A silicon singlet-triplet qubit driven by spin-valley coupling
Ryan M. Jock, N. Tobias Jacobson, Martin Rudolph, Daniel R. Ward, Malcolm S. Carroll, Dwight R. Luhman
AA silicon singlet-triplet qubit driven by spin-valley coupling
Ryan M. Jock, ∗ N. Tobias Jacobson, Martin Rudolph, DanielR. Ward, † Malcolm S. Carroll, ‡ and Dwight R. Luhman Sandia National Laboratories, Albuquerque, NM 87185, USA Center for Computing Research, Sandia National Laboratories, Albuquerque, NM 87185, USA
Spin-orbit effects, inherent to electrons confined in quantum dots at a silicon heterointerface, pro-vide a means to control electron spin qubits without the added complexity of on-chip, nanofabricatedmicromagnets or nearby coplanar striplines. Here, we demonstrate a novel singlet-triplet qubit oper-ating mode that can drive qubit evolution at frequencies in excess of 200 MHz. This approach offersa means to electrically turn on and off fast control, while providing high logic gate orthogonality andlong qubit dephasing times. We utilize this operational mode for dynamical decoupling experimentsto probe the charge noise power spectrum in a silicon metal-oxide-semiconductor double quantumdot. In addition, we assess qubit frequency drift over longer timescales to capture low-frequencynoise. We present the charge noise power spectral density up to 3 MHz, which exhibits a 1 /f α dependence, with α ∼ .
7, over 9 orders of magnitude in noise frequency.
INTRODUCTION
Qubits based on the spins of electrons confined togate-defined quantum dots (QDs) in silicon metal-oxide-semiconductor (MOS) structures have developed into apromising platform for quantum information processing.High-quality single-qubit [1, 2] and two-qubit gates [3–5]have been demonstrated, and device manufacture is gen-erally compatible with available silicon microelectronicsfabrication methods. Qubit control techniques demon-strated in silicon MOS have utilized electron spin reso-nance (ESR) with microwave strip-lines [1, 2, 6], electricdipole spin resonance (EDSR) using micromagnets [5] orthe intrinsic spin-orbit coupling (SOC) at the Si/SiO interface [7–9]. Making use of interfacial SOC has theappeal of driving qubit evolution with electrical-only con-trol without reliance on the added fabrication constraintsof micromagnets or on-chip microwave strip-lines.Confining electrons to quantum dots at the Si/SiO interface has been shown to produce spin-orbit couplingthat is stronger than that of bulk Si [7, 9–12]. Recentobservations have demonstrated that the broken crystalsymmetry at the silicon heterointerface and interactionswith excited valley states lead to this enhanced SOC.These effects contribute to variation of the g -factor inQDs [7, 9–13]. While the g -factor difference betweenneighboring QDs may lead to problems with single spinaddressability, it has also proved to be a valuable re-source, able to drive the evolution of spin qubits encodedinto a singlet-triplet subspace [7, 9]. Additionally, thespin-valley “hot spot” is known to enhance electron spinrelaxation (shorter spin T ) when the valley splitting,∆ v , is comparable to the electronic Zeeman splitting, E Z = gµ B B , in a QD, where µ B is the Bohr magne-ton and B is the applied external magnetic field [14–17].This enhanced relaxation mechanism has been used tostudy valley splitting [14, 16, 17] and intervalley spin-orbit coupling in silicon QD devices [14, 17, 18].In this work, we utilize the intervalley spin orbit interaction near the spin-valley hot spot in a siliconMOS QD and demonstrate the ability to drive singlet-triplet rotations in excess of 200 MHz using the inter-valley spin orbit interaction. We exploit these fast ro-tations near the hot spot to enable unique qubit oper-ation with high-speed all-electrical modulation betweenqubit logic gates and high orthogonality of control axesthrough electrical control of the valley splitting. We thentake advantage of this novel operating mode to investi-gate the charge noise power spectral density (PSD) inthis device. We use the noise filtering properties of aCarr–Purcell–Meiboom–Gill (CPMG) dynamical decou-pling pulse sequence to decouple the qubit from chargenoise experienced during the spin-spin exchange inter-action. Combined with long-timescale measurements ofdrift in the frequency of exchange driven ST qubit ro-tations, we find that charge noise in this device exhibitsa S ( f ) ∼ f − . power spectrum over 9 decades of fre-quency. RESULTS
The silicon MOS double-quantum dot (DQD) used inthis work is illustrated in Fig. 1(a). We operate thedevice near the (N
QD1 ,N QD2 ) = (4,0)-(3,1) charge tran-sition. Two electrons on QD1 form a spin paired closedshell [19–21]. The interaction between the remaining twoelectrons is electrically controlled via the detuning bias, (cid:15) , between the QDs. For shallow detuning, there is sig-nificant electronic wave function overlap between the twoelectrons and the exchange energy, J ( (cid:15) ), is the dominantinteraction. When the two electrons are well separated inthe deep tuning regime, J ( (cid:15) ) is small and the dominantinteraction is set by the interfacial SOC, which results indistinct Zeeman energies in each QD [7, 9, 11–13, 22].The system is initialized by loading a (4,0) singletground state, then quickly transferring an electron tothe (3,1) charge configuration to produce a (3,1) sin- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Figure 1. (a) Scanning electron micrograph of the gate structure of a device similar to that measured. The shaded regionsindicate estimated areas of electron accumulation. The red and blue circles represent the locations of QD1 and QD2, respectively.We sense QD charge state transitions using a nearby single electron transistor (SET) in the lower right corner. (b) Schematiclateral view of the device structure and representation of the electron spin filling in each QD. (c) Energy level diagram of thesinglet-triplet system in the DQD. The orange region represents the Pauli blockade window, with the singlet-triplet splittingin the (N
