Crossover from exciton polarons to trions in doped two-dimensional semiconductors at finite temperature

As discussed above, solving the eigenvalue problem (6) allows us to evaluate the exciton Green’s function. It turns out that it is numerically convenient to instead consider the exciton self-energy $\Sigma(\omega)$, which is related to the Green’s function via

The expression for the exciton self-energy can be derived by manipulating the eigenvalue problem (6) Liu et al. (2019), following the same procedure valid at zero temperature Combescot et al. (2007) — for completeness we describe this in Appendix B. In this way, the exciton self-energy reads

where the inverse of the $T$ matrix is defined as

The same expression (10) can also be derived by using a diagrammatic expansion within the ladder approximation Combescot et al. (2007); Liu et al. (2019). Thus, our variational approach provides an additional theoretical foundation for the ladder diagrams.

It is profitable to separate the vacuum contribution to the $T$ matrix, describing the electron-exciton scattering in the absence of a surrounding Fermi gas, from the many-body contribution. To this end, we note that the logarithmic divergence of the second term in (11) cancels with that of the inverse contact interaction constant $v^{-1}$ (3), allowing the vacuum contribution $\mathcal{T}_{0}$ to be calculated analytically Levinsen and Parish (2015):

where $m_{T}=m+m_{X}$ is the trion mass and $m_{r}=mm_{X}/m_{T}$ is the exciton-electron reduced mass. Note that the vacuum $T$ matrix (12b) has a pole at the bound state, i.e., the trion energy, $\omega=-\varepsilon_{T}+\epsilon_{T{\bf q}}$, where $\epsilon_{T{\bf q}}=\frac{q^{2}}{2m_{T}}$ is the trion kinetic energy (for a summary of the trion properties at finite momentum and finite doping, see Appendix C). The many-body contribution $\Pi_{mb}({\bf q},\omega)$ to the $T$ matrix has to be evaluated numerically—for details on the numerical procedure see Appendix D.

The optical absorption coincides with the exciton spectral function (up to a frequency-independent prefactor):

Indeed, in the linear-response regime, the spectral function is equivalent to the transfer rate from an initial state $|n\rangle$ containing no excitons (the impurity vacuum) to a final state $|\nu\rangle$ containing a single exciton. Here, the impurity vacuum and single-impurity states are eigenstates of the Hamiltonian, i.e., $\hat{H}|n\rangle=\hat{H}_{0}|n\rangle=E_{n}|n\rangle$ and $\hat{H}|\nu\rangle=E_{\nu}|\nu\rangle$. Using Fermi’s golden rule, we have

where $\hat{\rho}_{0}$ is the medium-only partition function introduced above. By using the completeness of the $\{|\nu\rangle\}$ basis, one can easily show that Eq. (14) coincides with (13) — see Ref. Liu et al. (2020b). Using the definition (14), it is now straightforward to show that the exciton optical absorption satisfies the sum-rule:

In order to evaluate the photoluminescence, we instead consider the opposite situation, i.e., an initial state $|\nu\rangle$ containing the medium and the exciton, and a final state $|n\rangle$ after the exciton has recombined to emit a photon. Here, we have assumed that the exciton density is low enough such that each exciton can be treated individually. The transfer rate at thermal equilibrium is then given by

where $\hat{\rho}=e^{-\beta\hat{H}}/Z_{int}$ is the density matrix associated with the interacting exciton and medium system (1) and $Z_{int}=\sum_{\nu}\expectationvalue*{e^{-\beta\hat{H}}}{\nu}$ the associated partition function. It is straightforward to show that the photoluminescence satisfies the following sum rule

Using the properties of the delta function, the absorption $A(\omega)$ and photoluminscence $P(\omega)$ can be related by a detailed balanced condition Liu et al. (2020b):

The thermodynamic, Boltzmann-type scaling between absorption and emission profiles (18) is also known as the Kubo-Martin-Schwinger relation Kubo (1957); Martin and Schwinger (1959), the Kennard-Stepanov relation Kennard (1918); Stepanov (1957); McCumber (1964) or the van Roosbroeck-Shockley relation van Roosbroeck and Shockley (1954), depending on the context within which it has been studied, and it applies to a broad range of systems, including semiconductors Band and Heller (1988); Chatterjee et al. (2004); Ihara et al. (2009). It relies on the assumption that the population of excited states, here excitons, has thermalized at a temperature $T$ before the emission and that they are otherwise uncorrelated. Note also that the thermalization temperature $T$ can be different from the system lattice (cryostat) temperature.

