Lecture X

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 partially filled bands conduct

Band TheoryThe calculation of the allowed electron states in a solid is referred to as band theory or band structure theory.

Free electron model

U(r)U(r)

Neglect periodic potential & scattering (Pauli)

Reasonable for “simple metals” (Alkali Li,Na,K,Cs,Rb)

Band Theory

The calculation of the allowed electron states in a solid is referred to as band theory or band structure theory.

Free electron model:

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

Band theory ctd.

To obtain the full band structure, we need to solve Schrödinger’s equation for the full lattice potential. This cannot be done exactly and various approximation schemes are used. We will introduce two very different models, the nearly free electron and tight binding models.

We will continue to treat the electrons as independent, i.e. neglect the electron-electron interaction.

Influence of the lattice periodicityIn the free electron model, the allowed energy states are

 

where for periodic boundary conditions

 

nx , ny and ny positive or negative integers.

L- crystal dimension

)(2

2222

zyx kkkm

E

L

nk

L

nk

L

nk z

zy

yx

x

2;

2;

2 E

k0

-5

-4

-3

-2

-1

0

r

E

Periodic potential

Exact form of potential is complicated

Has property V(r+ R) = V(r) where

R = m1a + m2b + m3c

where m1, m2, m3 are integers and a ,b ,c are the primitive lattice vectors.

Tight Binding Approximation

Tight Binding Model: construct wavefunction as a linear combination of atomic orbitals of the atoms comprising the crystal.

Where (r)is a wavefunction of the isolated atom

rj are the positions of the atom in the crystal.

) ( c = )( j

jj

r-r r

The tight binding approximation for s states

)aos(kc 2- -)e ae a( - - = k)( E xxx k i-k i

+ + + + +aNuclear positions

Solution leads to the E(k) dependence!!

1D:

E(k) for a 3D lattice Simple cubic: nearest neighbour atoms at 

So E(k) = 2(coskxa + coskya + coskza)

Minimum E(k) = 6for kx=ky=kz=0

Maximum E(k) = 6for kx=ky=kz=+/-/2

Bandwidth = Emav- Emin = 12

For k << acos(kxx) ~ 1- (kxx)2/2 etc.

E(k) ~ constant + (ak)2/2

c.f. E = (k)2/me

),,();,,();,,( a000a000a

-4 -2 0 2 4-18

-16

-14

-12

-10

-8

-6

-4

-2

0

F1

k [111] direction

/a/a

E(k)

Behave like free electrons with “effective mass” /a2

Each atomic orbital leads to a band of allowed states in the solid

Band of allowed states

Band of allowed states

Band of allowed states

Gap: no allowed states

Gap: no allowed states

Reduced Brillouin zone scheme

The only independent values of k are those in the first Brillouin zone.

Results of tight binding calculation

Discard for |k| > /a

The number of states in a bandIndependent k-states in the first Brillouin zone, i.e. kx < /a etc.

Finite crystal: only discrete k-states allowed

Monatomic simple cubic crystal, lattice constant a, and volume V.

One allowed k state per volume (2)3/V in k-space.

Volume of first BZ is (2/a)3

Total number of allowed k-states in a band is therefore

.etc,....2,1,0,2

xx

x nL

nk

N

aV

Va

3

3322

Precisely N allowed k-states i.e. 2N electron states (Pauli) per band

This result is true for any lattice:

each primitive unit cell contributes exactly one k-state to each band.

Metals and insulatorsIn full band containing 2N electrons all states within the first B. Z. are occupied. The sum of all the k-vectors in the band = 0.

A partially filled band can carry current, a filled band cannot

Insulators have an even integer numberof electrons per primitive unit cell.

With an even number of electrons perunit cell can still have metallic behaviourdue to band overlap.

Overlap in energy need not occurin the same k direction

E

k0 a

EF

Metal due to overlapping bands

Full Band

Empty Band

Energy Gap

Full Band

Partially Filled Band

Energy GapPart Filled Band

Part Filled Band EF

INSULATOR METAL METAL or SEMICONDUCTOR or SEMI-METAL

E

k0 a

EF

E

k0 a

E

k0 a

Insulator -energy band theory

Covalent bonding

Atoms in group III, IV,V,&VI tend to form covalent bond

Filling factor

T. :0.34 F.C.C :0.74

Covalent bonding

Crystals: C, Si, Ge

Covalent bond is formed by two electrons, one from each atom, localised in the region between the atoms (spins of electrons areanti-parallel )

Example: Carbon 1S2 2S2 2p2

C C

Diamond: tetrahedron, cohesive energy 7.3eV

3D 2D

Covalent Bonding in Silicon

•Silicon [Ne]3s23p2 has four electrons in its outermost shell

•Outer electrons are shared with the surrounding nearest neighbor atoms in a silicon crystalline lattice

•Sharing results from quantum mechanical bonding – same QM state except for paired, opposite spins (+/- ½ ħ)

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

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