Lecture 1
Periodicity and Spatial Confinement
Crystal structure
Translational symmetry
Energy bands
k·p theory and effective mass
Heterostructures
J. Planelles
SUMMARY (keywords)
Lattice → Wigner-Seitz unit cell
Periodicity → Translation group → wave-function in Block form
Reciprocal lattice → k-labels within the 1rst Brillouin zone
Schrodinger equation → BCs depending on k; bands E(k); gaps
Gaps → metal, isolators and semiconductors
Heterostructures: EFA k  pˆ  i
QWell QWire QDot
confinement  Vc  band offset
Crystal structure
A crystal is a solid material whose constituent atoms,
molecules or ions are arranged in an orderly repeating pattern
+
+
Crystal = lattice + basis
Bravais lattice: the lattice looks the same when seen from any point
a1
a2
P
a1
P
a2
P’ = P + (m,n,p) * (a1,a2,a3)
integers
Unit cell: a region of the space that fills the entire crystal by translation
Primitive: smallest unit cells (1 point)
Unit cell: a region of the space that fills the entire crystal by translation
Primitive: smallest unit cells (1 point)
Wigner-Seitz unit cell: primitive and captures the point symmetry
Centered in one point. It is the region which is closer to that point than to any other.
Body-centered cubic
Translational symmetry
We have a periodic system.
Is there a simple function to approximate its properties?
a
 (x)
 ( x)   ( x  na) | f ( x) |2  | f ( x  na) |2  f ( x  na)  ei f ( x)
Tn f ( x)  f ( x  na) 
Tn  ei a n pˆ
(i a n) q q

pˆ
q!
q
Tn   Translation
Group
[Tn , Tm ]  0  Abelian Group
ianpˆ
Tn f ( x)  e
q
(ian)q
qd f
f ( x)  
(i )
 f ( x  an)
q
q!
dx
q
Eigenfunctions of the linear momentum (Also Bloch Functions)
are basis of translation group irreps
linear momentum
Tneikx  eianpˆ eikx  eiank eikx
Bloch functions
Ψ k ( x)  ei k  x u ( x); u ( x  a)  u ( x)
Tn Ψ k ( x)  Tn eikxu ( x)  eianpˆ eikxu ( x)  eiankΨ k ( x)
E 
Tn





k
1  e
ikna
 e
ikx








Range of k and 3D extension
i[k 
2
k ~ k' k 
m, m  Z same character: e
a
 
k  [ , ] ; k  0 labels the fully symmetric A1 irrep
a a
x  r
3D extension 
k  k
2π
m]na
a
 k (r )  ei kr u(r ); u(r  a)  u(r )
i k a i k  r
Ta k (r )  ei k ( r  a)u (r  a )  e
e u (r )
character
 eikna
Reciprocal Lattice
2π
1D : k ~ k '  k 'k  K 
: eiKa  1
a
3D : k ~ k'  k' k  K : ei K ai  1, ai  a, b, c (lattice vectors)
K?
K  p1k1  p 2 k2  p3k3 ,
K  ai  2π pi
 : k  0,
ki  2
k1 , k2 , k3 
k  xk1  yk 2  zk3 ,
(a j  ak )
(a j  ak )ai
,
pi  Z
 reciproca l lattice
x, y, z  (-1/2,1/2)
First Brillouin zone: Wigner-Seitz cell of the reciprocal lattice
Solving Schrödinger equation: Von-Karman BCs
Crystals are infinite... How are we supposed to deal with that?
We use periodic boundary conditions
Group of translations:
k is a quantum number due to
Ta k (r )  e i k a  k (r )
 k ( -a / 2)  e i  k (a / 2),
translational symmetry
  [ ,  ]
1rst Brillouin zone
We solve the Schrödinger equation for each k value:
The plot En(k) represents an energy band
How does the wave function look like?
[Tˆ , Hˆ ]  0
Hamiltonian eigenfunctions are basis of the Tn group irreps

ik t
We require Tˆ   e




ik r
k (r )  e uk (r )
Envelope part
Envelope part
Periodic part
Bloch function
Periodic (unit cell) part
 

uk (r  t )  uk (r )
Energy bands
How do electrons behave in crystals?
quasi-free electrons
Vc (x)
ions
 p2


 Vc ( x)  k ( x)   k k ( x)
 2m

BC : Ψ k ( x  a )  e ika Ψ k ( x)
 k Vc (x)
k ( x)  N e
ik x
k
Empty lattice
2 p
a
Plane wave
Vc (x)
pˆ k ( x)   k k ( x)
2 k 2
k 
2m
k
T=0 K
Fermi energy
F
 kF
kF
k
2 k 2
k 
2m
Band folded into the first Brillouin zone
2 k 2
k 
2m
 (k): single parabola
BC : Ψ k ( x  a )  e ika Ψ k ( x)
k ~ k '  k ' k  K 
2π
: eiKa  1
a
folded parabola
Bragg difraction
 p2


