Controlling photocurrent channels in scanning tunneling microscopy

We investigate photocurrents driven by femtosecond laser excitation of a (sub)-nanometer tunnel junction in an ultrahigh vacuum low-temperature scanning tunneling microscope (STM). The optically driven charge transfer is revealed by tip retraction curves showing a current contribution for exceptionally large tip-sample distances, evidencing a strongly reduced effective barrier height for photoexcited electrons at higher energies. Our measurements demonstrate that the magnitude of the photo-induced electron transport can be controlled by the laser power as well as the applied bias voltage. In contrast, the decay constant of the photocurrent is only weakly affected by these parameters. Stable STM operation with photoelectrons is demonstrated by acquiring constant current topographies. An effective non-equilibrium electron distribution as a consequence of multiphoton absorption is deduced by the analysis of the photocurrent using a one-dimensional potential barrier model.

Alongside this instrumental progress, a detailed understanding of the properties of a tunnel contact during and after fs-laser illumination remains of interest, involving linear and nonlinear absorption mechanisms, transient modifications of the local field distribution, and the diverse pathways of excited charge carriers. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Specifically, the energetic distribution of the tunneling electrons due to the optical excitation and the roles of different photocurrent channels is of particular relevance.
In this work, we study the generation of photocurrents by fs-laser pulses in an ultrahigh vacuum (UHV) lowtemperature STM. The light-driven electron transport manifests itself in modified current-distance dependencies characterized by a much larger decay length compared to regular tunneling. While the magnitude of the photocurrent can be controlled by the incident laser power and the bias voltage, its decay length is largely unaffected by these parameters. Thus, the size of the tunneling gap can be used to vary the ratio of regular tunneling to optically driven electron transfer which allows for stable laser based constant-current imaging of a Cu(100) surface. The observed decay lengths of the photocurrent cannot be directly attributed to the spatially dependent field enhancement of a plasmonic gap mode. In order to identify the mechanism underlying of these enhanced photocurrents, we performed simulations based on a one-dimensional transport model and an effective electron occupation. From these simulations, we identify the major contribution to the photocurrent with transfer channels for hot electrons with energies near the potential barrier maximum.

Methods
The experiments were performed with a home-built UHV low-temperature STM at a base pressure of 5 × 10 −11 mbar and a base temperature of 80 K. Depending on the chosen bandwidth of the measurement electronics and the stability of the tip-sample contact, a current resolution of 50-200 fA is achieved in our setup. The bias voltage U B is applied to the sample while the tip is virtually connected to ground via the current amplifier. Electrochemically etched gold tips and a Cu(100) crystal have been utilized as the probe and surface material, respectively (figure 1(a)).
A mode-locked Ti:Sapphire laser oscillator with a center wavelength of 785 nm and 80 MHz repetition rate is used for optical excitation. Pulse duration and focus diameter in the STM chamber are estimated to 70 fs and 18 μm full-width at half-maximum (FWHM) (see appendix 'Interferometric autocorrelation'). The light polarization was chosen to be aligned along the tip's symmetry axis (unless otherwise stated). An overlap of the tunnel contact and the laser focus is achieved by a plano-convex lens ( f=200 mm) mounted outside the STM chamber on a 3D-translation stage (see focal raster scan in the left inset of figure 1(c)). The optical table and floating STM platform are mechanically decoupled; relative movements of the focus to the tunnel gap are compensated by an active beam stabilization system. Experimental details are found in the appendix 'Methods'.

Experiments
The fundamental ability of resolving single atoms in STM is based on the exponential decay of the tunnel current I upon retracting the tip by the displacement z from the sample. For our system-without illumination-a standard I z ( ) curve is plotted in figure 1(b) (black line) 4 showing a slope of 0.8 decades per Ångström, corresponding to an apparent barrier height (ABH) of 3.2 eV for the tunneling electrons (for a definition of the ABH see appendix 'Apparent barrier height'). With a setpoint current of = I 500 pA, SP the tunnel current drops below a noise level upon retracting the tip by ∼0.5 nm.
A striking change of the retraction curves is observed when the junction is illuminated with fs-laser pulses ( figure 1(b), red to yellow lines). Whereas the current closely follows the (unilluminated) reference at small distances, illumination of the gap greatly enhances the current for increasing displacements. For these larger displacements, the curves again decay as a single exponential. Increasing the laser power to 4.3 mW the photodriven contribution raises to the 100 fA level up to a distance of 2.3 nm.
