Statistical analysis of the figure of merit of a two-level thermoelectric system: a random matrix approach
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Statistical analysis of the figure of merit of a two-level thermoelectric system: arandom matrix approach
A. Abbout, ∗ H. Ouerdane,
2, 3 and C. Goupil Physical Science and Engineering Division, King Abdullah Universityof Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia Russian Quantum Center, 100 Novaya Street, Skolkovo, Moscow Region 143025, Russia UFR Langues Vivantes Etrang`eres, Universit´e de Caen Normandie, Esplanade de la Paix 14032 Caen, France Laboratoire Interdisciplinaire des Energies de Demain,LIED/CNRS UMR 8236 Universit´e Paris Diderot; Bˆat. Lamarck B 35 rue H´el`ene Brion 75013 Paris (Dated: July 31, 2018)Using the tools of random matrix theory we develop a statistical analysis of the transport proper-ties of thermoelectric low-dimensional systems made of two electron reservoirs set at different tem-peratures and chemical potentials, and connected through a low-density-of-states two-level quantumdot that acts as a conducting chaotic cavity. Our exact treatment of the chaotic behavior in suchdevices lies on the scattering matrix formalism and yields analytical expressions for the joint prob-ability distribution functions of the Seebeck coefficient and the transmission profile, as well as themarginal distributions, at arbitrary Fermi energy. The scattering matrices belong to circular ensem-bles which we sample to numerically compute the transmission function, the Seebeck coefficient, andtheir relationship. The exact transport coefficients probability distributions are found to be highlynon-Gaussian for small numbers of conduction modes, and the analytical and numerical results arein excellent agreement. The system performance is also studied, and we find that the optimumperformance is obtained for half-transparent quantum dots; further, this optimum may be enhancedfor systems with few conduction modes.
I. INTRODUCTION
Low-dimensional systems offer a wealth of technolog-ical possibilities thanks to the rich variety of artificialcustom-made semiconductor-based structures that nowa-days may be routinely produced. The properties ofthese systems, including confinement geometry, densityof states, and band structure, may be tailored on de-mand to control the transport of heat and the trans-port of confined electrical charges, as well as their cou-pling. In the current context of intense research in en-ergy conversion physics, this is crucial to further im-prove the performance and increase the range of oper-ation of present-day thermoelectric devices. These latterare characterized by their so-called figure of merit, which,in the frame of linear response theory, may be expressedas: ZT = σs T /κ , where s is the Seebeck coefficient, T is the average temperature across the device, and σ and κ are the electrical and thermal conductivities respec-tively, with κ = κ e + κ lat , accounting for both electronand lattice thermal conductivities.Amongst the various low-dimensional thermoelectricsystems that have been studied, quantum dots, whichare confined electronic systems whose size and shapeare controlled by external charged gates, keep attractingmuch interest because of their narrow, Dirac-like, elec-tron transport distribution functions or, equivalently,sharply peaked energy-dependent transmission profiles T , which permit obtainment of extremely high valuesof electronic contribution to ZT . There is a simple rela-tionship between the electrical conductivity σ appearingin the definition of the figure of merit and the trans-mission function T : σ = (2 e /h ) T , the proportionality factor being the quantum of conductance ( e being theelectron charge, and h being Planck’s constant).Models of quantum dots form two broad categories,namely interacting and noninteracting models. Thoseaccounting for electron-electron interactions permit anal-ysis of a rich variety of physical phenomena governed byelectronic correlations like, e.g., Coulomb blockade andthe related conductance oscillations vs. gate voltage ,peak spacing distributions , and phase lapses of thetransmission phase . The use of non-interacting modelsystems for quantum dots or very small electronic cavi-ties may be justified if these are strongly coupled to thereservoirs, i.e. if the confinement yields a mean levelspacing that is large compared to the charging energy e /C , C being the capacitance of the dot . So, thoughnoninteracting dot models may be seen as toy models,these may provide in some cases a number of insightfulresults without the need to resort to advanced numeri-cal techniques such as, e.g., the density matrix numeri-cal renormalization group . This is illustrated by, e.g.,the resonant level model of thermal effects in a quan-tum point contact , Fano resonances in the quantum dotconductance , and simple models of phase lapses of thetransmission phase .Interesting physics problems in a quantum dot alsostem from the chaotic dynamics that may be triggeredbecause of structural disorder, or as the quantum dot it-self behaves as a driven chaotic cavity because its shapevaries as one of the gates generates a random potential.In these cases, as explained in Ref. , one may be in-terested in the statistics of the system’s spectrum ratherthan the detailed description of each level. Random ma-trix theory provides tools of choice for this purpose;and, for quantum dots in particular, one may constructa mathematical ensemble of Hamiltonians that satisfiesessentially two constraints: the Hamiltonians belong tothe same symmetry class and they have the same av-erage level spacing, while the density of states must bethe same for the physical ensemble of quantum dots . Inthe context of thermoelectric transport, measurements ofthermopower and analysis of its fluctuations based onthe random matrix theory of transport demonstratedthe non-Gaussian character of the distribution of ther-mopower fluctuations.In this article, we concentrate on the statistical anal-ysis of the thermoelectric properties of noninteractingquantum dots connected to two leads that serve as elec-tron reservoirs set at different temperatures and electro-chemical potentials. We analyze the statistics of ZT ina two-level system connected to two electronic reservoirsset to two slightly different temperatures so that the tem-perature difference, ∆ T , is small enough to remain in thelinear response regime: the voltage induced by thermo-electric effect, ∆ V , is