Figure 1

(a) Sketch of a rolled-up Si-Ge tube; $R_1$ is the radius of the first shell of Si atoms. (b) Chosen unit cell for the tube with two atoms on each plane (yellow balls) evidencing the discrete rotational symmetry around the $z$ axis. Each red ball represents an atom that can be reached from the appropriate yellow ball on the same shell by means of discrete translations along (or discrete rotations around) $z$. (c) Conventional unit cell for a bulk diamond lattice (black cube) with eight atoms (yellow balls), from which the curved tube cell (b) originates. The green shaded region represents the primitive cell of the diamond lattice, containing only two atoms. Planes defined by couples of first-neighbor bonds are shaded in pink (see text).Reuse & Permissions

Figure 2

(a) Brillouin zone (black lines) of the bulk diamond primitive cell. The shaded cube with vertices at the $L$ points is the BZ corresponding to the conventional unit cell (see text). The dashed lines are a guide to the eye. (b) Folding of the bulk BZ onto the two-dimensional BZ (dark orange rectangle) for the planar slab, or equivalently for the tube structure. (c) Ellipsoids of constant energy around the six $\Delta$ minima of the bulk Si (or Ge) crystal structure. The two conjugated ${\Delta }_{\rho }$ minima (red) fold into the $\Gamma$ point of the 2D BZ of the tube. (d) Schematic of the CB electronic structure of a Si-Ge tube, adopting the same colors of panel (c). The two conjugated ${\Delta }_{\rho }$ valleys fold into $\Gamma$ and their degeneracy is resolved due to the VS interaction. The two conjugated ${\Delta }_{\theta }$ valleys remain unfolded and do not interact; the same holds for the ${\Delta }_z$ valleys.Reuse & Permissions

Figure 3

Conduction-band structure of a Si${}_{16}$/Ge${}_9$ tube (black solid lines) along the $\Theta -\Gamma -$$Z$ path. The internal radius of this tube is $R_1=49.1$ nm. The band structure of a Si${}_{16}$/Ge${}_9$ planar slab ($R_1\to \infty$) grown on a Si buffer is also reported with red dashed lines.Reuse & Permissions

Figure 4

(a)–(c) Squared moduli of the wave functions, calculated at the $\Gamma$ point, for tubes with $N_{\text{Si}}=40$, and internal radius $R_1=R_{\min }$, $R_1=2R_{\min }$, and $R_1=3R_{\min }$, respectively. The corresponding values for $N_{\text{Ge}}$ are 22, 85, 123. The triangular band profiles for the ${\Delta }_{\rho }$ valleys are also shown. The white central area represents the Si region; the left (inner side) is made of the H passivation layer; the Ge region in on the right side. (d) VS of the ground-state doublet at $\Gamma$ as a function of $N_{\text{Si}}$. The value of $N_{\text{Ge}}$ is chosen in order to obtain the minimum internal radius of the tube $R_1=R_{\min }$ (black dots), $R_1=2R_{\min }$ (red triangles), or $R_1=3R_{\min }$ (blue squares); see text. The lines, connecting points with even or odd $N_{\text{Si}}$, are only a guide to the eye. The vertical orange line indicates the value of $N_{\text{Si}}$ for which panels (a)–(c) are calculated. (e) VS of the ground-state doublet at $\Gamma$ as a function of the tube internal radius $R_1$, for three different values of $N_{\text{Si}}$. Each curve starts from the minimum attainable value for $R_1$. The black dots labeled A, B, and C refer to the systems of panels (a), (b), and (c), respectively.Reuse & Permissions

Figure 5

(a)–(c) Diagonal strain components ${\epsilon }_{\theta \theta }\left(\rho \right)$, ${\epsilon }_{\rho \rho }\left(\rho \right)$, ${\epsilon }_{zz}\left(\rho \right)$, respectively, as a function of the radial coordinate $\rho$, for a Si${}_{16}$/Ge${}_9$ tube with internal radius $R_1=49.1$ nm. (d)–(f) Band-edge profiles (thick lines) for the ${\Delta }_{\theta }$, ${\Delta }_{\rho }$, ${\Delta }_z$ valleys, respectively, and square moduli of their wave functions, calculated at the $k$ points corresponding to the valley minima (see Fig. 3).Reuse & Permissions