Negative Refractive Index Ferromagnetic Materials with Negative Permeability at Zero Applied Magnetic Field

Transcription

1 Appl Magn Reson (2009) 36:69 80 DOI /s Applied Magnetic Resonance Negative Refractive Index Ferromagnetic Materials with Negative Permeability at Zero Applied Magnetic Field E. Demirel Æ M. Ozdemir Æ B. Aktas Received: 19 August 2008 / Revised: 3 February 2009 / Published online: 12 August 2009 Ó Springer 2009 Abstract Negative magnetic permeability of ferromagnetic materials is studied in the microwave frequency region. The magnetic permeability is analyzed by employing the dynamic equation of motion of magnetization for different directions of the applied magnetic field with respect to crystalline axes. In the calculations, a very thin ferromagnetic layer with the cubic magneto-crystalline anisotropy energy is considered. It is found that there can be multiple regions of the field value in which the real component of the dynamic permeability becomes negative. In addition, the low-field region for negative permeability can be extended even to the zero external static magnetic field. 1 Introduction Recently, there has been a great interest in negative refractive index materials. Negative refractive index materials were proposed by Veselego four decades ago [1]. Analyzing Maxwell Equations, Veselago theoretically suggested that refractive index can be negative when magnetic permeability and dielectric permittivity are simultaneously negative, since the refractive index is square-root of product of these two parameters. If only the dielectric permittivity or the magnetic permeability is E. Demirel B. Aktas (&) Department of Physics, Gebze Institute of Technology, Cayirova-Gebze, Kocaeli, Turkey aktas@gyte.edu.tr E. Demirel TUBITAK-UEKAE, Gebze, Kocaeli, Turkey M. Ozdemir Department of Physics, Faculty of Sciences and Letters, Marmara University, Goztepe, Istanbul, Turkey

2 70 E. Demirel et al. negative, then the electromagnetic wave can not propagate more than skin depth like in bulk metals. It is a well-known fact that the realization of negative dielectric permittivity is much easier compared to that of negative magnetic permeability, since metals have negative dielectric permittivity up to THz frequencies. The magnitude of the refraction index should be close to unity for easy propagation of electromagnetic waves. However, while the real part of dielectric permittivity is negative, its magnitude is far from unity at much lower frequencies compared to the plasma frequency, which is of the order of THz frequencies for metals. Nevertheless, Pendry et al. succeeded in lowering the plasma frequency to the GHz range by introducing a photonic crystal lattice that is made of thin metallic wires [2, 3]. Then, Pendry et al. showed that negative refractive index materials can be artificially obtained by making a metamaterial consisting of periodically arrayed split-ring resonators and thin wires [4]. Following these studies, microwave transmission experiments through metamaterial structures of one and/or two dimensional thin metallic wires and split-ring resonators were realized, and a remarkable increase in transmission at the theoretically proposed negative refraction frequency band was observed [5, 6]. Then, the first refraction angle measurement in two dimensional metamaterial structures with an overall shape of a prism was done by Shelby et al. and a supporting evidence for negative refraction index was obtained in expected frequency band [7]. The complete review about the negative refraction including original article of Veselago can be found in the article by Veselago et al. [8]. In experimental researches that have been done so far, the split-ring resonator structures were mostly used to obtain negative magnetic permeability for negative refractive index metamaterials. A split-ring resonator structure produces artificial magnetic resonance for electromagnetic waves. That is, magnetic permeability of metamaterial becomes negative at a certain frequency range where minimum and maximum frequency limits of negative band depend on both the inner and the outer radii of split-ring resonators. The main disadvantage of the split ring resonator structures is the absence of an external tuning parameter for magnetic permeability. On the other hand, the magnetic permeability can be made negative near the ferromagnetic resonance condition for certain combination of internal magnetic parameters, external static magnetic field, and electromagnetic field frequency values [9 11]. Obviously, external static field values and dynamic field frequencies can be practically used as tuning parameters. Using interferometric measurement techniques at 150 GHz frequency, Pimenov et al. observed negative refractive index for certain external static magnetic field regions close to the ferromagnetic resonance in metallic ferromagnets [12]. In this study, we have studied the magnetic permeability of ferromagnetic materials having cubic anisotropy for different directions of external static magnetic field. We have calculated the magnetic permeability as a function of external static magnetic field. By sweeping external static magnetic field in different directions with respect to crystalline axes, we observed that more than one resonance condition can be satisfied for some combinations of anisotropy parameters, external field, and microwave frequencies. Thus, we have also showed that for some orientations of external static magnetic field, one can obtain negative magnetic permeability, even

