Light-hole spin confined in germanium

Hole spins in group-IV planar gated quantum dots are promising candidates for robust and scalable qubits [1, 2, 3, 4, 5, 6, 7]. Developing these qubits has been so far exclusively based on heavy-hole (HH) states, as the materials currently used are restricted to compressively strained germanium (Ge) heterostructures. Interestingly, the advent of the germanium/germanium-tin (Ge/Ge_1-xSn_x) material system provides an additional degree of freedom to tailor the valence band character in tensile-strained Ge, thus paving the way to implement silicon-compatible platforms for light-hole (LH) spin qubits [8, 9]. These LH qubits share many of the advantages benefiting the HH ones but also bring about other attractive characteristics pertaining to LHs and Ge/Ge_1-xSn_x heterostructures. These include a strong Rashba-type spin-orbit interaction (SOI) [9] and an efficient coupling with proximity-induced superconductivity [10] in addition to the bandgap directness [11] relevant to coupling with optical photons. These characteristics can expand the functionalities of hole spin qubits by facilitating the implementation of hybrid superconductor-semiconductor devices and photon-spin interfaces.

Although Ge_1-xSn_x semiconductors have been the subject of extensive studies in recent years, research in this area has mainly been focused on integrated photonics and optoelectronics leveraging Ge_1-xSn_x strain- and composition-dependent bandgap [11] leaving their spin properties practically unexplored [12, 13, 14, 9]. As a matter of fact, studies on hole spin in Ge/Ge_1-xSn_x are still conspicuously missing in the literature. Herein, this work addresses the dynamics of LH spin confined in Ge_1-xSn_x/Ge/Ge_1-xSn_x heterostructures and elucidates the key parameters affecting its behavior as a function of strain, well thickness, Sn content, and magnetic field orientation. First, the article describes and discusses the electronic structure of tensile-strained Ge on Ge_1-xSn_x. Second, the parameters defining the band alignment of the Ge_1-xSn_x/Ge/Ge_1-xSn_x quantum well are introduced and the criteria for LH confinement in Ge are established. The third section outlines the Hamiltonian of the in-plane motion of LHs for out-of-plane and in-plane magnetic fields yielding LH parameters such as the effective mass, the $g$-tensor, and the Rashba-SOI parameters. LH-HH mixing within the LH ground state is investigated in the fourth section. It is important to note that the focus here is on LH properties in the planar system without the effects of electrostatic in-plane confinement introduced in quantum dot systems. [9]

The preceding results demonstrate that the lattice mismatch between Ge_1-xSn_x alloys and Ge provides an additional degree of freedom to engineer the tensile strain required to confine LHs in Ge. According to Fig. 2(a), the region of interest, as defined by the parameters $(x,\varepsilon_{\text{BR}})$, lies in the range where $x$ is above $0.12$ and the compressive strain in the barriers $\varepsilon_{\text{BR}}$ is below $-0.4\%$ (i.e., $|\varepsilon_{\text{BR}}|\lesssim 0.4\%$). In this range, all band offsets $\Delta E_{i}$ are positive and the Ge layer is of direct bandgap. Ge_1-xSn_x layers at Sn content in the proposed range have already been demonstrated experimentally [11]. However, the addition of a highly tensile strained Ge layer on top of strain-relaxed Ge_1-xSn_x is still under development. For instance, the authors in Ref. [8] reported a $1.65\%$ tensile-strained Ge quantum well on partially relaxed Ge_0.854Sn_0.146 barriers with a residual strain to $\varepsilon_{\text{BR}}\approx-0.54\%$. This system would be located near the crossing between the $0\,\text{meV}$ $\Delta E_{1}$ line and the $50\,\text{meV}$ LBO line in Fig. 2(a), very close to the optimal region of interest mentioned earlier. Strain relaxation in the barriers is necessary to enhance confinement in Ge, while relaxing the criterion of minimal well thickness $w_{0}$ required for a LH-like valence band edge ($\Delta E_{2}>0$). The ideal amount of strain relaxation for a given barrier Sn content can be estimated from Fig. 2(b). For example, a barrier with $x=0.14$ does not allow a LH-like valence band edge for $|\varepsilon_{\text{BR}}|>0.542\%$ compressive strain. The additional requirement $w_{0}<h_{c}$, where $h_{c}$ is the critical thickness of Ge, further reduces the range of $|\varepsilon_{\text{BR}}|$ to around $<0.5\%$ compressive strain. Reducing the amount of Sn in the barriers relaxes the limit imposed by the critical thickness $h_{c}$, but at the cost of a smaller LBO. In contrast, increasing $x$ to $0.18$ for instance slightly increases the range for $|\varepsilon_{\text{BR}}|$ to around $<0.525\%$, and increases significantly the LBO and $\Delta E_{1}$ [Fig. 2(a)] but at the expense of a smaller $h_{c}$ and a narrower window $w_{0}<w<h_{c}$, but still in the range typically achievable in epitaxial growth experiments.

