Interface excitons at lateral heterojunctions in monolayer semiconductors

We study the interface exciton at lateral type II heterojunctions of monolayer transition metal dichalcogenides (TMDs), where the electron and hole prefer to stay at complementary sides of the junction. We find that the 1D interface exciton has giant binding energy in the same order as 2D excitons in pristine monolayer TMDs although the effective radius (electron-hole separation) of interface exciton is much larger than that of 2D excitons. The binding energy, exciton radius, and optical dipole strongly depends on the band offset at the junction. The intervalley coupling induced by the electron-hole Coulomb exchange interaction and the quantum confinement effect at interfaces of a closed triangular shape are also investigated. Small triangles realize 0D quantum dot confinement of excitons, and we find a transition from nondegenerate ground state to degenerate ones when the size of the triangle varies. Our findings may facilitate the implementation of the optoelectronic devices based on the lateral heterojunction structures in monolayer semiconductors.

Figure 1

(a),(d) Schematics of the single and double heterojunctions formed by ${\mathrm{MoSe}}_2$ and ${\mathrm{WSe}}_2$. The blue, red, and yellow spheres, respectively, denote $\mathrm{W},\mathrm{Mo}$, and $\mathrm{Se}$ atoms. The purple shadow regions denote the type II interfaces. (b),(e) Schematics of the monolayer TMDs p-n junction defined by the electrostatic gating. The red and blue regions denote the n and p regions, respectively. (c),(f) Illustration of the band edge profile across the single lateral junction in (a) or (b) and the double lateral junctions in (d) or (e). Here, $w$ is the width of the interface, which is zero for the case of (a) and (d), and has a finite value for the gate defined junctions in (b) and (e). $V_0$ is the average lattice potential of electron and hole in (a) or (d) and gate voltage in (b) or (e). The $\delta$ characterizes the difference of the band offsets for electron and hole.

Figure 2

(a) The binding energy $E_b$, (b) the exciton radius $a_b$, and (c) optical dipole ratio $\left|D\left(V_0\right)/D_0\right|$ versus the strength of the band offset $V_0$ for different on-site Coulomb potential $U$. The red sphere, blue triangle, and magenta diamond symbols, respectively, represent results for different on-site electrostatic energy $U=-0.79,-1.19$, and $-2.98\mathrm{eV}$. The physical observables converge to the same value at high voltage $V_0$ regardless of $U$, showing the spatial separation nature of the interface exciton for large $V_0$. There are two characteristic behaviors, in the regime of small band offset $(V_0\ll 0.1$ eV) and large interface potential $(V_0>0.1$ eV), which reflects the competition between Coulomb interaction and the band offset. Here, $D_0$ is the value of $D\left(V_0\right)$ at $V_0=0$ (i.e., that of the 2D exciton in the absence of interface). See text for details.

Figure 3

(a) Center-of-mass wave function and (b) relative motion spatial probability distribution at different $V_0$. The white dashed line denotes the central position of the interface. From (a) we see that the wave function undergoes a transition from an extended state to a localized state as $V_0$ increases, demonstrating the competition between the Coulomb interaction and the band offset. From (b), we see that for larger $V_0$ the electron-hole pair tends to be farther apart and electron-hole overlap is greatly reduced.

Figure 4

(a) Left: hole, and right: electron reduced wave function versus $V_0$. Since the translational symmetry is preserved along the $y$ direction, we only show the cross section of the reduced wave function along the $x$ direction at $y=0$. Apparently the electron-hole pair is effectively separated at high $V_0$. However, for $V_0\sim 0.1–0.2$ eV a finite tail into the barrier region remains which gives rise to a considerable electron-hole overlap. (b) Schematics of an interface exciton for $V_0>0.1$ eV. A considerable overlap between the electron and hole wave functions survives even for a great spatial separation of the electron and hole. Here, the thin lines denote the band offsets. The dark blue and dark red regions, respectively, denote the cross section of the electron and hole's reduced wave functions along the $x$ direction at $y=0$.

Figure 5

(a) Illustration of the intervalley exchange interaction. Mediated by the exchange part of the Coulomb interaction (the orange arrow), the exciton may change their pesudospin from $-K$ to $+K$ as indicated by the green arrow, effectively resulting in a valley exchange channel. (b) The exchange interaction $J$ leads to a splitting between the degenerate $\pm K$ states. The ${\sigma }_{+},{\sigma }_{-},{\sigma }_x,{\sigma }_y$, respectively, denotes the polarizations of the light fields which can pump the corresponding states. The $x$ and $y$ directions are shown in Fig. 1. (c) The valley coupling strength $J$ versus $V_0$ for a symmetric and asymmetric heterojunction, respectively. Rotational symmetry requires vanishing $J$ at $V_0=0$ for a symmetric heterojunction, in contrast to an asymmetric heterojunction for which $J$ increases at low $V_0$. At high voltage valley coupling of an interface exciton is small regardless of symmetry of the heterojunction because of the reduced electron-hole overlap.

