电荷密度波与Grüner方程的向量场分析
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摘要
准一维材料中电荷密度波现象是电子在强关联作用下的一种集体凝聚的表现。人们在铜氧化合物高温超导陶瓷材料以及很多其他材料中都可以发现电荷密度波的存在。
电荷密度波作为一种电子集体凝聚状态,在导电特性方面,会表现出非线性电导、窄带噪声、存在非线性电导的启动阈场等现象。在电荷密度波理论研究方面, Grüner及其合作者提出所谓单粒子模型,并给出一个非线性微分方程,即Grüner方程,来描述电荷密度波的导电行为。由于Grüner方程是一个非线性微分方程,人们无法得到它的严格的解析解。Grüner等人运用过阻尼近似方法,略去方程中的二阶微分项,得到比较满意结果。但是,Grüner得到的电导公式的导数在阈场处是发散的,这与实验结果不太符合。通过分析可以看到,这种缺陷并非Grüner方程本身所造成的,是由于对方程的近似处理所造成的。
本篇文章中,运用微分方程向量场理论对Grüner方程进行了定性分析,得出了在外场大于一定的阈值情况下方程将具有稳定的、唯一的周期解的结论。本文还从数学上对外场与周期解依赖关系进行了严格证明,并得出了在一定条件下,使方程具有周期解的阈值将是恒定不变的。这一结果为我们在物理上的进一步分析,提供了可靠的数学基础。
在此基础上,我们可以从Grüner方程导出,单段电荷密度波在外场作用下将会产生滑移速度叠加周期变化速度的运动,周期变化速度形成窄带噪音,滑移速度形成直流电流分量,并且遵循线性的欧姆定律。我们将这些结果与Portis多分段模型相结合,提出各段电荷密度波之间“弹性串联”机制,将分段与分段之间的相互作用转化为内力处理,并指出单段CDW的周期变化速度中的直流分量将转换为分段之间的相互之间挤压或拉伸的内力而失去对电流直流分量的贡献。最终我们可以导出与Fleming经验公式完全一致结果。
Charge density-waves is a type of electron collective behaviors in a condensed mode. In the CDW materials there are a lot of anomalous characters with it. Nonlinear conductivity, narrow-band noise, threshold field and so on are found by experiments. Grüner and his collaborators first proposed the single-particle model to explain them. In the model they gave a nonlinear differential equation to describing the behaviors of electronic transportation. Because this differential equation which is named as Grüner’s equation doesn’t have analytical solution, they derived the conductivity formula by over-damping approximation. Although this formula satisfied with the experiences very well in some degree, the derivative of conductivity is divergent at the threshold. This is contradictory with the result of experiment, but we think that the contradictory is due to neglecting the second order differential term of the equation.
In this paper, we analyze the equation by the means of the rotating vector field theory. Using the mathematical analysis method we obtain the relation between the applied field E and the periodical solution of the equation. It is concluded that in the over-damping situation the E0 in Grüner’s equation is just the critical value of the applied field E. we also obtain a conclusion that the equation has a unique and stable periodical solution when the applied field E is large than the critical value of the applied field E. According to the conclusion, it could be naturally derived that the sliding current of a segment of charge density wave obeys the Ohm’s law. Sequentially we apply these conclusions with Portis’s multiple segment model. We put forward a idea of elastic connection accounting for the connection of different segments. The theoretical result we derive from these for describing nonlinear conductivity of charge density wave is consistent with the experiential formula given by Fleming.
