Probing two-level systems with electron spin inversion recovery of defects at the Si / SiO 2 interface

The main feature of amorphous materials is the presence of excess vibrational modes at low energies, giving rise to the so-called “boson peak” in neutron and optical spectroscopies. These same modes manifest themselves as two-level systems (TLSs) causing noise and decoherence in qubits and other sensitive devices. Here, we present an experiment that uses the spin relaxation of dangling bonds at the Si / (amorphous)SiO 2 interface as a probe of TLSs. We introduce a model that is able to explain the observed nonexponential electron spin inversion recovery and provides a measure of the degree of spatial localization and concentration of the TLSs close to the interface, their maximum energy, and its temperature dependence.

Several properties of amorphous materials can be explained by assuming the presence of additional low energy vibrational modes on top of the usual phonon density of states.In neutron scattering and Raman spectroscopy these modes appear as a universal boson peak with average energy increasing with temperature [1,2].At low temperatures, these modes give rise to an anomalous contribution to the specific heat.A convenient assumption is to model the excess modes at the low energy tail of the boson peak as an ensemble of tunnelling twolevel systems (TLSs), each with energy splitting E. Assuming their energy density scales as a power law with exponent α (ρ ∝ E α ) leads to specific heat scaling as T 1+α [3,4].The coefficient α gives a measure of the degree of amorphousness of the material.
The TLSs are often responsible for the origin of noise, decoherence, and dielectric energy loss in all kinds of devices for solid state quantum computation, including superconducting Josephson devices [5,6] and spin qubits [7].As these devices are generally made from high quality materials, the TLSs usually appear close to surfaces and interfaces, where the degree of crystallinity is quite hard to control.In addition several architectures are based on semiconductor/amorphous oxide interfaces.
In contrast most experimental measurements of the boson peak and TLSs are "bulk probes".To date the few experiments that are able to probe TLSs at thin films or interfaces still lack the energy resolution to measure properties such as the exponent α and the temperature dependence of the boson peak [8,9].As devices become smaller, the interface plays an increasing role; this, combined with the fact that the interface is inevitably amorphous, motivates the development of a method that detects TLSs at surfaces and interfaces.
Here we describe an experiment that uses danglingbond spins as a probe of TLSs at the Si/SiO 2 interface.Unsaturated dangling bonds (DBs), generically called P b -centers, appear at the Si/SiO 2 interfaces.[10][11][12] Their structure is quite well understood.[13] We mea-sure spin-lattice relaxation of the DB spin magnetization, S z (t) , using inversion-recovery experiments with echo detection.We show that S z (t) approaches thermal equilibrium in a highly non-exponential fashion, leading to a wealth of information on the spatial distribution and energetics of TLSs nearby the DB spin.The signal intensity is measurable thanks to a nanostructuring of the interface into nanowires, instead of a flat surface, greatly increasing the surface-to-volume ratio.[14,15] It is in fact well known that DBs can act as a probe of TLSs because their spin relaxation rate 1/T 1 is strongly dominated by TLS dynamics, even at higher temperatures [16,17].However, previous experiments [17] were unable to interpret the long time non-exponential decay.Below we describe our experiment and propose a theoretical model based on a Poisson distribution of TLSs within a radius of each DB.This model is able to capture the long time non-exponential dynamics thus allowing the extraction of much more information on TLS parameters than previous approaches.As a result we are able to obtain a clear picture of TLSs at the interface, including the measurement of their degree of spatial localization, one of the most important unsolved problems in the physics of the boson peak.
Experiment.-Silicon nanowires were prepared by a metal-assisted chemical etching (MACE) process starting from two different types of seed metal deposited on intrinsic [001] silicon.Details of the two samples under investigation are reported in Table I.For sample A, pinholes in a 3.8 nm thick gold layer were used to realize the nanowires.For sample B, the seed metal consisted in Ag nanoparticles, deposited by electroless deposition, as explained in Refs.15 and 18, where also details of the etching process are reported.At the end of the MACE process and complete metal removal a dense carpet of straight silicon nanowires, fully passivated with H and with structural parameters dependent on the process details, cover the whole sample surface (Figure 1).The two samples were chosen out of many different batches for the present investigation.Both samples were annealed in vacuum for 15 minutes at 550 • C to induce depassivation of surface defects from the naturally hydrogen-passivated state formed during the MACE process.Details of the depassivation process have been reported elsewhere.[14,15] The samples were then characterized by continuouswave and pulsed electron paramagnetic resonance (Bruker Elexsys E580 system, X-band).After the depassivation treatment the typical electron paramagnetic resonance signal of DB defects at the interface, the so called P b defects, was observed to increase.[14,15] Inversion recovery experiments, with the magnetic field H [001], were performed in the temperature range 4-300 K to obtain information on the relaxation rate.A typical example of an inversion recovery curve is reported in Figure 2, together with a fit attempt with a single exponential recovery.
Such a model evidently fails, especially at low temperature, though the resulting thermal trend of the spinlattice relaxation rate determined assuming a single exponential recovery, may allow comparisons with data re-Inversion Recovery Delay (s) ported in the literature.Generally, the inefficiency of a single exponential recovery fit is neglected and the analysis focuses on the thermal variation of the resulting spin-lattice relaxation rate 1/T 1 , which is reported to follow a power-law trend ∝ T 2+α with α in the range ∼ 0.3 − 1.5.[17,19] The lowest value reported to the authors' knowledge is α = −0.2, in Ref. 16.In our case, the fit would result in even lower α values ranging from −0.45 to −0.55, which are quite unusual, at least for bulk materials.Fitting attempts with a stretched exponential recovery model were attempted and seemed indeed more successful, though they essentially shift the whole information on the dominating relaxation mechanisms into the temperature dependence of other two physical quantities: the stretched relaxation rate and the stretched exponent, which require further interpretation.We think that a deeper understanding of the non-exponential recovery is necessary.The temperature dependence of the spin lattice relaxation rate, obtained either by a single exponential or by a stretched inversion recovery, was attributed to the presence of TLSs.We need therefore a broader theoretical framework modelling the role of the TLSs already at the level of the recovery curves.
