrc propertes based frmly o multple mbad trasport. For smplcty ad order to study maly the effect of superlattce mbads, we wll ot clude the electro scatterg of ErAs aopartcles. The IGaAs wells are assumed doped uformly whle the IGaAlAs barrers are ot doped. Whe there s o exteral electrcal feld ad temperature gradet, the Ferm level teds to be the same everywhere at equlbrum. Thus, there s charge trasfer from well regos wth hgher Ferm level tha barrer regos before charge trasfer to barrer regos ad the resultg buld- feld modfes the bad profle. The bad profle affects the quatzed levels ad dsperso curves of mbads. Ths also modfes how these mbads are flled wth electros ad chages the Ferm level ad the charge redstrbuto space. So a self-cosstet calculato s eeded for the smulato of the mbad structure ad the Ferm level. It s terestg to ote that besdes dopg cocetrato, the temperature chage also affects the bad profle for a gve superlattce structure. The bad profle ad mbad structures calculated from a self-cosstet soluto of the coupled Schrödger ad Posso equatos are show Fg. 1 for superlattces wth well wdth 2 m, barrer wdth 1 m, ad coducto bad offset.2 ev. The temperature s 3 K ad the well regos are -type doped wth a total electro cocetrato 1 19 cm 3. It ca be see that the effectve barrer heght becomes larger due to the charge trasfer from the wells to the barrers. The ampltude of group velocty s larger for a hgher mbad, dcatg the electro eergy flterg whch s cosstet wth the thermoc emsso model

4 BIAN et al. At room temperature ad above, polar optcal phoo scatterg ad ozed mpurty scatterg are the two major electro scatterg mechasms IGaAs ad IGaAlAs alloys. The electro mometum relaxato tme due to polar phoo scatterg s 22 1 = e2 / 1 p 4 2E/m *N 1+ E + N +11 E N E + N +1 E sh 1 E 1/2 sh 1 E 11/2, 16 where s the agular frequecy of the optcal phoo, ad are the statc ad hgh frequecy delectrc costats, ad N s the umber of optcal phoos gve by Bose- Este statstcs: Seebeck coeffcet (µv/k) (a) PHYSICAL REVIEW B 76, fxed barrer self-cosstet Dopg cocetrato (cm -3 ) N = 1 exp /k B T The electro mometum relaxato tme due to ozed mpurty scatterg s 22 = 16 2m 2 * 2 2 N 1 e 4 l E 3/2, where N 1 s the cocetrato of ozed mpurtes, ad 2 8m * EL 2 D / 2. The Debye screeg legth L D s k B T/e 2 1/2 for odegeerate semcoductors, ad 2 E f /3e 2 1/2 for degeerate semcoductors, where s the charge carrer cocetrato. The electro mometum relaxato tme used Eq. 11 ca be calculated by =1/ p +1/ 1. Wth the chages of barrer wdth ad Al composto barrers, the scatterg rate should be chaged accordgly for a accurate calculato. Approxmately, the electro mometum relaxato tme of bulk IGaAs wth =13.9, =11.6, ad =34 mev are used the subsequet calculatos of thermoelectrc trasport coeffcets. Frst, we calculate the Seebeck coeffcet ad the Lorez umber of the superlattces structure descrbed above as fuctos of dopg cocetrato the well regos. I Fg. 2, we compare the results for two cases: fxed barrer heght.2 ev ad self-cosstet solutos. The comparso of these two cases has two purposes: oe s to verfy the self-cosstet calculato, ad the other s to study the effects of charge trasfer ad bad bedg. It ca be see Fg. 2a that the Seebeck coeffcet for the case of self-cosstet soluto s larger tha that for a fxed barrer heght. Ths ca be explaed by the charge trasfer ad bad bedg ad the resultg crease of the effectve barrer heght for electro eergy flterg. It s also terestg to see Fg. 2b that although there s some slght dfferece utless Lorez umber for the fxed barrer ad self-cosstet solutos of superlattces, they have the same tred wth the chage of carrer cocetrato. Surprsgly, we fd that the Lorez Utless Lorez umber (b) fxed barrer self-cosstet Dopg cocetrato (cm -3 ) FIG. 2. Color ole a The Seebeck coeffcets ad b the Lorez umbers