# Introduction to Nanomechanics (2024)

#### Overview

This course (not surprisingly) focuses on the mechanics of very small objects. In particular, we will discuss the regime of nanometer-scale objects where classical theory begins to break down and quantum mechanical behavior emerges. In order to do so, we will touch on continuum mechanics, statistical mechanics, opto-mechanics, and quantum mechanics. After covering the fundamentals of the field, we will study its most important results up to and including contemporary work. We will discuss efforts to bring nanomechanical resonators into the quantum mechanical regime. We will also emphasize the most practical and successful applications of these devices, i.e. as sensors of force, mass, and displacement.

The main topics to be covered include: mechanical sensors, cantilever mechanics (statics and dynamics), dissipation and noise in mechanical systems, nanomechanical transducers, cooling mechanical resonators, the standard quantum limit on displacement measurement, nanomechanical mass and force sensing, and current trends and applications.

#### Format and Requirements

The course consists of one 2-hour lecture per week and one 1-hour exercise session per week. Exercise sessions will be a forum to discuss and resolve assigned exercises. Exercises will not be graded, but their content will form the basis for the final exam and their completion is the best way to prepare for it. The final exam will be oral and will cover the major topics of the course. Grades will be on a scale of 1 to 6 based on this exam. The course will be conducted in English.

This course will be aimed at 3rd-year bachelor and master students in physics and nanoscience. Physics III is a prerequisite. Previous course-work in solid-state physics and statistical mechanics is expected.

Most of the source material and reading in this class will be drawn from original papers in scientific journals and will be provided in class. Some reading will be based on short sections of *Foundations of Nanomechanics*, A. N. Cleland (Springer, 2003), *Fundamentals of Nanomechanical Resonators*, S. Schmid, L. G. Villanueva, and M. L. Roukes (Springer, 2016), and *Fundamentals of Statistical and Thermal Physics*, F. Reif (McGraw-Hill, 1965). Copies of these readings will be distributed in class. A script for the full course is available for download here. Videos of all the lectures are available here.

#### Schedule

Lectures: Wednesdays, 10.00-12.00, 1.09

Exercise Session: Tuesdays, 13:00-14:00, 1.09

##### 18.09.2024

#### Preliminary Logistics and Introduction

Course outline and expectations; What is nanomechanics? Why study nanomechanics? Cantilever basics (static case). Lecture videos (Part 1, Part 2, Part 3).

**Exercise Session: **None.

**Reading: ***Foundations of Nanomechanics*: p. 191-211.

**Downloads:** Lecture notes, Lecture slides.

##### 25.09.2024

#### Cantilever Dynamics

Cantilever basics (dynamic case). Zener’s model for an anelastic solid. Lecture videos (Part 1, Part 2, Part 3, Part 4).

**Exercise Session: **Problem Set 1, harmonic oscillator (on 01.10)

**Reading: ***Foundations of Nanomechanics*: p. 233-237; *Fundamentals of Nanomechanical Resonators: p. 1-54.*

**Downloads:** Lecture notes.

##### 02.10.2024

__Cantilevers as Harmonic Oscillators__

Driven cantilevers; cantilevers as harmonic oscillators. Lecture videos (Part 1, Part 2, Part 3, Part 4).

**Exercise Session: **Problem Set 2, bending of beams (on 08.10)

**Reading: ***Foundations of Nanomechanics*: p. 277-301.

**Downloads:** Lecture notes.

##### 09.10.2024

__Dissipation and Noise in Mechanical Systems I__

Power spectral density; Fluctuation-dissipation theorem; thermal displacement. Lecture videos (Part 1, Part 2, Part 3).

**Exercise Session: **None

**Reading: ** *Fundamentals of Statistical and Thermal Physics: p. 560-573; Fundamentals of Nanomechanical Resonators: p.151-154.*

**Downloads:** Lecture notes, excerpt from thesis of Dr. Nicola Rossi about flexural motion of a beam.

##### 16.10.2024

__Dissipation and Noise in Mechanical Systems II__

Thermal displacement, phase, and frequency noise; thermally limited mechanical transducer; thermal limits on force, mass, and force gradient sensitivity. Lecture videos (Part 1, Part 2, Part 3, Part 4).

