Structural and Electrical Properties of Bilayer SiX (X= N, P, As and Sb)
aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Structural and Electrical Properties of Bilayer SiX (X= N, P, As and Sb)
Nayereh Ghobadi
Department of Electrical Engineering, University of Zanjan, Zanjan, Iran
Shoeib Babaee Touski ∗ Department of Electrical Engineering, Hamedan University of Technology, Hamedan, Iran
In this work, the structural, electrical, and optical properties of bilayer SiX (X= N, P, As, and Sb)are studied using density functional theory (DFT). Five different stacking orders are considered forevery compound and their structural properties are presented. The band structure of these materialsdemonstrates that they are indirect semiconductors. The out-of-plane strain has been applied totune the bandgap and its electrical properties. The bandgap increases with tensile strain, whereas,compressive strain leads to semiconductor-to-metal transition. The sensitivity of the bandgap tothe pressure is investigated and bilayer SiSb demonstrates the highest bandgap sensitivity to thepressure. These structures exhibit Mexican hat-like valence band dispersion that can be approvedby a singularity in the density of states. The Mexican-hat coefficient can be tuned by out-of-planestrain. Optical absorption of these compounds shows that the second and lower valence bands dueto the high density of states display a higher contribution to optical transitions.
PACS numbers:
I. INTRODUCTION
Two dimensional (2D) materials have becomethe head of research after exfoliation of graphene .The 2D structures of the other members ofthe group-IV atoms such as silicene, germanene,stanene, and plumbene have been reported theo-retically and experimentally . All these mono-layers demonstrate a Dirac cone with a near to zerobandgap. On the other hand, group-V monolayershave been extensively studied in both theoreticaland experimental works . Among them, phos-phorene attracts huge research interest in 2D ma-terials due to its proper bandgap, high carrier mo-bility, and excellent transport properties . Afterthat, antimonene was introduced as an interesting2D material with air stability .The combinations of group-IV and V atoms canundertake superior electrical properties of bothgroups. Barreteau et al have built the bulk struc-ture of layered SiP, SiAs, GeP, and GeAs. Thelayered configurations of these materials demon-strate that these materials can be exfoliated into2D structures. The easy exfoliation of these mate-rials have been approved experimentally . Themonoclinic crystal of GeAs and SiAs has layeredstructure with C2/m space group . Monolayersof GeAs and SiAs can be exfoliated from the bulkcounterparts due to low interlayer energy . Bothmonolayers demonstrate a bandgap around 2 eV .The group IV-V monolayer compounds demon- strate a hexagonal lattice (V-IV-IV-V) with P6m2space group . These hexagonal compounds aresemiconductors expect CBi and PbN with metal-lic phase. The structural stabilities and electronicproperties of IV-V monolayers with A B formula(A=C, Si, Ge, Sn, Pb; B=N, P, As, Sb, Bi)have been analyzed theoretically . Single-layergroup IV-V compounds demonstrate fascinatingphotocatalytic activity , thermoelectric ? ,mechanical , and electrical properties . Theelectrical properties of SiX (X=N, P, As, andSb) monolayers demonstrate that these materialsare semiconductors with an indirect band gap .These compounds have been reported as promisingcandidates for efficient thermoelectric applications.Li et al for the first time have exfoliated 2DGeP from the bulk monoclinic structure. Chenget al have reported the exfoliation energy of SiP,SiAs, GeP and GeAs which are of about 0.26 J / m ,0.27 J / m , 0.34 J / m and 0.37 J / m , respectively.The exfoliation energy of SiP and SiAs are lowerthan graphite (0.32 J / m ), which confirms the ex-perimental feasibility of their monolayers.Field effect transistors (FET) based on IV-Vhave been introduced as a candidate for nano-electronic applications, however, their performanceis limited by their low mobility. Guo et al havereported that the hole mobility of GeAs basedFET at room temperature can reach 100 cm / Vs.Monolayer GeP based FET is a p-type transistorand exhibits a I on /I off ratio in the range of 10 . AA1 AA2 AB1 AB2 AB3(a)(a)(b)
FIG. 1: Bilayer SiX from (a) side, and (b) top view. The Si and X atoms are indicated by blue and yellow colors,respectively.
