Common nonlinear features and spin-orbit coupling effects in the Zeeman splitting of novel wurtzite materials

The response of semiconductor materials to external magnetic fields is a reliable approach to probe intrinsic electronic and spin-dependent properties. In this study, we investigate the common Zeeman splitting features of novel wurtzite materials, namely, InP, InAs, and GaAs. We present values for the effective $g$ factors of different energy bands and show that spin-orbit coupling effects, responsible for the spin splittings, also have noticeable contributions to the $g$ factors. Within the Landau level picture, we show that the nonlinear Zeeman splitting recently explained in magnetophotoluminescence experiments for InP nanowires by D. Tedeschi et al. [Phys. Rev. B 99, 161204 (2019)] is also present in InAs, GaAs, and even the conventional GaN. Such nonlinear features stem from the peculiar coupling of the A and B valence bands as a consequence of the interplay between the wurtzite crystal symmetry and the breaking of time-reversal symmetry by the external magnetic field. Moreover, we develop an analytical model to describe the experimental nonlinear Zeeman splitting and apply it to InP and GaAs data. Extrapolating our fitted results, we found that the Zeeman splitting of InP reaches a maximum value, which is a prediction that could be probed at higher magnetic fields.

Figure 1

(a) Schematic band structure for bulk WZ crystals around the $\mathrm{\Gamma }$ point, indicating the different energy bands: CB, A, B and C (from top to bottom). The energies at the $\mathrm{\Gamma }$ point are indicated by the arrows. (b) Scheme of the magnetic field configurations parallel ($B_{\text{z}}$) and perpendicular ($B_{\text{x}}$) to the NW axis in typical magneto-PL experiments. The laser excitation and the collected PL signal are parallel to the NW axis. The inset shows the WZ crystal structure with the orientation of the $c$ axis in the [0001] direction.

Figure 2

Calculated Landau level spectra for the A band at $k_B=k_A=0$ as a function of the magnetic field for (a) InP, (b) InAs, and (c) GaAs. The color code indicates the contribution of the B valence band to the total state. The upper (lower) branch of the topmost Landau level is indicated by 1+ ($1-$). (d) Zeeman splitting for the topmost Landau level. We denote with positive (negative) values on the $x$ axis the magnetic field configuration $B_{\text{z}}$ along the $z$ ($B_{\text{x}}$ along the $x$) direction.

Figure 3

Probability density of the envelope functions at 30 T for the first LL (indicated by the labels 1+ and $1-$ in Fig. 2) along $B_{\text{x}}$ and $B_{\text{z}}$ for (a)–(d) InP, (e)–(h) InAs, and (i)–(l) GaAs. The B band envelope functions are multiplied by a factor of 8, as indicated in the legend. The spin notation for the energy bands is identified with respect to the leading contribution of the bulk basis states [48].

Figure 4

(a) Calculated Landau level spectra for the A band at $k_B=k_A=0$ as a function of the magnetic field for GaN. (b) Zeeman splitting for the topmost Landau level. The color code and axis notation follow Fig. 2. (c)–(f) Probability density of the envelope functions at 30 T for the first LL, indicated by the labels 1+ and $1-$ in (a), along $B_{\text{x}}$ and $B_{\text{z}}$. The B band envelope functions are multiplied by a factor of 4, as indicated in the legend.

Figure 5

Comparison of the fitting of Eq. (12) to the experimental Zeeman splitting for (a) InP and (b) GaAs. We use the calculated $g$ factors of Table 1 and the theoretical values of $E_{\text{B}}=-35.4\text{meV}$ for InP [39] and $E_{\text{B}}=-99.4\text{meV}$ for GaAs [42]. The fitting procedure provides $d_{-}=0.16\mathrm{meV}/\mathrm{T}$, $|d_{\text{AB}}|=0.43\mathrm{meV}/\mathrm{T}$ for InP and $d_{-}=17.63\mathrm{meV}/\mathrm{T}$, $|d_{\text{AB}}|=2.82\mathrm{meV}/\mathrm{T}$ for GaAs. The experimental data for InP are taken from Ref. [23], and those for GaAs are from Ref. [7].

Figure 6

Calculated Landau level spectra for a valence band at $k_B=k_A=0$ as a function of the magnetic field for (a) InP, (b) InAs, and (c) GaAs with a zinc-blende structure. The upper and lower branches of the first (thick solid lines) and second (thick dashed lines) topmost Landau levels are indicated by $1\pm$ and $2\pm$, respectively. Zeeman splitting for the first (solid lines) and second (dashed lines) topmost Landau levels for (d) InP, (e) InAs, and (f) GaAs. Probability densities at 30 T for LL = 1 and LL = 2 for (g)–(j) InP, (l)–(n) InAs, and (o)–(r) GaAs.