Single-electron
Transistor
As Fast and Ultra-Sensitive
Electrometer
Francesco Maddalena
Abstract
The single-electron transistor (SET) is a nanodevice that can control the transport of single elementary charges on and off a metallic island. It can also function as transistor similarly to a nowadays FET. The principles of the operation of the SET is determined by the Coulomb blockade, an energy barrier that determines the current flow through the device and the charge placed on the metallic island. Regulating the gate charge of the device can modify the Coulomb blockade. The SET can also be used as an ultra-sensitive electrometer in DC ad RF mode. Theoretical calculations show a charge sensitivity h values lower than for the SET and experimental research gives values of . The experimental value for the SET is 1000 times better than the field-effect transistor used as an electrometer. The SET can thus be used as ultra-sensitive electrometer and will be used in the future in the study of charged nanoscale systems.
Contents
Page
1. Introduction
/ 32. Principles of the Single Electron Transistor / 4
2.1 The Coulomb Blockade / 4
2.2 Performance of the SET / 5
2.3 Parameters that Influence the Performance of the SET / 8
3. Ultra-sensitive Charge Measurements with the SET / 10
3.1 The SET as electrometer / 10
3.2 Noise that can affect the SET / 12
3.3 The Sensitivity of the SET / 14
4. Conclusions / 17
5. References / 18
16
1. Introduction
Miniaturization has brought today’s electronic devices in scale very close to the size where quantum phenomena will dominate the operation of the device hence changing its whole properties. However quantum effects are not necessarily a downside to the electronic industry since they can be used to create new devices. Science has already entered new fields of technology that will be the future of modern electronics. One of these fields is nanoscience and the design of nano-scale devices called nanodevices. Nanodevices can often perform the same tasks as today’s micro-scale devices such as FETs, yet the physical principles under which they work are very different. One of the most fascinating nanodevices is the single electron transistor (SET). This device exploits the quantum mechanical phenomenon of tunneling and it can perform as a switch or as an amplifier, similarly to common FETs, but it can also control the transport of single electrons. It is thus interesting to examine the working an properties of this device and to look into what its best applications could be.
The SET, similarly to the FET, can be used in its amplifier mode as an electrometer to measure the charge or charge variations of a specific system. Devoret and Schoelkopf (Nature 2000) claim that the SET is actually very suited to be an electrometer having a very high energy and charge sensitivity that comes close to the theoretical quantum limit. A very interesting question is if it is indeed true or feasible that the SET can reach such high values of charge sensitivity or if in the experimental setting the SET is not as a good candidate for charge measurements after all. The objective of this paper is to answer this question, if indeed the SET is an ultra-sensitive electrometer and if the optimal value set by Devoret and Schoelkopf is actually a feasible reality.
This paper will first treat the principles of the working of SETs. Then the application of the SET as ultra-sensitive electrometer will be discussed. In here the review by Devoret et. al. (Nature 2000) will be taken in consideration and viewed critically, focusing on the charge sensitivity of the device. Finally a conclusion will be given with some future considerations over the applications of the SET.
2. Principles of the Single Electron Transistor
2.1 The Coulomb Blockade
The single-electron transistor consists of a metallic island, placed between two tunneling junctions connected to a drain and a source and has a gate electrode as in a normal field-effect transistor. The tunneling junctions are simply a thin (<10 nm) oxide layer between the island and the electrodes. Quantum dots have also been used as islands for the SET. The schematics of the SET are given in figure 1. Each tunneling junction in the SET has intrinsic tunneling resistance and capacitance (parallel to each other). Yet, before we can fully understand the working of a SET we must first understand the concept of Coulomb blockade.
Fig. 1: Left: schematic circuit representation of the single-electron transistor. Right: a more realistic representation of what the ‘core’ of the single-electron transistor looks like.
The island of the single-electron transistor, even if very small (nanometric scale) still contains a very large number of electrons (109). Yet, through tunneling, one can add or subtract electrons from the island charging it either negatively or positively. The extra electrons that charge the island are called excess electrons and their number is designed by n. The number of excess electrons can also be negative, meaning that electrons have been removed leaving a positive charge on the island (one could talk of excess holes in this case). The presence of excess electrons affects the electrostatic energy of the system, which depends on the charging energy of the SET:
(1)
where Qisl is the charge on the island, n the number of excess electrons, e the charge of one electron and CS the total capacitance of the island which is equal to: (CG, CL and CR are the gate capacitance and the intrinsic capacitances of the left and right tunneling junctions respectively). The energy scale applied when working with the SET is usually defined on the charging energy itself and the unit taken is usually: . The energy does not only depends on Qisl, but also on the charge induced by the gate, the gate charge QG=VGCG where VG is the gate voltage. The electrostatic energy of the system is equal to , where n is the number of excess electrons of the island and ng the number of elementary gate charges. The expression for the electrostatic energy of the system then becomes:
(2)
This energy determines if tunneling through a junction is forbidden or allowed: if the adding of an extra excess electron causes the energy of the system to increase then tunneling will be energetically forbidden and the Coulomb charging energy will act as a blockade. This is known as the Coulomb blockade.
