E.E. 820

ELECTRICAL MATERIALS SCIENCE

TITLE: FIELD ASSISTED THERMIONIC

EMISSION, FIELD EMISSION, AND

APPLICATIONS

Name: ADEAGBO EMMANUEL BAMISE

Student Number: 11192805

Date: December 21, 2015.

Lecturer: Professor Safa Kasap

1. INTRODUCTION

Vacuum are generally viewed as electrical antiques but their basis principle of operation, which is the emission of electrons from a heated cathode, still finds application in various electrical gadgets such as the cathode ray tubes, X-ray tubes and various RF microwave vacuum tubes, such as triodes, tetrodes, klystron, magnetron, and travelling wave tubes and amplifiers. [1] These pose a need to study and examine how electrons are emitted from the surface a heated metal. This text gives a brief explanation of the concept of thermionic emission, field assisted thermionic emission which is also known as Schottky emission, field emission also known as cold emission. The scope will also explain briefly the applications of these concepts.

2. THERMIONIC EMISSION: RICHARDSON – DUSHMAN EQUATION

Thermionic emission is one of the fundamental emission processes in modern electronics and it is the thermally induced flow of charge carriers from a surface or over a potential-energy barrier. This occurs because the thermal energy given to the carrier overcomes the work function of the material. Metals which have been demonstrated to contain mobile electrons which gives them the ability to conduct electric current although it is worth noting that most electrons in the metals, particularly the core electrons closest to the nucleus, are tightly bound to individual atoms; it is only the outermost valence electrons that are somewhat free.[2] So when a metal is heated, the electrons become more energetic as the Fermi-Dirac function extends to higher temperature. Some of these electrons now have sufficiently enough energies to escape or leave the metal and become free. Although this phenomenon has a limitation in the sense that the free electrons accumulate outside the metal forming a form of clouds of electrons which now prevents more electrons from leaving the metal in other words, emitted electrons leave a net positive charge behind which tends to pull the electrons in. One of the implications of this is that the ‘lost’ electrons needs to be replaced and the emitted electrons also must be collected which is done most conveniently using the vacuum tube arrangement in a closed circuit as shown in Figure 2.1a.[1]

Figure 2.1 [1]

Although not all the electrons that have energy above the vacuum level can escape. They also need to be in the right direction. This is because in a planar barrier, the lattice cannot take up momentum, except in the Bragg condition where a reciprocal lattice vector is exchanged. The momentum of the electron parallel to the surface is conserved that is pp = pp’. The planar barrier acts on the momentum transversal to the surface, pt, the energy outside is reduced by the barrier,

E’ = E – I = (1/2m)(pp2 + pt’2). (2.1)

The transverse momentum inside is pt = p cos (q). [3]

Thus the electron is refracted by the surface. Only electrons which have sufficient transverse momentum can escape to vacuum:

pt2/2m = (p cos(q))2/2m = E cos2(q) > I (2.2)

This defines an "escape cone" of angle:

qmax = cos-1 (I/E)½ = cos-1(I/(E’+I))½ (2.3)

Figure 2.2

This means that when q < qmax the electrons suffer total internal reflection. [3] The vacuum inside the tube ensures that the escape electrons do not collide with the air molecules and become dispersed with some even being returned to the cathode by collision. This makes the vacuum essential. The current due to the flow of emitted electron from the cathode to the anode depends on the anode voltage as shown in Figure2.1b. The current increases with the anode voltage until all emitted electrons are collected by the anode and the current saturates at sufficiently high voltages. [1]

2.1 RICHARDSON – DUSHMAN EQUATION

The emitted electrons from the hot filament cathode are drawn to the anode and the current saturates. At this point, the anode current is controlled primarily by the filament temperature and practically independent of the potential. Richardson’s law explains the current density Jx [A/m2] of thermally escaped electrons in the direction perpendicular to heated metal.[4] Considering the flow of thermal electrons with charge q in the x direction, the current density of the electrons at the empty space in front of the metal plate is given as Jx = ∫qn(E)vx(E)dE, (2.4) where n(E) is the density of electrons in units of [J-1m-3]. vx(E) is the speed of electrons in x direction. The density of particle from statistical physics is known as n(E) = g(E)f(E), (2.5) where g(E) is the density of state and f(E) is the probability for a taken state with energy E. From Fermi-Dirac statistic,

g(E) = 8√2𝜋

ℎ3𝑚3/2√𝐸 (2.6)

and f(E) = 1

1+𝑒𝑥𝑝(𝐸−𝐸𝐹

𝑘𝑇) (2.7)

