Increasing thermoelectric efficiency towards the Carnot limit
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Increasing thermoelectric efficiency towards the Carnot limit
Giulio Casati , , Carlos Mej´ıa-Monasterio , and Tomaˇz Prosen Center for Nonlinear and Complex Systems, Universit`a degli Studi dell’Insubria, Como Italy CNR-INFM and Istituto Nazionale di Fisica Nucleare, Sezione di Milano D´epartement de Physique Th´eorique, Universit´e de Gen`eve andPhysics Department, Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia (Dated: November 20, 2018)We study the problem of thermoelectricity and propose a simple microscopic mechanism for the increaseof thermoelectric efficiency. We consider the cross transport of particles and energy in open classical ergodicbilliards. We show that, in the linear response regime, where we find exact expressions for all transport co-efficients, the thermoelectric efficiency of ideal ergodic gases can approach Carnot efficiency for sufficientlycomplex charge carrier molecules. Our results are clearly demonstrated with a simple numerical simulation ofa Lorentz gas of particles with internal rotational degrees of freedom. PACS numbers: 72.15.Jf, 05.70.Ln, 05.45.-a
Although thermoelectricity was discovered almost 200years ago, a strong interest of the scientific community aroseonly in the 1950’s when Abram Ioffe discovered that dopedsemiconductors exhibit relatively large thermoelectric effect.This initiated an intense research activity in semiconductorsphysics which was not motivated by microelectronics but bythe Ioffe suggestion that home refrigerators could be built withsemiconductors [1, 2]. As a result of these efforts the thermo-electric material Bi Te was developed for commercial pur-poses. However this activity lasted only few years until themid 1960’s since, in spite of all efforts and consideration ofall type of semiconductors, it turned out that thermoelectricrefrigerators have still poor efficiency as compared to com-pressor based refrigerators. Nowadays Peltier refrigeratorsare mainly used in situations in which reliability and quietoperation, and not the cost and conversion efficiency, is themain concern, like equipments in medical applications, spaceprobes etc. In the last decade there has been an increasingpressure to find better thermoelectric materials with higherefficiency. The reason is the strong environmental concernabout chlorofluorocarbons used in most compressor-based re-frigerators. Also the possibility to generate electric powerfrom waste heat using thermoelectric effect is becoming moreand more interesting [1–3].The suitability of a thermoelectric material for energy con-version or electronic refrigeration is evaluated by the thermo-electric figure of merit Z , Z = σS κ , (1)where σ is the coefficient of electric conductivity, S is theSeebeck coefficient and κ is the thermal conductivity. TheSeebeck coefficient S , also called thermopower, is a measureof the magnitude of an induced thermoelectric voltage in re-sponse to a temperature difference across the material.For a given material, and a pair of temperatures T H and T C of hot and cold thermal baths respectively, Z is related to the efficiency η of converting the heat current J Q (between thebaths) into the electric power P which is generated by attach-ing a thermoelectric element to an optimal Ohmic impedance. Namely, in the linear regime: η = PJ Q = η carnot · √ ZT + 1 − √ ZT + 1 + 1 , (2)where η carnot = 1 − T C /T H is the Carnot efficiency and T = ( T H + T C ) / . Thus a good thermoelectric device ischaracterized by a large value of the non-dimensional figureof merit ZT .Since the 1960’s many materials have been investigatedbut the maximum value found for ZT was achieved for the ( Bi − x Sb x ) ( Se − y Te y ) alloy family with ZT ≈ . How-ever, values ZT > are considered to be