WX2 (X = S, Se, and Te) van

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Surfaces, Interfaces, and Catalysis; Physical Properties of Nanomaterials and Materials 2

Nonlinear Optical Response in Graphene/WX (X=S, Se and Te) van der Waals Heterostructures Chuan He, Qiyi Zhao, Yuanyuan Huang, Lipeng Zhu, Sujuan Zhang, Jintao Bai, and Xin Long Xu J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.9b00217 • Publication Date (Web): 11 Apr 2019 Downloaded from http://pubs.acs.org on April 12, 2019

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The Journal of Physical Chemistry Letters

Nonlinear optical response in Graphene/WX2 (X=S, Se and Te) van der Waals Heterostructures Chuan Hea, Qiyi Zhaob, , Yuanyuan Huanga, Lipeng Zhuc, Sujuan Zhanga, Jintao Baia, Xinlong Xua, d, a..Shaanxi

Joint Lab of Graphene, State Key Lab Incubation Base of Photoelectric Technology and

Functional Materials, International Collaborative Center on Photoelectric Technology and Nano Functional Materials, Institute of Photonics & Photon-Technology, Northwest University, Xi'an 710069, China. b.

School of Science, Xi’an University of Posts & Telecommunications, Xi’an 710121, China.

c.

School of Electronic Engineering, Xi’an University of Posts & Telecommunications, Xi’an 710121, China.

d.

Guangxi Key Laboratory of Automatic Detecting Technology and Instruments, Guilin University of Electronic Technology, Guilin 541004, China.

ABSTRACT Light-frequency conversion based on two-dimensional (2D) materials is of great importance for modern nano and integrated photonics. Herein, we report both the intrinsic (from the pure WX2 (X=S, Se and Te)) and extrinsic (from the interface of graphene/WX2) second order nonlinear coefficient tensor from graphene/WX2 van der Waals (vdW) heterostructures by first-principles calculations. The prominent peaks in the dispersion relation of the intrinsic second order nonlinear coefficient in monolayer WX2 are due to the Van Hove singularity in the high symmetry point or



Corresponding author. Tel: +86 029-88303667.

E-mail: [email protected] (X. Xu) , [email protected] (Q. Zhao)

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along the high symmetry line with high joint density of states. The enhanced nonlinear optical response in the infrared band can be achieved in graphene/WS2 vdW heterostructures, resulting from the interlayer charge transfer between graphene and WS2. The value of the intrinsic second order nonlinear coefficients of graphene/WSe2 vdW heterostructures is 1.5 times larger than that of pure monolayer WSe2 at the bandgap energy of monolayer WSe2 due to the enhanced carrier generation after the heterostructures formation. Different from pure monolayer WX2, azimuthal angle dependent second harmonic generation from graphene/WX2 vdW heterostructures exhibits extraordinary rotational symmetry at different photon energy, which can be used to deduce the extrinsic second order nonlinear coefficient. These results pave the way for the nonlinear optical coefficient design based on 2D heterostructures for the nonlinear nano photonics and integrated devices.

TOC Graphic


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Two-dimensional

(2D)

layered

semiconducting

materials

such

as

transition

metal

dichalcogenides (TMDs) demonstrate remarkable optical properties1-4 due to strong light-matter interaction, strong excitonic effects5-6, strong valley dependent polarization7-9. Recently, research on the nonlinear optical effect on the TMDs materials has suggested that these 2D materials exhibit large nonlinear optical response compared with the traditional nonlinear materials10. For example, the second order optical nonlinear coefficient of MoS211-12, WS213, WSe214-15, ReS216, GaSe17 shows 2~3 orders larger than that of the common used nonlinear crystals. This large nonlinear coefficient can also be tuned by the hot carriers and electric field18-20, and enhanced by the excitonic resonance21-23. Due to the 2D characteristic, these TMDs materials are desirable for on-chip frequency conversions such as second harmonic generation (SHG)14,

19-20

and terahertz (THz)

generation24-28 for integrable devices. These 2D materials have another outstanding features as the advances in designing heterostructures29-30 due to the weak van der Waals (vdW) interaction. These heterostructures are readily to achieve “one plus one better than two” functionalities31 due to the interlayer charge-transfer transitions and coupling32-34. Furthermore, the nonlinear absorption properties of graphene/MoS2 nanocomposite films are superior to pure MoS2 films35. Meanwhile, the enhanced nonlinear absorption was achieved in MoTe2/MoS2 heterostructures film compared with that of pure MoTe2 and MoS2 nano-films36. Further physical properties in heterostructures can be governed through the interface engineering37-39. For example, efficient electron−hole separation can be occurred in vdW heterointerfaces through ultrafast charge transfer32, 37; the optical and electrical properties of graphene/TMDs heterostructures are strongly modulated by interlayer coupling and effective charge transfer40-41; and efficient interfacial carrier generation can be achieved in WS2/graphene heterostructures34. Due to these excellent properties, 2D heterostructures can be



