- Record: found
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Probing optical anisotropy of nanometer-thin van der waals microcrystals by near-field imaging
research-article
Author(s):
Debo Hu
1 ,
Xiaoxia Yang
1 ,
Chi Li
1 ,
Ruina Liu
1 ,
Ziheng Yao
2 ,
Hai Hu
1 ,
Stephanie N. Gilbert Corder
2 ,
Jianing Chen
3 ,
Zhipei Sun
4 ,
Mengkun Liu
2
,
,
Qing Dai
1
,
Publication date (Electronic):
13 November 2017
Journal:
Nature Communications
Publisher:
Nature Publishing Group UK
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Abstract
Most van der Waals crystals present highly anisotropic optical responses due to their strong in-plane covalent bonding and weak out-of-plane interactions. However, the determination of the polarization-dependent dielectric constants of van der Waals crystals remains a nontrivial task, since the size and dimension of the samples are often below or close to the diffraction limit of the probe light. In this work, we apply an optical nano-imaging technique to determine the anisotropic dielectric constants in representative van der Waals crystals. Through the study of both ordinary and extraordinary waveguide modes in real space, we are able to quantitatively determine the full dielectric tensors of nanometer-thin molybdenum disulfide and hexagonal boron nitride microcrystals, the most-promising van der Waals semiconductor and dielectric. Unlike traditional reflection-based methods, our measurements are reliable below the length scale of the free-space wavelength and reveal a universal route for characterizing low-dimensional crystals with high anisotropies.
Abstract
The optical response of van der Waals layered crystals is strongly anisotropic. Here, the authors develop a nano-imaging technique to determine the in-plane and out-of-plane components of the anisotropic dielectric tensors in MoS 2 and hBN, two representative van der Waals crystals.
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Most cited references39
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Two Dimensional Atomic Crystals
D Jiang, K. S. Novoselov, T. Booth … (2005)
We report free-standing atomic crystals that are strictly 2D and can be viewed as individual atomic planes pulled out of bulk crystals or as unrolled single-wall nanotubes. By using micromechanical cleavage, we have prepared and studied a variety of 2D crystals, including single layers of boron nitride, graphite, several dichalcogenides and complex oxides. These atomically-thin sheets (essentially gigantic 2D molecules unprotected from the immediate environment) are stable under ambient conditions, exhibit high crystal quality and are continuous on a macroscopic scale.
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Van der Waals heterostructures
A. K. Geim, I. Grigorieva (2013)
Research on graphene and other two-dimensional atomic crystals is intense and likely to remain one of the hottest topics in condensed matter physics and materials science for many years. Looking beyond this field, isolated atomic planes can also be reassembled into designer heterostructures made layer by layer in a precisely chosen sequence. The first - already remarkably complex - such heterostructures (referred to as 'van der Waals') have recently been fabricated and investigated revealing unusual properties and new phenomena. Here we review this emerging research area and attempt to identify future directions. With steady improvement in fabrication techniques, van der Waals heterostructures promise a new gold rush, rather than a graphene aftershock.
