Diamond in a Nanopocket - American Chemical Society

This is an open access article published under an ACS AuthorChoice License, which permits copying and redistribution of the article or any adaptations for non-commercial purposes.

Article Cite This: ACS Omega 2018, 3, 4733−4742

Diamond in a Nanopocket: A New Route to a Strong Purcell Effect Gandhi Alagappan,*,† Leonid A. Krivitsky,‡ and Ching Eng Png† †

Institute of High Performance Computing, Agency for Science, Technology, and Research (A-STAR), Fusionopolis, 1 Fusionopolis Way, #16-16 Connexis, 138632, Singapore ‡ Data Storage Institute, Agency for Science, Technology, and Research (A-STAR), Fusionopolis, 2 Fusionopolis Way, #08-01 Innovis, 138634, Singapore ABSTRACT: Light emission from the color centers in diamonds can be significantly enhanced by their interaction with optical microcavities. In the conventional chip-based hybrid approach, nanodiamonds are placed directly on the surface of microcavity chips created using fabrication-matured material platforms. However, the achievable enhancement due to the Purcell effect is limited because of the evanescent interaction between the electrical field of the cavity and the nanodiamond. Here, we propose and statistically analyze a diamond in a nanopocket structure as a new route to achieve a high enhancement of light emission from the color center in the nanodiamond, placed in an optical microcavity. We demonstrate that by creating a nanopocket within the photonic crystal L3 cavity and placing the nanodiamond in, a significant and a robust control over the local density of states can be obtained. The antinodes of the electric field relocate to the nanosized air gaps within the nanopocket, between the nanodiamond and the microcavity. This creates an elevated and uniform electric field across the nanodiamond that is less sensitive to perturbations in the shape and orientation of the nanodiamond. Using a silicon nitride photonic crystal L3 cavity and aiming at silicon-vacancy and nitrogen-vacancy color centers in diamond, we performed a statistical analysis of light emission, assuming random positions of color centers and dipole moment orientations. We showed that in cavities with experimentally feasible quality factors, the diamond in the nanopocket structure produces Purcell factor distributions with mean and median that are tenfold larger compared to what can be achieved when the diamond is on the surface of the microcavity. Thus, a hybrid system18−23 is often explored. Here, the diamonds are either in nanodiamond 18,19,21−23 or in unpatterned blanket forms,20 and their optical emission is evanescently coupled to the optical microcavities, created using fabrication-friendly material platforms. Importantly, with fabrication-matured material platforms, such as gallium phosphide (GaP)18−20- and silicon nitride (SiN)21,22-based platforms, the hybrid approach allows the development of much needed, fully integrable, and scalable on-chip quantum optical devices that consist of waveguides, microcavities, integrated beam splitters, and other optical components. Examples of on-chip microcavities include toroidal,23,24 disk,25 ring,26,27 and a variety of one-dimensional28−30 and twodimensional31−38 photonic crystal resonators. The major bottleneck of the hybrid approach is only that the evanescent portion of the electric field couples to the diamond. Thus, the amount of LDOS that the external microcavity can offer to color centers in diamonds is very limited. In this article, we introduce and explore the idea of having the diamond in a nanopocket (DINP) within the core of the on-chip microcavity.

1. INTRODUCTION Single-photon sources1−3 have paramount importance in the quantum technology, as they constitute the fundamental building blocks for quantum networks and information processing and sensing. One of the most promising candidates for single-photon sources in nanophotonic systems is optically active impurities (color centers) in diamonds.3−13 Color centers in diamonds, at their zero-phonon line (ZPL) frequencies, act as photostable and coherent single-photon emitters. In a bulk crystalline diamond, the emission rate of a ZPL transition only accounts for a fraction of the entire emission. For instance, in a negatively charged nitrogen-vacancy (NV−) center, only 4% of the emitted light is originated from the ZPL transition.9−11 The percentage of light emission into the ZPL frequency can be significantly enhanced by means of an optical cavity.14−17 The cavity modifies the emission rate, by changing the available local density of states (LDOS) for light emission. A progress has been achieved on the fabrication of resonant photonic structures out of diamond.14,15 However, the hardness of the diamond material makes shaping and molding of cavities at nano- or microscales in crystalline diamonds a strenuous process and limits the achievable performance level. © 2018 American Chemical Society

Received: January 23, 2018 Accepted: March 22, 2018 Published: May 1, 2018 4733

DOI: 10.1021/acsomega.8b00139 ACS Omega 2018, 3, 4733−4742

Article

ACS Omega We show that deterioration to the quality factor (Q) of the cavity due to the presence of the nanopocket is recovered by filling the nanopocket with the nanodiamond. The nanodiamonds can be positioned in the nanopocket using the pick and place technique that make uses either an atomic force microscopy tip39−41 or an scanning electron microscope with a nanomanipulator.42,43 With the DINP structure, the antinodes of the cavity relocate to the nanosized air gap regions within the DINP structure and elevate the electric field across the nanodiamonds. The DINP structure thus becomes an essential part of the cavity, creating a prime influence on the mode volume and enabling strong coupling of an electric field to the color centers in the nanodiamond. To illustrate the idea of DINP, we explicitly considered a geometrical structure as in Figure 1aa SiN photonic crystal

Figure 2. (a) Three-dimensional illustration of the L3 cavity with the DINP structure. The inset (red box) distinguishes the pocket and the hole. The holes surround the cavity, and the pocket is located right at the (x, y) center of the L3 defect. (b) Enlarged view of the DINP structure. The cube represents the nanodiamond. The insets depict an arbitrarily positioned color center within the nanodiamond. The arrow represents the randomly oriented, symmetrical axis of the color center (axis of orientation). (c) Cross section of the DINP structure on the x = 0 plane. The cross-sectional plane shown by the dashed line represents the xy plane that cuts through the center of the nanodiamond.

imentally achievable figure of merits are shown for cavities with experimentally feasible quality factors and ideal quality factors. Section 3.3 addresses the robustness of the DINP structure with respect to perturbations in the shape, orientations, and the position of the nanodiamond. In section 3.4, a statistical analysis of Purcell factors with realistic dipole configurations for negatively charged silicon-vacancy (SiV−) centers9−11 and NV− centers12,13 is presented. We present achievable statistical quantities, assuming the positions of the color centers, and their axis orientations as random variables. Finally, section 4 concludes the paper.

