Growth Of Ag Films Research Articles

Oxygen-induced Ag-based binary structure for efficient heat-regulating windows

Heat-regulating (HR) windows are essential for net-zero-energy building approaches. Silver (Ag) thin films are preferred for developing high-functioning HR windows because of their unique optoelectrical properties. The quality of the HR windows depends on the tradeoff between the transparency and conductivity of the Ag film. Ultrathin Ag (thickness < 10 nm) obtained using the wetting technique has been universally adopted to overcome this trade-off. However, the use of a wetting layer can hinder the optimum transparency of HR windows because of the possible optical loss associated with the wetting layer. Herein, the partial oxidation technique was thoroughly investigated for fabricating one-step grown Ag used for high-functioning HR windows. The incorporation of oxygen (5.7 % volume) during Ag sputtering allowed the direct growth of continuous Ag film on substrates with a percolation threshold below 4 nm. Partial oxidization of Ag resulted in modulated optical coefficients of Ag, thereby facilitating the ease of developing AZO/Ag (6.5 nm)/glass with optimum transparency of 98 % and sheet resistance of 12 Ω sq−1. AZO/Ag/glass possessing a high Haacke figure of merit 68.1 (10−3 Ω−1) was developed for HR windows, with the capability of providing a temperature discrepancy of 2 °C for different simulated weather conditions.

Stable Ultrathin Ag Electrodes by Tailoring the Surface of Plastic Substrates for Flexible Organic Light-Emitting Devices.

Flexible transparent metal electrodes (FTMEs) have significant application potentials in the fields of flexible optoelectronic devices due to their outstanding optical transmittance and electrical conductivity. However, obtaining excellent optoelectrical properties and mechanical flexibility of FTMEs is challenging because ultrathin metal layers usually follow an island growth mode. In this paper, flexible transparent ultrathin Ag electrodes with high mechanical stability and good optoelectrical properties were exploited by tailoring the surface properties of plastic substrates with ultraviolet-ozone (UVO) treatment for regulating the nucleation and growth kinetics of Ag films. The composite transparent electrodes of Ag (9 nm)/MoO3 (20 nm) fabricated on the UVO-treated polyethylene terephthalate (PET) substrates possess a low sheet resistance of ∼7.9 Ω/sq, a high optical transmittance of ∼87.2% at 550 nm, a long-period environmental stability of 30 days (∼65 °C, ∼80% humidity), and excellent mechanical flexibility of 100,000 bending cycles at a bending radius of 1.5 mm. These properties are derived from the surface treatment of PET substrates by UVO, which increases substrate surface energy and produces chemical nucleation sites of the phenolic hydroxyl groups. The phenolic hydroxyl groups generated on the PET surface not only provided efficient nucleation sites for subsequent Ag film growth but also formed C-O-Ag bonds between the substrate surface and the Ag layer, which act as "anchor chains" to fix firmly the Ag atoms on the substrate surface. As a universal applicability strategy, the composite electrodes on the UVO-treated polyethylene naphthalate (PEN) and norland optical adhesive 63 (NOA63) substrates also possess excellent optoelectrical properties and mechanical flexibility. Based on the ultrathin Ag composite electrodes, the flexible white organic light-emitting devices with PET, PEN, and NOA63 as substrates present the maximum current efficiencies of 53.0, 77.0, and 65.2 cd/A, respectively.

Low-E glass improvement by the understanding and control of the Ag growth

Ag thin films are widely used in Low-E glass. High conductivity and low emissivity require a film thickness of at least 10 nm, which is costly for the industry. The cost of Low-E coating could be drastically reduced by lowering the amount of Ag without decreasing the conductivity. For that purpose, it is necessary to understand Ag film growth and related macroscopic properties. This can be studied by thin film growth simulations. In this work, we present results of Ag growth on ZnO by magnetron sputtering and related properties with multiscale simulations (DFT, Molecular Dynamics, and kinetic Monte Carlo). A new method to take into account defects generated by magnetron sputtering has been developed. Simulated conductivity has been compared with experiments and shows the same trend with a difference attributed to electron scattering at grain boundaries. To go further, we propose patterned substrates with and without defects to find an original tuneable process to increase the conductivity of ultrathin metallic films. We show that a patterned substrate does not increase the conductivity. Instead, it leads to a controlled growth of small islands which could be used for other applications requiring plasmonic properties. Finally, conductivity is higher for the same deposited thickness when the substrate has a higher defect density.

