Abstract
Ultrafast control of currents on the nanoscale is essential for future innovations in nanoelectronics. Recently it was experimentally demonstrated that strong nonresonant fewcycle 4 fs laser pulses can be used to induce phasecontrollable currents along gold–silica–gold nanojunctions in the absence of a bias voltage. However, since the effect depends on a highly nonequilibrium state of matter, its microscopic origin is unclear and the subject of recent controversy. Here we present atomistically detailed (timedependent nonequilibrium Green’s function) electronic transport simulations that recover the main experimental observations and offer a simple intuitive picture of the effect. The photoinduced currents are seen to arise due to a difference in effective silicametal coupling for negative and positive field amplitudes induced by lasers with low temporal symmetry. These insights can be employed to interpret related experiments, and advance our ability to control electrons in matter using lasers.
Introduction
A general goal in our quest to control matter and energy is the design of strategies to control electronic properties and electron dynamics using coherent laser sources^{1,2,3}. In addition to its interest at a fundamental level, lasers permit manipulation on an ultrafast timescale opening the way to control the ability of matter to chemically react, conduct charge, absorb light, or other properties, in a femto to attosecond timescale.
In a recent experiment, Schiffrin et al.^{4} demonstrated how a strong (I ~ 10^{13}–10^{14} W cm^{−2}) nonresonant fewcycle 4 fs laser pulse can be used to induce currents along gold–silica–gold nanojunctions in the absence of a bias voltage. Phenomenologically, these currents arise due to the nonlinear interaction of the junction with a laser pulse that has a low temporal symmetry^{5}. By varying the carrier envelope phase it is possible to vary the degree of time asymmetry of the incident radiation, and thus the direction and magnitude of the photoinduced current.
The experiment marks a new frontier in laser control of electronic dynamics and is the fastest existing method for the generation of currents. However, the microscopic origin of this rather spectacular effect is unclear and the subject of recent controversy. This is because the applied electric field is just below the dielectric breakdown of the silica, and the laserfrequencies are far detuned from the electronic transitions in the system such that Stark effects, the shifts in energy levels due to application of an electric field, can play an important role.
In a broader context, this experiment falls into a general class of symmetry breaking lasercontrol scenarios known to induce phasecontrollable currents in the absence of a bias voltage^{1,6,7,8,9}. In the traditional version of this laser control, laser pulses E(t) = \(\epsilon _\omega\) cos(ωt + ϕ_{ ω }) + \(\epsilon _{2\omega }\) cos(2ωt + ϕ_{2ω}) with frequency components ω and 2ω are used to photoexcite a spatially symmetric system from a bound state to a given energy in the continuum by means of a near resonance onephoton and twophoton excitation. Since oddphoton processes connect states with opposite parity while evenphoton process connect states with the same parity, simultaneous photoexcitation via a one and twophoton process creates a state in the continuum of no definitive parity. This breaks the spatial symmetry of the system and generates a net phasecontrollable current \(I\sim \overline {E(t)^3}\) ~ \(\epsilon _\omega ^2\epsilon _{2\omega }{\kern 1pt} {\mathrm{cos}}\left( {2\phi _\omega  \phi _{2\omega }} \right)\), where the overbar denotes time averaging. More generally, the scenario applies under resonant and nonresonant condition by using lasers, such as those employed in the experiment, that are neither symmetric nor antisymmetric with respect to sometime t′ [i.e., E(t − t′) ≠ E(−(t − t′)) and E(t − t′) ≠ −E(−(t − t′))]^{5}. The nonlinear response of matter mixes the frequencies and harmonics of the laser field and leads to the generation of a zerofrequency DC component in the current I. Such symmetry breaking component arises due to excitation via an even n and odd m photon process, appears in the \(I\sim \left E \right^{n + m}\) order in the response^{1,10}, and is often referred to as n vs. m multiphoton interference effect.
The experiments are performed by shining a strong 4 fs fewcycle laser pulse of varying amplitude (0.4–1.7 V Å^{−1}) and fixed carrier envelope phase φ to a 50 nm metalfused silicametal junction. The laser polarization is chosen to be along the junction direction, there is no bias voltage across the junction, and the laser central frequency (ħω = 1.7 eV) is far detuned from the electronic transitions across the 9 eV gap of the silica. The laser irradiates both the metal and the silica, and any response that is not dependent of the carrier envelope phase is experimentally eliminated.
Currently, there are four theories that seek to explain the microscopic origin of the effect. The mechanism proposed in refs. ^{4,11} is based on Zener bandtoband tunneling^{12,13} induced by an electric field and a theory for metallization of dielectrics through Stark shifts^{14}. This mechanism is also used to interpret the simulations in ref. ^{15}. In turn, ref. ^{16} adopts a more traditional perspective and argues that the spectral bandwidth of the pulse can sustain a resonant 5 vs. 6 photon absorption process that creates real carriers and induces symmetry breaking. A third interpretation^{17} is based on the idea of creating virtual carriers in the conduction and valence bands through nonresonant 1 vs. 2 and 2 vs. 3 multiphoton quantum interference. The possibility of generating virtual carriers through Stark effects was also suggested in ref. ^{11}.
