Solid State Ionics 1 (1980) 481-489 0 North-Holland Publishing Company

THE ROLE OF DISORDER IN SUPERIONIC CONDUCTORS

Marco VILLA Istituto di Fisica “A. Volta”e Gruppo Nazionale di Struttura della Mate& del C.N.R., 27100 Pavia, Italy

and

John L. BJORKSTAM

Department of Electrical Engineering, University of Washington, Seattle, WA 98195, USA

Received 5 May 1980

This paper illustrates from a phenomenological point of view why the study of superionic conductors is essentially a study of disorder. Crystals that are good ionic conductors lack a long-range order in their mobile-ion sublattice. Moreover, low-temperatuie anomalies typical of amorphous materials appear to be rather common in superionic crystals. In several solid electro- lytes the coupling between “disorder modes” typical of glasses and translational degrees of free- dom of the ions can be shown to enhance ionic diffusion. The observed, or expected, properties of these superionic conductors are briefly discussed. The hypothesis that disorder may often play a dynamic role in ion transport in solids suggests ways to synthesize materials of technolog- ical interest.

1. Introduction

Recently, considerable interest has been focused upon the solid materials called superionic conductors (SC). These materials reach an ionic mobility comparable to, or higher than, that found in the melt at temperatures below their melting point. The purpose of this article is to give an overview of some problems and perspectives of the current research while discussing the relationship between disorder and fast ion transport in solids.

From the structural point of view, the role of the disorder in SC is well under- stood as will be illustrated in section 2. Section 3 will review the evidence for

“anomalies” typical of amorphous materials that have been found, to date, in super- ionic crystals. Section 4 illustrates the evidence that “disorder modes” * enhance

* By “disorder modes” we mean the excess, compared to that of ideal crystals, in the density of states at low frequencies due either to a configurational motion or to anomalies of the phonon dispersion.

482 M. Villa, J.L. Bjorkstam / The role of disorder in superionic conductors

ionic mobility. This section also comments upon the properties of such a mechanism of diffusion and its most significant implications.

Liquid-like aspects of the ion dynamics and similarities with amorphous materials indicate that general problems of condensed matter physics enter the study of SC. In the last section we discuss how the recent literature approaches these problems.

2. Phenomenological aspects of SC

A few years ago, the general theoretical models for the phenomenon of fast ion transport in solids were developed. Instead, today the emphasis of research is more in characterizing particular classes of solid electrodes and electrolytes than in defin-

ing the “universal” behavior of superionic materials. In particular, it was the un- successful search for a “superionic transition” or the “ionic sublattice melting” in materials like Li3N and /3-aluminas that led some authors [ 1 ] to wonder whether a

qualitative difference exists between SC and ordinary ionic compounds. For this reason, it seems appropriate to start a discussion of the factors influencing the dc-

ionic conductivity a(O) from the elementary expression

a(O) = (6kT)-1c(Ze)2Z2v. . J

This formula is valid in three dimensions (3D) for ions performing random jumps of length I and frequency Vj. Ze is the mobile ion charge and c its concentration. In

most SC * c is equal to, or of the order of, the concentration of the ionic species responsible for conductivity. In other words, there is no production of charge car- riers driven by a temperature increase. This fact has a structural origin: the density

of the possible positions for an ion is considerably higher than c. This implies that the mobile ion sublattice is intrinsically disordered. Recent studies of such disorder [2,3] have used techniques such as EXAFS, molecular dynamics, neutron diffrac- tion, and X-ray diffuse scattering analysis.

An elementary statistical model predicts that, at sufficiently low temperatures, the jump frequency Vj is

Vi = ~0 exp(-E/kT)

* An

M, Villa, J.L. Bjorkstam / The role if disorder in superionic conductors 483

tion [6], experimental evidence suggests that the absolute rate theory is not univer-

sally valid. The height of the potential barrier separating two nearest lattice positions is often

associated with the activation energy E. For the best SC, E/k c lo3 K, while for normal ionic crystals E/k z lo4 K. An energy in the lo4 K range would be expected

from electrostatic calculations of barrier heights [7]. For Na P-alumina one finds E/k c 2 X lo3 K. For this compound the calculations of Wang et al. [8] show that it is the presence of two Na+ ions in the same planar cell which decreases the poten- tial barrier for Na+ diffusion. However, over the range of SC, various mechanisms seem to enhance the conductivity and it appears difficult to assess their relative im- portance. This fact, together with our still partial understanding of the relationship between vibrational and diffusional states, appears to be a major problem in the study of SC.

