First-Principles Prediction of the Electronic Structure and Carrier

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First-Principles Prediction of the Electronic Structure and Carrier Mobility in Hexagonal Boron Phosphide Sheet and Nanoribbons Bowen Zeng, Mingjun Li, Xiaojiao Zhang, Yougen Yi, Liping Fu, and Mengqiu Long J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b07048 • Publication Date (Web): 10 Oct 2016 Downloaded from http://pubs.acs.org on October 11, 2016

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The Journal of Physical Chemistry

First-Principles Prediction of the Electronic Structure and Carrier Mobility in Hexagonal Boron Phosphide Sheet and Nanoribbons Bowen Zeng a, Mingjun Li a, b, Xiaojiao Zhang c, Yougen Yi a, Liping Fu a, and Mengqiu Long a* a

Hunan Key laboratory of Super Micro-structure and Ultrafast Process, School of Physics and

Electronics, Central South University, Changsha 410083, China b

School of Material Science and Engineering, Central South University, Changsha 410083, China

c

Physical Science and Technology College of Yichun University, Yuanzhou, Yichun 336000, China

*E-mail:

[email protected], Telephone:+86-0731-88830323

1

ACS Paragon Plus Environment


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Abstract: Using density functional theory coupled with Boltzman transport equation with relaxation time approximation, we study the electronic structure and carrier mobility of graphene-like hexagonal Boron Phosphide (h-BP) monolayer and H-terminated armchair Boron Phosphide nanoribbons (ABPNRs). Our results show that the carrier mobility can reach over 104 cm2 V-1 s-1 for electron and 5×103 cm2 V-1 s-1 for hole in monolayer sheet. The carrier mobility in the ABPNRs is in the range of 103 ~ 104 cm2 V-1s-1 , we find that the width of nanoribbon plays an important role in tuning the polarity of the carrier transport, which exhibit a distinct 3p (p is a positive integer) alternating behavior. And the staggering oscillating behavior of mobility should be attributed to different bond characteristics of the edge states in the ABPNRs. Moreover, the H-terminated zigzag Boron Phosphide nanoribbons (ZBPNRs) have the characteristics of p-type semiconductors in electrical conduction, and the carrier mobility is increased with the width of the nanoribbons and no alternating size-dependent carrier polarity is found. The high carrier mobility and adjust ability polarity of transport suggest h-BP is promising candidate material for application in future nanoelectronic devices.

2


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Introduction In the past decades, atomically thin two-dimensional (2D) materials have attracted enormous interest because of their unique physical properties and potential applications in nanoscale electronics and photonic devices.1-7 However, the utility of these 2D materials is limited by some inherent weaknesses. For example, graphene is a semimetal and lacking of an intrinsic band gap,8 boron nitride (BN) sheet behaves more nearly as insulator owing to much too wide band gap,9 which are not able to function as switch in transistor devices. There are moderated band gaps for MoS2 (about 1.8 eV)10 and phosphorene (about 2.0 eV).11,

12

Nevertheless, the carrier

mobility of MoS2 calculated by DFT13 or measured by experiment14 is relatively low, thereby limiting its wide application in nanoelectronics. Phosphorene has high carrier mobility in nanosheets and nanoribbons,15 but the poor chemical stability in phosphorene is a particular disadvantage for device application.16,

17

Hence great

efforts have been devoted to exploring new 2D materials which have moderate band bap and high carrier mobility, as well as high thermal and chemical stability. Recently, one of 2D class of group- Ⅲ / Ⅴ elements, h-BP, has attracted considerable attentions.18-24 Theoretically, h-BP monolayer has been reported it is a semiconductor with direct band gap in the range of 0.81~1.81eV9, 18 and its effective mass,24 whether of electron or hole, is significantly less than monolayer MoS225 and even comparable with single layer phosphorene,26 which means that moderate band gap and high carrier mobility can coexist in h-BP. The calculation of phonon modes with no imaginary frequencies by Sahin et al18 and B−P bonds are reserved in the Born-Oppenheimer Molecular Dynamics (BOMD) simulation at high temperature of 2500 K for 5 ps by Wanget al,20 suggested the mechanically stability and high thermal stability of h-BP, respectively. All those theoretical results indicate h-BP is a promising candidate material for using in future electronic devices. Experimentally, Boron

