# Temperature-induced absolute energy shift of the electronic spectrum in HgTe nanocrystals

Tommaso Gemo, Dario Mastrippolito, Mariarosa Cavallo, Giorgia Strobbia, Albin Colle, Marco Paye, Jiho Roh, Adrien Khalili, Clement Gureghian, Erwan Bossavit, et al.

► To cite this version:

Tommaso Gemo, Dario Mastrippolito, Mariarosa Cavallo, Giorgia Strobbia, Albin Colle, et al. Temperature-induced absolute energy shift of the electronic spectrum in HgTe nanocrystals. Applied Physics Letters, 2025, 127 (26), pp.263302. 10.1063/5.0308025. hal-05440715

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Temperature-induced Absolute Energy Shift of the Electronic Spectrum in
HgTe Nanocrystals

Tommaso Gemo<fnref n="1" />,<fnref n="2" />$, Dario Mastrippolito<fnref n="1" />,<fnref n="3" />$, Mariarosa Cavallo<fnref n="1" />, Giorgia Strobbia<fnref n="1" />, Albin Colle<fnref n="1" />,
Marco Paye<fnref n="1" />, Jiho Roh<fnref n="1" />, Adrien Khalili<fnref n="1" />, Clement Gureghian<fnref n="1" />, Erwan Bossavit<fnref n="1" />,<fnref n="3" />, Sandrine
Ithurria<fnref n="4" />, Yoann Prado<fnref n="1" />, Sébastien Sauvage<fnref n="5" />, Mathieu G. Silly<fnref n="3" />, Nicolas Péré-Laperne<fnref n="2" />, Debora
Pierucci<fnref n="1" />, Emmanuel Lhuillier<fnref n="1" />*

**Abstract:** Mercury telluride (HgTe) nanocrystals (NCs) are promising materials for infrared photodetection due to their size-tunable band edges and compatibility with colloidal synthesis. While optical studies have established the temperature dependence of their relative band gaps, the evolution of their absolute electronic spectrum with cooling remains poorly understood, despite its critical role in the device operation. In this work, we investigate *in situ* the temperature-induced absolute energy shift of the electronic states of HgTe NCs with varying band edge energies, using X-ray photoemission spectroscopy. We observe a systematic and reversible rigid shift of core levels toward lower binding energies upon cooling, reaching approximately 80 meV per 100 K, four times larger than the corresponding optical band edge shift (≈20 meV). This behavior is consistent across different NC sizes, making the impact more dramatic for the narrower band gap NCs. Such absolute energy shifts can significantly alter carrier densities and interfacial band alignments, potentially creating Schottky barriers and reducing extraction efficiency in photodiode architectures. Our findings highlight the necessity of accounting for temperature-induced absolute energy shifts in the design of next-generation HgTe NC-based infrared detectors.

**Keywords:** HgTe, nanocrystals, temperature, X-ray photoemission, electronic structure.

$\$ these two authors have equal contributions

*To whom correspondence should be sent: el@insp.upmc.fr

<footnote n="1">¹ Sorbonne Université, CNRS, Institut des NanoSciences de Paris, 4 place Jussieu, 75005 Paris,
France.</footnote>

<footnote n="2">² LYNRED, Actipole - CS 10021, 364 route de Valence, 38113 Veurey-Voroize, France</footnote>

<footnote n="3">³Synchrotron SOLEIL, L'Orme des Merisiers, Départementale 128, 91190 Saint-Aubin, France.</footnote>

<footnote n="4">⁴Laboratoire de Physique et d'Etude des Matériaux, ESPCI, PSL Research University, Sorbonne Université, CNRS, 10 rue Vauquelin, 75005 Paris, France.</footnote>

<footnote n="5">⁵Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, C2N, Palaiseau 2110, France.</footnote>
Semiconductors, through their strong interaction with light and their ability to be electrically driven, are central building blocks of optoelectronics. In the visible range, despite its indirect band gap, which leads to weak absorption and prevents efficient light emission, silicon plays a key role in CMOS sensors and solar cells. The massive integration of Si-based technology has significantly advanced the understanding of this material. Its structural and electronic parameters are known with great accuracy, enabling modelling of the electronic structure across the entire Brillouin zone using 30-band *k*·*p* methods.¹⁻³ When it comes to infrared wavelengths, silicon, with its 1.12 eV band gap, becomes ineffective, and alternatives must be used.

