Research ArticleCONDENSED MATTER PHYSICS

Chemical identification through two-dimensional electron energy-loss spectroscopy

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Science Advances  08 Jul 2020:
Vol. 6, no. 28, eabb4713
DOI: 10.1126/sciadv.abb4713

Abstract

We explore a disruptive approach to nanoscale sensing by performing electron energy loss spectroscopy through the use of low-energy ballistic electrons that propagate on a two-dimensional semiconductor. In analogy to free-space electron microscopy, we show that the presence of analyte molecules in the vicinity of the semiconductor produces substantial energy losses in the electrons, which can be resolved by energy-selective electron injection and detection through actively controlled potential gates. The infrared excitation spectra of the molecules are thereby gathered in this electronic device, enabling the identification of chemical species with high sensitivity. Our realistic theoretical calculations demonstrate the superiority of this technique for molecular sensing, capable of performing spectral identification at the zeptomol level within a microscopic all-electrical device.

INTRODUCTION

The spectral determination of vibrational and electronic excitations in molecules provides the means to detect and chemically identify small amounts of analyte (1), opening exciting applications in areas such as medical diagnosis (2, 3) and detection of hazardous substances (4). In particular, all-optical techniques based on Raman and infrared (IR) absorption spectroscopies are widely used for this purpose (5, 6) and can reach single-molecule sensitivity (7, 8) when enhanced by the near-field light amplification of optical hot spots associated with plasmons—conduction electron excitations—localized at corrugated metal surfaces (9). Noninvasive far-field optical approaches are however limited in spatial resolution by diffraction (10), while near-field optics can resolve areas of a few tens of nanometers (11).

Atomic-scale spectral imaging is made possible by resorting to electron beams in electron microscopes, where localized optical excitations are revealed by analyzing the energy losses experienced by the electrons upon interaction with the sample. Recent advances in electron energy-loss spectroscopy (EELS) have pushed the combined space/energy resolution to unprecedented levels in the few milliectronvolts per subangstrom range (1214), enabling the detection and spatial imaging of nanoscale optical excitations such as plasmons (1518) and phonon polaritons (1214). However, the use of swift electrons requires high vacuum and produces structural damage, which hinders applications to in vivo samples. In addition, damage also limits the acquisition of visible and IR spectra from structures consisting of only a few molecules (19, 20). Aloof electron excitation can prevent this problem by probing excitations in a material without physically traversing it (21), an approach that has been used to map optical modes in metallic nanoparticles (17), including those confined to narrow gaps (22), but has not yet been applied to resolve localized IR molecular fingerprints for chemical analysis.

The spatial range of interaction between an electron moving with velocity v and an excitation of frequency ω is dictated by v/ω (23). For small electron-sample separations R compared with this distance, the interaction strength is dominated by the radial component of the electric field carried by the electron, the intensity of which scales as 1/(vR)2. Besides the advantage of reduced damage, low electron energies favor the interaction strength, as it is well known in the context of low-energy electron microscopy (24). However, an integrable approach without the need of vacuum technology would be desirable to perform spectral optical analysis as an alternative to the existing bulky electron microscope setups. Ballistic electrons moving in two-dimensional (2D) materials such as graphene (2527) and monolayer transition metal dichalcogenides (TMDs) (2836) could be used for this purpose, as they already feature vertical layer confinement without the need of a vacuum chamber, and their energies lie in the subelectronvolt regime, thus anticipating large inelastic interaction with neighboring molecules. In a parallel effort, transport and steering of ballistic electrons have been recently explored in graphene (3740) and other 2D systems (41) as a platform for 2D electron microscopy. In addition, gas sensors have been realized using graphene by exploiting the variation in electrical resistivity induced by charge transfer from the molecules (42, 43). However, these schemes lack the spectral resolution needed to identify molecular fingerprints for chemical analysis.

Here, we propose a disruptive approach toward chemical identification of zeptomol amounts of analyte through an all-electrical device capable of quantitatively resolving the spectral fingerprints of the sampled molecules. We rely on ballistic electrons moving within a 2D semiconductor, in which they are injected with well-defined energy through an engineered gate. Inelastic losses produced by the molecules are then resolved as features in the current transmitted after the electrons are energy filtered by an additional tunable exit gate. Our realistic simulations reveal a sensitivity down to the zeptomol level within a device of ~1 μm2 footprint, which could be integrated for massive multiplexing using currently available technology.

