# Spin relaxation in a Si quantum dot due to spin-valley mixing

###### Abstract

We study the relaxation of an electron spin qubit in a Si quantum dot due to electrical noise. In particular, we clarify how the presence of conduction-band valleys influences spin relaxation. In single-valley semiconductor quantum dots, spin relaxation is through the mixing of spin and envelope orbital states via spin-orbit interaction. In Si, the relaxation could also be through the mixing of spin and valley states. We find that the additional spin relaxation channel, via spin-valley mixing and electrical noise, is indeed important for an electron spin in a Si quantum dot. By considering both spin-valley and intra-valley spin-orbit mixings and Johnson noise in a Si device, we find that the spin relaxation rate peaks at the hot spot, where the Zeeman splitting matches the valley splitting. Furthermore, because of a weaker field dependence, the spin relaxation rate due to Johnson noise could dominate over phonon noise at low magnetic fields, which fits well with recent experiments.

###### pacs:

72.25.Rb, 03.67.Lx, 03.65.Yz, 73.21.La[Peihao Huang]PHblue

## I Introduction

A spin qubit is a promising candidate as an information carrier for quantum information processing,Hanson et al. (2007); Morton et al. (2011); Zwanenburg et al. (2013) and silicon is one of the best host materials for a spin qubit.Kane (1998); Morello et al. (2010); Xiao et al. (2010); Pla et al. (2012); Maune et al. (2012); Yang et al. (2013); Zwanenburg et al. (2013); Muhonen et al. (2014) Specifically, the low abundance of isotopes with finite nuclear spins (Si) in natural Si significantly reduces the hyperfine interaction strengthAssali et al. (2011) and the spin dephasing.Pla et al. (2012) Isotopic purification further suppresses this decoherence channel, so that Si behaves as if it is a “semiconductor vacuum” for a spin qubit. Muhonen et al. (2014) Spin-orbit (SO) interaction in Si is also weak because of the lighter mass of Si atoms and the lattice inversion symmetry in bulk Si.Xiao et al. (2010); Yang et al. (2013) Therefore, as has been calculated theoretically and measured experimentally, (donor-confined) spin dephasing and relaxation times are extremely long in bulk Si. Gumann et al. (2014); Muhonen et al. (2014); Wolfowicz et al. (2013)

But Si is not perfect. The existence of multiple conduction-band valleysYu and Cardona (2010) gives additional phase factors to the electron wave function, so that interaction between donor electron spins becomes sensitively dependent on the donor positions.Koiller et al. (2001, 2004); Wellard and Hollenberg (2005); Salfi et al. (2014); Gonzalez-Zalba et al. (2014) While interface confinement and scattering can lift this degeneracy, details at the interface, whether it is surface roughness or steps, play important roles in determining the magnitude of the valley splitting ,Boykin et al. (2004); Friesen et al. (2007); Culcer et al. (2010); Friesen and Coppersmith (2010); Saraiva et al. (2010, 2011); Rahman et al. (2011); Jiang et al. (2012); Gamble et al. (2013); Dusko et al. (2014) so that device variability is large. Experimentally measured ranges from vanishingly small, to several hundreds of eV,Shaji et al. (2008); Yang et al. (2013); Hao et al. (2014) to possibly a few meV.Takashina et al. (2006) Furthermore, to achieve controllability, spin qubits are generally located near or at the interface between the host and the barrier materials. Dangling bonds, charge traps, and other defects are inevitably present at the many interfaces of a semiconductor heterostructure, and the coherence properties of a spin qubit in a nanostructure are not as clearly understood and measured as in bulk Si.

With pure dephasing strongly suppressed in Si, spin relaxation becomes an important indicator of decoherence for a spin qubit. Spin relaxation could come directly from magnetic noise in the environment, or from electrical noise via spin-orbit or exchange interaction. Indeed, for a single spin in a quantum dot, we have shownHuang and Hu (2014) that electrical noise from the circuits or surrounding traps could be an important cause for spin relaxation, particularly at a smaller qubit energy splitting. In this previous study, however, we only considered intra-valley orbital dynamics for an electron in Si. On the other hand, it has been shown experimentally and theoretically that the presence of valleys in Si can significantly modify spin relaxation through spin-valley mixing, and a relaxation hot spot appears at the degeneracy point where the Zeeman splitting matches the valley splitting.Yang et al. (2013); Tahan and Joynt (2014)