Among the most mature platforms, HgTe is highly valued because of its ability to cover the entire infrared range, either through alloying⁴ or quantum confinement.⁵,⁶ This spectral tunability is enabled by the specific electronic structure of bulk HgTe (see **Figure 1a**). Compared to other II–VI semiconductors, the heavier atomic mass increases the spin–orbit coupling and generates an inverted band ordering. As a result, the Γ<sub>6</sub> symmetry band is not the conduction band but lies deep in the valence band. In the literature, the negative band gap of HgTe (ranging from –0.15 eV at room temperature to –0.3 eV at 4 K, see **Figure 1e**) describes how far the Γ<sub>6</sub> band is located below the Fermi energy. The Γ<sub>8</sub> bands, which usually act as valence bands (heavy hole, HH, and light hole, LH), now lie closest to the Fermi level and become the optically active bands. HgTe is also a semimetal (i.e., a zero-band gap semiconductor) because the LH and HH bands touch at the Brillouin zone center. Optical transitions in HgTe occur between the HH and LH bands. Under quantum confinement, the transition shifts from the Γ point to a larger wavevector given by π/L, with L being the confinement length. The curvature of the HH and LH bands is highly asymmetric (unlike PbS, where they are similar). The HH band has an effective mass close to that of the free electron, while the LH band disperses much more strongly. This results in a conduction band effective mass typically 2 orders of magnitude lighter than that of the valence band.⁷ For bulk and thin films, this yields high electron mobility (>10⁴ cm²·V⁻¹·s⁻¹ range at room temperature⁸,⁹). In quantum-confined HgTe, quantization produces a stronger confinement energy for electrons than for holes. Combined with the absence of a bulk band gap and the large dielectric constant of the material (ε<sub>r</sub> = 10–20)¹⁰, which weakens Coulomb corrections, the energy of an optical transition in HgTe quantum dots is dominated by electron confinement energy. As shown by the dotted line in **Figure 1a**, it is also worth noting that the LH band dispersion is extremely non-parabolic and significantly differs from a dotted line parabolic dispersion at only ≈1% of the Brillouin zone, while its dispersion is mostly linear across the most relevant wavevector range.⁷ Figure S1 quantitatively illustrates this big difference both near the Brillouin zone center and farther away.

Colloidal nanocrystals (NCs) provide a strategy to grow HgTe with quantum confinement.⁵,⁶,¹¹ Over the past decade, they have emerged as a promising platform for the development of infrared sensors and imagers. In nanoparticle form, the electronic spectrum results from the interplay of bulk electronic structure, quantum confinement, and surface chemistry, making their spectra more complex. However, advanced integration cannot be achieved without a thorough understanding of the electronic structure. The first step in this direction is the choice of particle size, which, through quantum confinement, determines the band edge energy. Here, sizes are chosen to be large enough that their band gap is clearly below that of silicon and rather matches the range covered by InGaAs technology and mid-IR detectors for the narrowest band gap. HgTe, by tuning the particle size in the 6–12 nm range, the band edge can be adjusted from the short-wave infrared (SWIR, 1.8 µm cut-off) to the extended SWIR (2.5 µm cut-off) and up to the mid-wave infrared (MWIR, 3 µm cut-off) for room-temperature absorption, as shown in **Figure 1b**. The HgTe NCs are grown using the procedure given in ref.¹² and will be later called by their room temperature band edge given in wavenumber. For example, HgTe 4k refers to HgTe NCs with a 4000 cm⁻¹ (2.5 µm or 0.5 eV) band edge.
<figure><img src="image_2.png" /><figcaption>**Figure 1** Electronic structure of HgTe and thermal dependence of the optical spectrum. a. Electronic structure of bulk HgTe at 300 K for states around the Fermi level using a 14-band k.p formalism. The dotted line is the effective mass parabolic dispersion. b. Absorption spectra of HgTe NCs with three different sizes associated with HgTe 6k, 4k, and 3k studied throughout the text. The peak at 0.36 eV corresponds to the C-H resonance from ligands. c. Absorption spectra from HgTe 4k at various temperatures showing a redshift of the band edge as the temperature is reduced. d. Band edge energy normalized by its low temperature (10 K) value for the three sizes of HgTe NCs under consideration. e. Electronic structure of bulk HgTe at 300 K and 0 K. Data from parts a, d, and e are modified, with permission from Ref. 7. Copyright 2020 American Chemical Society.</figcaption></figure>