essential for thermo-electrics to compete in efficiency with mechanical power gen-eration and refrigeration at room temperatures. The effortsrecently focused on a bulk of new advanced thermoelectricmaterials and on low-dimensional materials, and only a smallincrement of the efficiency, ZT . . , has been obtained [3].One of the main reasons for this partial success is a limitedunderstanding of the possible microscopic mechanisms lead-ing to the increase of ZT , with few exceptions [4]. From adynamical point of view, cross effects in transport have beenbarely studied [5, 6]. So far, the challenge lies in engineering amaterial for which the values of S , σ and κ can be controlledindependently. However, the different transport coefficientsare interdependent, making optimization extremely difficult.In this paper we take a completely different approach. In-spired by kinetic theory of ergodic gases and chaotic billiards,we show that large values of ZT , in principle approachingto Carnot’s efficiency, can be obtained when the energy ofthe carrier particles does not depend on the thermodynamicforces.In the linear response regime (see e.g. [7]), one writesthe general expressions for the heat current J Q and the elec-tric current J e through an homogeneous sample subjected toa temperature gradient ∂ x T and a electrochemical potentialgradient ∂ x ¯ µ as J Q = − κ ′ ∂ x T − T σS ∂ x ¯ µ ,J e = − σS ∂ x T − σ ∂ x ¯ µ . (3)Here and in what follows, we assume that the transport oc-curs along the x -direction and the temperature is given in unitswhere the Boltzmann constant k B = 1 .The electrochemical potential is the sum of a chemical andan electric part ¯ µ = µ + µ e , where µ is the chemical poten-tial of the particles and, if e is the particle’s charge, µ e = eφ is the work done by the particles against an external electricfield E = − ∂ x φ . From (3) the usual phenomenological rela-tions follow: if the thermal gradient vanishes, ∂ x T = 0 , then J e = − σ ∂ x φ = σ E , since for an isothermal homogeneoussystem µ is uniform. If the electric current vanishes, J e = 0 ,then ∂ x ¯ µ = S ∂ x T , which is the definition of the Seebeckcoefficient, and J Q = − κ ∂ x T where κ = κ ′ − T σS is theusual thermal conductivity (see e.g. [9]).From the theory of irreversible thermodynamics, µ and µ e cannot be determined separately; only their combination in ¯ µ appears in (3) [8]. Based on this equivalence, in what followswe take into account the chemical part only, i.e. , ¯ µ = µ .Our aim is to study thermoelectricity from an “energy trans-port” point of view. To linear order, the energy and parti-cle density currents J u and J ̺ respectively, can be written interms of the Onsager matrix L [9, 10] as J u = L uu ∂ x (cid:18) T (cid:19) + L u̺ ∂ x (cid:16) − µT (cid:17) ,J ̺ = L ̺u ∂ x (cid:18) T (cid:19) + L ̺̺ ∂ x (cid:16) − µT (cid:17) , (4)where J e = eJ ̺ . In the absence of magnetic fields, the On-sager reciprocity relations states that L is symmetric, L u̺ = L ̺u . From the entropy balance equation for open systems J u = J Q + µJ ̺ , (5)and substituting J Q in (3) in favor of J u and comparing theresulting equations with (4) it follows that the transport coef-ficients can be written in terms of the L -coefficients as σ = e T L ̺̺ , κ = 1 T det L L ̺̺ , S = 1 eT (cid:18) L u̺ L ̺̺ − µ (cid:19) . (6)Eqs. (4) and (6) are completely equivalent to the description(3). Furthermore, from Eq. (1), we obtain for the figure ofmerit ZT = ( L u̺ − µL ̺̺ ) det L . (7)Expressions (4) and (7) provide a very convenient way fornumerical or analytical evaluation of ZT for different kindsof dynamical models.The second law of thermodynamics only requires that L ispositive definite. Therefore, from (7) it is clear that the secondlaw does not impose any upper bound on the value of ZT .Furthermore, the crucial observation is that the Carnot’s limit ZT = ∞ is reached when the energy density current and theelectric current become proportional, since then det L = 0 .Suppose for example that both energy and charge are carried only by non-interacting particles, like in a dilute gas. Then themicroscopic instantaneous currents per particle at position x ∗ and time t , are j u ( x ∗ , t ) = E ( t ) v x ( x ( t ) , t ) δ ( x ∗ − x ( t )) ,j e ( x ∗ , t ) = ev x δ ( x ∗ − x ( t )) , (8)where E is the energy of the particle, x its position and v x its velocity along the field. The thermodynamic averages ofthe two currents (appearing in Eq. (4)) become proportionalprecisely when the variables E and v x are un-correlated J u = h j u i = h E ih v x i = h E i e h j e i = h E i J ̺ . (9)Therefore, ZT = ∞ follows from the fact that the averageparticle’s energy h E i does not depend on the thermodynamicforces. In the context of classical physics this happens forinstance in the limit of large number of internal degrees offreedom ( d.o.f. ), provided the dynamics is ergodic .We consider an ergodic gas of non-interacting, electricallyneutral particles of mass m with d int internal d.o.f. (rotationalor vibrational), enclosed in a d dimensional container. Tostudy the non-equilibrium state of such dilute poly-atomicgas we consider a chaotic billiard channel (Fig. 1) connectedthrough openings of size λ to two reservoirs of particles whichare idealized as infinite chambers with the same poly-atomicgas at equilibrium density ̺ and temperature T . From thereservoirs, particles are injected into the channel at a rate γ obtained by integration over energy of the appropriate canon-ical distribution to give γ = λ (2 πm ) / ̺T / . (10)The particle injection rate γ is related to the value of thechemical potential µ at the reservoirs which, for polyatomicmolecules with a total of D = d + d int d.o.f. , reads µ = T ln (cid:16) c D γT ( D +1) / (cid:17) , (11)where c D is a D -depending constant. Furthermore, averagingthe energy of the injected particles over the canonical distri-bution, denoted as h E i , we obtain the rate at which energy isinjected from the reservoirs as ε = γ h E i = γT ( D + 1) / .Let p t ( l ) be the transmission probability of the channel oflength l . For a billiard system of noninteracting particles [11],the density currents J u , J ̺ assume a simple form: they are p t ( l ) times the difference between the left and right corre-sponding injection rates, ε, γ , respectively, namely J ̺ = p t ( γ L − γ R ) , J u = p t ( ε L − ε R ) . (12)Using (11) to eliminate γ in favor of µ we obtain, J ̺ = − λp t ( l ) l (2 πm ) / ∂ x (cid:16) T ( D +1) / e µ/T (cid:17) ,J u = − λp t ( l ) l (2 πm ) / D + 12 ∂ x (cid:16) T ( D +3) / e µ/T (cid:17) , (13) T L , µ L T R , µ R FIG. 1: The open Lorentz gas system and a typical particle’s tra-jectory. The composite particle (schematically represented as amolecule) is scattered from fixed disks of radius R disposed ina triangular lattice at critical horizon, i.e. , the width and heightof the cells are ∆ x = 2 R and ∆ y = 2 W respectively, where W = 4 R/ √ is the separation between the centers of the disks. Thechannel is coupled at the left and right boundaries to two thermo-chemical baths at temperatures T L and T R and chemical potentials µ L and µ R , respectively. Taking total differentials of (13) in the variables /T and µ/T and comparing the resulting expression with Eq. (4) we obtainexact microscopic expressions for the Onsager coefficients,namely L ̺̺ = λp t ( l ) l (2 πm ) / ̺T / ,L ̺u = L u̺ = λp t ( l ) l (2 πm ) / (cid:18) D + 12 (cid:19) ̺T / ,L uu = λp t ( l ) l (2 πm ) / ( D + 1)( D + 3)4 ̺T / . (14)Note that for a chaotic billiard channel with a diffusive dy-namics, the transmission probability decays as p t ( l ) ∝ l − which means that all the elements of the Onsager matrix L become size independent.Finally, plugging (14) into (7) and noting that c ∗ V = D/ isthe dimensionless heat capacity at constant volume of the gas,we obtain ZT = 1ˆ c V (cid:16) ˆ c V − µT (cid:17) , (15)where for simplicity we have called ˆ c V = c ∗ V + 1 / . A partic-ular case of (15) was previously obtained, for noninteractingmonoatomic ideal gases in dimensions [12].In absence of particles’ interaction, ZT is independent ofthe sample size l and depends on the temperature only throughthe chemical potential term. This is due to the fact that with nointeractions, p t depends on the geometry of the billiard only.From a physical point of view this means that the mean freepath of the gas particles is energy independent. Were particlesinteracting, p t would depend on the local density and tem-perature of the gas, leading to a more realistic situation [13]. Interestingly enough, we have found that Eq. (15) is an up-per limit of the interacting case when the interaction strengthvanishes [14].We shall now confirm Eq. (15) with a very simple numericaldemonstration of a -dimensional chaotic Lorentz gas channelof particles elastically colliding with circular obstacles of ra-dius R . In what follows, we fix the unit length setting R = 1 .The geometry of the model is shown in Fig.1.We consider composite particles with d int ≥ internal ro-tational d.o.f. . Each “particle” of mass m can be imagined asa stack of d int small identical disks of mass m/d int and ra-dius r ≪ R , rotating freely and independently at a constantangular velocity ω i , i = 1 , . . . d int . The center of mass of theparticle moves with velocity ~v = ( v x , v y ) .At each collision of the particle with the boundary of thebilliard (either one of the circular obstacles or the outer wall)an energy exchange among all the D d.o.f. occurs accordingto the following collision rules v ′ n = − v n ,v ′ t = 1 − ηd int ηd int v t + 2 η ηd int d int X k =1 ω k ,ω ′ i = 21 + ηd int v t d int X k =1 (cid:18) δ ik − d int (1 + ηd int ) (cid:19) ω k , (16)where ( v n , v t ) are the normal and tangent components ofthe particle velocity at the collision point and the parameter η = Θ /mr , with Θ being the moment of inertia of eachsmall fictitious internal disk. The primed (unprimed) quan-tities refer to their values after (before) the collision. Thesecollision rules are a generalization of the ones introduced in[6]. Thus, they are deterministic, time reversible and preservethe energy and local angular momentum. The derivation ofthe collision rules will appear elsewhere [14].First we considered a closed system in a finite containerand we checked energy equipartition among all d.o.f. . Thenwe have opened the system from both ends and allowed it toexchange particles with the two baths at temperatures T L , T R and with chemical potentials µ L , µ R . The coupling among thesystem and the baths is defined as follows: whenever a parti-cle in the system crosses the opening which separates it fromthe bath, it is removed from the system. On the other hand,with frequency γ , particles are injected into the system, with avelocity distributed according to the canonical distribution atthe corresponding temperature. P n ( v n ) = mT | v n | exp (cid:18) − mv n T (cid:19) ,P t ( v t ) = r m πT exp (cid:18) − mv t T (cid:19) ,P ( ω i ) = r m DπT exp (cid:18) − mω i DT (cid:19) , (17)reflecting the assumption that the bath is an ideal gas at equi-librium temperature T . D ZT l ZT FIG. 2: Figure of merit ZT as a function of the number of d.o.f. , at µ = 0 . For each D , the transport coefficients were obtained from twodifferent simulations in a channel of cells, at mean particle density ̺ = 0 . and mean temperature T = 1000 with: a ) ∆ T /T =0 . , ∆( µ/T ) = 0 , and b ) ∆ T = 0 , ∆( µ/T ) = 0 . . The injectionrates γ L and γ R are obtained from ∆( µ/T ) , using (11) and (10). Thedashed line corresponds to ( D + 1) / . In