Excitonic transport driven by repulsive dipolar interaction in a van der Waals heterostructure.

Dipolar bosonic gases are currently the focus of intensive research due to their interesting many-body physics in the quantum regime. Their experimental embodiments range from Rydberg atoms to GaAs double quantum wells and van der Waals heterostructures built from transition metal dichalcogenides. Although quantum gases are very dilute, mutual interactions between particles could lead to exotic many-body phenomena such as Bose-Einstein condensation and high-temperature superfluidity. Here, we report the effect of repulsive dipolar interactions on the dynamics of interlayer excitons in the dilute regime. By using spatial and time-resolved photoluminescence imaging, we observe the dynamics of exciton transport, enabling a direct estimation of the exciton mobility. The presence of interactions significantly modifies the diffusive transport of excitons, effectively acting as a source of drift force and enhancing the diffusion coefficient by one order of magnitude. The repulsive dipolar interactions combined with the electrical control of interlayer excitons opens up appealing new perspectives for excitonic devices.

Van der Waals (vdW) heterostructures with a type-II band alignment enable the formation of long-lived interlayer excitons (IXs) composed of charges that are spatially separated in distinct layers[1]. In contrast to intralayer excitons in monolayer transition metal dichalcogenides (TMDCs), the spatial separation of charges gives rise to a sizable permanent out-of-plane electrical dipole moment, which makes IXs a promising platform to realize electrically controlled excitonic devices[2]. These are a new class of solid-state devices[3,4] for information and signal processing, analogous to electronic or spintronic devices, but based on encoding information in the amplitude and/or pseudo-spin of exciton currents, which can be controlled using electrical fields. As a result, exciton transport in vdW heterostructures has recently attracted a growing interest[5-8]. Many basic questions however remain open, such as the nature of mutual interactions between IXs. Moreover, exciton transport in vdW heterostructures has so far been attributed only to exciton diffusion currents and the observation of long IX transport distances is usually assigned to a large effective diffusion coefficient, a phenomenological parameter that conceals the role of the repulsive dipolar interaction on IX transport. In analogy with semiconducting devices, it would also be advantageous to introduce exciton drift currents in order to increase the range of exciton motion in a device and enhance control over exciton transport.

Here, by imaging the temporal evolution of the IX cloud, we reveal repulsive dipolar exciton-exciton interactions as the driving force behind IX transport, acting as an effective source of a drift field. Concurrently, we deduce the exciton mobility directly from power-dependent drift velocities. Our findings, combined with the electrical control of IXs, pave the way to controlling the motion of IXs over long distances.

In our work, instead of investigating IXs in a heterobilayer, we introduce a monolayer hBN (1L-hBN) as a thin spacer between monolayers of WSe2 (1L-WSe2) and MoSe2 (1L-MoSe2). The motivations are twofold: firstly, the spacer can increase the separation between the electrons and the holes, enhancing the size of the electrical dipole by a factor of ~1.5[5]. Secondly, the presence of a moiré potential with an amplitude on the order of 100 to 200 meV[9-11] in MoSe2/WSe2 heterostructures localizes IXs[12] by effectively decreasing the diffusion coefficient[7,13]. The spacer weakens the moiré potential and reduces its period due to the large lattice mismatch between hBN and MoSe2/WSe2 while retaining a sufficiently strong transition dipole moment for hosting bright IXs[14].

Figures 1a and 1b show an optical image and a schematic of the device. It consists of a WSe2/hBN/MoSe2 heterotrilayer encapsulated in hBN, with a transparent global top gate and several local back gates. Multiple local back gates allow us to control the exciton flux by applying a laterally modulated vertical electric field (Figure 1b). The yellow-shaded area in Figure 1a indicates the heterotrilayer region where we perform the spatial and time-resolved photoluminescence (PL) measurements at 4.6 K. We use a sub-picosecond 725 nm pulsed laser with an 80 MHz repetition rate to excite the left part of the heterostructure and to generate an initial population of IXs (red spot in Figure 1a). Figure 1c presents the PL intensity of IXs at the excitation spot as a function of emission wavelength and average laser power P. In Figure 1d we show the peak intensity and peak energy extracted from Figure 1c. With increasing laser power, the peak intensity first increases linearly and then begins to saturate at high laser powers (> 300 μW).

