Chapter 1. Fundamentals of Thermoelectrics

Transcription

1 Chater Fudametals of Thermoelectrics Thermoelectric coolig is the direct coversio of electrical voltage to temerature differece, i which electros or holes act as the workig fluids to carry eergy from heat source to sik uder a exteral electric field. This chater discusses thermoelectric effects ad trasort roerties o both macroscoic ad microscoic levels. It starts with a itroductio to thermoelectrics, which defies three basic thermoelectric effects: the Seebeck effect, Peltier effect ad Thomso effect, ad their corresodig coefficiets. The Peltier effect is the ricile at work behid thermoelectric coolig. It is followed by the aalysis of coolig erformace ad figure of merit ZT of sigle- ad multi-stage thermoelectric modules. Thermoelectric trasort roerties i bulk materials, icludig the Seebeck coefficiet, electrical resistivity, electro ad hoo thermal coductivities, have bee derived usig the Boltzma trasort equatios. Naostructured materials rovide additioal arameter sace that ca be egieered to steer the thermoelectric roerties for a ehaced figure of merit ZT. Electro ad hoo thermoelectric trasort i aostructures have also bee discussed i the last art of this chater. Cotets. Thermoelectric Effects..... Seebeck Effect Peltier Effect Thomso Effect Kelvi Relatio Joule Heatig ad Heat Coductio Pricile of Thermoelectric Coolig Thermoelectric Coolig ad Figure of Merit Effects of Electrical Cotact Resistace Effects of Temerature-Deedet Proerties Performace of Multistage Thermoelectric Coolers Other Forms of Figure of Merit Thermoelectric Trasort Proerties i Bulk Materials Electro Trasort Proerties Phoo Trasort Proerties Thermoelectric Figure of Merit Thermoelectric Trasort i Naostructured Materials... 3

2 Thermoelectric Microcoolers.4.. Formatio of Eergy Bad i Naostructured Materials Icoheret Electro ad Phoo Trasort: Classical Size Effects Coheret Electro ad Phoo Trasort: Quatum Size Effects Trasort i Partially Coheret Regime Summary Refereces Thermoelectric Effects Thermoelectric coolig is the direct coversio of electrical voltage to temerature differece, i which electros or holes act as the workig fluids to carry eergy from oe lace to aother uder a exteral electric field. There exist three basic thermoelectric effects: the Seebeck effect, Peltier effect, ad Thomso effect. These three effects describe thermodyamically reversible rocesses, i which o etroy is geerated. Other ievitable effects i thermoelectric rocess iclude Joule heatig ad thermal coductio, which lower the erformace of thermoelectric devices to less tha the thermodyamic limit, i.e., Carot efficiecy... Seebeck Effect The Seebeck effect describes the heomeo of voltage geeratio whe a material is subject to a temerature gradiet uder the oe circuit coditio. This effect was discovered by the Estoia hysicist Thomas Joha Seebeck i 8. A alied temerature gradiet causes charged carriers (electros or holes) to diffuse from the hot to cold side i the material, as show i Fig..a. At the steady state, the iteral field (Seebeck voltage), V, geerated from the differece i charged carrier cocetratio, balaces the drive force for thermal diffusio. The absolute Seebeck coefficiet for a idividual material is defied as V S =, (.) T where V ad T are the voltage ad temerature differece betwee the hot ad cold sides of the material, resectively. The Seebeck coefficiet has a uit of V/K, but microvolt er Kelvi ( / ) is ofte used for thermoelectric materials. The Seebeck coefficiet is sometimes called the thermal electromotive force (emf) coefficiet or thermoelectric ower. It is the ricile at work behid

3 Fudametals of Thermoelectrics 3 thermoelectric ower geeratio which coverts temerature differece to electricity. A voltage ca be established betwee two juctios of two differet materials A ad B if there is a temerature differece betwee them, as show i Fig..b. This cofiguratio is a basic thermocoule. The Seebeck coefficiet of the A-B coule is determied by V ( S A S B ) = (.) T This equatio rovides a way to determie a ukow Seebeck coefficiet from a kow oe. The absece of Seebeck effect for suercoductors has also made it ossible to defie a absolute Seebeck coefficiet for idividual materials. The Seebeck coefficiet S tyically has a ositive value whe the electrical carriers are holes, that is, the voltage differece is i the oosite directio of the temerature gradiet. The Seebeck coefficiet S is egative for materials with egative majority carriers, electros. I geeral, the Seebeck coefficiets are oliear as a fuctio of temerature, ad deed o the materials' absolute temerature, chemical comositio, ad molecular structure. Metals could have Thermal diffusio Iteral electric field V T cold V T hot Temerature Gradiet (a) S A T cold T hot S B V (b) Fig... Seebeck effect i a sigle material (a) ad a circuit cosistig of two differet materials A ad B (b).

4 4 Thermoelectric Microcoolers ositive or egative Seebeck coefficiets, deedig o their electroic bad structures. 3 The magitude of the Seebeck coefficiet is small i metals, tyically less tha 0 microvolt er Kelvi, due to the fact that most metals have halffilled electroic bads ad therefore the thermoelectric voltages iduced by electros ad holes cacel each other to some degree. I semicoductors, the Seebeck coefficiet varies with doig. The -tye semicoductors have egative Seebeck coefficiet while the -tye oes are ositive. The magitude of the Seebeck coefficiet ca be very large i semicoductors, e.g., larger tha 00 microvolt er Kelvi i Bi Te 3 based alloys,, 4 due to their excessive electros or holes geerated by doig... Peltier Effect The Peltier effect is the reverse of the Seebeck effect, which was discovered by Frech hysicist Jea Peltier i The Peltier effect measures the heat carried by electrical carriers (electros or holes), as show i Fig..a. The Peltier heat eters from oe ed of the material ad is released at the other ed. The absolute Peltier coefficiet of a material defies the relatioshi betwee the heat ower Q ad the electric curret I throughout the material at a costat temerature, Q = I Π, (.3) where Π is the absolute Peltier coefficiet of a idividual material. For a circuit cosistig of two differet materials A ad B, as show i Fig..b, the electric curret must be cotiuous across the juctios; the imbalace of the heat i ad out at each juctio creates coolig or heatig effects. The Peltier heat Q absorbed at the juctio is equal to Q = Π AB I = ( Π A Π B ) I, (.4) where I is the electric curret, Π AB is the Peltier coefficiet of the juctio betwee the materials A ad B, ad Π A ad Π B are the absolute Peltier coefficiet of the materials A ad B, resectively. The absolute value of the Peltier coefficiet for a idividual material ca be determied whe oe of the two braches is a suercoductor. The directio of heat flow at the juctio, i.e., coolig or heatig, is thus determied by the choice of materials ad the directio of the assig electric curret. The Peltier coefficiet has a uit of volt. It varies with the material's temerature ad secific comositio. For examle, -tye