We describe the distance-dependent current as the sum of a regular tunneling contribution and a photocurrent, fitting the expression to the experimental data, where z 0 is the tip-sample distance at which the setpoint is reached without laser illumination. We extract the photo-driven current fraction I I , For the regular tunneling contribution, we find the ABH to be independent of the applied laser power; the value of F~3.2 eV tc agrees well with that of the reference curve. In contrast, the ABH of the photo-induced 4 Note that throughout this paper, the given displacements z are relative to a starting point z 0 (U B = 2 V, I SP ), i.e. the initial tip-sample distance, defined by the bias voltage U B (set to 2 V) and the set point current I SP without laser illumination. Usually, z 0 attains values between 0.7 and 0.9 nm for typical tunnel parameters [61]. Additionally, z 0 must be modified by Δz(U B , P) when changing the bias voltage or laser power (see appendix 'Start point correction'). current is F~0.2 eV.
pc Interestingly, it also shows no dependency on the laser power. The 16 fold reduction of the ABH is an indication of tunneling electrons excited to higher energy levels, close to the vacuum edge. The fraction of the photocurrent prefactors I I pc SP / changes from <1% for the lowest to 60% for the highest measured laser power. Due to the additional photocurrent, for laser illumination, the setpoint is established at an offset distance D > z P 0 ( ) from z , 0 determined from the condition + D = I z z P I total 0 SP ( ( ) ) (see appendix 'Start point correction'). Note that these offsets are of minor magnitude.
The high stability of our setup allows for an investigation of the nonlinearity of the photocurrent (figure 1(c)). As reference, we measured the photo-emitted current for the retracted tip (∼1 μm distance to the sample). Laser-driven electron currents from free-standing gold tips previously revealed multiphoton photoemission (MPPE) processes [58,[62][63][64][65][66]. This is described by a generalized Fowler-DuBridge theory connecting the current with the average laser power P by a power law, I~P n [67]. The effective nonlinear order n is a measure of the number of photons per electron involved in the photoemission process. We observe a nonlinearity of 3.9 (purple dots), close to the expected value for an Au tip with a work function of~5 eV. Generally, the measured current is a function of gap width, bias voltage and laser power (indicated in the zoom-in). Laser illumination leads to an enhanced near-field E nf in the tunnel junction. (b) Current-distance dependencies without (black dots) and with gap illumination (increasing power from red to yellow dots: 1.0, 1.3, 1.8, 2.5, 3.3 and 4.3 mW) (logarithmic scale). The inset schematically demonstrates the measurement: Starting from the setpoint of 500 pA reached at + D z z P , 0 ( ) the tip is retracted while measuring the current. (c) Double-logarithmic plot of the current as a function of average laser power for different tip displacements (0.03, 0.3, 0.5, and 0.9 nm (yellow to blue)). The effective nonlinearities n resulting from linear fits (solid lines) are indicated near the curves. For comparison, the current-power dependency of the tip retracted~1 μm from the surface is plotted in purple. (Right inset) Nonlinearity as a function of tip displacement. For > z 0.5 nm the nonlinear order attains a nearly constant value of ∼2.5. (Left inset) Focal raster scan across the tunnel gap for a tip displacement of 0.9 nm (scale bar: 5 μm) demonstrating a more strongly confined photocurrent (∼20 μm 2 ) than the focus spot size (∼250 μm 2 ).
For different tips, we find values of n between 3.5 and 4.5, consistent with earlier results for free-standing tips [58,62,64]. For the tip-sample contact, the nonlinear order is greatly suppressed: n attains a constant value of~2.5 for all displacements > z 0.5 nm (right inset in figure 1(c)), which is in accordance with a previous result [36]. Importantly, this nonlinearity indicates lower-order emission processes for the photon-driven current contribution compared to the free-standing tip. For < z 0.5 nm (green and yellow line), the found values are further reduced by the additional regular tunneling, which starts to dominate upon approaching the setpoint. Hence, this reduction in nonlinearity is not linked to a change of the electron transfer process. Interferometric autocorrelation measurements of the photocurrents emitted from a free-standing tip and in the tunnel contact confirmed the general trend of a reduced nonlinear order for the gap illumination (see appendix 'Interferometric autocorrelation').
