Anisotropic buckling of few-layer black phosphorus

When a two-dimensional material, adhered onto a compliant substrate, is subjected to compression it can undertake a buckling instability yielding to a periodic rippling. Interestingly, when black phosphorus flakes are compressed along the zig-zag crystal direction the flake buckles forming ripples with a 40% longer period than that obtained when the compression is applied along the armchair direction. This anisotropic buckling stems from the puckered honeycomb crystal structure of black phosphorus and a quantitative analysis of the ripple period allows us to determine the Youngs's modulus of few-layer black phosphorus along the armchair direction (EbP_AC = 35.1 +- 6.3 GPa) and the zig-zag direction (EbP_ZZ = 93.3 +- 21.8 GPa).

Interestingly, when black phosphorus flakes are compressed along the zig-zag crystal direction the flake buckles forming ripples with a 40% longer period than that obtained when the compression is applied along the armchair direction.This anisotropic buckling stems from the puckered honeycomb crystal structure of black phosphorus and a quantitative analysis of the ripple period allows us to determine the Youngs's modulus of few-layer black phosphorus along the armchair direction (EbP_AC = 35.1 ± 6.3 GPa) and the zig-zag direction (EbP_ZZ = 93.3± 21.8 GPa).
Since its isolation in 2014, [1][2][3][4][5][6] few-layer black phosphorus (bP) keeps attracting the interest of scientific community because of its remarkable electronic (i.e.ultrahigh charge carrier mobility, ambipolar field effect, etc) [7][8][9][10] and optical (i.e.0][31][32][33][34][35][36][37][38]  https://pubs.rsc.org/en/content/articlelanding/2019/nr/c9nr03009c#!divAbstract ambient conditions. 32,39,40In the previous works, the mechanical testing methods used require exposing the flakes to air for relatively long periods of time and in some cases the studied flakes have been even subjected to several wet-chemistry microfabrication steps 28 or exposed to electron beam irradiation. 24Therefore, there is a need for a technique that allows to study the intrinsic mechanical properties of pristine black phosphorus flakes right after their exfoliation. 6][47][48][49] Interestingly, when the black phosphorus flakes are subjected to uniaxial compression, they tend to buckle forming ripples predominantly parallel to the zig-zag (referred to as ZZ hereafter) crystal axis of the black phosphorus lattice, allowing us to determine the Young's modulus along the arm-chair (referred to as AC hereafter) direction.Then, we performed control experiments where the same black phosphorus flake is compressed along the AC and ZZ directions finding that, when compressed along the ZZ direction, the flake buckles forming ripples with a period about ~ 1.4 times longer that obtained for compression along the AC direction.Note that in previous buckling metrology measurements on other 2D materials, like graphene or transition metal dichalcogenides, the controlled compression aligned along well-defined crystal directions were not reported.A quantitative analysis of the black phosphorus buckling allows us to determine the Youngs's modulus of few-layer black phosphorus along the AC direction (EbP_AC = 35.1 ± 6.3 GPa) and the ZZ direction (EbP_ZZ = 93.3± 21.8 GPa).
Black phosphorus flakes have been buckled by transferring them onto a compliant elastomeric substrate that has been initially stretched.After the flake transfer the substrate pre-stress is released exerting a uniaxial compression on the black phosphorus flakes.As elastomeric material we employ Gel-Film (WF x4 6.0 mil by Gel-Pak) and we apply the pre-stress by bending it (See Figure 1a).Black phosphorus flakes are exfoliated from bulk black phosphorus crystals (HQ Graphene) with Nitto tape (Nitto Denko corp.SPV224) and directly transferred onto the surface of the stretched Gel-Film substrate.After the transfer of the flake, we release the stress by unbending the Gel-Film substrate, and the sample is inspected under an optical microscope to find the buckled black phosphorus flakes.Figure 1b shows a transmission mode optical microscopy image of a few-layer https://pubs.rsc.org/en/content/articlelanding/2019/nr/c9nr03009c#!divAbstract black phosphorus flake displaying a clear periodic ripple pattern that arises from the interplay between the flake buckling and the flake-substrate adhesion interaction.
Using a (linear) polarization analyser during the optical microscopy inspection one can easily determine the crystal orientation of the flakes due to the strong linear dichroism of black phosphorus.1][52][53] Figure 1c shows the dependence of the red, green and blue image channels transmittance (calculated as the light intensity transmitted though the black phosphorus flake divided by the light intensity transmitted though the bare substrate) as a function of the angle between the linear polarizer direction and the horizontal axis (i.e.0º means polarization parallel to the horizontal axis and 90º means polarization parallel to the vertical axis).The angular dependence of the transmittance follows the Malus' law (T ∝ cos 2 (θ+δθ)) as expected for a material with a strong linear dichroism.From the angle values where the transmittance reaches the maximum and the minimum one can determine the ZZ and AC directions of the black phosphorus flake respectively.These directions are displayed in Figure 1b making it possible to determine the relative orientation of the ripples with respect to the crystalline directions: the ripples are oriented almost parallel to the ZZ crystal orientation.
According to theoretical calculations, the Young's modulus of black phosphorus is expected to be ~3-4 times larger along the ZZ than along the AC [54][55][56][57] and thus it results energetically favourable to bend the black phosphorus lattice along the ZZ axis.Moreover, a lower Young's modulus along the AC direction also means that the compression stress needed to buckle the black phosphorus along that direction is also lower than that needed to buckle it along the ZZ direction. 42,43,58These reasons explain why the black phosphorus flakes preferentially buckle forming ripples parallel to the ZZ direction.Figure 1c presents a polar histogram of the relative angle between the ZZ direction and the ripple orientation where a marked preferential alignment along the ZZ direction is seen.
