Spatially resolved optical sensing using graphene nanodisk arrays

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Spatially resolved optical sensing using graphene nanodisk arrays Lauren Zundel, and Alejandro Manjavacas ACS Photonics, Just Accepted Manuscript • Publication Date (Web): 13 Jun 2017 Downloaded from http://pubs.acs.org on June 13, 2017

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Spatially resolved optical sensing using graphene nanodisk arrays Lauren Zundel and Alejandro Manjavacas∗ Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, United States E-mail: [email protected] Abstract The ability of graphene nanostructures to support strong plasmonic resonances in the infrared part of the spectrum makes them an ideal platform for plasmon-enhanced spectroscopy techniques. Here we propose to exploit the exceptional tunability of graphene plasmons to perform infrared detection of molecules with subwavelength spatial resolution. To that end, we investigate the optical response of finite arrays of graphene nanodisks that are divided into a number of identical subarrays, or pixels, each of them with a uniform level of doping. Using realistic conditions, we show that, by adjusting individually the doping level of each of these pixels, it is possible to bring them sequentially into resonance with the vibrational spectrum of the analyte. This enables the identification of the analyte and the simultaneous detection of its spatial location with a resolution determined by the size of the pixels. Our work brings new possibilities to plasmon-enhanced infrared sensing by combining the already demonstrated sensing abilities of graphene nanostructures with subwavelength spatial resolution. This could be exploited to develop actively tunable substrates for multiplexed sensing, which could be used to analyze the chemical composition of complex biological systems and to follow their temporal evolution with spatial resolution.

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Keywords sensing, infrared, surface-enhanced, spectroscopy, graphene, plasmons, SEIRA, spatial resolution The detection of chemical and biological species using light is a technological challenge of paramount importance for society, with major implications in healthcare 1–4 and security, 5,6 among other areas. Optical sensing techniques usually rely on the measurement of the vibrational spectrum of the molecular species under analysis, since this is uniquely determined by the structure and chemical composition of the molecule, and therefore constitutes a fingerprint or barcode that allows its univocal identification. 7 Unfortunately, the large mismatch between the usual molecular sizes, typically below 10 nm, and the wavelength of their vibrational resonances, which lie in the mid-infrared part of the spectrum (≈ 2 − 16 µm), results in very weak absorption and scattering cross-sections that impose a threshold to the minimum quantities of analyte required for detection, as well as hinder the miniaturization of these sensing techniques. One way of overcoming these limitations is to exploit the extraordinary properties of the surface plasmons supported by metallic nanostructures. 8 These excitations, arising from the collective oscillations of conduction electrons, couple strongly to light, confining it into subwavelength volumes, 9,10 and therefore provide large near-field enhancements that can be used to amplify the inherently weak cross-sections of molecules placed in the vicinity of the nanostructures. 11 This is the basic mechanism behind surface-enhanced Raman scattering (SERS) spectroscopy, 12–15 in which the enhancement of the inelastic scattering cross-section of the molecule provided by surface plasmons allows pushing of the detection limits to the single molecule level. 16–19 A similar procedure can be employed to amplify the absorption cross-section of the molecule, leading, in this case, to surface-enhanced infrared absorption (SEIRA) spectroscopy, 20–22 a technique that enables the detection of molecules with optically-active vibrational modes (i.e. those that involve a change in the dipole moment of the molecule). 23–27 SERS and SEIRA are methodologies with complementary selection rules that permit 2


