Uniaxial stress controlled anisotropic Rashba effects and carriers-based currents in BiTeI monolayer semiconductor
Shi-Hao Zhang Bang-Gui Liu
cond-mat.mes-hall





Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China

[]bgliu@iphy.ac.cn
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China




Manipulation of Rashba effects in two-dimensional (2D) electron systems is highly desirable for controllable applications in spintronics and optoelectronics. Here, by combining first-principles investigation and model analysis, we use uniaxial stress to control BiTeI monolayer as a Rashba 2D semiconductor for useful spin and transport properties.
We find that the stress-driven electron system can be described by an effective anisotropic Rashba model including all the three Pauli matrixes, and uniaxial stress allows an out-of-plane spin component.
When appropriate electron carriers are introduced into the monolayer, an in-plane electric field can induce a charge current and three spin current components (including that based on the out-of-plane spin) because of the reduced symmetry. Therefore, uniaxial stress can be used to control such Rashba 2D electron systems as the BiTeI monolayer for seeking promising devices.








Uniaxial stress controlled anisotropic Rashba effects and carriers-based currents in BiTeI monolayer semiconductor
    Bang-Gui Liu
    December 30, 2023
==================================================================================================================








§ INTRODUCTION


Since the advent of graphene, it becomes clear that two-dimensional (2D) materials can be used to realize high-performance devices for next-generation electronic and optical applications. Various important electronic, magnetic, optical, and mechanical phenomena and effects have been observed. As an important effect,
Rashba effect is a spin splitting phenomenon, originating from the spin-orbit coupling (SOC) in electronic systems without out-of-plane mirror symmetry <cit.>. In a 2D electron gas system, this phenomenon can be descried by the famous Rashba model <cit.>.




The energy bands of this two-band model have a crossing point at the Γ point.
The extremum of the two bands is located along a circle of radius k_0 around the Γ point, and the energy difference between the extremum and the crossing point is defined as Rashba splitting energy ℰ_R <cit.>. It has been proved through first-principles or experimental investigations that there exist strong Rashba effects in some monolayers, multilayer, and heterostructures, such as Janus transition-metal dichalcogenide monolayers <cit.>, BiSb monolayers <cit.>, PbX monolayers (X=S, Se, Te) <cit.>, Ag_2Te monolayer <cit.>, gated multilayer InSe <cit.>, GaX/MoX_2 (X=S, Se, Te) heterostructures <cit.>.
The BiTeI bulk, as a polar crystal with layered crystal structure, has attracted more and more attention because of the strong Rashba splitting <cit.> , optical response <cit.>, and topological physics <cit.>. The BiTeI monolayer is also predicted as a polar material with giant Rashba effect <cit.>. Recently, the BiTeI monolayer was synthesised experimentally <cit.>. It is highly desirable to manipulate the BiTeI monolayer by applying uniaxial stress in order to take full potential of the BiTeI monolayer for controlling the Rashba effects and seeking promising charge/spin currents.

Here, we use uniaxial stress to manipulate the BiTeI monolayer semiconductor through combining first-principles investigations and theoretical model analyses. Our first-principles results reveal that uniaxial stress can cause strong anisotropy in the energy bands near the Γ point, and makes out-of-plane spin component occur. We obtain an effective two-band Hamiltonian to describe the stress-dependent conduction bands and spin texture well. After introducing electron carriers of low concentration, an in-plane electric field can induce a charge current and three nonzero spin current components, including an out-of-plane spin current component, in the monolayer because of the broken inversion symmetry. Controllable anisotropic Rashba effects and carriers-based charge/spin currents can be realized in this way. More detailed results will be presented in the following.





§ METHODOLOGY


The first-principles calculations are performed with the projector-augmented wave (PAW) method within the density functional theory <cit.>, as implemented in the Vienna ab initio simulation package software (VASP) <cit.>. The kinetic energy cutoff of the plane waves is set to 400 eV. All atomic positions are fully optimized until the energy difference between two successive steps is smaller than 10^-6 eV and the Hellmann-Feynman forces on each atom are less than 0.01 eV/Å. The generalized gradient approximation (GGA) by Perdew, Burke, and Ernzerhof (PBE) <cit.> is used as the exchange-correlation functional. The Brillouin zone integration in the self-consistent calculation is carried out with a 42×28×1 special Γ-centered k-point mesh following the convention of Monkhorst-Pack <cit.>. The spin-orbit coupling (SOC) effect is taken into consideration in all calculations including structure optimizations, self-consistent calculations, and energy-bands calculations. In order to calculate Rashba coefficients, we take 1001 k-points along each high-symmetry line in the energy band calculations to avoid the errors coming from the discreteness of k-points and ensure the convergence of Rashba parameter along different directions.



§ RESULT AND DISCUSSION




 §.§ Uniaxial stress and anisotropic Rashba effects



The structure of BiTeI monolayer, without external stress, is presented in Fig. <ref>. The red solid line represents the unit cell of BiTeI monolayer. It obeys the P3m1 space group and shows the C_3v symmetry. For convenience, we take the rectangular cell remarked by black dash line to calculate the effect of uniaxial stress. Through applying the uniaxial tensile stress upon the monolayer along x (or y) direction, the C_3v symmetry will be removed and the monolayer keeps the M_x  mirror symmetry only, and there appears substantial structural anisotropy between the x and y directions. When a tensile uniaxial stress is along the x (y) axis, there will be a tensile strain along the x (y) axis and a compressive strain along the y (x) axis due to Poisson effect. Actually, we allow the tensile x (y) strain to change actively and determine the compressive y (x) strain by structural optimization, and then we estimate the tensile x (y) stress  by ∂ℰ/A∂η where ℰ is the total energy of the monolayer under η strain along the x (y) direction and A is the area of the stretched unit cell. Our study show that the tensile 10% strain along the x or y axis can be achieved by applying 1.71 N/m or 1.82 N/m as the uniaxial stress along the same axis, which indicates that the uniaxial stress on the BiTeI monolayer is experimentally accessible.