The experimental results demonstrate that whilethe mobility of GeP can be enhanced when thethickness rises from ultrathin to bulk, the I on /I off ratio is reduced .Tuning the electrical and optical properties ofthe multilayer structures for their potential appli-cation in electro-mechanical devices, tunable pho-todetectors, and lasers is a challenge and can bedone by changing stacking order, interlayer spac-ing, applying strain and electric field . Ithas been shown that applying a vertical elec-tric field can open a small bandgap even in bi-layer graphene . Furthermore, it has been re-ported that a vertical electric field in the rangeof 0 . − . / ˚A leads to a semiconductor-to-metaltransition in bilayer TMDs . While this methodis promising, it has practical problems such as theneed for a very large electric field. On the otherhand, it has been shown that the band structureof bilayer TMDs can be effectively modified by theapplication of vertical strain To the best of our knowledge, there is not a com-prehensive study on the electrical and optical prop-erties of different stacking orders of bilayer SiX(X=N, P, As, and Sb). The bilayer SiX will surelyenrich the family of the 2D materials with fascinat-ing electrical properties and So it is necessary toinvestigate the electrical and optical properties in these materials. In addition, applying out-of-planestrain is a powerful method to tune the bandgapand electrical properties of 2D materials. There-fore, in this work, the effect of vertical strain onthe electrical properties of bilayer SiX (X=N, P,As, and Sb) is studied using density functionaltheory. Five different stackings are investigatedfor every compound and their electrical propertiesare discussed. The band gaps of these materialsdecrease gradually with out-of-plane compressivestrain and semiconductor-to-metal transition oc-curs at a specific pressure. This transition pressuredepends on the stacking order of layers. This widerange (1.7–0.0 eV) bandgap tuning can be utilizedin various applications.
II. COMPUTATIONAL DETAILS
Density functional calculations are performedusing the SIESTA package . The generalizedgradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) functional is utilized forthe exchange-correlation term. A Monkhorst-Packk-point grid of 21 × × ζ plus polarization basis-set is used. Thetotal energy is converged to better than 10 − eV. TABLE I: The lattice constant ( a ), the interlayer distance ( d int ), the distance between Si atoms ( d Si − Si ), thedistance between X atoms ( d X − X ), Si-X ( d Si − X ) bond length and elastic constants ( C and C ) of SiX bilayerswith different stacking orders.Stackingorder a (˚A) d int (˚A) d Si − Si (˚A) d X − X (˚A) d Si − X (˚A) E b (eV) C (N/m) C (N/m)SiN AA1 2.908 3.222 2.396 3.572 1.778 -1.341 558.45 136.12AA2 2.909 2.716 2.401 3.569 1.78 -1.439 554.42 123.27AB1 2.907 3.188 2.398 3.574 1.779 -1.338 557.48 135.65AB2 2.907 2.932 2.403 3.583 1.78 -1.383 519.66 123.91AB3 2.909 2.743 2.399 3.568 1.779 -1.416 555.37 138.46SiP AA1 3.536 3.596 2.357 4.416 2.287 -1.51 288.98 57.09AA2 3.537 3.039 2.357 4.415 2.287 -1.609 287.86 56.79AB1 3.537 3.587 2.356 4.412 2.287 -1.509 288.72 56.01AB2 3.541 2.998 2.361 4.417 2.289 -1.639 279.62 55.97AB3 3.54 3.002 2.359 4.416 2.289 -1.629 288.36 53.94SiAs AA1 3.683 3.71 2.348 4.576 2.402 -1.809 254.50 54.86AA2 3.686 3.106 2.346 4.576 2.403 -1.936 252.78 51.62AB1 3.683 3.713 2.347 4.576 2.402 -1.809 254.34 54.42AB2 3.693 2.994 2.347 4.565 2.405 -1.965 241.08 54.63AB3 3.688 3.065 2.347 4.574 2.404 -1.953 252.88 48.73SiSb AA1 3.978 4.074 2.342 4.819 2.609 -2.042 197.94 49AA2 3.982 3.273 2.341 4.814 2.611 -2.228 194.86 41.31AB1 3.978 4.069 2.342 4.819 2.609 -2.043 197.63 49.19AB2 3.994 3.145 2.342 4.799 2.614 -2.277 190.01 32.75AB3 3.987 3.222 2.34 4.81 2.612 -2.252 196.06 34.95 The geometries are fully relaxed until the force oneach atom is less than 0.01 eV/˚A. A vacuum re-gion of 30 ˚A is added to avoid interactions in thenormal direction. The van der Waals (vdW) inter-action between layers is treated using Grimme’scorrection to the PBE functional . To visual-ize the atomic structures, XCrySDen package hasbeen used . The vertical strain is defined as, ε = ( d − d ) d (1)where d and d are the equilibrium and de-formed interlayer distances, respectively. The ap-plied pressure (P) is