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nondestructive label-free optical sensing. However SEIRA, unlike SERS, requires the surface plasmon to be resonant with the frequency of the vibrational mode of the molecule. For this purpose, a wide variety of metallic structures supporting mid-infrared plasmons have been proposed and characterized, ranging from nanoparticles with simple shapes 28–32 to more complicated antenna designs. 33–39 The majority of these examples involve the use of noble metals, although structures made of other materials such as aluminum, 40,41 indium tin oxide, 42 polaritonic crystals, 43 and different doped semiconductors 44,45 have also been studied. More recently, the emergence of graphene as an alternative plasmonic material has opened new avenues in surface-enhanced sensing. 46,47 Doped graphene nanostructures support surface plasmons that provide unparalleled levels of light confinement and whose frequencies naturally lie in the mid-infrared part of the spectrum for reasonable levels of doping. 48,49 Furthermore, in contrast to plasmons supported by traditional plasmonic materials, which can be tuned only through composition and geometry, graphene plasmons can be actively controlled via electrostatic gating. 50,51 These extraordinary properties make graphene nanostructures ideal candidates for infrared sensing applications, as has been theoretically proposed 52–54 and experimentally demonstrated. 55–57 In this paper we propose to take advantage of the electrical tunability of graphene plasmons to perform infrared sensing with subwavelength spatial resolution. For this purpose, we consider finite arrays of graphene nanodisks that are divided into a number of identical subarrays, or pixels, each of which displays a doping level that can be independently controlled. Analyzing the optical response of these structures, we show that, by tuning the doping of each pixel in the array, it is possible to selectively switch on the interaction between the analyzed molecules and the different pixels. This allows us to identify the vibrational spectrum of molecules placed near the system, while simultaneously detecting their spatial location. We explore the robustness of this methodology by studying the effect that the different design parameters have on the sensing performance of the system. The results presented in this work help to set the foundations to develop novel label-free infrared sensors,

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which will open doors for new applications in analyzing the chemical composition of complex biological structures with temporal and spatial resolution.

Results and Discussion The system under study is depicted in Figure 1(a). It consists of a square array of identical nanodisks, carved out from a single graphene monolayer, with a diameter D = 60 nm, separated by a distance d = 1.5D measured from the center of the disks. The array is divided into N × N subarrays, or pixels, each containing n × n nanodisks, as shown in Figure 1(a). We assume that all nanodisks in a given pixel possess the same doping level, which we characterize through the corresponding Fermi energy EF , and that the doping level of the different pixels in the array can be controlled independently. The large mismatch between the size of the nanodisks and their resonant wavelengths (≈ 10 µm) allows us to safely work within the electrostatic limit, and hence model the optical response of an individual nanodisk employing a dipolar polarizability

α=

D3 ζ12 . (−2/η1 )/(ε1 + ε2 ) − iωD/σ

(1)

This expression is obtained using the plasmon wave function (PWF) formalism introduced by García de Abajo 58–60 (see the Supporting Information for details). Within this formalism, the polarizability of any graphene nanostructure can be completely characterized using two parameters: ζ1 and η1 , whose values only depend on the nanostructure shape, and can be calculated from the fitting of the polarizability to the corresponding results obtained from the numerical solution of Maxwell’s equations. In the particular case of a nanodisk, these parameters take the following values: ζ1 = 0.840 and η1 = −0.072 61 (see Figure S2 of the Supporting Information). Naturally, these values change for other nanoisland shapes, such as squares or triangles. The polarizability given in Eq. (1) also depends on the environment surrounding the nanodisk through the factor 2/(ε1 + ε2 ), as derived from the potential of 4


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a coupled dipole approach, 65–67 in which the dipole moment pi , induced by an external field in a nanodisk located at ri can be written as

pi = αi Ei + αi

2N 2 nX

Gij pj ,

j6=i

where αi is the disk polarizability defined in Eq. (1), Ei is the external electric field at ri , √ which we take to be polarized horizontally, Gij = [k 2 + ε−1 ∇∇] exp(ik ε|ri − rj |)/|ri − rj | is the dipole-dipole interaction tensor, ε is the dielectric function of the medium in which the array is embedded, and k = ω/c is the wavenumber. This equation can be solved to obtain the self-consistent dipole induced at each nanodisk

pi =

2N 2 nX

Mij−1 αj Ej ,

j=1

where Mij = δij − αi Gij . Using this solution we can write the absorption cross-section of the whole array as 68 2

2

n N pi · E∗i 4πk X Im A= √ . |Ei |2 ε i=1 Figures 1(b) and (c) show the absorption cross-section per nanodisk, normalized to the disk area πD2 /4, for a N = 4, n = 3 array in which only one pixel is active. An active pixel is characterized by a Fermi level EF1 , which is different from the background level of the remaining nonactive pixels, EF0 , and that is chosen to produce a plasmon resonance in the frequency range of interest. For simplicity, we assume the environment around the array to be vacuum (i.e., ε1 = ε2 = 1). Notice that choosing a different dielectric environment has only the effect of shifting the plasmon resonance of the array, a change that can be offset by readjusting the doping level of the system, as shown in Figure S3 of the Supporting Information. Panel (b) displays the results corresponding to EF1 = 300 meV and EF0 = 1 meV. The colors of the different curves indicate the pixel that is active (see Figure 1(a)). Incidentally, due to the large contrast between EF1 and EF0 , all of them present an identical 6


