Baixe Designer spoof surface plasmon structures collimate terahertz laser beams e outras Notas de estudo em PDF para Engenharia Elétrica, somente na Docsity! LETTERS PUBLISHED ONLINE: 8 AUGUST 2010 | DOI: 10.1038/NMAT2822 Designer spoof surface plasmon structures collimate terahertz laser beams Nanfang Yu1*, Qi Jie Wang1†, Mikhail A. Kats1, Jonathan A. Fan1, Suraj P. Khanna2, Lianhe Li2, A. Giles Davies2, Edmund H. Linfield2 and Federico Capasso1* Surface plasmons have found a broad range of applications in photonic devices at visible and near-infrared wavelengths. In contrast, longer-wavelength surface electromagnetic waves, known as Sommerfeld or Zenneck waves1,2, are characterized by poor confinement to surfaces and are therefore difficult to control using conventional metallo-dielectric plasmonic struc- tures. However, patterning the surface with subwavelength periodic features can markedly reduce the asymptotic surface plasmon frequency, leading to ‘spoof’ surface plasmons3,4 with subwavelength confinement at infrared wavelengths and be- yond, which mimic surface plasmons at much shorter wave- lengths. We demonstrate that by directly sculpting designer spoof surface plasmon structures that tailor the dispersion of terahertz surface plasmon polaritons on the highly doped semiconductor facets of terahertz quantum cascade lasers, the performance of the lasers can be markedly enhanced. Using a simple one-dimensional grating design, the beam divergence of the lasers was reduced from ∼180◦ to ∼10◦, the directivity was improved by over 10 decibels and the power collection efficiency was increased by a factor of about six compared with the original unpatterned devices. We achieve these improvements without compromising high-temperature performance of the lasers. Metamaterials and transformation optics offer major opportu- nities for the control of electromagnetic fields5–8. The underlying paradigm is to design spatial variations of the magnitude and sign of the effective refractive index; thus, the optical path, or more generally the ‘optical space’, can be engineered in a continuous and almost arbitrary way. One can extend the concept to surface plasmon (SP) optics where the dispersion properties of SPs are tailored by nanostructuringmetallic surfaceswith designer patterns. In this context ‘metasurfaces’ or ‘metafilms’ have found interesting applications, such as subwavelength imaging9, waveguiding10,11 and the localization10,11, confinement12 and slowing of light13. Consider a structure composed of arrays of grooves with subwavelength periodicities textured on the surface of a plasmonic material (metals or highly doped semiconductors, which behave as metals in the terahertz regime; see Fig. 1a). Such a structure supports strongly confined surface waves with a dispersion relation ω(β) similar to SPs on a planar metal surface in the visible regime, as calculated by Pendry, Martín-Moreno, and García-Vidal3,4 and observed on structured metals at terahertz frequencies14. The asymptotic frequency, ωspoof, is not solely determined by properties of the interface materials and can be designed over an extremely wide range by engineering the subwavelength pattern on the interface3. If the metal can be treated as a perfect electric conductor, 1School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA, 2School of Electronic and Electrical Engineering, University of Leeds, Leeds LS2 9JT, UK. †Present address: School of Electrical and Electronic Engineering & School of Physical and Mathematical Sciences, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore. *e-mail: nyu@fas.harvard.edu; capasso@seas.harvard.edu. ωspoof = πc/2h, where h is the groove depth and c is the speed of light in vacuum4. Physically, this corresponds to first-order standing waves along the depth of the grooves. As ωspoof is primarily determined by surface texturing, one can engineer the spoof SP dispersion curve and obtain a sizeable deviation between the curve and the light line at terahertz frequencies; that is, β(ωTHz) > ko(ωTHz) (refs 3,4,14; see Fig. 1b). Here β is the in-plane wave vector of the SPs and ko is the free-space wave vector. As a result, the out-of-plane wave vector κ(ωTHz) = i √ β2(ωTHz)−k02(ωTHz) can be considerable, corresponding to confined SPs with a 1/e decay distance in the air normal to the interface equal to 1/|κ| (ref. 14). In this Letter, we demonstrate the great design potential of spoof SP structures for active photonic devices by markedly improving the performance of