calculated from the energycost per unit area for decreasing the interlayer dis-tance by following equation , P = ( E − E )( d − d ) A (2)where A is the area of the unit cell, and E and E are the energies of the equilibrium and deformed structures. The effective mass of the carriers iscalculated by using the following equation , m ∗ = ~ / (cid:0) ∂ E/∂k (cid:1) (3)Here, ~ is the reduced Planck constant, E and kare the energy and wave vector of conduction bandminimum and valence band maximum. The ab-sorption coefficient is calculated using the energydependent dielectric functions. In the optical ab-sorption simulation, the absorption coefficient canbe calculated by α ( ω ) = √ ωc [ q ε ( ω ) + ε ( ω ) − ε ( ω )] (4)where ω , c , ε ( ω ), and ε ( ω ) are the angular fre-quency of light, the speed of light, and the real andimaginary parts of the dielectric function, respec-tively. M K M K-3-2-10123 E - E F [ e V ] SiAs-AB2
M K M K-3-2-10123 E - E F [ e V ] SiN-AB2
M K M K-3-2-10123 E - E F [ e V ] SiP-AB2
M K M K-3-2-10123 E - E F [ e V ] SiSb-AB2
PDOS
TotalSi-3sSi-3pN-2sN-2p
SiN-AB2
PDOS
TotalSi-3sSi-3pP-3sP-3p
SiP-AB2 PDOS
TotalSi-3sSi-3pAs-4sAs-4p
SiAs-AB2 PDOS
TotalSi-3sSi-3pSb-5sSb-5p
SiSb-AB2
FIG. 2: The band structures and PDOS of bilayer SiX with AB2 stacking order.
III. RESULTS AND DISCUSSION
The schematic of five different stackings hasbeen displayed in top and side views in Fig. 1.Two stacking categories are AA and AB which inAA stackings, the top layer is located exactly onthe underlying layer, and in AB stackings, the toplayer is shifted relative to the bottom layer. Thestructural properties of the bilayers are listed inTable I. The lattice constants are almost equal forthe five stackings of each material which impliesthat stacking order has a negligible effect on thelattice constant. On the other hand, interlayerdistance highly depends on the stacking configu- ration. The AA2 stacking of SiN has the lowestinterlayer distance, whereas, the lowest one is forAB2 stacking in other compounds. The highest d int is for AA1 and AB1 stackings and the inter-layer distance is approximately the same in thesetwo stackings. d Si − Si , d Si − X and d X − X similar tolattice constant demonstrate a low dependency onthe stacking configuration. Binding energy can becomputed as , E b = E Bilayer − × E Monolayer (5)where E Bilayer and E Monolayer are the total energyof bilayer and monolayer SiX, respectively. Thebinding energy decreases with the atomic number
TABLE II: The electrical properties of the different stacking orders of bilayer SiX. The band gap ( E g ) is in eVunit. The strain ( ε tran ) and pressure ( P tran ) required for semiconductor-to-metal transition are in percent andGPa, respectively. The effective masses at Γ-point of valence band, and K- and M-valleys of the conduction bandare in m unit. The Mexican-hat energy ( E M Γ ) and coefficient ( M Γ ) at Γ-point of the valence band are in eVand eV˚A unit, respectively.Stackingorder E g ε tran P tran m c, ∗ K → M m c, ∗ K → Γ m c, ∗ M → Γ m c, ∗ M → K m v, ∗ Γ → K m v, ∗ Γ → M E M Γ M Γ SiN AA1 1.401 -18 16.14 0.585 0.553 1.081 0.383 1.091 0.989 0.331 0.696AA2 1.232 -36 42.47 0.538 0.535 1.045 0.371 1.143 1.051 0.206 0.506AB1 1.413 -20 19.73 0.588 0.56 1.081 0.377 1.099 0.988 0.324 0.681AB2 1.434 -20 16.91 0.669 0.57 1.264 0.416 1.163 1.079 0.264 0.596AB3 1.262 -14 10.82 0.634 0.589 1.09 0.388 1.19 1.087 0.221 0.527SiP AA1 1.064 -9 4.04 0.434 0.461 2.321 0.141 1.729 1.402 0.021 0.134AA2 0.768 -6 2.50 0.382 0.423 2.839 0.142 1.436 1.289 0.002 0.025AB1 1.05 -8 3.42 0.434 0.467 2.144 0.14 1.683 1.364 0.021 0.138AB2 0.742 -6 1.14 0.501 0.496 2.412 0.134 1.577 1.318 0.001 0.017AB3 0.731 -6 1.96 0.484 0.52 2.338 0.138 1.459 1.282 0.001 0.019SiAs AA1 1.258 -10 4.47 0.428 0.46 6.237 0.133 1.848 1.461 0.009 0.096AA2 0.92 -9 4.1 0.379 0.427 2.804 0.136 1.004 0.884 0.008 0.101AB1 1.263 -10 4.36 0.427 0.463 5.468 0.134 1.829 1.451 0.01 0.097AB2 0.773 -6 1.65 0.503 0.518 2.953 0.125 0.978 0.822 0.005 0.068AB3 0.879 -8 2.87 0.47 0.506 5.621 0.132 0.999 0.86 0.007 0.089SiSb AA1 0.995 -10 3.99 0.411 0.446 0.411 0.2 0.445 0.459 0.0 0.0AA2 0.476 -4 1.36 0.351 0.408 0.38 0.176 0.782 0.685 0.019 0.225AB1 1.086 -11 4.54 0.419 0.451 0.443 0.243 0.444 0.459 0.0 0.0AB2 0.524 -4 0.92 0.392 0.653 0.875 0.131 0.707 0.59 0.012 0.155AB3 0.568 -5 