Ultrafast dynamics of hot carriers in a quasi–two-dimensional electron gas on InSe

Significance

Two-dimensional electron gases are an essential building block of today’s technology and attract broad interest in the context of material science or nanoengineering. Their widespread applications in many strategical sectors call for a direct visualization of hot electrons relaxation in an accumulation layer. In this work, we make use of time-resolved photoelectron spectroscopy to acquire snapshots of the electronic distribution after a strong and impulsive drive. We identified and quantified the remote coupling of the confined electrons to phonon modes of the adjacent polar material. The far-reaching outcomes provide insights on the screening of electron–phonon coupling in constrained dimensions and will be of high relevance for the development of aggressively downscaled circuits.

Abstract

Two-dimensional electron gases (2DEGs) are at the base of current nanoelectronics because of their exceptional mobilities. Often the accumulation layer forms at polar interfaces with longitudinal optical (LO) modes. In most cases, the many-body screening of the quasi-2DEGs dramatically reduces the Fröhlich scattering strength. Despite the effectiveness of such a process, it has been recurrently proposed that a remote coupling with LO phonons persists even at high carrier concentration. We address this issue by perturbing electrons in an accumulation layer via an ultrafast laser pulse and monitoring their relaxation via time- and momentum-resolved spectroscopy. The cooling rate of excited carriers is monitored at doping level spanning from the semiconducting to the metallic limit. We observe that screening of LO phonons is not as efficient as it would be in a strictly 2D system. The large discrepancy is due to the remote coupling of confined states with the bulk. Our data indicate that the effect of such a remote coupling can be mimicked by a 3D Fröhlich interaction with Thomas–Fermi screening. These conclusions are very general and should apply to field effect transistors (FET) with high-$\kappa$ dielectric gates, van der Waals heterostructures, and metallic interfaces between insulating oxides.

The low-energy consumption in integrated circuits would not be possible if the carriers of field effect transistors (FET) did not show exceptional mobility. By the same token, the two-dimensional electron gas (2DEG) in accumulation layers has been an ideal platform from which to observe the quantum Hall effect (1). Nowadays, 2DEGs are employed for computing (2), metrology (3), spin to charge conversion (4), and optoelectronics (5). Usually, these low-dimensional channels spontaneously form or can be electrostatically induced at the interface of polar materials. Some notable examples are Si/${\mathrm{S}\mathrm{i}\mathrm{O}}_2$/${\mathrm{H}\mathrm{f}\mathrm{O}}_2$ (6), InGaAs/${\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3$ (7), ${\mathrm{L}\mathrm{a}\mathrm{A}\mathrm{l}\mathrm{O}}_3$/${\mathrm{S}\mathrm{r}\mathrm{T}\mathrm{i}\mathrm{O}}_3$ (8), doped oxides (9, 10), and ${\mathrm{M}\mathrm{o}\mathrm{S}}_2$/${\mathrm{S}\mathrm{i}\mathrm{O}}_2$ (11). At these interfaces, the longitudinal optical (LO) phonons interact with carriers via long-range Fröhlich coupling (12). The polar scattering can limit the electron mobility if electrons attain a temperature comparable to the LO phonon frequency. Even if electronic screening drastically reduces the strength of the Fröhlich interaction (13), the quasi-2DEG can still suffer from Coulomb scattering with dipoles of the surrounding medium. Such a remote interaction is known to affect the mobility of few-layer devices covered by charged impurities (14), and it can also act on LO phonons of a polar medium (14, 15). The remote LO coupling is of major concern in gate insulators with a high relative dielectric constant. New FET devices based on high-$\kappa$ gates are partially replacing silicon dioxide (6) and have been already integrated in aggressively scaled complementary metal oxide semiconductors.

