---
title: Donor Spin Qubits in Silicon
url: https://www.emergentmind.com/topics/donor-spin-qubits
type: topic
---

# Donor Spin Qubits in Silicon

Searching arXiv for recent and foundational papers on donor spin qubits in silicon and related donor-based platforms.
Donor spin qubits are qubits encoded in the spin degrees of freedom of donor impurities in semiconductors, most extensively group-V donors in silicon and, in related photonic proposals, singly-ionized chalcogen donors. In the canonical silicon realization, a neutral donor binds one electron whose spin is hyperfine-coupled to the donor nucleus, so the electron spin, the nuclear spin, or a composite electron–nuclear manifold can serve as the computational degree of freedom. The platform combines isotopically enriched $^{28}$Si, which suppresses nuclear-spin noise, with atomistic placement by scanning-tunnelling-microscope lithography or deterministic ion implantation, and it supports architectures based on single donors, donor molecules, exchange-coupled donor pairs, donor chains, flip-flop qubits, and photonic or resonator interfaces [2009.04081][1807.10290][2602.12248].

## 1. Physical basis and donor species

The fundamental object is a donor-bound electron in a semiconductor host. For a neutral group-V donor in silicon, the spin Hamiltonian in frequency units can be written as
$$
H = (\gamma_e S_z - \gamma_n I_z)\,B_0 + A\,S\cdot I + H_Q + (\gamma_e S_x - \gamma_n I_x)\,B_1\cos(2\pi f t),
$$
where $A$ is the isotropic hyperfine constant and $H_Q$ is the nuclear electric-quadrupole interaction for nuclei with $I>1/2$ [2009.04081]. In the high-field regime, the eigenstates approximate the product basis $\lvert \uparrow,m_I\rangle,\lvert \downarrow,m_I\rangle$, and ESR transitions occur at $\nu_e(m_I)=\gamma_e B_0 \pm (A/2)m_I$ [2009.04081].

Phosphorus remains the reference donor species. In isotopically enriched $^{28}$Si, $^{31}$P provides an electron bound state with Bohr radius $a_0\approx1.22$ nm and binding energy $\approx45$ meV, while its nuclear spin $I=1/2$ couples strongly to the bound electron with $A\approx117.5$ MHz or, in the review tabulation, $A=117.53$ MHz [2308.04117][2009.04081]. The same review lists $^{75}$As with $I=3/2$ and $A=198.35$ MHz, $^{121}$Sb with $I=5/2$ and $A=186.80$ MHz, $^{123}$Sb with $I=7/2$ and $A=101.52$ MHz, and $^{209}$Bi with $I=9/2$ and $A=1475.4$ MHz [2009.04081]. The heavy donors add quadrupole physics and nuclear electric resonance, while Bi combines a very large hyperfine interaction with strong Stark tunability [1504.06370].

The donor-qubit concept also extends beyond shallow group-V donors. In $^{28}$Si, singly-ionized chalcogen donors such as $^{77}$Se$^+$ bind one electron with binding energy $\simeq593$ meV and hyperfine constant $A/2\pi\approx1.66$ GHz, while parity-allowed $1s\!:\!A\leftrightarrow1s\!:\!\Gamma_7$ optical transitions at $\lambda\approx2.9\,\mu$m create a cavity-QED-compatible donor-spin platform [1606.03488]. This broader donor taxonomy matters because different donor species emphasize different control channels: shallow donors favor electrical and exchange-based control, whereas chalcogen donors emphasize optical initialization, readout, and photon-mediated coupling.

## 2. Qubit encodings and effective Hamiltonians

The simplest encoding uses the donor electron spin. In isotopically purified $^{28}$Si, a single substitutional P atom supports an electron spin qubit, while the nuclear spin provides a long-lived memory; one summary gives nuclear $T_2> \text{minutes}$ and electron $T_2\sim \text{milliseconds}$ for a single P donor in a $^{28}$Si host [1807.10290]. A complementary encoding uses the donor nucleus itself. For a single $^{31}$P donor near a Si/SiO$_2$ interface, the combined orbital, electron-spin, and nuclear-spin Hamiltonian allows the qubit splitting
$$
\delta_q \approx B_0\gamma_n + \tfrac12\langle A\rangle,
$$
with $\langle A\rangle$ tuned electrically through donor–interface hybridization [2001.10029].

