Rectangular barrier model

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.^[15][5] 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 functions ${\displaystyle \psi (z)}$ that satisfy Schrödinger’s equation

${\displaystyle -{\frac {\hbar ^{2}}{2m_{e}}}{\frac {\partial ^{2}\psi (z)}{\partial z^{2}}}+U(z)\,\psi (z)=E\,\psi (z),}$

where ħ is the reduced Planck’s constant, z is the position, and m_e is the electron mass. In the zero-potential regions on two sides of the barrier, the wave function takes on the forms

${\displaystyle \psi _{L}(z)=e^{ikz}+r\,e^{-ikz}}$ for z < 0,

${\displaystyle \psi _{R}(z)=t\,e^{ikz}}$ for z > w,

where ${\displaystyle k={\tfrac {1}{\hbar }}{\sqrt {2m_{e}E}}}$. Inside the barrier, where E < U, the wave function is a superposition of two terms, each decaying from one side of the barrier:

${\displaystyle \psi _{B}(z)=\xi e^{-\kappa z}+\zeta e^{\kappa z}}$ for 0 < z < w,

where ${\displaystyle \kappa ={\tfrac {1}{\hbar }}{\sqrt {2m_{e}(U-E)}}}$.

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 ${\displaystyle j_{i}=\hbar k/m_{e}}$ only ${\displaystyle j_{t}=|t|^{2}\,j_{i}}$ is transmitted, as can be seen from the probability current expression

${\displaystyle j_{t}=-i{\frac {\hbar }{2m_{e}}}\left\{\psi _{R}^{*}{\frac {\partial }{\partial z}}\psi _{R}-\psi _{R}{\frac {\partial }{\partial z}}\psi _{R}^{*}\right\},}$

which evaluates to ${\displaystyle j_{t}={\tfrac {\hbar k}{m_{e}}}|t|^{2}}$. 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). This gives ${\displaystyle |t|^{2}={\big [}1+{\tfrac {1}{4}}\varepsilon ^{-1}(1-\varepsilon )^{-1}\sinh ^{2}\kappa w{\big ]}^{-1},}$ where ${\displaystyle \varepsilon =E/U}$. 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 eV.^[15] 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 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 V are used, so ${\displaystyle \kappa }$ is of the order of 10 to 12 nm⁻¹, while w is a few tenths of a nanometer. The barrier is strongly attenuating. The expression for the transmission probability reduces to ${\displaystyle |t|^{2}=16\,\varepsilon (1-\varepsilon )\,e^{-2\kappa w}.}$ The tunneling current from a single level is therefore^[15]

${\displaystyle j_{t}=\left[{\frac {4k\kappa }{k^{2}+\kappa ^{2}}}\right]^{2}\,{\frac {\hbar k}{m_{e}}}\,e^{-2\kappa w},}$

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

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 Å (0.1 nm).^[5] 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.^[15]

Tunneling between two conductors

Negative sample bias V raises its electronic levels by e⋅V. Only electrons that populate states between the Fermi levels of the sample and the tip are allowed to tunnel.

As a result of the restriction that the tunneling from an occupied energy level on one side of the barrier requires an empty level of the same energy on the other side of the barrier, tunneling occurs mainly with electrons near the Fermi level. The tunneling current can be related to the density of available or filled states in the sample. The current due to an applied voltage V (assume tunneling occurs from the sample to the tip) depends on two factors: 1) the number of electrons between the Fermi level E_F and E_F − eV in the sample, and 2) the number among them which have corresponding free states to tunnel into on the other side of the barrier at the tip.^[5] The higher the density of available states in the tunneling region the greater the tunneling current. By convention, a positive V means that electrons in the tip tunnel into empty states in the sample; for a negative bias, electrons tunnel out of occupied states in the sample into the tip.^[5]

For small biases and temperatures near absolute zero, the number of electrons in a given volume (the electron concentration) that are available for tunneling is the product of the density of the electronic states ρ(E_F) and the energy interval between the two Fermi levels, eV.^[5] Half of these electrons will be travelling away from the barrier. The other half will represent the electric current impinging on the barrier, which is given by the product of the electron concentration, charge, and velocity v (I_i = nev),^[5]

${\displaystyle I_{i}={\tfrac {1}{2}}e^{2}v\,\rho (E_{F})\,V.}$

The tunneling electric current will be a small fraction of the impinging current. The proportion is determined by the transmission probability T,^[5] so

${\displaystyle I_{t}={\tfrac {1}{2}}e^{2}v\,\rho (E_{F})\,V\,T.}$

In the simplest model of a rectangular potential barrier the transmission probability coefficient T equals |t|².

