Electronic and optical responses of quasi-one-dimensional phosphorene nanoribbons to strain and electric field

Electronic and optical responses of zigzag- and armchair-edge quasi-one-dimensional phosphorene nanoribbons (Q1D-PNRs) to strain and external field are comparatively studied based on the tight-binding calculations. The results show that: (i) Zigzag-edge Q1D-PNR has the metallic ground state; applying global strains can not open the gap at the Fermi level but applying the electric field can achieve it; the direct/indirect character of the field-induced gap is determined by the electron-hole symmetry; an electric-field-enhanced optical absorption of low-energy photons is also predicted. (ii) Armchair-edge Q1D-PNR turns out an insulator with the large direct band gap; the inter-plane strain modulates this gap non monotonically while the in-plane one modulates it monotonically; in addition, the gap responses to electric fields also show strong direction dependence, i. e., increasing the inter-plane electric field will monotonically enlarge the gap but the electric field along the width direction modulates the gap non monotonically with three characteristic response regions.

∑ < > runs over the considered hopping integrals. Operator † c i (c i ) is to create (annihilate) a p z orbital electron with spin s at site i; t i,j represents the hopping integral between sites i and j. The value of t i,j depends on the relative angle and distance between site i and j 17 . For the fully relaxed puckered structure, t i,j is determined by reproducing the first-principle calculations. We set the hopping integral between the nearest-neighboring (NN) sites as t 1 = −1.22eV. Meanwhile, the other parameters are set for: t 2 = −2.5t 1 , t 3 = 0.17t 1 , t 4 = 0.01t 1 , t 5 = 0.05t 1 , t 6 = 0.02t 1 23-26 . Obviously, it can be a reasonable approximation to further set t 4 = t 5 = t 6 = 0 because they are much smaller than t 1 , t 2 and t 3 . Indeed, we numerically checked that no qualitative difference was induced by this approximation. U denotes the on-site e-e interaction while the inter-site ones are modeled in 2 , where κ = 1.5 reads as a dielectric parameter and r i,j the distance between the sites i and j. Referring to Fig. 1, we set the structural parameters as follows: the distance between the in-plane  NN sites a 1 , and that between the inter-plane NN sites a 2 , are equally set as a a 2 16Å 1 2 = = . 26 ; the angular parameters are set as β = 104°, α = 98°. E-e interactions are treated by the Hartree-Fock (HF) approximations.

Results
4z-Q1D-PNR. By is the average charge density at site m′. The off-diagonal matrix elements include two types, the effective in-plane hopping integrals X m,n and the inter-plane ones Y m,n . X m,n are written in with (m, n) denoting the site couples of (1, 2), (3,4), (5, 6) and (7,8). Y m,n are derived as † where (m, n) represents the site couples of (2, 3), (4,5) and (6,7). We calculate the local density of state (LDOS) at site i in the momentum representation by , , x where E λ is the energy of the λ-th HF level and Ψ λ,s (k x ,i) denotes the wavefunction of wavevector-k x , spin-s and energy-E λ . Figure 3 presents the ground-state dispersion relations and LDOS of 4z-Q1D-PNR. As shown, the molecular orbital of the p z electrons are unfolded into eight band branches, among which the ones above (below) the Fermi level (E F = 0.24t 1 ) are numbered as J c (J v ) = 1.2,  . The electronic state turns out metallic meanwhile the adhesively paired bands are formed around the Fermi level. This band structure is characteristic for zigzag-edge PNRs and have been repeatedly revealed 25,28,29 . By observing the LDOS spectra, one can learn that the electronic state corresponding to the adhesively paired bands are mainly distributed along the ribbon edges, so to say, they are the so-called edge states. The point is that for a zigzag-edge Q1D-PNR, even its ribbon width is extremely narrow, the edge state should still be formed. In the following, we will reveal that the key factor to determine the formation of the paired edge state is the magnitude of the inter-plane hopping integrals.