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absorption resonance, with a quality factor of ≈ 70, arising from the collective interaction of the dipolar plasmon of the nanodisks in the active pixel. Some differences become appreciable when this contrast decreases, as shown in panel (c), where we study the same array as in panel (b) but with EF0 = 200 meV. Interestingly, arrays with a larger number of nanodisks display a similar behavior. In particular, panels (d) and (e) show the absorption cross-section for a N = 4, n = 11 array with the same Fermi levels as those chosen in panels (b) and (c). The only noticeable difference arising from the increase of the number of nanodisks is a small redshift of the peak position. However, this redshift saturates for n ≈ 10 as discussed in Figure S4 of the Supporting Information. The collective plasmon resonance responsible for the peak in the absorption spectrum, which arises from the coupling of the plasmons of the individual nanodisks, generates a strong near-field that can be exploited to detect the vibrational resonances of molecules placed in the vicinity of the active pixel. This is possible because the coupling of the molecules with the plasmon mode of the active pixel results in a Fano resonance 69 and a frequency shift that modify the absorption spectrum of the array. These changes in the spectrum signal the presence of the molecules, which are otherwise too small to directly couple to the external illumination. Here, we model this effect as follows: the presence of molecules on top of a nanodisk changes the dielectric function of the medium above it (i.e. ε2 ), which, in turn, results in the modification of the nanodisk polarizability and, consequently, of its absorption cross-section (see Eq. (1)). Then, nanodisks with molecules deposited over them are characterized by ε1 = 1 and ε2 = 1 + δε, while those without them have ε1 = ε2 = 1. We calculate δε using the Clausius-Mossotti relation 68

δε =

12παm s , 3 − 4παm s

(2)

where s is the concentration of molecules and αm is their polarizability. This expression is exact for three-dimensional systems and a very good approximation for layers with thicknesses

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larger than the molecular diameter. 70 For the polarizability of the molecule we use

αm =

β/~2 + αm0 . 2 − ω 2 − iωγ ωm m

(3)

The first term of this expression explicitly accounts for a single vibrational mode with energy ~ωm , width ~γm , and polarizability strength β, while the second describes the contribution of the rest of the nonresonant modes. Therefore, this expression allows us to model the Fano resonance produced by the vibrational mode, as well as the frequency shift induced by the nonresonant contributions. If necessary, the explicit modeling of more vibrational modes would be direct, only requiring the addition of extra Lorentzian terms to Eq. (3). It is also important to remark that situations for which surface-enhanced sensing techniques are relevant satisfy αm s ≪ 1, and therefore δε ≪ 1. This allows us to neglect the direct interaction of the molecules with the external illumination, as well as the effect of δε in the dipole-dipole interaction tensor Gij , when modeling the response of the array. With all these ingredients, we can analyze the variation of the optical response of the array induced by the presence of the molecules. This is done in Figures 2(a) and (b), where we plot the absorption spectrum of a N = 4, n = 3 array with (green curves) and without (red curves) molecules deposited over the active pixel, which, in this case, corresponds to the one on the corner (i.e. the blue pixel in Figure 1(a)). In both cases we choose EF0 = 1 meV, whereas EF1 is set to 300 meV in panel (a) and to 280 meV in panel (b). The concentration of molecules is s = 0.01 nm−3 (see Figure 5(b) for other values), while ~γm = 0.5 meV and β = 10 meV2 nm3 , which are reasonable values for mid-infrared vibrational modes. 53 Green solid curves show results obtained with αm0 = 0, while for dashed curves αm0 = 0.15 nm3 . As expected, the presence of molecules over the active pixel results in the appearance of a Fano dip, as well as in a shift when αm0 6= 0, that modify the absorption spectrum of the array. In all cases, we have chosen ~ωm to be resonant with the plasmon of the active pixel. Obviously, in an experimental realization, the resonance of the active pixel is what would be

