Relaxons in superionic conductors

Solid State Ionics 40/4 1 ( 1990) 244-249 North-Holland

RELAXONS IN SUPERIONIC CONDUCTORS Tadao ISHII Faculty of Engineering,

Characteristic respectively.

Okayama University, Okayama 700, Japan

behaviors

of diffusive

and non-diffusive

1. Introduction In the investigation of hopping conduction, the relaxon scheme (relaxation mode theory) is one of the promising methods. The relaxons are elementary relaxation modes, introduced first in the system of independent hopping particles on a regular lattice, which represent eigenvalues of the master equation of a Shriidinger form [ 11. ’ The relaxons consist of two kinds of branch modes, diffusive and non-diffusive. If the unit cell is formed of K different sites, then there exist one diffusive and ~-1 non-diffusive branches [ 11. The two branch modes contribute to physical phenomena in a different manner [ l-31. The name and concept of “relaxation mode” have been utilized, explicitly or implicitly in superionic conductors, for example, by Klein in the discussion of quasi-elastic light scattering [ 4,5 1, or by Walstedt et al. [6] and Richards [7] in the NMR analyses. Here we discuss it in a definite and systematic way.

relaxons

are illustrated,

which are of extended

and localized

nature,

where P,(t) is the probability of finding a particle on site i at time t, Pi, eq is the probability in an equilibriumstateand IY(t))=CYi(t)li) with{]i)}being i

b)

Eq(c)

I

2(r0+r, 1

1

non-diffusive

2. One-body system on regular lattice [l] The master equation

is rewritten

in the form,

$(I))=-HIP(f))

>

(1)

Y(t)=@-‘P(t),

qF=Peq,

(2)

0

_L

’ Eq. (17) in ref. 0167-2738/90/$03.50 ( North-Holland )

2a

Fig. 1. (a) l-d lattice of double-well type and (b) relaxons: fusive (t = - 1) and non-diffusive (t = + 1)

[ 1] should read eq. (A.2). 0 Elsevier Science Publishers

0

B.V.

dif-

245

T. Ishii / Relaxons in superionic conductors

the basis set. The matrix elements of the hermitian hopping operator H are given by Hii= 1 rji j#i

(3)

H,= -(rjirij)"2, rji being the transition probability rate from site i to _i. In the double periodic lattice of fig. 1a, two sets of eigenvalues E,(e) (fig. 1b ) and eigenfunctions !J$( e ) (Appendix) are obtained straightforwardly, in the stationary state, from HI ‘Y,(e)> ‘We)

I %(E)>

(4)

7

with q and t being wave number and branch index, respectively. The value at q=O in the diffusive branch (E = - 1) corresponds to the equilibrium state (condensation at q= 0). If the transition probability rate r,, approaches 0 (fig. la), only the non-diffusive relaxons (EC + 1) survive with a dispersionless nature. This simply implies that the non-diffusive relaxons are of localized nature and therefore do not contribute to the dc conductivity. The analytical formulas of the conductivity and dynamical structure factor in this lattice system are obtained as

(5)

which is q-dependent. Here 8, n and e are inverse temperature, density and charge of mobile ions, respectively. The first term in each equation of (5) and (7) is the contribution of the diffusive relaxons, and the second is from the non-diffusive relaxons. It is seen for q-0 that S/q2 consists of a sharp component at w-t0 and a broad component. It is also important to note that the conductivity (5) comes from two relaxons at q=O.

3. Non-diffusive relaxons 3.1. One-body problem in l-d lattice with one impurity Here we consider an example of one particle problem on the lattice with only one different barrier height (fig. 2). In the case of y/r> 1, we obtain the localized relaxon: ul,=Y6(-1)“exp(-lm-41~+~/2},

(8)

t=ln(%ff-

(9)

1); k=w,

E=40k02kr-

which implies that the eigenfunction Y localizes. The conductivity in this system with a dilute number of independent particles is therefore given by a(o~)=/h(eu)~- ioy%

(T-1)

= (&’ +rr’)/2


=vo+r,)P

Ar

=r,

(10)

1) ,

4rh-02 M

2~2-i

( 2~,ff- 1)

(11)

x4r$fl/(2y,,-l)-iw’

)

(6)

3

-r,,

with M the site number. Other extended modes (diffusive modes) may give rise to the conductivity de-

and S(q,o)=i

@exp(-iqR)&exp(iqR) (I

@ I>

(r-l)-’ = (qa)2 q-0 --ER ( (r-1)-‘q2a2)2+rB2 Ar2 4<0 + ( 4(r) > (4(r))2+02

1’

-M/2+1

(7)

-1

0

1

2

M/2

Fig. 2. Linear lattice with one isolated barrier.

*nl

T. Ishii / Relaxons in superionic conductors

246

rived by Blender et al. [ 81. In addition, the “resonances” can also exist in other cases as in the lattice dynamics [ 9 1.

