A Computational Study on the Electronic Properties of

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Japanese Journal of Applied Physics 51 (2012) 035101 DOI: 10.1143/JJAP.51.035101

A Computational Study on the Electronic Properties of Armchair Graphene Nanoribbons Confined by Boron Nitride Maziar Noei
, Morteza Fathipour, and Mahdi Pourfath Department of Electrical & Computer Engineering, University of Tehran, Tehran 14395-515, Iran Received November 19, 2011; revised December 25, 2011; accepted January 14, 2012; published online March 2, 2012 In this paper, we present a computational study on the electronic and charge transport properties of armchair boron nitride-confined graphene nanoribbon structures. We compare the electronic bandstructure of hydrogen passivated armchair graphene nanoribbons (AGNRs) with the bandstructure of boron nitride-confined AGNRs. Our study reveals that due to the energy gap opening in (3p+2) AGNRs in these novel hybrid structures and the possibility of realizing parallel arrays of semiconducting and isolating nanoribbons in them, they can be considered as better candidates for electronic applications than hydrogen passivated AGNRs. We also calculate the charge transmission probability and density of states in these nanostructures and investigate their behavior under different biases. In doing so, we have used the non-equilibrium Green’s function formalism to solve the Schro¨dinger equation and have coupled it to a two-dimensional Poisson-solver for treating the electrostatics of the system. # 2012 The Japan Society of Applied Physics

1. Introduction

Low-dimensionality leads to impressive electronic properties, since material characteristics are improved due to certain limitations in the device geometry. In the recent years, two-dimensional (2D) materials have gained extensive attention due to both their interesting basic physics and their possible applications in future generation devices. Among such possibilities, graphene, a 2D lattice of sp2 bonded carbon atoms, has been proposed as an excellent potential candidate for next-generation electronic applications.1) Graphene’s intriguing electronic properties, such as its extremely high carrier mobility,2) long phase coherent lengths and linear energy dispersion relation at Dirac points3) have brought this structure a lot of attention from the device research community. On the other hand, single-layer hexagonal boron nitride (h-BN) is known to be the III–V analog of the widely studied graphene structure and has been fabricated recently.4–6) In this structure, alternating boron and nitrogen atoms closely imitate the way carbon atoms are positioned in graphene where two B3 N3 hexagons are placed above and below each N3 B3 hexagon. The relatively large ionicities of boron and nitrogen atoms cause the optical and electronic properties of graphene and h-BN to be substantially different.7,8) Unlike graphene, which is a zero-gap material, h-BN has a wide band gap (of approximately 5.9 eV) and shows good insulating behavior.8) In order to have a reasonable value of bandgap for electronic applications, graphene and h-BN can both be truncated in one of the dimensions by lithographic patterning9,10) or chemical methods11) to obtain quasi-1D materials known as graphene nanoribbons (GNRs) and boron nitride nanoribbons (BNNRs), respectively. Depending on the direction in which the truncation is performed, nanoribbons can be classified as either armchair nanoribbons or zigzag nanoribbons. Similar to armchair GNRs, armchair BNNRs (ABNNRs) display semiconducting behavior independent of their width.12,13) Contrary to the ZGNRs, however, zigzag BNNRs (ZBNNRs) can be either magnetic or nonmagnetic determined by the details of their edge passivation.14,15) Different nanostructures of graphene such as nanoribbons and quantum dots of graphene have been widely investigated


