Review on structural and dynamical properties of silver chalcogenides

Solid State lonics 39 (1990) 121-149 North-Holland

REVIEW

REVIEW ON STRUCTURAL AND DYNAMICAL P R O P E R T I E S

OF SILVER C H A L C O G E N I D E S

Michisuke KOBAYASHI Department of Physics, Faculty of Science, Niigate University, lkarashi, Niigata 950-21, Japan

Received 28 August 1989; accepted for publication 13 February 1990

Structural and dynamical properties of silver chalcogenides are reviewed. Since Rahlfs and Strock proposed that the four Ag ions in a unit cell of ct-Ag2S are distributed with equal probability over the 42 crystallographic sites of 6 (b), 12 (d) and 24 (h), many measurements have been performed. Recent precise measurements of neutron scattering, EXAFS, and X-ray diffraction have suggested that Ag ions in ct-Agl type superionic conductors are distributed over 12 (d) sites with large asymmetric anharmonic thermal vibrations. The transport and electrochemical properties of the electronic and ionic conductivities and the self- diffusion coefficient and mobility of mobile ions have been investigated also extensively. These measurements have shown a remarkable deviation from the Einstein relation in Ag-chalcogenides. To interprete this behavior, Yokota presented a new diffusion theory, caterpillar mechanism. To understand the liquid-solid duality of an unusual state of materials in which some atoms have nearly liquid like properties while other atoms retain their regular crystalline arrangement, a continuum model has been presented. It is supposed that the lattice composed of anions is immersed in the cation liquid. The structure factors with which electrons are connected are expressed symmetrically in terms of the structure factors of ions in the long wavelength limit using the fluctuation-dissipation theorem and the Kramers-Kronig relation. The calculated conductivity satisfies the f-sum rule and its ionic part has a broad peak at the optical-phonon frequency, which is the similar frequency dependence to that of a-AgI. The frequency- and wave vector-dependent conductivity and the dispersion relations of collective modes are calculated. In particular the plasmon-LO phonon coupling is investigated in detail. A molecular dynamics method is applied for the further study of Ag- diffusion in ct-Ag2Te for several temperatures. The density distribution of Ag ions suggests that a Ag ion locates at tetrahedral site for the most of time and then moves to its neighboring tetrahedral site through the vicinity of the octahedral site. The activation energy for an ionic diffusion also is obtained from the Arrhenius plotting of the self-diffusion coefficient of Ag ions. Its value of 0.17 eV is obtained, which is in good agreement with the experiment.

1. Introduction

Superionic conductors (SIC) are crystalline materials which exhibit extremely high values of ionic conductivi ty comparable to those of l iquid electrolytes at relatively low temperatures. Many works have been performed on these materials [ 1-13 ]. Physical and chemical properties of SIC have been investigated by physicists and chemists since early this cen- tury. In 1899 Nernst [ 14 ] suggested that the current in a zirconia was conducted by O ions. The discovery of typical ionic conductor a-AgI was done by Tubandt [ 15 ], and Tubandt and Reinhold [ 16 ]. Just after the discovery ofct-AgI, Joffe [ 17 ] presented the impor tant notions such as lattice defect and interstitial ion. His studies were succeeded by Frenkel [ 18], Schottky and Wagner [ 19] and Wagner [20].

Then a theory of ionic conduction, which was based

upon thermodynamics, was accomplished in the

1940's. In 1934, Rahlfs [21 ], from an X-ray powder dif-

fraction study, proposed that the four Ag ions in a

Ag2S uni t cell were distributed with equal probabil-

ity over the 42 crystallographic sites of the octahe- drally coordinated 6 (b) (0 1/2 1/2) , tetrahedrally

coordinated 12 (d) ( 1/4 0 1/2 ) and triangularly co-

ordinated 24(h) (0xx) positions of Im3m. This

model was proposed as a ref inement of the model for

~t-AgI by Strock [ 22 ]. Rahlf 's proposed structure for I3-Ag2S is shown in fig. 1.

In 1957, Hoshino [23] measured the Bragg peaks and the diffuse background of X-ray diffraction in u- AgI and confirmed the liquid-like Strock model based upon his data. Hoshino et al. [24] reexamined the

122 M. Kobayashi/Structural and dynamical properties of silver chalcogenides

A6(b) I12(d) 024(h)

E

Fig. 1. Strock model [ 22 ] for 0t-Ag2S. The large spheres represent the sulfide ions in a bcc aray. Four Ag ions are randomly distributed over 42 crystallographic sites in a unit cell.

cation distribution in ct-AgI using neutron as well as X-ray diffraction data. A least-squares analysis showed that the experimental data are very well ex- plained by a structure model in which two Ag ions are distributed over 12 (d) sites of the space group Im3m with large asymmetric anharmonic thermal vibrations. The structure feature adopted in their study was consistent with the EXAFS results by Boyce et al. [25] and the single crystal diffraction study by Cava et al. [26 ].

Sakuma et al. [27 ] analyzed X-ray diffraction data of ct-Ag=Se by asuming a liquid-like distribution of Ag ions. The 42 sites of Rahlfs mode did not need to be assigned as the special positions for Ag ions in ct-Ag2Se. The diffraction patterns for Ag-chalcogenides were of special interest because these materials showed a coexistence of both Debye lines and liquid- like diffuse patterns. Tsuchiya et al. [28 ] estimated the effective pair potentials ~Ag_S(F) and ~Ag_Ag(r) from the calculated partial structure factors according to the method proposed by Ballentine and Jones, based upon the Percus-Yevick theory. Their results are shown in fig. 2. The vertical lines correspond to the uncertainty of each structure factor. ~Ag-S (r) has a minimum at about the Ag-S distance assuming that Ag ions distributed at randam over the 12(d) sites.

After Cava et al. [26] showed that the Ag distribution could be described to significantly greater precision by large anisotropic and anharmonic vibrations of Ag ions occupying solely the tetrahedral

r(~)

Fig. 2. The effective partial pair potentials of (a) Ag-Ag and (b) Ag-S pairs in ct-Ag2S (from ref. [28] ).

12(d) sites in a single-crystal neutron diffraction study of a-AgI, they undertook to ascertain whether their model would be valid for I]-Ag2S as well [29]. Their results are shown in fig. 3. Fig. 3 (a) represents a partial Fourier synthesis of the Ag ion density p(xyO) in one face of the Ag2S cell at 186°C. This section contains the 6(b), 12(d) and 24(h) sites in the bcc S array. The Ag scattering density is confined to surprisingly uniform, continuous channels which connect tetrahedral and octahedral sites along [ 100 ]. Weak local maxima occur in the density at the tetrahedral and octahedral interstices. Fig. 3 (b) represents the partial silver density p(xyO) at 325°C. The density map reveals further delocalization of the Ag density at 186°C.

In 1983, Hoshino et al. [30] measured neutron diffuse scattering from powder samples of ct-AgI type superionic conductors AgI, Ag3SI and Ag=S. The disorder scattering curves were analyzed under the assumption that Ag atoms distribute over 12 (d) sites among the bcc lattice formed by anions. These were found to be in fairly good agreement with the calculated curves when the appropriate values of probability functions were chosen.

M. Kobayashi/Structural and dynamical properties of silver chalcogenides 123

. - - . ' / llllllllYl((l(fll ,, - -

~ , , " l,,',/HIIIII/A\ltlllllI,, I ,

~,~~~~'2T--"-', / ,\iX ", , .... -22"

• , i/: lllllllll

/ . . . .

I X /

Fig. 3. Sections p (xy0) of the scattering density in/Y-Ag2S: (a) partial Fourier synthesis of Ag scattering density at 186 °C (contribution of S subtracted). Contour integral 0.0057 X 10-12 cm/ A, negative contours broken; (b) partial Fourier synthesis of Ag scattering density at 325 ° C. Contour interval 0.0048 X 10- ~2 cm/ • ~ (from ref. [29] ).

The transport and electrochemical properties of Ag-chalcogenides were investigated by Miyatani [ 31- 38], Okazaki [39,40] and Yokota [41-44] . One-dimensional transport o f charge carriers in an ionic- electronic conductor can be described by the following transport equations [45-47 ]:

aeO~e a e i ~ Je= e 0x ze 0 x ' (1)

O"i 0 / ~ i O"ie 0 / ~ e

Ji - - ze 0x + " (2) e 0x

Here fie and/ii are the electrochemical potentials of electrons and mobile ions, respectively, z is the val- ency of mobile ions, ae and a~ are the electronic and ionic conductivities, and o'ei and cri~ are the cross conductivities. According to the Onsager reciprocal relation [48], these two cross conductivities should be equal to each other:

O'i e = O 'e i . (3)

Miyatani [36] measured the values o f aei and aie for ct-Ag2Se at 162 °C and found that the Onsager reciprocal relation is satisfied in the whole measured range o f the nonstoichiometry. Fig. 4 shows his experimental results [36 ].

Okazaki [39,40] made measurements on the self

'E ,-7-0

v

¢o

b-

xlO 2 2

6 1 2 3x10 -3 I I I

/ c% /

/ Uie (Ji = O) ? " dO 3

3 ? ~ (j~= o) .,

.~"

/ /

• T

0.2 E (v.O.1) 0

Fig. 4. Cross coductivities aie and aei, electronic conductivity ae, and ionic conductivity al are plotted for the EMF, E, of the system PtlAg2SelAgllAg (from ref. [36] ).


diffusion coefficient D and the mobility # of Ag ions in Ag-chalcogenides using a tracer technique. He found a remarkable deviation from the Einstein relation. His experimental values of Haven's ratio defined by

f= De/12kT (4)

are plotted in fig. 5. Mekata and Mizoh [ 49 ] and Seguchi and Mekata

[ 50,51 ] investigated the ionic motion in AGES, Ag2Se and Ag3SI by making use of the 7-7 perturbed an- gular correlation method. Their results for the Ag-ion diffusion in ct-Ag2S and ct-Ag2Se were in good agreement with the experiments of Okazaki [39,40 ].

The heat capacity of ct-Ag2S was measured by Per- rott and Fletcher [52 ] over the range 50-400°C using an adiabatic calorimeter. At the l]-ct phase transition temperature, about 180°C, tile abrupt fall in heating rate was observed. For stoichiometric material, the latent heat at this phase transition was found to be 8 H = 9 5 0 + 100 cal/mole. They found that there was another anomalous transition at 350 °C for stoichiometric material. However for small deviation from stoichiometry in the sample, the heat capacity was found to be constant from 180°C to 400oc.

Abundant and detailed information on the structural and dynamical properties of SIC have been provided by the modern techniques of neutron scattering, EXAFS, X-ray scattering, NMR, and so on. These developments have made many scientists urge

3

f-~

2

. . ! j J ;o2s I

--0-----0---- O-----O---

I ~T~I

)25

330

f 340

].50

3.70

1.00

1.6 2 0 2.4 103/T ( K -1)

Fig. 5. The reciprocal of Haven 's ratio f - ~ and also the f-value are plotted against T - 1 for Ag-chalcogenides ( f rom ref. [ 40 ] ).

to study SIC. They have been attracted by the liquid-solid duality of an unusual state of materials in which some atoms have nearly liquid like properties while other atoms retain their regular crystalline arrangement. In order to understand the various physical and chemical properties of SIC, several theoretical models have been proposed. Yokota [43] suggested a kinematic model allowing the cooperative diffusion of two or more mobile ions which is called caterpillar mechanism, as is discussed in the following section. Coming to the 1970's, studies on SIC developed rapidly. The influence of correlation on the static conductivity has been studied by Sato and Kikuchi [53] in ~-alumina using the path probability method. Recently Ishii et al. [ 54] have applied their method to get the frequency dependent conductivity.

Rice and Roth [55] proposed the free-ion like model. The essence of the free-ion like model of ionic transport processes is the assumption that there exists in SIC an energy gap, above which ions can be thermally excited from localized ionic states to free- ion like states in which an ion propagates throughout the solid.

When we study the frequency-dependent response of SIC, it is convenient at first to ignore the detailed phonon spectrum of the rigid cage and to treat the mobile ions as hopping particles through some fixed sites, while executing finite amplitude oscillations around the potential minima provided by the lattice. This approach was developed by Huberman and Sen [56 ] in order to calculate the temperature and frequency-dependence of the ionic conductivity. The frequency and temperature-dependent conductivity is obtained by calculating the velocity-velocity correlation function in terms of Kubo's fluctuation-dissipation theorem [ 57 ]. The mobile ions are assumed to move successively in the cage potential. The vibrations of the cage are treated as a bath and provide random forces and also friction for the motion of the mobile ions. They introduced the Langevin equation for one-dimensional Brownian motion of the mobile ion to calculate velocity-velocity correlation function. So their model is called as continuous stochastic model. In their theory, the mobile ions are assumed to have two basic degrees of freedom: an oscillatory motion in the harmonic potential provided by the rigid lattice and a random-walk process


through which they diffuse through the crystal. Bruesch et al. [58] and Dieterich et al. [59] introduced a generalized Langevin equation and obtained the correlation functions by making use of Mori's continued-fraction expansion method [60]. Geisel [ 61 ] also studied a general expression for the ac and dc conductivities by introducing a model of inter- acting Brownian particles in a periodic potential. These works based upon a stochastic model and at- tendant studies were reviewed in [6,8]. Recently Munakata [62] presented a generalized Langevin equation approach to impurity diffusion in solids.

