Thermodynamic calculation of the ionic conductivity of LiPON glasses and solid electrolytes †

LiPON ﬁ lms are one of the most relevant solid electrolytes in Li-ion micro-batteries for low power applications, and the relationships between composition, processing parameters and ionic conductivity have challenged the materials community for the last three decades. Although ﬁ lms have almost exclusively been processed by deposition of Li 3 PO 4 polycrystalline pellets, the amorphous nature of LiPON has permitted the study of compositions with glass-forming ability where the e ﬀ ect of improving the conductivity by nitrogen doping is similar. The aim of this work is to form a general model that allows us to calculate the ionic conductivity of LiPON solid electrolytes for whichever composition. The modelling is based on the application of the fundamental laws of chemical equilibrium to elucidate the di ﬀ erent chemical species present for each given system composition. For this purpose, a program in the Matlab software was written, capable of solving simultaneous and competitive chemical equilibria. The method, known as the associated solutions model, was applied to the study of the Li 2 O – P 2 O 5 system, which is the starting point of the present work. The model assumes that the properties of a given glass or amorphous solid, such as the simulated LiPON ﬁ lms, can be obtained as a function of the weighted contributions from each stable compound in the respective composition diagram. Here, Li 2 PO 2 N oxynitride is used as the main phase and, to validate the computed chemical equilibria, nitridation of crystalline Li 4 P 2 O 7 has been performed by thermal ammonolysis. The combination of the calculations of chemical equilibria together with the structural characterization by XRD and NMR of the products of nitridation from Li 4 P 2 O 7 allowed us to predict both the formation of Li 2 PO 2 N and the existence of a second oxynitride stoichiometric compound with formula LiPO 1.5 N that would be obtained by nitridation of LiPO 3 . The solution of the system of equilibria was made possible on development of the Gibbs free energy functions of formation of both oxynitride phases, and the calculation of the activation energy and ionic conductivity in either LiPON thin ﬁ lms or oxynitride glasses, o ﬀ ering comparable values to those found in the literature.


Introduction
The development of solid-state lithium-ion electrolytes has been an issue of the utmost relevance in the eld of energystorage technologies during recent decades due to the possibility of solving the safety problems related to liquid electrolytes, such as thermal and electrochemical stability, packaging (liquid leakage) and ammability. 1 Among solid electrolytes in use nowadays, amorphous lithium phosphorus oxynitride (LiPON) thin lms have been proven as competitive electrolyte materials for micro-power applications due to their chemical stability against the Li metallic anode and the wide electrochemical window versus Li/Li + (0-5.5 V). 2 Amorphous LiPON was rstly produced at Oak Ridge National Laboratory by radiofrequency magnetron sputtering deposition from a Li 3 PO 4 target in a N 2 atmosphere. 3 The resulting compound, Li 3.3 PO 3.9 N 0.17 , showed an ionic conductivity (s) of 2.2 × 10 −6 S cm −1 , a factor of 30 greater than that of polycrystalline Li 3 PO 4 . 3 Numerous subsequent studies by other authors have corroborated the increase in conductivity and the decrease in the activation energy (E a ) of Li diffusion in LiPON thin lms under variable conditions of deposition. 4 However, this observed increase in conductivity on introducing nitrogen in LiPON electrolytes still remains controversial and several hypotheses have been debated.
Based on previous results of oxynitride silicate glasses, Wang et al. concluded that the decrease of the electrostatic energy aer substitution of P-O for P-N bonds in the amorphous phosphate network results in a decrease in the activation energy of conduction, thus increasing the ionic conductivity. 5 It was postulated that since N increases the cross-linking density of the network through new -Nh bonds with neighbouring P atoms, this may favor the formation of (unspecied) conduction pathways for Li diffusion. [6][7][8] Actually, both types of nitrogen-tophosphorus bonds, -N] and -Nh, are generally found in LiPON thin lms, regardless of the method of deposition, and it has been proposed that conductivity is enhanced when the relative content of -Nh is higher. 9 However, recent simulations of LiPON lms by Lacivita et al. indicated that lms with very high lithium contents would present two types of nitrogen atoms, di-coordinated nitrogen N d (-N]) bonded to two phosphorus, and apical nitrogen atoms or N a (-N) bonded to a single phosphorus atom, 10,11 with the former N d nitrogen originating the increase of conductivity by nitrogen in LiPON lms.
Due to the amorphous character of LiPON electrolytes, another approach that has been used to nd a correlation between structure and conduction is through the characterization of lithium phosphorus oxynitride bulk glasses. Lithiumphosphate glasses have a glass-forming range up to nearly 60 mol% Li 2 O in the Li 2 O-P 2 O 5 system, corresponding to a LiPON composition of Li 1.5 PO 3.25 , which is beyond the typical LiPON electrolyte compositions obtained by physicaldeposition methods. Nitrogen is instead incorporated into the phosphate glasses by reacting the phosphate melt under a NH 3 ow, giving rise to glasses of general composition Li a PO b−3x/ 2 N x , according to the substitution of 3O 2− by 2N 3− . 12 A number of studies have been performed in recent years on the structural modication of the glass network that accompanies nitrogen incorporation and how this inuences the increase of ionic conductivity. 4,5,[12][13][14][15][16] One such study indicated that the substitution of oxygen for nitrogen is intimately related to the increase of the relative content of non-bridging oxygens in the glasses. Therein, Mascaraque et al. found that the net increase attained in the conductivity for a xed nitrogen content was lower with increasing lithium content, indicating that the original structure of the given glass composition limits the maximum effect on conductivity. 15 We have recently used a new methodology for modelling the structure and conductivity of Li 2 O-P 2 O 5 glasses, 17 in which the thermodynamic model of associated solutions 18 and the Anderson and Stuart model of ionic conductivity have been employed. 19 The main objective of the present work is to extend such modelling to the study of LiPON compositions, either in the form of glasses or amorphous lms. Our starting point is the Li 2 O-P 2 O 5 -P 3 N 5 system, for which it is necessary to know the stable oxynitride compositions. To date, only one oxynitride with a stoichiometric composition is known, Li 2 PO 2 N, rst predicted and then synthesised and characterised by Senevirathne et al. 20 This oxynitride phase derives from lithium pyrophosphate Li 4 P 2 O 7 and it was proposed that its structure is built up of [PO 2 N 2 ] tetrahedra, grouped in the form of chains through P-N]P bonds in which all nitrogen atoms are of the N d type.
