ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 322 (2010) 603–608

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Journal of Magnetism and Magnetic Materials

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Regular Submission Influence of external magnetic field on – in negative-index antiferromagnet– superlattices

R.H. Tarkhanyan a,n, D.G. Niarchos b, M. Kafesaki a a Institute of Electronic Structure and Laser-FORTH, Heraklion, Crete 71110, Greece b Institute of Materials Science, NCSR ‘‘Demokritos’’, Athens 15310, Greece article info abstract

Article history: The peculiarities of the negative refraction in periodic multilayered antiferromagnet–semiconductor Received 31 July 2009 nanostructures are investigated in the presence of an external magnetic field parallel to the plane of the Received in revised form layers. Effective material tensors are obtained using method of anisotropic homogeneous medium. 30 September 2009 Dispersion and energetic relations for the mixed magnon–plasmon polaritons are investigated in the Available online 22 October 2009 case of the Voigt geometry. Frequency regions of anomalous dispersion are found and studied in various Keywords: regimes of the applied magnetic field. The necessary conditions are found under which the structure Negative refraction behaves as a left-handed negative-index metamaterial. Analytical expressions for the frequency- Magnon–plasmon polaritons dependent and group refractive indices are obtained. Superlattice & 2009 Elsevier B.V. All rights reserved. Semiconductor Antiferromagnet

1. Introduction The properties of the magnetic plasmon polaritons in homo- geneous negative-index metallic antiferromagnets have been During the last years, the exotic behavior of electromagnetic studied in [14]. (EMW) in so called left-handed media became an The aim of this work is an investigation of the effect of an object of many interesting and outstanding investigations external static magnetic field on the behavior of EMW in periodic (see, for example, [1–11] and citations therein). In particular, a multilayer nanostructures consisting of alternating layers of great effort has been made to predict theoretically and observe antiferromagnetic (for example, Cd1xMnxTe) and non- the negative refraction (NR) of the waves in various artificial magnetic semiconductor (CdTe). When the characteristic dimen- materials including periodic multilayer nanostructures, where the sions of the layers are much smaller than the internal wavelength, waves have many unusual properties due to their specific the coexistence of both magnetic and plasmonic properties in dispersion characteristics. In Ref. [12], the problem of anomalous such structures leads to the interesting peculiarities of the waves refraction has been studied in ferrite-semiconductor superlattices propagating in the backward regime. We shall examine the (SL) in the presence of an external magnetic field and has been anomalous refraction of the coupled magnon–plasmon polaritons, shown that the latter influences considerably on the dispersion that is, coupled with antiferromagnetic and properties of the waves as well as on the number of the spectral magnetoplasmons. As it is well known [15,16], this is branches and frequency regions of existence for the backward especially strong for the waves in Thz region, when frequencies of waves. On the other side, in the last several years there has been the photons are comparable with the characteristic frequencies of enormous interest in antiferromagnetic substances, which pos- the magnons and magnetoplasmons. sess special optical and spintronic properties in terahertz The paper is organized in five main sections. Section 2 frequency region. In spite of that, up to now there are only a describes the effective permittivity and permeability tensors of few studies on the possibility of NR in periodic structures SL used for the analysis of the dispersion relations for the containing antiferromagnetic layers. In [13], the characteristics propagating waves. Section 3 discusses mixed magnon–plasmon of the bulk EMW have been discussed in antiferromagnet/ polaritons in the case of the Voigt geometry when TE- and TM- semiconductor SL in the absence of an external magnetic field. type polarized waves propagate separately. The dispersion curves and frequency regions of existence are determined for TE waves, which can exhibit anomalous dispersion. In Section 4, the

n necessary conditions are obtained for absolute left-handed Corresponding author. Present address: 5 Arkadiou (Pellinis), Piraeus 18534, Greece. Tel.: +30 2104179379. behavior of the medium with respect to the TE waves. In Section E-mail address: [email protected] (R.H. Tarkhanyan). 5 the concluded remarks are highlighted.