Then we proceed to discuss the electronic/optical responses of 4z-Q1D-PNR to strains. The optical response is examined by calculating the polarized optical conductivities whose real part is derived as is the HF ground state and |0〉 is the true electron vaccum. Following the way of defining the current operator in ref. 30 , J x and J y are respec- Because PNRs demonstrate superior flexibility and can withstand high tensile strain up to 40% 31 , in this work, we adjust t 2 and t 3 in the large scales. Figure 4 presents 4z-Q1D-PNR's electronic/optical responses to the inter-plane strain, which is simulated by adjusting the magnitude of |t 2 /t 1 |. Dispersion relations for three typical valuse |t 2 /t 1 | = 0.5, 2.5 and 4.0 (respectively correspond to the tensile, none, and compressive inter-plane strain) are demonstrated. Comparing them three, it is found that the formation of the adhesively paired bands strongly depends on the magnitude of |t 2 /t 1 |. A larger |t 2 /t 1 | tends to separate the paired bands from the bulked ones and thus eventually leads to the edge state. On the other hand, the optical response to the inter-plane strain is significant [see Fig. 4(b)]. When |t 2 /t 1 | = 0, 4z-Q1D-PNR can not absorb the x-direction polarized photons, but it is easy to absorb the y-direction polarized photons at the low-energy regime. With increasing |t 2 /t 1 |, the absorption of x-polarized photons becomes allowed and the corresponding spectrum becomes narrower and higher. When |t 2 /t 1 | = 4.0, namely, under the strong compressive strain, the optical absorption has already become sharply peaked.
Obviously, the optical selection rules between the x and y direction is quite distinct. This can be explained by a similar mechanism which we revealed in the previous work 30 . 4z-Q1D-PNR exhibits the C 2x symmetry and there are the relations On the other hand, the spatial wave functions of the bands alternatively exhibit symmetrical and antisymmetrical parity. In order to ensure the inner product among the initial and the final state on J x(y) is nonzero, the transitions between the same-parity bands should be only allowed for the x-polarized photons but forbidden for the y-polarized ones and vice versa. The paired bands near the Fermi level are of opposite parity and therefore we only observe the Drude-like absorptions for the y-direction polarized photons. Figure 5 shows the in-plane strain's effects on the electronic/optical properties of 4z-Q1D-PNR. The in-plane strain will significantly change the angle α, therefore it primely influences the magnitude of t 3 /t 1 . In this work, we examine t 3 /t 1 in the range from 0 to 0.75 with the nominal value being t 3 /t 1 = 0.17. For the limiting case t 3 /t 1 = 0, the valence and conduction bands show exact electron-hole (e-h) symmetry, indicating the magnitude of t 3 /t 1 is the key factor to determine the e-h asymmetry of PNRs. This conclusion agrees with the experience in graphene nanoribbons that a finite next-nearest-neighbor (NNN) hopping integral will break the e-h asymmetry 32 . Furthermore, with increasing t 3 /t 1 , the bands tend to show the strong dispersive character, implying the effective mass of electrons/holes will be significantly modulated by the in-plane strains. On the other hand, for the optical properties [ Fig. 5(b)], it seems that although the e-h asymmetry is enhanced by increasing t 3 /t 1 , the optical conductivity spectra are almost not affected. We have checked that the energy difference between the band branches at all the k points are almost unchanged for any t 3 /t 1 , meanwhile the optical selection rules are also maintained against varying t 3 /t 1 . We notice that for both applying the in-plane and inter-plane strains, the adhesively paired bands do not split to form a gap. This is due to a global strain itself can not break the original symmetry. But applying an external electric field can achieve this goal. For example, Ezawa theoretically revealed the metal-insulator (M-I) transitions induced by applying the y-direction electric field 25 . In this paper, we demonstrate that same effects can be also achieved by applying the z-direction electric field. If only considering the unscreened electric field, the field induced