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We have experimentally and numerically investigated photo-induced voltage (PIV) effect across a Au film with a dielectric grating. We observed strongly enhanced voltage when surface plasmon polariton (SPP) is excited. It was found that electrons in the Au film are driven to the propagation direction of SPP. We have numerically shown that dissipative force called as scattering force well elucidates the experimental result for the first time. It is also clarified that this effect can be attributed to the momentum transfer from SPP to free carriers in the Au film. Thus the effect we observed can be called as surface plasmon drag effect in analogy with the photon drag effect.
© 2012 Optical Society of America
1. Introduction
When a photon propagates in a material, carriers in the material can be dragged along its propagation direction, which results in a voltage. The effect is called as a photon drag effect (PDE). In terms of momentum conservation law, the voltage is attributed to the momentum transfer from light to free carriers in the material. Historically such effect is investigated in semiconductors [1]. Vengurlekar and Ishihara extended the PDE to metallic thin film with the use of field enhancement effect of surface plasmon polariton (SPP) excited with a prism coupler [2]. Recently Noginova et al. investigated the PDE in silver thin film under off resonant condition as well as on resonant condition in the Kretschmann geometry. They also demonstrated that the SPP enhanced PDE can be controlled by an external dc current propagating in metallic film [3]. Hatano et al. investigated the PDE in metallic grating slabs and evaluated the voltage numerically in terms of momentum conservation law [4]. When voltage is observed normal to the direction of incident light and the origin of the voltage can be attributed to the asymmetry of the structure, PDE investigated in Ref. [4] can be regarded as optical rectification (OR) effect extended to nanostructure. OR effect is one of 2nd order nonlinear effects which generates DC polarization in a material without space inversion symmetry. When a material with free carrier is irradiated by light, voltage is generated in the material due to the DC polarization induced by OR effect. Although numerical calculation in Ref. [4] obtained qualitatively good agreement with experiment, detailed features such as magnitude of the voltage and sharp resonance structure were not reproduced. This indicates that in some cases momentum conservation method used in Ref. [4] is insufficient to evaluate photo-induced voltage (PIV). More recently English et al. investigated PIV in Ref. [5] with hydrodynamic theory of electromagnetic field studied by Goff and Shaich [6]. Although they obtained qualitative good agreement with experiment, the role of SPP in PIV is not clearly revealed due to the complexity of the hydrodynamic theory.
From Ref. [4] and Ref. [7] one can find Eq. (1) suitable for the numerical evaluation of the PIV when the area of structure that contributes to the voltage is constant.
where C is a constant depending on material and structure parameters, λ wavelength of light, A(λ) absorption coefficient, θ incident angle, Λ pitch of the sample, Dg(λ) diffraction coefficient of diffraction order g. On the one hand Eq. (1) is simple to discuss the physics of PIV, it is not obvious that near field effect such as SPP enhancement is correctly included in Eq. (1) because it is composed of optical spectra which are evaluated far away from a sample. On the other hand the hydrodynamic equation used in Ref. [5] is not preferable to discuss the physics of PIV because of its complexity even though it can handle the near field effect correctly. So in order to obtain deep understanding of the generation mechanism of PIV, a simpler model is profitable.In this paper, we apply a model called as a dipole approximation model to the numerical calculation of PIV in a periodic structure for the first time, which can correctly deal with near field effect and is simpler than the hydrodynamic equations. In the dipole approximation model, materials are regarded as the mass of a dipole and electromagnetic DC force acting on the dipole is generally given as [8, 9]
