c].ectricnl conductivity C()'!Jrynrf•ble to those of liquid

el'.,ctrolytes. These ·nn t!•r1.els are Ill so termed as "sunerionic

solids" or "ff' st ion con(!uctors". Typically a sunerionic

solid hv!! the following cht:rf;lcteristics (Ch<1ndra 1 , to be

oublished) :-

( i)

(11)

(iii)

crystal bon•Ung is ionic

~aectricol conductivity is high

( -1o·1 -10·4 ohm-1 cm-1 )

Prine ill" 1 chtl rge CF rri.ers ~Jre ions 1.1hi.ch

me1ms thet the ionic trvnsference nu':llber (t10n) is almost

equal to 1. Here t 10n refers to the frt~ction or charge

trc"lsferred by all tyoes of ions ta>ten together 'Aith

reference to the total charge trc n5ferred.

(iv) '!"hH electronic conductivity is s'!!all.

Gener<-lly '1111 ter ie.ls 1ofi th electronic tre nsference numbo:r (te)

less than 10-4 are considererl satisfectory superionic solids.

'l'he values r)f electrical c::mductivity of a few

ionic and su:oerionic solids ~lT!:! sho>-rn 1!1 Figure 1.1. 'l'he

highest corductivity at roont tem'1orature obtained so fer

is for RbAg<IS 11hich is 0.27 ohl!l-1 cm-1 • This is mFny orders

J ,f: ( c ;·

II l ,:: ' "

,I ,~ .. ; I

f' I ' d ' I i '~! I .

f I ~-q ; !/ ; -II r.· ;; i it .~~ /

I ,')

.¥ "

' ;

i i .-;

' .;" ·t" ~i-' f;

' I ' {: j f f I ' ~ ' I '

" i ' ... f r{_]

.,

/ ,)

11'

I

' .'

I ' '

' .

l ' ~ .,, -l [·

l .. .u ;

,, ,, t

~ ' '_+.' l ,-.: p, \ ' .:._·, )

''J I ,, . / i}) /. .,

::·; , .. ,1.~ '

.,,

·''

~: -·l,

';;.; -~""<

(''1

,I)

., fJ-'

' il.' 1/! :)

~· .:. . .:)

1.},\ ~ ;/> ;:;:;

' ~;.. i',,,J ' ,,

;; .....

'~3 ,. ~4<>•

~' '--:~ v"'1

:..\: , .. .C '"'

~~ ,., r:n . ' '\.~- rj

, :n '1-1;

"' ',' t .. t"1 ·:s :.:J :0 ·"' .I'J-

•:! ' 1:· ~ 5;.., r.::;_·

;:~ "' ,, ~

:.~ . ...... , .. t.. .:: ~

" .~ C:I.J L .... ~-~ >

of rnrenitude hir.hor ttwn those of the more com"lonly knolill

ioPiC solids KC1 1 l-'E,Cl otc. 1hich hr,ve room te'll'lerature

conductivity"-' lo-12 -lo-16 ohm-1 C'll-1• It rncy be noted

that most of the S'.m·rionic <1nteriols vttain high elG<'trical

conductivity r:fter undergoing a "lhasa transition at c, certain

tem'·>er~ture wl1ich !D1lY or 'lll>y not be WE!ll defined.

(1.1) Typ_g s of ionic solid~

f'rom the cryst!: llogra ohic point of view a ne:rfect

crystel of on ionic com;)~'lllnd would be an insulator (Lidit'rd 2).

~'he nr·esencc or defect or disorder is a necessity t·J sustr: in

mcr.:ningful ionic trE:n:1port. It is notural to cla ssi:f'y the

ionic solids tccor•Ung to the tyoos of defect or disorder

resnonsible for ionic transoort. '"hese solids f.re prinCi'Ofilly

of t;vo tyoos (Cht"ndra 1 , to be publ:ished):

(1) Point defect tyoo

(2) ?~olten subl!"ttice type

In the 'noint defect type' solids, tho tr&ns l)>:>rt is through

~ren!{el or '·ichottky defect cmirs which Pre th,.;rmally

generated. f,s & result, the nu.'Tiber of' defects (and hence

charge carriers) ore D function or tem'?'"r' ture. 'l'he acti-

vation energy is genernlly high rvl eV or :nora. 'Point

defect tyoe' solids c& r. further be subdivided according to

defect concentration dun:;ity as follows a

!)1] ute:

(b) ConcentrRtadl l•:xnm::>la - stabilised .~irCOl'it'

or Hafnia, l'Al· 2 etc.

In 'molten suU.nttice ty:'le 1 solids, the nu·~ber of

&. X•rticul: r ion io less ti1rm the nu'llb-:•r of sitas flWJilt ble

for it in its sublattice. f. s :;. result, ell these ions cen

'ho~• or ·~ova like a free-ion' from one position to another.

inca a 11 these ions o.rc avr.iie 1Jle for tr~msport, the

conrluctivity 1s l;o\rgc~ and ~:ctiwtion energy is low. 'I'hese

<itt terials/ions possess an 1n veragc structure 1 rether than

fl rigid structure.

···ice & Roth 3 hns suggested another scheone of

cla ssHication of ionic solids lm sed on the '"'Srameters of

his 'free-ton like theory' of super ionic solids wr.ich is

esr:ef'tit.lly the snme as 1.: bovG. f,ccording to this scheme,

ionic soli'ls <J:rs of three ty·1es :-

":yoe I: ~hese ere conventionPl ionic solids

in 11hich the defect concentntion is low. The nQ'Ubcr or

~obile defects isrvlo18 /cm3 or less. This is the c~se of 'dilute noint defect type' ionic solid discussed above.

