Leaky-mode resonances in nanopatterned thin-film structures are of interest owing to their great potential for creating integration-compatible, multifunctional devices harnessing desired spectral, polarization, intensity, and phase properties1,2,3. Guided-mode-resonance elements1, high-contrast gratings2, and plasmonic metasurfaces3 have been extensively studied within this context. Adding active tunability to these device classes for applications in practice, various approaches have been suggested using thermo-optic effects4, micro-electro-mechanical system architectures5, and liquid-crystal-based index-tuning methods6. Further expanding the application areas and innovating classical device counterparts, a major line of research is presently pursuing higher tuning speed, smaller device footprint area, and better long-term stability in order to secure essential requirements in potential application areas including ultra-broadband optical signal processing, high-power laser machining, and compact LIDAR systems.

To this end, free-carrier-induced electro-optic (EO) effects in heavily doped semiconductors and transparent conducting oxides at epsilon-near-zero (ENZ) conditions have been extensively studied as an efficient tuning mechanism. In particular, ENZ nanofilms incorporated in metal-oxide-semiconductor (MOS) capacitor arrays have showed remarkable intensity and phase modulation properties driven by field-effect free-carrier accumulation and depletion7,8,9,10. In this approach, highly dissipative, deep-subwavelength plasmonic resonances are necessary to induce significant optical interaction with sub-10-nm-thick EO-active layers. Consequently, strong absorption and low resonance Q factor result in performance restrictions such as shallow signal modulation depth, low absolute efficiency, and poor high-power durability. In addition, it is presently unclear whether or not the plasmonic MOS capacitor approach using highly reflective metallic components as an indispensable constituent material can operate in the transmission mode which is desirable for variety of applications.

Pursuing high-performance tunable leaky-mode resonance devices operating in the transmission mode in this paper, we propose an approach based on high-Q guided-mode resonances (GMRs) in low-loss semiconductor nanogratings. The proposed device class consists of moderately doped, low-loss semiconductor p-n junctions in a resonant subwavelength grating structure as shown in Fig. 1(a–d) where basic operation scheme, device structure, and electrical connections, and a possible architecture for integrated modulator arrays are illustrated, respectively. This structure is designed such that a high-Q GMR is supported in the optical domain while in the electrical domain bias voltage across the multiple p-n junctions effectively control density of mobile electrons and holes, resulting in the associated tuning of the Drude-type optical dielectric constant11. The major advantage of this configuration over the plasmonic tunable metasurfaces is to sustain a low-loss, high-Q resonance feature that experiences a remarkable resonance center shift under a favorable bias-voltage region. Hence, desired properties such as high efficiency and high on/off extinction ratio can be supported in the transmission mode operation. While the proposed operation principle is applicable for various group IV and III-V-compound semiconductor materials, here we select an interleaved Si p-n junction nanograting architecture as one promising example. Using well-established electrical and optical modeling methods, we theoretically demonstrate a robust transmission modulator with on/off power ratio of 18.9 dB, on-state efficiency of 85.2%, and modulation bandwidth of 54.3 GHz at an operation wavelength of 1550 nm. These performance characteristics are driven by favorably small bias voltage values in a range of −4 V ~ +1 V and possibly maintained in the high-speed operation regime up to 50 GHz when appropriate in-plane miniaturization schemes are incorporated.

Figure 1: Schematic of a Si p-n junction subwavelength-grating intensity modulator.
figure 1

(a) Device operation concept and parameters. (b) Multiple p-n junction configuration embedded in the grating structure. Va denotes applied bias voltage for inducing the free-carrier-induced EO effect. (c) Bias-voltage injection scheme for simultaneously driving constituent Si lines in an isolated device region. (d) Illustration of the proposed EO modulator arrays integrated on a single substrate surface.


