Abstract
The future of modern optoelectronics and spintronic devices relies on our ability to control the spin and charge degrees of freedom at ultrafast timescales. Rashba spinsplit quantum well states, 2D states that develop at the surface of strong spinorbit coupling materials, are ideal given the tunability of their energy and spin states. So far, however, most studies have only demonstrated such control in a static way. In this study, we demonstrate control of the spin and energy degrees of freedom of surface quantum well states on Bi_{2}Se_{3} at picosecond timescales. By means of a focused laser pulse, we modulate the bandbending, producing picosecond timevarying electric fields at the material’s surface, thereby reversibly modulating the quantum well spectrum and Rashba effect. Moreover, we uncover a dynamic quasiFermi level, dependent on the Lifshitz transition of the second quantum well band bottom. These results open a pathway for lightdriven spintronic devices with ultrafast switching of electronic phases, and offer the interesting prospect to extend this ultrafast photogating technique to a broader host of 2D materials.
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Introduction
Quantum well (QW) states in semiconductors are a result of a confining potential that traps electrons (typically in a 2D plane) quantizing their electronic states. These states are easy to produce, have tunable band gaps, and are conceptually simple in terms of quantum systems. For these reasons, they have yielded a plethora of applications such as highefficiency solar cells, infrared lasers, and qubits^{1,2,3,4}. Suitable confinement potentials can be achieved by constructing a multilayer heterostructure, but they can also occur spontaneously on surfaces from an adsorbateinduced localized surface potential^{5,6,7}. When these states appear on the surface, the breaking of inversion symmetry removes the spindegeneracy resulting in spinpolarized QW states^{8,9,10,11,12,13}. Combining quantum confinement with strong spinorbit coupling joins the malleable electronic dispersions of QW states with the robust spinpolarization of Rashba states, making them extremely promising candidates for spinbased devices, such as the Datta Das spintransistor^{14,15}.
Since the Rashba interaction and QW spectrum are both predicated on the underlying electric potential, tuning the surface potential becomes a powerful method to manipulate both their spin and energy degrees of freedom^{7,16,17,18}. Such control has been achieved via insitu surface chemical doping (irreversible and difficult to implement in devices)^{9,12,19,20} and electrostatic gating (ideal for devices but often limited by the magnitude of the QW potential at the surface of bulk samples, 10^{7}−10^{13} V m^{−1})^{21,22,23}. The real downfall of both methods is in the difficulty to dynamically tune the QW states on ultrafast timescales, restricting the ability to study the fundamental mechanisms of electronic relaxation and limiting device application in the time domain.
Here, by combining the surface photovoltage effect (SPV), a wellstudied mechanism for producing reversible surface potentials in many semiconductors, with time and angleresolved photoemission spectroscopy, we demonstrate an ultrafast and reversible control of Rashba splitting and energy level spacing, and subsequent quasiequilibrium charge storage in correspondence with a photoinduced Lifshitz transition. This is in contrast to proposed methods of spincurrent manipulation in topological insulators using the SPV to solely change the local surface potential of the Dirac surface state^{24,25,26}. These results present an allinone approach for lightdriven spinorbit modulation and charge manipulation by directly controlling the density of states on picosecond timescale. Experiments were performed on ptype Bi_{2}Se_{3}, a promising candidate for spinbased devices^{14,15}, and ideal since it exhibits both a strong SPV effect in the bulk^{27,28} and 2D QW Rashba surface states, with the latter readily produced through various surface dopants such as carbon monoxide, water, potassium, rubidium, or even long exposure to vacuum^{11,12,13,20,29,30,31,32}.
Results
Gating surface QWs with light
Figure 1 presents a schematic of the experimental setup and the effect of infrared light pulses on the bandbending and QW state. In a semiconductor, the SPV effect occurs when ingap surface states lead to a redistribution of charge between the bulk and surface at equilibrium. This nonuniform charge distribution is called the space charge region and produces an internal electric field that ’bends’ the bands near the surface, as schematically shown in Fig. 1a. In the case of downward bandbending, illumination of the surface via an infrared pulse generates photoexcited electrons and holes that drift to the surface and bulk, respectively. These carriers partially cancel the internal electric field thereby reducing the bandbending and altering the surface potential (solid yellow region)^{33,34}. Equivalently, the downward bandbending potential results in an internal electric field at equilibrium, (E_{BB}), and this electric field is reduced by an opposing electric field generated by the separated photoexcited charges. The consequence of this is a timevarying electric field in the bulk that decays according to the recombination of separated photoexcited electron holes. Hence, an SPV semiconductor substrate can serve as a timedependent electric gate to 2D electronic states at the surface, and the infrared pulse can be focused to gate a specific area of interest. The Bi_{2}Se_{3} samples have metallic surfaces due to a topological surface state and host QW states which can be modified by the pumpinduced timedependent charge separation. In this way, the surface of the SPV material itself is the subject of the electric field gating and is modified by the timevarying internal field. An added consequence of the photoinduced charge separation is that the resulting finite dipole at the surface also produces an external electric field between the sample and the detector, E_{vac}, that affects the detected kinetic energy of the photoelectron, resulting in a rigid energy shift of the measured dispersion, allowing us to track the SPV in realtime^{27,35,36,37,38}.
