Optical determination of thermoelectric transport coefficients
in a hot carrier absorber

Thomas Vezin,* Hamidreza Esmaielpour,† Laurent Lombez,‡ Jean-François Guillemoles,§ and Daniel Suchet§

(Dated: July 27, 2024)

Hot carrier solar cells offer potential for enhancing the energy conversion efficiency of photo-voltaic devices. Transport properties of such systems remains largely unexplored, while it could hinder their efficiency. In this work, we develop a suitable framework for describing the thermoelectric ambipolar transport of photogenerated hot carriers and derive analytical expressions for the ambipolar transport coefficients valid even in the degenerate case. We demonstrate that these transport coefficients can be measured from hyperspectral photoluminescence imaging. We validate this experimental determination by showing its consistency with Boltzmann transport equation in the relaxation time approximation.

I. INTRODUCTION

"Hot carriers" denote carrier populations that deviate from thermal equilibrium with the surrounding lattice. Such populations arise following intense excitation, under the condition of strong intraband carrier interactions and weak coupling to the lattice [1-3]. Hot carriers are detrimental to microelectronic devices like transistors [4]. However, they have been thought to be beneficial to energy conversion devices, particularly photocatalysis [5, 6] and solar cells [7-11].

In the context of solar cells, hot carriers may lead to a voltage boost $\Delta V$ compared to a single junction solar cell. Assuming isentropic extraction through perfectly selective contacts [7]

$$e\Delta V = \frac{E_{ext} - \Delta\mu}{T_H}\Delta T \quad (1)$$

where $E_{ext}$ represents the energy at which electron-hole pairs are extracted by the selective contacts, $\Delta\mu$ is the quasi-Fermi level splitting (QFLS) in the absorber, $T_H$ is carrier temperature and $\Delta T = T_H - T_L$ is the difference between carrier and lattice temperatures.

This equation showcases the similarities between a hot carrier solar cell and a thermoelectric generator. This parallel has provided an insightful framework to account for the extraction of hot carriers from the absorber towards a cold metal through energy selective contacts [12-15]. An overlooked consequence is the influence of inhomogeneous temperature on coupled carrier and heat transport within the absorber [16-18].

To investigate such in-plane thermoelectric transport, photoluminescence (PL) imaging is a promising tool [19-21]. Broadband PL imaging has first been utilized to measure isothermal ambipolar carrier diffusion in doped GaAs [22]. Then, the use spectral resolution through hyperspectral imaging enabled the study of thermoelectric transport of hot carriers [16] and hot trions [18]. However, a comprehensive model of thermoelectric transport for hot carriers in the context of solar cells has not yet been established.

This is because this system explores unusual conditions for thermoelectricity. First, many thermoelectric investigation focus on determining material properties. In contrary, we study thermoelectric properties of photogenerated populations which depend on excitation condition, as in photo-Seebeck experiments [23]. Second, the intense optical excitation necessary for hot carrier generation results in high carrier densities and strong electrostatic coupling between electrons and holes, leading to ambipolar diffusion [21]. Finally, we consider a situation with inhomogeneous optical excitation, which cannot be described with a constant QFLS, as typically assumed in most ambipolar derivations [24-26].

This article presents a novel framework for describing thermoelectric ambipolar transport of hot carriers, valid even in the degenerate regime. We show that this framework resembles the usual thermoelectric transport equation, provided we use effective ambipolar transport coefficients. Experimentally, we use point-like excitation to generate local hot carrier distributions. We probe hot carrier diffusion by mapping carrier densities and temperatures with micrometric resolution thanks to hyperspectral imaging. We show that the ambipolar Seebeck coefficient can be directly measured from this optical experiment. Finally, we perform numerical estimations based on Boltzmann transport equation and show that they are consistent with our measurements.

II. AMBIPOLAR SEEBECK TRANSPORT

The situation considered in this work is pictured in Fig. 1. A focused continuous wave laser beam photo-

<footnote>* Institut Photovoltaique d'Ile de France, UMR-IPVF 9006, CNRS, Ecole Polytechnique IPP, EN-SCP PSL, Palaiseau, France.; Corresponding author @thomas.vezin.x15@polytechnique.edu</footnote>

<footnote>† Walter Schottky Institute and Physics Department, TUM School of Natural Sciences, Technical University of Munich, Garching, Germany</footnote>

<footnote>‡ Universite de Toulouse, INSA-CNRS-UPS, LPCNO, Toulouse, France</footnote>

<footnote>§ Institut Photovoltaique d'Ile de France, UMR-IPVF 9006, CNRS, Ecole Polytechnique IPP, ENSCP PSL, Palaiseau, France.</footnote>
<figure><img src="image_1.png" /></figure>

<figure><img src="image_2.png" /><figcaption>FIG. 1. (a) InGaAsP quantum well structure studied in this work and schematic of our method to determine Seebeck coefficient from hyperspectral PL measurements. NR stands for "not represented". (b) Laser, carrier concentration and temperature normalized profiles measured at excitation intensity $I_{exc} = 5.7 \times 10^4$ W.cm$^{-2}$. Vertical lines highlight three regimes: (III) isothermal diffusion, where electrical conductivity may be measured, (II) thermoelectric diffusion, where Seebeck coefficient may be measured, (I) region dominated by generation and not used in this study.</figcaption></figure>

generates a hot carrier distribution in a limited region of a hot carrier absorber. These carriers move away from the generation region under the influence of both concentration and temperature gradients. To account for this transport and quantify the corresponding transport coefficients, we derive a framework appropriate to hot carriers solar cells.

