Investigations into the nature of spontelectrics: nitrous oxide diluted in xenon

The recent discovery of a new class of solids displaying bulk spontaneous electric fields as high as 10 8 V/m, so-called ‘spontelectrics’, poses fundamental and unresolved problems in solid state physics. The purpose of the present work is to delve more deeply into the nature of the interactions which give rise to the spontelectric effect in films of nitrous oxide (N 2 O), by observing the variation of the spontaneous field as the N 2 O molecules are physically removed from one another by dilution in Xe. Data, obtained using the ASTRID storage ring, are presented for films diluted by factors  = Xe/N 2 O of 0.9 to 67, at deposition temperatures of 38K, 44K and 48K, where 10 films are laid down by deposition from a gas mixture. Results show that the spontelectric field decreases as  increases and that at  = 67 for 44K deposition, the spontelectric effect is absent. Reflection-absorption Infrared spectroscopy (RAIRS) data are also reported, providing insight into the structure of Xe/N 2 O films and specifically showing that N 2 O remains dispersed in the Xe/N 2 O films prepared here. A simplified theoretical model is developed which illustrates that electric fields can be understood in terms of dilution-dependent dipole orientation. This model is used to reproduce experimental data up to an average molecular separation, s, of  1.25 nm apart, ~4 15 times that associated with pure solid N 2 O. The disappearance of the spontelectric effect at larger average distances of separation, between s = 1.25 nm and s = 1.75 nm, is a phenomenon which cannot be described by any existing model but which shows that dipole-dipole interactions are an essential ingredient for the creation of the spontelectric state.


Introduction
Spontelectrics form a new group of solid state materials, [1][2][3][4][5][6] whose salient characteristic is that they exhibit a spontaneous electric field within the bulk of the solid.The spontelectric state is unique in the physics of solids and shows characteristics quite distinct from any other known solids, such as the ferroelectric state of matter. 1 The fundamental interactions which govern the nature of the spontelectric state and the processes whereby the spontelectric state is created remain however elusive.It is the purpose of this paper to shed additional light on the nature of the spontelectric state.Experimental results reported here for the spontelectric material N 2 O diluted in Xe, further characterize the spontelectric state and place additional constraints on any future ab initio models of the structure or dynamical models of the creation of this state.
For the present, it is evident that the properties of spontelectrics arise through collective effects resulting in the orientation of permanent dipole moments of the constituent species.However it remains unclear whether dipole-dipole interactions between the active species are an essential ingredient of the physics which leads to this orientation.From hereon, dipole orientation refers to the tendency of dipoles to align in energetically unfavourable configurations in which + lies on average closer to another + and correspondingly to -, where + and refer to opposite dipolar ends of the species.45   Here we address the following questions: (i) how does the degree of dipole orientation change as the spontelectric species are removed physically from one another by dilution in a solid matrix of an inert gas? (ii) Is there some average physical separation between species at which the spontelectric effect can no longer be 50 observed?The latter is the major qualitative aim of the present work and essentially tests the hypothesis that dipole-dipole interactions are necessary to the spontelectric effect -and it is found that they are necessary.55 Questions (i) and (ii) are addressed by studying films of N 2 O diluted in Xe in which, with increasing dilution, the N 2 O molecules are increasingly further apart and are thus more weakly interacting.N 2 O is known to exhibit the spontelectric effect and it has been investigated in considerable detail in earlier work. 1,2,4Indeed it was 60 the first species which was shown to exhibit this effect. 2parameterized model is developed which adequately addresses question (i) posed above.This model quantitatively reproduces observations of a reduction in the spontaneous electric field in 65 films of N 2 O with increasing dilution in Xe.The answer to question (ii), as we have indicated, is that when the dilution and the corresponding average separation of dipolar species is sufficient, there is no evidence of a spontaneous field within the film.This occurs for a factor of dilution of between 26 and 67 for N 2 O in Xe deposited at 44K.We estimate the critical separation of N 2 O species for the loss of the spontelectric effect to lie between 1.25 and 1.75 nm.
The characteristics of spontelectrics in pure films have recently been reviewed in [1].In brief, the spontelectric effect is experimentally characterized by the following properties:


On forming a film by gas phase deposition under ultrahigh vacuum (UHV) of a spontelectric material, a polarization potential, typically of several volts, appears spontaneously on the surface of the film.


This potential, linearly proportional to the thickness of the film, may be either positive or negative and gives rise to a corresponding spontelectric field in the film.


The spontelectric field depends on both the nature of the material which is deposited and on the temperature at which the film is deposited.The field for any substance 20 is in general less for higher temperatures (save for methyl formate deposited at 80K). At greater than a certain temperature of deposition, no spontelectric effect can be observed.


Warming of a spontelectric film causes at first little 25 change in the spontaneous potential on the surface of the film, but a critical temperature is reached at which the spontelectric effect decays abruptly.


The nature of the substrate surface, upon which spontelectric films are deposited, has essentially no 30 bearing on the value of the spontelectric field within the film.
Materials which show the spontelectric effect are very diverse 1 and include species such as nitrous oxide, propane, toluene or chlorotrifluoromethane.Films of material may contain electric fields which can exceed 10 8 V/m, noting that breakdown fields of solids lie between 10 8 to 10 9 V/m.These fields are spontaneous, created without any external stimulus, such as an applied electric field; hence the term 'spontelectric'.
Spontelectric films constitute a form of the solid state in which both non-linear effects and non-local, collective effects operate.The spontelectric effect is non-linear in the sense that in the steady state the degree of dipole orientation and the spontelectric field are related in a manner such that one depends upon the other: that is, the degree of dipole orientation depends upon an expansion in powers of the local field at any dipole and the local field itself depends on the degree of dipole orientation (see for example reference 1 and Eqs. 1 and 2 in section 3.1).

