High sensitivity of the Attenuated Total Reflectance technique for exciting transverse magnetic surface plasmons in free-standing doped graphene is reported; complete agreement with the electromagnetic dispersion relation is numerically demonstrated in the terahertz regime. By reducing the air gap between prism and graphene in the Otto configuration we found that the surface plasmon excitation is weakened, but interference effects arise producing perfect absorption. At 5 THz two dips of zero-reflection were found, one of them with residual plasmonic contribution. Consequently, the reflection can be suppressed by changing the separation between prism and graphene; it is not needed to modify the graphene doping level. Conditions for destructive interference leading to complete absorption are presented and a particular behavior of the evanescent magnetic fields just at perfect absorption is reported

## I. INTRODUCTION

During the past few years doped graphene has attracted a great deal of research interest. The reason is the expectation of graphene as a promising new material for opto-electronic applications such as photovoltaic devices,^{1} photodetectors,^{2} optical sensing^{3} and metamaterials.^{4} Graphene, a flat monolayer composed solely of carbon atoms arranged in a hexagonal honeycomb lattice, is a gapless semiconductor with linear energy dispersion relation. Near the Fermi energy, at the cone linking the valence and conduction bands, the electronic properties of graphene are described by Dirac theory where the phenomenon of massless electrons results relevant.^{5} This extraordinary property gives place to a particular graphene optical conductivity σ(ω) = σ_{r} + *i*σ_{i}, with *intraband* and *interband* contributions.^{6–9} The interesting point is that the imaginary conductivity σ_{i} of doped graphene (graphene with chemical potential μ ≠ 0) can take positive or negative values leading to the occurrence of transverse magnetic (TM) and transverse electric (TE) surface plasmons (SPs).^{10,11} Experimentally, TM SPs have been detected in the terahertz regime in graphene micro-ribbon arrays,^{12} in graphene monolayers by IR tip-based nanoscopy^{13} and nano-imaging,^{14} and in graphene monolayers resting on subwavelength gratings.^{15,16} On the other hand, TE SPs in graphene have been only predicted theoretically.^{17}

For plasmonics, graphene is a suitable alternative to noble metals due to the possibility of tuning its SP spectrum by chemical or electrical doping. By use of the electrolyte gate technique, for example, carrier densities as high as *n* = 4 × 10^{14}cm^{−2} have been obtained for electrons and holes.^{18} Thus, Fermi energy as high as *E*_{F} = 2.3 eV is now available for experimental performance [*E*_{F} = ℏν_{F}(*n*π)^{1/2} where ν_{F} = 1 × 10^{6} m/s is the Fermi velocity of graphene]. Graphene can also be doped *in situ*^{12} and even by sections; novel graphene-based *flatland* metamaterials of patterned conductivities obtained by electrical gating were proposed recently in the IR regime.^{19} All these efforts to control the doping levels of graphene are made essentially to move the *transition energy* ℏω = 2*E*_{F} to frequency regions of interest. Aside from other physical properties, the threshold energy separates the region where SPs can exist. At energies ℏω > 2*E*_{F} the universal absorption of 2.3% of graphene is observed. SPs exist in graphene only for ℏω < 2*E*_{F}. In particular, in the THz regime they can be excited only for samples of high doping level, with a carrier density of order of 10^{12} cm^{−2} or higher. An important theoretical result establishes that TE and TM SPs in graphene cannot coexist in the same range of frequencies; the confinement of fields associated to the charge vibration requires negative (positive) the imaginary σ_{i} for TE (TM) SPs.

