Apr. 15, 17 Midterm Exam I Chap. 7: Elasticity /Phonons
Jun. 10, 12 Final Exam (review & perspective)
Apr. 22, 24 Chap. 8: Particles & Waves
2013. Spring: Introduction to Electro-Physics (Prof. Sin-Doo Lee, Rm. 301-1109); http://mipd.snu.ac.kr Electronic Properties of Engineering Materials James D. Livingston: MIT Series (John Wiley, New York, 1999)
Fig. 1.5: (a) Covalent bonding in methane, CH4, involves fourhydrogen atoms sharing electrons with one carbon atom. Eachcovalent bond has two shared electrons. The four bonds areidentical and repel each other. (b) Schematic sketch of CH4 onpaper. (c) In three dimensions, due to symmetry, the bonds aredirected towards the corners of a tetrahedron.
Fig. 1.7: In metallic bonding the valence electrons from the metalatoms form a "cloud of electrons" which fills the space between themetal ions and "glues" the ions together through the coulombicattraction between the electron gas and positive metal ions.
0.3 Metallic Bonding: electron gas or cloud (collective sharing of electrons)
Instantaneous electron (negative charge)distribution fluctuates about the nucleus.
Fig. 1.13: Induced dipole-induced dipole interaction and the resultingvan der Waals force.
Lattice
Basis
Crystal
Unit cell Unit cell
a
a
(a) (b) (c)
90
x
y
Basis placement in unit cell(0,0)
(1/2,1/2)
(d)
Fig. 1.70: (a) A simple square lattice. The unit cell is a square with aside a. (b) Basis has two atoms. (c) Crystal = Lattice + Basis. Theunit cell is a simple square with two atoms. (d) Placement of basisatoms in the crystal unit cell.
0.6 The Crystalline State
0.6.1 Type of Crystals : periodic array of points in space - lattice
Face centered cubic
Simple cubic Body centered cubic
Simplemonoclinic
Simpletetragonal
Body centeredtetragonal
Simpleorthorhombic
Body centeredorthorhombic
Base centeredorthorhombic
Face centered orthorhombic
RhombohedralHexagonal
Base centeredmonoclinic
Triclinic
UNIT CELL GEOMETRY
Fig. 1.71: The seven crystal systems (unit cell geometries) and fourteen Bravaislattices.
CUBIC SYSTEMa = b = c 90°
Many metals, Al, Cu, Fe, Pb. Many ceramics andsemiconductors, NaCl, CsCl, LiF, Si, GaAs
TETRAGONAL SYSTEMa = b - c = = = 90°
In, Sn, Barium Titanate, TiO2
ORTHORHOMBIC SYSTEMa - b - c = = =90°
S, U, Pl, Ga (<30°C), Iodine, Cementite(Fe3C), Sodium Sulfate
Fig. 1.30: (a) The crystal structure of copper is Face Centered Cubic(FCC). The atoms are positioned at well defined sites arranged periodicallyand there is a long range order in the crystal. (b) An FCC unit cell withclosed packed spheres. (c) Reduced sphere representation of the FCC unitcell. Examples: Ag, Al, Au, Ca, Cu, -Fe (>912°C), Ni, Pd, Pt, Rh
Fig. 1.31: Body centered cubic (BCC) crystal structure. (a) A BCCunit cell with closely packed hard spheres representing the Featoms. (b) A reduced-sphere unit cell.
(c) (d) Examples: Be, Mg, -Ti ( < 882°C ), Cr, Co, Zn, Zr, Cd
Fig. 1.32: The Hexagonal Close Packed (HCP) Crystal Structure. (a) TheHexagonal Close Packed (HCP) Structure. A collection of many Zn atoms.Color difference distinguishes layers (stacks). (b) The stacking sequence ofclosely packed layers is ABAB (c) A unit cell with reduced spheres (d) Thesmallest unit cell with reduced spheres.
- Hexagonal Closed Packed (HCP) Structure:
a
C
a
a
Fig. 1.33: The diamond unit cell is cubic. The cell has eightatoms. Grey Sn (-Sn) and the elemental semiconductorsGe and Si have this crystal structure.
The FCC unit cell of theBuckminsterfullerene crystal. Eachlattice point has a C60 molecule
(c) BuckminsterfullereneFig. 1.42: The three allotropes of carbon.
0.6.2 Three Phases of Carbon
Chap. 1. Conductors and Resistors
1.1. Ohm's Law and Conductivity
- Electrical conductivity electrical resistivity
- Insulator, semiconductor, conductor(metal)
- Ohm's law: constant (in ohms, in volts, in amps) at constant temperature
- Varistor: a very strong nonlinear V-I curve with a greatly reduced resistance at high voltages (variable resistor)
- Rectifier: AC -> DC, nonlinear V-I curve (nonohmic)
- Electrician's Ohm's law constant ⇒ Scientist's Ohm's law at an atomic scale
material's resistivity (in ohm-m), A=cross-sectional area, L= length Electric field E= V/L (in V/m) and the current density J = I/A (in A/m2). ⇒ Ohm's law:
constant or conductivity (ohm-m)-1 or siemens/m
- For an anisotropic conductivity as in a randomly oriented polycrystal,
→
- Metals, semiconductors, and insulators have different temperature-dependence of conductivity. The conductivity of a metal decreases and that of a nonmetal increases with increasing T.
