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https://aip.scitation.org/doi/full/10.1063/1.5049683
DOI: 10.1063/1.5049683
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DIRETORIA DE TRATAMENTO DA INFORMAÇÃO Cidade Universitária Zeferino Vaz Barão Geraldo
CEP 13083-970 – Campinas SP Fone: (19) 3521-6493 http://www.repositorio.unicamp.br
microwaves cavity: In
fluence of skin depth and of photo-injected carriers
T. C. Cordeiro,1M. E. Soffner,1A. M. Mansanares,2and E. C. da Silva1,a)
1
Laboratório de Ciências Físicas, Universidade Estadual do Norte Fluminense Darcy Ribeiro (UENF), 28013-602 Campos dos Goytacazes, Rio de Janeiro, Brazil
2
Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, Unicamp, 13083-859 Campinas, São Paulo, Brazil
(Received 24 July 2018; accepted 26 September 2018; published online 18 October 2018)
This study reports on the modulation frequency dependence of the photothermally modulated mag-netic resonance signal of a set of magmag-netic samples in the form of foils, layers, and thin films: 50μm Fe and Ni foils; 5 μm magnetic layer of γ-Fe2O3 in a cassette tape; and 150 nm Co and Permalloy films deposited on glass and Si (111) substrates, besides the naked Si substrate. It is shown, both by analytical calculation and by measurements, that the skin depth of the microwaves deeply influences the signal behavior by selecting the portion of the sample that is probed. Clear dif-ferences in the frequency dependence are observed between the metallic Ni and Fe foils and the dielectricγ-Fe2O3cassette tape. Furthermore, the thermal mismatch between the magneticfilms (Co and Permalloy) and substrates (glass and Si) also plays a crucial role, once the modulation of the temperature is strongly dependent on the substrate thermal parameters at low modulation frequen-cies. The non-resonant signal from the diamagnetic Si is also analyzed. It is produced by the absorp-tion of microwaves by the photo-injected free carriers and presents characteristic behavior in the investigated frequency range. Published by AIP Publishing.https://doi.org/10.1063/1.5049683
I. INTRODUCTION
A photothermally modulated magnetic resonance (PMMR) signal originates in the sample temperature varia-tion produced by the absorpvaria-tion of a laser beam.1–11 The magnetic properties of the sample (magnetization, spin-lattice and spin-spin interactions, anisotropy energies) change with temperature, thus producing a distortion in the magnetic resonance line by shifting the resonance external magnetic field, and by attenuating and broadening the line. At each magnetic field value, the small change in the magnetic resonance signal due to the temperature variation is lock-in detected at the modulation frequency of the laser beam.
If the laser is tightly focused in a given point of the sample, only the adjacent region is heated, and magnetic imaging becomes possible since the absorption of micro-waves at the frequency of the laser is restricted to the heated region.1,3,5,7 In the same way, even for a non-focused laser, by increasing the laser modulation frequency, the depth of penetration of the modulated heat decreases. Therefore, superficial layers are selected by the modulated microwaves absorption, and a depth profile analysis can be done.9,11
Previous works investigated several magnetic bulk mate-rials1,2 and films,3–11 characterizing their spectroscopic response2–8 and imaging their magnetic heterogene-ities.1,3,5,7,9,11However, few works were dedicated to investi-gating the frequency dependence of the PMMR signal,7,9,11 and a more comprehensive study is still lacking.
In this article, we followed the frequency dependence of the PMMR signal in a wide range of frequencies (100 Hz to 100 kHz) in several samples in the form of foils (Fe and Ni), layers (γ-Fe2O3 cassette tape), thinfilms (Co and Permalloy deposited on glass and Si substrates), and the Si substrate itself. Besides the thermal wave behavior as a function of the laser modulation frequency in the sample/substrate/backing arrangement, we explored the depth of penetration of the probing microwaves that selects specific portions of the sample, thus markedly influencing the frequency response. We also observed that the mismatch between the thermal properties of the sample layer and the substrate plays a funda-mental role in the temperature of the sample at low modula-tion frequencies and therefore in the PMMR signal. Additionally, the existence of a thermal contact resistance between the film and substrate influences the signal in the case of large heat flux from the sample to the substrate (low frequencies and high thermal conductivity of the substrate).12 Finally, we also observed and modeled the non-resonant signal from the diamagnetic Si substrate, which originates in the microwaves absorption by photo-injected free carriers, and can play a significant role in the case of semi-transparent films.
II. EXPERIMENTAL ASPECTS A. Samples
1. Ni and Fe foils
Nickel and iron foils were manufactured by the cold lamination procedure. Both samples are 50 μm thick. The sample holders (backing) were made of polyvinyl chloride (PVC) for all samples, except the cassette tape one.
a)
E-mail: ecorrea@uenf.br
2. Cassette tape
The sample was a commercial cassette tape type I manu-factured by Nipponic, not recorded, which is made up of 15μm polyester backing and a 5 μm magnetic layer of γ-Fe2O3 needle shaped particle immersed into a polyester matrix. In this case, the used sample holder was made of Tecnil.
3. Co and Permalloyfilms
Cobalt and Permalloy films, with 150 nm thickness, were deposited by sputtering (sputter system model ATC2400) on 150μm corning glass and 535 μm Si (111) wafer substrates, which were kept at room temperature. Working pressure, dynamic deposition rate, and discharge current used were 2:3 103Torr, 1:18 A=s, and 97 mA, respectively. Base pressure used in deposition on corning glass was 1:5 107Torr and on Si (111) wafer was 1:0 107Torr. All films have a 10 nm Ta (tantalum) pro-tective cover layer on top. The 535μm thick undoped Si (111) wafers were supplied by University Wafer Inc., MA, USA.
