A model for confined Tamm plasmon devices

It is shown that cavities formed between a multilayer quarter-wave Bragg reflector and a metal mirror which support Tamm plasmons can be modelled by using a hard-mirror approximation including appropriate penetration depths into the mirrors. Results from this model are in excellent agreement with those found by numerical methods. In addition Tamm modes that are laterally confined by the presence of a metallic disc deposited on the Bragg reflector can be described by the effective index model that is commonly used for vertical-cavity surface-emitting lasers (VCSELs). This enables the lateral modes confined by a circular disc to be found from conventional weakly-guiding waveguide theory similar to that used for optical fibres. The resonant wavelengths of these linearly-polarised (LP) guided modes are calculated as functions of disc diameter and other parameters.


INTRODUCTION
Electromagnetic modes bounded by a distributed Bragg reflector (DBR) and a metal mirror are termed Tamm plasmon-polaritons [1], or simply Tamm plasmons, by analogy to the electronic states that can appear at the surface of a crystal [2]. In the present contribution, the term "Tamm modes" will be used. Confined Tamm modes have been demonstrated with lateral confinement limited to the size of a metallic disk deposited on the DBR [3]. Lasing has been reported with Tamm modes for both large-area metal layers [4] and with confinement provided by micrometerscale metal disks [5]. Replacing the disks by microrectangles with an aspect ratio of 2 has resulted in polarization-controlled Tamm lasers [6]. Tamm single-photon sources have been demonstrated with InGaAs/GaAs quantum dots (QDs) emitting at 910 nm [7] and InP/GaInP QDs emitting at 656 nm [8]. Other proposed device applications of Tamm modes include all-optical bistable logic [9], multichannel filters [10], and novel forms of sensors [11,12].
Analysis of one-dimensional Tamm modes is conventionally performed using the transfer matrix method (TMM; see, for example Refs. [1,4,9,[12][13][14]). For three-dimensional confined Tamm modes, the modeling has used coupled wave analysis [5], the aperiodic Fourier modal method (a-FMM) [6], and the finite-difference time-domain (FDTD) method [3,14]. All these methods involve numerical computation and do not afford direct physical insight into modal behavior or identification of trends with variation of structural and material parameters. In the present contribution, drawing on concepts well established for modeling vertical-cavity surface-emitting lasers (VCSELs) and resonant-cavity light-emitting diodes (RCLEDs), we offer a novel approach to modeling one-dimensional and threedimensional Tamm modes that includes all the relevant physics and offers a simpler approach to studying trends.
In the next section, the hard-mirror model for a onedimensional cavity is applied to study Tamm modes with the aid of the concept of the penetration depth of the field into the DBR [15]. The following section extends this approach to describe three-dimensional confined Tamm plasmon modes by using the effective index model [16]. After that, a numerical example is given and compared with results from the literature where numerical methods have been used, and finally, we summarize our conclusions and the outlook for further applications of our approach.

