Interferometric Motion Detection in Atomic Layer 2D Nanostructures: Visualizing Signal Transduction Efficiency and Optimization Pathways

Atomic layer crystals are emerging building blocks for enabling new two-dimensional (2D) nanomechanical systems, whose motions can be coupled to other attractive physical properties in such 2D systems. Optical interferometry has been very effective in reading out the infinitesimal motions of these 2D structures and spatially resolving different modes. To quantitatively understand the detection efficiency and its dependence on the device parameters and interferometric conditions, here we present a systematic study of the intrinsic motion responsivity in 2D nanomechanical systems using a Fresnel-law-based model. We find that in monolayer to 14-layer structures, MoS2 offers the highest responsivity among graphene, h-BN, and MoS2 devices and for the three commonly used visible laser wavelengths (633, 532, and 405 nm). We also find that the vacuum gap resulting from the widely used 300 nm-oxide substrate in making 2D devices, fortunately, leads to close-to-optimal responsivity for a wide range of 2D flakes. Our results elucidate and graphically visualize the dependence of motion transduction responsivity upon 2D material type and number of layers, vacuum gap, oxide thickness, and detecting wavelength, thus providing design guidelines for constructing 2D nanomechanical systems with optimal optical motion readout.


Reflectance and Responsivity.
illustrates the device structure in our model. As in most experiments, we consider normal laser incidence from vacuum onto this tetra-layer structure composed of 2D crystal (subscript "2D" in equations), the vacuum gap, the remaining SiO 2 layer, and the Si wafer (treated as semi-infinite medium). The reflectance R of the structure (total fraction of light reflected) is determined by the interference of the reflected light from all interfaces. Analysis of the multiple reflections inside this tetra-layer structure gives 33 2  12  3  2  3  2  3  1  3   1  12  12  3  2  3  2  3  1  3 where r 1 through r 4 are the reflection coefficients at the vacuum-2D, 2D-vacuum, vacuum-SiO 2 , and SiO 2 -Si interfaces, respectively, and ϕ 1 , ϕ 2 , and ϕ 3    where d 1 is the 2D crystal thickness, d 2 is the vacuum gap depth, d 3 is the SiO 2 thickness, and λ is laser wavelength. Note that bold fonts indicate complex variables. As the 2D layer assumes flexural motion, the vacuum gap depth d 2 changes, leading to modification of the reflectance R (Eq. 1). Here we use MoS 2 as an example to illustrate this signal transduction process, as the real part of its index of refraction is much higher than other 2D materials (see Methods), resulting in much stronger reflection from the crystal surfaces (r 1 and r 2 in Eq. 2) and consequently stronger interferometric motion transduction, particularly important for ultrathin samples which transmit most of the light. Figure 1b shows the reflectance R (at 633 nm laser illumination) as a function of d 2 for mono-, bi-, and trilayer (1L, 2L, and 3L) MoS 2 devices with d 2 = 250 nm and d 3 = 50 nm, showing that R varies smoothly with d 2 , and increase with number of layers (or thickness d 1 ) over this range of d 2 .
The photodetector measures motion-induced changes in reflectance. Therefore, the greater change in R per unit device motion, the more efficient the signal transduction. This motion-to-reflectance responsivity ℜ is defined as ℜ = ∂ R/∂ d 2 : thus the slope of R(d 2 ) at d 2 = 250 nm (Fig. 1b) represents the values of ℜ in these devices. Note that ℜ can be negative (and its magnitude, not sign, determines the effectiveness in detecting device motion), and in Fig. 1c the magnitude |ℜ | is plotted. It can be seen that in this range, ℜ increases roughly linearly with device layer number and d 2 .
Visibility vs. Responsivity. The 'visibility' 33,35,36 and motion responsivity in 2D materials are related but different. While rooted from the same formalism (Eq. 1), they focus on different aspects of the equation. For visibility, a larger optical contrast (greater change in R) between two locations: one with 2D crystal (d 1 ≠ 0) and one without (d 1 = 0) gives better visibility; whereas for motion detection, a greater responsivity (steeper slope in R vs. d 2 ) leads to greater signal given the same motion amplitude. The optimal detection scheme is also different: when searching for 2D flakes in microscope, as the human eye is highly color-sensitive, it is desirable to use a multi-color (such as white light) illumination, and when R increases for one color and decreases for another it gives enhanced color contrast. In motion detection, the photodetector measures the change in light intensity, and using a monochromatic light source is more desirable as it removes the possible cancellation between different wavelengths (i.e., R increases for one λ and decreases for another). Therefore, instead of calculating over a range of continuous λ values (as in works focusing on visibility of on-substrate materials 35 ), we focus on monochromatic illumination, mostly λ = 633, 532, and 405 nm, which are widely used in 2D structures motion detection. Figure 2 shows the calculated motion responsivity ℜ (633 nm illumination) of 1L, 2L, and 3L MoS 2 over a large range of d 2 values, for devices fabricated on 300 nm SiO 2 substrate followed by oxide etch (thus d 2 + d 3 = 300 nm). Note that over large range of d 2 , ℜ can cross 0 (where the motions lead to no reflectance variation) and change sign.
