Bias Dependence

6 Two Parallel Quantum Point Contact Constrictions

ature is slightly higher than in the previous case. Again the amplitude of the modulation degrades homogeneously on all peaks with increasing temperature until they are no longer visible at a temperature above T > 500 mK. The dependence of the amplitudes of the conductance modulation is shown in figure 6.37. The data points are fitted by an exponen-tial function. A similar exponenexponen-tially decreasing amplitude of single-electron tunneling peaks in quantum dot structures is expected for the temperature range below T < 1 K for the case k_BT '∆E e²/C of comparable thermal energy and level spacing and has been predicted by I.O. Kulik and R.I. Shekhter [62] and C.W.J. Beenacker [9], respectively.

The distinctive single-electron charging levels of the compressible annulus around the antidot that reduces periodically the conductance as the magnetic flux is varied were unable to be resolved within the temperature range the experimental set-up offers. For low temperatures a series of well isolated dips in the conductance are expected. However, for the investigated range the conductance modulations take a sinusoidal appearance as the single-electron resonances are broadened, leading to a large overlap. The interplay of the thermal broadening and tunnel coupling strength of the compressible region of the antidot to the edge of the sample determines the width of the single-electron tunneling dips. But even for a low coupling strength, by setting the conductance value in both constrictions close to a value of I/V_DS = 1e²/h and at the lowest accessible temperature the appearance of the conductance modulations remains sinusoidal, as seen in figure 6.9 close to the conductance plateau.

6.6 My Experimental Findings

biasvoltageVDS[µV]

magnetic field[T]

-200 -100 0 100 200

6.25 6.26 6.27

-0.075 0 0.075

DCcurrentdeviation∆IDS[nA]

(a)

biasvoltageVDS[µV]

magnetic field[T]

-200 -100 0 100 200

6.25 6.26 6.27

0.18 0.27 0.36

differentialconductancedI/dVDS

h e2 h

i

(b)

Figure 6.39 Color-coded bias dependence of the conductance modulations for the fun-damental periodicity. (a) Relative deviation in the current. The average of the I/V curve has been subtracted for a better visibility of the modulation.

(b) Differential conductance measured by a lock-in technique.

6 Two Parallel Quantum Point Contact Constrictions

By taking the finite bias range where the conductance modulation appear the charg-ing energy of the antidot can be extracted. At a higher bias the systems is no longer Coulomb-blockaded and undergoes multi electron tunneling events. The conductance is no longer modulated as a function of magnetic flux density or by area variation. At this limit in the bias voltage with V_DS =±170µV where the modulation in figure 6.39 ceases, the charging energy yields about E_C = 85µeV for the compressible annulus area around the antidot¹⁰.

The bias windows of the h/2e conductance modulations was investigated with a focus of whether the halved periodicity reduces to the fundamental at first before vanishing.

This behavior would point to two separate mechanisms. A color-coded map of the modu-lated current is displayed in figure 6.40a. The mean on the current-voltage characteristic of the device has been subtracted as previously to highlight the modulation. The mag-netic field range sets the bulk of the sample to the range of filling factor ν = 3. The conductance modulations are not strictly opposite in sign with respect to zero bias. The working point degraded slightly during the measurement in the range around zero bias and shifted the resonance condition as it can be seen by the slight curvature at zero bias in data of figure 6.40b. The amplitude of the conductance modulations increases with larger source-drain bias voltage and reaches a maximum peak-to-peak signal of about δI = 280 pA at a bias voltage of VDS =±50µV. Thereafter it decreases and vanishes at around a bias of V_DS ≈ ±200µV. The modulation periodic stays the same in the whole bias range. A reemergence of any periodic modulation for a higher bias was not observed.

The limits of the bias range yields a charging energy in the range of EC = 100µeV for the compressible ring. It is slightly larger compared with the case of the fundamental h/e period, but the overall bias dependence is the same. In the differential conductance shown in figure 6.40b theπ-phase jump is more pronounced than for the fundamental periodicity displayed in figure 6.39b. It even exhibits a visibility dip in the signal with completely diminishing conductance modulations. The regions next to the vanishing modulations are shifted in phase byπ. The phase jumps occur for both bias directions, around a bias value of V_DS = ±52µeV. Close to the visibility dip for positive bias, the differential data are affected by a large disturbance (black arrow). Here, the lock-in ran into an error during the continuous data acquisition.