ONE-DIMENSIONAL APPROXIMATION
A general one-dimensional structure to support Tamm modes is shown schematically in Fig. 1(a). It consists of a spacer layer of thickness L S and refractive index n S , between a multilayer quarter-wave Bragg reflector and a metal whose complex refractive index is n M ik M . Figure 1(b) shows the hard-mirror model for this cavity, allowing for penetration of the field into the DBR and the metal.
From standard VCSEL and RCLED cavity modeling (see, e.g., Refs. [17,18]), we approximate the confinement of the Tamm mode between the DBR and the metal layer as a cavity formed by two hard mirrors, each of which is positioned at a penetration depth into the appropriate medium. The penetration depths are denoted by L M and L BR , respectively, for the metal and Bragg reflector, as shown in Fig. 1 Rigorous expressions for the penetration depth L BR are derived in Ref. [15] in terms of refractive indices, number of layers in the DBR, and wavelength. The expression for L BR [see Eq. (7) below] is found from the phase delay τ m on reflection from an m-layer quarter-wave Bragg stack. A recursion relation for τ m is derived by adding a single quarter-wave layer to the top of an (m − 1)-layer stack whose delay is τ m−1 . Full details of the derivation are given in Ref. [15]. Simpler analytical expressions that are very accurate for high numbers of layers when the penetration length saturates are given in Ref. [19]. The penetration depth into the metal, L M , at the center (Bragg) angular frequency ω 0 of the DBR can be calculated from the phase change, β, on reflection [20]: where c is the speed of light. An explicit (but lengthy) expression for tan β is given in Ref. [21] in terms of refractive indices, thickness of the metal, and wavelength; this expression is cast in the present notation in Appendix A. In the limit of large layer thickness, this expression reduces to the simple form obtained from the phase change on reflection from an isolated metal surface [22]. Usually β lies in the second quadrant. The longitudinal modes of the structure occur at frequencies for which the round-trip phase is an integer multiple of 2π. Allowing also for the reflection delay τ from the Bragg mirror [15], the sum of phase terms in the cavity round trip gives where ω N is the resonant angular frequency of mode N 1, 2, 3, … and the average refractive index in the Bragg mirror is defined [19] as with n H ,L and d H ,L as the refractive indices and thicknesses of the high and low index layers in the Bragg mirror. The reason for the term (2N − 1) in Eq. (2) is that the combination of the spacer layer and the metal thickness should behave as a quarterwave layer [23]. The delay time τ is related to the penetration depth in the DBR by Using Eq. (4), Eq. (2) can be rewritten in the form Equation (5) is the cavity resonance condition in terms of the phase changes in the spacer and on reflection from the metal and the DBR. It shows explicitly that when the mode frequency occurs at the center frequency of the grating, there is no dependence on the penetration depth L BR . This is because in this case the phase on reflection from the grating is either zero or π. Equation (5) here is equivalent to Eq. (5) in Ref. [1], with the difference that in Ref. [1], the refractive index of the metal is given by the Drude model, and there is no spacer layer L S 0. From Eq. (5), it follows that the mode frequency ω N is given by From Ref. [15], the expression for L BR is where m is the number of layers in the DBR and a, p, q are refractive index ratios (low to high) of the three types of interfaces that characterize the mirror: a n LE n HE p n L n H q n LI n HI : The ratio a applies to the interface between the last DBR layer and the exit medium (subscript E), and the ratio q refers to the interface between the incident medium (subscript I ) and the first DBR layer. We note here that the simpler expressions for penetration depth [15,19] are not sufficiently accurate for the numerical example considered below because the index differences are too large.
In the numerical work below, the full expression [21] for the phase β will be used in Eq. (1) for the penetration depth of the metal in order to include the effect of a finite layer.

THREE-DIMENSIONAL APPROXIMATION
To model confinement in the lateral direction, we follow the effective index approach proposed in Ref. [16] and now widely used for VCSEL design. The essence of this method is that the lateral effective index profile is determined by the local changes of the Fabry-Perot cavity resonance wavelength. So, for a where n e is the effective index of the cavity given by the mean value of indices in the spacer and the Bragg mirror: with the total effective cavity length, L eff , defined as The resulting radial index profile is illustrated in Fig. 2(a), with a schematic of the physical structure in Fig. 2(b).
For the region outside that beneath the metal disk, the upper mirror of the effective Fabry-Perot cavity is formed by the semiconductor-air interface. Hence in this region, Eq. (5) is replaced by where ω N c is the mode frequency in this region. It follows that the quantity Δλ, required in Eq. (9), is given by Thus, solutions of Eqs. (5) and (12) can be used in Eqs. (9) and (13) to calculate the effective index profile in the radial direction.
The resulting circular waveguide of diameter d, core index n e , and core-cladding index difference Δn e ≪ n e supports modes that are, to a very good approximation, linearly polarized (LP) by analogy with the modes of weakly guiding optical fibers [24]. These LP modes are characterized by the index eigenvalues, n, of the wave equation such that their cutoff frequency is given by n n e − Δn e . These modes are characterised by two subscripts, one for each of the radial and azimuthal coordinates. We will use l as the azimuthal subscript and p as the radial subscript, so that the mode is labeled LP lp . Mode solver software for the LP modes is widely available for optical fibers and can simply be used for the corresponding confined Tamm modes with appropriate definitions of core and cladding indices, as described above.
Assuming the solutions for n have been found, the frequency of the confined mode, ω lp , can be calculated by using again the cavity resonance condition in the form: Equation (14) indicates that the confined Tamm resonant wavelength is directly proportional to the index eigenvalue for the corresponding LP mode.

