Patent Application
20180131311
ThermoPhotoVoltaics (TPV) is a heat-to-electricity conversion mechanism, wherein Thermal radiation is absorbed by a semiconductor PhotoVoltaic (PV) cell. It is very favorable, as it involves no moving parts, allowing the possibility for compact, light (thus portable), quiet and long-lived generators, powerable from numerous sources, such as high-energy-density hydrocarbon or nuclear fuels, or solar irradiation. Like any heat engine, a TPV system has the Carnot efficiency limit, which can only be achieved with monochromatic radiation matched to the semiconductor electronic bandgap. Absorbed thermal radiation below the bandgap is completely lost and far above it suffers thermalization losses. Reaching this limit in practical implementations has been challenging.
By far the most developed TPV systems use the far-field radiation of the hot emitter to transfer thermal energy across a vacuum gap to a semiconductor PV-cell absorber. To enhance efficiency, systems with emitter emissivity of both a Lorentzian (resonant) shape and a step-function shape above the semiconductor bandgap have been suggested and built. As the emission bandwidth above the bandgap increases, more output power is produced, but the system suffers from thermalization losses. These losses can be reduced, while maintaining large output power, by utilizing tandem PV-cell absorbers, however the cost then increases significantly. Although such selective-emitter TPV systems are designed to suppress photons emitted below the bandgap, unfortunately, at high emitter temperatures, this suppression is not really effective, as the emitter-material losses are so high that both the Lorentzian and step transitions are very broad and a considerable number of photons is still emitted below the bandgap, with a great associated hit in efficiency. This can be inhibited via the use of reflectors on the cold absorber to recycle below-bandgap photons back to the emitter, either via rugate filters on the PV-cell front surface, but again cost increases drastically, or via back reflectors, but only if the substrate has low free-carrier absorption.
There is in recent years a huge interest in near-field TPV systems, primarily because utilization of not only the radiative modes but also the evanescent modes for thermal energy transfer can lead to significantly increased output power.
Systems with emitters employing a Surface Plasmon Polariton resonance tuned above the semiconductor bandgap of a thick PV cell, spaced across an extremely narrow vacuum gap, without and with a metal back-surface, have been analyzed. Indeed, much more useful power is transferred to the PV cell at a given emitter temperature, and therefore the efficiency is also improved. However, as we show, these systems not only experience the same spectrum broadening due to the increased losses of the emitter at high temperatures, but also suffer from large absorption losses in the conducting carriers of the PV-cell electrode. These two mechanisms keep the efficiency away from the ideal Carnot limit.
Systems with semiconductor emitters matched to the semiconductor absorbing PV cell have also been proposed. The case of thick emitter and absorber at nanoscopic vacuum spacings offers a n2 increase in transmitted power over the far-field equivalent, where n is the index of refraction of the semiconductors. The resonant-system case of a thin film emitter and absorber, supporting coupled electromagnetic modes, has also been investigated recently and shown to have better performance at larger gap spacings. The main drawback though of semiconductor-emitter TPV systems is that semiconductors have relatively low melting temperatures compared to metals and therefore a sacrifice is made in both efficiency (from the Carnot limit 1−Ta/Te) and power (from the Stefan-Boltzmann law ˜Te4). The efficiency also suffers from losses due to the thermal carriers in the hot semiconductor emitter and the shift of its bandgap edge at high temperatures.
A TPV system including a polaritonic emitter and a thin-film semiconductor absorber has also been studied in the past (surface-phonon emitter and tungsten emitter). However, those systems were not designed so that the emitter polaritonic mode and the absorber dielectric-waveguide-type mode cross, couple and are impedance-matched just above the bandgap. This is why the reported efficiencies in those studies are significantly lower than those reported here.
In most cases of TPV studied so far, the free-carrier absorption losses associated with the necessary partially-absorbing conducting electrode on the emitter side of the PV cell have been greatly ignored or not sufficiently studied. Using graphene layers on the front of the PV cell has been suggested in R. Messina and P. Ben-Abdallah, “Graphene-based photovoltaic cells for near-field thermal energy conversion,” Scientific Reports, vol. 3, p. 1383, 11 Mar. 2013, and in V. B. Svetovoy and G. Palasantzas, “Graphene-on-Silicon Near-Field Thermophotovoltaic Cell,” Physical Review Applied, vol. 2, no. 3, p. 034006, 11 Sep. 2014, but, in both cases, the PV cell had a bulk geometry.
Embodiments of the present technology include a thermophotovoltaic apparatus that comprises a thermal emitter and a photovoltaic cell. The thermal emitter comprises at least one material having a relative dielectric permittivity with a real part of −1 at a frequency between cog and about 1.7ωg. The photovoltaic cell comprises at least one semiconductor material having an electronic bandgap of energy ℏωg, where ℏ is Planck's constant and ωg is an angular frequency of the electronic bandgap. The semiconductor material has a thickness of less than λg=2πc/ωg, where c is the speed of light in vacuum. The photovoltaic cell further comprises a back reflector having a reflectivity at ωg of at least 90%,
In operation, the thermal emitter is placed in thermal communication with a heat source and receives heat from the heat source. Similarly, the photovoltaic cell is placed in electrical communication with an electrical load and delivers power to the electrical load. The photovoltaic cell is separated from the thermal emitter by a vacuum gap, which has a width of less than λg, and receives thermally emitted photons from the thermal emitter across the vacuum gap.
In other examples of the thermophotovoltaic apparatus, the thermal emitter comprises at least one polaritonic material and supports, on an interface of the polaritonic material, at least one surface polaritonic mode having (i) an upper cutoff frequency between ωg and about 1.7 ωg and (ii) a field that extends inside the vacuum gap.
The thermal emitter may also comprise at least one polaritonic material and support, on an interface of the at least one polaritonic material, a surface polaritonic mode having a resonant frequency between ωg and about 1.5ωg. In these cases, the photovoltaic cell supports a photonic mode having a resonant frequency between ωg and about 1.5ωg. And the surface polaritonic mode of the thermal emitter couples with the photonic mode of the photovoltaic cell with a coupling coefficient κ, where κ/ωg is larger than 0.01.
Other examples of the present technology include methods of fabricating thermophotovoltaic apparatuses. An example method includes holding a thermal emitter and a photovoltaic cell together with a temporary thin film to form a connected structure. A casing is formed around the connected structure. This casing is connected to the thermal emitter and to the photovoltaic cell and has at least one opening. The temporary thin film is removed through the casing's opening(s) to create a gap between the thermal emitter and the photovoltaic cell. The casing is closed. The air inside the casing is removed to create a vacuum in the gap between the thermal emitter and the photovoltaic cell.
It should be appreciated that all combinations of the foregoing concepts and additional concepts discussed in greater detail below (provided such concepts are not mutually inconsistent) are contemplated as being part of the inventive subject matter disclosed herein. In particular, all combinations of claimed subject matter appearing at the end of this disclosure are contemplated as being part of the inventive subject matter disclosed herein. It should also be appreciated that terminology explicitly employed herein that also may appear in any disclosure incorporated by reference should be accorded a meaning most consistent with the particular concepts disclosed herein.
The skilled artisan will understand that the drawings primarily are for illustrative purposes and are not intended to limit the scope of the inventive subject matter described herein. The drawings are not necessarily to scale; in some instances, various aspects of the inventive subject matter disclosed herein may be shown exaggerated or enlarged in the drawings to facilitate an understanding of different features. In the drawings, like reference characters generally refer to like features (e.g., functionally similar and/or structurally similar elements).
Consider an isotropic object of a relative dielectric permittivity, ε, that can absorb electromagnetic fields or photons, when at zero absolute temperature and without any voltage across it. Using exp(−iωt) time dependence for the fields, this implies Im{ε}>0. Then, if it is brought at a non-zero absolute temperature, T, and optionally has a voltage, V, across it, then the thermally excited molecules, atoms and electrons act as randomly fluctuating sources of electromagnetic fields or photons. These fluctuating sources can be modeled via the fluctuation-dissipation theorem (FDT) as current sources, J(r, ω), at position r and angular frequency ω, with spatial correlation function
is the mean number of generated photons of frequency ω in thermo-chemical quasi-equilibrium at voltage V and temperature Tεo is the dielectric permittivity of free space, ε″ is the imaginary part of ε, ℏ is the Planck constant divided by 2π, q is the electronic charge of an electron, kB is the Boltzmann constant, δ(r′−r″) is the Dirac delta function and δαβ is the Kronecker delta. In Θ the term 1/2 that accounts for vacuum fluctuations is omitted, since it does not affect the exchange of energy between objects.
Transmission of Electromagnetic Power (Photons) from a Hot Object to a Cold Object
Poynting's theorem is a statement of conservation of energy, stating that the electromagnetic power exiting a closed surface area, A, is equal to the electromagnetic power generated due to sources, J, inside the enclosed volume, V, minus the electromagnetic power absorbed by the volume, V.
If we denote as j an object at a non-zero absolute temperature, T, and optionally with a voltage, V, across it, to calculate the net power, Pij, absorbed by another object i at a zero absolute temperature and voltage due to the thermal current-sources inside object j, one needs to integrate over all frequencies the power per unit frequency, pij(ω), absorbed at each frequency ω. In Poynting's theorem, Eq. (3), we identify pij(ω) as
where here an extra factor of 4 has been added in the definition of the time-averaged power, since only positive frequencies are considered in the Fourier decomposition of the time-dependent fields into frequency-dependent quantities.
The electric field, Ei(ri), due to sources, Jj(rj), can be calculated via the Green's function for the electric field, GE(ω; ri, rj), from the convolution integral
E
i(ri)=iωμo∫V
where μo is the magnetic permeability of free space.
Using Eq. (5), the power per unit frequency Eq. (4) can be written
Using the FDT from Eq. (1), we get the average power per unit frequency
where ko=ω√{square root over (εoμo)}=ω/c is the wavevector of propagation in free space and c the speed of light in free space.
We define the thermal transmissivity, ∈ij(ω), of photons from object j to object i via the net power absorbed by object i due to thermal sources in j
therefore, from Eqs. (7) and (8), we get
∈ij(ω)=4ko4∫V
From the expression of the thermal transmissivity, Eq. (9), it can be seen that it is dimensionless (G has units of m−1) and it depends only on the geometry and material selection and not on the temperatures or voltages of the two objects. Furthermore, when the system is reciprocal, then G(ω;ri,rj)=GT(ω;rj,ri) and, as a consequence, from Eq. (9)
∈ij(ω)=∈ji(ω) (10)
One could, in principle, calculate the transmissivity via Eq. (9), however, this expression needs the calculation of two volume integrals (in Vi and Vj), which make it cumbersome either analytically or numerically. Instead, there is a simpler method to calculate the same quantity with only one integral. From Poynting's theorem, Eq. (3), and from Eq. (4), the power per unit frequency absorbed in object i due to the thermal current-sources inside object j can be written as
where εαβγ is the Levi-Civita symbol for outer product between vectors.
The first term on the right-hand side of Eq. (11) is non-zero, only in the case where we are calculating the power absorbed inside an object due to the thermal sources inside the same object, namely i and j coincide, which we denote with a Kronecker delta δij. For thermal power transmission between different objects, this term is zero.
