ISSN 10637761, Journal of Experimental and Theoretical Physics, 2015, Vol. 120, No. 6, pp. 1041–1054. © Pleiades Publishing, Inc., 2015. Original Russian Text © A.B. Drovosekov, N.M. Kreines, A.O. Savitsky, E.A. Kravtsov, D.V. Blagodatkov, M.V. Ryabukhina, M.A. Milyaev, V.V. Ustinov, E.M. Pashaev, I.A. Subbotin, G.V. Prutskov, 2015, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2015, Vol. 147, No. 6, pp. 1204–1219.
ORDER, DISORDER, AND PHASE TRANSITION IN CONDENSED SYSTEM
Interlayer Coupling in Fe/Cr/Gd Multilayer Structures A. B. Drovosekova*, N. M. Kreinesa, A. O. Savitskya, E. A. Kravtsovb, D. V. Blagodatkovb, M. V. Ryabukhinab, M. A. Milyaevb, V. V. Ustinovb, E. M. Pashaevc, I. A. Subbotinc, and G. V. Prutskovc a
Kapitsa Institute for Physical Problems, Russian Academy of Sciences, ul. Kosygina 2, Moscow, 119334 Russia *email:
[email protected] b Institute of Metal Physics, Ural Branch, Russian Academy of Sciences, ul. S. Kovalevskoi 18, Yekaterinburg, 620990 Russia c Russian Research Centre Kurchatov Institute, pl. Akademika Kurchatova 1, Moscow, 123182 Russia Received December 25, 2014
Abstract—The effect of the chromium layer thickness on the magnetic state of an [Fe/Cr/Gd/Cr]n multilayer structure is studied. A series of Fe/Cr/Gd structures with Cr spacer thicknesses of 4–30 Å is studied by SQUID magnetometry and ferromagnetic resonance in the temperature range 4.2–300 K. The obtained experimental results are described in terms of an effective field model, which takes into account a biquadratic contribution to the interlayer coupling energy and a nonuniform magnetization distribution inside the gado linium layer (which was detected earlier). Depending on the magnetic field and temperature, the following types of magnetic ordering are identified at various chromium layer thicknesses: ferromagnetic, antiferro magnetic, and canted ordering. A comparison of the experimental and calculated curves allowed us to deter mine the dependence of the bilinear (J1) and biquadratic (J2) exchange constants on chromium layer thick ness tCr. Weak oscillations at a period of about 18 Å are detected in the J1(tCr) dependence in the range 8– 30 Å. The interlayer coupling oscillations in the system under study are assumed to be related to the RKKY exchange interaction mechanism via the conduction electrons of Cr. DOI: 10.1134/S1063776115060059
1. INTRODUCTION The interlayer exchange coupling in magnetic lay ered structures attracts attention of researchers due to potential possibilities of practical application and from a fundamental standpoint. Although it has been actively studied in layered systems of various composi tions, most researchers were interested in the case of ferromagnetic (FM) layers of the same material sepa rated by a certain nonferromagnetic layer. As FM lay ered materials, transition 3d metals (e.g., Fe/Cr/Fe) and rareearth 4f metals (e.g., Gd/Y/Gd) were used. The Fe/Cr/Fe system is one of the most compre hensively studied layered structures [1–8]. An important specific feature of this system is the pres ence of interlayer coupling oscillations as a function of the chromium layer thickness. In most works, the socalled long oscillation period (about 17 Å) is observed; here, a 10Åthick Cr layer provides anti ferromagnetic (AFM) interlayer coupling and a 20Åthick Cr layer corresponds to interlayer FM coupling. These oscillations are considered to be related to the RKKY exchange mechanism by means of conduction electrons in chromium. Apart from the long period, the most perfect samples also exhibit a short oscillation period (about 3 Å), which is caused by the AFM structure of chromium [2, 5, 7]. Along with the conventional “Heisenberg”
exchange (FM or AFM), a substantial contribution from the “biquadratic” coupling of Fe layers through a Cr layer, which can result in a noncollinear mag netic structure, was also detected in most works [3, 4]. This unusual contribution is also associated with the AFM structure of the layer [7, 9]. The interaction in transition metal–rareearth metal (3d/4f) systems, among which the Fe/Gd and Co/Gd systems have received the most study, was investigated in a series of works. It was found that a strong AFM interaction takes place between 3d metal and Gd layers. A magnetic compensation point can exist in these systems due to the relatively low Curie temperature of gadolinium (290 K) relative to Fe and Co. Thus, such layered structures represent artificial ferrimagnets. These systems have a complex magnetic phase diagram, which includes collinear ordering regions with various Gd magnetization orientations with respect to a magnetic field and a canted state [10]. According to [11–13], strong exchange at the Fe–Gd interface leads to a substantially nonuniform magneti zation distribution inside Gd layers. Despite the long history of studying the interlayer magnetic interaction in layered structures, the exchange between the FM layers of different types, i.e., 3d and 4f metals, through a nonferromagnetic metallic layer is still poorly understood. The substan tially different structures of electron shells and, hence,
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different characters of magnetism in 3d and 4f ele ments can favor the manifestation of new interlayer coupling mechanisms. For example, the results obtained in [14–17] showed that the presence of the same layer (X) in 3d/X/4f and 3d/X/3d layered struc tures led to different types of the dependence of inter layer exchange on layer thickness X. To the best of our knowledge, the causes of this difference are still unknown. Therefore, it seems interesting to study the effect of a chromium layer on the interlayer coupling in the Fe/Cr/Gd system. This magnetic system was first proposed in [18] to form a structure with a high magnetic moment and a high FM ordering temperature. The authors of [18] assumed that the AFM sign of the interlayer coupling between Fe and Gd can be changed by introducing a spacer consisting of an odd number of Cr atomic lay ers. The numerical simulation of this system per formed in [18] supported this possibility. The results of experiments on studying Xray magnetic circular dichroism showed that the projections of the magneti zation of Gd and Fe layers on a magnetic field direc tion in a Fe/Cr/Gd structure had positive signs at a chromium layer thickness of 5 monolayers, which was attributed to an interlayer FM coupling. However, AFM ordering was retained at a thickness of 3 chro mium monolayer, which was explained by imperfect interfaces between the layers. Note that this behavior can also be explained by a rapid decrease in the AFM exchange with increasing chromium layer thickness even in the absence of interlayer coupling oscillations. Recently [19], a series of Fe/Cr/Gd superlattices was studied by magnetometry and reflectometry of polarized neutrons. It was found that a nonuniform magnetization profile, which is characterized by an increase in the magnetic moment near interfaces and its decrease at the center of the layer, formed in the Gd layers at room temperature and a small chromium layer thickness. As noted above, an analogous result was obtained for the Fe/Gd system [11–13]. The effect of the Cr layer thickness on interlayer coupling was also studied in the FeCo/Cr/Gd [20] and Co/Cr/Gd [21] superlattices, which are similar to the Fe/Cr/Gd system. However, the existence of only magnetometric data in those works makes it difficult to draw unambiguous conclusions regarding the charac ter of interlayer exchange. The purpose of this work is to comprehensively investigate the magnetic properties of Fe/Cr/Gd structures to determine the dependence of interlayer coupling on the Cr layer thickness. It is interesting to reveal the possibility of exchange oscillations, by anal ogy with the Fe/Cr/Fe system, in particular, the change of AFM exchange into FM exchange at a cer tain layer thickness. It was also interesting to check the necessity of taking into account the biquadratic con
