Confined Electron and Hole States in Semiconducting Carbon Nanotube sub-10 nm Artificial Quantum Dots
We show that quantum confinement in the valence and conduction bands of semiconducting single-walled carbon nanotubes can be engineered by means of artificial defects. This ability holds potential for designing future nanotube-based quantum devices such as electrically driven room-temperature single-photon sources emitting at telecom-wavelength. Using Ar and N ion-induced defects, intrananotube quantum dots with sub-10 nm lateral sizes are created, giving rise to quantized electronic bound states with level spacings of the order of 100 meV and larger. Using low-temperature scanning tunneling spectroscopy, we resolve the energy and real space properties of the quantized states and compare them with theoretical model calculations. By solving the Schrödinger equation over a one-dimensional piecewise constant potential model, the effects of inhomogeneous defect scattering strength as well as surface variations in the Au(111) substrate on the quantized states structure are remarkably well reproduced. Furthermore, using ab-initio calculations, we demonstrate that defect structures such as vacancies, di-vacancies and chemisorbed nitrogen ad-atoms constitute strong scattering centers able to form quantum dots with clear signatures of discrete bound states as observed experimentally. The ab-initio simulations also allowed to study the scattering strength profile as a function of energy for different defect combinations, supporting the potential of highly stable double vacancies for practical applications at room temperature.
IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain \alsoaffiliationFreiburg Materials Research Center FMF, University of Freiburg, Stefan-Meier-Strasse 21, D-79104 Freiburg, Germany
Here comes a graphic for the Table of Contents (a thumbnail). A graphic for the Table of Contents must be supplied with each Letter. This graphic, in conjunction with the manuscript title, should give the reader a summary of the research described in the paper. The TOC graphic will also be published in the Abstract of the manuscript. This graphic can be no wider than 8 cm and no taller than 4 cm and should be prepared following the specifications under Artwork. Text should be limited to essential technical labels.
Despite the rise of graphene and other 2D materials, semiconducting single-walled carbon nanotubes (SWNT) are still regarded as strong candidates for the next generation of high-performance ultrascaled transistors 1, 2 as well as for opto-electronic devices 3, 4 such as chip-scale electronic-photonic platforms 5 or low-threshold near-infrared tunable micro-lasers 6.
Engineering a quantum dot (QD) along a (suspended) semiconducting SWNT foreshadows promising opportunities in the field of quantum information processing and sensing through recently proposed schemes such as: detection and manipulation of single spins via coupling to vibrational motion 7, optomechanical cooling 8 as well as all optical manipulation of electron spins 9. Furthermore, the quasi one-dimensional (1D) geometry of SWNTs allows for defining tunable p-n junctions induced by electrostatic doping through local gates 10, 11. Combining a well-defined QD within such a p-n junction structure could constitute a crucial building-block for the realization of highly desirable electrically driven, on-demand single photon emitters operating at telecom wavelength, based on a turnstile device architecture 12, 13.
In practice, QDs in carbon nanotubes have been reported predominantly under two different confinement structures: i) Engineered tunneling barriers at metal-nanotube contacts 14 and/or by gate electrodes, used e.g. to manipulate single electron spins 15, ii) Unintentional localization potentials stemming from environmental disorder 16, allowing for single-photon emission mediated by localization of band-edge excitons to QD states4, 17, 18, 19, 20. Both structures are usually operated at cryogenic temperature due to small energy scales ranging from a few to few tens of millielectronvolts.
Another technique for achieving confinement in SWNTs makes use of artificial defects such as covalently bound oxygen or aryl functionalization groups on the side walls of semiconducting SWNTs, inducing deep exciton trap states allowing for single-photon emission at room temperature 21, 22. Also, carrier confinement between defect pairs acting as strong scattering centers has been reported for mechanically induced defects 23 as well as for ion-induced defects with reported level spacings up to 200 meV in metallic SWNTs. 24. The latter technique, combined with recent progress in controlling defects structure and localization 25, 26, 27 offers a high potential for engineering a broad set of SWNT-based quantum devices operating at room temperature.
