-
用于自旋探测的超灵敏悬臂梁,其端头带有球形微磁体作为探针,与样品中的自旋发生相互作用。对于球形磁探针,球内、外均无自由电流分布,设球外为真空,球内均匀磁化,设球形磁探针半径为
${R_0}$ ,双臂差动放大式悬臂梁的球形磁探针中磁场分布满足如下方程:$$\left\{ \begin{array}{l} {\nabla ^2}{\varphi _{m1}} = 0,R > {R_0} \\ {\nabla ^2}{\varphi _{m2}} = 0,R < {R_0} \\ {\left. {{\varphi _{m1}}} \right|_{R \to \infty }} = 0 \\ {\left. {{\varphi _{m2}}} \right|_{R = 0}} < \infty \\ {\left. {{\varphi _{m1}}} \right|_S} = {\left. {{\varphi _{m2}}} \right|_S} \\ {\mu _0}{\left. {\dfrac{{\partial {\varphi _{m1}}}}{{\partial n}}} \right|_S} = {\mu _0}\left({\left. {\dfrac{{\partial {\varphi _{m2}}}}{{\partial n}}} \right|_S} + {\overrightarrow M _{OR}}\right) \\ \end{array} \right.$$ (1) 方程的通解为:
$$\left\{ \begin{array}{l} {\varphi _{{{m1}}}} = \displaystyle\mathop \sum \limits_n [{a_n}{R^n} + {b_n}{R^{ - (n + 1)}}]{P_n}(\cos \theta ) \\ {\varphi _{{{m2}}}} = \displaystyle\mathop \sum \limits_n [{c_n}{R^n} + {d_n}{R^{ - (n + 1)}}]{P_n}(\cos \theta ) \end{array} \right.$$ (2) 对于球形磁探针外部的磁场,有:
$$ \begin{split} {\overrightarrow{H}}_{1}=&-\left(\dfrac{\partial }{\partial R}\overrightarrow{{\rm e}_{R}}+\dfrac{1}{R}\dfrac{\partial }{\partial \theta }\overrightarrow{{\rm e}_{\theta }}+\dfrac{1}{R\mathrm{sin}\theta }\dfrac{\partial }{\partial \varphi }\overrightarrow{{\rm e}_{\varphi }}\right)\dfrac{{R}_{0}{}^{3}{M}_{0}\mathrm{cos}\theta }{3{R}^{2}}=\\ &\dfrac{1}{4\pi }\left[\dfrac{3(\overrightarrow{m\cdot }\overrightarrow{R})\overrightarrow{R}}{{R}^{5}}-\dfrac{\overrightarrow{m}}{{R}^{3}}\right] \end{split}$$ (3) 磁感应强度为:
$${\overrightarrow B _1} = {\mu _0}{\overrightarrow H _1} = \frac{{{\mu _0}}}{{4\pi }}\left[\frac{{3(\overrightarrow {m \cdot } \overrightarrow R )\overrightarrow R }}{{{R^5}}} - \frac{{\overrightarrow m }}{{{R^3}}}\right]$$ (4) 对于球形磁探针内部的磁场,有:
$$ \begin{split} \overrightarrow{{H}_{2}}=&-\nabla {\varphi }_{\rm{m2}}=-\left(\frac{\partial }{\partial R}\overrightarrow{{\rm e}_{R}}+\frac{1}{R}\frac{\partial }{\partial \theta }\overrightarrow{{\rm e}_{\theta }}+\frac{1}{R\mathrm{sin}\theta }\frac{\partial }{\partial \varphi }\overrightarrow{{\rm e}_{\varphi }}\right)\cdot\\ &\frac{R{M}_{0}\mathrm{cos}\theta }{3}=-\frac{1}{3}{M}_{0}\overrightarrow{{\rm e}_{{\textit{z}}}} \end{split}$$ (5) 磁感应强度为:
$$\overrightarrow {{B_2}} = {\mu _0}({\overrightarrow H _2} + {\overrightarrow M _2}) = {\mu _0}\left( - \frac{1}{3}\overrightarrow {{M_0}} + \overrightarrow {{M_0}} \right) = \frac{2}{3}{\mu _0}\overrightarrow {{M_0}} $$ (6) 如图3建立的坐标系,以球形磁探针a的球心所在处为坐标原点,两探针球心之间的距离为d,则在z轴上距离坐标原点为R处的磁感应强度
