Polarization sensitive black phosphorus nanomechanical resonators

Arnob Islam; Anno van den Akker; Philip X.-L. Feng

doi:10.1364/OME.9.000526

1. Introduction

Black phosphorus (P) has recently been rediscovered as an exciting layered semiconductor for making two-dimensional (2D) devices. It offers very high hole mobility (up to ~1,000 cm²V⁻¹s⁻¹), layer thickness-dependent and tunable direct band gap (~0.3 eV in bulk, up to ~2 eV in its single-layer limit) that covers an unusually wide range of spectrum from visible to mid-infrared (IR) [1,2]. Black P also distinguishes itself from other 2D crystals for its strong, intrinsic in-plane anisotropic properties along armchair (AC) and zigzag (ZZ) directions, originated from its corrugated crystal structure (Fig. 1(a)). Due to the property of linear dichroism in black P, it has been observed that optical absorption for incident light polarized along AC direction is significantly higher than that along ZZ direction (Fig. 1(b)) [3].

Fig. 1 Black phosphorus (P) crystal and nanomechanical devices. (a) 3D illustration of crystal structure of black P. (b) Top view of the crystal showing in-plane armchair (AC) and zigzag (ZZ) directions and the nature of optical anisotropy. A_AC and A_ZZ denote absorbance when incident light is polarized along AC and ZZ directions, respectively. (c) Fabrication process of making black P drumhead resonators. (d) 3D representation of a black P circular drumhead resonator.

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Linear dichroism in black P has been utilized to realize polarization sensitive black P photodetectors [4,5], polarizer [6], optical waveplates [7], nanoantennas [8], etc. As linear dichroism of black P causes polarization dependent absorption, which can translate into polarization dependent heating in black P. Now, if we consider a black P flake mechanically suspended on a circular microtrench to form a drumhead membrane structure free to deflect and vibrate in its nanomechanical resonance modes [9], then this polarization dependent heating will cause different amount of heating induced surface tension. As the flexural resonance frequencies of a drumhead resonator are very responsive and sensitive to the change of its surface tension, we can develop a polarization sensitive black P drumhead nanomechanical resonator. Polarization responsivity and sensitivity of black P resonator can be used to realize polarimetric nanodevices. One type of application for polarimetric devices is the detection of the optical activity of chiral molecules. Optical activity in chiral solutions (e.g., glucose, amino acids, etc.) originates from interactions between the electric and magnetic dipoles in the molecules present in these solutions [10]. When a linearly polarized light passes through such solutions, the plane of polarization rotates (left or right) depending on the nature of the chiral molecules. The amount of rotation for a specific substance depends on the concentration and the length of optical path. Therefore, this can be used to characterize chiral molecule solutions and determine the specific rotation of new substances with known concentration, or the concentration or degree of purity of a known solution of chiral molecules [11]. This has significant importance and applications in toxicology, chemical industry, and drug discovery. In addition, polarimetric devices can be used for measurement of magneto-optical phenomenon named “Faraday rotation”, which has many scientific and technological applications [12].

In this study, we fabricate black P nanomechanical drumhead resonators to demonstrate the polarization responsivity and sensitivity. We observe that resonance frequencies (f_res values) of the drumhead resonators shift with the change of incident polarization. In addition, we develop a multiphysics model for polarization responsivity of the black P resonator and present the design procedure to optimize the device structure in order to maximize the polarization responsivity. Moreover, we discuss the viability of employing these resonators to probe or detect the optical rotation of chiral or optically sensitive substances.