The effect of the uniaxial stress on energy bands of the BiTeI monolayer is shown in Fig. 2 (a), where the left part describes the bands without stress, the middle part those for 10% tensile strain (stress 1.71 N/m) along the x axis, and the right part those for 10% tensile strain (stress 1.82 N/m) along the y axis. When no stress is applied on the monolayer, the conduction bands near the Γ point behave isotropic, and the calculated Rashba parameters (ℰ_R, k_0, and α) are 38.4 meV, 0.042 Å^-1, and 1.82 eV Å, in agreement with experimental values <cit.>. When the tensile x-strain reaches 10% (x-stress 1.71 N/m), the conduction band minima along the k_y direction become much lower than those along the k_x direction, which means that the conduction bands near the Γ point become strongly anisotropic. As a result, the calculated Rashba splitting energy ℰ_R and k-vector off-set k_0 are equivalent to 27.5 meV and 0.035 Å^-1 along the k_x direction, but they become 69.6 meV and 0.060 Å^-1 along the k_y direction. When the tensile y-stress 1.82 N/m (10% tensile strain) is applied, the evaluated parameters ℰ_R and k_0 are 23.6 meV and 0.031 Å^-1 along the k_y direction, but 67.7 meV and 0.060 Å^-1 along the k_x direction. These results reveal that the  uniaxial x (y) stress makes the lowest conduction bands around the Γ point become two minimum points along the ± k_y (± k_x) direction, leading to strong anisotropy in the Rashba parameters.


When the uniaixial stress is applied, the rotational symmetry  no longer exists in the stretched BiTeI monolayer, and there remains only ℳ_x mirror symmetry. Under the mirror transformation ℳ_x: x → -x, (k_x,k_y) → (-k_x,k_y), and (σ _x,σ _y,σ _z) → (σ _x,-σ _y,-σ _z). Then we can construct the two-band Hamiltonian for the lowest conduction bands near the Γ point,

    Ĥ = ℰ_k+α_1k_xσ_y+α_2k_yσ_x+α_3k_xσ_z,

where ℰ_k=ħ ^2/2(k^2_x/m_x+k^2_y/m_y), and the crossing point of the two bands is located at ℰ_k=0. The three Rashba parameters α _1, α _2, and α _3 in the Hamiltonian (1) can be obtained through fitting the conduction bands of strained monolayer, and the strain dependences of the parameters for the x/y stress are shown in Fig. <ref> (b,c). It is interesting that α_3  is negative for the uniaxial x-stress, but becomes positive for the uniaxial y stress. We can see that the uniaxial stress allows the emergence of new term k_xσ_z in the Hamiltonian. It is a new physical phenomenon arising from structural symmetry ℳ_x. Considering the Rashba Hamiltonian with rotational group, there is no σ _z term up to the third order of k when the symmetry obeys the C_2, C_2v, C_4 and C_4v <cit.>. When the point symmetry is C_3 or C_3v, there is no σ _z term in the first order of k, but some σ _z terms in the third order of k are allowed, for example, (k_x^3-3k_xk_y^2)σ _z term <cit.>. Thus we can conclude that uniaxial stress removes the C_3v symmetry and thus leads to the emergence of the k_xσ_z term, which can cause the inversion of out-of-plane spin polarization.

The eigenvalues of the two-band Hamiltonian (1) can be written as

    ℰ_k^λ=ℰ_k+λ√((α _1^2+α _3^2)k_x^2+α _2^2k_y^2),

where  λ = ± 1 refers to the two bands. It should be noted that ℰ_k=0 means ℰ^λ_k=0. We can obtain the spin expectation values at the momentum 𝐤 near the Γ point,

    ⟨𝐤λ|σ _x|𝐤λ⟩=λα _2k_y/√((α _1^2+α _3^2)k_x^2+α _2^2k_y^2),
    ⟨𝐤λ|σ _y|𝐤λ⟩=λα _1k_x/√((α _1^2+α _3^2)k_x^2+α _2^2k_y^2),
    ⟨𝐤λ|σ _z|𝐤λ⟩=λα _3k_x/√((α _1^2+α _3^2)k_x^2+α _2^2k_y^2).

The expression (5) means that there emerges a spin z component, in addition to the in-plane components. It is also indicated that the spin y-component and z-component along the k_x = 0 line are zero, which is in agreement with our effective Hamiltonian. This phenomenon can be regarded as the result of ℳ_x mirror symmetry.
Our first-principles investigation about the spin texture of the lowest conduction band with 10% strain along both x and y axes shows that the out-of-plane spin component has important effect.


Furthermore, we present both the spin textures and energy contours of the lower Rashba band (λ=-1) near the Γ point for 0, 5%, and 10% x-strain in Fig. <ref> (d-f). When the Fermi level is shifted downward from the crossing point, the Fermi lines will finally become two separated closed curves, even reduce to two points at the conduction band edge. The change of the Fermi line topology makes a Lifshitz transition. For low electron concentration, there are two Fermi pockets near the conduction band edge, as shown in Fig. <ref> (e,f).