Subbaswamy [63] and Huberman and Martin [ 64 ] discussed a theory of light scattering from long- wavelength thermal fluctuations using a model of a crystalline cage immersed in a viscous liquid. The same picture was used to study long wavelength collective motions [65]. Zeyher [66] presented a rig- orous hydrodynamic theory for the corrective excitations at low frequencies and long wavelengths. Pardee and Mahan [67] attempted to account for phase transitions by making use of the lattice gas model which is one of the hopping models. The continuous motion of the mobile ions was replaced by discontinuous motion from site to site.

Rice et al. [68 ] suggested that it is the interaction of interstitial cation defects with the strain field they induce which is primarily responsible for the phase transition to the cation disordered state. Lam and Bunde [69] tried to build a theory which can explain the different types of phase transitions, and structural changes. Hayashi [ 70 ] also studied the direct transition from 7 phase to ct phase in CuI in terms of the dependence of the free energy on the order parameter and temperature. Mahan [ 71 ] proposed the lattice gas model. It assumes the ions as the lattice gas in which hops between sites occur through in- teractions with the reservoir provided by the crystalline cage. By using the lattice gas model, it is possible to obtain explicit forms for the various correlation functions and transport coefficients in terms of the pair interaction energy. In order to get an ionic transport, a somewhat artificial transfer term is introduced, which produces jumps from one site to another.

When the coupling between mobile ions and cage displacements (phonons) is considered, it is called the ionic small polaron model. By treating the ion-

phonon coupling in SIC, Hinkelman and Huberman [ 72 ] have shown that the conductivity displays both the behavior of a dressed ion moving in a band-like state and that of a hopping particle that performs jumps leaving the phonon cloud behind it.

Pietronero and Stfiissler [ 73 ] introduced a master equation approach for a many particle hopping system [74]. Hayes and Boyce [75] thought that the diffusion of ions is determined predominantly by the very short-range repulsion between ion cores. Owing to these forces, the ions are constrained to flow through a confining network of voids and narrow channels.

2. Theoretical works of dynamical properties

2. I. Introduction

In 1961, a macroscopic theory of mixed conduction, which implies electronic and ionic conduction, was developed by Yokota [41 ]. For the case where the current density is so small that the linear approximation is valid, i.e., it is sufficient to take into account the variation of the concentration of stoichiometric excess N of mobile ions only up to the first order, he derived the following diffusion equation:

ON aioaeo (d_~) 02N Ot - aoe 2 o Ox2 ' (5)

with the boundary conditions

ON 1 ej at x = 0 , L

0x aeo (d/2/dN)o

for the electronic electrodes, and

ON - 1 ej at x = 0 , L ,

0x Oio (d l t / dN)o

for the ionic electrodes. Here ae and ai are the electronic and ionic conductivities, L is the length of species, j is the total current density and/z is the chemical potential of neutral Ag atoms, which is given by the sum of the electrochemical potentials of electrons and of Ag ions. The suffices 0 signify to take the value in the unpolarized state, which is the state before switching on and is in equilibrium. After


switching on, one can solve these boundary value problems. For the electronic electrodes, we have

ejL N = N o +

Go( d#/dN)o

X - ~ + , , (6)

where

4 o~ 1 q)(~, r/) = ~ ~--o (2n+ I )2

Xexp{- (2n+ 1 )2r/} cos{ (2n+ 1 )Try}

and

r - l = L2 ao e2 o"

z is a measure of the time at which the stationary polarization is attained. Yokota also derived the general expressions for potential distributions by making use of eq. (6). Two kinds of potentials were distinguished. One is the potential as measured by the use of probes consisting of electronic conductor such as Pt, and the other as measured by the use of probes consisting of ionic conductor such as AgI for Ag chalcogenides. The former reveals the relative electrochemical potential of electrons and the latter that of ions. For example, the potential for the formation process of the polarization was obtained as

Go ao aeo k , L (7)

for the electronic electrodes. His results of potential distributions for formation and decay process of stationary polarizations were compared with experiments for Ag2Te showing satisfactory agreements.

Dudley and Steele [76] attempted to produce a completely general treatment of the technique, draw- ing on the ideas of Yokota [41] and Wagner [77]. They described the theory and experiment of a solid state electrochemical technique which allows measurement of the chemical diffusion coefficient and partial conductivities of the mobile ions in a mixed conductor. Their theory incorporated non-ideal thermodynamic behavior of the carriers as well as cross terms in the flux equations.

Weppner [78] presented a steady state electro-

Ao A1 A2 An-1

0 1 2 3 n-1 n(vacant)

Fig. 6. Cooperative jumps of n ions associated with the spontaneous jump 0--, 1 tracer Ao (from ref. [43] ).

chemical method for measurement of the chemical diffusion coefficient, ionic mobility, partial ionic conductivity, enhancement factor and related kinetic quantities of mixed electronic and ionic con- ducting solids as a function of the deviation from the ideal stoichiometry. This method has the advantage of translating kinetic quantities, which are otherwise often difficult to obtain, into readily and precisely accessible electrical currents and voltages.

To interprete the remarkable deviation from the Einstein relation, Yokota [43] presented a new diffusion mechanism, which is called as the caterpillar mechanism. He supposed that an ion on a site is able to jump not only into a vacant neighbouring site but also into an occupied one, inducing the ion on the latter site to make a jump. He performed his calculation to the one-dimensional system. Fig. 6 shows one of cooperative jumps. Suppose that ion Ao on site 0 makes a spontaneous jump to site 1 which is neighbouring to site 0 and is occupied by ion A2. Then ion Al will be induced to make a jump to site 2. If site 2 is also occupied by ion A2, A2 will be induced to make a jump to site 3. If all the sites from 0 to n - 1 are occupied and n is vacant, the successive jumps will continue until ion An_ l on site n - 1 makes a jump to site n. Such n successive jumps will con- tribute n steps to the electrical conduction, while contributing only one step to the self diffusion of ion Ao. It is also assumed that the diffusing ions are randomly distributed over the available sites with occupation probability p. The probability per unit time that an ion makes a spontaneous jump to a neighbouring site will depend on whether the latter site is vacant or occupied. We denote the jump frequency to a vacant neighbouring site by ut and that to an occupied one by u2. In general u2 will be smaller than Pl.

The probability per unit time that the tracer Ao on site 0 makes a spontaneous jump 0--* 1 is evidently equal to

Vsp = u l ( l - - p ) + u2p. (8)


A0 • ~ ,

-m-m., -1 -1 0 1 n-1 n ( v a c a n t )

Fig. 7. Cooperative jumps of (m+n) ions associated with the induced jump 0~ 1 of tracer Ao (from ref. [43] ).

The tracer Ao can make an induced jump 0--, 1 as well as a spontaneous one. The probability per unit time that Ao makes an induced jump 0 ~ 1 can be calculated as follows. In fig. 7, let all the sites f rom - m to - 1 be occupied. Then a spontaneous jump to the right of the ion on site - m will induce a jump 0 ~ 1 of Ao. The probability that all the sites from - m to - 1 are occupied isp m. Hence the probability per unit time that the tracer Ao makes an induced jump 0--, 1 is

/2ind= ~ t12pm~-pzp/(1--p) . (9) m=|

The total jump frequency u is the sum of vsp and vi.o:

v = Vsp + rind = (Vl -- V2) ( 1 --p) + V2/ ( 1 - -p) . (10)

The self diffusion coefficient o f the tracer can be expressed in terms of u as

D = v a 2 , (11)

where a is the distance between two neighbouring sites.

On the other hand, the charge transport per unit time associated with the spontaneous jump 0- , 1 o f tracer Ao is given by

jsp=u, a e ( 1 - p ) + u 2 a e { 1 / ( 1 - p ) - l + p } , (12)

where e is the charge of the ions. The second term is the sum over n of the contribution from n successive jumps starting from the spontaneous jump 0 ~ 1 of Ao:

vz ~ n a e p n - ~ ( 1 - p ) = u 2 a e { 1 / ( 1 - p ) - l + p } . n=2

The charge transport per unit t ime associated with the induced jump 0--, 1 of Ao is the sum over m and n of the contribution from the m + n successive jumps starting from the spontaneous jump - m--, - m + 1 and ending at site n:

Jind--~/)2 ~ ~ (ITl'{'H)aepm+n--l(1--p) m=l n~l

= 2 v 2 a e { l / ( 1 - p ) 2 - 1 / ( 1 - p ) } . (13)

The total charge transport per unit time associated with the jump 0-~ l of the tracer Ao is simply the sum ofjsp and Jinj:

J =Jsp +Jind

=ae{ (ut - v2) ( 1 - p ) + v 2 ( l + p ) / ( 1 - p ) 2 , (14)

and it is easy to see that the mobility/~ o f the ion can be expressed in terms o f j as

I~=aj / kT . (15)

From these equations we have the expression of the f-value as follows:

f = ( v ' - V 2 ) ( t - - p ) 2 + V z ( I + P ) / ( 1 - - P ) (Ul __/22) ( 1 __p)2+/22 , (16)

which is seen to be always larger than unity as far as v2 does not vanish. I f v2 = 0, the f-value is reduced to unity.

I f Ul = v2, the f-value is reduced to

f = ( l + p ) / ( 1 - p ) . (17)

Theoretical works on the thermoelectric power also have been performed. Haga [ 79 ] studied the thermoelectric power by calculating the potential difference between two Ag electrodes in an arrangement: Ag(T~ ) I AgI (Tl) I AgzX I AgI (T2) I Ag(T2), where X = S , Se, Te. Girvin [80] calculated the thermoelectric power when the concentration is 1/2, using the lattice gas model. Yonashiro et al. [81,82] have studied the ionic conductivity, the thermoelectric power and thermal conductivity for the case of an arbitrary concentration by making use of Mahan's method [71]. Recently Tanaka et al. [83] and Sharma and Tanaka [ 84 ] have developed the lattice gas theory to the honeycomb lattice.

Ramasesha [85] presented a phenomenological model for semiconductor-metal transition in mixed conductors. The model predicted that an abrupt or gradual increase in the electronic carrier concentration accompanies the transition to the superionic phase.

The electronic structure of Ag chalcogenides in ct


phase was discussed on the basis of an energy-band structure calculation by Hasegawa [86]. As a sim- plest way of simulating the effect of the Ag ions on the electronic states, he used some hypothetical crystalline compounds. The magnitude of his calculated conduction electron mass was quite small irrespec- tively of the structures, about 10% of the free electron mass, and was in semiquantitative agreement with experiments. His calculation also found that a deviation from an effective mass approximation near the conduction bottom is appreciable. An origin of these features of the conduction band was a rather strong hybridization of the Ag 5s band and the chalcogen s band.

As discussed in the preceding chapter, a remarkable deviation from the Einstein relation has been found in Ag chalcogenides. Tomoyose [ 87 ] tried to explain the temperature dependence of this deviation modeling a diffusion theory of muon in metals. The ion hopping model dressed by conduction electrons led to the power low dependence on temperature of the deviation from the Einstein relation. However his theory has not considered some kind of cooperative motion of ions such as a caterpillar mechanism, which would be effective to interpret the remarkable deviation from the Einstein relation.

Attempts to understand the distinct structural and transport properties in terms of microscopic inter- ionic potentials have so far been based only upon the molecular dynamics (MD) technique [ 88]. Roman and Dieterich [89] presented a simple approach based upon a classical density functional theory, aiming at a microscopic explanation of the mobile ion density distribution. They regarded the anion host lattice as rigid, assuming that the thermal fluctuations of lattice are of minor significance with respect to the equilibrium density distribution p(r) of the diffusing cations, p(r) was determined by a static potential v(r) due to the host lattice and by the in- teractions within the subsystem of diffusing ions. According to classical density functional theory, p(r) could be computed by minimishing the grand free energy functional. Their results were applied to a- Ag2S. Their Ag density profile is shown in fig. 8 and are in almost quantitative agreement with the MD studies of Vashishta et al. [ 90 ].

