Direct Rashba spin-orbit interaction in Si and Ge nanowires with different growth directions

We study theoretically the low-energy hole states in Si, Ge, and Ge/Si core/shell nanowires (NWs). The NW core in our model has a rectangular cross section, the results for a square cross section are presented in detail. In the case of Ge and Ge/Si core/shell NWs, we obtain very good agreement with previous theoretical results for cylindrically symmetric NWs. In particular, the NWs allow for an unusually strong and electrically controllable spin-orbit interaction (SOI) of Rashba type. We find that the dominant contribution to the SOI is the “direct Rashba spin-orbit interaction” (DRSOI), which is an important mechanism for systems with heavy-hole–light-hole mixing. Our results for Si NWs depend significantly on the orientation of the crystallographic axes. The numerically observed dependence on the growth direction is consistent with analytical results from a simple model, and we identify a setup where the DRSOI enables spin-orbit energies of the order of millielectronvolt in Si NWs. Furthermore, we analyze the dependence of the SOI on the electric field and the cross section of the Ge or Si core. A helical gap in the spectrum can be opened with a magnetic field. For this gap, we obtain the largest $g$ factors for magnetic fields applied perpendicular to the NW and parallel to the electric field.

Figure 1

The basis states of Eq. (29) and their couplings. At $k_z=0$ and $E_x=0$, the states $|g_{\pm }\rangle$ and $|e_{\pm }\rangle$ are the two ground states and first excited states, respectively, of the Hamiltonian. They differ in energy by $\mathrm{\Delta }$ and the contained spin states are shown in parentheses. The couplings proportional to $E_x$ result directly from the electric-field-induced shift $-e{E}_{x}x$ (spin conserving) in the potential energy and therefore cannot occur between states that have different spins. The cross couplings proportional to $k_z$ result from the LK Hamiltonian. The combination of the shown couplings within the low-energy subspace results in an unusually strong SOI of Rashba type, the DRSOI [69]. The cross couplings would be absent if the spin states for “$+$” and “$-$” were solely $|3/2\rangle$ and $|-3/2\rangle$, see Eq. (27) [or solely $|-1/2\rangle$ and $|1/2\rangle$, see Eq. (28)], and so in many 2D-like systems the DRSOI is strongly suppressed by a large HH-LH splitting. The additional energies ${\hslash }^2{k}_z^2/\left(2m_{g,e}\right)$ in Eq. (29) are not shown in the sketch. Details are provided in the text and in Refs. [69, 70].

Figure 2

Sketch of the NWs and the coordinate system considered in this work. The core of the NW has a rectangular cross section which lies in the $x-y$ plane. The NW axis is the $z$ axis. In the case of a Ge/Si core/shell NW, the Ge core is compressively strained by the Si shell. Since standard silicon-on-insulator (or bulk Si [59]) wafers have a (100) surface, the $x$ axis in the sketch usually corresponds to a main crystallographic axis when a CMOS-compatible NW is fabricated as in Refs. [59, 84, 85, 86, 87, 88, 89]. We find that a much stronger SOI can be induced in Si NWs when $x\parallel \left[110\right]$ and $z\parallel \left[001\right]$ (see Secs. 5 and 6).

Figure 3

Low-energy hole spectrum of an unstrained Ge NW. Each line corresponds to two degenerate subbands. (Top) Spectrum for a square cross section, calculated with the model of Sec. 3. The two eigenstates of type A consist predominantly of the basis states $|1,1,0,\pm \frac{1}{2}\rangle$ ($96.4\%$). Those of type B have $|1,1,0,\pm \frac{3}{2}\rangle$ ($51.6\%$) as their largest contribution. For more information on the properties of A, B, and C, we refer to Sec. 5b. (Bottom) Spectrum for a circular cross section, adapted from Ref. [69]. Due to the cylindrical symmetry, the subbands can be classified via the total angular momentum $F_z$ along the NW axis [110, 111]. Despite the different cross sections, the spectra in the upper and lower panels closely resemble each other. Around $k_z=0$, the effective mass for the degenerate subbands of lowest energy is negative. We note that ${\hslash }^2/\left(m{R}^2\right)=3.05\text{meV}$ for $R=5\text{nm}$. The label $E$ at the vertical axis stands for “Energy.”

