Magnetic gradient free two axis control of a valley spin qubit in SiGe
Y.-Y. Liu, L. A. Orona, Samuel F. Neyens, E. R. MacQuarrie, M. A. Eriksson, A. Yacoby
MMagnetic gradient free two axis control of a valley spin qubit in SiGe
Y.-Y. Liu, L. A. Orona, Samuel F. Neyens, E. R. MacQuarrie, M. A. Eriksson, and A. Yacoby Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA University of Wisconsin-Madison, Madison, 53706, Wisconsin, USA (Dated: January 26, 2021)Spins in SiGe quantum dots are promising candidates for quantum bits but are also challengingdue to the valley degeneracy which could potentially cause spin decoherence and weak spin-orbitalcoupling. In this work we demonstrate that valley states can serve as an asset that enables two-axiscontrol of a singlet-triplet qubit formed in a double quantum dot without the application of a mag-netic field gradient. We measure the valley spectrum in each dot using magnetic field spectroscopyof Zeeman split triplet states. The interdot transition between ground states requires an electronto flip between valleys, which in turn provides a g-factor difference ∆ g between two dots. This∆ g serves as an effective magnetic field gradient and allows for qubit rotations with a rate that in-creases linearly with an external magnetic field. We measured several interdot transitions and foundthat this valley introduced ∆ g is universal and electrically tunable. This could potentially simplifyscaling up quantum information processing in the SiGe platform by removing the requirement formagnetic field gradients which are difficult to engineer. INTRO
Quantum algorithms have been proposed to solve prob-lems that are formidable for classical computers [1–4] butrequire quantum hardware to be implemented. Electronspins confined to gate defined semiconductor quantumdots hold promise as quantum bits (qubits) due to thepromise of long coherence times and the localized natureof their control, making them promising for scaling up tothe large number of qubits required for real algorithms[5–8]. These qubits have demonstrated fast initialization,high fidelity readout[9, 10], and the possibility of operat-ing at temperatures above 1 Kelvin [11, 12]. Single spinqubits have demonstrated high fidelity single qubit gatesof 99.9% [13] and two qubit gates above 98% [14]. Whilethe difficulty in addressing single spin quibts might bean obstacle for scaling up, singlet-triplet qubits have theadvantage of having a Hamiltonian whose magnitude anddirection can be electrically tuned from the exchange en-ergy axis to an additional control axis generated by amagnetic field gradient, ∆ B Z [15, 16].Among all semiconductor platforms, silicon/silicon-germanium (Si/SiGe) is appealing because its weak nu-clear spin background minimizes the decoherence causedby magnetic field fluctuations [8, 17]. Moreover, weakspin-orbit coupling further reduces the spin relaxationcaused by charge fluctuation. This, however, is a dou-ble edged sword, because spin-orbital coupling can alsobe used for electrical spin control. In the absence of fullelectrical control, other works have used micromagnetsplaced near the qubit to create a local magnetic fieldgradient for an effective spin drive [7, 13, 16, 18]. Strongfield gradients, however, are hard to create over large ar-eas of the sample, posing challenges for scaling up thisscheme. Another challenge to developing a high fidelityspin qubit in Si/SiGe is the valley degeneracy, which mayalso contribute to spin decoherence [19]. This degener- acy can be measured using magneto spectroscopy, HallBar measurements and microwave readout in hybrid cav-ity quantum electrodynamic (cQED) systems [20–22] andcan be removed by lattice strain and electrostatic confine-ment [21, 23].In this work we investigate