A fourth possibility arises from an early proposal^{18} to induce currents in nanojunctions through Stark shifts. The basic idea is to use a field with a low temporal symmetry with frequencies far detuned from electronic transitions in the photoactive material such that Stark effects, and not nearresonance photon absorption, dominate the dynamics. Through changes in the metalsemiconductor band alignment via Stark shifts, the time asymmetry of the pulse leads to a difference in effective metalsemiconductor coupling for positive and negative laser field amplitude, ultimately leading to phasecontrollable currents.
While the experiments are performed in a nanojunction, the proposals in refs. ^{4,11,15,16,17}. consider the silica as a periodic solid and do not take into account any effects that the gold–silica interface may play in the production of currents. Thus, even if virtual or real carriers are created through Stark or multiphoton interference, it is unclear if these carriers can be collected by the metallic contacts to form a measurable current. In turn, ref. ^{18}, while it does take into account the interface, does not consider that the electric field of light also induces an AC bias voltage across the junction (a basic feature recognized by Tien and Gordon^{19} that leads to photonassisted tunneling). Further, simulations in ref. ^{18} were performed using a 1.3 eV photoactive material using much weaker ~10^{9} W cm^{−2} lasers and, as a consequence, it is not clear if the identified mechanism is at play in the experiments.
What is then the microscopic origin of the experimental observations?
In this paper, we present stateoftheart atomistic simulations of the laserinduced timedependent electronic transport along gold–silica–gold nanojunctions that recover the experimental observations and offer a simple intuitive picture for the origin of the effect. The simulations explicitly take into account the nanoscale nature of the experimental setup and the crucial role of the silica–gold interface in the current rectification. This contrasts with previous efforts^{4,15,16,17,20} that model the silica as bulk matter. Using them we assess the feasibility of mechanisms that have been proposed^{4,11,15,16,17,18,20} to explain the observed currents.
Results
Computational approach
The simulations are based on solving the singleparticle Liouville von Neumann equation for the nanojunction in the presence of radiation using a nonequilibrium Green’s function method^{21} (TDNEGF) for timedependent electronic transport in the wide band limit. While the experiments are performed on fused silica, the simulations employ a quantitative atomistic model of αquartz that is developed from density functional theory (DFT). Since the symmetry breaking effect occurs along the direction of laser polarization, we focus on quasi onedimensional model junctions of varying length as they capture the essential physics of the effect. Specifically, we consider onedimensional slabs of N unit cells of αquartz along a given crystallographic direction \(\widehat {\boldsymbol{d}}\) (a, b, or c) connected by its ends to macroscopic metallic contacts. We focus on the a (E  a) and c (E  c) directions since the b direction is equivalent to a.
A schematic representation of the metalsilicametal nanojunction is shown in Fig. 1a. The Wannier functions in the terminal unit cell are assumed to couple identically to the metallic contact at their terminal end. The quantity ħ/Γ determines the rate of charge exchange between the silica and contacts and is fixed at a model but realistic value of Γ = 0.1 eV. The Fermi energy μ_{F} of the gold contacts is taken to be in the band gap and its exact value is a modeling parameter. The lasermatter interactions is considered in dipole approximation and the metallic contacts are assumed to behave like perfect metals that completely reflect the incident radiation. The vector potential A(t) associated with the electric field \(E(t) =  \dot A(t)\) employed in the simulations is of the form A(t) = (E_{0}/ω) exp[−(t − t_{c})^{2}/(2σ^{2})] sin(ω(t − t_{c}) + φ). This form guarantees that the fewcycle E(t) of amplitude E_{0} is an AC source, as \({\int}_{  \infty }^\infty {\kern 1pt} E(t){\mathrm{d}}t\) = A(−∞) − A(∞) = 0. To mimic the experiments, we choose a laser pulse centered at t_{c} = 0 fs of width σ = 2 fs and central frequency ħω = 1.7 eV. The carrier envelope phase φ is the main source of control of the electronic currents in the experiment. In the model, the effect of E(t) on the metallic contacts is to setup a total potential drop across the junction V(t) = eDE(t), where D is the total length of the junction and e the electronic charge, by rigidly shifting the energy levels of left and right contacts^{19}. In the wide band limit, this shift is computationally captured by a timedependence of the chemical potentials of the left μ_{L}(t) = μ_{F} − \(\frac{1}{2}eDE(t)\) and right μ_{R}(t) = μ_{F} + \(\frac{1}{2}eDE(t)\) contacts.
The net current passing through the nanojunction is calculated as the average current flowing into the two leads I(t) = (I_{L}(t) − I_{R}(t))/2, where I_{ α }(t) is the current entering lead α. The total charge transferred between two leads at time t is given by Q(t) = \({\int}_{  \infty }^t {\kern 1pt} I(t{\prime}){\rm d}t{\prime}\), while the accumulated charge in the silica region is Q_{acc}(t) = \({\int}_{  \infty }^t {( I_{\mathrm{L}}(t{\prime}) + I_{\mathrm{R}}(t{\prime})}){\kern 1pt} {\rm d}t{\prime}\).