for the case of fxed barrer heght.2 ev ad the case of self-cosstet solutos cludg bad bedg. umber ca be much larger tha 3 at low dopg cocetratos, whch s clearly dfferet from bulk materals ad caot be predcted tutvely. It s dffcult to verfy the modelg wth expermets because t s very challegg to measure the thermoelectrc propertes of th flms the cross-plae drecto very accurately. The curret or temperature uformty ad thermal leakage are bg ssues for thermoelectrc measuremets of th flms due to ther hgh aspect ratos of devce area to flm thckess. Parastc effects such as electrcal ad thermal cotact resstaces are typcally aoyg extractg trsc propertes of the thermoelectrc th flm because ther ampltudes ad ther varatos from devce to devce are ot eglgble. To measure the cross-plae Seebeck coeffcet of IGaAs/ IGaAlAs superlattces, mesa structures were formed usg covetoal mcroelectrocs processg techques ad mcroheaters were tegrated o the top of the devces to create the temperature dfferece the crossplae drecto. The detals of the devce structures were descrbed the Fg. 1 of Ref. 6. We measured the Seebeck coeffcet of superlattce materals usg two dfferet methods. Oe s at Uversty of Calfora Sata Barbara SB whch the temperature s measured at steady state by a m

5 CROSS-PLANE SEEBECK COEFFICIENT AND LORENZ Seebeck coeffcet (µv/k) modelg expermet-sb expermet-sc Dopg cocetrato (cm -3 ) FIG. 3. Color ole The comparso of the modelg ad expermetal data of the cross-plae Seebeck coeffcet for IGaAs/ IGaAlAs superlattces. crothermocouple ad the Seebeck coeffcet of superlattces s extracted usg the calbrated property of IP substrate ad the fte elemet thermal aalyss of the temperature dstrbuto across multple layers. 6 The secod measuremet s doe at Uversty of Calfora Sata Cruz SC usg the 3 method. A susodal curret at frequecy s set to the heater, the Seebeck voltage across the actve semcoductor materals s measured at 2, ad the temperature across the whole structure s calculated from the rato of 3 to 1 compoets of the heater voltage. A separate calbrato s doe to extract the temperature depedece of the heater resstace. The same electrcal power s appled to the heater o the top of a referece devce wth the same geometry but wthout the th flm layer. The the et temperature ad Seebeck voltage across the top th flm layer ca be extracted from the total by deductg the substrate ad other parastc cotrbutos usg the measuremet data of referece devce. So there s o eed of modelg of temperature drop across the electrcal sulato layer ad thermal terface. The the Seebeck coeffcet of superlattces s calculated from the rato of the et Seebeck voltage to the et temperature drop across the th flm. I Fg. 3, the measured Seebeck coeffcets usg the two methods descrbed above are plotted together wth the modelg. Compared to the plot Fg. 2a for the case of self-cosstet soluto, oparabolc bad dsperso s added here the modelg for a more accurate aalyss. It ca be see that they agree wth each other very well eve wthout ay fttg parameters the modelg. The slght dscrepacy betwee modelg ad expermets at low carrer cocetratos mght be due to eglectg ErAs partcle scatterg of electros, whch s comparable to mpurty scatterg at low dopg cocetratos. Now, we assume the coducto bad s parabolc the followg dscussos ad focus o how the superlattce perod ad the potetal profle ca modfy the thermoelectrc trasport. We expect that the temperature depedece of the thermoelectrc propertes of superlattces ca be dfferet from bulk materals because of the relatve algmet of Seebeck coeffcet (µv/k) (a) Utless Lorez umber (b) e e e e e e Temperature (K) PHYSICAL REVIEW B 76, e e e e e e19 Degeerate bulk Nodegeerate bulk Temperature (K) FIG. 4. Color ole The temperature depedeces of a the Seebeck coeffcet ad b the Lorez umber of superlattces the