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Optical Anisotropy of Few-Layer Black Phosphorus Visualized by Scanning Polarization Modulation Microscopy Hu Jiang, Hongyan Shi, Xiudong Sun, and Bo Gao ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b00341 • Publication Date (Web): 02 May 2018 Downloaded from http://pubs.acs.org on May 5, 2018

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Optical Anisotropy of Few-Layer Black Phosphorus Visualized by Scanning Polarization Modulation Microscopy Hu Jiang1, Hongyan Shi1,2, Xiudong Sun1,2, Bo Gao1,2* 1

Institute of Modern Optics, Department of Physics, Key Laboratory of Micro-Nano

Optoelectronic Information System, Ministry of Industry and Information Technology, Key Laboratory of Micro-Optics and Photonic Technology of Heilongjiang Province, Harbin Institute of Technology, Harbin 150001, China 2

Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan

03006, China

KEYWORDS: Black Phosphorus, Optical Anisotropy, Scanning Polarization Modulation Microscopy, Complex Refractive Index.

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Abstract

In-plane anisotropy, as one of the most intriguing properties of newly rediscovered 2D black phosphorus (BP), supplies another degree of freedom to design novel optical and optoelectronic devices. Here, the optical in-plane anisotropy of few-layer BP was directly visualized and studied by scanning polarization modulation microscopy (SPMM). The polarization analysis and experiment showed that the SPMM signal was dependent on the polarization angle of the laser beam in a sinusoidal way, by which the crystallographic orientation of BP was determined. The differential reflectance (∆R = Rzz - Rac) showed the excitation wavelength and thickness dependence, which was well explained by considering the anisotropic multi-reflection effect. The intrinsic anisotropic complex refractive indices (n + iκ) were also derived on the basis of the Fresnel equation.


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Black phosphorus (BP), the most stable allotrope of phosphorus, has been rediscovered as a new 2D layered material due to its superior electrical and optical properties such as its high carrier mobility and tunable moderate direct band gap.1-3 Compared to most other 2D materials, BP possesses the unique in-plane anisotropic nature arising from its puckered atom structure with two non-equivalent directions in the layer plane: armchair (ac) and zigzag (zz). Previous studies have revealed a range of anisotropic behaviors of BP in terms of the optical,1,

4-5

electrical,2-3,

6-7

thermoelectric,8-9 and mechanical aspects6, 10-11. In addition, these in-plane anisotropic properties can be effectively modulated by applying biaxial or uniaxial strain12-13 or electric field14-15 to the BP sample. Therefore, the unique anisotropic nature of this intriguing material is expected to play a critical role in designing multifunctional and adjustable 2D novel electronic, optoelectronic, and nano-mechanical devices that are not possible with other isotropic 2D layered materials. The comprehensive exploration of the anisotropic optical properties of few-layer BP is needed to lay a solid foundation for the application of this novel 2D material in photonics and optoelectronics. The refractive index, including real and imaginary parts, is the most important optical constant. BP is a birefringence and linear dichroism material, in which the birefringence means the dependence upon polarization of the real part of the refractive index and the dichroism is the variation of the imaginary part. Despite extensive theoretical predictions4, experimental investigations

7, 19-20

16-18

and some

on the optical anisotropy of BP, the quantitative

complex optical constants of few-layer BP still exist controversial. Many sets of 3 ACS Paragon Plus Environment

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theoretical values of permittivity can be found for bulk17-18 and mono or few-layer BP4, 16, 21 which led to an unexpectedly high refractive index at 532 nm, n ≈ 3.5-4.5,21 compared with the values measured on few-layer BP (n ≈ 3-3.420). These studies motivate us to develop new methods to interrogate the optical anisotropy of few-layer BP and to obtain more precise refractive index. Traditionally, reflected or transmitted optical signals at two crystal axes directions were collected separately and then compared to obtain the differential reflectance or transmittance, which may have large errors due to the extremely weak optical signal of 2D materials.7, 19, 20, 22 To decrease the errors, new methods were developed. E.g., Reflectance anisotropy microscopy (RAM), also called reflectance difference microscopy (RDM), measures the differential reflectance normalized to the mean reflectance, not the differential reflectance itself.6 Phase interference technology, based on phase modulation via a liquid crystal variable retarder19 or a Mirau interferometer23, was used to identify the crystalline axes and measure optical phase anisotropy of BP. However, this technology, which can only be implemented at low frequencies f, suffers from 1/f noise. As it has been demonstrated extensively in the optical metrology, high frequency modulation techniques could achieve ultrahigh sensitivity optical signal particularly in the weak signal detection. Therefore, it will be very necessary to integrate the high frequency modulation technique into those new methods. Scanning polarization modulation microscopy (SPMM) is one of these methods, in which high frequency (f = 50 kHz) polarization modulation is achieved using photoelastic modulator (PEM).24-25 Besides, there are some other advantages. 4 ACS Paragon Plus Environment

