rethfeld_2004 ============= B\. Rethfeld, "Unified Model for the Free\-Electron Avalanche in Laser\-Irradiated Dielectrics", Phys\. Rev\. Lett\. 92, 187401 \(2004\) `Source record `__ · :doc:`Catalogue ` · :doc:`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 \+ \\) with \ = 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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 \\. rethfeld\_2004/model\_impact\_ionization\_probability ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "critical_energy", "material": "SiO2", "equation": "Eq. (8), p. 187401-3", "printed": "eps_crit = (1 + mu/m_VB)(E_gap + ), = 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 :doc:`review register ` for the checks and decisions.