Current-Voltage Characteristics of Borophene and Borophane Sheets
aa r X i v : . [ phy s i c s . c h e m - ph ] J un Current-Voltage Characteristics of Borophene andBorophane Sheets
Sahar Izadi Vishkayi a , and Meysam Bagheri Tagani ∗ a Motivated by recent experimental and theoretical research on a monolayer of Boron atoms,Borophene, current-voltage characteristics of three different Borophene sheets, 2Pmmn, 8Pmmnand 8Pmmm, are calculated using density functional theory combined with nonequilibrium Greenfunction formalism. Borophene sheets with two and eight atoms in a unit cell are considered andbandstructure, electron density and structural anisotropy of them are analyzed in details. Re-sults show that 8Pmmn and 8Pmmm structures which have eight atoms in the unit cell have lessanisotropy than 2Pmmn. In addition, although 8Pmmn shows a Dirac cone in the bandstructure,its current is lower than two others. We also consider a fully hydrogenated Borophene, Boro-phane, and find that the hydrogenation process reduces the structural anisotropy and the currentsignificantly. Our findings reveal that the current-voltage characteristics of the Borophene sheetscan be used to detect the kind and the growth direction of the sample because it is strongly de-pendent on the direction of the electron transport, anisotropy and details of the unit cell of theBorophene.
Discovery of graphene opens a window for fabrication of two-dimensional (2D) materials which possibility of their synthesiswas not predicted before. Graphene has attracted a lot of at-tention in recent years due to its unique properties such as highelectron and thermal conductivity, high mechanical strength, andchemical inertness . After successful synthesis of Graphene, sci-entists searched both experimentally and theoretically to obtainnew 2D materials. Silicene- a silicon analogue to Graphene- wassuccessfully synthesized by Vogt et al. in 2012 . In 2014, the firstreport about the synthesis of Germanene- a Germanium analogueto Graphene- was published by Davila and coworkers . One yearlater, Zhu et al. synthesized a monolayer of Tin, Stanene, underultra-high vacuum condition at room temperature .Mannix and co-workers synthesized Borophene, a 2D-Boronsheet, on Ag (111) under ultrahigh vacuum condition in 2015 .The synthesized structure exhibited a strong anisotropy, further-more, mechanical properties of the sample were substantial. Atthe same time, Feng et.al synthesized 2D-Boron sheets by us-ing molecular beam epitaxy method and observed χ and β phases with triangular lattices. Nonetheless, there is a fun-damental difference between these outstanding works: the ob-tained structure by Mannix is corrugated, whereas, the structureprepared in the second work was completely planner. Accord-ing to the results of Ref. , it was predicted that the freestand- a Address, Department of Physics, Computational Nanophysics Laboratory(CNL), University of Guilan, Po Box:41335-1914, Rasht, Iran; E-mail: m _ bagheri @ guilan . ac . ir ing Borophene has space group of Pmmn and two atoms in theunit cell introduced as 2Pmmn. Freestanding Borophene is notstable due to the existence of a soft mode in long wavelengthin its phonon band structure . Zhou and co-workers showed that two freestanding allotropes of the Borophene, as8Pmmn and 8Pmmm, can be stable. 8Pmmn Borophene hasmassless Dirac fermions which make it similar to graphene butwith anisotropy Fermi velocity of electrons. Afterward, it wasrevealed that the 8Pmmn Borophene can be considered as thefirst single element two-dimensional structure with ionic bond-ing . Borophene has attracted a lot of attention in last yearand its optical properties , electronic properties of its nanorib-bons , transport properties , oxidization effect , surfacehydrogenation , superconductivity , mechanical prop-erties , phonon transport , and its application in batter-ies have been searched recently.In this article, we study the electronic properties of four 2D-Boron structures as corrugated Borophene, 2Pmmn, fully hy-drogenated Borophene, 2HPmmn, stable Borophene with Diraccones, 8Pmmn, and a bilayer Borophene sheet known as 8Pmmm.Their structure differences are examined using density functionaltheory and their responses to external bias voltages are analyzedusing nonequilibrium Green function formalism. To explore therole of the structural anisotropy in details, we consider two dif-ferent scenarios for the current. Indeed, the currents in x- andy-directions are computed and their differences are investigated.Results show that how anisotropy in the structure affects thecurrent-voltage characteristics of the samples. We show that thecurrent of the sample can be used as an analysis tool to verify the RESULTSAND DISCUSSIONS growth direction of the sample and even the kind of the sample.Section 2 is devoted to the method of the simulation and param-eters used to optimize the structures and calculate the current.Results are presented in section 3. Some sentences are given as asummary in the end.
