The so called 0.7 plateau, an unexpected conductance plateau observed around 0.7 × G₀ (G₀ = 2e²/h, where e is the electron charge and h is Planck’s constant) in a quantum point contact (QPC) has been a hot topic for decades. The existence of this anomalous plateau is of great importance because this phenomenon was first described as spontaneous spin polarization in zero external magnetic field, which could be very useful for developing spintronic devices. It was first observed by Thomas et al. [1], shortly after the discovery of conductance quantization in units of G₀ in QPCs [2,3]. Even though it was reported more than 30 years ago, debate about the origin of this phenomenon is on-going. Among numerous explanations, spontaneous spin polarization [1,4], Kondo effect via quasi-bound state in QPC [5], and a smeared van Hoff singularity in QPC [6] are popular, while ferromagnetic spin coupling [7], electron interference caused by backscattering in QPC [8], and other explanations [9–11] are less popular. Regardless of their popularity and the number of proposed theories, no consensus about the phenomenon’s origin has been achieved. In this work, we report experimental results in which Fano resonance [12–14] between bound state in quantum dot (QD) and continuum state in QPC channel can cause the 0.7 anomaly. The conductance through the device is calculated numerically with the KWANT quantum device simulator [15]. The results show good agreement with experimental results. This suggests that Fano resonance is a candidate for the origin of the 0.7 conductance anomaly observed in QPCs. The many popular explanations mentioned above rely on many-body interactions under specific conditions in QPCs, which are sometimes difficult to occur. In contrast, our results are based on rather simple and robust single particle interference, something that is more likely to happen.

Device was fabricated on a GaAs/AlGaAs heterostructure 2-dimensional electron gas (2DEG) wafer with electron density of 3.1 × 10¹¹ cm⁻² and mobility of 2.9 × 10⁶ cm²V⁻¹s⁻¹ at 4.2 K. The 2DEG was buried 63.5 nm below the surface of the wafer. The device was fabricated by optical and e-beam lithography processes on the GaAs/AlGaAs heterostructure [16,17]. All measurements were done at 80 mK in a dilution refrigerator. Conductance measurements were performed by applying 10µVrms sinusoidal voltage at a frequency of 487 Hz to the device and measuring the modulated current through the device. Figure 1(a) provides a scanning electron microscopy (SEM) picture of the device used in the experiment. Devices with two different QD diameters (300 and 600 nm) were used for the experiment. The QPC channel (green arrow) is defined between the 1D gate and two QD gates in the Fig. 1(a). The confinement state in the QD was controlled by the plunger gate. The right side of the QD (a small gap between two QD gates) was coupled to the QPC channel. The 1D gate controls the number of conducting modes in the QPC channel. The conductance through the device was measured between the source, S, and the drain, D, as shown in the figure. Figure 1(b) shows the measured Fano resonance through the device. The plunger gate voltage changed the energy states in the QD. The conductance through the QPC slowly decreased as the plunger gate voltage of the QD became more negative and finally pinched off at around −0.2 V. On top of slowly decreasing conductance curve, sharp asymmetric resonance peaks were observed, typical signatures of Fano resonance. Fano resonance can be regarded simply as interference between electrons flowing through the wire and electrons flowing through QD [14]. Since the electronic phase changes by π before and after the QD’s resonance state, these phase changes caused constructive and destructive interference of electrons flowing through the device, resulting in Fano resonance. The observed peaks were fitted with the theoretical expression of the Fano resonance. The conductance G of the Fano resonance is given by

Figure 1. (a) SEM picture of device. QD is side-coupled to the QPC channel. The conductance is measured between the source (S) and the drain (D) of the device. Electrons conduct through the QPC channel (green arrow) formed between the QD and the 1D gate. (b) Fano resonance measured as a function of plunger gate of QD. The red and blue traces are theoretical fits to the experimental data. (c) Measured charging energy of QD (300 nm in diameter).

where $ϵ=\alpha \left(V_g-V_R\right)/\left(\text{Γ}/2\right)$, α is the gate voltage to energy conversion ratio, q is Fano’s asymmetry parameter, G_bg is the incoherent background conductance, A is the amplitude, and V_R and Γ are the position and the width of the resonance, respectively [12,18]. The red and blue traces in the Fig. 1(b) are fit to the experimental data with above equation. The asymmetry parameters for the red and blue traces were 3.5 and −4, respectively. The perfect fit to the experimental data confirmed that the observed peaks were due to Fano resonances, as previously reported by other researchers [12–14]. The energy level spacings in the QD were measured using the 2D conductance map (as a function of bias and the gate voltage) of the Fano resonances, shown in Fig. 1(c). The measured charging energies (energy level spacing plus dot charging energy) of the QD were estimated to be around 0.6–0.9 meV, which is typical for the QD (inner diameter ~ 300 nm) used in the experiment.