In this letter, we demonstrate confinement of electrons and holes in sub-10 nm QD structures defined by ion-induced defect pairs along the axis of semiconducting SWNTs. Using low temperature scanning tunneling microscopy and spectroscopy (STM/STS), bound states with level spacings of the order of 100 meV and larger are resolved in energy and space. By solving the 1D Schrödinger equation over a piecewise constant potential model, the effects of inhomogeneous defect scattering strength as well as surface variations in the Au(111) substrate on the bound states structure are remarkably well reproduced. By means of ab-initio calculations based on density functional theory and Green’s functions, we find that single (SV) and double vacancies (DV) as well as chemisorbed nitrogen ad-atoms are good candidates in terms of scattering strength to produce QDs with the experimentally observed features. These simulations also allow to study the scattering profile as a function of energy for different defects combination.
The experiments have been performed in a commercial (Omicron) low temperature STM setup operating at K in ultra high vacuum. Topography images have been recorded in constant current mode with a grounded sample, using mechanically cut Pt/Ir tips. Differential conductance spectra, proportional in first approximation to the local density of states (LDOS) 28 have been recorded using a lock-in amplifier technique. The LDOS spatial evolution along a nanotube axis is obtained by maps built by a series of equidistant spectra. Spatial extent mismatches between topography images and consecutive maps have been systematically corrected 29, and the metallic nature of the tip has been systematically checked on the gold substrate to prevent any tip artefacts before recording STM or/and STS data sets.
Nanotube samples were made of extremely pure high-pressure CO conversion (HiPCo) SWNTs 30 with a diameter distribution centered around 1 nm, FWHM 0.3 nm. The measured intrinsic defect density was below one defect every 200 nm. SWNTs were deposited on atomically flat Au(111) surfaces from a 1,2-dichloroethane suspension, followed by an in-situ annealing process 31, 29.
Local defects in SWNTs have been created in situ by exposure to: (i) Medium energy 200 eV argon ions (Ar) produced by an ion gun 29, 24, (ii) Low energy (few eV’s) nitrogen ions (N) produced by a 2.45 GHz ECR plasma source 31, 32. In both cases, the exposure parameters have been calibrated to reach an average defect separation along the SWNTs of about 10 nm 29, 31.
In Fig. 1 (a)-(b), we show 3D STM images of the same semiconducting SWNT (referred as SWNT I in the following) with Arions-induced defect sites labeled . Panel (d) shows a 3D STM image of a second semiconducting SWNT (referred as SWNT II) with N ions-induced defect sites labeled . In both cases, defect sites typically appear as hillock-like protrusions with an apparent height ranging from 0.5 Å to 4 Å and an apparent lateral extension varying between 5 Å and 30 Å 32, 29, 33.
The resulting maps recorded along the horizontal dashed line drawn in panel (b) for SWNT I and (d) for SWNT II are displayed in panels (c) and (e) in Fig. 1, respectively. Defect signatures in the LDOS in both cases are characterized by deep in-gap states at the defects positions. This is consistent with the expected defect structures, mainly SVs, DVs and combinations thereof for collisions with Ar ions 29 and bridgelike N ad-atom for collisions with N ions 33, 34. Note that gap states at energy levels 0.2 eV and 0.05 eV in panels (c) and (e), respectively, are shifted to the right from by about 1 nm and to the right from by about 2 nm. This indicates the presence of intrinsic or ion-induced defects on the lateral or bottom side wall of the SWNTs 35, not visible in the topographic images. These defects are labelled and , respectively.
Remarkably, the maps in Fig. 1 display several broad discrete states in the conduction bands of SWNT I, II (white dashed boxes in panel (c) and (e), respectively) and in the valence band of SWNT I (white dashed box in panel (c)), characterized by a modulation of the signals in the spatial direction between pairs of consecutive defect sites and . Enlarged plots of these boxed regions are displayed in Fig. 2(a)-(b) and Fig. 3(a) for SWNTs I and II, respectively. In the conduction bands, cross-sectional curves recorded along the black horizontal dashed lines labelled m1–m3 in Fig. 2(a) and m1–m4 in Fig. 3(a) are plotted below the LDOS panels. These clearly reveal one to three and respectively one to four spatially equidistant maxima. The number of maxima increases for increasing and the measured level spacings between consecutive discrete states is of the order of 100 meV and larger for both cases. This indicates that defect sites and , respectively separated by 12.1 nm and 11.2 nm, act as strong scattering centers able to confine carriers in semiconducting SWNTs 24, 36. Such intrananotube QD structures will be referred as QD I (in SWNT I) and QD II (in SWNT II) in the following. We estimated the level spacings in the conduction band of QD I to 98 meV (m1-m2) and 116 meV (m2-m3). For QD II, we measured about 122 meV (m1-m2), 185 meV (m2-m3) and 210 meV (m3-m4).
