d波超導之磁性性質

Abstract

京茲堡-朗道理論已經成功地預言許多有關d波超導體的有趣性質。至目前為止，大部分被提出的理論均假設序參量具有單純的d波對稱性或s-d波混合對稱性。這篇論文是在惟象的和微觀的京茲堡-朗道理論架構下，對d波超導體的磁性性質作更深入研究。 首先，我們以質量為非均向性的京茲堡-朗道理論為基礎，推導由外加場引起的非零、次要序參量，其振幅與超流之振幅及方向有關。在某些特殊情況下，被引起的s波分量之極值產生。不同於質量均向性結果，我們也發現超流密度與波向量不平行。 利用相同理論，我們研究外加磁場對渦旋結構影響，我們發現s波的彎曲數與向量位能無關。我們也推導d波超導體所對應之廣義倫敦方程式，忽略高階項，倫敦方程式被解。此外，我們計算兩個平行渦漩之交互作用，並且證明之間存在力矩，這個重大發現指出因質量非均向性所造成的力矩，可能改變晶格結構。 在京茲堡-朗道理論範圍內，d波超導體是由a-b平面超導層偶和由有效質量近似之垂直方向組成，此外，要求垂直方向之相干長度大於層與層間距離，在此模型基礎下，我們計算出上臨界場，根據結果，上臨界場相對溫度之曲線的斜率和外加磁場與c軸之夾角有關。最有興趣的是c方向參數對a-b平面比值會影響上臨界場，當此比值遞減，上臨界場遞增。我們也發現在上臨界場範圍內，s波分量不存在，這意味上臨界場相對溫度之曲線圖會朝上是d波超導體特性之一。 最後我們以依賴時間的京茲堡-朗道理論為出發，研究d波超導體的渦旋運動方程式，考慮複數緩和時間，結果發現虛部導致霍爾效應變號。京茲堡-朗道自由能參數會因非磁性雜質而不同，此改變也會引起霍爾角度變號，這意味著非磁性雜質與異常霍爾效應有關。

The Ginzburg-Landau theory has successfully predicted many interesting properties of $d$-wave superconductors near $T_c$. Most of the theories proposed so far assumed the order parameter with purely $d_{x^2-y^2}$-wave symmetry or mixed $s$+$d$ symmetry. Within the context of the relevant phenomenological and microscopical Ginzburg-Landau theory, this thesis is concerned with studies of the magnetic properties for a $d$-wave superconductor. Underlain by the Ginzburg-Landau theory of a $d$-wave superconductor with mass anisotropy $\lambda = m_x/m_y$, we have derived the non-zero, subsidiary order parameter induced by an external current. The amplitude of induced $s$-component depends on the amplitude and direction of the current. The extreme values appear in some specific conditions. Unlike the isotropic case, we find that the current density is not parallel to the wave vector any more except $M_a = 1$. The single vortex structure is derived in the presence of applied magnetic field. According to our results, the winding number of the $s$-wave does not change regardless of the vector potential. The generic London equation for a $d_{x^2-y^2}$-wave superconductor with mass anisotropy is expressed. By neglecting higher order terms, this work analyzes the magnetic-field distribution with and without a vortex. The interaction force between two parallel vortices is derived as well. Our results further reveal the presence of a torque between vortices irrespective of $s$- or $d$-wave order parameter, which is expected to vanish for isotropic cases. This implies that a torque between vortices due to mass anisotropy can change the lattice structure. A $d_{x^2-y^2}$ superconductor is modeled as the superconducting layers in the $a$-$b$ plane, whose coupling in the $c$-direction is approximated by the effective mass, within the Ginzburg-Landau theory. In this work, this model is applied to the system where the coherence length along the $c$-direction is greater than the layer spacing. Based on our model, we calculate the upper critical field in a magnetic field lying in $a$-$c$ plane and tilted by an angle from the $c$-axis. According to our results, the curvature of $H_{c2}(T)$ is upward, and the slope $-dH_{c2}(T)/dT$ depends on the angle between the c-axis and the external field. It is worthwhile to note that the ratio of the $c$-direction parameter related to the effective mass to the $a$-$b$ plane parameter connected with the effective mass can make enormous influence $H_{c2}$. As this ratio decreases, $H_{c2}$ becomes increasing. We also find that there is no admixture of $s$-wave component in the critical regime and believe that the upward curvature of the $H_{c2}(T)$ is illuminated as the characteristic property of a $d$-wave superconductor. In the context of the time-dependent Ginzburg-Landau theory for a $d_{x^2-y^2}$-wave superconductor, we investigate the energy theorem that demonstrates that the rate of increase of the total free energy plus the rate of dissipation equals the inflow of energy current. In addition, we derive an equation of motion for a single vortex $(h \ll H_{c2})$ in the presence of an applied transport current. Our results indicate that the imaginary parts of the relaxation times for $s$- and $d$-wave order parameters can change a sign of the Hall effect. We also find that the change of the parameters in the Ginzburg-Landau free energy functional, due to the nonmagnetic impurities, can affect the anomalous Hall effect. The tangent of Hall angle is investigated and found that the negative part of the tangent of Hall angle is essentially due to the imaginary parts of the relaxation time.