Anisotropy with respect to the applied magnetic field of spin qubit decoherence times

The relaxation rate of a spin qubit in the noise magnetic field depends on the noise correlation function $\langle B_{i}^{\rm{(eff)}}(t)B_{j}^{\rm{(eff)}}(0)\rangle_{\omega_{op}}$ where $\mathbf{B}^{(\rm{eff})}$ is the effective noise magnetic field, $\omega_{op}$ is the operating frequency of the qubit. The effective noise magnetic field is any time-dependent field that couples to the spin in the usual way. Hence this could be a physical magnetic field, a field that comes from the motion of the qubit in an inhomogeneous field, a field that results from phonons mediated by spin-orbit coupling, etc.

$T_{1}$, the relaxation time, depends only on the transverse components of the correlation function. If we define $T_{1}^{(i)}$ as the relaxation time when the applied field is in the $i$-direction, then:

$\mu_{B}$ is the Bohr magneton. We take $g=2$.

If the applied field is in an arbitrary direction $\hat{n}=\sin\theta\cos\phi\,\hat{x}+\sin\theta\sin\phi\,\hat{y}+\cos\theta\,\hat{z}$, where $\theta$ is the polar angle and $\phi$ is the azimuthal angle, then the relaxation rate becomes

where

with $\{x,y,z\}$ as the basis for the matrix $Q^{(1)}_{ij}$. Note that if there are any nonzero off-diagonal correlation functions $\langle B_{i}^{\rm{(eff)}}B_{j}^{\rm{(eff)}}\rangle_{\omega}$ ($i\neq j$), they are also needed in the expression for the relaxation time.

The calculation of the dephasing time is more complicated than that for the relaxation time, since it depends more sensitively on the full frequency spectrum of the noise and higher-level correlation functions. For the purposes of this paper only ratios of $T_{2}$ for different applied field angles are important. Hence the specific approximation used to compute $T_{2}$ is not so crucial. It will be sufficient to assume that the field fluctuation obey Gaussian statistics. Then if the applied field is in the $z$-direction, the off-diagonal components of the density matrix of the qubit decay according to the expression $\exp[-\Gamma(t)]$ with

The dephasing time $T_{\phi}$ of the qubit is obtained by solving the transcendental equation $\Gamma(T_{\phi})=1$. Here the sinc function is defined by $\textrm{sinc}(x)=\sin x/x$. If the applied field is in the $(\theta,\phi)$ direction, then

where

in the same basis as used for $\mathsf{Q}^{(1)}$.

The results of this subsection and the previous one make it clear that the entire tensor structure of the noise correlation function is in principle accessible simply by measuring $T_{1}$ and $T_{2}$.

Single-qubit logic gates in spin systems are often implemented using a micromagnet to set up a magnetic field gradient. This has the unwanted complication that electric field noise moves the spin up and down the gradient, causing a time-dependent magnetic field $\mathbf{B}^{(E)}$ that can decohere the qubit. Here we outline how this plays into the anisotropy effect.

We will take a simple model of a quantum dot in a harmonic potential. The Hamiltonian is

where $i$ is a Cartesian index, $m_{i}$ and $k_{i}$ are the effective mass and spring constant of the electron in the $i$-direction, $q$ is the electric charge of the electron and $E_{i}(t)$ is the noise electric field component in the $i$-direction. The frequency of the electric noise is much smaller than the natural frequencies of the harmonic motion, so the Born-Oppenheimer approximation applies and the effect of $E_{i}(t)$ is to shift the position of the minimum of the potential by $\Delta x_{i}(t)=qE_{i}(t)/2k_{i}$. The confinement in the $z$-direction is much stronger, though the effects of excited states in this potential are measurable [30, 31]. To the level of approximation needed in this paper, we can drop the $z$ term in Eq. 7 in the potential and treat the dot as two-dimensional.

The micromagnet sets up a static field $\mathbf{B}^{m}$ that varies strongly in space. The associated effective noise field that acts on the spin is

The field gradient and the spring constant are device parameters. This equation already shows that considerable device modeling is necessary to extract any interesting information about the noise field $E_{i}(t)$.

