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Monte Carlo calculation for equilibrium oxygen content of YBa2Cu3Ox
Solid State Communications, Printed ih Great Britain.
Vol. 81, No. 6, pp. 537-539, 1992.
MONTE CARLO CALCULATION
FOR EQUILIBRIUM
0038-1098/92 $5.00 f .OO Pergamon Press plc OXYGEN CONTENT
OF YBa,Cu,O,
M. Ohkubo Toyota Central R&D Labs., Inc., Nagakute-cho, (Received
Aichi-ken, 480-l 1, Japan
17 October 1991 by T. Tsuzuki)
The dependencies of oxygen content of YBa,Cu,O, on both temperature and oxygen pressure have been simulated by the Monte Carlo method based on a lattice gas model with three interactions between oxygen ions and chemical potential for environmental molecular oxygen gas.
1. INTRODUCTION
the oxygen ions are allowed to be incorporated into or released from the Cu( l)Ozc square lattice only through the four edges of the lattice, and to randomly walk inside the lattice, as studied by Andersen et al. [9].
IT IS WELL known that oxygen ions play an important role in the occurrence of superconductivity in YBa,Cu,O,. It has been recognized that the electronic properties of YBa,Cu,O, are dependent not only on 2. MODEL AND CALCULATIONS oxygen content but also on oxygen ordering in the An essential feature of the lattice gas model is Cu( 1) basal plane between BaO, planes [l]. The ordered phases such as Ortho-I and Ortho-II, which three interactions between the oxygen ions in the Cu( l)O*, plane: repulsive nearest-neighbor interaction were observed experimentally have been successfully next-nearest-neighbor interaction reproduced by a 2-D anisotropic lattice gas model [2]. (V), attractive Additionally, further superlattice structures were mediated by a Cu ion (V,), repulsive next-nearestneighbor interaction without a Cu ion (V3) [2]. The studied by a 1-D model with long range interactions [3]. In the 2-D lattice gas model, the Cu(l)O,, plane Hamiltonian may be expressed by has been treated as an Ising square lattice. The c is the H= probability of oxygen ions occupying available oxygen ij ij sites. o-v-o Most calculations based on the lattice gas model - & C ninj - P C (1) have been performed with virtual oxygen chemical ij i where potential at arbitrary constant values. The dependence of chemical potential of molecular oxygen gas on p~-7/2 temperature has been ignored. This makes it difficult kTIn (2) 3(1 _ e-2wT) - E* to calculate oxygen configuration in the Cu(1) plane with oxygen pressure and temperature. In addition, The occupation variable ni is 1 if the oxygen site is the site occupancy by the oxygen ions has been often occupied, and 0 if the site is vacant, P is oxygen treated by a spin flip-flop. This means that oxygen pressure in atm, and T is temperature in K. The caldiffusivity along the c-axis (0,) is abnormally large, culations were performed with V, = - 0.376eV, while experimental results of the oxygen diffusivity v,= 0.131 eV and V3 = -0.060eV [lo]. The ,u is the have showed that DC is more than 105-- lo6 times chemical potential of the molecular oxygen gas [6-81. smaller than Dab [4, 51. The heat of solution E has been given by In this paper, we employ the formula for the E(eV) = -0.818 - 0.307T/lOOO + f(c) [8]. The oxygen chemical potential by Salomons et al. [6], f(c) term depending on oxygen content was neglected Bakker et al. [7], and Shaked et al. [8]. In the formula, as studied by Baumglrtel et al. [I I]. The Cu(l)O,, the oxygen chemical potential depends on both tem- square lattice used for the calculations had 40 x 40 perature and oxygen pressure. This allows one to oxygen sites. The oxygen content x was determined by calculate oxygen configuration in the Cu( 1) plane with x = 6 + 2c. real physical quantities instead of the virtual chemical The calculations were performed by the Monte potential. Moreover, in the present calculations, in Carlo method with the Metropolis algorithm. The order to take into account the fact that D,,b/DC N 106, initial condition taken for all calculations was 4,
P=
537
l/2 1
MONTE CARLO CALCULATION
538 6.9
I
A 6.4
A t
6.3
.