Composite encodings widen the design space. In zero magnetic field, the phosphorus donor Hamiltonian reduces to $H\to A\,S\cdot I$, so the eigenstates become the singlet $\lvert F=0,m_F=0\rangle$ and triplet manifold $\lvert F=1,m_F\rangle$, with the $S\leftrightarrow T_0$ transition forming a clock transition because $\partial f_{S\to T_0}/\partial B_0|_{B_0=0}=0$ [1801.10265]. In donor-based flip-flop qubits, the computational basis is encoded in electron–nuclear states such as $\lvert \uparrow\Downarrow\rangle$ and $\lvert \downarrow\Uparrow\rangle$ when the electron is partially displaced toward an interface or shared within a donor double dot [2009.04081][2602.12248]. In the donor–donor flopping-mode proposal, a 2P–1P system yields an effective Hamiltonian
$$
H_{\rm flop}= \tfrac12\Omega_z\sigma_z + \epsilon\tau_z + t_c\tau_x + (\Delta\Omega_\parallel \sigma_z + \Delta\Omega_\perp \sigma_x)\,\tfrac{\tau_z}{4},
$$
with $\Delta\Omega_\perp$ providing the electrically driven flip-flop coupling and $\Delta\Omega_\parallel$ acting as the unwanted longitudinal gradient [2105.02906].

Multi-donor structures generate further encodings. Exchange-coupled donor pairs obey
$$
H = g_e\mu_B\mathbf{B}_0\!\cdot\!(\mathbf{S}_1+\mathbf{S}_2)+J\,\mathbf{S}_1\!\cdot\!\mathbf{S}_2+A(\mathbf{S}_1\!\cdot\!\mathbf{I}_1+\mathbf{S}_2\!\cdot\!\mathbf{I}_2),
$$
and in the weak-exchange regime $J<A$ they support frequency-selective conditional rotations [2006.04483]. Odd-sized donor chains can behave as a spin-$1/2$ “extended qubit” with low-energy gap $\Delta_c\approx(\pi^2/2N)J_c$, providing a transport bus between distant source and target donors [1602.07058]. A further extension uses nearby $^{29}$Si nuclei as a register around the donor electron; this register exploits the donor-induced “frozen-core” environment rather than treating all residual $^{29}$Si spins only as a decoherence source [1505.02057].

## 3. Placement precision and device fabrication

Two fabrication paradigms dominate: STM-defined donors and deterministic implantation. Scanning-tunnelling-microscope hydrogen-resist lithography permits patterning of exposed silicon dimers with a resolution below the lattice constant. In the 1P–2P addressability experiment, PH$_3$ dosing at room temperature followed by a $350\,^\circ$C anneal embeds exactly one P atom per three exposed dimers, while slightly larger lithographic patches incorporate pairs of P atoms into a single quantum-dot site. After incorporation, a 55 nm epitaxial silicon overgrowth encapsulates the donors, and delta-doped regions define reservoirs and gates; a nearby highly doped Si SET acts as both local charge sensor and reservoir [1807.10290].

Deterministic ion implantation trades atomic STM placement for CMOS compatibility and wafer-scale processing. In the PF$_2^+$ strategy, the molecule ion dissociates on impact so that one P and two F atoms are co-implanted at the same lateral coordinate, but the bystander F ions increase the ion-beam-induced-charge signal and therefore the single-ion detection confidence. The reported values are $c\approx98.6\%$ for $9$ keV P$^+$ alone and $c>99.95\%$ for PF$_2^+$ at $E_{\rm tot}=19.5$ keV, with placement uncertainty $\Delta x_P\approx8$ nm for P in PF$_2^+$ versus $\approx11.5$ nm for P$^+$ alone at the same detection confidence [2308.04117]. Secondary ion mass spectrometry and ESR further show that after donor activation anneal the F diffuses away from the active region and no $^{19}$F hyperfine splitting remains in the donor spectrum [2308.04117].