We have thus conducted nitridation reactions by ammonolysis in Li 4 P 2 O 7 and Li 3 PO 4 to explore the stability domain of Li 2 PO 2 N and to determine whether another oxynitride phase can be obtained. Subsequently, since the model of associated solution requires the use of the Gibbs free-energy function of every compound within the composition system of interest, a theoretical derivation of the G(T) of the oxynitride compounds is presented. The work is concluded with modelling of the ionic conductivity for all compositions as a function of both the lithium and nitrogen contents, from which the value for any given oxynitride composition may be predicted.

Synthesis and characterization of nitrided phosphates
Lithium pyrophosphate, Li 4 P 2 O 7 , and orthophosphate, Li 3 PO 4 , were obtained by crystallization of their melts from batches prepared mixing reagent-grade materials Li 2 CO 3 (99% ACS. Reagent, Aldrich) and NH 4 H 2 PO 4 (99% ACS. Reagent, Merck) in stoichiometric amounts. The batches were calcined in porcelain crucibles up to 400°C for 24 h, then melted during 2 h at 950°C; polycrystalline pyro-and orthophosphate were obtained upon freely cooling the melts. Lithium pyrophosphate was subsequently recrystallized at 550°C for 24 h to increase its crystallinity.
Nitridation was performed by thermal ammonolysis using a constant NH 3 ow of 0.5 mL min −1 on ca. 200 mg of powdered crystalline phosphates. Some samples were sieved under 63 mm to study the inuence of the particle size on the reaction. The ammonolysis temperatures varied between 700°C and 800°C for Li 4 P 2 O 7 and from 800°C to 1100°C for Li 3 PO 4 . Steps of heating to the reaction temperature and cooling back to room temperature were performed under a N 2 ow.
Nitrogen content in the samples was determined in a nitrogen/oxygen LECO CHNS-932 elemental instrument by the inert gas fusion method.
X-ray diffraction (XRD) patterns were obtained in Bragg-Brentano geometry using a Bruker D8 Advance diffractometer (Bruker, Germany) equipped with a lynx-eye detector and a germanium monochromator providing monochromatic Cu Ka 1 radiation (l = 1.5406 Å). Diffraction data of powders were collected in the range 10 # 2q # 70°using a step width of 0.02°a nd a step counting time of 1 s. 31 P magic-angle spinning nuclear magnetic resonance (MAS-NMR) was performed in a Bruker ASX 400 spectrometer at a frequency of 161.96 MHz, using a spinning frequency of 10 kHz in 4 mm rotors. Spectra were acquired with 4 ms pulse length, and a radio-frequency (rf) strength of 60 kHz using recycle delays of up to 3000 s. Chemical-shi values were referred to (NH 4 ) 2 HPO 4 at 0.82 ppm with respect to H 3 PO 4 (85%).

Estimation of Gibbs free-energy functions of oxynitride compounds
In the thermodynamic model of associated solutions, a glass can be considered as a solution formed from the glass-oxide components plus the products of their interaction. These products are the so-called salt-like groupings and it is assumed that both their stoichiometry and structure are similar to those of the crystalline compounds that exist in the given system.
Thus, a glass of any composition can be represented by a superposition of various chemical groupings provided the system guarantees the minimum Gibbs energy. 18 The equilibrium molar amount, n, of each salt-like product (j) and unreacted oxide (k) is obtained by solving a set of nonlinear equations composed of the mass balance and mass-action laws of each reaction (j) that takes place in the system: 8 > > > < > > > : The stoichiometric coefficients are denoted by the Greek letter n and the equilibrium constants by the letter K. The superscript 0 denotes the initial molar amount of each unreacted oxide. There are many numerical algorithms capable of efficiently approximating the solution of this system. In this work, the Matlab internal function fsolve, which computes a numerical solution of a general nonlinear system of equations, has been used. Some mathematical manipulations of equation system (1) have been performed to facilitate its solution, which are related to the difference of several orders of magnitude between the equilibrium molar amounts of each product. These manipulations have been detailed in a previous work. 17 In the case of LiPON compositions, in either glass or amorphous thin-lm form, Li 2 O, P 2 O 5 and P 3 N 5 were chosen as the reactants to simulate the system in chemical equilibrium. Thus, the system of reactions to be solved is composed of the reactions of formation of the three lithium phosphates, LiPO 3 , Li 4 P 2 O 7 and Li 3 PO 4 , plus those of formation of the lithium oxynitride phosphates to be identied below.
The value of the Gibbs free-energy function of each compound is required to calculate the equilibrium constant of the corresponding formation reaction, K = exp(−G/RT). The Gibbs functions of meta, pyro and orthophosphate compounds to be used herein are the same as those employed in the solution of the system published in ref. 17; however, the G functions of the oxynitride compounds are unknown, so a method of estimation is required. Any derivation of G must involve calculation of enthalpy and entropy contributions.
The core of the present approach is the Gibbs-Helmholtz equation from which, if H(T) is known, it is possible to nd the G(T) function as given below: The procedure for integration of G(T) is the following: for a small interval of T, H can be considered as a constant. Thus, by integration of eqn (2) from T 1 to T 2 (any two temperatures close enough to consider H as constant) the following relation is obtained: If we work with small steps of temperature, G(T i+1 ) can be obtained from the previous value G(T i ), and the initial value (i = 1) corresponding to T = 298 K: To use this procedure, the H(T) function and the S 0 f value of the oxynitride compounds should be estimated.