1.51 0.415 0.445 0.626 0.166 0.748 0.644 0.016 0.188 of group-V elements and the heavier compoundsdisplay lower binding energy. The lowest bindingenergy which corresponds to the most stable struc-ture is for AB2 configuration in all compounds ex-cept SiN that AA2 has the lowest binding energy.As one can observe, the binding energy highly de-pends on the interlayer distance. The lower inter-layer distance results in the lowest binding energy.The elastic constants, C , C and C are alsostudied. C is the same as C and have not beenwritten in the table. The stability of these con-figurations is approved by the born stability crite-ria as: 0 < C , 0 < C and C < C , C .The values of C and C decrease with increas-ing the atomic number. Bilayer SiN demonstratesthe highest elastic constants that are about 50%and 300% larger than bilayer graphene and MoS ,respectively . C shows a dependency with in-terlayer distance and binding energy in most casesand the highest C belongs to the stacking with the highest interlayer distance and largest bindingenergy.The band structures of the bilayer SiX with AB2configuration are depicted in Fig. 2. The AB2stacking which is the most stable configuration isselected as a sample. All of them are indirectsemiconductor where conduction band minimum(CBM) is located at M-valley and valence bandmaximum (VBM) happens at Γ or a point near toΓ (we have called this point as Γ ∗ ). The creation ofMexican-hat is the reason of the moving of VBMfrom Γ- to Γ ∗ -point. The values of the band gapsfor different stackings of SiX compounds are listedin Table II. The size of the band gaps of the bi-layer SiN are distributed from 1.232eV in AA2 to1.434eV in AB2 stacking. The band gap of BilayerSiN displays a low dependency on the stacking or-der. On the other hand, the band gap of bilayerSiSb highly depends on the stacking orders. Theband gap of AB1 stacking of SiSb is 1.086 eV that FIG. 3: Absorption coefficient and imaginary part ofthe dielectric function of AB2 stacked bilayer SiX. is approximately two times as large as the bandgap of the AA2 stacking. The band gap of heav-ier compounds have a higher dependency on thestacking order. We also observed a high depen-dency of the band gap on the stacking order inbilayer antimonene . The highest band gap in Bi-layer SiN is 1.434eV for AB2 stacking. After AB2stacking, AB1 and AA1 stackings have the high-est band gap. In three other materials, SiP, SiAsand SiSb, AA1 and AB1 stackings approximatelyhave the same bandgap and demonstrate the high-est band gap. The lowest band gap is one of theAA2, AB2 or AB3 stackings that is different forvarious compounds. For example, the lowest bandgap in SiN is 1.232 eV in AA2 stacking, whereas,in SiAs is for AB2 stacking.All structures exhibit the Mexican-hat disper-sion in the top of the valence band that ismore noticeable in bilayer SiN. The value of theMexican-hat coefficient can be obtained with : M = ∆ E/ ∆ K , where ∆ E and ∆ K are the energyand momentum difference between the Γ-point andthe valence band maximum. The values of the Mexican-hat energies and coefficients for differentstackings of SiX compounds are listed in Table II.Bilayer SiN demonstrates the highest Mexican-hatcoefficient of 0.696 eV˚A at AA1 stacking. The dif-ferent stackings display different Mexican-hat co-efficients. Mexican-hat energies for bilayer SiN aredistributed from 0.206 to 0.324eV. On the otherhand, despite the negligible E M of other com-pounds, some stackings demonstrate a relativelyhigh Mexican-hat coefficient.For better understanding of the contribution ofthe atoms and orbitals on the band structure, par-tial density of states (PDOS) for AB2 stackings areplotted along with the band structures in Fig. 2.The singularity in the valence band especially inSiN is obvious. There also exist the singularity inthe valence band of the other compounds but theirvalues are small. These singularities are come fromthe Mexican-hat in the VBM. As one can observe,the p orbitals of two atoms have the main contri-bution to the both conduction and valence bandedges.While the conduction band minimum is locatedat M-valley, energy of the K-valley in some of theconfigurations is close to M-valley and K-valleycontributes to the conduction band minimum. Onthe contrary, the valence band maximum is lo-cated at Γ ∗ -point and the