1. Single Electron Devices
Transistors
Single-electron Transistors

2. Transistors
What are transistors?
How do they work?
A transistor is a device that functions only in one direction, in which it draws current from its load resistor. The transistor is a solid state semiconductor device which can be used for amplification, switching, voltage stabilization, signal modulation and many other functions. It acts as a variable valve which, based on its input current (BJT) or input voltage (FET), allows a precise amount of current to flow through it from the circuit's voltage supply.

3. Fig 1. NPN Transistor using two diodes and connecting both anodes together

4. One cathode is tied to common (the emitter); the other cathode (the collector) goes to a load resistor tied to the positive supply. For understanding, the transistor is configured to have the diode signal start up unimpeded until it reaches ~ 0.6 volts peak. At this point the base voltage will stop increasing. No matter how much the voltage applied from the generator increases (within reason), the "base" voltage appears to not increase. However, the current into that junction (two anodes) increases linearly: I = [E - 0.6]/R. 

5. Fig 2. Graph of how the base voltage acts with increasing input voltage.
As the voltage increases from 0 to 0.5 volts there is no current. However, at 0.6
a small current starts to show which is drawn by the base. The voltage at the base
stops increasing and remains at 0.6 volts, and the current starts to increase along
with the collector current. The collector current will slow down at some point until it
stops increasing. This is where saturation occurs. If this transistor was being used
as a switch or as part of a logic element, then it would be considered to be switched
on.

6. Single-electron Transistor
- what problem does it help solve?
- what is its operation?

7. Problem of Making More Powerful Chips
Intel co-founder Gordon Moore that the number of transistors on a chip will approximately double every 18 to 24 months. This observation refers to what is known as Moore’s Law.
This law has given chip designers greater incentives to incorporate new features on silicon.
The chief problem facing designers comes down to size. Moore's Law works largely through shrinking transistors, the circuits that carry electrical signals. By shrinking transistors, designers can squeeze more transistors into a chip. However, more transistors means more electricity and heat compressed into an even smaller space. Furthermore, smaller chips increase performance but also compound the problem of complexity.

8. Fig 3. A single-electron transistor
To solve this problem, the single-electron tunneling transistor - a device that exploits the quantum effect of tunneling to control and measure the movement of single electrons was devised. Experiments have shown that charge does not flow continuously in these devices but in a quantized way.
Fig 3. A single-electron transistor

9. The single-electron tunneling (SET) transistor consists of a gate electrode that electrostaticaly influences electrons traveling between the source and drain electrodes. The electrons in the SET transistor need to cross two tunnel junctions that form an isolated conducting electrode called the island. Electrons passing through the island charge and discharge it, and the relative energies of systems containing 0 or 1 extra electrons depends on the gate voltage. At a low source­drain voltage, a current will only flow through the SET transistor if these two charge configurations have the same energy
The SET transistor comes in two versions that have been nicknamed metallic and semiconducting.

10. SET Transistor Function
The key point is that charge passes through the island in quantized units. For an electron to hop onto the island, its energy must equal the Coulomb energy e2/2C. When both the gate and bias voltages are zero, electrons do not have enough energy to enter the island and current does not flow. As the bias voltage between the source and drain is increased, an electron can pass through the island when the energy in the system reaches the Coulomb energy. This effect is known as the Coulomb blockade, and the critical voltage needed to transfer an electron onto the island, equal to e/C, is called the Coulomb gap voltage.

11. What is this “island”? Fig 4
(a) When a capacitor is charged through a resistor, the charge on the capacitor is proportional to the applied voltage and shows no sign of quantization. (b) When a tunnel junction replaces the resistor, a conducting island is formed between the junction and the capacitor plate. In this case the average charge on the island increases in steps as the voltage is increased (c). The steps are sharper for more resistive barriers and at lower temperatures.

12. Here n1 and n2 are the number of electrons passed through the tunnel barriers 1 and 2, respectively, so that n = n1 - n2, while the total island capacitance, C∑, is now a sum of CG, C1, C2, and whatever stray capacitance the island may have.
Left: Equivalent circuit of an SET
Center: Energy states of an SET. Top Coulomb blockade regime, bottom transfer regime by application of VG=e/2CG
Right: I-(Va )-characteristic for two different gate voltages. Solid line VG= e/2CG, dashed line VG =0

13. The most important property of the single-electron transistor is that the threshold voltage, as well as the source-drain current in its vicinity, is a periodic function of the gate voltage, with the period given by
∆Qe = e, ∆U= e/C0 = const.
The effect of the gate voltage is equivalent to the injection of charge Qe = C0U into the island and thus changes the balance of the charges at tunnel barrier capacitances C1 and C2, which determines the Coulomb blockade threshold Vt. In the orthodox theory, the dependence Vt (U) is piece-linear and periodic.

14. The expression for the electrostatic energy W of the system:
W = (ne - Qe)2/2CS - eV[n1C2 + n2C1]/CS + const
The external charge Qe is again defined by Qe = C0U and is just a convenient way to present the effect of the gate voltage U.
The Coulomb blockade threshold voltage Vt as a function of Qe at T -> 0.

15. At a certain threshold voltage Vt the Coulomb blockade is overcome, and at much higher
voltages the dc I-V curve gradually approaches one of the offset linear asymptotes:
I -> (V +sin(V)´e/2C∑)/(R1+R2). On its way, the I-V curve exhibits quasi-periodic oscillations of its slope, closely related in nature to the Coulomb staircase in the single-electron box, and expressed
especially strongly in the case of a strong difference between R1 and R2.
Source-drain dc I-V curves of a symmetric transistor for several values of the Qe, i.e. of the gate voltage

16. Applications of SETs Quantum computers Microwave Detection
1000x faster
Microwave Detection
Photon Aided Tunneling
High Sensitivity Electrometer
Radio-Frequency SET

17. Tunneling Probability
Fabrication of the
Ti/TiOx SET
Barrier Height 285meV
Er = 24
18nm wide junction

18. Choose doping of 10^18 and 10^23
W=[2Er*Eo*Vbi/eNd]^1/2
W(n=10^18)= 2.75e-6 cm^-3
W(n=10^23)= 8.69e-9 cm^-3
To find E field divide Barrier height by Width
E(n=10^18)= 1.03e5 V/cm
E(n=10^23)= 32.8e6 V/cm

19. Substituting E into Tunneling Prob Equation
T= exp{ (4* (2m*)^1/2*phi^3/2 )/ (3eEhbar)}
T(n=10^18) = exp{-3187}=0
T(n=10^23) = exp{ }=
Conclusion:
At higher dopings, the tunneling probability starts to get better and electrons can move across the junction.

20. Main Problems with SETs
Operation at Room Temp
Capacitor Size
Fabrication
Chemical Fabrication
Couloumb Islands
Tunneling junctions
Gate between substrate and Coulomb islands
Charge Offset
1 electron at a time

21. Single Electron Devices
Group Members
Jonathan Sindel
Latchman,Kamivadin
Wayne Lyon