^{†}Corresponding author. Email: xzwang696@126.com
^{*}Project supported by the National Natural Science Foundation of China (Grant No. 11074061).
We present a modified method to solve the surface plasmons (SPs) of semiinfinite metal/dielectric superlattices and predicted new SP modes in physics. We find that four dispersionequation sets and all possible SP modes are determined by them. Our analysis and numerical calculations indicate that besides the SP mode obtained in the original theory, the other two SP modes are predicted, which have either a positive group velocity or a negative group velocity. We also point out the possible defect in the previous theoretical method in accordance to the linear algebra principle.
Metallic superlattices (SLs) composed of alternating layers of two different metals first appeared in the late 1980s, ^{[1– 3]} and scientists found their unique magnetic property and conductivity (the interlayer antiferromagneticexchange coupling and giant magnetoresistance).^{[2, 4– 6]} These findings have opened up a new technological field and laid the foundation of spin electronics. Metal photonic crystals or metamaterials began to attract the attention of scientists in the 1990s.^{[7]} Their surprising optical properties can lead to a large number of technological applications in optical– electron and optical– detection fields, so these metal structures have been received increasing attention.^{[8]} It has been wellknown that these properties mainly originate from the interaction between incident electromagnetic waves and plasmons in these structures.^{[9– 11]} Onedimensional photonic crystals, or say the superlattices, composed of alternating layers of metals and dielectrics are the simplest example of these structures. It is just a metallayer array in air when the dielectric layers are of air. The earliest research about plasmons of the metal/dielectric SLs was done by Mills and Camley.^{[12, 13]} For the semiinfinite metallayer arrays, the conclusion obtained by them is that the surface plasmons (SPs) exist only if the metal layers are thicker than air spacers and the SP frequency is the same as that of the semiinfinite metal. After which, the plasmons in the finitethickness SL were discussed^{[14]} and then the magnetoplasmons and plasmon polaritons in the same structure were investigated.^{[15– 17]} However, we find that there is a defect in the previous SP theory for the semiinfinite SLs, and so a rediscussion is necessary. In fact, the similar problem also exists in the previous theory of the surface plasmon polaritons in this SL. In this paper, a modified theory is presented to resolve the SPs of the semiinfinite metal/dielectric SLs and new SP modes are predicted.
The SL structure and the coordinate system are shown in Fig. 1, where d_{m} and d_{d} are the thicknesses of metal and dielectric layers, and the SL period L = d_{m} + d_{d}. We propose that the layers are thick enough to be characterized by bulk dielectric function ε _{m} and bulk dielectric constant ε _{d}, respectively, ε _{c} is the dielectric constant in the upper half space.
In this paper, we discuss only the SPs and do not discuss the SP polaritons, so the retarded term in the Maxwell equations is ignored. As a result, the plasmonwave equation is ∇ · D = 0, where the electric displacement D = ε E and electric field E = − ∇ φ with the electrostatic potential φ . We propose that the SP propagates along the y axis and exponentially attenuates along the x axis. According to the wave equation, we directly write the electrostatic potential in the upper semispace and the first SL period as follows:
where the indexes m, d, and c indicate the metal and dielectric layers, and upper medium, respectively, k is the wavenumber along the y axis, A_{j} and B_{j} (j = 0, + , − ) denote the amplitudes of the electrostatic potential. We should note that the potential in the second period is equal to the product of the firstperiod potential and e^{− Γ L}. Using the boundary conditions of φ and D_{x} continuous at the surface, the first and second interfaces, we find the following relations among the potential amplitudes:
where Γ is the attenuation coefficient of the SPs and must be real and positive when the metal dielectric function is considered as a real number. The six equations make up a set of homogeneous linear equations of the amplitudes. It should be mentioned that this system contains only five amplitudes (or five linear unknowns). Indeed, for calculating the SP dispersion curves (ω ∝ k), there are six unknowns to be eliminated, but Γ is not a linear unknown. Thus the SP dispersion relations must be carefully derived to guarantee the existence of the SP wave solutions. In order to simply and intuitively obtain the features of the SPs and compare with the given results, ^{[12]} we set ε _{c} = ε _{d} = 1 and then the SL is just a metallayer array in air.
The existence of nonzero solutions of unknowns A_{± } and B_{± } first require that the determinant of their coefficient matrix must be equal to zero, so according to Eqs. (3) and (4), we find a necessary dispersion equation
but it is not sufficient for solving the dispersion relation (ω ∝ k). The most critical question is how to find an additional equation to determine the attenuation factor Γ in Eq. (5). Although the above equation system include six unknowns (A_{0}, A_{± }, B_{± }, and Γ ), Γ is not a linear unknown and is included in the coefficients of A_{± }. Consequently, according to the linearalgebra principle, we can use only five equations in the system to determine the expression of Γ . According to Eq. (2), we obtain
Selecting three equations from Eqs. (3) and (4), we have four selections. The first selection, using Eqs. (3) and (4a), leads to
and the second choice is to take Eqs. (3) and (4b), we obtain
We obtain the third expression from Eqs. (3a) and (4), that is
and finally we find the forth expression from Eqs. (3b) and (4) that
Now, we discuss the above four equations. First, we consider two important limit cases, d_{d} → 0 and d_{d} → ∞ . In the first case, the array becomes a semiinfinite bulk metal. It is obvious that equations (7a)– (7d) all can be changed into
since equation (5) leads to Γ = k in this case. It is just the SP dispersion relation of the bulk metals. In the second case, equations (7a)– (7d) directly lead to
or equivalently
which is just the dispersion relation of the SPs in an isolated metallic film. Therefore, we conclude that our theory is available in the limit cases.
