FEATURE ARTICLE pubs.acs.org/JPCC
Theoretical Design of Nanomaterials and Nanodevices: Nanolensing, Supermagnetoresistance, and Ultrafast DNA Sequencing Seung Kyu Min, Yeonchoo Cho, Daniel R. Mason, Ju Young Lee, and Kwang S. Kim* Center for Superfunctional Materials, Department of Chemistry, Pohang University of Science and Technology, Hyojadong, Namgu, Pohang 790-784, Korea ABSTRACT: We briefly discuss theoretical studies of the optical properties of self-assembled nanolenses, spintronics of carbon-based materials, and molecular electronics for fast DNA sequencing. We address the superresolution phenomena of nanolenses, the supermagnetoresistance phenomena of graphene nanoribbon spin valves, and ultrafast DNA sequencing based on the 2-dimensional conductance of a graphene nanoribbon on a nanochannel.
1. INTRODUCTION In the early 21st century, various functional molecules, nanomaterials, and nanodevices have been designed, synthesized, and characterized.13 On the other hand, the theoretical understanding of the physical properties of nanomaterials remains a challenge due to their strong dependence on a range of electron and photon confinement effects. A more complete theoretical understanding of diverse nanoscale phenomena is important to assist in the design/prediction of new functional nanomaterials.4,5 With the aid of understanding the individual interaction forces, we can better design novel molecular architectures that exhibit intriguing functionality. Indeed, as manifestations of cooperative effects and competitive effects in intermolecular interactions, we were able to synthesize diverse organic, polymeric, inorganic, metallic, and hybrid nanostructures.6 We have designed ionophores/ receptors for highly selective nanorecognition/sensing, hybrid nanomaterials for effective hazardous materials removal, molecular flippers for nanomechanical devices, and dynamical molecular switches for nonlinear optics.7,8 Some intriguing structures of organic nanotubes, nanospheres, and nanolenses have been designed.9,10 We investigated new nano-optical phenomena of nanolenses, which show near field focusing and magnification beyond the diffraction limit.10,11 The quantum transport calculation for the graphene nanoribbon (GNR) shows intriguing supermagnetoresistance properties promising a future spintronic device,12 and the conductance calculation of a graphene nanoribbon beneath which a single stranded DNA (ssDNA) passes along a nanochannel shows strong potential for a reliable and fast DNA sequencing device.13 In this feature article, we discuss the implementation of nanolenses for near-field imaging beyond the diffraction limit,10,11 the “supermagnetoresistance” phenomena in GNR,12 and finally the “ultrafast DNA sequencing” device based on GNR. 13 r 2011 American Chemical Society
2. NANOSCALE LENS AND NANO-OPTICS Here, we focus our attention on the self-assembly phenomena of calix-4-hydroquinone (CHQ) in the presence and absence of water. The CHQ is a calixarene-based molecule which is considered as a good building block for supramolecular selfassembly. A CHQ composed of four hydroquinone subunits has both hydrogen bond donors and ππ stacking pairs. The four inner OH groups stabilize the cone shape through the circular proton tunneling resonance, and other OH groups produce infinitely long hydrogen bond arrays along the longitudinal directions. On the other hand, the ππ stacking interactions14 take part in the growth of lateral directions leading to the formation of tubular bundles. To understand the self-assembly phenomena, we investigated the assembling phenomena of CHQs with density functional theory (DFT) calculations of various possible combinations of assembled structures derived from calixarene-based N-mers.15 As a result, a hexamer structure would be formed in the absence of water, while in the presence of water a linear tubular polymeric chain is highly stabilized by the formation of H-bonded bridges between repeating tubular octamer units. Indeed, in experiments, thin needle-like nanotube bundles were formed by addition of water molecules to CHQ in acetone.9,10 As the strength of the one-dimensional short H-bonding interaction (∼10 kcal/mol)16 is stronger than the strength of the ππ stacking interaction, the assembly along the 1-dimensional short H-bond relay is much more favored energetically. CHQ nanoscale lenses are formed through self-assembly of CHQ molecules by supersaturation.10 The small lenses are usually obtained during the self-assembling process of nanospheres. The nanospheres appear to be anisotropically springing from the surfaces of CHQ nanostructures of various morphology Received: April 22, 2011 Revised: June 17, 2011 Published: June 22, 2011 16247
