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[J. R. Stellhorn](https://orcid.org/0000-0002-5579-6902), [A. Masuno](https://orcid.org/0000-0003-0667-9782), [Y. Onodera](https://orcid.org/0000-0002-3080-6991), [S. Kohara](https://orcid.org/0000-0001-9596-2680), K. Yoshida, [Y. Yanaba](https://orcid.org/0000-0003-1608-2146), H. Inoue, [T. Ohkubo](https://orcid.org/0000-0001-8187-1470), [H. Taniguchi](https://orcid.org/0000-0002-1773-7856)

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[Atomic-scale insights into the high dielectric permittivity of bismuth silicate glass](https://mdr.nims.go.jp/datasets/36a46a46-03fb-44b2-8668-b9a95130e4db)

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APS/123-QEDAtomic-Scale Insights into the High Dielectric Permittivity ofBismuth Silicate GlassJ. R. StellhornCo-Creation Institute for Advanced Materials,Shimane University, Shimane 690-8504,Japan & Department of Material Chemistry,Graduate School of Engineering, Kyoto University, Kyoto 615-8520, Japan ∗A. MasunoDepartment of Material Chemistry, Graduate School of Engineering,Kyoto University, Kyoto 615-8520, JapanY. Onodera and S. KoharaCenter for Basic Research on Materials,National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, JapanK. YoshidaGraduate school of Science and Technology,Hirosaki University, Aomori 036-8561, JapanY. Yanaba and H. InoueInstitute of Industrial Science, The University of Tokyo, Tokyo 153-8505, JapanT. OhkuboGraduate School of Engineering, Chiba University, Chiba 263-8522, JapanH. TaniguchiDepartment of Physics, Nagoya University,Furo-cho, Nagoya 464-8602, Japan(Dated: September 27, 2025)1AbstractThe amorphous phase of bismuth silicate (Bi2SiO5) is characterized by an exceptionally largedielectric permittivity over a wide temperature range. This study explores the relationship betweenthis remarkable property and the material’s atomic-scale structure, which has been modeled fromexperimental X-ray and neutron scattering as well as EXAFS and NMR spectroscopy data in aReverse Monte-Carlo approach. The resulting structural model is analyzed to reveal short- andintermediate-range features on the atomic scale.Our results show that the exceptional dielectric performance stems from the asymmetric coor-dination of Bi-Ox polyhedra as well as a nano-segregation induced by SiO4 chains, which togetherboost local polarizability. These findings establish a direct link between disordered atomic struc-tures and enhanced dielectric properties, and sets a new benchmark for amorphous materials.I. INTRODUCTIONAmorphous compounds and glasses are not typically associated with high a dielectricpermittivity. Notable applications in this area, for instance as capacitor materials, aretherefore largely limited to some specialized devices that demand extremely high stability,reliability, and/or tolerance to high voltages and temperatures. In this article, we reporton a bismuth silicate glass that possesses a high dielectric permittivity and a low dielectricloss over a wide temperature range from 200 to 600 K. Such properties make it a promisingcandidate for more widespread applications.The dielectric permittivity, usually denoted by the dielectric constant ε, is a dimen-sionless property that quantifies a material’s ability to store electrical energy in responseto an applied electric field. In crystalline materials, a high permittivity is often achievedthrough collective atomic displacements or long-range polar order, whereas in glasses suchmechanisms are generally suppressed by structural disorder. The vast majority of knownamorphous materials therefore exhibit values of ε lower than 12. Adding heavier elements,in particular Ba or Pb, can increase this value to around 20 [1–3], or even 30 with furtheraddition of Bi [4, 5]. An suitable overview of the distribution glasses taken from the SciGlassNext database [6] with reported ε values is given in Figure 1 A. The newly investigated bis-∗ jrstellhorn@mat.shimane-u.ac.jp2muth silicate glass phase, a-Bi2SiO5 (or a-BSO), prominently stands out, reaching a value of56. For comparison, this value is even larger then the reported value for crystalline bismuthoxide (Bi2O3, with ε ≈ 33[7]) and just short of that of crystalline bismuth silicate (Bi2SiO5,with ε ≈ 90 [8]). The latter is a wide-gap semiconductor, with a reported band gap of about3.8 eV [8]. Both c-Bi2O3 and c-Bi2SiO5 contain Bi in the nominal 3+ valence state, wherethe presence of a stereochemically active lone