In qubit experiments carried out to date, the micromagnet is made of pure cobalt [32, 33]. Co is hexagonal with the easy axis for magnetism along the $z$-direction. The anisotropy parameter (the difference in energy between the $z$-axis and the $x$-$y$ plane) is about $10^{6}\,$J/m³, which corresponds to an anisotropy field $H_{A}$ of about 0.5 T when the magnetization is saturated, as it surely is at the low temperatures of the experiments. This and shape anisotropy will mainly determine the coercive field, though sample-dependent domain wall pinning will also contribute. The anisotropy parameter quoted above is consistent with the results of Ref. [34], though it is sometimes assumed that the magnet will rotate freely [35, 36]. In any case, at the fields of present-day experiments (a fraction of 1 T), the magnet cannot be considered to be hard and some rotation of the direction of the magnetization of the Co micromagnet is certainly to be expected. This will change the field gradient tensor that is used to calculate $T_{1}$ and $T_{2}$.

There are three possible solutions to this problem, the first two of which involve changing the magnet.

The first is to use a softer ferromagnet with a cubic crystal structure such as Fe, where one could expect that the magnetization follows the applied field. This would erase much though not all of the anisotropy in the decoherence times. It is certainly not ideal but might still give useful information about the noise sources.

The second option is to use a harder ferromagnet, so that the field gradient tensor is fixed once and for all when the magnet is cooled in a field. There are many possibilities, but one that suggests itself is SmCo₅, very close chemically to the Co magnets in current use. The anisotropy field can be as high as $H_{A}=$55T, an order of magnitude higher than pure Co [37]. The magnetization would not be significantly affected by the rotation of the external field. Then the anisotropy in $T_{1}$ and $T_{2}$ is more pronounced and more information can be extracted from it.

The third possibility is to stick with the Co magnet but to use an external field that is considerably smaller ( $<0.05$ T, say) so that the field from the micromagnet is fixed. This has the problem that the energy level splitting of the qubit becomes smaller than $k_{B}T$, and initializing the spin for measurements is problematic. It is possible only at temperatures of the order of 1mK, considerably lower than the temperatures in use today for these experiments. One might still be able to do spin blockade-based measurements, however.

In this paper, we shall assume that the second alternative is chosen, since this gives the richest phenomenology, and seems feasible with fairly modest changes in fabrication techniques. So we take the field gradients to be fixed. Of course this means that the experiment cannot be carried out with the sample of Ref. [33], which uses a Co magnet and electron temperatures of order 150 mK.

Evanescent-wave Johnson noise (EWJN) is due to the random motion of charges in the metallic elements of the device. This motion produces random electric and magnetic fields on the qubits in the vicinity of the metal. For the discussion of this effect, let us take the growth direction for the device to be the $z$-direction, the distance of the qubit from the gate layer as $d$, the gate thickness as $w$, and the dielectric constant of the intervening insulating material as $\epsilon_{d}$.

For the case of noise from a conducting half-space, rather simple formulas are available [38, 9]. In most Si/SiGe heterostructure and Si MOS devices the gates form sheets of metal that can be approximated as a uniform layer from the standpoint of noise production. Thus the theory of EWJN from a film with a finite $w$ is more appropriate. It has been worked out in detail [39], though the results are somewhat complicated, and depend on whether we consider electric field noise or magnetic field noise. For the values of $d$ of interest to us, electric field noise is slightly enhanced for the film case as compared to the half-space case, while the opposite is true for the magnetic field noise, and the effect of finite $w$ is larger. A very good approximation is to use the half space formula for the electric noise and a modified formula for the magnetic case.

Given these considerations, the noise correlation functions for the electric field are

Here $\sigma$ is the conductivity. The other elements of the noise tensor are $\langle E_{x}E_{x}\rangle_{\omega_{op}}=\langle E_{y}E_{y}\rangle_{\omega_{op}}=(1/2)\langle E_{z}E_{z}\rangle_{\omega_{op}}$, while the off-diagonal elements of the tensor vanish. This electric noise is converted into effective magnetic field noise using the techniques of the previous section.