Accurate generalized tightbinding model for the silica
To be able to describe the dynamics of a junction with hundreds of atoms driven by strong laser fields, we developed an accurate and computationally efficient generalized tightbinding model (GTB) of the silica from first principle computations. For this, we computed the Bloch states and band structure of αquartz using DFT (modified BeckeJohnson metaGGA functional^{22}), and used the results to generate an orthonormal basis of maximally localized Wannier functions (MLWFs) via unitary transformation^{23}. The matrix elements between these Wannier functions are then employed to build a Hamiltonian for the silica and its interaction with the laser. The resulting basis consist of 27 MLWFs per unit cell, that capture 18 Valence Bands (VB) and 9 Conduction Bands (CB). Figure 1b shows the resulting groundstate band structure of αSiO_{2}^{24} (a = b = 4.9137 Å, c = 5.4047 Å, α = β = 90°, and γ = 120°) computed with DFT (solid lines) and the GTB model (open circles). Figure 1c shows the isosurface contours of four representative MLWFs in αSiO_{2} (red for positive value and blue for negative). The MLWFs that compose the VB correspond to the p_{ x }, p_{ y }, and p_{ z } orbitals of 6 O atoms, while the CB MLWFs involve contributions from both Si and O atoms.
As shown in Fig. 1b, the GTB accurately reproduces the firstprinciplebased band structure in a wide energy range and the bulk 9 eV band gap. The onedimensional model slabs employed in the transport simulations have a 16–18% larger band gap [10.4 eV (E  a) and 10.6 eV (E  c)] because they neglect tightbinding couplings in directions perpendicular to the junction. The GTB has no adjustable parameters and retains the atomistic detail of first principle approaches. In addition, it allows one to select the number and type of bands that participate in the dynamics. In this way, it offers a powerful theoretical tool to interpret the experiments. Additional details of the simulation approach are included in the Methods section.
Phase and size dependence of the photocurrents
Figure 2 shows the currents induced by the 4 fs laser pulse on a N = 6 junction (E  a). The laser (Fig. 2a, b) transiently generates large currents (Fig. 2c, d). Figure 2e, f shows the net charge transferred across the junction (solid line) and the accumulated charge in the silica (dashed line). As shown, the laser photoejects electrons and leaves the silica charged (Fig. 2e). Only a fraction of the photoejected electrons form part of the net current. After the laser, the metallic contacts inject charge back into the system and restore charge neutrality in 20–30 fs (Fig. 2f).
Figure 3a shows the dependence of the net charge extracted Q = Q_{m} + Q_{c} after the system has equilibrated on the carrier envelope phase φ. The effect of φ on the laser pulse is shown in Fig. 3b. There are two components to the response: a component Q_{m} that is independent of φ that arises because of the inherent spatial asymmetry of αSiO_{2}, and a phasecontrollable component Q_{c}. The experiments are designed to only capture Q_{c}. The simulations capture the experimentally observed sinusoidal dependence of the magnitude and sign of Q_{c} on φ. The slight discrepancy in the control map between theory and experiment arises because the experiment exhibits dispersion effects as φ is varied that are not included in the simulations. Importantly, the simulations show that the effect is largely independent on junction size N, as the net extracted charge observes essentially no dependence with the number of unit cells for N = 6, 10, 20, 24. This observation is consistent with experiments performed with 50 and 500 nm junctions in wedged and flat geometries, respectively, that suggest a mild dependence of the effect on junction size (Fig. S8 in ref. ^{4}). Below, we focus on N = 6 as it describes well the behavior of the longer N = 24 junction and is expected to be representative of the N ~ 100 experimental setup.
Dependence of the photocurrents on laser intensity
Figure 4 shows a comparison between experimental and computational maximum extracted charge \(Q_{\mathrm{c}}^{{\mathrm{max}}}\) and φ^{max} as a function of laser amplitude E_{0}. For a given laser amplitude, \(Q_{\mathrm{c}}^{{\mathrm{max}}}\) is extracted by scanning the dependence of the extracted charge Q_{c} on φ and recording the maximum charge \(Q_{\mathrm{c}}^{{\mathrm{max}}}\) in the control map. To compare with experiments, the simulation results are scaled by a factor η which represents the illumination crosssection area which is an experimentally unknown parameter. Simulations correctly capture the intensity dependence of the effect up to a laser amplitude of 2 V Å^{−1}. To capture observations beyond this laser amplitude, the generalized tightbinding model would require a larger number of bands. Note that the intensity and phase dependence of the control is approximately independent of the crystallographic direction in the model junction. This insensitivity makes the model of αquartz useful in the description of the experiment.
An unknown variable in the simulations is the position of the Gold’s Fermi energy μ_{F} with respect to the silica. To test the dependence of the results on μ_{F}, this parameter was varied between the top of the VB (1.8 eV) and the bottom of the CB (10.7 eV). As shown in Fig. 5, the results are approximately independent of μ_{F} under a wide range of values. Disagreement with experiment starts to emerge when the μ_{F} is chosen to be close to the band edges.
To examine possible effects of the 16–18% overestimation of the energy gap E_{g} by the model nanowires, we performed simulations with a larger central frequency ω such that the experimental E_{g}/ħω = 5.3 ratio is maintained. As shown in Supplementary Fig. 1, increasing ω changes the illumination crosssection η but leaves the qualitative features of the control map unchanged.