Free-carrier-induced EO effect in multiple p-n junction layers

The EO effect of our interest in this paper is based on change in mobile electron and hole densities that we denote by Ne and Nh, respectively, in response to an applied bias voltage Va across the p-n junctions. Dielectric constant εSi of Si in the optical domain is determined by a Drude formula

where ωe,h and Γe,h denote plasma and collision frequencies, respectively, for electron (e) and hole (h) plasmas, and ε = 11.7. Following the canonical Drude model, ωe,h2(Ne,h) = Ne,he2(ε0m*e,h)−1 with the elementary electric charge e, vacuum permittivity ε0, and effective mass m*e,h. In moderately doped Si with donor and acceptor doping concentration of N0 = 1018 cm−3 (=Ne = Nh), for example, values for these parameters are m*e = 0.27me, m*h = 0.39me, ωe = 0.449 eV/ħ, Γe = 4.95 × 10−2 eV/ħ, ωh = 0.374 eV/ħ, and Γh = 5.36 × 10−2 eV/ħ12. The two frequency-dependent parts in the Drude-part dielectric constant εDrude describe optical response of the mobile electron and hole plasmas, respectively. Under applied bias voltage Va across the p-n junctions in the configuration in Fig. 1(b,c), mobile electrons and holes are redistributed to modify Ne and Nh, and hence the ωe and ωh values until the internal electric field energy becomes minimal. Therefore, the desired bias-voltage-dependent dielectric constant is obtained in this interaction process. Previously, a similar effect provides a robust tuning mechanism for high-speed, low power consuming in-line Si-photonic modulators with device footprint length scales in the order of 1 mm to 100 μm13,14. In our case, the effect is used to create an electrically tunable low-loss GMR element taking advantage of the resonant light confinement in a subwavelength-thick zero-order grating layer.

Rigorously treating this free-carrier-induced EO effect, the carrier-density Ne and Nh distributions in thermal equilibrium follow the Poisson-Boltzmann distribution15. We use a 2-dimensional finite-element-method model16 of the Poisson-Boltzmann equation to calculate bias-voltage-dependent Ne(y, z) and Nh(y, z). We assume d1 = 55 nm, d2 = 2d1 = 110 nm, and identical donor (Nn) and acceptor (Np) concentrations such that Np = Nn = N0 = 1018cm−3 in this calculation. In this multiple-junction structure, we include five pairs of alternating 110-nm-thick p-n junction cells in the order of np-pn-np-pn-np from bottom. The thickness of 110 nm for a single p-n junction unit cell is chosen such that the depletion layer at a bias voltage Va = −4 V well below the breakdown voltage of −5.54 V for the given doping concentration completely covers the entire device volume and consequently the mobile-carrier-induced EO effect occurs over the entire device. The peculiar feature of d2 = 2d1 is a natural consequence of this layer-design rule to maximize the EO effect with a favorably minimal material embodiment in the proposed device concept.

Results for three bias-voltage values of Va = −4 V (reverse bias), 0 V (neutral), and +1 V (forward bias) are shown in Fig. 2(a–c), respectively. Therein, we indicate distribution of an effective compound-carrier density N* = (m*em*h)−1/2(m*hNe + m*eNh) that we define as an intuitive, single density parameter directly relevant to the free-carrier EO effect. This definition is followed by a simpler expression for εDrude as

Figure 2: Free-carrier induced electro-optic effect in the proposed multiple p-n junction structure.
figure 2

Effective compound-carrier density N*(y, z) distribution at room temperature (300 K) for Va = (a) −4 V, (b) 0 V, and (c) +1 V. Electrode connection for bias-voltage application and p/n layers are indicated in (a). We assume d1 = 55 nm and d2 = 110 nm. Drude-part dielectric constant εDrude(z) profiles for Va = (d) −4 V, (e) 0 V, and (f) +1 V. Blue solid and red dashed curves correspond to Re(εDrude) and Im(εDrude) profiles, respectively, and the p/n layers are indicated on the background of each panel.

