# Crystalline silicon Crystalline silicon (`Si`) has one recommended model of its complex refractive index, from the extreme ultraviolet to the far infrared, at 20 degrees C. This page gives: - the model and what it rests on; - how well it is known in each range; - how it compares with the published alternatives. The manual for solids is {doc}`solid`. The register entries are TC-1075 to TC-1087 (the sources) and TC-1088 (the fit). ```python from optmat.solid.models.registry import default_model, models silicon = default_model("Si") # fit/Si, at 20 degrees C n = silicon.refractive_index(1550e-9) # complex n + ik beta2 = silicon.gvd(2000e-9) # s^2/m cold = models("Si")["frey_2006/sellmeier_Si"].at_temperature(120.0) # 20-300 K ``` ## The complex index **The recommended model** is `fit/Si` (TC-1088). It is Franta et al.'s universal dispersion model of float-zone silicon, refitted to the measurements, in the same way as {doc}`silica`'s `fit/SiO2`. - **Range:** 31 nm to 143 µm, at 20 degrees C. - **Physics:** it is causal, passive and normalised to its transition strengths, with an exact d eps/d omega. - **The start:** the newAD2 parameter file of float-zone c-Si (`franta_2017/udm_c_si`), which the authors' site describes as unpublished. optmat reproduces the authors' own table of it to the last printed digit (TC-1075). It is 0.016 above the absolute refractometry in the near infrared, and its k at 633 nm is 2.6 times the measured. - **What it rests on:** - in the transparent window (1.2-5.5 µm), the absolute indices of Frey, Leviton & Madison (CHARMS, ±1e-4), taken from 295 K to 293.15 K with their own measured dn/dT; - on the absorption edge (250-1200 nm), Schinke et al.'s absorption coefficient, whose uncertainty analysis gives 0.2-13 % (k = 2); - elsewhere, Franta et al.'s temperature-dependent model at 20 degrees C, the numbers the authors gave to the refractiveindex.info database (TC-1075). Its ultraviolet comes from their ellipsometry and its vacuum ultraviolet (to 40 eV) from their synchrotron reflectance. Its infrared k is their measured lattice absorption. Its window n is tied to Frey et al.'s and is not fitted beside it. - **What it changed:** the interband, higher-energy and Urbach strengths, the localised-state strengths at the E1 and E2 critical points, the four excitons, and one factor on the phonon bands (x1.006). | lambda (nm) | n | k | |---|---|---| | 50 | 0.7456 | 0.0542 | | 100 | 0.2444 | 0.970 | | 150 | 0.5827 | 1.81 | | 200 | 0.9932 | 2.76 | | 250 | 1.6372 | 3.62 | | 300 | 5.0216 | 4.12 | | 370 | 6.8785 | 2.10 | | 400 | 5.6010 | 0.331 | | 500 | 4.2961 | 0.0474 | | 633 | 3.8704 | 0.0180 | | 800 | 3.6776 | 0.00598 | | 1000 | 3.5744 | 0.000491 | | 1064 | 3.5540 | 8.99e-05 | | 1550 | 3.4783 | 6.43e-10 | | 2000 | 3.4532 | 1.40e-10 | | 5000 | 3.4226 | 1.72e-07 | | 10000 | 3.4182 | 6.05e-05 | | 20000 | 3.4176 | 0.000268 | | 100000 | 3.4173 | 0.000202 | ## The transparent window The absolute refractometry and the models at 20 degrees C: | | Frey 2006, table | Frey 2006, formula | Li 1980 | Franta 20 C | newAD2 | fit/Si | |---|---|---|---|---|---|---| | n at 1.5 µm | 3.48268 | 3.48281 | 3.47990 | 3.48262 | 3.49947 | 3.48269 | | n at 2.0 µm | 3.45321 | 3.45320 | 3.45100 | 3.45323 | 3.46896 | 3.45323 | | GVD at 2.0 µm (fs^2/mm) | | 817.9 | | | 844.7 | 814.8 | - **Frey et al.