Semiconductor spin qubits

The spin degree of freedom of an electron or a nucleus is one of the most basic properties of nature and functions as an excellent qubit, as it provides a natural two-level system that is insensitive to electric fields, leading to long quantum coherence times. This coherence survives when the spin is isolated and controlled within nanometer-scale, lithographically fabricated semiconductor devices, enabling the existing microelectronics industry to help advance spin qubits into a scalable technology. Driven by the burgeoning field of quantum information science, worldwide efforts have developed semiconductor spin qubits to the point where quantum state preparation, multiqubit coherent control, and single-shot quantum measurement are now routine. The small size, high density, long coherence times, and available industrial infrastructure of these qubits provide a highly competitive candidate for scalable solid-state quantum information processing. Here the physics of semiconductor spin qubits is reviewed, with a focus not only on the early achievements of spin initialization, control, and readout in GaAs quantum dots but also on recent advances in Si and Ge spin qubits, including improved charge control and readout, coupling to other quantum degrees of freedom, and scaling to larger system sizes. First introduced are the four major types of spin qubits: single spin qubits, donor spin qubits, singlet triplet spin qubits, and exchange-only spin qubits. The mesoscopic physics of quantum dots, including single-electron charging, valleys, and spin-orbit coupling, are then reviewed. Next a comprehensive overview of the physics of exchange interactions is given, a crucial resource for single- and two-qubit control in spin qubits. The bulk of the review is centered on the presentation of results from each major spin-qubit type, the present limits of fidelity, and an overview of alternative spin-qubit platforms. A physical description of the impact of noise on semiconductor spin qubits, aided in large part by an introduction to the filter-function formalism, is then given. Last, recent efforts to hybridize spin qubits with superconducting systems, including charge-photon coupling, spin-photon coupling, and long-range cavity-mediated spin-spin interactions, are reviewed. Cavity-based readout approaches are also discussed. The review is intended to give an appreciation for the future prospects of semiconductor spin qubits while highlighting the key advances in mesoscopic physics over the past two decades that underlie the operation of modern quantum-dot and donor spin qubits.

Figure 1

The four major qubit types covered in this review, with images depicting the original proposals, early devices, and modern devices. (a) Loss-Divincenzo (LD) single spin qubits ([320]; [147]; [354]). (b) Donor spin qubits ([257]; [359]; [215]). (c) Singlet triplet (ST) spin qubits ([308]; [403]; [152]). (d) Exchange-only (EO) spin qubits ([141]; [341]; [201]).

Figure 2

(a) Spin configurations, (b) Bloch spheres, and (c) energy-level diagrams associated with LD single spin qubits, two-spin singlet triplet (${\mathrm{ST}}_0$) qubits, and three-spin EO spin qubits. Donor spin qubits also rely on single spins, in a manner similar to the LD case. We conventionally identify the north pole of the Bloch sphere with the qubit $|0⟩$ state and the south pole with $|1⟩$, irrespective of which state is lower in energy. For the LD qubit, a static magnetic field $B_{\mathrm{eff}}^z$ defines the quantization axis of the single spin, while a transverse (and smaller) ac magnetic field $B_{\mathrm{eff}}^x(t)$ drives coherent spin rotations between spin-up and spin-down. We identify $|0⟩=|\downarrow ⟩$ and $|1⟩=|\uparrow ⟩$ and note that the level ordering in (c) holds for $g>0$, as in Si. For the ${\mathrm{ST}}_0$ qubit, exchange coupling $J$ and a longitudinal magnetic-field gradient $\mathrm{\Delta }{B}^z$ provide two orthogonal control axes. For the EO spin qubit, nearest-neighbor exchange couplings $J_{12}$ and $J_{23}$ provide two control axes that are separated by 120° on the Bloch sphere.

Figure 3

Spin-qubit configurations grouped by the number of spin-$1/2$ particles per qubit and number of sites (usually QDs). Spins are indicated by the small gray dots (electrons or holes) or small white dots (nuclei), which are numbered to adhere to the basis description of Fig. 4. Sites are indicated by the large (pink) circles; their overlap indicates “always-on exchange,” meaning that the spins contained are somewhat delocalized across the site even for the idle qubit.

