- Single-Electron Tunneling Regime
7.5 Influence of the Electron Spin on the Electric Transport
7.5.3 No Appearance of Spin Restrictive Transport Channels in the Measurementsthe Measurements
In order to shed some light on the presence of spin restrictive channels of the two quan-tum dots in the parallel arrangement the mean peak-to-peak magnitude of the periodic conductance modulations is analyzed for several adjacent single-electron tunneling in-tersections. By going from one intersection of the single-electron tunneling peaks to its direct neighbor the presence of a restriction to a certain spin orientation in one transport channel might change, while the one of the other channel is conserved and vice versa, depending on the choice of the detuned quantum dot. Based on the previous discussion of spin-restrictive channels through the quantum dots, one might be able to draw inferences for the restriction from changes in the magnitude of the periodic modulations or from even the absence of the modulations.
A charge-stability diagram taken at a large magnetic field of B = 10 T is displayed in figure 7.25. Reference for the labels is the intersection I,1 to the lower left of the diagram (red circle) with the lowest plunger gate voltage for both quantum dots. The operating point is thereby chosen directly at the center of the intersection (red arrow). The labels of neighboring intersections expand to the right and upward to higher plunger gate voltages determining an array of 4×4 intersections. These are labeled in columns with Roman numerals for the upper quantum dot and in rows with Arabic numerals for the lower quantum dot, respectively. The numerals increase to more positive plunger gate voltages.
The conductance modulations around the magnetic flux density of B = 10 T were taken right at the center of the apex of the intersections of the single-electron tunneling peaks.
However,an absence of the periodic modulations was neither observed here nor in any other measurement among various samples and for different magnetic field strengths13, that would point to an opposite spin orientation in both transport channels. The peak-to-peak values of the conductance modulations14of all intersections is
13This includes also the capped intersections atI/VDS= 1e2/h, as for these the conductance modulations are always present in the area of the flanks of the intersections in the charge-stability diagram.
14The single traces are displayed in figures C.13 to C.16 in chapter C.2 of the appendix.
144
7.5 Influence of the Electron Spin on the Electric Transport
VLG[V]
VUG [V]
X
I,1 II,1 I,2
X
-0.9 -0.8 -0.7
-0.9 -0.8 -0.7
0 0.5 1
conductanceI/VDS
h e2 h
i
Figure 7.25 Combined conductance as a function of the gate voltages of both quantum dots. Reference is the intersection I,1 marked by the red circle in the charge-stability diagram. The working points are chosen directly at the center of the crossing (red arrow). Columns expand to the right and rows upwards, labeled with increasing Roman and Arabic numerals, respectively. The data are taken at a source-drain bias ofVDS= 25µV at a magnetic field ofB = 10 T.
Two regimes of common Coulomb-blockade of both dots are marked by a red and cyan cross, respectively.
summarized in table 7.2. The conductance modulations feature similar mean conductance values in the range ofI/VDS= 0.4e2/htoI/VDS= 0.6e2/h. The rows and columns of the table are analyzed in the following for the presence of a regular pattern in the magnitude of conductance modulations. The notion of a higher or lower modulation magnitude in the discussion is related to the relative difference with directly neighboring operating points rather than with the absolute values found:
• Row 1:
A strictly alternative behavior of high and low modulation magnitudes is found throughout all columns, starting with a high value in column I.
• Row 2:
As previously in row 1, starting from a high value in column I a strictly alternating magnitude is found.
• Row 3:
The alternative pattern starting from a large magnitude is found again in this row, but only well pronounced from column I to III. There is only a small difference in the magnitude between column III and IV.
• Row 4:
This row features the same pattern as found within row 3, starting again from a high magnitude and with a small change between column III and IV.
7 Two Parallel Quantum Dots - Single-Electron Tunneling Regime
Table 7.2 Comparison of the magnitude of the conductance modulations in units ofe2/h for the single-electron tunneling intersections from figure 7.25.
column I column II column III column IV row 4 0.157±0.025 0.066±0.013 0.229±0.021 0.201±0.005 row 3 0.196±0.017 0.106±0.014 0.245±0.012 0.198±0.013 row 2 0.172±0.011 0.085±0.009 0.231±0.009 0.107±0.008 row 1 0.228±0.016 0.062±0.007 0.196±0.014 0.131±0.015
• Column I:
An alternating behavior, starting with a high value in row 1, of the modulation magnitude is found within this column. However, the pattern is less pronounced than the one found within the rows.
• Column II:
The modulation magnitude starts at a low value in row 1 and increases steady towards row 3. Then it drops back at row 4 to a value comparably large as in row 1.
• Column III:
Within the variance, the magnitude of all rows is comparably large.