energy of the M- andK-points are much lower. The effective masses forthe M- and K-valleys of the conduction band andΓ ∗ -point of the valence band are listed in TableII. Two effective masses at K-valley in the conduc-tion band ( m cK → Γ and m cK → M ) demonstrate al-most the same values in the most of configurations.On the other hand, M-valley presents two differ-ent effective masses, m cM → Γ and m cM → K . m cM → Γ is approximately three times as large as m cM → K in bilayer SiN. Their ratio reaches to more thanone order of magnitude in SiP, and SiAs showsthe highest difference between these two effectivemasses. On the other hand, SiSb behaves differ-ently and the difference between these two effec-tive masses highly depends on the stacking order.Two effective masses are also calculated at Γ-pointof the valence band that are approximately thesame. These materials exhibit a high hole effec-tive mass. SiP displays the highest hole effectivemass, whereas, SiSb has the lowest one close to itselectron effective mass.The imaginary part of the dielectric functions( ε ) determines the optical absorption . Forthis reason, ε as a function of energy for four com- -10 0 10 Strain [%] E g [ e V ] AA1AA2AB1AB2AB3
SiN -10 0 10
Strain [%] E g [ e V ] AA1AA2AB1AB2AB3
SiP -10 0 10
Strain [%] E g [ e V ] AA1AA2AB1AB2AB3
SiAs -10 0 10
Strain [%] E g [ e V ] AA1AA2AB1AB2AB3
SiSb
FIG. 4: The band gap variation of different stacking orders of bilayer SiX as a function of vertical strain.
P [GPa] E g [ % ] SiNSiPSiAsSiSb
P [GPa] E g [ e V ] AB3 FIG. 5: Percentage change in bandgap with the appliedpressure for AB3 stacked bilayer SiX as a function ofthe applied pressure. The inset figure shows the valueof the bandgap versus the applied pressure. pounds are depicted in Fig. 3. ε is zero for en-ergy lower than 2 eV and is enhanced after that.SiSb with lower bandgap demonstrates ε at lowerenergy whereas in SiN with the highest bandgap, ε starts at higher energy. As one can observe,the band gaps of these compounds are distributedin the range of 0.476 to 1.434eV that are lowerthan those energies that ε rises. The peaks inthe conduction and valence bands of DOS deter-mines optical absorption. In SiN, ε is compatiblewith absorption coefficient and is associated withthe peaks of DOS in the valence band and the con- duction band. Three other compounds, SiP, SiAsand SiSb, have a single band in the valence bandedge that limits the band gap. As one can see,this band doesn’t create a high DOS and doesn’thighly contribute to the optical absorption. DOSdisplays a maximum after second band and thesebands contribute to ε . Bilayer SiP, SiAs and SiSbdemonstrate the same maximum value of ε witha shift in energy, whereas, SiN shows much lowervalue. The optical absorption as a function of en-ergy for all bilayers are shown in Fig. 3. Theoptical absorption is compatible with ε .Out-of-plane strain has been proposed as a pow-erful method to modify the electrical properties ofbilayers or hetero-structures . We have ap-plied the out-of-plane strain to all stackings andtheir electrical properties are studied. The varia-tion of the band gap of the compounds as a func-tion of vertical strain is plotted in Fig. 4. Theband gaps are enhanced with tensile strain. On theother hand, vertical compressive strain decreasesthe band gap values and semiconductor-to-metaltransition occurs at a specific strain in compres-sive regime. The required strain for phase transi-tion ( ε trans ) and its counterpart pressure ( P trans )are listed in Table II. ε trans is too high for bilayerSiN,for example, it reaches to -36% for AA2 stack-ing that needs 42.47 GPa of pressure. ε trans de-pends on the value of the band gap and a largerband gap results in a higher ε trans . Only AA2stacking of SiN doesn’t obey from this theorem. P trans of bilayer SiN are distributed between 10and 20 GPa, except AA2 stacking. SiP, SiAs and -10 0 10 Strain [%] -4-2024 E C - E F [ e V ] E CM E CK E C E VK E V SiN-AB2 -10 0 10
Strain [%] -2-1012 E C - E F [ e V ] E CM E CK E C E VK E V SiSb-AB2 -10 0 10
Strain [%] -3-2-10123 E C - E F [ e V ] E CM E CK E C E VK E V SiP-AB2 -10 0 10
Strain [%] -3-2-1012 E C - E F [ e V ] E CM E CK E C E VK E V SiAs-AB2
FIG. 6: The energy of the valleys in the conduction and valence bands for various strain in AB2 stacking of bilayerSiX.