In this paper, we study spin relaxation of a single QD-confined electron in Si due to the presence of electrical noises (including Johnson noise, phonon noise, and the charge noise). \added[]One relaxation mechanism involves the mixing of spin and valley states, which should be particularly important when Zeeman energy is comparable with valley splitting . Another mechanism involves the mixing of spin and orbital states within one conduction band valley, which is important at high magnetic fields. By considering both of these mechanisms and various electrical noises, such as phonon noise and Johnson noise, we find that the spin-valley mixing is indeed an important spin relaxation channel for an electron spin in a Si quantum dot. We also find that, because of a weaker field-dependence, spin relaxation due to Johnson noise through the mixing of spin and valley states could dominate over phonon noise and intra-valley scattering (relaxation due to mixing of spin and higher orbital states) at low magnetic fields. Our numerical results fit quite well with recent experimental measurements.

The rest of the paper is organized as follows. In Sec. II we set up the system Hamiltonian and describe the mechanism of spin-valley mixing. In Sec. III we derive explicitly the spin relaxation rate due to spin-valley mixing and electrical noise. In Sec. IV we evaluate the spin relaxation rates due to Johnson and phonon noises, and we compare the different spin relaxation mechanisms. Finally, conclusions are drawn in Sec. V. In the Appendices we discuss the field dependence of the spin relaxation, the effects of noise, and the phonon noise spectrum in more detail.

## Ii System Hamiltonian

We consider an electron in a gate-defined quantum dot in a Si heterostructure (whether a Si/SiO or a Si/SiGe structure). The growth-direction ([001]-direction in this paper) confinement is taken to be very strong, so that we focus on the in-plane dynamics of the confined electron. The strong field and strain at the interface lower the degeneracy of the Si conduction band by raising the energy of four of the valleys relative to the other two (in this case and valleys). Moreover, scattering off the smooth interface further mixes and splits the two low-energy valleys. We label the two valleys as and , with valley splitting . At this smooth-interface limit, and without considering the spin-orbit interaction, the valley degrees of freedom and the intra-valley effective-mass dynamics can be separated, so that the electron wave function can be written as , where is the index for the two lowest-energy eigen-valleys, is the orbital excitation index within an eigen-valley, and or is the spin index.

[]In the following, we first consider explicitly spin relaxation due to spin-valley mixing, which is important when . Later, in Sec.III, we compare these results with spin relaxation due to intra-valley spin-orbital mixing, which is a spin relaxation mechanism that is well known in the literature. By considering various electrical noises, we can then identify the dominant spin relaxation mechanism in different regimes.

We consider a quantum dot for which the lateral confinement is sufficiently strong ( meV), so that the intra-valley orbital level spacing is larger than the valley splitting (which is up to a fraction of 1 meV in general). In this case, we can focus on the effects of spin-valley mixing, and we neglect the intra-valley excitation, \added[]particularly when the Zeeman energy is close to the valley splitting, and is much less than intra-valley orbital level spacing. In this limit, only the lowest four spin-valley states are relevant, all having the intra-valley ground orbital state. These four states (with an implicit common orbital index ) are denoted as , , , and . Within the space spanned by these four lowest-energy spin-valley product states, the total Hamiltonian for the QD-confined electron is given by

(1) | |||||

Here contains valley and Zeeman splitting, with being the energies of the product states in the absence of SO interaction and the environmental noise. Specifically, , . represents spin-valley (SV) mixing due to the SO interaction, with and the SV mixing energy: and . Here the SO interaction is , with the interaction strength , and the and axes along the [] and [] directions (which also define the plane of the quasi-2D quantum dot).Golovach et al. (2004); Huang and Hu (2013) Here and are the Dresselhaus and Rashba SO interaction constants. The Dresselhaus SO interaction arises from the bulk inversion asymmetry, which in a Si QD could be from the interface disorder, while Rashba SO interaction arises from the structure inversion asymmetry and is tunable through the electric field across the QD. Lastly, contains the electrical noise from the environment, with the noise electric field. It could come from Johnson noise, charge noise, phonon noise, etc.. Here is the electric dipole matrix element between different valley states, which could arise from disorders at the interfaces of the QD.Gamble et al. (2013)