electronic potential difference between the two planes can be simply estimated as eE z z, where E z represents the field strength and z = a 2 cosβ is the vertical distance between the two planes. The results show that increasing the z-direction electric field will indeed trigger the M-I transitions [see Fig. 6(a)] with the transition threshold about eE z z ≈ 0.2t 1 . The mechanism of the M-I transition is as follows: z-direction electric field causes the on-site energy difference between the two planes, which induces the charge redistribution and therefore breaks the original symmetrical properties of bands J c = 1 and J v = 1. These two bands will repel each other due to the so-called field-induced anticrossing effect [33][34][35] meanwhile the on-site energy difference provides the splitting energy to separate them. It is worth noting that the electric-field-induced gap here is indirect. We find that whether this gap is direct or indirect is dependent on that whether the e-h symmetry of the original zPNR is breaking or not. For example when t 3 /t 1 = 0, the e-h symmetry is completely satisfied. In this case, applying the electric field will result in the directly gapped state (Fig. 7). Besides, we also find an interesting phenomenon of the optical response to electric field: the absorption of the low-lying (resonant to the gap size) x-polarized photons becomes allowed due to the existence of the electric field. Further increasing the field strength will continuously enhance this absorption. Meanwhile, as the gap is enlarged with increasing the field strength, the low-lying absorption peak behaves a blue shift [see Fig. 6 We briefly discuss the Coulomb interaction's effects on the electronic structure of 4z-Q1D-PNR. It is found that increasing Coulomb interaction results in the long-range charge-order-waves (CDWs) along two zigzag edges, while charges almost keep averagely distributed over the inner phosphorus atoms. Even though CDWs 5a-Q1D-PNR. Generally, armchair-edge PNRs are insulators and thus their electronic properties are qualitatively distinct to those of zigzag-edge ones. In this section, we theoretically investigate 5a-Q1D-PNR's electronic/ optical responses to strain and electric field.
The k x -block Hamiltonian of 5a-Q1D-PNR in the momentum-representation is: 5,9 7 ,9 9,9 9 ,10 6,10 8,10 9,10 10,10 [see Fig. 1(b)]. The diagonal elements in the above matrix read as The off-diagonal matrix elements corresponding to the in-plane NN effective-hopping-integrals are defined as m n x x m n s , m n s k k n s k m s x , , , and (m, n) represents the site-couples of (1, 3), (5, 3), (5, 7), (9, 7), (4,2), (4,6), (8,6) and (8,10) = ⁎ . The in-plane NNN effective hopping integrals are defined as where (m, n) represents the site-couples of (1, 5), (2,6), (3,7), (4,8), (5,9) and (6,10). The inter-plane effective hopping integrals are defined as and (m, n) = (2, 1), (3,4), (6,5), (7,8) and (10,9). Figure 8 presents the dispersion relations and LDOS of 5a-Q1D-PNR. As shown, 5a-Q1D-PNR has the direct band gap at k = 0. The gap size is about Δ g = 1.7t 1 ≈ 2.1eV. The scaling rule of armchair-edge PNRs's band-gap with changing the ribbon width has been previously revealed 26,36 . The gap is monotonically reduced with increasing the ribbon width by Δ g ~1/d 2 (d the ribbon width). It seems that the band-gap scaling rule of aPNRs does not follow the so-called 3n-rule which was found in aGNRs (ribbons with width 3n + 2 are nearly metallic) 37,38 . In addition, two flat bands are formed in the 5a-Q1D-PNR's band structure. Notice that the flat bands are not formed in the gap region but embedded in the bulked valance and conduction bands, therefore, they do not significantly contribute to additional effects on the electronic properties unless it is doped at a proper concentration. This band structure somehow seems similar to that of polyphenanthrene (PPN) 39 , which is the narrowest armchair-edge graphene nanoribbon. There was a famous story that PPN turned out a BCS-type superconductivity with the Curie temperature about 10 K by doping alkali 40 . We notice that a DFT calculation has predicted that similar superconductivity mechanism seems also realizable in aPNRs when it was doped by electrons 41 .