Figure 6

The binding energy $E_b$ versus the cutoff of principal quantum number $n$ for different potential strength $V_0$. The dotted red line with circle symbol, dot-dashed blue line with triangle symbol, magenta short-dashed line with pentagon symbol, and olive solid line with diamond symbol represent the ground state energies with potential strength $V_0=0.3,0.4,0.5,0.6$ eV, respectively. The other parameters are chosen as $\varepsilon \approx 1.1{\varepsilon }_0,m_e=0.434m_0,$ and $m_h=0.533m_0$ with vacuum dielectric constant ${\varepsilon }_0$ and free electron mass $m_0$. It is clear that the ground state energies converge very quickly along with increasing principal quantum number even for relative large potential strength.

Figure 7

(a) The binding energy $E_b$, (b) the exciton radius $a_b$, and (c) the optical dipole ratio $\left|D\left(V_0\right)/D_0\right|$ versus the strength of the band offset $V_0$. The blue triangle symbols with solid line and the red sphere symbols with dashed line, respectively, represent the numerical results obtained from the TB model and continuous model, which are respectively denoted as "TB" and "2D" in the plot.

Figure 8

(a) The binding energy of the interface exciton at lateral double heterojunctions versus the width of the double heterojunctions $L$. (b) The typical reduced wave function of electrons (blue solid lines) and holes (red dashed lines) for different widths of double heterojunctions. The three different widths are $L\approx 1.3\text{nm},3.3\text{nm},10\text{nm}$. The corresponding central regions are denoted by the shadow areas. See text for the details. (c) The binding energy of the interface exciton versus the potential strength of the double heterojunctions $V_0$ for $L\approx 3.3\text{nm}$.

Figure 9

Schematics of the triangular heterostructure formed by monolayer ${\mathrm{WSe}}_2-{\mathrm{MoSe}}_2$. The blue, red, and yellow spheres, respectively, denote $\mathrm{W}$ atoms, $\mathrm{Mo}$ atoms, and $\mathrm{Se}$ atoms. The red shadow region denotes the region of triangular band offset.

Figure 10

The binding energy of the exciton of the closed triangular interface $E_b$ versus the quantum number $m$ and $n$. The parameters are chosen as $R=30a,L_{\mathrm{SC}}=60a,W_{\mathrm{SC}}=36\sqrt{3}a,V_0=0.3\text{eV}$. The binding energy converges quickly along the quantum numbers. In the following calculation, we set the same cutoff $n_{\mathrm{cutoff}}=m_{\mathrm{cutoff}}=15$ for different $V_0$.

Figure 11

The binding energy versus (a) the band offset $V_0$ and (b) the size of the quantum dot $R$. The parameters for (a) are chosen as $R=21a$. The potential strength is chosen as $V_0=0.3\text{eV}$. The size of the supercell is sufficiently large.

Figure 12

The reduced wave function of the electron (left panel) and hole (right panel) for closed triangular interface with different sizes $R=21a,30a,45a$, namely $R\approx 7\text{nm},10\text{nm},15\text{nm}$ from the top to bottom. The potential strength is chosen as $V_0=0.3\text{eV}$.

Figure 13

(a) The energy level schemes for $t<0$ and $t>0$ when $\theta =\mp 2\pi /3$ for $\tau =\pm 1$ valleys. The optical selection rules are also depicted. Here, $\tau =\pm 1$ are valley index denoting $K$ and $-K$ valleys. The light red and light blue arrows, respectively, denote the right $\left({\sigma }_{+}\right)$ and left $\left({\sigma }_{-}\right)$ polarizations of the light fields coupling to the corresponding exciton states. (b) The numerical results of the eigenenergies $E$ and (c) the absolute value of the transition coefficient $t$ versus the size of the triangular quantum dot $R$. Obviously, there is a transition when the ground state varies from a nondegenerate state to a degenerate one. Additionally, the transition coefficient changes from a negative value to a positive one.

Figure 14

(a) The intervalley intraedge hopping $p$ versus the size of quantum dot $R$. The magnitude of $p$ almost decreases exponentially as the size of the quantum dot increases. (b) The energy level schemes for $t<0$ and $t>0$ for ${\theta }_{+}=-{\theta }_{-}=-2\pi /3$. The optical selection rules are also depicted. The light red and light blue arrows, respectively, denote the right $\left({\sigma }_{+}\right)$ and left $\left({\sigma }_{-}\right)$ polarizations of the light fields coupling to the corresponding exciton states. The different sizes of the arrows denote the coupling strengths between the exciton states and the light field.