5 Fudametals of Thermoelectrics 5 semicoductors tyically have ositive Peltier coefficiet, but -tye semicoductors tyically have egative oe. Electric charge flow, I Heat flow, Q (a) I, Q A Q Q I, Q B (b) Fig... Peltier effect i a sigle material (a) ad a circuit cosistig of two differet materials A ad B (b). Thermoelectric coolig devices exloit this heomeo, as do thermoelectric heat ums. I these devices, the directio of heat trasfer ca be cotrolled by the olarity of the electric curret; reversig the electric olarity will chage the directio of trasfer ad thus the sig of the heat absorbed/evolved. A thermoelectric module usually cosists of may idividual coules that are coected i series to augmet the coolig/heatig effect...3. Thomso Effect The Thomso effect describes the heatig or coolig heomeo alog a material which a electric curret is assig through ad is subject to a temerature gradiet, as show i Fig..3. This effect was discovered by Thomso (Lord Kelvi) i The Thomso coefficiet τ is defied as

6 6 Thermoelectric Microcoolers dq dx dt = τ I (.5) dx where dq/dx is the rate of the heatig er uit legth, I is the electric curret, ad dt/dx is the temerature gradiet alied to the material. The directio of heat flow dq is determied by the choice of materials, the directio of the electric curret, ad the temerature gradiet. The Thomso coefficiet τ ca be ositive, egative, or zero. For examle, i metals such as cobalt, ickel, ad iro, there is absortio of heat whe the temerature is iversely roortioal to the electrical otetial ad the electric curret moves from the hot to the cold ed. So these materials have egative Thomso coefficiets. The Thomso effect is tyically small comared to other heatig or coolig effects such as Peltier coolig ad Joule heatig, ad ofte eglected i device aalysis. dq T cold Electric charge flow, I T hot Fig..3. Thomso effect i a sigle material...4. Kelvi Relatio The aforemetioed three thermoelectric effects are due to flow of electros or holes ad their iteractio with the lattice ad thus the thermoelectric coefficiets are iterrelated. Thomso (Lord Kelvi) foud relatioshis betwee the three coefficiets i 854, imlyig that oly oe could be cosidered uique. 7 The first Kelvi relatio gives relatio betwee the Thomso coefficiet ad the temerature derivative of the Seebeck coefficiet, ds τ = T, (.6) dt where T is the absolute temerature i Kelvi, τ is the Thomso coefficiet, ad S is the Seebeck coefficiet. The secod Kelvi relatio is Π = ST, (.7)

7 Fudametals of Thermoelectrics 7 where Π is the Peltier coefficiet. These two equatios rovide a fudametal lik betwee thermoelectric coolig ad ower geeratio. The Kelvi relatios ca be derived rigorously usig irreversible thermodyamics...5. Joule Heatig ad Heat Coductio Joule heatig ad heat coductio are two irreversible rocesses that lower the erformace of thermoelectric devices to less tha the thermodyamic limit, i.e., Carot efficiecy. The Joule heatig Q is give by Q=I R, (.8) where I is the electric curret ad R is the electrical resistace. Because Joule heatig is roortioal to the square of the electric curret while Peltier coolig is oly liear with electric curret =Π I, oe caot icrease the temerature gradiet idefiitely by icreasig the electric curret. The heat coductio for a material uder a temerature gradiet is give by dt Q = Ak, (.9) dx where A is the cross-sectio area of the material, k is its thermal coductivity, ad dt/dx is the temerature gradiet. I ay material, a temerature gradiet leads to a irreversible heat flow which ooses the temerature gradiet. A good thermoelectric material must have a large Seebeck coefficiet, a low electrical resistivity ad a low thermal coductivity to reduce the effects of Joule heatig ad heat leakage by coductio... Pricile of Thermoelectric Coolig... Thermoelectric Coolig ad Figure of Merit A thermoelectric thermocoule cosists of a -brach with a ositive Seebeck coefficiet ad a -brach with a egative Seebeck coefficiet, as show i Fig..4. These two braches are joied by a metal itercoect which forms a lowresistivity ohmic cotact. A thermoelectric device is tyically made of multile thermocoules that are coected such that the electric curret flow is i series while the heat flow is i arallel. The reaso for the use of both ad braches is because their Seebeck ad Peltier coefficiets are of oosite sig, such that both braches cotribute to the desired thermoelectric effect. These braches are electrically i series due to the very small electrical resistace of each brach. Desite such a cofiguratio, the total electrical resistaces of bulk

8 8 Thermoelectric Microcoolers thermoelectric devices are still small, ad these devices usually oerate at electric curret o the order of ams. The coolig erformace of a thermocoule show i Fig..4 is aalyzed i the followig sectios. The objective of this aalysis is to fid the coefficiet of erformace of the thermoelectric device as a fuctio of the temerature differece betwee the source ad the sik. As see i Fig..4, electric curret is assed from the - to -tye braches such that electros i the -tye brach ad holes i the -tye brach both move away from the cold juctio to the hot juctio, thus carryig heat from heat source to sik. The Peltier coolig is give by (S S )TI. I additio to the Peltier coolig, there is heat leakage from the hot juctio to the cold juctio due to heat coductio i both ad -tye braches. A certai fractio of Joule heat geerated iside the ad -tye braches is also coducted back to the cold juctio. The coolig ower withi each brach is ad dt Q = S TI k A, (.0) dx x=0 dt Q = S TI k A, (.) dx x=0 + Heat Source Heat Source (Cold Juctio) eee Heat Sik hhh Heat Sik (Hot Juctio) Heat Sik x=0 x=l _ Fig..4. Schematic of a thermocoule cosistig of - ad -tye braches. 8

9 Fudametals of Thermoelectrics 9 dt where A is the sectio area of each brach, is the temerature gradiet, ad dx k is the thermal coductivity. The total coolig ower Q C is give by: Q = ( Q + Q ). (.) C Uder the boudary coditios i which the cold ad hot sides are at T C ad T H, resectively, the total coolig ower of the thermoelectric coule Q C reduces to Q C x= 0 ( S S ) I T K( T T ) I R = C H C, (.3) where the total thermal coductace K (arallel arragemet) ad electrical resistace R (series arragemet) are ad k A k A K + L L R =, (.4) L ρ L ρ + A A =. (.5) The factor ½ i Eq..3 imlies that half of Joule heat coducts back to the cold side. The electric ower W cosumed by the thermocoule icludes two arts: the Joule heatig ad the work agaist the Seebeck voltage, ( S S )( T T )I W = I R +. (.6) The coefficiet of erformace φ for refrigeratio or coolig is give by ( S S ) I T K( T T ) ( S S )( T T ) I + I R H H Q C H C I R C φ = =. (.7) W If there were o irreversible Joule heatig ad heat coductio, oe would fid TC φ =, (.8) T T H which is the Carot efficiecy for coolig. For the coolig alicatio, three secial cases are of iterest: ). maximum coolig ower, ). maximum coefficiet of erformace at a give temerature differece, ad 3). maximum temerature differeces; these cases are discussed below. C C C

