Statistical Analysis of the Figure of Merit of a Two-Level Thermoelectric System:A Random Matrix Approach

Adel Abbout1+, Henni Ouerdane2,3, and Christophe Goupil4

1Physical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST),Thuwal 23955-6900, Saudi Arabia

2Russian Quantum Center, 100 Novaya Street, Skolkovo, Moscow Region 143025, Russian Federation3UFR Langues Vivantes Etrangères, Université de Caen Normandie, Esplanade de la Paix 14032 Caen, France4Laboratoire Interdisciplinaire des Energies de Demain (LIED), UMR 8236 Universitié Paris Diderot, CNRS,

5 Rue Thomas Mann, 75013 Paris, France

(Received February 10, 2016; accepted July 7, 2016; published online August 5, 2016)

Using the tools of random matrix theory we develop a statistical analysis of the transport properties of thermoelectriclow-dimensional systems made of two electron reservoirs set at different temperatures and chemical potentials, andconnected through a low-density-of-states two-level quantum dot that acts as a conducting chaotic cavity. Our exacttreatment of the chaotic behavior in such devices lies on the scattering matrix formalism and yields analyticalexpressions for the joint probability distribution functions of the Seebeck coefficient and the transmission profile, as wellas the marginal distributions, at arbitrary Fermi energy. The scattering matrices belong to circular ensembles which wesample to numerically compute the transmission function, the Seebeck coefficient, and their relationship. The exacttransport coefficients probability distributions are found to be highly non-Gaussian for small numbers of conductionmodes, and the analytical and numerical results are in excellent agreement. The system performance is also studied, andwe find that the optimum performance is obtained for half-transparent quantum dots; further, this optimum may beenhanced for systems with few conduction modes.

1. Introduction

Low-dimensional systems offer a wealth of technologicalpossibilities thanks to the rich variety of artificial custom-made semiconductor-based structures that nowadays may beroutinely produced. The properties of these systems, includ-ing confinement geometry, density of states, and bandstructure, may be tailored on demand to control the transportof heat and the transport of confined electrical charges, aswell as their coupling. In the current context of intenseresearch in energy conversion physics, this is crucial tofurther improve the performance and increase the range ofoperation of present-day thermoelectric devices. These latterare characterized by their so-called figure of merit, which, inthe frame of linear response theory, may be expressed as:ZT ¼ �s2T=�, where s is the Seebeck coefficient, T is theaverage temperature across the device, and σ and κ are theelectrical and thermal conductivities respectively, with� ¼ �e þ �lat, accounting for both electron and lattice thermalconductivities.

Amongst the various low-dimensional thermoelectricsystems that have been studied, quantum dots, which areconfined electronic systems whose size and shape arecontrolled by external charged gates, keep attracting muchinterest because of their narrow, Dirac-like, electron transportdistribution functions1) or, equivalently, sharply peakedenergy-dependent transmission profiles T , which permitobtainment of extremely high values of electronic contribu-tion to ZT.2) There is a simple relationship between theelectrical conductivity σ appearing in the definition of thefigure of merit and the transmission function T : � ¼ð2e2=hÞT , the proportionality factor being the quantum ofconductance (e being the electron charge, and h beingPlanck’s constant).

Models of quantum dots form two broad categories,namely interacting and noninteracting models. Those ac-

counting for electron–electron interactions permit analysis ofa rich variety of physical phenomena governed by electroniccorrelations like, e.g., Coulomb blockade and the relatedconductance oscillations vs gate voltage,3) peak spacingdistributions,4) and phase lapses of the transmission phase.5)

The use of non-interacting model systems for quantum dotsor very small electronic cavities may be justified if these arestrongly coupled to the reservoirs, i.e., if the confinementyields a mean level spacing that is large compared to thecharging energy e2=C, C being the capacitance of the dot.6,7)

So, though noninteracting dot models may be seen as toymodels, these may provide in some cases a number ofinsightful results without the need to resort to advancednumerical techniques such as, e.g., the density matrixnumerical renormalization group.5,8) This is illustrated by,e.g., the resonant level model of thermal effects in a quantumpoint contact,9) Fano resonances in the quantum dotconductance,10) and simple models of phase lapses of thetransmission phase.11)

