Ultrafast response of monolayer molybdenum disulfide photodetectors

The strong light emission and absorption exhibited by single atomic layer transitional metal dichalcogenides in the visible to near-infrared wavelength range make them attractive for optoelectronic applications. In this work, using two-pulse photovoltage correlation technique, we show that monolayer molybdenum disulfide photodetector can have intrinsic response times as short as 3 ps implying photodetection bandwidths as wide as 300 GHz. The fast photodetector response is a result of the short electron–hole and exciton lifetimes in this material. Recombination of photoexcited carriers in most two-dimensional metal dichalcogenides is dominated by nonradiative processes, most notable among which is Auger scattering. The fast response time, and the ease of fabrication of these devices, make them interesting for low-cost ultrafast optical communication links.


Junction
The nature of the ultrafast photoresponse of the metal-MoS 2 junction is discussed here. Figure   3(a) in the article depicts the band diagram of the metal-MoS 2 junction (plotted in the plane of the MoS 2 layer) after photoexcitation with an optical pulse. Given the Schottky barrier height of 100-300 meV [1], the width of the MoS 2 region near the metal with a non-zero lateral electric field is estimated to be to ∼100-300 nm [2]. Note that the lateral electric field right underneath the metal is expected to be very small. As a result of light diffraction from the edge of the metal contact, light scattering from the substrate, and plasmonic guidance, a portion of the MoS 2 layer of length equal to a few hundred nanometers is photoexcited even underneath the metal (see the discussion above). Assuming similar mobilities and diffusivities of electrons and holes in MoS 2 , the photoexcited carriers, both free and bound (excitons), move, either by drift in the junction lateral electric field or by diffusion, less than ∼10 nm in 5 ps before they recombine and/or are captured by defects. The photoexcited carrier distributions therefore do not change significantly in space during their lifetime.
The ultrafast current response I 2 (t, ∆t) of a short circuited junction in response to two timedelayed optical pulses is expected to be fairly complicated. In TPPC experiments the quantity measured is the time integral I 2 (t, ∆t) dt (∝ V c (∆t)). The motion of the photoexcited electrons and holes in a short circuited junction causes capacitive (i.e. displacement) currents in the external circuit in order to keep the potential across the shorted junction from changing in accordance with the Ramo-Shockley theorem [3,4]. However, if the photoexcited carriers recombine before they make it out into the circuit then the net contribution of the capacitive currents to the integral I 2 (t, ∆t) dt is identically zero.
Photoexcited electrons and holes can be separated before they form excitons by the lateral electric field in the junction and this constitutes the standard drift current contribution to the detector short circuit current response I 2 (t, ∆t) dt. A photoexcited electron and a hole (free or belonging to an exciton) in the MoS 2 layer underneath the metal can also be separated at the metal-MoS 2 heterojunction. The hole can tunnel into the metal leaving behind the electron which is then swept by the lateral electric field to the opposite side of the junction. The electron can also tunnel into the metal leaving behind the hole which will then have a difficult time traversing the lateral field region (moving against the electric field) and making it to the opposite side of the junction. This argument shows that even if the probabilities of the electron and the hole tunneling into the metal are similar, the lateral field in the junction ensures that the process in which the hole tunnels into the metal makes the dominant contribution to the short circuit current response The experimentally measured sign of the photovoltage, and the photocurrent (see Figure 1(b) in the article), agrees with the above arguments.
The discussion above shows that the short circuit current response I 2 (t, ∆t) dt is proportional to the junction lateral electric field strength, and to the time integral of the photoexcited free electron and hole densities as well as to the time integral of the bound (exciton) electron and hole densities. Assuming similar electron and hole mobilities, one may write, Here, n f/b (t, ∆t) and p f/b (t, ∆t) are the spatially-averaged free/bound (f/b) photoexcited electron and hole densities in the junction, respectively. Since photoexcited electrons and holes don't have time to move much before they recombine and/or are captured by defects, spatial dynamics of the carrier densities are not important. The constants k f and k b capture the difference in the relative contributions from free and bound carriers to the current response. If one assumes that k f ≈ k b then, where n (t, ∆t) and p (t, ∆t) are the total photoexcited electron and hole densities in the junction, respectively, including carriers both free and bound (excitons).
The assumption k f ≈ k b will hold if the short circuit current is dominated by the free and bound electrons and holes that get separated at the metal-MoS 2 heterojunction. Since the junction resistance R j is expected to be largely determined by the transport across the metal-MoS 2 heterojunction rather than by the transport across the MoS 2 region, the assumption k f ≈ k b is a decent approximation if not an excellent one.

