Chapter 5 - Scanning tunneling microscopy

Scanning Tunneling Microscopy (STM) is a type of microscope utilized for imaging surfaces at the atomic level. It is based on the principle of quantum tunneling and was invented by Heinrich Rohrer and Gerd Binnig in 1981, for which they were awarded the Nobel Prize in Physics in 1986 [1](#Bibliography). Its widespread applications across physics, chemistry, and materials science have revolutionized our understanding of surface structures, electronic properties, and the intricate world of nanoscale phenomena. Operating as a non-optical technique, STM stands as an indispensable tool for the precise investigation and manipulation of matter, opening new frontiers in the exploration of nanomaterials and nanodevices [2](#Bibliography).

STM operates through three primary modes:

Imaging mode: this mode encompasses two scanning regimes for achieving atomic resolution of the sample surface morphology:
- Constant-current mode(Fig. 1(b)): here, a feedback loop maintains a constant tunneling current between the tip and sample at each lateral position $(x,y)$. Consequently, the tip's $z$-position requires adjustment during scanning.
- Constant-height mode (Fig. 1(c)): when the tunneling current is measured at a fixed vertical position $z$ of the tip.

Spectroscopy mode: in this mode, the resulting tunelling current $I$ is monitored as a function of changing parameters $z$ or $V$:
- $I(z)$ spectroscopy: involves ramping the distance $z$ at a fixed voltage $V$, useful for characterizing the quality, sharpness, and cleanliness of the STM tip.
- $I(V)$ spectroscopy: measures the tunelling current as a function of voltage $V$, offering crucial insights into the surface electronic structure, including barrier heights, local density of states, and analysis of modes of molecular motion.

Manipulation mode: This mode allows for the deliberate manipulation of atoms and molecules on the sample surface using the STM tip. By applying voltage pulses or carefully adjusting the tip-sample distance, it becomes possible to pick up, move, and position atoms or molecules with precision. This capability opens avenues for constructing nanostructures, investigating surface reactions, and studying the behavior of individual atoms and molecules in real-time [3](#Bibliography).

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Tunneling a single particle throught a one-dimentional potential barrier

In quantum mechanics, the electron penetration through a potential barrier, $U(z)$ is described by a wave function $\psi(z)$ that can be obtained from the Schrödinger equation: $$ \frac{-\hbar^2}{2m}\frac{d^2\psi(z)}{dz^2} + U(z)\psi(z) = E\psi(z) $$ where $\hbar$ is the reduced Planck’s constant, $m$ is the mass of the electron, $E$ is its energy and $z$ its position.

Schema of a one-dimentional potential barrier

In the case of a rectangular potential barrier $U(z)$ the solution to the wave equation for each region , sample (s), barrier (b) and tip (t) are written as:

$z >=0$ and $U(z) = 0$:

$\psi_s(z) = e^{ikz} + Ae^{-ikz} $ - incoming wave + reflected wave;

$0 < z < d$ and $U(z) = U$

$\psi_b(z) = Be^{-\varkappa z} + Ce^{\varkappa z}$ - - decaying in barrier;

$z < d$ and $U(z) = 0$:

$\psi_t(z) = De^{ikz} $ - transmitted wave;

where $k = \frac{\sqrt{2mE_0}}{\hbar}$ and $\varkappa = \frac{\sqrt{2m(U - E_0)}}{\hbar}$

The coefficients A, B, C and D result from the wave function continuity condition at two interface.

The following variables were used for calculations:

$U = $ eV
$E_{0} = $ eV
$d_p= $ Å

$A$

$B$

$C$

$D$

So, in this case, for low particle energy ($E_0 << U$) and wide rectangular barrier ($\varkappa d >> 1$), the probability of transmission simplifies to:

$$T(E) = T_0 \exp{\left(-\frac{2d}{\hbar}\sqrt{2m(U - E_0)}\right)}$$ where $T_0 = 4\left[1 + \frac{1}{4} \left(\frac{\varkappa}{k} - \frac{k}{\varkappa}\right)^2\right]^{-1}$.

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Tunneling current density in metal-insulator-metal system

Let's consider a metal-insulator-metal (MIM) system with an arbitrary shape of potential barrier. The applied bias $V$ leads to the initiation of a tunneling current which depends on the energy density of states $n(p_z)$ and the probability $T(E_z)$ of electron transmission through the barrier of height $U(z)$. $N_1$ and $N_2$ are the number of electrons tunneling through the barrier in two directions, which determine the value of current.
Using the WKB approximation, we can derive an expression for the density of tunneling curren:

Model of MIM system with an arbitrary shape of potential barrier

$$ J = \frac{J_0}{\delta_z^2}\left\{\bar \phi \exp\left[{-A\delta_z\sqrt{\bar \phi}}\right] - (\bar \phi + eV) \exp\left[-A\delta_z\sqrt{\bar \phi + eV}\right] \right\}$$
$ J_0 = \frac{e}{4\pi^2\beta^2\hbar} $ and $ A = 2\beta\sqrt{\frac{2m}{\hbar^2}}$,

where
$e$ is the elementary charge,
$δ_z$ is the distance between two contacts,
$\bar \phi$ - average barrier height,
$\beta$ - the correction factor that depends on the bais, for more details refer to[7](#Bibliography).

The following variables were used for calculations:

$\Phi_s = $   eV; $\Phi_t = $ eV;
$d = $   Å ; $S = $   nm$^2$;
$V = $   V;

To plot the dependence of the tunneling current $I$ on the applied voltage $V$ determine the barrier parameters and an appropriate range of operating voltages:

Low-Voltage range
Intermediate-Voltage range
High-Voltage range

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Modeling the operation of STM

Open notebook

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