二维磁振子晶体带隙优化及缺陷态性质的研究
摘要
磁振子晶体作为一种人工微纳米周期性复合材料,其最基本特点在于自旋波在其中传播时由于受周期性结构的调制而产生带隙,处于带隙范围内的自旋波不能在晶体中传播。该性质使磁振子晶体具有不同于一般材料的特殊物理性质,人们可以根据自己的需要设计不同的带隙,使自旋波局域在特定区域或沿着特定的方向传播,实现对自旋波传播的可控操作。因此,自旋波在磁振子晶体中的传播便成为当前人们研究的热点,且这些研究在微波领域具有潜在广泛的应用价值,如设计微波滤波器、导波器、转换开关、电流控制延迟线、传感器、自旋波逻辑器件及信号处理器等。
本论文主要围绕二维磁振子晶体带隙结构的优化及缺陷态性质进行了研究,主要内容如下。
首先,本文从自旋波波动方程入手,利用布洛赫定理及傅里叶级数展开详细地讨论了平面波展开法及短波微扰下改进的平面波展开法的理论,通过比较两种方法发现后者在计算过程中更为可行。因此,本文采用改进的平面波展开法把自旋波波动方程化解成一个本征方程,数值求解本征值,即可得到自旋波的K～Ω色散关系,此即磁振子晶体自旋波带结构。
第二,利用改进的平面波展开法,理论上计算了由Fe柱体正方周期排列在EuO基底中所构成二维磁振子晶体的自旋波带结构,并分别讨论了旋转截面为正方形、长方形和六边形的非圆散射柱体对带隙的影响。结果表明,通过旋转非圆柱散射体可使带隙的宽度增加或产生新的带隙,且当柱体正方排列在基底中时旋转正方柱体在某一体积填充率下可产生最宽的带隙。这种方法为设计及优化二维磁振子晶体带隙打开新的视野。
第三,利用改进的平面波展开法及超原胞近似,研究了二维磁振子晶体中具有单个点缺陷的缺陷态局域化性质,详细地讨论二维磁振子晶体中其它正常散射柱体取不同形状(正方柱体和圆柱体)时,缺陷柱体的形状(正方柱体、圆柱体及长方柱体)及其尺寸大小对缺陷态的影响。研究表明,缺陷对自旋波有局域限制作用,使其不能沿着晶体内其他方向传播。此外,我们发现对于正方或圆缺陷柱体而言,缺陷体体积填充率的大小直接决定着缺陷体的形状是否影响缺陷模性质,对体积填充率较小情形缺陷模的性质与缺陷体的形状无关,而对于体积填充率较大情形,则缺陷模的性质随缺陷体形状的变化而变化。然而,对于长方缺陷柱体而言,不论体积填充率大小,缺陷模的性质均随缺陷体形状的变化而变化,而且当缺陷体截面边长比值达到一定值时,某些双重简并缺陷模将劈裂为非简并模。
最后,在单个点缺陷研究的基础上,二维磁振子晶体中引入多个点缺陷,包括两点缺陷、多点缺陷及线缺陷,并分别对点缺陷间的能量耦合及线缺陷的缺陷态性质进行研究。对点缺陷间耦合而言,缺陷模间的耦合导致缺陷模的劈裂,且劈裂程度随两缺陷体间距离的减小而增加,缺陷模磁化强度分布的相位旋进也随之变化；而且,缺陷体沿着(10)方向缺陷模间的耦合较(11)方向强。对多点缺陷及线缺陷而言,某些频率的自旋波可以沿着缺陷方向传播。具有该性质的磁振子晶体可被用作方向性滤波器或窄带自旋波波导器件的制作材料。
本论文的研究工作均为数值计算研究,所得研究结果可为实际应用中自旋波器件材料的设计提供具有一定意义的理论指导。
Magnonic crystals (MCs) can be regarded as an artificial micro-nano periodic composite material, and its fundamental feature is that band gaps exists in the band structures of spin waves because of the modulation of the periodicity, so as to the spin waves which frequencies locate in the band gaps are forbidden to propagate in the magnonic crystals. The special property distinguishes the magnonic crystals from other general materials, and people can design different band gaps according to their requirements to make the spin wave locate in some certain areas or propagate along some certain directions so that the control for spin wave propagating in the magnonic crystals can be realized. Thus, it has been become a hotspot of current research, and these researches possess potential applications for the design of microwave filters, waveguide, switches, current controlled delay lines, sensors, spin-wave logical device, signal processor and so on.