tribution to the coupling, which is known to take place in Fe/Cr/Fe structures. We are also going to determine the temperature and field regions of various magnetic phases in the structure under study. To solve these problems, we apply ferromagnetic resonance (FMR) along with magnetometric meth ods. This method is widely used to study the interlayer exchange in layered structures of various composi tions, including the Fe/Cr system. Nevertheless, it has been little used to analyze 3d/4f and 3d/X/4f struc tures. As for the application of this method to gadolin iumbased structures, let us note the following. As was shown in the middle of the last century [22], the FMR line of a gadolinium film broadens significantly when this film undergoes a transition to an FM phase, which is likely to be related to the short time of magnetic relaxation of the FM moment of gadolinium [23]. Therefore, FMR is inconvenient to study the Gd films. However, 3d/X/Gd systems can have a reso nance mode with predominant precession of the mag netization of a 3d layer and a low Gd precession ampli tude. A relatively narrow resonance line can be expected for this oscillation mode, and the position of this line can be used to estimate the interlayer cou pling. Such investigations were performed in, e.g., [24], where FMR was used to investigate Co/Gd structures with thin (1–2 monolayers) Gd layers. As the temper ature decreased from room temperature, the absorp tion line broadened weakly and the resonance field decreased slightly. To explain these effects, the authors of [24] simulated this system in terms of the mean field approximation. As a result, they were able to describe the experimental results qualitatively and to estimate the AFM exchange at the Co–Gd interface. 2. EXPERIMENTAL We studied a set of [Fe(31.5 Å)/Cr(tCr)/Gd(45 Å)/ Cr(tCr)]12 superlattices with various chromium layer thicknesses tCr = 0–30 Å. The samples were grown by highvacuum magnetron sputtering on silicon and glass substrates and had a polycrystalline structure. As was shown in [19], the Gd layers of these structures had an (0001) hcp texture in the direction normal to the layer plane. Structural investigations were carried out by Xray reflectivity, which has a high sensitivity to the period icity of superlattices. Experiments were carried out on a laboratory rotating anode XRD diffractometer Rigaku SmartLab at the National Research Center “Kurchatov Institute” using monochromatized char acteristic Cu Kα1 radiation. The experimental scheme included a 4xGe(220) Bartels monochromator. To separate the coherent component of scattered radia tion, a specular beam was recorded by a detector with a narrow receiving slit. A sample was scanned in the
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θ–2θ geometry in the stepbystep mode near total external reflection critical angle in the wide angular range. The error of angular displacements did not exceed 0.1% over the entire scanning range. The sta tistical error of measuring the diffracted intensity was at most 15% in tails and up to 5% in the angular range up to 3°. The magnetic properties of the system were ana lyzed by SQUID magnetometry and FMR in the tem perature range 4–300 K. As a SQUID magnetometer, we used a standard MPMS (Quantum Design) device. The homemade FMR spectrometer detected the power of the microwave signal passed through a reso nator containing a sample. A resonance signal was detected at the fixed frequency of the excited resona tor when a field was changed in the range 0–10 kOe. The available frequency range was 7–37 GHz. In all experiments, a magnetic field was applied in the film plane. 3. EXPERIMENTAL RESULTS Static magnetization and FMR spectra were stud ied on samples prepared on both silicon and glass sub strates. The results obtained for both types of sub strates coincide with each other well. 3.1 XRay Diagnostics Figure 1 shows the reflectivity spectrum and the results of its processing for the [Fe(31.5 Å)/Cr(18 Å)/ Gd(45 Å)/Cr(18 Å)]12 sample. When processing reflectometry curves, we used the approach in which the layers in a superlattice are represented as a set of sublayers 3–5 Å thick and their thicknesses and optical constants are varied. Moreover, we assumed localiza tion of atoms of a certain kind across the structure depth. With this approach, we were able to achieve good agreement between the calculated and experi mental Xray reflectivity curves (Fig. 1a). The inset shows the segment demonstrating a high fitting qual ity. The calculated curves qualitatively describe all spe cific features of the experimental curves. The obtained profile of distribution of the optical constants is depicted in Fig. 1b (only the distribution profile of the real part of the complex refractive index, i.e., dispersion, is shown). The profile obtained for one period is separately illustrated in the inset to Fig. 1b (vertical lines bound the region of one period). We can only approximately divide the regions in the superlat tice period, since characteristic copper radiation can not be used to reliably separate iron and chromium rich regions because of the fact that the optical con stants of iron and chromium for this radiation energy are close. Nevertheless, these results agree well with the magnetic properties of the structures. Based on the Xray reflectivity data, we can con clude that the samples under study have a wellformed layered structure. The layer boundaries are not sharp:
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they are characterized by transition layers 2–3 atomic monolayers thick. The sample with a Cr layer thick ness of 4 Å has no continuous Cr layer because of interdiffusion Fe–Cr and Gd–Cr at the thicknesses that exceed the nominal Cr layer thickness. The sam ples with a Cr layer thickness larger than 4 Å have a continuous Cr layer. 3.2. Static Magnetization Measurements Figure 2 shows the temperature dependence of the magnetic moment per unit area, m(T), measured in a field of 500 Oe for a number of structures with various Cr layer thicknesses. For all samples, the magnetiza tion reaches a constant value above 250 K, which is obviously related to the transition of the main part of the Gd layer to a paramagnetic phase. In this case, the remanent magnetization is mainly caused by the mag netic moment of Fe layers. Below this temperature (250 K), gadolinium transforms into an FM phase and the m(T) curves for samples with different values of tCr behave differently depending on the type of interlayer ordering. Note that, according to [19], the interface regions of Gd layers retain certain magnetization at above 250 K, which is caused by exchange with neigh boring Fe layers. Nevertheless, it is obvious that the main part of the Gd layer undergoes FM transition at T ≈ 250 K. As follows from the temperature dependence of magnetization, all samples can be divided into the fol lowing two groups: samples with tCr < 9 Å and samples with tCr > 9 Å. The magnetization of the samples from the first group first increases as the temperature decreases below 300 K and then decreases at a certain temperature. Thus, there is a maximum in the temper ature dependence of magnetization. As the Cr layer thickness increases, the temperature corresponding to the maximum decreases and the magnetization of the superlattice increases at any fixed temperature. There fore, we can assume that the interlayer AFM coupling in the system decreases with increasing tCr. The magnetization of the samples from the second group (tCr > 9 Å) increases monotonically as the tem perature decreases. The shape of the m(T) curve and the magnetization cease to change as the layer thick ness increases. This behavior indicates weak (or absent) interlayer exchange and, hence, FM ordering in an applied magnetic field. The Fe/Gd sample without a chromium spacer has a compensation point at T ≈ 100 K. The lowtemperature magnetization curves m(H) support the assumption of a rapid decrease in the AFM exchange as the Cr layer thickness increases. Figure 3 shows the m(H) curves for a set of samples with tCr in the range 0–18 Å that were measured at T = 20 K. The m(H) curves for samples with tCr < 7 Å have a small remanent moment, which indicates AFM cou pling between Fe and Gd layers. As tCr increases from
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Intensity, c.p.s.