Due to the preferential alignment of the ripples parallel to the ZZ direction, the Young's modulus of black phosphorus along the AC direction can be determined by a quantitative analysis of the buckling-induced rippling period for black phosphorus flakes with different thicknesses.Indeed, there is a linear relationship between the ripple period and the flake thickness given by: 42,43,58 where h is the flake thickness, νs and νf are the Poisson's ratio of substrate and flake and Es and Ef are the Young's modulus of the substrate and flake respectively.Therefore, provided that one knows the values of the Poisson's ratio of black phosphorus (νbP_AC = 0.40, νbP_ZZ = 0.93), 59 and substrate (νs = 0.5) 60 and the Young's modulus of the substrate Es = 492 ± 11 kPa, 45 the Young's modulus can be readily determined from the slope of the linear relationship between the ripple period and flake thickness: Figure 2a shows four examples of rippled flakes with different thicknesses (ranging from 9 nm to 23 nm) that display a marked thickness-dependent period of ripple.The thickness of the flakes has been determined through quantitative analysis of the transmission mode optical images of the flakes (see the Electronic Supporting Information Figures S1 and S2 for details about the thickness determination procedure).As the whole measurement is carried out by optical microscopy, one can determine both the thickness of the flakes and the period of the ripples very quickly right after the exfoliation of black phosphorus (samples are exposed to air less than a 1 minute from their exfoliation to their experimental characterization) ensuring that this method provides the mechanical properties of pristine (not environmentally degraded black phosphorus).Note that in previous works on the mechanical properties of black phosphorus the studied flakes have been exposed to air for longer periods (atomic force microscopy was used to locate the flakes and to make forceindentation measurements) 24,25,28 and in some cases the flakes have even been subjected to several wet chemistry steps or electron beam irradiation (involved in the fabrication of freely suspended black phosphorus beams). 24,28We point the reader to the Electronic Supporting Information for a study about the role of environmental exposure on the buckling (and its in-plane anisotropy) of black phosphorus (Figures S3 and S4).In order to get an insight about the anisotropic mechanical properties of black phosphorus we transfer black phosphorus flakes onto a flat (unstrained) Gel-Film substrate, we determine their thickness and crystal orientation by optical microscopy (as detailed above) and we subject the same flakes to compressive strain along the AC and ZZ directions by pinching the surface of the Gel-Film with two glass slides as illustrated in Figure 3a.We compressed the Gel-Film substrate until the black phosphorus flake just starts to buckle and we stop compressing it further at that point (the approximate compression value is ~10%).Note that due to the large Young's modulus mismatch between the Gel-Film and black phosphorus this compression would translate into a much lower compressive strain on the flake.Flakes with relatively large areas (~4000 μm 2 ) with homogeneous thickness are selected for this experiment to facilitate the analysis.Figure 3b shows an example where the same flake is subjected to compressive strain along the AC and ZZ directions yielding to almost perpendicular ripple pattern with a sizeable different period (see the comparison between two line-profiles, for strain along ZZ and AC directions, in Figure 3c).Figure 3d summarizes the measured period in 4 different flakes upon compression along their ZZ and AC directions.Using Equation [2], one can determine the Young's modulus of black phosphorus along the ZZ direction from the Young's modulus value along the AC direction and the ratio between the ripple period for compression along the AC and ZZ directions (λAC/λZZ =1.39 ± 0.15): We obtain EbP_ZZ = 93.3± 21.8 GPa.Note that EbP_ZZ • νbP_AC ≈ EbP_AC • νbP_ZZ, as expected from the symmetries of the compliance/stiffness tensor in the strain vs. stress relationship.
In order to compare our results with the ones reported in the literature, we have summarized in Table 1 both theoretical and experimental results from the literature.Our 5][56] Regarding the previously reported experimental works, we first have to note that in references 25,27 the measurements were carried out by atomic force microscopy (AFM) indentation on circular drumheads and therefore it was not possible to probe the direction dependence of the Young's modulus.
In references 24,28 , on the other hand, the measurements were carried out by AFM indentation on freely-suspended doubly clamped beams which allowed to extract the Young's modulus along the AC and ZZ directions.In these works, however, the reported values (EbP_AC ~ 27 GPa and EbP_ZZ ~ 58-65 GPa) are noticeably lower than our obtained values.We must note that in these previous works, during the fabrication of the freelysuspended beams the black phosphorus flakes were exposed to several steps of wetchemistry and/or electron beam exposure which could effectively induce the degradation of black phosphorus thus affecting their mechanical properties.Alternatively to these nanoindentation works a recent work extracted the Young's modulus of a 95 nm thick black phosphorus of a circular drumhead mechanical resonator by comparing the frequencies and mode shapes of high order mechanical resonances. 26Interestingly, this nanomechanical resonator measurement on a 95 nm thick black phosphorus flake provides Young's modulus values (EbP_AC ~ 46 GPa and EbP_ZZ ~ 116 GPa) close to our results (obtained by buckling-metrology method) for 9-23 nm thick flakes.Moreover, among the experimental results on few-layer black phosphorus the one in reference 26 and our results based on the buckling metrology method are the ones that provide Young's modulus closer to those obtained from ultrasound velocity measurements in bulk black phosphorus crystals (EbP_AC ~ 55 GPa and EbP_ZZ ~ 179 GPa). Figure 4 shows a graphical comparison between the different reported values of Young's modulus and our experimental values.