0304-8853/$ - see front & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2009.10.023 ARTICLE IN PRESS

604 R.H. Tarkhanyan et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 603–608

x materials, this characteristical transverse magnon frequency is in the far infrared region. Inserting, for example, the following values: z HE ¼ 550kG; HA ¼ 3:8kG; g ¼ 2:87GHz=kG ðMnF2Þð4dÞ

from Eq.(4c) we obtain oT=185.874 GHz. Assuming that the period of the structure d=l1+l2 is much y smaller than the internal wavelength and using method of effective anisotropic homogeneous medium [19,20,12], we obtain constitutive relations in the form

D ¼ e0e^ef E; B ¼ m0m^ ef H; ð5Þ Substrate where relative effective permittivity and permeability tensors read 0 1 0 1 e11 iea 0 m im 0 Fig. 1. Geometry of the problem. The static magnetic field is parallel to the z-axis. B C B 11 a C @ A @ A e^ef ¼ iea e22 0 ; m^ ef ¼ ima m22 0 ; ð6Þ 2. Effective material tensors 00e33 001 "# !  The biasing magnetostatic field B 99z is assumed to be parallel 1 e e l e 1 0 e ¼ l e þl e 1 2 a ; e ¼ e 1þ 2 1 ; ð6aÞ to the plane of the layers and along the positive z-axis (see Fig. 1). 11 1 1 2 A 2 a 2 d eA l1eA We will make simplifying assumptions that B is less than the 0  -flop field and the medium is lossless. Also we assume that: 1 1 l1 l2 1 e22 ¼ þ ; e33 ¼ ðl1e3 þl2eAÞ; ð6bÞ (1) semiconductor layers contain only one type of free charge d e1 eA d carriers with a parabolic conduction band; (2) antiferromagnetic layers with two antiparallel magnetic sublattices have an ‘easy- m11 ¼½l2m1 þl1ð1 m2maÞ=d; ma ¼ l2m2=ðl2 þl1m1Þ; axis’ type magnetic anisotropy, so as the saturation magnetization 1 1 m22 ¼ðl1 þl2m1 Þ=d: ð6cÞ MS, anisotropy field HA and exchange field HE are oriented in the same direction along the static field H . Then, an individual 0 The behavior of a monochromatic plane electromagnetic nonmagnetic semiconductor layer of thickness l can be described 1 E,Hexp[i(krot)] in the medium described by Eq. (6) can be by the following permittivity tensor: 0 1 investigated using the Maxwell equations in the form   e1 ie2 0 B C k E¼oB; k H¼ oD: ð7Þ e^ ¼ @ ie2 e1 0 A; ð1Þ Eliminating the magnetic field H ¼ðom m^ Þ1k E from 00e3 0 ef Eq. (7), for an arbitrary direction of the wave propagation s=k/k where we obtain a set of equations ! ! 2 2 2 2 o e1oco o ðn Z^ l e^ ÞE ¼ 0; ð8Þ e ¼ e 1 p ; e ¼ p ; e ¼ e 1 p : m ef 1 1 o2 o2 2 oðo2 o2Þ 3 1 o2 c c where n=ck/o is the refractive index, ð2Þ l m m m2; ð9aÞ 2 1/2 m 11 22 a op=(Ne /me0eN) is the frequency, oc=eB0/m is the cyclotron frequency, e, m and N are the charge, effective mass and and 0 1 density of the conduction , respectively, B0 =m0H0, eN is m s2 þl s2 ðl s s þim s2Þs ðm s im s Þ B 11 z m y m x y a z z 11 x a y C the high-frequency dielectric constant, e0 and m0, respectively, are ^ ¼ B l s s þim s2 m s2 þl s2 s ðm s þim s Þ C: permittivity and permeability of the free space. Z @ m x y a z 22 z m x z 22 y a x A 2 2 szðm sx þim syÞszðm sy im sxÞ m s þm s An individual antiferromagnetic layer of thickness l2 is 11 a 22 a 11 x 22 y described by a scalar dielectric constant eA and permeability ð9bÞ tensor [17,18]. 0 1 The system of equations (8) has a nontrivial solution only if the m im 0 determinant of the coefficient matrix vanishes B 1 2 C m^ ¼ @ im2 m1 0 A; ð3Þ 001 : 2 : Det n Zij lmðeef Þij ¼ 0: ð10Þ where ^ "#Using Eqs. (9b) and (6) for the tensor eef , we can write Eq. (10) 1 1 in the form m ¼ 1þoAoS þ ; ð4aÞ 1 2 ð þ Þ2 2 ð Þ2 4 2 2 oT o o0 oT o o0 An lmBn þe33lelm ¼ 0; ð11Þ "# where 1 1 ¼ ð Þ m2 oAoS 2 2 ; 4b 2 2 2 2 2 2 2 2 A ¼ lmðZzz þsz Þðe11sx þe22sy þe33sz Þ2eamaZzzsz ; le ¼ e11e22 ea ; oT ðoþo0Þ oT ðo o0Þ ð12aÞ o0 =gH0, oA=gHA, oS =gMS, g is the , o is the angular frequency of the waves and 2 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B ¼ e33lmðe11sx þe22sy ÞþleZzz þe33ðe11m22 þe22m11 þ2eamaÞsz : oT ¼ g HAðHA þ2HEÞ; ð4cÞ ð12bÞ is the antiferromagnetic resonance frequency in the absence of an It follows that in the medium, two distinct extraordinary waves external magnetic field. For most of known antiferromagnetic can travel whose refractive indices are given by solutions of ARTICLE IN PRESS

