Electromechanical strain response of phosphorene nanotubes

Nanomaterials that undergo structural or other property changes upon application of external stimuli are called stimuli responsive materials and are particularly suited for drug delivery, biosensing or artificial muscle applications. Two-dimensional (2D) black phosphorus is an ideal material for such applications due to its remarkable electromechanical response. Given that one-dimensional (1D) black phosphorus nanotubes (PNTs) are calculated to be energetically stable, it is possible that they can undergo similar electromechanical responses to their 2D counterparts, allowing their potential application as nanochannel devices for drug delivery. Using first-principles density functional theory, we investigated the electromechanical response of different-sized PNTs upon charge injection. Upon hole injection, the diameter of the PNTs expands up to a maximum of 30.2% for a (0,15) PNT that is 0.24 nm in diameter. The PNTs become highly p-doped as the valence band maximum crosses the Fermi level and undergoes switching from a direct to indirect band gap. The mechanism behind the electromechanical response was determined through analysis of the structural deformations, charge density distribution and Bader partial charges. It was shown that injection of charge alters the Young’s Modulus of the PNTs, as hole injection weakens the structural integrity of the nanotube, allowing a greater electromechanical response, with PNT-15 showing the largest decrease in the Young’s Modulus of 15.34%. These findings show that 1D PNTs are promising materials for the development of nanoelectromechanical actuators which could be used for drug delivery, energy harvesting or similar applications.


Introduction
Many important devices used in our everyday lives require some form of mechanical response to function, such as sensors, switches, pacemakers, and prosthetics.As many of these devices are becoming smaller in size and more flexible, it is essential to discover novel materials that can maintain their functionality and efficiency.Stimuli responsive materials (SRM) are materials that can generate mechanical responses by undergoing dimensional changes when exposed to an external stimulus.For example, exposure of a SRM to high temperatures can activate a sensor that rings an alarm bell in the event of a fire [1].SRMs can also respond to electrical energy by converting it into mechanical energy, allowing the material to be used in nanoactuator devices as artificial muscles, for example [2][3][4].
A number of one-dimensional (1D) and two-dimensional (2D) nanomaterials have been shown to act as SRMs, undergoing mechanical strain in response to external stimuli [2,5,6].This allows them to be used in a diverse range of applications such as artificial implants [7], hydrogen storage devices [8], drug delivery systems [9], sensors [10], nanotweezers [11] and nanoprobes [12].However, a key requirement for these nanomaterials to be used in such applications is that they provide sufficient maximum strain responses while retaining their structural stability and integrity.
Early theoretical studies using computational methods showed that nanomaterials can achieve substantial strain responses upon application of charge injection.2D graphene can display reversible strain up to ∼0.2% as a result of electron injection [13], with oxygen doping significantly enhancing its response, achieving a maximum actuation strain of 28% upon hole injection [14].1D carbon nanotubes (CNTs) can also undergo expansions and contractions of up to ±1.2% upon charge injection [15].Other 1D materials can also act as SRMs, including MXene nanotubes that can achieve diameter expansions as high as 26.8% in response to hole injection [16].
In more recent studies, 2D black phosphorus (BP) was shown to generate strain responses greater than graphene and the MXenes, with maximum values of 36.6% along the armchair direction, due to its puckered orthorhombic structure, allowing for greater flexibility [17,18].However, the electromechanical response of phosphorene nanotubes (PNT) as a function of charge injection is not known.While PNTs have not yet been synthesized, their predicted stability and strain energies are comparable with CNTs [19], indicating they could be synthesized.
In this study, first principles calculations are employed to investigate the structural stability and electromechanical response of PNTs under applied charge injection.The effect of nanotube diameter on the electromechanical response is also determined to ascertain the ability of these materials to be used in nanoelectromechanical actuator applications.