In the valence band of SWNT I, discrete states with level spacings in the order of 80-90 meV, with one clear maximum at the level m-1, can also be distinguished between defect sites in Fig. 2(b). The discretization of the states indicates that this QD structure also confines holes. Discrete states starting from m-2 and lower show less well defined structures compared to the conduction band states. In the case of SWNT II, no clear discrete states are observed in the valence band (see supporting information). These observations are most probably the result of an energy dependent scattering strength of the defects, respectively - and -, leading here to a weaker confinement in the valence band. Such energy dependence is well known for metallic SWNTs 37, 38, 39, 40 and is consistent with ab-initio calculations presented further in this work. Note that mixing effects with defect states and substrate-induced effects 41 cannot be ruled out.
Another remarkable feature in the LDOS is the strong spatial asymmetry of the lowest energy states m1 and m-1 in QD I and m1 in QD II. In QD I, m1 is shifted to the right side of the dot while m-1 is shifted to the left side. Higher states m2 and m3 show more symmetry in terms of position of the maxima relative to the center of the QD. In QD II, m1 is shifted to the right side of the QD. We attribute the observed lowest energy states asymmetry (for electrons as well as for holes) in part to their strong sensitivity to weak potential modulations within the QD structure (we will corroborate this point in detail when discussing the simulation results). For QD I, this assertion is supported by the observation of a 0.25 nm high Au(111)terrace edge located around the center of the QD, leading to a supported-suspended interface (see white dashed lines in Fig. 1(b) and more topographic details in Fig. S2(a)-(d) in supplementary information). Such configurations have been reported to induce a rigid shift in the SWNT bands 42, for instance here a down-shift in the right side of QD I corresponding to the "suspended" portion between two terraces. In QD II, we attribute the spatial shift of m1 to a potential modulation induced by a layer of disordered impurities, most probably residua from the 1,2-dichloroethane suspension, lying between the gold substrate and the SWNT (see Fig. 1(d) and Fig. S2(e)-(h) in supplementary information).
Also, the LDOS in QD I and II (Fig. 2(a) and Fig. 3(a), respectively) reveals asymmetric patterns with curved stripes oriented from top left to bottom right for QD I and from bottom left to top right for QD II. These are characteristic signatures for defect pairs with different scattering strengths 36, 24. For instance here, the left defect in QD I () has a larger scattering strength than the right one (), while the right defect in QD II () has a larger scattering strength than the left one ().
The exact atomic structure of the defects could in principle be determined from a comparison of spectra with simulated first-principle LDOS signatures of expected defect types. In reality, this is hampered by the large number of possible geometries to simulate, including complex multiple defect structures 29, together with the large unit cells of the semiconducting chiral SWNTs studied here.
To better understand the real space asymmetry of the quantized states in terms of potential modulation and defect scattering strength, we solved the 1D time independent Schrödinger equation over a piecewise constant potential model of QD I and QD II. The scattering centers are approximated by semi-transparent rectangular tunneling barriers leading to a square confinement potential. This is supported by previous results on defect-induced confinement in metallic SWNTs using the same experimental conditions 24 and is consistent with ab-initio simulations presented later in this work. The potential modulation within the QD is approximated by a potential step. The resulting potential geometries are illustrated with gray shaded areas in Fig. 2 (c) and (d) and Fig. 3(b). Dispersion relations can be extracted experimentally from the quantized states wavefunctions by measuring the energy and corresponding momenta in the left and right sides of the QDs. The wavevectors are determined using stationary wave-like fitting functions 24 displayed with dashed red curves in Figs. 2(a)-(b) and 3(a)). From this procedure, the potential step height and position can be estimated (see supplementary information). The experimental data sets are plotted in the right panels of Figs. 2(a) and 3(a) together with dispersion relations from a third-nearest neighbor tight-binding calculation closely approximating ab-initio results 43. These chirality-dependent tight-binding dispersion relations, calculated within an extended Brillouin zone resulting from the defect-induced breaking of the translation invariance 36, are used in the Hamiltonian of our 1D model. Taking into account the measured chiral angle, diameter distribution and measured bandgaps, we find the best match with chiralities for QD I and for QD II (see supplementary information).
