Charge noise, spin-orbit coupling, and coherence of single-spin qubits
CCharge noise, spin-orbit coupling and dephasing of single-spin qubits
Adam Bermeister, Daniel Keith, and Dimitrie Culcer
School of Physics, The University of New South Wales, Sydney 2052, Australia (Dated: August 22, 2018)Quantum dot quantum computing architectures rely on systems in which inversion symmetry isbroken, and spin-orbit coupling is present, causing even single-spin qubits to be susceptible to chargenoise. We derive an effective Hamiltonian for the combined action of noise and spin-orbit couplingon a single-spin qubit, identify the mechanisms behind dephasing, and estimate the free inductiondecay dephasing times T ∗ for common materials such as Si and GaAs. Dephasing is driven by noisematrix elements that cause relative fluctuations between orbital levels, which are dominated byscreened whole charge defects and unscreened dipole defects in the substrate. Dephasing times T ∗ differ markedly between materials, and can be enhanced by increasing gate fields, choosing materialswith weak spin-orbit, making dots narrower, or using accumulation dots. Developments in quantum computing hold consider-able promise in the progress of modern information pro-cessing, and this has spurred a large experimental andtheoretical effort investigating two-level systems that canbe used as quantum bits (qubits). The need for scalabil-ity and long coherence times has led naturally to solidstate spin-based devices, such as quantum dot spin sys-tems, as ideal candidates for scalable qubits. The focushas been on single-spin [1] and singlet-triplet qubits. [2]While GaAs quantum dots have been studied for manyyears, a substantial effort is also underway researching Sispin quantum computing architectures, [3–5] motivatedby their compatibility with Si microelectronics and longcoherence times. [6–12] Recently, much effort has alsobeen devoted to quantum dot systems with spin-orbitinteractions, [13–15] where spin manipulation could inprinciple be achieved entirely by electrical means. [16–18]The coherence of a solid-state spin qubit is quantifiedby the relaxation time T and the dephasing time T ∗ ,both of which are determined by mechanisms that cou-ple up spins with down spins. This coupling can eitherbe direct, through the hyperfine interaction [19–22] andfluctuations in the g -factor, [23] or indirect, through thejoint effect of hyperfine or spin-orbit coupling and fluctu-ating electric fields, such as those due to phonons [24–30]or charge noise. [31–33] Inversion symmetry breakingnear an interface makes spin-orbit coupling unavoidable,even in materials such as Si in which it is weak. [34]Hyperfine effects typically occur on long time scales,the nuclear bath is relatively well known and can be con-trolled through feedback mechanisms [35] while in ma-terials such as Si hyperfine coupling can be eliminatedaltogether through isotopic purification. [36, 37] Thespin relaxation rate due to phonons is proportional tothe fifth power of the magnetic field in zinc-blende mate-rials, in which piezoelectric electron-phonon coupling isoften dominant, and to the seventh power of the magneticfield in Si, in which there is no piezoelectric coupling.[28]Hence phonon effects become less pronounced at lowmagnetic fields. They also become weaker at low tem-peratures. [25]Noise is a well-known source of dephasing in charge qubits. [38–41] Experiments on quantum dots and pointcontacts have shown noise to be strong even at dilutionrefrigerator temperatures. [38–46] Noise sources include P b centers, which may act as traps that charge and dis-charge, and tunneling two-level systems, which can bemodeled as fluctuating charge dipoles. [31, 47–52] Noiseand spin-orbit coupling give rise to nontrivial physics in2D and 1D structures. [53–55] In quantum dot spinqubits, Ref. 33 has already shown that spin-orbit andnoise lead to spin relaxation, and that noise and phononeffects in general become comparable at low-enough mag-netic fields. Hence, at dilution refrigerator temperaturesthe interplay of spin-orbit and