Charge carrier complexes in monolayer semiconductors

The photoluminescence (PL) spectra of monolayer (1L) semiconductors feature peaks ascribed to different charge-carrier complexes. We perform diffusion quantum Monte Carlo simulations of the binding energies of these complexes and examine their response to electric and magnetic fields. We focus on quintons (charged biexcitons), since they are the largest free charge-carrier complexes in undoped and low doping transition-metal dichalcogenides (TMDs). We examine the accuracy of the Rytova-Keldysh interaction potential between charges by comparing the binding energies (BEs) of charge-carrier complexes in 1L-TMDs with results obtained using ab initio interaction potentials. Magnetic fields $<8\mathrm{T}$ change BEs by $\sim 0.2\mathrm{meV}{\mathrm{T}}^{-1}$, in agreement with experiments, with BE variations of different complexes being very similar. Our results will help identify charge complexes in the PL spectra of 1L semiconductors.

Figure 1

(a) Upper spin-split VB and spin-split CB for 1L-${\mathrm{MoSe}}_2$ and 1L-${\mathrm{WSe}}_2$. The spin-split VB is $>150\mathrm{meV}$ [7], so we only show the upper VB. [(b) and (c)] Classification of quinton recombination processes in 1L-Mo and 1L-W-TMDs. $E_{\delta }=E_{{\mathrm{XX}}^{-}}-E_{{\mathrm{X}}^{-}}$ is the difference between the total energies $E_{{\mathrm{XX}}^{-}}$ and $E_{{\mathrm{X}}^{-}}$ of ${\mathrm{XX}}^{-}$ and ${\mathrm{X}}^{-}$. $\hslash \omega$ indicates the photon energies at which ${\mathrm{XX}}^{-}$ peaks in PL spectra are expected. ${\left({\mathrm{XX}}^{-}\right)}_{k_4{\sigma }_4{k}_5{\sigma }_5}^{k_1{\sigma }_1{k}_2{\sigma }_2{k}_3{\sigma }_3}$ denotes a quinton consisting of CB e in valleys $k_1, k_2$, and $k_3$ with spins ${\sigma }_1, {\sigma }_2$, and ${\sigma }_3$ and VB h in valleys $k_4$ and $k_5$, with spins ${\sigma }_4$ and ${\sigma }_5$. For example, the quintons in (a) are both ${\left({\mathrm{XX}}^{-}\right)}_{K\uparrow K^{\prime }\downarrow }^{K\uparrow K\downarrow K^{\prime }\downarrow }$. Unlike Figs. 1 and 2 of Ref. [29], we only show complexes with distinguishable charge carriers, because they are stable and should be experimentally observable. Note that in (b) recombination may be followed by momentum exchange, giving an additional contribution of $2{\mathrm{\Delta }}^{\prime }$.

Figure 2

Classification of e-h recombination processes, showing examples of [(a) and (b)] dark excitons, where recombination is prohibited, (c) bright excitons, where recombination is allowed, and (d) semidark excitons, where recombination is assisted by phonon emission/absorption $E_{ph}$.

Figure 3

(a) DMC BEs of ${\mathrm{XX}}^{-}$ as a function of $r_{*}/(r_{*}+a_0^{*})$. (b) DMC BEs of ${\mathrm{XX}}^{-}$ as a function of $\sigma /(\sigma +1)$. (c) ${\mathrm{XX}}^{-}$ BEs as a function of rescaled susceptibility and mass ratio. Above the yellow line $E_{{\mathrm{XX}}^{-}}^{\mathrm{DE}}<$ XX electron affinity, so that X and ${\mathrm{X}}^{-}$ are the most energetically competitive. Below the yellow line the situation is reversed, so that XX and free e are the most competitive. The white stars show the mass ratios and in-plane susceptibility of 1L-${\mathrm{MoS}}_2$ at (0.45,0.93), 1L-${\mathrm{MoSe}}_2$ at (0.46, 0.94), 1L-${\mathrm{MoTe}}_2$ at (0.50,0.98), 1L-${\mathrm{WS}}_2$ at (0.46,0.91), 1L-${\mathrm{WSe}}_2$ at (0.46,0.93), 1L-${\mathrm{WTe}}_2$ at (0.41,0.95), where the first and second numbers in brackets are $\sigma /(\sigma +1)$ and $r_{*}/(r_{*}+a_0^{*})$, respectively. ${\mathrm{XX}}^{-}$ BEs are between $0.00736\left(5\right)R_{\mathrm{y}}^{*}$ and $0.0288\left(1\right)R_{\mathrm{y}}^{*}$, with the numbers in brackets the BE error bars.