Figure 1. (a) Schematic of the backbone SiN L3 Cavity. (a) Geometry of the L3 cavity on the xy plane. The photonic crystal slab has periodicity a = 290 nm and slab thickness a. The holes away from the cavity have radii 0.3a (white holes). To optimize Q, the hole radii in the immediate surrounding of the cavity are adjusted to 0.25a (orange holes) and 0.2a (purple holes). Additionally, the purple-colored hole is shifted to 0.2a horizontally, away from the cavity. (b) Cross section (xy plane, z = 0) of the localized cavity mode profile, |E|.

2. BACKBONE CAVITY AND THE DINP STRUCTURE Figure 1a shows the schematic of the photonic crystal L3 cavity. The L3 cavity is created by removing three central holes from a slab patterned with holes in a triangular lattice. The slab has a thickness a, and it is surrounded by air. The Q of the resulting cavity is optimized22 by adjusting the radii and the positions of the surrounding cavity holes (see Figure 1a). The optimized Q is ∼4700. Figure 1b displays the electric field cross section of the fundamental mode profile on the z = 0 plane. The fundamental mode of the L3 cavity has a negligible z component, and the electric field around the central antinode (center of the L3 cavity) is polarized toward the y-direction. Figure 2 exhibits the L3 cavity with a DINP structure. The DINP structure is created right at the center of the L3 cavity on the x−y plane. The coordinate coincides with the (x, y) position of the antinode in the fundamental mode (see Figure 1b). The DINP structure consists of a nanopocketnanohole of diameter dp and depth t, and a nanodiamondmodeled as a cube44 of edge d. We coined the term “pocket” to distinguish this special hole which has a finite depth and holds the nanodiamond, to the rest of the holes protruding through the entire SiN membrane that makes up the L3 cavity (see the inset in Figure 2a). The t = 0 situation recovers the case18−23 where the nanodiamond is positioned on the surface of the cavity. We

L3 cavity membrane. L3 cavities have been proven to be very useful in various cavity quantum electrodynamics experiments in both strong35,36 and weak coupling regimes.37,38 The choice of the SiN material for making the cavity has the following basis: SiN has a refractive index of 2.0, and it is transparent for visible light. The refractive index of SiN is closer to the refractive index of nanodiamond (2.4), and hence, the losses due to the index mismatch can be alleviated. The fabrication method is highly compatible with the conventional CMOS silicon technology, and good quality L3 cavities made of SiN have been experimentally demonstrated for light at visible wavelengths.22,33 Figure 2 manifests the geometry of the DINP structure. The details of the L3 cavity and the DINP structure will be further elaborated in section 2. As we shall show, the proposed DINP structure is robust with respect to perturbations in the shape, orientations, and the position of the nanodiamonds within the nanopocket. The paper is organized as follows: after the introduction, section 2 describes the geometry of the L3 cavity and the DINP structure. Section 3.1 presents the quality factors of the DINP structure. In section 3.2, the details of Purcell enhancement using DINP are elaborated. Theoretical limits and exper4734


Article

ACS Omega

Figure 3. (a) Quality factors of the DINP structure as a function of nanodiamond size. The purple line represents the quality factor of the backbone cavity. When pockets of diameters dp = 50, 70, and 100 nm of the same thicknesses (t = 100 nm) created, quality factors drop to the levels represented by blue-, red-, and dark yellow-colored lines, respectively. When the respective pocket is filled with the nanodiamond of size d, the quality factor (circle) increases as a function of d. (b) Quality factors of the DINP (dp = 100 nm and d = 60 nm) structure as a function of pocket depth. (c) First derivative of the quality factor in (b) with respect to the pocket depth. (d) Second derivative of the quality factor in (b) with respect to the pocket depth.

When the pocket is filled with a nanodiamond of size d (i.e., the DINP structure), the reduced Q is partly recovered. The fraction of Q that is recovered increases as a function of d (see Figure 3a). Figure 3b presents the Q of the DINP structure as a function of pocket depth t, assuming a fixed size d = 60 nm and a fixed dp = 100 nm. The curve in Figure 3c represents the general trend of the Q as a function of t in DINP structures. As t increases, Q decreases. The curve Q versus t exhibits an inflection point at the critical depth, tc = 70 nm. The inflection point occurs when the second derivative of Q with respect to t vanishes. In Figure 3c,d, we show the first and second derivatives of Q with respect to t, respectively. For t < tc, the Q drops slowly, and for t > tc, the Q drops rapidly. If the pocket diameter is small, a large critical depth is required to inflect the Q with respect to t. In other words, when the opening of the pocket is small, it takes a larger pocket depth to deteriorate the quality factor of the cavity. In this article, for the purpose of illustration, we considered three DINP structures with small, medium, and large diameters of dp = 50, 70, and 100 nm, respectively. The small, medium, and large diameter pockets are filled with commercially available diamonds of d = 30, 40, and 60 nm, respectively. The diamonds sizes are chosen such that the cross-sectional filling ratio (i.e., the cross-sectional area of diamond/area of circle; see Figure 2c) at the center planes of the diamonds is nearly equal. 3.2. Purcell Enhancements with DINP Structures. To analyze the emission rate of the color center within the DINP structure, let us analyze the optical properties of the DINP structure as a function of t. Figure 4 shows electric field distributions of the cavity mode on the x = 0 plane for the y component of the field. x and z are components that are relatively very weak and thus are not shown. The electric field distributions are shown for the backbone cavity (i.e., without

