Hole-spin qubits in Ge nanowire quantum dots: Interplay of orbital magnetic field, strain, and growth direction

Hole-spin qubits in quasi-one-dimensional structures are a promising platform for quantum information processing because of the strong spin-orbit interaction (SOI). We present analytical results and discuss device designs that optimize the SOI in Ge semiconductors. We show that at the magnetic field values at which qubits are operated, orbital effects of magnetic fields can strongly affect the response of the spin qubit. We study one-dimensional hole systems in Ge under the influence of electric and magnetic fields applied perpendicular to the device. In our theoretical description, we include these effects exactly. The orbital effects lead to a strong renormalization of the $g$ factor. We find a sweet spot of the nanowire (NW) $g$ factor where charge noise is strongly suppressed and present an effective low-energy model that captures the dependence of the SOI on the electromagnetic fields. Moreover, we compare properties of NWs with square and circular cross sections with ones of gate-defined one-dimensional channels in two-dimensional Ge heterostructures. Interestingly, the effective model predicts a flat band ground state for fine-tuned electric and magnetic fields. By considering a quantum dot (QD) harmonically confined by gates, we demonstrate that the NW $g$-factor sweet spot is retained in the QD. Our calculations reveal that this sweet spot can be designed to coincide with the maximum of the SOI, yielding highly coherent qubits with large Rabi frequencies. We also study the effective $g$ factor of NWs grown along different high-symmetry axes and find that our model derived for isotropic semiconductors is valid for the most relevant growth directions of nonisotropic Ge NWs. Moreover, a NW grown along one of the three main crystallographic axes shows the largest SOI. Our results show that the isotropic approximation is not justified in Ge in all cases.

Figure 1

(a) Sketch of a rectangular Ge NW with side lengths $L_x$ and $L_z$. The NW can be covered by a Si shell which induces strain in the Ge core. The NW including shell has total side lengths $L_{S,x}$ and $L_{S,z}$. The gates (gray) can be used to apply electric fields and to define an elongated QD. (b) Sketch of a planar Ge/SiGe heterostructure with a gate-defined one-dimensional channel. The height of the Ge layer in the center is $L$ and the channel is electrostatically confined in the $x$ direction with a harmonic confinement length $l_x$.

Figure 2

Solutions of $H_{zz}+H_E$ for the one-dimensional problem in the $z$ direction. In all plots the solid lines are the numerically exact Airy function solutions. The dash-dotted cyan line at $E=2\text{V}\mu {\text{m}}^{-1}$ marks the transition from the weak to strong electric field regime. (a) Lowest-energy solutions for ${\varepsilon }^{\mathrm{H}\left(\mathrm{L}\right)}\left(n_z\right)$ (independent of $B$) as a function of electric field $E$ applied in the $z$ direction (perpendicular to the NW axis). The dashed lines correspond to the analytical solution defined by Eq. (25) (weak electric field) and the dotted lines to the one defined by Eq. (28) (strong electric field). (b) Overlap between wave functions of HH and LH states. The dashed line corresponds to the analytical solution given by Eq. (26) (weak electric field). The dotted lines represent the result obtained using the wave functions defined in Eq. (29). For weak electric field $\langle {\varphi }_{n_z^{\prime }\ne n_z}^{\mathrm{H}}|{\varphi }_{n_z}^{\mathrm{L}}\rangle \ll \langle {\varphi }_{n_z}^{\mathrm{H}}|{\varphi }_{n_z}^{\mathrm{L}}\rangle$. (c) Matrix elements of $k_z$ (in dimensionless units) between the lowest-energy HH and LH states. The dashed line (weak electric field) corresponds to the analytical solution given by Eq. (27) and the dotted line (strong electric field) represents the result obtained using the wave functions in Eq. (29). In all plots we use $L_z=22\text{nm}$. Generally, we find that both analytical approximations agree well with exact numerical results shown by the solid lines. These results are used in Sec. 3c to estimate the $g$ factor in Eq. (38) and the SOI in Eq. (43) in the weak electric field case.