The Bi_{2}Se_{3} samples in this study are ptype doped with a holelike majority carrier concentration of ≈1 × 10^{19} cm^{−3} (see Supplementary Fig. 3), yielding a fairly conductive bulk that generates a narrow SCR at the surface of ~10 nm (see discussion). In panel b, we have modeled the effect of an optical pump on the bandbending potential and QW energy levels. The lower plot shows the distribution of fixed charges that form the equilibrium bandbending potential (black line) and separated charges following photoexcitation (red gradient). Photoexcited electrons are distributed equally in three parts between the surface and two QWs (using the QW wavefunctions to determine the density profile) and an equal number of photoexcited holes piled at the edge of the SCR. The bandbending lengthscale is similar to the wavefunction spatial extension near the surface, such that only sufficiently separated charges on either side of the SCR can avoid high recombination rates in order to form a persisting SPV effect^{39}. The amplitudes of the excited charges with respect to the fixed charges have been exaggerated by 2× for visibility. The effect of the population of separated photoexcited charges on the QW potential is seen in the upper plot (gray to orange). A secondary well forms at the bulk edge of the SCR (see dotted curve in panel b) that limits the ability to trap holes in the bulk such that the QW potential is not quenched after SPV saturation at the highest fluence (saturation is shown later in Fig. 5b). The squared modulus of the wavefunctions and corresponding energies have been calculated for the equilibrium and maximally pumped condition shown as black and dark orange curves, demonstrating compression of the energy eigenvalues.
Figure 1c presents ∂^{2}/∂E^{2} ARPES spectra demonstrating the compression of the QWs after pump excitation as also schematically summarized in Fig. 1d. The strong downward band bending at equilibrium leaves the surface ntype doped, filling a large portion of the upper Dirac cone, and the quantized conduction band^{10,12,27}, despite the pdoping in the bulk. The first and second QWs, QW1, and QW2 are derived from the bulk conduction band (gray parabolic region in panel d) and exhibit nearparabolic dispersions. The strong spinorbit coupling inherent to Bi_{2}Se_{3} as well as inversion symmetry breaking at the cleaveplane generates a Rashba effect that duplicates the QW states into momentumseparated spinpolarized pairs (shown in red and blue in panel d)^{8,9,10}. Furthermore, the momentum splitting transforms the band minimum from pointlike to a onedimensional ring, resulting in a van Hove singularity below the intersection of the Rashba bands^{40} for both QW1 and QW2 (see DOS in panel d). At equilibrium, with no pump beam, only QW1 is populated, exhibiting strong Rashba splitting. After pumping with a 51.8 μJ cm^{−2} 820 nm pulse, QW1 and QW2 shift downwards in energy relative to the bulk conduction band minimum. QW1 shifts downward by ~25 meV and QW2 emerges below E_{F} and is subsequently populated resulting in a sudden increase of the DOS. Therefore, the lightdriven modification and subsequent filling of QW2 drives a van Hove singularity below the Fermi level, inducing a Lifshitz transition^{41}.
Dynamic QW spectrum and Lifshitz transition
Figure 2 demonstrates the evolution of the QW spectrum after ultrafast illumination. Since the SPV decays in time and the effective gating field depends on the SPV strength, a single timeresolved measurement records a continuum of the field effect on the QW.
Figure 2a illustrates the field effect on surfaceconfined QW as a function of delay. The gray region denotes the QW SCR confinement at the surface and the white region represents the nearsurface bulk. At equilibrium (I), the bandbending is maximally resulting in an enhanced slope for the QW potential and can be represented by an internal electric field E_{BB}. The maximum reduction in the bandbending is the potential difference between the surface and deep within the bulk at equilibrium, labeled V_{BB,0}. At τ = 0 (II), when the pump pulse arrives, separated photoexcited carriers generate an opposing electric field to E_{BB}, labeled E_{eff}. The internal field from the separated carriers acts as an effective applied field on the QW potential, driving the timedependent modification to the spectrum. In the illustrated scenario, the pump pulse is significantly strong as to saturate the available hole trap sites altering the surface potential as a function of delay by ΔV_{SPV}(τ) = V_{BB,0} − V_{BB}(τ), resulting in reduced but nonzero bandbending. At more positive delays (III–IV), the excited charge recombines thus reducing the effective field E_{eff}, reviving the bandbending potential, and bringing the surface potential back to its equilibrium value.