Counted heavy-donor implantation extends the same logic. Focused-ion-beam implantation of Sb$^{2+}$ at 120 keV, combined with a planar p–n diode detector adjacent to the quantum dot construction zone, produces $\sim3150$ electron–hole pairs per ion and a detector signal of $\sim8$ mV above baseline noise. The reported single-ion detection efficiency is $\gtrsim95\%$, and self-alignment to $\sim60$ nm wide polysilicon gates reduces the lateral uncertainty to $\lesssim15$ nm at 95% confidence [1512.04593]. A related hybrid donor–quantum-dot architecture proposed aligned single-ion implantation through a dynamic shadow mask with real-time single-ion detection, targeting $\lesssim50$ nm lateral precision relative to predefined quantum dots [1110.2228].

## 4. Initialization, control, and readout

The mature control stack is based on ESR, NMR, and spin-dependent tunnelling. In the implanted-donor review, electron spins are initialized by spin-selective tunnelling to a cold SET reservoir at $T_{\rm el}\sim100$ mK, with electron spin-to-charge conversion yielding fidelities $>90\%$. Nuclear spins are initialized and read out by hyperfine-selective mapping to the electron, giving repetitive near-QND readout with fidelities $>99.8\%$; coherent control uses ESR at $\nu_e\approx6$–40 GHz with $\pi$-pulses as short as 150 ns, neutral-donor NMR at $\sim10$–200 MHz with $\pi$-pulses $\sim25$–30 $\mu$s, and, for high-spin nuclei, nuclear electric resonance at $\sim5$–10 MHz [2009.04081].

Built-in frequency addressability has been demonstrated by hyperfine engineering. A 1P donor next to a 2P molecule at $16\pm1$ nm separation shows two hyperfine peaks separated by $96.5\pm2.5$ MHz for the single donor and three peaks with average splitting $261\pm10$ MHz for the donor molecule. Because the two qubits differ in ESR frequency by $\sim165$ MHz, the device achieves individual addressability without strong magnetic or electric field gradients or micromagnets; the same experiment used single-shot, energy-selective spin readout at $T\sim50$ mK and $B_0=1.35$ T with a $\pm20$ MHz chirp over 150 $\mu$s, and reported readout fidelity $>95\%$ [1807.10290].

Electrical control can replace oscillating magnetic fields. Wang et al. proposed all-electrical control in a 2P–1P donor double dot, where the difference in total hyperfine coupling across the $(1,0)\leftrightarrow(0,1)$ transition induces an electric dipole and electrically driven spin resonance. Near charge degeneracy, the single-photon Rabi frequency is
$$
\Omega_R=\frac{1}{2\hbar}\,\bigl|\Delta A'(E_0)\bigr|\,E_{ac},
$$
and the specific design example with $\Delta A'(E_0)\approx15$ MHz/(kV/m) and $E_{ac}=15$ kV/m gives $\Omega_R/2\pi\approx110$ MHz, corresponding to $\pi$-pulses in $\sim5$ ns [1703.05370]. A related donor–donor flopping-mode proposal uses multi-donor occupation with antiparallel nuclear spins to suppress the longitudinal magnetic-field gradient and reports $\pi/2$-$X$ gate error rates of $\sim10^{-4}$ under realistic noise models [2105.02906].

Optical approaches provide a distinct control layer. Zero-field optical magnetic resonance on phosphorus donors in enriched silicon exploits donor-bound-exciton transitions to hyperpolarize the donor and access the $S\leftrightarrow T_0$ clock transition, with initialization, manipulation, and readout fidelities $\sim99.9\%$ and Hahn echo coherence around $10$ s [1801.10265]. For group-V donors more generally, far-IR and near-IR spin-selective transitions have been analyzed for P, As, Sb, and Bi, including optical pumping via $1sA_1\to1sT_2$ and two-photon $\Lambda$ schemes through donor-bound excitons [1507.07929]. In the chalcogen-donor platform, optical pumping yields near-unit hyperpolarization in $\sim50$ ms at 4.2 K and cavity-QED readout is projected to exceed $99.9\%$ fidelity without exciting the donor [1606.03488].

## 5. Coherence, relaxation, and measurement backaction

Coherence metrics depend strongly on isotopic purity, donor environment, and encoding. For single P donors in natural Si, one summary gives $T_2^*\approx55$ ns with linewidth $\sim10$ MHz, extended to $>100\,\mu$s in $^{28}$Si, while electron $T_1>1$ s at low temperature [1807.10290]. In a PF$_2$-implanted $^{28}$Si qubit device, the measured electron coherence values are $T_2^*=20.5\pm0.5\,\mu$s and $T_2^{\rm Hahn}=424\pm5\,\mu$s, comparable to previous P-implanted devices and with no detected nearby $^{19}$F nuclear spins [2308.04117]. Ionized donors in bulk $^{28}$Si reach much longer nuclear times, with $T_2^*\sim T_2^{\rm Hahn}\gg100$ ms and $T_2^{\rm Hahn}\sim$ seconds under clock-transition conditions [2308.04117].

