Origin of Rashba-Dresselhaus effect in the ferroelectric nitride perovskite ${\mathrm{LaWN}}_3$

First-principles electronic-structure calculations have been employed to investigate the Rashba-Dresselhaus spin splitting of bands in the recently predicted ferroelectric nitride perovskite ${\mathrm{LaWN}}_3$. Our first-principles results are supplemented with an effective $\mathbf{k}·\mathbf{p}$ model analysis. A systematic study of orthorhombic and rhombohedral phases of this system reveals the importance of symmetry in realizing the nature of the splitting of bands around the time-reversal invariant $k$ points. Orthorhombic ${\mathrm{LaWN}}_3$ shows linear Rashba-Dresselhaus splitting where nonsymmorphic symmetries play an important role in enhancing the band splitting in the $k_z=\pi /c$ plane, while the rhombohedral phase shows the existence of a unique higher order Rashba-Dresselhaus term which mixes with the linear Rashba-Dresselhaus term to produce an unusual pattern of out-of-plane spin components in the spin texture.

Figure 1

Crystal structure of ${\mathrm{LaWN}}_3$ (a) orthorhombic $Pna{2}_1$ phase and (b) rhombohedral $R3c$ phase. La, W, and N atoms are indicated by red, blue, and brown colors, respectively. (c) Phonon band structure for orthorhombic $Pna{2}_1$ phase of ${\mathrm{LaWN}}_3$.

Figure 2

(a) Electronic band structure of ${\mathrm{LaWN}}_3$ for orthorhombic $Pna{2}_1$ phase in absence of SOC. The conduction bands which we have considered in the discussion are indicated in blue color. Inset: First Brillouin zone for orthorhombic $Pna{2}_1$ structure with chosen $k$ path of calculated band structure. (b) Projected density of states (DOS) of W-$5d$ and N-$2p$ states indicated by black and red colors, respectively. Fermi energy of the system is set to zero in the energy axis.

Figure 3

(a) Electronic band structure of ${\mathrm{LaWN}}_3$ for orthorhombic $Pna{2}_1$ phase in the presence of SOC. The conduction bands which we have considered in the discussion are indicated in blue color. (b) Projected density of states (DOS) of W-$5d$ and N-$2p$ states indicated by black and red colors, respectively. Fermi energy of the system is set to zero in the energy axis.

Figure 4

(a), (b) Electronic band structure of $Pna{2}_1$ phase in absence of SOC and presence of SOC around the $\mathrm{\Gamma }$ point, respectively. Band structure is plotted along $(\frac{2\pi }{a}0.19,0,0)-$ (0,0,0) $-(0,\frac{2\pi }{b}0.24,0)$ direction of the momentum space, which is along X-$\mathrm{\Gamma }$-Y direction. DFT band structure is plotted with dots and $\mathbf{k}·\mathbf{p}$ band structure is plotted with dashed lines. Black color indicates outer branch while red color indicates inner branch of the band structure. Spin texture of (c) inner and (d) outer branch around the $\mathrm{\Gamma }$ point of the conduction band minimum (CBM).

Figure 5

(a), (b) Electronic band structure of the $Pna{2}_1$ phase in absence of SOC and in presence of SOC around the Z point, respectively. Band structure is plotted along $(\frac{2\pi }{a}0.14,0,\frac{\pi }{c})-(0,0,\frac{\pi }{c})-(0,\frac{2\pi }{b}0.15,\frac{\pi }{c})$ direction of the momentum space which is along U-Z-T direction. DFT band structure is plotted with circles and $\mathbf{k}·\mathbf{p}$ band structure is plotted with dashed lines. Black color indicates outer branch while red color indicates inner branch of the band structure. Spin texture of doubly degenerate [(c), (d)] inner and [(e), (f)] outer branches around the Z point of the conduction band minimum (CBM).

Figure 6

(a) Electronic band structure of ${\mathrm{LaWN}}_3$ for rhombohedral $R3c$ phase in absence of SOC. The valence band which we have considered in the discussion are indicated in blue color. Inset: First Brillouin zone for the rhombohedral $R3c$ structure where the $k$ paths for the calculated band structure are shown. (b) Projected DOS of W-$5d$ and N-$2p$ states indicated as black and red colors, respectively. Fermi energy of the system is set to zero in the energy axis.

Figure 7

(a) Electronic band structure of ${\mathrm{LaWN}}_3$ for rhombohedral $R3c$ phase in the presence of SOC. The valence bands which we have considered in the discussion are indicated in blue color. Inset: Splitting of the valence band maximum of the chosen band around the $\mathrm{\Gamma }$ point. (b) Projected DOS of W-$5d$ and N-$2p$ states indicated as black and red colors, respectively. Fermi energy of the system is set to zero in the energy axis.

Figure 8

Spin texture of (a) outer and (b) inner branches around the $\mathrm{\Gamma }$ point of the valence band maximum of $R3c$ phase. (c), (d) Electronic band structure around the $\mathrm{\Gamma }$ point in absence and presence of SOC, respectively. Band structure is plotted along $(\frac{2\pi }{a}0.25,0,0)-$ (0,0,0) $-(\frac{2\pi }{a}0.16,\frac{2\pi }{b}0.16,0)$ direction of the momentum space which is along M-$\mathrm{\Gamma }$-K direction. DFT band structure is plotted with dots and $\mathbf{k}·\mathbf{p}$ band structure is plotted with dashed lines. Black color indicates the inner branch while red color indicates the outer branch of the band structure.