Effective out-of-plane g-factor in strained-Ge/SiGe quantum dots
Andrew J. Miller, Mitchell Brickson, Will J. Hardy, Chia-You Liu, Jiun-Yun Li, Andrew Baczewski, Michael P. Lilly, Tzu-Ming Lu, Dwight R. Luhman
EEffective out-of-plane g-factor instrained-Ge/SiGe quantum dots
Andrew J. Miller, ∗ , † Mitchell Brickson, † Will J. Hardy, † Chia-You Liu, ‡ Jiun-YunLi, ‡ Andrew Baczewski, † Michael P. Lilly, † Tzu-Ming Lu, † and Dwight R.Luhman † † Sandia National Laboratories ‡ National Taiwan University ¶ Center for Integrated Nanotechnologies
E-mail: [email protected]
Abstract
Recently, lithographic quantum dots in strained-Ge/SiGe have become a promis-ing candidate for quantum computation, with a remarkably quick progression fromdemonstration of a quantum dot to qubit logic demonstrations. Here we present ameasurement of the out-of-plane g -factor for single-hole quantum dots in this material.As this is a single-hole measurement, this is the first experimental result that avoidsthe strong orbital effects present in the out-of-plane configuration. In addition to ver-ifying the expected g -factor anisotropy between in-plane and out-of-plane magnetic( B )-fields, variations in the g -factor dependent on the occupation of the quantum dotare observed. These results are in good agreement with calculations of the g -factorusing the heavy- and light-hole spaces of the Luttinger Hamiltonian, especially thefirst two holes, showing a strong spin-orbit coupling and suggesting dramatic g -factortunability through both the B -field and the charge state. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b emiconductor quantum dots have proven to be a promising platform for scalable spin-based qubit operations. Much research has been performed using silicon-based electronquantum dots. Holes in germanium exhibit many of the desirable properties of electrons insilicon, with some beneficial differences: Absence of degenerate valley-states, the atomic p -orbital character that leads to a natural suppression hyperfine induced decoherence, and a large spin-orbit coupling that allows for control of qubits using electric dipolespin resonance. Because of these potential advantages, strained-germanium quantum dotshave been under investigation recently. Remarkably, hole-spin qubits in strained-Ge/SiGeheterostructures have seen a quick progression from demonstration of a quantum dot to qubitlogic.
The large separation between the heavy- and light-hole subbands in this material, which isenhanced by the strain, leads to a reduction in mixing between these subbands. This resultsin reduced effective off-diagonal terms in a single-band model of the heavy-hole space. Thus,the primarily heavy-hole character of holes in Ge/SiGe quantum wells is expected to showlarge g -factor anisotropy between magnetic fields applied in-plane and out-of-plane. This has been confirmed in one-dimensional channels and two-dimensional (2D) holesystems where the anisotropy was observed to be density dependent. For applications ofsingle spin qubits in planar quantum dots, the magnitude of the anisotropy remains unclear.The in-plane g -factor observed in single-hole qubit experiments is tunable with a value of < . Measurements of the g -factor of a single hole confined to a quantum dot with themagnetic ( B )-field directed out-of-plane have not been reported. For many-hole quantumdots with an applied out-of-plane magnetic field, the bare g -factor can be obscured by orbitaleffects. A highly anisotropic g -factor introduces additional opportunities for optimizationnot present in other qubits. For example, the B -field needed to create sufficient Zeemangap for spin readout would be significantly reduced for the direction with a large g -factor.This may provide a trade-off for integration with superconducting elements, especially if theanisotropy is significant. 2n this paper, we study the g -factor of a quantum dot in Ge/SiGe down to the last holeusing magnetospectroscopy with the B -field pointed out-of-the-plane of the sample. We findan out-of-plane g -factor of 15 . g k = 0 .
