Lessons from Nanoelectronics

The following sections are included:

• Dedication
• Preface
• Acknowledgments
• List of Available Video Lectures
• Constants Used in This Book
• Some Symbols Used
• Contents

The following sections are included:

• Conductance
• Ballistic Conductance
• What Determines the Resistance?
• Where is the Resistance?
• But Where is the Heat?
• Elastic Resistors
• Transport Theories
  • Why elastic resistors are conceptually simpler
• Is Transport Essentially a Many-body Process?
• A Different Physical Picture

It is a well-known and well-established fact that when the two terminals of a battery are connected across a conductor, it gives rise to a current due to the flow of electrons across the channel from the source to the drain.

If you ask anyone, novice or expert, what causes electrons to flow, by far the most common answer you will receive is that it is the electric field. However, this answer is incomplete at best. After all even before we connect a battery, there are enormous electric fields around every atom due to the positive nucleus whose effects on the atomic spectra are well-documented. Why is it that these electric fields do not cause electrons to flow, and yet a far smaller field from an external battery does?…

We saw in the last chapter that the flow of electrons is driven by the difference in the “agenda” of the two contacts as reflected in their respective Fermi functions, f₁(E) and f₂(E). The negative contact with its larger f(E) would like to see more electrons in the channel than the positive contact. And so the positive contact keeps withdrawing electrons from the channel while the negative contact keeps pushing them in. This is true of all conductors, big and small. But in general, it is difficult to express the current as a simple function of f₁(E) and f₂(E), because electrons jump around from one energy to another and the current flow at different energies is all mixed up…

We saw in the last chapter that the resistance of an elastic resistor can be written as

The following sections are included:

• Introduction
• Current Fluctuations in an Elastic Resistor
  • One-level resistor
  • Multi-level resistor

The following sections are included:

• Introduction
• E(p) or E(k) Relation
• Counting States
  • Density of states, D(E)
• Number of Modes
  • Degeneracy factor
• Electron Density, n
  • n-type conductors
  • p-type conductors
  • “Double-ended” density of states
• Conductivity versus n

Our “field-less” approach to conductivity comes as a surprise to many since it is commonly believed that currents are driven by electric fields. However, we hasten to add that the field can and does play an important role once we go beyond low bias and our purpose in this chapter is to discuss the role of the electrostatic potential and the corresponding electric field on the current-voltage characteristics beyond low bias…

The following sections are included:

• Introduction
  • A disclaimer
• Electrochemical Potentials Out of Equilibrium
• Current from QFL’s

The following sections are included:

• Introduction
• BTE from “Newton’s Laws”
• Diffusion Equation from BTE
  • Equilibrium fields can matter
• The Two Potentials

The following sections are included:

• Introduction
• The Landauer Formulas (Eqs. (10.1) and (10.2))
• Büttiker Formula (Eq. (10.3))
  • Application
  • Is Eq. (10.3) obvious?
  • Non-reciprocal circuits
  • Onsager relations

The following sections are included:

• Introduction
• Why n- and p-type Conductors are Different
• Spatial Profile of Electrochemical Potential
  • Obtaining Eq. (11.14) from BTE
• Edge States

A key insight of modern nanoelectronics is the concept of an interface resistance R_B that is in series with the standard length-dependent resistance Eq. (1.8):

The interface resistance depends solely on the properties of the channel and cannot be eliminated even with an ideal contact…

The following sections are included:

• Introduction
• Seebeck Coeffcient
• Thermoelectric Figures of Merit
• Heat Current
  • Linear response
• The Delta Function Thermoelectric
  • Optimizing power factor

The following sections are included:

• Introduction
• Phonon Heat Current
  • Ballistic phonon current
• Thermal Conductivity

The following sections are included:

• Introduction
• Asymmetry of Absorption and Emission
• Entropy
  • Total entropy increases continually
  • Free energy decreases continually
• Law of Equilibrium
• Fock Space States
  • Bose function
  • Interacting electrons
• Alternative Expression for Entropy
  • From Eq. (15.24) to Eq. (15.25)
  • Equilibrium distribution from minimizing free energy

The following sections are included:

• Introduction
• Information-driven Battery
• Fuel Value Comes from Knowledge
• Landauer’s Principle
• Maxwell’s Demon
• Suggested Reading

The following sections are included:

• Appendix A Derivatives of Fermi and Bose Functions
  • Fermi Function
  • Bose Function
• Appendix B Angular Averaging
  • One Dimension
  • Two Dimensions
  • Three Dimensions
  • Summary
• Appendix C Current at High Bias for Non-degenerate Resistors
• Appendix D Semiclassical Dynamics
  • Semiclassical Laws of Motion
    • Proof
• Appendix E Transmission Line Parameters from BTE

The following section is included:

• Index