## Singleelectron Tunneling

We have been discussing quantum dots, wires, and wells in isolation, such as die ones depicted in Figs. 9.1-9.3. To make them useful, they need coupling to their surroundings, to each other, or to electrodes that can add or subtract electrons from them. Figure 9.16 shows an isolated quantum dot or island coupled through tunneling to two leads, a source lead that supplies electrons, and a drain lead that

Source Quantum Drain

Lead Dot Lead

Source Quantum Drain

Lead Dot Lead

removes electrons for use in the external circuit. The applied voltage V^ causes direct current I to flow, with electrons tunneling into and out of the quantum dot. In accordance with Ohm's law V — IR, the current flow / through the circuit of Fig. 9.16 equals the applied source-drain voltage V^ divided by the resistance R, and the main contribution to the value of R arises from the process of electron tunneling from source to quantum dot, and from quantum dot to drain. Figure 9.17 shows the addition to the circuit of a capacitor-coupled gate terminal. The applied gate voltage Kg provides a controlling electrode or gate that regulates the resistance R of the active region of the quantum dot, and consequendy regulates the current flow I between the source and drain terminals. This device, as described, functions as a voltage-controlled or field-effect-controlled transistor, commonly referred to as an FET. For large or macroscopic dimensions die current flow is continuous, and die discreteness of the individual electrons passing through the device manifests itself by the presence of current fluctuations or shot noise. Our present interest is in the passage of electrons, one by one, through nanostructures based on circuitry of the type sketched in Fig. 9.17.

For an FET-type nanostructure the dimensions of the quantum dot are in the low nanometer range, and the attached electrodes can have cross sections comparable in size. For disk and spherical shaped dots of radius r the capacitance is given by disk

sphere

Source Lead

Quantum Dot

Drain Lead

Source Lead

Quantum Dot

### Drain Lead

Figure 9.17. Quantum dot coupied through source and a drain leads to an external circuit containing an applied bias voltage V^, with an additional capacitor-coupled terminal through which the gate voltage V„ controls the resistance of the electrically active region.

where e/£o is the dimensionless dielectric constant of the semiconducting material that forms the dot, and % = 8.8542 x 1012 F/m is the dielectric constant of free space. For the typical quantum dot material GaAs we have e/Sq = 13.2, which gives the very small value C = 1.47 x 10-lir farad for a spherical shape, where the radius r is in nanometers. The electrostatic energy E of a spherical capacitor of charge Q is changed by the amount AE ~ eQ/C when a single electron is added or subtracted, corresponding to the change in potential AV = AE/Q

where r is in nanometers. For a nanostructure of radius r = 10 nm, this gives a change in potential of 11 my which is easily measurable. It is large enough to impede the tunneling of the next electron.

Two quantum conditions must be satisfied for observation of the discrete nature of the single-electron charge transfer to a quantum dot One is that the capacitor single-electron charging energy e2/2C must exceed the thermal energy kBT arising from the random vibrations of the atoms in the solid, and the other is that the Heisenberg uncertainty principle be satisfied by die product of the capacitor energy e2/2C and the time r = RTC required for charging the capacitor

where is the tunneling resistance of the potential barrier. These two tunneling conditions correspond to

where A/e2 = 25.813 kil is the quantum of resistance. When these conditions are met and the voltage across the quantum dot is scanned, then the current jumps in increments every time die voltage changes by the value of Eq. (9.13), as shown by the /-versus- V characteristic of Fig. 9.18. This is called a Coulomb blockade because die electrons are blocked from tunneling except at the discrete voltage change positions. The step structure observed on the I-V characteristic of Fig. 9.18 is called a Coulomb staircase because it involves the Coulomb charging energy e2/2C of Eq. (9.15a).

An example of single-electron tunneling is provided by a line of ligand-stabilized Au55 nanoparticles. These gold particles have what is referred to as a structural magic number of atoms arranged in a FCC close-packed cluster that approximates the shape of a sphere of diameter 1.4nm, as was discussed in Section 2.1.3. The cluster of 55 gold atoms is encased in an insulating coating called a ligand shell that is adjustable in thickness, and has a typical value of 0.7 nm. Single-electron

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