QD1 ,N QD2 ) = (4,0) charge region denoted by J . (inset) Energy level diagram of the m = 0 qubit subspace in the(N QD1 ,N QD2 ) = (3,1) charge region. glet state (i.e. rapid adiabatic passage). Here, SOC inthe DQD will drive rotations between (3,1) singlet andtriplet states. We then rapidly return the system to the(4,0) charge sector, where Pauli spin blockade, combinedwith an enhanced latching mechanism [23], is used toread out the spin state of the two-electron system inthe single-triplet basis. In Fig. 2(b) we show the fastFourier transform (FFT) of SOC-driven rotations as theexternal magnetic field is swept along the [010] crystal-lographic direction. For low field strengths (
B < . g -factor between the two QDs which lifts the degeneracy ofthe m =0 states, |↑↓(cid:105) and |↓↑(cid:105) , driving rotations between(3,1) S and (3,1) T [7]. As B is further increased, weobserved an unexpected rapid rise in rotation frequencywith a sharp peak near B = 0.64 T. As discussed furtherbelow, these fast rotations are driven by an intervalleyspin-orbit interaction which involves a coupling betweendistinct valley states having opposite spin. The peak po-sition corresponds to the magnetic field at which the ex-cited valley state, T (1) − = |↓↓ (1) (cid:105) , crosses the ground state m =0 manifold of the two-electron system, B c, , as illus-trated in Fig. 2(c), often referred to as the spin-valley“hot spot”.Previous work has studied this regime in silicon QDsthrough single spin relaxation rates [14–18]. Our ap-proach of studying coherent rotations driven by the in-tervalley spin-orbit interaction yields new insight into theintervalley spin-orbit interaction and its dependence onapplied magnetic field. A detailed description is providedin Ref. [24]. Here we provide an intuitive three-level pic-ture of the system that describes the physics of the fre- quency dependence around the hot spot in Fig. 2(b) andexplains how the system can be used for a novel qubitoperating mode.In the two-electron DQD system, the intervalley hotspot corresponds to a distortion of the m =0 subspace {|↓↑(cid:105) , |↑↓(cid:105)} of the ST qubit due to coupling to |↓↓ (1) (cid:105) ,the down-polarized triplet state for which the electronin QD2 is in its excited eigenvalley. This hybridizes the |↓↑(cid:105) and |↓↓ (1) (cid:105) states, while |↑↓(cid:105) remains unperturbed.In the basis of {|↑↓(cid:105) , |↓↑(cid:105) , |↓↓ (1) (cid:105)} , this interaction canbe represented by an effective three-level system with aHamiltonian of the form [24] H = Bδ − Bδ γ γ ∗ ∆ v , QD2 − g ∗ µ B B , (1)where δ = µ B ∆ g/
2, with ∆ g = g − g the differencein g -factors between the QDs arising from variability ofinterfacial SOC, γ is the intervalley coupling strength,∆ v , QD2 is the valley splitting for the QD associated withthe |↓↓ (1) (cid:105) state, and µ B is the Bohr magneton. The g -factor governing the Zeeman shift of |↓↓ (1) (cid:105) is g ∗ =( g + g (1)2 ) /
2, the average of the g -factors of the groundvalley of QD1 and excited valley of QD2. The eigenstatesof this three-level system are | + (cid:105) = w − |↓↑(cid:105) + w + |↓↓ (1) (cid:105) (2) |↑↓(cid:105)|−(cid:105) = w + |↓↑(cid:105) − w − |↓↓ (1) (cid:105) where w ± = (cid:113) ± η/ (cid:112) η + 4 | γ | / √ η = ∆ v , QD2 + ( δ − g ∗ µ B ) B, (4) Figure 2. (a) Schematic of the pulse sequence used to to interrogate the magnetic field and detuning voltage dependencesof intervalley spin-orbit driven singlet-triplet rotations. (b) FFT of singlet-triplet rotations at fixed detuning in (3,1) as theexternal magnetic field is swept along the [010] crystallographic direction, with superimposed model fit (red dotted line). Theorange dashed line indicates the spin-valley hot spot. We fit an intervalley SOC strength of 0 . ± . µ eV and a valleysplitting of 73 . ± . µ eV, with uncertainty reported here as 95% confidence intervals (see Supplementary Information).(c) Magnetic field dependence of the system energy levels. The orange dashed arrow illustrates the change in the hot spotcritical field as the QD-QD detuning is increased. (d) Singlet-triplet rotations at a fixed magnetic field of 0.645 T as a functionof QD-QD detuning voltage. (e) Measured singlet-triplet qubit rotation frequency, f , as a function of QD-QD detuning voltage(black circles), with superimposed model fit (red curve). We estimate a valley splitting lever arm of 46 . ± . µ eV / V (seeSupplementary Information). (f) Black circles are extracted inhomogeneous dephasing times, T ∗ , as a function of QD-QDdetuning. The red dashed line is proportional to | df/dV | − , the expected dependence for quasi-static charge noise, where | df/dV | is found from a numerical derivative of the data in (e). (g) The black circles are the calculated quality factor of qubitrotations ( Q = f × T ∗ ) as a function of QD-QD detuning. The red dashed line is the expected quality factor from quasi-staticcharge noise found from the data in (e) and the red dashed line in (f). with eigenenergies E ± = − Bδ + 12 (cid:16) η ± (cid:112) η + 4 | γ | (cid:17) (5) E ↑↓ = Bδ The three-level Hamiltonian has three distinct energygaps (∆ + = E + − E ↑↓ , ∆ − = E ↑↓ − E − , ∆ + − = E + − E − )and, in principle, three frequencies corresponding to therate of dynamical phase accumulation for each of thesegaps could be present in the measured spectrum. How-ever, we observe only a single rotation frequency compo-nent in Fig. 2(b). This can be understood by the follow-ing physical picture. Supposing that the system is ini-tially tuned away from the spin-valley anticrossing, theinitial state prepared at the beginning of the evolutionis close to | S (cid:105) = √ ( |↑↓(cid:105) − |↓↑(cid:105) ). If the valley splittingis changed to bring the system closer to the spin-valleyhot spot, the |↓↑(cid:105) state adiabatically deforms into either | + (cid:105) or |−(cid:105) . The energy gap dictating the evolution fre-quency is the difference between E ↑↓ and the level (either E + or E − ) that is adiabatically connected to the initial |↓↑(cid:105) state. When operating on the low-field (high-field) shoulder of the hot spot peak, the measured frequenciesin Fig. 2(b) are dominated by rotations within the sub-space spanned by {|↑↓(cid:105) , |−(cid:105)} ( {|↑↓(cid:105) , | + (cid:105)} ), thus creatinga two-level qubit system. Qubit measurement amountsto projecting back onto | S (cid:105) , with any support in the spanof {| T (cid:105) , |↓↓ (1) (cid:105)} read out as triplet.We realize this novel operating mode in the experi-ment by controlling the valley splitting of QD2 throughmodulation of the electric field [14, 16, 17, 25] at a con-stant magnetic field. We apply a field of B = 0 .
