Coulomb blockade

Observation

In order for the Coulomb blockade to be observable, the temperature has to be low enough so that the characteristic charging energy (the energy that is required to charge the junction with one elementary charge) is larger than the thermal energy of the charge carriers. In the past, for capacitances above 1 femtofarad (10⁻¹⁵ farad), this implied that the temperature has to be below about 1 kelvin. This temperature range is routinely reached for example by helium-3 refrigerators. Thanks to small-sized quantum dots of only few nanometers, Coulomb blockade has been observed next above liquid helium temperature, up to room temperature.

To make a tunnel junction in plate condenser geometry with a capacitance of 1 femtofarad, using an oxide layer of electric permittivity 10 and thickness one nanometer, one has to create electrodes with dimensions of approximately 100 by 100 nanometers. This range of dimensions is routinely reached for example by electron beam lithography and appropriate pattern transfer technologies, like the Niemeyer–Dolan technique, also known as shadow evaporation technique. The integration of quantum dot fabrication with standard industrial technology has been achieved for silicon. CMOS process for obtaining massive production of single electron quantum dot transistors with channel size down to 20 nm x 20 nm has been implemented.

Single-electron transistor

The simplest device in which the effect of Coulomb blockade can be observed is the so-called single-electron transistor. It consists of two electrodes known as the drain and the source, connected through tunnel junctions to one common electrode with a low self-capacitance, known as the island. The electrical potential of the island can be tuned by a third electrode, known as the gate, which is capacitively coupled to the island.

In the blocking state no accessible energy levels are within tunneling range of an electron (in red) on the source contact. All energy levels on the island electrode with lower energies are occupied.

When a positive voltage is applied to the gate electrode the energy levels of the island electrode are lowered. The electron (green 1.) can tunnel onto the island (2.), occupying a previously vacant energy level. From there it can tunnel onto the drain electrode (3.) where it inelastically scatters and reaches the drain electrode Fermi level (4.).

The energy levels of the island electrode are evenly spaced with a separation of ${\displaystyle \mathrm{\Delta }E.}$ This gives rise to a self-capacitance ${\displaystyle C}$ of the island, defined as

${\displaystyle C=\frac{e^2}{\mathrm{\Delta }E}.}$

To achieve the Coulomb blockade, three criteria have to be met:

1. The bias voltage must be lower than the elementary charge divided by the self-capacitance of the island: ${\displaystyle V_{\text{bias}}<\frac{e}{C}}$ ;
2. The thermal energy in the source contact plus the thermal energy in the island, i.e. ${\displaystyle k_{\mathrm{B}}T,}$ must be below the charging energy: ${\displaystyle k_{\mathrm{B}}T<\frac{e^2}{2C},}$ or else the electron will be able to pass the QD via thermal excitation; and
3. The tunneling resistance, ${\displaystyle R_{\mathrm{t}},}$ should be greater than ${\displaystyle \frac{h}{e^2},}$ which is derived from Heisenberg's uncertainty principle.

Coulomb blockade thermometer

A typical Coulomb blockade thermometer (CBT) is made from an array of metallic islands, connected to each other through a thin insulating layer. A tunnel junction forms between the islands, and as voltage is applied, electrons may tunnel across this junction. The tunneling rates and hence the conductance vary according to the charging energy of the islands as well as the thermal energy of the system.

Coulomb blockade thermometer is a primary thermometer based on electric conductance characteristics of tunnel junction arrays. The parameter V_½ = 5.439 Nk_BT/e, the full width at half minimum of the measured differential conductance dip over an array of N junctions together with the physical constants provide the absolute temperature.

Ionic Coulomb blockade

Ionic Coulomb blockade (ICB) is the special case of CB, appearing in the electro-diffusive transport of charged ions through sub-nanometer artificial nanopores or biological ion channels. ICB is widely similar to its electronic counterpart in quantum dots,^[1] but presents some specific features defined by possibly different valence z of charge carriers (permeating ions vs electrons) and by the different origin of transport engine (classical electrodiffusion vs quantum tunnelling).

In the case of ICB, Coulomb gap ${\displaystyle \mathrm{\Delta }E}$ is defined by dielectric self-energy of incoming ion inside the pore/channel $${\displaystyle \mathrm{\Delta }E=\frac{z^2{e}^2}{2C}}$$and hence $\mathrm{\Delta }E$ depends on ion valence z. ICB appears strong $(\mathrm{\Delta }E\gg k_{\mathrm{B}}T)$, even at the room temperature, for ions with ${\displaystyle z>=2}$, e.g. for ${\displaystyle {\text{Ca}}^{2+}}$ ions.

ICB has been recently experimentally observed in sub-nanometer ${\displaystyle {\text{MoS}}_2^{}}$ pores.

In biological ion channels ICB typically manifests itself in such valence selectivity phenomena as ${\displaystyle {\text{Ca}}^{2+}}$ conduction bands (vs fixed charge ${\displaystyle Q_{\mathrm{f}}}$) and concentration-dependent divalent blockade of sodium current.