Editing Scanning tunneling microscope (section)

=== Bardeen's formalism ===
[[File:Scanning tunneling microscope - tip, barrier and sample wave functions.svg|thumb|300x300px|Tip, barrier and sample wave functions in a model of the scanning tunneling microscope. Barrier width is ''w''. Tip bias is ''V''. Surface work functions are ''ϕ''.]]

A model that is based on more realistic wave functions for the two electrodes was devised by [[John Bardeen]] in a study of the [[metal–insulator–metal]] junction.<ref name="Bardeen">{{cite journal |vauthors = Bardeen J |year=1961 |title=Tunneling from a many particle point of view |journal=Phys. Rev. Lett. |volume=6 |issue=2| pages=57–59 |bibcode=1961PhRvL...6...57B |doi=10.1103/PhysRevLett.6.57}}</ref> His model takes two separate orthonormal sets of wave functions for the two electrodes and examines their time evolution as the systems are put close together.<ref name="Chen" /><ref name="Lounis" /> Bardeen's novel method, ingenious in itself,<ref name="Chen" /> solves a time-dependent perturbative problem in which the perturbation emerges from the interaction of the two subsystems rather than an external potential of the standard [[Perturbation theory (quantum mechanics)|Rayleigh–Schrödinger perturbation theory]].

Each of the wave functions for the electrons of the sample (S) and the tip (T) decay into the vacuum after hitting the surface potential barrier, roughly of the size of the surface work function. The wave functions are the solutions of two separate Schrödinger's equations for electrons in potentials ''U''<sub>S</sub> and ''U''<sub>T</sub>. When the time dependence of the states of known energies <math>E^\text{S}_\mu</math> and <math>E^\text{T}_\nu</math> is factored out, the wave functions have the following general form
: <math>\psi^\text{S}_\mu(t) = \psi^\text{S}_\mu \exp\left(-\frac{i}{\hbar} E^\text{S}_\mu t\right),</math>
: <math>\psi^\text{T}_\nu(t) = \psi^\text{T}_\nu \exp\left(-\frac{i}{\hbar} E^\text{T}_\nu t\right).</math>

If the two systems are put closer together, but are still separated by a thin vacuum region, the potential acting on an electron in the combined system is ''U''<sub>T</sub> + ''U''<sub>S</sub>. Here, each of the potentials is spatially limited to its own side of the barrier. Only because the tail of a wave function of one electrode is in the range of the potential of the other, there is a finite probability for any state to evolve over time into the states of the other electrode.<ref name="Chen" /> The future of the sample's state ''μ'' can be written as a linear combination with time-dependent coefficients of <math>\psi^\text{S}_\mu(t)</math> and all <math>\psi^\text{T}_\nu(t)</math>:
: <math id="ansatz">\psi(t) = \psi^\text{S}_\mu(t) + \sum_\nu c_\nu(t) \psi^\text{T}_\nu(t)</math>
with the initial condition <math>c_\nu(0) = 0</math>.<ref name="Chen" /> When the new wave function is inserted into the Schrödinger's equation for the potential ''U''<sub>T</sub> + ''U''<sub>S</sub>, the obtained equation is projected onto each separate <math>\psi^\text{T}_\nu </math> (that is, the equation is multiplied by a <math>{\psi^\text{T}_\nu}^* </math> and integrated over the whole volume) to single out the coefficients <math>c_\nu.</math> All <math>\psi^\text{S}_\mu</math> are taken to be ''nearly orthogonal'' to all <math>\psi^\text{T}_\nu</math> (their overlap is a small fraction of the total wave functions), and only first-order quantities retained. Consequently, the time evolution of the coefficients is given by
: <math>\frac{\mathrm{d}}{\mathrm{d}t} c_\nu(t) = -\frac{i}{\hbar} \int \psi^\text{S}_\mu \,U_\text{T}\, {\psi^\text{T}_\nu}^* \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\exp\left[-\frac{i}{\hbar} (E^\text{S}_\mu - E^\text{T}_\nu) t\right].</math>

Because the potential ''U''<sub>T</sub> is zero at the distance of a few atomic diameters away from the surface of the electrode, the integration over ''z'' can be done from a point ''z''<sub>0</sub> somewhere inside the barrier and into the volume of the tip (''z''&nbsp;>&nbsp;''z''<sub>0</sub>).

