⇦Chinese Physics Letters, 2017, Vol. 34, No. 2, Article code 027302 Ballistic Transport through a Strained Region on Monolayer Phosphorene * Yi Ren(任裔), Fang Cheng(程芳)** Affiliations Department of Physics and Electronic Science, Changsha University of Science and Technology, Changsha 410004 Received 12 November 2016 *Supported by the National Natural Science Foundation of China under Grant No 11374002, the Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province, and the Construct Program of the Key Discipline in Hunan Province.
**Corresponding author. Email: chengfangg@gmail.com Citation Text: Ren Y and Cheng F 2017 Chin. Phys. Lett. 34 027302 Abstract We investigate quantum transport of carriers through a strained region on monolayer phosphorene theoretically. The electron tunneling is forbidden when the incident angle exceeds a critical value. The critical angles for electrons tunneling through a strain region for different strengths and directions of the strains are different. Owing to the anisotropic effective masses, the conductance shows a strong anisotropic behavior. By tuning the Fermi energy and strain, the channels can be transited from opaque to transparent, which provides us with an efficient way to control the transport of monolayer phosphorene-based microstructures. DOI:10.1088/0256-307X/34/2/027302 PACS:73.22.-f, 73.63.-b, 68.65.-k © 2017 Chinese Physics Society Article Text Recently, layered black phosphorus has attracted intensive attention due to its unique electronic properties and potential applications in nanoelectronics.[1-4] Different from graphene in the electronic band structure, black phosphorus is a semiconductor with a sizeable gap. Phosphorene-based field effect transistors (FETs) show a higher ON/OFF ratio compared with graphene[5,6] and have higher carrier mobility with respect to two-dimensional (2D) transition metal dichalcogenides,[7] which make black phosphorus a favorable material for next-generation electronics. To date, various interesting properties for phosphorene have been predicted, particularly those related to tunable optical properties,[8-11] mechanical anisotropy,[12] and thermal conductivity.[13] Strain engineering is an efficient mechanical approach to manipulating the physical properties in quasi-2D nanostructures. It has been shown in a number of previews works that mechanical strain is an effective means to generate a finite electronic bandgap for graphene.[14] The in-plane uniaxial strains along the armchair and zigzag directions have also been used to modify the bandgap of black phosphorus.[15,16] First-principles calculations have shown that both biaxial and uniaxial strains rotate the preferred electrical conducting direction by 90$^{\circ}$.[17] The method of invariants has been applied to investigate the electronic band structure of black phosphorus with external fields including the strain field.[18] A large uniaxial strain in the direction normal to the black phosphorus plane can even induce a semiconductor-metal transition.