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considered for the design and fabrication in optoelectronic devices42-47, such as field-effect transistors. However, few theory works on the nonlinear optical response of these 2D vdW heterostructures have been reported. And the second order nonlinear coefficients of the pure 2D TMDs in the infrared (IR) band is still unsatisfactory12-13. Thus, it is desirable to understand how the interface in vdW heterostructures affects the nonlinear properties as well. In this paper, we choose WX2 (X=S, Se and Te) as typical TMDs materials and graphene as the most extraordinary Dirac materials to form vdW heterostructures in order to understand nonlinear optical effect in these heterostructures. We calculate the second order nonlinear coefficients of both pure monolayer WX2 and graphene/WX2 heterostructures by first-principles calculations. The formation of graphene/WX2 heterostructures was determined through the band structure and the charge density differences. We found additional second order nonlinear coefficients due to the formation of graphene/WX2 heterostructures, which have been enhanced in graphene/WS2 and graphene/WSe2 vdW heterostructures. Typical azimuthal angle dependent SHG has also be calculated for the graphene/WX2 heterostructures, which can be used to deduce both symmetry and second order nonlinear coefficient in graphene/WX2 heterostructures. The second order nonlinear optical polarization (P) of a material under high intensity pump can be described as

P (2) (3 )    0  ijk(2) E j (1 ) Ek (2 ) j ,k

with 3  1  2 .48 Where

 ijk(2)

is the

element from the second order nonlinear susceptibility, which is a third rank tensor. Typically, for the SHG as we discussed in this paper, the light-frequency conversion satisfies 3  21  22 . For

simplicity,

the

second-order

nonlinear

coefficient

( d l )

is

usually

used

as

 ijk(2) ( ,  )  2d l ( ,  ) 48. Where d l is the element from the second-order nonlinear coefficient tensor (d), which is directly related to crystal symmetry for nonlinear optical materials. The monolayer WX2 (X=S, Se and Te) belongs to the non-centrosymmetric D3h1 space group, in which


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the corresponding nonzero d l elements are d16  d 21  d 22 .48 We calculate the absolute value of d l for monolayer WX2 by the first-principles calculations. In the calculations, both the pure interband processes and mixed interband and intraband processes49-50 have been taken into account in the second order nonlinear optical response (details in Computational methods). As shown in Figure 1, the photon energy-dependent nonzero second order nonlinear coefficients ( d l ) of monolayer WX2 also obey the relationship as d16  d 21  d 22 , with the other elements almost zero (Figure S3 in Supporting Information). The results are consistent with the crystal symmetry (D3h1 space group) of the monolayer WX2.



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Figure 1 Photon energy-dependent nonzero second order nonlinear coefficients dμl of (a) monolayer WS2, (c) monolayer WSe2 and (e) monolayer WTe2. Band structure of (b) monolayer WS2, (d) monolayer WSe2 and (f) monolayer WTe2. It is evident that there are several prominent peaks for monolayer WS2, which are labeled as I, II, III, IV in Figure 1(a). We obtain d22=141 pm/V, 587 pm/V, 414 pm/V, 1557 pm/V at 1 eV (peak I), 1.39 eV (peak II), 1.57 eV (peak III), 2.03 eV (peak IV) for monolayer WS2 as shown in Table 1, respectively. These peaks are due to the Van Hove singularity in the high symmetry point or along the high symmetry line. To understand these peaks, we calculate the band structure of monolayer WS2 as shown in Figure 1(b). The I peak for monolayer WS2 in Figure 1(a) corresponds to the transition with two photons at the K point. Due to the parallel band effect51-53, the joint density of states will be enhanced, resulting in the II peak in Figure 1(a) for WS2, which is along the high symmetry line (K to Γ) in Figure 1(b). It is also clear that the peak III for WS2 comes from transition at the Γ point as shown in Figure 1(b). Apart from the highest valence band to the first conduction band, the highest valence band to the second conduction band can also make the contribution to the enhancement of nonlinear optics as shown in shaded area (IV) in Figure 1(b). This enhancement at IV could come from two facts. One is the parallel effect of the two bands (the highest valence band and the second conduction band) and the other is the resonant effect due to the energy matching contribution from the intermediate transition (I) at K point in the band structure of monolayer WS2 as shown in Figure 1(b). These results are consistent with the recent report13. Several peaks above 2 eV in the ultraviolet region are due to the transition between high order valence band and conduction band. For monolayer WSe2, several prominent peaks as labeled I, II, III, IV at 0.88 eV, 1.35 eV, 1.76 eV, 1.86 eV are shown in Figure 1(c), respectively. The values of the d22 at these energies are