3, the anisotropy of the g -factor for a single hole confined to alateral quantum dot is found to be over 50. With increased hole occupancy in the quantumdot ( n >
2) we observe large changes in the apparent g -factor due to orbital effects. Inaddition, we perform calculations using the Luttinger Hamiltonian with a realistic quantumdot potential and find remarkable agreement with the single hole g -factor and also capturethe trend for higher occupancy up to six holes.In this work, we used a strained-Ge quantum well as our starting material. The SiGebarriers were composed of 27 % Si, with a 62 nm barrier above the strained-Ge well. Thestrained-Ge well was 20 nm thick, and showed a mobility of 1 . × cm V − s − in Hall barmeasurements. Details regarding the growth of the Ge/SiGe heterostructure are reported inRef. 24.A lithographic quantum dot was patterned onto the strained-Ge quantum well. A scan-ning electron microscope (SEM) image of a device made using the same process as themeasured device can be seen in Fig. 1, along with a schematic of the material stack. Bymanipulating the voltages of the patterned gate electrodes, charge-carriers (holes) are drawninto the strained-Ge layer, producing a hole gas and hole quantum dots.This electrode design defines two single quantum dots located next to each other onthe device, labeled as upper and lower, each controlled by five gate electrodes. Prior workleading up to this design can be found in Ref. 25. Details regarding the device tuning andmeasurements can be found in the supporting information.The data presented here were taken with an applied out-of-plane magnetic field in adilution refrigerator with a base temperature of 30 mK. In this configuration, magnetospec-troscopy measurements were made up to 3 T, as well as capacitance measurements betweenthe upper dot and each electrode. Through an evaluation of the thermal broadening of the3 iGes-GeSiGe1. Isolation gates2. Accumulation gates3. Barrier gates ALD oxide
500 nm
ULAG URAG U do t Iso UIso CIso L
Ldo t LLAG LRAG a) b)
20 nm
62 nm
Figure 1: SEM image of a device and the material stack. Subfigure a) shows an artificiallycolorized SEM image of the device design used in this work. Subfigure b) presents thematerial stack, using the same color scheme as a). Charge carriers are acculumated in thestrained-Ge layer, and quantum dots are formed under the UDot and LDot electrodes, inthe space between the isolation gates. 4harge sensing lines (see supporting information), the two-dimensional hole gas (2DHG) basetemperature was found to be 417(32) mK.The lower dot was used as a charge sensor, while the upper dot was tuned to the single-hole regime. The coupling between the upper dot gate, Udot, and the quantum dot wasmeasured to be α = 75 . − . Fig. 2 shows clearly resolved charge sensing lines. gradient(log |𝑖|) Figure 2: Stability plot of the quantum dot. The voltage applied to the dot gate is shownon the vertical axis, while the barrier gates (plungers) are displayed horizontally. Colorindicates the gradient of the charge sensor current, i . Horizontal labels indicate the voltageof URP at each point. ULP was kept 0 . n holes present in the dot are labeled.Magnetospectroscopy was used to determine the g -factor with the magnetic field orientedout-of-plane. Fig. 3 displays typical results. Multiple data sets were taken by sweeping theUdot gate (as in Fig. 3a) and also combinations of ULP and URP with Udot constant. Nosignificant differences were seen in the g -factors obtained by these two types of scans.The data show various kinks as expected for orbital crossings in magnetospectroscopy.Additional near-vertical features near B = ± . g -factor near B = 0 T, away fromthese resonances. n=1n=2 n=3 n=4 a) b) gradient(log |𝑖|) Figure 3: Magnetospectroscopy of the first several hole transitions. Ten magnetospec-troscopy scans such as the one shown on subfigure a) were taken, although not all scans wereable to resolve as far beyond the first two lines. This scan was taken using the Udot voltageto scan across states, while three of the ten scans were taken using the plungers. Greyscaleindicates the gradient of the charge sensor current, i . Each line’s position was extracted byfitting a Gaussian and is shown by the red overlay. These points were converted to energyas a function of B , and the region near B = 0 was fit using a linear, absolute value function,as shown in subfigure b). Each line has been shifted in energy to 0 energy at B = 0 in orderto easily visualize the difference in slopes. The lower plot shows the first two lines, fromthe s -like orbital, while the upper shows the next four lines, from the p -like orbitals. Thedifference between the first and second line was used to extract the effective g -factor.Magnetospectroscopy data were analyzed by fitting Gaussians to trace each charge-transition line. The maximum of each Gaussian fit is highlighted as a red overlay in Fig. 3a.Uncertainty in the position of each Gaussian was determined from the 95 % confidence in-terval of the fit. Gate voltage is converted to energy using α and set to zero at B = 0 T, asshown in Fig. 3b.The out-of-plane g -factor, g ⊥ , is extracted from each data