50
The spontelectric effect is non-local since it is a bulk phenomenon which cannot be described in terms of local interactions alone.Rather, the spontelectric field permeates the entire thickness of the film and assumes a value which is due to a combination of interactions which extend both over the entire film as well as through pair-wise or few-body local intermolecular interactions.In this connection, we observe experimentally that in general a certain thickness of the sample is required before the spontelectric effect develops.It is found for example that isoprene deposited at 70K 60 requires >50 monolayers (ML) or toluene at 75K requires >100 ML to develop a substantial electric field (figures 4 and 5 of [1]).In this respect the spontelectric effect is unique in solid state physics since the steady-state structures with which it is associated depend upon effects of long range feedback.This feedback results

65
in such counter-intuitive behaviour as the increase in dipole orientation, seen in methyl formate, with increasing temperatures of deposition above 80K. 5r current qualitative understanding of the spontelectric 70 phenomenon is as follows.On deposition of a dipolar gas to form a film, species adjust such that dipoles should tend to be configured on average in head to tail configurations, that is, + to -.However thermal fluctuations may give rise to temporary configurations in which are found local average excess of + close to + rather than 75 and correspondingly an excess of close to -, that is, net dipole orientation.Here, fluctuations play the role of the externally applied field in the standard description of dielectric polarization.
Such dipole aligned configurations are evidently accompanied by a 80 local energy deficit due to repulsions between molecules.These configurations will create a fleeting and spatially localised electric field which tends to oppose the orientation.There are now two possibilities.In the first place, the effect may remain a local fluctuation which continually forms, dies away and is resurrected, 85 with dipole orientation averaging to zero over time and space.In the second, given a large enough local fluctuation, there may be a macroscopic switch to a dipole orientation opposing that set up through thermal fluctuation.This opposing dipole orientation then propagates throughout the system via the corresponding opposing 90 electric field.This is posited to occur since, outside the immediate region of the fluctuation, there is instantaneously no, or less, net dipole orientation.In this model, the opposing field, in a region exterior to the fluctuation, sets up dipole orientation in the opposite sense to the fluctuation itself.It is dipole orientation in this 95 opposite sense which gives rise to the observed spontelectric field.Note that the film is not itself an isolated system but is fed by an external cooling system.Moreover the spontelectric state would not appear to be a minimum energy state, but rather a metastable state into which the system becomes lodged in its exploration of 100 the phase space on its journey towards the lowest energy configuration.Having entered the spontelectric state, there are substantial local barriers to the equilibrium state, much as there are in, say, amorphous water ice whose minimum energy configuration, unattainable on any practical timescale at low 105 temperature, is hexagonal ice.
With regard to this qualitative model of the generation of spontelectrics, an electric field must of course be described by two regions of established potential and, within the above model, these 110 locations may instantaneously be within a region of a fluctuation and outside of it.As an important, perhaps crucial, special case, one of these regions may be the surface-vacuum interface, the nature of which may determine the sign of the spontelectric field.Thus N 2 O at the surface is configured with the terminal N atom, which is +, pointing into the vacuum whereas, for example, CF 3 Cl is configured with the Cl atom pointing outwards, giving a spontelectric field of the opposite sign. 6 addition, the dynamical model put forward above implies that the presence of thermal fluctuations alone may be sufficient in causing the spontelectric phenomenon.However the dynamics of 10 the constituent species will be modulated by local dipole-dipole interactions.Our present work shows that such dipole-dipole interactions play a fundamental role in creating the spontelectric effect and that a model based solely upon independent fluctuations of the active species is insufficient.
2 Experimental method and results

The experimental method 20
Surface potential measurements were performed on the SGM2 beamline at the ASTRID synchrotron storage ring at Aarhus University (www.isa.au.dk).The experimental chamber operates under ultra-high vacuum conditions, with a base pressure of ~2  25 10 -10 mbar.Films were grown on a polycrystalline Au substrate which was cooled via a closed cycle He cryostat to a minimum temperature of 38K.Before film growth the substrate was cleaned by annealing to 700K using a cartridge heater.Background vapour deposition was used to grow all films with the substrate fixed at a 30 chosen temperature.
Throughout, film thicknesses are described in monolayers (ML), where one ML of coverage was calibrated from temperature programmed desorption (TPD) experiments.A single ML of nitrous oxide (or xenon) refers to the number of species required to saturate adsorption sites on the polycrystalline Au surface.The residual gas pressure in the chamber during dosing was measured using a quadrupole mass spectrometer, so that a given background pressure could be correlated to a dosing rate.The residual gas 40 pressure method was used to fine-tune actual final dosing rates while dosing gas mixtures for the creation of diluted films.Dilution ratio is defined here as the effective number of ML of Xe divided by the number of ML of nitrous oxide.
Gas mixtures were prepared in an adjacent gas-line, also under UHV, before admission to the experimental chamber.Nitrous oxide (≥99.998%) and Xenon (99.99%) were used as received from Sigma Aldrich.Nitrous oxide was loaded into an initially isolated gas-line, using a continuous flow of gas, until the pressure required 50 to maintain the dosing flux for a required dose rate (ML min -1 ), known from our calibration experiments, was achieved.Xenon gas was then mixed into the continuous flow of nitrous oxide until the required dilution, again estimated from initial calibration experiments, was achieved.Thus the gases were pre-mixed in a known ratio in a flowing system.This flow was then exposed to the cold substrate in the experimental chamber for an appropriate time to create a mixed film corresponding to any selected dilution.Films were built up by the sequential deposition of layers of the xenonnitrous oxide matrix.The rate of deposition was typically 5 to 10 Details of the experimental method for measurement of film 85 surface potentials have been described elsewhere. 1In brief, using synchrotron radiation from ASTRID, low energy electrons were produced by photoionization of Ar gas at ~5 meV above threshold (78.56 nm) in a source chamber and with an energy spread given by that of the photoionizing radiation, set here to be ~1.5 meV.

