When the negative voltage is applied to a pair of split gates, the potential depletes 2DEG to form the 1D channel resulting in the typical quantized conductance. As shown in Fig. 6.2, there are conductance plateaus in units of 2e2/h due to the transmission of 1D subbands for QPC1 of Device A. As known, 1D transport is sensitive to the carrier density which can be effectively tuned by the biasing top gate voltage Vtg. The more negative Vtg is, the less carrier density n is and vice versa. The more positive Vtg is, the more carrier density n is. Quantized conductance for a series of Vtg is also demonstrated in Fig. 6.2. The carrier density is smoothly decreased from left to right corresponding to
that the threshold Vsg for pinching off becomes less negative continuously. The number of plateau is determined by the number of subband with energy less than Fermi energy EF. As seen in the inset of Fig. 6.2, the reduction of Vtg leads to the decrease of the observed plateau number implying the decrease of EF. Besides, when carrier density is large the conductance plateaus are more clear. Once carrier density is severely reduced, the plateau disappears.
Figure 6.2: Quantized conductance of QPC1 of Device A as a function of split gate voltage at T=0.3 K for different top gate voltages. From left to right: Vtg decreases from +0.4 V to −1 V in steps of 0.2 V. Inset: Selected data curves are shown in larger y-scale.
It has been well known that the carrier density depends on Vtg in heterostructures in either 1D or 2D.[85] Fermi energy is determined by carrier density and dimensionality following EF= π2~2n21D/8m∗ in 1D and EF=πn2D/2m∗ in 2D where n1D and n2D are carrier densities in 1D and 2D, respectively. m∗ is the effective mass of carrier. Here, the transconductance as a function of Vtg against Vsd referred to as the half-plateau method1 based on Glazman and Khaetskii model[26] is used to obtain the subband energy level spacing and the corresponding Fermi energy.[27] Both calculated carrier densities versus Vtg in 1D and 2D models are plotted in Fig. 6.3. As seen, carrier density is indeed effectively changed by tuning Vtg.
As illustrated in Fig. 6.1(a), all pairs of split gates were made nearly identical. Features
1See section 1.3.3 for more detailed discussion.
Chap. 6 Ballistic transport in double quantum point contacts in series 73
Figure 6.3: (a) Calculated two dimensional carrier density versus top gate voltage for QPC1 of Device A at 0.3 K. (b) Calculated one dimensional carrier density versus top gate voltage for QPC1 at T=0.3 K. Subband index of QPC1 is confined at N =4 (black squares) and N =6 (red circles), respectively.
of ballistic transport through each QPC, e.g. quantized conductance versus gate voltage, is nearly the same. In the following, we focus on the transmission through serially connected QPC1 and QPC2. Conductance as a function of split gate voltage applied to one QPC was measured with the other kept in some specific 1D subbands N . In Fig. 6.4, we show a typical result of the conductance Gseriesof the serially connected pair of QPCs versus Vsg1 against a series of Vtg while QPC2 is confined to a particular subband number N =2 (Vsg2 is held constant). The arrangement of gate voltage is also sketched in Fig. 6.1(a). From the adiabatic transmission point of view, the device conductance is completely determined by the narrowest QPC with the least number of subbands for transmission.[105] Only two plateaus are expected since QPC2 is confined at N =2. However, there are more than two plateaus for all curves in Fig. 6.4. In addition, the series conductance at the plateau is less than the quantized conductance value of single QPC at the corresponding N. These differences indicate that the transmission through this device is not completely adiabatic.
The ballistic transport can be characterized by the direct transmission probability Td which reflects the probability of electrons traveling directly from QPC1 to QPC2 without momentum relaxation and loss of coherence. The completely adiabatic transport of Td=1
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
0.4 0.8 1.2 1.6
G series
(2e
2 /h)
V sg1
(V)
Figure 6.4: Conductance of two serially connected QPCs of Device A as a function of Vsg1 against a series of Vtg at 0.3 K. Vtg decreases from +0.4 V to −0.8 V in steps of 0.1 V.
QPC2 is confined at N =2.
was reported by van Wees et al. in high magnetic fields such that electrons have a high degree of collimation.[135] In the absence of magnetic field, the completely adiabatic transport was rarely observed. With the consideration of real geometry that two identical quantum point contacts facing each other at opposite boundries of 2DEG by Beenakker et al. and Beton et al.[136, 137], the series conductance through serially connected two identical QPCs of conductance G can be written as
Gseries= 1
2(G + 2e2
h Td) (6.1)
Therefore, we can obtain Td from series of traces as those shown in Fig. 6.4. For instances, we find the value of Gseries at the second plateau in Fig. 6.4 and substitute in Eq. 6.1 to calculate Td for N=2. We demonstrate all calculated values of Td in Fig.
6.5. The normalized transmission probability Td/N decreases continuously from 0.6 to 0.1 with decreasing top gate voltage (carrier density) when both QPCs are confined at only one 1D subband (N=1). The transport is partially adiabatic in high electron densities and transits to nearly complete ohmic (Td ∼0) in low densities. When both QPCs are confined at more than one 1D subband (N>1), Td/N is smaller than that for N=1, but seems to be insensitive to N for N=2, 3, and 4. Moreover, Td/N saturates and is about
Chap. 6 Ballistic transport in double quantum point contacts in series 75 0.3 for positive top gate voltages. When the top gate voltage is negative, Td/N decreases with decreasing Vtg similar to that for N=1.
