Kinetics of electron-phonon scattering in silicon resolved by Rydberg transitions of donors
Nils Dessmann, Aidan G. McConnell, Sergey G. Pavlov, Guy Matmon, Gabriel Aeppli, Nikolay V. Abrosimov, Hari Paudyal, Michael E. Flatté, Benedict N. Murdin
quant-phcond-mat.mes-hallSubmitted Jun 27, 2026• Announced Jun 30, 2026
Rydberg states of atoms in vacuum are now well recognized as a resource for quantum technologies. Donors in semiconductors also display analogous states, which have been proposed for similar applications. While they benefit from permanent locations in their host crystals, electron-lattice coupling leads to much shorter excited-state lifetimes than for neutral atoms in vacuum. Here we provide a quantitative description of donor-phonon kinetics, creating a basis for engineering donor systems in realistic material stacks for quantum devices. Our theory incorporates both form factors for the Rydberg states, which given their large extents in real space provide strong selectivity in momentum space, and tabulated deformation potentials for all six phonon branches throughout the Brillouin zone. By confronting this framework with carefully controlled time-resolved free electron laser measurements, we show that the widely quoted position of the silicon conduction band minimum, $k_0$, is inconsistent with observed donor relaxation rates and that quantitative agreement is obtained for a value further from the X-point than commonly assumed. This stringent experiment-theory comparison establishes donor relaxation as a precision metrology for conduction band parameters and scattering processes in silicon, with consequences spanning from quantum devices to classical electronics.
Rydberg states of atoms in vacuum are now well recognized as a resource for quantum technologies. Donors in semiconductors also display analogous states, which have been proposed for similar applications. While they benefit from permanent locations in their host crystals, electron-lattice coupling leads to much shorter excited-state lifetimes than for neutral atoms in vacuum. Here we provide a quantitative description of donor-phonon kinetics, creating a basis for engineering donor systems in realistic material stacks for quantum devices. Our theory incorporates both form factors for the Rydberg states, which given their large extents in real space provide strong selectivity in momentum space, and tabulated deformation potentials for all six phonon branches throughout the Brillouin zone. By confronting this framework with carefully controlled time-resolved free electron laser measurements, we show that the widely quoted position of the silicon conduction band minimum, k0, is inconsistent with observed donor relaxation rates and that quantitative agreement is obtained for a value further from the X-point than commonly assumed. This stringent experiment-theory comparison establishes donor relaxation as a precision metrology for conduction band parameters and scattering processes in silicon, with consequences spanning from quantum devices to classical electronics.
One of the great successes of quantum physics is the description of the long-lived Rydberg states of atoms and ions. The Bohr model is equally applicable to donor impurity atoms in semiconductor physics, where the conduction band corresponds to the vacuum, and the loosely bound electron orbiting a singly charged core has a hydrogen-like spectrum according to the usual Bohr–Sommerfeld formula, shifted to the far-infrared because of the small effective mass and high dielectric constant. Manipulation of Rydberg states in free atoms and ions by single and multiphoton processes has been tremendously productive since the development of pulsed visible laser spectroscopy. The analogous manipulations have not been conducted for donor impurities in silicon. Here, we use the FELIX pulsed free electron laser to perform time-domain measurements of the Rydberg state dynamics in phosphorus- and arsenic-doped silicon and we have obtained lifetimes consistent with frequency domain linewidths for isotopically purified silicon. This implies that the dominant decoherence mechanism for excited Rydberg states is lifetime broadening, just as for atoms in ion traps. The experiments are important because they represent a step toward coherent control and manipulation of atomic-like quantum levels in the most common semiconductor and complement magnetic resonance experiments in the literature, which show extraordinarily long spin lattice relaxation times—key to many well known schemes for quantum computing qubits—for the same impurities. Our results, taken together with the magnetic resonance data and progress in precise placement of single impurities, suggest that doped silicon, the basis for modern microelectronics, is also a model ion trap.