noise may set the definingbound on spin qubit coherence.In this paper we build on previous decoherence work[56–62] and devise a theory of dephasing due to the com-bined effect of charge noise and spin-orbit interactions,with two aims in mind. The first is to understand concep-tually how spin-orbit and noise cause dephasing. For ex-ample, noise can give relative fluctuations between levels,virtual transitions between levels, as well as fluctuationsin spin-orbit constants. We wish to isolate the terms thatare responsible for dephasing. The second aim is to studythe sensitivity to spin-orbit coupling across common ma-terials with similar noise profiles. We study a samplequbit with the same specifications in different materials,we determine sample T ∗ s due to common noise sources,discuss the variation in T ∗ across materials, and seekmethods to improve T ∗ generally.We consider a single-spin qubit implemented in a sym-metric, gate-defined quantum dot, located at a sharp flatinterface (Fig. 1) in a dilution refrigerator at 100mK. Thequbit is described by the Hamiltonian H = H QD + H Z + H SO + H N . The kinetic energy and confinement term H QD = − (cid:126) m ∗ (cid:18) ∂ ∂x + ∂ ∂y (cid:19) + (cid:126) m ∗ a ( x + y ) , (1)where a is the effective dot radius and m ∗ the effec-tive mass. The eigenstates of H QD are the Fock-Darwin a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov states, with the ground and first excited states given byΦ ( x, y ) = a √ π e − (cid:16) x y a (cid:17) Φ ± ( x, y ) = a √ π ( x ± iy ) e − (cid:16) x y a (cid:17) . (2)These have energies ε = (cid:126) / m ∗ a for the orbitalground state and ε = 3 (cid:126) / m ∗ a for the twofold degen-erate first orbital excited state. The orbital level splittingis assumed to be the dominant scale, so that only theground and first excited states are considered. The Zee-man Hamiltonian H Z = gµ B σ · B , with σ the vector ofPauli spin matrices. Since B is constant, the orbital ef-fect of B can be absorbed into the effective dot radius a .We have also not taken into account multi valley effectsin Si. For a certain interaction to couple valley statesappreciably, it must be sufficiently sharp in real space.Neither the spin-orbit coupling due to the interface fieldnor the electric field of the defect satisfy this require-ment – even though these interactions are important inrelaxation in particular around hot spots. [63]The spin-orbit term H SO = H R + H D . The Rashbaterm H R = α ( t ) σ · ( k × ˆ z ), stems from structure in-version asymmetry, where k = − i ∇ here is an operatorin real space, ˆ z is the unit vector perpendicular to theinterface, and α is determined by a material specific pa-rameter as well as the interface electric field E z . [64]Thus α is also sensitive to stray electric fields and fluc-tuates in time, thus we let α ( t ) = α [1 + λ ( t )] where λ (cid:28)
1. For a quantum dot on a (001) surface the lin-ear Dresselhaus term H D = β ( σ y k y − σ x k x ) is usuallythe dominant bulk inversion asymmetry contribution,[64]where β = β ( π/w ) , with β a material-specific param-eter and w the width of the z -confinement perpendicularto the interface. Since H D can be obtained from H R bya π/ β = 0 due to inversion symmetry, whereasRashba spin-orbit coupling is expected generally in a 2Delectron gas near an interface, and should be present inall gate-defined dots. In zincblende structures H R and H D may comparable in magnitude in certain parame-ter regimes, though for E z ≈ Vm − ( ≡ H N ( t ) is a random function oftime. We do not include gate noise in our model, andwe first consider random telegraph noise (RTN). In thesimplest case, in which the qubit is only sensitive to onedefect, H N represents a fluctuating Coulomb potential,screened by the nearby 2D electron gas. The 2D screenedCoulomb potential U scr is written in terms of its Fouriertransform, which is a function of momentum q [65] U scr ( r ) = e (cid:15) (cid:15) r (cid:90) k F d q (2 π ) e − i q · r q + q T F , (3)with (cid:15) r the relative permittivity, q T F the Tomas-Fermiwave vector, and k F