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shown in Table 1. Similarly, the origin of these peaks happen in the Van Hove singularity in the high symmetry point or along the high symmetry line with high joint density of states as shown in Figure 1(d). The I peak for monolayer WSe2 in Figure 1(c) originates from the transition at the K point in Figure 1(d). The II peak in Figure 1(c) arises from high symmetry line from K to Γ as shown in Figure 1(d), while IV peak in Figure 1(c) originates from the transition at the Γ point in Figure 1(d). The high order valence band and conduction band transition also contributes to the second order nonlinear optical response, such as the III peak in Figure 1(c), which can be enhanced by the intermediate transition (I) at K point as shown in Figure 1(d). For monolayer WTe2, several prominent peaks at 0.62 eV, 0.98 eV, 1.14 eV, 1.98 eV are labeled as I, II, III, IV in Figure 1(e), respectively with the values shown in Table 1. We can also utilize nearly parallel band effect and resonant effect in the band structure of monolayer WTe2 to clarify the origin of these peaks as shown by the shaded areas in Figure 1(f). The I peak for monolayer WTe2 in Figure 1(e) originates from the transition at the K point in Figure 1(f). The contribution of peak II in Figure 1(e) originates from the high symmetry line along K and Γ points as shown in Figure 1(f). Meanwhile, III and IV (Figure 1(e)) originate from transitions at the M point and Γ point in Figure 1(f). However, in contrast to monolayer WS2 and WSe2, the resonant effect can not be observed in WTe2 at the K point from the highest valence band to the second conduction band (red area V in Figure 1(f)) as the energy difference between the first and the second conduction band is not equal to the band gap of monolayer WTe2.



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Table 1 Second order nonlinear coefficient element d22 at the prominent peaks in Figure 1 d22 (pm/V) @ I

d22 (pm/V) @ II

d22 (pm/V) @ III

d22 (pm/V) @ IV

Monolayer WS2

141 @ 1 eV

587 @ 1.39 eV

414 @ 1.57 eV

1557 @ 2.03 eV

Monolayer

152 @ 0.88 eV

670 @ 1.35 eV

690 @ 1.76 eV

334 @ 1.86 eV

383 @ 0.62 eV

588 @ 0.98 eV

982 @ 1.13 eV

506 @ 1.98 eV

WSe2 Monolayer WTe2

As shown in Table 1, we find extraordinary second order nonlinear coefficients of monolayer WX2 with the values nearly three orders in magnitude larger than those of other traditional nonlinear crystals

48(such

as BaB2O454, LiNbO354 and III-V semiconductors(e.g., InAs, InSb,

GaAs)55). Large second-order nonlinear coefficient is not only beneficial to SHG10,

13,

but also

desirable for THz generation25-26 for on-chip light-frequency conversion. As the TMDs show ultrahigh surface atoms without dangling bonding, they also provide an ideal platform to study the nonlinear optics at surface and interface. As such, there are some questions such as how the interlayer charge transfer gives the influence to the second order nonlinear effect and how the interfacial nonlinear optical effect gives the contribution to the IR band of TMDs materials. The value of the second order nonlinear coefficients of monolayer WX2 (Figure 1) are not satisfactory within the IR range (0.1 eV-1 eV). This could limit the development of the IR nonlinear devices based on TMDs materials. However, the IR nonlinear properties of materials play an increasingly important role in various technological applications due to the heavy demanding in the military and medical science56-57. Although graphene demonstrate a whole spectrum response ranging from microwave region to ultraviolet region, the second order nonlinear optical response of monolayer ACS Paragon Plus Environment

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graphene is forbidden (Figure S1 in Supporting Information) due to the central-symmetry58. However, the graphene/WX2 vdW heterostructures can not only break the symmetry but also introduce the charge transfer through layers, which could be utilized to make up the disadvantage of nonlinear optical response of pure monolayer WX2.

