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{{Short description|Electronic device}}
[[File:SET schematic2.jpg|thumb|Schematic of a basic SET and its internal electrical components
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 electrical potential of the island can be tuned by a third electrode, known as the gate, which is capacitively coupled to the island. The conductive island is sandwiched between two tunnel junctions<ref>{{cite journal|last1=Mahapatra|first1=S.|last2=Vaish|first2=V.|last3=Wasshuber|first3=C.|last4=Banerjee|first4=K.|last5=Ionescu|first5=A.M.|title=Analytical Modeling of Single Electron Transistor for Hybrid CMOS-SET Analog IC Design|journal=IEEE Transactions on Electron Devices|volume=51|issue=11|year=2004|pages=1772–1782|issn=0018-9383|doi=10.1109/TED.2004.837369|bibcode=2004ITED...51.1772M|s2cid=15373278}}</ref> modeled by capacitors, <math>C_{\rm D}</math> and <math>C_{\rm S}</math>, and resistors, <math>R_{\rm D}</math> and <math>R_{\rm S}</math>, in parallel.
== History ==
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The first single-electron transistor based on the phenomenon of Coulomb blockade was reported in 1986 by Soviet scientists {{ill|K. K. Likharev|ru|Лихарев, Константин Константинович}} and D. V. Averin.<ref name=":1">{{Cite journal|
== Relevance ==
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The increasing relevance of the [[Internet of things]] and the healthcare applications give more relevant impact to the electronic device power consumption. For this purpose, ultra-low
Applicable areas
== Device ==
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=== Principle ===
[[File:Set schematic.svg|thumb|right|Schematic diagram of a single-electron transistor
[[File:Single electron transistor.svg|thumb|right|Left to right: energy levels of source, island and drain in a single-electron transistor for the blocking state (upper part) and transmitting state (lower part).]]
The SET has,
The current, <math>I,</math> from source to drain follows [[
In the blocking state all lower energy levels are occupied at the QD and no unoccupied level is within tunnelling range of electrons originating from the source (green 1.). When an electron arrives at the QD (2.) in the non-blocking state it will fill the lowest available vacant energy level, which will raise the energy barrier of the QD, taking it out of tunnelling distance once again. The electron will continue to tunnel through the second tunnel junction (3.), after which it scatters inelastically and reaches the drain electrode Fermi level (4.).
The energy levels of the QD are evenly spaced with a separation of <math>\Delta E.</math> This gives rise to a self-capacitance <math>C</math> of the island, defined as: <math>C=\tfrac{e^2}{\Delta E}.</math> To achieve the Coulomb blockade, three criteria need to be met:
|last1=Poole
|first1=Charles P. Jr.
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# The bias voltage must be lower than the [[elementary charge]] divided by the self-capacitance of the island: <math>V_\text{bias} < \tfrac{e}{C}</math>
# The thermal energy in the source contact plus the thermal energy in the island, i.e. <math>k_{\rm B}T,</math> must be below the charging energy: <math>k_{\rm B}T \ll \tfrac{e^2}{2C},</math> otherwise the electron will be able to pass the QD via thermal excitation.
# The tunnelling resistance, <math>R_{\rm t},</math> should be greater than <math>\tfrac{h}{e^2},</math> which is derived from Heisenberg's [[uncertainty principle]].
If the resistance of all the tunnel barriers of the system is much higher than the quantum resistance <math>R_{\rm t} = \tfrac{h}{e^2} = 25.813~\text{k}\Omega,</math> it is enough to confine the electrons to the island, and it is safe to ignore coherent quantum processes consisting of several simultaneous tunnelling events, i.e. co-tunnelling.
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The background charge of the dielectric surrounding the QD is indicated by <math>q_0</math>. <math>n_{\rm S}</math> and <math>n_{\rm D}</math> denote the number of electrons tunnelling through the two tunnel junctions and the total number of electrons is <math>n</math>. The corresponding charges at the tunnel junctions can be written as:
:<math>q_{\rm S} = C_{\rm S} V_{\rm S}</math>
:<math>q_{\rm D} = C_{\rm D} V_{\rm D}</math>
:<math>q = q_{\rm D} - q_{\rm S} + q_0 = -ne + q_0,</math>
where <math>C_{\rm S}</math> and <math>C_{\rm D}</math> are the parasitic leakage capacities of the tunnel junctions. Given the bias voltage, <math>V_{\rm bias} = V_{\rm S} + V_{\rm D},</math> you can solve the voltages at the tunnel junctions:
:<math>V_{\rm S} = \frac{C_{\rm D} V_{\rm bias} + ne - q_0}{C_{\rm S} + C_{\rm D}},</math>
:<math>V_{\rm D} = \frac{C_{\rm S} V_{\rm bias} - ne + q_0}{C_{\rm S} + C_{\rm D}}.</math>
The electrostatic energy of a double-connected tunnel junction (like the one in the schematical picture) will be
:<math>E_C = \frac{q_{\rm S}^2}{2 C_{\rm S}} + \frac{q_{\rm D}^2}{2 C_{\rm D}} = \frac{C_{\rm S} C_{\rm D} V_{\rm bias}^2 + (ne - q_0)^2}{2(C_{\rm S} + C_{\rm D})}.</math>
The work performed during electron tunnelling through the first and second transitions will be:
:<math>W_{\rm S} = \frac{n_{\rm S} e V_{\rm bias} C_{\rm D}}{C_{\rm S} + C_{\rm D}},</math>
:<math>W_{\rm D} = \frac{n_{\rm D} e V_{\rm bias} C_{\rm S}}{C_{\rm S} + C_{\rm D}}.</math>