We first find the eigenstates of the confined electron in the presence of spin-valley mixing but without environmental noises. As indicated in Fig. 1, states and are always well separated energetically, by both the valley and the Zeeman splitting, so that we neglect the mixing of states and by in this study. On the other hand, near , states and are strongly mixed by the spin-valley coupling . This degeneracy point is called a spin relaxation hot spot. Fabian and Das Sarma (1998); Stano and Fabian (2006) is in general a complex number, and it can be written as and . The eigenstates for are thus {, , , }, where

(2) | |||||

(3) |

Here and . The energy splitting between and is . When the magnetic field is along the [110] axis as in Ref. Yang et al., 2013, the spin-valley mixing matrix element can be expressed as (see Appendix A) Tahan and Joynt (2014)

(4) |

where the relationship has been employed.

## Iii Spin relaxation

### iii.1 Spin relaxation due to spin-valley mixing

With states and being spin-valley mixed, and assuming that the electric dipole matrix element between the two eigen-valleys is non-vanishing, any electrical noise, which couples states with the same spin orientation, can induce transitions between them and from them to the other two eigenstates. The transition rate is proportional to the amount of spin-valley mixing, and to the spectrum of the noisy electric field , where captures the electrical potential of the noise in the system, such as Johnson noise, charge noise or phonon noise, which will be discussed later.

Experimentally, in the preparation of a spin-up initial state, the electron orbital and valley states are kept in the lowest eigenstates in order to avoid the unnecessary mixing of the spin and orbital dynamics. The most relevant spin relaxation processes involve the relaxation of either state or state because of the experimental difficulty in making measurements close to the spin-valley crossing point, the small magnitude of spin-valley mixing ( neV), and the energy-selective nature of resonant tunneling Yang et al. (2013). Specifically, we consider the following situations: when , a spin-up electron is loaded only into the energy eigenstate ; while when , it is only loaded into the energy eigenstate .

In the low-field regime when , spin relaxation occurs from state to the ground state . The spin relaxation rate is Huang and Hu (2014)

(5) |

where is the energy difference between state and , and means an average of with respect to the noise electric field. In the case of quantum noise, this should be an ensemble average. Separating the noise electric field from the coupling matrix element, the spin relaxation rate can also be expressed as

(6) |

where is the noise spectrum (), and we have assumed that noise in different directions are not correlated. The relevant transition matrix element in this case is , which is proportional to the transition matrix elements between the valleys.

In the high-field regime when , state is loaded initially. The electron can then relax to both and due to spin-valley mixing and inter-valley transitions. Both these processes involve an apparent spin flip. The relaxation rates are

(7) | |||||

(8) |

where the relevant matrix elements are , . Below we will focus on the spin-valley transition from to when .

Since the SO mixing element is much less than Zeeman energy, , spin-valley relaxation rates and take the same algebraic form, \added[]and the energy transfer involved, and , can both be approximated by in their respective field regime. We thus use to denote spin relaxation rate due to SV mixing \added[](in the next subsection we will discuss the spin relaxation rate due to intra-valley SO mixing, which involves higher electron orbital states but within the same valley), so that when ; and when . The resulting spin relaxation rate is,

(9) | |||

(10) |

where is from the dipole matrix elements such as when . In other words, spin relaxation is now allowed because allows inter-valley charge transitions, while allows spin and valley-charge states to mix. More specifically, contains the field dependence of the spin-valley mixing. Its dependence comes directly from the applied field. As shown in Eq. 10, peaks at the degeneracy point , where and has a width of because of the maximum mixing of the valley states at the degeneracy point. Away from it, when ,

(11) | |||||

On the low-energy side of the peak, with , approaches a small constant that is ; On the high-energy side, with , , which again approaches 0 as increases. This clear peak structure means that the spin-valley mixing induced spin relaxation is the most significant near the degeneracy point between and .

In the cases when for the transition matrix elements between the valley states,Yang et al. (2013) which implies that valley energy shift due to the electrical noise is the same in both valleys, the relaxation rate vanishes. is then the only spin relaxation channel due to spin-valley mixing.