given by s = − ∆ V / ∆ T . The mu-tual dependence of the transport coefficients that define ZT (e.g., the Wiedmann-Franz law for metals) raises theproblem of finding which configuration of the mesoscopicsystem may yield the largest values of ZT , and henceoffer optimum performance.The two-level model presented in this article is theminimal model pertinent for the description of a cav-ity with two conducting modes presenting a completelychaotic behavior . We consider Fermi energies lying inthe vicinity of the spectrum edge of the system’s Hamil-tonian. This situation is in contrast with the case ofhigh level cavities for which the typical transmission pro-files vary much and exhibit numerous maxima and ex-tinctions, which precludes an analytical formulation ofthe Seebeck coefficient probability distribution since theCuttler-Mott formula does not apply. Although one mayconsider a very low temperature regime to concentrateonly on a small interval of energies, the need of a largenumber of levels to ensure the properties of the bulk uni-versality in chaotic systems makes this window of ener-gies so small that the corresponding temperature tendsto become insignificant. Conversely, at the edge of theHamiltonian spectrum, the typical profile of the trans-mission is smooth and the analytical treatment of theprobability distribution of thermopower is much easier .Indeed, at this class of universality (spectrum edge) thedescription of this kind of systems may rest on the equiva-lent minimal chaotic cavity defined as the system witha number of levels N equal to the number of the con-ducting modes M . By “equivalent” we mean that, at thespectrum edge, the original system and the correspondingminimal chaotic cavity lead to the same statistics thoughthey certainly have different results for one realization.The article is organized as follows: In the next sec-tion, we introduce the model, the main definitions andnotations we use throughout the paper. In Sec. III, wepresent two derivations of the Seebeck coefficient in the (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:5)(cid:1)(cid:7)(cid:4)(cid:11)(cid:4)(cid:7)(cid:12)(cid:8)(cid:10)(cid:7)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:5)(cid:1)(cid:7)(cid:4)(cid:11)(cid:4)(cid:7)(cid:12)(cid:8)(cid:10)(cid:7) (cid:13) (cid:14) (cid:13) (cid:15) FIG. 1. (Color online) Schematic figure of the two level-system consisting of negatively charged top gates (yellow) on atwo-dimensional electron gas. The red (blue) part representsthe hot (cold) electronic reservoir connected to the cavity.The inset shows the system as modeled by the Hamiltonian. scattering matrix framework: one is general, while theother is made under the restrictive assumption of left-right spatial symmetry. The formal identity of the two re-sults serves as a basis for the statistical analysis presentedin Sec. IV, which is the core of the article. We analyzethe numerical statistical results obtained for the proba-bility distributions of the figure of merit, thermopower,and power factor. In Sec. V, we analyze the relation-ship of the Seebeck coefficient to the density of states ofthe system. In Sec. VI, we extend the discussion to thecase of a lattice. The article ends with a discussion andconcluding remarks. A Supplemental Material detailingsome of the numerical aspects of the work as well as somederivations, accompanies the article . II. MODEL
We consider a low-density-of-states two-level quantumdot depicted in Fig. 1. The potential on the right threetop gates may be varied randomly in order to slightlymodify the shape of the cavity and therefore obtain astatistical ensemble . A. Preliminary remarks on the minimal chaoticcavities
Studies of confined systems may be achieved withoutloss of pertinence by means of simple models, which cap-ture the essential physics of actual systems. A chaoticcavity with N energy levels is modeled with an N × N Hamiltonian. In the frame of random matrix theory, toobtain a scattering matrix from circular ensembles, theHamiltonian is distributed from Lorentzian ensembles;other distributions, e.g., the Gaussian distribution, maylead to the same results only for large N in the bulkspectrum. For a fixed value of N , if the Fermi energy ap-proaches the bottom of the conduction band (continuumlimit) we enter a new universality class: the edge of theHamiltonian spectrum . The key point here, which willprove extremely useful as shown in the present article, isthat at the edge of the spectrum, all the distributions ob-tained with an N × N Hamiltonian may also be found byconsidering a simpler case with M × M matrices, with M ( < N ) being the number of conducting modes. This typeof cavity is called the minimal chaotic cavity , and permitsto lower N down to N = M to simplify the mathematicaland computational treatment of the problem, and derivephysically meaningful results. As we show, it allows ob-tainment of very simple formulas for the transport coeffi-cients because the Hamiltonian and the scattering matrixthen have the same size. It is important because, at theedge of the spectrum (large level spacing), it gives thesame result as for the original system ( N > M ). Thisapproach is justified for probing the spectrum edges ofmesoscopic systems where the physics is differentfrom that of the bulk spectrum . B. Scattering matrix
The two-level system tight-binding Hamiltonian readsas the sum of three contributions: H = H ℓ + H s + H c ,where H ℓ = P k ǫ k c † k c k is the Hamiltonian of the two (i.e.,left and right) leads; c † k and c k are the second-quantizedelectron creation and annihilation operators in the state k . The two energy levels and their coupling are charac-terized by the Hamiltonian H s : H s = (cid:18) v v v v (cid:19) (1)where the random on-site potentials v , v and coupling v describe the chaotic behavior of the system. Weassume that the left (right) lead is only connected tothe site v ( v ). The contribution H c involves the cou-pling matrix W entering the definition of the scatteringmatrix : S = 1 − πi W † ǫ − H s − Σ W (2)where Σ is the self-energy of the two leads. To keepour analysis on a general level, we only give its form:Σ = Λ − iπ WW † , with Λ being the real part. Withthe assumption of symmetric reservoirs, the self-energyis proportional to the identity, in which case W = p Γ / π × , where Γ = − ℑ Σ characterizes the broad-ening. Throughout the article, we adopt the same no-tations for scalars and their corresponding matrix formwhen this latter is proportional to the identity.