assume that the nanodiamond carries a single color center, at an arbitrary location and at an arbitrary axis orientation. Figure 2b illustrates the color center and its axis within the nanodiamond. The axis of the color center represents the symmetry axis of the defects. Though the nanodiamond is modeled as a cube, the shape of the diamond has little relevance to the conclusion made in this article and evidence supporting this statement will be presented in section 3.3. For concreteness, throughout this article, the dimensions of the L3 cavity and the DINP structure are based on the ZPL emission wavelength of the SiV− color center (λSiV = 738 nm). Nevertheless, the results are equally applicable for NV centers, provided a scaling in the geometry by a factor of λNV/λSiV, where λNV = 638 nm is applied. This advantage due to scaling is the result of negligible dispersion in refractive indices of SiN and nanodiamond across the wavelengths, λSiV and λNV. In this article, we assumed that the emission frequency of the color center is resonant with the frequency of the cavity mode. Practically, a discrepancy between the resonance frequencies of the emission and the mode of the fabricated cavity is expected. Therefore, a tuning mechanism of the cavity resonance is necessary to achieve the maximum Purcell enhancement. Tuning mechanisms such as gas injection and local heating have been part of many Purcell enhancement experiments reported previously (e.g., see refs 14 and 18).

3. RESULTS AND DISCUSSION 3.1. Quality Factors with DINP Structures. Figure 3 showcases the Q of L3 cavities with DINP structures. The purple line in Figure 3a represents the Q of the L3 cavity without the DINP structure. As expected, when a pocket is created, the Q drops. In Figure 3a, the dropped Q levels for pockets with t = 100 nm are shown. Specifically, the horizontal solid lines in blue, red, and dark yellow represent the Q levels of the empty pockets with dp = 50, 70, and 100 nm, respectively. 4735


Article

ACS Omega

the interior of the nanodiamond (see t = 70 and 120 nm cases in Figure 4). As we will show later, in the statistical analysis, this is favorable as the LDOS will be less sensitive to the random position of the color centers which can take a position anywhere within the volume of the nanodiamond. The Purcell factor45,46 represents the normalized emission rate (or decay rate) of the emitter when it is placed inside the optical cavity. The rate is normalized with respect to the rate in the bulk homogeneous medium. The Purcell factor also represents the normalized LDOS when the emitter is placed inside the optical cavity. As long as Purcell enhancements are of the interest, classical electromagnetism is sufficient to describe the physics of light emission from the color center.47 Let us first evaluate the emission rate of a dipole emitter in an optical microcavity. The power P dissipated by a current density J(r) is 1 given by the integral − 2 ∫ Re{J* (r) ·E(r)}d3r , with E(r) being the electric field. If the position of the dipole is rc, then the resulting current density can be expressed as J(r) = −iωpδ(r − rc) with ω and p being the angular frequency and dipole moment, respectively. Substituting this into the expression for P and dividing by the power emitted by the dipole in the bulk diamond of refractive index n = 2.4, we have

Figure 4. Cross sections of the y component of the electric field |Ey|/ max(|Ey|) on the x = 0 plane. The x and z electric field are components that are relatively very weak, and they are not shown.

the DINP) and t = 0, 70, and 120 nm. As we can clearly see from this figure, when t = 0, the field distribution is quite similar to the backbone cavity. The antinode position for t = 0 is right at the center of the SiN membrane. However, with the DINP structure (i.e., t = 70 and 120 nm cases), the electric field distribution is strongly modified and the antinode position moves to the nanosized air gap regions between the nanodiamond and the L3 cavity. This relocation of the antinode is the direct consequence of electric field discontinuity between the SiN−air−nanodiamond interfaces.29,30 The redistribution of the electric field also elevates the amount of field that penetrates the nanodiamond quite significantly. The region around the antinode has a very sensitive variation in the electric field values. Thus, the occurrence of the antinode in the air gap regions leaves a more uniform field to prevail through

F=

P 6πc =1+ Im{p*·Es(rc)} P0 μ0 n|p|2 ω3

(1)

where Es(rc) is the scattered electric field with the dipole as a point source.47,48 For the sake of illustration, and as to establish a figure of merit, in this section, we defined a Purcell factor, F0, which assumes that the dipole is located right at the center of the nanodiamond (r = r0) and the dipole moment is oriented along the dominant direction of the cavity’s electric field (i.e., ydirection). In section 3.4, we relaxed these assumptions and presented a statistical analysis to show how the presented results vary for specific color centers with random distributions of locations and orientations. To evaluate eq 1, we solved three-dimensional (3D) Maxwell equations using the finite difference time domain method49,50

Figure 5. Optical properties of the L3 cavity with the DINP structure (a) figure of merit (F0) at the cavity resonance wavelength, as a function of the pocket depth. Blue circleseq 1. Red asteriskseq 2. (b) Cavity resonance wavelength as a function of pocket depth. (c) Wavelength response of F0. Blue circleseq 1. Red lineeq 2. (d) Mode volume (eq 3) as a function of pocket depth. 4736


Article

ACS Omega

Figure 6. (a) Quality factors and (b) F0 (assuming no losses in fabrication) as a function of pocket depth for three different DINP structures (small DINPblue; medium DINPred; and large DINPgreen). (c) Quality factors and (d) F0 (with fabrication losses phenomenologically included) as a function of pocket depth for three different DINP structures (small DINPblue; medium DINPred; and large DINPgreen).