Figure 1

a, Optical image of the device, highlighting the regions for different materials. Scale bar, 10 μm. b, Schematic of the encapsulated heterotrilayer device with the global top and bottom split gates, as well as electrical connections. c, IX PL emission intensity as a function of laser average power P and energy. The sharp peak at 1.46 eV originates from the imperfect filtering of the laser line at 850 nm (Supplementary Note 3). d, PL peak intensity (red squares) and peak energy (black squares) extracted from c. The arrows point out the estimated exciton density at three different powers which will be used in the data in Figure 2. e, CCD image of the excitation laser spot. f & g, CCD images of the normalized IX PL intensity, acquired for P = 50 μW and 200 μW. The yellow solid lines indicate the shape of the heterostructure.

The observed blue-shift of IX energy with increasing laser power[5,15] is due to the repulsive exciton-exciton interaction, which can be decomposed into two terms: the dipolar repulsion, which is valley-independent, and the exchange interaction, which is determined by the valley indices[16]. The dipolar interaction is purely repulsive and can be estimated using a parallel-plate capacitor model[17-19]. This gives a lower bound on the exciton density, as it does not account for a reduction in interaction energy due to a rearrangement of the interlayer excitons, caused by Coulomb repulsion. Following the rearrangement and reduction in interaction energy, a larger density of IXs will be required to achieve the same energy shift. The exchange interaction has a more complex dependence on the electron-hole separation, since it can change from repulsive to attractive. As the vertical separation of electrons and holes becomes larger, the exchange interaction decreases and becomes negative when the vertical separation is larger than the Bohr radius of IX a 0 [19,20]. In our case, since the separation between the 1L-WSe2 and 1L-MoSe2 is around 0.9 nm and similar to the Bohr radius of IX (a 0 ~ 1 nm), we can neglect the exchange interaction and only consider the dipole-dipole interaction.

In order to quantify the influence of dipole-dipole interactions on exciton transport using time-resolved imaging, we first estimate the exciton initial density n 0 from the blue-shift δE, via the parallel plate capacitor model[17-19]: where d = 0.9 nm is the out-of-plane dipole size of IXs, ε 0 is the vacuum permittivity, ε = 6.26 is the effective relative permittivity of the WSe2/hBN/MoSe2 heterotrilayer, and U ~ 2.6 μeV·μm2 is the exciton-exciton interaction strength (see Supplementary Note 1). We deduce an exciton density at P = 50, 100, 200 μW of about 2 × 1011, 4 × 1011, 8 × 1011 cm-2 respectively. The exciton densities we extract from the spectral shift are consistent with the values estimated from the applied laser powers (see Supplementary Note 2).

To image the spatial and temporal distribution of IXs, we use the setup depicted in Figure S3, in which the emitted photons are filtered (< 1.45 eV) and sent to either a charge-coupled device (CCD) camera or to a homemade scanning avalanche photodiode (APD) system (see Supplementary Note 3). Figures 1f and g show CCD images of the normalized PL emission intensity from IXs, acquired for different excitation powers. Compared with the CCD image of the focused excitation spot (Figure 1e), the spatial profile of IXs extends farther and exhibits a growing size with increasing excitation power, signaling the presence of strong repulsive exciton-exciton interactions. A comparison between the power-dependent PL spectrum for MoSe2 intralayer excitons and IXs in the heterotrilayer region also proves the existence of strong interactions between IXs (see Supplementary Note 4).

