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Strong electron-phonon coupling among the electrons in 2D electron lattice
Physiea C 235-240 (1994) 2393-2394 North-Holland
PHYSiCA
Strong electron-phonon coupling among the electrons in 2D electron lattice W. Kinase, K. Takahashi and S. Kuwata Department of Physics, Waseda university, 3-Okubo, Shinjuku-ku, Tokyo 169, Japan
The interactions among the electrons in the two-dimensional lattice are considered for the several modes of the electron vibration. The relationship between the hole conduction and the vibration of electron lattice is discussed by the local fields due to the polarization wave caused by these modes.
1. INTRODUCTION Following the model of the two-dimensional electron (or hole) lattice presented by authors in M2S-HTSC-III in Kanazawa [1] and previous w o r k [2-5] we discuss here the strong interaction among the constituent electrons of the two-dimensional (2D) lattice. In the layered perovskite-type High Tc superconducting crystals such as LaSrCuO, YBaCuO and BiSrCaCuO the 2D electron system is consideIed to exist upon CuO plane, which is stabilized by neutralizing the local polarization accompanied by the spontaneous shift of the cations in the c-direction. The incommensurately (IC) modulated structure is frequently foun3 in these crystals.
other electrons than the one electron under consideration remain stationary. Thus in this article, we consider the whole electron shifts and various 2D electron-lattice vibrations in the xy-plane, as shown in Fig. 1. Here, the mode (a) corresponds to the acoustic mode in a large wavelength limit, where no potential change is found, because the relative shift among the electrons does not exist in this case. In general, however, a strong electron-phonon interaction within the electron lattice can be caused by the strong local field due to the polarization wave. In this article, we tentatively adopt the mode (c) in Fig. 1, because the local field is most violent in this case. ---.It,
(a) 2.2D ELECTRON-LATTICE VIBRATION The potential difference of one electron from the equilibrium point was calculated as [4] 2.26-~-T r2
(e: elementary charge),
(1)
where r represents the distance from the lattice point, and D, the electron lattice constant, is related to a local polarization P through the neutralizing condition [2] D2P = e .
~
--, --, -, --, acoustic
---'I,
'-~
~
(b) ,- ,- ,-- ,-stripes-type I
(c)
._ _, ,_ __, (d) ,_. -., ,- -, stripes-type II checkerboard-type Figure 1. Vibration modes in the 2D electron lattice. In a way similar to eq. (1), tlle potentials difference acting on one electron can be calculated for the various modes as V,, - 0,
Vl, = - 0 . 291( e: / D ~ )x z ,
V c = 5 . 2 8 ( e 2 / D ~ ) x 2, V,t = ,~..92(e2/D-~)x ~
(2)
In deriving eq.(1), we have assumed that the
.--t,
(3)
Thus from eq. (3), the corresponding zero-point
0921-4534/94/507.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0921-4534(94)01762-X
W. Kinase et al./Physica C 23.$-240 (1994) 2393-2394
2394
frequenciesare derived as co, ~ O,
cob = ( u n s t a b l e ) ,
co~ = 0.293 eV,
coa = 0.218 eV
(4)
for the probable value of D=15A [6], which is estimated from the modulation period.
3. D Y N A M I S M OF HOLE C O N D U C T I O N IN THE 2D ELECTRON SYSTEM . . . . ° . . . . . .
....
. . . . . .
:, e':.:, e':.:.:e'.:-.'e" :.:e" :. .*'.'.*'.*.'.
: e'7.e'i.:.:e'.."e"7e":.
:.e'X.:e'i.:.ie'..'.:e"Y.eq.
•"~':'e':'.:::',e',:e'."
.'e':'.:::'e':'.:'.:'.e',"
.'.".:'.':.:'2.'.:".'.:'.:'.:'
".'.'.'.'.".:L:L**'
.'e" " e " : ' e - : ' e ' . ' . e ' : ......-.'..-.*,'.'..:....;'.-.....
.'..'.".
• .:'.".*.'.'..*.%',
• ."
.'.:'.::5::'.*.:".'.'.:'.'.*.:' .'..'.'.',*.*.".'o
.','.:'.'.':..V
:::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::: ::::::::::::::::::::::::::: .,
...
° . . . . . . . , . . . . . . . . . . , . . . . ,,. ,.
.
•. e. . ' .. .v. .e" :. e'....e" ...e': . ° . . . . ° . ° . . . . .
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
(A)
. . . . . , . . . . . . . . . . . . . . .
(B)
, . . . . . . . . . . . . . . . . . . . .