Figure 4

Low-energy hole spectrum of a Ge/Si core/shell NW. The cross section of the Ge core is a square with side length $s=10\text{nm}$ and the relative shell thickness is $\gamma =0.4$. Each line represents two degenerate subbands. Compared with the case of a bare Ge NW in Fig. 3 (top), the contribution of $|1,1,0,\pm \frac{1}{2}\rangle$ ($97.8\%$) to the A-type eigenstates is greater, which is a consequence of the compressive strain in the Ge core that is caused by the Si shell. Moreover, the strain significantly increases the gap (referred to as $\mathrm{\Delta }$ in the text) between the eigenstates of type A and C, leading to an electronlike parabolic dispersion relation with positive effective mass for the subbands of lowest energy. At $k_z=0$, the two energetically lower states in the enlarged frame do not contain the basis states $|1,1,0,\pm \frac{3}{2}\rangle$ at all. The two energetically higher states originate from the B-type states of Fig. 3, but the contribution of $|1,1,0,\pm \frac{3}{2}\rangle$ decreased to only $14.5\%$.

Figure 5

Subbands of lowest energy in a Ge/Si core/shell NW with relative shell thickness $\gamma =0.3$. Each line represents a single subband, because the degeneracy is lifted by an electric field $E_x=6\mathrm{V}/\mu \mathrm{m}$ and a magnetic field $B_x=1\text{T}$ applied along the $x$ axis (see Fig. 2). The spectrum in the upper panel was calculated with the model of Sec. 3 using $s=10\text{nm}$ (square cross section). The lower panel was adapted from Ref. [69] (circular cross section). In both cases, the spin-orbit energy is approximately one millielectronvolt. The Zeeman gap at $k_z=0$ corresponds to a $g$ factor of 8.3 (top) and 5.2 (bottom), respectively. When the magnetic field is applied along $z$ instead of $x$, our calculation for the square cross section yields a $g$ factor of 1.8. As expected [66, 69, 75], this value is smaller than the one for the perpendicularly applied magnetic field.

Figure 6

Spin-orbit energy $E_{\mathrm{SO}}$ as a function of the electric field $E_x$ for three Ge/Si core/shell NWs. The NWs differ in the side length $s$ of the Ge core, the relative shell thickness is always $\gamma =0.4$. In each of the three cases, a black dot marks the point where the maximal spin-orbit energy $E_{\mathrm{SO}}^{\max }$ is reached. The sketched $E-k$ diagram in the inset illustrates how we extract the spin-orbit energy $E_{\mathrm{SO}}$ from a low-energy hole spectrum. At a given set of parameters, we obtain $E_{\mathrm{SO}}$ from the numerically calculated spectrum, similar to that of Fig. 5 (top), setting all magnetic fields to zero. In the absence of magnetic fields, there is a degeneracy at $k_z=0$, as sketched in the diagram.

Figure 7

Maximal spin-orbit energy $E_{\mathrm{SO}}^{\max }$ as a function of the side length $s$ for Ge/Si core/shell NWs with relative shell thickness $\gamma =0.4$. Details for the three data points at $s=6$, 10, and $14\text{nm}$ are shown in Fig. 6.

