Lecture 1
Periodicity and Spatial Confinement
Crystal structure
Translational symmetry
Energy bands
k·p theory and effective mass
Theory of invariants
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
Machinery: kp Theory → effective mass
J
character table
Theory of invariants: G GA1; H =  NiG kiG
Heterostructures: EFA k  pˆ   i 
QWell QWire QDot
confinemen t  V c  band offset
Crystal structure
A crystal is 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
How many types of lattices exist?
C4?
How many types of lattices exist?
C6?
How many types of lattices exist?
14 (in 3D)
C5?
Bravais Lattices
Translational symmetry
We have a periodic system.
Is there a simple function to approximate its properties?
a
 ( x)
 ( x )   ( x  na )  | f ( x ) |  | f ( x  na ) |  f ( x  na )  e
2
T n f ( x )  f ( x  na ) 
2
T n   Translatio
i
f ( x)
n Group
[T n , T m ]  0  Abelian Group
Tn  e
i a n pˆ

q
(i a n )
q!
q
pˆ
q
Tn f ( x )  e
ian pˆ
f ( x) 

q
( ian )
q!
q
( i)
q d
q
dx
f
q
 f ( x  an )
Eigenfunctions of the linear momentum
basis of translation group irreps
Tn e

Tn  e
i a n pˆ


q
(i a n )
q!
q
q
pˆ
ikx
e
ian pˆ ikx
e
e
iank
e
ikx
E

Tn






ikna

ikx


k
1




e
e

Range of k and 3D extension
k ~ k' k 
2
i[k 
m, m  Z
same character: e
2π
m]na
a
a
k  [
 
,
];
k  0 labels the fully symmetric
a a
x  r
3D extension 
k  k
Bloch functions as basis of irreps:
 k (r )  e
Ta  k ( r )  e
i k  ( r  a)
A 1 irrep
i k r
u ( r ); u ( r  a )  u ( r )
i k a
u ( r  a )  e e
character
i k r
u (r )
e
ikna
Reciprocal Lattice
1 D : k ~ k '  k ' k  K 
2π
: e
iKa
1
a
3 D : k ~ k'  k'  k  K : e
K ?
K  p1k 1  p 2 k 2  p 3 k 3 ,
K  a i  2π pi
G : k  0,
i K a i
 1,
a i  a, b, c (lattice vectors)
k i  2
k 1 , k 2 , k 3 
k  x k 1  yk 2  z k 3 ,
(a j  a k )
(a j  a k )a i
,
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
T a k ( r )  e
i k a
i
 k (r )
 k ( - a / 2 )  e  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
We require
Tˆ   e

ik t



ik r
k (r )  e u k (r )
Envelope part
Envelope part
Periodic part

Bloch function
Periodic (unit cell) part
 

u k (r  t )  u k (r )
Energy bands
How do electrons behave in crystals?
quasi-free electrons
Vc ( x )
ions
 p2



 2 m  Vc ( x )  k ( x )   k  k ( x )


BC : Ψ k ( x  a )  e
ika
Ψ k ( x)
 k  V c ( x )
k ( x)  N e
ik x
k
Empty lattice
2 p
Plane wave
a
Vc ( x )
2
pˆ  k ( x )   k  k ( x )
k 
 k
2m
2
k
T=0 K
Fermi energy
F
 kF
k
kF
2
k 
 k
2m
2
Band folded into the first Brillouin zone
2
k 
 k
2
2m
 (k): single parabola
BC : Ψ k ( x  a )  e
k ~ k '  k ' k  K 
ika
2π
Ψ k ( x)
: e
iKa
a
folded parabola
1
 p2



 2 m  Vc ( x )  k ( x )   k  k ( x )


Bragg diffraction
a k  2  n,
gap
 2 / a
0
a
2 / a
4 / a
k
nZ
 p2



 2 m  Vc ( x )  k ( x )   k  k ( x )


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 
 V c ( r ) 
 2m

e

i k r
Tight-binding
Pseudopotentials
k·p theory



ik r
k (r )  e u k (r )




i
k
r
ˆ
H k (r )   k e
k (r )
 
2
2
2
 p



 k
kp


 Vc (r ) 

u k (r )   k u k (r )
 2m

2m
m 

The k·p Hamiltonian
MgO
CB
uCBk
HH
uHHk
LH
uLHk
uSOk
Split-off
k=0
k=Γ
k
 

2

2
2
n
n 
 p


 k
k  p  u ( r ) 

 u k (kr )   k u
 Vc (r ) 

(c
r nk
) u0 (r )
k
 2m

n
2
m
m


gap
n
n'
u 0 Hˆ kp u 0


2
 n  |k | 
k
n 
n'


 0 



u
p
u
0
0

 n ,n '
2
m
m


2
Kane
parameter
n
n'
u 0 Hˆ kp u 0


2
 n  |k | 
k
n 
n'


 0 



u
p
u
0
0

 n ,n '
2
m
m


2


n
n 
u k ( r )   c nk u 0 ( r )
8-band H
CB
uCBk
n
s
1-band H
HH
uHHk
s
LH
uLHk
k
4-band
H
s
s
1-band H
2
0
P k
2m
uSOk
Split-off
3 3
,
2 2
s
k
0
2
0
2m
k=0
8-band H
3 3
,
2 2
k
P k
3 1
,
2 2

3 1
,
2 2

2
3
1
3
3 3
,
2 2
0
1 1
,
2 2
1 1
,
2 2
1
3

3
3

P k
2
3
P kz
P k
P kz
2
1
P kz
P k
 0 
0
P k
2
3
1
3
0
0
0
P k
0
P kz
0
k
s
3 1
,
2 2
s
2