2. Fabrication and characterization

To investigate the in-plane optical anisotropy of black P in the resonator platform, we fabricate 2D black P drumhead resonators (Fig. 1(c)). We start with bare 290nm-thick SiO₂-on-Si substrates. By using reactive ion etching (RIE) we create circular microtrenches with 900 nm depth. We then transfer exfoliated black P crystalline flakes on top of the microtrenches by using a dry-transfer method [13]. Two fabricated devices, D#1 (thickness, t_BP ≈90nm and diameter, d =9µm) and D#2 (t_BP ≈27nm and d =8µm) are shown in Fig. 2(a) and 2(b) respectively. As thinner black P could be oxidized and degraded in ambient condition with light and moisture [14], it often requires to use a protective layer to prevent degradation [14]. Important encapsulation methods using h-BN and Al₂O₃ (e.g., h-BN, Al₂O₃, O₂ plasma followed by Al₂O₃, etc.) [14–17] have been proposed to prevent degradation in ambient conditions. For mechanical resonators, these methods may help enable robust heterostructure resonators with preserved polarization-sensitive characteristics and anisotropic properties of black P. Moreover, there is another method named benzyl-viologen (BV) doping [18], which is promising for prevention of degradation in the case of black P resonators. In addition, hermetic vacuum packaging techniques well established in MEMS industry can be exploited to significantly improve the stability and lifetime of black P resonators in future feasible applications. For devices D#1 and D#2 here, nonetheless, we have not used any protective layer to prevent degradation of black P, as it has been found in literature that for multilayer or thicker black P (>7nm), the mechanical properties (e.g., Young’s modulus and resonances) do not change much due to the formation of a very thin passivation layer by self-oxidation [9,19]. Thicknesses of these resonators have been determined by using atomic force microscopy (AFM, Agilent 5500) and the corresponding traces of the measurements are shown in Figs. 2(a) and 2(b) with optical images of the devices. We also obtain crystal orientation of black P flakes used in these two devices by using polarized reflectance measurement (Figs. 2(a) and 2(b)) [20].

Fig. 2 Black P nanomechanical resonators and polarization dependent optical absorption. (a) and (b) Two black P resonators fabricated on SiO₂-on-Si substrate with known crystal orientation (AC and ZZ directions). Insets show AFM measurement results with corresponding traces marked by yellow dashed lines drawn on the optical images of devices. (c) Ellipse of imaginary part (k) of complex refractive index (n) indicating the optical anisotropy of black P. (d) Illustration of multilayer reflection and interference effect. (e) and (f) Calculated variation of optical absorbance (A_BP) and reflectance (R_BP) of D#1 and D#2, respectively, for 633nm wavelength, with respect to polarization angle (ϕ) (defined as the angle between the electric field ( $\vec{E}$ ) and the AC direction) (shown in (c)).

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3. Theoretical modeling and measurements

Owing to the in-plane optical anisotropy (linear dichroism) of black P, optical constants (i.e., refractive indices n = n-ik) are different along AC and ZZ directions. It has been found that the imaginary part (k) of n is about 10 times higher along AC direction compared to that along ZZ (Fig. 2(c)) for visible wavelength, which significantly dictates higher absorption upon linearly polarized light oriented along AC [3]. In order to model the optical anisotropy of black P, we write the refractive index (n_BP) as follows [20],

n_{BP}^{2} (ϕ) = \frac{n_{AC}^{2} n_{ZZ}^{2}}{n_{AC}^{2} \sin^{2} ϕ + n_{ZZ}^{2} \cos^{2} ϕ},

where ϕ = the angle between the electric field and AC direction, n_AC and n_ZZ are refractive indices along AC and ZZ directions, respectively. To calculate polarization dependent absorbance (A_BP) in black P, we employ a Fresnel-law based optical model by taking into account both optical anisotropy and multilayer optical interference effects (Fig. 2(d)) [21,22]. The black P resonator is modeled as a 4-layer system, vacuum-black P-vacuum-Si with a cavity between black P and Si substrate (Fig. 2(d)). For normal incidence, the reflectance (R_BP) of the device (defined as the ratio of total reflected light intensity to the incident light intensity) is calculated by the interference of reflected light from all of the material-vacuum interfaces in the model [21],