 §.§ Carrier charge and spin currents through electric field


Now we address electric-field-induced transport phenomena in the stretched BiTeI monolayer in the presence of electron carriers, which can be achieved by charge transferring between van der Waals layers, electric gating, or chemical doping. We suppose that the in-plane electric field is so weak and the electron concentration is so small that there are little changes in the crystal structure and lower conduction bands of the BiTeI monolayer. By keeping in the regime of low electron concentration, we fix the Fermi level near the conduction band edge, as shown in Fig. <ref>, and the occupied electrons are located in the lower Rashba band in the neighborhood of the Γ points. When a small in-plane electric field 𝐄 is applied, the relaxation time approximation for the Boltzmann equation of the distribution function f  can be expressed as
-e𝐄/ħ·∂ f/∂𝐤=-f-f_0/τ,
where τ is electron relaxation time and here Hall effect is not taken into consideration <cit.>.
As the response to the external electric field, the distribution function can be expanded as f=f_0+f_1+f_2+⋯, where f_0 is the distribution function of the system in the absence of 𝐄, and f_n = (eτ/ħ𝐄·∂/∂𝐤)^nf_0 comes from the iterative substitution of the Boltzmann equation <cit.>.

Consequently, the charge current 𝐣_c can be written as <cit.>

    𝐣_c   =-eτ/ħ ^2∫ d𝐤δ (ℰ^λ_k-ℰ_F)(𝐄·∇ _kℰ^λ_k)∇ _kℰ^λ_k,

where λ=-1.
We can define a normalized charge current as the ratio of current magnitude to doping electron concentration to describe the current-production efficiency. When the electric field is applied along the x direction, the normalized charge current j_cn as the function of electron concentrations under 10% strain along x/y direction are shown in Fig. <ref> (a). We can explain this phenomenon with the Drude model j = ne^2τ E/m^* where n being the concentration of doped electrons. When Fermi level is shifted upward from the band edge, ∂ ^2 E/∂ k_x^2=ħ ^2/m_x-(α _1^2+α _3^2)α _2^2k_y^2/[(α _1^2+α _3^2)k_x^2+α _2^2k_y^2]^3/2 will be increasing, which reduces the m^* value in the system. Thus the normalized current in eτ E will be monotonically increasing at this circumstance.

As for the spin current, we take the conventional definition of spin-current operator as
Ĵ_μ ,s_ν=1/4{∂Ĥ/∂ k_μ,σ _ν},
which refers to the spin component s_ν flowing along the μ direction <cit.>. It should be especially noted that this definition of spin-current operator will be invalid when the third-power k terms are included in the Hamiltonian <cit.>. The k terms in our Hamiltonian are only to the first order in the presence of small electron concentration (ℰ_F < 0).

Then we obtain the three non-zero components of the spin current near the Γ point,

    ⟨𝐤λ |Ĵ_y,s_x|𝐤λ⟩   =λħ ^2α _2k_y^2/2m_y√((α _1^2+α _3^2)k_x^2+α _2^2k_y^2)+α _2/2,
    ⟨𝐤λ |Ĵ_x,s_y|𝐤λ⟩   =λħ ^2α _1k_x^2/2m_x√((α _1^2+α _3^2)k_x^2+α _2^2k_y^2)+α _1/2,
    ⟨𝐤λ |Ĵ_x,s_z|𝐤λ⟩   =λħ ^2α _3k_x^2/2m_x√((α _1^2+α _3^2)k_x^2+α _2^2k_y^2)+α _3/2.

For the isotropic Rashba model, we have m_x = m_y, α _1 = -α _2, and α _3=0, and then derive ⟨Ĵ_y,s_x⟩ = -⟨Ĵ_x,s_y⟩, which is the same as Rashba's result <cit.>. The non-zero spin-current expectation will not lead to transport and accumulation of spin under the condition of thermodynamic equilibrium, and it can be connected to a non-Abelian SU(2) field generated by the spin-orbit coupling <cit.>. It can have real spin transport  when the Rashba medium is constructed with a spatially modulated spin-orbit parameter <cit.>. For this reason, the zero-order spin current will be addressed no more in the following.


It has been shown that odd (even) orders of the spin (charge) current become zero in the presence of the time-reversal symmetry, and the broken inversion symmetry ensures the existence of non-zero even spin current <cit.>. So the second-order spin current in our system can occur responding to the electric field. The second-order spin current is defined by <cit.>

    j^(2)_μ ,s_ν    = ∫d^2 𝐤/(2π)^2⟨𝐤λ |Ĵ_μ ,s_ν|𝐤λ⟩ f_2 
        = e^2τ ^2/4π ^2 ħ ^3∫ d 𝐤δ (ℰ^λ_k-ℰ_F)(𝐄·∇ _kℰ^λ_k)𝐄·∇ _𝐤⟨𝐤λ |Ĵ_μ ,s_ν|𝐤λ⟩

When the electric field 𝐄 is applied along the x or y direction, there are only three non-zero components for the spin currents, j^(2)_x, s_y, j^(2)_x, s_z and j^(2)_y, s_x. Returning to the isotropic Rashba system, our numerical calculations give that j^(2)_x, s_y=5j^(2)_y, s_x and j^(2)_x, s_z=0, which is in agreement with spin current results of isotropic Rashba model <cit.>. In Fig. <ref> (b), we present the spin current j^(2)_x, s_z under uniaxial stress (0∼10%) along x/y direction when the Fermi level is 10 meV lower than crossing point. When the x-strain (y-strain) reached 10%, the magnitude of j^(2)_x, s_z is equivalent to |j^(2)_x, s_z| ≈ 0.6|j^(2)_x, s_y|. It is clear that the uniaxial stress leads to remarkable spin s_z current j^(2)_x, s_z, in contrast to the isotropic case.

