Diffusion quantum Monte Carlo study of excitonic complexes in two-dimensional transition-metal dichalcogenides

Excitonic effects play a particularly important role in the optoelectronic behavior of two-dimensional semiconductors. To facilitate the interpretation of experimental photoabsorption and photoluminescence spectra we provide (i) statistically exact diffusion quantum Monte Carlo binding-energy data for a Mott-Wannier model of (donor/acceptor-bound) excitons, trions, and biexcitons in two-dimensional semiconductors in which charges interact via the Keldysh potential, (ii) contact pair-distribution functions to allow a perturbative description of contact interactions between charge carriers, and (iii) an analysis and classification of the different types of bright trions and biexcitons that can be seen in single-layer molybdenum and tungsten dichalcogenides. We investigate the stability of biexcitons in which two charge carriers are indistinguishable, finding that they are only bound when the indistinguishable particles are several times heavier than the distinguishable ones. Donor/acceptor-bound biexcitons have similar binding energies to the experimentally measured biexciton binding energies. We predict the relative positions of all stable free and bound excitonic complexes of distinguishable charge carriers in the photoluminescence spectra of ${\mathrm{WSe}}_2$ and ${\mathrm{MoSe}}_2$.

Figure 1

(a) Spin-split valence and conduction bands for ${\mathrm{MoSe}}_2$ (left) and ${\mathrm{WSe}}_2$ and ${\mathrm{WS}}_2$ (right). We only show the spin splitting of the conduction band; the spin splitting of the valence band is much larger, so that no holes in the lower spin-split valence band are expected at room temperature [3]. (b) and (c) Classification of biexciton recombination processes in molybdenum and tungsten dichalcogenides, respectively. $\mathrm{\Delta }$ is the band gap, while ${\mathrm{\Delta }}^{\prime }$ is the spin splitting of the conduction band. $E_{\delta }^{\prime }\equiv E_{\mathrm{XX}}-E_{\mathrm{X}}$ is the difference between the total energies $E_{\mathrm{XX}}$ and $E_{\mathrm{X}}$ of a biexciton and an exciton. $\hslash \omega$ indicates the photon energies at which peaks in photoluminescence spectra are expected to appear. ${\mathrm{XX}}_{k_3{\sigma }_3{k}_4{\sigma }_4}^{k_1{\sigma }_1{k}_2{\sigma }_2}$ denotes a biexciton consisting of conduction-band electrons in valleys $k_1$ and $k_2$ with spins ${\sigma }_1$ and ${\sigma }_2$ and valence-band holes in valleys $k_3$ and $k_4$ with spins ${\sigma }_3$ and ${\sigma }_4$. For example, the biexcitons shown in (a) are both denoted by ${\mathrm{XX}}_{K\downarrow K^{\prime }\uparrow }^{K\downarrow K^{\prime }\uparrow }$.

Figure 2

As Fig. 1, but for negative trions in molybdenum and tungsten dichalcogenides. $E_{\delta }\equiv E_{{\mathrm{X}}^{-}}$ is the total energy $E_{{\mathrm{X}}^{-}}$ of a negative trion. ${\mathrm{T}}_{k_3{\sigma }_3}^{k_1{\sigma }_1{k}_2{\sigma }_2}$ denotes a trion consisting of conduction-band electrons in valleys $k_1$ and $k_2$ with spins ${\sigma }_1$ and ${\sigma }_2$ and a valence-band hole in valley $k_3$ with spin ${\sigma }_3$.

Figure 3

Dimensionless interaction potential between charge carriers in a 2D semiconductor, as defined in Eq. (11). The inset shows the percentage error in different approximations [Eqs. (13), (14), and (15)] to the Keldysh interaction of Eq. (12).