a valley-assisted spin qubitformed in SiGe double quantum dots which does not re-quire a magnetic field gradient to achieve full two axissingle qubit control. For interdot transitions where theelectron number is (4n,4m) - (4n ± ∓ B Z rotation whose precessionrate increases linearly with increasing magnetic field. Im-portantly, this valley introduced g-factor gradient is alsoelectric field dependent. The combined dependence onthe magnetic and electrical fields enables a tunable ∆ B Z rotation. This could potentially simplify scaling up quan-tum information processing as it removes the need for anexternal magnetic field gradient. DEVICE AND METHOD
Figure 1(a) shows a scanning electron microscope im-age of a typical device that utilizes an overlap gate ge-ometry to achieve quantum dot confinement [25]. Thebarrier gates, B, create potential barriers for controllingtunneling rates and plunger gates, P, select the chargestate in each dot and tune their chemical potential. Forthis experiment, a double quantum dot (DQD, markedin yellow) is formed at the left two plunger gates P to P , while P to B control the tunneling rate to a fermi a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n e S(3,1)
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Figure 1. (a) SEM of an overlap style device. The DQD and sensor location are labeled as yellow and orange dots, respectively.(b) Charge stability diagram measured by performing charge readout as a function of plunger gate voltages after being preparedin a random (3,1) state. Spin blockade will keep triplet in configuration (3,1) and singlet in configuration (4,0) in the brightarea in the (4,0) region, giving rise to the readout spot M . The spot I is for the initialization of a singlet and spot E is forqubit gate operation. (c) Schematic level diagram as a function of (cid:15) with the singlet plotted as solid lines and triplets plottedas blue dashed lines. Inset: four possible (3,1) states interact with S . (d) Spin funnel obtained by measuring the cross overof singlet to triplet states as a function of B and ˜ (cid:15) . Two sets of curvatures are observed, as emphasized by the red and yellowarrows. The valley states of the triplets are labeled accordingly. sea. We also form a sensor dot (marked in orange) toperform charge detection and use RF-reflectometry forfast readout.Figure. 1(b) demonstrates Pauli-blockade at the (4,0)-(3,1) transition, as required for forming a singlet-tripletqubit. The energy states of the DQD are dependent onwhich of the two valley eigenstates, K or K’, the electronsoccupy. Here we note that valley states K and K’ canbe different between dots, and thus there is no orthogo-nality, which allows valley states to flip during interdottransitions. In the (4,0) charge state the four spin valleycombinations of the ground orbital state are completelyfilled. The spin blockade region shown in Fig. 1(b) is cut-off at low (cid:15) by transitions into the excited orbital state,which is 200 µ eV higher than the ground state and allowsthe triplet states to decay into (4,0) states. This energyis large enough to be ignored in the spin dynamics dis-cussed below.The insets of Fig. 1(c) show the four possible (3,1)states that the ground (4,0) state can transition to with-out a spin flip. We use the notation (3 , ij where thesuperscript i represents the valley of the vacancy in theleft dot and j represents the valley of the electron in theright dot. The ground (3,1) charge state is then (3 , K (cid:48) K and the transition from (4,0) to (3 , K (cid:48) K requires flippingan electron from the K’ valley to the K valley, suggestingthat there is a valley difference between these two states.Assuming that the valley splitting in the left(right) dot is ∆ L ( R ) , the three excited (3,1) valley states would be(3 , K (cid:48) K (cid:48) , (3 , KK and (3 , KK (cid:48) and are ∆ L , ∆ R and∆ L +∆ R higher in energy compared to the (3 , K (cid:48) K state.Figure 1(c) plots the energy diagram of all relevantstates as a function of detuning (cid:15) between the chemicalpotential of ground (4,0) and (3,1) charge states. Theinterdot coupling t c opens up the avoided crossing of all(4,0)-(3,1) singlet transitions, and leads to the hybridiza-tion of different S(3 , ij singlets when 2 t c (cid:38) ∆ L ( R ) asshown by the solid lines in Fig. 1(c). For all (3,1) tripletstates, the Pauli blockade forbids the interdot transitionand give rise to the energy levels as shown by the bluedashed lines. An external magnetic field B z will furtherintroduce Zeeman splitting E Z = ± gµ B B z that lift thedegeneracy of triplet states (not shown in the figure). SPIN FUNNEL
Spin funnel measurements can extract the exchangeenergy J ( (cid:15) ) between the singlet and zero angular momen-tum triplet state, T , by detecting the S- T + degeneracyas a function of magnetic field B and (cid:15) [9]. This allowsus to map J ( (cid:15) ) to the Zeeman energy E cZ = gµ B B cz . Inorder to detect this degeneracy, we first prepare the DQDin a singlet by loading the ground (4,0) state at the spot I in Fig. 1(b) and then park at spot M. We then abruptlypulse the DQD to the exchange location E and evolvefor a time of τ E = 1 µ s. Finally, we perform readoutof the spin state at position M. During the process wekeep the pulse along the diagonal ( (cid:15) ) direction such that∆ V P = − ∆ V P = ˜ (cid:15) with a lever arm that converts fromgate potential to energy given by (cid:15) = 0 . (cid:15) meV/mV.At a detuning where E Z = J ( (cid:15) ), an interaction betweenthe singlet state and the degenerate T + state results ina finite probability P T of flipping the state into a tripletstate [15, 26]. Fig. 1(d) demonstrates the funnel dataover ˜ (cid:15) = 0 - 3 mV and a magnetic field range B = -0.9 ∼ E cz (˜ (cid:15) ) is quickly changing with ˜ (cid:15) . The 1st pairapproaches zero field at large ˜ (cid:15) and maps out the theground state singlet energy that is plotted as the orangeline in Fig. 1(c). This results from the crossing of thisground singlet state and T + (3 , K (cid:48) K . The other 3 pairsare parallel to the first spin funnel and offset by 12 µ eV,33 µ eV and 45 µ eV, which corresponds to the intersectionbetween the ground singlet and T + (3 , K (cid:48) K (cid:48) , T + (3 , KK and T + (3 , KK (cid:48) . We have also noticed that the 2nd and3rd pair are brighter, which suggests a larger P T at thesetransitions. This is expected because flipping the sin-glet S (4 ,
0) to the T + (3 , K (cid:48) K (cid:48) or T + (3 , KK states doesnot require flipping the valley of the transitioning elec-tron (Fig. 1(c) insets) suggesting the valley states maybesimilar between dots. From these observations we findvalley splittings of 12 µ eV and 33 µ eV but we cannot tellwhich corresponds to ∆ L or ∆ R . We take ∆ L > ∆ R forconvenience in the following discussion.Our experiments take place with an electron tempera-ture of around 100 mK, which causes us to load the ex-cited singlet state, S (3 , K (cid:48) K (cid:48) , with approximately a 10%probability. The energy spectrum of this state is plottedas the red curve in Fig. 1(c). We observe signatures ofthe degeneracy between this exited singlet state and the4 triplet states as an additional four sets of curvaturesindicated by the red arrows in Fig. 1(d). The curvaturesnear B = 0 maps the degeneracy between S (3 , K (cid:48) K (cid:48) and T + (3 , K (cid:48) K (cid:48) . The crossing between S (3 , K (cid:48) K (cid:48) and T + (3 , KK and T + (3 , KK (cid:48) leads to parallel curves withoffsets