Phosphide

(BP)

films

have

been

synthesized

on

aluminum

nitride(0001)/sapphire substrate by chemical vapor deposition.27 The BP films is known as wurtzite structure, but it’s believe that BP will adopts hexagonal 3



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graphene-like lattice under suitable experimental condition like monolayer BN,28, 29 Aluminum Nitrogen (AlN),30 Zinc oxide(ZnO)31 and soon on. Owing to carrier mobility is the central issue in microelectronic semiconducting materials. And charge transport properties in h-BP would become the center of interests, because of its unique physical, chemical, structural properties and potential applications in next generation electronic devices.20, 24, 32 Therefore, in this work, we have calculated the electronic structure and carrier mobility of h-BP sheet by using first principles method combined with Boltzman transport equation (BTE) with deformation potential (DP) theory. Furthermore, reduction of the dimensionality from 2D to one-dimension (1D) can bring different properties and various adjustment method. For example, it has been found that the carrier mobility of graphene nanoribbons (GNRs) significantly less than those of graphene sheet due to lacking of the massless Fermi-Dirac characteristic.33 Interestingly, when GNRs is passivated by hydrogen, it can perform size-dependent carrier mobility and polarity.34 Whereas in the case of MoS2 nanoribbons, the carrier mobility can be regulated by different edge chemical modification owing to the changes in the electronic structure.13 To understand the charge transport properties and the effects of the edge states in ABPNRs, the electronic structure and carrier mobility of ABPNRs have been calculated.

Methods Our calculations are performed with density function theory (DFT) as implemented in Vienna ab-initio simulation package (VASP),35 we adopt the generalized gradient approximation(GGA)36 for the exchange-correlation potential. The ion-electron interaction is treated with the projected-augmented wave (PAW)37 and the plane-wave cutoff energy for the wave function is set to 550 eV. The criterion of convergence for structure relaxation is the change of total energy less than 10-7 eV and the residual force on atom less than 0.001 eV/Å. The Brillouin zone (BZ) is 4


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sampled by 17 × 17 × 1, 15 × 25 × 1 and 17 × 1 × 1 for static calculation of unit cell, rectangle supercell and nanoribbon, respectively. The vacuum space between two neighboring nanostructure is set as 15 Å to decrease the interactive effect, we also adopt DFT-D238 of Grimme for the long-range dispersion correction. In present work, within the Boltzman transport method, the carrier mobility u can be expressed as39, 40 r  εi k r 2 r ∑ i∈CB (VB ) ∫ τ α (i, k )vα (i, k ) exp  m k T B e  = r k BT  εi k  r exp ∑ i∈CB (VB ) ∫  m k T  d k B  

( )  d kr

uα e ( h )

()

 

（1）

Where α denotes the direction of external field and the minus(plus) sign is for electron (hole). τ（ α i, k） is the relaxation time, ε i (k ) and vα (i, k ) are the band energy and the component of group velocity at k state of the i-th band, respectively. The summation of band was carried out over valence band (VB) for hole and conduction band (CB) for electron, the integral of k states is over the first BZ. To obtain the carrier mobility, three key quantities τ（ , ε i (k ) and vα (i, k ) α i, k） must be determined. For the band energy ε i (k ) calculation,

the BZ is sampled by

more intensive k-points with 47×47×1 for rectangle supercell and 300×1×1 for nanoribbons to ensure converged relaxation time and carrier mobility. The group velocity vα (i, k ) is defined as the gradient of band energy ε i (k ) in k-space,

()

vα (i, k ) = ∇ε i k / h . The relaxation time τ（ α i, k） is calculated by the collision term in the BTE method. Following the argument of Shockley and Bardeen,41 in inorganic semiconductors, as the wavelengths of thermally activated electrons is close to the acoustic phonon wavelength and much larger than the lattice length, the electron-acoustic coupling become the dominant scattering mechanism of electron. Within the DP theory, relaxation time can be written as2, 42 5



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r uur   vα k ' 1 2π E1  1− r r r = k BT ∑ k '∈BZ   hC vα ( k ) τ α i, k  

( )  δ ε

2

( )

  

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uur r  k ' − ε k   

( ) ()

(2)

Where the delta function denotes that the scattering process is elastic and occurs between the band states with the same band index. E1 is the DP constant of electron or hole and C is the elastic constant along the deformation direction. In principle, the relation of total relaxation time and different scattering channels can be expressed as

1

τ

=

1

+

1

1

+

+ ......

