Charge-noise resilience of two-electron quantum dots in Si/SiGe heterostructures

When quantum dots contain more than one electron, new possibilities emerge for defining and controlling qubits. Theoretical studies have shown that GaAs dots with multiple electrons may be inherently protected from charge noise Barnes et al. (2011); Mehl and DiVincenzo (2013); Bakker et al. (2015), and recent experiments confirm some of these predictions Kodera et al. (2009); Higginbotham et al. (2014); Chen et al. (2017). Recent progress in Si-based multi-electron qubits Shi et al. (2012); Kim et al. (2014); Harvey-Collard et al. (2017); Leon et al. (2020); Yang et al. (2020); Petit et al. (2020) brings renewed attention to such noise-reduction schemes. However, the theoretical description of GaAs dots is not applicable to Si, due to the presence of both orbital and valley degrees of freedom for the electrons. While the orbital energies are determined by electrostatic confinement, similar to GaAs, the conduction-band valley splitting is determined by details of the quantum-well interface Friesen et al. (2007). It is technically challenging to describe such behavior because of the strong electron-electron (e-e) interactions that must be treated nonperturbatively, and because a minimal model of Si must include details of the Si band structure, as well as atomic-scale disorder at the quantum well interface, which gives rise to valley-orbit coupling (VOC) Friesen and Coppersmith (2010).

Here we develop a complete theoretical toolbox for describing two-electron dots in Si. We first apply these tools to study low-energy spin singlet and triplet states. Solving the two-electron wavefunctions as a function of orbital confinement energy $E_{\text{orb}}\sim\hbar\omega$ reveals two fundamentally different triplet excitations, based on their valley or orbital character, as illustrated in Figs. 1(a)-1(c). These excitations also have different coherence properties. For small $\hbar\omega$, the low-energy states that define the qubit are the singlet (S) and orbital triplet (T${}_{\text{orb}}$). Since these states have dissimilar charge distributions, they couple differently to electrical fluctuations [e.g., a nearby charge trap (CT), as shown in Fig. 1(d)], resulting in dephasing. For stronger confinement (larger $\hbar\omega$), the low-energy states are S and the valley triplet (T${}_{\text{val}}$). In this case, the charge distributions are very similar, and they respond similarly to electrical fluctuations [Fig. 1(e)], yielding qubits that are resilient to dephasing.

Theoretical methods. We compute two-electron wavefunctions in two steps. First, we use a tight-binding (TB) approach to obtain single-electron wavefunctions Boykin et al. (2004a, b). This method accounts for the essential features of the Si bandstructure and allows an atomistic description of disorder at the quantum well interface. Second, we incorporate these single-electron wavefunctions into a full configuration interaction (FCI) Szabó and Ostlund (1982) scheme for computing two-electron wavefunctions, nonperturbatively, while accounting for strongly interacting electrons. The full method is summarized in Fig. 2; additional details are given in Ref. Ercan et al. .

In step one, the single-electron Hamiltonian for the 3D heterostructure is assumed to be separable in terms of the $(x,z)$ vs. $y$ variables, where $\hat{x}$, $\hat{y}$, and $\hat{z}$ are the crystallographic axes. We consider atomistic disorder only in the $x$-$z$ plane. This simplification allows us to treat the $(x,z)$ variables using TB methods, while solving the separable $y$ wavefunctions using continuum effective-mass theory, which allows us to achieve full convergence on a more practical time scale. The single-electron Hamiltonian can then be written as

Here, the kinetic energy is given by

where the integer indices $i_{x}$ and $i_{z}$ refer to TB sites along the $\hat{x}$ and $\hat{z}$ axes, respectively. We have suppressed the spin index here, because our Hamiltonian is independent of spin, and the effects of Pauli exclusion become important only at the FCI stage of the calculation. The hopping parameters $t_{1}=0.68$ eV and $t_{2}=0.61$ eV are chosen to reproduce the key features of the two-fold degenerate structure at the bottom of the Si conduction band: (1) valleys centered at $\mathbf{k}=\pm k_{0}\hat{z}$ in reciprocal space, where $k_{0}=\pm 0.82(2\pi/a)$, $a=5.43$ Å is the cubic lattice constant, and $\Delta z=a/4$ is the grid spacing, and (2) longitudinal effective mass, $m_{l}=0.916\ m_{0}$. The hopping parameter $t_{3}=-0.026$ eV gives the correct transversal effective mass, $m_{t}=0.191\,m_{0}$, for the grid spacing $\Delta x=2.79$ nm Abadillo-Uriel et al. (2018). We note that $\Delta x$ can be much larger than $\Delta z$, because there are no fast valley oscillations along $\hat{x}$. The vertical (quantum well) confinement potential is given by