R.H. Tarkhanyan et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 603–608 605 biquadratic equation (11). In general, for an arbitrary direction of in the form propagation, both these waves are elliptically polarized and e~ðo2 $2Þðo2 O2 Þðo2 O2 Þ correspond to the coupled magnon–plasmon polaritons. In the n2 ¼ p þ ; ð21Þ 2 2 2 2 2 following, we will restrict ourselves by consideration of the o ðo o þ Þðo oÞ propagating waves in the case of the Voigt geometry. where l e þl e e~ ¼ 1 1 2 A ; ð22Þ 3. Coupled magnon–plasmon polaritons in the case of the l1 þl2 Voigt configuration  1=2 l2eA $p ¼ op 1þ ; ð23Þ Consider the propagation of the waves in the xy-plane l1e1 perpendicular to the magnetic field, so as sz=0. In this case, the is the effective plasma frequency, solutions of Eq. (11) are split in TM and TE waves with dispersion rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi relations 2 2 2 7 2 2 2 2 2 o 7 ¼ o0 þoT þoAoS ðoAoSÞ 1 4l1l2d sin bÞþ4o0ðoT þoAoSÞ; l 2 ¼ e ð Þ ð24Þ nTM 2 2 13 e11s þe22s x y b is the angle between the wave vector k and the positive y-axis: and tan b=kx/ky, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 lme33 2 2 2 7 2 2 2 2 n ¼ ; ð14Þ O 7 ¼ o0 þoT þoAoSð1þl2=dÞ ðoAoSÞ ðl1=dÞ þ4o ½oT þoAoSð1þl2=dÞ: TE m s2 þm s2 0 11 x 22 y ð25Þ respectively. Let us consider first the TM wave in which the Using Eq. (21), it is easily seen that there are three branches in magnetic field vector H is parallel to the external field: H={0,0,H}. the spectrum of the TE waves, and in distinct ranges of the applied The electric field and time-averaged Poynting vector are given by magnetic field these branches have different dispersion behaviors. 1 E ¼ðoe0leÞ fe22ky ieakx; e11kx ieaky; 0gH; ð15Þ In general, the magnetic fields can be divided into three distinct 0 0 0 regions: ‘‘low’’ field ðo0 ooÞ, ‘‘middle’’ ðo ro0 ro þ Þ and 2 0 1 ‘‘high’’ field ðo0 4o Þ regions. For a given direction of the wave S ¼fe11kx; e22ky; 0gð2oe0leÞ H9 ; ð16Þ þ propagation, the limits of these regions are defined by resonance where frequencies at zero magnetic field À qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 1 ky ¼ k0le e11kx Þe22 ; k0 ¼ o=c: ð17Þ 0 2 2 7 2 2 ðo 7 Þ ¼ oT þoAoSð1 1 4l1l2d sin bÞ: ð26Þ Using Eqs. (16) and (17) we conclude that the dot product 2 In the ‘‘low’’ and ‘‘high’’ field regions, the wave has two kS=om09H9 /2 is positive. It means that propagating TM waves resonance frequencies (n-N)ato=o+ and o=o while in the are always forward, but it does not mean that the phenomenon of ‘‘middle’’ field regime there is only one resonance at o=o+ . negative refraction is impossible. Indeed, consider the case when With increase in magnetic field, o+ increases monotonically e11 40 while e22o0, so as le is negative. It is not difficult to see while o decreases in the range of the ‘‘low’’ fields, vanishes at that if TM wave is incident from the vacuum onto the interface of 0 0 o0 ¼ o , appears again at o0 ¼ o and then increases approach- the medium, the transmitted bulk wave exists for all possible þ ing asymptotically the straight line o=o0 in the range of the angles of incidence and the Poynting vector of the refracted wave ‘‘high’’ fields (see Fig. 2, lines). Besides, the wave has three is in the same side of the normal to the interface as that in the cutoff frequencies (n=0) at o=$p and o=O 7. The dependence of incident wave. Therefore, the negative refraction of the TM waves O7 on the magnetic field is shown schematically in Fig. 2 (dashed occurs without backward waves. Such a possibility of negative curves). refraction has been predicted in [21] for uniaxial nongyromag- Note that the ‘‘middle’’ field region is very narrow: for MnF2, netic dielectrics with negative dielectric permittivity along the inserting in Eq. (26) l1=1 mm, l2=0.2 mm, b=p/6, oS =1.722 GHz anisotropy axis. It is important to note that TM waves correspond to the coupled plasmon polaritons without any mixture of the magnons. ω That is, why in the following we shall consider the peculiarities of the TE waves corresponding to the coupled magnon–plasmon polaritons, which are of interest in this work. The field structure and time-averaged Poynting vector of the TE Ω+ wave are given by ωL 1 0 ω E ¼f0;0;Eg; H ¼ðom0lmÞ fm22ky þimakx; m11kx þimaky; 0gE; ω + + ω − ð18Þ 0 Ω− 19 92 S ¼fm11kx; m22ky; 0gð2om0lmÞ E ; ð19Þ 0 Ω where ω− − 2 2 1 2 Ω− ky ¼ k0e33mv m11m22 kx ð20Þ and ω−

