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The stacking dependent electronic structure and optical properties of bilayer black phosphorus† Huabing Shu,a Yunhai Li,a Xianghong Niua and Jinlan Wang*ab By employing density-functional theory, the G0W0 method and Bethe–Salpter equation, we explore quasi-particle energy bands, optical responses and excitons of bilayer black phosphorus (BBP) with four different stacking patterns. All the structures are direct band gap semiconductors and the band gap is highly dependent on the stacking pattern, with a maximum of 1.31 eV for AB-stacking and a minimum of
Received 27th December 2015, Accepted 25th January 2016
0.92 eV for AD-stacking. Such dependence can be well understood by the tight-binding model in terms
DOI: 10.1039/c5cp07995k
energy are observed in BBPs. For x-polarized light, more red-shift of optical adsorption appears in
of the interlayer hopping. More interestingly, stacking sensitive optical absorption and exciton binding AA-stacking and the strong exciton binding energy in the AA-stacking bilayer can be as large as 0.82 eV,
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that is B1.7 times larger than that of AD-stacking bilayer.
1 Introduction Recently, a new member of the 2D materials family, few-layer black phosphorus (BP) has been successfully exfoliated.1–5 In few-layer BP, the phosphorus atom is bonded with three adjacent phosphorus atoms, forming a puckered peculiar structure with parallel zigzag rows, which is different from other 2D materials such as graphene6 and MoS2.7 In addition to their considerable direct band gaps (1.06 eV for tri-layer8 and 2 eV for monolayer8), few-layer BP exhibits strong excitonic effects with large exciton binding energies (490 meV for tri-layer8 and 760 meV for monolayer9). Moreover, the carrier effective mass,4 optical conductivity10 and absorption,11 excitons,8 spatial thermal conductance12,13 and Raman response14 of fewlayer BP are highly anisotropic, due to their special puckered structure. The carrier mobility of few-layer BP is also highly thickness-dependent.1,4,15 Specially, the hole mobility can be as large as 1000 cm2 V1 s1 for a thickness of about 10 nm and the on/off current ratios can exceed 105 for samples thinner than 7.5 nm. Significant blue shift of luminescence with decreasing layer thickness has been observed as well.14 These unique electronic and optical properties endow few-layer BP with great potential for electronic,1,2,4,16 and optoelectronic devices.17,18 For example, a few-layer BP based photovoltaic device has been fabricated which allows power generation for
a
Department of Physics, Southeast University, Nanjing 211189, China. E-mail: [email protected] b Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China † Electronic supplementary information (ESI) available. See DOI: 10.1039/ c5cp07995k
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illumination wavelengths up to 940 nm.17 The BP based photodetector is capable of acquiring high-contrast images in the visible and near-infrared spectral regime.18 Additionally, fewlayer BP based absorbers show saturable absorption at 532 nm,19 1030 nm19 and 1550 nm20 and can be used for optical pulse generation in Q-switching and mode-locking regime in the fiber laser setup.21 On the other hand, it has been reported that the stacking pattern has great influence on the properties of 2D layered materials.22–27 For example, the Bernal stacked bilayer graphene possesses unique electronic and optoelectronic properties,24,25 which are different from that of its twisted bilayer counterpart.22,23 h-BN bilayer structures with different interlayer symmetries show distinct optical second harmonic generation intensities.26 Also, different stacking patterns in the MX2 (M = Mo, W; X = S, Se) result in significantly different optical excitation and exciton binding energies.27 However, in bilayer black phosphorus, the relationship between the different interlayer interaction induced by different stacking patterns and properties has rarely been discussed and is still unclear. In this paper, we study the electronic and optical properties of bilayer black phosphorus considering different stacking patterns within many-body Green’s function and Bethe–Salpeter equation (BSE) formalism. The structure–property relation has been revealed and understood by a tight-binding model.
