# Hot-carrier thermophotovoltaic systems

Kartika N. Nimje  Maxime Giteau<EMAIL_ADDRESS>Georgia T. Papadakis
<EMAIL_ADDRESS>ICFO - Institut de Ciencies Fotoniques, The
Barcelona Institute of Science and Technology, Castelldefels (Barcelona)
08860, Spain

###### Abstract

A thermophotovoltaic (TPV) energy converter harnesses thermal photons emitted
by a hot body and converts them to electricity. When the radiative heat
exchange between the emitter and PV cell is spectrally monochromatic, the TPV
system can approach the Carnot thermodynamic efficiency limit. Nonetheless,
this occurs at the expense of vanishing extracted electrical power density.
Conversely, a spectrally broadband radiative heat exchange between the emitter
and the cell yields maximal TPV power density at the expense of low
efficiency. By leveraging hot-carriers as a means to mitigate thermalization
losses within the cell, we demonstrate that one can alleviate this trade-off
between power density and efficiency. Via detailed balance analysis, we show
analytically that one can reach near-Carnot conversion efficiencies close to
the maximum power point, which is unattainable with conventional TPV systems.
We derive analytical relations between intrinsic device parameters and
performance metrics, which serve as design rules for hot-carrier-based TPV
systems.

Detailed balance, hot carriers, thermophotovoltaic systems, thermal photonics,
energy harvesting

##  I Introduction

Thermophotovoltaic (TPV) energy conversion recycles heat into electricity via
a system consisting of a thermal emitter coupled to a photovoltaic (PV) cell.
The emitter, which is externally heated, radiates thermal photons toward a PV
cell, which, in turn, converts them into electricity [1, 2, 3, 4, 5]. In
recent years, various TPV systems have been realized, reporting record-high
conversion efficiencies [6, 7, 8]. The remarkable progress in TPV performance
has attracted applications in low-grade waste heat recovery [9], intermittent
power generation [10], and heat-to-energy conversion at ultra-high
temperatures [11, 6].

Nonetheless, unless sophisticated design takes place, for example, by placing
the thermal emitter in the thermal near-field of the cell [12, 13], reported
high TPV efficiencies come at the cost of considerably compromised extracted
electrical power densities. In fact, the extracted electrical power density
and the conversion efficiency are by nature competing performance TPV metrics.
The extracted electrical power density, termed $P_{\mathrm{el}}$ henceforth,
expresses the power consumed by a load connected to the TPV module. The
conversion efficiency, defined as $\eta=P_{\mathrm{el}}/P_{\mathrm{exc}}$,
expresses the portion of the radiative heat exchange between the emitter and
the cell, $P_{\mathrm{exc}}$, that is converted into electricity by the cell.
Various recent approaches have explored ways to simultaneously increase both
the conversion efficiency and the power density of TPV devices, for example,
by operating at very high temperatures [11, 6], optimizing photon recycling
[14, 15, 16], enhancing the absorption view factor [17], and leveraging the
thermal near-field [18, 5, 19, 20]. Nevertheless, the trade-off between power
density and conversion efficiency remains a major challenge in TPV design.

In an ideal TPV device, in the limit of spectrally monochromatic radiative
heat exchange between the emitter and cell, one can approach the Carnot
thermodynamic efficiency limit, denoted as
$\eta_{\mathrm{C}}=1-T_{\mathrm{C}}/T_{\mathrm{H}}$, where $T_{\mathrm{C}}$ is
the temperature of the cell and $T_{\mathrm{H}}$ is the temperature of the
emitter. Approaching the Carnot efficiency, however, leads to negligible power
output. On the other hand, broadening the spectrum of radiative heat exchange
improves $P_{\mathrm{el}}$ but comes at the cost of lower $\eta$. In [10], we
presented the explicit relation between the fundamental limits of
$P_{\mathrm{el}}$ and $\eta$ for radiative energy conversion. In the present
work, we propose a strategy to overcome the power-efficiency trade-off in
single-junction TPVs.

The notion of maximum extracted electrical power density in a single-junction
TPV cell is analogous to the Shockley-Queisser limit [21] for single-junction
solar cells. Multiple strategies have been proposed to overcome the Shockley-
Queisser limit in solar PVs [22]. The most widespread and record-holding
approach is to employ multi-junction solar cells, where each cell converts a
portion of the solar spectrum, thereby improving the current-voltage
characteristics of the cell [23, 24, 25]. This concept has also been explored
for TPV systems [26, 6]. In this case, the TPV power density is increased, and
as a consequence, the TPV efficiency follows.

Here, we explore an alternative approach to improve $P_{\mathrm{el}}$ and
$\eta$ simultaneously by leveraging hot-carriers, in analogy to their
implementation in hot-carrier solar cells (HCSC) [27]. In HCSC systems, the
excess heat generated through absorption of high-energy photons improves the
voltage characteristics of the cell, upon the extraction of electron-hole
pairs prior to their thermalization. This can be achieved by extracting the
hot carriers using energy-selective contacts (ESC)[28, 29], and results in a
theoretical efficiency limit of 85$\%$ for zero bandgap under full
concentration [22]. Although the experimental efficiency of HCSC remains
currently limited due to strong thermalization rates in most materials,
promising avenues have been identified [30, 31], using ultrathin [32] and low-
dimensional absorbers, such as quantum wells [33, 34, 35, 36], nanowires [37]
and dots [38], or hybrid hot-carrier multi-junction devices [39, 40].