using a dipole source. The field created by the dipole consists of its own field and the scattered field. The scattered field [Es(r)] is calculated by Fourier transforming the portion of the electric field after the dipole excitation is switched off (for more details please see ref 51). Figure 5a shows the computed results for the DINP with dp = 100 nm and d = 60, as a function of pocket depth, at the resonance wavelength of the system comprising the L3 cavity−DINP structure. The resonance wavelengths are shown in Figure 5b. As it can be seen from Figure 5a, F0 increases as a function of t, reaches the maximum when t = 80 nm, and decreases for t > 80 nm. For t = 80 nm, F0 is about five times larger when compare to F0 in the case where the nanodiamond is on the surface of the cavity (i.e., t = 0). The wavelength response F0 is non-Lorentzian. As an example, Figure 5c shows the wavelength response of F0 for t = 130 nm. As we can see from Figure 5c, the shape of the wavelength response is non-Lorentzian. To validate the first-principles calculation from eq 1 and to gain a more intuitive perspective on the Purcell enhancement, we revisited Sauvan’s theory of spontaneous emission52 for leaky optical cavities. The theory assumes that the electric field in eq 1 can be expanded in terms of quasi-normal modes of the cavity and defeats the earlier formulations,45,46 in which all neglect the leaky component of the cavity eigenmode. In the case of nanodiamond, the Purcell factor from Sauvan’s original formulation can be expressed as

LA (λ) =

⎡ λ − λc Im(V0) ⎤ ⎢1 − 2Q ⎥ λ Re(V0) ⎦ ⎣

F0 =

Q V0

is the interested Purcell factor at the

real

3 4π 2

λc 3 n

( ){ } Q V0

helps to understand

real

the trend observed in Figure 5a. For a shallow t, Q drops slowly (Figure 3b) but V drops rapidly, resulting in an increasing Q/V and hence an increasing F0. For a large t, Q drops rapidly and V drops slowly, resulting in a decreasing Q/V and hence a decreasing F0. Therefore, there should be an optimal t = to where Q/V is maximized. Figure 6a,b compares Q and F0 of the three DINP structures. The blue line corresponds to a small-sized DINP with dp = 50 nm and d = 30 nm. The red line corresponds to a mediumsized DINP with dp = 70 nm and d = 40 nm. The dark green line corresponds to a large-sized DINP with dp = 100 and d =

(2)

∫ {E(r)·ε(r)E(r) − μ0 H(r)·H(r)}d3r 2ε0n2[E(r0)]2

λc 3 n

( ){ }

The expression F0 =

where p̂ is the unit vector of the dipole moment. Here, the mode volume V0 is a complex quantity, and the line-shape function LA(λ) is non-Lorentzian. The expressions for V0 and LA(λ) are V0 =

3 4π 2

cavity resonance wavelength. The Purcell factor and the wavelength response evaluated using the analytical F0 and LA(λ) are shown in Figure 5a,c, respectively. As one can see, a very good agreement is found between the analytical F0 (from eq 1; shown as red asterisks in Figure 5a) and the numerical F0 (from eq 2; shown as blue circles in Figure 5a). Throughout this article, unless stated otherwise, results are obtained using eq 1. Figure 5d plots the mode volume (eq 3) as a function of t. The presence of the DINP structure shrinks the mode volume significantly. The mode volume decreases as t increases, but the amount of decrease in the mode volume reduces as a function of t.

LA (λ) real

(4)

ε(r) and H(r) are the position-dependent relative permittivity and magnetic field, respectively. To validate the results in Figure 5a,c, let us take rc = r0 and assume p̂ is oriented along the direction of E(r0). This yields F = F0LA(λ), where

3⎧

2⎫ 3 ⎛ λc ⎞ ⎪ Q ⎡ E(r0) ⎤ ⎪ ⎨ ⎬ ⎜ ⎟ F= ⎢ ⎥⎪ 4π 2 ⎝ n ⎠ ⎪ ⎩ V0 ⎣ p̂ ·E(rc) ⎦ ⎭

⎞ λ2 ⎛ λ2 ⎟ ⎜ 2 2 2 2 λc ⎝ λ + 4Q (λ − λc) ⎠

(3) 4737


Article

ACS Omega

Figure 7. (a) F0 as a function of pocket depth (reproduced from Figure 5a, blue circles). (b) Effect of asymmetry in the nanodiamond orientation and position to the LDOS. (c) Electric field distribution (on the xy plane that cut through the center of the nanodiamond) as a function of orientation.

Figure 8. Effect of the nanodiamonds shape to (a) Q and (b) F0 as a function of pocket depth. The circle and asterisk represent cube and spherical shapes, respectively.

the optimal depths are 160, 140, and 100 nm for the small-, medium-, and large-sized DINP structures, respectively. 3.3. Effect of Nanodiamond Orientation and Shapes. In sections 3.1 and 3.2, we assumed that the nanodiamond takes the shape of a cube of edge d. The orientation of the cube is assumed such that the symmetrical axis of the cube matches with the symmetrical axis of the underlying L3 cavity. Here, we would like to describe the influences of cube orientations and the shapes to the level of LDOS that can be attained. Let us perturb, the orientation of the cube, by rotating the cube around the z-axis by an angle θ. For the sake of discussion, let us evaluate the changes in the results of Figure 5a which represents the case, θ = 0 (reproduced in Figure 7a). Figure 7b shows the calculated F0 for nonzero values of θ. The figure also illustrates a special case, R. In this special case, the cube origin on the x−y plane is displaced arbitrarily from the center of the nanopocket. Figure 7c displays the x−y cross-sectional electric field distributions (along the plane that cut through the center of the nanodiamond) for various θ and in the special case R. As we can clearly see from these figures, the orientations and the random position have almost no effect to F0. We would like to point out the fact that the LDOS can be enhanced by DINP, and the enhancement can be maximized at the critical depth, which is generally true for all shapes of nanodiamonds. As evidence, in Figure 8, we show the result of recalculations of Q and F0, with a volume-wise equivalent,

