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Reading Quiz
Midterm 3 Nov. 29 G1B30, 7:30-9:00pm
• Confined space: Energy levels quantized
• Free particle: Energy levels are NOT
quantized
I have to go to Japan next week.
Ivan Smalyukh will cover for me Mon,
Wed.
Reading Quiz
• Confined space: Energy levels quantized
• Free particle: Energy levels are NOT
quantized
Quantized means:
a. Discrete
b. Continuous
c. Depends on h (Plank’s constant)
d. E=ħω
Quantized means:
a. Discrete
b. Continuous
c. Depends on h (Plank’s constant)
d. E=ħω
Nanotechnology: how small does a wire have to be
before movement of electrons starts to depend on size
and shape due to quantum effects?
How to start?
Which of these effects will alter the behavior of your circuit
first as you make the structures smaller and smaller?
a. Size of wire compared to size of atom
b. Size of wire compared to size of electron wave function
c. Spacing between wires compared to wavelength of ed. Energy level spacing compared to thermal energy, kBT.
e. Something else (what?) or more than one of the above.
Nanotechnology: how small does a wire have to be
before movement of electrons starts to depend on size
and shape due to quantum effects?
When is small ‘small’?
(When does quantum mechanics matter?)
How to start?
Need to look at Energy level spacing compared to
thermal energy, kBT ≈ 0.025eV. kB=Boltzmann’s constant
What happens
when we make
stuff ~5 times
smaller than on
this chip?
Typically focus on energies in QM.
Electrons, atoms, etc. hopping around with random energy kBT.
kBT >> than spacing, spacing irrelevant. Quantum does not
play a big role. Quantum effects = notice the discrete energy
levels.
Quantum effects
critical
Quantum effects
not critical
How short does a wire have to be before the motion
of electrons is affected by quantum effects?
Look at energy level spacing compared to thermal energy,
kBT= ~0.025 eV at room temp. kB= Boltzmann’s constant
Calculate energy levels for an electron in wire of length L.
We know the spacing is big for e- in atom. What length L do we
need to get kBT between energy levels? (i.e. E2-E1=25 meV)
…
0
L
Want this to be
~25 meV. What L
do we need?
L
Short copper wire, length L.
What is V(x)?
Use time independ. Schrod. eq.
E3
E2
E1
0
Remember photoelectric effect?
 Electron wants to be inside the wire. ⇒ inside is lower
PE (i.e. inside lower V(x)).
Figure out V(x), then figure out
how to solve, what solutions
mean physically.
Everywhere inside the same?
PE
V(x) for electrons with highest PE.
PE
+
+
1 atom
+
+
+
+
+
+
+
How could you find out how deep the pit is for the
top electrons in copper wire?
many atoms
V(x) for top electrons
PE
+
work function of
copper = 4.7 eV
Lots of electrons
(several from
each atom)
It is just the energy needed to remove them from the
metal. That is the work function!!
Electrons repel each other but they are all stuck inside wire.
Electrons of lower energy “smooth out” the potential for
electrons of higher energy
 Potential energy V(x) for electrons with highest energy
looks very flat (like the surface of the sea  “Fermi sea”)
Step 1 (solving Schr. eqn): Figure out V(x)
What is V(x)
of this wire?
Simplification: What if the potential would be infinity
outside the wire? (Very, very large work function)
∞
mathematically
V(x) = ∞ for x<0 and x>L
V(x) = 0 eV for 0<x<L
0 eV
V
0
x
Depth of the potential = Energy needed to remove
the electron from the metal = work function Φ.
(Φ ~ 4.7 eV for copper)
L
What can we say about ψ(x) in this situation?
A. ψ(x) is about the same everywhere
B. ψ(x) ≈ 0 everywhere
C. ψ(x) ≈ 0 everywhere, except for 0<x<L
D. ψ(x) = 0 everywhere, except for 0<x<L
E. Can’t tell anything yet. Need to find ψ(x) first.
ans D: “Infinite square-well potential” (aka. “rigid box”)
Chance of electron being outside: = 0
Approximation: the infinite square well
The “infinite square well”
Exact Potential Energy curve (V)
“Finite square well” or “non-rigid box”
small chance electrons get out of wire
ψ(x<0 or x>L)~0, but not exactly 0!
0
0
L
Good Approximation:
“Infinite square well” or “rigid box”
Electrons never get out of wire
ψ(x<0 or x>L) =0.
x (OK when Energy << work function)
V(x)
with boundary conditions,
ψ(0)=ψ(L) =0
Energy
V(x)
V(x)
So, a clever physicist (or
mathematician) just has to
solve (for 0<x<L)
x ≤ 0, V(x) = ∞
x ≥ L, V(x) = ∞
0<x<L,V(x) = 0
0
L
0
x
NOTE:
Book uses “rigid box” for “infinite square well”
∞
0 eV
0
∞
k = nπ /L
L
€
functional form of solution:
ψ (x) = Bsin(kx)
What is E?
a. can be any value (not quantized).
E quantized by B. C.’s
b.
En =
€
Apply boundary conditions:
x=L 
 A=0
why no ‘0’?
 k=nπ/L
d.
e.
€
⇒? kL=nπ (n=1,2,3,4 …)&
Does this L dependence make sense?
…
x=0  ?
c.
n=2
n=1
What value of L for E2 - E1 = kT?
You should check,
I estimate L ~6.4nm (~50 atoms)
Why bother? Remember Moore’s law?
2009: 32nm, 2011: 22nm, 2013: 16nm,
2015: 11nm, 2017: 8nm, 2019: 6nm
 2 k 2 n 2π 2 2
=
2m
2mL2
Finishing the story: Solving for everything there is to
Results:
know about an electron in a small metallic object (“rigid box”).
Ψ(x,t) = ψ (x)φ (t) =
2
nπx −iE n t / 
sin(
)e
for 0 < x < L;
L
L
Quantized: kn= nπ/L
Quantized En :
= 0 otherwise;
This fulfills the Schr. eqn. and the boundary conditions. But
we still need to… normalize the wavefunction!
€
Probability of finding electron between -∞ and ∞ must be 1.
Real(Ψ)&
0
(Do in Homework)
n=1,2,3,….
n=1
L