Topological phase transition in GeSnH induced by biaxial tensile strain: A tight-binding study
An effective tight-binding (TB) Hamiltonian for monolayer GeSnH is proposed which has an inversion-asymmetric honeycomb structure. The low-energy band structure of our TB model agrees very well with previous ab initio calculations under biaxial tensile strain. We predict a phase transition upon 7.5% biaxial tensile strain in agreement with DFT calculations. Upon 8.5% strain the system exhibits a band gap of 134 meV, suitable for room temperature applications. The topological nature of the phase transition is confirmed by: 1)the calculation of the topological invariant, and 2)quantum transport calculations of disordered GeSnH nanoribbons which allows us to determine the universality class of the conductance fluctuations.
Topological insulators (TIs) have attracted a lot of attention in condensed matter physics and from the materials science community during the past decade Qi and Zhang (2011); Hasan and Kane (2010); Moore (2010); Kane and Mele (2005a, b). TIs are fascinating states of quantum matter with insulating bulk and topologically protected edge or surface states. In two-dimensional (2D) TIs, also known as quantum spin Hall (QSH) insulators Kane and Mele (2005a), the gapless edge states are topologically protected by time reversal symmetry (TRS) and are spin-polarized conduction channels that are robust against non-magnetic scattering. They are more robust against backscattering than three-dimensional (3D) TIs, making them better suited for coherent transport related applications, low-power electronics, and quantum computing applications Chuang et al. (2014).
There are currently only a few 3D TI compounds that are experimentally realized such as BiSe, BiTe, and SbTe Zhang et al. (2009); and only HgTe/CdTe König et al. (2007) and InAs/GaSb Knez et al. (2011) quantum wells have been realized as 2D TIs. Also, due to the very small bulk band gaps (on the order of meV), these 2D exhibit TI only at ultra-low temperatures. Therefore, there is a great need to find new 2D TIs with large energy band gaps. Following the advancements in graphene and similar materials, intensive efforts have been devoted to explore 2D group-IV and V honeycomb systems, which can harbor 2D topological phases.
In terms of crystal structure, TIs can be generally divided into inversion-symmetric TIs (ISTIs) and inversion-asymmetric TIs (IATIs). Inversion asymmetry introduces many additional intriguing properties in TIs such as pyroelectricity, crystalline-surface-dependent topological electronic states, natural topological p-n junctions, and topological magneto-electric effects Ma et al. (2015); Zhang et al. (2016); Li et al. (2016). Therefore, 2D IATI materials that are stable at room temperature would be highly promising candidates for future spintronics and quantum computing applications.
Also, it would be interesting if we could implement such features in group-IV honeycomb systems for the integration of devices that use other carbon-IV honeycomb elemens. This would avoid issues such as contact resistances and their integration within the traditional Si or Ge based devices.
An efficient method for the synthesis of suitable 2D nanomaterials with honeycomb structure is chemical functionalization. Hydrogenation and halogenation of the above systems for the realization of topological phases have been extensively explored.
Here we consider GeSnH, which is an inversion asymmetric hydrogenated bipartite honeycomb system. ab initio calculations have shown that monolayers (ML) of GeSn halide (GeSnX, XF, Cl, Br, I) are large band gap 2D TIs with protected edge states forming QSH systems Ma et al. (2015). On the other hand hydrogenated ML GeSn (GeSnH) is a normal band insulator which can be transformed into a large band gap topological insulator via appropriate biaxial tensile strain Ma et al. (2015).