We can monitor the photoexcited charge separation through the energy shift of the ARPES spectrum induced by the in vacuum field produced by ΔV_{SPV}. The change in the surface potential has the effect of increasing the photoelectron kinetic energy at the detector with respect to the equilibrium energy position (ΔKE). Figure 2b presents the effect of the SPV rigid shift on the topological surface state at the Fermi wavevector as a function of delay. The rigid shift of the spectrum extracted from the shift of the Dirac dispersion at fixed momenta well below E_{F} is plotted in white. The Fermi edge as determined by fitting a Lorentzian multiplied by a FermiDirac distribution is plotted in red and denotes the quasiFermi level, \({E}_{{{{\rm{F}}}}}^{{{{\rm{* }}}}}\) (See Supplementary Fig. 2). The rigid energy shift is a direct measure of the surface potential as a function of delay since ΔKE(τ) =−eΔV_{SPV}(τ), where e is the charge of the electron.
Under the center of charge approach^{42,43}, the SPV signal can be described by an average displacement of electron and hole sheet charges induced by the band banding such that:
where n_{sep} and d are the charge separation density and separation distance, respectively, and ϵ_{r}ϵ_{0} is the dielectric constant of Bi_{2}Se_{3}. With the assumption that the separation distance is fixed and approximately equal to the width of the bandbending region, then the measurement of ΔV_{SPV} is proportional to the charge separation concentration n_{sep}, and therefore the charge at the surface^{42,44}. To be clear, this is different than the total photoexcited carriers generated by the pump since significant recombination has taken place after 10 ps, with only sufficiently separated charges surviving (see Supplementary Fig. 5).
Moreover, if the charge is sufficiently separated and localized (a fit assumption based on the long lifetime of the SPV), then the photoinduced charge separation is analogous to a parallel plate capacitor within the probe beam spot and the effective applied field is also proportional to ΔV_{SPV} and the surface charge. Therefore, the ΔKE shift (panel b for τ > 0) decays proportionally to the separated charge density (panel b for τ > 0), enabling us to map the delay axis to a surface voltage and ultimately an applied field strength (discussed in Fig. 3).
At negatives delays (photoemission occurs before the pump) there is a rigid shift of the recorded ARPES spectrum. This effect is due to the photoelectron interacting with an SPVgenerated field induced by a pump pulse that arrives after photoemission^{35,36,37,38}. The rigid energy shift in the negative delay is therefore shifting the unpumped spectrum and can be ignored for the purposes of this study.
In the positive delay, a maximum ΔKE shift of 72.0 meV is reached at 10 ps after 51.8 μJ cm^{−2} illumination. Between 10 ps and 125 ps, the SPVinduced shift follows an unusual linear decay profile. After 125 ps the SPV decays exponentially with a time constant of ~53 ps. This apparent discrete transition from a linear decay mechanism to an exponential decay is associated with the transit of QW2 above the Fermi level, τ_{cross}. In the absence of surface QWs, Bi_{2}Se_{3} exhibits a simple exponential decay with no linear region or kink^{27,38}. Panel c shows the DOS at the Fermi level, E = (−0.005, 0.005) eV, and at zero momentum, k = (−0.005, 0.005) Å^{−1}, demonstrating the crossing of the QW2 band bottom and indicating the Lifshitz transition as QW2 relaxes.
The continuous photoinduced modification of the QWs is observed in Fig. 2d which displays four spectra of the Bi_{2}Se_{3} QW spectrum at equilibrium and at three separate delay values and then schematically simplified in Fig. 2e. The spectra have been shifted to compensate for the extrinsic delaydependent rigid energy shift associated with the SPV effect (blacklined white dots in panel b)^{35,36,37,38}. At equilibrium (far left), only the topological surface state and QW1 are present and occupied below the Fermi level, E_{F}. The full bandbending is present creating a steep QW potential as we described in scenario I of panel a. This results in a large energy spacing for QW1 and QW2, leaving QW2 unpopulated above E_{F}.