Figure 3

Energy levels $E_{n_x,n_z}^{\uparrow ,\downarrow }$ of $H_0$, defined in Eq. (33), of a Ge NW as a function of the perpendicular magnetic field $B$ in the $z$ direction. Here the parity-mixing term [see Eq. (30)] is neglected. The dashed lines correspond to Eq. (37), in which we also neglect orbital effects. The solid lines are obtained by the approach developed in Sec. 3d, where we include orbital effects. The Zeeman splitting is large ($\approx 1\text{meV}$) at $B=1\text{T}$. The strong renormalization of the $g$ factor due to orbital effects can be observed by comparing the solid and dashed lines. Here $L_x=L_z=22\text{nm}$ and $E=1\text{V}\mu {\text{m}}^{-1}$. We also note that because $L_x=L_z$ the states shown with red and green lines are close in energy at $B=0$. This quasidegeneracy is a result of the small value of $E$ used here and is lifted in the strong $E$ field limit.

Figure 4

(a) Color code and contour plot of the SOI strength ${\alpha }_{\text{so}}$ obtained from Eqs. (41) and (42) as a function of the perpendicular electric field $E$ and the confinement length $L_x/L_z$. We fix $L_z=22\text{nm}$ and $B=1\text{T}$. The red dots mark the maximal SOI and the solid orange line shows the fitting function discussed in the text. With the dotted cyan line we mark the case $L_x=L_z$, where the cross section is a square. The plot shows a strong dependence on the geometry of the cross section, where the SOI increases for decreasing ratio $L_x/L_z$ and increasing $E$. (b) Cuts of the plot in (a) for specific side length ratios $L_x/L_z$. The solid lines show the SOI strength according to Eqs. (41) and (42) and the dashed lines depict the small electric field approximation according to Eq. (43). The approximation captures very well the linear dependence on $E$ at low field and yields the correct slope for all the considered ratios of side lengths.

Figure 5

Effective SOI strength ${\alpha }_{\text{so}}$ of a Ge NW with square cross section, $L_x=L_z=22\text{nm}$, as a function of the electric field $E$ for different values of the magnetic field. The solid lines are numerical solutions obtained with the discrete basis given in Eq. (58), while the dashed lines show the results from the semianalytical formulas in Eqs. (56) and (57). For weak electric field the analytical result compares to the numerics very well. The SOI decreases with increasing magnetic field due to orbital effects. The dots mark the points where the ground-state dispersion relation becomes flat and where one needs to include in the effective theory terms that are of higher order in momentum (see Sec. 6).

Figure 6

Energy levels of the Hamiltonian in Eq. (1) calculated numerically by using the discrete basis ($0<n_x,n_z\le 16$) defined in Eq. (58) with (solid lines) and without (dashed lines) orbital effects as a function of the magnetic field. We note that the energy spectrum is substantially modified if orbital effects are taken into account. We consider a square cross-section NW with $L_x=L_z=22\text{nm}, E=1\text{V}\mu {\text{m}}^{-1}$, and ${\epsilon }_s=0$. Note that, in contrast to Fig. 3, here the parity-mixing term defined in Eq. (30) (without orbital effects) or in Eq. (46) (with orbital effects) is included.

Figure 7

(a) Effective $g$ factor $g_{\mathrm{eff}}$ from Eq. (60) and (b) average inverse effective mass $1/\overline{m}$ from Eq. (66) of a Ge NW with square cross section, $L_x=L_z=22\text{nm}$, as a function of the electric field $E$ at $k_y=0$ for different values of the magnetic field $B$. The $g$ factor is large even at the minimum ($g_{\mathrm{eff}}>5$), but it decreases with increasing magnetic field. As described in Sec. 6, the minimum is preserved in a QD, resulting in a sweet spot where charge noise is suppressed. The averaged inverse effective mass $1/\overline{m}$ starts from a small value at weak electric fields and approaches a value close to the average HH-LH inverse mass ${\gamma }_1/m$ at large $E$. The dots mark the points where the ground-state dispersion relation becomes flat and where one needs to include in the effective theory terms that are of higher order in momentum (see Sec. 6).