NUMERICAL EXAMPLE
The method outlined above will be illustrated by applying it to the structure designed for Tamm modes in Ref. [14], which has a GaAs spacer, an AlAs/GaAs DBR with 35 layers, and a GaAs substrate. Hence the incident and exit media have the same (high) refractive index [25], and the first and last layers of the DBR have the same (low) index [26]. The metal is gold with n M 0.38 and k M 8.7 in the wavelength range of interest [27]. Numerical values of the semiconductor parameters are given in Table 1. The stopband of the Bragg mirror lies between 1205 and 1375 nm [14], so this defines the value of the Bragg frequency. Figure 3 shows the variation of phase with thickness of the gold layer calculated using the expression in Ref. [21].
Using Eq. (6) with N 1 for a 25 nm gold thickness and 75 nm spacer gives ω 1 1.4487 × 10 15 rad∕s, corresponding to a Tamm wavelength of 1301.1 nm. This compares well with the value of 1304 nm reported for this structure in Ref. [14], where the TMM was used. It is interesting also to estimate the Tamm resonance from the approximation used in Ref.  where the Drude model for the metal is used. Adapting the result from Ref. [1] to allow for the presence of a spacer layer gives where Δn n H − n L , ω p is the plasma frequency, and ε b is the background dielectric constant. Using our parameter values in Eq. (15) with a plasma energy of 8.9 eV [1] yields a frequency of 1.441 × 10 15 rad∕s, corresponding to a Tamm wavelength of 1308 nm. This level of agreement may be fortuitous, since the approximation in Eq. (15) takes no account of the gold thickness nor of the number of layers in the Bragg mirror, effectively assuming both these quantities are approaching infinity. Since the penetration depth into the DBR is approaching saturation for 35 layers, we do not expect additional layers to affect the resonant wavelength from Eq. (6) significantly. However, it is clear from Fig. 3 that the phase on reflection from the mirror is some way from saturation at a thickness of 25 nm, and hence, a change of the Tamm wavelength is expected, as we demonstrate below. Figure 4 shows the Tamm resonance wavelength as a function of (a) spacer thickness with the gold thickness constant at 25 nm, and (b) gold thickness with the spacer thickness kept constant at 75 nm. Results from the present model are compared with those from the TMM [14], and again the agreement is very good. The small departures between the two are attributed to inclusion of the dispersion of the refractive indices in the TMM.
Turning to the two-dimensional model, for the case of a 75 nm spacer and 25 nm of gold, in the region outside that beneath the metal disk, Eq. (12) is used with N 1 to calculate a wavelength of 1245.8 nm. Hence, we can use Eq. (9) with λ 1307.7 nm, Δλ 55.3 nm, and n e 3.170 [from Eq. (10)] to calculate the value of Δn e as 0.135. For these parameter values, the cutoff values (d c ) of the disk diameter for the LP modes can be calculated. Table 2 gives the values of d c for some lower-order LP modes using the effective index and index difference values calculated for this example. Each LP mode corresponds to hybrid (HE or EH), transverse electric (TE), or transverse magnetic (TM) modes, as indicated in Table 2. More details of these waveguide modes are given in many textbooks; see, for example, [28,29]. Mode profiles are illustrated in several good tutorial websites; see, for example, [30,31]. The number of lateral modes increases rapidly with disk diameter. Single-mode operation is only found for a diameter less than 1.1 μm.
The calculated variation of resonant wavelength with disk diameter, using Eq. (14) for the first 12 LP modes, is shown in Fig. 5. The trends here are similar to those measured and calculated in Ref. [3] (albeit expressed there in terms of photon energy rather than wavelength) for a somewhat similar structure.

CONCLUSION
We have shown that concepts widely used in design and analysis of VCSELs and RCLEDs can be successfully applied to model Tamm modes confined between a metal layer and a Bragg reflector. The hard-mirror approach, where an effective cavity is defined to include penetration depths of the optical wave into the metal and the Bragg mirror, successfully predicts the resonant wavelengths of one-dimensional Tamm modes. It is worth noting that this approach uses the calculated phase on reflection from the metal, using measured real and imaginary parts of the refractive index rather than invoking the Drude model. Three-dimensional Tamm modes can be modeled using the effective index approach, where the radial effective index profile is found from the difference in resonant wavelengths between the one-dimensional mode beneath a circular metal disk and that outside this region. This enables the number of transverse modes and their wavelengths to be determined as functions of disk diameter and other parameters, a task which is not straightforward with purely numerical computational approaches. It is clear that the methods proposed here can be applied to a wider range of structures than those currently of interest. While attention has been confined here to circular disks, the method proposed could just as well be applied to disks of other shapes, e.g., elliptical, racetrack, or rectangular, since the basic principle of decoupling the vertical field component from the lateral components applies independently of the geometry. Thus, for example, the method could be used to study polarization properties of noncircularly symmetric disks, such as the microrectangles used in Ref. [6], or more complicated shapes of the metal layer. In this context, it is worth noting that approximate but accurate solutions are available for some shapes, such as elliptical [32] or rectangular [33]. Further developments could include analysis of the effects of metallic layers and/or spacers of differing thicknesses in the lateral directions, thus opening up the prospect of integrating novel Tamm modal devices on a single Bragg mirror in order to produce multifunctional components for a variety of potential applications.