The first term (A), where i=j, can be calculated using the Green's function for the electric field, Eq. (5)
while the second term (B) needs also the Green's function for the magnetic field
Using the FDT from Eq. (1), we get the average power per unit frequency
Therefore the transmissivity, from Eq. (8) and (15), is
∈ij(ω)=4ko2∫VdAi,αGβμE(ri,rj)GγμH*(ri,rj)} (16)
where, again, the first term is only present if we are calculating the power absorbed inside an object j due to the thermal sources inside the same object. Both terms involve now only one volume integral and the second term also a simpler surface integral, so Eq. (16) should be the preferred method of transmissivity calculation.
We also define the thermal emissivity, ∈j(ω), of object j via the net power emitted outwards by object j due to its thermal sources. Clearly, the net power emitted outwards by object j must be the sum of the powers transmitted to all other absorbing objects and the power radiated into infinity. If radiation is also regarded as an “object”, then ∈j(ω)=Σi≠j∈ij(ω). In general, it is straightforward to show that
∈j(ω)=4ko2∫VdAj,α′GβμE(rj′,rj)GγμH*(rj,′rj)} (17)
Using Eq. (16) with i=j and (17), one can write a “thermal form” of the power conservation Poynting's theorem Eq. (3) for an object j, stating that the power generated by the thermal sources inside j is either emitted outwards (∈j) or reabsorbed inside it (∈ij):
∈jj(ω)=4ko2∫V
Note that, if the object j is of infinite extent, the integral term in Eq. (18) (same as first term in Eq. (16)) also diverges to infinity. Therefore, when in need to calculate the self-reabsorption term ∈jj, we typically restrict ourselves to a finite object j, which is also practically reasonable to assume.
Note that, similar to Eq. (8), also the net rate of photons, Rij, emitted from object j and absorbed by object i can be found by simply removing the energy per photon ℏω, therefore
Consider a thermal-power transmission system that varies only in two (xy) directions and is uniform and of infinite (practically very large) extent in the third (z) direction. Then, the Green's function between the emitter linear object j and the absorber linear object i can be written via its Fourier transform
where ρi=(xi,yi). Then in Eq. (7), if kz and kz′ are the integration variables for the two instances of the Green's function, the integral ∫dzj gives a result δ(kz−kz′) and the integral ∫dzi gives the total linear (large) length L of the system. Therefore, the power per unit length becomes
If we define again the dimensionless thermal transmissivity, ∈ij(ω,kz), via
then from Eq. (28) and (29) we get
∈ij(ω,kz)=4ko4∫A
which is the same as Eq. (9), only with surface instead of volume integrals and the Green's functions replaced by their Fourier transforms. Similarly, Eq. (16) can be used for the calculation of transmissivity for linear systems with the same modifications.
∈ij(ω,kz)=4ko2∫AdLi,αgβμE(ω,kz;ρi,ρj)gγμH*(ω,kzρi,ρj)} (24)
The thermal emissivity, ∈j(ω,kz), of object j is, for a linear system:
∈j(ω,kz)=−4ko2∫AdLj,α′gβμE(ω,kzρj′,ρj)gγμH*(ω,kz;ρj,′ρj)} (25)
The “thermal form” of Poynting's theorem for an object j in a linear systems is:
∈jj(ω,kz)=4ko2∫A
Consider a thermal-power transmission system that varies only in one (z) direction and is uniform and of infinite (or practically very large) extent in the other two (xy) directions. Let this planar system include N layers, of dielectric permittivities εn and thicknesses dn, stacked down-up. Let the structure be rotated, so that the emitting layer j is below or the same as the absorbing layer i, 1≤j≤i≤N. The layers 1 and N might be semi-infinite (unbounded) or bounded by a Perfect Electric Conductor (PEC) or a Perfect Magnetic Conductor (PMC).
A PEC can be implemented practically over a large range of frequencies by a metal with very large plasma frequency, low loss rate and thickness much larger than the skin depth or over a smaller range of frequencies and wave vectors by a photonic-crystal structure with a bandgap (such as an omni-directional mirror. A PMC can be implemented practically over a small range of frequencies by structured metallo-dielectric photonic crystals with a bandgap designed to reflect an incident electric field without any phase-shift.
For the planar system under investigation, the Green's function between the emitter layer j and the absorber layer i can be written via its Fourier transform
where kxy=√{square root over (kx2+ky2)}. Then in Eq. (7), if kx, ky and kx′, ky′ are the integration variables for the two instances of the Green's function, the integral ∫dxjdyj gives a result δ(kx−kx′)δ(ky−ky′) and the integral ∫dxidyi gives the total transverse (large) area A of the system. Therefore, using also that ∫∫−∞+∞dkxdky=2π∫∫0∞kxydkxy, the power per unit area becomes
If we define again the dimensionless thermal transmissivity, ∈ij(ω,kxy), via
then from Eq. (28) and (29) we get
∈ij(ω,kxy)=4ko4∫d
which is the same as Eq. (9), only with linear instead of volume integrals and the Green's functions replaced by their Fourier transforms. Similarly, Eq. (16) can be used for the calculation of transmissivity for planar systems with the same modifications.
∈ij(ω,kxz)=4ko2∫d
where β, γ run now only through x, y in Cartesian coordinates or ρ, θ in cylindrical coordinates.
The thermal emissivity, ∈j (ω,kxy), of object j is, for a planar system:
∈j(ω,kxy)=−4ko2∫d
The “thermal form” of Poynting's theorem for an object j in a planar systems is:
∈j(ω,kxy)=−4ko2∫d
We see that in planar systems, the surface integral of Eq. (16) and (17) conveniently converts in Eq. (31) and (32) to two simple evaluations at the limiting z coordinates of the layer i or j boundaries. These two limits capture the net power exiting or entering layer i, of thickness di=zi,max−zi,min, and are not always both present (non-zero).
When calculating the transmissivity ∈ij between two different layers i≠j, the first term in Eq. (31) is zero. With the emitter layer j below the absorber layer i:
When calculating the emissivity ∈j of layer j via Eq. (32), then:
When calculating the self-reabsorption term ∈ij of layer j via Eq. (33), again, we typically assume that layer j is finite, so that the integral in Eq. (33) does not diverge.
For planar systems, in order to calculate ∈ij(ω,kxy) from Eq. (31), one can construct a semi-analytical expression for the Green's functions, g, using a scattering matrix formalism, and then perform the integration in zj analytically. We use the procedure outlined by M. Francoeur, M. P. Mengüç and R. Vaillon, “Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green's functions and the scattering matrix method,” Journal of Quantitative Spectroscopy & Radiative Transfer, vol. 110, p. 2002, 2009, with the modification that we use the canonical scattering-matrix formulation:
For a two-port with ports I and II, waves incoming to the ports have amplitudes aI, aII—for example (E, H)I,in=aI(ϕE, ϕH)I—and waves outgoing from the ports have amplitudes bI, bII, where the wavefunctions ϕE, ϕH are normalized so that, for example for an incoming wave at port I (propagating along+{circumflex over (z)}):
Then the scattering matrix is defined by
With this definition, the scattering matrix is symmetric (
For calculation of a scattering matrix at the interface between layers i and j, we use the sign convention that the reflection coefficient of a wave incident from layer i is rij=−rij=(Xi−Xj)/(Xi+Xj), where the admittance of a transverse electric (TE) wave in layer n is Xn=kz,n/ω∈n, and the impedance of a transverse magnetic (TM) wave in layer n is Xn=kz,n/ωμo. Note that, with this convention, at kxy=0, where the TE and TM waves are identical, the reflection coefficient has opposite sign for the two polarizations.
For the thermal system of planar layers, we define the amplitude coefficients of the forward (upward) and backward (downward) propagating (or evanescent) waves inside each layer at the middle of the layer, except for the cases of a semi-infinite bottom layer 1 or a semi-infinite top layer N, for which they are defined at their only interface.
The first step is to find the amplitude coefficients at the layer j. Let
if j=1 and the layer is semi-infinite), and
Note that:
Using the Green's functions expansion in M. Francoeur, M. P. Mengüç and R. Vaillon, “Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green's functions and the scattering matrix method,” Journal of Quantitative Spectroscopy & Radiative Transfer, vol. 110, p. 2002, 2009, the first term in Eq. (31) or (33), denoted by ∈J,j is calculated as
where kz,j=√{square root over (ko2εj−kxy2)}=kz,j′+ikz,j″ the complex wavevector z-component, uj=kz,jdj=uz,j′+iuz,j″ the complex propagation phase, and p=1 for TE waves or p=(ko2−kz,j2)/(ko2+kz,j2) for TM waves. Again, for this term we are really only interested in the situation where layer j is finite (dj<∞). Note that, if a value is needed for a semi-infinite layer, one can provide one by removing the divergent parts and keeping only the structure-related terms:
To calculate the second expression in Eq. (31), we then find the amplitude coefficients at layer i relative to those at layer j. Let
Let xn=Xn/|Xn|=xn′+ixn″, and Rn=2xn′ sin h(uz,n″), In=2xn″ sin (uz,n′), if layer n is finite, or Rn=xn′, In=xn″, if it is semi-infinite. Then we calculate
Note that if layer j=1 and is semi-infinite, then Ci1=Di1=0.
If, additionally, layer i=N and semi-infinite, then AN1=S1N(2,1) and BN1=0, and
∈N1(ω,kxy)=x1′xN′|
A similar procedure can be followed if j>i.
Therefore, using Eq. (35)-(39), the transmissivity ∈ij(ω,kxy), as defined in Eq. (29), can be fully calculated for planar systems in a semi-analytical way.
The transmissivity, ∈ij(ω,kxy), physically quantifies how many of the thermal photons per mode of frequency ω and wavevector kxy, emitted from the emitter j, are absorbed by the absorber i. Therefore, it cannot be larger than 1, for each one of the two decoupled (for isotropic media) TE and TM polarizations. This can easily be seen for the special case of Eq. (40).
The emissivity, ∈j (ω,kxy), physically quantifies how many of the thermal photons per mode of frequency ω and wavevector kxy, emitted from the emitter j, are absorbed by all other absorbing objects or radiated and lost into the infinite background. Therefore, it also cannot be larger than 1, for each one of the two decoupled (for isotropic media) TE and TM polarizations.
If the infinite background has refractive index nb, the modes of the emitter layer j, which are evanescent in it (kxy≥ωnb/c), do not radiate into the background. Instead, only radiative modes (kxy<ωnb/c) can reach the far field. Consider the case where all emission goes into radiation (no absorption) and the emissivity is the same for both polarizations and has the form:
Then, from Eq. (29), the kxy-integral gives the background density of states ρb(ω)=ω2nb2/4πc2, and we get
where the factor of 2 comes from accounting for both polarizations. We see that Eq. (42) coincides exactly with Planck's law of thermal radiation by a body with the commonly-defined emissivity {tilde over (∈)}(ω), which also by definition is not larger than 1.