Two cases are thus possible. The first case is the one when we consider n excess electrons on the island and tunneling of one electron would cause the energy of the system increases (see also equation (2)). The system having n+1 excess electrons on the island will be then energetically forbidden. No tunneling will occur through the junctions. This is the Coulomb blockade that, in this case, said to be active. In the second case tunneling of an extra electron on the island will lower the energy of the system, hence there will be no Coulomb blockade and tunneling will happen adding an excess electron to the island. The same principle applies if we wish to subtract electrons from the island, charging it positively. The drain-source voltage, VDS, determines the energy of the electrons before the junction. When this energy is higher than the Coulomb blockade, the electrons will overcome the blockade and tunneling will occur. The height of the blockade is determined by the number of excess electrons on the island and the gate charge.
2.2 Performance of the SET
The SET controls the flow of electric current between the drain and the source through the gate electrode similarly to a FET. Unlike in a normal FET however the behavior of the SET (whether if it allows or not a current between the source and the drain) depends on the transport of single elementary charges and the gate voltage controls this via the Coulomb blockade. When the Coulomb blockade is overcome one electron will tunnel from the source to the island, adding one extra excess electron. Similarly a tunneling process will occur from the island to the drain. The energy of the source (Esource) and drain (Edrain) depend on their respective potentials, VS and VD. The change of the charging energy of the system if we go from n to n+1 excess electrons will depend on the gate voltage VG according to:
(3)
So the adding of an excess electron on the island will be either favorable or unfavorable depending on the gate charge which on its turn depends on the gate voltage. We can se from equation (3) that if the gate charge is equal to integer values of elementary charge e the Coulomb blockade will be active and there will be no conduction. That is because the system has minimum energy when the island has a well defined number of electronic charges and the tunneling of an electron will only increase the global energy of the system (see figure 3a). The Coulomb blockade will be active and no tunneling of electrons in or out the island will occur (figure 2a). The transistor will be in the conducting state (meaning that tunneling will lower the energy of the system) if the gate charge is equal to values that are half integers of the electron charge, that is: where N is an integer (recalling that e is the charge of one electron). Then the system have an energy minimum which is in between two states with well-defined elementary charges (see figure 3b) on it (where N is an integer number). This will cause a cascade of tunnel events, involving the two junctions sequentially, giving rise to a current between the drain and the source. This can be seen graphically in figure 2b. The another way to view the energy picture of the blockade and conduction modes of the SET is given in figure 3.
Figure 2: Graphical representation of the Coulomb blockade in a) blocking state and b) in conducting state for the single electron transistor. The transistor will be in the conducting state if the gate charge is equal to e(2N+1)/2 where N is an integer. If the gate charge is equal to integer values of e the Coulomb blockade will be active and there will be no conduction. In the picture N1 and N2 denote the numbers of electrons having tunneled through the junctions and p is an integer. (Source: Devoret & SchoelKopf, Nature 2000)
Figure 3: Charging energy against the charge present on the island n. If the charge is integer the Coulomb blockade will be active (adding or subtracting an electron to the island will increase the charging energy). If the charge is ½ of an integer the SET will be in full conduction mode (adding an electron will not cause increase of energy).
We can now write down the voltage conditions that have to be met to make tunneling possible. First we examine the case when the gate voltage VG is zero. If there are no excess electrons on the island (n = 0) and no gate voltage then the threshold voltage, VDS,T, must be equal to:
(4)
where e is the charge of one electron, CL and CR are the (internal) capacitances of the left and right tunneling junction respectively, CG is the gate capacitance and CS is the total capacitance. The threshold voltage is an important quantity. It is the minimal drain-source voltage necessary if we want and electron to overcome the Coulomb blockade, hence have a current flow through our SET. That means that to achieve current flow the potential energy eVDS must be higher than the charging energy e2/2CS. If we want to lower the threshold voltage down to zero we can achieve that by increasing the gate voltage and thereby lowering the Coulomb blockade. The voltage between drain and the source, VDS, depends on the gate voltage VG itself (assuming that VDS is close to zero):
(5)
The current will flow only if the voltage described in (5) is equal or higher than the threshold voltage given in (4). Hence the gate voltage needed to lower the threshold voltage to zero is equal to:
(7)
Looking at equation (7) we see that it meets the condition mentioned above that the gate charge (QG=VGCG) must be one half the value of the charge of the electron in agreement with the conditions we set above for current conduction through the SET. From the above relations we can derive the IV-characteristics of the SET that are depicted in figure 4. The characteristic, as it can be seen from the picture, is symmetrical around the zero point. When VG = 0 the current will start flowing only after the threshold voltage has been reached. Between VDS=e/2CS and VDS=-e/2CS there is a gap where no current will flow. This is the so-called Coulomb gap and it is the direct consequence of the Coulomb blockade. When VG = e/2CG the threshold voltage is equal to zero and the IV-characteristic presents no gap since there is no coulomb blockade.
Fig. 4: IV-characteristics of the SET. The DS-current is plotted against the DS-voltage.
The current IDS is given by the flow of electrons from the source to the drain (or vice versa for reversed bias). The electrons will tunnel from the source to the island to the drain, creating a current. One must keep in mind that even if the current is flowing, the number of excess electrons on the island can remain constant. Electrons will tunnel in the island from the source, also electrons will tunnel from the island to the drain, keeping the number of excess electrons unchanged, but creating a current. This process is not the same as co-tunneling, where electrons tunnel from the source to the island and from the island to the drain simultaneously. +