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Introduction to Quantum Properties
of Solid State Devices
Principles of Quantum Mechanics
1) The principle of Energy Quanta:
Experiments which showed inconsistency
between experimental results and classical
theories:
Principles of Quantum Mechanics
Thermal Radiation
• Classical Physics: the frequency of the radiation
of a heated cavity should increase monotonically
as the thermal energy of the cavity increases.
• Observations: the frequency of the radiation of a
heated cavity increases only up to a certain
energy then decreases.
Principles of Quantum Mechanics
When a particle jumps from one
discrete state to another, it either
absorbs or emits energy quanta (most
frequently it is photon) which
corresponds to a particular wavelength
associated with the difference of the two
states’ energy
Max Planck’s postulate (1900):
explains blackbody radiation
E = hυ
Principles of Quantum Mechanics
Hydrogen Atoms:
•
Classical Physics: The atomic spectra should be
continuous.
•
Observations: The atomic spectra contain only some
discrete lines.
Photoelectric effect:
•
Classical Physics: The absorption of optical energy by
the electrons in a metal is independent of light
frequency.
•
Observations: The absorption of optical energy by the
electrons in a metal depends on optical frequency.
Principles of Quantum Mechanics
• Planck’s postulate in 1900 that radiation
from a heated sample is emitted in
discrete units of energy, called quanta.
E =hυ
h is the Planck’s constant, υ is frequency of the radiation
Principles of Quantum Mechanics
• Photoelectric Effect
• Einstein in 1905 took a
step further to interpret
the photoelectric results
by suggesting that the
electromagnetic radiation
exists in the form of
packets of energy. This
particle-like packet of
energy is called photon.
Photoelectric Effect
KE =
1 2
mv = hν − hν 0 = hν − Φ
2
Principles of Quantum Mechanics
2) Wave-Particle Duality Principle
•
•
•
•
Photoelectric Effect (particle)
Compton Effect (particle)
Diffraction pattern by electrons (wave)
Since waves behave as particles, then particles
should be expected to show wave-like
properties.
• de Broglie hypothesized that the wavelength of a
particle can be expressed as λ=h/p
• The momentum of a photon is then: p = h/λ =ħk
Principles of Quantum Mechanics
3) The Uncertainty Principle (Heisenberg)
• First statement:
• It is impossible to simultaneously describe with absolute
accuracy the position and momentum of a particle.
• Δx Δp ≥ ħ
• ħ is very small so this principle only applies to very small
particles
• Second statement:
• It is impossible to simultaneously describe with absolute
accuracy the energy of a particle and the instant of time
the particle has this energy.
• ΔE Δt ≥ ħ
A Particle in a Box
Classical Mechanics vs. Quantum Mechanics
•
CM: The particle remains trapped inside the box and can acquire any
energy available to it .
•
QM: A particle trapped in such a box cannot have any energy, and will have
only certain allowed energy.
•
CM: If no other forces are in the system, then the particle should have
Brownian motion and has an equal probability of being anywhere and
everywhere in the box.
•
QM: The particle has certain preferable points, where the probability of
finding the particle is more than in other parts.
•
CM: The exact position and momentum of the particle should be obtainable
simultaneously for the particle in question.
•
QM: We cannot measure the momentum and the position of the particle in
the box simultaneously without some uncertainty.
A Particle in a Box
Classical Mechanics vs. Quantum Mechanics
•
CM: At a particular moment, the exact energy of the particle should
be measurable.
•
QM: At any instant of time, the energy is not measurable above a
certain extent.
•
CM: The particle must be confined in the box and has no chance of
escaping the box as long as its energy is lower than that of the box.
•
QM: The particle has a finite chance of escaping the system even
with energy lower than the box.
•
CM: If by any chance the particle attains energy larger than that of
the box, the particle will certainly leave the box.
•
QM: If by any chance the particle attains energy larger than that of
the box, there remains certain probability that particle might still be
trapped inside.
Principle of Quantum Mechanics
Basic Postulates
– Late 1920’s Ervin Schrodinger developed
wave mechanics based on Planck’s theory
and de Broglie’s idea of the wave nature of
matter.
– Warner Heisenberg formulated an alternative
approach in terms of matrix algebra called
Matrix Mechanics.
Principle of Quantum Mechanics
Basic Postulates
• Dynamical Variables in Physics: position, momentum,
energy
• In classical mechanics the state of a system with a
number of particles at any time is defined by designating
the particle and momentum coordinates of all particles.
• In quantum mechanics the state of a system is defined
by a state function Ψ that contains all the information we
can obtain about the system.
• To approach quantum mechanics we consider several
postulates that are assumed to be true
Principle of Quantum Mechanics
Basic Postulates
• Postulate I:
– There exists a state function Ψ(r;t) which
contains all the measurable information about
each particle of a physical system. Ψ(r;t) is
also called wave function.