the Fermi wave vector (the con- Figure 1: Defect locations with respect to the gate-definedquantum dot projected onto the xz -plane, with ˆ z normal tothe interface. In general a top gate is also present (not shown).The red area represents the region of the quantum dot. tribution from q > k F is negligible [62]). The ma-trix elements entering H N are v = (cid:104) Φ | U scr | Φ (cid:105) , v = (cid:104) Φ ± | U scr | Φ ± (cid:105) , v = (cid:104) Φ | U scr | Φ ± (cid:105) ≈ (cid:104) Φ ± | U scr | Φ (cid:105) and v = (cid:104) Φ ± | U scr | Φ ∓ (cid:105) ≈ (cid:104) Φ ∓ | U scr | Φ ± (cid:105) . For RTN we canwrite v i ( t ) = v i ( − N ( t ) for i = 0 , ,
2, and N ( t ) = 0 , τ . [73]Additional (extrinsic) spin-orbit coupling arises fromthe electric field of the defect itself. Yet for a charge de-fect located 40 nm away from the dot this field is severalorders of magnitude smaller than the interface electricfield E z . Because the matrix element involved is secondorder in v i , the contribution this makes to dephasing ismany orders of magnitude smaller than the Rashba in-teraction due to E z , and will not be considered further.In the basis { Φ ↑ , Φ ↓ , Φ + ↑ , Φ + ↓ , Φ −↑ , Φ −↓ } , with ↑ , ↓ representing up and down spins, the Hamiltonian reads H = ε +0 v s R v is D ε − is D v − s R v v − is D ε +1 v s R v ε − v v − s R v ε +1 − is D v v ε − (4)where ε ± ( t ) = ε + v ( t ) ± ε Z and ε ± ( t ) = ε + v ( t ) ± ε Z are the Zeeman-split orbital levels including the noiseterms, the Zeeman energy ε Z = gµ B B , and the spin-orbit terms s D = β/a and s R ( t ) = s R [1 + λ ( t )], with s R = α/a (not a function of time).The qubit subspace is simply the Zeeman-split orbitalground state { Φ ↑ , Φ ↓ } , which has been singled out inthe top left hand corner of Eq. 4. These two states arecoupled by H N to spin-aligned orbital excited states andby H SO to orbital excited states with anti-aligned spin.By projecting H onto this subspace we encapsulate thecombined effect of spin-orbit coupling and noise in an effective qubit Hamiltonian H qbt . To achieve this, wecarry out a Schrieffer-Wolff transformation, eliminatinghigher orbital excited states. [25, 64, 67, 68] Keepingterms up to the second order in this transformation, H qbt ( t ) = H Z − ε Z { v ( t )[ s R ( t ) σ x + s D σ y ] + [ s R ( t ) + s D ] σ z } [ δε + δv ( t )] (5)where δε = ε − ε (not a function of time) and δv ( t ) = v ( t ) − v ( t ). We retain only terms of first order in ε Z and δv . Equation (5) implies that, in addition to H Z ,there exists an effective Zeeman term σ · V ( t ), where V ( t ) represents an effective fluctuating effective magneticfield due to the combined action of spin-orbit and noise.For convenience V has units of energy and, for RTN, V ( t ) = V ( − N ( t ) . We will also use V ( t ) = | V ( t ) | forthe magnitude of V . Since the Rashba and Dresselhauscontributions are added in quadrature, there is no sweetspot for dephasing.The noise matrix elements appearing in H qbt maybe divided into two categories. The diagonal elements v ( t ) , v ( t ) cause different orbital levels to fluctuate bydifferent amounts, while the off-diagonal element v ( t )causes transitions between different orbital levels. If thequbit is initialized in an off-diagonal state, the diago-nal elements ( σ z ) in H qbt give dephasing. These termsinvolve the intraband matrix elements v ( t ) , v ( t ) of thedefect potentials. An additional contribution comes fromfluctuations in α , which lead to fluctuations in s R it-self. These fluctuations can be interpreted as a modu-lation of the g -factor, and are expected to come fromdefects in the substrate right above the dot, which mod-ify E z . Since the dot region is depleted, whole chargedefects cannot fluctuate, except in the very special casein which the defect lies right above the dot. Hence de-fects contributing to E z are expected to be mostly chargedipoles, stemming for example from passivated traps. Al-though these are weaker than whole charge defects, theyare unscreened, leading to a subtle competition. Thus,generally, dephasing