3. Low-temperature anomalies in superionic crystals

The amorphous state is characterized by low-temperature properties markedly

different from those of the crystalline state [9]. Recent investigations revealed a number of glass-like behaviors in crystalline @rluminas. Below 1 K, the heat capac-

ity is almost linear with T [lo] and, up to 40 K, it shows large deviations from a Debye behavior [ 111. Saturation effects in the dielectric response are noted below

2 K [12] and e’ decreases with increasing T below 10 K [ 12,131. Below 2 K, the thermal conductivity is proportional to T2 [ 141. Below 20 K a fast, non-vibrational motion is revealed by spin-lattice relaxation measurements of a color center in /3-

alumina [ 151. All these facts have been interpreted as evidence for a broad spectrum of localized excitations phenomenologically described with a distribution of tunnel- ling states.

Another landmark of the amorphous state at low-temperatures, i.e., the existence

of short NMR spin-lattice relaxation times in quadrupole perturbed spin systems, has also been observed in superionic Ag,SbS, [ 161, Cu3VS4 [ 171, and fl-aluminas [ 181. High dielectric losses at low temperatures have been observed in Li,N [19]. The ultrasonic behavior of SC at low temperatures is discussed by Doussineau et al.

PO1 . The experimental evidence collected to date may suggest that a low temperature

glass-like behavior is the rule rather than the exception in superionic conductors. As for amorphous compounds, the problem is obtaining a physical picture for the low- energy excitations that appear to accompany the fast ion transport phenomena.

4. Dynamic coupling between disorder modes and ion diffusion

We will begin by discussing some NMR and ionic conductivity experiments per-

484 M. Villa, J.L. Bjorkstam / The role of disorder in superionic conductors

formed on crystalline SC (/l-aluminas) and amorphous SC (quenched borates of nom-

inal composition (Agl),(Ag20 * B203)l_X, with x < 0.8). Some of the experimen- tal results we will quote appeared in the Proceedings of a recent Conference [2 l-231 Our aim is to show experimentally the existence of a coupling between non-vibra-

tional motion of the rigid lattice and ionic diffusion. In the superionic /3-aluminas. the mobile ion belongs to mirror planes separated

by spine1 blocks containing aluminum oxide. Relaxation times for the m = $ ++ -k transitions of both 27Al and 23Na are shown in fig. 1 as a function of T-l at 21 MHz. At this orientation the 23Na relaxation is dominated by the Am = k2 relaxa- tion rate W,. For both 23Na and 27Al the spin-lattice relaxation process is due to the interaction between the nuclear quadrupolar moments and the electric field gradient (efg) fluctuations at their positions. The solid lines in fig. 1 have been drawn using the experimental values for the Na+ diffusion coefficient and a continuum dif- fusion model [21]. Details of these calculations will be presented elsewhere. Here we focus attention upon the most remarkable features of the 27Al spin-lattice relaxa- tion times T,. Because of the shortness of the 27 Al T, , as well as its frequency and temperature dependence, it is clear that normal lattice vibrations do not provide the dominant relaxation mechanism over the temperature range explored. Secondly, the

500 X)0 200 150 125 100 K I I

L- I I I 1

-IO2

I I I I 2 4 6 8 IO

I OOO/ T

Fig. 1. Spin-lattice relaxation times in a melt-grown crystal of p-alumina. Filled and open circles represent, respectively, 2W;’ of 23 Na and the spin-lattice relaxation time T, [22] of 27A1. The magnetic field was parallel to the crystal c axis and the Larmor frequency was 21 MHz.

M. Villa, J.L. Bjorkstam f The role of disorder in superionic conductors 485

efg fluctuations that drive the T, process cannot be due to the direct influence of

diffusing ions. Aluminums are surrounded by shells of oxygen and electrically shielded from moving charges. Moreover, for orientations where the NMR spectral

lines are resolved, 27Al relaxation times for aluminums at different distances from

the conducting plane are almost the same. Therefore, 27A1 is sensitive to a deforma- tion motion that affects the whole spine1 block. A quantitative comparison between

the 27Al T1 and Na+ diffusion data in fl-aluminas shows that the spine1 block defor-

mation occurs when a Na+ “jumps” to a neighboring planar cell.