where α(ω) = αR(ω) + iαI(ω) is a complex polarizability of the material, Ẽ is a complex amplitude of electric field:E(t) = Re{Ẽe−iωt} and j = x,y,z. Equation (2) can handle the microscopic phenomena because it is composed of local electromagnetic field and is less complex than the hydrodynamic equation used in Ref. [5]. The first term in Eq. (2) is called as gradient force, while the 2nd term is called as scattering force. The same or quite similar theoretical treatment of plasmonic drag effect are presented in Ref. [10] and Ref. [11]. In Ref. [10] Hatano et al. reported that circularly polarized light induces voltage perpendicular to the incident plane and the voltage flips its polarity when the sense of the polarization is reversed. They analytically discussed the origin of the voltage from the view point of second order DC force and its symmetrical property. In Ref. [11] Durach et al. investigated PDE in plasmonic nanowires theoretically and numerically and predicted that PDE is gigantically enhanced due to the high confinement of SPP in nanowires. In their paper detailed forms of the Lorentz force acting on dipoles are presented, which are numerically estimated for actual nanostructures. In the terminology used in Ref. [11] gradient force corresponds to striction force. In order to achieve simpler configuration for the clarification of the physical picture of PIV we can eliminate the contribution from the gradient force to PIV, combining the periodic boundary condition with plane film geometry. Gradient force acting on a structure can be reduced to the surface integral as follows: where we have used one form of Gauss’ theorem [12] to convert the volume integral to the surface one. Equation (3) corresponds to the macroscopic force acting on dipole moments induced in the structure. If the structure in periodic systems does not have surfaces normal to x axis (x direction is the direction of periodicity), the x component of Eq. (3) is written as where dx is the period of the structure in the x direction. Now it is clear that the right-hand side of Eq. (4) vanishes due to the periodic boundary condition: Note that the gradient force gives finite contribution if the structure is not plane film shaped because the surface integral doesn’t always satisfy the specific condition that gives naught. For example, the gradient force doesn’t vanish in case of metallic grating slabs due to the grooves. Here we propose dielectric grating on metallic plane film structure that can satisfy zero integral condition for the gradient force. Since only the metallic film region contribute to the PIV, the condition can be satisfied in this structure.The purpose of this study is to demonstrate the PIV in the new structure presented here and to clarify the role of SPP in PIV. In the previous studies [2, 3] it is clarified that SPP plays the role of enhancement effect in PDE. However it is not clear whether SPP enhances the momentum transfer efficiency from incident light to free carriers due to increasing light absorption or SPP itself transfers its momentum to free carriers, because the direction of horizontal component of incident wavevector and the SPP propagation direction are in the same direction in the Kretschmann geometry. Furthermore in order to understand the microscopic generation mechanism of PIV under the SPP excitation we focus on a method for the numerical calculation of PIV from a microscopic point of view that is simpler than the hydrodynamic theory.
2. Structure and experimental setup
A 50 nm-thick Au film was prepared by electron beam evaporation on a cleaned quartz substrate. In order to achieve better adhesion of the Au film with the substrate, 3 nm-thick Cr was evaporated before the evaporation of Au. Then electron beam resist (ZEP520A, ZEON Corporation) was spin-coated on the Au film. And the film was baked on a hotplate. Using electron beam (EB) lithography, we fabricated a periodic structure composed of 200 nm-thick resist with the periodicity 700 nm on 50 nm-thick Au film as seen in the schematic in Fig. 1(a). In this case the EB resist pattern serves as a dielectric grating. The patterned area is 1 mm × 1 mm.
The experimental setup is shown schematically in Fig. 1(b). Idler light from an optical parametric oscillator (OPO) pumped by a tripled Nd:YAG laser (Surelite Continuum) was sent to the structure with p-polarization, for which surface plasmon polariton is excited. The laser pulse width and repetition rate were 5 ns and 10 Hz, respectively. We measured photo-induced voltage across the structure with an oscilloscope (Tektronix TDS3012B) through a high-speed amplifier with a gain of 125. In order to reduce the pulse to pulse fluctuation in the signal, we averaged it for 64 pulses. The maximum value of the laser intensity was 2.6 MW/cm2. We normalized the experimental data of PIV by the laser power intensity in order to eliminate the wavelength dependence of the laser power intensity. This normalization can be justified because the voltage is proportional to the intensity of incident light.