- Electrons were discovered by Thomson in 1897, and soon after Drude developed "the classical free-electron theory of metals" to explain the electrical conductivity of metals.
See Fig. 1.1.
1.2. Mobility and Relaxation Time
- Drude's classical free-electron theory (the first theory of materials science) The modified form in the quantum-mechanical free-electron theory of metals (Ch.12) & in the modern treatment of conduction electrons in semiconductors (Ch.15).
- In Drude's model, electrons' average kinetic energy () ⇒ per degree of freedom:
in a 3-dim. free-electron "gas".
average thermal velocity
;
For 300K, ≈ msec, but in metals, the velocity of the conduction electrons
is much faster than this. It is a fair approximation for conduction electrons in semiconductors.
- Under an electric field E, the free electron gas achieves the average drift velocity and a net current density J; Denoting the charge carrier density by ,
In terms of the mobility of free electrons, and (The unit for the mobility: m2V-1s-1, m/s per V/m)
- The relation between the mobility and the relaxation time; From Newton's law of motion, the acceleration
but in metals, the free electrons have frequent collisions with the lattice ion.
Therefore, the acceleration occurs only between collisions during the average time (the collision time or the relaxation time) :
≡ →
, then
- Suppose that the conductivity of a typical metal is about ≈ S/m; With the approximation above, ≈
m2V-1s-1, which in turn ≈ sec and the mean free path ≈ m (See the textbook).
1.3. Resistance as Viscosity
- Newton's law incorporating the collision processes in the form of a viscous drag (friction) force;
Under a DC electric field, the electrons with an initial velocity of zero will accelerate, in a time on the order of the collision time , up to a terminal drift velocity of
In this "steady state" ( ), we have Ohm's law.
1.4. The Hall Effect - Counting Free Electrons
- E. H. Hall in 1879 (graduate student)
In a magnetic field, ×, perpendicular to both and B.
→ the transverse force on electrons moving with an average drift velocity leads to a build-up of net charge on the longitudinal surfaces of the sample (an excess of electrons on one side, a deficiency on the other)
→ This charge build-up creates a transverse electric field (Hall field) .
Metal n [m-3]
RH (Experimental) [m3 A-1 s-1]
H = | RH| [m2 V-1 s-1]
1028 10–11 10–4 Ag
5.85
-9.0
57
Al 18.06 -3.5 13 Au 5.90 -7.2 31 Be 24.2 +3.4 ?
Cu 8.45 -5.5 32 Ga 15.3 -6.3 3.6 In 11.49 -2.4 2.9
Mg 8.60 -9.4 22
Na 2.56 -25 53
- In equilibrium, the two transverse forces are equal and opposite:
In terms of the current density J,
and
- For divalent zinc, the calculated value = × , but the measure one = ×
Fig. 2.26: Hall effect for ambipolar conduction as in asemiconductor where there are both electrons and holes. Themagnetic field Bz is out from the plane of the paper. Both electronsand holes are deflected toward the bottom surface of the conductorand consequently the Hall voltage depends on the relative mobilitiesand concentrations of electrons and holes.(E is the electric field.)
- A transverse electric field in the sample exists in a direction perpendicular to both and (along the y-axis); Lorentz force with = the drift velocity.
The accumulation of electrons generates an internal electric field in the -y direction.
Force balance (no further accumulation of electrons)
Hall coefficient
→
1.7. AC/DC
- Time-varying magnetic fields yield space-varying electric fields and vice versa;
Will still gold locally from the microscopic viewpoint?
Suppose that the local electric field is varying as sin and consider the collision processes, a collision between the lattice and an electrons occurs about 1014 times per second. → even a million times per second (freq = 1 MHz) is sluggish.
- As long as the period of the time-varying field is long compared to the collision time, or the product is much less than one,
AC: sin provided ≪
→ The local current density will vary as sin, in phase with the electric field, and the local AC conductivity will be equivalent to the DC conductivity, provided ≪ .
For of the order of 10-14 seconds, the above will be satisfied at frequencies up to as high as 1012 Hz.
1.8. Heat Conductivity and Heat Capacity - The equivalent to Ohm/s law in 1-dim. heat flow;
Denote the rate of heat flow per unit area (W/m2) as JQ, the temperature gradient (K/m) as dT/dx, and the material's thermal conductivity (Wm-1K-1) as ;
, and Ohm's law:
→ constant (Wiedermann-Franz law); an "empirical" law