Once chiralities together with potential step heights and positions are optimized, one needs to adjust the height and width of the rectangular tunneling barriers in order to reproduce the experimental level spacings and general LDOS patterns. On a qualitative ground, a symmetric double barrier system results in the formation of spatially symmetric discrete bound states. Increasing both barrier heights simultaneously shifts the bound state energy levels up (and at the same time the level spacings). This leads to sharper bound states as the confinement in the QD is made stronger thus increasing the lifetime of the confined electrons. Increasing the barrier thickness (while keeping the QD length, the distance between the barriers edges, constant) does not affect much the level spacings but further sharpens the bound states. Any asymmetry introduced by a change in the width or height of one single barrier leads to broader bound states. The presence of a potential step modifies the LDOS in lifting the levels of the bound states, with a more pronounced effect on the lower states. In QD I and II, the center of each barrier is aligned with the center of the gap states (- for QD I and - in QD II) and the width ratio is kept proportional to the ratio of the spatial extent of the gap states. Thus, by increasing the width of the barriers, we decrease the length of the QD leading to higher level spacings, and vice versa. The experimental level spacings can then be approximated by tuning both widths in the same ratio and the barriers heights individually, knowing that the scattering strength of () is larger than () according to the observed asymmetry in the LDOS described above 44.
For QD I, we find a good match in the conduction band for the barrier heights eV and eV, widths 3.4 nm, and potential step 60 meV. With these parameters, the spatial profile of the obtained quantized states (see lower subpanels in Fig. 2(a) and (c)) reproduces the experimental modulation features remarkably well. Also, the simulated LDOS displays a pattern with curved stripes oriented from top left to bottom right, as observed experimentally, due to a left barrier with a stronger scattering strength. In the valence band, although modes m-2 and lower do not show a well defined structure in the spatial direction, thinner barriers with dimensions 2.5 nm, eV, leading to a slightly longer QD length (9.6 nm compared to 8.7 nm in the conduction band) can reproduce the measured level spacings very well.
For QD II, we observed that the measured energy levels are overestimated by a factor 1.29, presumably due to a voltage division effect induced by the impurity layer mentioned above (see details in supplementary information). We find a good agreement with the experimental LDOS with the parameters: 0.47 eV, 1.5 nm, 2.6 nm and 47 meV. Note that in Fig. 3(b) the barrier and potential heights are multiplied by to allow a direct comparison with the experimental LDOS. The simulated LDOS shows a pattern with curved stripes oriented from bottom left to top right, as observed experimentally, due to a right barrier exhibiting a stronger scattering strength. Also, the spatial profile of the obtained bound states (see lower subpanels in Fig. 3(a) and (b)) reproduces the experimental features quite well. Note also that one can distinguish an isolated state in the experimental LDOS at an energy level between m1 and m2, about in the middle of the QD. This state that prevented an accurate fit of the state m2 in the right QD part is attributed to a spatial feature visible in the STM topography image in Fig. 2(d) (see also supplementary information, Fig.S2(f)), probably a physisorbed impurity which does not affect the LDOS significantly.
In order to elucidate the physical nature of the electron/hole confining scattering centers, we performed ab-initio simulations based on a combination of density functional theory 45, 46, 47, 48, maximally localized Wannier orbitals 49 and Green’s functions (see supporting information). Without loss of generality, we have simulated short unit cell semiconducting zigzag SWNTs with different combinations of the most probable defect structures. Results for vacancy defects likely being induced by 200 eV Ar ions, separated by about 11 nm in a SWNT are shown in Fig. 4(a)-(c) with DV-DV, DV-SV and SV-SV pairs, respectively. The LDOS displays midgap states at the defect positions as expected as well as defect states in the valence band 29. Most importantly, clear quantized states with a number of maxima increasing with energy are observed between the defects in the conduction band, emphasizing the ability of SVs and DVs to confine carriers. For the asymmetric configuration DV-SV, one can distinguish faint curved stripe patterns oriented from top left to bottom right, indicating a stronger scattering strength for DVs compared to SVs. This is consistent with observations in transport experiments 50. On the other hand, the patterns in the valence band strongly depend on the defect types. Discrete states can be distinguished for the DV-DV case, with m-2 being mixed with defect states. For the DV-SV case, clear curved stripe patterns oriented from bottom left to top right indicate again a stronger scattering strength for DV. Also, broader states are observed, indicating that the scattering strength of DVs and SVs is weaker in the valence band compared to the conduction band.