It is necessary to clarify the real-space texture of the electrically-generated spin current. The magnitude of the spin current along θ direction (θ=0 means the x direction) is given by

    𝐣_s =√((j^(2)_x, s_y+j^(2)_x, s_z)^2 cos ^2 θ + (j^(2)_y, s_x)^2 sin ^2 θ)

When the Fermi level is lower than the crossing point by 60 meV, 50 meV, 40 meV, or 30 meV, our numerical results of the spin currents as functions of θ  are shown in Fig. <ref> (c). The spin currents show giant anisotropy in the real space. They increase due to enlarged electron concentration when the Fermi level moves upwards from the conduction and edge.
To show the concentration dependence of the maximal spin currents per electron, j_sn, we present in Fig. <ref> (d) the normalized maximal spin currents as functions of electron concentration for the two cases of the 10% strain along the x and y directions. Here the definition of normalized maximal spin currents is similar to that of normalized charge current. Fig. <ref> (d) indicates that the different orientations of uniaxial stress, even under the same electric field (along the x direction), will lead to very different normalized maximal spin currents.

The lifetime τ in the BiTeI bulk measured by experiment is 3.9×10^-14 s <cit.> and then the relation time τ in the monolayer is assumed as this value. Assuming that the Fermi level is 60 meV lower than the crossing point  and the electric field is 1 V μm^-1 along the x direction, the ratio of the spin current to the charge current is equivalent to 0.09, and the spin current is 1.28×10^9 Å^-1s^-1 for the x-strain 10%.



 §.§ Further discussion for realization


It is useful to show the relationship between the Fermi lvel and the electron concentration. In Table <ref> we present the corresponding electron concentrations for the two 10% strains when the Fermi level is lower than the crossing point by 30 meV, 40 meV, 50 meV, and 60 meV.
Because doped carrier concentration 10^13 cm^-2 have been achieved by back-gate gating in transition metal dichalcogenide monolayers <cit.> and 10^14 cm^-2 by ion liquid gating in graphene <cit.>, the electron concentrations ranging from 0.20 to 1.20 ×10^11 cm^-2 should be experimentally accessible.





For such 2D materials as the BiTeI monolayer, the electron doping can be achieved by inter-layer charge transferring through van der Waals heterostructure formed with appropriate 2D semiconductors. PtSe_2 monolayer is good because it has been synthesised by molten-salt-assisted chemical vapour deposition in the experiment <cit.>. We can choose the 3×3 supercell of BiTeI monolayer and 2√(3)×2√(3) supercell of PtSe_2 monolayer to model the heterostructure. The lattice mismatch of the BiTeI/PtSe_2 heterostructure is 3%. The side view of the heterostructure and the calculated energy bands are shown in Fig. <ref>. The conduction band edge of the PtSe_2 monolayer is higher than that of the BiTeI monolayer, and the chemically-doped electrons in the PtSe_2 monolayer can easily transfer to the BiTeI monolayer. In such ways, the BiTeI monolayer can be doped with electrons, without changing its structure and energy bands.




§ CONCLUSION


In summary, we use uniaxial stress to manipulate the BiTeI monolayer semiconductor through first-principles calculations and theoretical analyses. Because the uniaxial stress destroys the C_3v symmetry, the monolayer has only ℳ_x mirror symmetry, which allows emergence of α _3k_xσ _z term in the effective model. We obtain effective anisotropic Rashba Hamiltonian through fitting the Rashba bands with the lowest conduction bands near the Γ point up to the stress of 1.71 N/m for the x axis or 1.82 N/m for the y axis.
We find the out-of-plane spin component  in the stretched BiTeI monolayer, in addition to usual in-plane spin components in isotropic Rashba model. When electron carriers of low concentration are introduced into the monolayer, an in-plane electric field can induce the first-order charge current and second-order spin currents including a s_z spin current increasing with the uniaxial stress. Such electron carriers can be realized through electron transferring from n-type doped monolayer to the BiTeI monolayer in appropriate van der Waals heterostructures, electric gating, or chemical doping. These make us believe that uniaxial stress and electric field together can open the door for controllable spintronic applications on the basis of good experimentally-realizable 2D materials such as the BiTeI monolayer.



This work is supported by the Nature Science Foundation of China (No.11574366), by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No.XDB07000000), and by the Department of Science and Technology of China (Grant No.2016YFA0300701). The calculations were performed in the supercomputer system at Institute of Physics and the Milky Way #2 supercomputer system at the National Supercomputer Center of Guangzhou, Guangzhou, China.