In general case, we find that equation (8) can always satisfy Eqs. (5) and (7) for Γ = k(d_{m} − d_{d})/L simultaneously. As a result, in the condition of d_{m} > d_{d}, there is such a SP mode the same as that of the bulk metals, which is a given SP mode predicted in the previous work.^{[12]} We study the four dispersion equation sets, composed of Eqs. (5) and (7a)– (7d), respectively. All of the possible SP modes of the array are determined by them. Equation (7) cannot be simultaneous, otherwise it will be inconsistent with the linear algebra principle.
Next, we will discuss the previous theory.^{[12]} Although the authors corrected some shortcomings, ^{[13]} there is still a defect. Their equations (2.35) and (2.36) in Ref. [12] are the dispersion relations of the SPs. If we imagine that the dielectric layers in and medium above the SL are the same, equation (2.35) has no significance both in mathematics and physics, successively, the final dispersion equation (2.38) is also questionable. In addition, for the metallayer arrays (or ε _{B} = ε _{c} = 1), it is difficult to understand why there is only an SP mode the same as that of the bulk metals. The SPs of the array should originate from the coupling of SPs in metal layers. As we know, the SPs of a single metal film has two modes changing with wave number, so we believe that there should be SP modes changing with wave number in the array.
The bulkplasmon dispersion relation can be obtained by the replacement Γ → iQ in Eq. (5), where Q is the Bloch wave vector. The dispersion relation is written as
which is useful for discussing the SPs of the array.
In this section, we solve the four dispersionequation sets numerically. Though the dielectric function of metals is usually expressed as
which seems better for actual metals.^{[18, 19]} The physical parameters are taken as ω _{p} = 1.419 × 10^{16} rad/s (75280 cm^{− 1}), ε _{∞ } = 4.017, and ε _{0} = 4.896, which are closed to those of sliver.^{[18, 19]} The array period is fixed at L = 1.0, equivalent to about 133 nm, k in the dispersion relations always appears in the style kd_{m} or kd_{d}, so the increase of k should cause the same effect as that resulting from the enlargement of L. The wave number and frequency are measured in ω _{p}.
In calculations, besides the given SP mode existing under the condition of d_{m} > d_{d}, we find that the four equation sets always offer two additional SP modes. For example, there are three SP modes for d_{m} ≥ d_{d}, as illustrated in Figs. 2(a) and 2(b). The two new SP modes seen here are situated above and below the given SP mode, respectively. The higherfrequency and lowerfrequency modes have a negative group velocity and a positive group velocity, respectively, whose frequencies approach to 0.418, that of the bulkmetal SP, in the largewave number limit. For d_{m} < d_{d}, only two additional SP modes are shown in Figs. 2(c) and 2(d), where the given PS mode disappears. Comparing the curves obtained in the cases of d_{m} > d_{d} and d_{m} < d_{d}, we find that the two additional modes possess the invariance as the air and metallayer thicknesses are exchanged. For example, the two modes in the cases of d_{m} = 0.2 and d_{m} = 0.8 are the same.
We also compare the SP modes of the array with those of the single metal layer (metal film), as illustrated in Fig. 3. It is indicated that the SP higher and lowerfrequency modes in the SL are closed to the top and bottom SP modes (ω _{± }) of the film, respectively. From this figure, we estimate that the lowfrequency mode comes from the coupling of the bottom surface modes in metallic layers and the highfrequency mode comes from the coupling of the top surface modes.
Finally, we examine the relation between the SPs and bulk plasmons. Their dispersion properties are illustrated in Fig. 4 with the same parameters as those in Fig. 3. We see that the two additional modes of the SPs both appear in the forbiddenband of bulk plasmons, extending out from the boundaries of the bulk continua.
We rediscussed the surface plasmons of the metallayer array and corrected the given theory and obtained new dispersion equations and numerical results. The four simultaneous dispersion equation sets, composed of Eqs. (5) and (7a)– (7d), respectively, determine the dispersion properties of the surface plasmons. Unlike the results given in the previous work, where only the surface plasmon mode exists in the condition of d_{m} > d_{d} and its frequency is equal to that of the surface plasmons of the bulk metal, in this paper the surface plasmons of the array possesses two additional modes. The three surface plasmon modes exist in the case of d_{m} > d_{d}, and the two modes appear in the situation of d_{m} < d_{d}. The surface plasmons of the array are a kind of collective excitation and originate from the coupling of surface plasmons in adjacent metal layers in the array. The additional surface plasmon modes have the invariability as the metallayer and the airlayer thicknesses d_{m} and d_{d} are exchanged.
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