It is well-known that SPs are characterized by wavelengths lower than the wavelengths of light in vacuum. Their direct excitation by photons is forbidden due to the mismatch between their wave vectors. Nonconventional spectroscopies as infrared nanoscopy and scattering-type SNOM that include a sharp tip as scatter element have been implemented to increase the wave vector of photons to facilitate the coupling in the mid-IR region.^{13,14} With the same purpose, other techniques employ diffractive gratings of lattice period *a*;^{15,16} the wave vector *k*_{SP} of SPs is reached by selecting the wavelength λ of the incident radiation, angle of incidence θ_{i} and order of diffraction *p*: *k*_{SP} = (2π/λ)sin θ_{i} + 2π*p*/*a*. Furthermore, several theoretical works have argued that it is feasible to excite SPs in graphene by use of the Attenuated Total Reflectance (ATR) technique, where a prism of high refractive index n_{p} is used to increase the wave vector of the incident radiation. Recently, SPs in doped monolayer (and few-layer) graphene mounted on a dielectric slab were studied by ATR method in the THz region.^{20} Also, ATR graphene-based structures with SPs tunable by an applied gate voltage have been proposed as efficient polarizers^{21} and switches.^{22} Further, theoretical ATR studies have demonstrated the existence of a cutoff thickness of the prism-graphene separation related to the possible excitation of TE SPs in graphene.^{23} In one way or another, the optical response of all these ATR graphene-based systems depends on the thickness of the air gap *d* that separates the base of the prism from the graphene sheet. There exists an implicit phenomenon of interference that makes the optical response sensitive to *d*.

ATR graphene-based systems can also be used as efficient components for enhancing optical absorption at specific wavelengths; in fact, the excitation of SPs is not required for this purpose.^{24} The phenomenon, proved in the visible frequency region, has to do with the graphene losses and interference effects suffered by the waves in the ATR layered configuration. In addition, extraordinary ability of graphene to harvest light making use of plasmonic resonances was recently reported; it has been predicted that, under condition of total internal reflection (thus, evanescent fields reaching the sample), extraordinary absorption is displayed by a planar periodic array of graphene nanodisks.^{25} Perfect absorption at specific wavelengths is a desirable effect in the technology of photodetectors and spectrometers involving ultra-thin materials. Thus, the ATR graphene-based geometry could be and important piece for improving these devices which may find applications in the high speed optical communications, interconnects, terahertz detection, etc.

In this paper we describe theoretically the excitation of TM surface plasmons in free-standing graphene by the ATR technique in the THz region. We shall demonstrate that once the SP is excited it remains stable within a short range of variations of the air gap *d*; *d* is the distance between prism and graphene. Then, for a fixed frequency, we show that both angle of incidence and depth of the reflection dips are strongly sensitive to the variations of *d.* In fact, two dips of zero reflectance are found at 5 THz when a prism of refractive index n_{p} = 4 is used for the ATR simulations. With our numerical results we establish that perfect absorption can exist with or without SP assistance. The precise separation *d* between prism and graphene that produces the required destructive interference is obtained from the reflection coefficient which involves the optical conductivity of doped graphene.

## II. ATR GRAPHENE-BASED CONFIGURATION AND BASIC EQUATIONS

In Fig. 1 we show the layered system under study. It consists of a graphene sheet bounded by air layers of thicknesses *d* and *l.* The graphene and the air layers are sandwiched between two dielectric, homogeneous media of dielectric constants ε_{p} and ε_{t}. Note that the Otto configuration for graphene on a substrate of dielectric constant ε_{t} is obtained when *l* = 0. Also, by choosing ε_{t} = 1 the SP excitation in a free-standing graphene can be studied. With this layered array we can also study the energy leakage through the prism by doing ε_{t} = ε_{p} and selecting properly the values of *d* and *l*. For calculations we employ the graphene conductivity derived within the random-phase approximation at zero temperature.^{26,27} It is written as

where*τ* is the phenomenological relaxation time, *e* is the charge of an electron, ℏ is the reduced Planck's constant and *μ* is the chemical potential. The first (second) term in this equation corresponds to the intraband (interband) contribution. For high doped graphene, in the limit of low energies (ℏω ≪ μ) Eq. (1) reduces to the Drude-like formula

The relaxation time *τ* takes into account losses due to electron-impurity, electron-defect, and electron-phonon scattering. It takes the value τ = 6.4 × 10^{−13}s when the electron mobility and doping level are μ_{e} = 10000 cm^{2}/Vs and *μ* = 0.64 eV, respectively.^{11} For our calculations with doping levels of *μ* = 0.8 eV and *μ* = 1.2 eV we use Eq. (2) with τ = 1 ps. We shall see that the angle of incidence of the SP resonance moves slightly when τ changes one order of magnitude.

