Strain-engineered inverse charge-funnelling in layered semiconductors

The control of charges in a circuit due to an external electric field is ubiquitous to the exchange, storage and manipulation of information in a wide range of applications, from electronic circuits to synapses in neural cells. Conversely, the ability to grow clean interfaces between materials has been a stepping stone for engineering built-in electric fields largely exploited in modern photovoltaics and opto-electronics. The emergence of atomically thin semiconductors is now enabling new ways to attain electric fields and unveil novel charge transport mechanisms. Here, we report the first direct electrical observation of the inverse charge-funnel effect enabled by deterministic and spatially resolved strain-induced electric fields in a thin sheet of HfS2. We demonstrate that charges driven by these spatially varying electric fields in the channel of a phototransistor lead to a 350% enhancement in the responsivity. These findings could enable the informed design of highly efficient photovoltaic cells.

The development of novel ways to tailor the electrical and optical properties of atomically thin semiconductors, for example by local modification of their composition 6,7 or structure, 8 holds the promise to explore new implementations of electric fields and unveil novel mechanisms of charge transport which can boost the efficiency of opto-electronic devices. The application of strain is one original way to engineer electric fields in semiconducting materials through a varying energy gap. However, common bulk semiconductors can only sustain strain of the order of ∼ 0.1 − 0.4 % without rupture, 9 a value which limits the range of novel physical phenomena and applications that can be accessed. On the contrary, layered semiconductors, such as graphene 10 and transition metal dichalcogenides (TMDs), 11 are theoretically predicted to be able to sustain record high levels of strain > 25 % 12,13 expected to lead to an unprecedented tunability of their energy gap by more than 1 eV. 14 One tantalizing charge transport phenomenon which could be accessible owing to large values of strain is the funneling of photoexcited charges away from the excitation region and towards areas where they can be efficiently extracted. 4,15,16 Such effect is heralded as a gateway for a new generation of photovoltaic devices with efficiencies that could approach the thermodynamic limit. 2,4 In general, strain-induced gradients of energy gaps create a force on (neutral) excitons that pushes them towards the regions with the smallest energy gap. In direct gap semiconductors, this area corresponds to that of maximum tension. Hence, the strain pattern generated by simply poking a sheet of direct gap TMD would normally funnel the charges towards the apex of the wrinkle. 4,15,17,18 Consequently, the extraction of the charges for energy harvesting or sensing poses considerable technological challenges and for this reason the funneling effect has not yet been observed experimentally in electrical transport. On the other hand, the opposite behaviour is theoretically expected in some indirect gap semiconductors (e.g. HfS 2 and HfSe 2 ) and in black-phosphorus, where the energy gap increases in the regions of tension. 14 This would allow the exploitation of the so-called inverse charge funnelling 5 whereby a strain pattern generated by poking a sheet of these materials would push the charges away from the apex of the wrinkle, making them readily available for energy harvesting or computing purposes to an external circuit.
In traditional semiconductors such as Si and Ge, strain is typically introduced at the growth stage by dislocations or elemental composition. 19 These techniques do not easily allow the creation of complex planar strain patterns, forbidding the development of ultra-thin charge-funnel devices. These limitations can be overcome by using atomically thin semiconductors, such as HfS 2 . In this case, specific strain patterns can potentially be engineered in the plane of the TMDC by exploiting the lattice mismatch between the semiconductor and its in-situ grown oxide, see figure 1a. Ab initio DFT calculations suggest that the [1 1 1] cleavage plane of monoclinic HfO 2 has a spatial arrangement of Hf atoms commensurate to that of the basal plane of HfS 2 , with an Hf-Hf distance of 3.426Å. Since the Hf-Hf distance in HfS 2 is 3.625Å, a transition between these two structures is likely to introduce an average 2.7 % compressive strain in the semiconductor at the interface with its oxide. Hence, anchoring the TMD at the edges, for example by depositing electrical contacts, 20 would allow the same amount of strain to be induced away from the oxidised area in the opposite direction (green arrows in figure 1a). Such tensile strain pattern results in the spatial modulation of the bandgap of HfS 2 and therefore the creation of spatially varying electric fields, which are the key ingredients to observe the inverse charge funnel effect. The magnitude of these electric fields can be determined from the change in the energy gap with strain. This has been calculated using DFT and the results, shown in figure 1b-c, predict an increasing (decreasing) value of the direct gap (Γ → Γ) with compressive (tensile) strain whilst the indirect gap (Γ → M) behaves the opposite.
To engineer strain-induced electric fields through lattice mismatch we employ a spatially resolved photo-oxidation technique. Upon exposure to a focussed laser (λ = 375 nm, P = 1 MW/cm 2 ) we find that HfS 2 is readily oxidized, becoming invisible to the naked eyes (see figure 2a). Surprisingly, topographic atomic force microscopy (AFM) measurements show no ablation of the material in the laser-exposed area while the tapping phase image clearly reveals a change in its viscoelastic properties (see also supplementary section S2.1). The energy dispersive X-ray microanalysis (EDXMA) shows the absence of the S peaks (K lines) and the appearance of an O peak (K α line) in the laser-irradiated areas. This is in stark contrast to the spectrum of the pristine HfS 2 where the expected S peaks are clearly measured and no O peak is resolved (figure 1b). No change is observed in the Hf and substrate peaks.
Quantitative analysis shows that, upon laser irradiation, the weight ratio of S decreases from ∼ 20% to ∼ 1% of the total, while the O content increases from ∼ 1% to ∼ 20%, indicating the formation of HfO 2 . Furthermore, the oxidized area is compatible with the diffractionlimited spot size of our laser system (see figure S1d,e), indicating that a photon-assisted oxidation process, as opposed to a thermally driven one, is taking place. To fully capture the role of the inverse charge funneling effect on the measured photoresponse we develop a one-dimensional analytical model. For simplicity we assume that the strain gradient induces a built-in potential which decays linearly with the distance from the strain junction, creating a local built-in electric field E 0 (see supplementary equation (S11)).
By solving the charge continuity equation (see supplementary section S5), assuming the rate of carrier generation to be a delta function at the illumination point x 0 , we find that the charge density as a function of position x is given by: where ∆n 0 is the excited carrier density at the injection point, T is the temperature, k B is the Boltzmann constant, q is the electron charge, τ is the carriers lifetime, D is the diffusion coefficient and E sd is the electric field due to the applied bias. The plus (minus) sign applies to the left (right) strain-engineered region, respectively and E 0 = 0 outside those regions.
Calculating the current generated by this charge density distribution (see ??) by scanning the laser along the channel, we can describe experimentally measured SPCM (see figure 4f).
Although the strain gradient, and thus the built-in field, should be treated as ∝ 1/x, our simple assumptions allow the derivation of an analytical result which is still able to reproduce well the experimental data with τ as the only free fitting parameter. In our case we find a value of τ 10 −10 s outside the strain region, which is typical of multi-layer semiconducting TMDs. 27 In the strain region we find τ 10 −6 s, which translates in a carrier diffusion Furthermore, the spatial modulation of the semiconductor bandgap could be used to create an effective tandem solar cell, which would be able to absorb a much larger portion of the solar spectrum compared to a single bandgap device, 2 see also discussion in Supplementary section S6.
In summary, in this work we report the first experimental observation of the inverse charge funneling effect, that is a novel microscopic charge transport mechanism enabled by strain-induced electric fields. By developing a unique technique of photo-oxidation, we are able to engineer deterministic and spatially resolved strain patterns in ultra-thin films of