电荷密度波作为一种电子集体凝聚状态,在导电特性方面,会表现出非线性电导、窄带噪声、存在非线性电导的启动阈场等现象。在电荷密度波理论研究方面, Grüner及其合作者提出所谓单粒子模型,并给出一个非线性微分方程,即Grüner方程,来描述电荷密度波的导电行为。由于Grüner方程是一个非线性微分方程,人们无法得到它的严格的解析解。Grüner等人运用过阻尼近似方法,略去方程中的二阶微分项,得到比较满意结果。但是,Grüner得到的电导公式的导数在阈场处是发散的,这与实验结果不太符合。通过分析可以看到,这种缺陷并非Grüner方程本身所造成的,是由于对方程的近似处理所造成的。
本篇文章中,运用微分方程向量场理论对Grüner方程进行了定性分析,得出了在外场大于一定的阈值情况下方程将具有稳定的、唯一的周期解的结论。本文还从数学上对外场与周期解依赖关系进行了严格证明,并得出了在一定条件下,使方程具有周期解的阈值将是恒定不变的。这一结果为我们在物理上的进一步分析,提供了可靠的数学基础。
在此基础上,我们可以从Grüner方程导出,单段电荷密度波在外场作用下将会产生滑移速度叠加周期变化速度的运动,周期变化速度形成窄带噪音,滑移速度形成直流电流分量,并且遵循线性的欧姆定律。我们将这些结果与Portis多分段模型相结合,提出各段电荷密度波之间“弹性串联”机制,将分段与分段之间的相互作用转化为内力处理,并指出单段CDW的周期变化速度中的直流分量将转换为分段之间的相互之间挤压或拉伸的内力而失去对电流直流分量的贡献。最终我们可以导出与Fleming经验公式完全一致结果。
Charge density-waves is a type of electron collective behaviors in a condensed mode. In the CDW materials there are a lot of anomalous characters with it. Nonlinear conductivity, narrow-band noise, threshold field and so on are found by experiments. Grüner and his collaborators first proposed the single-particle model to explain them. In the model they gave a nonlinear differential equation to describing the behaviors of electronic transportation. Because this differential equation which is named as Grüner’s equation doesn’t have analytical solution, they derived the conductivity formula by over-damping approximation. Although this formula satisfied with the experiences very well in some degree, the derivative of conductivity is divergent at the threshold. This is contradictory with the result of experiment, but we think that the contradictory is due to neglecting the second order differential term of the equation.
In this paper, we analyze the equation by the means of the rotating vector field theory. Using the mathematical analysis method we obtain the relation between the applied field E and the periodical solution of the equation. It is concluded that in the over-damping situation the E0 in Grüner’s equation is just the critical value of the applied field E. we also obtain a conclusion that the equation has a unique and stable periodical solution when the applied field E is large than the critical value of the applied field E. According to the conclusion, it could be naturally derived that the sliding current of a segment of charge density wave obeys the Ohm’s law. Sequentially we apply these conclusions with Portis’s multiple segment model. We put forward a idea of elastic connection accounting for the connection of different segments. The theoretical result we derive from these for describing nonlinear conductivity of charge density wave is consistent with the experiential formula given by Fleming.
引文
[1] 冯端,金国均,凝聚态物理新论,上海:上海科学技术出版社,1992,232-237
[2] 孙鑫,维度和凝聚态物理,物理学进展,1993,13:92-96
[3] V J Emery, S A Kivelson and O Zachar. Spin-gap proximity effect mechanism of high-temperature superconductivity. Phys Rev, 1997, B56: 6120-6147
[4] G Orlandi, F Zerbetto. The infrared spectrum of a non conjugated conducting polymer. Chem Phys Lett, 1991, 187: 642-648
[5] M Tikumoto. Superconductivity in BEDT-TTF based organic metals: An overview. G Saito, S Kagoshima. The physics and chemistry of organic superconductors: Proceedings of the ISSP international symposium, Berlin: Springer-Verlag, 1990:116-121
[6] J T Gammel, A Saxena, I Batistic & A R Bishop. Two-band model for halogen-bridged mixed-valence transition-metal complexes. Phys Rev, 1992, B45: 6408-6434
[7] R E Peierls. Quantum Theory of Solid. London: Oxford University Press, 1955, 108-111
[8] C G Kuper. On the thermal properties of Fr?hlich’s one-dimensional superconductor. Proc R Soc London, 1955, Ser A 227:214-228
[9] H. Fr?hlich. On the theory of superconductivity: the one-dimensional case. Proc Roy Soc London, 1954, A223: 296-305
[10] W Fogle & H perlstein. Semiconductor-to-Metal Transition in the Blue Potassium Molybdenum Bronze K0.30 MoO3: Example of a possible excitonic Insulator. Phys Rev, 1972, B6:1402-1412
[11] L Degiorgi, B Alavi, G Mihály & G Grüner. Complete excitation spectrum of charge-density waves: Optical experiments on K0.3MoO3. Phys Rev, 1991, B44:7808-7819
[12] 冯 天, 王楠林, 陈兆甲 等,电荷密度波材料 K0.3MoO3 及 W 掺杂样品的红外光学响应的研究,物理学报,2002,51:2113-2116
[13] A Meerschaut, J Rouxel. Selenide NbSe3—Preparation and structure. J Less-Common Met, 1975, 39:197-203.
