**Double Spectral Electromagnetically Induced Transparency Based on Double-Bar Dielectric Grating and Its Sensor Application**

**Guofeng Li 1,2, Junbo Yang 2, Zhaojian Zhang 2, Kui Wen 2, Yuyu Tao 1, Yunxin Han 2,\* and Zhenrong Zhang 1,\***


Received: 26 March 2020; Accepted: 24 April 2020; Published: 27 April 2020

**Abstract:** The realization of the electromagnetically induced transparency (EIT) effect based on guided-mode resonance (GMR) has attracted a lot of attention. However, achieving the multispectral EIT effect in this way has not been studied. Here, we numerically realize a double EIT-ike effect with extremely high Q factors based on a GMR system with the double-bar dielectric grating structure, and the Q factors can reach 35,104 and 24,423, respectively. Moreover, the resonance wavelengths of the two EIT peaks can be flexibly controlled by changing the corresponding structural parameters. The figure of merit (FOM) of the dual-mode refractive index sensor based on this system can reach 571.88 and 587.42, respectively. Our work provides a novel method to achieve double EIT-like effects, which can be applied to the dual mode sensor, dual channel slow light and so on.

**Keywords:** guided-mode resonance; electromagnetically induced transparency; high quality factor; double spectral; refractive index sensor

#### **1. Introduction**

In the past few decades, the electromagnetically induced transparency (EIT) effect caused by quantum destructive interference between two different excitation paths in a three-level atomic system has attracted extensive and in-depth research [1,2]. The EIT effect is that under the effect of strong resonance coupling light, the opaque medium becomes transparent for weak probe light and is accompanied by extraordinarily steep dispersion. In other words, the resonance probe light can propagate through the medium without being absorbed [3]. EIT has potential applications in many fields, such as sensing [4–6], slow light [5,7], nonlinear optics [8] and cavity quantum electrodynamics [9]. However, the realization of the EIT effect in atomic systems requires extremely strict experimental conditions, which limits its practical application. Therefore, researchers have turned to classical optical systems to achieve EIT-like effects. G. Shevts and J.S Wertele described the EIT analog in a classic plasma [10], and more of an EIT-like effect was subsequently observed in the plasmonic metamaterial [11,12]. The EIT-like effect has also been successfully achieved in other optical systems, such as waveguide cavity structures [13,14] and photonic crystals [15]. Sun-Goo Lee's group realized the EIT-like effect based on the guided-mode resonance (GMR) effect for the first time in 2015 [16], the system consists of two planar dielectric waveguides and a subwavelength grating and a narrow transparent window appears in the transmission dip when high Q and low Q resonant waveguide modes are coupled.

In recent years, the realization of the EIT-like effect based on GMR has attracted more and more interest of researchers. Sun-Goo Lee's group successfully realizes the polarization-independent EIT-like effect in two photonic systems, which are both composed of two planar dielectric waveguides and a two-dimensional photonic crystal [17]. Sun Y and Chen H et al. reported a planar metamaterial based on GMR achieve the EIT-like effect with a Q value exceeding 7000 [18]. Han Y and Yang et al. reported that a GMR system with two subwavelength silicon grating waveguide layers achieves the EIT-like effect with an ultra-high Q factor of 288,892 [19]. However, as is known to us, the realization of multispectral EIT effects by the subwavelength grating structure based on GMR has not been studied according to available works.

In this work, we numerically simulate the double spectral EIT-like effect in a coupled GMR system that consists of two silicon grating waveguide layers (GWLs) on a SiO2 substrate. In addition, both GWLs contain double-bar dielectric gratings with unequal lengths in a cycle. When the distance between the two GWLs meets the phase matching condition, the GMR system works as a Fabry–Perot (F–P) cavity and GWLs as the reflection boundary of the cavity thus two EIT peaks appear in the near infrared band due to the top GMR mode and the bottom GMR mode are intercoupled. The effects of two grating widths and the separation between the two gratings are also studied. After the structural parameters optimization, two EIT peaks present ultra-narrow FWHMs of 10−<sup>2</sup> nm and ultra-high Q factors of 104. In terms of the sensor, this system has ordinary sensitivity and high figure of merit (FOM) with compact structure and a simple manufacturing process. This double spectral EIT-like effect may not only be applied to the sensor, but also to slow light and nonlinear optics.

#### **2. Structure and Simulation**

The 3D finite-difference time-domain (FDTD) method was used to study the phenomenon of this optical system. During the simulation with FDTD, the mesh accuracy was set to λ/34 to ensure the convergence of the simulated results. The schematic of the GMR system with a double-bar dielectric grating structure is shown in Figure 1a. In terms of boundary conditions, X and Y were set to periodic, and Z was the perfectly matched layer (PML). The mesh accuracy in GWL1 and GWL2 were Δx = Δy = 5 nm and Δz = 10 nm, respectively. The inset of Figure 1b shows the real part of the refractive index value for SiO2 [20] and Si [21] as a function of the wavelength. The background index of the system was n*<sup>s</sup>* = 1.

**Figure 1.** (**a**) Schematic of the guided-mode resonance (GMR) system with double-bar dielectric grating structure and geometrical parameters: d1 = 1885 nm, D = 55 nm, t = 435 nm, *d*<sup>2</sup> = 2137 nm, *W*<sup>1</sup> = 100 nm, *W*<sup>2</sup> = 230 nm, *g* = 30 nm and P = 500 nm. (**b**) Transmittance spectra of the GMR system with only grating waveguide layer (GWL)1 (GWL2) and both GWLs. The inset shows the real part of the refractive index for SiO2 and Si as a function of the wavelength. (**c**) Magnified view of the transmission features of electromagnetically induced transparency (EIT)1. (**d**) Magnified view of the transmission features of EIT2.

Light polarized in the X direction (TM polarization) incidents vertically above the system in the near-infrared spectral band. The period P of grating was 500 nm. A SiO2 layer with a thickness of 1885 nm was designed on the top of the GMR structure in order to reduce the reflection of light. The distance (*d*2) between the two GWLs was designed to be 2137 nm. Gratings G11 and G12 had the same depths of D = 55 nm, and different widths of *W*<sup>1</sup> = 100 nm and *W*<sup>2</sup> = 230 nm, respectively. The separation between two Si gratings was *g* = 30 nm. The Si waveguide thickness was t = 435 nm. Meanwhile, structural parameters of GWL1 and GWL2 were consistent.

#### **3. Results and Discussion**

We first calculated the transmission spectrum of the structure with only single GWL1 (GWL2), as the green solid line (blue solid line) shown in Figure 1b. It is clear that the two different transmission dips appeared at the wavelengths of 1547.97 nm and 1553.08 nm. GWL1 and GWL2 had the same structure, but their optical properties were easily affected by the surrounding medium, thus the transmission spectra were slightly different. The SiO2 cover layer had an influence on the resonance

frequency. Adding a SiO2 cover layer can reduce the impact of the surrounding medium on its optical characteristics to reach the same resonance frequency [18].

When GWL1 and GWL2 are both present, and their distance (*d*2) met the phase matching conditions, the top GMR mode (in GWL1) and the bottom GMR mode (in GWL2) were intercoupled, Therefore, two different sharp EIT resonances appear in the two resonance dips respectively due to destructive interference [12,19,22]. The transmission spectrum is shown by the solid red line in Figure 1b. We refer to the two EIT peaks as EIT1 and EIT2. The transmission characteristics of their magnified views are shown in Figure 1c,d, respectively. The resonance wavelengths of EIT1 (λ1) and EIT2 (λ2) were 1548.12 nm and 1553.36 nm, respectively, and their corresponding full width at half maximum (FWHM) was 0.0441 nm (Δλ<sup>1</sup> in Figure 1c) and 0.0636 nm (Δλ<sup>2</sup> in Figure 1d). The Q factor was an important parameter of the resonant cavity, which is defined as follows:

$$\mathbf{Q} = \frac{\lambda}{\text{FWHM}}\tag{1}$$

There, their corresponding Q factors reached 35,104 and 24,423, respectively. Obviously, our work has two higher Q factors than another double EIT-like effects work, which has two Q factors of 950 and 216 [6].

The analogy between our system and the atomic EIT system can help us understand the physics of the double EIT-like effect [23,24]. A double-Λ five-level model of the double EIT-like effect in our system is illustrated in Figure 2a. In the system, |1 represents the ground state, the field of the bottom GMR mode corresponds to the probability amplitudes of atoms in the metastable states |2 and |3-, the field of the top GMR mode corresponds to the probability amplitudes of atoms in the excited states |4- and |5- [19,23,25]. The control field refers to the coupling between the top GMR mode and the bottom GMR mode, and the probe field refers to the input of the top GMR mode. We observed that two sharp EIT-like windows appeared in the probe area due to the introduction of the control field. The energy level of |5- — |1 was higher than |4- — |1-, hence EIT1 corresponded to the destructive interference between two different transition pathways |1- → |5 and |1- → |5- → |3- → |5-, EIT2 corresponded to the destructive interference between two different transition pathways |1- → |4 and |1- → |4- → |2- → |4-.