2 0 0 0 lm ma ω− Ω− ω+ ωL mv ¼ m11 ð20aÞ ω m22 m22 0 is the effective Voigt permeability. Using Eqs. (6b,6c), (2), Fig. 2. Schematic plot of the resonance (solid lines) and cutoff (dashed lines)

(4a, 4b) and (9a), the dispersion relation (14) can be rewritten frequencies versus o0. ARTICLE IN PRESS

606 R.H. Tarkhanyan et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 603–608 and using the parameter values given in Eq. (4d), we obtain spectrum, and low-frequency branch possess anomalous dispersion only if the condition 0 0 o ¼ 185:877GHz; o þ ¼ 185:971GHz; ð26aÞ $p 4o þ ð29Þ that is, 64.76 kGoH0 o64.80 kG. In spite of that, the ‘‘middle’’ field region should be divided in two parts is fulfilled (see Figs. 5b, c). Note that in part I of the ‘‘middle’’ field range, there is only one branch with anomalous dispersion for 0 0 ðIÞ o ro0 oO ð27aÞ op oO (Fig. 4a) and two such branches forop 4o+ (Fig. 4b). and Note also that unlike the cases shown in Figs. 3 and 5, the wave numbers for the low-frequency branch in Fig. 4a, b are restricted: 0 0 ðIIÞ O ro0 ro þ ; ð27bÞ kok , where where ðo2 o2Þðo2 þ2o o l d1 o2Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ ¼ ð ~ Þ1=2 ¼ L 0 T A S 2 0 k $p e0m0er ; r 2 : 0 2 2 2 2 2 2 2 2 ðoL o0Þðo0 oT Þ4oAoS l1l2d sin b O ¼ oT þ2oAoSl2=d ð28Þ ð30Þ is the cutoff frequency at zero magnetic field. An estimation for 0 0 However, the last statement is only true if o0 4o.At MnF2 gives O ¼ 185:891GHz. In part I, the wave has three cutoff ¼ 0 , the low-frequency branch exists for all possible wave frequencies while in part II there are only two such frequencies. o0 o numbers and approaches asymptotically to zero when k-N. In Fig. 3, the dispersion curves in the ‘‘low’’ field regime are Finally, ‘‘high’’ field region should be divided in two subregions shown schematically for three different values of the plasma 0 too: o þ oo0 roL and o04oL, where frequency: $p oo (Fig.3a), o o$poO (Fig.3b) and o+ o$p oO+ (Fig.3c). The wave exists, as in the absence of the 2 1=2 oL ¼ðoT þ2oAoSÞ ð31Þ magnetic field, in three frequency regions, the limits of which are different for these three cases and also depend essentially on the is the frequency of the longitudinal optical magnons at zero magnetic field. Note that the dispersion of the high-frequency magnetic field. Substituting in Eq. (31)oT =185.874 GHz, branch is normal in all these three cases while that for the low- oA =10.906 GHz and oS=1.722 GHz (MnF2), we obtain frequency branch is anomalous ð@o=@ko0Þ in cases (b) and (c). As oL =185.975 GHz. The wave has two cutoff frequencies in the first to the middle branch, it is normal in cases (a), (b) and anomalous part and three in the second one. In Fig. 6, the dispersion curves in case (c). It is evident that left-handed behavior of the medium are shown in both these subregions for the case when can only be expected in the frequency ranges corresponding to the o+ o$poO + . Note that in the frequency range o+ ooo$p branches with anomalous dispersion. there is only one branch with anomalous dispersion, and low- In Figs. 4 and 5, the dispersion curves of the TE wave are shown frequency branch in the case of Fig. 6a exists only if the wave in the ‘‘middle’’ field regime for different values of the plasma number k4k , where k is given by Eq. (30). frequency in part I [see Eq. (27a), Fig. 4] and part II [Eq. (27b), The estimated values of the resonance, cutoff and cyclotron Fig. 5). In the latter case, there are only two branches in the frequencies are shown in Table 1.

