ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 322 (2010) 603–608 Contents lists available at ScienceDirect Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm Regular Submission Influence of external magnetic field on magnon–plasmon polaritons in negative-index antiferromagnet–semiconductor 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 phase 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]. waves (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 insulator (for example, Cd1ÀxMnxTe) 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, photons coupled with antiferromagnetic magnons and properties of the waves as well as on the number of the spectral magnetoplasmons. As it is well known [15,16], this coupling 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 matter & 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 Àiea e22 0 ; m^ ef ¼ Àima 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 À1 0 e ¼ l e þl e 1 À 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 spin-flop field and the medium is lossless. Also we assume that: À1 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 À m2maÞ=d; ma ¼ l2m2=ðl2 þl1m1Þ; axis’ type magnetic anisotropy, so as the saturation magnetization À1 À1 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 wave nonmagnetic semiconductor layer of thickness l can be described 1 E,Hexp[i(krÀot)] 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 ÀoD: ð7Þ e^ ¼ @ Àie2 e1 0 A; ð1Þ Eliminating the magnetic field H ¼ðom m^ ÞÀ1k  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^ À l e^ ÞE ¼ 0; ð8Þ e ¼ e 1 À p ; e ¼ p ; e ¼ e 1 À p : m ef 1 1 o2 À o2 2 oðo2 À o2Þ 3 1 o2 c c where n=ck/o is the refractive index, ð2Þ l m m À m2; ð9aÞ 2 1/2 m 11 22 a op=(Ne /me0eN) is the plasma 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 electrons, respectively, B0 =m0H0, eN is m s2 þl s2 ðl s s þim s2Þs ðm s À 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 Àl s s þim s2 m s2 þl s2 Às 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 Àszðm sx þim syÞszðm sy À 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 @ Àim2 m1 0 A; ð3Þ 001 : 2 : Det n Zij À 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 À Þ2 4 2 2 oT o o0 oT o o0 An À lmBn þe33lelm ¼ 0; ð11Þ "# where 1 1 ¼ À 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 À ea ; oT ðoþo0Þ oT ðo À o0Þ ð12aÞ o0 =gH0, oA=gHA, oS =gMS, g is the gyromagnetic ratio, 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.