stems from noise matrix elementsthat cause relative fluctuations between orbital levels. Incontrast, if the qubit is initialized in the spin-up state,the off-diagonal elements ( σ x ) in H qbt give relaxation ( T processes), which was studied in detail in Ref. 33. Theseelements are of first order in α and involve the interbanddefect matrix element v ( t ). [74]In order to study dephasing further and obtain quan-titative estimates of T ∗ , we focus on a single-spin qubitdescribed by a spin density matrix ρ ( t ). The spin densitymatrix satisfies the quantum Liouville equation dρdt + i (cid:126) [ H qbt , ρ ] = 0 . (6)The spin density matrix ρ ( t ) = σ · S ( t ). Any spincomponent S i can be found as S i ( t ) = tr [ σ i ρ ( t )], withtr the matrix trace. We restrict our attention to RTNfor the time being. Using the time evolution operator e − ( i/ (cid:126) ) (cid:82) t H qbt ( t (cid:48) ) dt (cid:48) , we obtain the general time evolution of the spin as S ( t ) = S cos h − ( S × ˆ h ) sin h + ˆ h (ˆ h · S )(1 − cos h ) , (7)where we have defined S ≡ S ( t = 0) and, for RTN, h ( t ) = ( V / (cid:126) ) (cid:82) t ( − N ( t (cid:48) ) dt (cid:48) , with h ( t ) = | h ( t ) | . Thetwo components of h have exactly the same time evo-lution. Since | B | (cid:29) | V , if S = S x ˆ x is initialised, S x ( t ) ≈ S x cos [ h ( t )]. Averaging over noise realisations[56, 59, 62] (cid:104) cos [ h ( t )] (cid:105)(cid:105) = e − t/τ (cid:18) sinh Ξ t Ξ τ + cosh Ξ t (cid:19) , (8)where Ξ = (cid:112) ( (cid:126) /τ ) − V / (cid:126) . All systems of interest inthis work satisfy V (cid:28) ( (cid:126) /τ ) , in which case we may ap-proximate (cid:113)(cid:0) (cid:126) τ (cid:1) − V ≈ (cid:126) τ (cid:16) − V τ (cid:126) (cid:17) . When the de-nominator of the sinh is expanded in ( V τ / (cid:126) ) , only theleading term in the expansion may be retained. Physi-cally, in this case, the time dynamics of h ( t ) are a randomwalk in time, and the spread in cos h ( t ) leads to motionalnarrowing. As a result, the initial spin decays exponen-tially as S x ( t ) ≈ S x e − t/T ∗ , where (cid:18) T ∗ (cid:19) RT N = V τ (cid:126) . (9)For whole charge defects, where dephasing is dominatedby fluctuations in the orbital energy, we may set λ ( t ) = 0and retain V wh ( t ) = 8 ( s R + s D ) ε Z δv ( t ) / ( δε ) . For dipolecharge defects we have V dip ( t ) = 8 ( s R + s D ) ε Z λ ( t ) / ( δε ) .We turn our attention next to 1 /f noise. In semicon-ductors 1 /f noise is Gaussian [70] and is fully describedby its spectral density S ( t − t (cid:48) ) = (cid:104) H N ( t ) H N ( t (cid:48) ) (cid:105) . TheFourier transform of this spectral density has the form S ( ω ) = γk B Tω , where γ is a parameter typically inferredfrom experiment. Based on our estimates for RTN abovewe expect whole charge defects to dominate dephasing.Hence, for the effective fluctuating magnetic field V ( t )acting in the qubit subspace, we may write approximately S V ( ω ) ≈ (cid:20) s R + s D ) ε Z ( δε ) (cid:21) S ( ω ). To study dephasing, wewrite S x ( t ) = S x e − χ ( t ) , where χ ( t ) = 2 γk B T (cid:126) (cid:20) s R + s D ) ε Z ( δε ) (cid:21) (cid:90) ∞ ω dω sin ωt/ ω . (10)The low-frequency cut-off ω is usually taken to be theinverse of the measurement time. At times t (cid:28) /ω such as we consider here, we can approximate χ ( t ) ≈ (cid:18) tT ∗ (cid:19) ln 1 ω t , (11)where the dephasing time is estimated by (cid:18) T ∗ (cid:19) /f ≈ (cid:114) γk B T (cid:126) (cid:20) s R + s D ) ε Z ( δε ) (cid:21) . (12) Table I: Sample T ∗ for a quantum dot with a = 20 nm, λ =4 × − , τ = 1 µ s and the defect distance is 40 nm (for RTN), E z = 20 MV/m, ε Z = 60 µ eV, T = 0.1 K, α from Refs. 8, 64, β from Ref. 64 and S ( ω ) for 1 /f noise estimated from Refs.38, 42. Following Ref. 8, the confinement perpendicular tothe interface ( (cid:107) ˆ z ) is represented by a square well of width 15nm. For Si the valley splitting is assumed large. α (peV m) β (peV m) ( T ∗ ) RTNwh ( T ∗ ) RTNdip ( T ∗ ) /fwh Si/SiGe 0 .
02 0 3 ms 18 s 20 µ sGaAs 1 . .
12 60 ns 280 µ s 20 nsInAs 23 0 .
