 Home Mathematics  # THE VACUUM TRANSISTOR

The monumental technological progress that has been achieved in the second half of the 20th century and the first two decades of the 21st century can be attributed to a large extent to the micro- and nano-electronics technologies and these, in turn, to the invention of the transistor and the subsequent invention of the integrated circuit whose inventors won Nobel prizes for their work. The transistor was a transformation in solid state form of the vacuum triode or vacuum valve, although the term “transformation” does not exactly do justice to the transistor inventors. It was the possibility of the miniaturization and durability of the transistor that led to the microelectronics and nanoelectronics era and the subsequent technological development. The vacuum valve was pushed during this time into a very limited number of applications. Any transistor (there are a few types) works on the following principle: an electrode injects electrons into a channel made of a semiconductor material which travel along the channel and reach another electrode at the other end of the channel. A third electrode called a gate may or may not impose a barrier to the electron flow depending on the voltage attached to it. Depending on this voltage, electrons may flow or not flow in the channel. The current involved is a drift-diffusion current over the barrier, the sort of current that we discussed in chapter 3. FIGURE 6.16 Various forms of the newly invented vacuum transistor. The flow of electrons in space is shown by an arrow.

Very recently, a so called vacuum transistor has been manufactured by a few laboratories which have replaced the channel with vacuum. Furthermore, the injection into vacuum is accomplished by field emission. So this transistor is a true revival of the vacuum valve with the major difference that it is miniaturized, the cathode-anode distance is several nanometers, and modern lithography is used for its production. Several versions of the vacuum transistor made of semiconductor materials are shown in figure 6.16. The advantage of electron flow in vacuum should be obvious: there are no collisions with a lattice. Furthermore, the device is immune to radiation.

A simplification or idealization of the vacuum transistor for which calculations have been performed by the author is shown in figure 6.17, where two pin-like metallic structures FIGURE 6.17 Picture of a typical vacuum transistor. For the purpose of efficient calculations, the emitting tip is simulated by a stack of spheres as shown in the inset.

act as anode and cathode while a planar metal gate lies above them. The structure as a whole seems to be standing in vacuum while the anode and cathode in real devices lie on a Si/ Si02 substrate, as shown in figure 6.16b. For the purpose of simplifying calculations we will ignore the difference in the dielectric constant of the substrate and vacuum (er (Si02) = 3.9). This allows us to illuminate the functioning of the vacuum transistor with a relatively simple theory that is accurate enough.

The current Id between the pin-like structures, the cathode and the anode, is determined by the tunnelling-forbidden region, adjacent to the cathode. This region is created by the applied voltage Vd between anode and cathode but also by the voltage VG applied to the gate. Under the simplification discussed above, the electrostatic potential around the cathode can be evaluated if we approximate the anode and cathode as piles of spheres as shown in the inset of figure 6.17. Since the potential of an isolated sphere uUph- , where i denotes the site on which the sphere is located, is a well-known problem, the total potential in the space between anode and cathode is just the sum over i of these u^spi, plus the linear one created by the planar gate. More details can be found in .

If the electrostatic potential, a solution of the Laplace equation, is evaluated by this indirect method, the tunnelling potential energy Uton(r,0,(p) can be calculated according to the prescription where uL is the solution of the Laplace equation and wim is the image potential usually taken to be the spherical image potential. Note that we have changed the symbol for the potential energy from V to U since we will need V to denote voltages on electrodes in accordance with standard practice in transistor nomenclature. Due to presence of the gate plane, the system no longer has rotational symmetry, so in spherical coordinates the tunnelling potential energy is a function of

G to the gate is shown in figure 6.18. It can be seen that the gate voltage VG diminishes significantly both the height and length of the barrier, thus allowing a lot more current to flow between anode and cathode.

This is shown in figure 6.19 where we plot Id with respect to VD for various values of VG. The current ID can be obtained by a surface integration of the current density /(0,ф) at temperature T = 0. The latter has been calculated numerically by obtaining the transmission coefficient along radial lines as outlined in the previous section. It can be seen that VG has a catalytic effect on the current Id- The latter increases by orders of magnitude with a small increase by 1-2 volts in VG. Note that due to the logarithmic scale on the current axis in figure 6.19, the (0,0) point cannot be portrayed. It is rather early (2020) to decide whether the recently developed vacuum transistor will play any role in the field of nanoelectronics. We have presented it here mainly because it illustrates in more concrete term the “transistor action”, i.e. the control of the flow of current in a channel by a voltage applied to a third electrode. The whole area of present-day nanoelectronics mainly relies on the operation of FIGURE 6.18 Potential energy outside the emitter of a vacuum transistor. The gate voltage decreases both the length and height of the barrier.

the MOSFET (the Metal Oxide Semiconductor Field Effect Transistor), a Si-based device. Also, by an increasing degree, possible successors to the MOSFET based on III—V or other materials play a significant role. The operation of the semiconductor FETs is not as simple as the vacuum transistor, so we allocate a separate chapter for the Si devices and another for what is usually termed “post Si” devices or technologies. These form the content of the next two chapters. FIGURE 6.19 Current-voltage ID -VD characteristic (logarithmic scale) for various values of the gate voltage. The latter seems to be very efficient.

PROBLEMS

• 6.1 Obtain an equation for g2, i.e. the third order exponent, in the WKB expansion of a wavefunction.
• 6.2 Use the polynomial expansion 6.35b for the (correcting) field emission function u(y) to estimate the percentage average deviation from the straight line of the Fowler-Nardheim plot of the triangular barrier.
• 6.3 Derive equations 6.46,6.47 for the current densities JA and Jc in the allowed regions in the 3-dimensional WKB problem.
• 6.4 Show that the enhancement factor of a protrusion in the form of a hemisphere on a cathode is only 3.
• 6.5 Prove that the enhancement factor у of a sphere inside a parallel plate capacitor connected with a very thin wire to the cathode (i.e. equipotential to the cathode) is where h is the height of the centre of the sphere above the cathode, R is the radius of the sphere and c is a constant of the order of 1.

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• 7. M.S. Tsagarakis and J.P. Xanthakis AIP Advances 9, 105314(2019)

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