Supporting Information. Anisotropic Electron-Phonon Interactions in Angle- Resolved Raman Study of Strained Black

Transcription

1 Supporting Information Anisotropic Electron-Phonon Interactions in Angle- Resolved Raman Study of Strained Black Phosphorus Weinan Zhu,* 1 Liangbo Liang,* 2 Richard H. Roberts, 3 Jung-Fu Lin, 3,4 and Deji Akinwande 1,3 1 Microelectronics Research Center, Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78758, USA 2 Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA 3 Department of Materials Science and Engineering, Texas Materials Institute, The University of Texas at Austin, TX , USA 4 Department of Geological Sciences, Jackson School of Geosciences, 2305 Speedway Stop C1160, The University of Texas at Austin, Austin, TX , USA *Authors are equally-contributed deji@ece.utexas.edu 1

2 Figure S1. Mechanical robustness of polyimide substrate under uniaxial tensile strain up to 20%. Figure S2. Evolution of the Raman spectra of Ag 1, B2g, and Ag 2 modes in BP thin films under tensile strain along (a) zigzag and (b) armchair direction. The black dash lines are guides of the peak position. The 473 nm Raman system is in polarized configuration during the measurement. 2

3 Figure S3. Atomic force microscopy (AFM) image of a few layer BP sample after Raman measurement under strain. Here, tensile strain was released to take the AFM image. Buckles formed perpendicular to the strain loading direction owing to the crystal lattice contraction. Parallel cracks were resulted from the tensile strain loading. 3

4 Figure S4. DFT calculated frequencies of Ag 1, B2g, and Ag 2 modes as a function of the applied tensile strain along (a, c) ZZ and (b, d) AC directions of monolayer BP. In (a) and (b), the effect of the Poisson s ratio is considered, which corresponds to the experimental scenario. However, such effect is not considered in (c) and (d). 4

5 Figure S5. DFT calculated frequencies of Ag 1, B2g, and Ag 2 modes as a function of the applied tensile strain along (a, c) ZZ and (b, d) AC directions of bulk BP. In (a) and (b), the effect of the Poisson s ratio is considered, which corresponds to the experimental scenario. However, such effect is not considered in (c) and (d). 5

6 Figure S6. BP flakes with thickness of 3-5 nm showing anisotropic angle-resolved Raman characteristics. Incident laser wavelength is 473 nm. Dots are experimental data and black lines are the model fitting. Table S1. c/a ratio extracted for strained and unstrained BP thin films with different thickness. 6

7 INTERFERENCE EFFECT IN BLACK PHOSPHORUS Due to the sample configuration which consists of a thin black phosphorus (BP) layer (~10-15 nm) transferred onto a nano-polyimide (NPI, ~60nm) / Pd (~50nm) / bulk polyimide (PI) substrate and capped with PMMA (~200nm) optical interference effects will be present due to the varying refractive indices of the thin films. This effect is well documented in Ref. S1 for graphene on Si/SiO2 substrates. Ref. S2 performs a similar analysis for BP on Si/SiO2, while also considering the birefringence of the material (i.e. differing refractive indices along the zigzag (ZZ) and armchair (AC) directions). The measured Raman signal can be significantly impacted by this interference effect, which is dependent on the incident laser wavelength and the wavenumber of the measured Raman modes, in addition to the thin-layer thicknesses and refractive indices. In polarized Raman measurements of birefringent materials such as BP, the interference effect differs along different crystallographic orientations of the material, and must be considered to determine the intrinsic anisotropy of the material measured by polarized Raman. In this work, we have generally followed the approach detailed in Ref. S2, in which the interference factor, F, is given by the absolute square of the scattering and absorption terms (Fsc and Fab, respectively), integrated over the thickness of the BP sample, and multiplied by a normalization factor, as follows: F = N d 1 0 F ab (x)f sc (x) 2 dx [S1] (1+r 12 r 23 e F ab (x) 2iβ ex 2 )e iβ ex x +(r12 +r 23 e iβ 2 ex )e i(2β 1 ex β ex x ) = t 01 1+r 12 r 23 e 2iβ 2 ex +(r 12 +r 23 e 2iβ 2 ex )r 01 e 2iβ 1 ex [S2] F sc (x) = t 10 (1+r 12 r 23 e 2iβ 2 sc )e iβ x sc +(r12 +r 23 e iβ 2 sc )e i(2β 1 sc β x sc ) 1+r 12 r 23 e 2iβ 2 sc +(r 12 +r 23 e 2iβ 2 sc )r 01 e 2iβ 1 sc [S3] 7

8 where N is the normalization factor given by the reciprocal of the interference factor for a freestanding BP sample (only air on both sides), and t ij = 2n i n i +n j ; r ij = n i n j n i +n j [S4] are the Fresnel coefficients for the interface of layers i and j, and β x ex = 2πxn 1 λ ex ; β i ex = 2πd in i λ ex ; β x sc = 2πxn 1 λ sc ; β i sc = 2πd in i λ sc [S5] where ni is the complex refractive index for the i th layer, di is the thickness of the i th layer, x denotes the depth in the sample layer (i = 1), and λex and λsc denote the wavelength of incident and Raman scattered light, respectively. For BP, we consider two scattering factors: one calculated using the refractive index along the ZZ-direction and another using the refractive index along the AC-direction (FZZ and FAC, respectively). The ratio of these two interference factors indicates the degree to which polarized Raman plots are artificially enhanced/diminished when light is polarized parallel one of these directions. A ratio of 1 indicates that the interference effect contributes equally in both crystallographic directions, and thus interference effects on the anisotropic Raman response of the material can be neglected. Equations [S2] and [S3] take into account three interfaces: the interface between the environment and BP (i,j = 0,1), between BP and the layer below it (i,j = 1,2), and one additional interface (i,j = 2,3). Our sample configuration has five interfaces (Air/PMMA, PMMA/BP, BP/NPI, NPI/Pd, and Pd/PI). Deriving equations analogous to [S2] and [S3] for this number of interfaces is nontrivial, and some simplifying assumptions were made to reduce the problem to only three interfaces. Firstly, we assume that all the light that reaches the thin, metallic Pd layer is reflected, 8

