Tunable perfect absorption at infrared frequencies by a graphene-hBN hyper crystal

Jipeng Wu; Leyong Jiang; Jun Guo; Xiaoyu Dai; Yuanjiang Xiang; Shuangchun Wen

doi:10.1364/OE.24.017103

1. Introduction

Over the past several years, perfect absorption (PA) at infrared frequencies has been widely investigated [1–3]. Since it is very vital in many promising applications, such as solar cells [4], plasmonic sensors [5], molecular dectors [6], and selective thermal emitters [7]. Most of the studies on PA are focused on broadband and wide-angle. For example, Deng et al. studied the broadband perfect absorber in the visible and near-infrared range by using a ultrathin layer of the refractory metal chromium (Cr) embedded between two silica (SiO2) layers [8]. Pu et al. proposed an approach for designing a wide-angle perfect absorber at infrared frequencies based on plasmonic structure [9]. Although the broadband and wide-angle perfect absorbers can be applied in the field of solar power harvesting [10] and electromagnetic wave spatial filter [11], the flexible control of PA is also very important for applications in optical devices. It is desired to tune PA by using an external bias voltage. For this purpose, introducing graphene into the perfect absorber seems to be a good approach for controlling PA. Up to now, a lot of researches on controlling the PA by varing the Fermi energy of graphene have been investigated [12–15]. Recently, Baranov et al. firstly predicted the interferenceless perfect absorption by a strongly anisotropic metamaterial [16], Baranov et al. demonstrated both theoretically and experimentally that PA at infrared frequencies can be realized by using a half-space of hexagonal Boron Nitride (hBN) crystal [17]. It is interesting to investigate the interferenceless perfect absorber based on the graphene-hBN hyper crystal.

hBN, an interesting optical material has attracted considerable interest in infrared ranges in photonics due to its high optical anisotropy [18,19]. Graphene, on the other hand, has found significant applications in both photonics and plasmonics in a broad frequency (visible to mid infrared to THz) range due to its unique and tunable optical properties [20–23]. hBN is widely used as an excellent substrate material to form a hyper crystal with graphene as it can provide an amazing clean environment [24–28]. So when constructing a graphene-hBN hyper crystal, it will have both hyperbolic and tunable characteristics.

In this article, we theoretically analyze the condition of total absorption at infrared frequencies by a graphene-hBN hyper crystal. It has been found that PA can be realized for TM waves. While PA by a half-space of hBN crystal can be achieved at certain fixed wavenumber and incidence angle. But using a graphene-hBN crystal, we are able to control PA that it can be realized at different wavenumbers and incidence angles by varing the Fermi energy of graphene. We believe this work will be very useful for future investigations on applications of graphene-hBN based hyper crystals and could find practical applications in optical devices and bio-sensor devices.

2. Theory

The graphene’s surface conductivity can be determined by using the Kubo formalisms [29–31]. Without considering the external magnetic field, the isotropic surface conductivity of graphene can be written as the sum of the intra-band and the inter-band terms. But when working in the infrared ranges, we can neglect the inter-band term, so the conductivity of graphene in the infrared ranges can be written as

σ = \frac{i e^{2} E_{F}}{π ℏ^{2} (ω + i τ^{- 1})},

where

ω

is the frequency of the incident light,

E_{F}

is the Fermi energy,

τ

is the relaxation time. e and

ℏ

are the universal constants related to the electron charge and Boltzmann constant, respectively. As we know that the relaxation time

τ = μ E_{f} / e v_{f}^{2}

, in which

μ = 10^{4} {cm}^{2} V^{-1} s^{-1}

and

v_{f} = 10^{6} m/s

are the mobility and Fermi velocity. We can use a conservative value (

τ = 10^{- 13}

at E_f = 0.1 eV) extracted from the measured, impurity-limited dc mobility [32,33]. In this article, we choose

τ = 5 \times 10^{- 13}

for simplify at different Fermi energies.