III. DERIVATION OF THE SEEBECKCOEFFICIENT
It is instructive to derive an analytic expression of theSeebeck coefficient under the restrictive assumption ofspatial left-right symmetry, on the one hand, and com-pare the result to the that obtained assuming arbitraryenergy and arbitrary leads.
A. Left-right spatial symmetry
The scattering matrix S is unitary. In the absenceof a magnetic field, time reversal symmetry is preservedand the matrix S is therefore symmetric, which implies v = v . Moreover, for simplicity, we consider in thefirst part of this article, systems with left-right spatialsymmetry ( v = v ), so that the reflection from left isthe same as that from right. The matrix S may thusread: S = (cid:18) r tt r (cid:19) (3)where r and t are the reflection and transmission ampli-tudes respectively. With these definitions, the transmis-sion of the system and the Seebeck coefficient read: T = | t | , and s = ∂ ln( T ) ∂ǫ (4)The Seebeck coefficient is obtained at low temperatureswith the Cutler-Mott formula . Here, it is expressed inunits of π e k T (ommiting the sign). The transmissionof the system can then directly be obtained using theFisher-Lee formula : T = Γ G Γ G † (5)where the off-diagonal element of the Green’s matrix is G =
12 ( ǫ − ǫ )( ǫ − ǫ − Σ)( ǫ − ǫ − Σ) , with ǫ and ǫ being the eigen-values of H s . We may thus write: T ( ǫ ) = Γ ǫ − ǫ ) | ( ǫ − ǫ − Σ)( ǫ − ǫ − Σ) | (6)which is valid if the left/right symmetry condition is sat-isfied. Now, combination of Eqs. (4) and (6) yields anexpression of the Seebeck coefficient containing the scat-tering matrix S : s = α (cid:18) e i Θ S − e − i Θ S † i (cid:19) (7)with α = 4 | − ˙Σ | / Γ, Θ = Arg(1 − ˙Σ), and ˙Σ ≡ ∂ ǫ Σ. Foran energy ǫ = ǫ , which corresponds in general to themiddle of the conduction band of a semi-infinite lead (orhalf-filling limit), the imaginary part of the self-energyderivative vanishes: ℑ ˙Σ( ǫ ) = 0. B. General derivation
The starting point is the transmission of the two-levelsystem: T = Γ v | ( ǫ − ǫ − Σ)( ǫ − ǫ − Σ) | (8)This formula applies both in presence or absence of theleft/right spatial symmetry. The Seebeck coefficient inunits of π k T ) e reads: s = ∂ ǫ ln( T ). The application ofthis formula to T leads to: s = ∂ ǫ ln(Γ ) − − ˙Σ ǫ − ǫ − Σ + 1 − ˙Σ ǫ − ǫ − Σ + H.c. ! (9)The relation between the scattering matrix and theHamiltonian is simple in the case of a two-level nonin-teracting system: S = 1 − i Γ 1 ǫ − H − Σ → ǫ − H − Σ = 1 − S i Γ (10)Combination of the above expression with Eq. (9) gives: s = ∂ ǫ ln(Γ ) − (1 − ˙Σ)Tr (cid:18) − S i Γ (cid:19) +(1 − ˙Σ ⋆ )Tr (cid:18) − S † i Γ (cid:19) (11)where the star symbol denotes the complex conjugate.Then we may rewrite s as: s = ∂ ǫ ln(Γ ) − − ˙Σ)Tr (cid:18) S i Γ (cid:19) − (1 − ˙Σ ⋆ )Tr (cid:18) S † i Γ (cid:19) (12)which reduces to Eq. (7). This expression has been ob-tained because the scattering matrix and the Hamilto-nian have the same size (hence the interest in using theequivalent 2-level system); it corresponds to a system forwhich, in the continuum limit and at low Fermi energies,one deals with a universality class different from that ofthe bulk. We stress that Eq. (7) constitutes an importantresult upon which all the subsequent statistical analysisis based. IV. STATISTICAL ANALYSISA. Statistics of the Seebeck coefficient
In this work, the scattering matrix S is a randomvariable, which we assume to be uniformly distributed; as such, S belongs to circular orthogonal ensembles(COE) . The use of the circular ensemble is based on theequal a priori probability ansatz . This is a naturalchoice when there is no reason to privilege any scatter-ing matrix, in which case the mean of the distributionis hSi = 0. For a general case, obtained for different pa-rameters than those leading to CE , we obtain a moregeneral distribution called the Poisson kernel uniquelydetermined by its non vanishing mean scattering ma-trix hSi 6 = 0 . Since it is always possible to define anew unitary matrix uniformly distributed using matrixtransformations , the subsequent calculations are de-veloped considering only the circular ensemble.The statistics of the Seebeck coefficient thus follows: P ǫ ( s ) = Z δ (cid:26) s − α (cid:18) e i Θ S − e − i Θ S † i (cid:19)(cid:27) δ H S (13)where δ H S denotes the Haar measure, which is invariantunder the transformation S → V S V T for any arbitraryunitary matrix V (the superscript T denotes the trans-pose of a matrix). With V = e i Θ / × , we obtain: P ǫ ( s ) = Z δ ( s − α ˜ S − ˜ S † i !) δ H ˜ S = 1¯ α P ǫ = ǫ ( s/ ¯ α )(14)where ¯ α = α/α with α = α ( ǫ ). Equation (14) showsthat the general probability distribution of the Seebeckcoefficient at arbitrary energy can be deduced directly from the simpler one obtained at ǫ . Note that for easeof notations, we retain P as a generic notation for all theprobability distributions in the subsequent parts of thearticle. B. Joint probability distribution function
