Outlook
J. Planelles
Is this a lattice?
Is this a lattice?
Unit cell: a region of the space that fills the entire crystal by translation
Primitive?
Primitives
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
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
Positive /negative, lighter/heavier, effective mass
electron
hole
Translational symmetry
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  e
i a n pˆ
(i a n) q q

pˆ
q!
q
Tn   Translation Group
[Tn , Tm ]  0  Abelian Group
Bloch functions basis of irreps:  k (r )  ei kr u(r ); u(r  a)  u(r )
i kna i k r
Tn k (r )  ei k( r  na)u (r  na )  e

 e u (r )
character

k

Tn  e
i a n pˆ
(i a n) q q

pˆ
q!
q
E

1

 Tn


 eikna


 basis


 u (r ) eikr


Solving Schrödinger equation for a crystal: BCs?
For crystals are infinite we use periodic boundary conditions:
Group of translations:
Ta k (r )  ei k a  k (r )
 k ( - a / 2)  ei  k (a / 2),
  [ ,  ] ; k  1rst Brillouin zone
(Wigner-Seitz cell of the reciprocal lattice)
We solve the Schrödinger equation for each k value:
The plot En(k) represents an energy band
How does the wave function look like?



ik r
k (r )  e uk (r )
Envelope part
Envelope part
Periodic part
Bloch function
Periodic (unit cell) part
 

uk (r  t )  uk (r )
k·p Theory
2

 
p
ˆ
H  
 Vc ( r ) 
 2m

e

i k r



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



V
(
r
)



u
(
r
)


u
(
r
)
k
k k
 2m c

2m
m 

n 
uk ( r )


n
n 
cnk u0 (r )
The k·p
Hamiltonian
Expansion in a basis AND
perturbational correction
One-band Hamiltonian for the conduction band
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
Crude approximation... including remote bands perturbationally:
2
2
2

|
k
|


 kcb   0cb  
 2 | k |2
2m
m
  x, y , z
k
CB

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*
Theory of invariants
(Determining the Hamiltonian (up to constants) by symmetry considerations)
3
1. Second order perturbation: H second order in k: H   M ij ki 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 T2  T2  A1  E  T2  [T1 ]
3. Character Table: A1  k x2  k y2  k z2

E  2k z2  k x2  k y2 , k x2  k y2
T2  k x k y , k x k z , k y k z 

notation: elements
of these basis: k i .
T1  NO (ki k j symmetric tensor )
irrep
4.Invariant: sum of invariants: H 
dim ( )
  a Ni ki
i

basis element
fitting parameter
(not determined by symmetry)
Machinery (cont.)
How can we determine the N i 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.)
Form the following invariants
And build the Hamiltonian
Luttinger parameters: determined by fitting
Heterostructures: e.g. QW
How do we study it?
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”
 





i k r
ik r
 ( z ) uk ( r )
k (r )  e uk (r ) k (r )  e
Project Hkp onto {Ψnk}, considering that:


Envelope part
Periodic part
1
f (r ) unk (r ) dr 
unk (r ) dr  f (r ) dr 

 unit cell

Heterostructures
In a one-band model we finally obtain:
B
A
z
B
 2 d 2
 2 k 2

 V ( z) 
 2m dz2
2m


  ( z)    ( z)


 B , k 0
V(z)
 A, k  0
1D potential well: particle-in-the-box problem
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
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:  A1; H =  Ni ki
Heterostructures: EFA k  pˆ  i
QWell QWire QDot
confinemen t  Vc  band offset
Magnetisme
Newton’s Law
Lagrange
equation
Conservative systems
kinematic momentum:
canonical momentum:
Canonical momentum
Newton’s Law
Lagrange
equation
Velocity-dependent potentials: the case of the magnetic field:
kinematic
momentum:
canonical
momentum:
Hamiltonian:
Conservative systems
kinetic + potential energy
Hamiltonian:
Free particle in a magnetic field
Just kinetic energy!
Particle in a potential and a magnetic field:
Gauge
;
Coulomb Gauge :
Always!
Electron in an axial magnetic field
Rosas et al. AJP 68 (2000) 835
Magnetic confining potential
• Landau levels E(B)
• No crossings!
Electron in a spherical QD pierced by a magnetic field
Small QD
Landau
levels
limit
AB crossings
Competition: quadratic vs. linear term
Aharonov-Bohm Effect
E
Φ/ Φ0
• Periodic symmetry changes of the energy levels
•Energetic oscillations
• Persistent currents
Fractional Aharonov-Bohm Effect
1 electron
2 electrons coulomb
interaction
J.I. Climente , J. Planelles and F. Rajadell,J. Phys. Condens. Matter 17 (2005) 1573
The energy spectrum of a single or a many-electron system in a QR (complex topology) can
be affected by a magnetic field despite the field strength is null in the region where the
electrons are confined
It is not the case for a QD (simple connected topology confining potential)
Translations and Magneto-translations
Two-fold periodicity: magnetic and spatial cells
Summary
No magnetic
monopoles:
No conservative field:
Lagrangian:
Canonical
momentum:
Hamiltonian:
Coulomb gauge:
Hamiltonian operator:
vector
potential
velocity-dependent
potential:
kinematic momentum
Magnetic field: summary (cont.)
axial symmetry
Relevant at soft
confinement
(nanoscale and bulk)
Aharonov-Bhom
oscillations in nonsimple topologies
dominates at strong
confinement
(atomic scale)
Spatial
confinement
Magnetic field: summary (cont.)
Periodicity and homogeneous magnetic field
Magneto-translations and Super-lattices
B-dependent (super)-lattice constant
Fractal spectrum (Hofstadter butterfly)