2 Computational details The first-principle calculations were carried out within the framework of many-body perturbation theory. First, the electronic ground states were obtained by density functional theory (DFT)
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with the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional28 including interlayer van der Waals interaction via the vdW-DF approach,29–31 as implemented in the QUANTUMESPRESSO code.32 Plane-wave basis sets with a kinetic energy cutoff of 60 Ry, Monkhorst–Pack33 k-grid of 14 10 1 and norm-conserving pseudopotentials34,35 were adopted. A sufficiently large vacuum layer of 20 Å was used to avoid spurious interactions between periodic images. Cell parameters and atomic positions were fully relaxed until the residual forces on each atom were smaller than 0.01 eV Å1 and the total energy difference was less than 104 eV. Then, single-shot G0W0 approximation36,37 was employed to obtain quasi-particle (QP) band structures. Finally, optical excitation energies and exciton wave functions were determined by solving the Bethe–Salpeter equation.38–40 The convergence of quasi-particle band gap with respect to the number of empty bands, the size of dielectric matrix and the Monkhosrt–Pack grid were carefully examined and a convergence within 0.1 eV was assured. 10 valence and 15 conduction bands were included to build the electron–hole interaction kernel. The involved unoccupied band number (480) was employed to obtain the converged dielectric function. A box-shape truncated coulomb interaction was adopted after 25 bohr along the direction perpendicular to the surface. A denser 56 40 1 k-grid was used to converge the optical spectra. The G0W0 and BSE calculations were performed using the YAMBO program package.41
3 Results and discussion We consider four possible high-symmetric stacking structures, namely AA-, AB-, AC- and AD-stacked bilayer black phosphorus. AA: the top layer is vertically stacked on the bottom layer with the symmetry of Pmma; AB: the top layer of AA-stacking is shifted by half of one unit cell along the x or y direction with the symmetry of Pbcm and the stacking pattern matches the structure of bulk BP; AC: the top layer of AA-stacking is shifted by one unit cell along the x or y direction with respect to the bottom layer and the top and the bottom layers are mirror
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images of each other with the symmetry of Pmma; AD: the top layer of AC-stacking is shifted by a half of one unit cell along the x or y direction with respect to the bottom layer with the symmetry of Pccm. Their optimized structures and lattice parameters are presented in Fig. 1(a)–(d) and Table 1. The interlayer distance d increases from 3.580 Å for AB-stacked bilayer to 4.033 Å for AC-stacked bilayer significantly and the largest bond angle difference is about 1 degree. The structural differences stem from the different interlayer interaction caused by different stacking patterns. Among the four stacking structures, AB-stacked bilayer is the most energetically favorable, which is 2.9 meV, 6.4 meV and 7.3 meV per atom lower than that of AA-, AD- and AC-stacked bilayer, respectively. To determine the stability of the four stacking structures, we have also calculated the formation energy (Ef), defined as Ef = (Etotal Etp Ebm)/2, where Etotal is the total energy of whole BBP and Etp (Ebm) is the total energy of the top layer (the bottom layer). The formation energies are 24, 12, 9 and 5 meV per layer for AB-, AA-, AD- and AC-stacked BBP, respectively, indicating that in addition to the AB-stacked bilayer which has already been fabricated, other three stacking structures are also feasible experimentally. The electronic band structures of BBP obtained from the DFT-PBE and the G0W0 method are presented in Fig. 2(a)–(d). Their band structures are similar to that of the monolayer black phosphorus (Fig. 3(a)), and show clearly asymmetric dispersion around G point along G–X and G–Y directions, indicating the anisotropic nature of BBP. For all four stacking patterns, the band gaps are always direct and locate at the G point, with the values of 0.52 eV for AA-, 0.55 eV for AB-, 0.43 eV for AC- and 0.33 eV for AD-stacked BBP at the DFT-PBE level (Table 2), respectively. Compared to DFT-PBE band structures, the conduction bands of QP band structures from G0W0 calculations are shifted up while the valence bands are shifted down due to many-body effects (Fig. 2(a)–(d)). Hence, the QP band gaps of AA-, AB-, AC- and AD-stacked BBP increase to 1.21, 1.31, 1.15 and 0.92 eV (Table 2), respectively. This trend is in line with DFT-PBE results, although the G0W0 method reproduces appreciable larger band gaps. Such significant QP corrections are the
Fig. 1 (a–d) Top and side views of optimized four stacking structures. a, b and d are lattice constants and interlayer distance, respectively. y1 and y2 denote bond angles while R1, R2 and R3 indicate bond lengths. For each layer of the bilayer black phosphorus, phosphorus atoms belong to different sub-layers are colored in yellow and green for clarity.
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Table 1 Optimized lattice parameters a, b and interlayer distance d; the bond lengths R1, R2 and R3; the bond angles y1 and y2. Lattice parameters and bond lengths are in angstroms; bond angles are in degrees
Type
a
b
d
R1
R2
R3
y1
y2
AA AB AC AD
4.541 4.517 4.548 4.540
3.315 3.327 3.323 3.331
3.816 3.580 4.033 3.742
2.224 2.231 2.227 2.228
2.280 2.250 2.276 2.252
2.228 2.233 2.230 2.227
96.135 97.087 96.753 96.766
103.206 102.442 103.303 103.401
consequence of reduced electronic screening. Additionally, our calculated QP band gap of the AB-stacking structure is in good agreement with the other theoretical result (1.32 eV).8 To further clarify the effects of interlayer interaction on the electronic structure of BBP, we examine the dependence of the band gap on the interlayer distance for four different stacking structures. As shown in Fig. S1 (ESI†), the band gap increases with the increase in interlayer distance, arising from the decrease in weak interlayer interaction. We also plot the alignment of
valence band maximum (VBM) and conduction band minimum (CBM) with respect to the vacuum level and the corresponding charge densities, as shown in Fig. 3(b) and (c). It is evident that VBMs are slightly changed for all four stacking patterns while CBMs for AC- and AD-stacked BBP are significantly shifted downward compared to that of the AB bilayer. Also, the overlap of interlayer charge density of CBM emerges in AC- and AD-stacked BBP, which is absent in AA- and AB-stacked bilayers, suggesting the stronger interlayer interactions in AC- and AD-stacked BBP. Furthermore, based on a tight-binding model, we can clearly demonstrate that the dominant interlayer hopping term in every stacked pattern of BBP will determine the size of band gap and larger dominant interlayer hopping term will lead to smaller band gap. The dominant interlayer hopping terms (ti0 , i = 1, 2,. . ., 6) follow the order of t40 (AB) o t10 (AA) o t10 (AC) o t10 (AD). The details are shown in the ESI.† For AB-stacked BBP, three distinct Raman peaks corresponding to three optical modes (A1g, B22g and A2g), have been
Fig. 2 (a–d) Band structures of AA-, AB-, AC- and AD-stacked bilayer black phosphorus. The DFT-PBE and G0W0 results are depicted by black and red solid lines, respectively. The valence band maximum (VBM) of the DFT-PBE band structure is shifted to zero.