.

OI ~ 0

o.6

o_.~ 04

02

.S(ooo)

i o I

o o o °

o ~ ,

,,'7\ / \

M(~o) o(2i 12o) t(~. ~ 0) M( Distance

Fig. 8. The density of Ag ions normalized by the density at tetrahedral interstitial sites in ct-Ag2S at T= 464 K along various di- rections within the cubic unit cell. Broken curve: the bare density. Full curve: the theory including correlations. Circles: molecular dynamics results obtained by [ 90 ] (from ref. [ 89 ] ).

2.2. Continuum model and structure factors

Kobayashi et al. [ 91 ] investigated the dynamical properties ofAg chalcogenides using the same model as that of Hayashi et al. [ 92 ]. They suppose that the cage composed of anions is immersed in the cation liquid [63]. The equations of motion for the longitudinal cage displacement field ~(r, t) can be written as

~ ¢ ( r , t ) - VE+ F VEt(r, t) + - m A T

--oo

X{O ~ '(1", t' )--l~(r, l' )}--2AqEind(¥'mA t)

1 + V~,xt (r, t ) = 0 , (18)

m A

with

m*-I = m ~ l + m f f l ( 2 + 5 ) - l

and

M( t) =m*o92 e x p ( - t / z c ) .


Here mA and ma are the anion and cation masses, respectively, ZA is the valence of an anion, q is the elementary charge, F is the phenomenological damping coefficient for the motion of the cage composed of anions and z - ~ is that for the relative motion of the two ionic components. VL is the longitudinal sound velocity in the cage. v(r, t) is the velocity field of mobile ions. M(t ) is the memory function, which describes the relaxation of the restoring force in the relative motion of the two ionic components, with the oscillator frequency 090 and the viscoelastic relaxation time rE. O~ ~t (r, t) is the weak space- and time-dependent external potential acting on the A- component, i.e., an anion. ~ denotes the deviation from the stoichiometry.

The dimensionless number density of cage ions nA(r, t) is related to the cage displacement field ~(r, t) through the equation

nA(r, t ) = - V.~(r, t) . (19)

The dimensionless number density fluctuations ni (r, t) ( i= A, B, e) are defined by

ni(r, t)={Ni(r, t)--Noi)/Noi ,

where N~(r, t) are the number densities and Noi their mean values (NoA=No, NoB= (2+O)No, No~=ONo).

The motion of mobile ions can be described by the velocity field v(r, t) and the dimensionless number density fluctuation nB(r, t). The linearized hydrodynamic equations for these field are

O nB(r, t) + V-v(r, t) = 0 , (20)

which is the continuity equation, and

0 v(r, t) + C 2 VnB(r, t) - ctV2v(r, t)

+ 2 ~ a r { r ( r , t ) - O ~(r,t)}

l i d t ' M ( t - t ' ) { v ( r , t ' +~-~m~ )- O'(r't')} --Go

ZaqEind(r, t )+ 1 Vq~Xt(r, t ) = 0 , (21) mB mB

which is the Navier-Stokes equation. Co and ot de-

note the sound velocity in the fluid and the viscosity, respectively.

The number density of conduction electrons Ne (r, t) satisfy the continuity equation

~tNe(r, t ) - ~ V'je(r, t)=O . (22)

Hereje(r, t) is the electronic current density. By using the space-time dependent electronic conductivity ae(r, t) it is given by

L ( r , t ) = f d r ' i d t ' ao ( r - r ' , t - r )

The induced electric field Eina(r, t) obeys the Poisson's equation

V.Eind(r, t )= 4~Noq

× {zanA(r, t) + (2+~)zBnB(r, t) --6n~(r, t )} , (24)

where fi denotes the deviation from the stoichiometry. The system is assumed to be under the weak, space-

and time-dependent external potentials Oy x' acting on the three components 0 = A, B, e). We define the longitudinal density response function of the system by

No, ni(k, co)= ~ Z,j(k, og)0yxt(k, 09) (25) J

in the form of Fourier transforms. The structure factors So(k) are directly related to

the response functions by the fluctuation-dissipation theorem and the Kramers-Kronig relation by the following equations [ 93 ]:

kBr S~j(k) = - ~ xo(k, 0 ) . (26)

The structure factors S¢,, SA~, and SBe can be expressed in terms of the structure factors of the ions, SAA, SBB, and SAB in the long wavelength limit. Tak- ing the Fourier transforms of eqs. (18 ) - (24 ) and using eqs. (2 5 ) and (26), we have the following simple relations:


lim SA~(k) = lim / "~/N°A/'/2

X {ZASAA(k) "NoB "1/2" +ZBSA.(k)(~--~o~) }, (27)

/ . .112 -NoB

lim Sr~(k) = lira | ~ - J k+o k+O k Oe/

{ X ZBS~B(k)+ZASAB(k) , (28)

and

t lim S~(k) = lim Z A ~ SAA(k) k~O k~O t \ oe/

( NoA NoB ) ' /2 +2ZAZB SAB(k)

Noo

2 NOB

These relations of eqs. (27 ) - (29 ) have been given by March et al. [94] by making use of the condition of perfect screening in a liquid metal alloy. Calcu- lated values of the long wavelength limit (k--,0) of So(k) are shown in table 1 for a-Ag2Se at Noe = 1019 cm -3 and at 423 K. The numerical values of parameters used are shown in table 2.

2.3. Electronic relaxation time

The relaxation time z~ for electric conduction is given in a classical limit (hflog<< 1 ) by [95,96]

2kF

m~ f dkk3p(k) (30) r e 1 - 12Z(Tt./t ) s 0

where z is the number of conduction electrons per ion and p(k) is the probability per unit time that an electron gives up momentum hk to the system. We

assume that the concentration of conduction electrons are lower than 1020 cm-3. In this case we have 2kF<n/a, where a lattice constant a is 4.99 A at 170 ° C. Then we need zone center values for k. These things certify that a continuum model is applicable to the calculation of an electronic relaxation time by using eq. (30). p(k) is given by [97]

p(k) = { ¢(k)} 2SNN (k) + { VATS) - VB (~) } 2Scc (k)

+ 2 l?(k){ VA ( k ) - ~'a(k)}SNc(k) (31)

with

l?(k) =CA I?A (k) +CB VB(k). (32)

Here lP~(k) ( i=A, B) is the screened pseudo potential matrix element [97 ] of the ith kind of ions in a system. SNr~(k), Scc(k) and SNc(k) in eq. (31) are given by the partial structure factors as follows [ 98 ]:

SNN (k) =CA SAA ( k ) + cBSaB( k )

+2 C~/~%AC%SAB(k), (33)

Scc(k) =CACB{CnS~(k) +cASBB(k)

- 2 CX/~AC~SAB(k) } (34)

and

SNc(k) =CA CB {S~, ( k ) -SBB( k )

+ ~ S A B ( k ) I , (35) N/CA Ca J

where CAam =NoAaom / (NoA+NOa) are the number concentrations. The expression of the pseudo potential is given by [97]

4;Tziq 2 l?,-(k) = E~(k2+k~ ) cos(kR~). (36)

Here R~ is the core radius of ith kind of ions. In the long wavelength region where kR, << 1, we can ex-

Table 1 Calculated values of the long wavelength limit (k--*0) of Se(k) are shown for ~-Ag2Se at Noe= 1019 cm -3 and at 423 K (from Kobayashi etal. [91]).

S~ SAB &B SAc &, See

0.00665 0.00863 0.0186 -0.0432 0.510 55.0


Table 2 Values of physical parameters of tt-Ag2Se. We have taken a value of 105 cm-1 for k. The numeral in a parenthesis shows a reference number (from ref. [91 ] ).

coo high freuqency dielectric constant 15 (10) No number density of anions 1.6 × 1022 cm- 3 mA mass of Se 2- ion 1.321 X 10 -22 g mB mass ofAg + ion 1.79× 10 -22 g m* reduced mass 0.965 × 10 -22 g me mass of conduction electron me = 0.2 m ( 10 ) 090 oscillator frequency 105 cm- ~ ( 11 ) g2A plasma frequency of selenium ions 51.4 cm- 12~ plasma frequency of silver ions 31.2 cm- t2p effective ion plasma frequency 60.1 cm- ogp plasma frequency of conduction 547 cm-

electrons (for No~= 10 ~9 cm -a) relaxation time z - ~ = 45 cm- ~ ( 11 )

re viscoelastic relaxation time r~- ~ = 53 cm- a ( 11 ) VL longitudinal sound velocity (cage) vk= 0.8 cm- ~ (2) Co sound velocity (fluid) Cok=0.2 cm - l (2) F damping coefficient Fk 2 = 0.1 vk (6) a viscosity ak2=0.05 cm -j (6) RA core radius of Se 2- ion 1.98 A (12) RB core radius ofAg + ion 1.26 A (12)

Table 3 Calculated values of the relaxation time for electric conduction at 423 K are given for ct-Ag2Se at two values of electron concentrations (from ref. [91 ] ).

No¢ (cm -3 ) kF(Cm -j ) ks(cm - l ) re(s)

10 t9 6.67 × 106 4.64 X 10 6 5.22 × 10-14 5X 1019 1.14)< 107 6.07)< 106 8.18X 10 -14

p a n d c o s ( k R i ) in t e r m s o f kRi to t he s e c o n d order .

T h e c a l c u l a t i o n is p e r f o r m e d fo r a -Ag2Se at 423

K. T h e n u m e r i c a l va lues o f t h e p a r a m e t e r s u s e d a re

s h o w n in t ab l e 2. T h e n u m e r i c a l c a l c u l a t i o n is d o n e

for two cases o f e l e c t r o n c o n c e n t r a t i o n , N0e= 1019

c m -3 a n d 5 × 1019 c m -3, so as to c o m p a r e w i t h t he

e x p e r i m e n t a l v a l u e b y M i y a t a n i et al. [ 3 4 ] . Resu l t s

a re s h o w n in t ab l e 3. T h e o b t a i n e d va lues o f re a re

in good a g r e e m e n t w i t h t he e x p e r i m e n t a l r e su l t s [ 3 4 ] , r e = 4 . 1 - 4 . 5 × 10 -14 S.

2.4. Conductivity

We d e f i n e t he cha rge d e n s i t y - c h a r g e d e n s i t y re-

s p o n s e f u n c t i o n b y t a k i n g a l i n e a r c o m b i n a t i o n o f

t he r e s p o n s e f u n c t i o n s d e f i n e d in eq. ( 2 5 ) as fo l lows

[ 9 9 ] :

3 ZQQ(k, 0 9 ) = ~ ~,(i) t t - 09) ( 3 7 )

i=1

w h e r e

Z(')~k, co) =q2(Z~ZAA(k, to) QQx

+ Z~Znn( k, 09) + 2ZA ZBXAB( k, tO)} , ( 3 8 )

Z ( 2 ) t k 09) = q E z e e ( k , tO) ( 3 9 ) QQ \ '~'

a n d

Z(3) (k , tO) = --2qZ{ZhXhe(k, tO)+ZnZne(k, tO)} QQ ( 4 0 )

T h e l o n g i t u d i n a l c o n d u c t i v i t y o f a s y s t e m is d e f i n e d as t h e r e s p o n s e o f all t h e cha rges to a s c r e e n e d elec-

t r i c p o t e n t i a l b y

a(k, tO) =Re itOZQQ(k' tO) ( 4 1 ) k2 + 47tZQQ( k, tO ) l Eoo "


Now for any wave number and for any frequency X~3~ (k, to) cancels half and half with both parts of X ~ (k, to) and Z~2~ (k, 09) exactly. Therefore we introduce the following new charge response functions Fj(k, to) ( j=i , e):

Fi(k, to) = g ~ ( k , t o ) " 1,,(3)t/,- to~ (42a) / 2A, Q Q ~ '~ ,'

and

l v ( 3 ) ( / , . Fe(k, to)_-,,(2)t/,- t o )+ to) (42b) - - ~ , Q Q '~'~, 2 A Q Q k ' ~ , •

These functions Fi ( k, to) and Fe ( k, to) correspond to the ionic charge response function and to the electronic charge response function, respectively.

Next we define the conductivities aj(k, to) ( j= i , e) as follows:

i to~(k, to) (43) aj( k, to) =Re k2_l_4~ZZQQ( k, to)/Coo "

Then eq. (41 ) is rewritten as

or(k, to)=ai(k, to)+ae(k, to). (44)

It is ascertained analytically that ai(k, to) is the same as the longitudinal ionic conductivity obtained in ref. [ 100 ]. The numerically calculated ai (k, to) and ae (k, to) for u-Ag2Se are shown in fig. 9 for two wave numbers, k = 0 and 5>( 105 cm -1 at Noe= 1019 cm -3.

