Decoupling between propagating acoustic waves and two-level systems in hydrogenated amorphous silicon

Abstract

Specific heat measurements of hydrogenated amorphous silicon prepared by hot-wire chemical vapor deposition show a large density of two-level systems at low temperature. Annealing at 200 ◦C, well below the growth temperature, does not significantly affect the already-low internal friction or the sound velocity, but irreversibly reduces the non-Debye specific heat by an order of magnitude at 2 K, indicating a large reduction of the density of two-level systems. Comparison of the specific heat to the internal friction suggests that the two-level systems are uncharacteristically decoupled from acoustic waves, both before and after annealing. Analysis yields an anomalously low value of the coupling constant, which increases upon annealing but still remains anomalously low. The results suggest that the coupling constant value is lowered by the presence of hydrogen. PACS numbers: 61.43.Dq, 62.40.+i, 65.60.+a Amorphous solids show anomalous thermal, elastic and dielectric properties at low temperatures. The observation of anomalous thermal properties, through thermal conductivity and specific heat measurements [1], led in 1972 to development of the standard tunneling model (STM) [2, 3], in which atoms or groups of atoms tunnel between nearly degenerate configurations with a distribution of asymmetries, ∆, that are separated by energy barriers on the order of tens of Kelvin with a distribution of tunneling parameter, λ, resulting in tunneling-induced states with splittings < 1 K. The simplest description of these configurations and the resulting states are two-level systems (TLSs). The STM was further developed during the following years, providing an explanation for various anomalous low temperature properties [4–10]. The specific structures that enable the TLSs are generally not known, but have been proposed for some specific materials, such as silica [11]. Recently, we suggested that TLSs in hydrogenated amorphous silicon (a-Si:H) are primarily due to clustered atomic H in low density regions [12]. The STM assumes that the asymmetry between states, ∆, is small, and that the tunneling parameter, λ, is uniformly distributed over a range of values, which allows the TLS distribution function, P (∆, λ), to be written as P (∆, λ) ≈ P̄ , where P̄ is the TLS density. Under this assumption, the specific heat, CP (T ), of an ensemble of TLS has a linear term, ∗ Corresponding author: [email protected] † Present address: Department of Physics, California Polytechnic University, San Luis Obispo, California 93407, USA ‡ Present address: Northrop Grumman Corp., Linthicum, Maryland 21090, USA 2 c1, associated with the density of TLS, n0, that equilibrate with the phonon bath on the time scale of the measurement by CP (T ) = c1T = π2 6 k2 BNA nat n0T (1) where nat is the total atomic density, kB Boltzmann’s constant, and NA Avogadro’s number. The specific heat derived TLS density, n0, is proportional to the TLS density, P̄ , through a relationship first proposed by Black and Halperin [6, 7], and later experimentally verified [13, 14], which establishes n0 = 1 2 P̄ ln ( 4t τmin ) (2) where t is the measurement time (≈ 10 ms for the nanocalorimetric system used in the present work at low T) [15], and τmin is the TLS minimum relaxation time. At low temperatures, typically below 10 K, the main TLS relaxation mechanism is the one-phonon process, in which τmin is minimized when the energy difference between two TLS states E = √ ∆2 + ∆0 = ∆0, where ∆0 is the TLS tunnel splitting. This condition implies that the minimum relaxation time is achieved by symmetric TLSs (∆ = 0). In dielectric glasses, the main contribution to specific heat and mechanical loss comes from TLSs with energy splitting E = kBT , and their minimum relaxation time is given by [16–18] τmin = ( γ2 ` v5 ` + 2 γ2 t v5 t )−1 πρ~4 k3 B T−3 = aT−3 (3) where subscripts ` and t denote the longitudinal or transverse wave polarization, respectively, which from now on and as a generalization are denoted by the subscript α. γα is the coupling constant between TLS and phonons, vα the sound velocity, ρ the mass density, and T the system temperature. Typically, a ≈ 10−8 s K [16, 19], and therefore τmin ≈ 1 ns at 2 K. Note that because of the logarithmic time-dependence in Eq. 2, the ratio n0/P̄ depends only weakly on the measurement time, t, and on the TLS minimum relaxation time, τmin, such that n0/P̄ ≈ 10 for essentially all experimentally realized measurements [20]. The interaction between phonons and TLS at low temperature leads to energy loss, which may occur by resonant or relaxation mechanisms. At temperatures above∼ 10 K, losses arise from thermal activation over the energy barriers separating the states. At the experimental temperatures and excitation frequencies, ω, used in the present work, relaxation dominates over resonant mechanisms through TLSs with ωτmin = 1. The interaction between a phonon and a TLS is given by the deformation potential, γi,α = ∂∆i/2∂uα, where uα is the strain