### A. Bipolar diffusion

In semiconductors, carrier diffusion is bipolar as, in general, carriers in the conduction and valence band must both be considered. In this work, we describe carriers in the valence band as holes. To treat holes as real quasiparticles, symmetrical to electrons in the conduction band, we consider that hole energy is the opposite of the energy

<figure><img src="image_3.png" /><figcaption>FIG. 2. Schematic of a hot carrier absorber in the parabolic band approximation and energy conventions used in this work. Carrier densities are represented in the case where holes are heavier than electrons. Note that the chemical potential of holes ($\mu_h$) is the opposite of the one of electrons in the valence band ($\mu_{e,V}$), such that $\Delta\mu = \mu_e + \mu_h = \mu_{e,C} - \mu_{e,V}$.</figcaption></figure>

of the corresponding electron in the valence band. This is represented schematically in Fig. 2, which defines the energy conventions used in this work.

Carrier diffusion is described through steady-state carrier conservation:

$$0 = \partial_t n_i = G_i - R_i - \nabla \cdot \mathbf{J}_{N,i} \quad (2)$$

where the subscript i denotes either electrons (e) or holes (h). $n_i$ is the concentration of carrier i, $G_i$ (resp. $R_i$) the generation (resp. recombination) rate and $J_{N,i}$ is the particle current. In the linear regime, this current writes [25, 27]

$$\mathbf{J}_{N,i} = \frac{\sigma_i}{q_i} \mathbf{E} - \frac{\sigma_i}{e^2} \nabla \mu_i - \frac{\sigma_i S_i}{q_i} \nabla T \quad (3)$$

where $\sigma_i$ is the electrical conductivity, $S_i$ is the Seebeck coefficient, $q_i = \pm e$ is the algebraic charge, while $e = 1.6 \times 10^{-19}$ C is the absolute elementary charge, $\mu_i$ is the chemical potential, and $\mathbf{E}$ is the electrical field.

The electrical field has a contribution from (i) the possible external bias applied to the cell and (ii) from local charge concentration trough Poisson equation. The latter contribution allows to describe the interaction between electrons and holes. In this study, we consider the limiting case of strong interaction: the ambipolar diffusion regime.

### B. Ambipolar diffusion in the non-degenerate regime

In ambipolar diffusion, the electrostatic attraction between electrons and holes is strong enough to bind both
populations. Therefore the electron and hole particle currents are equal.

$$J_{N,e} = J_{N,h} \qquad (4)$$

In addition, generation and recombination processes are symmetrical for electrons and holes, i.e. $G_e = G_h$ and $R_e = R_h$. Finally, at high injection levels, we can assume that doping is negligible as compared with photogenerated charge concentrations. The combination of these three effects implies that local electroneutrality must hold, such that $n_e(r) = n_h(r)$.

Note that the system investigated in this paper is a quantum well, in which the well and the barrier may exchange carriers. This may disrupt the electroneutrality assumption. However, it was shown that, for this particular structure, the band-edge mismatch between the well and the barrier is almost the same for electrons and holes [11]. In addition, at the high carrier densities reached in the following experiment ($n \sim 10^{18}$ cm$^{-3}$), a discrepancy of carrier density would be prevented by the creation of a strong transverse electrical field. Therefore we expect that electroneutrality holds in this case.

In the effective mass approximation, electroneutrality takes the following dimensionless form

$$ \begin{aligned} & m_e^{D/2} \int_0^\infty \frac{x^{D/2-1} dx}{1 + \exp\left(x + \frac{E_g/2 - \mu_e}{k_B T}\right)} \\ &= m_h^{D/2} \int_0^\infty \frac{x^{D/2-1} dx}{1 + \exp\left(x + \frac{E_g/2 - \mu_h}{k_B T}\right)} \end{aligned} \qquad (5) $$

Where $m_e$ (resp. $m_h$) is the effective mass of electrons (resp. holes), $D$ is the dimensionality of the system, $E_g$ is the bandgap and $k_B$ is Boltzmann constant. This relation can be written in differential form

$$\nabla\mu_h = g_1(\mu_e, \mu_h, T)\nabla\mu_e + g_2(\mu_e, \mu_h, T)\nabla T \qquad (6)$$

See Appendix D for details on the derivation of $g_1$ and $g_2$.

Using this simplification, one can evaluate the electrostatic field necessary to ensure that transport is ambipolar (Eq. 4) [24]

$$ \begin{aligned} E = & \frac{1}{e} \frac{g_1 \sigma_h - \sigma_e}{\sigma_e + \sigma_h} \nabla \mu_e \\ & + \left( \frac{\sigma_e S_e + \sigma_h S_h}{\sigma_e + \sigma_h} + \frac{\sigma_h}{\sigma_e + \sigma_h} \frac{g_2}{e} \right) \nabla T \end{aligned} \qquad (7) $$

Note that we would find a similar expression with $\nabla\mu_h$.