Here we have used the expression

Ee(k , CO)=e~-E~to~/{to(o)+i /ve)} , (45)

as the dielectric function of conduction electrons. As the numerical value of rE 1, we have used %- 1 = 638 c m - 1, which is the result obtained in the foregoing paragraph.

The calculated values of ai(k, to) have the broad peak (at about 100 cm-1 ) which is attributed to optical phonons. The calculated frequency dependence of the ionic conductivity of ct-Ag2Se is similar to that of ct-AgI.

Next, we refer to the dc conductivity. It is given by

aa~- lim lim Re itoXQQ (k' to) (46) o,~o k~o k2+4nXQQ(k, ¢ 0 ) / ~ "

This is calculated as follows:

ade = ~ ~l._-CT'+----~02 r c +to2"t e / . (47)

5.0

2.5

>-

F

o 2.0 r~ z o

1.5

1.0

I

o'i(k:O)/Gi(d:t.c. ~

/o'i(d.c.)

/ O-e (k = 0, 5 x 105)/O-e (d.c.)

I I I 50 I00 150

oJ(cm -I)

Fig. 9. Ionic conductivity ai(k, 09) and electronic conductivity a, (k, ~o) of ~t-Ag2Se for electron concentration No.--- 1019 cm-3. Solid curves for k=0 and dotted curve for k= 5 × 105 cm- ~ (from ref. [91]).

The first and second terms of this equation express the contribution to the ionic conduction and the electronic conduction, respectively.

It is easy to prove that the conductivity given by eq. (41) satisfies the following f-sum rule by the an- alytical calculation:

f dto R itoZQQ(k, 09) do)g a(k, to) = - ~ e k 2 + ~ , ~/~<~ - - ~ - - o o

=q2( NOAZ2 -]- NoBZ2 + N°e~. (48) \ mA ma me /

2.5. Collective motion

The dispersion equation of the longitudinal collective modes is given by


47~ 1 - . ~r~(k, 0 ) ) = 0 . ( 4 9 )

117~ 0)

For the long wavelength limit (k-+0) seven solutions of eq. (49) can be obtained as follows:

2 2 1/2 + mA V L "~- 2mB Co •

0 ) 3 = - i ~ (mA+2mB)

2 2 2 2 2 2 2 QACo +QB VL + VLCoks × (z_~+0)ZoZ~)(mAVz+2maC2) (51)

and

2 2 --1 +Q2p__i0)_+{l+l ' l k z ~ c , ] 20) 2_7 =0)p +0)o + ZZc

1

2 2 __1 - i , (52) + 4 ~ p 0)o + zz~

We have no heat flow mode, since we have no energy-transport equation here for simplicity.

For the purpose of obtaining the dispersion curve of the coupled plasmon - LO phonon modes in detail we introduce the following spectral density function (SDF) of the coupled modes:

fl(k, 0))= --Im 0)o(k) 0)2_0)2(k) . (53)

Here 0)o(k) is determined so that the integral

i fl(k, 0)) do) 2~

- - o o

may be independent of k. 0)(k) in eq. (53) is the complex frequency of the coupled modes which is given by solving the dispersion equation (49). The calculation is carried out for a-Ag2Te. However the numerical parameters used are the same as those used in [ 101 ] except for the following parameters:

e~=15 [32] , m ¢ = 0 . 9 1 × 1 0 - 3 ° g [ 3 2 ] ,

ml =2 .12× 10-22 g, m2 = 1.79× 10-22 g.

where

0)p = (4nJNoq2/E<~me)1/2 , ~ p =~r~A ..t_ ~ B 2 2 2

and

1 2 2 0)_+ = {0),> + 0 ) 0 +

In a derivation of these dispersion equations we have used the collisionless free electronic conductivity. These solutions 0)1,2, o)3 and 0)4-7 represent longitudinal acoustic phonons, the diffusion mode and coupled modes of plasmons and damped LO-phonons, respectively. The long wavelength limit solutions of the dispersion equation of collective motion in a-AgI have been investigated by some authors [65,66,92,101 ]. There exist conduction electrons in a-Ag2Se. Some of the present solutions, eqs. ( 50 ) - (52), are different from those obtained in the case of a-AgI on account of electron interaction. Com- paring our present solutions with those obtained in [ 92 ], we see that damped optical phonons form the coupled modes with plasmons and the relaxation mode has been turned into the diffusion mode by taking into account the effect of electron interaction.

2.5.1. Carrier concentration dependence We take k = 100 cm -1 and T=450 K. We are in-

terested in the case where the crossing of plasmons and LO-phonons would occur. We artificially con- trol the nonstoichiometric value ~ to examine this case. When the carrier concentration amounts to 10~ 7 cm -3, the frequency of plasmons becomes of the same order of magnitude as that of LO-phonons. This concentration leads to the low degeneracy temperature of electrons, about 90 K, which is lower than the temperature in our consideration, 450 K. There- fore we assume that the electrons obey the Maxwel- lian. For 0)>>kVth, where V~h= (kBT/me) ~/2 is the electron thermal velocity, the dielectric function of Maxwellian plasmas is given by [ 102 ]

e~e)(k, 0)) = 1 - 0)_2 - 3 0) 2

--i 4 2 ~ , T ) ~-~th expt - - 2 \ ~ ] J" (54)

The SDFs of the coupled modes are calculated numerically at various carrier concentrations and are shown in fig. 10. We see that optical phonons are


._c

0

L_

6

,~6 - " ~ . ~ " i ~ ' ~ 150/2/3 ~'~- ~ ( cm -1 )

s ~ 6

50 100 150 200 W (cm -1 )

Fig. 10. Spectral density function of coupled modes at various carrier concentrations; (a) lower mode (b) upper mode (from ref. [ 100 ] ).

much damped in superionic conductors. The dispersion curves of the coupled modes are obtained by plotting the peak positions of the SDFs and are shown in fig. 11. The vertical bars give the imaginary part of the coupled mode frequency. We see that the plasmon-LO phonon coupling is not strong on account of heavily damped LO-phonons.

2.5.2. Wave number dependence We now examine the wave number dependence of

the coupled modes for the carrier concentration of

ne(cm-1 ) 1016 1017 3xld 7 5x1017

I I I I

200

15C

310(

5O

0 L i I I 0 2 4 6 8

xlO -8

Fig. 11. Carrier concentration dependence of the coupled modes. Vertical bars give the imaginary part of the coupled mode frequency (from ref. [ 100] ).

1019 c m -3 , which is a proper concentration for ct type silver chalcogenides. Since the degeneracy temperature of this concentration is about 2 × 103 K, we may assume that the electrons are degenerate. Now we use the following expression as the electronic conductivity with a finite life time [ 103,104 ]

O9 a~e)(k, to)= i4~

×(1 + -~){, ~°)(k, to+i7) - 1 }

× [ 1 + i7 ¢¢°'(k' t o + i 7 ) - 1] -1 to e~O)(k, 0 ) _ ~ . (55)

Here the quantities 7 and ~ co) (k, to) denote the electron collision frequency and the longitudinal dielectric function of Lindhard [ 105], respectively. The SDFs of the coupled modes are calculated numerically at various wave number. Fig. 12 shows the SDFs of the lower mode, L_. Fig. 13 shows the dispersion curves of the coupled modes, obtained by plotting the peak value of the SDF. The line widths of L_ spectra are mainly related to z and re but not to the real part of a~ e~ (k, to) in the single panicle excita-

M. Kobayashi/Structural and dynamical properties of silver chalcogenides 13 5

~E

o

~ L_ ~ c m 3

/

50 I00 150 200 W(cm -1 )

Fig. 12. Spectral density functions of the lower mode at various wave number (from ref. [ 100] ).

,100p 1000

k.' ~ ne =I019(cm-3) oof')/i , 8°°~' I !

3 1 0 0 ~

,

0 1 k/k F

/

2

Fig. 13. Dispersion curve of the coupled modes (from ref. [ 100 ] ).

tion region. Accordingly we cannot recognize the effect of Landau damping in the L spectra, since L_ mode is much damped even in the outside of Landau damping region in superionic conductors. These cir- cumstances differ very much from the case of polar semiconductors [106,107]. In the latter case, L_ mode is a well defined excitation and exists as a fairly well defined excitation even in the Landau damping region.

3. Molecular dynamics method

3. I. Introduction

The molecular dynamics (MD) method is effective and powerful to extract the definite structural and transport properties of SIC [88]. The first MD simulation to the Ag chalcogenides, a-Ag2S, was performed by Ihara and Suzuki [ 108 ]. They used a simple ionic force as follows:

F(ru)= qgqjrij { l +sign(qiqj)(ai+aJ~'~ r3 \ ~ u / J ' (56)

where qi and ai are the effective charge and radius of ion i. They assumed n = 6. The calculated pair distribution function (PDF) for 1296 particle-system are shown in fig. 14. We see that S ions form a well

v . ~ - _

3.0

~- 0 !

1.C

3C

0 2

'- 2.0 v

OT 1.0

3.0

v 2.0 c~

1.0

1 (Ag-S)

i I I I I I

(S- Ag)

8

I # I i #

! (s-s) 26

4 6 B 10 12 14 r (~)

Fig. 14. The radial distribution functions gu(r) for Ag-Ag, Ag- S, S-Ag, and S-S in ct-Ag2S. Solid line: molecular dynamics results. Broken line: experimental results of ref. [28] (from ref. [~o81).


defined bcc structure and Ag ions are at the tetrahedral sites. The density distribution on the (100) face in the 1296 system is shown in fig. 15. The probability density o f Ag ions is much more delocalised in ct-Ag2S than in ct-AgI. It should be noted that the density of the 12(d) sites and 24(h) sites increase with increasing temperature, whereas the density of 6(b) sites decreases. These are in good agreement with the neutron scattering by Cava et al. [29] .

Vashishta et al. [90 ] also performed the MD simulation of the (t-Ag2S. They used the following in- terionic potential

(~h + ~ ) ~j z~zje 2 (o~z~ +o~jz~)e 2 V°(r)=Au - - - 7 + r 2r 4 '

(57)

(b)

,'" (2) i (1) o

( a )

/ I

,, L. ; .~._'_~ , / : i " - -

I

0 : 0 6(b) I : T 12(d) A:S 24(h)

I I I I I I I , I

1.0 ## "~I"/#~'~ ~,,o.. ~d ~ ( C ) ;.-m~

: / / v , O.5

3

I

I 0 T 0 S I (1) (2) (3)

distance

Fig. 15, (a) bcc unit cell ofeL-Ag2S; (b) the (100) face ofbcc unit cell. The points I, O, T, and S represent a corner (S ion), 6(b) sites, 12(d) sites and 24(h) sites, respectively. Segments ( 1 )-(3) show the lines along I-O, O-T-O, and O-S-I, respectively; (c) the density distribution of Ag ions on the (100) face at 450 K (dash-dotted line), 530 K (dashed line) and 611 K (solid line), respectively (from ref. [ 108] ).

where i, j denote the type o f ion, A denote the repulsive strengths, a denote the particle radii and c~ denote the electronic polarizabilities. They take Au=A and nu=n=7. The PDF are shown in fig. 16. They are almost the same as those in fig. 14. Fig. 17 shows the S and the Ag densities along various paths on the (100) face. The sulphur density is well localized on the bcc lattice points and the width o f the density distribution gives a measure of the mean square displacement. The Ag density is plotted along the paths S - M - o - t - M - t (see inset in fig. 17). The point M corresponds to the point S in fig. 15. The Ag density at M is 0.2 for T = 6 4 6 K, while it is 0.7 at S for T = 611 K. The MD simulutions which were performed by two groups produced much different density at M (S). Cave and McWhan [ 109 ] observed in the superionic phase anisotropic discs of intensity in the vicinity o f the point Q = (1.6, 1, 0). This unusual feature is not observed in (t-AgI in which the anion lattice has the same bcc structure as in [3-Ag2S. Vashishta et al. concluded that these intensities in the X-ray diffuse scattering were due to the collective motion of Ag ions. In recent inelastic neutron scattering experiment on Ag2S, Grier et al. [ 110] observed anomalous intensities at Q = ( 1.6, 1, 0) similar to those observed in X-ray diffuse scattering by Cava and McWhan [ 109]. In addition, Grier et al. [ 110] also observed an excitation at ap- proximately 2 eV in the vicinity o f this Q vector, but only at temperatures below the supersonic transi-

v

[ T r T

- :! t I I I i I

: i I

0 2 4 6 8 10 12 r(~,)

Fig. 16. Pair distribution functions go(r) for S-S (solid line), S- Ag (dashed line), and Ag-Ag (dash-dotted line) in Ag2S at T=464 K (from ref. [90] ).


