Analytic treatment of the thermoelectric properties for two coupled quantum dots threaded by magnetic fields

Coupled double quantum dots (c-2QD) connected to leads have been widely adopted as prototype model systems to verify interference effects on quantum transport at the nanoscale. We provide here an analytic study of the thermoelectric properties of c-2QD systems pierced by a uniform magnetic field. Fully analytic and easy-to-use expressions are derived for all the kinetic functionals of interest. Within the Green 0 s function formalism, our results allow a simple inexpensive procedure for the theoretical description of the thermoelectric phenomena for different chemical potentials and temperatures of the reservoirs, different threading magnetic fluxes, dot energies and interdot interactions; moreover they provide an intuitive guide to parametrize the system Hamiltonian for the design of best performing realistic devices. We have found that the thermopower S can be enhanced by more than ten times and the figure of merit ZT by more than hundred times by the presence of a threading magnetic field. Most important, we show that the magnetic flux increases also the performance of the device under maximum power output conditions.


I. INTRODUCTION
Quantum dot systems have attracted enormous interest as workable thermoelectric device candidates for the study of electronic and thermal quantum transport at the nanoscale.
The origin of such an interest both from the theoretical and the experimental side, resides in the potential they offer, as artificial nanoscale junctions, to explore a large variety of thermoelectric effects. Relevance of nanostructures as performing energy harvesting devices was envisaged in the pioneering paper of Hick and Dresselhaus 1 . Since then nanoscale thermoelectricity has been addressed by an increasing number of theoretical and experimental works; a perspective of the field can be found in the focus point collection in Ref. [2].
The system composed by two single-level quantum dots coupled to each other (c-2QD) via metallic leads, in two terminal or multiterminal setups 20 , and via an interdot tunneling are most appropriate to probe how the Hamiltonian system parameters and external conditions can be varied to optimize the energy conversion efficiency and the output power of the thermoelectric device. This is a demanding task because such parameters often play conflicting roles in the optimization process. Strategies for increasing thermoelectric performances utilizing a steep slope in the transmission function T (E), or its specific shape, or its resonances, have been well described in Ref. [21] where also a comparison between the thermoelectric efficiency of inorganic and organic materials is discussed.
Enhancing thermoelectric performance in linear regimes, requires maximization of the dimensionless thermoelectric figure of merit ZT = σS 2 T /κ where σ is the electrical conductance, S the thermopower (Seebeck) coefficient, T is the temperature and κ = κ e + κ p is the thermal conductance (which includes electronic and lattice contributions). In the search of optimal thermoelectric response of the device, most important quantities are its maximum efficiency as thermoelectric generator, and the efficiency at the maximum of the output power.
A crucial aspect for the evaluation of the thermoelectric response of a device, is the wide parameters range to be explored simultaneously to determine its optimal functioning. In this context, the possibility of using analytic expressions for all the involved thermoelectric functions greatly simplifies the task. In the literature, the analytic treatment of the c-2QD is confined at sufficiently small temperatures by means of the Sommerfeld expansion, extended when necessary to fourth order in k B T in the evaluation of kinetic parameters. 8 In the case of Lorentzian shape of the transmission function, analytic expressions of the thermoelectric transport coefficients have been obtained in terms of digamma functions 22  We adopt the convention that the left reservoir is the hotter one (T L > T R ) while no a priori assumption is done on the relative position of the chemical potentials µ L and µ R of the left and right reservoirs. We consider a two-terminal quantum dot setup, stationary transport conditions, absence of lattice contributions to thermal conductivity (k ≈ k e ), and no electronic correlation effects. The general expression for thermoelectric transport charge current I through the c-2QD, in stationary conditions, is given by 25 where f L,R denote the Fermi functions of the two reservoirs. The electric power output (P(E) > 0) is given by where ∆V = (µ L − µ R )/(−e) is the voltage drop and e = |e| is absolute value of the electron charge.
The thermoelectric efficiency of the device is given by the ratio between the work done and the heat extracted from the high temperature reservoir: In steady state conditions the heats per unit time are the thermal currents and W per unit time is the output power P. Then Expressions from (1) to (4) depend on the thermodynamic parameters µ L , T L , µ R , T R and by the c-2QD transmission function T (E), and hold in the linear and nonlinear regimes. In this paper we are interested in the linear response of the system so that ∆µ = µ L − µ R and ∆T = T L − T R are infinitesimal quantities. To first order in ∆T and ∆µ, we can write Expressions (1), (2) and (4) For convenience, in Eqs.(5) the thermodynamic parameters µ L , T L and the Fermi function f L are denoted dropping the now inessential subscript L.
In Section II we report details on the c-2QD system and its description in terms of localized functions. In Section III we provide our novel analytic expressions of the transport parameters relevant to control and design of the thermoelectric response of the c-2QD, in the linear response regime. Application of the above expressions and discussion of the results are reported in Section IV where contour plots are reported to better evidence the energy and magnetic field values eventually responsible of efficiency at the maximum output power.
We have found that the thermopower S may be enhanced by more than ten times and the figure of merit ZT by more than hundred times due to a threading magnetic field. We red look for chemical potential and magnetic flux values which give the maximum output power and demonstrate that the magnetic flux also increases the corresponding efficiency. Section V contains our conclusions.