We conclude that the emissivity, ∈j(ω,kxy), defined in this specification for planar systems is a generalization of the commonly-defined emissivity, {tilde over (∈)}j(ω), with values between 0 and 1. It is wavevector-dependent and it depends on the geometry and materials, not only of the emitter but of the entire photonic system.
Consider a system of multiple objects, where each one may be at a non-zero absolute temperature, Tj, and may have a voltage, Vj, across it. The thermally-excited sources inside each object j generate photons with mean number Θ(ω,Vj,Tj). To find the net rate of photons Ri (or power Pi equivalently) emitted by object i, one must successively set to zero the temperatures of all objects but one each time, and then apply the superposition principle.
where the emissivity ∈j(ω)=Σi≠j∈ij(ω). If the system is reciprocal, then ∈ij(ω)=∈ji(ω) and therefore Eq. (43) can also be written as
Temporal Coupled-Mode Theory (CMT) is a powerful semi-analytical tool, providing good approximate results for resonant physical systems, that has been used primarily in the field of photonics and recently to predict far-field thermal radiation emission from resonant thermal emitters and near-field thermal energy exchange between resonant surface states. Here, we extend these thermal-radiation CMT models to show how to easily calculate the thermal transmissivity ∈ij between two objects i and j or the thermal emissivity ∈i of an object i. We demonstrate the model use for both orthogonal and coupled resonances, and in the presence of background losses.
Consider a system of multiple objects, where each object n is at a non-zero absolute temperature Tn and potentially has a voltage Vn across it. At this thermal and chemical equilibrium, the excited sources inside lossy object n generate photons at every frequency ω with mean number Θn(ω)=Θ(ω,Vn,Tn). To calculate the radiative thermal energy exchange between two lossy objects i, j typically requires complex and numerically intensive calculations, which can be hundreds of hours long in 3D geometries. However, if, in the appropriate frequency regime, the system supports a few well-defined resonant electromagnetic modes, which a typical mode solver can calculate in minutes at most, a CMT formalism can be used to provide an approximate, yet reliable, solution.
Consider first the case where, for some frequency regime of interest, the system of multiple objects supports and can be approximately described by predominantly one resonant electromagnetic mode at frequency ωo and with loss rates Γn associated with different loss mechanisms of the various system objects and/or radiation into the background. Essentially, the Green's function has, in that frequency regime, a dominant single pole at ωo and can be approximated by it.
Let a represent the amplitude of the mode, normalized such that |a|2 equals the total “number of photons per unit frequency” stored in the mode. (Note the difference here with the standard definition where |a|2 equals the total energy stored in the mode.) Then we can write the CMT equations:
where θ+nθ+n*
=Θn(ω) is the mean number of ω-photons injected by the thermochemical reservoir at Tn and Vn into loss mechanism n, and similarly
θ−nθ−n*
is the mean number of w-photons reflected from loss mechanism n back into the reservoir. Since different reservoirs have independent statistics,
θ+nθ+i*
=0.
In steady state at frequency ω, we substitute in Eq. (45) da/dt=−iωa, so the amplitude is
Then, the net mean number of photons emitted by loss mechanism i is the difference of those injected into the mode minus those reflected back, namely
Since the net rate of photons emitted by object i is Ri=∫Ni(ω)dω/2π, comparing with Eq. (44), we confirm that the system is reciprocal and we calculate the thermal transmissivity between two objects i and j solely from the resonant properties of the electromagnetic mode:
Furthermore, the thermal emissivity of an object i can be found from the net number of photons it emits at w, if all other objects are at zero temperature, Ni(ω)|T
In this case of a single resonance, both the emissivity ∈i and the transmissivity ∈ij have a Lorentzian spectrum and are smaller than 1. The emissivity ∈i is maximized to 1 at resonance ω=ωo and when the emitter loss rate Γi is “matched” to the sum of the other loss rates Γi=Σj≠iΓj, two conditions collectively known as “impedance matching” in electrical engineering and “Q-matching” in physics. The transmissivity ∈ij approaches 1, when no other lossy objects exist and the impedance matching conditions (ω=ωo and Γi=Γj) are satisfied.
Let now the system support two resonant electromagnetic modes, one at frequency ω1 with loss rates Γ1n and one at frequency ω2 with loss rates Γ2n, where the two modes are spatially separated and coupled to each other via their field tails with coupling coefficient κ. With θ+ and θ− having the same interpretation as before, the CMT equations are now written as
In steady state at frequency ω, if we denote δ1=ω−ω1+iΣnΓ1n and similarly for δ2, then
and the net number of photons emitted by the lossy object 1i is
Therefore, for an absorber in the vicinity of the second resonant mode 2j,
This transmissivity spectrum shape is now a double-Lorentzian and it is still smaller than 1. At resonance ω=ωo≡ω1=ω2 and in the absence of other loss mechanisms in the system, the “impedance matching” condition, which makes the transmissivity equal 1 and locally flat, becomes
Γ1iΓ2i=η2 (54)
For a symmetric system, where also Γ1i=Γ2j≡Γ, the transmissivity ∈1i,2j=1 (but not flat) at the two distinct frequencies ω=ωo±√{square root over (κ2−Γ2)}. As expected, these two frequencies merge to a double solution at ω=ωo, when κ=Γ, namely Eq. (54) is satisfied.
Note that, if the absorber was also in the vicinity of the first resonant mode 1j, just like the emitter i1, then
which correctly reduces to Eq. (48), if the two modes are decoupled (κ=0) and thus they can be treated separately.
Finally, the emissivity of object 1i is
The CMT model presented here can be applied also to a larger number of coupled resonant electromagnetic modes of the system.
When the electromagnetic system is invariant in one or two dimensions, the modes of the system can be enumerated via a conserved wavenumber kz and kxy respectively, where two modes of different wavenumbers are orthogonal to each other. Therefore, one can use superposition over the wavenumber and arrive at Eqs. (22) and (29), by redefining the transmissivity (and similarly the emissivity) from Eqs. (48) or (53) via the substitution
for linear systems and
for planar systems. The wavenumber dependence of the transmissivity and emissivity calculated via CMT stems from the wavenumber dependence of the resonant frequencies, loss and coupling rates.
The impedance matching condition is then also wavenumber-dependent. Typically, a main parameter that can be used to achieve impedance matching between the emitter and absorber, either for a single or for two coupled resonances, is the distance between the two objects. For smaller distances, it can be accomplished at higher wavenumbers and vice versa.
To test the accuracy of the CMT model, we analyze a planar system of a semi-infinite plasmonic (e.g., metallic) emitter e separated by a small vacuum gap of thickness d from a semi-infinite semiconductor absorber a (
In
As another planar-system example, we analyze thermal energy exchange between two semiconductor thin films (emitter-absorber) of relative dielectric permittivity εa(ω) as above and thickness t=0.04λg, both backed by a perfect metal, and separated by a small vacuum gap of thickness d=λg (
In the examples above, the coupling coefficient κ between two resonant modes was calculated via the splitting of the final modes of the coupled system. However, one can also use CMT to derive a formula for calculating the coupling coefficient κ using the electromagnetic fields of the two resonant modes. Let ε1 and μ1 be the (generally anisotropic tensor) dielectric permittivity and magnetic permeability, respectively, describing the first resonator in the absence of the second resonator, where the first resonant mode has eigenfrequency ω1 and eigenfields (E1, D1, H1, B1), and vice versa for the second resonator. Let ε and μ be the dielectric permittivity and magnetic permeability, respectively, describing the whole structure in the presence of both resonators. Then the coupling coefficient κ21 from the first resonant mode to second resonant mode is given by:
where (E2−T, D2−T, H2−T, B2−T) are the time-reversed and transposed eigenfields of the second resonator, the integral in the numerator is only over the volume of the first resonator, the integral in the denominator is over the entire volume of the structure (and should be treated carefully in the case of leaky modes) and only first order perturbation terms have been kept. κ12 is given by the same expression with 1↔2 exchanged. κ21 and κ12 should be equal from reciprocity, but may differ numerically due to approximation errors, so we take the coupling coefficient κ to be κ=√{square root over (κ12κ21)}.
When the system has linear or planar symmetry, the first and second resonant modes can be characterized via a conserved wavenumber kz1 and kz2 or kxy1 and kxy2 in one or two dimensions, respectively. The integral in the numerator is zero unless kz1=kz2 or kxy1=kxy2, in which case the integrals become dxdy-area or dz-line integrals respectively. So two guided modes have zero coupling unless their wavevectors are the same.
A typical ThermoPhotoVoltaic (TPV) system comprises an emitting object or emitter, e, at a high temperature Te, and an absorbing PhotoVoltaic (PV)-cell absorber, a, which may typically be a semiconductor with an electronic bandgap of energy Eg=ℏωg, at a lower temperature Ta<Te, surrounded by a background, b, at the same absorber-temperature Ta. The background may be free space or it may include any set of other absorbing or lossless, finite or infinite-extent objects, at Ta. To quantify the statistics of thermal emission/absorption of each object, we need to examine each object separately, “set the temperature” of the other objects to zero and apply the superposition principle.
Based on the FDT in Eq. (1), the thermally-excited sources in the emitter generate photons with mean number Θ(ω,0,Te). Some of those photons that leave the emitter can be absorbed by the absorber, some by the background and some can be radiated into the background.
The PV-cell absorber, with a relative dielectric permittivity, εa has several mechanisms by which it absorbs photons. One absorption mechanism, g, associated with interband absorption through the semiconductor electronic bandgap, leads to a coherent conversion of an absorbed photon to a photo-generated electron-hole pair, which separates to a free electron and a free hole under a built-in chemical potential and leads to a voltage differential, V, across the PV cell, and a photo-generated current, I, when an external load l is connected to the PV-cell output. Other absorption mechanisms in the PV cell include free-carrier absorption, inter-valence-band absorption, inter-valley-conduction-band absorption and non-radiative recombination in semiconductors, and do not lead to photo-generated voltage and current. We characterize the various absorption mechanisms via the imaginary part of the relative dielectric permittivity of the absorber, the voltage/current-generating one as εg″ and the ones leading to lost absorbed photons collectively as εl″, such that εa″=εg″+εl″. Using Eq. (9), one can see that, for any lossy object or loss mechanism i,
∈ia(ω)=∈ig(ω)+∈il(ω) (57)
At the temperature Ta of the absorber, randomly fluctuating sources, associated with all its different absorption mechanisms, may lead to photon emission, but with different statistics. To quantify the statistics of thermal emission of each such mechanism, again we need to examine each mechanism separately, “set the temperature” of the other mechanisms to zero and apply the superposition principle. Based on the FDT in Eq. (1), the thermally-excited sources in the absorber associated with the absorption mechanism g generate photons with mean number Θ(ω, V, Ta), when the PV cell operates with an induced voltage V across it. Note that the induced voltage is smaller than the bandgap energy, qV<ωg, so that the mean number of photons Θ(ω, V, Ta) is positive for frequencies ω>ωg, at which the bandgap absorbs/emits and its emissivity is non-zero. Some of those photons emitted by the bandgap can be absorbed by the emitter, some by the background, some can be radiated into the background and some can be re-absorbed inside the absorber by a different absorption mechanism (free-carrier, inter-valence-band, inter-valley-conduction-band, etc.). Similarly, the thermally-excited sources in the absorber associated with the other absorption mechanisms generate photons with mean number Θ(ω,0,Ta). Some of those photons can be absorbed by the emitter, some by the background, some can be radiated into the background and some can be re-absorbed inside the absorber by the electronic bandgap.