Injecting this expression in Eq. 2, we find that the ambipolar hot carrier current writes

$$J_{N,i} = - \frac{\sigma_{amb,i}}{e^2} \nabla \mu_i - \frac{\sigma_{amb,i} S_{amb,i}}{q_i} \nabla T \qquad (8)$$

where $\sigma_{amb,i}$ (resp. $S_{amb,i}$) is the effective ambipolar electrical conductivity (resp. Seebeck coefficient), defined as

$$ \left\{ \begin{aligned} \sigma_{amb,e} &= \frac{\sigma_e \sigma_h}{\sigma_e + \sigma_h} (1 + g_1), & \sigma_{amb,h} &= \frac{\sigma_e \sigma_h}{\sigma_e + \sigma_h} (1 + 1/g_1) \\ S_{amb,e} &= \frac{e(S_e - S_h) - g_2}{e(1 + g_1)}, & S_{amb,h} &= \frac{e(S_h - S_e) - g_2/g_1}{e(1 + 1/g_1)} \end{aligned} \right. \qquad (9) $$

This is the main theoretical result of this work, which does not depend on the specificity of the experimental conditions. It shows that in the context of hot carrier solar cells, a full description of the ambipolar transport resembles the usual thermoelectric equations with effective ambipolar coefficients.

Note that in our case, contrary to the textbook treatment of homogeneous thermoelectric transport [26], it is not possible to attribute one single ambipolar electrical conductivity nor one single ambipolar Seebeck coefficient to electrons and holes. This is a consequence of electroneutrality, which imposes that electron and hole chemical potentials differ by a temperature-dependent term (see Eq. 6).

### III. HYPERSPECTRAL IMAGING OF CARRIERS DISTRIBUTION

In order to measure transport coefficients defined in the previous section, we perform a hyperspectral PL measurement with a punctual illumination, as described in Fig. 1. This allows us to access temperature and QFLS profiles [28] required to compute the particle current.

#### A. Materials and methods

We use a 980 nm laser focused to a Gaussian profile of radius 1.6 µm at $2\sigma$ to generate hot carriers in a $In_{.78}Ga_{.22}As_{.81}P_{.19}/In_{.8}Ga_{.2}As_{.435}P_{.565}$ single quantum well (QW). Further details regarding the sample are published in ref. [11].

PL emitted from the surface of the sample is collected using a NIR hyperspectral imager equipped with a Mitutoyo APO NIR x50 objective lens (0.41 NA). It has a spatial resolution (sparrow criterion) of 1.7 µm and a spectral resolution of 2 nm. Taking advantage of the cylindrical symmetry of the problem, we integrate angularly to obtain PL spectra as a function of the distance from the laser spot (up until ~ 13 µm) for different laser central intensities ranging from $6.3 \times 10^3$ to $2.3 \times 10^5$ W.cm$^{-2}$. Our dataset is therefore composed of 270 PL spectra corresponding to hyperspectral acquisitions at 9 different excitation intensities and 30 different radii (from 0 to 13 µm) for each intensity.

Few PL spectra collected at different radii are reported in Fig. 3. The peak around 0.82 eV is attributed to the emission of the quantum well, while the peak at 1.05 eV is consistent with emission from the barrier [11]. The shape
<figure><img src="image_4.png" /><figcaption>FIG. 3. Angularly integrated PL spectra obtained at different radii for a single excitation intensity of $I_{exc} = 4.1 \times 10^4$ W.cm$^{-2}$. Pale envelope represents the dispersion of the spectra aggregated during integration. Dashed lines represent the result of a full fit (see Appendix A).</figcaption></figure>

of quantum well emission spectrum depends strongly on the distance to the laser, which is indicative of a stronger hot carrier effect and band filling effect in the center.

## B. Hot carrier spectroscopy

We probe local carrier distributions by analyzing the local PL emission. According to the Generalized Planck Law [29], PL spectrum is related to the carriers temperature and chemical potential as

$$I_{PL}(E) = \frac{2}{h^3 c^2} A(E, \Delta\mu, T) \frac{E^2}{\exp\left(\frac{E - \Delta\mu}{k_B T}\right) - 1} \quad (10)$$

where $I_{PL}$ is the emitted flux in s$^{-1}$.m$^{-2}$.sr$^{-1}$.J$^{-1}$, A is the sample absorptivity (which may include population dependent effects, see Appendix A), $\Delta\mu$ is the quasi-Fermi level splitting (QFLS), T is the carrier temperature, h is Planck constant, c the speed of light in vacuum.

Estimating carrier distribution from a PL spectrum requires absolute calibration of the setup, and a good knowledge of the absorptivity so as to perform a full fit of the spectra [11]. An example of this full fit analysis is shown in Fig. 3. Such advanced fitting methodologies are particularly critical in the context of hot carriers in nanostructures, where the strong excitation regime leads to a significant band filling [30, 31]. The details of the full fit model are presented in Appendix A and B.

The uncertainty on the determination of the absorptivity coefficient (see Appendix B) leads to an estimated $\pm 20$ meV uncertainty in the QFLS. Additionally, noise in the PL spectra and partial overfitting introduce a random uncertainty of approximately 1% for both variables.

To distinguish hot carriers from the laser-induced heating of the sample, we assess the lattice temperature by monitoring the PL peak according to Varshni's law [32]. The absence of peak shift indicates that the lattice remains at ambient temperature, while carriers can reach temperatures higher than 1000K (see Fig. 4).

## C. Carrier distribution profiles

From the measurement of $T(r)$ and $\Delta\mu(r)$, and owing to local electroneutrality, we deduce the local carrier density $n(r) = n_e = n_h$ and the corresponding chemical potentials $\mu_e(r)$ and $\mu_h(r)$. These measurements are all reported in Fig. 4. Note that carrier density and chemical potentials are computed using Fermi-Dirac distributions.

Fig. 4a. shows the spatial distribution of carrier temperature for increasing laser intensities. Temperature is shown to increase with increasing excitation intensity, as we expect from a hot carrier absorber [33] – except for a few incoherent spectra at highest intensity and close to the center. In this range of power, carrier temperature reaches more than 1000 K. It is also shown to decrease as we look further and further away from the center. At distances larger than 10 µm, carriers are thermalized with the lattice at room temperature regardless of the excitation intensity.