90
These low energy electrons were extracted, via a series of lenses, and formed into a beam.The potentials on the film surfaces are estimated using the principle that an electron beam formed at the potential of the centre of the photoionization chamber can just reach another point, for example the surface of a film or of clean 95 gold, at the same potential, potential barriers apart.Thus if electrons are formed at a known potential, the potential of their destination can be measured by the bias that must be applied, either to their point of formation or their destination, such that a current just begins to flow. 100 The spontaneous potential appearing on the surface of spontelectric films is measured by applying such a bias potential to the system and effectively adjusting this bias in order to null the current.Since electrons are formed at nominally zero volts with an energy of ~5 105 meV, the beam should therefore just be able to reach a clean Au target when the target is itself at this same nominal zero (less 5 mV).If the target were covered with a film of spontelectric material and its surface were not at zero but rather, say, at +5 volts, due to the spontaneous formation of an electric field within the 110 film, then in order to ensure that the electrons only just reach the target, it would be necessary to bias the target, or the point of formation of the electrons, by 5 volts.Measurement of the bias required to locate the onset of a measureable current (~2 fA) therefore gives the potential on the surface of the film.A small correction is made to establish a true zero difference between the clean gold substrate and the potential of formation of the electrons.This arises from the difference between the work function of the gold substrate and the graphite-covered walls of the photoionization chamber in which electrons are formed.This difference is measured for each series of experiments; a typical value of ~0.25 V is found, in agreement with standard figures.As emphasized in earlier work, we interrogate the material while perturbing it to a minimum.This was ensured by using very small currents, typically of 100-200 fA, in order not to reduce significantly the spontaneous polarization of the film surface or introduce negative charging.These low currents were detected using a Keithley 6340 femtoammeter.

Results
Experimental data are shown in Fig. 1 and 2 for deposition of diluted films at temperatures of 44K and 48K respectively.Fig. 3 shows data at 38K, 44K and 48K with dilutions of 2.1, 2.     3).The thickness of the layer is expressed as the equivalent number of ML of N2O deposited, excluding Xe.  9 These confirm that nitrous oxide molecules were dispersed in the film on deposition of the gas mixture and did not segregate from the Xe matrix when laid down from a gas mixture.Experiments also demonstrate the juxtaposition, and thus association, to form dimers, trimers etc. of N 2 O within Xe.

40
RAIRS experiments were performed as follows.Mixtures with  = 5 were prepared in a gas line maintained under vacuum and in dark conditions to avoid photodissociation.The mixing ratio was determined using a capacitance manometer and confirmed using a quadrupole mass spectrometer (Hiden Analytical Ltd, HAL301) inside the chamber.Thick films were grown by background dosing of the Xe/N 2 O mixture onto a copper substrate coated with 300 nm silica.Film thicknesses in ML were determined from the TPD of N 2 O from the silica-coated substrate.RAIR spectra were recorded over the range 400-4000 cm -1 at a resolution of 1 cm -1 using a Fourier-transform infrared spectrometer (Varian 670-IR) set at a grazing incidence of 75° with respect to the normal of the substrate.We note that the spontelectric effect does not depend on the nature of the substrate on which the films are deposited. 155 Therefore the RAIRS results on a silica substrate remain relevant to the surface potential measurements on polycrystalline Au performed at the ASTRID laboratory.
Initially a  = 5 film containing the equivalent of 48 ML of N 2 O 60 was deposited at 20 K on silica and monitored with RAIRS.This film gives rise to a single partially resolved set of features around 2220 cm -1 belonging to the adsorbate, over the 400-4000 cm -1 range investigated.Fig. 4 shows this set of features, where the lowest frequency, centered at 2214.5 cm -1 , corresponds to the NN 65 stretching mode of the isolated N 2 O monomer in Xe as discussed below. 7,8Spectra of films deposited at 29 K and 38 K are also shown in Figure 4. We note that TPD experiments on Xe films adsorbed on silica show that Xe does not stick efficiently to silica above 42 K, precluding deposition studies at higher temperatures. 70 Figure 4 shows that as the deposition temperature is raised, the intensity on the low wavenumber side of the band (2210 -2220 cm - 1 ) decreases, while intensity is gained on the high wavenumber side (2220 -2245 cm -1 ).A similar change of the RAIR spectra is 75 observed when the film is deposited at 19 K and annealed to 49 K, just below the onset of Xe desorption at 50 K.Significantly however, the νNN bands of a pure N 2 O multilayer, characterized by bands at 2260 and 2235 cm -1 in RAIR spectra recorded on nonmetallic surfaces, [10][11][12] are absent from the spectra in Figure 4 or in 80 similar spectra for annealing of mixtures up to 49 K (not shown).This demonstrates the major result of the RAIRS study that N 2 O does not separate from Xe to form a pure solid layer but remains dispersed within the xenon matrix for  = 5 and temperatures between 20 K and 38 K for deposition, and 49 K for annealing.