terahertz quantum cascade lasers (QCLs). Terahertz QCLs have undergone rapid development recently and have significant potential for applications in sensing, imaging and heterodyne detection of chemicals15–18. Terahertz QCLs with the highest operating temperature and lowest threshold current so far take advantage of the high optical confinement (near 100%) and heat removal properties of a double-metal waveguide design, in which the laser active region is located between a metal strip and a metal plane19–21. However, this leads to non-Fresnel reflection at the subwavelength laser apertures (as small as one-tenth of λo, the free-space wavelength), which gives rise to inefficient power out-coupling (power reflectivity of laser modes at the aperture up to 90%) and poor beam quality (characterized by a divergence angle ∼180◦ perpendicular to the laser material layers)20,21. The last of these is a particularly serious problem for the far-infrared heterodyne detection of chemicals because the output of terahertz QCLs (local oscillator) must be focused into a small-area Schottky diode mixer15. A number of schemes have been demonstrated to increase beam directionality and/or power out-coupling efficiency of terahertz QCLs (refs 22–27). One approach is to attach a silicon microlens22 or a metallic horn antenna23 onto one of the facets of the laser waveguide to reduce the mode impedance mismatch at the laser aperture, and thereby enhance the power output. However, this method requires meticulous manipulation and alignment of small optical components, which affects device yield and robustness. A monolithic approach would alleviate these problems. Another approach relies on processing terahertz lasers into surface-emitting structures with higher-order gratings24–26 or photonic crystals27; this approach relies on constructive interference between multiple surface emissions or a large emission area to reduce beam divergence. However, this results in devices with reduced mode confinement and therefore increases the laser threshold current 730 NATURE MATERIALS | VOL 9 | SEPTEMBER 2010 | www.nature.com/naturematerials NATURE MATERIALS DOI: 10.1038/NMAT2822 LETTERS Light line k0 ( THz) πc/2h=spoof THz ( )ω ω a h 1ω β( ) 2ω ωβ( ) 3 β( ) r 67 55 73 73 73 73 0.8 3.4 3.2 3.0 2.8 2.6 70 25 20 15 10 5 0 60 50 1/ | | ( μm ) 40 30 20 10 1.0 1.2 Re ( )/ ko Fr eq ue nc y (T H z) 1.4 1.6 1.8 β β Im ( ) (cm ¬ 1) 4 6 8 10 h (μm) 12 14 β ω β βω ( THz)ω a c d e f b 2/7/8/16 b/t/p/h = 2.5/6.5/8/12 2/7/8/16 2/7/8/16 2/7/8/16 2/7/8/16 2/7/8/16 2/4/8/8.5 2/4/8/7 2/4/8/7 2/4/8/7 2/4/8/7 (μ m ) (μ m ) κ Laser waveguide oλ Laser aperture Laser substrate Figure 1 | Terahertz plasmonic collimator design. a, By texturing a metal or a metallic semiconductor surface with subwavelength structures of various geometries, one can engineer the dispersion of SPs. In this way, complex designer plasmonic structures can be constructed to greatly improve device performance or to realize new functionalities. b, Schematic dispersion curve for terahertz spoof SPs on a perfect metal. The asymptote of the curve, ωspoof, the in-plane wave vector, β , and the out-of-plane wave vector, κ = i √ β2−k20, can be tailored by changing the geometry of the subwavelength grooves. c, Schematic of a terahertz QCL patterned with a spoof SP collimator. The plasmonic patterns are directly sculpted on the highly doped GaAs facet of the device. Artificial colouring in the figure indicates deep and shallow spoof SP grooves. The ‘blue’ grooves adjacent to the laser aperture increase device power throughput by coupling more laser output into spoof SPs on the facet; the deep ‘pink’ grooves modulate the dispersion properties of SPs on the facet, creating a second-order grating for power out-coupling; the shallow ‘blue’ grooves contribute to SP confinement. d, Cross-section of the design for a λo= 100 µm device. All of the grooves have trapezoidal cross-sections to resemble structures fabricated by FIB milling. The bottom and top of the grooves, their period and depth are labelled as b, t, p and h, respectively. e, The black curve is the dispersion diagram of surface waves on the planar semiconductor/air interface. In this frequency range, it is essentially linear with a slope extremely close to the speed of light in vacuum, a manifestation of the poor confinement of surface waves at terahertz frequencies. The red curves are the dispersion diagrams corresponding to the different sections of the collimator. Red solid curve: b/t/p/h= 2/4/8/7 