DOI: 10.1039/C9NR05660B
(Paper)
Nanoscale, 2019, Advance Article

Huanhuan Ma,
Wei Hu* and
Jinlong Yang*

Hefei National Laboratory for Physical Sciences at Microscale, Department of Chemical Physics, and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China. E-mail: whuustc@ustc.edu.cn; jlyang@ustc.edu.cn

Received
4th July 2019
, Accepted 11th October 2019

First published on 14th October 2019

Tailoring the electronic anisotropy of two-dimensional (2D) semiconductors with strain-engineering is critical in nanoelectronics. Recently, 2D tellurene has been predicted theoretically and fabricated experimentally. It has potential applications in nanoelectronics, in particular, β-phase tellurene (β-Te) shows a desirable direct band gap (1.47 eV), high carrier mobility (2.58 × 10^{3} cm^{2} V^{−1} s^{−1}) and high stability under ambient conditions. In this work, we demonstrated, with first-principles density functional theory calculations, that the highly anisotropic electron mobility and electrical conductance of β-Te can be controlled by strain-engineering. The direction of electrical conductance of β-Te can be changed from the armchair to the zigzag direction at the strain between −1% and 0%. Meanwhile, we found that the bandgap of β-Te under strain experiences an indirect-direct transition with a conduction band minimum (CBM) shift from the X to Γ point. The significant dispersion of the bottom of the conduction bands along the Γ–Y direction switches to the X–Γ direction under uniaxial or biaxial strain which makes the rotation of the effective masses tensor. The qualitative rotation of the spatial anisotropic electron effective masses tensor by 90° also rotates the direction of the electrical conduction as the carrier mobility is inversely dependent on the effective masses. On the another hand, we also found that the deformation potential constant also plays an important role in the rotation of electrical conductance anisotropy. While anisotropic conductance of hole is impregnable under strain. In order to verify that β-Te can sustain large strain, we studied its stability and mechanical properties and found that β-Te shows superior mechanical flexibility with a small Young's modulus (27.46 GPa (armchair)–61.99 GPa (zigzag)) and large anisotropic strain-stress (12.89 N m^{−1} at the strain of 38% along armchair direction and 25.72 N m^{−1} at the strain of 26% along zigzag direction). The high anisotropic carrier mobility and superior mechanical flexibility of β-Te make it a promising candidate for flexible nanoelectronics.

On the other hand, the electronic properties of most traditional semiconductors are poor under large strain, which makes it difficult to apply them directly in flexible electronics.^{26} Therefore, the search is urgent for promising semiconductors with desirable electronic and mechanical properties when they are under strain. Flexibility is a physical quantity which describes the ability of materials to resist deformation in response to an applied force, and is an important factor in flexible electronics. Compared with bulk materials, the counterparts of nanostructures can withstand larger strain and bending,^{27–29}and have potential applications in nanoelectronics. Thus, we search for promising 2D semiconductors for use in flexible nanoelectronics. To develop potential flexible nanoelectronics, 2D semiconductors must have some essential properties. Firstly, 2D materials must have a suitable bandgap and a high carrier mobility. Secondly, semiconductors can withstand room temperature and ambient conditions. However, for flexible electronic devices, 2D materials must sustain large strain. Unfortunately, despite the fact that a large number of 2D materials have been synthesized, there is a scarcity of materials which meet all of these properties. For example, graphene is stable under ambient conditions and has a high carrier mobility,^{30} but doesn't have a bandgap.^{14} MoS_{2} is stable under ambient conditions^{31} and has a suitable bandgap,^{32} but the Young's modulus is medium^{33} and the carrier mobility is too low.^{34} Black phosphorene has a small Young's modulus,^{35} a suitable bandgap, highly anisotropic carrier mobility and strain-engineered anisotropic electrical conductance,^{3,22} but is unstable under ambient conditions.^{36}