Reflection spectra were measured automatically on a θ–2θ rotational stage controlled by a computer. The incident angle was varied from −30 ° to 30 ° at the interval of 2 ° (reflection spectrum at normal incidence was not measured). From the wavelengths 700 nm to 900 nm spectrometer (ACTON SpectraPro 2300i) with a liquid-nitrogen-cooled CCD camera (Roper Scientific) was used, while from the wavelengths 900 nm to 1400 nm the one with a liquid-nitrogen-cooled InGaAs diode array (Roper Scientific) was used in the reflection measurement.
3. Results and discussion
First, fundamental optical property of the structure is studied. Angle resolved reflection spectra of experimental and numerical calculation results are shown as a function of wavelength in Fig. 2(a) and (b), respectively. Numerical calculation was carried out by scattering matrix method[13], which is improved to achieve faster convergence as is shown in Ref. [14] and Ref. [15]. Experimental value of the permittivity for Au was taken from Ref. [16]. There are some dips that shift in association with the change of incident angle, which corresponds to the SPP excitation. To identify this feature as the SPP excitation, we calculated magnetic field distribution in case of normal incidence at the wavelengths 885 nm and 1046 nm in Fig. 3(a) and (b), respectively. In Fig. 3(a), the magnetic field concentrates at the interface between the dielectric grating and the Au thin film, while in Fig. 3(b) it concentrates at the interface between the Au thin film and the quartz substrate.These are the characteristic feature of SPP excitation. In the experimental result of reflection spectra the dips associated with the SPP excitation at Au/substrate interface is not seen, while in the calculation result that is clearly seen (for example, at the incident angle ±4° and the wavelength 1000 nm). This difference can be attributed to the Cr layer for the better adhesion. Since Cr has relatively big imaginary part in its permittivity in the near infrared regime compared to that of noble metals such as Au and Ag, the Cr layer can reduce the SPP resonance. Furthermore the 3 nm-thick Cr layer is island shaped, which makes the boundary between Au film and the substrate rough. The roughness is not included in our calculation and it can reduce the SPP excitation at the Au/substrate interface in the experiment.
Fig. 2 (a): Experimental result of reflection spectra. (b): Numerical calculation of reflection spectra.
Fig. 3 (a): Distribution of magnetic field intensity at 885 nm. (b): Distribution of magnetic field intensity at 1046 nm. In both cases calculation was done at normal incidence. The white lines depict the shapes of the dielectric grating and the Au film.
Next, we show angle resolved PIV spectra in Fig. 4(a). The characteristic feature of PIV spectra shifts in association with the SPP excitation as in the reflection spectra. Thus SPP excitation dependence of PIV is clearly shown in Fig. 4(a) and it is considered that the generation mechanism of PIV here is attributed to the SPP resonance. Moreover PIV always exhibits positive polarity when the group velocity of SPP is negative and vice versa. This feature indicates that electrons in the Au film are driven to the propagation direction of SPP.
Fig. 4 (a): Experimental result of PIV measurement. (b): Calculation result of PIV evaluated by the microscopic model.
Here we try to reproduce the experimental result with the dipole approximation model which is shown in the introduction. Before discussing the calculation result, we briefly show the calculation flow and conversion procedure from the microscopic force to voltage. First we calculated a local complex electric field amplitude (Ẽ(r)) at the frequency ω by the scattering matrix method. Next using the dipole approximation model we constructed the DC electromagnetic force from the local electric field, which corresponds to the conversion from the frequency ω to DC (ω – ω = 0). In the conversion procedure from the microscopic to the macroscopic force, x component of space averaged scattering force F̄Scat,x was calculated:
where v is the region of Au film. Integrating Eq. (6) along x direction and dividing the result by the elementary charge e = −1.6 × 10−19C, we obtain the result: where the line integral along x direction was performed from 0 to the length of the structure (about 1 mm). Note that we have used the polarizability of Au divided by its electron density (5.9 × 1022/cm3) to convert the polarizability per volume to the one per one electron for the calculation of the scattering force.Figure 4(b) shows the microscopic calculation result of PIV spectra. Numerical calculation readily reproduced the experimental result. So near field effect, the SPP field enhancement, is correctly included in the calculation and scattering force is the origin of the PIV here. As in the case with the reflection spectra, PIV due to the SPP excitation at Au/substrate interface is not observed for the experiment.