Establishing Substoichiometric Ag Oxidation and Its Physicochemical, Optoelectrical, and Structural Consequences for Ag Electrodes

Oxidation is adversely associated with inducing catastrophic degradation of the electrical and optical properties of Ag film electrodes. Existing information is largely distributed across thermodynamically favorable, stoichiometric, and higher oxidation states. However, this information is elusive in affording a better understanding of substoichiometric oxidation in extremely oxygen-deficient states because conventional perspectives disregard the plausibility of substoichiometric oxidation. This study presents computational and experimental evidence for the implementation of substoichiometric Ag oxidation with unique physicochemical properties that are clearly discordant with those of the stoichiometric and higher oxidation states. Further, characterization results confirmed the favorable contribution of extreme substoichiometric oxidation to morphological/structural improvements by hindering the oxidation-induced optoelectrical degradations of Ag film electrodes. The unexpected outcomes of substoichiometric oxidation can be attributed to the spontaneous reduction of the oxide phases under ambient atmosphere conditions at 20 °C and 40% relative humidity, owing to their thermodynamic instabilities, while successfully implementing the oxidation-mediated, two-dimensional growth of Ag films. These findings provide a comprehensive understanding beyond the conventional conclusions drawn from stoichiometric and higher oxide phases. Therefore, the precise control of substoichiometric oxidation facilitates the preparation of highly efficient Ag electrodes by successfully implementing ultralow optoelectrical losses and structural deficiencies for Ag films simultaneously.

Manipulation of thin metal film morphology on weakly interacting substrates via selective deployment of alloying species

We demonstrate a versatile concept for manipulating morphology of thin (≤25 nm) noble-metal films on weakly interacting substrates using growth of Ag on SiO2 as a model system. The concept entails deployment of minority metallic (Cu, Au, Al, Ti, Cr, and Mo) alloying species at the Ag-layer growth front. Data from in situ and real-time monitoring of the deposition process show that all alloying agents—when deployed together with Ag vapor throughout the entire film deposition—favor two-dimensional (2D) growth morphology as compared to pure Ag film growth. This is manifested by an increase in the substrate area coverage for a given amount of deposited material in discontinuous layers and a decrease of the thickness at which a continuous layer is formed, though at the expense of a larger electrical resistivity. Based on ex situ microstructural analyses, we conclude that 2D morphological evolution under the presence of alloying species is predominantly caused by a decrease of the rate of island coalescence completion during the initial film-formation stages. Guided by this realization, alloying species are released with high temporal precision to selectively target growth stages before and after coalescence completion. Pre-coalescence deployment of all alloying agents yields a more pronounced 2D growth morphology, which for the case of Cu, Al, and Au is achieved without compromising the Ag-layer electrical conductivity. A more complex behavior is observed when alloying atoms are deposited during the post-coalescence growth stages: Cu, Au, Al, and Cr favor 2D morphology, while Ti and Mo yield a more pronounced three-dimensional morphological evolution. The overall results presented herein show that targeted deployment of alloying agents constitutes a generic platform for designing bespoken heterostructures between metal layers and technologically relevant weakly interacting substrates.

Mesoporous Ag–TiO2 based nanocage like structure as sensitive and recyclable low-cost SERS substrate for biosensing applications

We fabricated a porous Ag–TiO2 film with a nanocage (NC) like structure as a sensitive, recyclable, and low-cost SERS substrate using facile methods. The developed nanocage structure of TiO2 film was first coated on a glass plate by doctor-blade method, and then thin film of silver with an optimum thickness of 10 nm was deposited onto it using thermal evaporation method. The unique mesoporous cage-like structure of TiO2 film leads to the exceptional and non-uniform growth of Ag film. The developed Ag nanostructure is also porous, caps spherical in shape, and the average separation between two consecutive nanostructures is 10 nm. These closely decorated nanostructures of Ag produce hotspots by localising the incident electric field within the minimal volume. The unique nanocage structure of TiO2 film provides a larger effective surface area of adsorption to the analytes. Thus, the synergistic effects of Ag–TiO2 NC structure exhibit significant enhancement of 108 considering rhodamine-6g (R6G) molecule as a probe using a portable handheld Raman spectrometer. The amount of electromagnetic enhancement was further investigated by 3D FDTD simulation that matches well with the experiments. The recyclability of the developed SERS substrate was explored by placing the SERS substrate under UV illumination for 130 min. The enhanced photocatalytic activity of Ag–TiO2 NC SERS film degrades the dye molecules under UV radiation multiple times successively. Further, the Ag–TiO2 SERS substrate is highly sensitive to detect various concentrations of urea up to 1 mM covering the vital range of blood urea level. Thus, the optimised, low-cost, reusable SERS substrate can be used as level free biosensors for routine analysis of biomolecules in clinical applications.