Several solid-state quantum information processor proposals are based on silicon [16-18, 20, 52], including the original architecture conceived by Kane in 1998 [22], in which the quantum information was stored in the spin states of 31P donors. The interest raised by silicon in the field of quantum information is twofold. On the one hand, it represents the main semiconductor used for large- scale fabrication. On the other hand, it owns interesting physical properties suitable for preserving andmanipulating quantum states, such as long coherence times and relatively low disturbance of nuclear spins, which are further enhanced by employing purified 28Si. The employment of individual donors in silicon covers all the aspects of quantum information encoding and processing, like quantumbit (qubit) storage, implementation of quantum logic gates, coherent transfer of quantum states, and readout schemes. This chapter is devoted to the physics of few dopants in nanometric silicon devices and their applications to quantum information processing purposes. In the first part, an overview of the physics of themost common donors and acceptors in silicon is presented. Next, the quantum information key concepts and their implementation in donor-based architectures are discussed. Finally, decoherence effects and the concept of measurement of quantum states are presented for the case of electron states bound to donors in silicon. Between the sixties and the early eighties of the twentieth century, all the theoretical and experimental aspects related to the doping of bulk silicon have been investigated and explained. A silicon crystal consists of a diamond lattice constituted by two interleaved cubic lattices (face-centered cubic, FCC), the second of which has the origin in the center of the tetrahedron given by the origin (0,0,0) and the centers of the faces 100. Each Si atom has four valence electrons, which create a covalent bond with the shared electrons of four other Si atoms. The substitution of silicon atoms with atoms of a group V element (indicated as donors, typically P, As, and Sb in silicon) generates an n-doped silicon crystal (n-Si). Each donor introduces an eccess electron in the crystal, and it provides new energy levels in the band gap, close to the conduction band edge. Similarly, substituting silicon atoms with atoms of a group III element (an acceptor, typically B in silicon) generates a p-doped silicon crystal (p-Si). Each acceptor introduces an electron hole in the crystal, and it provides new energy levels in the band gap, close to the valence band. The diffusion of impurities in semiconductors alters the conduction properties from an insulating to a metallic regime. Depending on the different doping concentration, the wavefunction of the electron (hole) states introduced by each donor (acceptor) may overlap negligibly or substantially with those of neighboring sites. Such a transition from low doping to high doping as a function of the average distance between neighboring sites is described in terms of an Anderson-Mott transition and produces additional impurity bands (Hubbard bands) below the conduction band edge at sufficiently high concentration [1, 34]. The transport is governed by mechanisms based on localized states at low density, while it is based on delocalized states at high density. Four conventional regimes of impurity concentration are defined [50]. The dilute concentration of impurities holds for n < 1 · 1016 cm−3. There, the problem of the donor is that of a hydrogen atom with a scaled Rydberg and radius. Between the densities n of 1 ·1016 cm−3 and 2 · 1017 cm−3 (rNN = 11.9 nm), respectively, the regime is called semidilute, and it is characterized by formation of pairs. Above 2 · 1017 cm−3 and below the metallic behavior, which occurs at 3.7 · 1018 cm−3, the regime is called intermediate, and the formation of random clusters leads to effects generally accounted by the Hubbard band formation. Above n > 3.7 · 1018 cm−3, silicon is treated as a metal. In this section the physics of the first three regimes of donor concentration and the Anderson-Mott transition are described. The Anderson-Mott transition has been observed down to microscopic scale by employing arrays of few deterministically implanted As ions (see chapter 5) in Si transistors [39]. Analogous arguments and treatment can be given to acceptor concentration regimes.
It has long been known that shallow donors such as phosphorous and the other group-V elements, have a hydrogen-like optical spectrum. The main difference is that while the spectrum of atomic hydrogen lies in the visible band, the spectrum of shallow donors in silicon is downshifted to the THz frequency band. This is a direct consequence of the reduced Coulomb attraction seen by the loosely bound electron because the core electrons shield the positive donor atom nucleus, and because the electron is now moving in a dielectric material. While spectroscopy has already revealed much about the energy level structure, very little was known about the temporal dynamics of the system until now. We have used THz pulses from the FELIX free electron laser to probe these hydrogen-like levels. By exploiting the well-known pump-probe technique we have measured the characteristic lifetimes of the excited Rydberg states and found them to be of the order 200 ps. Then, by making subtle changes to the geometry of the pump-probe experimental setup we demonstrate the existence of a THz photon echo. The photon echo is a purely quantum phenomenon with no classical analogue, and it allows us to study the quantum state of the donor electron. We then show, using the photon echo, that it is possible to create a coherent superposition of the ground and excited state of the donor. Measuring the photon echo is important because it can also be used to measure a second important characteristic lifetime of the silicon-donor system, the phase decoherence time.