Type III They contain a large concentration

of defects which hllve COf;lesced into coherent substructure

or 'exterded defects' of sub~icroscooic dimension. These

are the smne as 'concontreted point defect type' solids.

'l'he de fact concentration is "'1020 /cm3 •

'!'yl:le III a This is the case of 'liquid like

molten subl1'ttice 1 in ·,thich nractict.lly v.ll the ions in a

sublnttice ere available for :-~ovement. 22 :l mobile charge carriers is ,v 10 /crtr.

The number of

These are generally

mar!ted by a channelled or layered structure.

Types II & III a.re sunerionic solids or solid

electrolytes while Type I are essentially Insulating Ionic

.3ol1ds like alkali halides.

(1.2) Insulating Ionic Polids

They possess conductivities lower than lo-10 ohm-1cn)1

at roO!Il te'llpersture. A great majority of the ionic solids

belong to this class viz. alkali halides, silver halides

(t,gel and tgl2r) etc.

In an ideally perfect ionic crystal the movement

of ions under the action of an electric field is not

oossible if we r. ssume thct trt nsoort or charge could only

take nlr ce vio <·nion - Cl·tion exchange. This nrocess involves

r very high onorgy of th::J order of 15 eV i.e. in one gram­

molecule auch nn event would ::~ccur only once in 1030 years.

Prom extons~ ve series of 'llef.· sUI'e'"1er:t s of ionic conductivity

cs 11 function of te~·1orntur"', t:.vo f<,cts stand out :Jlrinly;

(i) ~hG existence of two main regions in

conductivity curve r,s n function of

(11) In both the regions tho log (di') vs T-l

is e. :>.,rox1mt1 tely linear.

'Phe high t<l'n'lerature conductivity 12bove the transition

rl'!gi::m or ';mae' .in genefal is an intrinsic prooerty of

the crystal nnd ·~easure;'!lents in this region ere quite

reproducible. On the other hand, low te:nnerature conducti­

vity (o:r extrinsic conductivity) dis'Olf·ys a smaller slope

-1 in log o-'i.' vs T curve but it denends on the nurity of

the scmnle. 'i'o ~xnlllin the lou but finite te"!'>8rature

denendent ionic conductivity the existence ~f thermally ,.,_ I!L s ~ e-'TV

gener~1ted noint defect ,..invoked.

In general electrical conductivity is given by

cr = (1.1)

where ni is the nu~'ber of chorge carriers, 'i is the

charge and p1 is the :nobility of the churge carrier of

ith soccies. It is no11 wall recognized th!!t rrodomiMnt

tyt)os of chB.rge Clorriers in ionic solids ere either Frenkel

defects or Schottky defects.

ln case rJf T'renkel defect, an ion (cation or

onion) from 11 norrnttl lnttice stte is activsted to a

ne~rby interstitial site, cfoat1ng an interstitial ion

llnd &!" ion 1n1cancy. Jf this 'Jair of defect becomes thermelly

dissoci~ted to beyond the range of their Debye - Huckel

charge clouds, they lofill migrEtte indeoondently in an

electric field, in general, with different mobilities. If

the te'!lnerature is high enough to o ssure ion defect

equilibrium (and impurity effects be ignored), there will

be a fixed and equal concentration of the two types of

defects given by the following expression.

. 1/2 gf (NN') exp (- 2Kr) , (1. 2)

\'there gr is the energy required to form a. thermally disso­

ciated Frenkel defect pair, N is the number or normal

lattice sites !lnd r~· is the number of interstitial

sites per unit volume. Silver bromide and chloride are

the classic examnles of this class.

:;ohottky defects (cation and 11n1on vacenc1es)

are '' ssumed to bo introduced in ooirs into the crystal

le. tt ice frorr the surface sites or from vr C!l ncy sinks

such es dtslocvtions or grl>in bound~>ries. Fixed and

eC'u.al concentrr.tions of the tt~o ty:Jes of vac&ncies result

ct ench tenmercture assuming ionic ec,uilibrium and negli•

giblo impurity effects. i-. go in dissociH ted defects may

migrate indeoendontly in an electric field. In general

'chottky defects occur in crystals with Co'lloorable

cHtionic end f>nionic rr1di1. The number of vacancies (ns)

of ef:.ch tyoe is given by

= gs

N exo ( - 2kT ) , (1.3)

where g8

is the amount or energy required to form n

~~chottky defect ne ir.