**Figure 2.** (**a**) Double-Λ five-level model of the double EIT-like effect in this GMR system. (**b**) The electric field distribution diagrams of the GMR system correspond to the wavelengths indicated by the blue dash lines in Figure 1c,d, and I at the off-resonant wavelength of EIT1 1547.98nm, II at the EIT1—resonant wavelength of 1548.12 nm, III at the off-resonant wavelength of EIT1 1548.24 nm, IV at the off-resonant wavelength of EIT2 1553.12 nm, V at the EIT2—resonant wavelength of 1553.36 nm and VI at the off-resonant wavelength of EIT2 1553.54 nm.

In order to help us understand the origin of the double EIT-like effect in this GMR system, electric field distribution diagrams of the GMR system near two EIT peaks (such as the transmission spectra of Figure 1c,d are given in Figure 2b. Near the EIT1, electric field distribution at the off—resonant wavelength of EIT1 1547.98 nm (the blue dash I in Figure 1c) and 1548.24 nm (the blue dash III in Figure 1c) was very easily observed and excitation mainly occurred in GWL1 (I in Figure 2b) or GWL2 (III in Figure 2b), which was consistent with the characteristics of GMR. The corresponding transmittance was very low since the GMR mode was easily coupled to electromagnetic waves in free space. Near the EIT2, the electric field distribution at the off—resonant wavelength of EIT2 1553.12 nm (the blue dash IV in Figure 1d) and 1553.54 nm (the blue dash VI in Figure 1d) corresponded to IV and VI in Figure 2b. The principles were the same as near EIT1.

At the EIT1—resonant wavelength of λ<sup>1</sup> = 1548.12 nm (the blue dash II in Figure 1b) and the EIT2—resonant wavelength of λ<sup>2</sup> = 1553.36 nm (the blue dash V in Figure 1d), the electric field distribution diagrams were II and V Figure 2b, respectively. Obviously, very strong oscillations occurred simultaneously in GWL1 and GWL2. When the incident light was at two EIT wavelengths, the light was reflected back and forth between GWL1 and GWL2, and electromagnetic energy was coupled into the two GWLs, so strong oscillations were excited through coupling. Once the top GMR mode and the bottom GMR mode are intercoupled, and this system works like as an F–P cavity [19,26,27], therefore, two very narrow transmission peaks could be obtained at the wavelengths of 1548.12 nm and 1553.36 nm in the transmission spectrum, where transmissions reached 96.23% and 94.48%, respectively.

The transmission spectra of the coupled GMR system with different *d*<sup>2</sup> are shown in Figure 3, and other parameters are the same as Figure 1. In the range of *d*<sup>2</sup> from 1970 to 2220 nm, with the increase of *d*2, the two resonance wavelengths (λ<sup>1</sup> and λ2) were both red-shifted. When *d*<sup>2</sup> was increased from 2020 (in Figure 3b) to 2120 nm (in Figure 3d), λ<sup>1</sup> was increased from 1547.67 to 1548.06 nm and λ<sup>2</sup> was increased from 1552.51 to 1553.22 nm. This verified that the GMR system worked like an F–P cavity and GWLs as the reflective boundary of the cavity when *d*<sup>2</sup> met the phase matching condition. Obviously, for the bottom GMR mode in GWL2, a displacement of about one hundred nanometers could achieve a favorable coupling with the top GMR mode in GWL1, two EIT peaks appeared in the two transmission dips when the F–P cavity was introduced. Generally, a phase matching condition corresponds to a resonance wavelength when the distance and refractive index are determined [19]. Here, it is worth mentioning that when the determined *d*<sup>2</sup> satisfies the phase matching condition, two EIT peaks with different resonant wavelengths can be generated simultaneously because the refractive index of Si is different at different wavelengths.

**Figure 3.** Transmittance spectra of the coupled GMR systems with different *d*2. (**a**) Transmission spectrum with *d*<sup>2</sup> = 1970 nm. (**b**)Transmission spectrum with *d*<sup>2</sup> = 2020 nm. (**c**) Transmission spectrum

with *d*<sup>2</sup> = 2070 nm. (**d**) Transmission spectrum with *d*<sup>2</sup> = 2120 nm. (**e**) Transmission spectrum with *d*<sup>2</sup> = 2170 nm. (**f**) Transmission spectrum with *d*<sup>2</sup> = 2220 nm.

The transmission spectra and resonance wavelengths under different *W*<sup>1</sup> and *W*<sup>2</sup> are shown in Figure 4, and other parameters are the same as Figure 1. Distinctly, with the increase of *W*<sup>1</sup> from 85 to 145 nm or the increases of *W*<sup>2</sup> from 180 to 240 nm, both λ<sup>1</sup> and λ<sup>2</sup> appeared red shifted. For a clearer observation, we show two resonance wavelengths under different *W*<sup>1</sup> and *W*<sup>2</sup> in Figure 4c,d. When *W*<sup>1</sup> increased from 85 to 145 nm, λ<sup>1</sup> moved from 1547.33 to 1549.93 nm and λ<sup>2</sup> moved from 1552.16 to 1558.13 nm. When *W*<sup>2</sup> increased from 180 to 240 nm, λ<sup>1</sup> moved from 1545.44 to 1549.2 nm and λ<sup>2</sup> moved from 1549.41 to 1554.54 nm. Thus we could easily control the two resonance wavelengths by adjusting the parameters of *W*<sup>1</sup> or *W*2. The reason for the red shift of the two resonance wavelengths is that as *W*<sup>1</sup> or *W*<sup>2</sup> increase, the fill factor increases, which changes the average refractive index of the grating layer [28].

**Figure 4.** (**a**) The transmission spectra under different *W*<sup>1</sup> and *W*<sup>1</sup> is equal to 85 nm, 100 nm, 115 nm, 130 nm and 145 nm, respectively. (**b**) The transmission spectra under different *W*<sup>2</sup> and *W*<sup>2</sup> is equal to 180 nm, 195 nm, 210 nm, 225 nm and 240 nm, respectively. (**c**) Resonance wavelengths of two EIT peaks at different *W*1. (**d**) Resonance wavelengths of two EIT peaks at different *W*2.

The gap between the gratings G11 and G12 (gratings G21 and G22) is also an important parameter, being defined as "*g*". The transmission spectrums under different *g* are shown in Figure 5 and other parameters are the same as Figure 1. Obviously, as *g* varied from 10 to 50 nm in steps of 10 nm, the resonance wavelength of EIT1 (λ1) appeared blue shifted, while the resonance wavelength of EIT2 (λ2) appeared red shifted. Two resonance wavelengths and two resonance wavelengths difference (λ2-λ1) when *g* changed from 10 to 50 nm are shown in Figure 5f. As *g* increased, λ<sup>1</sup> moved from 1549.93 to 1547.19 nm and λ<sup>2</sup> moved from 1552.55 to 1553.93 nm (as shown on the left vertical axis in Figure 5f), and "λ2-λ<sup>1</sup> " increased from 2.62 to 6.74 nm (as shown in the right vertical axis in Figure 5f). The resonance wavelength of the two EIT peaks could also be flexibly adjusted by adjusting the parameter *g*.

**Figure 5.** The transmission spectra under different *g*, other parameters are the same as Figure 1. (**a**) *g* = 10 nm. (**b**) Transmission spectrum with *g* = 20 nm. (**c**) Transmission spectrum with *g* = 30 nm. (**d**) Transmission spectrum with *g* = 40 nm. (**e**) Transmission spectrum with *g* = 50 nm. (**f**) Two resonance wavelengths and two resonance wavelengths difference (λ2-λ1) under different *g*.

#### *Sensing Performance*

Considering the resonance wavelength difference between the two EIT peaks and the transmittance of the two EIT peaks (not shown here), we chose a structure with *W*<sup>1</sup> = 100 nm, *W*<sup>2</sup> = 230 nm and *g* = 30 nm to study the corresponding sensing performance. At the same time, the SiO2 between GWL1 and GWL2 changed to the dielectric sample. In order to make the refractive index change of the dielectric more sensitive, *d*<sup>2</sup> was set to 3740 nm. The performance of the sensor was evaluated by two factors, sensitivity (S) and FOM [29]:

$$\mathbf{S} = \frac{\Delta\lambda}{\Delta\mathbf{n}}\tag{2}$$

$$\text{FOM} = \frac{\text{S}}{\text{FWHM}} \tag{3}$$

Here, S refers to the resonance wavelength shift caused by the change in the refractive index unit of the dielectric, and FOM represents the optical resolution of the sensor.

