Question

A surface is examined using a scanning tunneling microscope (STM). For the range of the working gap, \(L\), between the tip and the sample surface, assume that the electron wave function for the atoms under investigation falls off exponentially as \(|\Psi|=e^{-\left(10.0 \mathrm{nm}^{-1}\right) a}\). The tunneling current through the STM tip is proportional to the tunneling probability. In this situation, what is the ratio of the current when the STM tip is \(0.400 \mathrm{nm}\) above a surface feature to the current when the tip is \(0.420 \mathrm{nm}\) above the surface?

Short answer

Answer: The ratio of the current when the STM tip is 0.4 nm above a surface feature to the current when the tip is 0.42 nm above the surface is approximately 1.221.

Step-by-step solution

Determine the wave function at 0.4 nm distance

We have to determine the wave function of the electron at a distance of 0.4 nm. We'll use the given exponential function with \(a=0.4 \mathrm{nm}\): \(|\Psi_{0.4}|=e^{-\left(10.0 \mathrm{nm}^{-1}\right) (0.4 \mathrm{nm})}\)

Determine the wave function at 0.42 nm distance

Next, we need to determine the wave function of the electron at a distance of 0.42 nm. We'll use the given exponential function again with \(a=0.42 \mathrm{nm}\): \(|\Psi_{0.42}|=e^{-\left(10.0 \mathrm{nm}^{-1}\right) (0.42 \mathrm{nm})}\)

Calculate the ratio between the wave functions

To find the ratio between the wave functions, we need to divide the value of the wave function at 0.4 nm by the value of the wave function at 0.42 nm: \(\frac{|\Psi_{0.4}|}{|\Psi_{0.42}|}=\frac{e^{-\left(10.0 \mathrm{nm}^{-1}\right) (0.4 \mathrm{nm})}}{e^{-\left(10.0 \mathrm{nm}^{-1}\right) (0.42 \mathrm{nm})}}\)

Simplify the ratio

We can simplify this expression by using the exponent rules: \(\frac{|\Psi_{0.4}|}{|\Psi_{0.42}|}=e^{\left(10.0 \mathrm{nm}^{-1}\right)(0.42 \mathrm{nm} - 0.4 \mathrm{nm})}\)

Calculate the ratio numerically

Now, we just need to plug in the numbers and compute the ratio: \(\frac{|\Psi_{0.4}|}{|\Psi_{0.42}|}=e^{\left(10.0 \mathrm{nm}^{-1}\right)(0.02 \mathrm{nm})}=e^{0.2}\)

Conclude the result

Since the tunneling current ratio is equal to the wave function ratio, the ratio of the current when the STM tip is 0.4 nm above a surface feature to the current when the tip is 0.42 nm above the surface is: \(\frac{I_{0.4}}{I_{0.42}}=e^{0.2} \approx 1.221\)

Key concepts

Tunneling Current

The scanning tunneling microscope (STM) relies on the concept of tunneling current, a phenomenon that emerges from the principles of quantum mechanics. Tunneling current occurs when electrons move through a barrier that, according to classical physics, would be insurmountable – in this case, the vacuum between the STM tip and the sample surface. This current is not a flow in the traditional sense, but a manifestation of the probability of an electron being found across the barrier due to its wave-like properties.

The strength of the tunneling current is heavily dependent on the distance between the STM tip and the surface feature because it is proportional to the probability of electron tunneling. The closer the tip to the surface, the higher the tunneling probability, and hence, the greater the tunneling current. The ability of the STM to produce atomic-scale images is primarily owed to the sensitivity of the tunneling current to the distance between the tip and the surface, allowing researchers to map the surface with incredible precision by scanning the tip across it and recording the changing current.

Electron Wave Function

An electron wave function, denoted as \(|\text{\Psi}|\), is a mathematical function that describes the quantum state of an electron in a system. This function is central to understanding various quantum phenomena, including the operation of a scanning tunneling microscope (STM).

The amplitude of this wave function is a critical factor in determining the probability of finding an electron in a certain position. In the context of the STM, the wave function's amplitude is used to calculate the tunneling current. Significantly, this amplitude is not constant but changes with distance from the atomic surface, exhibiting exponential decay. Due to this decay, the tunneling current drops rapidly as the STM tip moves away from the feature on the surface, thus affecting the resolution and the ability to visualize the surface effectively. Understanding electron wave functions enables scientists to predict and manipulate the chances of an electron's location, which is foundational for imaging in STM.

Exponential Decay

The concept of exponential decay is prevalent in many areas of science, including the study of electron wave functions in the context of STMs. When we express the fall-off of an electron wave function as an exponential function of distance, we observe that the wave function’s amplitude decreases rapidly as the electron moves further away from the source (atom's surface or STM tip).

In our example, the electron wave function falls off exponentially with the distance \(a\) from the surface as \(|\Psi|=e^{-\left(10.0 \mathrm{nm}^{-1}\right) a}\). This signifies that with each incremental increase in distance, the amplitude of the wave function is multiplied by a constant factor, here represented by \(e^{-10.0 \mathrm{nm}^{-1}}\). This results in the rapid drop-off of the probability for electron tunneling, which is directly associated with the measurable tunneling current. Hence, even small changes in distance can lead to noticeable changes in the current, allowing the STM to discern very fine surface details. The exponential nature of this decay enforces the precision in distance control necessary for effective STM imaging.