Cross conductivity in ionic-electronic mixed conductors

Solid State Ionics 111 (1998) 53–58

Cross conductivity in ionic-electronic mixed conductors a, b H. Ogawa *, M. Kobayashi a

Graduate School of Science and Technology, Niigata University, Niigata 950 -2181, Japan b Department of Physics, Niigata University, Niigata 950 -2181, Japan Received 20 November 1997; received in revised form 22 March 1998

Abstract On account of the electron-ion scattering force, the electron-ion cross conductivity of silver chalcogenides is investigated theoretically and the results agree with experiments. The equations of motion are prepared so that the Onsager reciprocal relation may be satisfied. The analytical calculation confirms that the total conductivity satisfies the f-sum rule.  1998 Elsevier Science B.V. All rights reserved. Keywords: Cross conductivity; Mixed conductor; Superionic conductor; Silver chalcogenide

1. Introduction Some kinds of superionic conductors show both mobile ions and conduction electrons, which contribute to the total electrical current. According to the general principle of irreversible thermodynamics, the electronic and ionic current densities, j e and j i can be described as [1,2]

se ≠ m¯ e sei ≠ m¯ i j e 5 ] ]] 1 ] ], e ≠x ze ≠x sie ≠ m¯ e si ≠ m¯ i j i 5 2 ] ]] 2 ] ]. ze ≠x e ≠x

(1)

Here, m¯ k (k5e,i) are the electrochemical potentials of electrons and ions, z is the valence number of mobile ions, s k (k5e,i) are the electronic and ionic conductivities, and s ei and s ie are the cross conductivities. The principle of the symmetry of the kinetic *Corresponding author.

coefficients or Onsager reciprocal relation tells us that these two cross conductivities should be equal to each other, s ei 5s ie . Miyatani [3] measured these cross conductivities in a-Ag 2 Se and b-Cu 2 S and found that the Onsager relation is really satisfied over the wide range of a non-stoichiometry and s ei is about 10 22 smaller compared with s i . We calculated to interpret the ratio of each conductivity by including only the electronion Coulomb interaction. We could derive the electronic conductivity and the ionic conductivity in agreement with experiments. However the cross conductivity s ie was in the same order as the ionic conductivity in disagreement with experiments. ¨ Hansch and Mahan [4] investigated electrical currents in semiconductor devices, which are often carried simultaneously by both electrons and holes. They also started from Eq. (1). They treated the electron (n) and hole (p) currents as independent, except for the recombination process. They indicated a possibility that the electrical current of one species

0167-2738 / 98 / $ – see front matter  1998 Elsevier Science B.V. All rights reserved. PII: S0167-2738( 98 )00157-X

54

H. Ogawa, M. Kobayashi / Solid State Ionics 111 (1998) 53 – 58

may exert a dragging effect upon the other, through their mutual Coulomb interaction. Their results were that the cross conductivity s np was very small compared to the majority carrier conductivity s nn and was in the same order as the minority carrier conductivity s pp for low doping. Apart from carriers, the circumstance is very similar to the above electron-ion cross conductivity. When we calculate the cross conductivity, it may be important to consider the scattering force of the electron wind on a mobile ion and vice versa. Sinha [5,6] offered a simple explanation for the observed reversal in the direction of electromigration in the Na-K system which at the same time provides a quantitative theory of electrotransport. He gave the force of the electron wind on a migrating atom as

≠ 2 u(r,t) ≠ ]]] 2 V L2 1 G ] = 2 u(r,t) 2 ≠t ≠t

F 5 "k F Sj e /e,

Here mA and mB are the anion and cation masses, respectively, ZA is the valence of an anion, e is the elementary charge, G is the phenomenological damping coefficient for the motion of the lattice anions and t 21 is that for the relative motion of the two ionic components. VL is the longitudinal sound velocity in the lattice. y(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 oscillatory frequency v 0 and the viscoelastic relaxation time t c . fA (r,t) is the weak space- and time-dependent external potential acting on the A component, i.e. an anion. d denotes the deviation from the stoichiometry. The dimensionless number density fluctuation of lattice ions nA (r,t) is related to the lattice displacement field u(r,t) through the equation

(2)

where " k F is the Fermi momentum, j e the electronic current density, and S the electron-atom scattering cross-section. We will discuss this scattering force in Section 4. In this paper we investigate the transport properties of mixed conductors, in particular the electron-ion cross conduction [7]. Then we calculate the density response functions in silver chalcogenides. Our calculation is based upon a model of a crystalline cage immersed in a viscous liquid [8,9]. We introduce the force of the electron wind on a mobile ion and vice versa. The form of the force is made ready so that the Onsager relation is satisfied. We neglect the force of electron wind on a lattice ion, because the contribution to the mobile ionelectron cross conductivity is negligible. We expect these forces play an important role to get a proper expression of cross conductivity. In Section 2 our model is introduced and Section 3 is devoted to calculate the longitudinal conductivity of the system.