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First, extremely low laser fluence is needed, thus avoiding heat damage. Second, SPMM offered much higher sensitivity due to background-free detection. Third, it is a micro-zone method, suitable for small samples, for example fabricated by mechanical exfoliation. Fourth, the differential reflectance or transmittance images can be directly collected.24-25 Therefore, SPMM could be a potential method to study the optical anisotropy of few-layer BP. Here, few-layer BP with thickness of couples to tens of microns was imaged and its optical anisotropy was studied by SPMM. The polarization analysis and experiment showed that the SPMM signal was dependent on the polarization angle of the laser beam in a sinusoidal way, by which the crystallographic orientation of BP was determined. Thickness and wavelength dependence of differential reflectance was found after imaging many different BP samples at four different excitation wavelength, and was well explained by considering the anisotropic multi-reflection effect. The intrinsic anisotropic complex refractive indices (n + iκ) were derived on the basis of the Fresnel equation. Experimental Section Figure 1a shows the diagram of SPMM setup. The 532/1064 nm output of a DPSS continuous-wave diode pumped laser (DPL) (Cobolt Rumba™ 3000) and 400/800 nm output of Ti-Sapphire oscillator were used. The 45° polarized laser beams generated by passing through a Glan-Taylor polarizer went through PEM (Hinds Instruments, I/FS50, 50 kHz), which was operated in half-wave retardation mode to switch the polarization of the light beam from + 45° to - 45° at 100 kHz. After that, the laser 5 ACS Paragon Plus Environment

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beam was guided into an inverted optical microscope (Nikon Ti-U) and focused onto the few-layer BP sample using an oil immersion objective (Olympus 100x, 1.4 NA). The reflected beam was detected by an amplified Si or InGaAs photodiode (Thorlabs, PDA36A, PDA20CS) and fed into a lock-in amplifier (Stanford Research Systems, SR830), which was referenced to the 2f output of the PEM controller (Hinds Instruments, PEM-100). A half-wave plate was placed before the microscope to change the polarization of the laser beam. The few-layer BP sample was scanned by using a closed-loop 3D piezo scanning stage (MCL, NanoDrive).

Figure 1. (a) Schematic diagram of scanning polarization modulation microscopy (SPMM) setup. The sample coverslip was mounted on the piezo stage at the focus of a 100x, 1.4 NA oil objective. The objective is also used to collect the reflected light beam sent to a photodiode (PD). PEM = photoelastic modulator; Pol = Glan-Taylor polarizer; λ/2 = half-wave plate; BS = beam splitter. (b) Coordinate system for the analysis of the SPMM experiments. i and j are parallel to the axes of PEM. Crystallographic directions are defined as follows: eac is along the armchair (puckering) direction, and ezz is along the zigzag direction. E+ is the polarization


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direction of the incident laser field. β is the angle of armchair direction measured anticlockwise from the i axis. Few-layer BP was mechanically exfoliated from bulk BP crystals (Smart Elements) by using polydimethylsiloxane (PDMS). To minimize the influence of Fabry–Perot interference of the substrate, 0.17 mm glass coverslip was used to support few-layer BP. In our experiment, few-layer BP sample was not isolated by nitrogen or other inert materials, because no detectable change of signal was found when exposed to air. The thickness of BP samples was measured by atomic force microscope (AFM, NT-MDT NEXT). The Raman spectra were collected with a NT-MDT NTEGRA Spectra Raman system with 532 nm excitation and laser power at sample below 0.2 mW to avoid laser induced heating. All measurements were performed at ambient temperature and humidity smaller than 10%. Results and Discussion Scanning Polarization Modulation Microscopy. To interpret the following results of these experiments, we first analyze the polarization state of laser beam before and after interacting with the BP samples, which has been used in molecules26 and 1D nanomaterials.27 In order to better describe the polarization, incident light was considered as a superposition of two orthogonal polarized lights. In these analyses, a coordinate system (i, j) was chosen (shown in Figure 1b), where i and j are parallel to the axes of PEM, respectively. eac and ezz denote the armchair and zigzag directions of BP crystal, respectively. Consider a BP sample whose armchair axis eac has an