,
I
6
Vol. 81, No. 6
’
0
A~AA~AAAAA
A "AA A &b A A, AA A", A AA AA A A AAA A AA AAA A A"A A 1
I 10 000
20 000
30 000
40 000
50 000
TIbfE [MCS/s]
Fig. 1. Oxygen content as a function of time in a unit of MSC s- ’ . The oxygen incorporation at P = 0.001 atm and T = 400°C and the oxygen release at P = 0.001 atm and T = 600°C are denoted by the squares and the triangles. x = 6.67. This condition was chosen to shorten calculation times at low temperatures. The system was equilibrated at 400-900°C in oxygen gases of O.OOOl1 atm. Since diffusivity (or jump frequency) of the oxygen ions strongly depends on temperature, the calculation time required to obtain almost steady states varies with temperature; for example 5 x IO4 Monte Carlo steps per site (MCS s-‘) at 4OO”C,while lo4 MCSs-’ at 900°C. During the calculations, the incorporation and release of oxygen ions took place depending on T and P. 3. RESULTS AND DISCUSSION Typical examples of the incorporation at P = 0.001 atm and T = 400°C and of the release at P = 0.001 atm and T = 600°C are shown in Fig. 1, in which the x’s are plotted against time in a unit of MCSs-‘. As the time goes on, in the incorporation, the x rapidly increases at the early stage, and then increases gradually. At 5 x lo4 MCSs-‘, the x reaches a value of 6.865 which we presume as an equilibrium X. A similar behavior is observed in the release except that the reduction of x is saturated quickly because of large oxygen diffusivity at the high temperature. A reason for the rapid x increase at the early stage of the incorporation in Fig. 1 is that most sites near the edges of the lattice are easily occupied by the oxygen ions because of a large driving force for the incorporation. This rapid incorporation results in the formation of a shell structure of the Ortho-I, that is, the Ortho-I phase is formed near the edges. Since the Ortho-I shell acts as a barrier against the incorporation, which originates from the large repulsive interaction of P’,, further incorporation is slowed down. In contrast with
Fig. 2. Oxygen configuration at P = 0.001 atm, T = 400°C and 5 x IO” MCSs-‘: oxygen (0), copper (0). the incorporation, there is no such barrier for the release in the present calculation. Figure 2 shows the oxygen configuration at 5 x lo4 MCSs-‘. The lattice is in the single domain state of the Ortho-I with oxygen vacancies which cause the fragmentation of the CuO chains and form oxygen-empty chains. These empty chains tend to be located inside the lattice because the oxygen ions are supplied from the four edges. In the present calculations, these vacancies inside the lattice cause a difficulty, which we are experimentally confronted with, in obtaining a phase with x = 7.0 at atmospheric pressure. The oxygen vacancies inside the lattice can hardly reach the edges of the lattice and disappear, because it is necessary that the stable CuO chains are destroyed many times. It seems that the energy barrier is so high that the system stays at a quasi-equilibrium state and cannot readily reach the most stable state beyond the energy height within a practical calculation time. In fact, it is considered that the Ortho-I phase with x = 7.0 is the most stable state at P = 0.001 atm and 400°C because if we take an initial condition of the Ortho-I with x = 7.0, the x continues to be 7.0 for 5 x lo4 MCSs-’ at least. Therefore, the calculated x’s at low T are quasiequilibrium values rather than equilibrium values. However, since it is suggested that experiments have the same problem as the present calculations, we call the x’s in Fig. 1 equilibrium values. The barrier of the shell structure of the Ortho-I and the unmobile oxygen vacancies in the lattice are not the case with the calculations in the high T region over about 600°C. Since the number of the oxygen
MONTE CARLO CALCULATION
Vol; 81, No. 6
539
on both temperature and oxygen pressure have been simulated by the Monte Carlo method. The model can simulate the dependencies of the equilibrium x on both P and T, and the experimental fact that it is difficult to obtain the phase with x = 7.0. We believe that better agreement between experiment and calculation will be obtained by adjusting parameters in equations (1) and (2). REFERENCES J.D. Jorgensen, S. Pei, P. Lightfoot, H. Shi, A.P. Paulikas & B.W. Veal, Physica C167,571 (1990). L.T. Wille, A. Berera & D. de Fontaine, Phys.
10’ OXYGEN
PRESSURE
[ah]
Rev. Left. 60, 1065 (1988).
