Enhanced spin hall ratio in two-dimensional semiconductors

ABSTRACT The conversion efficiency from charge current to spin current via the spin Hall effect is evaluated by the spin Hall ratio (SHR). Through state-of-the-art ab initio calculations

involving both charge conductivity and spin Hall conductivity, we report the SHRs of the III-V monolayer family, revealing an ultrahigh ratio of 0.58 in the hole-doped GaAs monolayer. In

order to find more promising 2D materials, a descriptor for high SHR is proposed and applied to a high-throughput database, which provides the fully relativistic band structures and Wannier

Hamiltonians of 216 exfoliable monolayer semiconductors and has been released to the community. Among potential candidates for high SHR, the MXene monolayer Sc2CCl2 is identified with the

proposed descriptor and confirmed by computation, demonstrating the descriptor validity for high SHR materials discovery. SIMILAR CONTENT BEING VIEWED BY OTHERS EFFECT OF HUBBARD

U-CORRECTIONS ON THE ELECTRONIC AND MAGNETIC PROPERTIES OF 2D MATERIALS: A HIGH-THROUGHPUT STUDY Article Open access 24 January 2025 HIGH-THROUGHPUT INVERSE DESIGN AND BAYESIAN OPTIMIZATION

OF FUNCTIONALITIES: SPIN SPLITTING IN TWO-DIMENSIONAL COMPOUNDS Article Open access 29 April 2022 HIGH-THROUGHPUT COMPUTATIONAL SCREENING FOR TWO-DIMENSIONAL MAGNETIC MATERIALS BASED ON

EXPERIMENTAL DATABASES OF THREE-DIMENSIONAL COMPOUNDS Article Open access 21 October 2020 INTRODUCTION The Hall effect encompasses a wide range of phenomena which realize the conversion

between various physical properties1,2. The strength of the Hall effect can be denoted by \(\beta =\tan ({\theta }_{{{\rm{H}}}})={E}_{{{\rm{H}}}}/E\) where _θ_H is the Hall angle, _E_H is

the transverse Hall field, and _E_ is the longitudinal electric field3. Correspondingly, the strength of the spin Hall effect (SHE) is given by the spin Hall ratio (SHR) as \(\xi =\tan

({\theta }_{{{\rm{SH}}}})=\frac{2e}{\hslash }\left\vert \right.\frac{{J}_{{{\rm{s}}}}}{{J}_{{{\rm{c}}}}}\left\vert \right.\) where _θ_SH is the spin Hall angle, _J_s is the transverse spin

Hall current density, and _J_c is the longitudinal charge current density. SHR is often used as a proxy to indicate the charge-to-spin conversion efficiency, which is crucial for

low-power-consumption spintronic applications4,5,6. Indeed, when _θ_SH is small, the first-order Taylor polynomial gives _ξ_ ≈ _θ_SH, which is a good approximation for the bulk

semiconductors and metals where _ξ_ ~ 0.017,8,9. Recently, enhanced SHR has been found in van der Waals materials with strong spin-orbit coupling (SOC). Huge SHRs over 10 are reported in

topological insulators10,11 while large SHR ~ 0.5 in MoTe2 and WTe2 Weyl semimetals have also been theoretically and experimentally identified12,13,14,15. However, the relative abundance of

topological materials is around 1%16,17, limiting material options for device manufacturing. The various two-dimensional (2D) materials enable a preferable compatibility with the integrated

circuit18 with desirable properties. For instance, the MoS2 monolayer can exhibit _ξ_ = 0.14 induced by the Rashba-Edelstein effect19. Note that large _ξ_ will break the approximation _ξ_ ≈

_θ_SH, and therefore, the spin Hall ratio rather than the spin Hall angle should be used to denote the ratio of spin current to charge current. 2D materials composed of heavy atoms are

promising for SHE20,21 since the strong SOC can induce a large spin Hall conductivity (SHC), and doping is an effective way to manipulate the transport behaviors in semiconductors. Both

factors can promote SHR enhancement in 2D semiconductors. Although charge transport and SHC have been separately investigated in 2D materials22,23,24,25,26, the study of SHR remains elusive

due to the multidisciplinary complexity involving the electron-phonon interaction (EPI) for electron motion27,28 and SOC for spin transport4,5. In the present work, we report the spin Hall

ratios in monolayer semiconductors using density functional theory (DFT)29, density functional perturbation theory (DFPT)30, and Wannier functions31. The family of III-V monolayer

semiconductors (MX, M=Ga, In, and X = As, Sb) are investigated. In the hole-doped regimes, the charge conductivities are significantly suppressed by the strong inter-peak scattering, while