In treating graphene as a component of a layered structure for optical applications there are two approximations that, at least in the THz region where we have proof, lead practically to the same results. On the one side, graphene can be considered as a zero thickness layer characterized by the two dimensional conductivity. On the other hand, graphene can be seen as a thin layer of effective dielectric constant ε_{g} = 1 + *i*σ/ωε_{0}*t*_{g}, where *t*_{g} = 0.5 nm is the thickness of the layer.^{28} The two approximations differ from the most accepted thickness of the monolayer graphene that is *t*_{g} = 0.345 nm. The wavelengths in the THz region are four orders of magnitude larger than *t*_{g}. Therefore, for our purposes, the use of one or the other approximation is only for numerical convenience and the same results are expected.

In order to obtain the reflection spectra of TM waves we need to resolve the system of equations that results from the ordinary boundary conditions that the fields satisfy at each interface. They are eight coupled equations when graphene is represented by a thin homogeneous film or six equations when graphene is a layer of zero thickness (See Fig. 1). We shall proceed following the second of these approaches. The two equations at *z* = *d* are the relevant equations because they include the optical conductivity which is driving the main physical properties of the system. For TM waves of fields |$\vec E = ( {E_x,0,E_z } )$|$E\u20d7=(Ex,0,Ez)$ and |$\vec H = ( {0,H_y,0} )$|$H\u20d7=(0,Hy,0)$, the equations representing the continuity of the tangential electric field and the jump of the tangential magnetic field through the graphene sheet are

where the sign > (<) means positive (negative) *z* direction. The wave vectors correspond to fields in air: |$q_{z2} = q_{z3} = \sqrt {q_0^2 - q_x^2 }$|$qz2=qz3=q02\u2212qx2$ with *q*_{0} = ω/*c* and the wave vector parallel to the interfaces is defined by the incident wave in the prism; it takes the form *q*_{x} = *q*_{0}n_{p}sin θ_{i} where |${\rm n}_{\rm p} = \sqrt {\varepsilon _{\rm p} }$|$np=\u025bp$. In this relations ε_{p} and θ_{i} are the prism dielectric constant and angle of incidence, respectively. We are also using the speed of light in vacuum |$c = {1 \mathord{/ {\vphantom {1 {\sqrt {\varepsilon _0 \mu _0 } }}}\kern-\nulldelimiterspace} {\sqrt {\varepsilon _0 \mu _0 } }}$|$c=1/\u025b0\mu 0$. Equation (4) results from the Ampere equation |$\vec \nabla \times \vec H = \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over J}$|$\u2207\u20d7\xd7H\u20d7=J\u21c0$ whose integration along a rectangular circuit of transverse section Δ*y*Δ*z*, perpendicular to the axis *x* and the sides of length Δ*y* in media 2 and 3, gives place to the relation *H*_{2y} − *H*_{3y} = σ*E*_{2x}, when the limit Δ*z* → 0 is taken. The complete system of six equations (not presented here) is obtained when we apply the boundary conditions at *z* = 0 and *z* = *d* + *l*.

By following this criterion for writing the boundary conditions involving the graphene sheet, we can obtain the dispersion relation of the TM surface plasmons. Assuming that ε_{p} = ε_{t} = ε_{0} we can write the next two equations for the free-standing graphene monolayer:

Then, by dropping the incident field and taking into account that |$q_{z1} = q_{z2} = \sqrt {q_0^2 - q_x^2 }$|$qz1=qz2=q02\u2212qx2$, we found that the resonant TM modes satisfies the equation

In this equation the solutions with evanescent fields on the perpendicular direction to the graphene sheet require *q*_{x} > *q*_{0}. On the other hand, by doing *l* = 0 in Fig. 1 we can obtain the reflection coefficient for the Otto geometry with graphene mounted on a dielectric substrate of dielectric constant ε_{t}. In terms of the partial reflection coefficients

where we are using ε_{p} = ε_{0}ε_{pr} and ε_{t} = ε_{0}ε_{tr}, the reflection coefficient for the Otto configuration takes the form

where φ = *q*_{z2}*d*.