85
To obtain the relative contributions from N 2 O monomer, dimer etc., the band centered at 2220 cm -1 for both deposition data (Figure 4) and annealing data was fitted using the addition of varying proportions of four bands in the form of Gaussian 90 functions, where each Gaussian centre corresponds to an absorption frequency of (N 2 O) x, where x ranges from 1-4.For the monomer in a Xe matrix, the relevant frequency is 2214.5 cm -1 . 7,8o the best of our knowledge, the absorption frequencies for x = 2, 3 and 4 in a Xe matrix are not available and we set out to estimate 95 them here.Kudoh et al. 13 isolated N 2 O in an argon matrix and observed the monomer band at 2218.6 cm -1 .From this value, the difference between the matrix shifts in Xe and Ar for the νNN band of N 2 O is taken to be -4.1 cm -1 in Xe relative to Ar.This gives access to estimates of dimer and trimer band frequencies in Xe 100 since the bands of N 2 O dimer and trimer in Ar have been reported. 13,14Given our estimated matrix shift difference of -4.1 cm -1 , frequencies of 2218.0 and 2223.4 cm -1 may be assigned to N 2 O dimers and trimers in Xe respectively, using the available Ar data.In addition, the N 2 O tetramer has been observed in a 105 supersonic helium jet expansion with a strong Q-branch at 2237.43 cm -1 . 15Given the gas phase nature of the jet expansion environment, a matrix shift for Xe of Δν (gas -Xe) = 8.9 cm -1 for the νNN band of N 2 O 8 may be applied and gives an expected band at 2228.5 cm -1 for (N 2 O) 4 in Xe, the value therefore assigned to this tetramer.
Adopting the above values for vibrational frequencies of (N 2 O) x , respectively 2214.5 cm -1 , 2218 cm -1 , 2223.4 cm -1 and 2228.5 cm -1 , and FWHMs of 9.4 cm -1 for all bands except the monomer band (FWHM = 3.5 cm -1 ), a fit is shown in the inset of Figure 4 for data obtained for deposition at 19 K, noting that this case involves negligible tetramer contribution.We note the appearance of a shoulder at 2216.5 cm -1 .This does not relate to any known N 2 O-N 2 O interaction and may be due to an impurity.This shoulder is not found in spectra taken for higher temperature films.Fits of the RAIR spectra with the contributions assigned to N 2 O monomer, dimer, trimer and tetramer can be adjusted to give good agreement at 20 K of deposition and up to 30 K of annealing.For deposition at 29 K and above however, a better fit may be obtained by including an additional contribution on the high wavenumber side.This contribution, composed of two peaks at 2234 and 2240 cm -1 (FWHMs = 9.4 cm -1 ), most likely arises from associations (or 'clusters') larger than the tetramer.There remains however, as noted above, no contribution from a pure N 2 O phase since the characteristic doublet bands at 2260 and 2235 cm -1 are absent.
The area of the gaussians used to fit the RAIR spectra is taken to be proportional to the amount of N 2 O in each configuration, that is, monomer, dimer etc., on the assumption that the strength of the νNN band in each configuration is similar.To allow the comparison between data from experiments performed on films held at different temperatures, the areas were normalised to the total area of the νNN band for every deposition temperature.The resulting quantities, proportional to the number of N 2 O molecules in each configuration, are given in Figure 5.This shows the prominence of isolated N 2 O monomers and dimers in the  =5 mixtures deposited at 20 K. Increasing the deposition temperature 45 of the film leads to an increase of the amount of trimers, tetramers and larger associations at the expense of monomers and dimersbut not to the segregation of N 2 O from Xe.This reiterates the most significant result of the RAIRS data, that is, the absence of segregation.Figure 6 shows equivalent data for annealing.Our present results are consistent with the conclusions drawn in [7]  and [8] and extend these conclusions to much higher mixing ratios.N 2 O monomers, dimers, trimers and tetramers are observed to remain dispersed in the lattice, while a higher deposition 70 temperature (or annealing) allows more association and cluster formation.The latter effect reflects the increased ability for mutual diffusion of Xe and N 2 O at higher temperatures but does not suggest any fundamental structural changes as the temperature of deposition is raised.

5
A mean-field model is used here to describe the spontelectric steady state.This model was originally introduced in [4] and is set out in detail in [1].The model is based on the concept that the net z-component of the electric field within a spontelectric film and 10 normal to the plane of the film, E z , is composed of a local symmetrical part, defining the interactions which bind layers to one another, and an asymmetrical part due to the infinite range field which permeates the film.The symmetrical part is expressed as a constant term plus a dipole-dipole term, proportional to (< z >/) 2 and representing average intermolecular dipole-dipole interactions.
Here < z >/, the degree of dipole orientation, is the ratio of the average z-component of the dipole moment and the total dipole moment of the molecular species in the solid state.7][18] The ζ(< z >/) 2 term in eq. 1 below may be interpreted as a measure of the tendency of one dipolar species to restrict the angular motion of another, a 'locking' term or, as it is sometimes called, a 'frustration' term.
The symmetrical part of the contribution to E z is related to the 'local field' at any molecular site as defined in standard texts. 19he asymmetrical part, <E asym >< z >/, is found only in the description of spontelectrics and has no direct counterpart for any other form of material.This asymmetrical part however resembles the Weiss field in ferromagnetism, which is assumed to be proportional to the magnetisation. 20Here, read degree of dipole orientation for magnetisation and read polarisation field for the Weiss field.We emphasise that the polarisation field, that is, the spontelectric field, is self-generated within the spontelectric material and takes the place of an external field imposed on the film.The polarization field acts in opposition to the symmetrical part and represents the infinite range field created by the average 40 dipoles and experienced by an average dipole.
Hence, using atomic units throughout, where <E sym >, <E asym > and ζ are taken to be temperature independent parameters.Mean field theory gives an implicit expression for < z >/ yielding the familiar Langevin function for orientational interactions: where T is the deposition temperature of the layer of material.The dipole moment of N 2 O in the pure solid state is reduced from that 55 in the gas phase through depolarization in the environment of other N 2 O species according to: where s pure is the average spacing between successive layers (= 0.32 nm for pure N 2 O), 1,4  is the molecular polarizability of N 2 O 60 (3.03  10 -30 m 3 ), k = 11.034 21and  0 is the gas phase dipole moment of N 2 O (= 0.166 D).This theoretical model is now adapted to cover the case in which N 2 O is diluted in Xe.The RAIRS data in Figs. 5 and 6 reveal the 65 complication that associations of N 2 O of larger size increasingly form as the temperature is raised.However for simplicity we ignore any compositional changes associated with different temperatures and we describe the films as two component mixtures of Xe and an average N 2 O species, without explicit reference to 70 temperature.Thus the composition of the film is treated as an average over the data between 38K and 49K in Figure 6.We therefore derive below expressions for the variation of the parameter ζ (Eqn.1), the dipole moment of N 2 O and the degree of dipole orientation which contains only dilution dependence, 75 effectively averaging over the temperature dependence of composition.The justification for this approach is that, though inexact, it turns out to give considerable insight into the nature of diluted spontelectrics, with a minimum of parameterization.