In order to uncover the relevance of remote phonon coupling, we perform here the ultrafast spectroscopy of an accumulation layer. The quasi-2DEG is obtained by evaporating cesium (Cs) atoms on the surface of indium selenide (InSe) at low temperature and in ultrahigh vacuum conditions. This doping method simulates, with good accuracy, the electrostatic gating and can be easily implemented in our experiment (16). The choice of polar material is motivated by the fact that InSe is one of the best van der Waals structures for the fabrication of FET devices (17, 18). It has an electronic gap comparable to silicon (19, 20), small effective mass ($m_{e\mathit{ff}}$) (21), layered structure (22), and carrier mobility higher than transition metal dichalcogenides (18). The electronic states and distribution function of hot electrons in the accumulation layer is directly monitored by time- and angle-resolved photoelectron spectroscopy (tr-ARPES). This approach offers exceptional advantages such as that 1) the probing depth is comparable to the localization length of the quasi-2DEG (23–25); 2) the momentum selectivity reveals whether excited electrons are in extended bulk states or in confined 2D states (26); 3) the electronic distribution discriminates between the semiconducting and metallic regime of hot carriers; and 4) the electronic cooling is slower than the typical pulse duration of mode-locked laser pulses (24, 27–29). As an example, previous applications of the tr-ARPES technique to quantum well states with high excess energy have already highlighted a 3D screening of the electron–electron interaction (24). In the following, we will investigate the low-energy excitations of a quasi-2DEG with variable carrier density. The comparison of the experimental results to model calculations of screened Fröhlich interaction allow us to identify and quantify the effects of a remote electron–phonon coupling.

Results and Discussion

Due to a downward band bending of roughly 150 meV (SI Appendix, Fig. S4), the chemical potential of a freshly cleaved InSe crystal lies 10 meV above the conduction band minimum. Accumulation layers with different charge concentration are obtained via subsequent exposure of the surface to Cs vapor. By increasing the density of the absorbed alkali atoms, the Fermi Energy $E_F$ of the electrons can be varied from 10 meV up to 200 meV (Fig. 1A). These 2D states penetrate the interface over a distance of several nanometers but are highly dispersive in the surface plane. At the highest doping level, an additional feature can be observed near zero wavevector and zero energy. It could be the bottom of the underlying 3D conduction band or the second subband of the quasi-2DEG. In one case as in the other, the occurrence of an extra band is in agreement with the 2D quantization of the electronic states in the confining potential. It is instructive to estimate the maximal carrier density ${\rho }_M$ in the accumulation layer. Since the $m_{e\mathit{ff}}$ of the quasi-2DEG is 10 times smaller than the one of a free electron $m_e$, we find ${\rho }_M=9\times 10^{12}\hspace{0.17em}{\text{cm}}^{-2}$ for $E_F=200$ meV. The ${\rho }_M$ value is comparable to the maximal doping level achieved via electrostatic gating (18). At even higher alkali concentration, the surface of InSe becomes unstable, and the quasi-2DEG disappears (20).

Fig. 1.

(A) Photoelectron intensity maps acquired at negative pump–probe delay. (B) Photoelectron intensity maps acquired at delay time of 0.1 ps. (C) Wavevector integrated intensity at negative delay time (blue curve) and 0.1 ps after photoexcitation (red curve). The black dashed curve is the estimated Fermi–Dirac distribution. We indicate, with $\gamma$, the ratio between the $E_x$ and the $E_F$. Here $E_F$ is the energy distance between the bottom of the dispersing parabola and the chemical potential in the low-temperature limit (zero of the energy axis). Panels of each line have been acquired for a given exposure to the Cs vapor. From Top to Bottom, the $E_F$ of the electron gas is $E_F=\mathrm{10,\; 40,\; 65,\; 95,\; 130,\; 200}$ meV.

