Semiconductor

Nanomaterials (I):

Quantum Dots

• In bulk crystalline lattices the Schroedinger equation for

the periodic potential V(r) = V(r+a), where a is the lattice

spacing, is of the form

• The time-independent wave-function is

• u(r) is periodic with periodicity a, and the energy

• eigenvalues are

• me,h is the effective mass of the electron or hole

Electrons in Solids

y r( ) = eikru r( )

Schematic plot of the single particle energy spectrum in a bulk

semiconductor for both the electron and hole states on the left

side of the panel with appropriate electron (e) and hole (h)

discrete quantum states shown on the right. The upper

parabolic band is the conduction band, the lower the valence.

Consider a spherical crystal with a diameter D (=2R). For a quantum

structure D ≤

λ = h/p (de Broglie’s wavelength).

• At T = 300 K, take E=(3/2)kBT=p2/2me to get λ ~ 6 nm.

• Now the Schroedinger equation for the spherical quantum dot (QD) is of

the form

where

• The wavefunction may be written in terms of Associated Legendre R

polynomials, Bessel functions, and spherical harmonics. The energies of

electrons or holes are

where n and l are the principal and angular momentum quantum numbers

and are the zeroes of the Bessel functions.

• The photon energy required to produce the electron-hole pair (exciton) is

Quantum Confinement – 3D QD

Vi ri( ) =0 for ri < R

¥ for ri > Ri = e,h y re, rh( ) = ye re( )yh rh( )

Solutions of quantum dots of varying size. Note the variation

in color of each solution illustrating the particle size

dependence of the optical absorption for each sample. Note

that the smaller particles are in the red solution (absorbs

blue), and that the larger ones are in the blue (absorbs red).

Nanoparticles and

Quantum Dots

Using oxidation/reduction in solutions to

create nano‐elements such as quantum dots,

metal particles, and insulator particles

Nanoparticle Chemical Growth:

Solution/Colloidal Chemistry

The principle for colloidal synthesis of semiconductor

nanocrystals is based on a study by Le Mur and Dinegar, which

showed that a temporally short cluster nucleation event

followed by controlled slow growth on the existing nuclei results

in the formation of monodispersed colloids

Colloidal Growth of Nanocrystals

- In practice, reagents are rapidly injected to a

vessel charged with hot coordinating solvent, thus

raising the concentration above nucleation

threshold. This is possible when the temperature is

sufficient to decompose the reagents forming a

supersaturated solution species.

- Supersaturation is followed by a short nucleation

period that partially relieves the supersaturation

- This results in a drop in concentration of species

below the critical concentration for nucleation, and

the clusters “bang” out of solution. As long as the

rate of addition of precursor does not exceed the

rate at which it is consumed by the growing

nanocrystals, no additional nuclei form.

- Since the growth of the nanocrystals is similar, the

size distribution is mainly governed by the time

over which the nuclei are formed and continue to

grow. For increasing reaction time the larger and

more

uniform the nanocrystals become and thus gives

one control over size.

From Whitesides and Love, Scientific American Sept 2001

Wolf, Edward L., “Introduction”, Nanophysics and Nanotechnology: An Introduction to Modern Concepts in Nanoscience, 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Functionalization

• The synthesis of group II-VI semiconductors ME, where M

= zinc, cadmium or mercury and E = sulfur, selenium, or

tellerium are prepared using metal alkyl group II sources, e.

g, dimethylcadmium, diethylzinc, and dibenzylmercury, and

organophosphine chalcogenides (R3PE) or

bistrimethylsilylchalogenides (TMS2E), where E = S, Se or

Te. The coordinating solvents (150 – 350 °C) are often long

chain alkylphosphines (R3P), alkylphosphine oxides

(R3PO).

• For group III-V semiconductors InE, where E = phosphorus

and arsenic the In precursors {InCl(C2O4)} is already present

in the coordinating solvent [R3P/R3PO] prior to injection of

the TMS2E, where E = P or As.

Colloidal Growth of Compound

Semiconductor QDs

• Another method of synthesis of quantum dots is through epitaxial growth

(from vapor or liquid) of clusters on a substrate.

• The epitaxial growth method allows for a wide range of control of

principally the order of the quantum dots on a substrate since through this

method a regular array is achievable through selective growth conditions.

• Once the material gets to the substrate and if there is sufficient energy

and number density, the atoms on the surface can move in 2D across the

surface and agglomerate into either; a dilute array of well ordered small

clusters ( # of atoms per cluster < 20) or a random agglomeration of

clusters that can range in size from < 1 nm to many microns.

• In the first case, a well ordered array of small clusters on a substrate

requires especially stringent growth conditions in order to achieve the dilute

film morphology.

• Quantum dot size control is achieved by keeping the amount of material

on the substrate low and ambient conditions pristine.

Epitaxial Growth of Quantum Dots

Atomic Force Microscope images of Ge clusters on two types

of surfaces. Graphite in the left two panels, and SiO2 in the

right. The line plots on the figure give vertical profiles of line

cuts through the AFM images directly above and give the

quantitative size information

Illustration of a cross sectional view of Si quantum dots

formed in a glass matrix via ion implantation. Note that the

random arrangement and spherical shape of the quantum dot

particles is expected for quantum dots implanted in an

amorphous media.

QDs Formed by Ion Implantation

Photoluminescence spectra from Si (400 keV, 1.53 x 1017 cm-

2) implanted SiO2 as implanted and after annealing at 950 and

1100 °C. (by permission of the American Institute of Physics.)

Scanning electron micrograph of quantum dot patterns on a

GaSb surface induced by Ar-ion sputtering with an ion energy

of 500 eV. The dots show a hexagonal ordering with a

characteristic wavelength that depends on ion energy. The

insets show the corresponding distribution of the nearest-

neighbor distance. (by permission of the American Physical

Society.)