τ ac τ op τ imp

(3)

Where ac, op, imp stands for acoustic, optical phonon and impurity respectively. Here we only consider the contribution of acoustic phonon scattering. Furthermore, to better understand the transport behaviors of sheets or nanoribbons, we calculated the carrier mobility by using effective mass approximation with DP theory for comparison, the u in 2D sheet43 and 1D nanoribbons2, 44 are expressed with following forms: u2 D =

u1D =

2eh 3C

(4)

2

3k BT m * E12 eh 2C

( 2π k BT )

1/2

m*

3/ 2

(5)

E12

[ () ]

2 2 2 Where m * is the effective mass, which is calculated by m* = h ∂ ε i k / ∂k

is the temperature.

6


−1

, T

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Results and Discussions The atomic structure of the h-BP is shown in Fig. 1(a). The optimized lattice constants of unit cell is a1 = a2 = 3.209 Å, in good agreement with previous theoretical study.18 The phonon dispersion curve of h-BP is presented in Fig. S1 and no imaginary frequencies are observed, which suggests our proposed h-BP model is mechanical stable. Band structure is shown in Fig. 1(c), we can see that h-BP monolayer is semiconductor with direct band gap of 0.90 eV at the K-point. However, the band gap of semiconductor materials generally is underestimated by DFT calculation, when calculated with Heyd-Scuseria-Ernzerhof (HSE)45 hybrid function the band gap increase to 1.35 eV. It should be noted that bulk zinc-blende BP is an indirect gap semiconductor with 2.02 eV band-gap.46,

47

Reduction of the

dimensionality from three-dimension (3D) to 2D not only decreases the band gap, but also achieves indirect band-gap transition to direct band-gap.

Fig. 1

Model of h-BP (a) unit cell and (b) rectangle supercell. (c) Band structure of h-BP unit cell along the

high symmetry points of the hexagonal BZ, calculated with DFT-GGA(black lines) and HSE06 (red lines). Fermi level is set to 0 eV and denoted by pink dashed line.

For a more intuitive description in the carrier transport calculations, we build a rectangle supercell where the vector a, b along the armchair and zigzag directions, respectively, which is shown in Fig. 1(b). In the 2D case, the stretching modulus is defined as C 2 D = (∂ 2 E / ∂δ 2 ) / S 0 , where the E and S0 is the total energy and the area 7



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of supercell , respectively, the δ = ∆l / l0 , ∆l is the deformation of lattice constant by uniaxial strain and l0 is the value at equilibrium geometry. The DP constant E1 is calculated by E1 = ∆Eedge / ∆δ , where the ∆ E edge is

the energy shift of band edge

of VB or CB with respect to lattice dilation along the direction of external strain. Additional calculation of free energy and energy of band edge at lattice constants of 0.990 l0 , 0.995 l0 , 1.005 l0 ,and 1.01 l0 , which simulate the deformation, as shown in Fig. 2(a-b).

Fig. 2 (a) Energy-strain relationship along the a and b direction. (b) Shifts of the conduction band and valence band edges under a uniaxial strain.

The values of the C 2 D along the a and b directions are 147.360 J m-2 and 147.375 J m-2, respectively, which are consistent with the previous report.9 The DP constant E1 of electron (hole) along the a and b directions are -2.223 eV and -2.288 eV (-3.856 eV and -3.762 eV), respectively, which means that the E1 doesn’t depend on transport direction but the carrier type. After substituting the C 2 D and E1 into equation (1) and (2), the acoustic-phonon-limited relaxation time and the carrier mobility are obtained, all those calculated results are shown in Table 1. We also have listed the carrier mobility, that is calculated by effective mass method as equation (4), for comparison. It is worth noting that , when calculated with the HSE06 hybrid function, the band gap will be increased as mentioned before, the band curvature changes with increasing of band gap, thus it would exert influence on effective mass of band edge. However, it does not make an appreciably difference to the transport property 8