645 Talong the [010] crystallographic direction, such that weare on the high magnetic field side of the hot spot peak( gµ B B > ∆ v ). In this case, an increase in applied elec-tric field in QD2 will increase the valley splitting ∆ v , QD2 ,shifting the location of the spin-valley hot spot to highermagnetic field. We assume a linear dependence of valleysplitting as a function of gate voltage away from a refer-ence voltage V , ∆ v , QD2 ( V ) = ∆ v , QD2 | V + λ v ( V QD2 − V ).We refer to λ v as the valley splitting lever arm. Since weare operating at constant magnetic field, we would ex-pect an increase in rotation frequency in the {|↑↓(cid:105) , | + (cid:105)} subspace as we drive up the flank of the hot spot peak.In Fig. 2(d) we show the singlet return signal as afunction of time spent at the manipulation point in (3,1)as the QD-QD detuning, (cid:15) , is varied along ∆ V QD2 = − ∆ V QD1 . The state is prepared in the same way asdescribed above. For shallow detuning, we do not ob-serve rotations since the exchange interaction, J ( (cid:15) ), islarge and (3,1)S is nearly an eigenstate of the system. Atmoderate detuning we begin to see oscillations betweensinglet and triplet states due to the spin-orbit interac-tion, indicating a relative reduction in J ( (cid:15) ). As we pulseto deeper detuning, the voltage on the QD2 plunger in-creases. This enhances the vertical electric field confin-ing QD2, resulting in an increase in valley splitting andhot spot critical field, B c , = ∆ v , QD2 /g ∗ µ B , and an in-crease in rotation frequency (Fig. 2(d)). In Fig. 2(e)we plot the rotation frequency as a function of QD-QDdetuning. Here, we demonstrate a rotation frequencyin excess of 200 MHz, illustrating the ability to electri-cally control the intervalley spin-orbit driven frequencyover a span of two orders of magnitude. Our model withan assumed linear dependence of valley splitting on gatevoltage fits the data well, giving a valley splitting leverarm of 46 . ± . µ eV / V.Next, we fit the decay in oscillations of measured sin-glet probability as a function of wait time, t , for a givendetuning to a Gaussian envelope, exp( − ( t/T ∗ ) ). Fromthis, we extract an inhomogeneous dephasing time, T ∗ ,as a function of detuning, shown in Fig. 2(f). When theinteraction with the excited valley is weak, we expectthe dephasing to be dominated by the hyperfine inter-action with residual Si [7, 26–30]. As the interactionstrength increases, the coupling to nearby electric fieldswill be enhanced, increasing sensitivity to charge noise.For deeper detuning, we observe a decrease in T ∗ , whichfollows a T ∗ ∝ | df /dV | − dependence, depicted as a reddashed line in Fig. 2(f), which is expected for quasi-staticcharge noise [7, 31]. At frequencies above 100 MHz ( (cid:15) >
65 mV), T ∗ is lower than the expected fit for quasi-staticcharge noise. The spin-valley hot spot is known to leadto an enhanced spin relaxation rate [14–18], and mayproduce a T limited dephasing as the system is tunedcloser to the S - T (1) − crossing. The quality of rotations, Q = f × T ∗ , which compare the rotation frequency tothe dephasing time, are plotted in Fig. 2(g). We observethat, while the dephasing is faster at deeper detunings,the rotation frequency grows more quickly and the qual-ity factor increases to Q ∼
20 at rotation frequenciesabove 100 MHz. We have observed hot spot driven rota-tion frequencies near 400 MHz in a separate nat
Si device,albeit with lower quality factors (see Supplementary In-formation). This highlights the dependence of the rota-tion quality on the interplay of the device tuning and thedetails of the intervalley coupling. Control of these pa-rameters may provide a path to improving the rotationquality to produce higher-fidelity gate operations.The logical basis for singlet-triplet qubits is generally represented by the linear combination of |↑↓(cid:105) and |↓↑(cid:105) states (e.g., | S (cid:105) and | T (cid:105) ). During operation, the qubitstates will rotate on the Bloch sphere about the vec-tor sum of the Z-axis governed by the exchange energy, J ( (cid:15) ), and the X-axis dictated by the difference in Zeemansplitting between the two QDs, ∆ E Z . Logic gates areperformed by electrically pulsing between regions