Silicon/silicon-germanium heterostructures have many important advantages for hosting spin qubits. However, controlling the valley splitting (the energy splitting between the two low-lying conduction-band valleys) remains a critical challenge for ensuring qubit reliability. Broad distributions of valley splittings are commonplace, even among quantum dots formed on the same chip. In this work, we theoretically explore the interplay between quantum-well imperfections that suppress the valley splitting and cause variability, such as broadened interfaces and atomic steps at the interface, while self-consistently accounting for germanium concentration fluctuations. We consider both conventional and unconventional approaches for controlling the valley splitting, and present concrete strategies for implementing them. Our results provide a clear path for achieving qubit uniformity in a scalable silicon quantum computer.
We report measurements consistent with the valley Kondo effect in Si/SiGe quantum dots, evidenced by peaks in the conductance versus source-drain voltage that show strong temperature dependence. The Kondo peaks show unusual behavior in a magnetic field that we interpret as arising from the valley degree of freedom. The interplay of valley and Zeeman splittings is suggested by the presence of side peaks, revealing a zero-field valley splitting between 0.28 to 0.34 meV. A zero-bias conductance peak for non-zero magnetic field, a phenomenon consistent with valley non- conservation in tunneling, i…
We have demonstrated few-electron quantum dots in Si/SiGe and InGaAs, with occupation number controllable from N = 0. These display a high degree of spatial symmetry and identifiable shell structure. Magnetospectroscopy measurements show that two Si-based devices possess a singlet N =2 ground state at low magnetic field and therefore the two-fold valley degeneracy is lifted. The valley splittings in these two devices were 120 and 270 μeV, suggesting the presence of atomically sharp interfaces in our heterostructures.
Spin and valley-orbit splittings are calculated in SiGe/Si/SiGe quantum wells (QWs) by using the tight-binding approach. In accordance with the symmetry considerations an existence of spin splitting of electronic states in perfect QWs with an odd number of Si atomic planes is microscopically demonstrated. The spin splitting oscillates with QW width and these oscillations related to the inter-valley reflection of an electron wave from the interfaces. It is shown that the splittings under study can efficiently be described by an extended envelope-function approach taking into account the spin- a…
Electron states are studied for quantum dots in a strained Si quantum well, taking into account both valley and orbital physics. Realistic geometries are considered, including circular and elliptical dot shapes, parallel and perpendicular magnetic fields, and (most importantly for valley coupling) the small local tilt of the quantum well interface away from the crystallographic axes. In absence of a tilt, valley splitting occurs only between pairs of states with the same orbital quantum numbers. However, tilting is ubiquitous in conventional silicon heterostructures, leading to valley-orbit co…
We observe minima of the longitudinal resistance corresponding to the quantum Hall effect of composite fermions at quantum numbers $p=1$, 2, 3, 4, and 6 in an ultraclean strongly interacting bivalley SiGe/Si/SiGe two-dimensional electron system. The minima at $p=3$ disappear below a certain electron density, although the surrounding minima at $p=2$ and $p=4$ survive at significantly lower densities. Furthermore, the onset for the resistance minimum at a filling factor $ν=3/5$ is found to be independent of the tilt angle of the magnetic field. These surprising results indicate the intersection …
Enhanced intrinsic spin-orbit driving of a Loss-DiVincenzo qubit near the spin-valley hotspot in Si/SiGe
RWTH Aachen University（亚琛工业大学）；Forschungszentrum Jülich GmbH（于利希研究中心有限公司）；Helmholtz Nano Facility (HNF)（亥姆霍兹纳米设施）；IHP - Leibniz Institute for High Performance Microelectronics（高性能微电子莱布尼茨研究所）；ARQUE Systems GmbH（ARQUE系统有限公司）
We have investigated the valley splitting of two-dimensional electrons in high quality Si/Si$_{1-x}$Ge$_x$ heterostructures under tilted magnetic fields. For all the samples in our study, the valley splitting at filling factor $ν=3$ ($Δ_3$) is significantly different before and after the coincidence angle, at which energy levels cross at the Fermi level. On both sides of the coincidence, a linear density dependence of $Δ_3$ on the electron density was observed, while the slope of these two configurations differs by more than a factor of two. We argue that screening of the Coulomb interaction f…
Spin splitting of conduction electron states has been analyzed for all possible point symmetries of SiGe quantum well structures. A particular attention is paid to removal of spin degeneracy caused by the rotoinversion asymmetry of a (001) heterointerface between two diamond-lattice materials. The asymmetry is shown to result in spin splitting of both Rashba and Dresselhaus types in symmetrical SiGe quantum wells. Consequences of the spin splitting on spin relaxation are discussed.