Figure 6 shows the transmission spectrum with a different refractive index of the dielectric, the insert shows the resonance wavelengths of two EIT peaks at a different refractive index of the dielectric. It is obvious that the resonance wavelengths of the two modes had a linear relationship with the refractive index of the dielectric. The changes in EIT1 and EIT2 with the refractive index of the dielectric are referred to as mode 1 and mode 2, respectively. Therefore, the average FWHM of the mode 1 and mode 2 from the refractive index of the dielectric from 1.440 to 1.452 was 0.051 nm and 0.061 nm. The S of mode 1 and mode 2 were 29.166 nm/RIU and 35.833 nm/RIU, respectively, so FOM of mode 1 was 571.88, and FOM of mode 2 was 587.42. Compared with the other sensors, this sensor had ordinary sensitivity due to the narrow line width of two EIT peaks [30]. However, it had higher FOM than other previous sensors [5,6,31,32], as shown in Table 1. So, this sensor had a super high optical resolution.

**Figure 6.** Transmission spectrum of the refractive index of the dielectric between the GWL1 and the GWL2 changed from 1.440 to 1.452. The inset shows the resonance wavelengths of the EIT1 peak (mode 1) and the EIT2 peak (mode 2) when the refractive index of the dielectric changes from 1.440 to 1.452.


**Table 1.** Figure of merit (FOM) compared with previous sensors.

#### **4. Conclusions**

In summary, we proposed a GMR system with a double-bar dielectric grating structure to achieve a double spectral EIT-like effect. The F–P cavity was introduced due to the distance (*d*2) between the two GWLs satisfying the phase matching condition, and two EIT peaks with ultra-narrow FWHM and ultra-high Q factors could be obtained. The influences of the width and separation of the double-bar gratings on two EIT peaks were also investigated. Meanwhile, the performance of the ultra-high FOM sensors based on this system was also studied. This work provided a way to achieve double spectral EIT-like effect, which has potential applications in the dual mode sensor, dual channel slow light and so on.

**Author Contributions:** G.L. completed structural design, simulation calculation and writing of articles. Y.H. provided the research ideas and paper revision of this article. J.Y. and Z.Z. (Zhenrong Zhang) determined the research direction, revised papers and provided fund support. Z.Z. (Zhaojian Zhang), K.W. and Y.T. contributed to

the software setting and analysis of simulation results. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** This work was supported by the National Natural Science Foundation of China under Grant 61661004, Guangxi Science Foundation (2017GXNSFAA198227), National Natural Science Foundation of China (60907003, 61805278), the China Postdoctoral Science Foundation (2018M633704), the Foundation of NUDT (JC13-02-13, ZK17-03-01), the Hunan Provincial Natural Science Foundation of China (13JJ3001), and the Program for New Century Excellent Talents in University (NCET-12-0142).

**Conflicts of Interest:** The authors declare that they have no competing interests.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Design and Modelling of a Novel Integrated Photonic Device for Nano-Scale Magnetic Memory Reading**

**Figen Ece Demirer 1,\*,†, Chris van den Bomen 1,†, Reinoud Lavrijsen 1, Jos J. G. M. van der Tol <sup>2</sup> and Bert Koopmans <sup>1</sup>**


Received: 30 October 2020; Accepted: 15 November 2020; Published: 21 November 2020

#### **Featured Application: On-chip optical reading of magnetic memory processed as ultrathin magnetic claddings on photonic waveguides.**

**Abstract:** Design and simulations of an integrated photonic device that can optically detect the magnetization direction of its ultra-thin (∼12 nm) metal cladding, thus 'reading' the stored magnetic memory, are presented. The device is an unbalanced Mach Zehnder Interferometer (MZI) based on InP Membrane on Silicon (IMOS) platform. The MZI consists of a ferromagnetic thin-film cladding and a delay line in one branch, and a polarization converter in the other. It quantitatively measures the non-reciprocal phase shift caused by the Magneto-Optic Kerr Effect in the guided mode which depends on the memory bit's magnetization direction. The current design is an analytical tool for research exploration of all-optical magnetic memory reading. It has been shown that the device is able to read a nanoscale memory bit (400 × 50 × 12 nm) by using a Kerr rotation as small as 0.2◦, in the presence of a noise ∼10 dB in terms of signal-to-noise ratio. The device is shown to tolerate performance reductions that can arise during the fabrication.

**Keywords:** integrated photonics; InP; magneto-optic; MZI; mode conversion; PMA; multi-layered thin film; magnetic memory; MOKE; Fourier transformation

#### **1. Introduction**

In the modern world, exponentially increasing generation of data and its handling require novel technologies that perform faster and more energy efficiently. To answer this need, optical components are being used in combination with electronic circuitry to improve the speed and bandwidth of data communication and telecommunication. For example, optical interconnections that were once a conceptual design suggestion [1] are currently being used in commercial products replacing slow and heat-dissipating electrical signal communication channels [2,3]. Researchers continue to demonstrate the superior performance circuitry achieved through the integration of photonics into electronics [4–7]. Yet, these advances require back-and-forth signal conversion between optical and electrical domains, which happens to be the new bottleneck in data communication and processing. Addressing this problem requires establishing novel functionalities in photonic devices that will enable a seamless conversion. Furthermore, (integrated) photonics is lacking a simple and fast non-volatile memory function. A huge potential is anticipated for future devices that enable direct inter-conversion of data between the photonic and magnetic (memory) domain without any intermediate electronics steps, cutting down on time and energy costs. This study works with existing non-volatile magnetic memory

material technology used in electronics: multilayered ferromagnetic thin-film layers. When the multilayered magnetic material is used as memory material, writing bits into the magnetic memory could be facilitated by recent advances in so-called all-optical switching of magnetization [8,9]. Reading out magnetic bits back into the photonic domain could be achieved via a nonreciprocal magneto-optical process [10–12], while dynamic, on-the-fly reading of magnetic bits could be facilitated by racetrack memory concept [9,13]. In a racetrack memory, magnetic domains (memory bits) move while the material that carries the magnetic domains remain stationary [14,15]. Previously, domain wall velocities up to 1000 ms−<sup>1</sup> were demonstrated [16,17]. It is in this spirit that our paper focuses on the functionality of on-chip optical reading of magnetic memory processed as ultrathin magnetic claddings on photonic waveguides. To our best knowledge, this is the first study which explores the possibility of on-chip, all-optical magnetic memory reading functionality.

State-of-the-art non-volatile magnetic memory such as spin-transfer torque magnetic random-access memory (STT-MRAM) relies on ferromagnetic multilayered ultrathin films with perpendicular magnetic anisotropy (PMA), in which the magnetization vector is perpendicular to the film plane [18]. Such PMA films turn out to be essential for the advanced schemes used to electrically control the magnetic memory elements but are also known for their relatively large magneto-optical efficiency. A simple layer stack that hosts all relevant physical mechanisms is Ta(4)/Pt(2)/Co(1)/Pt(2)/Co(1)/Pt(2) where the numbers in parenthesis are thickness in nm. Bringing this memory component to the proximity of light confined in a waveguide in a photonic device setting gives rise to magneto-optic interactions, specifically the Magneto-optic Kerr Effect (MOKE). MOKE causes a change in the polarization state of light (Kerr rotation and ellipticity), which changes sign when the magnetization direction of the memory component is flipped [11,12]. In a photonic waveguide context, this gives rise to partial mode conversion between TE and TM modes, which potentially enables reading of the memory bit. However, the MOKE signal is intrinsically small in amplitude, a typical Kerr rotation is around 0.05◦ for films with an in-plane magnetization in free-space optics [19,20]. In order to increase the efficiency of the mode conversion, we propose the use of PMA magnetic claddings, which have not been seriously addressed yet in a photonic perspective. Such claddings with a perpendicular magnetic orientation are expected to display larger amplitude magneto-optical effects, yet still small quantitatively. This calls for developing novel approaches to amplify the magneto-optical effects while showing the importance of on-chip analytical tools to explore the fundamental mode conversion properties of photonic waveguides with PMA claddings.

To assess the feasibility of using MOKE for on-chip all-optical magnetic memory reading functionality, as well as using it as an analytical tool to quantitatively measure magnetization-induced mode conversion, we investigated specially designed photonic devices whose waveguides are cladded with ultra-thin (12 nm), nano-scale (50 × 400 nm) PMA magnetic memory bits, of the composition mentioned before. By using mathematical models of the designed photonic devices, whose building block performance parameters are chosen according to the InP Membrane on Silicon (IMOS) platform [21], the accuracy of the memory-bit read-out, optical loss and tolerance to noise are tested. It has been shown that the device is able to read a nanoscale memory bit (400 × 50 × 12 nm) by using a Kerr rotation as small as 0.2◦, in the presence of a ∼10 dB noise in terms of signal-to-noise ratio (SNR). This paper is structured in the following way. In Section 2 materials and methods are given. Device designs, magneto-optic simulation, mathematical modelling and data analysis topics are covered. In Section 3 the results obtained via the mathematical model are presented for devices with varying degrees of performance parameters. A data analysis technique using Fourier transformation is presented. Lastly, in Section 4, the conclusions are given.

#### **2. Materials and Methods**

In this section, materials and geometries of the parts that contribute to the overall device are explained. In addition, the device concept, optical simulation and mathematical modelling methods are explained in the subsections.