10 0 Thermoelectric Microcoolers ). Maximum Coolig Power For the first case, the curret I Q for maximum coolig ower Q max ca be determied by settig dq C /di=0, which gives ad where the figure of merit Z is defied as I Q ( S S ) TC =, (.9) R ( T T ) 0.5ZTC H C φ Q =, (.0) ZTHTC ( S S ) Therefore, the maximum coolig ower Q max is Q Z =. (.) KR max ( S S ) TC =. (.) R The figure of merit Z i Eq.. is ot a fixed quatity for a give air of thermoelectric materials but deeds o the relative geometries of the ad braches. It has its maximum value whe the roduct of K ad R i Eqs..4 ad.5 is a miimum, which occurs whe L L A A ρ k =. (.3) ρ k Whe the dimesios of the ad braches are otimized, the figure of merit becomes ( S S ) ( k ρ ) / + ( k ρ ) [ ] / Z =. (.4) As see i Eq..4, the figure of merit Z of a thermoelectric cooler deeds o the roerties of the materials i both braches, but it is coveiet to defie a figure of merit for a sigle material as S Z =. (.5) k ρ

11 Fudametals of Thermoelectrics This figure of merit Z is a measure of the material s otetial for makig a efficiet thermoelectric device. It icludes electrical resistivity ρ due to Joule heatig. It has thermal coductivity k i its deomiator because of the heat leakage from the hot to cold side through coductio. The figure of merit Z has a uit of iverse Kelvi. The dimesioless form ZT, a roduct of Z ad absolute temerature T, is commoly used i device aalysis. From a microscoic oit of view, the figure of merit is iflueced by charge ad heat trasort as well as their coulig i thermoelectric materials, as discussed i Sectios 3 ad 4 of this chater. The best ZT materials are foud i heavily doed semicoductors. The commercially available bulk Bismuth Telluride Bi Te 3 alloy has a ZT of aroud room temerature. ). Maximum Coefficiet of Performace For the secod case, the curret at maximum coefficiet of erformace is determied by dφ/di=0, which leads to ( S S )( TH TC ) I φ =, (.6) R + ZT Coefficiet of Performace, T h =300K ( ) Cold Side Temerature, T c (K) M Carot Cycle ZT M = ZT M = Fig..5. Coefficiet of Performace (COP) of thermoelectric coolers with various values of figure of merit ZT. The hot side of the thermoelectric cooler is at 300K. 8

12 Thermoelectric Microcoolers ad max TC ( + ZTM TH / Tc ) ( T T )( + ZT + ) φ =, (.7) H C where T M = ( TH + TC )/ is the average temerature of the hot ad cold sides. The maximum coefficiet of erformace of thermoelectric coolig devices φ max strogly deeds o the ZT of the thermoelectric materials used, as show i Fig..5. φ max aroaches the coefficiet of erformace of a Carot cycle as the figure or merit Z goes to ifiity. Thermoelectric coolig devices based o Bi Te 3 alloys (ZT~) have a efficiecy of about 0% of Carot efficiecy. 8, 9 3). Maximum Temerature Differece Aother imortat arameter is the maximum temerature differece that a thermoelectric cooler ca reach. If the heat source is removed, i.e., =0, the coefficiet of erformace falls to zero ad the temerature differece (T H - T C ) rises to its maximum value. The maximum temerature differece ca be obtaied from Eq..0 as ( T T ) = ZT The corresodig electric curret is give by H M C max C. (.8) ( S S ) TC I T max =. (.9) R The maximum temerature differece of a thermoelectric cooler is a fuctio of the figure of merit Z ad the cold side temerature T C.... Effects of Electrical Cotact Resistace The aforemetioed aalysis o thermoelectric module erformace has eglected the electrical cotact resistace betwee the metal itercoect ad the thermoelectric braches. The electrical cotact resistace may be comarable to the electrical resistace of the thermoelectric braches whe the legth of these braches becomes small, for examle, thi-film thermoelectric devices. 0- I this case, the coefficiet of erformace of the thermoelectric devices should take ito accout the electrical cotact resistace. If the electrical cotact resistace for oe brach is R c, the effective electrical resistivity ρ is give by

13 Fudametals of Thermoelectrics 3 A ρ eff = Rc + ρ, (.30) L where ρ is the electrical resistivity of thermoelectric material, ad A ad L are the cross-sectio area ad legth of the brach, resectively. I thermoelectric microcoolers with short elemets, the cotact resistace is erhas the most critical roblem to deal with. The cotact resistace lowers the coefficiet of erformace of the thermoelectric devices, ad uts a lower limit o the elemet thickess ad a uer limit o their maximum coolig ower. More discussio o effects of cotact resistace o device erformace ca be foud i Chater Effects of Temerature-Deedet Proerties The aalysis show above assumes that thermoelectric roerties, such as the Seebeck coefficiet, electrical resistivity ad thermal coductivity, are all temerature ideedet. These thermoelectric roerties i fact vary with temerature. The errors origiatig from this assumtio are likely to be imortat for devices that are oerated with large temerature differeces betwee the heat source ad sik. A reasoable estimatio of the device 3, 4 erformace ca be made usig the temerature averaged roerties, ( S S ) ( k ρ ) / + ( k ρ ) [ ] / Z =, (.3) where the bars idicate the temerature averaged quatities. This aroach ca give a good aroximatio for thermoelectric coolers where the temerature differece betwee the hot ad cold sides is geerally a small fractio of the averaged absolute temerature...4. Performace of Multistage Thermoelectric Coolers A multi-stage thermoelectric cooler is eeded whe a sigle-stage cooler is uable to geerate the required temerature differece. The multi-stage cooler ca be viewed as two or more sigle stages stacked o to of each other. The costructio of a multi-stage module is usually of a yramidal shae; each lower stage is larger tha the uer stage, as show i Fig..6. I a -stage uit, the first stage must have a coolig ower that is equal to the sum of the coolig ower at the source ad the electric ower used i each of the stages, (-),, ad. The total heat delivered to the sik from the first stage is