B. Description of the setup
The experimental setup was based on a Bruker Elexys 500 Electron Spin Resonance spectrometer operating in microwaves X-band. For conventional magnetic resonance experiments, we mostly used a 5 Oe modulationfield at 100 kHz and a standard Bruker ER4102ST X-band cavity with a Q-factor of 6000. The resonance frequency of this cavity varied in a range from 9.82 to 9.85 GHz, and experiments were held at room temperature. For the PMMR experiments, an optical access ER4104OR cavity with a Q-factor of 7000 was used. In this case, the cavity resonance frequency varied from 9.40 to 9.65 GHz. Hence, the values of the external magnetic fields for resonance shift to higher fields for the conventional magnetic resonance experiments relatively to the PMMR ones in a maximum of 5%. The high quality factor of the cavity used in the PMMR experiments ensured a much better signal to noise ratio compared to previously reported measurements using an open microwaves cavity.2,7,9 PMMR experiments were also performed at ambient temper-ature. However, one should consider that for the PMMR, the laser absorption increases the sample dc temperature in a few degrees Celsius, shifting slightly the resonance line position to higherfield values. The operation mode is TE102for both
cavities.
The spectra obtained in both configurations are strongly dependent on the sample orientation with respect to the exter-nal magnetic field.6 In the present work, the external mag-netic field is always in the plane of the samples. Figure 1
shows the experimental arrangement [Fig.1(a)], the external magneticfield, sample surface, and laser incidence direction alignments [Fig.1(b)].
A Newport LQA660-110C laser, 110 mW nominal power, 660 nm wavelength, and pump beam radius of 1.0 mm (no focusing lens), was used as a periodic intensity light source by modulating its electronic power source. The
amplitude and phase of the microwaves absorption signal from the diode-detector external output, in the microwaves bridge, were analyzed in a Stanford lock-in amplifier, model SR830, for modulation frequencies in the range from 100 Hz to 100 kHz.
The transfer function of the whole experimental appara-tus is essentially flat: the microwaves diode-detector has a bandwidth of 6.0 MHz; the lock-in amplifier operates between 1.0 mHz and 102.4 kHz; and the laser intensity sine-modulation was found to vary less than 8% in the range from 100 Hz up to 100 kHz, as measured by a high bandwidth photodetector. This variation is much smaller than those observed in our experiments, and no data correction was per-formed in the curves presented below.
III. THEORY
A. PMMR signal from resonant absorption of microwaves by a magnetic material
As mentioned above, the PMMR signal originates in the sample temperature variation produced by the absorption of the laser beam. Indeed, it is proportional to the variation of the imaginary part of the magnetic susceptibility, χ00, due to the temperature oscillation T, multiplied by the microwaves magnetic field intensity, Hmw, integrated over the whole volume of the sample.
SignalPMMR/ ð vol @χ00(~r) @T T(~r)Hmw(~r)dv: (1)
Let us consider the one-dimensional treatment of a mag-netic sample of thickness ‘, which is valid in the cases when the dimension of the illuminated (heated) area of the sample (the radius of the laser Gaussian beam, for instance, which is 1.0 mm in our case) is much larger than the sample thickness or than the sample thermal diffusion length.13 Both condi-tions are fulfilled in our experiments. In this case, if the microwaves skin depth is δ (see Fig. 2) so that Hmw(x)¼ Hmw(0)ex=δ, we can write
SignalPMMR/ TH¼ ð‘
0
ex=δT(x)dx, (2) with TH being the temperature variation induced by the laser absorption integrated over the sample thickness and weighted by the microwaves magnetic field amplitude in the sample. Such characteristic average temperature, TH, actually is a temperature multiplied by a characteristic length.
The temperature profile of a sample of thickness ‘ (medium 1), on an infinite substrate or backing (medium 2; see Fig. 2), heated by the absorption of a laser beam on the sample front surface (x¼ 0), is obtained by solving the heat diffusion equation with the appropriate boundary conditions, i.e., the adiabatic condition at x¼ 0 and the continuity of temperature and heatflux at x ¼ ‘, and given by11
T1(x)¼ I0 k1σ1 (1þ b)eσ1(‘x)þ (1 b)eσ1(‘x) (1þ b)eσ1‘ (1 b)eσ1‘ for 0 x ‘: (3)
In Eq. (3), I0 is the absorbed laser beam intensity, b¼ e2=e1¼ k2σ2=k1σ1 is the ratio between the thermal effusiv-ities ei,σi¼ (1 þ i)=μiwith μi¼
ffiffiffiffiffiffiffiffiffiffiffi αi=πf p
being the thermal diffusion length, ki(αi) is the thermal conductivity (diffusiv-ity) of medium i, and f ¼ ω=2π is the modulation frequency of the laser beam intensity.
Putting Eq.(3)into Eq.(2), and usingγ ¼ δ1, the char-acteristic average temperature THbecomes
TH¼ I0 k1σ1(σ21 γ2) 1 (1þ b)eσ1‘ (1 b)eσ1‘ (1 þ b)(σ1 γ)eσ1‘ (1 b)(σ1þ γ)eσ1‘ þ 2(γ bσ1)eγ‘ 8 > < > : 9 > = > ;: (4) 1. Limiting cases
The expression for TH, Eq. (4), can be simplified in those limits where δ ‘ (the microwaves do not penetrate the sample-conducting, metallic samples) and δ ‘ (dielectric samples). On the other hand, ifδ ‘, simplified expressions can also be found in the cases in which the tem-perature oscillation is almost uniform throughout the whole sample thickness (thermally thin limit, μ1 ‘), as well as those for which the temperature oscillation is restricted to a region close to the sample surface (thermally thick limit, μ1 ‘). These simplified expressions are listed in Table I, Eqs.(5a)–(5d). The additional conditionδ μ1 was used to reach Eq.(5a).