80
In keeping with the mean field approach which allows for only one type of active species, we treat the system of N 2 O configurations (monomers, dimers, timers etc.) as consisting of a set of randomly dispersed and identical N 2 O species.A dimer, (N 2 O) 2 in figures 5 and 6, is viewed as forming by a chance juxtaposition of two N 2 O 85 monomers, a trimer, (N 2 O) 3 in figures 5 and 6, as a chance juxtaposition of three N 2 O monomers and so on.Pure N 2 O is the limit in which all N 2 O neighbours are themselves N 2 O molecules.Given this, the dipole moment of N 2 O in the solid mixture should be expected to increase towards the gas phase value for N 2 O as  90 increases, on the assumption that Xe perturbs N 2 O only weakly.For lack of other information, and again in the interests of keeping the number of adjustable parameters to a minimum, N 2 O in its various mutual configurations is assumed to have the molecular volume and the related polarizability associated with N 2 O 95 monomers.
On this basis, there are three interrelated parameters which need to be adjusted according to the experimental dilution of N 2 O in Xe.These are the average distance apart of the N 2 O species for a 100 specific dilution, the value of  (equation 1), which determines the strength of the interaction between adjacent N 2 O species, and the value of the average dipole moment of N 2 O species relevant to the diluted solid mixtures,   , where the subscript  refers throughout to the value of any parameter associated with a particular dilution.
Recollect here that we define the dilution in the film as the ratio of the number of atoms of Xe and of molecules of N 2 O condensed upon the Au substrate ratio, that is,  = n Xe /n N2O .The fraction of the total volume of the film taken up by N 2 O is then given by n N2O  N2O /( n N2O  N2O + n Xe  Xe ) and the total volume of the film taken up by Xe is given by where  N2O is the volume taken up by an N 2 O molecule and  Xe by a Xe atom in the solid film.In pure N 2 O, the volume occupied by the film is given by n N2O 4/3  (r N2O ) 3  From this it follows the average distance apart of two N 2 O molecules, 2r  , previously s pure (Eq.3), but now s  , is given by 2(r xe 3 + r N2O 3 ) 1/3 .The lattice constant of pure solid Xe on a FCC lattice yields r Xe = 0.219 nm and, for pure N 2 O, r N2O has been estimated to be 0.16 nm, 4 giving values of s  as a function of .
We turn now to the evaluation of   .The potential between two dipoles, at an angle  to one another, is given by -  2 /8r  3 (1-3 cos 2 ) where  is some permittivity.The latter quantity, at the low temperatures encountered here, should to a good approximation be independent of .It then follows that   / N2O , that is, the ratio of the value of  in the mixture of monomers, dimers, trimers etc. of N 2 O and that in the pure state, is given by where  N2O is the dipole moment of N 2 O in the pure solid state and   is defined above.The variation with  of the term multiplying <E sym > in Eq. 1 will therefore be given by a product of the varying values of   (< z >/ )  2 with the corresponding values of <E sym >  .
Since the parameter <E sym >  is associated with polarisation, dispersion and covalent forces, we do not attempt to estimate the variation of this term with .Instead <E sym >  remains as a variable parameter in subsequent fitting of theory to experiment.
Using Eq. 3, which relates the solid state dipole moment to the gas phase value, with Eq. 4 and inserting s  = 2r  = 2 (r xe 3 + r N2O 3 derived above, we obtain where  and k are defined above in connection with Eq. 3. Implicit in Eq. 5 is that the variation of the average effective dipole moment 55 of N 2 O in Xe is given by inserting s  = 2 (r xe 3 + r N2O 3 Values of the separation of N 2 O molecules, s  , the observed electric field based upon this value of s  , and of   are shown in Table 4, both for the three different temperatures of N 2 O/Xe mixtures investigated, 38K, 44K and 48K, and for different dilutions at 44K and 48K.Parameters for pure N 2 O ( = 0) are 65 included for comparison, for which values were obtained from a fit between theory and experiment reported in [1] and [4].Values in Table 4 reflect our intuition that the value of   drops 75 with increasing dilution, , in the limit that, as dilution tends to infinity, so dipole-dipole interactions tend to zero.Again the increasing average distance between N 2 O molecules, associated with increasing dilution, creates a dipole moment in the solid which approaches that in the gas phase of 0.0684 au (0.167 D), as 80 indicated through Eq. 6.This incidentally exposes an additional hidden assumption that the presence of Xe perturbs negligibly the electronic structure of N 2 O, an assumption supported for example by results in [8].In this connection one may note that although Xe is highly polarizable atom, with  = 4.01 x 10 -30 m 3 , it can readily 85 be shown that the dipole generated in Xe by the presence of N 2 O is only of the order of 0.002D, making a negligibly small contribution to the field at N 2 O.