## III. RESULTS AND DISCUSSION

We begin establishing consistence between the surface resonances detected by the ATR method and the SPs for free-standing graphene predicted by the electromagnetic dispersion relation - Eq. (7). Samples of two chemical potentials μ = 0.8 eV and μ = 1.2 eV were studied. In Fig. 2 we plot the curves of the corresponding dispersion relations that show how the speed of the TM SP increases when the doping level also increases. We are superimposing on Fig. 2 two series of reflection dips |$R(\omega,\theta _i,d) = | {{{E_{1x}^ < } \mathord{/ {\vphantom {{E_{1x}^ < } {E_{1x}^ > }}} \kern-\nulldelimiterspace} {E_{1x}^ > }}} |^2$|$R(\omega ,\theta i,d)=|E1x</E1x>|2$ corresponding to different separations *d*. For μ = 0.8 eV at *f* = 5 THz (*λ* = 60 μm) the three dips maintain the angle of incidence θ_{i}∼ 30.5° when the separation prism-graphene changes from *d* = 15 μm to *d* = 25 μm (this is almost 10 μm without significant angular variation). On the other hand, less stable is the case for μ = 1.2 eV at *f* = 0.8 THz where the positions of the reflection minima remain at θ_{i}∼ 32.2° in a shorter separation range, from *d* = 13 μm to *d* = 17 μm, approximately. The important point is that these series of reflection dips have wave vectors |$k_x = \frac{\omega }{c}{\rm n}_{\rm p} {\rm sin}\theta _i$|$kx=\omega cnp sin \theta i$ that practically satisfy the electromagnetic dispersion relation. The wave vectors are *k*_{x} = 0.21 μm^{-1} and *k*_{x} = 0.36 μm^{-1} for the two excited SPs at 5 THz and 8 THz, respectively. Therefore, TM SPs in graphene can be excited efficiently with the ATR technique with imperceptible deviation from the values predicted by the electromagnetic dispersion relation. In terms of the wavelength the range of prism-graphene separation ensuring this coupling runs from *d* = λ/4 to *d* = λ/2, approximately. For separations larger than *d* = λ/2 the excitation vanishes. However, below the lower limit (separations shorter than *d* = λ/4) the minima become deeper with angle of incidence very sensitive to *d*.

Two additional numerical evidences that the reflection dips of Fig. 2 are produced by the SP excitation are presented in Fig. 3. As can be seen the variation of the relaxation time τ within one order of magnitude changes slightly the angle of incidence of the reflection minima (less than one degree). Figure 3 shows that both breadth and depth of the reflection dips have the typical behavior of a vibrational normal mode as function of the absorption: by decreasing the relaxation time (thus, increasing the frequency of collisions) the graphene absorption increases broadening the dips leaving less deep the reflection minima. As second evidence, the inset in Fig. 3 presents the normalized magnetic field amplitude |$H_y = ( {{{\omega \varepsilon } \mathord{/ {\vphantom {{\omega \varepsilon } {q_z }}} \kern-\nulldelimiterspace} {q_z }}} )E_x$|$Hy=(\omega \u025b/qz)Ex$ in the graphene vicinity. We are plotting the reflected field from the graphene sheet in direction to the prism and the transmitted field beyond the graphene; they are the fields |$H_{2y}^ < e^{ - iq_{2z} z}$|$H2y<e\u2212iq2zz$ and |$H_{3y}^ > e^{iq_{3z} z}$|$H3y>eiq3zz$ implicit in Eqs. (2) and (3). The profile indicates field confinement of length *d*_{c}∼λ/4 in the perpendicular direction from the graphene sheet.

Now, one may expect that once the SP is optimally excited, a continuous variation of *d* will produce reflectivity profiles showing a coupling-strength of gradual weakening. Furthermore, additional dips associated to resonant TM SPs are not expected because they are not predicted by the dispersion relation. Contrarily to this expectation, we have found that additional dips exist; in fact, they can reach the limit of zero reflectivity. In order to have a complete picture of the problem, we plot in Fig. 4 the three more important reflection dips that we found at 5 THz by sweeping all the angles of incidence greater than the critical angle and for different separations *d*. They occur at the angles of incidence θ_{1} ∼ 30.5°, θ_{2} ∼ 34.8° and θ_{3} ∼ 78°. The first minimum is associated to the *pure* SP excitation. The other two dips of total absorption result from a complex mechanism involving a) the SP resonance, b) interference effects and c) the graphene losses.