The magnetic EWJN correlation function is given by

This is reduced from the half-space result by a factor of $w/d$. The other elements of this noise tensor are $\langle B_{x}B_{x}\rangle_{\omega_{op}}=\langle B_{y}B_{y}\rangle_{\omega_{op}}=(1/2)\langle B_{z}B_{z}\rangle_{\omega_{op}}$, while the off-diagonal elements of the tensor vanish. Unlike the electric noise, magnetic EWJN acts directly on the qubit spin to produce decoherence.

The exact nature of the low-frequency charge noise remains controversial. There are two leading models for the source of the two-level systems (TLS) that give rise to this noise.

The first model is the TLS proposal of Anderson, Halperin and Varma, and independently Phillips, of 1972 [40, 41]. The picture, shown in Fig. 1, is that the noise source is the motion of some atom or group of atoms in a potential that supports bistability in some range of parameters. The motion also produces a fluctuating dipole moment, which we take as $\pm\mathbf{p}$ in the two stable positions. For the most part, this idea has formed the conceptual background of the field in physics experiments for the last half century. We will assume that the orientation of these dipoles is uniformly random on the unit sphere, and that they are distributed uniformly in space in the oxide layer above the qubits. We call this the random dipole model.

The second model is the related but physically quite distinct idea of McWhorter [42], in which a conducting layer serves as a reservoir for electron traps near the surface of the layer whose energies are close to the Fermi level of the layer. Electrons from the reservoir can hop on and off, again changing the distribution of charge in the system, the change being well approximated by a fluctuating dipole perpendicular to the layer. In this case the fluctuation is between a zero and a fixed nonzero value of the dipole moment, while in the dipole model the fluctuation is between two different nonzero values. The trap-type TLS is in fact widely thought to be the most important for the noise in field-effect transistors in the engineering community. However, as in the case of the dipole model, real proof of the details of the model is hard to come by. We call this the trap model. An illustration of the model is given in Fig. 1(b).

It is evident that the two models are not easy to distinguish experimentally, since they will both give random telegraph noise with a distribution of switching rates, and reasonable assumptions about the distribution will lead to $1/f$ type noise. They differ in the orientation of the effective dipoles, however, suggesting that an experiment that can detect anisotropy in the noise correlation tensor will distinguish the two models. This forms a chief motivation for the current work.

Anisotropy in the decoherence rates can come from several sources. We have seen that the noise tensor from EWJN can itself be anisotropic, and the dipole and trap models also have characteristic anisotropy signatures. In addition to this, it is possible for charge noise sources to be clumped, either from a tendency to adhere to different device elements, or, particularly in the case of only a few sources, to cluster by random chance. Of course if the noise is coming from a certain direction this is also a source of anisotropy.

This leads to distinguishing four models all together, which we call the uniformly distributed dipole model (UD), uniformly distributed trap model (UT), localized cluster dipole model (CD) and localized cluster trap model (CT). The U-type models assume that the sources are many in number and uniformly distributed, while the C-type models assume that the sources are relatively few in number. The total number of sources is of course also very important to determine [13].

The U- and C- type models represent limiting cases of very many and just a few closely spaced noise sources, respectively. Of course it is not possible to rule out in advance a lumpy set of say 10 to 100 noise sources. The current computational method would need to be developed considerably further to become a useful characterization tool in this difficult intermediate case. In particular, multiple qubits and cross-correlation functions among them would most likely be needed.

The noise correlation functions for all models are calculated as follows.

First we note that because of the metallic elements in the device, it is important to include screening of the electric noise. Electric dipoles near a metal surface at $z=0$ are screened in an anisotropic fashion. A dipole oriented perpendicularly to the surface is anti-screened, since the image dipole is in the same direction as the original one. This is in sharp contrast to a dipole oriented parallel to a metal surface, which is strongly screened, with the image dipole opposite in direction to the original one [13]. These image charge effects are taken into account in our calculations by adding an image dipole $\mathbf{p}_{im}$ to the bare dipole $\mathbf{p}_{0}$. If the bare dipole $\mathbf{p}_{0}=(p_{x0},p_{y0},p_{z0})$ is located at $\mathbf{r}=(x,y,z)$ relative to a metallic layer at $z=0$ the image dipole is located at $(x,y,-z)$ and its moment is given by $\mathbf{p}_{im}=(-p_{x0},-p_{y0},p_{z0})$. Examples of a bare dipole and corresponding image dipole for the random dipole model and trap model are shown in Fig. 1(c) and Fig. 1(d).