Conclusions
In summary, we subjected few-layer black phosphorus flakes, deposited onto compliant elastomeric substrates, to uniaxial compressive strain yielding to their buckling and inducing a periodic rippling.The quantitative analysis of the period of the ripples for flakes with different thickness allows us to determine the Young's modulus of black phosphorus along the AC and ZZ directions: EbP_AC = 35.1 ± 6.3 GPa and EbP_ZZ = 93.3± 21.8 GPa.

Determination of the Young's modulus of the Gel-Film® substrate
The Young's modulus of the elastomeric substrate has been previously determined in reference [29] by force vs. elongation experiments (see the Supporting Information of reference [29]).

Optical microscopy
Optical microscopy images have been acquired with an AM Scope BA MET310-T upright metallurgical microscope equipped with an AM Scope mu1803 camera with 18 megapixels.The calibration of the optical magnification system has been carried out by imaging standard samples: one CD, one DVD, one DVD-R, and two diffraction gratings with 300 lines/mm (Thorlabs GR13-0305) and 600 lines/mm (Thorlabs GR13-0605).

Image analysis
The quantitative analysis of the transmittance of the flakes and the rippling wavelength has been carried out using Gwyddion® software.[39]   Thickness determination The thickness determination has been carried out by extracting the transmittance of the red, green and blue channel of the transmission mode optical microscopy images and comparing it with the results of reference (not-buckled) samples.See the Supporting Information for more details.riccardo.frisenda@csic.es, andres.castellanos@csic.es

Thickness determination through quantitative analysis of the transmission mode optical images of the flakes
The determination of the thickness of the black phosphorus flakes studied in the main text has been made by quantitative analysis of the transmission mode optical images of these flakes.
In order to determine the thickness through optical microscopy we acquire transmissionmode microscopy images of several flakes with different thicknesses on the surface of a Gel-Film substrate.Then these flakes are transferred onto a SiO2 substrate (performing AFM characterization on the Gel-Film substrate is rather challenging because of the large tip-Gel-Film adhesion force) and their thickness is measures with atomic force miscroscopy (AFM).All this process is summarized in Figure S1.With the above data collected, a thickness versus normalized transmission graph can be made for the three channels (red R, green G, blue B) of the camera.Then, a polynomial fit to the data is used to obtain a relationship between thickness and transmission.The results are shown in the Figure S2.By the graphs of Figure S2 we can obtain the thickness of a flake knowing the value of its normalized optical transmission, which makes this method quite comfortable.The error made by this approach is about ± 2 nm in the thickness.

