Nanofabrication by magnetic focusing of supersonic beams
Robert J. Clark, Thomas R. Mazur, Adam Libson, Mark G. Raizen
NNanofabrication by magnetic focusing of supersonic beams
Robert J. Clark, Thomas R. Mazur, Adam Libson, and Mark G. Raizen
Center for Nonlinear Dynamics and Department of Physics,The University of Texas at Austin, Austin, TX, 78712, USA
We present a new method for nanoscale atom lithography. We propose the use of a supersonicatomic beam, which provides an extremely high-brightness and cold source of fast atoms. The atomsare to be focused onto a substrate using a thin magnetic film, into which apertures with widths onthe order of 100 nm have been etched. Focused spot sizes near or below 10 nm, with focal lengthson the order of 10 µ m, are predicted. This scheme is applicable both to precision patterning ofsurfaces with metastable atomic beams and to direct deposition of material. Nanoscale fabrication is a critical tool for realizingmuch of modern technology, including information pro-cessing, biomedical research, and photonics [1–3]. Op-tical lithography, the current method for chip mass pro-duction, is used to produce many features in parallel, buthas a limited resolution due to the wavelength of lightused. Electron beam (e-beam) lithography, a method forproducing much smaller (order of 1 nm) features, is aserial, rather than parallel, method, meaning that it ismuch more time-consuming than optical lithography. E-beam, however, is frequently used to fabricate the masksthat are required by the latter. Additional methods, us-ing vacuum ultraviolet or X-ray radiation [4], or usingfocused ion beams [5], are being developed in an effortto achieve high-resolution and high-throughput nanofab-rication.One area of great potential for nanofabrication is atomlithography [6, 7]. The deBroglie wavelength of atomsis typically much less than an optical wavelength, po-tentially resulting in a much smaller diffraction-limitedspot size. Additionally, fabrication operations are paral-lelizable; much work has focused on depositing multiplelines and dots of atoms by focusing from a standing lightwave [8–12]. In addition, atom lithography is versatile inthe sense that one may either directly write structuresonto a substrate [13–17], or may pattern a resist priorto etching, as in traditional lithography, using a beam ofmetastable atoms [18]. The primary limitation of opticalfocusing is that it is difficult to focus some of the atomswithout simultaneously defocusing others, leading to sig-nificant aberrations and an unwanted underlayer of ma-terial. A method of mitigating this effect was proposed[19] and demonstrated [20], but is challenging to imple-ment in practice. Alternative approaches, such as focus-ing of atoms using macroscopic magnetic lenses [21] oran “atom pinhole camera” [22] have also been explored.All the above techniques have used effusive beams, limit-ing atomic density to around 10 atoms/cm , and haveachieved controlled feature sizes (at best) on the order of100 nm.In this Letter, we propose a new approach to atomlithography that should enable much smaller feature sizesand larger throughput. Our method consists of magnetic focusing of a supersonic beam through a nanofabricatedmagnetic mask. As nearly all atoms are paramagnetic ineither the ground state or an accessible metastable state,magnetic focusing is a very general approach. Further-more, the supersonic beam provides both high atomicflux and a low temperature T ≈
100 mK, a unique com-bination. This paper is organized as follows. We firstpresent details of our proposed method, including the su-personic source and focusing apparatus. We then providesimulation results, including spot sizes and focal lengthsfor a number of atomic species. Finally, we discuss thescalability of this scheme.Our proposed apparatus (see Fig. 1) consists of twomain components: a supersonic beam of spin-polarizedatoms, and a thin magnetic mask into which an arrayof holes of O (100 nm) width is etched, through whichthe atoms are focused onto a substrate. Depending onwhether one wishes to deposit material or pattern usingexcited-state atoms, one either entrains atoms into thebeam from one or more ovens, or excites the carrier gasto a metastable level in a discharge. The basic principle ofour method is that the very high magnetic field gradientsdue to the tiny holes in the magnetic material will beable to focus atoms, even though they are travelling athundreds of meters per second.A continuous-wave supersonic beam [23, 24] will pro-vide an atomic flux on the order of 10 atoms/sr/s [25].A typical fraction of either metastable or entrained atomsin the beam is 10 − . Following a skimmer, atoms in aspecific internal magnetic sublevel m J will be selectedby some method of magnetic filtering; either magneticguiding or deflection may be used. Alternatively, opticalpumping could be implemented on many atomic species.For some species, especially metastable noble gases, lasercollimation (by transverse laser cooling) may be appliedto increase the beam brightness by a factor of 10 orhigher, while reducing the velocity spread in the radialdirection [13, 26, 27].The focusing mask consists of a thin film of a magne-tized material, deposited on a substrate, with the mag-netization vector pointing out of the plane of the film.Such a film (specifically, an FePt film) was recently usedto build a permanent-magnet chip trap for atoms [28, 29]. a r X i v : . [ phy s i c s . a t o m - ph ] M a y FIG. 1: (a) Schematic (not to scale) of our nanolithography process. The supersonic nozzle produces a bright atomic beam,which may either be excited to a metastable level in a discharge or have atoms entrained into it