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The estimate, based on calculation, of the tunnel coupling energy quoted refers to the energy splitting between corresponding states when the ladders of single-particle energy levels for two identical dots are exactly aligned with each other. Using the atomic-orbital-like notation introduced in Sec. , ΔSAS would thus refer to the bare tunnel coupling at the 1s→1s, 2p→2p,... resonances, for example. However, as explained more fully in Sec. , the resonances of interest in this paper involve the ground single-particle state of one dot and a higher-energy single-particle state of the other dot, namely, the 1s→X resonances, where X is any single-particle dot state other than 1s. The small, though not insignificant, tunnel couplings at these resonances are not determined by ΔSAS but are related to natural perturbations in the nonideal dot confinement potentials. In Sec. , these interdot tunnel couplings are characterized by the s parameters in our model.
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The estimate, based on calculation, of the tunnel coupling energy quoted refers to the energy splitting between corresponding states when the ladders of single-particle energy levels for two identical dots are exactly aligned with each other. Using the atomic-orbital-like notation introduced in Sec., Δ S A S would thus refer to the bare tunnel coupling at the 1 s → 1 s, 2 p → 2 p,... resonances, for example. However, as explained more fully in Sec., the resonances of interest in this paper involve the ground single-particle state of one dot and a higher-energy single-particle state of the other dot, namely, the 1 s → X resonances, where X is any single-particle dot state other than 1 s. The small, though not insignificant, tunnel couplings at these resonances are not determined by Δ S A S but are related to natural perturbations in the nonideal dot confinement potentials. In Sec., these interdot tunnel couplings are characterized by the s parameters in our model.
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Self-consistent calculations (not shown) reveal that for our triple-barrier double-quantum-well resonant-tunneling structure, the confinement energy in each well along the z axis, ωz, is ∼70meV. This value is ten to fifteen times larger than the lateral confinement energies, ωx and ωy, in typical vertical QDs
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Self-consistent calculations (not shown) reveal that for our triple-barrier double-quantum-well resonant-tunneling structure, the confinement energy in each well along the z axis, ωz, is ∼70meV. This value is ten to fifteen times larger than the lateral confinement energies, ωx and ωy, in typical vertical QDs.
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The lateral confinement potentials of vertical QDs, though not infinitely high, are certainly sufficiently high that the employment of the Fock-Darwin spectrum and the elliptical parabolic spectrum for our basic arguments is quite justified provided one does not go too high up in energy (or equivalently let x and y get too large). This certainly holds even for the single-particle energy levels which are ∼25meV up in the probed dot spectra (see Fig.)
-
The lateral confinement potentials of vertical QDs, though not infinitely high, are certainly sufficiently high that the employment of the Fock-Darwin spectrum and the elliptical parabolic spectrum for our basic arguments is quite justified provided one does not go too high up in energy (or equivalently let x and y get too large). This certainly holds even for the single-particle energy levels which are ∼ 25meV up in the probed dot spectra (see Fig.).
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That the ellipticity of vertical QDs in circular mesas is never exactly unity and varies considerably from device-to-device (typically in the range of ∼1.05-2) is well documented for measurements in the few-electron regime. In addition to Refs., see also, 10.1103/PhysRevB.65.085325Natural perturbations in the confinement potentials of vertical QDs caused by local randomness and imperfections are ultimately responsible for this, as well as for the anticrossing behavior in the measured energy spectra
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That the ellipticity of vertical QDs in circular mesas is never exactly unity and varies considerably from device-to-device (typically in the range of ∼ 1.05 - 2) is well documented for measurements in the few-electron regime. In addition to Refs., see also P. Matagne, J. P. Leburton, D. G. Austing, and S. Tarucha, Phys. Rev. B 65, 085325 (2002). 10.1103/PhysRevB.65.085325
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(unpublished). Single-electron tunneling with longitudinal-optic-phonon emission only influences the measured single-particle spectra above the energy windows shown in Figs. , and so plays no role in the experiments we describe
-
K. Ono (unpublished). Single-electron tunneling with longitudinal-optic- phonon emission only influences the measured single-particle spectra above the energy windows shown in Figs., and so plays no role in the experiments we describe.