Fig. 3. Equilibrium oxygen content as a function of oxygen pressure at various temperatures. The squares connected by the dashed lines are the calculated values. The experimental values by Kishio et al. [12] are shown by the solid lines. vacancies is high at the edges of the lattice, the vacancies can migrate and the oxygen ions can be incorporated and released through the edges readily. In the high T region, there appear shortly fragmented CuO chains. The dependence of equilibrium x on P as a parameter of T is shown in Fig. 3 with experimental curves by Kishio et al. [ 121.The calculated equilibrium x’s shown by the squares are average values near at 400°C 3 x lo4 MCS s-’ at 5 x 104MCSs-’ 500°C 2 x IO4MCS s-’ at 600 and 700°C and 1 x IO4MCS s-’ at 800 and 900°C. It is seen that the calculated equilibrium x’s adequately agree with the experimental ones except considerable discrepancies in the regions at high P and low T, and at low P and high T. As a result, we conclude that the model by equations (1) and (2), with the restriction that the oxygen ions are allowed to be incorporated and released only through the four edges of the square lattice, may describe the real system properly. In summary, the dependencies of oxygen content
D. de Fontaine, G. Ceder & M. Asta, Nature 343, 544 (1990).
S.J. Rothman, J.L. Routbort, J.-Z. Liu, J.W. Downey, L.J. Thompson, Y. Fang, D. Shi, J.E. Baker, J.P. Rice, D.M. Ginsberg, P.D. Han & D.A. Payne, Proceedings of the Symposium on Atomic
5.
Migration
and Defects
in Materials,
Indianapolis (1989). S. Tsukui,T. Yamamoto, M. Adachi, Y. Shono, K. Kawabata, N. Fukuoka, S. Nakanishi, A. Yanase & Y. Yoshioka, Jpn. J. Appl. Phys. 30, L973 (1991).
6. 7.
E. Salomons, N. Koeman, R. Brouwer, D.G. de Groot & R. Griessen, Solid State Commun. 64, 1141 (1987). H. Bakker, D.O. Welsh & O.W. Lazareth, Jr., Solid State Commun.
8.
64, 237 (1987).
H. Shaked, J.D. Jorgensen, J. Faber, Jr., D.G. Hinks & B. Dabrowski, Phys. Rev. B39, 7363 (1989).
9.
J.V. Andersen,
H. Bohr & O.G. Mouritsen,
Phys. Rev. B42, 283 (1990).
10.
E. Salomons & D. de Fontaine, Phys. Rev. B42,
11.
10152 (1990). G. Baumgartel, P.J. Jensen & K.H. Bennemann, Phys. Rev. B42, 288 (1990).
12.
K. Kishio, J. Shimoyama, T. Hasegawa, K. Kitazawa & K. Fueki, Jpn. J. Appl. Phys. 26, L1228 (1987).
Vol. 81, No. 6, pp. 537-539, 1992.
MONTE CARLO CALCULATION
FOR EQUILIBRIUM
0038-1098/92 $5.00 f .OO Pergamon Press plc OXYGEN CONTENT
OF YBa,Cu,O,
M. Ohkubo Toyota Central R&D Labs., Inc., Nagakute-cho, (Received
Aichi-ken, 480-l 1, Japan
17 October 1991 by T. Tsuzuki)
The dependencies of oxygen content of YBa,Cu,O, on both temperature and oxygen pressure have been simulated by the Monte Carlo method based on a lattice gas model with three interactions between oxygen ions and chemical potential for environmental molecular oxygen gas.
1. INTRODUCTION
the oxygen ions are allowed to be incorporated into or released from the Cu( l)Ozc square lattice only through the four edges of the lattice, and to randomly walk inside the lattice, as studied by Andersen et al. [9].