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tuned to match ~ωm by selecting the appropriate Fermi level. This, incidentally, allows us to offset the effect of αm0 and therefore, in the following, we take αm0 = 0. We quantify the change in the absorption spectrum by introducing the absorption crosssection variation ∆A, defined as ∆A = 1 − A/A0 , where A and A0 are the values of the array absorption at resonance with and without molecules, respectively. Clearly, the measurement of ∆A serves to reveal the presence or absence of molecules over the active pixel. Therefore, by activating each of the pixels of the array (i.e. turning its Fermi level from EF0 to EF1 ) one at a time, and then measuring the corresponding value of ∆A, we can detect the presence of molecules and, at the same time, know their position with an accuracy determined by the pixel size. The proposed methodology is illustrated in Figure 2(c), where we plot ∆A as a function of the active pixel for a N = 4, n = 3 array with molecules deposited over the pixels 1-4, 2-1, and 3-3, as defined in panel (d). The red bars represent the results corresponding to EF1 = 300 meV and EF0 = 1 meV. Examining these results, we see that, as expected, ∆A = 0 when a pixel without molecules is active. However, if, on the contrary, the active pixel has molecules deposited on it, ∆A ≈ 36%, thus allowing the detection of the presence and the location of the molecules. This is also the case when EF0 is increased to 200 meV (green bars), although now the smaller contrast between the active and the background Fermi levels results in a small negative value of ∆A for the pixels without molecules, and a nonuniform value of ∆A for those with molecules. Using different EF1 levels, it is possible to detect more than one vibrational resonance belonging either to the same or to different types of molecules. This possibility is explored in panel (e), where we plot ∆A as a function of the active pixel for a N = 4, n = 3 array with two different types of molecules deposited on it. The first type, whose resonance is ~ωm = 176 meV, is placed over pixels 1-2 and 4-4, while the second one, with ~ωm = 170 meV, is located at pixels 1-3 and 3-2, as shown in panel (f). These two molecules are exactly the same ones used in panels (a) and (b), so, when EF1 is set to 300 meV (blue bars) we detect the molecules resonant at ~ωm = 176 meV, while if EF1 = 280 meV (yellow bars) we identify 10


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and the inset. In all cases, we keep EF1 = 300 meV, while the parameters of the molecules are the same as in Figure 2(c). Examining these results, we observe that ∆A remains almost constant (≈ 36%) for all pixels when EF0 is below 200 meV. However, a nonuniform behavior emerges above that value, with ∆A dropping rapidly to zero as EF0 approaches EF1 . It is also important to analyze the situation in which the active pixel is not the one having the molecules. This is investigated in panel (b), where each plot contains 15 curves corresponding to all the possible active pixels without molecules, and the color of the curves signals the pixel that has the molecules deposited on it, as indicated in the inset of panel (a). As expected, ∆A remains close to zero for EF0 up to ≈ 200 meV. However, larger background Fermi levels result in finite values of ∆A that become comparable to those of panel (a), thus hindering the detection of the molecules. This is caused by the decrease in the contrast between the absorption cross-section of the active and the nonactive pixels produced by the larger background Fermi level. Similar results are found for arrays with a larger number of nanodisks, as shown in panels (c) and (d). There we repeat the calculations of panels (a) and (b) for an array with N = 4 and n = 11. Comparing both systems, we observe that larger arrays display a more uniform response, although the value of ∆A remains almost unchanged. Indeed, as discussed in Figures S5 and S6 of the Supporting Information, ∆A is rather independent of n, d, and D, which allows us to generalize our predictions to other array configurations. The variation of the absorption cross-section ∆A, which ultimately determines the sensitivity of the array, also depends on the characteristics of the molecular layer, as we discuss below, as well as on the quality of graphene. The latter is controlled by the electron mobility µ, which, throughout this work, is taken to be 104 cm2 /(V s). 62 Higher values of µ result in narrower and stronger plasmon resonances and therefore larger values of ∆A, as shown in Figure 4, however values of µ below 104 cm2 /(Vs) still produce noticeable changes in the absorption. In addition to the mobility, the properties of the molecular layer also impact ∆A. These include its thickness, its separation from the nanodisks, and the molecule concentra-