Dependence of the Generation Behavior of Charged Nanoparticles and Ag Film Growth on Sputtering Power during DC Magnetron Sputtering

Effects of sputtering power on the deposition rate and microstructure, crystallinity, and electrical properties of Ag films during direct current (DC) magnetron sputtering are investigated. Thin films (~ 100 nm) are deposited at sputtering powers of 10, 20, 50, 100, 200 and 300 W and analyzed by field-emission scanning electron microscopy (FESEM), X-ray diffraction (XRD), transmission electron microscopy (TEM) and a four-point probe. The film deposited at a sputtering power of 10 W has the lowest growth rate, but the highest crystalline quality, with the lowest full width at half maximum (FWHM) and the lowest resistivity. The film deposited at a sputtering power of 200 W has the highest growth rate, and the second best crystalline quality in view of FWHM and resistivity. The film deposited at a sputtering power of 50 W has the moderate growth rate, and the worst crystalline quality in view of FWHM and resistivity. High-resolution TEM observations reveal that films deposited at sputtering powers of 10 and 200 W have far fewer defects, such as grain boundaries, dislocations and stacking faults than those deposited at a sputtering power of 50 W. Such deposition behavior could be explained by sputtering power, which affected the generation of the charged nanoparticles. And the high quality of films could be obtained at a high deposition rate, in which charge plays an important role.Graphic

Influence of aluminum interlayer on optical properties of very thin silver thin film

Generally, Ag thin films on glass substrates grow three-dimensionally because of the high Ag/glass interface free energy. However, flatter growth has been reported to be achieved by introducing an appropriate interlayer between the substrate and Ag film. With flatter growth, enhanced optical properties are expected. In the present study, very thin Ag films were deposited on an Al interlayer and their optical properties over a wide wavelength range from the visible-light region to the infrared region were investigated. Ag/Al films and Ag films were formed on substrates via resistance-heating vacuum evaporation. The Al interlayer thickness was fixed at 1 nm, and the Ag thickness was varied from 6 to 15 nm. The surface morphology was observed by atomic force microscopy, and the sheet resistance was measured by the four-point-probe method. The visible-light transmittance was evaluated using a spectrophotometer, and the infrared reflectance was evaluated using Fourier transform infrared spectroscopy. The surface roughness of a 6 nm-thick Ag film was found to be reduced from 1.0 nm to 0.6 nm by the introduction of the interlayer. In the absence of the interlayer, the sheet resistance was too high to be measured; however a sheet resistance of 16.2 Ω/sq. was obtained when the interlayer was present. Similar effects were observed for a 10 nm-thick Ag film. Ag films with thicknesses of 6 and 10 nm showed improved visible-light transmittance and infrared reflectance upon introduction of the interlayer. However, when the thickness of the Ag film was 15 nm, no difference in the sheet resistance or optical properties was observed between samples with and without the interlayer. As a consequence, we found that introducing an Al interlayer is effective for two-dimensional-like growth of Ag films, leading to enhanced visible-light transmittance and infrared reflectance.

Ultrathin continuous silver film for efficient infrared emission

An ultrathin silver (Ag) film can behave as an effective medium for enhancing infrared emission due to the tunneling effect and high extinction coefficients in the infrared range. Owing to a fully planar structure, the ultrathin Ag film is free from the complicated fabrication process and is available to practical application. Generally, the infrared emission efficiency of the ultrathin Ag film is deeply affected by film continuity. Herein, by applying plasma treatment and increasing the deposition rate, our work provides an effective solution to fabricating the ultrathin continuous Ag film. Plasma treatment can improve the wetting ability of Ag atoms on substrates and facilitate the lateral growth of Ag film, while high deposition rate can alter the traditional nucleation process. Moreover, the thickness required to obtain an ultrathin continuous Ag film is simultaneously reduced to 5 nm. Additionally, the influence of the ultrathin Ag film on the infrared emission performance of a dielectric-metal-dielectric multilayered infrared emitter is studied. The improvement in continuity and the reduction in thickness of the ultrathin Ag film facilitate the emitter to obtain a desired and efficient infrared emission performance. Our study provides an effective method to fabricate ultrathin continuous Ag film, which enables broadband and highly efficient infrared emission.