high SHCs occur due to the strong SOC, yielding an ultrahigh SHR of _ξ_ = 0.58 in the GaAs monolayer. Taking the transport behaviors of III-V monolayers as a prototype, we propose a general

descriptor for the high SHR based on the electronic structures. To validate the generality of this descriptor, we create a high-throughput database by performing fully relativistic DFT

calculations and Wannierizations on 216 monolayer semiconductors, whose electronic band structures, effective masses, and SHCs have been calculated. The database is screened by the

descriptor and suggests two MXene candidates, Sc2CCl2 and Y2CBr2 monolayers. The charge conductivities are investigated in both materials, confirming high SHR and validating the proposed SHR

descriptor. RESULTS AND DISCUSSIONS CHARGE TRANSPORT The phonon-limited charge conductivity in a doped 2D semiconductor is calculated as32 $${\sigma }_{\alpha \beta

}=\frac{-e}{{S}^{{{\rm{uc}}}}}{\sum}_{n}\int\frac{{{{\rm{d}}}}^{2}{{\bf{k}}}}{{\Omega }^{{{\rm{BZ}}}}}{v}_{n{{\bf{k}}}\alpha }{\partial }_{{E}_{\beta }}{f}_{n{{\bf{k}}}},$$ (1) where _α_ and

_β_ are Cartesian directions, _S_uc is the unit cell area, _Ω_BZ is the first Brillouin zone area, and _v__n_K_α_ = _ℏ_−1∂_ε__n_K/∂_k__α_ is the band velocity, _n_ is the band index. The

linear variation of the electronic occupation function _f__n_K in response to E, \({\partial }_{{E}_{\beta }}{f}_{n{{\bf{k}}}}\), can be obtained by solving the Boltzmann transport equation

given by Supplementary Equation 1, which induces the scattering rate given by $$\begin{array}{ll}{\tau }_{n{{\bf{k}}}}^{-1}\,=\,\frac{2\pi }{\hslash }\mathop{\sum}\limits_{m\nu

}\,\int\,\frac{{{{\rm{d}}}}^{2}{{\bf{q}}}}{{\Omega }^{{{\rm{BZ}}}}}| {g}_{mn\nu }({{\bf{k}}},{{\bf{q}}}){| }^{2}\\\qquad\quad\times \left[({n}_{{{\bf{q}}}\nu

}+1-{f}_{m{{\bf{k+q}}}}^{0})\delta ({\varepsilon }_{n{{\bf{k}}}}-{\varepsilon }_{m{{\bf{k+q}}}}-\hslash {\omega }_{{{\bf{q}}}\nu })\right.\\ \,\,\left.\qquad\quad+\,({n}_{{{\bf{q}}}\nu

}+{f}_{m{{\bf{k+q}}}}^{0})\delta ({\varepsilon }_{n{{\bf{k}}}}-{\varepsilon }_{m{{\bf{k+q}}}}+\hslash {\omega }_{{{\bf{q}}}\nu })\right],\end{array}$$ (2) where _g__m__n__ν_(K, Q) is the

electron-phonon matrix element with phonon frequency _ω_Q_ν_, _ε__n_K and _ε__m_K+Q are eigenvalues, _f__n_K is the Fermi-Dirac distribution, _n_Q_ν_ is the Bose–Einstein distribution. The

drift mobility _μ_ is given by $${\mu }_{\alpha \beta }={\sigma }_{\alpha \beta }/(e{n}^{{{\rm{c}}}})$$ (3) where we use a carrier concentration of _n_c = 1010 cm−2 which is the intrinsic

regime where _μ_ is independent of _n_c. Due to crystal symmetry, _μ_ = _μ__x__x_ = _μ__y__y_, _σ_ = _σ__x__x_ = _σ__y__y_ in all the III-V monolayers. More details about mobility

calculations are given in a related work33. Here, we focus on the conductivity of doped systems. When a heavy doping such as _n_c = 1013 cm−2 is applied, the EPI would be significantly

modified compared with the pristine regime. Thus, in the heavy doping cases, structural optimizations, electron and phonon dispersions, as well as conductivities are calculated

independently. In the following, _σ_c is used to denote the charge conductivity. Figure 1 presents the room temperature mobilities of the pristine monolayers and the conductivities of doped

systems. All the materials exhibit high electron mobilities, thanks to the small electron-effective mass and the single valley in the conduction bands. More variations in the hole mobility

are observed due to the multi-peak band structures. The two arsenides present different mobilities of 14 and 289 cm2/Vs, while two similar values, 940 and 904, are observed in the

antimonides. A detailed analysis of mobility mechanisms is given in another work33. In this paper, the conductivities in doped III-V monolayers are extensively explained. Doping is a

practical method to tune the transport properties of semiconductors34. Sufficient carriers are induced by heavy doping, which turns semiconductors into metallic systems where SHE can occur.

