Fast ion transport in solids induced by an electric field

Solid State Ionics 2 (1981) 301-308 North-Holland Publishing Company

FAST ION TRANSPORT IN SOLIDS INDUCED BY AN ELECTRIC FIELD Yu.I. KHARKATS The Institute of Electrochemistry of the USSR Academy o f Sciences, Moscow V- 71, USSR

Received 8 April 1981; in final form 13 June 1981

A theoretical model is suggested to describe the abrupt transition, induced by an electric field, of a cation sublattiee in the ionic crystal bulk, as a result of which the crystal passes into the superionie conductivity state. The concepts which underlie the model are that the Frenkel defect interactions play an essential part in crystal, and the electric field greatly infiuences the defect formation energy. Expressions for the free energy of disordering, for the interstitial population in the crystal as a function of the strength of the electric field applied and for the critical value of the electric field which induces phase transitions have been obtained in terms of the mean field approximation. The first experimental data on the discovery and analysis of the jump in the ionic conductivity induced by an electric field, descr~ed previously, are discussed.

1. Introduction

One of the specific features o f many superionic conductors is the abrupt increase in the ionic conductivity when their temperature reaches a certain critical point. This behaviour is explained by a disordering phase transition, "melting", of some crystal sublattice (usually of the cation sublattice). The theory o f disordering of the cationic sublattice in superionic conductors has been developed in refs. [ 1 - 1 0 ] and is based on the idea o f interacting Frenkel defects in a crystal which arise when cations are transferred from lattice sites to interstitials, the motion on the latter ones being characterized by a relatively low activation energy. When an electric field is applied to a crystal directed motion o f interstitial cations takes place. At the same time, the electric field may give rise to qualitatively new effects and, in particular, it can induce disordering of one of the crystal sublattices at temperatures much lower than the critical temperature for transition to the superionic conductivity state and also at values of crystal parameters for which thermal transition to the superionic conductivity state is impossible. In the present paper a theoretical model is proposed for the transition o f an ionic crystal into the superionic conductivity state induced by an electric 0 1 6 7 - 2 7 3 8 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 0.75 © 1981 North-Holland

field. In this model the Frenkel defect interaction in the crystal is supposed to play an essential part and the electric field to influence the defect formation energy. The first experimental data on the discovery of these transitions and their studies are discussed.

2. Sublattice disordering o f a superionic conductor in the absence o f an electric field

A model o f a crystal consisting of a rigid anion framework and a partially disorder cation sublattice was used in refs. [ 1 - 1 0 ] to describe the thermodynamics of thermal disordering in superionic crystals. Some cations occupy the cation sublattice sites, while other cations are distributed over interstitial sites of higher energy level. In the framework of the model in which a crystal consists of N identical cells, each cell containing g equivalent interstitital positions and one lattice site o f a disordering cation sublattice over which one cation may be distributed, the free energy of disordering in the mean field approximation is given by the expression F d = U ( n ) - TS(n) = U(n)- nkTlnx-

kTlnDV!gn/(N - n)!n!].

(1)

Here n is the number of cations transferred from the

Yu.L Kharkats / Fast ion transport in solids induced by an electric field

302

lattice sites to the interstitial positions, U(n) the internal energy of formation o f n defects, T the temperature, S the entropy, k the Boltzmann constant, and k In X the change in vibrational entropy during the formation of a defect. The last term in (1) describes the contribution of the configurational entropy which corresponds to the distribution of n arbitrarily chosen cations over the interstitial sites within their unit cells. Using the Stiriing formula in eq. (1), from the condition of free energy minimuni 3Fd/~n = 0, one can obtain the equation for the equilibrium concentration of defects. Introducing a dimensionless variable x = n/N, i.e. the fraction of cations transferred to the interstitial sites (0 ~
free energy F(x), andx(2) to its maximum. The absolute minimum o f F ( x ) corresponds to the equilibrium state of the thermodynamic system; in a certain temperature range, it is reached when x = x(1)(T), and at higher temperatures when x = x(3)(T). At the critical temperature determined from the condition F(x(1)(T), T) = F(x(3)(T), T),

