Transcript Prezentacja programu PowerPoint

Quantum Dots in Photonic Structures
Lecture 6: Electronic band structure of solids
Jan Suffczyński
Wednesdays, 17.00, SDT
Projekt Fizyka Plus nr POKL.04.01.02-00-034/11 współfinansowany przez Unię Europejską ze środków Europejskiego
Funduszu Społecznego w ramach Programu Operacyjnego Kapitał Ludzki
Plan for today
1.
Reminder
3.
Band structure
of solids – tight
bindning
approximation
2.
Band structure
of solidsNearly free
electron model
Reminder: Fundamental postulates of the
quantum mechanics
Postulate 1: All information about a system is provided by
the system’s wavefunction
𝜌 x dx = Ψ(𝑥, 𝑡) 2dx
∞
∞
ρ x dx =
Ψ(𝑥, 𝑡) 2 dx = 1
−∞
−∞
Postulate 2: The motion of a nonrelativistic particle is
governed by the Schrodinger equation
d

Time-independent S.E.:  
2
2
m
dx

2
2

ˆ
 V ( x )   ( x) 

E  ( x)
Postulate 3: Measurement of a system is associated with a
linear, Hermitian operator
Oˆ   Oˆ 
Oˆ   dx * ( x) Oˆ ( x)( x)
De Broglie’s Hypothesis
• ALL material particles possess wave-like
properties, characterized by the wavelength λB,
related to the momentum p of the particle in the
same way as for light
de Broglie
wavelength of the
particle
Frequency:
E
f 
h
h
B 
p
Planck’s Constant
Momentum of the
particle
h
E  hf 
2f  
2
Wave Picture of Particle

( x, t )   A(k ) cos(kx  t )dk
0
1.
A(k) is spiked at a given k0, and
zero elsewhere
Only one wave with k = k0 (λ = λ0)
contributes; thus one knows
momentum exactly, and the
wavefunction is a traveling wave –
particle is delocalized
2.
A(k) is shaped as a bell-curve
Gives a wave packet – “partially”
localized particle
3.
A(k) is the same for all k
No distinctions for momentums, so
particle’s position is well defined - the
wavefunction is a “spike”, representing
a “very localized” particle
Crystal lattice
+
+
We define lattice points; these are points with identical environments
Crystal = lattice + basis
Wigner-Seitz cell
The smallest (“primitive”) cell which displays the full symmetry
of the lattice is the Wigner-Seitz cell.
Construction method: surfaces passing through the
middle points to the nearest lattice points
Wigner-Seitz Cell construction
Form connection to all neighbors and span a plane normal
to the connecting line at half distance
Wigner-Seitz cell construction
The smallest (“primitive”) cell which displays the full symmetry
of the lattice is the Wigner-Seitz cell.
Wigner-Seitz cell construction
The smallest (“primitive”) cell which displays the full symmetry
of the lattice is the Wigner-Seitz cell.
First Brilluoin zone
Body-centered cubic
Realistic Potential in Solids

 
U ( r )  U( r  T )




T  n1a  n2 b  n3 c
– ni are integers
• Example: 2D Lattice



T  n1a  n2 b
n1  2; n2  3
Bloch theorem
Felix Bloch
1905, Zürich 1983, Zürich
Bloch waves
Bloch’s theorem:
Solutions of the Schrodinger equation
 2 d2 ˆ 
 
 V(r ) Ψ k (r )   k Ψ k (r )
2
 2m dr

for the wave in periodic potential U(r) = U(r+R) are:
Bloch function:
k (r )  e
i k r
uk (r )
Envelope part
Periodic (unit cell) part
uk (r )  uk (r  R )
Felix Bloch
1905, Zürich 1983, Zürich
Bloch waves
Two equivalent views on the Bloch wave (1D example):
modified slide from Rob Engelen
Bloch’s Theorem
• What is probability density of finding particle at coordinate x?
P( x)   ( x)   ( x) ( x)
2
P( x)  [uk ( x)e
P( x)  uk ( x)uk ( x)e
*
*
 ikx *
] [uk ( x)e
 ikx  ikx
e
P( x)  uk ( x)
 ikx
]
 uk ( x)uk ( x)
*
2
• But |uk(x)|2 is periodic, so P(x) is as well
Probability of finding matter at position x scales with ||2 or ×*
Compare: probability of detecting light scales with |E(x,t)|2 or E×E*
Bloch’s Theorem
P( x)  P( x  d)
The probability of finding an electron at any atom in
the solid is the same!
 Each electron in a crystalline solid “belongs” to each
and every atom forming the solid
Bloch theorem
Remark:
(in 1D case)
If V(x) has lattice periodicity [“translational invariance”, V(x)=V(x+a)]:
• the electron density r(x) has also lattice periodicity, however,
• the wave function does NOT:
r ( x )  r ( x  a )  * ( x )  ( x )
( x  a )  ( x )   *  1
Periodicity in reciprocal space
Reciprocal lattice vector
but :
Electron in a crystal periodic potential
Vc (x)
ions
If:
 k Vc (x)
The case of the „empty” lattice
Nearly free electron model
 2 d2 ˆ 
 