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arbitrary angle β with respect to the PEM i axis. The laser beam was adjusted to be polarized at 45° to the PEM axis, so the electric field E+ after the PEM is

E+ =

(

E0 i + e− iφ (t ) j 2

)

(1)

where E0 is the electric field intensity before PEM and ∅(t) is the phase shift modulated by the PEM. So the beam after the PEM can be regarded as a combination of a time-modulated beam (along j axis) and a constant polarization beam (along i axis). Afterwards, the laser beam passed through a half-wave plate to change its polarization. For simplicity, we define the polarization angle θ as the changed angle of the constant polarization components by clockwise rotating the half-wave plate. Then the electric field before the BP can be split into ezz and eac polarized components: E + = Ezz+ + Eac+ =

(

)

(

)

E0  sin(β + θ ) + e-iφ (t )cos(β + θ ) ezz + cos(β + θ ) − e-iφ ( t )sin(β + θ ) eac    2

(2)

The total intensity of reflected light field E- is the superposition of the intensity of reflected light field at zigzag axis and armchair axis, and can be simply expressed as E − = E zz− + Eac− . Superscript “+”/“-” represent the incident and reflected beam.

Because of the birefringence property of BP crystals, reflected light will transform from one elliptical polarization to another elliptical polarization. The light-BP interaction is remarkably different between zigzag and armchair crystal axes, which leads to polarization-dependent absorption. Therefore, BP has different reflectivity along the two principle crystal axes. The ratio between the reflected and incident electric fields is the reflection coefficient ( rzz = E zz− / E zz+ , rac = E ac− / E ac+ ) and the square of the ratio is the reflectivity Rzz = ( rzz ) ， Rac = ( rac ) . Thus, the total 2

2

intensity of reflected light field can be simply expressed by: 8 ACS Paragon Plus Environment

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I ( t ) = ( E zz− + Eac− )( E zz− + Eac− )

*

2

= rzz E zz+ + rac Eac+ =

2

(3)

I0 I ( Rzz + Rac ) + 0 sin2( β + θ ) × ( Rzz − Rac ) × cosφ ( t ) 2 2

where I0 is incident laser intensity. To calculate the time dependence of the detector signal,

cosφ ( t )

needs

to

cosφ ( t ) ≈ J 0 (φ0 ) − 2J 2 (φ0 ) sin 2ωmt

be

expanded

in

a

Fourier

series

as

utilizing Bessel functions.26 By setting the

lock-in amplifier reference as double modulated frequency, the first term can be removed and only the 2ωm time-dependent component of intensity can be detected:

Vdet ( 2ωm ) = kI 0 J 2 (φ0 ) × sin[2( β + θ )] × ∆R

(4)

where ∆R ≡ Rzz − Rac is the differential reflectance of BP and kI 0 J 2 (φ0 ) is an instrumental factor that can be measured by placing a polarizer between the PEM and the detector. Specifically, when this polarizer is rotated to -45° with respect to the axis of PEM, then Vref ( 2ωm ) = Rs kI 0 J 2 (φ0 ) . Rs is the reflectivity of the glass substrate. Then the normalized signal Vd = RsVdet / Vref can be simply expressed as:

Vd = sin[2( β + θ )] × ∆R

(5)

It can be seen that Vd periodically changes with θ, and equals ∆R when β + θ = 45°. Thus, by measuring Vd at different θ, we can obtain ∆R and the orientation of BP. The Differential Reflectance Image of Few-layer BP. Figure 2a and 2b shows the optical and AFM images of a piece of few-layer BP on glass substrate. It can be seen from both images that there are two different thicknesses. The AFM section analysis indicates that the thickness of the BP sample was approximately 14 nm and 30 nm, respectively. Angle-resolved polarized Raman spectroscopy (ARPRS) was carried out 9 ACS Paragon Plus Environment