domi-nated by J ( (cid:15) ) and regions dominated by ∆ E Z . In otherimplementations of singlet-triplet qubits, ∆ E Z is fixed[20, 29, 32, 33]. In contrast, by utilizing the electri-cally controlled intervalley interaction described above,we are able to independently implement high frequencyspin-orbit driven gates at deep detuning, where the ex-change interaction is weak, and exchange driven rotationsat shallow detuning, where the intervalley interaction isweak and J ( (cid:15) ) dominates.In Fig. 3 we demonstrate simultaneous two-axis con-trol of the intervalley driven singlet-triplet qubit. Weoperate on the high-field shoulder of the spin-valley hotspot and define the qubit basis in terms of the |↑↓(cid:105) and | + (cid:105) states, where | ˜ S (cid:105) = 1 √ |↑↓(cid:105) − | + (cid:105) ) (6) | ˜ T (cid:105) = 1 √ |↑↓(cid:105) + | + (cid:105) ) , (7)with w + ≈ | ˜ S (cid:105) ≈ | S (cid:105) and | ˜ T (cid:105) ≈ | T (cid:105) . Pulsing to shallow detun-ing drives exchange rotations, Fig. 3(b), while for deepdetunings the intervalley spin-orbit interaction is turnedon, Fig. 3(d). Furthermore, at moderate detuning (Fig.3(c)), both the exchange and intervalley interactions areweak and spin interaction is dominated by the intravalleyspin-orbit interaction [7]. This provides a regime wherequbit dephasing times are limited by the hyperfine inter-action with residual Si in the host lattice and decoupledfrom charge noise. The ability to rapidly toggle betweenthe two control axes by pulsing to detuning regions withlarge (small) exchange and small (large) spin-valley cou-pling, respectively, provides for high-orthogonality qubitcontrol.We can infer a qualitative measure of orthogonality ofcontrol over this qubit from the measurements shown inFigs. 3(e,f) and referring to an effective qubit Hamilto-nian H = h z σ z + h x σ x . Since the exchange, J ( (cid:15) ), governsthe h z component, while the intravalley and intervalleySOC control the h x component, as shown schematicallyin Figs. 3(b-d), the relative magnitudes of h z and h x dic-tate the axis about which the qubit rotates on the Blochsphere. For moderate detuning (middle dashed line inFig. 3(f)), where the intravalley SOC contribution, ∆ SO ,dominates the h x component, we observe a qubit rota-tion frequency of ∼ ∼
20 MHz. This correspondsto h z /h x ≈
10. Conversely, since the exchange J ( (cid:15) ) de- Figure 3. (a) Schematic of the pulse sequence used to demonstrate all-electrical modulation between exchange-dominated andspin-orbit-dominated qubit control axes. Here rapid adiabatic passage is used to transfer the qubit to a (3,1)S state at moderatedetuning, such that the strong spin-orbit effect is turned off, then the qubit is allowed to evolve for a π/ π/ cays quickly with detuning (cid:15) , the point of high interval-ley SOC (right dashed line in Fig. 3(f)) corresponds toa region where the residual exchange is negligible. Here h x /h z (cid:29) Si, charge noise has been identified as a dominantsource of error [34]. Here, charge noise may have theeffect of increasing dephasing rates for one- or two-qubitgates involving the exchange interaction or when the ar-chitecture employs a magnetic field gradient from a mi-cromagnet for spin control. CPMG pulse sequences area well-established technique for mitigating the effects ofqubit dephasing by applying a series of refocusing control pulses [35, 36] and has been successfully demonstratedwith silicon spin qubits [37–40]. In Fig. 4 we demon-strate the ability to use a CPMG pulse sequence to pro-long the qubit coherence time. We apply a string of in-tervalley spin-orbit driven pulses to decouple the qubitfrom charge noise during the spin-spin exchange inter-action. Fig. 4(b) shows qubit exchange rotations forthree QD-QD detuning voltages. We see that for fasterexchange pulses the qubit dephases more quickly, as ex-pected for quasistatic charge noise dominated inhomoge-neous dephasing in qubit exchange gates [7, 31, 41–43].Fig. 4(c) shows the CPMG coherence time, T CPMG2 , ver-sus the number of refocusing pulses, N π , for the threedetuning values. We observe an increase in coherencetime with increasing N π , which follows a power-law de-pendence with T CPMG2 ∝ N βπ . We find exponents of β ≈ . , .
40, and 0 .