RESULTS AND DISCUSSION

We present a sketch of our proposed device in Fig. 1A. Ballistic electrons are injected by an emitter into the conduction band of a 2D semiconductor with well-defined energy Ei. The interaction of these electrons with analyte molecules placed in the vicinity of the semiconductor produces energy losses ℏω associated with the excitation fingerprints of those molecules. The electron-molecule inelastic interaction strength is then spectrally resolved by recording energy-filtered electrons through a narrow-band transmission gate (electron detector) as a function of their final energy Ei − ℏω. EELS spectra are acquired by sweeping ω, therefore providing spectral identification of the molecules. TMD monolayers are good candidates to be used as the 2D semiconductor for the present application, as they feature direct bandgaps of ∼2 eV (2933), which can be tuned by several methods (3436). Specifically, we consider monolayer MoS2, which features a direct gap energy Eg = 1.58 eV and a nearly parabolic conduction band with effective mass m* = 0.55 me (44). We consider the semiconductor to lie on top of a substrate of permittivity ϵs = 4, which provides the electrical isolation needed to define gate potential profiles (see below), for example, through laterally patterned back gates. We further restrict the injected electron energy relative to the conduction band bottom (CBB) to satisfy the condition Ei < Eg, which guarantees that electron-electron scattering associated with interband transitions is energetically forbidden (Fig. 1B), thus resulting in longer phonon-limited mean free paths.

Fig. 1 Two-dimensional electron energy-loss spectroscopy (2D-EELS).

(A) Sketch of the proposed device. An electron emitter (tunable through a voltage VbE) injects ballistic electrons of well-defined energy Ei into the conduction band of a 2D semiconductor (MoS2). These electrons interact inelastically with a neighboring molecule (analyte) placed outside the material, producing characteristic excitations (energy ℏω) at the expense of the electron energy. Detection and spectral analysis of the transmitted electrons (through an analyzer tunable through a voltage VbA) reveal the chemical identity of the analyte by resolving its excitation fingerprints. (B) Schematic band structure of a MoS2 monolayer semiconductor (effective mass m* = 0.55 me, gap energy Eg = 1.58 eV) with Ei referred to the CBB and satisfying Ei < Eg to prevent electron decay by coupling to interband transitions. (C) Spectral cross section for the inelastic interaction between an electron plane wave of energy Ei = 12 meV moving in monolayer MoS2 and a single 9-cis retinal molecule for two molecule-semiconductor distances z0.

The theory needed to calculate EELS probabilities is well established in the context of transmission electron microscopy and generally leads to excellent agreement with experiment by simply describing the electron-sample interaction within first-order pertubation (23). Here, we adapt such EELS formalism to deal with the particular situation of 2D ballistic electrons and calculate the tunneling current using a well-established procedure based on the Landauer formalism (45). In this context, inelastic electron-molecule interactions play a similar role in our EELS derivation (see details in Methods) as electron-phonon scattering during electron transport in 2D materials (46), and the current approach is valid for high-quality samples characterized by large ballistic mean free paths. Specifically, we describe the molecules in terms of their polarizabilities. For concreteness, we consider 9-cis retinal (47), which features a strong optical response in the far-IR spectral region, where the device is intended to operate for chemical identification. The calculated inelastic scattering cross section of a single molecule (averaged over molecular orientations) is spectrally resolved in Fig. 1C for Ei = 12 meV electrons and molecule-surface separations z0 = 1 and 2 nm. The cross section takes substantial values of a fraction of the molecular lateral size when integrated over the strong ∼6.77 meV absorption features. As anticipated above, the cross section increases with decreasing electron energy (roughly as 1/v2 ∼ 1/Ei) down to the threshold Ei = ℏω (see fig. S1), an effect that could be exploited to maximize the inelastic signal when targeting specific molecular fingerprints.