Figure 2 Band structure of (a) graphene/WS2 heterostructures (at the HSE06 level), (b) charge ACS Paragon Plus Environment


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density differences and planar-averaged electron density difference Δρ(z) for graphene/WS2; (c) and (d) for graphene/WSe2 heterostructures; (e) and (f) for graphene/WTe2 heterostructures. The yellow and cyan areas indicate charge accumulation and depletion, respectively. The vertical lines indicate the positions of the C atoms in graphene and the X (X=S, Se and Te) and W atoms in the WX2 monolayer, respectively. To understand the above questions on the nonlinear optical effects in vdW heterostructures, the band structure and the charge density differences of graphene/WX2 heterostructures are calculated by density functional theory with hybrid functionals (Figure 2). For convenience, we also utilize the plane-averaged electron density difference along vertical axis to predict the electron redistribution upon formation of the interface from graphene/WX2 as follows:  (z)   graphene /WX 2   graphene  WX 2

(1)

where  graphene /WX , WX , and  graphene signify the plane-averaged densities of the graphene/WX2 2

2

heterostructures, isolated WX2 monolayer, and isolated graphene monolayer, respectively. As shown in Figure 2(a), different colors are employed to label the contributions of each single layer to the electronic band structures of graphene/WS2 vdW heterostructures. In this scheme, colors vary from blue (contribution from WS2 layer only) to red (contribution from graphene layer only), while the intermediary shades of purple depict the degree of delocalization of wave functions on the two layers. The wave function mixing is not very serious as most conduction band and valence band relies only on either WS2 or graphene. Moreover, it is evident that the band gap of monolayer WS2 (~ 2 eV) at K point in graphene/WS2 vdW heterostructures (Figure 2(a)) is almost equal to the band gap in pure monolayer WS2 (Figure 1(b)). The conduction band of WS2 is situated at approximately 0.7 eV above the Dirac point of graphene. Moreover, the parallel band appears on both sides of the valence band and the conduction band from WS2 for graphene/WS2.


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Figure 2(b) demonstrates the charge density differences of graphene/WS2 heterostructures with the yellow and cyan sections for the charge accumulation and charge depletion, respectively. As shown in Figure 2(b), the charge accumulation occurs around graphene, while the charge depletion occurs around WS2. It suggests that the electrons could be transferred from the WS2 to the graphene through the interface. Planar-averaged electron density difference Δρ(z) in the bottom of Figure 2(b) suggests that a built-in electric field along the direction from WS2 to graphene could be formed at the interface of graphene/WS2 vdW heterostructures, which is beneficial to the separation of photoexcited electrons and holes. Quite different electronic band structures of graphene/WSe2 vdW heterostructures can be observed in Figure 2(c). The movement of the Brillouin zones to each other in graphene/WSe2 heterostructures is caused by a lattice mismatch. Besides, the Dirac point of graphene locates at the position between the valence band and the conduction band of WSe2. In previous reports, WSe2 can form an n-type Schottky barrier when contacting with graphene59 and the Dirac point tends to stay near the conduction band of the semiconductor in the vdW heterostructures60. The Dirac point of graphene with respect to the band structure of WSe2 is related to the interlayer distance of vdW heterostructures and the perpendicular electric field41, 61-62. As the interlayer distance decreases from 4.5 Å to 2.5 Å, the relative position of the Dirac cone shifts downward toward the valence band maximum (VBM) of the WSe2 layer41. Similar to graphene/WS2, the band gap of WSe2 (~ 1.76 eV) at Γ point in graphene/WSe2 heterostructures is almost equal to that of pure monolayer WSe2. As shown in Figure 2(d), charge redistribution occurs in both interface and inside of each material. In contrast with graphene/WS2, Δρ(z) in the bottom of Figure 2(d) displays that the charge accumulation mainly occurs in the middle of the interface and the charge depletion gathers at both WSe2 and graphene surfaces from the interface. This suggests that some of the conduction band and



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valence band in graphene/WSe2 rely on both graphene and WSe2, and the wave function mixing is higher in graphene/WSe2 with stronger interlayer interaction in the heterostructures. Therefore, more photo-generated carriers can be shared between graphene and WSe2 after photoexcitation. Figure 2(e) demonstrates the electronic band structures of graphene/WTe2 vdW heterostructures. Similar to graphene/WS2, almost every band relies only on one material (either graphene or WTe2) for graphene/WTe2. Meanwhile, the band gap of WTe2 (~ 1.24 eV) at K point in graphene/WTe2 vdW heterostructures is almost equal to that of pure monolayer WTe2. Moreover, the Dirac point of graphene locates at the Γ point of WTe2 as shown in Figure 2(e), which could be caused by the lattice matching in graphene/WTe2. The charge accumulation occurs at the surface of WTe2 near the interface region, while the charge depletion gathers at the surface of WTe2 near interface as shown in Figure 2(f). The Δρ(z) as shown in the bottom of Figure 2(f) suggests a built-in electric field along the direction of graphene surface to WTe2 surface could be formed at the interface.