Roughly speaking, the surface temperature governs the signal when the microwaves do not penetrate the sample (case i), while the average temperature plays the role in the case of a sample nearly transparent to the microwaves (case iv). The intermediate cases, i.e., when the microwaves are partially attenuated while crossing the sample, are also gov-erned by the surface temperature (case ii, for which the con-dition μ1 ‘ ensures an almost uniform temperature across the sample) or by the average temperature (case iii, for which the conditions μ1 ‘ and μ1 δ result in an almost uniform Hmwacross the heated portion of the sample).
The above expressions were derived using a one-dimensional model with both the laser beam and the micro-waves directed onto the sample front surface, at x¼ 0. Actually, the sample is immersed in the standing microwaves in the cavity, so the microwaves reach both the front and the rear surface in the case of a dielectric, non-conducting, backing. Therefore, an accurate expression for the PMMR signal must take into account the true dependence of Hmw(x) on x (see Eq.(1)for a general dependence on~r). For the par-ticular case in which δ ‘ (metallic samples, for instance), we can write TH¼ ð‘ 0 ex=δþ e(‘x)=δT1(x)dx: (6) Taking the limit ofμ1 δ, we find
TH¼ δ[T1(0)þ T1(‘)]: (7) In order to see the general behavior of the PMMR signal, the calculated amplitudes of the functions T1(0) and Tav are
FIG. 1. (a) Block diagram of the experimental setup. (b) Details of the microwaves cavity with the intensity modulated laser beam, the dc magnetic field, and the sample surface alignments.
FIG. 2. Schematic diagram of the sample (medium 1) of thickness‘ on an infinite substrate or backing (medium 2). The microwaves skin depth, δ, and the thermal diffusion length in the sample,μ1, are also depicted.
TABLE I. Simplified expressions for THvalid for specific cases. Forδ ‘
(case i), see also Eq.(6)and related discussion.
(i) δ ‘, δ μ1 T H¼ δT1(0) (5a) (ii) δ ‘, μ1 ‘ TH¼ δ(1 e‘=δ) T1(0) (5b) (iii) δ ‘, μ1 ‘ T H ¼ ‘ Tav (5c) (iv) δ ‘ TH ¼ ‘ Tav (5d)
shown in Fig.3, for b¼ 0, b ¼ 1, and b 1, as a function of the relative frequency, f=fc, fc¼ α1=π‘2 being the fre-quency at which μ1 ¼ ‘. Here, T1(0) was calculated using Eq.(3), while Tavis given by
Tav¼ 1 ‘ ð‘ 0 T(x)dx ¼ I0 k1σ21‘ 1 2b (1þ b)eσ1‘ (1 b)eσ1‘ : (8) The frequency dependences of the curves are indicated in Fig.3and summarized in Table II, for the low and high frequency limits. Well above fc, T1(0) goes as f1=2 and Tav as f1, and their phases are45and90, respectively. At low frequencies, for both T1(0) and Tav, the slope of the amplitude shifts from f1 to f0 as b varies from zero to very large values, while the phase shifts from 90 to zero. Indeed, for f fc, Tav=T1(0)¼ (1 þ bσ1‘)1, and provided the frequency is low enough to ensure bσ1‘ 1, the average temperature equals the surface temperature.
As a final analysis of the above expressions, Fig. 4
shows the calculated curves for amplitude and phase of THas a function of f=fc, forδ ‘, δ ¼ ‘, and δ ‘. As one can see, the curves forδ ‘ and δ ‘ present the same behav-ior as those of Fig.3for T1(0) and Tav, respectively, in the entire range of frequency. The intermediate curves, for δ ¼ ‘, on the other hand, follow T1(0) at low frequencies and Tav at high frequencies, in accordance to the limiting expressions in TableI. Such a transition from T1(0) to Tav is clearly evidenced by the phase curve for b¼ 0, which starts
following the phase of T1(0) and turns back to the value of the phase of Tav[bottom long-dashed line in Fig.4(b)].
B. Photo-modulated signal from non-resonant absorption of microwaves by free carriers
Besides the resonant absorption by the magnetic sample, free carriers also absorb the incident microwaves. Actually, this is the mechanism which prevents the microwaves from penetration in metallic materials, for instance, thus originat-ing the tinny skin depth. Such absorption of microwaves by free electrons in metals does not change significantly with the temperature oscillation caused by the laser absorption, once the density of free electrons almost does not change (very small variation caused by the lattice expansion). However, when dealing with semiconductors, as is the case of silicon often used as magnetic thin films substrate, the laser beam absorption creates electron-hole pairs which diffuse throughout the sample before recombining. Therefore, the density of free carriers significantly oscillates at the fre-quency of the modulated laser beam. The electrical conduc-tivity, proportional to the density of carriers, also oscillates. For small variations induced by the laser beam absorption, all the quantities dependent on the electrical conductivity will oscillate at the same frequency. Finally, there will be absorption of microwaves at the modulation frequency, detected by the magnetic resonance spectrometer as well. Such a signal is independent of the applied dc magneticfield (non-resonant absorption). The frequency dependence of the photo-modulated signal, in this case, will be that of the density of injected carriers by the laser beam absorption.
Here, the one-dimensional treatment will be used again as an approximation to describe the main feature of the fre-quency dependence of the charge carrier density. The condi-tions of validity of the one-dimensional treatment are not accomplished so comfortably here, in opposition to the case of the thermal problem discussed in Sec.III A. Although the sample thickness and the carrier diffusion characteristic length (above a few kHz) are smaller than the pump beam radius, actually they are not very much smaller in general, as we can see by using the sample parameters described below.
FIG. 3. Amplitude of (a) the surface temperature, T1(0), and (b) the average temperature, Tav, as a function of f=fc, as calculated using Eqs.(3)and(8).