pair gives rise to a large inherent polarizability[9, 10].We measured the dielectric response of the Bi2SiO5 glass in temperature range between200 and 600 K (Figure 1 B). In this wide temperature region, ε varies only in a narrowregion between 54 and 64. The dielectric loss is small in the range up to about 500 K with avalue of about 0.5%, but starts to increase above that temperature. Note that the materialitself is stable until about 750 K before it starts to crystallize.Further exceptional optical properties of bismuth oxide-based glasses have been reportedin various reviews, such as a high linear refractive index, low glass transition temperature,and a high third-order nonlinear optical susceptibility[10, 11]. Additionally, their largeelectronic polarizability has been characterized using the concepts of optical basicity andthe interionic interaction parameter [10]. It was found that Bi2O3-based glasses exhibit andistinctly strong basic nature and weak chemical bond strength among the various oxideglasses. These properties are attributed to the high cation polarizability of Bi3+ and thepresence of an electron lone pair in the valence shell [9].While the dielectric properties of crystalline materials, and their relationship with theatomic structure, have been extensively studied, amorphous compounds present a distinctivechallenge due to their structural disorder and lack of long-range order. Understanding andcharacterizing their dielectric behavior is essential both for fundamental scientific explorationas well as for uncovering novel technological applications.One of the few notable examples in which the intricate correlation between the atomicstructure of an amorphous compound and their optical properties has been thoroughlyinvestigated are chalcogenide glasses for their application as phase-change materials. Inthese systems, the dielectric permittivity is closely tied to intermediate-range atomic order:the theoretical framework described by Huang and Robertson [12] presents a link betweenthe optical properties of Ge-Te glasses and angular order at the second-neighbor level , whichplays a crucial role in aligning bond orbitals. This alignment can facilitate the formation3FIG. 1. (A) Histogram of all tabulated values of dielectric constants ε (at 1 MHz and room tem-perature) in the SciGlassNext database [6]. (B) Measurement of the dielectric properties, relativepermittivity ε and dielectric loss function tan(δ), of Bi2SiO5 glass as a function of temperature.of ’resonant’ [13] bonding configurations. In contrast, the absence of resonant bondingin the crystalline phase of Cu2GeTe3 (a novel phase-change material), where tetrahedralcoordination dominates, leads to the unusual scenario where the amorphous phase exhibits ahigher dielectric permittivity than the crystal. This phenomenon results in a negative opticalcontrast between the crystalline and amorphous phase [14, 15]. Even though the exactmechanism is likely to be different in oxide glasses, such findings emphasize the importanceto look beyond the atomic short-range order to understand the optical properties of glasses.The corresponding crystal structure of Bi2SiO5 (c-BSO) in itself has been a subject ofextensive research, especially with regard to its dielectric properties. The c-BSO structureconsists of one-dimensional chains of SiO4 tetrahedra running along the c axis, which aresandwiched by a double layer of BiO6 polyhedra [8, 16]. This stoichiometric crystal structureexists only in the non-equilibrium phase diagram; it can be fabricated from a crystallizationof the glass. In this process, the system initially crystallizes at about 470 ◦C into a phase thatcontains crystalline bismuth oxide (c-Bi2O3) and amorphous silica (a-SiO2), from which thec-BSO phase is formed at about 540-600 ◦C [8, 16]. Furthermore, dielectric measurements ofc-BSO have demonstrated that its ferroelectricity disappears in association with disorderingof the SiO4 chains through La doping, highlighting the critical role of these chains in the4material’s ferroelectric behavior [16].Other comparable crystalline phases with a nominal Bi3+ state, like c-Bi2O3, also consistof interconnected BiO5 or BiO6 polyhedra. The geometry of these polyhedra is determinedby the Bi lone pair, resulting in an asymmetrical bonding configuration in which all oxygenatoms are basically constrained to one hemisphere around Bi. The opposite side is occupiedby the electron lone pair. This arrangement leads to substantial