54
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[Rashba and Sheka(1959)]Rashba1959
  author author E. I. Rashba and author V. I. Sheka, @noop  journal journal Fiz.
  Tverd. Tela.: Collected Papers volume 2, pages 162 (year 1959)NoStop
[Rashba(1960)]Rashba1960
  author author E. I. Rashba, @noop  journal journal Fiz.
  Tverd. Tela. volume 2, pages 1224
  (year 1960)NoStop
[Dyakonov (ed.)(2017)]SpinSec2017
  author author M. I. Dyakonov (ed.), 10.1007/978-3-319-65436-2_1 title Spin Physics in Semiconductors, series
  Springer Series in Solid-State Sciences, Vol. volume 157  (publisher Springer International Publishing AG, year 2017)NoStop
[Bychkov and Rashba(1984)]bychkov1984properties
  author author Y. A. Bychkov and author É. I. Rashba, @noop  journal journal JETP
  Lett. volume 39, pages 78 (year 1984)NoStop
[Manchon et al.(2015)Manchon, Koo, Nitta, Frolov, and Duine]Manchon2015
  author author A. Manchon, author H. C. Koo,
  author J. Nitta, author S. M. Frolov,  and author R. A. Duine, https://doi.org/10.1038/nmat4360 journal journal
  Nat. Mater. volume 14, pages 871
  (year 2015)NoStop
[Hu et al.(2018)Hu,
  Jia, Zhao, Wu, Stroppa, and Ren]PhysRevB.97.235404
  author author T. Hu, author F. Jia, author G. Zhao, author
  J. Wu, author A. Stroppa,  and author W. Ren, 10.1103/PhysRevB.97.235404
  journal journal Phys. Rev. B volume 97, pages 235404 (year
  2018)NoStop
[Singh and Romero(2017)]PhysRevB.95.165444
  author author S. Singh and author A. H. Romero, 10.1103/PhysRevB.95.165444 journal journal Phys. Rev. B volume
  95, pages 165444 (year 2017)NoStop
[Yuan et al.(2018)Yuan,
  Cai, Shen, Xiao,
  Ren, Wang, Feng, and  Yan]YUAN2018163
  author author J. Yuan, author Y. Cai, author L. Shen, author
  Y. Xiao, author J.-C. Ren, author A. Wang, author Y. P.  Feng,  and author X. Yan, https://doi.org/10.1016/j.nanoen.2018.07.041 journal journal Nano Energy volume 52, pages 163  (year 2018)NoStop
[Lu et al.(2017)Lu,
  Zhu, Xiao, Chuu,
  Han, Chiu, Cheng,
  Yang, Wei, Yang,
  Wang, Sokaras, Nordlund,
  Yang, Muller, Chou,
  Zhang, and Li]Lu2017
  author author A.-Y. Lu, author H. Zhu, author J. Xiao, author
  C.-P. Chuu, author
  Y. Han, author M.-H. Chiu, author C.-C.  Cheng, author C.-W. Yang, author K.-H. Wei,
  author Y. Yang, author
  Y. Wang, author D. Sokaras, author D. Nordlund, author P. Yang, author D. A.  Muller, author M.-Y.  Chou, author X. Zhang,  and author L.-J. Li, https://doi.org/10.1038/nnano.2017.100 journal
  journal Nature Nanotechnology volume
  12, pages 744 (year 2017)NoStop
[Hanakata et al.(2017)Hanakata, Rodin, Carvalho, Park, Campbell, and Castro Neto]PhysRevB.96.161401
  author author P. Z. Hanakata, author A. S. Rodin,
  author A. Carvalho, author H. S. Park, author
  D. K. Campbell,  and author
  A. H. Castro Neto, 10.1103/PhysRevB.96.161401 journal journal Phys.
  Rev. B volume 96, pages 161401
  (year 2017)NoStop
[Hanakata et al.(2018)Hanakata, Rodin, Park, Campbell, and Castro Neto]PhysRevB.97.235312
  author author P. Z. Hanakata, author A. S. Rodin,
  author H. S. Park, author D. K. Campbell,  and author A. H. Castro Neto, 10.1103/PhysRevB.97.235312 journal journal Phys. Rev. B volume 97, pages 235312 (year 2018)NoStop
[Noor-A-Alam et al.(2018)Noor-A-Alam, Lee, Lee, Choi, and Lee]ag2te
  author author M. Noor-A-Alam, author M. Lee,
  author H.-J. Lee, author K. Choi,  and author
  J. H. Lee, http://stacks.iop.org/0953-8984/30/i=38/a=385502 journal
  journal J. Phys. Condens. Mat. volume
  30, pages 385502 (year 2018)NoStop
[Premasiri et al.(2018)Premasiri, Radha, Sucharitakul,
  Kumar, Sankar, Chou,
  Chen, and Gao]acs.nanolett.8b01462
  author author K. Premasiri, author S. K. Radha, author S. Sucharitakul,
  author U. R. Kumar, author R. Sankar, author
  F.-C. Chou, author
  Y.-T. Chen,  and author
  X. P. A. Gao, 10.1021/acs.nanolett.8b01462 journal journal
  Nano Lett. volume 18, pages 4403
  (year 2018)NoStop
[Zhang and Schwingenschlögl(2018)]PhysRevB.97.155415
  author author Q. Zhang and author U. Schwingenschlögl, 10.1103/PhysRevB.97.155415
  journal journal Phys. Rev. B volume 97, pages 155415 (year
  2018)NoStop
[Ishizaka et al.(2011)Ishizaka, Bahramy, Murakawa, Sakano, Shimojima, Sonobe, Koizumi, Shin, Miyahara, Kimura, Miyamoto, Okuda, Namatame, Taniguchi, Arita, Nagaosa, Kobayashi, Murakami, Kumai, Kaneko, Onose, and Tokura]Ishizaka2011