Interesting physics problems in a quantum dot also stemfrom the chaotic dynamics that may be triggered becauseof structural disorder, or as the quantum dot itself behaves asa driven chaotic cavity because its shape varies as one ofthe gates generates a random potential. In these cases, asexplained in Ref. 12, one may be interested in the statisticsof the system’s spectrum rather than the detailed descriptionof each level. Random matrix theory13) provides tools ofchoice for this purpose; and, for quantum dots in particular,one may construct a mathematical ensemble of Hamiltoniansthat satisfies essentially two constraints: the Hamiltoniansbelong to the same symmetry class and they have the sameaverage level spacing, while the density of states must bethe same for the physical ensemble of quantum dots.12) Inthe context of thermoelectric transport, measurements ofthermopower and analysis of its fluctuations14) based on therandom matrix theory of transport15) demonstrated the non-

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Gaussian character of the distribution of thermopowerfluctuations.

In this article, we concentrate on the statistical analysis ofthe thermoelectric properties of noninteracting quantum dotsconnected to two leads that serve as electron reservoirs set atdifferent temperatures and electrochemical potentials. Weanalyze the statistics of ZT in a two-level system connected totwo electronic reservoirs set to two slightly different temper-atures so that the temperature difference, �T, is small enoughto remain in the linear response regime: the voltage inducedby thermoelectric effect, �V, is given by s ¼ ��V=�T. Themutual dependence of the transport coefficients that define ZT(e.g., the Wiedmann–Franz law for metals) raises the problemof finding which configuration of the mesoscopic system mayyield the largest values of ZT, and hence offer optimumperformance.

The two-level model presented in this article is theminimal model pertinent for the description of a cavity withtwo conducting modes presenting a completely chaoticbehavior.16) We consider Fermi energies lying in the vicinityof the spectrum edge of the system’s Hamiltonian. Thissituation is in contrast with the case of high level cavities forwhich the typical transmission profiles vary much and exhibitnumerous maxima and extinctions, which precludes ananalytical formulation of the Seebeck coefficient probabilitydistribution since the Cuttler-Mott formula does not apply.Although one may consider a very low temperature regime toconcentrate only on a small interval of energies, the need ofa large number of levels to ensure the properties of thebulk universality in chaotic systems makes this window ofenergies so small17) that the corresponding temperature tendsto become insignificant. Conversely, at the edge of theHamiltonian spectrum, the typical profile of the transmissionis smooth18) and the analytical treatment of the probabilitydistribution of thermopower is much easier.19) Indeed, at thisclass of universality (spectrum edge) the description of thiskind of systems may rest on the equivalent minimal chaoticcavity16) defined as the system with a number of levels Nequal to the number of the conducting modes M. By“equivalent” we mean that, at the spectrum edge, the originalsystem and the corresponding minimal chaotic cavity lead tothe same statistics though they certainly have different resultsfor one realization.

The article is organized as follows: In the next section,we introduce the model, the main definitions and notationswe use throughout the paper. In Sect. 3, we present twoderivations of the Seebeck coefficient in the scatteringmatrix framework: one is general, while the other is madeunder the restrictive assumption of left–right spatialsymmetry. The formal identity of the two results serves asa basis for the statistical analysis presented in Sect. 4, whichis the core of the article. We analyze the numericalstatistical results obtained for the probability distributions ofthe figure of merit, thermopower, and power factor. InSect. 5, we analyze the relation of the Seebeck coefficientto the density of states of the system. In Sect. 6, we extendthe discussion to the case of a lattice. The article ends witha discussion and concluding remarks. A SupplementalMaterial detailing some of the numerical aspects of thework as well as some derivations, accompanies thearticle.20)

2. Model

We consider a low-density-of-states two-level quantum dotdepicted in Fig. 1. The potential on the right three top gatesmay be varied randomly in order to slightly modify the shapeof the cavity and therefore obtain a statistical ensemble.14)

2.1 Preliminary remarks on the minimal chaotic cavitiesStudies of confined systems may be achieved without loss

of pertinence by means of simple models, which capture theessential physics of actual systems. A chaotic cavity with Nenergy levels is modeled with an N � N Hamiltonian. In theframe of random matrix theory, to obtain a scattering matrixfrom circular ensembles, the Hamiltonian is distributed fromLorentzian ensembles; other distributions, e.g., the Gaussiandistribution, may lead to the same results only for large N inthe bulk spectrum. For a fixed value of N, if the Fermi energyapproaches the bottom of the conduction band (continuumlimit) we enter a new universality class: the edge of theHamiltonian spectrum.19) The key point here, which willprove extremely useful as shown in the present article, is thatat the edge of the spectrum, all the distributions obtained withan N � N Hamiltonian may also be found by considering asimpler case with M �M matrices, with M (< N) being thenumber of conducting modes. This type of cavity is called theminimal chaotic cavity, and permits to lower N down toN ¼ M to simplify the mathematical and computationaltreatment of the problem, and derive physically meaningfulresults. As we show, it allows obtainment of very simpleformulas for the transport coefficients because theHamiltonian and the scattering matrix then have the samesize. It is important because, at the edge of the spectrum

