Entropy production in photovoltaicthermoelectric nanodevices
from the nonequilibrium Green’s function formalism
Abstract
We derive the expressions of photon energy and particle currents inside an open nanosystem interacting with light using nonequilibrium Green’s functions. The model allows different temperatures for the electron reservoirs, which basically defines a photovoltaicthermoelectric hybrid. Thanks to these expressions, we formulate the steadystate entropy production rate to assess the efficiency of reversible photovoltaicthermoelectric nanodevices. Next, quantum dot based nanojunctions are closely examined. We show that entropy production is always positive when one considers spontaneous emission of photons with a specific energy, while in general the emission spectrum is broadened, notably for strong coupling to reservoirs. In this latter case, when the emission is integrated over all the energies of the spectrum, we find that entropy production can reach negative values. This result provides matter to question the second law of thermodynamics for interacting nanosystems beyond the assumption of weak coupling.
I Introduction
At the end of the last century, structures grown or synthesized on a nanoscale have revealed new insights and potential applications which have profoundly altered our vision of the technology. Indeed, nanostructures integrated into a device fundamentally change the behavior of electrons due to quantum effects like tunneling, confinement or entanglement. Much hope has been focused on the dramatic potential of nanosciences and nanotechnologies. At the same time, these hopes have stimulated the use of manybody quantum methodologies to model electronic devices, like transistors datta (), but also concepts of energy conversion like in photovoltaic aeberhardJCEL11 (), optomechanic marquardt09 () or thermoelectric dubi11 () devices.
Among the methodologies of quantum statistics, nonequilibrium Green’s function (NEGF) formalism is probably the most convenient one to deal with particle and energy transport in open interacting systems datta (). In comparison, quantum master equation formalism is suited for regimes of weak coupling to electron reservoirs roy15 () or optically driven systems, while quantum cascade laser simulations have been carried out from NEGF formalism wacker02 (). The community of quantum thermodynamics, and hence thermoelectricity, has developed experience in NEGF methodology esposito15 (), which has permitted to develop new insights in timeresolved conversion crepieux11 (); dare16 (), electronelectron interacting systems zang15 (); azema12 (), fundamental laws yamamoto15 (); benenti15 (); whitney13 (), and potential new paradigms crepieux15 (); esposito15_prb (). For quantum optoelectronics and photovoltaics henrickson02 (); aeberhard08 (); berbezier13APL (); cavassilas15 (), NEGF framework is more and more used, due to the fact that more and more applications involve electron transport at a nanoscale and quantum effects.
Novel technical directions are now being explored thanks to these powerful methodologies. One possible direction follows the idea of cogeneration, which combines outputted energy forms from a single sustainable energy source, like the simultaneous production of electrical and thermal energies from a single light source. From a fundamental point of view, this direction is related to the thermodynamics of light markvart16 () which has naturally emerged in photovoltaics regarding the photon source as a thermal bath wurfel (), and the thermoelectricity at contact between absorber and lead humphrey05 (); rodiere15 (). Closer to the commercial level, the creation and performance analysis of stacked photovoltaicthermoelectric modules have been reported park13 (); bjork15 (). Incidentally, the promises of perovskites at the same time as photovoltaics even15 () and thermoelectrics mettan15 () also suggests that a combined energy conversion with these materials could be a success, using materials or nanostructures ning15 (). Down to the nanoscale, theoretical designs based on nanostructures have been recently proposed for cooling rey07 (); cleuren12 (); mari12 (); wang15 () or joint cooling and electrical energy production from a photon source entin15 (). These proposals suggest to address the idea of cogeneration inside a unique module conceived at the nanoscale, which requires a deeper look at the energetic aspects of the lightmatter coupling.
In this work, we derive the photon energy and particle currents in open nanosystems interacting with light using the framework of NEGFs. The Hamiltonian model is introduced in Sec. II, and the main lines of the derivation and results are given in Sec. III. This allows to calculate the entropy current flowing from the electron and photon reservoirs to the absorbing region of the device in Sec. IV: we reshape and discuss the entropy production in terms of efficiencies for photovoltaicthermoelectric nanodevices. Finally, we thoroughly examine a quantumdot based architecture described with a twolevel model in Sec. V: we show that the entropy production is always positive at any coupling to electron reservoirs as long as one considers a unique photon energy for the emission process, but the entropy production can reach negative values if modifications are made on the model as it is traditionally done.