As yet another point of contact with experiment^{20}, we examined the dependence of the carrier envelope phase required to achieve the maximum charge φ^{max} on the laser amplitude. Figure 4 shows such a dependence when the junction is constructed along the a (Fig. 4d) and c (Fig. 4e) crystallographic directions. For weak fields (region I) the simulations recover the experimentally observed φ^{max} ≈ 0 phase required for maximum current. For intermediate intensities (region II), we observe that φ^{max} is sensitive to the particular crystallographic direction. Since the experiments are performed in fused silica, while the simulations are done in αquartz, the disagreement between theory and experiment arises because the microscopic model in the simulations does not coincide exactly with the material employed in the experiment. Importantly, in the most relevant region where appreciable currents are observed (\(\left {E_0} \right > 1.4\) V Å^{−1}, region III), the simulations recover the experimentally observed φ^{max} ≈ π and the independence of φ^{max} on the laser intensity^{20}.
Microscopic origin of the effect
As shown above, the simulations recover the main experimental observations including the phase dependence, intensity dependence and size independence of the effect. We are now thus in a position to examine the microscopic origin of the effect, and the relevance of previously proposed mechanisms.
Importance of the 5 vs. 6 photon absorption coherent control scenario
To test this possible mechanism^{16}, we examined the experimental power r dependence of the effect on the electric field amplitude \(Q_{\mathrm{c}}^{{\mathrm{max}}}\sim \left {E_0} \right^r\). The 5 vs. 6 scenario should exhibit an \(\left {E_0} \right^{11}\) dependence. Figure 6 shows fits of the experimental data to \(Q_{\mathrm{c}}^{{\mathrm{max}}}\sim \left {E_0} \right^r\) that take into account different sets of consecutive experimental points. The regime of the response that can be captured by a single power law (points 2–7, R^{2} = 0.99, error 5.3%) offer an r = 7 which is inconsistent with a 5 vs. 6 scenario. Only the fit with points 1–5 offer an r = 10.4 consistent with a 5 vs. 6 scenario and that fit is, statistically, a poorer representation of the data (R^{2} = 0.84, error 21.2%). In the simulations r ≈ 7 (Fig. 4b) in agreement with experiments.
Further note that a resonant 5 vs. 6 scenario is expected to exhibit a strong dependence on the length of the material N because the magnitude of the transition dipoles between energy eigenstates (and the number of available transitions) increase with N. Such length dependence is not observed in experiments nor simulations (cf. Fig. 3a and Fig. S8 in ref. ^{4}).
These experimental and numerical observations suggest that 5 vs. 6 control is not the dominant mechanism underlying the effect.
Importance of Stark effects
An additional aspect that requires clarification is whether the laserinduced dynamics is due to nearresonance multiphoton absorption or due to Stark shifts. While the central frequency of the pulse is far detuned from the bandgap, the laser is intense enough that competition between these two effects is possible. This distinction is important because multiphoton excitation will generate real charge carriers, while Stark shifts will reversibly deform the electronic structure of the material generating vastly different mechanisms for the response.
To address this, in Fig. 4c we examine the power dependence of the uncontrollable part of the response, \(Q_{\mathrm{m}}\sim \left {E_0} \right^r\). To photoexcite electrons across the energy gap, 4–6 photons from the pulse need to be absorbed which implies that 8 ≤ r ≤ 12 if multiphoton absorption plays a role. As shown in Fig. 4c for laser amplitudes \(\left {E_0} \right\) < 1 V Å^{−1}, Q_{m} scales with r ≈ 5 and then saturates, which is a power dependence that is considerably below the threshold for multiphoton absorption. We thus conclude that Stark effects due to nonresonant lasermatter interactions dominate the dynamics.
Importance of WannierStark metallization
In refs. ^{4,11,15} the effect is interpreted through WannierStark metallization effects that require Zener tunneling to emerge. To test this interpretation, we performed numerical experiments (Fig. 7) in which Zener tunneling pathways are eliminated completely or partially from the dynamics. Specifically, we examined the control map (E_{0} = 1.7 V Å^{−1}) under circumstances in which the CB and VB are completely decoupled from one another (Fig. 7a, red line in c and d) eliminating Zener tunneling effects from the dynamics. This is achieved by setting the Hamiltonian matrix elements between the Wannier basis states that form the VB and those that form the CB to zero. We also examined a case (Fig. 7b, blue line in c and d) where Zener tunneling is maintained but transport is assumed to go through the VB (hole transport) or the CB (electron transport) independently, with no transport pathways that involve both bands as required for mechanisms based on Zener tunneling. This is done by performing two separate simulations in which either the VB or the CB is disconnected from the leads and adding their two separate contributions to the current. If Zener tunneling is an essential component of the dynamics, case (i) should exhibit no net currents while case (ii) should exhibit suppressed currents.
As shown in Fig. 7, eliminating completely Zener tunneling shifts slightly (by ~ 0.25π) the control map but has no appreciable incidence on the magnitude of the effect. Similarly, considering that transport does not involve pathways that involve both bands has a minor impact on the control map. We are thus forced to conclude that Zener tunneling is not essential for the description of the experimental observations and, thus, that WannierStark metallization^{14} does not underlie the effect.
The relevance of the mechanism proposed in ref. ^{18} is discussed below.