Figure 8

Spin-dependent masslike term $\beta$ defined in Eq. (67) as a function of the magnetic field $B$ at $k_y=0$ for different values of the electric field. At $B=0, \beta$ is zero due to time-reversal symmetry and it increases linearly for small $B$. At large $E, \beta$ increases very slowly. The dots mark the points where the ground-state dispersion relation becomes flat and where one needs to include in the effective theory terms that are of higher order in momentum (see Sec. 6). Here we use $L_x=L_z=22\text{nm}$.

Figure 9

Comparison of the effective parameters of Ge NWs with different geometries with (dashed lines) and without (solid lines) strain, obtained by the numerical diagonalization of the Hamiltonian in Eq. (1), as described in the text. Here $\square$ denotes a NW with a square cross section (blue) and $◯$ a NW with a circular cross section (orange); ch denotes the one-dimensional gate-defined channel (green). With s we label strained devices ($|{\varepsilon }_s|=0.62\%$ [51], ${\varepsilon }_s>0$ for the NW, and ${\varepsilon }_s<0$ for the channel). (a) Effective $g$ factor $g_{\mathrm{eff}}$. In the NW, this quantity has a minimum that persists even in the presence of strain. (b) SOI strength ${\alpha }_{\text{so}}$. The maximum value of ${\alpha }_{\text{so}}$ is reached at comparably weak electric field without strain. With strain a stronger electric field is required to reach the largest SOI. (c) Average effective mass $1/\overline{m}$. This quantity tends to converge to a value close to ${\gamma }_1/m$ and in unstrained NWs with circular cross section, it is negative at small $E$. Here $B=2\text{T}, L=22\text{nm}, R=L/\sqrt{\pi }\approx 12.4\text{nm}$, and $l_x=L/\pi \approx 7\text{nm}$.

Figure 10

Comparison of the spin-dependent masslike term $\beta$ in Ge NWs with different geometries with (dashed lines) and without (solid lines) strain as a function of perpendicular magnetic field $B$ in the $z$ direction. These results are obtained by the numerical diagonalization of the Hamiltonian in Eq. (1). Here $\square$ denotes the square cross section, $◯$ the circular cross section, and ch the one-dimensional gate-defined channel. With s we label the lines with strain ($|{\varepsilon }_s|=0.62\%$ [51], ${\varepsilon }_s>0$ for the NW, and ${\varepsilon }_s<0$ for the channel). Time-reversal symmetry at $B=0$ demands $\beta =0$. At small $B$ we observe a linear increase regardless of the geometry and strain. Here $E=2\text{V}\mu {\text{m}}^{-1}, L=22\text{nm}, R=L/\sqrt{\pi }\approx 12.4\text{nm}$, and $l_x=L/\pi \approx 7\text{nm}$.

Figure 11

Dispersion relations of a NW with square cross section of side length $L=22\text{nm}$ at $B=2\text{T}$ for (a) $E=0$, (b) $E=0.49\text{V}\mu {\text{m}}^{-1}$, and (c) $E=3\text{V}\mu {\text{m}}^{-1}$. The figures compare the exact numerical solution (black solid line) obtained by the diagonalization of the Hamiltonian in Eq. (1), the effective model quadratic in $k_y$ (red dashed line) in Eq. (59), and the effective model quartic in $k_y$ (blue dash-dotted line) discussed in Sec. 6b. We observe that the ground-state dispersion is well described by both effective models around $k=0$. Only when the condition from Eq. (72) is fulfilled as in (b), the $\mathcal{O}\left(k_y^2\right)$ model gives rise to a flat lowest-energy band and quartic corrections become more relevant.

Figure 12

Effective $g$ factor $g_{\mathrm{QD}}$ of a Ge NW QD according to Eq. (74) as a function of the electric field $E$ at $k_z=0$ for different values of the magnetic field $B$. The NW has a square cross section of side length $L=22\text{nm}$ and the QD confinement length is $l_y=35\text{nm}$. (a) The sweet spot is at very small electric fields, where the NW is unstrained, and (b) it is shifted to larger electric fields when strain is included. We consider here a strain tensor element ${\varepsilon }_s=0.62\%$.