The valley splitting in Si two-dimensional electron systems is studied using Si/SiGe single quantum wells (QWs) with different well widths. The energy gaps for 4 and 5.3 nm QWs, obtained from the temperature dependence of the longitudinal resistivity at the Landau level filling factor $ν=1$, are much larger than those for 10 and 20 nm QWs. This is consistent with the well-width dependence of the bare valley splitting estimated from the comparison with the Zeeman splitting in the Shubnikov-de Haas oscillations.
Spin splitting of conduction electron states has been analyzed for all possible point symmetries of SiGe quantum well structures. A particular attention is paid to removal of spin degeneracy caused by the rotoinversion asymmetry of a (001) heterointerface between two diamond-lattice materials. Consequences of the spin splitting on the electron spin relaxation time is discussed.
In this theoretical study we qualitatively and quantitatively investigate the electric dipole spin resonance (EDSR) in a single Si/SiGe quantum dot in the presence of a magnetic field gradient, e.g., produced by a ferromagnet. We model a situation in which the control of electron spin states is achieved by applying an oscillatory electric field, inducing real-space oscillations of the electron inside the quantum dot. One of the goals of our study is to present a microscopic theory of valley dependent $g$-factors in Si/SiGe quantum dots and investigate how valley relaxation combined with a vall…
The authors investigate the magnetic field dependence of the energy splitting between low-lying valley states for electrons in a Si/SiGe quantum well tilted with respect to the crystallographic axis. The presence of atomic steps at the quantum well interface may explain the unexpected, strong suppression of the valley splitting observed in recent experiments. The authors find that the suppression is caused by an interference effect associated with multiple steps, and that the magnetic field dependence arises from the lateral confinement of the electronic wave function. Using numerical simulati…
We present the theory and measurement of valley splitting in a quantum point contact (QPC) in a modulation doped Si/SiGe heterostructure. Our measurements are performed on a submicron Schottky-gated device. An effective mass theory is developed for a QPC formed in a quantum well, grown on a miscut substrate. Both theory and experiments include a perpendicular magnetic field. Our results indicate that both QPC and magnetic confinement can enhance the valley splitting by reducing the spatial extent of the electronic wavefunction. Consequently, the valley splitting can be much larger than the spi…
We examine energy spectra of Si quantum dots embedded into Si_{0.75}Ge_{0.25} buffers using atomistic numerical calculations for dimensions relevant to qubit implementations. The valley degeneracy of the lowest orbital state is lifted and valley splitting fluctuates with monolayer frequency as a function of the dot thickness. For dot thicknesses <6 nm valley splitting is found to be >150 ueV. Using the unique advantage of atomistic calculations we analyze the effect of buffer disorder on valley splitting. Disorder in the buffer leads to the suppression of valley splitting by a factor of 2.5, t…
Spins in SiGe quantum dots are promising candidates for quantum bits but are also challenging due to the valley degeneracy which could potentially cause spin decoherence and weak spin-orbital coupling. In this work we demonstrate that valley states can serve as an asset that enables two-axis control of a singlet-triplet qubit formed in a double quantum dot without the application of a magnetic field gradient. We measure the valley spectrum in each dot using magnetic field spectroscopy of Zeeman split triplet states. The interdot transition between ground states requires an electron to flip bet…
We present a calculation of the wavevector-dependent subband level splitting from spin-orbit coupling in Si/SiGe quantum wells. We first use the effective-mass approach, where the splittings are parameterized by separating contributions from the Rashba and Dresselhaus terms. We then determine the parameters by fitting tight-binding numerical results obtained using the quantitative nanoelectronic modeling tool, NEMO-3D. We describe the relevant parameters as a function of applied electric field and well width in our numerical simulations. For a silicon membrane, we find the bulk Rashba paramete…
Effect of electric field on spin splitting in SiGe quantum wells (QWs) has been studied theoretically. Microscopical calculations of valley and spin splittings are performed in the effective $sp^3d^5s^*$ tight-binding model. The splittings oscillate as a function of the QW width due to inter-valley reflection of the electron wave off the interfaces. In accordance with the symmetry considerations additional electric-field-induced terms appear in the electron spin-dependent Hamiltonian. The oscillations of splitting are suppressed in rather low electric fields. The tight-binding calculations hav…
Valley-orbit coupling is a key parameter for a silicon quantum dot in determining its suitability for applications in quantum information processing. In this paper we study the effect of interface steps on the magnitude and phase of valley-orbit coupling for an electron in a silicon quantum dot. Within the effective mass approximation, we find that the location of a step on the interface is important in determining both the magnitude and the phase of the valley-orbit coupling in a Si/SiGe quantum dot. Specifically, our numerical results show that the magnitude of valley orbit coupling can be s…