E-mail address: [email protected]

in the literature.1,16,17) However, recently the discovery of hybrids where graphene and h-BN structures occur as phase separated domains has attracted a lot of attention. These BN–C hybrid nanoribbons are found to be stable at room temperature due to their approximately matched lattice constant18) and exhibit various electronic properties.18,19) It is shown that the electrical properties of these hybrid structures can be easily controlled from insulator to highly conducting films by just tuning the carbon concentration in these nanoribbons.19) Unlike H-terminated nanoribbons, these kinds of structures can form a continuous 2D layer which does not require breaking of bonds.19) This material also provides a natural way of realizing a densely packed parallel array of semiconducting GNRs, which are considered necessary to provide large enough on-currents for circuit applications. The semiconducting characteristics and its extremely high mobility makes armchair GNR a promising structure for many future electronics applications. Studies on semiconducting armchair GNRs have shown that its energy gap is different among three subfamilies due to different bond length relaxations at the interfaces, and in general is inversely proportional to the nanoribbon’s width.9) The (3p+2) AGNRs (where the index refers to the number of honeycombs along the width of nanoribbon) show a very small energy gap and are considered as semi-metallic GNRs. However, it is shown that (3p+2) armchair nanoroads of graphene confined by BNNRs can have a considerable band gap.20) Other differences from H-passivated GNRs are also expected in these structures due to the charge redistribution at the edges of AGNRs confined by BNNRs. This charge redistribution is due to the large difference in electron affinity of nitrogen and boron atoms. In this paper, we present a tight-binding (TB) description of armchair BN–C hybrid nanoribbons and investigate their electronic and charge transport properties for the first time. In order to do so, after proposing the simulated nanostructures and the simulation formalisms in x2, we compare the energy gaps and band structures of two different armchair-edged BN–C nanoribbons and a hydrogen passivated armchair GNR in x3 and show that these hybrid nanostructures can be regarded as appropriate candidates for the next-generation electronic devices. We then study the carrier transport properties of these nanostructures by investigating their transmission probability function (x4) and

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density of states (x5) for a metal–nanoribbon–metal system. The non-equilibrium Green’s function (NEGF) method21) coupled to a 2D Poisson solver will be used to calculate these physical quantities of interest. Finally, we will provide the conclusions in x6. 2. Methodology

The crystal structures for three types of armchair GNRs are shown in Fig. 1. In terms of edge passivation, we have divided an armchair structure into three subgroups: (1) both edges of the GNR are passivated with hydrogen atoms (2H-passivated AGNR), (2) only one edge is passivated with hydrogen, while the other edge is matched to an armchair BNNR (H-BNNR-confined AGNR), and (3) the GNR is completely confined by two BNNRs (2BNNR-confined AGNR). In calculating the bandstructure of these three different structures, we have used the Hubbard model22) and by choosing the unit cells depicted with dashed lines in Fig. 1, we have taken advantage of the simple relations of a 1D chain of unit cells. Since the
bands are far away from the Fermi energy, it is a good approximation to take into account only the pz orbitals of boron, nitrogen and carbon atoms for investigating electronic band structure and electronic properties of nanoribbons.23) The index of each nanoribbon represents the number of honeycombs along its width, i.e., Fig. 1(a) shows a 3-AGNR. The wave function of the whole system can be constructed as 1 X X ikRij ji ¼ pffiffiffiffi e C j j j i; ð1Þ  i i; j

(a)

where  is the normalized coefficient, j j i is the atomic orbital of the ith atom and C j s are the expansion coefficients. The first summation runs over all the unit cells in onedimension infinite system, and Rij shows the position of the jth atom in the ith unit cell. In our system, the functions j j i correspond to the pz orbital of each atom. The band structure can be calculated by solving a matrix eigenvalue equation of the form E¼ n ¼ ½hðkÞ¼ n ;

(b)

ð2Þ

where in matrix representation, ¼n is a N  1 column vector denoting the wavefunction in unit cell number n, and we have24) X ½hðkÞ ¼ ½Hmn eikðdm dn Þ : ð3Þ m

In eq. (3), the matrix ½hðkÞ is N  N in size. dm and dn are the distance of mth and nth unit cell from the reference point, and the wave vector k is defined along the length of the nanoribbon. By only taking the interactions between adjacent unit cells into account, we can rewrite ½hðkÞ as ½hðkÞ ¼ Hself þ Hleft e

ika

þ Hright e

ika

:

ð4Þ

The distance a shown in Fig. 1 is found to be  25) In order to construct the above Hamiltonian 3  1:45 A. ^ pi matrices, we use the transfer integrals  lp ¼ hl jHj (where l and p can represent any boron, nitrogen, or carbon atom). The values of these integrals are chosen according to ref. 18. In our tight-binding description of the presented structures, the relaxations of carbon–carbon bonds at the hydrogen-passivated edges of a GNR are also taken into account accprding to ref. 24. As reported in several previous

(c) Fig. 1. (Color online) The structures of a (a) 2BNNR-confined AGNR, (b) H-BNNR-confined AGNR, and (c) 2H-passivated AGNR. the dashed rectangles show the unit cells chosen for calculation of band structure.

works on GNRs, this edge bond relaxation has a significant influence on the bandgap, effective mass and other qualitative properties of nanoribbons.26,27) We consider a change in  CC equal to  CC ¼ 0:12 CC for edge carbon atoms to model their relaxation.28) Note that graphene has a smaller equilibrium interatomic  than h-BN (1.45 A).  Thus, in the hybrids of distance (1.42 A) graphene and h-BN, the carbon bonds will usually be under tensile stress, which ensures overall planarity of a hybrid C–BN nanostructure.20) This change in the bond lengths of carbon atoms is accounted for in the respective C–C tight binding parameters as well.20)

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(a)

(b)

(c)

Fig. 2. (Color online) Energy dispersion relations vs wave vector for a (a) 2BNNR-confined 3AGNR, (b) H-BNNR-confined 3AGNR, and (c) 2H-passivated 3AGNR.

In order to investigate the nonequilibrium electron density in these nanostructures, we have used Green’s function formalism for the system of a nanoribbon attached to two metallic semi-infinite electrodes. The central nanoribbon is 15 nm long. In calculating the transport characteristics of such a system, a 15-nm-long central nanoribbon is sufficiently long to suppress direct contact tunneling.29) Longer nanoribbons do not change the results obtained in this work, since we have not considered scattering mechanisms in our simulations and the charge transport is assumed to be of ballistic nature.30) The retarded and advanced Green’s functions for this system are defined as GðEÞ ¼ ½ðE þ iÞI  Hribbon  L  R 1 ;

ð5Þ

ð6Þ Gy ðEÞ ¼ ½ðE  iÞI  Hribbon  L  R 1 ; where  is an infinitesimal positive number which moves the poles of G and Gy to the lower half-plane in complex energy. L and R in eqs. (5) and (6) are the self-energy functions, describing the interaction between nanoribbon and the semi-infinite leads. These functions can be calculated as follows: L=R ¼ hyL=R  gL=R  hL=R :

ð7Þ

hL=R corresponds to the transfer matrix between the nanoribbon and the leads, while gL=R represents Green’s function of the leads. The leads’ Green’s function is of a large size, if not infinite, and in refs. 31 and 32 algorithms are presented for reducing these functions to finite-sized matrices. The transmission probability function is then calculated as24) T ðEÞ ¼ trðL GR Gy Þ; ð8Þ L=R ¼ i½L=R  yL=R :

ð9Þ

3. Band Structure of a BN–C Nanoribbon

Figures 2(a)–2(c) show the typical tight-binding band structures of the three types of armchair nanoribbons demonstrated in Figs. 1(a)–1(c). All three structures show semiconducting properties with a direct energy gap. The band structures are calculated for nanoribbons in which the GNR part is 3 honeycombs wide. Our results indicate that

Fig. 3. (Color online) Energy gap comparison for the three presented structures as a function of AGNR width.