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High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus
Jingsi Qiao, Xianghua Kong, Zhi-Xin Hu … (2014)
The discovery of graphene laid the foundations for many new areas of research. One of the most important foundation is the intensive investigation of two-dimensional (2D) atomic-layer systems, including graphene itself1 2 3 4 5, transition metal dichalcogenides (TMDCs)5 6 7 8, silicene9 10 and germanane10 11, as candidate materials for future electronics applications12 13 14 15 16 17 18 19. A high-performance device such as a field-effect transistor (FET) requires a moderate electronic band gap, a reasonably high carrier mobility of the channel material and excellent electrode-channel contacts4 5 8 13 14 15 16 17 18 19. Graphene offers extremely high mobilities, due to its very low carrier effective mass, and thus is considered to be a promising candidate for high-speed FET devices, but its intrinsic dispersion is gapless2 3 4 12 13 14. Despite extensive efforts following a wide variety of approaches to the problem of opening a gap in different graphene nanostructures, all devices to date have a relatively large ‘off’ current and thus a low ‘on–off’ ratio13. For this reason the emergence of monolayer TMDs has attracted substantial research interest, and in particular MoS2 has recently been used to fabricate a FET15. Unlike graphene, monolayer MoS2 is a direct-bandgap semiconductor with a carrier mobility of ~200 cm2 V−1 s−1 (ref. 15), improvable up to 500 cm2 V−1 s−1 (ref. 8), a value which is not unreasonable for applications but remains orders of magnitude lower than that of graphene13 14. Thus the entire community working on atomic-layer transport continues to search for a 2D material which is semiconducting, preferably with a direct gap, has high carrier mobility and has the potential to form excellent contacts with known electrode materials. Here we present the theoretical discovery of a new category of layered direct-bandgap semiconductor, few-layer black phosphorus (BP), with high mobility, high in-plane anisotropy and linear dichroism. BP is an allotrope of phosphorus, whose crystal structure is a strongly folded honeycomb sheet with ‘troughs’ running along the y-axis (b direction). We demonstrate using density functional theory (DFT) calculations that few-layer BP systems, from the monolayer up to five-layer structures, are thermally stable with an interlayer interaction energy of −0.46 eV and a bandgap that falls exponentially with the thickness. Carrier mobilities at room temperature are high, and furthermore exhibit strongly anisotropic behaviour. The BP monolayer has an exceptionally high hole mobility, which we predict to lie in the range from 10,000 to 26,000 cm2 V−1 s−1, and which we explain by visualizing the wavefunctions. We find an explicit linear dichroism in the computed absorption spectra, whereby the positions of the lowest-energy absorption peaks for the two in-plane directions differ strongly. These results demonstrate that few-layer BP is a new category of 2D semiconductor with high potential for novel applications in nanoelectronics and optoelectronics. Results Geometric and electronic properties of bulk BP Because no experimental data are available for the atomic structure and electronic properties of few-layer BP, we begin by considering bulk BP to refine the accuracy of our theoretical predictions. We have performed DFT calculations using a number of different functionals to gauge which provide the best fit to experiment; a full list may be found in Supplementary Table 1 and we summarize our findings here. If one considers only the lattice geometry then the PBE-G06 (refs 20, 21) and optB86b-vdW22 methods produce the best results. However, neither local density approximation-modified Becke–Johnson (LDA-mBJ)23 24 nor HSE06 (refs 25, 26) calculations based on these ‘best’ geometries can provide a satisfactory value for the bulk bandgap, as shown in Supplementary Fig. 1. To optimize both sets of properties simultaneously, we found that the optB88-vdW functional22 27, combined with either the LDA-mBJ or HSE06 method to predict the electronic bandstructure, gave the best fits. Details of choosing the best-fit functional are available in Supplementary Methods. Figure 1a shows the fully relaxed atomic structure of bulk BP, where the equilibrium lattice constants obtained from optB88-vdW are only 1–2% larger than the experiment28 29 30. The accompanying Brillouin zone (BZ) and electronic bandstructures, labelled mBJ (optB88-vdW) and HSE06 (optB88-vdW) are shown in Fig. 1b,c. Both