More insight on the energy dependent scattering strength for each defect pair configurations can be obtained by extracting the wavevector for each resonant state. This data set is plotted in Fig. 4(d) for the conduction and valence bands together with the dispersion relations calculated from the 3rd n.n. TB model and from the ab-initio calculation for the pristine nanotube. A first observation is the excellent agreement between TB and ab-initio results, further validating the method used in Figs. 2(a)-(b) and 3(a). The vertical dashed lines indicate the limiting values corresponding to the closed system (infinite hard walls potential) with nm being the defect-defect distance. In the conduction band, we find that , indicating that the effective lengths of the QD are larger than ( the resonant states wavefunctions are characterized by penetrating evanescent modes inside the defect scattering potential), as expected for an open system. The shortest are obtained for the DV-DV configuration with 12.1 nm (m1), 13.1 nm (m2) and 12.9 nm (m3), which we attribute to wider scattering potential profiles for DVs compared to SVs. In the valence band, we find that , with values between 7.9 nm (DV-DV, m-1) and 9.66 nm (DV-SV, m-2). We attribute this pronounced QD shortening to wider scattering potential profiles of both DVs and SVs in the valence band, probably due to mixing with wide spread defect states in the valence band.
Ab-initio calculations for different defect pairs combinations containing at least one N-adatom, N-DV, N-SV and N-N, are presented in Fig. 4(e)-(h) for a SWNT, along with details on the defects geometries. Remarkably, clear QD states are generated for all three configurations, underlining the potential of N-adatoms to confine carriers in semiconducting SWNTs and thus to generate intrananotube QDs.
Note that the choice of the zigzag SWNT chiralities in the two different ab-initio scenarios is motivated by the different effective masses of both chiralities () which is typical for chirality families and 51. Taking advantage of recent reports on SWNT chirality control 52, 53, 54, this property could be used in practice to design QDs with different level spacings for the same QD length. From an application point of view, however, QDs generated by DVs will have far superior stability at room temperature due to their high migration barrier above 5 eV ( 1 eV for single vacancy) 55. This value drops down by at least 2 eV for N adatoms depending on their chemisorption configuration 34, 56.
The above ab-initio simulations do not take into account any substrate effect. In the experimental case, the carriers can decay through the substrate, thus limiting their lifetime. This leads to state broadening, measured between about 60 meV up to 120 meV in QD I and II, while the quantized states widths in ab-initio simulations vary between about 5 meV and 45 meV. This suggests that a better contrast of the experimental quantized states, especially in the valence band, could be achieved by lowering the nanotubes-substrate interaction through the insertion of atomically thin insulating NaCl films 57. This would allow to gain more insight on the electronic structure of the QDs as well as in the associated scattering physics at the confining defects 24.
In summary, using low-temperature STM/STS measurements supported by an analytical model and ab-initio simulations, we have demonstrated that intrananotube quantum dots with confined electron and hole states characterized by energy level spacings well above thermal broadening at room temperature can be generated in semiconducting SWNTs by structural defects such as vacancies and di-vacancies, as well as nitrogen adatoms. These results, combined with recent progresses in type and spatial control in the formation of defects 25, 26, 27 as well as chirality control 22, hold a high potential for applications in the design of SWNT based quantum devices. These include electrically driven single-photon emitters operating at room temperature and telecom wavelength. In this context, the observation of quantum confinement effects in the emitted light of cut, sub-10 nm, semiconducting SWNTs 58 shall be seen as an additional motivation for investigating the optical properties of our "QD with leads" building-blocks. These would include studying optical transitions selection rules for different types and configurations of defect pairs 59 associated with experimental studies such as photoluminescence 60 combined to correlation measurements 19 in suspended SWNT devices as well as photocurrent imaging 61 and spectroscopy 62.
Supporting information includes atomic resolution STM images for chiral angle determination, detailed 3D topographic images of the QD structures, additional maps as well as detailed descriptions of the analytical 1D piecewise constant potential model and ab-initio simulations.
The authors thank Ethan Minot, Jhon Gonzalez, Andres Ayuela, Omjoti Dutta and Arkady Krasheninnikov for fruitful discussions. The work of DB is supported by Spanish Ministerio de Economía y Competitividad (MINECO) through the project FIS2014-55987-P and by the (LTC) QuantumChemPhys. LM acknowledges support from the BMBF-project WireControl (FKZ16ES0929K) and computing time for the supercomputers JUROPA and JURECA at the Jülich Supercomputer Centre (JSC).
