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. 2015 Jun 2;5:9961. doi: 10.1038/srep09961

Theoretical predictions on the electronic structure and charge carrier mobility in 2D Phosphorus sheets

Jin Xiao 1, Mengqiu Long 1,2,a, Xiaojiao Zhang 1, Jun Ouyang 1, Hui Xu 1,b, Yongli Gao 1,3
PMCID: PMC4451805  PMID: 26035176

Abstract

We have investigated the electronic structure and carrier mobility of four types of phosphorous monolayer sheet (α-P, β-P,γ-P and δ-P) using density functional theory combined with Boltzmann transport method and relaxation time approximation. It is shown that α-P, β-P and γ-P are indirect gap semiconductors, while δ-P is a direct one. All four sheets have ultrahigh carrier mobility and show anisotropy in-plane. The highest mobility value is ~3 × 105 cm2V−1s−1, which is comparable to that of graphene. Because of the huge difference between the hole and electron mobilities, α-P, γ-P and δ-P sheets can be considered as n-type semiconductors, and β-P sheet can be considered as a p-type semiconductor. Our results suggest that phosphorous monolayer sheets can be considered as a new type of two dimensional materials for applications in optoelectronics and nanoelectronic devices.

Since the successful preparation of graphene1, two-dimensional(2D) atomically-thick materials, such as graphdiyne sheet2, boron nitride sheet3, silicene4 and layered transition-metal dichalcogenides5,6 have attracted intensive attention owing to their unique physical properties and potential applications in nanoscale devices. Recently, due to the synthesis of few layers black phosphorus (BP)7,8,9,10, named phosphorene, 2D phosphorous materials have become the focus of science community11,12,13,14,15,16,17,18,19,20,21,22. BP is the most stable phosphorus allotrope under normal conditions with a direct band gap of about 0.3 eV23,24,25. The direct band gap will increase to ~2.0 eV26,27 as BP reduces to a monolayer, which opens doors for applications in optoelectronics. Furthermore, bulk BP is found to have high carrier mobility in the order of 105 cm2V−1s−1 at low temperatures28,29. The field-effect carrier mobility of few-layer BP is measured to be still higher, up to 1000 cm2V−1s−1 for electron7 and 286 cm2V−1s−1 for hole8 at room temperature. Also, few-layer BP exhibits ambipolar behavior with drain current modulation up to 1057. Owing to the direct gap and high mobility, there is a high potential for BP thin crystals to be a new 2D material for applications in optoelectronics, nanoelectronic devices and so on30,31,32,33,34,35,36.

So far, there have been few reports about the mobility of monolayer BP experiment researches. Theory studies based on effective mass calculation25 have shown that the room temperature electron mobility of monolayer BP is over 2000 cm2V−1s−111 or 5000 cm2V−1s−119. However, the method is subject to the parabolic properties of the energy bands. In this paper, both of electron and hole mobilities are investigated with Boltzmann transport equation (BTE) method beyond the effective mass approximation. Furthermore, besides the monolayer BP (marked as α-P), other three stable 2D phosphorus allotropes, namely β-P, γ-P and δ-P based on theoretical predictions20,21, have also been investigated. We therefore report the first theoretical prediction on the charge mobility of those 2D phosphorus allotropes in this work.

Results

The atomic structures of four different types of phosphorus sheets are shown in Fig. 1. In order to get an intuitive demonstration of carrier conduction along the armchair and zigzag directions, an orthogonal supercell covered by a green shadow is used in Fig. 1. The lattice length is shown in Table 1, and is in agreement with previous studies. There are four phosphorus (P) atoms in the supercell of α, β, γ phosphorus (α-P, β-P, γ-P), and eight P atoms in the supercell of δ phosphorus (δ-P). There are two phosphorus sub-layers in each phosphorus sheet. The distance of two P sub-layers (d) is shown in Table 1. The energy per atom indicates that α-P sheet is the most stable. The average P bond length is about 2.23 ~ 2.27 Å.

Figure 1.

Figure 1

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The models of phosphorus sheets: (a) α-P, (b) β-P, (c) γ-P and (d) δ-P.

Table 1. The lattice length, distance of two P atom sub-layers (d), the energy per atom and average P-bond length in phosphorus sheets.

graphic file with name srep09961-i1.jpg
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aTheory results from Ref. 21.

