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field dependence

Field emission

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.]

History

In 1897, Robert W. Wood became the first person to describe the phenomenon of field emission, which he observed during experiments with a discharge tube [3]. Walter Schottky was the first to provide theoretical insight into the process, proposing that the electrons were emitted over a potential barrier at the metal surface reduced by the presence of an external field [4]. In Schottky’s model, the peak in the potential barrier is located at a distance z_0 from the metal surface where the image force, e^2/4z_0^2, equals the field force, eF, where -e is the electron charge and F is the field strength. These forces lower the work function by Deltaphi=frac{e^2}{4z_0}+eFz_0, which becomes Deltaphi=esqrt{eF} upon eliminating z_0. Within the framework of this model, the complete reduction of the potential barrier at the surface was assumed to be the mechanism of field emission from cold cathodes, which would require fields on the order of 10^8 V/cm for a work function of 4.5 eV. However, field emission had already been experimentally observed with fields on the order of 10^6 V/cm [3]. In 1926, Robert A. Millikan, C. F. Eyring, and B. S. Gossling observed that temperatures up to 1500 K did not affect emission currents [4]. Shortly thereafter, in 1929, Millikan and C. C. Lauritsen showed that the observed emission current was exponentially dependent on the applied field [3].

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

where -e is the electron charge, h is the Planck constant, mu is the Fermi level relative to the bottom of the conduction band, and phi 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]:

  • The metal obeys the free electron model of Sommerfeld with Fermi-Dirac Statistics.
  • The metal surface is planar, reducing the problem to a one-dimensional one. So long as the potential barrier thickness is several orders of magnitude less than the emitter radius, this assumption is justified.
  • The potential in the metal, V_1left(zright), is a constant, -V_0. The potential barrier outside the metal is entirely due to the image forces, V_z=-e^2/4z; the applied electric field does not affect the electron states in the metal.
  • The temperature of the system is T=0 K.

Here the hat z -direction is normal to the metal surface, pointing away from the surface. The second phase is a vacuum. The origin of the applied electric field is the metal surface, and the field contribution to the potential energy is -eFz [3]. Thus, the effective potential energy is

Vleft(zright) = 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 E_z 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)

Let Nleft(E_zright)dE_z be the number of electrons per unit area per second with the z-component of their energy within dE_z of E_z incident on the surface potential barrier; and let Dleft(E_zright) be the tunneling probability, also known as the transmission coefficient. Thus, the product Dleft(E_zright)Nleft(E_zright)dE_z gives the number of electrons per unit area per second within dE_z of E_z emitted from the metal surface. Then the current density is

j=eint_{-V_0}^{infty}Dleft(E_zright)Nleft(E_zright),dE_z.qquad (4)

The electron flux incident on the metal surface is

Nleft(E_zright)dE_z=frac{2}{h^3}dE_zint_{-infty}^{infty}int_{-infty}^{infty}frac{dp_xdp_y}{1+expleft(frac{E_z-zeta}{kT}+frac{p_x^2+p_y^2}{2mkT}right)}=frac{4pi mkT}{h^3}logleft(1+e^{-left(E_z-zetaright)/kT}right)dE_z,; (5)

where h is the Planck constant, -zeta 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

Dleft(E_zright)=expleft[-frac{8pi left(2mright)^{1/2}}{3he}frac{|E_z|^{3/2}}{F}varthetaleft(yright)right],qquad (6)

where F is the applied electric field. The Nordheim function, varthetaleft(yright), is

varthetaleft(yright)=2^{-1/2}left[1+left(1-y^2right)^{1/2}right]^{1/2}cdot left[Eleft(kright)-left{1-left(1-y^2right)^{1/2}right}right]Kleft(kright),qquad (7)

where y=left(e^3 Fright)^{1/2}/ |E_z|. The complete elliptic integrals of the first and second kinds, Eleft(kright) and Kleft(kright), are given by

Eleft(kright)=int_0^{pi/2}left(1-k^2sin^2 alpharight)^{-1/2}dalpha, and Kleft(kright)=int_0^{pi/2}left(1-k^2sin^2 alpharight)^{1/2}dalpha,qquad (8)

where k^2=2left(1-y^2right)^{1/2}bigg / left(1+left(1-y^2right)^{1/2}right) [4]. Combining equations (5) and (6), the number of electrons within dE_z emitted per unit area per second is

Dleft(E_zright)Nleft(E_zright)dE_z=frac{4pi mkT}{h^3}expleft[-frac{8pi left(2mright)^{1/2}}{3he}frac{|E_z|^{3/2}}{F}varthetaleft(yright)right]lnleft(1+e^{-left(E_z-zetaright)/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 E_z=zeta, approximating the exponent in equation (9) with the first two terms of a power series expansion at E_z=zeta is valid. In this approximation, the exponent reduces to