Instantaneous level alignment as an interpretative tool
To understand the microscopic origin of the effect it is useful to interpret the quantum dynamics in terms of the instantaneous laserdressed singleparticle eigenstates of the silica and to examine how the energy of the laserdressed levels match the chemical potentials of the contacts. The laserdressed eigenstates are obtained by diagonalizing the Hamiltonian of the silica in the presence of the lasermatter interactions for a fixed electric field. Figure 8a–f shows the eigenenergies and probability density distribution of the eigenstates along the junction with the densities coarsegrained over unit cells. To enhance the interpretative value of the plots, the probability density of the eigenstates in each unit cell is divided into a contribution due to the Wannier states that form the VB (blue) and those that form the CB (red). The position of the chemical potential in the left and right contact, μ_{L} and μ_{R}, are indicated by dashed lines and vary with the laser amplitude. The effect of the static electric field is to localize the silica eigenstates into socalled WannierStark states. For the laser amplitudes in the experiment this localization is extreme, confining the eigenstates to 1–2 unit cells.
Since Stark effect dominate the dynamics and Zener tunneling does not play a significant role, in the absence of metallic contacts, the VB levels are occupied while those in the CB are empty during and after the laser pulse. Thus, for charge to flow between silica and contacts the WannierStark states at the terminal ends of the junction need to be in proper energetic alignment with the contact’s Fermi sea. Specifically, for charge to flow from the VB into the contacts, the terminal VB WannierStark states need to be above the contact’s chemical potential. Otherwise, hole transport is blocked. Similarly, for charge to flow from the contact into the CB the terminal CB WannierStark states need to be below the chemical potential. Otherwise, electron transport is blocked. This basic energetic alignment picture is used below to develop an intuitive interpretation of the experiment.
Origin of the threshold at \(\left {\boldsymbol{E}}_{0} \right\sim \mathbf{1.4}\) V Å^{−1} to generate sizable currents
In both theory and experiments sizable currents require electric field amplitudes \(\left {E_0} \right > 1.4\) V Å^{−1} (cf. Fig. 4a). To understand this threshold consider the laserdressed eigenstates of the material and the level alignment shown in Fig. 8. For \(\left {E_0} \right\) < 1.4 V Å^{−1} the CB levels at the terminal ends are above the chemical potential of the contacts and thus no charge can be transferred from the contacts into the CB. Similarly, the VB levels at the terminal ends are below the chemical potential of the contacts and charge transfer from the material into the leads is blocked by the Fermi sea. Thus, no significant currents are observed for \(\left {E_0} \right\) < 1.4 V Å^{−1} because electron and hole transport are energetically blocked. At \(\left {E_0} \right\sim 1.4\) V Å^{−1} this blocking of the charge transfer is removed and current starts to flow across the junction. For stronger fields \(\left {E_0} \right\sim 1.7\) V Å^{−1} more levels satisfy the energy level alignment conditions leading to a larger currents.
Mechanism for the generation of currents
To understand the origin of the currents, it is useful to focus on the separate electron and hole contributions to the total current. This can be done because the sum of these two individual contributions coincides with the full dynamics (Fig. 7). Figure 8g shows the current flowing through the CB under the influence of the laser pulse shown in gray (\(\left {E_0} \right = 1.7\) V Å^{−1}, φ = π). The current entering the right (left) contact is shown in red (black). An analogous diagram for the VB is shown in the topright inset. The two additional insets show the laserdressed singleparticle eigenstates for the maximum positive E = 1.7 V Å^{−1} and negative E = −1.4 V Å^{−1} field amplitudes.
The net currents across the junction arise because of a difference in effective lead–silica couplings for positive and negative field amplitudes. To see this, consider transport through the CB first. For negative amplitudes, the field is not strong enough to push the terminal CB WannierStark levels below the chemical potential of the contacts and little current is injected from the leads into the CB. By contrast, for a positive amplitude of E ~ 1.7 V Å^{−1} the electric field is strong enough to bring the rightend CB WannierStark levels below μ_{R} leading to a large burst of charge being injected from the right contact into the leads. This imbalance between the effective coupling of the CB to the leads for positive and negative field amplitudes leads to a net electron current.
A similar situation occurs for hole transport (inset Fig. 8g). In this case, the field amplitude is strong enough to push the terminal WannierStark levels above the chemical potential of the contact for the three central peaks, one positive and two negatives, of the pulse. For the other peaks the field is not strong enough to open significant channels for charge transport from the silica into the leads. This generates two bursts of charge injected into the right contact and one into the left one. The difference in effective leadsilica coupling for positive and negative field amplitude yields a net hole current along the device. The hole and electron current do not exactly cancel one another and give rise to a net current.
In this context, it becomes intuitively clear the origin of the phase control. When appreciable currents are observed (\(\left {E_0} \right > 1.4\) V Å^{−1}), the control maximum is achieved for φ ≈ 0, π. This is because this field maximizes the difference in laser intensity for negative and positive electric field amplitudes (Fig. 3b) and, thus, the difference in effective coupling between silica and right and left contacts. Since the identified mechanism is at play for a wide range of laser amplitudes, this is the origin of the insensitivity of φ^{max} on laser intensity identified in ref. ^{20}. By contrast, when φ ≈ π/2 the field is antisymmetric with respect to time inversion around some time t′, i.e., E(t − t′) = −E(−(t − t′)) and the controllable current is small. This is because this field will have equal intensity for positive and negative field amplitude.