Metal-MoS 2 Junction
A circuit model of the photodetector is shown in Supplementary Figure 2 Here, the total device resistance R d equals 2R j + R MoS 2 , T R is the period of the optical excitation, and the time integrals above are performed over one complete period. The approximate equality above follows from the fact that in our experiments, R ex >> R d . Note that all the capacitances drop out in the expression for V c (∆t). Therefore, one can use the low-frequency circuit model shown in Figure 1(c)(in the article) when calculating V c (∆t).
It is instructful to determine whether the intrinsic device resistances and capacitances could If one ignores the parasitic capacitance C p then the circuit bandwidth is set by the time constant R j C j . The junction resistance R j is dominated by the metal-semcionductor contact resistance. In the case of MoS 2 , the contact resistance values are in the 1-10 kΩ-µm range at room temperature [8].
Because of the 2D nature of the metal-semiconductor junction, the junction capacitance C j is very small and entirely due to the fringing fields. For a ∼50 nm thick metal contact layer, C j is estimated to be less than .03 fF/µm [2]. Therefore, the relevant time constant is estimated to be shorter than a picosecond. The short circuit current response is (assuming R ext = 0), In this case, the circuit bandwidth is set by the time constant C j R j R MoS 2 /(2R j + R MoS 2 ). If R j << R MoS 2 , as would be the case if the doping in the MoS 2 sheet is small, then the time constant equals ∼R j C j , which is the same as for the open circuit voltage response. If on the other hand R j >> R MoS 2 , then the time constant equals ∼0.5R MoS 2 C j . Assuming an electron doping of ∼2 × 10 12 cm −2 (as in our devices) and a modest electron mobility of ∼20 cm 2 /V-s (as in our devices), and a device length of 1 µm, the value of R MoS 2 comes out to be ∼150 kΩ-µm and the time constant comes out to be ∼2.25 ps. Finally, the capacitance C p could come from the fringing fields between the two metal contacts and therefore its effect on the open circuit voltage response ought to be considered. For example, consider ∼50 nm thick metal contact layers that are one micron apart. The capacitance C p is estimated to be less than .02 fF/µm [9]. The relevant time constant is C p (2R j + R MoS 2 ) and, assuming R j << R MoS 2 , the time constant is found to be ∼3.0 ps. Therefore, in all the cases the intrinsic device resistances and capacitances are not expected to fundamentally limit the speed of operation of the detectors considered in this work. would expect the measured photoresponse to also increase with the gate voltage since, as argued in this paper, the photoresponse is proportional to the in-plane electric field [5].

Supplementary
Supplementary Figure 4 shows the measured |∆V c (∆t)| plotted as a function of the time delay ∆t between the pulses for different gate bias values: -3, 0, 3, 6 V. T =300K. The pump fluence is 8 µJ cm −2 . As in Figure 2(b,c) in the article, two distinct time scales are observed in the dynamics and, within the accuracy of our measurements, these time scales are largely independent of the gate bias. As expected from our model, the overall signal level increases with the gate bias.

Supplementary Note 5: Theoretical Model for Carrier Capture and Recombination via Auger Scattering
We use the model for carrier capture by defects via Auger scattering in MoS 2 [6,7] to model our experimental TPPC results. The model assumes carrier capture by two different defect levels, one fast (f ) and one slow (s). Keeping only the dominant Auger capture processes in an n-doped sample, and ignoring carrier emission processes for simplicity, the rate equations for the carrier densities and defect occupation probabilities with photoexcitation by two time-delayed optical pulses can be written as follows [6], Here, n(t, ∆t) and p(t, ∆t) are the total electron and hole densities, respectively, including both free and bound (excitons) carriers and n(t, ∆t) = n o + n (t, ∆t), where n o is the doping density. The values of the fitting parameters used in the theoretical model to fit the experimental data (see Figure 3 in the article) are listed in Supplementary Table 1. These values are almost identical to the values extracted from direct optical pump-probe measurements of the carrier dynamics in monolayer MoS 2 [6]. The value of the doping density, n o ∼ 8 × 10 11 cm −2 , needed to obtain a good match with the experiments is much smaller than the doping density determined from electrical transport measurements. This difference is attributed to the fact that the carrier density in MoS 2 near the metal contact is indeed much smaller than in the bulk of the device (see the energy band diagram in Figure 3(a) in the article).
In the simulations, we assumed, for simplicity, that the defect occupation probability before photoexcitation is unity. This assumption might not always hold. The defect occupation probability could be a function of the temperature or the gate bias and this could have observable consequences. If the occupation probability of the slow defects before photoexcitation is smaller at higher temperatures, which is plausible, then the ratio of |∆V c (∆t = 0)| to |∆V c (∆t)|, when ∆t is at the boundary between the slow and fast time constant regions, will be larger at higher temperatures since fewer photoexcited holes would have been captured by the slow defects during the initial fast transient and, consequently, fewer photoexcited electrons would have remained in the conduction band after the fast transient is over. This could explain the small decrease in the ratio of |∆V c (∆t = 0)| to |∆V c (∆t)| at the boundary between the slow and fast time constant regions observed in our measurements at the lower temperature in Figure 2(b) in the article. Note also that the ratio of |∆V c (∆t = 0)| to |∆V c (∆t)|, when ∆t is at the boundary between the slow and fast time constant regions, is also smaller at more positive gate bias values (see Supplementary   Figure 4). This trend is similar to the trend observed in |∆V c (∆t)| when going to lower substrate temperatures, and, as before, we attribute this to a larger initial occupancy of the slow defect states just before photoexcitation at more positive gate bias values.