A similar analysis of the transmission T shows thatits distribution does not depend on energy . Since thetransport coefficients T and s are tightly related by thetransport equations, the knowledge of the joint probabil-ity distribution function (j.p.d.f) P ǫ ( s, T ) is required tostudy any observable depending on both T and s . Thestarting point for this is the rewriting of the scatteringmatrix using the following decomposition: S = R T (cid:18) e iθ e iθ (cid:19) R (15)where the rotation matrix R is defined as: R = 1 √ (cid:18) −
11 1 (cid:19) (16)The eigenphases θ and θ are independent and uniformlydistributed in [ − π, + π ] which is the consequence of S being taken from a COE with the left-right symmetry .Now, we analyze the j.p.d.f. of the Seebeck coefficientand transmission at the half-filling limit ǫ = ǫ , with: T = sin [( θ − θ ) /
2] (17)and s = α sin [( θ + θ ) /
2] cos [( θ − θ ) /
2] (18)The j.p.d.f. may now take the form: P ǫ = ǫ ( s, T ) = h δ (cid:0) T − sin u (cid:1) δ ( s − α sin v cos u ) i u,v (19)where two independent and uniformly distributed vari-ables u = θ − θ and v = θ + θ are introduced. As shownin the Supplemental Material, integration over these vari-ables and use of Eq. (14) yield: P ǫ ( s, T ) = 1 π p T (1 − T ) 1 π p α (1 − T ) − s (20)which constitutes one of the main results of this worksince it shows the possible values of the couple ( S, T ): P ǫ ( S, T ) is non-zero only if the following condition is sat-isfied: α (1 − T ) − s > , T < S matricesbelonging to the COE with the left/right symmetry ,from which the transport coefficients T and s are numer-ically computed with Eqs. (4) and (7). The parabolic lawis clearly shown on Fig. 2 where the j.p.d.f. P ǫ ( s, T ) isplotted against T and s .Note that if we had used the Poisson Kernel distribu-tion, which only changes the probability weight of eachconfiguration of the system, the condition (21) and theensuing conclusions would still hold. C. System performance
From a thermodynamic viewpoint, thermoelectric sys-tems connected to two thermal baths at temperatures T hot and T cold , use their conduction electrons as a work-ing fluid to directly convert a heat flux into electricalpower and vice-versa with efficiency η . As for all heat en-gines, one seeks to increase either their so-called efficiencyat maximum ouput power, η P max , as discussed in numer-ous recent papers (see, e.g., Refs. for mesoscopic sys-tems and Ref. for fundamental questions related to ir-reversibilities) or simply maximize the efficiency η . Forsimple models in thermoelectricity, an expression of themaximum of η , is related to the figure of merit of thesystem ZT : η max = η C √ ZT − √ ZT + T cold /T hot (22) FIG. 2. (Color online) Probability densities: 1)Top left: P ( T )for the transmission. 2) Bottom left: P ( s ) for the Seebeck co-efficient. 3) Top right: j.p.d.f. of the couple ( s, T ) at the half-filling limit. 4) Bottom right: j.p.d.f. of the couple ( T , ZT )at the half-filling limit. The blue curves in the left figures areobtained by sampling scattering matrices from circular or-thogonal ensembles and the orange dashed curves representsthe analytical result of the correponding physical observable.The matching is excellent. where η C = 1 − T cold /T hot is the Carnot efficiency.Equation (22) clearly shows that for given working con-ditions, ZT is as good a device performance measure asthe efficiency η is. The maximum of the figure of merit ZT is reached when the maximum of the power factor p = T s is reached. Satisfaction of this latter conditionlies on the existence of the probability P ǫ ( s, T )): Eq.(21) implies that α T (1 − T ) > T s , which in terms ofpower factor implies that max( α T (1 − T )) > p , so that: α/ > p (23)We thus find that ZT reaches its maximum if the trans-mission T = 1 / κ is constant; this is justified on thecondition that κ e ≪ κ lat42 ( κ lat is assumed to be con-stant). Relaxation of this assumption makes our systemperformance analysis applicable to the power factor only,but not to the figure of merit. D. Marginal distributions