(6)

the defect concentration changes abruptly from x(1) to x(3) (or vice versa) and a first-order phase transition occurs in the system. An analysis of eq. (4) shows that for 2 > 1, the function x(T) is represented by a curve which starts from x = 0 and T = 0 and tends asymptotically to x = x** as T-~ 0% where x** is defined by the equation (7)

x ~ = v/(1 + u). x/(1 - x) = v exp [ - W(x)/kT],

(2)

where W(x) = aU/bn is the Frenkel defect formation energy which depends on defect concentration x, and v = gx. In the simplest model the function W(x) is linear: W = W0 - Me, where W0 is the Frenkel defect formation energy in the absence of defect interaction, and X is the effective constant of defect interaction. The expression for F(x) = F d / N can be transformed to the form

The function x(T) may, in this case, be monotonic (fig. la) or have an S-shaped region (fig. lb) where the phase transition takes place. The condition for the formation of this S-shaped region determines the critical curve viQ7 ) on the v, 2 plane (fig. 2), Vl(2 ) = exp[4(2 - 1)].

xj

x X

I

F(x)--f w(x)a 0 - kT[x In v - x lnx - (1 - x)ln(1 - x ) ] .

o

(3)

Now we shall discuss in more detail the behaviour of the thermodynamic system described by eqs. (2) and (3) for W = W0 - k x [1,11], taking into account that, qualitatively, the results are valid for other types of function IV(x) [5,6,10] decreasing with increasing x. Introducing a dimensionless parameter £ = W0/k, transform (2) and (3) as

(g)

b

1

X/&k

I

i

T~k/4k

T

, i1

d

c

0

o TtX/&k

Tt~ ~/&k

T

1 o

kT -X-

~ -x - x)/x] v ) '

(4)

F(x(T)) = ½XxE(T) + kTln[1 - x ( T ) ] .

(5)

= ln([(1

o ~/Z.,k

In certain ranges of the parameters v, 2 and temperature T, eq. (4) has three solutions x (1) < x(2) < x(3), of which x(1) andre (3) correspond to minima in the

T

Fig. 1. A sketch ofx(T) curves fox values of the parameters v and ~"belonging to the a-e regions on the parameter plane shown in fig. 2. Axrows show the positions of temperature phase transitions; solid lines, the states for which field-induced transitions axe possible.

Yu.L Kharkats / Fast ion transport in solids induced by an electric field

303

3. Sublattice disordering o f superionic conductors in an electric field 10

d

y°

0.5

Fig. 2. Location of the critical curves vi(~') and ~i(~') on the v, ~' plane. For ½ < ~ < 1 the plot ofx(T) consists of two branches: a loop and a main branch which tends to x = x.. as T ~ oo. The loop may be located both above (for~ >x**, figs. lc and le) and below (for ~
(9)

The second branch x(T) may be monotonic (figs. 1d and le) or have an S-shaped region (fig. lc). From eqs. (4) and (5), it follows than when ~ > 1/2, that branch which starts from x(0) = 0 corresponds to the stable states of the system at T ~ 0. From topological considerations it is clear that a phase transition should take place on the curves shown in figs. l b - l d . Thus, different types of plots x(T) and different schemes of phase transitions shown in figs. I a - le correspond to different a - e regions on the v, ~ plane. The phase transition temperature is determined from eqs. (4)-(6). It is given by the expression

kTtr = (W0 -

{X)/ln v.

When an electric field is applied to a crystal, directed motion of interstitial cations occurs. Under this condition, the conductivity can be considered with sufficient accuracy to be proportional to the concentration x(T) of charge carriers, interstitial cations and vacancies, depending on temperature and the parameters determining formation of defects in a crystal. Defect-defect interaction responsible for the abrupt disordering when the crystal temperature changes may also cause a jump in the population of interstitial positions which is induced by the electric field applied at a fixed temperature [12]. Consider a model for the crystal disordering under an electric field, assuming that the thermodynamic equilibrium between the cations at lattice sites and at interstitials of a unit cell is established quickly as compared to the time of exchange of interstitial cations between the cells. The energy of cation transfer from a lattice site to its adjacent interstitial depends on the direction and the strength E of the electric field in the crystal. For the sake of simplicity, consider a model in which each cation lattice site is surrounded with interstitial sites which may be subdivided into two types: those along and those opposite to the electric field (fig. 3).