Fig. 3 (a) Band structure of monolayer black phosphorus. The DFT-PBE and G0W0 results are depicted by black and red solid lines, respectively. The valence band maximum of DFT-PBE band structure is shifted to zero. (b) Energy levels of valence band maximum (VBM) and conduction band minimum (CBM) of BBP. All energies have been corrected with respect to the vacuum level from DFT-PBE calculation. (c) Isosurface plot of partial charge density corresponding to VBM and CBM of BBP. The isovalue is 1.58 103 e Å3.
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Table 2 Band gap Eg-DFT (eV) at the DFT-PBE level, Eg-G0W0 (eV) at the G0W0 level, excitation energy Ee (eV) and binding energy Eb (meV) of the exciton with the largest oscillator strength. Binding energy is defined as the difference between the excitation energy and the QP energy difference. The x and y indicate the polarization direction of the incident light
Type
Eg-DFT
Eg-G0W0
Ee
AA
0.52
1.21
AB
0.55
1.31
AC
0.43
1.15
AD
0.33
0.92
ML
0.79
2.01
0.39 3.42 0.76 3.01 0.85 3.26 0.88 2.91 1.18 3.06
Eb (x) (y) (x) (y) (x) (y) (x) (y) (x) (y)
820 770 550 630 620 270 470 370 830 470
experimentally observed and locate at 362 cm1, 438.5 cm1 and 467 cm1, respectively.42 We calculate these three Raman modes at the G point for four stacked BBP. Their frequencies corresponding to three Raman active modes are summarized in Table S3 (ESI†) and their vibrational patterns are displayed in Fig. S5 (ESI†). Interestingly, the A1g and A2g Raman modes show some dependence on the stacking pattern in BBP; the energies of A1g and A2g Raman modes increase with increasing interlayer interaction (Fig. S6, ESI†), i.e., the corresponding Raman peaks blue-shift. While the B22g mode is not sensitive to the stacking patterns in BBP, which is possibly attributed to the large elastic constant (162 GPa)43 along the zigzag direction. The optical absorption spectra of AA-, AB-, AC- and ADstacked BBP for incident light polarized along armchair (x) and zigzag (y) directions of the lattice are shown in Fig. 4(a)–(d) and (f)–(i), respectively. The optical absorption spectra with electron–hole interaction included are dominated by excitonic states and the excitation energies are significantly altered by the stacking pattern. For example, for incident light polarized along the x direction, the first strong absorption peak which is mainly contributed by the inter-band VBM–CBM transition at the G point, is located around 0.39 eV and 0.76 eV in AA- and
AB-stacked BBP respectively, indicating an optically active direct band gap. For AC- and AD-stacked BBP, a similar peak is observed around 0.85 and 0.88 eV, respectively, which mainly arises from the transition between the second valence band and the first conduction band at the G point. The difference in transitions of the first absorption peak is possibly due to the mirror image in AC- and AD-stacked BBP. The first strong absorption peak corresponds to a bound exciton (B1) with an exciton binding energy of 0.82 eV for AA-stacked BBP, while the binding energies of the same bound exciton corresponding to AB-, AC- and AD-stacked BBP are 0.55, 0.62 and 0.47 eV, respectively, as listed in Table 2. These exciton binding energies are comparable to 0.83 eV in monolayer black phosphorus (Table 2). Moreover, the optical spectrum has a wide absorption range of 0.2–5 eV which covers the whole spectrum of visible light for four stacking patterns as shown in Fig. 4(a)–(d), which may make few-layer BP as potential candidates for photovoltaic devices. For light polarized along the y direction, the optical absorption is inactive around the band gap due to the forbidden transitions, leading to the absorption onset of around 3 eV. However, the absorption onset can be changed from 0.2 eV to 0.75 eV for the light polarized along the x direction. This shows that the absorption onset is tunable in a wide range for BBP. The exciton (B1) with the largest oscillator strength locates at 3.42, 3.01 and 3.26 eV and the corresponding binding energies are 0.77, 0.63 and 0.27 eV for AA-, AB- and AC-stacked BBP respectively. However, the resonant exciton (R1) possessing the largest oscillator strength is found at 2.91 eV and corresponds to a binding energy of 0.37 eV for AD-stacked bilayer (Table 2). Moreover, the four stacking bilayers are almost transparent to y-polarized light in the energy range of 0–2 eV as shown in Fig. 4(f)–(j), rendering them suitable for applications associated with long-lifetime dark excitation,44 while x-polarized light is strongly absorbed in this range. The stacking-dependent anisotropic optical response in BBP further suggests that they may form a different optical linear polarizer, which can be utilized in optical quantum computers45 and liquid-crystal display devices.46
Fig. 4 (a–e) Optical absorption spectra of AA-, AB-, AC- and AD-stacked BBP and monolayer black phosphorus (ML) for incident light polarized along the x direction; (f)–(j) absorption spectra for incident light polarized along the y direction. Absorption spectra with and without electron–hole interaction are indicated with black and red lines, respectively. A 0.1 eV Lorentzian broadening is applied in these plots.
