Length Scales in Physics, Chemistry, Biology,…

Length scales are useful to get a quick idea what will happen when making objects smaller and smaller. For example, quantum physics kicks in when structures become smaller than the wavelength of an electron in a solid. In that case,

the electrons get squeezed into a “quantum box” and have

to adapt to the shape of the solid by changing their wave function. Their wavelength gets shorter, and that increases their energy. Since the wave function of the outer electrons determines the chemical behavior, one is able to come close to realizing the medieval alchemist’s dream of turning one chemical element into another.

Fundamental Length Scales in Physics

Quantum Electric Magnetic

Quantum Well:Quantum Well Laser

Capacitor:Single Electron Transistor

Magnetic Particle:Data Storage Media

a = V1/3

Charging Energy

2e2/ d

Spin Flip Barrier

½ M2a3

Energy Levels

3h2/8m l2

d

E1

E0

l

l < 7 nm d < 9 nm a > 3 nm

Quantum Corral

48 iron atoms are assembled into a circular ring.

The ripples inside the ring are electron waves.

Building a Quantum Corral for Manipulating Electron Wave Functions

Crommie and Eigler

21

3 4

http://www.almaden.ibm.com/vis/stm/gallery.html

Kanji character for atom (lit. original child)

Carbon monoxide man

1 nm ≈ 5 atomsBetween an atom and a

solid

A chain of N atoms (each carrying

one electron) creates N energy

levels.

With increasing chain length these become so dense that they

form a band.

As the bands become wider, the energy gap between them shrinks.

Quantum Length Scale

Quantum Well, Corral:Quantum Well Laser

Energy Level

Spacing:

E1E0 = 3h2/8m l2

E1

E0

l

l < 7 nm

Consider the two lowest energy levels of an electron in a box (in one dimension):

The energy E of an electron is determined by its momentum p in classical physics: E = p2/2m (m = electron mass)

Quantum physics relates the

momentum p to the wavelength of

the electron: p = h/ (De Broglie) (h=Planck’s constant)

That produces an inversely quadratic

relation between E and :

E = h2/2m 2

The quantum box restricts :

1 = l

0 = 2 l

E1E0 > kBT

Electric Length Scale

Capacitor, Quantum Dot:Single Electron Transistor

Charging Energy

EC = 2e2/ d

d

d < 9 nm

EC > kBT

Consider a metallic sphere with a single electron spread out over its surface.

It is embedded into an substrate with

dielectric constant , forming a

capacitor with a positive countercharge at infinity.

The electrostatic energy stored in this capacitor is given by Coulomb’s law :

EC = 2e2/ d (e = electron charge)

(d = sphere diameter) ( =12 used, i.e. silicon)

Magnetic Length Scale

Magnetic Particle:Data Storage Media

a = V1/3

Spin Flip Barrier

EM = ½ M2a3

a > 3 nm

EM > kBT

Consider a needle-shaped magnetic particle with two possible magnetization directions:

The magnetic energy barrier is proportional to the volume of the particle, i.e. the third power of its average dimension a :

EM = ½ M2a3 (e = electron charge)

(a = average diameter) (cgs unit system) The magnetization M is estimated from the magnetic moment 2B = eh/2mc of

an iron atom in a magnet and the iron atom density.

Elastic Inelastic E = 0 E > 0

Scattering Potential Electron- Electron- Trapping atDiffraction, Phase Shift Electron Phonon an Impurity

Semicond: long long 10 nmMetal: long 1000 nm 100 nm

Consequences:• Ballistic electrons at small distances (extra speed gain in small transistors)• Recombination of electron-hole pairs at defects (energy loss in a solar cell)• Loss of spin information (optimum thickness of a magnetic hard disk sensor)

e-

e-e-h+

e- e-

e-phonon

(Room temperature,longer at low temp.)

Scattering Lengths

Screening Lengths

l ~ 1 / n (n = Density of screening charges)

Metals: Semiconductors: Electrolytes:

Electrons at EFermi Electrons, Holes Ions

Thomas-Fermi: 0.1 nm Debye: 1-1000 nm Debye-Hückel: 0.1-

100 nm

Exponential cutoff of the Coulomb potential (dotted) at the screening length l .

V(r) qe-r/l

r

V

rl

Length Scales in Electrochemistry

Screening Electric: ECoulomb = kBT

Debye-Hückel Length Electrolyte

Bjerrum Length, Gouy-Chapman

Length Dielectric

Pure H2O

lB = 0.7 nm

lB = e2 / kBT , lGC = 2 / lBe

= rCoulomb

lDH = ( kBT / 4 niqi2

) ½

= 1 / (4 lB nizi2

) ½

0.1 Molar Na+Cl-

lDH = 1.0 nm

ni,qi=ezi

lGC

- e

lB

-e e

Length Scales in Polymers (including Biopolymers, such as DNA and

Proteins)Random Walk, Entropy Stiffness vs. kBT

Persistence Length (straight segment)

lP = / kBT

DNA (double) Polystyrene

lP 50 nm lP 1 nm

lPcos = 1/e

a

Radius of Gyration(overall size, N straight

segments)

RG lP N

Copolymers

RG 20-50 nm

RG

Self-Organization via two Competing Length Scales

Short Range Attraction versus Long Range Repulsion

Ferromagnetic Exchange: Magnetic Dipole Interaction:

Ferromagnet Diblock Copolymer

Hydrophilic versus Hydrophobic

Depends on the relative block size