2D tellurene has recently been predicted theoretically and fabricated experimentally.^{37,38} Tellurene has three different structures with different electronic properties. α-phase tellurene (α-Te) and β-phase tellurene (β-Te) are both semiconductors with an indirect bandgap of 0.75 eV and a direct bandgap of 1.47 eV, respectively; while γ-phase tellurene (γ-Te) is a metal.^{37} The carrier mobilities and the on/off ratio of field-effect transistors made from a few layers 2D tellurene were found to be 700 cm^{2} V^{−1} s^{−1} and in the order of 10^{6} respectively, in recent experiment.^{38} In addition, air-stable quasi-2D tellurium nanoflakes were synthesized and used as short-wave infrared photodetectors in a previous experiment.^{39} Tellurene has also been found to show low lattice thermal conductivity,^{40} excellent thermoelectric performance^{41–43} and high thermal and chemical stability under ambient conditions.^{38,44} These results may make tellurene a promising candidate material for electronic, optoelectronic and thermoelectric devices. Therefore, a further study of the strain effect on electronic and mechanical properties of tellurene should be helpful to exploit its potential application in flexible electronics.

In this work, we examined the electronic and mechanical anisotropy of tellurene by using first-principles density functional theory calculations. We found that strain-engineering can be used to control the anisotropic electrical conductance of β-Te. Moreover, we demonstrated the strain effect on the carrier mobilities of β-Te by using the acoustic phonon limited method. The mobilities of holes and electrons have different responses to strain, which can control the direction of electrical conductance. We also explored the intrinsic mechanism of the anisotropic electron mobility of β-Te under strain. We showed the strain effect on the effective masses tensor and band structure of β-Te. The effective mass of the electrons of intrinsic β-Te in the zigzag direction is an order of magnitude smaller than that in the armchair direction, indicating that the zigzag direction is favored for electron transport. Furthermore, strain can rotate the spatial anisotropic effective masses tensor, which can be used to control the electrical conductance as the carrier mobility is inversely dependent on the effective mass. Finally, in order to verify that β-Te can sustain large strain, we studied the stability and mechanical properties of β-Te. β-Te has the smallest Young's modulus (27.46 GPa) and the largest elastic strain limit (38%) than most common 2D semiconductors and shows potential application in flexible devices.

By using the acoustic phonon limited method,^{52,53} we calculated the mobilities of α-Te and β-Te under strain,

(1) |

We used the finite distortion method to calculate the mechanical properties of 2D materials, such as elastic constants, Young's modulus and Poisson's ratio. For 2D materials, the elastic constants and modulus were obtained from the Hooke's law under plane-stress conditions^{54}

(2) |

(3) |

For 2D materials, the stress was calculated using the Hellmann–Feynman theorem, modified to be the force per unit length.^{56} Tensile strain was applied to the chosen crystalline direction, and the corresponding stress was calculated. For 2D materials, the tensile stress σ is given by

(4) |

On the other hand, the hole mobility along the armchair direction was significantly improved when biaxial and zigzag direction compressed strain were applied as shown in Fig. 2(d) and (e), but was slightly changed under armchair direction strain from Fig. 2(f). At the same time, hole mobility along the zigzag direction was almost constant under uniaxial or biaxial strain (expect for the −6% strain along zigzag direction). So, the strain effect can just enhance or weaken the conductance of the hole, but cannot change the direction.

As discussed in section 2, in addition to the influence of effective mass on the carrier mobility, two other parameters (elastic modulus (C_{2D}) and deformation potential constant (E_{1}^{i})) can also influence the mobility of electrons. As we see from Fig. S14(a),† the ratio of elastic modulus in the zigzag and armchair direction is changed slightly, so elastic modulus can slightly influence electronic mobility. On the other hand, we found that the ratio of the deformation potential constant of electrons along the zigzag and armchair directions was similar to that presented in Fig. S24(b),† so the deformation potential constant also plays an important role in the rotation of anisotropy.