60 (reproduced from Figure 3b and Figure 5a). As we have stated earlier, tc increases as dp decreases. This can be clearly seen from Figure 6a. The inflection points of Q occur at the critical depths of 90, 80, and 70 nm for the small-, medium-, and large-sized DINP structures, respectively. From Figure 6b, we can see that the small DINP structure exhibits the largest F0. The maximum F0 for each DINP occurs at its optimal depth. The optimal depths are 140, 100, and 80 nm for the small-, medium-, and large-sized DINP structures, respectively. Now let us ask the question, provided the Q of the fabricated structures will be lower than the ideal theoretical limit, what values of Purcell factors can be expected in the experimental conditions. For this reason, we revert to experimentally measured quality factors for SiN cavities. In the literature, quality factors on the magnitude of 1500 are consistently reported for SiN L3 cavities at the visible wavelength regime.22,33 To achieve the same quality factor in the numerical simulations, we phenomenologically include an imaginary part of 5.2 × 10−4 in the refractive index of SiN. This will account the intrinsic losses suffered because of the fabrication imperfections. The recalculated Q and F0 are displayed in Figure 6c,d, respectively. As expected, we see a surge in the initially calculated F0. Nevertheless, the F0 of DINP structures is still well above to that of the nanodiamond on surface cases (i.e., t = 0). With the loss included, the optimal depth (to) of the cavity is slightly increased. From Figure 6d, we can see that 4738


Article

ACS Omega spherical-shaped nanodiamond (shown as asterisks). The spherical shape is chosen because it represents a direct contrary to the cube as the sphere contains no sharp edges. For a comparison, in Figure 8, we reproduced the results for cube with d = 60 nm and with the nanopocket of dp = 100 nm from Figures 3b and 5a and indexed them with circles. On the other hand, the results with an equivalent spherical nanodiamond which has a diameter of 80 nm in the same-sized nanopocket are indexed with asterisks. As it is clear from Figure 8, both shapes yield similar Q and F0. As can be seen from Figure 7c, the antinode occurs in the nanosized gap regions between the nanodiamond and the L3 cavity (along the y-direction: the cavity field direction, see Figure 4). The maximum of electric field increases as the gap narrows. Thus, the occurrence of sharp edges (which induce narrow gaps) increases the strength of the antinodes in the gap regions. The number of antinodes will increase with the number of sharp edges occurring along the y-direction. However, these antinodes have less impact to the electric field in the interior of the nanodiamond (see Figure 7c), where the color center (which is considered as a deep-level defect) is located. Therefore, the exact shape has a minimal effect to the Purcell enhancement. This is also vivid from Figure 8 where we have compared F0 of cube (four edges) and sphere (no edge). The F0 of a cube is only slightly higher than that of a sphere for t < 90 nm and almost equal for t ≥ 90 nm. 3.4. Statistical Analysis of Color Center Positions and Their Orientations. In sections 3.2 and 3.3, the figure of merit, F0, represents the Purcell factor when a dipole-like emitter is placed at the center of the nanodiamond. We assumed that the dipole moment is in perfect alignment with the orientation of the local cavity field. These assumptions, although useful as a figure of merit to benchmark DINP structures with respect to their geometrical parameters, they may not represent the actual conditions of the color centers in nanodiamonds. First, the color center may be accompanied by more than one dipole. Second, the position of the color center and the dipole moment orientations can vary to a large extent. In this section, we address these issues statistically and demonstrate realistic Purcell factors that can be expected. Figure 9a,b exhibits the formalism of the statistical analysis. Assuming the shape of the nanodiamond as cubical, we selected 100 random positions of color centers (rc) with random axis orientations, within the interior of the cube (see Figure 9a, where one case of the color center with a random position and orientation is depicted). As color centers are considered as deep-level defects,9 we discarded any possibilities of having a color center within 5 nm distance from the faces of the cube.10 The general orientation of the color center axis with respect to the coordinate system of the L3 cavity is depicted in Figure 9b. The angles θ and ϕ in Figure 9b are treated as random variables. Figure 9b showcases the configurations of dipole moments with respect to the axis of the color center. As illustrated in this figure, depending on the nature of the color center, the number and the orientations of the dipole moments can vary. For simplicity, we considered three scenarios: A, B, and C (see Figure 9c). At cryogenic temperatures, SiV centers12 have four ZPL transitions. All of the four transitions can be modeled as single dipole emissions.12,13 Therefore, a configuration as in A is applicable for the SiV center. On the other hand, light emission from a strain-free NV center8−11 has a single ZPL transition. The transition is modeled using a pair of orthogonal

Figure 9. Statistical analysis of the Purcell factors assuming the positions of the color centers and their axis orientations as random variables. The considered DINP structure has dp = 50 nm and d = 30 nm. (a) Schematic of the randomly positioned and oriented color center within the nanodiamond. The definition for the axis orientation of the color center with respect to the xyz coordinate system is shown on the right panel. The angles θ and ϕ are treated as random variables. (b) Three different scenarios of dipole moment orientations with respect to the axis of the color center. (c) Results of the statistical analysis with experimentally L3 feasible cavities. (d) Results of the statistical analysis with ideal L3 cavities. The inset in (c) indexes the information that the box plot carry.

dipoles (with equal strength of dipole moment), 10,11 perpendicular to the NV axis. Thus, the configuration in B is applicable for the NV center. In addition to the configurations in A and B, we also considered the third scenario as in C, assuming we can adjust the dipole orientation (as it has been demonstrated in experimental cases39−43) such that it is parallel to the local electric field. Using eq 2, the Purcell factors are evaluated for 100 different combinations of rc, θ, and ϕ and the resulting distributions are statistically analyzed. In B, the Purcell factor represents the average Purcell factors of the two orthogonal dipoles (see 4739


