quantum size effects in nanostructures

8/23/2019 Quantum Size Effects in Nanostructures

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Quantum size effects in nanostructures

Lecture notes to the courseOrganic and Inorganic Nanostructures

Kjeld Pedersen

Department of Physics and NanotechnologyAalborg University

Aalborg, August 2006


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1. Introduction

One of the most direct effects of reducing the size of materials to the nanometer range is theappearance of quantization effects due to the confinement of the movement of electrons. This leads

to discrete energy levels depending on the size of the structure as it is known from the simplepotential well treated in introductory quantum mechanics. Following this line artificial structureswith properties different from those of the corresponding bulk materials can be created. Controlover dimensions as well as composition of structures thus makes it possible to tailor material

properties to specific applications.Both semiconductor and metal nanostructures have been investigated over the years. While theapplications of metal nanostructures are still very limited semiconductor electronic andoptoelectronic components based on structures with quantum size effects have been on the marketfor several years. The present notes will therefore treat the size effects with semiconductor materialslike GaAs in mind.It is very rare that the nanostructures appear as free standing structures. Rather, they are embeddedin a matrix material that necessarily has an effect on the electronic levels in the nanostructure. In therather simple treatments presented here it is not possible to take the matrix materials into account.However, the simple models still give good impressions of the basic principles behind quantum sizeeffects in various structures.The nanostructures treated here can be divided into the following classes:

Two-dimensional objects: Thin films with thickness of the order of a few nanometers are usuallydeposited on a bulk material. Their properties may be dominated by surface and interface effects orthey may reflect the confinement of electrons in the direction perpendicular to the film (a quantumwell). In the two dimensions parallel to the film the electrons behave like in a bulk material.One-dimensional objects: Cylinder-like objects like wires and tubes with diameters on thenanoscale and lengths typically in the micometer range. Also embedded structures with rectangularcross section exist. Confinement effects for electrons may appear in the transverse direction while

electrons are free to move in one dimension (along the structure).Zero- dimensional objects: Usual names are nanoparticles, clusters, colloids, nanocrystals, andfullerenes. They are composed of several tens to a few thousand atoms. Electrons are confined in allthree directions.Carbon nanotubes: They can be described as graphene sheets rolled into cylinders. Single walledtubes have diameters of the order of 1-2 nm. The confinement effects can here be seen as theadjustment of electronic wavefunctions to fit to the circumfence of the tube.

It should be noted that many applications of nanostructures do not depend directly on size effects onelectronic properties. Instead, the surface to volume ratio may lead to mew functionalities, forinstance in catalytic effects. Intensive research is directed toward introduction of newfunctionalitites to matrix materials without affecting other material properties by introduction of

nanosized structures.

At the end of these notes some useful references with more detailed treatments of nanostructuresand their optical properties are given.


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2. Quantum wells, wires, and dots

In this section we will go through a sequence of structures with increasing confinement of electronsfrom the potential barriers at the boundaries of the structures.

2.1 Two-dimensional objects

In this type of systems the electron are confined in their motion in one direction while they move asin the corresponding bulk material in the other two directions. Today these effects are widely usedin semiconductor lasers where the band structure of quantum wells are tailored to give the desiredemission wavelengths. The classic example of a semiconductor quantum well is the GaAs

AlxGa(1-x)As multilayer system. Here the depth ofthe well can be adjusted through the composition ofth e AlGaAs layer (Fig. 2.1). Generally, wedistinguish between quantum well structures andsuperlattices. The two structures are illustrated inFig. 2.2. If the distance between the potential wells isso large that the wave functions of the bound stateshave little overlap we talk about quantum wells. If,on the other hand the wave functions overlap thestructure is usually called a superlattice.

In the following we will consider a single quantum well as illustrated on Fig. 2.3. In order todescribe wave functions and energy levels we start with the Schrdinger equation:

)()()(

22

2

22

2

||

2

||

2

rErzV

zmrm z

rrhh =

+

. (2.1)

This is a second-order differential equation in space coordinates. Wewill treat the parallel and z- directions as independent and thus separatethe variables according to:

)()()( |||| zrr nrr

= . (2.2)

We will then have two independent differential equations:

Fig. 2.1 Quantum well structure formed in GaAslayers sandwiched between AlGaAs layers.

Fig. 2.2. Quantum well and super latticestructures.

Fig. 2.3. A simple potentialwell.


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)()()(2 2

22

zEzzVzm nnz

=

+

h

(2.3)

)()()(2

||||||||2||

2

||

2

rEErrm

n rrh = . (2.4)

Let us start by considering z-dependent part which includes the size-dependence in the electroniclevels. At first let us look at the solution for infinite barriers, that is EC. The wave functionsmust then be zero at the borders of the well (and outside the well): n(L/2)=n(-L/2)=0. These

boundary conditions are fulfilled by wave functions of the form

=

=

=

.,6,4,2sin2

,5,3,1cos2

oddnL

zn

L

evennLzn

L

n

(2.5)

For growing n the wave functions are alternatively even and odd functions of z (Fig. 2.4). Thecorresponding energies are given by

=

= ,3,2,1

2

22

nforL

nm

Ez

n

h. (2.6)

This gives a first impression of the symmetries of wave functions and the order of magnitude of thediscrete energy levels appearing as a result of the finite thickness of the well. Notice that separation

between energy levels decrease as the thickness of the well increases. The infinite wellapproximation is however usually far from useful predictions in real systems. We thus have to dealwith finite barrier wells.