Even though one only has to evaluate two momentum integrals to obtain the exciton Green’s function in Eq. (9), namely the integrals in Eqs. (10) and (12c), some comments about the numerical procedure are necessary. For the optical absorption (13), the numerical convergence of the integrals is much improved by shifting the frequency to the complex plane, $\omega\mapsto\omega+i\eta$. Apart from helping with convergence, this shift provides a simplified description of the exciton’s intrinsic broadening due to effects beyond those included in the Hamiltonian such as recombination and disorder. In the following, we have used the typical value $\eta=0.04\varepsilon_{T}\simeq 1$ meV.

Including $\eta$ implies that the exciton spectral function decays as a Lorentzian at low and high energies. However, one cannot evaluate the photoluminescence using this procedure. Indeed, by using the detailed balance condition (18), the photoluminescence diverges at infinitely low frequencies if one uses a finite value of $\eta$ to evaluate the absorption, since the Boltzmann occupation increases more rapidly than the Lorentzian decay of absorption. This means that photoluminescence needs to be evaluated by first calculating absorption at $\eta=0$ and then multiplying by the Boltzmann occupation. The effects of the exciton intrinsic decay time can be re-introduced at the end of the calculation by convolving the photoluminescence with a Lorentzian profile with broadening $2\eta$. This procedure is described in detail in Appendix D.

Numerous theoretical works have so far analyzed the zero-temperature properties of an impurity interacting with a free Fermi gas, in the context of both semiconductor heterostructures Suris (2003); Sidler et al. (2017); Efimkin and MacDonald (2017); Rana et al. (2020, 2021a, 2021b); Efimkin et al. (2021) and ultracold atomic gases Massignan et al. (2014); Scazza et al. (2022). The main polaron properties at $T=0$ are summarized in Fig. 1. In panel (a) we plot the doping and energy dependence of the spectral function. One observes that the optical absorption is dominated by two polaron quasiparticle resonances: the attractive polaron branch at lower energy $E_{A}$ and the repulsive polaron branch at higher energy $E_{R}$. The energies $E_{A,R}$ are evaluated from the positions of the absorption peaks in Fig. 1. Here, both polaron branches are quasiparticles in the sense that $E_{A,R}$ satisfy the condition Fetter and Walecka (2003)

which coincides with a pole of the exciton Green’s function when the corresponding imaginary part of the self-energy $\imaginary\Sigma(E_{A,R})$ becomes small. Note that the repulsive branch eventually stops satisfying this condition when $E_{F}\gtrsim 3\varepsilon_{T}$ Ngampruetikorn et al. (2012), and thus ceases to be a polaron quasiparticle. This large doping regime is not analyzed in this work.

In the limit of vanishing doping, the attractive branch energy recovers the trion energy $E_{A}\to-\varepsilon_{T}$, while the repulsive branch energy reduces to the exciton energy, $E_{R}\to 0$. (We remind the reader that we measure energies with respect to that of the exciton at rest). However, differently from the trion, whose energy blueshifts with doping due to its interactions with the surrounding electrons [see Ref. Parish (2011) and Eq. (61)], the attractive polaron branch redshifts with doping. At the same time, the repulsive polaron branch blueshifts with doping. Thus, at zero temperature, as soon as doping is finite, the attractive and repulsive polaron quasiparticles are no longer the trion and the exciton, respectively, even if they recover the corresponding energies at low doping. Furthermore, within the single particle-hole-excitation ansatz (4), the effective mass of the attractive polaron Schmidt et al. (2012); Parish and Levinsen (2013) does not evolve into the trion mass at low doping.

As far as the coupling to light of the polaron branches is concerned, at very low doping the repulsive branch retains all the spectral weight and the attractive branch is dark, as is also the case for the trion. However, when doping increases, there is a transfer of oscillator strength from the repulsive to the attractive branch. This is shown in Fig. 1(c), where we plot the doping dependence of the spectral weights $Z_{A,R}$ for each branch. At low doping $E_{F}\ll\varepsilon_{T}$, $Z_{A}$ grows linearly with $E_{F}$, which agrees with the doping dependence of the trion oscillator strength Glazov (2020).