Ii Hamiltonian model
The Hamiltonian of a quantum nanosystem in contact with electronic reservoirs and interacting with light reads
(1) 
where
(2a)  
(2b)  
(2c)  
(2d)  
(2e) 
Subscript stands for the noninteracting and isolated nanosystem, for the transfer to the electron reservoirs, for the interaction with light, for the the left () and right () electron reservoirs, and for the photon bath. These expressions use the electron creation (annihilation) operators, () in the central region and () in the electron reservoirs. On the other side, () and are photon operators of the photon bath, with the wave vector of the radiation, and one of the two directions of polarization perpendicular to the propagation. The interacting central region is coupled to the electron reservoirs via parameters while it is coupled to the light radiation via parameters . Usually, calculation for optoelectronics relies on the dipole approximation: where is the spatial coordinate mandelwolf ().
We introduce the notations used in this paper: the energy current, with and the particle current, and finally for the heat current crepieux11 (), where is the (electro)chemical potential of reservoir . In the case of electron reservoirs, we will also use for the electrical current. All currents are flowing from the reservoir to the central region.
Iii Photon currents
iii.1 Photon energy current
We derived the formal expression of the photon energy current inside an optoelectronic device following the first order Born approximation within the Keldysh’s formalism haug (); mahan (). From the Heisenberg equation , the energy current can be expressed in terms of expectation values on mixed operators which combine electron and photon operators
(3) 
where with . Here we use matrix forms to encode level and/or eventual spacediscretization indices. In the framework of the Keldysh formalism, we sought the expression of the contour ordered mixed Green’s function
(4) 
where is the timeordering operator. The main lines of the derivation follow the first order Born approximation mahan (), which consists in switching to the interaction picture, developing the time evolution operator up to the second order in the electronboson interaction parameter, using Wick’s theorem, verifying the cancellation of the disconnected graphs, including higher order contributions with selfconsistency and finally performing the Langreth’s rules for the analytic continuation. We thus obtain
(5)  
with
The expression of is then deduced from the Langreth’s rules, which finally provides the photon energy current from Eq. (3),
(7)  
These expressions use the standard Green’s functions for the electrons inside the central region , defined as . On the other side, we introduce the photon Green’s function . Function differs in a negative sign from which appears in the derivation of the Dyson’s equation for an electron interacting with bosons mahan ().
For steadystate devices, we obtain
(8) 
where
(9)  
and
(10)  
is the occupation number of the radiation modes . These modes form the photon bath which is assumed to be in an equilibrium or quasiequilibrium state of temperature and chemical potential .
In Eq. (9), we outline the three essential contributions for the radiative processes of the electronphoton interaction mandelwolf (): the two induced processes which include the absorption () and the stimulated emission (), and the spontaneous emission () which is independent of the occupation number of the photon bath and is nonzero in the vacuum state. The photon energy current can thus be split according to different viewpoints
(11)  
(12)  
(13) 
as the function which has the dimension of
(14) 
with
(15a)  
(15b)  
(15c) 
iii.2 Energy conservation
In the case of the selfconsistent Born approximation, the energy has to be conserved in the total system baym ()
(16) 
The two first terms are known from Ref. crepieux11, , with the reservoir selfenergies haug (). Energy currents related to the central region, the transfer process and the lightmatter coupling are zero for steadystate operating: , , and . We verified the energy conservation requirement starting with expressions (8) together with and from Ref. crepieux11, . For the calculations, we used the property with the total selfenergy haug (), in order to eliminate . Then, we evidenced the following quantity , in order to perform change of integration variables of type .
iii.3 Absorption and emission rates
Similarly, the photon current can be derived relying on previous mixed Green’s functions defined Eq. (4). We get
(17)  
with
For steadystate operation, we obtained
where
(20)  
and still
(21)  
It is worth comparing the function with the interaction selfenergy which happens in the Dyson equation for an electron interacting with the light radiation mahan (); aeberhard08 ()
(22)  
Sign changes between and are intuitive: absorption(emission) means that a photon is flowing from the photon bath(central region) to the central region(photon bath). Without these sign changes, (to be compared with Eq. (III.3)), which fulfills the condition of current conservation along the nanodevice haug ().
