Magnetic properties of individual Co2FeGa Heusler nanoparticles studied at room temperature by a highly sensitive co-resonant cantilever sensor

The investigation of properties of nanoparticles is an important task to pave the way for progress and new applications in many fields of research like biotechnology, medicine and magnetic storage techniques. The study of nanoparticles with ever decreasing size is a challenge for commonly employed methods and techniques. It requires increasingly complex measurement setups, often low temperatures and a size reduction of the respective sensors to achieve the necessary sensitivity and resolution. Here, we present results on how magnetic properties of individual nanoparticles can be measured at room temperature and with a conventional scanning force microscopy setup combined with a co-resonant cantilever magnetometry approach. We investigate individual Co2FeGa Heusler nanoparticles with diameters of the order of 35 nm encapsulated in carbon nanotubes. We observed, for the first time, magnetic switching of these nanoparticles in an external magnetic field by simple laser deflection detection. Furthermore, we were able to deduce magnetic properties of these nanoparticles which are in good agreement with previous results obtained with large nanoparticle ensembles in other experiments. In order to do this, we expand the analytical description of the frequency shift signal in cantilever magnetometry to a more general formulation, taking unaligned sensor oscillation directions with respect to the magnetic field into account.


A: Correction for non-aligned co-resonant oscillations
The theoretical description of the co-resonant sensor concept used in the present and previous work [1,2,3] is based on a one-dimensional coupled harmonic oscillator model. This simple model can only be applied, however, if the oscillation directions of the two beam-like structures of a co-resonant sensor are aligned to each other. In a real sensor, like the one used in the main publication, this is usually not the case. In order to still derive quantitative data with the simple harmonic oscillator model, a modification is necessary [4]. Essentially, the misalignment of the oscillation directions of the beam-like structures can be treated by using an angled coupled harmonic oscillator model (see figure 1 (a)). In this model the two masses (m 1 ,m 2 ) are still subject to one-dimensional movements individually that are, however, not aligned along the same direction. Consequently, the projection of a m 1 displacement on the m 2 oscillation direction is smaller than the m 1 displacement itself. The same applies to the projection of the m 2 displacement on the m 1 oscillation direction. Other than the projection of the m 1 displacement, the force generated by k 2 acts fully on m 2 . However, in addition to the projection of the m 2 displacement, only a projection of the k 2 -induced force acts on m 1 . All reductions by these projections can be described by the factor . Thus, misalignment of the two subsystems' oscillation directions does not change their individual resonance frequencies but leads to a reduction of the coupling of the two oscillators. If damping is neglected, the equations of motion for the thus modified problem can be written in matrix form: with the matrices the deflection of the masses from their equilibrium position s = (s 1 , s 2 ) and the excitation force vector F (t) = (F 0 , 0) cos (ω D t). Here, m i and k i are the effective masses and spring constants for the two partial mechanical resonators and k 3 describes an external force gradient acting on the smaller subsystem.
From these equations it is possible to calculate the amplitudes and phases of the partial systems as homogenous solutions by using a complex ansatz s (t) = (A 1 , A 2 ) exp (iωt). In this work, however, only the resonance frequencies of the coupled system are of interest. They can be derived by using this ansatz in equation (1), which results in a system of equations for the complex amplitudes. The resonance frequencies can be derived by demanding a non-trivial solution, therefore requiring the determinant of the coefficient matrix to vanish. This results in the resonance frequencies of the coupled system: with To derive these equations, With these results it can be understood that each misalignment of the oscillation directions leads to a change of the frequency shift of the respective mode ∆ω a/b which corresponds to adjusted effective spring constants k * a,b .
By introducing a correction factor this effect can be included into effective spring constants that result from calculations based on an aligned harmonic oscillator model (k a,b ). Please note that for sufficiently small k 3 this correction factor depends only on : For the co-resonant sensor used for the experiments in this work the geometric reduction factor can be described by = cos (ϕ) cos (ξ). The factor cos (ϕ) originates from the structural inclination of the long axis of the nanocantilever with respect to that of the microcantilever (see figure 1(b)) as can be derived from scanning electron microscopy (SEM) images taken from the side view perspective. This results in an angle of ϕ ≈ 31 • , which does not change during the experiment. The contribution cos (ξ) describes the non-aligned preferred oscillation direction of the nanocantilever with respect to that of the microcantilever due to its disturbed axial symmetry (see figure 1(c)). In order to derive the angle ξ, the sensor was excited to resonant oscillations and observed from side and top view inside the SEM with identical excitation parameters. From the SEM images the projection of the resonant oscillation amplitude of the nanotube in the microscope's imaging plane was measured for both views. The angle ξ was then calculated from these amplitude projections by simple trigonometry. Please note that this angle is different for both resonant modes of the coupled system and was measured independently for each respective mode. This resulted in ξ a ≈ 0 • and ξ b ≈ 35 • . Finally, the condition of a sufficiently small k 3 is well satisfied in these experiments, so the correction factors from equation (5) for both combined resonance modes (η ( a ) ≈ 1.24, η ( b ) ≈ 0.94) are constant and were used to calculate the adjusted effective spring constants k * a/b .