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Figure 3 Photon energy-dependent intrinsic nonzero second-order nonlinear coefficient of (a) graphene/WS2, (c) graphene/WSe2 and (e) graphene/WTe2. Photon energy-dependent additional nonzero second-order nonlinear coefficient of (b) graphene/WS2, (d) graphene/WSe2 and (f) graphene/WTe2. The photon energy-dependent nonzero second order nonlinear coefficients of graphene/WX2 vdW heterostructures are shown in Figure 3. It is significant that the number of nonzero second order nonlinear coefficient elements of graphene/WX2 heterostructures increase compared to that of ACS Paragon Plus Environment


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monolayer WX2, which can be determined by the symmetry properties of graphene/WX2 vdW heterostructures. According to our results, the second order nonlinear coefficients of graphene/WX2 vdW heterostructures could be expressed as:

d graphene /WS2

 0   d 22  d31

d graphene /WSe2

 0   d 22  d31

d graphene /WTe2

 0   d 22  0

0

0

0

d15

d 22 d31

0 d33

d15 0

0 0

0

0

0

d15

d 22 d31

0 d33

d15 0

0 0

0

0

0

d 22 0

0 d33

0 d34

d 22  0  0 

(2a)

d 22  0  0 

(2b)

0 d 22  0 0  0 0 

(2c)

For similar lattice constants (WS2 and WSe2), graphene/WSe2 has the same symmetry as graphene/WS2 (Equation (2a) and Equation (2b)). The intrinsic non-zero nonlinear coefficient elements (d16, d21, d22 in Figure 3(a)) inherited from WS2 in graphene/WS2 are reduced due to the loss of charge (Figure 2(b)) in WS2 after the formation of heterostructures. However, in contrast with pure WS2, the additional nonlinear coefficient elements (d15, d24, d31, d32, and d33 in Figure 3(b)) become quite impressive in graphene/WS2 heterostructures. The nonlinear coefficients (d15 and d24) in graphene/WS2 heterostructures are enhanced within the IR region as marked in Figure 3(b). The nonlinear coefficient d24 of graphene/WS2 heterostructures within IR region are about two orders in magnitude greater than that of famous nonlinear crystals (such as LiGaSe2, LilnS2, LiGaS2)57. We will discuss the mechanism of this IR enhancement later in Figure 4. Besides, the second order nonlinear coefficient elements d15 and d24, which are almost equal within the calculated range, can reach 1477 pm/V at 2.51 eV. Meanwhile, there are many peaks above the band gap of monolayer WS2 (~ 2 eV) in graphene/WS2 heterostructures, which is due to the high order conduction and valence band transition and parallel band effect in graphene/WS2 vdW


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heterostructures. Our results suggest that the stable graphene/WS2 heterostructures demonstrates

C3v space group by determining the nonzero second order nonlinear coefficient elements. As shown in Figure 3(c), the intrinsic non-zero nonlinear coefficient elements (d16, d21, d22) inherited from WSe2 in graphene/WSe2 heterostructures display an enhancement within the calculated range compared to those of pure monolayer WSe2 in Figure 1(c). It is prominent that the value of the second order nonlinear coefficient elements (d16, d21, d22) of graphene/WSe2 heterostructures, at 0.88 eV labeled as I and 1.76 eV labeled as III in Figure 3(c), are 1.5 times larger than that of pure monolayer WSe2 in Figure 1(c). The additional nonlinear coefficients (d15, d24, d31, d32, and d33 in Figure 3(d)) almost satisfy the relationship d15≈d24 and d31≈d32, even if these nonlinear coefficients are small. The value of the additional nonlinear coefficients d31 and d33 can reach 131 pm/V and 170 pm/V at 1.45 eV, respectively. Compared to the pure monolayer WTe2 in Figure 1(e), the intrinsic second order nonlinear coefficients (d16, d21, d22) values have not been enhanced in graphene/WTe2 as shown in Figure 3(e). This could be due to the reason that the charge density within WTe2 remains unchanged as shown in the charge density differences (Figure 2(f)) of graphene/WTe2 heterostructures. Because of the symmetry of graphene/WTe2 vdW heterostructures, the additional nonlinear coefficient elements only have d33 and d34, which can reach 70 pm/V at 3.15 eV and 219 pm/V at 3.21 eV as shown in Figure 3(f), respectively. Similar to graphene/WS2 and graphene/WSe2, many peaks appear above the band gap of monolayer WTe2 (~ 1.24 eV) in graphene/WTe2 heterostructures, which is due to the high order conduction and valence band transition and parallel band effect. Due to the d-orbit of heavy atoms in TMDs, spin-orbit coupling effect will result in spin polarization and band splitting in K/K’ points of the Brillouin zone. We have calculated the band structure of monolayer WX2 by PBE with spin–orbit coupling (PBE-SOC), which presents band