Ionic mobility in such solids occurs as a result

of random, tharmr,lly activated jumps, either of an inter­

stitiel ion into an adj~cent unoccupied interstitial site,

or of an ion on u norm[,l lcttice site into on adjacent

vace.ncy for th[·t t-ype of ion. Another migration mechanism

is that in which a.n interstitiHl ion replaces a~ike ion on

normnl lattice site, pushing it into on unoccuoied edje.cent

interstitial site. The lttter is known as 1 interstitialcy 1

mechanism. In eny event, each jumn nrocess is ~ssumed to

involve &n activrtion energy of L'l.g, such th£t the jumo

fro' uency 1s given by

w= 1 exp ( - ) , (1.4)

where y is the v1brationel frenuency. The above exnress1on

1s the ju:np freouency vthen there is no a nplied electric

field. , :unpose an uniform electric field B acts along

one of the sxes (say x - axis). The aryplied electric field

ccn be rcgr.rded as adding a term -eEx in the potential

energy of the interstitial. The net effect is that a

jump in the direction of field would take olace with an

increased probability

W' = '1 exp -1 -- ( kT

E - ! ea.E:) 2

and a jump in a direction opposite to the O.'Jplied field

takes nlace with & reduced prot~; bility

If ea.E (<( k1:', this would result in a current density

(L1diard 2 )

y exp (-.LlE ) ItT '

(1.5)

(1. 7)

The mobility ( Fil ir then given by

= ea 2 A& = kT- Y exp ( - kT ) '

!fence the Of•rtial conductivity (fro111 Eq. 1.1) due to

mechanism consHered ~Jb.:>ve is

( 6.& ) v exp -- , \<T

(1.8)

(1.9)

where e is the charge on :_ r.~.rticulnr ion oartic1nat1.ng

in conduction 'll<~chanis~ and n is given by Bq. 1. 2 or

Ect. 1.3. Hence a n1ot bet~;een log cr •.r and l/T would

be linear if one type of defect mechanis~ is resoonsible

for the conductivity.

In ~ctuel experiment, broadly speaking, the log

a- T vs l/l' plot shows t~ro distinct regions. The high

temperature conductivity is indeoondent or purity of the

crystal (if the i~purity concentr&tion is not too large)

but the low temnerature conductivity is strongly impurity

sensitive. The former is ;::no1r1n as intrinsic conductivity

&n'l the l~}er e s the extrinsic. Intrinsic conduct! vity

is due to thermally generated defects while the extrinsic

conductivity is due to vac~ncies created by imnurity doning

(nliovelent cation or anion). J. few '!lore veri&tions in

slOJ)e cen be seen in the conductivity curva due to associa­

tion and nrocipitc tion of hnurities. For alkali ht•lidos,

it is now well established tho.t both the positive and

negative ion vncencies nrc 11obile. The transnort oora-

meters cc n be obtainod either by folloving Chandra and

l'olfe's 4 1roccdure (from electrical conductivity measure­

~ent on oure and divalent cation and enion doned samples)

or by the method of Fuller et al 5 (fro11 diffusion and

electrical conductivity 'lll!lt;sure•nents on nure and divalent

cation doned sn"lloles).

In general both ce.tions e1:1d enions can move in

a solid lattice. 'l'he cations have reltetively small ionic

r!'ldii as co:noored to anions. t·or e.xa11ple, the Pfluling 0

ionic radii of rlknli metal ions are : Li+ (0.60 A), +, 0 + 0 + 0

ra i0.95 /.), K (1.33 A) and Rb (1.48 !>.)while the ionic 0 0

radii of halt de ions are : F'- (1. 3ti l\), Cl- (1.86 /,), 0 0

Br- (1.95 A) and I- (2.16 /\ ). f.•s such the number of

conducting solids vrith alkeli metal ions l'.re expected to

be more. The nossibility of halide ion conductors is much

less except tiwt for the s:nsllest 1dn F-. In fact, the

materiels aVt?ilable so far confir'1l this sttJtement.

The more cO''l"lOn ions showing high mobility are

i + r + + + + - d o2-. L , Jo , K , A g , Cu , F an ~~ost Of the CO'IIpOunds

of the latter two ions (1·'- t-nd o2-) show high conductivity

only nt high te'l!)'lerntures. Ler.ving Oxygen ion conductors,

most of the sunerionic solids inv·)lve D monovalent ion,

There ۥre no di- 11nd trivnlent ions reoorted showing high

mobility though renorts for S111lll Ionic conduction exist.

'T'hls 1s soMewhat axooctad since the coulombic energies

involved in jumoing/hopping of di- or trivr•lent ions would

be more thnn those for monov~; lent ions. !IOI<Tever, the

obove descrl~tion ia an oversimplification or the actual

sitUP.tion. The structure is probably the most important

f~ctor ( cJiedersich & Geller5~. Structural princiPles

governing the formation or solids showing fast ion

transport has not yet been studied in. detr.11 though the

beginning has been made by Phillips 6 and Paleigh 7 •

Table 1.1. gives some of the irnnortant high

conductivity solid electrolytes or sunerionic solids.

bmongst these, the highest conductivity has been reported

for silver ion conductors out or which F.bAg4 I 5 is the

best. ~his thesis is concerned with silver ian

conductors and hence some of the salient features of

silver ion conductors are reviewed below.

Te ble 1.1 :o:~~e 1mnortent ~:uner1on1c :;olids

(s~c Chnndrs 1 , to be published)

--------------------------------------------------------------------;.Jaterinl ~~obile

ion t:onducti- ?o:nnoratura i'.ctivntion vity _1 _1 (oC) energy (ohm em l (aV)

--------------------------------------------------------------------I II III IV v

-------------------------------~------------------------------------

Alkali metal iop conductors:

Lithium p-alu'!linn

··odium ~-nlu'rlina

Dod!U'll ~-gallate

:C:otassium f-4'lumim

Silver ~on conductors~

' '<!. g4 !5

u•

-1"" ,, Li+

,.~~ +

,.,,.,.., + dU

~~a+

,.+

1.2xl0-5

1.5xl0 -3

1.3xl0-4

1.4xlo-2

5.2xl.0-4

3.oxi.o-2

6.5x10 -5

0.21

0.21 (pellet)

0.27 (crystal)

0.19

25 0.43

400 o.ca

25 0.37

25 0.16

300 -300 0.27

300 0.29

22 0.1.