For the pumped spectrum, we first look at the data at 10 ps. By this time, nearly all hot carrier recombination has already taken place^{45,46,47}, leaving behind longlived trapped charges (see Supplementary Fig. 4). QW1 shifts downward in energy and QW2 falls below E_{F} as photoexcited charges have reduced the equilibrium bandbending modifying the QW potential; QW2 is subsequently filled below E_{F} and states above E_{F} are populated resulting in a new quasiequilibrium Fermi level. At τ = 80 ps, the QW spectrum drifts towards higher energies relative to the topological surface state and E_{F}. The competition between the rate of charge being pushed above E_{F} by the upward energy shifts (dotted horizontal lines in Fig. 2e) and the decay rate of charge above E_{F} allows charge to existing above the Fermi level during the course of the spectrum relaxation. This ultimately leads to excess charge remaining in QW2 at τ = 370 ps even when the QW energy spacing has returned to equilibrium. For each delay, we can determine the photoinduced change in the surface potential, ΔV_{SPV}(τ), from ΔKE in panel b, thereby tracking the QW modifications as a function of surface voltage.
Reduction of QW spacing and prolonged charge storage
We further investigate the delay dependence of the QW level spacing and charge occupation in order to better understand the pumpinduced QW state modification. In Fig. 2, we demonstrated that the surface potential is a function of delay, and so we are able to track the QW modification with respect to surface voltage. We can take this one step further and estimate the effective field, E_{eff}, from the energy positions of the QW1 and QW2 band bottoms.
Figure 3a displays momentum integrated ∂^{2}/∂E^{2} spectra for QW1 and QW2 band bottoms as a function of delay. The momentum integration windows for QW1 and QW2 band bottoms are centered at k = 0.000 Å^{−1} and k = 0.023 Å^{−1}, respectively, with a width of Δk = 0.002 Å^{−1}. The QW1 and QW2 band bottom positions, E_{1} and E_{2}, (blacklined white dots) track the evolution of the energy levels as the QW spectrum is modified by the field effect and subsequent relaxation. The positions of E_{1} and E_{2} are displayed in Fig. 2d for comparison with the full dispersion. After pump illumination, QW1 and QW2 shift toward higher binding energies, dropping QW1 by ~25 meV and QW2 by ~35 meV and bringing QW2 ~25 meV below E_{F}. Both QWs shift back towards lower binding energies linearly as a function of delay at different rates. For QW2, 126 ps corresponds to τ_{cross} (see vertical dashed line in panel a), the delay time in which the band bottom crosses E_{F}. After this point, the spectral intensity of the band bottom decreases at an accelerated rate.
An important property of QW systems is the intersubbandgap, defined as the difference in energy between the QW2 and QW1 band bottom energies ΔE = E_{2} − E_{1}, which sets crucial material properties including the optical response. The delay dependence of the QW band bottom positions, and therefore the surface potential dependence of the intersubbandgap, demonstrates ultrafastfield tunability. Panel b shows the potential dependence of the intersubbandgap ΔE for the two highest fluences. We focus on the bandgap values for τ < τ_{cross} where QW2 is fully populated and the position of the band bottom is not strongly influenced by the Fermi level.
The effective applied field on the QW spectrum can be extrapolated from the intersubband gap. The triangularwell potential is a widely used approximation to describe surface confinement, especially for lowenergy wavefunctions, and is suited for capturing the effective electric field experienced by the QWs (see Supplementary Fig. 7d). A more precise model of the confinement potential may be necessary for the condition of higherenergy wavefunctions (n > 2)^{12,48,49,50}.
The triangularwell potential is described by, V(z) = eFz, in which F represents the steepness of the well with units of the electric field. As such, the QW wavefunctions take on the form of Airy functions with energy eigenvalues^{48,50}:
The energy eigenvalues, ϵ_{n}, are referenced to the bulk conduction band minimum before quantization, which represents the bottom of the QW potential. By subtracting ϵ_{1} from ϵ_{2}, the equation can be rearranged to solve for F using only the intersubbandgap energy, and the known Airy coefficients, c_{2} ≈ 2.338 and c_{1} ≈ 4.088.
Utilizing the above equation and the average best fit lines from panel b, the x axis can be rescaled in terms of the effect applied field on the QW states. We use the average effective mass from dispersion fits of m^{*} = 0.45 m_{e}. The total electric field in the QW region at equilibrium (ΔV_{SPV} = 0) is 42.0 mV nm^{−1} and 55.7 mV nm^{−1} at the maximal SPV (ΔV_{SPV} = 78.5), reasonable electric field values for surface QW systems^{23}. This is equivalent to applying a field, E_{eff} between 0 mV nm^{−1} (ΔV_{SPV} = 0 mV) and 13.7 mV nm^{−1} (ΔV_{SPV} = 78.5 mV) opposite to the equilibrium bandbending field E_{BB}.
The field effect on the intersubbandgap shown in Fig. 3b exhibits a nearly linear dependence since the QW electric field, F, is always far from 0, existing in the range of (40–60 mV nm^{−1}). The applied field axis ticks are determined by the y axis values of the best fit line and are independent of the surface voltage axis. The even spacing of both x axis ticks demonstrates that the intersubband gap is nearlinear in both surface voltage and effective electric field, as expected from the center of charge separation model in Equation (2).