µm; red dashed curve: b/t/p/h= 2/4/8/8.5 µm; red dash–dotted curve: b/t/p/h= 2.5/6.5/8/12 µm; red dash–double-dotted curve: b/t/p/h= 2/7/8/16 µm. The horizontal dotted line indicates the lasing frequency. ko= 2π/λo, where λo= 100 µm. f, Black open circles: the 1/e decay length of the spoof SP electric field (|E|) normal to the interface into the air as a function of h. Red open triangles: imaginary part of the in-plane wave vector as a function of h, which characterizes propagation loss of spoof SPs. Other groove dimensions are fixed: b/t/p= 2/4/8 µm. density, which usually leads to reduced maximum operating temperatures in continuous-wave operation. Conventional metallo-dielectric plasmonic structures defined on laser facets have been used to shape the wavefront of mid- infrared semiconductor lasers28–31. Unfortunately thismethodology is not scalable to far-infrared wavelengths, because those structures do not significantly modify the SP dispersion properties and in particular the asymptotic SP frequency, thus providing limited control of terahertz surface waves. A schematic of our design for a 3 THz frequency (λo= 100 µm) laser and its cross-section are shown in Fig. 1c and d, respectively. The double-metal waveguide of the laser is defined on a 450-µm- thick highly dopedGaAs substrate. Two colours are used in Fig. 1c,d to identify shallow and deep spoof SP grooves. All of the grooves are defined directly on the GaAs substrate without any metal coating. We take advantage of the fact that in the terahertz regime the carrier concentration in highly doped semiconductors is sufficiently large that the semiconductor is ‘metallic’ with the real part of its dielectric permittivity being largely negative (see the Methods section for more details). At the aperture of the double-metal waveguide, the laser emits both directly into the far-field and also into surface waves on the device facet. In the original unpatterned device, both components have a wave vector close to ko. The wave vector of the laser mode in the waveguide is several times larger, ∼neffko (neff ≈ 3.5 is the effective mode index). Therefore, there is a wave vector mismatch of the modes at the aperture. In our collimator, the spoof SP grooves adjacent to the aperture increase the effective in-plane wave vector of the SPs, reducing the wave vector mismatch. More light is therefore coupled out from the laser cavity and a larger percentage of it is channelled into the spoof SP modes instead of being directly emitted into the far-field. In addition, the deep grooves (pink in Fig. 1d) periodicallymodulate the dispersion of the SPs on the device facet, creating an effective second-order grating that scatters the energy of the SPs into the far-field. Constructive interference between these scattered waves and the direct emission NATURE MATERIALS | VOL 9 | SEPTEMBER 2010 | www.nature.com/naturematerials 731 LETTERS NATURE MATERIALS DOI: 10.1038/NMAT2822 only the second-order grating (pink in Fig. 1d). This difference in measured power is primarily a result of the increased total power throughput originating from the reduced wave vector mismatch in the spoof SP collimator. Our power measurement apparatus with a collection cone of ∼50◦ captures the main lobe as well as a significant portion of the background light. Eventually, what matters most for applications is the power carried in the main lobe of a laser beam, because the optical background outside the main lobe is often lost during propagation in optical systems. On the basis of the far-field measurements, the power in the main lobe of the device with the spoof SP collimator is about two times larger than that of the device with only the second-order grating. The maximum operating temperature of the patterned device is 135K, the same as that of the original device. Figure 3f shows that the lasing threshold was not changed after defining the collimator. The threshold is proportional to the sum of waveguide loss, αw, mainly resulting from absorption in the waveguide material, and mirror loss, αm, resulting from out-coupling of laser light from the facets. In our devices, αw is about 30 times larger than αm, so even a significant increase in αm will not lead to a noticeable increase of the threshold. The ability to confine low-frequency surface waves to subwave- length scales can lead to their more efficient manipulation and stronger interaction with analytes on the surface; the capability to engineer at will the dispersion properties of SPs may facilitate optical