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Chapter Two <Fiecd~ Emission Theory 2 FIELD EMISSION

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Chapter Two <Fiecd~ Emission Theory 2 FIELD EMISSION Chapter Two <FieCd~ Emission Theory 2 FIELD EMISSION THEORY 2.1 Historical Introduction Field emission is the emission of electrons from the surface of 3 condensed phase (metal or semiconductor) into another phase (usually vacuum) under . influence of high electrostatic field. The electric field required for this process is 0.3 - 0.6 V/A° which is quite high. This phenomenon is different than extraction of electrons from the metal or semiconductor surface by photoelectric and thermal emission processes in that no energy is supplied to the electrons to overcome the potential barrier at the surface of the emitter. Electrons are defined to tunnel through the potential barrier, at the surface of the emitting material, after the applied electric field thins it. The tunneling process has no analogy in classical physics. It is rather defined as a complete quantum mechanical process. It is so because there is a probability of finding the electrons in the other side of the potential barrier though their energies are less than its height. This phenomenon is also known as cold emission as the temperature of the surface could be much lower than the room temperature while the emission takes place. The first probable indication to this phenomenon can be dedicated to the work done by Wood in 1897 on his x-ray discharge tube, in which a new form of cathode discharge was observed (as explained by him) [1]. Subsequent field emission studies have made a significant contribution to basic science since then [2-7]. These studies gave indications that electrons may be pulled out of the metal by intense electric field. The third decade of the last century witnessed the major evolution in the theories describing the field emission phenomenon [8-12]. Based on a work done by Oppenheimer in 1928 on the hydrogen atom in an external electric field, Fowler and Nordhiem (in the same year),dealt with the problem of field emission phenomenon theoretically after extending the results of Nordhiem (1928) to include the effect of an external field (image potential) [12], using Sommerfeld picture of metal and Fermi-Dirac statistics [13, 14]. They explained the field emission process quite satisfactorily and introduced an equation (known to this date by Fowler-Nordhiem equation) relating the total current density of the emitted electrons to the work function of the emitter and the electric field strength [12]. Fowler and Nordheim stated on the difference of field emission from other phenomena; 11 Chapter Two Tiedf'Emission Theory "It seems fair to conclude that the phenomenon of electron emission in intense fields is yet another phenomenon which can be accounted for in a satisfactory quantitative manner by Sommerfeld's theory". By rejecting the then proposed theory of distinction between thermions and field emitted electrons, they could explain that both types of emission is coming from a single electron band under different conditions of temperature and field. They, nevertheless, indicated that a more exact determination of their results is possible by using a well defined potential barrier and by reconsidering the approximations upon which their model was put up (one-dimensional barrier, zero temperature approximation, and the neglect of surface morphology effect on work function value). Fowler and Nordhiem main achievement was to predict weak temperature dependence in the field emission current, despite the fact that they did not determine it explicitly. W.V. Housten (1929) was the first to give an approximation for the temperature dependence of the field emission by combining the results of Fowler and Nordheim with the distribution of electrons velocities used by Sommerfeld [16]. Theoretical treatment of the field emission in the transition regions (intermediate values of temperature and applied field) was introduced by Sommerfeld and Bethe in 1933 [17], and by Guth and Mullin in 1942 [18] using analytical method and also by Nakai in 1951 [19] and by Dolan and Dyke in 1954 using numerical methods [20]. In 1956, Murphy and Good were the first to develop a mathematically correct and complete analysis of Nordheim's 1928 physical model, finding an equation valid at low to moderate temperatures [21]. This equation is now known by standard Fowler- Nordheim equation for metals. They treated the electron emission (thermal and field emission) from a unified point of view in order to establish the ranges of temperatures and fields for the two types of emission and to investigate the current in the region intermediate between thermionic and field emission. Since 1956 there have been numerous and continuous attempts to further improve the field emission theory and to relate to it the experimental observations for wide range of materials. Nevertheless, the theory was considered satisfactory at least for metallic, microscopic field emitters. 