Interface induced spin-orbit interaction in silicon quantum dots and prospects for scalability

TL;DR

This paper investigates how interface steps at Si/SiGe or Si/SiO2 interfaces affect spin qubit coherence in silicon quantum dots, and proposes methods to control and enhance dephasing times through spin-orbit interaction anisotropy.

Contribution

The study combines atomistic calculations and experiments to link interface steps with g-factor variations and dephasing, and shows how magnetic field orientation can optimize qubit coherence.

Findings

Interface steps cause significant g-factor and Stark shift variations.

Experimental observations align with theoretical predictions.

Aligning magnetic fields along specific crystal directions can improve dephasing times by over an order of magnitude.

Abstract

We identify the presence of monoatomic steps at the Si/SiGe or Si/SiO$_2$ interface as a dominant source of variations in the dephasing time of Si quantum dot (QD) spin qubits. First, using atomistc tight-binding calculations we show that the g-factors and their Stark shifts undergo variations due to these steps. We compare our theoretical predictions with experiments on QDs at a Si/SiO$_2$ interface, in which we observe significant differences in Stark shifts between QDs in two different samples. We also experimentally observe variations in the $g$-factors of one-electron and three-electron spin qubits realized in three neighboring QDs on the same sample, at a level consistent with our calculations. The dephasing times of these qubits also vary, most likely due to their varying sensitivity to charge noise, resulting from different interface conditions. More importantly, from our calculations we show that by employing the anisotropic nature of the spin-orbit interaction (SOI) in a Si QD, we can minimize and control these variations. Ultimately, we predict that the dephasing times of the Si QD spin qubits will be anisotropic and can be improved by at least an order of magnitude, by aligning the external DC magnetic field towards specific crystal directions.

Equations

$$\delta g_{\pm }^{[110]}=2\frac{\left|e\right|⟨z⟩}{{\mu }_{\text{B}}\hslash }\left(-{\alpha }_{\pm }+{\beta }_{\pm }\right)$$

$$\delta g_{\pm }^{[110]}\approx 2\frac{\left|e\right|⟨z⟩}{{\mu }_{\text{B}}\hslash }{\beta }_{\pm }$$

$$T_2^{\ast }=\frac{\sqrt{2}\hslash }{\Delta F_z|\frac{dg}{d{F}_z}|{\mu }_{\text{B}}{B}_{\text{ext}}}$$

$$\delta g_{\pm }\approx 2\frac{\left|e\right|⟨z⟩}{{\mu }_{\text{B}}\hslash }\left(-{\alpha }_{\pm }+{\beta }_{\pm }\sin 2\varphi \right)$$

$$\delta g_{\pm }^{[100]}\approx -2\frac{\left|e\right|⟨z⟩}{{\mu }_{\text{B}}\hslash }{\alpha }_{\pm }$$

$$\frac{\delta g_{\pm }^{[100]}}{\delta g_{\pm }^{[110]}}\approx \frac{{\alpha }_{\pm }}{{\beta }_{\pm }}$$

$$\frac{1}{T_2^{\ast }\left(v_{\pm }\right)}=\Delta F_z\frac{2\left|e\right|B_{\text{ext}}}{\sqrt{2}{\hslash }^2}\left|-\frac{d\left(⟨z⟩{\alpha }_{\pm }\right)}{d{F}_z}+\sin 2\varphi \frac{d\left(⟨z⟩{\beta }_{\pm }\right)}{d{F}_z}\right|$$

Full text

A scalable quantum computing architecture requires reproducibility and control over key qubit properties, such as resonance frequency, coherence time, etc. Variability in such parameters among qubits of a large-scale quantum computer would necessitate individual qubit characterization and controlVandersypen et al. (2016), while excessive variability could even make scaling impractical. In case of significant variability in the dephasing time, the performance of a large-scale quantum computer might be limited by the fidelity of the qubit that decoheres the fastest.

Spin qubits hosted in silicon (Si) quantum dots (QD)Loss and DiVincenzo (1998) have been showing promise as a potential building block for a large-scale quantum computerZwanenburg et al. (2013), because of their compatibility with already existing CMOS technology and the long coherence times available due to the presence of negligible nuclear spins in isotopically purified 28SiItoh and Watanabe (2014). SingleMaune et al. (2012); Kim et al. (2014); Wu et al. (2014); Veldhorst et al. (2014); Kawakami et al. (2014); Eng et al. (2015) and two qubitVeldhorst et al. (2015a) gates have been demonstrated already. To move forward with increasing numbers of qubitsZajac et al. (2016); Jones et al. (2016); Veldhorst et al. (2016); Vandersypen et al. (2016), we have to study possible sources that can cause variations in the coherence time and limit the performance of these qubits.