Donor molecules and donor–interface coupling introduce new relaxation channels. In natural Si, the 2P donor molecule exhibits linewidth $\sim72$ MHz and apparent dephasing $T_2^*<10$ ns because the larger envelope overlaps more $^{29}$Si nuclei, although isotopic purification is expected to restore ms-scale coherence as in single donors [1807.10290]. Near the Si/SiO$_2$ interface, donor states hybridized with interface orbitals show phonon-assisted spin relaxation with $\Gamma\propto B^5$ when the applied field is weak compared to the orbital spacing, together with spin-relaxation hot-spots where orbital states with opposite spin strongly hybridize and cool-spots where different relaxation channels interfere destructively [1710.07674].

Long-lived nuclear subspaces can be built around the donor rather than against it. Wolfowicz et al. showed that nearby $^{29}$Si nuclei in the donor “frozen core” reach coherence times in the second timescale: the donor nucleus has $T_2=1.10\pm0.10$ s, a nearby $^{29}$Si with $A=4.03$ MHz has $T_2=1.22\pm0.03$ s, and CPMG with $N=8$ extends $T_2$ to $3.7\pm0.2$ s [1505.02057]. Chalcogen donors provide another clock-transition route, with $T_1>6$ min, $T_2^*\approx1$ ms, and $T_2^{\rm Hahn}=2.14\pm0.04$ s at low field [1606.03488].

Measurement itself can limit nuclear lifetimes. In multi-donor qubits occupied by a single electron, electron tunnelling during readout turns the hyperfine interaction on and off non-adiabatically, giving single-spin flip probabilities $P_{j,\rm flip}\propto A_j^2/B^2$ and an additional nuclear spin flip-flop mechanism specific to multi-donor dots. Monir et al. show that increasing the hyperfine difference $\Delta A=\lvert A_1-A_2\rvert$ suppresses the flip-flop probability as $P_{\rm ff}\propto1/(\Delta A)^2$, and that the engineered-Stark 2P lifetimes exceed those of 1P once $\Delta A\gtrsim30$ MHz [2310.12656]. This corrects the common assumption that adding donors necessarily worsens nuclear backaction.

## 6. Coupling, transport, and scalable silicon architectures

Two-qubit logic in donor systems is not limited to the strong-exchange regime. In a MOS-compatible silicon device with two exchange-coupled implanted $^{31}$P donors, the measured coupling is $J=32.06\pm0.06$ MHz, while the individual hyperfine couplings are $A_t=97.75$ MHz and $A_c=87.57$ MHz. Because $J<A$, the target ESR line depends on the control-spin state, and a native controlled-rotation gate is obtained by a simple ESR pulse; the reported $\pi$-pulse durations are $\tau_\pi\approx60$–250 ns, with the gate remaining functional for any $J$ in the interval $[0.01\ \text{MHz},100\ \text{MHz}]$ [2006.04483]. This weak-exchange operating mode is explicitly designed to reduce sensitivity to donor-placement variation.

Hyperfine engineering provides an orthogonal route to addressability. The 1P–2P device cited above achieves $\sim165$ MHz built-in ESR detuning at only $16\pm1$ nm spacing, and the same work argues that donor molecule size and geometry can provide a library of distinct resonance frequencies; one stated example is that four distinct 2P configurations plus a single donor could yield five qubits addressable within a $\sim1$ GHz ESR bandwidth [1807.10290]. A plausible implication is that exchange-based connectivity and frequency-based addressability need not be engineered by the same physical knob.

Donor chains address on-chip transport. Mohiyaddin et al. modeled odd-sized chains with nearest-neighbor Heisenberg exchange and showed that operating in interface mode greatly relaxes donor placement tolerances because the exchange dependence becomes less steep. For $\sigma_d=2$ nm, interface mode permits a 7-donor chain $\sim115$ nm long at 90% yield, whereas bulk-like mode permits only $\sim13$ nm. Their adiabatic transport protocol uses $J_{\max}/h\sim1$–10 GHz and $T_{\rm AP}\sim10$–100 ns, and following the stated guidelines yields spin-state transport over $\sim100$ nm with fidelity $>99.9\%$ under realistic noise and fabrication constraints [1602.07058].