E=2(2?,,‘+q)T,

to give

=Q(W) 3

)~(-We);w)

(13)

Now let us consider the {4,2)-r& model introduced by Kimball and Adams [ lo- 12 1, being very simple but includes the nearest-neighbor interaction V between particles by q=exp(j?Y). In this model system, eigenvalues are obtained as and

a(o)

distribute

+Ide F(e)ldE,(e)WE,(c)

3.2. A few-body system

E=O

ons of long wavelength

(12)

where q> 1 and q< 1 are repulsive and attractive, respectively. On the other hand, the {6,2} and {6,4}-rings have a set of eigenvalues in common, E= 0 and two nonzero values [ 12 1. The {6,3}-ring gives E = 0 and three nonzero eigenvalues. 3.3. Many-body system In an interacting many-particle system, the nondiffusive modes may be considered to distribute, at least, at around q=O, even in an equivalent site lattice. The matter is how the eigenvalues distribute, which is unknown at present. Let us consider a half-filled square lattice with mobile ions. In this case, we have two particles a unit cell of length 2a on an average. Thus two average configurations shown in fig. 3 may play central roles. Then we have one diffusive and three non-diffusive eigenvalues which are each a function of nearestneighbor interacting potential. In general, we may have more eigenvalues centered around the above non-diffusive values, and hence, non-diffusive relax-

.

S(q,o)=S,(q,w)+ldtF(~)S,,(E,(t);w)

(14)

The first term in each equation comes from the diffusive relaxons and the second from a distribution of the non-diffusive relaxons (see eqs. ( 5 ) and ( 7) ), where D(E,( CZ) ) is the density of states for the branch E and F(E) expresses the distribution of E. This consideration leads us to think that also in an equivalent site system, we surely have diffusive and non-diffusive modes.

4. Relaxons in many-body systems The many-body master equation can be interpreted by the field theory. Let us confine ourselves to the diffusive relaxons in an equivalent site lattice. A single-hop approximation in the weak coupling limit gives rise to H=ChO(k)a:a, k

+

1 h k.k’,q

(k&‘;q)a:

a:

ak+&k’-q

,

(15)

where ho is the unperturbed relaxons and hi is the interaction term of relaxons and pairwise interaction energies. An application of a method of Bogoliubov [ 131 to eq. (15) results in

Eq=Mq)(l+BNoVa)

(16)

(b) Fig. 3. Two periodic configurations

including two particles a unit cell where (a) can directly transfer to (b) by a hop, and vice versa.

T. Ishii / Relaxons in superionic conductors

where No is the number of mobile ions, and V, is the interaction energy among mobile ions.

5. Possible

scope of relaxons view

5.1. Phonon damping by relaxons Suppose that a mobile ion, vibrating on site i, hops to one of its adjacent sitesj with the transition probability rate r. This means that the phonon is annihilated on i and created on i, which is naturally the damping of phonons. The master equation for this problem has previously been solved for the Einstein oscillator in a simplified fashion [ 11,141. Let us deal with this problem in the wave-number space version. The correlation functions of normal modes of phonons, Q(qA) with A a branch, in q-space are given by

SQQ= $1Q(d)*exp(

=

(Q(@)*Q(d)>

x

[l-D]-’

C (E,-io)-‘}‘=O, 4

(20)

the sum over q, exactly performed, results in fig. 4, which shows real and imaginary parts of the normalized eigenvalues o/w,_. In this evaluation has been taken that r= wexp ( - EJ T), and (0: ) = w:. In fig. 4, temperature is normalized by the activation energy E,. If we have two activation energies EaL) EaHjust below and above a phase transition temperature T,, respectively, but assume that w4s do not change, then we have the gaps in both parts of w at T, (fig. 4). The damping of phonons correlated to mobile ions have been reported experimentally by Brtiesch et al. [ 15 ] and in the MD calculation by Kaneko et al. [ 161, both in cr-AgI. A gap of phonon damping at T, has been found by Wakamura et al. [ 171. 5.2. One-body problem on a fractal lattice

--h-W

x~_~:+~, Q(q~)w(iq-W

l+(wi){M-’

241

If we know eigenfunctions, eigenvalues and density of states, all is understood in the system. Let us illustrate an example of the finite fractal lattice shown in fig. 5. One can obtain the eigenvalues E and den-

I@)

-e(O;qA)},

(17) 1.0

D= -M-l

2 e(O&)

(G>G(q)o(qJ)2e*(O;qA)

, (18)

where iL, is the Liouville operator for Q(ql), e( 0;qA) is the phonon amplitude for the sublattice K=O on which mobile ions sit, G(q) = [E,-io] -r and (G) =M-’ C G(q). In order to see a qualitative behavior for hopping of particles on one-dimensional regular sublattice, we take account of only one phonon branch, say an optical phonon of renormalized frequency o, with e(O;q), put e= 1 and neglect others. Equating [ 1 -D] = 0, the eigenvalues are obtained by 1 +M-’

c4$+4-‘&&=0.