We proceed to study 5a-Q1D-PNR's electronic/optical responses to strain and electric field. The real part of optical conductivities are calculated using Eq. (8), where the current operators along the x and y directions are now respectively defined as: with (m, n) representing for the site-couples of (1, 3), (4, 6), (5,6) and (8,10); (p, q) for (3, 5), (2, 4), (7,9) and (6,8); (j, l) for (1, 5), (2, 6), (5,9), (6,10), (3,7) and (4,8). Figure 9 demonstrates the electronic/optical responses to the inter-plane strain. Fundamentally, we conclude that the tensile strains (e. g. |t 2 /t 1 | = 0.5) tend to reduce the gap and the band width, while the compressive ones (e. g. |t 2 /t 1 | = 2.5) tend to enlarge them. Such a trend can be read out from the optical conductivity spectra σ x and σ y , where the absorption peaks show significant blue shift with increasing |t 2 /t 1 |. Furthermore, we find that the positions of the two flat bands in the band structure can be modulated by adjusting |t 2 /t 1 |. For example, when |t 2 /t 1 | = 0.5, the flat bands almost coincide to Fermi level. In such case, the flat bands may induce some novel phenomena caused from the Van Hove singularity near the Fermi level, e. g. ferromagnetism 42 , fractional Hall effect 43 , and superconductivity 44 . So to say, we predict that beside the way of electronic doping, a tensile strain may also possibly induce a superconductivity in armchair-edge Q1D-PNR.
On the other hand, by comparing the x-polarized optical conductivity spectra in Figs 4(b) and 9(b), it can be concluded that the selection rule of PNRs shows strong edge-dependence. This phenomenon can be understood by referring to the scenario in GNRs 45-47 . Lin et al. addressed that the selection rule of zGNRs satisfied ΔJ = |J c − J v | = odd while the armchair-edge ones were governed by the rule of ΔJ = 0 33,45,47 . By checking the dipole moment of interband transitions of 4z-Q1D-PNR and 5a-Q1D-PNR contributing to their main absorption peaks [red arrows in Figs 4(b) and 9(b)], we find the edge-dependent selection rules revealed by Lin also work for PNRs. Figure 10 demonstrates the effects of adjusting t 3 /t 1 in 5a-Q1D-PNR. Same to 4z-Q1D-PNR, a finite t 3 /t 1 will lead to the e-h asymmetry. In addition, the band gap is significantly decreased with increasing t 3 /t 1 . Unlike the case of varying |t 2 /t 1 |, increasing t 3 /t 1 will parallelly move the two flat bands in one direction. Meanwhile, the J c = 1 band becomes more dispersive while the J v = 1 band becomes more flat with increasing t 3 /t 1 . These bands' transitions can be identified from the optical conductivity spectra shown in Fig. 10(b). Increasing t 3 /t 1 until the low-energy flat band approaches to the J v = 1 band, one satellite absorption peak is separated from the continuum spectra of σ x . We have checked that this satellite peak corresponds to the transitions from the flat band to the J c = 1 band. On the other hand, because the band gap is continuously decreased with increasing t 3 /t 1 , one can observe a resultant red shift of the lowest-energy absorption peak in the σ y spectra.