In this letter, we provide a microscopic understanding of the dephasing time $T_{2}^{*}$ of Si QD spin qubits. We show that electrical noise modulates the electron $g$-factor through spin-orbit interaction (SOI) and causes dephasing. Moreover, the atomic scale details of the interface controls the sensitivity of the g-factor to the electric field or noise and hence introduce variability in the $T_{2}^{*}$ times. We experimentally observe variations in the $g$-factors, their gate voltage dependence and $T_{2}^{*}$ times among spin qubits hosted in gate-defined quantum dots formed at a Si/SiO2 interface. Finally we predict that, due to the anisotropic nature of the SOI in Si QDs, the $T_{2}^{*}$ times will be anisotropic and hence can be improved and their variability can be reduced as well by choosing the appropriate direction of the external magnetic field.

The energy levels of interest in a Si QD for qubit operations are two low lying conduction band valley states $v_{-}$ and $v_{+}$, each split in two spin levels in the presence of a DC magnetic field, $B_{\textup{ext}}$. Until recently, the main focus has been to increase the valley splitting energy to create well isolated states. In metal-oxide-semiconductor based Si QDs, this is now routinely achievedVeldhorst et al. (2014); Yang et al. (2013). However, it turns out that the spin splitting ($E_{\textup{ZS}}^{\pm}=g_{\pm}\mu_{\textup{B}}B_{\textup{ext}}$, where $\mu_{\textup{B}}$ is the Bohr magneton) and also the dephasing time $T_{2}^{*}$ are valley dependentKawakami et al. (2014); Veldhorst et al. (2015b); Scarlino et al. (2015, 2017); Ferdous et al. (2017) and, as we will show experimentally, is sample-to-sample dependent.

In a Si quantum well or dot, the presence of structure inversion asymmetry (SIA) introduces the Rashba SOIGolub and Ivchenko (2004); Nestoklon et al. (2006, 2008). Though it is known that due to the lack of bulk inversion asymmetry (BIA), the Dresselhaus SOI is absent from bulk Si, interface inversion asymmetry (IIA) contributes a Dresselhaus-like term in interface confined structures in SiGolub and Ivchenko (2004); Nestoklon et al. (2006, 2008). Both the Rashba and Dresselhaus SOI modify the electron $g$-factors in a Si QD, and enable the Stark shift of the $g$-factors through gate voltage tuningVeldhorst et al. (2014); Yang et al. (2013); Veldhorst et al. (2015b). The different sign of the Rashba ($\alpha_{\pm}$) and Dresselhaus coefficients ($\beta_{\pm}$) results in different g-factors among the two valley statesVeldhorst et al. (2015b). The Dresselhaus contribution is usually much stronger than the Rashba SOINestoklon et al. (2008); Ferdous et al. (2017), and dominates the $g$-factor renormalization Ferdous et al. (2017). These SOI effects also make the qubits susceptible to electrical noise.

In a Si QD with a strong vertical electric field, the electrons are usually confined to only one interface. A monoatomic shift in the location of this interface results in a sign inversion of the Dresselhaus coefficient ($\beta$), while the Rashba coefficient ($\alpha$) remains unchangedGolub and Ivchenko (2004); Nestoklon et al. (2006, 2008). In practice, Si/SiGe or Si/SiO2 interfaces certainly contain roughness, i.e monoatomic stepsZandvliet and Elswijk (1993); Friesen et al. (2006); Kharche et al. (2007). A non-ideal interface with monoatomic steps can be thought of as multiple smooth interface regions, where interfaces of neighboring regions are shifted by one atomic layer with respect to each other. Thus the neighboring regions will have opposite signs of $\beta$. An electron wavefunction spread over multiple such regions will witness multiple local $\beta$s and the effective $\beta$ will be a weighted average. Thus the presence of interface steps can change both the sign and magnitude of the effective Dresselhaus contribution to the electron $g$-factors in a Si QDFerdous et al. (2017). To accurately understand these atomic-scale physics of the interface, here we use spin resolved $sp^{3}d^{5}s^{*}$ tight-binding model, where the effects of the SOI comes out automatically based on the atomic arrangement of the QD, without any pre-assumption about the Rashba or Dresselhaus SOI.