Let the qubit be at the point $\mathbf{r}$, so we are interested in the correlation function $\langle E_{i}(\mathbf{r},t)E_{j}(\mathbf{r},0)\rangle$. Let the dipole be at $\mathbf{r}^{\prime}$ and the root-mean-square dipole strength be $p_{0}$. Define $\mathbf{R}^{\prime}=\mathbf{r}-\mathbf{r}^{\prime}$. Then

The dipoles, (except for direct and image pairs) are assumed to be statistically independent, so their correlation function is

where the model-dependent factor $\delta_{mod}=\delta_{ij}/3$ for the random dipole model, $\delta_{mod}=\delta_{iz}\delta_{jz}$ for the trap model, and $\delta_{ij}$ is the Kronecker delta. Here $g(t)$ is the time correlation function for a single dipole. We substitute Eqs. 10 and 11 into the definition of the electric field correlation function and, after some calculation and a time Fourier transform, find

for the UD model and

for the UT model. Here $\rho_{v}$ is the volume density of dipoles, $\rho_{a}$ is the areal density of dipoles, and $g(\omega)$ is the Fourier transform of $g(t)$. Both direct and image pairs are included in the rate formulas 3 and 4.

For charge noise $g(\omega)$ is often of the $1/f$ type. However, one of the important advantages of the experiments described in this paper is that we can investigate the sources of noise using spatial and geometric information alone, and the frequency spectrum of the noise is less important, a point we will return to below.

For the CD and CT models, the passage to an integral as in Eqs. 12 and 13 is not possible, and the sum in Eq. 10 must be performed explicitly.

We now use the above theory to make predictions for the coherence time anisotropy for a model device. We have chosen the device parameters from the sample used by Kawakami et al. [33] because it has been particularly well-characterized: $T_{1}$ and $T_{2}$ were measured, the distance from the qubits to gates is accurately known as $d=137\,$nm, and, most importantly, the magnetic field gradient tensor created by the micromagnet was simulated in detail. We take the tensor as fixed and independent of the direction of the applied field, though we repeat that this would be unlikely if a Co magnet is used. Parameters for the device are given in the Methods section.

The goal of this section is to determine signatures of the different types of noise in $T_{2}$ data. We first show that EWJN is not important for $T_{2}$. Then we point out that hyperfine noise is isotropic and estimate its magnitude.

In the experiment of Ref. [33], $T_{2}^{*}$ of the device is measured to be $840\pm 70\,$ns and corresponds to $T_{2}^{(x)}$ of our simulation because the applied magnetic field $B_{0}$ is in the $x$-direction. For EWJN, the dephasing rate in the $i$-direction is

This is the dephasing time for the exponential regime when the time $t\gg\hbar/k_{B}T.$ The calculated $T^{(x)}_{2}=1.19\,$s from EWJN is six orders of magnitude larger than the experimental one. Thus the dominant mechanism for the decoherence of the qubit should be charge noise, not EWJN.

The isotropic hyperfine noise is taken into account by estimating it from other experiments. It should not differ too much from one device to another. The material in the case study device is natural silicon. If the decoherence rate in isotopically purified silicon [2] is subtracted from that in natural silicon [43], we find a hyperfine contribution to the rate of $1/T_{2}^{hyper}=(1.83\upmu$s$)^{-1}-(20.4\upmu$s$)^{-1}=(2.01\,\upmu$s$)^{-1}$. $1/T_{2}^{hyper}$ is then simply added to the dephasing rates from charge noise. Because of its isotropy, its effect is to smooth the resulting plots.

