Field emission

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Field emission (FE) is the emission of electrons from the surface of a condensed phase into another phase due to the presence of high electric fields. In this phenomenon, electrons with energies below the Fermi level tunnel through the potential barrier at the surface, which the high electric field sufficiently narrows for the electrons to have a non-negligible tunneling probability. Variations in the emitted current are primarily due to the field dependence of this surface potential barrier [1]. Field (electron) emission, sometimes called cold emission or Fowler-Nordheim tunneling, is unique in comparison with thermionic emission or photoemission, phenomena in which only electrons with sufficient energy to surmount the surface potential barrier are able to escape the condensed phase [2].

[NOTE: the references cited in this article seem to be missing. Suggested Ref. [4]: Schottky, Z.fur Physik, vol.14, p.80, 1923.]

Contents

History

e^2/4z_0^2

Sir Ralph Fowler and Lothar Wolfgang Nordheim obtained the first accurate description of field emission, based on tunneling of electrons through the surface potential barrier, in 1928 [5]. Fowler and Nordheim assumed Fermi-Dirac statistics for the electron energy distribution in the metal, calculated the number of electrons impinging on the surface from the bulk for each range of energy, and solved the Schrödinger equation to find the fraction of electrons that penetrate the barrier. Upon integrating the product of the number of electrons arriving at the surface from the bulk and the tunneling probability over all energies, they obtained a formula for the current density given by

j=\frac{4\sqrt{\mu\phi}}{\mu+\phi}\frac{e^3 F^2}{8\pi h \phi}e^{-\frac{8\pi\sqrt{2m}\phi^{\frac{3}{2}}}{3heF}},\qquad (1)

where − e is the electron charge, h is the Planck constant, μ is the Fermi level relative to the bottom of the conduction band, and φ is the work function [3]. The Fowler-Nordheim theory accurately described the electric field and work function dependences of the emission current. Nordheim later refined the theory further to include the potential barrier deformation due to Schottky’s image force [6]. This refinement reduced the predicted field strength necessary for the same current density [3]. Furthermore, the prediction of extremely high FE current densities, far greater than those possible with thermionic emission, was one of the most important results of the Fowler-Nordheim theory [4].

Theory of Field Emission from Metals

The Fowler-Nordheim theory is generally used in order to quantitatively describe the FE process for metals, which requires calculating the FE current density as a function of the electric field. Since this process is essentially a tunneling process, the tunneling transition probability for the electron to tunnel through the potential barrier and the number of electrons incident on the potential barrier must be found. Integrating these over all energy values gives the desired current density. The assumptions of the Fowler-Nordheim theory are [4]:

\hat z
 V\left(z\right) = \begin{cases} -V_0, &  \mbox{for }z < 0 \\ -eFz-\frac{e^2}{4z}, & \mbox{for } z > 0 \end{cases}.\qquad (2)

Additionally, the model assumes that the electrons in the metal remain at equilibrium, despite the electrons escaping the metal surface. Integrating the product of the flux of electrons incident on the surface potential barrier and the tunneling probability over all electron energies. Define Ez to be the z-component of the electron energy:

 E_z = E - \frac{p_x^2}{2m} - \frac{p_y^2}{2m} = \frac{p_z^2}{2m} + V(z).\qquad (3)
N\left(E_z\right)dE_z
j=e\int_{-V_0}^{\infty}D\left(E_z\right)N\left(E_z\right)\,dE_z.\qquad (4)

The electron flux incident on the metal surface is

N\left(E_z\right)dE_z=\frac{2}{h^3}dE_z\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{dp_xdp_y}{1+exp\left(\frac{E_z-\zeta}{kT}+\frac{p_x^2+p_y^2}{2mkT}\right)}=\frac{4\pi mkT}{h^3}log\left(1+e^{-\left(E_z-\zeta\right)/kT}\right)dE_z,\; (5)

where h is the Planck constant, − ζ is the work function, k is the Boltzmann constant, T is the temperature, and m is the electron mass [3]. Using the semiclassical WKB approximation, the transmission coefficient is

D\left(E_z\right)=exp\left[-\frac{8\pi \left(2m\right)^{1/2}}{3he}\frac{|E_z|^{3/2}}{F}\vartheta\left(y\right)\right],\qquad (6)
\vartheta\left(y\right)
\vartheta\left(y\right)=2^{-1/2}\left[1+\left(1-y^2\right)^{1/2}\right]^{1/2}\cdot \left[E\left(k\right)-\left\{1-\left(1-y^2\right)^{1/2}\right\}\right]K\left(k\right),\qquad (7)
y=\left(e^3 F\right)^{1/2}/ |E_z|
E\left(k\right)=\int_0^{\pi/2}\left(1-k^2sin^2 \alpha\right)^{-1/2}d\alpha
k^2=2\left(1-y^2\right)^{1/2}\bigg / \left(1+\left(1-y^2\right)^{1/2}\right)
D\left(E_z\right)N\left(E_z\right)dE_z=\frac{4\pi mkT}{h^3}exp\left[-\frac{8\pi \left(2m\right)^{1/2}}{3he}\frac{|E_z|^{3/2}}{F}\vartheta\left(y\right)\right]ln\left(1+e^{-\left(E_z-\zeta\right)/kT}\right)dE_z.\qquad (9)