Figure 4

Expected photoemission spectra for (a) ${\mathrm{MoSe}}_2$ and (b) ${\mathrm{WSe}}_2$, showing lines for the different complexes studied in this work. $\mathrm{\Delta }$ and ${\mathrm{\Delta }}^{\prime }$ are the quasiparticle band gap and the spin splitting of the conduction band, respectively. The numerical values of ${\mathrm{\Delta }}^{\prime }$ are taken from density-functional-theory calculations with the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional [3]. $E_{\mathrm{X}}$ is the total energy of an exciton. The lines show the frequency relative to the bright exciton peak arising due to the recombination of a single electron-hole pair in each complex; see Sec. 3c. For example, the ${\mathrm{D}}^0\mathrm{X}$ line shows the frequency relative to the exciton peak of the process ${\mathrm{D}}^0\mathrm{X}\to {\mathrm{D}}^0+\gamma$. The trion and biexciton peaks labeled “SD” arise from semidark complexes, and are offset by $2{\mathrm{\Delta }}^{\prime }$, as explained in Sec. 2a; the exciton peak labeled “dark” arises from the process described in Sec. 2b; the other peaks arise from bright complexes. Donor- and acceptor-bound exciton peaks are shown with very low intensity due to the marginal stability of these complexes.

Figure 5

DMC ground-state (GS) energy of a biexciton with distinguishable particles at mass ratio $\sigma =1$ against time step, with the logarithmic interaction between charges. The configuration population was varied in inverse proportion to the time step. The reduced mass is $\mu =0.5m_0$ and $r_{*}=r_0$.

Figure 6

DMC ground-state (GS) energy of a negative trion at mass ratio $\sigma =1$ and $r_{*}=0.5a_0^{*}$ against time step, with the Keldysh interaction between charges. The configuration population was varied in inverse proportion to the time step.

Figure 7

DMC ground-state (GS) energy of a donor-bound biexciton at $\sigma =0.3$ and $r_{*}=0.077a_0^{*}$ against time step, with the Keldysh interaction between charges. The configuration population was varied in inverse proportion to the time step.

Figure 8

Exciton ground-state (GS) energy evaluated using DMC and a finite-element method. The plot also shows the approximations to the ground-state energy obtained by first-order perturbation theory about the Coulomb limit (green) and by using the logarithmic approximation to the Keldysh potential (red).

Figure 9

Difference of dimensionless exciton energies with the Keldysh interaction and the logarithmic approximation to the Keldysh interaction, calculated using first-order perturbation theory within VMC. The solid line is a fit of $a\sqrt{a_0^{*}/r_{*}}$ to the VMC data, with $a=0.871\left(2\right)$.

Figure 10

PDF of an exciton with $\sigma =0.3$ and $r_{*}=6.15a_0^{*}$. The contact PDF is extracted by fitting the numerical results to Eq. (44).

Figure 11

(a) Electron-hole contact PDFs of an exciton (in black) and a negative trion (in color). (b) Electron-electron contact PDFs of a negative trion. These data were presented in Ref. [25], and are shown here for completeness.

Figure 12

DMC binding energies of biexcitons with distinguishable electrons and distinguishable holes and biexcitons with distinguishable electrons and indistinguishable holes against mass ratio $\sigma$ with (a) the Coulomb interaction $\left(r_{*}=0\right)$, (b) the Keldysh interaction with $r_{*}=8a_0^{*}$, and (c) the logarithmic interaction [Eq. (14)] between charge carriers.

Figure 13

DMC Born-Oppenheimer potential energy of a heavy-hole biexciton with distinguishable electrons against the hole separation for (a) the Keldysh interaction with $r_{*}=a_0^{*},2a_0^{*},4a_0^{*},6a_0^{*}$, and $8a_0^{*}$; and (b) the logarithmic interaction between charge carriers. The zero of the Born-Oppenheimer potential energy in the plot is twice the isolated exciton energy.

Figure 14

(a) DMC binding energies of biexcitons with distinguishable particles against rescaled susceptibility $r_{*}/(a_0^{*}+r_{*})$. (b) DMC binding energies of biexcitons with distinguishable particles against rescaled mass ratio $\sigma /(1+\sigma )$. (c) DMC binding energies of biexcitons with distinguishable particles against rescaled susceptibility and rescaled mass ratio. The DMC results for distinguishable particles were reported in Ref. [25].