where $\Theta_{i_{x},i_{z}}$ is a step function that takes the value $1$ on a SiGe site and $0$ on a Si site, $V_{\mathrm{QW}}=150$ meV is the band offset between Si and SiGe, $e=|e|$ is the elementary charge, and $\bm{F}^{\mathrm{e}}=(F^{\mathrm{e}}_{x},0,F^{\mathrm{e}}_{z})$ is the electric field perpendicular to the interface due to the gate electrodes. All calculations assume 10 nm quantum wells. Interface disorder is implemented via the choice of $\Theta_{i_{x},i_{z}}$. In this work we consider an interface tilted slightly away from $\hat{z}$, with uniformly-distributed single-atom steps of height $a/4$ and width $W$, as depicted in Fig. 2(a). The field $\bm{F}^{\mathrm{e}}$ is taken to be perpendicular to the tilted interface. The lateral confinement potential is of electrostatic origin, and is taken to be parabolic, with the form

We solve $H^{\mathrm{1e}}\phi_{i}=\varepsilon_{i}\phi_{i}$ to obtain the single-electron basis states $\phi_{i}$ used in the FCI calculation. A typical energy level structure is shown in Fig. 2(b).

In step two, we solve the two-electron Hamiltonian, which includes the Coulomb interaction term,

where $\epsilon_{0}$ is the permittivity of free space, and $\epsilon_{r}=11.4$ is the dielectric constant of low-temperature Si Faulkner (1969). The Coulomb matrix elements are computed using a combination of numerical and analytical methods, taking advantage of the analytical wavefunction solutions along $\hat{y}$. We then solve for the eigenvalues and eigenstates of $H^{\mathrm{2e}}\Psi_{q}=E_{q}\Psi_{q}$ using FCI methods: $H^{\mathrm{2e}}$ is diagonalized in a basis of Slater determinants generated from spin orbitals $\chi=\phi\,\otimes\uparrow$ or $\chi=\phi\,\otimes\downarrow$, as shown in Figs. 2(c) and 2(d). A finite set of determinants is used and the convergence of the FCI method is checked, as in Ref. Ercan et al.

Results. To better understand the nontrivial behavior arising from e-e interactions and VOC, it is instructive to consider these effects one at a time, as summarized in Fig. 1. In the absence of interactions, the ground-state singlet (S) is formed from two electrons, each in the lowest orbital level. There are two different types of single-electron excitations above the ground state: valleys and orbitals, with corresponding single-electron excitation energies, $E_{\text{val}}$ and $E_{\text{orb}}$. Two-electron excitations take on the character of these single-electron excitations, yielding distinct valley (T${}_{\text{val}}$) and orbital (T${}_{\text{orb}}$) triplets, with excitation energies $E_{\text{ST}_{\text{val}}}=E_{\text{val}}$ and $E_{\text{ST}_{\text{orb}}}=E_{\text{orb}}$. Here we assume $E_{\text{val}}<E_{\text{orb}}$, as consistent with many qubit experiments Borselli et al. (2011); Mi et al. (2017); Schoenfield et al. (2017); Chen et al. (2021); Zajac et al. (2016); Dodson et al. ; Corrigan et al. . The charge distribution of T${}_{\text{val}}$ is identical to S; however T${}_{\text{orb}}$ is quite different, due to its $p$-orbital contribution. Now, introducing e-e interactions [Fig. 1(b)], we find that many additional Slater determinants contribute to the two-electron wavefunctions. To a very good approximation, the triplets retain their valley or orbital character; however the e-e interactions strongly suppress $E_{\text{ST}_{\text{orb}}}$ below $E_{\text{orb}}$, while having almost no effect on $E_{\text{ST}_{\text{val}}}\approx E_{\text{val}}$ Ercan et al. . The charge distributions for S and T${}_{\text{val}}$ take the form of Wigner-molecule doughnuts, with dimples at their centers Bryant (1987); Yannouleas and Landman (1999); Egger et al. (1999); Filinov et al. (2001); Reusch et al. (2001). For stronger interactions [i.e., smaller $\hbar\omega$, Fig. 1(c)], there is a crossover from T${}_{\text{val}}$ to T${}_{\text{orb}}$-dominated excitations. In this regime, the qubit states (S and T${}_{\text{orb}}$) have very different charge distributions.