TABLE II. Limiting frequency dependence and phase of T1(0) and Tav. At
very low frequencies, both T1(0) and Tav present similar behavior for a
given value of b. f fc(μ1 ‘) f fc(μ1 ‘) b¼ 0 f1 −90 deg T1(0) f1=2 −45 deg b¼ 1 f1=2 −45 deg b 1 f0 0 deg T av f1 −90 deg
In order to solve the one-dimensional carrier diffusion equation14 for a sample of thickness‘, the carrier generation was placed at the front surface x¼ 0 (the depth of penetra-tion of 660 nm light in silicon is 3 μm15), and the Robin boundary condition at the sample surfaces were employed
Front surface (x¼ 0): JP v0n(0)¼ D @n @x x¼0 (9a) and Rear surface (x¼ ‘): D @n @x x¼‘ v‘n(‘) ¼ 0, (9b) where n(x) is the carrier density, JP is the photonflux at the front surface (if we consider 1 photon = 1 electron-hole pair, then JP¼ I0=hν, with ν being the laser light frequency), D is the ambipolar carrier diffusivity, and v0 and v‘ are the surface recombination velocities at x¼ 0 and x ¼ ‘, respec-tively. The front surface carrier density, n(0), and the average carrier density along the sample thickness, nav, thus become
n(0)¼ JP (D=‘) (ξ‘ þ δ‘)eξ‘þ (ξ‘ δ‘)eξ‘ (ξ‘ þ δ0)(ξ‘ þ δ‘)eξ‘ (ξ‘ δ0)(ξ‘ δ‘)eξ‘ (10) and nav¼ (1=‘) ð‘ 0 n(x)dx ¼ JP (D=‘) 1 ξ‘ (ξ‘þδ‘)eξ‘(ξ‘δ‘)eξ‘2δ‘ (ξ‘þδ0)(ξ‘þδ‘)eξ‘(ξ‘δ0)(ξ‘δ‘)eξ‘: (11) Here,δ0¼ v0=(D=‘) and δ‘¼ v‘=(D=‘) are the surface recom-bination velocities normalized by a characteristic sample velocity, D=‘, and ξ is the complex carrier diffusion coeffi-cient, given by ξ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ iωτ Dτ r , (12)
withτ being the electron-hole bulk recombination time. In the case of silicon, D¼ 20 cm2=s16,17and for our Si (111) substrate, ‘ ¼ 535μm, so the characteristic velocity of our sample becomes (D=‘) 400 cm=s.
We define the characteristic frequencies fD¼ D=π‘2 and fτ ¼ 1=τ so that we can write ξ‘ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi( fτ=πfD)þ 2i( f = fD)
p
. For low modulation frequencies, well below fτ,ξ‘ ¼ ‘=pffiffiffiffiffiffiDτ, which is frequency independent. In this case, n(x) becomes independent of the modulation frequency, and the same applies to the surface carrier density, n(0), and to the average carrier density, nav. When f fτ, ξ‘ ¼ (1 þ i)
ffiffiffiffiffiffiffiffiffiffi f= fD p
. At intermediate frequencies, when fτ f fD (jξ‘j 1), if possible, the signal goes as f1 for δi
ffiffiffiffiffiffiffiffiffiffi f= fD p (i¼ 0, ‘) and as f0 for δ0 ffiffiffiffiffiffiffiffiffiffi f= fD p orδ‘ ffiffiffiffiffiffiffiffiffiffif= fD p . Finally, in the high frequencies limit, when f fD (jξ‘j 1), the signal goes as f1 forδ0 ffiffiffiffiffiffiffiffiffiffi f= fD p and as f1=2forδ0 ffiffiffiffiffiffiffiffiffiffi f= fD p (in this case, the signal is independent ofδ‘).
Here, again, the frequency behavior of the measured signal depends whether microwaves penetrate the sample or not. For silicon, for instance, the electrical conductivity is of the order of (103 104)(Ω m)1,18 thus making δ in the range of centimeters (δ ‘). In this case, the photo-modulated signal becomes proportional to nH¼ ‘nav.
Figure5shows the calculated amplitude of nHas a func-tion of f= fD, in a range that corresponds to our measure-ments, i.e., 100 Hz–100 kHz (fD¼ 2:25 kHz for our Si substrate). The ratio fτ= fD¼ 0:45 was used, which corre-sponds to a value for the bulk recombination time of 1:0 ms. Furthermore, the used values for the normalized surface recombination velocities were δi¼ 0, 2:5, and 25 (i ¼ 0, ‘). They correspond to velocities of zero, 103cm=s, and
FIG. 4. (a) Amplitude and (b) phase of the characteristic average temperature, TH, as a function of f=fc, calculated
using Eq. (4). Distinct values of the skin depthδ and of the thermal cou-pling parameter b are shown.
FIG. 5. Relative amplitude of the average carrier density, nH, as a function
of f= fD, calculated using Eq.(11), for fτ= fD¼ 0:45. The three solid curves
(bottom curves) were calculated usingδ0¼ 0 and δ‘¼ 0, 2:5, and 25. The
two dashed curves are the results forδ0¼ δ‘, withδ‘¼ 2:5 and 25. The f1
104cm=s, taking into account that (D=‘) ¼ 400 cm=s for our case. The amplitude curve for δ0¼ δ‘¼ 0 presents a f1 slope for almost the entire range of frequencies, except at the lower extreme. As the front and the rear surface recombina-tion velocities increase, a plateau is observed at low frequen-cies (frequency independent), thus shifting to higher values the range of frequencies at which the amplitude decays. Finally, at high frequencies, the slope shifts from f1 to f1=2 as the surface recombination velocities increase. All these features observed in the calculated curves endorse the analysis of the limiting cases described above.