Figure 2.3 Free electrons inside metal are treated as in potential well at constant confining potential U. They fill all available states up to the Fermi energy EF. Electrons with the highest energies lack exactly W (work function) to leave metal.

where h and k are Planck's and Boltzmann's constants. EF is Fermi energy that is the highest energy of electrons in constant confining potential U (Figure 2.3). Only electrons with the highest energies E >> EF can escape from metal, so f(E) approaches Boltzmann's distribution:

𝑓(E) = 𝑒𝑥𝑝 (𝐸−𝐸𝐹

𝑘𝑇) (2.8)

The minimum velocity needed to overcome the potential barrier EF + ф for the electron to escape is given by;

1

2𝑚𝑣𝑥

2 = 𝐸F + ф (2.9)

Solving equation (2.9) and substituting the solution and equations (2.5) to (2.8) into (2.4) and solving the resulting equation to yield;

𝐽 = 𝐵0𝑇2𝑒𝑥𝑝 (−ф

𝑘𝑇) (2.10)

where Bo = 4πemek2/ h 3. Equation 2.10 is called the Richardson–Dushman equation, and Bo is the Richardson–Dushman constant, whose value is 1.20 × 106 Am−2K−2 and ф is the work function of the metal whose value for some common elements expressed in eV is shown in table 2.1. We see from Equation 2.10 that the emitted current from a heated cathode varies exponentially with temperature and is sensitive to the work function ф of the cathode material. Both factors are apparent in Equation 2.10 [1]

Table 2.1 Work functions of some common elements expressed in units of eV

From equation (2.3), an electron has the probability of being reflected back into the material as it approaches the surface instead of being emitted over the potential barrier. As ф tends to infinity, in other words as the barrier becomes very large, the electrons are totally reflected and there is no emission. The thermionic emission equation can be modified, taking the reflection into consideration, as;

𝐽 = 𝐵𝑒𝑇2𝑒𝑥𝑝 (−ф

𝑘𝑇) (2.11)

where Be = (1-R)Bo is the emission constant and R is the reflection coefficient. The value of R will depend on the material and the surface conditions. For most metals, Be is about half of Bo, whereas for some oxide coatings on Ni cathodes used in thermionic tubes, Be can be as low as 1 × 102 Am−2 K−2. [1] It may be important to note that the term thermionic emission may be used more generally to indicate the flow of charge carriers, either electrons or ions, over a potential barrier. Even for standard thermionic emission, it should be cautioned that the work function depends critically on surface conditions. For example, surface pollution can dramatically change it. [5] 3. FIELD ASSISTED THERMIONIC EMISSION (SCHOTTKY EMISSION)

This is basically an extension of the thermionic emission phenomenon. In thermionic emission, the electrode must normally be at a high temperature which makes the electron to escape from the surface of the electrode having overcome the potential barrier. However, electrons can still be emitted at room temperature thermionically if a sufficiently high electric field act at the surface of the electrode to remove the electrons as they emerge.

The application of positive voltage to the anode with respect to the cathode makes the electric field at the cathode to enhance the thermionic emission process by lowering the potential energy barrier ф. This phenomenon is known as the Schottky effect. [1]

The theorem of image charges in electrostatics which says that an electron at a distance x from the surface of a conductor possesses a potential given by equation (3.1). This PE explains why the electron is pulled in by the effective positive charge left in the metal.

PEimage(x) = - 𝑒2

16𝜋𝜀𝑜𝑥 (3.1)

where 𝜀𝑜is the absolute permittivity.