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100 n=3 n=7 n = 11

80

û A (%)

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60

N=4 0

40

EF = 1 meV 1

EF = 300 meV

20 0 3 10

104

+ (cm2/(Vs))

105

Figure 4: Dependence of ∆A on the electron mobility µ. ∆A is plotted as a function of µ for three different arrays: N = 4, n = 3 (green curve), N = 4, n = 7 (yellow curve), and N = 4, n = 11 (red curve). In all cases, we have molecules deposited on an active pixel that corresponds to the one on the corner of the array (i.e. the blue pixel in Figure 1(a)). We choose EF1 = 300 meV for the active pixel and EF0 = 1 meV for the rest. The parameters of the molecular layer are the same as in Figure 2(c).

tion. So far we have considered a molecular layer with infinite thickness and concentration s = 0.01 nm−3 , placed directly over the graphene array. However, this configuration may not be the one used in an experimental realization. For this reason, it is crucial to analyze the effect that all of these parameters have on ∆A. For simplicity, we focus on the case of a single graphene nanodisk, although, due the invariance of ∆A with respect to n (see Figure S5 of the Supporting Information), we expect the conclusions to hold for any array size. This allows us to exploit the PWF formalism to obtain the following closed expression for the effective polarizability of the disk when placed a distance h from a molecular layer of thickness t (see the Supporting Information for the derivation of this expression)

α

eff

=

F D3

D3 ζ12 2l l=0 F [V11 (2h + 2lt) − V11 (2h + 2(l + 1)t)] −

P∞

1 η1

−

iωD σ

.

(4)

Here, F = −δε/(2 + δε), with 1 + δε being the dielectric function of the molecular layer, as indicated in the inset of Figure 5(a). On the other hand, V11 (z) is the dipolar component 13


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of the effective Coulomb potential (see the Supporting Information) that satisfies V11 (0) = −D3 /η1 , and therefore ensures that this expression of the polarizability reduces to Eq. (1) in the limit of h = 0 and t → ∞.

(a) 40 t

b¡

6A (%)

30

h

20

t: ∞ 10 n t = 40 n m t=4 m nm t = 1 nm t=

10 0 0

4

2

(b) 100

6

10

8

h (nm)

85

6A (%)

80

s (10-3 nm-3)

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75%

60 40

50%

20 25 % 0 0

0 10

20

30

t (nm)

40

50

60

Figure 5: Dependence of ∆A on the molecular layer properties. (a) ∆A for a single nanodisk with Fermi level EF = 300 meV plotted as a function of the distance h between the nanodisk and the molecular layer for different values of the layer thickness t (see the inset). We use s = 0.01 nm−3 and the same molecule parameters as in Figure 2(c) (notice that for these parameters δε = 0.014 at resonance). (b) ∆A for the same nanodisk of panel (a) plotted as a function of t and the molecule concentration s, assuming h = 0. The dashed line marks s = 0.01 nm−3 , while the solid lines indicate different values of ∆A.

Figure 5(a) shows ∆A as a function of h for a single nanodisk with EF = 300 meV, obtained using Eq. (4). Each solid curve in the plot corresponds to a different value of t, ranging from 1 nm to 40 nm. Not surprisingly, smaller separations and thicker molecular 14


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layers produce larger values of ∆A. However, the results for t = 40 nm are almost equal to those obtained for an infinite layer (cf. red and black curves). This behavior is a consequence of the strong field confinement provided by graphene plasmons, as shown in Figure S7 in the Supporting Information, and clearly supports the use of the approximation of an infinite molecular layer. Indeed, looking at panel (b), where we plot the variation of the absorption cross-section as a function of s and t, we observe that ∆A remains almost unchanged for t ' 25 nm, which is in excellent agreement with previous observations. 71 Figure 5(b) also serves to analyze the effect of the molecule concentration. The dashed line marks s = 0.01 nm−3 , which is the value of s used in panel (a) and in previous figures. As expected, ∆A increases with s, although the actual value of the former also depends on the characteristics of the molecule described by its polarizability. In fact, by examining Eq. (3) it directly follows that ∆A is determined by the product sαm , which, at resonance is proportional to sβ/(~2 ωm γm ) for αm0 = 0. This relation allows us to readily generalize the results of Figure 5(b), which are obtained for a molecule with β = 10 meV2 nm3 , ~ωm = 178 meV, and ~γm = 0.5 meV, to any other molecule with different polarizability parameters.