For infrared sensors, an important consideration is the operating temperature. As the targeted cut-off wavelength increases, the band edge energy decreases, and thus photoactivation of carriers to generate photocurrent increasingly competes with phonon activation. As a result, the operating temperature decreases with increasing wavelength. For example, room temperature is still suitable for SWIR sensing, but in the MWIR range, cryogenic conditions (80–200 K, depending on the cut-off and performance targets) remain dominant, though progress toward room temperature operation have been obtained recently.13–15 In a photoconductive device, the current density is given by Eq. (1)16:
$$J=e.n(T).\mu(T).F \quad (1)$$

where e is the elementary charge, n(T) the carrier density, μ(T) the carrier mobility, and the applied electric field. The carrier density is the sum of two terms:

$$n(T)=n_{\text{dark}}(T)+n_{\text{light}}(T) \quad (2)$$

$n_{\text{dark}}(T)$ is the density at equilibrium and is strongly thermally activated. For NCs, the activation energy usually ranges $E_{\text{BE}}/2^{17}$ to $E_{\text{BE}}/10$, where $E_{\text{BE}}$ is the optical band edge, which depends on the specific material, surface chemistry, and native doping. This typically corresponds to an activation energy in the 100-200 meV range (Figure S3). Reducing $n_{\text{dark}}(T)$ is the main motivation for cooling. $n_{\text{light}}(T)$ is the light-induced carrier density, which depends on the material's absorption cross section and the photocarrier lifetime. This term generally shows a much weaker thermal dependence, driven mainly through the temperature dependence of the carrier lifetime.¹⁴

In NCs, charge transport occurs via hopping conduction, which introduces a thermally activated behavior of the mobility. The associated activation energy is usually much smaller (on the order of tens of meV) than that of the carrier density. Nevertheless, it also influences the photocurrent. Consequently, an optimal operating temperature exists that minimizes the dark current without significantly degrading the photocurrent. As mentioned above.

$$n_{\text{dark}}(T) \propto \exp(-E_{\text{BE}}/[\gamma.k_{\text{b}}.T]) \quad (3)$$

with $k_{\text{b}}$ the Boltzmann constant and $\gamma$ a material-dependent term, but typically in the 2-10 range. In such an expression, the band gap is assumed to remain constant; however, in practice, it also shifts with temperature, as shown in **Figure 1c**. In bulk HgTe, cooling leads to a redshift.¹⁸,¹⁹ This behavior is also observed in the case of NCs⁷,²⁰ (**Figure 1c**), but the shift depends on the particle size and can even flip sign for the most confined form of HgTe.⁷ A misleading interpretation will be to assume that lattice contraction upon cooling leads to a change in NC size and therefore to quantum confinement. Such an effect appears negligible.²⁰,²¹ The thermal dependence of the optical energy transition actually results from the thermal dependence of the term $E_g$ (the $\Gamma_6-\Gamma_8$ gap, see **Figure 1e**), which results from electron-phonon coupling. With cooling $E_g$ increases, which affects the curvature of the LH band through the coupling term $E_p^{22}$ (also called Kane parameter). In other words, as the temperature changes, the effective mass is affected (**Figure 1e**), and this tunes the electron confinement energy and, therefore, the optical band gap magnitude. Since the optical probing only ascertains the difference of energy between the bands, it remains unclear how a given band shifts in absolute value, which is the focus of this paper. This dependence of the optical band gap generates a refinement to Eq.(3), since the band edge itself depends on the temperature: $n_{\text{dark}}(T) \propto \exp(-E_{\text{BE}}(T)/[\gamma.k_{\text{b}}.T])$.

Optical measurements are well-suited to capture changes in the energy difference between bands but miss possible variations in absolute energy. However, absolute band alignment is critical. The best-performing HgTe NC-based detection devices rely on photodiode stacks.²³⁻²⁸ In such devices, the HgTe NC absorbing layer is surrounded by charge transport layers. Their role is to selectively extract one type of carrier through resonant band alignment with the desired band, while maintaining a significant offset with the other band to act as a tunnel barrier, preventing the transfer of carriers through their limited transparency. The design of these diodes depends on accurately determining the energy of each band on an absolute energy scale—either relative to vacuum, as in photoemission measurements, or relative to a reference electrode potential, as in electrochemical studies. Measurements are generally performed at room temperature and are often assumed to remain only marginally affected under operating conditions. However, this assumption must be verified, since potential shifts in absolute energy can induce Schottky barriers that reduce photocarrier extraction efficiency or alter carrier density profiles. Addressing this issue requires in
situ measurements, which cannot be performed with electrochemistry at cryogenic temperatures, as electrolytes freeze and lose their ability to control the Fermi level. For this reason, we focus here on *in situ* X-ray photoemission spectroscopy (XPS) conducted across a temperature range relevant to infrared sensor operation with HgTe NCs.