(19)

Considering the diffusive relaxons for E, and further approximating eq. ( 19) by

Oi 0

0.2

0.4 UC)

0.6

0.8 UC)

1

Normalized Temperature T/E,

Fig. 4. Temperature dependence of real and imaginary parts of the eigenvalues: The gaps appear on both parts at r,, transition temperature, where T,/E,=O.4 and T,/E,=O.S are taken.

T. Ishii / Relaxons in superionic conductors

248

Fig. 5. A fractal lattice.

sity of states D(E) [18,19] as E=c, k’;

by the decimation

z=ln33/ln3,

D(E)=c~E~~/~-‘;

method

(21)

dFln5/ln3,

respectively, where df is the fractional dimension, z is the dynamical exponent, k is the wave number and es are the constants. If we take the same type of function given by Bernasconi, et al. [ 201 as I (9 IR I Y) I* then we have the frequency dependent aE_‘, conductivity: a(w)aw”;

v=dJz--z+

l< 1 .

[4] M.V. Klein, in: Light scattering in solids, eds. M. Balkanski, R.C.C. Leite and S.P.S. Port0 (Flammarion, Paris, 1976) p. 351. [5] T. Suemoto andT. Ishigame, Phys. Rev. B33 (1986) 2527. [ 61 R.E. Walstedt, R. Dupree, J.P. Remeika and A. Rodriguez, Phys. Rev. B15 (1977) 3442. [7] P.M. Richards, in: Physics of superionic conductors, ed. M.B. Salamon (Springer, Berlin, 1979) p. 14 1. [ 81 R. Blender and W. Dieterich, Solid State Ionics 18/ 19 (1986) 240. [ 91 P. Brtlesch, Phonons: theory and experiments III (Springer, Berlin, 1987). [lo] J.C. Kimball and L.W. Adams Jr., Phys. Rev. B18 (1978) 5851. [ 111 T. Ishii, Solid State Commun. 47 ( 1983 ) 7 17. [ 121 T. Ishii, Prog. Theor. Phys. 73 (1985) 1084. [ 131 C. Kittel, Quantum theory of solids (Wiley, New York, 1963). [14]T.Ishii,SolidStateIonics 18/19 (1986) 191. [ 151 P. Brtlesch, W. Btlhrer and H.J.M. Smeets, Phys. Rev. B22 ( 1980) 970. [ 161 Y. Kaneko and A. Ueda, Phys. Rev. B39 (1989) 10281. [ 171 K. Wakamura and K. Hirokawa, in: 3rd Int. Conf. Phonon Physics and 6th Int. Conf. Phonon Scattering in Condensed Matters (Heidelberg, 1989). [ 181 R.B. Stinchcombe, in: Static critical phenomena in inhomogeneous systems, eds. A. Pekalski and J. Sznajd (Springer, Berlin, 1984) p. 39. [ 191 R. Rammal and G. Toulouse, J. Phys. (Paris) 44 (1983) L13. [20] J. Bemasconi, W.R. Schneider and W. Wyss, Z. Phys. B37 (1980) 175.

(22) Appendix

6. Summary

We have two kinds of relaxons, diffusive and nondiffusive. Especially in a dilute limit on an equivalent site lattice, there exists only the diffusive branch. But even in the equivalent site lattice, but with many particles, we may have one diffusive branch and a distribution of non-diffusive branches. In this case, we have two kinds of physical properties such that we have seen in eqs. ( 5 ) and (7).

The eigenfunctions and eigenvalues in the double periodic lattices are shown: I~~(t)>=lCe~(t;K)exp(i~R~)In;K>, @%K eq(E;K)=

=

e,(-1;O) e,(+l;O)

e,(-1;l) e,(+l;l)

References [ 1 ] T. Ishii, Prog. Theor. Phys. 77 (1987) 1364. [2] T. Ishii, Solid State Ionics 28-30 (1988) 67. [ 31 T. Ishii, Appl. Phys. A49 ( 1989) 6 I.

1

1 (1 +M/JD,)‘/* (1 -fV/&)“*eiqO 2 -(1-M/@q)1/2(1+M/fiq)1/2eiqa

J[

Dq=4{(~)2-~O~lsin2qu}, Eq(t)=2(r)

+~fi~;

(A.l)

1’ (A.21 (A.3)

t= f 1 ,

(‘4.4)

T. Ishii / Relaxons in superionic conductors

for the inverse double-well type potential. For the double-well type: e&t; K) =

exp(iOJ2)

exp{-i(8,/2-qa)} exp{-i(8,/2-qa)}

8,=Tan-‘((4I’/2(r)tan

1’ (A.5)

249

qu} ,

with the same eigenvalues of eq. (A.4).

(A.61