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A generalized analytical approach is presented to derive closed-form formulae for the elastic moduli of hexagonal multiplanar nano-structures. Hexagonal nano-structural forms are common for various materials. Four different classes of materials (single layer) from a structural point of view are proposed to demonstrate the validity and prospective application of the developed formulae. For example, graphene, an allotrope of carbon, consists of only carbon atoms to form a honeycomb like hexagonal lattice in a single plane, while hexagonal boron nitride (hBN) consists of boron and nitrogen atoms to form the hexagonal lattice in a single plane. Unlike graphene and hBN, there are plenty of other materials with hexagonal nano-structures that have the atoms placed in multiple planes such as stanene (consists of only Sn atoms) and molybdenum disulfide (consists of two different atoms: Mo and S). The physics based high-fidelity analytical model developed in this article are capable of obtaining the elastic properties in a computationally efficient manner for wide range of such materials with hexagonal nano-structures that are broadly classified in four classes from structural viewpoint. Results are provided for materials belonging to all the four classes, wherein a good agreement between the elastic moduli obtained using the proposed formulae and available scientific literature is observed.
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2
2 2
2
2 2
2 2
2
2
E
EsEbEtEnb
E=Es+Eb+Et+Enb
Eb
Ebi Ebo
E
E=Es+Eb
=Es+Ebi +Ebo
=1
2kr(∆l)2+1
2kθ(∆θ)2+1
2kθ(∆α)2
lθα
krkθ
Es
Ebi
Ebo
krkθ
A l E I
N
Ua=1
2ZL
0
N2
EA dl=1
2
N2l
EA =1
2
EA
l(∆l)2
M
Ub=1
2ZL
0
M2
EI dl=1
2
EI
l(2∆φ)2
Es
Kr=EA
l
2∆φθα
Ebi Ebo
kθ=EI
l
krkθ
EA E I
E1
σ1δaH
δbHi δbH o
δH11 =δaH +δbHi +δbH o
=Hl cos2ψcos2α
AE +Hl3sin2ψ
12EI +H l3cos2ψsin2α
12EI
A=πd2
4I=πd4
64 H=σ1tl(1 + sin ψ) cos α l d
t
H
ψ α
H Hsinψ
Hcosψ
Hsinψ Hsinψ
Hi H i δbHi
δ=P L3
12EI L P
δbHi
δbHi = ∆H isinψ =Hsinψl3
12EI sinψ =H l3sin2ψ
12EI
Hcosψ
Hcosψcosα
Hcosψsinα Ho
Ho
δbHo
δbHo = ∆H osinαcosψ =Hcosψsinαl3
12EI sinαcosψ =Hl3cos2ψsin2α
12EI
Hcosψcosα
δaH
δaH =Hcosψcosαl
AE cosψcosα =Hlcos2ψcos2α
AE
krkθ
EA E I
11 =δH11
lcos ψcos α
=σ1tl(1 + sin ψ)
lcos ψl2
12kθsin2ψ+ cos2ψsin2α+cos2ψcos2α
kr
E1=σ1
11
ψ α
E1=cos ψ
t(1 + sin ψ)l2
12kθsin2ψ+ cos2ψsin2α+cos2ψcos2α
kr
ψ= 90θ
2θ
E2
σ2
δV1δV2
δaV 1δbV 1i
δbV 1o
δV1=δaV 1+δbV 1i+δbV 1o
=V l sin2ψcos2α
AE +V l3cos2ψ
12EI +V l3sin2ψsin2α
12EI
V=σ2tl cos ψcos α
δaV 2
δbV 2o
2V
δV2=δaV 2+δbV 2o
=2V l cos2α
AE +2V l3sin2α
12EI
V A I AE EI
δV22 =δV1+δV2
=σ2tl cos ψcos αl2
12kθcos2ψ+ sin2ψsin2α+ 2 sin2α+cos2α
krsin2ψ+ 2
22 =δV22
(l+lsin ψ) cos α
=σ2tcos ψ
1 + sin ψl2
12kθcos2ψ+ sin2ψsin2α+ 2 sin2α+cos2α
krsin2ψ+ 2
E2=σ2
22
E2=1 + sin ψ
tcos ψl2
12kθcos2ψ+ sin2ψsin2α+ 2 sin2α+cos2α
krsin2ψ+ 2
ψ= 90θ
2θ
kr
kθl θ α
ν12
ν12
ν12 =12
11
12 11
11
12
σ1
δbV i1δbV o1
δH12 =δbV i1+δbV o1
=Hl3sin ψcos ψ
12EI +H l3sin ψcos ψsin2α
12EI
=Hl3sin ψcos ψcos2α
12EI
kθ
EI
12 =δH12
(l+lsin ψ) cos α
=Hl sin ψcos ψcos α
12kθ(1 + sin ψ)
ν12