Article

ACS Omega Figure 9b). In all cases of A, B, and C, we evaluated the fivenumber summary, consisting of the minimum, first quartile (Q1, 25th percentile), median, third quartile (Q3, 75th percentile), and maximum values of the distributions. The maximum and minimum values represent the extreme values after the outliers are discarded. In statistics, outliers are the data points that are numerically distant from the rest of the points. They represent rarely occurring, extreme Purcell factors. Numerically, outliers are identified as data points below Q1 − 1.5(Q3 − Q1) and above Q3 + 1.5(Q3 − Q1). Let us consider the DINP structure with dp = 50 nm and d = 30 nm. The corresponding figure of merit (F0) as a function of t is presented in Figure 6b,d. What kind of actual Purcell factors (F) can be expected, given the randomness in the position of the color center, and the randomness of the color center orientation? The result of statistical analysis of F for the three different scenarios, A, B, and C, are shown in Figure 9c. In this figure, the five-number summaries are presented in the form of boxplots (see the inset in Figure 9c for information) as a function of t. Let us first take case A with the experimentally feasible L3 cavities (as considered in Figure 6d) and quantitatively examine F as a function of t. When t = 0, the median of F is 0.5, and the maximum (excluding outliers) of F is 2.5. In all of the three configurations, the statistical parameters increase as a function t and reach the optimal value at the optimal depth, t = t0 (this trend and the value of the optimal depth are the same as those for F0 in Figure 6). When t = to = 160, the median of F is 5.6 and the maximum is 35.5. The median of F at t = to for B and C is, respectively, 13.8 and 35.0. In ideal cavities, the statistical quantities are at least larger than twofold of those in experimentally feasible cavities (see Figure 9d). Comparing configurations A and B, B produces a more consistent Purcell factors (compare the length of the box in A and B which is a sign of data consistency). In A, the axis orientation is quite critical. As there is only one dipole in A, the misalignment of the dipole orientation with respect to the cavity field strongly affects the Purcell factor. This makes, the Purcell factor distributions for A to be skewed to the right. On the other hand, in B, as there are two orthogonal dipoles, when one dipole is completely misaligned with the orientation of the electric field, there is another dipole perpendicular to it which helps to elevate the Purcell factor. This makes the Purcell factor distribution of B to be more symmetric and consistent compared to that of A. Given the cons of A over B, there is one important advantage in configuration A. That is, the maximum achievable Purcell factors in A is always greater than those of B. The maximum Purcell enhancement occurs when the dipoles are in parallel orientation with the electric field direction. In B, there are two orthogonal dipoles of equal strength and only one dipole can be maximally aligned with the electric field. Thus, one dipole will have maximum enhancement, and the other will be minimally enhanced. In C, the dipoles are perfectly aligned with the orientation of the cavity field. Thus, the statistical parameters for configuration C substantially outperform configurations A and B, where the dipole orientations are random. The C scenario describes the distribution in the Purcell factors solely because of the randomness in the color center position within the interior of the nanodiamond. Although, we have a perfect alignment in C, the Purcell factors still vary as the electric field values vary from position to position in the interior of the nanodiamond (see Figures 4 and 7c).

Table 1 displays the numerical summaries (mean and median) of the Purcell factor distributions for two important Table 1. Mean and Median of Purcell Factor Distribution for the Dipole configurations A, B, and C for the Statistical Analysis in Figure 9a A t (nm) t=0 t = 160

mean median mean median

B

C

Expt

ideal

Expt

ideal

Expt

ideal

0.8 0.5 9.9 5.6

2.1 1.3 21.8 12.3

1.1 1.0 12.5 13.8

3.5 3.1 33.7 36.2

2.9 2.7 34.6 35.0

9.1 8.4 88.5 88.8

a

The t = 0 case represents diamond on the surface of the cavity. The t = 160 nm represents the DINP structure at its critical (optimal) pocket depth. The results are shown for experimentally feasible L3 cavities (Expt) and ideal L3 cavities (ideal).

cases. The t = 0 case represents the diamond on the surface of the cavity. The t = to = 160 nm represents the DINP structure at its optimal pocket depth. The results are shown for all three configurations of the dipole moments (A, B, and C) and for both experimentally feasible L3 cavities and ideal L3 cavities. Remarkably, in all cases, the numerical summaries for the DINP structure at its critical depth surpass the diamond-on-surface case nearly by a factor of 10.

4. CONCLUSIONS In conclusion, we have introduced a DINP structure as a new route to achieve strong enhancements of light emission rates from the colors centers of the nanodiamonds. The DINP structure consists of a nanopocket and a nanodiamond. As an illustration, we incorporate the DINP structure into a silicon nitride photonic crystal L3 cavity and show that the accessible LDOS increases as a function of pocket depth. The LDOS is maximized at a certain critical pocket depth and it is robust with respect to perturbations in the shapes and orientations of the nanodiamond. To analyze the feasible Purcell factors, we considered experimentally feasible L3 cavities and performed statistical analysis, assuming random positions and orientations of color centers within the nanodiamond. Using appropriate dipole models of SiV and NV centers, we statistically prove that the microcavity with DINP structures enables a significant Purcell enhancement compared to the microcavity without the DINP structure.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (G.A.). ORCID

Gandhi Alagappan: 0000-0002-4730-2503 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is supported by NRF-CRP14-2014-04, “Engineering of a Scalable Photonics Platform for Quantum Enabled Technologies”. 4740