II. SYSTEM DESCRIPTION AND MODEL
In this section we establish a localized basis model for the c-2QD electronic system in contact with the left and right reservoirs, in the presence of a threading magnetic field. To keep the model at the essential, we make some simplifications that could be dropped or better analyzed, when necessary.
Consider a double dot electronic system, with a single orbital per dot, described within the one-electron approximation in the tight-binding framework. The one-electron Hamiltonian can be partitioned in the left lead, central device, right lead, and coupling interaction The electronic system is schematically pictured in Fig.1, where the presence of a uniform magnetic field is also considered.
The central device, a double dot molecule, is described by the Hamiltonian of the type in the bra-ket notations where E d is the energy of both dots orbitals φ 1 , φ 2 , and t d (supposed real and negative) is the off-diagonal coupling between the two dots.
For what concerns the description of two electrodes not yet coupled to the dots, we can proceed as follows. Consider, for instance, the left lead and specifically the "left seed state" |φ a > that carries the coupling with the central device. The effect of all the other (infinite) degrees of freedom of the left electrode are embodied in the Green's function g aa on the end seed state. In principle, the Lanczos procedure can be applied to generate the Lanczos chain and, then, to determine the Green's function [see for instance Ref. [27]]. The same considerations apply for the right lead. We have is the flux of the magnetic field through the entire two-loop (φ a , φ 1 , φ b , φ 2 ) plaquette, and Φ 0 = hc/e is the quantum of flux. In the case of degeneracy Following the routinely adopted "wide-band approximation" we consider explicitly only the imaginary part of the above Green's functions and disregard the energy dependence. The leads are replaced by the corresponding end states, with the retarded and advanced Green's functions purely imaginary quantities, independent from energy. In a symmetric geometrical environment, we have where ρ = −(1/π) Im g R represents the local density-of-states, assumed to be constant in the typical energy region of actual interest.
The coupling between leads and central device in the absence of magnetic field is represented by a loop with nearest neighbor interaction t (taken as real for simplicity). In the presence of magnetic field, appropriate Peierls phases are introduced. The Berry phases corresponding to the magnetic field are set on the hopping parameters connecting the upper quantum dot φ 1 with the end orbitals φ a , φ b of the electrodes: We have now all the ingredients for the calculation of the Green's function and of the transmission function of the electronic device.

