PETER PAZMANY CATHOLIC UNIVERSITYConsortiummembersSEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben

***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.

Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework**

Consortium leader

PETER PAZMANY CATHOLIC UNIVERSITY

Consortium members

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***

PHYSICS FOR NANOBIO-TECHNOLOGY

Principles of Physics for Bionic Engineering

Chapter 9.

Heuristic Models for the Structure of Matter

(A nanobio-technológia fizikai alapjai )

(Heurisztikus modellek az anyagszerkezetekhez)

Árpád I. CSURGAY,

Table of Contents

9. Heuristic Models for the Structure of Matter 1. Structure of Matter

2. Classical Electrodynamics – MaxwellLorentz equations 3. Solutions of the Single-Electron Problem –

Band Structure of Matter

4. The Effective Mass Schrödinger Equation 5. Quantum Well, Quantum Line, Quantum Dot

1. Structure of Matter

We envision the structure of matter for the physics of nanobio- technology as a physical system composed of

 vacuum;

 electromagnetic field (photons);

 nuclei with mass m_Z , charge Z·e, spin 1/2;

 electrons with m_e, charge e, spin 1/2.

We assume that gravity and electromagnetic interactions hold the components together, the energy level is limited, thus relativistic effects can be neglected. Nor strong neither weak nuclear

interactions have any impact.

First, we consider the models of non-relativistic classical mechanics and classical electrodynamics, and introduce

MaxwellLorentz equations and classical conservation laws.

Next, quantization of the electromagnetic cavity modes leads us to the physics of vacuum, to vacuum fluctuations and photons.

Then, a nanoparticle is considered as a quantum mechanical system with band structure and electron population following the FermiDirac statistics, and its interaction with the quantized electromagnetic field is defines our heuristic physical model.

Nanoparticle

(atoms, molecules, metal, dielectric, semiconductor) Closed physical system in

stationary state

Electromagnetic field (vacuum, photons, cavity

modes)

Quantization

0

0 0 0

0

0 0 0

1 2

0 0 0

1 2

, , ,

, , , ,...

n n n

n n

E

E E E

 

  



 



H H

2

2 2

, ) 1 , )

, ) ( , )

, ) , )

t t

c t

t t

t t

t

  



  

    

A(r A(r

B(r A r

E(r A(r

1

n 2

E   ^n  ^ Interaction  Perturbation

Heuristic Models

1. Solution of the single electron problem for a closed system; Stationary states – no radiation;

2. Adding electron population (FermiDirac statistics);

3. Cavity electrodynamics – Modal expansion and quantization;

4. Quantum-electromagnetic interaction (multipole

representation, perturbation; photon absorption, and emission);

5. Model simplification – input/output models, „circuit‟

paradigm.

2. Classical Electrodynamics – Maxwell Lorentz equations Let us start with empty space (classical vacuum), in which nuclei

and electrons as point-like moving bodies generate and are subject of classical electromagnetic field. The particles are characterized by their mass m_α and charge q_α, position and velocity, thus the charge and current densities are

   

     

, ;

, .

t q t

t q t t

 



  



 



    

    





r r r

j r v r r

r

v



 q m ,

In vacuum

Thus the Maxwell – Lorentz equations are

The force acting on each particle is the Lorentz force

 

0 ; 1/ 0 ; 0 0 1/ c .

   

  

D E H B

       

   

     

2 0

0

, 1 , ,

, , ,

, 1 , , 0.

t t q t t

c t

t t

t

t q t t

  



 



 

 

       

   



         





B r E r v r r

E r B r

E r r r B r

   

( ) ( ) , thus if .

q q c

           

F E r v B r v



 



m_{ }r  q_ E r_ t  v_ B r_ t   

State variables are Initial conditions are Conservation Laws Energy

Momentum

Angular momentum

( ),t ( , ),t ( , ),t

rα E r B r

       





     

2 0 2 2 2

1 , , d .

2 2

H m_{ } t t c t V



 _{ }







 

   

m_{ } t t t V











P v E r B r

 

 

   

   







J r v r E r B r

1, 2,3,...