Computational methods
The calculations in this work were performed using DFT as implemented in the Vienna ab initio Simulation Package [20][21][22][23][24].The generalised gradient approximation (GGA) and Perdew Burke-Ernzerhof (PBE) exchange-correlation functional was adopted [25].The ion-electron interaction was defined using the project augmented wave (PAW) method [26] and the plane-wave energy cut-off was set to 600 eV.All atoms in the models were allowed to relax until the total energy was converged to 1 × 10 −6 eV and the Hellman-Feynman force on each relaxed atom was less than 0.001 eV Å −1 .The calculated lattice parameters of monolayer phosphorene were a (zig-zag direction) = 3.30 Å and b (armchair direction) = 4.47 Å, which agree with previous experimental (a = 3.31 Å and b = 4.38 Å) [27] and other theoretical values [18,19,28].To model the nanotubes a vacuum region of 20 Å was applied along the x-and y-lattice directions to avoid interactions between adjacent nanotubes (see supplementary note 1.).The atomic structure, key bond lengths and angles of the nanotubes are shown in figure 1.
A Monkhorst-Pack k-point mesh of 9 × 9 × 1 and 1 × 1 × 9 was used for the BP monolayer and nanotubes, respectively.For the nanotube models, five armchair PNT configurations, in the range of (0, 9) to (0, 17) were considered.The zig-zag configurations were not considered due to the fact that they are energetically unfavourable and therefore not stable [19].The Bader partial charges were calculated using the procedure as described by Henkelmen et al [29].The thermodynamic stability of the PNTs was investigated by performing ab initio molecular dynamics (AIMD) simulations.The simulations were run at 298 K with temperature control using the Nosé thermostat [30] and a time step of 1 fs.To avoid the structure drifting within the cell, one atom in each nanotube was fixed for the duration of the simulation.
The strain energies of the armchair PNTs were calculated as the total energy difference between the energy per atom of a nanotube and a BP monolayer as follows: where, E Tot(1D) is the total energy of an armchair PNT, E Tot(2D) is the total energy of the BP monolayer and n P is the total number of phosphorene atoms that comprise the PNT and BP monolayer.The Young's modulus, Y, for the PNTs was calculated through the second order derivative of the total energy, E Tot , with respect to axial strain [31]: where V 0 is the nanotube volume at ε = 0 and is defined as V 0 = 2π RL 0 ∂; R is the radius of the nanotube, L 0 is the nanotube length along its transport direction at ε = 0 and ∂ (5.2 Å) is the shell thickness chosen as the van der Waals distance for black phosphorene [32].The total energies over uniaxial strain along the transport direction are fitted to a second order polynomial to obtain the second derivative.

Atomic structure of PNTs
Before calculating the electromechanical strain response of the PNTs, it is important to understand the structural parameters that can be affected by the injection of charge (figure 1).Five different sized armchair PNTs were considered here, with chiral indices (0, n), where n = 9, 11, 13, 15 and 17.The optimised lattice parameters and bond lengths (B1 and B2) and bond angles (α and β) of the nanotubes are presented in table 1, along with those of the 2D monolayer for comparison.The diameter (D) of the PNTs was measured as the average of the inner and outer P-P distances as shown in figure 1(b).
For the (0, 9) PNT, both the inner and outer B1 bond lengths contract and expand, respectively, compared to the 2D monolayer.As the diameter of the PNT increases, both B1 and B2 approach the values of the monolayer and the α and β angles increase, approaching the values of the monolayer.The length of the transport direction of the nanotubes is 3.30 Å and lies along the zig-zag direction.The structural parameters obtained in this work agree with previous theoretical studies on armchair PNTs [19,31].

Strain energies and stability of PNTs
A key criterion before measuring the electromechanical response of the PNTs is to determine their energetic stability relative to the 2D monolayer.Figure 2 shows the strain energies of each PNT investigated as calculated according to equation (2).The strain energies were calculated to decrease as the diameter of the nanotubes increase indicating that the larger sized nanotubes are more energetically stable, with the strain energy of PNT-17 being 15 meV.This follows the trend calculated previously for nanotubes with a diameter smaller than or equal to PNT-15 [19].Therefore as the nanotube diameter continues to increase, the strain energy decreases as the geometry approaches that of the 2D monolayer.When we compare the strain energies of the PNTs to CNTs [33] of a similar diameter (1.15-2.3nm), the values are approximately 30% smaller, indicating that the nanotubes with diameters studied here are likely to be stable and grown experimentally.
The thermodynamic stability of PNT-9 and PNT-15 was evaluated using AIMD simulations and the evolution of the total energy and P-P bond lengths (as described by B1 and B2), as a function of time, are presented in figure 3.Both the B1 and B2 bond lengths were shown to fluctuate by 0.1 and 0.2 Å, respectively, throughout the whole simulation, indicating that no major structural deformations occurred at 298 K (see movie S1).The total energy for both nanotubes was found to fluctuate by less than 0.001 eV, which reinforces the stability of both PNTs at 298 K and also that the diameter does not significantly affect their stability.