Based on the FDT in Eq. (1), the thermally-excited sources in the absorbing objects of the background generate photons with mean number Θ(ω,0,Ta). If the TPV system extends to infinity, so that photons can be radiated away from the system, then, at thermal quasi-equilibrium at the background temperature Ta, photons are also radiated into the system from infinity, again with mean number Θ(ω,0,Ta)
The efficiency, η, of the TPV system is defined as the ratio of the power, Pl=V·I, fed into the load l connected to the PV-cell output over the net power outflow from the emitter, Pe, so that η=Pl/Pe.
The net power outflow from the emitter is the integral over all frequencies of the energy per photon ℏω times the net number of photons flowing out of the emitter. Using Eq. (8) and the superposition principle Eq. (43), the net number of photons flowing out of the emitter is the total number of photons emitted by the emitter towards the absorber and background, assuming Ta=0, minus the number of photons absorbed by the emitter, assuming Te=0, and emitted by absorption mechanism g of the absorber and all other absorption or radiation mechanisms in the absorber and background.
where, using also Eq. (57),
∈e(ω)=∈ge(ω)+∈le(ω)+∈be(ω)=∈ae(ω)+∈be(ω) (59)
and ∈ij(ω) is the thermal transmissivity of object or mechanism j into object or mechanism i. Using now Eq. (59) and reciprocity Eq. (10), we can rewrite Eq. (58) as
The current fed into a load connected to the PV cell output is the integral over all frequencies of the electron charge q times the net number of photons absorbed by the electronic bandgap of the PV cell and successfully converted to free electrons and holes flowing through the load. Using Eq. (19) and the superposition principle Eq. (43) (with a minus sign for absorption vs emission), the net number of photons absorbed by mechanism g is the number of photons emitted by the emitter and absorbed by g, assuming Ta=0, plus the number of photons emitted by all other absorption or radiation mechanisms in the absorber and background and absorbed by g, minus the total number of photons emitted by the mechanism g towards the emitter and all other absorption or radiation mechanisms in the absorber and background.
where
∈g(ω)=∈eg(ω)+∈lg(ω)+∈bg(ω) (62)
Using now Eq. (62) and reciprocity Eq. (10), we can rewrite Eq. (61) as
As can be seen from Eq. (9), if the absorptive object j is homogeneous, namely its dielectric permittivity does not depend on position r, then ∈ij(ω)∝∈j′(ω). If the absorber is homogeneous, it is then convenient to define the internal quantum efficiency, ηiq, of the absorbing material of the PV cell as
which may be approximately zero at frequencies below the bandgap ωg. Using Eq. (64), we have
∈ig(ω)=ηiq(ω)∈ia(ω) and ∈il(ω)=[1−ηiq(ω)]∈ia(ω) (65)
and therefore we can rewrite
with
∈a(ω)=∈ae(ω))+εal(ω)+∈ab(ω)=∈ae(ω)+[1−ηiq(ω)]∈aa(ω)+∈ab(ω) (68)
Eq. (60) and (63) can be simplified, if ℏωg−qV>kBTa, as one can then use the (equivalent to, but slightly better than, the Boltzmann) approximation
Both of these equations are written as a sum of a term, which is non-zero at short circuit (V=0), and a term, which is non-zero at thermal equilibrium (Te=Ta). Under the approximation of Eq. (69), they can be written as
Eqs. (70) and (71) are general and contain simple subcases:
(i) No emitter (∈en=0, ∈gl, ∈gb≠0): A simple diode. If losses into the background include radiation, it is a Light-Emitting Diode (LED).
(ii) No absorber (∈an=0, ∈eb≠0): A hot resonant emitter emits thermal photons into the cold background, where they are absorbed and/or radiated away.
(iii) No background losses (∈bn=∈ln=0, ∈eg≠0): Ideal TPV case.
In the case of a linear system, the Eqs. (60), (63), (66), (67) and (72)-(75) for the input power and output current of the TPV system should be reinterpreted as input power per unit length and output current per unit length, with the substitution of all transmissivity terms as Eqs. (8) and (22) suggest
In the case of a planar system, the Eqs. (60), (63), (66), (67) and (72)-(75) for the input power and output current of the TPV system need to be reinterpreted as input power per unit area and output current per unit area, with the substitution of all transmissivity terms as Eqs. (8) and (29) suggest
The operating voltage of a TPV system can be chosen via the choice of the load impedance, R=V/I, connected at the output of the TPV system.
In contrast to PhotoVoltaic (PV) systems, both the emitted power Pe and load power Pl=V·I(V) of a TPV system depend on the operating voltage, V, as can be seen from Eq. (60) and (63). Therefore, the output power Pl and system efficiency η=Pl/Pe are not maximized at the same voltage, with the efficiency maximized at a larger voltage than the power. In
The operating voltage, V, can also be chosen to achieve a desired output power level, Pl. In this case, since two values of the voltage can give the same output power, it may be preferable to choose the larger value, since that corresponds to higher efficiency. The efficiency advantage of the higher-voltage solution can be seen in
If the approximation of Eq. (69) applies, one can use the Eq. (70)-(75) to determine the desired (optimal-efficiency, optimal-power or target-power) voltage without requiring multiple frequency integrations.
In some embodiments of a TPV system, one can choose the operating voltage to increase or maximize the efficiency. In some embodiments, one can choose the operating voltage to increase or maximize the output power. In some embodiments, one can choose the operating voltage to achieve a compromise between efficiency and output power. In some embodiments, one can choose the operating voltage to achieve a target output power, wherein the chosen voltage can be larger than the voltage which maximizes output power.
It is instructive to know what the highest achievable limit is for the efficiency. Losses arise, first, from photons that are emitted by the emitter and absorbed by non-voltage-generating absorption mechanisms inside the absorber or in the background and, second, from photons that are emitted by the emitter with frequency above the absorber bandgap, as these photons generate one electron-hole pair and lose their excess (over the bandgap) energy by thermalization. The highest efficiency may be possible, first, when no non-voltage-generating loss mechanisms exist inside the absorber and in the background, namely only ∈eg(ω)≠0, and, second, when all emitter photons are emitted exactly at the bandgap frequency, namely ∈eg(ω)˜∈gωgδ(ω−ωg). Then, Eq. (60) and (63) become
Therefore the efficiency is
and is maximized, when the voltage is maximized. This occurs at open circuit (R=∞), which implies, from Eq. (79) and (80),
which is, as one would expect, the Carnot limit of efficiency of a Carnot engine between a hot object at Te and a cold object at Ta. However, in this maximum efficiency limit, the output power tends to zero, since the output current tends to zero.
In a practical non-ideal TPV system, there may be some photons absorbed by the TPV cell that are not converted into useful electron-hole pairs, such as below-bandgap photons. Since these photons correspond to some power lost, the optimal emission profile deviates from a sharp peak right above the bandgap frequency, since such a sharp peak implies very little output power generated, so efficiency would be low. A larger emission bandwidth above the bandgap is necessary, so that the output power can overcome the power lost, but not too large, to avoid the increasing thermalization losses.
Based on the previous sections, the efficiency of a planar layered TPV system should be high if the thermal transmissivity, ∈ea(ω,kxy), between the emitter and absorber layers is close to 1 (its maximum per polarization) for a small bandwidth of frequencies right above the bandgap of the semiconductor absorber and close to zero everywhere else. Put differently, if the emitter-absorber thermal transmissivity has a “top-hat” profile above the semiconductor bandgap, the TPV system should be very efficient. A common problem with most prior-art systems, especially non-resonant ones, in implementing such a transmission profile is the large broadening of the emitter electromagnetic modes at elevated temperatures. The result is typically unavoidable losses due to below-bandgap emitted photons absorbed by free carriers in the PV cell, either at its electrodes or its main body.
In this disclosure, we describe and show several versions of a TPV system whose thermal transmission profile increases efficiency by suppressing below-bandgap and high-frequency transmitted photons. The underlying principle is to make both the emitter and absorber support resonant electromagnetic modes that are substantially different (in wave vector kxy) at all frequencies except for a single frequency above the PV-cell bandgap, where they cross and couple appropriately to achieve impedance matching. The selective property of coupled impedance-matched resonances means that the thermal transmissivity is high at the resonant crossing frequency, and very low at other frequencies. Mathematically, we accomplish for the transmissivity, at the coupling wave vector kxy, the double-Lorentzian shape of Eq. (53), which can have very sharp transitions compared to the single Lorentzian of Eq. (48), even if one of the resonances is very broad (that of the emitter).
Physically, one can understand the advantage of two coupled resonances versus a single resonance in the following way. One typically aims to achieve impedance matching at a frequency above the semiconductor PV cell bandgap. For a single-Lorentzian TPV system without a resonant absorber, to satisfy the condition γe=γa, since the modal loss rate in the absorber scales quadratically with the emitter field γa˜|Ee|2, where the emitter modal field |Ee| decays exponentially inside the vacuum gap, one typically has to bring the absorber in very close proximity to the resonant mode of the emitter. At that close proximity the emitter photons of all (also below-bandgap) frequencies are seen by the absorber, thus the transmissivity profile is broad. For a double-Lorentzian system, the impedance matching condition between the two (emitter and absorber) resonances at the frequency of the resonant modal crossing is approximately γeγa=κ2. In this case, the coupling constant scales linearly with the fields κ˜|Ee|, so, given their exponential decay, the PV cell has to be kept at a large distance from the hot emitter. At that distance, the evanescent tails of the two photonic modes overlap sufficiently to couple at resonance, while the penetration of the emitter-mode tail inside the absorber (and vice versa) is very weak, so transmission is low away from resonance.
An additional advantage of the present systems of coupled resonances is that the vacuum gap between the emitter and absorber can be kept relatively large. One bottleneck for practically implementing near-field TPV systems has so far been mainly the extremely small (tens of nm) gaps that need to be accomplished and maintained. That distance can be much larger with the present designs.
One way to accomplish a resonant crossing between two electromagnetic modes is to use modes that, at the same frequency, have substantially different group velocities.
In some embodiments of the present TPV system, such as the system 400 shown in
Therefore, to get a mode crossing with clear separation between the emitter and absorber modes at frequencies other than the desired one, an emitter resonant mode with group velocity close to zero or even negative would be best. Photonic modes that exhibit such dispersion are surface polaritons on the interfaces of a polaritonic material with a dielectric. Such modes usually exist only for Transverse Magnetic (TM) polarization, where the magnetic field is parallel to the uniform symmetry plane and perpendicular to the direction of the mode propagation. Although any type of polaritonic material (e.g., exciton, phonon) can be used, we focus on plasmonic materials and Surface Plasmon Polariton (SPP) modes. On their interface with dielectric materials, such as air or vacuum, they are well known to support slow-light surface states, whose dispersion can thus be designed to cross the dispersion of a dielectric waveguide mode. Therefore, in some embodiments of the present TPV system, the thermal emitter 410 can be a slab of plasmonic material, as shown in
Note that a TPV system including a polaritonic emitter and a thin-film semiconductor absorber has been studied in the past (surface-phonon emitter and tungsten emitter). However, those systems were not designed so that the emitter polaritonic mode and the absorber dielectric-waveguide-type mode cross, couple and are impedance-matched just above the bandgap. This is why the reported efficiencies in those studies are significantly lower than those reported here. Furthermore, free-carrier absorption was not modeled properly in those studies.