IV. RESULTS AND DISCUSSION A. Fe and Ni 50μm foils
Figure 6 presents the conventional ferromagnetic reso-nance (FMR) and the PMMR spectra for the samples of Fe and Ni. The used experimental conditions were 5.0 mW microwaves power, 113 mW laser beam power, and 100 Hz laser modulation frequency. The FMR lines are centered at 0.56 kOe and 1.36 kOe and have peak-to-peak linewidths of 0.92 kOe and 0.91 kOe, for Fe and Ni, respectively. The PMMR spectra are centered at 0.67 kOe and 1.53 kOe, i.e., external magnetic fields slightly higher than those for the FMR lines, with linewidths of the same order. The PMMR signal phase was almost constant along the magnetic field scan for both samples. This particular shape of the PMMR spectra was previously observed2,11 and indicates that the temperature increase produced by the laser absorption results mainly in attenuation and broadening of the FMR line, instead of shifting it to higher magneticfields.
The shape of the PMMR spectrum did not change with the modulation frequency, indicating uniformity of the Fe and Ni samples across its thickness. By fixing the external magnetic field and sweeping the laser power, the PMMR signal amplitude varied linearly with the power, as expected. On the other hand, byfixing the laser power and the external magnetic field, the PMMR signal amplitude showed to be proportional to the square root of the microwaves power, i.e., proportional to Hmw. Similar responses were generally observed for the other samples discussed below.
The measurements of the PMMR signal as a function of the modulation frequency of the laser intensity were per-formed by setting the external magnetic field to a value around the PMMR amplitude peak (we used 0.70 kOe for Fe and 1.45 kOe for Ni). Figure 7(a) (symbols) shows the PMMR signal amplitude for both samples in the range from 100 Hz to 100 kHz, and Fig. 7(b) presents the measured signal phase for the Fe sample (similar result was found for Ni). The observed behavior of the signal amplitude goes as f1 at low frequencies and changes to f1=2 at high frequen-cies, for both samples. Also, as the frequency increases, the signal phase shifts up almost 45° in the measured range.
Using‘¼50 μm, αFe¼ 0:20 cm2=s, and αNi= 0.23 cm2/s,19
we find fc,Fe¼ 2:55 kHz and fc,Ni¼ 2:93kHz (indicated by arrows in Fig. 7). Moreover, the electrical conductivity of these metallic samples is σFe¼ 1:0 107(Ω m)1 and σNi¼ 1:4 107(Ω m)1,20which make the microwaves skin depths δ smaller than one tenth of micrometer. Therefore,
δ ‘ and δ μ1 in the entire range of frequencies. In this case, at afirst glance, the behavior of the PMMR signal must be dictated by Eq. 5(a), i.e., proportional to TH ¼ δ T1(0). Indeed, the overall behavior of the measured PMMR signal is consistent with T1(0) for b ¼ 0 (see TableIIand Figs. 3 and 4). The value of b in our experiment is in fact much smaller than one, once we used a backing of PVC. Taking the literature values kFe¼ 800 mW=cm K, kNi¼ 900 mW=cm K, αPVC¼ 0:7 103cm2=s, and kPVC=
1:4 1:6 mW=cm K,20–23 the calculated thermal coupling parameters are bFe bNi 0:03.
Although the observed experimental behavior fulfills the prediction in the limiting frequencies, the calculated curve using Eq. 5(a)does notfit very well the experimental data in the transition between thermally thin and thermally thick regimes (dashed lines in Fig. 7). Definitely, the appropriate description of the signal in this case is given by Eq. (7), which takes into account the incidence of microwaves both at the front and at the rear surface of the sample. Indeed, the calculated curves using Eq. (7) (solid lines in Fig. 7) fit much better the experimental data in the intermediate fre-quency range than those obtained using Eq.(5a).
B. γ-Fe2O3cassette tape
In this case, the orientation of the sample is so that not only the surface of the tape is parallel to the external mag-netic field, ~H, but also the recording direction has the same direction of the magnetic field. It is important to state this because theγ-Fe2O3 tape presents anisotropy not only in the direction perpendicular to the plane, which is associated to the demagnetization field, but also in the plane itself due to the orientation of theγ-Fe2O3 needle shaped particles along the recording direction. In this configuration (~H parallel to the recording direction), the external resonance magnetic field is minimum, shifting always up by rotating the sample to any other direction.
The FMR spectrum (not shown) measured under the configuration described above and using 5 mW microwaves power has a line centered at 1.38 kOe, with a peak-to-peak linewidth equal to 0.71 kOe. The PMMR spectrum (not shown) has two non-symmetrical amplitude peaks close to the external resonance magnetic field, accompanied by a 180° phase shift between them. The shape of the PMMR curve in this case was also observed before,3,6,7,11and it indi-cates that the main mechanism of signal modulation is by shifting the resonance magnetic field with temperature, in opposition to that observed above in the case of Fe and Ni foils.
For the measurements of the PMMR signal as a function of the modulation frequency, the external magneticfield was set to 1.2 kOe, with 20 mW microwaves power and 113 mW laser beam power. Figure 8(a) (symbols) shows the PMMR signal amplitude in the range from 100 Hz to 100 kHz, and Fig.8(b)presents the measured signal phase. As the modula-tion frequency increases, the slope of the measured signal amplitude changes from f1=2 to f1, while the signal phase shifts down almost 45°.
As mentioned above, the cassette tape is made up of 15μm polyester backing and a 5 μm magnetic layer of γ-Fe2O3 needle shaped particle immersed into a polyester matrix. Using a characteristic value for the thermal diffusivity of polyester, αpolyester¼ 1:2 103cm2=s24 and ‘ ¼ 5 μm for the thickness of theγ-Fe2O3 layer (medium 1),3wefind fc¼ 1:53 kHz for the magnetic layer in the cassette tape (indicated as arrows in Fig.8). On the other hand, by taking the 15μm polyester layer as medium 2, one can see that it is thermally thick above 170 Hz. Therefore, for frequencies well above this value, the semi-infinite approximation for the substrate becomes fully applicable.