Article

ACS Omega

■

(23) Schell, A.; Kaschke, J.; Fischer, J.; Henze, R.; Wolters, J.; Wegener, M.; Benson, O. Three-dimensional quantum photonic elements based on single nitrogen vacancy-centres in laser-written microstructures. Sci. Rep. 2013, 3, 1577. (24) Chang, L.; Jiang, X.; Hua, S.; Yang, C.; Wen, J.; Jiang, L.; li, G.; Wang, G.; Xiao, M. Parity−time symmetry and variable optical isolation in active−passive-coupled microresonators. Nat. Photonics 2014, 8, 524−529. (25) Hosseini, E. S.; Yegnanarayanan, S.; Atabaki, A. H.; Soltani, M.; Adibi, A. High Quality Planar Silicon Nitride Microdisk Resonators for Integrated Photonics in the Visible Wavelength Range. Opt. Express 2009, 17, 14543−14551. (26) Cai, D.-P.; Lu, J.-H.; Chen, C.-C.; Lee, C.-C.; Lin, C.-E.; Yen, T.J. High Q-factor microring resonator wrapped by the curved waveguide. Sci. Rep. 2015, 5, 10078. (27) Ma, P.; Choi, D.-Y.; Yu, Y.; Yang, Z.; Vu, K.; Nguyen, T.; Mitchell, A.; Luther-Davies, B.; Madden, S. High Q factor chalcogenide ring resonators for cavity-enhanced MIR spectroscopic sensing. Opt. Express 2015, 23, 19969−19979. (28) Khan, M.; Babinec, T.; McCutcheon, M. W.; Deotare, P.; Lončar, M. Fabrication and characterization of high-quality factor silicon nitride nanobeam cavities. Opt. Lett. 2011, 36, 421−423. (29) Robinson, J. T.; Manolatou, C.; Chen, L.; Lipson, M. Ultrasmall Mode Volumes in Dielectric Optical Microcavities. Phys. Rev. Lett. 2005, 95, 143901. (30) Ryckman, J. D.; Weiss, S. M. Low mode volume slotted photonic crystal single nanobeam cavity. Appl. Phys. Lett. 2012, 101, 071104. (31) Akahane, Y.; Asano, T.; Song, B.-S.; Noda, S. High-Q photonic nanocavity in a two-dimensional photonic crystal. Nature 2003, 425, 944−947. (32) Song, B.-S.; Noda, S.; Asano, T.; Akahane, Y. Ultra-high-Q photonic double-heterostructure nanocavity. Nat. Mater. 2005, 4, 207−210. (33) Gopinath, A.; Miyazono, E.; Faraon, A.; Rothemund, P. W. K. Engineering and mapping nanocavity emission via precision placement of DNA origami. Nature 2016, 535, 401−405. (34) Barth, M.; Nüsse, N.; Stingl, J.; Löchel, B.; Benson, O. Emission properties of high-Q silicon nitride photonic crystal heterostructure cavities. Appl. Phys. Lett. 2008, 93, 021112. (35) Englund, D.; Faraon, A.; Fushman, I.; Stoltz, N.; Petroff, P.; Vučković, J. Controlling cavity reflectivity with a single quantum dot. Nature 2007, 450, 857−861. (36) Majumdar, A.; Rundquist, A.; Bajcsy, M.; Vučković, J. Cavity quantum electrodynamics with a single quantum dot coupled to a photonic molecule. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 86, 045315. (37) Englund, D.; Fattal, D.; Waks, E.; Solomon, G.; Zhang, B.; Nakaoka, T.; Arakawa, Y.; Yamamoto, Y.; Vučković, J. Controlling the Spontaneous Emission Rate of Single Quantum Dots in a TwoDimensional Photonic Crystal. Phys. Rev. Lett. 2005, 95, 013904. (38) Strauf, S.; Hennessy, K.; Rakher, M. T.; Choi, Y.-S.; Badolato, A.; Andreani, L. C.; Hu, E. L.; Petroff, P. M.; Bouwmeester, D. SelfTuned Quantum Dot Gain in Photonic Crystal Lasers. Phys. Rev. Lett. 2006, 96, 127404. (39) Schell, A. W.; Wolters, J.; Kewes, G.; Schröder, T.; Aichele, T.; Benson, O. Assembly of Quantum Optical Hybrid Devices via a Scanning Probe Pick-and-Place Technique. Conference on Lasers and Electro-Optics 2012; OSA Technical Digest (Optical Society of America), 2012; p QW3H.2. (40) Schietinger, S.; Barth, M.; Aichele, T.; Benson, O. PlasmonEnhanced Single Photon Emission from a Nanoassembled Metal− Diamond Hybrid Structure at Room Temperature. Nano Lett. 2009, 9, 1694−1698. (41) Barth, M.; Nüsse, N.; Löchel, B.; Benson, O. Controlled coupling of a single-diamond nanocrystal to a photonic crystal cavity. Opt. Lett. 2009, 34, 1108−1110. (42) Ampem-Lassen, E.; Simpson, D. A.; Gibson, B. C.; Trpkovski, S.; Hossain, F. M.; Huntington, S. T.; Ganesan, K.; Hollenberg, L. C.