SiSb demonstrate a lower ε trans and P trans . Onecan observe, the band gap of all stackings of bi-layer SiN are close to each other under differentstains and just AB3 stacking exhibits a lower bandgap at compressive strain. In SiP, SiAs and SiSb,the band gap of AA1 and AB1 configurations varyclose to each other. Furthermore, the three struc-ture with more stability (AA2, AB2 and AB3) varysimilarly with strain. The difference between thesetwo groups increases for heavier compound. As onecan observe, SiSb demonstrates a large differencebetween these two groups.These structures can be introduced as a pres-sure sensor. In order to investigate the feasibilityof semiconductor to metal transition and possibleapplication of the structures as a pressure sensor inexperiments, the applied pressure (P) is calculatedand the sensitivity of the band gap on the appliedpressure is studied. The band gap variation versusthe applied pressure is plotted in Fig. 5 for AB3stacking. The plot is almost linear for all of thestructures and the pressure range is easily achiev-able experimentally. SiSb displays the highest sen-sitivity to the pressure and the bandgap closes ata low pressure. The band gap of SiSb closes at apressure lower than 2GPa. The sensitivity of theband gap decreases with decrement of X atomicnumber. SiN demonstrates the lowest band gapsensitivity but remains a semiconductor until thepressure of 10GPa. The transition pressure de-creases as the atomic number of X atom increases. This is due to the increased delocalization of theatomic orbitals, which leads to reduced interactionbetween Si and X atoms and a lower transitionpressure. This trend is also observable in the bandgap variation with X atomic number [See Table II].The energy of the effective valleys in the con-duction and valence bands for various strains areplotted in Fig. 6. The behavior of all stackings arethe same and AB2 stacking is plotted as a sample.The figures demonstrate the CBM and VBM arelocated at M- and Γ-points, respectively. The ener-gies of K-valleys in the conduction band ( E CK ) be-come closer to E CM under tensile strain. However,the energy of Γ-valley in the conduction band andK-point in the valence band are far from the CBMand VBM, respectively. Therefore, these two val-leys don’t contribute to electrical properties. E CM decreases with applying compressive strain and inthe same time, E V Γ rises. These two bands inter-sect at ε trans . Therefore, the CBM and VBM getcloser to Fermi level and electron and hole densityincrease exponentially that results in increment ofthe current.The compounds demonstrate Mexican-hat dis-persion at the valence band that can be affectedby the strain. Mexican-hat energy and coefficientas a function of strain are plotted in Fig. 7. SiSb,SiAs and SiSb display a comparable Mexican-hatcoefficient with SiN, whereas, their Mexican-hatenergies are much lower than SiN. One can ob-serve that the value of the Mexican-hat energy in -10 0 10 Strain [%]
AA1AA2AB1AB2AB3
SiN -10 0 10
Strain [%]
AA1AA2AB1AB2AB3
SiP -10 0 10
Strain [%]
AA1AA2AB1AB2AB3
SiAs -10 0 10
Strain [%]
AA1AA2AB1AB2AB3
SiSb -10 0 10
Strain [%] E M [ e V ] AA1AA2AB1AB2AB3
SiN -10 0 10
Strain [%] E M [ e V ] AA1AA2AB1AB2AB3
SiP -10 0 10
Strain [%] E M [ e V ] AA1AA2AB1AB2AB3
SiAs -10 0 10
Strain [%] E M [ e V ] AA1AA2AB1AB2AB3
SiSb
FIG. 7: The Mexican-hat energy and coefficient of SiX bilayers as a function of out-of-plane strain. The top andbottom rows stand for energy and coefficient of the Mexican-hat, respectively.