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Fig. 5 (a) and (b) Optical absorption spectra of BBP and monolayer black phosphorus with electron–hole interaction included for incident light polarized along x and y directions, respectively. A 0.1 eV Lorentzian broadening is applied in these plots.
Fig. 6 (a–d) Real-space distribution of typical bright excitons for four different stacking structures. The upper (lower) panels are for light polarized along the x(y) direction. The holes indicated by red filled circles are fixed on phosphorus atoms.
Additionally, compared to that of monolayer black phosphorus, the optical absorption spectra of BBP for x-polarized light, are clearly red-shifted and show more intense absorption in the infrared region (Fig. 5(a)), which are attributed to the decreased band gap induced by the interlayer interaction. For y-polarized light, significant redshift only occurs in the optical absorption spectrum of AA-stacked BBP. Also, the AC- and AD-stacked BBP have stronger absorption in the high energy region than AA- and AB-stacked BBP for y-polarized light, and the optical spectrum of AC-stacked bilayer has more significant blue-shift than that of the AA-stacked bilayer (Fig. 5(b)). To gain insight into the optical excitations of four stacking structures, we plot the real-space distribution of typical bright excitons, i.e. the bound excitons correspond to first absorption peaks for x-polarized light and the excitons with the largest oscillator strength for y-polarized light, in Fig. 6(a)–(d). The wave function of an exciton can be expressed as S c ðre ; rh Þ ¼ P AS cck ðre Þ cvk ðrh Þ, where re(rh) and c are cvk
cvk
the real space electron (hole) coordinate and the quasiparticle wave function, respectively. The wave function distribution all exhibit an obvious decaying nature, indicating the binding between quasi-electron and quasi-hole confined by the Coulomb interaction. Also, for x-polarized light, the electron spatial distribution of these bound excitons is extended along the armchair direction of the lattice and these excitons usually form striped patterns. This is similar to the case that was found
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in bundles of silicon nanowire,47 arising from the near-isotropic Coulomb interaction. Furthermore, the electron and the hole are confined within a more narrow area in the AA-stacked BBP due to its relatively larger exciton binding energy (0.82 eV). In contrast, for light polarized along the y direction, the spatial distribution of the bright exciton wave functions spreads over a large area mainly along the zigzag direction of the lattice for all four stacking structures. The spatial distribution of the bound or resonant exciton wave function in the same stacking pattern exhibits a highly anisotropic exciton as presented in Fig. 6(a)–(d).
4 Conclusions In summary, we have systematically investigated the electronic structure and optical properties of bilayer black phosphorus in four possible stacking patterns within the framework of density functional theory, the G0W0 method and Bethe–Salpeter equation. Both the electronic structure and optical properties show evident dependence on the stacking pattern. AD-stacked bilayer black phosphorus has the smallest band gap of 0.92 eV at the G0W0 level, while AB-stacked BBP has the largest band gap of 1.31 eV. Such dependence is attributed to the different interlayer interaction due to different stacking pattern. The exciton binding energy of the AA-stacked BBP can be as large as 0.82 eV, much larger than that (0.55 eV) of AB-stacked BBP. In addition
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to the stacking pattern dependence, BBP also shows excellent optical anisotropy, where infrared and visible light polarized along the armchair direction are strongly absorbed while it is ultraviolet light polarized along the zigzag direction that is strongly absorbed.
Acknowledgements This work is supported by the NSFC (21525311, 21173040, 21373045), NSF of Jiangsu (BK20130016), SRFDP (20130092110029) and the Scientific Research Foundation of Graduate School of Southeast University (YBJJ1521) in China. The authors thank the computational resources at the SEU and National Supercomputing Center in Tianjin.