Similarly to the case of the photon energy current, it is meaningful to distinguish between the three radiative processes of the electronphoton interaction throughout
(23) 
with
(24a)  
(24b)  
(24c) 
The derivation of Eq. (III.3) provides general expressions for the radiative rates in the stationary case. Indeed, we decompose and then identify (using the cycling property of the trace)
(25a)  
(25b)  
(25c) 
where
Expressions (25a25c) reiterate the formula provided by Aeberhard in Ref. aeberhard11, from analogy between the Boltzmann and Dyson equations.
iii.4 Spectral photon currents
From decomposition Eq. (23), it is possible to derive the three spectral photon currents using the photon density of states
(27a)  
(27b)  
(27c) 
We have introduced the direction of light propagation and abbreviated the frequency by writing .
Within theses derivations, the radiation is treated as a third terminal, in contrast with other developments where the photon Green’s functions are fully taken into accounts with their own dynamics richter08 (). However, these derivations allow us to provide radiation properties from the knowledge of the matter, in terms of electron Green’s functions, via the trace of (see Eq. (III.3)). This function depends on both the electron and photon energies, and it is connected to the polarization insertion of the interaction dynamics fetterwalecka (). It is also interesting to introduce the induced spectral current given by
(28) 
where
(29) 
is a rate of net absorption (if ) or gain (if ) in the optoelectronic device. Taking advantage of the equality of , the spectral current is shaped into
(30)  
with
(31)  
Using the dimensionless function , we finally formulate the photon particle and energy currents as follows
(32)  
where is the elementary solid angle around the direction along which the light propagates. It is interesting to point out the similar expressions we have for these currents: they are both written as the product of a twodimensional spectral quantity multiplied by the photon energy at the power zero for the particle current, and at the power one for the energy current.
iii.5 Quasiequilibrium limits
Within NEGF formalism, the electronphoton interaction is described using the selfconsistent Born approximation in terms of electron and photon Green’s functions. The approach is original in the sense that it is in fact not necessary to define local thermodynamic parameters to obtain particle, energy or entropy currents which flow outside the outofequilibrium central region. In devices where the central region reaches the nanoscale, particles experience nonthermal states while the device is working. It is not a simple task to define local temperature, and electrochemical potential in the interacting central region whitneyPE16 (); meair14 (). Indeed, all the particle statistics is encoded in NEGF formalism bruus (). However, if NEGFs can be represented by quasiequilibrium Green’s functions, they will verify a KuboMartinSchwinger relation balzer () , where and represent the electronic chemical potential and temperature respectively. This relation generalizes the following properties of the Fermi and Bose functions: and .
More generally, let us consider the case of a semiconductor in which electrons inside the conduction band, and holes inside the valence band experience separate quasiequilibrium states characterized by two different chemical potentials and temperatures, and . In that case, the diagonal components of Green’s functions follow local KuboMartinSchwinger relations
(34) 
where refers to the band index. Thanks to these relations, simplifications occur in the expression of , Eq. (31). In particular for , no longer depends on the electron energy , and it follows
(35) 
in which one can define and , the temperature and chemical potential of the spontaneously emitted radiation wurfel82 (). Hence we obtain the full BoseEinstein statistic function that happens in the socalled generalized Planck’s law for the emission lasher64 (); wurfel82 (), that was also discussed in photovoltaic cells of quantum dot arrays using NEGFs berbezier15 (), and notably used to determine the thermopower from optical measurements gibelli16 ().
In the quasiequilibrium limit, does not depend on , which allows us to write the spectral emission current as
(36) 
Using Eqs. (28) and (29), the two quasiequilibrium limits of the photon particle and energy currents finally read as
(37)  
(38)  
Expressions (37) and (38) are similar to the ones obtained in Ref. richter08, dealing with nonequilibrium photon Green’s functions. Interestingly, our approach suggests that in the case of a nonequilibrium nanosystem given by Eqs. (32) and (III.4), a generalized energy flow law would involve twodimensional spectral functions, as with , following the idea of Ref. richter08, .
Iv Entropy current
Equation (8), which gives the energy current from the photon bath to the dot, allows us to calculate the entropy current flowing from the central region to the reservoirs in terms of Green’s function and selfenergies.
iv.1 Spectral entropy current
The device is an open interacting nanosystem connected to three reservoirs: the two electron left and right reservoirs, and the photon bath. In this threeterminal configuration, the entropy current flowing from the central region to the three reservoirs is defined as
in which we use the relation guaranteed by the charge conservation.