Fig. 2 gives, as a function of the ionic radii, the temperatures of the 27Al T, minima and the relaxation rates at the minimum for some partially substituted Na

p-alumina crystals: Na(lOO%), Na(lO%)-K(90%), Na(40%)-Tl(60%), Na(45%)- Rb(55%). The relaxation rates and the temperatures of the minima increase when the radius of the substituted cation is increased. These facts indicate that the defor- mations of the spine1 block are larger for larger cations and that E increases mono-

tonically with the mobile ion radius. As in fi-aluminas, in the superionic glasses (AgI),(AgzO - B203)1_x the dynam-

ics of the “rigid” substrate has been studied through the NMR of a nucleus with a quadrupolar moment (l 1 B). Boron atoms are shielded from the diffusing Ag+ ions by distorted tetrahedra of oxygens. For all values of “x”, a structureless and narrow

(AH% 1.5 G at 20 MHz) llB central line is observed which can be assigned to borons of BO, groups. The absence of three-coordinated borons may be due to the quenched nature of these borate glasses. In the composition range 0.1 <x < 0.8, l1 B T, minima are observed which always occur when the jump frequency Vj, as estimated through conductivity data with eq. (l), becomes of the order of the I1 B Larmor frequency [23]. Again, it can be shown that these minima are due to fluc-

tuations of the BO, units that have a large amplitude and low frequency when compared to the vibrational motion. Very interestingly, a frequency-dependent T, minimum is also observed in the compound with the nominal composition Ag20 - B,O, which is a very poor ionic conductor. This minimum signals that BO4 fluc-

800

I ; 600 Na+

A

A0

I I.1 1.2 1.3 1.4 1.5

Radius (A )

800

600 +- t

400 5

Fig. 2. Temperatures of the 27A1 T,-minima (filled circles) and relaxation rates (T;‘) at the minima (open triangles - right scale) versus the radius of the substituted cation (see text).

486 M. Villa, J. L. Bjorkstam / The role of disorder in superionic conductors

tuations also occur in the absence of the ionic diffusion process.

The NMR evidence presented in this section indicates that localized disorder modes of the “rigid” substrate which have frequencies characteristic of the diffusion

process may exist in SC. The concept of a configurational mode, or a “tunneling

state”, is often attached to a localized disorder mode [24]. Notice that, according

to our definition, the hopping of an ion corresponds to a disorder mode. Theories

concerning the cooperative aspects of ionic diffusion in SC are nothing but models

of interaction among configurational motions of the ions. However, here we will discuss the interaction between disorder modes of the substrate and the mobile ion

sublattice because they may be separately analyzed.

To discuss the interaction between two disorder modes we will use the following

model. Let A and B be the energies needed to activate the configurational motion of the ion and of the substrate, respectively, when the energy (C) of interaction between these modes is zero. Ion and substrate are both idealized as two-level sys- tems described by pseudospin variables. With the high energy wavefunction we re- present the ion and the substrate “in flight” towards a new equilibrium contigura- tion. The interaction will be represented with an exchange-type hamiltonian that allows the energy to be traded between “AI” and “B” systems. We further assume that the degrees of freedom not described by the pseudospin variables constitute a thermal bath responsible for keeping the Boltzmann equilibrium. Fig. 3 gives a graphical representation of the energy levels with and without the interaction. The exchange term creates an excited level with energy E, less than that of either A or B. For C > IA - B I (strong coupling) and at temperatures where, among the ex- cited states, only 11) has a non-negligible occupation probability, the fluctuation in “,4” is almost always started by one in “B” and vice versa. The two disorder modes

occur at rates following the same activated law with activation energy E,. When A

is significantly larger than B, the disorder mode of the substrate can be considered the main cause of the ionic mobility at temperatures sufficiently lower than A/k.

mode’A’t mode’B’t exchange I coupled system E

13> - E3=A+B

12) -E2:$A+B+~~)

-A

-B

II> -E, =J~(A+B-~)

'-0 -0 IO> - EO= 0

Fi k

3. Schematic representation of the energy levels for the hamiltonian H = Aa+a- + Bb+b- + 5 C(a+b- + a-b+) where a’, b’ are Fermion raising and lowering operators. At the left side,

we show the levels in the absence of the exchange term (C = 0). The levels of the coupled sys- tem are shown at the right side.

M. Villa, J. L. Bjorkstam / The role of disorder in superionic conductors 481

For weak coupling (C < ]A - B I) and for A > B, the exchange model predicts that

the probability of finding the ion in the diffusional state scales as C2/(A - B)2 at sufficiently low temperatures. In such a case the attempt frequency deduced from ionic conductivity data may be expected to be substantially smaller than 1012-1013 s-1.