S

m* 1 ]] t mA

F

D

≠u(r,t) ]] 2 y (r,t) ≠t

F

G

t

1 1] mA

E dt9M(t 2 t9) 2`

G

≠u(r,t9) 3 ]] 2 y (r,t9) ≠t9 ZA e 1 1 ] E(r,t) 1 ] =fA (r,t) 5 0 mA mA

(3)

with (m * )21 5 m A21 1 (2 1 d )21 m 21 B and M(t) 5 m * v 20 exp(2t /tc ).

nA (r,t) 1 =u(r,t) 5 0

(4)

The dimensionless number density fluctuations n i (r,t) (i5 A,B,e) are defined by

2. Model

n i (r,t) 5 hNi (r,t) 2 N0i j /N0i ,

We suppose that the lattice composed of anions is immersed in the cation liquid [8] and use the same model as our former one [9]. The equations of motion for the longitudinal lattice displacement field u(r,t) can be written as

where Ni (r,t) are the number densities and N0i their mean values (N0 A 5N0 , N0B 5(21d)N0 , N0e 5dN0 ). The motion of mobile ions can be described by the velocity field y(r,t) and the dimensionless number density fluctuation nB (r,t). The linearized hydrodynamic equations for these fields are

H. Ogawa, M. Kobayashi / Solid State Ionics 111 (1998) 53 – 58

≠nB (r,t) ]]] 1 =? y (r,t) 5 0 ≠t

(5)

4p e =? E(r,t) 5 ]] h 2 ZA NA (r,t) 1 ZB NB (r,t) 2 Ne (r,t) j e` (9)

which is the continuity equation, and ≠y (r,t) ]] 1 C 20 =nB (r,t) 2 a= 2 y (r,t) ≠t m* ≠u(r,t) 1 ]]]] y (r,t) 2 ]] ≠t t (2 1 d )mB

F

3. Conductivity

G

The system is assumed to be under the weak, space- and time-dependent external potentials f j (r,t) acting on the three components ( j5 A, B, e). We define the longitudinal density response function of the system by

t

E dt9M(t 2 t9) ≠u(r,t9) j 3 F y (r,t9) 2 ]] G 2 ] y (r,t) ≠t9 m 1 1 ]]] (2 1 d )mB

2`

e

n i (k,v ) 5 xij (k,v )fj (k,v )

B

ZB e 1 2 ] E(r,t) 1 ] =fB (r,t) 5 0, mB mB

(6)

which is the Navier–Stokes equation. C0 and a denote the sound velocity and the viscosity in the cation liquid, respectively. The term which includes jye (r,t) is the force of the electron wind on a mobile ion. The adjustable parameter j is determined by experimental values of cross conductivity. On the other hand, the equation of motion for electrons is described by the hydrodynamic equation of motion and the equation of continuity, which are simplified in the linear approximation as

S

D

≠ 1 h e ] 1 ] ye (r,t) 2 ] y (r,t) 1 ] E(r,t) ≠t te me me 1 1 ] =fe (r,t) 5 0 me

(7)

(8)

Here ye (r,t) is the velocity field of electrons, t e the relaxation time and f e (r,t) the weak space- and time-dependent external potential acting on electrons. The term hye (r,t) is the force of the mobile ion stream on an electron. Both parameters j and h are not independent and are made ready so that the Onsager reciprocal relation is satisfied in the next section. The induced electric field E(r,t) obeys the Poisson equation

(10)

in the form of Fourier transforms. The mutual relation between j and h is determined so as to satisfy the Onsager reciprocal relation

xij (k,v ) 5 x ji (k,v ) (i ± j).

(11)

Then we get

d j 5 ]] h. 21d

(12)

We define the charge density-charge density response functions by taking a linear combination of the density response functions defined in Eq. (10) as follows [10] 1 x QQ 5 e 2 [Z 2A x A A 1 Z 2B xBB 2 2ZA ZB x AB ],

x 2QQ 5 e 2 xee ,

and ≠n e (r,t) ]] 1 =? ye (r,t) 5 0. ≠t

55

(13)

3 x QQ 5 2e 2 hZA x Ae 2 ZB xBe j.