Simultaneous achievement of high visible transmission and near-infrared heat shielding in flexible liquid crystal-based smart windows via electrode design

Light and heat management are the two vital requirements for smart window towards ideal solar control. However, as a representative polymer dispersed liquid crystal (PDLC) device, little attention has been paid to heat management. Herein, a TiO2/Ag(Cu)/TiO2 (TCAT) electrode was designed and optimized to achieve the tunability of transmission in visible range and near-infrared heat shielding simultaneously. The continuous growth of ultrathin Ag films with low percolation threshold was achieved on TiO2 underlayer by doping a certain amount of Cu. Then the trade-off of PDLC devices’ optoelectrical properties was realized. The average ON state specular transmittance of TCAT-based PDLC devices reached 81.9%, 84.9%, 81.9% and 79.9% for Ag(Cu) thicknesses of 7 nm, 9 nm, 11 nm and 13 nm, respectively. In addition, TCAT electrodes exhibited satisfactory ratio of light to solar gain by taking visible transmittance and solar heat gain coefficient into comprehensive consideration. Compared with ITO-based PDLC counterpart, a notable temperature-decreases of 4.6 °C was observed at ON state of TCAT-based device, indicating an excellent heat shielding performance. The proposed TCAT electrode design is promising in smart window application especially for building energy conservation in hot areas or the summer season.

Growth of Ag(1 1 1) on Si(1 1 1) with nearly flat band and abrupt interface

Growth of Ag films of up to 30 nm thickness on Si(1 1 1) 7 × 7 at room temperature is investigated by low energy electron diffraction (LEED), X-ray photoelectron spectroscopy (XPS) and scanning tunneling microscopy (STM). LEED revealed the coexistence of Ag and Si spots starting with 1 monolayer (ML) of Ag deposited. The Ag lattice constant, starting with 25 ML, is slightly higher than for bulk Ag and increase linearly with Ag thickness, reaching about 4.2 nm for the thickest films. The average terrace widths detected from LEED spot profile analysis are about 30 nm for clean Si(1 1 1) 7 × 7 and about 5.5 nm for the thickest Ag(1 1 1) film, in agreement with STM observations. The intensity variation of core levels analyzed by XPS is taken into account by a model assuming the initial formation of Ag islands with linear variation of coverage vs. the amount of Ag deposited, followed by growth in a quasi layer-by-layer mode. The interface barrier is in the range of 0.4 eV, lower than all values reported previously. Ag deposited on Si(1 1 1) 7 × 7 at room temperature provides flat Ag(1 1 1) for synthesis of 2D materials, and may be used for low barrier Schottky diodes.

Initial growth and microstructure feature of Ag films prepared by very-high-frequency magnetron sputtering**Project supported by the National Natural Science Foundation of China (Grant Nos. 11675118 and 11275136).

The initial growth and microstructure feature of Ag films formation were investigated, which were prepared by using the very-high-frequency (VHF) (60 MHz) magnetron sputtering. Because of the moderate energy and very low flux density of ions impinging on the substrate, the evolutions of initial growth for Ag films formation were well controlled by varying the sputtering power. It was found that the initial growth of Ag films followed the island (Volmer—Weber, VW) growth mode, but before the island nucleation, the adsorption of Ag nanoparticles and the formation of Ag clusters dominated the growth. Therefore, the whole initial stages of Ag films formation included the adsorption of nanoparticles, the formation of clusters, the nucleation by the nanoparticles and clusters simultaneously, the islands formation, and the coalescence of islands.

