rethfeld_2004#
B. Rethfeld, “Unified Model for the Free-Electron Avalanche in Laser-Irradiated Dielectrics”, Phys. Rev. Lett. 92, 187401 (2004)
Source record · Catalogue · Review register
DOI: 10.1103/physrevlett.92.187401
Source kind: theory
Recorded technique: The multiple rate equation (MRE, Eq. 2): conduction electrons on discrete energy levels spaced by the photon energy, climbing by one-photon intraband absorption (probability W_1pt) to the critical energy for impact ionization, then impact-ionizing (probability alpha~); analytic asymptotic solution (Eqs. 4-7) and numerical examples compared with the kinetic calculation of Kaiser et al. (2000) and the single rate equation.
Materials in structured entries: SiO2
Dataset quantities: avalanche_coefficient, band_gap, effective_mass
Contents: 4 datasets; 3 formulas.
Curation notes#
Journal pages 187401-1 to -4 = PDF pages 1-4 (‘publisher error corrected 11 May 2004’ on page 1). Identity validated.
MATERIAL. The paper names no material for its examples: ‘For the following examples we choose parameters similar to those applied in Ref. [6] in order to allow direct comparison with the kinetic approach’ (Kaiser et al., Phys. Rev. B 61, 11437 (2000)), and ‘W_1pt = 3.5 x 10^-7 E_L^2 m^2/V^2 s ... compares well with the mean value of the one-photon absorption probability for SiO2 in Refs. [6,15]’ (p. 187401-3). The inputs are recorded for SiO2 on that basis (assumed, check): E_gap = 9 eV; m_VB = m_CB = the free-electron mass (so mu = m_e/2); alpha~ ~ 1 x 10^15 s^-1 ‘estimated from the corresponding collision term given in Ref. [6]’. The example wavelength is 500 nm (hbar omega = 2.48 eV), fields up to 1e10 V/m.
CHECKS (2026-09-26): hbar omega = 2.4797 eV at 500 nm; eps_crit = (1 + mu/m_VB)(E_gap + <eps_osc>) with <eps_osc> = e^2 E_L^2/(4 mu omega^2) is 13.50 eV at E_L -> 0 and 14.43 eV at 1e10 V/m, ‘between 13.5 and 14.5 eV’; k = ceil(eps_crit/hbar omega) = 6, ‘k + 1 = 7 equations’. From Eq. (7) with k = 6: alpha_asymp = 1.07e12 s^-1 and t_MRE = 933 fs at 50 MV/cm; 2.1e12 s^-1 and 476 fs at 70 MV/cm (not printed; Figs. 2-3 show the evolution; the text says the stationary regime takes ‘several hundreds of femtoseconds’).
FIGURES ONLY: Fig. 2 (alpha~ n_k/n_total against time for several fields, with Kaiser’s kinetic result at 50 MV/cm), Fig. 3 (the impact-ionization share of the density for a 300-fs, 70 MV/cm Gaussian pulse by four models).
REPRODUCED BY: gulley_2012 (the MRE for the carrier density), couairon_2005 (cited as refinement), jing_2012 (cited), bourgeade_2012 (cites Rethfeld 2006, Phys. Rev. B 73, 035101, not in the library).
Recorded bibliographic identity status: VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
rethfeld_2004/model_band_gap#
{
"id": "model_band_gap",
"material": "SiO2",
"table": "text",
"quantity": "band_gap",
"unit": "eV",
"abscissa": "none",
"use": "check",
"parameters": {
"kind": "assumed",
"assumed": true
}
}
1 inline points in the source record.
p. 187401-3: ‘We assume the effective masses m_VB and m_CB both equal to the free-electron mass and a bandgap of E_gap = 9 eV.’ The material is not named (source notes).
rethfeld_2004/model_electron_mass#
{
"id": "model_electron_mass",
"material": "SiO2",
"table": "text",
"quantity": "effective_mass",
"unit": "m_e",
"abscissa": "none",
"use": "check",
"parameters": {
"carrier": "electron",
"assumed": true
}
}
1 inline points in the source record.
p. 187401-3: m_CB equal to the free-electron mass (conduction band).
rethfeld_2004/model_hole_mass#
{
"id": "model_hole_mass",
"material": "SiO2",
"table": "text",
"quantity": "effective_mass",
"unit": "m_e",
"abscissa": "none",
"use": "check",
"parameters": {
"carrier": "hole",
"assumed": true
}
}
1 inline points in the source record.
p. 187401-3: m_VB equal to the free-electron mass (valence band); the reduced mass mu = m_e/2 enters eps_crit and <eps_osc>.