 Vc ( x)  k ( x)   k k ( x)
 2m

Bragg diffraction
a k  2  n, n  Z
gap
 2 / a
0
a
2 / a
4 / a
k
 p2


 Vc ( x)  k ( x)   k k ( x)
 2m

Types of crystals
k
k
Conduction Band
Fermi
level
k
few eV
fraction eV
gap
Valence
Band
k
k
Metal
Insulator
• Empty orbitals
available at low-energy:
• Low conductivity
high conductivity
k
Semiconductor
• Switches from conducting
to insulating at will
k·p Theory
How do we calculate realistic band diagrams?
2
p
 
ˆ
H  
 Vc (r ) 
 2m

e

i k r
Tight-binding
Pseudopotentials
k·p theory



ik r
k (r )  e uk (r )




i
k
r
ˆ
H k (r )   k e
k (r )
 
2
2 2
 p
  k


k  p


 2m  Vc (r )  2m   m  uk (r )   k uk (r )


The k·p Hamiltonian
MgO
CB
uCBk
HH
uHHk
LH
uLHk
uSOk
Split-off
k=0
k=Γ
{ 0n , u0n }
1. Solving Hkp at  point (k=0)
2. Expanding Hkp in this basis
k
n 
uk ( r )


n 
cnk u0 (r )
n
gap
u0n Hˆ kp u0n '
2

 n  |k | 
k n  n'
  n,n '   u0 p u0
   0 

2
m
m


2
Kane
parameter
8-band H
s
s
s
k2
2m
0
s
0
k2
2m
CB
uCBk
1-band H
1-band H
HH
uHHk
LH
uLHk
4-band H
uSOk
Split-off
k=0
k
s
s
3 3
,
2 2
s
k2
2m
0
P k
s
0
k2
2m
0
P k
0
2
P kz
3
1
 P k
3
1
P k
3
2

P kz
3
0
P k
8-band H
3 3
,
2 2
3 1
,
2 2
3 1
,
2 2
3 3
,
2 2
1 1
,
2 2
1 1
,
2 2

1
P kz
3
2

P k
3
2
P k
3
1
P kz
3
 0 
k2
2m
3 1
,
2 2
2

P kz
3
1
P k
3
3 1
,
2 2
1
 P k
3
2

P kz
3
3 3
,
2 2
0
0
0
k2
2m
1 1
,
2 2
1
P kz
3
2
P k
3
1 1
,
2 2
2

P k
3
1
P kz
3
0
0
0
0
0
0
0
0
0
0
0
0
P k
0
 0 
0
0
 0 
0
0
0
 0 
0
0
0
0
0
0
0
0
k2
2m
k2
2m
 0   
0
k2
2m
0
 0   
k2
2m
Influence of the romote bands: one-band Hamiltonian for conduction
u0n Hˆ kp u0n '
2

 n  |k | 
k n  n'
  n,n '   u0 p u0
   0 

2
m
m

 
2
2

|
k
|
 kcb   0cb 
2m
2
This is a crude approximation... Let’s include remote bands perturbationally
 kcb
k
CB
2
2
2

|
k
|


  0cb  
 2 | k |2
2m
m
  x, y , z

ncb
u0cb
p u0n
2
 0cb   0n
1/m*
Free electron
m=1 a.u.
negative mass?
k
InAs
m*=0.025 a.u.
1
1  kcb
 2
m *  k 2
Effective mass
2 2

k
cb
cb
k  0 
2m*
Heterostructures
Brocken translational symmetry
How do we study this?
B
A
B
B
A
B
A
z
z
z
quantum well
quantum wire
quantum dot
Heterostructures
How do we study this?
If A and B have:
B
A
z
B
• the same crystal structure
• similar lattice constants
• no interface defects
...we use the “envelope function approach”



ik r
k (r )  e uk (r )
 


i k  r
k (r )  e
 ( z ) uk ( r )
Project Hkp onto {Ψnk}, considering that:


Envelope part
Periodic part
f (r ) unk (r ) dr 
1
unk (r ) dr  f (r ) dr 

 unitcell

Heterostructures
In a one-band model we finally obtain:
B
A
z
B
 2 d 2
 2 k 2

 V ( z) 
 2m dz 2
2m


  ( z)    ( z)


 B , k 0
V(z)
 A, k  0
1D potential well: particle-in-the-box problem
Quantum well
Most prominent applications:
• Laser diodes
• LEDs
• Infrared photodetectors
Image: CNRS France
Image: C. Humphrey, Cambridge
Quantum well
B
A
B
z
Image: U. Muenchen
2 2
 2

kx 
2
2

  ( y, z )    ( y, z )
( y   z )  V ( y, z ) 
 2m
2m 

Most prominent applications:
• Transport
• Photovoltaic devices
Quantum wire
A
z
 2 2

 
  V ( x, y, z )   ( x, y, z )    ( x, y, z )
 2m

Most prominent applications:
• Single electron transistor • In-vivo imaging • Photovoltaics
• LEDs
• Cancer therapy
• Memory devices
• Qubits?
Quantum dot