The material which stores the magnetic information (memory bit) is a multi-layered ferromagnetic metal thin-film structure, whose stack order is given in the previous section. These multi-layers display PMA, where the magnetization vector is perpendicular to the film plane [22]. PMA is highlighted due to its relatively large magneto-optical efficiency [23]. The multi-layers are placed on top of the waveguides as the top cladding. The rest of the photonic device is fabricated on InP membranes since the devices are based on the IMOS platform [21]. The waveguides have a cross-section of 300 × 400 nm (height and width) and the multi-layered top claddings have the dimensions of 400 × 50 × 12 nm (width, length and height).

#### *2.1. Optical Simulation and Device Concept*

Before describing the optical simulation method to quantify the MOKE in waveguides, a brief overview is given on MOKE and its impact on the light confined in waveguides. Following this, the devcie concept is introduced.

MOKE is a type of magneto-optic interaction that takes place when the light reflects from a magnetized material. In polar configuration, the effect causes a change in the light's polarization state which is quantified by Kerr rotation and ellipticity (in angles). Typically, in the literature, MOKE is reported for single reflections. Comparing a single reflection case with our work, more interaction, thus a larger MOKE are expected in waveguides with magnetized top claddings. To our best knowledge, there is no prior work that quantifies the Kerr rotation in a waveguide setting. Therefore, finite-difference time-domain (FDTD) simulations [24] of the waveguides with top-claddings are conducted to estimate the MOKE in the guided modes. In the simulation, multi-layer cladding material is defined by using the magneto-optic constant obtained from the literature [25]. It is seen that the Kerr effect causes conversion between *TE* and *TM* modes in the waveguide, comparable to the polarization rotation in free-space optics. The resulting Kerr rotation (*θ*), ellipticity (*φ*) and optical loss (*Lossclad*.) values obtained for a single memory bit are listed in Table 2. These values are used as inputs for the mathematical model explained in Section 2.2.

The device design is done by considering the key enabler of the magnetic memory reading functionality: a change in the sign of the Kerr rotation upon flipping of the magnetization direction of the memory bit (memory bit "1" and "0"). Assuming the confined light is initially in *TE* mode, the Kerr rotation ±*θ* (*θ* 1) leads to an emergent *TM* mode whose field amplitude is proportional to *θ* for bit 1 and −*θ* for bit 0. Therefore, devices which can probe the phase of the emergent *TM* mode are explored. Mach-Zehnder Interferometers (MZI) are chosen due their ability to convert the phase difference (between the interfering branches) into intensity difference. Balanced and unbalanced MZI are considered as two candidates for the final design. An unbalanced MZI, which has a defined path length difference between the two branches is chosen due to the noise related issues that cannot be addressed in a balanced MZI. This is further elaborated when the presented results are discussed in Section 3. Since at an initial stage, a device is designed for research and exploration purposes, on-chip light source or detector are not considered. To couple an off-chip laser source and an off-chip detector to the device, mode-selective grating couplers are added to the design.

The device design is shown in Figure 1. In this device, *TE* mode-selective grating coupler is used to couple the light in. Later, a multi-mode interferometer (MMI) is used to split the light equally into two branches. On the upper branch, the *TE* mode is converted into *TM* via the polarization converter. The propagation continued (in *TM* mode) and a delay line is crossed. On the lower branch, the memory bit (magnetic cladding section) caused the *TE* mode to partially convert into *TM* mode due to Kerr rotation (*θ*). The light from the two branches are merged via another MMI. After interference took place, the resulting intensity is picked up via a *TM*-selective grating coupler.

**Figure 1.** An unbalanced MZI. *TE* and *TM* mode selective grating couplers are used to couple the light in and out. The polarization converter is taken from [26].

#### *2.2. Mathematical Modelling and Fourier Transformation*

A mathematical model is built in order to simulate the output light intensity vs. light wavelength for the designed devices. The model is given input parameters that are based on IMOS building block performances [21] and FDTD magneto-optical simulations [24] (see Section 2.1). An overview of the model parameters and their brief descriptions are given in Table 1. Additionally, reduced-performance devices with and without noise are simulated with the model to compare the magnetic memory reading capabilities of the devices. These parameters—some standard for all devices and some changing according to the performance levels—are summarized in Tables 2 and 3, respectively.

Using the mathematical model, equations which determine the electric field (E-field) components of *TE* and *TM* modes in branches 1 and 2, are obtained. For simplicity, coefficients addressing the losses of mode propagations, grating couplers and magnetic cladding are combined into the terms *Bn*. For description of other parameters please refer to Table 1.

$$\begin{aligned} E\_{TE,1} &= B\_1 \cos\left(\alpha\right) e^{-i\frac{2\pi}{\lambda} n\_{TE} L\_1} \\ E\_{TM,1} &= B\_2 \sin\left(\alpha\right) e^{-i\frac{2\pi}{\lambda} n\_{TM} \left(L\_1 - \text{xPC}\right)} \\ E\_{TE,2} &= B\_3 \cos\left(\theta\right) e^{-i\frac{2\pi}{\lambda} n\_{TE} L\_2} \\ E\_{TM,2} &= B\_4 \sin\left(\theta\right) e^{-i\left(\frac{2\pi}{\lambda} n\_{TM} \left(L\_2 - \text{x\_{\text{lead}}}\right) + \phi\right)} \end{aligned}$$

(1)




**Table 2.** Showing generic parameters that are valid for all devices.

**Table 3.** Showing parameters that are dependent on the device performance.


It is important to recall that the interference takes place between the modes whose *E*-fields lay in parallel planes and the output light intensity (*I*) from devices can be calculated via *<sup>I</sup>* <sup>=</sup> <sup>|</sup>*E*<sup>|</sup> 2 2 *Z*<sup>0</sup> , where |*E*| is the total E-field amplitude and *Z*<sup>0</sup> is the impedance of the vacuum. The presented equations for *E*-field amplitudes reveal that a wavelength sweep of the input light will result in oscillations in intensity. Recall that the information regarding the magnetization direction of the cladding (memory bit type) can be retrieved from the sign of the Kerr rotation and ellipticity (*θ*, *φ*). As seen from the equations above, when *TE* mode input light is used, information of the memory bit type is visible only in the phase of the *TM* mode output light. For an output light intensity vs. wavelength plot that is obtained upon interference of both *TE* and *TM* modes, two oscillation frequencies, *ν*TE and *ν*TM that correspond to these modes are observed.

$$\begin{split} \nu\_{\rm TE} &= \frac{n\_{\rm TEg}(L\_1 - L\_2)}{\lambda^2}, \\ \nu\_{\rm TM} &= \frac{n\_{\rm TMg}(L\_1 - L\_2 + x\_{\rm clad.} - x\_{\rm PC})}{\lambda^2}. \end{split} \tag{2}$$

*n*TEg and *n*TMg in Equation (2) indicate group indices of the respective modes. A Fourier transformation can be applied to the resulting output light intensity vs. wavelength data to separate the *TM* mode contribution. Thanks to this technique, the amplitude and phase of the *TM* mode component can be found. In order to separate the *TE* and *TM* mode contributions, non-overlapping peaks in the Fourier transform is required. Therefore, at the design stage, it is vital to choose *x*clad. and *x*PC parameters (see Table 1) accordingly.

#### **3. Results and Discussion**

In order to demonstrate the magnetic memory reading capabilities of our devices, the mathematical model described in Section 2.2 was used. As explained in Section 2.1, the chosen

devices were unbalanced interferometers that contain built-in ferromagnetic memory components as their top claddings. The model predicted the output light intensity vs. wavelength plots of the devices with opposing memory bits (bit '1' and '0'). Later these plots were analyzed by the Fourier transformation technique to determine the memory bit type, thus realize 'reading' of the magnetic information. Recall that since the magneto-optic interaction which enables the determination of the memory bit type is only extractable from the phase of the *TM* mode (when *TE* mode is used as input), Fourier technique greatly reduced the noise and enhanced the sensitivity.

In Figure 2, the left column plots present output light intensity vs. wavelength data. Note that plots depict the intensity after a windowing function is applied. The right column plots show the Fourier transformation of the left column in blue color and the phase difference between two memory bit states for each oscillatory components in red color. Figure 2a,c,e represent the standard, reduced-performance and noisy reduced-performance devices, respectively. The standard device shown in Figure 2a demonstrate a clear 180◦ phase shift between the two signals which correspond to the opposite memory states. The Fourier transformation in Figure 2b (right side y-axis) show a single peak which correspond to the *TM* mode (see Equation (2)). The fact that there is only *TM* mode is thanks to the well-performing *TM*-selective out-couplers in the standard devices that have a negligible out-coupling of the *TE* mode. As expected, the phase difference plot in Figure 2b (left side y-axis) indicate 180◦ difference at the region which correspond to *TM* peak. Note that the plots depicting phase difference between two memory states convey meaningful information only at the locations where a correspondent Fourier peak is present. To stress this aspect visually in the graph, the points corresponding to a peak are shown in black, whereas the rest is left grey. In Figure 2c, the 'reduced-performance device' is seen. This device has only 45◦ conversion at the polarization converter and the *TE*-mode couples out from the *TM*-selective out-coupler (see Table 3). Due to coupling out of the *TE* mode that does not carry information on the memory bit's state, it impossible to observe a 180◦ phase shift in the intensity vs. wavelength plot upon a change in the memory bit type. As expected, Figure 2d reveals two Fourier peaks that correspond to *TE* and *TM* modes. As seen from the peak intensities, despite the use of *TM*-selective out-couplers, the *TE* mode dominates. Undeterred by the *TE* mode dominance, the phase difference plot in Figure 2e indicates a phase of 180◦ at the position corresponding to the *TM* peak. The phase shift corresponding to the *TE* mode reads 0◦. Testing the device design further by addition of a noise that described in Section 2.2, Figure 2e,f are obtained. The 'noisy and reduced-performance device' demonstrates that, even though the intensity vs. wavelength plot is dominated by noise and mixed modes, it is still possible to determine the magnetic memory type via the Fourier transform technique.

