1. 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.

2. Fundamental Length Scales in Physics Quantum Electric Magnetic Quantum Well: Quantum Well Laser Capacitor: Single Electron Transistor Magnetic Particle: Data Storage Media a = V 1/3 Charging Energy 2e 2 /  d Spin Flip Barrier ½ M 2 a 3 Energy Levels 3h 2 /8m l 2 d E1E1 E0E0 l l 3 nm

3. Quantum Corral 48 iron atoms are assembled into a circular ring. The ripples inside the ring are electron waves.

4. Building a Quantum Corral for Manipulating Electron Wave Functions Crommie and Eigler 21 3 4

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

7. Kanji character for atom (lit. original child) Carbon monoxide man

8. 1 nm ≈ 5 atoms Between 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.

9. Quantum Length Scale Quantum Well, Corral: Quantum Well Laser Energy Level Spacing: E 1  E 0 = 3h 2 /8m l 2 E1E1 E0E0 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 = p 2 /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 = h 2 /2m 2 The quantum box restricts : 1 = l 0 = 2 l E 1  E 0 > k B T 

10. Electric Length Scale Capacitor, Quantum Dot: Single Electron Transistor Charging Energy E C = 2e 2 /  d d d < 9 nm  E C > k B T 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 : E C = 2e 2 /  d (e = electron charge) (d = sphere diameter) (  =12 used, i.e. silicon)

11. Magnetic Length Scale Magnetic Particle: Data Storage Media a = V 1/3 Spin Flip Barrier E M = ½ M 2 a 3 a > 3 nm  E M > k B T 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 : E M = ½ M 2 a 3 (e = electron charge) (a = average diameter) (cgs unit system) The magnetization M is estimated from the magnetic moment 2  B = eh/2  mc of an iron atom in a magnet and the iron atom density.

12. ElasticInelastic  E = 0  E > 0 Scattering Potential  Electron- Electron- Trapping at Diffraction, Phase Shift Electron Phonon an Impurity Semicond: long long  10 nm Metal: 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-e- e-e- h+h+ e-e- e-e- e-e- phonon (Room temperature, longer at low temp.) Scattering Lengths

13. Screening Lengths l ~ 1 /  n ( n = Density of screening charges) Metals: Semiconductors: Electrolytes: Electrons at E Fermi 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)  q e -r/l r V r l

14. Length Scales in Electrochemistry Screening Electric: E Coulomb = k B T Debye-Hückel Length Electrolyte Bjerrum Length, Gouy-Chapman Length Dielectric Pure H 2 O l B = 0.7 nm l B = e 2 /  k B T, l GC = 2 / l B e  = r Coulomb l DH = (  k B T / 4   n i q i 2 ) ½ = 1 / (4  l B  n i z i 2 ) ½ 0.1 Molar Na + Cl - l DH = 1.0 nm n i,q i =ez i l GC -- e lBlB -e e

15. Length Scales in Polymers (including Biopolymers, such as DNA and Proteins) Random Walk, Entropy Stiffness  vs. k B T Persistence Length (straight segment) l P =  / k B T DNA (double) Polystyrene l P  50 nm l P  1 nm lPlP cos  = 1/e a Radius of Gyration (overall size, N straight segments) R G  l P  N Copolymers R G  20-50 nm RGRG

16. Self-Organization via two Competing Length Scales Short Range Attraction versus Long Range Repulsion Ferromagnetic Exchange:  Magnetic Dipole Interaction:  FerromagnetDiblock Copolymer Hydrophilic versus Hydrophobic Depends on the relative block size