106 103
(a)
102 104
101 1.2
1.4 1.6 θ, deg
2
10
1.8
100 σ 5 0 −5
0
0 Ren × 105 2.5
1
2
3
4 θ, deg
1
2
3
4 θ, deg
(b) 2.0 Ren × 105 2.5
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1.0
2.0
0.5
0
1.5 62
20
40
66
60
70
74 78 Depth, nm
80
100
120 140 Depth, nm
Fig. 1. (a) (points) Experimental and (solid curve) calculated reflectivity curves for a multilayer [Fe(31.5 Å)/Cr(18 Å)/Gd(45 Å)/ Cr(18 Å)]12 sample. σ is the deviation of the calculated curve from the experimental curve. (b) Distribution profile of the real part of the refractive index in the structure.
0 to 7 Å, the susceptibility of the samples in a zero field increases rapidly, which indicates a decrease in the AFM exchange. The m(H) curves of samples with tCr ≈ 9 Å demonstrate a relatively high remanent moment. As tCr increases above 9 Å, the magnetization value and the shape of the m(H) curve change weakly, which is likely to point to a weak interlayer coupling as compared to the coercive force at large chromium layer thicknesses. Moreover, the m(H) curves demon strate a smooth approach to saturation, which is likely
to be related to peculiarities of film magnetization processes (in particular, to polycrystallinity effects). This circumstance makes it difficult to determine the exchange coupling from the saturation field directly. The hysteresis loops of the samples with tCr > 10 Å almost coincide. Their shape can be qualitatively explained by independent magnetization reversal of Fe and Gd layers, which have different coercive forces; therefore, we cannot speak about the manifestation of interlayer coupling (inset to Fig. 3).
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Using these experimental data, we can estimate a number of magnetic parameters of the grown Fe and Gd layers. As noted above, the Curie temperature of the gadolinium layers estimated from the m(T) curves C is T Gd ≈ 250 K, which is significantly lower than TC of bulk gadolinium (290 K). Such a decrease in the Curie temperature is almost always detected for the thin Gd layers in the Fe/Gd system [11, 25–27]. The magnetic moment at room temperature can be used to estimate s the saturation magnetization of the iron layers ( M Fe ≈ 1500 G), and the magnetic moment of saturation at low temperatures can be used to estimate the satura s tion magnetization of the gadolinium layers ( M Gd ≈ 900 G). These values are well below their bulk analogs, 1700 and 2060 G, respectively. A decrease in the mag netization of FM layers was often observed in the Fe/Gd system [12, 25]. The reduction of layer magne tization is usually at most 20–30%. A similar strong decrease in the magnetization was detected in [28] for s the Fe/Gd/Fe system: it was found that M Fe ≈ 1545 G
m, 10−3 emu/cm2 H = 500 Oe 55 Å 18
8 10.5 6
s MFe ≈ 1500 G
6.5
4 5.2 4.0
2
tCr = 0 0
100
200
s
and M Gd ≈ 1030 G. A more than twofold decrease in the magnetization of thin Gd layers as compared to the tab ulated value was also detected for the Gd/Cr multilayer system [29]. The suppression of the magnetization of Gd in the Gd/Cr/FeCo system in [20] was related to the presence of a paramagnetic crystalline fcc phase, which is nonequilibrium in bulk gadolinium. An analysis of the temperature and field depen dences of static magnetization obtained in this work for a series of Fe/Cr/Gd samples with various values of tCr allows us to speak only about a rapid decrease in the interlayer AFM exchange coupling when the chro mium layer thickness increases. No signs of oscillation of this coupling with the Cr layer thickness were detected. This is likely to be caused by weak exchange effects for the samples with tCr > 9 Å as compared to the parasitic changes in the shape of hysteretic magne tization curves that are caused by crystallite anisot ropy, a domain structure, the heterogeneity of inter faces, and so on. In this case, FMR can be more sensitive, since a change in the interlayer exchange leads to a shift in resonance peaks irrespective of the applied magnetic field. Therefore, such an exchange shift can be observed in rather high fields outside a hysteresis loop, where Fe and Gd layers are uniformly magnetized. 3.3. Ferromagnetic Resonance The investigation of FMR at room temperature demonstrates that the spectra of all samples have one absorption line. Its field position is well described by the Kittel formula for an FM film, f = γ H ( H + KM ),
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Fig. 2. Magnetic moment per unit area as a function of temperature for several samples with various Cr layer thicknesses: (points) experiment and (curves) simulation.