of ∆ R − ∆ L = 21 µ eV and ∆ L = 33 µ eV real-tive to zero field. The crossing between S (3 , K (cid:48) K (cid:48) and T − (3 , K (cid:48) K gives rise to curvatures with an offset of∆ R = 12 µ eV relative to zero field and a flipped directionof curvature in response to the magnetic field because itis from the opposite Zeeman branch.Figure 2(a) overlaps fits for the ground and first ex-cited singlet energies (red and orange curve in Fig. 1(c))with the funnel data in Fig. 1(d). The fitting finds rea-sonable agreement between the model and data using theinterdot coupling as the only free parameter with a bestfit of t c = 15 µ eV. We further utilize this spectroscopy (b) -0.2-0.100.1 -4 Z:\qDots\data\silicon\HarvardFab7Dev1Data2\sm_STPMagnet_A_0620.mat -0.14 -0.12 -0.10-0.08 -4 e ̃ (mV) B ( T ) (a) e ̃ (mV) B ( T ) Figure 2. (a) Fit to the first order spin funnel lines inFig. 1(d). (b) Funnel measurements when the magnetic fieldzooms in the second branch and ˜ (cid:15) extend the over large rangeto explore the gate dependence of the valley splitting. T T S S T T S T - T S e E n e r g y J( e ) (a) (c) e ̃ ( m V ) Z:\qDots\ data\silico n\Harvard Fab7Dev1 Data2\sm _STP2D_A _0562.mat .5 1.0 1.5 -4 t E ( m s) (b) (d) ST J( e ) D B z W
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Z:\qDots\data\silico n\HarvardFab7Dev1 Data2\sm_RamseyA _0574.mat e ̃ ( m V ) t E (ns) -4 Figure 3. (a) level diagram of relevant states for qubit. Theground singlet (orange) and T (blue) states are the base ofthe singlet-triplet qubit. (b) The Bloch sphere for singlet-triplet qubit. (c-d) the charge readout as a function of ˜ (cid:15) andexchange time after the ∆ B Z rotation (c) and the exchangerotation (d). method to measure the gate dependence of ∆ L ( R ) . Fig.2(b) shows the second funnel in greater detail by reduc-ing the range of E cz and enlarging the range of ˜ (cid:15) . We fitthe curve assuming ∆ R = ∆ R + ˜ (cid:15) ∆ (cid:48) R and find ∆ (cid:48) R = 0 . (cid:48) L = − .
09 meV/V. [27] As-suming that ∆ L ( R ) is only dependent on P we have ∂ ∆ L/R /∂V P = 0.3(0.09) meV/V, which is comparableto results reported in previous works [19, 21, 23, 24]. TUNABLE TWO AXIS CONTROL
Coherent qubit control is explored in the subspacespanned by S (4 ,
0) and (3 , K (cid:48) K , as demonstrated inFig. 3(a). The energy gaps between the other (3,1)valley states and (3 , K (cid:48) K are much larger than thequbit energy scales and can be neglected when consid-ering qubit operations. The ground singlet (orange) and T (3 , K (cid:48) K (blue) constitute the basis of the qubit andthe Hamiltonian in this subspace can be approximatedby H = J ( (cid:15) ) σ z + ( g L B zL − g R B zR ) µ B σ x . Here σ x,z arethe Pauli operators, g L ( R ) and B zL ( R ) are the g-factorand external field at left(right) dot. This system canachieve flexible two axis control, as illustrated by theBloch sphere in Fig. 3(b) [8, 16]. Rotation along the Zaxis can be performed near the interdot transition where J ( (cid:15) ) is large and { S , T } are the eiganstate of the sys-tem Hamiltonian. X axis rotations (∆ B z rotations) areachieved at large (cid:15) where J( (cid:15) ) is negligible and the energysplitting is set by µ B ( g L B zL − g R B zR ) and the eigenstatesare | ↑↓(cid:105) and | ↓↑(cid:105) [15, 16].We characterize the ∆ B Z rotations by initializing thesystem in S (4 ,
0) and then pulsing the DQD to large (cid:15) to turn off J , allowing the qubit to rotate around the∆ B Z axis into superpositions of S and T . We thenpulse back to M for readout [15, 16]. Figure 3(c) demon-strates charge readout as a function of exchange time τ E and location (˜ (cid:15) ) when we set the field B z = 1 T. Forlarge ˜ (cid:15) >