For the sake of completeness, we now consider the relaxation of state . The relaxation of state has two possible origins: The first is the relaxation to and . This is valley relaxation due to electrical noise, with a relaxation rate that is proportional to , so that . The spin-valley relaxation of state to is identical to relaxation from to , because the transition matrix elements, the degree of spin-valley mixing, and the energy splitting are all the same for these two transitions. The second relaxation mechanism for state is the relaxation due to spin-valley mixing of and , which has been omitted at the beginning, since the effect of mixing is suppressed by the large energy separating and . However, when considering relaxation of state , this particular spin valley mixing could certainly lead to additional relaxation. In the following, we focus on the spin relaxation of states and with the flipping of spin-up state to spin-down state.

The spin relaxation mechanism discussed here is a consequence of spin-valley mixing and finite electric dipole matrix elements between the valley states. Therefore, as shown in the Eq. (9), the relaxation rate is proportional to the matrix elements and the function , which captures the extent of SV mixing. Finally, the dependence of is given by , which depends on the specific noise spectrum .

### iii.2 Spin relaxation due to intra-valley SO mixing

\added[]Spin relaxation due to spin-valley mixing is particularly important when is comparable with and is much less than orbital level spacing . As B-field increases, higher-energy orbital states also start to contribute to spin relaxation significantly. For comparison, we include in our discussion below spin relaxation due to intra-valley SO mixing \added[](higher energy p-orbitals are involved), which has been studied extensively in the literature, especially for spin qubit in GaAs QD. Khaetskii and Nazarov (2001); Erlingsson and Nazarov (2002); Golovach et al. (2004); Tahan and Joynt (2005); Marquardt and Abalmassov (2005); San-Jose et al. (2006); Trif et al. (2008); Yang et al. (2013); Tahan and Joynt (2014); Huang and Hu (2014); Jing et al. (2014) For spin qubit in Si QD, this intra-valley SO mixing induced spin relaxation is also present, and is important in high B-field due to the stronger B-field dependence. Tahan and Joynt (2014); Huang and Hu (2014) \added[]We use the existing results in the literature, and the corresponding spin relaxation rate is Golovach et al. (2004); Tahan and Joynt (2014); Huang and Hu (2014)

(12) |

where is the lateral confinement strength of QD, is the Zeeman frequency, and is the Fourier spectrum of the correlation of in-plane electric field fluctuations (in-plane electrical noise is assumed to be isotropic, and out of plane electrical noise is neglected because of the strong vertical confinement at the interface). contains the dependence on the SO interaction strength and the orientation of magnetic field. For a magnetic field along [110] direction as in Ref. Yang et al., 2013, we have . Huang and Hu (2014)

[]In a general calculation of spin relaxation in a Si QD, both spin relaxation mechanisms, namely relaxation due to spin-valley mixing () and relaxation due to intra-valley SO mixing (), need to be accounted for. We consider both in our calculations below in order to achieve a comprehensive understanding of spin relaxation.

## Iv Results

In this section, we present spin relaxation rates for different noises, and we compare the spin relaxation channels due to SV mixing and intra-valley SO mixing. We mainly focus on the electrical noise from Johnson noise and phonon noise. Although charge noise is ubiquitous as well in semiconductor material, we do find that spin relaxation due to noise is much slower compared to that due to Johnson and phonon noise. Thus we only give a brief discussion on noise in Appendix B.

### iv.1 Johnson noise

Johnson noise is the electromagnetic fluctuations in an electrical circuit. For a gate-defined QD, Johnson noise inside the metallic gates, such as the source and drain circuits, could give rise to strong electrical fluctuations acting on the QD, and it could induce spin decoherence for the electron confined in the QD.

The spectrum of Johnson noise is given by Weiss (1999)

(13) |

where is the spectrum of electrical voltage , is a dimensionless constant, k is the quantum resistance, and is the resistance of the circuit. is a natural cutoff function for Johnson noise, where is the cutoff frequency, and is capacitors in parallel with the resistance .

As shown in Fig. 2, the Johnson noise of the circuits outside the dilution refrigerator is generally well-filtered. Thus we consider only Johnson noise of the low-temperature circuit inside a dilution refrigerator. The corresponding spectrum of electric field is , where is the length scale between the source and drain. Accordingly, the spin relaxation rate due to SV mixing and Johnson noise is

(14) |

where is given by Eq. (10). The small capacitance of source and drain leads means that the cutoff frequency satisfies , so that the cutoff function . The low temperature environment ensures . Therefore, the dependence of is determined by .