-frac{8pi left(2mright)^{1/2}}{3he}frac{|E_z|^{3/2}}{F}varthetaleft(yright)approx-c+frac{E_z-zeta}{d};qquad (10)

where

begin{array}{lcl}
c & = & frac{8pi left(2mright)^{1/2}}{3he}frac{|E_z|^{3/2}}{F}varthetaleft(yright), d & = & frac{heF}{4pi left(2mphiright)^{1/2}tleft(left(e^3Fright)^{1/2}/phiright)}, tleft(yright) & = & varthetaleft(yright)-frac{2}{3}yfrac{dvarthetaleft(yright)}{dy}, end{array}

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

kTlnleft(1+expleft(-left(E_z-zetaright)/kTright)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:

Dleft(E_zright)Nleft(E_zright)dE_z=begin{cases} 0, & mbox{for }{E_z} > {zeta} {frac{4pi m}{h^3}expleft(-c+frac{E_z-zeta}{d}right)left(zeta-E_zright)}, & mbox{for } {E_z} < {zeta} end{cases}.qquad (12)

Integration of equation (12) will give the current density. Assuming that -V_0ll zeta, the Fermi energy, the lower limit of the integral can be set to -infty. The current density obtained via these assumptions is [3]

begin{align}
j & = eint_{-infty}^{zeta}frac{4pi m}{h^3}left(zeta-E_zright)expleft(-c+frac{E_z-zeta}{d}right)dE_z=frac{4pi med^2}{h^3}e^{-c} & = frac{e^3F^2}{8pi hphi t^2left(left(e^3Fright)^{1/2}big / phiright)}expleft(frac{4left(2mright)^{1/2}phi^{3/2}}{3heF}varthetaleft(frac{left(e^3Fright)^{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

For finite, non-zero temperatures, the Fermi-Dirac distribution that applies to electrons indicates that there will be electrons in the metal with energies greater than the Fermi level [4]. Since the transmission coefficient increases with the particle’s incident energy, these electrons with energies greater than the Fermi level are more likely to tunnel through the potential barrier at the metal surface. The emission current changes only slightly from that at T=0 K for small temperatures, but in the high temperature limit, which is thermionic emission, the electrons with energies greater than the barrier height constitute the majority of the current [3].

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

lnleft(1 + expleft(-left(E_z-zetaright)/kTright)right)approx expleft(-left(E_z-zetaright)/kTright).qquadqquad (14)

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

Dleft(E_zright)Nleft(E_zright)dE_z=frac{4pi m k T}{h^3}expleft[-c+left(E_z-zetaright)left(frac{1}{d}-frac{1}{kT}right)right].qquadqquad (15)

Integrating equation (15) gives the following expression for the current density for non-zero temperature, valid only for Tle 1000 K:

jleft(Tright)=jleft(0right)frac{pi kT/d}{sinleft(pi kT/dright)},qquadqquad (16)

where jleft(0right) is the current density at zero temperature shown in equation (13) [3]. Using equation (16) for a field of 4.7times 10^7 V cm^{-1} and a work function of 4.5 eV, the current densities at room temperature and at 1000 K are jleft(300 Kright)approx 1.03jleft(0right) and jleft(1000 Kright)approx 1.5jleft(0right) [2][3]. Treatment of the transition region between field emission and thermionic emission requires a more rigorous analysis, such as that presented by Murphy and Good [1], or numerical calculations.

The assumption of a planar, or very smooth, surface of the emitter is one of the primary assumptions of the Fowler-Nordheim theory. This assumption is fairly accurate for atomically smooth emitters with a radius of curvature approximately greater than 0.1 mu m, for which the surface potential barrier width is much less than the radius of curvature. However, the planar assumption, which reduces the problem to a one-dimensional one, is not valid for field emitters with a radius of curvature around 1-20 nm, which is on the order of the barrier width. A rigorous treatment of such systems requires solving the three-dimensional Schrödinger equation with an asymmetric barrier potential [4]. Additionally, the field at the apex of emitters with small radii of curvature will be larger than those with larger radii of curvature for the same applied bias, with the field inversely proportional to the radius of curvature [2].

Compared to FE from metals, the process of FE from semiconductors is much more complicated. A qualitative theory of the field emission process from semiconductors has yet to be developed, despite the plentiful amount of experimental data. Experimental results have shown that the relationship between lnleft(jright) and left(1/Fright) is nonlinear for semiconductors with low conduction band carrier concentrations. This is in contrast to metals, for which the relationship between lnleft(jright) and left(1/Fright) is linear over a wide range of field strengths [4]. 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 phi^{3/2} [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].

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