Basics of Quantum Point Contacts 4
4.2 Quantum Dots
4.2.1 Electrical Transport Through a Single Quantum Dot
In order to enable electrical transport through quantum dots, the Coulomb-blockade has to be lifted. This can be achieved by tuning the electrostatic potential of the quantum dot via a plunger gate shifting the charging thresholds up or down in energy. Alternatively by applying a finite source-drain bias between two reservoirs the quantum dot is coupled to also allows to give rise to a charge exchange from and onto the dot.
The mechanism lifting the Coulomb-blockade for both tuning the gate voltage and the DC source-drain bias is summarized in figure 4.5. The main panel shows the interplay of both tuning parameters (charge-stability diagram) with regions of Coulomb-blockade and adjacent regimes of single-electron tunneling, for which an electron enters the quantum dot from one reservoir and subsequently leaves to the other depending on the sign of the voltage bias. Several cross-cuts for either the gate voltage VG or the source-drain bias VDS are shown, including the corresponding energy ladder diagram and the current respectively the conductance as a function of the tuning parameter chosen in the cross-cut:
• Single-Electron Tunneling Peaks:
In the vertical cross-cut, tuning the gate voltage at zero source-drain bias yields a series of single-electron tunneling peaks, displayed in the upper right panel of the figure. Their spacing on the gate voltage axis is determined by the electro-static contribution plus the eigenenergy of the state used in the quantum dot. The conductance can reach a value of I/VDS = 1e2/h at the apex of the peak [9].
• Current-Voltage Characteristic:
In the second method for a fixed gate voltage the source-drain bias is detuned, see panel to the lower right of the figure. The energy difference between the charging threshold and the electro-chemical potential of the leads reduces with increasing bias, but the current remains zero in the regime of Coulomb-blockade. At a certain bias valueVth(0) for either bias direction the charging level aligns with the respective electro-chemical potential of either lead and current can flow. It leads to a non-linear current-voltage characteristic varying with the gate voltage.
The combination of both tuning parameter limits the regime of the Coulomb-blockade, leading to the diagram in the main panel of the figure. The central yellow almost rhombic shaped areas are the regime of Coulomb-blockade with a fixed number of electrons confined on the quantum dot. For an increasing bias value the energy difference needed to be tuned by the plunger gate of the quantum dot reduces, defining the boundary of the Coulomb-blockade regime in the interplay of both parameters. From the extremal extension of the Coulomb-diamond on both the DC source-drain bias and the gate voltage axis one 30
4.2 Quantum Dots
N N-1 N+1
I
VDS
I
VDS
VDS I/VDS
VG VG
N+1
e/C e/CG
drain source
eVDS N
drain source
E
-eVDS
drain source
eVDS
N-1 N N+1
E E
N-1 N N*
N+1
1 e²/h
~
Vth -V‘th
single-electron tunneling peaks
current-voltage characteristic
ground-state transition ground-state transition excited-state transition
N N+1
Figure 4.5 Electrical transport through a quantum dot for the interplay of the source-drain bias and the plunger gate voltage (charge-stability diagram). Line-cuts in the source-drain bias at different gate voltage settings on the right hand side illustrate the non-linear current-voltage characteristic. Another line-cut along the gate voltage axis is displayed for zero bias that features a series of single-electron tunneling peaks. The ground-state charging thresholds of the quantum dot define the regime of Coulomb-blockade (yellow) in the charge-stability diagram. In the adjacent regime the fluctuations betweenN ↔N+1 electrons on the quantum dot are energetically allowed. In addition to the ground-state, excited states can be present in the confining potential of the quantum dot, leading to additional lines in the charge-stability diagram.
4 Basics of Quantum Point Contacts and Quantum Dots
obtains [113]
|VDS|= e
CΣ and (4.3)
|VG|= e
CG . (4.4)
It allows to calculate the charging energy and the conversion factor for a gate voltage shift as resulting shift of the electrostatic potential of the quantum dot. In the neighboring areas (light red and blue) for a positive and negative source-drain bias, respectively, a fluctuation by one electron is energetically allowed.
The electrical transport through a quantum dot structure used in this work, obtained for the interplay of both the plunger gate voltage and the DC source-drain bias, is dis-played in figure 4.6. In the upper panel of the figure the DC current data is disdis-played, while the lower displays the differential conductance dI/dVDS acquired simultaneously. In total five Coulomb-diamonds are present showing zero current or a zero differential con-ductance, respectively. Their border line becomes blurry as the gate voltage is increased to a more positive value thereby increasing the tunnel coupling strength of the dot to the leads. For a weak tunnel coupling strength at a more negative gate voltage the sharp diamond shape is recovered. From the largest Coulomb-diamond in the plot the charging energy of the quantum dot is determined with EC = 1 meV. The shape of the diamond also reflects the symmetry of the tunneling barriers to either reservoir, id est for a regular diamond shape as obtained here, the strength of the tunnel coupling to both the source and the drain reservoir is similar. In addition to the ground-state transition defining the Coulomb-diamonds, excited states are present leading to steps in the current-data and to peaks in the differential conductance map in the regimes of single-electron tunneling.
In summary, tuning both the source-drain bias and the plunger gate voltage allows to extract valuable information about the quantum dot, such as the charging energy.
However, it is evident that the pure electrostatic model fails to comprise all interactions present in a real device, but catches reasonably well the experimental observations in the electric transport. The electrons on the dot have rather to be considered as an interacting many-electron system in the confinement potential. But already in the simple consideration of a harmonic trapping potential the addition of a surplus electron results in a rich spectrum of ground and excited states due to the electron-electron interaction that increases in complexity with an increasing number of confined electrons [84]. However, the exact shape of the confinement potential plays a minor role in transport measurements which can sufficiently be characterized by the charging thresholds as the difference in energy required to add or remove an electron to the ground state of the system. Additional transitions of charges were observed in small quantum dots that can be attributed to excited states in the confinement potential [111, 113]. These form additional charging thresholds that can be used to both add or remove an electron to or from the quantum dot, respectively. One has to note that a ground and excited state belonging to the same number of electrons confined on the dot cannot be occupied simultaneously, prohibited by the Coulomb-blockade effect.
32
4.2 Quantum Dots
-3 -2
-1 0
1 2
3
bias voltage
VDS[mV] -0.9
-0.8 -0.7
-0.6
gatevoltage [V]
-20 -10 0 10 20
currentIDS[nA]
-20 -10 0 10 20
currentIDS[nA]
(a)
gatevoltage[V]
bias voltageVDS[mV]
e/C
e/C
G-0.9 -0.8 -0.7 -0.6
-3 -2 -1 0 1 2 3
0 0.3 0.6
differentialconductancedI/dVDS
h e2 h
i
(b)
Figure 4.6 Electrical transport data of a quantum dot for the interplay of the source-drain bias and the plunger gate voltage (charge-stability diagram). (a) Current flowing through the quantum dot. (b) Differential conductance of (a) with dI/dVDS obtained via lock-in technique with a small AC modulation applied to the source-drain bias. Symmetric diamond shaped regions of Coulomb-blockade are obtained. In addition, a fair amount of excited states is present in the regime of single-electron tunneling. The charging energy of the quantum dot yields EC = 1 meV. Taken at base temperature of T = 50 mK in the cryostat.
4 Basics of Quantum Point Contacts and Quantum Dots