Handout 32. Electronic Energy Transport and Thermoelectric Effects
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- Cynthia Freeman
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1 Haut lti y aspt a hmlti ts I is ltu yu will la: hmal y taspt by lts hmlti ts b t Plti t hmlti ls hmlti pw ts Las Osa (9-976) C 47 pi 9 aha Raa Cll Uisity Nt Ntati I is haut ulss stats wis w will assum a uti ba wi a ispsi i by: M M I ps a lti il: wh: M C 47 pi 9 aha Raa Cll Uisity
2 C 47 pi 9 aha Raa Cll Uisity hmlti ts h a tw imptat ts i matials at lat ltial uts hat lw ( mal uts) lta aits ( lti ils) a tmpatu aits: ) b t ) Plti t h b t is imptat lially si it xpsss hw tmpatu is a b us t at lta is h Plti t xpsss hw ut lw a b us t at tmpatu is C 47 pi 9 aha Raa Cll Uisity ltial Cuts a hmal Cuts lts I mst al as wh lti il sity ait a/ a tmpatu aits a all pst ltial a mal uts a b witt as O i matix m as: h ab quatis shw at a tmpatu ait a at a ltial ut a a lti il ( a ai sity ait) a at a mal ut h ab quatis a b us t aluat matial spss i it situatis patial itst NO: h tibuti phs ( latti) t mal ut will b i h
3 ltial Cut m mpatu Gait tmpatu ait i uti matial a aus a lti ut Csi lts i uti ba a -p smiut a mtal h is appli il h is a tmpatu ait ssum lt sity: BZ h lal quilibium istibuti uti is: C 47 pi 9 aha Raa Cll Uisity mpatu is psiti pt Physial xplaati lt iusi lts wi is hih a mi ll ius m i hih tmpatu t i lw tmpatu lts wi is lw a mi ll ius m i lw tmpatu t i hih tmpatu h hih y lts us wi a ut is i iti tmpatu ait (Q:What will happ i a p-p smiut?) mi ll a als ha wi tmpatu but w will assum at it s t C 47 pi 9 aha Raa Cll Uisity
4 4 C 47 pi 9 aha Raa Cll Uisity tat m Bltzma quati assumi appli il: Multiply b sis by a itat -spa t t: LH: BZ ltial Cut m mpatu Gait: Bltzma quati C 47 pi 9 aha Raa Cll Uisity RH: BZ BZ BZ Nt at: h RH bms: ltial Cut m mpatu Gait: Bltzma quati
5 ltial Cut m mpatu Gait: Bltzma quati lt iusi ially putti LH a RH t w ha: BZ lts wi is hih a mi ll ius m i hih tmpatu t i lw tmpatu lts wi is lw a mi ll ius m i lw tmpatu t i hih tmpatu -p smiut: p-p smiut: C 47 pi 9 aha Raa Cll Uisity ltial Cut m mpatu Gait: miuts BZ xampl -p smiut at hih tmpatus: Csi a smiut at hih tmpatus a assum at Maxwll-Bltzma statistis apply: uti ba a smiut wi llwi ispsi: M M W t (assumi a y ipt satti at ): M C 47 pi 9 aha Raa Cll Uisity 5
6 6 C 47 pi 9 aha Raa Cll Uisity xampl mtal a -p smiut at lw tmpatus: I is as: Hw usi ab appximati will i a z s has t b m aul uti ba wi llwi istpi ispsi: BZ m O btais at a m aul mputati ab ital: D D D D 6 ' ' ltial Cut m mpatu Gait: Mtals C 47 pi 9 aha Raa Cll Uisity hmpw: h b t a b s Csi a pi mtal ( smiut) wi its tw s pt at it tmpatus by sm xtal mas i ut a lw i xtal iuit a lti il will buil up isi matial i sps t tmpatu ait sulti i a lta i btw tw utput tmials (is is b t ) h ttal ut sity i matial a b witt as: h hmpw ts b ts is i as: iaam ab: x x xx xx x y
7 7 C 47 pi 9 aha Raa Cll Uisity h b s: Mtals a miuts x y C I - I slab was a -p smiut (a Maxwll Bltzma statistis appli): l N xx N xx l N N l C II - I slab was a haily -p smiut ( a mtal): D D xx 6 ' xx Lss: mpa t mtals (i whih >> ) p smiuts will pu a la pttial i a i tmpatu i Mtt s mula C 47 pi 9 aha Raa Cll Uisity Masumt b s a hmupl m a is i masumt b t Csi a stup t masu b t matial by tati it wi las ma matial B as shw: + - h tmpatu tw s matial a pt at a It is t iiult t shw at i abs ut lw pttial masu i xtal iuit is: B B B h b tss matials a B t b siiiatly it i t btai a la pttial i I B ltas at i ah matial al wh i au lp h b t is piipl bhi pati tmpatu ss all mupl