To acquire the map of PL intensity as a function of time and position I (x, y, t), the APD is scanned in the image plane across the emission spot[21,22]. We use a time-correlated photon counting module (TCPCM) to record the photon clicks. In order to rule out the decay of PL intensity induced by radiative emission, the raw data I(x, y, t), proportional to the exciton density distribution n(x, y, t), are normalized at each recorded time t to obtain (see Supplementary Note 5). We present 2D spatial profiles of IXs at different times in Figure 2a for P = 200 μW. To further analyze the expansion of the spatial profile, the area of IXs at each time is extracted by counting the number of pixels for which the normalized PL intensity is higher than 0.2. This is shown in Figure 2c for three different excitation powers. In the absence of spatial constraints[23,24], the exciton cloud is expected to grow linearly as a function of time, with the evolution of the exciton density n(x, y, t) due to diffusion described by , where D denotes the diffusion coefficient and τ the exciton radiative lifetime.

Figure 2

a, 2D PL images for P = 200 μW acquired by the scanning APD system at different times after the excitation. The yellow solid lines indicate the shape of the heterostructure. b, Simulated exciton area as a function of time for different diffusion coefficients D. The fitting parameters are as follows. Dashed lines: U = 0, τ = 3.5 ns; solid lines: U = 2.6 μeVμm2, n 0= 1 × 1011 cm-2, τ = 3.5 ns. c, Exciton area as a function of time for different excitation powers. The gray line shows the area of the excitation laser spot size (see Figure S6). The solid lines are fits using equation (2) for n 0= 1 × 1011, 2 × 1011 and 4 × 1011 cm-2 respectively. The fitting parameters are U = 2.6 μeV·μm2, τ = 3.5 ns, D = 0.15 cm2/s and T = 4.6 K. The black dashed line indicates the simulated exciton area without considering U The red dashed line indicates the slope of the area increasing for P = 200 μW at t < 1 ns.

We find however that the area occupied by the IX cloud initially (t < 2 ns) grows at a higher, power-dependent speed, but then slows down to a speed which is independent of the excitation power and initial exciton density. Even though in our case the motion of IXs is limited by the finite size of the heterostructure, this cannot explain the significant power dependency at early times, when the expansion is not constricted by the edges. Instead, we attribute the observed exciton cloud dynamics to dipolar interactions.

A sublinear increase of the IX cloud area with time has been observed before[13,25] and has been attributed to two possible mechanisms. First is the effect of the strong moiré potential introducing a modification of diffusivity , where D 0 denotes the bare diffusivity and U moire the depth of the moiré trapping potential[13]. The second is the generation of electron-hole plasma at exciton densities exceeding the Mott transition density n ~ 10[13] cm-2 25,26. Neither of these reports however show a large area of transport (> 5 μm2) in dilute excitonic gases (n 0 ⩽ n). We emphasize that in our work, in contrast with previous results, we observe a significant power-dependent expansion of the IX cloud in the dilute regime. In addition, in the heterotrilayer, the 1L-hBN spacer between WSe2 and MoSe2 is expected to weaken the trapping due to the moiré potential by increasing the spatial separation between the electron and hole wavefunctions, thereby facilitating the propagation of IXs[7]. Numerical simulations that include the effect of the moiré potential fail to reproduce our data (Supplementary Note 6). We therefore explain exciton transport using repulsive dipolar interactions only, decreasing in strength as the IXs cloud expands and decays radiatively.

To identify the contribution of exciton-exciton interactions, we introduce a power-related term into a 2D drift-diffusion equation and solve it numerically (see Supplementary Note 7). The drift-diffusion equation describes the spatial and temporal distribution of exciton density n(x, y, t): where μ is the exciton mobility which can be expressed using D and temperature T via Einstein relation μ = De / (k) and δE is the total potential energy of IXs. The first term on the right-hand side of equation (2) is the diffusion term, while the second term denotes the drift term. Here, δE = δE = n(x, y, t) · U, leading to an exciton potential energy which varies both in time and space. The excitation power enters the equation via the initial exciton density n 0(x, y), which is expected to be of the same order of magnitude as the exciton density determined from the interaction-induced energy shift (Figure 1d). We determine the lifetime of IXs to be about τ = 3.5 ns from time-resolved PL measurements. Since the IX lifetimes in the studied power range do not decrease with increasing exciton density, we neglect the effect of exciton-exciton annihilation[23] (see Supplementary Note 8).