Fig. 1. (Color online) Schematic figure of the two level-system consistingof negatively charged top gates (right gates) on a two-dimensional electrongas. The top left (bottom left) part represents the hot (cold) electronicreservoir connected to the cavity. The inset shows the system as modeled bythe Hamiltonian.

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(large level spacing), it gives the same result as for theoriginal system (N > M). This approach is justified forprobing the spectrum edges of mesoscopic systems16,19,21)

where the physics is different from that of the bulkspectrum.19,22)

2.2 Scattering matrixThe two-level system tight-binding Hamiltonian reads as

the sum of three contributions: H ¼ H‘ þHs þHc, whereH‘ ¼

Pk �kc

ykck is the Hamiltonian of the two (i.e., left and

right) leads; cyk and ck are the second-quantized electroncreation and annihilation operators in the state k. The twoenergy levels and their coupling are characterized by theHamiltonian Hs:

Hs ¼v1 v12

v21 v2

!; ð1Þ

where the random on-site potentials v1; v2 and coupling v12describe the chaotic behavior of the system. We assume thatthe left (right) lead is only connected to the site v1 (v2). Thecontribution Hc involves the coupling matrix W entering thedefinition of the scattering matrix:23)

S ¼ 1 � 2�iWy 1

� �Hs � �W; ð2Þ

where Σ is the self-energy of the two leads. To keep ouranalysis on a general level, we only give its form: � ¼� � i�WWy, with Λ being the real part. With the assumptionof symmetric reservoirs, the self-energy is proportional to theidentity, in which case W ¼

ffiffiffiffiffiffiffiffiffiffiffi�=2�

p12�2, where � ¼ �2=�

characterizes the broadening. Throughout the article, weadopt the same notations for scalars and their correspondingmatrix form when this latter is proportional to the identity.

3. Derivation of the Seebeck Coefficient

It is instructive to derive an analytic expression of theSeebeck coefficient under the restrictive assumption of spatialleft–right symmetry, on the one hand, and compare the resultto the that obtained assuming arbitrary energy and arbitraryleads.

3.1 Left–right spatial symmetryThe scattering matrix S is unitary. In the absence of a

magnetic field, time reversal symmetry is preserved and thematrix S is therefore symmetric, which implies v12 ¼ v21.Moreover, for simplicity, we consider in the first part of thisarticle, systems with left–right spatial symmetry (v1 ¼ v2), sothat the reflection from left is the same as that from right. Thematrix S may thus read

S ¼r t

t r

!; ð3Þ

where r and t are the reflection and transmission amplitudesrespectively. With these definitions, the transmission of thesystem and the Seebeck coefficient read

T ¼ jtj2; and s ¼ @ lnðT Þ@�

: ð4Þ

The Seebeck coefficient is obtained at low temperatures withthe Cutler–Mott formula.24) Here, it is expressed in units of�2

3e k2BT (ommiting the sign). The transmission of the system

can then directly be obtained using the Fisher–Lee for-mula:25)

T ¼ �G12�Gy12; ð5Þ

where the off-diagonal element of the Green’s matrix isG12 ¼ 1

2�1��2

ð���1��Þð���2��Þ, with �1 and �2 being the eigenvaluesof Hs. We may thus write

Tð�Þ ¼ �2

4

ð�1 � �2Þ2

jð� � �1 � �Þð� � �2 � �Þj2; ð6Þ

which is valid if the left=right symmetry condition issatisfied. Now, combination of Eqs. (4) and (6) yields anexpression of the Seebeck coefficient containing the scatter-ing matrix S:

s ¼ �

2Tr

ei�S � e�i�Sy

2i

� �ð7Þ

with � ¼ 4j1 � _�j=�, � ¼ Argð1 � _�Þ, and _� � @��. For anenergy � ¼ �0, which corresponds in general to the middle ofthe conduction band of a semi-infinite lead (or half-fillinglimit), the imaginary part of the self-energy derivativevanishes: = _�ð�0Þ ¼ 0.