Graphene is an optically uni-axial anisotropic material due to its 2D nature, whose permittivity tensor can be given by,

ε_{g r a p h e n e} = [\begin{matrix} ε_{g, t} & 0 & 0 \\ 0 & ε_{g, t} & 0 \\ 0 & 0 & ε_{g, ⊥} \end{matrix}],

Graphene’s tangential permittivity can be given by,

ε_{g, t} = ε_{g} = 1 + \frac{i σ}{ε_{0} ω t_{g}},

where

ε_{0}

is the free space permittivity and t_g is the thickness of monolayer graphene. As graphene is a two dimensional material, the normal electric field cannot excite any current in the graphene sheet. So the normal component of the permittivity is given by

ε_{g, ⊥} = 1

[34].

hBN is a van der Waals crystal with two kinds of IR active phonon modes relevant to hyperbolicity: 1) out of plane A_2u phonon modes which have $ω_{T O} = 780 c m^{- 1}$ , $ω_{L O} = 830 c m^{- 1}$ and 2) in-plane E_1u phonon modes which have $ω_{T O} = 1370 c m^{- 1}$ , $ω_{L O} = 1610 c m^{- 1}$ [35–37]. These lead to two distinct Reststrahlen (RS) bands, which the lower frequency RS band corresponds to type I hyperbolicity ( $ε_{x} = ε_{y} > 0$ , $ε_{z} < 0$ ) and the upper RS band shows type II hyperbolicity ( $ε_{x} = ε_{y} < 0$ , $ε_{z} > 0$ ). The hBN permittivity is given by

ε_{u} = ε_{\infty, u} (1 + \frac{{(ω_{L O, u})}^{2} - {(ω_{T O, u})}^{2}}{{(ω_{T O, u})}^{2} - ω^{2} - i ω γ_{u}})

where

u = x, y

represents the transverse (a, b crystal plane) and

u = z

represents the z (c crystal axis) axes.

ε_{\infty}

and

γ

represent the high-frequency dielectric permittivity and the damping constant respectively,

ε_{\infty, x} = ε_{\infty, y} = 4.87

,

γ_{x} = γ_{y} = 5 c m^{- 1}

,

ε_{\infty, z} = 2.95

, and

γ_{z} = 4 c m^{- 1}

.

We plot the permittivities of monolayer graphene and hBN as a function of wavenumbers as shown in Fig. 1. Figures 1(a) and 1(b) show the real and imaginary parts of the permittivity of monolayer graphene at different Fermi energies. It is seen that the real part of the permittivity of graphene is negative and decreases with the increasing Fermi energy, while the imaginary part of the permittivity of graphene is positive and increases as increasing the Fermi energy. Figures 1(c) and 1(d) show the real and imaginary parts of the permittivity of hBN as a function of wavenumbers. It is abvious that the real part of $ε_{t} > 0$ and $ε_{z} < 0$ in type I, while the real part of $ε_{t} < 0$ and $ε_{z} > 0$ in type II. It is known from Fig. 1(d) that the imaginaty part $ε_{z}$ is large positive in type I, while the imaginaty part $ε_{t}$ is large positive in type II.

Fig. 1 (a) and (b) are the real and imaginary parts of the permittivity of monolayer graphene as a function of the wavenumber for various chemical potential of graphene. (c) and (d) are the real and imaginary parts of the tangential and perpendicular permittivity of hBN.

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Our graphene-hBN hyper crystal is representatively shown in Fig. 2. It is a graphene multilayer structure in which separations between graphene sheets are assumed to be filled with the hBN crystal. Based on the effective medium theory [38,39], the permittivity of the graphene-hBN hyper crystal can be given by

ε_{t e f f} = ε_{x e f f} = ε_{y e f f} = f_{g} ε_{g, t} + (1 - f_{g}) ε_{x},

ε_{z e f f} = {(\frac{f_{g}}{ε_{g, ⊥}} + \frac{1 - f_{g}}{ε_{z}})}^{- 1},

where

f_{g} = t_{g} / t

is the filling ratio of graphene sheet and

t = t_{g} + t_{h B N}

. Here, we assume

t_{g} = N * 0.35 n m

, N = 1 and

t_{h B N} = 100 n m

.

Fig. 2 The interferenceless perfect electromagnetic absorber based on the graphene-hBN hyper crystal.

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Figure 3 shows the real part of the permittivity of the graphene-hBN hyper crystal as a function of wavenumbers at different Fermi energies. It is seen that the real part of $ε_{t e f f}$ can be positive at different Fermi energies, while it only can be negtive at $E_{f} =1 eV$ . And when increasing the Fermi energy, the real part $ε_{t e f f}$ decreases. So if we want to get the negative real part of $ε_{t e f f}$ , we should increase the Fermi energy at near 1 eV or optimizing other parameters, such as the filling ratio of graphene sheet. It is also seen that the real part of $ε_{z e f f}$ is fixed as varing the Fermi energy. It can be understand from Eq. (6) that the filling ratio of graphene sheet is very small and $ε_{g, ⊥} = 1$ , so $ε_{z e f f}$ is almost equal to the hBN’s permittivity $ε_{z}$ . It means that the graphene has no influence on the real part of $ε_{z e f f}$ .