Figure 3.1 (a) PE of the electron near the surface of a conductor. (b) Electron PE due to an applied field that is between cathode and anode. (c) The overall PE is the sum

The image charge theorem explains that an electron at a distance x from the surface of a conductor experiences a force as if there were a positive charge of +e at a distance 2x from it. The force is e2/[4πεo(2x)2] or e2/[16πεox2]. Integrating the force gives the potential energy in Equation 3.1 [1]

PE inside the metal is defined to be zero but since we are also considering the emitted electrons from the surface of the metal, Equation 3.1 needs to be modified. As shown in Figure 3.1athat the image PE varies with x and far away from the surface, the PE is expected to be (𝐸F + ф) so equation 3.1 becomes;

PEimage(x) = (EF + ф) - 𝑒2

16𝜋𝜀𝑜𝑥 (3.2)

This implies that PEimage(x) ranges from 0 to (𝐸F + ф) along Equation 3.1 which agrees with the thermionic emission analysis because the electron must overcome a PE barrier of (𝐸F + ф) to escape. [1] Figure 3.1b gives the variation of the PE gradient just outside the surface of the metal expressed as;

PEapplied(x) = -exE (3.3)

where E is the applied field and is assumed to be uniform. The total PE(x) of the electron outside

the metal is the sum of Equations 3.2 and 3.3 as depicted in Figure 3.1c,

PE(x) = (EF + ф) - 𝑒2

16𝜋𝜀𝑜𝑥 - exE (3.4)

The implication Equation 3.4 is that PE(x) outside the metal will not reach (𝐸F + ф) and the PE barrier against thermal emission is reduced to (𝐸F + ф eff), where ф eff is a new effective work function taking the effect of the applied field into consideration. ф eff is determined by differentiating Equation 3.4 and setting it to zero.

ф eff = ф - (𝑒3𝐸

4𝜋𝜀𝑜)1/2 (3.5)

Equation 3.5 verified the lowering of the work function by the applied field which is known as Schottky effect. So the current density given by Richardson-Dushman equation is now modified

using ф eff instead of ф,

𝐽 = 𝐵𝑒𝑇2𝑒𝑥𝑝 (−ф− 𝛽𝑠𝐸1/2

𝑘𝑇) (3.6)

where 𝛽𝑠= [e3/4𝜋𝜀𝑜]1/2 is the Schottky coefficient, whose value is 3.79 × 10-5 (eV/√Vm-1).[1]

4. FIELD EMISSION(COLD EMISSION)

So far we have examined how electrons can be emitted from the surface of a hot metal having acquire sufficient energy in order to overcome the natural barrier which is known as the thermionic emission explained by Richardson-Dushman equation. And has gone further to discuss the effect of applying the electric field to reduce the barrier in order to emit electron in the phenomenon known as Schottky effect or field assisted thermionic emission.

However, we will move another step further in studying the effect of increasing the electric field because some materials cannot withstand the heat in thermionic effect as their melting temperature is very low. Also some other materials do not have reasonably low work function which means more thermal energy is required to enhance emission of electrons in thermionic emission.

So increasing the magnitude of the electric field, for example, 𝐸 > 107 makes the PE(x) outside the metal surface to bend sufficiently steeply resulting in a narrow barrier. As a result of this, there is a distinct probability that an electron at an energy EF will tunnel through the barrier and into the vacuum, as shown in Figure 4.1. Tunneling likelihood depends on the effective work function or the height of the effective barrier to overcome and the width xF of the barrier at the energy level EF. However, because the tunneling of the electron doesn’t depend on temperature, this concept is known as field emission or cold emission. (Cold because it is temperature independent). The tunneling probability P in elementary quantum physics which depends on xF

and ф eff as shown in Equation 4.1.

P ≈ exp [−2(2𝑚𝑒фeff)1/2𝑥𝐹

ℎ] (4.1)

xF can easily be expressed from Figure 4.1 because at x = xF, PE(xF) is at the same level as EF. At this point, the second term of Equation 3.4 becomes negligible putting x = xF and PE(xF) = EF yields ф = e𝐸xF. Substituting xF = ф/e𝐸 in Equation 4.1 yields the tunneling probability P in Equation 4.2.