Figure 4

Unscreened Coulomb interaction potential: $v_{\mathrm{Coul}}\left(r\right)=1/r$; RKI: $v_K\left(r\right)=V(r/r_{*})/r_{*}$, with $r_{*}$ of 1L-${\mathrm{MoS}}_2$ in vacuum from Table 1; RPAI: $v_{\mathrm{RPA}}\left(r\right)$. $a_{{\text{1L-MoS}}_2}=3.15$ Å [7].

Figure 5

BE of a donor atom in 1L-${\mathrm{MoS}}_2$ as a function of donor impurity charge $Z$. VMC and DMC results are compared with the numerical data of Ref. [60].

Figure 6

Theoretical BEs of (a) X, (c) ${\mathrm{X}}^{-}$, and (e) XX as a function of perpendicular magnetic field for 1L-${\mathrm{WSe}}_2$ in vacuum. We use the ab initio mass and $r_{*}$ parameters of Table 1. The CoM contribution for X is $E_{\mathrm{X}}^{\mathrm{b},\mathrm{CoM}}={\left(E_{\mathrm{X}}^{\mathrm{b}}\right)}_{B=0}+E_{\mathrm{e}}+E_{\mathrm{h}}-E_{\mathrm{X}}^{\mathrm{CoM}}$; for ${\mathrm{X}}^{-}$ is $E_{{\mathrm{X}}^{-}}^{\mathrm{b},\mathrm{CoM}}={\left(E_{{\mathrm{X}}^{-}}^{\mathrm{b}}\right)}_{B=0}+E_{\mathrm{e}}-E_{{\mathrm{X}}^{-}}^{\mathrm{CoM}}$; and for XX is $E_{\mathrm{XX}}^{\mathrm{b},\mathrm{CoM}}={\left(E_{\mathrm{XX}}^{\mathrm{b}}\right)}_{B=0}+2E_{\mathrm{X}}^{\mathrm{CoM}}-E_{\mathrm{XX}}^{\mathrm{CoM}}$. Experimental BEs of (b) X, (d) ${\mathrm{X}}^{-}$, (f) XX for 1L-${\mathrm{WSe}}_2$ encapsulated in hBN, compared with DMC ones using $\epsilon ={\epsilon }_0$ and $r_{*}=48$ Å and the fit to Eq. (21).

Figure 7

DMC BE shift for (a) X, (b) XX, (c) donor atoms as a function of $F^2$ for different 1L-TMDs in vacuum and encapsulated in hBN. Error bars in (a) and (c) are smaller than the symbols. The solid and dashed lines are BEs determined by the polarizabilities in Table 6 for 1L-TMDs in vacuum and encapsulated by hBN. The vertical dotted lines correspond to $F=50\mathrm{mV}{\mathrm{nm}}^{-1}$, beyond which VMC energy minimization does not result in bound-state wave functions.

Figure 8

DMC BE shifts for (a) ${\mathrm{X}}^{-}$ and (b) ${\mathrm{X}}^{+}$ as a function of $F^2$ for different 1L-TMDs in vacuum and encapsulated in hBN. Where error bars are not visible they are smaller than the symbols. The solid and dashed lines show BEs determined by the polarizabilities in Table 6

Figure 9

(a) Dependence of $C_{\mathrm{X}}$ on susceptibility. The markers show fitted $C_{\mathrm{X}}$. The line is a quadratic fit to the points as for Eq. (22). (b) Shift in energy of X with equal $m_{\mathrm{e}}^{*}$ and $m_{\mathrm{h}}^{*}$ due to external magnetic field, after subtracting the large-$\tilde{B}$ behavior from Eq. (19), for several ${\tilde{r}}_{*}$. Markers indicate finite-element method results, while lines show the fit of Eq. (21).