41 for the detuning points (cid:15) , (cid:15) ,and (cid:15) , respectively. CPMG prolongs qubit coherence byrefocusing noise for time scales longer than the time be- Figure 4. (a) Schematic for CPMG pulses. We initializethe qubit into the (4,0)S ground state and ramp adiabati-cally, such that the qubit transfers to the ground state ( ↑↓ or ↓↑ ) in the (3,1) charge sector at moderate detuning, awayfrom the spin-valley hot spot. This acts as an effective π/ S - T qubit basis. A fast pulseto and from a detuning, (cid:15) , where exchange is substantial,drives coherent rotations around an axis dominated by theexchange interaction. Here, charge noise drives qubit dephas-ing. A series of π pulses are then applied to decouple thequbit from charge noise. Here we operate with an intervalleyspin-orbit driven rotation frequency of 20MHz. A final waittime, τ (cid:48) , at the end of the sequence allows for the observa-tion of the free induction decay of the refocused echo. Re-turning to the (4,0) charge sector adiabatically produces aneffective − π/ basis for measurement. (b) Qubit exchange rota-tions at three QD-QD detuning points. Black lines are fits tooscillating Gaussian decay envelopes ∝ exp (cid:0) − ( t/T ∗ ) (cid:1) . (c)Qubit CPMG coherence time as a function of the number ofrefocusing pulses N π for three QD-QD detuning points whereexchange is the dominant spin interaction. Dashed lines arefits to the form T CPMG2 ∝ N βπ . tween refocusing pulses. For a given total time exposedto exchange, τ total , more N π pulses will decrease the timethe qubit is exposed to noise before being refocused. Assuch, the effectiveness of CPMG to mitigate charge noisewill largely be determined by the noise spectral density, S ( f ). For colored noise of the form S ( f ) ∝ f − α , we ex-pect T CPMG2 ∝ N α α π [37–40]. A fit to the data in Fig.4(c) indicates a noise spectrum with α ≈ . S ( f N π ) = π · T CPMG2 ,N π , (8)where f N π is the relevant noise frequency being inter-rogated and is given by the time between pulses whenrefocused echo intensity drops to 1 /e , f N π = N π T CPMG2 ,N π . (9)The noise PSD is given in terms of fluctuations in ex-change rotation frequency, which will be dependent onthe strength of the exchange interaction at each detun-ing value. By using the gradient of the qubit frequency ateach detuning point, df ( (cid:15) ) /dV ( (cid:15) ), we convert the spec-trum to voltage noise on the QD-QD detuning, whichprovides a means to compare the three detuning points.The combined data are plotted in Fig. 5(a), where astrong agreement in the noise PSD for all three detuningvalues is observed. The blue dashed line is a power lawfit, which gives S ( f ) ∝ f − . .Next, we examine the low-frequency portion of chargenoise spectrum in this system. In Fig. 5(b) we plot thesinglet return probability for repeated exchange rotationexperiments near detuning (cid:15) . Fig. 5(c) shows the slowdrift in the extracted exchange rotation frequency. Usinga periodogram method and df ( (cid:15) ) /dV ( (cid:15) ) at this tuning,we plot the low frequency noise PSD in Fig. 5(d) along-side the high frequency results. A power law fit to the lowfrequency data (gray dashed line), extracted out to highfrequency shows a S ( f ) ∝ f − α dependence of the chargenoise PSD with α ≈ . /f -like, with α near 1 [39, 40, 47–53] and presumed tobe caused by a distribution of charge fluctuators. SUMMARY
Interfacial spin-orbit interactions are known to play asignificant role in the control of spin qubits in silicon QDs.In this work, we observe a rapid increase in the singlet-triple rotation frequency near the spin-valley hot spotand develop a simple three state model to explain the ob-servations. We utilize this effect to demonstrate an inter-valley driven singlet-triplet qubit with high-orthogonalityand fast electrical-only qubit control. We show the abilityto electrically tune the intervalley spin-orbit interaction,enabling high-speed modulation between three qubit con-trol regimes: (1) large exchange interaction, (2) smalleffective magnetic field gradient between QDs, and (3)hot spot driven qubit rotations with operational rotationfrequencies exceeding 200 MHz. When the intervalley
Figure 5. (a) Noise spectral density for charge noise experienced by the qubit during exchange pulses for three QD-QD detuningvalues. The blue dashed line is a power law fit to the data, S ( f ) ∝ f − α . The red dashed line is a fit to a 1 /f noise spectrum asa guide to the eye. (b) Repeated experiment of singlet return probability versus wait time for an exchange pulse near detuning (cid:15) over the course of 10 minutes. (c) Extracted qubit frequency for data in (b) as a function of experimental measurementtime. (d) combined low- and high-frequency measurements of the noise spectral density. The blue dashed line is a power lawfit to the high frequency data and the gray dashed line is a power law fit to the low frequency data extracted from (c). Thered dashed lines are fits of 1 /f spectra to the low- and high-frequency data sets, respectively, as guides to the eye. spin-orbit or exchange interactions are weak, qubit de-phasing is dominated by the hyperfine interaction withthe residual Si in the isotopically enriched substrate.However, for strong exchange or intervalley spin-orbitcoupling, quasi-static charge noise becomes the dominantdephasing mechanism.Furthermore, we utilize this high-speed control to im-plement dynamical decoupling techniques to extend thequbit coherence when exposed to charge noise during aspin-spin exchange interaction. Using CPMG sequences,we increase the qubit coherence by an order of magni-tude. We then exploit the filter function properties ofCPMG to extract the noise power spectrum of the chargenoise in this device without the added complexity of on-chip, nano-fabricated micromagnets or nearby co-planarstriplines that may otherwise be needed for qubit con-trol. The fast hot spot refocusing pulses and strong cou-pling to charge noise when the exchange interaction isturned on allows for a probe of the noise power spectraldensity at high frequencies. These experiments, com-bined with low frequency drift measurements, reveal a S ( f ) ∝ f − . noise spectrum for frequencies between 3mHz and 3 MHz. METHODSDevice overview
The double quantum dot studied in this work was real-ized in a fully foundry-compatible, single-gate-layer, sili-con metal-oxide-semiconductor (MOS) device structurecontaining an epitaxially-enriched Si layer with 500ppm residual Si at the Si/SiO interface. The confine-ment and depletion gates are defined by electron beamlithography followed by selective dry etching of the poly-silicon gate layer, which produces the pattern shown inFig. 1(a). Electrons are confined at the Si/SiO inter-face and relevant biasing of the poly-silicon gates createquantum dot potentials under the tips of gates QD1 andQD2. The tunnel rate to the electron reservoirs under thelarge gates in the bottom left and top right corners of thedevice is controlled by the applied voltage to the reser-voir gates [54]. We operate with the bottom left electronreservoir receded such that the DQD system is coupledonly to the top right reservoir through QD1. A singleelectron transistor (SET) in the lower right corner of thedevice is used for charge sensing. The number of electronsin each QD is inferred from changes in current throughthe SET as well as by magneto- and pulsed-spectroscopymethods. Measurements
Measurements were performed in a He/ He dilutionrefrigerator with a base temperature of around 8 mK.The effective electron temperature in the device was 150mK. Gates QD1 and QD2 are connected to cryogenicRC bias-T’s, which allow for the application of combinedDC bias voltages and fast gate pulses. An external mag-netic field is applied using a 3-axis vector magnet. Weperform cryogenic preamplification of the charge sens-ing SET current using a heterojunction bipolar transistor(HBT) [55].