SiN is approximately one order of magnitude largerthan the others that is compatible with Table II.Mexican-hat energy and coefficient behave simi-larly in a compound. For example, in the bilayerSiN Mexican-hat energy and coefficient increasewith the tensile strain and decrease in compressiveregime. However, Mexican-hat properties in othercompounds show a minimum at equilibrium andincrease with applying both compressive and ten-sile strains. Mexican-hat vanishes at tensile strainfor SiAs and SiSb bilayers. AA1 and AB1 stack-ings demonstrate a larger Mexican-hat propertiesthan the other stackings in the SiN and SiP andtheir differences rises at compressive stain.
IV. CONCLUSION
The structural, electrical and optical proper-ties of five different stacking orders of bilayer SiX(X=N, P, As and Sb) are studied. All these mate-rials are indirect semiconductors where the CBMand VBM are located at M- and Γ ∗ -points, respec-tively. The Mexican-hat is obvious from the bandstructures and the singularity in the DOS confirmsthe existence of the Mexican-hat. SiN stackingsdemonstrate a considerable Mexican-hat disper-sion, whereas, it is negligible for other compounds.In the following, out-of-plane strain has been ap-plied to tune the electrical properties. The bandgap increases with a rise in the tensile strain and asemiconductor-to-metal transition occurs at com-0pressive strains. SiSb demonstrates the highestband gap sensitivity to the pressure, whereas, SiNhas the lowest band gap sensitivity and closes athigher level of pressure. At high tensile strain, theenergy of the K-valley gets closer to the M-valleyand contributes to the CBM. SiN displays a highMexican-hat coefficient that increases with ten-sile strain and decreases with compressive strain. Other compounds also indicate a high Mexican-hat coefficient but the energy of the Mexican-hatis small for them. The optical absorption is alsostudied where the peaks in the conduction and va-lence bands of DOS determines optical absorption.The single band in the valence band edge of SiP,SiAs and SiSb has a low DOS and a little contri-bution to the optical absorption. ∗ Electronic address: [email protected] K. S. Novoselov, A. K. Geim, S. V. Morozov,D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva,and A. A. Firsov, science , 666 (2004). S. Cahangirov, M. Topsakal, E. Akt¨urk, H. S¸ahin,and S. Ciraci, Physical review letters , 236804(2009). Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu,P. Tang, W. Duan, and S.-C. Zhang, Physical re-view letters , 136804 (2013). X.-L. Yu, L. Huang, and J. Wu, Physical Review B , 125113 (2017). P. Vogt, P. De Padova, C. Quaresima, J. Avila,E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet,and G. Le Lay, Physical review letters , 155501(2012). J. Yuhara, B. He, N. Matsunami, M. Nakatake, andG. Le Lay, Advanced Materials , 1901017 (2019). H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu,D. Tom´anek, and P. D. Ye, ACS nano , 4033(2014). A. Castellanos-Gomez, L. Vicarelli, E. Prada, J. O.Island, K. Narasimha-Acharya, S. I. Blanter, D. J.Groenendijk, M. Buscema, G. A. Steele, J. Alvarez,et al., 2D Materials , 025001 (2014). S. Zhang, Z. Yan, Y. Li, Z. Chen, and H. Zeng,Angewandte Chemie , 3155 (2015). J. Ji, X. Song, J. Liu, Z. Yan, C. Huo, S. Zhang,M. Su, L. Liao, W. Wang, Z. Ni, et al., Nature com-munications , 1 (2016). M. Pumera and Z. Sofer, Advanced Materials ,1605299 (2017). C. Barreteau, B. Michon, C. Besnard, and