The reduced barrier involved in the photo-induced electron transfer suggests a further investigation of the photocurrent dependency on the barrier shape and height, which can be adjusted by the bias voltage ( figure 2(a)). While the data at a low bias voltage ( < U 5 V B ) can be described by the above-mentioned biexponential behavior, additional features are observed for These are attributed to field emission resonances (FER), also evident as peaks in the Dz figure 2(a)) (details are found in the appendix 'Field emission resonances'). For measurements in the interval from 2 to 8 V, we extract regular tunneling ABHs ranging from 3.0 eV down to 0.5 eV, respectively. These values match to the found F tc without illumination (a comparison of dark versus illuminated data is given in figure A3 in the appendix 'Bias voltage dependent measurements'). Interestingly, the determined ABHs of the photocurrents do not exhibit such a trend, with values weakly varying around a few to a few tens of meV.
The transition to the negative bias voltages regime - ) reveals positive, photo-driven currents (figure A4 in the appendix 'Bias voltage dependent measurements'): Although the negative setpoint results in a negative regular tunneling (from the sample to the tip) for very small displacements, we find optically driven electron transfer reverse to the static electric field (from the tip to the sample) for larger tip displacements. Note that for the positive photocurrent is compensated by negative photo-driven currents from the surface.
Laser-driven STM Controlling the photocurrent fraction I I pc SP / allows for a transition from regular to photon based imaging. To investigate the impact of surface features on the photocurrent and on topographic information, we measured  We note that multiple sequentially measured topographies with and without laser illumination show no indication of a tip-or laser-induced surface modification. We can therefore rule out previously observed changes in surface morphology [72], induced by thermal tip expansion and penetration into the surface [1,73,74].

Modeling
In the following, we address the mechanism underlying the observed current-distance characteristics I z ( ) for the optically excited tunnel junction. Generally, the electron transport is determined by two major quantities. Firstly, the charge carrier has a transfer probability T to transmit from one electrode to the other. Specifically, T is determined by the potential barrier formed between both electrodes. Hence, it is a function of the electron energy E, the gap widthz z gap and the bias voltage U . B Secondly, the number of transmitting charge carriers is given by the initial occupied and by the final empty states. In an elastic process, this number is a function of the occupation distribution and density of states of the tip and sample at the energy E [61].
Under fs-laser excitation both the transmission probability and the electron population can be transiently changed due to photon absorption or local field modifications. However, for moderate excitation intensities (perturbative regime), we can exclude strong-field effects on the potential landscape determining the transmission probability (see discussion) [36,64]. Therefore, the impact of the laser excitation on the electron population can be modeled by an effective time-averaged occupation function f eff [62,75]. Based on the Bardeen model for tunneling, we calculate the current I by an energy (E) integral over the product of the electron occupation f eff and the transmission probability T [61]: assuming a constant density of states for the tip and the sample. The temperature of the sample is set to 0 K, hence, the electron occupation is unity up to the Fermi level on the sample side. Importantly, the electron population in the tip f eff is given by the absorption of photons from the enhanced near-field E nf in the tunnel gap (see zoom-in in figure 1(a)), which depends on the laser power, the tip-sample geometry and the dielectric response of the materials. Especially, for gold nanostructures excited with near-infrared light, we expect a strong enhancement of E nf due to a local surface plasmon (gap plasmon) [29,41,76,77]. Explicitly, both the occupation f eff and transmission T are functions of the tip-sample distancez z, gap which is given by the tip displacement z in the experiment.
We first consider the possibility of the local field z E nf ( ) responsible for the measured photocurrent spatial decay. Since the plasmonic enhancement is a function of the system's geometry and the dielectric properties of the materials, a strong modification of E nf is expected when sharp features on the surface or different materials are present in the gap [55,60,[77][78][79]. This should lead to different topographic heights when imaging the surface with photo-driven electrons compared to the regular tunneling. Yet, we find the same topographic profiles for both cases (see figure 3(c)).