set for the range B = − . . E for the n = 1and n = 2 lines to a linear function and relating the slope to the g -factor using ∆ E = gµ B B ,where µ B is the Bohr magneton. A final value of g ⊥ is determined through a weighted6verage of the results from each data set and the uncertainty in g ⊥ is calculated using thestandard deviation among the values of g ⊥ from each of the 10 scans. The results of eachscan individually are displayed in the supporting information. Using this method, we find g ⊥ = 15 . E . Small scan-to-scan variation in individual values of g ⊥ was random. Therewas no apparent correlation with tuning adjustments or whether Udot or a combination ofULP and URP were used to sweep through the dot occupancy levels. This suggests thatchanges in electric field did not alter the observed value of g ⊥ in these experiments. Saiddifferently, any change in g ⊥ due to electric field is not resolved within the uncertainty ofour experiments.These out-of-plane measurements yield the first experimental result of g ⊥ in lithographic,strained-germanium quantum dots in the single-hole regime. The value of the in-plane g -factor in similar devices used for qubit demonstrations has been reported as g k = 0 . . Taking the upper bound on g k , and using our value for g ⊥ of15 . g -factor anisotropy of g ⊥ /g k = 52, much greater than the anisotropy of < To our knowledge, this is the largest g -factoranisotropy observed among quantum dots that might be used as qubits. It is important tonote that our measurement of g ⊥ was done in the s -orbital of the quantum dot and representsthe bare Zeeman g -factor of a hole in a quantum dot with perpendicular applied field.To further emphasize this, consider a similar analysis using holes in the p -orbital. If onewere to ignore orbital effects and extract the g -factor from the difference between the n = 2and n = 3 states, the effective perpendicular g -factor is − . n = 3 and n = 4, the effective g -factor is 4 . B = − . . g -factor, and is7ikely the source of the weaker g -factor anisotropy seen in previous transport results obtainedwith out-of-plane B -fields. With measurements down to the last hole, these results showthe expected dramatic anisotropy with respect to the in-plane g -factor, and also indicate astrong variability in the effective g -factor between different charge states. This variation ineffective g -factor as a function of charge state is also due in part to the strong spin-orbitcoupling, an effect which has been seen in other hole-based qubits. To better understand the mechanisms at play in the determination of g ⊥ , calculationsof the quantum dot energy spectrum were performed based on the heavy- and light-holesubbands of the Luttinger Hamiltonian. Barrier gatesIsolation gatesAccumulation gatesQD location −0.02 𝑉−0.10 𝑉−0.18 𝑉
Figure 4: A wireframe design of the gate layout is included in a COMSOL model of thedevice. This model allows an electrostatic simulation of the accumulation of the quantumdot and the charge in the leads using the Thomas-Fermi approximation for the charge densityvia a nonlinear Poisson solve. The resulting potential is used as an input to our model ofthe electronic structure of our hole quantum dot to calculate the eigensates of the potentialusing an effective mass model.The quantum dot potential was generated using a COMSOL Multiphysics calculationwith the actual physical device geometry. The potential was then further tuned to theexperiment to match the crossing observed between the first and second excited states at B = 0 . ≈ .
25 meV. This was accomplishedby rescaling the potential in the x- and y-dimensions before solving for eigenstates without8isturbing the potential in the z-direction.We numerically study the B -field dependence of the spectrum of single-particle statesfor a realistic electrostatic model of the device to produce theoretical estimates for the g -factor. Our framework first considers the light- and heavy-hole bands in isolation and solvesfor the spectrum of eigenstates of the associated effective mass Hamiltonians including theCOMSOL-generated electrostatic potential (Fig. 4). We do so using an interior penaltydiscontinuous Galerkin discretization of the associated partial differential equations imple-mented in an in-house solver. These numerically generated eigenstates were used as a basisfor a Luttinger Hamiltonian, which explicitly treats the mixing of these bands due to spin-orbit coupling. The impact of strain due to the lattice mismatch between the Ge and SiGelayers was added in the form of a Bir-Pikus Hamiltonian. This leads to a splitting of thesubbands in energy, in which the lower energy single-particle states are dominated by theheavy-hole band.The energy spectrum from the simulation is shown in Fig. 5a. Due to the band gap instrained-germanium, these states are all spin ± /
2. The average slope of each line from 0 Tto 0 . . s - and p -orbitalnature these are the slopes of individual lines, not the difference in lines used to calculate g ⊥ above. The experimental slopes shown are taken from the data in Fig. 3b.From these simulations, g ⊥ was calculated from the first two levels at 0 . E/µ B B and found to be 21 .