atomic-level polarizability,contributing to the large dielectric constant observed in the crystal phase.Previous structural characterizations of the amorphous phase of the Bi2O3-SiO2 systemare reported by Witkowska et al. [17], and for low contents of Bi2O3 by Ohkura et al. [18].However, these works are mainly based on X-ray Absorption Fine Structure spectroscopy(XAFS), and do not propose a detailed structural model, since this method typically probesonly the immediate short-range order. In these glasses, the XAFS investigations observeonly the nearest O atoms around Bi [17, 18], with a reported coordination number of 3.0at a bond distance of 2.26 Å for the Bi2SiO5 glass [17]. Beyond the short-range order,an investigation by 29Si Magic-Angle Spinning Nuclear Magnetic Resonance (MAS-NMR)spectroscopy by Todea et al. [19] confirmed that most SiO4 units are part of chain struc-tures. Infrared and Raman spectroscopy data [20] further indicate that Bi maintains a 3+valence state in amorphous Bi-O systems, analogous to the crystalline phases. This suggeststhe persistence of a stereochemically active lone pair, which would provide a natural linkbetween local structure and the pronounced dielectric properties of a-BSO. However, themuch greater configurational freedom to arrange Bi-O units in the amorphous state makesit highly challenging to determine the actual local geometry.Our new study on a-BSO is based on a more comprehensive experimental approach. Wethereby establish a reliable model of the atomic structure, and shed light on the originof the materials’ distinct physical properties. In addition, we compare the local atomicgeometry of the amorphous phase with that of the crystal, in order to assess how the largerconfigurational freedom of Bi-O units influences the material’s dielectric response.II. EXPERIMENT AND ANALYSISa. Sample preparation To prepare the Bi2SiO5 glass samples, first Bi2O3 and SiO2powders were mixed in equimolar portions and sintered at 800 ◦C for 12 h. Then, small5beads of amorphous samples were synthesized by rapid quenching of the laser-heated meltsusing a containerless aerodynamic levitation technique [21].b. Composition analysis The composition of the sample was checked by InductivelyCoupled Plasma Optical Emission Spectroscopy (ICP-OES). We found the following ratiosof Bi and Si: Bi 79.0 wt.%, Si 5.60 wt.%. This indicates that a small amount of Bi2O3is evaporated during the sample preparation, and the actual composition is Bi1.9SiO4.84.However, for simplicity, we still refer to the nominal composition in the main text.c. Optical transmittance measurements We performed optical transmittance measure-ments of the Bi2SiO5 glass. From these, we can estimate that the optical band-gap energyis 3.0 eV. For reference, crystalline BSO was reported to have a bandgap of approximately3.8 eV[22]. Note that the bandgap of glasses is generally smaller than that of comparablecrystals because of their amorphous nature.d. Scattering experiments Total X-ray scattering experiments were performed atSPring-8 beamline BL04B2 at an energy of 113 keV. The weighting factors for the par-tial S(Q)’s in X-ray scattering depend on the scattering vector. Approximate values at thestructure factor maximum (Q=2 Å−1) are 0.60, 0.09, 0.26, 0.003, 0.02 and 0.03 for the Bi-Bi,Bi-Si, Bi-O, Si-Si, Si-O and O-O correlations, respectively.Additionally, neutron scattering experiments were performed at the high intensity totaldiffractometer NOVA, installed at BL21 of the Materials and Life Science (MLF) Experi-mental Facility at the J-PARC spallation neutron source, with a reciprocal space range ofup to 30 Å−1. The weighting factors depend on the scattering length of the atomic core,and are 0.1154, 0.0562, 0.3925, 0.0068, 0.0954 and 0.3337 (same order as above).e. EXAFS experiments Extended X-ray absorption spectroscopy experiments at theBi L3 edge were performed at the KEK-PF beamline BL27B in transmission mode. Thenear-edge region of both crystalline and amorphous Bi2SiO5 coincides well, and there is noobservable shift between them, indicating that both systems share a similar electronic stateof Bi3+.f. 29Si MAS NMR The 29Si MAS NMR spectroscopy measurements were performedon a JEOL JNM-ECA 500 spectrometer equipped with a MAS probe head (zirconia rotorwith 4 mm diameter) at 11.74 T (500 MHz). The 29Si NMR spectra were recorded using π/6pulses (1.5 µs) at a spinning rate of 10 kHz, a relaxation delay of 1 s, and 50,000 accumulated6signal