zhokhov_2014
============

P\. A\. Zhokhov and A\. M\. Zheltikov, "Field\-cycle\-resolved photoionization in solids", arXiv\:1402\.5530v2 \[physics\.optics\] \(2014\)

`Source record <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/zhokhov_2014.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.48550/arxiv\.1402\.5530 <https://doi.org/10.48550/arxiv.1402.5530>`__

**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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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 :doc:`review register </register>` for the checks and decisions.