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splitting at K point (Figure S2 in Supporting Information). Such band splitting can introduce the spin or pseudo-spin dependent effects under circular polarization excitation, which arise an interesting issue about the nonlinear optical effect under the circular polarization due to the SOC effect. However, recent calculated second-order nonlinear coefficient without SOC is consistent with the experimental data under the linear polarization excitation13. SOC would give the influence (2) (2) ( ) and  RCP ( ) ) under to the dispersion relation of the second-order nonlinear coefficients (  LCP

left and right circularly polarized excitations (LCP and RCP)19, especially at the K point of the Brillouin zone. However, according to the previous reports12-13, the calculated second-order nonlinear coefficient without SOC shows reasonable order-of-magnitude agreement with the experimental data under the linearly polarization excitation. For all graphene/WX2 vdW heterostructures, the value change of the second-order nonlinear coefficient elements is mainly due to the redistribution of the charge at the interface, while the numbers of the elements can be determined by the symmetry of heterostructures. The charge redistribution of graphene/WX2 in Figure 2 can result in the electronic band structures change, which could affect the carrier generation, resulting in the nonlinear coefficient change. To better understand the enhancement of second-order nonlinear coefficients, we give the schematic of carrier generation within the IR band for graphene/WS2 in Figure 4(a) and the enhanced carrier generation at K point for graphene/WSe2 in Figure 4(b). Because the conduction band of WS2 is approximately 0.7 eV above the Dirac point of graphene (Figure 2(a)), the effective broadband carrier generation in graphene/WS2 heterostructures could be achieved by the with 0.35 eV at K point as shown in Figure 4(a). The charge density differences of graphene/WSe2 vdW heterostructures reveal some of the conduction band and valence band in graphene/WSe2 rely on both graphene and WSe2. This indicates that the wave function mixing is higher in graphene/WSe2


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with stronger interlayer interaction in the heterostructures. As such, more photo-generated carriers can be shared between graphene and WSe2 after photoexcitation with 0.88 eV or 1.76 eV at Γ point from graphene/WSe2 as shown in Figure 4(b). Our results suggest that electronic band engineering by heterostructures could be used to control second order nonlinear optical response. And the 2D heterostructures can be usually composed of more than two layers63. Meanwhile, the nonlinear optical response from 2D heterostructures is not only related to thickness, but also has dispersion relation40, 64. Hence, both the nonlinear optical response enhancement and decrease can occur for 2D heterostructures. In general, the optical second harmonic generation depends on the stacking angle for TMDCs/TMDCs heterostructures65. However, the stacking angle in the TMDCs/graphene heterostructures has a weak effect on nonlinear optical properties64.

Figure 4 Schematic of (a) carrier generation within the infrared band for graphene/WS2 and (b) enhanced carrier generation for graphene/WSe2. In order to further investigate the second order nonlinear optical response feature of WX2 (X=S, Se and Te) and graphene/WX2 heterostructures, we perform the SHG intensity from monolayer WX2 and graphene/WX2 heterostructures as a function of the crystal’s azimuthal angle. As shown in Figure S6 in Supporting Information, the SHG components with an incident angle θ and



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linearly polarized along the X direction can be described as (details in Section 6 in Supporting Information):   Ex2 ( ,  )   E y2 ( ,  )    Px (2 )  2     E ( , ) z  P (2 )   2 d  ( )    y 0 l   2 E y ( ,  ) Ez ( ,  )    Pz (2 )   2 E ( ,  ) E ( ,  )  x  z   2 Ex ( ,  ) E y ( ,  ) 

(3)

where  0 is the permittivity of the space;  denotes the angle between the mirror plane in the crystal structure and the polarization of the pump beam; d l ( ) is the transformation of the second-order nonlinear coefficient tensor by the rotation operation T ( ) (details in Section 6 in Supporting Information). Thus, the dependence of the parallel ( I / / ) and perpendicular ( I  ) components of SHG response with the sample orientation can be expressed as:

I / /  [ Px (d l ,  ) cos[ ]  Pz (d l ,  ) sin[ ]]2 I   Py 2 (d l ,  )

(4)

Based on the calculated second order nonlinear coefficient nonzero elements of monolayer WX2 (d16=d21=d22), we can further achieve the incident angle dependent SHG intensity from monolayer WX2 as a function of the crystal’s azimuthal angle as

//  d 22 2 sin 2 3  cos 4 [ ] IWX 2

and

  d 22 2 cos 2 3  cos 4 [ ] . When the incident angle θ is at 0°, the SHG responses with a IWX 2

//  polarization parallel IWX (perpendicular IWX ) exhibit a six-fold rotational symmetry as shown in 2 2