Considering the densities of states (DOS), an electron doping of 1 × 1013 cm−2 and a hole doping of 2 × 1013 cm−2 are respectively applied to the III-V monolayers, whose structures have been

relaxed again. The main impact of such doping is the shift of Fermi energy (EF) by a few hundred meV, leaving the crystal structure and electronic bands nearly unaffected as verified by

Supplementary Fig. 3. Interestingly, an electron-hole asymmetry occurs in the phonon dispersions of the III-V monolayers: the phonon dispersion is weakly affected by electron doping, while

the softening of longitudinal optical (LO) modes specifically occurs in the hole doping cases as shown in Fig. 2. The asymmetry can be explained by the difference in conduction and valence

bands. For electron doping, the Γ valley is mainly occupied, leading to a limited change compared with the conduction bands in the pristine case. In contrast, for hole doping, multiple

inequivalent peaks around Γ and K points enter into the transport window, enhancing the effective scattering \(\frac{-\partial {f}_{n{{\bf{k}}}}^{0}}{\partial {\varepsilon

}_{n{{\bf{k}}}}}{\tau }_{n{{\bf{k}}}}^{-1}\). Besides, attributed to the higher carrier concentration of 2 × 1013 cm−2 and consequent stronger screening effect, the hole doping induces a

stronger phonon softening effect, which also occurs in other hole-doped III-V monolayers, see details in Supplementary Fig. 4. In a nutshell, the asymmetry in the conduction and valence

bands leads to a limited scattering in electron-doped systems as verified by the high \({\sigma }_{{{\rm{c}}}}^{{{\rm{e}}}}\) in Fig. 1a and an enhanced scattering in hole-doped systems as

verified by the low \({\sigma }_{{{\rm{c}}}}^{{{\rm{h}}}}\) in Fig. 1b. It is expected that the charge conductivity should be proportional to the carrier mobility. However, Fig. 1a

illustrates that GaAs present a much lower _μ_e but higher \({\sigma }_{{{\rm{c}}}}^{{{\rm{e}}}}\) than GaSb with the same electron doping. In Fig. 1b, GaSb shows a high _μ_h but a low

\({\sigma }_{{{\rm{c}}}}^{{{\rm{h}}}}\) with the hole doping. These unusual behaviors are attributed to the heavy doping, which leads to a distinct transport from the intrinsic case. In

intrinsic semiconductors, the mobility is independent of _n_c. However, in heavy doping cases, semiconductors acquire a metallic character, and the mobility in doped systems depends on the

band structure as well as EF location determined by doping concentration. As a result, the linearly proportional relation between undoped mobility and heavily doped conductivity is no longer

valid. The effect on transport induced by doping will be interpreted within the self-energy relaxation time approximation32, where the conductivity is inversely proportional to the

scattering rate and directly proportional to carrier velocity. Considering the Fermi-Dirac distribution at equilibrium and 300 K, we define the K-resolved scattering rates as

\({\tau}_{{{\bf{k}}}}^{-1}={\sum}_{n}\frac{-\partial {f}_{n\bf{k}}^{0}}{\partial {\varepsilon }_{n{{\bf{k}}}}}{\tau }_{n{{\bf{k}}}}^{-1}\), and K-resolved velocities as _v_K = ∑_n__v__n_K,

where _n_ denotes the number of bands involved in the transport. Figure 3 compares the electronic structures, \({\tau }_{{{\bf{k}}}}^{-1}\), and _v_K of doped GaAs and GaSb. The

broken-inversion symmetry in pristine GaAs and GaSb monolayers results in a Rashba splitting35 in the conduction band minimum (CBM) which can be regarded as a single valley. After electron

doping, Fig. 3a shows that for GaAs, the single valley is preserved in the Fermi surface window [EF − 0.3 eV, EF + 0.3 eV], while Fig. 3d shows that for GaSb, more states around M and K

points enter into the window, leading to enhanced scatterings with states that possess low velocities. As a result, \({\sigma }_{{{\rm{c}}}}^{{{\rm{e}}}}\) in GaSb is reduced, as shown in