ν12 =sin ψcos2ψcos2αl2
12kθ(1 + sin ψ)l2
12kθsin2ψ+ cos2ψsin2α+cos2ψcos2α
kr
ψ= 90θ
2θ
ν21
ν21
ν21 =21
22
21 22
22
21
σ2
δbHi2δbH o2
δH21 =δbHi2+δbH o2
=V l3sin ψcos ψ
12EI +V l3sin ψcos ψsin2α
12EI
=V l3sin ψcos ψcos2α
12EI
kθ
EI
21 =δH21
lcos ψcos α
=V l sin ψcos α
12kθ
ν21
ν21 =sin ψ(1 + sin ψ) cos2αl2
12kθl2
12kθcos2ψ+ sin2ψsin2α+ 2 sin2α+cos2α
krsin2ψ+ 2
ψ= 90θ
2θ
E1ν21 =E2ν12 =
sin ψcos ψcos2αl2l2
12kθsin2ψ+ cos2ψsin2α+cos2ψcos2α
kr1
12kθtl2
12kθcos2ψ+ sin2ψsin2α+ 2 sin2α+cos2α
krsin2ψ+ 2
E1E2ν12
ν21
˜
E1=cos ψ
(1 + sin ψ)λsin2ψ+ cos2ψsin2α+ cos2ψcos2α
˜
E2=1 + sin ψ
cos ψλcos2ψ+ sin2ψsin2α+ 2 sin2α+ cos2αsin2ψ+ 2
˜ν12 =sin ψcos2ψcos2αλ
(1 + sin ψ)λsin2ψ+ cos2ψsin2α+ cos2ψcos2α
˜ν21 =sin ψ(1 + sin ψ) cos2αλ
λcos2ψ+ sin2ψsin2α+ 2 sin2α+ cos2αsin2ψ+ 2
λ(= l2
12
kr
kθ
0.4 2.8
λ4
3l
d2
krkθ
l d
λ˜
E1=E1t
kr
˜
E2=E2t
kr
˜ν12 =ν12
˜ν21 =ν21
α= 0
E1=cos ψ
t(1 + sin ψ)l2
12kθ
sin2ψ+cos2ψ
kr
E2=1 + sin ψ
tcos ψ l2
12kθ
cos2ψ+sin2ψ+ 2
kr!
θ
120ψ= 30
E1=E2=43krkθ
tkrl2
4+ 9kθ
α= 0
ν12 =sin ψcos2ψl2
12kθ(1 + sin ψ)l2
12kθ
sin2ψ+cos2ψ
kr
ν21 =sin ψ(1 + sin ψ)l2
12kθ l2
12kθ
cos2ψ+sin2ψ+ 2
kr!
θ
120ψ= 30
ν12 =ν21 =1
1 + 36kθ
krl2
2
¯
E1=E1×t¯
E2=E2×t t
E1E2
¯
E1¯
E2
t
krkθkr= 938
12= 6.52 ×1071kθ= 126 12= 8.76 ×1010
2α= 0 θ= 120
ψ= 30
E1=E2= 1.0419
krkθ
kr= 4.865×1071kθ= 6.952×1010
2α= 0 θ= 120
ψ= 30
¯
E1=¯
E2= 0.2659
¯
E1=E1×t¯
E2=E2×t t
¯
E1=¯
E2
¯
E1= 0.3542
¯
E2= 0.3542
±
±0.01
¯
E1= 0.2659
¯
E2= 0.2659
±
¯
E1= 0.0545
¯
E2= 0.0643
2
¯
E1= 0.1073
¯
E2= 0.2141
± ±
ν12 = 0.2942
ν21 = 0.2942
ν12 = 0.2901
ν21 = 0.2901
ν12 = 0.1394
ν21 = 0.1645
2ν12 = 0.0690
ν21 = 0.1376
kr
kθkr= 0.85 ×1071kθ= 1.121 ×109
2α= 17.5
θ= 109ψ= 35.5
F=EA
LL=kL k (= EA
L)LL
E
A
k1=E1ta
b=¯
E1a
bk2=E2tb
a=¯
E2b
a
a b
a= 0.61 b= 0.46
E1= 0.3166 E2= 0.3736
¯
E1= 0.0545 ¯
E2= 0.0643
k1= 0.0723
k2= 0.0484
2
krkθ
kr= 1.646 ×1071kθ= 1.677 ×1092
2
α= 48.15θ= 82.92ψ= 48.54
¯
E1= 0.1073 ¯
E2= 0.2141
2
ν12 =ν21
2ν12 < ν21
˜
E1˜
E2
θ α λ
λ0.4 2.8
λ= 1.2507,2.495,0.5061,0.479 2
λ˜
E1˜
E2
θ α
λ=l2kr
12kθ
˜
E1=E1t
kr
˜
E2=E2t
kr
l t
λ ν12 ν21
˜ν12 =ν12 ˜ν21 =ν21 θ
α λ =l2kr
12kθ
E1=E2ν12 =ν21 2 E1< E2ν12 < ν21
θ= 120α= 0
2
2
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mose2mos2wse2
2
c2F,
c2F,
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The generation of sustainable and stable semiconductors for solar energy conversion by photoredox catalysis, for example, light-induced water splitting and carbon dioxide reduction, is a key challenge of modern materials chemistry. Here we present a simple synthesis of a ternary semiconductor, boron carbon nitride, and show that it can catalyse hydrogen or oxygen evolution from water as well as carbon dioxide reduction under visible light illumination. The ternary B-C-N alloy features a delocalized two-dimensional electron system with sp(2) carbon incorporated in the h-BN lattice where the bandgap can be adjusted by the amount of incorporated carbon to produce unique functions. Such sustainable photocatalysts made of lightweight elements facilitate the innovative construction of photoredox cascades to utilize solar energy for chemical conversion.