combinations of methods predict that bulk BP is a semiconductor with a direct bandgap at the Z point of 0.31 eV (mBJ) or 0.36 eV (HSE06), both values being fully consistent with the experimental value of 0.31–0.35 eV (refs 31, 32, 33, 34). At the bandgap wavevector, which is the Z point, one valence band (VB) and one conduction band (CB) disperse rather strongly along the Z–Q and Z–G directions (Fig. 1c), indicating very small effective masses. A fit of these bands using the nearly-free electron model gives effective carrier masses along the Z–Q direction which are rather small and similar, namely 0.12 m 0 for electrons and 0.11 m 0 for holes, whereas these along Z–G are slightly larger, taking the respective values 0.15 m 0 and 0.30 m 0 (all masses taken from HSE06). Considerably larger values are found along Z–T′–A′, where the carrier effective masses are 1.15 m 0 and 0.71 m 0, respectively. All of these results are fully consistent with the experimental values for bulk BP35. The corresponding mBJ results are within 0.04 m 0 of the HSE06 values along Z–Q, Z–G and Z–T′–A′ except for the electron effective mass along Z–T′–A′, which is larger than the HSE06 value by 0.13 m 0. Values for the direction Z–Q are similar to other high-mobility semiconductors, such as AlGaAs (0.1 m 0)36, and four times smaller than those of MoS2 (ref. 37). Geometric and electronic properties of few-layer BP The similarity between the bandstructures of bulk and monolayer BP38, and the low effective masses of bulk BP35, suggest that few-layer BP is likely to be a high-mobility, direct-bandgap 2D semiconductor. We have thus performed direct calculations of the geometric and electronic properties for few-layer BP systems. Table 1 summarizes the changes of geometrical properties as a function of the layer number from 1 to 5. The lattice parameter a increases by 0.11 Å on passing from bulk to monolayer BP, whereas b grows only by 0.02 Å. There is an abrupt reduction of a between the monolayer and the bilayer, which we believe to be the primary consequence of interlayer interactions (0.46 eV) in the bilayer. The significant stretching of a in few-layer BP systems is caused almost exclusively by an increase of the bond angle θ1 but not of the bond lengths. Results from our bandstructure calculations for the five few-layer BP systems are shown in Fig. 2. In the monolayer, the original Z point of the bulk BZ folds back to the Gamma (G) point, so that the original Z–Q and Z–T′–A′ directions of the bulk BZ project onto the G–X and G–Y directions of the monolayer, which correspond respectively to the a and b directions in real space (Fig. 2a,b). Results obtained from the mBJ method are quantitatively the same as those obtained from HSE06. Supplementary Figure 2 and Supplementary Table 2 show that the predicted bandgap is rather less sensitive than the unit-cell dimensions to the functional used for optimizing the atomic structures, with similar values emerging for all few-layer BP systems from the best-performing functionals. However, other physical quantities, notably the location of band maximum (minimum) and the carrier effective mass (and hence the mobility) are in fact very sensitive to the choice of the functional, and hence we have focused primarily on our optB88-vdW results for the discussion of these. Monolayer BP is indeed a direct-bandgap semiconductor, as shown in Fig. 2c, with the gap value of 1.51 eV obtained at the G point. Phosphorus is a heavier element than carbon and therefore should have stronger spin-orbit coupling (SOC) in its 2D forms than graphene. We have considered full SOC effects in calculating the electronic bandstructure of bulk BP. Supplementary Figure 3 shows that inclusion of SOC terms has no appreciable effect on the primary features of the bandstructure, indicating that phosphorus is ‘not sufficiently heavy’ to cause any qualitative changes. Only a very small separation of the formerly fourfold degenerate bands, into two band pairs, can be found from X along N (or M for few-layer BP) to Y, which reaches a maximum of 23 meV around the N (M) point. Along G–X and G–Y, these bands are already separated by bandstructure effects and SOC causes no further splitting. That the spin-orbit splitting energy is so small is a consequence of the small effective nuclear charge (Z eff) of the P atom and the weak variation of the charge gradient in an elemental system such as BP. When two monolayers are combined to form a bilayer (Fig. 2d,e), the gap is reduced to 1.02 eV and two additional bands emerge around the gap at the G point. Together with the original VB and CB, we denote