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The stacking dependent electronic structure and optical properties of bilayer black phosphorus† Huabing Shu,a Yunhai Li,a Xianghong Niua and Jinlan Wang*ab By employing density-functional theory, the G0W0 method and Bethe–Salpter equation, we explore quasi-particle energy bands, optical responses and excitons of bilayer black phosphorus (BBP) with four different stacking patterns. All the structures are direct band gap semiconductors and the band gap is highly dependent on the stacking pattern, with a maximum of 1.31 eV for AB-stacking and a minimum of
Received 27th December 2015, Accepted 25th January 2016
0.92 eV for AD-stacking. Such dependence can be well understood by the tight-binding model in terms
DOI: 10.1039/c5cp07995k
energy are observed in BBPs. For x-polarized light, more red-shift of optical adsorption appears in
of the interlayer hopping. More interestingly, stacking sensitive optical absorption and exciton binding AA-stacking and the strong exciton binding energy in the AA-stacking bilayer can be as large as 0.82 eV,
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that is B1.7 times larger than that of AD-stacking bilayer.
1 Introduction Recently, a new member of the 2D materials family, few-layer black phosphorus (BP) has been successfully exfoliated.1–5 In few-layer BP, the phosphorus atom is bonded with three adjacent phosphorus atoms, forming a puckered peculiar structure with parallel zigzag rows, which is different from other 2D materials such as graphene6 and MoS2.7 In addition to their considerable direct band gaps (1.06 eV for tri-layer8 and 2 eV for monolayer8), few-layer BP exhibits strong excitonic effects with large exciton binding energies (490 meV for tri-layer8 and 760 meV for monolayer9). Moreover, the carrier effective mass,4 optical conductivity10 and absorption,11 excitons,8 spatial thermal conductance12,13 and Raman response14 of fewlayer BP are highly anisotropic, due to their special puckered structure. The carrier mobility of few-layer BP is also highly thickness-dependent.1,4,15 Specially, the hole mobility can be as large as 1000 cm2 V1 s1 for a thickness of about 10 nm and the on/off current ratios can exceed 105 for samples thinner than 7.5 nm. Significant blue shift of luminescence with decreasing layer thickness has been observed as well.14 These unique electronic and optical properties endow few-layer BP with great potential for electronic,1,2,4,16 and optoelectronic devices.17,18 For example, a few-layer BP based photovoltaic device has been fabricated which allows power generation for
a
Department of Physics, Southeast University, Nanjing 211189, China. E-mail: [email protected] b Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China † Electronic supplementary information (ESI) available. See DOI: 10.1039/ c5cp07995k
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illumination wavelengths up to 940 nm.17 The BP based photodetector is capable of acquiring high-contrast images in the visible and near-infrared spectral regime.18 Additionally, fewlayer BP based absorbers show saturable absorption at 532 nm,19 1030 nm19 and 1550 nm20 and can be used for optical pulse generation in Q-switching and mode-locking regime in the fiber laser setup.21 On the other hand, it has been reported that the stacking pattern has great influence on the properties of 2D layered materials.22–27 For example, the Bernal stacked bilayer graphene possesses unique electronic and optoelectronic properties,24,25 which are different from that of its twisted bilayer counterpart.22,23 h-BN bilayer structures with different interlayer symmetries show distinct optical second harmonic generation intensities.26 Also, different stacking patterns in the MX2 (M = Mo, W; X = S, Se) result in significantly different optical excitation and exciton binding energies.27 However, in bilayer black phosphorus, the relationship between the different interlayer interaction induced by different stacking patterns and properties has rarely been discussed and is still unclear. In this paper, we study the electronic and optical properties of bilayer black phosphorus considering different stacking patterns within many-body Green’s function and Bethe–Salpeter equation (BSE) formalism. The structure–property relation has been revealed and understood by a tight-binding model.