Figure 8

Low-energy hole spectra of Si NWs with different orientations of the crystallographic axes (see labels at the top of each panel and Fig. 2 for an illustration of the axes $x$ and $z$). In each case, the cross section of the NW is a square with side length $s$. The spectra in the top, middle, and bottom panel were calculated with $\varphi =0, \varphi =\pi /4$, and $\xi =0$, respectively (see Secs. 3c1 and 3c2). Every line corresponds to two degenerate subbands. The marked eigenstates of type A, B, and C are discussed in Sec. 5b. The relative contribution of the basis states $|1,1,0,\pm \frac{1}{2}\rangle$ to the two eigenstates of type A is $97.7\%$ (top), $99.9\%$ (middle), and $95.0\%$ (bottom), respectively. That of $|1,1,0,\pm \frac{3}{2}\rangle$ to the two eigenstates of type B is $95.9\%$ (top), $99.8\%$ (middle), and $85.5\%$ (bottom). Despite the absence of strain, the effective mass of the lowest-energy subbands is always positive.

Figure 9

Subbands of lowest energy in a Si NW with side length $s=10\text{nm}$. An electric field $E_x=6\mathrm{V}/\mu \mathrm{m}$ is applied along the $x$ axis (see Fig. 2). The plotted spectra were calculated with $\varphi =\pi /4$, i.e., the $x$ axis corresponds to the [110] direction and the NW axis $z$ corresponds to the [001] direction. (Top) The electric field leads to a SOI and shifts the two originally degenerate subbands in opposite directions along the $k_z$ axis. This lifts the degeneracy, except at $k_z=0$. Ground (excited) states are represented by the solid blue (dashed red) lines. The spin-orbit energy $E_{\mathrm{SO}}$ is of the order of millielectronvolt. (Bottom) A magnetic field $B_x$ along $x$ opens a Zeeman gap at $k_z=0$ with a $g$ factor of approximately 2.

Figure 10

Dependence of the spin-orbit energy $E_{\mathrm{SO}}$ on the electric field $E_x$ for three Si NWs with identical cross sections ($s=10\text{nm}$). For each of the three NWs, the orientation of the crystallographic axes is different and provided by the labels of the respective lines. The blue, red, and orange solid lines were calculated analogously to Fig. 6, using $\varphi =\pi /4, \varphi =0$, and $\xi =0$, respectively. Black dots mark the points with maximal spin-orbit energy $E_{\mathrm{SO}}^{\max }$. The dashed lines are plots of Eq. (85) with $\varphi =\pi /4$ (black) and $\varphi =0$ (gray). In the regime of small $E_x$, they almost coincide with the numerically calculated solid lines. For details on Eq. (85) and the underlying model, see Sec. 6.

Figure 11

Maximal spin-orbit energy $E_{\mathrm{SO}}^{\max }$ as a function of the side length $s$ for Si NWs with different orientations of the crystallographic axes (see explanations at the top right of the figure). Details for the three data points at $s=10\text{nm}$ are shown in Fig. 10, the other data were calculated analogously. We note that $E_{\mathrm{SO}}^{\max }\propto s^{-2}$ in good approximation. As explained in Sec. 7, we estimate that the spin-orbit split-off band may become important in Si NWs when $s<10\text{nm}$, which might lead to strong deviations from $E_{\mathrm{SO}}^{\max }\propto s^{-2}$ in this regime. We believe that if all bands of the semiconductor were fully taken into account for $s<10\text{nm}, E_{\mathrm{SO}}^{\max }$ would still increase with decreasing $s$.

Figure 12

Dependence of the derived coefficients ${\overline{\delta }}_{\mathrm{DR}}\left(\varphi \right)$ [see Eqs. (78) and (79)] and ${\overline{\alpha }}_{\mathrm{DR}}\left(\varphi \right)$ [see Eqs. (80) and (79)] on the angle $\varphi$. For detailed information, we refer to Sec. 6. We use the Luttinger parameters ${\gamma }_1=4.22, {\gamma }_2=0.39$, and ${\gamma }_3=1.44$ for Si (Sec. 5a) [92]. The Si NW has a square cross section with side length $s$ and the electric field, applied along the $x$ axis (Fig. 2), is assumed to be small.