3 Negative Refractive Index Ferromagnetic Materials 71 at zero external static magnetic field. That is, the external static magnetic field value and its direction can be used as tuning parameters for negative magnetic permeability. 2 Theory It should be recalled that the negativity of refractive index requires simultaneous negativity of both dielectric permittivity and magnetic permeability at the frequency of exciting electromagnetic wave. Obtaining negative dielectric permittivity is easy because the dielectric permittivity of metals is negative below the plasma frequencies. However, the number of substances having negative magnetic permeability is limited by superconducting materials showing Dynamic Meissner Effect at low temperatures, which are impractical for applications. However, in case of ferromagnetic materials, l can become negative in the presence of external (or even internal anisotropy field for some particular cases) static magnetic field close to ferromagnetic resonance [12].Therefore, we have focused on investigating the conditions required for negative magnetic permeability in a ferromagnetic thin film having thickness L smaller than its skin depth, d. Suppose, an external static magnetic field, H, is applied in a general direction with respect to the single crystalline axes of the film having a saturation magnetization M. The relative orientations of reference axes and relevant vectors used in the calculations are illustrated in Fig. 1. Here, h H (and h) and u H (and u) are polar and azimuthal angles of external static magnetic field (and magnetization) vectors respectively. The lateral dimensions of the plate (film) in x and y directions are assumed to be much larger than the thickness of the film, L. Therefore, the plate can practically be considered as endless in x and y directions. As well known, the time rate of change of magnetization M, is simply given by the torque equation Fig. 1 The relative orientations of magnetic field, H, and magnetization, M, with respect to the coordinate system used in calculations

4 72 E. Demirel et al. 1 dm~ c dt ¼ ^s þ M~ ~ h M ~ x;y ct 2 Here, c is the gyro-magnetic ratio and s is the total static torque on M due to external static magnetic field and magnetic anisotropy fields. The second term in the parenthesis represents the dynamic torque due to the external microwave field h, which is taken to be parallel to the sample plane and perpendicular to the static magnetic field in all calculations in this study. The relaxation of transverse M is simply assumed to be Bloch type (last term) represented by relaxation time T 2. A value of 50 G is assumed for the resonance line-width, DH, which is related to the relaxation parameter as DH = 2/(cT 2 ) = 50 G. In spherical polar coordinate system, the static torque in Eq. 1 is expressed as [13]. oe T ^s ¼ ^e u oh þ ^e 1 h sinðhþ where E T represents total magnetic energy including Zeeman E z, shape anisotropy E d, magneto-crystalline E c terms which can be written as where E z ¼ M j jjh E T ¼ E z þ E d þ E c oe T ou ð1þ ð2þ ð3þ j½sin h sin h H sinðu u H Þþcos h cos h H Š ð4þ E d ¼ 2pM 2 a 2 3 h i E c ¼ K 1 ða 1 a 2 Þ 2 þða 1 a 3 Þ 2 þða 2 a 3 Þ 2 Here, K 1 denotes cubic crystalline anisotropy constants, and a 1, a 2, a 3 are direction cosines of M, given in terms of polar angles of M as a 1 ¼ sinðhþ cosðuþ ð7þ a 2 ¼ sinðhþ sinðuþ ð8þ a 3 ¼ cosðhþ ð9þ Here, one of the cubic crystal axis, a 3, is chosen perpendicular to the film plane. While the analytic form of the magneto-crystalline anisotropy varies depending on crystalline symmetry, here we assumed cubic anisotropy term for simplicity in our calculations. The static equilibrium direction of M is obtained from the conditions qe T /qh = 0, and the qe T /qu = 0. By the application of a sufficiently small microwave magnetic field in addition to the external static magnetic field as in Fig. 1, the magnetization M precesses about equilibrium orientation (about the effective field) with the following frequency [14]. x 2 0 ¼ 1 o 2 E T c M 0 oh 2 1 M 0 sin 2 ðhþ o 2 E T ou 2 1 o 2 2 E T þ 1 M 0 sin h ohou c 2 T2 2 ð5þ ð6þ ð10þ