Here, we propose a tight-binding (TB) model for a better understanding of the electronic band structure of GeSnH near the Fermi level. The band structure of our TB model is fitted to the ab initio results, where we consider the cases with spin-orbit coupling (SOC) and without SOC. Within the linear regime of strain the band gap of our TB model agrees very well with the DFT results Ma et al. (2015). This TB model when including SOC predicts a band inversion at 7.5% biaxial tensile strain in agreement with DFT calculations Ma et al. (2015). We show the topological nature of the phase transition by the calculation of the topological invariant. Quantum transport of this system is calculated in order to examine the protection of the edge states against nonmagnetic scattering. It is shown that the conductance fluctuations of disordered nanoribbons for energies near the band gap belong to the universality class of the circular unitary ensemble (), while for high energies and strong disorder the fluctuations follow the circular orthogonal ensemble ().
This paper is organaized as follows. In Sec. II, we introduce the crystal structure and lattice constants of monolayer GeSnH. Our proposed TB model is introduced in Sec. III, and the effect of strain on the electronic properties of GeSnH is examined. In Sec. IV, the topological phase transition under strain is examined by looking at the nano-ribbon band structure and by determining the topological invariant. In Sec. V electronic transport in disordered GeSnH nanoribbons is examined and the protection of the chiral edge states against nonmagnetic scatterings is verified. Also, we discuss the universality class of the conductance fluctuations. Our results are summarized in Sec. VI.
Ii lattice structure
The ML GeSnH prefers a buckled honeycomb lattice, analogous to its homogeneous counterparts germanene Liu et al. (2011a) and stanene Xu et al. (2013). Figs. 1(a) and 1(b) show the atomic structure of ML GeSnH and its geometrical parameters. The 2D honeycomb lattice consists of two inequivalent sublattices made of Ge and Sn atoms, which are named A and B sublattices, respectively. Both Ge and Sn atoms exhibit hybridization. One orbital is passivated by a hydrogen atom and the other three are bonded to three neighboring Ge or Sn atoms. Thus, the unit cell of GeSnH consists of four atoms: one Ge, one Sn, and two Hydrogen atoms that are right above (below) the Ge (Sn) atoms. The lattice translation vectors are with the lattice constant Åand the buckling height is Å Ma et al. (2015); Zhang et al. (2016). Note that the and axes are taken to be along the armchair and zigzag directions, respectively; and the axis is in the normal direction to the plane of the GeSnH film.
Iii Low-energy effective TB Hamiltonian for monolayer GeSnH
The electronic structure and topological properties of 2D honeycomb ML GeSnH were studied in Ref. Ma et al. (2015) using first principle calculations based on DFT. The DFT calculations including spin-orbit interaction predicted that a topological phase transition is induced in ML GeSnH through the application of biaxial in-plane tensile strain. It was shown that the low-energy electronic structure of ML GeSnH is determined exclusively by using , and atomic orbitals of Ge and Sn atoms Ma et al. (2015); Zhang et al. (2016). However, in order to examine the effect of random disorder on the electronic properties of this system and to confirm the topological nature of the phase transition, large system sizes are required. A limitation of the DFT calculations is that only small system sizes are managable. Therefore, we need a TB model for ML GeSnH that is able to describe the low-energy electronic structure of large system sizes.
In the following we will derive a low-energy TB model including spin-orbit coupling (SOC) and we will show that the results of the proposed TB model even in the presence of biaxial strain is in very good agreement with previous DFT calculations.
|Hopping parameters||Without strain||With biaxial strain|
iii.1 Tight-Binding model Hamiltonian without SOC
To describe the low-energy spectrum and the electronic properties of ML GeSnH, we propose a TB model hamiltonian involving the three outer-shell , and atomic orbitals of Ge and Sn atoms. The effective TB Hamiltonian without SOC can be written in the second quantized representation as
Here, , are the orbital indices and denotes the nearest-neighboring th and th atoms. is the on-site energy of th orbital of th atom, is the creation (annihilation) operator of an electron in the th orbital of the th atom, and is the nearest-neighbor hopping parameter between th orbital of th atom and th orbital of th atom.
The hopping parameters of Eq. (1) are determined by the Slater-Koster (SK) Slater and Koster (1954) integrals as shown in the second column of table 1, where and are direction cosines of the angles of the vector connecting two nearest-neighboring atoms with respect to and axes, respectively.