8 hmyamis a hmal Cuts i Matials h ist law myamis lats ha U i ital y a systm t hat y ita Q mhaial w by systm P a patil umb ha N: U Q P N lts i smiuts mtals mhaial w tm a b lt a hmial pttial quals mi ll : Q U N Csi a slab matial i whih hat y ai by lts is lwi m lt t iht as shw: upps ah ai has y U U = N N upps hat y lux (uits: Watts/m ) is ital y lux is U (uits: Watts/m ) a ai umb lux is N (uits: #/m ) : U N N h ab lati is us t mput mal y lw u t lts i matials C 47 pi 9 aha Raa Cll Uisity hmal Cut m mpatu Gait tmpatu ait i a uti matial sults i hat lw (mal ut) baus lt lw Csi lts i uti ba a -p smiut a mtal h is appli il but is a tmpatu ait s lts m m ht si t l si y als tas mal y W ha alay sl istibuti uti: h tibuti t hat lw by lts a b btai by multiplyi istibuti uti by a summi all stats: BZ H is hmal Cutiity ts lts C 47 pi 9 aha Raa Cll Uisity 8
9 9 C 47 pi 9 aha Raa Cll Uisity xampl -p smiut at hih tmpatus: Csi a smiut at hih tmpatus a assum at Maxwll-Bltzma statistis apply: uti ba a smiut wi llwi ispsi: M M h mal utiity lts ms ut t b: BZ hmal Cut m mpatu Gait: miuts C 47 pi 9 aha Raa Cll Uisity BZ xampl mtal a -p smiut at lw tmpatus: I is as: Hw usi ab appximati xpssi will i a z s has t b m aul uti ba wi llwi istpi ispsi: m D hmal Cut m mpatu Gait: Mtals h mal utiity lts ms ut t b: Wima az Law mtals
10 C 47 pi 9 aha Raa Cll Uisity hmal Cuts m lti ils a Dsity Gaits Csi lts i uti ba a -p smiut a mtal h is tmpatu ait but is a appli il a pssibly a ai sity ait as wll s lts m y als tas mal y W ha alay sl lat istibuti uti: h tibuti t hat lw by lts a b btai by multiplyi istibuti uti by a summi all stats: BZ H is sam ts u ali whih lat ltial ut t a tmpatu ait C 47 pi 9 aha Raa Cll Uisity ltial Cuts a hmal Cuts I mst al as wh lti il sity ait a/ a tmpatu aits a all pst ltial a mal uts a b witt as O i matix m as: h ab quatis a b us t aluat matial spss i it situatis patial itst
11 h Plti t a Plti s Csi a matial i whih mal ( sity) aits a t pst W ha: is all Plti ts a is lat t b ts h lati a istpi matial: implis at a mal ut ampais a ltial ut Nw si ut lw i a ubl juti matials a B as shw blw a supps at < B h ltial ut is stat ywh B B B B B B i matial B ais m mal ut a matial sam ltial ut xta mal ut s t b xtat ut m lt juti wis mal y will pil up at at juti a ma it ht imilaly hat must b pi t iht juti wis it will ls hat a bm l his piipl is us i lti mlti ls ( Plti ls) B B C 47 pi 9 aha Raa Cll Uisity hmlti Cls Riati Cl ua Bi Bi R Rp p p h G G N Numb p uits i sis p h Ht ua I R G ltial sista hmal uta (lti as wll as latti tibutis) hmal y absb m -smiut a tp mtal juti: m I hmal y absb m p-smiut a tp mtal juti: p m I tal mal y absb m tp mtal i sil ll: I I Nt at b a Plti iits a ati -smiuts t tai it aut ul lsss a hat uta iit pma (COP) li is: Hat m m l by p I G Gp I R Rp COP W by ut su p I I R Rp COP h Cat limit iats wh R G p p C 47 pi 9 aha Raa Cll Uisity
12 h hmltis Pw Gati Ht ua Bi Bi R Rp p p h G G N Numb p uits i sis p R ltial sista Cl ua R G hmal uta (lti xt I as wll as latti tibutis) mlti l pat i s ats li a hat i h pw si iiy is i by: Rx p I Pw li t xtal la N R Rp Rx Hat lst t l by h p I G Gp I R Rp wh: p I N R R h h p Rx Cat limit hat is wh R G mmly us iu mit a p mlti is: B COP a appah Cat alus as Z Z p R R G G p p C 47 pi 9 aha Raa Cll Uisity hmlti iu Mit a D Paabli Ba Limit h OM is usually xpss as imsilss put Z: Z I ial sai wh latti tibuti t mal utiity is muh small mpa t lti tibuti a smiut is asably wll p ( - ~ 5 ) : 5 5 Z 78 5 * 5 5 * I xpimts lti is masu u itis z ut whih is: masu Z Ipt mst matial paamts! alus Z aius Matials h bst masu alu Z t b xpt is masu Z Z 55 Z Z a alu ~-4 is maximum upp limit Z D paabli ba matials a typially it is -4 tims small u t mstly latti mal utiity mpatu () C 47 pi 9 aha Raa Cll Uisity
13 C 47 pi 9 aha Raa Cll Uisity C 47 pi 9 aha Raa Cll Uisity
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