A. Green's function of the degenerate double dot in magnetic fields
The central part of the device is constituted by the two orbitals of the two quantum dots, coupled one to the other. We can use the renormalization-decimation procedure to fully eliminate the degrees of freedom of the leads, now represented by the end seed states |φ a > and |φ b > [see for instance Ref. [27]]. The retarded self-energies produced by the left lead on the central device become Similar procedures can be followed for the right lead and for the advanced self-energies.
It is convenient to define the real and positive quantity γ/2 = πρ t 2 > 0 , that encompasses two parameters of the structure into a single one. Using Eqs. (11), the retarded (advanced) self-energy matrix produced by the left lead in the central device can be cast in the form (with γ > 0). Similarly, for the retarded and advanced self-energies produced by the right lead, we have The total self-energies of the left and right leads are then Finally the coupling parameters are given by the expressions It should be noticed that the self-energies Σ and the broadening parameters Γ depend on the applied magnetic field, but are completely independent from the energy variable. This nice feature is a consequence of the wide band approximation and fosters the possibility of a fully analytic treatment of transport parameters, which is a key aspect of this article.
The retarded effective Hamiltonian for the double-dot in the central device, after the full decimation procedure of the leads, is given by the expression It follows The inversion of the above matrix provides the retarded Green's function, represented by the symmetric matrix where The advanced Green's function is the hermitian conjugate of the retarded one. Since the matrix G R (E) in Eq. (14) is symmetric, it follows In the present case, the advanced Green's function is the complex conjugate of the retarded one.

B. Transmission function of the symmetric double dot in magnetic fields
We can now proceed to the explicit calculation of the transmission function T (E) of the double dots, coupled one to the other and immersed in magnetic fields. Using the general Keldysh nonequilibrium formalism (applicable to interacting or noninteracting systems) or the Landauer-Büttiker procedure (specific for the latter case) [see for instance Refs. 28,29], we have that the transmission coefficient of the non-interacting nanostructure is given by the familiar relation where we have taken notice that, in the wide band approximation, the left and right coupling are independent from energy.
To perform the product of the four matrices in Eq. (16), we begin to consider the product of the first two matrices. Using Eq.(12d) and Eq. (14) one obtains From Eq.(12d) and Eq.(15), we also have Multiplication of the matrix of Eq.(17) by its complex conjugate matrix, followed by the trace operation, gives the transmission function.
After somewhat lengthy but straight manipulations one obtains the expression of the transmission function of a coupled double quantum dot in a uniform magnetic field and symmetrical geometry: Whenever necessary, some of the approximations done for sake of simplicity and for making transparent the main guidelines can be overcome at the modest cost of some further manipulation. For instance the same procedure can be exploited in the case the dot levels are non degenerate, or the geometric environment is non-symmetric, the magnetic field is nonuniform, for multilevel dots, and other similar situations.
For instance, in the case of a non-degenerate double quantu.epsm dot, with levels E 1 = E 2 in a symmetric geometrical environment the transmission function becomes where In the case of degeneracy E 1 = E 2 = E d , one recovers back Eqs. (18).

C. Magnetic field effects on the transmission function
In the following we keep on focusing on the degenerate double dots. The deep interference effects of the magnetic field on the transmission function, with the introduction of sharp resonances and anti-resonances, make these and similar nano-structures appealing candidates for thermoelectric applications.
According to Eqs.(18) the transmission function of the double quantum dot system, as a function of the energy variable and of the magnetic phase variable, takes the form The transmission function versus θ is periodic with period 4π, corresponding to two additional flux quanta, or equivalently to one flux quantum for each of the two loops of Fig.1.
In the absence of magnetic fields (or in the presence of an even number of flux quanta), from Eq.(20) one obtains which is just a Lorentzian function centered at and effective width Γ ef f = 2γ. In the presence of one flux quantum (or any odd integer number of flux quanta) Eq. (20) gives which is a Lorentzian function centered at and effective width Γ ef f = 2γ. At semi-integer flux quanta θ = π (or any odd integer number of π) the transmission function versus E takes the symmetric structure with respect to the For γ << |t d | (including also γ ≤ |t d |) the transmission function of Eq.(23) exhibits two peaks at ±(t 2 d − γ 2 ) 1/2 , and a valley around E = 0. The two peaks are well separated if |t d | >> γ.
It is of much importance to notice that, apart the special values θ = 0, π, 2π, 3π (modulus 4π) discussed above, for finite values of E, the transmission function of Eq.(20) has a unique zero; namely: Thus the antiresonance is at the right of the anti-bonding state for 0 < θ < π, while it is at the left of the bonding state for π < θ < 2π.
From the above discussion, it is seen how the application of the magnetic field may trans-