[14] P Monceau, N P Ong, A M Portis, A Meerschaut & J Rouxel. Electric field breakdown of charge-density-wave—Induced anomalies in NbSe3. Phys Rev Lett,1976, 37:602-606
[15]KTsutsumi, T Takagki, M Yamamato, et al. Direct electron-diffraction evidence of charge-density-wave formation in NbSe3. Phys Rev Lett, 1977, 39:1675-1676
[16] N P Ong and P monceau. Anomalous transport properties of a linear-chain metal: NbSe+3. Phys Rev, 1977, B16: 3443-3455
[17] R M Fleming. Electric-field depinning of charge-density waves in NbSe3. Phys Rev, 1980, 22B: 5606-5612
[18] R M Fleming & C C Grimes. Sliding-mode conductivity in NbSe3: Observation of a threshold electric field and conduction noise. Phys Rev Lett, 1979, 42:423-1426
[19] G Grüner, A Zawadowski & P M Chaikin. Nonlinear conductivity and noise due to charge-density-wave depining in NbSe3. Phys Rev Lett, 1981, 46(7): 511-515
[20] P A Lee & T M Rice. Electric field depinning of charge density waves. Phys Rev, 1979, 19B: 3970-3980
[21] A M Portis. Class model of charge density wave transport. Mol Cryst Liq Cryst, 1982, 81: 59-72
[22] John Bardeen. Theory of Non-Ohmic conduction from charge-density wave in NbSe3. Phys Rev Lett, 1979, 42:1498-1500
[23] He Haifeng, Zhang Dianlin. Charge density wave gap formation of NbSe3 detected by electron tunneling. Phys Rev Lett, 1999, 82:811-814
[24] A Meerschaut, J Rouxel, P Haen, P Monceau, et al. Observation of a new phase of the one-dimensional compoumd TaS3:X-Ray characterization and electrical measurements. J Phys, 1979, 40:L-157-159
[25] E Bjerkelund and A. Kjekshus. On the properties of TaS3, TaSe3 and TaYe4. Z Anorg Allgem Chem, 1964, 328:235-242
[26] E Bjerkelund, J H Fermor and A Kjekshus. On the properties of TaS3 and TaSe3. Acta Chem Scand, 1966, 20:1836-1842
[27] C Roucau, R Ayroles, Pmonceau, et al. Electron diffraction and resistivity measurements on the One-Dimensional orthorhombic and monodimic Structure of TaS3:Comparsion with NbSe3. Phys Status Solidi (a), 1980, 62:483-493
[28] T Sambongi, K Tsutsumi, Y Shiozaki, et al. Peierls transition in TaS3. Solid State Commun, 1977, 22:729-731
[29] M Ido, K Tsutsumi,T Sambongi, et al. Pressure dependence of the metal-semiconductor transition in TaS3. Solid State Commun, 1979, 29:399-402
[30] T Takoshima, M Ido, K Tsutsumi, et al. Non-Ohmic conductivity of TaS3 in the low-temperature semiconducting regime. Solid State Commun, 1980, 35:911-915
[31] A H Thompson, A. Zettl and G. Grüner. Charge-density-wave transport in TaS3. Phys Rev Lett, 1981, 47:64-67
[32] K Hasegawa, A Maeda, S Uchida & S Tanaka. Non-linear conductivity of monoclinic TaS3. Solid State Commun,1982, 44:881-883
[33] C M Jachson, A Zettl & G Grüner, A H Thompson. High frequency conductivity in Charge Density Wave semiconductor TaS3. Solid State Commun, 1981, 39:531-534
[34] G Grüner, A Zettl, W G Clark, et al. Observation of narrow-band Charge-Density-Wave noise in TaS3. Phys Rev, 1981, 23B:6813-6815
[35] Wang Z Z, P Monceau, M Renard, P Gressier, L Guemas and A Meerschaut. Charge Density Wave transport in a novel halogened transition metal tetrachalcogenide (NbSe4)3.33I. Solid State Commun, 1983, 47: 439-443