[19] In this Letter, we study the quantum transport of carriers through a strained region on monolayer phosphorene. The tunneling is forbidden when the incident angle exceeds a critical value. The critical angle for electrons tunneling is different for transport along different directions as the different modifications of the bandgap by the $x/y$ strain. The channels can be transited from opaque to transparent by tuning the Fermi energy and strain. The puckered atomic structure of phosphorene and its geometrical parameters are shown in Fig. 1, where the $x$ and $y$ axes are the armchair and zigzag directions, respectively, and the $z$ axis is in the normal direction to the plane of phosphorene. The components of the geometrical parameters as shown in Figs. 1(a) and 1(b), $(r_{1x},r_{1y},r_{1z})=(1.503,1.660,0)$ and $(r_{2x},r_{2y},r_{2z})=(0.786,0,2.140)$. The bond lengths are $r_{1}=\sqrt{r_{1x}^2+r_{1y}^2+r_{1z}^2}=2.240$ Å, $r_{2}=\sqrt{r_{2x}^2+r_{2y}^2+r_{2z}^2}=2.280$ Å, and $r_{3}$, $r_{4}$ and $r_{5}$ are simply defined by parameters of $r_{1}$ and $r_{2}$. The two in-plane lattice constants are $a=4.580$ Å, $b=3.320$ Å  and the thickness of a single layer due to the puckered nature is $r_{2z}=2.140$ Å. The structure parameters have been taken from Ref. [20], which are very close to experimentally measured parameters[21] for its bulk structure. It has been shown that the bond lengths and bond angles of phosphorene both change under axial strains.[22,23] According to the Harrison rule,[24,25] the hopping parameters for $p$ orbitals are related to the bond length as $t_{i}\propto 1/r_{i}^{2}$ and the angular dependence can be described by the hopping integrals along the $\pi$ and $\sigma$ bonds. Though the changes in angles are almost noticeable, the modification of the hopping parameters due to them is much smaller than the effect of changes of bond lengths.[26] Hence, we consider only changes of the bond lengths in the hopping modulation. When an axial strain is applied to phosphorene, the rectangle shape of the unit cell with lattice constants of $a_{0}$ and $b_{0}$ remains unchanged. Therefore the initial geometrical parameter $r_{i}^{0}$ is deformed as $(r_{ix},r_{iy},r_{iz})=((1+\varepsilon _{x})r_{ix}^{0},(1+\varepsilon _{y})r_{iy}^{0},(1+\varepsilon _{z})r_{iz}^{0})$, where $\varepsilon _{j}$ is the strain in the $j$-direction, and $r_{i}$ is a deformed geometrical parameter. In the linear deformation regime, expanding the norm of $r_{i}$ to the first order of $\varepsilon _{j}$ gives $r_{i}=(1+\alpha _{x}^{i}\varepsilon _{x}+\alpha _{y}^{i}\varepsilon _{y}+\alpha _{z}^{i}\varepsilon _{z})r_{i}^{0}$, where $\alpha _{j}^{i}=(r_{ij}^{0}/r_{i}^{0})^{2}$ are the coefficients related to the structure of phosphorene, which are simply calculated via the special geometrical parameters given above. Using the Harrison relation, we obtain the strain effect on the hopping parameters as $$ t_{i}\approx (1-2\alpha _{x}^{i}\varepsilon _{x}-2\alpha _{y}^{i}\varepsilon _{y}-2\alpha _{z}^{i}\varepsilon _{z})t_{i}^{0}.~~ \tag {1} $$ These hopping parameters are $t_{1}^{0}=-1.220$ eV, $t_{2}^{0}=3.665$ eV, $t_{3}^{0}=-0.205$ eV, $t_{4}^{0}=-0.105$ eV and $t_{5}^{0}=-0.055$ eV.[27]
cpl-34-2-027302-fig1.png
Fig. 1. The lattice geometry of phosphorene. The two different colors of the P atoms refer to upper and lower chains. (a) The hopping parameters $t_1,t_2,\ldots,t_5$ are indicated in the figure. The blue dashed rectangle shows the unit cell of phosphorene. (b) Lattice constants and the components of geometrical parameters describing the structure of phosphorene. (c) Schematic diagram of a strained region geometry on monolayer phosphorene. (d) The energy dispersions of monolayer phosphorene with different strains. The dashed lines indicate the positions of the Fermi energies chosen for the calculations in Fig. 2. The green dashed line is for $E_{\rm F}=0.9$ eV, and the black dashed line is for $E_{\rm F}=-1.5$ eV, respectively.