Polyester is highly transparent to the microwaves, since its electrical conductivity isσpolyester 1015(Ω m)1.25This means that we are truthfully working in the limit ofδ ‘. Furthermore, in view of the similar thermal properties (α and k) of the magnetic layer and backing, as considered before, b 1. Under these circumstances, we can predict that the PMMR signal must follow the sample (γ-Fe2O3 layer) average temperature, whose amplitude is represented by the long-dashed line in Fig.3(b)for b¼ 1, with a slope of f1=2 at f fcand of f1 at f fc. Such a behavior is quite well replicated in the experimental data of Fig.8(a). Additionally, the measured shift down of about 45° in the phase [see Fig.8(b)] closely follows the predicted behavior for δ ‘ and b¼ 1 depicted in Fig.4(b). Finally, one should mention that similar frequency dependence was observed in Gdfilms on fused quartz substrates reported in Ref.11.
C. Co and Permalloy thinfilms on glass substrate
The film of Co on glass substrate presents an almost symmetrical FMR line centered at 0.50 kOe, with a peak-to-peak linewidth equal to 0.18 kOe. The corresponding PMMR spectrum has non-symmetrical amplitude peaks close
to the external resonance magnetic field (center of the FMR line), accompanied by a 180° phase shift between them, similar to the case of the γ-Fe2O3 cassette tape discussed above. The FMR spectrum of the Permalloy film on glass substrate presents spin wave modes (the uniform mode, n = 0, and the non-uniform modes, n = 1 and n = 2, are resolved in our measurements). The FMR line for n = 0 is centered at 1.094 kOe, with a peak-to-peak linewidth of 0.037 kOe. The associated PMMR spectrum also presents non-symmetrical amplitude peaks with 180° phase shift, as in the case of the Co film. The FMR and PMMR spectra described above are not shown.
For the measurements of the PMMR signal as a function of the modulation frequency, the external magneticfield was set to 0.51 kOe for the Co film and to 1.12 kOe for Permalloy. Figure 9(a) (symbols) shows the PMMR signal amplitude for both samples in the range from 100 Hz to 100 kHz, and Fig.9(b)presents the measured signal phase for the Co film (similar phase result was found for Permalloy). The observed behavior of the signal amplitude goes as f1=2 in the entire range of frequencies for both samples.
The thicknesses of the magnetic films are ‘Co¼ ‘Permalloy¼ 150 nm, making the characteristic frequency fcof the order of tens to hundreds of megahertz for both films. Here, we used the literature bulk values αCo¼ 0:27 cm2=s andαPermalloy¼ 0:12 cm2=s.19,20,26Therefore, both layers are thermally very thin in the used range of frequencies (f fc). Besides, their optical absorption coefficient at 660 nm are in the range of (7:7 8:1) 105cm1,27,28
resulting in optical absorption lengths of about 10 nm. Hence, the surface absorption condition used in our model is still roughly valid. Furthermore, and more important, since the films are ther-mally very thin, their average temperatures (which are very close to the front surface temperatures) are independent on the location of a plane heat source within medium 1, from
FIG. 6. FMR and PMMR spectra for the 50μm foils of (a) Fe and (b) Ni. Experimental conditions: 5.0 mW micro-waves power, 113 mW laser beam power, and 100 Hz laser modulation frequency.
FIG. 7. (a) PMMR signal amplitude for the 50μm foils of Fe and Ni as a function of the modulation frequency for 20 mW microwaves power and 113 mW laser beam power. (b) PMMR signal phase for the Fe foil. Symbols represent the experimental points, while lines are the calculated curves using Eq. (5a) (dashed) and Eq. (7)
x¼ 0 up to x ¼ ‘ (the same is valid for a superposition of sources). Such a result can be rigorously demonstrated by simplyfinding the Green function of the problem and taking the thermally thin limit.
The microwaves skin depths, on the other hand, are cal-culated using the electrical conductivity (σCo¼ 1:7 107(Ω m)1 and σPermalloy¼ 5 106(Ω m)1)20,29 and esti-mated values of the relative magnetic permeability (μrel,Co[ [100 102] and μrel,Permalloy 104 105),
30
resulting in δCo and δPermalloy in the order of 101 102nm. Hence, our experimental conditions here (δ ‘ and μ1 ‘) fit those embodied in Eq. (5b), which predicts that the PMMR signal is proportional to the surface temperature of the sample, T1(0).
For f fc, the frequency dependence of T1(0) is strongly dependent on the value of the thermal coupling parameter, b, as one can see from Fig.3(a). Indeed, Eq.(3), with x¼ 0 and f fc, yields
T1(0)¼ I0 k1σ1 1þ bσ1‘ σ1‘ þ b , (13)
making explicit its dependence on b. Literature values for the thermal conductivities of bulk Co and Permalloy are kCo¼ 1000 mW=cm K and kPermalloy¼ 460 mW=cm K.20,21,26 Actually, measurements of the thermal conductivity in 75 nm and 167 nm Co films showed values 5 times and 20 times smaller than those for bulk Co, respectively.31 Furthermore, for the glass substrate, we have the characteristic thermal transport parameters kglass¼ 15 mW=cm K and αglass¼ 5:0 103cm2=s.32Therefore, using the bulk values for the thermal conductivity and diffusivity of the metallic films, the thermal coupling parameters become bCo=glass 0:11 and bPermalloy=glass 0:16. However, by using the films
values of Ref. 31, larger values of b can be estimated (bCo=glass[ [0:22 2:2]). Nonetheless, in any case, it is easy to recognize that jσ1‘j b and jbσ1‘j 1 in the entire range of frequencies of our measurements (a direct conse-quence of jσ1‘j ¼
ffiffiffiffiffiffiffiffiffiffi f= fc p
1 and b 1 or less). Taking this specific limit in Eq. (13), we find T1(0)¼ I0=bk1σ1¼ I0=k2σ2, i.e., the temperature of the magnetic film is essen-tially governed by the thermal properties of the 150μm glass substrate (medium 2) (here, we must state that the expression for T1(0) obtained in this specific limit could be reached directly by ignoring the metallic thin layer, i.e., by removing medium 1 from the calculation, and taking the surface tem-perature of the semi-infinite substrate-medium 2-heated at its own surface). The glass substrate, in its turn, is thermally thick for frequencies above fc,glass¼ 7 Hz, thus ensuring the validity of the semi-infinite medium 2 considered in the temperature calculation.