At high excitation intensities, carrier temperature remains above room temperature for typically 6 – 8 µm, much larger than the laser radius. Therefore we report hot carrier diffusion length in the order of few µm, several times the value previously reported in monolayers (~ 500 nm) [34, 35], in perovskites thin films (~ 230 nm) [36] and in nanowires (~ 300 nm) [37].

Carrier density profiles are presented in Fig. 4b. It is increasing with increasing excitation intensity, as absorbing more photons will promote more carriers in the bands. Additionally, carrier density decreases with increasing distance to the center. This is expected from a punctual illumination experiment, as carriers are photo-generated at the center, and then diffuse to further distances while recombining. Note that in exotic cases with strong enough thermoelectric transport, it is possible to observe a depletion of carrier density close to the center [18]. Such an effect did not occur in our experiment.

Electron (resp. hole) chemical potential profiles are presented in Fig. 4c (resp. Fig. 4d). At high excitation intensities, the electrons form a completely degenerate electron gas as their chemical potential is well above the conduction band ($E_C \approx 0.41$ eV). However, the hole distribution remains non degenerate due to the discrepancy in effective masses. Surprisingly, at high intensity, hole chemical potential decreases as we come closer to the center. This can be explained by the large increase of carrier temperature in this region. Therefore, despite the drop of chemical potential, carrier density is still increasing when we go closer to the center (see Fig. 4b).
<figure><img src="image_5.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>(d)</figcaption></figure>

<figure><img src="image_8.png" /></figure>

FIG. 4. In reading order: (a) estimated carrier temperature profile in the quantum well at all excitation intensities. (b) Carrier density estimated from PL measurements. Dashed line indicates room temperature $T_{amb} = 293$ K. (c) (resp. (d)) Chemical potential of electrons (resp. holes) in the quantum well. Dashed line represents the electron (resp. hole) band-edge. Dots represent the boundaries of region II defined in Fig. 1 and Sect. IV. Pale green region indicates the estimated errorbars (Sect. III B) for the highest excitation intensity. Similar errorbars are expected at all intensities but are not represented to enhance readability.

## IV. AMBIPOLAR TRANSPORT COEFFICIENTS DETERMINED BY PHOTOLUMINESCENCE

In the ambipolar configuration, electrons and holes diffuse at the same rate such that it is equivalent to study one or the other. In this section, we choose to study electrons, and we distinguish three different spatial regions (see Fig. 1).

In the central region (I), close to the laser beam, the generation term is significant in the continuity equation (Eq. 2). We estimate that this region spans until the laser intensity reaches 5 % of its maximal intensity, namely $r = 2.7$ µm. In a complex stack such as this quantum well, obtaining a precise estimate of the generation rate is not easy. Therefore we simply discard this region.

Far from the center (III) (typically 3.6–7.7 µm), carriers have relaxed towards the lattice temperature. In this region, concentration profile is determined by isothermal

diffusion. This allows us to measure the *electrical conductivity*.

In between these regions (II), carriers experience a significant temperature gradient. Therefore their transport is induced not only by the charge concentration gradient, but also by thermoelectric effects. This region grants access to *Seebeck coefficient*.

The boundaries of these three regions are represented for every excitation intensity in Fig. 4.

### A. Determination of the electrical conductivity

In region III, the continuity equation is simplified by the absence of generation and of temperature gradient. In this context, the only terms left in this equation are the isothermal diffusion term and the recombination term, and Eq. 2 becomes $R = \nabla \cdot (\sigma_{amb,e} e^2 \times \nabla \mu_e)$.

The electrical conductivity can be written in terms of
carrier mobilities $\bar{\mu}_i$ as $\sigma_{amb,e} = \gamma_e(1 + g_1)n$, where we introduce the parameter $\gamma_e = e\bar{\mu}_e\bar{\mu}_h/(\bar{\mu}_h - \bar{\mu}_e)$. By assuming that carrier mobilities are constant, and noticing that $g_1$ is slowly varying with the position, we can take the factor $\gamma_e(1 + g_1)$ out of the gradient in the equation above.

To measure the conductivity, we need an estimate of the recombination rate. In this study, carrier density is around $10^{18} \text{ cm}^{-3}$ (see Fig. 4b) such that we assume that recombination rate is dominated by Auger process $R = Cn^3$ (see Appendix C for details). Therefore, in the isothermal region, Eq. 2 becomes

$$ \frac{C}{\gamma_e} = \frac{1 + g_1}{n^3} \nabla \cdot \left( \frac{n}{e^2} \nabla \mu_i \right) \quad (11) $$

The determination of the Auger scattering rate $C$ is beyond the scope of this article [38]. Therefore we can only access the ratio $C/\gamma_e$ and estimate the electrical conductivity up to a multiplicative constant. We find that the ratio $C/\gamma_e$ decreases with increasing excitation intensity, and has typical order of magnitude $10^{-22} - 10^{-20} \text{ m}^4 \cdot \text{V} \cdot \text{s}^{-1} \cdot \text{A}^{-1}$. This is consistent with typical values of the Auger coefficient in InGaAsP quantum wells $C \sim 10^{-27} \text{ cm}^6 \cdot \text{s}^{-1}$ [39] and mobility $\bar{\mu}_e \sim 10^{3-4} \text{ cm}^2 \cdot (\text{Vs})^{-1}$ [38].