P ≈ exp [−2(2𝑚𝑒фeff)1/2ф

𝑒ℎ𝐸] (4.2)

Equation 4.2 is known as the field assisted tunneling probability which is the representation of the probability that an electron will tunnel out from the metal at 𝐸F. The current density J, which is the number of electron moving toward the surface per unit area, is expressed as the product of the number of electron moving toward the surface per second per unit area, which is the electron flux, and the probability that they will tunnel out.

The result gives the Fowler-Nordheim equation, as shown in Equation 4.3, which depends on the exponential field and is based on the following assumptions:

I. The metal is assumed to obey sommerfield free electron model with Fermi-Dirac statistics.

II. The metal surface is taken to be planar, that is the one-dimensional problem is considered. This assumption is accurate because in most cases the thickness of the potential barrier ф of 107 - 108 V/cm is several orders of magnitude less than the emitter radius. Thus, the external field can be taken to be uniform along the surface.

III. The potential within the metal is considered constant. The potential barrier outside the metal is entirely regarded as due to the image force with the externally applied electric field having no effect on the electron states inside the metal.

IV. The calculation is performed for the temperature T = 0K. [10]

Jfield-emission ≈ 𝐶𝐸2𝑒𝑥𝑝 (−𝐸𝑐

𝐸) (4.3)

where C and Ec are temperature-independent constants but depend on the work function of the metal.

𝐶 = 𝑒3

8𝜋ℎф and 𝐸𝑐 =

8𝜋(2𝑚𝑒ф3)1/2

3𝑒ℎ (4.4)

Figure 4.1 (a) Field emission is the tunneling of an electron at an energy EF through the narrow PE barrier induced by a large applied field. (b) For simplicity, we take the barrier to be rectangular. (c) A sharp point cathode has the maximum field at the tip where the field emission of electrons occurs

It should be noted that one of the major contributing factor to the concept of field emission is high electric field, [7] which can be enhanced by shaping the cathode into a cone with a sharp point where the field is maximum and electron emission occur as shown in Figure 4.1c. [1] This tip has apex radius of curvature ranging from tens of angstroms to several microns while the field E is the effective field at the tip of the emitter. [7][1]

As an extension of Equation 4.3, the emission current or the anode current IA can be expressed in Equation 4.5 because E ∝ VG, where VG is the potential difference between the gate and the cathode (or emitter) which controls field emission.

IA = 𝑎𝑉𝐺2𝑒𝑥𝑝 (−

𝑏

𝑉𝐺) (4.5)

where a and b are constants that depend on the particular field-emitted structure and cathode material.

Figure 4.2 (a) (b)

(a) Emission (anode) current versus gate voltage. (b) Fowler–Nordheim plot that confirms field emission

Figure 4.2a shows how IA depends on VG as there is a very rapid increase with the voltage after the threshold voltage, which is around ∼ 45V in Figure 4.2a, are reached to start emitting electron. Once emission is in full operation, IA vs VG follows the Fowler-Nordheim emission and the plot of ln(IA/VG

2) vs 1/VG gives a straight line as shown in Figure 4.2b. [1]

However, in field emission, high emission current densities can be predicted, which is one of the most remarkable result of the theory. These are possible because;

A very large deposit or reservoir of electrons exist near the Fermi level of a metal and

The emitter does not require to be heated, irradiated or otherwise excited by external energy source in order to maintain the emission process because electrons exit the metal by tunneling.[7]

The following resume qualify field emission for its several applications in modern micro and nano-electronics:

1. Field emission (FE) cathode is the most pointed, most bright, and most monochromatic of electron emitter. Usual dimensions of a field electron emitter lie within tenths of a micrometer. With special methods, one can reduce the linear dimension of an emitting spot down to 106-10-7 cm, or even make the emitter atomically sharp in some particular cases.

2. Tip FE cathodes are employed in the industrially manufactured transmission and scanning electron microscopes of super high resolution, in Auger-electron spectrometers, and in the electron holography.