Energy band structures and Fermi surface heat map of the phosphorus sheets are shown in Fig. 2. All four types of phosphorus sheets are semiconductors. For α-P, β-P, γ-P and δ-P, as shown in Table 2, the energy band gaps based on PBE (HSE06) calculation are 0.91(1.70), 1.93(2.64), 0.42(1.03) and 0.10(0.78) eV, respectively. The energy band structures and band gaps are consistent to those reported in previous studies21. Our calculations indicate that only δ-P is a direct gap semiconductor. The other three type sheets are indirect semiconductors. These results are in good agreement with previous study20,21. Interestingly, when we zoom in the energy band spectrum around the Γ point in α-P, as shown in the inset of Fig. 2 we can find that the top of the valence band is located at (0, 0.035) K point (skewed slightly along the zigzag direction) and is about 0.75 (0.53) meV higher than the Γ point based on the PBE (HSE06) calculation. This tiny skewing away Γ point on the top of the valence band is in good agreement with Ref. 37 and has also been demonstrated in α-P zigzag nanoribbon16. The optical characteristics should be influenced only slightly in α-P sheets due to such tiny skewing.

Figure 2.

Figure 2

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Energy band structures, valance band edge surface heat map and conduction band edge surface heat map of α-P, β-P, γ-P and δ-P. Red arrows are the direct gaps at Γ point. The black lines and red lines in band structures are calculated by PBE and HSE06 respectively. K point: Γ(0,0,0), Μ(0.5, 0, 0), Ν (0, 0.5, 0), Τ(0.5, 0.5, 0).

Table 2. The energy gap of phosphorus sheets:

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ais the PBE results;

bis the HSE06 results.

Based on the band structures, we calculate the effective mass of the charge carrier by parabolic fitting near the Fermi surface, which is presented in Table 3. It can be found that most of |m*| is smaller than the mass of the free electron (me), which means that the phosphorus sheets have considerably high carrier mobility. Our results show that the m* for electrons and holes in α-P are 0.1382 and 1.2366 me, respectively, which are in good agreement with Yang’s report11. Furthermore, it is clearly seen that the |m*| of electron or hole along the armchair direction over an order of magnitude smaller than that along the zigzag direction. It indicates that the carrier transport is anisotropic and the armchair direction is the main transport direction in α-P. The case in β-P is the opposite. The |m*|of electron or hole along the zigzag direction is three times larger than that along the armchair direction, which means that the carrier transport ability is stronger along the armchair than the zigzag direction in β-P. It is easily to see that the |m*| of hole along the zigzag direction in α-P and γ-P is much larger than others’, which result from the almost flat valence band in those materials.

Table 3. The effective mass (m*) of carriers in phosphorus sheets.

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The variation of total energy (E) with uniaxial strain (δ) applied along the armchair and zigzag directions are shown in Fig. 3. Based on those energy-strain curves, the in-plane stretching modulus C2D can be obtained. In α-P sheets, we can also find that C2D is obvious anisotropic, and it is about four times larger along the zigzag direction (103.278 N/m) than along the armchair direction (24.255 N/m). These are in good agreement with Qiao’s report (101.60 and 28.94 N/m)25. In general, the three-dimensional Young’s modulus can be estimated as C3D = C2D/t0. Based on the optB86b van der Waals functional, the interlayer separation of α-P, β-P, γ-P and δ-P have been calculated as 5.30, 4.20, 4.21 and 5.47 Å21, respectively. By assuming a finite thickness (t0 = 5.30, 4.20, 4.21 and 5.47 Å) for α-P, β-P, γ-P and δ-P sheet, the Young’s modulus along the armchair and zigzag direction are shown in Table 4. The previous theoretical study has shown the Young’s modulus of monolayer α-P sheet to be 44 GPa (armchair direction) and 166 GPa (zigzag direction)31.

Figure 3.

Figure 3

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Energy−strain relationship along armchair (a) and zigzag (b) directions.

Table 4. Three-dimensional Young’s modulus (C 3D ) of phosphorus sheets.

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Figure 4 shows the shifts of band edges as a function of strain along the armchair and zigzag directions. Through dilating the lattice along the armchair and zigzag directions, the DP constant E1 is then calculated as dEedge/dδ, equivalent to the slope of the fitting lines, where Eedge is the energy of the conduction (valence) band edge. Each line is fitted by 11 points. The E1 values of phosphorus sheets are shown in Table 5. The standard error of all E1 values is smaller than 1% excluding three valuses marked in Table 5.

Figure 4.

Figure 4

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Shifts of conduction band and valence band under uniaxial strain: conduction band along (a) armchair and (b) zigzag direction; valence band along (c) armchair and (d) zigzag direction. The balck dashed line is the linear fitting.

