Theoretical Estimation of Characteristics of Thermoelectric Materials Made of Nanopowders

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1 Theoretical Estimation of Characteristics of Thermoelectric Materials Mae of Nanopowers A.I. Holopkin V.N. Abrutin V.A. Romanko P.V. Miroevskiy I.A. Zotov AV-Engineering Co. Lt. Office 6 Builing 5 B. Tolmachevskiy per. 97 Moscow Russia info@av-engineering.ru phone number: Scientific an Research Institute of Vacuum Technique Abstract The results of theoretical estimations of characteristics of thermoelectric materials compose of nanometer particles that are connecte with each other through point contacts are presente in this paper. The goal of conucte calculations is the verification of possibility of the increasing of thermoelectric material efficiency Z = α σ/(κel + κph) where α Seebeck coefficient σ electric conuctivity κel heat conuctivity associate with electrons an κph heat conuctivity associate with phonons. Possibility of Z increase is base on very simple iea of rastic κph ecrease cause by the presence of nanometer gaps between separate thermoelectric particles that stop free photon propagation in the sample an the conservation of α an σ thanks to the tunneling of electrons through these gaps. Results of calculations are iscusse. Introuction One of the first attempts to estimate possibility of increasing the figure of merit of the flat thermoelectric vacuum structures (TVS) i.e. the structures compose of semiconuctor layers separate by very thin vacuum gaps was mae in 969 []. Term vacuum can be use here so as the nanometer gap with is much less than the free path of molecules in gases. The thermoelectric figure of merit is etermine as Z = α σ/(κ el + κ ph ) where α is the Seebeck coefficient σ is the electrical conuctivity an κ el an κ ph are the electron an phonon thermal conuctivities. Possibility of Z increase is base on very simple iea of rastic κ ph ecrease cause by the presence of nanometer gaps between separate thermoelectric layers that stop free photon propagation in the sample an with the conservation of α an σ thanks to the tunneling of electrons through these gaps. It was shown that the Seebeck coefficient in such structure coul be more than in the bulk thermoelectric in efinite range of Fermi level that shoul lea to Z increase too. However there were not taken into account the thermoelectric material characteristics as functions of Fermi energy an influence the transmission coefficient of electron tunneling through vacuum potential barrier. Now in connection with rapi evelopment of nanotechnologies the iea of TVS creation shoul be raise up an iscusse once more. One of the stimulative reasons for conucting present theoretical estimations was the appearance of high efficient installation for proucing inexpensive nonopowers in Scientific an Research Institute of Vacuum Technique. We can anticipate that TVSs mae of nanopowers shoul have better basic thermoelectric characteristics than ones in bulk thermoelectric materials. Estimation of vacuum gap characteristics Here we consier the moel of the thermoelectric vacuum structure (TVS) compose of two flat thermoelectric plates of thickness separate by vacuum gap of with. We assume that thermoelectric material in plates have simple parabolic ban structure electron energy is counte off from the bottom of the conuction ban electrons can pass through the gap thanks to tunneling effect an phonon thermal conuctivity in vacuum gap is absent. The electric an energy flux ensities of electrons through vacuum gap between two semi-infinite thermoelectric layers ere given by js qe hp vx ( fc fh) TEx p () qe qs hp vx ( Ex+ Et) ( fc fh) TEx p () where V x x-component of electron velocity in irection perpenicular to thermoelectric surface f c an f h Fermi- irac istribution functions in col left an hot right plates respectively q e electron charge m effective electron mass h p Plank constant T(E x ) transmission coefficient through potential barrier p =p x p y p z p x p y an p z - components of electron momentum V x = p x /m E = p x /m E t = (p x +p z )/m an E = E x + E t. Fermi-irac istribution functions in thermoelectric plates are given by fc Ex Et exp ( + Ef) + T () fh ( Ex+ Et Ef q U) exp + ( T + T) where E f Fermi level T an T+ T absolute temperature at col an hot plates respectively U potential ifference between plates K b Boltzmann constant. Introucing imensionless variables x=e x /K b T t=e t / K b T an = E f /K b T (reuce Fermi level) equations () an () can be written as js( ) qe m ( T) Tx ( fc fh) t x () π hp