22 0.1

22 0.1

Table 1.1 (contd)

-------------------------------------------------------------------I II III IV v

·------------------------------------------------------------------q ..t.gl

Copper ion conductors:

··cu " ·. 4 ... 5

CX: -cu2 ;e

oxrgen ion conductors:

Th02 : 8;i Y 2o3

C I) 7 ·. ~ 1 e 2 : ... r ..

Fluorine ion cQnductor§a

+ ·g

' + •'· g

~·­•

1.0

0.02

0.09

0.09

0.6

5.5xl.o-2

-3 4.B:xJ.O

1.lxl.O-l

1

6:xl0-4

4xlo·2

4:x:lo·3

150

25

450

280

150

1000

1000

1000

500

200

700

100

0.16

0.11

0.19

0.05

1.10

1.12

o.e 0.26

0.2

0.38

(1.4) .;..LUyor Iop Conductors

·uver ion conductors llre the most extensively

studied croup of su'mrionic solids and hnvo beon very

'"!Jll revietrad by :tedersictl ti Geller 6 , 7't•iwhnshi 8 ;

J·unke 9 1 hnhi 10 ond Ch1.ndra 1 (to be :mbl1shed).

"'hn enrliest good high ternnerntu:re silver ion conductor

"ltOb<'lbly rol'lorted in litor~'ture ls silver sulfide,

r g2:>, by f'r,rr,dny 11 ·.mo had also Slll'Tllised the ~rcser,ce

of many othel' such compounds -- "t.here is no other body

'•lith \·rhich I rm PCqWlinted., that like sul,buret or silver cz;n compere 1.rith metr.ls in conducting pot.,er for

eleot:r1c1ty when hot, b~t which •••• during cooling

lose in nower • • • • Probably, hot/ever, l!lllny others may,

Hhcn sought for, be found". 1 g2se and Ag21'e, compounds

similar to l'g2u, lu:ve been lr.ter found to have high

conduct tvity at high te:nrJerr. tures (Tubandt 12).

f{ovrever, tl1e ~e<'sure:nent of conductivity in silver 13 halides by 'uoondt ''- r,o:renz F~&y ~-tell be taken as

the starting no1nt ror the uresent day effort to

underst~:nd <:n.d se:>rch for mnr silver ion conduction.

'rhey renorted th~:t the conductivity of fo.gi abruntly

increases by more than 3 orders of 'ilagnitude at l47°C.

This high conductivity Ph11se is known e.s O{•Agi (cubic)

nnd the low conductivity obese is f3-!.gi (wurtzite).

C)1nce then, a lr,rge nu:nber of silver ion conducting

solids ht•Ve be'ln disc11vnred.

''o~Jt of tho llilver ion conductors ere b;. ;_.od on

i ei. "'hcse conductive nhn~~cs hove been obtr-ined by

rx: rt!e lly :

( 1} + ubstitutin.<; different cntions for f,g

(11) 'ubstltutin;: different anions for x- or

(iii) oUbstit·uting both Cf-tions and anions

'"'he fir t>t re.,:rrtod ro•)'IJ! twl'l ,ero ture good silver ion

conductor is cetion substltuted end >1as obteined by

solid solution }H + 4 ;;g I ('" = ~, Hb, ~rrr4 ) independently

by <r£ldley ._-. ;;reant'l 14 and -~v1ens <;; .\rguo 15• ':'he

compounds '·l!;g4I 5 are known to decompose ~ccording to the

fol101~ing solid st£te disnr'1'Jortionation reection 16 (owens ) belo'" o certain tomnerature :-

7 /;g I

':'he dis'Orooortionation te1!:Jeraturcs for

1'ifg4 I 5 , H~\g4I5 Rnd ~H4r.g4r5 are 37, 28, 32°c

tesnectively. ~'he nresenca of water vapour seems to

eatalyse the disproportionation reaction rate.

Chcndra & :~ohabey 17 hr.ve studied the e.bove reaction by

observing the rP.te of growth of t gi exciton neak.

'rhese results are givAn in Cha;,ter VI es a subsidif;ry

matter of this thesis.

Ion trnns'lOrt apd dywmicsh!!vt been :Jtudied in

silver ion conctuctors usinl': ~ \·tide V!•riety of techni<'uas.

's n r<"sult of these studios, more or lass a common

nicture is e·noreing out for the ~~ gl- ty'le su:>erionic

solids (l·unl<o 9 ; Chnndra 1 to be nubl1shed).

(a) '!'he c1 tions ~.J.re structurally disordered

e.r•d tho cntion sublrtticc is "liouid-lilte". The nurnber

of c1 tioTls is less tlm·~ tho nucnlJ.::lr of ~itos (voids)

avr• ilfcble for thecn nnd those site;; ~;~rc OOCU;'lied at

(b) '"he cnions vre arrenged in such a \·my

that the l•:>eal potenti&ls are rather flat r.long certain

lines whtch intercor~ect neighbourine sites. Along

these lines the difference in notentiul energy is of the

order of thermal energy. i ceording to Geller 13 "the

common structural motiff in the conductive comryounds

is the existence of pcssege 1<1ays of face•sherod iodide

ion polyhadr!'! n.