Theoretical model.-Previousmodels for DB spin relaxation [7,17] assumed a low disorder prescription: Each DB spin was assumed to relax through cross-relaxation with only one TLS.For the case of an interface we generalize this model to account for an arbitrary number of TLSs surrounding each P b DB.A key aspect is that different P b spins may be surrounded by different numbers of TLSs, leading to a much greater level of disorder.In this high disorder prescription the fraction of DBs with n TLSs within a "coupling radius" follows a Poisson dis-tribution, where N is the average number of TLSs coupled to each DB.
We assume that the TLSs do not interact with each other and are independently distributed with energy splitting in the interval E ∈ [0, E max ], each with density proportional to E α .The average spin magnetization for a DB interacting with n TLSs is given by where Γ(E i , T ) is the spin relaxation rate for a DB interacting with one TLS of energy E i , and the associated magnetization decay.Taking an average over the number n of TLSs nearby each DB leads to (4) This expression shows that the Poisson distribution of TLSs makes DB spin relaxation highly non-exponential in time.
In order to complete the model we need to obtain a suitable expression for the relaxation rate Γ(E, T ), the rate for a DB spin to achieve thermal equilibrium with a single TLS of energy E. The mechanism is based on spinorbit induced cross-relaxation [7].When a TLS switches, the local environment around the DB spin fluctuates; spin-orbit coupling translates this switch into a fluctuating magnetic field that flips the spin.
The energy eigenstates of each TLS are denoted | ± with energies ±E/2, and the transition rate for a TLS to switch from state | ± to state | ∓ is denoted r ± (the subscript refers to the initial state in the transition).
When the magnetic field is low so that DB Zeeman energy is much less than both k B T and E, the rate for crossrelaxation is well approximated by [7] with A ≪ 1 a dimensionless spin-orbit coupling parameter.Here Γ +↑ denotes the rate for a cross switch from the TLS-DB state |+, ↑ into the state |−, ↓ .These cross rates are much stronger than non-cross spin flips because they couple states that are not the time reversal of each other (|+, ↑ is not the time reversal of |−, ↓ ).
The thermalized rate Γ(E, T ) is obtained by averaging over TLS and DB spin states with their corresponding Boltzmann occupations.Denote p(i|σ) the probability of finding TLS in state i = +, − given that the DB spin is known to be in state σ =↑, ↓.In the limit of DB Zeeman energy much smaller than k B T, E we get p(i| Note how the DB spin relaxation rate Γ(E, T ) is solely determined by the TLS rates r ± .To describe the physics up to quite high temperatures we generalized the theory for r ± described in [7] to processes involving one and two acoustic phonons.The final result is where β = 1/(k B T ), and a and b model the linear and quadratic dependence of TLS parameters on the phonon dilation strain, respectively.Therefore, a models the efficacy of TLS flipping following the emission/absorption of a single acoustic phonon and b models the same effect involving two phonons.Equation ( 7) was plugged into Eqs.( 3) and ( 4) leading to an explicit analytic expression for the measured inversion recovery curve, We stress again the highly non-exponential form of the model.This expression has five fitting parameters: a, b, α, N , E max .The fitting was done by assuming the first four parameters independent of temperature, with E max temperature dependent according to The assumed polynomial fit for the T dependence of E max may be seen as an approximation to the complex TLS thermal activation.The fit results are reported in Table II.The common value of N ≈ 0.7, i.e. less than one, reveals either a relatively diluted distribution of TLSs at or close to the interface and/or strong space localization for the TLS atomic configuration.The differences between the two samples may be tentatively ascribed to their different structural characteristics -mainly diameter and nanowire density on the surface.This applies to a and b values, which have a higher impact on the effective typical times of the inversion recovery.Differences can be observed also in the parameters describing the thermal evolution of E max , though the two leading terms, c and d, are relatively similar.This implies that the difference between the two samples is more relevant in the higher temperature range.
Conclusions.-Weexploited the high interface area of silicon nanowires to detect, with good signal-to-noise ratio, the electron spin inversion recovery of P b centers at the Si/SiO 2 interface.A novel model was developed to describe the non-exponential character of the inversion recovery, attributed to a relatively dilute density of TLSs.The fact that each dangling-bond center interacts, on the average, with less than one TLS ( N < 1) indicates the TLSs are highly localized and/or dilutely distributed across the interface.The proposed method provides information on the TLSs and can be extended to other relevant systems.A comparison of the results with theoretical models of specific TLSs, such as amorphous modes associated to hydrogen or other point defects in the oxide, may lead to the still missing identification of the microscopic nature of the TLSs.
M.F. and M.B. acknowledge financial support from the CARIPLO Foundation (Italy), ELIOS project, and the Italian Ministry of Defense, QUDEC project.R.d.S. acknowledges financial support from NSERC (Canada) through its Discovery (RGPIN-2015-03938) and Collaborative Research and Development programs .We thank J. Fabian for useful discussions.

FIG. 1 .
FIG. 1. Scanning Electron Microscope images of the two systems under investigation.Images on the left refer to sample A, while images on the right refer to sample B. The images were taken on twin samples obtained from the same batches of the ones used for magnetic resonance investigations.

FIG. 2 .
FIG. 2. (Color online)Comparison between a single exponential inversion recovery fit at 5 K for sample B and a fit according to the model outlined in Eq. (8).