Relation between polarization, spin splitting, and spin textures. (a) In-plane electric polarization P of the GaTeCl ML as a function of ferroelectric distortion τ is shown. The insert shows the optimized structure of the GaTeCl ML in the ferroelectric phase with P and − P polarization. τ is defined as the magnitude of the distortion vector | r| of the systems given in Eq. (1) normalized by the magnitude of the distortion vector of the optimized ferroelectric phase, | r0|. Here τ = 0 represents the paraelectric phase and τ = 1 shows the optimized ferroelectric phase. (b) Band structure of the GaTeCl ML calculated along Y − Γ − X line around the VBM as a function of the ferroelectric distortion τ is presented. (c) The SOC strength α of the GaTeCl ML as a function of the ferroelectric distortion τ is presented. Reversible out-of-plane spin orientation in GaTeCl ML calculated at constant energy cut of 1 meV below the degenerate state at the VBM around the Γ point for the optimized ferroelectric phase with opposite in-plane electric polarization: (d) − P and (e) P .

Contexts in source publication

Context 1

... summarizing, we highlighted the interplay between the in-plane ferroelectricity, spin splitting, and the spin textures in the GaXY ML compounds. Fig. 5(a) displayed the in-plane electric polarization as a function of the ferroelectric distortion, τ . Here, τ is defined as the magnitude of the distortion vector | r| of the systems defined by Eq. (1), which is normalized by the magnitude of the distortion vector of the optimized ferroelectric phase, | r 0 |. Therefore, τ = 0 represents the ...

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... τ . Here, τ is defined as the magnitude of the distortion vector | r| of the systems defined by Eq. (1), which is normalized by the magnitude of the distortion vector of the optimized ferroelectric phase, | r 0 |. Therefore, τ = 0 represents the paraelectric phase, while τ = 1 shows the optimized ferroelectric phase as shown by the insert of Fig. 5(a). We can see that it is possible to manipulate the in-plane electric polarization P by distorting the atomic position [see Fig. 1(a)]. The dependence of the in-plane polarization on the ferroelectric distortion τ sensitively affects the spin-split bands at the VBM around the Γ point as shown in Fig. 5(b). It is found that the splitting ...

Context 3

... phase as shown by the insert of Fig. 5(a). We can see that it is possible to manipulate the in-plane electric polarization P by distorting the atomic position [see Fig. 1(a)]. The dependence of the in-plane polarization on the ferroelectric distortion τ sensitively affects the spin-split bands at the VBM around the Γ point as shown in Fig. 5(b). It is found that the splitting energy and the position of the VBM around the Γ point strongly depend on the ferroelectric distortion, i.e., a decrease in τ substantially reduces the spin splitting energy while the position of the VBVM shifts up to be higher in energy around the Γ point. Accordingly, the significant change of the SOC ...

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... strongly depend on the ferroelectric distortion, i.e., a decrease in τ substantially reduces the spin splitting energy while the position of the VBVM shifts up to be higher in energy around the Γ point. Accordingly, the significant change of the SOC strength α is achieved, in which a linear trend of α as a function of τ is observed as shown in Fig. 5(c). Importantly, our results also show that the SOC strength α changes sign when the direction of the in-plane ferroelectric polarization P is switched, resulting in a full reversal of the out-of-plane spin textures shown in Figs. 5(d)-(e). Such reversible spin textures are agreed well with our symmetry analysis given by Eq. (15), putting ...

Context 5

... summarizing, we highlighted the interplay between the in-plane ferroelectricity, spin splitting, and the spin textures in the GaXY ML compounds. Fig. 5(a) displayed the in-plane electric polarization as a function of the ferroelectric distortion, τ . Here, τ is defined as the magnitude of the distortion vector | r| of the systems defined by Eq. (1), which is normalized by the magnitude of the distortion vector of the optimized ferroelectric phase, | r 0 |. Therefore, τ = 0 represents the ...

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... τ . Here, τ is defined as the magnitude of the distortion vector | r| of the systems defined by Eq. (1), which is normalized by the magnitude of the distortion vector of the optimized ferroelectric phase, | r 0 |. Therefore, τ = 0 represents the paraelectric phase, while τ = 1 shows the optimized ferroelectric phase as shown by the insert of Fig. 5(a). We can see that it is possible to manipulate the in-plane electric polarization P by distorting the atomic position [see Fig. 1(a)]. The dependence of the in-plane polarization on the ferroelectric distortion τ sensitively affects the spin-split bands at the VBM around the Γ point as shown in Fig. 5(b). It is found that the splitting ...

Context 7

... phase as shown by the insert of Fig. 5(a). We can see that it is possible to manipulate the in-plane electric polarization P by distorting the atomic position [see Fig. 1(a)]. The dependence of the in-plane polarization on the ferroelectric distortion τ sensitively affects the spin-split bands at the VBM around the Γ point as shown in Fig. 5(b). It is found that the splitting energy and the position of the VBM around the Γ point strongly depend on the ferroelectric distortion, i.e., a decrease in τ substantially reduces the spin splitting energy while the position of the VBVM shifts up to be higher in energy around the Γ point. Accordingly, the significant change of the SOC ...

Context 8

... strongly depend on the ferroelectric distortion, i.e., a decrease in τ substantially reduces the spin splitting energy while the position of the VBVM shifts up to be higher in energy around the Γ point. Accordingly, the significant change of the SOC strength α is achieved, in which a linear trend of α as a function of τ is observed as shown in Fig. 5(c). Importantly, our results also show that the SOC strength α changes sign when the direction of the in-plane ferroelectric polarization P is switched, resulting in a full reversal of the out-of-plane spin textures shown in Figs. 5(d)-(e). Such reversible spin textures are agreed well with our symmetry analysis given by Eq. (15), putting ...