**Exercise Session: **Paper Discussion 1, mass sensing (on 22.10)

**Reading:** *Fundamentals of Statistical and Thermal Physics: p. 560-573; Fundamentals of Nanomechanical Resonators: p.151-154.*

**Downloads:** Lecture notes.

##### 23.10.2024

__Nanomechanical Measurements__

Measurement of displacement, frequency, and dissipation; types of transducers for nanomechanics; focus on optical interferometry. Lecture videos (Part 1, Part 2, Part 3, Part 4).

**Exercise Session: **Paper Discussion 2, cantilever magnetometry (on 29.10)

**Reading:** *Fundamentals of Nanomechanical Resonators: p.115-147.*

**Downloads:** Lecture notes.

##### 30.10.2024

__Cooling Mechanical Resonators I__

Freezing out thermal motion; motivations for ground-state cooling; quantum harmonic oscillator; cryogenic ‘brute-force’ cooling; feedback cooling. Lecture videos (Part 1, Part 2).

**Exercise Session:** None

**Reading:**M. Poggio et al., *Phys. Rev. Lett.* **99**, 017201 (2007).

**Downloads:** Lecture notes.

##### 06.11.2024

__Cooling Mechanical Resonators II__

Feedback cooling: practical considerations, measurement bandwidth, dynamic range, sensitivity. Lecture videos (Part 1, Part 2, Part 3).

**Exercise Session:** Paper Discussion 3, brute-force ground-state cooling (on 12.11)

**Reading:** F. Marquardt et al., *J. Mod. Opt.* **55**, 3329 (2008).

**Downloads:** Lecture notes.

##### 13.11.2024

__Cooling Mechanical Resonators III__

Cavity or optical cooling: semi-classical picture. Lecture videos (Part 1, Part 2, Part 3).

**Exercise Session:** Problem Set 3, quantum harmonic oscillator (on 19.11)

**Reading: **

**Downloads:** Lecture notes.

##### 20.11.2024

__Cooling Mechanical Resonators IV__

Cavity or optical cooling: quantum picture; quantum fluctuation-dissipation theorem. Lecture videos (Part 1, Part 2, Part 3).

**Exercise Session:** Paper Discussion 4, cavity-assisted ground-state cooling (on 26.11)

**Reading:** A. A. Clerk et al., *Rev. Mod. Phys.* **82**, 1155 (2010) and Appendix.

**Downloads:** Lecture notes.

##### 27.11.2024

__Standard Quantum Limit and Ground State Cooling__

Back-action; standard quantum limit (SQL); approaching SQL. Lecture videos (Part 1, Part 2, Part 3, Part 4).

**Exercise Session:** None

**Reading:**

**Downloads:** Lecture notes.

##### 04.12.2024

__Mechanics of Nanowires__

We will discuss nanowires (NWs) as nanomechanics resonators, including applications as mechanical sensors. Mechanical modes, non-linearity, mode coupling will be discussed. Lecture videos (Part 1, Part 2, Part 3).

**Exercise Session:** Paper discussion 5 (on 10.12)

**Reading:** N. Rossi, Ph.D. Thesis in Physics (2019)*.*

**Downloads:** Lecture slides; N. Rossi, *Ph.D. Thesis in Physics *(2019).

##### 11.12.2024

#### Sources of Mechanical Dissipation and its Reduction

We will discuss the origin of mechanical dissipation in nanomechanical elements and explain how this dissipation can be minimized. Lecture videos (Part 1, Part 2, Part 3).

**Exercise Session: **Problem Set 4 (on 17.12)

**Reading: ***Fundamentals of Nanomechanical Resonators*: p.57-90.

**Downloads:** Lecture notes.

##### 18.12.2024

#### Applications and Review

We will discuss applications of nanomechanical resonators. In particular, we will cover devices used for doing torque magnetometry and torque-detected magnetic resonance. We will review the major concepts covered in the course in view of the final exam. Lecture video (Part 1).

**Exercise Session: **None

**Reading:** None

**Downloads:** Lecture notes.