Atomic force microscopy and energy-dispersive X-ray microanalysis
Atomic force microscope (AFM) topography and phase image were acquired with a Bruker Innova AFM system, operating in the tapping (or dynamic) mode to avoid damage to the sample while maintaining a high spatial resolution. The measurements were done using a highly doped silicon tip acquired from Nanosensors with a nominal resonance frequency of 330 kHz, and a sharp radius of curvature (< 10 nm). Energy dispersive X-ray microanalysis In order to determine with great accuracy the Raman shift of the A 1g mode of HfS 2 we calibrated the acquired spectra relying on the presence of two fixed peaks which were acquired in the same spectrum: the silicon peak (from the substrate) and the spurious laserline peak (L) which appear at 520 cm −1 and 316 cm −1 , respectively (see figure S1b and S2h).
Since these two peaks do not belong to the HfS 2 they will not shift with the strain applied to the semiconductor and can be used to correct for instrumental shifts of the frequency.

Determination of the absorption coefficient
The absorption coefficient α (λ) is defined as the fraction of the power absorbed per unit length in the medium, and it is a strong function of the incident wavelength λ. We used the formulation by Swanepoel 34 to calculate the absorption coefficient of HfS 2 and HfO 2 . 25 In order to account for the interface between the HfS 2 and the substrate we used the measured reflectance of a thick HfS 2 , so that we can ignore multiple reflections from the substrate, to compute the refractive index n of HfS 2 . We found that n ∼ 2.5 across the measured range and, thus R 2 ∼ 5.6% (reflectance at air/medium interface). Since R 3 = 5.0% (air/quartz interface), we assumed R 2 = R 3 in equation (A3) in reference. 34 The same result can be obtained by computing n from the measured transmittance curve, using equation (20) in reference. 34 The bandgap of a semiconductor is related to the absorption coefficient by: