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Sep 12, 2012 - A giant magnetoresistance (GMR) effect with 200% change in magnetoresistance and large magneto-dielectric response are observed at room...
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Giant Magnetoresistance and Magneto-Dielectric Effects in a Charge Transfer Metal Complex for Multiferroic Applications Bikash Kumar Shaw and Shyamal K. Saha* Department of Materials Science, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India ABSTRACT: The study of charge transfer complexes is a fascinating area of research due to their unique electronic and magnetic properties. During the last two decades, several interesting properties like superconductivity, charge transport, and magnetic and ferroelectric properties have been investigated in detail except the magneto-transport and magneto-dielectric effects which have potential applications in multiferroic devices. In the present work, magneto-transport and magneto-dielectric properties have been investigated in the [{Mn III (CH 2 (CN) 2 ) 2 (H 2 O) 2 }{Chloranil •− }]·(H 2 O) x charge transfer complex to elucidate multiferroic applications in these types of charge transfer complexes. A giant magnetoresistance (GMR) effect with 200% change in magnetoresistance and large magneto-dielectric response are observed at room temperature. We believe this new finding makes the charge transfer complexes a potential material for multiferroic devices.

1. INTRODUCTION

2. EXPERIMENTAL METHODS 2.1. Preparation of Metal Complex. Manganese chloride (0.162 g, 1 mmol) is dissolved in 2 mL of deionized water taken in a 50 mL beaker. Acetonitrile solvent (0.5 mL) is added to it and stirred gently. Malononitrile (0.066 g, 1 mmol) is dissolved in 2 mL of acetonitrile. The first solution is mixed with the second, and a few drops of water are added to make the solution clear. Chloranil (2,3,5,6-tetrachloroparaquinone, 0.025 g, 0.1 mmol) is dissolved in acetonitrile (6 mL), from which 1 mL is added to the main solution dropwise, slowly under continuous stirring for 15 min with a magnetic stirrer (if needed, a few drops of water are added to homogenize properly). A deep blue solution appears which is kept for 2 h and then dried using a freeze drier. Finally, the freshly synthesized compound is dried and collected for structural, chemical, and magnetic characterizations. 2.2. Characterizations. The individual properties are demonstrated by a spectroscopic method with a Cary 5000 UV−vis NIR spectrophotometer (solution state and solid state spectra with diffuse reflectance accessory using the BaSO4 matrix) measured in the range of 200−1000 nm and Fourier Transform Infrared spectrometer using KBr pellets (NICOLET MAGNA IR 750 System). The XRD pattern is investigated using a powdered sample with an X-ray diffractometer (RICH SEIFERT-XRD 3000P, wavelength 1.54 Å). The EPR spectrum is obtained by a Jeol (JES FA200) spectrophotometer. Electrochemical measurement is carried out with a CH Instrument (model chi720d). We have used a SQUID magnetometer (Quantum Design MPMS) to investigate the

Charge transfer (C-T) metal complexes have been investigated extensively during the last two decades owing to their interesting physical properties like unique charge transport,1 superconductivity,2−6 and superior magnetic7−11 and ferroelectric properties.12,13 In spite of possessing good magnetic and ferroelectric properties, surprisingly, the multiferroic behavior in these complexes, which is currently the most active area of research, has not yet been resolved. Although exhaustive studies on electronic transport, magnetic, and ferroelectric properties in these materials have already been explored, the detailed charge transport in the presence of magnetic field (magneto-transport) and magnetodielectric effect, i.e., the coupling between magnetism and dielectric properties, which have potential applications in multiferroic devices still remains in its infant stage. In the present work, we discuss the interesting interplay of the ligand to metal charge transfer phenomenon (LMCT). We choose manganese(II) (S = 5/2) and chloranil (2,3,5,6-tetrachloroparaquinone), which after taking an electron from the metal ion forms a linear building block of anion radical−trivalent metal (chloranil•−−Mn3+) deep blue complex exhibiting good electronic properties. We have investigated magnetic hysteresis including detailed magneto-transport behavior and magnetodielectric effect in the [{Mn III (CH 2 (CN) 2 ) 2 (H 2 O) 2 }{Chloranil•−}]·(H2O)x charge transfer complex to explore whether the charge transfer complexes exhibiting several functional properties could be used as potential materials for multiferroic devices. A giant magnetoresistance (GMR) effect of 200% change in magnetoresistance and a fairly strong coupling between the magnetic field and dielectric function are observed in the present C-T complex. © 2012 American Chemical Society