We finally consider a realistic dot model including e-e interactions and VOC. Typical results for excitation energies are shown in Fig. 3(a). To begin, we consider wide steps, $W=101$ nm, to clearly demonstrate the types of behavior observed as a function of $\hbar\omega$. We also position the dot as far as possible from a step, with $d=W/2$, as depicted in Fig. 2(a). For small $\hbar\omega$ (weak confinement), T${}_{\text{orb}}$ is the dominant excitation, with an excitation energy, $E_{\text{ST}_{\text{orb}}}\lesssim k_{B}T$, that is typically too small to enable high fidelity qubit initialization or readout. For larger $\hbar\omega>600$ $\mu$eV, there is a crossover to T${}_{\text{val}}$-dominated behavior. If the valley splitting is also $\gtrsim 100$ $\mu$eV, the energy $E_{\text{ST}_{\text{val}}}$ will certainly be large enough for practical applications. (For quantum-dot hybrid qubits, slightly smaller $E_{\text{ST}_{\text{val}}}$ are preferred Shi et al. (2012); Kim et al. (2014).) For this calculation, we used $\absolutevalue{\bm{F}^{\mathrm{e}}}$=0.6 MV/m, which gives $E_{\text{val}}\sim 105$ $\mu$eV. For $\hbar\omega$ values below the triplet crossover, $E_{\mathrm{ST_{val}}}$ drops quickly, as the dot (with diameter $D=2\sqrt{\hbar/m_{t}\omega}$) begins to strongly overlap with different interface steps, suppressing the valley splitting Friesen et al. (2007), and causing VOC Friesen and Coppersmith (2010). The value of $W=2D$ used in Fig. 3(a) was chosen such that the ST splitting is not significantly affected by VOC. We can also explore other regimes by computing the excitation energies as a function of $\hbar\omega$ while holding $W=2D$ fixed, as shown in Fig. 3(b), to ensure that the dot does not interact excessively with the steps. (This requires simultaneously changing $W$ as $\hbar\omega$ is varied.) Here, we again observe a crossover from T${}_{\text{orb}}$ to T${}_{\text{val}}$-dominated behavior. However, because the wavefunction remains far from the nearest step, $E_{\mathrm{ST_{val}}}$ is not suppressed for low $\hbar\omega$. In contrast, the insets of Fig. 3(b) show that smaller $W/D$ ratios cause significant reductions in ST splittings, making it more difficult to achieve acceptable values for qubit applications. In the Supplementary Materials we further discuss interface profiles that give small ST splittings.

The crossover from T${}_{\text{orb}}$ to T${}_{\text{val}}$-dominated behavior has a strong effect on qubit coherence, because the T${}_{\text{orb}}$ and T${}_{\text{val}}$ charge distributions couple differently to charge fluctuations. For the quantum-dot hybrid qubit Shi et al. (2012); Kim et al. (2014), for example, $E_{\text{ST}}$ determines the qubit energy, and fluctuations of $E_{\text{ST}}$ lead directly to dephasing Thorgrimsson et al. (2017). The insets of Fig. 3(a) show typical in-plane electron densities for $\hbar\omega=550$ and 650 $\mu$eV, which bracket the crossover between low-energy triplet states. As noted above, the charge distribution of S is very similar to that of T${}_{\text{val}}$ but not T${}_{\text{orb}}$. These distinctions are still valid when the VOC, which mixes the valley and orbital character of the wavefunctions, is weak but nonzero. Consequently, charge noise affects S and T${}_{\text{val}}$ similarly, yielding weak fluctuations of $E_{\mathrm{ST_{val}}}$, but much larger fluctuations of $E_{\mathrm{ST_{orb}}}$. The high-$\hbar\omega$ regime is therefore expected to yield qubits with much better coherence properties.

To quantify these claims, we consider the effect of a charge trap, as depicted in Figs. 1(d) and 1(e). First-order perturbation theory is used to estimate the shifts in $E_{\text{ST}}$ due to the electrostatic potential $V_{\text{CT}}$ of the trap,