SIESTA package was used to optimize and compute electronicproperties of the considered structures. Perdew-Burke-Ernzerhof(PBE) generalized gradient approximation (GGA) was em-ployed as exchange-correlation functional and norm-conservedTroullier-Martins pseudopotentials were applied to describe theinteraction between the valence and core electrons. A doublezeta-single polarized basis set (DZP) was used and density meshcut-off was set to Ry . The integration in the k-space was per-formed using a × × Monkhorst- Pack k-point mesh centeredat Γ -point. The forces on all atoms were less than − eV / Å, and Å vacuum was considered to eliminate interlayer interactions.Transport properties of the structures were studied usingnonequilibrium Green function formalism. The current was com-puted using Landauer-Büttiker formula as I = q ¯ h Z dET ( E )( f L ( E ) − f R ( E )) , (1)where f α ( E ) = + exp ( E − µα kBT ) is Fermi-Dirac distribution functionof α -electrode, µ L ( R ) = E f ± qV / describes the chemical poten-tial of left (right) lead and V denotes the external bias volt-age. T ( E ) = Tr ( Γ L ( E ) G A ( E ) Γ R ( E ) G R ( E )) is the transmission co-efficient of the structure which is dependent on the couplinggeometry, shape of the central region, and external bias. To com-pute the transmission coefficient, TRANSIESTA code was used.We set k-points perpendicular to the transmission directionand k-points parallel to the transport direction to take intoaccount the two-dimensional structures of Borophene and Boro-phane perfectly. Fig.1 shows four considered structures. 2Pmmn has two atoms ina unit cell that are placed in two different plans, shown by redand blue balls in Fig.1 (a). According to the Mulliken analysis, noelectron transfers between the Boron atoms in the unit cells, sothey are coupled to each other by covalent bonding.Fig.1 (b) shows 2HPmmn structure which Hydrogen atomsare in opposite side of the adjacent horizontal line of Boronatoms. The hydrogenation of Borophene increases the lattice vec-tor along a direction, see table 1, about percent which is consis-tent with results of ref. and reduces the buckling height. Fromthe Mulliken analysis, we find that . electron charge is trans-ferred in B-H bonding. In Fig.1 (c), another structure with highsymmetrical space group Pmmn and 8 Boron atoms in a unit cellis drown. 8Pmmn Borophene was predicted to be stable in free-standing condition . Bezanilla et al. showed that the Boronatoms are classified into two types, inner atoms (red balls) andridge atoms (blue balls). The ridge atoms gain negative charge,about 0.14 e, thus, the inner and ridge atoms are placed next (a) (b)(c) (d)y x y z Fig. 1
The schematic of in-plane (left) and side (right) views of fourconsidered structures. a) 2Pmmn, b)2HPmmn, c)8Pmmn, andd)8Pmmm. The left (right) axis shows the direction of atoms in in-plan(side) view. (color on-line) to each other by ionic bonding. Another structure with 8 Boronatoms in a unit cell with high symmetry space group Pmmm isshown in Fig.1 (d) and known as 8Pmmm. Three types of Boronatoms exist in the structure, ridge atoms ( B R , blue balls), middleatoms ( B m , red balls) and inner atoms ( B I , green balls). B I atomsare pillars of the structure jointing two planes of the Borophene,so that the structure can be thought of as a hexagonal bilayerBorophene. A similar structure has been recently introduced asP6/mmm . Mulliken analysis shows that the B R − B m bondingis covalent whereas, B R − B I bonding is ionic. 0.15e is gainedby each B R atoms while each B I loses 0.3e. Lattice constants(a,b), buckling height (h), total energy ( E cell ) and binding en-ergy ( E B = E freeatom − E cell N , where N is the number of atoms in theunit cell) of the considered structures are listed in table 1. Thedata shows that the 8Pmmm structure has the lowest binding en-ergy which is in good agreement by the achievements of ref. .2Pmmn has lower energy than 2HPmmn. The Borophene syn-thesized by Mannix et.al. was just partially hydrogenated overtimes which can be explained by the different binding energy be-tween 2Pmmn and 2HPmmn. Table 1
The structural properties of the considered structures. a and bare the lattice vectors in x and y direction, respectively, h is the bucklinglength
Structure a(Å) b(Å) h(Å) total energy(eV) binding energy( eVatom )2Pmmn 1.6 2.82 0.91 -158.25914 -6.515592HPmmn 1.87 2.84 0.84 -190.27067 -5.07598348Pmmn 3.26 4.52 2.25 -634.66187 -6.7187548Pmmm 2.89 3.28 4.15 -634.94009 -6.753531
