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Dimensional crossover of collective modes in layered superionic conductors
Solid State Ionics 40/4 1 ( 1990) 2 13-2 15 North-Holland
DIMENSIONAL CROSSOVER OF COLLECTIVE LAYERED SUPERIONIC CONDUCTORS
MODES IN
T. TOMOYOSE, K. YONASHIRO Physics Department, General Education Division, Ryukyu University, Okinawa 903-01, Japan
and M. KOBAYASHI Department of Physics, Niigata University, Niigata 950-21, Japan
Using the continuum model, we have studied the dimensional crossover property of longitudinal collective modes in a layered superionic conductor (LSIC) which is composed of alternating layers of superionic conductor (SIC) and ionic crystal. The ionic plasma mode of the LSIC has changed from the three-dimensional to the two-dimensional plasma mode with increasing distance between SIC layers. As a result of this dimensional crossover of the plasma mode, the collective modes have shown various dispersion relations: the coupled optical phonon-plasma mode, the damped optical phonon mode, the relaxation mode, the diffusion mode and the acoustic phonon mode.
1. Introduction The artificial substances with a layered structure of charged particles have attracted much interest in recent years in connection with the dimensional crossover property of the system. We will be able to make a superlattice composed of alternating layers of superionic conductor (SIC) and ionic crystal. We call it the layered SIC (LSIC). In general, the plasma mode of the layered system changes its dispersion relation depending on the distance between layers as found in the layered electron gas [ l-41. Since the LSIC system is the layered system, the plasma mode of the system will change its dispersion relation from the two-dimensional (2D) like to the three-dimensional (3D) like with decreasing distance between layers. This dimensional crossover of plasma mode presumably affects the dynamical properties of the LSIC system. The details for the dynamical properties have been given elsewhere [ 5 1. In the present paper we study the dimensional crossover property of the longitudinal collective excitation in the LSIC system. We assume that the SIC layer is regarded as a 2D SIC plane. Our model struc0167-2738/90/S 03.50 0 Elsevier Science Publishers B.V. (North-Holland )
ture of the system consists of an infinite array of parallel SIC planes with a distance d apart. We take the z axis to be perpendicular to the SIC planes. The SIC planes are located at z=zi where Zi= id (i=O, f 1, + 2,... ). The ionic crystal between SIC planes is regarded as a background medium with the static dielectric constant e,,, which does not affect dynamical properties of the SIC planes. Each SIC plane consists of mobile ions and lattice ions. Following the continuum model [ 5-91, we assume that the crystalline cage composed of lattice ions is immersed in the viscous fluid of mobile ions. The dynamics of the lattice ions is described by the 2D lattice displacement field in the ith SIC plane, while the motion of mobile ions is treated as that of a viscous fluid of cations. The details for the equation of motion of ions are given in elsewhere [ 5,6].
2. Dielectric response function
Using the linear response theory, the longitudinal dielectric function is obtained as
T. Tomoyose et al. /Layered superionic conductors
214
k*Eintk,kz,o)
E~(k,kr,W) = l-
k.E(k,k,,w)
(1)
’
where Ei”(k,kz,w) is the induced electric field, E(k,k,,o) the total electric field, k the 2D wave vector in the SIC plane, k, the wave number perpendicular to the plane, and w is the frequency. We consider only the case of the non-retarded limit. The dielectric function is given by [ $61,
In the strong coupling limit of de k-‘, we can regard the LSIC system as a 3D-like SIC system. However, the plasma mode shows not only a 3D-like dispersion relation but also an intermediate one depending on the correlation between SIC planes, that is, m,(k>k,) sl, = (4nNoq2/m*E,,d)“‘,
for k,d=O,
= { L&,(k) = (xN,,q2d/m*eo)“2k,
for k,d=n,
~,(k,kw)
(6) w;(k,k,)
= l-
[wz- I’f(o)k211H(k,w)
,
(2)
where o, (k,k,) is the plasma mode of the LSIC system. The plasma mode o,(k,k,) is given by oE(k,k,)
=@,(k)sinh
M/(cosh
kd-cos
k,d) , (3)
where G&,(k) is the 2D ionic plasma frequency. The function H(k,o) is defined as H(k,o)
= [w2- V;(u)k2
-a;(w)
[02-
V:(o)k’]
] [w2-c:(w)k2] ,
(4)
where Vt(o)k’ and c$(w)k2 correspond to the damped acoustic phonon mode of the lattice ion system and that of the mobile ion system, respectively; Vf ( o)k2 and Qg( w) correspond to the damped acoustic phonon mode and the optical phonon mode of the LSIC system, respectively.