The PV cell 430 in
In some embodiments, a front transparent electrode 432 can be formed by very highly doping a portion of the thin-film semiconductor, as shown in
In a typical thin-film PV cell, the semiconductor film thickness may be so small that the film should be doped at significantly high levels for both the depletion and electrode regions to fit within the thin film, as depicted in
Throughout this specification, we model free carriers, for example, in the plasmonic material or in the PV-cell electrodes. The Drude model is an approximate but realistic and convenient model, described by the relative dielectric permittivity
The free-carrier plasma frequency and loss factor can be calculated by
where N is the carrier density (from doping), m* the effective mass of the carriers (electrons or holes), μ the carrier mobility and p the dc resistivity. Note that the plasma frequency ωp is approximately the frequency at which Re{ε}=0.
At the interface of such a Drude-type material with a dielectric of relative permittivity ∈d, a SPP surface mode is supported with upper cutoff frequency (as k→∞) at
The optimal choice of semiconductor bandgap Eg=ℏωg relates to the operating temperature Te of the emitter. The short-circuit power emitted by the emitter is given by Eqs. (72) and (77). If we ignore the term Θ(ω,0,Ta) for high enough emitter temperature Te>Ta, normalize the integration frequency by u=ω/ωg and the integration wave vector by nxy=kxy/kg, then
Since, by design, the emissivity is high close to u=1 and as small as possible at other frequencies, we can approximate the slowly-varying term Θ(u,0,Te/(ω)≈Θ(1,0, Te/ωg), therefore the emitted power is maximized with respect to ωg along with ωg4Θ(1,0, Te/ωg), which happens when
E
g=ℏωg≈4kBTe (85)
As a guide, an emitter at 3000° K would require a bandgap of approximately 1 eV to maximize emitter power, while at 1200° K approximately 0.4 eV. In all subsequent example designs, we use the choice of Eq. (85). Note, however, that different bandgap frequencies can also be used with the current method to improve TPV efficiency, only the power level may be smaller than the maximum.
A consequence of the choice of Eq. (85) for the bandgap is that the power emitted by the emitter (and also that absorbed by the PV cell) scales as Pe,A˜Te4, similarly to the Stefan-Boltzmann law for far-field thermal radiation.
A very large subset of available semiconductors falls inside the above spectrum of bandgaps. The semiconductor material can be a group-IV semiconductor, such as germanium or even silicon, although they do not have a direct bandgap and silicon has just a bit higher bandgap than the 1 eV suggested above. The semiconductor material can be a III-V semiconductor, such as GaSb, InAs, InN, ternaries GaInSb, GaInAs, InAsSb, GaAsSb, and quaternaries GaInAsSb, GaInPAs, GaInPSb, GaInNAs. The semiconductor material can be a II-VI semiconductor ternary, such as HgCdTe, HgCdSe.
For the semiconductor absorber, we use a high-frequency relative dielectric constant ε∞,a=14, approximately appropriate for both direct- and indirect-bandgap materials stated above.
The interband absorption profile for a direct-bandgap bulk semiconductor scales as αg(ω)≈ωεg″(ω)/c√{square root over (εa″(ω))}˜√{square root over (ω−ωg)}, therefore εg″(ω)≈M√{square root over (ε∞,aωg(ω−ωg))}/ω, since εa′(ω)≈ε∞,a at ω>ωg. The dimensionless proportionality constant M depends on the conduction-band electron and valence-band hole effective masses, and we found it to be around 0.7 for many semiconductors used for TPV, such as GaInAsSb. In embodiments, quantum wells (step-function absorption profile), quantum wires (inverse-square-root absorption profile) or quantum dots (Lorentzian absorption profile) can also be used for the PV cell. The approximate functional form of the interband absorption αg(ω) may also be one of the above forms convoluted with a Lorentz functional to account for homogeneous and inhomogeneous broadening.
The free-carrier absorption of the semiconductor absorber is modeled via the Drude model in Eq. (82). Again, we initially assume uniform average dielectric properties for the semiconductor absorber, including the electrode region and entire pn junction. In this initial analysis, we stipulate that the doping concentration (and thus free-carrier density) ND increases as ωg increases (and thus film thickness da decreases) to maintain low electrode square resistance, so we use scaling ND˜ωg2ωp,a˜ωg. With guidance typical values (for GaInAsSb semiconductors) for the electron and hole effective masses me*≈0.035me and mh*≈0.4me, where me the electron mass, and typical carrier-density levels from previously reported TPV cells, we use √{square root over (ε∞,a)}ωp,a=0.4ωg. For example, this value corresponds to Ne≈7×1017 cm−3 electrons or Nh≈8×1018 cm−3 holes at Te=1200° K. Assuming furthermore, for simplicity, an electron mobility scaling μ˜1/√{square root over (ND)}, which is a fair approximation for large ranges of ND in semiconductors and matching typical mobility values, we use γa=0.05 √{square root over (ε∞,a)}ωp,a=0.02ωg. Therefore, in our first analysis, we model the absorber with dielectric permittivity
The internal quantum efficiency of the electrode region of the semiconductor is
Other loss mechanisms, which reduce the internal quantum efficiency, could be included in εa″, but they are ignored here, for simplicity and to determine optimal performance limits.
The back metallic electrode/reflector should have the lowest possible loss to reduce or minimize the loss of the absorber mode due to its penetration into the metal. Therefore, we can use silver (Ag) as the electrode material and the Drude approximation for silver with ε∞,Ag=4, √{square root over (ε∞,Ag)}ωp,Ag=9.3 eV and
γAg=0.023 eV. Other metals can also be used, such as gold, aluminum or copper, but efficiency may be lower.
The emitter plasmonic material should support a SPP mode with a cutoff frequency ωc,e slightly above the semiconductor bandgap ωg, say ωg<ωc,e≤1.6ωg (e.g., ωc,e≤1.2ωg). As discussed earlier, ωg typically lies in the range (0.2-1) eV/, so ωc,e should fall roughly in the range (0.24-1.2) eV/
. Since the SPP mode predominantly “sees” the vacuum gap εvac=1, its cutoff frequency occurs when Re{∈e(ωc,e)}≈−1 (Eq. (84)). Therefore, suitable materials may have high melting temperature, with ε′(ω)=−1 for ω roughly in (0.24-1.2) eV/h. This range is confirmed by our simulations, as shown later in
In Table 1, we show a list of such materials. In general, there are several refractory metals, metal carbides (e.g., ZrC, VC and TiC), nitrides and silicides that can be used in embodiments of the present technology as the emitter material. Also the most common TPV emitter material, tungsten W, can be used for very high emitter temperatures, perhaps even matched with a silicon thin-film PV cell. Thick enough films of these refractory materials can be grown epitaxially on a substrate to ensure minimum defects and surface variations on the films.
The thermal emitter material may have a high melting temperature and a relative dielectric permittivity with a real part that is −1 at infrared frequencies. It can be but does not have to be a plasmonic material. Phononic materials have negative dielectric permittivities within a finite bandwidth rather than all the way down to zero frequency (like plasmonic materials). Therefore, in example embodiments, a phononic material (e.g., SiC and BN) can be used for the thermal emitter.
A metamaterial is a periodic arrangement (e.g., a photonic crystal) of at least two different materials with a period substantially smaller than the wavelength of operation. Such a metamaterial exhibits effectively a uniform (but generally anisotropic) dielectric permittivity and magnetic permeability to electromagnetic waves at that wavelength of operation. If at least one of the constituent materials in a metamaterial is a polaritonic material, then the metamaterial can have an effective permittivity with a negative real part. Therefore, in example embodiments, a metamaterial can be used for the thermal emitter, where at least one of the constituent materials is a polaritonic material, such as a plasmon polaritonic (plasmonic) or phonon polaritonic (phononic) material. A metamaterial comprising a polaritonic material and a dielectric material can have an effective plasma frequency that is smaller than the plasma frequency of the constituent polaritonic material, and thus it can have an effective εeff′=−1 at a lower frequency than the frequency at which the constituent polaritonic material has ε′=−1. Therefore, in some example embodiments, metamaterials comprising a refractory polaritonic material of high plasma frequency (e.g., W, Ta, Hf, Re, Os, Ir, Mo, Nb, Ru) or alloys thereof (e.g., Tantalum Hafnium Carbide) and a refractory dielectric (e.g., thoria ThO2, hafnia HfO2 and zirconia ZrO2) can be used for the thermal emitter.
The plasmonic emitter is also modeled via the Drude permittivity. In our design optimizations that follow, we have left ωp,e as an optimization variable and we have assumed ε∞,e=1 for simplicity. In order to quantify the losses of the plasmonic emitter in the Drude model, we have to take into account how the losses increase with temperature. F. A. Modine, M. D. Foegelle, C. B. Finch and C. Y. Allison, “Electrical properties of transition-metal carbides of group IV,” Physical Review B, vol. 40, no. 14, pp. 9558-9564, 15 Nov. 1989, describes a model for the temperature dependence of resistivity, which should also hold for the Drude loss factor γ in Eq. (83), since N and m* do not significantly depend on temperature for metallic-type materials. We use the simplified model
From parameters and measurements for ZrC and TiC, we deduce and use approximate values γo=0.05ωp,e, α=0.002/° K and γ∞=2γo. Note that, compared to silver (Ag) with γ≈0.0025√{square root over (ε∞)}ωp, these are much lossier metallic-type materials and they become even lossier (almost twice) at very high temperatures. The associated broadening has been a problem for prior TPV systems with metallic emitters, however, here we show how a coupled-resonant system circumvents this problem.
In order to evaluate the performance of the present mechanism for enhanced TPV efficiency, we perform an optimization procedure to calculate the maximum efficiency that can be accomplished. The optimization results throughout this specification are not guaranteed to be global optima, but we believe they are very close. For our first optimization, we consider the emitter at temperature Te=1200° K and the absorber at room temperature Ta=300° K. The PV cell bandgap is selected from Eq. (85) as Eg≈0.414 eV. For the structure of
In
The resonant crossing (“X-shape”) of the emitter SPP mode and the absorber dielectric-waveguide-type mode is clearly visible in
Because of the presence of the metallic back electrode, the emitter SPP mode and the absorber first dielectric-waveguide-type mode couple in a way that, of the two resulting system “super-modes”, the higher-frequency one has a (kxy=0) cutoff and then (for large kxy) gets close to the light-line of the absorber material, which is shown in
One exception, an absorber mode that is not removed by the metal back electrode at low (below bandgap) frequencies, is a SPP mode on the interface between the vacuum gap and the doped semiconductor absorber, due to the free carriers of the latter inducing plasmonic-material behavior with √{square root over (ε∞,a)}ωp,a=0.4ωg, as seen in Eq. (86). Using Eq. (84), the upper-cutoff frequency for this SPP mode is ωp,a/√{square root over (1+1/ε∞,a)}≈0.1ωg, at which below-bandgap frequency this absorber free-carrier SPP also couples to the emitter SPP and there is an associated undesired emissivity peak. This front-electrode loss mechanism can have a large impact on efficiency in different prior-art topologies.