Received: May 15, 2012 Revised: August 31, 2012 Published: September 12, 2012 20700

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magnetic properties (FC-ZFC and M-H measurements), and the magnetoresistance is measured by the two-probe method using the PPMS (Cryogenic, model No. J 2468). Temperaturedependent permittivity over 190−300 K and magneto-dielectric measurements (using an electromagnet supplied by M/S control systems and devices; Mumbai, India) have been carried out with the Agilent LCR meter (model: E4980A).

3. RESULTS AND DISCUSSION 3.1. UV−Visible Spectra. Figure 1a shows the diffuse reflectance UV−visible spectra of the metal complex in the

Figure 2. Left image: Molecular orbital diagram representing the ligand to metal charge transfer (LMCT) phenomenon with the d−d transition (CTTM: charge transfer to metal). Right image: Tunabe− Sugano energy diagram corresponding to the d−d transition.

transition occurs from the filled bonding orbital (π) of the ligand (chloranil•−) to the half-filled orbitals of metal (Mn3+) attributed as a ligand to metal charge transfer (LMCT) transition, which gives an intense peak at 21 880 cm−1, respectively, as a result of the t2g(π) → t2g*(π*) transition.14−16 However, the d−d transition from t2g* → eg* for strong charge transfer complexes gives rise to a fairly weak crystal field band in which an electron is excited from a weak antibonding orbital.17 The low energy peak at 535 nm (υ1 = 18 690 cm−1) corresponds to the d−d transition that occurred from the 5Eg ground state to the excited state 5T2g.14−16 In the case of d4 high spin octahedral configuration (5D), the crystal field splitting energy (Δ0) equals to υ1. The decrease of the value of crystal field strength (Δo) to 18 690 cm−1 instead of 21 000 cm−1 (250 KJ/mol) for the Mn3+ hexaaqua system (Orgel1966) is the effect of LMCT interaction which destabilizes the t2 g set of bonding orbitals of metal with an increase in energy resulting in a decrease in Δ0 value. After taking the value of Δ = 18 690 cm−1, the Δ/B ratio becomes 19.4 (B = 965 cm−1 for the Mn3+ system), which agrees well in the high spin side of the Tanabe−Sugano diagram, shown in the right side of Figure 2 indicating the d−d transition from 5Eg → 5T2g. 3.3. Powder XRD. In spite of providing serious efforts, we were not able to obtain single crystals of the present complex because of the formation of an infinite chain between metal and ligands. Therefore, we have performed the powder diffraction technique to investigate the crystal structure. Figure 3a shows the XRD pattern of the present complex measured over the range of 2θ values lying between 0 and 80° which indicates the polycrystalline nature of the sample. 3.4. FT-IR Spectra. The appearance of 1518 and 1571 cm−1 peaks in the Fourier transform infrared spectra of the metal complex (Figure 3b) indicates the presence of aromatic group frequency (υCC) due to the resonance of the carbonyl group with the benzenoid ring.18 The 1114 cm−1 peak in the metal complex spectra appears as a result of conversion of the ligand from a paraquinonoid (1689, 1679 cm−1 carbonyl stretching frequencies) to an anionic system (υC−O). The peak at 2217 cm−1 shows that cyanide (−CN) of malononitrile is coordinating to the metal center (υCN = 2274 cm−1 for free malononitrile). The broad (s) peak at 3400 cm−1 arises due to

Figure 1. (a) Diffuse reflectance UV−vis spectra of the Mn(III)− chloranil charge transfer complex using a BaSO4 matrix. Inset shows the enlarged picture of the strong charge transfer band. (b) UV− visible absorption spectra of the charge transfer metal complex in solution phase.

BaSO4 matrix. From the spectra, we have seen that an intense peak appeared at 457 nm along with the higher energy transitions at 350, 304, 248, and 225 nm. A weak absorption band at 535 nm (18 690 cm−1) is revealed at lower energies. An intense broad hump at 675 nm in the liquid phase absorption spectra (Figure 1 b) gives a clear indication of generation of radical species in the system. The charge transfer bands were seen at higher energies. 3.2. MO Theory. Correlating the UV−vis spectroscopy (Figure 1 a) with the molecular orbital (MO) theory, it is easier to demonstrate the electronic picture of the charge transfer metal complex (Figure 2). The charge transfer transitions usually result in substantial shifting of charge densities giving rise to intense absorptions at higher energies (UV region) compared to d−d transitions. Here the charge transfer 20701

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Figure 3. (a) Powder X-ray diffraction pattern of the synthesized complex. (b) Comparison of FT-IR spectra of the metal complex with a chloranil ligand. (c) EPR spectra of the charge transfer anion radical metal complex showing a singlet line at g ≈ 2.002 for both liquid and solid phases. (d) Cyclic voltammogram of the metal complex using the Ag/AgCl reference electrode measured in ambient conditions.

Scheme 1. Reaction Mechanism for the Formation of a Charge Transfer Complex

the −OH stretching vibration of water attached to manganese ion.19 3.5. ESR Study. EPR spectroscopy is a powerful technique which is sensitive toward the electronic structure of paramagnetic species. The X-band (9.5 GHz) EPR spectra (Figure 3 c) of the sample in the liquid phase with concentration 1 mM is run at 77 K in a liquid nitrogen finger dewar. The same is done for the solid sample at room temperature. Both experiments show only one singlet line of S = 1/2 signal with resonance at g ≈ 2.002.20 The appearance of a derivative singlet peak is evidence for the formation of anionic radical species after single electron transfer9 from manganese(II) to chloranil. The same g value at 2.002 indicates that the anionic radical is quite9 stable in both the solution and solid phases due to a strong charge transfer interaction with the trivalent cation. This confirms the formation of trivalent manganese (high spin)

which is a non-Kramers EPR silent species resulting in no signal in the EPR spectra. EPR line broadening effects21 occur in this charge transfer complex due to the equivalent coordination sites for one unpaired electron, present in this paramagnetic radical species. 3.6. Cyclic Voltammetry. Electrochemical study of the metal complex has been performed on 0.5 mM of complex (per manganese) in acetone/aqueous (6/4 v/v) solvent in the range of +1.0 to −0.8 V, using Pt, glassy carbon, and Ag/AgCl as the counter, working, and reference electrode, respectively, containing KPF6 as the supporting electrolyte, under a nitrogen atmosphere in ambient conditions. The voltammogram (Figure 3 d) displays two new waves as compared to the metal-free ligand (chloranil).22 After application of voltages from negative to positive, the Mn(III) ion reduces at a potential of −0.20 V, while oxidation takes place at +0.54 V (vs Ag/AgCl). For the 20702

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first scan (green cycle), only the trivalent manganese ion which is present in the complex reduces to the bivalent state (at negative potentials) and starts to oxidize a little to the tetravalent state (at positive potentials). As the scan proceeds further, with increasing scan rates (from 0.3 to 1.0 V/s), the Mn(IV) ion which is generated earlier in the first scan showing sharp reduction and oxidation peaks for the (III/IV) couple. It is predicted from the voltammogram that the corresponding half wave potential (E1/2) values at −0.15 V (E11/2; peak to peak separation, ΔEp = 69 mV; number of electron transfer, n = ∼1) and at +0.57 V (E21/2; peak to peak separation, ΔEp = 60 mV, number of electron transfer, n = 1) refer to the Mn(III)/(II) and Mn(III)/(IV) couples, respectively. The half wave potentials (E11/2 and E21/2) of the trivalent manganese ion for this charge transfer system are a little bit shifted, due to a partial change in redox peaks from the reported values for different Mn(III) systems.23,24 3.7. Mechanism. On the basis of the experimental results (XRD, IR, UV−vis, EPR spectroscopy, and cyclic voltammetry), the reaction mechanism is proposed (Scheme 1). The initial electron transfer from manganese(II) to the chloranil ligand (π acceptor), i.e., oxidation of bivalent metal, is generally not possible from the hexaaqua manganese ion. A potential π mediator is needed to facilitate the electron transfer from manganese(II) to the vacant π antibonding orbital of chloranil, making the path for back charge transfer. Therefore, we use malononitrile which by −N coordination with the bivalent metal increases the energy of the dz2 orbital, resulting in the oxidation of a metal ion. The ligand to metal charge transfer complex comprised of an anion radical as the donor and a trivalent metal as the acceptor is formed after the oxidation of malononitrile-coordinated bivalent manganese. 3.8. McConnell Model. The extended McConnell model and its mathematical embodiment as the generalized Hubbard model offer a convenient guide to explore ferro-, antiferro-, and ferrimagnetic phenomena in charge transfer molecular systems. The McConnell model is applicable for a chain structure consisting of alternating radical cation donors (D) and radical anion acceptors (A), i.e., D•+A•−D•+A•−D•+A•− or vice versa. According to this model, the magnetic coupling can be predicted if the direction of charge transfer is known. For homospin systems (mSD = mSA), it behaves like ferromagnetic if the spin multiplicity (MS) is one and antiferromagnetic if it is zero in both the ground and excited states, respectively. In our case, magnetic coupling indicated in Figure 4 also shows ferromagnetism, which confirms the formation of a ligand to metal charge transfer (back charge transfer) complex between Mn3+ (S = 2) and singly occupied chloranil (S = 1/2) as already predicted earlier by MO theory.11 3.9. Magnetic Study. The temperature-dependent magnetization of the compound has been measured in zero-fieldcooled (ZFC) and field-cooled (FC) processes from 2 to 300 K using a magnetic field of 100 Oe (Figure 5a). Both FC and ZFC data show a similar behavior, and magnetization increases exponentially upon cooling.25 However, below 40 K, a splitting is detected between the field cooled and the zero field cooled curves down to 2 K, and MFC is slightly larger than MZFC. After 40 K, this difference seems to vanish in the high temperature limit. The magnetizations vs field data (inset of Figure 5b) are measured at temperatures 2, 10, 50, 100, 200, and 300 K, respectively. The most striking feature in magnetization data is the appearance of inverted hysteresis curves,25 which have been

Figure 4. McConnell mechanism supports the back charge transfer LMCT process.

Figure 5. (a) Temperature dependence of magnetization measured at 100 Oe. Inset reveals the splitting of ZFC and FC curves. (b) The enlarged view of the hysteresis curves for different temperatures. The complete hysteresis curves are shown in the inset.

observed recently in a few samples containing both the ferromagnetic as well as superparamagnetic phases. The superparamagnetism in ferromagnetic systems usually arises due to size reduction; viz., for larger particles it is ferromagnetic, and below a critical size it is superparamagnetic.26 In the present system, the donor−acceptor charge transfer compound leads to the formation of ferromagnetic domains, while due to lack of high crystallinity, some smaller superparamagnetic domains are also formed. The coexistence of these two phases in the material, both ferromagnetic and superparamagnetic, leads to inhomogeneity in the system, giving rise to inverted hysteresis loops as a result of magnetic anisotropy. Another interesting behavior is that the hysteresis 20703

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Figure 6. (a) Histogram of magnetoresistance with field. (b) Represents the variation of conductivity as a function of temperature with and without fields. (c) Gives the variation of magnetoconductance with field. (d) Shows the H1/2 dependence of magnetoconductivity in the high field region showing the electron−electron interaction.

loops are symmetrical until 50 K with a coercive field almost constant at about 20 Oe, whereas a strong exchange bias is observed at higher temperatures (Figure 5 b). At 300 K, the loop is entirely shifted to the negative field region. The existence of a strong exchange bias as a result of asymmetry in the hysteresis loop indicates that a remarkable magnetoresistance behavior comes into play in the present charge transfer complex at room temperature. 4.0. Magnetoresistance Study. The magnetoresistance behavior as a function of temperature is measured using a pelleted sample. A remarkable change in positive magnetoresistance of about 200% is observed at room temperature. Usually, the charge transfer complexes are arranged as donor− acceptor polymeric chain like systems. The charge transport (electronic transport) in this complex is very similar to the doped organic conducting polymers. Our charge transfer donor−acceptor complex shows a positive change in magnetoresistance similar to that of doped conducting polymers which often show positive magnetoresistance. With this idea, we have explained the magnetoresistance behavior of this complex with the help of the charge transfer theory of conducting polymers and observed a good correlation between the experimental results with the theoretical predictions as illustrated below. The histogram shows the changes in MR behavior in different field regions with 146%, (0T-1T) up to 24% (1T-2T), 15% (2T-3T), 9% (3T-4T), and 3% (4T-5T), respectively (Figure 6 a). Here the electron−electron interactions play the key role for this positive MR which is theoretically verified from the mentioned theory given below.27 The magnetic field dependence of the conductivity (MC) is the effect of several facts like interplay of weak localization (positive and negative MC for weak and strong spin−orbit

coupling, respectively), electron−electron interactions (gives negative MC), and anisotropic diffusion coefficient contributions to the magneto conductance. For the disordered donor−acceptor quasi one-dimensional system where the electron−electron interactions play an important role (negative MC), the conductivity (without and with field) is expressed by the equations as σI(T ) = σ(0) + mT1/2 + BT p /2

(1)

σI(H , T ) = σ(H , 0) + m′T1/2 + BT p /2

(2)

1/2

where the term T results from thermally induced electron diffusion near the Fermi energy; the third term on the right is the correction of the zero temperature metallic conductivity, σ(0), due to disorder; and m and m′ are given by ⎛4 3γFσ ⎞ ⎟ m = α⎜ − ⎝3 2 ⎠

(3)

and ⎡4 ⎛ F ⎞⎤ m ′ = α ⎢ − γ ⎜ σ ⎟⎥ ⎝ 2 ⎠⎦ ⎣3

(4)

where the Hartree factor (F) is the screened interaction averaged over the Fermi surface; α is a parameter that depends on the diffusion coefficient (D); and γFσ is the interaction parameter. Using the above two eqs 3 and 4, the value of γFσ has been expressed as γFσ = 20704

⎛ 3 ⎞ m′ − m ⎜ ⎟ ⎝ 8 ⎠ 3m′ − m

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Table 1. Close Matching of the αγFσ Value Obtained from Both Conductivity Temperatures (With and Without Field) and Magnetoconductivity Fitting Procedures obtained from R-T R-H

parameters of temperature

calculated values (from eqs 3 and 5)

m

m′

γFσ

α

−0.00449

−0.00513

0.022

−0.0035

product of α and γFσ −0.76 × 10−6

αγFσ −0.99 × 10−6

−0.99 × 10−6

assuming that α, γ, and Fσ are independent of the magnetic field. The magnetic field dependence of the contribution to the MC from the electron−electron interactions can be written as Δ∑(H,T) = σ(H,T) − σ(0,T). At sufficiently high fields (gμBH ≫ kBT), the equation is written as ⎛ g μ ⎞1/2 Δ ∑ (H , T ) = αγFσ T1/2 − 0.77α⎜ B ⎟ γFσ H1/2 ⎝ kB ⎠

(6)

Thus, for R−H measurements up to high fields, Δ∑(H,T) is proportional to H1/2. The measured conductivity as a function of temperature with and without field is shown in Figure 6b, in which the points are the experimental data, and the red curves represents the theoretical expressions given by eqs 1 and 2, respectively. From the fitting procedure, the values of γFσ and α are extracted using eqs 3 and 5. The difference in magnetoconductance data with and without field as a function of field (Figure 6c) has also been fitted by the expression given in eq 6, and the extracted values of the parameters α, γ, and Fσ are also in good agreement with the values obtained from eqs 1 and 2 (Table 1). The close agreement between the variation of Δσ with H1/2 also supports the origin of negative magnetoconductance in the present charge transfer complex as a result of electron−electron interaction (Figure 6d).27 4.1. Temperature-Dependent Dielectric Study. The dielectric modulus spectra (real and imaginary) for the powdered sample, represented in Figure 7(a) and 7(b), have been analyzed over the temperature range from 190 to 300 K. The values of m′ increase with frequency and reach a rather constant value, while values of ε′, as expected, decrease to almost the same value. The dielectric permittivity values measured from the modulus spectra are plotted as a function of frequency at different temperatures (Figure 7c), and the same values of permittivity as a function of temperature with different frequencies are shown in the inset of this. The peak value obtained at around 265 K indicates the ferroelectric transition temperature. It is therefore seen that the present charge transfer complex shows a ferroelectric ordering below the transition temperature at 265 K. 4.2. Magneto-Dielectric Study. To investigate the magneto-dielectric effect in this charge transfer complex, we have measured directly the modulus spectra using the magnetic field varying from 0T to 0.5T and frequency range from 100 Hz to 2 MHz. The dc resistance values at different fields are obtained from the Cole−Cole diagram (Figure 8a) to compare the magnetoresistance behavior, measured directly using a source meter in the PPMS system. Close agreement between the results obtained from the two independent measurements (both dc and ac) is noticed as shown in the inset of the Cole− Cole diagram. To analyze the dielectric modulus data in the