Fig. 1a. The surprising behaviors of _μ_h and \({\sigma }_{{{\rm{c}}}}^{{{\rm{h}}}}\) of GaSb in Fig. 1b can be attributed to the doping-induced EF shift. The valence band maximum (VBM) of

pristine GaAs locates at K points, leading to a multi-peak band structure and a high DOS around VBM, thus the hole doping can only induce a small EF shift, as shown in Fig. 3g. Besides, the

spin-orbit splitting at K in the electronic band is 12 meV, which matches well with the phonon energy at K in the phonon dispersion [see Fig. 2c]. Considering momentum and energy

conservations, the strong inter-peak scatterings between the K and K’ peaks are allowed in hole-doped GaAs. Differently, the VBM in pristine GaSb is located at Γ, which is a single peak for

the hole mobility and leads to a high _μ_h. The single-peak band structure gives a small DOS around VBM, thus a large EF shift of 0.21 eV is induced by the hole doping as shown in Fig. 3j,

leading the states around K to dominate the scattering. Considering the low velocities at K points, \({\sigma }_{{{\rm{c}}}}^{{{\rm{h}}}}\) is greatly reduced compared with _μ_h in GaSb. It

should be noted that for GaSb, the spin-orbit splitting at K is 40 meV, mismatching the phonon energy [see Fig. 2f], thus the inter-peak scattering between K and K’ is weakened compared with

GaAs, as shown by the colors in Fig. 3h and k. The discussions above demonstrate that doping is an effective way to manipulate the electronic structure, further controlling the EPI and

charge conductivity in semiconductors. Analyses of scatterings in other III-V monolayers are similar, as given in Supplementary Fig. 5. SPIN HALL CONDUCTIVITY SHE is composed of intrinsic

and extrinsic contributions. The former is defined by the Kubo formula, and the latter is due to the skew-scattering and side-jump mechanisms, strongly dependent on disorder36,37,38. In this

work, we focus on the intrinsic SHC in a weak scattering limit. With spin current along _x_, electric field along _y_, and spin orientation along _z_ direction, the intrinsic SHC is

calculated using Kubo formula39,40: $${\sigma }_{{{\rm{s}}}}=\frac{\hslash }{2e}\frac{{e}^{2}}{\hslash }{\int}_{{{\rm{BZ}}}}\frac{{{{\rm{d}}}}^{2}{{\bf{k}}}}{{(2\pi)}^{2}}{\Omega

}_{{{\rm{s}}}}({{\bf{k}}}),$$ (4) where Ωs(K) = ∑_n__f__n_KΩs,_n_(K) is the spin Berry curvature (SBC) with _f__n_K at 300 K and the band-resolved SBC as $${\Omega

}_{{{\rm{s}}},n}({{\bf{k}}})={\hslash }^{2}\mathop{\sum}\limits_{m\ne n}\frac{-2\,{{\rm{Im}}}[\left\langle n{{\bf{k}}}\right\vert {\hat{j}}_{z}\left\vert m{{\bf{k}}}\right\rangle

\left\langle m{{\bf{k}}}\right\vert {\hat{v}}_{y}\left\vert n{{\bf{k}}}\right\rangle ]}{{({\varepsilon }_{n{{\bf{k}}}}-{\varepsilon }_{m{{\bf{k}}}})}^{2}+{\eta }^{2}},$$ (5) where

\({\hat{j}}_{z}=\frac{1}{2}\{{\hat{\sigma }}_{z}{\hat{v}}_{x}+{\hat{v}}_{x}{\hat{\sigma }}_{z}\}\) is the spin current operator, \({\hat{\sigma }}_{z}\) is the Pauli operator,

\({\hat{v}}_{x}\) and \({\hat{v}}_{y}\) are velocity operators. Considering the SOC strength in all the materials, a broadening of _η_ = 2 meV is used as a weak scattering induced by

extrinsic factors41, and the _η_ value has been validated in Supplementary Fig. 6. Due to the low conductivities, the hole-doped systems are more promising for high SHRs, thus III-V

monolayer with 2 × 1013 cm−2 hole doping are targeted for SHCs, and this doping level has been realized in 2D systems42,43. Figure 4a presents the energy-dependent SHCs of hole-doped

materials at room temperature. GaAs exhibits \({\sigma }_{{{\rm{s}}}}^{{{\rm{h}}}}=0.6\) (_ℏ_/2_e_)_e_2/_h_ due to the small EF shift induced by doping. More prominently, SHCs of InAs, GaSb,

and InSb can reach up to 1.5, 2.0, and 1.8 (_ℏ_/2_e_)_e_2/_h_, respectively. For comparison, MoS2 monolayer only reaches \({\sigma }_{{{\rm{s}}}}^{{{\rm{h}}}}\approx 0.2\)

(_ℏ_/2_e_)_e_2/_h_ 22. The high SHCs in hole-doped III-V monolayers are attributed to the in-gap EF location, as shown in Fig. 3j. The spin-orbit gap separates the positive and negative SBC.