3.2 General derivationThe starting point is the transmission of the two-level

system:

T ¼ �2 v212jð� � �1 � �Þð� � �2 � �Þj2

: ð8Þ

This formula applies both in presence or absence of the left=right spatial symmetry. The Seebeck coefficient in units of�2

3

k2BT

e reads: s ¼ @� lnðT Þ. The application of this formula toT leads to

s ¼ @� lnð�2Þ � 1 � _�

� � �1 � �þ 1 � _�

� � �2 � �þ H:c:

� �: ð9Þ

The relation between the scattering matrix and theHamiltonian is simple in the case of a two-level non-interacting system:

S ¼ 1 � i�1

� � H � �! 1

� � H � �¼ 1 � S

i�: ð10Þ

Combination of the above expression with Eq. (9) gives

s ¼ @� lnð�2Þ � ð1 � _�ÞTr 1 � Si�

� �þ ð1 � _�

?ÞTr 1 � Sy

i�

� �;

ð11Þwhere the star symbol denotes the complex conjugate. Thenwe may rewrite s as

s ¼ @� lnð�2Þ � 2_�

�þ ð1 � _�ÞTr S

i�

� �� ð1 � _�

?ÞTr Sy

i�

� �;

ð12Þwhich reduces to Eq. (7). This expression has been obtainedbecause the scattering matrix and the Hamiltonian have thesame size (hence the interest in using the equivalent two-levelsystem); it corresponds to a system for which, in thecontinuum limit and at low Fermi energies, one deals with auniversality class different from that of the bulk. We stressthat Eq. (7) constitutes an important result upon which all thesubsequent statistical analysis is based.

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4. Statistical Analysis

4.1 Statistics of the Seebeck coefficientIn this work, the scattering matrix S is a random variable,

which we assume to be uniformly distributed; as such, Sbelongs to circular orthogonal ensembles (COE).26) The useof the circular ensemble is based on the equal a prioriprobability ansatz.27,28) This is a natural choice when there isno reason to privilege any scattering matrix, in which case themean of the distribution is hSi ¼ 0. For a general case,obtained for different parameters than those leading toCE,16,19) we obtain a more general distribution called thePoisson kernel uniquely determined by its non vanishingmean scattering matrix hSi ≠ 0.26) Since it is always possibleto define a new unitary matrix uniformly distributed usingmatrix transformations,29,30) the subsequent calculations aredeveloped considering only the circular ensemble.

The statistics of the Seebeck coefficient thus follows:

P�ðsÞ ¼Z

� s � �

2Tr

ei�S � e�i�Sy

2i

� �� ��HS; ð13Þ

where �HS denotes the Haar measure, which is invariantunder the transformation S ! VSVT for any arbitrary unitarymatrix V (the superscript T denotes the transpose of a matrix).With V ¼ ei�=212�2, we obtain

P�ðsÞ ¼Z

� s � �

2Tr

~S � ~Sy

2i

!( )�H ~S

¼ 1

��P�¼�0 ðs= ��Þ; ð14Þ

where �� ¼ �=�0 with �0 ¼ �ð�0Þ. Equation (14) shows thatthe general probability distribution of the Seebeck coefficientat arbitrary energy can be deduced directly from the simplerone obtained at �0. Note that for ease of notations, we retainP as a generic notation for all the probability distributions inthe subsequent parts of the article.

4.2 Joint probability distribution functionA similar analysis of the transmission T shows that its

distribution does not depend on energy.31) Since the transportcoefficients T and s are tightly related by the transportequations, the knowledge of the joint probability distributionfunction ( j.p.d.f.) P�ðs; T Þ is required to study anyobservable depending on both T and s. The starting pointfor this is the rewriting of the scattering matrix using thefollowing decomposition:

S ¼ RT ei�1 0

0 ei�2

!R; ð15Þ

where the rotation matrix R is defined as

R ¼ 1ffiffiffi2

p1 �11 1

!: ð16Þ

The eigenphases �1 and �2 are independent and uniformlydistributed in ½��;þ�� which is the consequence of S beingtaken from a COE with the left–right symmetry.32) Now, weanalyze the j.p.d.f. of the Seebeck coefficient and trans-mission at the half-filling limit � ¼ �0,33) with