from some number of ovens.Following this, the beam is collimated by passing through a skimmer. Magnetic filtering ensures that only the correct m J statearrives at the mask, which then focuses the atoms onto a substrate. (b) Schematic (not to scale) of a single hole in the maskwith proposed dimensions given. A Si N substrate supports a 300 nm-thick FePt mask with perpendicular magnetization of670 kA/m. Only small regions of the mask and substrate are shown; the actual length of the substrate is on the order of 1 µ m.The magnitude of the magnetic field within the hole in the mask is plotted (color online). Holes of diameter on the order of 100 nm will be etchedinto the film and into the substrate that supports it(Fig. 1). This substrate could be made of one of manymaterials, such as silicon nitride (Si N ). The holes inboth layers may be fabricated by conventional e-beamlithography, or possibly by optical lithography. Althoughthe e-beam process is time-consuming, it only needs tobe done once to create a “master” mask that can be usedmany times, as in the case of optical lithography.The atoms are focused by the force (cid:126)F due to the in-teraction of the atomic magnetic dipole moment (cid:126)µ withthe magnetic field (cid:126)B of the magnetized mask: (cid:126)F = ∇ (cid:16) (cid:126)µ · (cid:126)B (cid:17) . Assuming the atomic magnetic dipole adi-abatically follows the magnetic field, we may write theforce in the radial direction (normal to the propagationdirection) as F r = − µ B g J m J ( ∂ | B | /∂r ), where µ B isthe Bohr magneton and g J is the Land´e g-factor. Forthe simulations that follow, we assume a material thick-ness of 300 nm and a perpendicular magnetization ofM = 670 kA/m [28, 29]. We also choose a hole diameterof 150 nm. The magnetic fields are computed numeri-cally; the peak field near such a hole is | B | ≈ . (cid:15) represents the fraction ofatoms emitted from a skimmer of radius r s that will passthrough the focusing aperture; it is calculated as the ratioof the area of the circular hole in the mask to the areaof the atomic beam at the position of the mask. Writingthe beam divergence angle as θ , the distance from theskimmer to the mask as d , and the radius of the holeas r m , (cid:15) = r m / ( r s + d tan θ ) . Typical numbers for ourchosen conditions are r s = 125 µ m, r m = 75 nm, θ = 7 ◦ , FIG. 2: (a) Cross-section of 250 trajectories for Ne with acenter-of-mass speed of 400 m/s and a collimation distance of d = 2 m. The front face of the mask is located on z = 0, andthe center of the hole lies on r = 0. The spot size w is 8.5 nmand the focal length is f = 34 µ m. The color of the trajecto-ries changes from black to blue at the focal plane. This spotsize is less than the diffraction-limited spot size w d = 13 . w d with the distribution of atoms foundvia ray tracing. (b) Density of atoms as a function of radiusat the focal point for the data in (a). and d = 2 m, leading to (cid:15) ≈ × − . Given the beambrightness of 10 atoms/sr/s, a discharge or entrainmentefficiency of 10 − , and a loss of one in ten atoms due tomagnetic filtering, we estimate a flux through each holeof ≈ atoms/s.The primary goal of our simulations is to calculate thefocused spot sizes as the parameters of the problem, in-cluding the atomic species and the amount of collima-tion, are varied. Unless stated otherwise, we assumethat the beam is moving with a center-of-mass speed of v = 400 m/s, an appropriate value for a neon beam at77 K. We also assume the radius of the (circular) skim-mer is r s = 125 µ m. We assign each atom a randomradial velocity, due to geometric collimation, within therange ∆ v r ≈ r s v /d , where d is the distance from theskimmer to the mask. The spread in velocities along thepropagation axis ∆ v z is a property of the beam that isdue to the supersonic expansion (not to collimation). Itis assumed, based on measurements in our laboratory,to be fixed at ∆ v z = 14 m/s for neon at 77 K. Simula-tions are performed by numerically integrating the equa-tions of motion for a particle moving through the holein the mask. The spot size w is calculated as twicethe average value (cid:104) r (cid:105) of the radius of atoms at the focalpoint, weighted by the density of atoms at a given ra-dius: (cid:104) r (cid:105) = (cid:82) ( ρ ( r ) r dr ) / (cid:82) ( ρ ( r ) dr ), where ρ ( r ) is thedensity of atoms at radius r and the integral is evaluateddiscretely using dr = 1 nm.We present simulation results in Fig. 2 showing focus-ing of metastable neon. Even when many atoms reachthe substrate far from (cid:104) r (cid:105) , the density of atoms there isorders of magnitude less than near the focus. There aretwo dominant mechanisms, apart from limited collima-tion, by which the spot is broadened: aberrations, andvan der Waals attractions within the substrate and mask.Aberrations appear in the numerical solution to the fieldswithin the mask. Van der Waals forces are modelled byincluding a force term in the equations of motion that wascalculated numerically and is well-approximated as beingproportional to D − − D − , where D is the distance tothe nearest edge of the tube and D the distance to thefurthest edge. For atoms that are close enough to the wallof the tube, this force either causes the atom to collidewith the tube or to strike the substrate far from the fo-cused spot. Atoms that strike the tube walls are removedfrom the simulation, because they will, with high prob-ability, either release their internal energy, making themuseless for patterning, or scatter inelastically and not befocused. Although we obtain a simulated spot size of w = 8 . w d for ourparameters is w d = 1 . λ dB f / (2 r m ) = 13 . λ dB = h/ ( mv ) ≈ .