Variation of temperature with approximately constant dilution
The purpose of this section is to use the above model to fit the three values of electric field for 38K, 44K and 48K, with respective dilutions, , of 2.1, 2.2 and 2.8 (Table 3) using the parameters set out in Table 4. Recollect for this purpose that (i) the total column densities of nitrous oxide are close to being equal in all three cases, respectively N = 240 ML, 245 ML and 227 ML and (ii) the electric field, E obs , is given by the measured voltage on the surface of the film, V, divided by the estimated thickness of the film, Ns  .
Results in Table 5 give the values of parameters required exactly to reproduce the values of electric field, E obs in rows 4,5,6 of Table 4 15 (also shown in Table 5 for convenience).This is achieved by varying only the value of <E sym >, where <E sym >, or more properly <E sym >  , is defined in connection with Eq. 1. Recollecting that <E asym >  =   / 0  where  is the molecular volume (= 37.79 Å 3 ) 1 a value of the degree of dipole orientation < z >/  follows, since 20 E obs = <E asym >  < z >/  .As seen in Table 5, values of < z >/  , corresponding to the values of the fitting parameter <E sym >  , lie between 0.01 to 0.02 with a dilution of 2 to 3, very much less than the value for the pure material 1 of 0.08 to 0.15 shown in the last column of Table 5.In connection with values in Table 5, E obs is found to be only very weakly sensitive to variation of the value of   which have therefore been held at the values computed from eq. 5 and given in Table 4.

Variation of dilution at constant deposition temperature
The purpose here is to fit data for E obs at 44K and 48K (rows 7-12 of Table 4) for different dilutions for similar column densities of N 2 O. Once more it turns out to be possible exactly to reproduce the 40 experimental value of electric field, E obs in Table 3, by varying only the value of <E sym >  .The resulting values of <E sym >  and < z >/  are shown in Table 6, with the value of the degree of dipole orientation for pure N 2 O given in the last column for comparison.
In connection with results in Tables 4, 5 and 6, we note that for the pure material it proved possible to fit the variation of the observed spontelectric field with temperature of deposition with a single value of <E sym > = 5.430.2 10 8 V/m. 1,4Results in  4) as function of dilution, , as shown.Corresponding values of the degree of dipole orientation are also shown, with values for pure N2O taken from [1] and [4] for comparison.Note that Eobs = 0 for  = 67 at a deposition temperature of 44K.

Power law variations of the degree of dipole orientation with dilution.
Values for < z >/  presented in Table 6 show that this quantity may be represented by a power law in the dilution , omitting the 70 case of  = 67 for 44 K. Taking data for 48 K as an example, we find that 75 where  = 0.025 and  = 0.65 in the range 0.9    23.It may be shown, again empirically, that   <E sym >  may be represented by a power law in  of the form for the same range of , where  = 1.51 x 10 -5 ,  = 0.45 and   is given by eq. 6.Note that   <E sym >  is an energy, which is related to the depth of the well restricting molecular rotation in the film (see Section 4).

85
Recollecting that values of < z >/  and   <E sym >  are themselves derived from a model represented by Equations 1 to 6 above, our purpose here is to show that the relationships (7) and ( 8) are as far as possible in algebraic agreement with the model embodied in 90 Eqs. 1 to 6.This acts as a check on the self-consistency of our analysis.
We now introduce two approximations both of which may be justified numerically in the present case.The first is that coth(x) -1/x = 1/3 x in Eq. 2 and the second derives from an algebraic analysis, based on the first approximation, which shows that the degree of dipole orientation < z >/  may be expressed as given that numerically d 3 /d 2 2 <<1.
We then find that: We need now to show that the rhs of Eq. 9 can be written in the form   - as found by fitting to experimental values (Eq.7).However we do not know, save from experiment, how <E sym >  20 varies with , that is essentially how the field in the environment of an average N 2 O species depends on the degree of dilution of N 2 O in Xe.To proceed, we therefore introduce the observation that   <E sym >  can be replaced by   - , Eq. 8, noting also that d 1 in Eq. 9 contains .We therefore obtain: (10)   This shows that the variation of < z >/  with dilution, , indeed follows an exponential form, as required by Eq. 7, but modified by 30 the denominator in Eq. 10.
For Eqs. 7 and 8 to be consistent as desired, it should be possible approximately to represent Eq. 10 by  - such that  ~ 0.0246 and  ~ 0.6533 when  ~ 1.51  10 -5 and  ~ 0.45, as derived 35 empirically.If for example we hold  and  at the empirical values and allow  and  to vary, we find that for  = 1.45  10 -5 and  = 0.59 we obtain < z >/  vs  in better than 1% agreement at low dilution rising to ~6.5% for  = 23.Alternatively, consistency of Eqs. 7 and 8 can for example be achieved within 3% for all values of  by varying all four parameters, , ,  and  with  = 0.028 and  = 0.6 when  =1.43 x 10 -5 and  = 0.5.This brief analysis demonstrates that the parameterization introduced intuitively into the theoretical model, in particular in developing Eqs. 5 and 6 for   and   , is consistent with the observed exponentially decreasing behaviour of the degree of dipole orientation with dilution.
Eq. 9 can also be used for comparison with data for the temperature 50 dependence of the degree of dipole orientation for a similar value of .This is shown in Table 7.For 38 K, an average value of  for 44 K and 48 K has been used and a smaller value of  has been estimated, referring to parameters in Eqs.7

The high dilution case: absence of the spontelectric effect
65 Table 1 shows that at 44K for a dilution of 67, that is,  = 67, no electric field can be detected in the solid film.This is a significant result of the present work, since it leads directly to the conclusion that some mutual degree of restriction of rotational motion is essential to the many body collective physics which leads to the 70 formation of the spontelectric state.We show below that the lack of detection of an electric field at high dilution is not due to lack of sensitivity of the system but is a real result lying well within the capabilities of the experimental system.