The bandstructure of bulk silicon has a six-fold valley degeneracy. Strain in the Si/SiGe quantum well system partially lifts the valley degeneracy, but the materials factors that set the splitting of the two lowest lying valleys are still under intense investigation. We propose a method for accurately determining the valley splitting in Si/SiGe double quantum dots embedded into a superconducting microwave resonator. We show that low lying valley states in the double quantum dot energy level spectrum lead to readily observable features in the cavity transmission. These features generate a "fin…
We show that Ge concentration oscillations within the quantum well region of a Si/SiGe heterostructure can significantly enhance the spin-orbit coupling of the low-energy conduction-band valleys. Specifically, we find that for Ge oscillation wavelengths near $λ= 1.57~\text{nm}$ with an average Ge concentration of $\bar{n}_{\text{Ge}} = 5\%$ in the quantum well region, a Dresselhaus spin-orbit coupling is induced, at all physically relevant electric field strengths, which is over an order of magnitude larger than what is found in conventional Si/SiGe heterostructures without Ge concentration os…
The presence of non-degenerate valley states in silicon can drastically affect electron dynamics in silicon-based heterostructures, leading to electron spin relaxation and spin-valley coupling. In the context of solid-state spin qubits, it is important to understand the interplay between spin and valley degrees of freedom to avoid or alleviate these decoherence mechanisms. Here we report the observation of relaxation from the excited valley state to the ground state in a Si/SiGe quantum dot, at zero magnetic field. Valley state read-out is aided by a valley-dependent tunneling effect, which we…
The effects of alloy disorder on strongly-driven flopping mode qubits in Si/SiGe
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Laboratory for Physical Sciences（物理科学实验室）；Department of Physics, University of Maryland（马里兰大学物理系）；School of Physics, University of New South Wales（新南威尔士大学物理学院）
Electron spins in Si/SiGe quantum wells suffer from nearly degenerate conduction band valleys, which compete with the spin degree of freedom in the formation of qubits. Despite attempts to enhance the valley energy splitting deterministically, by engineering a sharp interface, valley splitting fluctuations remain a serious problem for qubit uniformity, needed to scale up to large quantum processors. Here, we elucidate and statistically predict the valley splitting by the holistic integration of 3D atomic-level properties, theory and transport. We find that the concentration fluctuations of Si …
Silicon quantum dot qubits must contend with low-lying valley excited states which are sensitive functions of the quantum well heterostructure and disorder; quantifying and maximizing the energies of these states are critical to improving device performance. We describe a spectroscopic method for probing excited states in isolated Si/SiGe double quantum dots using standard baseband pulsing techniques, easing the extraction of energy spectra in multiple-dot devices. We use this method to measure dozens of valley excited state energies spanning multiple wafers, quantum dots, and orbital states, …
Spectroscopy of low-lying valley states in hot Si/SiGe quantum dots
University of California, Los Angeles（加州大学洛杉矶分校）；Center for Quantum Science and Engineering（量子科学与工程中心）；Department of Physics and Astronomy（物理与天文系）；Department of Electrical and Computer Engineering（电气与计算机工程系）
We measure the excited-state spectrum of a Si/SiGe quantum dot as a function of in-plane magnetic field, and we identify the spin of the lowest three eigenstates in an effective two-electron regime. The singlet-triplet splitting is an essential parameter describing spin qubits, and we extract this splitting from the data. We find it to be tunable by lateral displacement of the dot, which is realized by changing two gate voltages on opposite sides of the device. We present calculations showing the data are consistent with a spectrum in which the first excited state of the dot is a valley-orbit …
We report on the fabrication and electronic transport characterisation of Schottky-gated strongly confined Si/SiGe quantum point contacts (QPC). At zero magnetic field and T=450mK the QPC conductance as a function of gate voltage shows a quantization in units of e^2/h, indicative of transport through 1D modes which appear to lack both spin and valley degeneracy.
Silicon-germanium heterostructures have successfully hosted quantum dot qubits, but the intrinsic near-degeneracy of the two lowest valley states poses an obstacle to high fidelity quantum computing. We present a modification to the Si/SiGe heterostructure by the inclusion of a spike in germanium concentration within the quantum well in order to increase the valley splitting. The heterostructure is grown by chemical vapor deposition and magnetospectroscopy is performed on gate-defined quantum dots to measure the excited state spectrum. We demonstrate a large and widely tunable valley splitting…
The valley degree of freedom presents challenges and opportunities for silicon spin qubits. An important consideration for singlet-triplet states is the presence of two distinct triplets, comprised of valley vs. orbital excitations. Here we show that both of these triplets are present in the typical operating regime, but that only the valley-excited triplet offers intrinsic protection against charge noise. We further show that this protection arises naturally in dots with stronger confinement. These results reveal an inherent advantage for silicon-based multi-electron qubits.
