Scattered points in Fig. 5.3(a) are the typical result of the measured dc current as a function of the phase difference. Idc as a function of ∆ϕ has the sinusoidal form with a small offset in the period of 2π. To systematically characterize the current amplitude, the
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Figure 5.2: Dot resistance as a function of gate voltage Vqpc2 when QPC1 is confined at 4.3, 6.5, and 12.7 kΩ. The mode number of QPC1, Nqpc1, is 3, 2, and 1 respectively.
The mode number of QPC2 was counted and labeled on the trace by the numbers. Inset:
Resistance of the independent quantum point contacts vs. gate voltage. The red(black) trace is for QPC1(2).
data is numerically fit with the form Idc(∆ϕ) = Ip(r)sin (∆ϕ + ϕ0) + Ip(r)0 . The solid line in the panel is the fit to the dataset by least square regression. The current generated in setup 1, Ip, as a function of voltage amplitude for N=(2,2) and f=7 MHz is presented in Fig. 5.3(b). Equal voltage amplitude is applied on the side gates, i.e. Vs1,ac = Vs2,ac = Vac. In this log-log plot, the data points follow a straight line of slope of 2 indicating a bilinear dependence, Ip ∝ (Vac)2. Ipincreases from as low as tens of pA for Vac=1 mV to tens of nA for Vac=40 mV. For Vac >40 mV, nonlinear effect becomes significant and Idc(∆ϕ) does not follow the sinusoidal form. The inset of Fig. 5.3(b) shows Ir(Vsd,ac, Vs2,ac = 15mV ) in setup 2 for N =(2,2), f=5 MHz. The slope of the trace is∼ 1.05, also suggesting a bilinear dependence of the current on the voltage amplitude.
Although the phase difference and amplitude dependences of Ip and Ir are similar, the relations with frequency, coupling strength and magnetic fields diversify. In Fig. 5.4(a), the five datasets represent Ip as a function of frequency for various coupling strengths between the dot and the reservoirs. From top down, the mode number of QPC1 and QPC2 are (1, 1), (1, 2), (1, 3), and so on. Ip is about linearly dependent with frequency for f >8 MHz. The current begins to saturate for higher frequencies. The dc current is the largest for N = (1, 1) and decreases by increasing the mode number. There is one thing worth
Chap. 5 Time dependent electric fields generated DC currents in a large gate-defined open dot 65
Figure 5.3: (a) Typical plot of the measured Idc(∆ϕ)(squares) along with the fit of the form Idc(∆ϕ) = Ip(r)sin (∆ϕ + ϕ0) + Ip(r)0 (solid curve). (b) Logarithmic plot of Ip(Vac) with a power law fit. The least square root fit gives a power of 1.99±0.06. In this device, the channel number of both entrance and exit of the dot is controlled to be 2 and the frequency of the ac voltage is 7 MHz. Inset: logarithmic plot of Ir(Vsd,ac) for N=(2, 2), f=5 MHz and Vs2,ac=15 mV for setup 2.
mentioning: Ip drops rapidly as soon as the dot is closed when Rdot is slightly larger than h/e2 while N = (0, 0). Although the measured Rdot is still finite∼ 100 kΩ, the measured current reduces to almost beyond the experimental resolution. Interestingly, Ip is found to follow a specific relation with the total mode number of the dot Ntot = Nqpc1+ Nqpc2. The ratio of Ntot for the last four datasets to N = (1, 1) is 3/2, 2, 2, and 3 respectively.
Fig. 5.4(b) shows the rescaled data of Ip multiplied by the factor of the ratio. e.g.
Ip*[N = (1, 2)] = 32Ip[N = (1, 2)] . All datasets fall onto one trace. Ip not only decreases with increasing Ntot (or the increasing coupling strength between the dot and reservoirs), it is proportional to Ntot−1 as demonstrated in the inset of Fig. 5.4(b). The results imply that the generated current is a type of charge pump as argued in the following. (1) The escape rate Γesc of electron in an open quantum dot is determined by the total channel number Ntot and follows Γesc ∼ N2πtot2∆ . The escaping or leaking of electrons from the dot increases with the increasing coupling strength which also reduces the time electrons relax to the reservoirs. One would expect a the performance of pumping to be suppressed by an increased coupling strength due to the opening of the dot [128, 129, 130, 131], or the dephasing due to the inelastic scattering from a third fictitious voltage probe[132]. (2) If the current is a result of any circuitry effect (e.g. classical dependence of the resistance on
the applied electric fields or cross talk between the electric fields), Ip would increase with increasing the dot resistance regardless of the mode number N, and Ip[N = (0, 0)] should be greater than Ip[N = (1, 1)]. But this scenario does not apply to our results here.