We calculated the hopping parameters and on-site energies of the above Hamiltonian using the method of minimization of the least square difference between the DFT obtained band structure based on the Heyd-Scuseria-Ernzerhof (HSE) approximation Ma et al. (2015) and the band structure of our TB model. Our TB model Hamiltonian has nine fitting parameters; namely, four on-site orbital energies and five SK parameters related to the hopping energies . Note that is the hopping integral between orbitals of atoms in sublattice A (B) and orbitals of atoms in sublattice B (A).
Table 2 presents the obtained numerical values of the SK parameters. Using the optimized parameters, we can reproduce the three low-energy bands near the Fermi level ( and bands). Fig. 2(a) shows the TB low-energy bands of ML GeSnH that is in good agreement with the DFT results. The band structure has a direct band gap of 1.155 eV at the point.
iii.2 Spin-orbit coupling in ML GeSnH
In general, spin-orbit interaction can be written as Liu et al. (2011b)
where, is Planks constant, is the rest mass of an electron, is the velocity of light, is potential energy, is momentum, and is the vector of Pauli matrices. The major part of SOC in systems that consist of heavy atoms comes from the orbital motion of electrons close to the atomic nuclei. In such systems and within the central field approximation, the crystal potential can be considered as an effective spherical atomic potential located at th atom. Therefore, by substituting and terms into Eq. (12) the SOC term takes the form Liu et al. (2011b),
The above equation can also be expressed in the form
where, is the effective atomic SOC constant whose value depends on the specific atom. , are the operators for angular momentum and spin, respectively. In the basis set of , the matrix elements of the on-site SOC Hamiltonian for ML GeSnH are given by
Here and are the atomic orbitals and is the on-site SOC strength of the th atom.
Note that since the two atoms in the unit cell of ML GeSnH are different, we have two distinct SOC strengths
and for atoms in sublattice A (Ge atoms) and sublattice B (Sn atoms), respectively.
The resulting SOC Hamiltonian matrix in the above basis is given by
where, the elements are 66 matrices
The value of the on-site effective SOC strength is determined by fitting the TB energy bands to the DFT results. Here we fitted the energy bands obtained from our TB model to the one from the DFT+HSE approach Ma et al. (2015). The optimized numerical results using the hopping parameters from Table 2 are eV, eV for Ge and Sn atoms, respectively. The TB energy bands of ML GeSnH in the presence of spin-orbit interaction using the mentioned values of SOC strengths are in good agreement with the results as shown in Fig. 2(b).
Notice from Fig. 2(b) the spin splitting in the doubly degenerated bands. The splitting is more clear for the upper valence band close to the K symmetry point which is 92 meV. For the conduction band the spin-splitting is 241 meV at the K symmetry point. All the energy states of a system with inversion symmetry will be spin-degenerate while the TRS is held. The degeneracy can be lifted by breaking the inversion symmetry in such a system. The band gap of the system is reduced to 0.977 eV, demonstrating that ML GeSnH is still a normal semiconductor.
iii.3 The effect of strain
The electronic properties of a system can be affected significantly by applying strain Bir and Pikus (1974); Sun et al. (2010). This is due to the fact that strain changes both the bond lengths and the bond angles. This in turn changes the SK parameters and hopping integrals that further affect the electronic band structure.
Based on knowledge from our previous works, we expect a topological phase transition upon the application of biaxial tensile strain Ma et al. (2015). Further, since the electronic band structure of this system under strain is available from calculations, the comparison of our TB band structure with DFT results will be another verification for the correctness of our TB model Hamiltonian.
Here, we investigate the effect of biaxial tensile strain on ML GeSnH. In our previous work we studied the effect of biaxial tensile strain on a related system where the low-energy electronic properties were dominated by , and atomic orbitals Rezaei et al. (2017). We follow the approach of Ref. Rezaei et al. (2017) and obtain the effect of strain on the hopping parameters which we list in table 1. Note that indicates the unstrained hopping parameters.