The second TM dielectric-waveguide-type mode of the thin-film absorber is also evident at higher frequencies and also the high emitter emissivity associated with this mode's exponential tails reaching the emitter. This emissivity is also undesired, as it can be associated with large thermalization losses.
In
The optimization procedure is repeated for emitter temperatures Te from 600° K to 3000° K. At each emitter temperature, the PV-cell semiconductor bandgap is selected from Eq. (85). The resulting optimized efficiency as a function of Te is shown in
The power density at the output load of the PV cell as a function of temperature, for this optimal design, is also shown in
We show the optimal values of the parameters ωp,e in
Performance examples of our structures include: at Te=1200° K, output 2 W/cm2 with ˜47% efficiency (e.g., using ZrC emitter and 160 nm-thick InAs absorber at 300 nm vacuum gap), at Te=2100° K, 24 W/cm2 with ˜57% efficiency (e.g., using TiC emitter and 80 nm-thick GaSb absorber at 200 nm gap), and, at Te=3000° K, 115 W/cm2 with ˜61% efficiency (e.g., using W emitter and 50 nm-thick Ga0.3In0.7P0.6As0.4 absorber, grown on InP, at 140 nm gap).
The performance of any electronic power converter, including a TPV, is often judged by the efficiency achievable as a function of the power delivered to the load. Therefore, in
The optimization results are shown in
The results can be understood as follows: As the desired output power level increases,
From that point on, the absorber should be so close to the emitter that the emitter SPP has substantial energy inside the absorber. Thus, it is beneficial to redesign this single mode to get an efficient impedance-matched single-Lorentzian emissivity profile (Eq. (49)). So one optimal solution for ωp,e/ωg and da/λg has an abrupt change from P2 to P3 (
It is instructive to compare how this prior-art system performs compared to an example of the current system. Consider the structure 900 in
We take the doping level of the “base” region 938 in
We keep the silver back electrode as it helps prevent radiation of photons within the semiconductor light-line. We optimize again the efficiency vs load power, at Te=1200° K with fixed da=0.07λg and at Te=3000° K with fixed da=0.061λg, to compare with the thin-film optimized structures at high power. The optimization parameters are ωp,e/ωg, dvac/λg, qV/Eg.
The results are shown in
In prior art with thick absorbers in both far-field and near-field TPV systems, a metallic back reflector has been shown to slightly improve efficiency by reflecting back into the emitter and recycling the undesired below-bandgap photons. However, this improvement was not significant and we see here that true removal of the radiation modes can only happen with a thin-film PV cell. Notice also how, at the same power level P1, the coupled-resonances structure of
Note that, for both the current thin-film-absorber and the prior-art bulk-absorber TPV systems, the thermalization losses (
Another prior-art system, which does use the concept of resonance, is that of two coupled semiconductor thin-films, shown in
We perform the optimization at Te=1200° K, which is higher than most semiconductor melting temperatures (exceptions Ge: ˜1211° K and Si: ˜1638° K), but this temperature helps us compare to the structures already discussed. For the PV cell, we assume the same semiconductor permittivity Ea from Eq. (86) and a back silver electrode. For the emitter, we assume an undoped semiconductor with matched bandgap ωg and permittivity εe of the form of Eq. (86). However, at that close-to-melting temperature the thermally-excited intrinsic carriers may be extremely high and the mobility of those carriers greatly reduced. It is not easy to exactly predict those quantities, but based on estimations (using ni˜T3/2e−E
Finally, the emitter thin film is backed by tungsten W. By fitting the Drude model to room-temperature experimental data for tungsten in the range (0.1-1) eV, we get ε∞,W=36, √{square root over (ε∞,W)}ωp,W=5.88 eV and ℏγW(300° K)=0.058 eV; at 1200° K, we increase the loss rate in proportion to the increase of the dc resistivity, so
γW(1200° K)=0.33 eV. Since the real parts of εa, εe are quite similar and Ag, W performs similarly for modal confinement, we take, for simplicity, equal thickness da of the two thin films, which is such that the first dielectric-waveguide-type modes of the emitter and absorber couple just above bandgap. Our optimization parameters are thus dvac/λg, da/λg and qV/Eg.
The results are shown in
It is certainly difficult to predict accurately the performance of this system, as accurate modeling of the semiconductors (especially at very high temperature) is non-trivial, but qualitatively one can expect a lower load-power limit for these systems.
It can be beneficial to design a TPV system where the emitter and absorber support separate resonances, which couple in the near field and are impedance matched at a crossing frequency just above the semiconductor bandgap frequency. However, it was seen that, for one embodiment of a SPP emitter coupled to a thin-film absorber with a silver back electrode/reflector, there was an upper load-power limit, above which it is impossible to maintain resonant coupling, since the dielectric-waveguide-type absorber mode has a kxy-limit, the light line of the absorber material. There is another way, however, to achieve resonant coupling between emitter and absorber at any power, if also the absorber supports a SPP mode. Note that this typically requires the semiconductor film to be even thinner.
One such resonant absorber-SPP implementation is to make the back electrode of a plasmonic material (ε∞,m,ωp,m,γm) with a small enough ωp,m that the SPP supported at its interface with the semiconductor has a cutoff (≈ωp,m/√{square root over (1+εa/ε∞,m)}) just above the semiconductor bandgap ωg. There are quite a few metals or metallic-type materials with relatively small ωp,m (for example, some of those presented in Table 1 for the emitter) that can be used as a plasmonic back electrode. Another choice is a very highly-doped semiconductor or a highly-doped conducting oxide (a few of which we see below) whose ωp,m can be tuned via the doping concentration. However, we saw (in
Therefore, we disclose a way to get an effective plasmonic material with parameters in the desired frequency range using silver. It has been shown that metallo-dielectric photonic crystals, which are metals with periodic geometry patterning, can exhibit plasmonic-type effective behavior with a plasma frequency that is lower than that of the metal used to make the photonic crystal and tunable via the fill factor of the patterning. In some sense, one gets a material that is a (fill-factor weighted) average of the metal and dielectric material (often air/vacuum). The period of the geometric patterning is a few times smaller than the SPP propagation wavelength 2π/kxy for the metamaterial effective medium approximation to hold. These effective SPP modes are called “spoof” plasmons for the subcase of a metal with a 2D periodic patterning of holes on its flat interface.
In performing optimization of efficiency vs load power for this system, the properties of the plasmonic emitter and semiconductor absorber are the same as in the case of γAg=0.023 eV, independent of the patterning. In reality, the silver losses may additionally depend on its grain size, with a dependence of the form
γAg=0.023 eV+const/dpattern, where dpattern is a characteristic size of the patterning.
The results for the best-case scenario are shown in
Therefore, in some embodiments of the near-field TPV systems, patterning the silver back electrode may increase TPV efficiency at high emitter temperatures and load power levels unless the patterning-dependent silver losses are very large. In this case, the back electrode losses may increase so much that it may not be beneficial anymore to use a patterned back electrode instead of plain silver and the implementation of a single emitter-SPP resonance extending into the absorber.
Another way to implement a resonant absorber SPP with strictly (instead of effectively) a planar structure using silver is to use, as back electrode, an ultra-thin silver film on a dielectric substrate. The film supports an SPP on each of its two interfaces (with the semiconductor film and the dielectric substrate). These SPPs couple more strongly as the silver thickness gets smaller and one of the two resulting coupled-SPP modes is shifted to lower frequencies. For some ultralow silver-film thickness, this coupled-SPP mode can be shifted from ≈ωp,Ag/√{square root over (1+εa/ε∞,Ag)}≈2.2 eV (for εa≈14) down to 1.2 eV and thus become resonant with the emitter SPP, for an emitter at 3000° K.
Our simulations have shown that such an implementation can be efficient, although for the simulation parameters chosen it performed worse than simply a bulk semiconductor absorber. This is because such an absorber-SPP implementation relies heavily on the ultrathin silver film for confinement and is therefore very lossy. In addition, the square resistance of such an ultrathin back electrode may not be adequate for the electrical operation of the device. However, this implementation is practically much simpler and may provide improved performance in certain operational regimes (e.g., if a very low front-electrode resistance is required).
As discussed earlier, the PV cell has a front electrode that is conductive and transmits hot photons to from the emitter to the thin film of semiconductor material. One way to accomplish this is via a medium-conductive thin film electrode, which allows photon passage, assisted by a (parallel or rectangular) nanowire mesh of a highly conducting material (e.g., silver), which would itself be opaque to photons. Such embodiments are shown in
The current due to photo-generated carriers, upon exiting the depletion region and entering the thin-film electrode, “turns” to run parallel to this thin film in order to make it to the highly conducting nanowires. How high the DC conductivity σDC,el=σel (ω=0) and how large the thickness del of the thin-film electrode should be depends on its necessary square resistance Rel=1/σDC,eldel. The efficiency drop due to the finite Rel can be estimated by Δηel=IARelw2/3V=ηPe,ARelw2/3V2. If we assume the previously explained scaling Eg≈4kBTe and qV≈Eg(1−Ta/Te)˜Te, for Te>>Ta, and Pe,A˜Te4, one can estimate that the efficiency drop Δηel may be approximately independent of emitter temperature, if w˜λg˜1/Te and Rel is a constant. Therefore, in our first subsequent simulations, we impose the condition that Rel=60Ω, a reasonable value for the square resistance also in solar PV-cell electrodes.
The conductivity can be calculated using the Drude model for free-carriers: we equate the permittivity of Eq. (82) to ε∞+iσ(ω)/ωεo to find
where ζo=√{square root over (μo/εo )} the impedance of free space. For each candidate electrode material the mobility μel of free carriers depends on their density N; since γ(N)=q/μ(N)m* and ωp=q√{square root over (N/εoε∞m*)} from Eq. (83), each electrode material can be described either by μel(N) or, equivalently, by a function γel(ωp,el), where N is an underlying tunable-via-doping parameter. Therefore, a given desired electrode square resistance imposes the condition
Typically, the higher the carrier density N (and thus ωp,el), the more conductive the thin-film electrode material is, and thus the thinner it should be.
However, a material with high DC conductivity σDC typically also has high AC conductivity Re{σ(ω)}, thus absorbing more impingent photons instead of transmitting them. If we consider the ratio Re{σ(ω)}/σDC, then, using Eq. (90),
Therefore, one may be tempted to conclude that we are looking for the smallest possible loss rate γel. Certainly, given a free-carrier plasma frequency ωp,el, the material with a small or minimum Yet (namely least possible carrier scattering rates and thus highest mobility) may be the best. However, given a certain material, it is not straightforward that one should choose the doping level (ωp,el) that minimizes γel, as Eq. (91) poses an additional restriction. Therefore, for a given electrode material, it is not clear what is the optimal doping level (N) under condition Eq. (91).