P kz
3
1
P k
3
k
s
2
0
2 m3  1
,
2
2 2k
0


1 2m
P k
3
2
P kz
3
3 3
,
2 2
1 1
,
2 2
1 1
,
2 2
1
0
3
2
P k
3

P kz
P k
2
P k
3
1
P kz
3
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2m
 0 
0
0
0
0
k
2
2m
 0 
0
0
0
k
2
2m
 0 
0
0
k
2
2m
 0   
0
k
2
0
2m
 0   
k
2
2m
n
u0
One-band Hamiltonian for the conduction band


2
2
 n  |k | 
k
n
'
n 
n'
ˆ
  n ,n '  
H kp u 0    0 
u0 p u0


2
m
m



2
2

|
k
|
cb
cb
k  0 
2m
This is a crude approximation... Let’s include remote bands perturbationally
2
k  0 
cb
k
CB

cb
  x, y ,z
 | k |
2
2m


2
m
2
| k |

2
n  cb
cb
u0
n
p u 0
2
0  0
cb
n
1/m*
Free electron
m=1 a.u.
negative mass?
InAs
m*=0.025 a.u.
1
k
m*
1  k
cb


2
k
2
Effective mass
2
 k
cb
cb
k  0 
*
2 m
2
Theory of invariants
1. Perturbation theory becomes more complex for many-band models
2. Nobody calculate the huge amount of integrals involved

grup them and fit to experiment
Alternative (simpler and deeper) to perturbation theory:
Determine the Hamiltonian H by symmetry considerations
Theory of invariants (basic ideas)
1. Second order perturbation: H second order in k:
H 
3
 M ij k i k j
i j
2. H must be an invariant under point symmetry (Td ZnBl, D6h wurtzite)
A·B is invariant (A1 symmetry) if A and B are of the same symmetry
e.g. (x, y, z) basis of T2 of Td: x·x+y·y+z·z = r2 basis of A1 of Td
Theory of invariants (machinery)
1. k basis of T2
2. ki kj basis of T 2  T 2  A1  E  T 2  [T1 ]
3. Character Table:
A1  k x  k y  k z
2

2
2
E  2k z  k x  k y , k x  k y
2
2
2
2
2

T 2  k x k y , k x k z , k y k z 
T1  NO ( k i k j symmetric
notation: elements
of these basis: k iG .
tensor )
irrep
dim( G )
G G
4.Invariant: sum of invariants: H    a G N i k i
i
G
basis element
fitting parameter
(not determined by symmetry)
Machinery (cont.)
How can we determine the N iG matrices?
( J x , J y , J z ) basis of T1 ,
and
T2  T2  T1  T1
 we can use symmetry-adapted JiJj products
Example: 4-th band model: | 3 / 2 ,3 / 2  , | 3 / 2 ,1 / 2  , | 3 / 2 ,  1 / 2  , | 3 / 2 ,  3 / 2  
Machinery (cont.)
We form the following invariants
Finally we build the Hamiltonian
Luttinger parameters: determined by fitting
Four band Hamiltonian:
Exercise: Show that the 2-bands {|1/2,1/2>,|1/2,-1/2>} conduction band k·p Hamiltonian reads
H= a k2 I, where I is the 2x2 unit matrix, k the modulus of the linear momentum and a is a fitting
parameter (that we cannot fix by symmetry considerations)
Hints: 1. T 2  T 2  T1  T1  A1  E  T 2  [T1 ]
2. Angular momentum components in the 1/2 basis: Si=1/2 si, with
3. Character tables and basis of irreps
A1  k x  k y  k z
2
2

 k k
 k k '
2
E  2k z  k x  k y , k x  k y
2
T2
T1
2
2
2
2


x
y
 k ykx , kxkz  kzk x , k ykz  kzk y
x
y
 k y k 'x , k x k 'z  k z k 'x , k y k 'z  k z k ' y

A1  k x  k y  k z
2
Answer:
2

 k k
 k k '
2
E  2k z  k x  k y , k x  k y
2
T2
T1
2
2
2. Disregard E:
2
x
s
2
y
s
2
z
4. A1:
 s
2
x
s
2
y
2
z
x
y
 k y k 'x , k x k 'z  k z k 'x , k y k 'z  k z k ' y
s
2
z
 3I
2


 k ykx , kxkz  kzk x , k ykz  kzk y
 I(2 x 2)
 k  k  k  k
2
y

y
3. Disregard T2: s is j  s js i  0
2
x
2
x
1. Disregard T1: k i k j  k j k i  0
s
2
H  ak I
2

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 u k (r )
 


i k  r
k (r )  e
 ( z ) u k (r )
Project Hkp onto {Ψnk}, considering that:


Envelope part
Periodic part
f ( r ) u nk ( r ) dr 
1

u
unit cell
nk
( r ) dr   f ( r ) dr 

Heterostructures
In a one-band model we finally obtain:
B
A
z
B
2
2
 2 d 2
 k

 V (z) 
 2 m 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


k
2
2
x

  ( y, z)    ( y, z)
( y   z )  V ( y , z ) 
 2m
2 m 

Most prominent applications:
• Transport
• Photovoltaic devices
Quantum wire
A
z
 2

2

  ( x, y, z )    ( x, y, z )


V
(
x
,
y
,
z
)
 2m



Most prominent applications:
• Single electron transistor • In-vivo imaging • Photovoltaics
• LEDs
• Cancer therapy
• Memory devices
• Qubits?
Quantum dot
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
Machinery: kp Theory → effective mass
J
character table
Theory of invariants: G GA1; H =  NiG kiG
Heterostructures: EFA k  pˆ   i 
QWell QWire QDot
confinemen t  V c  band offset
Thanks for your attention