Compared with the intra-valley SO mixing mechanism, where shows an dependence,Huang and Hu (2014) is linearly dependent on at low fields, when so that . Because of this weaker field dependence, the spin relaxation rate would dominate over at very low magnetic fields. On the other hand, at high fields, when , we have , then : the relaxation rate is slower as the external field increases. Thus, at high fields the intra-valley spin relaxation should dominate over inter-valley spin relaxation.

Below we carry out numerical calculations of the spin relaxation rate in a small Si/SiO QD. Based on the parameters of Ref. Yang et al., 2013, the valley splitting here is set as meV, the dot confinement energy is meV, and the electric dipole matrix elements for the valley states are set as and nm. The magnetic field is along the direction, and the SO interaction strength for Si is set as m/s and m/s.Wilamowski et al. (2002); Tahan and Joynt (2005); Prada et al. (2011); Yang et al. (2013) We use the bulk g-factor , and in the lowest two valleys the electron effective mass is , where is the free electron rest mass. For Johnson noise parameters, we choose the resistance k, length scale nm and temperature K. The magnitude of the chosen resistance allows us to obtain the best numerical fit to the experimental data (with the rest of the parameters chosen according to Ref. Yang et al., 2013). While resistances of the thin metallic gates at low temperatures are generally much smaller than 2 k, the resistance of other elements such as 2DEG channels can easily be in this order.

Figure 3 shows the spin relaxation rates through SV mixing (red dashed line), through SO mixing (blue dash-dotted line), and the total spin relaxation rate (black solid line) as a function of the applied magnetic field due to Johnson noise. As shown in Fig. 3, the relaxation rate through the intra-valley SO mixing is dominant in the high-field regime, showing a dependence. The relaxation due to SV mixing peaks at the degenerate point (), and it dominates in the low-magnetic-field regime due to a linear dependence. The relaxation time due to the Johnson noise is about s when T, and about 0.01 s when T.

### iv.2 Phonon noise

Phonon noise is the most studied spin relaxation source, and it is usually the dominant source of spin relaxation in the strong-magnetic-field regime because of the higher phonon density of states at high frequency.Khaetskii and Nazarov (2001); Golovach et al. (2004); Amasha et al. (2008); Yang et al. (2013); Tahan and Joynt (2014); Huang and Hu (2014) Although results for spin relaxation due to SV mixing and phonon noise have been obtained in Ref. Yang et al., 2013, we include this spin relaxation channel here for completeness. Furthermore, a unified treatment is given here for both phonon and Johnson noise, and the phonon bottleneck effect is taken into account in a simplified manner.Tahan and Joynt (2014)

To obtain the results for phonon noise, we need the correlation of the electric field , which can be derived based on the electron-phonon interaction potential ,Golovach et al. (2004); Tahan and Joynt (2014)

(15) |

where ( ) creates (annihilates) an acoustic phonon with wave vector , branch index , and dispersion ; is the sample density (volume is set to unity). The factor equals unity for and vanishes for , where is the characteristic size of the quantum well along the axis. Here we consider the deformation potential electron-phonon interaction, with being the deformation potential constants (piezo-electric interaction vanishes in Si due to the non-polar nature of the lattice). In Si, the deformation potential strength for different branches is (LA), (TA) and (TA), where , and are the dilation and uniaxial shear deformation potential constants.Yu and Cardona (2010)

To calculate spin relaxation due to the phonon noise, we first need to obtain the phonon correlation functions, which are discussed in detail in Appendix C. Substituting the correlation functions into Eq. (9), we find that the dependence of on the applied magnetic field is determined by the factor . , on the other hand, is proportional to . Both rates are proportional to the deformation potential strength and inversely proportional to the seventh power of phonon velocity .