To highlight the effect of the drift term on exciton transport, we show on Figure 2b the simulated exciton area as a function of time with and without δE (solid and dashed lines). We use U = 2.6 μeV·μm2, n 0 = 1 × 1011 cm-2 and do not take the finite size of the heterotrilayer into consideration. When U is neglected, the exciton area increases linearly with time and the slope is proportional to D. With U included, the area first increases sublinearly while at later times (t > 2.5 ns) when the drift term almost vanishes, the slope becomes the same as in the case of neglected U. Similar simulations have successfully reproduced the experimental results of the indirect exciton diffusion in GaAs double quantum wells with strong dipole-dipole interactions[27]. We fit our data in Figure 2c using the same model and take the boundaries of the heterostructure into account (see Supplementary Note 9). To be consistent, we employ the same condition to calculate the exciton area. In Figure 2c, we use D = 0.15 cm2/s and U = 2.6 μeV·μm2 as parameters and treat the initial exciton density n 0 as a free variable to fit out data. The deduced exciton densities n 0 are similar to the values which we determined from the blue-shift energy and the applied laser power, confirming the consistency of our model. Considering the effective diffusion coefficient, this is enhanced by a factor of ~ 12 for P = 200 μW when t ≤ 1 ns (the ratio of slope between the red dashed line and the black dashed line).

We use the Einstein relation first to estimate the exciton mobility from the diffusion constant, finding μ ~ 380 cm2/(V·s), similar to the low-temperature mobility of single charge carriers in monolayer TMDCs[28-30]. This indicates that at this temperature, exciton transport could be limited by the same mechanism, namely charged impurity scattering.

We now turn towards the drift term and quantify the relationships between exciton drift velocities and mobility, in analogy with transport of charge carriers in semiconductors. To visualize this relationship, we use an alternative way to normalize the raw data I(x, y, t) by normalizing at each spatial coordinate (x) by (see Supplementary Note 5). This allows us to observe the spatial distribution of IX lifetime as well as the spatially and temporally resolved IX transport. By scanning the APD along0 x = 0 in Figure 2a with a finer step, we obtain one-dimensional (1D) normalized PL data and for different excitation powers, as shown in Figures 3a, b and c. From , we observe that IXs propagate about y ~ 15 μm in the +y direction, as indicated by the white dashed lines in Figure 3a, b and c. The lower panels distinctly show the time delay at y > 0 generated during IX propagation in the +y direction. Figure 3d shows the simulated normalized exciton density distribution and for n 0 = 4 × 1011 cm-2 using equation (2). allows us to extract the effective speed v of IXs at different powers (see Supplementary Note 10). The black dashed lines in the lower panels of Figure 3a, b and c are guides for the eye which highlight the effect of excitation power and initial exciton density on v. We can further decompose v into its diffusion and drift components as v = v + v. At a higher excitation power, due to the repulsive interactions, IXs experience a stronger effective drift field F, which can be deduced from the spectral blue-shift δE and drift distance y via F = δE. In analogy with the transport of charge carriers in an electric field, the neutral exciton mobility μ can be approximately expressed using μ = v, allowing us to directly estimate the exciton mobility without using the Einstein relation[31]. We extract v at different powers from and plot them with the spectral blue-shift δE in Figure 3e. The error bars of v are given by the linear fits along the black dashed lines. By applying a linear fit to the data in Figure 3e, we estimate μ ~ 440 cm2/(V·s) which is consistent with the value calculated from the diffusion coefficient. The presence of the power-dependent excitonic drift force shows that the repulsive dipolar interactions can be used to control IX transport.

