Electrical Detection Of Ferromagnetic Resonance

Yang Ge and Yang Xiao

University Of Minnesota

Methods of Experimental Physics Spring 2013

Introduction

In the presence of external magnetic field, the magnetization inside ferromagnetic materials would process. When the precession is driven by external oscillating torque, ferromagnetic resonance (FMR) will occur. Electrical detection of FMR relies on the a DC photovoltage generated by the driving microwave. Electrical detection of FMR provides an easy way to probe the magnetic and crystalline properties of a ferromagnet.

In this experiment, we detected ferromagnetic resonance (FMR) by measuring the associated photovoltage. The sample used is Fe, deposited on !GaAs substrate. In FMR, the resonance frequency is given by [[#YyVonsovskii][Vonsovskii’s formula]], as a function of applied static field and, indirectly, the anisotropic constants hence the orientation of sample. Initially we planned to extract anisotropic constants via anisotropic magnetoresistance (AMR) measurements, and use the constants to plot the theoretical curves for FMR at different applied field and sample orientation. Then we would measure FMR response and compare. In the end we could not get a very accurate data from AMR measurements, but the FMR ones turns out fine.

We measured FMR in X-band microwave, 6.6-12GHz. The resonance frequency and applied field relation is determined by sweeping magnetic field at each microwave driving frequency. When one sweeps the applied field, FMR response would conform to asymmetric Lorentz distribution, with a characteristic Gilbert damping parameter α_G. The damping parameter is usually between 0.007 and 0.01, even for FMR different sample structures. We get a α_G of 0.00721 ± 0.00018, which confirms that an FMR response is observed. Then, by adjusting anisotropy constants used to generate theoretical curve, we found approximate values for uniaxial and cubic anisotropy, which are k_u ~ -365000 erg/cc and k_v ~ -34000 erg/cc, respectively.

Background

Free Energy

The magnetization of ferromagnet depends on crystalline anisotropy inside the material, which is captured by anisotropy terms in the free energy. The free energy is minimized at equilibrium, which gives the equilibrium magnetization for an externally applied field.

Our sample is a rectangular chip, about 7 mm long by 0.6 mm wide, cleaved from a wafer with following structure

Two anisotropy terms are the dominant ones in our sample: a cubic term due to crystal symmetry and a uniaxial term due to the interface of GaAs surface [Sjö02]. In our sample, the hard axis is along the longer dimension of the rectangle, and the easy axis is along the shorter one. Hence both axes are parallel to the layered planes.

The free energy can be formulated as follows, under an applied field H [Isak01]. Here, φ denotes the azimuthal angle of magnetization from the easy axis, and $theta; denotes the out of plane polar angle.

At equilibrium, ∂F/∂φ = 0 and ∂²F/∂φ² > 0.

Magnetization

Magnetization at a given applied field can be derived from minimization conditions of free energy above. Since the free energy depends on current magnetization, equilibrium magnetization is path dependent. This results in hysteresis loops in magnetization. Below is one example with applied field at 45° from the easy axis and external field swept from +∞ to -∞ to +∞. Note that there is more than one minimum in the free energy curve.

If the field is applied along the hard axis, there would be no hysteresis and the magnetization would point along the hard axis above a saturation field. This saturation condition imposes a constraint on k_u and k_v:

H_s = -2 (k_u + k_v) / M_s

where M_s is the saturation magnetization, which is about 1671.1 emu/cc for iron. The saturation field strength H_s can be derived from, say, AMR measurements.

A piece of ferromagnet might be multi-domain, meaning it consists of regions where magnetization is uniform inside but different across regions. In the following discussion we assume the ferromagnet consists of a single domain unless otherwise noted.