In addition to enabling an ultrafastfield effect on the QW energy positions and intersubbandgap, the ultrafast gating grants access to the charge dynamics of the QW spectrum. In particular, we uncover quasiequilibrium charge occupation and determine recombination timescales, not accessible by any static gating technique. Furthermore, we observe that the crossing of the QW2 band bottom above E_{F} (vertical dashed line in Fig. 3a, τ_{cross} = 126 ps at 59.1 μJ cm^{−2}) corresponds to the transition of the SPV relaxation from a linear to an exponential decay mechanism as observed in Fig. 2b. Figure 3c displays the integrated spectral intensity of the QW2 band bottom as a function of delay (shaded region in panel a) for fluences between 1.5 and 59.1 μJ cm^{−2}, showing stable charge storage before fluencedependent τ_{cross} values that then depopulate exponentially. At the lowest fluences, there is no stable region and only exponential behavior. The decay region in red is fitted with exponential fits, with the start of the fitting range (red dots) set by τ_{cross} + 10 ps and a minimum cutoff of 10 ps.
As a function of fluence, the resulting timescales from the exponential fit, τ_{decay}, are plotted in Fig. 3d (diamonds) alongside τ_{cross} (red dots). The decay rate of charge in QW2, τ_{decay}, is independent of fluence with an average decay time of 53 ps. On the other hand, τ_{cross} increases logarithmically for fluences above 4 μJ cm^{−2}, marking a critical fluence for the Lifshitz transition of QW2. For fluences in which the τ_{decay} = 0, QW2 never dips below E_{F} after the pump, and therefore does not undergo a Lifshitz transition. All together, this demonstrates that the charge stored in QW2 is stable while it is below E_{F} and charge above E_{F} decays with a timescale of ~53 ps. Moreover, the timescale for stable stored charge in QW2, τ_{cross}, is a consequence of the dynamic Lifshitz transition and can be controlled by varying the pump fluence, equivalent to increasing the SPV strength.
Ultrafast reversible reduction of Rashba splitting in QW1
Another appealing capability of timeresolved photogating is the ultrafast control and tunability of the Rashba splitting since the Rashba effect depends on the inversion symmetry breaking electric field at the surface and can therefore be tuned by an applied field^{51,52,53}. The Rashba effect is an energy splitting that is, in the simplest form, linear in momentum with ΔE_{R} = α_{R}k where α_{R}k is known as the Rashba parameter. In the case of parabolic bands, this leads to a rigid splitting in momentum k_{0} = α_{R}m/\(\hbar\)^{2}^{29,54}. By directly fitting the bands, we can obtain an accurate determination of α_{R}. In Fig. 4a, we present the band fitting of the Rashbasplit QW1 band at equilibrium and τ = 30 ps after the pump. The momentum positions of the Rashba bands as a function of energy (red and blue dots) were determined by peak fitting the momentum distribution curves along delaydependent binding energy ranges and avoiding the Rashba crossing at Γ.
In panel b, the delay dependence of the Rashba splitting is clearly seen in the QW1 band bottom momentum positions captured by integrating the second derivative in energy spectra from panel a over a narrow energy range 15 meV below the dispersion, avoiding the increased intensity at the Rashba intersection at Γ. The black dashed curve serves as a guide to the eye, indicating a 15% reduction in k_{0} at τ = 0 that returns to equilibrium. The peak positions for the negativemomentum band bottom are plotted in pink and closely follow the 15% reduction.
Figure 4c presents the resulting change in Rashba parameter (yellow dots) as a function of delay at high pump fluence (59.1 μJ cm^{−2}) as determined by the dispersion fits in panel a. In the negative delay, the Rashba parameter remains constant at its equilibrium value. Upon illumination the Rashba parameter drops by ~15% and then linearly returns towards equilibrium up until ~125 ps (vertical arrow). From then on, the Rashba parameter exponentially returns to the equilibrium value on a timescale of 53 ps. The decay behavior of the Rashba parameter closely matches the SPV decay (black open dots) in which linear decay transitions into an exponential decay near 125 ps. This demonstrates a linear relationship of the Rashba effect with respect to the surface photovoltage and therefore the applied effective field, and subsequently dependent on the dynamic Lifshitz transition.
Figure 4d plots the percent change of the Rashba parameter as function of the SPV at each delay as well as the extrapolated effective field on the QW. The curves are spaced artificially by 5% intervals for clarity. The field effect behavior of the Rashba parameter is linear for all fluences, decreasing as the applied field is driven to more negative values with ~15% reduction in Rashba splitting at a maximal applied field of 13.5 mV nm^{−1}. From the average of the fluence measurements, the response of the Rashba parameter versus field strength is 1.26 ± 0.21% nm mV^{−1}. This ultimately demonstrates precise control of the Rashba splitting of the QW states with ultrafast photogating, with the Rashba parameter tuned by the pump pulse intensity and dependent on the Lifshitz mediated relaxation.
