Schrödinger Equation - WordPress.com · Schrödinger Equation 2S h! Schrödinger Equation is the...
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0)(2 2
22
=−+
VEdx
d
m
Potential Energy
Wave function
The mathematical form of photon-
wave duality of de Broglie
Total Energy
Schrödinger Equation
2
h=
Schrödinger Equation is the basic of Quatum Theory
It can not be proven but works very well.
Erwin Schrödiner
Nobel Prize in Physics, 1933
(1)
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Electron in a Box
➢ Since the wavefunction VANISHES in the barrier regions (this is because V → , hence
the damping is infinitely fast), we conclude that:
* The ALLOWED electron wavelengths within the box are QUANTIZED
* This is similar to the case of a vibrating STRING that is TIED at both ends (the
allowed wavelengths in the string depend on the DISTANCE BETWEEN the
two points where the string is tied)
B
C
D
E
F
G
L
• IN A VIBRATING STRING TIED AT TWO ENDS A DISTANCE L
APART ONLY A LIMITED SET OF VIBRATIONAL MODES ARE
POSSIBLE
• THESE MODES HAVE QUANTIZED WAVELENGTHS THAT ARE
GIVEN AS
,3,2,1,2
== nn
L
It will turn out that our problem is precisely the same, and
that these quantized modes will produce a limited set of
“allowed” energies for the wave.
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Electron in a Box
• At this point we can reach a number of GENERAL conclusions:
* A trapped particle CANNOT have arbitrary energy like a free particle
This is a consequence of the WAVE properties of the particle
* A trapped particle CANNOT have ZERO energy since this implies an
INFINITE wavelength for the particle (which CONTRADICTS the notion the
particle is trapped)
2
222
2mL
nEn
=
NOTE HOW THE SPACING OF
SUCCESSIVE LEVELS GETS BIGGER
AS L GETS SMALLER
FREE
ELECTRON
ELECTRON
IN A BOX
ELECTRON
IN A SMALLER
BOX
AL
LO
WE
DE
LE
CT
RO
N E
NE
RG
IES
Confinement
costs energy!
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Electron in a Box
• EXAMPLE
* Compute the permitted energies for an electron in a 1 Å box and a 10-g
marble in a 10 cm box
* Note how the quantization is only SIGNIFICANT for the electron
* Quantum effects are not important for MACROSCOPIC object
The quantization scale is too SMALL to observe
J105.51001.08
104.4
8:
eV38J100.610101.98
104.4
8:
2642
2
67
2
22
22182
2031
67
2
22
nnmL
hnEMarble
nnnmL
hnEElectron
n
n
−
−
−
−
−−
−
=
==
==
==
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Calculate the emission wavelenght of an electron from excited state to the ground
state for a GaAs potential well with 10.0 nm in length
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-electrons in conjugated molecules
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-electrons in conjugated molecules
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pmpmpmCCdCCd
dort 5,1442
135154
2
)()(=
+=
=+−=
For HOMO n=2 and for LUMO n=3
Calculate the HOMO-LUMO energy difference for an electron
in butadiene by using the Particle in a box model.
average bond length
Total length by 4 bonds of butadiene = 578 pm
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ExcitationExcitation in a Semiconductor
✓The excitation of an electron from the valance band
to the conduction band creates an electron hole pair
h e−(CB)+ h+(VB)
E=h
optical
detector
semiconductor
E
EVB
CBE h=Eg
Creation of an electron hole pair where h is the photon energy
exciton: bound electron and hole pair
usually associated with an electron trapped in a
localized state in the band gap
Band Gap
(energy barrier)
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E
EVB
CBE
band-to-band
recombination
recombination
atinterband trap states
(e.g. dopants, impurities)
E
EVB
CBEE=h
radiative
recombination
non-radiative
recombination
recombination processes
radiative recombination → photon
non-radiative recombination → phonon (lattice
vibrations)
e−(CB)+ h+(VB) → h
ReleaseRecombination of Electron Hole Pairs
✓Recombination can happen two ways:
radiative and non-radiative
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Exciton
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• Size of semiconductor
crystal on the order of
Exciton Bohr Radius
– Discrete energy levels
→Tunable band gap
• Quantum Confinement
• Light-Emitting Diode
(LED) is a PN junction
– Recombination of an
electron and hole
– Electron-hole pair known
as an exciton
e- h+
Exciton Bohr Radius
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Quantum confinement
In small nanocrystals, the electronic energy levels are not continuous as
in the bulk but are discrete (finite density of states), because of the
confinement of the electronic wavefunction to the physical dimensions of
the particles. This phenomenon is called quantum confinement and
therefore nanocrystals are also referred to as quantum dots (QDs).