For charge noise, $\Gamma(t)$ for the applied field along the $(\theta,\phi)$ direction can be written as

where $\gamma_{2x}(t)$ and $\gamma_{2y}(t)$ are the prefactors related to the gradients in $x$- and $y$-direction respectively.

Now we assume that the TLS noise is a Poisson process with an exponential time correlation functions with characteristic relaxation time $\tau$ and carry out the necessary integrations. For the UD model we have

For the UT model we find

The temporal part of $\gamma_{2x}$ and $\gamma_{2y}$ results from the integration of the product of a Lorentzian $g(\omega)=2\tau/(1+(\omega\tau)^{2})$, and $\mathrm{sinc}^{2}(\omega t/2)$. Those equations are obtained by converting electric field noise correlations (Eq.12 or Eq.13) into effective magnetic field correlation using Eq.7 (See details in Supplementary Notes). It is important to note first that the details of the noise spectrum and thus the choice of an exponential correlation are not crucial for the anisotropy patterns, since they depend only on ratios of noise strengths. On the negative side, if some parameter of the noise such as $\tau$ itself depends on position in the sample, then the extraction of useful information from the analysis of the data would become far more complicated.

$\rho_{v}$ and $\rho_{a}$ are poorly known, so we use them as fitting parameters. $T_{2}^{*}=840\,$ns was measured for only a single direction of the field, indicated by the red dots in Fig. 2. This yields $\rho_{v}=2.93\times 10^{20}\,\mathrm{m}^{-3}$ and $\rho_{a}=2.66\times 10^{11}\,\mathrm{m}^{-2}$ for Fig. 2(a) and Fig. 2(b), respectively.

The anisotropy maps of $T_{2}$ for the various models are shown in Fig. 2. There is some redundancy in the maps since they are symmetric under the transformation $\theta\to\pi-\theta$ and $\phi\to\pi+\phi$, stemming from $T_{2}(\mathbf{B}_{0})=T_{2}(-\mathbf{B}_{0})$. This same redundancy also arises in the anisotropy maps of $T_{1}$ in Fig. 3. We have chosen to show the full angular ranges since in some instances the topology of the function is clearer this way. The number of maxima ($N_{max}$), minima ($N_{min}$), and saddle points ($N_{s}$) in each map are shown in Table I. When the dephasing and relaxation times in the maps are continuous functions without higher-order critical points, Morse theory can be applied. In particular for such functions on a 2D sphere (genus zero), they follow the relation: $N_{max}+N_{min}=N_{s}+2$. Table I shows how the topology of the function can change when parameters affecting the noise are varied. For the maps of 2(e) and 2(f), Morse theory cannot be applied because the ridges in the maps correspond to a line of higher-order critical points.

The background $T_{2}^{hyper}=2.01\,\upmu$s sets an upper bound on the plotted values in Fig. 2. The white regions in Fig. 2(a) and Fig. 2(d) represent angular regions where the charge noise contribution is negligible and this upper bound is reached.

Fig. 2(a) shows the results for the UD model and Fig. 2(b) for the UT model. The horizontal (vertical) axis denotes polar (azimuthal) angle with respect to the device’s $z$-direction. The key feature of these two models is that anisotropy of $T_{2}$ results only from the magnetic field gradients. The patterns are not too dissimilar, with the ratio between maximum and minimum values being around 3 for the UD model and 2 for the UT model. The main difference between the UD and UT models is that the peaks and valleys are broader in the UT model. In the UD model, the dipoles are oriented randomly, while in the UT model they are in the $z$-direction. The differences in the anisotropy maps between UD and UT can be traced back to the different behavior of electric field lines from these two different types of sources. However, this does not manifest itself in a simple way because of the complexity of the gradient tensors that mediate the electric noise. Because of that, it is difficult to develop much physical intuition about the distinction between UD and UT charge noise sources from inspection of the anisotropy maps, and it appears that a full calculation is necessary to test the differences between the two noise models.