5 Negative Refractive Index Ferromagnetic Materials 73 By using the well-known definition of dynamic magnetic susceptibility function v ¼ m x ¼ m u ¼ v h x h 1 iv 2 ð11þ u u¼p=2 one can obtain the real v 1 and imaginary v 2 components of dynamic magnetic susceptibility v as [15, 16]. v 1 ¼ x0 1 o 4pM 2 E T 0 M 0 oh 2 v 2 ¼ x 0 c 2 x c 2 2x 1 4pM 0 c 2 T 2 x 0 c 2 x c 2 2 x c c 2 þ 4x2 o 2 E T M 0 oh 2 c 4 T þ 4x2 c 4 T 2 2 ð12þ ð13þ Having obtained the dynamic susceptibility function, real (l 1 ) and imaginary (l 2 ) components of magnetic permeability are simply obtained from the basic definitions, l 1 ¼ l 0 ð1 þ v 1 Þ ð14þ l 2 ¼ l 0 v 2 ð15þ where l 0 is the permeability of the free space. 3 Results and Discussions The precession frequency for magnetization vector in Eq. 1 and the dynamic permeability components were calculated by using the Eqs for several experimental conditions. Figure 2 shows the external field dependence of precession frequencies of M and permeability components. The chosen values of the parameter sets used in Eqs. 10, 12, 13 are given in the figure. It should be noted that selected values of polar angles for the external static magnetic field correspond to the easy direction of magnetization in the film plane. This figure shows that the precession frequency of magnetization varies almost linearly with respect to the external static magnetic field for our selected values of the parameter. It is revealed from the field dependence of x 0 /c that an applied external microwave field at a certain frequency causes the magnetization vector to resonate once at corresponding single resonance field. Thus, the real and imaginary components of l have resonance curves as shown in Fig. 2. However, the field dependence of these curves dramatically changes depending on the values of the magnetic parameters. Although the general shape of the resonance frequency versus external static magnetic field curve is approximately linear for two different selected values of cubic anisotropy parameter (K 1 /M 0 = 400 G and K 1 /M 0 = 1,500 G), the external static magnetic field dependence of magnetic permeability components at constant

6 74 E. Demirel et al. Fig. 2 The field dependence of the precession frequency x 0, real (l 1 ) and imaginary (l 2 ) parts of magnetic permeability, for the case of H along easy axes direction of magnetization for anisotropy parameters (a) K 1 /M 0 = 400 G, (b) K 1 /M 0 = 1,500 G microwave frequency, let say x/c = 3,500 G, is strongly anisotropy-dependent for easy direction of magnetization as seen in Fig. 2(a, b). Since the resonance (precession of M) frequency for zero external static magnetic field is larger than x/c = 3,500 G for K 1 /M 0 = 1,500 G, there is no resonance for any positive values of external static magnetic field. That is, because the theoretical resonance field would had already taken place at negative field (anti-parallel to M which is not stable state) region, as can be inferred from the behavior of the curves in Fig. 2b. For sufficiently small values of the cubic anisotropy parameter (K 1 /M 0 = 400 G see Fig. 2a), the field dependent curve for the real component of magnetic permeability l 1 initially starts with a positive unit value, posses a positive maximum then decreases sharply to have a negative minimum. Then it starts to increase again, pass through the zero value and then slowly approaches positive unit value as shown in Fig. 2a. On the other hand, the imaginary part of magnetic permeability l 2 always takes positive values. It should be noted that the curves of both components of l in Fig. 2a exhibit sharp resonance behavior at the same static field region for sufficiently small relaxation time T 2. However, if the relaxation process gets faster, these resonance curves will broaden into smooth curves. Another interesting point is that, the real part of magnetic permeability l 1 remains negative for a wide range of positive value of external static magnetic field in case of higher values of the cubic anisotropy parameter (K 1 /M 0 = 1,500 G). While any positive value of external static magnetic fields is already well-above the resonance field, H 0, for the anisotropy parameter K 1 /M 0 = 1,500 G, the imaginary component of magnetic permeability l 2 still takes significant positive values even for zero external static magnetic field and decreases exponentially with increasing external static magnetic