impedance matching between different components in an optical system. Furthermore, we plan to build devices with reconfigurable functions by using doped semiconductors as the plasmonic media, because their optical properties (for example, permittivity, anisotropy) can be tuned by applying optical exci- tation, electrical potentials or magnetic fields, or by controlling the temperature. We believe that these functionalities will lead to developments in plasmonics and metamaterials where key prob- lems in imaging, sensing, wavefront engineering, slow light and photovoltaics will be addressed. Methods Devices and fabrication. The QCL material was grown by molecular beam epitaxy on an undoped GaAs substrate. The growth sequence started with a 250-nm-thick undoped GaAs buffer layer, and was followed by a 300-nm-thick Al0.5Ga0.5As etch-stop layer, a 75-nm-thick layer of GaAs n-doped to 5×1018 cm−3, the active region and finally a 50-nm-thick GaAs layer n-doped to 5×1018 cm−3. The active region consists of 170 periods of a two-phonon-resonance active region design similar to that of ref. 36, with a doping sheet density of ns = 3.65×1010 cm−2. The material was processed into copper–copper waveguides using the following procedure19. First, a square centimetre of QCL material was cleaved and sputter-coated with Ta/Cu/Au (15/500/500 nm). The material was then wafer-bonded to a highly doped (1.6×1018 cm−3) GaAs substrate coated with sputtered layers of Ti/Au (15/500 nm). The bonded QCL wafer was next polished and wet-etched down to the etch-stop layer with a hydrogen peroxide/ammonium hydroxide solution (19:1 in volume), and the etch-stop layer was stripped with concentrated hydrofluoric acid. The laser ridges, with widths ranging from 25 to 150 µm, were then defined using dry etching with an SU-8 2005 photoresist mask. After SU-8 removal, metal (Ta/Cu/Au, 15/100/30 nm) was sputtered on top of the laser ridges. A gold capping layer was added to the waveguides with copper cladding to avoid copper oxidation and to facilitate wire bonding. The processed wafers were finally cleaved into 1–2-mm-long bars and indium-mounted onto copper blocks. The back facets of QCLs were coated with 300-nm Al2O3 and 100-nm gold so that the measured far-field profiles are solely due to emission from the front facets. We note that in devices without back-facet coating and front-facet patterning, the highly divergent emissions from the front and the back apertures interfere with each other, which gives rise to unpredictable far-field patterns that depend sensitively on the geometries of the laser ridges37,38. To suppress higher-order transverse modes in our wide-ridge devices, side-absorbers39 were defined by removing ∼3–4-µm-wide strips of metal along the edges of the device top contact. Devices fabricated in this way have maximum operating temperatures ∼10◦–20◦ lower compared with devices without side-absorbers. The plasmonic patterns were sculpted into the semiconductor substrate using FIB milling (Zeiss NVision 40) at a high ion current of 13 or 27 nA to limit fabrication time. The diameter of the Ga ion beam at this current is ∼1 µm, which gives rise to a trapezoidal cross-section of the fabricated grooves. Optical constants. The highly doped semiconductor substrate supports surface waves in the terahertz spectral range. We used the Drude model to calculate the dielectric permittivity of highly doped GaAs at terahertz frequencies40,41 and found that the real part of the permittivity is negative. For example, εGaAs ∼−200+42i for our device and measurement conditions (λo = 100 µm, GaAs n-doped to 1.6×1018 cm−3, T = 80K). The propagation distance of surface waves (defined as the 1/e decay length of the electric field) along a planar GaAs/air interface under these conditions corresponds to ∼3 cm (∼300λo), which is about 10% of that of surface waves on a planar gold surface. Therefore, the highly doped GaAs functions like a poor metal at terahertz frequencies, which is unlike the situation for visible and near-infrared wavelengths. It is extremely beneficial that metals are not needed for plasmonics in the terahertz regime, because this greatly simplifies device fabrication. In our system (surface-corrugated GaAs n-doped to 1.6× 1018 cm−3 at T = 80K), the determination of the asymptotic frequency of the spoof SP dispersion curve is quite complicated because it involves three factors: the subwavelength pattern, the free-carrier concentration and the restrahlen band of GaAs. If one assumes that the