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The Journal of Physical Chemistry C including nanotubes. These intermediate nanostructures of the nanosphere growth process are small sized nanolenses. The formation process of larger CHQ nanolenses is different from the small-sized ones. When the CHQ solution is evaporated, film-like structures of CHQ cover the surface of the crystals. In this case, the CHQ molecules released from the surface cannot diffuse into the solution directly. Instead, they accumulate in a small volume under the film, which induces high chemical potentials leading to the nucleation and growth of two-dimensional disk-shaped structures. Spherical curvatures are then gradually formed, as more of the released CHQ molecules reassemble in three dimensions. These lenses are “planospherical convex” (PSC) structures with a spherical face on one side and a flat face on the other side. In numerous optical characterization experiments, we have demonstrated the functionality of these nanolenses for both regular far-field imaging and immersion microscopy. Indeed, we were able to capture high-quality far-field images using a CCD camera of various shaped objects placed behind the nanolens from which the magnification and focal length were determined.10 The measured focal length of the nanolens, being much shorter than that predicted by geometrical optics, has been referred to as near-field focusing. Further, we demonstrated that line patterns with a pitch of 250 and 220 nm, beyond the diffraction limit of a conventional optical microscope, could be clearly resolved when implementing the nanolens as a nanoscale solid immersion lens (nSIL).10 The fundamental restriction on the resolution in conventional far-field optical imaging systems is the diffraction limit, such that two objects located at a distance apart less than about half the wavelength of light cannot be resolved.17 Therefore, many attempts have been made to overcome this limitation, and some of them show remarkable results, for example, superlens/hyperlens systems driven by surface-plasmon excitation18 and fluorescence microscopy driven by molecular excitation.19 While the immersion technique using a macroscopic solid-immersion lens (SIL) has been proven as a means of surpassing the free-space diffraction limit, a regular hemispherical SIL does not itself perform a role in increasing the imaging resolution aside from simply increasing the refractive index near the object surface.20 On the other hand, we have predicted that the nanolens, aside from the regular immersion effect, plays an additional role in further increasing the imaging resolution when implemented as an immersion lens. Here we discuss such unique nano-optical phenomena of the CHQ self-assembled nanoscale lens based on both optical experiments and exact finite-difference timedomain21 (FDTD) electromagnetic (EM) simulations.10,11 When the size of an optical element is much larger than the wavelength of light, wavelike features such as interference and diffraction are negligible, and the optical imaging properties can be reasonably described by geometrical optics. If the size of an optical lens decreases to the wavelength scale, it is well-known that geometrical optics is not applicable, and the wave nature of light must be considered. In the case of the nanolens, diffraction of waves through the lens aperture, diffraction at the lens edges, as well as the complicated interference of waves within the nanolens itself make a significant contribution to the optical properties of the nanolens, and as such numerical methods based on accurate solution of Maxwell’s equations become essential. Until recently, it was still not known what would happen in nanoscale lenses because no ideal nanoscale lens was available in the past. Our self-assembly approach, as a pathway to the
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Figure 1. (a) Ray-tracing simulation result of the nanolens calculated by OSLO program packages (Sinclair Optics, Inc.). (b) A finite-difference time-domain (FDTD) simulation result (Ex) obtained by FullWAVE 4.0 program (RSoft Design Group). D = 800 nm, H = 280 nm, and λ = 365 nm. (Reproduced with permission from the Nature Publishing Group (ref 10).)