The marginal distributions P ǫ ( T ) and P ǫ ( s ) are ob-tained from the integration of P ǫ ( s, T ) over s and T re-spectively: P ǫ ( T ) = 1 π p T (1 − T ) (24) P ǫ ( s ) = 2 απ K (cid:16)p − s /α (cid:17) (25)where K is the complete elliptic integral of the first kind.It is interesting to note that P ǫ ( T ) does not depend onthe energy since it derives from the sole assumption thatthe scattering matrix is uniformly distributed (Dyson’sCE) with the left/right spatial symmetry . As for all thephysical observables with no energy derivative in theirexpression (shot noise for example), their distributionsdo not depend on the number of levels N in the system:for all N ≥
2, we obtain the same distribution, as it canbe seen with the decimation method on the conditionthat
S ∈ { CE } .The case of the Seebeck coefficient is different: its dis-tribution P ǫ ( s ) depends on the number of levels in thesystem since its expression contains an energy deriva-tive. The only situation where the Seebeck coefficient,scaled with the appropriate parameter ( α in our case),is independent on the number of levels is at the edge ofthe Hamiltonian spectrum where the result can be ob-tained by assuming the simpler two-level system consid-ered as the minimal cavity for the two-mode scatteringproblem . The two-level model provides simple equa-tions but yields general results, which apply to the N -level model at low density.All the results presented so far in this article were ob-tained and discussed assuming a two-level system withleft/right spatial symmetry and S ∈ { CE } . We must nowsee whether these hold when the assumption of left/rightspatial symmetry is relaxed. E. Relaxation of the spatial symmetry assumption
Assuming that there is no left/right symmetry, the twoeigenphases θ and θ of Eq. (15) are no longer indepen-dent, and we must consider a distribution of the form: P ( θ , θ ) ∝ | e iθ − e iθ | . The treatment is the same asthat for the previous case, and while we obtain a differentj.p.d.f. for the Seebeck and the transmission coefficients: P ǫ ( s, T ) = 1 απ p T (1 − T ) K (cid:16)p − s /α (1 − T ) (cid:17) (26)the mathematical constraint on the couples ( T , s ) to ob-tain a non-vanishing joint probability distribution is ex-actly the same as Eq. (23); this implies that the best sys-tem performance is obtained when it is half-transparent: T = . The marginal distributions of s and T are also differ-ent: P ǫ ( T ) = 12 √T (27) P ǫ ( s ) = − απ ln | s | /α p − s /α ! (28)Equation (27) is consistent with results of Refs. ob-tained with different methods and Eq. (28) confirms theresult of Ref. . We see in both distributions, Eqs. (20)and (26), that the variables ( s, T ) are not independentbut it is interesting to note that if we define a new vari-able X = s/ p (1 − T ) then we have two independentvariables, and the j.p.d.f. becomes multiplicatively sepa-rable: P ǫ ( X, T ) = P ǫ ( T ) × P ǫ ( X ) (29)where we have P ǫ ( X ) = 1 /π √ α − X in the symmetriccase and P ǫ ( X ) = απ K ( p − X /α ) when the symme-try is relaxed; P ǫ ( T ), given above, is the correspondingmarginal distribution. It is worth mentionning that S and X also constitute a couple of independent variables. F. Statistics of the density of states
The density of states in the two-level quantum dot maybe expressed with the usual formula: ρ ( ǫ ) = − π ℑ Tr G = − π ℑ Tr 1 ǫ − H − Σ (30)which, using the scattering matrix, thus reads: ρ ( ǫ ) = − π ℑ Tr 1 − S i Γ = − π i Tr (cid:18) − S i Γ − − S † − i Γ (cid:19) (31)and simplifies to: ρ ( ǫ ) = − π i (cid:20) i Γ − Tr (cid:18) S + S † i Γ (cid:19)(cid:21) (32)so that δρ ( ǫ )¯ ρ ( ǫ ) = 12 Tr (cid:18) S + S † (cid:19) (33)where ¯ ρ ( ǫ ) = 2 /π Γ. Here, we used the fact that forcircular ensembles we do have ¯ S = ¯ S † = 0 (where theover bar denotes the mean). The local density of states inthe central system, when connected to the leads, reads : ρ ( ǫ ) = − π ℑ Tr 1 ǫ − H s − Σ (34)which combined with Eq. (2) yields: δρ ( ǫ )¯ ρ ( ǫ ) = 12 Tr (cid:18) S + S † (cid:19) (35) FIG. 3. (Color online) Probability density of the power factor p for different values of the number of conduction modes: bluecurve M = 4, purple curve M = 8, and brown curve M = 16.The number of energy levels in the cavity equals the totalnumber of modes. In the inset, the standard deviation ofthe power factor is shown as a function of M . The resultsare fitted with the function p δ p ∼ . /M . The numericalresults are obtained by sampling 2 . × scattering matrices. where δρ = ρ − ¯ ρ . The analysis of this expression showsthat the relative change in the density of states differsfrom the Seebeck coefficient, Eq. (7), but the interest-ing result is that both δρ ( ǫ ) / ¯ ρ ( ǫ ) and s/α have exactlythe same distribution, which may be seen by using theinvariance of the Haar measure under the transforma-tion S → − i S . We finally add to this set of variablesthe scaled Wigner time δτ w /α which is related to thetime spent by a wavepacket in the scattering region: τ w = − i ~ Tr( S † ∂ S /∂ǫ ) . V. GENERALIZATION TO M MODES OFCONDUCTION