B: Derivation of cantilever frequency shift from magnetic energy
The following derivation is mostly based on [5,6,7].
We define the external magnetic field to be oriented along the x-axis, i.e. H = H ext e x , and the sensor equilibrium position to be in the x-z-plane. The sensor oscillation plane is tilted by an angle α with respect to the x-z-plane (see figure 2 in the main paper for angle definitions). Furthermore, we assume the magnetic energy of the system to be dominated by Zeeman and shape anisotropy energy. The Zeeman energy is given by: If the easy axis of the magnetic sample is not aligned with the external magnetic field but tilted by an angle γ, the magnetization M already cants away from the easy axis by an angle Θ at the equilibrium position. With a sensor oscillation of an angle β in the x-z-plane, the magnetization vector is: To take a rotation of the oscillation plane into account, M is multiplied with a rotation matrixR which is defined by the rotation angle α and an axis of rotation. The latter is given by the unit vector of the equilibrium position of the magnetization: M /M s = cos γ e x + sin γ e z , with γ = γ − Θ. Hence: Application ofR on M (no y-component), scalar multiplication with H (only x-component) and employing the result into equation (7) yields the Zeeman energy: (10) with the saturation magnetization M s and the sample volume V .
For the shape anisotropy energy of prolate spheroids, only the angle between the easy axis and the magnetization is relevant, hence: with the demagnetization factors N 1 , N 3 and, therefore, the total magnetic energy is: First, the angle Θ needs to be expressed as a function of β. In order to achieve this, the derivative of the magnetic energy with respect to Θ is set to zero: Furthermore, the second derivative of the magnetic energy with respect to Θ, ∂ 2 E mag /∂Θ 2 , needs to be positive.
Applying addition theorems for sin (γ ± β − Θ), sin (±β − Θ) and cos (±β − Θ) to isolate γ and furthermore using a small angle approximation for the oscillation angle β which yields sin β ≈ β and cos β ≈ 1, leads to an expression for β (Θ): Finding an expression for Θ by using equation (14) is only possible, if a small angle approximation is applied for Θ. Since Θ denotes the deviation of the magnetization direction from the easy axis, this is only valid for small external magnetic fields. In that case, sin Θ ≈ Θ and tan Θ ≈ Θ are used. If a more accurate solution is desired, a numerical solution is necessary at this point.
With the above approximations, equation (14) becomes: Now, Θ can be expressed as a function of β by using a Taylor series expansion: Evaluation of equations (15) and (16) finally yield: Equation (17) can be inserted into expression (12) for the magnetic energy. Calculating the second derivative of the magnetic energy with respect to β leads to the frequency shift for cantilever magnetometry as presented in the main paper.