(c) f1s a conSG1;uenca of (a) and (b) all or

e. large frnction of the crctior:s cDn move from one site

to anothe:r site ~lith a. very low activation energy of the

order of ther:ne.l em!rgy and thus, in genore.l, partici!Ja•

ting in the cation diffusion nrocess. The diffusion ryath

is channel like cni not e_:xrctly liquid-like.

(d) The disordering process may be first-

orlier or Mcond-order. ".t the first-order transition

t;e;.,ere.ture the conductivity changes nbruptly and generally

involves o chnng'3 in J.ntt1co symmetry G l!tant hent. Jn

s0cond order ptwse tr11nsltion, there is no Abruot change

in conductivity -- only the slone changes slightly.

i1cnerally, thore i!:l no or S'nall chEmge in lattice

sym'l!etry and is c·c~omut.nied by & power le.w divergence in

soecific heat.

(e) Ion-Ion correlr>t1on is en imoortnnt

f&ctor in brin,:;ing !'bout disorder (e.g. Uubel'!/IS.n 18;

Pice et al 19 i ''ardee & t~aha.n 20; ;fohnn 21 ; vergas et al

~iolomon 23; :eloh & Dienes et al 24 ; Lederman et al 25 ).

(f) The ion transr.>ort is -oossibly by a

ju'!lp·diffusi<m n:rocess on which an additional cn tion

local motion is superimposed (diffUsive and/or overda~,ad

oscillatory local motion).

Unusual struc1lure is tho single ~ost imnortent

factor responsible for high l~g+ ion trfinsport. There are

:nore sites avl'ilable than the nll'llber of ions par unit

cell giving the ions e 1feee-ion like state•. ~he

nU'llber of available sites end nu'llber of ions ner unit

cell ere given in Table 1.2 to illustrate this point.

Table 1.2

SUperionic Solld

I

o<' -.Agi

CCHs)4N 2.Agl.3ILS

<<:siisNH) Agsis

RbAg415

r/ -Ag3~U

f3 -Ag3

SI

Ag3

::mr

v<. -Ag2S

o( -Ag2

se

«-Ag2Te

Structural parameters or some Ag ion conductors

structure

II

b.e.c.

HeXagonal

HIOO'lgonal

CUbic

b.c.c. (disordered)

b.c.e. (ordered)

b.c.c. {ordered)

b.e.c.

b.c.e.

r.c.c.

Lattice constant

Ao

III

5.06

a 12.77 c 26.54

a 11.97 c 7.41

11.24

4.99

4.90

4.81

4.88

5.0

6.57

lbmbfllr or .Ag pl'!r unit call

No. or .Ag s1 tes available

Stab111 ty range/ Rmarks

IV v VI

2 12• or 147-555° 42

*J>ref'erl"ntial

39 123

lD 34

16 56

3 12

3 3

3 3

4 6

4 -8 *4 are at ~ah('!dral 179°C

sites and the rP.maining mixf!ld conductor 4 distributf!ld are 20 posi-tions 3-,4- and G-coordin~ t~1.

l~

c

(1.5) Superionic pht,se trrnsitions

It bE• s been found durinr. the electrical

conductivity Vs tem))erpturc studic s for most SU:JCrionic

solir'ls thllt the high ionic conductivity is ottninad only after

the mstariDl has undergone s001e phase trcnsition. 'f\~O tynicol

e:xemples ore givan in figure (1.2) for Rbllg4I5

l!.nd Agi. It

is seen thr.t at T1

the conductivity rises by order of mBg­

nitudc • 'i'hhJ tr: nsition is rcferr()d to FlS Class I (or Insu­

lc.toY'-Electrolyte phase trensition). '?he w lues of T1 for

RbAg4

I 5 and Agi respectively E:re -151 °c end l4?0c. There is

yet another type of transition in which the conductivity does

not cilf·nee repidly but at the trvnsition only the slope of

()Vs"f curve changes. The temper" ture T2 (•-64°C) merked

in Figure 1.2 for Rbllg4t5

refers to such a trrnsition. l.ft;tny

theo1•ies have been :out forward to account for tho above two

types of trt·nsitions outlined below.

A phenomenologiccl model was proposed aL~ost simul­

taneousi;Jin 1974 by Huberman18 and Rice, Stressler and Tocxnbs19

(hereafter referred as RST model). <4elch & D1enes24 have

suggested a generd phenomenological model or which Huber-

man's and RGT models come out as a speclrl case. It is well

kn01m thlct the ionic conductivity dcpends U')On the number of

t'renkel or Schott'{' defect p~\irs which is an exnonential

function of the ectivrtion energy for their formetion. In

IE 0

Agl

F,t~ 1'2: ELECTHICA L CONDUCTIVITY OF RbAq4

15

AND A.:jl AS A FUNCTION OF TEMPERATURE

SHOWtrJG THE VARIATION OF CONDUCTIVITY

AT PHASE TRANSITION. AT T1 , THE COND\..'CI-·

IVITY CHANC,ES ABRUPTLY WHILE AT T2 ,THE

CHANGE IS SLOW AND ONLY THE SLOPE CHANGES.