Tessier et al. studied the enthalpy of formation of oxynitride phosphate glasses and obtained a general expression for the standard formation enthalpy against the nitrogen content in oxynitride glasses by comparing the difference in bond energy with the oxidation reaction. 21 The known H(T) of the corresponding phosphate composition is employed, assuming that H(T) for the oxynitride is the value of H(T) of the phosphate phase coupled with the change in the bond energies on substituting oxygen for nitrogen. Thus, herein the values of the bond energies of O-O, N-N, P-N and P-O are used, as well as the value of the enthalpies of formation of the lithium phosphate compounds from which the oxynitrides are to be obtained.
Considering the oxidation reaction of the Li 2 PO 2 N oxynitride to form Li 4 P 2 O 7 : the enthalpy of oxidation can be calculated from the change in binding energies: In the case of the oxidation of Li 2 PO 2 N to pyrophosphate, the reaction enthalpy is −380.53 ± 3.67 kJ mol −1 , so the enthalpy of the nitridation reaction will be the same.
The enthalpy of formation of pyrophosphate must then be added to this enthalpy of reaction to obtain the enthalpy of formation of the oxynitride: This method can be generalised for any lithium and nitrogen contents through the following general oxidation reaction: In this case, the enthalpy of oxidation is: As seen from eqn (9), the change in H(T) does not depend on the amount of lithium present in the compound. The reaction described in eqn (8) can be generalised to any nitridation reaction from a given phosphate composition, irrespective of the type of cation X of charge n or the central atom of the oxide anion M of charge m: Finally, the difference in bond energy can be calculated using the following approximation: 22 Hence, the function of enthalpy against temperature can be obtained for the given oxynitride.
The standard entropy of formation employed in eqn (6) is derived using Latimer's method, 23 which establishes that the entropy of a crystalline solid can be calculated by adding the contributions of the cation and the anion from which it is formed. As stated in the aforementioned article, the entropy of a solid compound is a function of the masses of the constituent atoms and of the forces acting between these atoms: the greater the mass and the lower the force, the greater the entropy.
In accordance with the above, the entropy of an ionic solid depends on the magnitude of the ionic charges. To illustrate this, Latimer calculated the contribution of the chloride ion in combination with +1, +2 and +3 ions by subtracting the entropy of the cations from the experimental salt entropy value, assuming the entropy of the cations equal to that of their corresponding elements, estimated previously. 24 The agreement in entropy values of Cl − calculated for the three types of salt (Cl − plus cation +1, Cl − plus cation +2 and Cl − plus cation +3) was remarkable. A close examination of the data for a large number of anions showed that a similar loading effect existed in all cases.
Consequently, Latimer proposed to arbitrarily assign all the variation of the entropy with the charge to the negative ion and to tabulate said entropy as a function of the charge of the cation from the experimental values. The entropy of the salt is then calculated as the sum of the entropy of the monatomic cation and that of the corresponding anion.
For example, for the compound Ag 2 SO 4 = 2Ag + SO 4  The problem with this method is that several measurements are needed for each type of anion to be able to tabulate the average entropy as a function of the charge of the cation. Hence, for compounds in which the entropy of their anion has not been measured, it is not possible to use this method, as is the case for the PO 3 − anion.
To solve this problem, it is necessary to return to the initial statement made in Latimer's article: 23 "the entropy of a solid compound is a function of the masses of the constituent atoms and of the forces acting between these atoms: the greater the mass and the smaller the force, the greater the entropy". Therefore, to estimate the entropy of the unknown anions, an attempt has been made to establish a relationship of the type S 0 f f mass × 1/(interaction forces) from the data of known anions, 23,25 which are tabulated in Table S1. † Using Latimer's method, the unknown entropy of other anions has also been calculated from entropy data of different salts. 26 These entropies are tabulated in Table S2. † To quantify the effect of mass, the molecular mass of the anion will be taken as a variable. To quantify the interaction forces, the binding energy central atomsecondary atom (P-O for PO 4 3− , Si-F for SiF 6 3− .) will be used. This correlation has been established in the present work for oxide anions, and it can be extrapolated to LiPON (PO x N y anions) for two reasons: rst, the amount of nitrogen in these compounds is very low in comparison with the amount of oxygen; second, only a rough estimate of entropy is needed to start the procedure, as will be seen in the results section. The bond energy in PO x N y anions to use the entropy correlation of oxides can be estimated as The correlation between entropy, molecular weight and binding energy can be seen in Fig. S1 of the ESI † le, where experimental data for the entropy of anions have also been plotted.
The binding energy shows the highest correlation with the entropy, and the most precise expression found so far to estimate the entropy through the molecular weight (M m ) and the binding energy (B e ) is the following: where the entropy is expressed in J mol −1 K −1 , the molecular mass in g mol −1 and the binding energy in kJ mol −1 . Using these units for oxides, A = 0.757815094 and B = 1.62345464; the strength of the correlation is R 2 = 69.01% (Fig. S2 †). From the estimation of the standard entropy (eqn (13)) and the enthalpy (eqn (7) and (8)), G(T) has been calculated for the oxynitride compounds, LiPO 1.5 N and Li 2 PO 2 N as will be seen below, by using eqn (4). Fig. S3 to S5 of ESI † show a comparison of the calculated Gibbs free energy functions using the above method and the experimental ones, as obtained from the FactSage database, where a very good agreement can be seen. On the other hand, Fig. S6 † expresses the calculated G functions for the two oxynitride phases that will be used in the calculations below.