In order to understand the origin of the perfect (total) absorption phenomenon we present in Fig. 5 the absolute values of the numerator *r*_{num} and denominator *r*_{den} of Eq. (10). The resonant behavior associated to the SP excitation is predominant in the figure; there exist also two angles of incidence at which the condition of total reflection, abs(*r*_{num}) = abs(*r*_{den}), is broken. In fact, at these angles the numerator of Eq. (10) satisfies strictly the condition *r*_{num} = 0 and the corresponding graphene-prism separations are given by

Hence, in terms of the components of Eq. (10) the condition of zero reflection is that both *r*_{12} and |$r_{23} e^{2iq_{2z} d}$|$r23e2iq2zd$ be two complex quantities with the same amplitude but with a phase difference of 180°. It is the condition of destructive interference between the reflected wave at the prism-air interface and the reflected wave at the graphene sheet. In addition, we note that the angle of incidence of the first dip of perfect absorption is very close to the angle of the SP excitation. In fact, the curvature of abs(*r*_{den}) indicates that it is occurring at the border of the resonance region (the shaded region in Fig. 5) whose breadth is defined by the graphene losses. Thus, we can conclude that the first dip of zero reflection results from a combined effect of plasmonic resonance and wave interference.

Figure 5 shows that the effects of the SP resonance are negligible at the angular region of the second dip of perfect absorption. Thus, a SP resonance is not the main requirement for the perfect absorption phenomenon. Equation (11) establishes that now the destructive interference occurs for a separation *d* ∼ 0.8 μm which is almost six times smaller than the separation needed to produce the first dip of zero reflection, *d* ∼ 4.6 μm. Without a mechanism for wave guidance (as is the case of a SP) the energy is being dissipated completely by the currents induced in the graphene sheet.

It is instructive to analyze the field distributions of the three excitations presented in Fig. 4. Figure 6 presents the magnetic field map when the SP in graphene is excited. Note that a well-confined surface wave is observed along the graphene layer (which is placed at *z* = 17 μm away from the prism) with the attached fields having minimum interaction with the prism (the region *z* < 0). Consistence between the ATR results and the electromagnetic dispersion relation is found: at 5 THz the wave length of the SP is *λ*_{SP} = 29.5 μm. In the upper panel of Fig. 6 we plot the fields along the *z* direction for *x* = 0. We call the attention to the cross suffered by the forward and backward evanescent fields in the air region between prism and graphene. In some way the position of this crossing point is related to interference effects (its *z* position depends on *d*). We shall see below that coherent interference leading to perfect absorption occurs when this crossing point coincides with the prism basis. It is notorious the phase shift between the fields in the prism and the fields in the graphene layer. The reason is that we are plotting the magnetic field *H*_{y}. Independently of the change of amplitude introduced by the evanescent behavior (exponential variation), what one expects is the phase matching of the tangential electric fields *E*_{x}. However, in graphene the complex impedance introduces an additional phase between *H*_{y} and *E*_{x}: |$H_y = \frac{{E_x }}{{| \eta |}}e^{ - i\phi _\eta }$|$Hy=Ex|\eta |e\u2212i\varphi \eta $ where ϕ_{η} is the phase of the complex impedance η.