To initiate this study, we use a series of HgTe NCs, as previously introduced, which present a band edge in the SWIR, extended SWIR, and MWIR. The electronic spectrum of these NCs is studied using XPS (PHI GENESIS setup, source Al Kα at 1486.6 eV with 100 μm spot size), in which a high-energy photon source (i.e., far above the material's work function) is shone on the sample and promotes an electron out of the sample. Its energy is measured with an electron analyzer, which enables determination of its binding energy through energy conservation law. The binding energy provides direct evidence on the semiconductor doping and chemical environment. To conduct such a measurement, the samples are deposited on a gold-coated silicon substrate and are ligand-exchanged. The procedure presents a dual benefit: (i) by increasing the film conductance, we reduce the risk of photo-induced shift of the photoemission spectrum; moreover, (ii) the procedure is made the same as the one used to deposit the active layer in a photodiode, so that the NCs experience the same dielectric environment. The survey spectrum acquired at room temperature (Figure 2a) displays contributions from Hg and Te, Au (from substrate), and C. The latter directly relates to the ligand capping the NCs: mercaptoethanol in this case. It is worth stressing that despite air preparation and lack of annealing step, the films do not display any contribution from O 1s, which is expected to have a binding energy (BE) of 532 eV, consistent with the previously reported robustness of the material toward oxidation.²⁹

<figure><img src="image_3.png" /><figcaption>**Figure 2** X-ray photoemission spectra from HgTe NCs. a. Survey spectra acquired at 1486.6 eV for HgTe 4k NCs. b. (resp c.) Hg 4f (resp Te 4d) core level from HgTe 4k NCs. Data have been acquired at room temperature.</figcaption></figure>
We then further track the Hg 4f and Te 4d core levels, acquiring high-resolution spectra. Hg 4f (Figure 2b) comes as a doublet due to the spin-orbit (SO) coupling of the 4f orbital ($4f_{5/2}$ and $4f_{7/2}$, SO splitting = 4.1 eV), with each peak slightly asymmetric toward higher binding energy. Decomposition of the spectrum reveals a main contribution (Hg $4f_{7/2}$ BE = 100.3 eV) related to Hg bound to Te, while a second, weaker contribution (Hg $4f_{5/2}$ BE = 101.0 eV) presents a relative weight that depends on the incident photon energy.²⁹ The latter is more intense using surface-sensitive conditions (i.e., low photon energy) and was therefore associated with Hg surface atoms bonded to thiols (present in the ligands). In the case of Te 4d state (Figure 2c), a single contribution appearing at BE=40.2 eV is attributed to Te bound to Hg, with a 1.5 eV SO splitting.

We then cool sample in a range relevant for infrared detection and matching our instrument capability (i.e., down to 173 K in our laboratory setup, and down to 80 K, as shown in the Supporting Information, at TEMPO beamline at synchrotron SOLEIL) and observe a shift of the core level toward lower binding energies, see **Figure 3a**. The shift is observed for all sizes of NCs (Figure 3b and S4). Notably, the shift is fully reversible, and warming the sample restores the spectrum to its initial BE value (Figure S5).

<figure><img src="image_4.png" /><figcaption>**Figure 3** Temperature-induced shift of the electronic spectra. a. Hg 4f core level acquired at 1486.6 eV for HgTe 4k NCs under various temperatures. The inset is a zoom on the peak relative to Hg 4f 7/2 state. b. Temperature-induced shift of the Hg 4f core level, showing a drop of the binding energy, for 3 sizes of HgTe NCs. Data relative to Te 3d, Te 4d and valence band can be found in Figures S6-7.</figcaption></figure>

The shift appears as a rigid shift of the whole photoemission spectrum, since similar shifts are also observed for Te 4d and Te 3d (Figure S6) core levels and valence band, see Figure S7. We confirmed that the observed shift is not related to any artifacts of our setup, as the same behavior was consistently observed in measurements performed on a different setup at a synchrotron facility (Figure S8). Upon cooling, the drop in the carrier density is expected to make the sample more resistive, which may favor photocharging (i.e., accumulation of photoholes). However, the latter should lead to an increase in binding energy, whereas we observe the opposite trend. Therefore, we can exclude that photocharging is at the origin of the observed behavior.
It is interesting to note that the BE redshift is quite large, reaching around 80 meV already at 173 K, and saturates at about 100 meV at 80 K (see Figure S8). This shift magnitude must be compared with the tuning of the optical band gap under the same cooling (Figure 1c-d), which redshifts by about 20 meV only ($\approx$200 $\mu$eV·K$^{-1}$,⁷ slightly larger for large NCs). This suggests that, on an absolute energy scale, both the valence and conduction bands shift with temperature, and that the difference in their relative shift rates is smaller than their overall energy displacement.

To finish, we propose a reconstructed energy diagram for the three sizes of HgTe NCs under cryogenic conditions (Figure 4a), accounting for both the change in the optical band gap and the shift of the bands relative to the Fermi level. To determine this graph, the signal relative to the valence band is measured at room temperature and in cryogenic conditions (Figure S9), and is used to determine the energy of the valence band maximum with respect to the Fermi level. Then, the energy of the conduction band minimum is determined relative to the valence band by adding the optical band gap, which assumes that there is no coulombic correction for the excitonic structure, an assumption motivated by the large dielectric constant of HgTe¹⁰. For the low temperature data set, we account for the size dependence of the band gap,⁷, and the shift of the valence band.

