7.6 The quantum tunneling of particles through potential barriers  (Page 7/22)

Suppose a conduction electron has a kinetic energy E (the average kinetic energy of an electron in a metal is the work function $\varphi$ for the metal and can be measured, as discussed for the photoelectric effect in Photons and Matter Waves ), and an external electric field can be locally approximated by a uniform electric field of strength $E_g$ . The width L of the potential barrier that the electron must cross is the distance from the conductor’s surface to the point outside the surface where its kinetic energy matches the value of its potential energy in the external field. In [link] , this distance is measured along the dashed horizontal line $U(x)=E$ from $x=0$ to the intercept with $U(x)=\text{−}e{E}_{g}x$ , so the barrier width is

We see that L is inversely proportional to the strength $E_g$ of an external field. When we increase the strength of the external field, the potential barrier outside the conductor becomes steeper and its width decreases for an electron with a given kinetic energy. In turn, the probability that an electron will tunnel across the barrier (conductor surface) becomes exponentially larger. The electrons that emerge on the other side of this barrier form a current (tunneling-electron current) that can be detected above the surface. The tunneling-electron current is proportional to the tunneling probability. The tunneling probability depends nonlinearly on the barrier width L , and L can be changed by adjusting $E_g$ . Therefore, the tunneling-electron current can be tuned by adjusting the strength of an external electric field at the surface. When the strength of an external electric field is constant, the tunneling-electron current has different values at different elevations L above the surface.

The quantum tunneling phenomenon at metallic surfaces, which we have just described, is the physical principle behind the operation of the scanning tunneling microscope (STM)    , invented in 1981 by Gerd Binnig and Heinrich Rohrer. The STM device consists of a scanning tip (a needle, usually made of tungsten, platinum-iridium, or gold); a piezoelectric device that controls the tip’s elevation in a typical range of 0.4 to 0.7 nm above the surface to be scanned; some device that controls the motion of the tip along the surface; and a computer to display images. While the sample is kept at a suitable voltage bias, the scanning tip moves along the surface ( [link] ), and the tunneling-electron current between the tip and the surface is registered at each position. The amount of the current depends on the probability of electron tunneling from the surface to the tip, which, in turn, depends on the elevation of the tip above the surface. Hence, at each tip position, the distance from the tip to the surface is measured by measuring how many electrons tunnel out from the surface to the tip. This method can give an unprecedented resolution of about 0.001 nm, which is about 1% of the average diameter of an atom. In this way, we can see individual atoms on the surface, as in the image of a carbon nanotube in [link] .