There are a few applicable simplifications for field emission assumptions listed above. Since field-emitted electrons have energies near Ez = ζ, approximating the exponent in equation (9) with the first two terms of a power series expansion at Ez = ζ is valid. In this approximation, the exponent reduces to

-\frac{8\pi \left(2m\right)^{1/2}}{3he}\frac{|E_z|^{3/2}}{F}\vartheta\left(y\right)\approx-c+\frac{E_z-\zeta}{d};\qquad (10)

where

\begin{array}{lcl}
c & = & \frac{8\pi \left(2m\right)^{1/2}}{3he}\frac{|E_z|^{3/2}}{F}\vartheta\left(y\right), \\
d & = & \frac{heF}{4\pi \left(2m\phi\right)^{1/2}t\left(\left(e^3F\right)^{1/2}/\phi\right)},\\
t\left(y\right) & = & \vartheta\left(y\right)-\frac{2}{3}y\frac{d\vartheta\left(y\right)}{dy},  
\end{array}

and the work function is φ = − ζ [3]. For sufficiently low temperatures, the temperature dependent part of equation (5) reduces as follows:

kTln\left(1+exp\left(-\left(E_z-\zeta\right)/kT\right)\right)=\begin{cases} 0, &  \mbox{for }{E_z} > {\zeta} \\ {\zeta-E_z}, & \mbox{for } {E_z} < {\zeta} \end{cases}.\qquad (11)

Upon substituting (11) into equation (9), the following is obtained:

D\left(E_z\right)N\left(E_z\right)dE_z=\begin{cases} 0, &  \mbox{for }{E_z} > {\zeta} \\ {\frac{4\pi m}{h^3}exp\left(-c+\frac{E_z-\zeta}{d}\right)\left(\zeta-E_z\right)}, & \mbox{for } {E_z} < {\zeta} \end{cases}.\qquad (12)
-V_0\ll \zeta
\begin{align}
j & = e\int_{-\infty}^{\zeta}\frac{4\pi m}{h^3}\left(\zeta-E_z\right)exp\left(-c+\frac{E_z-\zeta}{d}\right)dE_z=\frac{4\pi med^2}{h^3}e^{-c} \\
& = \frac{e^3F^2}{8\pi h\phi t^2\left(\left(e^3F\right)^{1/2}\big / \phi\right)}exp\left(\frac{4\left(2m\right)^{1/2}\phi^{3/2}}{3heF}\vartheta\left(\frac{\left(e^3F\right)^{1/2}}{\phi}\right)\right) \\
\end{align}. \qquad (13)

The dependences of the emission current on the work function and the field strength in the above expression match those observed experimentally [4].

Extending the Field Emission Theory from Metals

 T=0\ K

Using the expansion around the Fermi level in equation (10) in equation (9) and approximating the natural logarithm term for Ez > ζ gives

\ln\left(1 + \exp\left(-\left(E_z-\zeta\right)/kT\right)\right)\approx \exp\left(-\left(E_z-\zeta\right)/kT\right).\qquad\qquad (14)

Substituting equations (10) and (14) in equation (9), the expression becomes

D\left(E_z\right)N\left(E_z\right)dE_z=\frac{4\pi m k T}{h^3}\exp\left[-c+\left(E_z-\zeta\right)\left(\frac{1}{d}-\frac{1}{kT}\right)\right].\qquad\qquad (15)
T\le 1000\ K
j\left(T\right)=j\left(0\right)\frac{\pi kT/d}{\sin\left(\pi kT/d\right)},\qquad\qquad (16)
j\left(0\right)0.1\ \mu m\ln\left(j\right)

The Morgulis-Stratton theory [7] adequately describes experimental results for FE from semiconductors for relatively small currents. The theory describes the linear increase in the natural logarithm of the current with the inverse of the applied bias, the lack of photosensitivity, and the constant emission image size. The theory assumes that the electron gas is degenerate due to penetration of the electric field into the emitter near the surface, increasing the free electron concentration near the surface. Additionally, the theory assumes that the tunneling transmission coefficient is small. Calculation of the emission current density follows the method of the Fowler-Nordheim model [4].

Applications

The field emission microscope (FEM), invented in 1936 by E. W. Müller, is one of the primary applications of the field emission phenomena. The introduction of commercial FEMs enabled more accurate field calculations, which confirmed the validity of the Fowler-Nordheim theory within experimental error and the exponential dependence of the emission current density on φ3 / 2 [3]. A fluorescent screen anode is placed at a macroscopic distance from the field emitter cathode. The image that appears on the screen is a projection of the emitter apex produced by the impinging field-emitted electrons. Due to the parabolic trajectories of the emitted electrons, the magnification is proportional to the quotient of the anode-cathode distance and the emitter radius of curvature [4]. Due to the work function change induced by adsorption of molecules on surfaces and the sensitivity of the emission current on the work function, the FEM is effective for studying adsorption phenomena such as diffusion on surfaces and adsorption-desorption kinetics [3][8].

In addition to the applications of FE to surface science studies, FE is used in vacuum microelectronic devices, which rely on electron transport through vacuum rather than carrier transport in semiconductors. Displays based on field emitter arrays are by far the most common use of vacuum microelectronic devices. These field emission displays generally replace the thermal cathodes of traditional cathode ray tube displays with arrays of field emitters. In addition to applications in display technology, several other vacuum microelectronic devices have been demonstrated, including FE triodes and amplifiers and microwave frequency devices [4].