The in-plane effective mass $\gamma$ [Fig. 3(a)] shows a strong dependence on both $x$ and $w$, with small $\gamma$ expected from the general rule that LHs are heavier in the plane than HHs. There is also an interesting feature where the dispersion changes from a hole-like ($\gamma<0$) to an electron-like ($\gamma>0$) curvature at $\mathbf{k}_{\parallel}=0$. In the hole-like regime, the valence band edge is formed by a single valley located at $\mathbf{k}_{\parallel}=0$. In contrast, in the electron-like regime, the valence band edge consists of four valleys, each located a distance $k_{0}^{*}$ from $\mathbf{k}_{\parallel}=0$ along the four equivalent $\left<110\right>$ crystallographic directions in the QW plane. For instance, for a $5\,\text{nm}$ well at $x=0.13$ and $\varepsilon_{\text{BR}}=0$, the valley minima are located at $k_{0}^{*}\approx 0.0813\,\text{nm}^{-1}$ away from $\mathbf{k}_{\parallel}=0$. For larger wavevectors, the dispersion goes away from the bandgap as required, owing to $k^{4}$-terms or higher that are not taken in account by the effective Hamiltonians (7) and (9). According to Fig. 3(a), electron-like subbands occur for small $w$ and large $x$. This effect takes place for two reasons. The first is when the LH subband anticrosses a neighboring HH subband such that the curvature is inverted at $\mathbf{k}_{\parallel}=0$. This is typical in systems where the ground state is HH-like and the first LH subband is allowed to mix strongly with the first excited HH subband [20], or when the LH subband is close to a HH continuum (e.g., when $w$ is small). When a LH is far from neighboring HH levels (e.g., when tensile strain is large), mixing decreases, as illustrated in Fig. 4(b), and becomes too weak to invert the curvature. The second reason for a curvature change is related to the anticrossing between the LH and the SO bands in the bulk dispersion of Ge for $k_{z}>0$ [Fig. 1(a)]. This anticrossing results in a curvature sign change of the LH band at some point $k_{z}^{*}$ such that the bulk energy dispersion $E(k_{x},k_{y};k_{z})$ has a hole-like (electron-like) curvature when $k_{z}$ is fixed to a value smaller (larger) than $k_{z}^{*}$ and $k_{x,y}$ are close to zero. For a quantum well along the $z$ direction, the reciprocal-space envelope functions $\tilde{\psi}(k_{z})$ become wider for thinner wells and thus get a larger contribution from the electron-like regions in $k$-space, resulting in a dispersion with inverted curvature at $\mathbf{k}_{\parallel}=0$.