transients. The 29Si chemical shift was referenced to an external standard, sodium3-(trimethylsilyl) propionate 2.2.3.3-d4 (+1.445 ppm).From the 29Si MAS NMR data, the ratios Qn of the connection between SiO4 tetrahedracan be extracted (where n is the number of bridging oxygen). The result is about 90% forQ2 species, i.e. corner-sharing tetrahedra arranged in a 1-D chain, and 10% for Q3 species,i.e. forming a 2-D network of corner-sharing tetrahedra. This information was included inthe RMC modeling procedure.g. Measurement of dielectric constants Dielectric measurements were performed in atemperature range from 200 to 600 K using a lab-made system equipped with a precisionLCR meter Keysight E4980A and a temperature controller Linkam THMS600.h. RMC analysis We employed the RMC POT package for the Reverse Monte Carlo(RMC) modeling [23, 24], with a box containing 2455 Bi, 1292 Si and 6253 O atoms (corre-sponding to the actual composition) in the appropriate density (0.06685 atoms/Å3). Nearestneighbor distances were defined as (in Å) 3.3, 3.3, 2.0, 3.1, 1.4 and 2.5 for the Bi-Bi, Bi-Si,Bi-O, Si-Si, Si-O and O-O pairs, respectively. Furthermore, constraints on the Si-O corre-lation were included to realize an environment in which every Si is surrounded by exactly4 O neighbors in a tetrahedral coordination, and to suffice the second-nearest neighborconstraints that were determined experimentally from the 29Si NMR, with Q2 = 90% andQ3 = 10% of all SiO4 units. This coordination geometry was retained during the entiremodeling procedure by fixed neighbor constraints (FNC) and second-neighbor constraints,respectively. An additional coordination constraint was placed on Bi atoms to exclude ex-tremely under-coordinated cases, i.e. Bi-O coordination numbers of 0, 1 and 2.The experimental data from X-ray and neutron scattering, as well as the Bi L3 edgeEXAFS data were used as input. Additionally, the Fourier transform of the neutron struc-ture factor was used to improve the shape of the pair correlation functions in the rangebelow 5.4 Å. For the EXAFS data, a fit of the energy shift value ∆E0 was allowed, anddetermined to be 2.3 eV. The starting configuration consisted of SiO4 units with appropri-ate Qn connections, while Bi and the remaining O atoms were added at random positions.The final RMC fit qualities were determined with Rneutron g(r) = 0.037, Rneutron S(Q) = 0.061,RX−ray S(Q) = 0.073 and REXAFSχ(k) = 0.259.i. Persistence Homology For the Persistence Homology (PH) analysis, the HomCloudpackage was used [25]. The PH analysis offers a possibility to extract multi-scale information7about topological features of an atomic configuration. The homology is expressed as a 2Dhistogram called persistence diagram (PD) [26–29], typically illustrated with a ’birth’ and’death’ coordinate of a feature of interest. For the input configuration, we used the RMC-derived model with 10,000 atoms of a-BSO, and compare it with supercells of the crystalphases of BSO and Bi2O3 containing a comparable amount of atoms.This approach allows for an extended view into the intermediate-range order of a system.More detailed descriptions of this method can be found in the indicated references, alongwith examples for Cu-Zr [27] and SiO2 [28] glasses.j. Cluster analysis The ’cluster approach’ considers the immediate environment arounda specific element. In this approach, the nearest neighbors around each atom of the sameelement (Si and Bi respectively, in our case) are determined within a certain cut-off distance,which was set to 2.0 Å for Si and 2.85 Å for Bi, respectively, according to the extent of thefirst coordination shell from the partial pair distribution functions. Then, this cluster isisolated from the rest of the configuration. Its geometric center, i.e. the center of gravityof the oxygen atoms, is then translated to the origin of the coordinate system. The shiftof the central atom from the origin then includes some information on the symmetry of thecluster: in a symmetric case, e.g. a regular SiO4 tetrahedron, the Si atom will be located atthe center of gravity, but any distortion will shift it away from this position.Using the same clusters, another analysis can be performed: determining the centralatom’s chemical bonds’ spherical harmonics parameters. The concept of using sphericalharmonics (SH) parameters is an established approach to characterize bond-orientationalorder in liquids and glasses [30, 31]. They can be considered as quantitative descriptors