Effective detection of spatio-temporal carrier dynamics by carrier capture

Detecting the spatio-temporal dynamics of generic (quasi-)particles [1–11] is challenging when a subpicosecond temporal scale has to be combined with a nanometer spatial resolution [4, 8]. Nevertheless it is of high interest to unravel the spatio-temporal carrier dynamics to use it, e.g. in quantum information protocols, and several studies have investigated how small wave packets evolve or can be controlled [12–16].

In this paper we propose a mechanism to detect and reveal the spatio-temporal dynamics of an electronic wave packet moving in a semiconducting two-dimensional (2D) system. The detection mechanism relies on the phonon-induced carrier capture [16–29] from the delocalized states in the 2D host material into the bound states of a quasi-zero dimensional quantum dot (QD) embedded in this material. The captured carriers can then recombine on a longer time scale resulting in an optical signal spectrally well separated from the host material's optical signals, providing information on the time-integrated carrier capture into the QD. Using ultrafast pump-probe spectroscopy on a single QD [30, 31], time-resolved information on the QD population, and thus on the capture process, can also be obtained [27]. The crucial aspect for the detection is that the carrier capture happens locally [16, 25–29], i.e. only when the electronic density is close to the QD it can be captured. Here we analyze the question, whether this locality is pronounced enough to describe where the electronic wave packet was distributed. In other words, we examine, if the phonon-induced carrier capture can be used to spatially resolve efficiently the ultrafast dynamics of the electronic density.

As a testbed, we will consider the diffraction of an electronic wave packet moving in a monolayer of a transition metal dichalcogenide (TMDC) at an obstacle. TMDCs represent a class of 2D semiconducting materials which has recently attracted lots of attention [32–38], mostly due to their excitonic properties, although a mixture of bound and unbound electron–hole pairs could be present [39, 40]. In these materials QDs can be created efficiently, e.g. by applying a tensile strain [41–48], resulting in clearly resolved optical signals. Alternatively, a localized attractive potential for a single type of carriers—electrons or holes—can be achieved by applying a strongly localized gate voltage, e.g. by means of a tip of a scanning tunneling microscope. Such tip-induced QDs have already been realized in graphene [49], and tip-induced band-bending effects in $\mathrm{MoSe_2}$ monolayers of few tens of meV have been estimated [50]. While QDs result from an attractive localized potential in the monolayer, an obstacle is formed by a repulsive localized potential. For tip-induced potentials this could be reached by reversing the voltage (see, e.g. [51]). Also, a strongly localized compressive strain, which increases the band gap of TMDCs [52, 53], would result in an obstacle for the carriers. When an electronic density travels inside the TMDC monolayer and gets diffracted at such an obstacle, the spatial dynamics can be theoretically calculated exactly in well specified points. We compare this exact quantity with the predictions provided by the proposed carrier-capture detection scheme, which can be experimentally realized. This will give insight in the efficiency of the proposed scheme.

To be specific, we will consider an electronic wave packet moving in a monolayer of MoSe₂, which gets diffracted by an obstacle. The resulting spatial distribution evolves in fringes which, in view of their amplitude and separation, may be challenging to address.

The initial state is assumed to be a pure state with wave function $\newcommand{\im}{{\rm Im}} \newcommand{\e}{{\rm e}} \psi(\mathbf{r})\propto \sum_{k_x} {\rm exp} \left[-\left(\epsilon_{k_x} - E_0 \right){}^2/(2\Delta_E^2)\right] {\rm exp}\left[\imath k_x (x-x_{{\rm WP}})\right]$. The energy has a Gaussian distribution in x-direction with $\newcommand{\e}{{\rm e}} \epsilon_{\bf{k}}$ being the energy of the electronic eigenstate in the bare monolayer labeled by the 2D wave vector $\newcommand{\e}{{\rm e}} \mathbf{k}\equiv (k_x,k_y)$, $\Delta_E$ is the energy width and E₀ the excess energy. The energies $\newcommand{\e}{{\rm e}} \epsilon_{\bf{k}}$ for TMDC monolayers are essentially parabolic close to the band gap [28]. In the y -direction the wave packet is homogeneous. Taking E₀  =  26.8 meV and $\Delta_E=4$ meV, this results in a spatial charge density $n(\mathbf{r})$ (with $\newcommand{\e}{{\rm e}} \mathbf{r}\equiv (x,y)$ being the 2D space vector) behaving as a wavefront with initial FWHM of about 40 nm in x-direction. The initial position of the wave packet is given by $x_{{\rm WP}}=-50$ nm and we assume the wave packet moves to the right. A sketch of the initial wave packet is seen in figure 1(a) on the left side. Such a wave packet could be considered as the far field of a localized optical excitation [28] induced by, e.g. near-field or plasmonic excitation [30, 54–59] with ultra-fast nano-optics techniques [60].