In order to clarify the physical meaning of the scattering force more precisely, we decompose it to the form described in Eq. (8) [17, 18]:
where ρ̃ is a complex amplitude of polarization charge density given by ρ̃ = (1/(4π))∇ · Ẽ. In the right hand side of the Eq. (8) the first term is responsible for conventional radiation pressure due to the transfer of wavevector because it is proportional to the time averaged Poynting vector, the second term is associated with the spin and angular momentum of light because it vanishes when the light is linearly polarized and the third term is responsible for the electric force acting on a polarization charge in a material. In the terminology used in Ref. [11] the first term corresponds to Abraham force and the third term corresponds to pressure force. Since we only deal with the case of p-polarization and 1D periodic structure in this paper, the second term vanishes. Intuitively the first term is considered to be dominant in the scattering force because it is explicitly proportional to the wavenumber (ω/c) and the Poynting vector of electromagnetic field that can drive electrons to its direction. To affirm this intuition, we fix the incident angle from here after to discuss the electromagnetic near field.Figure 5 and Fig. 6 show the microscopic calculation result of PIV spectrum with experimental result at 6 ° and pseudo color plot of absorption spectra which shows dispersion diagram of this structure as a function of wavelength and incident angle, respectively. In Fig. 5 the polarity of PIV can be positive and negative and there are two main feature in the spectrum: one, at the longer wavelength, corresponds to the SPP excitation at the interface between the dielectric grating and the Au film due to the fold back of reciprocal lattice vector −G and the sign of PIV is positive, the other, at the shorter wavelength, corresponds to that of +G and the sign of PIV is negative. In Fig. 6, there are two groups of features: one is the dispersion curve crossing at 1070 nm, which corresponds to the SPP mode excited at the Au/substrate interface (Metal/Substrate (MS) Mode), the other is the one which has plasmonic band gap around 850 nm, which corresponds to the SPP mode excited at the grating/Au interface (Grating/Metal (GM) Mode). The upper band of the latter SPP mode is forbidden at normal incidence due to the symmetry and the shape of the structure [19, 20]. This feature was confirmed in the experimental result of transmission spectrum at normal incidence (not shown). Figure 6 shows that the sign of group velocity of SPP is negative and positive at the longer and shorter wavelength, respectively (see at 6°), which coincides with the sign of the direction where electrons in Au are driven. At the longer wavelength the direction of the horizontal component of incident wavevector and the SPP propagation direction is in opposite direction and the electrons in the Au film is clearly driven to the opposite direction of the incident wavevector. Therefore we can conclude that SPP doesn’t enhance the momentum transfer efficiency from incident light to free carriers due to its high absorption.
Fig. 5 Calculation result of PIV evaluated by the microscopic model with experimental result at 6°. The order of the voltage in experiment is tenfold bigger than that in calculation.
Fig. 6 Pseudo color plot of absorption spectra. GM mode and MS mode are the SPP modes excited at the grating/metal and metal/substrate interface, respectively.
In Fig. 5 the intensity peak at the shorter wavelength is larger than that at the longer wavelength in the experiment. In comparison, the calculation shows a roughly similar magnitude for the same two wavelengths. This difference can be attributed to the number of the Drude oscillators at the two wavelengths. Au has the interband transition from d band to sp band around the wavelengths 500 nm. The influence of the interband transition cannot be neglected below the wavelength 1000 nm because reflection from a Au film decreases roughly below the wavelength. The number of the Drude oscillators at the shorter wavelength is less than that at the longer wavelength. Therefore the magnitude of the voltage at the shorter wavelength is higher than that at the longer wavelength because PIV is inversely proportional to the number of free carriers in a material. In our numerical calculation we assumed the number of Drude oscillators is constant, in other words we neglected the contribution from the interband transition, which results in the difference between the numerical calculation and experiment. Although we cannot completely include the influence of the interband transition in the calculation, the main contribution to the voltage is surely included. As for the difference of the magnitude of voltage between the numerical calculation and the experiment, the order of the voltage in experiment is tenfold bigger than that in calculation. This difference can be attributed to the deviation of the optical constant and the thickness of the Au film used in the calculation from that of experiment. Moreover the number of the Drude oscillators can be different between the numerical calculation and the experiment. Despite the difference of the magnitude of voltage, this calculation result is the first numerical demonstration which shows that the magnitude of PIV can be reproduced within the tenfold difference.