If the tunneling matrix element is defined as
: <math>M_{\mu\nu} = \int_{z > z_0} \psi^\text{S}_\mu \,U_\text{T}\, {\psi^\text{T}_\nu}^* \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z,</math>
the probability of the sample's state ''μ'' evolving in time ''t'' into the state of the tip ''ν'' is
: <math>|c_\nu (t)|^2 = |M_{\mu\nu}|^2 \frac{4 \sin^2\big[\tfrac{1}{2\hbar}(E^\text{S}_\mu - E^\text{T}_\nu )t\big]}{(E^\text{S}_\mu - E^\text{T}_\nu)^2}.</math>

In a system with many electrons impinging on the barrier, this probability will give the proportion of those that successfully tunnel. If at a time ''t'' this fraction was <math>|c_\nu (t)|^2,</math> at a later time ''t''&nbsp;+&nbsp;d''t'' the total fraction of <math>|c_\nu(t + \mathrm{d}t)|^2</math> would have tunneled. The ''current'' of tunneling electrons at each instance is therefore proportional to <math>|c_\nu(t + \mathrm{d}t)|^2 - |c_\nu(t)|^2</math> divided by <math>\mathrm{d}t,</math> which is the time derivative of <math>|c_\nu(t)|^2,</math><ref name="Lounis" />
: <math>\Gamma_{\mu \to \nu}\ \overset{\text{def}}{=}\ \frac{\mathrm{d}}{\mathrm{d}t} |c_\nu(t)|^2 = \frac{2\pi}{\hbar} |M_{\mu\nu}|^2\frac{\sin\big[(E^\text{S}_\mu - E^\text{T}_\nu) \tfrac{t}{\hbar}\big]}{\pi(E^\text{S}_\mu - E^\text{T}_\nu)}.</math>

The time scale of the measurement in STM is many orders of magnitude larger than the typical [[femtosecond]] time scale of electron processes in materials, and <math>t/\hbar</math> is large. The fraction part of the formula is a fast-oscillating function of <math>(E^\text{S}_\mu - E^\text{T}_\nu)</math> that rapidly decays away from the central peak, where <math>E^\text{S}_\mu = E^\text{T}_\nu</math>. In other words, the most probable tunneling process, by far, is the elastic one, in which the electron's energy is conserved. The fraction, as written above, is a representation of the [[Dirac delta function#Oscillatory integrals|delta function]], so
: <math>\Gamma_{\mu \to \nu} = \frac{2\pi}{\hbar} |M_{\mu\nu}|^2 \delta(E^\text{S}_\mu - E^\text{T}_\nu).</math>

Solid-state systems are commonly described in terms of continuous rather than discrete energy levels. The term <math>\delta(E^\text{S}_\mu - E^\text{T}_\nu)</math> can be thought of as the [[Density of states#Definition|density of states]] of the tip at energy <math>E^\text{S}_\mu,</math> giving
: <math>\Gamma_{\mu \to \nu} = \frac{2\pi}{\hbar} |M_{\mu\nu}|^2 \rho_\text{T}(E^\text{S}_\mu).</math>