<figure><img src="image_5.png" /><figcaption>**Figure 4** Effective electronic spectra for HgTe NCs. a. Valence band (VB), conduction band (CB), and Fermi level ($E_F$) for three sizes of HgTe NCs, measured at 173 K (i.e., lowest temperature of the XPS setup). b. Band gap energy for the HgTe 4k at room temperature and at 173 K, highlighting the formation of a Schottky barrier for the electron.</figcaption></figure>

In **Figure 4** the relative nanocrystal band gap shifts with lowering temperature correspond to the shifts predicted by the 14-band k.p model, as illustrated in Fig. S2. The energies of the valence band maxima are within the accuracy of the measurement, all the same. This observation fits with the picture of the valence band being based on a quasi-flat (i.e., heavy mass) band as shown in **Figure 1a**. This makes the hole confinement energy weak and poorly dependent on particle size. Conversely, the conduction band accounts for most of the size-induced shift, as expected from a light strongly dispersive band.

Since the shift results in a decrease in binding energy and appears rigid, the valence band moves closer to the Fermi level, making the material more *p*-type (or less *n*-type). Because the shift is similar for all NC sizes (Figure 3 and S5), the effect is more pronounced for narrower band gaps. From the perspective of a photodiode, temperature reduction can induce a 100 meV shift (**Figure 4b**) in the energy of one of the carriers within a 350 meV optical band gap. Such a Schottky barrier
becomes even more difficult to overcome at reduced operating temperatures. Assuming the Schottky barrier is overcome through an Arrhenius process, the efficiency drops, resulting from the temperature-induced absolute energy shift ($Φ_B$), can be estimated as $exp(-/Φ_B/k_BT) ≈ 10⁻³$, for operation at 173 K. This may require a substantial redesign of the photodiode and illustrates the specific challenges relative to mid-IR sensitive NCs. Charge transport layers developed for the visible and even near-infrared ranges are difficult to transfer, not only because of the presence of a massive offset but also because the design strategy must account for the actual targeted operating temperature. Absolute positioning of the band edges constitutes a level of refinement certainly necessary for the development of the next generation of MWIR NC-based photodetectors.

In conclusion, we have investigated *in situ*, using X-ray photoemission spectroscopy, the absolute energy shift of the electronic spectrum of HgTe NCs of various sizes relevant for infrared sensing. Upon cooling to temperatures typical of infrared sensor operation, we observe a rigid shift of the electronic spectrum, reaching about 100 meV at 80 K. This shift appears barely dependent on particle size or band edge energy. Consequently, the resulting corrections in carrier density and the formation of Schottky interface barriers are more severe for narrow-bandgap materials, which are particularly important for mid-infrared sensing.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon request.

## ACKNOWLEDGMENTS

The project is supported by ERC grant AQDtive (grant n°101086358) and from French National Research Agency (ANR) through the grants MixDferro (ANR-21-CE09-0029), Quicktera (ANR-22-CE09-0018), Operatwist (ANR-22-CE09-0037-01), E-map (ANR-23-CE50-0025), DIRAC (ANR-24-ASM1-0001), camIR (ANR-24-CE42-2757) and Piquant (ANR-24-CE09-0786). We acknowledge the use of clean-room facilities from the "Centrale de Proximité Paris-Centre" and support from Renatech+ for micro and nanofabrication. This project has also received financial support from the CNRS through the MITI interdisciplinary programs (project WITHIN) and Region Ile de France through Sesame project INSIDE.

## COMPETING INTEREST

The authors declare no competing interests.

## Supporting Information

Supporting Information includes (i) method for sample growth and preparation, (ii) activation energy of the dark current (iii) additional data documenting the thermal shift of the photoemission spectra.

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Supporting information

Temperature-induced Absolute Energy Shift of the Electronic
Spectrum in HgTe Nanocrystals

Tommaso Gemo<fnref n="1" />,<fnref n="2" />$, Dario Mastrippolito<fnref n="1" />,<fnref n="3" />$, Mariarosa Cavallo<fnref n="1" />, Giorgia Strobbia<fnref n="1" />, Albin Colle<fnref n="1" />, Marco Paye<fnref n="1" />, Jiho Roh<fnref n="1" />, Adrien Khalili<fnref n="1" />, Clement Gureghian<fnref n="1" />, Erwan Bossavit<fnref n="1" />,<fnref n="3" />, Sandrine Ithurria<fnref n="4" />, Yoann Prado<fnref n="1" />, Sébastien Sauvage<fnref n="5" />, Mathieu G. Silly<fnref n="3" />, Nicolas Péré-Laperne<fnref n="2" />, Debora Pierucci<fnref n="1" />, Emmanuel Lhuillier<fnref n="1" />*