In any material, substantial variation of fundamental electrical and optical
properties with reduced size will be observed when the energy spacing
between the electronic levels exceeds the thermal energy (kT).
Moreover, nanocrystals possess a high surface are and a large fraction
of the atoms in a nanocrystal are on its surface. Since this fraction
depends largely on the size of the particle (30% for a 1-nm crystal, 15%
for a 10-nm crystal), it can give rise to size effects in chemical and
physical properties of the nanocrystal
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3D (no confinement)
)(2
1)
2(
222
*
2
zyx kkkm
hE ++=
Bulk Materials
E
Ev
Ec
k2 = kx2+ky
2+kz2
k
E
Continuum of states in 3D
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2D (confinement in 1
dimension
])([2
1)
2( 222
*
2
d
nkk
m
hE yx
++=
Quantum Well
n = 1,2,3, …
E Ev
Ec
d
k2 = kx2+ky
2
k
E
Subbands form in quantum
confined directions
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1D (confinement in 2 dimensions
)
Nanowire])()([
2
1)
2( 2
2
2
1
2
*
2
d
n
d
mk
m
hE x
++=
m, n = 1,2,3, …
E Ev
Ec
∞ ∞
∞ ∞
d2
d1k2 = kx
2
k
E
continuum of states along nanowire direction
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0D (confinement in 3 dimensions
)
Quantum Dot
])()()[(2
1)
2( 2
3
2
2
2
1
*
2
d
n
d
m
d
l
m
hE
++=
l, m, n = 1,2,3, …
d2
d1
d3
k
E
E Ev
Ec
∞ ∞
∞ ∞
All states are discrete: no continuum of states
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•What is the Effective Mass
t
vmqF
d
d0=−= Ε
t
vmqF *
d
dn=−= Ε
An electron in crystal may behave as if it had a mass different from the
free electron mass m0. There are crystals in which the effective mass of
the carriers is much larger or much smaller than m0. The effective
mass may be anisotropic, and it may even be negative. The important
point is that the electron in a periodic potential is accelerated relative
to the lattice in an applied electric or magnetic field as if its mass is
equal to an effective mass.
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Band gap of spherical particles
The average particle size in suspension can be obtained from the absorption onset using the effective mass model where the band gap E* (in
eV) can be approximated by:
Egbulk - bulk band gap (eV), h - Plank’s constant (h=6.626x10-34 J·s)
r - particle radius e - charge on the electron (1.602x10-19 C)me - electron effective mass - relative permittivity
mh - hole effective mass 0 - permittivity of free space (8.854 x10-14 F cm-1)
m0 - free electron mass (9.110x10-31 kg)
Brus, L. E. J. Phys. Chem. 1986, 90, 2555
E*
= Egbulk
+22
2er2
1
mem0
+1
mhm0
−
1.8e
40r−
0.124e3
240( )
2
1
mem0
+1
mhm0
−1
Effective Mass Model
✓Developed in 1985 By Louis Brus
✓Relates the band gap to particle size of a spherical
quantum dot
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Brus, L. E. J. Phys. Chem. 1986, 90, 2555
Term 2
✓The second term on the rhs is consistent with the particle in a
box quantum confinement model
✓Adds the quantum localization energy of effective mass me
✓High Electron confinement due to small size alters the effective
mass of an electron compared to a bulk material
Consider a particle of mass m confined
in a potential well of length L. n = 1, 2, …
En =n2 22
2mL2=
n2h2
8mL2
For a 3D box: n2 = nx2 + ny
2 + nz2
0 Lx
Pote
ntia
l E
nerg
y
•
E* = Egbulk +
h2
8r2
1
mem0
+1
mhm0
−
1.8e2
40r−
0.124e4
h2 20( )2
1
mem0
+1
mhm0
−1
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Brus, L. E. J. Phys. Chem. 1986, 90, 2555
Term 3
✓ The Coulombic attraction between electrons and holes lowers
the energy
✓Accounts for the interaction of a positive hole me+ and a negative
electron me-
E* = Egbulk +
h2
8r2
1
mem0
+1
mhm0
−
1.8e2
40r−
0.124e4
h2 20( )2
1
mem0
+1
mhm0
−1
Electrostatic force (N) between two charges (Coulomb’s Law):
Consider an electron (q=e-) and a hole (q=e+)
The decrease in energy on bringing a positive
charge to distance r from a negative charge is:
E =e2
40r2dr = −
e2
40r
r
F =q1q2
40r2 Work, w = F·dr
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Term Influences
✓The last term is negligibly small, Spatial correlation effect (independent of radius) and
significant only in case of semiconductor materials with low dielectric constant.