There are also important differences between the attractive and repulsive polaron branches at zero temperature, even though they both satisfy Eq. (19) for values of doping up to $E_{F}\sim\varepsilon_{T}$. The attractive polaron is always a sharp resonance, with a Lorentzian broadening coinciding with the intrinsic broadening $2\Gamma_{A}=2\eta$ — see Fig. 1(d). By solving the eigenvalue problem (6) at zero temperature, one finds that the attractive branch energy coincides with the lowest eigenvalue energy, $E_{A}=E^{(n=1)}$, which is separated from the energy of the excited states $n>1$ by a gap. By contrast, the repulsive branch does not correspond to a specific eigenstate; rather it is composed of a continuum of eigenstates with closely spaced eigenvalues, resulting in a polaron quasiparticle with finite lifetime. As a consequence, its broadening $2\Gamma_{R}$ grows monotonically with $E_{F}$ and only coincides with $2\eta$ at small doping — see Fig. 1(d).

In between the attractive and repulsive branches, one can observe in Fig. 1(a) a continuum of trion and Fermi-sea-hole states with a well-defined relative momentum ${\bf q}$ in the variational function $\varphi_{{\bf k}{\bf q}}$. The boundaries of this so-called trion-hole continuum can be evaluated from the energy of a trion (see Appendix C) plus a hole separately. If the hole has zero momentum ${\bf q}={\bf 0}$, then, because of momentum conservation, the trion is also at zero momentum and the upper boundary of the trion-hole continuum is

Conversely, the energy of the trion-hole lower bound is

where both the hole and trion are at ${\bf q}={\bf k}_{F}=k_{F}\hat{\textbf{n}}$, with $\hat{\textbf{n}}$ an arbitrary direction, and $E_{T}^{({\bf k}_{F},E_{F})}$ is the trion energy [which can be evaluated from Eq. (58)]. Thus, at zero temperature and finite doping, the attractive branch is separated from the trion-hole continuum by an energy gap. Note that, while this is a consequence of considering a single excitation of the medium, our results are consistent with those of diagrammatic quantum Monte Carlo Goulko et al. (2016) which demonstrated that there is a region of anomalously low spectral weight (a “dark continuum”) above the narrow attractive polaron branch. Furthermore, calculations that incorporate Coulomb interactions between charges also obtain a suppression of spectral weight of this continuum Rana et al. (2020).

We now discuss the effect of temperature on the optical response. Figure 1(b-d) allows a comparison between the doping-dependent properties of optical absorption at zero and finite temperature. Both the energies and spectral weights of attractive and repulsive branches have a very weak dependence on temperature in the regime $T\lesssim\varepsilon_{T}$. The branch energies are slightly redshifted, while $Z_{A}$ ($Z_{R}$) is slightly smaller (larger) compared to the zero-temperature case. This small variation with temperature is also observed in Fig. 2(a,b) for a fixed doping. The most important difference at finite temperature is the behavior of the trion-hole continuum, which subsumes the attractive branch when $T\gtrsim E_{F}$, i.e., at sufficiently low doping or sufficiently high temperature. This can be clearly seen from Fig. 1(b), where we no longer observe a sharp lower bound of the trion-hole continuum at low doping because, at finite temperature, the unbound hole belonging to the trion-hole continuum can thermally occupy any momentum state. However, the upper bound of the trion-hole continuum is still clearly visible and approximately follows the zero-temperature expression $E_{+}$ (20). Similarly, for a fixed doping in Fig. 2(a), we observe that the trion-hole continuum is only well-separated from the attractive branch at low temperatures.

Since the spectral weight of the trion-hole continuum is small, its merging with the attractive branch only slightly affects the attractive peak energy. However, the disappearance of the attractive polaron quasiparticle strongly modifies the attractive-branch linewidth $2\Gamma_{A}$. In particular, we observe in Fig. 1(d) that $\Gamma_{A}$ has a striking non-monotonic dependence at low doping, while it decreases towards its zero-temperature value (corresponding to the intrinsic broadening $\eta$) when $E_{F}$ increases. Likewise, increasing the temperature at fixed doping can substantially increase $\Gamma_{A}$ from $\eta$, as shown in Fig. 2(c). As we will discuss in Sec. IV.3, this behavior signals a crossover from a coherent Fermi polaron regime to an incoherent trion-dominated regime, where there no longer exists a well-defined attractive quasiparticle that is separated from the trion-hole continuum.