IT IS WELL known that oxygen ions play an important role in the occurrence of superconductivity in YBa,Cu,O,. It has been recognized that the electronic properties of YBa,Cu,O, are dependent not only on 2. MODEL AND CALCULATIONS oxygen content but also on oxygen ordering in the An essential feature of the lattice gas model is Cu( 1) basal plane between BaO, planes [l]. The ordered phases such as Ortho-I and Ortho-II, which three interactions between the oxygen ions in the Cu( l)O*, plane: repulsive nearest-neighbor interaction were observed experimentally have been successfully next-nearest-neighbor interaction reproduced by a 2-D anisotropic lattice gas model [2]. (V), attractive Additionally, further superlattice structures were mediated by a Cu ion (V,), repulsive next-nearestneighbor interaction without a Cu ion (V3) [2]. The studied by a 1-D model with long range interactions [3]. In the 2-D lattice gas model, the Cu(l)O,, plane Hamiltonian may be expressed by has been treated as an Ising square lattice. The c is the H= probability of oxygen ions occupying available oxygen ij ij sites. o-v-o Most calculations based on the lattice gas model - & C ninj - P C (1) have been performed with virtual oxygen chemical ij i where potential at arbitrary constant values. The dependence of chemical potential of molecular oxygen gas on p~-7/2 temperature has been ignored. This makes it difficult kTIn (2) 3(1 _ e-2wT) - E* to calculate oxygen configuration in the Cu(1) plane with oxygen pressure and temperature. In addition, The occupation variable ni is 1 if the oxygen site is the site occupancy by the oxygen ions has been often occupied, and 0 if the site is vacant, P is oxygen treated by a spin flip-flop. This means that oxygen pressure in atm, and T is temperature in K. The caldiffusivity along the c-axis (0,) is abnormally large, culations were performed with V, = - 0.376eV, while experimental results of the oxygen diffusivity v,= 0.131 eV and V3 = -0.060eV [lo]. The ,u is the have showed that DC is more than 105-- lo6 times chemical potential of the molecular oxygen gas [6-81. smaller than Dab [4, 51. The heat of solution E has been given by In this paper, we employ the formula for the E(eV) = -0.818 - 0.307T/lOOO + f(c) [8]. The oxygen chemical potential by Salomons et al. [6], f(c) term depending on oxygen content was neglected Bakker et al. [7], and Shaked et al. [8]. In the formula, as studied by Baumglrtel et al. [I I]. The Cu(l)O,, the oxygen chemical potential depends on both tem- square lattice used for the calculations had 40 x 40 perature and oxygen pressure. This allows one to oxygen sites. The oxygen content x was determined by calculate oxygen configuration in the Cu( 1) plane with x = 6 + 2c. real physical quantities instead of the virtual chemical The calculations were performed by the Monte potential. Moreover, in the present calculations, in Carlo method with the Metropolis algorithm. The order to take into account the fact that D,,b/DC N 106, initial condition taken for all calculations was 4,
P=
537
l/2 1
MONTE CARLO CALCULATION
538 6.9
I
A 6.4
A t
6.3
.
,
I
6
Vol. 81, No. 6
’
0
A~AA~AAAAA
A "AA A &b A A, AA A", A AA AA A A AAA A AA AAA A A"A A 1
I 10 000
20 000
30 000
40 000
50 000
TIbfE [MCS/s]
Fig. 1. Oxygen content as a function of time in a unit of MSC s- ’ . The oxygen incorporation at P = 0.001 atm and T = 400°C and the oxygen release at P = 0.001 atm and T = 600°C are denoted by the squares and the triangles. x = 6.67. This condition was chosen to shorten calculation times at low temperatures. The system was equilibrated at 400-900°C in oxygen gases of O.OOOl1 atm. Since diffusivity (or jump frequency) of the oxygen ions strongly depends on temperature, the calculation time required to obtain almost steady states varies with temperature; for example 5 x IO4 Monte Carlo steps per site (MCS s-‘) at 4OO”C,while lo4 MCSs-’ at 900°C. During the calculations, the incorporation and release of oxygen ions took place depending on T and P. 3. RESULTS AND DISCUSSION Typical examples of the incorporation at P = 0.001 atm and T = 400°C and of the release at P = 0.001 atm and T = 600°C are shown in Fig. 1, in which the x’s are plotted against time in a unit of MCSs-‘. As the time goes on, in the incorporation, the x rapidly increases at the early stage, and then increases gradually. At 5 x lo4 MCSs-‘, the x reaches a value of 6.865 which we presume as an equilibrium X. A similar behavior is observed in the release except that the reduction of x is saturated quickly because of large oxygen diffusivity at the high temperature. A reason for the rapid x increase at the early stage of the incorporation in Fig. 1 is that most sites near the edges of the lattice are easily occupied by the oxygen ions because of a large driving force for the incorporation. This rapid incorporation results in the formation of a shell structure of the Ortho-I, that is, the Ortho-I phase is formed near the edges. Since the Ortho-I shell acts as a barrier against the incorporation, which originates from the large repulsive interaction of P’,, further incorporation is slowed down. In contrast with