75
As shown in Table 4,   has been estimated to fall to 1.119 at  = 67.This may be compared with the value of   = 2.801 for the  = 26 or   = 43.2 in the pure material.Thus we suggest that somewhere between these two figures of   of 1.119 and 2.801, the strength of the dipole-dipole interaction becomes sufficiently weak, 80 at 44K deposition temperature, that collective dipole orientation becomes unattainable.In this connection, the increase in the strength of the dipole is negligible, lying at 1% over the range of the layer spacing s  of 1.304 nm for  = 26 to 1.782 nm for  = 67 (Table 4).

85
To substantiate the absence of a spontelectric field, it is now shown, on the basis of the theoretical description given above, that the observed electric field for  = 67 should be quite substantial and a potential should readily be measured, if the material were 90 spontelectric in the manner described for lower dilutions.First we note that the observed spontelectric field, E obs , is given by the second term in Eq. 1 , that is, <E asym >< z >/, (see section 4.1 of [1]) and thus: 95 Using atomic units, 4/and k = 246.466and inserting values of  0 , r Xe (0.219 nm) and r N2O (0.16 nm), we obtain: where for simplicity we have used r Xe 3 >> r N2O 3 introducing an error of 4% for  = 67.Thus for  = 67, E obs should take the value 0.0033 < z >/ or 1.7 x 10 9 < z >/ V/m.For 44K, experimental data in Table 4 show that < z >/ = 0.0258  -0.585 , yielding a value 10 of dipole orientation of 0.0022 for  = 67, a figure which may be compared with 0.0037 for  = 26 at 44K in Table 6.Thus if the material remained spontelectric at  = 67, one should measure an electric field of 3.74 x 10 6 V/m.
Experiments have for example been performed for a total column density of 175 ML of N 2 O equivalent to a total film thickness of 175 x 1.782 = 312 nm (see table 2, 44K,  = 67).This would yield a potential on the surface of ~1.16 V given a field of 3.74 x 10 6 V/m.We find a measurement of zero within experimental error, 20 whereas if the material remained spontelectric at this dilution, the above estimate shows that a value in excess of 1 V should be observed.Hence the lack of detection of a spontelectric effect at  = 67 clearly shows that the spontelectric effect is switched off at sufficiently high dilution.

Concluding remarks
At the outset two questions were posed whose answers had as their 30 object the provision of greater insight into the nature of the spontelectric state.These questions were (i) how does the degree of dipole orientation change as the spontelectric species are removed physically from one another?(ii) is there some physical separation between species or species dilution at which the spontelectric effect 35 can no longer be observed?
With respect to the first question, we find that dipole orientation is reduced on dilution of N 2 O in Xe.Thus for the same column density of material, a smaller spontaneous electric field is generated when the N 2 O molecules are positioned on average further apart.With respect to the second, we find that given sufficient dilution, the spontelectric effect is no longer present.
There are two factors at play here in the context of both the 45 questions posed and the answers given.First, we recollect that the dipole term in Eq. between random thermal motion and the restriction of rotational motion, it is therefore evident why < z >/ falls to small values at high dilution.For  = 67 for 44K, <E sym >  . is ~1.3% of kT indicating that at some point where the <E sym >  . electrical interaction lies between 1 and 3% of kT, the spontelectric effect is 80 altogether inhibited.The conclusion is that a local field of sufficient strength to restrict rotation is necessary to the development of the non-local many body collective physics associated with spontelectrics and that the local field-dipole interaction must be at least 1 to 3% of kT.This is not reflected in 85 the theoretical description of spontelectrics presented here, which makes the assumption that a spontelectric state may always be achieved.However it sets a goal for molecular dynamics simulations of the spontelectric effect.
Second, results may be understood in the context of the model involving fluctuating motions of dipoles, put forward in the introduction, to illustrate how the spontelectric state may form dynamically.Some fluctuation, whose average physical extent depends on the temperature of deposition and the degree of 95 dilution, leads to local dipole orientation < z >/ , which we note will be in the opposite sense to that encountered in the spontelectric state.A fluctuation is therefore able to create a fleeting polarisation, (< z >/)  .(  /), and thus a field will briefly arise.We suggest that the more dilute the material the less the value of 100 dipole orientation which will be formed through fluctuations.This arises because of the greater freedom of rotational movement associated with greater dilution, as described above in relation to the spontelectric state but equally applicable here.The compensating field, the spontelectric field, will be correspondingly 105 less as dilution is increased.
2 and 2.8 respectively.Corresponding voltages per ML of N 2 O added are shown in Tables 1, 2 and 3.

Figure 1 .Table 1 :
Figure 1.Surface potentials measured for mixtures of N2O and Xe deposited at 44K for the dilutions shown.The thickness of the layer is expressed as the equivalent number of ML of N2O deposited, excluding Xe.Note the absence of a spontelectric effect at a dilution, , of 67.