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calculation by BTE method since only the shifts of VB and CB is taken into account in DP theory. It is found that the mobility of electron ( µ e ) can reach 104 cm2 V-1 s-1 which is about double than the mobility of hole ( µ h ), and those values are significantly larger than MoS213 even comparable with phosphorene.6 The acoustic phonon scattering relaxation times is calculated to be 0.7 ps for hole and 1.9 ps for electron, all of which are larger than those in phosphorene.6 Table 1 DP constant E1 , in-plane stretching modulus C 2 D , relaxation time τ , carrier mobility µ at 300 K, effective mass m * and carrier mobility µ * , which based on effective mass method for h-BP.

Carrier type

E1

C2D

τ

µ

m∗

µ∗

(eV)

(Jm-2)

(ps)

(cm2V-1s-1)

( me )

(cm2V-1s-1)

armchair

-2.223

147.3595

1.90

1.361×104

0.192

1.154×104

zigzag

-2.288

147.3746

1.80

1.017×104

0.198

1.023×104

armchair

-3.856

147.3595

0.70

5.045×103

-0.180

4.331×103

zigzag

-3.762

147.3746

0.74

4.493×103

-0.186

4.286×103

Direction

electron

hole

Subsequently, we examine the carrier mobility of ABPNRs. As the bare zigzag BPNRs (bZBPNRs) is metallic [see Fig. S2(b)], we first consider the semiconducting bare ABPNRs (bABPNRs), as shown in Fig. S2(c-d)), the unpaired electrons on the edge are highly reactive and will make edge atoms bond with impurities. Therefore, in our model, the edges are passivated with hydrogen. The ABPNRs are classified by the number of B-P dimer lines (N) across the nanoribbon width as shown in Fig. 3(a), its width is in the range of 6 ~15. The representative band structures of ABPNRs with N=11, 12, 13 are shown in Fig. 5, we can see that all the H-termination ABPNRs is predicted to be direct gap semiconductor. The stretching modulus in the 1D case is defined as C 1 D = (∂ 2 E / ∂δ 2 ) / L0 , L0 is the lattice constant of the optimized ABPNRs 9



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in the armchair direction as shown in Fig. 3(b). Owing to the unpaired electron saturated by hydrogen, the L0 doesn’t show much difference when compared with the lattice constant of supercell. The stretching modulus C 1D increase successively with N as shown in Fig. 4(a), considering the width of nanoribbons W0 , it can be evaluated C 2D via equation

C 2 D = C1D / W0 , which in the range of 99.1 ~ 120.9 N m-1. Compared with the 2D sheet, the phonon mode quantization and the emergence of edge phonon modes lead to softening of the lattice modes and reduction of the elastic modulus.48 Fig. 4(a) shows the DP constants E1 of electron and hole. It can be seen that the E1 of electron or hole is oscillating with N, and the latter one shows larger amplitude. With the same theoretical methods employed for h-BP, we calculate the carrier mobility of ABPNRs as shown in Fig. 4(b). It is found that the mobility of ABPNRs can be divided into three distinct group (N = 3p, N = 3p +1 and N = 3p + 2). The first group, where the width of nanoribbon is N=3p, has the largest µ e and smallest µ h , this group can be considered as n-type semiconductor. The opposite trend can be found in the N=3p +1 group, which has the largest µ h and the smallest µ e , which can be considered as

p-type semiconductor, the carrier mobility of the N = 3p +2 group is between those of the former two groups, the behavior of µ e is very similar to the group of N = 3p +1 and µ h is more like the group of N = 3p. Moreover, we also present the mobilities for ABPNRs with N = 30, 31 and 32 (the width is about 5 nm) in Table S1, the similar 3p alternating behavior of charge mobility can be observed obviously. Furthermore, as a comparison, the effective mass of nanoribbon and the corresponding carrier mobility (using equation 5) are shown in Fig. 4(c-d). It can be seen that the width of nanoribbon plays an important role in tuning the polarity of the carriers transport, and the carrier mobility calculated by the above-mentioned two method shares the same magnitude and similar oscillation behavior. 10


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Fig. 3 (a) Model of ABPNRs. (b) Optimized lattice constant along the armchair direction.