25, in reasonable agreement with the experimental result of 15 . p -orbital states, the simulations correctly reproduce the signs, butoverstimate the magnitude. This may suggest that the interactions between holes have alarge effect, as those are not accounted for in this single-particle simulation.General trends of the experimental data are reproduced by this simulation, includingthe signs of the slopes, as well as the ordering of their magnitudes. The curvature of eachstate due to increasing magnetic confinement can also be seen in both the experiment and9 ) b) -3/2+3/2 Figure 5: Theoretical calculation of the single-particle energy spectrum for a dot in apotential that is isotropic in the xy-plane shown in a). This plot shows both the orbital andspin effects on the energy simultaneously. The dominant spin eigenstate is shown by theindicated colors. The splitting between the lowest two states is dominated by the Zeemansplitting, but the upper four states have both Zeeman and orbital effects that determinethe slopes of the lines. b) Comparison between the experimental and theoretical slopes ofthe energy levels with respect to magnetic field strength. The simulation shows an excellentagreement regarding the first two states in the s -like orbital, however only presents thecorrect sign of the next 4 p -like states. 10imulation. One possible explanation for the reduced slopes seen in experiment may be dueto leakage of the wavefunction into the SiGe barriers. The Luttinger parameter is muchlower in Si than Ge ( | κ | ≈ .
42, compared to ≈ . It is expected that | g | = 6 | κ | , andexperimentally g ≈ This was suggested in Ref. 23 as an explanationfor the small value obtained for g ⊥ . Although our work suggests that orbital effects are aprimary cause of their reduced effective g -factor, the g -factor in the SiGe barriers may stillinfluence the measurements presented here.It is also important to note that although the slopes show repeated signs between states3 and 4 as well as the states 5 and 6, the spin filling follows an alternating pattern, as shownby the color axis in Fig. 5a. These repeated signs are due to the p -like nature of theseorbitals. Angular dependence of the phase of the wavefunction in each orbital is alignedwith or against classical cyclotron motion, causing the B field to raise or lower the energy ofeach state. Stated another way, the orbital angular momentum alignment (anti-alignment)with the B field causes states 3,4 (5,6) to decrease (increase) with the increasing B -field,overpowering the Zeeman splitting due to the spin state.Strained-germanium continues to stand out as a promising material for realizing qubitswith lithographic quantum dots. This experimental verification of the large g -factor anisotropyin the single-hole regime suggests that a very weak B -field can provide a sufficient Zeemangap in the out-of-plane configuration. Future designs may be able to take advantage ofthis fact, allowing integration with superconducting elements and the ability to performspin readout using a B -field produced by on-chip components if desired. These results alsohighlight the strong orbital-effects present in the out-of-plane case for p -like