Figure 4

Spin-qubit encodings. The first column $N$ is the number of spin-$1/2$ particles per qubit, followed by a named “type” of qubit discussed in this review. The two-qubit states $|0⟩$ and $|1⟩$ are then specified in terms of both a conserved and a qubit-dependent “$q$ number” describing the total angular momentum; here $m$ always refers to the total spin projection, whereas $S_{jk\cdots }$ refers to the combined total spin angular momentum of spins $j,k,\dots$. Clebsch-Gordan coefficients translate these spin angular momentum combinations into “states.” For the three-spin case, $m$ may take either value $\pm 1/2$ in the encoded subspace. The final column shows the encoded Pauli operators $\mathit{\sigma }$ of the qubit in terms of the spin operators ${\mathbf{S}}_j$ of each spin-$1/2$ particle $j$. The qubit states are the $\pm 1$ eigenstates of ${\sigma }^z$. Degeneracies in these eigenstates indicate gauge freedom, and the null spaces of these operators are leakage states. For the LD qubit, the constant relating the logical qubit to the spin changes with the $g$ factor. The minus value shown here is consistent with the Si $g>0$ choice used in Fig. 2.

Figure 5

Device designs commonly used to confine electron spins. Vertical confinement is illustrated in the plots of $E(z)$ and lateral confinement is illustrated in the $x\text{−}y$ plane. (a) Donor electrons are confined by the positive potential of the donor atom and manipulated with gates defined through conventional or STM lithography. (b) Depletion-mode device design commonly used in early GaAs experiments. Modern Si MOS and $\mathrm{Si}/\mathrm{SiGe}$ devices utilize overlapping gate architectures to achieve tight control of QD electrons. (c) In Si MOS, electrons are localized at the ${\mathrm{SiO}}_2$/Si interface. (d) In $\mathrm{Si}/\mathrm{SiGe}$, the electrons reside in a buried QW. (e) Single-layer etch-defined gate electrode (SLEDGE) devices utilize a single layer of gates patterned on the top surface of a $\mathrm{Si}/\mathrm{SiGe}$ heterostructure. The gates are contacted from above using vias, which allows gate wiring to fan out away from the active area of the device in multiple planes. (f) FinFETs use a combination of dry etching and electrostatic gating to confine QD electrons. (c)–(f) Quantum-dot electrons are accumulated beneath the dark (orange) plunger gates, while the tunnel barrier gates are indicated with lighter shading (cyan). Gates are electrically isolated from one another using a thin dielectric. In (c) and (d), the plunger and barrier gate layers are isolated from the semiconductor in inactive regions of the device using screening gates.

Figure 6

Bulk Brillouin zone (upper panels) and band structure (lower panels) as a function of $k$ along the $⟨100⟩$ and $⟨111⟩$ directions for (a) GaAs and (b) Si. The nondegenerate conduction band minimum in GaAs is centered at the $\mathrm{\Gamma }$ point ($k=0$), while Si has six equivalent conduction band minima along the high-symmetry $⟨100⟩$ ($\mathrm{\Delta }$) directions and an anisotropic effective mass. The heavy-hole (HH) and light-hole (LH) valence bands for both materials are separated in energy from the split-off (SO) band by the spin-orbit splitting.

Figure 7

Electrostatic confinement. (a) 1D states can be formed in a QPC due to the potential constriction from a split gate. (b) Electrostatic confinement in both in-plane directions of a QW lead to 0D QD states. (c) Two QDs placed in series form a DQD. Depletion-mode devices are shown.

Figure 8

(a) DQD confinement potential. (b) DQD charge stability diagram. From [577]. (c) DQD energy levels near the $(1,0)$-$(0,1)$ interdot charge transition. (d) DQD energy levels in the two-electron regime showing the crossover from the $(2,0)\to (1,1)\to (0,2)$ charge state.

Figure 9

Low-energy orbital spectrum of a one- and two-electron QD in the limit of an infinitesimal magnetic field. (a) A one-electron QD with a parabolic potential has excited orbital states equally spaced by $E_{\mathrm{orb}}$ (only excitations along one dimension are shown for simplicity, and a small Zeeman splitting illustrates the spin degeneracy). (b) For two electrons, the total energy is increased by $E_{\mathrm{add}}$ and the lowest spin-singlet and spin-triplet eigenstates are shown with the combinations of the orbital wave functions $|g,e⟩$ that dominate each state. Note that antisymmetric spin singlets have spatially symmetric orbital wave functions, and vice versa, to satisfy the Pauli exclusion principle. The singlet triplet splitting $J$ is due to the triplet occupation of the excited orbital, though the energy of the latter is lowered from the one-electron orbital splitting by direct exchange $2\mathcal{J}$.