Control of growth and structure of Ag films by the driving frequency of magnetron sputtering

The growth and structural properties of Ag films prepared by radio-frequency (2, 13.56 and 27.12 MHz) and very-high-frequency (40.68 and 60 MHz) magnetron sputtering were investigated. Using 2 MHz sputtering, the Ag film has a high deposition rate, a uniform and smooth surface and a good fcc structure. Using 13.56 and 27.12 MHz sputtering, the Ag films still have a high deposition rate and a good fcc structure, but a non-uniform and coarse surface. Using 40.68 MHz sputtering, the Ag film has a moderate deposition rate and a good fcc structure, but a less smooth surface. Using 60 MHz sputtering, the Ag film has a uniform and smooth surface, but a low deposition rate and a poor fcc structure. The growth and structural properties of Ag films are related to the ions’ energy and flux density. Therefore, changing the driving frequency is a good way to control the growth and structure of the Ag films.

Optimizing ultrathin Ag films for high performance oxide-metal-oxide flexible transparent electrodes through surface energy modulation and template-stripping procedures

Among new flexible transparent conductive electrode (TCE) candidates, ultrathin Ag film (UTAF) is attractive for its extremely low resistance and relatively high transparency. However, the performances of UTAF based TCEs critically depend on the threshold thickness for growth of continuous Ag films and the film morphologies. Here, we demonstrate that these two parameters could be strongly altered through the modulation of substrate surface energy. By minimizing the surface energy difference between the Ag film and substrate, a 9 nm UTAF with a sheet resistance down to 6.9 Ω sq−1 can be obtained using an electron-beam evaporation process. The resultant UTAF is completely continuous and exhibits smoother morphologies and smaller optical absorbances in comparison to the counterpart of granular-type Ag film at the same thickness without surface modulation. Template-stripping procedure is further developed to transfer the UTAFs to flexible polymer matrixes and construct Al2O3/Ag/MoOx (AAM) electrodes with excellent surface morphology as well as optical and electronic characteristics, including a root-mean-square roughness below 0.21 nm, a transparency up to 93.85% at 550 nm and a sheet resistance as low as 7.39 Ω sq−1. These AAM based electrodes also show superiority in mechanical robustness, thermal oxidation stability and shape memory property.

Sputtered Ag thin films with modified morphologies: Influence on wetting property

Silver thin films with thickness ranging from 3nm to 33nm were sputter deposited onto silicon wafers and tungsten layers. Those W layers were previously synthesized in the same DC magnetron sputter deposition system with various experimental conditions (argon pressure, target to substrate distance) in order to stabilize different surface morphologies. SEM observations and AFM images showed that the growth mode of Ag films is similar on Si substrates and on the smoothest W layers, whereas it is modified for rough W layers made of sharp grains. The effect of the W layer morphology on Ag film growth was clearly evidenced when the deposition took place at high temperature. It is seen that performing the deposition onto substrates of various morphologies allows tailoring the wetting property of the Ag deposit.

Growth of Atomically Flat Ultra-Thin Ag Films on Si(111) by Introducing a √3 × √3-Ga Buffer Layer

It is known that, when Ag is deposited on Si(111)-7×7 substrates in a conventional growth procedure at room temperature, no atomically flat Ag film could be obtained. We use scanning tunneling microscopy and low-energy electron diffraction to investigate the growth of ultra-thin Ag films on the Si(111) substrates at room temperature. Our study reveals that, upon introducing a Si(111)-√3 × √3-Ga buffer layer, atomically flat Ag films can easily grow on Si(111) with a critical thickness of two monolayers. Moreover, Ag film growth follows a layer-by-layer mode with further deposition. This novel growth behavior of Ag can be explained in terms of a free electron model (i.e., particle in a box) and kinetic Monte Carlo simulations.

Improvement of transparent silver thin film anodes for organic solar cells with a decreased percolation threshold of silver