12 40 ps 65 ns 900 psInSb 105 3 . a (cid:61) (cid:72) (cid:144) f (cid:76) a (cid:61) (cid:72) (cid:144) f (cid:76) a (cid:61) (cid:72) RTN (cid:76) a (cid:61) (cid:72) RTN (cid:76) t (cid:72) Μ s (cid:76) S x (cid:72) t (cid:76)(cid:144) S x (cid:72) (cid:76) Figure 2: Time evolution of the initated spin for differentdot radii a in Si/SiGe, ω =1 s and other values as in Table I. Since this definition of T ∗ is approximate, we plot thefull time evolution of S x ( t ) in Fig. 2.We consider a sample dot with radius a = 20 nm lo-cated at x = y = 0, and α as calculated in Refs. 8, 64.For a defect in the plane of the dot with x = 40 nm, v = 23 µ eV, v = 71 µ eV and v = 31 µ eV. Next we es-timate the change in α due to a dipole defect right abovethe dot ( x = y = 0) and z = 3 nm away from it. [31, 47]The potential of an unscreened charge dipole located adistance R D away from the dot, is U dip ( R D ) = p · ˆ R D πε ε r R D .The charge dipole has dipole moment p = − e l , where l = ( l x , l y , l z ). We take the expectation value of U dip ( R D )using Φ , and compare it with the matrix element of eE z z , yielding λ = 4 × − . We use this figure in allour estimates since ε r for all materials considered areof very similar magnitudes. For 1 /f noise we extract γ from experiment. For Si/SiGe we use Ref. 42, and forGaAs Ref. 38, while for InAs and InSb, in the absence ofexperimental data, we use the same S ( ω ) as for GaAs. The results are listed in Table I, which is the centralresult of this paper. For all materials, whole charge de-fects dominate dephasing. Table I shows that terms ofsecond-order in spin-orbit are effective in causing dephas-ing, and the dependence on α causes vast differences indephasing times T ∗ between materials. Hence, using ma-terials with a small α such as Si can improve coherenceenormously. If spin-orbit coupling is needed for electricdipole spin resonance, increasing E z will align the chargedipoles. Although that increases α and with it dephas-ing, it also reduces the gate time by an equal amount.Moreover, for 1 /f noise, T ∗ ∝ a − , so by halving the dotradius the dephasing time can be increased by an orderof magnitude (Fig. 2; for RTN, T ∗ ∝ a − ). One can alsouse pulse sequences, [43] lower the temperature to reduce S ( ω ), use accumulation dots, in which there is no nearby2DEG, or focus on reducing charge noise. [42, 44, 45]Following existing calculations of α , [8] we have takenthe ˆ z -confinement in the form of a square well, whereassemiconductor interfaces are more accurately describedby a triangular well. Nevertheless, since the form of H R and H D is dictated by symmetry, they will be identi-cal in structure for triangular confinement, thus we maysimply treat α and β as phenomenological parameters.Finally, fluctuations in w affect β . Although this effect,likewise driven by fluctuating dipoles, can be calculatedin the same way as the renormalization of α by λ ( t ),we expect its contribution to be minor, in exact analogywith H R .In summary, we have shown that spin-orbit couplingand charge noise are an effective source of dephasingin single-spin qubits even in materials such as GaAs inwhich spin-orbit coupling is weak. Based on realistic ex-perimental parameters vast differences in spin dephasingtimes exist between common materials. In the future wewill devise a full model of 1 /f noise [71] as an ensemble ofincoherent RTNs, [46] where qubit dynamics is nontriv-ial. [72] Dephasing of hole spin qubits, in which spin-orbitinteractions are also strong but the heavy hole-light holecoupling cannot be ignored, will likewise be studied in afuture publication. Acknowledgments