Calculation of ionic conductivity
The ionic conductivity (s) is calculated as a function of temperature (T) from the following Arrhenius equation, which assumes that a glass is a weak electrolyte: 27 where n is the total cation concentration (Li + in this case), e is the fundamental charge, l is the jump distance between cation sites, n 0 is the jump attempt frequency, E a is the activation energy and K B is Boltzmann's constant. Charge-carrier concentration, n, can readily be calculated subsequently once the relative content of each phase in the system is known, considering the stoichiometric number of Li in each phase, as shown previously. 17 Assuming isotropy (as stated in the aforementioned paragraph), the average jump distance, l, can then be calculated from the charge-carrier concentration as l ¼ ffiffiffiffiffiffiffiffi 1=n 3 p . Hopping frequency has been taken from far-infrared studies of Exharhos et al. as reported earlier. 17 The activation energy can be calculated through the Anderson & Stuart model as described in ref. 17. One can approximate the electrostatic contribution (binding energy) by using Coulomb's law for one ion pair with the addition of a covalence parameter g. Anderson and Stuart justied the introduction of this parameter by claiming that, in the materials that they studied, the direct use of Coulomb's law led to excessive activation-energy values due to the fact that the electron cloud of O 2− anions is rather easily deformed by the cation, lowering the binding energy. 19 Coulomb's law must, therefore, be modi-ed to take this into account as follows: where R Li + and R O 2− are the ionic radii of lithium and oxygen atoms, 1.4 Å and 0.6 Å, respectively. According to their experiments, Anderson and Stuart concluded that the covalence parameter can be approximated by the dielectric constant. 17,19 However, a review conducted by the authors of the present work has brought to light that g is, in fact, equivalent to the dielectric constant. The reason for this confusion is that the authors used the CGS system in their work, so the constant term in Coulomb's law equals 1. However, this is only true in a pure vacuum. In this case, 3 r = 1, and in the CGS system we have: Nevertheless, for a dielectric material such as glass, 3 r has to be considered, and eqn (15) becomes: For lithium phosphate glasses, a value of 3 r = 10 can be taken as an approximation. 28,29 Hence, to evaluate the inuence of N on the dielectric behavior of the material, a 10% increase of the dielectric constant for each 10% wt. of N added into the composition can be considered as a rst approximation, considering data of dielectric constants in alkali-and alkalineearth-containing oxynitride silicate glasses. 30 The dielectric constant for Li 2 PO 2 N would thus reach a value of 11.5. It should be remarked that even if 10 is only an approximation of the dielectric constant for Li 4 P 2 O 7 , this may not be that relevant since what we seek is the effect on E a and s due to the increase of the dielectric constant while introducing N with respect to the value of the oxide composition. However, taking a value of 10 for the dielectric constant has worked well in the previous work on the calculation of ionic conductivity in lithium phosphate glasses. 17 The strain-energy contribution to the activation energy can be neglected due to the small size of the Li + cation. 11 Finally, to determine n and l for eqn (14), the molar volume (V m ) of the material is required, and can be calculated from the equation of state of the SVTD model: 19 where g i are the Gibbs free energies of the phases of the system's phase diagram. From eqn (17), molar volume can be obtained as: where V mi are the molar volumes of the phases. With the exception of LiPO 1.5 N, most values can be taken from experimental data. 26,31,32 For LiPO 1.5 N, Korotkov's method can be used to estimate the molar volume on establishing a relationship between certain metric properties of the material and the macroscopic density: 33 where k is a "lling factor" that accounts for the fraction of the total space occupied by the atoms of the phase; l i and V i are, respectively, the stoichiometric number and the volume of each atom.
The lling factor typically drops in the range 0.4-0.6 for inorganic compounds, so a value of 0.5 is usually taken as an average. However, lithium phosphorus oxynitride compounds have two elements in their structure (Li and N) which contract the network, so this number is expected to be higher than usual and may be estimated from volume experimental data of the oxynitride Li 2 PO 2 N, 229.47 Å 3 according to Senevirathne et al. 20 Knowing V m (ref. 20) and M m , we can use eqn (19) to calculate the lling factor, which gives k = 0.66. Using this value as a rst estimate for lithium phosphorus oxynitride compounds and eqn (19) again, we obtain V m (LiPO 1.5 N) = 28.46 cm 3 mol −1 , which is, as expected, below the value of 34.62 cm 3 mol −1 of LiPO 3 , the phosphate from which LiPO 1.5 N is formed.

Results and discussion
3.1. Nitrogen incorporation in lithium phosphates, Li 4 P 2 O 7 and Li 3 PO 4 Fig. 1 shows the results of the ammonolysis reactions on lithium pyrophosphate plotted as wt% N against temperature for samples without and with sieving below 63 mm (1a), reaction time (1b) and the number of consecutive reactions for temperatures between 700°C and 800°C, each for a reaction time of 3 h (1c). Fig. 1a shows that particle size has a marked inuence on the amount of N that can be introduced aer a single treatment of 3 h, particularly below 800°C. This strong dependence on particle size results from the solid-gas reaction, in which the available surface area of the solid reactant is a crucial variable. Furthermore, higher temperatures give rise to a higher content of nitrogen for the rst 3 h of ammonolysis, while at 800°C the effect is reversed, likely due to the faster sintering of the powdered solid that reduces the surface area in contact with NH 3 .
The plateau for reaction times up to 12 h at 750°C (Fig. 1b) indicates that there is a limit to the content of nitrogen that can be reached. As in Fig. 1a, the maximum content that can be reached in these conditions is ca. 10 wt%. Fig. 1c shows the content of N analyzed in samples submitted to consecutive reactions of 3 h at 750°C; samples were re-ground aer each reaction to study the newly introduced nitrogen into deagglomerated samples. Even if the temperature is increased to 800°C or the number of reaction steps increases, the maximum N content in the samples lies between 10 and 11 wt%.  Fig. 2a, it is concluded that only crystalline phases of Li 4 P 2 O 7 and Li 3 PO 4 are present and that nitrogen should enter into an amorphous phase. Fig. 2b, however, exhibits the XRD patterns of the nitrided products obtained for two nitrogen contents, 4.34 and 7.98 wt%, that were submitted to ammonolysis reactions at 675°C for 6.5 h and at 750°C for 3 h aer 3 consecutive reactions, respectively, and which reveal the crystallisation of additional phases, particularly that of Li 2 PO 2 N (ICSD188493). Some additional reections are observed at 13.2°and 26.5°, which could not be identied with known phases in the ICSD and NIST databases. For Li 2 PO 2 N, we note that although the dspacings of its principal peaks match those of the XRD pattern published by Senevirathne et al. 20 the intensities do not match to the same extent, which may indicate some compositional variation or preferred orientation.