R_{B P} (ϕ) = {| \frac{r_{1} e^{i (φ_{1} + φ_{2})} + r_{2} e^{- i (φ_{1} - φ_{2})} + r_{3} e^{- i (φ_{1} + φ_{2})} + r_{1} r_{2} r_{3} e^{i (φ_{1} - φ_{2})}}{e^{i (φ_{1} + φ_{2})} + r_{1} r_{2} e^{- i (φ_{1} - φ_{2})} + r_{1} r_{3} e^{- i (φ_{1} + φ_{2})} + r_{2} r_{3} e^{i (φ_{1} - φ_{2})}} |}^{2},

where

r_{1}

,

r_{2}

, and

r_{3}

are the refractive indices at the vacuum-black P, black P-vacuum, and vacuum-Si interfaces, respectively,

r_{1} = \frac{n_{v a c u u m} - n_{B P} (ϕ)}{n_{v a c u u m} + n_{B P} (ϕ)}, r_{2} = \frac{n_{B P} (ϕ) - n_{v a c u u m}}{n_{B P} (ϕ) + n_{v a c u u m}}, r_{3} = \frac{n_{v a c u u m} - n_{S i}}{n_{v a c u u m} + n_{S i}}

. The refractive index of Si is n_Si = 3.8789-0.0017i. The refractive index of black P is n_BP,AC = 2.732-0.526i with AC polarized incident light and n_BP,ZZ = 2.669-0.040i for ZZ polarized incident light (633 nm) [20]. The corresponding phase shifts at the black P suspended region and the vacuum cavity are

φ_{1} = \frac{2 π n_{B P} (ϕ) t_{B P}}{λ}

and

φ_{2} = \frac{2 π n_{v a c u u m} g}{λ}

, where t_BP is the Black P resonator thickness, g is the vacuum gap depth, and λ is the wavelength of the laser. The transmittance (T_BP) for the black P resonator (defined as the ratio of total transmitted light intensity to the incident light intensity) by taking into account the multilayer interference for normal incidence, can be expressed as below [22]:

T_{B P} (ϕ) = | \frac{n_{S i}}{n_{v a c u u m}} | \cdot {| \frac{t_{1} t_{2} t_{3} e^{- i (φ_{1} + φ_{2})}}{1 + r_{1} r_{2} e^{- 2 i φ_{1}} + r_{1} r_{3} e^{- 2 i (φ_{1} + φ_{2})} + r_{2} r_{3} e^{- 2 i φ_{2}}} |}^{2},

where t₁, t₂, and t₃ are the transmission coefficients of the vacuum-black P, black P-vacuum, and vacuum-Si interfaces, respectively, with

t_{1} = \frac{2 n_{v a c u u m}}{n_{v a c u u m} + n_{B P} (ϕ)}

,

t_{2} = \frac{2 n_{B P} (ϕ)}{n_{v a c u u m} + n_{B P} (ϕ)}

,

t_{3} = \frac{2 n_{v a c u u m}}{n_{v a c u u m} + n_{S i}}

. Finally, the absorbance (A_BP) of the suspended black P flake can be written as below:

A_{B P} (ϕ) = 1 - R_{B P} (ϕ) - T_{B P} (ϕ) .

Figures 2(e) and 2(f) show the calculated A_BP and R_BP with respect to the polarization angle (ϕ) for the two devices D#1 and D#2. It shows significant variation of A_BP and R_BP with respect to ϕ. Polarization change leads to variation of absorption, and consequently heating (i.e., photothermal effect from light absorption) induced surface tension variation in the 2D resonator, which in turn shifts the resonance frequency (f_res) of the drumhead resonator (Fig. 3(a)).

Fig. 3 Illustrations of measurement scheme and transduction diagram of polarization responsivity. (a) Transduction diagram of resonance frequency shift upon the change of polarization of normally incident laser illumination. (b) Scheme of thermomechanical resonance measurement for studying black P resonators by using optical interferometric readout and controlling the polarization to monitor the resonance frequency shift.