In Fig. 7, distribution of the x component of the scattering force at the longer wavelength (925 nm) is shown. The magnitude of the scattering force is indicated by pseudo color, where the logarithmic scale is used and the orientation is indicated by an arrow whose length is normalized to unity. As is seen in Fig. 7(a), the scattering force is directed to the negative direction in the Au film everywhere. In Fig. 7(b) and (c) the second term in Eq. (8) and the third term in Eq. (8) at the longer wavelength (925 nm) are shown, respectively. Although electrons in the Au film are driven to the negative direction in experiment (PIV exhibits positive sign), Poynting vector flow in the Au film is positive.
Fig. 7 (a): Distribution of x component of scattering force. (b): Distribution of x component of Poynting vector force.(c): Distribution of x component of force on polarization. Calculation was done at the wavelength 925 nm and the incident angle 6 °. The maximum absolute value of each force is also shown.
Clearly the first term in the Eq. (8) cannot explain the experimental result, i.e. the third term is dominant in our experiment. This result can be understood from the view point of the group velocity (∂ω/∂kx) of SPP. Generally total energy flow of SPP is given as
where S denotes Poynting vector and v1 and v2 denotes the grating region and the Au film region, respectively. The sign of Poynting vector flow in v1 is opposite to that in v2, while its intensity in v1 is much stronger than that in v2. The reversal of Poynting vector flow at the interface between metals and dielectrics is a general property of an electromagnetic field derived from the boundary condition [21]. So total Poynting vector flow is negative, which is consistent with the sign of group velocity of SPP. In other words Poynting vector flow must be positive in the Au film region so as to be consistent with the sign of group velocity of SPP. Thus the first term in the Eq. (8) cannot explain the experimental result from not only the microscopic point of view but also the viewpoint of dispersion relation. At the shorter wavelength (802 nm), where the group velocity of SPP is positive, the same discussion can be applied to the situation. So we only show the calculation result of scattering force in Fig. 8(a), the second term in Eq. (8) in Fig. 8(b) and the third term in Eq. (8) in Fig. 8(c) at the shorter wavelength (802 nm), respectively.Fig. 8 (a): Distribution of x component of scattering force. (b): Distribution of x component of Poynting vector force.(c): Distribution of x component of force on polarization. Calculation was done at the wavelength 802 nm and the incident angle 6 °. The maximum absolute value of each force is also shown.
The third term contains the operator Im{} that means the imaginary part of a quantity. This means that the electric force acting on a polarization charge which is out of phase, in other words the loss of the oscillating polarization charge, transfers the momentum of light to free carriers in a material, while the force which is in phase is responsible for the optical radiation such as reflection and transmission. Polarization charge density is strongly enhanced when SPP is excited. So SPP excitation makes the force dominant near the metal surface and drags electrons along the surface via polarization charge enhanced by SPP. In other words SPP transfers its momentum to electrons in the Au film when propagating with dissipation. Thus this phenomena should be called as surface plasmon drag effect in analogy with the photon drag effect.
In the end of this section we would like to refer to the momentum conservation model used in Ref. [4]. At 6 ° the first term in Eq. (1) is negligible because of the small incident angle ((1/2)sin2θ≈ 0). So the second term should explain the experimental result in terms of diffraction spectra. Figure 9(a) shows the absorption and diffraction spectra at 6 °.