where γ is the gyromagnetic ratio (see Fig. 4). Here, KM corresponds to the effective demagnetization field s s 4π M Fe with M Fe ≈ 1500 G, which agrees well with the static magnetization measured at room temperature. As the temperatures decreases, the spectra of the samples with tCr > 8 Å change weakly: only small shifts (at most 100 Oe) of resonance lines in the field are observed (see Figs. 4, 6). This finding supports our assumption concerning a weak interlayer coupling in these samples. In contrast, the spectra of the samples with tCr < 8 Å change strongly as the temperature decreases: two absorption branches are observed at low temperatures (see Figs. 4, 5). As the Cr layer thickness increases, the lowfield branch shifts toward high fields and the high field branch shifts toward low fields. The highfield resonance branch disappears at tCr > 8 Å. Figure 5a illustrates the characteristic behavior of the FMR spectrum of the sample with tCr = 4.0 Å at a frequency of 26 GHz when the temperature changes. One resonance line is observed at room temperature. Upon cooling below a certain temperature, this line begins to broaden and shift toward low fields. Upon further cooling in higher fields, a second broad absorption line appears in higher fields. Figure 5b shows the resulting temperature dependence of the resonance field for both lines. Note a small maximum in the Hres(T) dependence for the lowfield absorption line below the Curie temperature TC of gadolinium (also see Fig. 6).
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10 10.5 8
6.5 5.2
4.0
6 0.01 4 tCr = 0
s ≈ 1500 G MFe
10.5 18
0
2 −0.01 −1 0
5
10
15
0 20
1 25 H, kOe
Fig. 3. Magnetization curves m(H) for several samples with various Cr layer thicknesses obtained at T = 20 K: (points connected by solid lines) experiment and (dashed lines) simulation in the effective field approximation. (×) The magnetization curve mea sured on a sample with tCr = 18 Å at room temperature is additionally showed. (inset) Lowtemperature hysteresis loops for sam ples with chromium layer thicknesses of 10.5 and 18 Å.
Qualitatively similar behavior of the resonance line was observed at all frequencies in the range 7–37 GHz for the samples with thin chromium layers. As the layer thickness increases, the maximum in the Hres(T) curve shifts toward low temperatures (Fig. 6a). At a thick ness of 10–15 Å, this maximum disappears and the resonance field decreases monotonically upon cooling (Fig. 6b). However, the maximum again appears at a thickness of 18–20 Å (Fig. 6c) and then disappears when tCr increases further. The smallness of the tem perature shift of the FMR lines of the samples with tCr > 8 Å indicates a weak interlayer coupling in this layer thickness range. Nevertheless, the curves shown in Fig. 6 suggest a periodic change in the magnetic properties of the system as a function of the layer thickness. To interpret these experimental data, we use an effective field model. 4. MODEL CALCULATION For theoretical analysis of the magnetic properties of systems similar to Fe/Gd and Co/Gd, many researchers use the meanfield method [10, 24–26]. As noted above, the exchange interaction at the inter face between 3d/Gd layers is high and comparable with the exchange inside the Gd layers. This property of the system leads to a substantially nonuniform mag netization distribution across the Gd layer thickness at
temperatures comparable with TC of gadolinium and to the appearance of a twisted state inside the FM (both Gd and 3d metal) layers when a magnetic field is applied [12, 27]. To calculate this nonuniform magnetization distri bution, researchers simulate FM layers by dividing them into individual atomic planes. In this case, the average magnetization of atoms inside each plane is determined by the effective exchange fields from the nearest neighbors inside a plane and in the neighbor ing planes. This simulation can describe the magnetic properties of the system on a qualitative level. How ever, for a better quantitative description, additional assumptions regarding the structure of Gd layers are to be applied. For example, to calculate the magnetiza tion curves of multilayer Fe/Gd structures, the authors of [26] use the nonuniform magnetization distribution across the Gd layer thickness that was experimentally determined. To describe the magnetic properties of Gd/Fe superlattices, the authors of [27] assumed the presence of interfacial regions of Gd layers with a higher Curie temperature as compared to the central part. As will be shown below and as follows from [21], the introduction of a Cr spacer thicker than 4 Å between 3d metal and Gd layers leads to a substantial (at least by an order of magnitude) decrease in the interlayer exchange interaction. Therefore, we can assume that the magnetization distribution inside the
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can significantly influence the properties of the sam ples because of surface defects and interface imperfec tion. According to [19], it is this situation that takes place in the samples under study. Since the FMR spectra have at most two absorp tion lines, we can restrict ourselves to the consider ation of one structural superlattice element that determines all properties of the system in order to describe the temperature dependences of magneti zation and resonance frequencies. We first consider two exchangecoupled uniformly magnetized iron and gadolinium layers. The layer exchange energy per unit area Eex is written as the sum of bilinear and biquadratic contributions, as is usually done for the Fe/Cr/Fe system,
f, GHz
30
20
tCr = 4.0 Å tCr = 5.2 Å
10
tCr = 10.5 Å
2 M Fe ⋅ M Gd Fe ⋅ M Gd⎞ ⎛M E ex = – J 1 + J , 2 s s ⎝ Ms Ms ⎠ M Fe M Gd Fe Gd
tCr = 18 Å 0
2
4
6
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8 H, kOe
Fig. 4. Resonance frequencies vs. the magnetic field for several samples at (solid symbols) room temperature and (open symbols) T = 30 K. (curves) Theoretical model (solid curve corresponds to the Kittel formula for an FM film).
FM Gd layers is more uniform, which makes it possi ble to avoid calculations with division into separate atomic planes. Nevertheless, the interfacial regions in the Gd layers can have the magnetic properties that differ substantially from those of the inner volume and Absorption, arb. units
(1)
where MFe and MGd are the magnetization vectors of s
s
the Fe and Gd vectors, respectively; M Fe and M Gd are the corresponding saturation magnetizations; and J1 and J2 are the bilinear and biquadratic exchange con stants, respectively. Note that the biquadratic contri bution to the layer interaction energy was also taken into account in [21] for Co/Cr/Gd structures. To describe the temperature dependences of mag netization and FMR, we use an effective field method. It is known that, in the temperature range 4–300 K, the exchange constants of the Fe/Cr/Fe system change relatively weakly and, in contrast, MGd changes Hres, kOe
(a)
(b) 37 K
6
44
75
4
115 134
TC
2
220 295 0
2
4
6
8
10 H, kOe
0
100
200
300 T, K
Fig. 5. (a) Series of resonance signals for a sample with tCr = 4.0 Å at a frequency of 26 GHz and various temperatures. (b) Result ing temperature dependence of the resonance field for this sample: (points) experiment and (curves) simulation. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS
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Hres, kOe
(a)
(b)
Hres, kOe (c)
16 Å
27 Å
tCr = 6.5 Å
2.2
2.10 14
2.0
22
5.2 10
2.0 1.9
2.05 4.0 20
1.8
1.8 2.00 0
100
200
300 T, K
0
100
200
300 T, K
0
100
200
300 T, K
Fig. 6. Temperature dependences of the resonance field Hres(T) for a series of samples with various chromium layer thicknesses, 4 Å < tCr < 27 Å, at a frequency of 17 GHz: (points) experiment and (curves) simulation. For clarity, the curves are shifted verti cally with respect to each other.