fabrication of nanoscale dielectric optical elements such as the nanolens, has provided a useful means for optical studies of nanoscale lenses. Unusual magnification properties of nanolenses that were revealed in the far-field imaging experiments have been described as due to near-field focusing. This phenomenon is clearly supported by our theoretical calculations based on numerical solution of Maxwell’s equations using FDTD, which demonstrate that the nanolens focal length F is shorter than Fparaxial obtained from the ray tracing method (Figure 1). Our calculations show that the focal length tends to decrease as the lens size is decreased toward the light wavelength. Due to the high curvature of the nanolens surface, the wave path must have a shorter focal length than that of a regular macroscopic lens. Indeed, the curved surface of the lens leads to a much more deflected beam pathway than the geometrical ray path to match the amplitude and phase inside and outside the lens. As a result, it forms a small focal spot at a very short nearfield focal distance. The FDTD simulation for comparison of wave intensities depending on different positions of the light source shows that the interferences of waves through the CHQ lens leads to the short focal length (Figure 2). The beam pathways visualized along the highest intensity profiles (Figure 2b) show the case for the simple sum of two symmetric wave intensities without interference. However, the actual intensity is formed by the interference of two waves from each source and increases due to the enhancement of the amplitude by superposition (Figure 2c). The focal length tends to be much shorter for the light sources closer to the edges of the lens. Since the solid angles around the edges are much larger than that around the center, the focal length obtained by the plane wave is close to the focal length obtained by the two symmetric sources near the edges. Consequently, the focusing phenomenon of a nanolens is due to the near-field focusing whose origin arises from the stringent phase matching phenomenon at the interface of a nanolens with extremely high curvature. We have further investigated numerically using FDTD the situation when the nanolens is implemented as a nanoscale SIL (nSIL). A schematic of the numerical scheme is shown in Figure 3 (a). The focal region of a high numerical aperture (NA) objective lens (OL) of focal length f0 is determined analytically from the Richards and Wolf (R&W) vector diffraction theory as an input to the simulation—the FDTD algorithm propagates the field 16248
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The Journal of Physical Chemistry C
Figure 2. FDTD simulation results of light intensities (|Ex|2) through a CHQ lens using point sources at different positions (x = 100700 nm). D = 1.5 μm, H = 0.4 μm, and λ = 472 nm. (a) Variation of refraction angle depending on the position of a light source. (b) Simple sum of two wave intensities from each source, assuming that the waves are independently propagating without interference (|Ex|2 = |Ex1|2 + |Ex2|2). (c) Intensities of two interfering waves through the lens (|Ex|2 = |Ex1 + Ex2|2). The arrows denote the highest-intensity spots. (Reproduced with permission from the Nature Publishing Group (ref 10).)
from the source plane onto the nSIL with the refractive index n = 1.6. In practice, the object (the lithographic line pattern in our experiments) is placed just beneath the nSIL, such that an upper limit to the imaging resolution will be the full width at half maximum (fwhm) of the focal spot (i.e., the electric field intensity |E2| along this plane, referred to as the plane of interest (POI)). Our main interest was to optimize the relative position (Δz) of the OL with respect to the nSIL (this is akin to focusing the microscope) to determine the smallest focal width possible. Figure 3(b) shows an example FDTD calculation of the distribution of |E2| corresponding to where the base of the nSIL (the POI) is a distance of f0 from the OL (this corresponds to Δz = 0). The inset shows a plot of |E2| along the POI, the fwhm being the imaging resolution at this particular position of the OL. Figure 3(cd) shows the fwhm along the POI corresponding to a number of different nSILs with different diameter D, though maintaining the same height to diameter (H:D) aspect ratio of 0.375, in both a (c) face-down and (d) face-up (where the flat surface of the lens faces the OL) implementation. The solid horizontal line marks the best possible resolution one could obtain with a NA = 0.9 OL focused onto a macroscopic hemispherical SIL with refractive index n = 1.6. The presence of data points beneath this line indicates that beyond the regular immersion effect the complex nature of the interaction of the focused beam with the wavelength scale nSIL provides an additional mechanism by which even further resolution increase is achieved. Particularly, in the case of the face-up nSIL, the resolution is improved a further ∼25%11 beyond the regular immersion effect. That the nSIL provides a
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Figure 3. (a) Numerical scheme for determining the lateral focal spot width of a nanolens implemented as an nSIL. A linearly polarized (E parallel to the x axis) plane wave (λ = 532 nm) incident onto an OL with NA = 0.9 is focused onto a face-down nSIL of refractive index n = 1.6. The fwhm of the electric field intensity |E2| on the plane of interest (POI) is measured at different relative positions (Δz) of the OL. (b) |E2| on the yz plane showing the OLnSIL (D = 1 μm, H = 0.375 μm) combined focal region (scale in micrometers) at Δz = 0. [(b) Inset] |E2| on the POI indicating the fwhm (arrows). (cd) Focal spot fwhm's of faceup and face-down nSILs (aspect ratio H:D = 0.375:1) of varying diameters. The solid horizontal line corresponds to the smallest fwhm in the focal region of a NA = 1.44 OL as predicted by the Richards and Wolf vector theory. Different-shaped data points correspond to different positions of the objective lens: Δz = 0.5λ (O), 0 (b), 0.5λ (Δ), and λ (2). (See ref 11.)
resolution close to that of a macroscopic SIL is in itself interesting, considering that the complex interaction of the focused beam with the nSIL could be expected to result in severe aberrations that would rather reduce the resolution.