cross-plae drecto. mbads, Ferm eergy, ad potetal barrers. Ths s demostrated Fg. 4, where the temperature depedeces of Seebeck coeffcets ad Lorez umbers are plotted for dfferet dopg cocetratos for the above superlattces structure. The Seebeck coeffcet for the two small dopg cocetratos creases frst ad decreases later whe the temperature creases. Oe should ote that crease temperature has two effects o electro trasport: oe s the broadeg of Ferm wdow f /E, ad the other s the crease of Ferm eergy. The former ca expla the crease of Seebeck coeffcet wth temperature. The latter s resposble for the decrease of Seebeck coeffcet wth temperature at low carrer cocetratos ad hgh temperatures regme where the Seebeck coeffcet s sestve to the posto of Ferm eergy. It ca also be see that compared to the costat values of the Lorez umbers of degeerate ad odegeerate bulk semcoductors, the Lorez umber of superlattces chages qute a lot wth temperature. The Lorez umbers are surprsgly large for low dopg cocetratos at low temperatures, whch the Ferm eergy s deep the wells. These observatos dcate a sgfcat devato from the Wedema-Fraz law cross-plae drecto of superlattces. For bulk materals, the product of the three terms v z 2 EE f E/E makg up the dfferetal coduc

6 BIAN et al. PHYSICAL REVIEW B 76, Seebeck coeffcet (µv/k) Seebeck coeffcet Lorez umber Lorez umber Seebeck coeffcet (µv/k) well 6m, barrer 3m, 3K well 6m, barrer 5m, 3K well 6m, barrer 8m, 3K well 6m, barrer 1m, 3K well 6m, barrer 3m, 5K well 2m, barrer 3m, 3K Dopg cocetrato (cm -3 ) FIG. 5. Color ole The Seebeck coeffcet ad the utless Lorez umber of a short perod superlattce structure well 6 m, barrer 3 m, at 3 K. tvty see Eq. 1 becomes more symmetrc relatve to the Ferm eergy so that the Seebeck coeffcet usually decreases mootocally whe the dopg cocetrato ad Ferm eergy crease. For superlattces, however, ths product could become more asymmetrc because of the step crease of desty of states whe the Ferm eergy moves up ad approaches a hgher mbad. The resultg omootoc behavor of the cross-plae Seebeck coeffcet versus dopg was observed expermetally ad explaed by the mbad formulato ad a modfed Boltzma trasport model. 18 However, oly the electro wave couplg betwee two adjacet perods was used to calculate the resoat trasmsso of electros ad the dsperso curves of mbads were ot cluded the aalyss. Accordgly, the group veloctes were ot calculated from the dsperso curves ad the dfferece betwee dfferet mbads could ot be cosdered. As we have dscussed, t s ecessary to use dfferet group veloctes for dfferet mbads because a relatvely larger group velocty a hgher mbad s equvalet to the electro eergy flterg whch ca crease the Seebeck coeffcet. Furthermore, the strog modfcato group velocty has a bg mpact o the oscllatory behavor of Seebeck coeffcet versus dopg. I Fgure 5, we plot the Seebeck coeffcet ad the utless Lorez umber at 3 K usg the curret model for a short perod superlattce structure well wdth 6 m, barrer wdth 3 m. Both the Seebeck coeffcet ad the Lorez umber oscllate wth dopg cocetrato. Although the cross-plae Seebeck coeffcet has bee aalyzed ths paper ad prevous lteratures oly for short perod superlattces, accordg to the above dscussos we ca fd that the strog couplg betwee adjacet wells ad tur a wde mbad formato are ot ecessary for the omootoc behavor of Seebeck coeffcet. I fact, a thcker barrer ad the formato of arrow mbads ca crease oscllatos of the Seebeck coeffcet versus dopg because the chage of desty of states wth eergy s more abrupt ths case. The group velocty of a lower mbad s reduced more tha a hgher oe f the barrer s thcker. Ths Dopg cocetrato (cm -3 ) FIG. 6. Color ole The cross-plae Seebeck coeffcet versus dopg for several structures wth dfferet well thckesses, barrer thckesses, ad temperatures. causes the