1.0. , ~8~ I! I ~ ' 1

8

l l / 1\ °31-t 1 ,/

v -I 0.11"- ~ /~-I-i:i-.~, ' ~ q

', °%0' 71K)' ' I

Fig. 17. Density distribution of S 2- and Ag + on the (100) face ofbcc at T=646 K. In the inset S, o, and t mark the corner (sulphur), octahedral and tetrahedral sites respectively. M is the mid- point of the t-t line. Full circles are MD calculations; lines are drawn for clarity. The broken curve denotes Ps/Ps (000) along the path S-M-o. Ag density PAg/Pk, along paths S-M-o, o-t and t- M-t. Maximum Ag density at t and at point (7/16, 0, 7/16) along S-M-o. A 10% dimple in Ag density at o. The inset shows the temperature dependence of the normalized Ag density, p~(T)/ pi(Tc),fori=t,o, M, with Tc=450 K (from ref. [90]).

0 4 I I ' i ~ i ~ I f i , i i i (ca)

0

0 .02~

0 , I , I , I , I , f , i i - -

1 3 5 Energy ( meV )

Fig. 18. S(Q, co) of Ag2S for Q= ( 1.8, 1, 0) ( I QI =2.663/~-~ ) at temperatures of (a) 268 K and (b) 339 K, below the superionic transition, and at (c) 470 K, above the transition. The curves include experimental resolution of FWHM = 0.7 MeV. An excitation at a low energy of -~ 3 MeV is clearly seen below the superionic transition (from ref. [90] ).

tion. Such an excitation is common to many SIC and have been found for Na-[3 alumina [111 ], 13-AgI [ 112 ], RbAgals [ 113 ] and 7-Ag3SI [ 114 ]. The MD study by Ebbsj6 et al. [ 115 ] was an at tempt to explain the anomalous neutron scattering intensities and the nature of the low energy excitation in Ag2S. Their calculation of the intermediate scattering function showed that (S, S), (S, Ag) and (Ag, S) correlations were negligible in the vicinity o f ( 1.6, 1, 0) and that (Ag, Ag) correlation only was dominant. They found that the weighted sum of the dynamical structure factors, S(Q, o9), was maximal at Q = ( 1.6, 1, 0) and also the neutron intensity, S(Q, o9=0) , formed anisotropic discs around Q = ( 1.6, 1, 0).

To investigate the low-lying ( = 2 MeV) excitation, Ebbsj0 et al. [ 115 ] "'cooled" the Ag2S system below the superionic transition temperature, keep- ing the bcc structure. Their results for S(Q, 09) for Q = (1.8, 1, 0) are shown in fig. 18 at 268 and 339 K, which are both below the transition temperature,

and at 470 K, which is just above the transition temperature. A low energy peak is evident in the 268 K results at an energy transfer o f about 3 MeV. They identified this excitation as an Einstein-like mode of the Ag ions at their residence sites. The fact that they found the maximum intensity of the excitation at ( 1.8, 1, 0) rather than at ( 1.6, 1, 0) might be related to the restrictions in their model which did not allow the S ions to transform into a monoclinic structure below the transition temperature.

Rino et al. [ 116 ] have studied the structural properties, single-particle dynamics, and the charge transport in ct-Ag2Se and molten Ag2Se using the method of MD. Diffuse neutron and X-ray scattering have been calculated and investigated in detail in the vicinity ofqo( 1.6, 1, 0). Let us define the partial structure factors, total structure factors, and give the expressions for the calculated neutron and X-ray intensities.

The intermediate scattering function F,~B(q, t) is defined as


F,~p(q, t)

=<p~(q, t)p*p(q, 0 ) ) , (a , fl)e{Ag, Se},

where the qth Fourier component of the density of the a-type particles p~(q, t) is

1 p~( q, t)= -~ ~ ) exp [ iq'ri( t ) ] ,

where N~ is the total number of a-type particles and r~(t) is the position of the ith particle. The ( ) bracket denotes the thermal average. The partial static structure factors are defined as

S,~ , (q ) = (Po,(q, O)p~(q, O ) )

=F~,(q, t=O ) .

The total structure factor S(q), neutron and X-ray intensities S,(q) and Sx(q) are given as

S ( q ) = ~ (¢aQa)l/2Sao(q),

&(q)= ~ b,~bp(c,~cp)'/2S~p(q) ,

& ( q ) = ~f~(q)fp(q) (c~cp)'/2&~(q), ap

where c~=N~/ZNp is the concentration of a-type particles, b~ the coherent neutron scattering length, andf~(q) the X-ray form factor. Using the MD trajectories, the partial structure factors Ss~-s~(q), SSe-Ag(q), and SA,-A,(q) have been calculated. To investigate anisotropic disks of diffuse scattering observed at qo=(1.6, 1, 0) by Cava and McWhan [ 109 ], Rino et al. [ 116 ] have calculated the above structures for q in the vicinity ofqo= ( 1.6, 1, 0). Their results for S,~#(q), S(q), S,(q), S~(q) for wave vec- tors q along (h,/, 0) are shown in fig. 19. The partial structure factors Ss,_s~(q) and SS~-Ag(q) are practi- cally zero in the vicinity of qo. The behavior of Ag- Ag correlation is quite different from that of the other two correlations. SAg-Ag(q) has a maximum for h = 1.6. It is the highly correlated motions of Ag ions which lead to this maximum in Ag-Ag partial structure factor at qo = ( 1.6, 1, 0). Rino et al. [ 116] have concluded that the structure factors S(q), S,(q), and Sx(q) merely reflect the behavior of SAg_Ag(q ).

Rino et al. [ 116 ] also have obtained the self-diffusion coefficient DAg of Ag by calculating the normalized velocity autocorrelation function. The temperature dependence of DAg (T) in ct-Ag2Se is shown

1.88

1.48

~-1 08 t t ' )

0.68

0.28

-01 1.0

(a) ---o- Sse-se * Sse-~j

S~_~

, , , J . . . , . . . ,

1.4 1.8 2. 2 ( h , l , 0 )

1.8

1.4

1.0

0.6

0.2 1.0'

* S (b) * Sn

I.; I'8 ' ' 2'2 ( h , l , 0 )

Fig. 19. (a) Partial structure factors S~(q) versus q along (h, l, 0) at 435 K; (b) total structure factor S(q), neutron structure factor S,(q) X l0 -24 cm 2, X-ray structure factor Sx(q)× l0 -3 versus q along (h, l, 0). SAg-gg(q), S(q), S,(q), and Sx(q) have maxima at qo= (1.6, 1,0) (from ref. I l l 6 ] ) .

~ 6

E 4

%

E

,} .... .o---- Experiment .o

• M D J~" '" f -

¢-

400 6oo 8oo lOOO 1200 T ( K )

Fig. 20. Temperature dependence of the self-diffusion constant for Ag + in o,-AgESe. The experimental results are taken from ref. [39] (from ref. [ l l 6 ] ) .

in fig. 20. The agreement with the experimental measurements of Okazaki [ 39 ] for Dkg (T) is quite good in the entire temperature range.

Vashishta et al. [ I 17 ] observed fractal behavior in MD simulations of ionic trajectories in AGES. For Ag ions in the superionic state and Ag ions and S ions in the molten phase, fractal dimensionalities of Brownian trails and isosets were found to be 2 and


0.5, respectively. These results confirm the predic- tions of Mandelbrot [ 118 ].

We have studied structural and dynamical properties of the superionic conductor AgzTe using MD method [ 119,120], which are reviewd in the following sections. For Ag2Te we used the following effective pair wise potential:

~ a ( r) = A o + ziz je2 , ( 58 ) r

where i, j describe the type of ions, A o the repulsive strength, ai, aj the particle radii, z~, zj the effective valence, and e the elementary charge. We assume Aa=A. The values of axe and aA~ are determined so as to avoid the overlapping of ions, that is, these are defined as follows:

the + axe = Te-Te nearest neighbor distance,

trTe + trAg = Te-Ag nearest neighbor distance.

If the fcc lattice constant is taken to be 6.6/k, we get axe=2.21 A and trAg=0.65 A, assuming the Ag positions to be the tetrahedral sites of the fcc lattice. We took n = 7. The values of other parameters used as as follows:

A=0.1 eV, ZTe ~-.~- - -2ZAg = -- 1.1 .

These values are chosen so as to reproduce the diffusion constant at 550 K and the ionic conductivity of AgETe measured by Miyatani [32,33 ] and Oka- zaki [ 121 ]. The calculations were performed on a 324-particle system in a cubic cell 19.8 /~ long in which the Te ions form a fcc lattice with the lattice constant 6.6 A. The integration time step was 9 .3×10 -~5 s.

3.2. Density distribution of Ag ions

Since the Te ions form a well-defined fcc structure, it is reasonable to use its lattice as a reference frame to see the position of Ag ions. The most interesting region is a (110) plane of the lattice and the pres- ence of Ag ions in a block of thickness x/~ a~ 16 (=0 .58 ]~, a is the lattice constant). The block is shown in the solid lines in fig. 21 and is divided into 32 × 32 rectangular pieces with the (x /~a/32) × (a/ 32) as shown in fig. 21. The number of Ag ions in- cluded in each piece was calculated and was aver-

aged over times. Points B and C are the tetrahedral (T) and the octahedral (O) sites, respectively. The Ag-ion density distribution on (110) plane at 850 K is shown in fig. 22. Letters A, B, C, D and E express the corresponding positions shown in fig. 21.

Fig. 22 suggests that Ag-ion stays at the T-site for the most of time and moves to its neighbouring T- site through the vicinity of the O-site. The calculated Ag distribution supports the jump-diffusion mechanism of cations along the zig-zag lines consisting of alternating T- and O-sites in (110) plane, as suggested by Okazaki [39,40]. This result is in good agreement with the X-ray data by Sakuma and Sai- toh [ 122 ].

Koto et al. [123 ] measured the X-ray diffraction on PbF2. They found that the diffusive F - ions left from one of the T-sites through the tetrahedral faces and moved through the octahedron in the center of the unit cell and then returned to a T-site. There was no pronounced local potential minima along the diffusion path of F - ions. Our results of the diffusion path of Ag + were in good agreement with their results.

The neutron diffraction work by Dickens et al. [ 124] and simulation studies by Walker et al. [ 125 ] and Gillan [ 126] showed that the diffusion of the F - ion in PbF2 occured by direct jumps between the T-sites. PbF2 and Ag2Te have the same structure. However PbF2 differs from Ag2Te in that the immobile sublattice is the same in the low- and high- temperature phases and mobile sublattice disordering occurs gradually with increasing temperature. On the other hand there is a rearrangement of the immobile-ion sublattice at the superionic phase-transition temperature, Tc, in addition to the disordering of the mobile ion sublattice in AgI-type superionic conductors. Moreover, the mobile ions in PbFz are F - anions and those in Ag2Te are Ag + cations. This difference yields, consequently, a different path for the diffusion of mobile ions. In addition to these it is worth mentioning that the effective sizes of these mobile ions are quite different, as discussed below. The ratio y of ionic radius is defined as follows: 7= ionic radius of immobile ion/ionic radius of mobile ion. If we use the values of the ionic radii by Pauling, the ratio 7 is about unity for PbF> and 1.6 for Ag2Te. We took y = 3.4 for our simulation. A high value of 7 will offer a space which is available to the


L- a

~ .ff, m

Fig. 21. Anti-fluorite structure. White circles show the fcc anion (Te) lattice sites and black circles tetrahedral sites. The points B and C are the tetrahedral (T) and the octahedral (O) sites, respectively. The parameter a is the lattice constant.

B B

B A

A

Fig. 22. Ag-ion density distribution in (110) plane. Notations are the same as those in fig. 21 (from ref. [ 119] ).

diffusion. Furthermore, the mass ratios of these materials are mAJmxe=0.845 and mv/rnpb=O.092. Then it is reasonable that very great differences in both the ratio 7 and mass ratio have a great influence upon the dynamical properties of these materials: diffusion path, ionic conductivity, etc. The same dis- cussion was given in the simulation work on CaF2 by Kaneko and Ueda [ 127].

3.3. Mean-square displacement and diffusion coefficient

The mean-square displacement (MSD) for anions and cations is defined by

1 No~ (r2(t)),=~,=,~ (tri(t)-ri(O)12), a=+,

(59)

where the time averaging (.. .) is to be understood as an average over time and the summations are over either anions or cations. The mean-square displacement was calculated for Te and Ag ions and is shown in fig. 23 for several temperatures. The MSD of Ag ions increases with time, whereas that of Te ions re- mains nearly constant. The linear regions in the functions (rZ(t))a may be related to the diffusion coefficients D. by the well-known equation

(rZ(t) ),=6DM+C~, (60)

where C, is a constant term. The large-time behavior of the MSD shows that Ag ions have a large liquid- like self-diffusion coefficient. The asymptotic value of the MSD of Te ions is a measure of the Debye- Waller factor. From the gradient of the straight line in fig. 23, the self-diffusion coefficient DAg of Ag ions is obtained for each temperature. These are shown in fig. 24 with the experimental results [ 121 ]. The temperature dependence agrees well with experiments.