E. Gian-nini, J. Cryst. Growth , 75 (2016). D. Kim, K. Park, F. Shojaei, T. T. Debela, I. S.Kwon, I. H. Kwak, J. Seo, J. P. Ahn, J. Park, andH. S. Kang, Journal of Materials Chemistry A ,16526 (2019). C. S. Jung, D. Kim, S. Cha, Y. Myung, F. Shojaei,H. G. Abbas, J. A. Lee, E. H. Cha, J. Park, andH. S. Kang, Journal of Materials Chemistry A ,9089 (2018). S. Yang, Y. Yang, M. Wu, C. Hu, W. Shen, Y. Gong,L. Huang, C. Jiang, Y. Zhang, and P. M. Ajayan,Advanced Functional Materials , 1707379 (2018). L. Li, W. Wang, P. Gong, X. Zhu, B. Deng, X. Shi,G. Gao, H. Li, and T. Zhai, Advanced materials ,1706771 (2018). A.-Q. Cheng, Z. He, J. Zhao, H. Zeng, and R.-S.Chen, ACS Appl. Mater. Interfaces , 5133 (2018). Y. Jing, Y. Ma, Y. Li, and T. Heine, Nano letters , 1833 (2017). P. Wu and M. Huang, Phys. Status Solidi B ,862 (2016). L. Zhou, Y. Guo, and J. Zhao, Physica E , 149(2018). B. ¨Ozdamar, G. ¨Ozbal, M. N. C¸ ınar, K. Sevim,G. Kurt, B. Kaya, and H. Sevin¸cli, Phys. Rev. B , 045431 (2018). J.-H. Lin, H. Zhang, X.-L. Cheng, andY. Miyamoto, Physical Review B , 035438(2017). M. Ashton, S. B. Sinnott, and R. G. Hennig, Appl.Phys. Lett. , 192103 (2016). F. Shojaei and H. S. Kang, The Journal of PhysicalChemistry C , 23842 (2016). R. N. Somaiya, Y. A. Sonvane, and S. K. Gupta,Phys. Chem. Chem. Phys. , 3990 (2020). B. Mortazavi and T. Rabczuk, Physica E: Low-dimensional Systems and Nanostructures , 273(2018). A. Grillo, A. Di Bartolomeo, F. Urban, M. Pas-sacantando, J. M. Caridad, J. Sun, and L. Camilli,ACS Applied Materials & Interfaces , 12998(2020). F. Shojaei, B. Mortazavi, X. Zhuang, and M. Azizi,Materials Today Energy , 100377 (2020). K. Zhang and N. Li, RSC Advances , 14225(2020). B. Mortazavi, M. Shahrokhi, G. Cuniberti, andX. Zhuang, Coatings , 522 (2019). J. Guo, Y. Liu, Y. Ma, E. Zhu, S. Lee, Z. Lu,Z. Zhao, C. Xu, S.-J. Lee, H. Wu, et al., Advancedmaterials , 1705934 (2018). Y. Guo, W. Guo, and C. Chen, Applied PhysicsLetters , 243101 (2008). N. Ghobadi, Physica E , 158 (2019). M. Shamekhi and N. Ghobadi, Physica B ,411923 (2020). S. B. Touski and N. Ghobadi, Physica E: Low-dimensional Systems and Nanostructures ,114407 (2020). E. McCann, Phys. Rev. B , 161403 (2006). E. V. Castro, K. Novoselov, S. Morozov, N. Peres,J. L. Dos Santos, J. Nilsson, F. Guinea, A. Geim,and A. C. Neto, Physical review letters , 216802(2007). A. Ramasubramaniam, D. Naveh, and E. Towe,Phys. Rev. B , 205325 (2011). S. Bhattacharyya and A. K. Singh, Phys. Rev. B , 075454 (2012). J. M. Soler, E. Artacho, J. D. Gale, A. Garc´ıa,J. Junquera, P. Ordej´on, and D. S´anchez-Portal, J.Phys.: Condensed Matter , 2745 (2002). J. P. Perdew and A. Zunger, Phys. Rev. B , 5048(1981). S. Grimme, , 1787 (2006). A. Kokalj, , 155 (2003). N. Ghobadi and S. B. Touski, Journal of Physics:Condensed Matter , 085502 (2020). S. B. Touski, M. Ariapour, and M. Hosseini, PhysicaE: Low-dimensional Systems and Nanostructures , 113875 (2020). A. Huang, W. Shi, and Z. Wang, J. Phys. Chem. C , 11388 (2019). F. Mouhat and F. X. Coudert, Phys. Rev. B ,224104 (2014). C. Guoxin, Polymers , 2404 (2014). K. A. N. Duerloo, M. T. Ong, and E. J. Reed, J.Phys. Chem. Lett , 2871 (2012). M. Ariapour and S. B. Touski, Materials ResearchExpress , 076402 (2019). O. Madelung,
Introduction to solid-state theory ,vol. 2 (Springer Science & Business Media, 2012). S. Soleimani-Amiri and S. G. Rudi, Optical Mate-rials , 110491 (2020). T. Zhao, Y. Sun, Z. Shuai, and D. Wang, Chemistryof Materials29