Conclusions In summary, we have characterized the optical response of finite arrays of graphene nanodisks with the goal of using these systems to perform infrared sensing with spatial resolution. The arrays under consideration are divided into a number of identical subarrays, or pixels, whose doping levels can be controlled individually. This tunability allows us to selectively activate each pixel by raising its Fermi level from a background to an active value, adequately chosen to produce a plasmon resonance that overlaps with the vibrational modes of the analyzed molecules. Then, by sequentially activating each pixel in the array and measuring the change in the absorption spectrum, it is possible to identify the presence of the molecules and, at the same time, detect their position over the array in real time with a precision determined

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by the size of the pixel. Through the study of arrays with different numbers of nanodisks we have verified the functionality of the proposed sensing mechanism over a wide range of pixel sizes. Furthermore, the proposed methodology is remarkably robust with respect to the contrast between the active and the background Fermi levels; ratios as small as 1.5 result in variations of the absorption cross-section similar to those obtained for ratios of 300. By analyzing the effect of the molecular layer properties on the sensing performance, we have shown that layer thicknesses in excess of several nanometers and separations from the array below 10 nanometers produce substantial variations of the absorption spectrum for realistic concentration of molecules, which is a direct consequence of the strong near field produced by graphene plasmons. In fact, this strong near-field can be exploited in combination with the concept proposed in this paper, to design optically-driven localized heat sources with spatial tunability that can be used to control chemical reactions. The implementation of the proposed sensing mechanism requires the individual manipulation of the Fermi level of each pixel in the array. This can be achieved by employing the standard schemes for electrostatic gating of graphene nanostructures 72,73 with the caveat that, in this case, each pixel must have its own separate backgate. A simplified alternative would involve the use of arrays of graphene ribbons. These structures support localized plasmons similar to those of nanodisks for illumination polarized across their length, but with the advantage that they can be electrically contacted far from the sensing region, thus requiring less demanding fabrication standards. 55 However, using a single array of ribbons would reduce the spatial resolution to one dimension. This limitation could be overcome by using a second array of ribbons as a backgate, oriented perpendicularly to the sensing array, as schematically shown in Figure S8 of the Supporting Information. Then, by controlling the potential difference between pairs of backgate and sensing ribbons, it would be possible to confine the doping charge along the direction parallel to the ribbons, since it would tend to accumulate over the crossing regions. These charge accumulations would effectively function as finite graphene nanostructures that can be selectively switched, thus mimicking the

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behavior of the graphene nanodisk arrays proposed in this work. The results described here bring a new perspective to surface-enhanced infrared spectroscopy techniques by adding subwavelength spatial resolution to the exceptional sensing abilities of graphene nanostructures, and thereby pave the way for the development of tunable substrates for multiplexed sensing without the need of chemical functionalization of the nanostructures. The proposed system may be exploited to design a platform for label-free analysis of complex biological structures including cells, viruses, and membranes. This platform would be capable of determining the chemical composition of these structures in real time with subwavelength resolution, and therefore would enable the monitoring of the dynamics of biological processes in which they are involved.

Acknowledgement We acknowledge financial support from the Department of Physics and Astronomy and the College of Arts and Sciences of the University of New Mexico, and the UNM Center for Advanced Research Computing for computational resources used in this work. We are also grateful to Prof. Javier García de Abajo and Prof. Peter Nordlander for valuable and enjoyable discussions and for their critical reading of the manuscript. L.Z. acknowledges support from the Rayburn Reaching Up Fund.

Supporting Information Available Derivation of Eqs. (1) and (4) using the PWF formalism. Comparison of the PWF formalism with the rigorous solution of Maxwell’s equations. Analysis of the effect of a substrate with ε1 > 1 on the response of the nanodisks. Analysis of the dependence of the array absorption spectrum on n. Analysis of the dependence of ∆A on n. Analysis of the dependence of ∆A on d and D. Analysis of the vertical decay of the electric field intensity induced by a nanodisk. Discussion of a potential implementation of the proposed system. 17


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Spatially resolved optical sensing using graphene nanodisk arrays

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