The various origins of a frequency-dependent conductivity in amorphous semiconductors are reviewed, stressing particularly recent advances and the influences that factors such as correlation and non-random spatial distributions of electrically active centres can have on the a.c. conductivity. A comprehensive survey is given of the experimental a.c. data for two types of amorphous semiconductor, namely chalcogenide and pnictide materials. It is concluded that the a.c. behaviour at intermediate to high temperatures is well accounted for by the correlated-barrier-hopping model, whereas the low-temperature behaviour is probably due to atomic tunnelling.

A.c. conduction in amorphous chalcogenide and pnictide semiconductors

This review covers a wide range of experimental and theoretical studies of two-level or tunnelling states in glasses. Emphasis is on fundamental physics rather than a detailed comparison of experiment and theory. Sections cover the static and dynamic properties of tunnelling states, their contribution to thermal properties and their response to weak and strong electric and acoustic fields, both steady state and pulsed. A section on metallic glasses focuses on the importance of electron tunnelling-state interactions, and a final section illustrates approaches to a microscopic description by means of selected examples.

https://iopscience.iop.org/article/10.1088/0034-4885/50/12/003/pdf

Two-level states in glasses

In order to test whether the low-energy excitations explored extensively in amorphous solids are indeed universal, all measurements published on the low-temperature thermal conductivity and acoustic attenuation in these solids have been reviewed, on a total of over 60 different compositions. The ratio of the phonon wavelength \ensuremath{\lambda} to the phonon mean free path l has been found to lie between ${10}^{\ensuremath{-}3}$ and ${10}^{\ensuremath{-}2}$ in almost all cases, independent of chemical composition and frequency (wavelength) of the elastic waves, which varied by more than nine orders of magnitude in the different experiments. When the data were fitted with the tunneling model, which is based on the assumption of atomic or molecular tunneling states with a certain spectral distribution, the tunneling strength C, which describes their coupling to the lattice, was found to range from ${10}^{\ensuremath{-}4}$ to ${10}^{\ensuremath{-}3}$ in almost all cases. The only exceptions reported so far are certain films of amorphous silicon, germanium, and carbon. In these films, low-temperature acoustic attenuations over two orders of magnitude smaller have been observed compared to all other amorphous solids. Another remarkable observation is that a large number of disordered crystals, and even a thermally equilibrated quasicrystal, have low-energy lattice vibrations that are quantitatively indistinguishable from those of amorphous solids. Their tunneling strengths also range from ${10}^{\ensuremath{-}4}$ to ${10}^{\ensuremath{-}3}.$ These measurements have also been reviewed. It is concluded that the absence of long-range order is neither sufficient nor necessary for the existence of the low-energy excitations.

Low-temperature thermal conductivity and acoustic attenuation in amorphous solids

Amorphous solids show surprisingly universal behaviour at low temperatures. The prevailing wisdom is that this can be explained by the existence of two-state defects within the material. The so-called standard tunneling model has become the established framework to explain these results, yet it still leaves the central question essentially unanswered-what are these two-level defects (TLS)? This question has recently taken on a new urgency with the rise of superconducting circuits in quantum computing, circuit quantum electrodynamics, magnetometry, electrometry and metrology. Superconducting circuits made from aluminium or niobium are fundamentally limited by losses due to TLS within the amorphous oxide layers encasing them. On the other hand, these circuits also provide a novel and effective method for studying the very defects which limit their operation. We can now go beyond ensemble measurements and probe individual defects-observing the quantum nature of their dynamics and studying their formation, their behaviour as a function of applied field, strain, temperature and other properties. This article reviews the plethora of recent experimental results in this area and discusses the various theoretical models which have been used to describe the observations. In doing so, it summarises the current approaches to solving this fundamentally important problem in solid-state physics.

https://iopscience.iop.org/article/10.1088/1361-6633/ab3a7e/pdf

Towards understanding two-level-systems in amorphous solids: insights from quantum circuits.

A new model for the relaxation of a potential perturbation in dielectric materials is developed based on the correlated properties of a two-level system containing two types of decay mechanism. The time and frequency behaviour of the general relaxation process is shown to be in accord with recent experimental analyses. The technique developed to analyse the model is of general applicability in the solid, and liquid, state.

Non-exponential decay in dielectrics and dynamics of correlated systems

A complete theoretical description of the effect of the relaxation of two-state structural defects on the elastic properties of glasses at low temperatures is given. New experimental data on the ultrasonic attenuation in vitreous silica and borosilicate glass in the temperature range between 0.3 and 4 K and at frequencies between 30 and about 500 MHz are presented and explained by this theory. We propose that over the entire temperature range from our lowest temperature of 0.3 to about 100 K the attenuation is due to the relaxation of defects with similar atomic structure.

Elastic effects of structural relaxation in glasses at low temperatures

Experimental evidence for the direct interaction between two-level systems in glasses at very low temperatures

Propagation of transverse sound waves in glasses at very low temperatures

“Glassy” properties of crystalline Li3N at low temperatures

A new highly sensitive method is described for the measurement of optical absorption and its wavelength dependence in extremely low loss materials (e.g. starting materials for optical fibres). The optical absorption — as distinct from scattering losses — is measured calorimetrically at low temperatures via the temperature rise due to the absorbed heat. Low temperature as compared with room temperature calorimetry is several orders of magnitude more sensitive due to the drastically decreased specific heat and the higher resolution of the temperature measurements. We have measured an optical absorption coefficient as low as 3 dB km −1 (7 × 10 −6 cm −1 ) in a Suprasil I glass rod of only 5 cm in length. In principle, by our method absorption coefficients even smaller than 10 −6 cm −1 can be determined in a sample of the same length. Since the optical power necessary for this method amounts to only a few mW no single strong laser lines are required as in room temperature calorimetry and also low power light sources which are continuously tunable can be used. Our method is also sensitive enough to distinguish between absorption in the bulk and at the surfaces. Measurements on Suprasil I samples show an absorption peak around 600 nm probably due to OH-impurities in the bulk and a background absorption caused by dissipative processes at the two surfaces.

S. Hunklinger

Papers

Elastic effects of structural relaxation in glasses at low temperatures

Experimental evidence for the direct interaction between two-level systems in glasses at very low temperatures

Propagation of transverse sound waves in glasses at very low temperatures

“Glassy” properties of crystalline Li3N at low temperatures

New method for measuring extremely low optical absorptions