Figure S7 in Supporting Information. This suggests that the monolayer WX2 has a three-fold rotational symmetry, which is consistent with the previous experimental reports10, 13-14, 19. //  The parallel I graphene and perpendicular I graphene components of the SHG intensity from /WX 2 /WX 2

graphene/WX2 heterostructures as a function of the crystal’s azimuthal angle and the incident angle (details in Section 6 in Supporting Information) can be expressed as:


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

// 2 I graphene /WS2    2d15 cos   sin    d 22 cos   sin  3   cos  

  d31 cos 2    d33 sin 2   sin    2 I graphene /WS2    d 22 cos   cos  3  



2

(5a)

2



// 2 I graphene /WSe2    2d15 cos   sin    d 22 cos   sin  3   cos  

  d31 cos 2    d33 sin 2   sin    2 I graphene /WSe2    d 22 cos   cos  3  



2

(5b)

2



// 2 I graphene /WTe2     d 22 cos   sin  3   cos  

  d33 sin 2    2d34 cos   sin   sin   sin    2 I graphene /WTe2    d 22 cos   cos  3  



2

(5c)

2

It is obvious that the parallel components of the SHG intensities of graphene/WX2 heterostructures are not only related to second order nonlinear coefficient element d22, but also depended on the second order nonlinear coefficient elements d15, d31, d33, and d34 compared with the monolayer WX2. Unexpectedly, the perpendicular component ( I  ) of the SHG intensities of graphene/WX2 heterostructures is still only related to d22 as that of the monolayer WX2. On one hand, at 0° incident angle, the parallel component

// 2 2 I graphene /WX 2  d 22 sin  3 

and the

 2 2 perpendicular component I graphene/ are only associated with d22. On the other WX  d 22 cos  3  2

hand, at the 45° incident angle, the perpendicular component ( I  ) of the SHG intensities shows a six-fold rotational symmetry, which are similar to that of monolayer WX2 (Figure S7). This feature is determined by the Equation (5) as the perpendicular component are only related to d22 for all graphene/WX2 heterostructures. However, the parallel component ( I / / ) of the SHG intensities of graphene/WS2 exhibits a three-fold rotational symmetry from graphene/WX2 heterostructures at different photon energy (Table S1 in Supporting Information) at 0.48 eV, 1.01 eV, and 2.01 eV as shown in Figure 5(a)-(c). It is worth noting that the intensity of the parallel component is much bigger than that of the perpendicular component at 0.48 eV (Figure 5(a)) as the value of d15 is bigger than the value of d22 (Table S1 in Supporting Information). The parallel component of the ACS Paragon Plus Environment


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SHG intensities exhibits different rotational symmetry with different single photon energy for graphene/WSe2 as shown in Figure 5(d)-(f), which is due to the different contributions of second order nonlinear coefficients elements (d22, d33) (Table S1 in Supporting Information). Similarly, the parallel component of the SHG intensities exhibit different rotational symmetry with different photon energy for graphene/WTe2 due to the d22, d33 and d34 contributions (Table S1 in Supporting Information), respectively, as shown in Figure 5(g)-(i). Symmetry analysis as done by Heinz et al.11 can be used to determine the orientation of the heterostructures as well as the value of the intrinsic and additional nonzero second-order coefficient elements by the fitting with Equation (5). The crystal’s azimuthal angle study of graphene/WX2 heterostructures is of great significance to nonlinear nano-photonics and nano-devices based on 2D heterostructures materials.


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Figure 5 Polar plots of the SHG intensity from (a)-(c) graphene/WS2 heterostructures, (d)-(f) graphene/WSe2 heterostructures and (g)-(i) graphene/WTe2 heterostructures as a function of the crystal’s azimuthal angle at different single photon energy. In summary, we report the dispersion relation of the second order nonlinear coefficients of monolayer WX2 (X=S, Se and Te) and graphene/WX2 vdW heterostructures by first-principles calculations. We observed prominent peaks from the second order nonlinear coefficient in monolayer WX2 at the high symmetry point and high symmetry line. Relatively large second-order nonlinear coefficient from monolayer WX2 could be beneficial to the second order nonlinear optical ACS Paragon Plus Environment


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process (e.g. SHG and THz generation). Moreover, interlayer charge transfer allows effective broadband carrier generation and enhanced second order nonlinear coefficients in the IR band could be achieved in graphene/WS2 heterostructures. Besides, the value of second order nonlinear coefficient d 22 of graphene/WSe2 heterostructures is 1.5 times larger than that of pure monolayer WSe2 at the bandgap resonance energy of monolayer WSe2. Finally, we find that the crystal’s azimuthal angle dependent SHG intensity from graphene/WX2 heterostructures exhibit extraordinary rotational symmetry at different photon energy. Our study could be extended to other 2D materials and be used to design the nonlinear heterostructures.