Fig. 4 (a) The elasticity modulus ( C 1 D ) and deformation potentials (DP) constant of hole and electron for ABPNRs. (b) Mobility of hole and electron. (c,d) Effective mass and correspond mobility of hole and electron.

To better understand the staggering oscillating behavior of carrier mobility, we turn to analyze the band-decomposed charge density of the state of band edge at Γ point, which are equivalent to the frontier molecule orbitals [i.e., the highest occupied molecular orbital (HOMO) for hole and the lowest unoccupied molecular orbit (LUMO) for electron] responsible for transport. Since the ABPNRs in the same group have similar band structures, the width of N = 11, 12 and 13 are chosen as the representative, and the band structures and the corresponding HOMOs and LUMOs of 11



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11-ABPNR, 12-ABPNR and 13-ABPNR are shown in Fig. 5(a-c), respectively. For N = 12 (3p), the bond direction is vertically localized perpendicular to the stretching direction for the HOMO, Whereas delocalized along the stretching axis for the LUMO. The band-edge shifting would come from the site energy with the ribbon stretching, thus it’s expected that the HOMO should experience more scattering by the acoustic phonon than the LUMO. However an opposite trend is found for N = 13 (3p + 1), where LUMO is scattered strongly more than HOMO due to the spatial distribution of the HOMO perform a horizontally connecting bond, Whereas for the LUMO is vertically localized. For N = 11 (3p + 2), the bond characters of both HOMO and LUMO are localized in the direction of horizontal and vertical, thus the transport properties of this group is between the former two groups.

Fig. 5 Band structure and band decomposed charge density of ABPNRs with (a) N = 11, (b) N = 12 and (c) N = 13.

We next examined the band structure and carrier mobility of ZBPNRs with width of N = 8, 9, 10 as shown in Fig. 6(a-b). It should be noted that H-terminated ZBPNRs 12


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is semiconductor and with band gap of ~ 1.0 eV. As shown in Fig. 6(b), the carrier mobility show the ZBPNRs have the characteristics of p-type semiconductors in electrical conduction, the calculated hole mobility can reach the order of 0.5 × 104 cm2 V-1 s-1 at room temperature, while the electron mobility is about one order of magnitude lower. Furthermore, we also find the carrier mobility is increased with the width of the nanoribbons and no alternating size-dependent carrier polarity.

Fig. 6 (a) Band structure of ZBPNRs. (b) Carrier mobility of ZBPNRs.

Conclusions In conclusion, we have calculated the electronic structure and carrier mobility of

h-BP and ABPNRs by using first-principles methods coupled with BTE methods and relaxation time approximation. Our results show that the monolayer h-BP is semiconductor with a moderate band gap of 0.9 eV. The numerical results indicate that µ h can reach 5 × 103 cm2V-1s-1 at room temperature for h-BP sheet, and the µ e almost double than µ h . Regarding the ABPNRs, we have found that the width of nanoribbon plays an important role in tuning the polarity of the carrier transport, which exhibit a distinct 3p alternating behavior. It can lead to p-type semiconductor (N = 3p), n-type semiconductor(N = 3p + 1), and conventional semiconductor(N = 3p + 2) successive transitions by controlling the width of nanoribbons. Moreover, we 13



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also find the ZBPNRs have the characteristics of p-type semiconductors in electrical conduction, and the carrier mobility is increased with the width of the nanoribbons and no alternating size-dependent carrier polarity is found. In short, we can come to the conclusion that there are many prefect properties can coexist in h-BP, such as hexagonal structure, moderate band gap, high mobility in nanosheets and oscillation behavior of carrier mobility in nanoribbons, which suggest it is a good candidate material for the next generation of electronic devices.

Acknowledgments: This work is supported by the National Natural Science Foundation of China (Nos. 21673296, 61306149 and 11334014).

Supporting Information: Vibrational band structures of monolayer h-BP; model of bZBPNRs, ZBPNRs and bABPNRs and band structure of bZBPNRs and bABPNRs; carrier mobility of 30-ABPNR, 31-ABPNR, and 32-ABPNR. This material is available free of charge via the Internet at http://pubs.acs.org.

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First-Principles Prediction of the Electronic Structure and Carrier

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