substantially. Therefore, to a first approximation, we neglect the temperature dependences of exchange constants J1 and J2 and assume that the magnetiza tions of the Fe and Gd layers depend on temperature according to Brillouin functions B Si for the corre sponding values of spin Si, s
eff ⎛ μi Hi ⎞
M i = M i B Si , ⎝ kB T ⎠
(2)
where subscript i means the Fe or Gd layer. As Si, we used the following tabulated values: SGd ≈ 3.5 for gad olinium and effective value SFe ≈ 1 for iron. The mag netic moment per atom is determined from the expres sion μi = gμBSi, where the gyromagnetic ratio is g ≈ 2 for Fe and Gd, μB is the Bohr magneton, and kB is the Boltzmann constant. eff
Effective field H i acting on layer i is determined from the expression eff
ex
Hi = λi Mi + H + Hi .
(3)
Here, λiMi is the internal molecule Weiss field, which determines the Curie temperature of the magnetic layer (in the absence of an external field and interac tion with the neighboring layer). Meanfield constant C λi is related to Curie temperature T i by the formula C
3k B T i S i λ i = . s μi Mi Si + 1
(4)
eff
Expression (3) for H i also contains magnetic field H ex
and H i , which is the effective field of interaction with the neighboring layer. This field is determined through the derivative of energy (1), ex 1 ∂E H i = – ex, t i ∂M i
(5)
where ti is the layer i thickness. The eigenfrequencies of the magnetic oscillations of the system (i.e., FMR frequencies) are determined from linearized Landau–Lifshitz equations without a dissipative term, ∂M i eff z = γ i [ ( H i – K i m i ) × M i ]. ∂t
(6)
Because of a large value of the demagnetizing factor, the magnetic moments of the layers in equilibrium always lie in the film plane; however, nonzero compo z nent m i of a magnetic moment, which is normal to the film plane, appears in the case of its precession. There z fore, the addition Ki m i to the effective field appears; here, Ki is the easyplane anisotropy coefficient (including the demagnetizing factor). With Eqs. (1)–(6), we can calculate both the static magnetization curves and the field dependences of the FMR frequencies of the model structure under study at any temperatures. The disadvantage of the model is C a large number of fitting parameters. However, T Gd , s
s
M Fe , and M Gd are known from an analysis of experi
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mental magnetization curves. We can also neglect a possible deviation of the Curie temperature of iron C ( T Fe = 1043 K) from its value for the bulk substance, since the temperature range under study is well below this temperature. Thus, most parameters in the set of Eqs. (1)–(6) turn out to be well determined. The exception is the following two most important param eters: J1 and J2, which should determine all substantial features of the magnetization curves and the FMR spectra. The discontinuous curves in Fig. 2 illustrate the calculated dependences m(T) for various values of exchange parameters J1 and J2. The dashed line describes the m(T) dependence in the absence of interlayer coupling. Its shape is seen to differ substan tially from the experimental m(T) curves for the sam ples with tCr > 8 Å (with weak exchange). The m(T) curve of the sample with tCr = 4 Å and strong AFM exchange is also poorly described (dotanddash curve in Fig. 2). We also cannot describe the complex non monotonic m(T) dependence for the samples with intermediate layer thicknesses. For example, the dot ted curve in Fig. 2 is the attempt of the best approxi mation of the experimental data obtained for the sam ple with tCr = 6.5 Å. The values of constants J1 and J2 for the samples with tCr < 8 Å are about 1 erg/cm2. When varying these values, we were not able to achieve better agreement with the experimental data. The observed deviation of the calculated curves from the experimental m(T) dependences can be ascribed to a nonuniform magnetization distribution across the Gd layer thickness. To take into account this effect, (by analogy with [11, 27]) we introduce addi tional rough division of the Gd layer into internal and in interfacial (external) regions with thicknesses t Gd and ex
in
ex
t Gd (here, the relation t Gd + t Gd = tGd takes place), C
which have different Curie temperatures ( T Gd, in , C
T Gd, ex ). In this case, the expression for the interlayer coupling energy (Eq. (1)) can be rewritten as ex ⎛ M Fe ⋅ M ex ⎞ M Fe ⋅ M Gd Gd E ex = – J 1 + J ⎜ ⎟ 2 s s ⎝ M sFe M sGd ⎠ M Fe M Gd
–
2
(7)
in M Fe
ex ⋅ M Gd A , 2 s ( M Gd )
where additional parameter A, which characterizes the energy of interaction between the internal (in) and external (ex) regions in the gadolinium layer, appears. In a rough approximation, this parameter takes into account the energy of the nonuniform magnetization across the Gd film thickness. On the order of magni tude, parameter A is connected to the exchange stiff
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ness of gadolinium (AGd ≈ 3 × 10–7 erg/cm) by the rela tion A ~ AGd/tGd ~ 1 erg/cm2. Taking the aforesaid into account, we calculated the temperature and field dependences of magnetiza tion and resonance frequencies and compared them with the experimental data. 5. COMPARISON WITH EXPERIMENT. PHASE DIAGRAM The solid curves in Fig. 2 show the calculated dependences m(T) with allowance for the nonuniform magnetic moment distribution across the Gd layer in ex thickness at t Gd = 2 t Gd = 22.5 Å, A = 5.7 erg/cm2, C
C
T Gd, in = 250 K, and T Gd, ex = 100 K. The calculated curves corresponding to structures with different chro mium layer thicknesses differ only in the values of exchange constants J1 and J2 (for more detail, see Sec tion 6). As is seen in Fig. 2, the assumption of the existence of interfacial regions in Gd makes it possible to describe the temperature dependences of the magneti zations of the samples (solid curves) better than the model of a uniformly magnetized Gd layer (discontin uous curves) does. In particular, the experimental nonmonotonic m(T) dependences for the samples with thin Cr layers are described at least qualitatively. Note that, in contrast to the Fe/Gd system (where the interfacial Gd region was assumed to have a higher Curie temperature) [11, 27], we assume that the inter facial gadolinium region in the Fe/Cr/Gd system has a lower Curie temperature in order to describe the experimental data better. This difference is likely to be associated with a substantial weakening of the inter layer coupling in our system as compared to the Fe/Gd system. As will be shown below (Section 6), the obtained values of the exchange constants do not exceed 1 erg/cm2, which is at least an order of magni tude lower than the exchange in the Fe/Gd system. In Fig. 3, we compare the calculated and experi mental m(H) magnetization curves. To approximate the FMR data, we need to take into account the additional parameters of Eq. (6), namely, gyromagnetic ratio γi and easyaxis anisotropy coefficient Ki of the Fe and Gd layers. As the gyromag netic ratio for both types of layers, we use its value for bulk Fe and Gd, which corresponds to g factor g = 2. As noted above, the easyaxis anisotropy coefficient of iron (KFe) corresponds to demagnetizing factor 4π within a good accuracy. The approximation of the field dependences of the resonance frequencies at low tem peratures gives a significantly higher value KGd ∼ 19 for Gd layers, which can be related to the hcp texture of the gadolinium layers [19] and the related additional anisotropy (see Fig. 4). Figure 4 also shows the experimental and calcu lated curves for the field dependences of the resonance
