## Introduction

Spintronics has the potential to deliver computational devices that are less volatile, faster, and more energy efficient with respect to their electronic counterparts1. However, the need to control the spin degree of freedom in a fast and efficient manner is challenging, as the field required to flip the electron’s spin in magnetic materials is often prohibitively high. Spin-orbit coupling (SOC) effects, such as the Rashba effect, allow the formation of spin-polarized electron states without a magnetic moment, thereby circumventing this limitation. In particular, the Rashba effect manifests as a broken spin degeneracy at semiconductor interfaces, resulting in quasi-particle bands of opposite spin texture that are offset in momentum2,3. The Rashba effect has long been a staple in the field of spintronics owing to its superior tunability, which allows the observation of fully spin-dependent phenomena, such as the spin-Hall effect, spin-charge conversion, and spin-torque in semiconductor devices4,5. An example of a Rashba-split quasi-free electron state with effective mass m* is shown in Fig. 1a. To the first order, its dispersion relation is given by:

$$E=\frac{{\hslash }^{2}{k}^{2}}{2{m}^{* }}\pm {\alpha }_{R}k.$$
(1)

Here, the parameter αR is the strength of the Rashba SOC (RSOC) in the system, and it depends on the atomic SOC as well as the electric field perpendicular to the surface (E). Experimentally, αR can be extracted from the detailed dispersion of the spin-split subbands: the energy splitting of the subbands is given by ΔER = 2αRk, and can be seen as a momentum-dependent Zeeman splitting caused by the pseudo-magnetic field – or Rashba field – BRk × E; correspondingly, the momentum splitting is given by ΔkR = 2αRm*/2.

We illustrate the inner workings of these parameters with the paradigmatic example of the spin field-effect transistor (spinFET), depicted in Fig. 1b. In the pioneering concept of Datta and Das6, a Rashba-split two-dimensional electron gas (2DEG) in a channel of length L is sandwiched between two spin-polarized leads. As electrons transit the 2DEG in the direction perpendicular to the Rashba field BR, their spin precesses, acquiring a phase ΔΘ = ΔkRL (assuming the chemical potential lies above the bands’ degeneracy point). Switching between the 0/1 logic operation - corresponding to the low/high resistance state in the device—is achieved by tuning ΔkR such that the electron spin at L aligns to that of the drain lead. As ΔkR is proportional to αR, the operation of such a device relies primarily on the possibility to tune the RSOC, typically realized by gating the 2DEG7.

In spintronic devices such as the spinFET, the prospect to replace the gate with an optical field prompts the development of even faster and more efficient hybrid opto-spintronics. To this end, previous works have demonstrated the generation of spin-polarized currents in Rashba and topological states through the photogalvanic effect, as well as the ultrafast switching of spin orientation in antiferromagnets8,9,10,11,12,13,14, however little is known about the direct optical control of the intrinsic spin splitting15,16. Here, we show that light can change the RSOC strength, effectively manipulating the Rashba spin-transport properties on an engineered 2DEG. The proposed mechanism is as follows: in the presence of a band-bending surface potential, an above-gap optical excitation drives a charge redistribution along the axis perpendicular to the surface. This charge redistribution creates an ultrafast photovoltage, which then reliably alters the RSOC strength (αR) of the 2DEG system on a sub-picosecond timescale. We employ time- and angle-resolved photoemission spectroscopy (TR-ARPES) to track the evolution of the RSOC strength through the dispersion of the Rashba 2DEGs. By directly measuring ΔkR and ΔER as a function of pump-probe delay, we unambiguously extract the evolution of αR.

## Results

Among the materials that can host Rashba-split 2DEGs, bismuth-based topological insulators (TI) are an ideal platform: 2DEGs can be induced on the surface of TIs by applying a positive surface bias or chemical gating17,18,19,20. The combination of the strong atomic SOC in TIs with surface gating generates a substantial Rashba effect in the 2DEGs, allowing one to finely resolve the spin splitting. In an ideal TI, only the topological surface state (TSS)—recognizable by its linear dispersion across the bandgap—crosses the Fermi level (EF), and all charge carriers belong to the TSS21. As represented in Fig. 2a, the application of a sufficient positive bias at the surface induces a strong band bending, leading to the creation of 2DEGs in the form of surface confined quantum well states (QWSs). While the TSS wavefunction extends only within a few layers from the surface and does not depend on the shape of the surface potential, the wavefunction of the QWS does, and extends comparatively deeper into the bulk18. The difference in spatial extent between the TSS and QWS wavefunctions allows us to extract the behavior specific to QWSs, as opposed to the behavior of surface states in general.