The repulsive branch, on the other hand, remains a polaron quasiparticle with a finite lifetime (broadening) for the dopings considered in this work ($E_{F}\lesssim\varepsilon_{T}$). In particular, we see that temperature does not change the nature of the repulsive branch in Figs. 1 and 2, but it can lead to a faster increase of the half linewidth $\Gamma_{R}$ with increasing doping [Fig. 1(d)].

We now discuss the shape of the optical response profiles and how they evolve with either doping or temperature. Figures 3 and 4 display both the absorption and the Lorentzian convolved photoluminescence (see Appendix D) at fixed temperature and fixed doping, respectively. The repulsive polaron quasiparticle shows up approximately as a Lorentzian symmetric profile in both absorption and photoluminescence spectra, with a full width at half maximum (FWHM) that increases with increasing $E_{F}$. Note that the FWHM of the repulsive branch only has a weak dependence on temperature in Fig. 4 since $E_{F}$ has been fixed to a low value such that the intrinsic width $2\eta$ dominates [see Fig. 2(c)]. By contrast, the shape of the attractive branch is strongly modified by temperature: in the low-temperature (high-doping) regime $E_{F}>T$, it is described by a Lorentzian with FWHM $2\eta$, while for $E_{F}<T$, it develops a strongly asymmetric shape with an exponential tail below the trion energy. This evolution in the asymmetry of the attractive branch once again indicates a crossover from a Fermi-polaron quasiparticle to a continuum of trion states.

In Fig. 5 we further analyze the shape of the attractive peak at low doping $E_{F}\lesssim T$, comparing the Lorentzian convolved photoluminescence $\bar{P}(\omega)$ with the “bare” photoluminescence $P(\omega)$, where we have removed the effects of any intrinsic exciton broadening. In panel (a), we observe a sharp onset of the photoluminescence which approximately coincides with the upper boundary of the trion-hole continuum at zero temperature, $E_{+}$. Thus, according to Eq. (20), it blueshifts with increasing doping. As shown in Fig. 5(b), any intrinsic broadening $\eta$ only smooths out the sharp onset, while it has little effect on the position of the peak. When $E_{F}$ increases, the sharp onset tends to disappear as the attractive peak redshifts and detaches from the trion-hole continuum.

Our calculated profiles for the attractive branch are in excellent quantitative agreement with recent experiments in the high-temperature (low-doping) regime Zipfel et al. (2022), as discussed in the accompanying paper Mulkerin et al. (2022). The exponential tail of the asymmetric attractive peak has previously been modelled within a trion picture for the case of photoluminescence Esser et al. (2000, 2001); Ross et al. (2013); Christopher et al. (2017). There, the tail is ascribed to the kinetic energy of remaining electrons after the exciton within each trion has decayed into a photon. This description in terms of electron recoil can be formally derived from our theory in the limit of weak interactions, as we will discuss in Sec. IV.4.

In this section, we use the pole condition in Eq. (19) to characterize the crossover from a well-defined polaron quasiparticle to a trion-hole continuum with increasing temperature (decreasing doping). In order to find the values of temperature and doping at which this crossover occurs, we plot in Fig. 6 the local maximum value of the function $\omega-\real\Sigma(\omega)$ for $\omega<0$ and identify the curve of doping versus fugacity $z=e^{\beta\mu}=e^{\beta E_{F}}-1$ at which this maximum value is zero. For $E_{F}\lesssim\varepsilon_{T}$, we find that this occurs roughly when $z\sim 1$ and thus $E_{F}\sim 0.7T$. On the left of this curve, we lose the attractive polaron quasiparticle, as the condition (19) cannot be satisfied, i.e., $E_{A}-\real\Sigma(E_{A})\neq 0$. On the right of this curve, instead, the system is in the polaron regime where (19) is satisfied.