A major difference with $\eta$ subbands is the large $g_{\parallel}$ compared to HH systems [21, 22, 23, 6, 7]. A comparison between $g_{\parallel}$ and $g_{\perp}$ reveals an anisotropy for well thicknesses away from ${\sim}10\,\text{nm}$. Both components have a stronger dependence on $w$ than on $x$ but have opposite behavior with $w$ due to how they couple with neighboring subbands. For the out-of-plane component, $g_{\perp}\sim 2{\color[rgb]{0,0,0}\kappa}$ for large $w$ because the coupling with neighboring levels becomes weaker as LH1 gets further from the continuum. In contrast, the in-plane $g$-factor does not depend on couplings with neighboring HH levels [c.f. (10)] and is instead more influenced by the spatial distribution of the envelopes across the layers. Thus, for large (small) $w$, $g_{\parallel}\sim 4{\color[rgb]{0,0,0}\kappa}$ with $\kappa$ being that of Ge (Ge_1-xSn_x).

Another peculiar feature associated with $\eta$ subbands is the absence of direct connection between $\gamma$ and $g_{\perp}$ in contrast with HHs (i.e., see Eq. (5) in Ref. [24]). Perturbation theory gives the following for $\gamma$ and $g_{\perp}$:

However, the result is different for $\eta$ subbands due to the additional $D$ term in (19). The latter is also related to the nonzero $\beta_{1}$ coefficient of $\eta$ subbands [9].

Rashba parameters follow the general behavior $\alpha_{i}\to 0$ as $w$ increases. This is caused by a reduced sensitivity of the wavefunction to electric fields when the level does not spread as extensively into the barriers. Although the QW is characterized by relatively small LBOs ($\lesssim 100\,\text{meV}$) and small out-of-plane LH effective masses, the device operation can comfortably sustain realistic DC electric fields along the growth direction regardless of the well thickness and LBO without inducing any envelope leak into the barriers. In devices where space inversion symmetry needs to be broken, such as in electric dipole spin resonance (EDSR) experiments, this should not be an issue as the relevant Rashba parameter $\alpha_{3}$ for EDSR, (which is proportional to ${\color[rgb]{0,0,0}\gamma_{2}}+{\color[rgb]{0,0,0}\gamma_{3}}$) is one order of magnitude larger than the $\alpha_{2}$ Rashba parameter (proportional to ${\color[rgb]{0,0,0}\gamma_{2}}-{\color[rgb]{0,0,0}\gamma_{3}}$), thus requiring smaller out-of-plane fields [9].

This work demonstrates how Ge_1-xSn_x/Ge/Ge_1-xSn_x heterostructures can be tailored to achieve a selective confinement of LHs in Ge while pushing HHs in to the Ge_1-xSn_x barriers. For a sufficiently large Sn content ($x>0.12$), small residual compressive strain in the barriers ($|\varepsilon_{\text{BR}}|<0.4\%$) and a well thickness $w>w_{0}$, the LH ground state emerges from within the HH continuum, thus yielding a pure LH-like valence band edge ($\Delta E_{2}>0$). This regime also corresponds to a direct bandgap in both the well and its barriers, owing to the large tensile strain in the Ge well and the high Sn content in the barriers. Satisfying the condition of $\Delta E_{2}>0$ imposes a threshold for residual strain in the barriers beyond which a LH-like VB edge becomes virtually impossible.

The in-plane effective mass, the out-of-plane and in-plane $g$-factor, and the Rashba parameters $\alpha_{1,2,3}$ were computed by explicitly taking in account the spread of the LH envelopes into the barriers and the coupling with the neighboring HH continuum. Small inverse effective masses $\gamma$ are obtained. A peculiar sign change in $\gamma$ appearing for small well thicknesses ($w\lesssim 7\,\text{nm}$) is observed and attributed to the proximity of the LH to the HH continuum (larger LH-HH mixing) and the contribution of the SO band in the LH spinor. An increasingly strong anisotropy in the $g$-factor components is also observed for well thicknesses away from ${\sim}10\,\text{nm}$. Most notably, the in-plane component of the $g$-tensor is significantly larger than what is expected in HH systems. A nonzero linear Rashba parameter $\alpha_{1}$ was obtained, as anticipated for LH systems, with an $\alpha_{3}$ coefficient one order of magnitude larger than $\alpha_{2}$.