where m* = (m*em*h)1/2 and Γ′ = (m*hNeΓe + m*eNhΓh)(m*hNe + m*eNh)−1 in the weak collision regime where Γe,hω and Γe,hωe,h. Obviously, εDrude is linearly proportional to N*. The Va-dependent N* distributions in Fig. 2(a–c) show that N* is effectively modified from 0 to 2 × 1018cm−3 over the whole structure under Va adjustment in a range from −4 V to 1 V. In addition, N* is almost independent of in-plane position y, confirming that the electrical signal-injection configuration envisioned in Fig. 1(c) should be feasible for efficiently controlling N* over the whole isolated device region. Subsequent change in εDrude at a wavelength of 1,550 nm is in a range from 0 to −1.3 × 10−2 for Re(εDrude) and from 0 to 0.86 × 10−4 for Im(εDrude) as shown in Fig. 2(d–f). Therefore, the proposed configuration provides a 10−2-order EO change in Re(εDrude) over a 550-nm-thick Si film under a favorably low bias-voltage tuning range from −4 V to +1 V while keeping acceptably low material absorption levels of Im(εSi) < 10−3.

EO-tunable GMR excitation

We apply the obtained bias-voltage-induced mobile-carrier effect to an example GMR element optimized for the transmission-mode optical modulation in the telecommunications C-band around 1,550 nm. Figure 3(a) shows the Va–dependent transmission spectra in dB (a linear scale in the inset) under transverse-electric (TE) polarized planewave incidence at surface-normal angle (θ = 0). The optimized grating-design parameter values are given in the caption. Following the standard convention, the TE polarization refers to electric field oscillating in the axis of grating lines (y-axis). We use the finite-element method16 in this calculation involving εSi(z) profiles obtained by the method described in the previous section. The transmission spectra show an asymmetric Fano-resonance profile as a result of the configuration interference between resonant and non-resonant pathways17. The resonant pathway is created by coupling of the incident wave with a leaky TE0 mode and its radiation decay toward the transmitted zero-order planewave channel through dominant first-order diffraction processes. This resonance feature possesses remarkably high resonance Q factor ~3.69 × 103. Consequently, the design yields a very high field enhancement factor ~1.2 × 103 in the 550-nm-thick EO-active Si-p-n-junction layers as confirmed in Fig. 3(b) showing an electric-field intensity distribution at the resonance center wavelength.

Figure 3: EO-tunable high-Q GMR excitation.
figure 3

(a) Transmission spectra of an example device for Va = −4.0 V (blue), 0 V (black), 0.9 V (brown), and 1.0 V (red). The optimized device parameters are Λ = 730 nm, F = 0.647, h = 550 nm. Normal incidence (θ = 0) of TE-polarized planewave is assumed. The p-n junction configuration is identical to that used in Fig. 2. Inset shows the transmission spectra in linear scale. (b) Electric field intensity |Ey|2 distribution at the resonance center (λ0 = 1550.02 nm) for Va = 0 V. (c) Resonance-center shift Δλc as a function of applied bias voltage Va. (d) Average effective compound carrier density 〈N*〉 (solid curve, right-vertical axis) and depletion layer thickness (dashed curve, left-vertical axis) and in response to the applied bias voltage control. Two inset diagrams illustrate carrier distribution statics in the open (lower left) and closed (higher right) depletion-layer regimes.

Subtle interaction between the highly enhanced resonant optical fields and bias-voltage-induced mobile-carrier effect results in a resonance-center (λc) shift Δλc as shown in Fig. 3(a). Basically, the observed resonance shift is directly resulting from the Va-dependent change in εSi. Although there is no exact closed-form expression for the dielectric-constant-dependent resonance shift known in general, a plausible estimation can be found by taking the Wentzel-Kramers-Brillouin (WKB) approximation on the resonance condition. In our case, the change in εSi leads to the change in the optical path length inside the Si grating bars while there is no optical-path-length difference outside. For a small dielectric-constant change, i.e., ΔεSi 1, that further implies no significant modification in the field distribution of the leaky resonance mode, the WKB approximation for the eigenvalue determination18 dictates that the optical phase accumulation inside the Si bars should remain constant under small change in εSi. This condition directly yields a relation (nSi + ΔnSi)−1c + Δλc) = nSi−1λc, where nSi = εSi1/2 and consequently ΔnSi = (2nSi)−1Re(ΔεSi) = (2nSi)−1 Re(ΔεDrude) as the dielectric constant change is solely in the Drude part. Including Eq. (2) and the standard phase-matching condition Λ−1 = neffλc−1 for a GMR at normal incidence, where neff is effective index of the leaky guided mode, the constant optical-phase-accumulation condition is rewritten by

where g(ω) = e2[ε0m*(ω2 + Γ′2)]−1. Obviously, increase in N*N* > 0) with Va leads to a corresponding linear blue shift of the resonance feature or vice versa.