$ these two authors have equal contributions

*To whom correspondence should be sent: el@insp.upmc.fr

Table of contents

<table><tr><td>1. Methods</td><td>2</td></tr><tr><td>2. 14-band k.p modeling of the electronic structure</td><td>4</td></tr><tr><td>3. Temperature dependence of the current</td><td>5</td></tr><tr><td>4. Thermal shift of the absolute energy electronic structure</td><td>6</td></tr><tr><td>5. References</td><td>9</td></tr></table>

<footnote n="1">¹ Sorbonne Université, CNRS, Institut des NanoSciences de Paris, 4 place Jussieu, 75005 Paris,
France.</footnote>

<footnote n="2">² LYNRED, Actipole - CS 10021, 364 route de Valence, 38113 Veurey-Voroize, France</footnote>

<footnote n="3">³Synchrotron SOLEIL, L'Orme des Merisiers, Départementale 128, 91190 Saint-Aubin, France.</footnote>

<footnote n="4">⁴Laboratoire de Physique et d'Etude des Matériaux, ESPCI, PSL Research University, Sorbonne Université, CNRS, 10 rue Vauquelin, 75005 Paris, France.</footnote>

<footnote n="5">⁵Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, C2N, Palaiseau 2110, France.</footnote>
# 1. Methods

**Chemicals:** Mercury chloride ($HgCl_2$, Alfa Aesar), tellurium powder (Te, Alfa Aesar, 99.99%), trioctylphosphine (TOP, thermofisher scientific, 90%), oleylamine (OLA, Acros, 80-90%), octadecene (ODE, Acros Organics, 90%), dodecanethiol (DDT, Sigma-Aldrich, 98%), methanol (MeOH, Carlo Erba, 99.8%), acetone (VWR), absolute ethanol (EtOH, VWR), isopropanol (IPA, VWR), toluene (Carlo Erba, 99.3%), N,N dimethylformamide (DMF, VWR), 2-mercaptoethanol (MPOH, Merck, >99%). All chemicals are used as received, except oleylamine, which is centrifuged before use. **Mercury compounds are highly toxic. Handle them with special care.**

**1 M TOP:Te precursor:** 6.35 g of Te powder was mixed in 50 mL of TOP in a three-neck flask. The flask was kept under vacuum at room temperature for 5 min, and then the temperature was raised to 100 °C. Furthermore, degassing of the flask was conducted for the next 20 min. The atmosphere was switched to Ar, and the temperature was raised to 275 °C. The solution was stirred until a clear orange coloration was obtained. The flask was cooled down to room temperature, and the color switched to yellow. Finally, this solution was transferred to a nitrogen-filled glove box for storage.

**HgTe 6k:** synthesis is inspired by Ref.¹. In a 100 mL three neck flask, 540 mg of $HgCl_2$ (2 mmol) and 50 mL of oleylamine are degassed under vacuum at 110 °C for 1h. Then, the atmosphere is switched to $N_2$, and the temperature stabilizes at 58 °C. Meanwhile, 2 mL of TOP:Te (1 M) is extracted from the glove box and mixed with 8 mL of oleylamine. Then, the TOP:Te solution is quickly injected. After 3 min, 10 mL of a mixture of 10% DDT in toluene is injected, and a water bath is used to quickly decrease the temperature. The content of the flask is split over 4 centrifuge tubes. MeOH is added, and the solution is sonicated for 3 min. After centrifugation, the formed pellets are redispersed in one centrifuge tube with toluene. The solution is precipitated a second time with absolute EtOH and sonicated for 3 min. Again, the formed pellet is redispersed in toluene. Then, the NCs are centrifuged in pure toluene to get rid of the lamellar phase. The solid phase is discarded.

**HgTe 4k:** In a 100 mL three-neck flask, 540 mg of $HgCl_2$ (2 mmol) and 50 mL of oleylamine are degassed under vacuum at 110 °C for 1h. Then, the atmosphere is switched to $N_2$, and the temperature stabilizes at 82 °C. Meanwhile, 2 mL of TOP:Te (1 M) is extracted from the glove box and mixed with 8 mL of oleylamine. Then, the TOP:Te solution is quickly injected. After 3 min, 10 mL of a mixture of 10% DDT in toluene is injected, and a water bath is used to quickly decrease the temperature. The content of the flask is split over 4 centrifuge tubes. MeOH is added, and the solution is sonicated for 3 min. After centrifugation, the formed pellets are redispersed in one centrifuge tube with toluene. The solution is precipitated a second time with absolute EtOH and sonicated for 3 min. Again, the formed pellet is redispersed in toluene. Then, the NCs are centrifuged in pure toluene to get rid of the lamellar phase. The solid phase is discarded.