The longitudinal conductivity of a system is defined as the response of all the charges to a screened electric potential by ivxQQ s (k,w) 5 Re ]]]]. 4p k 2 1 ] xQQ E`

(14)

It is easy to prove that the conductivity given by Eq. (14) satisfies the following f-sum rule by the analytical calculation:

H. Ogawa, M. Kobayashi / Solid State Ionics 111 (1998) 53 – 58

56 `

Z Ne (2 1 d )Z N e s (k,v ) E dv ]] 5 H]] 1 ]]]] p m m 2 A

2

2 B

0

A

2`

2

dh 2 m *te G 5 1 2 ]]]]]]]. 1 (2 1 d )mB m e ] 1 v 20tc t

0

S

B

2

J

d N0 e 1 ]] . me

(15)

D

When we put the usual values into the parameters, G is almost unity.

When we put 1 3 1 2 Fi 5 x QQ , Fe 5 x QQ , Fie 5 Fei 5 ] x QQ , 2

4. Discussion (16)

the ionic conductivity, electronic conductivity, and cross conductivity are given by iv Fi si (k,w) 5 Re ]]], 4p k2 1 ] E` iv Fe se (k,w) 5 Re ]]], 4p k2 1 ] E`

(17)

iv Fie sie (k,w) 5 Re ]]]. 4p k2 1 ] E` The denominator and numerator of s(k,v) are expressed by the power series of v. The lowest 2 power term is v . But this term does not give us a physically insignificant result. Then we disregard a 2 term of v and pick up the next important term of 3 v . Taking the limit of k→0, v→0

s (k,v ) 5 si (k,w) 1 se (k,w) 1 2sie (k,w), si (k,w) 5

3S

D

Z 2A N0 e 2 (2 1 d )Z 2B N0 e 2 1 ]] 1 ]]]] ]]] mA mB 1 ] 1 v 20tc t

d 2 m * N0 e 2te 1 ]]]] mA mB (2 1 d )

4

1 ], G

d N0 e 2 1 ]] se (k,w) 5 t ], me e G d ZB Chm *te 1 sie (k,w) 5 ]]]]] ]. 1 G mB m e ] 1 v 20tc t

S

Here G is given by

D

¨ Buhrer et al. [11] determined the dispersion relation for the normal modes of vibration in hexagonal b-AgI at 160 K along three principal symmetry directions from coherent neutron inelastic scattering data using a triple-axis spectrometer. They used 0.56 for the valence number of the valence-shell model. Vashishta and Rahman [12] used 0.6 for that value for their computer simulation of superionic conductor a-AgI. With changing the values of valence numbers, the value of each conductivity is shown in Table 2. In Table 2, ZB 50.5 is tried and is in good agreement with experiments. To preserve the charge neutrality, the correspondent valence number of anion ZA 51.0 is taken. The values of parameters used are as follows: N0 51.6310 22 cm 23 (d510 23 ), mA 51.3207310 222 g, mB 51.79310 222 g, m e 5 1.822310 228 g [13], e ` 515 [13], t e 21 5744.6 cm 21 [13], t 21 545 cm 21 [14], t c 21 553 cm 21 [14], v 0 5 105 cm 21 [14], VA 512.9 ZA 2 cm 21 , V B 531.2 ZB 2 cm 21 , v p 5547 cm 21 , j 52.08310 214 dyne s / cm. Table 1 and Table 2 show that on the whole each value of electronic, ionic, and cross conductivities is in good agreement with experiments. In addition to the normal case of ZA 52, ZB 51, we also have studied the case of ZA 51.0, ZB 50.5. As mentioned above, the latter case is due to the fact that analysis of phonon dispersion data of AgI indicated Z.0.6 to be an appropriate value. The latter case may connect with the time fluctuating local change of the type of

Table 1 Experimental values of the conductivity for a-Ag 2 Se in units of (V cm)21 [2]

(18)

d

se 23

1.0310 2.0310 23 3.0310 23

3

1.6310 2.7310 3 3.6310 3

si

s ie

2.991 2.997 3.001

0.9310 22 1.4310 22 1.8310 22

H. Ogawa, M. Kobayashi / Solid State Ionics 111 (1998) 53 – 58

57

Table 2 Calculated values of the conductivity for a-Ag 2 Se in units of (V cm)21 d