Lifshitzdependent surface charge trapping
Finally, in Fig. 5 we report a nonmonotonic dependence of the quasiFermi level, \({E}_{{{{\rm{F}}}}}^{{{{\rm{* }}}}}\), with respect to fluence that demonstrates the significance of the Lifshitz transition on the infrared response of the QW system.
The delay dependence of \({E}_{{{{\rm{F}}}}}^{{{{\rm{* }}}}}\) for all measured fluences is plotted in Fig. 5a. \({E}_{{{{\rm{F}}}}}^{{{{\rm{* }}}}}\) is measured in reference to the equilibrium Fermi level (i.e. the difference between the red fit and white fit from Fig. 2b). For fluences above 4 μJ cm^{−2}, \({E}_{{{{\rm{F}}}}}^{{{{\rm{* }}}}}\) is offset from equilibrium starting in a plateau region for delays less than τ_{cross} preceding a small peak after τ_{cross} that leads into exponential decay. Curiously, the offset at early delays is nonmonotonic with fluence, peaking at ~15 meV for 5.7 μJ cm^{−2} and leveling off towards ~5 meV at high fluence. The encircled points indicate the values that are averaged to generate the \({E}_{{{{\rm{F}}}}}^{{{{\rm{* }}}}}\) offset values in panel e. The hump that occurs near τ_{cross}, may be indicative of a large charge transfer from QW2 to the TSS following the reverse Lifshitz transition of the QW2 singularity. This may additionally suggest that the sustained \({E}_{{{{\rm{F}}}}}^{{{{\rm{* }}}}}\) plateau region is also from the charge in the QW states leaking into the TSS during relaxation. For the purposes of this study, we focus on \({E}_{{{{\rm{F}}}}}^{{{{\rm{* }}}}}\) at early delays. The relatively small offset of \({E}_{{{{\rm{F}}}}}^{{{{\rm{* }}}}}\) is in contrast to previous timeresolved studies on ptype Bi_{2}Se_{3}related samples without QWs^{24}, emphasizing the influence of QW1 and QW2 on the relaxation dynamics.
The behavior of \({E}_{{{{\rm{F}}}}}^{{{{\rm{* }}}}}\) at early delays with respect to pump fluence can be explained within the framework of the stored equilibrium charge in QW1 and QW2 described in Fig. 3. To start, in Fig. 5b we display the fluence dependence of the maximal SPV, ∣ΔV_{SPV}(τ = 10 ps)∣, which is proportional to the initial net charge at the surface after hot carrier relaxation, Δn_{surf} = n_{sep}. Next, the fraction of charge stored below E_{F} in QW1 and QW2 is calculated. To do this, we use linear extrapolations of the QW1 and QW2 band bottom positions as a function of SPV from the high fluence data in Fig. 3a and the cumulative DOS of Rashbasplit parabolic bands shown as dashed blue curves (see Supplementary Fig. 8f). The band bottom positions, and therefore the cumulative DOS, are functions of the SPV strength and are calculated from the phenomenological fit of the SPV maximal values (dashed red curve). The sum of the QW1 and QW2 cumulative DOS corresponds to the total available quasiequilibrium charge states for photoexcited charge (solid blue curve).
Figure 5c depicts two photoexcitation regimes that summarize the role of the Lifshitz transition on \({E}_{{{{\rm{F}}}}}^{{{{\rm{* }}}}}\). The brackets on the left indicate the pumpinduced QW energy shift from equilibrium positions, the blue and red regions correspond to excited charge below and above E_{F}, respectively. At moderate fluences of 3−7 μJ cm^{−2} QW2 undergoes a Lifshitz transition, however, only QW1 has significant available states below E_{F} leaving excess separated charge to populate above E_{F}. At high fluence, QW2 has shifted well below E_{F} enabling a large degree of quasiequilibrium charge storage reducing the burden on states above E_{F}.
To further quantify the effect, it is more useful to reparameterize the curves of panel b in terms of the surface voltage to avoid reliance on the SPV phenomenological fit. In this way, the total net charge concentration is a linear function and the QW1 and QW2 band bottoms are linearly dependent on the x axis. Because the QW1 and QW2 band bottom positions are empirically determined, the only consequential free parameter is the relative amplitude of the total available quasiequilibrium charge states (blue regions) with respect to n_{surf} (dashed black line). The difference between the aforementioned populations represents the excess photoexcited charge at the surface (lightred region), which describes the charge remaining after QW1 and QW2 are filled below E_{F} and that must occupy aboveE_{F} states. From panel d, it is clear that the emergence of the QW2 band bottom below E_{F}, which acts to enhance the DOS contribution of QW2 below E_{F}, necessitates a peaked behavior in the excess charge population near the Lifshitz transition, regardless of the fluence dependence of the SPV effect.