Table 5. The in-plane elastic constant (C 2D ), deformation potential (E 1 ), electron relaxation time (τ e ), hole relaxation time (τ h ), electron mobility (μ e ) and hole mobility (μ h ) in phosphorus sheets.

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The temperature is 300K.The Standard Error is a1.01%, b1.78%, c8.06% and others smaller than 1%.

On the basis of our energy band spectrum, we calculated E1 and C2D, the acoustic phonon-limited mobility (using Eq. 1) and relaxation time (using Eq. 2) at room temperature (300 K). The results are shown in Table 5. It can be seen that the electron relaxation time (τe) in phosphorus sheets is much longer than the hole relaxation time (τh), excluding β-P. The electron mobilities of α-P, β-P, γ-P and δ-P sheets are about 1.1 × 104, 4.7 × 102, 2.9 × 105 and 3.0 × 103 cm2V−1s−1, respectively. The corresponding τe’s are about 1.29, 0.09, 70.73 and 0.61 ps. The hole mobilities of α-P, β-P, γ-P and δ-P sheets are about 2.0 × 102, 1.7 × 103, 7.3 × 101 and 5.9 × 102  cm2V−1s−1, respectively. The corresponding τh’s are about 0.02, 0.86, 0.24 and 0.09 ps. It can be found that all four phases have higher mobility than MoS2 monolayer sheet (the hole mobility is 86 cm2V−1s−1 and electron mobility is 44 cm2V−1s−1)38. Due to the very small conduction band deformation potential (0.187 eV), the electron mobility along the zigzag direction in γ-P sheet is as high as ~3 × 105 cm2/Vs, which is in the same order of magnitude of that in graphene42,47, silicone39 and germanene40. The minimum is the electron mobility along the zigzag direction in β-P sheet, which is about 47.32 cm2/Vs. The electron carriers move faster than the hole ones in α-P, γ-P and δ-P sheet. Only in β-P sheet, the hole mobility is higher than the electron mobility. γ-P sheet has the best electron carrier transmitting capacity and the biggest difference between the electron and hole mobility in four type sheets. Moreover, the obvious anisotropy in carrier mobility can be found. The charge carriers move faster along the armchair direction than the zigzag direction in α-P, β-P and δ-P sheet. While in γ-P sheet, the zigzag direction is preferred.

Discussion

It must be noted that the mobility in our calculation is a theoretical value. Only the acoustic phonon scattering mechanism is considered. Actually, there are inevitably impurities and defects in the vast majority of materials, and they have a great influence on the charge transport properties, especially at low temperatures where phonon has little effect47. For example, in a MoS2 sheet, owing to scattering from charged impurities, the mobility at low-temperatures signally decreases with temperature41. So the mobility of phosphorus sheets measured experimentally can be much smaller than theoretically predicted.

The band decomposed charge density around the Fermi level of phosphorus sheets is shown in Fig. 5. The composition of the top valence and the bottom conduction band are shown in Table 6. Atomic orbital analysis shows that the top of valence states in phosphorus sheets are mainly composed of 3pz orbits. Our calculations indicate that, in α-P, the conduction bands are mainly composed of pz orbitals with mixed s and px orbits. In β-P, the conduction bands are hybridization orbitals with mixed s and p orbits. In γ-P, the conduction bands are mainly composed of py and pz orbitals. In δ-P, the conduction bands are major composed of pz and s orbitals.

Figure 5.

Figure 5

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Band decomposed charge density of phosphorus sheets: (a)–(d) is the valance band edge for α-P, β-P, γ-P and δ-P respectively; (e)–(h) is the conduction band edge for α-P, β-P, γ-P and δ-P respectively. The isosurface value is 0.01. Drawings are produced by VESTA software42.

Table 6. The percentage (%) of each orbit in the top of valance and the bottom of conduction.

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In α-P, β-P and γ-P, owing to the valence bands partly composed of in-plane p orbits, the top of the valence band is skewing awaly the Γ point. Due to the distribution of the mainly charge density of the valence is along the armchair direction, the hole carrier will move faster along the armchair direction than the zigzag direction in α-P, β-P and δ-P. While in γ-P, due to the contributions of s and pz orbitals, the distribution of valence band charge density is along the zigzag direction (Fig. 5c). So the hole mobility in γ-P is slightly higher along the zigzag direction than the armchair direction. For the conduction band, the distribution of charge density is along the armchair direction, as shown in Fig. 5e,f, and g. This is identical with the electron mobility except γ-P. The orbital analysis shows that the proportion of px orbital is larger than that of py orbit except γ-P and δ-P. Due to the contribution of px orbitals, the electron mobility is higher along the armchair direction than the zigzag direction in α-P and β-P. In γ-P, the contribution of py orbits is over 50%. At the same time, there is much lower deformation potential. So the electron mobility along zigzag is surperisely high in γ-P.