** measured in vacuum, from 30 to 295 K and 1.1 to 5.5 µm (TC-1079). The columns are their 295 K table taken to 20 degrees C with their dn/dT (Table 4, -3.3e-4 at 2 µm), and their temperature-dependent Sellmeier formula (`frey_2006/sellmeier_Si`). The formula reproduces their own tables only to about 1e-4 (rms 1.35e-4, up to 1e-3 at 1.1 µm), so the fit uses the tables. - **The fit** follows Frey's tables to within 1.4e-4 from 1.2 to 5.5 µm. The largest departure is at 1.2 µm, on the absorption edge, and it is +0.6-1.2e-4 at 2.5-5.5 µm, where the model's shape does not quite follow the data. - **Li's** critical review (1980) lies 2-4e-3 below Frey's, beyond its own stated ±2e-3 (TC-1080). It is a check, not fitted. Li does not say whether his index is absolute or relative to air; read as relative to air, it is still 1.3-3.3e-3 low. - **Green's** self-consistent table (300 K, 2008) lies 1.7-3.3e-3 below the fit at 0.93-1.45 µm. From Frey's dn/dT (1.8e-4 per K) it should lie 1.2e-3 above at 300 K, so it is 3-4.5e-3 below Frey's level (TC-1081). - **GVD:** the fit's GVD is 1112 fs^2/mm at 1550 nm and 815 at 2 µm. That is within 4 % of Frey's formula from 1.55 to 5 µm, and 6 % above it at 1.2 µm. The newAD2 file is 3-6 % higher. - **Temperature:** dn/dT is about 1.8e-4 per K at 2 µm, so 1 K moves n by twice Frey's uncertainty. For other temperatures in the window, use Frey's formula `frey_2006/sellmeier_Si` with `at_temperature` (20-300 K, 1.1-5.6 µm). It carries the formula's own 1e-4 misfit. Their dn/dT at all twelve temperatures is curated as datasets. ## The absorption edge Silicon's gap is indirect, at 1.12 eV (1.1 µm). Below the direct transitions at 3.4 eV (365 nm), its absorption rises by six decades through phonon-assisted transitions. | lambda (nm) | Schinke 2015 (295 K) | Green 2008 (300 K) | Franta 20 C | newAD2 | fit/Si | |---|---|---|---|---|---| | 400 | 0.326 | 0.296 | 0.327 | 0.335 | 0.331 | | 500 | 0.0485 | 0.0442 | 0.0523 | 0.0693 | 0.0474 | | 633 | 0.0158 | 0.0161 | 0.0152 | 0.0398 | 0.0180 | | 800 | 0.00527 | 0.00541 | 0.00598 | 0.0166 | 0.00598 | | 900 | 0.00211 | 0.00217 | 0.00260 | 0.00702 | 0.00218 | | 1000 | 0.000490 | 0.000509 | 0.000464 | 0.00143 | 0.000491 | | 1100 | 3.02e-05 | 3.06e-05 | 3.05e-05 | 3.44e-05 | 3.04e-05 | | 1200 | 1.39e-07 | 2.10e-07 | 1.16e-07 | 5.18e-07 | 3.48e-07 | The columns give k; the tables of alpha are converted with k = alpha lambda / (4 pi). - **The fit's k** is within 4 % of Schinke's below 370 nm, within -8 to +13 % at 370-550 nm and 810-1100 nm, 5-19 % high at 550-810 nm, and up to 27 % low at 1100-1150 nm. At 550-810 nm Schinke's own uncertainty reaches 13 % (k = 2), and the values come from a Tauc fit. The model's few smooth terms cannot follow the shape more closely. - **Green's** table is a Kramers-Kronig re-analysis of others' measurements, including Jellison's and Aspnes's (TC-1081). It is a check, so that those measurements are not counted twice. It agrees with Schinke's within 10 % to 1.1 µm. - **Beyond 1.15 µm** the band-to-band absorption falls by five decades in 0.15 eV: alpha < 0.1 cm^-1, k < 1e-7. The model's Urbach tail cannot follow it. It is 2.5 times Schinke's at 1.2 µm and levels off at 1e-10 to 1e-9 in the window. **The model's k beyond 1.15 µm is an upper bound, not a measurement.