25 nm.Therefore, our simulations suggest that we can focus toa diffraction-limited spot size.Highlighting the generality of our method, our simula-tions show that the same mask can focus a very widerange of atomic masses with spot sizes of O (10 nm).Fig. 3 shows the spot sizes as a function of collimationdistance d . For low d , the radial velocity spread domi-nates, while for high d , van der Waals forces and sphericalaberrations dominate. In Table I, we report the expectedspot size and focal length for several species of interestat a fixed value of d = 2 m. For some atoms, laser exci-tation at a single frequency will be necessary, since theyare either non-paramagnetic or have too small a magneticmoment in the ground state. For these species, a singlephoton will suffice to pump the atom (with some proba-bility) into a suitable metastable state, which must havea lifetime larger than the time it takes for the atom to bedeposited (typically a few milliseconds). Two notableexamples are indium and gallium, which each occupy FIG. 3: Simulated spot size w as a function of collimationdistance d for four atomic species. All species except phospho-rus are in a metastable excited state (given in Table I). Theselarge- d spot sizes are smaller than the diffraction-limited spotsizes, also reported in Table I. Helium has the largest spot sizeat large d because it is affected more, due to its low mass, byvan der Waals forces, while at small d , the spot is smaller dueto its relatively high ratio of magnetic moment to mass.Species | (cid:126)µ | ( µ B ) w (nm) w d (nm) f ( µ m) State He 2 8.5 14.6 7.2 S Ne 3 10.0 13.2 32.6 P P 3 12.1 11.7 44.6 S / Ga 6 11.7 6.2 53.7 P / In 6 17.1 5.9 82.3 P / TABLE I: Spot sizes ( w ) and focal lengths ( f ) for a vari-ety of species. All are travelling at 400 m/s and have tra-versed 2 m following a 250 µ m skimmer. The approximatediffraction-limited spot size w d is also given; the actual spotsize is expected to be the convolution of the atomic distri-bution of waist w and the Airy disc of diameter w d . Themagnetic moment ( | (cid:126)µ | ), and atomic state are given for refer-ence. All except P are in metastable states. The width ofboth the substrate and the mask is 150 µ m. a P / ground state, with maximal magnetic moment µ B /3. A single photon, at 410 nm for In and at 403 nmfor Ga, would pump the atoms to a metastable P / state, with maximal magnetic moment 6 µ B , a state thatis focused very well. The branching ratio into the desiredstate is 38% for In and 67% for Ga [34].For several of the species we examined, the spotsize is limited by diffraction from the mask aperture.Since the diameter of the Airy disc is given by w d =1 . λ dB f / (2 r m ), a decrease in the ratio f /r m will re-duce the diffraction limit. Simulations with multiple lay-ers of thin film magnetic material show a decreased fo-cal length with identical mask apertures, resulting in asmaller diffraction limit with similar or better spot sizes.Alternatively, advances in the science of thin film mag-netic materials may provide for greater magnetizations,which would similarly reduce the focal length. As such,we do not regard the calculated diffraction limits forour simulated apparatus as being the best that can beachieved with this method.Among the many potential applications of our method,one of the most intriguing is the fabrication of quan-tum dots. Currently, quantum dots are most frequentlyfabricated by molecular beam epitaxy, which results inquantum dots of random size and location. Our schemecould produce dots with a position known to within afew nm and a size limited only by the Poisson fluctu-ations. Mounting the mask on a nanometer-resolutiontranslation stage increases the versatility of our method.For instance, the quantum dots could be combined withnanofabricated wires or mirrors, producing electrical oroptical interconnects between the dots. This could en-able one vision of quantum computation with quantumdots [30, 31]. Our method has many other potential uses,as well: applications to basic science include plasmon-ics [32], metamaterials [33], and quantum photonics [2],while applications of commercial interest include photo-voltaics, light sources, and light sensors.In conclusion, we have outlined a new method forfabricating a wide variety of nanoscale devices with anunprecedented combination of nanometer precision andhigh throughput. Our method relies on technologies thatare well understood, including supersonic beams andmagnetic filtering, and has no strenuous laser require-ments. 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