Given the standard definition of free energy in the form:
:<math>F = E_{\rm tot} - W,</math>
where <math>E_{\rm tot} = E_C = \Delta E_F + E_N,</math> we find the free energy of a SET as:
:<math>F(n, n_{\rm S}, n_{\rm D}) = E_C - W = \frac{1}{C_{\rm S} + C_{\rm D}} \left( \frac{1}{2} C_{\rm S} C_{\rm D} V_{\rm bias}^2 + (ne - q_0)^2 + e V_{\rm bias} C_{\rm S} n_{\rm D} + C_{\rm D} n_{\rm S} \right).</math>
For further consideration, it is necessary to know the change in free energy at zero temperatures at both tunnel junctions:
:<math>\Delta F_{\rm S}^{\pm} = F(n \pm 1, n_{\rm S} \pm 1, n_{\rm D}) - F(n, n_{\rm S}, n_{\rm D}) = \frac{e}{C_{\rm S} + C_{\rm D}} \left( \frac{e}{2} \pm (V_{\rm bias} C_{\rm D} + ne - q_0) \right),</math>
:<math>\Delta F_{\rm D}^{\pm} = F(n \pm 1, n_{\rm S}, n_{\rm D} \pm 1) - F(n, n_{\rm S}, n_{\rm D}) = \frac{e}{C_{\rm S} + C_{\rm D}} \left( \frac{e}{2} \pm (V_{\rm bias} C_{\rm S} + ne - q_0) \right),</math>
The probability of a tunnel transition will be high when the change in free energy is negative. The main term in the expressions above determines a positive value of <math>\Delta F</math> as long as the applied voltage <math>V_{\rm bias}</math> will not exceed the threshold value, which depends on the smallest capacity in the system. In general, for an uncharged QD (<math>n = 0</math> and <math>q_0 = 0</math>) for symmetric transitions (<math>C_{\rm S} = C_{\rm D} = C</math>) we have the condition
:<math>V_{\rm th} = \left|V_{\rm bias}\right| \ge \frac{e}{2 C},</math>
(that is, the threshold voltage is reduced by half compared with a single transition).
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When the applied voltage is zero, the Fermi level at the metal electrodes will be inside the energy gap. When the voltage increases to the threshold value, tunnelling from left to right occurs, and when the reversed voltage increases above the threshold level, tunnelling from right to left occurs.
The existence of the Coulomb blockade is clearly visible in the [[
In the case where the permeability of the tunnel barriers is very different <math>(R_{T1} \gg R_{T2} = R_T),</math> a stepwise I-V characteristic of the SET arises. An electron tunnels to the island through the first transition and is retained on it, due to the high tunnel resistance of the second transition. After a certain period of time, the electron tunnels through the second transition, however, this process causes a second electron to tunnel to the island through the first transition. Therefore, most of the time the island is charged in excess of one charge. For the case with the inverse dependence of permeability <math>(R_{T1} \ll R_{T2} = R_T),</math> the island will be unpopulated and its charge will decrease stepwise.
:<math>q = -ne + q_0 + C_{\rm G}(V_{\rm G} - V_{2}).</math>
Substituting this value into the formulas found above, we find new values for the voltages at the transitions:
:<math>V_{\rm S} = \frac{(C_{\rm D} + C_{\rm G}) V_{\rm bias} - C_{\rm G} V_{\rm G} + ne - q_0}{C_{\rm S} + C_{\rm D}},</math>
:<math>V_{\rm D} = \frac{C_{\rm S} V_{\rm bias} + C_{\rm G} V_{\rm G} - ne + q_0}{C_{\rm S} + C_{\rm D}},</math>
The electrostatic energy should include the energy stored on the gate capacitor, and the work performed by the voltage on the gate should be taken into account in the free energy:
:<math>\Delta F_{\rm S}^{\pm} = \frac{e}{C_{\rm S} + C_{\rm D}} \left( \frac{e}{2} \pm V_{\rm bias}(C_{\rm D} + C_{\rm G}) - V_{\rm G} C_{\rm G} + ne + q_0 \right),</math>
:<math>\Delta F_{\rm D}^{\pm} = \frac{e}{C_{\rm S} + C_{\rm D}} \left( \frac{e}{2} \pm V_{\rm bias} C_{\rm S} + V_{\rm G} C_{\rm G} - ne + q_0 \right).</math>
At zero temperatures, only transitions with negative free energy are allowed: <math>\Delta F_{\rm S} < 0</math> or <math>\Delta F_{\rm D} < 0</math>. These conditions can be used to find areas of stability in the plane <math>V_{\rm bias} - V_{\rm G}.</math>
With increasing voltage at the gate electrode, when the supply voltage is
=== Temperature dependence ===
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Various materials have successfully been tested when creating single-electron transistors. However, temperature is a huge factor limiting implementation in available electronical devices. Most of the metallic-based SETs only work at extremely low temperatures.
[[File:TySETimage.png|thumb|right|Single-electron transistor with [[niobium]] leads and [[aluminium]] island
As mentioned in bullet 2 in the list above: the electrostatic charging energy must be greater than <math>k_{\rm B} T</math> to prevent thermal fluctuations affecting the [[Coulomb blockade]]. This in turn implies that the maximum allowed island capacitance is inversely proportional to the temperature, and needs to be below 1 aF to make the device operational at room temperature.
The island capacitance is a function of the QD size, and a QD diameter smaller than 10
=== CMOS compatibility ===
[[File:SETFET schematic.jpg|thumb|Hybrid
The level of the electrical current of the SET can be amplified enough to work with available [[CMOS]] technology by generating a hybrid
The EU funded, in 2016, project IONS4SET (#688072)<ref>{{cite web|url=http://www.ions4set.eu|title=IONS4SET Website|access-date=2019-09-17}}</ref> looks for the manufacturability of
== See also ==
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