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frequencies f(H) for several samples at room tempera ture and at T = 30 K. The calculated m(H) and f(H) dependences agree qualitatively with the experimental data (Figs. 3, 4; dashed and dotted curves). Note that taking into account the biquadratic contribution to the interlayer interaction energy leads to the experimentally observed nonlinear approach of the m(H) curves to saturation and the initial increasing segment in the f(H) dependence in the highfrequency FMR branch. Similar behavior of magnetization and FMR spectra was observed earlier for the Fe/Cr/Fe system and was also described in terms of the biquadratic model [4]. The lowfrequency (highfield) FMR branch corre sponds to the inphase precession of the Fe and Gd layers, and the highfrequency (lowfield) branch cor responds to outofphase precession (internal and interfacial regions in the gadolinium layer precess almost codirectionally). The calculated temperature dependence of the res onance field Hres(T) agrees qualitatively with the experimental data for the sample with tCr = 4.0 Å (Fig. 5b, solid curve). The nonideal quantitative agreement between the calculated and experimental curves can be attributed to the roughness of the theoretical model (Figs. 2–4, 5b). A comparison of the calculated m(T), m(H), f(H), and Hres(T) dependences with the corresponding experimental data obtained for the entire series of samples allowed us to draw the following conclusions regarding the types of magnetic states in the system at various values of H and T. For example, the decrease of the m(T) value for the sample with tCr = 4.0 Å upon cooling below 250 K is associated with the antiparallel ordering of Fe and Gd layers (see Fig. 2); in this case, the magnetic moment of iron is directed along a magnetic field. In the range T ⱗ 50 K (where the magnetization of the sample begins to grow), a canted magnetic phase takes place. For the samples with tCr = 5.2 and 6.5 Å, the phase with FM ordering of the Fe and Gd layers first occurs below 250 K; as a result, the magnetization of the sam ples increases. As the temperature decreases, the canted state appears and the magnetization begins to decrease. At the lowest temperatures, the magnetiza tion again increases, which is caused by an increase in the relative contribution of the biquadratic term to the interlayer coupling energy as compared to the bilinear contribution due to a higher degree in the magnetiza tion of Gd. The weak shift of the absorption line in the temper ature dependence of the resonance field Hres(T) C
toward high fields at T < T Gd ≈ 250 K is related to the formation of an FM phase at these magnetic fields (Fig. 5b). The sharp decrease in the resonance field below a certain temperature is caused by the transition into the canted phase. For comparison, the dotted line
in Fig 5b shows the results of calculation disregarding the interfacial regions in Gd layers with a low Curie temperature. It is seen that the resonance field C decreases sharply below T Gd and the experimental points are poorly described. To generalize all obtained data, we calculated an H–T diagram for the magnetic states of the sample with tCr = 4.0 Å, which indicates the existence of an FM phase, an AFM phase, and a canted state and determines boundaries between them (Fig. 7a, solid lines). The points correspond to the experimental Hres(T) data obtained at frequencies of 17, 20, and 26 GHz for this sample. This diagram again illustrates the fact that the rapid shift of the FMR line toward low fields as the temper ature decreases begins at the boundary of transition into the canted state. The weak increase of Hres below C
T Gd, in corresponds to the FM phase, where the mag netizations of the Fe and Gd layers are directed along the field. Figure 7b shows the magnetization profiles across the sample thickness at tCr = 4.0 Å calculated for sev eral characteristic points in the phase diagram in Fig. 7a. Note that a weak magnetization is retained in C the Gd layer at temperatures above T Gd, in = 250 K (profiles c and e). Here, the magnetic moment of the interfacial region in the Gd layer is opposite to the magnetization of the Fe layer because of interlayer AFM coupling. As the temperature decreases below about 250 K, the magnetization of the internal region in the Gd layer increases substantially (profiles b, d). The sign of the magnetization of the interfacial layer ex ( M Gd ) changes between points b and c in the phase diagram. The dashed line in the phase diagram between the corresponding regions illustrates the van ex ishing of M Gd . The dotted line between points d and e in the AFM region in the phase diagram corresponds to the equalization of the magnetizations of the inter nal and interfacial regions in the Gd layer. Note that the calculated magnetization profile at point e corre sponds qualitatively to that obtained in [19]. 6. EXCHANGE CONSTANTS VERSUS THE CHROMIUM LAYER THICKNESS The comprehensive experimental data obtained when Hres(T) (FMR) was investigated at a frequency of 17 GHz for all samples under study over a wide tem perature range and a comparison of them with the cal culated curves allowed us to make certain conclusions regarding the behavior of the exchange constants as functions of the chromium layer thickness (see Fig. 6). When the layer thickness increases in the range 4–8 Å, the interlayer AFM coupling decreases. At high tem peratures, the FM state becomes favorable (weak
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M, G H, kOe
1000
(a) 4
FM
Canted state
(b) a(M||)
26 GHz
0 Fe
−500 3
a
b
a(M⊥)
500
c
Gd Cr
1500
b
c
1000 20 2
×10
500 0
17 1 d
AFM 0
100
200
1500 1000 500 e 0 −500 300 T, K
d
e ×20
−25
0
25
50
−25
0
25
50 t, Å
Fig. 7. (a) Calculated phase diagram and (b) magnetization profiles across the Fe and Gd layer thicknesses for several character istic points in the diagram (letters a–e) for a sample with tCr = 4.0 Å. The profiles of the magnetization components along and across a field are shown for the canted state. Zero on the thickness scale corresponds to the position of the Cr layer. Points in the phase diagram illustrate the experimental Hres(T) dependences at frequencies of 17, 20, and 26 GHz.
