洞摻雜(p-type)較小能帶寬度之化合物La1-xCaxMnO3，因其在外加磁場下具有龐大磁阻之特性引起廣泛的注意，此類材料稱為龐磁阻(colossal magnetoresistance，CMR)。這種龐磁阻材料有著特殊的磁性和電性性質。其電荷有序情形只發生在當 x> 0.5時，此時物質的相隨溫度的降低即從「順磁絕緣相」轉變到「電荷有序相」再到「反鐵磁相」。而在我們的研究當中，我們所採用格林函數的電子關聯函數，將理論推導所得的「電化率」以及「磁化率」函數，分別代入實驗所得的相轉變溫度，然後即可求得內部「庫侖」斥力的臨界值 (產生相轉變)。這樣的庫侖斥力就是造成電荷有序的關鍵，並且還可獲得在電荷有序情況下，charge gap的大小。經由得知內部庫侖斥力的以及charge gap，我們即可得知電荷有序相轉變與內部庫侖斥力以on-site庫侖斥力的定性上的關係。這裡的on-site庫侖斥力指的即為Hund’s coupling，也就是說透過我們的研究可以得知在龐磁阻材料中造成如此特殊現象之內部物理圖象。

Hole-doped maganite with middle to narrow bandwidth La1-xCaxMnO3 was extensively studied because of its colossal magnetoresistance (CMR) characteristic under a magnetic field. These kind of materials show un- common magnetic and electric properties. The charge order phase only happens to the region x> 0.5, and along with decreasing temperature, its phase goes from para-insulator to charge-ordered then to antiferromagne-
tism. In our studies, we apply correlation function of Green’s function to LCMO and get susceptibility of charge and spin. Then we can get the cri-
tical value of Coulomb repulsion inside the material by substituting the
experimental values of phase transition temperature. This critical values is the key point of charge-ordered. Then we can also get the size of char- ge gap which decides the stability of charge-ordered phase. After know- ing the Coulomb repulsion and charge gap, we can picture the relation of inside and on-site Coulomb repulsion qualitatively while the transition happens. Here the on-site Coulomb repulsion means to the Hund’s coupl- ing between d electrons. And by this we’ll understand the physics inside
CMR materials.

第一章 緒論
1-1 歷史演進………………………………………………….05
1-2 La1-xCaxMnO3 性質介紹…………………………………10
第二章 理論研究
2-1 理論架構………………………………………………….20
2-2 結果與討論……………………………………………….24
2-3 結論……………………………………………………….30
第三章 理論背景
3-1 二次量子化算符………………………………………….31
3-2 Interaction representation………………………………....37
3-3 Green’s function…………………………………………..39
第四章 完整推導
4-1 Susceptibility of charge………………………………...…41
4-2 Susceptibility of spin……………………………………...48
參考資料……………………………………………………………..56

[1] Jonker, G.H., Van Santen, J.H., Physica (Utrecht) 16, 337, (1950).
[2] Wollan, E.O., Koehler, W.C.,Phys. Rev. 100, 545, (1955).
[3] Zener, C.,. Phys. Rev. 81, 440, (1951a). Zener, C., Phys. Rev. 82, 403, (1951b).
[4] Anderson, P.W., Hasegawa, H., Phys. Rev. 100, 675, (1955).
[5] Goodenough, J., Phys. Rev. 100, 564,(1955).
[7] Kasuya, T., Yanase, A., Rev. Mod. Phys. 40, 684,(1968).
[8] Holstein, T., Ann. Phys. 8, 343, (1959).
[9] Kubo, K., Ohata, N., J. Phys. Soc. Japan 33, 21, (1972).
[10] Jahn, H.A., Teller, E., Proc. Roy. Soc. London A 161, 220, (1937).
[11] Moreo, A., Yunoki, S., Dagotto, E., Science 283, 2034, (1999a).
[12] Matsumoto, G., J. Phys. Soc. Japan 29, 606,( 1970a).
[13] Jirak, Z., Krupicka, S., Simsa, Z., Dlouha, M., Vratislav,
Z., J. Magn. Magn. Mater. 53, 153,( 1985).
[14] Ramirez, A.P., Schi!er, P., Cheong, S-W., Chen, C.H., Bao,
W., Palstra, T.T.M., Gammel, P.L., Bishop, D.J., Zegarski,
B., Phys. Rev. Lett. 76, 3188,(1996).
[15] Cheong, S.-W., Hwang, H.Y.,In: Tokura, Y. (Ed.), Contribution to Colossal Magnetoresistance Oxides,
Monographs in Condensed Matter Science.. Gordon & Breach,
London. (1999).
[16] Zhao, G.-M., Conder, K., Keller, H., MuK ller, K.A., Nature 381, 676,(1996).
[17] Zhao, G.-M., Conder, K., Keller, H., MuK ller, K.A., Phys. Rev. B60, 11,914,(1999.)
[18] Radaelli, P.G., Cox, D.E., Capogna, L., Cheong, S.-W.,
Marezio, M., Phys. Rev. B 59, 14,440,(1999).
[19] M. Imada, A. Fujimori, Y. Tokura, Rev. Mod. Phys. 70, 1039
(1998); Y. Tokura, Y. Tomioka, J.Mag. Mag. Mat. 200,1
(1999); E. L. Nagaev, Phys. Rep. 346, 387 (2001);
E. Dagotto. T. Hotta, A. Moreo, Phys. Rep. 344, 1 (2001).
[20] A. J. Millis, B. I. Shraiman and R. Mueller, Phys. Rev.
Lett. 77, 175 (1996); J. Zang, A. R. Bishop, and H. Roder,
Phys. Rev. B 53, 8840 (1996)
[21]A. J. Millis, Nature 392, 147 (1998)
[22] S. W. Cheong and H. Y. Hwang: Ferromagnetism vs
Charge/Orbital Ordering in Mixed-Valent Manganites, in
Colossal Magnetoresistance Oxides, edited by Y. Tokura
(Gordon & Breach, Monographs in Condensed Matter Science,
London 1999)
[23] Tran Minh-Tien, Phys. Rev. B 67, 144404-1 (2003); T.
Mizokawa and A. Fujimori, Phys. Rev. B 56, R493 (1997);
Priya Mahadevan, K. Terakura, and D. D. Sarma, Phys. Rev.
Lett. 87, 0664041 (2001); I. V. Solovyev, and K. Terakura,
Phys. Rev. Lett. 83, 2825 (1999); T. Mizokawa, D. I.
Khomskii and G. A. Sawatzky, Phys. Rev. B 61, R3776 (2000)
[24] Y. Murakami, et al., Phys. Rev. Lett. 80, 1932 (1998); Y.
Tomioka, et al., Phys. Rev. B 53, R1689 (1996)
[25] P. SchiRer, A. P. Ramirez, W. Bao, and S. W. Cheong, Phys.
Rev. Lett. 75, 3336 (1995)
[26] J. C. Loudon, N. D. Mathur, and P. A. Mldgley, Nature 420,
797 (2002).