In
mesoscopic physics, a Coulomb blockade (CB), named after
Charles-Augustin de Coulomb
Charles-Augustin de Coulomb (; ; 14 June 1736 – 23 August 1806) was a French officer, engineer, and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, the description of the electrostatic force of attra ...
's
electrical force
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventio ...
, is the decrease in
electrical conductance
The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallel ...
at small
bias voltages of a small electronic device comprising at least one low-
capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
tunnel junction
In electronics/spintronics, a tunnel junction is a barrier, such as a thin insulating layer or electric potential, between two electrically conducting materials. Electrons (or quasiparticles) pass through the barrier by the process of quantum ...
.
Because of the CB, the conductance of a device may not be constant at low bias voltages, but disappear for biases under a certain threshold, i.e. no current flows.
Coulomb blockade can be observed by making a device very small, like a
quantum dot
Quantum dots (QDs) are semiconductor particles a few nanometres in size, having optical and electronic properties that differ from those of larger particles as a result of quantum mechanics. They are a central topic in nanotechnology. When the ...
. When the device is small enough,
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
s inside the device will create a strong
Coulomb repulsion preventing other electrons to flow. Thus, the device will no longer follow
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equa ...
and the current-voltage relation of the Coulomb blockade looks like a staircase.
Even though the Coulomb blockade can be used to demonstrate the
quantization of the electric charge, it remains a
classical effect and its main description does not require
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. However, when few electrons are involved and an external static
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
is applied, Coulomb blockade provides the ground for a
spin blockade (like Pauli spin blockade) and
valley blockade, which include quantum mechanical effects due to
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally ...
and
orbital interactions respectively between the electrons.
The devices can comprise either metallic or
superconducting
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
electrode
An electrode is an electrical conductor used to make contact with a nonmetallic part of a circuit (e.g. a semiconductor, an electrolyte, a vacuum or air). Electrodes are essential parts of batteries that can consist of a variety of materials ...
s. If the electrodes are superconducting,
Cooper pairs (with a
charge of minus two
elementary charge
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundam ...
s
) carry the current. In the case that the electrodes are metallic or ''normal-conducting'', i.e. neither
superconducting
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
nor
semiconducting,
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
s (with a charge of
) carry the current.
In a tunnel junction
The following section is for the case of tunnel junctions with an insulating barrier between two normal conducting electrodes (NIN junctions).
The
tunnel junction
In electronics/spintronics, a tunnel junction is a barrier, such as a thin insulating layer or electric potential, between two electrically conducting materials. Electrons (or quasiparticles) pass through the barrier by the process of quantum ...
is, in its simplest form, a thin insulating barrier between two conducting electrodes. According to the laws of
classical electrodynamics
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fi ...
, no current can flow through an insulating barrier. According to the laws of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, however, there is a nonvanishing (larger than zero)
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
for an electron on one side of the barrier to reach the other side (see
quantum tunnelling
Quantum tunnelling, also known as tunneling ( US) is a quantum mechanical phenomenon whereby a wavefunction can propagate through a potential barrier.
The transmission through the barrier can be finite and depends exponentially on the barrie ...
). When a
bias voltage is applied, this means that there will be a current, and, neglecting additional effects, the tunnelling current will be proportional to the bias voltage. In electrical terms, the tunnel junction behaves as a
resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active e ...
with a constant resistance, also known as an
ohmic resistor. The resistance depends
exponentially on the barrier thickness. Typically, the barrier thickness is on the order of one to several
nanometer
330px, Different lengths as in respect to the molecular scale.
The nanometre (international spelling as used by the International Bureau of Weights and Measures; SI symbol: nm) or nanometer (American and British English spelling differences#-re, ...
s.
An arrangement of two conductors with an insulating layer in between not only has a resistance, but also a finite
capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
. The insulator is also called
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the ma ...
in this context, the tunnel junction behaves as a
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
.
Due to the discreteness of electrical charge, current through a tunnel junction is a series of events in which exactly one electron passes (''tunnels'') through the tunnel barrier (we neglect cotunneling, in which two electrons tunnel simultaneously). The tunnel junction capacitor is charged with one elementary charge by the tunnelling electron, causing a
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
build up
, where
is the capacitance of the junction. If the capacitance is very small, the voltage build up can be large enough to prevent another electron from tunnelling. The electric current is then suppressed at low bias voltages and the resistance of the device is no longer constant. The increase of the
differential resistance around zero bias is called the 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
The farad (symbol: F) is the unit of electrical capacitance, the ability of a body to store an electrical charge, in the International System of Units (SI). It is named after the English physicist Michael Faraday (1791–1867). In SI base unit ...
), this implied that the temperature has to be below about 1
kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ...
. This temperature range is routinely reached for example by
Helium-3
Helium-3 (3He see also helion) is a light, stable isotope of helium with two protons and one neutron (the most common isotope, helium-4, having two protons and two neutrons in contrast). Other than protium (ordinary hydrogen), helium-3 is the ...
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
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
10 and thickness one
nanometer
330px, Different lengths as in respect to the molecular scale.
The nanometre (international spelling as used by the International Bureau of Weights and Measures; SI symbol: nm) or nanometer (American and British English spelling differences#-re, ...
, 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
A single-electron transistor (SET) is a sensitive electronic device based on the Coulomb blockade effect. In this device the electrons flow through a tunnel junction between source/drain to a quantum dot (conductive island). Moreover, the electric ...
. 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
This gives rise to a self-capacitance
of the island, defined as
:
To achieve the Coulomb blockade, three criteria have to be met:
# The bias voltage must be lower than the
elementary charge
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundam ...
divided by the self-capacitance of the island:
;
# The thermal energy in the source contact plus the thermal energy in the island, i.e.
must be below the charging energy:
or else the electron will be able to pass the QD via thermal excitation; and
# The tunneling resistance,
should be greater than
which is derived from Heisenberg's
uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
.
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
A thermometer is a device that measures temperature or a temperature gradient (the degree of hotness or coldness of an object). A thermometer has two important elements: (1) a temperature sensor (e.g. the bulb of a mercury-in-glass thermometer ...
based on electric conductance characteristics of tunnel junction arrays. The parameter V
½=5.439Nk
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,
">/sup> 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 is defined by dielectric self-energy of incoming ion inside the pore/channel and hence depends on ion valence ''z''. ICB appears strong , even at the room temperature, for ions with , e.g. for Ca^2+ ions.
ICB has been recently experimentally observed in sub-nanometer MoS2 pores.
In biological ion channels ICB typically manifests itself in such valence selectivity phenomena as conduction bands (vs fixed charge ) and concentration-dependent divalent blockade of sodium current.
References
;General
* ''Single Charge Tunneling: Coulomb Blockade Phenomena in Nanostructures'', eds. H. Grabert and M. H. Devoret (Plenum Press, New York, 1992)
* D.V. Averin and K.K Likharev, in ''Mesoscopic Phenomena in Solids'', eds. B.L. Altshuler, P.A. Lee, and R.A. Webb (Elsevier, Amsterdam, 1991)
*
External links
Computational Single-Electronics book
Coulomb blockade online lecture
{{DEFAULTSORT:Coulomb Blockade
Nanoelectronics
Quantum electronics
Mesoscopic physics