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Figure 5.4: (a) DC current generated in setup 1, Ip, as a function of frequency for a an open dot having different transmission mode numbers in QPC1 and QPC2. Each data point are averaged over five dataset of Idc(∆ϕ). N are (1, 1), (1, 2)H, (1, 3)N, (2, 2) and (3, 3)u respectively. Vac=15mV. (b) Rescaled current, Ip∗, vs. f for the same mode numbers.Ip∗(N ) = (Nqpc1+N2 qpc2)Ip. Ip of N=(1, 2), (1, 3), (2, 2), and (3, 3) are multiplied by the total mode number ratio to N=(2, 2). Inset: Ip vs. Ntot−1 for 5 MHz.
The current generated in setup 2 has completely different features. As shown in Fig.
5.5, Ir decreases with increasing frequency for f∼100 kHz–1 MHz, and then increases very slightly from 1 MHz to 5 MHz for various mode numbers. For comparison, Ip(f ) for N=(2, 2) is also plotted in the same panel as the starry points. Ir is presenting a very dissimilar frequency dependence compare to Ip. In addition, as opposed to setup 1, Ir is the smallest for N=(1, 1) at a fixed frequency, increases with N, and saturates at higher mode number for N> (3, 3). No specific relation between the current and the mode number is noticed. In this setup, the source-drain voltage Vsd, and the dot conductance G = R−1dot is expected to oscillate with the same frequency. Similar to the rectification mechanism, the current Ir = 2πω ∫2π/ω response to f compared with Ip for 1 MHz<f<6 MHz. Besides, the rate of increase in Ir with respect to f is almost the same for different transmission modes N.
Chap. 5 Time dependent electric fields generated DC currents in a large gate-defined open dot 67
Figure 5.5: DC current Iras a function of frequency for various mode number. N=(1, 1)u, (1, 2)H, (2, 2)N, (3, 3)# and (4, 4)2 respectively. Vsd,ac=1 mV and Vs2,ac=25 mV. For comparison, Ip(f ) for Vac=15 mV and N=(2, 2)I and its linear fit (dashed line), are also plotted.
To further investigate the characteristics of the two dc currents, transport in perpen-dicular magnetic fields B was also studied. The magnetic field is normal to the plane of the 2DEG (open dot). The dot resistance as a function of B is presented in Fig. 5.6(a) for N = (2, 2) and (3, 3). Both traces are symmetric in magnetic field. Rdot decreases with increasing B due to weak localization for|B| >130 mT. [133, 134] Rdot decreases from∼12 kΩ at B=0 to∼10.8 kΩ for N=(2, 2) and from ∼8.4 kΩ at B=0 to ∼7.3 kΩ for N=(3, 3).
Then resistance increases with increasing field for |B| ?130 mT. Figs. 5.6(b) and 5.6(c) show the magnetic field dependence of Ip and Ir for various frequencies. Compared with Fig. 5.6(a), Ip(B) and Ir(B) show no direct correlation with Rdot(B). Additionally, Ir becomes asymmetric with respect to B with the increased frequency while the magne-toresistance of the dot is symmetric about B. E.g. the 3 MHz trace is more asymmetric than the 100 kHz trace. The results again suggest that the possibility of the circuitry effects can be ruled out here for both Ip and Ir. On the other hand, the magnetic field dependences of the current show discrepancies in Ip and Ir. Ip overall increases but Ir decreases with increasing |B|. This difference further support that Ip and Ir are indeed generated by different mechanisms.
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Figure 5.6: (a) Dot resistance as a function of perpendicular magnetic field for N=(2,2) and (3,3). (b) Ip vs. B for three frequencies at N=(2,2) for Vac= 30 mV. (c) Ir vs. B for four frequencies at N=(3,3) for Vs2,ac= 25 mV and Vsd,ac=1.25 mV.