states. Whilethe simulations presented show good agreement in the single-hole case, they also highlightthe complexity of the higher-order states. These behaviors allow for dramatic tunability ofthe g -factor via the charge occupation and the B -field, and highlight the difference between g -factor measurements in the single-hole and many-hole regime.11 eferences (1) Petta, J. R.; Johnson, A. C.; Taylor, J. M.; Laird, E. A.; Yacoby, A.; Lukin, M. D.;Marcus, C. M.; Hanson, M. P.; Gossard, A. C. Coherent Manipulation of CoupledElectron Spins in Semiconductor Quantum Dots. Science , , 2180–2184.(2) Veldhorst, M.; Hwang, J. C. C.; Yang, C. H.; Leenstra, A. W.; de Ronde, B.; Dehol-lain, J. P.; Muhonen, J. T.; Hudson, F. E.; Itoh, K. M.; Morello, A.; Dzurak, A. S. Anaddressable quantum dot qubit with fault-tolerant control-fidelity. Nature Nanotech-nology , , 981–985.(3) Kim, D.; Shi, Z.; Simmons, C. B.; Ward, D. R.; Prance, J. R.; Koh, T. S.; Gamble, J. K.;Savage, D. E.; Lagally, M. G.; Friesen, M.; Coppersmith, S. N.; Eriksson, M. A. Quan-tum control and process tomography of a semiconductor quantum dot hybrid qubit. Nature , , 70–74.(4) Eng, K.; Ladd, T. D.; Smith, A.; Borselli, M. G.; Kiselev, A. A.; Fong, B. H.;Holabird, K. S.; Hazard, T. M.; Huang, B.; Deelman, P. W.; Milosavljevic, I.;Schmitz, A. E.; Ross, R. S.; Gyure, M. F.; Hunter, A. T. Isotopically enhanced triple-quantum-dot qubit. Science Advances , .(5) Maurand, R.; Jehl, X.; Kotekar-Patil, D.; Corna, A.; Bohuslavskyi, H.; Lavi´eville, R.;Hutin, L.; Barraud, S.; Vinet, M.; Sanquer, M.; De Franceschi, S. A CMOS silicon spinqubit. Nature Communications , , 13575.(6) Harvey-Collard, P.; Jacobson, N. T.; Rudolph, M.; Dominguez, J.; Ten Eyck, G. A.;Wendt, J. R.; Pluym, T.; Gamble, J. K.; Lilly, M. P.; Pioro-Ladri`ere, M.; Carroll, M. S.Coherent coupling between a quantum dot and a donor in silicon. Nature Communica-tions , , 1029.(7) Zajac, D. M.; Sigillito, A. J.; Russ, M.; Borjans, F.; Taylor, J. M.; Burkard, G.;12etta, J. R. Resonantly driven CNOT gate for electron spins. Science , , 439–442.(8) Watson, T. F.; Philips, S. G. J.; Kawakami, E.; Ward, D. R.; Scarlino, P.; Veldhorst, M.;Savage, D. E.; Lagally, M. G.; Friesen, M.; Coppersmith, S. N.; Eriksson, M. A.; Van-dersypen, L. M. K. A programmable two-qubit quantum processor in silicon. Nature , , 633–637.(9) Huang, W.; Yang, C. H.; Chan, K. W.; Tanttu, T.; Hensen, B.; Leon, R. C. C.; Fog-arty, M. A.; Hwang, J. C. C.; Hudson, F. E.; Itoh, K. M.; Morello, A.; Laucht, A.;Dzurak, A. S. Fidelity benchmarks for two-qubit gates in silicon. Nature , ,532–536.(10) Moriya, R.; Sawano, K.; Hoshi, Y.; Masubuchi, S.; Shiraki, Y.; Wild, A.; Neumann, C.;Abstreiter, G.; Bougeard, D.; Koga, T.; Machida, T. Cubic Rashba Spin-Orbit Interac-tion of a Two-Dimensional Hole Gas in a Strained-Ge/SiGe Quantum Well. Phys. Rev.Lett. , , 086601.(11) Bulaev, D. V.; Loss, D. Electric Dipole Spin Resonance for Heavy Holes in QuantumDots. Phys. Rev. Lett. , , 097202.(12) Testelin, C.; Bernardot, F.; Eble, B.; Chamarro, M. Hole-spin dephasing time associatedwith hyperfine interaction in quantum dots. Phys. Rev. B , , 195440.