The proportionality factor between the charge stored in QW1 and QW2 with respect to n_{surf} is chosen such that the excess charge best fits the \({E}_{{{{\rm{F}}}}}^{{{{\rm{* }}}}}\) offset plotted in panel e. The excess charge from the model in panel d closely captures the nonmonotonic behavior in the measured \({E}_{{{{\rm{F}}}}}^{{{{\rm{* }}}}}\) values.
Discussion
Using the reported dielectric constant of Bi_{2}Se_{3} of a nominal ϵ_{r} ≈ 30−100^{55,56}, we can extract an approximate SCR width, describing the spatial extent of the bandbending utilized in Fig. 1b, \(d=\sqrt{\,2\,{{\Delta }}{V}_{{{{\rm{BB,0}}}}}\,{\epsilon }_{{{{\rm{r}}}}}{\epsilon }_{0}\,/\,e\,{n}_{{{{\rm{A}}}}}}\approx 10\) nm^{34}, where the acceptor ion concentration, n_{A}, is approximately equal to the bulk majority carrier density in the SCR and V_{BB,0} ~ 300 mV based on the observation of the QW3 energy level at short delay and the zeropoint energy of the QW spectrum (see Supplementary Fig. 6). A second independent method is to use charge conservation at the SPV saturation, since bandbending cancellation occurs when the total surface charge is equal to the volumetric bulk charge times the depth. The maximal surface charge can be calculated by Luttinger’s theorem^{57} on the QW1, QW2, and TSS bands giving 5 × 10^{12} cm^{−2} (see Supplementary Fig. 8), and since only ~1/4 (from ΔV_{SPV}/V_{BB,0}) of the charge is screened in the SCR we get, d = 5 × 10^{12} cm^{−2}/(1 × 10^{19} cm^{−3}/4) ~10 nm. These estimates ignore possible variations in both the dopant concentration and dielectric constant between surface and bulk, as well as hexagonal deformation and possible nonisotropic charge storage at the Fermi surface^{20,58}. However, the approximate SCR width is consistent with the bandbending lengthscale observed previously in Bi_{2}Se_{3}^{59}.
Initial reports of the Rashba effect on metallic surfaces suggested that applied fields on the order of 1000 mV nm^{−1} are insufficient to produce the observed Rashba effect, and that the nuclear core potentials are responsible for large splittings^{52,60}. In our experiment we observe a 15% reduction on an ~1 eV Å Rashba effect with fields near 10 mV nm^{−1} only. There is a large body of evidence demonstrating small changes to surface or interface fields generate large changes to the Rashba parameter^{51,61,62}. A reasonable explanation that has been presented is that the applied field plays an indirect role on the Rashba effect by shifting electrons to and from the nuclear cores at the surface plane^{53,61,63,64,65}. With respect to Bi_{2}Se_{3}, Eremeev et al. presents a debated hypothesis on the origin of the QWs from surface modulated van der Waal’s gaps^{66}. Although the origin of the QWs has been further argued to be from surface bandbending^{20,30}, their study indirectly demonstrates the role of wavefunction overlap with the surfacemost quintuple layer as a key ingredient to the Rashba effect. We have shown through our simplified bandbending calculation of the QW wavefunctions in Supplementary Fig. 7e that the shift in QW position is fieldlinear and nearly 2 Å over 10 mV nm^{−1}, a significant fraction of the quintuple layer thickness. In totality, our results are consistent with a picture that the origin of the Rashba splitting is largely controlled by wavefunction overlap with asymmetric nuclear potentials within the first quintuple layer, controlled linearly by the field.
One consequence of the ultrafast QW modification is a large rapid and reversible change in the DOS at the Fermi level. This is made possible by the emergence of QW2 below E_{F} after pump excitation; changing the topology of the Fermi surface and triggering a reversible Lifshitz transition. The kink in the relaxation of the SPV as well as the Rashba modification is correlated with τ_{cross}, indicating that the reversal of the Lifshitz transition plays a key role in the relaxation. Furthermore, the new states near the Fermi level imply an increase in conductivity at the surface affecting spindependent transport properties, and occupation of the QW2 band bottom would influence intersubband optical absorption^{67}. As such, this ultrafastfield effect device provides many opportunities for applications that are sensitive to the DOS at the chemical potential and requires further attention exploring the DOSdependent scattering dynamics.