7 Negative Refractive Index Ferromagnetic Materials 75 field as shown in Fig. 2b. It should be emphasized that we used the same value for intrinsic relaxation parameters in all cases throughout the calculations given in this manuscript. Figure 3 shows the calculated values of precession frequency of magnetization vector for various magnetic anisotropy parameters indicated above corresponding curves as a function of external static magnetic field that was applied along the hard direction of magnetization in plane of the film. As seen from this figure, the behavior of the precession (excitation) frequency strictly depends on magnetic anisotropy parameters. It should be noted that, we have assumed the cubic crystalline anisotropy and the applied external static magnetic field to be along the magnetically hard direction of the sample plane. For isotropic case (where the cubic anisotropy parameter was chosen to be zero, K 1 /M 0 = 0), the precession frequency starting from zero value at zero external static magnetic field first increases with square root of the value of the external static magnetic field, and then, continues to increase linearly at sufficiently higher external static magnetic field regions. However, for nonzero cubic anisotropy cases, this curve starts at a finite value at zero external static magnetic field and decreases as external static magnetic field increases, it assumes a sharp minimum and then increases first nonlinearly and then linearly for sufficiently larger external static magnetic field values. As mentioned above, when the exciting microwave field frequency, x, is chosen to be equal to the precession frequency, x 0, the resonance condition occurs for magnetization vector. As a result, the imaginary component of the dynamic permeability reaches its maximum value and the real component of the dynamic permeability almost vanishes at the resonance. As can be seen in Fig. 3, drawing a horizontal straight line corresponding to the constant chosen value of the external microwave frequency, this curve intersects the precession frequency curve once, twice or three times depending on value of the anisotropy parameters and external static magnetic field scan ranges. This means that, by choosing different values for microwave frequency, one can have single, double or triple resonance peaks at the static magnetic field values corresponding to these intersection points in permeability versus external static magnetic field curve. It should be recalled that one could get only a single resonance for the field along magnetically easy direction in thin film with cubic anisotropy energy as seen in Fig. 2. The imaginary component remains always positive and gets maximum value Fig. 3 The calculated values of the precession frequencies as a function of external static magnetic field applied along hard direction of magnetization for different values of anisotropy parameters

8 76 E. Demirel et al. at exact resonance field while the real component changes sign at about the resonance field. In general, the real component starts with a positive value and increases first slowly and then sharply as the field approaches the resonance for a small field interval and then sharply decreases, passes zero value at around resonance field and then receives negative value, reaches minimum value and then sharply increases to get positive value again. As mentioned above, single, double or triple peaks can occur in the curves for high frequency permeability components, depending on the magnitude of anisotropy parameters, applied external static magnetic field and the direction of the microwave and static component of the external field as seen in Fig. 4. The horizontal line in the figure represents the dynamic field frequency. For small anisotropy parameter values, a single resonance peak occurs when external static magnetic field is applied along even the hard direction as shown in Fig. 4a. However multi resonance can take place for the cases of large anisotropy parameters when the external static magnetic field is applied along the hard direction. Moreover, the second (intermediate) peak occurred for multi-resonance case is inverted with respect to the horizontal axis near the resonance point, that is, the lower and higher field sides of real component of the permeability with respect to the resonance field change sign as seen in Fig. 4b. On the other hand, the imaginary component of the dynamic permeability remains always positive and assumes its maximum value at the resonance point under all experimental conditions as shown in Fig. 4b. This is consistent with the fact that, imaginary component represents the absorption of microwave by the sample (negative value of imaginary component would be unphysical). As in Fig. 4b, there exist regions (or field band) in external static magnetic field range for the negative Fig. 4 The calculated values for precession frequencies, real (l 1 ) and imaginary (l 2 ) components of dynamic permeability for two different sets of values for anisotropy parameters: (a) single resonance case, (b) multiple resonance case. The horizontal line represents the dynamic field frequency