When the EF locates inside the spin-orbit gap, SHC, as the integration over the Fermi sea of SBC, can be maximized by the sign-invariant SBC over all the K-points. The SBC decompositions of

all the materials are presented in Supplementary Fig. 8. For GaAs, the SBC originates from both K and Γ points, while for the others, the SBC mainly stems from the Γ point due to the in-gap

EF at Γ. The discussions above highlight that the doping in semiconductors can yield large SHCs in III-V monolayers. SPIN HALL RATIO With the charge conductivities _σ_c and spin Hall

conductivities _σ_s, the spin Hall ratio \(\xi =\frac{2e}{\hslash }\left\vert \right.\frac{{\sigma }_{{{\rm{s}}}}}{{\sigma }_{{{\rm{c}}}}}\left\vert \right.\) can be discussed in hole-doped

systems. It has been illustrated in Supplementary Table 1 that the computational and experimental SHR values coincide with each other, i.e., both present _ξ_ ~ 0.01 in heavy metals,

demonstrating the effectiveness of our calculations. Figure 4b shows that large \({\sigma }_{{{\rm{s}}}}^{{{\rm{h}}}}\) values are found in antimonides which also possess fairly large

\({\sigma }_{{{\rm{c}}}}^{{{\rm{h}}}}\), as a result, GaSb and InSb exhibit _ξ_h = 0.23 and 0.13, respectively. More importantly, hole-doped arsenides are perfect candidates with high SHCs

and low charge transports, yielding exceptional _ξ_h = 0.58 and 0.34 in GaAs and InAs, respectively. Compared with heavy metals where _ξ_ ~ 0.01, the hole-doped GaAs and other III-V

monolayers exhibit great potential for efficient charge-to-spin conversion. SPIN HALL RATIO DESCRIPTOR Taking the transport behaviors in III-V monolayers as a prototype, a descriptor to

enhance SHR can be proposed. The idea is to decrease the charge conductivity as well as increase SHC. Large effective mass reduces _σ_c via lowering down the carrier velocity. As shown in

Fig. 3, the band structure with multiple extrema around EF can enhance scattering. Besides, the energy-dependent SHCs demonstrate that SHC can be maximized by locating EF in the spin-orbit

gap. Overall, a band structure descriptor for high SHR can be proposed as depicted in Fig. 5: (1) large effective mass \({m}_{{{\rm{K}}}}^{* }\) around EF ; (2) multiple K-inequivalent

extrema around EF ; (3) a EF located inside the spin-orbit splitting \(\Delta {\varepsilon }_{{{\rm{K}}}}^{mn}\). It should be stressed that within a limited doping concentration, there is a

competition between the first two conditions and the last one, since flat and multiple band extrema would induce a large DOS, which hinders the tuning of EF. A small effective mass at Γ

point, \({m}_{\Gamma }^{* }\), is also preferred for tuning EF when _ε_Γ > _ε_K. In short, a delicate balance between _σ_c and _σ_s is essential for SHR enhancement. HIGH-THROUGHPUT

CALCULATIONS Potential candidates for high SHR can be found by screening a database with the aforementioned descriptor. Focusing on the 2D materials, we perform fully relativistic

high-throughput calculations for exfoliable monolayers. From the MC2D database44,45, all the rare-earth free materials with up to 6 atoms per unit cell are considered, yielding 216

semiconductors which are employed for the fully relativistic DFT calculations and Wannierizations. The band interpolation quality of Wannier Hamiltonian is measured by the average band

distance, given as46,47 $${\eta }_{\nu }=\sqrt{\frac{{\sum }_{n{{\bf{k}}}}{\tilde{f}}_{n{{\bf{k}}}}{\left({\varepsilon }_{n{{\bf{k}}}}^{{{\rm{DFT}}}}-{\varepsilon

}_{n{{\bf{k}}}}^{{{\rm{Wan}}}}\right)}^{2}}{{\sum }_{n{{\bf{k}}}}{\tilde{f}}_{n{{\bf{k}}}}}}$$ (6) where K is the eigenstate along the high-symmetry K-path for band structure,

\({\tilde{f}}_{n{{\bf{k}}}}=\sqrt{{f}_{n{{\bf{k}}}}^{{{\rm{DFT}}}}\left({{{\rm{E}}}}_{{{\rm{F}}}}+\nu ,\sigma \right){f}_{n{{\bf{k}}}}^{{{\rm{Wan}}}}\left({{{\rm{E}}}}_{{{\rm{F}}}}+\nu