Figure 10

(a)–(c) Few electron single, double, and triple QDs ([98]; [146]; [456]). (d) Eight-site 1D QD array ([535]). (e) $3\times 3$ QD array ([360]). (f),(g) Si MOS single and DQD devices ([7]; [295]). (h) Donor device fabricated using STM lithography ([215]). (i) Single-layer etch-defined $1\times 6$ QD array in $\mathrm{Si}/\mathrm{SiGe}$ ([201]). (j) $1\times 9$ QD array fabricated using overlapping Al gates on $\mathrm{Si}/\mathrm{SiGe}$ ([576]). (k) Enhancement-mode $\mathrm{Ge}/\mathrm{GeSi}$ structure ([220]). Holes are confined in (k), while the remaining devices isolate electrons. Images are sized to share common dimensional scales.

Figure 11

(a) QPC charge detector to probe the charge occupation of a single QD ([158]). (b) rf QPC for fast sensing of a DQD ([423]). (c) Fast charge sensing of a DQD using a rf-QD charge sensor ([23]). (d) Donor device fabricated using STM lithography and probed using rf reflectometry ([266]). (e) cQED device for detecting charge and spin states in a cavity-coupled InAs nanowire DQD ([398]). (f) Dispersive gate sensing of charge states in a FinFET device ([189]).

Figure 12

(a) Spin-orbit interactions in QDs arise microscopically from the inversion asymmetries due to the bulk crystal structure (BIA), structural effects (SIA) like external fields, and interfaces (IIA). Under an applied magnetic field into the page, the local momentum of the electron wave function rotates (as depicted by arrows), causing local couplings to the atomic-scale gradients induced by these asymmetries that sum to the effective couplings in Eq. (7). (b) Effective spin-orbit field direction for Dresselhaus- and Rashba-type interactions as a function of in-plane momentum at the Fermi surface $k=k_F$; see Eq. (8).

Figure 13

(a) The valley splitting of donors in bulk Si from the admixture of the sixfold degenerate valleys (depicted in the Brillouin zone) leads to three sets of states. (b) In Si QDs, the electric fields in MOS and strain in $\mathrm{Si}/\mathrm{SiGe}$ raises the energy of the four in-plane valleys, and the relevant valley splitting is between the two out-of-plane valleys. (c) The admixture of valley states leads to rapidly varying modulations in the donor ground state, taken from an effective-mass calculation presented by [178]. (d) The full ground- and excited-state wave functions in Si QDs oscillate rapidly due to the intervalley phase. Interference of the valley Bloch functions minimizes the interface overlap for the ground state.

Figure 14

Energy levels, exchange coupling $J$, and wave functions in a DQD with two particles. (a) Two-particle energy levels as a function of level detuning $ϵ$. Tunnel coupling $t_c$ leads to level repulsion between the singlet states (blue curves) where the on-site Coulomb energy $U$ equals $\pm ϵ$. $J$ is the energy difference between the low-energy spin singlet and the spin triplets (red curves). Wave functions for (b) the symmetric (i.e., $ϵ=0$) and (c) detuned DQDs.

Figure 15

(a)–(d) Simulation of the change in the gray DQD potential (light shading) and pink electron density (dark shading) as the interdot barrier is lowered to increase exchange from approximately 10 kHz to 1 GHz. The potential is generated by solving the Poisson equation for a representative $\mathrm{Si}/\mathrm{SiGe}$ DQD, which is then used in FCI calculations to obtain the wave functions and $J$ ($K=30$ single-particle eigenstates are used to construct the basis). At practically useful multimegahertz levels of exchange, the electrostatic barrier vanishes and the electrons shift closer together, separated primarily by their Coulomb repulsion.