the phenomenologicnl mod9ls, it is assumed that tho acti•

v: tion energy for formation is ooncentr1, tion deoondent due

to defect interactions. \ofithout going into thO deteils of

defect interaction, rlOlch & Dienas24 WTote the concentration

(c) denendent ter~ for frue energy per ion, F(o), as under •-

F(c),. B(c) - kT £ -2o ln c - (1-c) ln (1-o)

< o( -c> 1n col-e> ~lno(s-Tsv1b<c> c1.1o>

Where E (c)-,

~s the concentra.tion dependent enere;y required

pushing an otam into an 1nterst1t1el site;

o( is the rttio of nWilber of interstitial

"cells" to the nUIIlber of ions;

svib is the vibrPtionnl contribution to the

entropy.

for

delch & Dienes24 considered t~•o e pproximate forms

for E(c). In the 'Quttdrl tic form•,

B(c) = E1c - ~2c2 0~ (1·1i)

or' 0 c • El-2E 2c J

2 and, svib (c) ·~ c -.LJ.s2c

In thi3 approximation, Welch & Dienes model reduces to

Huberman or RST models. welch & Dienes formulation is

quite arbitrcry while Huberman & RST arrived at their

equations using & physiaFl model for defect interaction.

Huberli!B.n assumed an •attre.ctive intGraction• between an

interstial ion and a vacancy while RST considered the

interaction of intHrstitinl C(ltion defects with the str:.in

field they indue A,

One more form or ~:(c) other than eqtmt1on 1.11

hE•s been tlltmn by ·telch & :)ienes \-lhich is bt sod on 1!4ott­

Littleton tyne• o::>lculf: tions. This is ~~iven o s,

(1.12)

The ecu111br1uln Jei'ect concentrr tion (on whase

numb<n.' the conductivity del<mds) cun he obt[1n:Jd by minim!-

Sill_; F{c) with NSJ.l<lCt to c, i.e.

~(~0 (30

where F(c) is ,siven by equution 1.10 under approxillWtion of

equation 1.11 or equation 1.12. Theoretical CB1culations

show th&.t both class I (1'emper:·ture T1 of Fig. 1.2) or

cl:rcss II (Te:npe:rature T2 of Fig. 1.2) transitions or.n be

simulated by t&king different strength of interaction

terms (i.e. valt.es or E1 , E2, f2' etc.).

Lettice g:, s theories form another approach to

understr-nd the superionic phflse trDnsition. In these

theories it is ~:.cssumed th~t the mobile ions within the

suporionic solids can be considered as a 1lattice-gas'

hopping from one htticc point to another. 'l.'he hopping

'lntice-gas1 p&rticles interDct with each other and modify

the ion diffusion or tr£•ns1'lort act1vnt1on energy. This is

referred to fis 1 cooptJrl·t1vo hopping' of 1ntcrnct1ng ions.

:>o:Jto & Kikuchi26 did fir:;t str t15t1ct.l c:.lcUV-·tion for

{?> -alumim• using the Pc.th Prob: bility !{et:hod. Moro

recently Murch &: Thorn<:? reported the above:: cclcubtion

using a '-!onto Cc rlo Method. Both the above cnlculr• tions

could simulate cln ss II type phE--se tr£,nsitions success-

fully.

f..nother interesting !?.pproech has bo'm suggo::tod

by Ptn•dee & l{e.han20 and Mahen21 which is referr:d as Ionic

Polsron Theorz. It. is considerad thvt en ion nolarises

the host crystal, and when hopping must cerry this polari­

sing cloud with it cr,llcd 1 1on1c polaron•. This is

similar to the theory or conduction by hopping electrons

(small polaron) developed for metf ls and semiconductors.

The ionic poleron int•eraction hns been considered by

Mahan2l. quantum meeh&nim<lly through explicit use of

phonon co-ordinate and introducing coupling parameter

for the coupling or ion hop to the surrounding lattice

vibrut1ons. Ionic polaron thaory could also simulate

class II transitions but fails to account for class I

trf'nsit1ons.

(1.6) D'>Jlll:J!lics of ion transport

'l'he v•rious microscooic theories

proposed for the ion dynamics in super1onic solids can

broadly be clDssified e.sJ

(1) !roo-ion !IIOOGl

(2) Dalll in modol

(3) :i )!)'>in•; or jump Lli:'l'uaion Dl•>dol

In frtjo-ion modol 1 it is ,,s:;umod th: t en ion is

thormslly excited fr'Jill a locnlis•-d st. tu to '' !>t: te in which

tho ion mov"s tr,nsln tionally tllrou~_;h tho aolid to anothor

loorlisod st1.V1. i!istoricnlly1 this WbS the first thoory

proposod by nico & noth3 but it is no more acc:N':ltt~c1 (see

aaas2B). It has been shown th[tt it is not n<Jcessnry to

invoke 'free-ion theory' &nd tho s<mo rosults cnn be obtained

by convvntionnl 'hopping l!IO(i;Jls 1 •

Van 3ool29 end Van Gool & iJottelberghs30 have

given a 'domain-model'. As nointed out er:rliar, the high

conductivity in suporionic solids is marked by low values

of activations. 'I'here are two imnortr:.nt possibilities for

low VF-•luos of activction energya (a) coonorrtiva jump or

a number of ions es tCtken by Sato & Kikuchi26 or (b) move­

ment of walls between domains or orderad configurations.