  author author K. Ishizaka, author M. S. Bahramy, author H. Murakawa,
  author M. Sakano, author T. Shimojima, author
  T. Sonobe, author K. Koizumi, author S. Shin, author H. Miyahara, author A. Kimura,
  author K. Miyamoto, author T. Okuda, author
  H. Namatame, author
  M. Taniguchi, author
  R. Arita, author N. Nagaosa, author K. Kobayashi, author Y. Murakami, author R. Kumai, author Y. Kaneko, author Y. Onose,  and author Y. Tokura, https://doi.org/10.1038/nmat3051
  journal journal Nat. Mater. volume 10, pages 521 (year 2011)NoStop
[Crepaldi et al.(2012)Crepaldi, Moreschini, Autès,
  Tournier-Colletta, Moser, Virk, Berger, Bugnon, Chang, Kern, Bostwick, Rotenberg, Yazyev, and Grioni]PhysRevLett.109.096803
  author author A. Crepaldi, author L. Moreschini, author G. Autès,
  author C. Tournier-Colletta,
  author S. Moser, author N. Virk, author
  H. Berger, author P. Bugnon, author Y. J.  Chang, author K. Kern, author A. Bostwick,
  author E. Rotenberg, author O. V. Yazyev,  and author M. Grioni, 10.1103/PhysRevLett.109.096803 journal journal
  Phys. Rev. Lett. volume 109, pages
  096803 (year 2012)NoStop
[Landolt et al.(2012)Landolt, Eremeev, Koroteev, Slomski, Muff, Neupert, Kobayashi, Strocov, Schmitt, Aliev, Babanly, Amiraslanov, Chulkov, Osterwalder, and Dil]PhysRevLett.109.116403
  author author G. Landolt, author S. V. Eremeev, author Y. M. Koroteev, author B. Slomski,
  author S. Muff, author
  T. Neupert, author M. Kobayashi, author V. N.  Strocov, author T. Schmitt, author Z. S. Aliev,
  author M. B. Babanly, author I. R. Amiraslanov, author E. V. Chulkov, author
  J. Osterwalder,  and author
  J. H. Dil, 10.1103/PhysRevLett.109.116403 journal journal
  Phys. Rev. Lett. volume 109, pages
  116403 (year 2012)NoStop
[Eremeev et al.(2012)Eremeev, Nechaev, Koroteev, Echenique, and Chulkov]PhysRevLett.108.246802
  author author S. V. Eremeev, author I. A. Nechaev, author Y. M. Koroteev, author P. M. Echenique,  and author E. V. Chulkov, 10.1103/PhysRevLett.108.246802 journal journal Phys. Rev. Lett. volume 108, pages 246802 (year
  2012)NoStop
[Sakano et al.(2013)Sakano,
  Bahramy, Katayama, Shimojima,
  Murakawa, Kaneko, Malaeb,
  Shin, Ono, Kumigashira,
  Arita, Nagaosa, Hwang,
  Tokura, and Ishizaka]PhysRevLett.110.107204
  author author M. Sakano, author M. S. Bahramy,
  author A. Katayama, author T. Shimojima, author
  H. Murakawa, author
  Y. Kaneko, author W. Malaeb, author S. Shin, author K. Ono, author H. Kumigashira,
  author R. Arita, author N. Nagaosa, author
  H. Y. Hwang, author
  Y. Tokura,  and author
  K. Ishizaka, 10.1103/PhysRevLett.110.107204 journal journal
  Phys. Rev. Lett. volume 110, pages
  107204 (year 2013)NoStop
[Monserrat and Vanderbilt(2017)]PhysRevMaterials.1.054201
  author author B. Monserrat and author D. Vanderbilt, 10.1103/PhysRevMaterials.1.054201
  journal journal Phys. Rev. Mater. volume 1, pages 054201 (year
  2017)NoStop
[Bordács et al.(2013)Bordács, Checkelsky, Murakawa,
  Hwang, and Tokura]PhysRevLett.111.166403
  author author S. Bordács, author J. G. Checkelsky, author H. Murakawa, author H. Y. Hwang,
   and author Y. Tokura, 10.1103/PhysRevLett.111.166403 journal journal Phys. Rev. Lett. volume 111, pages 166403 (year 2013)NoStop
[Sakano et al.(2012)Sakano,
  Miyawaki, Chainani, Takata,
  Sonobe, Shimojima, Oura,
  Shin, Bahramy, Arita,
  Nagaosa, Murakawa, Kaneko,
  Tokura, and Ishizaka]PhysRevB.86.085204
  author author M. Sakano, author J. Miyawaki,
  author A. Chainani, author Y. Takata, author
  T. Sonobe, author T. Shimojima, author M. Oura, author S. Shin, author M. S. Bahramy,
  author R. Arita, author N. Nagaosa, author
  H. Murakawa, author
  Y. Kaneko, author Y. Tokura,  and author K. Ishizaka, 10.1103/PhysRevB.86.085204
  journal journal Phys. Rev. B volume 86, pages 085204 (year
  2012)NoStop
[Butler et al.(2014)Butler,
  Yang, Hong, Hsu,
  Sankar, Lu, Lu, Yang, Shiu, Chen, Kaun,
  Shu, Chou, and Lin]Butler2014
  author author C. J. Butler, author H.-H. Yang,
  author J.-Y. Hong, author S.-H. Hsu, author
  R. Sankar, author C.-I. Lu, author H.-Y.  Lu, author K.-H. O. Yang, author H.-W. Shiu,
  author C.-H. Chen, author C.-C. Kaun, author
  G.-J. Shu, author F.-C. Chou,  and author M.-T. Lin, https://doi.org/10.1038/ncomms5066
  journal journal Nat. Commun. volume 5, pages 4066 (year 2014)NoStop
[Kohsaka et al.(2015)Kohsaka, Kanou, Takagi, Hanaguri, and Sasagawa]PhysRevB.91.245312