FMR

Under the torque exerted by a steady magnetic field, magnetic moments would precess around their equilibrium position. The precession frequency is in GHz range, which corresponds to a microwave. Thus microwave with specific frequencies can excite the resonance in this precession motion, by exerting oscillating torque through its magnetic field, and results in FMR. The resonance frequency for an applied field is given by Vonsovskii’s formula [<!---->[[#RefFmr][Morr65]]<!---->]:

where γ is the effective electron gyromagnetic ratio e/m for electron.

The fundamental equaltion of motion is [<!---->[[#RefFmr][Mor65]]<!---->]:

Here H_T denotes the total field. α_G is the Gilbert damping constant.

AMR

In AMR effect, the resistance of a ferromagnet is dependent on the angle &delta; between magnetization and electric current, to the lowest order given by [OHan99]

R = R₀ + ΔR cos²δ

In principle, one can determine the orientation of magnetization from AMR measurements, hence the saturation field H_s and anisotropy constants k_u, k_v.

AMR effect plays an important role in the generation of photovoltage, described below.

Photovoltage and Lorentz line shape

Simply put, microwave electric field generates an oscillating electric current in the sample, or the photocurrent. The oscillating magnetization due to microwave magnetic field results in oscillating angles between current flow and magnetization. Hence a time-varying photoresistance is created. The product of the two results in a DC photovoltage.

When sweeping the field at a given microwave frequency, one will observe the following line shape

In the equation, H_r gives the resonance frequency. ΔH is the line width, related to dampling parameter α_G as below:

The line shape consists of a symmetric and anti-symmetric part with respect to the resonance field. Their relative strength depends on phase difference in microwave electric and magnetic field.

A detailed discussion of DC photovoltage and Lorentz line shape can be found in [Meck07] and[Hard11]

Experimental Methods

Sample

Much of the sample is mentioned in Free energy part of Background. The sample is fixed on a PCB board using rubber cement. It has four strips of gold deposited on it at different position along the longer dimension, two at one end and two at the other. This would allow us to do 4-wire measurements for AMR.

The resistance of the sample needs to be double checked to make sure all gold protection are removed. Otherwise signal would be attenuated if current runs parallel to the layers.

AMR

Since the AMR measurements did not turn out well, we just briefly describe it. We measure the resistance of our sample using 4-wire measurement. The outer two lead are used to send current, whereas the inner two are used for voltage reading. The sample is in series with a metal-film resistor shielded in an aluminum box. We read the voltage across this resistor to determine current flowing through the sample. The resistor has a much larger resistance than our sample, thus AMR in the sample has little effect on the current. The whole resister ladder is driven by a function generator. The voltage across the sample is first feed to the differential input of an SR-560 preamp, then send to an SR-830 lock-in amplifier.

We did both an angular sweep at high field, and a field sweep at fixed sample orientation. The orientation chosen are first with the field parallel to the hard axis of the sample and then the easy axis. In the hard axis field sweep, one would see a continues increase of resistance from zero field up through saturation field H_s; after that the resistance would be invariant of field strength. In an easy axis sweep, if the sample is single-domain, there would be a sudden jump of resistance as one increases field from -∞ to +∞ at a single point. The resistance to the left and right of this point should be the same. Our measurements shows a peak rather than a point, thus our sample is multi-domain. Thus the results gathered with lower than saturation field should be treated cautiously.

Our AMR data is rather noisy, and shows a steady increasing of resistance with time that is beyond temperature effects. These two effects are huge, even to the point that the data is not quite repeatable. Possible improvements may include better shielding the sample, e.g. position it inside a slotted waveguide, use a well matched Wheatstone bridge or apply lock-in measurement across reference resistor. Another issue is to avoid attenuation due to residue gold, mentioned above. If desired, one can switch to a vertical current flow configuration.

FMR

The X-band microwave setup is shown in below. In the microwave connection, a microwave synthesizer connects to a isolator, which absorbs reflecting waves, and then a slotted waveguide. A step motor and translational platform is mounted on the slotted waveguide to move the sample. At the end is a tunable short, which sets up a standing wave.