Bardeen's formalism

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.^[16] 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.^[5][15] Bardeen's novel method, ingenious in itself,^[5] 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 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_S and U_T. When the time dependence of the states of known energies ${\displaystyle E_{\mu }^{S}}$ and ${\displaystyle E_{\nu }^{T}}$ is factored out, the wave functions have the following general form

${\displaystyle \psi _{\mu }^{S}(t)=\psi _{\mu }^{S}\exp \left(-{\frac {i}{\hbar }}E_{\mu }^{S}t\right),}$

${\displaystyle \psi _{\nu }^{T}(t)=\psi _{\nu }^{T}\exp \left(-{\frac {i}{\hbar }}E_{\nu }^{T}t\right).}$

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_T + U_S. 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.^[5] The future of the sample's state μ can be written as a linear combination with time-dependent coefficients of ${\displaystyle \psi _{\mu }^{S}(t)}$ and all ${\displaystyle \psi _{\nu }^{T}(t)}$:

${\displaystyle \psi (t)=\psi _{\mu }^{S}(t)+\sum _{\nu }c_{\nu }(t)\psi _{\nu }^{T}(t)}$

with the initial condition ${\displaystyle c_{\nu }(0)=0}$.^[5] When the new wave function is inserted into the Schrödinger’s equation for the potential U_T + U_S, the obtained equation is projected onto each separate ${\displaystyle \psi _{\nu }^{T}}$ (that is, the equation is multiplied by a ${\displaystyle {\psi _{\nu }^{T}}^{*}}$ and integrated over the whole volume) to single out the coefficients ${\displaystyle c_{\nu }.}$ All ${\displaystyle \psi _{\mu }^{S}}$ are taken to be nearly orthogonal to all ${\displaystyle \psi _{\nu }^{T}}$ (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

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}c_{\nu }(t)=-{\frac {i}{\hbar }}\int \psi _{\mu }^{S}\,U_{T}\,{\psi _{\nu }^{T}}^{*}\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z\,\exp \left[-{\frac {i}{\hbar }}(E_{\mu }^{S}-E_{\nu }^{T})t\right].}$

Because the potential U_T 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 z₀ somewhere inside the barrier and into the volume of the tip (z > z₀).

If the tunneling matrix element is defined as

${\displaystyle M_{\mu \nu }=\int _{z>z_{0}}\psi _{\mu }^{S}\,U_{T}\,{\psi _{\nu }^{T}}^{*}\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z,}$

the probability of the sample's state μ evolving in time t into the state of the tip ν is

${\displaystyle |c_{\nu }(t)|^{2}=|M_{\mu \nu }|^{2}{\frac {4\sin ^{2}{\big [}{\tfrac {1}{2\hbar }}(E_{\mu }^{S}-E_{\nu }^{T})t{\big ]}}{(E_{\mu }^{S}-E_{\nu }^{T})^{2}}}.}$

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 ${\displaystyle |c_{\nu }(t)|^{2},}$ at a later time t + dt the total fraction of ${\displaystyle |c_{\nu }(t+\mathrm {d} t)|^{2}}$ would have tunneled. The current of tunneling electrons at each instance is therefore proportional to ${\displaystyle |c_{\nu }(t+\mathrm {d} t)|^{2}-|c_{\nu }(t)|^{2}}$ divided by ${\displaystyle \mathrm {d} t,}$ which is the time derivative of ${\displaystyle |c_{\nu }(t)|^{2},}$^[15]

${\displaystyle \Gamma _{\mu \to \nu }\ {\overset {\text{def}}{=}}\ {\frac {\mathrm {d} }{\mathrm {d} t}}|c_{\nu }(t)|^{2}={\frac {2\pi }{\hbar }}|M_{\mu \nu }|^{2}{\frac {\sin {\big [}(E_{\mu }^{S}-E_{\nu }^{T}){\tfrac {t}{\hbar }}{\big ]}}{\pi (E_{\mu }^{S}-E_{\nu }^{T})}}.}$

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 ${\displaystyle t/\hbar }$ is large. The fraction part of the formula is a fast-oscillating function of ${\displaystyle (E_{\mu }^{S}-E_{\nu }^{T})}$ that rapidly decays away from the central peak, where ${\displaystyle E_{\mu }^{S}=E_{\nu }^{T}}$. 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 delta function, so

${\displaystyle \Gamma _{\mu \to \nu }={\frac {2\pi }{\hbar }}|M_{\mu \nu }|^{2}\delta (E_{\mu }^{S}-E_{\nu }^{T}).}$

Solid-state systems are commonly described in terms of continuous rather than discrete energy levels. The term ${\displaystyle \delta (E_{\mu }^{S}-E_{\nu }^{T})}$ can be thought of as the density of states of the tip at energy ${\displaystyle E_{\mu }^{S},}$ giving

${\displaystyle \Gamma _{\mu \to \nu }={\frac {2\pi }{\hbar }}|M_{\mu \nu }|^{2}\rho _{T}(E_{\mu }^{S}).}$

