II–2 Microscopic View of Electric Currents

1. II–2 Microscopic View of Electric Currents

2. Main Topics • The Resistivity and Conductivity. • Conductors, Semiconductors and Insulators. • The Speed of Moving Charges. • The Ohm’s Law in Differential Form. • The Classical Theory of Conductivity. • The Temperature Dependence of Resistivity

3. The Resistivity and Conductivity I • Let’s have an ohmic conductor i.e. the one which obeys the Ohm’s law: V = RI • The resistance R depends both on the geometry and the physical properties of the conductors. If we have a conductor of the length l and the cross-section A we can define the resistivity r and its reciprocal the conductivity  by: R = rl/A = l/A

4. The Resistivity and Conductivity II • The resistivity is the ability of materials to defy the electric current. Roughly a stronger field is necessary if the resitivity is high to reach a certain current. The SI unit is 1 m. • The conductivity is the ability to conduct the electric current. The SI unit is 1 -1m-1. There is a special unit siemens 1 Si = -1.

5. Mobile Charge Carriers I • Generally, they are charged particles or pseudo-particles which can move freely in conductors. They can be electrons, holes or various ions. • The conductive properties of materials depend on how freely their charge carriers can move and this depends on deep intrinsic construction of the particular materials.

6. Mobile Charge Carriers II • E.g. in solid conductors each atom shares some of its electrons, those least strongly bounded, with the other atoms. In zero electric field these electrons normally move chaotically at very high speeds and undergo frequent collisions with the array of atoms of the solid. It resembles thermal movement of gas molecules electron gas.

7. Mobile Charge Carriers III • In non-zero field the electrons also have some relatively very low drift speed in the opposite direction then has the field. The collisions are the predominant mechanism for the resistivity (of metals at normal temperatures) and they are also responsible for the power loses in conductors.

8. Differential Ohm’s Law I • Let us again have a conductor of the length l and the cross-section A and consider only one type of charged carriers and a uniform current, depends on their: • density n i.e. number per unit volume • charge q • drift speed vd

9. Differential Ohm’s Law II • Within some length x of the conductor there is a charge: Q = n qx A • The volume which passes some plane in 1 second is Ax/t = vd A so the current is: I = Q/t = n q vd A = j A • Where j is so called current density. Using Ohm’s law and the definition of the conductivity: I = j A = V/R = El  A/l  j = E

10. Differential Ohm’s Law III j = E • This is Ohm’s law in differential form. It has a similar form as the integral law but it contains only microscopic and non-geometrical parameters. So it is a the starting point of theories which try to explain conductivity. • Generally, it is valid in vector form: j = E

11. Differential Ohm’s Law IV • Its meaning is that the magnitude of the current density is directly proportionalto the field and that the charge carriers movealong the field lines. • For deeper insight it is necessary to have at least rough ideas about the magnitudes of the parameters involved.

12. An Example I • Let us have a current of 10 A running through a copper conductor with the cross-section of 3 10-6 m2. What is the charge density and drift velocity if every atom contributes by one free electron? • The atomic weight of Cu is 63.5 g/mol. • The density  = 8.95 g/cm3.

13. An Example II • 1 m3 contains 8.95 106/63.5 = 1.4 105 mol. • If each atom contributes by one free electron, this corresponds to n = 8.48 1028 electrons/m3. vd = I/Anq = 10/(8.48 1028 1.6 10-19 3 10-6) = 2.46 10-4 m/s

14. The Internal Picture • The drift speed is very low. It would take the electron 68 minutes to travel 1 meter! In comparison, the average speed of the chaotic movement is of the order of 106 m/s. • So we have currents of the order of 1012 A running in random directions and so compensating themselves and relatively a very little currents caused by the field. It is similar as in the case of charging something a very little un-equilibrium.

15. A Quiz • The drift speed of the charge carriers is of the order of 10-4 m/s. Why it doesn’t take hours before a bulb lights when we switch on the light?

16. The Answer • By switching on the light we actually connect the voltage across the wires and the bulb and thereby create the electric field which moves the charge carriers. But the electric field spreads with the speed of light c = 3 108 m/s, so all the charges start to move (almost) simultaneously.

17. The Classical Model I • Let’s try to explain the drift speed using more elementary parameters. We suppose that during some average time between the collision  the charge carriers are accelerated by the field. Using what we know from electrostatics: vd = qE/m

18. The Classical Model II • We substitute the magnitude of the drift velocity into the formula for the current density: j = n q vd = n q2  E/m • So we obtain conductivity and resistivity:  = n q2  /m r = 1/ = m/nq2

19. The Classical Model III • It may seem that we have just replaced one set of parameters by another. But here only the average time is unknown and it can be related to mean free path and the average thermal speed using well established theories similar to those studying ideal gas properties. • This model predicts dependence on the temperature but not on the electric field.

20. Temperature Dependence of Resistivity I • In most cases the behavior is close to linear. • We define a change in resistivity in relation to some reference temperature t0 (0 or 20° C): r = r(t) – r(t0) • The relative change of resistivity is directly proportional to the change of the temperature: r/r(t0) = (t – t0) =  t  r(t) = r(t0)(1 +  t)

21. Temperature Dependence of Resistivity II •  [K-1] is the linear temperature coefficient. It is given the temperature dependence of n and vd. It can be negative e.g. in the case of semiconductors (but exponential behavior). • If it doesn’t work we have to add a quadratic term etc. r/r(t0) = (t – t0) =  t +  (t)2 + …  r(t) = r(t0)(1 +  t +  (t)2 + …)

22. Homework • Please, try to prepare as much as you can for the midterm exam!

23. Things to read • Chapter 25 – 8 and 26 – 2 • See demonstrations: http://buphy.bu.edu/~duffy/semester2/semester2.html