A design of an electron emitter gate is shown in Fig. 2A, consisting of a potential barrier that hosts a central well. Tunneling through the barrier can be resonantly assisted by coupling to trapped quantum well states (QWSs) (48), leading to full transmission maxima (Fig. 2B). In a related context, gates of larger size have been experimentally explored with lower-energy electrons in semiconductors (49, 50). The more deeply bound the trapped state, the weaker its leakage to the surrounding regions and, consequently, the sharper the transmission resonance. The present gate hosts a single QWS of energy Ew ∼ 12 meV. We further avoid coupling to higher-energy resonances (the ~14-meV feature) by placing the Fermi energy of the region to the left of the gate slightly above Ew (see below). The energy of transmitted electrons can be directly raised by increasing the well voltage VbE, with higher energies resulting in broader transmitted spectra, in accordance with the mentioned increase in width produced by leakage out from the QWS when its energy gets closer to the barrier height (Fig. 2C).

Fig. 2 Electron emission gate.

(A) Schematic 1D potential configuration for quasi-monochromatic electron injection. The potential energy (vertical direction) defines the position-dependent CBB. Tunneling through the double potential barrier is assisted by resonant coupling to an intermediate quantum well state. (B) Electron transmission as a function of electron energy for gate parameters d = 11 nm, d′ = 16 nm, VgE=13meV, and VbE=11.2meV. (C) Energy and full width at half maximum of the first transmission peak as a function of bias voltage VbE for fixed VgE=13meV.

Similarly, the analyzer gate must feature monochromatic transmission over a wide energy range. In addition, it is desirable that it blocks unscattered electrons to enhance the relative inelastic signal in the electron transmission through the device. A tunable analyzer potential configuration that satisfies both of these conditions is presented in fig. S2.

Coming back to the sketch of our 2D EELS device illustrated in Fig. 1A, it incorporates emitter and analyzer gates, the transmission of which can be tuned by varying their well voltages VbE and VbA, respectively. Electrons are injected with energy Ei from region I (where the Fermi energy EF is moved into the conduction band at an energy slightly above the first QWS of the analyzer) into region II (width L, where the molecules are deposited). We consider a molecular density nmol = 103 μm−2 and a 2-nm separation between molecules and the monolayer MoS2 semiconductor. This short separation could be realized, for instance, by passivating the latter with a six-monolayer boron nitride film, as done in state-of-the-art experiments (51). We note that the overlap between the out-of-plane component of the electron wave function and the molecule is significantly reduced for separation distances z0 ≥ 1 nm (52). After inelastic scattering by the molecules, electrons are finally transmitted with tunable energy Ei − ℏω, thus revealing the molecular excitation spectrum as a function of excitation frequency ω.

The gate potentials are assumed to have translational invariance along the surface direction perpendicular to the plane of Fig. 2A, so we eventually quantify the transmitted electron current normalized per unit of length along that direction. We compute the resulting current density J(ω) ∝ nmol × L from the product of transmission factors associated with the two gates and the frequency-resolved molecule inelastic cross section, after integrating over all possible electron energies Ei < EF (see region I in Fig. 1A) and in-plane directions (see Eq. 8 in Methods). The result is plotted in Fig. 3A as a function of energy loss ℏω and analyzer well voltage VbA for a width L = 1 μm of region II. Two bright features are observed, corresponding to the two inelastic scattering peaks of 9-cis retinal in the selected energy loss range (cf. Figs. 1C and 3A). We find an almost linear correlation between the peak energy loss contribution and the gate voltage ω=7.72 meVe VbA [Fig. 3A, green dashed line, which is slightly redshifted with respect to the spectral maximum of J(ω) due to the strong spectral asymmetry of this function]. This enables us to retrieve spectra by measuring the ω-integrated transmitted current as a function of VbA. Actually, the device can only provide information as a function of the applied gate voltages, so the measured signal corresponds to the ω-integrated current.

Fig. 3 EELS probability in a 2D electron microscope.