Dependence on Device Structure. Recently emerging transfer techniques 38 make it possible to fabricate suspended 2D devices on arbitrary substrate structures. Therefore, in analysis below we vary d 1 , d 2 , and d 3 independently. Figure 3a illustrates the results for 1L, 2L and 3L MoS 2 devices. We make the following observations. First, in this comprehensive parameter space (both d 2 and d 3 covering multiples of λ), periodic behavior is evident. Second, these devices exhibit similar locations (d 2 d 3 combinations) for the ℜ peaks (both positive and negative). Third, the magnitude of ℜ (and thus the amplitude of ℜ variation) increases with number of layers for 1-3L devices.
We first focus on the |ℜ | peak locations by using 2D color plots (Fig. 3b,c). We find the same ℜ periodicity (along both d 2 and d 3 ) for all device thicknesses. In d 2 direction, the period is exactly λ/2 (λ = 633 nm). This is because when d 2 changes by λ/2, the total optical path for rays 3&4 (Fig. 1a) changes exactly by λ, keeping the interferometric condition unchanged. The periodicity along the d 3 axis is λ/2n SiO 2 , which can be understood in a similar pattern by considering the optical path inside SiO 2 . Note that the 0 optical absorption of SiO 2 at 633 nm 39 helps ensure the perfect (i.e., including both periodicity in d 3 and identical amplitude along this periodic pattern).
We thus plot gridlines at d 2 = nλ/2 and λ = n d m /2 SiO 3 2 (n, m are integers) to better visualize the periodicity. We find that all the |ℜ | peaks are located along In contrast, their d 2 values gradually vary with d 1 , more apparent in the positive peaks (dark red). By compiling the results for 1L to 50L MoS 2 devices into a 3D stack (Fig. 4a), with device thickness d 1 in the third (vertical) dimension, we confirm that  change with device motion. Therefore, the d 3 value for the highest responsivity is the one that maximizes the total reflected light intensity underneath the 2D crystal (rays 3&4), which interferes with the lights reflected from the 2D crystal surfaces (rays 1&2). This condition is met when the sacrificial SiO 2 layer is completely removed (d 3 = 0), or gives completely constructive interference between rays 3&4 (d 3 = mλ/2n SiO 2 ).
Dependence on Crystal Thickness. We now focus on the dependence of |ℜ | peaks on d 1 and d 2 , assuming d 3 = 0 and hereafter. Figure 4b shows that the |ℜ | peak values, both positive (wine) and negative (blue), vary non-monotonically from 1L to 50L devices: the projections onto the two vertical planes shows that |ℜ | (for an optimized structure) increases with thickness, reaching the highest value at 12-layer (with d 2 ≈ 330 nm), while further increase in thickness decreases |ℜ |. Note that the projection on the bottom d 1 -d 2 plane reproduces the front view (along d 3 ) of Fig. 4a.
To understand this, we examine its root in reflectance R. We again use 2D color plots (Fig. 5e,f) to examine the periodic variations of R and |ℜ |. As in Fig. 3, the period along the d 2 axis is strictly λ/2. In the d 1 direction, R oscillates between ~0-0.8, with the oscillation amplitude decrease as d 1 increases, approaching R ≈ 0.5 for large d 1 . This period is ~60 nm, or λ/(2n MoS2 ), as expected from thin film optics. The magnitude of R variation decays with d 1 due to increased absorption for thicker crystals. As d 1 becomes very large, the MoS 2 crystal becomes semi-infinite, and the optical process reduces to a single reflection and |ℜ | approaches 0 (Fig. 5c,d).
