Single-PbS-CQD transistor. Transport measurements were performed on single PbS CQDs in contact with nanogap source-drain electrodes, as shown in Figs. 1a and 1b. A back-gate electrode was used to tune the electronic state of the PbS CQDs. Figure 1c shows the source-drain current, ISD, as a function of the source-drain voltage, VSD, of sample A with CQD diameter d ~ 3.6 nm, measured for different back-gate voltages, VG. The sample exhibits a strongly suppressed current near VSD = 0 V, followed by a step-like current increase at high |VSD|. The voltage width of the suppressed conductance is modulated by changing VG. The suppressed current at VSD ~ 0 V is a manifestation of the strong Coulomb interaction in this system and is called the Coulomb blockade effect. A current step occurs at a large |VSD| at which an extra electron is energetically allowed to enter the CQD. The observed ISD–VSD characteristics clearly indicate that this sample operates as a single-electron transistor (SET). The Coulomb stability diagrams were obtained by plotting the differential conductance, dI/dVSD, as a function of VSD and VG, as shown in Fig. 2.
Coulomb stability diagrams for various CQD sizes. Figure 2 shows Coulomb stability diagrams obtained for three samples with different CQD sizes (see also Supplementary Fig. S1 for more data). The dark-colored diamond-shaped areas (Coulomb diamonds) correspond to the Coulomb blockade regions. Each Coulomb diamond is associated with a well-defined number of confined electrons, n, in the CQD. Although the number of electrons in the CQD could not be determined, we can tell the parity of n from the size of the Coulomb diamonds. Especially in samples A and B, the Coulomb diamond for n = N + 1 electrons is notably larger than that for n = N and N + 2. The (N + 1) diamond contains an even number of electrons and has a larger addition energy, Eadd, which consists of the Coulomb charging energy, EC, and difference in orbital quantization energy, ΔE, in the CQD. The diamonds labeled N and N + 2 are smaller because the addition energies are only due to EC. Therefore, N is an odd number in these samples. In large CQDs (sample C, d ~ 8.7 nm), such even-odd behavior of the Coulomb diamonds is not clearly observed, as shown in Fig. 2c, most likely because of the very different numbers of electrons n in large CQDs; a larger PbS CQD has a narrower bandgap, as well as smaller EC and ΔE, leading to electrical access to the larger n regime. In Figs. 2a and 2b and Supplementary Fig. S1a, the sizes of the Coulomb diamonds for n = N and N + 2 are not the same and significantly differ, indicating that EC strongly depends on the number of electrons in these samples. Since similar even-odd behavior of the Coulomb diamonds and n-dependent charging energy are reported in many other few-electron quantum dot systems [38–42], samples A, B, and D are in the few-electron regime, while samples C and E are in the many-electron regime.
Charge addition energy and conductance. In Fig. 3a, we plot the Eadd of electrons for each sample as a function of the number of electrons labeled in Fig. 2 and Supplementary Fig. S1. In small CQD samples (samples A, B and D), Eadd for n = N (odd) is slightly larger than that for n = N + 2 (odd), indicating that EC decreases with increasing n. These results indicate that the electron wavefunction in the PbS CQD becomes more extended in space with increasing n, leading to larger capacitive coupling with the electrodes and reduced electron-electron Coulomb repulsion in the CQDs. Such n dependence of Eadd is not clear for large CQD samples with d ~ 8.7 nm; Eadd shows much smaller and almost constant values in samples C and E, indicating that the spatial size of the electron wavefunction in the PbS CQDs does not significantly change and that the quantized energy levels are almost degenerate. Concerning the large peak in Eadd at n = N + 1 for sample D, it reflects a large orbital quantization energy difference ΔE, which strongly depends on n as well as the size and shape of measured single PbS CQDs [43]. The number of electrons in sample D might be slightly different from those in samples A and B.