  author author Y. Kohsaka, author M. Kanou,
  author H. Takagi, author T. Hanaguri,  and author T. Sasagawa, 10.1103/PhysRevB.91.245312 journal journal Phys.
  Rev. B volume 91, pages 245312
  (year 2015)NoStop
[Lee et al.(2011)Lee,
  Schober, Bahramy, Murakawa,
  Onose, Arita, Nagaosa, and  Tokura]PhysRevLett.107.117401
  author author J. S. Lee, author G. A. H. Schober,
  author M. S. Bahramy, author H. Murakawa, author
  Y. Onose, author R. Arita, author N. Nagaosa,  and author Y. Tokura, 10.1103/PhysRevLett.107.117401 journal journal
  Phys. Rev. Lett. volume 107, pages
  117401 (year 2011)NoStop
[Demkó et al.(2012)Demkó, Schober, Kocsis, Bahramy, Murakawa, Lee, Kézsmárki, Arita, Nagaosa, and  Tokura]PhysRevLett.109.167401
  author author L. Demkó, author G. A. H. Schober, author V. Kocsis,
  author M. S. Bahramy, author H. Murakawa, author
  J. S. Lee, author I. Kézsmárki, author R. Arita, author N. Nagaosa,  and author Y. Tokura, 10.1103/PhysRevLett.109.167401 journal journal
  Phys. Rev. Lett. volume 109, pages
  167401 (year 2012)NoStop
[Bahramy et al.(2012)Bahramy, Yang, Arita, and Nagaosa]Bahramy2012
  author author M. S. Bahramy, author B.-J. Yang,
  author R. Arita,  and author N. Nagaosa, https://doi.org/10.1038/ncomms1679 journal journal Nat. Commun. volume 3, pages
  679 (year 2012)NoStop
[Eremeev et al.(2017)Eremeev, Nechaev, and Chulkov]PhysRevB.96.155309
  author author S. V. Eremeev, author I. A. Nechaev,  and author E. V. Chulkov, 10.1103/PhysRevB.96.155309 journal journal Phys. Rev. B volume
  96, pages 155309 (year 2017)NoStop
[Facio et al.(2018)Facio,
  Efremov, Koepernik, You,
  Sodemann, and van den
  Brink]PhysRevLett.121.246403
  author author J. I. Facio, author D. Efremov,
  author K. Koepernik, author J.-S. You, author
  I. Sodemann,  and author
  J. van den Brink, 10.1103/PhysRevLett.121.246403 journal journal
  Phys. Rev. Lett. volume 121, pages
  246403 (year 2018)NoStop
[Qi et al.(2017)Qi,
  Shi, Naumov, Kumar,
  Sankar, Schnelle, Shekhar,
  Chou, Felser, Yan, and  Medvedev]10.1002/adma.201605965
  author author Y. Qi, author W. Shi, author P. G. Naumov, author
  N. Kumar, author R. Sankar, author W. Schnelle, author C. Shekhar, author F.-C.  Chou, author C. Felser, author B. Yan,  and  author S. A. Medvedev, 10.1002/adma.201605965 journal journal
  Adv. Mater. volume 29, pages 1605965
  (year 2017)NoStop
[Ma et al.(2014)Ma,
  Dai, Wei, Li, and  Huang]ma2014emergence
  author author Y. Ma, author Y. Dai, author W. Wei, author
  X. Li,  and author
  B. Huang, @noop  journal journal Phys. Chem. Chem. Phys. volume 16, pages 17603 (year
  2014)NoStop
[Fülöp et al.(2018)Fülöp, Tajkov, Pető,
  Kun, Koltai, Oroszlány,
  Tóvári, Murakawa, Tokura, Bordács et al.]fulop2018exfoliation
  author author B. Fülöp, author Z. Tajkov, author J. Pető,
  author P. Kun, author
  J. Koltai, author L. Oroszlány, author E. Tóvári, author H. Murakawa, author Y. Tokura, author S. Bordács,  et al., @noop  journal journal 2D Mater. volume
  5, pages 031013 (year 2018)NoStop
[Blöchl(1994)]PhysRevB.50.17953
  author author P. E. Blöchl, 10.1103/PhysRevB.50.17953 journal journal Phys. Rev. B volume
  50, pages 17953 (year 1994)NoStop
[Kresse and Hafner(1993)]PhysRevB.47.558
  author author G. Kresse and author J. Hafner, 10.1103/PhysRevB.47.558 journal
  journal Phys. Rev. B volume 47,  pages 558 (year 1993)NoStop
[Perdew et al.(1996)Perdew,
  Burke, and Ernzerhof]PhysRevLett.77.3865
  author author J. P. Perdew, author K. Burke,   and author M. Ernzerhof, 10.1103/PhysRevLett.77.3865 journal journal Phys. Rev. Lett. volume 77, pages 3865 (year 1996)NoStop
[Monkhorst and Pack(1976)]PhysRevB.13.5188
  author author H. J. Monkhorst and author J. D. Pack, 10.1103/PhysRevB.13.5188 journal
  journal Phys. Rev. B volume 13,  pages 5188 (year 1976)NoStop
[Vajna et al.(2012)Vajna,
  Simon, Szilva, Palotas,
  Ujfalussy, and Szunyogh]PhysRevB.85.075404
  author author S. Vajna, author E. Simon,
  author A. Szilva, author K. Palotas, author
  B. Ujfalussy,  and author
  L. Szunyogh, 10.1103/PhysRevB.85.075404 journal journal Phys.
  Rev. B volume 85, pages 075404
  (year 2012)NoStop
[Fu(2009)]PhysRevLett.103.266801
  author author L. Fu, 10.1103/PhysRevLett.103.266801 journal journal Phys. Rev. Lett. volume 103, pages 266801 (year
  2009)NoStop
[Yu et al.(2014)Yu,
  Wu, Liu, Xu, and  Yao]PhysRevLett.113.156603