Referring back to Section 2.1 and clarifying the reason for the choice of an unbalanced MZI design over a balanced one, as seen in Figure 2a, if the device is performing at a fixed wavelength, the change in the light intensity upon changing the memory bit type corresponds to only 0.3% of the total light intensity. This observation indicates that the magnetic memory reading functionality of the device can be obstructed by the noise when operating at a single wavelength. Sweeping of a range of wavelengths accompanied by the Fourier transformation method are the key concepts for eliminating sensitivity to noise and increasing memory reading accuracy. Since the wavelength sweep technique is not successful without the specific frequency oscillations that the added delay line provides, an unbalanced MZI is preferred over a balanced one.

Note that for an ideal device depicted in Figure 2a, the difference in light intensity between the two memory states is proportional to the strength of the Kerr rotation. Therefore, if a calibration by using a material with known Kerr rotation and optical loss is done, very small Kerr rotations can be measured quantitatively by using the same design.

**Figure 2.** (**a**,**c**,**e**) Output light intensity vs. wavelength plots for standard, reduced-performance, and noisy reduced-performance devices (see Tables 2 and 3), The light intensities are shown in arbitrary units and is normalized assuming initial intensity (*I*0) is 1. (**b**,**d**,**f**) In blue, Fourier transformations of the intensity vs. wavelength plots are shown. The normalization is done assuming the highest intensity Fourier peak has amplitude 1. In black, the phase differences between the memory bit "1" and "0" are shown for each wavenumber. The data-points which correspond to a Fourier peak are shown in black while the rest is shown in grey. This is done for guidance to eye for separation of statistically significant result (black) and fitting procedure noise (grey).

#### **4. Conclusions**

An integrated photonic device specially designed to perform memory reading functionality is presented. The functionality is achieved through detection of the magnetization direction of an ultra-thin memory bit. The device is shown to operate despite performance reductions in the contributing building blocks and noise levels which correspond to ∼10 dB in terms of SNR. Post-processing of the intensity signal via Fourier transformation method stressed that the device is suitable as an analytical tool for research purposes. It is highlighted that the quantitative measurement

of very small magneto-optic Kerr rotation (0.2◦) is possible after a calibration which also considers optical loss.

**Author Contributions:** F.E.D. designed and directed the project, did FDTD simulations, contributed to the interpretation of the results and took the lead in writing the manuscript. C.v.d.B. developed the mathematical model, obtained and analyzed the results and prepared a draft manuscript. R.L. aided in interpreting the results, helped with methodology and provided input on the structure of the manuscript. J.J.G.M.v.d.T. and B.K. supervised the project, provided conceptualization of the work and reviewed/edited the work. All authors provided critical feedback and helped shape the research, analysis and manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is part of the Gravitation program 'Research Centre for Integrated Nanophotonics', which is financed by the Netherlands Organisation for Scientific Research (NWO).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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## *Article* **Theoretical Investigation of Responsivity/NEP Trade-off in NIR Graphene/Semiconductor Schottky Photodetectors Operating at Room Temperature**

**Teresa Crisci 1,2, Luigi Moretti <sup>1</sup> and Maurizio Casalino 2,\***


**Abstract:** In this work we theoretically investigate the responsivity/noise equivalent power (NEP) trade-off in graphene/semiconductor Schottky photodetectors (PDs) operating in the near-infrared regime and working at room temperature. Our analysis shows that the responsivity/NEP ratio is strongly dependent on the Schottky barrier height (SBH) of the junction, and we derive a closed analytical formula for maximizing it. In addition, we theoretically discuss how the SBH is related to the reverse voltage applied to the junction in order to show how these devices could be optimized in practice for different semiconductors. We found that graphene/n-silicon (Si) Schottky PDs could be optimized at 1550 nm, showing a responsivity and NEP of 133 mA/W and 500 fW/√Hz, respectively, with a low reverse bias of only 0.66 V. Moreover, we show that graphene/n-germanium (Ge) Schottky PDs optimized in terms of responsivity/NEP ratio could be employed at 2000 nm with a responsivity and NEP of 233 mA/W and 31 pW/√Hz, respectively. We believe that our insights are of great importance in the field of silicon photonics for the realization of Si-based PDs to be employed in power monitoring, lab-on-chip and environment monitoring applications.

**Keywords:** graphene; silicon; photodetectors; internal photoemission effect; near-infrared

#### **1. Introduction**

Silicon (Si) Schottky photodetectors (PDs) have attracted the interest of the scientific community due to the possibility of making Si suitable for detecting infrared (IR) radiation, which is the range of wavelengths included in the spectrum where Si has a negligible optical absorption due to its bandgap of 1.12 eV (1.1 μm). Schottky Si PDs are metal/Si junctions whose detection mechanism is based on the internal photoemission effect (IPE), that is, the photo-excitation of charge carriers in the metal and their emission into Si over the Schottky barrier of the junction [1–3]. In other words, in Si Schottky PDs the metal and not the Si is the active material absorbing the incoming optical radiation. In this context, both palladium silicide (Pd2Si) and platinum silicide (PtSi) Schottky PDs have been extensively investigated for the realization of infrared CCD image sensors. Pd2Si/Si Schottky PDs were developed for satellite applications showing the ability to detect a spectrum ranging from 1 to 2.5 μm when cooled to a temperature of 120 K [4,5]. On the other hand, PtSi/Si Schottky PDs were developed for operation at longer wavelengths ranging from 3 to 5 μm [6,7], although they require a lower temperature of 80 K. A focal plane array (FPA) constituted by an array of 512 × 512 PtSi/Si pixels was realized, demonstrating the first spectacular convergence between Si photonics and electronics [8]. Unfortunately, these devices can only work at cryogenic temperature. Indeed, the low Schottky barrier height (SBH) required to achieve an acceptable efficiency (0.21 eV for PtSi [7] and 0.34 eV for Pd2Si/Si [4]) is comes at the cost of PD noise (dark current), which must be reduced by lowering the working

**Citation:** Crisci, T.; Moretti, L.; Casalino, M. Theoretical Investigation of Responsivity/NEP Trade-off in NIR Graphene/Semiconductor Schottky Photodetectors Operating at Room Temperature. *Appl. Sci.* **2021**, *11*, 3398. https://doi.org/ 10.3390/app11083398

Academic Editor: Maria Ferrara

Received: 4 March 2021 Accepted: 8 April 2021 Published: 10 April 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

temperature. PD noise affects the noise equivalent power (NEP), that is, the minimum detectable optical power, which has a huge impact on both the device sensitivity and the bit error rate (BER) of a communication link. Higher Schottky barriers make it possible to achieve low noise, but they unfortunately also lead to low efficiencies. This efficiency–noise trade-off is a peculiar characteristic of the Schottky PDs based on the IPE.

In 2006, for the first time, it was theoretically proposed to use Schottky PDs for the detection of near-IR (NIR) wavelengths at room temperature [9], taking advantage of the interference phenomena occurring inside a high-finesse Fabry–Pérot microcavity. The main idea was to work with metal/semiconductor junctions characterized by higher SBHs in order to reduce the dark current and then to recover the device efficiency by increasing the metal absorption through the multiple reflections of the optical radiation inside the microcavity. Later, many other strategies were pursued to enhance the efficiency of these devices; indeed, surface plasmon polaritons (SPPs) [10,11], Si nanoparticles (NPs) [12], metallic antennas [13], and gratings [14] were proposed and investigated. In any case, the measured responsivity was lower than 30 mA/W [12] and 5 mA/W [15] for waveguide and free-space Schottky PDs, respectively. More important, the efficiency–noise tradeoff of these Schottky PDs has never been optimized in terms of SBH for achieving high efficiency and low noise at the same time. The low responsivity (i.e., the ratio between the photogenerated current and the incoming optical power) of the Schottky PDs based on metals is mainly due to the small emission probability of the photo-excited carriers from the metal to the Si, related to the momentum mismatch.