Strain response as a function of charge injection
The strain response of the PNTs upon electron and hole injection, up to a maximum of ± 0.125 e/atom, is shown in figure 4. Charges beyond ± 0.125 e/atom for these systems were not investigated due to the nanotubes reaching a rupture point (see figure S2).As some of the nanotubes changed shape after charge injection (figures S3-S7) their diameter was calculated as the average of all the inner and outer P-P distances, as labelled in figure 1.
From figure 4, it is clear that the structural deformations are greater for hole injection than electron injection.For electron injection, the diameter strain does not change significantly as a function of NT diameter, with all PNTs expanding by less than 5% at the largest electron injection value of −0.125 e/atom.PNT-9 expands the most (by 3.70%) while PNT-17 only expands by 1.03% with the order following the chirality.
For hole injection, the PNTs show an exponential increase in diameter strain as the concentration of injected holes increases.Interestingly, the trend does not follow the chirality of the NT, with PNT-15 showing the greatest response, expanding by up to 30.2%.This is larger than what has been achieved for other nanotubes, including Sc 2 C nanotubes (26.8% [16]), CNTs (1.2% [15]) and MoS 2 /WS 2 MWNTs (16.0%[34]) (see table 2), suggesting that PNTs may be ideal materials for transporting large volumes of drugs, for example.
From figure 5, the structure of the PNTs upon electron or hole injection is shown at the maximum values.For electron injection, there is little change to the NT shape, consistent with the small change in diameter strain.For hole injection, the changes to the NT shape are more noticeable such that the B1 bonds along both the x-and y-axes at the centre of the cell become parallel to each other on alternate sides of the nanotube.As a result, the PNT adopts a square-like shape with 'corners' and 'sides' .The change in shape is also evident by the different diameter values measured within the NT.As shown in figure S8, the diameter between the corner atoms of PNT-15 is 26.8 Å while the diameter between the side atoms is 25.2 Å.The structural deformations that result in the formation of this shape will be discussed in more depth in the next section.

Structural deformations and charge density distributions of the PNTs upon charge injection
To investigate the mechanism behind the high electromechanical strain values for the PNTs upon hole injection in particular, the change in key interatomic and projected deformations (see figure 1) as a function of charge injection have been calculated and are summarised in figure 6.The excess charge density distribution (CDD) (figure 7) and Bader partial charges (figures 8 and 9) have also been calculated to assist with the analysis of the Coloumb interactions that occur within the nanotubes.Upon hole injection, we find that the changes in the bond lengths and angles generally correspond to the diameter strain responses calculated previously, for example, PNT-15 has the largest change in B1 (outer) and B2 as shown in figure 6.From the CDD plots (figure 7(a)), there are large amounts of excess hole (cyan) that form on the inner and outer ring of the NT between each P-P bond pairings, which results in strong repulsive forces that cause B1 (inner/outer) to contract.In the case of B2, the bond length expands instead, which is due to both attractive and repulsive forces acting on the bond.The attractive forces arise from the small amounts of excess electron (yellow) that form on the inner and outer ring of the NT, which elongates the bond.The repulsive forces come from the large amounts of excess hole that form on B2 which strongly repel with the hole formed on the outer and inner ring of the NT.The change in B2 directly corresponds to the change in the bond angle, β.
The difference in strain response for the NTs can also be explained by the Bader partial charges.As shown in figure 8(a) there is a uniform distribution of electrons removed from the (0,9) NT, with an average of −0.055 e and −0.194 e removed from the inner and outer P-P bonds, respectively.This suggests that the repulsive forces acting on the bond are uniform causing them to contract by an equal amount.For the larger (0,15) NT, there is greater variability in the amount of electrons that were removed, ranging from −0.073 to −0.091 e and −0.141 to −0.201 e for both inner and outer P-P bonds, respectively.Therefore the repulsive forces are not uniform within the inner ring leading to varying amounts of contraction that give the nanotube its square-like shape upon hole injection.
For electron injection, where there is little deformation of the NTs, the accumulation of charge on B1 (outer) is less than for hole injection (figure 7(b)) and the number of electrons distributed within the inner ring are uniform (figure 9).As a result the NTs are not deformed to the same extent as for hole injection.