Figure 4 shows the spin relaxation rates through SV mixing (red dashed line), through SO mixing (blue dash-dotted line), and the total spin relaxation (black solid line) as a function of the applied magnetic field due to phonon noise. The parameters are kg/m, m/s, m/s (data for SiO), eV, eV, K, and the other parameters are the same as before. Similar to Johnson noise, the relaxation through the SV mixing dominates in the low-magnetic-field regime, and it peaks at the degeneracy point. The relaxation rate through the intra-valley SO mixing is dominant in the high-magnetic-field regime, which shows a dependence before the phonon bottleneck takes effect and the curves bend downward from the line. Golovach et al. (2004); Tahan and Joynt (2014); Huang and Hu (2014) The phonon bottleneck effect is due to the averaging of electron-phonon interaction matrix element for high-frequency phonons. This reduction in the effective coupling strength causes the spin relaxation rate to decrease from the curve in Fig. 4, and it could even lead to a suppression of spin relaxation,Huang and Hu (2014) as has been observed experimentally for a spin singlet-triplet qubit.Meunier et al. (2007) Quantitatively, the relaxation time due to phonon noise is s in a 1 Tesla field, and ms in a 10 Tesla field.

### iv.3 Comparison of Johnson and phonon noises

In this section, we compare the magnetic-field dependence of the spin relaxation rate for Johnson noise and phonon noise. The effects of other noises, such as noise, are relatively small, as shown in Appendix B. Since the magnetic-field dependence of spin relaxation is different for different noises, the dominant source of relaxation could be different in different regimes.

Figure 5 shows the spin relaxation due to phonon noise (red dashed line) and Johnson noise (blue dash-dotted line) as a function of the applied magnetic field with a valley splitting meV. The other parameters are the same as in the previous two subsections, namely a QD confinement of meV, the dipole matrix elements nm, the SO interaction strengths m/s and m/s, and the resistance for Johnson noise at k. The red triangles are experimental results from Ref. Yang et al., 2013. For comparison, the phonon-induced relaxation rates (black dotted line) obtained without considering the phonon bottleneck effect (without the cutoff function) are also presented, which reproduces the original fitting in Ref. Yang et al., 2013. There are three interesting features to this figure: the spin hot spot, which we have discussed extensively in previous subsections, the high-field trend, and the low-field trend. Below we examine the later two features in more detail.

Figure 5 shows that, at high B-field, spin relaxation due to phonon noise dominates over relaxation due to Johnson noise, as expected from the spectral densities of these two noises. At the highest magnetic fields in the figure, the curve without phonon bottleneck effect looks more consistent with the experimental data. This is because we are using the parameters from Ref. Yang et al., 2013 instead of refitting the parameters such as the SO coupling and the dipole matrix element . We emphasize that the only fitting parameter in our case is the resistance . If we want more consistent results with experimental data, one needs to (i) increase the spin-orbit coupling to have faster spin relaxation due to spin-orbit mixing; (ii) reduce the dipole matrix element , so that the width of the spin relaxation peak, which is determined by , does not change; and (iii) increase the resistance to get the same magnitude of spin relaxation at low fields. Since a slight variation of these parameters does not have much of an impact on the understanding of the system, and the parameters differs for different materials, we prefer using the parameters given by Ref. Yang et al., 2013, and changing only the resistance of Johnson noise to make sure that the low-frequency regime is well understood. We also note that the measured relaxation rate seems to increase faster at very high fields ( 4 T) than both theoretical calculations, with or without the phonon bottleneck effect.Yang et al. (2013) This discrepancy could be due to another level crossing (and the associated spin hot spot) at a higher field that is not taken into consideration in the current study, or a reflection of non-parabolic features of the QD confinement.

At low magnetic fields, the dominant spin relaxation channel crosses over from phonon noise to Johnson noise (around 2 T). As discussed in Sec. IV.1, the dominant relaxation mechanism at low magnetic field is due to Johnson noise and SV mixing. By considering the Johnson noise, the theoretical results of total spin relaxation (black solid line) are now more consistent with the experimental measurements in Ref. Yang et al., 2013, where the relaxation rate at T is around 0.1 s.

Figure 6 shows the spin relaxation rate due to phonon noise and Johnson noise as a function of the applied magnetic field at a valley splitting of meV. The other parameters are the same as in Fig. 5, except the dipole matrix elements are a bit larger at nm.Yang et al. (2013) In essence, throughout the whole field range in this figure, the system is on the low-energy side of the degeneracy point or the spin hot spot. As shown in Fig. 6 (a), at higher magnetic fields, the dominant relaxation source is phonon noise and SV mixing. For lower fields, the dominant relaxation channel changes over to Johnson noise and SV mixing. Figure 6 (b) shows that, similar to Fig. 5, after including the effects of Johnson noise, the theoretical results of total spin relaxation (black solid line) are now more consistent with the experimental measurements (green triangles) at lower magnetic fields,Yang et al. (2013) where the relaxation rate at T is around 0.3 s.