9 Negative Refractive Index Ferromagnetic Materials 77 l 1 and this field range can be broadened by suitable choice of magnetic anisotropy parameter. The negative l 1 implies that there is a 180 degrees phase shift between excitation microwave field and precession of the magnetization vectors. Near zero field, it should be noted that there is a low field region in which real component of l becomes negative (shaded area) as seen in Fig. 4b. It should be kept in mind that the condition for both the dielectric permittivity and the magnetic permeability values to have the same sign provides a good transmission for electromagnetic waves. However, when either one of the dielectric permittivity or the magnetic permeability gets negative value, then the transmission of electromagnetic waves will be very low. As shown in Fig. 4b, there is a positive magnetic permeability band between the regions for negative magnetic permeability for multi-resonance peak conditions. This result is important because the material can be used as a band-pass filter when the material has positive dielectric permittivity or band-stop filter when the material has negative dielectric permittivity. In both case, external static magnetic field will be utilized as a tuning parameter as well. That is when the material has positive dielectric permittivity and the condition shown in Fig. 4b is satisfied then the material can be used as band pass filter within the static magnetic field range between two resonance fields. However, when the material has negative dielectric permittivity then the material can be used as band stop filter within the static magnetic field range limited by two resonance fields as shown in Fig. 4b. Applications of negative refractive index materials were summarized in the literature [17, 18]. As mentioned above, behavior of the real part of dynamic magnetic permeability strictly depends on anisotropy parameter K 1 /M 0. For sufficiently small anisotropy values, usually single resonance peak is observed in field dependent permeability curve in microwave frequency band. As the anisotropy value increases, the number of resonances increases up to two or three. As seen in Fig. 5, for K 1 /M 0 = 700 G, two well-resolved peaks are obtained at 9.8 GHz (x/c = 3,500 G). While the resonance peak on the curve for l 1 at low field side is broader and always positive, the resonance peak at high field is relatively narrower and anti-symmetric with respect to the resonance field point on this curve. That is real component of the permeability receives positive values until resonance field and it changes sign to get negative values at high field side of the resonance curve. Nevertheless, as the external static magnetic field is increased further, the permeability crosses horizontal axis and approaches a limiting value of positive 1 at higher field values. However, as the anisotropy value is slightly increased beyond 700 G, the asymmetric(distorted) resonance peak at low field values (close to zero) starts to split into two distorted peaks so as the value of l 1 becomes negative in a significantly wide field band between two peaks for K 1 /M 0 = 750 G. As the value of anisotropy is increased further, these two resonance peaks at lower field values separate more and more so that the resonance peak at lowest field shifts towards the zero field and eventually disappears to go negative fields. Thus, the field band for negative l 1 is broadened so that l 1 remains negative even from zero fields until second resonance field point. However, the symmetric character of third peak at higher field still remains almost the same for selected values of K 1 /M 0 = 775 G and K 1 /M 0 = 900 G. The critical value for anisotropy parameter which determines the

10 78 E. Demirel et al. µ K 1 /M 0 =700G µ 1 µ 1 µ K 1 /M 0 =750G K 1 /M 0 =775G K 1 /M 0 =900G H(kOe) Fig. 5 The calculated values for real (l 1 ) component of dynamic permeability for four different values of anisotropy parameter at the microwave frequency of 9.8 GHz shape of real part of magnetic permeability versus external static magnetic field can be evaluated by using the curves for l 1, which are shown in Fig. 3. As for the out of plane geometry, where the external static magnetic field is rotated from sample plane towards film normal, a single resonance peak is obtained and the value of the resonance field monotonically increases as the microwave frequency increases if the magneto crystalline anisotropy energy is chosen to be zero as shown in Fig. 6. However, if one includes cubic or axial magneto-crystalline anisotropy, the number of resonance peaks increases as well and the resonance field value also varies depending on the anisotropy parameters and the direction of external field. It should be noted that we have used the same relaxation parameter for magnetization for all the experimental conditions. However the simulated resonance curve exhibits dramatic changes in the behavior of resonance field and especially in the resonance curve width. As the direction of the external field comes closer to the film normal, the resonance curve drastically broadens as seen in Fig. 6. This is caused by the fact that, even the direction of the external field is unchanged, the equilibrium direction of static magnetization vector changes continuously with the increasing value of the external field, which results in longer lasting resonance during the field scan processes. Of course, we cannot include all the calculated results here in order to be economical for the volume of the manuscript. Multi-peaks case can also arise for perpendicular geometry depending on the anisotropy parameters and microwave frequency as seen in Fig. 6b. We have shown that, the real component of dynamic magnetic permeability exhibit quite rich properties. The number of resonances can be increased depending on magnetic anisotropy, microwave frequency and the relative directions between