Note that, due to noise on the determination of the right-hand side of Eq. 11, fitted values of $C/\gamma_e$ have a typical uncertainty of $\pm 15\%$.

## B. Determination of the Seebeck coefficient

a. Computation We now turn to the analysis of region II. We assume that the variations of $S_{amb,e}$ are sufficiently small, such that the contribution of $\nabla S_{amb,e}$ to transport is negligible. Under this assumption, Eq. 2 can be rewritten to express the ambipolar Seebeck coefficient from measured quantities as:

$$ S_{amb,e} \approx - \frac{\frac{C}{\gamma_e} n^3 + \frac{1 + g_1}{q_e^2} \nabla (n \cdot \nabla \mu_e)}{\frac{1}{q_e} \nabla (n \cdot \nabla T)} \quad (12) $$

where the quantity $C/\gamma_e$ has been estimated from Eq. 11.

To compute uncertainty on this ambipolar Seebeck coefficient, we consider the following error sources: (i) random noise on the temperature profiles, (ii) random shift of the QFLS profiles as explained in Sect. III B and (iii) an uncertainty on the estimate of $C/\gamma_e$ as explained in Sect. IV A.

b. Result and qualitative discussion Experimental determination of the ambipolar Seebeck coefficient at all the relevant excitation conditions is presented in Fig. 5. The ambipolar Seebeck coefficient takes values between -4 and 1 $k_B/e$, which is the typical order of magnitude of Seebeck coefficients in QWs [40]. Points with an uncertainty larger than $4k_B/e$ are not represented.

<figure><img src="image_9.png" /><figcaption>FIG. 5. $S_{amb,e}$ obtained from Eq. 12 plotted versus the corresponding reduced QFLS $\eta = \frac{\Delta\mu - E_g}{k_B T}$ deduced from PL measurements. Grey region indicates the values modelled from Boltzmann transport equation (see Eq. 13) with $-2 \le r_0 \le 0$.</figcaption></figure>

Such a large value is surprising however for an ambipolar Seebeck coefficient. That is because the usual derivation of ambipolar Seebeck coefficient is done in dark conditions and leads to $S_{amb} = (\sigma_e S_e + \sigma_h S_h)/(\sigma_e + \sigma_h)$ [26]. In such case, the Seebeck coefficients of electrons and holes compensate each other. However, this article treats the case of strong photogeneration, and therefore our ambipolar Seebeck coefficient is different in nature. In our case, Seebeck coefficients of electrons and holes add up, as shown in Eq. 9. Therefore, we expect that our ambipolar Seebeck coefficient should be of the same order of magnitude than the one of electrons and holes.

c. Comparison with Boltzmann transport equation For a more quantitative discussion, we make use of Boltzmann transport equation in the relaxation time approximation. This theory provides a model for Seebeck coefficient of a gas of particles with charge $q_i$ [41]

$$ S_i = - \frac{1}{q_i T} \left[ \mu_i - \frac{E_g}{2} - \frac{\int_0^\infty D_i(\epsilon) \epsilon^{r_0+2} \frac{\partial f_i}{\partial \epsilon} d\epsilon}{\int_0^\infty D_i(\epsilon) \epsilon^{r_0+1} \frac{\partial f_i}{\partial \epsilon} d\epsilon} \right] \quad (13) $$

where $\mu_i - E_g/2$ is the difference between the chemical potential of carrier $i$ and its band-edge (see Fig. 2), $\epsilon = E - E_g/2$ is the kinetic energy, $D_i$ is the density of state, $f_i$ is the occupation function (Fermi-Dirac in this case), and $\tau(\epsilon) \propto \epsilon^{r_0}$ is a standard parametrization of the relaxation time. We assume that $r_0$ is the same for electrons and holes. From this formula, the ambipolar Seebeck coefficient can be computed with Eq. 9. Note that in this approach, the ambipolar Seebeck coefficient $S_{amb,e}$ is parametrized by one single free parameter $r_0$.

Because of the dispersion on our data, it is not possible to determine an exact value for $r_0$. However, our measured values of the ambipolar Seebeck coefficient are
consistent with $-2 \le r_0 \le 0$ (see Fig. 5). This confirms that our method allows to measure meaningful ambipolar Seebeck coefficients.

The comparison of our results with Boltzmann transport equation also allows us to study the asymptotic behavior of the ambipolar Seebeck coefficient. In the non-degenerate regime $\Delta\mu \ll E_g - k_B T$ (i.e. $\eta \ll -1$), Eq. 13 predicts that $S_{amb,e} = k_B/2e \times [\eta - (r_0 + 2) + \ln(m_h/m_e)]$ is linear in the reduced QFLS with a universal slope $k_B/2e$ [26]. This means in particular that the ambipolar Seebeck coefficient should be larger at lower $\eta$, hence for lower excitation intensity, as is the case with photo-Seebeck [23]. However, if the excitation is not high enough, carriers will stay cold and out optical method will not permit to measure the Seebeck coefficient. In this study for instance, acquisitions with intensity $I_{exc} < 2 \times 10^4$ W.cm$^{-2}$ did not exhibit large enough temperature variations to allow for a precise determination of $S_{amb,e}$, such that they are not reported in Fig. 5.

In the degenerate regime, Eq. 13 predicts that $S_i = \pi^2 k_B (r_0 + 1)/3q_i \eta$ which tends to 0 as $\eta \to \infty$. In this regime, $S_{amb,e} \simeq -g_2/(1 + g_1)$ has no clear asymptotic behavior (see Appendix Fig. D.9).