According to the computational results of Li 2 O-P 2 O 5 -P 3 N 5 equilibria, there should not be any reections of Li 2 PO 2 N below ∼7.3 wt% N. The Li 2 PO 2 N peaks observed in the pattern with 4.34 wt% N may arise from nitridation of the g-Li 3 PO 4 phase that forms on nitriding Li 4 P 2 O 7 , in accordance with computational results that predict the decomposition of g-Li 3 PO 4 into Li 2 PO 2 N and Li 2 O with increasing N content. We speculate that the additional unidentied reections may be attributable to a LiPO 1.5 N phase predicted to form by computational results of chemical equilibria, as shown below. Fig. 3a shows the 31 P MAS NMR spectra of two samples with nitrogen contents of 5.86 and 10.08 wt% N, as obtained by nitridation at 750°C for 1 h and 775°C for 3 h, respectively. The sample with 5.86 wt% N shows two very intense and narrow resonances at −3.9 and −6.2 ppm with Lorentzian proles, which are attributable to the two crystallographic sites of phosphorus in Li 4 P 2 O 7 . The spectrum also shows an even more intense peak at 9.45 ppm, attributed to Li 3 PO 4 phase detected in the XRD patterns (Fig. 2), together with a very broad resonance resembling an amorphous background that may be ascribed to the presence of phosphorus in an amorphous phase. For the second sample with ca. 10 wt% N, the broad resonance with a large chemical-shi distribution and the peak attributed to Li 3 PO 4 again appears, but the peaks typical of Li 4 P 2 O 7 are absent. Fig. 3b shows the 31 P NMR spectra of the samples with 4.34 and 7.98 wt% of nitrogen prepared with longer ammonolysis treatment in a single reaction at 675°C for 6.5 h and for the sample aer three consecutive reactions at 750°C for 3 h each, respectively. The sample of lowest nitrogen content clearly shows the two resonances of Li 4 P 2 O 7 (−3.9 and −6.1 ppm) and that of Li 3 PO 4 (9.45 ppm); however, four more signals with Lorentzian prole are also present at 4.74, 10.79, 11.94 and 14.42 ppm, indicating the presence of further phosphorus atoms in one or more crystalline environments. The sample of nitrogen content 7.98 wt% shows the same resonances as those of the 4.34 wt% N sample, except for those of Li 4 P 2 O 7 . Both simulations include a broad Gaussian resonance at ca. 0 and 2 ppm arising from phosphorus in an amorphous phase that had not yet crystallised aer the given reactions. Marple et al. have very recently performed ab initio calculations of the chemical-shi tensor of 31 P for a number of oxynitride and nitride compounds and reported a value of 14.949 ppm for the isotropic chemical shi of 31 P in Li 2 PO 2 N, 34 which has a single phosphorus crystallographic site. 20 Given that the calculated value of chemical shi would not be precisely the experimental one of 31 P in Li 2 PO 2 N, it is reasonable to suppose that the resonance at 14.4 ppm in the NMR spectra of Fig. 3b is attributable to Li 2 PO 2 N, in agreement with the presence of reections of Li 2 PO 2 N in the XRD pattern of the same sample (Fig. 3a). The three other resonances at ca. 4, 10 and 11 ppm may thus correspond to the unidentied crystalline phase appearing in both XRD patterns.
The crystalline phases identied by XRD and NMR for the different N contents suggest that the rst reaction to consider during nitridation is the simultaneous decomposition of pyrophosphate into ortho-and metaphosphate phases, with the latter undergoing the partial substitution of oxygen for nitrogen:

Li 4 P 2 O 7 + NH 3 / Li 3 PO 4 + LiPO 1.5 N + 3/2H 2 O
The maximum incorporation of nitrogen into the LiPO 3 structure would be obtained for the stoichiometry LiPO 1.5 N, in which every phosphorus atom is bonded to two nitrogen atoms via P-N]P bonds, as in the structure of its homologous phase for Li/P = 2 (Li 2 PO 2 N), 20 and where all tetrahedral building units are of the type [PO 2 N 2 ]. This would correspond to the stoichiometry of a metaphosphate glass where the maximum nitrogen content of 18.4 wt% is attained. However, considering the decomposition of the pyrophosphate into Li 3 PO 4 and the nitrided metaphosphate amorphous phase, the nitrogen content where the reaction stops corresponds to only 7.3 wt%. Above this, the XRD patterns do not show evidence of Li 4 P 2 O 7 ; therefore, as the reaction still proceeds up to the maximum observed N content of ca. 10% (Fig. 1), the formation of a second oxynitride phase should be considered. This phase appears to be closely related to Li 2 PO 2 N, which is the nitrided product of Li 4 P 2 O 7 for 15.4 wt% nitrogen content. In fact, if the known oxynitride Li 2 PO 2 N is considered as a new reaction product, the calculated nitrogen content that corresponds to an equimolar system composed of the sum of Li 3 PO 4 , LiPO 1.5 N and Li 2 PO 2 N is 9.9 wt%. This value closely approximates the maximum obtained in the nitridation experiments (Fig. 1).
Nitridation of Li 3 PO 4 has also been carried out to examine the formation of Li 2 PO 2 N. Fig. 4 shows the XRD patterns of a set of nitrided samples for increasing nitrogen contents, all of which exhibit reections of Li 3 PO 4 .
For the highest nitrogen contents, reections attributable to Li 2 PO 2 N phase are clearly visible. This suggests that Li 3 PO 4 decomposes to Li 2 O and a nitrided amorphous phase from which crystallization of Li 2 PO 2 N takes place. As will be shown below from thermodynamic calculations, this oxynitride phase is consistent with the presence of Li 2 PO 2 N.  Li 2 O is not observed in the XRD patterns, which could be due to the very small amount formed and because Li 2 O can simultaneously react with water forming Li(OH) of high volatility.