Here, we generalize and investigate the performance ofa system with a large number of conduction modes M .We concentrate on systems with 2 N levels connected to2 M independent and equivalent leads ( M on the left sideand M on the right side). To facilitate both the numericaland computational works, we make use of the model ofminimal chaotic cavities and we concentrate on the edgeof the Hamiltonian spectrum , so that we can set N = M , and obtain results which are the same as those whichwould be produced for the general case N > M (at thespectrum edge). We also assume no spatial symmetry.The form of the scattering matrix which is now of size2 M × M remains unchanged [see Eq. (2)], and the trans-mission is given by T = Tr( tt † ). For large values of M ,the shape of the p.d.f. of the transmission tends to that ofa Gaussian distribution for the typical values of T aroundits mean, which for M large is given by ¯ T ∼ M . Thevariance of this Gaussian is equal to h δ T i =1/8 whichdoes not depend on M because of the universal character FIG. 4. (Color online) Standard deviation of the Seebeckcoefficient as a function of the number of modes for a minimalchaotic cavity ( M = N ). The Seebeck coefficient scales as1 /M for large numbers of modes. of the conductance fluctuations . It is worth mention-ning that the tail of the distribution, which describes theatypical values of the conductance has a non-Gaussianform . At first glance, this scaling of the mean transmis-sion seems to favour an enhancement of the power factorand hence of the figure of merit; however, our study ofthe Seebeck coefficient yields a different conclusion.The Seebeck coefficient may be expressed by writingthe derivative of the scattering matrix and using the ex-pression s = 2 ℜ (Tr( ˙ tt † ) / T ) (where the dot refers to theenergy derivative and ℜ is the real part); then it is com-puted numerically assuming modes with a self-energyΣ = ǫ/ − i p − ( ǫ/ . Still considering the half-filling limit since the generalization to arbitrary energypresents no particular difficulty, we find that the meanvalue of the Seebeck coefficient is always zero: h s i = 0because the probability to obtain a positive thermopoweris the same as that for a negative one. The most inter-esting result is the standard deviation which appears toscale as: h δs i / = O (1 /M ). This clearly demonstratesthat increasing the number of conduction modes in thesystem yields a decrease of the thermopower as shownin Fig. 4 where the standard deviation of thermopower isrepresented as a function of the number of modes on eachside (left and right). The numerical results were obtainedby sampling scattering matrices of size 2 M × M from thecircular orthogonal ensemble and computing a histogramfrom which the standard deviation was obtained.Since p ∝ s , the lowering of the system performanceinduced by a decrease of the thermopower, which is fasterthan the increase due to conductance, is a reflection ofthe fact that the typical power factor is thus estimatedto scale as 1 /M . We deduce from this analysis that thedistributions P ( T − ¯ T ) and P ( M s/α ) do not dependon M for large values of M . In Fig. 5, we see a verygood agreement between the distribution P ( M s/α ) and
FIG. 5. Orange dashed line: Gaussian fit of the distributions P ( T − ¯ T ) compared to the numerical result (blue curve). Theagreement is good. P ( MS/α ) (left panels), and j.p.d.f. forthe couples ( T , ZT ) and ( T , MS/α ).FIG. 6. Statistics of the Seebeck coefficient (Blue curve) in arealistic cavity simulated on a lattice. The sample is truncatedand the very large values of S were ommited due to numericalderviation over energy. (Red line), represents simulation usingEq. (28) with α as an ajustement parameter. The kind ofcavity we used is shown in the left inset. The law P ǫ ( T ) = √ T is verified in the right inset. The width of the leads are5 a and the Fermi energy is E = − . t the Gaussian centered on zero. We also see from thej.p.d.f of ZT and T that high values of the figure of meritare obtained for T ∼ M/
2, which generalizes the resultobtained for the two-level system.