ofthe angular distribution of interatomic vectors [32–34]. We denote the SH functions as Yl,mwith order l and degree m for the angular coordinates.The sum the of spherical harmonics is represented by the coefficient ql,m:ql,m =1NN∑iY il,m (1)Averaging over all allowed values m for a given index l leads to a quantitative descriptorthat is independent of any set of reference axes:8ql =√√√√ 4π2l + 1(l∑m=−lql,m)2(2)In the structure analysis using these rotationally invariant descriptors ql, higher-symmetryarrangements (e.g. cubic, tetrahedral) exhibit a higher degree of variability than lowersymmetry environments (e.g. triangular or random)[32], so that the variance of ql can beused as a quantitative descriptor of local symmetry.While the reference crystal structures possess a limited number of unique ql values foreach atomic site, the atomic configuration obtained from RMC modeling comprises one valuefor each central atom, and necessitates a statistical analysis.III. RESULTSAs outlined in the Experimental section, we use a combination of several experimentalcharacterizations (X-ray and neutron scattering, Bi XAFS, Si MAS NMR) integrated in aReverse Monte Carlo modeling approach to extract information on the atomic-scale structureof amorphous bismuth silicate.One of our main research objectives is to clarify whether the atomic structural informationallows to draw conclusions regarding the exceptionally large dielectric permittivity of a-Bi2SiO5. The corresponding geometries in Bi-O crystal phases represent suitable startingpoints for further considerations. They indicate that the free electron pair at Bi, and theasymmetrical bonding configuration connected with it, is one of the most important factorsleading to a significant polarizability. One major piece of this puzzle is therefore to analyzeif similar local environments can be identified in the amorphous phase on the short-rangeorder (SRO) scale. Furthermore, as noted in the introduction, there can be a significantdiscrepancy in the dielectric permittivity even for similar local building blocks, as is thecase in c-Bi2SiO5 vs. c-Bi2O3. Their main difference lies in the atomic intermediate-rangeorder (IRO) - i.e. the separation of the BiO6 double layers in c-BSO. Therefore, it stands toreason that a characterization of the IRO of the amorphous phase may also provide furtherhelpful information.Moreover, we also aim to elucidate why this material follows a 2-step crystallizationprocess (from a-BSO via a Bi2O3 phase to c-BSO), instead of a direct transformation (from9a-BSO to c-BSO). Understanding this pathway is crucial for elucidating the crystallizationkinetics, which play a vital role in controlling grain size, phase composition, and defectdensity, thereby enabling the optimization of the material for advanced device applications.FIG. 2. (A) Final configuration of the RMC model with SiO4 tetrahedra shown in blue and BiOx(x = 3− 7) polyhedra in purple. (B) Fits (red lines) to the experimental data (circles). .The initial configuration at the start of the modeling procedure consists of appropriatelyconnected SiO4 units in a box, with the remaining Bi and O atoms placed randomly. Fromthis initial guess, the RMC algorithm optimizes a structure that simultaneously satisfiesall available experimental information. The resulting configuration from RMC modeling isdisplayed in Figure 2 A. In the modeling procedure, the tetrahedral coordination of theSiO4 units and their connectivity (predominantly Q2) is constrained, and it is therefore wellrecognizable in the configuration. The fit results to the experimental data are shown inFigure 2 B.Looking at the SRO, we first analyze the partial pair correlation functions from the RMCmodel, as illustrated in Figure 3 A along with running values of the coordination numbers(red lines). All distributions are, naturally, more disordered compared to the crystal phase,however there are several remarkable similarities in the overall short-range order. Notably,the coordination numbers around Bi are 5.3 O atoms (in c-BSO: 6) in the range up to 3.0 Å,2.4 Si (cryst.: 2), and about 6.5 Bi (cryst.: 6) atoms, all depending on the exact cut-off10FIG. 3. (A) Partial pair correlation functions (black) along with running coordination numbers(red) of the final model, together with a comparison of c-BSO (gray bars). (B) The bond angledistributions O-Bi-O obtained from the RMC model (lines) in comparison with the Bi2SiO5 crystalstructure (bars).value for the