PHYS3004 Crystalline Solids

PHYS3004Crystalline Solids

Prof. P.A.J. de Groot

1. BONDING IN SOLIDS

• Born-Oppenheimer Approximation

• Linear Combination of Atomic Orbitals

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0-0.5

0.0

0.5

1.0

AttractiveInteraction

RepulsiveInteraction

BindingEnergy

Equ

ilibr

ium

Sep

erat

ion

Pot

entia

l Ene

rgy

(arb

. uni

ts)

Seperation of Atoms (nm)

1. BONDING IN SOLIDS

• Covalent bonding

• Ionic bonding

• Metallic bonding

• Van der Waals bonding

1. BONDING IN SOLIDS• Covalent bonding: LCAO

• Bonding and anti-bonding states

2. CRYSTAL LATTICES• Lattice & basis

• Wigner-Seitz cell

X

T1R1

(i) (ii)(iii)

3. RECIPROCAL LATTICE• Diffraction of waves (x-rays)

• Reciprocal lattice

dVereEUnitCell

rkkie

R

Rkki outin

n

noutin

)).(().(

det

0.,0.,2. 312111 aaaaaa

321 alakahQ

3. RECIPROCAL LATTICE• First Brillouin zone

• Bloch theorem:

rkierur .)()( )()( ruRru

4. FREE ELECTRON MODEL – JELLIUM

• Electrons in a box

• Time independent Schrödinger eq.

• Plane wave solutions

• Boundary conditions (box is finite) E

m Ln n n

kmx y z

2 2

22 2 2

2 2

2 2

( )

V

V=0

E

z L


• Density of states

• Fermi energy

Ekm m

nFf

2 2 22 2 3

2 23

/

dEEmVdEED2/3

222

2)(

kx

ky

kz

kx

D(k)

Energy

kx

kx

ky

E F


• Charge oscillations – plasmons• Electrical transport (relaxation time)

• Quantum Jellium• Hall effect• Breakdown of Jellium Model

eEvmdt

dvm dd j nev

nemd

2E

5. NEARLY-FREE ELECTRON MODEL

Perturbation theory – only significant changes in E(k) near degenerate points

V(x)

Crystal edge

Jellium potential

EVm

2

2

2

midgapenergy

kx/a /a

k

k-g

Energy

bandgap

Brillouin zone

Extended zonescheme

5. NEARLY-FREE ELECTRON MODEL

• Effective mass

• Electrons and holes

kx/a /a

E

EF

kx/a /a

E

EF

kx/a /a

E

EF

Metallic MetallicInsulating/

Semiconducting

1

2

22*

dkEdm

Energyelectrons

Energy

kx

Valenceband holes

kx

6. TIGHT BINDING

)3()2()1(atomatomatommolecule cba

N EX TA TO M EE 2

A T O ME

N EX TATO M EE 2

Ene rg y

N EX TA TO M EE 2

A T O ME

N EX TATO M EE 2

jellium nearly-free electrons tight binding atoms

6. TIGHT BINDING

• LCAO• Bloch theorem

• Crystal momentum

dVeRrRrHeRrN

dVHE Rki

RS

Ratom

Rki

RScrystal

..** )()(ˆ)(1ˆ

P

S

Ener

gy

1/a

gas

sem

icond

ucto

r

met

al

Gkkkk outoutinin 2121

7. MAGNETIC PROPERTIES OF CRYSTALS

• Paramagnetism – partly filled shells

• Curie’s law

)( JBavg JBgm

TkJBg

B

B

TCJJ

TkNg

B

B )1(3

022


• Pauli paramagnetism

• Ferromagnetism & mean field theory

B

BBBB

E

MBB appliedlocal 0


Do m a in 1

Do m a in 2NS N S N S N S

WriteRead

8. SEMICONDUCTORS• Intrinsic

• Doping (donors/acceptors)

TkE

npnB

gii 2

exp0

Donors-extraelectrons

EF

EG

-

+ + + ++

- - - -

EF

-

+ + + ++

- - - -

Acceptors-fewerelectrons

n-type p-type

Valence band

Conduction band

8. SEMICONDUCTORS• Transport (electrons/holes)

• Einstein relation

nEenEme

ej ee

ee

*

V+

n

x

drift

diffusionV+

n

x

drift

diffusion

Tke

D B

8. SEMICONDUCTORS• MOSFET

• Depletion width• Gate-controlled

conduction channel

• 2D electron gas – quantum Hall effect

Vs

metal

SiO2

x p-Si

xEF

depletion width

yp-typeE

n+n+

Energy

Vgate >0

PHYS3004 Crystalline Solids

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