Seebeck coeffcet to crease more whe the dopg creases. However, f the barrer s too thck ad the trasmsso of lower mbads s eglgble, the oscllato of Seebeck coeffcet wth dopg ca dsappear. Ths s because the coductve hgher mbads are more cotuous ad the desty of states chages smoothly wth eergy. Ths ca be see Fg. 6, whe the barrer thckess s chaged from 3 m to 5, 8, ad 1 m, respectvely. Besdes the barrer thckess, there are other factors affectg the oscllatory behavor of the cross-plae Seebeck coeffcet. For example, f the temperature creases, the Ferm wdow f /E becomes wder ad blurs the dfferetal coductvty, whch wll suppress the oscllato of Seebeck coeffcet. Ths ca be see by the comparso of 3 ad 5 K Fg. 6, whle keepg well 6 m thck ad barrer 3 m thck. Equvaletly mportat, the well thckess determes the spacg of mbads ad cotuty of desty of states. Thcker wells gve smaller mbad spacg ad suppress the oscllato of cross-plae Seebeck coeffcets. Ths ca be see Fg. 6 whe oly the well thckess s chaged from 6 to 2 m. IV. SUMMARY We vestgated the Seebeck coeffcet ad Lorez umber of superlattces the cross-plae drecto. Charge trasfer ad bad bedg were cluded by a self-cosstet soluto of coupled Schrödger ad Posso equatos. The thermoelectrc propertes were the calculated accordg to the dsperso curves of multple mbads. The modelg of Seebeck coeffcet fts wth expermetal data very well for a specfc superlattce structure at varous dopgs. I geeral, the smulato shows that the Seebeck coeffcet of superlattces the cross-plae drecto has dfferet temperature depedece compared to bulk materals. The Lorez umber of superlattces the cross-plae drecto ca be larger tha that predcted by the Wedema-Fraz law by a factor of 2. I some cases, the superlattce Lorez umber s

7 CROSS-PLANE SEEBECK COEFFICIENT AND LORENZ reduced by 5%. The oscllato of Seebeck coeffcet ad Lorez umber wth the chage of dopg cocetrato s explaed usg the multple mbad trasport model. A low temperature, a small well thckess, ad a moderately thck barrer are favorable to crease the oscllatos of cross-plae Seebeck coeffcet. PHYSICAL REVIEW B 76, ACKNOWLEDGMENTS Ths work was supported by the Offce of Naval Research through MURI Thermoc Eergy Coverso Ceter program maaged by Mhal Gross. Z.B. would lke to thak K. Esfarja ad W. Km for frutful dscussos. 1 A. Shakour ad J. E. Bowers, Appl. Phys. Lett. 71, G. D. Maha ad L. M. Woods, Phys. Rev. Lett. 8, Z. Ba ad A. Shakour, Appl. Phys. Lett. 88, D. Vashaee ad A. Shakour, Phys. Rev. Lett. 92, J. M. O. Zde, D. Vashaee, Z. X. Ba, G. Zeg, J. E. Bowers, A. Shakour, ad A. C. Gossard, Phys. Rev. B 74, G. Zeg, J. M. O. Zde, W. Km, J. E. Bowers, A. C. Gossard, Z. Ba, Y. Zhag, A. Shakour, S. L. Sger, ad A. Majumdar, J. Appl. Phys. 11, R. Vekatasubramaa, E. Svola, T. Colptts, ad B. O Qu, Nature Lodo 413, T. C. Harma, P. J. Taylor, M. P. Walsh, ad B. E. LaForge, Scece 297, W. Km, S. Sger, A. Majumdar, D. Vashaee, Z. Ba, A. Shakour, G. Zeg, J. E. Bowers, J. M. O. Zde, ad A. C. Gossard, Appl. Phys. Lett. 88, M. V. Smk ad G. D. Maha, Phys. Rev. Lett. 84, R. Vekatasubramaa, Phys. Rev. B 61, G. Che, Phys. Rev. B 57, Y. Che, D. L, J. R. Lukes, Z. N, ad M. Che, Phys. Rev. B 72, W. Km, J. Zde, A. Gossard, D. Kleov, S. Stemmer, A. Shakour, ad A. Majumdar, Phys. Rev. Lett. 96, D. Vashaee ad A. Shakour, J. Appl. Phys. 95, L. W. Whtlow ad T. Hrao, J. Appl. Phys. 78, Z. Tao ad L. Fredma, J. Phys. C 18, L D. Vashaee, Y. Zhag, A. Shakour, G. Zeg, ad Y. J. Chu, Phys. Rev. B 74, E. L. Ivcheko ad G. E. Pkus, Superlattces ad other Heterostructures Sprger-Verlag, Berl, A. M. Cruz Serra ad H. Abreu Satos, J. Appl. Phys. 7, G. S. Nolas, J. Sharp, ad H. J. Goldsmd, Thermoelectrcs: Basc Prcples ad New Materals Developmets Sprger-Verlag, Berl, M. Ludstrom, Fudametals of Carrer Trasport Cambrdge Uversty Press, Cambrdge,