,\sigma \right)}\) where \(f\left({{{\rm{E}}}}_{{{\rm{F}}}}+\nu ,\sigma \right)\) is the Fermi-Dirac distribution. Here EF + _ν_ and _σ_ respectively represent energy position and smearing

value for the distribution, thus we can focus on the bands below EF + _ν_ in energy. EF is set as the CBM of semiconductor, and _σ_ is fixed to be 300 K. Since the Wannier Hamiltonian

describes the low-energy valence electrons, it is expected that the Wannier-interpolated band will deviate more from the DFT results in the higher conduction band region. Therefore, the

higher _ν_ is, the larger _η__ν_ is expected to be. The symbols _η_1 and _η_2 denote the band distance for the bands below EF + 1 eV and EF + 2 eV, respectively, and the band distances of

216 materials are present in Fig. 6. For the bands below EF + 1 eV, 97% (210/216) of materials show a small band distance below 10 meV, and 93% (200/216) of materials show a very small band

distance below 2 meV. For the bands below EF + 2 eV, 90% (195/216) of materials show a distance below 10 meV, and 65% (141/216) of materials show a distance below 2 meV, highlighting the

overall quality of Wannier Hamiltonians in our database. Figure 7 shows the electron and hole effective masses of all the semiconductors, as well as SHC values in the doped systems where a

doping concentration of 2 × 1013 cm−2 is employed for both carriers. The effective mass is calculated by \({m}_{\alpha \alpha }^{* }={(\frac{1}{{\hslash }^{2}}\frac{{d}^{2}\varepsilon

}{d{k}_{\alpha }^{2}})^{-1}}\) (_α_ = _x_, _y_) through the interpolation around CBM and VBM, respectively, for electron and hole. Overall, the hole presents a larger effective mass than the

electron, and hole-doped materials exhibit larger SHC values. Nevertheless, many materials with high SHCs present small effective masses, populated at the left lower corner in Fig. 7b.

Still, by setting the screening conditions as \({m}_{xx}^{* }\approx {m}_{yy}^{* } > 0.5\,{m}_{0}\), multiple band extrema around EF, and SHC >0.4 (_ℏ_/2_e_)_e_2/_h_, two MXene

monolayers are selected as candidates, Sc2CCl2 and Y2CBr2. All the effective masses, doped SHCs, energy gaps, and band structures of all the 216 semiconductors are reported in Supplementary

Table 5 and its followings. RESULTS OF MXENE MONOLAYERS Sc2CCl2 and Y2CBr2 present similarities since they are composed of homologous elements. After a full relaxation considering the hole

doping, both structures present a P-3m1 space group, as shown in Fig. 8d. Figure 8a and e illustrate the hole-doped band structures of Sc2CCl2 and Y2CBr2, respectively. The SOC splitting

induces two peaks around Γ point, and the top band presents an effective mass \({m}_{xx}^{* }={m}_{yy}^{* }\approx 0.7\,{m}_{0}\). EF locates inside the gap, leading to high \({\sigma

}_{{{\rm{s}}}}^{{{\rm{h}}}}\) = 0.5 (_ℏ_/2_e_)_e_2/_h_. It has been verified that SHCs in both materials are robust against broadening, and the main SBC contributions originate from the Γ

point, details are given in Supplementary Fig. 12. Compared with Y2CBr2 possessing an SOC gap of 75 meV at Γ, Sc2CCl2 shows a smaller gap of 20 meV due to lighter elements. Apart from the Γ

point, eigenstates around the M point also enter into the Fermi surface window. Figure 8b, f validate the mechanical stabilities of hole-doped Sc2CCl2 and Y2CBr2 monolayers by showing phonon

dispersions without imaginary frequency. Phonon dispersions illustrate that Sc2CCl2 presents higher-energy modes than Y2CBr2. Figure 8c and g show the K-resolved scattering rates of both

materials. Sc2CCl2 exhibits dominant scattering as high as 836 THz around the Γ point. This high rate is attributed to its small SOC gap of 20 meV which matches the phonon energy _ω__n_Γ. On

the other hand, the scattering in Y2CBr2 is limited to 370 THz, since its large gap of 75 meV goes beyond its phonon energy range. Compared with the Γ point, scatterings around M points are

quite weak. This is attributed to _ε_M = − 273 meV for Sc2CCl2 and −57 meV for Y2CBr2, thus the scattering contributions are greatly limited by the large distance from EF. Overall, Sc2CCl2

presents lower \({\sigma }_{{{\rm{c}}}}^{{{\rm{h}}}}\) = 2.2 _e_2/_h_ while Y2CBr2 shows \({\sigma }_{{{\rm{c}}}}^{{{\rm{h}}}}\) = 3.9 _e_2/_h_ in Fig. 8h. Benefiting from the low charge

conductivity, the SHR in Sc2CCl2 is computed to be _ξ_h = 0.22, and Y2CBr2 possesses _ξ_h = 0.12. The discovery of an MXene monolayer with high SHR validates the effectiveness of the SHR

descriptor. Discussions of III-V monolayers and MXene monolayers indicate that compared with the materials composed of heavy elements, the light materials can be more promising for high SHR.