When we plot log DAg against 1/T, we see that it takes an Arrhenius-type behavior with straight line. Then we can express the diffusion coefficient as follows:


v

[ i i

1

4 2

3

2 4 _

..~_1 2

00 I 2 t ( p s )

Fig. 23. Mean square displacements of Ag ions (solid line) and Te ions (dotted line) at (1) T=850K, (2) T=750 K, (3) T=650 K, and ( 4 ) T= 550 K as a function of time (from ref. [ 119 ] ).

E 6 u

16 5 / f

400 800 1200 T ( K )

Fig. 24. The temperature dependence of the self-diffusion coefficient DAs of Ag ions. The MD results and experimental values [ 116 ] are shown by circles and black circles, respectively (from ref. [119]).

DAg =Do exp( --e/kB T) . (61)

Here, Do is a constant with the d imens ion o f diffusion coefficient and ka is the Bol tzmann constant . The quant i ty E denotes the act ivat ion energy for an

ionic diffusion. F r o m the gradient o f the straight line o f log DAg with eq. (59) , an e value o f 0.17 eV is obtained, which is in good agreement with the exper- imenta l value, 0.14 eV [ 21 ].

3.4. Current-current correlation function and conductivity

In order to obta in the ionic conduct ivi ty, it is nec- essary to calculate the cur ren t -cur ren t correlat ion function. The normal ized cur ren t -cur ren t correla- t ion function is defined by

( J ( t ) . J ( O ) ) ~v(t) = (62) > '

where J( t ) is the charge current and is given by

J ( t ) = zAgvi(t) + ~ zrevi(t . (63) I . i=, I i = 1

The Four ie r t ransform ~u(tn) is shown in fig. 25. The ionic conduct ivi ty a ( to ) is given by the Kubo for- mula as follows [83]:

o o

a ( e ) ) - 3VkB~T ( J ( t ) . J ( O ) ) exp( io) t ) dt, (64) 0

1 5

10 3

v

' I r I

t f °" !

/;

1 2 ~ o ( l O ~ 3 r a d / s )

Fig. 25. Fourier transforms of current-current correlation functions at T=550 K (solid line) and at T=850 K (dotted line) (from ref. [119]).

142 M. l(obayashi/Structural and dynamical properties of silver chalcogenides

where Vis the volume. The temperature dependence of static ionic conductivity a ( t o = 0 ) is obtained, which is shown in fig. 26 with experimental values [ 32,33,121 ]. The tendency of temperature dependence is in good agreement with experiments.

Next we calculate the Haven's ratio HR, which is defined by

H R =DAg/Do. (65)

Here, D~ is the diffusion constant which is obtained by using the Nernst-Einstein relation. It is written by

Do = kB Ta(O)/nAg(ZAge) 2 , (66)

where nAg is the Ag-ion density, NAg/V. We show Ha- ven's ratio in fig. 27 as a function o f temperature.

if"6.0 E

4.0

b 2.0

1.0

06

= Miyatani I I • 0kazaki _ .#.-tJ---

400 800 1200

T(K)

Fig. 26. The temperature dependence of static ionic conductivity a(0) is shown by circles. Experimental values measured by Mi- yatani [32,33] (square) and Okazaki [ 121 ] (black circle) also are shown (from ref. [ 119 ] ).

H i i

0.8

0.6

0.4

- - - , O - . . . . " O - . . . . . O - . . . . . O . _ _

51)0 ' 7()0 ' 900 T ( K )

Fig. 27. The calculated temperature dependence (circle) of Ha- ven's ratio are plotted with experiments [39,40] (black circle) (from ref. [119]).

Experimental values of Okazaki [39,40] are also plotted. The MD results agree well with experiment. The difference from unity in Haven 's ratio implies that this system has a deviation from the Nerns t - Einstein relation.

The peak of a(co) is attributed to the transverse optical-phonon mode in et-AgI-type superionic conductors. Fig. 25 gives 82 cm-~ as the transverse optical-phonon frequency.

3.5. Polyhedron analysis and residence t ime

For a microscopic study of Ag-ion diffusion, Tachibana et al. [120] introduced a polyhedron analysis, which divides the fcc sublattice into tetrahedra and octahedra. As shown in fig. 28, each tetrahedron shares faces with four octahedra, while each octahedron shares faces with eight tetrahedra. The tetrahedra and octahedra share edges with each other. Thus a cation inside a tetrahedron diffuses into an octahedron through the triangular interface shared by them.

From the results derived in the preceding section, it is clarified that a Ag-ion moves from a tetrahedron into its neighbouring octahedron through the triangular interface between them and vice versa. In fig. 29, the diffusion traces at 850 K are shown for several Ag ions. These ions are picked up randomly. The full line and the broken line show that a Ag ion is in a tetrahedron and in an octahedron, respectively. We see that a diffusion path of a Ag ion is composed of

A A

N':J/I/

Fig. 28. Tetrahedra and octahedra in fcc anion sublanice. Points O and T denote the octahedral- and tetrahedral sites, respectively. Point P lies on the triangular interface between a tetrahedron and an octahedron. The distance AP is two-third of AO (from ref. [ 120] ).


5

4

:D

2 ....... I---t l I.I .<

1 -]...]].. F - l J - -

.r--L..lrLn.....J-lq .................... .q_.......Fu---qj--Ul...J--i. J-- l . . .rqJ-[ ......

~ J~L....FI ................ j-L....E._. _r---TI-I..I--UI Jq J ' - - I . ~ J----LLi L... r-I.II ..

t.._n.~n.J--L~ tn.I]J ..................... r - - l . . . r - q r i ~ n . . . I--m.....rq.._F

u L . ~ . r-qJq..]]..._.~ t .n....nF-qn.~.7

Lrl .......i--l.Fl.. [ ~ l I-I L. . . r ' -q . . .J -~q J ~ ......

' ' ' 2--000 1000 Time Steps

Fig. 29. Trace of the diffusion ofa Ag ion for 2500 time steps at 850 K. Full and broken lines mean that each Ag ion resides in a tetrahedron and in an octahedron, respectively (from ref. [ 120] ).

t - O

5

L- tn

t 3

1>-, ,,,-,

2~ o

I:1.

0.5

4 5 0 K .......... 5 5 0 K .... 6 5 0 K -- 750 K . . . . 8 5 0 K . . . . . 1000K

0 0

/i ~ , l l i

/ I i \/~..;< ~, .... / ..(. f ' / , . ~ . X . G ,

0.5 1.0

• T ( i I / I :

Fig. 30. Probability distribution which each silver ion resides in the tetrahedron in all time steps, z, at six temperatures (from ref. [ 1 2 0 ] ) .

temperature . The tempera ture dependence o f zx/Zo is the same as that o f Ag-ion density, which was obta ined in Kobayashi et al. [ l 19 ].

When we denote the probabi l i ty that a Ag ion is found in the te t rahedra as PT and in the octahedra as Po, respectively, the ratio Po/Px is equal to ZO/ZT and is p ropor t iona l to e x p ( - A E / k B T ) . Here A~ denotes the potent ia l difference between O- and T-sites. The change of cat ion densi ty was monotonic along the T - O line. Then the potent ia l difference Ae may be regarded as nearly the "ac t iva t ion energy" for ionic conduct ion.

In fig. 31, log(zo/Zx) versus 1/T is shown at six temperatures . F rom the gradient of the straight line in the figure, the value of AE is es t imated as 0.18 eV, which is almost the same as that der ived from the tempera ture dependence of the diffusion constant [ 119,120 ]. This value is in good agreement with the exper imenta l value, 0.14 eV [ 121 ].

3.6. Jump frequency of silver ions for diffusion in ot-Ag2 Te

We studied the diffusion process of silver ions in ct-AgETe by applying the caterpi l lar mechanism theory by Yokata [43] to interpret the Haven ' s rat io obta ined by the M D calculat ion [128,129]. We assumed that two kinds of sites, te t rahedral site and octahedral site, were avai lable for silver ions. These sites are shown in fig. 32. We also assumed that silver ions occupy these two sites with a different prob- abili ty, considering the results of the M D calculation and the X-ray experiment .

alternating tetrahedra and octahedra. We also see that the residence t ime in te t rahedra is relat ively longer than that in octahedra. There are a few blackened areas in fig. 29. In these sections, a Ag ion vibrates frequently between a te t rahedron and its nearest neighbour oc tahedron in an extremely short t ime interval.

Next Tachibana et al. [ 120] calculated the residence t ime o f ith cat ion in the te t rahedra , zx( i ) , in all t ime steps z and also that in the octahedra, zo(i). In fig. 30, the d is t r ibut ions o f ZT(i) are shown for six temperatures . When the mean value o f ZT(i) and zo(i) are denoted as zx and Zo, respectively, it is found that ZT > ZO and ZT decreases with increasing

0 - r-~_--"~U~850 K

-1 ' ~ - . ~ O K

I

1 2

1 0 3 / T ( K -1 )

Fig. 31. From the gradient of the straight line, "activation energy" A~ is estimated as 0.18 eV (from ref. [ 120 ] ).


I I t

I , , , ¢ 1 I I . . . . . ,, ..-. ~ . . . . . . / / / ~ [ ] - - I . . . .

Fig. 32. Configuration of Ag2Te. Squares and circles show the tetrahedral and octahedral sites, respectively. Triangles show the Te ions which construct the fcc lattice.

A B A

0 1 2 n-1 n

Fig. 33. Cooperative jumps associated with the spontaneous jump 0--, 1 of traser ion T. The nth site is vacant. This is the case in which the 0th site is on a site A.

For simplicity we considered a system which had at even intervals alternating A and B sites which cor- responded to a tetrahedral site and an octahedral site, respectively, on the straight line. We denote the jump frequency f rom B site to a vacant neighbouring site A by l'aA( 1 ) and that to an occupied site A by VaA(2). Similarly we denote the j u m p frequency f rom A site to a vacant neighbouring site B by PAB( 1 ) and that to an occupied site B by VAn(2). It is required that silver ions can occupy site A with probabil i ty P , and site B with probabil i ty Pn. The cooperative j u m p shown in fig. 33 is considered.

While the j u m p frequency of Ag ion is proportional to e x p ( - E a / k T ) , the occupation probabil i ty is proport ional to e x p ( - on site energy/kT). Then we have the following relation:

VAB( 1 )/VBA ( 1 ) =Pa/PA. (67)

By applying the caterpillar theory, the total j u m p frequency v of a tracer ion is given by

v,~. r Pa PA P~ v=~-L1-Pa+-ot +~(1-PA)+flPB

+ ( 1 - P A P a ) PAPa + flPBJ + '

(68)

with

or= VAB(I )/VAB(2) , f l= VBA(1 )/VBA(2) •

The self-diffusion coefficient D of the tracer ion can be expressed in term of the j u m p frequency v as

D=va 2 , (69)

where a is the distance between two neighbouring sites.

Next we calculated the current. The calculation is a little more complicated than that o f the j u m p frequency. The total current J is made up of the folow- ing two parts: one is from the spontaneous j ump 0 - , 1 of the tracer ion and the other is f rom the induced j u m p 0--, 1 of the tracer ion on the 0th site.

Finally the total current J associated with the j u m p 0-+ 1 of tracer ion T is obtained by summing up ,/so and Jind as follows:

j = aevaa (1 ) I i _ p a + PA 1 2 ~ (1 --PA) + 1 -PAP~

P~ +ca)} × { I ( 2--PAPa + PA)Pa + -~-~B ( 2--PAPa

4PA {P~ + ( 1 _ P A p a ) 3 ( I - - P A ) + ~ (1 - -PB)}

PA(PA +Pa) ( 4 --1) + (1--PAPB) 2 1 - P A P s

X{( 1 -Pn) / a + ( 1 --PA) /fl}

2PA ( 2 ) + (1--PAPB) 2 1 - P A P s 1

× { ( 1 - -PA)Ia+ ( 1 -Pa)l f l} ] • (70)

The residence t ime of silver ions at the tetrahedral site and the octahedral site have been calculated by a MD method [ I 19,120 ]. The residence t ime can be expressed by the t ime steps taken for its residence. The t ime steps which have been calculated by MD are shown in table 4. We can define the ratio of res-


Table 4 The temperature dependence of the Ag ion density in ct-Ag2Te by MD calculations [ 119,120 ]. Yx and Yo are the rates of time steps of how long Ag ions stay at tetrahedral sites and octahedral sites, respectively. The rate ofyo is taken as unity (from ref. [ 128] ).