2 Computational details The first-principle calculations were carried out within the framework of many-body perturbation theory. First, the electronic ground states were obtained by density functional theory (DFT)
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with the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional28 including interlayer van der Waals interaction via the vdW-DF approach,29–31 as implemented in the QUANTUMESPRESSO code.32 Plane-wave basis sets with a kinetic energy cutoff of 60 Ry, Monkhorst–Pack33 k-grid of 14 10 1 and norm-conserving pseudopotentials34,35 were adopted. A sufficiently large vacuum layer of 20 Å was used to avoid spurious interactions between periodic images. Cell parameters and atomic positions were fully relaxed until the residual forces on each atom were smaller than 0.01 eV Å1 and the total energy difference was less than 104 eV. Then, single-shot G0W0 approximation36,37 was employed to obtain quasi-particle (QP) band structures. Finally, optical excitation energies and exciton wave functions were determined by solving the Bethe–Salpeter equation.38–40 The convergence of quasi-particle band gap with respect to the number of empty bands, the size of dielectric matrix and the Monkhosrt–Pack grid were carefully examined and a convergence within 0.1 eV was assured. 10 valence and 15 conduction bands were included to build the electron–hole interaction kernel. The involved unoccupied band number (480) was employed to obtain the converged dielectric function. A box-shape truncated coulomb interaction was adopted after 25 bohr along the direction perpendicular to the surface. A denser 56 40 1 k-grid was used to converge the optical spectra. The G0W0 and BSE calculations were performed using the YAMBO program package.41
3 Results and discussion We consider four possible high-symmetric stacking structures, namely AA-, AB-, AC- and AD-stacked bilayer black phosphorus. AA: the top layer is vertically stacked on the bottom layer with the symmetry of Pmma; AB: the top layer of AA-stacking is shifted by half of one unit cell along the x or y direction with the symmetry of Pbcm and the stacking pattern matches the structure of bulk BP; AC: the top layer of AA-stacking is shifted by one unit cell along the x or y direction with respect to the bottom layer and the top and the bottom layers are mirror
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images of each other with the symmetry of Pmma; AD: the top layer of AC-stacking is shifted by a half of one unit cell along the x or y direction with respect to the bottom layer with the symmetry of Pccm. Their optimized structures and lattice parameters are presented in Fig. 1(a)–(d) and Table 1. The interlayer distance d increases from 3.580 Å for AB-stacked bilayer to 4.033 Å for AC-stacked bilayer significantly and the largest bond angle difference is about 1 degree. The structural differences stem from the different interlayer interaction caused by different stacking patterns. Among the four stacking structures, AB-stacked bilayer is the most energetically favorable, which is 2.9 meV, 6.4 meV and 7.3 meV per atom lower than that of AA-, AD- and AC-stacked bilayer, respectively. To determine the stability of the four stacking structures, we have also calculated the formation energy (Ef), defined as Ef = (Etotal Etp Ebm)/2, where Etotal is the total energy of whole BBP and Etp (Ebm) is the total energy of the top layer (the bottom layer). The formation energies are 24, 12, 9 and 5 meV per layer for AB-, AA-, AD- and AC-stacked BBP, respectively, indicating that in addition to the AB-stacked bilayer which has already been fabricated, other three stacking structures are also feasible experimentally. The electronic band structures of BBP obtained from the DFT-PBE and the G0W0 method are presented in Fig. 2(a)–(d). Their band structures are similar to that of the monolayer black phosphorus (Fig. 3(a)), and show clearly asymmetric dispersion around G point along G–X and G–Y directions, indicating the anisotropic nature of BBP. For all four stacking patterns, the band gaps are always direct and locate at the G point, with the values of 0.52 eV for AA-, 0.55 eV for AB-, 0.43 eV for AC- and 0.33 eV for AD-stacked BBP at the DFT-PBE level (Table 2), respectively. Compared to DFT-PBE band structures, the conduction bands of QP band structures from G0W0 calculations are shifted up while the valence bands are shifted down due to many-body effects (Fig. 2(a)–(d)). Hence, the QP band gaps of AA-, AB-, AC- and AD-stacked BBP increase to 1.21, 1.31, 1.15 and 0.92 eV (Table 2), respectively. This trend is in line with DFT-PBE results, although the G0W0 method reproduces appreciable larger band gaps. Such significant QP corrections are the
Fig. 1 (a–d) Top and side views of optimized four stacking structures. a, b and d are lattice constants and interlayer distance, respectively. y1 and y2 denote bond angles while R1, R2 and R3 indicate bond lengths. For each layer of the bilayer black phosphorus, phosphorus atoms belong to different sub-layers are colored in yellow and green for clarity.
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Table 1 Optimized lattice parameters a, b and interlayer distance d; the bond lengths R1, R2 and R3; the bond angles y1 and y2. Lattice parameters and bond lengths are in angstroms; bond angles are in degrees
Type
a
b
d
R1
R2
R3
y1
y2
AA AB AC AD
4.541 4.517 4.548 4.540
3.315 3.327 3.323 3.331
3.816 3.580 4.033 3.742
2.224 2.231 2.227 2.228
2.280 2.250 2.276 2.252
2.228 2.233 2.230 2.227
96.135 97.087 96.753 96.766
103.206 102.442 103.303 103.401
consequence of reduced electronic screening. Additionally, our calculated QP band gap of the AB-stacking structure is in good agreement with the other theoretical result (1.32 eV).8 To further clarify the effects of interlayer interaction on the electronic structure of BBP, we examine the dependence of the band gap on the interlayer distance for four different stacking structures. As shown in Fig. S1 (ESI†), the band gap increases with the increase in interlayer distance, arising from the decrease in weak interlayer interaction. We also plot the alignment of
valence band maximum (VBM) and conduction band minimum (CBM) with respect to the vacuum level and the corresponding charge densities, as shown in Fig. 3(b) and (c). It is evident that VBMs are slightly changed for all four stacking patterns while CBMs for AC- and AD-stacked BBP are significantly shifted downward compared to that of the AB bilayer. Also, the overlap of interlayer charge density of CBM emerges in AC- and AD-stacked BBP, which is absent in AA- and AB-stacked bilayers, suggesting the stronger interlayer interactions in AC- and AD-stacked BBP. Furthermore, based on a tight-binding model, we can clearly demonstrate that the dominant interlayer hopping term in every stacked pattern of BBP will determine the size of band gap and larger dominant interlayer hopping term will lead to smaller band gap. The dominant interlayer hopping terms (ti0 , i = 1, 2,. . ., 6) follow the order of t40 (AB) o t10 (AA) o t10 (AC) o t10 (AD). The details are shown in the ESI.† For AB-stacked BBP, three distinct Raman peaks corresponding to three optical modes (A1g, B22g and A2g), have been
Fig. 2 (a–d) Band structures of AA-, AB-, AC- and AD-stacked bilayer black phosphorus. The DFT-PBE and G0W0 results are depicted by black and red solid lines, respectively. The valence band maximum (VBM) of the DFT-PBE band structure is shifted to zero.