,e

=,

r

i

I

21i

I

0

¢ ' i I

:1 I j

I

o E

(10)

In the case of non-symmetrical disordering systems discussed in refs. [ 3 - 5 ] , pairs of genetically connected transitions (hi-transitions) with the participation of the loop solution branch could occur in certain range of the parameters.

Fig. 3. Scheme of interstitial site arrangement (solid rings) near the lattice site of the cation sublattice and the energy profile for the cations located at the lattice site and interstitials.

Yu.L Kharkats / Fast ion transport in solids induced by an electric field

304

For the interstitial sites of the first type the defect formation energy is under-rated W1 = Wl(Xl, x2) deE, while for the interstitials of the second type, over-rated W2 = W2(Xl, x2) + deE, where e is the cation charge and d is of the order of the characteristic size of the unit cell. Now consider the thermodynamics of disordering in a unit cell containing both types of interstitial sites- with a lower and higher energy of defect formation. Denoting by x 1 and x 2 the populations of the interstitial sites o f the first and second type, in the framework of the mean field approximation, for the free energy of disordering, we obtain

ysis shows that for sufficiently large X/kT the functions x I(E) and x2(E ) have a multi-valued region where at the critical strength of the electric field a phase transition accompanied by a jump in x I(E) and x2(E ) may occur. For W1,2 = W0 - Xx, on adding eqs. (12) and (13) we obtain

F ( X l , x 2 ) = U(Xl, X 2 ) - (x 1 - x2)deE

F(E, T) = {Lx2(E) + kTln[1 - x(E)],

- kT[(x 1 + x2)ln(½ v) - x 1 lnx 1 - x 2 lnx 2 -

(1 - x 1 - x2)ln(1 - x I - x2) ] .

(11)

From the conditions of thermodynamic equilibrium OF/3x 1 = 0 and aF/ax 2 = 0 and from eq. (2) we obtain: Xl 1 -

x 1 -

~

x2

= ~vexp

[_ Wl(Xl, X2)-deE] kT

X2 1 [__ W2(Xl'X2) + deE 1 i --Xl--X 2 = ~ v e x p kT ],

(12) '

x =vch dee l----Z-~ ~

where Wl(Xl, x2) = ~U/Ox 1 and W2(Xl, x 2 ) = ~U/ax 2. In the framework of a model in which the defect formation energy is a linear function of the defect concentrations, the functions Wl(Xl, x2) and W2(x 1, x2) can be written as W1 = W0 -- XlX 1 -- X12x2,

(14)

W2 = W0 -- k2x 2 -- kl2Xl,

(15)

where X1, )'2 and ~k12are the effective constants of defect interactions. From the symmetry of a system we have k 1 = )'2 = X. In the absence of an electric field x 1 = x2 =x/2, where x is the total population of interstitial sites. In this case k = ~k12. In an electric field, however, X and ~k12may, in general, differ. Below we shall consider for simplicity of the analysis the simplest case ofX = XI2 when W1 = W2 = W0 - X(x 1 +x2). The system of equations (12) and (13) determines the equilibrium populations of the interstitial sites of the first and secon~t t y p e s x l ( E ) andx2(E). An anal-

( WOL_Xx.~ kT 1'

(16)

where x is the total population of the interstitial sites. Using eqs. (12) and (13) the expression for F [eq. (11)] can be reduced to a simpler form (17)

where x(E) is given by eq. (16). The value o f x at zero electric field strength x(E = 0) = x 0 is given by eq. (4). From (16) and (4) when x = x 0 we obtain

deE ch ~

[x exp(--Xx/kT)l

=L

J

1-x

x [ x° exp_(_~dk:~l--i L

(13)

exp

J

1 -x 0

(18) "

The right-hand side of eq. (18) is a function f(x), its character determines the form of the x(E) dependence. For small ~/kT, the function / ( x ) increases monotonically and has a maximum and a minimum at sufficiently large X/kT (fig. 4). The values OfXmax and Xmin at w h i c h / ( x ) is at a maximum or minimum f(x)

fmax

.....

fmin

0

Xmo x

Xmi n

x

Fig. 4. The form of the function f(x) for h/kT < 4 (curve 1) and for k/kT > 4 (curve 2).