Gate control of donor electrons near interfaces is a generic ingredient of donor-based quantum computing. Here, we address the question: how is the phonon-assisted qubit relaxation time T 1 affected as the electron is shuttled between the donor and the interface? We focus on the example of the 'flip-flop qubit' (Tosi et al arXiv:1509.08538v1), defined as a combination of the nuclear and electronic states of a phosphorus donor in silicon, promising fast electrical control and long dephasing times when the electron is halfway between the donor and the interface. We theoretically describe orbital relaxation, flip-flop relaxation, and electron spin relaxation. We estimate that the flip-flop qubit relaxation time can be of the order of 100 μs, 8 orders of magnitude shorter than the value for an on-donor electron in bulk silicon, and a few orders of magnitude shorter (longer) than the predicted inhomogeneous dephasing time (gate times). All three relaxation processes are boosted by (i) the nontrivial valley structure of the electron-phonon interaction, and (ii) the different valley compositions of the involved electronic states.
Real-time time-dependent density functional theory provides a first-principles description of coherent electron dynamics in laser-driven solids, but its unitary formulation cannot capture the irreversible scattering, relaxation, and decoherence processes that drive excited carriers toward equilibrium. Here, we develop a dissipative rt-TDDFT framework in which first-principles electron-phonon interactions enter the evolution of the reduced one-body density matrix through self-energy-derived collision integrals within the Born-Markov approximation. The approach retains the quantum-coherent real-time propagation of the electronic system while introducing phonon-mediated transitions that redistribute carriers in energy and crystal momentum, thereby incorporating the microscopic momentum-transfer processes responsible for relaxation in real materials. The resulting framework provides a practical first-principles route to simulate relaxation, decoherence, and time-resolved spectroscopic signatures in realistic crystalline materials.
In this thesis the reasons for pursuing an optically addressable quantum object
in silicon are outlined before introducing some possible technological applications.
Chalcogen donor electron systems in silicon are then discussed as a candidate for
these quantum technologies. The chalcogen donor complexes in silicon allow the
solid-state analogues of hydrogen and helium to be studied. The time-resolved
Fourier transform technique was employed to measure the photoluminescence from
a selenium doped silicon sample when pumped above band gap at the temperatures
10 K, 80 K and 300 K. The excited state lifetime of the 2p0 state was found to
be 28.2 ns for atomic centres and 66.4 ns for molecular centres at 10 K. The lifetimes
were seen to be similar across large emission bands indicating a rate limiting
step. A multi-colour pump-probe experiment, using time-resolved Fourier transform
spectroscopy, was performed at the FELIX institute to determine the excited state
lifetimes of the 2p0 and 1s(T2) to be <2.5 ns and 10.62 ns respectively. A system of 5
coupled rate equations was then implemented to verify the experimental results. The
lifetimes were theoretically determined from three sets of initial conditions. Carriers
were initially positioned in the conduction band, modelling the photoluminescence
experiment, the 2p and 1s states, imitating the FELIX experiment. The 2p state
was shown to have two decay lifetimes, one 15.1 ps and the second of 7.24 ns. The
first fast decay is due to the pure 2p state relaxation while the second long lifetime is
a result of the 1s(T2) state in
uencing the 2p excited state lifetime. The 1s(T2) excited
state lifetime that back-fills the 2p state elongates the 2p excited state lifetime,
causing the lifetime to appear longer than the true lifetime. This research implies
that silicon has the potential to be an accessible test bed for quantum experiments
by showing that it could be possible to create isolated non-interacting donor centres
that are individually optically addressable by mid-IR laser light.