(13) Keane, Z. K.; Godfrey, M. C.; Chen, J. C. H.; Fricke, S.; Klochan, O.; Burke, A. M.;Micolich, A. P.; Beere, H. E.; Ritchie, D. A.; Trunov, K. V.; Reuter, D.; Wieck, A. D.;Hamilton, A. R. Resistively Detected Nuclear Magnetic Resonance in n- and p-TypeGaAs Quantum Point Contacts. Nano Letters , , 3147–3150, PMID: 21714512.(14) Prechtel, J. H.; Kuhlmann, A. V.; Houel, J.; Ludwig, A.; Valentin, S. R.; Wieck, A. D.;Warburton, R. J. Decoupling a hole spin qubit from the nuclear spins. Nature Materials , , 981–986. 1315) Chou, C.-T.; Jacobson, N. T.; Moussa, J. E.; Baczewski, A. D.; Chuang, Y.; Liu, C.-Y.; Li, J.-Y.; Lu, T. M. Weak anti-localization of two-dimensional holes in germaniumbeyond the diffusive regime. Nanoscale , , 20559–20564.(16) Hardy, W. J.; Harris, C. T.; Su, Y.-H.; Chuang, Y.; Moussa, J.; Maurer, L. N.; Li, J.-Y.;Lu, T.-M.; Luhman, D. R. Single and double hole quantum dots in strained Ge/SiGequantum wells. Nanotechnology , , 215202.(17) Hendrickx, N. W.; Franke, D. P.; Sammak, A.; Scappucci, G.; Veldhorst, M. Fast two-qubit logic with holes in germanium. Nature , , 487–491.(18) Hendrickx, N. W.; Lawrie, W. I. L.; Russ, M.; van Riggelen, F.; de Snoo, S. L.;Schouten, R. N.; Sammak, A.; Scappucci, G.; Veldhorst, M. A four-qubit germaniumquantum processor. 2020.(19) Nenashev, A. V.; Dvurechenskii, A. V.; Zinovieva, A. F. Wave functions and g factorof holes in Ge/Si quantum dots. Phys. Rev. B , , 205301.(20) Mizokuchi, R.; Maurand, R.; Vigneau, F.; Myronov, M.; De Franceschi, S. BallisticOne-Dimensional Holes with Strong g-Factor Anisotropy in Germanium. Nano Letters , , 4861–4865, PMID: 29995419.(21) Lu, T. M.; Harris, C. T.; Huang, S.-H.; Chuang, Y.; Li, J.-Y.; Liu, C. W. Effectiveg factor of low-density two-dimensional holes in a Ge quantum well. Applied PhysicsLetters , , 102108.(22) Hendrickx, N. W.; Lawrie, W. I. L.; Petit, L.; Sammak, A.; Scappucci, G.; Veldhorst, M.A single-hole spin qubit. Nature Communications , , 3478.(23) Hofmann, A.; Jirovec, D.; Borovkov, M.; Prieto, I.; Ballabio, A.; Frigerio, J.;Chrastina, D.; Isella, G.; Katsaros, G. Assessing the potential of Ge/SiGe quantumdots as hosts for singlet-triplet qubits. arXiv , , [cond–mat.mes–hall].1424) Su, Y.-H.; Chuang, Y.; Liu, C.-Y.; Li, J.-Y.; Lu, T.-M. Effects of surface tunneling oftwo-dimensional hole gases in undoped Ge/GeSi heterostructures. Phys. Rev. Materials , , 044601.(25) Hardy, W. J.; Su, Y.-H.; Chuang, Y.; Maurer, L. N.; Brickson, M.; Baczewski, A.;Li, J.-Y.; Lu, T.-M.; Luhman, D. Gate-Defined Quantum Dots in Ge/SiGe QuantumWells as a Platform for Spin Qubits. ECS Transactions , , 17–25.(26) Liles, S. D.; Li, R.; Yang, C. H.; Hudson, F. E.; Veldhorst, M.; Dzurak, A. S.; Hamil-ton, A. R. Spin and orbital structure of the first six holes in a silicon metal-oxide-semiconductor quantum dot. Nature Communications , , 3255.(27) Winkler, R.; Papadakis, S.; De Poortere, E.; Shayegan, M. Spin-Orbit Coupling inTwo-Dimensional Electron and Hole Systems ; Springer-Verlag Berlin Heidelberg, 2003.(28) COMSOL, Inc., COMSOL Multiphysics Reference Manual, version 5.2. 2016.(29) Voisin, B.; Maurand, R.; Barraud, S.; Vinet, M.; Jehl, X.; Sanquer, M.; Renard, J.;De Franceschi, S. Electrical Control of g-Factor in a Few-Hole Silicon Nanowire MOS-FET.