3.1. Experimental results for 0.7 structure

In Fig. 2, the conductance through the QPC as a function of 1D gate voltage was measured for various QD gate voltages. A 0.7 plateau was observed for the QPC coupled with a 300 nm diameter QD. The plunger gate voltage was set to −0.05 V to pinch off all conduction channels under the plunger gate. (Device was cooled down with +0.3 V voltage on all the gates). When the QD gate voltage was set to −0.69 V, sharp conductance dips (denoted by a red arrow) in the QPC plateau region were observed. A dip in the QPC conductance plateau is a typical sign of Fano resonance. However, as the QD gate voltage changed, these dips weakened and a conductance plateau began to appear around 0.7 G₀, indicated by the blue arrow in the Fig. 2(a). A few broad resonance-like peaks also followed the 0.7 plateau. The existence of such resonance-like peaks and dips, and the plateau, suggests that Fano resonance is a possible cause of the 0.7 conductance plateau. If a broad Fano resonance peak exists around the conductance transition region (e.g., between zero and the first plateau), the resulting conductance (the Fano resonance peak added to the QPC conductance) may look like a plateau. For the QPC coupled with a 600 nm diameter QD, it was difficult to observe the 0.7 conductance plateau, as shown in Fig. 2(b); instead, we observed many small resonance peaks added to the quantized conductance of the QPC. In Fig. 2(c), the charging energy of the 600 nm dot device was measured and found to be approximately 0.3–0.5 meV, or roughly half that of the 300 nm QD device. We suspect that the small charging energy of the QD overlaps the Fano resonance peaks, making the 0.7 plateau difficult to observe.

Figure 2. Conductance through the QPC was measured as a function of 1D gate voltage for various QD gate voltages. The plunger gate voltage was fixed at −0.05 V. The same voltage V_up = V_down = V_c was applied to the upper and lower QD gates. (a) The inner diameter of the side-coupled QD was 300 nm. V_c was varied in 15 steps from −0.69 to −0.9 V. A small conductance bump (grey arrow) in the transition region evolved into a plateau (blue arrow) as the 1D gate voltage changed. (b) The inner diameter of the side-coupled QD was 600 nm. V_c was varied in 15 steps from −0.42 to −0.7 V. (c) The charging energy of the 600 nm QD was measured. The charging energies was estimated to be around 0.3–0.5 meV.

3.2. Numerical simulation

The conductance through the device was calculated numerically using KWANT [15], a software package for quantum transport. The electrostatic potential profile of the device was calculated by analytically solving the Laplace equation [19]. Figure 3(a) shows results calculated for a device coupled with a 300 nm diameter QD. These results are consistent with the experimental results shown in Fig. 2(a). It was found that the results can only be obtained when the gate voltages of the upper and lower QD gates are not the same. The lower QD gate voltage was shifted 30 mV negative from the upper QD gate voltage (i.e., V_up = V_c, V_down = V_c − 0.03 V for various V_c) to obtain the results shown in the Fig. 3(a). A small conductance bump (grey arrow) was observed in the transition region (between zero and the first plateau) when QD gate voltage was less negative, i.e., when the QD was more open to the QPC channel. As the QD gate voltages became more negative, the bump was enhanced and turned into a plateau for a QD gate voltage of -0.066 V. The plateau changed to a broad peak when the QD gate voltage increased further. The overall characteristics are very similar to the experimental results shown in Fig. 2(a).

Figure 3. Conductance through device was calculated numerically using KWANT simulator. (a) Calculated results for 300 nm QD device. The upper and lower QD gate voltages were set to be different. The upper and the lower QD gate voltages were set to V_up = V_c, V_down = V_c − 0.03 V, respectively, while V_c was varied in 15 steps from -0.69 to -0.895 V. A small conductance bump (grey arrow) in the transition region evolved into a plateau as the 1D gate voltage changed. (b) Calculated results for 600 nm QD device. The upper and lower QD gate voltages were set to be identical, V_up = V_down = V_c, while V_c was varied in 15 steps from −0.42 to −0.7 V.

Figure 3(b) shows results calculated for the device with 600 nm QD. Many small resonance peaks repeating one after another were observed, instead of a plateau. We believe that the frequent repetition of Fano peaks hindered the observation of the 0.7 plateau in our device. A detailed explanation is provided in the discussion. To obtain results similar to those of the experiment, the voltages of the upper and lower QD gates were assumed to be the same. The symmetrical gate voltages on the upper and lower QD gates made the resonance peaks stronger, making it difficult to create a broad resonance peak, which seems to be a necessary condition to form the 0.7 plateau.