In this paper, the optimization of band gaps and the properties of defect states in two-dimensional (2D) magnonic crystals are mainly investigated, and the detailed contents are listed as follows:
Firstly, based on spin-wave dynamics equation, the traditional plane-wave expansion (PWE) method and the improved plane-wave expansion method have been compared in detail by using Bloch theorem and Fourier expansion, and we found the latter is more practicable under the short-wavelength perturbations. So, we change the dynamics equation into the eigen equations of spin waves by using the improved plane-wave expansion method, and then the eigenvalues can be obtained numerically, and the spin-waves dispersions relation of K～Ω are also gained, which are the band structures of spin waves in magnonic crystals.
Secondly, spin-wave band structures of two-dimensional magnonic crystals composed of Fe rods squarely arranged in a EuO matrix are theoretically calculated using the improved plane-wave expansion method, and the effects of rotating noncircular scattereres (square, rectangle and hexagon) on band gaps are discussed. The results indicate that it is possible to increase the width of the band gaps or to create new band gaps by rotating the noncircular rods, and the width of the band gaps is the largest under a certain filling fraction by rotating the square rods when the rods squarely arranged in a matrix. Such an approach may open up a new scope for engineering and optimizating band gaps of two-dimensional magnonic crystals.
Thirdly, using the improved plane-wave expansion method under supercell approximation, localized properties of defect states in two-dimensional magnonic crystals with single point defects are studied, and the effects of shape (square, circular and rectangular rods) and size of the defect rods on defect states are also discussed when other regular scattereres of magnonic crystals have different shapes, such as square rods and circular rods. The studies show that spin wave is localized around the defect body and forbidden from propagating along other direction in the magnonic crystals. Furthermore, we have found that for the square or circular defect, the influence of the point-defect shape to the properties of the defect modes directly depends,on the magnitude of filling fraction of the point defect body, that is to say, the properties of the defect modes are independent of the shape of the defect for a smaller point defect body, contrarily, they are relevant for a bigger defect filling fraction. However, for the rectangular defect, the defect modes show a dependence on its shape regardless of the defect filling fraction, moreover, the double degenerate defect modes will split into two nondegenerate modes when the ratio of edge widths of the defect reaches a certain value.
Finally, based on the study of single point defect, two point defects, multi-point defects and line defects also can be introduced into two-dimensional magnonic crystals, and the properties of the these defect states are investigated, respectively. For the case of coupling between two point defects, the defect modes are split due to the coupling between defect modes, and the splitting degree increases with the decreasing the distance between two defects, and the phase of precession of defect modes'magnetization distributions also varies as the distance. Moreover, the coupling of defect modes is stronger for the defects along (10) direction than the case of the (11) direction. For multi-point defects and line defects, spin waves of certain frequency can propagate along the direction of defects. The magnonic crystals with such properties can be used as the fabricating materials of the directional spin-wave filters or narrow band spin-wave waveguides.
The above mentioned numerical results can provide theoretically some values for the applications of magnonic crystals into the fabricating materials of spin-wave devices.
本论文主要围绕二维磁振子晶体带隙结构的优化及缺陷态性质进行了研究,主要内容如下。
首先,本文从自旋波波动方程入手,利用布洛赫定理及傅里叶级数展开详细地讨论了平面波展开法及短波微扰下改进的平面波展开法的理论,通过比较两种方法发现后者在计算过程中更为可行。因此,本文采用改进的平面波展开法把自旋波波动方程化解成一个本征方程,数值求解本征值,即可得到自旋波的K～Ω色散关系,此即磁振子晶体自旋波带结构。
第二,利用改进的平面波展开法,理论上计算了由Fe柱体正方周期排列在EuO基底中所构成二维磁振子晶体的自旋波带结构,并分别讨论了旋转截面为正方形、长方形和六边形的非圆散射柱体对带隙的影响。结果表明,通过旋转非圆柱散射体可使带隙的宽度增加或产生新的带隙,且当柱体正方排列在基底中时旋转正方柱体在某一体积填充率下可产生最宽的带隙。这种方法为设计及优化二维磁振子晶体带隙打开新的视野。
第三,利用改进的平面波展开法及超原胞近似,研究了二维磁振子晶体中具有单个点缺陷的缺陷态局域化性质,详细地讨论二维磁振子晶体中其它正常散射柱体取不同形状(正方柱体和圆柱体)时,缺陷柱体的形状(正方柱体、圆柱体及长方柱体)及其尺寸大小对缺陷态的影响。研究表明,缺陷对自旋波有局域限制作用,使其不能沿着晶体内其他方向传播。此外,我们发现对于正方或圆缺陷柱体而言,缺陷体体积填充率的大小直接决定着缺陷体的形状是否影响缺陷模性质,对体积填充率较小情形缺陷模的性质与缺陷体的形状无关,而对于体积填充率较大情形,则缺陷模的性质随缺陷体形状的变化而变化。然而,对于长方缺陷柱体而言,不论体积填充率大小,缺陷模的性质均随缺陷体形状的变化而变化,而且当缺陷体截面边长比值达到一定值时,某些双重简并缺陷模将劈裂为非简并模。
最后,在单个点缺陷研究的基础上,二维磁振子晶体中引入多个点缺陷,包括两点缺陷、多点缺陷及线缺陷,并分别对点缺陷间的能量耦合及线缺陷的缺陷态性质进行研究。对点缺陷间耦合而言,缺陷模间的耦合导致缺陷模的劈裂,且劈裂程度随两缺陷体间距离的减小而增加,缺陷模磁化强度分布的相位旋进也随之变化；而且,缺陷体沿着(10)方向缺陷模间的耦合较(11)方向强。对多点缺陷及线缺陷而言,某些频率的自旋波可以沿着缺陷方向传播。具有该性质的磁振子晶体可被用作方向性滤波器或窄带自旋波波导器件的制作材料。
本论文的研究工作均为数值计算研究,所得研究结果可为实际应用中自旋波器件材料的设计提供具有一定意义的理论指导。