Figure 3

a, b and c, 1D normalized PL intensity along x = 0 in Figure 2a for P = 100 μW, 200 μW and 350 μW. Upper panel: ; lower panel: . White dashed lines enclose the region of IXs transport. Black dashed lines are guides for the eye for v. d, Simulation of 1D normalized exciton distribution along x = 0 using equation (2) for n 0 = 4 × 1011 cm-2. Upper panel: ; lower panel: . e, Exciton effective velocity v as a function of the spectral blue-shift δE extracted from Figure 1d. The error bars of v are given by the linear fits along the black dashed lines in a, b and c (see Figure S12). The red solid line is a linear fit to the data.

Next, we present how combining the repulsive interactions together with an external electric field can be used to control the motion of IXs. We generate a modulated electrostatic potential δE(x) along the x direction using one of the local back gates, which creates a spatially varied but time-independent energy profile acting on IXs: [5]. The upper panels in Figure 4a and b show the schematics of the energy profiles as well as the expected exciton motion for a back-gate voltage V = −2 V and 2 V. The middle panels in Figure 4a and b present CCD images of the normalized IX PL intensity using P = 200 μW. The region enclosed by the black dashed lines indicates the position of the local back gate. By tuning the gate region higher or lower in energy with respect to its surroundings, we generate a potential barrier or a trap, effectively controlling the propagation distance of IXs along the +x direction. We measure the 1D normalized PL intensity along y = 0 using the scanning APD system. As presented in the lower panels in Figure 4a and b, we clearly observe the process of IX gas moving into the back-gate region as we adjust the electro-static potential configuration from a barrier to a trap. Figure 4c shows instead the 1D normalized PL intensity for V = –1 V, i.e. a barrier configuration of lower amplitude with respect to Figure 4a. Here the excitons first flow towards the gate region of lower potential, but then surprisingly move away from the gate region after ~ 1 ns. This phenomenon is another manifestation of the repulsive exciton-exciton interaction, and its interplay with the electrostatic potential in the device. In order to explain this observation, in Figure 4e we show a schematic of the energy profile of the dipolar repulsion δE(t) = n(t) · U and the electro-static energy δE. Initially, when the exciton density n 0 is high, δE dominates over δE such that the exciton flux is pushed towards the gate region. As the exciton density decreases due to the spatial expansion and the radiative emission, δE becomes higher than δE, resulting in the exciton flux leaving the gate region. The observation of this competitive phenomenon requires that n 0 U is larger than but comparable to δE. In this measurement, we create such condition by using P = 200 μW, which corresponds to , and δE ~ 9 meV for V = –1 V (see Supplementary Note 1). We introduce δE into the simulation to better evaluate the interplay between δE and δE. Here we assume that the electro-static potential has a form of an harmonic potential δE ∝ (x − x)[2] with height (insert in Figure 4f). Figure 4f shows the experimental results as well as the simulations of the propagation distance L along the +x direction as a function of (see Supplementary Note 11). Therefore, adjusting the ratio allows us to precisely control the propagation distance of IXs.

Figure 4

a, b, Effect of electro-static potential applied to the back-gate (V) on the exciton spatial and temporal distribution. Upper panel: schematic of the energy profiles as well as the expected exciton motion; middle panel: CCD images of the normalized IX PL intensity; lower panel: 1D normalized PL intensity along y = 0. The black dashed lines enclose the region of the local back gate. The yellow solid lines indicate the shape of heterostructure. c, 1D normalized PL intensity along y = 0 for V = -1 V. d, Simulation of 1D normalized exciton distribution along y = 0 for . e, Schematic of the energy profile of δE(t) = n(t)U and δE. f, Exciton propagation distance L as a function of . Insert: schematic of the energy profile in simulations. Red: δE; yellow: δE(t = 0).

