Quantum dots

Adapted from Jana’s thesis.

Quantum dots are artificially structured systems confining charges within regions small enough to make their quantum mechanical energy levels observable (Hanson et al. 2007, Ihn book chapter 18). While it is a physical system made up of atoms, each having many electrons, the relevant particles of interest are the free electrons which can be manipulated and probed with electric fields. Quantum dots confining a small number of free electrons can be formed in many different systems, but the main system of interest for charge, spin and topological qubits are gate-defined quantum dots. Here, lithographically fabricated gate electrodes in proximity of a nanowire or on top of the surface of a heterostructure are used to create a potential landscape with wells and barriers. Together with the system’s geometry, this potential landscape confines charges in all three dimensions. As a result of this confinement and opposed to the continuum of states that typically occur in macroscopic systems, quantum dots feature a discrete energy spectrum. For this reason, quantum dots are sometimes called artificial atoms (Kouwenhoven et al. 2001). In what follows, we focus on quantum dots formed in two-dimensional electron gas (2DEG) in a GaAs/AlGaAs heterostructure, illustrated in figure Fig. 3.

Dots in a two-dimensional heterostructure (2DEG)

Fig. 3 GaAs/AlGaAs heterostructure hosting a double quantum dot. Mobile electrons from the thin silicon donor layer are attracted by the lower energy arising due to a smaller band gap in GaAs, all while being held close to their ionised donors. As a result, electrons are confined along the z axis, creating a two-dimensional electron gas (2DEG). Wells and barriers in the potential profile created by voltages applied to electrostatic gates further localize electrons within the 2DEG.

Mobile electrons from a thin silicon donor layer within  AlGaAs layer are attracted by GaAs - AIGaAs interface, marked in red in figure Fig. 3. This is due to the lower energy (energy profile depicted in black) they carry there. This effectively confines charges on a 2D plane.

Control of the charges is achieved by applying voltages to electrostatic gates at the surface of the heterostructure. The electric field created by these gates repels electrons from underneath, creating a potential profile with barriers and wells and thus localising electrons. An illustration of electron confinement in one or several directions is depicted in Fig. 4. By choosing appropriate gate voltages, electrons can be confined in two directions, creating a one dimensional channel, or in all three directions, resulting in an effectively zero dimensional system - the quantum dot. Note that in the case of a nanowire, charges are already confined in two directions and gate voltages only need to create barriers to localize them in the third direction. If the one dimensional confinement is very narrow, with a width comparable to the electron’s wavelength, the system is called a quantum point contact (QPC).

Fig. 4 Dimensions of electron gases.

Transport through quantum dots

The physics of quantum dots can be studied based on the dots’ transport properties, i.e. by measuring current through the system. To this end, a small bias voltage is applied to metallic reservoirs on both sides of the quantum dot device, allowing electrons to move between reservoirs and dots via tunnelling processes. Current probes attached to the reservoirs are used to measure the resulting current. The fundamental transport phenomenon in quantum dots is the Coulomb blockade (Van Houten et al 1992). The Coulomb blockade is a classical effect arising due to the Coulomb repulsion between electrons, resulting in a finite energy cost when adding an extra electron onto a dot. At low enough temperatures, tunnelling of electrons between dots or dots and adjacent reservoirs can be suppressed and the device’s current-voltage relation no longer follows Ohm’s law.

Once formed, quantum dots can be modelled as conducting islands connected to source (S) and drain (D), illustrated in Fig. 5 and Fig. 6. Here, the conducting island is coloured grey, while source and drain are shown in blue. Couplings between dots and reservoirs are modelled as a resistor in parallel to a capacitor.

Fig. 5 Schematic of a single dot, modelled as a conducting island connected via tunnel junctions to source and drain reservoirs. A nearby plunger gate is capacitively coupled to the dot and used to tune the its energy levels.

Fig. 6 Schematic of a double dot, modelled as two conducting islands in series with source and drain reservoirs, coupled by tunnel junctions.

When a quantum dot is in the Coulomb blockade regime, transport through the dot-reservoir system only occurs when the dot’s energy level :math:epsilon falls between the energy levels of the reservoirs held at different bias potentials, also called the bias window.

Transport through a double dot system is illustrated in figures Fig. 7 and Fig. 8. A measurable current arises when one or both dot energy levels are within the energy levels of the reservoirs (fig. Fig. 7). To be precise, resonant tunnelling occurs when both dot levels are within the bias window. A so-called co-tunnelling process via virtual states takes place when only one potential is within the bias window (De Franceschi et al. 2001). The resonant current is typically higher in amplitude than the off-resonant current. The double-dot states are probed by stepping over the voltages of two nearby gates, resulting (in the most typical and ideal case) in the charge transition pattern illustrated in Fig. 9.

Note that one often omits the small bias and talks about the alignment of the dots energy levels. If the levels don’t align, as in Fig. 8, the dot is in the Coulomb blockade regime. Here, Coulomb repulsion between electrons prevents multiple electrons to occupy the same energy level and transport is suppressed.

Fig. 7 Electron transport via resonant tunnelling occurs from source to drain when both dot energy levels are within the bias window.

Fig. 8 Electron tunnelling is suppressed whenever the dots’ energy levels are not within the energy levels of the reservoirs. However, if one energy level is within the bias window, a measurable current arises due to a so-called co-tunnelling processes via virtual states.

Fig. 9 Schema of a so-called charge diagram showing the charge transition pattern of a double dot.

Fig. 10 Illustration of charge transitions measured by varying both plunger gate voltages and monitoring current from source to drain. Four types of charge transitions are observed, exchanging charges between dots, a dot and its adjacent reservoir, or allowing a current to flow from source to drain.

Transport features of dots with well-localized and weakly coupled charges can be explained and qualitatively reproduced by the classical capacitance model, which represents gates, dots and reservoirs as conductors connected through resistors and capacitors. It also allows to capture the so-called gate cross-talk, i.e. the effect of capacitive couplings of all gates to each dot. The capacitance model is discussed in Capacitance model.