We consider the device depicted in Fig. 1A with region I doped to a Fermi energy EF = 12.1 meV, the emitter gate of Fig. 2B, and the analyzer gate (see details in fig. S2) filtering the transmission of inelastically scattered electrons (energy Ei − ℏω) from II to III by varying the voltage VbA. (A) Spectral current density J(ω) of electrons transmitted through the entire device, plotted as a function of electron energy loss and analyzer gate voltage VbA. We assume a molecular layer density nmol = 103 μm−2 in region II, spanning a width L = 1 μm and a molecule MoS2 distance z0 = 2 nm. (B) Total ω-integrated transmitted current density J as a function of analyzer voltage VbA, compared with the IR optical absorption spectrum of 9-cis retinal. The upper and lower horizontal axes are related through ω=7.72 meVeVbA, which corresponds to the dashed line in (A). Vertical dotted lines mark the two dominant resonances of the molecule.

The total current J = ∫ dω J(ω) is plotted in Fig. 3B (blue curve, left vertical axis) as a function of analyzer gate voltage VbA (lower horizontal axis). We take EF = 12.1 meV in region I slightly above the Ei = 12 meV transmission peak so that electrons injected by the emitter gate (120 nA/μm current) are mainly directed along the in-plane normal to the gate. In addition, inelastic scattering by the molecule is mainly focused along the forward direction (see fig. S3), thus introducing only a minor contribution to spectral broadening of the signal transmitted through the analyzer gate (i.e., for a selected transmitted energy along the in-plane gate-normal direction). For comparison, we provide the dominant energy loss ℏω according to the above linear relation in the upper horizontal axis. The EELS spectrum thus generated bears a close correlation with the optical absorption spectrum of 9-cis retinal molecules (Fig. 3B, red curve, right vertical axis). This corroborates the spectral resolution of our proposed device, which, in particular, is sufficient to distinguish between different molecules, such as 9-cis and all-cis retinal (see fig. S4). Incidentally, the inelastic current is reasonably large, translating into measurable values of 0.6 pA per zeptomol at the peak of Fig. 3B. Incidentally, the traveling time for a width L = 1 μm of the sampling region II is L/2m*Ei20ps, compatible with pulsed injection for lock-in detection using currently available technology.

CONCLUSIONS

In summary, on the basis of realistic theoretical simulations, we have demonstrated the viability of a disruptive approach toward chemical detection and identification, in which molecular fingerprints associated with far-IR optical excitations are resolved as spectral features in the current transmitted through our proposed device as a function of applied gate voltages. This strategy circumvents the long-standing terahertz gap problem by resorting to electrons rather than photons as probes of molecular spectra. In this study, we have engineered emitter and analyzer gates to provide a direct reading of the molecular spectrum through the voltage-dependent profile of the current transmitted through the device. Nevertheless, simpler gate designs (e.g., single barrier potentials) could yield similar information by inspecting the derivatives of the transmitted current with respect to the emitter and analyzer barrier voltages. For simplicity, we have ignored multiple scattering by the potential barriers, which, in the absence of intrinsic inelastic attenuation, is an unavoidable effect. In practice, the intrinsic inelastic mean free path is finite, so we could adjust the size of the sampling region (II in Fig. 1A) to such length scale, thus avoiding multiple reflections while optimizing the signal due to the molecules. In addition, an out-of-plane magnetic field could be used to selectively steer the electrons, as already demonstrated in experiment (3741); actually, this approach provides an alternative way of resolving the transmitted electron energy if an array of exit gates is used to detect different energy-dependent deflections or just using a single gate and scanning the strength of the magnetic field. Magnetic fields and lateral bias could also be used in combination with the lock-in injection noted above to drain electrons that are not scattered by the molecules without altering the in-plane normal energy that conveys the targeted spectral information. We note that a conducting tip could be used as an alternative electron source instead of electrical gates, as routinely done in the context of scanning tunneling microscopy, where elastic electrons are injected with atomic spatial resolution, thus suggesting a promising approach to improve imaging capabilities in the concept that we are exploring. A critical aspect of the present device is the ability of the 2D material to propagate hot electrons without inelastic losses over sufficiently long distances between the emitter and analyzer gates. The required electron energy has to exceed the sampled molecular modes, lying in the far-IR for the 9-cis retinal molecule here considered. For other macromolecules, where resonances can be localized at specific functional groups (e.g., vibrations in specific bonds in proteins), our formalism could be adapted by associating an effective polarizability with those groups rather than the entire molecule. As an alternative to 2D semiconductors, high-quality graphene is predicted to display mean free paths reaching hundreds of nanometers for the electron energies that we target here, according to detailed many-body theory (53). In addition, instead of monolayer MoS2, thin MoS2 multilayers should be explored because of their higher mobility, although stronger screening is also expected. Indirect bandgap materials could equally be advantageous to avoid interband transitions and minimize electron-electron scattering. Yet another alternative consists in exploiting ballistic transport in carbon nanotubes, in which light emission has been observed by electron-hole recombination after propagation over long mean free paths (54). We remark that the proposed chemical identification approach relies on planar solid-state devices, which due to their small footprints ∼1 μm2 could be integrated in a millimeter setup comprising millions of sampling elements for massive multiplexing. Spectral microscopy could thereby be performed with a spatial resolution limited by the footprint.

