Theoretical model for torque differential magnetometry of single-domain magnets

Akashdeep Kamra, Michael Schreier, Hans Huebl, and Sebastian T. B. Goennenwein

Phys. Rev. B 89, 184406 – Published 12 May 2014

Abstract

We present a generic theoretical model for torque differential magnetometry (TDM)—an experimental method for determining the magnetic properties of a magnetic specimen by recording the resonance frequency of a mechanical oscillator, on which the magnetic specimen has been mounted, as a function of the applied magnetic field. The effective stiffness change, and hence the resonance frequency shift, of the oscillator due to the magnetic torque on the specimen is calculated, treating the magnetic specimen as a single magnetic domain. Our model can deal with an arbitrary magnetic free-energy density characterizing the specimen, as well as any relative orientation of the applied magnetic field, the specimen, and the oscillator. Our calculations agree well with published experimental data. The theoretical model presented here allows one to take full advantage of TDM as an efficient magnetometry method.

Received 28 March 2014
Revised 23 April 2014

DOI:https://doi.org/10.1103/PhysRevB.89.184406

Authors & Affiliations

Akashdeep Kamra^1,2, Michael Schreier¹, Hans Huebl^1,3, and Sebastian T. B. Goennenwein^1,3

¹Walther-Meissner-Institut, Bayerische Akademie der Wissenschaften, Walther-Meissner-Strasse 8, D-85748 Garching, Germany
²Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
³Nanosystems Initiative Munich (NIM), Schellingstrasse 4, 80799 Munich, Germany

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Issue

Vol. 89, Iss. 18 — 1 May 2014

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Images

Figure 1
Comparison between (a) cantilever torque magnetometry (CTM) and (b) torque differential magnetometry (TDM). The magnetic specimen (light blue) is mounted at the tip of the cantilever (red), the motion of which can be modeled by an effective mass and spring system. (a) In CTM, one measures the static equilibrium deflection of the cantilever, which translates to the torque (modeled as an effective force $F_{m}$ ) exerted on the magnetic specimen by the applied magnetic field. (b) In TDM, one measures the magnetic field dependent resonance frequency shift of the cantilever which translates to a stiffness change due to magnetic torque $k_{m}$ [cf. Eq. (3)]. In both (a) and (b), the physical quantity that the experiment measures is shown in blue color.
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Figure 2
Schematic of the magnetic specimen (light blue) mounted on a cantilever (red) in an applied magnetic field. ${\hat{r}}_{c}$ (red arrow), ${\hat{r}}_{m} = \vec{M} / | M |$ (blue arrow), and ${\hat{r}}_{h} = \vec{H} / | H |$ (green arrow) denote the unit vectors along the oscillator axis, magnetization, and applied magnetic field, respectively. The angles characterizing the unit vectors are depicted in the specimen frame of reference.
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Figure 3
Frequency shift vs applied magnetic-flux density for two special cases of interest. Red open circles depict experimental data taken from Ref. [18] while the blue solid line is the frequency shift calculated from Eq. (21), using the oscillator and free-energy density parameters presented in Table 2. The configurations are depicted in the corresponding insets. The uniaxial easy axis is along the longer dimension of the specimen and the green dotted arrow represents the applied magnetic field. The magnitude of the uniaxial anisotropy field $B_{u}$ is indicated on top of the figures by a black arrow. The base resonance frequency $f_{e l}$ is a few kHz.
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Figure 4
Frequency shift ( $Δ f$ ) vs polar angle of the applied magnetic field direction ( $θ_{h 0}$ ) for cases $ϕ_{h 0} = ϕ_{c 0} = 0$ (a) and $ϕ_{h 0} = ϕ_{c 0} - \frac{π}{2} = 0$ (b) from Eq. (29). The corresponding measurement configurations are depicted in the respective insets. The cubic thin-film sample (light blue) is mounted on an oscillator (red). We consider $K_{1} = 47.5 kJ m^{- 3}, K_{2} = 0.75 kJ m^{- 3}$ corresponding to magnetocrystalline anisotropy constants of iron [3] for different values of $K_{s}$ (also in units of $kJ m^{- 3}$ ). The qualitative shape of the curve depends upon the value of $K_{s}$ in relation to $K_{1}$ . The oscillator and free-energy density parameters are given in Table 2. The base resonance frequency $f_{e l}$ is about 2.8 kHz.
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Figure 5
Frequency shift vs applied magnetic-flux density. The configuration is depicted in the inset of the figure. The uniaxial easy axis is along the longer dimension of the specimen and the green dotted arrow represents the applied magnetic field. We consider a weak cubic anisotropy ( $K_{c} = 1 kJ m^{- 3}$ ) in addition. The magnitude of the uniaxial anisotropy field $B_{u}$ is indicated on top of the figure by a black arrow. The oscillator and free-energy density parameters used are quoted as the first set in Table 2. The base resonance frequency $f_{e l}$ is about 2.8 kHz. This measurement configuration allows for isolation of axially symmetric and polar anisotropies in a single measurement.
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Figure 6
Frequency shift vs applied magnetic-flux density for a thin-film sample with cubic magnetocrystalline anisotropy. Magnetic field and oscillator axis point in the out-of-plane direction. We consider $V = 10^{- 20} m^{- 3}$ , $K_{1} = 47.2 kJ m^{- 3}$ , and $K_{s} = 1846 kJ m^{- 3}$ corresponding to an iron thin film [3] and oscillator parameters quoted as the first set in Table 2. The base oscillator frequency $f_{e l}$ is about 2.8 kHz. Energetically equivalent magnetization directions $ϕ_{m 0}^{h} = 0 and π / 2$ can easily be distinguished using low fields. There is a unique energetically favorable equilibrium orientation at high fields. The critical field separating the two regimes $B_{s} - B_{1}$ is indicated by an arrow on the top. The frequency shift close to the critical field is not shown as the expressions given in Eq. (C13) are, strictly speaking, not valid in this region.
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