APPENDIX E: Proof of the Field Emission Formula

We begin with the expression for the current density /

We perform a change of variable u = dF(Ex - E?), and we define c = dFkT. We can assume that с < 1 and that the bottom of the conduction band is sufficiently deep, i.e. Eb F. We get

» CO ~u

We shall prove that the above integral, I = e"ln 1 + e c du, is equal to nlsin(nc).

We perform an integration by parts L

Now assuming c < 1, the first term is 0. For the second we perform the change of variable


v = ec and we get

This integral can be found from tables. It is equal to nlsin(nc).


This list of suggested books serves to expand ones knowledge in each of the areas under which each book is listed. The research articles quoted in the present book give a broader view of specific subjects treated in the text. They are not listed here but at the end of each chapter.


1) A. P. French and Edwin F. Taylor, An Introduction to Quantum Physics, The MIT Introductory Physics Series, 1978.

Although it seems introductory, it goes deeper than its title suggests. Written in the 1970’s it is still in use today with a kindle edition.

2) Peter LHagelstein, Stephen D. Senturia, and Terry Orlando, Introductory Applied Quantum and Statistical Mechanics, Wiley, 2004.

A recent book that includes chapters on statistical mechanics. Not so applied as it claims.

3) P. A.M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, 1976.

This is the standard book on the fundamental principles by one of the founders of Quantum Mechanics. Not easy because of the abstract notation, but probably the book with the clearest presentation of principles.


1) J.M. Ziman, Principles of the Theory of Solids, Cambridge University Press (2nd edition), 1972.

A classic and authoritative textbook on the foundations of solid state theory but it does not contain much about Quantum Transport. Though written decades ago, it is still useful. It has been translated in many languages.

2) Giuseppe Grosso and Giuseppe Pastori Parraviani, Solid State Physics, Academic Press, 2000.

A recent, comprehensive, and clear book.


1) YuanTaur and Так H. Ning, Fundamentals of Modern VLSI Devices (2nd edition), Cambridge University Press, 2009.

A comprehensive textbook mainly on devices and electronics.

  • 2) Donald A. Neamen, Semiconductor Physics and Devices (4th edition), McGraw-Hill, 2011
  • 3) Umesh K. Mishra and Jasprit Singh, Device Physics and Design, Springer, 2008.

It gives a comprehensive but classical coverage of devices and offers a selection of a few Quantum Transport cases in the appendices.


1) John H. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge University Press, 1998.

The book deals almost exclusively with Solid State Physics at the nano-level but contains a good account of Quantum Transport theory.

2) Vladimir V. Mitin, Viatcheslav A.Kochelap, and Michael A.Strossio, Quantum Heterostructures, Cambridge University Press, 1999.

A book with a good blend of Physics and Devices at the nano-level.

3) Mark Lundstrom, Fundamental of Nanotransistors, World Scientific Publishing, 2018.

A book by one of the main contributors in the field. It deals almost exclusively with the MOSFET but it will be a book to be consulted frequently.

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