spoof SP pattern is defined in a perfect electric conductor, the asymptotic frequency will ∼30–67 THz for the groove geometries used. However, it will be further lowered considering the actual dielectric permittivity of GaAs, which is affected by the free-carrier concentration and the restrahlen band of GaAs. On one hand, the Drude model predicts that the free-carrier concentration is able to support ‘metallic’ GaAs (that is, the real part of the permittivity <0) up to ∼10 THz. On the other hand, the restrahlen band of GaAs (∼7.5–10 THz; ref. 42) is going to introduce a large negative permittivity when an electromagnetic field strongly interacts with the optical phonons. The actual asymptotic frequency is therefore ∼7.5 THz primarily determined by the restrahlen band of GaAs. Measurements. All devices were tested in pulsed mode with 60-ns pulses at 0.3% duty cycle (60-ns pulses at a 100 kHz repetition rate, with an extra 1 kHzmodulation at 50% duty cycle for lock-in detection). Laser powers were measured using a Fourier transform infrared (FTIR) spectrometer with a calibrated helium-cooled bolometer using two 2′′-diameter parabolic mirrors: one with a 5 cm focal length to pass light from the devices to the input of the FTIR spectrometer, and the other with a 15 cm focal length to focus the light from the output of the FTIR spectrometer onto the bolometer. To map the 2D far-field emission profile of the devices, a cryostat, in which the lasers were mounted vertically (laser material layers normal to the horizontal plane), was placed on a rotation stage. Line-scans of the laser far-field along the θ direction were obtained as the stage was rotated in the horizontal plane. The relative height of the cryostat and the bolometer was adjusted to allow many line-scans of the device far-field to be obtained, and finally constructed into a 2D map. Received 10 March 2010; accepted 30 June 2010; published online 8 August 2010 References 1. Zenneck, J. Über die Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie. Ann. Phys. 23, 846–866 (1907). 2. Sommerfeld, A. Über die Ausbreitung derWellen in der drahtlosen Telegraphie. Ann. Physik 28, 665–736 (1909). 3. Pendry, J. B., Martín-Moreno, L. & García-Vidal, F. J. Mimicking surface plasmons with structured surfaces. Science 305, 847–848 (2004). 4. García-Vidal, F. J., Martín-Moreno, L. & Pendry, J. B. Surfaces with holes in them: New plasmonic metamaterials. J. Opt. A: Pure Appl. Opt. 7, S97–S101 (2005). 5. Engheta, N. & Ziolkowski, R. W. Metamaterials: Physics and Engineering Explorations (Wiley-IEEE Press, 2006). 6. Cai, W. & Shalaev, V. M.Optical Metamaterials: Fundamentals and Applications (Springer, 2009). 7. Pendry, J. B., Schurig, D. & Smith, D. R. Controlling electromagnetic fields. Science 312, 1780–1782 (2006). 8. Leonhardt, U. Optical conformal mapping. Science 312, 1777–1780 (2006). 9. Smolyaninov, I. I., Hung, Y-J. & Davis, C. C. Imaging and focusing properties of plasmonic metamaterial devices. Phys. Rev. B 76, 205424 (2007). 10. Beermann, J., Radko, I. P., Boltasseva, A. & Bozhevolnyi, S. I. Localized field enhancements in fractal shaped periodic metal nanostructures.Opt. Express 15, 15234–15241 (2007). 11. Radko, I. P. et al. Plasmonic metasurfaces for waveguiding and field enhancement. Laser Photon. Rev. 3, 575–590 (2009). 12. Navarro-Cía, M. et al. Broadband spoof plasmons and subwavelength electromagnetic energy confinement on ultrathin metafilms. Opt. Express 17, 18184–18195 (2009). 13. Gan, Q., Fu, Z., Ding, Y. J. & Bartoli, F. J. Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures. Phys. Rev. Lett. 100, 256803 (2008). 14. Williams, C. R. et al. Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces. Nature Photon. 2, 175–179 (2008). 734 NATURE MATERIALS | VOL 9 | SEPTEMBER 2010 | www.nature.com/naturematerials NATURE MATERIALS DOI: 10.1038/NMAT2822 LETTERS 15. Hajenius, M. et al. Surface plasmon quantum cascade lasers as terahertz local oscillators. Opt. Lett. 33, 312–314 (2008). 16. Kim, S. M. et al. Biomedical terahertz imaging with a quantum cascade laser. Appl. Phys. Lett. 88, 153903 (2006). 17. Lee, A. W. M. et al. Real-time terahertz imaging over a standoff distance (>25meters). Appl. Phys. Lett. 89, 141125 (2006). 18. Hübers, H-W. et al. High-resolution gas phase spectroscopy with a distributed feedback terahertz quantum cascade laser. Appl. Phys. Lett. 89, 061115 (2006). 19. Belkin, M. A. et al. High-temperature operation of terahertz quantum cascade laser sources. IEEE J. Sel. Top. Quantum Electron. 