The spin-orbital coupling (SOC) of the P atom is very small since the atomic SOC is proportional to $Z^4$, where $Z=15$ for the P atom. It was shown that the energy gap of a few-layers of black phosphorene can be closed under a huge external electric field or strain.[28,29] However, a topological nontrival energy gap of 5 meV[28] induced by spin-orbital coupling can be opened in few layers of black phosphorene under such a huge electric field. The spin-orbital coupling would not affect the conclusion qualitatively. All P atoms in a unit cell have the same on-site energy, thus we can project the position of upper and lower chains of phosphorene on a horizontal plane to reduce the spinless $4\times4$ Hamiltonian into a two-band TB model.[10] The new $K$-space Hamiltonian of the strained phosphorene is given by $$ H=\left(\begin{matrix} B_{k}e^{i(k_{a}-k_{b})/2} & A_{k}+C_{k}e^{i(k_{a}-k_{b})/2} \\ A_{k}^{\ast }+C_{k}^{\ast }e^{-i(k_{a}-k_{b})/2} & B_{k}e^{i(k_{a}-k_{b})/2} \end{matrix}\right),~~ \tag {2} $$ where $$\begin{align} A_{k}=\,&t_{2}+t_{5}e^{-ik_{a}}, \\ B_{k}=\,&4t_{4}e^{-i(k_{a}-k_{b})/2}\cos (k_{a}/2)\cos (k_{b}/2),\\ C_{k}=\,&2e^{ik_{b}/2}\cos (k_{b}/2)(t_{1}e^{-ik_{a}}+t_{3}),~~ \tag {3} \end{align} $$ with $k_{a}={\boldsymbol k\cdot a}$, $k_{b}={\boldsymbol k\cdot b}$ and $\theta =\arctan (r_{1y}/r_{1x})$. By expanding the structure factors around $k=0$ (${\it \Gamma}$ point) and retaining the terms up to the second order in $k$, one can write a long-wavelength approximation for the Hamiltonian as $$ H_{k}=\left(\begin{matrix} u_{0}+\eta _{x}k_{x}^{2}+\eta _{y}k_{y}^{2} & m_{12} \\ m_{21} & u_{0}+\eta _{x}k_{x}^{2}+\eta k_{y}^{2} \end{matrix}\right),~~ \tag {4} $$ where $m_{12}=\delta +\gamma _{x}k_{x}^{2}+\gamma _{y}k_{y}^{2}+i\chi k_{x}$, $m_{21}=\delta +\gamma _{x}k_{x}^{2}+\gamma _{y}k_{y}^{2}-i\chi k_{x}$, $u_{0}=4t_{4}$, $\eta _{x}=-\frac{1}{2}a^{2}t_{4}$, $\eta _{y}=-\frac{1}{2}b^{2}t_{4}$, $\delta =2t_{1}+t_{2}+2t_{3}+t_{5}$, $\gamma _{x}=-\frac{1}{4}a^{2}(t_{1}+t_{3})-\frac{1}{2}a^{2}t_{5}$, $\gamma _{y}=-\frac{1}{4} b^{2}(t_{1}+t_{3})$, and $\chi =-a(t_{1}-t_{3}+t_{5})$. The continuum approximation Hamiltonian can be rewritten in a more compact form as $$ H=\left(\begin{matrix} \epsilon _{1} & \epsilon _{2}e^{i\theta _{k}} \\ \epsilon _{2}e^{-i\theta _{k}} & \epsilon _{1} \end{matrix}\right),~~ \tag {5} $$ where $\epsilon _{1}=\frac{F_{+}+F_{-}}{2}$, $\epsilon _{2}=\sqrt{(\frac{F_{+}-F_{-}}{2}) ^{2}+(\chi k_{x})^{2}}$, $\theta =\arctan (\frac{2\chi k_{x}}{F_{+}-F_{-}})$. Here we defined $F_{\pm}=(u_{0}\pm \delta)+(\eta _{x}\pm \gamma _{x})k_{x}^{2}+(\eta _{y}\pm \gamma _{y})k_{y}^{2}$. At the ${\it \Gamma}$ point, the $F_{+}$ and $F_{-}$ expressions yield the dispersions for the conduction and valence bands, respectively. Thus using this polar notation, one can readily obtain the eigenstates as $$ {\it \Psi}=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ \lambda e^{-i\theta _{k}} \end{pmatrix},~~ \tag {6} $$ where $\lambda =\pm 1$ with the positive (negative) signs corresponding to electrons (holes). Dispersion relations for electrons and holes are then given by $$\begin{align} E=\,&u_{0}+\eta _{x}k_{x}^{2}+\eta _{y}k_{y}^{2}\\ &\pm \sqrt{(\delta +\gamma_{x}k_{x}^{2}+\gamma _{y}k_{y}^{2})^{2}+(\chi k_{x})^{2}}.