I. Neder et al. [75] report a ’lobe-pattern’ with respect to the bias voltage with the same behavior of diminishing differential conductance values with subsequent π phase jumps measured in a two-path arrangement. While the two parallel quantum point contacts enclosing an antidot used here allowed to observe two lobes, the measurements of I. Neder et al. showed up to three. The simultaneous data acquisition of differential and integrated data allows to uncover the origin of the ’lobe-pattern’ in the differential conductance map.

Three line-cuts marked in figure 6.40 at a fixed magnetic field for both the differential conductance and the deviation in the current from the I/V characteristic are shown in figure 6.41. The traces marked in red and blue color start out from a positive and negative amplitude, respectively. The trace marked in gray is taken at the boundary of the

10The coupling to a single edge already allows a charge exchange between the compressible antidot ring and the edge, even if the other constriction maintains its quantized value, see figure 6.8. The charge fluctuations on the antidot compressible region changes the electrostatic potential. Knowing the charging energy of the antidot its influence on the conductance can be estimated withδI/V_DS≈ 0.1e²/hfor the applied bias and fits to the noise level in the data of the figure.

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6.6 My Experimental Findings

biasvoltageVDS[µV]

magnetic field[T]

-300 -200 -100 0 100 200 300

4.39 4.40 4.41

-0.2 0 0.2

DCcurrentdeviation∆IDS[nA]

(a)

biasvoltageVDS[µV]

magnetic field[T]

-300 -200 -100 0 100 200 300

4.39 4.40 4.41

1.10 1.20 1.30

differentialconductancedI/dVDS

h e2 h

i

(b)

Figure 6.40 Color-coded bias dependence of the conductance modulations withh/2e pe-riodicity. (a) Relative deviation in the current. The average of the I/V curve has been subtracted for a better visibility of the modulation. (b) Differential conductance measured by a lock-in technique.

6 Two Parallel Quantum Point Contact Constrictions

DCcurrentdeviation∆IDS[nA] differentialconductancedI/dVDS

h e2 h

i

bias voltageV_DS[µV]

-0.2 -0.1 0 0.1 0.2

-300 -250 -200 -150 -100 -50 0

1.1 1.2 1.3

Figure 6.41 Line-cuts through the ’lobe-pattern’ at the positions marked in figures 6.40a for the deviation in the current of the I/V characteristic as lower curves and through figure 6.40b for the differential conductance as upper curves, respectively. Displayed are the data points with overlying lines as guide to the eyes.

conductance modulation in between the red and blue trace, where the I/V characteristic is not modified by the magnetic flux density. The lower curves of the graph correspond to the traces for the deviation of the I/V characteristic, the upper ones to the differential conductance, respectively. The graph shows the data points taken from the line-cut with superimposed lines as a guide to the eye. The deviation of the I/V curve of the red (blue) trace increases (decreases) with the applied bias, until it reaches a maximum (minimum), decreases (increases) thereafter and vanishes. Along the line-cut marked in gray the I/V characteristic is not affected by the magnetic flux density and its deviation is zero. The differential conductance corresponds to the derivative dI/dVDS of the I/V characteristic with respect to the bias voltage V_DS. Along the trace marked in gray, it corresponds to the average I/V characteristic of the device. At the bias values, where the modulation of the I/V characteristic is extremal in the red and blue trace, the derivative dI/dVDS

yields the same value as for the average non-modulated I/V characteristic. In the region around these bias values, the differential measurement is unable to resolve the deviation from the I/V characteristic by the way it is measured. With the magnetic flux dependent deviation of the I/V characteristic reaching a maximum or minimum with increasing voltage bias V_DS, the derivative dI/dVDS of the differential conductance changes its sign with the traversal of these extremal points and thus yields a jump by π in the differential data. The ’lobe-pattern’ in the differential conductance map simply arises due to the fact that the magnetic flux dependent deviation of the I/V characteristic is not monotonic with respect to the applied source-drain bias voltage. The lock-in measurement delivers the derivative of the I/V characteristic and hence causes the ’lobe-pattern’ with a vanishing visibility and a subsequent phase jump ofπfor the modulations as a measurement artifact.

106