FERROMAGNETIC RESONANCE - INTRODUCTION

 

Ferromagnetic Resonance (FMR) is a powerful technique that enables to determine the anisotropy of a magnetic samples, and to investigate magnetization dynamics via the determination of the Gilbert damping constant. The only limitation is that it does not allow to measure directly the magnetization density, unlike magnetometers. In order to perform FMR measurements, one needs a magnet to saturate the film under study, an ac field generator (coplanar waveguide, a cavity resonator or a stripline) to pump energy into the magnetic film, and a rf generator that provides the ac signal. Here I will be discussing the theory of FMR, and the technical approaches to measure the susceptibility of magnetic devices. Keep in mind that this section does not pretend to provide a complete picture of FMR but rather the must-know of the technique and theory. The section is by definition UNDER CONSTRUCTION, as I will try to add additional information.

The magnetization dynamics is usually well described by the LLG equation. It includes a gyroscopic term and a damping term:

 

 
 


When the magnetization is at equilibrium, dM/dt=0 and the magnetization points towards Heff.
If the magnetization is disturbed from equilibrium where (q is a small angle), the magnetization will precess at a given frequency w0 (in the GHz range) that depends on Heff and reach equilibrium after a time (relaxation time) that is inversely proportional to a. Ferromagnetic resonance consists in driving the precessional motion of the magnetization using a small ac field normal to the Heff. The resonance will be obtained when the ac field frequency matches the eigen frequency of precession of the magnetization for a given Heff.

 

M at equilibrium
M disturbed from equilibrium
Driven precessionnal motion

 

1. The principle of the measurements
 




Click on the image to watch a short animation

Microwaves irradiate the sample (magnetic film) and we monitor the absorption of the microwave signal. We apply a dc magnetic field which causes the spins to line up: the film is fully saturated. If the microwave magnetic field is perpendicular to the dc field, the spins precess at a certain frequency that depends upon the dc field, and if that frequency matches that of the microwaves, the film absorbs power, resulting in a resonance state. The occurence of resonance is due to the transition between the different energy levels of FM electron spin states. The resonance frequency and its linewidth give a great deal of information about the magnetic properties of the sample. FMR is the technique of choice to investigate thermodynamic ground-state properties.

 

2. The Theory

3.1. EPR -

Electron Paramagnetic Resonance (EPR): the difference in energies between the two spin states of each electron of an atom that is critical. The energy difference between the two electronic states is much larger than that of nuclear particles, so EPR resonance frequencies are typically much higher than NMR resonance frequencies, of the order of GHz/T. Thus for a spectrometer of a certain frequency, EPR generally requires a lower field strength than NMR. A radio wave applied at this frequency will induce transitions among the energy levels of the nucleus, and will lead to emission of a resonant signal at that same frequency which can then be detected. A common test substance for EPR is 2,2-diphenyl-1-picrylhydrazyl (DPPH), because it contains a de-localized unpaired electron, yet is still non-magnetic.

10GHz at 300K (P=-17dBm)

15GHz at 300K (P=-17dBm)

25 GHz at 300K (P=-17dBm)


Hres versus frequency shows a linear dependence as predicted by the theory:
 

3. The experimental technique

A CPW was made by photolithography and lift-off process, using a double-resist layer process. The metallic layer, 200 nm Cu, is thermally evaporated on a GaAs substrate. The CPW was designed for havng  50 W, impedance value that matches the impedance at the PNA port. Hence, reflections can be limited.

The CPW is placed in a brass cavity and connected to a Network Analyzer, a PNA (Fig.1). The PNA enables to measure the reflected, transmitted power through the CPW: S11 and S12. The CPW is used as an ac field generator and as an inductor sensor. The ac current in GHZ range from the PNA generate an ac field around the central line.