10 meV, we find an oscillation at a frequency f ∆Bz = µ B ( g L B zL − g R B zR ) / (cid:126) = 5.5 MHz. The decayin the oscillation amplitude indicates a T ∗ = 1 µ s [27].When ˜ (cid:15) <
10 mV, the oscillation rate is larger than f ∆Bz and the amplitude is smaller, indicating a finite J ( (cid:15) ),which contributes to the rotation rate and shifts the an-gle from the ∆ B Z axis. The coherent J rotation can becharacterized by a similar process by adding adiabaticramping before and after the exchange rotation to map Sto | ↓↑(cid:105) and T to | ↑↓(cid:105) [9]. The result is presented in Fig.3(d). For 1 < ˜ (cid:15) < (cid:15) .Here J is dominant, and the rotation rate is electricallytunable as expected.We emphasize that no micromagnet or other externalmagnetic field gradient source was added to this device.To explore the mechanism responsible for the field gra-dient, we measure f ∆B Z as a function of B z . Figure 4(a)plots charge readout as a function of τ E and B z after a∆ B Z rotation at ˜ (cid:15) = 10 mV such that J (˜ (cid:15) ) is negligible.The ∆ B Z rotation rate is then extracted and plotted as afunction of field in Figure 4(b). This result is consistentwith a difference in the g factor of the two dots of around∆ g = 3 . × − = 0 . g . We note that the two dots’ground states occupy different valley states and this ∆ g is consistent with the g factor difference between valleysas previously reported [28, 29].This valley introduced ∆ g allows the rate of the ∆ B Z rotations to be tuned by an external magnetic field. Weexpect the ∆ B Z rotation rate would be 14 MHz at 3 T(beyond the current limit of our magnet), comparableto the field gradient generated by a micro-magnet [16].This would potentially reduce design complexity for alarge array of spin qubits because it eliminates the needfor an artificially generated field gradient.In order to verify the generality of this phenomena, 'Z:\qDots\data\silicon\HarvardFab7Dev1Data2\sm_dBznofbAMagnet_A_1155.mat',...'Z:\qDots\data\silicon\HarvardFab7Dev1Data2\sm_dBznof bAMagnet_A_1307.mat',... 'Z:\qDots\data\silicon\HarvardFab7Dev1Data2\sm_dBznofbAMagnet_A_1310.mat' (a) (b) t E ( m s) -1.0-0.50.00.51.01.5 B z ( T ) -1 -0.5 0 0.5 1 1.5B z (T)246 f d B z ( M H z ) (0,1) V P2 V P (0,0) (1,1)(2,2) (3,3) (4,4)(1,2)(2,3) (3,4) (4,5) (5,4) (1,0) (2,1) (3,2)(4,3)(5,3)(2,0) (3,1)(4,2)(5,2) (3,0) (4,1)(5,1)(4,0)(5,0) (0,4) (0,5)(0,3) (0,2) (1,4) (1,5)(1,3) (2,4) (2,5)(3,5) (5,5) e ̃ (mV) t E ( m s ) Z:\qDots\data\silicon\HarvardFab7Dev1Data4\sm_STP2D_A_3059.mat (d)(c) (f) M H z M H z (e) -4 -3-2 -1
01 01 -4 Figure 4. (a) The charge readout as a function of exchangetime and magnetic field after a ∆ B Z rotation with ˜ (cid:15) = 10 mV(b) After ∆ B Z rotation rate as a function of magnetic field.(c) Schematic of a charge stability diagram. Colored circles la-bels the transitions that satisfy (4n, 4m)-(4n ±
1, 4m ∓ (cid:15) . (d-e) The ground statetransition at (d) (4,0)-(3,1) and (e) (0,4)-(1,3) charge config-uration and the associated f dBz /B z (f) The charge readout asa function of ˜ (cid:15) and exchange time after the δB z rotation at(4,4) transition. Dashed lines indicates the range of ˜ (cid:15) that Xaxis rotation is dominant and the rotation rate is labeled ontop. The gradient of ∆ B z rotation is 2MHz/10mV. we measured other (4n, 4m)-(4n ±
1, 4m ∓
1) transitionswhere the 4 valley spin states are all filled up in one dotand the relevant orbital of the other dot is empty. Aspreviously described, near these transitions the groundstates of the (4n, 4m) and (4n ±
1, 4m ∓
1) states oc-cupy different valley states, which introduce a ∆ g to thesinglet-triplet qubit Hamiltonian.Figure 4(c) labels these (4n, 4m)-(4n ±
1, 4m ∓
