PHYS3004 Crystalline Solids
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PHYS3004Crystalline Solids
Prof. P.A.J. de Groot
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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)
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1. BONDING IN SOLIDS
• Covalent bonding
• Ionic bonding
• Metallic bonding
• Van der Waals bonding
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1. BONDING IN SOLIDS• Covalent bonding: LCAO
• Bonding and anti-bonding states
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2. CRYSTAL LATTICES• Lattice & basis
• Wigner-Seitz cell
X
T1R1
(i) (ii)(iii)
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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
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3. RECIPROCAL LATTICE• First Brillouin zone
• Bloch theorem:
rkierur .)()( )()( ruRru
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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
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4. FREE ELECTRON MODEL – JELLIUM
• 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
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4. FREE ELECTRON MODEL – JELLIUM
• Charge oscillations – plasmons• Electrical transport (relaxation time)
• Quantum Jellium• Hall effect• Breakdown of Jellium Model
eEvmdt
dvm dd j nev
nemd
2E
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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
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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
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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
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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
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7. MAGNETIC PROPERTIES OF CRYSTALS
• Paramagnetism – partly filled shells
• Curie’s law
)( JBavg JBgm
TkJBg
B
B
TCJJ
TkNg
B
B )1(3
022
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7. MAGNETIC PROPERTIES OF CRYSTALS
• Pauli paramagnetism
• Ferromagnetism & mean field theory
B
BBBB
E
MBB appliedlocal 0
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7. MAGNETIC PROPERTIES OF CRYSTALS
Do m a in 1
Do m a in 2NS N S N S N S
WriteRead
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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
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8. SEMICONDUCTORS• Transport (electrons/holes)
• Einstein relation
nEenEme
ej ee
ee
*
V+
n
x
drift
diffusionV+
n
x
drift
diffusion
Tke
D B
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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