these bands as VB1, VB2, CB1 and CB2, as shown in Fig. 2f. In real space, the states from these four bands with k close to G are extended throughout the bilayer, as shown in Fig. 2g. States VB1 and VB2 differ in the interlayer region although they share the same origin in terms of the atomic orbitals. A clear bonding-like feature is visible in the interlayer region (marked by red rectangles) for VB1, which lies lower in energy, whereas VB2 shows an anti-bonding feature. Similar bonding and anti-bonding features are also found in CB1 and CB2, where they are observed not between the layers but across the troughs. These features indicate that wavefunction overlap, rather than van der Waals effects, plays the primary role in mediating the interlayer interaction, which explains the abrupt reduction of the lattice constant a from monolayer to bilayer. This interlayer interaction introduces band dispersions of VBs and CBs in the direction normal to the layers, which leads to the reduction of the bandgap (by 0.5 eV) from monolayer to bilayer. The thicker the few-layer BP system, the stronger is the interlayer interaction and thus the larger is the dispersion of VBs (CBs), resulting in a smaller bandgap. Thus the gap falls continuously on adding more layers to the bilayer, reaching 0.59 eV in five-layer BP, as illustrated in Fig. 2h. We fitted these values by an exponential decay relation and found that the corresponding bulk gap obtained by extrapolation is 0.53 eV, a value 0.17 eV larger than the one we calculated for bulk BP. We ascribe this difference to the elongated lattice parameter a in few-layer BP and results calculated with a series of constrained a values confirm our expectation; these results imply that the bandgap is very sensitive to the lateral strain along the a (x) direction and could be modulated by varying interatomic separations and angles. During the preparation of this manuscript, we became aware that a strain-induced bandgap modulation has recently been demonstrated theoretically39. The small effective masses remain in all of the few-layer BP systems (Table 2). For the G–X direction in the monolayer, equivalent to Z–Q in the bulk, the carrier effective masses are 0.15 m 0 (hole) and 0.17 m 0 (electron), only 0.04–0.05 m 0 larger than those in bulk BP. It is remarkable that in the G–Y direction, equivalent to Z–T′–A′ in the bulk, the VB appears to be nearly flat close to the G point, with an effective mass of 6.35 m 0 (see Fig. 2c), nine times its bulk value of 0.71 m 0. By contrast, the effective mass for the CB is 1.12 m 0, very close to its bulk value of 1.15 m 0. Unlike the bandgap, only the hole effective mass along G–Y shows a strongly layer-dependent evolution, in that it decreases from 6.35 m 0 for the monolayer to 1.81 m 0 for the bilayer and eventually to 0.89 m 0 in the five-layer system, close to the bulk value of 0.71 m 0. Carrier mobility The electronic properties of few-layer BP are governed to a large extent by the carrier mobilities, which are in turn strongly influenced (but not solely determined) by their effective masses. Thus we also provide theoretical predictions for the carrier mobilities, in both x and y directions, in few-layer BP systems by applying a standard 2D model as discussed under Methods below. We apply a phonon-limited scattering model, in which the primary mechanism limiting carrier mobility is scattering due to phonons36 40 41 42. Two properties of the few-layer BP lattice, namely the deformation potential E 1 and the elastic modulus C 2D in the propagation direction of the longitudinal acoustic wave, are then the most relevant factors determining the mobility43 44. These quantities were calculated using the optB88-vdW functional and the data are presented in Table 2. If few-layer BP is cut into ribbons, a one-dimensional model should be adopted for predicting the carrier mobilities. We present the results and discussion for a 10-nm-width BP ribbon in the Supplementary Discussion and Supplementary Table 3. Our predicted mobilities for the five few-layer BP systems (Table 2) are in general moderately large (hundreds to thousands of cm2 V−1 s−1), moderately anisotropic and asymmetric between electrons and holes, with the holes being more mobile in both directions. Concerning directional anisotropy, electron and hole mobilities for the x direction are nearly twice the values along y for holes and four times as large for electrons. However, the monolayer is exceptional in every way. The electron mobility along x is nearly 14, rather than four, times the value along y, that is, 1,100–1,140 cm2 V−1 s−1 compared with ~80 cm2 V−1 s−1, which is consistent with the trend of the other few-layer systems. To our great surprise, however, the hole mobility along x is 16–38 times smaller than that along y, that is, 640–700 cm2 V−1 s−1 versus 10,000–26,000 cm2 V−1 s−1, making y the direction of higher hole conductivity. This extraordinarily large value for monolayer BP is a consequence of the extremely small deformation potential, E 1y=0.15±0.03 eV, and occurs despite the fact that the carriers are very heavy (6.35 m 0). This ‘monolayer exception’ should be valuable in applications such as the separation of electrons and holes. The value of E 1y for holes in a monolayer is very striking in that it is an order of magnitude smaller than typical values of E 1, which are 5.0 eV for graphene44, 3.9 eV for MoS2 (ref. 37), and 3.7 eV for h-BN40 and 7 eV for AlGaAs quantum wells36. In few-layer BP systems, E 1y for holes has a conventional value 2.97±0.18 eV in a five-layer structure, decreasing smoothly to 1.63±0.16 eV in a bilayer. These results can be explained rather well by the form of the VB wavefunctions shown in Fig. 2g, where interlayer and stacking-induced intra-layer overlap increases with layer number. These effects are completely absent in monolayer BP, making the VB wavefunction quite isolated along y, as it is also for the VB2 (anti-bonding) state of a bilayer highlighted by the green rectangle in Fig. 2g. Thus small structural deformations due to longitudinal phonon oscillations along y have almost no effect on this wavefunction and cause little change in its energy, resulting in the very small deformation potential. The situation is quite different for the CB (electron) wavefunction, which has E 1x=2.72±0.02 eV in the monolayer and 5.02±0.02 eV in the bilayer. The in-plane bonding and anti-bonding features of the CB wavefunctions in few-layer BP systems cause a substantial enhancement of the energy-level perturbations due to structural deformations along y. The monolayer exception aside, all E 1 values for CB states are rather large (5.0–7.7 eV), for both x and y directions, and are almost independent of the number of layers, a result which can be explained by the strong CB wavefunction overlap, especially along y (Fig. 2g). For the VB states, the potentials are somewhat smaller (2–3 eV), precisely because their wavefunctions have less overlap between different P atoms. Optical absorption spectra and linear dichroism We have also predicted the optical absorption spectra of few-layer BP systems by computing the dielectric function. Two absorption spectra are shown in Fig. 3a,b for light incident along the z direction and linearly polarized in the x and y directions, respectively; while the results for incident light polarized in the z direction may be found in Supplementary Fig. 4. These results demonstrate a strong linear dichroism: for a dielectric polarization in the x direction, the band edge of the first absorption peak is found at the bandgap and thus falls rapidly with the thickness of the sample (Fig. 3a). By contrast, with y-polarized light this peak is found at 3.14 eV in the monolayer and its position falls only slightly with thickness, remaining at 2.76 eV in bulk BP (Fig. 3b). From the symmetries of the wavefunctions in Fig. 2g, clearly the (odd) dipole operator connects the VB and CB states for x-polarization, allowing the direct-bandgap process, but this is symmetry-forbidden for y-polarization and the transition occurs between VB and CB states elsewhere in the BZ. Discussion High carrier mobilities would be expected in BP systems due to the small effective masses of 0.1–0.2 m 0. However, our calculations reveal that the key issue is not the effective mass but the very small elastic modulus C 2D: all the BP systems we have studied are ‘very soft’ in the x direction, due to the fact that bond angles are changed easily (with little effect on bond lengths). When compared to other monolayer systems, the modulus of BP for the x direction is 29.0 J m−2, a value respectively 2, 10 and 11 times smaller than MoS2 (ref. 37), h-BN40 and graphene44. These results indicate that much higher mobilities could be most likely achieved if few-layer BP systems could be modified to become stiffer, that is, to have a larger elastic modulus for the x direction, for example by external stress, doping or substrate confinement. However, possible changes of the deformation potential as a result of modifying the elastic moduli are not yet clear and require further investigation. The modulus for the y direction is significantly larger than that for x, but the higher carrier effective