Further, we can now readily understand why the effect is independent of junction size. Due to WannierStark localization only the 1–2 unit cells at the junction boundaries determine the charge dynamics. Thus by increasing the size of the junction one is not affecting the effectiveness of the scenario.
It is qualitatively useful to consider this control scenario in a minimal singleband model with N sites. In the presence of a static electric field of amplitude E(t), the energy of the WannierStark states^{25} located at the terminal ends of the singleband material are \(\epsilon _{\mathrm{R}} \approx \epsilon _0\) + \(\frac{1}{2}eNdE(t)\) and \(\epsilon _{\mathrm{L}} \approx \epsilon _0\) − \(\frac{1}{2}eNdE(t)\), where \(\epsilon _0\) corresponds to the onsite energies of the pristine material. In turn the chemical potentials of the right and left contacts are μ_{R} = μ_{F} + \(\left( {\frac{{Nd}}{2} + u} \right)eE(t)\) and μ_{L} = μ_{F} − \(\left( {\frac{{Nd}}{2} + u} \right)eE(t)\) where Nd + 2u is the total length of the junction and u is the distance of the leadsilica interface. In this model, the threshold for charge transport between the band and the metallic contacts occurs when \(\epsilon _{\mathrm{R}}\) = μ_{R} or \(\epsilon _{\mathrm{L}}\) = μ_{L}. This yields a threshold electric field of \(E_{\mathrm{L}}^ \ast\) = \(\frac{{(\mu _{\mathrm{F}}  \epsilon _0)}}{{eu}}\) for the left contact and \(E_{\mathrm{R}}^ \ast =  E_{\mathrm{L}}^ \ast\) for the right one. The key to achieve a current is then to use a field for which, for instance, the threshold \(E_{\mathrm{L}}^ \star\) is contained in the pulse but \(E_{\mathrm{R}}^ \star\) is not such that a net difference in effective leadsilica couplings for positive and negative field amplitudes emerges. To guarantee that the intensity \(I\sim \left E \right^2\) is different for positive and negative amplitudes, E(t) cannot be antisymmetric with respect to time inversion around any given time. This can be achieved by using fewcycle pulses like the one employed in the experiment with φ ≠ (2n + 1)π/2 where n = 0, 1, 2, … or, alternatively, using two color nω + mω pulses where n is an even integer while m is odd.
This identified mechanism is reminiscent to the one proposed in ref. ^{18}. While the qualitative ideas are identical an important technical difference, however, is that in the early study^{18} it was supposed that the chemical potentials of the contacts did not change with the laser field. This supposition leads to an effect that arises at much weaker fields. As shown in Fig. 4b–d (green line), when such a time dependence is not taken into account the simulations cannot recover the experimental observations.
Discussion
We have presented atomistically detailed timedependent quantum transport simulations of experiments^{4,20} that induce currents along metalsilicametal nanoscale junctions using strong nonresonant fewcycle 4 fs laser pulses. The simulations are based on propagating the singleparticle von Neumann equation for the junction using a stateoftheart timedependent nonequilibrium Green’s function method that, contrary to previous simulation and interpretational efforts, explicitly take into account the nanoscale nature of the experiment and the crucial role of the metallic contacts on the emergence of the effect. The simulations do not take into account possible effects of plasmons, screening or other processes that require feedback between the electromagnetic field and the material response.
Under these conditions, the simulations recover the experimental observations and offer an intuitive picture of the effect in which the temporal asymmetry of the incident radiation generates a difference in effective coupling of the silica to the left and right metallic contact and leads to a net phasecontrollable current. Specifically, because the few cycle laser pulse in the experiment has different laser intensity for negative and positive field amplitude, through Stark shifts, such laser generates different metalsemiconductor band alignment for the left and right contacts leading to a net current across the nanojunction. Varying the carrier envelope phase controls the difference in intensity of the pulse for positive and negative field amplitudes and thus the sign and magnitude of the photoinduced currents. This identified mechanism is reminiscent to the early proposal in ref. ^{18}.
An analysis of both simulation and experimental results suggest that previously proposed resonant 5 vs. 6 coherent control do not underlie the experimental observations. In addition, WannierStark metallization and other possible mechanisms based on Zener tunneling do not underlie the simulated dynamics, and are thus not necessary for the emergence of the effect. Further, to explain the experimental observations it is not necessary to invoke mechanisms that involve the generation of virtual carriers. Additional progress in understanding the photoinduced dynamics in junctions requires experiments that address the relative importance of bulk and interfacial contributions^{26} and the length dependence of the effect at all relevant regimes of the lasermatter interaction.
Importantly, the simulations reveal that the experiment by Schiffrin et al.^{4,20} is based on Stark effects and not on near resonance multiphoton absorption. Thus, the experiment exemplifies the power of Starkbased strategies to control electronic properties and dynamics. These insights can be employed to interpret recent related experiments^{27,28,29,30} and to advance our ability to control electrons in matter using lasers.