(c)
Figure 2. Hole configuration in the 2D lattice. Figures 2(A), (B) and (C) show the hole configurations, namely, without hole, one hole and two holes respectively. The Hamiltonian of these systems can be expressed as H = H i + H2 + H s + H 4 +/'/5,
considered to be essential, because an electron e-(A) or e-(B) located at the nearest neighbour of a hole (Fig. 3) is affected by the strong local field E a o~ B ~ 51a/D3 for the mode (c) in Fig. 1, resulting in the state having the momentum conversion of +k and -k, as shown in Fig. 3(a). In the case of two holes, we further consider an intermediate electron e-(C), aside from the bordered electron e-(A) or e-(B), where the momentum conversions are +k', +k-k' and --k, as shown in Fig. 3(b). Although in the case of one hole [Fig. 3(a)] the electron transfer is not favorable due to the indetermination of the direction of the momentum conversion, in the case of two holes [Fig. 3(b)] the electron transfer is likely to occur with the aid of the i n t e r m e d i a t e electron (C); the matching between the electron movement and the phase of the electron-lattice vibration is expected to realize the easy electron-transfer in one direction. Further considerations on the pairing mechanism will be made elsewhere.
(5)
where Ha is Coulomb energy among electrons, and H 2 is the interaction between the electron and the neutralized positive background, which was calculated for the case (A) as [2] H l + H 2 =-3.90e2/D,
(6)
corresponding to the Madelung energy of the 2D electron lattice. Now/-/3 is the potential for the electron shift r on the xy-plane calculated from eq. (3), and H 4 is the kinetic energy of an electron. Thus, the total vibration energy of one electron in the ground state is evaluated as (7)
where cois given by eq. (4). In many HTSC, the state of conuuction is the p-type, so we now consider the behaviour of positive holes in the 2D electron-lattice. The dipole interaction H5 among the electrons is
e" _~
(B)
........
ei . - = I ~ - .......
e-
e-
-k
e-
e-
.......
(B)
+k
(A) :. . . . . . . : (C)
(b)
i
+k-k'
:
e-
e-
+k'
Figure 3. (a) One hole and (b) two holes.
REFERENCES 1. W. Kinase, S. Kuwata and X. P. He, Physica C 185 - 189 (19911781. 2. W o Kinase, W. Makino, K. TaKahashi and H. Naka, Jpn. J. Appl. Phys., 24, $24-2 (1985) 850. w ~c;n~e, l p 7,,,,~,;n K Takahashi and w Hirose, Jpn. J. Appl. Phys., 26 (1987) L745. 4. W. Kinase, K. Hirose and K. Takahashi, Ferroelectrics 92 (1989) 167. 5. W. Kinase, K, Takahashi and K. Hirose, Phase Transition., 23 (1990) 1. 6. S. Kuwata and W. Kinase, Jpn. J. Appl. Phys., 32 (1993) 764. . . . .
H s + H4 = hco,
(A)
(a)
~.
~
•
.~-,.
)" ' L , 4 ~ , , . L
'.!
=.
.
~.~..
PHYSiCA
Strong electron-phonon coupling among the electrons in 2D electron lattice W. Kinase, K. Takahashi and S. Kuwata Department of Physics, Waseda university, 3-Okubo, Shinjuku-ku, Tokyo 169, Japan
The interactions among the electrons in the two-dimensional lattice are considered for the several modes of the electron vibration. The relationship between the hole conduction and the vibration of electron lattice is discussed by the local fields due to the polarization wave caused by these modes.
1. INTRODUCTION Following the model of the two-dimensional electron (or hole) lattice presented by authors in M2S-HTSC-III in Kanazawa [1] and previous w o r k [2-5] we discuss here the strong interaction among the constituent electrons of the two-dimensional (2D) lattice. In the layered perovskite-type High Tc superconducting crystals such as LaSrCuO, YBaCuO and BiSrCaCuO the 2D electron system is consideIed to exist upon CuO plane, which is stabilized by neutralizing the local polarization accompanied by the spontaneous shift of the cations in the c-direction. The incommensurately (IC) modulated structure is frequently foun3 in these crystals.
other electrons than the one electron under consideration remain stationary. Thus in this article, we consider the whole electron shifts and various 2D electron-lattice vibrations in the xy-plane, as shown in Fig. 1. Here, the mode (a) corresponds to the acoustic mode in a large wavelength limit, where no potential change is found, because the relative shift among the electrons does not exist in this case. In general, however, a strong electron-phonon interaction within the electron lattice can be caused by the strong local field due to the polarization wave. In this article, we tentatively adopt the mode (c) in Fig. 1, because the local field is most violent in this case. ---.It,
(a) 2.2D ELECTRON-LATTICE VIBRATION The potential difference of one electron from the equilibrium point was calculated as [4] 2.26-~-T r2
(e: elementary charge),
(1)
where r represents the distance from the lattice point, and D, the electron lattice constant, is related to a local polarization P through the neutralizing condition [2] D2P = e .