Figure 3b shows the CQD size dependence of the experimentally obtained EC. Here, EC is plotted as the average of all Eadd for n = odd in each sample. The calculated EC for a metallic sphere quantum dot is shown as a solid line in Fig. 3b. The self-capacitance of a metallic sphere quantum dot surrounded by oleic acid is given by Cdot = 4πε0εrr, where ε0 is the vacuum permittivity, εr ~ 2.5 is the relative permittivity of oleic acid [44, 45], and r is the radius of the sphere. Then, EC is calculated as EC = e2/2Cdot, where e is the elementary charge. The experimentally obtained EC is found to be in reasonable agreement with the simple metallic sphere quantum dot model. The observation of a large EC of ~ 150 meV in small CQDs suggests that devices with small CQDs operate as SETs even at room temperature. Considering that PbS CQDs are excellent emitters and absorbers of light, our device is a potential candidate platform for room-temperature SETs with good optical properties, increasing the functionality and versatility of single-quantum-dot devices.
Next, we discuss the magnitude of the tunneling conductance. Figure 3c shows the differential conductance taken at VSD = 0 V as a function of VG plotted for samples with d = 4.8 nm (i.e., samples B and D), exhibiting clear Coulomb oscillation peaks. The conductance at the Coulomb peaks shows a strong dependence on n. The two paired N-th and (N + 1)-th Coulomb peaks exhibit almost the same conductance and a one order of magnitude lower conductance than the other paired (N + 2)-th and (N + 3)-th peaks located in the high VG region (this tendency is clearer in sample D). These results strongly suggest that the N and (N + 1) electrons ((N + 2) and (N + 3) electrons) in these CQDs are in spin-up and spin-down states while occupying the same orbital. As mentioned in the previous paragraph, the spatial size of the electron wavefunction becomes larger for higher orbitals. These facts lead to the conclusion that the tunneling barrier in the present metallic electrode-CQD system is formed not only by the capping materials (oleic acid) but also by the intrinsic gaps between the metal electrodes and the electron wavefunction in the CQD, as schematically illustrated in Fig. 3d. Since the tunneling conductance exponentially depends on the tunneling gap, even a small change in the size of the electron wavefunction gives rise to a large difference in the tunneling conductance.
Kondo effect in PbS CQDs. Next, let us examine the spin-dependent carrier transport observed in single PbS CQDs. Figure 4a shows a Coulomb stability diagram for a high-conductance sample with d ~ 4.8 nm (sample F) measured at 4 K. One can only observe the data near the charge degeneracy point of the Coulomb stability diagram for the addition of the N-th electron owing to the poor gate modulation efficiency in this sample. The Coulomb diamond for n = N shows a clear resonant enhancement of the zero-bias conductance. This behavior arises from the formation of a spin singlet state between an unpaired electron in the CQD and an electron with the opposite spin in the electrodes, i.e., the spin-half Kondo effect [46, 47]. The spin-half Kondo effect only appears in n = odd Coulomb diamonds, suggesting that N is an odd number in Fig. 4a.
In Fig. 4b, dI/dVSD is plotted as a function of VSD at VG = 25 V for various temperatures. The Kondo peak at VSD = 0 V grows with decreasing temperature. Figure 4c shows the temperature dependence of dI/dVSD at VSD = 0 V (i.e., at the Kondo peak) for various VG. With increasing temperature, dI/dVSD at VSD = 0 V decreases up to ~ 30 K because of the weakened Kondo effect then begins to increase at T > 30 K due to the increased background conductance induced by the thermal broadening of the adjacent tunneling peak. Although the background conductance due to the falling edges of the adjacent tunneling peaks hinders precise determination of the full-width at half-maximum of the Kondo peak, w, it is roughly estimated to be ew ~ 3 meV. The Kondo temperature, TK, is expressed as kBTK ~ ew [46, 48, 49], where kB is the Boltzmann constant. From this relation, TK in this sample is determined to be TK ~30 K, which is consistent with the enhanced conductance at VSD = 0 V for T < 30 K in Fig. 4c.