increase of Hres when T decreases) and the phase boundary of the canted state shifts toward low temper atures (the sharp drop of Hres when T decreases). In the layer thickness range 10–15 Å, the sign of the exchange coupling changes into a ferromagnetic one and Hres decreases monotonically as T decreases. For the samples with tCr in the range 20–25 Å, Hres increases again upon cooling, which can be related to transition into interlayer AFM coupling. On the whole, the considered model well describes the experimental Hres(T) dependences on a qualitative level; however, the bad agreement between the experi mental and calculated curves at low temperatures (below 30 K) is noteworthy. Obviously, the approach used here cannot be applied in this temperature range. One of the causes can be the essential change of the coupling constants at low temperatures; therefore, our analysis is only applicable in a sufficiently hightem perature range. The change in the sign of exchange interaction caused by a change in the chromium layer thickness in the system under study that was described above is more clearly illustrated in Fig. 8, which shows the temperature dependences of the shift in the FMR line relative to its position at 250 K for several sam ples. It is seen that the direction of shifting the absorption peak below 250 K depends on the Cr layer thickness. The fact that this shift is related to
the sign of interlayer coupling follows from the fol lowing considerations. At high temperatures, the detected FMR line cor responds to the precession of the magnetization vector of the Fe layer in the effective field, which includes the external field and the interlayercouplinginduced field. The magnetic moments of Fe and Gd are ordered along a magnetic field at a sufficiently high field and low interlayer coupling. In this case, the effective field at Fe decreases or increases with decreasing temperature, depending on the interlayer coupling sign. To meet the resonance conditions (at a fixed frequency), an external field should increase in the case of an AFM sign of interlayer coupling and should decrease at an FM sign. Thus, it follows from Fig. 8 that the samples with tCr = 5.2, 6.5, and 20 Å have an AFM interlayer coupling and the samples with tCr = 10 and 14 Å have an FM interlayer coupling. The inset to Fig. 8 shows the dependence of the interlayer couplinginduced contribution to the effective field at Fe on the Cr layer thickness. The effective field is obtained by the extrapolation of the linear segments in the temperature dependence of the resonance field at T = 0 K. Our calculation makes it possible to estimate exchange constants J1 and J2 and to construct their dependence on the Cr layer thickness (Fig. 9). The interaction constants are seen to decrease rapidly with increasing layer thickness in the range 0–8 Å (by
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Hres(T) − Hres(250 K), Oe 50 5.2 Å 6.5 Å 20 Å
J1, J2, erg/cm2 0.5 J2 0 J1, erg/cm2
16 Å 0 27 Å 14 Å
10 Å
−50 0
−0.5
ΔH, Oe 200 100 0
100
−100 0 200 T, K
J1
0.02 0.01
−1.0
0 −0.01
10
20
30 tCr, Å
300
Fig. 8. Temperature dependences of the resonance field shift in the initial segment when the temperature decreases from 300 K. (inset) Effective field in the Fe layer vs. the Cr layer thickness plotted from the slope of the linear segment in the temperature dependence of the resonance field.
approximately two orders of magnitude). When pro cessing the results obtained for the samples with tCr > 8 Å, we take into account only exchange constant J1 and neglect constant J2. In the onset to Fig. 9, on an expanded scale we can see weak J1 oscillations with layer thickness tCr in the range 8–30 Å. The detected interlayer coupling oscillation period in the Fe/Cr/Gd system as a function of the Cr layer thickness is about 18 Å. This value agrees with the long oscillation period in the Fe/Cr/Fe system, which is 16–18 Å [8]. Therefore, we can relate the detected oscillations to the RKKY exchange mechanism. In this work, we failed to detect shortperiod inter layer coupling oscillations, which are detected for the most perfect Fe/Cr/Fe samples and associated with an AFM structure of the Cr layer. This fact can be caused by the imperfection of the layer interfaces in the struc tures under study. The dotted line in Fig. 9 illustrates an RKKY curve that is proportional to sin(qt + ϕ)/t2 with a period of about 18 Å. This dependence qualitatively describes the experimental points at tCr > 8 Å; however, the detected AFM exchange at small layer thicknesses (tCr < 8 Å) is much higher than that predicted by the RKKY formula. One of the possible causes of the significant increase in the exchange at small Cr thicknesses can be the direct exchange induced by direct contact between the Fe and Gd layers due to the roughness of the Fe– Cr and Gd–Cr interfaces. To describe this effect, in
0
10
20
30 tCr, Å
Fig. 9. Exchange constants (solid symbols) J1 and (open symbols) J2 vs. the Cr layer thickness. The points were obtained by processing magnetization curves and FMR spectra. The lines correspond to the following model func tions: (dotted curve) J ∝ sin(qt + ϕ)/t2, (dashed curve) J ∝ exp(⎯t2/σ2), and (solid curve) J ∝ Aexp(–t2/σ2) + Bsin(qt + ϕ)/t2. Different types of points correspond to samples prepared on different substrates.
Fig. 9 we use a Gaussian distribution, which is propor tional to exp(–t2/σ2) and qualitatively describes the probability of contact of two rough boundaries. Roughness σ entering into this formula is estimated at about 4 Å. The exchange oscillation amplitude in the range 10 Å < tCr < 30 Å is lower than that detected in the Fe/Cr/Fe system by an order of magnitude, which can be caused by peculiarities of conduction elec tron scattering at the Gd–Cr interface. According to the Bruno calculations [30], the interlayer RKKY exchange oscillation amplitude in a layered structure is determined by both the energy spectrum of con duction electrons in the Cr layer and the difference ↑↓ between reflection coefficients Δ i of electrons with different spin directions from each layer boundary (in our case, Fe–Cr and Gd–Cr), sin ( qt + ϕ, ) J ∝ Δr Fe–Cr Δr Gd–Cr 2 t ↑↓
↑↓
↑↓ Δr i
↑
(8)
↓
ri – ri = . 2
↑↓
↑↓
The smallness of Δ r Gd–Cr as compared to Δ r Fe–Cr is assumed to result in substantial suppression of the exchange oscillations in the Fe/Cr/Gd system as com pared to Fe/Cr/Fe. Such an effect is likely to hinder the detection of exchange oscillations in other Co/X/Gd systems [14, 15].