Fig. 1 shows how a monoatomic step at the interface of a Si QD can affect the $g$-factors of the valley states and their electric field dependence, with an external magnetic field along the [110] crystal orientation, from atomistic tight-binding simulations. An electron wavefunction near an interface step is shown in Fig. 1a. The distance between the dot center and the location of the edge of the interface step is denoted by $x_{0}$. The dot radius is around 10 nm. So for $x_{0}<$-10 nm the dot is completely on the left side of the step and has different $g$-factors ($g_{-}>g_{+}$) compared to that ($g_{+}>g_{-}$) for $x_{0}>$10 nm, when the dot is completely on the right side of the step, as seen in Fig. 1b. For -10 nm$<x_{0}<$10 nm, the $g$-factors are a weighted average of those of the two sides based on the dot location. To understand this atomistic calculation we use an analytic effective mass model that relates $g_{\pm}$ in a Si QD, with the Rashba and Dresselhaus SOIVeldhorst et al. (2015b); Ferdous et al. (2017). We briefly summarize this model in the Supplemental Materialsupplementary . For $B_{\textup{ext}}$ along the [110] crystal orientation

[TABLE]

Here, $\left|e\right|$ is the electron charge, $\left\langle z\right\rangle$ is the spread of the electron wavefunction along the vertical direction z ([001]) and $\hbar$ is the reduced Planck constant. Now, in a Si QD, $\beta>>\alpha$Nestoklon et al. (2008); Ferdous et al. (2017), and so

[TABLE]

As previously discussed, $\beta$ has a different sign between the two sides of the step. When the location of the dot changes with respect to the step, the weighted average of the positive and negative $\beta$s change, which changes the g-factors.

Figs. 1c and 1d show that the Stark shift of the $g$-factors, as a function of the confining vertical electric field $F_{z}$, for both valley states are also affected by the presence of an interface step. The differential change in the $g$-factors with electric field, $\frac{dg_{\pm}}{dF_{z}}$, can vary in both sign and magnitude depending on the location of the step with respect to the dot center. This behavior can also be explained by equation 2, with the change in $\beta$ near an interface step. For example in Fig. 1c, for $x_{0}\approx$ -10 nm, the dot is completely on the left side of the step, where the $v_{-}$ valley state has positive $\beta$. Thus an increase in $\beta_{-}$ with increasing $F_{z}$ increases $g_{-}$ as well, hence a positive $\frac{dg_{-}}{dF_{z}}$. On the other hand, when the dot is completely on the right side of the step, at $x_{0}\approx$ 10 nm, $\beta_{-}$ is negative. Thus increasing $F_{z}$ increases $|\beta_{-}|$ but decreases $g_{-}$ and thus results in a negative $\frac{dg_{-}}{dF_{z}}$. For -10 nm$<x_{0}<$10 nm, $\frac{dg_{-}}{dF_{z}}$ changes gradually with $x_{0}$. We see a similar but opposite change for $g_{+}$ in Fig. 1d.

Similar variations in the $g$-factors, and their gate voltage dependence, are measured in gate-defined quantum dots formed at a Si/SiO2 interface for two different samples (A and B) with similar architecture. In sample A we operated spin qubits in three neighboring QDs while in sample B we studied a qubit in a single QD. In Fig. 2a, we show a schematic diagram of one of the devices (sample A) measured experimentally. In sample A, we observe variations in one-electron and three-electron $g$-factors among three neighboring QDs ($Q_{1}$, $Q_{2}$, $Q_{3}$), as shown in Fig. 2b. We understand that the one-electron (three-electron) qubit corresponds to an electron occupying the lower (higher) energy valley state $v_{-}$ ($v_{+})$Veldhorst et al. (2015b). We could not achieve three-electron spin resonance for $Q_{2}$ as it was strongly coupled to the other dots. Fig. 2c shows the Stark shift of $Q_{1}$ on sample A and one QD on another sample B. We see here that the $g_{-}$ of $Q_{1}$ has opposite dependence on $F_{z}$ compared to that of the QD in sample B. These observed variations in both the Stark shifts and the $g$-factors qualitatively agree with the theoretically predicted variations shown in Fig. 1. We therefore conclude that these experimentally observed variations are primarily due to different interface conditions associated with each of the QDs.

We also observe variations in the measured $T_{2}^{*}$ times for both valley states of the three QDs, as shown in Fig. 2d. The $T_{2}^{*}$ times were extracted by performing Ramsey experiments on all of the qubits in sample A, as shown in the Supplemental Materialsupplementary .

