Question

In nanoscale electronics, electrons can be treated like billiard balls. The figure shows a simple device currently under study in which an electron elastically collides with a rigid wall (a ballistic electron transistor). The green bars represent electrodes that can apply a vertical force of \(8.00 \cdot 10^{-13} \mathrm{~N}\) to the electrons. If an electron initially has velocity components \(v_{x}=1.00 \cdot 10^{5} \mathrm{~m} / \mathrm{s}\) and \(v_{y}=0\) and the wall is at \(45.0^{\circ},\) the deflection angle \(\theta_{\mathrm{D}}\) is \(90.0^{\circ} .\) How long does the vertical force from the electrodes need to be applied to obtain a deflection angle of \(120 .^{\circ} ?\)

Short answer

Answer: The vertical force should be applied for approximately \(9.86 \cdot 10^{-14}\,\mathrm{s}\) to achieve a deflection angle of 120 degrees.

Step-by-step solution

Convert angles into radians

In order to work more easily with trigonometric functions, we must first convert the given angles (45, 90, and 120 degrees) into radians. We can do that using the conversion factor \((\pi/180)\). \(45^{\circ} = 45(\pi/180) \approx 0.785 \mathrm{~rad}\) \(90^{\circ} = 90(\pi/180) = \pi/2\) \(120^{\circ} = 120(\pi/180) \approx 2.09 \mathrm{~rad}\)

Calculate the acceleration in \(x\) and \(y\) directions

Given the vertical force applied on the electron, we can calculate the acceleration in the \(y\) direction. To do that, we need to consider the electron's charge \(e = 1.6 \cdot 10^{-19} \mathrm{C}\) and its mass \(m = 9.11 \cdot 10^{-31} \mathrm{kg}\). The force is given by \(F_y = 8 \cdot 10^{-13} \mathrm{N}\). We can find the acceleration using the following equation: \(a = F/m\). \(a_y = F_y / m = (8.00 \cdot 10^{-13} \mathrm{N}) / (9.11 \cdot 10^{-31} \mathrm{kg}) \approx 8.78 \cdot 10^{17} \mathrm{m}/\mathrm{s}^2\) Since there is no horizontal force applied, the acceleration in the \(x\) direction will be \(0\).

Calculate the final velocities in the \(x\) and \(y\) directions

Using the desired deflection angle of \(120^{\circ}\) (or \(2.09\) rad), we can find the final velocities in the \(x\) and \(y\) directions: \(v_{x \: final} = v_{x \: initial} * \cos{(2.09)} \approx -5.00 \cdot 10^4 \mathrm{m}/\mathrm{s}\) \(v_{y \: final} = v_{x \: initial} * \sin{(2.09)} \approx 8.66 \cdot 10^4 \mathrm{m}/\mathrm{s}\)

Calculate the time it takes to achieve the desired deflection angle

Now we can use the equation \(v_f = v_i + a*t\) for the \(y\) direction to find the time required to reach the final deflection angle. Remember that \(v_{y \: initial} = 0\). \(v_{y \: final} = a_y * t\) Solving for \(t\): \(t = v_{y \: final} / a_y\approx (8.66 \cdot 10^{4}\mathrm{m}/\mathrm{s}) / (8.78 \cdot 10^{17}\mathrm{m}/\mathrm{s}^2) \approx 9.86 \cdot 10^{-14}\,\mathrm{s}\) So, the vertical force from the electrodes needs to be applied for approximately \(9.86 \cdot 10^{-14}\,\mathrm{s}\) in order to obtain a deflection angle of \(120^{\circ}\).

Key concepts

Nanoscale Electronics

Nanoscale electronics represents a field of technology concerned with the development of electronic components and devices at an extremely small scale, typically under 100 nanometers. At this scale, quantum mechanical effects become significant and the behavior of electrons can differ markedly from that seen in larger-scale circuits.

For example, in a ballistic electron transistor—an advanced nanoscale electronic device—an electron' trajectory inside the device can be reflected and deflected much like a billiard ball bounces off the sides of a pool table. The fascinating part is that at the nanoscale, traditional resistive effects that would slow down an electron in a normal circuit are minor or nonexistent. Instead, the electrons travel in straight lines until they strike a boundary or are deflected by electrostatic forces from electrodes, as outlined in the exercise.

Understanding the behavior of electrons at this scale is crucial for the design and optimization of nanoelectronic devices. With devices growing ever smaller, the ability to control electron flow with precision is key to improving performance and developing new technologies.

Electron Collision Physics

In electron collision physics, we study how electrons interact when they collide with other particles or obstacles. This is particularly relevant in the operation of devices like the ballistic electron transistor mentioned in our exercise. Normally, an electron moving within a material would experience numerous random collisions with atoms and other electrons, which scatter it in various directions. This scattering process can impede the electron's mobility, generate heat, and ultimately limit the efficiency of electronic components.

However, in an ideal ballistic transport scenario, collisions are either perfectly elastic or are entirely avoided, allowing electrons to move straight through a device without scattering. To manipulate the path of an electron in these collision-free environments, forces, such as the vertical force exerted by electrodes in the exercise, are applied. When this force is exerted, we need to understand the dynamics of the electron's motion, which includes its mass, velocity, and the time and magnitude of the applied force to predict how its trajectory will change after the collision with a wall.

Electrostatic Deflection

Electrostatic deflection refers to the bending of an electron's path due to the influence of an electric field. In the case of our exercise, this deflection is achieved by applying a vertical force by using green bars as electrodes. The force exerted on an electron by an electrostatic field is directly proportional to the charge of the electron and the magnitude of the electric field. In simpler terms, you can control where and how fast an electron moves by turning on and adjusting the strength of an electric field.

Applying a vertical electrostatic force perpendicular to the initial motion of an electron, as in our ballistic electron transistor, will cause the electron's path to curve. The trajectory will change based on how long the force is applied. The longer the force is active, the more pronounced the deflection. Calculating the precise deflection angle given in the textbook exercise involves using trigonometry and knowledge of the time over which the force acts on the electron. This concept is vital for steering electron beams in various devices, from classic cathode ray tubes to modern semiconductor components.