In such systems, it is expected that the buckling angle changes linearly with strain as Rezaei et al. (2017). is the strength of the applied biaxial strain and is the initial buckling angle. However, our calculations show that the changes in the buckling angle in this system produces negligible effect on the hopping parameters and therefore, we will use the initial buckling angle in the rest of our calculations.
The Hamiltonian for the strained system can be obtained by substituting the new hopping parameters (last column of table 1) in the original Hamiltonian Eq. (1). The calculated TB energy spectrum of ML GeSnH are shown in Figs. 3(a-c) without SOC and Figs. 3(d-f) with SOC in the presence of 0%, 4%, and 8% biaxial tensile strains. As shown, these results are in good agreement with the DFT calculations Ma et al. (2015).
In Figs. 4(a) and 4(b), we show the variation of the energy gap of ML GeSnH as a function of biaxial tensile strain without and with SOC, respectively. As shown in Fig. 4(b), by applying biaxial tensile strain in the presence of SOC, the band gap that is located at the point decreases steadily and eventually a band inversion occurs at the critical value of 7.5% strain. By increasing the strain the band gap reaches 134 meV at a reasonable strain of 8.5%. With increasing strain even further to 9.5%, the band gap with SOC becomes indirect. As shown in TI phase in Fig. 4(b), both and (global band gap) remain relatively unaffected by increasing the strain beyond 13%.
Remarkably, the value of the band gap is significantly larger than at room temperature ( 25 meV), and large enough to realize the QSH effect in ML GeSnH even at room temperature. The excellent agreement between the results of our TB model and the DFT calculations in predicting the band inversion in this system upon biaxial tensile strain, implicitly confirms the validity of our proposed TB model.
Iv Topological phase transition of monolayer GeSnH under strain
Using our TB model including spin orbit interaction, we showed that ML GeSnH is a NI with a direct band gap of 0.977 eV. We showed that applying biaxial tensile strain modifies its electronic spectrum and a band inversion is taking place at .
One way to show that there is a topological phase transition at the critical strain of , is to perform calculations of the 1D band structure of nanoribbons with zigzag edges in the presence of biaxial tensile strain, and verify the existence of gapless helical edge states. Here, we study the edge state energy bands by cutting a 2D film of ML GeSnH into nanoribbons. In our calculations, the Sn and Ge atoms at the edges of the nanoribbon are passivated by hydrogen atoms.
The width of a zigzag GeSnH nanoribbon (z-GeSnH-NR) is defined by N, which is the number of zigzag chains across the ribbon width. In order to reduce the interaction between the two edges, the width of the nanoribbons is taken to be at least 10 nm.
We calculated the band structure of the corresponding nanoribbons. Note that the change of the hopping parameters and the on-site energies of the edge Ge or Sn atoms caused by the passivation procedure is negligible. Therefore, we can use the results in Table 1 and 2 for the on-site energies of Ge and Sn atoms and also for the hopping parameters corresponding to Ge-Sn bonds in unstrained or strained systems. We have two different on-site energies of hydrogen atoms and , which are pertinent to the Hydrogen atoms that are introduced to passivate the Ge and Sn atoms on each edge, respectively. We can write the hopping parameters related to the H-Ge and H-Sn bonds as
where denotes the lower (upper) H-X edge bonds. The numerical value of the mentioned hopping parameters and on-site energies of the hydrogen atoms can be obtained by a fitting procedure. The results are shown in Table 3.
Figs. 5(a) and 5(b) show the energy bands of z-GeSnH-NR with in the presence of 5% and 8.5% biaxial tensile strains, respectively. Gapless conducting edge bands are seen for strain of . This is consistent with our proposal for the topological phase transition at .