Our experimental approach is depicted in Fig. 2b. We choose p-doped Bi2Se3 to host QWSs, as the hole doping provides a lower bulk conductivity in this material. The 2DEGs are prepared by depositing a controlled amount of alkali atoms on the surface, leading to a population of conduction band-derived states that are spin-split by the Rashba effect. Increasing the concentration of deposited atoms is analogous to raising the surface bias, which introduces a higher surface charge density and stronger band bending. The system is then optically excited with a near-infrared (1.55 eV) “pump” pulse and its response is probed by photoemission using a UV (6.2 eV) pulse at variable time delay Δt22,23. The result of such a TR-ARPES experiment is summarized in Fig. 2c over a long range of delays. The left and right panels show the ARPES spectra at negative delay (–100 ps), and 500 ps after the pump arrival, respectively. The central panel presents the evolution of the states at the Brillouin zone center (black dashed lines in the left panel). Before the pump arrival (–100 ps), the system is in equilibrium; the Fermi level is crossed by the linear topological surface state (TSS) and a single parabolic band, nominally the first quantum well state (QWS1). Here, the Rashba-splitting is just barely discernible, owing to the moderate chemical gating. At zero-delay, electrons are optically excited into unoccupied states and subsequently decay into a quasi-thermalized state24,25. Remarkably, we see that QWS1 is pushed to lower energies after the excitation, and a second band becomes populated. This second band (shown also in the spectrum at 500 ps) is in fact the second quantum well state (QWS2), which emerges following an increase in surface charge density. It is worth noting that the aforementioned photovoltage induced by the pump pulse also affect the kinetic energy of photoemitted electrons26,27,28. This manifests as a rigid shift of the ARPES spectra that can be accounted for by a simple subtraction; henceforth, we refer all energy scales to the electron quasi-Fermi level EFn, extracted by fitting a Fermi-Dirac distribution to the photoemission intensity around the TSS Fermi vector (details can be found in the supplementary information).

In pursuance of determining the impact of the optical excitation on the Rashba effect, we perform TR-ARPES on a sample with a higher concentration of deposited alkali atoms, so that energy and momentum splittings are better distinguished. The results of this experiment are shown in Fig. 3. In panel a, the dispersion is shown for three pump-probe delays (Δt = –0.5, 0, and 8 ps). We observe that both QWSs (parabolic bands) are populated before pump arrival, with energy minima at –114 and –13 meV, and QWS1 exhibits a visible and strong Rashba splitting. Differential ARPES maps [I(k, E, t) − I(k, E, − 0.5 ps)] of the 0 and 8 ps delays are also shown, highlighting the pump-induced modification of the QWSs. At time zero, the optical excitation creates an electron population (depletion) above (below) EFn, but shows no appreciable change in dispersion. However, at 8 ps, while the TSS shows no significant change, both QWSs shift downwards in response to an increase in surface charge, similarly to what was reported in Fig. 2(c). This is further emphasized in Fig. 3b where the time dependent energy shifts of the QWSs at Brillouin zone center are displayed. For both QWSs, the energy minimum shows a fluctuation at short timescales before eventually settling to lower energy. We fit the curves in Fig. 3b with a phenomenological model that includes two exponentially decaying processes, shown in purple and cyan respectively (note that the latter curve appears flat because of a long decay time). The first process acts to increase the QWSs’ energy, peaking at approximately 1.5 ps after the pump excitation, and decaying within 3 ps. The dynamics of this components follows the same temporal evolution of the electronic temperature in the system (shown in Supplementary Information), and the timescales are characteristic of the optically-driven electron population above the Fermi level in TIs24; therefore, we attribute this process to an effect caused by the presence of hot carriers (HC) close to EFn. The effect of HC on the surface potential can be extremely complex, but it is likely that, the mobile hot carriers further screen the built-in electric field, causing the QWSs’ to shift to higher energy. The second and more interesting process is a long-lasting shift of the QWSs to lower energy, that emerges as a consequence of an increase in the surface electron population and variation of the electrostatic environment. This process—as we will show in detail—is given by a photovoltage (PV) effect, and it is the central mechanism of this work. The effect of the PV arises within a few hundred femtoseconds and alters the energy and density of the QWSs over hundreds of picoseconds.