Figure 15

Binding energies of biexcitons with distinguishable particles of equal mass $\left(\sigma =1\right)$ against rescaled susceptibility $r_{*}/(a_0^{*}+r_{*})$, as calculated using DMC [25] and PIMC [20].

Figure 16

PDF $g_{\mathrm{XX}}\left(r\right)$ of a biexciton with distinguishable particles interacting via the logarithmic interaction plotted against interparticle separation at two different electron-hole mass ratios $\sigma$.

Figure 17

Electron-hole contact PDF of a biexciton with distinguishable particles against rescaled susceptibility. The black line indicates twice the exciton electron-hole contact PDF. The inset shows the electron-electron contact PDF. These data were presented in Ref. [25], and are shown here for completeness.

Figure 18

(a) DMC binding energies of trions with distinguishable particles against rescaled susceptibility $r_{*}/(a_0^{*}+r_{*})$. (b) DMC binding energies of trions with distinguishable particles against rescaled mass ratio $\sigma /(1+\sigma )$. (c) DMC binding energies of trions with distinguishable particles against rescaled susceptibility and rescaled mass ratio. These data were presented in Ref. [25], and are shown here for completeness.

Figure 19

(a) DMC binding energies of donor-bound excitons with against rescaled susceptibility $r_{*}/(a_0^{*}+r_{*})$. (b) DMC binding energies of donor-bound excitons against rescaled mass ratio $\sigma /(1+\sigma )$. (c) DMC binding energies of donor-bound excitons against rescaled susceptibility and rescaled mass ratio.

Figure 20

Electron-hole contact PDF of a donor-bound exciton. The solid lines were obtained using the fitting function reported in the Supplemental Material [53].

Figure 21

(a) DMC binding energies of donor-bound trions with distinguishable particles against rescaled susceptibility $r_{*}/(a_0^{*}+r_{*})$. (b) DMC binding energies of donor-bound trions with distinguishable particles against rescaled mass ratio $\sigma /(1+\sigma )$. (c) DMC binding energies of donor-bound trions with distinguishable particles against rescaled susceptibility and rescaled mass ratio.

Figure 22

Electron-hole contact PDFs of a donor-bound trion complex. The inset shows electron-electron contact PDFs. The solid lines were obtained using the fitting function reported in the Supplemental Material [53].

Figure 23

(a) DMC binding energies of donor-bound biexcitons with distinguishable particles against rescaled susceptibility $r_{*}/(a_0^{*}+r_{*})$. (b) DMC binding energies of donor-bound biexcitons with distinguishable particles against rescaled mass ratio $\sigma /(1+\sigma )$. (c) DMC binding energies of donor-bound biexcitons with distinguishable particles against rescaled susceptibility and rescaled mass ratio.

Figure 24

Born-Oppenheimer potential energy of a complex of three positive, fixed ions and two electrons, with the positive ions placed at the corners of an equilateral triangle. Example for $r_{*}/a_0^{*}=1$.

Figure 25

(a) Born-Oppenheimer potential energy of a complex of three positive, fixed ions and two electrons. We fix two of the ions and change the position of the third one. Example for $r_{*}/a_0^{*}=1$. (b) Vertex of the triangle of fixed, positive charges in greater detail.

Figure 26

(a) Electron-hole contact PDFs of a donor-bound biexciton. For comparison, the black line indicates twice the exciton contact PDF. (b) Electron-electron and (c) hole-hole contact PDFs of a donor-bound biexciton. The solid lines were obtained using the fitting function reported in the Supplemental Material [53].

Figure 27

DMC binding energies of negative trions $\left({\mathrm{X}}^{-}\right)$, biexcitons (XX), donor-bound excitons $\left({\mathrm{D}}^{+}\mathrm{X}\right)$, donor-bound trions $\left({\mathrm{D}}^0\mathrm{X}\right)$, and donor-bound biexcitons $\left({\mathrm{D}}^{+}\mathrm{XX}\right)$. Particles in the complexes interact via the logarithmic interaction. The ${\mathrm{X}}^{-}$ data were presented in Ref. [19], and are shown here for completeness.