Fig. 9 (a): Calculation result of absorption and diffraction spectra. (b): Schematic of the situation where the incident wavelength is 925 nm.
When the first term in Eq. (1) is negligible, Eq. (1) is reduced to the form:
where we used the approximation cos6° ≈ 1. First we focus on the wavelengths around 925 nm, where two diffraction channels are open. One is the diffraction in the substrate with diffraction order +1 (DS+1), the other is the one with the order −1 (DS-1). From Fig. 9(a) DS-1 is dominant compared to DS+1 because DS-1 is strongly coupled with SPP due to the fold back of the reciprocal lattice vector (−1)G. Therefore Eq. (10) can be reduce to the from: As is shown in Ref. [4] the constant C gives negative sign when the free carriers that contribute to voltage have negative charge, Eq. (11) gives negative sign around 925 nm. From the viewpoint of momentum conservation law electrons in the Au film are pushed to the positive direction in order to compensate the momentum carried away by the DS-1 which propagates in the negative direction. This situation is schematically described in Fig. 9(b). The same discussion can be applied to the situation where voltage estimated by the momentum conservation approach exhibits positive polarity around the wavelength 802 nm. These are the physical picture of PIV in terms of momentum conservation law between light and free carriers in the metallic thin film. However although the same feature, namely the SPP dependence, can be understood (diffraction is enhanced when SPP is excited), the polarity of PIV is reversed compared to that of experimental result. Thus momentum conservation approach to the evaluation of PIV fails to reproduce the experimental result in the structure.As is obvious from Eq. (1), approach used in Ref. [4] utilizes optical spectra such as transmission, reflection and diffraction, which is evaluated far away from the structure. However SPP strongly enhances electromagnetic near field. So the previous approach cannot include the near field enhancement effect properly. Moreover when using the momentum conservation approach we cannot distinguish the amount of momentum transfered from light to free carriers in the metal and to the bound electrons in the dielectric, while only the metallic thin film contribute to PIV. These are the possible reason why the previous approach cannot explain the experimental result in this paper. The primary difference between the momentum conservation model and the microscopic model presented here is that in the former model electromagnetic far field is used to calculate momentum transfer from light to the material system, while in the latter model microscopic electromagnetic field is used to calculate the force acting directly on an electron. When the internal momentum exchange process between SPP and free electrons can be negligible as is the case in Ref. [4], momentum conservation approach will elucidate the property of PIV well.
4. Conclusion
We have experimentally and numerically investigated PIV in a dielectrically modulated metallic thin film. While momentum conservation approach used in Ref. [4] gives wrong sign for the experimental result of PIV at the SPP excitation, microscopic calculation used in this paper correctly reproduced it for the first time. It is also clarified that gradient force acting on the whole structure vanishes due to the periodic boundary condition combined with the plane film geometry and scattering force is responsible for the generation of PIV in this structure. Moreover contrary to the intuition, it is numerically shown that not Poynting vector but the electric force acting on a polarization charge which is out of phase transfers its momentum to the electrons and drags them to the direction of group velocity of SPP along the metal surface. In the structure presented here the direction of the horizontal component of incident wavevector and the SPP propagation direction can be different. The experiment clearly showed that the polarity of PIV is determined by not the direction of incident wavevector but the propagation direction of SPP. So it can be concluded that SPP doesn’t enhance the momentum transfer efficiency from incident light to free carriers but SPP itself drags free carriers to the direction of its propagation direction, which results in the enhanced voltage. Thus this effect can be called as surface plasmon drag effect in analogy with the photon drag effect.
Acknowledgments
The authors would like to deeply thank Dr. Seigo Ohno and Dr. Kazuyuki Nakayama for the fruitful discussions and the help for publishing this manuscript. The authors would like to express our appreciation to Makoto Saito for his technical support. This study was partially performed at Junichi Nishizawa Memorial Research Center in Tohoku University. H. K. is supported by the Japan Society for the Promotion of Science (JSPS). This work has been partially supported by Grant-in-Aid for Scientific Research (B) 21340075 from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
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