The SPV photogating method can be extended to other semiconductor systems with 2D electron gases. In fact, a similar gating method to the one presented here has been utilized successfully in black phosphorous to drive gap renormalization at the surface^{68,69}. An example of a future application is a heterojunction system in which an exfoliated or MBEgrown 2D material is placed on top of an SPV substrate separated by a thin insulating layer. In this case, a pump pulse focused on the grounded top layer would result in transient gating of the illuminated surface region. Moreover, SPV time constants can vary significantly from picoseconds to milliseconds depending on the system allowing for a large dynamic range of study^{36,69,70}.
In conclusion, we have exposed complex QW dynamics using a timeresolved SPVbased gate. The phenomenology of the photoinduced QW modification is general for surface QW systems because it relies on the surface bandbending of not only topological insulators but many semiconducting systems^{34}. For this system, the unique decay profile of both the SPV and Rashba reduction is contingent on the dynamic DOS near the Fermi level and the existence of a van Hove singularity in the Rashbasplit QWs. Accordingly, we have demonstrated local control of the nearE_{F} DOS; a useful tool for driving electronic phases such as superconductivity^{71}, magnetism^{72}, and, more broadly, correlated phases^{73}. Furthermore, the ultrafast gating mechanism also revealed dynamically stored charge in QW1 and QW2 that ultimately controls the resulting quasiFermi level, impossible to observe using static gating techniques. Lastly, we have demonstrated ultrafast reversible modulation of the Rashba effect on 10–100 picosecond timescales opening up the possibility for GHz spincurrent manipulation. The temporary photoinduced Rashba tuning creates a platform for spinorbit modulation enabling lightdriven spintronic devices such as ultrafast spintransistors and photocontrolled Rashba circuitry^{14,74,75}.
Methods
Sample synthesis and preparation
The ptype Bi_{2}Se_{3} single crystals were grown via directional slow solidification with ≈1 at.% Mg substitution with bismuth in order to bring E_{F} close to the bulk valence band maximum. The crystals were coated in graphite spray and then cleaved in a vacuum along the (111) plane at a base pressure of 5 × 10^{−11} Torr. The sample experienced rapid dosing triggered by initial exposure to the infrared pump resulting in the formation of a clear QW spectrum similar to the observations by Lin et al.^{8}, with no further dosing occurring after subsequent pumping. This dosing is perhaps due to the presence of trapped gasses in the graphite spray or the crystal itself, and produced results similar to the effects of carbon monoxide and water surface dosing^{12,13}. We have observed this dosing effect multiple times but only after using graphite spray on specifically ptype Bi_{2}Se_{3} samples.
Measurement details
For the pumpprobe measurement, we used a pump of 1.48 eV ppolarized photons generated by a cavitydumped Ti:sapphire oscillator operating with repetition rates of 1357.50 kHz (51.8 μJ cm^{−2}) and 678.75 kHz (all other fluences). The probe beam consists of 5.94 eV ppolarized photons generated by frequency quadrupling of the 1.48 eV source in two BBO stages^{76}. The total time resolution from the two beams is ~300 fs. The pump and probe beam spot fullwidthhalfmaxima was 120 μm and 30 μm, respectively. The delay between the pump and the probe is achieved with a mechanical translation stage. Due to the long delay range, careful consideration was taken to ensure that the maximum pointing drift due to the delay stage was minimized to <15 μm. A variable neutral density filter applied to the pump beam enabled fluencedependent measurements. All reported fluence values indicate incident fluence on the Bi_{2}Se_{3} surface. All spectra were taken at the system’s base temperature with liquid nitrogen of ~80 K. Data were analyzed using PyARPES analysis framework^{77}.
Data availability
All data and code used for analysis are made available upon request to the corresponding author.
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Acknowledgements
The authors would like to thank Kayla Currier, Daniel Eilbott, ChiuYun Lin, Nicholas Dale, and Conrad Stansbury for critical discussions on the experiment and analysis. This work was supported by the Director, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, of the U.S. Department of Energy, under Contract No. DEAC0205CH11231, as part of the Ultrafast Materials Science Program (KC2203). The transport work performed by N.M. and J.G.A. was supported by the National Science Foundation under Grant No. 1905397.
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S.C. performed the ARPES measurements and the analysis. A.L. and S.C. designed the experiment and contributed to writing the manuscript. N.M. performed the transport measurements and analysis. J.G.A. supported the transport analysis and interpretation.
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Ciocys, S.T., Maksimovic, N., Analytis, J.G. et al. Driving ultrafast spin and energy modulation in quantum well states via photoinduced electric fields. npj Quantum Mater. 7, 79 (2022). https://doi.org/10.1038/s41535022004902
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DOI: https://doi.org/10.1038/s41535022004902
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