Before using the synthesizer, we used a gunn diode to make sure the system works. Also need to check is that there is no DC component of microwave set up in the waveguide, as it would damage the synthesizer. To measurement the microwave strength, we mount a diode on the translational platform. The voltage across the diode would reveal the magnitude of the microwave. A PIN diode is installed after gunn diode, which absorbs all microwaves when forward-biased. This allows lock-in measurement of voltage across the diode. A positional scan shows that the electric field intensity is like sin²_x_ and are very close to zero at nodes. Thus there is no DC component.

The sample sits between the magnets and is oriented along microwave electric field E. Its hard axis is along the same direction of electric field. The static magnetic field H runs parallel to the sample surface.

We controlled the translational platform with a step motor. The magnets are controlled with voltage regulated current source. The current supply is measured by taking the voltage drop across a shunt resistor, which is connected in series with the magnets. Finally, lock-in measurements are made with an SR-830 lock-in amplifier and setting amplitude modulation to the output of our synthesizer, Agilent !N5183A.

We completed field sweep at different frequencies, with sample's hard axis oriented both 0° and 45° with respect to applied field. At each frequency, we attempted to move sample to where largest photovoltage signal is obtained. We attempted sweeps near 90°, but could not get a sizable photovoltage curve.

The alignment of angle is critical, as the FMR frequency versus field curves depends on the angle φ. As shown in graph below, where φ, the angle from easy axis, varies from 0°, 30°, 45°, 60°, 84°, 87°, to 90°. In addition, in principle the photovoltage would vanish at 0° and 90°.

Results

Following are some snapshots of field sweep curves with sample hard axis 0° and 45° from H.

The sweep data was fitted to Lorentz line shapes, as below.

The asymmetric Lorentz curves are fitted around the peaks in the data. Note that the inversion of photovoltage at zero field is not incorporated in Lorentz distributions. It is the result of reverting precession direction, whereas Lorentz distributions only accounts for positive/negative branch of the photovoltage curve. The curve gives a Gilbert damping parameter α_G of 0.00721 ± 0.00018, within the accepted range 0.007~0.01.

Then the resonance field H_r at each microwave frequencies are plotted below. We tuned uniaxial and cubic anisotropy constants to best fit the data. The constraint that saturation field H_s is 500 oe is applied. The saturation field is derived from Changjiang Liu's TAMR measurement use a small sample, which is single-domain, and taken from the same wafer as our sample. The results are k_u ~ -365000 erg/cc and k_v ~ -34000 erg/cc. The plot is shown below.

From the figure above, it seems that the saturation field is lower than what we set. Also, the sample demonstrates rather low cubic anisotropy.

Future work

Other than those mentioned before, one could also try measure FMR at other frequencies, e.g. K-band, 18-27GHz. A smaller sample might also be used.

Acknowledgements

Many thanks to Prof. Crowell for all the comments, suggestions and guidance. Changjiang Liu and Eric Kamp have provided a great deal of assistance and help to our project. Our internal advisor, Prof. Pryke gave us much valuable advice. We would also want to thank Andrew Galkiewicz and Chad Geppert, who have repaired our samples countless many times.

References

[Sjö02] E. Sjöstedt, L. Nordström, F. Gustavsson, and O. Erriksson. Phys.Rev.Lett. 89, 267203 (2002)

[Isak01] A. F. Isakovic, J. Berezovsky, P. A. Crowell, L. C. Chen, D. M. Carr et al. J. Appl. Phys. 89, 6674 (2001)

[Morr65] A. H. Morrish, The Physical Principles of Magnetism (Wiley, New York, 1965)

[OHan99] R. C. O'Handley, Modern Magnetic Materials: Principles and Applications (Wiley, New York, 1999)

[Meck07] N. Mecking, Y. S. Gui, C.-M. Hu, Phys. Rev. B 76, 224430 (2007)

[Hard11] M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, and C.-M. Hu, Phys. Rev. B 84, 054423 (2011)