Fig. 3 The real part of the tangential and perpendicular permittivities of the graphene-hBN hyper crystal.

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Let us consider TM and TE waves incident from air to the surface of a half-space of graphene-hBN crystal, the reflection coefficients are given by

r_{T M} = \frac{\cos^{2} θ_{i} - \sqrt{(ε_{z e f f} - \sin^{2} θ_{i}) / (ε_{z e f f} ε_{x e f f})}}{\cos^{2} θ_{i} + \sqrt{(ε_{z e f f} - \sin^{2} θ_{i}) / (ε_{z e f f} ε_{x e f f})}},

and

r_{T E} = \frac{\cos^{2} θ_{i} - \sqrt{ε_{x e f f} - \sin^{2} θ_{i}}}{\cos^{2} θ_{i} + \sqrt{ε_{x e f f} - \sin^{2} θ_{i}}},

From Eq. (7), we can derive the condition for zero reflection for TM waves as

\cos^{2} θ_{0} = \frac{ε_{z e f f} - 1}{ε_{z e f f} ε_{x e f f} - 1},

where

θ_{0}

denotes the incidence angle at which the energy of the incident light are completely absorbed by the graphene-hBN crystal, meaning PA. When it meets

0 < Re (\cos^{2} θ_{0}) < 1

and

Im (\cos^{2} θ_{0}) = 0

at some wavenumbers, PA can be realized, where Re and Im represent the real and imaginary part. For TE waves, we also can know that when it meets

ε_{x e f f} = 1

, the reflection vanishes, however, this condition cannot be met with a lossy material at any frequency due to non-zero imaginary part of permittivity Im(

ε_{x e f f}

). Although the imaginary part of in-plane permittivity

ε_{x e f f}

can be very small far from phonon-polariton resonances of hBN, it is still finite and perfect absorption does not occur. Hence, we can obtain the nearly PA for the TE polarization.

3. Results and discussions

Firstly we discuss the absorption of our graphene-hBN hyper crystal based on both the effective medium theory and the transfer matrix method. In the transfer matrix method, we assume that graphene-hBN hyper-crystal has the periodic structure with (AB)^N, where A is graphene and B is hBN thin film. In order to guarantee the accuracy of the numerical results, we have adopted enough periodic number N (N≥1000) due to the infinite thickness of graphene-hBN hypercrystal in the effective medium theory. In Fig. 4, we plot the absorption based on the effective medium theory and the transfer matrix method at the incidence angle 60° and 80°, respectively. Other parameters are $τ = 500 fs$ and E_f = 0.25 eV. Figures 4(a) and 4(b) are the absorption based on the effective medium theory. Figures 4(c) and 4(d) show the absorption by using the transfer matrix method. It is obvious that Figs. 4(c) and 4(d) are in good agreement with Figs. 4(a) and 4(b). So the effective medium theory is credible in our model. Hence, in the following discussion, we will calculate the absorption by means of the effective medium theory.

Fig. 4 (a) and (b) are the absorption as a function of the wavenumber based on the effective medium theory at the incidence angle 60°, 80°. (c) and (d) are the absorption as a function of the wavenumber by using the transfer matrix method with enough period at the incidence angle 60°, 80°, where E_f = 0.25 eV.

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In order to examine the possibility of PA for TM waves, we plot in Fig. 5(a) the real and imaginary parts of $\cos^{2} θ_{0}$ deduced from Eq. (9) at E_f = 0.25 eV. It is clear that the condictions $Im (\cos^{2} θ_{0}) = 0$ and $0 < Re (\cos^{2} θ_{0}) < 1$ can be meet at some wavenumbers, so PA can be realized at those wavenumbers. Figures 5(b) and 5(c) show PA can be achieved at two points: point A and point B. It is seen in Fig. 5(b) that PA can be realized at point A which shows the wavenumber is at near 754 cm⁻¹ and the corresponding $θ_{0}$ is near 67°. Figure 5(c) shows that PA can be achieved at point B where the wavenumber is near 947 cm⁻¹ and the corresponding $θ_{0}$ is near 74°. Therefore, when a TM-polarized plane wave incident on a half-space of graphene-hBN hyper-crystal, the energy of the incident light can be totally absorbed at these two points.

Fig. 5 (a) The real and imaginary parts of squared cosine of the perfect absorption angle $θ_{0}$ . (b) and (c) show that the perfect absorption of TM polarized wave can be achieved at point A and B according to the condition of perfect absorption.