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An analytical framework has been developed for predicting the equivalent in-plane elastic moduli (longitudinal and transverse Young’s modulus, shear modulus, Poisson’s ratios) of irregular auxetic honeycombs with spatially random variations in cell angles. Employing a bottom up multi-scale based approach, computationally efficient closed-form expressions have been derived in this article. This study also includes development of a highly generalized finite element code capable of accepting number of cells in two perpendicular directions, random structural geometry and material properties of irregular auxetic honeycomb and thereby obtaining five in-plane elastic moduli of the structure. The elastic moduli obtained for different degree of randomness following the analytical formulae have been compared with the results of direct finite element simulations and they are found to be in good agreement corroborating the validity and accuracy of the proposed approach. The transverse Young’s modulus, shear modulus and Poisson’s ratio for loading in transverse direction (effecting the auxetic property) have been found to be highly influenced by the structural irregularity in auxetic honeycombs.
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Since the advent of graphene, other 2D materials have garnered interest; notably the single element materials silicene, germanene, and stanene. We investigate the ballistic current-voltage (I-V) characteristics of armchair silicene and stanene armchair nanoribbons (AXNRs with X = Si, Sn) using a combination of density functional theory and non-equilibrium Green's functions. The impact of out-of-plane electric field and in-plane uniaxial strain on the ribbon geometries, electronic structure, and (I-V)s are considered and contrasted with graphene. Since silicene and stanene are sp2/sp3 buckled layers, the electronic structure can be tuned by an electric field that breaks the sublattice symmetry, an effect absent in graphene. This decreases the current by. 50% for Sn, since it has the largest buckling. Uniaxial straining of the ballistic channel affects the AXN Relectronic structure in multiple ways: it changes the bandgap and associated effective carrier mass, and creates a local buckling distortion at the lead-channel interface which induces a interface dipole. Due to the increasing sp3 hybridization character with increasing element mass, large reconstructions rectify the strained systems, an effect absent in sp2 bonded graphene. This results in a smaller strain effect on the current: a decrease of 20% for Sn at 15% tensile strain compared to ∼75% decrease for C.
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2D crystals based on transition metal dichalcogenides (TMDs) provide a unique platform of novel physical properties and functionalities, including photoluminescence, laser, valleytronics, spintronics, piezoelectric devices, field effect transistors (FETs), and superconductivity. Among them, FET devices are extremely useful because of voltage-tunable carrier density and Fermi energy. In particular, high density charge accumulation in electric double layer transistor (EDLT), which is a FET device driven by ionic motions, is playing key roles for expanding the functionalities of TMD based 2D crystals. Here, we report several device concepts which were realized by introducing EDLTs in TMDs, taking the advantage of their extremely unique band structures and phase transition phenomena realized simply by thinning to the monolayer level. We address two kinds of TMDs based on group VI and group V transition metals, which basically yield semiconductors and metals, respectively. For each system, we first introduce peculiar characteristics of TMDs achieved by thinning the crystals, followed by the related FET functionalities.
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From first-principles calculations, the effects of h-BN and AlN substrates on the topological nontrivial properties of stanene are studied with different strains. We find that the quantum spin Hall phase can be induced in stanene film on a h-BN substrate under a tensile strain of between 6.0% and 9.3% with a stable state confirmed by the phonon spectrum, while for stanene on 5 × 5 h-BN, the quantum spin Hall phase can be preserved without strain. However, for stanene on a AlN substrate, the quantum spin Hall phase cannot be found under compressive or tensile strains less than 10%, while for 2 × 2 stanene on 3 × 3 AlN, the compressive strain needed to induce the quantum spin Hall phase is just 2%. These theoretical results will be helpful in understanding the effect of substrate and strain on stanene and in further realizing the quantum spin Hall effect in stanene on semiconductor substrates.