12 Chapter Two TieOf'Emission Theory 2.2 Field Emission: Theoretical Model Emission of the electron from the surface of metal or semiconductor is explained by its tunneling through the deformed potential barrier at the emitter surface. It is then supposed that no energy is supplied to the electrons inside the metal to let them overcome the potential barrier, the process that happens in the thermal and photoelectric cases. The common factor is that the details of the configuration of the surface potential are important so that the surface condition affects the emission profoundly [22]. The closely related phenomenon to field emission is field ionization consisting of electron tunneling from atoms or molecules under the action of even higher electric field than required for field emission. 2.3 Field Electron Emission in Metals In field emission the potential barrier is strongly deformed that enables unexcited electrons to leak out through it. The situation can be illustrated for metallic emitters as in figure 2.1. When a field F is applied to the surface of the metal the electrons inside, having kinetic energy E in the direction of emission, will be seeing a barrier of a height 0 + ju-Ex and a thickness (fi + jj-Ex )/Fe (where ^ is the work function value of the metal, // is the chemical potential of the conduction electrons inside the metal and e is the electron charge). If it is thin and low enough, penetration of the potential barrier will occur with finite probability. Due to applied electric field the thickness of the barrier will get reduced leading to an increase of the probability of barrier penetration. From the uncertainty principle, the required condition for field emission to take place is roughly [22]. MS)- « where m is the mass of free electron and h = h/2x is the Plankiconstant. 13 Chapter Two (Fie£d'Emission Theory Metal Vacuum level o -o- Figure2.1: The potential energy of an electron V(x) in eV as a function of its distance x (in A0) from the surface of the metal. $: work function, // : Chemical potential of electron in the metal and E is the electric field strength. The Green line is the narrower barrier due to application of higher electric field. 2.4 Fowler - Nordheim Model Fowler and Nordheim, in 1928, put up a mathematical model to relate the applied electric field (F), the work function (<f>) of the metal and the current density (/') of the emitted electrons [10]. Their calculations were closely related to previous work by Oppenheimer (in 1928) on the emission of electrons from the hydrogen atom influenced by high external electric field [11]. Their model was based upon the following assumptions: 1. The temperature of the metal isO K. So that no electrons with energies higher than Fermi energy exist in the metal and Fermi level will be the top most filled level with the electrons. The electronic charge distribution inside the metal, l therefore, follows Fermi-Dirac statistics: f(E)- \\ + [exp(£ - Ef )lkBT\] 2. Free electron approximation applies inside the metal (i.e. neglect of electron-ion and electron-electron interactions). 3. The surface is taken to be smooth and plane. It is so when irregularities are small compared to the width of the potential barrier. The tunneling occurs if the uncertainty in the position (Ax) of the electron is of the order of the barrier width (x = &/Fe); then the quantum mechanical tunneling mechanism will still be valid. 14 Chapter Two TieU. 'Emission Theory 4. Potential barrier close to the surface in the vacuum region consists of an image- force potential and a potential due to applied electric field. The image force is the Coulomb attraction towards the surface of an electron outside, due to its induced charge inside the metal. The first and the third assumptions, both combined, signify that the Fermi energy coincides with the maximum energy the electrons in the metal they can acquire and that the work function (of extended plane surface) is well defined by the energy required to raise an electron from the Fermi level to the vacuum level at infinity. If an electron of 5 A wavelength originates from the Fermi level, with work function of the metal at 0 K equals to 4.5 eV and applied field of 3.0 x 107 V/cm on the metal surface, the width of the potential barrier will be 15 A and the requirement to tunneling is fulfilled. The following definitions hold in the Fowler Nordheim Model: W = \p2 (x)/2m]+ V(x) Electron energy in the normal direction to the surface. n(E) = Number of electrons per unit volume within the metal between E and E+dE (total energy) according to Fermi-Dirac statistics. n(E)dE s Energy distribution in Fermi-Dirac electron gas measured relative to an electron at rest at oo. N(W, E)dW dE = Supply function: Number of electrons with energy within the range E+dE whose x-part of the energy lies in the range Wto W+dW, incident upon the surface z=0 per unit area per time.
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