Solids and conductivity

1
Solids and conductivity

• At long distances, the electric field from one
  atom polarizes another, that is distorts it
  slightly, inducing an electric dipole that
  attracts the other atom, and vice versa.
• At short distances, the electron wave functions
  start to overlap, and the Pauli exclusion
  principle kicks in, and the atoms repel each
  other- all this seen in molecules.
• If there are many atoms close to each other, and
  there is enough time for them to find the lowest
  energy state, there must always be some
  geometrical configuration that is most stable.

2

• If this configuration is most stable, it will be
  so everywhere, and we will have the same
  configuration over the whole sample called long
  range order.
• If we start in a small region with a relatively
  stable configuration, different from the lowest
  energy state, it will spread throughout the
  sample, and we have a metastabel state which will
  be stable for nearly infinite time- it would have
  to tunnel to the more stable state.
• If we start from a disordered state and cool
  quickly, we can get disordered cold states that
  are not as stable, but still live practically
  forever, like glass, amorphous with great
  viscosity of flow.

3

• We dont need to study the different
  configurations, leading to crystals of different
  symmetry. The only property we need for our
  study of conductivity is the fact that the
  molecules are fixed in a regular pattern.
• CONDUCTIVITY we will investigate
• electrical conductivity
• semiconductivity
• superconductivity
• Classical electrical conductivity (look back to
  Chapter 22-5 in the Tipler book, to combine with
  Chapter 39-2
• (Developed first in lectures at Columbia by H.
  Lorentz in 1909.)

4

• Assume that in some materials that there are some
  electrons that are not attached to a particular
  molecule, but can move more or less freely in the
  material. We call them non-localized. These
  must be electrons that sit in the outer shells of
  atoms, and cant decide which atom they are
  attached to.
• When an electric field is applied to the
  material, these electrons start to accelerate,
  with a force eE, so an acceleration aF/meE/m.
  If they were completely free, they would continue
  to accelerate across the whole material and the
  time to arrive at the far side would be
  proportional to ?1/E, from sat²/2 .

5

• Then the total current would be
  where A is the area of the conductor. This is
  not Ohms law, which tells us that the current
  varies as E. One concludes that electrons dont
  accelerate freely all the way across the
  material, but only for a short distance,
  presumably until they hit one of the fixed atoms
  and bounce off in a random direction (scatter)

6

• Then the average drift velocity of a random
  electron, making a collision after the average
  time ? is We can express ? in terms
  of the mean free path and the average electron
  velocity
• and can get the average velocity from the
  thermal velocity with the thermal kinetic energy,
  for three degrees of freedom (xyz) this is
  3/2(kT), so
• and it is striking that this average
  velocity is typically 1010 times vd, so the
  average is not affected by E

7

• Since vav doesnt change with E, neither will ?
  or ? and none of the constants in the equation
  for the drift velocity will
  depend on E, and we get the
  linear current with E that the experimental Ohms
  Law finds. To fill in the details, note

8

• From the picture of the crystal, it is clear that
  since the spacing of atoms is such that that just
  touch in some sense, the value of ? must be
  about equal to the lattice spacing between atoms
  in the crystal, a few tenths nanometer.
• This all seems pretty reasonable
• it gives Ohms Law
• it gives a resistivity that is too big, but only
  by an order of magnitude at room temperature, but
• it says that the resistivity should go down as
  the temperature is increased and the drift
  velocity increases, not so

9

• Look at resistivity over a wide range of
  temperature
• This looks seriously wrong. Also, the
  resistivity at low temperature is very dependent
  on tiny admixtures of impurities, a sign that the
  low resistivity in pure crystals is related to a
  very long mean free path- also, why are there
  insulators?

10

• Another question, is there channeling?
• If we rotate the crystal, we should see more
  conductivity when the direction of measurement
  lines up with a path down the channels in the
  crystal. We do not, but if we look at gt1000Volt
  particles, we do see channeling.

11

• Another hint as to what is wrong comes from the
  heat capacity of metals and insulators. In both,
  we should see the heat capacity of the nuclei
  vibrating around their locations on the
  crystalline lattice (3R in gas law constant
  units) and in insulators, no more, if there are
  no free electrons to move around. This is
  observed. In metals, we should expect an
  additional ½kT for motion in each direction
  (xyz). Not observed! There is an excess of only
  about 0.02R. Evidently, there are very few free
  electrons, but the conductivity at low
  temperatures is way greater than that given by
  our classical theory! Those few must be working
  very hard!

12

• The only way an electron can work harder to make
  more conductivity is to move further before it
  scatters, but it has to move in this forest of
  atoms. How?
• The wave packet of the electron diffracts through
  the forest because of the periodic nature of
  the lattice, it can go on forever, if the lattice
  is perfect. Thermal motions and impurities make
  the lattice imperfect.

13

• This long distance before the electron explains
  how we can get so much conductivity with few free
  electrons and wave doesnt channel. Now let us
  look at the distribution of the electrons. The
  key is going to be the Pauli Principle. Use the
  particle in a box again to get a
  one-dimensional version of the energy
  distributions of the electron.
• Recall that the energy levels in a box are
• and we know that we can only put two electrons in
  a state with a given n, one with spin up, one
  with spin down, according to the Pauli Principle

14

• Calculate the energy for N electrons in the box,
  which we call the Fermi energy
• and for copper this is 1.8eV, or about
  80,000Kelvin. Hot! So most electrons are frozen.
  At room temperature a few electrons are
  thermally excited

15

• Now we must look more carefully at the wave
  properties of the electrons in the periodic
  potential of the lattice of atoms, to find out
  the story on conductors, insulators and
  semiconductors. There are two ways 1. Start
  with free atoms and calculate what happens when N
  atoms are brought close together. (this is done
  in the book.) 2. Start with free electrons,
  calculate the solutions in a periodic potential.
• Free atom atomic levels with quantum numbers
  (n,l) are split up into N levels by the effects
  of distant atoms. Ngtgtgt1, so consider them a
  continuous band., labeled by (n,l). Example Na
  n3, l0.

16

• That n3, l0 band of the outermost electron
  could hold two electrons, one spin up, one spin
  down. Na has only one electron in that state, so
  only half the states are filled. We say that the
  band is half filled. Then, since the states in
  the band are essentially a continuum, the
  slightest increase in energy of the topmost
  electron in the band can move it into a new state
  with no objection by the Pauli Principle. It can
  be moved easily by an electric field in
  particular, so Na is a conductor. In contrast
  take NaCl in solid form. The outer electron goes
  in the outer shell of Cl for binding the crystal,
  so we have a Na ion and a Cl- ion. Both are
  filled shells, so there is no nearby state

17

• To move an electron up to the next state, or band
  in the crystal, we have to go to the state of
  (Na), and excited state, and that takes several
  eV. In between there are no states, and we say
  that there is a band gap or forbidden band. We
  draw the two energy level pictures
• Na (conductor) NaCl (insulator)

18

• The free electron method uses the Bloch-Floquet
  theorem which says that the solutions of the wave
  equation in a periodic potential must be of the
  form
• the u(x) must satisfy a boundary condition, we
  usually bend the crystal in a circle of N periods
  and demand that

19

• What is special about kan? ?
• This corresponds to electron wavelengths of 2a/n,
  or twice the lattice spacing coresponds to an
  integer multiple of the free particle wavelength,
  and a free particle with this wavelength would be
  reflected by the lattice.
• This corresponds to the stop bands in a
  periodically loaded waveguide. The width of the
  stop band depends on the details of the
  interaction.

20

• Semiconductors are just insulators with a small
  forbidden band gap, if they are intrinsic
  superconductors with NO impurities, or
  extrinsic with impurities giving states with
  energies in the forbidden band

21

• In the n-type a little thermal exitation of the
  impurity (often Arsenic) introduces an electron
  into the conduction band, so we get electron
  conduction
• In the p-type, an electron from the valence band
  is easily excited into the impurity (typically
  Gallium) leaving a little bubble in the filled
  valence band that acts like a positive
  electron--it is called a hole
• notice that the conductivity goes up with
  temperature in a semiconductor, unlike the metal
  dependence, because of the increased thermal
  exitation with temperature

22

• What if we make a junction of a p-type and an
  n-type??

23

• Field Effect Transitor
• The drain is reverse biased, just enough to
  nearly pinch off the current source-to-drain,
  while the gate voltage is trying to open it up.
  Very like a vacuum tube triode!
• Good because 1) gate controls s-to-d current
  with no dc current (high impedance) 2) easy to
  make by planar process

24

• Almost 100 years ago, K. Onnes succeeded in
  liquefying Helium, finding it boiled at 4.2K.
  Many remarkable discoveries of quantum properties
  of Liquid Helium followed, but one discovery was
  Superconductivity. He found that below a certain
  termperature, lead and tin, and some other metals
  lost all resistance. How can it be? BCS
  (three guys) explained it as one electron
  distorting the lattice slightly, and another one
  with opposite spin joins (and aids) it, to form a
  pair Cooper pair. Pauli Principle tells us
  that no more than a pair can play this game.
• Well exaggerate for a picture

25

• Distortion of the lattice creates a well
• The more wobbly the lattice, the easier it is
  distorted to form a well, and the more the
  electron pair is bound against thermal
  disturbances (high Tcritical) and against
  disruption by a magnetic field (high Bcritical),
  so you create alloys more wobbly until they
  become unstable gt Tcritical about 20K, with a
  well thousands of atoms across.

26

• Thousands of atoms across that is a macroscopic
  state. It is not so easy to break that pair of
  electrons apart, that is why the critical
  temperature is high. It is very hard for just
  one electron to be scattered by an irregularity
  in the lattice, since the electrons are averaging
  over billions of atoms in the lattice. We saw
  before in normal metals that the mean free path,
  and the conductivity, go to infinity except for
  scattering. There is no scattering in the
  superconducting state, so the resistance is zero.
  Experiments have looked for decay of currents
  for years and seen none in good superconductors.

27

• Since these Cooper pairs are both pure quantum
  states and macroscopic, it is easier to see real
  quantum states than with microscopic states, and
  relatively easy to build applied quantum devices.
• Brian Josephson, while a Cambridge undergraduate,
  realized one could observe interference between
  two Cooper pairs in a macroscopic junction
  tunneling between two superconducting metal
  samples.
• As the phase is changed between the two pairs,
  for example by putting on a small magnetic field,
  the tunneling rate changes in an oscillatory way

28

• Superconducting Quantum Interference Device
• or SQUID
• this gadget will detect magnetic field below
• 10-14 T 10-10 Gauss, if you keep the
  environmental noise down (special rooms) -see
  nice article in book on measure currents in the
  human brain

29

• This finishes our exploration of quantum physics
  and engineering. (Nuclei and particles have
  interaction that are quantal, but the particles
  usually have high energy and act like particles
  indeed.)
• We have tried to show how many common properties
  of atoms and solids have quantum effects which
  dominate their behavior, so the phenomena that
  seem strange are key to understanding much or our
  environment.