~~ \tag {7} \end{align} $$ At the ${\it \Gamma}$ ($k_{x}=k_{y}=0$) point, $E_{(c,v)}=u_{0}\pm \delta =4t_{4}\pm (2t_{1}+t_{2}+2t_{3}+t_{5})$. In terms of Eq. (1), we obtain the band gap $E_{\rm g}=(4t_{1}^{0}+2t_{2}^{0}+4t_{3}^{0}+2t_{5}^{0})- \sum_{j}(8\alpha _{j}^{1}\varepsilon _{j}t_{1}^{0}+4\alpha _{j}^{2}\varepsilon _{j}t_{2}^{0}+8\alpha _{j}^{3}\varepsilon _{j}t_{3}^{0}+4\alpha _{j}^{5}\varepsilon _{j}t_{5}^{0})$, where $j$ denotes the summation over $x$, $y$, $z$ components. The first bracket is the unstrained band gap, i.e., $E_{\rm g}^{0}=1.52$ eV and the second one indicates the structural dependent values of changes in the band gap due to the axial strains. Inserting the numerical values of the structural parameters to the band gap equation, we obtain $E_{\rm g}=E_{\rm g}^{0}-\sum_{j}\eta _{j}\varepsilon _{j}$, where $\eta _{x}=-4.09$ eV, $\eta _{y}=-5.73$ eV and $\eta _{z}=12.86$ eV. For in-plane axial strain, i.e., $\epsilon _{x}\neq 0$, $\epsilon _{y}\neq 0$, and $\epsilon _{z}=0$, we have the strain-induced modification in the bandgap $E_{\rm g}=1.52+4.09\epsilon _{x}+5.73\epsilon _{y}$. Due to the translation invariance in the $y$-direction, the solution of Eq. (2) in the $j$th region can be written as ${\it \Psi}_{j}(x,y)={\it \Psi}_{j}(x)\exp (ik_{y}y)$. We model the stain potential by a step-like potential. This idealized model can give the essential features of the quantum tunneling. A smoother experimentally realizable potential profile would not change the main results. The wavefunctions ${\it \Psi}_{j}(x)$ in the three different regions $x < 0$, $0 < x < D$ and $x>D$ are written as $$\begin{align} {\it \Psi}_{\rm I} =\,&\frac{1}{\sqrt{2}}\left(\begin{matrix}1\\ \lambda e^{-i\theta _{0}}\end{matrix}\right) e^{ik_{x}x} +\frac{r}{\sqrt{2}}\left(\begin{matrix}1\\ -\lambda e^{i\theta _{0}}\end{matrix}\right)e^{-ik_{x}x},\\ {\it \Psi}_{\rm I\!I} =\,&\frac{a}{\sqrt{2}}\left(\begin{matrix}1\\ \lambda _{1}e^{-i\theta }\end{matrix}\right) e^{iq_{x}x} +\frac{b}{\sqrt{2}}\left(\begin{matrix}1\\ -\lambda _{1}e^{i\theta }\end{matrix}\right)e^{-iq_{x}x},\\ {\it \Psi}_{\rm I\!I\!I} =\,&\frac{t}{\sqrt{2}}\left(\begin{matrix}1\\ \lambda e^{-i\theta _{0}}\end{matrix}\right)e^{ik_{x}x},~~ \tag {8} \end{align} $$ where $\theta =\arctan (k_{y}/q_{x})$, $$\begin{align} k_{y}=\,&\sqrt{\frac{(E-u_{0}^{0}\mp \delta ^{0})\tan ^{2}\theta _{0}}{(\eta _{x}^{0}\pm \gamma _{x}^{0}\pm \frac{(\chi ^{0})^{2}}{2\delta ^{0}})\pm (\eta _{y}^{0}\pm \gamma _{y}^{0})\tan ^{2}\theta _{0}}},~~ \tag {9} \end{align} $$ $$\begin{align} q_{x}=\,&\sqrt{\frac{(E-u_{0}\mp \delta )-(\eta _{y}\pm \gamma _{y})k_{y}^{2}}{ (\eta _{x}\pm \gamma _{x}\pm \frac{\chi ^{2}}{2\delta })}},~~ \tag {10} \end{align} $$ with $u_{0}^{0}$, $\delta ^{0}$, $\eta _{x}^{0}$, $\eta _{y}^{0}$, $\gamma _{x}^{0}$, $\gamma _{y}^{0}$ and $\chi ^{0}$ being parameters in the absence of the strain. The critical angle is $$ \theta _{\rm c}=\arctan\Big(\sqrt{\frac{(\eta _{x}^{0}\pm \gamma _{x}^{0}\pm \frac{(\chi^{0}) ^{2}}{2\delta^{0}})}{\frac{(E-u_{0}^{0}\mp \delta^{0})}{(E-u_{0} \mp \delta)}(\eta _{y}\pm \gamma _{y})-(\eta _{y}^{0}\pm \gamma _{y}^{0})}}\Big).