Recently, graphene/Si Schottky PDs have shown higher efficiencies with respect to the metallic counterpart and, even if the physical mechanism behind this enhancement is still under debate, it seems related to the increased emission probability due to the two-dimensionality of the material [16–18]. Although graphene is characterized by a low optical absorption (2.3%) many approaches based on resonant-cavity-enhanced (RCE) configurations [19,20], plasmonic structures [21], waveguiding structures [22], and quantum dots [23] have been proposed to overcome this drawback. At present, graphene/Si PDs [18,22,24] show superior performance to the corresponding metallic PDs, representing the most promising solution to realize low-cost Si PDs operating in the NIR regime. In addition, graphene offers a novel attractive possibility: the graphene Fermi level (i.e., the SBH with Si), can be simply modified by applying a bias to the junction, making it feasible to optimize the efficiency–noise trade-off.

In this work we theoretically investigated the responsivity/NEP trade-off in graphene/ semiconductor Schottky PDs operating at NIR wavelengths and at room temperature. First, we used the results of the recent literature to derive a responsivity/NEP analytical equation that can be maximized with an appropriate choice of SBH. Then, we reviewed the SBH dependence on the bias applied to the graphene/semiconductor junctions to show how the responsivity/NEP ratio could be maximized in practice. Finally, we numerically calculated both the responsivity and the NEP of graphene/semiconductor PDs discussing their possible applications and highlighting the validity limits of the proposed optimization process. Even if this work was carried out with the aim of gaining greater insight into graphene/Si PDs, it is worth mentioning that we trace here a general methodology which can also be applied to different semiconductors, such as: germanium (Ge), gallium arsenide (GaAs), and aluminum gallium arsenide (AlGaAs).

#### **2. Theoretical Background**

IPE theory was first developed by Fowler in 1931, and it was focused on the injection of electrons from a metal into vacuum [25]. Several authors have extended Fowler's theory to the emission of carriers into semiconductors, conceiving the modified Fowler theory [26–28] and providing the following expression for the internal quantum efficiency (IQE) *η*int of IPE-based PDs, defined as the number of charge carriers *Ne* produced per absorbed photons *N*ass [26]:

$$\eta\_{\rm int} = \frac{N\_{\rm \varepsilon}}{N\_{\rm ass}} = \frac{1}{8E\_F} \cdot \frac{(h\nu - q\Phi\_B)^2}{h\nu} \tag{1}$$

where *EF* represents the Fermi level, *hν* = *hc*/*λ* is the energy of the incident photon (*λ* is the wavelength and *c* the speed of light in a vacuum), *q* is the electron charge, and Φ*<sup>B</sup>* is the potential barrier at the interface between the metal and the semiconductor. This expression is derived by taking into account the ratio of charge carriers having kinetic energy *normal* to the surface of the junction, necessary to overcome the potential barrier. This mechanism usually leads to poor efficiency (about 1%) [29,30]; however, it has been demonstrated that two-dimensional materials replacing metals in the Schottky junctions provide an IQE enhancement [18]. In particular, in single-layer graphene (SLG)/semiconductor junctions a still higher ratio of photon conversion in charge carriers is observed. Regarding this, Amirmazlaghani et al. [18] explain how this can be ascribed to the molecular structure of the graphene. Indeed, the *π* orbitals are normal to the interface with the semiconductor, and the charge carriers' momentum can be directed only towards the semiconductor or in the opposite direction, leading to an enhancement of the emission probability up to <sup>1</sup> 2 . When SLG is used as active medium in an IPE-based PD, Equation (1) can no longer be applied due to the linearity of the dispersion relation near the Dirac point [31], different density of states, and probability of emission. However, the IQE of Schottky PDs based on SLG has been derived as [18]:

$$
\eta\_{\rm int}^{\rm SLG} = \frac{1}{2} \cdot \frac{(h\nu)^2 - (q\Phi\_B)^2}{(h\nu)^2}. \tag{2}
$$

The responsivity *R* is related to *η*SLG int by the following relation:

$$R = \frac{I\_{ph}}{P\_{inc}} = S \cdot \frac{1}{h\nu} \cdot \eta\_{\text{int}}^{\text{SL}, \text{G}} = \frac{S}{2} \cdot \frac{(h\nu)^2 - (q\Phi\_B)^2}{(h\nu)^3} \tag{3}$$

where *Iph* is the photogenerated current, *Pinc* is the incident optical power, and *S* is the graphene optical absorbance. It is worth mentioning that in Equation (3) the charge carrier *q* is been considered in order to express the responsivity in A/W. Graphene has an optical absorption related to the universal fine-structure constant *α* = *e*2/(*π* 0*hc*¯ ) [32] and independent of the frequency, *AG* = *πα* ≈ 2.3%. Here we focus our attention on devices that provide the complete absorption of the incident radiation such as long waveguides and resonant structures, thus we consider *S* = 1.

As the Schottky barrier Φ*<sup>B</sup>* decreases, more electrons can pass into the semiconductor, giving rise to higher responsivities, as shown in Equation (2). Unfortunately, the dark current *Id* of the junction also increases as Φ*<sup>B</sup>* diminishes due to thermal effects [33]:

$$I\_d = A\_j A^\* T^2 \cdot e^{-\frac{q\Phi\_B}{kT}} \tag{4}$$

where *Aj* is the area of the Schottky junction, *A*<sup>∗</sup> is the Richardson constant, *T* is the absolute temperature and *k* is the Boltzmann constant. Furthermore, there is a component of noise intrinsic to the photodetection mechanism: due to the quantized nature of the light, the current is constituted by a succession of random impulses, which cause fluctuations of the measured current (shot noise). The quadratic mean value of the fluctuations linked to both photocurrent *Iph* and dark current *Id* is the following:

$$\dot{d}\_s^2(\Phi\_B) = 2qB(I\_d(\Phi\_B) + I\_{pl}(\Phi\_B))\tag{5}$$

where *B* is the device bandwidth. In addition to the shot noise, there is a thermal noise (Johnson noise) with quadratic mean value:

*i*

$$
\sigma\_R^2 = \frac{4kTB}{R\_L},
\tag{6}
$$

where *RL* is the load resistance of the PD. Since the two contributions of the noise current are statistically independent, the total noise *in* is given by their squared sum:

$$\dot{q}\_{\rm li} = \sqrt{2qB(I\_d(\Phi\_B) + I\_{\rm ph}(\Phi\_B)) + \frac{4kT B}{R\_L}}.\tag{7}$$

At low signal levels *Iph* << *Id*, the condition to make the thermal noise negligible compared to the shot noise in Equation (7) is:

$$I\_d \gg \mathcal{D}V\_{\text{th}}/R\_{L\text{-}} \tag{8}$$

where the thermal voltage *V*th = *kT*/*q*. At room temperature, Equation (8) mainly depends on both SBH and *RL*. Of course, if the thermal noise dominates the shot noise, *in* does not depend on the SBH and the optimization procedure reported here can no longer be adopted. Compared to the absolute value of *in*, its magnitude compared to the generated signal *Iph*, defined as the signal-to-noise ratio *SNR* = *Iph*/*in*, is even more important.

In order to find the value of photogenerated current *Iph* that brings *SNR* = 1, we can take advantage of the definition of the SNR and considering Equations (7) and (8), we obtain:

$$SNR = \frac{I\_{ph}}{\sqrt{2qB\left(I\_d(\Phi\_B) + I\_{ph}(\Phi\_B)\right)}} = 1. \tag{9}$$

The square of the previous equation gives a quadratic form in the unknown *Iph*; by solving it we find:

$$I\_{ph} = qB\left(1 \pm \sqrt{1 + \frac{2I\_d}{qB}}\right). \tag{10}$$

This expression makes it possible to obtain the minimum incident optical power *Pinc* necessary to get *SNR* = 1 for a PD characterized by a responsivity *R*. Since the NEP is defined as the incident optical power *Pinc* necessary to get *SNR* = 1 divided by the square root of the bandwidth (*NEP* = *Pinc*/ <sup>√</sup>*B*), we numerically obtain NEP by considering *<sup>B</sup>* <sup>=</sup> <sup>1</sup> Hz in Equation (10) and dividing it by the responsivity *R*:

$$NEP = \frac{q\left(1 \pm \sqrt{1 + \frac{2I\_d}{q}}\right)}{R} \tag{11}$$

which reduces to the very well-known formula:

$$NEP \approx \frac{\sqrt{2qI\_d}}{R},\tag{12}$$

where 2*Id*/*q* is much larger than 1 in typical PDs. It is worth noting that in Equation (12) the sign of *R* follows the sign of *Iph*, as is clear when looking at Equation (3).