among these nanoribbons, the 2H-passivated 3AGNR has the largest energy gap of 1.52 eV, while a 2BNNR-confined 3AGNR has the medium bandgap of 1.31 eV and a HBNNR-confined 3AGNR has the smallest energy gap of all three, with Eg ¼ 1:07 eV. However, this hierarchy is not the same for all of the subfamilies of an AGNR. This can be inferred from Fig. 3, where the values of energy gap for different indices of the AGNR in these three structures are shown. Although the exact hierarchy of these energy gaps may seem insignificant, the important point to notice is the considerable energy gap opening for a 2BNNRconfined AGNR of index (3p+2). Figure 4 clearly shows that a 2BNNR-confined (3p+2)AGNR has an energy gap which can be three to four times larger than that of a 2H-passivated (3p+2)AGNR. In fact, this energy gap is approximately equal to the energy gap of a 2BNNR-confined (3p+4)AGNR. The ab initio simulations in ref. 18 have shown that this bandgap opening in 3p+2 category of 2BNNR-confined AGNRs can be attributed to the large ionic potential difference between boron and nitrogen atoms, which affects the on-site energy of those carbon atoms that are positioned at the edges of that AGNR.20) Opening of an energy gap in the band structure of (3p+2) categories of BNNR-confined AGNRs enables us to fabricate parallel arrays of ultra-thin nanoribbons in the channel region of a

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(a)

Fig. 4. (Color online) Transmission probability comparison for structures of H-passivated AGNR and BNNR-confined AGNR.

single GNRFET in order to obtain substantially larger oncurrents as well as sufficient gate control over the channel potential profile. 4. Electron Transmission through a BNNR-Confined AGNR

To delineate the conducting properties of these nanostructures we investigate their transmission probabilities. As mentioned in the previous section, this function can be derived directly from the NEGF formalism. Therefore, we will next study the electron transport in these nanostructures by studying the transmission probability function. In Fig. 4, the transmission functions for a 2BNNR-confined 3-AGNR and a 2H-passivated 3-AGNR [shown in Figs. 1(a) and 1(c)] are modeled and plotted versus electron energy. These curves are also compared with that of a 6-AGNR, the width of which is equal to the simulated 2BNNR-confined 3-AGNRs. The curves in Fig. 4 correspond to the case of semi-infinite electrodes as well as the central nanoribbon being connected to form a uniform nanoribbon that does not vary in the longitudinal direction and the leads are prolongations of the nanoribbon. The step-like behavior observed in these curves is expected due to the quasi-1D character of system. The transmission values increase sharply whenever a new subband in the infinite nanoribbons bandstructure is turned on. Therefore, the results in Fig. 4 can be verified by examination of the bandstructures of their corresponding nanoribbons. Since no interference effect caused by the nanoribbon-electrode transitions is taken into account, these results can be considered as the ideal case and the curves demonstrate the upper possible limits of charge transmission by the nanoribbons. However, the overall behavior in transmission probability of a finite sized nanoribbon will be irregular as a result of constructive and destructive interferences of electron waves which are reflected from the left and right nanoribbon/lead interfaces. Figure 5 compares the transmission probabilities of an infinite and a finite 2BNNR-confined 3-AGNR. The finite nature of the simulated nanoribbon is manifested by oscillations and peaks in its transmission spectra, the amplitude of which depend on the mismatch between the bandstructures of an infinite nanoribbon and an infinite electrode structure. We can see that by choosing a contact approximately matched with the central nanoribbon, we may obtain an electrode/nanoribbon/electrode system that ex-

(b) Fig. 5. (Color online) The transmission probabilities of a 2BNNRconfined 3AGNR for different electrode/nanoribbon matchings. The dashed curves correspond to the ideal case of completely matched electrodes. The filled transmissions in (a) and (b) are calculated for square lattices with different electrode/nanoribbon hopping integrals of 2:5 and 3 eV, respectively.