VI. LATTICE MODEL
The Seebeck distributions obtained using the 2 × N × N system’s Hamiltonian with N > . Increasing the number of levels N will push thedistribution towards that of the bulk spectrum wherethe mean level spacing is decaying as 1 /N . So for a given N , the Fermi energy has to be lowered in order to drawaway from the bulk spectrum towards the spectrum edge.On the other hand, the Fermi energy must remain signif-icant; that is why systems with a relatively small numberof levels N are very well suited to our approach. Indeed,The center of the Hamiltonian distribution is differentfrom the Fermi energy and this difference is larger at lowFermi energies. This can be understood and verified eas-ily in a Graph-Hamiltonian model since the Lorentziandistribtion can be numerically generated.The case of a lattice model is a more challenging situa-tion. We wish to verify the distribution of the transportcoefficients in a manner which can be fullfilled experi-mentally and not only numerically. For this purpose, westudy the transport in a cavity connected to two semi-infinite uniform leads. The cavity is subject to a smallpotential, uniform inside the cavity, but its value is ran-domly chosen in a small interval. This potential may begenerated by external gates. The Fermi energy is chosensmall yet significant to allow a unique conducting modein the leads. The simulation on such a lattice model canbe done using the Kwant software . For a suitable rangeof magnitude of the small random potential, the result ofthe statistics of the conductance verifies the law given inEq. (27) as it can be seen in the right inset of Fig. 6. Oncethe distribution of the conductance is verified, we looktowards the distribution of the Seebeck coefficient. Thiscoefficient is obtained using a numerical derivation of thetransmission with respect to energy. This numerical pro-cedure can be sensitive, especially where the transmissionand its derivative vanish at the same time . That is whywe truncate the sample of the Seebeck values and get ridof the very large results. We also vary slightly the shapeof the cavity used in the simulation as done in Ref. toavoid any spatial symmetries. The shape of the cavitywe use is shown in the left inset of Fig. (6). The resultof the simulation is shown in Fig. (6) and was fitted withthe analytical result of Eq. (28) with α as an ajustableparameter. VII. DISCUSSION AND CONCLUDINGREMARKS
We investigated the quantum thermoelectric transportof nanosystems made of two electron reservoirs connectedthrough a low-density-of-states two-level chaotic quan-tum dot, using a statistical approach. Assuming nonin-teracting electrons but accounting for the energy depen-dence of the self energy and retaining its real part , andusing the scattering matrix formalism, analytical expres-sions based on an exact treatment of the chaotic behaviorin such devices were obtained. Equation (7) in particu-lar is an exact and general mathematical result, whichcould be obtained because the scattering matrix and theHamiltonian have the same size (hence the interest inthe treatment of the equivalent 2-level system). Anal-ysis of the results provided the conditions for optimumperformance of the system. Optimum efficiency is ob-tained for half-transparent dots and it may be enhancedfor systems with few conducting modes, for which theexact transport coefficients probability distributions arefound to be highly non-Gaussian.To end this article, we comment on the fact that ourcalculations are based on sampling of the scattering ma-trix using the equal a priori probability ansatz , in-stead of considering the Hamiltonian as a random ma-trix. This point is of interest as, instead of dealing withscattering matrices, one could of course follow a Hamil-tonian approach with which integrations are performedover the eigenvalues instead of the eigenphases. The cor- respondence between the two formalisms is ensured byEq. (2) and the corresponding distribution for the Hamil-tonian, compatible with the circular ensembles is found,for a fixed N , to be Lorentzian. The equal a prioriprobability ansatz is therefore reflected in the relation-ship between the (center; width) of the distribution andthe (real; imaginary) parts of the self-energy of the leadswhich ensures h S i = 0. More precisely, the widthof the eigenvalues needs to be equal to the broadeningΓ of the leads. Using a distribution different from theuniform one would of course imply that different weightsare allocated to the possible configurations of the system,but while unlikely configurations would have no influencewhatsoever on the outcome, unlike those which are possi-ble, only these latter with larger weights would contributeto the maximization of the power factor, and the aboveconclusions would then remain essentially the same. ∗ [email protected] G. D. Mahan and J. O. Sofo, Proc. Natl. Acad. Sci. USA , 7436 (1996). P. Trocha and J.Barna´s, Phys. Rev. B , 085408 (2012). S. Datta,