Thermodynamic modelling of oxynitride systems
As discussed above, the system of compositions under consideration is Li 2 O-P 2 O 5 -P 3 N 5 . The reactions of formation to be considered are those that give rise to the three lithiumphosphate phases, LiPO 3 , Li 4 P 2 O 7 and Li 3 PO 4 , which were considered in the modelling of the Li 2 O-P 2 O 5 glass system, 17 and the reactions that form the two oxynitride phases derived from the proposed nitridation reactions above, LiPO 1.5 N and Li 2 PO 2 N: 8 > > > > > > > > < > > > > > > > > : The system of nonlinear equations is given by the constants of equilibrium of each reaction of formation and the mass balances of each of the reacting compounds: 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > : The system consists of 8 equations and 8 variables, each n i , given that each x i may be written as x i = n i / P n. Fig. 5 shows the solution of the chemical equilibria for the reaction of nitridation of Li 4 P 2 O 7 in two possible situations: (i) nitridation only produces the Li 2 PO 2 N oxynitride (Fig. 5a); (ii) nitridation results in the formation of both LiPO 1.5 N and Li 2 PO 2 N oxynitrides (Fig. 5b).
In the rst case, a very minor amount of Li 3 PO 4 would appear, which tends to disappear concomitantly with increasing oxynitride content during the course of the reaction; this scenario is not observed experimentally. In the second situation, the LiPO 1.5 N oxynitride and Li 3 PO 4 are both formed in similar amounts up to x = 0.2, corresponding to ∼7.3 wt% N, above which point no Li 4 P 2 O 7 remains. With greater N content, the concentrations of both LiPO 1.5 N and Li 3 PO 4 decrease and Li 2 PO 2 N instead forms. When the nitrogen content reaches ca. 10 wt%, x = 0.27, there are, in fact, equimolar amounts of Li 3 PO 4 and the two oxynitrides, as proposed above. Therefore, aer the formation of 1 mol of Li 3 PO 4 and LiPO 1.5 N from Li 4 P 2 O 7 , the reaction for the formation of Li 2 PO 2 N above x = 0.2 is as follows: The solution of chemical equilibria conrms, therefore, that the equimolar sum of Li 3 PO 4 and the two oxynitrides accounts for a 9.9 wt% of N, as observed in the nitridation from pyrophosphate. When the number of mols of Li 3 PO 4 and LiPO 1.5 N are smaller than that of Li 2 PO 2 N formed at 9.9 wt% N, the reaction between Li 3 PO 4 and LiPO 1.5 N becomes increasingly slower and the formation of further Li 2 PO 2 N much more difficult. This explains why it is not possible to increase the nitrogen content further than ∼10 to 11 wt%.
The simulations of the 31 P NMR spectra of the samples with 5.86 and 10.08 wt% N in Fig. 3 were performed with the Dmt analysis soware. 35 The quantication of NMR spectra of Fig. 3a gives orthophosphate and amorphous phase, corresponding to LiPO 1.5 N and Li 2 PO 2 N, with very similar values to those obtained in the estimated contents of the calculated proportions in Fig. 5b. The value of wt% N calculated from x in the system, according to the quantication of the different phases, gives 5.4% (x = 0.15) and 10.4% N (x = 0.28), in reasonable agreement with the analysed nitrogen contents of these samples, thereby conrming the validity of the proposed reaction scheme. On the other hand, the NMR spectra shown in Fig. 3b conrm the two scenarios resulting from calculations: (i) for the 4.34 wt% N sample, Li 4 P 2 O 7 is still present together with Li 3 PO 4 , in addition to the four resonances attributed to Li 2 PO 2 N and another crystallised phase; (ii) the sample with 7.98 wt% N only shows the resonance of Li 3 PO 4 and those of crystallised oxynitrides as observed in the corresponding XRD pattern. The presence of Li 4 P 2 O 7 or Li 3 PO 4 phases for the nitrogen contents 4.34 and 7.98 wt% (Fig. 3b) is thus supported by the calculated equilibria. The only point of difference between calculated and experimental results is that for 4.34% N (Fig. 5b) there is crystallisation of Li 2 PO 2 N seen by XRD (Fig. 2b) and NMR (Fig. 3b). This may be explained if part of the Li 3 PO 4 that originates from decomposition of Li 4 P 2 O 7 simultaneously reacts with NH 3 to produce Li 2 PO 2 N. As discussed below, modelling of the nitridation reaction of Li 3 PO 4 indeed produces the Li 2 PO 2 N oxynitride. Fig. 6a shows the solution of the chemical equilibria at the Li 2 O content of the metaphosphate line against the nitrogen content. The results clearly indicate the formation of the LiPO 1.5 N oxynitride phase in parallel with a decrease in the content of LiPO 3 .
The nitridation reaction that takes place in this case would be: We note that the above reaction is exactly that which should proceed from the nitridation of a lithium metaphosphate melt and which would continue until all the LiPO 3 is converted into the respective oxynitride, LiPO 1.5 N, corresponding to a N content of 18.4 wt%. To conclude, we postulate that the formation of an oxynitride of formula LiPO 1.5 N is coherent both experimentally and theoretically with the above proposed nitridation reactions. Nevertheless, a phase of a stoichiometry with Li/P and N/P = 1 is predicted to exist in the equilibrium phase diagram of the Li 2 O-P 2 O 5 -P 3 N 5 system. Further work is underway with the aim of synthesising the pure LiPO 1.5 N phase, which is expected to form from the nitridation of LiPO 3 . We have not been able to provide experimental details on the formation of LiPO 1.5 N as a pure phase, as we were not able to isolate it from the nitridation products of Li 4 P 2 O 7 , nor did we obtain conclusive results from the direct nitridation of LiPO 3 . We attempted to nitride the LiPO 3 metaphosphate in different ways to synthesise phase-pure LiPO 1.5 N; however, at this point, we have not observed conclusive proof of its synthesis. The nitridation of lithium metaphosphate via a similar route as that employed for the pyrophosphate Li 4 P 2 O 7 presents the inconvenience that its melting temperature is much lower, only 656°C, and the formation of only a small amount of liquid phase during the reaction considerably limits nitrogen incorporation because the surface area is very much reduced, giving rise to products of very low nitrogen content.