~~ \tag {11} $$ When $\theta _{0}$ exceeds the critical angle $\theta _{\rm c}$, $q_{x}$ is purely imaginary. To obtain the transmission probabilities through a strained region, we calculate the undetermined coefficients through the application of the boundary conditions, i.e., the wavefunctions should be continuous at the interfaces $x=0$ and $x=D$. The transmission probability is obtained as $$ T=\frac{1}{\cos ^{2}(q_{x}D)+\sin ^{2}(q_{x}D)\frac{(kk^{\prime }-\lambda \lambda _{1}k_{y}^{2})^{2}}{k_{x}^{2}q_{x}^{2}}}.~~ \tag {12} $$ Note that $T$ is not only a function of the energy but also of the incident angle $\theta _{0}$ of the incoming carriers. Based on the obtained results for the transmission probabilities $T$, one can find the two-terminal Landauer conductance $G$ for the finite structure. Within a linear regime on bias voltage at very low temperatures, the conductance is given by $$ G(E_{\rm F})=G_{0}(E_{\rm F})\int_{-\pi /2}^{\pi /2}T{\cos \theta _{0}d\theta _{0}},~~ \tag {13} $$ where $G_{0}=ge^{2}E_{\rm F}L_{y}/(h^{2}\upsilon _{\rm F})$, $L_{y}$ is the length of the slab in the transverse ($y$) direction, and $g$ equals 4 due to the transmission probability for spin-up (down) electrons. The transmission spectra for electrons traversing such a strained region are shown in Fig. 2. For a fixed incident energy $E_{\rm F}=0.9$ eV, the transmission declines sharply and then is blocked when the incident angle exceeds a critical value (see Figs. 2(a) and 2(b)), while the wavevector $q_{x}$ in the strained region becomes imaginary denoting the appearance of evanescent modes. Comparing Fig. 2(a) with Fig. 2(b), we find that the critical angle is different for the same strength and different directions in-plane strains, which can be understood from Eq. (11). It is interesting to notice that for the same strength strain the critical value $\theta _{\rm c}$ becomes smaller and the tunneling is forbidden for a wider region when the direction of the strain in the II region changes from $x$-direction to $y$-direction. As is seen, for the incident energy $E_{\rm F}=-1.5$ eV, the strained region becomes transparent at the incident angle $\theta _{0}=0$ (see Figs. 2(c) and 2(d)). The transmission probability displays a sharp peak at a specific incident angle, i.e., a strong momentum filtering. Here the resonant angles $\theta _{0m}$ are determined by $q_{x}D=m\pi $ ($m=1,2,\ldots $) and $\theta _{0m}\in $Reals. As shown in Figs. 2(c) and 2(d), the resonant angles for $\epsilon _{x}=0.05$ are different from that for $\epsilon _{y}=0.05$. The positions and number of resonant peaks are tuned by the strain and the incident energy.
cpl-34-2-027302-fig2.png
Fig. 2. (Color online) Transmission probability as a function of the incident angle for (a) the incident energy $E_{\rm F}=0.9$ eV and several representative $\epsilon _{x}$, (b) $E_{\rm F}$=0.9 eV and several representative $\epsilon _{y}$, (c) $E_{\rm F}=-1.5$ eV and several representative $\epsilon _{x}$, (d) $E_{\rm F}=-1.5$ eV and several representative $\epsilon _{y}$. The blue, red dashed and black dot-dashed lines correspond to the results for $\epsilon _{x/y}=0.05$, 0.1 and 0.17, respectively. The width of the strained region is $D=10$ nm.
cpl-34-2-027302-fig3.png
Fig. 3. (Color online) Transmission probability as a function of (a) $\epsilon _{x}$, (b) $\epsilon _{y}$ for two different incident energies $E_{\rm F}=0.9$ eV (dashed line) and $E_{\rm F}=1.1$ eV (solid line). The incident angle is fixed at $\theta _{0}=\pi/4$. The width of the strained region is $D=10$ nm.