COMPUTATIONAL METHODS The calculations on geometry optimization and optical properties of materials are performed using the plane-wave basis Vienna ab initio simulation package (VASP)66-67 based on universal Density Functional calculations (DFT) under the generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE)68. The accurate band alignments of the materials can be obtained by the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional69. For the calculations of the monolayer WX2 (X=S, Se and Te) and graphene/WX2 (X=S, Se and Te) structures, to avoid interactions between adjacent layers, a vacuum layer with a thickness of 20 Å is utilized. The cutoff kinetic energies for plane waves are set as 450 eV, and the relaxation of energy is taken as 1.0 × 10−5 eV. To satisfy the lattice matching, we used the 5×5 graphene/4×4 WS2, 4×4 graphene/3×3 WSe2 and 3×3 graphene/2×2 WTe2 to form the heterostructures, respectively (details in Section 4 in Supporting Information). The second-order nonlinear susceptibility

 abc (2 ,  ,  ) corresponding formalism was

derived in a simpler manner by Aversa and Sipe70, and was rearranged by Rashkeev et al50. The


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second-order nonlinear susceptibility  abc (2 ,  ,  ) , which consists of the contribution of the pure interband processes

 eabc (2 ,  ,  ) , and the mixed interband and intraband processes

 iabc (2 ,  ,  ) can be described as49-50:  abc (2 ,  ,  )   eabc (2 ,  ,  )   iabc (2 ,  ,  )

(6)

The contribution of these two processes could be obtained by 49-50:

 eabc (2 ,  ,  ) 

a (rmlb rlnc  rmlc rlnb )  2 f nm fln f ml  e3 1 rnm      2 h  nml ,k 2 (ln  ml )  mn  2 ln   ml   

 iabc (2 ,  ,  )  

(7a)

  i e3 2 a b c f nm  rnm (rnm  ;c  rmn ;b )  2 2 h  nm ,k  mn (mn  2 )  1

mn (mn   )

a b a c (rnm ;c rmn  rnm ;b rmn )

 a b c 1  1 4 c b  2    rnm (rmn  mn  rmn  mn ) mn  mn   mn  2  1 b c c b  (rnm ;a rmn  rnm ;a rmn ) 2mn (mn   )

(7b)

Where r is the position operator;  is the unit cell volume; superscripts a, b and c are Cartesians components; hmn  hm  hn , is the energy difference for the bands m and n; f mn  f m  f n is b the difference of the Fermi distribution functions; rmn ;a

is the generalized derivative of the

coordinate operator. b rmn ;a 

a b rnm  bmn  rnm  amn

nm



i

nm

 (

r r  nl rnlb rlma )

a b lm nl lm

(8)

l

a a Where  amn  ( pnn  pmm ) / m is the difference between the electronic velocities at the bands n and

m; p is the momentum matrix element. At the zero-frequency limit, the Equation (7) can be simplified as49-50:

 eabc   iabc 

a rnm (rmlb rlnc  rmlc rlnb ) e3 n f ml  m fln  l f mn   h2  nml ,k (ln  ml )

f nm a b i e3 c b a c c a b  r (r  rmn   ;b )  rnm ( rmn ;c  rmn ; a )  rnm ( rmn ;b  rmn ; a )  2 2  nm mn ;c 4 h  nm ,k mn


(9a)

(9b)


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To simplify the rank of tensors, the second order nonlinear susceptibility  ijk(2) is related to the second-order nonlinear coefficient d l as  ijk(2) ( ,  )  2d l ( ,  ) .

ASSOCIATED CONTENT Supporting Information The second-order nonlinear coefficient from graphene; The electronic band structures of monolayer WX2 (at the PBE-SOC level); The second-order nonlinear coefficient from monolayer WX2; The lattice parameters of WX2 and graphene/WX2; The second-order nonlinear coefficient from graphene/WX2 van der Waals heterostructures; The dependence of second harmonic generation as a function of azimuthal angle and incident angle; Second harmonic generation in monolayer WX2; The value of the second order nonlinear coefficients at different photon energies for graphene/WX2

AUTHOR INFORMATION Corresponding Author * Corresponding author: [email protected] (Q. Zhao), [email protected] (X. Xu) ACKNOWLEDGEMENTS This work was supported by National Natural Science Foundation of China (No. 11774288, 11374240), Key Science and Technology Innovation Team Project of Natural Science Foundation of Shaanxi Province (2017KCT-01).


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WX2 (X = S, Se, and Te) van

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