From Fig. 3(a), we find that the resonance-center shift in response to the applied bias voltage Va has two different regimes. In Fig. 3(c), we show Δλc(Va) that reveals slow blue shift with increasing Va in the low-voltage region of Va < 1 V and abrupt increase in the differential resonance shift ΔλcVa in the high-voltage region of Va > 1 V. According to Eq. (3), this peculiar property is associated with the dependence of N* on Va. Va–dependent volume average 〈N*(Va)〉 of the effective compound-carrier density is plotted in Fig. 3(d) and it is in exact correlation with −Δλc(Va) in Fig. 3(c). Explaining the dependence of 〈N*〉 on Va in Fig. 3(d), a key factor is built-in potential Vbuilt-in across the p-n junction. Applying the Poisson-Boltzmann equation for Va = 0 in our case with Np = Nn = N0 = 1018 cm−3, we obtain Vbuilt-in = 0.934 V. In the low bias-voltage region of Va < Vbuilt-in, increase in 〈N*〉 with Va is led by the decrease in the depletion layer thickness without significant growth in the carrier-density level. In contrast, in the high bias-voltage region of Va > Vbuilt-in, the depletion region is closed and the excessive electrons and holes injected from the electrodes lead to mobile-carrier density level growth in the whole Si regions to result in a more rapid increase in 〈N*〉 with Va. In Fig. 3(d), we provide the Va-dependent depletion layer thickness and illustrations showing the two regimes of carrier distribution statics.

Importantly, the obtained resonance-shift tunability in Fig. 3(c) implies a full λc tuning range over 2.3 nm that is remarkably larger than the resonance bandwidth of 0.43 nm by a factor 5.3. Therefore, we can fully utilize the spectral maximum and minimum as the on-state and off-state transmittance levels, respectively. For our particular design at an optimal wavelength of λ0 = 1550.02 nm which corresponds to the transmittance minimum at Va ≈ Vbuilt-in, sharp transmission modulation is obtained as shown in Fig. 4(a). In this case, the intensity modulation is induced mainly in the closed depletion-layer regime and thereby small bias-voltage tuning induces rapid intensity change as confirmed again in the total electric field patterns for Va = 0.9 V and −4.0 V in Fig. 4(b,c). Key performance parameters in this case are transmittance modulation depth of 83.9%, on-state efficiency of 85.2%, and on-off extinction ratio of 18.9 dB under remarkably low control bias-voltage signals within the −4 ~ +0.9 V range. Importantly, the obtained performance parameters are highly desirable for variety of applications when compared with LiNbO3-crystal-based EO modulators requiring 100 V-scale control signals for similar performances.

Figure 4: Transmission modulation properties.
figure 4

(a) Bias-voltage-dependent transmittance at operation wavelength of λ0 = 1550.02 nm (blue solid curve) and 1550.29 nm (red dashed curve) for an optimized device used in Fig. 3. Inset shows magnified plot for λ0 = 1550.02 nm. Total electric field Ey/E0 patterns for the (b) off-state and (c) on-state operation regimes at Va = 0.9 V and −4 V, respectively. We assume λ0 = 1550.02 nm for (b,c).

For a different operation wavelength which corresponds to the transmittance minimum at Va < Vbuilt-in, we have much slower intensity tuning as the device operates in the open depletion-layer regime. For example, we select λ0 = 1550.29 nm and the corresponding Va-dependent transmittance is indicated by red dashed curve in Fig. 4(a). Such slow intensity modulation is desirable for continuous modulation or precise control of light intensity for analog signal processing systems while the rapid, closed depletion-layer regime is more appropriate for digital signal processing applications.