EVALUATION OF THE KINETIC PARAMETERS
Once the transmission function is known, we can access the kinetic transport coefficients that control, in the linear approximation, the thermoelectric properties of the nanoscale device. The kinetic transport coefficients, in dimensionless form, are linked to the transmission function T (E) by the relations: where µ is the chemical potential, T the absolute temperature, and f (E, µ, T ) the Fermi function.
In the literature, the evaluation of the kinetic coefficients K 0,1,2 is in general carried out either with the Sommerfeld expansion, 30 where [. . .] stands for any arbitrary function of E for which the integral exists. Then, the expression of the kinetic parameters of the symmetric double dot reads where T (E) is the transmission function reported in Eq. (18).
Transmission function T (E) for the coupled degenerate double dot Dimensionless kinetic parameters for the degenerate double dot system in the linear regime: ) Im w ≶ 0; I 1 (w) = 1 + wI 0 (w); I 2 (w) = w + w 2 I 0 (w)  where the pole positions z 1,2 and the weighting factors A 12 are given by Eq.(A9).
Equation (28) is fully equivalent to Eq.(18), but it enjoys the invaluable advantage to put in evidence its two pole analytic structure. This permits the straight evaluation of the kinetic parameters: The analytic expressions of the kinetic functionals entering Eq. (29) are provided in Appendix B. The results for K 0 , K 1 , K 2 are given by Eqs. (B11,B12,B13) respectively. The transmission function and the corresponding kinetic integrals of the symmetric double dot are summarized in Table I, for immediate reference. The procedure here outlined is of value not only for the present problem, but also because it provides useful guidelines for a number of more complex situations.
Expressions of the thermoelectric functions in terms of the kinetic parameters  After achieving the task of a straight analytic evaluation of the kinetic parameters of the double quantum dot system as summarized in Table I, it becomes now routine to investigate the transport properties. Following closely Ref. 26, in Table II we report for sake of completeness the expressions of the electric and thermal conductances, of the Seebeck coefficient and the other transport parameters of interest, in terms of the kinetic coefficients K 0 , K 1, , and K 2 .

Expressions of the thermoelectric natural units for nanoscale devices
In the next section we evaluate magneto transport properties of specific double dot devices, and discuss the variety and wealth of effects occurring in spite of the reasonable simplicity of the model. n odd. We notice that T (E) is symmetric around E d for θ = π or θ = (2n + 1)π as required by Eq. (23). It is important to observe that |S| increases by more than 10 times and ZT by more than 100 times with respect to the case θ = 0, for specific values of the magnetic flux threading the c-2QD circuit as evidenced in the plots in the right side of Fig.3c and  Fig.3d shows that for the chosen T and γ parameters, ZT can reach values ≈ 6 in the regions θ ∼ π/2 and θ ∼ 3π/2. The above results evidence that temperature and magnetic flux can be exploited to increase the thermoelectric factor of merit .
Most interesting is the evaluation of the performance of the c-2QD as heat engine, in this case a study of the efficiency at the maximum output power is required. Several recent papers [32][33][34][35][36][37] have shown that the mere knowledge of the maximum efficiency of a heat engine is of limited importance since the useful operative information concerns the conditions corresponding to the maximum power output. 38,39 It is known in fact,that even if the figure of merit ZT of a thermoelectric device can assume large values (>>1) mainly for nanostructured systems 9,40,41 , what really matters is just the efficiency evaluated at the maximum power output. To better clarify this point, we report in Fig.4a the thermoelectric efficiency and in Fig.4b the output power, respectively, as function of µ and θ, as defined in Table II.
Once again we observe that the magnetic field strongly enhances the thermoelectric effi-  power. From Fig.5 we can observe that the maximum efficiency is higher than the efficiency at operating conditions where the maximum output power is realized We can see that the highest value of the power output P M /η 2 C is 16800 (in units k 2 B T 2 0 /h) for the values θ ≈ 1, and θ ≈ (4π − 1), at µ ≈ 1.062 eV, and for θ ≈ (2π − 1) and θ ≈ (2π + 1), at µ ≈ −1.062 eV.