The observed variations in $T_{2}^{*}$ can be explained from the measured (Fig. 2c) and predicted (Fig. 1c,d) variations in $|\frac{dg_{\pm}}{dF_{z}}|$ with interface steps. The larger the $|\frac{dg_{\pm}}{dF_{z}}|$, the more sensitive the qubit is to electrical noise. The dephasing time due to nuclear spin fluctuations is given in refs. Hanson et al. (2007); Merkulov et al. (2002) and in our samples, which employ an isotopically enriched 28Si substrate, these times are very long. In the absence of nuclear spin, we can relate $T_{2}^{*}$ times with electrical noise in a similar way,

[TABLE]

Here, $\Delta F_{z}$ is the standard deviation of the electric field fluctuation seen by the dot, due to electrical noise on the gate. For $Q_{1}$ in sample A, we extract $|\frac{dg_{-}}{dF_{z}}|\approx 1.1\times 10^{-10}$ mV*-1* from Fig. 2c and estimate $\Delta F_{z}\approx 870$ Vm*-1* for $T_{2}^{*}=120$ $\mu s$. When we compare the $T_{2}^{*}$ times between the two valley states of $Q_{1}$, we see $T_{2}^{*}(v_{-},Q_{1})\approx 1.7$ $T_{2}^{*}(v_{+},Q_{1})$ and from ref. Veldhorst et al. (2015b) we find, $|\frac{dg_{+}^{Q_{1}}}{dF_{z}}|\approx 2.2$ $|\frac{dg_{-}^{Q_{1}}}{dF_{z}}|$. This comparison also shows how $\frac{1}{T_{2}^{*}}$ almost linearly depends on $|\frac{dg_{\pm}}{dF_{z}}|$ as predicted by equation 3. Thus variations in $|\frac{dg_{\pm}}{dF_{z}}|$ due to different interface conditions among different qubits will result in a variability in the dephasing time.

The calculations of Fig. 1 and the experimental observations of Fig. 2 highlight the device-to-device variability issues that would require individual knowledge of each qubit, and impose a challenge to the implementation of a large scale quantum computer. Any possible way of reducing the variability is crucial to the scale up of Si QD spin qubits. Also an increase in $T_{2}^{*}$, regardless of the interface condition, will aid scalability. Next we investigate ways to improve these issues.

One obvious way to suppress these variabilities and gain more control over the g-factors, their tunability and the dephasing times, is to minimize interface roughness, which is a well known fabrication challenge. Here we propose an alternate approach. As predicted in ref. Ferdous et al. (2017), the $g$-factors in a Si QD are anisotropic. We can study the anisotropy from a simplified expressionFerdous et al. (2017); supplementary ,

[TABLE]

Here, $\phi$ is the angle of the external magnetic field with the [100] crystal orientation. From equation 4 we see that the contribution of the Dresselhaus SOI is anisotropic. Thus by changing the direction of $B_{\textup{ext}}$ we can tune the Dresselhaus contribution. For $\phi=0^{\circ}/90^{\circ}/180^{\circ}/270^{\circ}$, the Dresselhaus contribution will be negligible. For example, when $B_{\textup{ext}}$ is along [100], $\phi=0^{\circ}$ and

[TABLE]

Comparing equation 2 and 5 we see that,

[TABLE]

As the effect of the monoatomic steps are more dramatic on $\beta_{\pm}$, the change in $g_{\pm}$ and $\frac{dg_{\pm}}{dF_{z}}$ with interface steps should be smaller for $B_{\textup{ext}}$ along [100] compared to that for [110]. Moreover, since $\beta_{\pm}>>\alpha_{\pm}$ Ferdous et al. (2017); Nestoklon et al. (2008), $\frac{dg_{\pm}}{dF_{z}}$ itself will be much smaller for [100].

Fig. 3a shows variations in $\frac{dg_{\pm}}{dF_{z}}$ with $x_{0}$ for $B_{\textup{ext}}$ along [110] and [100]. Though there are variations in $\frac{dg_{\pm}}{dF_{z}}$ with $x_{0}$ for $B_{\textup{ext}}$ along [100], as shown by the inset of Fig. 3a, these variations and also $\frac{dg_{\pm}}{dF_{z}}$ themselves are negligible, when compared to that along [110]. Variation of $g_{\pm}$ with $x_{0}$, will also be negligible for $B_{\textup{ext}}$ along [100], as shown in the Supplemental Materialsupplementary . Such phenomena will have a critical impact on the realization of a large scale quantum computer made of Si QDs. If the external magnetic field is along the [100] crystal orientation, all the qubits will have negligible variations in $g_{\pm}$, $\frac{dg_{\pm}}{dF_{z}}$, and consequently in $T_{2}^{*}$ even in the presence of varying interface conditions. Very small $|\frac{dg_{\pm}}{dF_{z}}|$ along [100] would also result in very long $T_{2}^{*}$ times. From Fig. 3a, we calculate the average $|\frac{dg_{\pm}}{dF_{z}}|$ over different values of $x_{0}$ and find that, $avg\left(|\frac{dg_{-}}{dF_{z}}\left([110]\right)|\right)\approx 30$ $avg\left(|\frac{dg_{-}}{dF_{z}}\left([100]\right)|\right)$ and $avg\left(|\frac{dg_{+}}{dF_{z}}\left([110]\right)|\right)\approx 50$ $avg\left(|\frac{dg_{+}}{dF_{z}}\left([100]\right)|\right)$. If a linear relationship between $\frac{1}{T_{2}^{*}}$ and $|\frac{dg_{\pm}}{dF_{z}}|$ holds, as predicted by equation 3, the $T_{2}^{*}$ times can be longer than 1 ms, provided that other noise sources do not contribute significantly.