METHODS

Screened interaction

We consider a system formed by a 2D material (blue), lying at the interface between two homogeneous media (z = 0 plane) of permittivities ϵ1 and ϵ2, in the presence of a neighboring molecule (red) placed at position r0 in medium 1, as depicted in Fig. 4A. We work in the quasi-static limit because the size and surface separation of the molecule are small compared with the light wavelength associated with its excitation fingerprints. The response of the system is then captured by the screened interaction W(r, r, ω), which is defined as the electric potential produced at a position r by a point charge placed at r′ and oscillating over time with frequency ω as e−iωt. We treat the 2D material as an infinitesimally thin layer of wave vector– and frequency-dependent surface conductivity σ(k, ω), while the molecule is described as a point particle of 3 × 3 polarizability tensor α(ω). In the absence of molecule and interface, the Coulomb interaction in medium 1 reduces toW1(rr)=1ϵ11rr=1ϵ1d2k2πk eik·(RR)kzz(1)where we use the notation r = (R, z), with R = (x, y). The molecule and interface add contributions to W = W1 + Wref + Wmol arising from surface reflection (Wref, blue) and scattering by the molecule (Wmol, red), as illustrated in Fig. 4B. A self-consistent dipole p is induced at the molecule (green), which must incorporate its image self-interaction (thin green arrows).

Fig. 4 Elements involved in the dielectric response of the system under consideration.

(A) Optical response functions. (B) Decomposition of the screened interaction.

Reflection at the interface. In the quasi-static limit, the reflection at the planar interface is fully captured by the Fresnel coefficient for p polarization rp = (ϵ2 − ϵ1 + 4πiσk/ω)/(ϵ2 + ϵ1 + 4πiσk/ω), which bears a nonlocal dependence on k inherited from σ(k, ω). In the present analysis, we approximate the response of MoS2 by taking (55) σ = (iω/4π)(1 − ϵ)d with a far-IR in-plane permittivity (56) ϵ ∼ 16 and layer thickness (57) d = 0.65 nm, while we take ϵ1 = 1 and ϵ2 = 4 to represent a substrate of moderate permittivity. We note that electron motion in the semiconductor is driven by in-plane electric fields, which are continuous across the interface, thus removing any ambiguity regarding the location of induced charges in either medium 1 or 2. Following previously described methods (58), we consider a p-polarized plane wave eik·Rikzz (kkzk2ẑ) incident from z > 0 (medium 1) and its reflection rp eik·R+ikzz (kkzk2ẑ), where kz=k2ϵ1k2 and k = ω/c; in the quasi-static limit (c → ∞), we have kz → ik, so the normal electric field reduces to k2eik·R(ekz+rpekz), which corresponds to an electric potential keik·R(ekzrpekz); we then conclude that each component of externally incident potential eik·R+kz generates a reflected potential rpeik·Rkz. For a source at r′, this prescription allows us to directly write from Eq. 1 the reflection component of the screened interactionWref(r,r,ω)=1ϵ1d2k2πk eik·(RR)k(z+z) rp(2)which is valid for z, z′ ≥ 0 in medium 1.