Fig. 1. CPW in a brass cavity

The film is then mounted on top of the CPW (Fig. 2). Field sweep FMR measurements is conducted at fixed frequency. The applied field generated by an electromagnet, lies along the axis of the CPW, and therefore perpendicular to the ac field. For a specific (field,frequency), ferromagnetic resonance occurs: it is when the angle of precession of the magnetization is maximum, the magnetic susceptibility of the films changes.

The change in the magnetic susceptibility of the film induces a change in the impedance of the CPW. It must be kept in mind, that the technique is in the approximation of a perturbation measurement. This is to say that the change in impedance due to resonance is much smaller than the impedance of the line, that is mainly 50 Ohms. 




Fig. 2. Flip chip method: The magnetic film is placed metal face down on top of the CPW.


 

 


5. Typical Results: Resonance field

The resonance field Hres can be measured as a function of frequency (with broadband device like CPW) or as a function of the out of plane angle (cavity X-band). When dealing with a broadband technique, the resonance field of an infinite magnetic film (thickness much smaller than the lateral size) is given by the Kittel formula:

Happ (blue vector) is in the film plane:
 

Happ (blue vector) is normal to the film plane:

 


 

4pMeff is the effective demagnetization field, w is the frequency of the ac field, Hres is the resonance field and g is the gyromagnetic ratio (proportional to the Lande g-factor).

The condition of resonance becomes more complicated when the lateral sizes of the film are in the range of the film thickness.

By fitting the frequency dependence of the resonance field to the Euations above (depending on the measurement geometry), the parameters 4pMeff and g can be determined.

When the setup is limited to a single frequency, angular dependence of Hres can be used to determined 4pMeff and g.

The angle f between Happ (blue vector) and the film plane is changed and for every single angle, the field sweep absorption line is recorded. The data are then fitted to a "modified" form of the Kittel formula that includes the angle f , the effective demagnetization field and the gyromagnetic ratio. The condition of resonance can be calculated using Smith-Suhl Equation for which it is necessary to calculate the magnetic energy of the system: Zeeman energy+ shape anisotropy+crystalline anisotropy. The exchange anisotropy can be dismissed, if the magnetic film is considered as a macro-spin. When the exchange is very strong (for 3d magnetic materials, it is about 1000K), the spins are strongly coupled and precess all together. (here are the details of the calculation and the corresponding mathematica script)

Note that at every angle, the applied field is always perpendicular to the ac magnetic field.

 

6. Effective field and Surface anisotropy - Typical results

For relatively thick films (above 20 nm), the effective field 4pMeff = 4pMs which is the demagnetization field that favours the in-plane orientation of the magnetization. However, when the film thickness is reduced, the surface of the films play a predominant role in the direction of the magnetization. In the 50's, Néel have introduced the idea that at the surface of the films, because of the broken symmetry at the interface, the spins prefers to point out of the plane [J. Phys. Rad. 15 (1954) 376]. This is called the Néel surface anisotropy and is characterized by a surface anisotropy field Hs = 2Ks/(Ms t), where Ks is the surface anisotropy constant, Ms the magnetization saturation and t the thickness of the film. Hs increases with magnetic layer thickness decreasing.

The effective demagnetization field is given by:

 

100 nm Py film on glass substrate. The resonance field versus frequency measured with a CPW in the in-plane geometry.

 From the Equation above, one can see that there is a critical thickness tc where the effective demagnetization field 4pMeff  is zero.  For magnetic film with t> tc,  4pMeff is positive and the magnetization lies in the film plane. For very thick films, Hs is negligible and one find 4pMeff = 4pMs.  For magnetic film with t< tc,  4pMeff is negative and the magnetization lies normal the film plane. This has been proved by numerous experiment on crystalline thin films.

In addition of the Néel surface anisotropy, other mechanism can give rise to an anisotropy with 1/t thickness dependence: misfit strain anisotropy [Phys. Rev. Lett. 60 (1988) 2769, J. Magn. Magn. Mater. 93 (1991) 562], surface roughness [Appl. Phys. Lett. 64 (1988) 3153, J. Phys. F : Met. Phys. 18 (1988) 1291].