3. Dimensional
crossover of collective modes
The plasma mode changes its dispersion relation from the 2D- to the 3D-like with decreasing d. In the weak coupling limit of d> k- ‘, all SIC planes are independent of each other. Thus the collective mode of the system is reduced to that of 2D system and then the plasma mode o,(k,k,) is 2D-like: w,(k,kz)=122n,,(k)=
(2nN,,q2k/m*e,,)“2,
where Sz, is the effective 3D plasma frequency and Q=,(k) the acoustic plasma frequency. This dimensional crossover property of o, (k,k,) influences the dispersion relation of the collective mode in the LSIC system. As is well known, the dispersion relation of longitudinal collective excitation is derived from the equation EL(k,kr,m) = 0. Taking account of eqs. (2) and (4), we obtain the factorized equation where o+(w) and [w2-o:(o)][c02_(w)]=o, w_ (w) are related to the upper and the lower collective mode of the system, respectively. The functions ok (cc) are defined as
(5)
where N,, is the mean value of the 2D number density; m* is the reduced mass; q is the elementary charge.
-4Q;2: V:(w)k2-4V?(w)c;(o)k4]
“2,
(7)
with ti
=Qi%o)
v:
= vt(o)+cf(w)
,
+w; (k&J
.
As a result of the factorization, the collective modes are classified into two kinds of modes, the upper mode and the lower mode. Because the continuum model is valid in the long wavelength region, we will consider the collective modes only in the long wavelength limit (k-0). In this limit the lower mode satisfies the dispersion relation 02=lim
02_(w)=
Vz(w)k2.
k-0
This leads to the damped acoustic phonon of o,,~= +_f&-irk2 independent of d. The explicit expression of the sound velocity V (damping coefficient r) is omitted [ $61.
i? Tomoyose et al. /Layered superionic conductors
The upper mode is derived from the equation w’=!? o: (0). Taking account of the relation Gj(o)=wfw/(w+i/7),
we get
(8) where 7 is a relaxation time of the oscillation with a frequency o. in the relative ionic motion. In the limit of kd<< 1, we can get the high-frequency and low-frequency solutions of eq. (8) within the first order of o~(k,k,)/o~r, where the plasma mode satisfies w~(k,k,)w~7<< 1 (for example, the condition 52%-=Kr&r is fulfilled for a-AgI [ 91). The real part of high-frequency upper mode is given by Rew,,=+[og+wE(k,k,)-l/47*]“*,
w,=-iw~(k,k,)/[w?j+w~(k,k,)]~,
215
(10)
which is the coupled mode of plasma mode w,,(k,k,) and optical mode w,,. Only in the case of k_d= 0 this mode indicates the relaxation mode similar to that of 3D SIC system, [ 8,9] that is, the LSIC system can be regarded as an effective 3D SIC system. For k,# 0 it is transformed into a diffusion mode. The diffusion mode is due to the softening of the plasma mode, since its restoring force, namely the Coulomb field, is weakened by the out-of-phase density oscillation between the adjacent SIC planes. In the limit of kds 1, where the system can be considered as a single 2D SIC system, the relaxation mode is reduced to w5= -i/7.
(9)
which represents the coupled mode of optical phonons and plasma mode. Because of the dimensional crossover of w,( k,k,), this mode can take two kind of modes. In the case of kJ= 0 it is the coupled mode of optical phonons and 3D ionic plasma mode. In the case of kp’=n it represents the damped optical phonon mode. In the weak coupling limit (kd> 1), these highfrequency upper modes approach to the coupled optical phonon-2D plasma mode independent of k, which represents a collective mode in a single SIC layer. On the other hand, in the strong coupling limit (Me 1)) the low-frequency upper mode indicates the pure imaginary mode
References [ 1] A.L. Fetter, Ann. Phys. (NY) 88 (1974) 1. [2] S. Das Sarma and J.J. Quinn, Phys. Rev. B 25 (1982) 7603. [ 31 M. Kobayashi, J. Mizuno and I. Yokota, J. Phys. Sot. Japan 39 (1975) 18. [4] A.C. Tselis and J.J. Quinn, Phys. Rev. B 29 (1984) 3318. [ 5] T. Tomoyose, K. Yonashiro and M. Kobayashi, submitted for publication. [ 61 T. Tomoyose, K. Yonashiro and M. Kobayashi, to be published. [ 71 K.R. Subbaswamy, Solid State Commun. 19 (1976) 1157. [8] H. Hayashi, M. Kobayashi and I. Yokota, Solid State Commun. 31 (1979) 847. [9] J. Jackie, Z. PhysikB 30 (1978) 255.