Figure 7. (a) Real part (m′) of electric modulus spectra at different temperatures. (b) Variation of the imaginary part (m″) of the electric modulus with temperature. With an increase in temperature, saturation values of m′ and peak values of m″ shift toward the higher frequency side and then move back to lower frequencies after a certain limit. (c) Values of dielectric permittivity as a function of frequency at different temperatures. Ferroelectric transition temperature is shown in the inset.

present complex, we consider the stretched exponential relaxation function F(t) given by28 F(t ) = exp[−(t /τR )β ] 20705

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where Ms = 1/ε∞ is the saturation value of the real part of the electric modulus and ε∞ is the limiting value of dielectric permittivity and n

N *(ω) =

⎛

⎞ 1 ⎟ ⎝ 1 + jωτi ⎠

∑ gi ⎜ i=1

(10)

Putting the values of N*(ω) using eq 10 in eq 9, the theoretical expressions for the real and imaginary parts of the electric modulus are given by n

M′(ω) = Ms[1 −

∑ gi /(1 + ω2λi 2τR 2)] i=1

(11)

n

M″(ω) = Ms ∑ giλiωτR /(1 + ω 2λi 2τR 2) i=1

(12)

The experimental data are fitted by the theoretical expressions for real and imaginary parts of the electric modulus (Figure 8 b), by eqs 11 and 12. The relaxation parameter β is estimated from the fitting procedure using gi and λi as parameters to be about 0.42. The low β value is also indicative to the strong coupling between the relaxing unit which is in conformity with the Coupling Model of relaxation proposed by K. L. Ngai.29 From this electric modulus figure, it is also seen that the relaxation peaks shift to the lower frequency range with the increase in magnetic field, which also indicates the positive magnetoresistance in the present complex. The variation of dielectric permittivity with frequency (Figure 8c) shows that the dielectric function changes significantly with the magnetic field. The room-temperature permittivity value obtained from the magneto-dielectric path is almost the same as the temperature-dependent ac measurement. The permittivity value (ε∞) obtained from the saturation value of M′(ω) also shows a similar effect (Figure 8b). It is therefore concluded that the present charge transfer compound shows a giant positive magnetoresistance and a clear magneto-dielectric effect for multiferroic applications.

4. CONCLUSIONS In summary, magneto-transport and magneto-dielectric properties are investigated in the [{MnIII(CH2(CN)2)2(H2O)2}{Chloranil•−}]·(H2O)x charge transfer complex which shows a GMR (giant magnetoresistance) effect with 200% change in magnetoresistance and large magneto-dielectric response at room temperature as a result of coupling between dielectric function and magnetic field. We believe this new finding makes the charge transfer complexes a potential material for multiferroic devices.

Figure 8. (a) Cole−Cole diagram containing the real and imaginary parts of the complex impedance. The dc resistance values measured from the Cole−Cole diagram are shown in the inset. (b) Real (m′) and imaginary (m″) parts of the electric modulus spectra as a function of frequency. The experimental data are fitted by theoretical expressions given in eqs 11 and 12. (c) Variation of dielectric permittivity as a function of frequency at different fields. The corresponding loss factors are shown in the inset.

■

where τR is the conductivity relaxation time and the exponent β has a value between 0 and 1. Since any nonexponential decay function can be written as a summation of some Fourior components, we write eq 7 (shown in text) as

Corresponding Author

*E-mail: [email protected]. Phone: +91-33-2473 4971 (Ext 227). Fax: +91-33-2473−2805. Notes

n

F (t ) =

∑ gi exp(−t /τi),

with τi = λiτR

i=1

The authors declare no competing financial interest.

■

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ACKNOWLEDGMENTS BKS acknowledges CSIR, New Delhi, for awarding fellowship, and SKS acknowledges DST, New Delhi, for financial support. SKS acknowledges financial support from the DST-Centre for Nanotechnology, Government of India.

where gi and λi are the parameters. In the frequency domain, the electric modulus is given by − M *(ω) = 1/ε*(ω) = Ms[1 − N *(ω)]

AUTHOR INFORMATION

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