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on the BP sample to determine the crystalline orientation.28-32 Raman spectra was collected with 532 nm excitation under the parallel configuration. Figure 2c shows some representative polarized Raman spectra of BP. The intensities of the three characteristic Raman modes A (363 cm−1), B (438 cm−1) and A (466 cm−1) varied periodically with the change of the polarization of excitation laser, which is consistent with literature.28-32 The polar plot of the A mode in Figure 2d shows the maximum when the polarization angle of excitation laser was ∼-20°. When the thickness is 30 nm and the excitation wavelength is 532 nm, the intensity of the A mode would achieve the maximum when the excitation laser polarized parallel with the armchair direction of BP.28,30 Therefore, the armchair direction of the BP sample is ∼-20°. Accordingly, the zigzag direction is perpendicular to the armchair and is along the second maximum at ∼70°.28-32


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Figure 2. (a) Optical microscope (OM) (b) AFM images of a piece of few-layer BP on glass substrate. The dashed red arrows indicate the identified armchair and zigzag directions, respectively. (c) Polarized Raman spectra of the BP sample excited with different polarization angles. (d) Polar plot of the fitted intensity of A modes in (c). Figure 3 shows a series of Vd images of the BP sample (shown in Figure 2) with θ from 0° to 170°. It can be seen that Vd images are highly dependent on θ due to the anisotropic structure of BP. Vd is positively largest when the polarization angle between 60° and 70°. Vd is negatively largest when the polarization angle between 150° and 160°. To clearly see the variations of Vd, we plotted normalized Vd at the green point indicated in Figure 2a versus polarization angle (shown in Figure S1a of Supporting Information). According to Eq 5, Vd is a sinusoidal function of θ, so sinusoidal function was used to fit normalized Vd in Figure S1. It can be seen that the fit curve fits perfectly with the experiment data, being maximum at θ = 66°. Given that Vd is maximum when β + θ = 45°, we can infer that the armchair direction β is -21°, which is consistent with ARPRS result. By adjusting θ = 66°, we directly obtained the ∆R image, which has the largest contrast (shown in Figure S1b of Supporting Information).


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Figure 3. Vd images of BP with polarization angles of the laser beam from 0 to 170°. Wavelength and Thickness Dependence of ∆R. It can also be seen in Figure S1b that, ∆R was different for BP with thickness of 14 nm and 30 nm, indicating the thickness dependence of optical anisotropic. In order to reveal the relationship between the thickness and the optical anisotropic of BP, we investigated another piece of few-layer BP. Figure 4a shows the OM image of the BP, the crystalline orientation of which was identified by ARPRS and is indicated by the dashed red arrows. Figure 4b shows the AFM image, in which eight different thicknesses are indicated by numbers on the image. Figure 4c shows the ∆R image excited by 532 nm laser at β + θ = 45°. It can be seen that the BP of each thickness showed its pattern, meaning high dependence of ∆R on the thickness. Meanwhile, ∆R did not vary monotonously with the thickness, but changed in a complicated way. Particularly, BP flakes with size smaller than one micron can also be observed and investigated due to the high frequency modulation in SPMM, which is very difficult in previous studies.19-20


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The thickness dependence is an important property of the optical anisotropy of few-layer BP. The strong optical anisotropy is a direct consequence of the differently orientated transition dipole moments due to the puckered structure of BP. According to the symmetry of the point group of BP, the possible transition in Vis-NIR region only involved with armchair-polarized light. Therefore, optical absorption is different for different incident light polarizations in Vis-NIR region.32 The optical constant is the function of the dielectric constant which is often expressed in terms of the optical conductivity with an effective mass component.18, 33 The effective mass can usually be influenced by thickness.4,

18

Previous studies showed that interlayer interactions

reduce the anisotropy of few-layer BP.4, 19 In our experiment, the thickness of most BP samples ranged within tens of nanometers (hundreds of phosphorene layers). The interlayer interactions are thought to be not sensitive to the thickness and hence could be ignored. It is known that interference effects are remarkable due to multiple-reflection in optical experiments of 2D materials, which has been proved by lots of experiments.30,

34-36

Multi-reflection effect coming from multi-interfaces is

assumed to play a major role in the thickness dependence.