masses, result in carrier mobilities being smaller than those along x. Both the bandgap and the mobility depend strongly on the structural properties and this offers a number of possibilities for further control over the electrical and mechanical properties of few-layer BP. Further, the linear dichroism we find allows an electronic determination of the sample orientation. We propose an experimental setup to deduce the orientation of few-layer BP systems using optical spectroscopy, as illustrated in Fig. 3c. If the incident light is linearly polarized in a chosen orientation and is near-normally incident, the absorption spectrum should vary when the sample is rotated and such that the a and b directions can be identified by monitoring the absorption signal. Once the sample orientation is determined, it is far easier to fabricate electrodes or gates utilizing the highest-mobility direction of a few-layer BP sample in an FET-type device. In summary, we have shown theoretically that few-layer BP is a novel category of 2D semiconductor offering a direct bandgap, high carrier mobility and high transport anisotropy, all of which are tunable by controlling the layer thickness. The bandgap decreases from 1.51 eV for a monolayer to 0.59 eV for five-layer BP, a set of values fitting well into the gap between graphene nanoribbons and TMDCs. The anisotropy of electric and optical properties is another unique feature distinguishing BP from other 2D materials, such as silicene9 10, h-BN5 45 and germanane10 11. Higher conductivity is generally found in the direction oriented perpendicular to the troughs (the x direction) and holes are usually more mobile than electrons. For holes, the mobility in the x direction increases from 600 cm2 V−1 s−1 for a monolayer to over 4,000 cm2 V−1 s−1 for five-layer BP. We found that monolayer BP is a very special material, in which we predict an extremely high hole mobility of 10,000–26,000 cm2 V−1 s−1. All of our mobility results can be understood rather well from the form of the real-space wavefunctions, which show a much stronger interlayer coupling in few-layer BP than in graphene or TMDCs. All these results make few-layer BP a very promising candidate material for future applications in electronics and optoelectronics. Note added in proof: during the review process for this manuscript, we became aware that few-layer BP samples have very recently been exfoliated from bulk BP and fabricated into electronic devices46 47. Measurements on these samples verify the high hole mobilities46 47 and transport anisotropy46 we predict. Although preliminary optical characterizations have been performed46 47, linear dichroism has not yet been explored. Methods DFT calculation DFT calculations were performed using the generalized gradient approximation for the exchange-correlation potential, the projector augmented wave method48 49 and a plane-wave basis set as implemented in the Vienna ab-initio simulation package50. The energy cutoff for the plane-wave basis was set to 500 eV for all calculations. Two k-meshes of 10 × 8 × 4 and 10 × 8 × 1 were adopted to sample the first BZ of the conventional unit cell of bulk and few-layer BP, and the mesh density of k points was kept fixed when calculating bandstructures using primitive cells. In optimizing the system geometry, van der Waals interactions were considered by the vdW-DF level with the optB88 exchange functional (optB88-vdW)22 27. The shape and volume of each supercell were optimized fully and all atoms in the supercell were allowed to relax until the residual force per atom was <0.001 eV Å−1. Electronic bandstructures were calculated by the mBJ23 24 and hybrid functional (HSE06)25 26 methods based on the atomic structures obtained from the full optimization by optB88-vdW. The charge gradient used in the mBJ calculations, c=1.1574, was extracted from the bulk value. Carrier mobility calculation In 2D the carrier mobility is given by the expression40 41 42 where is the effective mass in the transport direction and is the average effective mass determined by . The term E 1 represents the deformation potential constant of the valence-band minimum for hole or conduction-band maximum for electron along the transport direction, defined by . Here ΔV i is the energy change of the i th band under proper cell compression and dilatation (calculated using a step of 0.5%), l 0 is the lattice constant in the transport direction and Δl is the deformation of l 0. The elastic modulus C 2D of the longitudinal strain in the propagation directions (both x and y) of the longitudinal acoustic wave is derived from (E−E 0)/S 0=C(Δl/l 