Figure 6 is essentially the low-energy side of Fig. 5, with a shift in the peak position and a slight increase in the peak width. The enlarged plot does reveal more clearly one important fact: with the given Si parameters the phonons provide a more important relaxation channel compared to Johnson noise at the spin hot spot. The transition of the dominant relaxation channel happens at a field significantly below the degeneracy point, at just below 2 T. Again the no-cut-off results seem to fit the experimental data better than results with the phonon bottleneck effect. This is due to our choice of parameters and , which are taken directly from Ref. Yang et al., 2013. Since a slight variation of these parameters does not change our understanding of spin dynamics, we use the values of these parameters from Ref. Yang et al., 2013, and we change only the resistance of Johnson noise to make sure that the data fit in the low frequency regime is optimized.

## V Conclusion

In conclusion, we have studied spin relaxation of an electron in a Si QD with valley splitting. In particular, we have clarified how the presence of conduction-band valleys influences spin relaxation. By considering both spin-valley mixing and intra-valley spin-orbit mixing in a Si QD, we find that spin relaxation due to Johnson noise is the dominant spin relaxation channel (as compared to phonons and other electrical noises) when the Zeeman splitting is much smaller than the valley splitting.

In our calculations, we have included both Johnson and phonon noises, and we incorporated both spin-valley and intra-valley spin-orbit mixings. For the various field regimes as compared with valley splitting we find the following. In the low-field regime, when Zeeman splitting is much smaller than the valley splitting , Johnson noise together with spin-valley mixing leads to the fastest spin relaxation because of a weaker field-dependence. As the magnetic field increases and the Zeeman splitting approaches the valley splitting, , spin-valley mixing together with both phonon noise and Johnson noise produces a sharp peak in the spin relaxation rate, though for Si with the parameters from experiments, phonon noise is now the most important source of spin relaxation (while Johnson noise also contributes significantly). When the applied field increases further, , the intra-valley spin-orbit mixing gradually becomes the dominant spin relaxation mechanism because of its stronger dependence on the external field, which is consistent with the existing literature. Using parameters obtained from an experimental measurement Yang et al. (2013), and a single fitting parameter of low-temperature circuit resistance, we obtain numerical results that fit the measurements well in the whole range of applied magnetic field.

We acknowledge the support of U.S. ARO (W911NF0910393) and NSF PIF (PHY-1104672). We also acknowledge useful discussions with Andrew Dzurak, Andrea Morello, Jason Petta, and Charles Tahan. XH would also like to acknowledge the hospitality of Kavli Institute of Theoretical Physics China, where part of this work was completed.

## Appendix A Effects of the magnetic field orientation

The spin relaxation mechanism we study in this paper involves the spin-orbit interaction. When both Dresselhaus and Rashba SO coupling are present in a system, such as in a Si heterostructure, the orientation of the applied magnetic field plays an important role in determining the amount of transverse magnetic noise and thus the relaxation rate. Here we discuss this field orientation dependence in detail.

Consider a magnetic field in an arbitrary direction, , where and are the polar and azimuthal angles of the magnetic field in the () coordinate system. By using the relationship , the spin-valley mixing matrix element can be expressed in terms of the electric dipole matrix element,Tahan and Joynt (2014)

(16) |

where is the spin flip matrix elements, and are the spin-orbit coupling constants.

In order to calculate the spin flip matrix elements , it is convenient for us to express the spin state (= or ), which are the eigenfunctions of ( axis along the magnetic field), in terms of the eigenstates of : , where is the finite rotation matrix,Landau and Lifshitz (1977)

(17) | |||||

(18) |

Therefore, spin-flip matrix elements are and , and the square of the magnitude of the SV mixing matrix element is

(19) | |||||

When is along the z-direction (, ),

(20) | |||||

When is in the plane of 2DEG (),

(21) | |||||

Therefore, the magnetic field orientation dependence of (or ) depends on the values of , , and , which is material- and device-specific. In particular, if the magnetic field is along the [110] crystal axis , as is the case in Ref. Yang et al., 2013, , , and . In our calculation, we used to reproduce the results of Ref. Yang et al., 2013.