ω Ω Ω Ω+ + + ϖ p ω + Ω Ω − Ω− − ϖ p ω− ϖ p ~ ck ε )( − 2/1

Fig. 3. Dispersion curves in ‘‘low’’ field regime for different values of the effective plasma frequency. (a) $p oo; (b) o o$poO; (c) o + o$p oO + .

ω ϖ p Ω+ ϖ p ω +

Ω− ϖ p

ρϖ ρϖ p ρϖ p p ~ ck ε )( − 2/1

Fig. 4. Dispersion curves in part I of the ‘‘middle’’ field regime. (a) $p oO; (b) o+ o$p oO+ ; (c) $p 4O + . ARTICLE IN PRESS

R.H. Tarkhanyan et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 603–608 607

ω ϖp Ω+

ϖp ω +

ϖ p

~ ck ε )( − 2/1

Fig. 5. Dispersion curves in part II of the ‘‘middle’’ field regime. (a) $p oo+ ; (b) o+ o$poO +; (c) $p4O + .

ω 4. Necessary conditions for the negative refraction of the TE Ω+ waves. Phase and group refractive indices Ω+ ϖ p Suppose now that TE type polarized plane wave

E,Hexp[i(k0xx+k0yyot)] is incident from the vacuum onto the ω+ ω+ interface of the medium. For a given value a of the angle of incidence, k0x =k0 sin a and k0y=k0 cos a. Inside the medium, the ω− refracted wave propagates as mixed magnon–plasmon ω− Ω of the same polarization. − Using Eqs. (19) and (20), for the dot product of the wave and ~ −1/2 Poynting vectors we obtain p ρϖ ck ε)( 2 2 ðm11kx þm22ky Þ 2 1 2 kS ¼ 9E9 ¼ oe0e339E9 : ð32Þ 2om0m22mv 2 Fig. 6. Dispersion curves in different parts of ‘‘high’’ field regime. (a) o0 o o . þ o 0 r L Taking into account that the flux for the refracted wave (b) o04oL. is always directed away from the interface (sy40), we conclude that the wave is backward with respect to the interface (i.e., the Table 1 phase velocity is directed to the interface) only if the condition

The estimated values of the resonance(o7), cutoff (O 7) and cyclotron (oc) m o0 ð33aÞ frequencies for SL MnF2 /CdTe at six different values of the field H0 (two values for v each field region). The effective mass of the conducting electrons in CdTe is fulfilled. At this end, the medium is left-handed with respect to m=0.11m ; eN=10. 0 the wave (i.e., kSo0), if H (kG s) o (GHz) o (GHz) o (GHz) O (GHz) O (GHz) o (GHz) 0 0 + + c e33 o0; ð33bÞ