Figure 2 .Table 2 :
Figure 2. Surface potentials measured for mixtures of N2O and Xe deposited at 48K for the dilutions shown.The thickness of the layer is 35

Figure 3 .
Figure 3. Surface potentials measured for mixtures of N2O and Xe deposited at 38K, 44K and 48K at dilutions of 2.1, 2.2 and 2.8 respectively 50

Figure 4 :
Figure 4: νNN region of the RAIR spectra of the silica substrate before (black line) and after deposition of a Xe/N2O ( = 5) solid mixture equivalent to 48 ML N2O at 20 K (red line), 29 K (blue) or 38 K (green).The inset presents the experimental spectrum of the mixture as deposited at 20 K (red circles).This spectrum was fitted with gaussian functions representing the contributions of N2O monomer, dimer and trimer (dotted lines); the overall fit is shown with a black line in the inset.

Figure 5 :Figure 6 :
Figure 5: Areas of the Gaussian functions used to fit the RAIR spectra as a function of the deposition temperature for the  = 5 films.The areas were 55 65 whereas, if diluted, the same number of molecules of N 2 O may be found in a volume n N2O  Xe + n N2O  N2O .Thus one molecule of N 2 O may be found in a volume of the solid film of 4/3  r  3 for a film composed of a mixture of N 2 O and Xe with a dilution .Hence n N2O 4/3  r  3 , the total volume of the film, may be expressed as n N2O  Xe + n N2O  N2O which yields r  = (r xe 3 65

Table 3 :
Values of mV of potential added per equivalent ML of N2O laid down at 38K, 44K and 48K, for dilutions, = Xe/N2O, as shown.We note that discrepancies between data at 44K for  = 1 and  = 2.2 in Tables1 and 3lie within the 20-30% accuracy attributed to absolute values of layer

The nature of Xe/N 2 O films; Reflection-absorption Infrared spectroscopy studies 10
7,8mentioned above, we have taken the view that films laid down from gas mixtures of Xe and N 2 O are themselves mixed and do not segregate into pure Xe and N 2 O phases.The only evidence for this is that previous studies7,8have shown that at low concentrations, that is,  = 729 to 1000 and temperatures of 16 to 45 K, N 2 O does not segregate from Xe matrices and remains in the form of monomers.However a simple argument, given below, shows the experimental data presented in Figures 1 to 3 illustrate that Xe and N 2 O remain mixed for the much lower values of dilution used here.
20 It is known from other work that deposition of N 2 O onto a Xe layer or deposition of Xe on top of a N 2 O layer has no measureable effect on the spontelectric field in the N 2 O. 1 Thus segregation into pure Xe and N 2 O phases in the present work would result in a lack of any dependence of the spontelectric field on the value of the 25 dilution, in clear contrast to observation.This provides conclusive evidence that Xe and N 2 O do not segregate in the films studied here.This qualitative result still leaves open the question of clustering in 30 mixed films.For lower values of  it is inevitable that N 2 O species must find themselves juxtaposed to form dimers, trimers etc.In order to investigate this more subtle effect on film structure with dilution, experiments were performed using Reflection-Absorption Infrared Spectroscopy (RAIRS) at Heriot-Watt University.

Table 4 :
the variation of s, , Eobs and  for both temperature and dilution 70 based upon experimental observations of the spontelectric potential (Tables1,2 and 3 and Figures 1,2 and 3) and equations given in the text.Note that to obtain the dipole moment in Debyes, divide the value in au by 0.409353.

Table 5 :
Values of the parameter <Esym> used to reproduce the experimental values of Eobs (also given) as function of temperature of deposition for similar dilution parameters, , as shown.Corresponding values of the degree of dipole orientation are also shown, with values for pure N2O taken from[1]and[4]for comparison.

Table 6 :
Table 5 tend to 50 confirm this for a diluted sample; the relatively small variation in <E sym >  may be put down to the variation in values of dilution, , encountered in these experiments.Results in Table 6 however show a large drop in <E sym >  as dilution is increased.This reflects the property that atoms of Xe create a much weaker local field than Values of the parameter <Esym> used to reproduce the 60 experimental values of Eobs (also in Table
measure of the tendency of one dipole to restrict the angular motion of another.The first term of Eq.1, simply <E sym >, also effectively attempts to hold the dipole in a vice-like grip, that is,  = 23 to 26, this reduces to 0.003% to 0.004% (and, at  = 67, to zero at 44K).At the same time the total symmetrical field falls by a factor of ~25 from between 7.13 -7.85 x 10 8 V/m for pure N 2 O 1 to 2.8 -3  10 7 V/m for  = 23 to 26, where figures span the range 48K to 44K.This reflects both the drop in dipole-60 dipole interactions and the weaker interaction between N 2 O with Xe, compared with N 2 O with itself, as dilution is increased.This is itself accompanied by an increase in effective dipole moment in the solid by a factor of ~2.1, resulting in a net decrease in <E sym >  . by a factor of ~12 on dilution from pure N 2 O to  = 23 -26.Thus 65 in diluted mixtures of N 2 O and Xe, the N 2 O molecules sit in a well which restricts rotation whose depth is an order of magnitude smaller at  = 23 to 26 compared with pure N 2 O or a factor of ~5 between  = 0.9 to 1 and 23 to 26. 70 At high dilution, rotational motion is therefore much less restricted and thermal energy can overpower <E sym >  . .For example for  = 26 at 44K, <E sym >  . is ~3% of kT whereas the figure for  = 1 is ~10% and for pure N 2 O is ~25%.Given that average dipole orientation in the steady state is in part related to competition 75 50from both sides, restricting any form of motion.The relative importance of these two terms, <E sym > and