In order to further illustrate this crossover, we compare the results for the polaron energies, spectral weights and linewidths extracted from the spectral function with those obtained by treating the polaron as a well-defined quasiparticle Fetter and Walecka (2003). In the latter case, the polaron properties can be obtained directly from the expression of the impurity self-energy. Close to a quasiparticle resonance, the exciton Green’s function can be approximated as

where $j=A,R$ is the two-branch index, and the quasiparticle energy $E_{j}$ is a solution of Eq. (19). The pole weight or residue $Z_{j}$ is

and the polaron damping rate is

We compare the results for $E_{j}$, $Z_{j}$, and $\Gamma_{j}$ obtained with both methods in Fig. 7. We observe that the positions of the poles coincide to high accuracy with those of the spectral function maxima — see panel (a). For the repulsive branch, both the spectral weight and the broadening from the quasiparticle theory are in good agreement with those evaluated from the spectral function, even when the linewidth is non-negligible. By contrast, for the attractive branch, the results depart from one another when we approach the (gray) region $E_{F}\lesssim 0.7T$ where there is no attractive quasiparticle, and the quasiparticle description breaks down since, according to Eq. (23), $1/Z_{A}\to 0$ when $\rm{max}[\omega-\real\Sigma(\omega)]=0$.

In the following section, we analyze the system properties well inside the trion-hole continuum (gray) regime, where the attractive branch is no longer a polaron quasiparticle. Here, for temperatures $T\gtrsim E_{F}$, we can apply a systematic quantum virial expansion. This connects the results of this work with those obtained in the accompanying paper Mulkerin et al. (2022).

As discussed, at high temperature or low doping such that the fugacity $z=e^{\beta\mu}\lesssim 1$, the attractive polaron quasiparticle disappears and the attractive branch only consists of a broad continuum. In this limit, one can formally apply a perturbatively exact quantum virial expansion in the fugacity Mulkerin et al. (2022). We will now briefly discuss how this expansion is related to the polaron theory, and how this allows us to demonstrate that the trion picture results of Refs. Esser et al. (2000); Zipfel et al. (2022) are contained within the polaron formalism.

When $T\gg E_{F}$ and $z\simeq\beta E_{F}\ll 1$, we can formally expand the Fermi occupation function which then coincides with the Boltzmann distribution $f_{\bf k}\simeq ze^{-\beta\epsilon_{{\bf k}}}$. Likewise, to leading order in $z$ we have ${\mathcal{T}}\simeq{\mathcal{T}}_{0}$ — see Eq. (12). Within this expansion, the exciton self-energy in Eq. (10) becomes

In other words, to leading order in the fugacity, the self-energy is determined by two-body interactions weighted by a Boltzmann distribution.

Focusing first on the attractive branch, the dominant contribution to the self-energy arises from the pole of the $T$ matrix at the trion energy. We therefore expand the $T$ matrix for $\omega\simeq-\varepsilon_{T}+\epsilon_{T{\bf q}}$, with the result

where $Z_{T}\equiv 2\pi\varepsilon_{T}/m_{r}$. The $T$ matrix can be expressed in terms of the vacuum ($E_{F}=0$) trion wave function $\tilde{\eta}_{{\bf q}_{r}}^{({\bf 0})}$ for a contact electron-exciton interaction (see Appendix F). In our case, the relative momentum is ${\bf q}_{r}={\bf q}m_{X}/m_{T}$, and therefore the kinetic energy of the relative motion is $\epsilon_{r{\bf q}_{r}}=\epsilon_{\bf q}-\epsilon_{T{\bf q}}$. Using the expression for the vacuum trion wave function in the center of mass frame, $\tilde{\eta}_{{\bf q}_{r}}^{({\bf 0})}=\displaystyle\frac{\sqrt{Z_{T}}}{\varepsilon_{T}+\epsilon_{r{\bf q}_{r}}}$ (see Appendix C), we have

Here, the numerator is derived by manipulating the relation between the trion wave function and $Z_{T}$, and is momentum independent. However, this is a special property of the contact electron-exciton interaction, and Eq. (27) in fact yields the correct generalization for an arbitrary electron-exciton interaction that leads to the formation of a trion. In other words, it can be applied for any realistic trion wave function. For details of the generalization to arbitrary interactions, see Appendix F.

Using the approximation for the vacuum $T$ matrix, Eq. (27), the self-energy in Eq. (25) becomes

where ${\mathcal{P}}$ denotes the principal value. This explicitly relates the self-energy calculated within the polaron theory to the trion wave function in the regime where the fugacity is small, and importantly it applies for any realistic trion wave function. A similar approach involving trion wave functions has been used to calculate absorption Bronold (2000). Note that the real part diverges logarithmically when $\omega+\varepsilon_{T}\to 0^{-}$, and therefore it cannot in general be neglected close to the onset of the attractive branch.