ACKNOWLEDGMENTS
Sandia National Laboratories is a multi-mission labo-ratory managed and operated by National Technologyand Engineering Solutions of Sandia, LLC., a whollyowned subsidiary of Honeywell International, Inc., forthe U.S. Department of Energy’s National Nuclear Se-curity Administration under contract DE-NA-0003525.This paper describes objective technical results and anal-ysis. Any subjective views or opinions that might beexpressed in the paper do not necessarily represent theviews of the U.S. Department of Energy or the UnitedStates Government.
AUTHOR CONTRIBUTIONS
R.M.J. performed the central measurements presentedin this work and analyzed results. N.T.J. carried outthe theoretical modeling and statistical analysis of mea-surement data. M.R. performed the initial measure-ments demonstrating the electrical control of valley hotspot rotations on a similar device. All authors dis-cussed central results throughout the project. D.R.W.and M.S.C. designed the process flow, fabricated devices,and designed/characterized the Si material growth forthis work. M.S.C. and D.R.L. supervised the combinedeffort, including coordinating fabrication and identifyingmodeling needs. R.M.J. and N.T.J. wrote the manuscriptwith input from co-authors. ∗ Corresponding author: [email protected] † Present Address: HRL Laboratories, LLC, Malibu, CA90265 ‡ Present Address: Princeton Plasma Physics Laboratory,Princeton, NJ 08543[1] M. Veldhorst, J. C. C. Hwang, C. H. Yang, A. W. Leen-stra, B. de Ronde, J. P. Dehollain, J. T. Muhonen, F. E.Hudson, K. M. Itoh, A. Morello, and A. S. Dzurak.An addressable quantum dot qubit with fault-tolerant control-fidelity.
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Applied Physics Letters , 108(6):063101,2016. SUPPLEMENTARY INFORMATIONMeasurement details
The DQD studied in this work was formed within a de-vice nominally identical to that shown in SupplementaryFig. 6(a,b). This device was fabricated in a fully foundry-compatible process using a single-gate-layer, metal-oxide-semiconductor (MOS) poly-silicon gate stack with anepitaxially-enriched Si epi-layer with 500 ppm residual Si. The device is operated in enhancement mode usingvoltage biasing of the highly doped n+ poly-silicon gatesto confine electrons to quantum dot (QD) potentials un-der gates RD and LD (QD1 and QD2, respectively). Thegates LR, RR, BLR, BRR, TLR, and TRR overlap withimplanted n+ ohmic contacts regions and are biased toaccumulate two-dimensional electron gas (2DEG) regionsunder each gate that serve as electron reservoirs for thequantum dots and single electron transistor (SET) chargesensors. The upper left and bottom right corners of thedevice may be used as charge-sensing SETs by confin-ing QDs under gates TQD and BQD, respectively. Inthe measurements presented in the main text, we utilizeonly the bottom right SET.The number of electrons in each QD may be inferredfrom changes in current through the SET, as depictedin Supplementary Fig. 6(c). The collection of red andorange parallel lines correspond to charge transitions inQD1 and QD2, respectively. We can infer the approxi-mate locations of QD1 and QD2 by measuring their ca-pacitances to nearby poly-silicon gates through scans ofvoltages applied to pairs of poly-silicon gate electrodes.From these measurements, we obtain the relative capac-itance of the QDs to each gate compared to their capaci-tance to RD or LD, the gates that have strongest capac-itive coupling to QD1 and QD2, respectively. We plotthe relative capacitances in Supplementary Fig. 7. Themeasured symmetric capacitance ratios of the two QDsindicate that they are well-formed lithographic quantumdots.We operate this system near the (N
QD1 ,N QD2 ) = (4,0)-(3,1), spin-blockaded charge anti-crossing. A charge sta-bility diagram for the double dot system is shown in Sup-plementary Fig. 6(c). The ground state charge configura-tion is determined by the detuning between dots, (cid:15) , whichis controlled by tuning the voltages on gates RD and LD.These gates are connected to cryogenic RC bias-T’s whichenable the application of fast gate pulses. A schematic ofthe cyclical pulse sequence used in qubit measurements isshown in Supplementary Fig. 6(d). The system is initial-ized in the (4,0) charge sector by first unloading an elec-tron from the DQD (point U). An energy-selective pulseinto the (4,0) charge state between the singlet and tripletenergy levels is applied to load a (4,0)S ground state(point L). Following that, the system is plunged (point P) to a detuning ( (cid:15) <
0) close to the charge anti-crossing.The electrons are then separated (point C) and qubitmanipulation pulse sequences are performed in the (3,1)charge region ( (cid:15) >
CPMG Analysis
The measured data for a CPMG echo experiment atdetuning (cid:15) with N π = 10 are shown in SupplementaryFig. 8(a). Here, we plot the singlet return probability fortime τ (cid:48) after the end of the CPMG sequence as the totalqubit evolution time τ total increases. The oscillations insinglet return represent the free induction decay, FID, ofthe refocused echo. Fitting the FID to a Gaussian enve-lope function for each τ total gives the echo amplitude asa function of total time exposed to charge noise. Supple-mentary Fig. 8(b) plots this as the number of refocus-ing pulses, N π , is increased. As described in the maintext, the relevant noise frequency being interrogated is f N π . The coherence time T CPMG2 ,N π , when the echo dropsto 1 /e , indicates the noise strength near that frequency[38–40, 45]. In Supplementary Fig. 8(c) the noise powerspectral density is plotted for the three detuning valuesshown in the main text.Supplementary Fig. 9 shows exchange dominated STqubit rotations, reflected in the singlet return probabilityas a function of time the exchange interaction is turnedon for a range of QD-QD detunings. Here, we initializethe qubit into the (4,0)S ground state and ramp adiabat-ically into the (3,1) charge region, such that it transfersto the ground state, |↑↓(cid:105) or |↓↑(cid:105) . A rapid pulse to andfrom a detuning, (cid:15) , where exchange is substantial drivescoherent rotations around an axis depending on bothexchange, J ( (cid:15) ), and the difference in Zeeman splitting,∆ E Z . Returning to the (4, 0) charge sector adiabaticallyprojects the states onto the (4,0)S and (3,1)T basis formeasurement. We then fit the rotation frequency, f , ateach detuning to a smooth function to find the derivative, df /dV . This is used to convert the noise power spectraldensity in Supplementary Fig. 8(c) from a frequency toa voltage fluctuation, which allows for a comparison of2 Figure 6. (a) A cartoon schematic of the MOS gate stack. (b) A top-down SEM of the single-layer poly-silicon gate design withthe gate names labeled in white.