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Thus in each panel of Fig. , along the vertical axis corresponding to energy (E), VSD, and VG are altered such that Δ VG /Δ VSD =λ, where λ is a constant. Also, from Sec. onward, when we use the term differential conductance in connection with strategy A measurements, strictly speaking we are referring to dI/dE but this is simply proportional to dI/d VSD
-
Thus in each panel of Fig., along the vertical axis corresponding to energy (E), V SD, and V G are altered such that Δ V G / Δ V SD = λ, where λ is a constant. Also, from Sec. onward, when we use the term differential conductance in connection with strategy A measurements, strictly speaking we are referring to d I / d E but this is simply proportional to d I / d V SD.
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48
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77956335670
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The data sets used to build up the parts of spectra shown in Figs. had no extra spectral features [see, for example, Fig. 2(a) of Ref., for data set related to Fig. ]. The data set used to build up Fig. did have some extra spectral features but it was straightforward to exclude the unwanted extra lines by comparison with the calculated spectrum in Fig. . Thus the vector-voltage lines employed were well set up for the measurements related to Figs. but not quite so well set up for the measurement related to Fig..
-
The data sets used to build up the parts of spectra shown in Figs. had no extra spectral features [see, for example, Fig. 2(a) of Ref., for data set related to Fig.]. The data set used to build up Fig. did have some extra spectral features but it was straightforward to exclude the unwanted extra lines by comparison with the calculated spectrum in Fig.. Thus the vector-voltage lines employed were well set up for the measurements related to Figs. but not quite so well set up for the measurement related to Fig..
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49
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77956326095
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(unpublished). Although not shown, the energy spectrum of dot 1 from device I (device II) is overall quite similar in general appearance, but not fine detail, to that of dot 2 from the same device (for example, the dot ellipticities are comparable)
-
D. G. Austing (unpublished). Although not shown, the energy spectrum of dot 1 from device I (device II) is overall quite similar in general appearance, but not fine detail, to that of dot 2 from the same device (for example, the dot ellipticities are comparable).
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Also, the position of the 1s→1s resonance line in the VSD - VG plane hardly moves on application of the B field (not shown). This is what we would expect for our measurement scheme. If the upstream and downstream dots are identical the diamagnetic shift of the ground state in both dots would be the same
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Also, the position of the 1 s → 1 s resonance line in the V SD - V G plane hardly moves on application of the B field (not shown). This is what we would expect for our measurement scheme. If the upstream and downstream dots are identical the diamagnetic shift of the ground state in both dots would be the same.
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The lower edge of the reverse bias SET region in Fig. actually weakens and eventually disappears at high bias (V SD < - 50 mV). The reason for this is not fully understood. However, in the vicinity of the γ crossing for each panel in Fig., this edge, if visible, would bisect the region of interest (that bounded by the two dashed lines in the 1.9 T panel) into two zones. The upper zone only is strictly part of the reverse bias SET region highlighted and marked at lower bias in Fig.. In the lower zone, double-electron tunneling can occur but apparently the additional tunneling processes are very weak as no extra spectral features (resonance lines) are evident in contrast to the region below this zone. Thus inside the extended region of interest consisting of both zones SET dominates, and so our characterization and discussion of the level mixing remains valid. The quarter (three quarters) points defined in the 2.4 T panel of Fig. lie in the lower (upper) zone. The half points lie close to the boundary between the two zones.
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For weakly coupled vertical QDs and experimental conditions different from those described in this paper, variations in the properties of resonance lines have recently been reported. Also evident in Fig. of this work is the weakening and eventual disappearance of the lower edge of the arc-shaped SET region at high bias described in Ref..
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For weakly coupled vertical QDs and experimental conditions different from those described in this paper, variations in the properties of resonance lines have recently been reported. Also evident in Fig. of this work is the weakening and eventual disappearance of the lower edge of the arc-shaped SET region at high bias described in Ref..
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10.1103/PhysRevB.81.035323In this recent interesting work, randomly distributed ionized donors in double-barrier resonant-tunneling structures are shown to lead to QD-like potential minima.
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