14 4 Thermoelectric Microcoolers Q ( + )( + )( + ) ( + ), (.3) φ φ φ φ where Q is the coolig rate for the th stage, ad the coefficiet of erformace for the ith stage is φ i. Therefore, the overall coefficiet of erformace of a - 5, 6 stage thermoelectric cooler ca be exressed i followig form, φ = ( + )( + φ φ )( + φ. (.33) ) ( + ) φ If the multi-stage thermoelectric cooler is desiged so that each stage oerates with the same coefficiet of erformace φ ', the the overall coefficiet of erformace reduces to φ =. (.34) ( + ) φ' The overall coefficiet of erformace of a multi-stage cooler is always smaller tha that the sigle-stage s oe. This is because the multi-stage cooler oerates at a much larger temerature differece. However, there is a gai i coefficiet of erformace for a fixed temerature differece if a multistage uit is used. The icreased coolig ower that is required for the high-temerature stages of a multi-stage module, i.e.,, is usually achieved by icreasig the umber of thermocoules rather tha reducig the thermocoule height. So the multi-stage modules look like yramids. For examle, a six-stage module could be comosed of stages with,, 4, 8, 6 ad 3 thermocoules. Six-stage thermoelectric coolers are commercially available as stadard roductio items. 7- Fig..6. Picture of sigle-stage ad three-stage thermoelectric coolers. 7

15 ..5. Other Forms of Figure of Merit Fudametals of Thermoelectrics 5 The thermoelectric coolers ca be cofigured differetly from Figs..4 ad.6. Oe examle is o-chi hotsot coolig, as show i Fig..7. I this cofiguratio, the silico chi is used as the thermoelectric microcooler, ad the hot sot ca be cooled by both thermoelectric coolig ad heat sreadig i the silico substrate. Modelig aalysis suggests that the traditioal thermoelectric figure of merit, Z, may ot be suited for this utraditioal thermoelectric cofiguratio, but the ower factor / would be a better measure of the material s otetial for makig a efficiet o-chi microcooler. Materials with high Seebeck coefficiet ad low electrical resistivity are desired i these alicatios. More detailed discussio o o-chi coolig ca be foud i Chater 5. SiN x Layer Metal Lead Metal Cotact Silico Ca Silico Substrate (-tye) Groud Electrode V=0 Fig..7. Schematic of a o-chi thermoelectric microcooler for hotsot suressio. The arrows idicate the directio for electric curret flow. (Coyright 006, America Istitute of Physics)..3. Thermoelectric Trasort Proerties i Bulk Materials This sectio discusses electrical ad thermal trasort withi bulk thermoelectric materials. Although there are differet simulatio aroaches available, the focus here is ut o classical trasort based o Boltzma s equatio. Boltzma s trasort equatio is a tool for aalyzig trasort roerties i systems that ivolve desity ad temerature gradiet. This equatio ca be alied to aalysis of thermoelectric trasort roerties, such as the Seebeck coefficiet, electrical resistivity ad thermal coductivity, ad therefore geeratig a microscoic descritio of the evolutio rocesses i the thermoelectric (electro

16 6 Thermoelectric Microcoolers ad hoo) systems. Boltzma s equatio is valid at time itervals loger tha the electro ad hoo iteractio duratio ad at distaces loger tha the size of the iteractio domai. The stadard form of the steady state, liear Boltzma trasort equatio, 3, 4 uder the relaxatio time aroximatio, is give by, F f f o v r f + k f =. (.35) ħ τ Here f(r,k) is the o-equilibrium distributio fuctio i hase sace that icludes both coordiates r ad wavevector k, ad it reresets the robability of a article (electro or hoo) withi the system to be at r ad to have k. f o is the local equilibrium distributio give by Fermi-Dirac (for electros) or Bose- Eistei (for hoos) distributios. F is the force F actig o the articles. τ is the relaxatio time, a time costat whe the o-equilibrium distributio fuctio f relaxes to its equilibrium state f o after removig all drivig forces.ħ is the Plack costat divided by. Figure.8 illustrates the equilibrium ad o equilibrium distributio fuctios for electros ear the eergy bad edge. I Boltzma s equatio, Heiseberg's ucertaity ricile is eglected, ad each article is exactly described by its mometum ad ositio (r,k) withi the 6 dimesioal sace. The relaxatio time τ deeds o the article eergy E ad is ofte modeled by a ower law, τ γ τ o E, (.36) where γ is a arameter deedig o article scatterig mechaisms Electro Trasort Proerties E(k) E(k) f 0 (k) f(k) k k Fig..8. Illustratio of equilibrium ad o equilibrium distributio fuctios for electros at the coductio bad.

17 Fudametals of Thermoelectrics 7 Here cosider electros i thermoelectric material uder a small electric field, temerature gradiet, ad cocetratio gradiet. The local equilibrium distributio f o for electros obeys the Fermi-Dirac distributio, f o ( k ) =, (.37) E( k) E f ex + k BT where the Fermi level E f is a fuctio of coordiate r ad carrier cocetratio ad k B is the Boltzma costat. 6 The electro eergy cosists of otetial ad kietic eergy; uder the sherical ad arabolic bad aroximatio, it ca be exressed as ħ k E( k ) = Ec +, (.38) * m where m* is the electro effective mass ad E c is the bottom of the coductio bad. I the absece of a magetic field, the force F actig o the electros is give by de de c q q F = ε = = c, (.39) q dx dx where is the electric field, q is the uit charge, ad q=-e for electros ad q=e for holes. I thermoelectric alicatios, the temerature gradiet ad electric field are small so the deviatio from equilibrium distributio is small, i.e. f f, r f r fo, (.40) f f. k o This is the so-called first order aroximatio to the Boltzma equatio. Uder this aroximatio, the solutio of the o-equilibrium distributio fuctio ca be obtaied, << k f o o E E f f o f = fo + τ v E f + T T. (.4) E The electric curret desity J e ad heat flux J q carried by charged carriers ca be exressed as 7 3 J e ( r) = qv( k)f( r, k)d k, (.4) 3 4π

