Editing Scanning tunneling microscope (section)

=== Rectangular barrier model ===
[[File:Scanning tunneling microscope - rectangular potential barrier model.svg|thumb|300x300px|The real and imaginary parts of the wave function in a rectangular potential barrier model of the scanning tunneling microscope]]
The simplest model of tunneling between the sample and the tip of a scanning tunneling microscope is that of a [[rectangular potential barrier]].<ref name="Lounis">{{cite arXiv |last=Lounis |first=Samir |name-list-style = vanc |date=2014-04-03 |title=Theory of Scanning Tunneling Microscopy |class=cond-mat.mes-hall |eprint=1404.0961}}</ref><ref name="Chen" /> An electron of energy ''E'' is incident upon an energy barrier of height ''U'', in the region of space of width ''w''. An electron's behavior in the presence of a potential ''U''(''z''), assuming one-dimensional case, is described by [[wave function]]s <math>\psi(z)</math> that satisfy [[Schrödinger's equation]]
: <math>-\frac{\hbar^2}{2m_\text{e}} \frac{\partial^2\psi(z)}{\partial z^2} + U(z)\,\psi(z) = E\,\psi(z),</math>

where ''ħ'' is the [[reduced Planck constant]], ''z'' is the position, and ''m''<sub>e</sub> is the [[electron mass]]. In the zero-potential regions on two sides of the barrier, the wave function takes on the forms

: <math>\psi_L(z) = e^{ikz} + r\,e^{-ikz}</math> for ''z'' < 0,
: <math>\psi_R(z) = t\,e^{ikz}</math> for ''z'' > ''w'',

where <math>k = \tfrac{1}{\hbar} \sqrt{2m_\text{e}E}</math>. Inside the barrier, where ''E'' < ''U'', the wave function is a superposition of two terms, each decaying from one side of the barrier:
: <math>\psi_B (z) = \xi e^{-\kappa z} + \zeta e^{\kappa z}</math> for 0 < ''z'' < ''w'',
where <math>\kappa = \tfrac{1}{\hbar} \sqrt{2m_\text{e}(U - E)}</math>.

The coefficients ''r'' and ''t'' provide measure of how much of the incident electron's wave is reflected or transmitted through the barrier. Namely, of the whole impinging particle current <math>j_i = \hbar k/m_\text{e}</math> only <math>j_t = |t|^2\, j_i</math> is transmitted, as can be seen from the [[probability current]] expression
: <math>j_t = -i \frac{\hbar}{2m_\text{e}} \left\{\psi_R^* \frac{\partial}{\partial z}\psi_R - \psi_R \frac{\partial}{\partial z}\psi_R^*\right\},</math>

which evaluates to <math>j_t = \tfrac{\hbar k}{m_\text{e}} \vert t \vert^2</math>. The transmission coefficient is obtained from the continuity condition on the three parts of the wave function and their derivatives at ''z'' = 0 and ''z'' = ''w'' (detailed derivation is in the article [[Rectangular potential barrier#Analysis of the obtained expressions|Rectangular potential barrier]]). This gives <math>|t|^2 = \big[1 + \tfrac{1}{4} \varepsilon^{-1}(1 - \varepsilon)^{-1} \sinh^2\kappa w\big]^{-1},</math> where <math>\varepsilon = E/U</math>. The expression can be further simplified, as follows:

In STM experiments, typical barrier height is of the order of the material's surface [[work function]] ''W'', which for most metals has a value between 4 and 6&nbsp;eV.<ref name="Lounis" /> The [[work function]] is the minimum energy needed to bring an electron from an occupied level, the highest of which is the [[Fermi level]] (for metals at ''T'' = 0&nbsp;K), to [[vacuum level]]. The electrons can tunnel between two metals only from occupied states on one side into the unoccupied states of the other side of the barrier. Without bias, Fermi energies are flush, and there is no tunneling. Bias shifts electron energies in one of the electrodes higher, and those electrons that have no match at the same energy on the other side will tunnel. In experiments, bias voltages of a fraction of 1&nbsp;V are used, so <math>\kappa</math> is of the order of 10 to 12&nbsp;nm<sup>−1</sup>, while ''w'' is a few tenths of a nanometre. The barrier is strongly attenuating. The expression for the transmission probability reduces to <math>|t|^2 = 16\,\varepsilon(1 - \varepsilon)\,e^{-2\kappa w}.</math> The tunneling current from a single level is therefore<ref name="Lounis" />
: <math>j_t = \left[\frac{4k\kappa}{k^2 + \kappa^2}\right]^2 \, \frac{\hbar k}{m_\text{e}}\,e^{-2\kappa w},</math>

where both wave vectors depend on the level's energy ''E'', <math>k = \tfrac{1}{\hbar} \sqrt{2m_\text{e}E},</math> and <math>\kappa = \tfrac{1}{\hbar}\sqrt{2m_\text{e}(U - E)}.</math>

Tunneling current is exponentially dependent on the separation of the sample and the tip, typically reducing by an order of magnitude when the separation is increased by 1&nbsp;Å (0.1&nbsp;nm).<ref name="Chen" /> Because of this, even when tunneling occurs from a non-ideally sharp tip, the dominant contribution to the current is from its most protruding atom or orbital.<ref name="Lounis" />