The number of energy levels in the sample between the energies <math>\varepsilon</math> and <math>\varepsilon + \mathrm{d}\varepsilon</math> is <math>\rho_\text{S}(\varepsilon)\,\mathrm{d}\varepsilon.</math> When occupied, these levels are spin-degenerate (except in a few special classes of materials) and contain charge <math>2e \cdot \rho_\text{S}(\varepsilon)\,\mathrm{d}\varepsilon</math> of either spin. With the sample biased to voltage <math>V,</math> tunneling can occur only between states whose occupancies, given for each electrode by the [[Fermi–Dirac distribution]] <math>f</math>, are not the same, that is, when either one or the other is occupied, but not both. That will be for all energies <math>\varepsilon</math> for which <math>f(E_\text{F} - eV + \varepsilon) - f(E_\text{F} + \varepsilon)</math> is not zero. For example, an electron will tunnel from energy level <math>E_\text{F} - eV</math> in the sample into energy level <math>E_\text{F}</math> in the tip (<math>\varepsilon = 0</math>), an electron at <math>E_\text{F}</math> in the sample will find unoccupied states in the tip at <math>E_\text{F} + eV</math> (<math>\varepsilon = eV</math>), and so will be for all energies in between. The tunneling current is therefore the sum of little contributions over all these energies of the product of three factors: <math>2e \cdot \rho_\text{S}(E_\text{F} - eV + \varepsilon)\,\mathrm{d}\varepsilon</math> representing available electrons, <math>f(E_\text{F} - eV + \varepsilon) - f(E_\text{F} + \varepsilon)</math> for those that are allowed to tunnel, and the probability factor <math>\Gamma</math> for those that will actually tunnel:
: <math>I_t = \frac{4 \pi e}{\hbar} \int_{-\infty}^{+\infty} [f(E_\text{F} - eV + \varepsilon) - f(E_\text{F} + \varepsilon)] \, \rho_\text{S}(E_\text{F} - eV + \varepsilon) \, \rho_\text{T}(E_\text{F} + \varepsilon) \, |M|^2 \, d \varepsilon.</math>

Typical experiments are run at a liquid-helium temperature (around 4&nbsp;K), at which the Fermi-level cut-off of the electron population is less than a millielectronvolt wide. The allowed energies are only those between the two step-like Fermi levels, and the integral becomes
: <math>I_t = \frac{4 \pi e}{\hbar} \int_0^{eV} \rho_\text{S}(E_\text{F} - eV + \varepsilon) \, \rho_\text{T}(E_\text{F} + \varepsilon) \, |M|^2 \, d \varepsilon.</math>

When the bias is small, it is reasonable to assume that the electron wave functions and, consequently, the tunneling matrix element do not change significantly in the narrow range of energies. Then the tunneling current is simply the convolution of the densities of states of the sample surface and the tip:
: <math>I_t \propto \int_0^{eV} \rho_\text{S}(E_\text{F} - eV + \varepsilon) \, \rho_\text{T}(E_\text{F} + \varepsilon) \, d \varepsilon.</math>

How the tunneling current depends on distance between the two electrodes is contained in the tunneling matrix element
: <math>M_{\mu\nu} = \int_{z > z_0} \psi^\text{S}_\mu \,U_\text{T}\, {\psi^\text{T}_\nu}^* \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z.</math>

This formula can be transformed so that no explicit dependence on the potential remains. First, the <math>U_\text{T}\, {\psi^\text{T}_\nu}^*</math> part is taken out from the Schrödinger equation for the tip, and the elastic tunneling condition is used so that
: <math>M_{\mu\nu} = \int_{z > z_0} \left({\psi^\text{T}_\nu}^* E_\mu \psi^\text{S}_\mu + \psi^\text{S}_\mu \frac{\hbar^2}{2m} \frac{\partial^2}{\partial z^2}{\psi^\text{T}_\nu}^*\right) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z.</math>

Now <math>E_\mu\, {\psi^\text{S}_\mu}</math> is present in the Schrödinger equation for the sample and equals the kinetic plus the potential operator acting on <math>\psi^\text{S}_\mu.</math> However, the potential part containing ''U''<sub>S</sub> is on the tip side of the barrier nearly zero. What remains,
: <math>M_{\mu\nu} = -\frac{\hbar^2}{2m} \int_{z > z_0} \left({\psi^\text{T}_\nu}^* \frac{\partial^2}{\partial z^2}{\psi^\text{S}_\mu} - {\psi^\text{S}_\mu} \frac{\partial^2}{\partial z^2}{\psi^\text{T}_\nu}^*\right) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z,</math>
can be integrated over ''z'' because the integrand in the parentheses equals <math>\partial_z\left({\psi^\text{T}_\nu}^* \, \partial_z \psi^\text{S}_\mu - {\psi^\text{S}_\mu} \, \partial_z {\psi^\text{T}_\nu}^*\right).</math>