DIMENSIONAL CROSSOVER OF COLLECTIVE LAYERED SUPERIONIC CONDUCTORS
MODES IN
T. TOMOYOSE, K. YONASHIRO Physics Department, General Education Division, Ryukyu University, Okinawa 903-01, Japan
and M. KOBAYASHI Department of Physics, Niigata University, Niigata 950-21, Japan
Using the continuum model, we have studied the dimensional crossover property of longitudinal collective modes in a layered superionic conductor (LSIC) which is composed of alternating layers of superionic conductor (SIC) and ionic crystal. The ionic plasma mode of the LSIC has changed from the three-dimensional to the two-dimensional plasma mode with increasing distance between SIC layers. As a result of this dimensional crossover of the plasma mode, the collective modes have shown various dispersion relations: the coupled optical phonon-plasma mode, the damped optical phonon mode, the relaxation mode, the diffusion mode and the acoustic phonon mode.
1. Introduction The artificial substances with a layered structure of charged particles have attracted much interest in recent years in connection with the dimensional crossover property of the system. We will be able to make a superlattice composed of alternating layers of superionic conductor (SIC) and ionic crystal. We call it the layered SIC (LSIC). In general, the plasma mode of the layered system changes its dispersion relation depending on the distance between layers as found in the layered electron gas [ l-41. Since the LSIC system is the layered system, the plasma mode of the system will change its dispersion relation from the two-dimensional (2D) like to the three-dimensional (3D) like with decreasing distance between layers. This dimensional crossover of plasma mode presumably affects the dynamical properties of the LSIC system. The details for the dynamical properties have been given elsewhere [ 5 1. In the present paper we study the dimensional crossover property of the longitudinal collective excitation in the LSIC system. We assume that the SIC layer is regarded as a 2D SIC plane. Our model struc0167-2738/90/S 03.50 0 Elsevier Science Publishers B.V. (North-Holland )
ture of the system consists of an infinite array of parallel SIC planes with a distance d apart. We take the z axis to be perpendicular to the SIC planes. The SIC planes are located at z=zi where Zi= id (i=O, f 1, + 2,... ). The ionic crystal between SIC planes is regarded as a background medium with the static dielectric constant e,,, which does not affect dynamical properties of the SIC planes. Each SIC plane consists of mobile ions and lattice ions. Following the continuum model [ 5-91, we assume that the crystalline cage composed of lattice ions is immersed in the viscous fluid of mobile ions. The dynamics of the lattice ions is described by the 2D lattice displacement field in the ith SIC plane, while the motion of mobile ions is treated as that of a viscous fluid of cations. The details for the equation of motion of ions are given in elsewhere [ 5,6].
2. Dielectric response function
Using the linear response theory, the longitudinal dielectric function is obtained as
T. Tomoyose et al. /Layered superionic conductors
214
k*Eintk,kz,o)
E~(k,kr,W) = l-
k.E(k,k,,w)
(1)
’
where Ei”(k,kz,w) is the induced electric field, E(k,k,,o) the total electric field, k the 2D wave vector in the SIC plane, k, the wave number perpendicular to the plane, and w is the frequency. We consider only the case of the non-retarded limit. The dielectric function is given by [ $61,
In the strong coupling limit of de k-‘, we can regard the LSIC system as a 3D-like SIC system. However, the plasma mode shows not only a 3D-like dispersion relation but also an intermediate one depending on the correlation between SIC planes, that is, m,(k>k,) sl, = (4nNoq2/m*E,,d)“‘,
for k,d=O,
= { L&,(k) = (xN,,q2d/m*eo)“2k,
for k,d=n,
~,(k,kw)
(6) w;(k,k,)
= l-
[wz- I’f(o)k211H(k,w)
,
(2)
where o, (k,k,) is the plasma mode of the LSIC system. The plasma mode o,(k,k,) is given by oE(k,k,)
=@,(k)sinh
M/(cosh
kd-cos
k,d) , (3)
where G&,(k) is the 2D ionic plasma frequency. The function H(k,o) is defined as H(k,o)
= [w2- V;(u)k2
-a;(w)
[02-
V:(o)k’]
] [w2-c:(w)k2] ,
(4)
where Vt(o)k’ and c$(w)k2 correspond to the damped acoustic phonon mode of the lattice ion system and that of the mobile ion system, respectively; Vf ( o)k2 and Qg( w) correspond to the damped acoustic phonon mode and the optical phonon mode of the LSIC system, respectively.