 

3. Solutions of the Single-Electron Problem – Band Structure of Matter

Let us assume that the solution of the single-electron problem (energy spectrum, stationary eigenstates) are already

determined, like in case of the potential boxes, harmonic

oscillator, hydrogen-like atoms, etc. We are going to rely upon the sophisticated results of quantum chemistry, „soft‟ and „solid‟

state physics.

In case of the hydrogen like atoms the energy-spectrum is

2 4 2

eV

2 2 2 2

0

e , in eV, 13.6 , 1, 2,

8 h

n n

mZ Z

E E n

n n

     

An the stationary states are the orbitals

The maximum number of electrons on an orbital is The Hamiltonian operator of an atom is

where the index i goes through the

orbitals to be considered in an interaction.

,...

, ,

, ,

, ,

, ,

, _{2 3 4 5 6 7 8 9 10}

1 E E E E E E E E E

E

,...

10 , 2 , 6 , 10 , 2 , 6 , 2 , 6 , 2 , 2

1 , 2 , 2s s p , 3 , 3s p , 4 , 3s d , 4p , 5 , 4s d ,...

atom i

i

E i i





H

1s 2s

2p 3s 3p

3d

Single atom Two atoms Energy ‘bands’

The solution of the single-electron problem in the potential field of two, three, …, N, … atoms, resembles the spectrum of one- atom. Each discrete energy level will split into two, three, …, N, … neighboring energy levels, and stationary states. If N increases, the discrete levels gradually become „quasi-

continuous energy „bands‟.

2s 2p 3s

1s 1s

2s 2p

3s 3p 3d

1s

2s 2p

3s 3p 3d

Qualitative View of the Solid State Band Structure

If the number of atoms in a single atom solid state crystal is N, and the number of protons of the atom is Z, then the total number of electrons in a neutral crystal are Z·N. Let the temperature be T = 0, and let us drop the Z·N electrons into the crystal one by one.

Every microstate will be populated from bottom up in energy. Into an s-band 2N, in a p-band 6N, in a d-band 10N electrons can be dropped.

E.g. in case of diamond (C) 6N, sodium (Na) 11N, aluminum (Al) 13N, silicon (Si) 14N, potassium (K) 19N, copper (Cu) 29N.

Let us look at Na in detail.

Example 1: Qualitative View of the Solid State Band Structure

 If in a sodium crystal the number of atoms is N , then the total number of electrons is 11·N.

 At absolute zero T = 0 we fill the microstates from bottom up in energy. The 1s band is filled with 2N, the 2s band 2N, 2p band 6N electrons. We could put 2N electrons into the 3s band, however, there is only N electrons left. Thus the 3s band will not be filled. Electrons can be moved with very small

electric field.

 The electrons in the 3s band obey the Pauli principle, as fermions they behave according to the Fermi–Dirac statistics. The Fermi-level in the 3s band is the highest occupied energy at T = 0 K. The behavior of the electrons in the 3s band, for temperatures T > 0, can be approximated by the behavior of electrons in a “big box” (no forces inside the box).

Qualitative conclusion: Sodium is a good conductor (metal).

Sodium, Na 11N

Filled bands

Partly populated

band 3p

3s

2p

2s

1s

1s 2s 2p 3s

Electron energy

6N

2N

2N 2N

Qualitative conclusion: Diamond is an insulator (dielectric).

Diamond 6N

Filled bands

Empty band 3p

3s

2p

2s

1s

1s 2s 2p 3s

Electron energy

6N

2N

2N 2N

Semiconductors are insulators with a narrow forbidden band gap between the „conduction‟ and „valence‟ bands.

overlap

metal semiconductor insulator

band gap Fermi level

Electron energy

conduction band

valence band

In solid state particles, the electrons move in periodic potential field. The solution of the single-electron problem provides us the band structure, i.e. the allowed and forbidden bands for occupation with electrons. In the bands the admitted energy levels are quasi-continuous, the number of discrete levels are equal to the number of atoms in the crystal.

The detailed analysis of the band structure provides the relation of the electron‟s energy to its wave number vector

In case of the KrönigPenney crystal we saw the E(k) diagrams (called „dispersion curves‟).

( ,_{x y}, _z).

E k k k

Example 2: The band structure of silicon

In case of metals and semiconductors we rely upon the available references.

E.g. the band structure of silicon in the <111> and <100> orientations Energy gap: 1.12 eV

Energy separation: 4.2 eV

1.12 eV 1.2 eV

<100>

0.044 eV

<111>

Split-off band

Heavy holes Energy

Wave vector 2.0 eV

3.4 eV 4.2 eV

300 K

Electrons in crystals

If we move electrons in a crystal with external forces, we can not neglect internal forces inside the crystal. The electron will be subject of internal and external forces:

Effective mass is defined as

For free electrons there are no internal forces, thus the effective mass is the mass of the electron.

external internal

d .

m d

  tv

F F

external eff

d .

m d

 tv

F

internal eff

If F  0,  m  m.

The group velocity of the electron‟s wave pocket

If the electron moves under the influence of external force F_external, its energy is changing as

The time-derivative of energy is thus

d 2π d

d h d . E

k k

   v

external external

d 2π d

. .

d h d

E E

F v F

t   k

d d d

d d d ,

E E k

t  k t

external

h d d

2π d d .

k k

F  t  t

The group velocity changes in time as

In three dimensions

2 2

external

2 2 2

d 1 d d 1 d d 1 d

d d d d d d

E E k E

t t k k t k F

 

    

v

2

eff external 2 2

eff

d 1 1 d

d d

m F E

tv   m  k

1

 

grad , .

E 2π d

 _k  kt

v k F

Effective mass tensor

 

d 1 d 1

d d grad E

t  t _k  _k ^k

v k T F

 

2 2 2

2

2 2 2

1

eff 2 2

2 2 2

2

1

x x y x z

y x y y z

z x z y z

E E E

k k k k k

E E E

k k k k k

E E E

k k k k k



    

 

    

 

    

 

      

 

  

 

     

 

m





1

  _k ^k

m T

4. The Effective Mass Schrödinger Equation

In a crystal the dynamics of an individual electron can be described by the single-electron Schrödinger equation.

In a solid-state material the potential can be separated into three parts

where is a perfectly periodic potential due to a perfect static lattice, is a random time-varying potential

representing the deviations, and is the macroscopic potential due to any externally applied voltage and/or due to macroscopic space charge.

 

 

 

 

pot , pot pot , pot ,

E r t  E r  E r t  E r t

 

L

Epot r

 

S

pot , E r t

 

E

pot , E r t

First, let us assume that the electron moves in a perfectly periodic and static potential , thus the Hamiltonian is

Let us assume that we know the band-structure of the material, thus we know the eigenvalues and eigenfunctions of H in each band.

If  denotes the index of a band ( = c in conduction band, v = h,  in valence band for heavy and light holes).

are the eigenvalues, and

the orthonormal eigenfunctions of the operator H, then

 

L

Epot r

2

 

L

2 Epot

  m  

H r

 

E_ k  _,_k

 

 

the solution of the wave equation can be expanded as

It is feasible to look for the solution by introducing an „envelope wave-function‟

where is periodic and follows the periodicity of the lattice, and is the „envelope‟ wave function.

 

 

,

,t _ t , .



 



k

r k

 

   

 r  _k r  r u_,k r

 

 r

 

E_ k are the eigenvalues, and ^ _,k

 

 

the orthonormal eigenfunctions of the operator H, then

It can be shown that in case the electron moves in a single band, the envelope function satisfies a wave equation

where m* is the electron‟s effective mass in the periodic potential field, U(r,t) potential of the external field and E_C0 is defined by the parabolic approximation of the dispersion relation E(k):

         

0

2

, C , , j , ,

2 t E U t t t

m^ t

  

  r   r  r   r    r

 

 

C C .

E E 2

m^

 

k r k

Note that this wave equation resembles the single-electron Schrödinger equation with a new potential, where E_C0 is a

material parameter. The effect of the periodic lattice is replaced by the effective mass and the constant E_C0. The effective mass is determined by the band structure, i. e. by the E(k) function of the lattice

In general, when the motion depends on k, the effective mass is a tensor.

 

1 E .

m k

 

 

 

 If the electron moves in “more than one” band (multi-band wave function), we have to introduce envelope functions and effective masses for each band, and the dynamics will be described by a set of coupled wave equations.

 In semiconductors 8 bands can have significant role: the

conduction band, the valence band of heavy and light holes, and the displaced valence band, each of them with electron spin +1/2 and 1/2.

The external potential-distribution in a device is not known because of the lack of information on the space-charge distribution inside the device. From the envelope function an electron density can be determined by summing the probability densities of all the electrons (ensemble averaging)

The current density J can be calculated from the envelope function

 

   

n r t   ^ r t  r t

 

   

2 t q

m

 

       

J r

Note that n and J calculated from the envelope function satisfies the continuity equation

The macroscopic potential due to any externally applied voltage and/or macroscopic space charge can be determined from the Poisson equation. If we consider as carriers only electrons in the conduction band, then

where U ^E is the electrostatic potential

is the ionized donor‟s density,  is the dielectric constant.

   

2 E +

/ D ,

U q  N n

   





+

N

n 0.

q t

    J 

The effective mass equation can be used as an approximate model of hetero-structures and super-lattices

 if we can neglect the spatial variation in the effective mass,

 and the different materials are modeled with their effective masses and

The potential in the effective mass equation can be determined by a self-consistent iteration. Assuming a given ,

(i) solve the effective mass equation,

(ii) calculate the electron density and the current density assuming charge continuity,

(iii) solve the Poisson equation for and compare the result with the assumption.

C0

 

E r

UE

U E

5. Quantum Well, Quantum Line, Quantum Dot  Confinement of the electron

A single-electron satisfying the effective mass Schrödinger equation where the Hamilton operator is

The available nanotechnologies (molecular beam epitaxy and electron beam lithography) can fabricate three dimensional structures with almost arbitrary spatial variation of

  E H

0

 

2

2

C .

2 E

m^

   

H r

C0. E

Quantum Well Wire Dot The density of quantum states for confined electrons,

(i) electron confined in a quantum well;

(ii) in a quantum wire, (iii) in a quantum dot.

 

     

 

2 2

2D 2 2

1

2 1D

, 1

0D

, , 1

2 2 2 2

2 2 2

1 π ,

π

2 π

, 1

π .

n z z

z y

n y z

n k x y z

z y x

m h n

g E E

h L m L

m L L

g E h

E E E

g E

L L L

h n k

E m L L L



 

 















 

    



  

 

  

       







2D

 

E E3D E⁰ E¹

1D

 

g E

E E00 E^nm

0D

 

g E

Enlm

Example 3: Thermal emission of electrons from metals (Sommerfeld model)

In metals conduction electrons are quasi-free electrons in a „big potential box‟.

Only the conduction band can be considered, and electrons follow the FermiDirac statistics. The band structure approximately

 



E EF0

0 T 

0 T 

Conduction band EB

Electrons with energy large enough, and

moving toward the surface will be emitted.

in 1 s those among the electrons having

velocity-component up to v_x , which are in a box of length v_xwill be emitted from the metal.

N is the number of electrons leaving the metal in 1 s through the unit surface

F B

3 3

k

2 d d d .

h ^{x x z} e 1

x

x y z

E E T

N m

  



  



  

v v vy v

v v v v

1 m2

m

x s v

B B

2

x

E

  m vx v

Approximation:

RichardsonDushman‟s Electron Emission Formula

F B F

B B

k k

e e 1

E E E E

T T

 

 

F0

B B

B

2 2 2

F0

B B B

B

3

k k

3

3

2k 2k 2k

k 3

2 e e d d d

h

2 e e d e d e d

h

x

x y z

x

E E

T T

x x y z

m m m

E

T T T

T

x x y z

N m

m

   

 

     

 

 

 

  

  

v

v v v

v

v v v v

v v v v

B F0 B

2

2 3

4π e e.

h

E E B k T

t

N k mT J N

 

  

B

2

2 3

4π e

e . h

EW

B k T T

J k m T

 