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The absorption of our graphene-hBN hyper-crystal as a function of wavenumbers depending on the incidence angle is shown in Fig. 6. It is seen in Fig. 6(a) that there is an absorption dip which the wavenumber is near 830 cm⁻¹, while both sides have a relatively high absorption, but the absorption varies at different incidence angles. Note in Fig. 6(b) that PA can be realized at the wavenumber near 750 cm⁻¹ and the incidence angle $θ_{0}$ near 67°, which corresponds with point A in Fig. 5(b). It is also seen that when the incidence angle is near $θ_{0}$ , the absorption is relatively high. When we increase the incidence angle from 65° to $θ_{0}$ , the absorption increases. While from $θ_{0}$ to 69°, the absorption decreases. Figure 6(c) shows PA can be achieved at the wavenumber near 950 cm⁻¹ and the incidence angle $θ_{0}$ near 74°, which corresponds with point B in Fig. 5(c).

Fig. 6 (a) calculated absorption spectra for TM polarized wave at different incidence angles. (b) shows the perfect absorption occurs at the angle 67°. (c) shows the perfect absorption occurs at the angle 74°.

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Through the above discussions, we know that PA can be realized at two different wavenumbers and two different incidence angles. But the condition discussed above is all at E_f = 0.25 eV, next we will discuss PA at different Fermi energies. We plot the real and imaginary parts of $\cos^{2} θ_{0}$ deduced from Eq. (9) at different Fermi energies, as shown in Fig. 7. It is seen from Fig. 7(a) that there are two parts that can meet the condition $Im (\cos^{2} θ_{0}) = 0$ . One part is at the wavenumber from 700 cm⁻¹ to 830 cm⁻¹ which we regard as part 1 and another part is at the wavenumber from 830 cm⁻¹ to 1000 cm⁻¹ which we regard as part 2. It is known that when we increase the Fermi energy in part 1, the position of PA is blue shift, while in part 2, the position of PA is red shift. Figure 7(b) shows that PA angle $θ_{0}$ decreases as increasing the Fermi energy both in part 1 and part 2, but PA angle changes greater in part 1 than in part 2.

Fig. 7 (a) and (b) show the perfect absorption condition for TM polarized wave as a function of the wavenumber at different Fermi energy E_f.

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For TE waves, we can know from Eq. (8) that the reflection vanishes when the in-plane permittivity component $ε_{x e f f}$ is equal to 1. We plot in Fig. 8(a) the reflection of our graphene-hBN hyper-crystal as a function of wavenumbers depending on the incidence angle at E_f = 0.25 eV and Fig. 8(b) the perfect transmittance condition at different Fermi energies. Figure 8(a) shows the reflection dip only occur at the wavenumber near 1700 cm⁻¹, which corresponds to the condition $ε_{t} = 1$ at E_f = 0.25 eV for TE waves as seen from Fig. 8(b), and it is therefore insensitive to the incidence angle. Figure 8(b) also shows that by tuning the Fermi energy, the wavenumber that meets $ε_{t} = 1$ changes and the position of the reflection dip is blue shift as increasing the Fermi energy.

Fig. 8 (a) calculated reflection spectra for TE polarized wave at different incidence angles for E_f = 0.25 ev. (b) the perfect transmittance condition for TE polarized wave at different Fermi energies E_f.

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4. Conclusions

In conclusion, we have studied PA in a half-space of graphene-hBN hyper-crystal. The permittivity of the hyper-crystal was approximated by the the effective medium theory. Through PA condition anlysis, we find that PA at infrared frequencies can be achieved and controlled. For TM waves, we can realize PA at two different wavenumbers and two different incidence angles, while for TE waves, we find that only the nearly PA can be realized due to non-zero imaginary part of permittivity of the graphene-hBN hyper-crystal. But both PA of TM polarization and nearly PA of TE polarization can be controlled by tuning the Fermi energy of graphene. We believe PA at infrared frequencies and the flexible control of PA is very interesting and will find potential applications in optical communications, photovoltaic and optical detectors.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 61505111, 61575127 and 61490713), the Guangdong Natural Science Foundation (Grant No. 2015A030313549), the Science and Technology Planning Project of Guangdong Province (Grant No. 2016B050501005), the Science and Technology Project of Shenzhen (grant Nos.JCYJ20140828163633996 and JCYJ20150324141711667), and the Natural Science Foundation of SZU (Grant Nos. 201517, 201452, 827-000051, 827-000052, and 827-000059).

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Tunable perfect absorption at infrared frequencies by a graphene-hBN hyper crystal

Abstract

Corrections

1. Introduction

2. Theory

3. Results and discussions

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Equations (9)

Optics Express