To show the switching behavior in a strained system more clearly, we plot the dependence of the transmission on the strengths of the in-plane strains in Fig. 3. The in-plane strain shifts the gap by modifying the hopping parameters. A sufficient high in-plane strain switches off the transport when the Fermi energy is the gap in the strained system. The transmitted channels are blocked at $\epsilon _{x}\approx 0.19$ or $\epsilon _{y}\approx 0.15$ for the fixed incident energy $E_{\rm F}=0.9$ eV and the incident angle $\theta _{0}=\pi /4$. The conduction band edge in the absence of the in-plane strain is $E _{\rm c}=u_{0}^{0}+\delta ^{0}=4t_{4}^{0}+(2t_{1}^{0}+t_{2}^{0}+2t_{3}^{0}+t_{5}^{0})=0.34$ eV. The conduction band edge in the presence of the in-plane strain in the II region is larger than that in the absence of the in-plane strain. The increased values of the conduction band edge for different directions of in-plane strains are different (see Fig. 1(d)). In the vicinity of the gap where the tunneling is forbidden, the critical value of the strain depends sensitively on the direction of the strain. When $k_{y}>k'$, the longitudinal wave vector $q_{x}$ in the middle region becomes imaginary, thus transmission is totally blocked. For a given incident angle, a larger Fermi energy corresponds to a larger value of $k'$. The transverse wave vector $k_{y}$ is conserved, while the Fermi wavevector is different in II and III regions as a consequence of strain. Therefore, the critical strength of the strain is larger for a larger Fermi energy. These results are consistent with that in Fig. 2.
cpl-34-2-027302-fig4.png
Fig. 4. (Color online) Conductance as a function of the incident energy for several representative strain (a) $\epsilon _{x}$, and (b) $\epsilon _{y}$. The red and black dashed lines are for $\epsilon_{x/y} =0.1$ and 0.17, respectively. The width of the strained region is $D=10$ nm.
Finally, we plot the conductance versus the Fermi energy for transport through a strained region of width $D=10$ nm with strain $\epsilon _{x/y}=0.1$ (solid line) and $\epsilon _{x/y}=0.17$ (dashed line) in Fig. 4. The width of the gap in the conductance decreases with the increasing strain. The critical incident energy for a fixed strength of the strain $\epsilon _{x}$ is different from that for a fixed strength of the strain $\epsilon _{y}$, as the different effective masses along $x$ and $y$ directions. The crossover from the zero to the plateau 2 occurs in a small region of the incident energy $E_{\rm F}$, indicating an electric switching effect. In summary, we have investigated quantum transport of carriers through a strained region on monolayer phosphorene theoretically. Generally, the transmission probability for transport through such structure declines sharply and then is blocked when the incident angle exceeds a critical value for a fixed incident energy in the conduction band. The critical angle for electrons tunneling is different for transport along different directions as the different modification of the bandgap by the $x/y$ strain. The transmission probability displays a sharp peak at a specific incident angle for a fixed incident energy in the valence band. The positions of resonant peaks depend on the strain and incident energy. The width of the gap in the conductance decreases with increasing the strength of the strains. This feature provides us with an efficient way to control the transport of monolayer phosphorene-based microstructures. References Mechanisms of current fluctuation in ambipolar black phosphorus field-effect transistorsTwo-dimensional crystals: Phosphorus joins the familyPhosphorene excites materials scientistsElectric field effect in ultrathin black phosphorusBlack phosphorus field-effect transistorsRediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronicsQuantum Hall effect in black phosphorus two-dimensional electron systemAnomalous magneto-optical response of black phosphorus thin filmsScaling laws of band gaps of phosphorene nanoribbons: A tight-binding calculationTopological origin of quasi-flat edge band in phosphoreneExcitons and optical spectra of phosphorene nanoribbonsDirectional-dependent thickness and bending rigidity of phosphoreneResonant bonding driven giant phonon anharmonicity and low thermal conductivity of phosphoreneThe electronic properties of grapheneModulation of electronic and mechanical properties of phosphorene through strainInverse Funnel Effect of Excitons in Strained Black PhosphorusStrain-Engineering the Anisotropic Electrical Conductance of Few-Layer Black PhosphorusEffective Hamiltonians for phosphorene and siliceneStrain-Induced Gap Modification in Black PhosphorusHigh-mobility transport anisotropy and linear dichroism in few-layer black phosphorusElectronic Structure of Black Phosphorus in Tight Binding ApproachStrain-induced gap transition and anisotropic Dirac-like cones in monolayer and bilayer phosphoreneStrain Engineering for Phosphorene: The Potential Application as a PhotocatalystA full spd tight-binding treatment for electronic bands of graphitic tubesStrain-induced topological phase transition in phosphorene and in phosphorene nanoribbonsQuasiparticle band structure and tight-binding model for single- and bilayer black phosphorusSwitching a Normal Insulator into a Topological Insulator via Electric Field with Application to PhosphoreneStacked bilayer phosphorene: strain-induced quantum spin Hall state and optical measurement
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