Optimized PDs are characterized by high responsivity and low NEP. However, by looking at Equations (3) and (12) it is clear that by increasing the SBH, the NEP improves at the expense of the responsivity. On the other hand, an SBH decrease is beneficial in terms of responsivity but it degrades the NEP. Hence, we sought investigate the Schottky barrier Φ*<sup>B</sup>* that maximizes the *R* to *NEP* ratio. Toward this aim we introduce the function *<sup>G</sup>*(Φ*B*) = *<sup>R</sup> NEP* using Equations (2)–(4), and (12):

$$G(\Phi\_B) = \sqrt{\frac{R}{NEP}} = \frac{R}{\sqrt[4]{2qI\_d}} = \mathbb{C} \cdot \frac{(h\nu)^2 - (q\Phi\_B)^2}{\sqrt{T}(h\nu)^3} \cdot e^{\frac{q\Phi\_B}{4kT}} = \mathbb{C} \cdot g(\Phi\_B) \tag{13}$$

where *C* = 1/(2 <sup>4</sup> 2*qAjA*∗) depends on the geometry through the junction area *Aj* and on the semiconductor through the Richardson constant *A*∗. Figure 1a displays the behavior of *g*(Φ*B*) at 300 K for three different wavelengths, 1.3 μm, 1.55 μm, and 2 μm, showing the presence of a peak. By calculating the first and second derivatives of *G*(Φ*B*) we can find the value Φ∗ *<sup>B</sup>* of SBH corresponding to this peak:

$$\Phi\_B^\* = -4kT \left[ 1 - \sqrt{1 + \frac{(h\nu)^2}{16(kT)^2}} \right]. \tag{14}$$

**Figure 1.** (**a**) Behavior of *g*(Φ*B*) at 300 K for three wavelengths: 1.3 μm, 1.55 μm, and 2 μm; (**b**) optimized responsivity *R* (blue solid line) and optimized Schottky barrier height (SBH) Φ∗ *<sup>B</sup>* (red dashed line) as a function of the wavelengths.

We define Φ∗ *<sup>B</sup>* as the optimized SBH because it is the value at which the *R*-to-*NEP* ratio is maximized. The dashed red line in Figure 1b shows the optimized Schottky barrier Φ∗ *B* as a function of the wavelength. This behavior can be explained by considering that when the wavelength is reduced the photon energy *hν* increases by diminishing the responsivity *R*, as shown in Equation (3), requiring a reduction of the *NEP* to maintain the maximized *R*/*NEP* ratio. In turn, the *NEP* reduction can be achieved by an increase of the optimized Φ∗ *<sup>B</sup>*, which decreases the amount of charge carriers able to overcome the Schottky barrier due to thermal effects. Even if the Φ∗ *<sup>B</sup>* increment also produces a decrease in responsivity, it is important to recall that while the *NEP* is characterized by an exponential decay as a function of Φ∗ *<sup>B</sup>* (*NEP* ∼ *e* <sup>−</sup> <sup>Φ</sup><sup>∗</sup> *<sup>B</sup>* <sup>2</sup>*Vt* ), the responsivity is characterized by a simple quadratic behaviour (*R* ∼ Φ<sup>∗</sup> 2 ).

*B* The substitution of Equation (14) in Equation (3) provides the responsivity when the ratio *R*/*NEP* is maximized (here we refer to it as the optimized responsivity), as shown by the blue solid line in Figure 1b. Note that this optimized responsivity depends only on the SBH of the junction. Figure 1b shows how the optimized responsivity is increased by increasing the wavelength owing to a drop in the optimized SBH Φ∗ *<sup>B</sup>*, providing values at room temperature of 0.10 A/W, 0.14 A/W, and 0.23 A/W at 1.3 μm, 1.55 μm, and 2 μm, respectively, as reported in Table 1. If higher responsivities are required, they can be achieved by lowering the SBH but at the expense of the *SNR*.

**Table 1.** Values of the Schottky barrier Φ∗ *<sup>B</sup>* optimizing the responsivity (*R*)/noise equivalent power (*NEP*) ratio at the three wavelengths of interest: 1.3 μm, 1.55 μm, and 2 μm at *T* = 300 K. The corresponding efficiency *η*SLG int and responsivity *R*, calculated respectively through Equations (2), (3) and (14), are also shown. SLG is the acronym of single layer graphene.


#### **3. Theoretical Results and Discussion**

In this section we theoretically derive the SBH dependence on the bias applied to the junction in order to show how the graphene Schottky PDs based on different semiconductors could be optimized.

It is well-known that the SBH Φ*<sup>B</sup>* of Schottky PDs can be determined by the two following equations (i.e., the Schottky–Mott relations) [33]:

$$q\Phi\_B^{(\text{n})}(V\_R) \quad = \quad q\Phi\_{\text{gr}}^0 - \Delta E\_\text{F}(V\_R) - q\chi\_{\text{sm}} \tag{15}$$

$$q\Phi\_{\rm B}^{(\rm p)}(V\_{\rm R}) \quad = \quad E\_{\rm \mathcal{S}} - (q\Phi\_{\rm \mathcal{S}^{\rm r}}^{0} - \Delta E\_{\rm F}(V\_{\rm R}) - q\chi\_{\rm sm}) \quad \text{(p-type)}\tag{16}$$

where *χ*sm and *Eg* are, respectively, the electron affinity and the bandgap of the semiconductor and *q*Φ<sup>0</sup> gr is the difference between the vacuum level *E*<sup>0</sup> and the Dirac point *E*<sup>0</sup> *F*, while the graphene Fermi level is *EF* (Figure 2). Therefore, <sup>Δ</sup>*EF* = *EF* − *<sup>E</sup>*<sup>0</sup> *<sup>F</sup>* can be expressed as [34]:

$$
\Delta E\_F = -\text{sgn}(n)\hbar v\_F \sqrt{\pi|n|}\tag{17}
$$

where *vF* = 1.1 × 108 cm/s is the Fermi velocity, *<sup>h</sup>*¯ is the reduced Planck constant, and *<sup>n</sup>* is the carrier density in graphene. The carrier density *n* not only depends on the graphene extrinsic doping *n*<sup>0</sup> (defined positive and negative for p-type and n-type graphene doping, respectively) but also on the thermal contact with the semiconductor. Indeed, when a p-doped graphene (*n*<sup>0</sup> > 0) is transferred onto the semiconductor, the space charge *Q*sm in the depletion region induces an opposite charge *Q*gr = −*Q*sm in the graphene layer. This creates additional charge carriers, modifying the carrier density, which becomes *<sup>n</sup>* <sup>=</sup> *<sup>n</sup>*<sup>0</sup> <sup>+</sup> *<sup>Q</sup>*gr *<sup>q</sup>* . The expression of the space charge *Q*sm when the region is completely depleted is *<sup>Q</sup>*sm <sup>=</sup> <sup>±</sup><sup>2</sup>sm*NqV*bi, where sm and *<sup>N</sup>* are the dielectric permittivity and the doping density of the semiconductor, respectively, while *V*bi is the built-in potential. Moreover, by applying a reverse voltage, the charge per unit area in the graphene becomes *Q*gr = ∓ <sup>2</sup>sm*Nq*(*V*bi + *VR*), providing a carrier density:

$$n = n\_0 \mp \sqrt{\frac{2\varepsilon\_{\rm sm}}{q} N(V\_{\rm bi} + V\_R)}\tag{18}$$

where the signs minus and plus are for n- and p-type semiconductors, respectively. Equation (18) replaced into Equation (17) and then in Equation (15) or (16) gives the desired dependence between the SBH and the reverse bias *VR*.

**Figure 2.** Band diagrams of (**a**) graphene/n-semiconductor and (**b**) graphene/p-semiconductor junctions at the thermal equilibrium and when a reverse bias *VR* is applied. At the thermal equilibrium, graphene has an initial carrier density *n*(*VR* = 0). After a reverse bias this charge density becomes *n*(*VR*). *E*<sup>0</sup> represents the vacuum energy level while *E*<sup>0</sup> *<sup>F</sup>* is the Dirac point. <sup>Φ</sup><sup>0</sup> gr, *χ*sm, *Eg*, *EC*, and *EV* are respectively the intrinsic graphene work function, electron affinity, conduction band, bandgap, and valence band. *E*sm *<sup>F</sup>* is the Fermi energy level in the semiconductor and *q*ΦB0 the Schottky barrier at zero bias. The values of the Schottky barrier *q*Φ<sup>B</sup> depend on the graphene Fermi energy level *EF* that shifts when a voltage is applied.

In Table 2 we report the bandgap energy and the electron affinity for various semiconductors and the SBH at zero bias ΦB0, calculated through Equation (15) or (16) when *VR* = 0. The values of ΦB0 were evaluated by considering a graphene work function Φ<sup>0</sup> gr = 4.6 eV [35,36], a built-in potential *V*bi = 0.6 V, an initial SLG extrinsic p-doping *n*<sup>0</sup> = 10<sup>12</sup> cm<sup>−</sup>2, and a low doping of the semiconductors *N* = 10<sup>16</sup> cm<sup>−</sup>3.

**Table 2.** Bandgap *Eg* and electron affinity *χ*sm of various semiconductors together with values of SBH when the Schottky junction is formed, calculated thanks to Equations (15)–(17) by taking into account an initial extrinsic p-doping *n*<sup>0</sup> = 1012 cm−<sup>2</sup> of the single-layer graphene (SLG) and the thermal equilibrium contact with the substrate. For the calculations we considered low-doped semiconductors (i.e., *N* = 10<sup>16</sup> cm<sup>−</sup>3).