Figure 16

PSB in a DQD. (a) Starting with a $(1,0)$ charge state, $(2,0)$ singlet initialization occurs by biasing the left QD such that ${\mu }_{S_{(2,0)}}<E_F<{\mu }_{T_{(2,0)}}$. Qubit operations and readout are then performed by changing bias positions along the $(2,0)$-$(1,1)$ detuning axis. Readout is implemented by detuning such that the singlet ground state is $(2,0)$. Interdot tunneling is prohibited by PSB for the spin-triplet state (lower right panel), keeping it in the $(1,1)$ charge state and allowing spin-to-charge conversion. (b) Charge stability diagram in the vicinity of the $(2,0)$-$(1,1)$ anticrossing.

Figure 17

Various approaches for achieving long-range spin coupling. (a) Surface acoustic waves ([35]). (b) Charge transport ([177]). (c) Superexchange ([13]). (d) Spin swaps ([255]). (e) Spin CTAP ([198]). (f) Capacitive coupling ([466]). (g) Coupling through a spin chain ([54]).

Figure 18

Energy-dependent tunneling for single-spin initialization and readout of LD qubits. Note that the ground state in GaAs is $|\uparrow ⟩$ due to its negative electron $g$ factor. (a) $|\uparrow ⟩$ can be initialized by emptying the dot (top panel) and then applying a positive voltage pulse, such that $E_{\downarrow }>E_F>E_{\uparrow }$ (bottom panel). With $E_{\downarrow }>E_F>E_{\uparrow }$ an electron can tunnel only into the spin ground state. (b) After spin manipulations, spin readout is performed by pulsing back to the initialization bias condition. In this example, the presence (absence) of a tunneling event during the measurement period indicates $|\downarrow ⟩$ ($|\uparrow ⟩$).

Figure 19

Single-spin rotations driven with (a) an ac magnetic field generated by a coplanar waveguide ([283]), (b) an ac electric field in the presence of intrinsic SOC ([380]), and (c) an ac electric field in the presence of synthetic SOC (a magnetic-field gradient) ([502]; [410]). (d) Low-power EDSR in a field gradient can be achieved in the flopping-mode regime of a DQD ([30]; [111]).

Figure 20

Coherent exchange oscillations as first observed in a DQD using PSB readout for (a) GaAs ([403]) and (b) SiGe ([337]).

Figure 21

(a) Single-shot readout of a donor-bound electron spin ([359]). (b) Rabi oscillations of a donor-bound electron spin ([412]). (c) Exchange oscillations from a two-donor device. ([215]). (d) Coherence of three-qubit Greenberger-Horne-Zeilinger states ([327]).

Figure 22

(a) Energy-level diagram for two electrons in a DQD. $ϵ$ is the energy-level detuning, and $(1,1)$ and $(2,0)$ indicate the DQD charge configurations. The Zeeman and exchange splittings are $g{\mu }_{B}B$ and $J(ϵ)$, where $B$ denotes the magnetic field. The spin states are $|\mathrm{S}⟩=(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩)/\sqrt{2}$, $|{\mathrm{T}}_0⟩=(|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩)/\sqrt{2}$, $|{\mathrm{T}}_{+}⟩=|\uparrow \uparrow ⟩$, and $|{\mathrm{T}}_{-}⟩=|\downarrow \downarrow ⟩$. Singlet triplet oscillations driven by (b) $g$-factor differences between dots ([316]), (c) micromagnets ([558]), and (d) dynamic nuclear polarization ([163]).

Figure 23

Two-qubit operations in ${\mathrm{ST}}_0$ qubits. (a) Bell-state fidelity during a capacitive entangling operation between two ${\mathrm{ST}}_0$ qubits. From [466]. (b) Concurrence during a two-qubit operation between capacitively coupled, resonantly driven ${\mathrm{ST}}_0$ qubits. From [373].

Figure 24

Gradient-free exchange oscillations from an isotopically enhanced $\mathrm{Si}/\mathrm{SiGe}$ TQD ([148]). At significantly negative detunings, dots 2 and 3 are exchange coupled and exchange oscillations are geometrically interpreted as a qubit rotation about $\widehat{n}$ [see Fig. 2]; at less negative detunings, dots 1 and 2 are coupled and geometrically interpreted as rotation about $\widehat{z}$. Exchange increases exponentially with detuning. At $ϵ=-7\mathrm{mV}$, both exchange couplings are active, as would be required for operation in the RX regime.