We note that recent studies that have also considered hot carriers in TPV
systems have primarily focused on the selection of materials comprising the
emitter and cell, and only pertain to near-field operation [41, 42]. By
contrast, in this work, we identify performance limits of hot-carrier-based
TPV (HCTPV) systems in the far-field. Using a detailed balance formalism [21],
we compute the extracted electrical power density and efficiency for any
combination of semiconductor bandgap and emitter temperature, both
analytically and numerically. As such, this work presents a general framework
describing the physics of HCTPV systems.

Figure 1: (a) Schematic of a hot-carrier thermophotovoltaic (HC-TPV) system: A
blackbody emitter at temperature $T_{\mathrm{E}}$ exchanges thermal radiation
with a PV cell. Perfect photon recycling is considered, represented visually
with a perfect backside reflector (grey) at temperature $T_{\mathrm{C}}$. The
radiative power densities, denoted by $P_{\mathrm{abs}}$ and
$P_{\mathrm{rad}}$, are associated with the absorption and emission of photons
by the PV cell respectively. Each absorbed photon generates high-energy _hot_
carrier. A HCTPV device extracts the energy of these hot-carriers as
electrical power density, $P_{\mathrm{el}}$, via the selective contacts shown
in cyan. (b) Thermodynamic representation of the HCTPV device using isentropic
extraction at energy $E_{\mathrm{ext}}$.

We demonstrate that a HCTPV system offers a significantly higher maximum power
density as compared to a standard TPV. At the maximum power point, efficiency
is also very high as compared to a standard TPV system. Finally, we show that,
at $V=V_{\mathrm{OC}}$, a HCTPV always reaches the Carnot thermodynamic
efficiency limit, a feature unattainable in standard TPV operation. As such, a
single-junction HCTPV offers a physical approach to reaching thermodynamic
limits of radiative heat engines, as described in [10] [10].

##  II Theoretical formalism

Our model is analogous to the one introduced by Ross and Nozik for HCSCs [27],
where the sun is replaced by a local blackbody source at temperature
$T_{\mathrm{E}}$. We consider that the PV cell has a temperature
$T_{\mathrm{C}}$ and a bandgap $E_{\mathrm{g}}$, as shown in Fig. 1(a). To
simplify the analysis, both the emitter and cell are modeled as semi-infinite
parallel slabs to eliminate any radiation leakage, ensuring a unity view
factor [43]. In addition, we consider that the absorptivity of the cell is
unity above the bandgap and that below-bandgap radiative heat exchange between
the emitter and the cell is completely suppressed. This can be achieved in
practice with an ideal backside reflector ensuring perfect sub-bandgap photon
recycling [7] as shown in Fig. 1(a). We also do not consider near-field
contributions to the radiative heat exchange between the emitter and the cell.
The analysis is conducted in the radiative limit, for which non-radiative
losses in the PV cell are neglected. Finally, we consider that each absorbed
photon generates a single electron-hole pair with the same energy.

Figure 2:  (a) Electrical output power density and, (b) efficiency,
analytically derived up to a second-order Maxwell-Boltzmann approximation, as
a function of normalised chemical potential and temperature of the hot-
carriers at $T_{\mathrm{E}}=850\text{K}$, $T_{\mathrm{C}}=300\text{K}$ and
$E_{\mathrm{g}}=0.74\text{eV}$. The red curve in (a) represents the
temperature as a function of chemical potential for which the power density is
maximum. The black point in (b) indicates the open-circuit voltage, that is,
$\left(\mu,T\right)=\left(0,T_{\mathrm{E}}\right)$ at which the efficiency
reaches Carnot. The gray regions correspond to $P_{\mathrm{el}}<0$ (not
relevant for TPV operation).

Since only photons with energy $\varepsilon\geq E_{\mathrm{g}}$ are exchanged,
the particle flux (in $\mathrm{m^{-2}s^{-1}}$) absorbed by the cell can then
be expressed as:

$\displaystyle N_{\mathrm{abs}}$
$\displaystyle=\frac{1}{4\mathrm{c}^{2}\pi^{2}\hbar^{3}}\int_{E_{\mathrm{g}}}^{\infty}\frac{\varepsilon^{2}}{\exp{{\left(\dfrac{\varepsilon}{k_{\mathrm{B}}T_{\mathrm{E}}}\right)}}-1}\,d\varepsilon$
(1)

where $\mathrm{c}$ is the speed of light in vacuum, $k_{\mathrm{B}}$ is the
Boltzmann constant, and $\hbar$ is the reduced Planck constant. We assume that
carrier-carrier scattering occurs sufficiently fast so that photo-generated
carriers follow a Fermi-Dirac distribution characterized by a quasi-Fermi
level splitting $\mu$ (hereafter referred to as _chemical potential_) and a
temperature $T$. Thus, the photon flux radiated by the cell follows the
generalized Planck law [44]:

$\displaystyle N_{\mathrm{rad}}\left(\mu,T\right)$
$\displaystyle=\frac{1}{4\mathrm{c}^{2}\pi^{2}\hbar^{3}}\int_{E_{g}}^{\infty}\frac{\varepsilon^{2}}{\exp{{\left(\dfrac{\varepsilon-\mu}{k_{\mathrm{B}}T}\right)}}-1}\,d\varepsilon.$
(2)

Under continuous illumination, steady-state is achieved when the rate of
photo-generation is balanced with the combined rate of recombination and
extraction of carriers. Based on the principle of detailed balance introduced
by Shockley and Queisser [21], the extracted current density,
$J_{\mathrm{ext}}$ (in $\mathrm{Am^{-2}}$), can be written as:

$\displaystyle J_{\mathrm{ext}}\left(\mu,T\right)$
$\displaystyle=\mathrm{q}\left[N_{\mathrm{abs}}-N_{\mathrm{rad}}\left(\mu,T\right)\right]$
(3)

where $\mathrm{q}$ is the electron charge. In the absence of hot carriers, in
other words, when carriers dissipate their excess kinetic energy through
thermalization, upon reaching thermal equilibrium with the lattice, it holds
that $T=T_{\mathrm{C}}$. In that case, and assuming ideal carrier transport,
the chemical potential $\mu$ is equal to $\mathrm{q}V$, where $V$ is the
applied voltage [28]. By solving Eq. (3) as a function of $\mu$ using Eqs. (1)
and (2), we obtain the well-known $J_{\mathrm{ext}}-V$ characteristic of a PV
cell. By contrast, in a hot-carrier PV cell, the carriers are not in thermal
equilibrium with the lattice ($T\neq T_{\mathrm{C}}$), and the carrier
population is a function of both $\mu$ and $T$.

We introduce the quantity $P_{\mathrm{th}}$, which represents the energy lost
in thermalization, upon the excitation of carriers. The extracted power
density of carriers equals
$P_{\mathrm{ext}}=P_{\mathrm{exc}}-P_{\mathrm{th}}$, where $P_{\mathrm{exc}}$
is the radiatively exchanged power density between the emitter and the cell,
expressed as:

$\displaystyle P_{\mathrm{exc}}\left(\mu,T\right)$
$\displaystyle=P_{\mathrm{abs}}-P_{\mathrm{rad}}\left(\mu,T\right),$ (4)

where $P_{\mathrm{abs}}$ and $P_{\mathrm{rad}}$ are the power densities
absorbed and radiated by the cell, respectively. These can be obtained by
multiplying the integrand in Eqs. 1-2 by $\varepsilon$.

To harness the energy of hot carriers with minimal loss, following [28], we
consider isentropic extraction, expressed through:

$\mathrm{q}V=\mu\frac{T_{\mathrm{C}}}{T}+\left(1-\frac{T_{\mathrm{C}}}{T}\right)E_{\mathrm{ext}}.$
(5)

In Eq. (5), the extraction energy, termed $E_{\mathrm{ext}}$, is the
difference between the energy levels of extraction for electrons and holes,
and satisfies $P_{\mathrm{ext}}=J_{\mathrm{ext}}E_{\mathrm{ext}}/\mathrm{q}$
[45]. As a reference, we note that, in the limit $T=T_{\mathrm{C}}$ for a
conventional PV cell, from Eq. (5), we indeed retrieve the standard expression
$\mu=\mathrm{q}V$.

The electrical power density, $P_{\mathrm{el}}$, is defined as
$P_{\mathrm{el}}=J_{\mathrm{ext}}V$ (Fig. 1(b)). Consequently, the conversion
efficiency in the general case of a HCTPV is given by

$\eta=\frac{P_{\mathrm{el}}}{P_{\mathrm{ext}}+P_{\mathrm{th}}}.$ (6)

In the limit where thermalization is completely suppressed and carriers remain
at temperature $T$, which is the case for HCTPV considered here,
$P_{\mathrm{th}}=0$, and $P_{\mathrm{ext}}=P_{\mathrm{exc}}$. From Eq. (6),
$\eta=P_{\mathrm{el}}/P_{\mathrm{exc}}$ as introduced above.

##  III Results and discussion

Figure 3:  (a) Electrical output power density and (b) efficiency calculated
as a function of the device parameters (extraction energy and voltage) for
$T_{\mathrm{E}}=1700\text{K}$, $T_{\mathrm{C}}=300\text{K}$ and
$E_{\mathrm{g}}=0.74\text{eV}$. The dashed line in (a) indicates the
extraction energy that maximizes the power density. The gray regions
correspond to $P_{\mathrm{el}}<0$ (LED operation).