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Figure 4. (a) OM and (b) AFM images of another piece of few-layer BP. The crystalline orientation is indicated by the dashed red arrows. Eight different thicknesses are indicated by numbers on (b). ∆R images excited by (c) 532 nm and (d) 1064 nm laser at β + θ = 45°. All scale bars are 5 µm. Step size in ∆R images is 0.1 µm.

Figure 4d shows the ∆R image excited by 1064 nm laser at β + θ = 45°. It can be seen that the BP showed different ∆R image compared to that excited by 532 nm in Figure 4c. For example, ∆R of the region with thickness of 96 nm was the biggest at 532 nm, but extremely small at 1064 nm. ∆R of the region with thickness of 77 nm was very small at 532 nm, but quite big at 1064 nm. This means ∆R is dependent on the excitation wavelength. The wavelength dependence of ∆R is believed to be originated from the frequency dependent dielectric function and different interference effects for


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different wavelengths experienced in thin films. Two more different excitation wavelengths were also used to explore the relationship between wavelength and ∆R (see Figure S2 of Supporting Information). It can be seen from Figure 4 and Figure S2, for larger wavelength, ∆R has more insignificant contrast, meaning that the optical anisotropy decreased with wavelength increasing. To reveal the origin of the thickness and wavelength dependence of the optical anisotropy of BP, we carried out in-depth analyses of ∆R using the Fresnel equation.35-36 In our experiments, light is always normal incidence. However, as shown in the schematic diagram of Figure 5a, the oblique incidence was drawn to show the light path more clearly. In Figure 5a, Eij+ is the electric field in the ith medium at the i-j interface, where the superscript “+”/“-” denotes the incident/reflected light. By imposing the electromagnetic field boundary conditions at each interface, the relation between the four electric fields ( Eij+ , Eij− , E +ji and E −ji ) under the normal incidence can be described as:  Eij+  1  1  −  =   Eij  tij  rij

rij   E +ji    1   E −ji 

(6)

where t ij = ( 2ni ) / ( ni + n j ) and rij = ( ni − n j ) / ( ni + n j ) are the transmission and reflection coefficients at the i-j interface, respectively. Subindex i, j = (0, 1, 2) represents the glass, BP and air layers. In the same medium, the relation between the four electric fields at the upper and bottom interfaces can be expressed by:

 Eij+   eiδ  −  =   Eij   0

+ 0   Eij +2   −  e− iδ   Eij +2 

(7)


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where δ = 2π n% d / λ is the phase factor and d is the thickness of BP. Because the refractive index of oil we used in oil immersion objective is close to that of glass, we simply consider glass, oil and objective as a whole. Then in our system there are just three media and two interfaces as shown in Figure 5a. Hence, the final formula that describes the incident and reflected electric fields on the three media is:  E01+  1  1  − =   E01  t01  r01

r01   eiδ  1   0

0 11  − iδ  e  t12  r12

r12   E21+    1   0 

(8)

Eliminating variable E21+ , we can get the relation of incident and reflected light field:

E01− =

r01eiδ + r12e− iδ + E01 eiδ + r01r12e− iδ

Note that the coefficient of the equation is just reflectivity r =

(9)

r01eiδ + r12 e− iδ . Let us eiδ + r01r12 e− iδ

just consider the zigzag polarized component: *

 r eiδ zz + r12 zz e− iδ zz  r01zz eiδ zz + r12 zz e− iδ zz  Rzz = rzz × r =  01iδzzzz − iδ zz  iδ zz − iδ zz   e + r01zz r12 zz e  e + r01zz r12 zz e  * zz

where r01 zz =

ng − nzz − iκ zz ng + nzz + iκ zz

and r12 zz =

(10)

nzz + iκ zz − na are the reflection coefficients of nzz + iκ zz + na

BP for zigzag polarized component relative to glass and air; δ zz = 2π n% zz d / λ is the phase factor for zigzag polarized component; ng and na is refractive index of glass and air, respectively. For armchair polarized component, Rac has the same form. Finally, we get the relation of d, λ, nzz, nac, κzz, κac and ∆R. Because ∆R is a function of thickness of BP and wavelength of light, for a given thickness (wavelength), ∆R will change with wavelength (thickness). This could explain the differences in ∆R of the different regions of the BP sample at same wavelength and the differences in ∆R of the same region of the BP sample at different wavelength in Figure 4. 16 ACS Paragon Plus Environment