- Cao et al. 2015 Cao, Q.; Han, S.-J.; Tersoff, J.; Franklin, A. D.; Zhu, Y.; Zhang, Z.; Tulevski, G. S.; Tang, J.; Haensch, W. Science 2015, 350, 68 – 72.
- Shulaker et al. 2017 Shulaker, M. M.; Hills, G.; Park, R. S.; Howe, R. T.; Saraswat, K.; Wong, H.-S. P.; Mitra, S. Nature 2017, 547, 74.
- Avouris et al. 2008 Avouris, P.; Freitag, M.; Perebeinos, V. Nature Photon. 2008, 2, 341 – 350.
- Yamashita et al. 2013 Yamashita, S.; Saito, Y.; Choi, J. H. Carbon nanotubes and graphene for photonic applications; Woodhead Publishing: Philadelphia, USA, 2013.
- Pyatkov et al. 2016 Pyatkov, F.; Fütterling, V.; Khasminskaya, S.; Flavel, B. S.; Hennrich, F.; Kappes, M. M.; Krupke, R.; Pernice, W. H. Nature Photon. 2016, 10, 420.
- Grad et al. 2017 Grad, A.; Held, M.; Zakharko, Y.; Tropf, L.; Gather, M. C.; Zaumseil, J. Nature Mater. 2017, advance online publication.
- Palyi et al. 2012 Palyi, A.; Struck, P.; Rudner, M.; Flensberg, K.; Burkard, G. Phys. Rev. Lett. 2012, 108, 206811.
- Wilson-Rae et al. 2012 Wilson-Rae, I.; Galland, C.; Zwerger, W.; Imamoglu, A. New J. Phys. 2012, 14, 115003.
- Galland and Imamoglu 2008 Galland, C.; Imamoglu, A. Phys. Rev. Lett. 2008, 101, 157404.
- Buchs et al. 2011 Buchs, G.; Barkelid, M.; Bagiante, S.; Steele, G. A.; Zwiller, V. J. Appl. Phys. 2011, 110, 074308.
- Liu et al. 2011 Liu, C.-H.; C.-C-Wu,; Zhong, Z. Nano Lett. 2011, 11, 1782.
- Imamoglu and Yamamoto 1994 Imamoglu, A.; Yamamoto, Y. Phys. Rev. Lett. 1994, 72, 210–213.
- J.Kim et al. 1999 J.Kim,; O.Benson,; H.Kan,; Y.Yamamoto, Nature 1999, 72, 500–503.
- Jarillo-Herrero et al. 2004 Jarillo-Herrero, P.; Sapmaz, S.; Dekker, C.; Kouwenhoven, L. P.; van der Zant, H. S. J. Nature 2004, 429, 389.
- Laird et al. 2015 Laird, E. A.; Kuemmeth, F.; Steele, G. A.; Grove-Rasmussen, K.; Nygård, J.; Flensberg, K.; Kouwenhoven, L. P. Rev. Mod. Phys. 2015, 87, 703.
- Hofmann et al. 2016 Hofmann, M. S.; Noé, J.; Kneer, A.; Crochet, J. J.; Högele, A. Nano Lett. 2016, 16, 2958 –2962.
- Högele et al. 2008 Högele, A.; Galland, C.; Winger, M.; Imamoglu, A. Phys. Rev. Lett. 2008, 100, 217401.
- Walden-Newman et al. 2012 Walden-Newman, W.; Sarpkaya, I.; Strauf, S. Nano Lett. 2012, 12, 1934 – 1941.
- Hofmann et al. 2013 Hofmann, M. S.; Glückert, J. T.; Noé, J.; Bourjau, C.; Dehmel, R.; Högele, A. Nature Nanotech. 2013, 8, 502 – 505.
- Khasminskaya et al. 2016 Khasminskaya, S.; Pyatkov, F.; Slowik, K.; Ferrari, S.; Kahl, O.; Kovalyuk, V.; Rath, P.; Vetter, A.; Hennrich, F.; Kappes, M. M.; Gol’tsman, G.; Korneev, A.; Rockstuhl, C.; Krupke, R.; Pernice, W. H. P. Nature Photon. 2016, 10, 727.
- Ma et al. 2015 Ma, X.; Hartmann, N. F.; Baldwin, J. K. S.; Doorn, S. K.; Htoon, H. Nature Nanotech. 2015, 10, 671 – 676.