The analysis considered here pertains to an idealized HCTPV system where
thermalization is completely suppressed. The temperature of the carriers is
$T$, whereas the temperature of the lattice is $T_{\mathrm{C}}$ (Fig. 1(b)).
From Eq. (4) and the expressions of $P_{\mathrm{ext}}$ and $P_{\mathrm{el}}$,
the conversion efficiency Eq. (6) reduces to:

$\eta=\frac{\mathrm{q}V}{E_{\mathrm{{ext}}}}.$ (7)

From Eq. (7), the maximum efficiency is obtained at $V=V_{\mathrm{oc}}$, for
which, here, we derive general analytical expressions for the efficiency and
open-circuit voltage of the HCTPV system. At $V=V_{\mathrm{oc}}$,
$J_{\mathrm{{ext}}}=0$ and $P_{\mathrm{{ext}}}=0$. Solving Eqs. (3)-(4) at
$V=V_{\mathrm{oc}}$ yields $\mu=0$ and $T=T_{\mathrm{E}}$. The open circuit
voltage Eq. (5) and the corresponding efficiency Eq. (7) are, respectively,
given by:

$V_{\mathrm{{oc}}}=\left(1-\frac{T_{\mathrm{C}}}{T_{\mathrm{E}}}\right)\frac{E_{\mathrm{{ext}}}}{\mathrm{q}}$
(8)
$\eta_{\mathrm{oc}}=1-\frac{T_{\mathrm{C}}}{T_{\mathrm{E}}}=\eta_{\mathrm{C}}$
(9)

Eq. (9) demonstrates that an ideal HCTPV system always reaches the Carnot
thermodynamic efficiency limit ($n_{\mathrm{C}}$). This is not the case for
conventional TPV systems, unless the bandwidth of the radiative heat exchange
between the emitter and the cell approaches zero [8]. In a conventional TPV
system, $\eta$ is also maximal at $V_{\mathrm{oc}}$, nonetheless this
maximized efficiency does not approach $\eta_{\mathrm{C}}$. At the same time,
near $V_{\mathrm{oc}}$, the power density of a conventional TPV system
vanishes, whereas in a HCTPV this is no longer true. As demonstrated in Fig. 4
(a) of the following analysis, in a HCTPV system, one approaches maximal power
density at near-Carnot efficiencies.

As a reference, in [10], by considering reciprocal radiative heat exchange
between a blackbody coupled to a Carnot engine, termed the endoreversible
approximation, we analyzed the thermodynamic performance bounds of radiative
heat engines. The analysis of a HCTPV system outlined above reduces to the
endoreversible approximation in the limit of $E_{\mathrm{G}}=\mu=0$. As a
result, a zero-bandgap HCTPV system approaches the thermodynamic bound of
radiative heat engines. By contrast, approaching this bound with conventional
TPV cells would require an infinite number of junctions (multicolor limit)
[22].

Next, we identify the optimal internal parameters (hot carrier temperature,
$T$, and chemical potential, $\mu$, in Fig. 2) as well as external device
parameters (extraction energy, $E_{\mathrm{ext}}$, and voltage, $V$, in Fig.
3) for optimal HCTPV performance. We note that, in the limit
$\left(E_{g}-\mu\right)/k_{\mathrm{B}}T\gg 1$, in other words, within the
Maxwell-Boltzmann approximation, it is possible to derive analytical
expressions. In the Appendix, Eqs. (A9), (A11), we present the relation
between $\mu$ and $T$ for optimal power density and efficiency, respectively,
up to a second-order approximation.

Figure 4: (a) Electrical power density ($P_{\mathrm{el}}$) versus conversion
efficiency in terms of the Carnot limit ($\eta/\eta_{\mathrm{C}}$) for
$(E_{\mathrm{g}}=0.74\text{eV})$. Red line: HC-TPV at power-maximizing
extraction energy $E_{\mathrm{ext}}=1.27E_{\mathrm{g}}$. Blue line:
conventional broadband TPV. Green line: conventional narrowband TPV (emission
bandwidth of 15 meV above the bandgap). (b) Power density normalised to
$\sigma T_{\mathrm{E}}^{4}$ ($\rho$) versus efficiency in terms of the Carnot
limit ($\eta/\eta_{\mathrm{C}}$) for $E_{\mathrm{g}}=0.74\text{eV}$,
$E_{\mathrm{g}}=0.35\text{eV})$ and a zero-bandgap cell (the optimal
extraction energies are $E_{\mathrm{ext}}=0.94\text{eV}$, $0.59\text{eV}$ and
$0.41\text{eV}$ respectively). The dashed line represents the performance of
an endoreversible engine, a good approximation for the performance bound of
TPV devices [10]. All curves are obtained for $T_{\mathrm{E}}=1700\text{K}$
and $T_{\mathrm{C}}=300\text{K}$.