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To calculate the real and imaginary parts of the anisotropic refractive index (nzz, nac, κzz and κac), ∆R for 532 nm and 1064 nm and corresponding thickness was extracted from Figure 4b, and ∆R for 400 nm and 800 nm was measured on other BP samples with known thickness (see Figure S3 of Supporting Information). Table 1 shows the complex refractive indices of few-layer BP along armchair and zigzag directions at 400, 532, 800 and 1064 nm. Both n and κ show a signiﬁcant diﬀerence between armchair and zigzag directions, indicating intrinsically large optical anisotropy of BP. n is larger along the zigzag direction than that along the armchair direction for all four wavelengths, while κ along armchair direction is always larger than that along zigzag direction. Moreover, both n and κ increase with wavelength decreasing. These results are reasonably comparable to the reported results (shown in Table S1 of Supporting Information).4, 20, 33

Table 1. Complex refractive index of few-layer BP at four different wavelengths

n% zz

n% ac

400 nm

532 nm

800 nm

1064 nm

3.94(±0.15) –

3.56(±0.12) –

3.26(±0.12) –

3.11(±0.13) –

0.307(±0.012)i

0.126(±0.004)i

0.102(±0.004)i

0.087(±0.004)i

3.55(±0.14) –

3.29(±0.12) –

3.19(±0.12) –

2.99(±0.13) –

0.958(±0.036)i

0.428(±0.015)i

0.291(±0.011)i

0.276(±0.012)i


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Figure 5. (a) Schematic diagram of optical reflection and transmission for thin multi-films system: BP on glass. (b) ∆R as a function of the thickness for four different wavelengths. Symbols are experiment data. Curves are plotted by using complex refractive index in table 1. Purple/green/pink/red colors are for the wavelength of 400/532/800/1064 nm.

By using the complex refractive index in table 1, ∆R was plotted as a function of the thickness for wavelength of 400, 532, 800 and 1064 nm in Figure 5b. It can be seen that ∆R oscillated with thickness increasing for 400 nm and 532 nm, but the oscillation did not occur for 800 nm and 1064 nm, which we believe will happen when further increasing the thickness. Meanwhile, ∆R is generally larger in visible region than near infrared region, which indicates larger optical anisotropy at visible 18 ACS Paragon Plus Environment

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region.4, 7, 20 We also measured more ∆R on BP with known thickness for the four wavelengths (see Figure S3 of Supporting Information), and plotted in Figure 5b. It can be seen that these experiment data well lied on the corresponding curves.

Conclusions In conclusion, optical anisotropy of few-layer BP was directly visualized by SPMM. The SPMM signal Vd was highly dependent on the polarization direction of the laser beam, and equals differential reflectance ∆R when β + θ = 45°, by which the crystallographic orientation was determined. The SPMM signal, including ∆R, was found to be dependent on the thickness of BP, based on anisotropic complex refractive indices along ac and zz directions were obtained via Fresnel equation. Meanwhile, SPMM images excited at four different wavelengths showed that ∆R is generally larger at visible region than near infrared region, indicating larger optical anisotropy in visible region. This study showed that SPMM could be used to explore the fundamental optical constants and properties of arbitrary anisotropic 2D materials with sub-micron size.

ASSOCIATED CONTENT

Supporting Information

The Supporting Information is available free of charge on the ACS Publications website at http://pubs.acs.org. Relationship between Vd and polarization angles of the 19 ACS Paragon Plus Environment

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laser beam (Figure S1). Excitation wavelength dependence of ∆R (Figure S2). Thickness characterization of BP samples (Figure S3). Comparison of refractive index of BP (Table S1).

AUTHOR INFORMATION

Corresponding Author *E-mail: [email protected]

Author Contributions B. G. initiated the idea. H. J. and H. S. performed the experiments. H. J., X. S., H. S. and B. G. analyzed data, interpreted data, H. J. and B. G. wrote the manuscript. B. G. supervised the project.

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENT This work was financially supported by the National Natural Science Foundation of China (Nos. 21473046 and 21203046).

ABBREVIATIONS BP

SPMM

black phosphorus

scanning polarization modulation microscopy


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ac

armchair

zz

zigzag

∆R

differential reflectance

ARPRS Angle-resolved polarized Raman spectroscopy.

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