The interpretative frame offered by the exchange model is very qualitative but it can be applied whenever a degree of freedom of the thermal bath can be considered as the main cause for the ion transition to the diffusional state. Both physical intui-

tion and the NMR evidence suggest that the substrate motion should have large am-

plitude (compared with harmonic vibrations) in order to couple with ionic diffusion. The diffusion enhancement mechanism outlined above is a form of “lattice in-

stability”, typical of disordered materials, in which small configurational changes of

the ion environments may take place. For glasses the balance of forces which sepa- rate different configurations is very delicate and likely to be affected by the method

of preparation and/or thermal treatment. These factors are known to influence, among other properties, the conductivity of vitreous electrolytes [25-271. For

crystalline /3-aluminas the question of lattice stability - or lack thereof - seems inter. woven with the origin of non-stoichiometry and departure of the aluminum oxide blocks from the ideal spine1 configuration [28]. The ion mobilities seem to be influ- enced more by preparation techniques than by compositional variations, provided that no 0” form is present. For example, the RT conductivities of melt-grown and flux-grown Na &alumina single crystals differ by an order of magnitude with Na+ concentration differences of only 6% [29]. Moreover, this variation in Na+ mobil- ity has not been identified with changes in structural or optical properties [29]. Thus, in both (3-aluminas and glasses, the relationship between structural and ion transport properties appears intriguingly ambiguous and dependent upon prepara- tion procedures.

5. Conclusions

We have shown that in /3-aluminas and in a class of ionic borates a disorder mode

of the substrate accompanies the phenomenon of ion transport and plays a dynamic role in enhancing the ionic diffusion. The evidence for such a role in other solids is mostly indirect, to date. The comparison between conductivities of several ionic phases and of the corresponding crystalline phases shows that the amorphous state favors ionic diffusion [2.5]. The rolling-quenching experiments of Glass et al. [27]

are an extreme example of this fact. A configurational low temperature motion of the “rigid” substrate is evident in the crystalline SC Cu,VS, [ 171. Observation of low temperature anomalies in some crystalline SC strongly favors the hypothesis that these SC have a disorder-enhanced ionic transport process.

Though rather speculative at the present time, we consider two subtle implica- tions of the exchange model discussed above.

488 M. Villa, J. L. Bjorkstam / The role of disorder in supen’onic conductors

(i) Low temperature anomalies are usually associated with broad distributions of localized excitations. #en T is decreased, the probability that high energy modes are excited goes rapidly to zero while modes with energies comparable to kT are still saturated. These low-energy modes are expected to couple weakly with the ion transport process and may be able to cause only a local motion of the ion. Never- theless, below a certain temperature and frequency (w) they may give a contribution

to a(o) that is larger than that of the high energy modes. The contribution to ionic diffusion of a mode with energy ZkT is nearly temperature independent. Therefore,

the exchange model offers a possible exp!anation for the low temperature flattening in the Arrhenius plots of U(W) reported for o-alumina [ 131 and for vitreous

Ag7I,AsO, [25]. (ii) Enhancement of ionic diffusion through coupling with a localized disorder

mode of the substrate occurs by lowering the energy otherwise required to reach the diffusional state. Optimization of this mechanism may be attempted by experi-

menting with different methods of synthesis and/or thermal treatment of a given compound. Moreover, an implication of the exchange model is that, if a suitable dis- order mode exists, the conductivity may be rather independent of the ionic species,

as long as the change in chemical composition does not alter the dynamics of the substrate. In (AgI),(AgzO - B203)1_x it has been observed that substitution of iodine with bromine does not appreciably modify either the Ag+ mobility or the

llB spin-lattice relaxation behavior [23]. From the technological point of view. it would be highly desirable to obtain glasses with Na+ or Li+ mobilities comparable to that observed in silver-based glasses. The exchange model suggests that these glasses may be found and their performances optimized with appropriate synthesis procedures.

Characterization of what we have termed disorder modes is today a very open problem of condensed matter physics. As a matter of fact, different authors refer to disorder modes in different ways: tunnelling states, off-center positions [ 171. corre- lated states [30]. However, to a large extent, these names refer to the same type of entity. The main implication of the exchange model is that this entity may be studied through ionic conductivity measurements in a number of SC. It may well be the case that experimc ts in SC will make fundamental contributions to our under- standing of disorder modes.

Acknowledgement

Financial support of the U.S. Department of Energy (Grant No. EY-76-06 2225) is gratefully acknowledged.

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