In Fig. 2, we show the electrical power density and efficiency of a HCTPV
system as a function of $\mu$ and $T$. We only consider positive values of
$P_{\mathrm{el}}$, as negative power density corresponds to LED operation and
is outside the scope of this paper. We consider $T_{\mathrm{E}}=850\text{K}$,
$T_{\mathrm{C}}=300\text{K}$ and $E_{\mathrm{g}}=0.74\text{eV}$, corresponding
to the bandgap of InGaAs lattice-matched to InP. The red curve in Fig. 2(a)
represents the analytically derived chemical potential $\mu$ that maximizes
$P_{\mathrm{el}}$ for any temperature $T$ Eq. (A9). This curve covers various
different combinations of ($\mu$, $T$) that would all yield a similar
$P_{\mathrm{el}}\approx 0.013\ \mathrm{W.cm^{-2}}$ and efficiency $\eta\approx
0.8\ \eta_{\mathrm{C}}$. In Fig. 2(b), we plot the corresponding HCTPV
efficiency with respect to $\left(\mu,T\right)$. We confirm from Fig. 2(b)
that at open circuit voltage (that is, for $\mu=0$ and $T=T_{\mathrm{E}}$),
the efficiency reaches the Carnot limit (See Eq. (A11)). We note that, unlike
for conventional PV systems, the chemical potential can take negative values.
Similarly, the carrier temperature can take values lower than $T_{C}$ or
larger than $T_{E}$.

Fig. 2 demonstrates the interplay between hot carrier temperature and chemical
potential that defines HCTPV operation. For instance, a small hot carrier
temperature only slightly above $T_{\mathrm{C}}$ would give negligible
performance metrics in a standard TPV system. Nonetheless, in a HCTPV system,
a large chemical potential $\mu$ can compensate for small $T$ and yield near-
Carnot efficiency. Inversely, a very large hot carrier temperature does not
necessarily improve power density as shown in Fig. 2 (a), as there is an
optimal combination of $\mu$ and $T$ for $P_{\mathrm{el}}$ optimization. At
the same time, from Fig. 2 (b) it is shown that, for all considered
temperatures, one can approach Carnot efficiency by adjusting the chemical
potential, a degree of freedom that is unavailable in standard TPV systems. We
note, finally, that operating at low emitter temperature with a large bandgap
strongly limits the benefits of hot carriers, because the cell may only absorb
a small fraction of the radiation (see Fig. A3 in Appendix).

In Fig. 2, we described HCTPV operation in terms the chemical potential $\mu$
and temperature $T$ of the hot carriers. These quantities, however, cannot be
controlled directly. Below, we describe the HCTPV system in terms of
$E_{\mathrm{ext}}$ and $V$, which are externally controlled quantities of the
device. Fig. 3 shows the figure of merits $P_{\mathrm{el}}$ and $\eta$,
numerically calculated, as functions of $\left(V,E_{\mathrm{ext}}\right)$. For
these calculations, we consider the same bandgap and lattice temperature, but
double the emitter temperature ($T_{\mathrm{E}}=1700\ \text{K}$). In Fig.
3(a), we demonstrate that maximum power density has improved by $500$ times
with respect to the same device operating at half the emitter temperature
(Fig. 2(a)). This is expected as increasing $T_{\mathrm{E}}$ not only enhances
total blackbody radiation by a factor $T_{E}^{4}$ but also increases the
average energy of these photons, meaning a larger fraction of them are above
the bandgap.

From Fig. 3(b), we confirm that the efficiency reaches the Carnot
thermodynamic efficiency limit at open-circuit voltage for any
$E_{\mathrm{ext}}$, as per Eqs. (8)-(9). By selecting
$E_{\mathrm{ext}}=1.27E_{\mathrm{g}}\approx 0.94\ \text{eV}$ such that the
power density is maximized, the efficiency remains very high, nearly
$0.8\eta_{\mathrm{C}}$. We note that this is an important distinction between
a standard TPV system and the HCTPV system considered here; having control
over the energy level at which hot carriers are extracted is essentially the
benefit of employing hot carriers in TPV systems. Finally, unlike for
conventional PV systems, the voltage can be higher than the bandgap energy of
the cell thanks to the temperature contribution of the hot carriers Eq. (5).

In a standard TPV system, optimal efficiency and optimal power density are
achieved at significantly different voltages [46]. By contrast, in Fig. 3 in
combination with Eqs. (8), we demonstrated that a HCTPV system reaches optimal
power density at a voltage for which the efficiency penalty is very small.

In Fig. 4, we vary the voltage applied to the HCTPV cell from $0$ to
$V_{\mathrm{oc}}$ and plot the power-versus-efficiency performance curves. In
Fig. 4(a), we present the performance of the optimal HCTPV system considered
above ($E_{\mathrm{g}}=0.74\ \mathrm{eV}$, $E_{\mathrm{ext}}=0.94\
\mathrm{eV}$). We compare this device to a conventional TPV cell with the same
bandgap, for two distinct cases: broadband (blackbody) and narrowband thermal
radiative exchange between the emitter and the cell. In the narrowband case,
we consider a bandwidth of $0.1k_{\mathrm{B}}T_{\mathrm{E}}\approx 15\
\text{meV}$ at frequencies above the bandgap. First, we note that the power-
efficiency trade-off between a broadband and a narrowband conventional TPV
system can be clearly seen in this figure: a spectrally broadband TPV device
reaches significantly higher power density, nonetheless this comes at the cost
of a conversion efficiency that can never reach the Carnot limit. On the other
hand, a spectrally narrowband TPV device does approach the Carnot limit,
nonetheless its corresponding power density vanishes. In both cases, the
power-versus-efficiency characteristic forms a loop, as the efficiency returns
to $0$ at $V_{\mathrm{oc}}$.