Yu.l. Kharkats / Fast ion transport in solids induced by an electric field

are found from the condition df/dx

=

0

Xmax = ~1 T- (½ - kT/X)l/2. min

(19)

The condition Xmax = Xmin determines the critical value ~/kT = 4, from whichf(x) changes non-monotonically. The condition k / k T > 4 is necessary for x0(T) to be multi-valued where x0(T) is given by eq. (4). When ~/kT < 4 and f ( x ) is increasing monotonically, the function x(E) determined by eq. (18) also increases monotonically from x(E = O) = x 0 to its asymptotical value x = 1. The character of the x(E) dependence when k/kT > 4 essentially depends on the relationship betweenfmax = f(Xmax), fmin = f(Xmin) and 1. If fmax < 1 then x(E) increases monotonically. If fmin ( 1 1 then x(E) has an S-shaped region (fig. 5). When x(E) is multi-valued, x(1)(E) < x(2)(E) < x(3)(E), that branch ofx(E) corresponds to a thermodynamically stable state for which the free energy is at a minimum. An abrupt transition from the state which corresponds to the x(1)(E) branch to the state corresponding to the x(3)(E) branch occurs at a critical electric field strength E found from the condition

F(x (t)(E)) = F(x (3)(E)).

(20)

305

x(E). If the loop is the multi-value region, then the critical field strength and, consequently, the phase transition occur only when the loop is so large that x~1) corresponds to the system's stable state. The x~l)(T) states are stable at temperatures less than the phase transition temperature Ttr in the absence of an electric field. Thus, phase transitions induced by an electric field can occur when T < ~ 4 k (figs. la and le) or when a more rigid condition T < Ttr < L/4k (figs. 1b - 1 d) is fulfilled. In the temperature regions where x0(T) is single-valued, the field-induced transitions on the x(E) dependence take place within the S-shaped interval. In the temperature regions where x0(T) is multi-valued, transitions induced by an electric field occur also within the loop branch x(E), The states ofx0(T) for which an abrupt transition induced by an electric field may occur are shown in fig. 1 by solid lines * The critical field strength Ecr at which a jump in the population of interstitial sites occurs is determined by condition (20) and eqs. (16) and (17). It is given by the relation [deEcr \

In c h t - - - ~ ] -

WO - ½X ~¢~ -lnu.

(21)

non-symmetricaldisordering systems [3-5] in which under certain conditions, double temperature transitions could take place at T = T~rl)and T = ~ ) , the phase transi-,. tions induced by an electric field can occur at 0 < T < T~) and T~r2)< T < Tmax, where Tmax is determined by the condition of origination of the S-shaped region of x(E).

* For

Condition (20) is necessarily fulfilled at one of the points inside the S-shaped multi-valued interval of

de Ecr/kTtr 3

In x(E)

0 2

-4 -6

1

-g -10 0

1

2

3

Z,

5

deE/kT

Fig. 5. Curve of ln(x(E)), determined by e£. (18) for ;V'kT=

10; (1) x o = 0.05, (2) Xo = 0.005, (3) x o = 0.001, (4) Xo = 0.0001.

0

0.5

1

T/mtr

Fig. 6. Temperature dependence of deEcr determined by eq. (23) for v = 27.

306

Yu.L Kharkats / Fast ion transport in solids induced by an electric field

If, besides, deEcr > kT, it follows from (21) that deEc r = WO_ I

I g 5, Ohm'!m "*

(22)

kTln(½v).

