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

Technical Summary: Coherent Electronic Raman Excitation of Valley-Orbit Split States in Phosphorus-Doped Silicon

Problem and Context
The electronic structure of dopants in semiconductors, particularly the valley-orbit splitting of donor ground states in multivalley materials like silicon, is critical for understanding electronic components and emerging fields like valleytronics. While the effective mass theory predicts a degenerate donor ground state, experimental observations in n-type silicon reveal a splitting into a singlet $1s(A_1)$, a doublet $1s(E)$, and a triplet $1s(T_1)$ due to intervalley mixing and central-cell corrections. Although incoherent electronic Raman scattering has previously identified the $1s(A_1) \to 1s(E)$ transition (approx. 13.1 meV), a detailed understanding of the ultrafast coherent dynamics, the role of carrier density, and the excitation of Raman-forbidden transitions remains limited.

Methodology
The authors employed a degenerate pump-probe technique using mid-infrared femtosecond pulses (0.62 eV, 40 fs duration) to investigate the coherent electronic Raman transition between the $1s(A_1)$ and $1s(E)$ states of phosphorus donors in crystalline silicon.

• Experimental Setup: Two monocrystalline silicon samples with different phosphorus concentrations (Q8S: high doping, $\sim 10^{17}-10^{18} \text{ cm}^{-3}$; O92: low doping, $\sim 10^{15}-10^{16} \text{ cm}^{-3}$) were cooled in a cryostat (12–295 K).
• Excitation Scheme: A strong-field infrared pump pulse generated a coherent wavepacket (linear superposition of bound electron eigenstates). A time-delayed probe pulse, linearly polarized at $45^\circ$ relative to the pump, monitored the evolution of this wavepacket via transient polarization anisotropy ($\Delta T/T_0$).
• Variables: The study systematically varied crystal temperature, pump pulse peak intensity, and crystal orientation relative to the pump polarization ([100] vs. [110]). Additionally, a pre-excitation pulse (1.2 eV, 170 fs) was introduced 100 ps prior to the pump to control the density of free carriers and study Coulomb screening effects.
• Analysis: The transient signals were analyzed using Fourier transforms and fitted with exponentially decaying cosine functions to extract oscillation amplitude, coherence time, transition energy, and initial phase.

Key Contributions and Theoretical Framework
The paper provides a theoretical derivation of the differential Stokes-Raman scattering cross-section for valley-orbit transitions within the effective mass approximation.

• Selection Rules: The theory confirms that the $1s(A_1) \to 1s(E)$ transition is Raman-allowed, while the $1s(A_1) \to 1s(T_1)$ transition is Raman-forbidden under standard conditions.
• Mechanism Identification: The authors distinguish between standard coherent Raman scattering and "displacive impulsive excitation," a mechanism where the system is excited to a higher electronic state, altering the equilibrium position and driving coherent oscillations even for forbidden transitions.

Results

1. Temperature Dependence:

   • Both oscillation amplitude and coherence time decreased with increasing temperature. This is attributed to the thermal population of the $1s(E)$ state (depleting the $1s(A_1)$ ground state) and increased incoherent thermal excitation.
   • Higher dopant concentrations (Q8S) resulted in faster decoherence due to increased wavefunction overlap between neighboring donors.
   • For [100] polarization, the transition energy remained stable up to ~30 K, corresponding to the $1s(A_1) \to 1s(E)$ transition.
   • For [110] polarization, above 30 K, the transition energy decreased, and the initial phase shifted, indicating a switch to a displacive impulsive excitation mechanism driving the Raman-forbidden $1s(A_1) \to 1s(T_1)$ transition. This was hypothesized to result from indirect dopant absorption generating electron-hole pairs.

2. Pump Intensity Dependence:

   • At 15 K, the oscillation amplitude increased with pump intensity until saturation. In the high-doped Q8S sample with [100] polarization, the amplitude rose nonlinearly before saturating, suggesting that all phosphorus dopants were coherently excited.
   • Higher pump intensities led to faster decoherence, attributed to high densities of electrons and holes generated via two-photon absorption.

3. Pre-excitation and Carrier Density:

   • Introducing a pre-excitation pulse increased the amplitude of coherent oscillations up to a threshold, after which it plateaued. This was attributed to the release of electrons trapped in deep-level defects (e.g., phosphorus-vacancy complexes) created by heavy doping.
   • The release of trapped electrons repopulated the $1s(A_1)$ ground state, enhancing the Raman signal.
   • Increased carrier density from pre-excitation led to stronger Coulomb screening, resulting in faster decoherence and slight broadening of the transition linewidth.

Significance
The paper demonstrates that ultrafast coherent time-resolved techniques can access bound electron dynamics in doped semiconductors with femtosecond resolution, offering a perspective distinct from traditional incoherent Raman or photoexcitation spectroscopy. The study successfully:

• Characterized the coherent wavepacket dynamics of valley-orbit split states as a function of temperature, doping, and pump intensity.
• Identified a mechanism allowing the excitation of Raman-forbidden transitions ($1s(A_1) \to 1s(T_1)$) via displacive impulsive excitation under specific polarization and temperature conditions.
• Showed that carrier pre-excitation can be used to manipulate the population of donor states by releasing electrons from deep-level traps, thereby enhancing coherent signals.

The authors conclude that this methodology provides a robust tool for studying electron dynamics in multivalley semiconductors, potentially enriching the field of valleytronics in bulk materials.