Fig. 3 (a) Band structure of monolayer black phosphorus. The DFT-PBE and G0W0 results are depicted by black and red solid lines, respectively. The valence band maximum of DFT-PBE band structure is shifted to zero. (b) Energy levels of valence band maximum (VBM) and conduction band minimum (CBM) of BBP. All energies have been corrected with respect to the vacuum level from DFT-PBE calculation. (c) Isosurface plot of partial charge density corresponding to VBM and CBM of BBP. The isovalue is 1.58 103 e Å3.
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Table 2 Band gap Eg-DFT (eV) at the DFT-PBE level, Eg-G0W0 (eV) at the G0W0 level, excitation energy Ee (eV) and binding energy Eb (meV) of the exciton with the largest oscillator strength. Binding energy is defined as the difference between the excitation energy and the QP energy difference. The x and y indicate the polarization direction of the incident light
Type
Eg-DFT
Eg-G0W0
Ee
AA
0.52
1.21
AB
0.55
1.31
AC
0.43
1.15
AD
0.33
0.92
ML
0.79
2.01
0.39 3.42 0.76 3.01 0.85 3.26 0.88 2.91 1.18 3.06
Eb (x) (y) (x) (y) (x) (y) (x) (y) (x) (y)
820 770 550 630 620 270 470 370 830 470
experimentally observed and locate at 362 cm1, 438.5 cm1 and 467 cm1, respectively.42 We calculate these three Raman modes at the G point for four stacked BBP. Their frequencies corresponding to three Raman active modes are summarized in Table S3 (ESI†) and their vibrational patterns are displayed in Fig. S5 (ESI†). Interestingly, the A1g and A2g Raman modes show some dependence on the stacking pattern in BBP; the energies of A1g and A2g Raman modes increase with increasing interlayer interaction (Fig. S6, ESI†), i.e., the corresponding Raman peaks blue-shift. While the B22g mode is not sensitive to the stacking patterns in BBP, which is possibly attributed to the large elastic constant (162 GPa)43 along the zigzag direction. The optical absorption spectra of AA-, AB-, AC- and ADstacked BBP for incident light polarized along armchair (x) and zigzag (y) directions of the lattice are shown in Fig. 4(a)–(d) and (f)–(i), respectively. The optical absorption spectra with electron–hole interaction included are dominated by excitonic states and the excitation energies are significantly altered by the stacking pattern. For example, for incident light polarized along the x direction, the first strong absorption peak which is mainly contributed by the inter-band VBM–CBM transition at the G point, is located around 0.39 eV and 0.76 eV in AA- and
AB-stacked BBP respectively, indicating an optically active direct band gap. For AC- and AD-stacked BBP, a similar peak is observed around 0.85 and 0.88 eV, respectively, which mainly arises from the transition between the second valence band and the first conduction band at the G point. The difference in transitions of the first absorption peak is possibly due to the mirror image in AC- and AD-stacked BBP. The first strong absorption peak corresponds to a bound exciton (B1) with an exciton binding energy of 0.82 eV for AA-stacked BBP, while the binding energies of the same bound exciton corresponding to AB-, AC- and AD-stacked BBP are 0.55, 0.62 and 0.47 eV, respectively, as listed in Table 2. These exciton binding energies are comparable to 0.83 eV in monolayer black phosphorus (Table 2). Moreover, the optical spectrum has a wide absorption range of 0.2–5 eV which covers the whole spectrum of visible light for four stacking patterns as shown in Fig. 4(a)–(d), which may make few-layer BP as potential candidates for photovoltaic devices. For light polarized along the y direction, the optical absorption is inactive around the band gap due to the forbidden transitions, leading to the absorption onset of around 3 eV. However, the absorption onset can be changed from 0.2 eV to 0.75 eV for the light polarized along the x direction. This shows that the absorption onset is tunable in a wide range for BBP. The exciton (B1) with the largest oscillator strength locates at 3.42, 3.01 and 3.26 eV and the corresponding binding energies are 0.77, 0.63 and 0.27 eV for AA-, AB- and AC-stacked BBP respectively. However, the resonant exciton (R1) possessing the largest oscillator strength is found at 2.91 eV and corresponds to a binding energy of 0.37 eV for AD-stacked bilayer (Table 2). Moreover, the four stacking bilayers are almost transparent to y-polarized light in the energy range of 0–2 eV as shown in Fig. 4(f)–(j), rendering them suitable for applications associated with long-lifetime dark excitation,44 while x-polarized light is strongly absorbed in this range. The stacking-dependent anisotropic optical response in BBP further suggests that they may form a different optical linear polarizer, which can be utilized in optical quantum computers45 and liquid-crystal display devices.46
Fig. 4 (a–e) Optical absorption spectra of AA-, AB-, AC- and AD-stacked BBP and monolayer black phosphorus (ML) for incident light polarized along the x direction; (f)–(j) absorption spectra for incident light polarized along the y direction. Absorption spectra with and without electron–hole interaction are indicated with black and red lines, respectively. A 0.1 eV Lorentzian broadening is applied in these plots.