It should be noted that the electrical transport behaviour in semiconductors is mainly dependent on the band edges of valence and conduction states, including the valence band maximum (VBM) and the conduction band minimum (CBM). The VBM and CBM of intrinsic β-Te respectively are determined by the p_{y} and p_{x} orbitals of Te at the Γ point, which can be found from the projected band structure (pband) of intrinsic β-Te and can also be seen from the projected density of states, pDOS (see ESI†). When a −6% or +6% biaxial strain was applied to β-Te, there was almost no change in the VBM and it was still mainly determined by the p_{y} orbitals of Te at the Γ point as shown in Fig. 4(a) and (b). In particular, when the −6% biaxial strain was applied in β-Te, the CBM shifted from Γ to X (0.5, 0, 0) and β-Te became an indirect semiconductor as shown in Fig. 4(a), which was validated by using the HSE06 + SOC (see ESI†). In this case, the CBM is mainly contributed by the p_{z} orbitals of Te at X point because two Te atoms become close to each other, resulting in p_{z} orbital overlap between two Te atoms, and p_{z} orbit is contributed from the two outer-layer Te atoms (1 and 3 in Fig. 1(a), see ESI†). While the +6% biaxial strain is applied to β-Te, the CBM almost has no change and is still mainly contributed by the p_{x} orbitals of Te at Γ point as shown in Fig. 4(b). The contribution of the p_{z} orbital of the outer-later Te atom will increase when the compressed strain was applied to β-Te, but the contribution decreased when tensile strain was applied (see ESI†). This is the main intrinsic mechanism of the anisotropic electron mobilities of β-Te.

In addition to the shift of CBM, another important feature of +6% biaxial strained β-Te is its highly anisotropic band dispersion around the CBM. The bottom of the conduction bands have much more significant dispersions along the Γ–Y direction, which is the zigzag direction in real space; however, along the Γ–X direction it is nearly flat, which is the armchair direction in real space. Such similar phenomenon can also be found in intrinsic β-Te. But when −6% strain is applied to β-Te rotation occurs, therefore, the conduction bands along the X–M direction are relatively flat compared to those along the X–Γ direction. This can also be confirmed by the effective mass of biaxially strained β-Te.

Using the orientation-dependent relation, the polar diagrams of Young's modulus E(θ) and Poisson's ratio υ(θ) for β-Te computed are plotted in Fig. 5(b) and (c). The Young's modulus represents the fully reversible stiffness response as in a linear elastic Hookean spring.^{65} As expected, the Young's modulus and Poisson's ratio of β-Te are highly anisotropic in Fig. 5(b) and (c). The maximal Young's modulus of 61.99 GPa is along the zigzag direction and the minimal value of 27.46 GPa is along the armchair direction of β-Te, which are consistent with previous theoretical work.^{66} But for multilayer β-Te, the maximal Young's modulus is along the armchair direction and the minimum is along the zigzag direction, because the structures of multilayer β-Te are similar to the bulk (see ESI†). We found that β-Te has the lowest Young's modulus compared to that of graphene,^{63,67} MoS_{2}^{64} and phosphroene,^{35} as presented in Table 1.

2D materials | E_{g} |
μ | E | ε | Stability |
---|---|---|---|---|---|

Graphene^{30,62,63} |
0 | 100 | 1000 | 30% | High |

MoS_{2}^{34,64} |
1.64 | 0.34 | 330 | 26% | High |

phosphorene^{17,22,35,36} |
1.61 | 2.2 | 44 | 30% | Low |

α-Te | 0.75 | 4.06 | 57.96 | 20% | High |

β-Te | 1.47 | 2.58 | 27.46 | 38% | High |

To further explore the mechanical properties of tellurene, the ideal tensile strength (the highest achievable stress of a perfect crystal at 0 K) and the critical strain (the strain at which ideal strength reaches)^{68} of β-Te were also studied. An in-plane tensile force was applied along the armchair, zigzag or diagonal direction to research the strain-stress relation. Using eqn (4), our calculated stress–strain relation is presented in Fig. 5(d) for β-Te. The applied strain ranges from 0 to 50% for β-Te. It shows that the ideal strengths of β-Te are up to 12.89 N m^{−1}, 14.92 N m^{−1} and 25.72 N m^{−1} in the armchair, zigzag and diagonal direction, respectively. The corresponding tensile strain limits are 38%, 26% and 30% along the armchair, zigzag, and diagonal directions, respectively, that's because β-Te has a puckered structure which can be seen from Fig. 1(a). This predicted elastic strain limit is larger than that of other 2D materials such as graphene,^{62,69} MoS_{2}^{70} and phosphorene,^{17} suggesting that tellurene is highly flexible and may have potential applications in flexible devices.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/C9NR05660B |

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