Temp. (K) Y-r Yo

550 64.75 1 650 18.45 1 750 7.17 1 850 5.29 1

idence time steps as follows:

ZT =Y'r/(YT +Yo) , Zo =Yo / (YT +Yo) • (71)

20

,-,10

i i i

', T=550K

I I

0 1'0 20 o~

It is clear that

zx +Zo = 1. (72)

There are eight Ag ions and eight tetrahedral sites in a unit cell. Therefore we can take the occupation probability PA to be equal to zx. When we take account of being four octahedral sites in a unit cell, we have

8 P k + 4 e a = 8 . (73)

Then we have

PB =2Zo. (74)

This is reasonable to consider that the number of Ag ions in a unit cell is twice the number of octahedral sites. Kobayashi et al. [ 119 ] and Tachibana et al. [ 120 ] have calculated Haven's ratio by a MD method and have had 0.5 as its value. Their result that Ha- ven's ratio of Ag ions in et-Ag2Te has not been dependent upon the temperature has been in good agreement with the experimental result [ 39,40]. Ha- ven's ratio is defined by

f n = e a v / J . (75)

By making use ofeqs. (68) and (70) - (74) , the values of parameters o~ and fl are estimated so that we may obtain the value of Haven's ratio by MD calculations, fn = 0.5. In fig. 34 the a - f l map forfH = 0.5 is shown in two temperatures. VAn(l) and VAn(2) are the jump frequencies from a site A to a vacant and an occupied neighbouring site B, respectively. Then the former will have a characteristic like a sin-

Fig. 34. a-fl map offr~ = 0.5 for two temperatures, 550 K and 850 K (from ref. [128]).

10

" 5

i i i i

+

+

+

+

I I i I

550 650 750 85O T(K)

Fig. 35. The temperature dependence of a ( = f l ) (from ref. [128]).

gle hopping and the latter has it like a cooperative jump. To realize fH=0.5, some cooperative jumps are required. For the case of small value of VAn (2), a large value of g a k ( 2 ) has to guarantee a cooperative jump. Fig. 34 shows that the smaller one of the two parameters, ot and fl, is dominant to get the value Offn=0.5 for the case of small value of VAn(2) or rag (2). There will be nothing wrong with taking a = fl to get the f-value of Haven's ratio. Fig. 35 shows the temperature dependence of ot for the case where the equality of or= fl is assumed. The value of ot increases with the temperature increase. The potential barrier will decrease relatively in comparison with


the kinetic energy of a mobile ion with the temperature increase and then the jumping process of Ag ions will be frequent. When the temperature increases, the randomness of a system enlarges and then the frequency of a single jump increases relatively, compared with a cooperative jump. This shows the temperature dependence of a in fig. 35.

4. Concluding remarks

Structural and dynamical properties of Ag-chalcogenides have been reviewd. More recent development of these materials have been produced by structural analysis through the modern techniques of neutron scattering, EXAFS, and X-ray scattering. The structure factors, the electronic relaxation time and the conductivity of silver chalcogenides have been investigated based upon a continuum model. The relaxation time of electrons which has been obtained by making use of the expression in the classical limit is in good agreement with the experimental results. The structure factors have had simple relations themselves in the long wavelength limit: the structure factors in which electrons participate, SeE, SAc and SBe, have been expressed symmetrically in terms of the structure factors of ions, SAg, SBa and SAB-

The collective motions in silver chalcogenides in a hydrodynamic description have been investigated. The frequency- and wave vector-dependent conductivity and the dispersion relations of collective modes have been presented. The conductivity satisfies the f-sum rule. The ionic part of the conductivity has a broad peak at the optical-phonon frequency, which is the similar frequency dependence to that of ct-AgI. Particularly we have investigated in detail the plasmon-LO phonon coupling in superionic conductors. It has been shown that the optical phonons are too much damped to distinguish LO-phonons and TO- phonons definitely. It may be difficult to occur the crossing of plasmons and LO phonons by controlling the value of a nonstoichiometricity of the materials, since silver chalcogenides have a high intrinsic carrier concentration, about 1019~20 c m -3.

To understand the various physical and chemical properties of SIC, we will make a great point of the study based upon the microscopic view-point. The microscopic data of neutron scattering, EXAFS, X- ray diffraction and NMR will perform an important

role to do it. Molecular dynamics techniques present a detail information on the structural and dynamical properties which include the structural phase transition.

Recently the atomic structure of the Si (001 ) sur- face has been examined with use of scanning tun- neling microscopy (STM) [ 130 ]. STM has made it possible to directly probe the atomic structures of surfaces with high lateral resolution. The atomic structure of the Ag-chalcogenide also may be examined possibly by making use of STM.

Nuclear magnetic resonance (NMR) can be a powerful method for studying motions of ions and therefore has had extensive application in the field of SIC [131]. NMR gives us both static and dynamic information of SIC. The relaxation process of the latter is due to the fluctuating field around an ob- servation nucleus, associated with the diffusion of ions.

The recent advance of molecular-beam epitaxy has made possible the growth of ultrathin layers controlled within the configuration of atomic size [ 132]. Today we can study various new materials with superlattice structure which are composed of alternating layers of different materials. In particular, semiconductor superlattices have been extensively studied. I f the electron density in a type I semiconductor superlattice (such as GaAs-A1GaAs) is not too high so that only the lowest subband is occupied by electrons, we can regard such a superlattice as a periodic system of two-dimensional electron gas. Thus as far as electronic properties are concerned, we can treat the type I semiconductor superlattice as a layered electron gas (LEG). The collective excitations in LEG have attracted much interest [ 133- 138 ]. The predicted dispersion relation of plasmon in LEG is quite different from that of those in a two- dimensional (2-D) or a three-dimensional (3-D) electron gas. The 2-D plasma frequency in a single charge layer is known to have a dispersion relation proportional to square root of wavenumber in the long wavelength limit [ 139]. On the other hand the 3-D plasma frequency has the constant value independent of wave number in the long wavelength limit [ 140 ]. The plasma mode of the LEG system changes its dispersion relation from the 2-D plasmon-like to the 3-D plasmon-like with decreasing the distance between the layers, and shows an intermediate char- acter between those in 2-D and 3-D electron gas sys-


tern depending on the layered structure [ 141 ]. These properties shown in the LEG system will be characteristic to a layered system of charged particles.

We will be able to make a periodic structure of a superlattice composed o f 2-D superionic conductors (SIC) planes and ionic crystals. We call it the layered SIC (LSIC). Because o f the layered structure in the LSIC system, the dispersion relation of the plasma modes will change from the 3-D-like to the 2-D-like with the decreasing distance between the layers as described above. Hence the collective modes of LSIC are expected to be different f rom that of the 3-D SIC system. Tomoyose et al. [ 142 ] have considered the infinite system of LSIC as shown in fig. 36 .2-D SIC layers are arranged in parallel to xy plane at regular intervals d. The space between the layers is filled with the ionic crystal of dielectric constant ~o. They obtain a sum rule for longitudinal ionic conductivity aL in the LSIC system as follows:

i ReaL(k ,k~ , to ) dto

= (Eo/4Zr)to2(k, k z ) { ( k 2 + k 2 z ) / k 2 } . (76)

Here top(k, kz) is given by

top(k, kz) =~22p( k )F( k, kz) '/2 ,

with

I22p (k) = (2nNoq2k /m*Eo) , /2 .

k and kz are a 2-D wave vector in the layered plane and a z component of wave number perpendicular to the layered plane, respectively. Q2p(k) denotes the 2-D ionic plasma frequency which is proportional to k '/2. F ( k , kz) is the structure factor which charac- terizes the layered system and is defined by

F ( k, kz ) = sinh ( kd ) / ( cosh ( kd ) - cos ( kzd ) } .

Recently the fast ion conductor superlattice has been studied with a consideration of the finite thickness by Aniya and Kobayashi [ 143 ]. The recent advance o f molecular beam epitaxy has made possible the growth of ultra-thin layers controlled within the configuration of atomic size. The study of materials with ultra-thin layers of SIC will lead to the elucidation o f conduction mechanism in layered materials and may be associated with the development of the new, superior materials.

Acknowledgement

Z

e0 I d

Eo !,9 SIC ~ , ~ - " ' " ' " " , ~ - ' , ' , i = 0

Co I d SIC " ' ' " " ' " / " / / / < ' i =-1

Fig. 36. The infinite system of model layered SIC. Two-dimensional SIC layers are arranged in parallel to xy plane with regular intervals d. The space between the layers is filled with the ionic crystal of dielectric constant Co.

The author is grateful to Dr. P. Vashishta for his kind interest. The author would like to thank Pro- fessors I. Yokota, H. Okazaki and S. Tamaki for their stimulating discussions, valuable advice and kind interest. The author also acknowledges fruitful discussions with our colleague Professor F. Tachibana. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture.

References

[ 1 ] W. van Gool, ed., in: Fast ion transport in solids (North- Holland, Amsterdam, 1973 ).

[2] A.S. Nowick and J.J. Burton, eds., in: Diffusion in solids (Academic, New York, 1975 ).

[3]G.D. Mahan and W.L. Roth, eds., in: Superionic conductors (Plenum, New York, 1976).

[4IS. Geller, ed., in: Topics in applied physics, Vol. 21 (Springer, Berlin, 1977).

[ 5 ] P. Hagenmuller and W. van Gool, eds., in: Solid electrolytes (Academic, New York, 1978 ).


[6] M.B. Salamon, ed., in: Physics of superionic conductors (Springer, Berlin, 1979 ).

[ 7 ] J.B. Boyce and B.A. Huberman, Phys. Rep. 51 ( 1979 ) 189. [ 8 ] W. Dieterich, P. Fulde and I. Peschel, Adv. Phys. 29 (1980)

527. [ 9 ] S. Chandra, Superionic solids (North-Holland,

Amsterdam, 1981 ). [ 10] H. Bottger and V.V. Bryksin, Hopping conduction in solids

(Academic, Berlin, 1985). [ 11 ] T. Kudo and K. Fueki, Kotai Ionics (Kodansha, Tokyo,

1986) [in Japanese]. [ 12 ] Yu.Ya. Gurevich and A.K. Ivanov-Shits, in:

Semiconductors and semimetals, eds. R.K. Willardson and A.C. Beer, Vol. 26 (Academic, San Diego, 1988).

[ 13 ] T. Takahashi, High conductivity solid ionic conductors: Recent trends and applications (World Scientific, Singapore, 1989).

[ 14] W. Nerunst, Z. Electrochem. 6 (1899) 41. [ 15 ] C. Tubandt, Z. Anorg. Allg. Chem. 115 ( 1921 ) 105. [ 16] C. Tubandt and H. Reinhold, Z. Electrochem. 29 (1923)

313. [ 17 ] A. Joffe, Ann. Phys. (Leibzig) 72 ( 1923 ) 461. [ 18] J. Frenkel, Z. Phys. 35 (1926) 652. [ 19] C. Wagner and W. Schottky, Z. Phys. Chem. B 11 (1930)

163. [ 20 ] C. Wagner, Naturwissenschaften 31 ( 1943 ) 265. [21 ] P. Rahlfs, Z. Phys. Chem. Abt. B 31 (1936) 157. [22] L.W. Strock, Z. Phys. Chem. Abt. B 25 (1934) 411; 31

(1936) 132. [23] S. Hoshino, J. Phys. Soc. Japan. 12 (1957) 315. [ 24 ] S. Hoshino, T. Sakuma and Y. Fujii, Solid State Commun.

22 (1977) 763. [25] J.B. Boyce, T.M. Hayes, W. Studius and J.C. Mikkelsen,

Phys. Rev. Letters 38 (1977) 1362. [26] R.J. Cava, F. Reidinger and B.J. Wuensch, Solid State

Commun. 24 (1977) 411. [27] T. Sakuma, K. Iida, K. Honma and H. Okazaki, J. Phys.

Soc. Japan 43 (1977) 538. [28] Y. Tsuchiya, S. Tamaki, Y. Waseda and J.M. Toguri, J.

Phys. C 11 (1978) 651. [29] R.J. Cava, F. Reidinger and B.J. Wuensch, J. Solid State

Chem. 31 (1980) 69. [30] S. Hoshino, T. Sakuma, H. Fujishita and K. Shibata, J.