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Fig. 5 (a) and (b) Optical absorption spectra of BBP and monolayer black phosphorus with electron–hole interaction included for incident light polarized along x and y directions, respectively. A 0.1 eV Lorentzian broadening is applied in these plots.
Fig. 6 (a–d) Real-space distribution of typical bright excitons for four different stacking structures. The upper (lower) panels are for light polarized along the x(y) direction. The holes indicated by red filled circles are fixed on phosphorus atoms.
Additionally, compared to that of monolayer black phosphorus, the optical absorption spectra of BBP for x-polarized light, are clearly red-shifted and show more intense absorption in the infrared region (Fig. 5(a)), which are attributed to the decreased band gap induced by the interlayer interaction. For y-polarized light, significant redshift only occurs in the optical absorption spectrum of AA-stacked BBP. Also, the AC- and AD-stacked BBP have stronger absorption in the high energy region than AA- and AB-stacked BBP for y-polarized light, and the optical spectrum of AC-stacked bilayer has more significant blue-shift than that of the AA-stacked bilayer (Fig. 5(b)). To gain insight into the optical excitations of four stacking structures, we plot the real-space distribution of typical bright excitons, i.e. the bound excitons correspond to first absorption peaks for x-polarized light and the excitons with the largest oscillator strength for y-polarized light, in Fig. 6(a)–(d). The wave function of an exciton can be expressed as S c ðre ; rh Þ ¼ P AS cck ðre Þ cvk ðrh Þ, where re(rh) and c are cvk
cvk
the real space electron (hole) coordinate and the quasiparticle wave function, respectively. The wave function distribution all exhibit an obvious decaying nature, indicating the binding between quasi-electron and quasi-hole confined by the Coulomb interaction. Also, for x-polarized light, the electron spatial distribution of these bound excitons is extended along the armchair direction of the lattice and these excitons usually form striped patterns. This is similar to the case that was found
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in bundles of silicon nanowire,47 arising from the near-isotropic Coulomb interaction. Furthermore, the electron and the hole are confined within a more narrow area in the AA-stacked BBP due to its relatively larger exciton binding energy (0.82 eV). In contrast, for light polarized along the y direction, the spatial distribution of the bright exciton wave functions spreads over a large area mainly along the zigzag direction of the lattice for all four stacking structures. The spatial distribution of the bound or resonant exciton wave function in the same stacking pattern exhibits a highly anisotropic exciton as presented in Fig. 6(a)–(d).
4 Conclusions In summary, we have systematically investigated the electronic structure and optical properties of bilayer black phosphorus in four possible stacking patterns within the framework of density functional theory, the G0W0 method and Bethe–Salpeter equation. Both the electronic structure and optical properties show evident dependence on the stacking pattern. AD-stacked bilayer black phosphorus has the smallest band gap of 0.92 eV at the G0W0 level, while AB-stacked BBP has the largest band gap of 1.31 eV. Such dependence is attributed to the different interlayer interaction due to different stacking pattern. The exciton binding energy of the AA-stacked BBP can be as large as 0.82 eV, much larger than that (0.55 eV) of AB-stacked BBP. In addition
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to the stacking pattern dependence, BBP also shows excellent optical anisotropy, where infrared and visible light polarized along the armchair direction are strongly absorbed while it is ultraviolet light polarized along the zigzag direction that is strongly absorbed.
Acknowledgements This work is supported by the NSFC (21525311, 21173040, 21373045), NSF of Jiangsu (BK20130016), SRFDP (20130092110029) and the Scientific Research Foundation of Graduate School of Southeast University (YBJJ1521) in China. The authors thank the computational resources at the SEU and National Supercomputing Center in Tianjin.
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