[36] A Zettl & G Grüner. Broad band noise associated with the current carrying Charge Density Wave state in TaS3. Solid State Commun, 1983, 46: 29-32
[37] J Epstein, J S Miller. Two band conductors with variable band filling: probing the properties of quasi-one-dimensional materials. L Alcácer. The physics and chemistry of low dimensional solid: Proceedings of the NATO Advance Study Institute. Dordrecht: D Reidel Publish Company, 1979, 339-351
[38] P A Lee, T M Rice & P W Anderson. Conductivity from charge or spin density waves. Solid State Commun, 1974, 14:703-709
[39] H Fukuyama & P A Lee. Dynamics of the Charge-density wave. Phys Rev, 1978, B17: 535-548
[40] J Bardeen, E Ben-Jacob, A Zettl &G Grüner. Currennt oscillations and stability of charge-density-wave motion in NbSe3. Phys Rev Lett, 1982, 49:493-496
[41] G.Grüner, A.Zettl. Charge density wave conduction: a novel collective transport phenomenon in solids. Phys Rep, 1985, 119:117-232.
[42] D S Fisher. Sliding charge-density waves as a dynamic critical phenomenon. Phys Rev, 1985, B31:1396-1427
[43] G.Grüner. The dynamics of charge-density wave. Rev Mod Phys, 1988, 60:1129-1181
[44] J Bardeen. Tunneling theory of charge-density-wave depinning. Phys Rev Lett, 1980, 45:1978-1980
[45] 李连钢,电荷密度波导电理论,[硕士论文],天津:南开大学,1990
[46] 李连钢,阮永丰,电荷密度波电输运方程-Grüner 方程解的周期性,天津大学学报,2007,40(2) (待发表)
[47] H Poincaré. Mémoire sur les courbes définies par une équation différentielle. Jour Math Pures et Appl, 1881,7(3):375-422; 1882,8:251-296; 1885,1(4):167-244; 1886, 2:151-217
[48] G F D Duff. Limit-cycles and rotated vector fields. Annals of Math, 1953, 57:15-31
[49] G Seifert. Contributions to the theory of nonlinear oscillations, 1958, Ⅳ:125-140
[50] 陈翔炎,广义旋转向量场,南京大学学报, 1975,1:100-108
[51] 马知恩,旋转向量场中奇异闭轨线的运动,西安交大学学报,1978,4:49-65
[52] 李连钢,阮永丰,电荷密度波经典模型的分析,物理学报,2006,55(1):441-445 [53a] 张芷芬,丁同仁,黄文灶 等,微分方程定性理论,北京:科学出版社,1985,51-55 [53b] 张芷芬,丁同仁,黄文灶 等,微分方程定性理论,北京:科学出版社,1985,1-1 [53c] 张芷芬,丁同仁,黄文灶 等,微分方程定性理论,北京:科学出版社,1985,7-8 [54a] 叶彦谦,极限环论,上海:上海科学技术出版社(第 2 版),1984,5-5 [54b] 叶彦谦,极限环论,上海:上海科学技术出版社(第 2 版),1984,29-29
[55] M Urabe. The least upper bound of a damping coefficient insuring the existence of a periodic motion of a pendulum under constant torque. Jour Sci Hiroshima Un,1954, 18A: 379-389
[56] H Fukuyama. Pinning in Peierls-Frohlich state and conductivity. J Phys Soc Jpn, 1976,41:513-520
[57] H Fukuyama, P A Lee. Dynamics of the charge-density wave. Phys Rev, 1978, B17:535-541
[58] M Qjani, A Arbaoui, A Ayadi et al. Numerical simulation of the one-dimensional incommensurate charge density wave dynamicas. 1998, Synthetic Metals, 96:137-140
[59] 李吉晓,石兢,秦晓岿等,钨参掺杂铊蓝青铜中电荷密度波的低频钉扎,低温物理学报,1997,19(5): 358-365
[2] 孙鑫,维度和凝聚态物理,物理学进展,1993,13:92-96
[3] V J Emery, S A Kivelson and O Zachar. Spin-gap proximity effect mechanism of high-temperature superconductivity. Phys Rev, 1997, B56: 6120-6147
[4] G Orlandi, F Zerbetto. The infrared spectrum of a non conjugated conducting polymer. Chem Phys Lett, 1991, 187: 642-648