Implementing results of Secs. III.1 and III.2 in Eq. (IV.1), we are hence able to derive the entropy current in terms of Green’s functions from the spectral entropy current as follows
(42) 
with
(43) 
and
(44)  
In nanosystems maintained in outofequilibrium steady states, the entropy current flowing from the central region to the reservoirs is equal to the rate of entropy production esposito10a ().
iv.2 Entropy production is recast in terms of efficiencies
For photovoltaicthermoelectric converters, we define the nanodevice efficiency as the ratio of the output electrical power or useful heat current to the input power in the form of light, which is given by the heat current of the absorbed photons, . This definition contrasts with “the maximal power conversion efficiency” defined in practice at maximal output power, and where the denominator is the incident radiant power; it thus does not depend on the processes undergone by the system einax11 ().
In this section, we focus on three devices based on a central region interacting with light: a photovoltaics (), a refrigerator based on a cooling by heating process () cleuren12 (), and finally a joint device which provides both cooling and electrical energy production () entin15 (). For the three nanodevices, the rate of entropy production is recast in terms of efficiencies according to the device, as Whitney proposed in Ref. whitney15, . Indeed, all nanodevices () provide the same formal rate of entropy production
(45) 
where is the efficiency of the reversible nanodevice, is the output power in the reversible nanodevice, and is the maximum rate of entropy production achievable in the nanodevice. Ratio reflects how close to the maximum efficiency the device is working.
Table 1 summarizes the definitions and notations of the relevant efficiencies discussed for the three nanodevices. These efficiencies are named thermodynamic efficiencies as they can be manipulated following the laws of thermodynamics.
Nanodevice (ND)  







Standard engine  


Refrigeration ()  
Heat pump ()  

iv.2.1 Standard photovoltaics
For , we can derive the photovoltaic case
(46) 
where , and . We always have while . The Carnot efficiency is defined Tab. 1.
Here, it can be worth deriving the related electroluminescent () case, for which implies ,
(47) 
where (with ) is the efficiency of the electroluminescent device, and replaces is the temperature of the photon bath formed by electroluminescence.
The efficiency of the reversible photovoltaic nanodevice is reduced compared to the Carnot limit: from expression (46), the maximum value of the efficiency is . This maximal efficiency may be compared to the Landsberg’s limit landsberg98 (); green (): ( stands for ambient, and it corresponds to in this work). Landsberg wanted to reconsider the limit of the Carnot efficiency as the upper limit for photovoltaics. Starting from the model of a dithermal engine, he included the energy and entropy fluxes related to the emission process. In the Landsberg’s approach, the central region is a converter in a state of equilibrium, and it behaves as a black body emitting photons at temperature ( stands for converter). Landsberg demonstrated that the maximal efficiency of the reversible device, , is reached when . The Landsberg’s approach ignores the details of the electron properties in the converter which is also assumed at equilibrium. However, despite these differences, NEGFbased expression of the entropy production Eq. (46) comes to similar conclusions to those of Landsberg: the maximum efficiency is always lesser than the Carnot limit of a heat engine producing work from the electron and photon reservoirs.
iv.2.2 Cooling by heating process
We discuss the coefficient of performance of a cooling by heating process as proposed in Ref. cleuren12, with and (see Tab. 1 for the efficiency definitions),
From this formula, we deduce for this original cooling process
(49) 
which meets Eq. (11) of Ref. cleuren12, with the additional reducing contribution to the coefficient of performance. Indeed, the emission processes were not included in the approach of Ref. rutten09, , which was developed in the strong optical coupling regime. Moreover, in the recent model proposed by Wang and coauthors in Ref. wang15, to verify the third law of thermodynamics in the refrigerator, the cooling regime includes a parasitic emission in the regime of weak coupling to the electron reservoirs, which involves a single emission wavelength.
iv.2.3 Joint cooling and energy production
For a more general case, but with a specific device objective, we examine the joint cooling and energy production proposed in Ref. entin15, . The joint process can be seen as a photovoltaic configuration with , or a cooling by heating configuration with . It follows two expressions for the rate of entropy production