2 qs( ) m ( T) π Tx ()x ( + y) ( fc fh) y x. (5) hp We introuce functions M() M() an M() by integrals: M( ) M( ) M( ) Tx exp x + t ( exp( x + t ) + ) Tx ()x ( + t)exp x + t ( exp( x + t ) + ) Tx ()x ( + t) exp x + t ( exp( x + t ) + ) t x (6) t x (7) t x. (8) Expaning (fc-fh) in series on small parameters T/T an q e V /K b T an leaving only the first expansion terms one can erive equations of the Seebeck coefficient electric an thermal conuctivities. The Seebeck coefficient αs() = U/ T is erive from (6) at conition j s () = : αs( ) qe M M (9) electric conuctivity of gap unit area σ s () = j s ()/V is erive from (6) at conitions T= an V : σs ( ) qe m T π hp M( ) () thermal conuctivity of gap unit area κ s () = qs()/ T is erive from (6) an (7) at conitions j s () = an T : κs( ) m T M M( ). () π hp M( ) Integrals (8) (9) rastically epen on the transmission coefficient of electron through vacuum gap T(E). In general T(E) can be presente as TE ( ) W sinh m( W( ) E) E( W( ) E) hp + W Wo.5 q ln( ) εo π () () where W() effective barrier height Wo barrier height counte off from bottom of conuction ban an approximately equal to work function of semiconuctor. The secon term in () takes into account the mirror forces that lower barrier height. T(E) ecrease very rapily with increase of approximately by 5 orers of value at increase on.5 nm. The integran functions in equations (6)- (8) have the sharp maximum at electron energy about.5-.8 K b T. The optimal value of opt at which the transmission coefficient reaches maximum value is erive from equation () at W() =. So opt =.6 nm for Wo= ev an opt =. nm for Wo= ev. Estimation of thermoelectric plates characteristics The basic characteristics of semiconuctors with any level of generation are given by following equations []: σ ( r) qe κe r αr ( ) αo( r) qe () r+ m ( Th) τo() r σo( r) (5) hp ( π).5 σ ( r) Th q κe( r) κph( r) κr Z r + αr ( ) σ ( r) L( r) κr (6) (7) where α(r) the Seebeck coefficient σ e (r) electric conuctivity κ e (r) electron component of thermal conuctivity κ ph - phonon component of thermal conuctivity κ(r) = κ e (r) + κ ph total thermal conuctivity Z(r) the figure of merit an imensionless functions α o (r) σ o (r) L(r) an F(r) are given by: 5 αo( r) r σo r L( r) F( r) r + F r + r + F r + + F r + Γ 7 r + F r 5 + r + F r + ε r x e x + 5 r + F r + r + F r + where Γ(r) Gamma function r exponent of electron energy in the relaxation time τ(e) = τo(r) E r τo(r) constant ifferent for ifferent mechanisms of the scattering of electrons r = l/ for electron scattering on thermal phonons an r = / for electron scattering on ionize atoms of impurities.

3 Estimation of flat TVSs effective characteristics It is evient that for calculation of basic characteristics of flat thermoelectric vacuum structure it is enough to consier only two semiconuctor plates with vacuum gap between them connecte in a series. In this case the effective characteristics of TVSs can be easily erive: K( ) qe α αeff( ) α (8) κs( ) + κ σeff κeff Zeff ( + ) σe σe( ) + σs κ κ ( ) ( + ) + κs σeff( ) αeff κeff (9) (). () For more easy comparison an analysis of results of calculations we introuce imensionless characteristics that are etermine as ratio of TVS effective characteristics to corresponing bulk thermoelectric characteristics by the following equations γ(): γα γσ γκ γz α ( ) ( ) κ αeff σeff σe κeff Z( ) Zeff. () where γα() γσ() γκ() an γz() correspon to the Seebeck coefficient electric an thermal conuctivity an the figure of merit respectively. Results of calculations of TVSs The calculations were conucte for two variants of the relaxation time for scattering electrons on ionize atoms of impurities (r=/) an on thermal phonons (r=.). We use the following characteristics of thermoelectric material: effective electron mass m =.5m o constants in the relaxation time τo( /) = s J.5 an τo(/) =. 7 s J -.5 for ion concentration of Ni= m - lattice thermal conuctivity κ ph = W/m K work function Wo= ev. At the room temperature T= K an for scattering electrons on thermal phonons the basic characteristics of thermoelectric material ha the following values: the Seebeck coefficient α= - V/K electric an thermal conuctivity σ e =9.8 /Ohm m an κ=.8 W/m K respectively an the figure of merit Z=.9 - /K. These characteristics are very close to characteristics of alloy of bismuth tellurie (Bi Te ). Relative characteristics γα() γσ() γκ() an γz() as functions of reuce Fermi level = E f /K b T thickness of thermoelectric layers in nm an with of vacuum gap in nm are presente in figures -5. Low inexes in expressions γ() i an p correspon to scattering electrons on ionize atoms of impurities an thermal phonons respectively. Calculations show that effective Seebeck coefficient ecreases an effective electric an thermal conuctivities increase with increase of reuce Fermi level. γα_i( γα_p ( γα_i( γα_p ( Fig.. epenence γα_i() an γα_p() on reuce Fermi level for =.6 nm an = an nm. Fig. shows that the effective Seebeck coefficient is equal to the Seebeck coefficient of bulk material γα_p()= for scattering electrons on thermal phonons so as coefficients have the same epenence on. In general α eff () is lower α() for scattering on ions an ecreases with increase. γσ_i( γσ_p( γσ_i( γσ_p( Fig.. epenance γσ_i() an γσ_p() on reuce Fermi level for =.6 nm an = an nm. Fig. shows that the relative electric conuctivities γσ_i() an γσ_p() ecrease roningly with increase. At the same time the relative thermal conuctivities γκ_i() an γκ_p() increase with increase an γκ_i() > γκ_p() for < an γκ_i() <