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Along the motion of the wave packet we consider an obstacle resulting in a non-trivial spatio-temporal distribution. The obstacle is modeled by a potential barrier $V_{{\rm B}}(\mathbf{r})=V_{{\rm B}}\,{\rm sech}\left(\vert\mathbf{r}\vert/a_{{\rm B}}\right)$ located at the origin. We take $V_{{\rm B}}=40$ meV as strength of the barrier and $a_{{\rm B}}=5$ nm as its size parameter. After hitting the barrier, the wave packet diffraction leads to fringes with different magnitude and separations as displayed at the right side of figure 1(a). These fringes are on a nanometer scale and we want to explore if our approach is appropriate to resolve them.

In order to understand the timescales, in figure 1(b) we show the temporal evolution of the spatial charge density $n({\bf r},t)$ along a vertical line at x  =  50 nm. Note that the electronic density is normalized to its maximal value. We find that the charge is different from zero only in a small time window centered around a time $t_0\approx 800$ fs and with a width $\tau$ of about 300 fs. The time $\tau$ quantifies how long the wave packet stays at a given point $\bf{r}$, i.e. the duration of the spatio-temporal dynamics we want to study. Its value indicates that this happens on a (strong) sub-picosecond timescale.

Although the spatio-temporal dynamics takes place on an ultrafast timescale, its memory is included in the time-integrated density $\bar{n}(\mathbf{r})$ defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \bar{n}(\mathbf{r}) = \int_0^{T\gg t_0+\tau} {\rm d}t \,\,n(\mathbf{r}, t), \nonumber \end{align} \tag{ 1 }$$

where we integrate up to a time T being much larger than $t_0+\tau$. The time-integrated density can be interpreted as the amount of charge which has crossed point $\mathbf{r}$. We show $\bar{n}(\bf{r})$ in figure 1(c). We find that the time-integrated density $\bar{n}(\bf{r})$ displays the non-trivial spatial behavior of the fringes in agreement with typical wave packet dynamics known, e.g. from light waves.

Now we want to explore if the carrier capture into a QD is able to resolve such nanometer-scale diffraction patterns. For this purpose, we introduce a QD described by the confinement potential

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{V_QD} V_{{\rm QD}}= - V_{{\rm QD}}\,\, {\rm sech}\left(\frac{\vert\mathbf{r}-\mathbf{r}_0\vert}{a_{{\rm QD}}}\right), \nonumber \end{align} \tag{ 2 }$$

describing a QD centered at $\mathbf{r}_0$ with depth $V_{{\rm QD}}=20$ meV and size parameter $a_{{\rm QD}}=3$ nm. Such a QD has a single bound state with energy $\newcommand{\e}{{\rm e}} \epsilon_1\approx -7.6$ meV independently of $\mathbf{r}_0$ (when the latter is big enough to avoid overlapping with the barrier, as it is the case here). The wave function of this bound state has a full width at half maximum (FWHM) of approximately $7\,$ nm and is in good agreement with the wave function of a cylindrical potential well with a diameter of about $10\,$ nm. This state can be populated by capturing carriers from the 2D states via interaction with phonons [16, 26, 28]. In the MoSe₂ system at low temperature, the capture of conduction-band electrons takes place mostly via the emission of longitudinal optical (LO) phonons [28] with energy $\newcommand{\e}{{\rm e}} E_{{\rm LO}} = 34.4 {\rm meV} \approx E_{0}-\epsilon_1$. In view of the dynamics shown in figure 1, the QD will be crossed by a different amount of charge depending on its position $\mathbf{r}_0$. To study the detection capability of the QD, we will compare the time-integrated density $\bar{n}(\mathbf{r})$ with the charge trapped in the QD at position $\mathbf{r}_0$ after the carrier capture. This is given by the quantity

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{f} f(\mathbf{r}_0) = \left.\sum_{\alpha=1}^{N_b} f_{\alpha}\right|_{\mathrm{QD\,\,in\,\,}\mathbf{r}_0; {T\gg t_0+\tau}}, \nonumber \end{align} \tag{ 3 }$$

with N_b being in general the number of bound states of the QD (for our QD we have N_b  =  1) and $\left.f_{\alpha}\right|_T$ being the population of bound state $\alpha$ at time T after the wave packet has crossed the QD area. In previous works [26, 28] we have shown that the carrier capture shows an interesting time evolution and the capture takes place only in a finite time-window. Therefore, the final captured density $\left.f_{\alpha}\right|_T$ is a well defined quantity for every T after this time-window. Thus, even if the readout of the occupation occurs in a time-integrated way, the signal provides information on the dynamics of the wave packet during a short time interval when it is in the region of the QD.

To describe the spatio-temporal carrier dynamics including the interaction with phonons, we use a recently introduced Lindblad single-particle approach within the density matrix formalism [26, 28]. It has been shown that this is a Markovian and trace-preserving [15, 61–64] description of carrier capture. The method is able to catch the locality of the scattering, while remaining computationally lighter than, e.g. full quantum kinetic equations [16, 26]. Details of the method can be found in [26, 28]. The resulting equations of motion for the density matrix elements are numerically integrated by standard finite time-difference techniques.