- Zhang et al. 2017 Zhang, S.; Kang, L.; Wang, X.; Tong, L.; Yang, L.; Wang, Z.; Qi, K.; Deng, S.; Li, Q.; Bai, X.; Ding, F.; Zhang, J. Nature Photon. 2017, adv. online.
- Postma et al. 2001 Postma, H. W. C.; Teepen, T.; Yao, Z.; Grifoni, M.; Dekker, C. Science 2001, 293, 76.
- Buchs et al. 2009 Buchs, G.; Bercioux, D.; Ruffieux, P.; Gröning, P.; Grabert, H.; Gröning, O. Phys. Rev. Lett 2009, 102, 245505.
- Robertson et al. 2012 Robertson, A. W.; Allen, C. S.; Wu, Y. A.; He, K.; Olivier, J.; Neethling, J.; Kirkland, A. I.; Warner, J. H. Nature Comm. 2012, 3:1144.
- Yoon et al. 2016 Yoon, K.; Rahnamoun, A.; Swett, J. L.; Iberi, V.; Cullen, D. A.; Vlassiouk, I. V.; adn S. Jesse, A. B.; Sang, X.; Ovchinnikova, O. S.; Rondinone, A. J.; Unocic, R. R.; van Duin, A. C. T. ACS Nano 2016, 10, 8376–8384.
- Senkovskiy et al. 2017 Senkovskiy, B. V. et al. Nano Lett. 2017, 17, 4029.
- Tersoff and Hamann 1985 Tersoff, J.; Hamann, D. R. Phys. Rev. B 1985, 31, 805–813.
- Tolvanen et al. 2009 Tolvanen, A.; Buchs, G.; Ruffieux, P.; Gröning, P.; Gröning, O.; Krasheninnikov, A. V. Phys. Rev. B 2009, 79, 125430.
- Chiang et al. 2001 Chiang, I. W.; Brinson, B. E.; Huang, A. Y.; Willis, P. A.; Bronikowski, M. J.; Margrave, J. L.; Smalley, R. E.; Hauge, R. H. J. Phys. Chem. B 2001, 105, 8297–8301.
- Buchs et al. 2007 Buchs, G.; Ruffieux, P.; Gröning, P.; Gröning, O. Appl. Phys. Lett. 2007, 90, 013104.
- Buchs et al. 2007 Buchs, G.; Krasheninnikov, A. V.; Ruffieux, P.; Gröning, P.; Foster, A. S.; Nieminen, R. M.; Gröning, O. New J. Phys. 2007, 9, 275.
- Buchs 2008 Buchs, G. Local modification and characterization of the electronic structure of carbon nanotubes. Ph.D. thesis, University of Basel, 2008.
- Lim et al. 2007 Lim, S. H.; Li, R.; Ji, W.; Lin, J. Phys. Rev. B 2007, 76, 195406.
- Krasheninnikov et al. 2001 Krasheninnikov, A. V.; Nordlund, K.; Sirviö, M.; Salonen, E.; Keinonen, J. Phys. Rev. B 2001, 63, 245405.
- Bercioux et al. 2011 Bercioux, D.; Buchs, G.; Grabert, H.; Gröning, O. Phys. Rev. B 2011, 83, 165439.
- Chico et al. 1996 Chico, L.; Benedict, L. X.; Louie, S. G.; Cohen, M. L. Phys. Rev. B 1996, 54, 2600–2606.
- Neophytou et al. 2007 Neophytou, N.; Ahmed, S.; Klimeck, G. Appl. Phys. Lett. 2007, 90, 182119.
- Mayrhofer and Bercioux 2011 Mayrhofer, L.; Bercioux, D. Phys. Rev. B 2011, 84, 115126.
- Bockrath et al. 2001 Bockrath, M.; Liang, W.; Bozovic, D.; Hafner, J. H.; Lieber, C. M.; Tinkham, M.; Park, H. Science 2001, 291, 283–285.
- Khomyakov et al. 2009 Khomyakov, P. A.; Giovannetti, G.; Rusu, P. C.; Brocks, G.; van den Brink, J.; Kelly, P. J. Phys. Rev. B 2009, 79, 195425.
- Clair et al. 2011 Clair, S.; Kim, Y.; Kawai, M. Phys. Rev. B 2011, 83, 245422.
- Reich et al. 2002 Reich, S.; Maultzsch, J.; Thomsen, C.; Ordejón, P. Phys. Rev. B 2002, 66, 035412.