For the repulsive branch, we can again apply Eq. (25) to find the leading contribution at small fugacity. In the regime where $E_{F}\ll\varepsilon_{T}$, the width of the repulsive branch is much smaller than the trion binding energy, and to lowest order in the fugacity, we can simply evaluate the repulsive polaron self-energy within the repulsive branch by taking $\omega=0$. We then use the fact that, to logarithmic accuracy, the logarithmic behavior at small momentum is generic for the vacuum $T$ matrix for any short-range interaction Adhikari (1986); Landau and Lifshitz (2013) and thus we have

where $\gamma_{\rm E}\simeq 0.5772$ is the Euler-Mascheroni constant. Here, we have evaluated the integral by noting that the integral is dominated by $\epsilon_{\bf q}\sim 1/\beta$ for small $\beta\varepsilon_{T}$, and the inclusion of $\gamma_{\rm E}$ originates from expanding the integral to the first subleading order in $\beta\varepsilon_{T}$.

The self-energies in Eqs. (28) and (29) can now be directly inserted into the Dyson equation (9) to yield the absorption in Eq. (13) or the photoluminescence in Eq. (18). To be explicit, within the virial expansion we have the exciton spectral function

This yields a broad continuum for the attractive branch, where the spectral weight vanishes at $\omega+\varepsilon_{T}\to 0^{-}$, and when we are far below this onset, we have $\Sigma_{A}(\omega)/\omega\to 0$. Therefore we have an exponential tail modulated by the trion wave function:

The peak of the absorption is between the onset and the tail, and therefore it will not correspond to the vacuum trion energy. This can lead to an overestimate of the trion binding energy in experiments Zipfel et al. (2022), as shown in the accompanying paper Mulkerin et al. (2022). From Eq. (29), we find that the repulsive branch is a Lorentzian of width $\Gamma_{R}=\pi(m/m_{r})E_{F}/[\pi^{2}+\ln^{2}(e^{\gamma_{\rm E}}\beta\varepsilon_{T})]$. We have compared in Fig. 7(c) this expression against the numerical evaluation of the repulsive branch half linewidth $\Gamma_{R}-\eta$ (where we have removed the effect of the intrinsic exciton lifetime), finding excellent agreement at low doping, inside the region of validity of the virial expansion.

For the photoluminescence, we find

where we have used the fact that the width of the repulsive branch is much smaller than the temperature, and thus the repulsive branch is very weakly modified by the Boltzmann factor.

Finally, we discuss how our theory relates to the calculation of electron recoil in previous trion-picture calculations such as by Esser et al., Refs. Esser et al. (2000, 2001). As we shall demonstrate, that theory corresponds to a weak-interaction limit of the low doping/high temperature version of our polaron theory. To see this, we take the limit of a small self-energy in the Dyson equation in Eq. (9) as follows:

Since the $1/\omega$ terms only have a pole at $\omega=0$ (i.e., at the bare exciton energy), the attractive branch is obtained from the imaginary part of the self-energy. The detailed balance equation (18) then implies that the corresponding PL from the attractive branch is

where we used Eq. (28). Up to frequency-independent prefactors, this precisely matches the result of Esser et al. Thus, the trion-picture PL is already contained within the polaron picture. However, as we have already argued, the trion picture calculation assumes that we are in a weakly interacting limit where the self-energy is much smaller than the frequency. This assumption manifestly breaks down close to the onset of the attractive branch, where the real part of the self-energy diverges, and hence the previous trion picture calculation fails to correctly describe the onset and shape of the attractive branch. On the other hand, it correctly identifies the exponential tail of the PL, which is dominated by the imaginary part of the self-energy.

For the repulsive branch, the trion picture again uses the weak-interaction limit of the Dyson equation (33), this time neglecting even the second term on the right hand side. This results in

with a small correction that counteracts the oscillator strength transferred to the attractive branch Esser et al. (2001). We see that the trion picture fails to describe the Lorentzian shape of the repulsive polaron, which arises from the exciton-electron scattering states as well as the full Dyson series.