By contrast, the HCTPV system not only yields significantly higher electrical
power density than conventional broadband TPV system, it also reaches the
Carnot limit at open-circuit voltage. As in the limiting cases of ultra-
narrowband and broadband TPV systems, we note that the power-versus-efficiency
characteristic curve of any conventional ($T=T_{\mathrm{C}}$) TPV system will
lie within the HCTPV curve. This means that an ideal HCTPV system will always
demonstrate an improved power-efficiency trade-off compared to a conventional,
single-junction TPV one.

In Fig. 4(b), we compare the performance of HCTPV systems to the thermodynamic
bound of reciprocal radiative heat engines, approximated by the endoreversible
model discussed in [10]. This is described analytically through:

$\rho=\eta\left[1-\left(\frac{T_{\mathrm{C}}}{T_{\mathrm{E}}}\right)^{4}\frac{1}{\left(1-\eta\right)^{4}}\right].$
(10)

where $\rho$ is the electrical power density, $P_{\mathrm{el}}$, normalised to
$\sigma T_{\mathrm{E}}^{4}$. Eq. (10) is plotted in Fig. 4 (b) with the dashed
black curve. We consider HCTPV systems with several bandgaps: 0.74 eV (InGaAs)
0.35 eV (InAs) and 0 eV, and plot their power-versus-efficiency characteristic
curves at the extraction energy, $E_{ext}$, for which their power is
optimized, respectively 0.94 eV, 0.59 eV and 0.41 eV. As shown, the power
density _for any efficiency_ increases with decreasing bandgap as more hot-
carriers contribute to the power density while no power gets dissipated. As a
limiting case, indeed, the zero-bandgap HCTPV system approaches the
endoreversible model (Eq. (10)), black dashed curve), and even reaches a
marginally higher maximum power point. This is because the endoreversible
model considers a purely thermal engine with $\mu=0$, while HCTPV systems
offer control over both the temperature and the chemical potential, which can
be leveraged to achieve slightly better performance.

##  IV Conclusion

We highlight the potential benefits of harvesting hot carriers in TPV devices.
Based on our theoretical formalism, we demonstrate that harnessing hot
carriers alleviates the trade-off between electrical power output and
conversion efficiency, allowing performance improvements in both metrics
relative to conventional TPV cells. Specifically, we show that an ideal
broadband hot-carrier TPV system can reach the Carnot efficiency limit at open
circuit voltage, and that the power density at the optimal extraction energy
is higher than that of conventional TPV devices, for any efficiency. We
demonstrate that the power density substantially increases with the emitter
temperature, whereas decreasing the bandgap consistently improves the
performance.

In this work, we extend conclusions derived by solar PV community for the case
of TPV systems. In particular, as demonstrated in the seminal HCSC analysis by
Ross and Nozik [27], the performance of a HCSC system is nearly identical to
that of an infinite-junction SC. In other words the performance limit of a
HCSC nearly overlaps with the multicolor limit. Analogously, in this work
(Fig. 4), we showed that a zero-bandgap hot-carrier TPV system is similar to
an endoreversible radiative heat engine, approaching the thermodynamic
performance bound of radiative heat engines. A multi-junction TPV system would
have the same performance to a zero-bandgap HCTPV system, in the limit of
infinite junctions.

As a final comment, we note that the results of this work can be further
extended to near-field TPV operation, where an increased thermal radiative
heat exchange can be exploited to saturate the thermalization channels [47,
35, 45]. Such a synergy between near-field radiative heat transfer and hot
carriers may enable substantial improvement in realistic TPV devices.

###### Acknowledgements.

This work is dedicated to the memory of John S. Papadakis. The authors declare
no competing financial interest. G.T.P. acknowledges financial support from
the la Caixa Foundation (ID 00010434). M. G. acknowledges financial support
from the Severo Ochoa Excellence Fellowship. This work was supported by the
Spanish MICINN (PID2021-125441OA-I00, PID2020–112625GB-I00, and
CEX2019-000910- S), the European Union (fellowship LCF/BQ/PI21/11830019 under
the Marie Skłodowska-Curie Grant Agreement No. 847648), Generalitat de
Catalunya (2021 SGR 01443) through the CERCA program, Fundació Cellex, and
Fundació Mir-Puig.