We suddenly photoexcite the sample via an ultrafast laser pulse centered at 1.55 eV, leading to an initial photoexcitation density of roughly $1\times 10^{12}\hspace{0.17em}{\text{cm}}^{-2}$ in the topmost layer. Since InSe has a direct electronic gap of $\sim$1.28 eV, the excited carriers attain a maximal excess energy of 0.3 eV. The electron–electron interaction efficiently redistributes the energy density in the accumulation layer. As shown in Fig. 1C, the electronic distribution at pump–probe delay of 0.1 ps matches well a thermalized state with effective electronic temperature $T_e$. The only notable exception is observed on the pristine surface ($E_F=10$ meV), where the distribution near the conduction band minimum differs from the one expected from a thermal gas. We estimate the $T_e$ for each $E_F$ by fitting the high-energy tail of the excited electronic spectrum with a Fermi–Dirac function, and we also extract, directly from the data, the parameter

$$E_x\left(t\right)=\frac{\int E|I\left(E,t\right)-I\left(E,-\right)|dE}{\int |I\left(E,t\right)-I\left(E,-\right)|dE},$$ [1]

where $I\left(E,t\right)$ is the angle-integrated spectrum at delay time $t$, while $I\left(E,-\right)$ is the angle-integrated spectrum at negative delay. The integration range is between –0.3 eV and 0.3 eV. The experimental $E_x\left(t\right)$ is an electronic excess energy (numerator in Eq. 1) normalized by the excitation density of the thermalized electron gas (denominator in Eq. 1). More details about this parameter are given in SI Appendix. Note that the excitation density after internal thermalization can be different from the one initially injected by the pump pulse because of an imbalance between impact ionization and Auger processes.

As shown in Fig. 2, the average excess energy $E_x$ at 0.1 ps scales as $\left(1.3\pm 0.1\right)k_b{T}_e$ within the entire doping range and decreases by 70% at high Cs concentration. Since the incident pump pulse always has the same fluence, this drop is due to the energy redistribution between a fixed amount of photoexcited carriers and the increasing number of electrons in the accumulation layer. As a consequence, the parameter $\gamma =E_x/E_F$ spans from the semiconducting regime ($\gamma =10$) to the metallic regime of a deeply degenerate quasi-2DEG ($\gamma =0.2$). When the electronic density in accumulation level increases, three effects reduce the energy dissipation of the photoexicted gas: 1) The metallic electrons efficiently screen the long-range Fröhlich interaction between electrons and LO phonons, 2) the Fermi statistics hinders dissipation channels near to the chemical potential, and 3) the $E_x$ of the electrons approaches the threshold value for an emission of LO phonons.

Fig. 2.

(A) $E_x$ (red circles) and $k_b{T}_e$ (gray squares) as a function of $E_F$. The shaded blue area indicates the region where the $E_x$ of electrons becomes lower than the $E_F$. The dashed blue line indicates the threshold electron energy for LO phonon emission. (B) Sketch of near-surface potential leading to the accumulation layer as a function of energy (solid line) and local probability density (red area) of the confined state at the bottom of the 2DEG dispersion as a function of distance from the surface. The chosen parameters are $E_F=50$ meV and $E_x=50$ meV. (C) Same as for B but with parameters $E_F=200$ meV and $E_x=40$ meV. (D) Sketch of the dispersion of quasi-2DEG for $E_F=50$ meV and $E_x=50$ meV. (E) Same as for D but with parameters $E_F=200$ meV and $E_x=40$ meV.

On equal footing, the surface doping also induces a dimensionality cross-over of the hot carriers (23). When $E_F$ is small compared to $E_x$, the confining potential plays the role of a small perturbation. Therefore, hot electrons occupy bulk-like states with 3D character. Conversely, the 2D confinement becomes effective once the $E_F$ is larger than the $E_x$. At $E_F>100$ meV, hot electrons with low $E_x$ accumulate in wavefunctions that extend only a few nanometers in the bulk. Although an effective dimensionality of hot electrons is not sharply defined, we remark that $\gamma =E_x/E_F$ is a meaningful measure of the 3D–2D cross-over.

The full dynamics of hot electrons elucidates the evolution of cooling process at different doping levels. Fig. 3A shows the differential signal obtained by subtracting the map acquired at negative to the one acquired at positive values of the pump–probe delay. Blue and red colors stand for photoexcited electrons and holes in the electronic system, respectively. These panels offer a visual guide of the electronic cooling for three indicative densities ($E_F=\mathrm{10,40,130}$ meV). A qualitative analysis of the differential intensity maps provides clear evidence of the dimensionality cross-over. Note that excited electrons are distributed in a broad area of reciprocal space when $E_F=10$ meV. This signal originates from many 3D states with different values of the perpendicular wavevector $k_z$. Since these 3D states disperse along the $z$ axis, the excited electrons cover part of the projected band structure. In contrast, when $E_F=130$ meV, the differential intensity is concentrated along a nearly free electron parabola that has no measurable dispersion in the $z$ direction. From this map, we deduce that hot electrons become confined in the high-doping regime.

Fig. 3.

(A) Differential intensity maps obtained by subtracting the photoelectron intensity at negative delay from the photoelectron intensity at positive delay. (Left to Right) Data acquired at increasing delay time after photoexcitation. (Top, Middle, and Bottom) An electron gas with $E_F=\mathrm{10,\; 40,\; 130}$ meV, respectively. (B) Temporal evolution of the $E_x$ in the electron gas with different $E_F$. The curves have been normalized to the maximal $E_x$ value for better comparison. (C) Initial decay time of $E_x$ versus the $E_F$. Filled and open symbols indicate the results of two independent set of measurements.

Fig. 3B plots the temporal evolution of the excess energy normalized to its maximal value. Initially, the excited electrons experience a subpicosecond relaxation due to emission of LO phonon with small momentum transfer. After 1 ps to 2 ps, such a subset of phonon modes enters into equilibrium with hot electrons. The subsequent dynamics is dictated by anharmonic decay of the hot optical phonons into lattice modes of lower energy. Here we are interested only in the relaxation at early delay, namely, when LO phonons are still near to equilibrium conditions. As shown in Fig. 3B, the initial cooling time $\tau$ becomes 2 times slower at high carrier concentration. Measurements on two different cleaves confirm that $\tau$ is roughly 0.4 ps in the pristine surface, while it saturates to 0.8 ps upon increasing the concentration of absorbed Cs (Fig. 3C). We estimate the cooling rate $\eta$ of the electronic system as the $E_x$ at 0.1 ps divided by cooling time $\tau$. Fig. 4A shows that this rate decreases by nearly one order of magnitude when moving from the semiconducting to the metallic regime.

Fig. 4.

(A) Electronic cooling rate $\eta$ as a function of $E_F$. Filled and open symbols represent data from two independent set of measurements. The black solid, dashed blue, and dotted red curves are the cooling rates calculated by an ab initio model with 3D, 2D, and no screening of the Fröhlich interaction, respectively. We show, in SI Appendix, that the different phase space for electron–phonon scattering plays a major rule in the dimensionality dependence of the cooling rate. (B) Calculated phonon dispersion of $\epsilon$-InSe in $\mathrm{\Gamma }-M$ direction of the Brillouin zone. Polar optical mode $E\prime$ is shown in red, while the best coupled nonpolar $A_1^{\prime }$ mode is shown in green. Red squares at $q=0$ stand for experimental from ref. 38. (C) Calculated electron–phonon deformation potentials as a function of the phonon wavevector $\mathbf{q}$ along the $\mathrm{\Gamma }-M$ direction. The initial electronic state is in the conduction band at $\mathbf{k}=\left(\mathrm{0,0.11},0\right)2\pi /a$. For the polar mode, both the results of the DFPT (black circles) and Vogl’s model (red dotted line) are shown. The numerical results were obtained with Quantum Espresso (39).

Since, at low doping levels, the hot electrons show 3D behavior, a less restrictive confinement could, in principle, open transport channels into the bulk conduction band. Transport effects can contribute to the energy relaxation of the electrons when the diffusion length $L=\sqrt{Dt}$ covered during the relaxation time $t$ is larger than the depth $d$ of the excitation profile (29). An upper bound of the out-of-plane diffusion constant $D$ can be estimated from the in-plane mobility (18) ${\mu }_e$ (the out-of-plane being lower) and the Einstein relation $D={\mu }_e{k}_b{T}_e/e$. With ${\mu }_e<\mathrm{1,000}\hspace{0.17em}{\mathrm{c}\mathrm{m}}^2\cdot$V⁻¹$\cdot$s⁻¹, $k_b{T}_e<100$ meV, and $t<1$ ps, we obtain $L<0.1\hspace{0.17em}\mathrm{\mu }$m. Since our pump pulse excites electrons just above the bandgap, the penetration depth $d$ of the pump pulse is particularly long. From the absorption coefficient (19) at 1.55 eV, we find $d=10\hspace{0.17em}\mathrm{\mu }$m. Being that $L$ is two orders of magnitude smaller than $d$, we conclude that diffusion effects can be safely ignored.

In order to simulate the cooling rate, we calculated the phonon spectrum by density functional perturbation theory (DFPT) (Fig. 4B). We identify the polar and the nonpolar branches for each direction in wavevector space. Since the $\mathrm{\Gamma }$ valley is very narrow ($m_{e\mathit{ff}}=0.1m_e$, where $m_e$ is the free electron mass) and there are no adjacent valleys, the energy and momentum conservation rules limit the phonon emission to wavevector transfer of $q<0.2$ Å⁻¹. In the 3D case, the unscreened scattering with polar optical phonons is long range in real space and diverges when the phonon wavevector $q$ tends to zero (30). As a consequence, and due to the absence of intervalley scattering, we expect LO emission to be the largely dominant electron–phonon scattering mechanism. Indeed, the preeminence of Fröhlich scattering at low excess energies is a general property of polar semiconductors, such as, for example, GaAs (31). To illustrate this point, we compare, in Fig. 4C, the strength of the electron–phonon matrix elements for all phonon modes of $\epsilon$-InSe in the $\mathrm{\Gamma }-M$ direction. Our numerical results show that the coupling of the $E\prime$ polar mode is, on average, more than 10 times larger than that of the best coupled nonpolar mode (the mode with $A_1^{\prime }$ symmetry). The contribution of the polar coupling to the energy relaxation rate scales as coupling matrix element squared, being at least 100 times larger than the $A_1^{\prime }$ nonpolar one. With respect to the other nonpolar modes, this ratio becomes even larger, so that the black curves in Fig. 4C collapse to values near to the ordinate axis. Fig. 4C also shows that the matrix elements of the polar $E\prime$ mode nearly follow the $q$ dependence predicted by Vogl’s model (30). This result allows us to use Vogl’s model for the electron–phonon coupling instead of the DFPT description.

We simulate the cooling rate $\eta$ due to Fröhlich interaction by calculating (32) as

$$\begin{array}{ccc}\hfill \eta =\left(1-f\left(E_x-\hslash {\mathrm{\Omega }}_{em}\right)\right){\mathrm{\Omega }}_{em}{\mathrm{\Gamma }}_{em}& \hfill & \hfill \\ \hfill -\left(1-f\left(E_x+\hslash {\mathrm{\Omega }}_{abs}\right)\right){\mathrm{\Omega }}_{abs}{\mathrm{\Gamma }}_{abs}.& \hfill & \hfill \end{array}$$ [2]