• Column IV:
This column features low values for the modulation magnitude in row 1 and 2 and high values in row 3 and 4, respectively.
From only the relative differences in the magnitudes of neighboring intersections of single-electron tunneling peaks no regular pattern is found, that is consistent for both directions in the charge stability diagram and would thus allow to deduce possible spin-restrictive channels. So far, the absolute value of the conductance modulations has been omitted from the discussion. It shall be considered in the following in a graphical rep-resentation of the data from table 7.2 displayed in figure 7.26. The scheme displays a mapping of high ’H’ and low ’L’ conductance values, with the threshold separating in between the high and low category chosen with a magnitude of ∆I/VDS = 0.150e2/h for the conductance modulations, respectively. With this choice of threshold, a strictly alter-nating pattern is found for the upper quantum dot (along the columns) with alteralter-nating high and low magnitudes for all combinations with single-electron tunneling peaks of the lower quantum dot. However, a pattern of changes for the lower quantum dot (along the rows) is not found. At best it can be considered to be absent, but even this statement cannot be made considering the value of intersection IV,3. A regular pattern sustaining the presence of restrictions to a certain spin orientation in the either transport channel would have to be fully consistent for both quantum dots (along the columns and rows of the figure), which is not found here.
Another thing to note, concerning the absolute values of the conductance modulations is the ratio in the magnitude from neighboring intersections from either one column or row to the next. In the simple picture discussed earlier, for the case of spin unpolarized 146
7.5 Influence of the Electron Spin on the Electric Transport
column I column II column III column IV
row 1
row 2
row 3
row 4
H H H
H H H
H H L
H H H
L
L
L
L
Figure 7.26 A possible mapping of the number of processes maintaining interference in the channels of the two parallel quantum dots based on the data from ta-ble 7.2. Two unrestricted channels are represented by ’H’, a single restriction by ’L’, respectively.
leads, a change from two unrestricted channels to a single restriction in either channel halves the number of interference maintaining processes. This would consequently cause a change by a factor of two in the magnitude of the conductance modulations. Here however, it is found to range from a factor of two to a factor of three, e.g. in row 4 the ratio yields for columns III/II = 3.2±0.1. These factors exceed the prediction of a factor of two by far. It cannot be expected within the simple model of transport chan-nels through the quantum dots being either restrictive to a certain spin orientations or not.
In summary, from the analysis at different single-electron tunneling intersections in the charge-stability diagram, changes in the magnitude of the periodic conductance mod-ulations are found in the experiment. If the conductance modmod-ulations were caused by interference, one could expect changes in the magnitudes of the conductance modulations for the assumption of possible restriction certain spin orientation in the transport channels through the quantum dots. These changes in the magnitude are predicted to arise from the change of the number of interference maintaining processes. There should even exist a configuration where interference is excluded from the electric transport. One has to stress the fact that the absence of the periodic conductance modulations was never observed in any sample or for any magnetic field strength examined in the course of this thesis. In addition, from the varying magnitude a regular consistent pattern for both quantum dot of restricted and unrestricted channels cannot be drawn. This points to an even richer and more complex behavior in the transport through the two parallel quantum dots, that cannot be covered in the simple model of restrictive and unrestrictive transport channels.
This also poses the question of the character of the transport channels involved in the
7 Two Parallel Quantum Dots - Single-Electron Tunneling Regime
process. In order to occupy the quantum dot with an additional electron two requirements have to be fulfilled:
1. The energy difference of the charging energyEC has to be overcome, id est by tuning the charging threshold in resonance via a gate voltage, and
2. a suitable quantum mechanical single-particle state has to be present on the quan-tum dot.
In the considerations of the transport channels, these single-particle states were only considered to pose a possible restriction to a certain electron spin, but their density of states has been neglected. For the case of dense spectrum of the single-particle levels accessible in the range of the charging energy of the quantum dot, the presence of a spin restriction might get lifted in any case. In consequence there will always be the possibility to pass either spin orientation in the transport channels. It hence allows to obtain the conductance modulations at each and every intersection of two single-electron tunneling peaks, as it is seen experimentally. For future experiments it might therefore be desirable to realize even smaller quantum dots where, in comparison with the quantum dots used here, the density of states of the single-particle levels exceeds the electrostatic charging energy. Ideally, an individual single-particle state is present as transport channel. By chance, it might give rise to the absence of the periodic conductance modulations15 or even a dip in the common conductance of a single-electron tunneling intersection as the smoking gun for the interference effect as the origin of the conductance modulations, if the adverse spin configuration of the two transport channels can be accessed experimentally.
15Including both the very apex and the flank regions of the intersection.
148