Scattering by the molecule. A dipole p is induced at the molecule (position r0) by the test charge at r′, giving rise to an induced potential Wmol(r, r′, ω) = p · ∇r0[W1(rr0) + Wref(r, r0, ω)]. Using Eqs. 1 and 2, we findWmol(r,rω)=1ϵ1d2k2πk eik·(RR0)kz0×(rpekzekz) (ik+kẑ)·p(3)for 0 ≤ z < z0. The dependence on r′ is carried by p, which must satisfy the self-consistent relationα1p=r0[W1(r0r)+Wref(r0,r,ω)]+G·pwhere the ∇r0 term represents the sum of direct and reflected potentials due to the charge at r′, whereas the rightmost term accounts for the dipole image self-interaction through (58)G(ω)=r1r2Wref(r1,r2,ω)r1=r2=r0=(G/2000G/2000G)withG=1ϵ10k2 dk rp e2kz0(4)

Solving for p and using Eqs. 1 and 2, we findp=1ϵ11(α1G)·d2k2πk eik·(R0R)kz0×(rpekzekz) (ikkzˆ)(5)which together with Eq. 3 permits calculating Wmol for 0 ≤ z, z′ < z0.

2D electron energy-loss spectroscopy

Following a previous analysis in the context of transmission electron microscopy (23), we use the expression Γfi=0 dω Γfi(ω) withΓfi(ω)=2e2d3rd3r ψf(r)ψi*(r)ψf*(r)ψi(r)×Im{W(r,r,ω)} δ(EfEi+ω)(6)to calculate the rate of transitions between initial and final electron states ψi and ψf of energies Ei and Ef, respectively. More precisely, Γfi(ω) is the loss rate associated with an energy transfer ℏω to the sample. We now apply this formalism to ballistic electrons moving in the 2D material by expressing their wave functions as ψi,f(r)=A1/2eiQi,f·Rψ(z), where A is the normalization area, Qi,f are the 2D electron wave vectors, and ψ(z) is the out-of-plane electron wave function, assumed to be state independent and subject to the normalization condition ∫dz ∣ψ(z)∣2 = 1. Inserting these wave functions into Eq. 6, we findΓfi(ω)=2e2A2d2Rd2R eik·(RR)×Im{W(R,0,R,0,ω)} δ(EfEi+ω)(7)where k = QiQf is the wave vector transfer, and we assume for simplicity that ψ extends a negligible distance away from the 2D material plane z = 0 (i.e., we only need to evaluate W at z = z′ = 0). In addition, Ei,f=2Qi,f2/2m*, where m* is the effective mass of the parabolic semiconductor conduction band. We further assume medium 1 to have real permittivity ϵ1, so that W1 is real and does not contribute to create inelastic transitions.

We now integrate Eq. 7 over final states by adopting the customary prescription ∑f → (A/4π2) ∫ d2Qf. Then, using Eqs. 2, 3, and 5, we findΓi=fΓfi=Γiref+ΓimolwithΓiref(ω)=e2m*πϵ12 θ(Eiω)dφk Im{rp},Γimol(ω)=2e2m*ϵ122A θ(Eiω)dφk2 e2kz0×Im{(rp1)2 (kikzˆ)·1α1G·(k+ikzˆ)}where k=(Qi2+Qf22QiQfcos φ)1/2, Qf=(Qi22m*ω/)1/2, and φ is the scattering angle.

The rate Γiref originates in intrinsic inelastic processes of the 2D material and surrounding homogeneous media, whereas Γimol results from the interaction with the molecule. In this study, we assume lossless dielectrics (i.e., real ϵ1,2 and imaginary σ), leading to Γiref=0. This is consistent with the condition Ei < Eg for ballistic propagation. A small inelastic scattering associated with phonons or disorder could be easily incorporated through a phenomenological intrinsic mean free path.

Electron scattering cross section. We find it convenient to extract an inelastic molecule cross section σimol=Γimol/(vi/A), where we divide by the electron flux vi/A given in terms of the initial electron velocity vi = ℏQi/m*. In the present study, we assume an isotropic polarizability (see below) and a lossless surface (real rp), so the scattering angle– and transition frequency–resolved differential cross section reduces to (for ℏω < Ei)σimol(ω,φ)=2e2m*ϵ122vi e2kz0 (rp1)2×Im{1α1G/2+1α1G}with G defined by Eq. 4. Plots of σimol(ω,φ) are provided in fig. S3. The angle-integrated spectrally resolved cross section is given by σimol(ω)=1dφ σimol(ω,φ), which is plotted in Fig. 1C and has units of length (i.e., the effective length scattered by the molecule from the incident 2D electron plane wave) per unit energy.

