Photoemission

Chapter II: Theoretical Background of Electron Emission

2.3 Photoemission

The first experiments on the photoelectric effect were done by H. Hertz in 1887, years before the discovery of the electron by J. Thompson. In his experiment, he generated electromagnetic waves by spark discharge using an induction coil connected to a pair of capacitor plates attached to two metal spheres. The primary spark generated by this apparatus would then induce a secondary spark in a receiving circuit consisting of a micrometer spark gap attached to an open ring of wire. To see the spark better, he placed the secondary spark gap in a dark box and noticed that the voltage at which the sparking took place changed. He concluded that light produced by the primary spark was affecting the receiving spark. By using a quartz prism, he dispersed the radiation from various light sources and discovered that ultraviolet radiation was the cause for the increase in sparking. At the time, he could not explain his observations [39, 40]. Further experiments were later carried out by W. Hallwachs and P. Lenard.

Yet, they were only explained in 1905 with A. Einstein’s quantum theory of the photoelectric effect, for which he received the Nobel Prize in 1921 [41, 42].

In his article, Einstein suggested that the energy of light is not distributed contin- uously in space as J. Maxwell’s theory dictated, but rather quantized [43]. The derivation of the concept of a quantum of light—the photon—was not based on experimental results but rather on statistical mechanics. Specifically, he considered that an isothermal change in volume from an initial volume𝑉₁to a final volume𝑉₂ would cause a change in entropy due to radiation equal to (𝑘_𝐵𝑑𝐸/ℎ𝜈)ln(𝑉₁/𝑉₂), where𝑑𝐸 is the total energy of the radiation in a frequency interval𝜈 to𝜈+𝑑 𝜈, in which Wien’s radiation law is satisfied. This expression showed the same logarith- mic dependence on volume as the equation for the entropy of a monoatomic ideal gas as long as the number of particles in the gas is𝑑𝐸/ℎ𝜈, leading Einstein to conclude

that light must consist of particles with energy ℎ𝜈. This concept allowed him to explain several experiments, such as photoluminescence, the ionization of gases by ultraviolet radiation, and black-body radiation, which could not be explained using continuous spatial functions to describe light. He also ventured to explain the energy transfer from photons to electrons, leading him to the fundamental equation that is now known as the “photoelectric effect,” given by

𝐸_{𝑘 𝑖𝑛} =ℎ𝜈−𝑃 (2.60)

where 𝐸_{𝑘 𝑖𝑛} is the maximum kinetic energy of the emitted electrons and 𝑃 is the amount of work an electron must do to leave a solid and is characteristic of that solid, which is currently known as the work function𝜙. Even though this linear frequency relation was consistent with Lenard’s experiments, which Einstein referred to in his publication, it was not until several years later that experiments conducted by A.

Hughes, O. Richardson, K. Compton, and R. Millikan finally led to Einstein’s model being generally accepted by the scientific community.

In the photoelectric effect, the absorption of photons by bound electrons in solids causes these electrons to transition to higher energy levels. If the energy of the photon is sufficiently large to overcome the material work function, the excited electrons can travel above the potential barrier and become free electrons. This is commonly referred to as photoemission [44, 45]. In the simplest case, one photon provides the required energy to overcome the work function, as shown in Fig. 2.5 (a). If multiple photons are absorbed to lift the electron over the potential barrier, the process is known as multi-photon emission. Moreover, in the special case where a greater number of photons than required for emission are absorbed, i.e.,𝑁 > 𝜙/ℎ𝜈, the process is termed above-threshold photoemission. Both cases of multi-photon emission are depicted in Fig. 2.5 (b). It has been shown that in multi-photon emission, the photoelectron current density 𝐽_{𝑚𝑢𝑙 𝑡𝑖} is proportional to the nth power of laser intensity𝐼, i.e.,𝐽_{𝑚𝑢𝑙 𝑡𝑖} ∝ 𝐼^𝑁, where 𝑁 denotes the total number of photons absorbed [46–48].

In addition, an electron can be emitted into free space even if 𝑁 ℎ𝜈 < 𝜙. In this case, the electron is promoted to a non-equilibrium electron distribution by the absorption of one or more photons and tunnels through the potential barrier, which has been narrowed by an applied DC bias [49, 50]. The emitted current density can

E_F

E E

vacuum

e-

(a)

E_F ^e-

(b)

Above-Threshold Photoemission

Photoemission

EF

E

vacuum

e-

(c)

Photo-Assisted Field Emission

EF

E

vacuum

e-

(d)

Optical Field Emission Multiphoton Emission

vacuum

Figure 2.5: Electron emission processes from metal to vacuum under optical illu- mination: (a) single photon photoemission (photoelectric effect), (b) multi-photon emission and above-threshold photoemission, (c) photo-assisted field emission, and (d) optical field emission.

be described using Eq. (2.43) with an effective work function 𝜙_{𝑒 𝑓 𝑓} = 𝜙− 𝑁 ℎ𝜈. This process is called photo-assisted field emission and is illustrated in Fig. 2.5 (c).

Furthermore, there is another way in which an electron can interact with an optical field. If the laser intensity is sufficiently high, the local electric field associated with the optical field modulates the potential barrier such that during part of the optical cycle, the barrier becomes narrow enough to allow an electron from the Fermi level to tunnel through it [51–53]. This process is known as optical field emission and is displayed in Fig. 2.5 (d). The emitted current density can also be expressed using Eq. (2.43), where the total field is now given by the sum of the DC and AC electric fields: 𝐹 → 𝐹+𝐹_{𝑙 𝑎 𝑠𝑒𝑟}.

To distinguish between emission regimes, L. Keldysh proposed a dimensionless adiabaticity parameter 𝛾_𝑘 given by the ratio of the optical driving frequency 𝜔 to the tunneling frequency𝜔_𝑡 [54]

𝛾_𝑘 = 𝜔 𝜔_𝑡

. (2.61)

The tunneling frequency can be determined by the mean free time of the elec- tron traversing a barrier of width 𝑙 = 𝜙/𝑒 𝐹_{𝑙 𝑎 𝑠𝑒𝑟} with an average electron velocity proportional to(𝜙/𝑚)^1/2, thus allowing𝛾_𝑘 to be expressed as

𝛾_𝑘 = 𝜔√︁

2𝑚 𝜙 𝑒 𝐹_{𝑙 𝑎 𝑠𝑒𝑟}

. (2.62)

In addition,𝛾_𝑘can be written in terms of the ponderomotive energy of electrons𝑈_𝑝, i.e., the cycle-averaged kinetic energy of a free electron in an alternating electric field

𝛾_𝑘 =

√︄

𝜙 2𝑈_𝑝

(2.63)

with𝑈_𝑝 = ^𝑒

2𝐹²

𝑙 𝑎 𝑠𝑒𝑟

4𝑚𝜔² . Therefore, for relatively weak fields, the field decay length is larger than the quiver amplitude, yielding 𝛾_𝑘 ≫ 1 [55]. In this regime, electron emission is governed by photoemission processes. In the strong field regime, the ponderomotive energy becomes comparable to the electron binding energy, and 𝛾_𝑘 < 1. In this limit, the electron tunneling adiabatically follows the instantaneous optical field, and the dominant process is optical field emission [56].