T ¼ sin2�1 � �2

2

� �ð17Þ

and

s ¼ �0 sin�1 þ �2

2

� �cos

�1 � �22

� �: ð18Þ

The j.p.d.f. may now take the form:

P�¼�0 ðs; T Þ ¼ h�ðT � sin2 uÞ�ðs � �0 sin v cos uÞiu;v; ð19Þ

where two independent and uniformly distributed variablesu ¼ �1��2

2and v ¼ �1þ�2

2are introduced. As shown in the

Supplemental Material, integration over these variables anduse of Eq. (14) yield

P�ðs; T Þ ¼ 1

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTð1 � T Þ

p 1

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2ð1 � T Þ � s2

p ; ð20Þ

which constitutes one of the main results of this work since itshows the possible values of the couple ðs; T Þ: P�ðs; T Þ isnon-zero only if the following condition is satisfied

�2ð1 � T Þ � s2 > 0; T < 1: ð21Þ

The relationship between the Seebeck coefficient and thetransmission function is thus constrained by a paraboliclaw. This may be checked by sampling the S matricesbelonging to the COE with the left=right symmetry,34) fromwhich the transport coefficients T and s are numericallycomputed with Eqs. (4) and (7). The parabolic law is clearlyshown on Fig. 2 where the j.p.d.f. P�ðs; T Þ is plotted againstT and s.

Note that if we had used the Poisson Kernel distribution,which only changes the probability weight of each config-uration of the system, the condition (21) and the ensuingconclusions would still hold.

4.3 System performanceFrom a thermodynamic viewpoint, thermoelectric systems

connected to two thermal baths at temperatures Thot and Tcold,use their conduction electrons as a working fluid to directly

Fig. 2. (Color online) Probability densities: 1) Top left: PðT Þ for thetransmission. 2) Bottom left: PðsÞ for the Seebeck coefficient. 3) Top right:j.p.d.f. of the couple ðs; T Þ at the half-filling limit. 4) Bottom right: j.p.d.f. ofthe couple ðT; ZTÞ at the half-filling limit. The noisy curves in the left figuresare obtained by sampling scattering matrices from circular orthogonalensembles and the dashed curves represent the analytical result of thecorresponding physical observable. The matching is excellent.

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convert a heat flux into electrical power and vice-versa withefficiency η. As for all heat engines, one seeks to increaseeither their so-called efficiency at maximum output power,Pmax, as discussed in numerous recent papers (see, e.g.,Refs. 35–38 for mesoscopic systems and Ref. 39 forfundamental questions related to irreversibilities) or simplymaximize the efficiency η. For simple models in thermo-electricity, an expression of the maximum of η, is related tothe figure of merit of the system ZT:40,41)

max ¼ C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ ZT

p� 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ ZTp

þ Tcold=Thot; ð22Þ

where C ¼ 1 � Tcold=Thot is the Carnot efficiency.Equation (22) clearly shows that for given working

conditions, ZT is as good a device performance measure asthe efficiency η is. The maximum of the figure of merit ZT isreached when the maximum of the power factor p ¼ Ts2 isreached. Satisfaction of this latter condition lies on theexistence of the probability P�ðs; T Þ: Eq. (21) implies that�Tð1 � T Þ > Ts2, which in terms of power factor impliesthat maxð�Tð1 � T ÞÞ > p, so that

�=4 > p: ð23ÞWe thus find that ZT reaches its maximum if the transmissionT ¼ 1=2.

Physically, this means that the best performance resultsfrom the conditions that make the system half-transparent;this is numerically checked on Fig. 2. Note that the mainassumption on the thermal properties of the system is that κ isconstant; this is justified on the condition that �e � �lat

42) (�latis assumed to be constant). Relaxation of this assumptionmakes our system performance analysis applicable to thepower factor only, but not to the figure of merit.

4.4 Marginal distributionsThe marginal distributions P�ðT Þ and P�ðsÞ are obtained

from the integration of P�ðs; T Þ over s and T , respectively:

P�ðT Þ ¼ 1

�

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT ð1 � T Þ

p ; ð24Þ

P�ðsÞ ¼2

��2Kð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � s2=�2

pÞ; ð25Þ

where K is the complete elliptic integral of the first kind. It isinteresting to note that P�ðT Þ does not depend on the energysince it derives from the sole assumption that the scatteringmatrix is uniformly distributed (Dyson’s CE) with the left=right spatial symmetry.32) As for all the physical observableswith no energy derivative in their expression (shot noisefor example), their distributions do not depend on the numberof levels N in the system: for all N � 2, we obtain thesame distribution, as it can be seen with the decimationmethod16,43) on the condition that S 2 fCEg.