The number of energy levels in the sample between the energies ${\displaystyle \varepsilon }$ and ${\displaystyle \varepsilon +\mathrm {d} \varepsilon }$ is ${\displaystyle \rho _{S}(\varepsilon )\,\mathrm {d} \varepsilon .}$ When occupied, these levels are spin-degenerate (except in a few special classes of materials) and contain charge ${\displaystyle 2e\cdot \rho _{S}(\varepsilon )\,\mathrm {d} \varepsilon }$ of either spin. With the sample biased to voltage ${\displaystyle V,}$ tunneling can occur only between states whose occupancies, given for each electrode by the Fermi–Dirac distribution ${\displaystyle f}$, are not the same, that is, when either one or the other is occupied, but not both. That will be for all energies ${\displaystyle \varepsilon }$ for which ${\displaystyle f(E_{F}-eV+\varepsilon )-f(E_{F}+\varepsilon )}$ is not zero. For example, an electron will tunnel from energy level ${\displaystyle E_{F}-eV}$ in the sample into energy level ${\displaystyle E_{F}}$ in the tip (${\displaystyle \varepsilon =0}$), an electron at ${\displaystyle E_{F}}$ in the sample will find unoccupied states in the tip at ${\displaystyle E_{F}+eV}$ (${\displaystyle \varepsilon =eV}$), 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: ${\displaystyle 2e\cdot \rho _{S}(E_{F}-eV+\varepsilon )\,\mathrm {d} \varepsilon }$ representing available electrons, ${\displaystyle f(E_{F}-eV+\varepsilon )-f(E_{F}+\varepsilon )}$ for those that are allowed to tunnel, and the probability factor ${\displaystyle \Gamma }$ for those that will actually tunnel:

${\displaystyle I_{t}={\frac {4\pi e}{\hbar }}\int _{-\infty }^{+\infty }[f(E_{F}-eV+\varepsilon )-f(E_{F}+\varepsilon )]\,\rho _{S}(E_{F}-eV+\varepsilon )\,\rho _{T}(E_{F}+\varepsilon )\,|M|^{2}\,d\varepsilon .}$

Typical experiments are run at a liquid-helium temperature (around 4 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

${\displaystyle I_{t}={\frac {4\pi e}{\hbar }}\int _{0}^{eV}\rho _{S}(E_{F}-eV+\varepsilon )\,\rho _{T}(E_{F}+\varepsilon )\,|M|^{2}\,d\varepsilon .}$

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:

${\displaystyle I_{t}\propto \int _{0}^{eV}\rho _{S}(E_{F}-eV+\varepsilon )\,\rho _{T}(E_{F}+\varepsilon )\,d\varepsilon .}$

How the tunneling current depends on distance between the two electrodes is contained in the tunneling matrix element

${\displaystyle M_{\mu \nu }=\int _{z>z_{0}}\psi _{\mu }^{S}\,U_{T}\,{\psi _{\nu }^{T}}^{*}\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z.}$

This formula can be transformed so that no explicit dependence on the potential remains. First, the ${\displaystyle U_{T}\,{\psi _{\nu }^{T}}^{*}}$ part is taken out from the Schrödinger equation for the tip, and the elastic tunneling condition is used so that

${\displaystyle M_{\mu \nu }=\int _{z>z_{0}}\left({\psi _{\nu }^{T}}^{*}E_{\mu }\psi _{\mu }^{S}+\psi _{\mu }^{S}{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial z^{2}}}{\psi _{\nu }^{T}}^{*}\right)\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z.}$

Now ${\displaystyle E_{\mu }\,{\psi _{\mu }^{S}}}$ is present in the Schrödinger equation for the sample and equals the kinetic plus the potential operator acting on ${\displaystyle \psi _{\mu }^{S}.}$ However, the potential part containing U_S is on the tip side of the barrier nearly zero. What remains,

${\displaystyle M_{\mu \nu }=-{\frac {\hbar ^{2}}{2m}}\int _{z>z_{0}}\left({\psi _{\nu }^{T}}^{*}{\frac {\partial ^{2}}{\partial z^{2}}}{\psi _{\mu }^{S}}-{\psi _{\mu }^{S}}{\frac {\partial ^{2}}{\partial z^{2}}}{\psi _{\nu }^{T}}^{*}\right)\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z,}$

can be integrated over z because the integrand in the parentheses equals ${\displaystyle \partial _{z}\left({\psi _{\nu }^{T}}^{*}\,\partial _{z}\psi _{\mu }^{S}-{\psi _{\mu }^{S}}\,\partial _{z}{\psi _{\nu }^{T}}^{*}\right).}$

Bardeen's tunneling matrix element is an integral of the wave functions and their gradients over a surface separating the two planar electrodes:

${\displaystyle M_{\mu \nu }={\frac {\hbar ^{2}}{2m}}\int _{z=z_{0}}\left({\psi _{\mu }^{S}}{\frac {\partial }{\partial z}}{\psi _{\nu }^{T}}^{*}-{\psi _{\nu }^{T}}^{*}{\frac {\partial }{\partial z}}{\psi _{\mu }^{S}}\right)\,\mathrm {d} x\,\mathrm {d} y.}$

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.^[5]

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.^[15] This leads to Fermi's golden rule for the transition probability ${\displaystyle \Gamma _{\mu \to \nu }}$ 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^[17][18][19] used Bardeen's theory and modeled the tip as a structureless geometric point.^[5] 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 nanometer, even though it predicted atomic-scale corrugations of less than a picometer. These are well below the microscope's detection limit and below the values actually observed in experiments.

In sub-nanometer-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 first principles.