- Single-Electron Tunneling Regime
7.3 Interplay of Two Parallel Quantum Dots at a Finite Magnetic Field
7.3.1 Common Conductance Value at the Single-Electron Tunneling Intersections: Presence and Absence ofTunneling Intersections: Presence and Absence of
7.3 Interplay of Two Parallel Quantum Dots at a Finite Magnetic Field edge thereby spatially confines the electrons and yields an upper bound for the expansion of the area (red dotted line) that enters the phase of the wave function of the electrons passing the device from the left to the right hand side. Electrons in the compressible strips lying further outwards are isolated by the potential barrier of the incompressible strip in between and do not contribute to the electric transport as long as the potential gradient in the 2DES in front of the quantum dots is shallow enough. This renders the details of the strip pattern insignificant in the bulk of the leads. With the periodicity of δB = (3.12±0.07) mT in the magnetic flux density the area enclosed in between the two quantum dots is estimated withA= (1.32±0.03)µm2, that is larger than the expectation with A= (1.12±0.06)µm2 from chapter 5.4, but can be attributed to the actual position of the compressible strips.
7.3 Interplay of Two Parallel Quantum Dots at a
7 Two Parallel Quantum Dots - Single-Electron Tunneling Regime
-0.84 -0.82
VUG -0.96 [V]
-0.92
VLG [V]
0 0.5 1 1.5 2 2.5
0 0.5 1 1.5 2 2.5
conductanceI/VDS
e2 h
(a) B = 0 T
-0.84 -0.82
VUG -0.96 [V]
-0.92
VLG [V]
0 0.5 1 1.5 2 2.5
0 0.5 1 1.5 2 2.5
conductanceI/VDS
e2 h
(b) B= 0.5 T
-0.84 -0.82
VUG [V]
-0.96 -0.92
VLG [V]
0 0.5 1 1.5 2 2.5
0 0.5 1 1.5 2 2.5
conductanceI/VDS
e2 h
(c) B = 1.0 T
-0.84 -0.82
VUG [V]
-0.96 -0.92
VLG [V]
0 0.5 1 1.5 2 2.5
0 0.5 1 1.5 2 2.5
conductanceI/VDS
e2 h
(d) B= 1.5 T
-0.84 -0.82
VUG [V]
-0.96 -0.92
VLG [V]
0 0.5 1 1.5 2 2.5
0 0.5 1 1.5 2 2.5
conductanceI/VDS
e2 h
(e) B = 2.0 T
-0.84 -0.82
VUG [V]
-0.96 -0.92
VLG [V]
0 0.5 1 1.5 2 2.5
0 0.5 1 1.5 2 2.5
conductanceI/VDS
e2 h
(f) B= 2.5 T
-0.84 -0.82
VUG [V]
-0.96 -0.92
VLG [V]
0 0.5 1 1.5 2 2.5
0 0.5 1 1.5 2 2.5
conductanceI/VDS
e2 h
(g) B= 3.0 T
-0.84 -0.82
VUG -0.96 [V]
-0.92 VLG
[V]
0 0.5 1 1.5 2 2.5
0 0.5 1 1.5 2 2.5
conductanceI/VDS
e2 h
(h) B= 3.5 T
Figure 7.6 (a)-(h) Evolution of a single-electron tunneling peak intersection for the pres-ence of an apex with an enhanced conductance value in an increasing magnetic field fromB = 0 T toB = 3.5 T. The common conductance value of the apex reduces with increasing magnetic field, but is always present. The data are taken at a bias of VDS= 25µV.
118
7.3 Interplay of Two Parallel Quantum Dots at a Finite Magnetic Field
-0.86 -0.84
VUG [V]
-0.84 -0.8
VLG [V]
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
conductanceI/VDS
e2 h
(a) B = 0 T
-0.86 -0.84
VUG [V]
-0.84 -0.8
VLG [V]
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
conductanceI/VDS
e2 h
(b) B= 0.5 T
-0.84 -0.82
VUG [V]
-0.84 -0.8
VLG [V]
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
conductanceI/VDS
e2 h
(c) B = 1.0 T
-0.86 -0.84
VUG [V]
-0.84 -0.8
VLG [V]
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
conductanceI/VDS
e2 h
(d) B= 1.5 T
-0.86 -0.84
VUG [V]
-0.84 -0.8
VLG [V]
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
conductanceI/VDS
e2 h
(e) B = 2.0 T
-0.86 -0.84
VUG [V]
-0.84 -0.8
VLG [V]
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
conductanceI/VDS
e2 h
(f) B= 2.5 T
-0.86 -0.84
VUG [V]
-0.84 -0.8
VLG [V]
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
conductanceI/VDS
e2 h
(g) B= 3.0 T
-0.84 -0.82
VUG -0.84 [V]
-0.8 VLG
[V]
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
conductanceI/VDS
e2 h
(h) B= 3.5 T
Figure 7.7 (a)-(h) Evolution of a single-electron tunneling peak intersection turning to an apex-less common conductance with increasing magnetic field from B = 0 T to B = 3.5 T. The enhanced conductance value at the apex reduces with increasing magnetic flux density and vanishes around B = 2.5 T. The data
7 Two Parallel Quantum Dots - Single-Electron Tunneling Regime
apex of significantly enlarged conductance value close to the sum of the contribut-ing values of both quantum dots is visible. The linewidth of both scontribut-ingle-electron tunneling peaks is rather small and the conductance drops rapidly to a value close to zero as a function of the gate voltage. With increasing the magnetic field from panel (b) to (h) both the linewidth and the overall conductance of the single-electron tunneling peaks reduces. A clear apex of enlarged conductance directly at the cross-ing is present for all magnetic field values. However, with increascross-ing magnetic flux density its peak conductance value reduces with respect to the contributions of the conductance values of the single quantum dots.