The case of the Seebeck coefficient is different: itsdistribution P�ðsÞ depends on the number of levels in thesystem since its expression contains an energy derivative.The only situation where the Seebeck coefficient, scaled withthe appropriate parameter (α in our case), is independent onthe number of levels is at the edge of the Hamiltonianspectrum19) where the result can be obtained by assuming thesimpler two-level system considered as the minimal cavityfor the two-mode scattering problem.16) The two-level model

provides simple equations but yields general results, whichapply to the N-level model at low density.

All the results presented so far in this article were obtainedand discussed assuming a two-level system with left=rightspatial symmetry and S 2 fCEg. We must now see whetherthese hold when the assumption of left=right spatialsymmetry is relaxed.

4.5 Relaxation of the spatial symmetry assumptionAssuming that there is no left=right symmetry, the two

eigenphases �1 and �2 of Eq. (15) are no longer independent,and we must consider a distribution of the form: Pð�1; �2Þ /jei�1 � ei�2 j.26) The treatment is the same as that for theprevious case, and while we obtain a different j.p.d.f. for theSeebeck and the transmission coefficients:

P�ðs; T Þ ¼ 1

��21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Tð1 � T Þp Kð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � s2=�2ð1 � T Þ

pÞ ð26Þ

the mathematical constraint on the couples ðT; sÞ to obtain anon-vanishing joint probability distribution is exactly thesame as Eq. (23); this implies that the best system perform-ance is obtained when it is half-transparent: T ¼ 1=2.

The marginal distributions of s and T are also different:

P�ðT Þ ¼ 1

2ffiffiffiffiffiT

p ; ð27Þ

P�ðsÞ ¼ � 1

��ln

jsj=�1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � s2=�2

p !

: ð28Þ

Equation (27) is consistent with results of Refs. 27 and 28obtained with different methods and Eq. (28) confirms theresult of Ref. 19. We see in both distributions, Eqs. (20) and(26), that the variables ðs; T Þ are not independent but it isinteresting to note that if we define a new variable X ¼s=

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 � T

pthen we have two independent variables, and the

j.p.d.f. becomes multiplicatively separable:

P�ðX; T Þ ¼ P�ðT Þ � P�ðXÞ; ð29Þ

where we have P�ðXÞ ¼ 1=�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � X2

pin the symmetric case

and P�ðXÞ ¼ 2��2 Kð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � X2=�2

pÞ when the symmetry is

relaxed; P�ðT Þ, given above, is the corresponding marginaldistribution. It is worth mentionning that S and X alsoconstitute a couple of independent variables.

4.6 Statistics of the density of statesThe density of states in the two-level quantum dot may be

expressed with the usual formula:

ð�Þ ¼ � 1

�=TrG ¼ � 1

�=Tr

1

� �H � �; ð30Þ

which, using the scattering matrix, thus reads

ð�Þ ¼ � 1

�=Tr

1 � Si�

¼ � 1

�

1

2iTr

1 � Si�

� 1 � Sy

�i�

� �ð31Þ

and simplifies to

ð�Þ ¼ � 1

�

1

2i

4

i�� Tr

S þ Sy

i�

� �� �ð32Þ

so that

�ð�Þ�ð�Þ ¼ 1

2Tr

S þ Sy

2

� �; ð33Þ

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where �ð�Þ ¼ 2=��. Here, we used the fact that for circularensembles we do have �S ¼ �Sy ¼ 0 (where the over bardenotes the mean). The local density of states in the centralsystem, when connected to the leads, reads44)

ð�Þ ¼ � 1

�=Tr

1

� �Hs � �; ð34Þ

which combined with Eq. (2) yields

�ð�Þ�ð�Þ ¼ 1

2Tr

S þ Sy

2

� �; ð35Þ

where � ¼ � �. The analysis of this expression shows thatthe relative change in the density of states differs from theSeebeck coefficient, Eq. (7), but the interesting result is thatboth �ð�Þ= �ð�Þ and s=� have exactly the same distribution,which may be seen by using the invariance of the Haarmeasure under the transformation S ! �iS. We finally addto this set of variables the scaled Wigner time ��w=� which isrelated to the time spent by a wavepacket in the scatteringregion: �w ¼ �iħTrðSy@S=@�Þ.16)