1) transi-tions with colored circles in the charge stability diagram.Fig. 4(d) demonstrates the ground states for the (4,0)-(3,1) transition, which is the symmetric configurationcompared to the (0,4)-(1,3) transition as demonstratedin Fig. 4(e). At the (1,3)-(0,4) transition, we measure f ∆Bz /B z = 8 MHz/T which is almost double the valuefor the (4,0)-(3,1) transition. V P1 and V P2 have beenchanged by 100 mV, which allows us to estimate that thegradient can be tuned at a rate of 35 MHz/(T · V). The(4,4)-(3,5) transition has the same filling as Fig. 4(d) and(5,3)-(4,4) transition corresponds to Fig. 4(e). Althoughthe two transitions are close in the charge stability di-agram, we find a dramatic change in the ∆ g . At the(5,3)-(4,4) we find f ∆Bz /B z = 0 and at the (4,4)-(3,5)transition f dBz /B z = 5.5 MHz/T at ˜ (cid:15) = 15 mV.This drastic change indicates a strong gate dependenceof ∆ g . Figure 4(f) demonstrates the ∆ B z rotation at the(4,4)-(3,5) transition using charge readout as a functionof ˜ (cid:15) and τ E . Unlike the (4,0)-(3,1) transition where theoscillation rate is a constant once ˜ (cid:15) >
10 mV, we find thisfrequency increases with ˜ (cid:15) once ˜ (cid:15) > J is always negligible suchthat the rotation is along X axis with the rate only de-termined by f ∆Bz . At ˜ (cid:15) = 15 mV f ∆Bz /B z = 6 MHz/Tand at ˜ (cid:15) = 25 mV, f ∆Bz /B z has increased by 2MHz/T.If we linearly interpolate f ∆Bz /B z for ˜ (cid:15) <
10 mV, weexpect the f ∆Bz /B z = 0 at ˜ (cid:15) = −
15 mV. The (5,3)-(4,4) transition is located at ∆ V P1 = − ∆ V P2 >
40 mVfrom the (4,4)-(3,5) transition and thus ˜ (cid:15) < ˜ (cid:15) . Hence f ∆Bz /B z = 0 at the (5,3)-(4,4) transition is not com-pletely surprising due to this rapid change in ∆ g at the(4,4)-(3,5) transition.The g factor difference between valleys has been ex-plained by spin-orbital coupling in previous works [28,29]. A gate dependent g factor has been predicted forSiGe [29] and observed in Si-MOS system [30]. The gatedependence of the g factor observed here could enablegate operations that require fast changes in ∆ B z . Com-pared to the field gradient generated by a micro-magnet, f ∆Bz by ∆ g can be arbitrarily high with increasing ex-ternal field. As with the valley splitting, a maximized∆ g is expected for an atomically flat Si/SiGe interfaceand high external electrical field. We thus expect a po-tentially higher ∆ B z gate fidelity with better substratesand smaller QDs. CONCLUSION
In this work we have investigated DQD transitionswhere the electron number is (4n,4m)-(4n ± ∓ B z ro-tation that is linear to an external magnetic field and isalso gate dependent. These two dependencies providea tunable ∆ B z rotation that does not require a micro-magent. This would potentially simplify scaling up tolarge arrays of spin qubits for quantum information pro-cessing. ACKNOWLEDGEMENT
We acknowledge Lisa Edge from HRL Laboratoriesfor the growth and distribution of the Si/SiGe heteros- tuctures that were used in this experiment. This workwas sponsored by the Army Research Office (ARO),and was accomplished under Grant Number W911NF-17-1-0274 and W911NF-17-1-0202. We acknowledge theuse of facilities supported by NSF through the UW-Madison MRSEC (DMR-1720415) and the MRI program(DMR-1625348). The views and conclusions containedin this document are those of the authors and shouldnot be interpreted as representing the official policies,either expressed or implied, of the Army Research Of-fice (ARO),or the U.S. Government. The U.S. Govern-ment is authorized to reproduce and distribute reprintsfor Government purposes notwithstanding any copyrightnotation herein. [1] R. D. Somma, S. Boixo, H. Barnum, and E. Knill, Quan-tum simulations of classical annealing processes, Physicalreview letters , 130504 (2008).[2] G. Carleo and M. 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