Quantum Transport: Atom to Transistor G. Usaj and H. U. Baranger, Phys. Rev. B , 201319(R)(2001). C. Karrasch, T. Hecht, A. Weichselbaum, J. von Delft, Y.Oreg and V. Meden, New J. Phys. , 123 (2007). P. W. Brouwer, S. A. van Langen, K. M. Frahm, M.B¨uttiker, and C. W. J. Beenakker, Phys. Rev. Lett. ,913 (1997). Y. Alhassid, Rev. Mod. Phys. , 895 (2000). A. Weichselbaum and J. von Delft, Phys. Rev. Lett. ,076402 (2007). A. Abbout, G. Lemari´e, and J.-L. Pichard, Phys. Rev.Lett. , 156810 (2011). M. Goldstein and R. Berkovits, New J. Phys. , 118 (2007). Y. Oreg, New J. Phys. , 122 (2007). R. Shankar, Rev. Mod. Phys. , 379 (2008). T. Guhr, A. M¨uller-Groeling, H. A. Weidenm¨uller, Phys.Rep. , 189 (1998). S. F. Godijn, S. Moller, L. W. Molenkamp, and S. A. vanLangen, Phys. Rev. Lett. , 2927 (1999). C. W. J. Beenakker, Rev. Mod. Phys. , 731 (1997). A. Abbout, Euro. Phys. J. B , 117 (2013). Indeed, a large number of levels generates many resonancesin a given energy window. By definition, at the edge of the spectrum, the levels aretypically rare. A. Abbout, G. Fleury, J.-L. Pichard, K. Muttalib, Phys.Rev. B , 115147 (2013). The Supplemental Material and the python code may besent to the interested reader upon request to Dr. Adel Ab-bout: abbout.adel @ gmail.com. A.Abbout and P. Mei, arXiv:1211.5169. S. A. van Langen, P. G. Silvestrov, and C. W. J. Beenakker,Superlat. & Microstruct. , 691 (1998). J. J. M. Verbaarschot, H. A. Weidenmuller, and M. R.Zirnbauer, Phys. Rep. , 367 (1985). M. Cutler and N. F. Mott, Phys. Rev. , 1336 (1969). D. S. Fisher and P. A. Lee, Phys. Rev. B , 6851 (1981). Mehta M. L.,
Random Matrices (New York: Academic,1991). R. A. Jalabert, J.-L. Pichard, and C. W. J. Beenakker,EPL , 255 (1994). H. U. Baranger and P. A. Mello, Phys. Rev. Lett. , 142(1994). J. P. Forrester,
Log-Gases and Random Matrices (Prince-ton University Press, 2010). P. W. Brouwer, Phys. Rev. B , 16878 (1995). To use the invariance of the Haar measure in the presenceof transmission, one may better use this expression: T =Tr( C SC S † ) with C ij = δ i δ j and C ij = δ i δ j . V. A. Gopar, M. Martinez, P. A. Mello, H. U. Baranger,J. Phys. A: Math. Gen. , 881 (1996). At ǫ , the conduction band may not be half-filled. Nev-ertheless, for the usual 1D, 2D , and 3D uniform perfectleads we usually consider, the band is half-filled so that weretain the expression “half-filling limit”. One would usually start by sampling the Hamiltonian firstand then obtain the scattering matrix, but since generat-ing Lorentzian ensemble uses unitary matrices, one mayskip this step and sample directly S . This simplificationis possible here because the expressions we obtain involveonly S in the minimal models. G. Benenti, K. Saito, and G. Casati, Phys. Rev. Lett. ,230602 (2011). K. Saito, G. Benenti, G. Casati, and T. Prosen, Phys. Rev.B , 201306(R) (2011). D. S´anchez and L. Serra, Phys. Rev. B , 201307(R)(2011). V. Balachandran, G. Benenti, and G. Casati, Phys. Rev.B , 165419 (2013). Y. Apertet, H. Ouerdane, C. Goupil, and Ph. Lecoeur,Phys. Rev. E , 031116 (2012). A. Ioffe,
Semiconductor thermoelements and thermoelectriccooling (London, Infosearch, ltd., 1957). C. Goupil, W. Seifert, K. Zabrocki, E. M¨uller, and G. J.Snyder, Entropy , 1481 (2011). K. Suekuni, K. Tsuruta, T. Ariga, and M. Koyano, APEX , 051201 (2012). A. Abbout, H. Ouerdane, and C. Goupil, Phys. Rev. B ,155410 (2013). If the system is not connected to the leads, then ρ ( ǫ ) = − π lim η → ℑ Tr ǫ −H s + iη = P i =1 δ ( ǫ − ǫ i ), which yields ¯ ρ ( ǫ ) = π Γ / / +Λ . For 1D uniform leads, ¯ ρ ( ǫ ) = Γ π . P. Vivo, S. N. Majumdar, and O. Bohigas, Phys. Rev. Lett. , 216809 (2008). K. Sasada and N. Hatano, J. Phys. Soc. Jpn. , 025003(2008). C. W. Groth, M. Wimmer, A. R. Akhmerov, X. Waintal,New J. Phys. , 063065 (2014).48