Long-range couplers increasingly rely on electric dipoles and resonators rather than direct exchange. For donor nuclear qubits, electrically induced donor–interface dipoles enable dipole–dipole cZ gates in $\sim500$ ns at $r=0.5\,\mu$m, while single-qubit $R_z(\pi)$ gates take $\approx14$ ns and composite $R_x(\pi)$ gates reach fidelity $>99.9\%$ at $\sigma_E=100$ V/m [2001.10029]. For donor-based flip-flop qubits coupled to superconducting resonators, the central trade-off is between spin–charge admixture and decoherence: with $\omega_r/2\pi=6.5$ GHz and $\omega_B/2\pi=6.6$ GHz, the simultaneous regime $g_s/\kappa\ge1$ and $SNR^2\ge282$ for $F\ge99\%$ is narrowly realized for $V_t\approx10$–16 GHz with sufficient $g_c$ and low $\kappa$, and squeezed input fields can mitigate charge-photon-coupling and photon-loss constraints [2602.12248].

## 7. Extensions beyond the standard phosphorus-in-silicon setting

The donor-spin category includes broader material and control proposals that illuminate the general design space. Atomistic tight-binding studies of P, As, Sb, and Bi donors under strain and electric field show that a hybrid control scheme based on (001) compressive strain and in-plane (100 or 010) fields results in higher gate fidelities and/or faster gate operations for all four donor species, while only Bi is predicted to benefit similarly from both in-plane and out-of-plane fields [1504.06370]. These calculations also show donor-dependent Stark curvatures and saturation behaviors of the hyperfine interaction, emphasizing that “donor spin qubit” is not a single fixed Hamiltonian but a family of donor-specific operating points.

Photonic donor platforms relax the requirement of atomically precise inter-donor spacing. In $^{28}$Si, singly-ionized chalcogen donors offer mid-IR electric-dipole transitions, optical initialization, $T_1>6$ min, $T_2^{\rm Hahn}=2.14\pm0.04$ s, cavity-QED single-shot readout projected above $99.9\%$, and coupling variations below 10% for donor-placement uncertainty plus annealing $\lesssim80$ nm; the same proposal notes that no atomically precise donor spacing is required because the optical interface tolerates $\sim100$ nm placement error [1606.03488]. This stands in deliberate contrast to exchange-only schemes.

Non-silicon donor-spin proposals test how much of the silicon donor toolkit is host-independent. In germanium, shallow donor spins in quasi-2D phononic crystals have been proposed as qubits whose Zeeman splitting lies inside a phonon bandgap, suppressing one-phonon relaxation so that $T_1=\nu_1^{-1}\gtrsim10^2$ s and enabling virtual-phonon-mediated couplings with $J/(\hbar\nu_1)\gtrsim10^5$ in structures with unit-cell sizes of 100–150 nm [1409.6285]. In ZnO, indium ion implantation followed by annealing produces neutral donors with donor-bound-exciton linewidth less than 10 GHz, hyperfine coupling $A/h\approx50$ MHz, and longitudinal relaxation times reaching $T_1\gtrsim20$ s at $B=1$ T, while two-laser CPT resolves the $I=9/2$ In nuclear manifold [2212.05230]. These systems remain less developed than the silicon platform, but they underscore that donor-spin qubits are a materials class rather than a single-device format.

Across these implementations, a recurring misconception is that donor-spin scalability is synonymous with atomically tuned exchange between nearest neighbors. The literature instead supports several non-exclusive strategies: weak-exchange conditional ESR, hyperfine-engineered donor molecules, interface-mode donor chains, donor–interface dipoles, optical chalcogen nodes, and superconducting-resonator coupling [2006.04483][1807.10290][1606.03488]. This suggests that the central problem in donor-spin quantum engineering is not merely donor placement, but the joint optimization of hyperfine structure, orbital admixture, readout backaction, and the coupling mechanism chosen for scale-up.

Source: https://www.emergentmind.com/topics/donor-spin-qubits