Fig. 2. Oxygen configuration at P = 0.001 atm, T = 400°C and 5 x IO” MCSs-‘: oxygen (0), copper (0). the incorporation, there is no such barrier for the release in the present calculation. Figure 2 shows the oxygen configuration at 5 x lo4 MCSs-‘. The lattice is in the single domain state of the Ortho-I with oxygen vacancies which cause the fragmentation of the CuO chains and form oxygen-empty chains. These empty chains tend to be located inside the lattice because the oxygen ions are supplied from the four edges. In the present calculations, these vacancies inside the lattice cause a difficulty, which we are experimentally confronted with, in obtaining a phase with x = 7.0 at atmospheric pressure. The oxygen vacancies inside the lattice can hardly reach the edges of the lattice and disappear, because it is necessary that the stable CuO chains are destroyed many times. It seems that the energy barrier is so high that the system stays at a quasi-equilibrium state and cannot readily reach the most stable state beyond the energy height within a practical calculation time. In fact, it is considered that the Ortho-I phase with x = 7.0 is the most stable state at P = 0.001 atm and 400°C because if we take an initial condition of the Ortho-I with x = 7.0, the x continues to be 7.0 for 5 x lo4 MCSs-’ at least. Therefore, the calculated x’s at low T are quasiequilibrium values rather than equilibrium values. However, since it is suggested that experiments have the same problem as the present calculations, we call the x’s in Fig. 1 equilibrium values. The barrier of the shell structure of the Ortho-I and the unmobile oxygen vacancies in the lattice are not the case with the calculations in the high T region over about 600°C. Since the number of the oxygen
MONTE CARLO CALCULATION
Vol; 81, No. 6
539
on both temperature and oxygen pressure have been simulated by the Monte Carlo method. The model can simulate the dependencies of the equilibrium x on both P and T, and the experimental fact that it is difficult to obtain the phase with x = 7.0. We believe that better agreement between experiment and calculation will be obtained by adjusting parameters in equations (1) and (2). REFERENCES J.D. Jorgensen, S. Pei, P. Lightfoot, H. Shi, A.P. Paulikas & B.W. Veal, Physica C167,571 (1990). L.T. Wille, A. Berera & D. de Fontaine, Phys.
10’ OXYGEN
PRESSURE
[ah]
Rev. Left. 60, 1065 (1988).
Fig. 3. Equilibrium oxygen content as a function of oxygen pressure at various temperatures. The squares connected by the dashed lines are the calculated values. The experimental values by Kishio et al. [12] are shown by the solid lines. vacancies is high at the edges of the lattice, the vacancies can migrate and the oxygen ions can be incorporated and released through the edges readily. In the high T region, there appear shortly fragmented CuO chains. The dependence of equilibrium x on P as a parameter of T is shown in Fig. 3 with experimental curves by Kishio et al. [ 121.The calculated equilibrium x’s shown by the squares are average values near at 400°C 3 x lo4 MCS s-’ at 5 x 104MCSs-’ 500°C 2 x IO4MCS s-’ at 600 and 700°C and 1 x IO4MCS s-’ at 800 and 900°C. It is seen that the calculated equilibrium x’s adequately agree with the experimental ones except considerable discrepancies in the regions at high P and low T, and at low P and high T. As a result, we conclude that the model by equations (1) and (2), with the restriction that the oxygen ions are allowed to be incorporated and released only through the four edges of the square lattice, may describe the real system properly. In summary, the dependencies of oxygen content
D. de Fontaine, G. Ceder & M. Asta, Nature 343, 544 (1990).
S.J. Rothman, J.L. Routbort, J.-Z. Liu, J.W. Downey, L.J. Thompson, Y. Fang, D. Shi, J.E. Baker, J.P. Rice, D.M. Ginsberg, P.D. Han & D.A. Payne, Proceedings of the Symposium on Atomic
5.
Migration
and Defects
in Materials,
Indianapolis (1989). S. Tsukui,T. Yamamoto, M. Adachi, Y. Shono, K. Kawabata, N. Fukuoka, S. Nakanishi, A. Yanase & Y. Yoshioka, Jpn. J. Appl. Phys. 30, L973 (1991).
6. 7.
E. Salomons, N. Koeman, R. Brouwer, D.G. de Groot & R. Griessen, Solid State Commun. 64, 1141 (1987). H. Bakker, D.O. Welsh & O.W. Lazareth, Jr., Solid State Commun.
8.
64, 237 (1987).
H. Shaked, J.D. Jorgensen, J. Faber, Jr., D.G. Hinks & B. Dabrowski, Phys. Rev. B39, 7363 (1989).
9.
J.V. Andersen,
H. Bohr & O.G. Mouritsen,
Phys. Rev. B42, 283 (1990).
10.
E. Salomons & D. de Fontaine, Phys. Rev. B42,
11.
10152 (1990). G. Baumgartel, P.J. Jensen & K.H. Bennemann, Phys. Rev. B42, 288 (1990).
12.
K. Kishio, J. Shimoyama, T. Hasegawa, K. Kitazawa & K. Fueki, Jpn. J. Appl. Phys. 26, L1228 (1987).