18 8 Thermoelectric Microcoolers ad 3 J q ( r) = [E( k) E f ( r)] v( k)f( r, k)d k, (.43) 3 4π resectively. Substitutig Eq..4 ito Eqs..4 ad.43 leads to ad q J e( r) = q L0 Φ + L ( T ), (.44) q T J q ( r) = ql Φ + L ( T ). (.45) q T The itegrals L (for =0,,) i the above equatios are exressed as L = 3 4π τ( k)v( k)v( k)(e( k) E F ) f o ( E ) d 3 k. (.46) The electrochemical otetial Φ i Eqs..44 ad.45 is comosed of two comoets a electrostatic comoet ad a chemical comoet, Φ=φ+E, (.47) where ϕ is the electrostatic otetial (also called electrical otetial), which ca be determied from the electric field, ε= φ. (.48) The electrical resistivity, Seebeck coefficiet ad electro thermal coductivity ca be determied from the electric ad heat curret fluxes, J e ad J q, T = 0 q f o ( q L ) = τv D( E) Φ / q ρ = = 0 d 3 E E, (.49) J e S ad Φ / q = T = qt = qt L0 L Je= 0 v τv f o ( E E ) D( E) τ f E f o D E ( E)dE de, (.50) J q L f k o e ( L ) ( E E f ) D( E) de T T L0 3 T = = = τν. (.5) Je= 0

19 Fudametals of Thermoelectrics 9 It may be ecessary to use umerical methods to fid the trasort roerties from the above equatios. The roduct S /ρ is ofte called the ower factor which deeds o the desity of states, the mobility, the scatterig arameter, ad the ositio of the Fermi level. The electrical resistivity ad electro thermal coductivity obey the Wiedema-Fraz law, where L is the Loretz factor, ρ k e = TL, (.5) k -8 L = π B =.45 0 ( WΩ ) 3e /K. (.53) The Lorez factor should be the same for all metals ad ot deed o the scatterig law for electros. I thermoelectric materials that are tyically doed semicoductors, the total thermal coductivity comes from two sources: electros (k e ) ad hoos (k ). The Lorez factor may vary with carrier cocetratio i these materials. Phoo thermal coductivity (k ) is ofte comuted as the differece betwee k ad k e (Eq..60) usig the exerimetal electrical resistivity. This stadard method could lead to a erroeous icrease i k as biolar thermal coductio is ot cosidered i the derivatio of the Wiedema Fraz law. The Wiedema-Fraz law rovides a coflict i otimizig thermoelectric roerties as a large figure of merit requires both a low electrical resistivity ad a low thermal coductivity..3.. Phoo Trasort Proerties Ulike electrical charges, there are o exteral forces, such as electric force ad magetic force, actig o hoos. So the steady state, liear Boltzma 3, 8, 9 trasort equatio reduces to f f v f = o r, (.54) τ where f is the o-equilibrium distributio fuctio for hoos ad f o is their equilibrium distributio fuctio. The local equilibrium distributio f o for hoos obeys the Bose-Eistei robability distributio, 5 fo ( k ) =, (.55) E( k) E f ex kbt

20 0 Thermoelectric Microcoolers which gives the average umber of hoos i a system i equilibrium at temerature T foud i a state of k. A lot of three distributio fuctios, the Fermi-Dirac distributio (for electros ad holes), the Maxwell-Boltzma distributio (for high temerature alicatios) ad the Bose-Eistei distributio (for hoos),is show i Fig..9. All three distributio fuctios are almost equal for large eergies (more tha a few k B T beyod the Fermi eergy). Uder this first order aroximatio, the solutio of the o-equilibrium distributio fuctio ca be exressed as, f = f o τ v r f o. (.56) Bose-Eistei Probability of Occuacy.5 Maxwell-Boltzma 0.5 Fermi-Dirac Eergy (ev) Fig..9. Comariso of the Fermi-Dirac (Gree curve), the Bose-Eistei (Red curve) ad the Maxwell-Boltzma (Blue curve) distributio fuctios. T = 300 K. The hoo heat flux ca thus be writte as, f o 3 J = T τ vv d k = k T. (.57) T For bulk homogeeous materials, the hoo thermal coductivity is give by, = C( ω) v ( ω) τ ( ω) dω, (.58) 3 k where ( ) is the secific heat of hoos at frequecy, is the hoo grou velocity, ad is the hoo relaxatio time. This equatio ca be simlified by usig a averaged quatities ν ad τ,

21 Fudametals of Thermoelectrics k ν = C τ, (.59) 3 which is essetially the Kietic theory exressio of the thermal coductivity. The total thermal coductivity of a thermoelectric material ca be comuted as the sum of the electroic comoet k e ad the hoo cotributio k, k = k e + k. (.60) The kowledge of the electroic comoet allows the hoo cotributio to be determied if the total thermal coductivity ca be measured. The biolar thermal coductivity should be cosidered at high temeratures. Measuremet of thermal coductivity for semicoductor materials with differet doig levels may rovide a way to searate the two comoets of thermal coductivity, as show i Fig Thermal Coductivity (W/m K).5.5 -tye U-doed Bi Te Electrical Resistivity (0-5 Ω m) Fig..0. Plot of thermal coductivity versus electrical resistivity i Bismuth Telluride Bi Te 3 at room temerature. (Data from Ref.[30] ) Thermoelectric Figure of Merit The figure of merit ZT for three-dimesioal bulk materials ca be obtaied usig the thermoelectric roerty exressios i Eqs ad.60,

22 Thermoelectric Microcoolers where the B factor is, Z 3D T = 5F * 3 / 3F ξ 3F/ 7F5 / 5F + B 6F 3D / 3 / /, (.6) B 3D * 3 / m = 3π k BT ħ ad the Fermi-Dirac itegral F i is [4, 3], 3 / k B ek Tµ, (.6) * ( ξ ) 0 i x dx F i =. (.63) ex + * ( x ξ ) * Here m = ( m ) / 3 xmymz is the effective desity-of-states mass of electros or holes i the eergy bad, is the electro mobility, is the chemical otetial ormalized by k B T, k B is the Boltzma costat, T is the absolute temerature, ad ħ is the Plack costat, ad k is the hoo thermal coductivity. The B factor was first itroduced by Chasmar ad Stratto i 959; 3 a high B factor traslates to a high figure of merit. I Eqs. (37) ad (38), the subscrit 3D is used to deote that those exressios are derived based o the desity-of-states of 3D bulk crystals. I low-dimesioal structures, these exressios must be reformulated. 3 The above exressios for thermoelectric roerties derived from the Boltzma equatio rovide guidace i search for thermoelectric materials with high figure of merit, as discussed below: ). The Seebeck coefficiet ad electrical coductivity both deed strogly o the roduct of the desity of states D(E) ad the derivative of the Fermi-Dirac distributio f o / E. However the derivative of the Fermi-Dirac distributio f o / E is ozero oly i the regio of several k B T ear the Fermi level E f. So icreasig the electro desity of states D(E) ad electro eergy E ear the Fermi level E f ca lead to a high ZT. Figure. shows a illustratio of the desity of states for the coductio ad valece eergy bads ad the Fermi- Dirac distributio fuctio. ). A large figure of merit is obtaied by makig the B factor as large as ossible. A large B factor requires a large electro (or hole) effective mass, a high electro (or hole) mobility ad a low hoo thermal coductivity. The requiremet for the effective mass of the charge carrier rovides a coflict as