We note that interfacial disorder breaks the circular symmetry, allowing us to use nondegenerate perturbation theory for the circular confinement potentials used in this work. We evaluate Eq. (6) for the geometry shown in the inset of Fig. 3(c), assuming a 10 nm Si quantum well, a SiGe barrier of width 40 nm, a 1 nm Si cap, a 5 nm layer of $\mathrm{Al_{2}O_{3}}$, and a metal top gate. For simplicity, we consider the gate to be an infinite plane giving rise to a uniform electric field, $\bm{F}^{\mathrm{e}}$, but not the dot confinement potential. The dot confinement is simply given by Eq. (4), and we assume the image potentials for the dot electrons are subsumed into this potential. We take the charge trap to be located $\sim$50 nm above the dot, inside the oxide layer, as suggested by recent experiments Connors et al. (2019). Due to its proximity to the top gate, the trap is strongly screened. Following Ref. Takashima and Ishibashi (1978), and using the dielectric constants of the different layers, we obtain the leading terms in the potential, $V_{\text{CT}}\approx\frac{1.13e}{4\pi\epsilon_{0}\epsilon_{r}}\left(\absolutevalue{\bm{R}-\bm{r}}^{-1}-\absolutevalue{\bm{R}_{\text{im}}-\bm{r}}^{-1}\right)$, where $\bm{R}$ is the position of the trap, $\bm{R}_{\text{im}}$ is the position of its mirror image inside the metal, and $\bm{r}$ is the position of the dot. To estimate the distance between the trap and the top gate, we consider a double-dot geometry with an interdot separation of 200 nm. The top inset of Fig. 3(c) shows the shift $\delta\varepsilon$ in the double-dot detuning parameter $\varepsilon$, caused by a charge trap separated from the dot by a lateral distance $x_{\text{CT}}$. For a trap located 0.1 nm below the gate, the resulting shifts fall into the range 4-9 $\mu$eV, as consistent with experimental measurements of detuning fluctuations $\sigma_{\varepsilon}$ Wu et al. (2014); Thorgrimsson et al. (2017); Mi et al. (2018); Watson et al. (2018).

Our perturbative results for the dominant ST splittings are shown in Fig. 3(c), as obtained on either side of the triplet crossover, at locations $\hbar\omega=550$ $\mu$eV (ST${}_{\text{orb}}$) or $\hbar\omega=650$ $\mu$eV (ST${}_{\text{val}}$). Here, $x_{\text{CT}}=0$ corresponds to a charge trap located directly above the dot. As expected, the energy fluctuations are strongly suppressed for ST${}_{\text{val}}$. We can also estimate dephasing rates for a double-dot qubit from the relation $\Gamma^{*}_{2}=\frac{\sigma_{\varepsilon}}{\sqrt{2}\hbar}\absolutevalue{\partial E_{Q}/\partial\varepsilon}$ Dial et al. (2013); Petersson et al. (2010); Abadillo-Uriel et al. (2018). Assuming the charge trap has equal switching rates between its empty and occupied states, we can approximate the standard deviation of $E_{\mathrm{ST_{val(orb)}}}$ fluctuations as $(1/2)\delta E_{\mathrm{ST_{val(orb)}}}$, so that $\sigma_{\varepsilon}\absolutevalue{\partial E_{Q}/\partial\varepsilon}\approx(1/2)\delta E_{\mathrm{ST_{val(orb)}}}$. Dephasing estimates obtained in this way are also reported in the main panel of Fig. 3(c). For the settings considered here, we see that the dephasing rates for ST${}_{\text{val}}$ vs. ST${}_{\text{orb}}$ can differ by a very large factor ($\sim 10$), depending on the lateral position of the trap. The lower curve, associated with $E_{\mathrm{ST_{val}}}$, appears to be more consistent with recent experimental measurements of $\Gamma^{*}_{2}=6$-210 MHz in a Si hybrid qubit Thorgrimsson et al. (2017). These results may also help to explain the much higher dephasing rates observed in a GaAs hybrid qubit Cao et al. (2016), $\Gamma^{*}_{2}=0.12$-1.4 GHz, which has no T${}_{\text{val}}$ state. (Note that the current results are obtained using Si materials parameters.)

Summary. Using a combination of tight-binding and full-configuration-interaction calculations, we have shown that an important crossover occurs in the low-lying triplet state of two-electron dots in Si/SiGe: for weak confinement, the orbital triplet is the dominant excitation, while for strong confinement the valley triplet is dominant. We find that strong e-e interactions and valley-orbit coupling (induced by atomic steps at the quantum-well interface) both play key roles in this behavior, in the physically relevant operating regime. We further show that the charge distribution of the valley triplet is similar to that of the singlet, but differs from the orbital triplet. Consequently, qubits based on valley-triplet excitations are much more resilient to charge noise. These results are crucial for successful implementations of multi-electron qubits in Si/SiGe dots.