~
--, --, -, --, acoustic
---'I,
'-~
~
(b) ,- ,- ,-- ,-stripes-type I
(c)
._ _, ,_ __, (d) ,_. -., ,- -, stripes-type II checkerboard-type Figure 1. Vibration modes in the 2D electron lattice. In a way similar to eq. (1), tlle potentials difference acting on one electron can be calculated for the various modes as V,, - 0,
Vl, = - 0 . 291( e: / D ~ )x z ,
V c = 5 . 2 8 ( e 2 / D ~ ) x 2, V,t = ,~..92(e2/D-~)x ~
(2)
In deriving eq.(1), we have assumed that the
.--t,
(3)
Thus from eq. (3), the corresponding zero-point
0921-4534/94/507.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0921-4534(94)01762-X
W. Kinase et al./Physica C 23.$-240 (1994) 2393-2394
2394
frequenciesare derived as co, ~ O,
cob = ( u n s t a b l e ) ,
co~ = 0.293 eV,
coa = 0.218 eV
(4)
for the probable value of D=15A [6], which is estimated from the modulation period.
3. D Y N A M I S M OF HOLE C O N D U C T I O N IN THE 2D ELECTRON SYSTEM . . . . ° . . . . . .
....
. . . . . .
:, e':.:, e':.:.:e'.:-.'e" :.:e" :. .*'.'.*'.*.'.
: e'7.e'i.:.:e'.."e"7e":.
:.e'X.:e'i.:.ie'..'.:e"Y.eq.
•"~':'e':'.:::',e',:e'."
.'e':'.:::'e':'.:'.:'.e',"
.'.".:'.':.:'2.'.:".'.:'.:'.:'
".'.'.'.'.".:L:L**'
.'e" " e " : ' e - : ' e ' . ' . e ' : ......-.'..-.*,'.'..:....;'.-.....
.'..'.".
• .:'.".*.'.'..*.%',
• ."
.'.:'.::5::'.*.:".'.'.:'.'.*.:' .'..'.'.',*.*.".'o
.','.:'.'.':..V
:::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::: ::::::::::::::::::::::::::: .,
...
° . . . . . . . , . . . . . . . . . . , . . . . ,,. ,.
.
•. e. . ' .. .v. .e" :. e'....e" ...e': . ° . . . . ° . ° . . . . .
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
(A)
. . . . . , . . . . . . . . . . . . . . .
(B)
, . . . . . . . . . . . . . . . . . . . .
(c)
Figure 2. Hole configuration in the 2D lattice. Figures 2(A), (B) and (C) show the hole configurations, namely, without hole, one hole and two holes respectively. The Hamiltonian of these systems can be expressed as H = H i + H2 + H s + H 4 +/'/5,
considered to be essential, because an electron e-(A) or e-(B) located at the nearest neighbour of a hole (Fig. 3) is affected by the strong local field E a o~ B ~ 51a/D3 for the mode (c) in Fig. 1, resulting in the state having the momentum conversion of +k and -k, as shown in Fig. 3(a). In the case of two holes, we further consider an intermediate electron e-(C), aside from the bordered electron e-(A) or e-(B), where the momentum conversions are +k', +k-k' and --k, as shown in Fig. 3(b). Although in the case of one hole [Fig. 3(a)] the electron transfer is not favorable due to the indetermination of the direction of the momentum conversion, in the case of two holes [Fig. 3(b)] the electron transfer is likely to occur with the aid of the i n t e r m e d i a t e electron (C); the matching between the electron movement and the phase of the electron-lattice vibration is expected to realize the easy electron-transfer in one direction. Further considerations on the pairing mechanism will be made elsewhere.
(5)
where Ha is Coulomb energy among electrons, and H 2 is the interaction between the electron and the neutralized positive background, which was calculated for the case (A) as [2] H l + H 2 =-3.90e2/D,
(6)
corresponding to the Madelung energy of the 2D electron lattice. Now/-/3 is the potential for the electron shift r on the xy-plane calculated from eq. (3), and H 4 is the kinetic energy of an electron. Thus, the total vibration energy of one electron in the ground state is evaluated as (7)
where cois given by eq. (4). In many HTSC, the state of conuuction is the p-type, so we now consider the behaviour of positive holes in the 2D electron-lattice. The dipole interaction H5 among the electrons is
e" _~
(B)
........
ei . - = I ~ - .......
e-
e-
-k
e-
e-
.......
(B)
+k
(A) :. . . . . . . : (C)
(b)
i
+k-k'
:
e-
e-
+k'
Figure 3. (a) One hole and (b) two holes.
REFERENCES 1. W. Kinase, S. Kuwata and X. P. He, Physica C 185 - 189 (19911781. 2. W o Kinase, W. Makino, K. TaKahashi and H. Naka, Jpn. J. Appl. Phys., 24, $24-2 (1985) 850. w ~c;n~e, l p 7,,,,~,;n K Takahashi and w Hirose, Jpn. J. Appl. Phys., 26 (1987) L745. 4. W. Kinase, K. Hirose and K. Takahashi, Ferroelectrics 92 (1989) 167. 5. W. Kinase, K, Takahashi and K. Hirose, Phase Transition., 23 (1990) 1. 6. S. Kuwata and W. Kinase, Jpn. J. Appl. Phys., 32 (1993) 764. . . . .
H s + H4 = hco,
(A)
(a)
~.
~
•
.~-,.
)" ' L , 4 ~ , , . L
'.!
=.
.
~.~..