According to Eq. 6.1, when Td = 1, Gseries = G indicating a completely adiabatic transport. When Td = 0, Eq. 6.1 leads to that Gseries = G/2 for a completely ohmic transport. In the theoretical model, the reality that the two QPCs are separated by a re-gion of large unrestricted 2D electron sea was taken into account.[136] For the completely adiabatic transport, the electrons should travel ballistically across the 2DEG region from one QPC to the other while preserving their 1D momenta. Hence, the transmission through a series of pairs of QPCs depends on the details of confinement potentials and 2DEG. Takagaki and Ferry predicted that Td/N becomes smaller with increasing N be-cause the opening angle of the constriction increases as W/L increases where W is the effective gap width and N=kFW/π.[105] However, the variation is less pronounced in the quantum-mechanical calculation since the lower-lying mode are most likely to be trans-mitted. Here, both QPCs are at a distance of L=600 nm that is much longer than the Fermi wavelength (λF=57 nm) at Vtg=0.4 V. Moreover, calculation of Td/N in terms of L/W in Ref.[139] gives much smaller value. For example, Td/N <0.05 using Takagaki’s method which is about 0.58 in Fig.6.5 for N=1 and Vtg=0.4 V. The effect of the electron collimation seems be either overestimated or less important in our case.
In our device, the negative split gate voltage was applied to produce electrostatic 1D confinement. The top gate voltage determined the carrier density of the unconstricted 2DEG as well as both quantum point contacts. In quantum devices, electron-electron interaction becomes more important in low carrier densities. As reported by Kane et al., both mobility and mean free path decrease when the top gate tuned carrier density is less than a critical value.[85] The trend is similar to the data curves of Td/N for N=2, 3, and 4 in Fig. 6.5. Mean free path is about 10 µm, much longer than some characteristic lengths such as L and W, and has no direct relevance with Td. On the other hand, the coherent length is closely related to mobility and mean free path. For completely adiabatic trans-port, the phase of scattering electrons should be preserved across the unconstricted 2DEG.
However, the electron-electron interactions introduce the effect of dephasing and hence,
0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2
Figure 6.5: Normalized transmission probability Td/N versus top gate voltage Vtg for serial QPCs of some identical confined subband indices N of device A. Dashed lines are guides to the eye.
transport approaches to ohmic regime. This may explains qualitatively the experimental findings that Td/N saturates in high carrier densities, decreases with decreasing carrier density in low carrier densities, and is insensitive to the effective gap width W (subband index N ). As for N =1, the large values of Td/N compared with other mode number is probably due to the lack of inter-mode scattering. Further investigation is required on this issue.
Direct transmission across two QPCs separated intermittedly by 2DEG is expected to decrease with increasing edge-to-edge distance L between two QPCs due to decreasing collimation angle.[136] When Td decreases, it implies that the ratio of coherent electrons traveling across two QPCs decreases as well. Fig. 6.6 shows the conductance of double QPCs after subtracting the resistance contributed from the second quantum point contact by aligning the last plateau to 2e2/h. The conductance of QPC2 is fixed at∼ 4e2/h with a mode number of two. G∗s as a function of Vqpc1, the gate voltage applied on QPC1, shows five robust quantization steps for L =2.4 µm in th left panel and six steps in the middle panel.
At higher subbands for N &5, conductance oscillations are observable suggesting quan-tum interference. The interference at higher subbands becomes stronger as L decreases,
Chap. 6 Ballistic transport in double quantum point contacts in series 77 selections with different separation distances in Device B. The measured conductance across two QPCs is subtracted by a serial resistance of ∼6 kΩ aligning the last plateau to 2e2/h to obtain G∗s.
e.g. the L =1.4 and 0.6 µm traces in the middle panel. The transport resolves from ohmic regime into adiabatic regime when L reduces down to L.0.5 µm. The adiabatic transport manifests as more missing plateaus and less well defined quantization as the G∗s traces in the right panel demonstrated. Notice that two steps are developed for L = 0.5 µm but only one step is observed for L = 0.3 µm.
Figure 6.7: Normalized direct transmission Td/N as a function of edge-to-edge distance between two QPCs for N =4 (leftmost) to N =1 (rightmost). (N = Nqpc1 = Nqpc2.) The dashed lines are the fit to Td/N ∝ L−α by least square regression. N=4–2 are plotted in logarithmic scale, while N=1 is plotted in linear scale.
The direct transmission probability Td can be analyzed in the same way as previously
discussed. Tdwas expected to be proportional to L−1since the collimation angle decreases as L increases.[136] Fig. 6.7 shows the normalized transmission factor Td/N as a function of L for mode number Nqpc1 = Nqpc2 = N =4 to N =1. Td decreases from ∼0.6–0.8 to less than 0.1 as L increases to ∼2.4 µm for N=4–2. The data in the logarithmic plot follows a relation of Td∝ L−αwhere α extracted from numerical fitting is∼0.95–1.3. The extracted value of α is comparable with the theoretical prediction of unity, however dephasing in 2DEG and the detailed geometrical structure may account for the deviation. For N =1, Td/N drops more rapidly with increasing L and reaches close to zero for L &0.6 µm.
Due to the insufficient resolution for L ≤0.5 µm, numerical fitting for determining α is unreliable. The mechanism of faster decay of Td for N = 1 than higher mode numbers remains to be explored.