Here, $f$ is a Fermi–Dirac distribution function which depends on the $T_e$ and on the chemical potential, ${\mathrm{\Gamma }}_{em}$ and ${\mathrm{\Gamma }}_{abs}$ are the total probabilities for emission and absorption, and ${\mathrm{\Omega }}_{em}$ and ${\mathrm{\Omega }}_{abs}$ are effective LO phonon frequencies for emission and absorption (see SI Appendix for details about the calculations). For each value of the $E_F$, we extract the $T_e$ from the experimental curve in Fig. 2 and derive the chemical potential on the base of a free electron model (33). The values of ${\mathrm{\Gamma }}_{em}$ and ${\mathrm{\Gamma }}_{abs}$ strongly depend on the electron–phonon coupling strength of the polar modes. The nonpolar modes would contribute to the cooling rate with an extra term below 0.01 eV ps⁻¹ and can be neglected with respect to the emission of LO optical phonons. First, we consider the case of a 3D electron gas with unscreened Fröhlich interaction. As already shown by our previous work on the pristine surface (21), this approach provides an estimated value of the cooling rate that is in agreement with tr-ARPES experiments. Upon increasing the doping level, the available momentum transfer increases by a small amount. On the other hand, the drop of excess energy combined with the Fermi–Dirac statistic generates a stronger Pauli blocking of the scattering channels. The effects of the quantum electron statistic on excitons has been evinced by pioneering experiments on GaAs Quantum Wells (27). In our case, the Pauli blocking alone would explain the moderate decrease of calculated $\eta$ for $\gamma \cong 1$. Nonetheless, this unscreened coupling badly overestimates the experimental behavior of a quasi-2DEG even at moderate carriers concentration (see red dotted curve in Fig. 4A).

In order to explain the experimental data, we consider a 3D model of the Fröhlich interaction (30) with Thomas–Fermi screening. We expect these simulations to be accurate for the regime $\gamma >1$, namely, when hot electrons retain their 3D character. Surprisingly, the black line in Fig. 4A shows that the screened 3D interaction matches well the experimental data even in the case of a quasi-2DEG. This result needs to be carefully addressed, since a dimensionality cross-over toward 2D states is observed in photoelectron intensity maps with $\gamma <1$. In this limit, a 3D treatment of the electronic degrees of freedom is clearly nonjustified. In order to gain more insights, we compute the $\eta$ expected in a strictly 2D case of 2D Fröhlich coupling (12) and 2D dielectric screening (13). Remark that such model assumes that all particles and interactions are fully constrained in a plane. Therefore, it differs profoundly from the real physical case in which a metallic 2D slab has a 3D coupling with a surrounding dielectric. The perfect confinement of the strictly 2D model does cut off the divergence of the polar interaction at small wavevector (see also SI Appendix, Fig. S1). Moreover, the electron–hole excitations that are responsible for the screening response acquire a higher density in the phase space (consider that, in a strictly 1D system, the collective excitations become so strong that even the Fermi liquid picture breaks down). Such an increase of phase space for electron–phonon scattering has strong effects, so that the $\eta$ of a strictly 2D model drops dramatically already by a small doping level. It is clear from the dashed blue line of Fig. 4A that such a model grossly underestimates the experimental cooling rate. The difference between the 2D model and experimental data is ascribed to a remote coupling between the quasi-2DEG and 3D phonons. This interaction could arise from an energy transfer mediated by surface plasmons–polaritons or any other emerging channel that couples the confined electrons with LO modes that are not strictly 2D. An accurate simulation of such a remote interaction would be a challenging task which is beyond the scope of the present work. However, Fig. 4A shows that a 3D model with Thomas–Fermi screening already reproduces remarkably well the experimental data.

