Electrical Properties of Materials 재료의 전자기적 성질 Part II. Electrical Properties of Materials

Electrical Conduction in Metals And Alloys 4 Chapter 7 in the textbook

7.1 Introduction and Survey Current density: Discuss flux equation: 3

Understanding of electrical conduction 7.2 Conductivity-Classical Electron Theory Understanding of electrical conduction As Drude did, A free electron gas or plasma ; valence electrons of individual atoms in a crystal What is plasma? For a monovalent metal, : # of atoms / cm3 : Avogadro constant (#/mole) d : density (gram/cm3) M : atomic mass of element (gram/mole) One calculates about 1022 to 1023 atoms per cubic centimeter, i.e., 1022 to 1023 free electrons per cm3 for a monovalent metal. Mobility: ~ 102 to 103 cm2/Vsec. Without electric field: the electrons move randomly so that no net velocity results. Thermal velocity: 4

If E is applied Contradiction: For a free electron (a) Acceleration (?), Constant velocity(?) Free electron model should be modified. Ballistic transport versus Drift velocity Drift velocity versus Thermal velocity 5

Friction force (or damping force) In a crystal, Friction force (or damping force) collision Equation of Motion Drift velocity 6

For the steady state Final drift velocity Where, t is a relaxation time: average time between two consecutive collisions 7

Mean free path between two consecutive collisions Final drift velocity Nf: number of free electrons Mean free path between two consecutive collisions 8

7.3 Conductivity-Quantum Mechanical Considerations Without field With an electric field Fermi velocity At equilibrium, no net velocity What difference between Class. and QM The maximum velocity that the electrons can have is the Fermi velocity (i.e., the velocity of electrons at the Fermi energy) Only specific electrons participate in conduction and that these electrons drift with a high velocity (vF) 9

Fig.7.4 Population density We now calculate the conductivity by quantum mechanical means and apply, as before, Ohm’s law. only specific electrons participate in conduction Fig.7.4 Population density 10

Population density at EF Fig.7.5 Population density at EF When the electric field vector points in the v(k)x direction: 11

Since, s = j/E vs Fig.7.6 EM: monovalent metals EB: bivalent metals For a spherical Fermi surface, Fermi velocity, relaxation time, population density at Fermi energy vs Since, s = j/E Classical expression Fig.7.6 Contains the information that not all free electrons are responsible for conduction, i.e., the conductivity in metals depends to a large extent on the population density of the electrons near the Fermi surface. EM: monovalent metals EB: bivalent metals EI: Insulators 12

Classical Concept: QM: vs Since, s = j/E Classical expression 13

7.4 Experimental Results and Their Interpretation 7.4.1 For Pure Metals Linear temperature coefficient of resistivity Matthiessen’s rule Ideal resistivity Residual resistivity Good indicator to check the quality of materials. One definition of metal – the resistivity goes up with the temperature 14

7.4.2 For alloys Linde’s rule ∝(Dvalence electrons)1/2 For dilute single-phase alloys Linde’s rule ∝(Dvalence electrons)1/2 Atoms of different size cause a variation in the lattice parameters local charge valence alter the position of the Fermi energy. 15

7.4.3 Ordering vs Disordering (Nordheim’s rule) Two phase mixture: 16

7.5 Superconductivity Superconductors are materials whose resistivities become immeasurably small or actually become zero below Tc. High Tc superconductors (Tc>77K) 77K: boiling point of liquid nitrogen 20K: boiling point of liquid hydrogen 4K: boiling point of liquid helium 17

7.5.1 Experimental Results Type I superconductors ma: atomic mass a: material constant (Isotope effect) Elimination of the superconducting state also occurs by subjecting the material to a strong magnetic field. Ceramic superconductors usually have a smaller Hc than metallic super conductors, i.e., they are more vulnerable to lose superconductivity by a moderate magnetic field. Type I superconductors 18

Type II superconductors: Transition metals and alloys consisting of Nb, Al, Si, V Pb, Sn, Ti, Nb3Sn, Nb-Ti, ceramic superconductors The interval represents a state in which superconducting and normal conducting areas are mixed in the solid. Contains small circular regions called vortices or fluxoids, which are in the normal state with a mixed superconducting and normal conducting area. Flux quantum: 19

7.5.2 Theory Postulate superelectrons that experience no scattering, having zero entropy (perfect order), and have long coherence lengths. BCS theory: Cooper pair (pair of electrons that has a lower energy than two individual electrons) Electrons on the Fermi surface having opposite momentum and opposite spin forms cooper-pair These electrons form a cloud of Cooper pairs which drift cooperatively through the crystal. The superconducting state is an ordered state of the conduction electrons. 10-4 eV 20

7.6 Thermoelectric Phenomena Thermoelectric power, or Seebeck coefficient Contact potential: metals with different EF Explain the band diagram of metals to explain the contact potential (based on workfunction difference) Peltier effect: a direct electric current that flows through the junctions made of different materials causes one junction to be cooled and the other to heat up.(The electrons having a large energy (higher EF) are caused by the current to transfer their extra energy into the materials having a lower EF, which in turn heats up.) 21