Coherent electronic Raman excitation of valley-orbit split states of phosphorus dopants in silicon

In this study, we demonstrate coherent optical excitation of the electronic Raman transition between the $1s\left(A_1\right)$ and $1s\left(E\right)$ split states of phosphorus donor in crystalline silicon. The dynamics of the generated wavepacket is characterized in the time domain using a degenerate pump-probe technique with mid-infrared femtosecond pulses via transient polarization anisotropy of the probe pulse. In addition, we study the role of resonantly excited carriers, and we show that the amplitude and coherence time of the electronic wavepacket depend on the pre-excited carrier density. Further, we demonstrate that under certain conditions, the Raman-type excitation changes to displacive impulsive excitation, which allows us to address the Raman-forbidden transition between $1s\left(A_1\right)$ and $1s\left(T_1\right)$.

💡 Research Summary

In this work the authors present a comprehensive study of coherent optical control of valley‑orbit split donor states in phosphorus‑doped silicon. The paper begins with a detailed theoretical background on the origin of the six‑fold degeneracy of the shallow donor ground state in silicon and how multivalley effective‑mass theory, central‑cell corrections, and valley‑orbit coupling lift this degeneracy into three distinct levels: the singlet 1s(A₁) ground state, the doublet 1s(E), and the triplet 1s(T₁). By expanding the donor wavefunction in terms of Bloch functions from each of the six equivalent X‑valleys and applying group‑theoretical analysis of the Td point group, the authors derive the symmetry‑adapted coefficients αj(ν) that dictate the selection rules for Raman transitions.

The Raman scattering formalism is then developed using second‑order time‑dependent perturbation theory. The interaction Hamiltonian is split into linear (H¹) and quadratic (H²) terms in the vector potential, and the differential Stokes‑Raman cross‑section is expressed in terms of a two‑photon matrix element M(2). By assuming that the dominant intermediate states lie in the valence band, the authors simplify M(2) to a form that depends on the bandgap Eg, the pump and Stokes photon frequencies, and the effective mass tensor components (m∥, m⊥). The resulting cross‑section (Eq. 22) shows that only the 1s(A₁)→1s(E) transition is allowed under the usual Raman polarization geometry, while the 1s(A₁)→1s(T₁) transition is symmetry‑forbidden.

Experimentally, the team uses a degenerate pump‑probe scheme with mid‑infrared femtosecond pulses (~12 µm wavelength, sub‑100 fs duration). The pump pulse, with peak intensities on the order of 10⁸ W cm⁻², excites the donor system non‑resonantly, creating a coherent superposition of the 1s(A₁) and 1s(E) states—a wavepacket that oscillates at the valley‑orbit splitting frequency (~13 meV). The probe pulse, delayed in time, interrogates the evolving wavepacket via transient polarization anisotropy: the orthogonal and parallel components of the probe’s electric field are detected on separate InGaAs photodiodes, and their difference yields a signal proportional to the coherence amplitude.

Key experimental findings include: (i) the wavepacket amplitude and dephasing time are strongly dependent on the density of carriers pre‑excited by an auxiliary above‑bandgap pulse; higher carrier densities lead to reduced amplitude and faster dephasing, indicating that free‑carrier scattering and screening accelerate loss of coherence. (ii) By increasing the pump intensity beyond a threshold, the excitation mechanism switches from a Raman‑type impulsive drive to a displacive (non‑Raman) impulsive drive. In this regime the donor electron cloud is suddenly displaced, breaking the symmetry that originally forbids the 1s(A₁)→1s(T₁) transition, and a weak but measurable signal corresponding to the triplet state appears. This observation demonstrates that strong‑field, non‑perturbative dynamics can lift Raman selection rules in solid‑state systems. (iii) The authors quantify the resonance enhancement factor R₁₂ = Eg²/(Eg²–(ħωL)²), showing that the Raman efficiency is maximized when the pump photon energy approaches, but does not exceed, the bandgap.

The paper concludes that coherent electronic Raman scattering provides a viable pathway for ultrafast manipulation of donor‑based quantum states in silicon, a platform of great interest for quantum information processing. The ability to generate, monitor, and control valley‑orbit superpositions on femtosecond timescales opens new avenues for spin‑valley qubits, electrically driven donor spin resonance, and the exploration of non‑linear light‑matter interactions in multivalley semiconductors. The demonstrated transition to a displacive regime further suggests that strong‑field optics can be employed to access otherwise forbidden transitions, expanding the toolbox for quantum control in silicon‑based architectures.