Figure 25

(a) Nanowire spin-orbit qubit. From [365]. (b) Spin-orbit qubit in a Si MOS DQD. From [246]. (c) Carbon nanotube qubit. From [112]. (d) Four-qubit quantum processor based on holes in $\mathrm{Ge}/\mathrm{SiGe}$. From [220].

Figure 26

Fidelities of (a) single-qubit and (b) two-qubit gates in Si as evaluated by randomized benchmarking (RB). In each experiment, an initial qubit state is prepared, random sequences of $N$ random Clifford gates $C_j$ are applied, a single Clifford recovery $C_R$ is applied to each random sequence that would, in the absence of error, return the qubit or qubits to their initial state, and readout is performed. The initial-state probability is plotted as a function of $N$. The experimental data shown use different initialization, readout, and Clifford implementations, but in all cases a least-squares fit to an exponential decay with $N$ provides a fidelity benchmark. A measure of SPAM fidelity is indicated by where each decay starts and saturates. Ideally each $n\mathrm{Q}$ RB curve would saturate to return probability $1/2^n$ as $N\to \infty$, but leakage and SPAM errors generally lead to other saturation values. Note that two-qubit Clifford operations generally involve multiple two-qubit entangling, swap, and/or single-qubit gates.

Figure 27

Decoherence and relaxation mechanisms for spin qubits in semiconductor QDs or donors. Energy levels are shown as horizontal dashed lines. Decoherence and relaxation mechanisms (a)–(c) for a LD qubit and (d)–(f) for singlet triplet qubits in DQDs. (a) Spin relaxation through emission (absorption) of energy quanta (such as phonons) to or from the environment. (b) Charge noise, as represented by the squiggly lines (cyan) leading to a fluctuating confinement potential and electronic wave function. When SOI or a magnetic-field gradient (vertical arrows, in green) is present, charge noise leads to spin dephasing. (c) Electron spin dephasing due to the hyperfine coupling to nuclear spins. (d) Singlet triplet spin relaxation. (e) Charge noise affecting detuning $ϵ$. (f) Charge noise affecting interdot tunneling $t_c$.

Figure 28

Cavity transmission for a charge qubit coupled to a superconducting microwave resonator. (a) As the double-dot detuning is swept across the tunneling transition, the charge-qubit frequency comes into resonance with the cavity frequency. As a result of the strong coupling between the cavity and charge qubit, the system hybridizes, and two distinct transmission peaks separated by the vacuum Rabi splitting are observed. The eigenenergies of the uncoupled system are shown as dashed lines, and the eigenergies of the coupled system are indicated as solid lines. (b) Cavity transmission at two different values of detuning, with theoretical predictions overlaid. From [347].

Figure 29

Cavity transmission for a single-spin qubit coupled to a superconducting microwave resonator. As the magnetic field is swept, the spin qubit comes into resonance with the cavity at about 6.03 GHz, and the qubit-cavity coupling splits the cavity resonance into two hybrid spin-photon modes. The characteristic vacuum Rabi splitting indicates the strong-coupling regime. From [441].

Figure 30

Resonant cavity-mediated spin-spin interactions. Measured cavity transmission $A/A_0$ as a function of the in-plane field angle $\varphi$, showing an avoided crossing between both qubits and the resonator at the expected angle. From [45].

Figure 31

Cavity-mediated single-spin readout. (a) Cavity response as a function of magnetic field showing the single-spin resonance frequency. (b) Electron-spin-resonance line. (c) Pulse sequence to detect single-spin Rabi oscillations via the cavity transmission. (d) Rabi oscillations measured through the cavity dispersive shift. From [346].

Figure 32

Cavity-mediated coupling between a triple-dot resonant-exchange qubit and a transmon superconducting qubit. Here a superconducting cavity is driven near its resonance frequency of about 5.6 GHz. The $y$ axis indicates the drive frequency of the resonant-exchange qubit, and the $x$ axis corresponds to changes in the electrochemical potential of the middle dot, which changes the overall energy of the spin qubit. The color scale indicates the transmission through the cavity. The small dashed (black) lines indicate the eigenenergies of the system in the absence of coupling, and the large dashed (red) lines indicate the eigenenergies of the system, including the coupling. From [301].