The 90ssibility (b) ~s considered by Van Gool et a130• In

most superionic solids, the number of sites or l~ttice

position is much larger than number of ions avr. ilable. It

is assumed that in one domain ions are ocupying one type of

positions while in an adjacent domain another set of

positions is occupied. Considering the domain as a. "defect",

the energy per ion neces~Wry to creste the "defect•• has been

found to be low in nn iderlised calculr.tion by Vnn Gool

et a130 for ~-eluminr-. 'l'hi.'l rtecounts for lou nctivr-tion

enorey or ion mL;rl·ti ·::m in tham. .'lo fr r, no '!X"'lcri-nentcl

ev1.iencA hf s been re'lorted ei th.:r to nrove c>r i!is'>rove

1 domain modal I.

It is no-.r wall rucognis0d thet d-~-'u":r nnd new

insi ~ht into ft> st diffusion in sun·~rionic conrluct·:)rs could

be gc:ined by invcst1g;-tirw the full dynmnics of i.onic motion

includin•::: the host lattice (see G; .;.ndr81 , to b•~ publishod).

':.'hG ~irst microscoryic theory in this direction l>'f s atte'!lpted

by Huberm€!n and l>en31 in which the mobil(l ions e:re assumed

to have two basic degree13 of freedom {1) an oscillfl.tory

motion in the hr.rmonic ,otentiel providod by the rip;id

lattice r:.nd (11) a random-wHlk 'roeass through which they

c~n diffuse throughout the crystcl. The ju~n process was

assU!'Ied as occuring "1nstante~neously" and heine; uncorNlated

with the oscillatory motion. This theory could successfully

oredict the oef:k in conductivity e.t 60 Clll•l for f:S·alumina

but det~iled structure could not be rJxolt;.inad. Racontly in

a series of uapers from

(Jrueseh et ~132, Fulde

Brown Baver! nose:,rch Gentre Group 33 34 et e.l , :t;eller et al , Bruesch

et a135) have investigc:.ted the Brownian motion in a nerio-

die lattice including the effects of pol~risability or lett ice and correle ted jum:;s of ions relevrnt to the problem

of super1on1c conductors.

Various physical mechanisms for diffusion are

shown in Fig.l.3. The physical jump diffusion model shovn

in Fig.l.3(b) has been proposed by Clemen & Funke36, Funke9

to explain the microwave conductivity data of superionic

conductors. The# ion jumps are characterised by a mean

duration or the jumps <'1i ) and a mean residence time (·'""( 0 ) •

In classical jump diffUsion L.1 ~ <. l.'0 but for superionio

solids the experimental results indicate that L 1 -;::; 't0 •

So, a super~position of jump diffusion and a local tyne 3t.i motion we s considered by Clemen & l''unke as a good first,

approximation. The concept developed by Clemen & Funke to

understa.nd physically the microwave conductivity data 1s

interesting and deserves some mention though it does not

give the microscopic detail of ion dynamics. During conduc­

tivity measurements, the applied elactrienl field is assumed

to give rise to two effects referred to as "start eff'ect" and

"acceleration effect". The "start effect" can be visualised

as in the conventional theory of conduct! vity of alkali halides.

The electrical field slightly supports or hinders the 'start'

of jumps in or a.geinst the instantaneously preferred direction.

At low frequencies ( '1:.1 .( ,Z 1/l.o.l ) the phase of the filld

is practically constant while an ion is performing a jump

giving a constant value of c-- (_o) or o- Cw) at low frequ­

encies. At slightly higher frequencies, a cation starting

jump with the help or the electrical field is still on its way

while the field 1s already changing its sign and this would try to

j

lll l ',!MI'l \ \ill fl~l',\UN 1~1 1\qU\P

{b) I OCI\l 1'<10TIOI'.1 \ .ilH>1P tllfrU SION

( 1''o "'~ l; ·:::.: \ o 1\ s e, . )

( C ) JUMP 1) \ F F II'> I 0 N ( 1· ( < j 0 )

FIG-. i) VARIOU'.i '-'OSSIBLE: i)IFFUSION

MECH/\N1c:Y1

reverse the ion jump direction consequently lending to

decrebse in conductivity. In the high frequency limit

, the applied field cnnnges its sign many

times during the flight of an ion &nd the overall conducti•

vity tends to zero. This exPlain~the decr~asing

experimental micrownve conductivity with the increase in

frequency but an "accelerotion effect" has to be invoked

to explain the conductivity pe~:k vl cm·1• During flight

or the !on, friction in expectod because the coulomb

1nternct1ons with the other moving cetions ceuse rspid

fluctuations of the local potenti~ls. Funke9 assumed that

the jump lengths of the ions ~re fixed, being given by the

geometry of the anion lattice and not being affected by the

electrical field. The jump distances being constrnt, the

effect of acceleration or retr,rdat!on of the moving ions

by the field changes their times of flight but cannot give

any contribution to (E) (0) , as this does not cho.nge the

jump rates.

The complex conductivity is given by,

"' 1\ ()' (w) = q • l(w,t)/(&

0e·iwt)

A

where I is the complex ionic flux 1n the d1roct1on of the

electric field &0

exp (•!tift), Clemen & Funke36 gave a

datE, iled expression for I = I start + lac eel • The flux

Isttrt was a function or change in ion number densities

while Incccll deoend unon velocity Vl'lriation Ett differant times.

Tho cnlculf ted vv luc s i',H ve " good fit with experi•nental cW ta

confirming thn t the .1ump diffusion r.todel sunerimpos,Jd by

locnl 'llOtion is 1' good phono'ller,ological descriotion of ion

dynr.mics in supcrionic conductors.

r:orerencos

1 nolievtl.>ns". (to be nublished by .Tohn :11ey & ..:ons).

