91Ó°ÊÓ

A free electron that contributes to the current in an ohmic material typically has a speed of \(10^{5} \mathrm{~m} / \mathrm{s}\) (much greater than the drift velocity). (a) Estimate its de Broglie wavelength, in \(\mathrm{nm}\).(answer check available at lightandmatter.com) (b) If a computer memory chip contains \(10^{8}\) electric circuits in a \(1 \mathrm{~cm}^{2}\) area, estimate the linear size, in \(\mathrm{nm}\), of one such circuit.(answer check available at lightandmatter.com) (c) Based on your answers from parts a and b, does an electrical engineer designing such a chip need to worry about wave effects such as diffraction? (d) Estimate the maximum number of electric circuits that can fit on a \(1 \mathrm{~cm}^{2}\) computer chip before quantum-mechanical effects become important.

Step by step solution

01

Calculate Electron's de Broglie Wavelength

First, we need to calculate the de Broglie wavelength of the electron using the formula: \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ m}^2\text{kg/s} \), \( m \) is the mass of the electron \( 9.109 \times 10^{-31} \text{ kg} \), and \( v \) is the speed of the electron \( 10^5 \text{ m/s} \). Substituting these values into the formula gives: \[ \lambda = \frac{6.626 \times 10^{-34}}{9.109 \times 10^{-31} \times 10^5} = 7.27 \times 10^{-9} \text{ m} = 7.27 \text{ nm}. \]

02

Estimate Linear Size of Single Circuit

The problem states there are \( 10^8 \) circuits in a \( 1 \text{ cm}^2 \) area. To find the linear size of one circuit, assume each circuit is approximately square. Taking the square root of the area occupied by one circuit will give its linear size. The area of one circuit is \( \frac{1 \text{ cm}^2}{10^8} = 10^{-8} \text{ cm}^2 \). The linear size is then \( \sqrt{10^{-8} \text{ cm}^2} = 10^{-4} \text{ cm} = 1000 \text{ nm}. \)

03

Assess Importance of Wave Effects

Compare the de Broglie wavelength of the electron with the linear size of a circuit. The de Broglie wavelength is approximately \( 7.27 \text{ nm} \), while the linear size of a circuit is about \( 1000 \text{ nm} \). Since the wavelength is much smaller than the size of the circuits, wave effects such as diffraction are not significant for circuit design at this scale.

04

Determine Circuit Density Before Wave Effects Matter

To estimate the circuit density limit before quantum-mechanical effects become significant, assume circuits must be at least the size of the electron's de Broglie wavelength (\( 7.27 \text{ nm} \)). From \( 1 \text{ cm}^2 = 10^8 \text{ nm}^2 \), the maximum number of circuits is \( \frac{10^8}{(7.27)^2} \approx 1.9 \times 10^6 \). Therefore, up to \( 1.9 \times 10^6 \) circuits could potentially fit before quantum effects become significant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

de Broglie wavelength

The concept of de Broglie wavelength forms the core of quantum mechanics, tying waves and particles into one framework. According to de Broglie, every moving particle or object has an associated wavelength, known as its de Broglie wavelength. This is given by the equation:\[ \lambda = \frac{h}{mv} \]where:

• \( \lambda \) is the de Broglie wavelength
• \( h \) is Planck's constant \(6.626 \times 10^{-34} \text{ m}^2\text{kg/s}\)
• \( m \) is the mass of the particle, for an electron it's \(9.109 \times 10^{-31} \) kg
• \( v \) is the velocity of the particle

By applying this equation, we can calculate the de Broglie wavelength for an electron. For example, with an electron speed of \(10^5\) m/s, we find the wavelength to be approximately \(7.27\) nm. This wavelength shows the scale at which quantum mechanical effects like diffraction can occur, even for tiny particles like electrons.

Ohm's Law

Ohm's Law is a fundamental principle in the world of electronics and electrical circuits. It relates the voltage (V), current (I), and resistance (R) in an electrical circuit. The law is expressed with the equation:\[ V = IR \]This tells us:

• The voltage across a resistor is equal to the product of the current flowing through it and the resistance.
• If you increase the voltage, the current will increase proportionally, provided the resistance remains the same.
• Similarly, if the resistance increases for a constant voltage, the current must decrease.

Ohm’s Law is pivotal in designing and analyzing circuits as it predicts how the electrical components of a circuit will interact. However, as circuit elements decrease in size to the nanoscale, quantum effects may begin to alter the straightforward application of Ohm's Law, challenging engineers to rethink traditional electrical designs.

Quantum Effects in Circuits

As electronic components shrink to the size of a few nanometers, quantum mechanics begins to dominate their behavior. Quantum effects in circuits become significant when the dimensions of circuit elements become comparable to the de Broglie wavelength of the electrons. These effects include:

• Quantum Tunneling: Electrons can pass through insulators even if they lack the energy required to go over the potential barrier according to classical physics.
• Wave-Particle Duality: Electrons can exhibit wave-like behavior, affecting how they move and interact within small circuit components.
• Discrete Energy Levels: At very small scales, electrons can occupy only certain energy levels, influencing how charge is stored and propagated.

Understanding and managing these effects is crucial for developing nanoscale technologies, which are the future of computing and electronics. As circuits approach the size of an electron’s wavelength, traditional engineering methods must adapt to accommodate the peculiarities of quantum physics.

Drift Velocity

In electrical circuits, drift velocity refers to the average velocity that free electrons gain in a material due to an electric field. Despite the electric field, these electrons do not move at their thermal speed. Instead, they acquire a much smaller drift speed. Drift velocity \( v_d \) is given by:\[ v_d = \frac{I}{nAq} \]where:

• \( I \) is the current flowing through the conductor.
• \( n \) is the number of charge carriers per unit volume.
• \( A \) is the cross-sectional area of the conductor.
• \( q \) is the charge of the electron, approximately \(1.6 \times 10^{-19} \text{ C}\).

The drift velocity is usually much lower than the individual electron's velocity within the conductor due to constant collisions with atoms. Understanding drift velocity is important for designing circuits, as it affects the current flow and the efficiency of electron transport within materials. As we venture into nanoscale circuits, controlling drift velocity becomes even more critical, combating issues like increased resistivity due to reduced size.