0)2/2, where E is the total energy and S 0 is the lattice volume at equilibrium for a 2D system. All structural properties in the calculation of carrier mobilities were obtained from optB88-vdW and properties related to the electronic structure were computed with the HSE06 functional. The temperature used for the mobility calculations was 300 K. Uncertainty of deformation potential In calculations of the electron and hole mobilities, the deformation potential is derived from linear fits to the respective energies of the conduction-band maximum and the valence-band minimum as functions of the lattice dilation or compression. Typical fitting results for E 1y (electron state) and E 1x (hole state) are shown in Supplementary Fig. 5a–c. The slope represents the deformation potential, which usually has an unavoidable uncertainty accounted for in the standard fitting error. Supplementary Figure 5d summarizes the relative errors for the deformation potentials of electrons and holes in few-layer BP for the x and y directions. For a given direction and type of carrier, the error is generally smallest for monolayer BP, while that for five-layer BP is the largest, except for the hole deformation potential along y. The error in E 1y for hole states appears rather larger than the others, especially for monolayer BP where it exceeds 20%, presumably due to the small absolute value of E 1y for this case. Supplementary Fig. 5c shows details of the fit. The absolute value of E 1y for hole states is only 0.15 eV, an order of magnitude smaller than the other energies; an error of 1 meV, believed to be the accuracy limit of DFT, may then lead to a fitting error of more than 10%. In this respect, a fitting error of 20% is definitely reasonable, and suggests that our calculation is highly accurate. Absorption spectra calculation Absorption spectra were calculated from the dielectric function using the expression A(ω)=1−e −α(ω)·Δz , where is the absorption coefficient, is the index of refraction, ε 1 and ε 2 are the real and imaginary parts of the dielectric function, ω is the light frequency, c is the speed of light in vacuo and Δz represents the unit-cell size in the c direction. The electronic structures were obtained from HSE06 results and the k-mesh was doubled in calculating dielectric functions. Excitonic contributions were not considered in our calculations. The total number of bands considered was set to be twice that used in the total-energy and bandstructure calculations. Because the dielectric function is a tensor, the absorption spectra along the three directions a (x), b (y) and c (z) were obtained separately. Author contributions W.J. conceived this research. J.Q., Z-X.H. and W.J. performed atomic and electronic structure calculations. X.K. calculated deformation potentials, elastic moduli and carrier mobilities. F.Y. and W.J. predicted the optical absorption spectra. J.Q., X.K. and W.J. wrote the manuscript and all the authors commented on it. Additional information How to cite this article: Qiao, J. et al. High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus. Nat. Commun. 5:4475 doi: 10.1038/ncomms5475 (2014). Supplementary Material Supplementary Information Supplementary Figures 1-5, Supplementary Tables 1-3, Supplementary Discussion, Supplementary Methods and Supplementary References
Author and article information
Contributors
Mengkun Liu: mengkun.liu@stonybrook.edu
Qing Dai: daiq@nanoctr.cn
Journal
Journal ID (nlm-ta): Nat Commun
Journal ID (iso-abbrev): Nat Commun
Title:
Nature Communications
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Nature Publishing Group UK
(London
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2041-1723
Publication date
(Electronic):
13
November
2017
Publication date PMC-release: 13
November
2017
Publication date Collection: 2017
Volume: 8
Electronic Location Identifier: 1471
Affiliations
[1
]ISNI 0000 0004 1806 6075, GRID grid.419265.d, Nanophotonics Research Division, CAS Center for Excellence in Nanoscience, , National Center for Nanoscience and Technology, ; Beijing, 100190 China
[2
]ISNI 0000 0001 2216 9681, GRID grid.36425.36, Department of Physics, , Stony Brook University, ; Stony Brook, NY 11794 USA
Author information
Stephanie N. Gilbert Corder
http://orcid.org/0000-0002-5041-8670
Article
Publisher ID: 1580DOI: 10.1038/s41467-017-01580-7
PMC ID: 5684389
PubMed ID: 29133779
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June
2017
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accepted
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October
2017
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