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)

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.

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!

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

−

−

−

−

−−

−

=

==

==

==

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

-electrons in conjugated molecules

-electrons in conjugated molecules

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

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)

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

Exciton

• 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

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

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

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

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

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

•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.

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

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

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

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

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

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

Band-gap energy of nanoparticle

according to the effective mass model

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

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

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)

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

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

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

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

Real-World Thermoelectric Devices

GALILEO Spacecraft Thermoelectric Power Unit

Temperature Regulated Laser Diodes

Personal / Mobile Refrigeration

Body-heat Powered Electronics

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

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

=

Improving the Thermoelectric Efficiency

TS

ZT

2

=

Try different material compositions…

…still ZT < 1 for 40 years!

Superlattice (2D) Nanowire (1D)

phonon

e-

New Possibilities: Superlattice (SL) Nanowires

Superlattice Nanowire

Quantum Dots

0D ?

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

Ternary (BixSb1-x)2Te3 Thin Films

İbrahim Erdoğan and Ümit Demir, Electrochimica Acta 55 (2010) 6402–6407

İbrahim Y. Erdogan, Ümit Demir,Journal of Electroanalytical Chemistry 633 (2009) 253–258

Sb2Te3 nanofilms

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

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

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