3. NANOSCALE ELECTRONIC MATERIALS 3.1. Transport Calculation. As the size of devices is reduced, the atomistic change in molecular junctions shows significant changes in transport phenomena. In this regard, a theoretical understanding of quantum mechanical transport phenomena is essential for the development of nanoscale devices. In the case of the ballistic transport regime, there is no scattering and no energy loss when electrons pass through the device. The conductance of the device is determined in terms of quantized conductance (G0 = 2e2/h). In the case of elastic/coherent transport, there is no energy loss of electrons since there are no interactions with the vibrational motion of nuclei (phonons). To study the nanoscale transport phenomena theoretically, we should deal with open boundary systems (stemming from the source and drain leads) under nonequilibrium conditions due to the external bias voltage. The most popular method is based on the Keldysh nonequilibrium Green’s function (NEGF) coupled to the DFT (NEGF-DFT) method.22 Using the converged electron density, we can calculate the transmission coefficient T as a function of energy (E) and the current (I) at the given bias voltage (Vb) through the LandauerB€uttiker formula23 as follows
TðE, Vb Þ ¼ Tr½ΓL ðE, Vb ÞGa ðE, Vb ÞΓR ðE, Vb ÞGr ðE, Vb Þ 16249
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The Journal of Physical Chemistry C and I ¼
2e h
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Thus, the total lesser (greater) self-energy (Σ) tot ) due to the metalmolecule contact and e-ph interaction is given by
Z ½TðE, Vb ÞffL ðE, Vb Þ fR ðE, Vb ÞgdE
where ΓL/R is the imaginary value of the self-energy of the left/ right junctions; fL/R is the FermiDirac distribution function for the left/right electrodes; and Ga/r is the advanced/retarded Green’s function. In the presence of a finite bias Vb, a system becomes nonequilibrium because of the difference in chemical potential between the left and right electrodes. The present NEGF-DFT method gives a steady-state current under the nonequilibrium condition. Using this Keldysh type NEGFDFT method, we have developed the POSTRANS program package.24 As the molecular device gets larger, the inelastic effects, such as electronphonon couplings, are important since the traversal time is long enough to interact with molecular vibrations. In the case of the inelastic/incoherent transport regime, the phonon and the electronphonon (e-ph) coupling term (Hph and He-ph, respectively) should be added to the molecular Hamiltonian He. Then, the total Hamiltonian is H ¼ He + Hph + He-ph The He-ph is given by He-ph ¼
∑R ∑i, j MijR c+i cjðb+R + bR Þ
where c+i (ci) and b+R (bR) are electron and phonon creation (annihilation) operators, respectively, and MRij is an e-ph coupling matrix element for the R phonon mode. Even though the molecular vibrations are dynamic properties and, thus, should be described in a real-time manner, the electron transport rate for a small molecule is much faster than the nuclei motion so that the e-ph coupling term can be obtained by Taylor expansion from the adiabatic BornOppenheimer approximation with respect to a given molecular coordinate. From the lowest-order expansion, the coupling matrix element is defined by sffiffiffiffiffiffiffiffiffiffiffiffiffi ∂He R p R jjæ vIν Mij ¼ Æij 2mI ωR ∂RIx Ix
∑
Ro
where νRIx is the nucleus displacement vector of the R phonon mode; ωR is the corresponding phonon frequency; mI is the mass of the I-th nucleus; and RIx is the relative coordinate along the x spatial direction of the I-th nucleus to its equilibrium position (Ro). Each component of the matrix element can be obtained from a series of DFT calculations. The additional phonon-related self-energy from the above Hamiltonian can be derived by the same renormalization step as the electrode part within the Keldysh NEGF formalism Z ∞ dε R Þ Þ M d ðR, E εÞG Þ ðEÞM R Σe-ph ðEÞ ¼ i 2π ∞ where G) represents the lesser (greater) Green’s function as the electron (hole) propagator and d) is the phonon propagator for the R mode. For the weak e-ph interaction regime, the Born approximation gives a systematic perturbation series of the phonon self-energy up to the infinite order. The lowest-order Born approximation has practically been quite successful for the description of the weak interactions in molecular systems.
Þ ðEÞ ∑tot ÞðEÞ ¼ ∑ Þ ðEÞ + ∑ ÞðEÞ + ∑e-ph
The current in the left/right contact is given by the Landauertype equation Z 2e ∞ > < IL=R ¼ Tr½ L=R ðEÞG< ðEÞ L=R ðEÞG> ðEÞdE h ∞
∑
∑
Finally, the total current in the left (or right) contact is just the sum of elastic and inelastic contributions IL ¼ Ielastic + Iinelastic where IðinÞelastic ¼ 2e=h
Z
∞ ∞
∑>L ðEÞG