We fabricated strongly confined Schottky-gated quantum point contacts by etching Si/SiGe heterostructures and observed intriguing conductance quantization in units of approximately 1e2/h. Non-linear conductance measurements were performed depleting the quantum point contacts at fixed mode-energy separation. We report evidences of the formation of a half 1e2/h plateau, supporting the speculation that adiabatic transmission occurs through 1D modes with complete removal of valley and spin degeneracies.
Intervalley mixing between conduction-band states in low-dimensional Si/SiGe heterostructures induces splitting between nominally degenerate energy levels. The symmetric double-valley effective mass approximation and the empirical pseudopotential method are used to find the electronic states in different types of quantum wells. A reasonably good agreement between the two methods is found, with the former being much faster computationally. Aside from being an oscillatory function of well width, the splitting is found to be almost independent of in-plane wave vector, and an increasing function o…
Non-adiabatic quantum effects, often experimentally observed in semiconductors nano-devices such as single-electron pumps operating at high frequencies, can result in undesirable and uncontrollable behaviour. However, when combined with the valley degree of freedom inherent to silicon, these unfavourable effects may be leveraged for quantum information processing schemes. By using an explicit time evolution of the Schrodinger equation, we study numerically non-adiabatic transitions between the two lowest valley states of an electron in a quantum dot formed in a SiGe/Si heterostructure. The pre…
We report the observation of the quantum-confined Stark effect in Ge/SiGe multiple quantum well heterostructures grown on Si(0.22)Ge(0.78) virtual substrates. The large compressive strain in the Ge quantum well layers caused by the lattice mismatch with the virtual substrate results in a blue shift of the direct absorption edge, as well as a reduction in the Γ-valley scattering lifetime because of strain-induced splittings of the conduction band valleys. We investigate theoretically the Γ-valley carrier lifetimes by evaluating the Γ-to-L and Γ-to-Δ scattering rates in strained Ge/SiGe semicond…
We study three-electron energy spectra in Si/SiGe single and vertically coupled double quantum dots where all the relevant effects, such as, the Zeeman splitting, spin-orbit coupling, valley coupling and electron-electron Coulomb interaction are explicitly included. In the absence of magnetic field, our results in single quantum dots agree well with the experiment by Borselli {\em et al.} [Appl. Phys. Lett. {\bf 98}, 123118 (2011)]. We identify the spin and valley configurations of the ground state in the experimental cases and give a complete phase-diagram-like picture of the ground state con…
Optimization of Si/SiGe Heterostructures for Large and Robust Valley Splitting in Silicon Qubits
Weierstrass Institute for Applied Analysis and Stochastics (WIAS)（魏茨塞克应用分析与随机过程研究所）；RWTH Aachen University（亚琛工业大学）；Forschungszentrum Jülich（于利希研究中心）；Institute of Quantum Information（量子信息研究所）；ARQUE Systems GmbH（ARQUE系统有限公司）
We develop a theory of effects of electron-electron collisions on the Dyakonov-Perel' spin relaxation in multi-valley quantum wells. It is shown that the electron-electron scattering rate which governs the spin relaxation is different from that in a single-valley system. The theory is applied to Si/SiGe (001)-grown quantum wells where two valleys are simultaneously populated by free carriers. The dependences of the spin relaxation rate on temperature, electron concentration and valley-orbit splitting are calculated and discussed. We demonstrate that in a wide range of temperatures the electron…
Conventional quantum transport methods can provide quantitative information on spin, orbital, and valley states in quantum dots, but often lack spatial resolution. Scanning tunneling microscopy, on the other hand, provides exquisite spatial resolution of the local electronic density of states, but often at the expense of speed. Working to combine the spatial resolution and energy sensitivity of scanning probe microscopy with the speed of microwave measurements, we couple a metallic probe tip to a Si/SiGe double quantum dot that is integrated with a local charge detector. We first demonstrate t…
Enhancing valley splitting in SiGe heterostructures is a crucial task for developing silicon spin qubits. Complex SiGe heterostructures, sharing a common feature of four-monolayer (4ML) Ge layer next to the silicon quantum well (QW), have been computationally designed to have giant valley splitting approaching 9 meV. However, none of them has been fabricated may due to their complexity. Here, we remarkably simplify the original designed complex SiGe heterostructures by laying out the Si QW directly on the Ge substrate followed by capping a (Ge4Si4)n superlattice(SL) barrier with a small sacrif…
We analyze spin splitting of the two-dimensional hole spectrum in strained asymmetric SiGe quantum wells (QWs). Based on the Luttinger Hamiltonian, we obtain expressions for the spin-splitting parameters up to the third order in the in-plane hole wavevector. The biaxial strain of SiGe QWs is found to be a key parameter that controls spin splitting. Application to SiGe field-effect transistor structures indicates that typical spin splitting at room temperature varies from a few tenth of meV in the case of Si QW channels to several meV for the Ge counterparts, and can be modified efficiently by …