## Appendix B charge noise and spin relaxation

The charge noise is quite common in semiconductor devices, and is often believed to be an important decoherence source for charge qubits. Here we explore how much it affects a spin qubit.

The charge noise is often measured via the fluctuations it causes in the energy levels in a quantum dot or a quantum point contact (QPC). Jung et al. (2004); Buizert et al. (2008); Hitachi et al. (2013); Takeda et al. (2013) Consider the current through a QPC connected to two leads. The current is sensitively dependent on the gate voltage applied to the QPC. By measuring the electric current fluctuations, the overall effect of the charge noise on the QPC can be measured. Normally, such an experiment has a finite frequency range, e.g. from a few Hz to hundreds of Hz. The measured energy level fluctuations actually depend on the frequency range of the measurement, and are thus dependent on the specific experiment. Thus here we first try to extract a quantity that is independent of the frequency range in these experiments.

We assume the current fluctuation spectral density due to the charge noise in a QPC to be . An integration of the spectrum yields

(22) |

Phenomenologically, the current fluctuation can be represented by an effective gate voltage fluctuation, Jung et al. (2004); Buizert et al. (2008); Hitachi et al. (2013); Takeda et al. (2013)

(23) |

where and are the lower and upper cutoff frequency (response frequency) in the experiment. is the effective differential conductivity, which represents the variation of the electric current through QPC due to the gate voltage difference. Therefore, the quantity represents the effective gate voltage fluctuation due to charge noise in the system. In order to get the effective electric field on the electron in the QD, we should also consider the screening effect of the gate voltage.

The quantity defined here is dependent on the frequency range of the measurement in the experiments,

(24) |

We define a quantity as the effective gate voltage fluctuation, which is independent of the frequency range. Take Ref. Takeda et al., 2013 as an example for the charge noise in Si/SiGe, where meV, Hz, Hz, and . Therefore, the effective gate voltage fluctuation due to charge noise is eV. Due to the screening of the gate voltage, the effective voltage fluctuation sensed by the electron in the QD is around eV.

With the knowledge of the magnitude of charge noise, we can calculate the corresponding spin relaxation. The spin relaxation due to the SV mixing and charge noise is given by

(25) |

where are the transition matrix elements between the two lowest valley states, is the charge noise amplitude, and is the Zeeman frequency. The dependence of on the applied magnetic field is , and the function is given by Eq. (10).

Figure 7 shows the spin relaxation rates through SV mixing (red dashed line), through SO mixing (blue dash-dotted line) and the total spin relaxation (black solid line) as a function of the applied magnetic field due to charge noise. The results of the spin relaxation rate due to charge noise and intra-valley SO mixing is from Ref. Huang and Hu (2014). As shown in the figure, the relaxation through the mechanism of SV mixing dominates in the low magnetic field regime, and it peaks at the degenerate point (). The relaxation rate through the intra-valley SO mixing is dominating in the high magnetic field regime. The relaxation time due to the charge noise is about s for a Si QD, when the Zeeman energy is away from the valley splitting.

## Appendix C Spectrum of Phonon Noise

The electron phonon interaction is given by Eq. (15). In the interaction picture, the electron phonon interaction acquires a time dependence, with and . The correlation of the electric force due to phonons, , is thus given by (x component),

(26) | |||||

We consider the adiabatic condition, where the energy scale of the noise is much less than the dot confinement energy and the valley splitting, so that the electron orbital state stays in the instantaneous ground state where is the effective radius. Then, we simplify the exponential terms by its mean-field value .

The summation in Eq. (26) for all possible in the momentum space can be expressed as integrals

(27) |

where is the density of states for phonons, and

(28) | |||||

In Eq. (28), is the phonon excitation number and the cutoff function is due to the suppression of the matrix element for the electron-phonon interaction in a large QD.Huang and Hu (2014)

The spectrum of the phonon noise in the -direction is therefore ()

Similarly, and

If the dipole approximation is employed (for most spin qubit applications, the dipole approximation should be valid), so that