5. Generalization to M Modes of Conduction

Here, we generalize and investigate the performance of asystem with a large number of conduction modes M. Weconcentrate on systems with 2N levels connected to 2Mindependent and equivalent leads (M on the left side and Mon the right side). To facilitate both the numerical andcomputational works, we make use of the model of minimalchaotic cavities and we concentrate on the edge of theHamiltonian spectrum,16) so that we can set N ¼ M, andobtain results which are the same as those which would beproduced for the general case N > M (at the spectrum edge).We also assume no spatial symmetry.

The form of the scattering matrix which is now of size2M � 2M remains unchanged [see Eq. (2)], and the trans-mission is given by T ¼ TrðttyÞ. For large values of M, theshape of the p.d.f. of the transmission tends to that of aGaussian distribution for the typical values of T around itsmean, which for M large is given by �T � M=2. The varianceof this Gaussian is equal to h�T 2i ¼ 1=8 which does notdepend on M because of the universal character of theconductance fluctuations.15) It is worth mentionning that thetail of the distribution, which describes the atypical values ofthe conductance has a non-Gaussian form.45) At first glance,this scaling of the mean transmission seems to favour anenhancement of the power factor and hence of the figure ofmerit; however, our study of the Seebeck coefficient yields adifferent conclusion.

The Seebeck coefficient may be expressed by writing thederivative of the scattering matrix and using the expressions ¼ 2<ðTrð_ttyÞ=T Þ (where the dot refers to the energyderivative and < is the real part); then it is computednumerically assuming modes with a self-energy � ¼ �=2 �iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � ð�=2Þ2

p.46) Still considering the half-filling limit since

the generalization to arbitrary energy presents no particulardifficulty, we find that the mean value of the Seebeckcoefficient is always zero: hsi ¼ 0 because the probability toobtain a positive thermopower is the same as that for anegative one. The most interesting result is the standarddeviation which appears to scale as: h�s2i1=2 ¼ Oð1=MÞ.This clearly demonstrates that increasing the number of

conduction modes in the system yields a decrease of thethermopower as shown in Fig. 3 where the standard deviationof thermopower is represented as a function of the number ofmodes on each side (left and right). The numerical resultswere obtained by sampling scattering matrices of size2M � 2M from the circular orthogonal ensemble andcomputing a histogram from which the standard deviationwas obtained.

Since p / s2, the lowering of the system performanceinduced by a decrease of the thermopower, which is fasterthan the increase due to conductance, is a reflection of thefact that the typical power factor is thus estimated to scale as1=M as we can check it on Fig. 4. We deduce from thisanalysis that the distributions PðT � �T Þ and PðMs=�Þ do notdepend on M for large values of M. In Fig. 5, we see a verygood agreement between the distribution PðMs=�Þ and theGaussian centered on zero. We also see from the j.p.d.f. ofZT and T that high values of the figure of merit are obtainedfor T � M=2, which generalizes the result obtained for thetwo-level system.

Fig. 3. (Color online) Standard deviation of the Seebeck coefficient as afunction of the number of modes for a minimal chaotic cavity (M ¼ N). TheSeebeck coefficient scales as 1=M for large numbers of modes.

Fig. 4. (Color online) Probability density of the power factor p fordifferent values of the number of conduction modes: right curve M ¼ 4,middle curve M ¼ 8, and left curve M ¼ 16. The number of energy levels inthe cavity equals the total number of modes. In the inset, the standarddeviation of the power factor is shown as a function of M. The results arefitted with the function

ffiffiffiffiffiffiffi�p2

p� 0:08=M. The numerical results are obtained

by sampling 2:1 � 105 scattering matrices.

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6. Lattice Model

The Seebeck distributions obtained using the 2 � 2

Hamiltonian reproduce those of the N � N system’sHamiltonian with N > 2 as a limit at the spectrum edge.16,19)

Increasing the number of levels N will push the distributiontowards that of the bulk spectrum19) where the mean levelspacing is decaying as 1=N. So for a given N, the Fermienergy has to be lowered in order to draw away from the bulkspectrum towards the spectrum edge. On the other hand, theFermi energy must remain significant; that is why systemswith a relatively small number of levels N are very wellsuited to our approach. Indeed, The center of the Hamiltoniandistribution is different from the Fermi energy and thisdifference is larger at low Fermi energies. This can beunderstood and verified easily in a Graph–Hamiltonian modelsince the Lorentzian distribution can be numericallygenerated.