** In a real wafer the free carriers of the doping usually absorb more. - **Temperature:** k on the edge changes by 0.5-2 % per K at 0.9-1.15 µm (Green's normalised coefficients, 54-210e-4 per K). Schinke's 295 K data are fitted at 20 degrees C with a 1 % allowance. ## The ultraviolet The ellipsometric dielectric functions bracket the model across the E1 (3.4 eV) and E2 (4.3 eV) critical points: | E (eV) | Jellison n | Jellison k | Aspnes n | Aspnes k | Franta n | Franta k | fit n | fit k | |---|---|---|---|---|---|---|---|---| | 3.0 | 5.226 | 0.196 | 5.222 | 0.269 | 5.239 | 0.222 | 5.243 | 0.219 | | 3.4 | 6.486 | 2.674 | 6.522 | 2.705 | 6.490 | 2.712 | 6.480 | 2.727 | | 4.0 | 5.077 | 3.557 | 5.010 | 3.586 | 5.027 | 3.571 | 5.014 | 3.577 | | 4.3 | 4.191 | 5.280 | 4.087 | 5.395 | 4.095 | 5.303 | 4.084 | 5.305 | | 5.0 | 1.683 | 3.608 | 1.570 | 3.565 | 1.634 | 3.548 | 1.627 | 3.555 | - **Aspnes & Studna's** (1983) n is 0.10-0.13 below Jellison's at 4.3-5.3 eV (TC-1083, TC-1084). Herzinger et al. (1998) found that Aspnes's table behaves as if 0.18 nm of oxide remained on the surface (TC-1085). Aspnes's k below 2.8 eV is Dash & Newman's absorption, not his measurement, and is left out. - **Jellison's** (1992) error limits are ±0.012-0.046 in n. Two misprinted n values (595 and 705 nm) are corrected as errata. - **The fit** follows Franta's ultraviolet n to 3e-3 on average above 120 nm and 7e-3 at 31-120 nm, and to 0.04 at the sharpest structure. It lies between Jellison's and Aspnes's n, nearer Aspnes's at 4-5 eV. In this range the spread of the measurements, 0.01-0.1 in n and a few per cent in k, is the uncertainty. - **In the vacuum and extreme ultraviolet** (31-200 nm) Franta's model is fitted to their own synchrotron reflectance (6-40 eV). Edwards's chapter in Palik's handbook compiles Philipp's (1972) Kramers-Kronig analysis and Hunter's films to 100 eV (TC-1086). Philipp's n is 23 % below Franta's at 150 nm, and 4 % above at 210 nm. ## The infrared From 5.6 to 143 µm the model's n is Franta's to 1.5e-4. Its k is Franta's measured lattice absorption, mostly two-phonon bands, from 6 to 64 µm: to 10 % rms, and up to 45 % on the sharpest bands. These are float-zone values. Czochralski silicon adds the 9 µm (1107 cm^-1) band of interstitial oxygen, and doped silicon adds free-carrier absorption, both sample-dependent, which the model does not include. ## Limitations - 20 degrees C only. The only temperature-dependent model is Frey's formula, in the window. - Franta et al.'s full temperature-dependent model (2017) follows the indirect edge's phonon structure from 10 to 700 K. It cannot be implemented: its parameters, more than a thousand, are not printed in the 2017, 2018 or 2019 papers (TC-1075, TC-1077, TC-1078). The refractiveindex.info tables at 10-500 K are its only numerical form. - The model is intrinsic, float-zone silicon. It has no oxygen, no doping and no free carriers.