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Next, we calculate the heating induced surface tension from the absorption of light in the drumhead, which will cause the shift of f_res. At first, we obtain the temperature (T) profile due to localized laser heating at the center of the drumhead resonator by solving the following simplified 2D heat transfer equation:

κ_{AC} \frac{\partial^{2} T}{\partial x^{2}} + κ_{ZZ} \frac{\partial^{2} T}{\partial y^{2}} = \frac{P_{l a s} A_{B P}}{t_{B P} π σ_{l x} σ_{l y}} e^{[\frac{- {(x - x_{0})}^{2}}{2 σ_{l x}^{2}} - \frac{{(y - y_{0})}^{2}}{2 σ_{l y}^{2}}]},

where κ_AC and κ_ZZ are components of the anisotropic thermal conductivity of black P, which are thickness dependent [23], the standard deviations of the laser beam distribution are σ_lx and σ_ly, P_las is laser power on the device, and the laser spot location is (x₀, y₀). We assume T = 300K or room temperature at the edge of the suspended part of the drumhead as boundary condition while solving the differential equation to compute the temperature distribution. We can then obtain thermal strain magnitude profile from this temperature profile, ε_th ≈-αΔT, where α is thermal expansion coefficient and ΔT is temperature rise (above the boundary temperature). From this, we can obtain heating induced surface tension,

γ_{th} = - \frac{t_{B P}}{π a^{2}} \iint E_{Y} (x, y) ε_{th} (x, y) d x d y,

where E_Y is anisotropic Young’s modulus, t_BP is thickness of black P and a is the radius of the circular suspended part. Finally, this surface tension (negative in polarity due to compressive strain) causes the resonance frequency (f_res) of the device to decrease, which can be approximated analytically by the following equation [24],

f_{res} = \frac{k_{2}^{m n} a}{2 π} \sqrt{\frac{D}{ρ a^{4}} [\frac{(γ_{0} + γ_{th})}{D} + {(k_{2}^{m n} a)}^{2}]},

where

D = \frac{E_{Y, a v g} t_{B P}^{3}}{12 (1 - ν^{2})}

is the flexural rigidity, E_Y,avg is average of anisotropic Young’s modulus, ρ is the density of black P, ν is the Poisson’s ratio of black P, γ₀ is the pre-tension, and

k_{2}^{m n}

is the eigenvalue of a particular mode, (m, n) .

In order to realize the concept experimentally, we employ polarized light with normal incidence on the resonator and detect the resonance by optical interferometric readout simultaneously utilizing the same light source. We use a polarizer to produce linearly polarized light from a He-Ne laser (633 nm) (unless mentioned otherwise) and a half-wave (λ/2) plate to control the in-plane polarization angle on the device. As no external excitation mechanism is employed to the nanomechanical resonators, spontaneous Brownian agitations and fluctuation-dissipation theorem dictate the thermomechanical motions of the drumhead structures such that their eigenmodes emerge as thermomechanical resonances in the frequency-domain, undriven noise spectra. More details can be found in earlier studies [22,24]. The thermomechanical resonances are detected by a custom-built ultrasensitive scanning laser interferometry system (Fig. 3(b)). Due to the fact that change in incidence angle of light affects the absorption in black P, in the measurement, we make sure that the laser is normally (perpendicularly) incident on the device to only detect the polarization dependent absorption change. All measurements are performed at room temperature under moderate vacuum (~20 mTorr). The laser is focused on the resonator by using a 50× microscope objective. Thermomechanical noise spectral density is recorded by a spectrum analyzer (Agilent E4440A). The details of the interferometry measurements can be found in prior works [22,24] and their supplementary materials.

4. Results and discussions

When the polarization of incident light is aligned with AC direction, absorption or heating is maximum. Due to heating, black P will expand (positive thermal expansion coefficient) and compressive strain induced surface relaxation will act against the built-in pre-tension, lower the net surface tension. Therefore, f_res will be the lowest in this case (given a fixed illumination intensity). Likewise, due to less absorption, f_res will be the highest when polarization is along ZZ direction. These trends are experimentally observed in measurements of two devices (Fig. 4(a) and 4(c)). For D#1 and D#2, f_res upshifts of ~0.5MHz and ~0.6MHz, respectively, when polarization orientation is changed from AC to ZZ. Further, resonance peak amplitude is the highest when light is ZZ-polarized, which is due to better responsivity of optical readout because of higher R_BP (Figs. 2(e) and 2(f)). From the polar plots of f_res values with respect to the pre-determined crystal orientation (illustrated by the ‘purple (ZZ)’ and ‘brown (AC)’ arrows), it can be inferred that main axes of the polar plots are always aligned with ZZ direction, which can be used as a criterion for determining crystal orientation of a given black P sample (Figs. 4(b) and 4(d)).