REFERENCES

(1) Yao, P.; Manga Rao, V. S. C.; Hughes, S. On-chip single photon sources using planar photonic crystals and single quantum dots. Laser Photonics Rev. 2010, 4, 499−516. (2) Toishi, M.; Englund, D.; Faraon, A.; Vučković, J. High-brightness single photon source from a quantum dot in a directional-emission nanocavity. Opt. Express 2009, 17, 14618−14626. (3) Hausmann, B. J. M.; Choy, J. T.; Babinec, T. M.; Shields, B. J.; Bulu, I.; Lukin, M. D.; Lončar, M. Diamond nanophotonics and applications in quantum science and technology. Phys. Status Solidi A 2012, 209, 1619−1630. (4) Aharonovich, I.; Greentree, A. D.; Prawer, S. Diamond photonics. Nat. Photonics 2011, 5, 397−405. (5) Aharonovich, I.; Neu, E. Diamond Nanophotonics. Adv. Opt. Mater. 2014, 2, 911−928. (6) Schröder, T.; Mouradian, S. L.; Zheng, J.; Trusheim, M. E.; Walsh, M.; Chen, E. H.; Li, L.; Bayn, I.; Englund, D. Quantum nanophotonics in diamond. J. Opt. Soc. Am. B 2016, 33, B65−B83. (7) Aharonovich, I.; Castelletto, S.; Simpson, D. A.; Stacey, A.; Mccallum, J.; Greentree, A. D.; Prawer, S. Two-Level Ultrabright Single Photon Emission from Diamond Nanocrystals. Nano Lett. 2009, 9, 3191−3195. (8) Inam, F. A.; et al. Modification of spontaneous emission from nanodiamond colour centres on a structured surface. New J. Phys. 2011, 13, 073012. (9) Doherty, M. W.; Manson, N. B.; Delaney, P.; Jelezko, F.; Wrachtrup, J.; Hollenberg, L. C. L. The nitrogen-vacancy colour centre in diamond. Phys. Rep. 2013, 528, 1−45. (10) Mohtashami, A.; Koenderink, A. F. Suitability of nanodiamond nitrogen−vacancy centers for spontaneous emission control experiments. New J. Phys. 2012, 15, 043017. (11) Alegre, T. P. M.; Santori, C.; Medeiros-Ribeiro, G.; Beausoleil, G. R. Polarization-selective excitation of nitrogen vacancy centers in diamond. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 76, 165205. (12) Hepp, C.; et al. Electronic Structure of the Silicon Vacancy Color Center in Diamond. Phys. Rev. Lett. 2014, 112, 036405. (13) Neu, E.; Fischer, M.; Gsell, S.; Schreck, M.; Becher, C. Fluorescence and polarization spectroscopy of single silicon vacancy centers in heteroepitaxial nanodiamonds on iridium. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 205211. (14) Faraon, A.; Santori, C.; Huang, Z.; Acosta, V. M.; Beausoleil, R. G. Coupling of Nitrogen-Vacancy Centers to Photonic Crystal Cavities in Monocrystalline Diamond. Phys. Rev. Lett. 2012, 109, 033604. (15) Faraon, A.; Barclay, P. E.; Santori, C.; Fu, K.-M. C.; Beausoleil, R. G. Resonant enhancement of the zero-phonon emission from a colour center in a diamond cavity. Nat. Photonics 2011, 5, 301−305. (16) Hausmann, B. J. M.; et al. Coupling of NV Centers to Photonic Crystal Nanobeams in Diamond. Nano Lett. 2013, 13, 5791−5796. (17) Babinec, T. M.; Hausmann, B. J. M.; Khan, M.; Zhang, Y.; Maze, J. R.; Hemmer, P. R.; Lončar, M. A Diamond nanowire single-photon source. Nat. Nanotechnol. 2010, 5, 195−199. (18) Wolters, J.; et al. Enhancement of the zero phonon line emission from a single nitrogen vacancy center in a nanodiamond via coupling to a photonic crystal cavity. Appl. Phys. Lett. 2010, 97, 141108. (19) Englund, D.; et al. Deterministic Coupling of a Single Nitrogen Vacancy Center to a Photonic Crystal Cavity. Nano Lett. 2010, 10, 3922−3926. (20) Thomas, N.; Barbour, R. J.; Song, Y.; Lee, M. L.; Fu, K.-M. C. Waveguide-integrated single-crystalline GaP resonators on diamond. Opt. Express 2014, 22, 13555−13564. (21) McCutcheon, W. M.; Loncar, M. Design of a silicon nitride photonic crystal nanocavity with a Quality factor of one million for coupling to a diamond nanocrystal. Opt. Express 2008, 16, 19136− 19145. (22) Barth, M.; Kouba, J.; Stingl, J.; Löchel, B.; Benson, O. Modification of visible spontaneous emission with silicon nitride photonic crystal nanocavities. Opt. Express 2007, 15, 17231−17240. 4741


Article

ACS Omega L.; Prawer, S. Nano-manipulation of diamond-based single photon sources. Opt. Express 2009, 17, 11287−11293. (43) van der Sar, T.; Heeres, E. C.; Dmochowski, G. M.; de Lange, G.; Robledo, L.; Oosterkamp, T. H.; Hanson, R. Nanopositioning of a diamond nanocrystal containing a single nitrogen-vacancy defect center. Appl. Phys. Lett. 2009, 94, 173104. (44) Kaupp, H.; et al. Purcell-Enhanced Single-Photon Emission from Nitrogen-Vacancy Centers Coupled to a Tunable Microcavity. Phys. Rev. Appl. 2016, 6, 054010. (45) Coccioli, R.; Boroditsky, M.; Rahmat-Samii, Y.; Yablonovitch, E.; Kim, K. W. Smallest possible electromagnetic mode volume in a dielectric cavity. IEE Proc.-Optoelectron. 1999, 145, 391−397. (46) Sanchis, L.; Cryan, M. J.; Pozo, J.; Craddock, I. J.; Rarity, J. G. Ultrahigh Purcell factor in photonic crystal slab microcavities. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 76, 045118. (47) Novotny, L.; Hecht, B. Principles of Nano-Optics; Cambridge University Press: New York, 2006. (48) Krasnok, A. E.; Slobozhanyuk, A. P.; Simovski, C. R.; Tretyakov, S. A.; Poddubny, A. N.; Miroshnichenko, A. E.; Kivshar, Y. S.; Belov, P. A. An antenna model for the Purcell effect. Sci. Rep. 2015, 5, 12956. (49) Taflove, A.; Hagness, S. C. Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed.; Artech House: Norwood, 2000. (50) Lumerical. FDTD Solutions. www.lumerical.com. (51) Ge, R.-C.; Hughes, S. Design of an efficient single photon source from a metallic nanorod dimer: a quasi-normal mode finitedifference time-domain approach. Opt. Lett. 2014, 39, 4235−4238. (52) Sauvan, C.; Hugonin, P. J.; Maksymov, I. S.; Lalanne, P. Theory of the Spontaneous Optical Emission of Nanosize Photonic and Plasmon Resonators. Phys. Rev. Lett. 2013, 110, 237401.

4742


Diamond in a Nanopocket - American Chemical Society

Recommend Documents