## Appendix

###  IV.1 Analytics in the Maxwell-Boltzmann approximation

Here, we derive the analytical expressions of the power density and the
efficiency using Eq. (1)-(2) in the main text. We consider the Maxwell-
Boltzmann (MB) approximation, namely, we approximate the occupation number of
photons by:

$\frac{1}{e^{(\varepsilon-\mu)/k_{B}T}-1}\approx e^{\mu-\varepsilon/k_{B}T}.$
(A1)

For the sake of algebraic simplicity, we normalize all the parameters to the
bandgap energy of the semiconductor:

$\displaystyle m$ $\displaystyle=\dfrac{\mu}{E_{\mathrm{g}}},$ $\displaystyle
t$ $\displaystyle=\dfrac{k_{B}T}{E_{\mathrm{g}}},$ $\displaystyle\tau_{E}$
$\displaystyle=\dfrac{k_{B}T_{E}}{E_{\mathrm{g}}},$ $\displaystyle\tau_{C}$
$\displaystyle=\dfrac{k_{B}T_{C}}{E_{\mathrm{g}}}.$

Retaining terms up to the second order in $t$, we obtain the following
expressions for radiated particle flux and power density:

$\displaystyle N^{\mathrm{MB}}_{\mathrm{rad}}\left(m,t\right)$
$\displaystyle=\frac{E_{\mathrm{g}}^{3}}{4\mathrm{c}^{2}\pi^{2}\hbar^{3}}\ t\
\left(1+2t\right)\ e^{\frac{m-1}{t}}$ (A2) $\displaystyle
P^{\mathrm{MB}}_{\mathrm{rad}}\left(m,t\right)$
$\displaystyle=\frac{E_{\mathrm{g}}^{4}}{4\mathrm{c}^{2}\pi^{2}\hbar^{3}}\ t\
\left(1+3t\right)\ e^{\frac{m-1}{t}}.$ (A3)

The absorbed fluxes, $N^{\mathrm{MB}}_{\mathrm{abs}}$ and
$P^{\mathrm{MB}}_{\mathrm{abs}}$, are obtained similarly by replacing
$\left(m,t\right)$ with $\left(0,\tau_{\mathrm{E}}\right)$ in Eq. (A2)-(A3).
From here, we have an analytical expression for the electrical power density
as a function of the internal parameters $m$ and $t$, using Eq. (5) in the
main text:

$P_{\mathrm{el}}^{\mathrm{MB}}=\frac{\tau_{\mathrm{C}}}{t}mE_{\mathrm{g}}N_{\mathrm{ext}}^{\mathrm{MB}}+\left(1-\frac{\tau_{\mathrm{C}}}{t}\right)P_{\mathrm{ext}}^{\mathrm{MB}},$
(A4)

where

$\displaystyle N_{\mathrm{ext}}^{\mathrm{MB}}$
$\displaystyle=\frac{E_{\mathrm{g}}^{3}}{4\mathrm{c}^{2}\pi^{2}\hbar^{3}}\left[\tau_{E}\left(1+2\tau_{E}\right)e^{-\frac{1}{\tau_{E}}}-t\left(1+2t\right)e^{\frac{m-1}{t}}\right],$
(A5) $\displaystyle P_{\mathrm{ext}}^{\mathrm{MB}}$
$\displaystyle=\frac{E_{\mathrm{g}}^{4}}{4\mathrm{c}^{2}\pi^{2}\hbar^{3}}\left[\tau_{E}\left(1+3\tau_{E}\right)e^{-\frac{1}{\tau_{E}}}-t\left(1+3t\right)e^{\frac{m-1}{t}}\right].$
(A6)

Figure A1: Absorbed power density ($P_{abs}^{\mathrm{MB}}$) normalized to
$\sigma T_{\mathrm{E}}^{4}$ versus bandgap energy in terms of
$k_{\mathrm{B}}T_{\mathrm{E}}$. The second-order Maxwell-Boltzmann
approximation is valid above 95$\%$ for
$E_{\mathrm{g}}/k_{\mathrm{B}}T_{\mathrm{E}}\geq 10$. For
$E_{\mathrm{g}}=0.74\text{eV}$ and $T_{\mathrm{E}}=850\text{K}$,
$E_{\mathrm{g}}/k_{\mathrm{B}}T_{\mathrm{E}}\approx 10$.

This allows us to write the extraction energy as:

$\displaystyle
E^{\mathrm{MB}}_{\mathrm{{ext}}}=E_{\mathrm{g}}\left[1+\Theta\left(m,t\right)\right],$
(A7)

where

$\displaystyle\Theta(m,t)=\frac{\tau_{\mathrm{E}}^{2}e^{-\frac{1}{\tau_{\mathrm{E}}}}-t^{2}e^{\frac{m-1}{t}}}{\tau_{\mathrm{E}}\left(1+2\tau_{\mathrm{E}}\right)e^{-\frac{1}{\tau_{\mathrm{E}}}}-t\left(1+2t\right)e^{\frac{m-1}{t}}}.$
(A8)

Figure A2: (a) Electrical output power density and, (b) efficiency numerically
calculated as a function of normalised chemical potential and temperature of
the hot-carriers at $T_{\mathrm{E}}=850\text{K}$, $T_{\mathrm{C}}=300\text{K}$
and $E_{\mathrm{g}}=0.74\text{eV}$. The red curve in (a) represents the
temperature as a function of chemical potential for which the power density is
maximum, given by Eq. (A9). The gray regions correspond to $P_{\mathrm{el}}<0$
(not relevant for TPV operation).