Fig. 4 Resonance frequency shift with respect to varying polarization. (a) Measured f_res upshifts upon change of polarization angle from 0° (AC) to 90° (ZZ) for D#1 (P_las = 0.7 mW). (b) Measured f_res values plotted in polar plot with respect to the pre-determined crystal orientation for D#1. (c) and (d) Similar f_res shift and polar plots for D#2 (P_las = 1.3 mW).

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In our sensing scheme, we envision that after calibrating the resonator for different polarization angles of incident laser illumination, we can detect the optical rotation of polarization of light while passing through any chiral molecule solution by monitoring the resonance frequency shift. If the initial polarization angle is calibrated at ϕ = 45°, left or right rotation of polarization (α) from chiral molecules can be detected by upshift and downshift of resonance frequency, respectively, where −45°≤ α ≤45°. We obtain optical rotation or polarization responsivity (defined as $ℜ = (f_{res}^{\vec{E} ||ZZ} - f_{res}^{\vec{E} ||AC}) / 90^{\circ}$ ) of 5.5kHz/degree (P_las = 0.7mW) and ℜ = 6.6 kHz/degree (P_las = 1.3mW) from D#1 and D#2, respectively. In addition to the devices presented in Fig. 4, we have measured two more devices, D#3 (t_BP≈20nm and d = 9µm) and D#4 (t_BP≈70nm and d = 8µm), which show ℜ ≈4.1kHz/degree (P_las = 0.2mW at λ = 532 nm) and ℜ≈6.4kHz/degree (P_las = 0.7mW for λ = 532 nm), respectively. Due to the excellent frequency resolution of nanomechanical resonators, we can envision highly sensitive polarization sensors. The ultimate polarization sensitivity or angle resolution for resonant polarimetric applications is determined by the minimum resolvable f_res shift of the resonator. Thanks to the excellent frequency stability of nanomechanical resonators, it is promising to achieve excellent angle resolution. If we assume fractional frequency resolution of <δf_res/f_res> ~10⁻⁵ to 10⁻⁶ corresponding to minimal resolvable frequency shift of δf_res,min ~10−100Hz for a resonator with f_res = 10MHz, which is in reality limited by frequency fluctuations from intrinsic thermomechanical noise and other external noise processes, then by using our calibrated responsivity (ℜ = 6.6kHz/degree for D#2), we can estimate sensitivity or angle resolution of δϕ_min = δf_res,min/ℜ ~0.0015°−0.015°. This is already comparable to, or even better than, the commercial benchtop polarimeters that exhibit angle resolution of ~0.001°−0.1°. By engineering the device structure and incident laser power, we can also further improve the angle resolution or sensitivity.

Now we focus on exploring the device structure to maximize responsivity ℜ. This responsivity can be increased by optimizing several parameters (e.g., thickness of black P, diameter and built-in tension of the resonator). According to the model, we can calculate the shift of f_res under the change of polarization of incident light (λ = 633nm, unless stated otherwise). At first, we present the thickness dependent (from 10 to 100nm black P) absorption in black P under incident polarization along AC and ZZ (Fig. 5(a)). It shows that for g = 900nm, highest difference in absorption is obtain near 85−90nm thickness of black P. However, optical responsivity of a drumhead resonator strongly depends on whether the resonator operates in membrane or plate regime. In membrane regime, f_res is highly sensitive to built-in tension, ℜ will be significantly higher. Therefore, the highest ℜ will be determined by both absorption difference depending on the polarization and operation regime of the resonator. From our calculation, we obtain the thickness dependent fundamental f_res scaling for 8 and 9 µm diameter microtrenches (Fig. 5(b)). These plots elucidate the membrane and plate regimes of the resonators, where D#2 and D#3 fall into membrane regime, while D#1 and D#4 fall into plate regime (Fig. 5(b)). Under fixed laser power, we calculate f_res with respect to the thickness of black P under two polarization conditions ( $\vec{E}$ || AC and $\vec{E}$ || ZZ) of incident light in Fig. 5(c). Moreover, Δf = $f_{res}^{\vec{E} ||ZZ} - f_{res}^{\vec{E} ||AC}$ also increases with P_las as absorption is higher when $\vec{E}$ || AC, which dictates higher amount of f_res shift upon the increase of P_las (Fig. 5(d)). Next, we investigate the dependence of Δf on the diameter of the microtrench and pre-tension level in the drumhead resonator (Figs. 5(e) and 5(f)). It has been found that Δf or ℜ is higher for a drumhead with larger diameter and lower pre-tension. From these results, it is also obvious that ℜ is significantly higher for thinner black P membrane resonator.