Discussion. In sample A, the slopes of the Coulomb diamond boundaries are not constant and depend on VG and VSD (see Fig. 2a and Supplementary Fig. S2a), indicating that the electron density profile in the CQD is affected by the potential gradient induced by VG and VSD. In this sample, the noise level is low enough to identify the excited-state lines in the Coulomb stability diagram [50]. Clear excited-state lines are observed, from which the orbital quantization energy differences for n electrons, ΔE(n), are determined to be ΔE(N) = 50 meV, ΔE(N + 1)low = 80 meV from the low VG region of the (N + 1) Coulomb diamond, ΔE(N + 1)high = 90 meV from the high VG region of the (N + 1) Coulomb diamond, and ΔE(N + 2) = 35 meV (see Supplementary Fig. S2). We simulated ΔE(n) for regular octahedral and cubic PbS quantum dots with similar volumes in the few-electron regime using the software QTCAD from Nanoacademic Technologies [51] and found that ΔE(n) is between 10 and 100 meV, which is in reasonable agreement with the experimentally observed ΔE(n). Although the observed ΔE(N + 1)high and ΔE(N + 1)low originate from the same excited state, ΔE(N + 1)high is larger than ΔE(N + 1)low, indicating stronger confinement of electrons at higher VG. Furthermore, ΔE(N) significantly increases with increasing VG (see Supplementary Fig. S2c), which again indicates stronger confinement of electrons in the CQD at higher VG. These results indicate the strong influence of the external electric field induced by VG and VSD on the confinement potential of electrons in single PbS CQDs, suggesting the tunability of the effective confinement size of electrons and the bandgap in CQDs by the external electric field.
Regarding the Kondo effect, we found a sample (sample G, d ~ 4.8 nm) that has higher conductance than sample F, as shown in Supplementary Fig. S3. The width of the Kondo peak in sample G is ew ~ 7 meV at VG = 40 V, suggesting a very high Kondo temperature of TK ~80 K. The TK in a SET is described by kBTK ~ (EC Γ\({)}^{\frac{1}{2}}\)exp[-π(µ-ε0)/2Γ], where Γ is the tunnel coupling energy between the quantum dot and the source-drain electrodes, µ is the electrochemical potential of the source-drain electrodes at VSD = 0 V, and ε0 is a spin-degenerate energy level in the quantum dot occupied by a single electron [46]. Therefore, TK is limited by both EC and Γ. Considering the large charging energy EC ~130 meV (corresponding to a thermal energy of 1500 K) in PbS CQDs with d ~ 4.8 nm, the small Γ limits the TK in samples F and G; this is reasonable because oleic acid (~ 2 nm long ligand) is known as a long-chain insulating ligand that blocks charge carrier transport through CQDs [34]. The observation of a high TK in oleic acid-capped PbS CQDs suggests the feasibility of realizing a much higher TK up to room temperature by replacing oleic acid with short-chain ligands, which may add the Kondo effect as one of the carrier transport mechanisms in CQD assemblies operated at room temperature.
In summary, we have demonstrated single-CQD transistors based on commercially available high-quality PbS CQDs. The transport characteristics strongly depend on the quantum dot size; a few-electron regime is observed in small PbS CQDs, while a many-electron regime is observed in large CQDs. From the n dependence of EC and the conductance, we demonstrated that the tunneling barrier in this system is formed not only by the capping material but also by the intrinsic gap between the electron wavefunction in the CQDs and electrodes. Analysis of the excited states indicates that the confinement potential of electrons in CQDs is strongly affected by the external electric field induced by VG and VSD. The Kondo effect is also observed for the first time in a single-CQD system; this indicates strong coupling between the electrodes and the CQDs despite the use of a long-chain insulating oleic acid ligand. These results provide nanoscopic insight into the carrier transport through CQDs at the single quantum dot level, which is the knowledge essential for developing CQD applications in optoelectronic devices, such as solar cells and photodetectors. Furthermore, the observation of a large charging energy suggests that devices with small CQDs could operate as SETs even at room temperature. Considering that PbS CQDs are excellent emitters and absorbers of light, our device is a potential candidate platform for room-temperature SETs with good optical properties, increasing the functionality and versatility of single-quantum-dot devices. It will bring about innovation in quantum information technology.