Methods
Hamiltonian
The Hamiltonian for the composite metalsilicametal junction is given by:
where H_{S}(t) describes the Hamiltonian of the silica, H_{G}(t) the leads and H_{SG}(t) the silicalead couplings. The composite system is assumed to be well described by an effective singleparticle Hamiltonian H(t) = \(\mathop {\sum}\nolimits_{\nu \mu } {\kern 1pt} h_{\nu \mu }(t)c_\nu ^\dagger c_\mu\) where the operator \(c_\nu ^\dagger\) (or c_{ ν }) creates (or annihilates) a fermion in a singleparticle state ν and satisfies the usual fermionic anticommutation relations. As such, the electronic properties of the composite system are completely determined by the singleparticle reduced density matrix ρ_{ νμ }(t) = \(\left\langle {c_\nu ^\dagger c_\mu } \right\rangle\).
Tightbinding model for laserirradiated silica nanostructures
To obtain a first principle description of the silica and its interaction with a laser field, we computed the Bloch states and the band structure of bulk αquartz and used that to construct an accurate generalized tightbinding model for the material and the transition dipoles required to capture the lasermatter interactions. Specifically, the groundstate band structure of αSiO_{2}^{24} was computed using DFT in the Vienna ab initio simulation package (VASP)^{31} with the modified BeckeJohnson (MBJ) metaGGA functional^{22}, and a planewave basis set with an energy cutoff of 650 eV. The calculated band gap of αSiO_{2} is about 9 eV, in good agreement with experiment^{32}. From the resulting Bloch eigenstates, an orthonormal basis of maximally localized Wannier functions (MLWFs) \(\left\{ {\left {\phi _n({\boldsymbol{r}},{\boldsymbol{R}}_l)} \right\rangle } \right\}\) was constructed via unitary transformation using Wannier90^{23,33}. Here r is the electron coordinate and \(\left\{ {\left {\phi _n({\boldsymbol{r}},{\boldsymbol{R}}_l)} \right\rangle } \right\}\) is the nth Wannier function localized on the lth unit cell associated with the realspace lattice vector R_{ l }. The MLWFs were chosen to reproduce the band structure in the ([−8 eV, 14.5 eV]) energy window which includes 18 valence bands (VB) and 9 conduction bands (CB). The procedure resulted in N_{ b } = 27 MLWFs per unit cell that quantitatively reproduce the band structure of αSiO_{2} in a wide range of energies.
In dipole approximation, the Hamiltonian for the onedimensional slab of αSiO_{2} in the presence of a laser field polarized along the junction direction is given by
where H_{0} is the Hamiltonian of the pristine silica, μ is the dipole operator and E(t) is the electric field of light. In the maximally localized Wannier basis, the Hamiltonian of N unit cells of αSiO_{2} along a given crystallographic direction is given by
where \(c_{n,l}^\dagger \left 0 \right\rangle = \left {\phi _n({\boldsymbol{r}},{\boldsymbol{R}}_l)} \right\rangle\) creates a fermion in MLWF \(\left {\phi _n({\boldsymbol{r}},{\boldsymbol{R}}_l)} \right\rangle\) and \(\left 0 \right\rangle\) is the vacuum state. Here \(h_{nl,n\prime l\prime }\) = \(\left\langle {\phi _n({\boldsymbol{r}},{\boldsymbol{R}}_l)} \rightH_0\left {\phi _{n\prime }({\boldsymbol{r}},{\boldsymbol{R}}_{l\prime })} \right\rangle\) are the matrix element of the Hamiltonian among the Wannier states. In this microscopic model of the junction, the \({\boldsymbol{R}}_l = ld\widehat {\boldsymbol{d}}\) are chosen to be collinear and defined along a particular crystallographic direction \(\widehat {\boldsymbol{d}}\) with lattice constant d. Since the \(\left\{ {\left {\phi _n({\boldsymbol{r}},{\boldsymbol{R}}_l)} \right\rangle } \right\}\) basis is maximally localized in real space (as shown in Fig. 1c) it suffices to only consider same cell l = l′ and nearest neighbor (\(\left {l  l\prime } \right = 1\)) contributions to the Hamiltonian. Hamiltonian matrix elements with unit cells that are located in directions perpendicular to \(\widehat {\boldsymbol{d}}\) are neglected.
The total dipole operator μ = (μ_{N} + μ_{ e }) along the junction direction that determines the lasermatter interactions is also obtained from firstprinciple computations. Specifically, the electronic component of μ is given by
where e is the magnitude of the electronic charge, and \({\boldsymbol{r}}_{nl,n\prime l\prime }\) = \(\left\langle {\phi _n({\boldsymbol{r}},{\boldsymbol{R}}_l)} \right{\boldsymbol{r}}\left {\phi _{n\prime }({\boldsymbol{r}},{\boldsymbol{R}}_{l\prime })} \right\rangle\) are the matrix elements of the position operator in the Wannier basis, which are computed with Wannier90. In turn, the nuclear dipole is constant throughout the simulation and given by μ_{N} = \(e\mathop {\sum}\nolimits_{l = 1}^N {\kern 1pt} \mathop {\sum}\nolimits_{A \in l} {\kern 1pt} eZ_A\widehat {\boldsymbol{d}} \cdot {\boldsymbol{R}}_{A,l}\), where the second sum runs over all atoms A in cell l with position R_{A,l} and atomic number Z_{ A }. The total junction length is taken to be D = Nd + d where Nd is the length of the silica and the extra d accounts for the approximate distance between the silica and the first layer of metallic atoms on each side of the junction.