Organic solar cells (OSCs) using thermally evaporated Ag thin films as the transparent anode are investigated with poly(3-hexylthiophene) (P3HT) and [6,6]-phenyl-C61-butyric acid methyl ester (PCBM) films as the active layer. The optimal power conversion efficiency (PCE) of 2.57% is obtained for the device with the 11nm thick Ag layer, namely the percolation threshold of Ag. Then an interlayer of MoO3 between the Ag anode and glass substrate is introduced. Two different thicknesses of MoO3 (2nm or 10nm) are chosen here, corresponding to the unclosed or closed MoO3 layer, respectively. It is observed that the introduction of the MoO3 interlayer can effectively improve the wetting of Ag on the substrate and reduce the percolation threshold of Ag. When the MoO3 thickness is 10nm, since the surface energy of Ag (γ=1.25Jm−2) is higher than that of MoO3 (γ=0.06Jm−2), the Ag–Ag interactions are stronger than the Ag-substrate interactions, weakening the surface-modifying effect. In contrast, since the 2nm thick unclosed MoO3 layer may create preferred nucleation sites on the substrate to enhance the lateral growth of Ag film effectively, a very low sheet resistance of 9.32 Ω/□ and a relatively high transmittance are obtained for MoO3 (2 nm)/Ag (9 nm) anode. The PCE of corresponding OSCs can be improved to 2.71%, comparable to that of ITO-based reference OSCs (PCE of 2.85%). From our results, it is concluded that the performance of the Ag thin film electrode can be improved by decreasing the percolation threshold of Ag with a MoO3 interlayer (especially when the MoO3 layer is thin). It may be instructive for the further research of metal thin film electrodes.

Composite PET Membrane with Nanostructured Ag/AgTCNQ Schottky Junctions: Electrochemical Nanofabrication and Charge-Transfer Properties

Large-area nanostructured Ag/Ag-tetracyanoquinodimethane (TCNQ) Schottky junctions are fabricated electrochemically on a mesoporous polyethylene terephthalate (PET) membrane-supported water/1, 2-dichloroethane (DCE) interface. When the interface is polarized, Ag(+) ions transfer across the PET membrane from the aqueous phase and are reduced to form metallic Ag on the PET membrane, which reacts further with tetracyanoquinodimethane (TCNQ) in the DCE phase to form nanostructured Ag/AgTCNQ Schottky junctions. Once the mesoporous membrane is blocked by metallic Ag, a bipolar mechanism is proposed to explain the successive growth of AgTCNQ nanorods and Ag film on each side of the PET membrane. Due to the well-formed nanostructure of Ag/AgTCNQ Schottky junctions, the direct electrochemical behavior is observed, which is essential to explain the physicochemical mechanism of its electric performance. Moreover, the composite PET membrane with nanostructured Ag/AgTCNQ Schottky junctions is tailorable and can be assembled directly into electric devices without any pretreatment.

Metal Contacts on Physical Vapor Deposited Monolayer MoS2

The understanding of the metal and transition metal dichalcogenide (TMD) interface is critical for future electronic device technologies based on this new class of two-dimensional semiconductors. Here, we investigate the initial growth of nanometer-thick Pd, Au, and Ag films on monolayer MoS2. Distinct growth morphologies are identified by atomic force microscopy: Pd forms a uniform contact, Au clusters into nanostructures, and Ag forms randomly distributed islands on MoS2. The formation of these different interfaces is elucidated by large-scale spin-polarized density functional theory calculations. Using Raman spectroscopy, we find that the interface homogeneity shows characteristic Raman shifts in E2g(1) and A1g modes. Interestingly, we show that insertion of graphene between metal and MoS2 can effectively decouple MoS2 from the perturbations imparted by metal contacts (e.g., strain), while maintaining an effective electronic coupling between metal contact and MoS2, suggesting that graphene can act as a conductive buffer layer in TMD electronics.

Roughening in Electronic Growth of Ag on Si(111)-(7×7) Surfaces

Roughening in the electronic growth of Ag films on Si(111)-(7×7) surfaces for a film thickness ranging from 1 to 30 monolayers is reported. Ag films exhibit the growth of flat-top plateaus of preferential heights due quantum electronic effect. We have observed roughening of the film growth due to instability with linear diffusion characterized by the ln(θ)(1/2) dependence of the local surface slope, where θ is the Ag coverage. The roughening of the surface morphology has been characterized by scaling exponents α, β and 1/z, which are determined using scanning tunneling microscopy. Increased value of α = 0.67 ± 0.04 at the early stage of the electronic growth with two atomic layer height flat-top isolated Ag mounds to 0.77 ± 0.06 at the later stage of the growth when isolated mounds coalesce and form percolated structures maintaining preferential heights of an even number of atomic layers in the Ag mounds indicates the instability in the electronic growth. As a result, interface width W increases as a power law of coverage (θ), W ∼ θ(β), with growth exponent β = 0.33 ± 0.03, and lateral correlation length ξ grows as ξ ∼ θ(1/z) with 1/z = 0.27 ± 0.05.