15, 952–967 (2009). 20. Williams, B. S. Terahertz quantum-cascade lasers. Nature Photon. 1, 517–525 (2007). 21. Scalari, G. et al. THz and sub-THz quantum cascade lasers. Laser Photon. Rev. 3, 45–66 (2009). 22. Lee, A. W. M. et al. High-power and high-temperature THz quantum-cascade lasers based on lens-coupled metal–metal waveguides. Opt. Lett. 32, 2840–2842 (2007). 23. Amanti, M. I., Fischer, M., Walther, C., Scalari, G. & Faist, J. Horn antennas for terahertz quantum cascade lasers. Electron. Lett. 43, 573–574 (2007). 24. Fan, J. A. et al. Surface emitting terahertz quantum cascade laser with a double-metal waveguide. Opt. Express 14, 11672–11680 (2006). 25. Mahler, L. et al. Vertically emitting microdisk lasers. Nature Photon. 3, 46–49 (2009). 26. Amanti, M. I., Fischer, M., Scalari, G., Beck, M. & Faist, J. Low-divergence single-mode terahertz quantum cascade laser. Nature Photon. 3, 586–590 (2009). 27. Chassagneux, Y. et al. Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions. Nature 457, 174–178 (2009). 28. Yu, N. et al. Plasmonics for laser beam shaping. IEEE Trans. Nanotech. 9, 11–29 (2010). 29. Yu, N. et al. Small-divergence semiconductor lasers by plasmonic collimation. Nature Photon. 2, 564–570 (2008). 30. Yu, N. et al. Semiconductor lasers with integrated plasmonic polarizers. Appl. Phys. Lett. 94, 151101 (2009). 31. Yu, N. et al. Multi-beam multi-wavelength semiconductor lasers. Appl. Phys. Lett. 95, 161108 (2009). 32. Stutzman, W. L. & Thiele, G. A. Antenna Theory and Design (John Wiley, 1981). 33. Wang, B., Liu, L. & He, S. Propagation loss of terahertz surface plasmon polaritons on a periodically structured Ag surface. J. Appl. Phys. 104, 103531 (2008). 34. Gaidis, M. C. et al. A 2.5-THz receiver front end for spaceborne applications. IEEE Trans. Microw. Theory Tech. 48, 733–739 (2000). 35. Siegel, P. H. & Dengler, R. J. The dielectric-filled parabola: A new millimetre/submillimetre wavelength receiver/transmitter front end. IEEE Trans. Antennas Propag. 39, 40–47 (1991). 36. Belkin, M. A. et al. Int. Workshop on Optical Terahertz Science and Technology Talk TuC5, Santa Barbara (2009). 37. Adam, A. J. L. et al. Beam patterns of terahertz quantum cascade lasers with subwavelength cavity dimensions. Appl. Phys. Lett. 88, 151105 (2006). 38. Orlova, E. E. et al. Antenna model for wire lasers. Phys. Rev. Lett. 96, 173904 (2006). 39. Fan, J. A. et al. Wide-ridge metal–metal terahertz quantum cascade lasers with high-order lateral mode suppression. Appl. Phys. Lett. 92, 031106 (2008). 40. Huggard, P. G. et al. Drude conductivity of highly doped GaAs at terahertz frequencies. J. Appl. Phys. 87, 2382–2385 (2000). 41. Walukiewicz,W., Lagowski, L., Jastrzebski, L., Lichtensteiger,M. &Gatos, H. C. Electron mobility and free-carrier absorption in GaAs: Determination of the compensation ratio. J. Appl. Phys. 50, 899–908 (1979). 42. Adachi, S. GaAs and Related Materials: Bulk Semiconducting and Superlattice Properties (World Scientific Publishing Company, 1994). Acknowledgements We gratefully acknowledge constructive and helpful discussions with R. Blanchard, C. Pflügl, L. Diehl and A. Belyanin. M.A.K. is supported by the National Science Foundation through a Graduate Research Fellowship. We would like to thank N. Antoniou for assistance in FIB milling. We acknowledge support from AFOSR under contract No. FA9550-09-0505-DOD and the EPSRC (UK). The authors acknowledge the Center for Nanoscale Systems (CNS) at Harvard University. Harvard CNS is a member of the National Nanotechnology Infrastructure Network (NNIN). The computations in this Letter were run on the Odyssey cluster supported by the Harvard Faculty of Arts and Sciences (FAS) Sciences Division Research Computing Group. Author contributions N.Y. designed the devices, in collaboration with J.A.F., and, with Q.J.W., fabricated them and carried out the experiments. M.A.K. participated in the device simulation and in the data analysis. S.P.K. and L.L. grew QCL material using molecular beam epitaxy. N.Y. and F.C. wrote the paper. F.C., A.G.D. and E.H.L. supervised the project. Additional information The authors declare no competing financial interests. Supplementary information accompanies this paper on www.nature.com/naturematerials. Reprints and permissions information is available online at http://npg.nature.com/reprintsandpermissions. Correspondence and requests formaterials should be addressed toN.Y. or F.C. NATURE MATERIALS | VOL 9 | SEPTEMBER 2010 | www.nature.com/naturematerials 735