The solution of the system if nitridation proceeds from Li 3 PO 4 is represented in Fig. 6b. In this case, nitridation involves reaction of Li 3 PO 4 and NH 3 with simultaneous formation of the Li 2 PO 2 N phase and Li 2 O: Therefore, we have no need to consider any other oxynitride phase apart from LiPO 1.5 N and Li 2 PO 2 N, and we believe that, due to structural reasons, the formation of a similar oxynitride to these two phases from Li 3 PO 4 would not be plausible. A stable structure made up of [PO 2 N 2 ] tetrahedra linked through P-N]P bonds, as in the two former oxynitrides, would have an excess of oxygens per phosphorus and an unbalanced charge. It appears, therefore, that Li 3 PO 4 decomposes into Li 2 O and the stable Li 2 PO 2 N phase on the substitution of oxygen for nitrogen. The rst point to note is that, whereas the conductivity calculated for the non-nitrided compositions (x = 0) increases from LiPO 3 to Li 4 P 2 O 7 and Li 3 PO 4 , in agreement with the expected variation of conductivity in Li 2 O-P 2 O 5 glasses as a function of Li 2 O, 17 the degree of increase from metaphosphate to pyrophosphate is notably higher than that from pyrophosphate to orthophosphate. This indicates that the mobility of Li + in the orthophosphate is smaller with respect to that in the other phases.

Ionic conductivity of LiPON glasses and amorphous lms
Previous studies of lithium-ion conductivity in single crystals and polycrystalline lithium pyro-and orthophosphates documented conductivity values that are, in general, much lower for Li 3 PO 4 than for Li 4 38 also presented a conductivity of 3 × 10 −5 S cm −1 , signicantly lower than the conductivity of Li 4 P 2 O 7 at a similar temperature. The calculated conductivities of the Li 4 P 2 O 7 and Li 3 PO 4 compositions presented here, Fig. 7a, correspond to the amorphous solids so the most valid comparison is with the corresponding amorphous thin lms; nevertheless, the relationship with the conductivity observed in crystalline ceramics is indicative that the orthophosphate composition presents a somewhat lower conductivity than that expected. The great majority of research on LiPON thin lms involves processing by deposition of Li 3 PO 4 targets, so it is not readily possible to make a direct comparison with values of the lms made up of Li 4 P 2 O 7 specically.
The conductivity for the metaphosphate compositions, in which LiPO 1.5 N forms by nitridation of LiPO 3 , exhibit a linear increase with increasing nitrogen content. This is consistent with an enhanced mobility of Li + ions in LiPO 1.5 N, such that a linear increase of the conductivity of the system occurs proportional to the linear increase of the molar fraction of LiPO 1.5 N, as observed in Fig. 7a. On the other hand, the conductivity of the nitrided systems obtained from Li 4 P 2 O 7 (n = 2) and Li 3 PO 4 (n = 3) show slightly different behavior, with some deviation from linearity to a smaller slope occurring at the highest nitrogen contents. This change of trend is more marked for Li 3 PO 4 than for Li 4 P 2 O 7 , where the conductivity difference of the former between x = 0.4 and x = 0 is of about an order of magnitude, 30% less than the increment calculated in the metaphosphate line. The curves of activation energy against nitrogen content (Fig. 7b) 15 showed a regular increase of conductivity with increasing nitrogen content, although with some dispersion in the data. This scattering may be because of heterogeneities in the bulk glass samples, which result in different behavior from that predicted in the present calculations. The values of log 10 s at 298 K are also in good agreement with experimental results for the compositions for which bulk glasses can be prepared: [14][15][16] both experimental and calculated values are in the range log 10 s = −8.5 to −6.5 in S cm −1 , the conductivity increasing with higher lithium and nitrogen contents. Furthermore, some representative experimental values of the ionic conductivity in LiPON thin lms obtained by deposition of Li 3 PO 4 have been compared with the values calculated in the present work in Table 1. It can be seen that calculated and experimental values are in good agreement. Fig. 8a shows the electrical conductivity against the Li/P ratio for nitrogen contents in the range x = 0-0.40. All conductivity curves in Fig. 8a exhibit an increase with both lithium and nitrogen; however, besides the smaller value of conductivity for Li 3 PO 4 as a function of lithium than that expected from a linear increase, the observed difference in the conductivity between x = 0.10 and 0.40 is clearly smaller at the n = 3 line than at n = 1. This difference has been plotted for each nitrogen content as a function of the Li/P ratio in Fig. 8b. For the lowest nitrogen content (x = 0.1), Dlog 10 is almost constant at 0.4 for all Li/P values. With higher nitrogen content, Dlog 10 becomes more signicant but decreases as the lithium content increases. At the highest nitrogen content of x = 0.4, the conductivity difference between the nitrided compositions is clearly higher at the metaphosphate line, with formation of LiPO 1.5 N, than for the orthophosphate one (Li 2 PO 2 N + Li 2 O). This means that the improvement of conductivity that can be attained through formation of the LiPO 1.5 N oxynitride phase can be as high as that obtained at any other lithium content.
One of the key observations in the work by Mascaraque et al. was that, in LiPON glasses, 15 the higher the lithium content, the smaller the augmentation of conductivity for a constant nitrogen. The authors pointed out that the ratio between the bridging oxygen atoms (BO) to non-bridging oxygens (NBO) in the network inuences the degree of augmentation of the conductivity aer nitrogen incorporation. In oxynitride glasses, the BO/NBO ratio decreases for increasing nitrogen contents as -N_ and -N] bonds are formed and substitute for both BOs and NBOs. According to the substitution rules rst established by Marchand, 39 more BOs will be substituted than NBOs and so the BO/NBO decreases upon nitridation. Part of the NBOs are also substituted, but considering that there are two types of NBOs, P]O and P-O-Li, the number of NBOs of the P-O-Li type would be constant and the P]O type mainly substituted so that charge neutrality is maintained. Lithium ions are coordinated by several non-bridging oxygens of [PO 4 ] tetrahedral units with a typical coordination number between 4 and 5 in lithiumphosphate glasses, 40 so if the relative number of negatively charged non-bridging oxygens increases, the mobility of lithium ions would be enhanced, increasing conductivity. The authors demonstrated that the smallest increments in conductivity were found for the smallest changes in the BO/NBO ratio, 15 which is corroborated in the present calculations.
