zhokhov_2014#
P. A. Zhokhov and A. M. Zheltikov, “Field-cycle-resolved photoionization in solids”, arXiv:1402.5530v2 [physics.optics] (2014)
Source record · Catalogue · Review register
Source kind: theory
Recorded technique: Keldysh’s two-band photoionization theory of a solid generalized to pulses of arbitrary waveform: the momentum integral is done first, for a periodic (cosine) band dispersion, giving the conduction-band population as a double time integral (Eqs. 6-9), with a phenomenological decoherence factor; a closed-form tunnelling rate in the strong-field limit (Eq. 12).
Materials in structured entries: None recorded.
Dataset quantities: None recorded.
Contents: 0 datasets; 2 formulas.
Curation notes#
THE LIBRARY COPY IS THE arXiv PREPRINT (arXiv:1402.5530v2, dated 4 July 2014; document_version ‘preprint’, 5 pages). A journal version is not in the library and the preprint does not cite one; pages below are the preprint’s.
NO MATERIAL, NO DATASET. The calculations are for a model solid: E_g/omega = 6.45 at lambda0 = 800 nm, band-width ratio delta = Delta/E_g = 1 or 0.1, one dimension, B = 1 (no decoherence) unless stated, FWHM 10 fs and 2.4 fs (Figs. 1-2). E_g/omega = 6.45 at 800 nm is a 10.0-eV gap (6.45 x 1.5498 eV = 9.996 eV; computed), not named as any material. Results are figures only: the conduction-band population against gamma^-2 and time, compared with Keldysh’s formula (agreement for gamma > 2/(pi sqrt(delta)) and long pulses; Franz-Keldysh steps), its carrier-envelope-phase dependence for 2.4-fs pulses, and the effect of decoherence.
Recorded bibliographic identity status: VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
No datasets are recorded.
Formulas#
zhokhov_2014/cycle_resolved_population#
{
"id": "cycle_resolved_population",
"equation": "Eqs. (4), (6)-(9), pp. 1-2",
"printed": "E(p) = E_g + Delta - (Delta/D) sum_{k=1}^{D} cos(d_k p_k) (4); W_c(t) = |N~|^2 int_-inf^t dt1 int_-inf^t dt2 E(t1) E(t2) G(t1, t2) (6); G(t1, t2) = (2 pi)^D exp{-i (E_g + Delta)(t2 - t1)} B(|t2 - t1|) prod_{k=1}^{D} J_0(|Phi_k|) (9); B(|t2 - t1|) = <exp{i phi(t2) - i phi(t1)}> (8); Phi_k = (Delta/D) int_{t1}^{t2} exp{i mu_k(tau)} dtau, mu_k = A_k d_k",
"variable": "E(t): the field of arbitrary waveform; A(t) = -int E dt the vector potential; d_k the lattice constants; Delta the band width, m_eff = (Delta sum_k d_k^2)^-1 the electron-hole effective mass; D the dimension; atomic units",
"coefficients": {
"E_g_over_omega": "6.45 (examples)",
"delta": "Delta/E_g = 1 or 0.1 (examples)"
},
"check": "No rate values printed; figures only."
}
Agrees with the Keldysh formula only where the cosine band is near-parabolic over the excursion, gamma > 2/(pi sqrt(delta)), with gamma = omega sqrt(m Delta)/E_0. Decoherence ansatz B(tau) = exp(-Gamma E_g tau), coherence time tau_c = (Gamma E_g)^-1 (p. 4).
zhokhov_2014/tunnelling_limit#
{
"id": "tunnelling_limit",
"equation": "Eqs. (10)-(12), pp. 3-4",
"printed": "G(t1, t2) = 2 pi E^2 exp{-2i ((E_g + Delta)/|eps|) xi} J_0((2 Delta/|eps|) |sin xi|), xi = (|eps|/2)(t2 - t1), eps = d_1 E (10); w(E) = dW_c/dt = 4 (|E|/d_1) |N|^2 Re int_-pi^pi d eta int_-inf^inf d xi exp{-i s nu(xi, eta)}, s = 2 Delta/|eps|, nu = (1/delta + 1) xi + sin xi sin eta (11); w(E) = K E^2 exp{-(4/3) sqrt(2/delta) E_g/|eps|} = K E^2 exp{-(4/3) (2m)^(1/2) E_g^(3/2)/|E|} (12)",
"variable": "K: a field-independent factor; one dimension; valid for 2/(pi sqrt(delta)) < gamma < 1, which needs delta > (pi/2)^2",
"check": "Nothing numeric to reproduce."
}
Recovers the tunnelling exponential and the E^2 pre-exponential scaling (refs. 17, 26).
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.