${{{B}}_{\textit{z}}}$ 为${{{B}}_0}$ 、${{{B}}_a}$ 和${{{B}}_b}$ 的矢量叠加场。由于:$$ \overrightarrow {{B_{a{\textit{z}}}}} = - \frac{{4{\mu _0}{R_0}^3{M_0}}}{{3{R^3}}}\overrightarrow {{{\rm e}_{\textit{z}}}} ,\;\overrightarrow {{B_{b{\textit{z}}}}} = \frac{{2{\mu _0}}}{3}\frac{{{R_0}^3{M_0}}}{{{{(d - R)}^3}}}\overrightarrow {{{\rm e}_{\textit{z}}}} $$ (7) 则样品实际感受到的磁感应强度为:
$$\overrightarrow {{B_{\textit{z}}}} = \overrightarrow {{B_{a{\textit{z}}}}} + \overrightarrow {{B_{b{\textit{z}}}}} + \overrightarrow {{B_0}}$$ (8) 以钐钴合金磁球作为磁探针,对双臂差动放大式悬臂梁的灵敏度进行数值模拟。根据参考文献[5, 7],单晶硅悬臂梁的尺寸取为
${\rm{465}}\;{\text{μ}} {\rm{m}} \times 10\;{\text{μ}} {\rm{m}} \times 0.5\;{\text{μ}} {\rm{m}}$ ,密度为$\;\rho = 2.33 \times {10^3}$ kg/m3,杨氏模量为$E = 168$ GPa,则悬臂梁的有效质量为:$${m_{can}} = \frac{1}{4}\rho wlt = 1.354 \times {10^{ - 12}}\;{\rm{kg}}$$ (9) 弹性系数为:
$$k = \frac{1}{4}Ew{\left(\frac{t}{l}\right)^3} = 5.221 \times {10^{ - 4}}\;{\rm{N/m}}$$ (10) 悬臂梁的共振频率值为:
$${f_0} = \frac{1}{{2\pi }}\sqrt {\frac{k}{{{m_{can}}}}} = 3\;125\;{\rm{Hz}}$$ (11) 首先考虑无质量加载的情况,依据参考文献[8],取品质因数
$Q = 23\;000$ ,并假设阻尼不变,则悬臂梁的最大测力灵敏度为:$${F_{\min }} = t{\left(\frac{w}{{lQ}}\right)^{1/2}}{(E\rho )^{1/4}}{({k_B}TB)^{1/2}} = 7.382 \;{\rm aN}$$ (12) 下面考虑有质量加载的情况,依据参考文献[8],如果在悬臂梁的一端加载质量为
$\Delta m$ ,钐钴合金磁球探针质量取${m_0} = 5.498 \times {10^{ - 13}}\;{\rm{kg}}$ ,在离磁探针1 μm的地方,磁场梯度为$\dfrac{{\partial B}}{{\partial {\textit{z}}}} = 5.0 \times {10^3}\;{\rm{T/m}}$ ,则悬臂梁的共振频率变为:$$f = \sqrt {\frac{1}{{7.561 \times (\Delta m + 5.498 \times {{10}^{ - 13}}) \times {{10}^4} + \dfrac{1}{{{{3\;125}^2}}}}}} $$ (13) 如果
$\Delta m$ 被加载在悬臂梁端头15.0 μm处,阻尼系数不变,悬臂梁的弹性系数取最大为$k{'_{\max }} = $ $ 5.743 \times {10^{ - 4}}\;{\rm{N/m}}$ ,则悬臂梁能探测到的最小力为:$${F_{\min }}' = 6.527\;{\rm aN}$$ (14) 通常,单电子自旋磁矩为
$9.27 \times {10^{ - 24}}\;{\rm{A}} \cdot {{\rm{m}}^2}$ [8],若取外加磁场梯度为$\dfrac{{\partial B}}{{\partial {\textit{z}}}} = 5.0 \times {10^3}$ T/m (实验上通常可以达到${\rm{2}}.0 \times {\rm{1}}{{\rm{0}}^{\rm{4}}}\;{\rm{T/m}}$ ),那么单个电子作用在悬臂梁上的力为$1.807\;{\rm{aN}}$ ,则理论上悬臂梁最小能够测得4个电子自旋反转的信号,从而显著提高了探测灵敏度。
Design of double-arm micro-cantilever beam of two-dimensional nanomaterial magnetic detection
-
摘要: 磁共振弱力探测技术能够对物质实现非破坏性的高精度结构信息探测,在物理、生物、医学等领域有着非常重要的应用。该技术中,超灵敏悬臂梁是实现弱力探测的核心组成之一。近年来,二维纳米材料由于其奇特的物理特性得到了越来越多的关注。为实现对二维纳米材料磁性的探测,基于单臂微悬臂梁模型,利用差动放大的方法,提出了双臂微悬臂梁的设计,并分析了双臂微悬臂梁中上下球形磁探针内外部的磁场分布,最后以单晶硅悬臂梁和钐钴合金磁球探针为例,对该悬臂梁进行数值模拟,发现该方案能够显著提高悬臂梁的探测灵敏度。