Note also that the ambipolar Seebeck coefficient depends directly on the electron and hole effective masses ratio $m_h/m_e$ through $g_1$ and $g_2$ (see Appendix Sect. D and Fig. D.9). In order to achieve a large negative $S_{amb,e}$, we should seek materials with $m_h/m_e$ as low as possible. However, we should keep in mind that a large negative $S$ will result in a small positive $S_{amb,h}$ and vice-versa, as shown on Eq. 9 and Appendix Fig. D.9.

## V. CONCLUSION

This article lays the theoretical framework necessary to study thermoelectric ambipolar transport of photo-generated hot carriers. We described this transport with effective ambipolar transport coefficients valid even in the degenerate case. We proposed a purely optical method to measure these coefficients. It relies on the use of absolutely calibrated hyperspectral photoluminescence. We validated this method by determining the ambipolar transport coefficients – and in particular Seebeck coefficient – in an InGaAsP quantum well. Results were shown to be consistent with Boltzmann transport equation in the relaxation time approximation.

This study opens new perspectives to understand the properties of non-ideal hot carrier solar cells, where trans-

port phenomena may limit device efficiency. Specifically, it enables the description of inhomogeneous hot carrier solar cells, where carrier transport is influenced by inhomogeneous hot carrier distributions.

## ACKNOWLEDGMENTS

The authors would like to acknowledge the French National Research Agency, specifically the grant ANR ICE-MAN 19-CE05-0019-01, as well as PEARL program of IPVF for providing funds for this research. We especially thank FOTON Institute for providing the sample. The authors also thank Christophe Goupil and Stanko Tomic for valuable discussions over the course of the project.

## Appendix A: Description of the photoluminescence model

In this appendix, we describe the model used to describe and fit our photoluminescence (PL) spectra. In the Generalized Planck Law, most of the difficulties in term of modelling come from the absorptivity $A$, which depends on many variables.

### 1. Description of a single layer

a. Absorptivity and absorption coefficient If we neglect reflexions at the different interfaces, the absorptivity $A$ of a single layer of thickness $d$ can be linked to its coefficient of absorption $\alpha$ by Beer-Lambert law

$$A = (1 - R(E))(1 - \exp(-\alpha d)) \quad (A1)$$

The absorption coefficient itself depends both on material parameters (band structure, defect densities, etc) and on carrier distribution through the effect commonly called "band-filling" (or "Moss-Burnstein effect") [42]. Other effects such as bandgap-renormalization and changes in reflectivity are disregarded here. Therefore, we write that

$$\alpha(E) = \alpha_0(E) \times BF(E, \Delta\mu, T) \quad (A2)$$

where $\alpha_0$ is the absorptivity coefficient measured in the dark, and $BF = 1 - f_h - f_e$ is the contribution of band-filling. Here we wrote explicitely the dependence of $BF$ with the carrier distribution, but it also depends on the band structure of the material. In an effective mass model including only contributions from the $\Gamma$ valley, one can show that [11]

$$BF(E, \Delta\mu, T) = \frac{\sinh\left(\frac{E - \Delta\mu}{2k_B T}\right)}{\cosh\left(\frac{E - \Delta\mu}{2k_B T}\right) + \cosh\left[\frac{m_h - m_e}{m_h + m_e}\frac{E - E_g}{2k_B T} - \frac{D}{4}\ln\frac{N_V}{N_C}\right]} \quad (A3)$$
where $E_g$ is the bandgap of the considered material, $m_i$ is the effective mass of particle $i$, and $D$ is the dimensionality of the considered layer. In our case $D = 2$ for the quantum well and $D = 3$ for the barrier.

Because we measured QFLS values close to the bandgap for both the quantum well and the barriers, we included this band filling contribution in the absorption of both layers. We consider that, in the quantum well, $m_{e,w} = 0.042 m_0$, $m_{h,w} = 0.41 m_0$, while in the barrier, $m_{e,b} = 0.061 m_0$ and $m_{h,b} = 0.54 m_0$ [11]. $m_0 = 9.1 \times 10^{-31}$ kg is the mass of one electron.

b. **Quantum well emission** Quantum wells have step-like density of states and therefore step-like absorptivity. In our case, two levels seem sufficient to explain our PL spectra. It's been shown that exciton states may also have a significant contribution even at room temperature [43, 44]. Therefore, we model our quantum well absorptivity with 9 parameters ($A_x, E_x, \Gamma_x, A_1, R_y, \Gamma_1, A_2, E_2, \Gamma_2$):

$$ \alpha_0^{QW} = A_x \exp \left[ - \left( \frac{E - E_x}{\Gamma_x} \right)^2 \right] + A_1 \frac{1}{1 + \exp \left[ - \left( \frac{E - E_1}{\Gamma_1} \right) \right]} \frac{2}{1 + \exp \left( -2\pi \sqrt{\frac{R_y}{|E - E_1|}} \right)} + A_2 \frac{1}{1 + \exp \left[ - \left( \frac{E - E_2}{\Gamma_2} \right) \right]} \quad (A4) $$

where $E_i$ denote the position of the energy level $i$, $A_i$ the amplitude of the transition and $\Gamma_i$ is a broadening parameter. $R_y = E_1 - E_x$ is the Rydberg binding energy of excitons. Note that we took into account the Sommerfeld enhancement factor in the emission of the first level.

c. **Barrier emission** For the absorptivity of the barrier layer, we use a phenomenological step-like absorption

$$ \alpha_0^B = A_b \frac{1}{1 + \exp \left( - \frac{E - E_b}{\Gamma_b} \right)} \quad (A5) $$

Note that it is not necessary to use a more advanced model for the barrier because we are solely interested in the carrier distribution in the quantum well, which is only weakly affected by the model used for the barrier emission.