[5] M Tikumoto. Superconductivity in BEDT-TTF based organic metals: An overview. G Saito, S Kagoshima. The physics and chemistry of organic superconductors: Proceedings of the ISSP international symposium, Berlin: Springer-Verlag, 1990:116-121
[6] J T Gammel, A Saxena, I Batistic & A R Bishop. Two-band model for halogen-bridged mixed-valence transition-metal complexes. Phys Rev, 1992, B45: 6408-6434
[7] R E Peierls. Quantum Theory of Solid. London: Oxford University Press, 1955, 108-111
[8] C G Kuper. On the thermal properties of Fr?hlich’s one-dimensional superconductor. Proc R Soc London, 1955, Ser A 227:214-228
[9] H. Fr?hlich. On the theory of superconductivity: the one-dimensional case. Proc Roy Soc London, 1954, A223: 296-305
[10] W Fogle & H perlstein. Semiconductor-to-Metal Transition in the Blue Potassium Molybdenum Bronze K0.30 MoO3: Example of a possible excitonic Insulator. Phys Rev, 1972, B6:1402-1412
[11] L Degiorgi, B Alavi, G Mihály & G Grüner. Complete excitation spectrum of charge-density waves: Optical experiments on K0.3MoO3. Phys Rev, 1991, B44:7808-7819
[12] 冯 天, 王楠林, 陈兆甲 等,电荷密度波材料 K0.3MoO3 及 W 掺杂样品的红外光学响应的研究,物理学报,2002,51:2113-2116
[13] A Meerschaut, J Rouxel. Selenide NbSe3—Preparation and structure. J Less-Common Met, 1975, 39:197-203.
[14] P Monceau, N P Ong, A M Portis, A Meerschaut & J Rouxel. Electric field breakdown of charge-density-wave—Induced anomalies in NbSe3. Phys Rev Lett,1976, 37:602-606
[15]KTsutsumi, T Takagki, M Yamamato, et al. Direct electron-diffraction evidence of charge-density-wave formation in NbSe3. Phys Rev Lett, 1977, 39:1675-1676
[16] N P Ong and P monceau. Anomalous transport properties of a linear-chain metal: NbSe+3. Phys Rev, 1977, B16: 3443-3455
[17] R M Fleming. Electric-field depinning of charge-density waves in NbSe3. Phys Rev, 1980, 22B: 5606-5612
[18] R M Fleming & C C Grimes. Sliding-mode conductivity in NbSe3: Observation of a threshold electric field and conduction noise. Phys Rev Lett, 1979, 42:423-1426
[19] G Grüner, A Zawadowski & P M Chaikin. Nonlinear conductivity and noise due to charge-density-wave depining in NbSe3. Phys Rev Lett, 1981, 46(7): 511-515
[20] P A Lee & T M Rice. Electric field depinning of charge density waves. Phys Rev, 1979, 19B: 3970-3980
[21] A M Portis. Class model of charge density wave transport. Mol Cryst Liq Cryst, 1982, 81: 59-72
[22] John Bardeen. Theory of Non-Ohmic conduction from charge-density wave in NbSe3. Phys Rev Lett, 1979, 42:1498-1500
[23] He Haifeng, Zhang Dianlin. Charge density wave gap formation of NbSe3 detected by electron tunneling. Phys Rev Lett, 1999, 82:811-814
[24] A Meerschaut, J Rouxel, P Haen, P Monceau, et al. Observation of a new phase of the one-dimensional compoumd TaS3:X-Ray characterization and electrical measurements. J Phys, 1979, 40:L-157-159
[25] E Bjerkelund and A. Kjekshus. On the properties of TaS3, TaSe3 and TaYe4. Z Anorg Allgem Chem, 1964, 328:235-242
[26] E Bjerkelund, J H Fermor and A Kjekshus. On the properties of TaS3 and TaSe3. Acta Chem Scand, 1966, 20:1836-1842