Figure 2(a) shows the captured density $f(\mathbf{r}_{0})$ as function of the QD center at x₀  =  50 nm and varying y ₀ (see vertical dashed line in figure 1(c)). For comparison we also show the time-integrated electron density $\bar{n}(\bf{r})$ at x  =  x₀  =  50 nm and y   =  y ₀. Both functions have been normalized to their center of the oscillations defined by the average between the second maximum and minimum. The agreement between these two curves is very good with $f(\mathbf{r}_0)$ being able to predict the locations of all the fringes displayed by $\bar{n}(\mathbf{r})$. A major difference is the amplitude of the spatial oscillations, which in the capture-induced results of $f(\mathbf{r}_0)$ are smaller than in the exact case of $\bar{n}(\mathbf{r})$. This deviation originates mostly from the character of the two quantities: while $\bar{n}(\mathbf{r})$ is defined at every single point $\bf{r}$, the bound state of the QD has a finite extension, therefore monitoring the density over an extended area. In order to show the contribution of the finite size of the receiving bound state, we consider the spatially averaged density

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:coarsegrain} \langle n \rangle(\mathbf{r}_0) = \int {\rm d}\mathbf{r} \, \bar{n}(\mathbf{r}) \vert\psi_1(\mathbf{r}-\mathbf{r}_0)\vert^2, \nonumber \end{align} \tag{ 4 }$$

i.e. we coarse grain $\bar{n}(\mathbf{r})$ by the squared modulus of the receiving bound-state wave function $\psi_1$. The resulting spatially averaged density $\langle n \rangle({\bf r}_0)$ is shown alongside $\bar{n}(\bf{r})$ and $f(\mathbf{r}_0)$ in figure 2(a). Note that $\langle n \rangle({\bf r}_0)$ is normalized by the same factor as $\bar{n}(\bf{r})$. The oscillations of $\langle n \rangle({\bf r}_0)$ now have a reduced amplitude very similar to the one of $f(\mathbf{r}_0)$, thus confirming that it is the size of the receiving state which mostly rules the magnitude of the signal provided by the capture-induced experiment. Larger QDs would therefore result in a loss of spatial resolution such that fringes of higher order become less visible. We nevertheless expect that especially the first fringe would still be well separated from the rest of the signal.

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To experimentally realize such a set-up this would require movable QDs, which could in principle be created, e.g. by charged tips exploiting the so-called tip-induced band bending [65–67]. Such tip-induced QDs have been created for several years [68–70], resulting in the creation of zero-dimensional confinement potentials few tens of nanometers wide and few tens of meV deep [70], resulting in a deepest bound state with FWHM of about 10 nm, i.e. not far from the FWHM of about 7 nm of the bound state $\psi_1$ here considered. We therefore expect that the reported spatial resolution can be achieved experimentally. Although mostly studied through electrical scanning-tunneling spectroscopy measurements, these tip-induced QDs could also be studied optically by putting a proper material under the studied surface [71, 72].

Another option to change the geometrical set-up is to keep the distance between the barrier and the QD fixed and rotate the sample by an angle $\theta$ around the barrier w.r.t. the incoming wave packet. In such a configuration the shape of the wave packet and the positioning of barrier and QD stay fixed. The corresponding results then lie on a circle with $\newcommand{\e}{{\rm e}} \mathbf{r}_0\equiv (r_0 \cos{\theta},r_0 \sin{\theta})$ and r₀ is the fixed distance between barrier and QD (see dashed-dotted line in figure 1(c)). The captured density $f(\mathbf{r}_{0})$ in comparison to the exact results $\bar{n}(\bf{r})$ and the averaged density $\langle n \rangle({\bf r}_0)$ for the rotation set-up are shown in figure 2(b). At larger angles the captured electron density $f(\mathbf{r}_{0})$ is not able to efficiently address the location of the fringes anymore, because their amplitude and separation (see figure 1(c)) become too small to overcome the previously-described weakening of the amplitude. Nevertheless, as far as one considers bigger peaks, also in the case of figure 2(b) the matching is high, showing that our detection scheme can be performed also without movable QDs.

In conclusion, we have proposed a detection method to resolve the nanometer scale electron dynamics in a 2D system. For this purpose we have exploited the phonon-induced carrier capture into the bound states of a QD and shown the efficiency of the relation between exact and captured dynamics. In view of the developments of experimental techniques able to create movable QDs, we believe that the results reported here pave the way to an efficient detection of the spatio-temporal dynamics of low-energy carriers in 2D materials.

FL and DER acknowledge financial support from the Deutsche Forschungsgemeinschaft via the Project No. RE4183/2-1.