Bandstructures and density of states (DOS) of the structuresare drawn in Fig. 2. 2Pmmn is an anisotropic metal becausethree bands cross the Fermi level in a direction (2 bands along G- RESULTSAND DISCUSSIONS
X and 1 band along Y-S) while it has a big gap in the corrugateddirection (along G-Y and X-S). Our results are consistent with pre-viously published works except that found four metallicbands. The metallic behavior of 2Pmmn is completely supportedby the DOS graph. At E − E F = − eV , a van Hove singularity isobserved in the DOS attributed to the saddle point of the bandstructure in the Γ point . Dirac cone is observed in the bandstructure of 2HPmmn. Although, the electronic configuration of B − H and its band structure look like Graphene, there are somesignificant differences between them . The Dirac cone of 2HP-mmn is anisotropic because of the difference of the band slopesin two sides of the Dirac point. Moreover, the velocity of Fermielectrons in 2HPmmn is 2-4 times more than Graphene ones. Xuet al. showed that the Dirac cone of 2HPmmn comes from thehybridization of p x and p y orbitals, but p z orbitals are responsiblefor the Dirac cone of Graphene. The presence of the Dirac pointis sustained by the DOS at E − E F = . G Y S X G-505 (a) E - E F ( e V ) G Y S X G-505 (b) E - E F ( e V ) G Y S X G-505 E - E F ( e V ) (c) G Y S X G-505 E - E F ( e V ) (d) Fig. 2
The bandstructures and density of states of a) 2Pmmn, b)2HPmmn, c)8Pmmn, and d) 8Pmmm. ) p z orbitals of inner Boron atomsthat make disorder hexagonal pattern . Fig.2 (d) shows theband structure and DOS of 8Pmmm. It is obvious that 8Pmmm isan isotropic metal with a lot of electronic states around the Fermilevel. Metallic behavior of the structure is supported by the DOS.No gap is observed in its DOS spectrum.Electron localization function (ELF) of 2Pmmn and 2HPmmnis drawn in Fig.3. It is clear that in corrugation direction (alongy), the electrons are completely localized, while, localization is a ) ( b ) y zx z Fig. 3
The electron localization function of a) 2Pmmn, and b) 2HPmmn. weaker for 2Pmmn structure. This strong localization of the elec-trons will lead to suppression of the current for 2HPmmn in ydirection (discussed later). In x direction, the electron is delocal-ized in both structures and
ELF = . can be interpreted as thequasi-free electron. Thus, we expect that two structures behaveas a metal in x direction. (a) V(V) I/ L ( A / m ) (b) V(V) I/ L ( A / m ) (c) V(V) I/ L ( A / m ) (d) V(V) I/ L ( A / m ) V(V) I x /I y V(V) I x /I y I x I y Fig. 4
The current-voltage characteristics of a) 2Pmmn, b) 2HPmmn(the inset shows the I x I y for 2Pmmn (solid line) and 2HPmmn (dashedline)), c)8Pmmn (the inset shows the I x I y for 8Pmmn (solid line) and8Pmmm (dashed line)) , and d) 8Pmmm. In order to more analyze the role of structural anisotropy of thesamples, the current-voltage characteristic of the four consideredstructures is shown in Fig.4. For each structure, we consider twoscenarios: the external bias voltage is applied along x-directionor along y-direction. In all cases, the current per length of thecentral region is computed. The current-voltage characteristic of2Pmmn is ohmic, Fig.4a, but its slope is direction-dependent sothat the slope is more for I x . The bonding length along x-directionis shorter than y-one, therefore, the electron transport is moredesirable in this direction. In the other word, the resistivity of the CONCLUSIONS .The current-voltage characteristic of 2HPmmn showsanisotropy. The current is more in x-direction which is ingood agreement with ELF prediction. In addition, it is clearthat the magnitude of the current reduces significantly whenthe Borophene is hydrogenated. The transition from a metal,2Pmmn, to a semi-metal, 2HPmmn, significantly reduces thecurrent. Unlike 2Pmmn case, the current-voltage characteristic of2HPmmn is not linear which is directly related to its bandstruc-ture. Although I x I y increases with voltage, the ratio is moderatedby hydrogenation, see the inset of Fig.4 b. It comes from theincrease of the lattice constant in x-direction. Ref. studiedthe same structure and found that hydrogenation reduces theanisotropy in very low bias voltages. Here, we considered highvoltage regime and found that the structural anisotropy increasesby voltage. (a)x(A) E ( e V ) (b)y(A) E ( e V ) (c)x(A) E ( e V ) (d)y(A) E ( e V ) T y T x T x T y Fig. 5