Although the heavy elements induce strong SOC for SHC enhancement, they also result in large band splitting and low-energy phonon mode simultaneously. By breaking the energy conservation

rule, these two factors reduce the electron-phonon coupling and hinder the scatterings in charge transport. As a result, the charge conductivity is also enhanced. Since the charge-to-spin

conversion is an interrelated process, balancing charge and spin transports is critical to enhance SHR rather than only maximize SHC. In addition, compared with the SHR _ξ_h = 0.58 in GaAs,

_ξ_h = 0.22 in Sc2CCl2 is relatively lower. Considering that the SHC values in the two materials are similar, the SHR difference is attributed to the high \({\sigma

}_{{{\rm{c}}}}^{{{\rm{h}}}}\) in Sc2CCl2. In GaAs, strong inter-peak scatterings occur [see Fig. 3h] thanks to the K eigenstates around EF. In contrast, Sc2CCl2 exhibits weak scatterings

around the M point [see Fig. 8c] due to the M eigenstates far away from EF. Besides, the inversion symmetry keeps the band degeneracy in Sc2CCl2, further reducing the scattering

possibilities. This comparison demonstrates the importance of multi-peaks around EF to reduce charge conductivity, further validating the SHR descriptor. By expanding the high-throughput

calculations database, materials with more ideal electronic structures and phonon frequencies could be found for SHR enhancement. EXPERIMENTAL FEASIBILITY Charge-to-spin conversion has been

realized in MoS2 and WSe2 monolayers grown by chemical vapor deposition19. Since the first synthesis of 2D AlN layers by metal organic deposition48, many efforts have been devoted to the

synthesis of other 2D III-V materials49,50. For example, GaSb films can be grown via a seeded lateral epitaxy, and the free-standing crystalline GaSb can be exfoliated from these films51.

Moreover, 2D InAs flakes with high crystalline quality have been synthesized through van der Waals epitaxy with a thickness down to 4.8 nm52. Due to chemical similarity in one family, we

expect similar techniques can be applied to the other III-V monolayers. Two promising materials, Sc2CCl2 and Y2CBr2, are easily exfoliable from their van der Waals bulk compounds45. Finally,

the doping levels proposed in this work can be realized via the advanced technique of electron beam, which implements the doping of 1.7 × 1013 cm−2 in 2D systems42. The doped state persists

even after removing the electron beam and back-gate voltage, and the process is reversible and repeatable42. Moreover, the doping level over 5 × 1013 cm−2 has been realized in MoS2

monolayers via the ionic-liquid gate43. In conclusion, 2D materials with high spin Hall ratios have been found using a multidisciplinary investigation involving charge transport, spin Hall

conductivity, and a high-throughput database. The hole-doped GaAs monolayer presents an ultrahigh SHR of _ξ_ = 0.58, attributed to the strong scattering and the high SHC. An SHR descriptor

is proposed and validated by a high-throughput database of 216 exfoliable monolayer semiconductors, suggesting a new promising material, Sc2CCl2. Besides, this database is fully released to

the community. This work reveals potential 2D materials for efficient charge-to-spin conversion, providing a guideline for materials discovery in spintronics. METHODS DFT CALCULATIONS DFT

and DFPT calculations are performed by the Quantum ESPRESSO package53. The fully relativistic norm-conserving pseudopotentials PseudoDojo54 within the Perdew–Burke–Ernzerhof (PBE)

parametrization of the generalized gradient approximation (GGA)54. The cutoff energy for wavefunctions is set to 120 Ry. We use a 2D Coulomb truncation55. Fully relativistic computations

have been performed for structural relaxation, self-consistent, non-self-consistent, and phonon calculations. Relaxations are performed respectively for pristine and doped materials.