Conclusions

In summarily, we have calculated the electronic structures and the intrinsic charge carrier mobility of four type phosphorus sheets (α-P, β-P, γ-P and δ-P), using first-principles density functional theory and the BTE with the relaxation time approximation. We find that α-P, β-P and γ-P are indirect gap semiconductors. The numerical results indicate that the electron mobility of α-P, γ-P and δ-P sheets at room temperature (about 1.107 × 104, 2.895 × 105 and 3.022 × 103 cm2V−1s−1, respectively) is much higher thanthe corresponding hole mobility (about 204.288, 72.645 and 586.339 cm2V−1s−1, respectively). Nevertheless, in β-P sheet, the hole mobility (1.711 × 103 cm2V−1s−1) is about four times of electron mobility (466.262 cm2V−1s−1). Owing to the huge difference mobilities in hole and electron, α-P, γ-P and δ-P sheets can be considered as n-type semiconductors, and β-P sheet can be considered as p-type semiconductors. All four types of phosphorus sheets present anisotropy in carrier mobility. Charge carriers move faster along the armchair direction than the zigzag direction in α-P, β-P and δ-P sheet. But in γ-P sheet, the more favorable charge transmission direction is along zigzag.

Methods

In this paper, the carrier mobility is calculated by BTE method beyond the effective mass approximation which is used to predict the mobility of semiconductor nanometerials, like graphene, carbon nanotubes and so on2,43,44,45,46,47,48. Within the BTE method, the carrier mobility μ in the relaxation time approximation can be express as Ref. 2 and 49:

graphic file with name srep09961-m1.jpg

Where the minus (plus) sign is for electron (hole).Inline graphic is the relaxation time, Inline graphic and Inline graphic are band energy and the component of group velocity at Inline graphic state of the i-th band, respectively. The summation of band was carried out over VB for hole and CB for electron. Furthermore, the integral of Inline graphic states is over the first Brillouin zone (BZ).

In order to obtain the mobility, three key quantities (Inline graphicand Inline graphic) must be determined. The coherent wavelength of thermally activated electrons or holes at room temperature in inorganic semiconductors, which is much larger than their lattice constant, is close to that of acoustic phonon modes in the center of the first BZ. The electron−acoustic phonon coupling can be effectively calculated by the deformation potential (DP) theory proposed by Bardeen and Shockley50. So, the relaxation time Inline graphicbased on DP theory can be expressed as2,48

graphic file with name srep09961-m10.jpg

Here the delta function denotes that the scattering process is elastic and occurs between the band states with the same band index. Inline graphicis the DP constant of the i-th band, and C is the elastic constant.

The band energy Inline graphic is calculated by the Vienna ab-initio simulation package (VASP)51. The Inline graphic -mesh is chosen as 11 × 11 × 1 for electronic structures calculation and 61 × 61 × 1 for band eigenvalue calculation, which is fine enough to give converged relaxation time and mobility. The generalized gradient approximation (GGA)52 with the Perdew-Burke-Ernzerhof (PBE)53 exchange correlation function is used with the plane-wave cutoff energy set at 600 eV for all calculations. The criterion of convergence is that the residual forces are less than 0.001 eV/Å and the change of the total energy is less than 10−7  eV. The vacuum space between two adjacent sheets is set at least 15 Å to eliminate the interactive effect on each other.

The group velocity of electron and hole carriers can be obtained from the gradient of the band energy Inline graphic in Inline graphic-space, Inline graphic.

Additional Information

How to cite this article: Xiao, J. et al. Theoretical predictions on the electronic structure and charge carrier mobility in 2D Phosphorus sheets. Sci. Rep. 5, 09961; doi: 10.1038/srep09961 (2015).

Acknowledgments

This work is supported by Hunan Key Laboratory for Super-microstructure and Ultrafast Process, and the National Natural Science Foundation of China (Nos. 61306149 and 11334014), the Natural Science Foundation of Hunan Province (No. 14JJ3026), Hong Kong Scholars Program (No. XJ2013003), ShenghuaLieying Scholarship by the Central South University and Hunan Provincial Innovation Foundation for Postgraduate.

Footnotes

Author Contributions J. X. carried out the first-principles calculations, prepared all figures and wrote the manuscript. M. L. directed this work and revised the manuscript. X. Z., J. O., H. X. and Y. G. involved in discussion. All authors analyzed the results and reviewed the manuscript.

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