  author author H. Yu, author Y. Wu, author G.-B. Liu, author
  X. Xu,  and author
  W. Yao, 10.1103/PhysRevLett.113.156603 journal journal
  Phys. Rev. Lett. volume 113, pages
  156603 (year 2014)NoStop
[Hamamoto et al.(2017)Hamamoto, Ezawa, Kim, Morimoto, and Nagaosa]PhysRevB.95.224430
  author author K. Hamamoto, author M. Ezawa,
  author K. W. Kim, author T. Morimoto,  and author N. Nagaosa, 10.1103/PhysRevB.95.224430 journal journal Phys.
  Rev. B volume 95, pages 224430
  (year 2017)NoStop
[Sodemann and Fu(2015)]PhysRevLett.115.216806
  author author I. Sodemann and author L. Fu,  10.1103/PhysRevLett.115.216806 journal
  journal Phys. Rev. Lett. volume 115,  pages 216806 (year 2015)NoStop
[Morimoto et al.(2016)Morimoto, Zhong, Orenstein, and  Moore]PhysRevB.94.245121
  author author T. Morimoto, author S. Zhong,
  author J. Orenstein,  and  author J. E. Moore, 10.1103/PhysRevB.94.245121 journal journal Phys. Rev. B volume 94, pages 245121 (year 2016)NoStop
[Rashba(2003)]PhysRevB.68.241315
  author author E. I. Rashba, 10.1103/PhysRevB.68.241315 journal journal Phys. Rev. B volume
  68, pages 241315 (year 2003)NoStop
[Sinova et al.(2004)Sinova,
  Culcer, Niu, Sinitsyn,
  Jungwirth, and MacDonald]PhysRevLett.92.126603
  author author J. Sinova, author D. Culcer,
  author Q. Niu, author
  N. A. Sinitsyn, author
  T. Jungwirth,  and author
  A. H. MacDonald, 10.1103/PhysRevLett.92.126603 journal journal
  Phys. Rev. Lett. volume 92, pages
  126603 (year 2004)NoStop
[Drouhin et al.(2011)Drouhin, Fishman, and Wegrowe]PhysRevB.83.113307
  author author H.-J. Drouhin, author G. Fishman,   and author J.-E. Wegrowe,  10.1103/PhysRevB.83.113307 journal journal Phys. Rev. B volume 83, pages 113307 (year 2011)NoStop
[Tokatly(2008)]PhysRevLett.101.106601
  author author I. V. Tokatly, 10.1103/PhysRevLett.101.106601 journal journal Phys. Rev. Lett. volume 101, pages 106601 (year
  2008)NoStop
[Sonin(2007)]PhysRevB.76.033306
  author author E. B. Sonin, 10.1103/PhysRevB.76.033306 journal
  journal Phys. Rev. B volume 76,  pages 033306 (year 2007)NoStop
[Sun et al.(2008)Sun,
  Xie, and Wang]PhysRevB.77.035327
  author author Q.-F. Sun, author X. C. Xie,  and  author J. Wang, 10.1103/PhysRevB.77.035327 journal journal Phys.
  Rev. B volume 77, pages 035327
  (year 2008)NoStop
[Martin et al.(2013)Martin,
  Mun, Berger, Zapf, and  Tanner]PhysRevB.87.041104
  author author C. Martin, author E. D. Mun,
  author H. Berger, author V. S. Zapf,  and author D. B. Tanner, 10.1103/PhysRevB.87.041104 journal journal Phys.
  Rev. B volume 87, pages 041104
  (year 2013)NoStop
[Mak et al.(2013)Mak,
  He, Lee, Lee, Hone, Heinz, and Shan]mak2013tightly
  author author K. F. Mak, author K. He, author C. Lee, author
  G. H. Lee, author J. Hone, author T. F.  Heinz,  and author J. Shan, @noop  journal journal Nat. Mater. volume 12, pages
  207 (year 2013)NoStop
[Zhang et al.(2014)Zhang,
  Oka, Suzuki, Ye, and  Iwasa]zhang2014electrically
  author author Y. Zhang, author T. Oka, author R. Suzuki, author
  J. Ye,  and author
  Y. Iwasa, @noop  journal journal Science volume
  344, pages 725 (year 2014)NoStop
[Efetov and Kim(2010)]PhysRevLett.105.256805
  author author D. K. Efetov and author P. Kim,  10.1103/PhysRevLett.105.256805 journal
  journal Phys. Rev. Lett. volume 105,  pages 256805 (year 2010)NoStop
[Ye et al.(2011)Ye,
  Craciun, Koshino, Russo,
  Inoue, Yuan, Shimotani,
  Morpurgo, and Iwasa]Ye13002
  author author J. Ye, author M. F. Craciun,
  author M. Koshino, author S. Russo, author
  S. Inoue, author H. Yuan, author H. Shimotani, author A. F. Morpurgo,  and author Y. Iwasa, 10.1073/pnas.1018388108 journal
  journal Proc. Natl. Acad. Sci. volume
  108, pages 13002 (year 2011)NoStop
[Zhou et al.(2018)Zhou,
  Lin, Huang, Zhou,
  Chen, Xia, Wang,
  Xie, Yu, Lei, Wu, Liu, Fu, Zeng,
  Hsu, Yang, Lu, Yu, Shen, Lin, Yakobson,
  Liu, Suenaga, Liu, and  Liu]Zhou2018
  author author J. Zhou, author J. Lin, author X. Huang, author
  Y. Zhou, author Y. Chen, author J. Xia, author H. Wang, author Y. Xie, author
  H. Yu, author J. Lei, author D. Wu, author F. Liu, author Q. Fu, author
  Q. Zeng, author C.-H. Hsu, author C. Yang, author L. Lu, author T. Yu, author Z. Shen, author
  H. Lin, author B. I. Yakobson, author Q. Liu, author K. Suenaga, author G. Liu,  and  author Z. Liu, 10.1038/s41586-018-0008-3 journal journal
  Nature volume 556, pages 355
  (year 2018)NoStop