Variable cell relaxation is performed with total energy convergence of 10−8 Ry, force convergence of 10−4 Ry/Bohr, and pressure convergence of 0.1 kbar. CHARGE CONDUCTIVITY CALCULATIONS The

drift mobility and charge conductivity are calculated using the EPW package56,57, where the electron-phonon coupling is interpolated from coarse K/Q-grids to fine K/Q-grids using Wannier

functions with the considerations of dipole, quadrupole, and gauge-covariant contributions58,59. A coarse 12 × 12 × 1 K/Q-grid is adopted in ab initio calculations, then a fine K/Q-grid of

720 × 720 × 1 is used for electron mobility, and K/Q-grid of 360 × 360 × 1 for hole mobility and charge conductivity of doped systems. Note that in the calculation at the zone-center Q = Γ,

a denser K-grid of 32 × 32 × 1 is used in DFPT calculation to get convergent electrostatic properties. A Fermi surface window of 0.3 eV has been validated for convergence and employed in all

calculations. An adaptive smearing60 is applied in the energy-conserving delta functions, and a phonon frequency cutoff of 1 cm−1 is employed. The calculations are performed at room

temperature. SPIN HALL CONDUCTIVITY CALCULATIONS The spin Hall conductivity is calculated by Wannier90 package39,61, with a 12 × 12 × 1 coarse K-grid in ab initio calculation, and a fine

K-grid of 360 × 360 × 1 for the Wannier interpolation. A Fermi-Dirac distribution function of 300 K and a broadening of 2 meV have been considered. HIGH-THROUGHPUT CALCULATIONS The fully

relativistic DFT calculations and Wannierizations have been performed using AiiDA62 and PseudoDojo54 on 216 materials from the MC2D database44,45. The initial guess for projectors is given

by the orbital information in PseudoDojo63. The semicore orbitals corresponding to deep bands are excluded from projections. A small trial step is used in the minimization of the spread

function to stabilize the Wannierization, and a large number of iterations is used to converge. The maximum of the frozen window is set as CBM + 2 eV, which is verified to be the optimal

value by Supplementary Fig. 13. The script for the automated generation of the Wannier90 input file is provided64. To provide a high-quality Wannier Hamiltonian database, failed cases

(23/216) have been manually corrected by tuning the frozen window, increasing the number of Bloch wavefunctions, or manually modifying initial projections. DATA AVAILABILITY For the III-V

and MXene monolayers investigated in this work, details of the relaxed atomic structures, effective masses, densities of states, doping levels, and electron and phonon dispersions are given

in Supplementary Material, and input and output files for computations are given on Materials Cloud Archive64. The input and output files, pseudopotentials, as well as Wannier Hamiltonians

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Cloud Archive. https://archive.materialscloud.org/deposit/records/2138 (2024). Download references ACKNOWLEDGEMENTS The authors would like to thank Xi Dai, Matteo Giantomassi, and Junfeng

Qiao for fruitful discussions. S.P. acknowledges the support from the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS). J.Z. and J.-C.C. acknowledge financial support from the

Fédération Wallonie-Bruxelles through the ARC Grant “DREAMS” (No. 21/26-116), from the EOS project “CONNECT” (No. 40007563), and from the Belgium F.R.S.-FNRS through the research project

(No. T.029.22F). Computational resources have been provided by the PRACE award granting access to MareNostrum4 at Barcelona Supercomputing Center (BSC), Spain and Discoverer in SofiaTech,

Bulgaria (OptoSpin project ID. 2020225411), and by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the F.R.S.-FNRS under Grant No. 2.5020.11 and by the Walloon Region, as

well as computational resources awarded on the Belgian share of the EuroHPC LUMI supercomputer. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Institute of Condensed Matter and Nanosciences

(IMCN), Université catholique de Louvain (UCLouvain), Chemin des Étoiles 8, B-1348, Louvain-la-Neuve, Belgium Jiaqi Zhou & Jean-Christophe Charlier * European Theoretical Spectroscopy

Facility and Institute of Condensed Matter and Nanosciences (IMCN), Université catholique de Louvain (UCLouvain), Chemin des Étoiles 8, B-1348, Louvain-la-Neuve, Belgium Samuel Poncé * WEL

Research Institute, Avenue Pasteur 6, B-1300, Wavre, Belgium Samuel Poncé Authors * Jiaqi Zhou View author publications You can also search for this author inPubMed Google Scholar * Samuel

Poncé View author publications You can also search for this author inPubMed Google Scholar * Jean-Christophe Charlier View author publications You can also search for this author inPubMed 

Google Scholar CONTRIBUTIONS J.Z. designed the study and performed calculations. S.P. and J.-C.C. supervised the project. All authors analyzed the results and contributed to writing the

paper. CORRESPONDING AUTHORS Correspondence to Jiaqi Zhou, Samuel Poncé or Jean-Christophe Charlier. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing interests.

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Zhou, J., Poncé, S. & Charlier, JC. Enhanced spin Hall ratio in two-dimensional semiconductors. _npj Comput Mater_ 10, 247 (2024). https://doi.org/10.1038/s41524-024-01434-z Download

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