coordination number.These parameters already hint at the existence (and prevalence) of BiO5 and BiO6 clus-ters. However, they do not yet provide evidence that their local geometry is indeed similarto the crystal - for instance, it is entirely possible that the oxygen atoms could be distributedin a random and/or symmetrical manner around Bi.A further look into the SRO around Bi can be extracted from the bond angle distributions(BAD) of O around Bi, as shown in Figure 3 B (with a comparison of the BAD of c-BSO).The crystalline BiO6 polyhedron has 3 main regions of O-Bi-O angles: a relatively widedistribution of around cos(θ) ≈ 0.3 (70◦), a narrow region around -0.4 (110◦), and a widerregion again around -0.8 (140◦) that is characteristic of the two O atoms facing the emptybond hemisphere. The RMC-derived BAD is naturally broad as a result of both the natureof the amorphous phase and the characteristics of the RMC technique, which tends to prefermore disordered configurations. However, it exhibits maxima in similar regions, with themost pronounced signal centered at 0.28 (73.7◦) and a another broad feature around -0.81(144.1◦).11FIG. 4. Analysis of local cluster geometries. (A) Shift of Bi [Si] from the cluster centroid in theBiOx (black squares) and SiO4 (red circles) clusters, and reference values for comparable crystallinephases. The dashed line is a guide for the eye and set at the average value of the RMC result ofBiOx. Also illustrated are some exemplary model structures (B) Analysis of the distribution ofbond spherical harmonics parameters for the BiOx [SiO4] clusters, comparing the RMC result(black) with c-BSO (red). The dashed line indicates the distribution for a theoretical completelyrandom case. In both cases, the graphs also show the standard deviations of the RMC model.For investigating the exact distribution of O around Bi, we analyze the local clustergeometries (see e.g. [32–34]) in the RMC model: For each BiOx (x ≥ 3) and SiO4 clusterin the model, the displacement rG,i of the central atom i from the cluster centroid (G) iscalculated, as illustrated in Figure 4 A. This distance provides a benchmark for the clustersymmetry, since in a perfectly symmetric cluster rG,i = 0. For instance, the SiO4 units evenin c-BSO are not perfectly tetrahedral, but slightly distorted, amounting to a small valueof rG,Si of 0.05 Å. In a similar manner, the SiO4 units in the RMC model are also sightlydistorted by a value of 0.08 Å on average, with a standard deviation of 0.05 Å.The displacement of Bi from the centroid of the BiOx clusters is rG,Bi = 0.57 with astandard deviation of 0.27 Å. The large standard deviation is owing to the fact that the RMCprocedure generates a wide range of possible local geometries. Exemplary configurations forthe smallest and largest deviations are also illustrated in Fig. 4 A. The average value isremarkably close to that found in c-BSO of 0.61 Å. For further comparison, Figure 4 A also12shows the displacements in two other bismuth oxides, c-Bi2O3 and c-BiO2. Both includetwo different types of BiOx clusters, as illustrated in the toy models in the figure. c-Bi2O3exhibits two local geometries, the first with a smaller displacement (0.32 Å), but the othercluster with an even larger displacement (rG,Bi = 0.68 Å) than c-BSO. We can explain thewide distribution in the RMC model by a continuous mix of such local geometries. On theother hand, more symmetrical distributions around Bi, as in the case of c-BiO2 with anoctahedral geometry (rG,Bi = 0.0 Å), can clearly be ruled out in a-BSO.Further details on the local symmetry of these clusters can be extracted from an analysisof the bond spherical harmonics (SH) parameters ql [32, 33] (see ’Methods’ section fordetails). They are displayed in Figure 4 B, again for BiOx as well as SiO4 clusters. Bydesign of the RMC model, the statistics of the ql parameters for the SiO4 clusters closelymatch those of the crystalline phase (shown in red). The standard deviations in each valuecorrespond to the variability in the Si-O bond of the RMC model. Both the crystal andamorphous distribution exhibit a large variance of the values: (var(ql) is 0.06 and 0.08 fora- and c-BSO, respectively). A large variance indicates a high symmetry of the cluster [32],and is a sign of the tetrahedral geometry.A lower symmetry (= smaller variance) is found in the case of the BiOx clusters. Notethat a theoretical limit for a completely disordered geometry would consist in every valueof ql being