On the other hand, Fig. 7 shows the magnetic field corresponding to the first excitation of perfect absorption, the dip at θ_{i} = 34.8° in Fig. 4. Although a residual behavior of surface wave in graphene is observed, clearly the field distribution has suffered a strong modification respect to the pure SP excitation. The evanescent field reflected from the graphene layer that reach the prism has become crucial for this excitation. In fact, just at the prism-air interface both evanescent fields take the value |$H_{2y}^ > (z = 0)$|$H2y>(z=0)$ = |$H_{2y}^ < (z = 0)$|$H2y<(z=0)$ = 0.5, which means a continuous exponential behavior without gap or cross at, or near, the prism-air interface (see the upper panel in Fig. 7). [It is important to remind that all the fields in our calculations are normalized to the incident field; under this condition, the incident magnetic field is the unity, the reflected field is zero and the total field in the air region just outside the prism must be also the unity as is required by the boundary conditions.] Our numerical calculations show that the coherent interference of the fields reflected by both the prism-air interface and the graphene sheet that lead to zero reflection is equivalent to ask a continuous exponential profile of the magnetic field in the air region between prism and graphene. The continuous profile satisfying the boundary condition at the prism-air interface is clearly observed by mirror reflection at *z* = 0 of |$H_{2y}^ > (z)$|$H2y>(z)$ or |$H_{2y}^ < (z)$|$H2y<(z)$ in the upper panel of Fig. 7.

For the second excitation of perfect absorption, the reflection dip at θ_{i} ∼ 78° in Fig. 4, we obtain the magnetic field shown in Fig. 8. The map shows that the system behaves practically as a cavity; there is not a field distribution associated to a vibrational mode confined to the graphene layer. Because of the fields are evanescent, wave guidance is not expected and all the energy should be dissipated by the currents induced in the graphene layer. The field profile in the upper panel of Fig. 8 shows that the boundary condition and the behavior of the evanescent fields are being satisfied.

We want to remark that interference effects are intrinsic in the ATR geometry. They contribute to the construction of the optical response not only in ATR graphene-based systems but also in ATR configurations that incorporate a thin metallic film. As we have plotted in the upper panels of Figs. 7 and 8, the occurrence of total destructive interference that leads to the perfect absorption phenomenon requires some degree of discontinuity (a jump) of the forward [|$H_{2y}^ > (d)$|$H2y>(d)$] and backward [|$H_{2y}^ < (d)$|$H2y<(d)$] evanescent fields at the graphene position. (In fact, under conditions of destructive interference the jump of the magnetic field at the graphene position Δ*H*_{2y}(*z* = *d*) and the air gap *d* are linked by the relation Δ*H*_{2y} = 0.25sinh α*d*, where |$\alpha = \sqrt {q_x^2 - q_0^2 }$|$\alpha =qx2\u2212q02$.) This jump of magnetic field is caused by the planar electrical current induced by the (evanescent) electric field |$E_{2x}^ > (d)$|$E2x>(d)$ on the graphene sheet. This current contributes to form the magnetic field profiles of Figs. 7 and 8, and is the only channel for energy dissipation via Joule effect when SPs are absent.

Finally, we are aware of recent reports about enhanced absorption in ATR graphene-based structures in the terahertz regime. The absorption was controlled by a variable voltage gate applied to modify the plasmonic response of graphene, but the studies were made for a fixed prism-graphene separation *d*.^{21,22} In the present paper we establish that the interference mechanisms, strongly dependent on *d*, are primordial to obtain perfect absorption in the ATR structure. The ability of graphene to dissipate the received energy, with or without plasmonic contribution, is essential to reach the full absorption.

## IV. SUMMARY

In conclusion, we have studied numerically three TM excitations occurred in a layered system resembling the ATR Otto configuration that incorporates a graphene monolayer. Trying to establish the response of graphene to incident evanescent fields, we present a theoretical study for graphene surrounded by air, a free-standing doped graphene. At 5 THz, we identify the SP resonance and two excitations of perfect absorption; these two excitations result from interference effects which are dependent on the prism-graphene separation *d*. In terms of the partial reflection coefficients at the interfaces of the ATR system we have given the semi-analytical expression for *d* that leads to total absorption. We have also analyzed the behavior of the two evanescent fields in the air gap, the forward and backward fields between prism and graphene, just at the conditions of perfect absorption. At the prism-air interface one field is continuation of the other one maintaining the exponential profile without jumps or crosses. Destructive interference is also expected for doped graphene mounted on a thick slab of dielectric constant lower than ε_{p}. However, perfect absorption (zero reflectivity) may not be reached.

## ACKNOWLEDGMENTS

This work was supported by SESIC México, PROMEP Grant FOFM-2010. JAHL wishes to thank to CONACYT - México for support.