The number density of Li + ions (n Li + ) is plotted in Fig. 9 as a function of the Li/P ratio and nitrogen contents in the range x = 0.00-0.40.
The curve for the density of lithium ions in the oxide compositions (x = 0) shows a near-linear behaviour, although it exhibits a slight curvature with a smaller slope near the orthophosphate line (n = 3). However, the non-linearity of the curves increases with greater nitrogen content. For compositions with Li/P ratios below 2, the increase of n Li + becomes steeper with nitrogen content with the greatest difference observed for the n Li + of the respective non-nitrided composition at Li/P = 2.
When Li/P approaches 3, the number density of lithium even decreases with respect to the non-nitrided compositions. Below 0.2 of Li/P, nitridation of any composition in the LiPO 3 -Li 4 P 2 O 7 region will mainly produce LiPO 1.5 N, and only Li 2 PO 2 N from Li 4 P 2 O 7 (Li/P = 2) at the highest nitrogen A direct inuence of lithium number density is seen in the plots of conductivity of Fig. 9. The conductivity increases with both Li/P ratio and nitrogen content; however, the degree of the increase from the non-nitrided composition varies signicantly with the lithium content. If only LiPO 1.5 N is formed, the much greater n Li + increases the conductivity most at the highest nitrogen content, but the dielectric constant of the nitride network also contributes to this improvement. On the other hand, the difference between the number density of ions of Li 2 PO 2 N and Li 3 PO 4 is less signicant, which, added to the fact that nitridation of the orthophosphate does not result in pure Li 2 PO 2 N but also some Li 2 O, contributes to a much smaller increase of conductivity with respect to that obtained on nitridation of LiPO 3 . Nevertheless, the stronger inuence of the higher dielectric constant in Li 2 PO 2 N still contributes to some positive change in conductivity with respect to Li 3 PO 4 .

Structural considerations of the modelled materials
Regarding the manner in which the oxynitride amorphous network could inuence ionic conduction, some further structural aspects must be considered. In view of the calculated ionic conductivity along with the Li/P ratio and incorporated nitrogen, we have demonstrated that the mobility of Li + ions will be inherently dependent on their density, as well as on the increased dielectric constant of the resultant oxynitride amorphous network, whether in the form of a glass or an amorphous lm. The network structure can thus be represented in terms of the different tetrahedra representing phosphate and nitrided phosphate units of the chemical groupings, on which the thermodynamic model of associated solutions is based. which to date has not been obtained. In fact, the oxynitride Na 2 Mg 2 P 3 O 9 N is a known crystalline oxynitride whose structure is based on [PO 3 N] tetrahedra linked by triply coordinated nitrogen atoms. 41 3.5. Further implications of the thermodynamic approach to calculate ionic conductivity Because of the experimental inconveniences to normalise the chemical composition of amorphous electrolytes, most particularly of those obtained via deposition by magnetron sputtering, neither the relevant experimental parameters to be considered in exploring a standardised conduction mechanism nor, more importantly, a way to predict the ionic conductivity for designing new electrolyte compositions can be easily found in the literature. Regardless of the method of synthesis of the LiPON electrolytes, the procedure presented here is a general means for the prediction of conductivity in terms of the chemical composition. Since the methodology can be of general application, it may also be extrapolated to the prediction of material properties of other chemistries. In fact, if the system of composition under study is based on oxides, and no oxynitride phases are to be considered, the solution to the problem can be even simpler, since, if all oxide components of the equilibrium phase diagram of the system are known, then, there is no need for estimation of new Gibbs energy functions. The method that we have developed here for the LiPON electrolytes may thus predict the ionic conductivity of any type of amorphous solid electrolyte material. Furthermore, because no adjustable parameters are considered, the method allows us to obtain a model of the ionic conductivity for any range of composition of interest. In turn, research may be focused on adapting the synthesis method to achieve the highest conductivity, which represents a great advantage compared to a trial-and-error approach considering many compositions and experimental procedures.

Conclusions
The application of the model of associated solutions is suitable for the calculation of the conduction properties of LiPON compositions for any lithium and nitrogen content in both glasses and amorphous thin lms. As a result of the theoretical derivation of the Gibbs formation energies developed for the two oxynitride compounds, we now have a powerful tool for prediction of the properties of amorphous solid electrolytes, in the form of bulk glasses or lms. The existence of a new oxynitride, LiPO 1.5 N, is proposed, based on the results of nitridation of lithium pyrophosphate. This phase also forms from the nitridation of lithium metaphosphate. The structure of this new compound consists of [PO 2 N 2 ] tetrahedral units, as is the case in the known phase of Li 2 PO 2 N. However, no other oxynitride is expected to appear aer oxygen-for-nitrogen substitution from Li 3 PO 4 . These results validate the scheme of reactions of formation of oxynitride compounds. Throughout the quantication of the molar amount of phosphate and oxynitride phosphate compounds at each composition, the number density of lithium ions that take part in ionic conduction of the modelled solids may be calculated. The conductivity increases monotonously with the lithium content from metaphosphate to orthophosphate compositions. Although the introduction of nitrogen increases conductivity, the magnitude of the increase reduces for the highest lithium contents. The greatest increase in conductivity aer nitridation was obtained in the metaphosphate composition, in correspondence with published experimental data for LiPON glasses. Furthermore, the values of log 10 s and E a obtained by the model presented in this work are very close to experimental values for LiPON thin lms.