(c) Charge stability diagram of the DQD. Here, we plot the gradient of the SET charge sensorcurrent as the gates QD1 and QDs are varied about fixed offset voltages (V
DC,QD1 = 3.4 V and V
DC,QD2 = 3.9 V). The broaddiagonal background features are due to Coulomb blockade peaks of the SET charge sensor. The sharp features correspond tocharge transitions in the QDs. The red and orange dashed lines are guides to the eye for QD and QD charge transitions. Theyellow circle represents the (N QD1 ,N QD2 ) = (4,0)-(3,1) charge region where this work was done. (d) A pulsed charge stabilitydiagram for the (4,0)-(3,1) anticrossing, showing the gradient of the charge sensor current. The red arrows depict a generalpulse sequence for controlling the qubit: The system is initialized by first unloading an electron from the DQD (point U). Anenergy-selective pulse is applied to load a (4,0)S ground state (point L). The system is then plunged (point P) near the chargeanti-crossing. The electrons are then separated (point C) and qubit manipulation pulse sequences are performed in the (3,1)charge region. The system is then pulsed back to point P where, due to Pauli spin blockade, a singlet spin state is allowedto transfer to the (4,0) charge state but a triplet spin state is energetically blocked and remains in a (3,1) charge state. Anenhanced latching mechanism is then utilized for a spin-to-charge conversion (point M). Here the qubit control point, C, mayconsist of a complex voltage detuning sequence for qubit manipulation. The black and white dashed lines correspond to thelocation of the singlet and triplet state inter-QD charge preserving lines, respectively.Figure 7. Gate capacitance to QDs relative to LD (blue) andRD (red) gates. noise power at the measured detuning points shown inthe main text.
Magnetic Noise
We used similar techniques to characterize the powerspectral density of magnetic noise in Device A. Here welook at singlet-triplet rotations at shallow detuning awayfrom the hot spot. In Supplementary Figs. 10(b,c) weshow singlet return probability for repeated experimentsof SOC-driven qubit rotations and their extracted rota-tion frequency over the course of 20 minutes. In Supple- mentary Fig. 10(c) we plot the noise PSD, which displaysa S ( f ) ∝ f − . power law dependence [30, 52].Next we utilize a CPMG sequence to decouple thequbit from magnetic noise and examine the PSD athigher frequencies. To do this, we use an exchange π pulse at a qubit frequency of 8.33 MHz, illustrated inSupplementary Fig. 11. When the number of refocus-ing pulses, N π , is increased, we observe no change inthe CPMG coherence time, T CPMG2 . This suggests aflat (white) noise mechanism, which is consistent withthe extracted noise power spectral density. In this low-frequency regime coherence may be limited by a T qubitrelaxation process, but to be definitive further studies areneeded. Supporting measurements
We independently observed valley hot spot-drivensinglet-triplet rotations in another silicon device, DeviceB, shown in Supplementary Fig. 12. This device wasfabricated similarly to the device presented in the maintext, but with two main differences: (1) This device hasa natural silicon substrate and (2) uses a single accu-mulation gate SET charge sensor design [20, 23, 61] forits bottom right charge sensor. The device was oper-ated in a similar fashion near the (N
QD1 ,N QD2 ) = (4,0)-(3,1) spin-blockaded charge anti-crossing. Supplemen-tary Figs. 12(c,d) show the magnetic field dependenceof spin-orbit driven qubit rotations as a function of mag-3
Figure 8. (a) CPMG echo at detuning (cid:15) with N π = 10. We plot singlet return probability as function of wait time after thequbit is refocused, τ (cid:48) , as a function of τ total , the total qubit manipulation time for a CPMG sequence. For each τ total , the echocan be fit to an oscillating Gaussian decay to extract the echo amplitude. (a,inset) Echo amplitude as a function of total timeexposed to charge noise, τ total , for N π = 10. Red line is a fit to a decay of the form exp( − ( t/T ∗ ) n ). (b) Echo amplitude as afunction of wait time, τ wait , as N π is stepped. (c) Frequency noise spectral density for charge noise experienced by the qubitduring exchange pulses for three QD-QD detuning values. The dashed lines are power law fits to the data. netic field applied along the [100] crystallographic direc-tion. The spin-valley hot spot at 0.22 T indicates a valleysplitting in this device of ≈ µ eV. Electrical control ofthe hot spot driven qubit frequency is shown in Supple-mentary Figs. 12(e,f), where a qubit drive frequency of400 MHz is achieved.In Supplementary Fig. 13 we show the electrical con-trol of the qubit frequency as a function of voltage appliedto QD2 for QD-QD detunings along three separate paths.The different paths behave similarly, yet show differencesin the plots. This suggests that while the vertical electricfield influences the valley splitting of QD2, the voltage onboth the QD1 and QD2 gates modify the the intervalleyspin-orbit interaction. Valley splitting lever arm
