Low-emissivity glazing: dielectric / silver / dielectric

A window should let daylight in and keep heat where it is. The two wavelength ranges are well separated: sunlight carries its energy between 0.3 and 2.5 µm, the thermal radiation of a room at 20 °C peaks near 10 µm. Silver does both jobs, if it is thin enough and dressed in the right dielectrics, and the physics is the same Drude response as in the beam-splitter section, read at two very different frequencies.

A metal at two frequencies

Below the plasma frequency a Drude metal has $\varepsilon \approx -\omega_p^2/\omega^2$: at 550 nm silver's $|\varepsilon|$ is about 13, at 10 µm it is about $4\,000$ with an imaginary part of similar size. The reflectance of the bulk metal is 97 % in the visible and 99.5 % in the thermal infrared, and the skin depth shrinks from 12 nm to a few nanometres. A film of 10–15 nm is therefore thin compared with the visible skin depth, so much of the visible leaks through, yet is already optically thick in the infrared, where it reflects as if it were bulk. That asymmetry is the whole principle; the dielectric layers only tidy up the visible.

Antireflecting a metal: induced transmission

Bare, a 12 nm silver film on glass still reflects about half of the visible, because its admittance (large and imaginary) is far from that of glass or air. Two dielectric layers of index around 2 (ZnO, TiO₂, SnO₂, Si₃N₄) on either side act as an induced-transmission filter (Macleod ch. 7): each is roughly a quarter wave at the design wavelength, and the pair rotates the stack's admittance locus so that the silver, seen from the glass, is matched to the air. Reflections from the stack then cancel across the visible, and transmittance rises from about 50 % to 80 %, with the remainder absorbed rather than reflected. The potential transmittance of a 12 nm silver film in the visible, the most any matching layers can extract, is about 85 %, and commercial coatings get within a few percent of it. In the infrared the silver's admittance is far too large for any dielectric of index 2 to match, the quarter-wave layers are a few hundredths of a wave thick anyway, and the stack reflects as bare silver would.

Emissivity is absorptance

By Kirchhoff's law a surface's spectral emissivity equals its spectral absorptance at the same wavelength and direction. For an opaque coated pane $T \approx 0$ in the thermal infrared, so

and the quantity glazing engineers quote, the hemispherical emissivity, is this absorptance averaged over the room-temperature black-body spectrum and over angle. Bare glass, which absorbs strongly through its Si–O vibrations across the whole window, has $\varepsilon \approx 0.84$; a single-silver low-e coating has 0.04–0.10, a triple-silver stack about 0.02. A pane that emits a twentieth of what bare glass does loses a correspondingly smaller share of a room's heat by radiation, which is about half of the loss through a window in winter.

For a good conductor the emissivity has a closed form. Hagen and Rubens (1903) showed that far below the plasma frequency, where $\varepsilon \approx i\sigma/(\varepsilon_0\omega)$ with $\sigma$ the dc conductivity, the normal emissivity of a thick metal is

about 0.01 for silver at 10 µm, falling as $\sqrt{\omega}$ toward longer wavelengths. For a thin film the same physics is expressed through its sheet resistance $R_\square = 1/(\sigma d)$: a film on a transparent substrate has, to a good approximation, $\varepsilon \approx 4 R_\square / Z_0$ with $Z_0 = 377\,\Omega$ the impedance of free space, that is about 0.01 per ohm per square. A 12 nm silver film has $R_\square \approx 3\,\Omega/\square$ and an emissivity near 0.03, and the rule explains at a glance why emissivity and electrical conductivity go together, why a thicker film is always a better emitter-suppressor, and why transparent conductive oxides (ITO at 20–30 Ω/□) make hard-coat low-e glass of emissivity 0.15–0.2 but can never reach silver's.

import numpy as np
from pystratify import planar
from pystratify.docs import material  # n + ik of a tabulated page (refractiveindex.info format)

n = lambda wl: [1.0, material('docs/data/tutorials/ZnO_Aguilar.yml', wl), material('docs/data/tutorials/Ag_Ciesielski.yml', wl), material('docs/data/tutorials/ZnO_Aguilar.yml', wl), 1.52]  # incident medium, layers, substrate
d = [np.inf, 0.04, 0.012, 0.04, np.inf]  # thicknesses, µm
lam = (300 + 20 * np.arange(111)) * 1e-3  # wavelength, µm
th = np.radians(0)
R, T = {}, {}
for pol in ('s', 'p'):
    rows = [planar.coh_tmm(pol, n(wl), d, th, wl) for wl in lam]
    R[pol], T[pol] = np.array([r['R'] for r in rows]), np.array([r['T'] for r in rows])
R_avg, T_avg = (R['s'] + R['p']) / 2, (T['s'] + T['p']) / 2  # unpolarized

Thicker silver raises the infrared reflectance and lowers the visible transmittance; the trade-off is the one number a glazing designer chooses. The silver data here extend to 20 µm (Ciesielski et al., 2017); the classic study of this design is Martín-Palma et al. (1998), and Granqvist's review (1991) is the standard account of spectrally selective coatings.

Figures of merit, and the edge

Three numbers describe a glazing: the visible transmittance $T_\mathrm{vis}$ (weighted by the eye's sensitivity), the solar transmittance or solar heat gain coefficient $g$ (weighted by the solar spectrum, including the fraction of absorbed heat re-emitted inward), and the emissivity. A winter window wants high $g$ and low $\varepsilon$; a summer window in a hot climate wants low $g$ with high $T_\mathrm{vis}$, that is a sharp cut-off at the end of the visible, near 700 nm, with the near infrared of sunlight (half its energy) reflected. The selectivity $T_\mathrm{vis}/g$ of a single silver film is limited, because its transmittance falls off gently through the near infrared. Repeating the dielectric/silver/dielectric unit two or three times makes the stack an induced-transmission filter with two or three coupled cavities, which, exactly as for the multi-cavity bandpass filter, squares the top and steepens the edge: triple-silver coatings reach $T_\mathrm{vis}/g \approx 2$, close to the theoretical limit for the solar spectrum. The price is a lower $T_\mathrm{vis}$ and a slight colour, and the design problem is to keep the reflected colour neutral, which the dielectric thicknesses control.