Optimization Pathways. The |ℜ | map (Fig. 5f) provides a design guideline for choosing the optimal device structure (in terms of interferometric detection of device motion) for any given MoS 2 crystal thickness. The plots show that for λ = 633 nm, the 300 nm-SiO 2 -on-Si substrate widely used in 2D crystal research is a good choice: it provides good optical contrast for identifying thin crystals 36 , and once etched out, the resulting ~300 nm vacuum cavity (dotted line in Fig. 5f) provides good responsivity for a wide range of MoS 2 thicknesses. One can further tune the "effective" cavity depth d 2 by not fully removing SiO 2 . As shown in Fig. 3b, ℜ varies slowly along d 2 + d 3 n SiO2 = constant (dashed lines in Fig. 3b,c). So for example, in a recent work 10 , when 250 nm of 290 nm SiO 2 is etched, the "effective" cavity depth is 250 + 40 × 1.457 = 308.3 nm, close to the optimal d 2 value, allowing the observation of thermomechanical motion in MoS 2 resonators with different thicknesses.

Dependence on Wavelength.
We now examine the effect from laser wavelength. We choose a few representative wavelengths: 633 nm (red), 532 nm (green), and 405 nm (blue), all among the most commonly used lasers in labs, and cover the visible range. Figure 6a-c shows the results for λ = 532 nm. The overall pattern is similar to λ = 633 nm, with the main difference in the spatial periodicity in both d 2 and d 1 directions (as λ changes). One consequence is that the magnitude of ∂ R/∂ d 2 (i.e., |ℜ |) increases as the d 2 axis effectively rescales (e.g., negative ℜ peaks in Fig. 6a bottom pane compared with those in Fig. 5b).
When λ further reduces to 405 nm (Fig. 6d-f), additional effect rises: the optical absorption becomes much stronger such that the decay in d 1 direction is significant, and beyond ~50-layer there is little responsivity (as little light can penetrate the 2D crystal).  the R modulation (with d 2 ) is most pronounced for devices below 50 nm (~150L), which consequently exhibit higher |ℜ |; while for h-BN, the R modulation (and thus |ℜ |) monotonically increases with d 1 within the entire plot range. This manifests the effect from band structure. For graphene (0 eV bandgap), optical absorption increases quickly with thickness, thus thinner devices (< 150L) exhibit higher responsivity. In contrast, h-BN has large bandgap (~5 eV) and thus minimal absorption at 633 nm; together with the relatively low refractive index (thus low reflectivity) 40 , there is little R modulation as h-BN device vibrates, unless the crystal is sufficiently thick to induce sizable absorption. The results show that for mono-and few-layer h-BN, (even with optimized device geometry) |ℜ | is orders of magnitude lower than for graphene and MoS 2 , and only multilayer (> 20L, green curve in Fig. 7d) h-BN has comparable |ℜ | values as monolayer MoS 2 or graphene.

Different 2D Crystals.
Quantitative Design Guideline for Optimizing Responsivity. We finally summarize the optimized responsivity and corresponding device structures for the different 2D materials for the three widely employed wavelengths. Figure 8(a) shows the highest achievable |ℜ | values for 1L to 200L graphene, h-BN, and MoS 2 devices under 405 nm, 532 nm, and 633 nm illumination. The line style represents different 2D material and line color corresponds to each wavelength. The results show that towards the monolayer limit, MoS 2 devices can have the highest |ℜ | among these 2D materials for the three wavelengths (1L to 14L for 405 nm; 1L to 33L for 532 nm; 1L to 37L for 633 nm). As the number of layers increases, the highest |ℜ | values are found in graphene structures (405 nm: 15L to 100L; 532 nm: 34L to 116L; 633 nm: 38L to 112L). In even thicker structures, up to 200L, h-BN devices offer the highest |ℜ | (405 nm: ≥ 101L; 532 nm: ≥ 117L; 633 nm: ≥ 113L). The color bars on the top of Fig. 8(a) summarizes the thickness range in which each particular 2D material exhibits the highest |ℜ | in their respective optimized device geometry. Figure 8(b) shows the optimal vacuum gap depth d 2 for achieving the highest |ℜ | values discussed above, providing a clear design guideline for NEMS devices based on these 2D materials.
The physical origin of the findings (that as thickness varies, different 2D crystals attain their highest motion transduction responsivity conditions with different device configurations) can be intuitively understood by considering two mechanisms of light-device interaction: reflection and absorption. Multireflection (Fig. 1a) causes interference between different optical paths, which generates the detailed interferometric effects and signals and thus determines the motion transduction responsivity. Inside the multilayer structure (see Fig. 1a), in the regime that the 2D material layer (thickness d 1 ) only reflects less than a few percent of incident light (as often found in ultrathin samples), if a 2D flake gives comparatively more reflection, it leads to stronger interference effects inside the vacuum gap (depth d 2 ) and thus higher motion transduction responsivity (as d 2 varies). Further, such multireflection of light within the layered structure (Fig. 1a) creates a spatially-varying optical field in the vacuum gap underneath the 2D crystal. As the vacuum gap depth varies, the light intensity at the 2D flake changes, and the finite absorption in the 2D material causes change in the total light intensity. In the regime that the absorption (here we specifically refer to the percentage of light intensity absorbed when passing through a 2D flake) of the 2D flake is only a few percent or less (as often found in ultrathin samples) such that sufficient light enters the vacuum gap (to form a spatially-varying optical field), 2D flakes with greater absorption can engender greater responsivity.