The anisotropy maps for the CD model, a localized dipole cluster, are shown in Fig. 2(c) and in Fig. 2(d). The cluster is located at $(x,y,z)=(37,0,37)\,$nm and $(x,y,z)=(0,37,37)\,$nm respectively. The maps for the CT model, a localized trap cluster, are shown in Fig. 2(e) and Fig. 2(f). The trap is located at $(x,y,z)=(37,0,137)\,$nm and $(x,y,z)=(0,37,137)\,$nm respectively. In both CD and CT model, the qubit is located at the origin. Thus Fig. 2(c) is directly comparable to Fig. 2(e) and Fig. 2(d) is directly comparable to Fig. 2(f). The overall dipole strength $p_{0}$ is used for the fitting parameter of these single cluster models, once more by using the experimental value measured at the red point. Figs. 2(c)-2(f) exhibit more anisotropy relative to the uniform distribution models. This is expected since the localization of the source itself introduces anisotropy. On the other hand, one might expect that C- and U-type models would be easy to distinguish because of a simpler azimuthal dependence for the latter. But once more because of the mediation of the noise by the complicated field gradient tensors, such simple expectations are not borne out.

The difference between the CD and CT models lies in the dipole orientation. In the CD model, it is assumed that the cluster contains dipoles of all orientations and the noise electric field is averaged over the solid angle. This washes out the anisotropy to some extent, but the pattern still depends on the direction of the line connecting the dipole and the qubit. The distance between the dipole and qubit just changes the overall magnitude of $T_{2}$. In the CT model, however, the trap generates a noise electric field with more directionality, so the overall anisotropy patterns are sharper and both the direction and the distance are important.

Comparing Fig. 2(c) to Fig. 2(d) and Fig. 2(e) to Fig. 2(f) indicates that the source position has a large effect. To understand this in more detail, let us focus on the CT model in Fig. 2(e) and Fig. 2(f). Note that $\partial B_{x}/\partial x$, $\partial B_{z}/\partial x$, and $\partial B_{y}/\partial y$ are an order of magnitude greater than the other gradient terms. In Fig. 2(e), the electric field at the qubit has only $x$ and $z$ components. The $x$ component contributes to $1/T_{2}$ after multiplication by $\partial B_{x}/\partial x$ and $\partial B_{z}/\partial x$. Thus small $T_{2}$ is expected when the applied field is in the $x$ and $z$ directions, which can be identified on the map with $(\theta,\phi)=(\pi/2,0)$ and $(\theta,\phi)=(0,0)$, respectively. On the other hand, in Fig. 2 (f), the electric field at the qubit has only $y$ and $z$ components. The leading contribution to $1/T_{2}$ is the result of the product of the $y$ component and $\partial B_{y}/\partial y$. A small $T_{2}$ is expected with the applied field is in the $y$-direction, which is seen at the point $(\theta,\phi)=(\pi/2,\pi/2)$ on the map. Thus for the distinction between CD and CT models, some relatively simple physical considerations can help to decipher the anisotropy map.

It is important to point out that if relatively few two-level systems contribute to the dephasing of the qubit, as is often hypothesized based on deviations for power-law spectra [13, 44], the anisotropy can be used to determine the source position and to distinguish between random dipole and trap models for the charge noise. The present method can be extended to models with very few sources by eliminating the averaging we have performed, but the analysis quickly becomes complicated.

In the experiment, $T_{1}$ of the device is in the order of $1\,$s. To estimate the contribution of charge noise, we use the results from II.1 together with the determination of densities from $T_{2}$. This leads immediately to an estimate in the range of $10^{9}\,$s so we conclude that charge noise is not important for spin relaxation in the single-qubit system considered here. To exclude phonon relaxation we need to stipulate for the moment that the external field strength is less than about $1\,$T. This leaves EWJN as the dominant mechanism.

The relaxation rate with applied field direction in $(\theta,\phi)$ for EWJN can be written as

where

The first term represents the direct effect of magnetic field noise while the second term is due to the electric field noise that is converted to effective magnetic field noise by the field gradient created by the micromagnet. The angular variation in $Q_{ij}^{(1)}$ and the rather anisotropic character of the magnetic field gradients imply that the anisotropy of $T_{1}$ will be intensified when the electric term is bigger than the direct magnetic term. Noting that $\alpha\sim\sigma$ and $\beta\sim 1/\sigma$, we see that varying $\sigma$ will change the anisotropy pattern of $T_{1}$. As noted above, $\sigma$ was not measured in the experiment and it makes sense to vary it to investigate the angular dependence of $T_{1}$.