Phys. Soc. Japan. 52 (1983) 1261. [31 ] S. Miyatani, J. Phys. Soc. Japan. 10 (1955) 786. [32] S. Miyatani, J. Phys. Soc. Japan. 13 (1958) 341. [33] S. Miyatani and I. Yokota, J. Phys. Soc. Japan. 14 (1959)

750. [ 34 ] S. Miyatani, Y. Toyoda, T. Yanagihara and K. Iida, J. Phys.

Soc. Japan. 23 (1967) 35. [35] S. Miyatani, J. Phys. Soc. Japan. 24 (1968) 328. [ 36 ] S. Miyatani, Solid State Commun. 38 ( 1981 ) 257. [ 37 ] S. Miyatani and A. Tabuchi, J. Phys. Soc. Japan. 50 ( 1981 )

2050. [ 38 ] S. Miyatani, J. Phys. Soc. Japan. 50 ( 1981 ) 3415. [39 ] H. Okazaki, J. Phys. Soc. Japan. 23 (1967 ) 355. [40] H. Okazaki, J. Phys. Soc. Japan. 43 (1977) 213. [41 ] I. Yokota, J. Phys. Soc. Japan. 16 ( 1961 ) 2213.

[42] I. Yokota and S. Miyatani, Japan. J. Appl. Phys. 1 (1962) 144.

[ 43 ] I. Yokota, J. Phys. Soc. Japan. 21 (1966) 420. [44] I. Yokota and S. Miyatani, Solid State Ionics 3 /4 ( 1981 )

17. [45 ] C. Wagner, Progr. Solid State Chem. 10 (1975) 3. [ 46 ] T. Ohachi, Thesis (Doshisha University, 1974 ). [47] G.J. Dudley and B.C.H. Steele, J. Solid State Chem. 31

(1980) 233. [48]S.R. de Groot and P. Mazur, Non-equilibrium

thermodynamics (North-Holland, Amsterdam, 1969). [49] M. Mekata and Y. Mizoh, Hyperfine Interact. 10 ( 1981 )

995. [ 50 ] Y. Segnchi and M. Mekata, J. Phys. Soc. Japan. 54 ( 1985 )

3447. [ 51 ] Y. Seguchi and M. Mekata, J. Phys. Soc. Japan. 54 ( 1985 )

3454. [52] C.M. Perrott, N.H. Fletcher, Phys. Rev. 50 (1969) 2344. [53] H. Sato and R. Kikuchi, J. Chem. Phys. 55 ( 1971 ) 677. [ 54 ] T. Ishii, H. Sato and R. Kikuchi, Phys. Rev. B 34 (1986)

8335. [55] M.J. Rice and W.L. Roth, J. Solid State Chem. 4 (1972)

294. [ 56 ] B.A. Huberman and P.N. Sen, Phys. Rev. Letters 33 (1974)

1379. [59] R. Kubo, J. Phys. Soc. Japan. 12 (1957) 570. [58] P. Bruesch, L. Pietronero, S. Strassler and H.R. Zeller,

Electrochim. Acta 22 (1977) 717. [59] W. Dieterich, I. Peschel and W.R. Schneider, Z. Physik B

27 (1977) 177. [60] H. Mori, Prog. Theor. Phys. 34 (1965) 399. [61 ] T. Geisel, Phys. Rev. B 20 (1979) 4294. [62] T. Munakata, Phys. Rev. B 33 (1986) 8016. [63] K.R. Subbaswamy, Solid State Commun. 19 (1976) 1157. [64] B. Huberman and R.M. Martin, Phys. Rev. B 13 (1976)

1498. [65] J. J~ickle, Z. Phys. B 30 (1978) 255. [66] R. Zeyher, Z. Phys. B 31 (1978) 127. [67] W.J. Pardee and G.D. Mahan, J. Solid State Chem. 15

(1975) 310. [ 68 ] M.J. Rice, S. Str~issler and G.A. Toombs, Phys. Rev. Letters

32 (1974) 596. [69] L. Lain and A. Bunde, Z. Physik. B 30 (1978) 65. [ 70 ] H. Hayashi, J. Phys. Soc. Japan. 51 ( 1982 ) 811. [71 ] G.D. Mahan, Phys. Rev. B 14 (1976) 780. [ 72 ] H. Hinkelmann and B.A. Huberman, Solid State Commun.

19 (1976) 365. [73] L. Pietronero and S. Strassler, Z. Physik B 36 (1980) 263. [74] T. Ishii, Solid State Commun. 33 (1980) 249. [ 75 ] T.M. Hayes and J.B. Boyce, Phys. Rev. B 21 (1980) 2513. [76] G.J. Dudley and B.C.H. Steele, J. Solid State Chem. 31

(1980) 233. [77] C. Wagner, Progr. Solid State Chem. 10 (1975) 3. [ 78 ] W. Weppner, Solid State Ionics 3/4 ( 1981 ) 1. [79] E. Haga, J. Phys. Soc. Japan. 15 (1960) 1949. [80] S.M. Girvin, J. Solid State Chem. 25 (1978) 65. [ 81 ] K. Yonashiro, T. Tomoyose and E. Sakai, J. Phys. Soc.

Japan. 52 (1983) 3837.


[82] K. Yonashiro, T. Tomoyose, E. Sakai, M. Yamashiro and M. Kobayashi, Solid State Ionics 27 (1988) 157.

[83] T. Tanaka, M.A. Sawatarie, J.H. Barry, N.L. Sharma and C.H. Munera, Phys. Rev. B 34 (1986) 3773.

[ 84 ] N.L. Sharma and T. Tanaka, Phys. Rev. B 28 ( 1983 ) 2146. [85] S. Ramasesha, J. Solid State Chem. 41 (1982) 333. [ 86 ] A. Hasegawa, Solid State lonics 15 ( 1985 ) 81. [87] T. Tomoyose, J. Phys. Soc. Japan. 54 ( 1985 ) 4232. [88] A. Rahman and P. Vashishta, The physics of superionic

conductors and electrode materials, ed. J.W. Perran (Plenum, New York, 1983 ) p. 93.

[ 89 ] H.E. Roman and W. Dieterich, J. Phys. C 19 ( 1986 ) L801. [90] P. Vashishta, I. Ebbsjr, R. Dejus and K. Skrld, J. Phys. C

18 (1985) L291. [91 ] M. Kobayashi, F. Sukegawa and I. Yokota, Solid State

lonics 2 (1981) 237. [92]H. Hayashi, M. Kobayashi and I. Yokota, Solid State

Commun. 31 (1979) 847. [93] N.H. March and M.P. Tosi, Atomic dynamics in liquids

(Wiley, New York, 1976). [94]N.H. March, M.P. Tosi and A.B. Bhatia, J. Phys. C 6

(1973) L59. [ 95 ] G. Baym, Phys. Rev. 135 (1964) A 1691. [96] G.D. Mahan, J. Phys. Chem. Solids 31 (1970) 1477. [97] N.W. Ashcroft, Phys. Letters 23 (1966) 48; J. Phys. C 1

(1968) 232. [98] A.B. Bhatia and D.E. Thornton, Phys. Rev. B 2 (1970)

3004. [99 ] M.P. Tosi, M. Parrinello and N.H. March, Nuovo Cimento

23B (1974) 135. [ 100] F. Sukegawa, M. Kobayashi and I. Yokota, Phys. Status

Solidi b 109 (1982) 717. [ 101 ] N. Mizoguchi, H. Hayashi, M. Kobayashi and I. Yokota,

J. Phys. Soc. Japan. 50 (1981) 2043. [ 102] S. Ichimaru, Basic principles of plasma physics (Benjamin,

Reading, 1973). [ 103 ] N.D. Mermin, Phys. Rev. B 1 (1970) 2362. [ 104] M.P. Greene, H.J. Lee, J.J. Quinn and S. Rodriguez, Phys.

Rev. 177 (1969) 1019. [ 105 ] For example, D. Pines, Elementary excitations in solids

(Benjamin, New York, 1964). .106 ] K. Murase, S. Katayama, Y. Ando and H. Kawamura, Phys.

Rev. Letters 33 (1974) 1481. • 107] A. Pinczuk, G. Abstreiter, R. Trommer and M. Cardona,

Solid State Commun. 21 (1977) 959. [108]S. lhara and K. Suzuki, J. Phys. Soc. Japan. 53 (1984)

3081. [ 109] R.J. Cava and D.B. McWhan, Phys. Rev. Letters 45 (1980)

2046. [ 110] B.H. Grier, S.M. Shapiro and R.J. Cava, Phys. Rev. B 29

(1984) 3810. [ 111 ] S.M. Shapiro, D. Semingsen and M.B. Salamon, Proc. Int.

Conf. Lattice Dynamics (Paris, 1977) p. 538. [ 112 ] W. Buhner, R.M. Nicklow and P. Bruesch, Phys. Rev. B

17 (1978) 3362. [ 113 ] S.M. Shapiro and M.B. Salamon, in: Fast ion transport in

solids, eds. P. Vashishta, J.N. Mundy and G.K. Shenoy (North-Holland, Amsterdam, 1979) p. 237.

[ 114] S. Hoshino, H. Fujishita, M. Takashige and T. Sakuma, Solid State Ionics 3/4 ( 1981 ) 35.

[ 115 ] I. Ebbsjr, P. Vashishta, R. Dejus and K. Skrld, J. Phys. C 20 (1987) L441.

[ 116] J.P. Rino, Y.M.M. Hornos, G.A. Antonio, I. Ebbsjr, R.K. Kalia and P. Vashishta, J. Chem. Phys. 89 (1988) 7542.

[ 117 ] P. Vashishta, I. EbbsjS, R.K. Kalia and S.W. de Leeuw, Solid State Commun. 59 (1986) 873.

[ 118 ] B.B. Mandelbrot, The fractal geometry of nature (Freeman, San Francisco, 1982).

[ 119 ] M. Kobayashi, K. Ishikawa, F. Tachibana and H. Okazaki, Phys. Rev. B 38 (1988) 3050.

[ 120] F. Tachibana, M. Kobayashi and H. Okazaki, Phys. Rev. B 40 (1989) 3360.

[ 121 ] H. Okazaki, private communication. [ 122] T. Sakuma and S. Saitoh, J. Phys. Soc. Japan. 54 ( 1985 )

3647. [ 123 ] K. Koto, H. Schulz and R.A. Huggins, Solid State Ionics 1

(1980) 355. [ 124] M.H. Dickens, M.T. Hutchings and C. Smith, J. Phys. C

15 (1982) 4043. [125]A.B. Walker, M. Dixon and M.J. Gillan, J. Phys. C 15

(1982) 4061. [ 126] M.J. Gillan, Physica 131 B (1985) 157. [ 127] Y. Kaneko and A. Ueda, J. Phys. Soc. Japan. 57 (1988)

3064. [ 128] M. Kobayashi, T. Tomari, F. Tachibana and H. Okazaki,

Phys. Rev. B 40 (1989) 9552. [129] T. Tomari, M. Kobayashi and H. Okazaki, Solid State

lonics 40/41 (1990) xxx. [ 130] R.J. Hamers, R.M. Tromp and J.E. Demuth, Phys. Rev. B

34 (1986) 5343. [ 131 ] P.M. Richards, in: Physics of superionic conductors, ed.

M.B. Salamon (Springer, Berlin, 1979) p. 141. [ 132 ] L.L. Chang and K. Ploog, eds., in: Molecular beam expitaxy

and hetrostructures (Martinus Nijhoff, Boston, 1985). [133] P.B. Visscher and L.M. Falicov, Phys. Rev. B 3 (1971)

2541. [134] A.L. Fetter, Ann. Phys. (N.Y.) 88 (1974) 1. [135] M. Kobayashi, J. Mizuno and I. Yokota, J. Phys. Soc.

Japan. 39 (1975) 18. [ 136 ] M. Apostol, Z. Phys. B 22 ( 1975 ) 13. [ 137] A. Caille, M. Banville and M.J. Zuckermann, Solid State

Commun. 24 (1977) 805. [ 138 ] G.F. Giuliani and J.J. Quinn, Phys. Rev. Letters 51 ( 1983 )

919. [139] F. Stern, Phys. Rev. Letters 18 (1967) 546. [ 140 ] C. Kittel, Introduction to solid state physics (Wiley, New

York, 1986) chap. 10. [ 141 ] D. Olego, A. Pinczuk, C. Gossard and W. Wiegmann, Phys.

Rev. B 25 (1982) 7867. [ 142 ] T. Tomoyose, K. Yonashiro and M. Kobayashi, Solid State

lonics 40/41 (1990), to be published. [ 143 ] M. Aniya and M. Kobayashi, Appl. Phys. A 49 (1989) 641;

Solid State Ionics 40/41 (1990), to be published.

Documents

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