3. Dimensional
crossover of collective modes
The plasma mode changes its dispersion relation from the 2D- to the 3D-like with decreasing d. In the weak coupling limit of d> k- ‘, all SIC planes are independent of each other. Thus the collective mode of the system is reduced to that of 2D system and then the plasma mode o,(k,k,) is 2D-like: w,(k,kz)=122n,,(k)=
(2nN,,q2k/m*e,,)“2,
where Sz, is the effective 3D plasma frequency and Q=,(k) the acoustic plasma frequency. This dimensional crossover property of o, (k,k,) influences the dispersion relation of the collective mode in the LSIC system. As is well known, the dispersion relation of longitudinal collective excitation is derived from the equation EL(k,kr,m) = 0. Taking account of eqs. (2) and (4), we obtain the factorized equation where o+(w) and [w2-o:(o)][c02_(w)]=o, w_ (w) are related to the upper and the lower collective mode of the system, respectively. The functions ok (cc) are defined as
(5)
where N,, is the mean value of the 2D number density; m* is the reduced mass; q is the elementary charge.
-4Q;2: V:(w)k2-4V?(w)c;(o)k4]
“2,
(7)
with ti
=Qi%o)
v:
= vt(o)+cf(w)
,
+w; (k&J
.
As a result of the factorization, the collective modes are classified into two kinds of modes, the upper mode and the lower mode. Because the continuum model is valid in the long wavelength region, we will consider the collective modes only in the long wavelength limit (k-0). In this limit the lower mode satisfies the dispersion relation 02=lim
02_(w)=
Vz(w)k2.
k-0
This leads to the damped acoustic phonon of o,,~= +_f&-irk2 independent of d. The explicit expression of the sound velocity V (damping coefficient r) is omitted [ $61.
i? Tomoyose et al. /Layered superionic conductors
The upper mode is derived from the equation w’=!? o: (0). Taking account of the relation Gj(o)=wfw/(w+i/7),
we get
(8) where 7 is a relaxation time of the oscillation with a frequency o. in the relative ionic motion. In the limit of kd<< 1, we can get the high-frequency and low-frequency solutions of eq. (8) within the first order of o~(k,k,)/o~r, where the plasma mode satisfies w~(k,k,)w~7<< 1 (for example, the condition 52%-=Kr&r is fulfilled for a-AgI [ 91). The real part of high-frequency upper mode is given by Rew,,=+[og+wE(k,k,)-l/47*]“*,
w,=-iw~(k,k,)/[w?j+w~(k,k,)]~,
215
(10)
which is the coupled mode of plasma mode w,,(k,k,) and optical mode w,,. Only in the case of k_d= 0 this mode indicates the relaxation mode similar to that of 3D SIC system, [ 8,9] that is, the LSIC system can be regarded as an effective 3D SIC system. For k,# 0 it is transformed into a diffusion mode. The diffusion mode is due to the softening of the plasma mode, since its restoring force, namely the Coulomb field, is weakened by the out-of-phase density oscillation between the adjacent SIC planes. In the limit of kds 1, where the system can be considered as a single 2D SIC system, the relaxation mode is reduced to w5= -i/7.
(9)
which represents the coupled mode of optical phonons and plasma mode. Because of the dimensional crossover of w,( k,k,), this mode can take two kind of modes. In the case of kJ= 0 it is the coupled mode of optical phonons and 3D ionic plasma mode. In the case of kp’=n it represents the damped optical phonon mode. In the weak coupling limit (kd> 1), these highfrequency upper modes approach to the coupled optical phonon-2D plasma mode independent of k, which represents a collective mode in a single SIC layer. On the other hand, in the strong coupling limit (Me 1)) the low-frequency upper mode indicates the pure imaginary mode
References [ 1] A.L. Fetter, Ann. Phys. (NY) 88 (1974) 1. [2] S. Das Sarma and J.J. Quinn, Phys. Rev. B 25 (1982) 7603. [ 31 M. Kobayashi, J. Mizuno and I. Yokota, J. Phys. Sot. Japan 39 (1975) 18. [4] A.C. Tselis and J.J. Quinn, Phys. Rev. B 29 (1984) 3318. [ 5] T. Tomoyose, K. Yonashiro and M. Kobayashi, submitted for publication. [ 61 T. Tomoyose, K. Yonashiro and M. Kobayashi, to be published. [ 71 K.R. Subbaswamy, Solid State Commun. 19 (1976) 1157. [8] H. Hayashi, M. Kobayashi and I. Yokota, Solid State Commun. 31 (1979) 847. [9] J. Jackie, Z. PhysikB 30 (1978) 255.