We thank R. Winkler, Sven Rogge, Joe Salfi, AndreaMorello, K. Takeda, Amir Yacoby, L. Vandersypen, NeilZimmerman, S. Das Sarma, Alex Hamilton, Xuedong Hu,Guido Burkard, Mark Friesen, Andrew Dzurak, MennoVeldhorst, Floris Zwanenburg, J. R. Petta, and MattHouse for enlightening discussions. [1] D. Loss and D. P. DiVincenzo, Phys. Rev. A , 120(1998).[2] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,and A. C. Gossard, Science , 2180 (2005).[3] J. Morton, D. McCamey, M. Eriksson, and S. Lyon, Nature , 345 (2011).[4] F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Sim-mons, L. Hollenberg, G. Klimeck, S. Rogge, S. Copper-smith, and M. Eriksson, Rev. Mod. Phys. , 961 (2013).[5] X. Hao, R. Ruskov, M. Xiao, C. Tahan, and H. Jiang,Nature Comm. , 3860 (2014).[6] G. Feher, Phys. Rev. , 1219 (1959).[7] E. Abe, K. M. Itoh, J. Isoya, and S. Yamasaki, Phys.Rev. B , 033204 (2004).[8] C. Tahan and R. Joynt, Phys. Rev. B , 075315 (2005).[9] A. M. Tyryshkin, J. J. L. Morton, S. C. Benjamin, A. Ar-davan, G. A. D. Briggs, J. W. Ager, and S. A. Lyon, J.Phys. Condens. Matter , S783 (2006).[10] L. Wang, K. Shen, B. Y. Sun, and M. W. Wu, Phys.Rev. B , 235326 (2010).[11] M. Raith, P. Stano, and J. Fabian, Phys. Rev. B ,195318 (2011).[12] J. Muhonen, J. Dehollain, A. Laucht, F. Hudson,T. Sekiguchi, K. Itoh, D. Jamieson, J. McCallum,A. Dzurak, and A. Morello, arXiv:1402.7140 (2014).[13] M. Friesen, C. Tahan, R. Joynt, and M. A. Eriksson,Phys. Rev. Lett. , 037901 (2004).[14] S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, andL. P. Kouwenhoven, Nature , 1084 (2010).[15] A. P´alyi, P. R. Struck, M. Rudner, K. Flensberg, andG. Burkard, Phys. Rev. Lett. , 206811 (2012).[16] D. Bulaev and D. Loss, Phys. Rev. Lett. , 97202(2007).[17] P. Szumniak, S. Bednarek, B. Partoens, and F. M.Peeters, Phys. Rev. Lett. , 107201 (2012).[18] J. C. Budich, D. G. Rothe, E. M. Hankiewicz, andB. Trauzettel, Phys. Rev. B , 205425 (2012).[19] I. A. Merkulov, A. I. Efros, and M. Rosen, Phys. Rev.B , 205309 (2002).[20] A. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. B , 195329 (2003).[21] C. Deng and X. Hu, Phys. Rev. B , 241303 (2006).[22] W. A. Coish and J. Baugh, Phys. Stat. Sol. B , 2203(2009).[23] E. Ivchenko, A. Kiselev, and M. Willander, Solid StateComm. , 375 (1997).[24] S. I. Erlingsson and Y. V. Nazarov, Phys. Rev. B ,155327 (2002).[25] V. N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev.Lett. , 016601 (2004).[26] D. Bulaev and D. Loss, Phys. Rev. Lett. , 076805(2005).[27] P. San-Jose, G. Zarand, A. Shnirman, and G. Sch¨on,Phys. Rev. Lett. , 076803 (2006).[28] M. Prada, R. H. Blick, and R. Joynt, Phys. Rev. B ,115438 (2008).[29] X. Hu, Phys. Rev. B , 165322 (2011).[30] J. I. Climente, C. Segarra, and J. Planelles, New. J.Phys. , 093009 (2013).[31] D. M. Fleetwood, S. T. Pantelides, and R. D. Schrimpf, Defects in Microelectronic Materials and Devices (CRCPress, Boca Raton, FL, 2008).[32] S. W. Jung, T. Fujisawa, Y. Hirayama, and Y. H. Jeong,Appl. Phys. Lett. , 768 (2004).[33] P. Huang and X.Hu, Phys. Rev. B , 195302 (2014).[34] Z. Wilamowski, W. Jantsch, H. Malissa, and U. R¨ossler,Phys. Rev. B , 195315 (2002).[35] H. Bluhm, S. Foletti, D. Mahalu, V. Umansky, andA. Yacoby, Phys. Rev. Lett. , 216803 (2010). [36] W. M. Witzel, X. Hu, and S. Das Sarma, Phys. Rev. B , 035212 (2007).