The anisotropy map of $T_{1}$ is shown in Fig. 3 with (a) $\sigma=2\times 10^{8}\,$S/m, (b) $\sigma=2\times 10^{7}\,$S/m, and (c) $\sigma=2\times 10^{6}$ S/m. The anisotropy pattern in Fig. 3(a) is fairly simple because the magnetic noise is dominant and the direct magnetic EWJN itself is not very anisotropic, as can be seen from Eq.  9 and the text following it. The anisotropy is increased as shown in Fig. 3(b) where the magnetic noise is somewhat more comparable to the electric noise. The anisotropy becomes even larger in Fig. 3(c) where the magnetic noise is one order of magnitude smaller than the electric noise. This pattern looks like the reversal of the anisotropy map of $T_{2}$ in Fig. 2(a) and Fig. 2(b). This is natural since $T_{2}$ of a spin qubit is due to longitudinal noise while $T_{1}$ is due to transverse noise. From practical point of view, the qubit performance would be improved when the applied field direction is set to the angles that give maximal $T_{2}$ (in the case of $T_{\phi}\ll T_{1}$).

When the strength of the applied magnetic field is increased, there is a crossover from EWJN- to phonon-dominated spin relaxation. The anisotropy maps for phonons was worked out in Ref. [15]. They are determined by the orientation of the field relative to the crystal axes, not axes coming from the the device geometry. Hence we expect sharp changes in the anisotropy map as the field is increased beyond about 3 T.

The parameters for the device of Ref.Fig. [33] are as follows. The field gradients at the qubit in units of mT(nm)^-1 are $\partial B_{x}/\partial x=-0.20$, $\partial B_{y}/\partial x=-0.05$, $\partial B_{z}/\partial x=-0.27$, $\partial B_{x}/\partial y=-0.03$, $\partial B_{y}/\partial y=0.18$, and $\partial B_{z}/\partial y=-0.02$. The $z$-direction of the device is taken to be the growth direction. The variation in the $z$-direction is not needed in the two-dimension approximation we are using. Other important parameters are the thickness of the aluminum oxide layer $l=100\,$nm, the dielectric constant $\epsilon_{d}=13.05$ for $\rm{Si}_{0.7}\rm{Ge}_{0.3}$ [47], and the transverse effective mass $m=0.19\,m_{e}=1.73\times 10^{-31}\,$kg. The lowest orbital excitation frequency is taken as $\omega_{orb}=6.84\times 10^{11}\,\mathrm{s}^{-1}$ and it is related to the spring constants by the equations $k_{x}=k_{y}=m\omega_{orb}^{2}$. As mentioned above, we take $k_{z}\rightarrow\infty$ since confinement is strong along the growth direction. The base temperature is $25\,$mK, while the electron temperature is about $150\,$mK, the value we use for the calculations.

Another parameter needed as input to the theory is the conductivity $\sigma$ of the Au gates. This was not measured in this device, but under similar growth conditions for Au films a value of $\sigma=2\times 10^{8}\,$S m^-1 was obtained at the temperatures of the experiment [48]. We should regard this as a probably somewhat high order-of-magnitude estimate of $\sigma$ in the actual device, and we used a range of values for $\sigma$. We take the gate thickness as $w=25\,$nm, and the distance from the qubit to the gates $d=137\,$nm. The qubit operating frequency is $\omega_{op}=2\pi\times 12.9\,$GHz $=8.11\times 10^{10}\,\mathrm{s}^{-1}$. Two micromagnets made of cobalt are defined on top of the gates, approximately $162\,$nm above the qubit. In the experiments reported in Ref. [33] the applied field $B_{0}$ was in the $x$-direction. This paper concerns what happens if this direction is varied (but with the caveats discussed in Sec. II.4) .