equal, as indicated by the dashed line (which is just the average of all ql). Thedistribution in a-BSO roughly follows that of c-BSO, with relatively high values at l = 1, 2,signifying the polar nature of the cluster. Both exhibit much smaller variances than theSiO4 units, with var(ql)=0.01 for the amorphous and 0.02 for the crystalline phase. The SHparameters of a-BSO therefore are in line with a cluster symmetry similar to that found inc-BSO, though with a more disordered arrangement.To characterize the IRO, we employ an analysis of the persistent homology (PH) of theRMC configuration. The PH analysis offers the possibility to extract multi-scale informationabout topological features. The homology is expressed as a 2D histogram called persistencediagram (PD) [26–29]. These diagrams pick up specific shapes in the data for multiplelength scales and for various dimensionalities Dn, e.g. one-dimensional linkages such as ringstructures (D1) or void regions (D2). The PD’s encode certain characteristics of such shapes,in particular the maximum distance between two adjacent atoms in a closed loop and the13FIG. 5. (A) Bi-centric persistence diagrams D2 from the RMC configuration of a-BSO, andcomparison with the data from crystalline references, c-BSO (B) and c-Bi2O3 (C).size of the ring/void.For our investigation, the Bi-centric D2 PD offers the most evident insight. It is displayedin Figure 5. The PD of the RMC configuration of a-BSO (a) is best understood by comparingit with the corresponding crystalline phases as shown for c-BSO (B) and c-Bi2O3 (C). Theirinterpretation is straightforward and illustrated by the toy models in the figure: c-BSOexhibits two regions; the first at a ’birth’ radius B of about 2.2 Å, which is related to thealmost tetrahedral arrangement of Bi inside a Bi2O2 layer, and the second region aroundB ≈ 3.2 − 3.6 Å is linked to the separation between the Bi2O2 layers. On the other hand,c-Bi2O3 only exhibits a single region at B ≈ 2.5 Å with a larger lifetime (= distance fromthe diagonal, corresponding to the size of the void), which is related to the nearly octahedralarrangement of Bi atoms in the unit cell (see inset illustrations in Figure 5). An octahedronencompasses a larger void area than a tetrahedron, and therefore exhibits a larger lifetimein the PD. The octahedra in c-Bi2O3 are also slightly distorted, which leads to a distributionof data points in a limited region.Amorphous BSO shows a mix of these characteristics: a separation into two distinctregions, with a wide-spaced first region with relatively large lifetime starting at B ≈ 2.2 Å,and a second region above B ≈ 3.2 Å. Owing to the nature of the amorphous phase [29],weak but long streaks towards long lifetimes are also observed, as well as a concentrationof most of the signal intensity close to the diagonal line, representing the overall disorderedarrangement of atoms.14FIG. 6. (A) View of the final RMC model for a-BSO, showing a slab with a thickness of 33% ofthe whole simulation box. Silica chains are illustrated as blue lines, and bonding polyhedra aroundBi in purple. (B) Detail of the c-BSO structure, illustrating that distortion vectors are pointed inseparate regions in space, in contrast to c-Bi2O3 (C), in which they pairwise point into the samespatial regions.IV. DISCUSSIONWe have established that the local environment of a-BSO is made up of the same buildingblocks as the reference crystal phases. With this thorough analysis, we are now able todiscuss the relationship between the atomic structure and the material properties of a-BSO.The fundamental structure of a-BSO is made up of SiO4 chains, between which a Bi2O3-rich phase is embedded. Note that this does not indicate a form of phase separation, sincethe Bi2O3-rich enclosures remain small, and there are not more than a few BiOx clustersbetween each strand of the SiO4 chains. An illustration of this architecture is given inFigure 6 A.Together with the PD analysis, which shows characteristics both from c-Bi2O3 and c-BSO, these findings shed light on the 2-step crystallization behavior [8, 16] that was notedin the introduction: upon heating, at around 470 ◦C, initially the Bi-rich regions start toform crystallization seeds for an expanded c-Bi2O3 phase; subsequently, (at around 540 ◦C)the remaining silica phase is incorporated to form c-BSO.15Furthermore, we are also in a position to explain the large dielectric permittivity of thebismuth silicate glass. The first point to emphasize is