Figure 13

Position of the sweet spot $E_{\mathrm{sw}}$ (black solid line) of the effective $g$ factor $g_{\mathrm{QD}}$ in a strained Ge NW QD as a function of the QD confinement length $l_y$. We consider here $B=0.1\text{T}$ and we calculated $g_{\mathrm{QD}}$ [see Eq. (74)] numerically by using the discrete basis defined in Eq. (58) for diagonalizing $H$ defined in Eq. (1). The NW has a square cross section of side length $L=22\text{nm}$. The blue solid line shows the value of $g_{\mathrm{QD}}$ at the sweet spot and the horizontal black dashed line marks the electric field at which the SOI is maximal. By changing the size of the QD, $l_y$, we can tune the position of the sweet spot of $g_{\mathrm{QD}}$ to be at the same value of the electric field at which SOI also achieves its maximum strength.

Figure 14

Qubit parameters of a Ge NW QD as a function of electric and magnetic field. The dashed vertical line marks $E=14.3\text{V}\mu {\text{m}}^{-1}$, where the minimum of $g_{\mathrm{QD}}$ and the maximum of $E_{\text{so}}$ are located. (a) Rabi frequency ${\omega }_R$ according to Eq. (76) for $E_{ac}=0.02\text{V}\mu {\text{m}}^{-1}$. The Rabi frequency increases linearly with $B$ due to the resonance driving frequency ${\omega }_D$ and is almost constant with $E$. (b) Resonance driving frequency ${\omega }_D$ according to Eq. (75). (c) Dephasing rate $1/T_2^{*}$ according to Eq. (77). Independently of the magnetic field, $1/T_2^{*}$ has a minimum at $E=14.3\text{V}\mu {\text{m}}^{-1}$. However, for stronger $B$ the dephasing rate increases faster when the electric field is not exactly at the sweet spot. (d) The spin-orbit energy $E_{\text{so}}=\overline{m}{\alpha }_{\text{so}}/2{\hslash }^2$, as well as the QD $g$ factor in Fig. 12, starts to depend on the magnetic field noticeably only at stronger $B$. We choose $L=22\text{nm}, l_y=35\text{nm}$, and $\sqrt{\langle \partial E^2\rangle }/E={10}^{-3}$.

Figure 15

Effective $g$ factor $g_{\mathrm{eff}}$ for different growth directions of the NW, obtained numerically from the diagonalization of the Hamiltonian given by Eq. (1) and the anisotropic LK Hamiltonian given by Eq. (80), using the discrete basis defined in Eq. (58). The four panels on top show the rotations of the coordinate system corresponding to the color code in the legend. In each case one axis is fixed and then we rotate by the angle $\phi$ around it. For the blue, green, and purple lines the NW axis ($y$) is fixed parallel to the crystallographic directions [001], [100], and [111], respectively. For the orange line, we fix $z\parallel \left[001\right]$. The dashed square indicates the NW cross section and how it is fixed with respect to the coordinate axes. (a) Without orbital effects, without strain; (b) without orbital effects, with strain; (c) with orbital effects, without strain; and (d) with orbital effects, with strain. The $g$ factor depends on the growth direction and this dependence changes when orbital effects are included. This leads to the conclusion that the isotropic approximation is not well justified in Ge NWs in a general case without strain. In core/shell NWs with strain the isotropic approximation is better justified. We choose a NW with a square cross section with side length $L_x=L_z=22\text{nm}, B=0.1\text{T}, E=0$, and strain tensor element ${\varepsilon }_s=0.62\%$.

Figure 16

Dispersion relation of the lowest-energy states of a circular NW calculated numerically by diagonalizing the Hamiltonian in Eq. (1) as described in Sec. 4 for (a) $E=0$ and $\overline{m}<0$ and (b) $E=1\text{V}\mu {\text{m}}^{-1}$ and $\overline{m}>0$. Here $B=2\text{T}$ and $R=22\text{nm}/\sqrt{\pi }$.