Abstract: Magnetic resonance weak force microscopy (MRWFM) can achieve non-destructive high-precision structural information detection of substances. This advantage makes MRWFM be widely applied in fields of physics, biology, medicine, and so on. The super-sensitive cantilever beam is one of core composition to realize weak force detection in this technology. In recent years, two-dimensional nanomaterials have attracted more and more attention due to their unique physical properties. In order to achieve the detection of the magnetism of two-dimensional nanomaterials, the design of double-arm micro-cantilever beams with differential amplification based on single arm micro-cantilever beam model was proposed. Then the magnetic field distributions inside and outside of the scan balls fastened on the double-arm micro-cantilever beams were analyzed. Finally, the numerical simulation of the cantilever beam was completed, taking the single-crystal silicon cantilever beams and CoSm magnetic ball probe as examples. It is found that the scheme can improve detection sensitivity of cantilever beam significantly.
-
Key words:
- magnetic resonance technology /
- weak force detection /
- cantilever beam /
- sensitivity
-
-
[1] Zhang J S. Magnetic resonance imaging technology and its advantages [J]. Medical Equipment, 2021, 34(5): 184-185, 187. (in Chinese) [2] Wang Y N. Advances in fluorescent probes based on the “on-off-on” mechanism of quantum dots [J]. Guangdong Chemical Industry, 2020, 47(10): 59-63. (in Chinese) [3] Mehlin A, Xue F, Liang D, et al. Stabilized skyrmion phase detected in mnsi nanowires by dynamic cantilever magnetometry [J]. Nano Letters, 2015, 15: 4839. doi: 10.1021/acs.nanolett.5b02232 [4] Cao G Y, Fu H. Magnetic force resonance microscope [J]. Journal of Chinese Electron Microscopy Society, 2011, 30(1): 81-90. (in Chinese) [5] Liu Y. Design, fabrication and characteristic research of highly sensitivecantilever beams[D]. Hefei: University of Science and Technology of China, 2011. (in Chinese) [6] Chui B W, Hishinuma Y, Budakian R, et al. Mass-loaded cantilevers with suppressed higher-order modes for magnetic resonance force microscopy[C]//12th International Conference on Solid-State Sensors, Actuators and Microsystems. Digest of Technical Papers (Cat. No. 03TH8664), 2003, 2: 1120-1123. [7] Rugar D, Budakian R, Mamin H J, et al. Single spin detection by magnetic resonance force microscopy [J]. Nature, 2004, 430: 329-332. doi: 10.1038/nature02658 [8] Yan M, Peng X L. Fundamentals of Magnetism and Magnetic Materials[M]. Hangzhou: Zhejiang University Press, 2006. (in Chinese)