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The experimental data at a small layer thickness (up to about 8 Å) are better described with the use of additional phenomenological biquadratic exchange constant J2. Although the physical nature of this con stant is unclear, the fact that the obtained results can not be described using only exchange constant J1 is likely to be related to substantial interlayer coupling fluctuations because of the roughness of the layer boundaries. At small layer thicknesses, these fluctua tions are most pronounced and the obtained values of J2 are maximal. As tCr increases, J2 decreases similarly to constant J1. 7. CONCLUSIONS The main results of this work are as follows. A series of Fe/Cr/Gd/Cr multilayer structures with various Cr layer thicknesses (4–30 Å) was studied by SQUID magnetometry and FMR over a wide temper ature range (4.2–300 K). The obtained experimental results can be qualitatively described in terms of an effective field model with allowance for the contribu tion of biquadratic exchange interaction. Using this model, we were able to identify the types of the magnetic states that appear in the system at var ious H and T and to construct a phase diagram for the system in these coordinates for a given chromium layer thickness. When comparing the experimental dependences and the curves calculated in terms of this model, we estimated the values of exchange constants J1 and J2 and plotted them as a function of the Cr layer thick ness. This function exhibits weak J1 oscillations at a period of about 18 Å when tCr changes in the range 8– 30 Å. This period agrees with the long oscillation period detected in the Fe/Cr/Fe system [1–3, 7, 8]. These oscillations are likely to be related to RKKY exchange via conduction electrons in the Cr layer. The exchange oscillation amplitude in the range 10–30 Å is an order of magnitude lower than that detected in the Fe/Cr/Fe system, which can be caused by peculiarities of conduction electron scattering at the Gd–Cr interface [30]. In conclusion, note that FMR is an effective method to study the magnetic states in complex mag netic superlattices. ACKNOWLEDGMENTS We thank A.A. Mukhin, V.Yu. Ivanov, and A.M. Kuz’menko for their assistance in performing measurements on a SQUID magnetometer. We are grateful to D.I. Kholin for many useful discussions of this work. We also thank S.N. Yakunin for providing us with the software modules to analyze the experimental data
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of Xray reflectivity with allowance for the error of the resulting profile. This work was supported by grants of the President of the Russian Federation (project no. NSh 1540.2014.2, NSh5517.2014.2) and by the Russian Foundation for Basic Research (project nos. 1422 01063, 100201110a). A.B. Drovosekov and A.O. Savitsky are grateful to the Russian Science Sup port Foundation for the financial support of their work. REFERENCES 1. S. S. P. Parkin, N. More, and K. P. Roche, Phys. Rev. Lett. 64, 2304 (1990). 2. J. Unguris, R. J. Celotta, and D. T. Pierce, Phys. Rev. Lett. 67, 140 (1991). 3. V. V. Ustinov, N. G. Bebenin, L. N. Romashev, V. I. Minin, M. A. Milyaev, A. R. Del, and A. V. Semer ikov, Phys. Rev. B: Condens. Matter 52, 15958 (1996). 4. A. B. Drovosekov, N. M. Kreines, D. I. Kholin, V. F. Meshcheryakov, M. A. Milyaev, L. N. Romashev, and V. V. Ustinov, JETP Lett. 67 (9), 727 (1998). 5. C. M. Schmidt, D. E. Bürgler, D. M. Schaller, F. Meis inger, and H.J. Güntherodt, Phys. Rev. B: Condens. Matter 60, 4158 (1999). 6. D. T. Pierce, J. Unguris, R. J. Celotta, and M. D. Stiles, J. Magn. Magn. Mater. 200, 290 (1999). 7. S. O. Demokritov, A. B. Drovosekov, N. M. Kreines, H. Nembach, M. Rickart, and D. I. Kholin, J. Exp. Theor. Phys. 95 (6), 1062 (2002). 8. Ultrathin Magnetic Structures III, Ed. by J. A. C. Bland and B. Heinrich (SpringerVerlag, Berlin, 2005). 9. V. N. Men’shov and V. V. Tugushev, J. Exp. Theor. Phys. 98 (1), 123 (2004). 10. R. E. Camley and R. L. Stamps, J. Phys: Condens. Matter 5, 3727 (1993). 11. D. Haskel, G. Srajer, J. C. Lang, J. Pollmann, C. S. Nelson, J. S. Jiang, and S. D. Bader, Phys. Rev. Lett. 87, 207201 (2001). 12. E. Kravtsov, D. Haskel, S. G. E. te Velthuis, J. S. Jiang, and B. J. Kirby, Phys. Rev. B: Condens. Matter 79, 134438 (2009). 13. E. A. Kravtsov and V. V. Ustinov, Phys. Solid State 52 (11), 2259 (2010). 14. K. Takanashi, H. Fujimori, and H. Kurokawa, J. Magn. Magn. Mater. 126, 242 (1993). 15. K. Takanashi, H. Kurokawa, and H. Fujimori, Appl. Phys. Lett. 63, 1585 (1993). 16. S. S. P. Parkin, Phys. Rev. Lett. 67, 3598 (1991). 17. R. ChaiNgam, N. Sakai, A. Koizumi, H. Kobayashi, and T. Ishii, J. Phys. Soc. Jpn. 74, 1843 (2005). 18. B. Sanyal, C. Antoniak, T. Burkert, B. Krumme, A. Warland, F. Stromberg, C. Praetorius, K. Fauth, H. Wende, and O. Eriksson, Phys. Rev. Lett. 104, 156402 (2010). 19. M. V. Ryabukhina, E. A. Kravtsov, D. V. Blagodatkov, L. I. Naumova, V. V. Proglyado, V. V. Ustinov, and Yu. Khaydukov, J. Surf. Invest. 8 (5), 983 (2014).
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Translated by K. Shakhlevich
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