11 Negative Refractive Index Ferromagnetic Materials 79 Fig. 6 The calculated values of the precession frequency, real (l 1 ) and imaginary (l 2 ) components of dynamic permeability for the case of H along the direction perpendicular to film plane for (a) x/ c = 1,250 G and (b) x/c = 880 G values the external fields and film geometry. It should also be noted that these multi peaks do not originate from spin wave resonance. Of course the spin-wave resonance can also be included in Ferromagnetic Resonance spectra. However, the relative amplitude of spin wave resonance peaks vanishes especially for higher order modes or one can use sufficiently thick films such that Spin Wave Resonance modes almost overlap. On the other hand, for very thin film case, higher order Spin Wave Resonance mode even can not be excited due to the need for a very high frequency relative to the excitation microwave frequency. Therefore we have assumed only uniform Ferromagnetic Resonance modes that can be easily realized experimentally. The additional modes arise in addition to the main mode at the highest resonance field due to the continuous evolution of the equilibrium direction of static magnetization to match the resonance condition twice or third times as the external field is scanned. It should also be noted that the multi peak cases in permeability curves can only occur for field scan cases. That is, if one keeps the field as constant and scans the frequency of the excitation microwave frequency, only one resonance peak occurs corresponding to the equilibrium direction of the magnetization in a constant external static magnetic field. However, by choosing a suitable crystal, one can get negative real permeability even at zero field regions. 4 Conclusions By changing the magnitude or direction of external static magnetic field one can change shape of magnetic permeability curve. Since multiple resonance peaks can

12 80 E. Demirel et al. be obtained, the material can be used as band pass or band stop filter as a function of external static magnetic field depending on positive or negative dielectric permittivity value that this material has. It should also be noted that the second (intermediate) peak for l 1 is inverted with respect to the horizontal axis compared to that of main mode at higher fields. When single resonance peak occurs, the material can be used as low pass or high pass filter if the value of external field is suitably chosen to get positive or negative dielectric permittivity value. This provides us a tool to tune the magnetic permeability and thus electromagnetic transmission level due to the change in refractive index by the external static magnetic field. As mentioned above, negativity of real component of permeability is crucial to obtain negative index of refraction for some applications. This study proves that negative magnetic permeability for negative refractive index can be obtained at x-band microwave frequencies by using moderate anisotropy values. For higher frequency bands, one should use a sample with strong anisotropy energy which is possible by using rare earth materials or antiferromagnetic samples with higher resonance frequencies. One can also combine both ferromagnetic and antiferromagnetic substance to get broader band negative permeability. Acknowledgments This work was supported by the Turkish Republic Ministry of Industry and Trade with San-Tez Project No: STZ , and Scientific and Technical Research Council of Turkey (Project No: 104T524). References 1. V.G. Veselago, Sov. Phys. USPEKHI 10, (1968) 2. J.B. Pendry, A.J. Holden, W.J. Stewart, I. Youngs, Phys. Rev. Lett. 76, (1996) 3. J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, J. Phys. Condens. Matter 10, (1998) 4. J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, IEEE Trans. Microw. Theory Tech. 47, (1999) 5. D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Phys. Rev. Lett. 84, (2000) 6. R.A. Shelby, D.R. Smith, S.C. Nemat-Nasser, S. Schultz, Appl. Phys. Lett. 78, (2001) 7. R.A. Shelby, D.R. Smith, S. Schultz, Science 292, (2001) 8. V.G. Veselago, L. Braginsky, V. Shklover, C. Hafner, J. Comput. Theor. Nanosci. 3, 1 30 (2006) 9. N. Garcia, E.V. Ponizovskaia, Phys. Rev. E 71, (2005) 10. Y.-S. Zhou, B.-Y. Gu, F.-H. Wang, Europhys. Lett. 75, (2006) 11. B. Aktas, Europhys. Lett. 77, (2007) 12. A. Pimenov, A. Loidl, K. Gehrke, V. Moshnyaga, K. Samwer, Phys. Rev. Lett. 98, (2007) 13. R.F. Soohoo, Magnetic Thin Films (Harper & Row Publishers, New York, 1965) 14. M. Ozdemir, B. Aktas, Y. Oner, T. Sato, T. Ando, J. Magn. Magn. Mater. 164, (1996) 15. B. Aktas, M. Ozdemir, Physica B 193, (1994) 16. B. Aktas, B. Heinrich, G. Woltersdorf, R. Urban, L.R. Tagirov, F. Yildiz, K. Ozdogan, M. Ozdemir, O. Yalcin, in Nanostructured Magnetic Materials and their Applications, ed. by B. Aktas, L. Tagirov, F. Mikailov. NATO Science Series II. Mathematics, Physics and Chemistry, vol 143 (Kluwer, Netharlands, 2004), pp G.V. Eleftheriades, IEEE Antennas Propag. Mag. 49, (2007) 18. S.A. Ramakrishna, T.M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (Taylor & Francis Ltd., London, 2008)