In the following we will consider bound states, that is E 23 222 h

.

For Al0.3Ga0.7As the dept of the confining potential meVEC 230= corresponding to a minimumsize of the box a=2.2 nm.

0D

Elmn

Fig. 2.15. Density of states for a 0D system.Fig. 2.17. Electron and hole states in thequantum dot.


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3. Optical properties I nterband transitions

Some of the most useful techniques for studies of the properties of nanostructures are based on theoptical transitions between the discrete electronic levels. Absorption spectroscopy, for instance, willreveal the direct band gap and thereby also an increase in gap energy (blue shift) due to

confinement effects.

Fig. 3.1. Direct (left) and indirect (right) band gaps. This could for instance be GaAs (left) and Si (right).

Figure 3.1 above illustrates two situations that occur in symmetry directions in semiconductors. Ifthe highest point in the (occupied) valence band lies directly under the lowest point in the (empty)conduction band we talk about a direct band gap and the gap energy corresponds to the onset ofoptical absorption. If, on the other hand, maxima and minima in the valence and conduction bandappear at different points in k-space we talk about an indirect gap. Since optical transitions arevertical in the E(k) diagram the absorption at the indirect gap will be weak and transitions at thedirect gap will dominate the absorption spectrum.

The transition rate from initial (i) to final (f) state is given by

)(2 2

hh

gMW fi = (3.1)

where the matrix element is given by

= rdrHM ifrr

)('* (3.2)

and )( hg is the joint density of states which is a folding of the density of states of the initial andfinal states. In order to calculate the transition matrix element an expression for the interaction

Hamiltonian H is needed. In classical physics the energy of an atom considered as a dipole in anelectric field r is rr

= pE . As a first approximation we can then use that to express theinteraction Hmiltonian on the form

rr+= reH ' . Let us consider a plane wave of the form

rkierrrrrr = 0)( (3.3)

and electronic wavefunction on the Bloch formrki

fififieru

Vr

rrrr = ,)(1

)( ,, (3.4)


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where the functions )()( ,, ruTru fifirrr

=+ are periodic in the crystal lattice described by translation

vectorsTr

. With this we get

( ) rderuereruVe

M rkiirkirki

fif

rrrrr rrrrr

r

= )()( 0* (3.5)

which by using the momentum conservation if kkk

r

h

r

h

r

h = is reduced to( ) rdrurru

Ve

M ifrrrrr

)()( 0* = . (3.6)

Figure 3.2. The transition from free atoms to atoms with bonds in a solid.

The transition matrix element thus only depends on the space coordinate through the cell-periodicpart of the Bloch function. These functions also contain the dependence on the material. Thecalculation ofM can thus be done through integration over a single Brillouin cell.In order to continue the evaluation of the optical transition rate we need to consider the symmetry ofwave functions in solids. Figure 3.2 illustrates the transition from discrete atomic levels tocontinuous bands of levels in Ge.

The scheme shown in Fig. 3.2 is in principle the same for III-V alloys such as GaAs or II-VI alloyslike CdTe. It is found that the transition matrix element is large for ps transitions, for instancefrom the valence band to the conduction band of semiconductors.

Near the band edge (Fig. 3.3) the 4 bands of a semiconductor are parabolic with the form

*2

22

mk

Eh

= (3.7)

where m* is the effective mass describing the curvature of the individual bands as can be seen from


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2

2

2

2

2

2 1

*

1

* dkEd

mmdkEd

h

h== . (3.8)

From Fig. 3.4 it can be seen that( )

2

223 2

42

1)(

kkkg == where as before3)2(

1

is the number of k-

points per unit volume. The relation to the density of states per energy unit is seen from

dkdE

kgEgdEEgdkkg

)(2)()()(2 == . (3.9)

Inserting the expression for g(k) and using the parabolic form E(k) described above we arrive at

Em

Eg2

3

22

*2

2

1)(

=h

. (3.10)

We will consider transitions between the bands

Fig. 3.3. The band near teh band gapin a semiconductor.

Fig. 3.4. The discrete values of the k-vector in reciprocal space.


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hh

egc

mk

kE

mk

EkE

2)(

2)(

22

22

h

h

=

+=

(3.11)

which gives a difference in energy between the two bands of the form

heg

heg mm

kE

mk

mk

E111

,222

222222

+=+=++=

hhh

h . (3.12)

To get an expression for the joint density of states we thus just have to replace the effective mass bythe reduced mass in Eq. (3.10) above

gg

g

EEgE

gE

=

=

quantum size effects in nanostructures

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