Moreover, given the experimental geometry, the very short decay lengths render the gap plasmon z-dependency an unlikely explanation. Specifically, the expected field-distance dependency of the signal can be estimated by a coupled dipole approximation, with the tip apex modeled as a sphere (see appendix 'Near-field enhancement') [80]. The associated electric field component in the ¢ z -direction E nf is given by an algebraic relation~+ ) with the tip radius R T and gap widthz z gap [29]. Estimating the distance dependency of a current driven by a nonlinear process (Ĩ z E z n nf 2 ( ) | ( )| ) for different tip radii and = n 2.5 (figure A6 in the appendix 'Near-field enhancement'), the observed decay lengths in our experiment could only be achieved for unrealistically small tip radii (5 nm). However, such radii would lead to a strong deviation from an exponential law, in contrast to our experimental findings. We estimate a signal reduction by a factor of up to~11 in the experimentally relevant regime of 0.7-3.2 nm for a tip radius providing a nearly exponential decay of the near-field (the actual reduction factor is expected to be even lower, since plasmon-driven tunneling reduces the field enhancement for very low distances [29,81]). In contrast, we find reduction factors of up to 10 4 in the related distance regime in our experiment (see figure 1(c)). Interestingly, a current-distance dependency measured for increased laser powers (~35 mW) strongly deviates from the low-power experiments (figure A7 in the appendix 'Near-field enhancement'). The setpoint current is purely laser-driven. Therefore, the tip-sample distance must be considerably larger 5 . In this case, it deviates from an exponential law with a decay length much larger compared to the curves in figure 1(b) and figure 2(a), and the current converges to a finite value of 0.4 pA at the distance of 10 nm. A tip radius of = R 28 nm T and a nonlinear order of = n 4.4 is extracted from a fit of the coupled dipole model to the data in figure A7. We attribute these results to a four-photon process dominating the intermediate distance regime with a current decay governed by z E . nf ( ) Thus, another mechanism must be responsible for the observed decay length scale for the short distance regime and the nearfield enhancement is assumed to be constant in the model discussed below. Consequently, the effective occupation distribution has only an explicit energy dependency, whereas the near-field enters as a parameter given by the laser power . We find the z-dependency of the transmission probability T E z U , , B ( )to explain the observed photocurrent spatial decay. We calculate T with a one-dimensional representation of the potential landscape including image potentials for both electrodes (figure 4(a) and panels (i)-(iv) in figure 2(b)). Field emission resonances and their spectral change due to the Stark shift is covered by the model as well. For this potential, we numerically solve the Schrödinger equation with the Numerov method (see schematic wave function in figure 4(a)) and extract the transmission probability from the found scattering parameters [82,83]. A detailed description is found in the appendix 'Transport model'. From equation (2) the current with its tip-sample distance and bias voltage dependency is simulated as a function of the excited electron population f E P ( ) , whereas the tunnel current reveals the general distribution of electrons, it is not necessarily sensitive to the exact locations of the intervals E . j Thus, for simplicity, we set the energy intervals of f E P ( ) to multiples of the photon energy w =  E j j · above the Fermi energy, with integer j and w =  1.55 eV. We find that one unexcited ( = j 0) and two higher-energy contributions ( = j 1, 2) are fully sufficient to describe the data. We note that there is no one-to-one correspondence between the energy intervals and the respective one-or two-photon absorption process. Specifically, the observed nonlinear order of 2.5 indicates that other factors, including lower-lying initial states and energy redistribution by thermalization, significantly affect the resulting carrier distribution. The parameters adjusted are the amplitudes A 1 and A 2 relative to A 0 (set to unity), the energy widths DE 1 and DE , 2 and the scaling constant C. The broadening D = E 7 meV 0 is set to correspond to the base temperature of 80 K.

Simulation results
The simulations yield a general agreement with the respective experimental curves demonstrating the broad applicability of the model (lines in figures 2(a) and 5(a)). One representative result is presented in figure 4(c) along with the respective occupation function in figure 4(b). Each individual current channel (black lines and colored areas) exhibits an almost ideal exponential decay over all displacements and justifies the previously applied multi-exponential fits. As found before, the short-and long-distance ranges are dominated to nearly 100% by the regular and high-energy contributions (I 2 ), respectively. The first photocurrent channel (I 1 ) contributes only in a narrow transition region with a few percent of the total current ( figure 4(d)).
We identify the electron energy regions from which the current channels are originating by calculating the product G = E Cf E T E ,  4(e)). Several conclusions can be drawn: (1) While I 2 is the dominant photo-driven current for all bias voltages, the relative fraction I I 1 2 / becomes more substantial at higher bias voltages. (2) The higher-energy contributions are always close to the potential barrier maximum (extracted from the simulation and indicated by dashed lines in figure 4(e)), which is consistent with the fitted ABH of a few tens of meV. However, there is always a significant above-barrier fraction (up to 80% (0.8 pA) for the 2 V case).  Analyzing the current composition as a function of laser power 6 ( figure 5(b)) shows that I 1 has a significant contribution for lower laser powers. The average charge transferred per channel can be increased up to a few tens of electrons per laser pulse by increasing the incident power (indicated as numbers in figures 4(e) and 5(b)).