23

1.0310 2.0310 23 3.0310 23

se

si

ZB 51 and 0.5

ZB 51

3

0.63310 1.26310 3 1.89310 3

s ie

22.0 22.0 22.0

5.5 5.5 5.5

bonding, ionic↔covalent which was suggested by Aniya [15]. Namely, the change of bonding is induced locally and fluctuates in time. If this process happens in a superionic material, the polarization field around a mobile ion may be changed locally to reduce an effective charge on average. We have chosen the values of effective valence of ions so as to reproduce the diffusion constant and the ionic conductivity of Ag 2 Te [16]. Based upon the viewpoint of a continuum and hydrodynamic picture, we have modeled a superionic conductor and have calculated a conductivity. It is important that the short range scattering force between mobile ions and electrons is introduced in our formulation, in addition to the Coulomb interaction. By introducing the scattering force, we have succeeded in getting the three kinds of conductivities in good agreement with experiments. When we have introduced only the electron-ion Coulomb interaction, without including the short range scattering term, the obtained cross conductivity was in the same order as the ionic conductivity which was a few orders different from experiments. The behavior is very similar to an electron(n)-hole(p) cross conductivity s np in semiconductors. The cross conductivity s np was very small compared to the majority carrier conductivity s nn and was in the same order as the minority carrier conductivity s pp for low doping [4]. When we introduce the force of scattering in the equation of motion, parameters j and h included there have been made ready so that the Onsager reciprocal relation is satisfied. The force of the electron wind on a diffusing ion may be given by [5] F 5 "k F Sj e /e,

ZB 50.5

(19)

as was introduced in Section 1. Using this expression

ZB 51

ZB 50.5 22

0.83310 1.67310 22 2.5310 22

0.42310 22 0.84310 22 1.25310 22

as the force of scattering, our parameter j is expressed by

j 5 "k F p r¯ 2 Ne .

(20)

Here r¯ is the radius of scattering cross section S and Ne is the electron density. If we take Ne 51.6310 19 ˚ we get j 52.083 ¯ cm 23 (d510 23 ) and r522.5 A, 214 10 dyne s / cm, which is the same value as we have just used. Then, the candidate of the force of the electron wind on a mobile ion in Eq. (6) may be originated by Eq. (19). In the discussion of electromigration in the liquid alloy system, Sinha emphasized that the force of the electron wind is given by Eq. (19) and the drift velocity is related to the force through the Einstein Debye relation [5]. The electronic and ionic conductivities in mixed conductors have been investigated in Ag chalcogenides mainly. The experiments of the electronicionic cross conductivity in Ag chalcogenides were conducted by Miyatani [3] very carefully. Recently, Yoo [17] has been studying the interference effect to a non-negligible degree in a semiconducting transition metal oxide, Co 12d O (d.0.1), which is a semiconductor. This material shows a high nonstoichiometry. If we conduct a measurement of conductivity of Co 12d O, a high value of cross conductivity will be expected, because d.10 21 for Co 12d O is extremely larger than d.10 23 for Ag 2 Se. Yoo’s interference effect may be the same as introducing the force of the electron wind on a mobile ion. In conclusion, we have calculated the conductivity of silver chalcogenides, considering the force of the electron wind on a mobile ion and vice versa. The obtained electron-ion cross conductivity is in good agreement with experiments and satisfies the Onsager reciprocal relation. The total conductivity satisfies the f-sum rule by the contour integral.

58

H. Ogawa, M. Kobayashi / Solid State Ionics 111 (1998) 53 – 58

Acknowledgements The authors (M.K.) are supported by a Grant-inAid for Scientific Research [09874077 from the Ministry of Education and Culture of Japan. This work is also funded in part by the Japan–US Cooperative Science Program: Molecular Dynamics Simulations of Synthetic Functional Materials on Parallel Computers, supported jointly by the Japan Society for the Promotion of Science and the US National Science Foundation.

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[5] O.P. Sinha, Phys. Lett. 38A (1972) 193. [6] M. Shimoji, Liquid Metals; an Introduction to the Physics and Chemistry of Metals in the Liquid State, Academic, London, 1977. [7] H. Ogawa, M. Kobayashi, submitted. [8] K.R. Subbaswamy, Solid State Commun. 19 (1976) 1157. [9] M. Kobayashi, F. Sukegawa, I. Yokota, Solid State Ionics 2 (1981) 237. [10] M.P. Tosi, M. Parrinello, N.H. March, Nuovo Cimento 23B (1974) 135. ¨ ¨ [11] W. Buhrer, R.M. Nicklow, P. Bruesch, Phys. Rev. B 17 (1978) 3362. [12] P. Vashishta, A. Rahman, Phys. Rev. Lett. 40 (1978) 1337. [13] S. Miyatani, Y. Toyoda, T. Yanagihara, K. Iida, J. Phys. Soc. Jpn. 23 (1967) 35. ¨ [14] P. Bruesch, L. Pietronero, H.R. Zeller, J. Phys. C 9 (1976) 3977. [15] M. Aniya, Solid State Ionics 50 (1992) 125. [16] M. Kobayashi, K. Ishikawa, F. Tachibana, H. Okazaki, Phys. Rev. B 38 (1988) 3050. [17] K.-C. Lee, H.-I. Yoo, Solid State Ionics 86–88 (1996) 757.