that the experimental data on theatomic SRO proves the existence of BiOx clusters (with predominantly x=5 and 6) in theamorphous phase, and that their local geometry resembles that of a (disordered) crystal. Inother words, the O atoms are largely pushed to one side of the bond hemisphere, leadingto a strongly polar atomic arrangement. This is a major driving force of the high dielectricpermittivity of this material.The IRO characteristics are another factor to increase the permittivity. Here, the con-nection of the SiO4 units play a vital role: These are mainly Q2-connected (i.e. silica chains,as they are also found in c-BSO) and, to a smaller amount, Q3-connected. A comparablebinary silicate glass is PbO-SiO2, in which a wide distribution of Qn values is found [35].This is associated with a much smaller dielectric constant, e.g. ε for binary PbO-SiO2 is only15.5 [36] and does not surpass 24.5 for any of the known binary PbO-SiO2 glasses [6, 37, 38].To explain the role of the silica chains in more detail, let us consider again the structuralproperties of c-BSO and c-Bi2O3, which posses strongly different dielectric constants (90vs. 33). The cluster polarization vectors, i.e. the vectors pointing in the direction of thedisplacement of Bi from the cluster centroid, are illustrated in Figure 6 B and C. Thesevectors essentially indicate the direction of the free electron pair of Bi. In c-BSO (B), thesevectors are oriented each in a separate spatial region within the Bi2O2 layer. This geometryleads to a strong anisotropy of the permittivity within the Bi2O2 layer versus out-of-plane[16], and enables a high polarizability of each BiO6 unit. In contrast, in c-Bi2O3 (C), thereis no separation between the BiOx clusters, and the Bi lone pairs are pairwise pointing intoa similar spatial region. This geometry causes a frustration in the polarizability of the BiOxclusters, and thereby reduces the permittivity of the entire system.An analogous effect takes place in the amorphous phase (a-BSO): the nano-separation bysilica chains of the Bi2O3-rich areas allows the displacement vectors of the BiOx clusters topoint away from each other, thus reducing the overall frustration in the cluster polarizability.Therefore, we can identify the micro-structure as another key factor to explain the materialsdielectric permittivity. In particular, it provides the basis to explain why ε of a-BSO is largerthan that of c-Bi2O3, but due to the inherent disorder it is smaller than that of c-BSO.16V. CONCLUSIONWe have conducted a comprehensive investigation of the atomic structure and dielectricproperties of amorphous bismuth silicate (a-BSO). By employing a combination of exper-imental techniques, we modeled both the short- and intermediate-range atomic structuresusing a Reverse Monte Carlo approach. The resulting model reveals BiO-rich regions em-bedded within SiO4 chains, exhibiting notable structural similarities to the crystalline phaseof Bi2SiO5 (c-BSO).Amorphous bismuth silicate exhibits a much larger dielectric permittivity than any otherknown glass, with a value of ε = 56 at ambient conditions. This high dielectric permittivityis primarily attributed to the polar nature of the BiOx polyhedra. In this configuration,oxygen atoms are largely confined to one bond hemisphere - an asymmetric arrangementthat induces a significant atomic-level polarizability. In addition, the intermediate-rangeseparation of BiOx clusters by SiO4 chains enables a flexibility in the local arrangements ofthe clusters. These two factors can be identified as the main driving forces of the material’sexceptional dielectric properties.ACKNOWLEDGMENTSThe present work was partially supported by JSPS Grants-in-Aid for TransformativeResearch Areas (A) “Hyper-Ordered Structures Sciences” (Grant Nos. 20H05878, 20H05879and 20H05880) as well as the JSPS KAKENHI grants 20H02429, 20K15027 and 21K18800,and by the MEXT Program for Creation of Innovative Core Technology for Power ElectronicsGrant Number JPJ009777. The project was conducted as an SDGs Research Project ofShimane University.The authors would like to thank Prof. T. Okabe in the University of Tokyo for the NMRexperiments, as well as Prof. L. Pusztai of the HUN-REN Wigner Centre for Physics forstimulating discussions on the topic.The synchrotron radiation experiments were performed at KEK-PF (Proposal Nos.2019G133, 2021G091, and 2023G040) and at SPring-8 (Proposal no. 2021A1297). 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