23 Fudametals of Thermoelectrics 3 heavy carriers will move with slower velocities, ad therefore small mobilities, which i tur leads to high electrical resistivity. A balace must be foud for the effective mass to form a comromise betwee high effective mass ad high mobility. The idea thermoelectric materials are called hoo-glass-electrocrystal by Slack. 33 3). A small hoo grou velocity ad a short hoo relaxatio time are eeded for a small hoo thermal coductivity. 9 The hoo grou velocity is tyically small i materials with high atomic mass ad/or with comlex uit cells. O the other had, the hoo relaxatio time ca be reduced by scatterig, such as through alloyig (e.g., Bi Te 3 based alloys) 4, ad addig hoo rattlers (e.g., clathrates 38 ad skutterudites ). f(e)=0 E C E f E v f(e)d C (E) D C (E C )=0 D V (E V )=0 f(e)d v (E) D C (E) f(e) f/ E D v (E) f(e)= Fig... Schematic illustratio of the desity of states D(E) (Blue curves), the Fermi-Dirac robability distributio (Black curves), ad their roduct (Red curves) for a semicoductor thermoelectric material..4. Thermoelectric Trasort i Naostructured Materials The thermoelectric trasort roerties (the Seebeck coefficiet, resistivity, ad thermal coductivity) are defied ad aalyzed i homogeeous bulk materials i the above sectios. However, these exressios may ot be valid i aostructured materials that are defied as ihomogeeous materials with at least oe legth scale i the aometer rage. 3, 44, 45 Figure. shows several

24 4 Thermoelectric Microcoolers examles of aostructures, icludig suerlattice, aowires, ad aocomosites. Ehaced thermoelectric figure of merit has bee redicted ad, 9, 44, realized i aostructured materials. The area of aostructured thermoelectric materials is develoig raidly. Thermoelectric trasort i such aostructures differs from the descritio for bulk materials because of the followig reasos: ). Iterfaces ad boudaries of aostructures imose costraits o the electro ad hoo waves if the characteristic legth of the aostructures is 9, 50, 5 comarable with the electro ad hoo wave legth. Therefore electroic eergy bad ad hoo disersio relatio ca be very differet i the aostructures from those i the corresodig bulk materials. (a) (b) (c) (d) Fig... Examles of aostructures for otetial thermoelectric alicatios. (a) suerlattices, (b) quatum dot suerlattices, (c) quatum wires, ad (d) aocomosites. 4 (Coyright 004, WIT Press).

25 Fudametals of Thermoelectrics 5 ). The hoo thermal coductivity ca be reduced through iterface scatterig ad through the alteratio of the hoo sectrum i aostructures. 4, 5, Formatio of Eergy Bad i Naostructured Materials A imortat but ofte igored questio is whe oe should use the eergy sectra of electros ad hoos i bulk materials ad whe oe must resort to the eergy sectra of aostructures. Suerlattice aostructures are used as a examle i search for the aswer to this questio. Geerally seakig, two imortat arameters eed to be evaluated: oe is the mea free aths for electros ad hoos i the aostructures, ad the other is the umber of eriods required to form ew eergy bads for electros ad hoos. There is o simle way to calculate the required eriods for bad formatio. The required eriods may vary deedig o materials ad structures but oe ca gai some ideas by examiig the quarter wavelegth stack (Bragg reflector) used i otical coatigs Takig a GaAs/AlAs quarter wavelegth stack as a examle, although the reflectivity at a idividual iterface betwee GaAs ad AlAs is small, a reflectivity close to uity ca be created with a small umber of eriodic quarter wavelegth layers, as show i Fig..3. The chage i trasmittace is small after about 0 eriods. Thus, oe ca ifer that if hoos (or electros) ca maitai their hase coherece over a few to tes of eriods of the uit cell, ew hoo (or electro) bads will form. I bulk solids, 0 uit cells are equivalet to aroud 50 Å, which is tyically shorter tha the mea free aths for electros ad hoos. Here it is assumed that all the scatterig rocesses that limit the mea free ath destroy hoo coherece. For bulk materials (excet amorhous materials), the mea free ath is tyically loger tha the miimum domai legth required for bad formatio, as show i Fig..4. For suerlattices, however, the miimum domai legth required for the bad formatio is much larger, comared to that i the bulk materials. I additio, diffuse iterface scatterig ca shorte the mea free ath i suerlattices. Cosequetly, the coditio for the hoo bad formatio i suerlattices is ot always easily satisfied. I this case, however, the bulk bad ca still be established i each layer as log as each layer still has eough umber of origial uit cells.

26 6 Thermoelectric Microcoolers Trasmittace Periods 5 Periods 0 Periods 0X0 GaAs/AlAs SL Frequecy ( T Hz ) (a) Trasmittace X0 GaAs/AlAs SL T=-/e Number of Periods Fig..3. Trasmittace of acoustic wave as a fuctio of frequecy (a) ad umber of eriods i GaAs/AlAs suerlattice (SL) structures. The trasmittace aears to be ideedet of the umber of eriods whe the total umber of eriods is larger tha 0. 58, 59 (Coyright 00, IEEE). (b)

27 Fudametals of Thermoelectrics 7 Mea Free Path i Bulk Material Uit Cell Miimum Size for Bad Formatio (a) Mea Free Path i Suerlattice Uit Cell Miimum Size for Bad Formatio (b) Fig..4. Characteristic legths determiig whether trasort is i the coheret regime or icoheret regime i bulk materials (a) ad i suerlattices (SLs) (b). The characteristic legths iclude ). the miimum domai size eeded to form a bad, ad ). the mea free ath limited by hase destroyig scatterig evets. 58, 59 (Coyright 00, IEEE). Mea free aths for electros ad hoos i aostructures ca be very differet from those i the corresodig bulk materials. For examle, hoo scatterig i suerlattices may hae at iterfaces due to the diffusive iterface scatterig (MFP d ) ad withi the layers due to iteral volumetric scatterigs (MFP i ) such as Umkla scatterig ad isotoe scatterig. Accordig to Mathiesse s rule, the total mea free ath (MFP SL ) i the suerlattice is. MFPSL = (.64) + MFP MFP i d