The gap size as functions of |t 2 /t 1 | and t 3 /t 1 are summarized in Fig. 11. As shown, adjusting |t 2 /t 1 | will modulate the gap non monotonically. As the nominal value is |t 2 /t 1 | = 2.5, fundamentally, we say that a compressive inter-plane strain tends to enlarge the gap while a tensile inter-plane strain tends to reduce the gap. The smallest gap reaches about 0.5t 1 at |t 2 /t 1 | = 1.5. On the other hand, for adjusting t 3 /t 1 , it is found that increasing t 3 /t 1 leads to the monotonic reduction of the band gap. The gap almost closes when t 3 /t 1 = 1.5t 1 . In a word, the electronic response of 5a-Q1D-PNR to strains shows strong direction dependence. At last, we discuss the modulation of 5a-Q1D-PNR's band gap by applying the external electric field. The cases of applying the y-and z-direction electric fields are respectively considered. The effect of applying the y-direction (along the width direction) electric field is to form the electronic potential difference between between the ribbon edges as = α V e E a 4 sin y y0 2 , while the z-direction field results in the potential difference between the two planes as V z = eE z a 1 cosβ. Modulations of the gap by applying electric fields also show strong direction dependence. The   dispersion-relations for |V y | = 0, 1.5t 1 , 2.6t 1 , 4.0t 1 , 5.6t 1 , 7.0t 1 and 8.0t 1 are respectively shown in Fig. 12(a-g). The electronic response is nonlinear and turns out three variation regions with increasing |V y |: (i) Firstly, the band gap decreases with increasing the field strength [see Fig. 12(a,b)]. (ii) The band gap almost closes at about |V y | = 3.6t 1 [see Fig. 12(c)], but interestingly, it opens again with further increasing |V y | [ Fig. 12(d)] and closes again at |V y | = 5.6t 1 [see Fig. 12(e)]. This variation was also previously predicted for 8a-Q1D-PNR by Sisakht 26 and it was addressed that such a novel trend was a character of extremely narrow aPNRs. The last achieved metallic state exhibits two Dirac-like points. These two points are pushed toward k = ±π with increasing |V y |. (iii) After the two Dirac-like points reached k = ±π, further increasing V y reversely increase the band gap [see Fig. 12(f,g)]. In contrast, when increasing the y-direction electric field strength [|V y | = 0, 1.5t 1 , 2.6t 1 , 4.0t 1 , 5.6t 1 , 7.0t 1 , 8.0t 1 are respectively shown in Fig. 12(a′-g′)], the gap size monotonically keeps increasing.

Discussions and Conclusions
In summary, based on the TB calculations, we comparatively studied the electronic and optical responses of 4z-Q1D-PNR and 5a-Q1D-PNR to strain and electric field. The results suggested that zigzag-and armchair-edge phosphorene nanoribbons had distinct response behavior and therefore they could be used as different functional devices.
Zigzag-edge Q1D-PNR exhibited the metallic ground state. The inter-plane strain played the central role to form adhesively paired bands near the Fermi level. Adjusting the magnitude of inter-plane strain would significantly influence the optical conductivity spectrum and induce the shift of the absorption peaks; on the contrary, the optical response to in-plane strain is relatively weak, but the bands' dispersive character is sensitive to inter-plane strains so that the effective mass of electrons/holes could be significantly affected. On the other hand, for armchair-edge Q1D-PNR, which was an insulator with the direct band gap, we found that applying the compressive inter-plane strain would enlarge the gap, while the compressive in-plane strain would decrease the gap. No matter zigzag-or armchair-edge ones, it seemed that the topology of the electronic state was preserved against any strains, so to say, one should not expect a strain induced M-I or I-M transition in a Q1D-PNR.
To break the symmetry preserved topology of the electronic state, one could consider applying an electric field. We addressed that for zigzag-edge Q1D-PNR, applying the electric field either along the y or z direction would induce the M-I transition when the strength of the field reaches a threshold. Furthermore, it was showed that the key factor to determine the direct/indirect character of the field-induced gap was the e-h asymmetry, which could be controlled by adjusting the in-plane strains. On the contrary, for armchair-edge Q1D-PNR, we showed that both the y-and z-direction external electric field could modulate the gap size, but they behaved quite distinct modulating rules: increasing the y-direction electric field led to the non monotonic response of the gap with three variation regions, while increasing the z-direction electric field monotonically enlarged the gap. Our theoretical work provides a fundamental understanding of the electronic and optical properties of zigazag-and armchair-edge Q1D-PNRs in presence of strains and electric fields. We believe these results should be meaningful for engineering BP based quasi-one-dimensional molecular devices in the future.