The achieved expression T1(0)¼ I0=k2σ2, valid for the Co and Permalloy thin films on glass substrate, has a fre-quency dependence of the form f1=2. This prediction fits precisely the measured signal amplitude, as shown in Fig.9(a).
D. Co and Permalloy thinfilms on silicon substrate
The FMR lines of Co and Permalloy films on silicon substrate present external magneticfields for resonance equal to 0.49 kOe and 1.117 kOe, respectively, very close to those obtained in the films on glass substrate. The peak-to-peak linewidth of Permalloy (0.038 kOe) is also almost the same; however, that of Co on silicon (0.34 kOe) is around twice the measured for Co on glass. The PMMR spectra also follow behaviors similar to those observed in the films on glass
FIG. 8. PMMR signal (a) amplitude and (b) phase for theγ-Fe2O3cassette
tape as a function of the modulation frequency for 20 mW microwaves power and 113 mW laser beam power. Symbols represent the experimental points, while lines are calculated using Eq.(5d).
FIG. 9. PMMR signal (a) amplitude for Co and Permalloy films on the glass substrate as a function of the modulation frequency for 20 mW microwaves power and 113 mW laser beam power. (b) PMMR signal phase for the Co on the glassfilm. Symbols represent the experimental points, while lines are the corresponding f1=2 power laws and the constant phase, as predicted by Eq.(13)using the appro-priate parameters.
observed. For both Co and Permalloy on the Si substrate, the signal amplitudes decay much slower than f1=2 in the range from 100 Hz to 100 kHz, as one can see in Fig.10: average slopes of f0 for Co (open squares) and of f0:3 for Permalloy (open circles).
Also in Fig. 10, the frequency dependence of the PMMR signal amplitude for the Si (111) sample is shown (used as a substrate for thefilms), with an average slope of f0:15(open triangles). Silicon presents diamagnetic behavior at ambient temperature and therefore does not exhibit magnetic resonance. However, the free charge carriers actu-ally do absorb incident microwaves, as already discussed in Sec.III B. Such absorption is, naturally, independent on the applied dc magneticfield, giving no signal at all in a conven-tional magnetic resonance measurement. On the other hand, upon modulated laser absorption, as it happens in PMMR measurements, the density of free carriers, n(x), becomes modulated. Under these conditions, a non-resonant photo-modulated signal is produced, whose general behavior is pre-dicted by Eq.(11)and shown in Fig.5. The observed signal amplitude dependence on the modulation frequency, with a slow decay slope, can be well described by Eq. (11) with appropriate values of surface recombination velocities, as depicted in Fig.5. Furthermore, the three-dimensional char-acter of the carrier diffusion, not totally negligible at low fre-quencies, would favor the small decay slope of the signal magnitude, as observed in our measurements.
The analysis of the frequency dependence of the Co and Permalloyfilms, conversely, is more complex and relies on the thermal mismatch between the film and substrate. The 150 nmfilms, covered by a 10 nm Ta protective layer, do not transmit the laser light to the Si substrate. Their optical absorption coefficient at 660 nm are in the range of (7:7 8:1) 105cm1,27,28
resulting in optical
generated in the substrate is negligible in these cases.
The temperature of the magneticfilm, which modulates the magnetic resonance, thus producing the PMMR signal, is strongly influenced by the substrate thermal properties, as discussed in Sect. IV C. The relatively higher thermal conductivity and diffusivity of the Si substrate (kSi¼ 1480 mW=cm K and αSi¼ 0:9 cm2=s)19–21,26 result in larger values of the b parameter, when compared to the glass substrate. Using the bulk values for the thermal conductivity and diffusivity of the metallicfilms presented in Sect.IV C, the thermal coupling parameters become bCo=Si 0:8 and bPermalloy=Si 1:7. However, by using the films values of Ref.31, even larger b can be estimated (bCo=Si[ [1:6 16]). One of the consequences of these larger values of b is that the temperature of the films can no longer be given exclusively by the thermal properties of the substrate, as in the case of glass substrate treated before. Instead, Eq. (13)
must be taken without any further simplification in the low frequency domain (f fc)-a minor simplification could be done in its denominator, but we will keep it in the unchanged form for further analysis. The calculated surface temperature T1(0), using Eq.(13), is represented in Fig. 10by the small closed symbols for three values of the parameter b (for now, b0¼ b), and by taking fc ¼ 2 107Hz, a value well-suited for the poorest value ofαCoin Ref.31. As one can see, only very large values of b, above the estimated ones for our samples, atfirst, significantly change the slope of the curves, dropping them close to the measured ones.