The proposed device is basically a 1D-periodic GMR element and thereby has a characteristic angular dispersion. In our case, a primary angular dispersion of the resonance location appears for the angle of incidence with respect to x axis on which the discrete light diffraction processes take place. As the resonance location follows the dispersion curve of the leaky guided mode, the angular shift of the resonance wavelength as a function of polar angle θ of incidence can be found from the definition of group velocity, i.e., Vx = ∂ω(k)/∂kx, where Vx, ω(k), and kx denote group velocity in x axis, dispersion frequency surface, and x-component wavevector of the leaky guided mode, respectively. A simple calculus with basic relations of ω(k) = 2πcλc−1, kx = 2πcλc−1sinθ, and the diffractive phase matching condition λc−1sinθ = neffλc−1−Λ−1 results in

where nG = c/Vx is group index of the leaky guided mode. For near-normal incidence (θ  1), Eq. (4) reduces to ∂λc/∂θ ≈ − nG−1neffΛ. This property explicitly appears in the angle-dependent transmission spectrum as shown in Fig. 5(a) where the angle dependent λc loci are identified as being along the dark transmission dip. Therein, ∂λc/∂θ = 0 at θ = 0 as the group index nG diverges to infinity for the laterally standing guided-mode with Vx = 0 at normal incidence. For off-normal incidence (θ ≠ 0) for which the two counter-propagating guided modes are not coincidental anymore and the resonance is driven by a single leaky-guided mode. Consequently, the group-to-effective-index ratio nG−1neff tends to a constant value and so is ∂λc/∂θ. Estimating from Fig. 5(a), the off-normal angle-tunability ∂λc/∂θ ≈ −2 nm/deg. at θ = 5°. Importantly, the angle-dependent shift of λc does not significantly affect the resonance profile and the EO-tunable resonance shift as shown in Fig. 5(b). In particular, the EO tunability of 0.17 nm/V persists for all three cases and the transmission modulation depth values are 85%, 83%, and 77% for θ = 1°, 3°, and 5°, respectively.

Figure 5: Angle dependence of the EO-tunable GMR.
figure 5

(a) Angle-dependent transmission spectrum for a fully depleted case at Va = −4 V. (b) Bias-voltage-dependent spectral profiles of the transmittance at angles θ = 1° (red), 3° (green), and 5° (blue). Solid and dashed curves indicate the spectral profiles for Va = −4 V and +1 V, respectively.

The angular dispersion of λc and persistent EO tuning properties suggest important information for operation of the proposed device class in practice. First, highly collimated light beam should be used to fully utilize the proposed device functionality. Angular full-width-at-half-maximum bandwidth for the GMR in our case is estimated from Fig. 5(a) as 2.6° at θ = 0 and 0.18° at θ = 5°. Divergence angle of the incident light beam should be well within these values. In another consideration, the angle tunability of λc provides an efficient way to precisely shoot the desired operation wavelength. In practice, fabrication errors and imperfections present. Although exact tolerance values depend on fabrication steps and specific tools selected for device production, one may accept a-few-nm scale errors in the spatial device parameters and grating period and a-few-Å scale errors in the layer thickness values. A combination of these errors in period, grating linewidth, and layer thicknesses might result in an imperfect λc off from the desired value by an amount even in a 10-nm scale. Considering 0.1 nm (distributed feedback type) ~10 nm (Fabry-Pérot type) for typical diode laser line width around 1,550 nm, the anticipated fabrication errors can be critically problematic if there is no tuning method available for matching λc to the source-laser wavelength. Suppose that we control θ within a ±10° range and with an accuracy of 10−3 deg. The angle tunability of 2 nm/deg. implies a full tuning range of ~40 nm with an accuracy of ~2 pm. We note that typical angle precision of commercially available rotary or tilting stages for optomechanical controls is in 10−4 deg. scales.