We find that the ability to electrically modulate theintervalley SOC is consistent with control of the valleysplitting, ∆ v , through the applied gate voltages. Sincethe valley splitting plays an important role in dictatingthe magnetic field at which the polarized triplet state T (1) − = |↓↓ (1) (cid:105) comes into resonance with the spin state |↓↑(cid:105) , we can use the voltage dependence of the qubit evo-lution frequency in the vicinity of the hot spot to probethe variation of valley splitting with gate voltage.To do this, we first fit to the qubit frequency versusmagnetic field data of Fig. 2(b) for Device A and 12(e)for Device B, respectively. We then fix these fit parame-ters and assume a linear variation of the valley splittingas a function of deviation of the gate voltage from the op-erating point at which the preceding measurements weretaken, ∆ v = ∆ + λ v ( V − V ) (10)We show fits to the model parameters in Figs. 14 and15, with parameter estimates in Table I. The reported un-certainties correspond to 95% confidence intervals. Notethat these measurements do not permit unambiguous de-termination of the valley-averaged g -factor g ∗ of Eq. 1,so for the purpose of these parameter estimates we en-force g ∗ = 2. The resulting valley splitting lever armsshown in Table I are comparable with other results insilicon MOS QDs [14, 17, 25].4 Figure 9. (a) Exchange rotations at a fixed magnetic field of 0.645 T as a function of QD-QD detuning voltage. (b) Extractedfrequency of qubit exchange rotations as a function of QD-QD detuning voltage. The blue and red circles are two experimentaldata sets and the black dashed line is a fit to the form f ( (cid:15) ) = (cid:112) J ( (cid:15) ) + ∆ E Z , with J ( (cid:15) ) ∝ t c (cid:15) .Figure 10. (a) Repeated experiment of singlet return probability versus wait time for spin-orbit driven singlet-triplet rotationsover the course of 20 minutes. (b) Extracted qubit frequency for data in (a) as a function of experimental measurement time.(c) The low frequency noise spectrum extracted using a periodogram method for magnetic (blue data) and charge (black data)noise. The red and green dashed lines are fits to S ( f ) ∝ f − α noise spectra.Device A Device BIntervalley SOC, γ ( µ eV) 0 . ± .
014 0 . ± . /h ) δ (MHz / T) 0 . ± .
15 0 ± v ( µ eV) 73 . ± .
033 26 . ± . . Valley splitting lever arm, λ v ( µ eV / V) 46 . ± .
85 188 ± J (neV) 0 ± . ± Si device of the main text) and Device B ( nat
Si device providing supportingindependent measurements). The reported valley splitting for each device is that of the quantum dot associated with themeasured spin-valley hot spot. The valley splitting lever arms correspond to collective variation of the dot gates (QD1,QD2)by ( − V , V ). Reported uncertainties are 95% confidence intervals based on a χ analysis using the estimated linewidths. Theparameter J accounts for incomplete vanishing of the exchange coupling J ( (cid:15) ) in the large positive detuning regime in whichthese measurements were taken. Figure 11. (a) Schematic for CPMG pulses to investigate magnetic noise. We initialize the qubit into the (4,0)S ground stateand transfer one electron to the neighboring dot using rapid adiabatic passage, such that the qubit remains a singlet in the(3,1) charge sector. The qubit is then allowed to evolve and dephase due to fluctuations in the Overhauser fields between thetwo QDs. A series of fast pulses to and from a detuning (cid:15) , where J is substantial, drive π/ τ (cid:48) at the end of the sequence allows for the observation of the free induction decayof the refocused echo. Returning to the (4,0) charge sector by rapid adiabatic passage projects the states onto the (4,0)S and(3,1)T basis for measurement. (b) Qubit CPMG coherence time as a function of the number of refocusing pulses, N π . (c) Highfrequency noise spectrum for magnetic noise experienced by the qubit. (d) Combined low- and high-frequency measurements ofthe magnetic noise power spectral density. The blue and red dashed lines are fits to a power law, S ( f ) ∝ f − . , and constant(white noise), respectively. (inset) Fluctuations in the nuclear spins of the residual Si in each QD will cause fluctuations intheir respective Zeeman splittings and cause dephasing of singlet-triplet qubit rotations.Figure 12. (a) Scanning electron micrograph of the gate structure of a device similar to that measured. The overlaid regionsindicate the estimated locations of electron accumulation (b) Change in singlet return as a function of X -rotation manipulationtime as the magnetic field is varied along the [100] crystallographic direction. (c) Change in singlet return as a function of X -rotation manipulation time as the QD-QD detuning is varied. (d) A cartoon representation of the electron spin filling ineach QD in this device. (e) The FFT extracted rotation frequency as a function of magnetic field for the data in (b). (f) TheFFT extracted rotation frequency as a function QD-QD detuning for the data in (c). Figure 13. (a) A pulsed charge stability diagram for the (N
QD1 ,N QD2 ) = (4,0)-(3,1) anti-crossing in Device B, showing thegradient of charge sensor current. The black circles represent the qubit reset (U), load (L), plunge (P), manipulation (C) andreadout (M) points. The qubit manipulation point is varied in three experiments along three detuning paths (black, blue, andred dashed lines). (b) Extracted rotation frequency vs voltage applied to QD2 for the three paths. (c) The extracted frequencyand dephasing times, T ∗ , give the Q-factors of the data for paths 1, 2, and 3. Figure 14. Fits to magnetic field dependence (above) andgate voltage dependence (middle and below, with same dataplotted with frequency on linear and logarithmic scales forclarity, respectively) for Device A. The vertical dashed line ineach plot corresponds to the parameter held fixed in the otherplot. Error bars represent ± σ for Gaussian fits to linewidths. Figure 15. Fits to magnetic field dependence (above) and gatevoltage dependence (below) for Device B. The vertical dashedline in each plot corresponds to the parameter held fixed inthe other plot. Error bars represent ± σσ