Discussion
The results presented in this paper, specifically the determination of the effective electron distributions, yield insights into the transport mechanism responsible for photocurrents in STM under fs-laser illumination. The main experimental features are reproduced, and the findings suggest multiphoton absorption processes leading to the population of higher-energy electron states (hot electrons) close to the potential barrier maximum. Open questions involve the possible participation of higher-order photon absorption, the role of lower-energy initial states (d-band), and transfer rate modifications due to quantum coupling of electronic states (quenching of radiative resonances). As former studies demonstrated, thermally induced tip expansion due to the pulsed illumination have been a major issue for combining STM and fs-laser excitation, since they can obscure the electronic signal by the oscillatory altering of the gap width by a certain amount of dz t exp ( ) and its strong impact on the exponential tunnel current [1]. The tip expansion can result in a mechanical tip-sample contact, which causes instabilities and tip and sample structuring [72]. However, for the low laser fluences used in this experiment, we can neglect any contact formation (as demonstrated in figure 3). The magnitude of dz t exp ( ) can be estimated from theoretical and experimental studies, which demonstrate monotonically decreasing values for high repetition rates 7 [73,74]. In addition, by assuming an exponential current-distance relation we see that for the measured time-averaged signal, only a modification of the amplitude by a constant factor » c 1 exp is present (assuming dz t exp ( ) independent of z). Hence, an oscillatory tip expansion dz t exp ( ) cannot explain the found reduced ABH. We conducted several validation experiments that exclude a strong thermal impact on the observed currentdistance dependencies. First of all, the negative bias measurements (figure A4 in the appendix 'Bias voltage dependent measurements') show a strong rectification effect, i.e. even for negative setpoint currents (electron transfer to the tip) we find a positive current contribution (electron transfer to the sample) when retracting the tip out of the regular tunneling regime. Secondly, we do not find any signal for laser s-polarization. Finally, the signal is confined to an area that is a factor of 5-6 smaller than the focal spot size (demonstrated by the focal scan in the left inset of figure 1(c)). This is a strong indication of the nonlinearity of the photo-driven current and contradicts a thermal expansion effect which, in contrast, is expected to be governed by linear absorption.
Our experiments have been operated in the perturbative regime with low-order nonlinear transitions. By contrast, strong-field effects are expected to play a major role for laser powers increased by about a factor of 10 compared to those in our experiments [36,64]. Performing STM measurements under such conditions, laserpower-dependent ABHs have been observed [55]. In the limit of much lower intensities, continuous-wave illumination may change the transfer mechanism to plasmon-assisted resonant tunneling, as recently demonstrated by FER shifts of one photon energy [84]. We do not observe such shifts, presumably due to a broader electron energy distribution and the smaller photo-driven contribution to the total current (see figure  A5 in the appendix 'Bias voltage dependent measurements').
Both the experimental and theoretical approach can be further extended. On the one hand, pump-probe schemes have the potential to give access to the temporal evolution of the electron distribution [85]. On the other hand, additional modeling, including the distance dependent plasmonic field, electronic band structures, the three-dimensional transient field distribution as well as the relaxation dynamics (Landau damping, electronelectron and electron-phonon scattering) promise further information on the specific electronic pathways under fs illumination [86,87].

Conclusion
In conclusion, we demonstrated photo-driven electron transfer through the tunnel junction of a scanning tunneling microscope. Under gap illumination, this current is evident by tip retraction curves with additional contributions distinguished by a strongly reduced apparent barrier height leading to a long decaying current compared to regular tunneling. The analysis of power dependent measurements suggests a multiphoton absorption mechanism where the electrons are excited to levels a few 100 of meV around the potential barrier maximum. Neither the laser power nor the bias voltage strongly affects the ABH in the measured range. The electron excitation to the high energies is provided by the plasmonically enhanced field, albeit its distance dependency does not explain the observed decay length scales. Simulations based on a one-dimensional potential barrier model and a time-averaged effective electron occupation are able to reproduce the central features of the current-distance dependencies. By this, we identify the involved energy domains from which the transfer channels are established and find a high-energy distribution in the vicinity of the potential barrier maximum to be the dominant contribution. Prospectively, this could provide an ultrafast excitation procedure with high-energy electrons in a nearly field-free environment, e.g. to disentangle fieldand particle-driven chemical reactions of molecules. 7 Most of the presented data was measured with laser average powers between 1 and 10 mW. The resulting fluences are several magnitudes below the contact formation threshold given in [72]. Moreover, the amplitude of the oscillating tip expansion is expected to be in the sub-Ångström regime as estimated from theoretical and experimental studies [73,74]. treatment comprises multiple cycles of argon ion sputtering (700 V) and annealing (350°C-400°C) of single crystals. Finally, 0.1-0.2 monolayers of germanium have been evaporated by electron beam evaporation.