The case of a lattice model is a more challenging situation.We wish to verify the distribution of the transport coefficientsin a manner which can be fullfilled experimentally and notonly numerically. For this purpose, we study the transport ina cavity connected to two semi-infinite uniform leads. Thecavity is subject to a small potential, uniform inside thecavity, but its value is randomly chosen in a small interval.This potential may be generated by external gates. The Fermienergy is chosen small yet significant to allow a uniqueconducting mode in the leads. The simulation on such alattice model can be done using the Kwant software.47) For asuitable range of magnitude of the small random potential,the result of the statistics of the conductance verifies the lawgiven in Eq. (27) as it can be seen in the right inset of Fig. 6.Once the 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 proce-dure can be sensitive, especially where the transmission andits derivative vanish at the same time.43) That is why wetruncate the sample of the Seebeck values and get rid of thevery large results. We also vary slightly the shape of thecavity used in the simulation as done in Ref. 28 to avoid any

spatial symmetries. The shape of the cavity we use is shownin the left inset of Fig. 6. The result of the simulation isshown in Fig. 6 and was fitted with the analytical result ofEq. (28) with α as an adjustable parameter.

7. Discussion and Concluding Remarks

We investigated the quantum thermoelectric transport ofnanosystems made of two electron reservoirs connectedthrough a low-density-of-states two-level chaotic quantumdot, using a statistical approach. Assuming noninteractingelectrons but accounting for the energy dependence of theself energy and retaining its real part,48) and using thescattering matrix formalism, analytical expressions based onan exact treatment of the chaotic behavior in such deviceswere obtained. Equation (7) in particular is an exact andgeneral mathematical result, which could be obtained becausethe scattering matrix and the Hamiltonian have the same size(hence the interest in the treatment of the equivalent two-levelsystem). Analysis of the results provided the conditions foroptimum performance of the system. Optimum efficiency isobtained for half-transparent dots and it may be enhanced forsystems with few conducting modes, for which the exacttransport coefficients probability distributions are found to behighly non-Gaussian.

To end this article, we comment on the fact that ourcalculations are based on sampling of the scattering matrixusing the equal a priori probability ansatz,27,28) instead ofconsidering the Hamiltonian as a random matrix. This pointis of interest as, instead of dealing with scattering matrices,one could of course follow a Hamiltonian approach withwhich integrations are performed over the eigenvaluesinstead of the eigenphases. The correspondence betweenthe two formalisms is ensured by Eq. (2) and the corre-sponding distribution for the Hamiltonian, compatible withthe circular ensembles is found, for a fixed N,49) to beLorentzian.19,30) The equal a priori probability ansatz istherefore reflected in the relationship between the (center;width) of the distribution and the (real; imaginary) parts ofthe self-energy of the leads which ensures hSi ¼ 0.16,19) Moreprecisely, the width of the eigenvalues needs to be equal to

Fig. 5. (Color online) Dashed line: Gaussian fit of the distributionsPðT � �T Þ compared to the numerical result (noisy curve). The agreementis good. PðMS=�Þ (left panels), and j.p.d.f. for the couples ðT; ZTÞ andðT;MS=�Þ.

Fig. 6. (Color online) Statistics of the Seebeck coefficient (noisy curve) ina realistic cavity simulated on a lattice. The sample is truncated and the verylarge values of S were ommited due to numerical derivation over energy. Thesmooth line, represents simulation using Eq. (28) with α as an adjustmentparameter. The kind of cavity we used is shown in the left inset. The lawP�ðT Þ ¼ 1=2

ffiffiffiT

pis verified in the right inset. The width of the leads are 5a

and the Fermi energy is E ¼ �3:43t.

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the broadening �=2 of the leads.19) Using a distributiondifferent from the uniform one would of course imply thatdifferent weights are allocated to the possible configurationsof the system, which means that the condition maximizingthe power factor remains the same while the probability toobtain that configuration changes.

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49) The Lorentzian is an exact result, even for small N. At large N, otherdistributions are compatible (Ex. Gaussian) but only at the bulkspectrum [mean level spacing = O(1=N)]. For details see Ref. 19.

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