 V(x) Ψ k ( x)   k Ψ k ( x)
2
 2m dx

 k Vc (x)
k (r )  e
i k r
uk (r )
k ( x)  ei k x
2
 k
k 
2m
2
Free electron energy
Nearly free electron model

2
electron
k
 k

2m
Remark:
2

E
Energy of an electron with 1 A wavelength  150 eV
Energy of a photon with 1 A wavelength  12 keV
photon
k
 kc
Brilluoin zones
2
 k
k 
2m
2
 (k): single parabola
folded parabola
Nearly free electron model
Consider a set of waves with +/ k-pairs, e.g.
1  e
ikx
e
ix / a
k
 

a
wave moving right
2  e ikx  e ix / a wave moving left
Superposition of these waves also a solution of Schr. equation:
ix / a
  1  2  e
   1   2  e
i x / a
e
ix / a
e
 i x / a
 x 
 2 cos 
a
x 
 2i sin 

 a 
Nearly free electron model
Origin of a band gap!
Kittel
allowed energy bands
Electronic energy bands
Effective Mass: m*
A method that the free electron model to work in the
situations where there are lattice crystal perturbations

2
2
 k

2 m*
Effective Mass m*
-- describing the balance between applied ext-E and lattice site reflections
1

m*
1  2
 2 k 2
m* a =
SF
ext
q Eext
2)
greater curvature, 1/m* > 1/m > 0,  m* < m 
net effect of ext-E and lattice interaction
provides additional acceleration of electrons
m = m*
greater |curvature| but negative,
At inflection pt
net effect of ext-E and lattice interaction
de-accelerates electrons
1)
No distinction between m & m*,
m = m*, “free electron”, lattice structure does
not apply additional restrictions on motion.
Isolated Atoms
Diatomic Molecule
Four Closely Spaced Atoms
Six Closely Spaced Atoms as fn(R)
the level of interest
has the same Energy in
each separated atom
Two atoms
Six atoms
ref: A.Baski, VCU 01SolidState041.ppt
www.courses.vcu.edu/PHYS661/pdf/01SolidState041.ppt
Solid of N atoms
Four Closely Spaced Atoms
conduction band
valence band
Solid composed of ~NA Na Atoms
as fn(R)
1s22s22p63s1
Schrödinger Equation Revisited
• If a wavefunctions ψ1(x) and ψ2(x) are
solutions for the Schrödinger equation for
energy E, then functions
– -ψ1(x), -ψ2(x), and ψ1(x)±ψ2(x) are also solutions
of this equations
– the probability density of -ψ1(x) is the same as for
ψ1(x)
2
 2 d  1, 2 ( x)

 U( x)[ 1,2 ( x)]  E 1,2 ( x)
2
2m dx
 2 d 2 [ 1 ( x)  2 ( x)]

 U( x)[ 1 ( x)  2 ( x)]  E[ 1 ( x)  2 ( x)]
2
2m
dx
Hydrogen molecule
• Consider an atom with only one electron in s-state
outside of a closed shell
• Both of the wavefunctions below are valid and the
choice of each is equivalent

 s (r ) 

 s (r ) 
 Af (r )e
Zr / na 0
 Af (r )e
Zr / na 0
• If the atoms are far apart, as before, the
wavefunctions are the same as for the isolated
atoms
Hydrogen molecule
•
Once the atoms are brought together the
wavefunctions begin to overlap
– There are two possibilities
1. Overlapping wavefunctions are the same (e.g., ψs+ (r))
2. Overlapping wavefunctions are different
• The sum of them is shown in the
figure
• These two possible combinations
represent two possible states of two
atoms system with different
energies
Covalent Bonding Revisited
• When atoms are covalently bonded
electrons supplied by atoms are shared
by these atoms since pull of each atom is
the same or nearly so
– H2, F2, CO,
• Example: the ground state of the
hydrogen atoms forming a molecule
– If the atoms are far apart there is very little
overlap between their wavefunctions
– If atoms are brought together the
wavefunctions overlap and form the
compound wavefunction, ψ1(r)+ψ2(r),
increasing the probability for electrons to
exist between the atoms
 1s 
1
a03
e  r / a0
Interatomic Binding
• All of the mechanisms which cause bonding between the atoms
derive from electrostatic interaction between nuclei and electrons.
• The differing strengths and differing types of bond are determined by
the particular electronic structures of the atoms involved.
• The existence of a stable bonding arrangement implies that the
spatial configuration of positive ion cores and outer electrons has less
total energy than any other configuration (including infinite separation
of the respective atoms).
• The energy deficience of the configuration compared with isolated
atoms is known as cohesive energy, and ranges in value from 0.1
eV/atom for solids which can muster only the weak van der Waals to
7ev/atom or more in some covalent and ionic compounds and some
metals.
• This typical curve has a
minimum at equilibrium
distance R0
• R > R0 ;
– the potential increases
gradually, approaching
0 as R∞
– the force is attractive
• R < R 0;
– the potential increases
very rapidly,
approaching ∞ at small
separation.
– the force is repulsive
V(R)
Repulsive
0
R0
R
Attractive
r
R
Force between the atoms is the negative of the slope of this curve. At
equlibrium, repulsive force becomes equals to the attractive part.
Electron vs photon