hibits good conducting behavior and a transmission curve that approximately follows the transmission curve trends in the ideal infinite nanoribbon. In Fig. 6, the transmission probabilities of a 2BNNRconfined 3-AGNR and a 2BNNR-confined 4-AGNR are compared for different gate voltage biases. Here, a gate electrode is assumed to be wrapped around central nanoribbon and its voltage affects the on-site potential of the atoms in the previously mentioned Hubbard model provided that under any VG the energy of all the atoms within the channel changes by the same value of qVG . We can see from Fig. 6 that the gate bias shifts the transmission curves. It also smoothes the abrupt quantization steps in the transmission function. The overall transmission also decreases by the gate bias due to the bandstructure mismatches between the channel and the electrodes when the nanoribbons band structure is shifted compared to that of the electrodes. Figure 6 indicates that these nanoribbons have the potential to be effectively used as the channel materials of field effect transistors like the previously studied hydrogen-passivated GNRs. 5. Density of States in a BNNR-Confined AGNR

Figure 7 shows the local density of states (LDOS) in a 2BNNR-confined 3-AGNR for different atomic sites in the middle of nanoribbons length, where the solid lines and the dashed lines correspond to the atoms in the GNR region and

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(a)

Fig. 8. (Color online) Density of states (DOS) of a 2BNNR-confined 3AGNR for different widths of the BNNR domains. The solid curve is calculated for a structure in which the BNNR domains are each approximately 0.5 nm wide, where in the dashed curve the BNNR domains are each 5 nm wide.

(b) Fig. 6. (Color online) Transmission probabilities of (a) a 2BNNRconfined 3AGNR and (b) a 2BNNR-confined 4AGNR for different gate biases. The dashed curves correspond to the unbiased nanoribbon, while the green (solid light gray) curves are calculated for VG ¼ 0:2 V and the red (solid dark gray) curves are calculated for VG ¼ 0:5 V.

like isolators. Figure 8 also shows the density of states (DOS) of the entire nanoribbon for different BNNR widths. From Fig. 8, it is clear that in the low-energy regimes, changing the BNNR indices has no effect on the DOS in the channel region. This can be explained by noticing that in the bandstructure of a 2BNNR-confined AGNR, subbands related to the BNNR parts of the nanoribbon are separated considerably from the subbands corresponding to the GNR part of the nanoribbon, and have a minimal impact on these low-energy subbands. Therefore, independent of their width, boron nitride nanoribbons can be utilized as isolating nanostructures between armchair GNRs modifying their energy gaps and realizing parallel arrays of nanoribbons in future integrated circuits. 6. Conclusions

Fig. 7. (Color online) Local density of states (LDOS) for different atomic sites in a 2BNNR-confined 3AGNR. The solid curves correspond to the LDOS for two atoms in the GNR domain of the structure, while the dashed curves correspond to the LDOS for two atoms arbitrary atoms in the BNNR domain of the structure. The black curves represent atoms on the edge of each domain and the red (gray) curves represent atom on the middle inner dimer (in the width direction) of each domain.

BNNR region of the nanoribbon, respectively. It is observed that the BNNR regions of a 2BNNR-confined AGNR provide a much larger energy gap in their LDOS than the GNR regions, and in the low-energy regime, where the gate voltage biases less than VG ¼ 1:5 V are applied to the structure, the electronic conduction is completely occurring through the AGNR part of the nanoribbon and in these lowenergy regimes, the boron nitride nanoribbons are behaving

We have investigated the band structure and electronic properties of BN–C hybrid armchair nanoribbons and have shown that due to the energy gap opening in (3p+2) categories of 2BNNR-confined AGNRs in these nanoribbons, they can be better candidates for the channel material of high-performance field effect transistors as compared to the hydrogen passivated AGNRs. We also addressed the electron transport in these nanoribbons by calculating their transmission probability function and their density of states using the non-equilibrium Green’s function formalism coupled to a two-dimensional Poisson-solver. These simulations can be regarded as the first steps in describing the potential electronic applications of these hybrid armchair nanoribbons. Future works can take into account some of the more realistic effects which we have ignored here for simplicity such as the substrate–nanoribbon interactions, different scattering mechanisms, the impact of vacancies and roughnesses in their structures.

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