## 2. Emission of a multilayer

In a multilayer, light emitted in a deep layer may be partially reabsorbed by shallower layers. By denoting $A_w$ (resp. $A_b$) the absorptivity of the quantum well (resp. the barrier), we can write that [11]

$$ I_{PL}(E) = \frac{2E^2}{h^3 c^2} \left\{ \frac{A_w(1 - A_b)}{\exp\left(\frac{E - \Delta\mu_w}{k_B T_w}\right) - 1} + \frac{A_b(1 - A_w)(1 - A_b) + A_b}{\exp\left(\frac{E - \Delta\mu_b}{k_B T_b}\right) - 1} \right\} \quad (A6) $$

## Appendix B: Fitting methodology

In total, our full fit model includes sixteen parameters. Four are related to the carrier distribution in the well ($T_w, \Delta\mu_w$) and in the barrier ($T_b, \Delta\mu_b$), and twelve are related to the material ($A_x, E_x, \Gamma_x, A_1, R_y, \Gamma_1, A_2, E_2, \Gamma_2, A_b, E_b, \Gamma_b$). Those sixteen parameters being partially redundant, we need a strategy to reduce the risk of overfitting our data and obtain meaningful values for the carrier distribution parameters that we are most interested in.

For the band-to-band transition, some of the material parameters remain constant when changing the excitation intensity. This is the case of the absorption amplitudes ($A_1, A_2, A_b$) and of the energies of optical transition ($E_x, R_y, E_2, E_b$). However, the linewidths ($\Gamma_x, \Gamma_1, \Gamma_2, \Gamma_b$) are power-dependent because it depends on the phonon density [31]. Similarly, the contribution of the excitonic transition $A_x$ can be screened by increasing carrier density [43].

External quantum efficiency (EQE) measurement can be used to determine the low-power absorptivity $A = (1 - R)(1 - \exp(-\alpha_0 d))$. Indeed, assuming that all photogenerated carriers are extracted in the contacts, the EQE spectrum equals the absorptivity.

<figure><img src="image_10.png" /><figcaption>FIG. B.6. EQE spectrum obtained at room temperature, along with a fit using the absorptivity model described in the sections above.</figcaption></figure>
Fig. B.6 shows the EQE spectrum measured with a ThermoFisher FTIR-FTPS using a 32 cm⁻¹ resolution. It was fitted against an absorptivity model derived from Eq. A6, assuming that $T_w = T_b = 293$ K and $\Delta\mu_w = \Delta\mu_b \ll E_x$. As explained above, seven of the twelve fitted parameters are material dependent only and should not depend on the excitation condition. Table I reports the fitted value of those parameters, which were used to model all PL spectra.

<table><caption>TABLE I. Absorption parameters determined by EQE measurement. Absorption coefficients are in m⁻¹ while energies are in eV. These values were used to fit all PL spectra in this study.</caption><thead><tr><th>Parameter</th><th><i>E</i><sub>x</sub></th><th><i>A</i><sub>1</sub></th><th><i>E</i><sub>1</sub></th><th><i>A</i><sub>2</sub></th><th><i>E</i><sub>2</sub></th><th><i>A</i><sub>b</sub></th><th><i>E</i><sub>b</sub></th></tr></thead><tbody><tr><td>Value</td><td>0.823</td><td>5.4 10<sup>5</sup></td><td>0.808</td><td>7.4 10<sup>5</sup></td><td>0.890</td><td>8.5 10<sup>5</sup></td><td>1.035</td></tr></tbody></table>

Five more power-dependent parameters were necessary to reproduce the EQE spectrum, and are reported in Table II. These parameters depend on excitation intensity and on position, such that they were kept as fitting parameters for the PL spectra [11].

<table><caption>TABLE II. Power-dependent absorption parameters necessary to reproduce the EQE spectrum. Absorption coefficients are in m⁻¹ while spectral bandwidth are in eV. These parameters were fitted again for each and every PL spectrum presented in the main text, as they depend on carrier population.</caption><thead><tr><th>Parameter</th><th><i>A</i><sub>x</sub></th><th><i>Γ</i><sub>x</sub></th><th><i>Γ</i><sub>1</sub></th><th><i>Γ</i><sub>2</sub></th><th><i>Γ</i><sub>b</sub></th></tr></thead><tbody><tr><td>Value</td><td>6.5 10<sup>5</sup></td><td>0.007</td><td>0.005</td><td>0.011</td><td>0.013</td></tr></tbody></table>

To summarize, fitting the EQE spectrum allowed us to determine 7 parameters prior to fitting PL spectra. Only nine parameters were left to fit for each PL spectrum ($T_w$, $\Delta\mu_w$, $T_b$, $\Delta\mu_b$, $A_x$, $\Gamma_x$, $\Gamma_1$, $\Gamma_2$, $\Gamma_b$). We used a differential evolution algorithm [45] to search for the global minimum in this parameters space with 9 dimensions. We used an objective function in log scale, as our PL spectra cover several orders of magnitude.

## Appendix C: Assessing the strength of Auger recombinations

In this appendix, we explain in details why Auger recombination is considered dominant in our study.