It is now well established that for all time reversal invariant 2D band insulators a change in the topological invariant from zero to one, indicates a topological phase transition from a NI phase to TI. In our previous worksSisakht et al. (2016); Rezaei et al. (2017), we successfully used the algorithm of Fukui and Hatsugai Fukui and Hatsugai (2007) in order to calculate the number to characterize the topology of the energy bands. In this work, we implemented the same procedure to confirm the existence of two distinct topological phases in the electronic properties of ML GeSnH. We found that the value of switches from zero to one at the critical strain of , which confirms the topological nature of the phase transition in the electronic properties of ML GeSnH.
V Electronic Transport in disordered GeSnH Nanoribbons
Transport measurement is a different approach to confirm the existence of helical gapless edge states, which is an important signature of the TI phase. Therefore, we next study the transport properties of ML GeSnH nanoribbons in the presence of strain To this end, we calculate the conductance of z-GeSnH-NR using the Landauer formalism Datta (1995, 2005). As is standard the z-GeSnH-NR is divided into three regions; the left and right leads and the middle scattering region. We initially assume the ribbon to be perfect in order to satisfy the condition of ballistic transport. In the Landauer approach, the total conductance per spin of a nanoscopic device at the Fermi energy is given by
where , with being the self energy of the left (right) lead. is the retarded Green’s function of the device and . The retarded Green’s function of the device is given by
Here is the Hamiltonian for the device region. The numerically calculated total conductance, in units of , for as a function of energy is shown in Figs. 5(c) and 5(d) in the presence of 5% and 8.5% biaxial tensile strain, respectively. For strains less than (), the z-GeSnH-NR has nonzero conductivity only above a threshold energy corresponding to the minimum energy of the conduction band, which opens a conducting channel with conductance in steps of . As shown in Fig 5(b), the total conductance at zero energy changes from 0 to 2 by applying biaxial tensile strains larger than (). The non-zero conductance in the gap originates from the zero energy edge states and also indicates a topological phase transition from NI to TI in ML GeSnH. These conducting edge states are protected by TRS leading to the robustness of the electronic quantized conductance against backscattering by disorder and therefore holds great promise for spintronics applications Gusev et al. (2011); Van Dyke and Morr (2017).
It would be helpful to further consider the effect of disorder on the electronic transport properties of this system. The disorder may be originated from unwanted dislocations or other defects. Such calculations are another proof for our proposal of strain-induced TI transition. In TIs, the edge states are robust against weak disorder, and only strong disorder can affect the electronic properties.
Here we introduce the disorder in our TB Hamiltonian model using the so-called Anderson disordered model Lee and Fisher (1981) as
where is a random number uniformly distributed over the range . We assume a disordered z-GeSnH-NR with ( 13 nm width) and 51 nm length as the middle region which is sandwiched between two semi-infinite perfect leads and calculate the conductance averaged over 100 different realizations. The width of the ribbon is chosen large enough in order to avoid finite-size effects. Fig. 6(a) shows the average conductance of z-GeSnH-NR as a function of disorder strength in the presence of biaxial strain at three different values of the Fermi energy. The energy eV corresponds to the energy in the middle of the band gap, while eV and eV are two energies in the bulk. It can be seen that the averaged quantized conductance at eV, which originates from the gapless edge states is insensitive to weak disorder. The mean conductance decreases only for strong disorder of strength eV which is a signature of a topological insulator.
