Lecture XV

Solid state

dr hab. Ewa Popko

Measured resistivities range over more than 30 orders of magnitude

Material Resistivity (Ωm) (295K)

Resistivity (Ωm) (4K)

10-12

“Pure”Metals

Copper

10-5

 

 

Semi-Conductors

Ge (pure) 5 102 1012

 

 

Insulators Diamond 1014  

Polytetrafluoroethylene (P.T.F.E)

1020  

1014

1020

Potassium

2 10-6 10-10

Metals and insulators

Metals, insulators & semiconductors?

At low temperatures all materials are insulators or metals.

Semiconductors: resistivity decreases rapidly with increasing temperature. Semiconductors have resistivities intermediate between metals and insulators at room temperature.

Pure metals: resistivity increases rapidly with increasing temperature.

1020-

1010-

100 -

10-10-R

esis

tivi

ty (

Ωm

)

100 200 3000Temperature (K)

Diamond

Germanium

Copper

Bound States in atoms

r4

qe = )r(V

o

2

Electrons in isolated atoms occupy discrete allowed energy levels E0, E1, E2 etc. .

The potential energy of an electron a distance r from a positively charge nucleus of charge q is

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

F6 F7 F8 F9

r

V(r)E2

E1

E0

r

0

Increasing Binding Energy

Bound and “free” states in solids

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

F6 F7 F8 F9

r

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

F6 F7 F8 F9

r

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

F6 F7 F8 F9

r

V(r)E2

E1

E0

The 1D potential energy of an electron due to an array of nuclei of charge q separated by a distance R is

Where n = 0, +/-1, +/-2 etc.

This is shown as the black line in the figure.

n o

2

nRr4

qe = )r(V

r

0

0

+ + + + +RNuclear positions

V(r) lower in solid (work function).

V(r)Solid

Energy Levels and Bands

+E

+ + + +position Electron level similar to

that of an isolated atom

Band of allowed energy states.

In solids the electron states of tightly bound (high binding energy) electrons are very similar to those of the isolated atoms.

Lower binding electron states become bands of allowed states.

We will find that only partial filled band conduct

Solid stateN~1023 atoms/cm32 atoms 6 atoms

Energy band theory

Metal – energy band theory

At a temperature T the probability that a state is occupied is given by the Fermi-Dirac function

1 +

Tk

-Eexp = f(E)

B

-1

EEF

N(E

) dE

kBT

T=0

T>0

n(E

)dE

The finite temperature only changes the occupation of available electron states in a range ~kBT about EF.

where μ is the chemical potential. For kBT << EF μ is

almost exactly equal to EF.

Fermi-Dirac function for a Fermi temperature TF =50,000K, about right for Copper.

The effects of temperature

Insulator -energy band theory

diamond

semiconductors

Intrinsic conductivity

kTEss

ge2/

0

ln()

1/T

1/T

ln()

kTEdd

de /0

Extrinsic conductivity – n – type semiconductor

Extrinsic conductivity – p – type semiconductor

Conductivity vs temperature

kTEss

ge2/

0

ln()

kTEdd

de /0

1/T

ActiniumAluminium (Aluminum)AmericiumAntimonyArgonArsenicAstatineBariumBerkeliumBerylliumBismuthBohriumBoronBromineCadmiumCaesium (Cesium)CalciumCaliforniumCarbonCerium

ChlorineChromiumCobaltCopperCuriumDarmstadtiumDubniumDysprosiumEinsteiniumErbiumEuropiumFermiumFluorineFranciumGadoliniumGalliumGermaniumGoldHafniumHassium

HeliumHolmiumHydrogenIndiumIodineIridiumIronKryptonLanthanumLawrenciumLeadLithiumLutetiumMagnesiumManganeseMeitneriumMendeleviumMercuryMolybdenumNeodymium

NeonNeptuniumNickelNiobiumNitrogenNobeliumOsmiumOxygenPalladiumPhosphorusPlatinumPlutoniumPoloniumPotassiumPraseodymiumPromethiumProtactiniumRadiumRadonRhenium

RhodiumRubidiumRutheniumRutherfordiumSamariumScandiumSeaborgiumSeleniumSiliconSilverSodiumStrontiumSulfur (Sulphur)TantalumTechnetiumTelluriumTerbiumThalliumThoriumThulium

TinTitaniumTungstenUnunbiumUnunhexiumUnunoctiumUnunpentiumUnunquadiumUnunseptiumUnuntriumUraniumVanadiumXenonYtterbiumYttriumZincZirconium