First, we measure high carrier densities, $n \sim 10^{18}$ cm⁻³, even far from the center (see Fig. 3d). Assuming an Auger coefficient of $C \sim 10^{-27}$ [39], we get that Auger recombination rate $Cn^3 \sim 10^{27}$ cm³·s⁻¹, which is comparable or higher than monomolecular recombination rate $R = n/\tau$ assuming a carrier lifetime of $\tau \sim 1$ ns.

Second, by modeling transport far from the center, we show that the measured profiles of carrier density and chemical potential can be best explained by Auger recombinations. Far from the center, generation is negligible,

<figure><img src="image_11.png" /><figcaption>FIG. C.7. Contribution of the different terms in transport equation (Eq. 2) in the isothermal regime, assuming various dominant recombinations mechanisms (monomolecular, bimolecular or Auger). Far from the center, carrier conservation imposes that yellow and green line should compensate each other. Profiles are established based on a high intensity acquisition ($I_{exc} = 1.1 \times 10^5$ W·cm⁻²).</figcaption></figure>

carriers are thermalized such that the recombination rate must balance isothermal particle current: $R = \frac{1}{e^2} \nabla \cdot (\sigma \nabla \mu_e)$ (see main text Sect. IV.A). Fig. C compares these two quantities using three different scaling for the recombination rate: monomolecular ($R = n/\tau$), bimolecular ($R = Bn^2$) and Auger ($R = Cn^3$). One sees that Auger recombination is the one that compensates isothermal particle current the best at large distances.

## Appendix D: Details on electroneutrality and on the derivation of $g_1$ and $g_2$

In the effective mass approximation, assuming a typical density of states $D_i(\epsilon) \propto m_i^{D/2} \epsilon^{D/2-1}$, electroneutrality takes the following form

$$m_e^{D/2} \int_0^\infty h(x, \mu_e, T) dx = m_h^{D/2} \int_0^\infty h(x, \mu_h, T) dx \quad (D1)$$

where we introduce the function

$$h(x, \mu, T) = x^{D/2-1} / \left( 1 + \exp \left( x + \frac{E_g/2 - \mu}{k_B T} \right) \right) \quad (D2)$$

Electroneutrality can be rewritten in the following differential form

$$\nabla \mu_h = g_1(\mu_e, \mu_h, T) \nabla \mu_e + g_2(\mu_e, \mu_h, T) \nabla T \quad (D3)$$

where
$$ \left\{ \begin{aligned} g_1(\mu_e, \mu_h, T) &= \left(\frac{m_e}{m_h}\right)^{D/2} \left(\int_0^\infty \frac{\partial h}{\partial \mu}(x, \mu_e, T)dx\right) / \left(\int_0^\infty \frac{\partial h}{\partial \mu}(x, \mu_h, T)dx\right) \\ g_2(\mu_e, \mu_h, T) &= \left[m_e^{D/2} \int_0^\infty \frac{\partial h}{\partial T}(x, \mu_e, T)dx - m_h^{D/2} \int_0^\infty \frac{\partial h}{\partial T}(x, \mu_h, T)dx\right] / \left[m_h^{D/2} \int_0^\infty \frac{\partial h}{\partial \mu}(x, \mu_h, T)dx\right] \end{aligned} \right. \quad (D4) $$

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>FIG. D.8. (a) $g_1$ and (b) $g_2$ computed as a function of $\eta = (\Delta\mu - E_g)/k_B T$ for different mass mismatch $m_e/m_h$ and assuming $T = 300$ K. Dashed lines represent the asymptotes in the non-degenerate case ($\eta \ll -1$).</figcaption></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>FIG. D.9. (a) $S_{amb,e}$ and (b) $S_{amb,h}$ computed from Boltzmann transport equation for different mass mismatch $m_e/m_h$. We assume $T = 300$ K and $r_0 = -2$.</figcaption></figure>

In the general case, these equations can only be computed numerically. Since PL measurements allow us to measure $\mu_e$, $\mu_h$ and $T$ (see main text), we can estimate these functions for all relevant experimental conditions. In the simpler case of a non-degenerate system, $h(x, \mu, T) \approx x^{D/2-1} \exp(-x) \exp(\frac{\mu-E_g}{k_B T})$ and electroneutrality takes the form

$$ \mu_h = \mu_e - k_B T \frac{D}{2} \ln \left( \frac{m_h}{m_e} \right) \quad (D5) $$

such that

$$ \left\{ \begin{aligned} g_1 &= 1 \\ g_2 &= -k_B \frac{D}{2} \ln \left( \frac{m_h}{m_e} \right) \end{aligned} \right. \quad (D6) $$

Fig. D.8 represents the values of $g_1$ and $g_2$ for varying reduced QFLS $\eta = (\Delta\mu - E_g)/k_B T$ and different value of the mass mismatch $m_h/m_e$. First, in the case $m_e = m_h$, we recover that $\mu_e = \mu_h$, i.e. $g_1 = 1$ and $g_2 = 0$. Otherwise, it appears that, for all values of $\eta$, $g_2$ decreases with increasing mass mismatch $m_h/m_e$. Note that in the non-degenerate regime ($\eta \ll -1$), $\mu_e = \mu_h + k_B T(D/2) \ln(m_h/m_e)$ [46], such that $g_1 = 1$ and $g_2 = -k_B(D/2) \ln(m_h/m_e)$.
From here, it is possible to compute the expected value of $S_{amb,e}$ and $S_{amb,h}$ as predicted from Boltzmann trans-

port equation (by combining Eqs. 9, 13 and S.10). These values are reported in Fig. D.9. Notice how both $S_{amb,e}$ and $S_{amb,h}$ increase with the mass mismatch $m_h/m_e$.

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