The universal conductance fluctuations (UCF) Lee and Stone (1985) are a mesoscopic phenomena, which is caused by the quantum interference of electrons. The UCF are of order and correspond to the deviation of the conductance from its ballistic value . The standard deviation of conductance in the diffusive regime and zero temperature depends only on the dimensionality and universality class of the disordered mesoscopic system and is independent of the details of the system such as the disorder strength , the conductance and the sample size. The UCF takes universal values for three types of symmetry classes corresponding to = 1, 2 and 4, respectively. In systems that preserve TRS and spin rotational symmetry (SRS), the symmetry index (orthogonal ensemble); and corresponds to the case that TRS is broken (unitary ensemble); while for systems that preserve TRS but with broken SRS (symplectic ensemble) Hu et al. (2017); Hsu et al. (2018). We study numerically the conductance fluctuations in the disordered z-GeSnH-NR in the presence of biaxial tensile strain. The standard deviation of the conductance of disordered z-GeSnH-NR is plotted as a function of disorder strength at three different Fermi energies in Fig. 6(b). There are no conductance fluctuations for energy eV in case of weak disorder strength, revealing the robustness of the helical edge states against disorder in the QSH phase. The approach the value , which corresponds to the UCF for the symmetry class . Our total model Hamiltonian preserves TRS and the reason that the UCF shows the symmetry class is the following: since there are no spin-flip terms (Rashba SOC term) in our model Hamiltonian, we can block diagonal the Hamiltonian matrix with respect to the spin degrees of freedom with zero off-diagonal terms. Then we are dealing with two isolated and identical Hamiltonians corresponding to spin up and spin down states. Each of these blocks lack TRS due to the intrinsic SOC terms, and consequently the total UCF follows the symmetry class. A similar discussion can be found in Refs. Hsu et al. (2018); Choe and Chang (2015), except that in Ref. Hsu et al. (2018) spin is not a good quantum number and the hamiltonian was block diagonalized with respect to the sublattice degrees of freedom. For high energies and strong disorder strengths, the standard deviation approaches the value , which belongs to the symmetry class . We will compare the localization length with the spin relaxation length (spin relaxation length originates from the intrinsic SOC terms) Kaneko et al. (2010). As long as the disorder is weak and the localization length is much larger than the spin relaxation length the SOC is significant and the system lacks SRS. But when disorder is strong enough and the localization length is much smaller than the spin relaxation length, we can ignore the SOC terms and SRS is preserved and the system follows the symmetry class. Finally, beyond eV, where approaches the value of the conductance fluctuations decreases and approaches the superuniversal curve Qiao et al. (2010) that is independent of dimensionality and symmetry. The superuniversal curve is beyond eV, which is not shown in Fig. 6.
In summary, we constructed an effective TB model without and with SOC for ML GeSnH, which is able to reproduce the electronic spectrum of this system in excellent agreement with the DFT results near the Fermi level. Including SOC decreases the band gap from 1.155 eV to 0.977 eV. In the presence of biaxial tensile strain, our proposed TB model predicts correctly the evolution of the band spectrum and also predicts a topological phase transition from NI to TI phase in ML GeSnH at 7.5% biaxial tensile strain. The global bulk gap, which is topologically protected, is 134 meV at a reasonable strain of 8.5%. This bulk gap exceeds the thermal energy at room temperature and is large enough to make ML GeSnH suitable for room-temperature spintronics applications.
The strong SOC and the applied mechanical strain are two essential factors that induce the topological phase transition from NI to QSH phase in ML GeSnH. More interestingly, ML GeSnH is a strain-induced TI with inversion asymmetry which makes it a promising candidate for understanding intriguing topological phenomena like magneto-electric effects. The TI nature of ML GeSnH for strain was confirmed by calculating the topological invariant. Also we showed the existence of topologically protected gapless edge states in a typical z-GeSnH-NR in the presence of biaxial strain .
In addition we found topologically protected gapless edge states in a typical z-GeSnH-NR for a biaxial strain of in the presence of disorder by calculating electronic transport. The conductance fluctuations reach the universal value of the unitary class . For high Fermi energies and strong disorder the conductance fluctuations follow the orthogonal ensemble .
Acknowledgements: This work was supported by the FLAG-ERA project TRANS-2D-TMD.
- Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
- Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
- Moore (2010) J. E. Moore, Nature 464, 194 (2010).
- Kane and Mele (2005a) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005a).
- Kane and Mele (2005b) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005b).