A.2. Apparent barrier height
In general, the work function of a material is the central quantity defining the potential barrier for an electron that transfers from the cathode to the anode. For sub-nanometer gaps between both electrodes this barrier is strongly modified in its shape and height. In this case, the characteristic quantity of electron transport is the apparent barrier height (ABH) which is a measure of the effective potential: where m e is the electron mass [61]. For an exponential current representation ( with the decay constant k.

A.3. Start point correction
In conventional STM, the absolute tip-sample distance z 0 is determined by the parameters bias voltage U B and setpoint current I . SP In our experiments, the start point is also a function of the laser power P, since the photodriven signal has a pronounced tip-sample distance dependency. We take this circumstance into account by introducing a distance Dz > 0. Qualitatively, for a given distance, the current increases when increasing U B or P and, in conclusion, the tip has to be retracted by Dz from the sample in order to keep the setpoint current constant.
In our experiments, we did power I z P ( ) and voltage I z U B ( ) dependent measurements for given setpoint currents and corrected the data sets by extracting Dz P ( ) from the bi-exponential fits and by using a separate Dz U B ( ) measurement for the power and bias voltage dependent measurements, respectively (figures 1(b) and 2(a)). While a minor correction of 0.05 nm is determined for the highest laser power compared to the lowest one,D = z 1.2 nm is found for a voltage change from 2 to 8 V (see top inset of figure 2(a)).

A.4. Interferometric autocorrelation
We measured interferometric autocorrelation traces for the two scenarios of a free-standing tip (∼1 μm tipsample distance) and for tunnel contact (figures A2(a) and (b)) by utilizing double pulses with a variable delay provided by a Michelson-type interferometer (see appendix A.1). Both interferometer arms have the same laser average power and are collinearly interfering at the apex or at the tunnel junction. In order to measure only the photo-driven current, the tip is retracted at each delay step by 0.7 nm with respect to the setpoint (no regular tunneling) and the photocurrent is recorded. I z ( ) curves demonstrate for a maximal pulse overlap and without a pulse overlap that the photocurrent is finite and regular tunneling is dominant providing a quasi-constant reference distance z 0 at each delay step (see inset in figure A2(b)). As in the power dependent measurement in figure 2, the setpoint distance z 0 only varies in a sub-Ångström regime for different delays. From the traces we found peak-to-background ratios (PBR) of ∼68 and ∼31 for the free-standing tip and for tunnel contact, respectively, which indicates the high nonlinearity n of both situations: under ideal experimental conditions the PBR is equal to -2 . n 2 1 This implies an effective nonlinearity of = n 3.54 for the free-standing tip and = n 2.97 for the tunnel contact. These values are within the variations, which we observed in the power dependent measurements and support that we have a lower nonlinear order in the tunnel contact compared to a Figure A3. Comparison of current-distance dependencies with (dots) and without (crosses) illumination of the tunnel junction for six different bias voltages. In the former case the average laser power is = P 3.4 mW (same data as in figure 2(a)). The y-axis has a logarithmic scale. For clarity the curves are shifted vertically and only data points exceeding the noise level are plotted. For simplicity, no displacement correction due to different start points is applied. Figure A4. Tip-distance dependencies for positive and negative bias voltages ranging from −2.2 to 2.2 V. The data were measured with laser excitation ( = P 8.4 mW) and a setpoint current of 500 pA. Importantly, I z ( ) curves for negative bias voltages show a pronounced positive current regime (shaded areas) evidencing electron transport from the tip to the sample, despite the negative setpoint. This rectification effect is clearly observable for bias voltages down to −1 V. At some point, a negative photocurrent contribution originating from the sample surface conceals the positive current from the tip resulting in a negative net photocurrent. For visibility, only a segment of the actual measured range is shown (no displacement correction is applied).
free-standing tip. The value of » n 3 for the tunnel contact might be somewhat overestimated due to thermal tip expansion changes induced by the intensity oscillations in the interfering pulses.