As the PV alters the electrostatic environment at the surface, we expect it will have an impact on the Rashba effect as well. For a quantitative look at the momentum splitting, we plot in Fig. 3c the momentum distribution curves (MDCs) of QWS1 at EFn, in equilibrium (before the pump arrival) and 8 ps after the excitation. The MDCs span the two spin-polarized bands on the right-hand side of QWS1 (see red dashed line in Fig. 3a) and are referenced to the Fermi momentum of the inner branch (kF1). The MDC peak locations are indicated by dashed lines; in comparing the equilibrium (purple) and post-excitation (blue) MDCs, we observe that the momentum splitting of the carriers is reduced from (26.0 ± 0.5) × 10−3 to (22.5 ± 0.5) × 10−3 Å−1. A similar result is observed for the energy splitting: in Fig. 3d, we plot the energy distribution curves (EDCs), for the same two delays, along the cut shown in Fig. 3a; we find that, whereas the outer branch of QWS1 maintains its position, the inner branch moves to lower energy, leading to a reduction of the energy splitting ΔER by approximately 13 meV. The additional shoulder observed in the EDC at 8 ps arises from QWS2, which also moves to significantly lower energy. The simultaneous reduction of both ΔkR and ΔER is a clear indication of an optically driven change of the Rashba spin-orbit coupling strength αR, and excludes modifications of the electron dispersion and effective mass as relevant contributions.

The full temporal dynamics of the Rashba effect in QWS1 under optical excitation is given in Fig. 3e, where the splitting in momentum ΔkR (in orange), as well as the RSOC strength obtained following Eq. (1), are shown on the left and right y-axis, respectively. We observe that ΔkR decreases immediately after the excitation, and—after less than 3 ps—is effectively reduced to a seemingly constant value. The change in the RSOC strength is about 15%, decreasing from a value of 0.76 ± 0.02 to 0.66 ± 0.02 eVÅ. Similar values of αR are obtained by performing an analogous analysis on ΔER(k), plotted in green; here, the MDC and EDC analysis between 0 and 2 ps could not be performed due to the presence of a highly non-thermal electronic distribution. Our data outline a scenario where an opportune optical pulse changes the RSOC strength, in a manner similar to a static electric field, on a picosecond time-scale.

The observation of an increased surface electron density in concert with a decrease in RSOC is, however, nontrivial, as the two quantities typically increase/decrease correspondingly. Thus, a satisfactory explanation for the observed effect of the pump excitation, requires one to consider the detailed variation of the surface electric field in relation to the spatial electron distribution. To this end, we build a model to capture the salient aspects of the experimental results. We begin by calculating the band bending of the system at equilibrium, which can be described by a one-dimensional model in the out of plane direction x. The potential profile V(x) is calculated by solving the Poisson equation within a modified Thomas-Fermi approximation29,30. The binding energy and wavefunction of the QWSs is computed a posteriori by solving the Schrödinger equation within the calculated V(x). For our simulation, all material-specific parameters for the calculation are taken from Refs. 31,32,33,34, and the surface potential V0 is determined empirically by the shift of the TSS Dirac point induced by chemical gating (see Supplementary Information).

We present the calculated equilibrium potential and QWSs’ energy minima in Fig. 4a. The space charge region (SCR) spans more than 30 nm, and QWS1 and QWS2 are partially populated, replicating the experimental observations of Fig. 3. Following an optical excitation across the band gap, the generated electron-hole pairs within the SCR are swept apart by the electric field ESCR, which pushes the negative charges towards the surface and the positive charges towards the bulk35. The electrons and holes become spatially separated over tens of nanometers, giving rise to a long-lasting photovoltage field EPV of the opposite sign to ESCR. This effectively softens the band bending, pushing the surface potential V0 to less negative values, as shown in Fig. 4b. The new shallower surface potential drives both the QWSs and the EFn to higher energy. To accommodate the surplus of surface electric charge, the EFn shifts further upwards, resulting in the QWSs moving to more negative energies when plotted with respect to EFn, as seen in the TR-ARPES data (details in Supplementary Information). It must be noted that, as small surface state-induced band bending is a common feature in semiconductors, surface PV has previously been observed in pristine TIs36,37,38. However, while this is technologically relevant for TIs—because it leads to spin-polarized diffusion currents—the TSS only undergoes a rigid shift in energy under the surface PV. The 2DEGs, on the other hand, are much more sensitive to the shape and magnitude of the confining potential V(x), and by extension, the PV.

