Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova

Lecture 11

GrapheneYet another amazing form of

carbon

Andre Geim Konstantin Novoselov

Nobel Prize in Physics 2010

Univ. of Manchester, UK

Carbon: the Element of LifeCarbon: the Element of LifeHas unique flexibility for bonding and ability to make complex

compounds

All life forms on Earth, from viruses to complex mammals (including humans) are based on carbon chemistry.

The Tobacco Mosaic Virus

contains a single strand

of RNA, about 0.1 mm long

This complex mammal contains about 3

billion miles of DNA.

Even pure carbon can be present in a variety of forms:

Graphene (top left) is a 2D honeycomb lattice of carbon atoms. Graphite (top right) can be viewed as a stack of graphene layers. Carbon nanotubes are rolled-up cylinders of graphene (bottom left). Fullerenes C60 (bottom right) are molecules consisting of wrapped graphene by the introduction of pentagons on the hexagonal lattice. (From Castro-Neto et al. 2006)

Diamond vs. graphite

Cubic Lattice,Very tight, inflexible

HoneycombSheets that easily slide (pencil)

How Geim and Novoselov produced it

• They used Scotch tape to repeatedly split graphite crystals into increasingly thinner flakes

• Then placed the flakes to a silicon dioxide substrate to prevent them from scrolling

• You could do this as well (if only you knew what to look for)

Graphene structure

A single layer of carbon atoms tightly packed into a honeycomb lattice

Electron dispersion (dependence of electron energy from its momentum)

Note linear dependence E(p) near E = 0!!

What does it mean?

• Ultrahigh mobility, low resistance (like in copper!)

• Unique optical properties (absorption independent on wavelength)

• Unique magnetic properties

• Penetration through energy barriers

p

E

Electron Dynamics in GrapheneElectron Dynamics in Graphene

Potential applications

• Transistors• Integrated circuits• Lasers• Detectors• memory

Work Energy Theorem

22

222

1

initialfinalr

rtotal

mVmVrdFW

22

2222

11

22

11

initialfinalyx

yx

totaly

yx

yx

totalx

mVmVdyFdxF

A person is pulling a crate of mass M along the floor with a constant force F over a distance d. The coefficient of friction is .

(a) Find the work done by the force F on the crate.(b) Same if F changes as F0(1+x2/d2). (c) Find the work done by the force of friction on the crate (F is constant). (d) Find the net work done on the crate if the crate is pulled with a constant

velocity. (e) Find the final velocity of the crate if the crate is pulled with a non-zero

acceleration starting from the rest.

M

A block of mass m starts at the top of an inclined plane. The coefficient of friction between the plane and the block is . Assuming the block slides down the plane calculate the work done by each force.

ixxkFLawsHooke)(:' 0

x

positionunstrechedisx0

Problem 2 p.122

A 3 slug mass is attached to a spring which is pulled out one foot. The spring constant k is 100 pounds/ft. How fast will the mass be moving when the spring is returned to its unstretched length? (Assume no friction.)

Problems from handout

Problem 3

A 5.00 kg block is moving at v0=6.00 m/s along a frictionless, horizontal surface toward a spring with constant k=500 N/m that is attached to a wall.

a) Find the maximum distance the spring will be compressed.

b)If the spring is to compress by no more than 0.150 m, what should be the maximum value of v0?

Have a great day!