Young's modulus of PNTs upon charge injection
To determine the structural integrity of the PNTs upon charge injection, in particular hole injection, the Young's modulus was calculated according to the energy-strain relationship as described in equation (2).For the charge-neutral systems, the Young's modulus was calculated to increase as the diameter of the nanotube increases, being 157.68, 167.44 and 168.21GPa for PNT-9, -15 and -17, respectively.This indicates that the structural integrity of the nanotubes increase with increasing diameter, converging to a value similar to that of the 2D monolayer [31].
In the case of hole injection (which had the greatest strain response), the Young's modulus decreases for each nanotube, indicating a weakening of the P-P bonds, consistent with the trend seen for straining the 2D monolayer [18].However, it is worth noting that the Young's modulus for PNT-15 decreases the most (−12.72%)compared to the charge-neutral nanotube, while PNT-9 and PNT-17 decreased by 4.92% and 7.31%, respectively.Therefore, the structural integrity of PNT-15 is significantly less and can therefore undergo the greatest diameter strain upon hole injection as it is more prone to structural deformations due to having weaker P-P bonds.This trend explains why the calculated ε max is greatest for PNT-15.
Given that the nanotubes undergo significant structural changes upon hole injection, defining the actual wall thickness (δ) is not necessarily straightforward as it will be different for the nanotubes that expand by different amounts, given they vary in their structural deformation.This ambiguity in defining the nanomaterial thickness can give rise to the so-called Yakobson's paradox, as has been described previously for CNTs [36].
In our work, we see a flattening of the NT wall upon increasing charge injection values, which is related to a change in the bond angle, β.For NTs with 'flatter' walls, the β value is largest.Given this is the case, we have determined the equilibrium distance between 2 BP monolayers, created from monolayers having a lattice constant (along the armchair direction) the same as for a non-charged system, and monolayers with expanded lattice constant that produces a β angle the same as seem for the 'flattest' charged NT.These two distances were calculated to be 5.62 and 5.76 Å, respectively, and represent the likely extremes for the wall thickness values in our work.Using these two values, the difference in the Young's modulus was calculated to be only ∼2 GPa (or ∼1.5% different).Given the difference in the calculated Young's moduli for any of the charged NTs, compared to their corresponding neutral systems, is greater than 20 GPa, this is insignificant.As a result, any differences in the wall thicknesses of the charged NTs does not affect the trend determined from our work, or the overall findings.

Band structures of PNTs upon charge injection
The band gap for each PNT was calculated using the GGA-PBE method and are shown in table S1.As GGA-PBE methods are known to underestimate the band gap compared to HSE methods, the GGA values have been shifted by 0.73 eV (see table 3), which is the shift seen in the band gap between GGA-PBE (0.75 eV) and HSE06 (1.49 eV) for the 2D monolayer.
For the charge neutral systems, the direct band gap widens by up to 0.46 eV as the diameter increases, in agreement with a previous study [19], and converges to that of the 2D monolayer.
Upon electron injection, spin-polarised states are introduced in the conduction band which cross the Fermi level so that the systems become highly n-doped.This behavior is similar to what was determined for  the 2D monolayer [18].There is also a narrowing of the band gap, compared to the charge neutral state, by up to ∼20%.For hole injection, the valence band crosses the Fermi level with the gap switching from direct to indirect (figure S9).Unlike the charge-neutral (and electron-injected) systems, as the diameter of the nanotube increases, the band gap decreases by up to ∼50% as the valence band maxmimum and conduction band minimum become very close.

Conclusion
Using density functional theory calculations, this work has shown that the electromechanical response of PNTs can be enhanced through charge injection.The calculated strain energies of the armchair PNTs are comparable with CNTs, which suggests they could be synthesized.A PNT-15 with a diameter of 24.38 Å and a (0,15) chirality can achieve a maximum diameter strain response of 30.2% upon hole injection, which is larger than other 1D materials, and is specifically 12.7% and 24.17% greater than reported for Sc 2 C nanotubes and CNTs, respectively, which may allow for a larger transport volume through the nanotube, making PNTs ideal for drug delivery applications.The greater response for this size NT is explained by the structural changes that occur upon charge injection which was analysed through changes in the charge density difference plots.The PNTs undergo significant dimensional changes and adopt a square-like shape upon hole injection.The calculated Young's modulus shows that hole injection has the greatest effect on the structural integrity of the (0,15) NT, which allows for its larger electromechanical response due to having weakened P-P bonds.Upon hole injection the nanotubes undergo a switching of the band gap from direct to indirect, allowing for potential application in optoelectronic devices.This work has shown that armchair PNTs can withstand diameter strains making them ideal materials for actuator devices in drug delivery and nanoprobe applications.

Figure 2 .
Figure 2. Calculated strain energies (Estr) as a function of armchair PNT diameter.

Figure 3 .
Figure 3. Evolution of (a) key bond lengths (B1 and B2) and (b) total system energy of PNT -9 and -15 during the AIMD simulations performed at 298 K for 5 ps.(See movies S1 and S2 for the animation).

Figure 4 .
Figure 4. Average diameter strain of armchair PNTs upon charge injection.

Figure 5 .
Figure 5. Side view of the optimised PNTs upon (a) 0.125 e/atom hole injection and (b) −0.125 e/atom electron injection.

Figure 7 .
Figure 7. Cross section of the PNTs, showing excess charge density distribution upon (a) 0.125 e/atom hole injection and (b) −0.125 e/atom electron injection at isosurface levels of 0.0006 e/Bohr 3 .The colours blue and yellow represent excess hole and electrons, respectively.

Table 2 .
Comparison of maximum electromechanical strain εmax of PNTs and other materials.

Table 3 .
Calculated band gaps (Eg) of the different PNTs upon charge injection.
a Indirect bandgap.