Larger index of refraction n leads to greater reflection. For all the wavelengths in this study, n MoS2 > n graphene > n h-BN . In the visible spectrum, MoS 2 has greater absorption than graphene (see Methods), while h-BN absorbs the least as its bandgap corresponds to ultraviolet. Therefore, mono-and few-layer MoS 2 (h-BN) devices have the greatest (smallest) responsivity as the flake is most (least) reflecting and absorbing (as shown in the left end of Fig. 8a).
As the thickness (and thus total absorption) increases beyond just a few percent (as often the case for many-layer or thin film samples), the dependence of responsivity on 2D flake reflectance and absorption can change. High reflection causes less light entering the vacuum gap (for interferometry), and stronger absorption leads to less total reflected light (thus less intensity available for modulation by device motion); both reduce the responsivity for multilayer 2D flakes. Therefore MoS 2 (h-BN), the most (least) reflecting and absorbing among the three crystals, is the first (last) to experience such effect: beyond a given thickness, responsivity starts to decrease as number of layers further increases, as seen in the solid (dashed) curves in Fig. 8a.
In conclusion, we have systemically investigated interferometric motion detection in 2D nanomechanical devices based on atomic layer crystals. We have quantitatively elucidated and graphically visualized the dependences of motion responsivity upon parameters in device structure, probing wavelength, and type of 2D material. We find that the highest responsivity may be achieved with no oxide layer at the bottom of the vacuum gap, and the optimal vacuum gap varies (with crystal thickness) around mλ/2; specifically, when using 633 nm He-Ne laser, the ~300 nm-SiO 2 -on-Si substrates commonly used in 2D research (and the resulting vacuum gap) offer close-to-optimal motion responsivity for several 2D crystals over a wide range of thickness. We also illustrate the trade-off between enhancing responsivity and increasing absorption when using short wavelengths, and show that different types of 2D layered materials exhibit different patterns in the same parameter space due to their different band structure and dielectric constants. The optimization pathways shown in our results provide a complete design guideline for building 2D nanomechanical devices with the highest achievable optical transduction efficiency, which can significantly improve the signal-to-noise ratio and enhance device performance such as dynamic range and sensitivity. This can in turn help enable new functions and high performance in emerging applications, e.g., future fiber-optic, near-field, and on-chip interferometric schemes with ultra-sensitive signal detection and processing using 2D nanodevices.

Methods
The indices of refraction (complex values) are obtained from a number of references. In some cases, instead of the complex refractive index n = n-iκ (κ is also called the "extinction coefficient"), the values are given in the form of complex dielectric constant, i.e. relative permittivity: ε = ε 1 + iε 2 , which is related to the index of refraction through ε = n 2 = (n− ik) 2 . One can calculate n and κ using = We note that the refractive index could be layer-dependent for certain materials, e.g., the value for monolayer could be different from that of the bulk 44 . Earlier study suggests that such difference is expected to be insignificant, as the optical response of 2D layered material with light incident normal to the basal plane is dominated by in-plane electromagnetic response, which is similar in few-layer structures and in bulk 33 . We therefore use the same value for all the device thicknesses in calculation, as in previous work 33,35,36 .
We also note that different sources in the literature may give different refractive index values 39,43,45 . While such quantitative differences can lead to changes in the numerical values (and causes the patterns in the figures to slightly shift), all the results remain qualitatively unchanged, and all the physical arguments and interpretations remain valid.
While monolayer graphene has zero bandgap and exhibit strong absorption (percentage of light intensity absorbed when passing through a 2D flake, also sometimes called "absorbance" in 2D literature) over a wide spectrum range 46,47 , monolayer MoS 2 has greater absorption in the visible range due to interband transitions and higher density of states 48 . This is also manifested in their complex indices of refraction: the absorption coefficient α = 4π κ/λ is proportional to the extinction coefficient κ, and in the ultrathin limit the exponential dependence of absorption on flake thickness d 1 reduces to a linear relation: absorption ∝ αd 1 ∝ κd 1 . Using monolayer thickness and κ values of MoS 2 and graphene, we estimate that monolayer absorption of MoS 2 is ~40% greater than that of graphene at 633 nm, consistent with measured values 46,48 .