3.3. Discussion

In Fig. 4, we tried to reconstruct the 0.7 conductance plateau by adding omit the Fano resonance peak to an ideal quantized conductance curve of a QPC; calculations were performed using Eq. (1) and the KWANT simulator. Several Fano peaks with different asymmetry parameters (q) and ideal conductance quantization curve of a QPC are plotted in Fig. 4(a). The resonance width Γ was set to 3, a relatively wide peak width, representing a situation in which the QD is relatively open to the 1D wire. The asymmetry parameter q was varied from 2.7 to 6.7, resulting in a resonant peak height of around a few 0.1e²/h and a relatively asymmetric peak. The added results are shown in Fig. 4(b). A conductance plateau at around 0.7 G₀ was observed when the Fano peak with q = 5.7 was added. From these results, it is obvious that the increasing peak conductance compensated for the decreasing QPC conductance [Fig. 4(c), red shaded region] and created an unexpected conductance plateau around 0.7 G₀. This simple analysis shows that Fano resonance in a 1D channel is a candidate for the origin of the 0.7 conductance plateau.

Figure 4. (a) Simple model explaining 0.7 conductance plateau by adding Fano resonance peak to QPC’s quantized conductance curve. (b) Several Fano peaks with different asymmetry parameters were added to the typical QPC conductance curve. The center of the Fano peak was placed at −0.0283 V. A 0.7-like conductance plateau was observed when the Fano peak with q = 5.7 and Γ = 3 was added. (c) In the red shaded region, the increase in peak conductance was compensated for by a decrease in QPC conductance, resulting in an unexpected conductance plateau at around 0.7G₀. (d) Position of Fano peak shifted to give a plateau with different plateau conductance. The peak centers were at −0.0292, −0.0287, −0.0284, and −0.0278 V from left to right. The asymmetry parameters of the peaks were 5, 6, 6, and 5, from left to right; Γ = 3 for all peaks.

Althoughthe unusual plateau is named a ‘0.7 plateau’, it does not necessary mean that the conductance value is always 0.7. The name comes from the first report of an unexpected conductance plateau [1], observed around 0.7 G₀. However, later reports showed various plateau values below G₀. Figure 4(d) shows plateaus with different conductance values when Fano peaks are positioned at different locations in the transition region. The center of the Fano peak positions is −0.0292, −0.0287, −0.0284, and −0.0278 V from left to right.

A few resonant peaks followed the 0.7 plateau in the experiment, as shown in Fig. 2. This repetition is obvious, since Fano resonance occurs whenever a QD’s energy state aligns with the QPC’s Fermi energy. The peak repeats more frequently in the 600 nm QD device than in the 300 nm QD device because energy level spacing between confined energy states is smaller for the 600 nm QD, as shown in Fig. 2(c). To observe the 0.7 plateau, the spacing between the peaks must be wide enough so that the plateau region is not affected by the following peak. For the 600 nm QD device, no 0.7 plateau was observed in the experiments. We believe that the relatively small charging energy of the 600 nm QD made gaps between resonance peaks small, causing the resonance peaks to overlap and eventually hindering plateau development.

A real QPC device consists of only two split gates and does not have a QD coupled to a 1D channel, as our device does. However, the impurity potential (caused by modulation doping of the GaAs/AlGaAs heterostructure) inside the 1D channel can create a QD-like potential puddle, and thus lead to a resonance-like characteristic in the QPC conductance [11]. The size of the localized state in the QPC cannot be bigger than the size of the QPC channel itself. Therefore, it is reasonable to assume that the diameter is less than 200 nm. Therefore, the energy level spacing is relatively larger than that in our device, resulting in less frequent resonance peaks. This partly explains why only a plateau or a peak [20] was observed in the real QPC device.

A ‘0.7-like’ conductance plateau was observed for a QPC device coupled with a QD on the side of the conduction channel. The origin of the 0.7 conductance plateau in our experiment can be explained by Fano resonance via the side-coupled QD. Numerical simulations show results similar to those of the experiment. The results were explained using a very simple model that adds the Fano resonance peak to the QPC’s quantized conductance curve. The conductance plateau resulted from the increase in Fano peak conductance compensating for the decrease in QPC conductance when the QPC gate was slowly pinched off. Although there are no side-coupled QDs in real QPC devices, an unexpected potential puddle caused by impurity potential inside the 1D channel can create a QD-like state. Therefore, we believe that Fano resonance via unexpected QD-like state in a QPC is a candidate for the origin of the 0.7 conductance anomaly.

This work was supported by a 2-year research grant from Pusan National University.

The authors declare no conflicts of interest.