rethfeld_2004/model_impact_ionization_probability#
{
"id": "model_impact_ionization_probability",
"material": "SiO2",
"table": "text",
"quantity": "avalanche_coefficient",
"unit": "1/s",
"abscissa": "none",
"use": "check",
"reproduced_from": [
"A. Kaiser, B. Rethfeld, M. Vicanek and G. Simon, Phys. Rev. B 61, 11437 (2000)"
],
"parameters": {
"definition": "alpha~: impact-ionization probability per unit time of the electrons above eps_crit in the MRE, Eq. (2)",
"assumed": true
}
}
1 inline points in the source record.
p. 187401-3: ‘The probability of impact ionization alpha~ can be estimated from the corresponding collision term given in Ref. [6] and lies in the range of alpha~ ~ 1 x 10^15 s^-1. As long as alpha~ >> W_1pt, the exact value of alpha~ plays no role for the total free-electron density’. Not the avalanche coefficient of the single rate equation: alpha = alpha~ n_k/n_total.
Formulas#
rethfeld_2004/multiple_rate_equation#
{
"id": "multiple_rate_equation",
"material": "SiO2",
"equation": "Eqs. (2)-(3), p. 187401-2",
"printed": "dn_0/dt = n_pi + 2 alpha~ n_k - W_1pt(eps_0) n_0; dn_1/dt = W_1pt(eps_0) n_0 - W_1pt(eps_1) n_1; ...; dn_(k-1)/dt = W_1pt(eps_(k-2)) n_(k-2) - W_1pt(eps_(k-1)) n_(k-1); dn_k/dt = W_1pt(eps_(k-1)) n_(k-1) - alpha~ n_k, with k = ceil(eps_crit/(hbar omega_L)) (2); dn_total/dt = n_pi(E_L) + alpha~ n_k (3)",
"variable": "n_j: density of electrons at eps_j = eps_0 + j hbar omega_L; n_pi the photoionization rate (Keldysh, ref. 7); W_1pt the one-photon intraband absorption probability; alpha~ the impact-ionization probability",
"coefficients": {
"W_1pt": "3.5e-7 E_L^2 m^2/(V^2 s)",
"alpha_tilde": "~1e15 s^-1",
"E_gap": "9 eV",
"m_VB": "m_e",
"m_CB": "m_e"
},
"check": "k + 1 = 7 at 500 nm for fields up to 1e10 V/m reproduced (k = 6; checked 2026-09-26)."
}
The single rate equation dn_total/dt = n_pi + alpha n_total (Eq. 1) is the special case of a stationary distribution, alpha = alpha~ n_k/n_total. n_total is printed as sum_{j=1}^{k} n_j (from j = 1, as printed).
rethfeld_2004/asymptotic_avalanche#
{
"id": "asymptotic_avalanche",
"material": "SiO2",
"equation": "Eqs. (5)-(7), p. 187401-2",
"printed": "n_k(t) = n_pi / (2k (1 - (1/2)^(1/k)) alpha~) exp[(2^(1/k) - 1) W_1pt t] + ... (5); n_total(t) = (n_pi/W_1pt) / (2k (2^(1/k) - 2 + (1/2)^(1/k))) exp[(2^(1/k) - 1) W_1pt t] for t >> t_MRE = [(2^(1/k) - 1) W_1pt]^-1 (6); alpha_asymp = (2^(1/k) - 1) W_1pt (7)",
"variable": "k: the number of photons to reach eps_crit; W_1pt constant in energy; valid for alpha~ >> W_1pt",
"coefficients": {
"W_1pt": "3.5e-7 E_L^2 m^2/(V^2 s)"
},
"check": "alpha_asymp -> ln(2) W_1pt/k as k -> infinity (p. 187401-4), as the formula implies; with k = 6: 1.07e12 s^-1 at 50 MV/cm (computed, not printed)."
}
The common estimate alpha_est = W_1pt hbar omega_L/eps_crit (or /E_gap) is compared in Fig. 3; the MRE corrects it by the factor ln 2 in the long-time limit.
rethfeld_2004/critical_energy#
{
"id": "critical_energy",
"material": "SiO2",
"equation": "Eq. (8), p. 187401-3",
"printed": "eps_crit = (1 + mu/m_VB)(E_gap + <eps_osc>), <eps_osc> = e^2 E_L^2/(4 mu omega_L^2)",
"variable": "mu: the reduced mass of m_VB and m_CB; E_L the laser field amplitude; omega_L the laser frequency",
"coefficients": {
"E_gap": "9 eV",
"m_VB": "m_e",
"m_CB": "m_e"
},
"reproduced_from": [
"A. Kaiser, B. Rethfeld, M. Vicanek and G. Simon, Phys. Rev. B 61, 11437 (2000)"
],
"check": "13.50 eV (E_L -> 0) to 14.43 eV (1e10 V/m) at 500 nm, printed 'between 13.5 and 14.5 eV' (checked 2026-09-26)."
}
The first factor accounts for momentum conservation (parabolic bands, ref. 8, Keldysh 1959).
Corrections#
No dataset errata are recorded in this source record.
Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the review register for the checks and decisions.