Fig. 5 Optimization of the polarization responsivity. (a) Dependence of absorbance with respect to t_BP for λ = 633 nm. (b) Scaling of f_res with respect to the t_BP for d = 8µm and d = 9µm. (c) Variation of f_res depending on the polarization for different thicknesses of black P. (d) Difference in f_res shift with respect to P_las depending on the polarization. (e) and (f) Plots depicting effects of d and γ₀ in the resonator on Δf versus t_BP, respectively.

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From the perspective of emerging and future applications, the polarization sensitive black P resonators have potential to be harnessed as sensors for detecting the optical activity or rotation of solutions of chiral molecules. By utilizing nanophotonics (e.g., metamaterials) and black P itself for fabricating on-chip polarizers [6] and implementing all-electrical readout of nanomechanical resonances [25], it is possible to realize highly miniaturized and ultrasensitive polarimeters, in contrast to the state-of-the-art large benchtop polarimeters that consists of costly and bulky optical components. In order to quantitatively estimate the optical rotation detection capability of the resonators mentioned here, we introduce the following equation to calculate the optical rotation, $α = {[α]}_{λ}^{T} c l$ , where, ${[α]}_{λ}^{T}$ is the specific rotation in a particular wavelength (λ) and temperature (T), c is concentration of the chiral molecules in solution and l is optical interaction length [11]. To provide a practical example, we consider a D-glucose solution with ${[α]}_{633nm}^{25^{\circ} C}$ = 4.4662°cm²g⁻¹ [Ref. 26], c = 0.9g/mL and l = 1 cm. From the responsivity measured from D#2, we shall be able to detect the optical rotation of 4.02° from a f_res shift of 26.53kHz, which is readily measurable in D#2.

5. Conclusions

In summary, we have designed, demonstrated and analyzed a new type of polarization sensitive devices based on ultrathin black P vibrating nanomechanical resonators, by harnessing the unique, strong intrinsic anisotropic properties of black P in a device platform. Enabled by the polarization sensitive nature of black P crystals and ultrasensitive optical interferometric readout of nanomechanical resonances, we have also demonstrated the potential for polarimetric applications in sensing optical rotation of optically active chiral molecules in solutions. This study provides a stimulating research direction to investigate miniaturized polarimeters, which can be an important tool in chemical and drug industry. From material science perspective, polarization sensitive resonance frequency shifts can also be exploited to identify unknown crystal orientation of black P unambiguously, and can be extended to nondestructively examining other anisotropic crystals.

Funding

National Science Foundation (ECCS-1454570, ECCS-0335765).

Acknowledgments

We thank the financial support from the National Science Foundation (NSF) through the CAREER Award (Grant #: ECCS-1454570). Part of the device fabrication was performed at the Cornell Nanoscale Science and Technology Facility (CNF), a member of the National Nanotechnology Infrastructure Network (NNIN), supported by the National Science Foundation (ECCS-0335765).

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Polarization sensitive black phosphorus nanomechanical resonators

Abstract

1. Introduction

2. Fabrication and characterization

3. Theoretical modeling and measurements

4. Results and discussions

5. Conclusions

Funding

Acknowledgments

References

Cited By

Figures (5)

Equations (7)

Optical Materials Express