Wurtzite structure effects on spin splitting in $\mathrm{GaN}∕\mathrm{AlN}$ quantum wells

A different mechanism (${\Delta }_{C1}\text{−}{\Delta }_{C3}$ coupling) accounts for the spin splitting of wurtzite $\mathrm{GaN}$, that originated from the intrinsic wurtzite effects (band folding and structure inversion asymmetry). The band-folding effect generates two conduction bands (${\Delta }_{C1}$ and ${\Delta }_{C3}$), in which $p$-wave probability has tremendous change when $k_{\mathrm{z}}$ approaches the anticrossing zone. The spin-splitting energy induced by the ${\Delta }_{C1}\text{−}{\Delta }_{C3}$ coupling and wurtzite structure inversion asymmetry is much larger than that evaluated by traditional Rashba or Dresselhaus effects. When we apply the coupling to $\mathrm{GaN}∕\mathrm{AlN}$ quantum wells, we find that the spin-splitting energy is sensitively controllable by an electric field. Based on the mechanism, we proposed a $p$-wave-enhanced spin-polarized field effect transistor, made of ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}∕{\mathrm{In}}_y{\mathrm{Al}}_{1-y}\mathrm{N}$, for spintronics application.

Figure 1

(Color) (a) The full band structure calculated by LCAO method. The band-folding effect generates two conduction bands (${\Delta }_{\mathrm{C}1}$ and ${\Delta }_{\mathrm{C}3}$), which are crossing at the saddle point. The inset shows the first Brillouin zone and its high symmetric points. (b) and (c) show the band-folding effect on ideal wurtzite $\mathrm{GaN}$ and $\mathrm{AlN}$, respectively. After the band folding, the section of zinc-blende ${\Lambda }_{\mathrm{C}1}$ band between ${\mathrm{A}}_{\mathrm{WZ}}$ and ${\mathrm{L}}_{\mathrm{ZB}}$ points is folded to be wurtzite ${\Delta }_{\mathrm{C}3}$ band, and ${\mathrm{L}}_{\mathrm{C}1}$ point turns into ${\Gamma }_{\mathrm{C}3}$ point.

Figure 2

(Color) The plots of ${\Delta }_{\mathrm{C}1}$ and ${\Delta }_{\mathrm{C}3}$ bands in wurtzite $\mathrm{GaN}$ against $k_z$ for different values of $k_x$ at $k_y=0$. The two bands cross each other at the saddle point $(k_x=k_y=0)$. Here, we use the same unit $(\pi ∕c)$ for $k_x$ and $k_z$ axes $(a∕c=0.6153)$, for convenience. As $k_x$ is not equal to $0$, the asymmetry generates an anticrossing between the two bands and creates an opening gap.

Figure 3

(Color) (a) The spin-splitting energies for ${\Delta }_{\mathrm{C}1}$ and ${\Delta }_{\mathrm{C}3}$ bands of wurtzite and ideal wurtzite $\mathrm{GaN}$ are plotted against $k_z$ for $k_x=0.12\pi ∕c$ and $k_y=0$. The band-folding effect for ${\Delta }_{\mathrm{C}1}$ and ${\Delta }_{\mathrm{C}3}$ in wurtzite $\mathrm{GaN}$ is also described by the arrows. (b) The probabilities of density of states for $s$ and $p$ waves in the two conduction bands. (c) The details of the spin-polarized ${\Delta }_{\mathrm{C}1}^{\pm }$ and ${\Delta }_{\mathrm{C}3}^{\pm }$ bands in the vicinity of the anticrossing zone.

Figure 4

(Color) The spin-splitting energies of ${\Delta }_{\mathrm{C}1}\left({\mathit{k}}^{\mathrm{F}}\right)$ are plotted against the well thickness from 1 to 10 layers at $k_{∕∕}^{\mathrm{F}}=0.12\pi ∕c$, for the cases with and without electric field in ideal wurtzite and wurtzite $\mathrm{GaN}∕\mathrm{AlN}$ QWs. The probabilities of density of states for $s$ and $p$ waves in ${\Delta }_{\mathrm{C}1}\left({\mathit{k}}^{\mathrm{F}}\right)$ and ${\Delta }_{\mathrm{C}3}\left({\mathit{k}}^{\mathrm{F}}\right)$ are also shown in the inset.