- Chuang et al. (2014) F.-C. Chuang, L.-Z. Yao, Z.-Q. Huang, Y.-T. Liu, C.-H. Hsu, T. Das, H. Lin, and A. Bansil, Nano Lett. 14, 2505 (2014).
- Zhang et al. (2009) H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Nat. Phys. 5, 438 (2009).
- König et al. (2007) M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766 (2007).
- Knez et al. (2011) I. Knez, R.-R. Du, and G. Sullivan, Phys. Rev. Lett. 107, 136603 (2011).
- Ma et al. (2015) Y. Ma, L. Kou, A. Du, and T. Heine, Nano Res. 8, 3412 (2015).
- Zhang et al. (2016) S.-j. Zhang, W.-x. Ji, C.-w. Zhang, S.-s. Li, P. Li, M.-j. Ren, and P.-j. Wang, RSC Adv. 6, 79452 (2016).
- Li et al. (2016) S.-s. Li, W.-x. Ji, C.-w. Zhang, P. Li, and P.-j. Wang, J. Mater. Chem. C 4, 2243 (2016).
- Liu et al. (2011a) C.-C. Liu, W. Feng, and Y. Yao, Phys. Rev. Lett. 107, 076802 (2011a).
- Xu et al. (2013) Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan, and S.-C. Zhang, Phys. Rev. Lett. 111, 136804 (2013).
- Slater and Koster (1954) J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954).
- Liu et al. (2011b) C.-C. Liu, H. Jiang, and Y. Yao, Phys. Rev. B 84, 195430 (2011b).
- Bir and Pikus (1974) G. Bir and G. Pikus, Symmetry and Strain-induced Effects in Semiconductors (Wiley, New York, 1974).
- Sun et al. (2010) Y. Sun, S. E. Thompson, and T. Nishida, Strain Effect in Semiconductors (Springer, US, 2010).
- Rezaei et al. (2017) M. Rezaei, E. T. Sisakht, F. Fazileh, Z. Aslani, and F. M. Peeters, Phys. Rev. B 96, 085441 (2017).
- Sisakht et al. (2016) E. T. Sisakht, F. Fazileh, M. H. Zare, M. Zarenia, and F. M. Peeters, Phys. Rev. B 94, 085417 (2016).
- Fukui and Hatsugai (2007) T. Fukui and Y. Hatsugai, J. Phys. Soc. Jpn. 76, 053702 (2007).
- Datta (1995) S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, Cambridge, England, 1995).
- Datta (2005) S. Datta, Quantum Transport: Atom to Transistor (Cambridge University Press, Cambridge, England, 2005).
- Gusev et al. (2011) G. M. Gusev, Z. D. Kvon, O. A. Shegai, N. N. Mikhailov, S. A. Dvoretsky, and J. C. Portal, Phys. Rev. B 84, 121302 (2011).
- Van Dyke and Morr (2017) J. S. Van Dyke and D. K. Morr, Phys. Rev. B 95, 045151 (2017).
- Lee and Fisher (1981) P. A. Lee and D. S. Fisher, Phys. Rev. Lett. 47, 882 (1981).
- Lee and Stone (1985) P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985).
- Hu et al. (2017) Y. Hu, H. Liu, H. Jiang, and X. C. Xie, Phys. Rev. B 96, 134201 (2017).
- Hsu et al. (2018) H.-C. Hsu, I. Kleftogiannis, G.-Y. Guo, and V. A. Gopar, J. Phys. Soc. Jpn. 87, 034701 (2018).
- Choe and Chang (2015) D.-H. Choe and K.-J. Chang, Sci. Rep. 5, 10997 (2015).
- Kaneko et al. (2010) T. Kaneko, M. Koshino, and T. Ando, Phys. Rev. B 81, 155310 (2010).
- Qiao et al. (2010) Z. Qiao, Y. Xing, and J. Wang, Phys. Rev. B 81, 085114 (2010).