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 Solids: conventions and models. The register entries are TC-1075 to TC-1087 (the sources) and TC-1088 (the fit).

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 Fused 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 • The start: the newAD2 parameter file of float-zone c-Si (franta_2017/udm_c_si), which the authors’ site describes as unpublished.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.