3. A new approach to the development of electron optical systems is being developed based on the vacuum microelectronics techniques, which permits forming of the cathode and of electron optics elements in a single block. [10]

5. APPLICATIONS

Both the field assisted thermionic emission and the cold emission primarily find applications where semiconductor devices fail or are not practical to use. These are areas where large power is required with respect to the size of the device. Field assisted thermionic emission and field emission have quite a number of applications in radio and television broadcasting, radars for space and aerospace applications, microwave communications, x-ray tubes and portable x-ray generators, microwave amplifiers, field emission microscopy, parallel electron beam microscopy, nanolithography, electronic holography, vacuum micro and nano-electronics and field emission displays (FEDs). [1][10]

As an example, vacuum micro-electronics (VME) is a new field in micro and nano-electronics that has been developed during the next few decades and has applications in the production of flat low-voltage displays of high brightness and high resolution based on field emission array (FEA), Auger electron spectrometer and tunnel microscopy, production of active elements for integrated circuits (diodes and transistors) and development of different types of new ion sources. [8]

Some of the latest achievement in VME which is one of the major applications of field emission include:

Miniature Field Emission Triodes and Amplifiers Based on Field Emission Arrays (FEAs) Microwave Devices for Millimeter Wave Amplification Nanolithography Vacuum Magnetic Sensors External pressure sensors

Mass-spectrometers with field emission cathodes Use of Field Emission in Gas Lasers

Also, field emission displays (FEDs) are thin flat display with low power consumption, quick start and a wide angle of about 1700. [1]

6. CONCLUSIONS

So far we have been able to learn the concept of thermionic emission, field assisted thermionic emission and field emission. In thermionic emission, electrons escape or are emitted from the surface of the metal due to heating of the metal which supplies thermal energy leading to the free electrons in the metal to gain enough energy above the Fermi energy. [9]

However, when electric field is applied in form of a large positive bias voltage to the anode with respect to the cathode in order to enhance thermionic emission process in the process known as field assisted thermionic emission. This is done in order to lower the potential energy barrier ф of the metal. This is a more improved thermionic emission because some metals have low melting temperature while some have high potential energy barrier ф, so in order to compensate for this demerit in such metals, fields assisted thermionic emission concept evolved.

But both concepts still involve supplying thermal energy which brings up the question of energy efficiency, limiting the practical applications of these concepts. But field emission has a number of advantages because it is more power efficient than the thermionic emission which requires heating the cathode to high temperature. Field emission also known as cold emission in principle can be operated at high frequencies which makes it very instrumental in fast switching devices. [1]

Furthermore, field emission devices are basically attractive owing to their low mass and power budgets but do have their drawbacks which however are that they are sensitive to atmospheric contamination and must be kept under vacuum or protective gas prior to operation. This requires hermetic vessels with automatic opening mechanisms, which partially offset their advantages. In cases where these problems have been overcome, field emission devices have proven useful and simple to operate in a space environment. [8]

REFERENCES

[1] S. . Kasap, Principles of Electronic Materials and Devices, 3rd ed., no. April. McGraw Hill, 1998.

[2] O. Richardson, “5 : Thermionic Emission,” pp. 101–130.

[3] “Thermionic Emission.” http://www.virginia.edu/ep/SurfaceScience/electron.html.

[4] M. Grilj, “Thermionic emission,” no. April 2008, pp. 1–13, 2008.

[5] https://www.eng.fsu.edu/~dommelen/quantum/style_a/cboxte.html [6] D. R. Lide, CRC Handbook of Chemistry and Physics, CRC Press/Taylor and Francis, Boca Raton, FL (2008). [7] G. N. Fursey, “Field emission in vacuum micro-electronics,” vol. 215, pp. 113–134, 2003. [8] G. R. Friedrich, “Field emission devices for space applications CONTROL,” pp. 116–122, 2007.

[9] V. K. M. and R. Mehta, “Principles of Electronics,” 10 th Edition., NEW DELHI: S. CHAND & COMPANY RAM NAGAR, NEW DELHI-110 055, 2004, pp. 28–37.

[10] George N. Fursey, “Field Emission in Vacuum Microelectronics,” Kluwer Academic/Plenum

Publishers, New York, 2005, pp 1.