• Postulate II:
– Every dynamical variable has a corresponding
operator. This operator is operated on the
state function to obtain measurable
information about the system.
Principle of Quantum Mechanics
Basic Postulates
In a one-dimensional system
Dynamical Variables:
• Position r
• Momentum p
• Total Energy E
• Potential Energy V(r)
Principle of Quantum Mechanics
Basic Postulates
Dynamical Variable (1-D & 3-D)
Position x , r
Momentum p (x; r)
Total Energy E
Quantum Mechanic Operator
∧
X ≡ X,
∧
r≡r
h ∂
Pˆx ≡
J ∂X
∧
Pr ≡
h ∂
j ∂r
h ∂
Eˆ ≡ −
J ∂t
Potential Energy V(x; r)
Vˆ ( x ; r ) ≡ V ( x ; r )
Principle of Quantum Mechanics
Basic Postulates
• Postulate III:
– The state function Ψ(r;t) and its space derivative
∂Ψ/∂r must be continuous finite and single-valued for
all values of r.
• Postulate IV
The state function must be normalized i.e
∞
∗
ΨΨ
dr = 1
∫
−∞
– Ψ* is the complex conjugate of Ψ. Obviously Ψ Ψ* is
a positive and real number and is equal to /Ψ/².
Principle of Quantum Mechanics
Basic Postulates
• Assume a single particle-like electron;
then Ψ Ψ* is interpreted as the statistical
probability that the particle is found in a
distance element dx at any instant of time.
Thus Ψ Ψ* represents the probability
density.
Principle of Quantum Mechanics
Basic Postulates
Postulate V.
•
The average value 〈Q〉 of any variable corresponding to the state
function Ψ is given by expectation value:
∞
Q =
∫Ψ
∗
∧
Q Ψ dr
−∞
•
•
〈Q〉 is the expected value of any observation.
Once the state function corresponding to any particle was found, it is
possible to calculate the average position, energy and momentum of
the particle within the limit of the uncertainty principle.
Principle of Quantum Mechanics
Schrodinger Equations
Assume:
p2
E=
+ V ( x, y, z )
2m
where
p 2 = p x2 + p 2y + p z2
Recall the operators were defined in postulate II;
p2 − h2 2
≡
∇
2m 2m
∧
V ( x, y, z) ≡ V ( x, y, z)
h ∂
E≡−
j ∂t
Schrodinger Equations:
⎛ h2 2
⎞
∂
∇ + V (r )⎟⎟ψ (r , t )
jh ψ (r , t ) = ⎜⎜ −
∂t
⎝ 2m
⎠
The potential energy term forms the necessary link between quantum
mechanics and the real world. It contains all environmental factors
that could influence the state of a particle. As an example for a
particle in a deep potential well:
En
2
n 2π 2
h
=
2 ma 2
2
ψn =
sin Kx
a
Note than p(r)= ΨΨ* = ψψ* i.e. probability density is time -independent and define an stationary state
Time dependant Schrödinger’s Equation
∂
j h ψ (r , t ) =
∂t
single-particle 3D time-dependent
Schrödinger equation
Let’s assume :
ψ (r , t ) = ψ (r )ψ (t )
Divide both sides by
ψ (r )ψ (t )
Time –dependent part:
jh dψ (t)
=E
ψ (t) dt
1 dψ (t)
jE
→
=−
ψ (t) dt
h
ψ (t ) = e
− jEt / h
⎛ h2 2 ∧ ⎞
⎜⎜ −
∇ + V (r )⎟⎟ψ (r , t )
2
m
⎠
⎝
⎡ h2 2 ∧
⎤
d ψ (t )
ψ ( r ) jh
= ψ (t ) ⎢ −
∇ + V ( r ) ⎥ψ ( r )
dt
⎣ 2m
⎦
∧
⎤
jh d ψ (t )
1 ⎡ h2
2
=
−
∇
+
V
(
r
)
⎥ψ ( r )
ψ ( t ) dt
ψ ( r ) ⎢⎣ 2 m
⎦
Time-independent part:
And
General
solution:
⎡ h2 2 ∧ ⎤
∇ + V ( r ) ⎥ψ ( r ) = Eψ ( r )
⎢−
⎣ 2m
⎦
ψ ( r , t ) = e − jEt / hψ ( r )
Some Remarks
2
ψ ( r , t ) = ψ ∗ ( r , t )ψ ( r , t ) = e − jEt / hψ ( r ) ∗ e − jEt / hψ ( r ) = ψ ( r ) ∗ψ ( r )
Probability density is time -independent and define an stationary state.