Magnonic crystals (MCs) can be regarded as an artificial micro-nano periodic composite material, and its fundamental feature is that band gaps exists in the band structures of spin waves because of the modulation of the periodicity, so as to the spin waves which frequencies locate in the band gaps are forbidden to propagate in the magnonic crystals. The special property distinguishes the magnonic crystals from other general materials, and people can design different band gaps according to their requirements to make the spin wave locate in some certain areas or propagate along some certain directions so that the control for spin wave propagating in the magnonic crystals can be realized. Thus, it has been become a hotspot of current research, and these researches possess potential applications for the design of microwave filters, waveguide, switches, current controlled delay lines, sensors, spin-wave logical device, signal processor and so on.
In this paper, the optimization of band gaps and the properties of defect states in two-dimensional (2D) magnonic crystals are mainly investigated, and the detailed contents are listed as follows:
Firstly, based on spin-wave dynamics equation, the traditional plane-wave expansion (PWE) method and the improved plane-wave expansion method have been compared in detail by using Bloch theorem and Fourier expansion, and we found the latter is more practicable under the short-wavelength perturbations. So, we change the dynamics equation into the eigen equations of spin waves by using the improved plane-wave expansion method, and then the eigenvalues can be obtained numerically, and the spin-waves dispersions relation of K～Ω are also gained, which are the band structures of spin waves in magnonic crystals.
Secondly, spin-wave band structures of two-dimensional magnonic crystals composed of Fe rods squarely arranged in a EuO matrix are theoretically calculated using the improved plane-wave expansion method, and the effects of rotating noncircular scattereres (square, rectangle and hexagon) on band gaps are discussed. The results indicate that it is possible to increase the width of the band gaps or to create new band gaps by rotating the noncircular rods, and the width of the band gaps is the largest under a certain filling fraction by rotating the square rods when the rods squarely arranged in a matrix. Such an approach may open up a new scope for engineering and optimizating band gaps of two-dimensional magnonic crystals.
Thirdly, using the improved plane-wave expansion method under supercell approximation, localized properties of defect states in two-dimensional magnonic crystals with single point defects are studied, and the effects of shape (square, circular and rectangular rods) and size of the defect rods on defect states are also discussed when other regular scattereres of magnonic crystals have different shapes, such as square rods and circular rods. The studies show that spin wave is localized around the defect body and forbidden from propagating along other direction in the magnonic crystals. Furthermore, we have found that for the square or circular defect, the influence of the point-defect shape to the properties of the defect modes directly depends,on the magnitude of filling fraction of the point defect body, that is to say, the properties of the defect modes are independent of the shape of the defect for a smaller point defect body, contrarily, they are relevant for a bigger defect filling fraction. However, for the rectangular defect, the defect modes show a dependence on its shape regardless of the defect filling fraction, moreover, the double degenerate defect modes will split into two nondegenerate modes when the ratio of edge widths of the defect reaches a certain value.
Finally, based on the study of single point defect, two point defects, multi-point defects and line defects also can be introduced into two-dimensional magnonic crystals, and the properties of the these defect states are investigated, respectively. For the case of coupling between two point defects, the defect modes are split due to the coupling between defect modes, and the splitting degree increases with the decreasing the distance between two defects, and the phase of precession of defect modes'magnetization distributions also varies as the distance. Moreover, the coupling of defect modes is stronger for the defects along (10) direction than the case of the (11) direction. For multi-point defects and line defects, spin waves of certain frequency can propagate along the direction of defects. The magnonic crystals with such properties can be used as the fabricating materials of the directional spin-wave filters or narrow band spin-wave waveguides.
The above mentioned numerical results can provide theoretically some values for the applications of magnonic crystals into the fabricating materials of spin-wave devices.
引文
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