Metallic contacts and lead–silica interactions
The leads are described by H_{G}(t) = \(\mathop {\sum}\nolimits_{\alpha = {\mathrm{L}},{\mathrm{R}}} \mathop {\sum}\nolimits_q {\kern 1pt} \varepsilon _{\alpha q}c_{\alpha q}^\dagger c_{\alpha q}\) where \(c_{\alpha q}^\dagger\) and c_{ αq } are the fermionic operators for the lead states of energy ε_{ αq } and α = L or R denotes the left or right contact, respectively. The leads are assumed to be in a state of thermal equilibrium with a density matrix described by the FermiDirac distribution at temperature T = 300 K and with chemical potential μ_{ α }.
The leads and their interaction with the silica are described in the wide band limit (WBL). In this limit, the density of states in the metal is assumed to be constant and the Wannier functions that couple to the leads are supposed to couple identically to all lead levels. From a dynamical perspective, in WBL the metallic contacts behave as a Markovian reservoir that exchanges particles and energy with the material. As a model of the lead–silica interactions, we suppose that only the Wannier states \(\left {\phi _n({\boldsymbol{r}},{\boldsymbol{R}}_l)} \right\rangle\) in the terminal unit cells (l = 1 or N) couple to its adjacent contact. Further, the couplings to each lead is taken to be independent of the nature of the Wannier state. Thus, the silicalead interaction is given by
where \(V_q^\alpha\) is the coupling between level q in lead α and the Wannier states \(\left {\phi _n({\boldsymbol{r}},{\boldsymbol{R}}_l)} \right\rangle\) in the unit cell adjacent to it, and H.c. denotes Hermitian conjugate. The effective coupling between Wannier state \(\left {\phi _n({\boldsymbol{r}},{\boldsymbol{R}}_l)} \right\rangle\) and lead α is specified by the spectral density \({{\Gamma }}_\alpha (\varepsilon )\) = \(2\pi \mathop {\sum}\nolimits_q \left {V_q^\alpha } \right^2\delta \left( {\varepsilon  \varepsilon _{\alpha q}} \right)\), a quantity that contains information about the characteristic frequencies of the leads and their coupling to the molecule. In the WBL the \(V_q^\alpha\) and the leads’ density of states ζ^{α} = \(\mathop {\sum}\nolimits_q {\kern 1pt} \delta \left( {\varepsilon  \varepsilon _{\alpha q}} \right)\) are assumed to be energy independent. In this case, the spectral density is also energy independent and given by
In this work, we use Γ_{L} = Γ_{R} = Γ = 0.1 eV. The quantity Γ dictates the characteristic timescale, ħ/Γ, for charge exchange between silica and contacts and generates an effective Lorentzian broadening of the silica energy levels by 2Γ.
Timedependent transport
The timedependent transport characteristics of the metalsilicametal junction are characterized via the nonequilibrium Green’s function method (NEGF) as developed and implemented by Chen and colleagues ^{21,34,35,36,37}. In TDNEGF the current is obtained by solving the Liouville von Neumann equation for the singleparticle electronic reduced density matrix in the presence of leads. In it, the current entering lead α is defined by I_{ α }(t) = −\(e\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}\left( {\mathop {\sum}\nolimits_q \left\langle {c_{\alpha q}^\dagger c_{\alpha q}} \right\rangle } \right)\) and the net current passing through the nanojunction is calculated as the average current flowing into the two leads I(t) = (I_{L}(t) − I_{R}(t))/2.
The employed method combines timedependent density functional theory and NEGF^{21,34,35,36,37}. Specifically, ref. ^{35} presents a computational efficient closed set of equations (Eqs. (3), (12) and (14) in ref. ^{35}) to capture timedependent transport by invoking the wide band limit and a Padé expansion of the Fermi distribution function. The former allows closing the resulting hierarchy of equations at first tier in the Hierarchical Equation of Motion sense. In turn, the Padé expansion allows for analytically solving the energy integrals that appear in the definition of the selfenergies. Here, results were checked for convergence on the number of Padé functions (50) required to represent the leads, and on the integration time step (0.002 fs) of the Runge Kutta method of order four employed in the numerical integration of the equations of motion.
Data availability
The data are available from the corresponding author upon reasonable request.
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Acknowledgements
This material is based upon the work supported by the National Science Foundation under CHE1553939. Y.Z. and G.C. thank Hong Kong Research Grant Council for support (HKU 700912P and AoE/P04/08). L.C. thanks Linjun Wang for illuminating discussions.
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G.C. and his group developed the timedependent transport methods and provided associated computer codes that were employed in this work. Y.Z. helped to carry out preliminary computations under G.C.’s supervision. L.C. adapted and augmented these codes, and performed all simulations in the paper under I.F.’s supervision. L.C. and I.F. designed the simulation strategy and performed the analysis. I.F. conceived the study, designed and overviewed the project, and wrote the paper using input from all authors.
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Chen, L., Zhang, Y., Chen, G. et al. Stark control of electrons along nanojunctions. Nat Commun 9, 2070 (2018). https://doi.org/10.1038/s41467018043934
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