< A >= ∫ψ ∗ ( r , t ) Aψ ( r , t ) = ∫ψ ∗ ( r )ψ (r )
The expectation value for any
time-independent operator is
also time-independent
plane wave
If there is no force acting on the particle, then the potential function V(r) will be
constant and we must have E>V(r).
assume
ψ (t ) = e
V(r)=0
− j ( E h )t
ψ ( x ) = Ae
in +x direction
∂ 2ψ ( x ) 2mE
+ 2 ψ (x ) = 0
2
h
∂x
time dependant part
(
)
⎛ j
⎞
⎜ x 2 mE − Et ⎟
⎝h
⎠
ψ
(x ) =
ψ ( x ) = Ae( jx
+ Be
Ae
(
)
⎛ j
⎞
⎜ − x 2 mE + Et ⎟
⎝ h
⎠
j ( kx − ω t
)
2 mE h
) + Be(− jx
2 mE h
)
k=2π/λ
or
λ=h/√(2mE)
λ=h/p
The probability density function ψ(x,t)ψ*(x,t)=AA*
a free particle with a well
defined energy will also
have a well defined
wavelength and
momentum.
is a constant independent of position.
a free particle with a well defined momentum can be
found anywhere with equal probability.
Particle in a box
classical 1D case
⎡ h2 2 ∧ ⎤
∇ + V (r )⎥ψ (r ) = Eψ (r )
⎢−
⎦
⎣ 2m
1D Infinite Quantum Well
U=8
U=8
V=0 inside the one dimensional box
Function f(x)
∂ 2ψ ( x ) 2mE
+ 2 ψ (x ) = 0
2
∂x
h
A particular form of
solution
X=1
X=0
U=0
Discrete X axis
ψ (x) = A1 coskx + A2 sinkx
X=n
X=n+1
k=
2mE
h2
is the wave number.
Particle in a box…
ψ (x ) = A2 sin kx
First boundary condition, x=0, ψ(0)=0 then A1=0
second boundary condition
ψ (a ) = A2 sin kx = 0
x=a, ψ(a)=0
a
from the normalization condition
∫ (A
2
2
)
sin 2 kx dx = 1
k=nπ/a
A2=√(2/a)
0
Final solution
ψ (x ) =
2 ⎛ nπx ⎞
sin⎜
⎟
a ⎝ a ⎠
h 2π 2 n 2
2mE n 2π 2
= 2 ⇒ E = En =
2
h
a
2ma 2
energy for
this system is
quantized
Particle in a box
n=4
E
n=3
ψ
|ψn|
2
n
n=2
n=1
(a)
x=
0
(b)
x=
a
x=
0
(c)
Particle in an infinite potential well:
a) Four lowest discrete energy levels;
b) Corresponding wave functions
c) Corresponding probability functions
x=
a
En
2
n 2π 2
h
=
2 ma 2
Reflection: Step potential
ψ 1 (x ) = A1e jk x + B1e − jk x (x ≤ 0)
1
T.I.S.E
∂ 2ψ 1 ( x ) 2mE
+ 2 ψ 1 (x ) = 0
2
∂x
h
k1 =
2mE
h2
vi and vr are the magnitude of the incident and reflected wave velocity.
1
region II the Schrödinger
equation
∂ 2ψ 2 ( x ) 2m
+ 2 (V0 − E )ψ 2 ( x ) = 0
2
∂x
h
ψ 2 (x ) = A2e − k x + B2e k x (x ≥ 0)
2
k2 =
2m(V0 − E )
h2
ψ 2 (x ) = A2e − k x
2
B2=0 is zero due to fact that function must be
finite everywhere and potential is a step function
boundary x=0
2
ψ1(0)= ψ2(0)
A1+B1=A2.
jk1A1-jk1B1=-k2A2
∂ψ 1
∂x
=
x =0
∂ψ 2
∂x
x =0
Continue
(
)(
B1 ⋅ B1*
k 22 − k12 + 2 jk1k2 k 22 − k12 − 2 jk1k 2
=
*
2
2 2
A1 ⋅ A1
k2 + k1
kinetic energy
K.E.=mv2/2
(
k1 =
)
2mE
=
2
h
2m
h2
(
(
1
2
mv
)
2
)
2 2
1
)
v 2 mv
= m 2 =
h
h
2
vr ⋅ B1 ⋅ B
k − k + 4k12 k 22
R=
=
= 1.0
2
2
2
vi ⋅ A1 ⋅ A
k 2 + k1
*
1
*
1
2
2
(
)
Barrier potential and tunneling
quantum particles can tunnel
trough the barrier and get out on
the other side
Tunneling
E<U0 for leftside incident
particle
k0 = p√(2mE)/ħ,
For wide barriers,
The tunneling rate is a very sensitive function of L