-5

And, if deEcr < kT, from (21) we obtain deEcr = [2(W0 - ½?OkT - 2 ( k r ) 2 In v] 1/2.

-4

(23)

An example of the temperature dependence of the critical transition field is shown in fig. 6. 4. Experimental data on the phase transition in ionic conductors induced by an electric field

The crystal sublattice disordering induced by an electric field must manifest itself as a jump in the ionic conductivity. This is connected, first of all, with the jump in the interstitial cation concentration. In addition, the activation energy of the interstitial cation motion may decrease, which was discussed in ref. [ 10]. The first experimental observation of the jump of the ionic conductivity induced by an electric field was described in ref. [13]. On the application of an electric field to a-AgSbS 2 crystal, its conductivity at room temperature smoothly increased and when the critical strength was reached it increased abruptly 620 times (fig. 7). Further investigations performed at various temperatures have shown [14] that the critical field and the conductivity jump decrease with increasing temperature (fig. 8). Studies of the dependence of the Hall mobility and the charge carrier concentration on the electric field strength in the region preceding the transition have shown that the conductivity increase in t~-AgSbS2 is caused mainly by the increase in charge carrier concentration. In ref. [ 15 ] the electrophysical properties of the

-6 -7 -8 -9

0

0.'5 1.'0 1.5 2:0E,kVcm-'

Fig. 8. Conductivity of oeAgSbS2versus electric field strength E at different temperatures [14]. (1) 300 K, (2) 345 K, (3) 390 K, (4) 440 K, (5) 500 K.

polycrystalline system Ag2S-Ag3SbS3 were studied and the conductivity induced by an electric field was also found to change abruptly. At T = 300 K the jump in conductivity is greater than two decimal orders (fig. 9). It was also found that the critical field strength and the conductivity jump diminish with temperature. An abrupt change in conductivity in an electric field was also found recently in fl-AgSbS2 crystals [16] (fig. 10). An interesting feature of field-induced phase transitions, which was found in refs. [ 13-16], is a "memory effect" - conservation of the low-resistant state of a sample after the electric field is switched off. The time of the transition from the low-resistant state to high-resistant one after the switching-off of the electric field is ~24 h for a-AgSbS 2 at T = 300 K. It is necessary to heat a crystal to accelerate its transition to the high-resistant state after switchingoff the electric field. The transition to a superionic state induced by an electric field was also recently observed in X-ray diffraction studies [ 16], by the abrupt change of the attenuation coefficient and the velocity of longitudinal acoustic waves [ 17], by the absorption edge shift Ig d,Ohm-~ m"

Ig ~" Oh rn~.m-~

-5

0 (

\\ \ \\

-6

\

-7

\

\

\

\

\

\\\ \

-B -9

0

05

1.0

1.5

2.0 E, kV.crfi ~

-3

0

0.1

0.2 E, kV.cm -~

Fig. 7. Conductivity of oeAgSbS2 versus electric field strength

Fig. 9. Conductivity of Ag2S-Ag3SbS a versus electric field

at T= 300 K [13].

strength at T= 300 K [15].

307

Yu.L Kharkats / Fast ion transport in solids induced by an electric field

Ig ff, Ohm!~'

J

i

i

Fig. 10. Conductivity of &AgSbS2 versus electric field strength Eat T= 420 K [16]. [17] and by dielectric permittivity and dielectric losses [ 15].

5. Conclusions

Our analysis shows that by applying an electric field of a certain critical strength to a crystal it is possible to attain an abrupt disordering in the cation sublattice which results in an increase in the concentration of interstitial cations in the crystal bulk. Possibility of disordering in the near-surface Debye layer of a crystal in an external field which is interlinked with the redistribution of mobile carriers (interstitial ions and vacancies) was considered in refs. [18,19]. From the present study we can conclude that the high-conductivity state of a crystal can be reached by "melting the cation sublattice by an electric field" without heating the crystal. Moreover, as follows from the behavior of a system in the regions a and e on the v, ~ plane, where thermal phase transitions were impossible and the field-induced phase transition might occur, one may expect that the superionic conductivity state can be induced in crystals which cannot be brought into high conducting state only by heating. For example, a- and/~-AgSbS 2 crystals are not good ionic conductors and have no pronounced abrupt changes in ionic conductivity typical of many superionic conductors in the absence of an applied electric field, but have some temperature anomalies. Critical values of the electric field found experimentally [ 1 3 17] were relatively small (Eer ~ 103 V/cm) and could correspond according to eq. (21) to a small variation of the transition temperature (if such thermal transitions existed), unless v is close to 1. In the last case