2. I.idihrd, / .•• : in .tndbuch dar flhysil<, Vol. 201 p. 246;

edited by .• l'luc~e (:''1ringer - verlog, 1157).

3. rlico, ·~ •. J. end roth, I.L.: .r. ::olid .~t&te Cham. ,i,

4. •:handr~J, :~.and rolfe, J: Cnne.d. J. 'hys., s·~, 412, (1070b).

5. Jiedersicl,, n. and .~oll•~r, :~ in "The Chemi.stry of

'~xtar.ded Defects in ~·ol:'-·:etallic ~~olids". Ed. L. f!:yrin.g

end '1.0' :·~eefe, North H·Jlland, Amsterdam, 1370 (p.629).

6. :'hilli.os, .r.c.: .r. F:lactrochem • .:;oc. Jl,i!, 934, (l'Y.76).

7 • Foleigh, D.O.; 'resented in "U. :' .;\. - .Tapan ,Joint •Jeminar

on do feet and diffusion in ··olids". ;;a saga we ~iemorial

Hall, 'I'okyo, Jnoan, Oct. 4-6, (1.:)'76).

B. 'l'e.imhashi 1 '1'.1 J. Appl. c~l::lctrochem. 3 1 791 (1973).

9.

10. hahi,

, . '.\. .. 7 • ... .

:'rog. ')olid :·ta.te ·":hem. lit 3451 (1976).

::>hys. :tat. :ol1d1(a) ~~ 11 (1977).

11. !'tJreday, ·:.: :·uoted by E. ·:iedersich and '. Geller in

reference (5).

12. 'i'ubandt, c. in ":Iandbuch ;)er Experimental Physik" XII,

oart 1 1 Elds. :. 'len,, f. ,'!arms, Acedem. Yerlag 1

Leiozig (1932).

13. '"ubf,ndt, c:. and Loror.z, • : ~. 11ysik. Chem. ci7 1

513-&;3, (1 '11.3).

14. :·.ro.dley, .r .~'• and Greene, P.D.: 'l'rans. rnrad •. :oc.

~. 424, (1967).

15. •)wens, L •• iO!nd l'rgue, G.n.: ~>cionce .!§1.1 30'~, (1967).

lG. "lwens, x.;.n.: J. c:loctrxhem. ·oc. 1171 15361 (1970).

17. C:!l~ndr11 1 • r.nd ·~ohDbey, "·':.: J. T'hys. D1 ~1 5761 (HJ75).

18. Huberman, ::.1 •• : "hys. r<ev. Lett. ~1 1000 (1'1'14).

19. race, ·: • .r., itrsssler, ;.;. :,nd~oombs 1 G.l,.: Phys.

Fev. Lett. ,;g, 5961 (1974}.

20. Pardee, iJ.J. fl!ld :raha.n, c. D.: J. Solid ;;tate Chem.

!§, 310, (1975).

21. 'laban, G. D.: "hys. :Rev. 1"' 1 .J.i1 7801 (l\n6).

22. lfargas1 R., f~&lemon, 'I.';. and Flynn, C.P.: Phys. nev.

,;u, 1550, (1976).

23. :~;alamon, ·.1. '1.: Phys. Fev. !J 1 151 22361 (1977).

24. :Jelch, D.O. and Dienes, G •. r.: .r. "hys. Chem. ~'!olids

~~ 311 1 (D77).

25. Lederman, J··.L., Salomon, '!.23. and .:hacklette, L.H.:

bys. Rev. 3 • .£!, 2981 1 (1974).

26a. seto, H. and :'<ikuchi, R.: .r. Chern. 'hys., .§21 6771 (l$l?la).

26b. ~·ato, :I. and '\ikuchi 1 n.: ibid 1 .§21 7021 (1~1b).

28. ::nts 1 c.'.: .r. olid tt.tt'l r:hem. 1 1 (15G) 1 (l'Y73).

2'1. •·en Gool 1 .'.: ,J. ·:ol1d ,;t-,: te -hem. 11 55 1 (1)73).

30. 1'nn Gool 1 .• end uottleberghs 1 P.H.: .r. i>ol1d ·tr.te Cham.

1, 59, (117~!).

31. lluberman 1 r:.t .• end ·:en1 ?.:·.: Phys. !lev. Lett.

~I 1379, (1974).

32. B(U:.esch 1 ~>., .:trnss1er1 s. and >:Oel1er 1 a.n.: Phys.

·:tet. ,<:ol. (n) 1 .i2!,1 2171 (1375).

33. Flllde 1 P. 1 :>ietronero, L. 1 <.chneider, :J.R. and !!trassler1 ~'•f

Phys. tev. Lett. ,;&1 1776 (1975).

34. 'eller, :I.n., Jruesch 1 r., ''ietronero, L. and :>trassler, s.:

in ".'uperionic ··~onductors" &ds. G.D. :'ahan ;. •,J.L. Roth,

Plenum Press, 1976.

35. "ruesch, P., Pietronero, t., ;>trassler, s. and Zeller, H.R.:

Phys. rev. B. 1§1 4631 1 (1977).

36. Clemen., c. and F'1mke 1 ~(.: Der. :'.unsenges. i'hys. Chem.,

,Zf!, 1119 (1975).