✓Term one, as expected, dominates as the radius is decreased
0
1E
nerg
y (
eV
)
0 5 10
d (nm)
term 3
term 2
term 1
Conclusion: Control over the particle’s
fluorescence is possible by adjusting the
radius of the particle M
od
ulu
s
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Quantum Confinement of ZnO & TiO2
✓ZnO has small effective masses → quantum effects can be
observed for relatively large particle sizes
✓Confinement effects are observed for particle sizes <~8 nm
✓TiO2 has large effective masses → quantum effects are nearly
unobservable
3
4
Eg (
eV
)
250
300
350
400
onset
(nm
)
0 5 10
d (nm)
ZnO
3
4
Eg (
eV
)
250
300
350
400
onset
(nm
)
0 5 10
d (nm)
TiO2
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Eg = h c / λ
h = 6.63x10-34 J s
c = 2.998x108 m/s
e = 1.60x10-19 C
ε0 = 8.85x10-12 C2/N/m2
CdSe
λbulk = 709 nm
ε = 10.6
me* = 1.18x10-31 kg
mh* = 4.10x10-31 kg
CdS
λbulk = 512 nm
ε = 5.7
me* = 1.73x10-31 kg
mh* = 7.29x10-31 kg
Journal of Chemical Education, 2007, 84, 709
Journal of Chemical Education, 2005, 82, 775
Calculate the diameter of CdS
nanoparticles by using the absorption
spectrum obtained for bulk (red) and
CdS nanoparticles (blue).
The x-intercept of lineer portion of absorption curve gives the
band gap value
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Band-gap energy of nanoparticle
according to the effective mass model
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www.reprap.org
Science 2008 319 1776
•Energy
efficient
•Long life
•Durable
•Small size
•Design
flexibility
Replacement for incandescent and
fluorescent lighting
Improve White LED performance
Quantum dot white LED
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InGaN-CdSe-ZnSe Quantum Dot White LEDs
InGaN CdSe-ZnSe
IEEE Photonics Technology Letters 2006 18 [1] 193
• Single-chip InGaN used
as excitation source
• CdSe-ZnSe QDs used as
phosphor
• Efficiency 7.2 lm/W at 20
mA
– Commercial WLEDs (15-
30 lm/W)
• CIE (0.33, 0.33)
• CRI = 91
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WLED from Ternary Nanocrystal
Composites
Advanced Materials (2006) 18 2545-2548
Charge transfer mechanisms:
-Charge trapping
-Forster energy transfer
QDs: CdSe/ZnS
-Red λ =618 nm
-Green λ =540 nm
-Blue λ =490 nm
At 13 V:
CIE (0.32,
0.45)
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RGB Colloidal Quantum Dot
Monolayer
Nano Letters (2007) 7 [8] 2196-2200
Electron transport layer
Cathode
Hole blocking layer
Quantum dot layer
Hole transport layer
Hole injection layer
Anode
Red: CdSe/ZnS (λ=620 nm)
Green: ZnSe/CdSe (λ=540 nm)
Blue: ZnCdS (λ=440 nm)
Charge injection into blue QDs more
efficient at higher applied biases
At 9V:
CIE (0.35, 0.41)
CRI = 86
Brightness: 92 cd/m2
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Introduction to Thermoelectricity
Seebeck Effect
Peltier Effect
–+V
T1T2
S −=V
TT −12
I
Heat QHeat Q
I
Q=
[V/°K]
[V]
S > 0 for p-type
S < 0 for n-type
S = Seebeck coefficient
π = Peltier coefficient
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Applications
p n I
I
I
Heat
p n
–+V
Heat Source
Refrigerator
(Cooling of electronics)
Power generator
(Waste heat recovery)
Efficiency determined by TS
ZT
2
=
: Electrical conductivity
S : Seebeck coefficient
: Thermal conductivity
ZT: Thermoelectric figure of merit