**HgTe 3k:** In a 100mL three-neck flask, 543 mg of $HgCl_2$ and 50 mL of oleylamine were degassed under vacuum at 110°C. Meanwhile, 2 mL of TOP:Te (1M) was extracted from the glove box and mixed with 8 mL of oleylamine. After the flask atmosphere was switched to $N_2$ and the temperature stabilized at 110°C, the TOPTe solution was quickly injected. After 3 mins, 10 mL of a mixture of DDT in toluene (10% of DDT) was injected, and a water bath was used to quickly decrease the temperature. The content of the flask was split over 3 centrifuge tubes, and MeOH was added. After centrifugation, the formed pellet was redispersed in one centrifuge tube with chloroform. The solution was precipitated a second time using ethanol. The formed pellet was redispersed in
chlorobenzene. At this step, the nanocrystals were centrifuged in pure toluene to get rid of the lamellar phase. The solid phase was discarded. The solution is filtered with a 0.2 µm filter.

**NC ink preparation:** Before deposition, the NCs are prepared into a concentrated ink. A volume of 0.65/OD of the prepared HgTe solution, 0.2 mL of DMF, and 0.8 mL of exchange solution (30 mg HgCl₂ and 2 mL of MPOH in 18 mL of DMF) were added to a centrifuge tube and agitated thoroughly with vortex mixing and sonication. 5 mL of hexane was added to wash the solution, agitated thoroughly without sonication, and then discarded after phase separation. The QDs were then precipitated by adding 4 mL of toluene and centrifuging at 6000 rpm for 3 min. The remaining pellet was dried for 60 s under vacuum, then redispersed in 250 µL of DMF by agitating and sonicating.

**NC film preparation:** The NCs are then deposited on a gold-coated silicon substrate. A few drops of filtered DMF were deposited on the substrate and left for 30 s, then dried by spin coating at 4000 rpm for 30 s. Then, 15 µL of previously prepared ink was deposited on the substrate and spin-coated at 1250 rpm for 180 s, then 6000 rpm for 60 s.

**X-ray photoemission spectroscopy (XPS):** Photoemission spectroscopy measurements were performed using a PHI GENESIS setup. The samples were introduced into the load-lock chamber and degassed until a vacuum of 2·10⁻⁷ mbar was reached. They were then transferred to the analysis chamber (base pressure 2·10⁻⁹ mbar). The spectra were recorded using a monochromatic Al Kα X-ray source (hv=1486.6 eV) with a spot size of 100 µm in a normal emission configuration at room temperature. No charge compensation was applied. Core-level spectra (Hg 4f, Te 3d, and Te 4d) were acquired with a pass energy of 27 eV. The zero-binding energy (BE) reference, corresponding to the Fermi level, was calibrated using the leading edge of a clean Ag foil.

**Synchrotron-based XPS:** For synchrotron-based XPS, the TEMPO beamline of synchrotron SOLEIL was used.² Samples were introduced in the preparation chamber and degassed until a vacuum below 10⁻⁹ mbar was reached. Then, samples were introduced into the analysis chamber. The signal was acquired by a MBS A-1 photoelectron analyzer, with acquisitions performed with a pass energy of 50 eV within the detector. Photon energies of 700 eV and 150 eV were used. The zero BE reference was taken at the leading edge of a clean Au film.

**XPS analysis:** XPS spectra were decomposed using pseudo-Voigt functions (20% weighted Lorentzian contribution) using the CASAXPS software package. A linear background was used for Hg 4f and Te 4d, while a Tougaard background was used for Te 3d.

To evaluate the shift of core levels spectra at low temperature relative to room temperature, the peaks are overlapped by aligning both the low-binding energy background and the peak positions. This ensures a consistent determination of the spectra shift across temperatures.

To fit the valence band data, we first subtract the background (typically the signal value near the Fermi energy) and normalize the signal. We then systematically perform a linear fit of the valence edge between 15 and 85% of the normalized magnitude.³ The intercept of this line with the background is taken as the valence band position relative to the Fermi level (V<sub>B</sub> – E<sub>F</sub>). To estimate the error of this procedure, we use a second method in which the low-temperature data are shifted along the x-axis to overlap the room-temperature data. From this comparison, we estimate the uncertainty to be approximately 15 meV.
## 2. 14-band k.p modeling of the electronic structure.

The 14-band k.p model (referred to as kp14 in what follows) used to describe the electronic structure is detailed in Ref. 4. The same single parameter set is used for all energies and temperatures. The consideration of 14 bands provides a more realistic description of the dispersion relation from the zone center to around 15% of the Brillouin zone, as compared to a simple effective mass parabolic dispersion approach.