The full temporal dynamics of the QWSs has been simulated by introducing a pair of effective photo-charges in the system, which approximates the collective charge motion via a center of mass approach39. At each time step, the charge distribution, electric field and EFn are reevaluated. The time evolution of the QWSs’ energy with respect to the EFn—shown in Fig. 4c—is in good qualitative agreement with experimental data. Both QWS1 and QWS2 fall to more negative values almost instantaneously at positive delays. Consistent with experimental observations, the energy shift of QWS2 is larger than that of QWS1. This is a manifestation of the shallower confining potential, which allows for a smaller energy difference between consecutive QWSs. Lastly, the RSOC strength and, by extension, Rashba-splitting for QWS1 are calculated at each step from Eq. (1). The evolution of ΔkR is also consistent with experimental findings (Fig. 4d), confirming that the reduction in the Rashba strength is due to the PV-induced softening of the surface potential.

With the exception of the fluctuation at early delays given by the presence of hot carriers, our simple model succeeds in reproducing all salient features of the experimental data, proving that a photovoltage is responsible for the observed behavior. The small quantitative deviations between the simulated and measured PV effect can be attributed to the model simplicity and approximations, such as the omission of changes in the screening and dielectric properties of the material induced by the PV. More complex simulations of the PV effect such as those recently developed in ref. 40 might further improve quantitative accuracy. Nevertheless, the model captures the fundamental observations of the experiment and provides a clear explanation of the underlying mechanisms. The simulated spectral function in the inset of Fig. 4d showcases the calculated dispersion of QWS1 at the two representative time delays, highlighting the faithful reproduction of the TR-ARPES data. Finally, the PV model can also account for the long timescale (950 ps) needed to recover equilibrium conditions (Fig. 2c), as the spatial separation of electrons and holes in the SCR drastically reduces the recombination rate. The same timescale is expected for the Rashba splitting to recover its initial value as it is modified by the same effect. Since the return to equilibrium is ultimately determined by the charge carriers’ diffusion from the illuminated area, the lifetime of the PV effect could in principle be tuned by varying the size of the pump beam.

In conclusion, we demonstrated that light can be used to control the Rashba spin splitting and, by extension, the spin transport properties in semiconductor devices. Specifically, an optically driven photovoltage can be used to manipulate the surface band dispersion and electron distribution at ultrafast (picosecond) timescales. The specific application of this technique on 2DEGs to tune the Rashba-splitting on a picosecond timescale is an important benchmark for the development of optically controlled spin devices. While the implementation of this effect in a working device is by no means trivial, it is informative to contextualize our finding within the framework of the spinFET discussed in Fig. 1b: the observed variation of ΔkR in QWS1, about 3.5 × 10−3 Å−1, translates into a difference in spin precession angle of π after < 100 nm travel distance, making this effect theoretically appreciable in devices of such length, where ballistic transport can be achieved. It is important to emphasize that, while this study is performed on a TI platform, the underlying physics does not require topological non-triviality and is universal to semiconductors. The effect of the PV on the Rashba strength can be enhanced producing 2DEGs with higher effective masses, while surface gating and pump fluence can be utilized as tuning parameters (see Supplementary Information).

## Methods

Samples of Ca-doped Bi2Se3 are synthesized as described in Ref. 41. Here, Ca acts as acceptor atom, positively doping Bi2Se3 which is normally found to be n-doped due to Se vacancies. The samples are cleaved in vacuum at pressures lower than 7  10−11 mbar, and kept at a temperature of 15 K during evaporation and measurements. 2DEGs are induced by evaporating K (Fig. 2) or Li (Fig. 3) in situ on the cleaved sample surface. The TR-ARPES experiments are performed at QMI’s UBC-Moore Center for Ultrafast Quantum Matter42,43, with 1.55 and 6.2 eV photons for pump and probe, respectively. Both pump and probe have linear horizontal polarization (parallel to the analyzer slit direction). The pump (probe) beam radius is 150 μm (100 μm), and the pump fluence is 40 and 80 μJ/cm2 for experiments represented in Figs. 2 and 3, respectively. Pump and probe were collinear with an incidence angle of 45 degrees with respect to the sample normal. Energy and temporal resolution are 17 meV and 250 fs, respectively, as determined by the width of the gold Fermi edge and of the combined pump-probe dynamics of the pump induced direct population peak in Bi2Se324. For the band bending model, the Poisson equation was solved numerically employing a modified Thomas-Fermi approximation, which intrinsically accounts for modulation of the charge density due to confinement-induced quantization, without the need for numerically heavy self-consistent calculations30,44. The Schrödinger equation was solved numerically with the Numerov algorithm45. The code is available at Ref. 46.