5. Quantum mechanics and electronic properties of nanostructures

5. Quantum mechanics and electronic properties ofnanostructures

We have seen that downscaling of purely classical bulk material properties can lead to dramatic

changes in behaviour in the nanoscale.

Nevertheless the most exciting effects in the nanorealm where quantum physics comes into play and

leads to completely new kinds of behaviour.

Here we will look into some of the simpler, yet much studied and even commercially widely used

quantum physical effects arising in nanostructures.

Introduction to Nanoscience, 2005 JJ J I II × 1

5.1. Quantum confinement

5.1.1. Reminder of the basics of quantum physics

[Any introductory structure of matter book, e.g. Eisberg-Resnick]

In quantum physics particles are not considered point-like, but instead have wave-like properties. In

1+1 dimensions (x, t) a particle is described by a wave function

Ψ(x, t) (1)

from which the probability that the particle is in a given region of space at a given time P (x, t) is

obtained as

P (x, t) = |Ψ(x, t)|2dx (2)

which requires that Ψ is normalized asZ ∞

−∞Ψ∗(x, t)Ψ(x, t)dx = 1. (3)


In the non-relativistic limit the development of the system is determined by the time-dependent

Schrodinger equation

−~2

2m

∂2Ψ(x, t)

∂x2+ V (x, t)Ψ(x, t) = i~

∂Ψ(x, t)

∂t(4)

where V is some potential energy function. If V is independent of time, one can separate the

position and time dependence as

Ψ(x, t) = ψ(x)e−iEt/~

(5)

and by insertion into the time-dependent equation 4 obtain the time-independent Schrodinger

equation

−~2

2m

∂2ψ(x)

∂x2+ V (x)ψ(x) = Eψ(x) (6)

By defining the Hamiltonian operator

Hop = −~2

2m

∂2

∂x2+ V (x, t) (7)

this can also be written as

Hopψ(x) = Eψ(x) (8)


ψ(x) is an eigenfunction and E an eigenvalue of Hop. Often this equation has several different

solutions with energies En, n = 1, . . .

5.1.2. Electron levels for atoms vs. solids

When the time-independent Schrodinger equation is applied to an individual atom, the potential

function V (x) contains the Coulomb interactions between negative electrons and the central

nucleus as well as the interaction between the electrons with each other. Qualitatively this leads to

a potential well for the electrons looking like

The solutions to this are of course a set of bound discrete energy levels for the electrons, which

then lead to the properties of atoms such as x-ray emission and absorption at given energy levels

and so on. This is the basis of atomic physics.


On the other hand, if one takes an inreasingly large collection of atoms, the potential energy

landscape for electrons changes as follows:


The Coulombic potential is valid only at the edges, while inside there are minima in the potential

energy at the atoms. However, the maxima inside are well below the level needed to remove an

electron to infinity. This energy difference is associated with the work function of the material (cf.

Einsteins explanation of the photoelectric effect).

When the number of atoms is sufficently large, one can forget about the boundaries and only

consider an infinite periodic array of atoms inside the material:

This is the basis for electronic structure in solid state physics.

The solid state quantum mechanical description of the electron behaviour inside this energy


landscape leads to the following main qualitative results [Any introductory textbook on solid state physics, e.g. Kittel,

Hook-Hall or Aschroft-Mermin]

- The presence of the other atoms broadens the energy levels from the atomic ones:

- The bottom electrons are hardly affected, though.

- The upper electron levels become overlapping energy bands.

- There are both filled and empty bands (depending on how many electrons the solid has), and

there may be band gaps between them.


- If the uppermost filled energy bands are not fully filled, the electrons can move quite freely in

them. Moreover, they are only loosely bound to the atoms in the lattice, and one can in fact show

that they behave very much like free electrons moving in a constant background potential. Such

materials are metals.

See sect. 9.3. in my Solid State Physics (FTF)-notes. Free electron parabola and halfilled bands in them!

- A material where the uppermost filled band is completely full and the lowermost unfilled band is

empty, and the bandgap is > 0, is either a semiconductor or an insulator.

.- It is a semiconductor if the energy gap Eg . 2 eV. In this case some electrons are still promoted

to the conduction band at room temperature so there is some electrical conductivity.

- Additional conductivity can be achieved by adding impurities, “doping” the semiconductor.

5.1.3. Electron levels for nanostructures

[Nanoscience aspects see Wolf 4.6- or Poole-Owens 9.3-]

Now for nanostructures the interesting case is the intermediate one: what happens after we have

’started’ to move to a solid?


- This is actually the bottom part of the figure above taken literally:

- What happens when we have a few tens or hundreds of atoms in a row?

(in the figure exactly 17)

- It is reasonable to assume (and tight-binding theory actually proves it) that already after

we have collected a few atoms the basic solid state physics result holds: the outermost

electrons in conduction bands are almost unbound to their atoms, and move almost as

if they were in a constant background potential.


- This is relevant for a metal (or semiconductor with some electrons excited to the

conduction band)

- But the difference is now that the edges are still fairly close, and we can not assume they have no

effect on the system!

- Now to figure out what happens, we can as a first approximation use the solid state insight that

atoms move almost as if in a constant potential to justify using an exactly constant background

potential, and approximate the outer edges as infinitely sharp.

- Then we arrive at the finite square well potential!

This is of course the very system for which students of quantum mechanics first solve the Schrodinger equation - although on

traditional courses it feels like an oversimplified model system, it is actually directly relevant to real-life nanoscience systems!

- For simplicity we consider it analytically only in 1D. But the 3D solution, although a bit more

complicated mathematically, qualitatively behaves the same, and is even quantitatively pretty close.

5.1.4. Solution of Schrodinger equation in the finite and infinite square well potentials

5.1.4.1. Analytical solution [Eisberg-Resnick sec. 6-7]


This potential is the simplest possible potential which gives bound state solutions to the Schrodinger

equation.

This potential can mathematically be written as

V (x) =

V0 when |x| > a/2

0 when |x| ≤ a/2(9)

If the energy of the particle E < V0 this gives bound state solutions with quantized energy levels.

In the well region the solutions are of the form for standing waves and can in general be written as

ψ(x) = A sin(kx) + B cos(kx) when |x| < a/2 (10)


Outside the well region the solutions must vanish with increasing |x| and are hence of the form

ψ(x) =

Cekx when x < −a/2De−kx when x > a/2

(11)

The values of the 4 constants A, B, C and D can be determined by requiring that both ψ(x)

and its first derivative are continuous at ±a/2. One then obtains solutions of the type

with a few quantized energy levels like


Unfortunately the solution leads to a transcendental equation which is not analytically solvable.

Here I will quote a sample numerical solution to be used below. If m, V0 and a are related such

that smV0a2

2~2= 4 (12)

then the ground state solution will be approximately

E0 = 0.0980V0. (13)

To get an analytical solution, consider the infinite square well potential, i.e. the limit V0 →∞.

Then states outside the box will be completely forbidden, and we are left with the solution inside


the box

ψ(x) = A sin(kx) + B cos(kx) (14)

By requiring that

ψ(x) = 0 when x = ±a/2 (15)

one can easily show that the exact solutions of the Schrodinger equation are simple sine and cosine

functions, and the lowest of these gives

ψ0(x) =

r2

acos

πx

a(16)

which has the ground state energy

E0 =π2~2

2ma2(17)

The full solution for all energy levels is

En = n2π

2 ~2

2ma2(18)

where

n = 1, 2, 3, . . . (19)


The three-dimensional (spherical symmetry) corresponding result is

Enl = β2nl

~2

2ma2(20)

where l is the angular quantum number, and βnl is the n:th zero of the l:th spherical Bessel

function jl(x).

- These are tabulated e.g. in Abramovitz and Stegun p. 467 (nowadays available in the web:

[http://www.convertit.com/Go/Maps/Reference/AMS55.ASP?Res=150&Page=467 )

- The first few zeroes zeroes are:

3.141593 (π); 4.493409, 5.762459, 6.283185 (2π), 7.72525, ...

- Comparison with the 1D solution shows that the first energy value is exactly the same, but above

that the 3D case has many more solutions.

The effect of having bound electron states (which are not atomic!) when electrons are confined in

nanosize systems is called quantum confinement.


5.1.4.2. Energy range

Now that we have the simple analytical solution, let’s plug in a few numbers and see what energy

range is relevant for this calculation.

- The work functions of metals are well known to be of the order of a few eV’s. Let’s assume it is 1

eV.

- Let us also assume a quantum well of size a = 1.55nm. Then the ground state electron energy

are from eq. 18 or 20:

E0 =π2~2

2mea2= 0.156 eV. (21)

This result already justifies using as first approximation the infinite well, since the energy is clearly

much less than 1 eV. Moreover, the values 1 eV and 1.55 nm were deliberately chosen such that

they about fulfill criterion 12 for the finite square well. Now we can test what the difference between

a finite and inifite well is; the above result for the finite well would give

E′0 = 0.0980V0 = 0.098 eV (22)

I.e. we see that there is a clear difference, but still the order of magnitude is the same.


5.1.4.3. Implications of quantum confinement

- Since the electron levels in a quantum well are discrete (rather than continuous as in a bulk metal),

this means the only allowed transitions of the outermost electrons are those between the discrete

levels.

- This is of course also true in single atoms. Hence nanostructures are sometimes called artificialatoms.

- For the infinite quantum well the possible transition energies ∆E are of the type

∆E = (m2 − n

2)π

2 ~2

2ma2(23)

where m and n are integers with m > n > 0. This means the lowest possible transition energy is

3π2 ~2

2ma2(24)

For the 3D infinite quantum well the possible transitions are more complicated, but the list of zeroes


of the Bessel function given above shows that they are of the order of

∆E ∼~2

2ma2(25)

Also the lowest levels in finite quantum wells are actually of the same order as this. So let’s use

this simple formula 25 to estimate the possible transition energies. Evaluating all the constants for

m = me gives

∆E ∼0.038 eV

(a in nm)2(26)

which means the transition would typically be 0.04 eV or less.

- The transitions between the levels can be generated by photons or phonons (quanta of lattice

vibrations), which means quantum confinement may have optical or vibrational/sound applications.

- But the most important results is that the 1/a2 dependence means that by changing the quantum

well size (i.e. the size of a thin film or nanoparticle) one may finetune the energy at which the

energy absorption or emission occurs!

5.1.5. Optical effects from nanoclusters

- Let us consider the optical effects


- 0.04 eV lies in the infrared light region, so quantum structures like this may directly be used in

infrared light applications!

- However, it is also possible to apply this in the visible light region.

- In semiconductors one has to account for the ordinary band gap Eg in addition to the quantum

confinement effect

- Moreover, in semiconductors conduction is possible both by electrons and holes (empty electron

states in an almost filled energy band). The hole levels are confined just like the electrons.

- In semiconductors there is the additional complication that the electrons and holes can

be treated better using so called effective masses m∗e and m∗

h which are not the same

as the real electron mass.

- This is a rather counterintuitive result, but derives from the semiclassical model for

electron dynamics which is treated on solid state physics courses.


Ener

gy

quantum well

filled valence band

empty conduction band

hole levels

quantum wellelectron levels

band gap

Hence for semiconductor quantum dots the possible transitions are

∆E = Eg + En,electron + En,hole (27)

Assuming again our simple order-of-magnitude estimate 25 one can quantify this as

∆E ∼ Eg +~2

2m∗ea

2+

~2

2m∗ha

2(28)

For many semiconductors the effective masses can be rather low, ∼ 0.1me or so. This gives then


for the lowest lying states

0.38 eV

(a in nm)2(29)

The band gaps on the other hand are typically some 1-2 eV. Visible light has the energy range of

about 1.8 - 3.1 eV.

This means that using semiconductors one can obtain quantum wells which work in the visible light

region, and where one can select to particle sizes a to finetune the optical absorption over much of

the visible light region!

A classic example of this is CdSe nanoparticles, which have Eg = 1.74 eV, m∗h = 0.45me and

m∗e = 0.13me. The above effect, plus the fact that also the band gap depends on nanoparticle

size (due to strain) allows for changing the light absorption maximum from about 1.9 eV to almost

2.6 eV with changing nanoparticle size:


NOTE added after course: in reality things are more complicated, since also the band gap size

changes strongly from the bulk value for small particles. Calculating this involves advanced electronic

structure calculations and is thus beyond the scope of this course.


See e.g. Wang and Zunger, J. Phys. Chem. 98 (1994) 2158 for a calculation in Si, showing that

the band gap changes from 1.1 eV up to some 4 eV for about 1 nm diameter particles!

Finally for the smallest nanoparticles the problem arises that as the large fraction of surface

atoms has dangling bonds, these dangling bonds induce additional states into the band gap. To

remove this effect, nowadays often the nanoparticles are manufactured surrounded by a shell of

another semiconductor, which passivates the dangling bonds. Such particles are called core-shellnanoparticles. Typical examples are CdSe/CdS, CdSe/ZnS. [Chen et al, Int. J. of Nanotechnology, 1 (2004) 105]

Here is a picture of colors from commercial CdSe and CdTe core/shell nanoparticles in solution

manufactured by Evident Technologies, obtained when they are illuminated by white light:

Shell was ZnS at least for CdSe


and here are the associated emission spectra:

These things have application e.g. as markers in medical applications: they can be made to stick


to medically important molecules, and then the positions of the molecules can be detected by

illumination.

5.1.6. Quantum corrals

[Wolf 4.6.4]

Another spectacular demonstration of quantum confinement can be obtained on surfaces.

- Certain metals, such as Cu, have electron states confined to the surface.

- Then if one used an atomic force microscope (AFM) tip to form a geometrical shape of some

other metal, e.g. Fe, this may act as a ’fence’ confining the surface electrons into a 2D quantum

well.

- The electron wave function on the surface will be a a superposition of one or a few allowed

electron states on the surface

- Reminder of what these looked like


- The fantastic thing is that using scanning tunneling microscopy, it is possible to directly measure

the electron wave function on the surface. One obtains something like the waves seen in the lower

right part of the atom movement figure shown in section 1 of this course:


With some image processing a figure like that may be made even nicer-looking (yes, this really is

an experimental image):


5.2. Density of states

[Poole-Owens 9.3.5, see also http://britneyspears.ac/physics/dos/dos.htm ]

5.2.1. Confinement and dimensions

- Until now we have been quite sloppy about in how many dimensions the quantum confinement

actually occurs.

- To understand this, we have to think a bit more about what the confinement means. For

confinement to occur, the nanostructure length scale has to be less than the characteristic length

scale l of the electron or hole behaviour in normal bulk matter.

- What this scale is, is a complicated issue and beyond the scope of this course. But e.g. for

semiconductors it is related to so called excitons, which have an associated length scale l of the

order of 10 nm.

- Sometimes one can also distinguish between strong confinement and weak confinement. These


are:a < l strong confinement

a > l weak confinement

a >> l no confinement

(30)

- Because confinement in a 3D nanoparticle means that the electrons essentially have no freedom to

move, one can consider this a zero-dimensional (0D) object with respect to the electronic properties.

Of course this does not conform to the mathematical definition of a point...

- On the other hand in a thin film the electrons are free to move in two dimensions, but are confined

in one, and so on. Hence one can distinguish between different kinds of quantum structures, as

follows:

Type Delocalized dimensions Confined dimensions

Quantum dot 0 3

Quantum wire 1 2

Quantum well 2 1

Bulk 3 0

- Notation is somewhat confused, though. Often quantum dots are understood as atom mounds on

a surface, but by the above definition also nanoparticles are quantum dots!


5.2.2. Density of states

[P-O 5.3.3 - ]

- We have already seen how quantum confinement changes things dramatically from conventional

bulk matter.

- There is another major reason why the electronic properties of nanostructures are different from

bulk matter

- This is the due to the density of states (DOS)

- Let’s quickly recall from structure of matter courses what the density of states is about:

- The Sommerfeld model of conduction states that at ordinary temperatures practically all of the

electrons in a solid have energies less than the so called Fermi energy EF .

- Correspondingly the wavenumber k are limited to the Fermi wavenumber kF where EF =~2k2

F

2m.

- The possible wavenumbers are due to quantum physics quantized in all dimensions (this is now


not related to quantum confinement but true in bulk as well) such that

kx =2πnx

L, ky =

2πny

L, kz =

2πnz

L, (31)

where L is the size of the structure and nx, ny, nz are integers.

This means that a certain region of space V = L3 can only contain a given number of possible

electron states. This density of points is for d dimensions

ρ = (2π

L)−d

(32)


In 3D the density of states is obtained as follows.

A thin shell between k and k + dk has the volume

4πk2dk (33)

and the number of points in it, the density of states D(k)dk, becomes

D(k)dk = 4πk2dk × ρ4πk

2dk ×

L3

8π3=V k2

2π2(34)

To get the DOS as a function of energy, accounting for the fact that there can be two electrons per

k state (spin up and down), we use

D(E)dE = 2D(k)dk (35)

and using

E =~2k2

2m=⇒ k =

√2mE

~(36)


do

D(E) = 2D(k)dk

dE= 2

V k2

2π2

! d

dE

√2mE

~

!=

V k2

π2

! 12

√2m

~√E

!(37)

=

„V 2mE

~2π2

« 12

√2m

~√E

!=

V

2π2~3(2m)

3/2√E (38)

On the other hand the density of states is just

D(E) =dN

dE(39)

where N(E) is the possible number of electrons. Thus N(E) can be obtained by straightforward

integration of D(E)

The above calculation of the density of states can of course be repeated for 1D and 2D systems.

The results will be, just in terms of the dependence on energy:


Type Delocalized dimensions D(E) N(E)

Quantum wire 1 ∝ E−1/2 ∝ E1/2

Quantum well 2 Independent of E ∝ E

Bulk 3 ∝ E1/2 ∝ E3/2

- This clearly shows that the dependence on energy is dramatically different.

- The DOS affects a huge lot of electronic properties of matter, and hence these differences make

for big changes in the electronic properties of 0, 1 and 2D confined electron systems as compared

to bulk!


5.2.3. Combination of density of states and quantum confinement

- The above derivation was for the unconfined dimensions of the

electrons.

- In the 0, 1 and 2D confined systems we saw earlier that the

electron levels will on the other hand be discrete in the confined

dimension.

- What happens when these results are combined?

- Physically speaking e.g. in a quantum well the electron levels are

confined in one dimension, leading to discrete states i. But then

for each state i they will delocalize in the two other directions by

populating levels according to the density of states.

- Thus one arrives at the adjacent picture of the number of

electrons and density of states in 0 - 3 dimensions.

- A few examples of material properties that depend on the DOS:

• The specific heat of solids cv depends on both the phonon and electron density of state


• Photon absorption and emission in optical devices.

• Magnetic susceptibility of a magnetic material ξ.

• Concentration of electrons and holes in semiconductors

• Superconducting energy gap

• The tunneling current in a scanning tunneling microscopy experiment


5.3. Quantum well lasers

[Mayer-Lau, http://britneyspears.ac/physics/fplasers/fplasers.htm]

- Probably the first widely spread commercial application of quantum confinement are the quantumwell light emitting diodes (LED’s) and lasers.

- These are based on the quantum confinement principle.

- One can achieve a quantum well by placing thin semiconductor films of different materials on top

of each other. Then if the films have different band gaps, confined electron and hole states can

appear in between the layers.


- A typical example of such a structure is the AlGaAs/GaAs/AlGaAs double heterojunction laser.

- In here a thin GaAs layer is sandwiched between n-doped and p-doped AlGaAs

Explain doping: n-type electrons...


- The n-AlGaAs layer feeds in electrons to the GaAs, while the p-AlGaAs feeds in holes

Fed in by diffusion

- Hence there is a population inversion in the GaAs layers: the electrons are in an excited state

and want to recombine with the holes.

- This enables laser operation, i.e. stimulated emission of photons: when a photon interacts with

an excited electron it can cause it to emit an additional photon, doubling the number of photons.


Mirrors are at its simplest just the air at the end of the structure

- The reason the quantum confinement is relevant for this is twofold:

1. The quantum confinement leads to discrete energy levels, and hence to mono-energetic

light which is the aim in a laser.

- This can be improved on by using quantum dots in the active thin film layer [Poole-Owens

9.25]


2. The quantum confinement makes the electrons and holes stay better in the GaAs layer, improving

on efficiency.

- Also the difference in referective indices between the GaAs and AlGaAs makes the active region

work as an optical waveguide, but this is not directly related to the quantum confinement.

- The ’mirrors’ at the end of the active region needed to create a resonant cavity for a laser can at

their simplest be just air, but are often nowadays other semiconductor heterostructures.


5.4. Single-electron transistor (SET)

[Poole-Ovens 9.5; http://physicsweb.org/articles/world/11/9/7/1]

Consider the following structure:

- If the source, dot and grain would be directly connected to each other, this would just be a regular

Field Effect Transistor (FET), the basic building block of modern electronics.

- In FET’s the gate voltage Vg is used to control the current running from source to drain through

the gate (gate is at the position of the quantum dot).


- Gate terminal and gate are separated by a good insulator, so there is practically no current flowing

through the gate. Hence there is an associated gate capacitance Cg.

- Now our interest is in the nanoscale device, where first of all the source, dot and drain are also

electrically isolated, but close enough that electron can tunnel quantum mechanically from source

to dot and from dot to drain. Such a structure is called a tunnel junction.

- The gate voltage can be used to adjust the resistance of the quantum dot, but is not really needed

for the operation.

- But the capacitance of the quantum dot structure plays a role!

- The capacitance for a spherical dot is given by

C = 4πε0

„ε

ε0

«r (40)

- Let’s see what this would mean for GaAs: GaAs has εε0

= 13.2 which gives C = 1.47×10−9r

F/m which for r = 10 nm gives 1.47×10−17 F. This is a very small value.

F is farad


- The relation between charge, capacitance and voltage change is from basic electrodynamics

Q = C∆V (41)

- Now since we are dealing with an nanoscale device, let’s use this for a single electron, which of

course has charge e, and our assumed GaAs dot of 10 nm:

e = C∆V =⇒ ∆V =e

C= 0.011 V (42)

Thanks to the fact that both e and our capacitance are very small, the answer actually becomes

quite sensible - 11 mV is a very easily measurable and/or controllable voltage!

- This means that the current flow will be directly affected by how many electrons are in the

quantum dot, since within every 11 mV voltage window no more electrons can fit in!

- Hence the current flow will become a staircase function in steps of


First electron comes in at e/2Cg, additional at steps of e/C. Reason: inhibited at e/Cg, thus comes in at about midpoint between

these!

- This has been experimentally observed. Here is one of the first observations:


[Fulton and Dolan, PRL 59 (1987) 109]

and here a later more staircase-like one (part A is experimental):


(Note that Poole-Owens has an error in their reference to FIG 9-18, the authors and ref. do not match

- Because of the relation of these things to single electrons, they are called single electrontransistors or SET’s.

- The gate voltage can be used to tune the conductance. Here is a 3D plot of this:


Be careful not to confuse gate voltage and bias voltage

- The energy associated with a capacitor is E = 12Q

2/C so in this case the energy needed to insert

an electron into the dot is

∆E =e2

2C(43)

Since electrons need to have this electrostatical energy to enter the dot, the effect of quantized

current is called Coulomb staircase and the energy barrier preventing insertion of electrons is called

Coulomb blockade.


- To be more precise, there are two criteria which need to be fulfilled for the staircase to appear:

1. The single-electron energy needs to be larger than the thermal fluctuations, otherwise thermal

excitations can put in electrons washing away the effect. I.e.

e2

2C kBT (44)

2. The product of the energy and the capacitor charging time needs to be larger than the Planck

constant h, otherwise the uncertainty principle will make atoms jump in and out of the capacitor

at will. The capacitor charging time is RTC where RT is the resistance of the tunnel junction

between the dot and the source and drain. I.e.

∆E∆t =

e2

2C

!RTC h (45)


Simplifying these one obtains

e2

2CkB T (46)

RT h

e2(47)

(48)

Here the quantity he2

is the quantum of resistance = 25.815 kΩ.

To see what the first criterion means in practice, consider our above-mentioned 10 nm dot. It had

C = 1.47×10−17 F, giving e2

2CkB= 63 K.

- This means that SET’s work usually only at low temperatures.

- There are nowadays room-temperature versions as well, though: see e.g. Y T Tan et al. 2003 J.

Appl. Phys. 94 633.


5. Quantum mechanics and electronic properties of nanostructures

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