A.5. Bias voltage dependent measurements A.5.1. Field emission resonances. A well-known phenomenon for large bias voltages U B is the contribution of image states in front of the surface of a conducting sample to the tunneling current [68,69,71,90] (see Dz U B ( ) spectra in figures 2(a) and A5). These field emission resonances (FER) are characterized by an increased conductivity for the bias voltage matching the FER energy [91]. Considering a V-shaped potential landscape, such as an image potential, the corresponding electronic states exhibit a hydrogen-like energy spectrum [92].
calculated with a coupled dipole approximation for tip radii between 5 and 100 nm. Dielectric functions for the gold tip (e t ) and copper sample (e s ) for a wavelength of 785 nm have been taken from [89]. Note that we incorporated the nonlinear order of n=2.5 extracted from the experimental data. The numbers indicate the reduction factor between 0.7 and 3.2 nm, which is the experimentally investigated interval.
STM studies found a modification of the image potential energies due to the Stark shift caused by the static electric field between tip and sample [68,70].
The FER appear at bias voltages of 5.0, 6.6, and 7.7 V, as evident from the peaks in the Dz U U d d B B ( ( ))/ spectrum (bottom inset of figure 2(a)). The increased conductivity at the resonances causes slight deviations from the typical exponential form in the regular tunneling regime (e.g. 6.8 and 8.0 V in figure 2(a)). The potential sensitivity of the FER spectrum (and as a consequence the dependency from the tip-sample distance) qualitatively explains the curve shape deviations.
A.6. Near-field enhancement The electromagnetic field enhancement of a tip-sample system illuminated with a plane wave E 0 propagating in ¢ x -direction and polarized along the z'-direction (inset figure A6) can be modeled with a sphere of radius R T representing the tip apex in front of a surface. The electromagnetic response of the sphere is described by a dipole   moment. This, in turn, induces its image dipole in the sample from which an effective dipole moment can be calculated [80]. The superposition of both dipolar fields with the incident plane wave gives the total field distribution [93]. Evaluating the field at the tip apex ( ¢ = ¢ = x z z 0, gap ) delivers an algebraic relation for the ¢ z -component of the total field~+ -E z R nf gap T 3 ( ) [29]. Figure A6 presents the tip-sample distance dependency calculated with the coupled dipole model (CDM) for a nonlinear process (~E n nf 2 | | ) of the order of = n 2.5 and tip radii between 5 and 100 nm.
A.7. Transport model For the calculation of the transmission probability T, a one-dimensional barrier model, composed of the three regions (tip, gap and sample) is used ( figure 4(a) and figure A8). The tip and sample are assumed to be field free, i.e. constant potentials of + U eU T 0, B and U 0,S for the tip and sample ( U j 0, is the inner potential), respectively. The total potential V(z′) inside the gap is the result of the superposition of the image potentials for both tip and sample, and the linear potential drop due to the bias voltage and the work function differences. Effective surface positions for the tip and sample are applied to fulfill continuous boundary conditions at ¢ = z 0 and ¢ = z z gap [91,94]. Within a scattering approach, the Schrödinger equation is solved numerically by the Numerov method with the usual assumption of continuously differentiable wave function transitions [82,91]. From the complex wave function amplitudes the transmission probability is calculated [83]. A schematic illustration of the real part of a wave function is given in figure 4(a): regions with E > V(z′) (tip and sample) are characterized by an oscillatory waveform while the wave function inside the gap (E < V(z′)) decays exponentially.
Both, tip and sample material, are assumed to have a constant local density of states. The sample temperature is set to 0 K.
The optimization procedure of the free parameters in the effective occupation function was implemented in Matlab. In advance, the absolute gap width z 0 was fitted for a representative data set and has been fixed for all following simulation iterations. In addition, a slight offset of the order of a few tens to a few hundreds of meV was added to the energy intervals E j in order to match the actual work function of the tip. We found that the energy widths DE 1 and DE 2 -corresponding to the temperature of the two photo-driven contributions to the effective occupation distribution-attain values of several tens to a few hundreds of meV, which is equivalent to 1000-2000 K. These high values are necessary to somewhat flatten out the effects of scattering and field emission resonances.