the right-hand side of eq. (21) is equal to ( l n v ) ( T t r / T - 1) and small at any Ttr - T ~ Ttr. In general the variation of the transition temperature may correspond to much higher critical field strengths. In the present work a relatively simple model for the distribution of cations over interstitial positions and the simplest interaction law in the Frenkel defect system were used. It is evident that all the conclusions found must be valid also for more complicated models in which the attraction among defects and a more complicated structure of the interstitial positions are taken into account. For example, if there are several (/) groups of interstifial sites in a crystal, which are differently oriented relative to the electric field, an analysis analogous to that performed above gives the following system of equations for the interstitial population

x,

[_ Wo-

1 - Y,lm=lX m = v-lexp[-

Z/=IXim l kT

]"

(24) The solutions of eqs. (24) have the same features as those of the system of equations (12) and (13). Just as in the case of the simplest model, application of an electric field to a crystal decreases the effective energy of defect formation and renormalizes the entropy factor. The system (24) also describes the situation where thermodynamic equilibrium is established quickly not only within one unit cell but also in a larger region of space. In the present paper the effect of an abrupt disordering of the crystal sublattice in an electric field was considered assuming that the rigid sublattice remains unchanged. Besides these transitions occurring in one crystal modification, field-induced phase transitions are also possible in which the rigid crystal sublattice undergoes rearrangement, i.e. polymorphous transitions may take place. A theoretical model for polymorphous phase transitions conjugated with the disordering occurring in the absence of an electric field was studied in refs. [20-22]. The approach developed above can also be used to describe the effect of an electric field on polymorphous transitions conjugated with disordering. If disordering takes place simultaneously with lattice rearrangement, for example, from t~ to/3 modification, the condition determining the critical field Ecr contains, besides the disordering free energies

Yu.l. Kharkats/Fast ion transportin solidsinducedby an electricfield

308

F,,,(x~(E)) and Fa(xfs(E)), the difference in the free energies of the defectless a and 13phases FO~ and F~3 per disordered cation:

FO~+ Fa(xa(E))= F~ + Fo(xt~(E)).

(25)

The functions Fa(E) and Fo(E) satisfyeq. (17) where xa,~(E) are determined from (16), each phase being characterised by the parameters Pa,¢~,Wa,t~and ~'a,O" For a "strong" transition from phase a to j3,for whichxa(Eer) < 1 and 1 - xo(Ecr ) < I, from (16), (17) and (25) we obtain approximately

I d~eEa\ W# - ~ ~ +~ - F 0 I n c h ~'---k-T--)=

leT

a - In va.

(26)

Here the subscript 13 shows that the parameters relate to the/3 modification corresponding to a high degree of disorder. If exp(dtseEer/kT) >> 1, from (26) it follows that

d~eEer=l,C~-½~,t~+F~-FO-kTln(½v3).

(27)

For a "weak" transition from phase ot to 13 for

whichx~(gcr) ~ 1, xa(Ecr) ¢ 1, andx~(Ecr) x3(Ecr ) from (16), (17) and (25) we obtain

\ kT

/ _ /

v ~ k ~ exp(W#/kT).

(28)

In a general case, in order that condition (25) is satisfied, it is not necessary that xa(E) or xo(E) should be multi-valued functions. Transition between the two modifications is also possible when the parameters characterizing the defect-defect interaction ;ka/kT and X#/kT are small, so that xc~(E) and x¢(E) are single-valued functions.

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