electrons in solids energy bands and resistance in conductors and semiconductors

Electrons in Solids

Energy Bands and Resistance in Conductors and Semiconductors

What Have We Learned About Electrical Storage

• The electric force FE on a charge q0 can be considered due to an electric field which is produced by other charges in the area

FE = q0 E• If moving a charge between two points requires work (or does

work), the charge gains (or loses) potential energy:V = – E dx = (for a constant field) Ex

• Capacitors store charge Q in proportion to the voltage V between the plates:

C = Q/V = C = 0 A/d• Capacitors are used in RAM

What Have We Learned About Magnetic Storage?

• Two domains magnetized in same direction is a 0• Two domains magnetized in opposite directions is

a 1• Direction of magnetization changes at start of new

bit.• Magnetic data is written by running a current

through a loop of wire near the disk• As magnetic data passes by coil of wire, changing

field induces currents according to Faraday’s Law:

dt

dBA

dt

diR B

What Have We Learned About Magnetoresistance?

• Charges traveling through magnetic field experience magnetic force (provided velocity and field are not aligned):

FB = qv x B = (if v perpendicular to B) qvB• In a current-carrying wire, this force results in more

frequent collisions and thus an increased resistance: Magnetoresistance

• Electrons traveling through magnetized material undergo spin-dependent scattering

• When magnetic field is present in magnetic superlattice, scattering of electrons is cut dramatically, greatly decreasing resistance: Giant magnetoresistanced

Stuff to remember about GMR

• Electrons (and other elementary “particles”) have intrinsic magnetic fields, identified by spin

• The scattering of electrons in a ferromagnetic material depends on the spin of the electrons

• Layers of ferromagnetic material with alternating directions of magnetization exhibit maximum resistance

• In presence of magnetic field, all layers align and resistance is minimized

What Have We Learned About Spectra?

• ENERGY LEVELS ARE QUANTIZED

• Different elements have different allowed energies (since different numbers of protons and electrons provide different structure of attraction

• Light emitted when electrons move from a high energy level to a lower energy level in an atom will have only certain, QUANTIZED, allowed energies and wavelengths.

• Those wavelengths depend solely on the element emitting the light and compose the characteristic emission spectrum for that element

Our Model of the Atom

• If the atom is in the “ground state” of lowest energy, electrons fill the states in the lowest available energy levels. The first shell has two possible states, and the second shell has eight possible states. Higher shells have more states, but we’ll represent them with the eight states in the first two sub-shells.

• Electrons in the outermost shell are called “valence” electrons. We’ll

make them green to distinguish from e- in filled shells

E=0 (unbound)

n=1

n=2n=3n=4

Really eight distinct states with closely spaced energies, since two electrons cannot occupy the same state.

The Hydrogen Atom• Has one electron, normally in the ground state n=1

• This electron can absorb energy and go to a higher state, like n=3

• The atom will eventually return to its ground state, and the electron will emit the extra energy in the form of light.

• This light will have energy E = (13.6 ev)(1/1 – 1/32) = 12.1 eV

• The corresponding wavelength is = hc/E = 1020 Å

E=0 (unbound)

n=1

n=2

n=3n=4

Other Atoms• Electrons can absorb energy and move to a higher level

– White light (all colors combined) passing through a gas will come out missing certain wavelengths (absorption spectrum)

• Electrons can emit light and move to a lower level

• Calculating the allowed energies extremely complicated for anything with more than one electron

• But can deduce allowed energies from light that is emitted

E=0 (unbound)

n=1

n=2

n=3n=4

Really eight closely spaced energies, since no two electrons can occupy same state

Atomic Bonding• Electrons in an unfilled valence shell are loosely bound• Atoms will form bonds to fill valence shells, either by

sharing valence electrons, borrowing them, or loaning them

• When atoms bond in solids, sharing electrons, each atom’s energy levels get slightly shifted

E=0 (unbound)

n=1

n=2

n=3n=4

Electron Motion• Electrons can only move to an open energy state• If the atom does not absorb energy, electrons can

only move to an open energy state in the same shell

(drawing is NOT to scale)E=0 (unbound)

n=1

n=2

n=3n=4

Electrons in Solids• The shifted energies in adjacent atoms combine to create a continuous

“band” of allowed energies for each original energy level; each band, however, has a finite number of states equal to the number in original atoms

• Electrons can move from the locality of one atom to the next only if an energy state is available within the same band

Conductors & Semiconductors

• In conductors, the valence band is only partially-full, so electrons can easily move from being near one atom to being near another

• In semiconductors and insulators, the valence band is completely full, so electrons must gain extra energy to move

• In semiconductors, extra electrons (or holes) can be introduced in a “controlled” way.