**Figure S 1** illustrates the strong non-parabolicities by comparing parabolic and a 14-band k.p $E_{LH}(k)$ dispersions. A parabolic dispersion is taken as $E(k) = \frac{\hbar^2}{2m^*}k^2$ where $m^*$ is the considered effective mass and $\hbar$ the reduced Plank constant. **Figure S 1a** shows that the very small effective mass at 293 K of $m^* = 0.0057m_0$, originating from the red-shift bandgap at this temperature, cannot describe the more realistic kp14 dispersion beyond around 1% only of the reciprocal space. The effective mass of $m^* = 0.0134m_0$ at 0 K, nearly twice as much as the one at 293 K, due to the blue-shift of the HH-$\Gamma_6$ energy difference, is also considered and shows an even worse discrepancy with kp14 model. Considering an effective mass of $m^* = 0.03m_0$ as considered in **Figure S 1b** is no longer suited to describe the quasi-linear dispersion relation over the investigated energy range. Quantitatively, the ratio between the parabolic and kp14 dispersions in this case is reported in **Figure S 1c**, showing a strong deviation away from 0.6 eV energy.

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

**Figure S 1** Comparison between parabolic and 14-band k.p $E_{LH}(k)$ dispersions. a. Dispersion $E_{LH}(k)$ close to the Brillouin zone center in the 14-band k.p model (blue full line) as compared to a single effective mass parabolic dispersion with room temperature 0.0057 $m_0$ (full gray line), low temperature 0.013$m_0$ (gray dashed line), and heavier 0.03$m_0$ (gray dotted line) effective masses. b. Comparison between a parabolic dispersion with 0.03$m_0$ effective mass and $E_{LH}(k)$ of the kp14 model away from the Brillouin zone center at larger energies than a.c. Ratio of the two energies plotted in b. quantifying the discrepancy between the two parabolic and kp14 dispersions.
<figure><img src="image_7.png" /></figure>

**Figure S 2 Measured and theoretical band gap shifts with lowering temperature.** Comparison between the band gap relative red-shifts with temperature from Fig. 4 (full squares) and the band gap shifts as calculated in the 14-band k.p model (full blue line). A value of 1, lower than 1, or larger than 1 indicates no shift with respect to the 293 K reference temperature, a redshift, and a blue shift, respectively.

**Figure S 2** illustrates the adequacy of the 14-band k.p model when confronted with the red-shifts observed experimentally in Figure 4, when the temperature is lowered from 293 K to 173 K. The experimental band gaps of three nanocrystal sizes are reported as full squares. The calculated $E_G(T) = E_{LH}(T) - E_{HH}(T)$ band gap at temperature $T$ is reported as a full blue line as a function of $E_G(293\text{K})$.

### 3. Temperature dependence of the current

To track the temperature dependence of the current, a film of NC has been deposited onto interdigitated electrodes (25 pairs, spaced by 10 µm, and 2.5 mm long). The sample is mounted onto the cold finger of a cryostat. A constant bias is applied, and the current is recorded thanks to a Keithley 2634 sourcemeter. Care is taken that sample resistance is not limited by contact.

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

**Figure S 3 Current as a function of temperature for HgTe 4k NC film.**
4. Thermal shift of the absolute energy electronic structure.

<figure><img src="image_9.png" /></figure>

**Figure S 4 Thermal Shift for HgTe 3k and 6k.** a. (resp b.) Hg 4f core level acquired at 1486.6 eV for HgTe 3k (resp 6k) NCs under various temperatures. Inset in both figures is a zoom on the Hg 4f 7/2 peak to better highlight its shift with temperature.

<figure><img src="image_10.png" /></figure>

**Figure S 5 Reversibility of the shift.** Hg 4f core level from HgTe 6k NCs at room temperature, before and after a cooling cycle.
<figure><img src="image_11.png" /><figcaption>**Figure S 6 Shift of the Te 3d core level as a function of temperature.** Te 3d core level acquired at 1486.6 eV for HgTe 6k (a), HgTe 4k (b.), HgTe 3k (c.) NCs under various temperatures. Inset in both figures is a zoom on the Hg 4f 7/2 peak to better highlight its shift with temperature.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure S 7 Rigid shift of the spectrum with temperature.** Shifts of the Hg 4f, Te 3d, and Te 4d core levels, and the valence band with temperature for HgTe 4k NCs.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Figure S 8** Confirmation of the effect on synchrotron setup. Hg 4f core level acquired at 150 eV for HgTe 6k (a), HgTe 4k (b.), HgTe 3k (c.) NCs acquired at room temperature and 80 K.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure S 9** Valence band signal. Photoemission signal relative to the valence band from HgTe 6k (a, resp. d), HgTe 4k (b, resp. e), HgTe 3k (c, resp. f). Data have been measured at room temperature (resp. 173 K).</figcaption></figure>
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