To examine this problem in more detail, we study a TPV system with a SPP emitter and the PV cell including a thin-film semiconductor absorber, a silver back electrode, and a thin-film front electrode described by the Drude model (γel˜ωp,el, and specifically γel=0.06ωp,el with ε∞,el=4, which resembles some real materials, as shown below. With Rel=60Ω, Eq. (91) is rewritten as del=0.015λp,el, where λp,el=2πc/ωp,el. Since a separate thin film takes care of the front-electrode functionality, this time we assume that the semiconductor's entire thickness is the depletion region with permittivity:
εdep(ω)=14+i0.7√{square root over (14ωg(ω−ωg)/ω)} (93)
and that the internal quantum efficiency in the depletion region is ηiq,dep(ω)=H(ω−ωg), where H the Heaviside step function, ignoring the very small free-carrier absorption and radiative recombination in it and all other recombination mechanisms. We then optimize the TPV efficiency at Te=3000° KEg≈1.03 eV with respect to ωp,e/ωg, dvac/λg, ddep/λg and qV/Eg, for different values of ωp,el.
The results are shown in
The corresponding TM emitter emissivity spectrum (
For all values of ωp,el, the optimized structures have electrode and depletion-region widths such that the first (kxy=0) cutoff of the system photonic modes lies at a frequency slightly above the bandgap (
Note also that, in some embodiments, if the electrode material is such that the carrier scattering rates are extremely small (for example, the electrode is a high-Tc superconductor), the triple resonance may not have such a negative effect on efficiency. In fact, in such embodiments, it can be beneficial to use the triple resonance, as the latter allows efficient transmission of photons also with an even larger vacuum gap, as seen in
In conclusion, a low-square-resistance front electrode can be achieved by doping its material at an appropriate level, so that the passage of hot photons happens either via propagation through a transparent (Re{εel}>0) layer or evanescent tunneling through an ultra-thin opaque (Re{∈el}<0) layer. In both cases, the free-carrier absorption of the electrode should be reduced or minimized, so materials with very low carrier scattering rates should be chosen.
We now calculate the optimal TPV efficiency for front electrodes made with real materials whose conductivity can be tuned via doping and with the same imposed square resistance Rel.
In some embodiments, part of the front electrode can be shaped by a thin film of a transparent conducting oxide (TCO), as shown in
Another oxide, Cadmium Oxide doped with Dysprosium (CDO), was recently discovered to have even higher mobility and can also be useful for infrared applications. We extracted and interpolated this mobility data μCDO(N) and, using ε∞,CDO=5.5, me,CDO=0.21me, we plot γCDO (ωp,CDO) in
In some embodiments, part of the front electrode can be shaped by highly doping a region of the same semiconductor thin film that also performs the photo-current generation, as discussed in the previous section and shown in
In some embodiments, part of the front electrode can be shaped by layers of graphene, as shown in
where Ta=300° K the absorber (PV cell) temperature, EF=vgr√{square root over (πN2D)} the Fermi level due to carriers of density N2D and with vgr=106 m/sec, and G(u)=sin h(
u/kBTa)/[cos h(EF/kBTa)+cos h(
u/kBTa)]. The loss rate γgr includes two terms due to scattering of free carriers with acoustic and optical phonons:
where ωgrOP=0.2 eV/ the optical phonon frequency in graphene. The second term is a very rough (and likely pessimistic) approximation of the optical-phonon-related scattering rate γgrOP and is dependent on the frequency ω of the photon. For the first term, it is believed that, in graphene monolayers at room temperature, carrier mobility due to acoustic-phonon scattering scales as μgr˜1/N2D˜1/EF2, so we use μgr(N2D)=3.47×1016/N2D/Vsec appropriate for the very high doping levels (N2D>1013 cm−2 ⇒EF>0.37 eV) used here.
With this scaling, the acoustic-phonon-scattering rate γgrAP becomes proportional to the Fermi level EF, and γgrAP(EF) is plotted in
For the assumed mobility scaling, σDC,gr2D is constant, so a square resistance Rel=60Ω gives Mgr=3 non-interacting graphene monolayers. For different values of Rel, we here allow Mgr to simply be a continuous number for continuity of the results and to fairly compare with other electrode materials.
We assume that the depletion region of the semiconductor extends throughout the entire thin region between the front and back electrodes, with permittivity εdep(ω) of Eq. (93) and thickness ddep, as shown in
We saw in
We perform a first optimization as a function of emitter temperature Te, for a fixed desired square resistance Rel=601. The optimization results are shown in
As expected, the optimal ωp,e (
In
Orange lines are for an ultra-thin silver film used as an opaque front electrode. Because of the large number of carriers (ωp,Ag) and small loss rate (γAg) of silver, the required thickness is extremely small (
We also perform another optimization to find optimal performance as a function of desired front-electrode square resistance Rel, for a fixed emitter temperature Te=3000° K. The results are shown in
In some embodiments, a hybrid design can have a combination of a TCO, graphene layers, an ultra-thin opaque plasmonic layer and some doping on the semiconductor to implement the front PV-cell electrode.
On the side of the thin-film PV cell opposite of the emitter and vacuum gap, a back electrode/reflector confines the resonant waveguide-mode. As we have used so far, in some embodiments of the current TPV systems, the back electrode/reflector can be achieved via a low-loss metallic material (typically silver) up against the semiconductor thin film. This can remove modes from the radiation cone and can position a system mode just above the semiconductor bandgap for increased power output and efficiency. However, we saw that, at the optimal efficiency operating point, most of the loss goes into this back electrode. Therefore, let us examine some alternatives.
In some embodiments, the back metallic electrode/reflector may be spaced apart from the thin-film semiconductor absorber via a thin spacer layer of a lower-refractive-index material. In some embodiments, this spacer layer can be formed of a lossless dielectric material and ultra-thin so that photo-generated carriers can tunnel to the back metallic electrode. In this case, there may still be significant penetration of the mode into the back metallic electrode, without substantial improvement in efficiency. In some embodiments, this spacer layer can be a bit thicker so that it can provide the dielectric-waveguide-type mode confinement. In this way, the absorber mode decays sufficiently by the point it reaches the back metal electrode. In this case, however, this material should be conductive for the electrons and holes photo-generated in the semiconductor to reach the metallic back electrode. Therefore, this thin film may be a TCO, such as those we examined in the previous section (IMO, CDO, etc.). It may be preferable for this film to be thin enough, so that its resistive loss is limited and the metal electrode still sufficiently removes modes from the semiconductor radiation cone.
In some embodiments, part of the back electrode can be shaped by depositing on the semiconductor absorber layers of graphene, such as non-interacting monolayers or coupled multilayers. In some embodiments, the graphene back electrode can be assisted by a (parallel or rectangular) nanowire mesh of a highly conducting material (e.g., silver). These nanowires may be thin so that they absorb few hot photons. In some embodiments, the graphene/nanowires back electrode may be spaced apart from a metallic back reflector by a thin lossless dielectric film, as shown in
In some embodiments, the PV-cell photo-diode, which converts absorbed hot photons to electron-hole pairs and then load voltage/current, can be a semiconductor p-n junction. In some embodiments, the absorbing region can include bulk semiconductor or quantum wells (QW) or quantum wires (QWR) or quantum dots (QD). Our simulations have shown that using quantum wells for our photonic topologies may not contribute a significant improvement in efficiency when the losses are dominated by the silver back-electrode absorption. However, a quantum-dot absorber may be beneficial in that it creates a triple resonance system: emitter SPP mode coupled to absorber photonic or SPP mode and coupled to electronic QD states.
In some embodiments, the PV-cell photo-diode, which converts absorbed hot photons to electron-hole pairs and then load voltage/current, can be a vertical graphene p-n junction, instead of a semiconductor p-n junction. In this case, the interband absorption of graphene due to the tunable-via-doping Fermi level EF can be used to create a photo-diode with a tunable-via-doping bandgap.
In some embodiments, a very thin dielectric film can be deposited on the interface between the plasmonic material and the vacuum gap. For example,
In some embodiments, the emitter plasmonic material itself can be a thin film, e.g., deposited on another support material of very high melting temperature. We found that, if the supporting material was a PEC, such a thin film plasmonic material would have superior TPV efficiency than a thick (bulk) plasmonic material. Unfortunately, real metals may not have low enough losses, especially at really high temperatures, for this implementation to be beneficial.
In some embodiments, the emitter can be a multilayered structure that includes many thin layers of dielectric and plasmonic materials. This multilayered structure can support at least one SPP resonant mode that couples to a resonant mode of the absorber. In some embodiments, the two coupled modes are impedance-matched at a frequency just above the absorber bandgap. In some embodiments, this multilayered structure can be designed such that its coupled SPP mode has a prescribed dispersion.
In planar systems, including those with metamaterials, a guided mode can be characterized by a wavevector k or kxy in the plane of uniform symmetry of the planar system. The upper cutoff frequency for the relevant surface polaritonic mode in a planar system is the highest value reached by the real part of the surface polaritonic mode eigenfrequency as k varies from 0 to ∞ (or π over the subwavelength period of a metamaterial) in the absence of the absorber.
So far, we have examined only (at least effectively) planar TPV systems. We saw that, at a given emitter temperature, efficiency can be optimized by coupling and impedance matching, just above the semiconductor bandgap, a planar emitter SPP mode with a planar absorber photonic mode. The emitter emissivity spectrum of
An example method includes patterning at least one of the emitter and the absorber in the xy plane so as to create in-plane localized resonances with lateral extent on the order of ˜(π/ko)2 in a way that only modes with kxy≈mko (m integer) survive in the emitter emissivity spectrum.
In some embodiments, the emitter plasmonic material can be patterned to create an array of pillars of size ˜(π/ko)2. In some embodiments, the space between the pillars can be filled with a high-melting-temperature dielectric material to provide structural support of the patterning at high temperatures. In some embodiments, the pillars can be spaced sufficiently, so that their in-plane localized resonant modes do not couple to each other significantly. The patterning is deep enough that modes generated at the bottom of the grooves do not couple efficiently to the distant absorber, rather simply circulate thermal energy inside the emitter.
In some embodiments, the absorber semiconductor material can be patterned to create an array of pillars of size ˜(π/ko)2. In some embodiments, the space between the pillars can be partially filled with a low-loss dielectric material to provide structural support for a front-electrode material on top of the low-loss dielectric material. In some embodiments, a uniform thin film of a transparent conducting oxide or graphene layers can be deposited on top of the semiconductor pillars to provide a front electrode, as shown in
In some embodiments, the absorber is patterned and the emitter is planar, so the emitter geometry can be stable at very high temperatures.
To estimate the performance of such systems, we start with a planar system and assume that, after patterning holes in it for in-plane confinement, the patterned system emissivities and transmissivities are the same as for the original planar system, but only the discrete set of wave vectors kxy mko survives. This would be exact if the lateral modal confinement was achieved with perfect boundaries (PEC or PMC). In the case of (potentially filled) holes, there may be some field penetration in the holes, but we assume that these holes are deep enough that only the system resonances at kxy mko contribute significantly to the transmissivity spectra. In fact, we let ko be an additional optimization parameter whose optimal value is expected close to the peak of the emitter emissivity (impedance-matching point). To further simplify the calculations, since the emitter emissivity at 2ko is negligible for the planar systems disclosed above, we approximately describe the class of patterned systems by analyzing the underlying planar systems only at the optimizable kxy≈ko.