Further considering applicability of the proposed device concept in practice, modulation bandwidth is an important measure. The GMR bandwidth and RC time constant are two major factors in this consideration. For the analyzed example device in Figs 3 and 4, the estimated GMR bandwidth is Δfopt ~ 54.3 GHz. Therefore, stable 10-GHz-scale optical signals can be generated in a purely optical property aspect. However, the final modulation bandwidth should be also restricted by electrical response characteristics, i.e., a bias-voltage signal modulation bandwidth Δfbias = τRC−1, where τRC denotes the RC time constant determined by the junction capacitance and termination impedance. Assuming the standard radio-frequency termination impedance of 50 Ω and the design configuration used in Figs 2 and 4, estimated Δfbias values are 14.7 MHz for a device footprint area of 1 × 1 mm2. Since Δfbias is inversely proportional to device footprint area, GHz-scale modulation bandwidth should be feasible for small devices with the footprint area <260 × 260 μm2. We note that Inoue et al.19 recently demonstrated a high-Q GMR filter with a device footprint area reduced down to 10 × 10 μm2 without significant degradation in the spectral performance characteristics by using graded-parametric design approach combined with integrated first-order Bragg reflection boundaries. Introducing such miniaturization strategy to the proposed concept, Δfbias > Δfopt and the full resonance bandwidth should be available for the final optical signal-modulation bandwidth.


In summary, we proposed a multiple-p-n-junction subwavelength grating structure that enables high-performance EO modulation in the transmission mode. The proposed device operates under high-Q GMRs interacting with electric signals through the Drude-type optical free-carrier effect. Using rigorous electrical and optical modeling methods, we theoretically demonstrated highly efficient transmission modulation generated by remarkably low-voltage control signals with modulation speed in possibly 10 GHz scales. Notably, the obtained properties are supported by the low-loss free-carrier-induced EO effect occurring in the whole device region with 500-nm-thick Si layers as opposed to the transparent-conducting-oxide-based plasmonic metasurface approaches involving a sub-10-nm-thick EO-active region and strong ohmic absorption. In another similar approach, a low-loss GMR modulator was suggested using a combination of the Burstein-Moss effect, Pockels effect, and Fraz-Keldysh effect in a weakly-modulated InGaAsP waveguide grating structure20,21. Therein, a robust reflection modulation with an extinction ratio of 17 dB and modulation bandwidth of 5 MHz was experimentally demonstrated.

Experimental realization of the proposed device concept is definitely the next step. In potential fabrication, crucial parts are to establish 100-nm-thick multiple p-n junction cells and subwavelength grating structure with a critical dimension in a few 100 nm scale. First, the multiple p-n junction cells can be generated by the standard deposition processes based on chemical vapor deposition and sputtering techniques. We note that the present state-of-the-art deposition methods easily produce such multiple-junction semiconductor layer structures as established well in tandem solar cells and in vertical-cavity surface-emitting lasers22,23,24. Second, the subwavelength periodic structure for a high-Q resonance excitation is also well-established using standard nanolithography techniques including the laser-interference lithography4 and electron-beam lithography24.

Considering further study, applicability and limitations of the proposed concept over other spectral domains are of key importance. Although direct application of the concept to the visible domain is unclear because of highly lossy nature of the group IV or III-V-compound semiconductors, longer-wavelength applications in the mid-infrared (mid-IR) and THz domains are intriguing in several aspects. First, we notice from Eq. (2) that the EO modulation of εDrude scales with λ2. This implies that the modulation amplitude ΔεDrude that is in the order of 10−2 around λ = 1.5 μm under ΔVa = 5 V is amplified up to the unity order around λ = 15 μm in the mid-IR domain and even further up to the order of 102 around λ = 150 μm in the THz domain. Therefore, the proposed device class can take advantage of the far stronger EO effect in the mid-IR and THz domains. In addition, fabrication errors and imperfections with respect to the operation wavelength are substantially lower and consequently precise device fabrication is much more feasible in the longer wavelength domains. Therefore, further in-depth study on the available materials, parametric optimization, and experimental realization in the telecommunications IR and longer wavelength domains is of great interest to develop compact, low driving power, and high-speed modulators for applications in telecommunications, optical information processing, LIDARs, laser machining, and many others.

Additional Information

How to cite this article: Lee, K. Y. et al. Multiple p-n junction subwavelength gratings for transmission-mode electro-optic modulators. Sci. Rep. 7, 46508; doi: 10.1038/srep46508 (2017).

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