28 8 Thermoelectric Microcoolers The ossibility that hoos or electros i suerlattices will ot be x cos(θ ) d diffusively scattered after travelig a distace x is P, where d is the sacig of iterfaces, is the secularity arameter of each iterface that reresets the fractio of secularly scattered hoos or electros, ad θ is the icidet agle. Therefore, the mea free ath caused by the iterface diffuse scatterig takes the form d MFP d =. (.65) cos ( θ )L(P) I the followig, the discussio o thermoelectric trasort i aostructures maily follows alog three lies. The first oe treats electros ad hoos as coheret carriers. This ca be thought of as the quatum size regime. The secod oe treats them as icoheret carriers, which ca be thought of as the classical size effect regime. The third oe treats them as artially coheret carriers, which is i the itermediate regio whe both quatum ad classical size effects exist..4.. Icoheret Electro ad Phoo Trasort: Classical Size Effects If the mea free aths of electros ad hoos are too short to eable the formatio of ew eergy bads i the aostructures, the electros ad hoos have the same eergy sectra as i their bulk materials. This is the icoheret regime, ad electros ad hoos ca be treated as classical articles i the Boltzma equatio, as discussed i Sectio. The Boltzma equatio ca be solved for the electro ad hoo distributio fuctios ad the thermoelectric trasort roerties i the aostructures. Boudaries ad iterfaces i aostructures are imosed oto the trasort rocesses as a additioal scatterig mechaism. Boudary scatterig is ot ew ad classical size effects o electros ad hoos have bee studied a log time ago. 5, 60, 6 There are two differet aroaches to deal with the boudary scatterig. ). Oe is to add a extra boudary scatterig term ito the relaxatio time through Mathiesse s rule, as described i the above sectio. This aroach is simle ad had bee widely used i dealig with low-temerature trasort roblems i bulk materials where size effects are imortat, but it is ot accurate because it treats the boudary scatterig rocesses equally with the iteral 6, 63 volumetric scatterig rocess. ). The other is to treat iterfaces ad boudaries through boudary coditios to the Boltzma equatio. 3, Electro (or hoo) reflectio ad trasmissio tyically are modeled as a diffuse or a secular rocess or a mixture of diffuse ad secular rocesses. The thermoelectric trasort roerties uder the classical size effect ca be calculated usig the Boltzma Trasort Equatios. Quatum wells are chose

29 Fudametals of Thermoelectrics 9 as a examle ad the results are lotted i Fig These figures show the modeled ower factor subject to quatum cofiemet oly [Fig..5a] ad combied quatum cofiemet ad diffuse boudary scatterig [Fig..5b]. It ca be see i these figures that the ower factor decreases, rather tha icreases, with icreasig diffuse boudary scatterig. Power Factor (0 - W/mK ) Power Factor (0 - W/mK ) Mobility: 440 cm / Vs 6m 0m Reduced Fermi Level (E F -E c )/kt (a) Mobility: 440 cm / Vs (b) Fig..5. Power factor for a silico quatum well calculated (a) based o quatum size effects at differet quatum well widths, ad (b) based o combied quatum cofiemet ad diffuse boudary scatterig effects for a m-wide quatum well. 68 (Coyright 00, Cambridge Uiversity Press). m BULK 4m BULK =.0 =0.9 = Reduced Fermi Level (E F -E c )/kt

30 30 Thermoelectric Microcoolers.4.3. Coheret Electro ad Phoo Trasort: Quatum Size Effects If the electro ad hoo mea free aths are log eough to form ew eergy bads i aostructures, their electros ad hoos are subject to ew disersio relatios differet from those i the bulk materials. This is the coheret regime, ad quatum size effects o electros ad hoos must be cosidered to aalyze thermoelectric trasort i these aostructures. Quatum size effects o electros ca be used to imrove the electro thermoelectric roerties. 44, For examle, i a ifiite quatum well, the Eige-states of electro eergy are give by, E ħ ħ π ( kx, k y, ) ( k k ) * x + y + * =, (.66) m m d where ħis the Plack costat, m* is the effective bad-edge mass, d is the quatum well thickess i z directio, =0,,, that reresets the quatized eergy i the z directio, ad k x ad k y are the wavevectors i the x ad y directios, resectively. Because of eergy quatizatio i the z-directio, electros ca oly have quasi cotiuum wavevectors i k x ad k y directios ad the subbads ca be thought of the rojectio of the stadig waves i the z-directio to the x-y lae. This eergy quatizatio has several cosequeces o the exressios for the Seebeck coefficiet ad electrical resistivity i bulk materials. Oe is that the itegratio of the wavevector should be erformed oly i the k x ad k y directios. I the k z directio, the itegratio should be relaced by a summatio. Equivaletly, the electro desity of states must be relaced by the ew desity of states iside the quatum wells. It is ossible to chage the doig level such that the Fermi level is laced close to oe subbad edge ad the ower factor ca be icreased comared to that i bulk materials. Such a idea has bee demostrated i PbTe quatum wells ad Si/SiGe quatum wells. 44 Modelig suggests that hoo trasort ca beefit tremedously from quatum size effects although oly moderate gai ca be achieved i the electro 75, 76 thermoelectric erformace. Quatum cofiemet o hoos ca be calculated usig lattice dyamics. 58, 77-8 For examle, the dislacemet of a atom from its equilibrium ositio, u lm, i a suerlattice structure ca be writte i the form: 8 u = u~ ex[ i(k x + qnd ωt)] (.67) lm where ~u is the comlex amlitude of the th atom i the Nth eriod of the suerlattice, k is the i-lae wavevector, q is the cross-lae wavevector ad d is the thickess of a uit eriod i the cross-lae directio. The equatio of motio lm

31 Fudametals of Thermoelectrics 3 for the atom u lm ca be solved to determie the hoo disersio relatio uder certai boudary coditios aroriate for suerlattices. Figures.6a ad b show the hoo disersio i a Si/Ge suerlattice alog the i-lae ad crosslae directios, cosiderig oly acoustic hoos i the bulk materials. Clearly, the cross-lae grou velocity has bee sigificatly reduced while the i-lae grou velocity reductio is just modest. The case i GaAs/AlAs suerlattices is similar to that i Si/Ge suerlattices excet that the former has much smaller reductio i grou velocity due to the smaller acoustic mismatch. 80 (a) (X)Si/Ge (b) (X)Si/Ge 80 Frequecy (0 rad/s) [ω S i ] m ax [ω Ge ] m ax [ω Si ] m ax [ω Ge ] m ax Frequecy (0 rad/s) k a/π x qd/π DOS (0-4 /Hz) Si G e ( 5X 5) Si/G e (c) (a) V X (5x5) V (5X 5) Z V (S i) V (G e ) (d) (b) <v (w)> (0 0 m - s - ) F requ ecy ( 0 ra d/s) F re quecy (0 ra d/s) Fig..6. Phoo behavior i Si/Ge suerlattices, (a) hoo disersio i the i-lae ad (b) the cross-lae directios, (c) desity of states showig shar features but is aroximately a average of these of Si ad Ge, ad (d) grou velocity i i-lae ad cross-lae directios. Cross-lae directio shows a large reductio while i-lae directio is roughly a average of their bulk materials. 8 (Coyright 00, Taylor & Fracis Ic.).