4 γκ_p() for >. It is interesting that the ecrease of relative thermal conuctivity is less than the ecrease of relative electric conuctivity in several times. γκ_i(..8.6 γκ_p( γκ_i( γκ_p(.. Fig.. epenence γκ_i() an γκ_p() on reuce Fermi level for =.6 nm an = an nm. γz_i( γz_p ( γz_i( γz_p ( Fig.. epenence γz_i() an γz_p() on reuce Fermi level for =.6 nm an = an nm. γz_i( γz_p( γz_i( γz_p( Fig. 5. epenence γz_i() an γz_p() on thermoelectric plate thickness for =.6 nm an = и. The relative figure of merit γz_i() an γz_p() ecrease roningly with increase. But if the γz_p()> in all range of variable then γz_i()> for <. an γz_i()< for >. i.e. Zeff_i of TVS is lower than Z of bulk thermoelectric material for >.. Fig. 5 shows that the relative figure of merit γz_i() an γz_p() increase with the ecrease of thickness of thermoelectric layers. The TVS characteristics strongly epen on with of vacuum gap. Zeff() increases an σ eff () rastically ecreases with increase. In orer to have TVS with electric conuctivity comparable with electric conuctivity of bulk material an Zeff>Z shoul have value close to opt. The gap with opt is inversely proportional to the work function W o. Therefore it is very important to use TE material or coating materials with low work function. Flat TVS is a goo moel for calculations of basic characteristics of other similar structures. Real materials with thermoelectric vacuum structure can be prouce from powers of ifferent semiconuctors. As a first approximation we can imagine power thermoelectric material as close-packe particles of spherical form that have a point contacts with each other. Then effective contact area for tunneling current can be evaluate as Sc( opt) π ( opt) () an effective unit area of sample for tunneling current can be presente as γs( opt ) π ( opt ). () Taking γ s =. we have maximum particle iameter max =9 nm for opt =.6 nm (W o = ev) an max =6 nm for opt =. nm (W o = ev). It is evient that energy structure of semiconuctor particles rastically changes with ecrease of its iameter besies particle surface conitions begin to play very important role in processes of electron tunneling. Estimation of influence of energy structure change an particle surface conitions on properties of nanopower thermoelectric materials shoul be mae in the nearest future. Conucte calculations show that thermoelectric nanopower materials mae of weakly egenerate semiconuctor materials ( = ) shoul have the electric conuctivity on - orer of value lower an the figure of merit in several times higher than ones in bulk materials an that these materials are promising in broa range of applications. Conclusions Flat thermoelectric vacuum structures (TVSs) have the following characteristics: - the figure of merit excees by -7 times an effective electric conuctivity is on - orers of value less than the same characteristics of bulk material in practically interesting range of Fermi level (= ) - the figure of merit increases an electric conuctivity

5 ecreases with ecrease of thermoelectric plate thickness - optimum with of vacuum gap at which figure of merit an electric conuctivity reach its maximums increases from.6 to. nm for ecrease of electron work function from to ev. For materials mae of nanopowers there exists the maximum particle iameter in range of - nm at which the figure of merit an electric conuctivity are close to the same characteristics of flat TVSs. The figure of merit of laminate power an porous thermoelectric materials can be significantly higher than the same characteristics of bulk materials. Mae estimations show the necessity of carrying out new calculations of characteristics of nanopower materials taking into account the change of energy structure an influence of surface state in nanoparticles as well as conuction of new experiments with thermoelectric nanopowers. From the practical point of view the increase of figure of merit in - times signifies the broa expansion of TE evice an system applications lets them to compete with an replace the existing compressor coolers freezers an air conitioners as well as gives unique opportunities to evelop high efficiency soli-state power generators using the waste heat of iesel engines metallurgical works an thermal power stations in temperature range of -9 K. References. Liorenko N.S. et al On influence of tunneling on the efficiency of thermoelectric evices Papers of Acaemy of Sciences of USSR Physics Vol. 86 No. 6 (969) pp L.I. Anatychuk Hanbook Thermo-elements an thermoelectric evices Kiev Naukova umka (979) p..