V. CONCLUSIONS
We have presented in this paper a systematic analytic study of the thermoelectric response functions of a coupled double quantum dot system, pierced by a magnetic field, where the A 1,2,3,4 constants (i.e. independent from the energy variable) have the expressions It is seen that the A i constant is the product of the differences of z i with all the other poles To demonstrate the identity (A1), suppose to multiply both members of Eq.(A1) by the expression Π i=1,4 (E − z i ). After multiplication, the first member becomes independent from E, and equal to unit. This occurs also for the second member. In fact after multiplication, the second member becomes a polynomial in E of order three, which takes the unity value at the four arguments E = z 1 , z 2 , z 3 , z 4 , and is thus unity everywhere.

Case of complex conjugate poles
In the case of complex conjugate poles, say (z 1 , z 2 , z 3 = z * 1 , z 4 = z * 2 ), it is seen by inspection that Eqs.(A1,A2) simplify in the form where the A 1,2 constants have the expressions .

(A4)
Pole structure of the double dot transmission function The expression of the transmission function of the symmetric double dot is provided in Eq. (18), and can be written in the form The analytic structure of the transmission function can be put in better evidence by expressing D R (E) in the form and similarly for its complex conjugate D A (E). Using Eqs.(A3,A4,A7) one obtains where The transmission function of the symmetric double can be expressed in the form Eq.(A10) shows explicitly the pole structure of the transmission function, and is ready for analytic evaluation of the corresponding kinetic integrals.

Appendix B. Kinetic functional for the analytic treatment of thermoelectricity in double dots
For the analytic treatment of thermoelectricity in double dots, it is convenient to define the kinetic functional as follows

Kinetic functional of a constant and of the variable itself
Due to the fact that the functional (B1) is linear, the functional of a constant equals the functional of unity times the constant. The functional of unity is Thus the kinetic functionals of unity equal the Bernoulli-like numbers. The first few Bernoulli-like numbers b n are [For details see Ref. 23].
The kinetic functional of the energy is easily obtained, using again the linear properties of the functional. It holds It follows Kinetic functional of a simple pole Consider the simple pole function of the form where z p is a given position in the upper or lower part of complex plane. The kinetic functional becomes As usual, it is convenient to introduce the dimensionless variables With the indicated substitutions, one obtains In summary, it holds where J n denote the complex functions Kinetic functional of a simple pole times the first and second power of the energy Consider the function of the form where z p is the position of the pole in the upper or lower part of complex plane, and E d is an arbitrary complex constant. We have

It follows
Another function to consider is where z p is a given position in the upper or lower part of complex plane. The kinetic functional can be cast in the form Using previous results we obtain We could proceed with higher powers along similar lines, whenever needed.
Analytic expression of the kinetic integrals for the symmetric double dot According to Eq.(29), the kinetic integrals for the symmetric double are given by the expression The first functional in the above equation, using Eqs.(B6),(B7),(B8), reads It is convenient to write more explicitly the first few values F 0,1,2 . Using Eqs.(B3) we obtain the expressions: F 0 [cos(θ/2) (E − E d ) + t d ] 2 E − z 1 = cos 2 (θ/2) (µ + z 1 − 2E d ) + 2 cos(θ/2) t d Similarly: It also holds Inserting the above result into Eq.(B9) provides the analytic expression of the kinetic parameters. It holds: K 0 = 8γ 2 Re 1 A 1 cos 2 (θ/2) (µ + z 1 − 2E d ) + 2 cos(θ/2) t d The next kinetic parameter reads The third kinetic parameter of interest is For convenience the basic results for actual simulations are summarized in Table I.