Nano Letters , , 88–92. Acknowledgement
This work was supported by the Laboratory Directed Research and Development Programat Sandia National Laboratories and was performed, in part, at the Center for IntegratedNanotechnologies, a U.S. DOE, Office of Basic Energy Sciences user facility. Sandia NationalLaboratories is a multimission laboratory managed and operated by National Technology andEngineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell InternationalInc., for the U.S. Department of Energy’s National Nuclear Security Administration undercontract DE-NA0003525. 15his paper describes objective technical results and analysis. Any subjective views oropinions that might be expressed in the paper do not necessarily represent the views of theU.S. Department of Energy or the United States Government.This work at National Taiwan University (NTU) has been supported by the Ministry ofScience and Technology (109-2112-M-002-030-,109-2622-8-002-003-).
Supporting Information Available
Additional details regarding the experimental methods and results are available.16 upporting Information
Andrew J. Miller, ∗ , † Mitchell Brickson, † Will J. Hardy, † Chia-You Liu, ‡ Jiun-YunLi, ‡ Andrew Baczewski, † Michael P. Lilly, † Tzu-Ming Lu, † and Dwight R.Luhman † † Sandia National Laboratories ‡ National Taiwan University ¶ Center for Integrated Nanotechnologies
E-mail: [email protected]
Experimental Methods
Measurements
Measurements were performed in a commercial dilution refrigerator at T ∼
30 mK. Duringthis cooldown, a slight leakage current was seen on the upper-left accumulation gate (ULAG),preventing transport measurements through the upper side. Because of this, the upperdot was tuned into the few-hole regime, while the lower dot was used as a charge sensor.Typical voltages which were found to produce good results were − − − . − . . .
75 V to 2 .
75 V were1 a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b sed on the lower, charge-sensing quantum dot, while voltages of 0 . . . g -factor from each scan can be seen in Fig. 1.Figure 1: Effective g -factors obtained from each scan. Scan number is shown on the hori-zontal axis, and corresponds to the chronological order in which they were taken. Scans 4,5, and 6 were taken by scanning URP, and using the relative capacitance between URP andUdot to determine the energy. All other scans were performed by scanning Udot. The threedata sets show the results of the effective g -factor using different pairs of states. Not all scanswere able to resolve four lines, so some points are not present. The solid lines indicate theweighted average of the results from each scan, and the dashed lines indicate the statisticaluncertainty from this weighted average. Standard deviation of each data set is indicated bythe colored band and is the dominant source of the uncertainty we report.2 ever arm calculation Figure 2: Line widths as a function of temperature. The fitting parameter c from thetransition was used to extract the base temperature of the 2DHG and lever arm, ( α = k B /a ),of the dot electrode. The inset plot shows an example of the transition width measured attwo different temperatures. To highlight the variation in width, the vertical axis of eachtransition has been normalized, and the horizontal axis has been shifted.To determine the lever arm of the dot electrode, the factor α representing the relationbetween the gate voltage and the potential of the dot, an analysis of the charge sensing tran-sition vs. temperature was performed. Due to the fact that one of the upper accumulationgates was not holding voltage well, it was not possible to determine the lever arm throughthe more traditional Coulomb diamonds, as transport could not be achieved through the3pper dot. With the upper dot in the single-hole regime, Udot was tuned to scan acrossthe first charge sensing line. This was scanned with the fridge held at a given temperature,and the current through the lower dot was measured. Temperatures ranged from 30 mK to700 mK with fifteen points being measured.In each scan, the charge-sensing signal was fit with the expression I = a (1 + exp [( V − b ) /c ]) + d where I is the current signal through the charge-sensor, and V is the scanning voltageapplied to Udot. From this equation, the fit parameter c determines the width, and wasplotted against the temperature, as shown in Fig. 2.From the relation to the Fermi-Dirac distribution, it can be seen that c = k B α q T + T where α has units of eV/V, T is the base temperature of the 2DHG, and T bath is thetemperature the fridge was held at for the measurement. From the fit shown in Fig. 2, T and α were extracted as 417(32) mK and 75 . − respectively. Beyond the fourth line