4.3 Coulomb blockade and single-electron transistors

Coulomb blockade and single-electron transistors are key concepts in molecular electronics. They involve controlling electron flow through tiny conducting islands. This phenomenon occurs when the energy needed to add an electron is greater than the thermal energy, blocking current flow.

Single-electron transistors use this effect to control electron transport with incredible precision. By manipulating gate voltages, we can switch between blocked and conducting states. This allows for ultra-sensitive charge detection and potential applications in quantum computing.

Coulomb Blockade Fundamentals

Charging Energy and Electron Addition Energy

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• Coulomb blockade occurs when the charging energy required to add an electron to a small conducting island is greater than the thermal energy, preventing electron transport through the island
• Charging energy $E_C = e^2/2C$ depends on the capacitance $C$ of the island, with smaller islands having higher charging energies due to reduced capacitance
• Electron addition energy $E_{add} = E_C + \Delta E$ includes both the charging energy and the quantum energy level spacing $\Delta E$ in the island
• Higher electron addition energies lead to more pronounced Coulomb blockade effects, requiring larger voltages to overcome the energy barrier and allow electron tunneling (metallic nanoparticles, quantum dots)

Tunnel Junctions and Islands

• Tunnel junctions are thin insulating barriers separating the island from the source and drain electrodes, allowing electrons to tunnel through the barrier quantum mechanically
• Resistance of the tunnel junctions $R_T$ must be much larger than the quantum resistance $R_Q = h/e^2 \approx 25.8 k\Omega$ to suppress quantum fluctuations and observe Coulomb blockade
• Islands are small conducting regions, such as metallic nanoparticles or semiconductor quantum dots, where electrons can be confined and their number precisely controlled
• Size and material of the island determine its capacitance and electron addition energy, with smaller islands exhibiting stronger Coulomb blockade effects (carbon nanotubes, graphene nanoribbons)

Single-Electron Transistors

SET Structure and Operation

• Single-electron transistor (SET) consists of a conducting island connected to source and drain electrodes through tunnel junctions, with a gate electrode capacitively coupled to the island
• Electron transport through the SET is controlled by the gate voltage, which shifts the energy levels of the island relative to the Fermi levels of the source and drain electrodes
• Quantum dot SETs use semiconductor nanostructures as the island, offering tunable electron addition energies and the ability to control the number of electrons in the dot precisely
• Gate electrode modulates the electrostatic potential of the island, allowing for switching between Coulomb blockade and single-electron tunneling regimes (electrometer, charge sensor)

Coulomb Staircase and SET Applications

• Coulomb staircase refers to the stepwise increase in current through the SET as the gate voltage is swept, with each step corresponding to the addition of a single electron to the island
• Height and width of the Coulomb staircase steps depend on the charging energy and the capacitance of the island, providing a means to probe the electronic structure of the island
• SETs are highly sensitive to charge and have been used as electrometers, capable of detecting sub-electron charge variations (charge qubits, quantum computing)
• Potential applications of SETs include single-electron memory devices, ultra-sensitive sensors, and quantum information processing (quantum dots, spin qubits)

Coulomb Blockade Characteristics

Coulomb Diamonds and Charging Energy

• Coulomb diamonds are diamond-shaped regions in the differential conductance plot of an SET as a function of source-drain and gate voltages, representing the Coulomb blockade regime
• Size of the Coulomb diamonds is determined by the charging energy of the island, with larger diamonds corresponding to higher charging energies and stronger Coulomb blockade
• Slopes of the diamond edges depend on the capacitances between the island and the source, drain, and gate electrodes, providing information about the device geometry
• Periodic nature of the Coulomb diamonds reflects the discrete electron addition energy spectrum of the island (quantum dots, metallic nanoparticles)

Electron Addition Energy and Tunnel Junction Resistance

• Electron addition energy can be extracted from the height of the Coulomb diamonds, as it represents the energy required to add an electron to the island at a given gate voltage
• Spacing between the Coulomb diamonds along the gate voltage axis is proportional to the electron addition energy, allowing for the determination of the energy level spectrum of the island
• Tunnel junction resistance affects the lifetime of the charge states on the island and the width of the conductance peaks in the Coulomb blockade regime
• Higher tunnel junction resistances lead to sharper conductance peaks and more well-defined Coulomb diamonds, as the charge states on the island are more stable against quantum fluctuations (oxide tunnel barriers, electromigrated junctions)