Furthermore, we observed in Fig. 2(a) (in the main text) that the power
density reaches its maximum value for particular values of
$\left(\mu,T\right)$, represented with a red line. Here, we derive the
chemical potential $m_{0}$ at which the power density is maximized for a given
$t$ by setting the derivative of $P_{\mathrm{el}}^{\mathrm{MB}}$ with respect
to $m$ equal to zero. We find:

$m_{\mathrm{0}}=\frac{\tau_{\mathrm{C}}-t\left[2(t-1)\tau_{\mathrm{C}}+3t+1\right]}{(2t+1)\tau_{\mathrm{C}}}+t\
\mathrm{W}\left[\alpha\right],$ (A9)

where $\mathrm{W}$ is the Lambert W function and

$\alpha=\frac{\tau_{\mathrm{E}}(2\tau_{\mathrm{E}}+1)}{t(2t+1)}\exp\left({\frac{2t\tau_{\mathrm{E}}\tau_{\mathrm{C}}+3t\tau_{\mathrm{E}}-2t\tau_{\mathrm{C}}+\tau_{\mathrm{E}}-\tau_{\mathrm{C}}}{2t\tau_{\mathrm{E}}\tau_{\mathrm{C}}+\tau_{\mathrm{E}}\tau_{\mathrm{C}}}}\right).$
(A10)

Finally, we derive the efficiency of the hot-carrier TPV system in the MB
approximation using Eq. (5)-(7), from the main text and Eq. (A7), leading to:

$\eta^{\mathrm{MB}}=1-\dfrac{\tau_{C}}{t}\left(1-\dfrac{m}{1+\Theta\left(m,t\right)}\right).$
(A11)

At open circuit voltage, that is, for $m=0$ and $t=\tau_{\mathrm{E}}$, we
recover $\eta^{\mathrm{MB}}=\eta_{\mathrm{C}}$.

###  IV.2 Validity of the MB approximation

We compare the power density obtained using the MB approximation,
$P_{abs}^{\mathrm{MB}}$, with the exact power density given by integrating
Planck’s law from $\omega_{\mathrm{g}}$ to $\infty$. In Fig. A1, we plot the
relative absorbed power as a function of
$E_{\mathrm{g}}/k_{\mathrm{B}}T_{\mathrm{E}}$. We observe that the power
absorbed using the MB approximation reaches 95$\%$ of its actual value for
$E_{\mathrm{g}}/k_{\mathrm{B}}T_{\mathrm{E}}\geq 10$, implying that the
second-order MB approximation holds true within the shaded region.

Furthermore, the validity of the MB approximation in the domain
$E_{\mathrm{g}}/k_{\mathrm{B}}T_{\mathrm{E}}\approx 10$ can be confirmed in
Fig. A2, where the exact numerical results closely match the analytical
results presented in Fig. 2 of the main text. The red curve, derived
analytically using the Maxwell-Boltzmann approximation (Eq. (A9)), coincides
with the region indicating the maximum electrical power density.

Figure A3: Absorbed spectral energy density as a function of normalized
absorbed energy ($E/k_{\mathrm{B}}T_{\mathrm{E}}$). The dashed line indicates
$E_{\mathrm{g}}/k_{\mathrm{B}}T_{\mathrm{E}}$ for
$E_{\mathrm{g}}=0.74\text{eV}$ and $T_{\mathrm{E}}=1700\text{K}$.

Finally, we show in Fig. A3 the spectral energy density radiated by the
emitter as a function of the photon energy (normalized to
$k_{\mathrm{B}}T_{\mathrm{E}}$). The absorbed particle flux (Eq. (1)) and
power density is obtained by integrating this quantity from
$E_{\mathrm{g}}/k_{\mathrm{B}}T_{\mathrm{E}}$ to $\infty$. We note that being
within the validity domain of the MB approximation
($E_{\mathrm{g}}/k_{\mathrm{B}}T_{\mathrm{E}}\geq 10$, see Fig. A1) implies a
very small portion of the spectrum is absorbed in the cell, strongly limiting
the power output. As a result, larger values of
$E_{\mathrm{g}}/k_{\mathrm{B}}T_{\mathrm{E}}$ restrict the potential benefits
of harnessing hot-carriers, as only the high-energy tail of the incident
spectrum can be absorbed. In fact, raising $T_{\mathrm{E}}$ not only shifts
the dashed line (represented by the dashed line in Fig. A3) to the left, but
also increases blackbody radiation, resulting in higher power density. Thus,
low-bandgap semiconductors and high emitter temperatures synergistically
contribute to the improved performance of hot-carrier TPV systems. As a
result, high-power HC-TPV is not compatible with the MB approximation.

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