[37] K. M. Itoh and H. Watanabe, MRS Communications (tobe published).[38] K. D. Petersson, J. R. Petta, H. Lu, and A. C. Gossard,Phys. Rev. Lett. , 246804 (2010).[39] E. Dupont-Ferrier, B. Roche, B. Voisin, X. Jehl, R. Wac-quez, M. Vinet, M. Sanquer, and S. D. Franceschi, Phys.Rev. Lett. , 136802 (2013).[40] O. E. Dial, M. D. Shulman, S. P. Harvey, H. Bluhm,V. Umansky, and A. Yacoby, Phys. Rev. Lett. ,146804 (2013).[41] E. Paladino, Y. M. Galperin, G. Falci, and B. L. Alt-shuler, Rev. Mod. Phys. , 86 (2014).[42] K. Takeda, T. Obata, Y. Fukuoka, W. M. Akhtar,J. Kamioka, T. Kodera, S. Oda, and S. Tarucha, Appl.Phys. Lett. , 123113 (2013).[43] H. Ribeiro, G. Burkard, J. R. Petta, H. Lu, and A. C.Gossard, Phys. Rev. Lett. , 086804 (2013).[44] C. Buizert, F. Koppens, M. Pioro-Ladrire, H.-P. Tranitz,I. T. Vink, S. Tarucha, W. Wegscheider, and L. Vander-sypen, Phys. Rev. Lett. , 226603 (2008).[45] K. Hitachi, T. Ota, and K. Muraki, Appl. Phys. Lett. , 192104 (2013).[46] J. M¨uller, S. von Moln´ar, Y. Ohno, and H. Ohno, Phys.Rev. Lett. , 186601 (2006).[47] S. M. Sze and K. K. Ng, Physics of Semiconductor De-vices (Wiley, New York, NY, 2006).[48] J. Zimmermann and G. Weber, Phys. Rev. Lett. , 661(1981).[49] J. Jang, K. Lee, and C. Lee, J. Electrochem. Soc. ,2770 (1982).[50] J. Reinisch and A. Heuer, J. Phys. Chem B , 19044(2006).[51] R. Biswas and Y.-P. Li, Phys. Rev. Lett. , 2512 (2006).[52] N. M. Zimmerman, W. H. Huber, B. Simonds, E. Hour-dakis, A. Fujiwara, Y. Ono, Y. Takahashi, H. Inokawa,M. Furlan, and M. W. Keller, JAP , 033710 (2008).[53] M. M. Glazov, E. Y. Sherman, and V. K.Dugaev, Phys-ica E , 2157 (2010).[54] M. Glazov and E. Sherman, Phys. Rev. Lett. , 156602(2011).[55] F. Li, Y. V. Pershin, V. A. Slipko, and N. A. Sinitsyn,Phys. Rev. Lett. , 067201 (2013).[56] R. de Sousa and S. Das Sarma, Phys. Rev. B , 115322(2003).[57] X. Hu and S. Das Sarma, Phys. Rev. Lett. , 100501(2006).[58] L. Chirolli and G. Burkard, Adv. Phys. , 57 (2008).[59] D. Culcer, X. Hu, and S. Das Sarma, Appl. Phys. Lett. , 073102 (2009).[60] R. de Sousa, in Electron Spin Resonance and RelatedPhenomena in Low-Dimensional Structures , edited byM. Fanciulli (Springer, 2009).[61] G. Ramon and X. Hu, Phys. Rev. B , 045304 (2010).[62] D. Culcer and N. M. Zimmerman, Appl. Phys. Lett. ,232108 (2013).[63] P. Huang and X. Hu, arXiv:1408.1666 (2014).[64] R. Winkler, Spin-orbit effects in two-dimensional electronand hole systems (Springer, 2003).[65] J. H. Davies,
The Physics of Low-Dimensional Semicon-ductors: An Introduction (Cambridge University Press,1998).[66] The effect of fluctuators with τ ≥ µ s can be eliminated in experiment through dynamical decoupling. Hence, onphysical grounds, we impose 1 µ s as a cutoff for theswitching time.[67] I. L. Aleiner and V. I. Falko, Phys. Rev. Lett. , 256801(2001).[68] P. Stano and J. Fabian, Phys. Rev. Lett. , 186602(2006).[69] We expect whole charge defect potentials to be dominantin relaxation since they are much stronger than dipolepotentials [62].[70] S. Kogan, Electronic noise and fluctuations in solids (Cambridge University Press, New York, 2008). [71] I. Martin and Y. M. Galperin, Phys. Rev. B , 180201(2006).[72] G. Burkard, Phys. Rev. B , 125317 (2009).[73] The effect of fluctuators with τ ≥ µ s can be eliminatedin experiment through dynamical decoupling. Hence, onphysical grounds, we impose 1 µµ