Tunneling current density

We consider the device of Fig. 1A and choose the x and y in-plane directions to be oriented perpendicular and parallel to the potential barriers, respectively. The left-to-right inelastic current density (per unit of barrier length) that is transmitted through the device can be calculated as J = −(2e/A)∑QivixT(Qi) (59), where the sum extends over electrons incident from region I with wave vector Qi, transversal velocity vix, and energy within the 0 ≤ EiEF range. The factor of 2 in this expression accounts for spin degeneracy, while A is a quantization area. Neglecting multiple scattering by molecules or potential barriers, the transmission coefficient can be written as the product T(Qi)=TE(Qix) [L nmoldωdφ σimol(ω,φ)] TA(Qfx), where nmol is the areal molecule density in region II, whereas the transmission coefficients of the emitter and analyzer barriers TE and TA, which we calculate using standard 1D scattering matrix theory, depend exclusively on the transversal wave vector (i.e., Qix for TE and Qfx for TA, before and after inelastic scattering by the molecule, respectively). In addition, we use the effective path length L′, which is increased with respect to the width L of region II according to L′ = L/ cos φi (i.e., taking into account the oblique in-plane incidence angle of the electron φi relative to the gate normal). The wave vector after inelastic scattering is Qf=(Qi2Qω2)1/2, where Qω=2m*ω/, while the transversal component depends on scattering angle φ as Qfx = Qf cos (φi + φ). Putting these elements together and making the substitution ∑Qi → (A/4π2) ∫ d2Qi, we find J = ∫ dω J(ω), whereJ(ω)=QωkFdQi J(ω,Qi)(8)withJ(ω,Qi)=eLnmol2π2m* Qi2 π/2π/2dφi TE(Qix)×π/2φiπ/2φidφ σimol(ω,φ) TA(Qfx),is the spectrally resolved inelastic current density, which is plotted in Fig. 3A, while J is shown in Fig. 3B. Here, Qix = Qi cos φi and kF = (2m*EF)1/2/ℏ is the Fermi wave vector in region I. Incidentally, we simplify the evaluation of the integral in Eq. 8 by realizing that the emitter gate produces a narrow transmission peak, which allows us to approximate TE(Qix) by an energy δ−function (peaked at the quantum-well energy Eicos2φi = Ew) times (π/2)ΔEw, where ΔEw ≈ 0.08 meV at Ei = 12 meV (see Fig. 1). Incidentally, the dependence on incident electron energy Ei=2Qi2/2m* is readily given by (m*/ℏ2Qi)J(ω, Qi), as shown in figs. S1 and S3.

Optical polarizability of 9-cis retinal

We model the orientation-averaged optical polarizability of this molecule as a scalar function formed by the sum of two Lorentzians, α(ω)=j=1,2Vjωj2/[ωj2ω(ω+iγj)], with parameters V1=0.3 Å3, ℏω1 = 5.39 meV, ℏγ1 = 0.94 meV, V2=0.7 Å3, ℏω2 = 6.68 meV, and ℏγ2 = 0.77 meV obtained by fitting measured IR absorption spectra (47). The optical absorption spectrum in Fig. 3B is obtained from (4πω/c) Im {α(ω)} (60).

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/28/eabb4713/DC1

https://creativecommons.org/licenses/by-nc/4.0/

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REFERENCES AND NOTES

Acknowledgments: Funding: This work has been supported in part by the European Research Council (ERC Advanced Grant 789104-eNANO), the Spanish MINECO (MAT2017-88492-R and no. SEV2015-0522), the European Commission (Graphene Flagship 785219), the Catalan CERCA Program, the Fundació Privada Cellex, and the Fundació Mir-Puig. Author contributions: F.J.G.d.A. conceived the research, developed the theoretical formalism, and wrote the manuscript. R.Y. performed numerical simulations. Both authors discussed the results. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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