32 3 Thermoelectric Microcoolers.4.4. Trasort i Partially Coheret Regime The regime where both coheret ad icoheret trasort rocesses exist is much more difficult to model. Models for such mixed trasort regimes are begiig 58, 8 to emerge for hoo. Oe aroach is to use a modified lattice dyamics model, i which a imagiary wavevector is icororated. 58, 79, 8 The mea free ath (MFP) caused by diffuse iterface scatterig is icluded i the comlex wavevector K, i K = k +,. (.68) MFP where i is the imagiary uit ad k is the real wavevector. The thermal coductivity k x ca be determied whe the hoo disersio 58, 8 relatio is kow, ω kx = C h( ωλ ) MFPSL ( θ, T), (.69) λ where λ deotes the suerlattice modes, k is wavevector, C h (ω λ ) reresets mode secific heat, x is the thermal coductio directio, ad T is temerature. This model combies the effects of hoo cofiemet ad diffuse iterface scatterig o the thermal coductivity i suerlattices, ad is alicable to hoo trasort i the artially coheret regime, where bulk ad suerlattice hoo modes mix u. 40 (a) (b) k x P=.0 P= π Fig..7. Disersio relatios of hoos i the cross-lae directio of the GaAs/AlAs suerlattice, calculated from the lattice dyamics model (a) without diffuse iterface scatterig (erfect iterfaces, the secularity arameter of each iterface P=) ad (b) with diffuse iterface scatterig (rough iterfaces, P=0.9). 59, 8 (Coyright 003, America Physical Society).

33 Fudametals of Thermoelectrics 33 Figure.7 shows the disersio curves for hoos roagatig i the crosslae directio i the GaAs/AlAs suerlattices, calculated from lattice dyamics models (a) without ad (b) with the additio of the imagiary wavevector. I both cases, the frequecy gas occur at the ceter ad boudary of the folded Brilloui zoe ad the disersio curves are flatteed, esecially at high frequecies. The hoo grou velocity i suerlattices is reduced, but the magitude of the reductio deeds o the bad ga width. Comarig Figs..7a ad b, it is see that the itroductio of the imagiary wavevector results i dimiishig bad gas. Figure.8 shows the calculated ad the exerimetal cross-lae thermal coductivities of GaAs/AlAs suerlattices as a fuctio of the eriod thickess at room temerature, alog with the i-lae thermal coductivity for comariso. As see i this figure, the cross-lae thermal coductivity, as well as the ilae thermal coductivity, decreases as the eriod thickess icreases, reaches a miimum, ad the starts to icrease towards the corresodig bulk values. I this artially coheret model, the article effect domiates i the large-eriod suerlattices while the wave effect gradually set i with the decreasig eriod thickess. Therefore, a miimum thermal coductivity occurs at the crossover betwee the wave effect ad the article effect. This aroach would be alicable to electro trasort i aostructured thermoelectric materials. Thermal Coductivity (W/mK) 00 0 I-Plae Cross-Plae P=0.83 Caiski,999 P=0.9 GaAs/AlAs, X P= Period Thickess (Å) Fig..8. Phoo thermal coductivity as a fuctio of eriod thickess i both i-lae ad 59, 8 cross-lae directios of GaAs/AlAs suerlattices, based o the combied wave-article model. The triagles are reorted i exerimetal data. 83 (Coyright 003, America Physical Society).

34 34 Thermoelectric Microcoolers.5. Summary This chater severs as a itroductory to thermoelectrics ad articularly thermoelectric coolig. Basic thermoelectric effects ad their uderlyig hysics have bee discussed first, ad a set of arameters characterizig thermoelectric materials ad devices have bee itroduced. The figure or merit Z is a good measure of the materials otetial for makig efficiet traditioal devices. However, the ower factor is a more aroriate figure of merit for utraditioal o-chi thermoelectric coolig devices. Thermoelectric trasort roerties have bee aalyzed i both bulk homogeeous ad aostructured materials. Quatum ad classical size effects ca be exloited to imrove eergy coversio efficiecy i aostructured materials. Refereces. T.J. Seebeck, Magetic olarizatio of metals ad mierals, Abhadluge der Deutsche Akademie der Wisseschafte zu Berli, 65 (8) H.J. Goldsmid, Thermoelectric refrigeratio. New York: Pleum Press, D.M. Rowe, CRC Hadbook of Thermoelectrics. Boca Rato: CRC Press, H.J. Goldsmid, Recet Studies of Bismuth Telluride ad Its Alloys, Joural of Alied Physics, 3 (96) J.C. Peltier, Nouvelles Exerieces sur la caloricite des couras electrique, A. Chim., LV (834) W. Thomso, O a mechaical theory of thermoelectric currets, i Proc.Roy.Soc., Ediburgh, 85, W. Thomso, Mathematical Physical Paers [Tras. R. Soc. Ediburgh, Vol, XXI, Pt., read May, 854] vol. : (Cambridge Press, 88). 8. B. Yag, H. Ahuja, ad T. N. Tra, "Thermoelectric techology assessmet: Alicatio to air coditioig ad refrigeratio," HVAC&R Research, vol. 4, , O Page 37, 0 chages to R. Radermacher, B. Yag, ad Y.H. Hwag, Itegratig alterative ad covetioal coolig techologies, Ashrae Joural, 49 (Oct 007) D.D.L. Wijgaards, S.H. Kog, M. Bartek, ad R.F. Wolffebuttel, Desig ad fabricatio of o-chi itegrated olysige ad olysi Peltier devices, Sesors ad Actuators, 85 (000) R. Vekatasubramaia, E. Siivola, T. Colitts, ad B. O'Qui, Thi-film thermoelectric devices with high room-temerature figures of merit, Nature, 43 (00) S. Ramaatha ad G.M. Chrysler, Solid-state refrigeratio for coolig microrocessors, Comoets ad Packagig Techologies, IEEE Trasactios o [see also Comoets, Packagig ad Maufacturig Techology, Part A: Packagig Techologies, IEEE Trasactios o], 9 (006) A.F. Ioffe, Semicoductor Thermoelemets ad Thermoeletric Coolig. Lodo: Iforsearch Limited, 957.