Thermal contact resistance between the film and sub-strate can play an important role in the temperature profile of the sample, especially in the cases when b. 1, i.e., when the heat flux to the substrate is substantial.12 The thermal contact resistance can originate in poor adhesion of the film or in the mismatch of the lattice parameters offilm and sub-strate atoms. Also, an interfacial alloy layer composed offilm and substrate atoms can be formed during the deposition process, as is the case of Co silicides,33 and could be modeled as a thermal contact resistance. The introduction of a thermal contact resistance RT at x¼ ‘ in our model (see Fig. 2) will result in discontinuity of temperature, propor-tional to the heat flux Φ ¼ k(dT=dx). In this case, the appropriate boundary condition to be used at x¼ ‘ becomes T1 T2 ¼ RTΦ (instead of T1¼ T2, previously employed). By introducing RT, the temperature profile T1(x) keeps the same general form of Eq. (3), by simply replacing the brack-ets (1+b) by (1 þ RTk2σ2+b). In the limit of very low mod-ulation frequencies, the analogous of Eq.(13)for the surface temperature becomes T1(0)¼ I0 k1σ1 (1þ RTk2σ2)þ bσ1‘ (1þ RTk2σ2)σ1‘ þ b ¼ I0 k1σ1 1þ b(RTk1σ1þ σ1‘) (1þ RTk2σ2)σ1‘ þ b: (14) Considering moderate values of RT so that j (1 þ RTk2σ2)σ1‘ j ,, b, the surface temperature can be simplified to T1(0)¼ (I0=k1σ1b) [1þ b(1 þ RTk1=‘)σ1‘].
FIG. 10. Relative PMMR signal amplitude for Co and Permalloyfilms on silicon substrate, and for naked Si, as a function of the modulation frequency for 20 mW microwaves power and 113 mW laser beam power. Open symbols represent the experimental points, while dashed lines are the f0 and f1=2
power laws. Closed small symbols correspond to the calculated T1(0) using
Eq.(13)for fc¼ 2 107Hz and three values of the modified thermal coupling
parameter b0¼ 2, b0¼ 10, and b0¼ 50. The measured signal amplitudes at 100 Hz were 4.5 mV for Co, 1.1 mV for Permalloy, and 0.7 mV for Si.
This expression becomes the same as Eq.(13)for b. 1, pro-vided b and I0 are replaced by b0¼ (1 þ RTk1=‘)b and I00¼ (b0=b) I0. In other words, the effect of the thermal contact resistance for b. 1 is the same obtained by enhanc-ing the value of b, i.e., by accentuatenhanc-ing the effect of the thermal mismatch between medium 1 and 2. For instance, the curve for b0¼ 50 shown in Fig. 10 can be achieved by taking b¼ 10, which can be reasonable for the Co film on Si, and RT¼ 4(‘=k1). Such a value of thermal contact resis-tance means that, under the same dc heatflux, it produces a temperature discontinuity equivalent to that of a layer of the material 1 with a thickness equal to 4‘. Finally, the effect of the thermal contact resistance saturates, i.e., the temperature discontinuity reaches its maximum by virtually vanishing the temperature of medium 2, when (1þ RTk1=‘) 1, i.e., RT ‘=k1.
V. CONCLUSIONS
In this study, we investigated a set of magnetic samples in the form of foils, layers, and thin films, using the photothermally modulated magnetic resonance technique. Specifically, the frequency dependence of the signal was explored. The signal amplitude and phase as a function of the modulation frequency presented markedly distinct behav-iors depending on the electrical conductivity of the sample, as well as on the thermal properties of both the sample and substrate. The depth of penetration of the microwaves, gov-erned by the sample’s electrical conductivity, selects the portion of the sample that is being probed, and hence the rel-evant region of temperature modulation that produces the signal. Furthermore, at low modulation frequencies, the mis-match between the thermal properties of the sample layer and the substrate plays a fundamental role in the temperature of the sample itself, thus strongly interfering in the signal behavior. Also, we observed and modeled the non-resonant signal from the diamagnetic Si, which originates in the microwaves absorption by photo-injected free carriers, and can play a significant role in the case of semi-transparent films.
In the case of 50μm Fe and Ni metallic foils, the small depth of penetration of the microwaves rendered the PMMR signal proportional to the sample surface temperature. The signal amplitude as a function of the frequency presented slopes that changed from f1 at low frequencies to f1=2 at high frequencies. Conversely, the dielectric cassette tape (γ-Fe2O3), which is traversed by the microwaves, presented slopes that changed from f1=2 at low frequencies to f1 at high frequencies, following the average temperature depen-dence on f.
The thinfilms of Co and Permalloy showed very distinct results for glass and Si substrates. Both films are thermally very thin in the range of frequencies investigated. In the case of the glass substrate, because of the small values of the thermal coupling parameter b, the films could be regarded simply as magnetic surface layers, with the temperature gov-erned by the glass substrate thermal properties. The predicted f1=2 signal amplitude dependence was observed in the whole range of frequencies. On the other hand, thefilms on
the Si substrate, with larger values of b, presented signal amplitude dependence with much smaller slopes. Values of b 1 tend to decrease the slope of the signal amplitude at low frequencies and can explain, in part, the measured signal. Additionally, the existence of a thermal contact resis-tance between thefilm and substrate, although does not play significant role for b , 1, accentuates the reduction of the slope for b 1, thus shifting the calculated curves to a behavior much closer to the experimental ones.
Finally, it is worth concluding that, by scanning the laser modulation frequency, the PMMR signal can reveal the key electrical, magnetic, and thermal characteristics of the sample, making the technique an important non-destructive tool to investigate local properties of layered magnetic struc-tures, including thin magnetic films. The good agreement between calculation and experiment observed in our study offers confidence for the use of the technique to extract sample parameters from the data fitting of the measured signal.
ACKNOWLEDGMENTS
The authors acknowledge the Brazilian agencies FAPERJ, FAPESP, CNPq, and CAPES forfinancial support. Isabel Merino and Elisa Saitovitch, from the Brazilian Center for Research in Physics (CBPF), are acknowledged for the preparation of the Co and Permalloy films on glass and Si substrates.
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