Next, we compare our result with complementary works on the same subject. Several authors investigated the Fröhlich interaction of a 2D electron gas with an LO mode of oxides such as ${\mathrm{T}\mathrm{i}\mathrm{O}}_2$ (9) and ${\mathrm{S}\mathrm{r}\mathrm{T}\mathrm{i}\mathrm{O}}_3$ (10). The ARPES maps have shown that shake-off replicas of the quasi-particle peak disappear above a critical density of electrons in accumulation layer. This cross-over from long-range to screened interaction takes place at an electron density which is 20 to 40 times higher in ${\mathrm{S}\mathrm{r}\mathrm{T}\mathrm{i}\mathrm{O}}_3$ ($\cong 4\hspace{0.17em}\text{to}\hspace{0.17em}8\times 10^{13}\hspace{0.17em}{\text{cm}}^{-2}$) than in InSe ($\cong 2\times 10^{12}\hspace{0.17em}{\text{cm}}^{-2}$). The same holds true in the case of ${\mathrm{T}\mathrm{i}\mathrm{O}}_2$. We ascribe such a disparity to the wide antiadiabatic regime occurring in the oxides. Indeed, the LO phonon energy and the 2DEG $m_{e\mathit{ff}}$ are $\hslash \mathrm{\Omega }\cong 100$ meV and $m_{e\mathit{ff}}\cong m_e$ in ${\mathrm{S}\mathrm{r}\mathrm{T}\mathrm{i}\mathrm{O}}_3$, whereas we found $\hslash \mathrm{\Omega }=24$ meV and $m_{e\mathit{ff}}=0.1m_e$ in InSe (10, 21). At equal carrier concentration, the Migdal–Eliashberg parameter $\hslash \mathrm{\Omega }/E_F$ is 40 times larger in ${\mathrm{S}\mathrm{r}\mathrm{T}\mathrm{i}\mathrm{O}}_3$ than in InSe. Since the screening becomes most effective in the adiabatic regime (i.e., $\hslash \mathrm{\Omega }/E_F<1$), the cross-over occurs at high critical density in the case of ${\mathrm{S}\mathrm{r}\mathrm{T}\mathrm{i}\mathrm{O}}_3$.

A second and related topic of interest is the electron–phonon coupling between graphene and a polar substrate (14, 34). Simulations of transport properties predicted that remote coupling limits the mobility of charge carriers in the 2D semimetal (34). A determination of such remote interaction by time-resolved techniques would likely be impossible. Indeed, the strongly coupled in-plane modes of graphene dominate the energy relaxation of photoexicted carriers (35). Nonetheless, the transport measurements of graphene (14) on ${\mathrm{S}\mathrm{i}\mathrm{O}}_2$ indicate that scattering with interface phonons has a high impact on the transport properties of the electrons.

Conclusions

In conclusion, hot electrons in quasi-2DEGs display a remote coupling to polar optical phonons persisting up to high electronic density. The accurate modeling of such interaction should include the wavefunctions of confined 2D electrons, dynamical screening effects, surface plasmons polaritons, and interface phonons. Nonetheless, the static screening of bulk phonons by 3D electrons can quantitatively reproduce the experimental cooling rate. This finding highlights that electrons in the accumulation layers or 2D conductors at the interface with a polar medium experience 3D dissipation channels. The outcome is of high relevance for the carriers’ mobility in FET devices with high $\kappa$ dielectric gates, van der Waals heterostructures, and 2DEGs at the interface between oxides.

Single crystals of $\epsilon$-InSe have been grown using the Bridgmann method from a nonstoichiometric melt (36). ARPES and photoluminescence spectra show that our bulk crystal is naturally $n$ doped and has a direct band gap of $\sim$1.28 eV (20). All samples have been cleaved at the base pressure of $8\times 10^{-11}$ mbar and exposed to an atomic jet of Cs. After each exposure to the alkali vapor, we performed tr-ARPES of the surface. For the entire duration of the experiment, the sample was kept at the base temperature of 40 K. This low temperature guarantees a higher concentration and higher stability of the absorbed Cs atoms. The photon source for tr-ARPES experiments is a Ti:Sapphire laser system delivering 6-$\mathrm{\mu }$J pulses with a repetition rate of 250 kHz. Part of the fundamental beam ($\omega =1.55$ eV) is employed to pump the sample with incident fluence 0.5 mJ cm⁻², while the rest is employed to generate the fourth harmonic beam ($4\omega =6.2$ eV) (37). The electrons are emitted by the $4\omega$ pulses incident on the sample at $45^{○}$ and p polarization. An electrostatic analyzer discriminates the photoelectrons in kinetic energy and emission angle with a resolution of 50 meV and $0.5^{○}$, respectively. Independently of the configuration of the experiment, the two beams generating the tr-ARPES signal display a cross-correlation with full width at half maximum of $<0.15$ ps.

Data Availability.

All study data are included in the article and SI Appendix.