9 and thus we ignore the Pd/PI interface. Secondly, we consider two situations: one in which the PMMA on top of BP is ignored ( air environment ) and another in which we treat the PMMA layer as infinitely thick ( PMMA environment ). We then calculate interference factors for both situations to determine the a worst-case scenario, in which the ratio of the interference factors is largest. Additionally, since there is some uncertainty in the thickness of the BP sample, we consider both 10-nm and 15-nm BP layers for both aforementioned scenarios to find the overall worst-case interference. For 10- and 15-nm BP in a PMMA environment, the difference in interference factors FZZ and FAC was determined to be ~1.5% and ~0.3%, respectively. For 10- and 15-nm BP in an air environment, the difference in interference factors FZZ and FAC was determined to be ~0.75% and ~4.0%, respectively. Note: these values differ between different Raman modes, but by less than 0.05% in each case. Since the worst-case scenario (15-nm BP sample in an air environment) has a difference of interference factors along different crystallographic directions of less than 5%, we determined interference effects in our sample configuration are mostly negligible. To illustrate this, we have fit our polarized Raman data to functions for the Raman intensity as a function of polarization angle based on the Raman tensor for Ag-symmetry modes in BP, again following the procedure outlined in Ref. S2. We then corrected these fits to reflect the interference factors calculated for the worst-case interference scenario, for comparison. It is clear from Figure S7 that the interference effect is negligible compared to experimental error and, importantly, the change in anisotropy of the polarized Raman plots due to strain. The interference effects in our sample configuration are much less pronounced than those calculated in Ref. S2 due to the difference of refractive indices of the substrate in capping layers used. 9

10 Refractive Index* Air Environment PMMA Environment Reference 2 Air 1 1 PMMA [S3] 1.49 BP (ZZ) [S4] 3.74-i i i0.52 BP (AC) [S4] 3.84-i i i0.08 PI [S5] Pd [S5] SiO2 [S6] 1.47 Si [S7] 4.46-i0.05 *Values either directly taken from or extrapolated from the references listed. Considering the non-bp layers, one would expect that those with refractive indices deviating most from the refractive index of the environment would have the largest contribution to the interference effect. In the case of Ref. S2, Silicon has a real part of the refractive index that is >300% larger than that of the environment (air). Contrast this with our configuration, where the non-bp layer with the largest refractive index is PI, which has a refractive index that is 70% larger than that of the air environment or 14% larger than the PMMA environment. 10

11 Figure S7. Polarized Raman plots for strained and unstrained Ag-type modes of BP. Black dots are actual data, red and blue lines are fits to the data using the Raman tensor for Ag-type BP modes before and after correcting for interference effects, respectively. Raman data were collected in backscattering configuration, with parallel incident and measured polarizations, selected such that 0 corresponded to polarization parallel to the armchair axis. 11

12 Figure. S8. Interference ratio for 30nm Al2O3/Si substrate vs. 300nm SiO2/Si substrate on BP samples with thickness ranging from 0 to 50 nm. Figure S9. For bilayer BP under no strain (black lines), 1.7% tensile strain along zigzag direction (blue lines), and 1.7% tensile strain along armchair direction (red lines), panels (a) and (b) show calculated Raman tensor element ratios c a as a function of excitation laser energy for Ag 1 and Ag 2 modes, respectively. Note that DFT tends to underestimate electronic band gaps and thus the computed laser energy here cannot be directly compared to the experimental value. The DFT computed changes of c a for Ag 1 and Ag 2 modes under strain provide a qualitative picture of how Raman tensor elements and electron-phonon interactions in BP change under strain. REFERENCES 12

13 S1. Li, Y.; Hu, Z.; Lin, S.; Lai, S.; Ji, W.; Lau, S. P., Giant Anisotropic Raman Response of Encapsulated Ultrathin Black Phosphorus by Uniaxial Strain. Adv. Funct. Mater. 2017, 27, S2. Kim, J.; Lee, J. U.; Lee, J.; Park, H. J.; Lee, Z.; Lee, C.; Cheong, H., Anomalous Polarization Dependence of Raman Scattering and Crystallographic Orientation of Black Phosphorus. Nanoscale 2015, 7, S3. Rollefson, R.; Havens, R., Index of Refraction of Methane in the Infra-Red and the Dipole Moment of the CH Bond. Phys. Rev. 1940, 57, S4. Asahina, H.; Morita, A., Band Structure and Optical Properties of Black Phosphorus. J. Phys. C: Solid State Phys. 1984, 17, S5. Heavens, O. S., Optical Properties of Thin Films. Rep. Prog. Phys. 1960, 23, S6. Malitson, I. H., Interspecimen Comparison of the Refractive Index of Fused Silica. J. Opt. Soc. Am. 1965, 55, S7. Vuye, G.; Fisson, S.; Nguyen Van, V.; Wang, Y.; Rivory, J.; Abelès, F., Temperature Dependence of the Dielectric Function of Silicon Using in-situ Spectroscopic Ellipsometry. Thin Solid Films 1993, 233,