In Fig. 3b, the angular dependence of $\frac{1}{T_{2}^{*}}$ for $x_{0}$=6 nm, is shown. From atomistic simulations we calculated $\frac{dg_{\pm}}{dF_{z}}$ for different $\phi$ and used equation 3 to calculate $\frac{1}{T_{2}^{*}}$ for $\Delta F_{z}\approx 870$ Vm*-1* and $B_{\textup{ext}}$=1.4 T. The anisotropy in Fig. 3b can be understood by connecting equation 3 and 4,

[TABLE]

As $\frac{d\left(\left\langle z\right\rangle{{\beta}_{\pm}}\right)}{d{{F}_{z}}}\gg\frac{d\left(\left\langle z\right\rangle{{\alpha}_{\pm}}\right)}{d{{F}_{z}}}$ a large increase in $T_{2}^{*}$ is achievable by orientating $B_{\textup{ext}}$ along $[100]$/$[010]$/$[\bar{1}00]$/$[0\bar{1}0]$.

Now, a decrease in $|\frac{dg_{\pm}}{dF_{z}}|$ would also mean a reduced tunability of the $g$-factors, which is necessary for selective addressing of individual qubits. However, an increase in $T_{2}^{*}$ times will result in a narrower electron spin resonance (ESR) linewidth, $\delta f_{\textup{FWHM}}=\frac{2\sqrt{\ln 2}}{\pi T_{2}^{*}}$Kawakami et al. (2014), which would then require a smaller difference in $g$-factors between qubits to individually address them. As the $g$-factors and $\frac{dg_{\pm}}{dF_{z}}$ have lesser variations with interface roughness along [100], the $g$-factor difference between qubits can be more controlled. Moreover, by tuning the direction of the external magnetic field a desired trade off between large $|\frac{dg_{\pm}}{dF_{z}}|$ and large $T_{2}^{*}$ times can be achieved.

Orienting the magnetic field along the [100] crystal orientation results in a Dresselhaus SOI with only off-diagonal components supplementary . Therefore, electric field fluctuations, to first order, contribute to spin dephasing through the weaker Rashba SOI, ensuring a long $T_{2}^{*}$ time. At the same time, a resonant oscillating electric field can induce electric dipole spin resonance (EDSR) through the off-diagonal Dresselhaus coupling. Since $T_{2}^{*}$ is long under these conditions, coherent operations can be expected even for relatively weak EDSR driving strength, and without invoking the use of micromagnetsKawakami et al. (2014).

To conclude, we experimentally observe variations in the electron $g$-factors and their sensitivity to electric field, which also leads to variability in the dephasing times among Si/SiO2 quantum dot spin qubits. We identify that the source of the variation is the presence of random monoatomic steps at the interface. To gain control over these key qubit parameters one has to minimize interface roughness, which is challenging to achieve. We show here that even in the presence of interface steps we can control and minimize these variations by taking advantage of the anisotropic SOI in a Si QD. Importantly, we can increase $T_{2}^{*}$ times if we align the external magnetic field along the [100] crystal orientation, rather than along [110]. Pointing the external B-field along [100] will also help to reduce the SOI induced dephasing in Si QD devices with integrated micro-magnets, as SOI also contributes to the $g$-factors in these devicesFerdous et al. (2017).

ACKNOWLEDGMENT

This work was supported by the U.S. Army Research Office (W911NF-13-1-0024,W911NF-12-0607), the Australian Research Council (CE11E0001017), and the NSW Node of Australian National Fabrication Facility. Computational resources on nanoHUB.org, funded by the NSF grant EEC-0228390, were used.