Dependence While Tuning the Quantum Dots Through Their ResonancesTheir Resonances

7.4 Magnetic Flux Dependence in the Vicinity of Single-Electron Tunneling Intersections

Table 7.1 Magnitude in units ofe²/hand the ratio of the conductance modulations taken on either flank of the single-electron tunneling intersection from figure 7.20 at the operating pointsi, j∈ {1, 2, 3, 4}. The magnitude for operating point no. 3 is determined for each measurement individually.

cross-cut −→

34 cross-cut −→

23 cross-cut −→ 13 OP no. 1 0.139±0.005

OP no. 2 0.078±0.004

OP no. 3 0.124±0.009 0.110±0.007 0.102±0.008

OP no. 4 0.068±0.005

ratio (OP no. 3/j) 0.89±0.10 1.42±0.17 1.50±0.24

This assumption is supported by the measured data that features a significant increase in the magnitude of the periodic conductance modulations for these type of operating points in comparison with the higher occupation probability at the opposite flanks of the single-electron tunneling peaks.

7.4.3 Dependence While Tuning the Quantum Dots Through

7 Two Parallel Quantum Dots - Single-Electron Tunneling Regime

In order to obtain the full evolution of the phase of the periodic conductance modula-tions the conductance map of the magnetic flux density and the plunger gate voltage of the quantum dot was acquired:

• Tuning Through Two Resonances:

A simultaneous transition through both single-electron tunneling peaks of an inter-section in the charge-stability diagram is displayed in figure 7.21. Panel (a) of the figure displays the conductance values, while panel (b) features solely the conduc-tance modulation from which the background of the single-electron tunneling peaks has been subtracted for a better lucidity. One has to note that during the measure-ment two sudden changes, marked by the black arrows in the gate voltage axis, are present. Here, two redistribution events of charges in the vicinity of a quantum dot shift the single-electron tunneling peak resulting in the abrupt change of the con-ductance value. The iso-phase lines are determined by the maximum or minimum of the magnetic flux dependent conductance modulations in the parameter space of the plunger gate voltage tuning the charging threshold of both quantum dots and the magnetic flux density. In the stable regions of the measurements they show a piecewise linear slope, see black solid lines in figure 7.22b. A comparable tilt of the iso-phase lines of the periodic modulations was obtained in the antidot system in chapter 6.3.2 while detuning the gate voltages, where the tilt was caused by a change in the magnetic flux threaded area. Here however, the magnetic flux density period yields the same values withδB = (3.55±0.04) mT andδB = (3.56±0.02) mT taken at either side of the resonance of the single-electron tunneling peak, respectively.

For a transition through a single-electron tunneling peak, that can be approximated by a Breit-Wigner resonance, the transmission phase is expected to change smoothly by π in an s-shaped manner. However, even for a broad resonance the evolution of the phase should occur mainly within the linewidth of the resonance. Here in-stead, a linear dependence in the phase of the conductance modulations is found. In addition the phase does not converge to a fixed value far away from the resonance peak. In consequence, by taking only two line-cuts for different gate voltage values, one could willingly claim any phase shift, id est a shift by π sketched by green solid line in figure 7.21. This emphasizes that the full evolution of the phase of the periodic conductance modulations as a function of the dot’s plunger gate voltage has to be known in order to make a reliable statement on the phase changes. A clear statement of the phase shift by tuning the quantum dot in this configuration can therefore not be made. Another thing to note, here both quantum dots were detuned and thus the change in transmission phase should be either additive yielding 2π or cancel out.

• Tuning Through a Single Resonance:

In a second approach, for a different sample, only one quantum dot was tuned through the single-electron tunneling peak, while the other one was kept at its operating point. This procedure offers the advantage that only one of the two resonances used in the electrical transport is detuned and the phase of the periodic conductance modulations should solely depend on the one that is detuned. The corresponding conductance map displayed in figure 7.22 shows both the conductance as well as solely the periodic modulations without the background conductance.

Here however, the evolution of the iso-phase lines (black dotted line) is inverted and 138

7.4 Magnetic Flux Dependence in the Vicinity of Single-Electron Tunneling Intersections

gatevoltage[V]

magnetic field[T]

-1.04 -1.02 -1 -0.98

6.59 6.6 6.61

0 0.5 1

conductanceI/VDS

h e2 h

i

(a)

gatevoltage[V]

magnetic field[T]

-1.04 -1.02 -1 -0.98

6.59 6.6 6.61

-0.15 0 0.15

deviationinconductance∆I/VDS

h e2 h

i

(b)

Figure 7.21 (a) Conductance map of the magnetic flux density versus the combined plunger gate voltage of both quantum dots. Flux dependent conductance modulations are present for the traversal through both single-electron tun-neling peaks. The tilt of the iso-phase lines (position of maximum or min-imum of the conductance modulations, marked exemplary in panel (b) by the black solid line) is not caused by a difference in the magnetic flux density period. It yieldsδB = (3.55±0.04) mT andδB = (3.56±0.02) mT at either side of the resonance, respectively. (b) Extracted modulations out of (a) by subtracting the mean conductance value. In between the endpoint of the green solid line a phase shift ofπ can be claimed from the data. Two sudden changes in the measurement are present, marked by the black arrows at the gate voltage axis. The data are taken at a source-drain bias of = 25 V.

7 Two Parallel Quantum Dots - Single-Electron Tunneling Regime

-0.98 -0.96

6.59 6.6 6.61

gatevoltage[V]

magnetic field[T]

0 0.5 1

conductanceI/VDS

h e2 h

i

(a)

-0.98 -0.96

6.59 6.6 6.61

gatevoltage[V]

magnetic field[T]

-0.15 0 0.15

deviationinconductance∆I/VDS

h e2 h

i

(b)

Figure 7.22 (a) Conductance map of the magnetic flux density versus the plunger gate voltage of a single quantum dot. Flux dependent conductance modulations are present during the traversal of the single-electron tunneling peaks, but diminish as the apex reaches a conductance of I/V_DS = 1e²/h. The (in-verse) slope of the iso-flux lines (highlighted in panel (b) as black solid line) may arise from a low occupation number of the quantum dot, see refer-ence [7]. There is no differrefer-ence in the periodicity of the modulations de-pending on the gate voltage, id est with δB|_V

G=−0.95 V = (3.45±0.03) mT and δB|_{VG=−0.99 V} = (3.43±0.04) mT, respectively. (b) Conductance mod-ulations extracted from (a) without the background conductance of the single-electron tunneling peak. The data are taken at a source-drain bias ofVDS= 25µV.

140

7.5 Influence of the Electron Spin on the Electric Transport shows the inverse tilt compared with the previous measurement from figure 7.21.

Based of the values taken for the more positive gate voltages to the apex of the single-electron tunneling peak, the evolution of the phase of the conductance modulations is again more linear than s-shaped, see solid black line in figure 7.22b. At the more negative gate voltage values to the apex of the peak the measurements is affected by too many instabilities to make out a clear trend of the phase, see black solid zig-zack line.

In summary, both measurements with either the transition through the resonance of a single quantum dot or the simultaneous transition through both resonances of the two dots feature a linear non converging evolution of the phase. This contrasts the findings of the typical s-shaped evolution of the transmission phase through the (Breit-Wigner type of) resonance of the single-electron tunneling peak of the quantum dot found in the two-path experiments with only a single quantum dot and a reference channel [91, 7]. With the linear behavior found here, a solid statement of a change in the phase of the periodic conductance modulations obtained for the combination of two parallel quantum dot cannot be made.

7.5 Influence of the Electron Spin on the Electric