Effect of the environmental degradation on the buckling of black phosphorus
2][3] This environmental degradation have been found to have an impact on the mechanical properties of black phosphorus but previous reports could not test the mechanical properties along different crystal orientations of the black phosphorus lattice. 4 have employed the buckling metrology method to test the effect of environmental exposure on the mechanical properties (and on its anisotropy) of black phosphorus.Figure S3(a) shows a sequence of transmission mode optical microscopy images of a buckled black phosphorus flake as a function of time.To get a deeper insight into the effect of the environmental induced degradation on the mechanical properties of black phosphorus we compare the results of the controlled buckling experiments (where the compression is applied in a controlled way along the ZZ and the AC directions) on pristine black phosphorus flakes and on aged flakes (after 9 days of air exposure).Figure S4(a) shows optical microscopy images of an aged black phosphorus flake (after 9 days of air exposure with an average relative humidity 20%)

Figure
Figure2bsummarizes the results, acquired for 22 flakes with thicknesses ranging 9 nm to 23 nm, that follows a linear relationship with a slope of 192 ± 9 from which one can determine the Young's modulus along the AC direction EbP_AC = 35.1 ± 6.3 GPa.

Figure 2 .
Figure 2. a) Optical microscopy images of ripples with different periods for 4 black phosphorus flakes of different thickness.b) Wavelength vs thickness graph for several black phosphorus flakes with different thickness, their error bars with 95% confidence curves.The solid blue line is the linear fit and the shaded area around indicates the uncertainty of the fit (95% confidence).

Figure 3 .
Figure 3. a) Process of ripples formation in the two directions (ZZ and AC) in a controlled way two manipulators to apply the compression along both directions.b) Optical microscopy images of black phosphorus compressed in both directions.The period of the ripple pattern sizeably depends on the direction where the compression is applied.The coloured arrows indicate the ZZ (blue) and AC (red) compression direction.c) Comparison between the ZZ and AC ripple period measured for different flakes.

Figure 4 .
Figure 4. Graphical comparison between the reported theoretical and experimental values of the Young's modulus in the literature (symbols) and our experimental results (solid horizontal lines, the shaded area indicates the uncertainty).The values along the AC direction are displayed at the left and along the ZZ at the right.

Figure S1. a )
FigureS1.a) Optical microscopy image of multi-layer black phosphorus flake.Different RGB channels for the same flake.b) AFM image of the same flake in a) after being transferred onto a SiO2/Si substrate.A topographic line profile over a region with some layers is included.

Figure S2 .
Figure S2.Thickness vs normalized optical transmission graph for some flakes and the polynomial fit with the 95% confidence curves for each channel a) Red b) Green c) Blue Figure S3(b) shows the line profile along the blue lines in (a) where the time evolution of the rippling can be easily observed.The degradation of the flake can be readily observed from the change in the flake transmittance as the generation of a phosphorus oxide crusts on the surface will reduce the optical absorption (while black phosphorus has a narrow gap, phosphorus oxides are wide band semiconductors).FigureS3(c) shows the line-profiles along the red lines in (a) where the time-dependent transmittance can be seen.FigureS3(d)summarizes the time evolution of the ripple wavelengths and transmittance measured in different spots of the sample.

Figure S3. a )
Figure S3.a) Sequence of transmission optical microscopy images of a buckled black phosphorus flake upon time exposure.b) Line profiles along buckling-induced ripples at different time exposures.c) Time evolution of the transmission of the black phosphorus flake.d) Summary of the time evolution of the ripple wavelength and the transmission (of the green channel) measured at different locations of the black phosphorus flake upon air exposure.Note: average relative humidity 20% during the experiment.
compressed along both AC (blue) and ZZ (red) directions.A direct comparison between the ripples profiles measured on a pristine flake and on the aged flake shown in Figure S4(a) is displayed in Figure S4(b).A summary of the anisotropy measured on pristine flakes and on two different aged black phosphorus flakes is shown in Figure S4(c).

Figure S4. a )
Figure S4.a) Optical microscopy images of an aged black phosphorus flake (after 9 days of air exposure with an average relative humidity 20%) compressed along both ZZ (red) and AC (blue) directions.b) Comparison between the ripples profiles measured on a pristine flake and on the aged flake.c) Comparison between the buckling anisotropy measured on pristine flakes and on two aged black phosphorus flakes.

Table 1 .
Summary of the reported valued (both theoretical and experimental) in the literature, indicating the method and conditions employed to obtain them.This experimental measurement did not allow to resolve the Young's modulus for different crystal orientations.