0.3 0.861 185.063 186.786 185.072 186.794 7.63 and the negative refraction occurs if 30 86.1 99.822 272.025 99.833 272.033 763.47 64.767 185.88 – 371.804 0.355 371.813 1648 m11m22 40: ð33cÞ 64.780 185.92 – 371.844 – 371.852 1648.59 Except that, the condition of existence for the refracted wave 64.799 185.973 0.764 371.911 – 371.919 1649 (k2 40) must be fulfilled 65.157 187 1.039 372.925 0.969 372.933 1658.19 y m11 2 e33mv 4 sin a: ð33dÞ m22 We see that all the values of the characteristic parameters are All these four conditions (33a–d) can only be fulfilled much far from the cyclotron frequency. Note also that the simultaneously within the frequency bands of anomalous disper- interesting frequency regions with anomalous dispersion in sion. The corresponding situation in Ref. [13] has been termed as Figs. 4c and 5c may be difficult to realize experimentally if the the case of absolute left-handed medium. The mutual orientation of the wave and Poynting vectors in this case is illustrated effective plasma frequency $p is below O +. Using the estimates schematically in Fig. 7. The angles of refraction for the wave and for O+ in Table 1 and substituting l2/l1=0.2, eN=10(CdTe) and Poynting vectors are given by Snell’s law sinb ¼ n1 sina and eA =5.5(MnF2) in the formula for $p [see Eq. (23)], we obtain that p 1 the situations shown in Figs. 4c and 5c( 4O ) can only be sing ¼ ng sina, respectively, where $p + qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi accessible if the density of the free charge carriers in semicon- 2 n ¼ e m þð1 m =m Þsin a ð34Þ ductor layers N45.3 1019 m 3. Moreover, the frequency regions p 33 v 11 22 shown in Fig. 6, where o+ o$p oO+ , can only be realized is the phase refractive index and experimentally if 5.332 1019 oNo5.333 1019(m3). However, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 in most of other cases shown in Figs. 3–6, the value of the plasma ng ¼ e33mvðm22=m11Þ þð1 m22=m11Þsin a ð35Þ frequency can be high enough and the corresponding frequency is the group refractive index. The angle f between the wave and regions are perfectly accessible. Taking into account that the Poynting vectors is given by density of the free charge carriers depends strongly on the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi temperature, as well as on the type of impurities and the doping 2 2 2 2 2 sin aþ ðnp sin aÞðng sin aÞ level and therefore can be varied in the wide interval, we can be cosj ¼ ð36Þ n n sure that the necessary values of the effective plasma frequency p g are fully reachable. and is always obtuse if the conditions (33a–d) are fulfilled. ARTICLE IN PRESS

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frequency ranges and different regimes of the applied field. In KR general, the behavior of the wave depends on the sign of the Voigt permeability, diagonal components of the effective permeability tensor in the plane perpendicular to the applied field and effective

permittivity e33 along the field direction. It is shown that the wave K 0 α α is backward and can be refracted negatively if the following conditions are fulfilled simultaneously: (a) e33 should be negative; (b) instead of the scalar permeability in isotropic case, the x effective Voigt permeability must be negative; (c) the additional K β conditions (33c,d) should be fulfilled too. We have shown also γ that the structure with negative permeability along the normal to the interface can achieve negative refraction with respect to the TE wave even when the wave is forward. The main properties of the wave, in particular, the number of the frequency bands of propagation, their limits and width, as well S as the number of the branches with anomalous dispersion are quite different in various ranges of the applied magnetic field and y therefore can be changed using the latter as a tuning parameter. It Fig. 7. Mutual orientation of the wave and Poynting vectors in the negatively is remarkable that the limits of the frequency regions of the refracted TE wave. K0 and KR are the wave vectors for the incident and reflected anomalous refraction are very sensitive not only to the applied waves, respectively. field but also to the effective plasma frequency. This fact opens possibility for control of the regions using the samples with It is remarkable that the TE-wave can be refracted negatively even different values of the free charge carrier’s density in the when it is forward (with respect to the interface) and the wave and semiconductor layers. Poynting vectors make an acute angle (kS40).Suchasituationis only possible if the following conditions are fulfilled simultaneously: Acknowledgement mv 40; e33 40; m11 40; m22 o0; ð37Þ Indeed, using Eq. (19), it is easily to see that in this case The authors would like to thank Professor C. Soukoulis for his the Poynting vector of the refracted wave is in the same side of encouragement and useful discussions. the normal to the interface as that in the incident wave, that is, the negative refraction of the TE waves occurs without backward waves. References

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