Figure 3a shows the intersections between these values of SBH ΦB0 for different semiconductors and the curve of the optimized Φ∗ *<sup>B</sup>*(*λ*) at room temperature (given by Equation (14)), suggesting the working wavelength to achieve the highest *R*/*NEP* ratio for each material. In the range of wavelengths where ΦB0 > Φ<sup>∗</sup> *<sup>B</sup>*(*λ*), the SBH can be lowered down to its optimal value as in Equation (14) by simply applying a specific reverse bias *VR* to the junction.

**Figure 3.** (**a**) Intersection between the curve Φ∗ *<sup>B</sup>*(*λ*) at 300 K and the values of SBH ΦB0 at the interface between graphene and several semiconductors in conditions of thermal contact (no voltage applied to the junction); (**b**) reverse voltage *VR* to apply to the graphene/semiconductor junction as function of the wavelength for maximizing the signal-to-noise ratio (*SNR*) (Φ*<sup>B</sup>* = Φ<sup>∗</sup> *<sup>B</sup>*) for various semiconductors. The values of ΦB0 = ΦB(*VR* = 0) were calculated through Equations (15)–(18) by considering an initial graphene p-doping of *n*<sup>0</sup> = 10<sup>12</sup> cm−<sup>2</sup> and a doping of *N* = 1016 cm−<sup>3</sup> for all the semiconductors reported in Table 2.

By inverting Equation (18) and using Equation (17) and Equation (15) or (16) it is possible to calculate, for each wavelength and for each semiconductor, the values of the reverse voltage *VR* such that Φ*<sup>B</sup>* = Φ<sup>∗</sup> *<sup>B</sup>*. We report this plot in Figure 3b by considering a maximum reverse bias of 20 V. It is interesting to observe that within this limit, graphene Schottky PDs based on p-Al0.3Ga0.7As and p-GaAs can be optimized only in a narrow window of the NIR spectrum, whereas n-Si can be optimized in a broader range, including at 1.55 μm where only a small reverse voltage *VR* = 0.66 V for maximizing the *R*/*NEP* ratio is required. Indeed, at a reverse voltage of 0.66 V the ΦB0 = 0.73 eV of the graphene/n-Si junction can be reduced to its optimum value of Φ∗ *<sup>B</sup>*(1.55 μm) = 0.71 eV. In contrast, p-GaAs requires a higher reverse voltage of 12 V to maximize *R*/*NEP*. Finally, n-Ge stands out among the analyzed semiconductors in view of the possibility to be employed over a region of the NIR spectrum above 2 μm. The range of wavelengths where *R*/*NEP* can be optimized for various semiconductors, by applying a reverse bias up to 20 V, is summarized in Table 3.


**Table 3.** Range of wavelengths in which the *R*/*NEP* ratio of the Schottky photodetectors (PDs) can be maximized by applying a reverse bias up to 20 V.

In Figure 4a,b we report the values of the quantities of interest in this work—the *R*/*NEP* ratio and the optimized *NEP*—for all the examined semiconductors by considering a graphene circular area with radius of 500 μm and a PD closed on a load resistance of 10 MΩ. We compute these optimized quantities through Equation (14) substituted into Equations (3) and (12). Recall that the results shown in Figure 4 are valid when the condition in Equation (8) is fulfilled. In order to verify it, we consider the dark current *Id* one order of magnitude higher than <sup>2</sup>*V*th *RL* (*Id* <sup>=</sup> <sup>10</sup>2*V*th *RL* ), and we calculate both optimized *R*/*NEP* and *NEP* by Equation (12). The solid black lines drawn in Figure 4a,b represent

the validity thresholds of our discussion: graphene Schottky PDs can be optimized in terms of *R*/*NEP* ratio at a given wavelength by means the use of semiconductors placed below and above the solid black lines drawn in Figure 4a,b, respectively. These thresholds depend on the load resistance *RL*, the SBH ΦB, and the graphene active area *Aj*, as is clearly shown by Equation (8). We discover that in the case analyzed here, only graphene/n-Si, graphene/n-Ge, and graphene/n-GaAs Schottky PDs can be suitable for this optimization procedure. Although Si is typically used for visible detection, analysis shows that graphene/n-Si Schottky PDs with a maximized *R*/*NEP* ratio could be adopted for detecting sub-bandgap NIR wavelengths with responsivity and *NEP* of 133 mA/W and 500 fW/√Hz at 1.55 <sup>μ</sup>m, respectively. These devices provide low NEP, enabling their employment for power monitoring and lab-on-chip applications. Note that the predicted responsivity of graphene/n-Si PDs is higher than that reported for NIR Si PDs based on bulk-defect-mediated absorption. Indeed, taking advantage of mid-gap defects introduced into Si ring and disk resonators, Ackert et al. reported a responsivity of only 23 mA/W at −5V[37] and 45 mA/W at −3V[38] at 1560 nm, respectively. On the other hand, if the inter-band absorption of Ge is typically used for detecting the wavelength of 1.55 μm for telecommunications applications, graphene/n-Ge Schottky PDs could allow the detection of wavelengths longer than 1.55 μm, where the the Ge inter-band absorption suddenly decreases. Indeed, graphene/n-Ge Schottky PDs with optimized *R*/*NEP* ratio show a responsivity and *NEP* of 227 mA/W and 31 pW/√Hz at 2 <sup>μ</sup>m, respectively, enabling their employment in environment monitoring applications. The predicted responsivity of graphene/n-Ge PDs is higher than that reported for NIR Ge PDs based on the introduction of tin (Sn) atoms in the Ge lattice. Indeed, with a substitutional Sn concentration of 6.5%, Ge-based PDs are able to absorb optical radiation at 2 μm but provide a limited responsivity of only 20 mA/W [39]. Note that NEP depends on many parameters, such as graphene's optical absorbance, the graphene area in contact with the semiconductor, and the temperature. Among these, particular attention should be paid to the temperature, which appears in the exponential argument of the dark current (Equation (4)), which in turn affects the NEP (Equation (12)). As an example, in a graphene/n-Si Schottky PD we evaluated this by increasing the temperature of 1 ◦C with respect to the room temperature, and an increase in optimized NEP below 5% could be achieved at any wavelength in the range of interest for this junction. As reported in Figure 4a,b, semiconductors such as n-Si, n-Ge, and n-GaAs can be exploited at room temperature for the realization of optimized graphene-based Schottky PDs in the spectral range from 1955 to 2080 nm with a responsivity from 219 to 245 mA/W (Figure 1b); however, while n-GaAs would be characterized by a lower NEP, n-Si and n-Ge would have the advantage of a better compatibility with CMOS technology.

**Figure 4.** (**a**) The optimized *R*/*NEP* and (**b**) the optimized *NEP* of the Schottky graphene-based PDs for various semiconductors as function of the wavelength range individuated in Table 3. All figures were obtained at room temperature and by considering a graphene circular area in touch with the semiconductor with radius of 500 μm and a load resistance of 10 MΩ. The arrows indicate the validity regions of the proposed optimization procedure.

#### **4. Conclusions**

In this work we theoretically investigated the responsivity/NEP trade-off of NIR graphene/semiconductor Schottky PDs at room temperature. An analytical expression of the SBH able to maximize the *R*/*NEP* ratio was derived. Furthermore, we discussed how the optimized SBH can be tuned by applying a reverse voltage to the junction in order to establish the best operation conditions to achieve higher responsivity as well as lower noise for various semiconductors. Toward this aim we have accounted for the physics behind the emission of photo-excited charge carriers from graphene to Si, the theory of the graphene/semiconductor Schottky junctions, and the properties of graphene related to its two-dimensionality.

Remarkably, we found that CMOS-compatible materials such as Si and Ge could be exploited for the realization of optimized graphene Schottky PDs able to detect wavelengths beyond the limit imposed by their inter-band optical absorption. Indeed, graphene/n-Si Schottky PDs with maximized *R*/*NEP* ratio showed responsivity and *NEP* of 133 mA/W and 500 fW/√Hz, respectively, at 1.55 <sup>μ</sup>m by applying a reverse voltage of only 0.66 V. On the other hand, graphene/n-Ge Schottky PDs with maximized *R*/*NEP* ratio showed the potential to work at wavelengths longer than 1.55 μm, being for instance characterized by a responsivity and *NEP* of 227 mA/W and 31 pW/√Hz at 2 <sup>μ</sup>m.

We believe that the insights reported in this work could be of paramount importance in silicon photonics for the realization of optimized PDs to be employed in power monitoring, lab-on-chip, and environment monitoring applications.

**Author Contributions:** Conceptualization, T.C. and M.C.; methodology, T.C., M.C., and L.M.; formal analysis, T.C. and M.C.; resources, L.M.; data curation, T.C., M.C. and L.M.; writing original draft preparation, T.C.; writing—review and editing, M.C. and L.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


*Review*