Bardeen's tunneling matrix element is an integral of the wave functions and their gradients over a surface separating the two planar electrodes:
: <math>M_{\mu\nu} = \frac{\hbar^2}{2m} \int_{z = z_0} \left(
 {\psi^\text{S}_\mu} \frac{\partial}{\partial z}{\psi^\text{T}_\nu}^* -
 {\psi^\text{T}_\nu}^* \frac{\partial}{\partial z}{\psi^\text{S}_\mu}
\right) \,\mathrm{d}x\,\mathrm{d}y.</math>

The exponential dependence of the tunneling current on the separation of the electrodes comes from the very wave functions that ''leak'' through the potential step at the surface and exhibit exponential decay into the classically forbidden region outside of the material.

The tunneling matrix elements show appreciable energy dependence, which is such that tunneling from the upper end of the ''eV'' interval is nearly an order of magnitude more likely than tunneling from the states at its bottom. When the sample is biased positively, its unoccupied levels are probed as if the density of states of the tip is concentrated at its Fermi level. Conversely, when the sample is biased negatively, its occupied electronic states are probed, but the spectrum of the electronic states of the tip dominates. In this case it is important that the density of states of the tip is as flat as possible.<ref name="Chen" />

The results identical to Bardeen's can be obtained by considering adiabatic approach of the two electrodes and using the standard time-dependent perturbation theory.<ref name="Lounis" /> This leads to [[Fermi's golden rule]] for the transition probability <math>\Gamma_{\mu \to \nu}</math> in the form given above.

Bardeen's model is for tunneling between two planar electrodes and does not explain scanning tunneling microscope's lateral resolution. Tersoff and Hamann<ref>{{cite journal| vauthors = Tersoff J, Hamann DR |date=1983-06-20 |title=Theory and Application for the Scanning Tunneling Microscope |journal=Physical Review Letters |volume=50 |issue=25 |pages=1998–2001 |doi=10.1103/PhysRevLett.50.1998 |bibcode=1983PhRvL..50.1998T |doi-access=free}}</ref><ref>{{cite journal | vauthors = Tersoff J, Hamann DR | title = Theory of the scanning tunneling microscope | journal = Physical Review B | volume = 31 | issue = 2 | pages = 805–813 | date = January 1985 | pmid = 9935822 | doi = 10.1103/PhysRevB.31.805 | bibcode = 1985PhRvB..31..805T | url = https://link.aps.org/doi/10.1103/PhysRevB.31.805 }}</ref><ref>{{cite journal |last1=Hansma |first1=Paul K. |last2=Tersoff |first2=Jerry |name-list-style = vanc |date=1987-01-15 |title=Scanning tunneling microscopy |url=https://aip.scitation.org/doi/10.1063/1.338189 |journal=Journal of Applied Physics |volume=61 |issue=2 |pages=R1–R24 |doi=10.1063/1.338189 |bibcode=1987JAP....61R...1H |issn=0021-8979}}</ref> used Bardeen's theory and modeled the tip as a structureless geometric point.<ref name="Chen" /> This helped them disentangle the properties of the tip—which are hard to model—from the properties of the sample surface. The main result was that the tunneling current is proportional to the local density of states of the sample at the Fermi level taken at the position of the center of curvature of a spherically symmetric tip (''s''-wave tip model). With such a simplification, their model proved valuable for interpreting images of surface features bigger than a nanometre, even though it predicted atomic-scale corrugations of less than a picometre. These are well below the microscope's detection limit and below the values actually observed in experiments.

In sub-nanometre-resolution experiments, the convolution of the tip and sample surface states will always be important, to the extent of the apparent inversion of the atomic corrugations that may be observed within the same scan. Such effects can only be explained by modeling of the surface and tip electronic states and the ways the two electrodes interact from [[Ab initio quantum chemistry methods|first principles]].