the inset, the dependenceof ZT on the length of the channel l is shown for D = 9 . The dashedline shows the corresponding expected value. By two different simulations with two linearly indepen-dent sets of thermodynamic forces we have numerically de-termined the Onsager’s matrix L and the value of ZT (seeFig.2). Numerical results excellently confirm the theoreticalprediction (15). We have also carefully checked that all On-sager coefficients, or conductivities, decay with the size l ofthe system as /l which indicates a diffusive transport.The simple mechanism for the growth of ZT with d int isnicely illustrated in Fig.3 which shows that the particle veloc-ity v x has a Maxwellian (Gaussian) distribution (inset), whilethe equilibrium distribution of the particle energy per degreeof freedom E D becomes more and more sharply peaked, andthus de-correlated from v x as d int grows.In conclusion, we have discovered a simple general theoret-ical mechanism which may find a way to applications of ther-moelectricity in real world materials. Even though the case ofan ionized polyatomic gas may seem a little artificial in thiscontext, there may be other important instances where eachcharge would be carried by many effectively classical d.o.f. .We have also performed the first numerical computation of ZT from deterministic microscopic equations of motion. Ourmethod can easily be implemented for more realistic modelswhere also quantum effects can be taken into account.The authors are indebted to H. Linke and C. Vining forenlightening discussions and correspondence, and thank thehospitality of the Institut Henri Poincar´e, Paris, where part ofthis work was done. TP acknowledges support from grantsP1-0044 and J1-7347 of Slovenian research agency. CMMacknowledges support from Fonds National Suisse and a La- grange fellowship from Fondazione CRT. E D /T P ( E D / T ) -100 0 100 ξ i P ( ξ i ) FIG. 3: Probability distribution function of the energy per degree offreedom E D = E/D determined from equilibrium simulation with T L = T R , µ L = µ R , for different number of freedoms: D = 3 (red), D = 13 (green) and D = 43 (blue). The dashed curves arethe theoretical (“Chi-square” χ D ) distributions of E D . In the inset,the corresponding probability distribution functions for the x - com-ponent of the velocity P ( v x ) (blue) and for the angular momentumof one of the particle’s disks P ( ω i ) (green) is shown, for D = 3 .The dashed curves are the theoretical Gaussian distributions.[1] G. Mahan, B. Sales, J. Sharp, Phys. Today , 42 (March 1997).[2] A. Majumdar, Science , 778 ( 2004).[3] M. S. Dresselhaus et al , Adv. Mater. , 1043-1053 (2007).[4] T. E. Humphrey and H. Linke, Phys. Rev. Lett. , 096601(2005).[5] C. Maes and M. H. van Wieren, J. Phys. A: Math. Gen. (2005) 1005; J. Vollmer, T. Tell, and L. M´aty´as, J. Stat. Phys. (2000) 79; C. Van den Broeck, in Advances in ChemicalPhysics
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Eds. S. A. Rice, (2007 John Wiley & Sons,Inc.).[6] C. Mejia-Monasterio, H. Larralde, and F. Leyvraz, Phys. Rev.Lett. (2001) 5417.[7] D. J. Bergman and O. Levy, Appl. Phys. (1991) 6821.[8] P. L. Walstrom, Am. J. Phys. (1988), 890. See also H. Lar-ralde, F. Leyvraz, and C. Mejia-Monasterio, J. Stat. Phys. (2003), 197, where the equivalence between µ and µ e has beenstudied for a similar model to the one we study here.[9] C. A. Domenicali, Rev. Mod. Phys. (1954), 237.[10] H. B. Callen, Phys. Rev. (1948) 1349; S. R. de Groot and P.Mazur, Non-equilibrium Thermodynamics (Dover, New York,1984).[11] Assuming that particles do not interact means that we fully ne-glect the phonon transport.[12] C. B. Vining, Mat. Res. Soc. Symp. Proc. (1997) 3.[13] J.-P. Eckmann, C. Mejia-Monasterio, and E. Zabey, J. Stat.Phys. (2006) 1339.[14] G. Casati, C. Mejia-Monasterio, and T. Prosen, to be publishedto be published