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Heat
p nII
Thermoelectric Refrigerator
Thermoelectric cooling offers
– No moving parts
– Environmentally friendly
– No loss of efficiency with size reduction
– Can be integrated with electronic
circuits (e.g. CPU)
– Localized cooling with rapid response
Thermoelectric Cooling vs. Mechanical
Refrigeration
ZTTE ~ 1
ZTMech ~ 3
Efficiency lags behind
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Real-World Thermoelectric Devices
GALILEO Spacecraft Thermoelectric Power Unit
Temperature Regulated Laser Diodes
Personal / Mobile Refrigeration
Body-heat Powered Electronics
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Motivation for Nanotech
Thermoelectricity
ZTS T
=2
Seebeck Coefficient ConductivityTemperature
Thermal
Conductivity
ZT ~ 3 for desired goal
Difficulties in increasing ZT in bulk
materials:
S
S and
A limit to Z is rapidly obtained in
conventional materials
So far, best bulk material
(Bi0.5Sb1.5Te3)
has ZT ~ 1 at 300 K
Low dimensional materials give additional control:• Enhanced density of states due to quantum confinement effects
Increase S without reducing
• Boundary scattering at interfaces reduces more than
• Possibility of materials engineering to further improve ZT
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Best alloy: Bi0.5Sb1.5Te3
ZT ~ 1 @ 300 K
To increase Z, we want
but
With known conventional
solids, a limit to Z is rapidly
obtained.
S , ,
S
Thermoelectric Properties of
Conventional Materials
Carrier Concentration
TS
ZT
2
=
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Improving the Thermoelectric Efficiency
TS
ZT
2
=
Try different material compositions…
…still ZT < 1 for 40 years!
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Superlattice (2D) Nanowire (1D)
phonon
e-
New Possibilities: Superlattice (SL) Nanowires
Superlattice Nanowire
Quantum Dots
0D ?
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ZT Enhancement in SL
NanowiresSegment Length Dependence
ZT for [001] n-type PbSe/PbS SL nanowires as a function of segment
length at 77 K. Greater enhancement is predicted for SL nanowires with
diameters of 5nm.
Wire diameter: 10 nm
PbSe/PbS
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Ternary (BixSb1-x)2Te3 Thin Films
İbrahim Erdoğan and Ümit Demir, Electrochimica Acta 55 (2010) 6402–6407
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İbrahim Y. Erdogan, Ümit Demir,Journal of Electroanalytical Chemistry 633 (2009) 253–258
Sb2Te3 nanofilms
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Murat Alanyalıoğlu, Fatma Bayrakçeken, Ümit Demir, Electrochimica Acta 54 (2009) 6554–6559
PbS Nanofilms
PbTe nanofilmsİbrahim Y. Erdoğan, Tuba Öznülüer, Ferhat Bülbül, Ümit Demir, Thin Solid Films 517
(2009) 5419–5424
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İlkay Şiman, Murat Alanyalıoğlu, and Ümit Demir,
J. Phys. Chem. C 2007, 111, 2670-2674
CdS Nanofilms
Tuba Öznülüer,İbrahim Erdoğan, and Ümit Demir,
Langmuir 2006, 22, 4415-4419
ZnS Nanofilms
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1 D Confinement
2D Confinement
2 D confinement
Bi2Te3
Electrodeposition
pH: 9 with EDTA
Nanofilm Nanobelt
Nanowire
pH: 1.5 without
EDTA
İbrahim Erdoğan, Ümit Demir, Electrochimica Acta, 2011, 56 2385–2393