The optimization results as a function of emitter temperature are shown in
The thickness of the semiconductor material and the width of the vacuum gap are measured along a direction perpendicular to the interface of the PV cell with the vacuum gap. For example, in
A surface polaritonic mode is a mode supported on an interface of a polaritonic material, in the sense that the mode field is evanescent in both directions away from the interface of the polaritonic material. In non-planar systems, a mode is to be understood as a resonant mode that may not be able to be characterized by a wavevector. Therefore, while in planar systems there may often be only a single surface polaritonic mode, guided on a planar interface of a polaritonic material, in non-planar systems there may be multiple resonant surface polaritonic modes, which may be localized at regions of a potentially non-planar interface of a polaritonic material and may not be able to be characterized by a wavevector. In this case, the upper cutoff frequency of at least one surface polaritonic mode is the highest value of the real parts of the eigenfrequencies of all related (at least one) surface polaritonic modes, where two surface polaritonic modes are related, when they are supported on roughly the same region of an interface of a polaritonic material and within the same continuous frequency range of negative real part of the permittivity of the polaritonic material. For example, if the permittivity of a polaritonic material has several frequency ranges of alternating negative and positive real part, the polaritonic material may support on an interface several surface polaritonic modes with eigenfrequencies in two different frequency ranges of negative real part of the permittivity; such two modes will not be considered related.
As already discussed, a ThermoPhotoVoltaic (TPV) system is a power converter, converting thermal power to electricity. Such a converter can be used in several applications, with different modes of operation.
In some embodiments, the TPV converter can be powering a varying load, such as an electronic device, a battery charger, an electrical machine, an automobile, etc. (
In some embodiments, the TPV converter can receive heat in a controllable way from some energy storage system. In some embodiments, the energy storage system can be fuel, such as gasoline or propane, which is burned to provide heat. As the input amount of heat is controlled, the emitter temperature can be controlled. In some embodiments, the amount of heat (and thus the emitter temperature) can be controlled by adjusting the flow of fuel into the burner. In some embodiments, the amount of heat (and thus the emitter temperature) can be controlled by adjusting the flow of oxygen or air into the burner. In some embodiments, the TPV converter can include a control feedback system, which can control the input amount of heat (and thus the emitter temperature) as the load power varies, in order to optimize efficiency (
In
In some embodiments, the TPV converter can receive heat from a varying but uncontrollable source. In some embodiments, the varying and uncontrollable heat source can be absorbed solar illumination (in which case, the TPV converter is called a solar TPV converter) (
In some embodiments, the TPV converter can be powering a controllable load, such as a battery charger. In some embodiments, the TPV converter can control the amount of power fed into the load, such as controlling the charge current of a battery charger. In some embodiments, the TPV converter can include a control feedback system, which can control the load power as the source power (and thus the emitter temperature) varies, in order to optimize efficiency (
In some embodiments, the TPV converter includes a component to tune the width of the vacuum gap between the emitter and the absorber. In some embodiments, the tuning component includes at least one MicroElectroMechanical Systems (MEMS) actuator device. Since the thin-film PV cell is ultra-thin, it is also light enough that it can be placed on top of at least one MEMS actuator, which can push the thin-film PV cell by nanometer-sized controllable amounts and thereby effectively tune the width of the vacuum gap. Since the PV-cell is also flexible, at least one MEMS actuator can be used to bend the PV cell surface closer to the emitter by nanometer-sized controllable amounts and thereby effectively tune the width of the vacuum gap. As discussed below, the TPV converter can include a control feedback system, which can control the width of the vacuum gap, as either the desired load power or the applied source power varies, in order to increase or optimize efficiency.
In some embodiments, the TPV converter can control both the impedance of the output load and the input amount of heat. For example, the impedance of the output load can be controlled by periodically opening an electronic switch (e.g., a FET) placed in series with the load or by periodically shorting an electronic switch (e.g., a FET) placed in parallel with the load. In some embodiments, the control feedback system can monitor the load voltage and/or current and control the impedance of the load to regulate the load voltage and/or current. In some embodiments, the control feedback system to regulate the load voltage and/or current can include a hysteresis mechanism to set maximum and minimum levels for the load voltage and/or current under regulation. In some embodiments, the control feedback system can simultaneously monitor the input power and control the input power and/or tune the width of the vacuum gap to increase the efficiency. The control loop for regulating the output voltage and/or current may be faster than the control loop for increasing the efficiency.
In some embodiments, the planar plasmonic emitter and thin-film PV cell structures described here can be fabricated separately and then placed at the desired distance within a vacuum. The emitter and PV cell can be fabricated with standard deposition and epitaxy techniques, such as physical vapor deposition (PVD) (e.g., evaporation or sputtering), chemical vapor deposition (CVD), atomic layer deposition (ALD), vapor phase epitaxy (VPE), liquid phase epitaxy (LPE), and molecular beam epitaxy (MBE). The thin-film PV cell can be fabricated on a reusable substrate and separated from it with an epitaxial lift-off process.
The heat-generating unit can be a micro-reactor, using carbohydrate or nuclear fuels, or a solar absorber, built out of a refractory material that can withstand the desired operating emitter temperature. It can be suspended from the walls of the vacuum chamber via long and low-thermal-conductivity (e.g., glass or other oxide) veins or capillary tubes that can also serve as fluidic connections for carbohydrate fuels and air/oxygen. In some embodiments, the emitter can be attached to the heat-generating unit. The attachment process can be welding or brazing or diffusion brazing. In some embodiments, the emitter can be built right on top of the heat-generating unit. In some embodiments, the heat-generating unit can itself be built out of the emitter refractory material, so that it can also function as the emitter.
Achieving and maintaining under operation a sub-Mm vacuum gap can be challenging. In some embodiments, isolated posts or pillars can separate the emitter and PV cell by the fixed desired gap. These posts should be narrow and sparse to provide large thermal resistance and thus to reduce or minimize thermal conduction between the emitter and the cell. In some embodiments, the posts can be narrower than the vacuum gap. In some embodiments, deep holes can be etched into the emitter. The posts can penetrate into these holes without touching the hole walls. In these cases, the length of the posts is approximately equal to the depth of the holes plus the desired vacuum gap.
In some embodiments, a temporary thin film of thickness equal to the desired vacuum gap can be deposited on either the emitter or the PV cell. The emitter and PV cell can be bonded together so that the heat-generating unit, the emitter, the temporary film, and the PV cell form a single structure (
In maintaining the desired vacuum gap during operation within a specific tolerance, thermal expansion of the hot emitter material may have to be taken into account. The expected expansion at the nominal operating emitter temperature can first be calculated or experimentally determined, so the vacuum gap can be fabricated to be larger by the predetermined amount than the desired gap under operation.
In
The same fabrication method can be used to fabricate solar TPV cells. In this case, the heat-generating unit is a solar absorber.
The term “vacuum gap” does not necessarily mean that absolute vacuum (zero air pressure) occupies the gap between the thermal emitter and the PV cell. Rather, “vacuum gap” means that there is sufficient vacuum (sufficiently low air pressure) in a volume between the thermal emitter and the PV cell such that the thermal power conducted from the emitter to the PV cell via this volume is much lower than the thermal power radiated to the PV cell via the photons emitted by the emitter and absorbed inside the PV cell.
While various inventive embodiments have been described and illustrated herein, those of ordinary skill in the art will readily envision a variety of other means and/or structures for performing the function and/or obtaining the results and/or one or more of the advantages described herein, and each of such variations and/or modifications is deemed to be within the scope of the inventive embodiments described herein. More generally, those skilled in the art will readily appreciate that all parameters, dimensions, materials, and configurations described herein are meant to be exemplary and that the actual parameters, dimensions, materials, and/or configurations will depend upon the specific application or applications for which the inventive teachings is/are used.
Those skilled in the art will recognize, or be able to ascertain using no more than routine experimentation, many equivalents to the specific inventive embodiments described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described and claimed. Inventive embodiments of the present disclosure are directed to each individual feature, system, article, material, kit, and/or method described herein. In addition, any combination of two or more such features, systems, articles, materials, kits, and/or methods, if such features, systems, articles, materials, kits, and/or methods are not mutually inconsistent, is included within the inventive scope of the present disclosure.
The above-described embodiments can be implemented in any of numerous ways. For example, embodiments of designing and making the technology disclosed herein may be implemented using hardware, software or a combination thereof. When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers.
This disclosure includes various explanations and theories about how and why the inventive devices and methods operate. These explanations and theories should be considered illustrative rather than binding. They are not intended to and should not limit the scope of the appended claims.
Also, various inventive concepts may be embodied as one or more methods, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.
All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and/or ordinary meanings of the defined terms.
The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.”
The phrase “and/or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with “and/or” should be construed in the same fashion, i.e., “one or more” of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and/or” clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to “A and/or B”, when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc.
As used herein in the specification and in the claims, “or” should be understood to have the same meaning as “and/or” as defined above. For example, when separating items in a list, “or” or “and/or” shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only terms clearly indicated to the contrary, such as “only one of” or “exactly one of,” or, when used in the claims, “consisting of,” will refer to the inclusion of exactly one element of a number or list of elements. In general, the term “or” as used herein shall only be interpreted as indicating exclusive alternatives (i.e. “one or the other but not both”) when preceded by terms of exclusivity, such as “either,” “one of,” “only one of,” or “exactly one of.” “Consisting essentially of,” when used in the claims, shall have its ordinary meaning as used in the field of patent law.
As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, “at least one of A and B” (or, equivalently, “at least one of A or B,” or, equivalently “at least one of A and/or B”) can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including more than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc.
In the claims, as well as in the specification above, all transitional phrases such as “comprising,” “including,” “carrying,” “having,” “containing,” “involving,” “holding,” “composed of,” and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases “consisting of” and “consisting essentially of” shall be closed or semi-closed transitional phrases, respectively, as set forth in the United States Patent Office Manual of Patent Examining Procedures, Section 2111.03.
This application is a bypass continuation application of International Patent Application PCT/US2017/038733, entitled “Highly Efficient Near-Field ThermoPhotoVoltaics Using Surface-Polariton Emitters and Thin-Film PhotoVoltaic-Cell Absorbers,” filed on Jun. 22, 2017 which claims the priority benefit, under 35 U.S.C. § 119(e), of U.S. Application No. 62/353,265, which was filed on Jun. 22, 2016. Each of these applications is incorporated herein by reference in its entirety.
This invention was made with Government support under Contract No. W911NF-13-D-0001 awarded by the Army Research Office. The Government has certain rights in the invention.
| Number | Date | Country | |
|---|---|---|---|
| 62353265 | Jun 2016 | US |
| Number | Date | Country | |
|---|---|---|---|
| Parent | PCT/US2017/038733 | Jun 2017 | US |
| Child | 15864634 | US |