[27] C Roucau, R Ayroles, Pmonceau, et al. Electron diffraction and resistivity measurements on the One-Dimensional orthorhombic and monodimic Structure of TaS3:Comparsion with NbSe3. Phys Status Solidi (a), 1980, 62:483-493
[28] T Sambongi, K Tsutsumi, Y Shiozaki, et al. Peierls transition in TaS3. Solid State Commun, 1977, 22:729-731
[29] M Ido, K Tsutsumi,T Sambongi, et al. Pressure dependence of the metal-semiconductor transition in TaS3. Solid State Commun, 1979, 29:399-402
[30] T Takoshima, M Ido, K Tsutsumi, et al. Non-Ohmic conductivity of TaS3 in the low-temperature semiconducting regime. Solid State Commun, 1980, 35:911-915
[31] A H Thompson, A. Zettl and G. Grüner. Charge-density-wave transport in TaS3. Phys Rev Lett, 1981, 47:64-67
[32] K Hasegawa, A Maeda, S Uchida & S Tanaka. Non-linear conductivity of monoclinic TaS3. Solid State Commun,1982, 44:881-883
[33] C M Jachson, A Zettl & G Grüner, A H Thompson. High frequency conductivity in Charge Density Wave semiconductor TaS3. Solid State Commun, 1981, 39:531-534
[34] G Grüner, A Zettl, W G Clark, et al. Observation of narrow-band Charge-Density-Wave noise in TaS3. Phys Rev, 1981, 23B:6813-6815
[35] Wang Z Z, P Monceau, M Renard, P Gressier, L Guemas and A Meerschaut. Charge Density Wave transport in a novel halogened transition metal tetrachalcogenide (NbSe4)3.33I. Solid State Commun, 1983, 47: 439-443
[36] A Zettl & G Grüner. Broad band noise associated with the current carrying Charge Density Wave state in TaS3. Solid State Commun, 1983, 46: 29-32
[37] J Epstein, J S Miller. Two band conductors with variable band filling: probing the properties of quasi-one-dimensional materials. L Alcácer. The physics and chemistry of low dimensional solid: Proceedings of the NATO Advance Study Institute. Dordrecht: D Reidel Publish Company, 1979, 339-351
[38] P A Lee, T M Rice & P W Anderson. Conductivity from charge or spin density waves. Solid State Commun, 1974, 14:703-709
[39] H Fukuyama & P A Lee. Dynamics of the Charge-density wave. Phys Rev, 1978, B17: 535-548
[40] J Bardeen, E Ben-Jacob, A Zettl &G Grüner. Currennt oscillations and stability of charge-density-wave motion in NbSe3. Phys Rev Lett, 1982, 49:493-496
[41] G.Grüner, A.Zettl. Charge density wave conduction: a novel collective transport phenomenon in solids. Phys Rep, 1985, 119:117-232.
[42] D S Fisher. Sliding charge-density waves as a dynamic critical phenomenon. Phys Rev, 1985, B31:1396-1427
[43] G.Grüner. The dynamics of charge-density wave. Rev Mod Phys, 1988, 60:1129-1181
[44] J Bardeen. Tunneling theory of charge-density-wave depinning. Phys Rev Lett, 1980, 45:1978-1980
[45] 李连钢,电荷密度波导电理论,[硕士论文],天津:南开大学,1990
[46] 李连钢,阮永丰,电荷密度波电输运方程-Grüner 方程解的周期性,天津大学学报,2007,40(2) (待发表)
[47] H Poincaré. Mémoire sur les courbes définies par une équation différentielle. Jour Math Pures et Appl, 1881,7(3):375-422; 1882,8:251-296; 1885,1(4):167-244; 1886, 2:151-217
[48] G F D Duff. Limit-cycles and rotated vector fields. Annals of Math, 1953, 57:15-31
[49] G Seifert. Contributions to the theory of nonlinear oscillations, 1958, Ⅳ:125-140
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