The projected local density of states and transmissioncoefficients of a) 2Pmmn along x-direction, b) 2Pmmn along y-direction,c)2HPmmn along x-direction , and d) 2HPmmn along y-direction at V = . V . Local density of states (LDOS) projected along transport direc-tion and transmission coefficient at V = . V are shown in Fig.5for 2Pmmn and 2HPmmn. The transmission coefficient in the x-direction is four times more than in the y-one leading to a signifi-cant increase of the current along x in 2Pmmn case. This phe-nomenon comes from the strong coupling of the Boron atomsalong x-direction as it was shown in the ELF analysis. It is ob-vious that the LDOS is more in the left side of the central channelwhich is coupled to the source. In addition, it is observed thatthe coupling of the transport channel to the source is stronger inthe x-direction. One can also observe that the LDOS of y case isnearly discontinuous along the transport direction attributed tothe long bonding length in the y-direction. About 2HPmmn, al-though the appearance of the transmission coefficient is the samein both directions, there are some differences between them. Re- sults show that the transmission coefficient is zero in ε = µ L and ε = µ R and increases between them by the increase of the voltageresulting in the nonlinear increase of the current. LDOS analysisshows that the electron states are more near the drain, right side.In addition, the LDOS spectrum exhibits there are more electronstates when the transport is in the x-direction.The current-voltage characteristic of 8Pmmn is also dependenton the direction of the transport channel. Unlike 2Pmmn case,the ratio of I x to I y is decreased by increase of the voltage so that I x / I y becomes constant in high voltages, inset of Fig.4 c. The cur-rent of the structure is not linear because it is a semimetal. Itis found that the ridge atoms have a better participation in thetransport than the inner ones, so I x > I y . Although both 8Pmmmand 2HPmmn have a Dirac cone, the current of 8Pmmn is morethan 2HPmmn. This phenomenon is related to the small band gapof 8Pmmn in S point of the brillouin zone. Unlike three previouscases, the current of 8Pmmm is nearly isotropic so that I x I y ≃ .Indeed, the results show that the transition from a Borophenewith two atoms in the unit cell to one with eight atoms signifi-cantly shrinks the anisotropy of the structure. There are signif-icant differences between the current of four considered struc-tures, hence, the current-voltage analysis can be used as a goodtool to verify the structure of the synthesized Borophene. (a) x(A) E ( e V ) (b) y(A) E ( e V ) x y Fig. 6
The projected density of states and the transmission coefficientsof 8Pmmn a) along x-direction and b) along y-direction at V = . V . The projected local density of states (LDOS) of 8Pmmn struc-ture is plotted in Fig.6. It is observed that there are more electronstates in the bias window when the transport channel is in thex-direction. Furthermore, T x is more than T y so I x > I y . To specifythe contribution of the inner, ridge and middle atoms in the cur-rent of the 8Pmmm structure, density of states (DOS) is plottedin Fig.7 for V = . V . It is found that the ridge atoms partici-pate in the transport more than the others. In addition, it is clearthat the inner and middle atoms have equal weight in the chargetransport. We have studied the current-voltage characteristics of three dif-ferent Borophene sheets using density functional theory com-bined with nonequilibrium Green function formalism. Resultsshow that the increase of the Boron atoms in the unit cell of theBorophene sheet significantly reduces the structural anisotropy.
EFERENCES REFERENCES -1 -0.5 0 0.5 100.511.522.533.54 E(eV) D O S ( / e V ) (a) -1 -0.5 0 0.5 101234567 (b) E(eV) D O S ( / e V ) innermiddleridge Fig. 7
The density of states of inner (solid lines), middle (dashed lines),and ridge (dash-dotted lines) atoms in 8Pmmm along a) x-direction andb) y-direction at V = . V . Dotted lines show bias window. In addition, a transition from covalent to ionic bonding is ob-served in the Borophene by increase of the unit cell atoms. Struc-tural anisotropy directly affects on the current of the sample sothat the current is dependent on the direction of the transport. Afully hydrogenated Borophene as Borophane is also considered.Our findings show that the hydrogenation can significantly mod-erate the anisotropy but it also reduces the current of the sample.Furthermore, it is observed that the Borophene sheets with Diraccone have lower current than the others.
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