Electrons in an Electric Field• Conduction electrons move randomly in all directions in

the absence of a field.• If a field is applied, the electric force results in

acceleration in a particular direction: F=ma= –eE a = –eE/m

• As the charges accelerate, the potential energy stored in the electric field is converted to kinetic energy which can be converted into heat and light as the electrons collide with atoms in the wire

• This acceleration produces a velocity

v = at = –eEt/m

Electrical current

• If an electric field points from left to right, positive charge carriers will move toward therightwhile negative charges will move toward theleft

• The result of both is a net flow of positive charge to the right.

• Current is the net change in positive charge per time

t

Qi

• Look at the movement of charges through a wire with a potential difference applied - animation

• The net velocity is called drift velocity vd

• The charge contained in a cross-sectional volume is q=Nq(Volume)=NqAx=NqAvdt

• So the charge per time crossing through A is

Electrical current

dd NqAv

t

tNqAv

t

Qi

• The drift velocity is the net velocity of charge carriers after collisions

• It is not equal to the velocity caused by acceleration of field on individual charge, but it certainly is proportional to it.

• Since current is proportional to vd, current is proportional to electric field

How do I figure the drift velocity?

dvtm

qEv

Answer the Ohm’s Law - Before You Start Questions in Today’s Activity,

Then Continue Through Question 4

• Electric field is proportional to potential difference or voltage

• So, . . .

Ohm’s Law

ViVEvvi d Ohm gets credit for being the first to notice that many materials displayed this proportionality. He defined resistance as the ratio of V to i.

V = iR• Ohm’s law only applies to materials in which the

behavior of charge carriers can be described statistically by the drift velocity

• In short, Ohm’s law only applies to ohmic materials.

• Were both of your resistors made of ohmic material?

Ohm’s Law - worth remembering

A Good Analogy to Remember

On what does resistance depend?

On What Does Resistance Depend?

• If I increase the length of a wire, the current flow decreases because of the longer path

• If I increase the area of a wire, the current flowincreases because of the wider path

R = L/A• If I change to a material with better conductivity, the

current flowincreases because charge carriers move better

• If I change the temperature, the current flowchanges

Conductors vs. Semiconductors

• Electrons are free to move in the conduction band• As temperatures rise, electrons collide more with vibrating

atoms; this effect reduces current• Conductors have a partially-filled valence band which

doubles as a conduction band at any temperature – The primary effect of temperature on resistance is due to more

collisions at higher temperatures

• Semiconductors have a completely-filled valence band, so electrons have to be “excited” to enter the conduction band– The primary effect of temperature on resistance is due to this

requirement: the higher the temperature, the more conduction electrons

What is a semiconductor?

• Metals– Many free electrons not tied up in chemical

bonds

• Insulators– All electrons (in intrinsic material) tied up in

chemical bonds

Crystal (Perfect)

Crystal (Excited)

Crystal (Excited)

Band Gap

Valence Band

Conduction Band

Band Gap Energy Eg

(Minimum Energy needed tobreak the chemical bonds)

Energy

Position

Valence Band

Conduction Band

Energy

Position

photon in gEh

Band Gap

Valence Band

Conduction Band

Energy

Position

photon out

Band Gap

Valence Band

Conduction Band

Energy

Position

photon out

Band Gap

What have we learned today?• When atoms bind, the energy levels in each atom get shifted

slightly (the size of the shift is very small compared with the energy difference between different levels)

• Atoms in solids form bands of closely-spaced energy levels• Electrons fill the lowest-energy bands first• The highest energy band with electrons in it is called the

valence band• If the valence band is not full, electrons can move from atom

to atom. This is the case for conductors• Electrons can not move in filled bands. Thus electrons in

semiconductors must gain energy (usually from thermal sources) to move from atom to atom.

What else have we learned today?

• In many, ohmic, materials, current is proportional to voltage:

V = iR• Resistance is proportional to the length of an object and

inversely proportional to cross-sectional area:R = L/A

• The constant of proportionality here is called the resistivity. It is a function of material and temperature.

• The resistivity of conductors increases with temperature since atomic vibrations increase.

• The resistivity of semiconductors decreases with temperature since more conduction electrons exist, and this effect overshadows the vibrations.

Before the next class, . . .

• Start (and finish?) Homework 16

• Do Activity 14 Evaluation

• Read Chapters 3 and start 6 in Turton

• Do Reading Quiz

Finish The ActivityRemember, i = NqAvd, so

R 1/N, 1/A, 1/q, 1/vd