• Evolution Type II:
The second type of evolution of a single-electron tunneling intersection is displayed in figure 7.7. The linewidth for zero magnetic field displayed in panel (a) of both single-electron tunneling peaks is considerably larger than in the previous case.
The conductance of the single-electron tunneling peaks thereby reduces to only a finite value as a function of the gate voltage. Directly at the intersection the common conductance features an apex with an enhanced value that is considerably lower than the sum of the contributions of either quantum dot. For an increasing magnetic field from panel (b) to (h) at low magnetic fields the apex in the combined conductance is well developed at first but reduces further and diminishes completely at large magnetic flux density values. With increasing magnetic field strength the linewidth of the single-electron tunneling peaks shrinks and the overall conductance reduces. The transition to the apex-less intersection occurs in the interval between B = 2.0 T and B = 2.5 T. Within this magnetic field range the transition of several intersections of single-tunneling peaks to an apex-less common conductance in the charge-stability diagram were observed. However, the flat-top behavior is only present if at least one of the quantum dots reaches a conductance ofI/VDS= 1e2/h.
In the figure panel (g) for B = 3.0 T the conductance of the quantum dot drops below I/VDS < 1e2/h for the larger single-electron tunneling peak contributing to the intersection and a slight apex at the center of the crossing is obtained, but diminishes again for B = 4.0 T at which the conductance of the single-electron tunneling peak returns to I/VDS = 1e2/h.
Both types of evolution of the common conductance values described above were found in the course of the same measurements of the charge-stability diagram. The difference in their behavior is emergent only for sufficiently large magnetic field strengths above B >2.5 T at which a well developed compressible and incompressible landscape is present in the 2DES in front of the quantum dots. The unique characteristic of the apex-less common conductance is puzzling and seems to be linked to the conductance value of I/VDS = 1e2/h of the single quantum dots. Reaching this conductance value at large magnetic field values without retuning the tunneling barrier of the single quantum dots can only be obtained by a sufficiently large initial linewidth of the single-electron tunneling peaks at zero magnetic field.
Comparison of Enhanced and Non-Enhanced Common Conductance
Both the enhanced conductance and the missing apex at the intersection of single-electron tunneling peaks are examined in large magnetic field strength, where the separation in both distinct types of behavior is well pronounced. Both types of the common conductance 120
7.3 Interplay of Two Parallel Quantum Dots at a Finite Magnetic Field
-1
-0.99
VUG [V]
-1.06 -1.04
-1.02 -1
VLG [V]
0 0.2 0.4 0.6
0 0.3 0.6
conductanceI/VDS
h e2 h
i
(a)
-0.84 -0.82
-0.8
VUG -0.98 [V]
-0.96 -0.94
VLG [V]
0 0.2 0.4 0.6 0.8 1
0 0.5 1
conductanceI/VDS
h e2 h
i
(b)
Figure 7.8 Comparison of two different intersections of single-electron tunneling peaks.
The data are taken at a bias of VDS = 25µV at base temperature and a magnetic flux density ofB = 6.6 T. (a) An enhanced conductance with a clear apex at the intersection is present. However, its peak value is significantly less than the contributing conductance values of each quantum dot taken separately. (b) A flat top behavior with a conductance capped at I/VDS = 1e2/h without a dominant increase of the conductance at the intersection point.
7 Two Parallel Quantum Dots - Single-Electron Tunneling Regime
at the intersection are displayed in figure 7.8. The data were taken on the same device with the tunneling barriers of the quantum dots defined at the particular large magnetic flux density of B = 6.6 T :
• Enhanced Common Conductance:
In figure 7.8a both conductance values of the single-electron tunneling peaks con-tributing to the intersection feature a comparably large conductance value in the range of I/VDS ≈ 0.4e2/h. At the center of the intersection a significant apex of enhanced conductance is present. However, its peak conductance value with I/VDS ≈ 0.6e2/h is well below the sum of the conductance contributing values of each quantum dot taken separately.
• Capped Common Conductance:
At the intersection shown in figure 7.8b the conductance values of each single-electron tunneling peaks are comparably large as well, but here with a conductance value of the order of I/VDS ≈ 1.0e2/h. Directly the crossing point a well defined apex of an enhanced conduction is absent. Instead, the conductance value of the quantum dots with I/VDS ≈ 1e2/h is maintained. This capping behavior of the common conductance value was observed as long as one quantum dot reaches the