Spatial and time-resolved PL imaging performed in the heterobilayer region of the same sample (see Supplementary Note 12) shows no exciton diffusion. Under low excitation intensities and in contrast to spectra from the heterotrilayer region, we find multiple narrow peaks, with linewidths of 0.5 - 1 meV, limited by the energy resolution of our spectrometer grating. The narrow peaks are sensitive to electrostatic doping and could originate from excitons localized by strain or moiré traps. Moiré trions have indeed recently been observed by several groups in WSe2/MoSe2 heterobilayers[32,33]. Exciton trapping explains the absence of a sizeable expansion of the IX cloud in the heterobilayer region. Moreover, this further demonstrates that the 1L-hBN spacer between the WSe2 and MoSe2 layers could strongly reduce the effect of the moiré potential and strain traps, resulting in an evident excitonic diffusion/drift.

Our results demonstrate that the repulsive dipolar interactions in dilute excitonic gases have a strong influence on exciton transport and that they act as a source of a drift force. Time-resolved PL imaging enables us to visualize the dynamic evolution of IXs in the WSe2/hBN/MoSe2 heterotrilayer and allows us to quantify the diffusion coefficient and exciton mobility which play a central role in the prospect for applications of excitonic devices. Our findings constitute a crucial step towards using spatial patterns of laser field to control the propagation of IXs[34,35], for example by using doughnut-shaped beams as simulated and discussed in more detail in Supplementary Note 13. The excitons that are driven away from the laser hot spot by the repulsive exciton-exciton interaction are expected to constitute a cold excitonic gas. Many exotic phases of matter and emergent phenomena might appear in it, including Bose Einstein condensation and high-temperature superfluidity[36-38]. Finally, a strong spatial confinement of IXs that experience strong repulsive dipolar interactions leads to nonlinearities in energy[39]. Due to the additional 1L-hBN spacer and thus stronger repulsive interactions, we expect the nonlinearity of localized excitons in the heterotrilayer to be more significant.

Methods

Device fabrication

Thin Cr/Pt (2-3 nm) local back gates were patterned using electron-beam lithography and deposited on a silicon substrate using electron-beam evaporation. The heterostructure was fabricated using a polymer-assisted transfer method. Flakes were first exfoliated on a polymer double layer. After monolayers were optically identified by photoluminescence, the bottom layer was dissolved with a solvent and free-floating films with flakes were obtained. These were transferred using a home-built setup with micromanipulators to carefully align flakes on top of each other. Polymer residue was removed with a hot acetone bath. Afterwards, the heterostructure was thermally annealed under high-vacuum conditions (10-6 mbar) for 6 h at 340 °C. Finally, electrical contacts (80 nm Pd for the contacts to the flakes, 8 nm Pt for the global top gate) were patterned using electron-beam lithography and deposited using electron-beam evaporation.

Time-resolved optical characterization

A confocal microscope is used to optically excite IXs and collect the emitted photons through the same objective with a working distance 4.5 mm and NA = 0.65. IXs are excited using sub-ps pulses with a repetition rate of 80 MHz from Ti:Sapphire laser. The collected photons are sent to a APD (Excelitas Technologies, SPCM-AQRH-16) mounted on a 2D motorized translational stage. The output of the APD is connected to a time-correlated photon counting module (TCPCM) with a resolution of 12 ps r.m.s. (PicoQuant, PicoHarp 300), which measures the arrival time of each photon. For the measurements in this work, we set the time bin to 64 ps to record the photon clicks. The single-photon timing resolution of the APD is ~ 350 ps, which is the main time-limitation for the setup. Technical details can be found in Supplementary Note 3.

Numerical simulations

The 2D drift-diffusion equation are solved numerically using Forward Time Centered Space (FTCS) method. The method is based on central difference in space and the forward Euler method in time. Details can be seen in Supplementary Note 7. Numerical simulations that include the effect of moiré potential is performed using the same method and are summarized in Supplementary Note 6.