- 44 Note that for a sufficiently small area under the barrier, it can be shown that the transmission probability is a function of the area such that two rectangular barriers with different geometries but with the same area lead to the same, or very close transmission probability . Indeed, the textbook expression for the transmission probability through a rectangular tunneling barrier is given by , where and are respectively the barrier height and width. For the argument in the sufficently small such that , it can be shown that , with . In our case, this condition is not satisfied and thus the barrier geometries are tuned iteratively on an empirical fashion.
- Perdew et al. 1996 Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865.
- Blöchl 1994 Blöchl, P. E. Phys. Rev. B 1994, 50, 17953–17979.
- Kresse and Joubert 1999 Kresse, G.; Joubert, D. Phys. Rev. B 1999, 59, 1758–1775.
- Kresse and Furthmüller 1996 Kresse, G.; Furthmüller, J. Phys. Rev. B 1996, 54, 11169–11186.
- Mostofi et al. 2008 Mostofi, A. A.; Yates, J. R.; Lee, Y.-S.; Souza, I.; Vanderbilt, D.; Marzari, N. Comput. Phys. Commun. 2008, 178, 685.
- Gómez-Navarro et al. 2005 Gómez-Navarro, G.; De Pablo, P. J.; Gómez-Herrero, J.; Biel, B.; Garcia-Vidal, F. J.; Rubio, A.; Flores, F. Nature Mater. 2005, 4, 534–539.
- Bulusheva et al. 1998 Bulusheva, L. G.; Okotrub, A. V.; Romanov, D. A.; Tomanek, D. J. Phys. Chem. A 1998, 102, 975.
- Sanchez-Valencia et al. 2014 Sanchez-Valencia, J. R.; Dienel, T.; Gröning, O.; Shorubalko, I.; Mueller, A.; Jansen, M.; Amsharov, K.; Ruffieux, P.; Fasel, R. Nature 2014, 512, 61.
- Zhang et al. 2017 Zhang, S.; Kang, L.; Wang, X.; Tong, L.; Yang, L.; Wang, Z.; Qi, K.; Deng, S.; Li, Q.; Bai, X.; Ding, F.; Zhang, J. Nature 2017, 543, 234.
- Powell et al. 2017 Powell, L. R.; Kim, M.; Wang, Y. J. Am. Chem. Soc. 2017, X, XX.
- Krasheninnikov et al. 2006 Krasheninnikov, A. V.; Lehtinen, P. O.; Foster, A. S.; Nieminen, R. M. Chem. Phys. Lett. 2006, 418, 132–136.
- Ma et al. 2005 Ma, Y.; Foster, A. S.; Krasheninnikov, A. V.; Nieminen, R. M. Phys. Rev. B 2005, 72, 205416.
- Ruffieux et al. 2016 Ruffieux, P.; Wang, S.; Yang, B.; Sánchez-Sánchez, C.; Liu, J.; Dienel, T.; Talirz, L.; Shinde, P.; Pignedoli, C. A.; Passerone, D.; Dumslaff, T.; Feng, X.; Mul̈len, K.; Fasel, R. Nature 2016, 531, 489.
- Sun et al. 2008 Sun, X.; Zaric, S.; Daranciang, D.; Welsher, K.; Lu, Y.; Li, X.; Dai, H. J. Am. Chem. Soc. 2008, 130, 6551.
- Barros et al. 2006 Barros, E. B.; Capaz, R. B.; nad G. G. Samsonidze, A. J.; Filho, A. G. S.; Ismail-Beigi, S.; Spataru, C. D.; Louie, S. G.; Dresselhaus, G.; Dresselhaus, M. S. Phys. Rev. B 2006, 73, 241406(R).
- Lefebvre et al. 2006 Lefebvre, J.; Austing, D. G.; Bond, J.; Finnie, P. Nanolett. 2006, 6, 1603.
- Buchs et al. 2014 Buchs, G.; Bagiante, S.; Steele, G. A. Nature Commun. 2014, 5:4987.
- Gabor et al. 2009 Gabor, N. M.; Zhong, Z.; Bosnick, K.; Park, J.; McEuen, P. L. Science 2009, 325, 1367.
- Horcas et al. 2007 Horcas, I.; Fernandez, R.; Gomez-Rodriguez, J. M.; Colchero, J.; J., G.-H.; Baro, A. M. Rev. Sci. Instrum. 2007, 78, 013705.